{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n" ], "text/plain": [ "" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from IPython.core.display import HTML, display\n", "css_file = './custom.css'\n", "HTML(open(css_file, \"r\").read())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# M62 DM 2020" ] }, { "cell_type": "markdown", "metadata": { "toc": true }, "source": [ "

Table of Contents

\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exercice : implémentation et comparaison de schémas\n", "\n", "Le déplacement $x(t)$ d’un système oscillant composé d’une masse et d’un ressort, soumis à une force de frottement proportionnelle à la vitesse, est décrit par l’équation différentielle du second ordre \n", "$$\n", "x''(t)+ 5x'(t)+6x(t)=0,\n", "$$\n", "avec $x(0) = 1$ et $x'(0) = 0$, pour $t \\in [0, 5]$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ ">**Q1 [1 point]** \n", "Écrire le système sous forme matricielle. Calculer ensuite la solution exacte avec `sympy`. Afficher $t\\mapsto x(t)$, $t\\mapsto x'(t)$ et $x\\mapsto x'(x)$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Si on note $y(t)=x(t)$ et $z(t)=x'(t)$ on a $y(0)=1$, $z(0)=0$ et\n", "$$\n", "\\begin{cases}\n", "y'(t)=z(t),\\\\\n", "z'(t)=-5z(t)-6y(t)\n", "\\end{cases}\n", "\\quad\\text{i.e.}\\quad\n", "\\begin{pmatrix}y\\\\z\\end{pmatrix}'(t)\n", "=\n", "\\begin{pmatrix}0&1\\\\-6&-5\\end{pmatrix}\n", "\\begin{pmatrix}y\\\\z\\end{pmatrix}(t)\n", "$$" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "%reset -f\n", "%matplotlib inline\n", "\n", "from matplotlib.pylab import *\n", "\n", "t0 = 0\n", "tfinal = 5\n", "\n", "tt = linspace(t0,tfinal,301)\n", "h = tt[1]-tt[0]\n", "\n", "y0 = 1\n", "z0 = 0\n", "\n", "phi1 = lambda t,y,z : z\n", "phi2 = lambda t,y,z : -6*y-5*z" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "La solution exacte est\n", "\\begin{align*}\n", "x(t)=y(t)&=3e^{-2t}−2e^{-3t},\\\\\n", "x'(t)=z(t)&=-6e^{-2t}+6e^{-3t}.\n", "\\end{align*}\n", "Vérifions-le avec le module `sympy`:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$\\displaystyle \\frac{d}{d t} y{\\left(t \\right)} = z{\\left(t \\right)}$" ], "text/plain": [ "d \n", "──(y(t)) = z(t)\n", "dt " ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": 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"text/latex": [ "$\\displaystyle \\frac{d}{d t} z{\\left(t \\right)} = - 6 y{\\left(t \\right)} - 5 z{\\left(t \\right)}$" ], "text/plain": [ "d \n", "──(z(t)) = -6⋅y(t) - 5⋅z(t)\n", "dt " ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle \\left[ y{\\left(t \\right)} = C_{1} e^{- 2 t} + C_{2} e^{- 3 t}, \\ z{\\left(t \\right)} = - 2 C_{1} e^{- 2 t} - 3 C_{2} e^{- 3 t}\\right]$" ], "text/plain": [ "⎡ -2⋅t -3⋅t -2⋅t -3⋅t⎤\n", "⎣y(t) = C₁⋅ℯ + C₂⋅ℯ , z(t) = - 2⋅C₁⋅ℯ - 3⋅C₂⋅ℯ ⎦" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle \\left\\{ C_{1} : 3, \\ C_{2} : -2\\right\\}$" ], "text/plain": [ "{C₁: 3, C₂: -2}" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle y{\\left(t \\right)} = 3 e^{- 2 t} - 2 e^{- 3 t}$" ], "text/plain": [ " -2⋅t -3⋅t\n", "y(t) = 3⋅ℯ - 2⋅ℯ " ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle z{\\left(t \\right)} = - 6 e^{- 2 t} + 6 e^{- 3 t}$" ], "text/plain": [ " -2⋅t -3⋅t\n", "z(t) = - 6⋅ℯ + 6⋅ℯ " ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import sympy as sym\n", "sym.init_printing()\n", "\n", "t = sym.Symbol('t')\n", "y = sym.Function('y')\n", "z = sym.Function('z')\n", "edo1 = sym.Eq( sym.diff(y(t),t) , phi1(t,y(t),z(t)) )\n", "edo2 = sym.Eq( sym.diff(z(t),t) , phi2(t,y(t),z(t)) )\n", "display(edo1)\n", "display(edo2)\n", "solgen = sym.dsolve([edo1,edo2],[y(t),z(t)])\n", "display(solgen)\n", "\n", "consts = sym.solve( [ sym.Eq( y0, solgen[0].rhs.subs(t,t0)) , sym.Eq( z0, solgen[1].rhs.subs(t,t0)) ] , dict=True)[0]\n", "display(consts)\n", "solpar_1=solgen[0].subs(consts)\n", "solpar_2=solgen[1].subs(consts)\n", "display(solpar_1)\n", "display(solpar_2)\n", "\n", "func_1 = sym.lambdify(t,solpar_1.rhs,'numpy')\n", "func_2 = sym.lambdify(t,solpar_2.rhs,'numpy')\n", "\n", "from matplotlib.pylab import *\n", "figure(figsize=(17,7))\n", "tt=linspace(t0,tfinal,501)\n", "yy=func_1(tt)\n", "zz=func_2(tt)\n", "subplot(1,2,1)\n", "plot(tt,yy,tt,zz)\n", "legend([r'$t\\mapsto y(t)$',r'$t\\mapsto z(t)$'])\n", "subplot(1,2,2)\n", "plot(yy,zz)\n", "xlabel(r'$y$')\n", "ylabel(r'$z$')\n", "axis('equal');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ ">**Q2 [2 points]** \n", "Calculer la solution approchée obtenue par la méthode d'Euler Progressif avec $301$ points. Afficher $t\\mapsto x(t)$, $t\\mapsto x'(t)$ et $x\\mapsto x'(x)$ en comparant solution exacte et solution approchée. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "On notera $u_n\\approx y_n=x(t_n)$ et $w_n\\approx z(t_n)$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Euler explicite**\n", "$$\n", "\\begin{cases}\n", "u_0=1,\\\\\n", "w_0=0,\\\\\n", "u_{n+1}=u_n+h\\varphi_1(t_n,u_n,w_n),\\\\\n", "w_{n+1}=w_n+h\\varphi_2(t_n,u_n,w_n).\n", "\\end{cases}\n", "$$" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "image/png": 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vqivI2sdpwLbMopDqs9oqQKQD7cq8bgtsByIr2H4Ea+1MYCbAwIEDbWpqqmPhRjOatLQ0BgwdxsUvfsmGog9YZhay2W7m8ZMep1/Lfo5dqyFIS0vDye9PQ6a+dI760jmh1Jdr1qwhLs79QvLUqVOZOnXqEdvvvfdeSkpKaNKkCVOmTGHKlCmH9r377rvcdNNNFeaPioo6tIilVKqiMUigxwDBHY84Ye2eC+m+bjontcrG9BjndpyQEEr/p9UF6q/AqL8Co/6quoydW+B7CI8ID6k+q61bMGYDl/hXmR4KHLTW7gCWAF2NMR2NMZHABP+xroiLiuAfl5xIt8gJ3NbnWSI9kVz+yeXM2zzPrUgiIlKPPfjgg2zZsqXcfb169WLUqFG1nKheqspYo6JxSsjZlZRKZuMObHnnLrwlJW7HERGRIDH+iXq21n6ld4YjMyCMMf8BUoHmxph04B4gAsBaOwOYC5wBbABygcv9+4qNMTcAnwBhwD+ttaucyFRdLeOieO+6YQCc3uN1bvziRqZ8NYW84jzO7lLr62OKiEg9NmjQoApnOEyaNKmW09RPFY01jDHX+vdXOE4JRdYTxrru1zFo2W38MO9f9D89ZL8UERGpjPUtcB1qyw47UoCw1l54lP0WuL6CfXPx/eCvU/KLSnhl4W4ePH469383hbsX3U1sRCyjOujdKBERkVBS3ljDX3go/bzCcUooOu6Mq9i8/BmaLXkc7+iJeMJr645bERGpNYeeeBRa6/2E1nyNWrTtQB5PfLqeVxft4KmRT9ErsRdTv5rKsl3L3I4mIiIiUqGw8HD2DJxMB+9WfvjoH27HERGRIEiI9hWXOzcJrSKzChAV6NwilnF9W/PK4l8pKAzn2VHPkhybzOS0yezK2eV2PBEREZEKDTjtUjZ5Umi57ClKiovcjiMiIg4L8998ERGmGRD1xvUju5BbWMJLizbTJKoJT6Y+SW5xLpO/nExhSaHb8URERETK5QkLw5t6O+3sdsJW/sftOCIi4rC8omIA9uaF1ioQKkBU4pikOE47NolZC38hr7CELk278MCwB1i5ZyUPf/ew2/FERCTEWWs5+eSTyczM5MCBAzz33HOH9u3Zs4cxY8a4mE5CXZfhF0CbgZD2MBTlux1HREQclFfoK0DsynM5SIBUgDiKK0/sRN92TcjIKQDg1JRTufzYy3lz/Zt8vuVzl9OJiEgo++STT+jbty/x8fFHFCBatGhBcnIyCxcudDGhhDRjKEz9E2Sm893bj7udRkREHFX6GM7QugUjtFascMHgjs145cohv9l2Y/8bWbxjMfctuo++LfrSPLq5S+lERCQgL409ctux42HwJCjMhdfOO3J/v4ug/8WQkwFvXvLbfZd/WOnlHnnkEaKiorjpppu45ZZbWLFiBV988QWff/45L730EgUFBVx/ve/hC9OmTWPjxo3069eP0aNH8+ijjzJ+/Hhee+01hg0bVt2vWBq4yK4j+alRP7qsm0FO5nXExDd1O5KIiDhBT8Go33YczGPrvlwAIsIieGj4Q+QW53LXwruwNrTuuxERkdoxYsQIFixYAMDSpUvJzs6mqKiIr7/+muHDh/Ptt98yYMAAAB566CE6d+7M8uXLefTRRwEYOHDgofNFqit89D00I5Mf39XtoyIi9YYtnQERWjQDogoKi72MeXIBp/ZM4tHz+gLQqUknbhlwCw999xDv/PwO5x5zrsspRUTkqCqbsRDZuPL9MYlHnfFwuAEDBvD999+TlZVFo0aNOO6441i6dCkLFixg+vTp7N+/n7i4uArPb9myJdu3bw/omiKH6z7wZJZ9MYxjN88iM2My8Ykt3Y4kIiI1ZEpLD0YzIOqdyHAPp/ZMYu6PO8grLDm0/aLuFzGo1SCeWPoEe3L3uJhQRETqooiICFJSUnjppZc44YQTGD58OPPnz2fjxo306NGD8PBwvF5vhefn5+cTHR1di4mlvmoy9l5ibD5r3rnf7SgiIuKAhCjfXIJuTUNrToEKEFX0u+PaklNYwrzVOw9tM8Zw99C7KSgp4K/f/dXFdCIiUleNGDGCxx57jBEjRjB8+HBmzJhBv379MMbQpUsXNm3aBEBcXBxZWVm/OXf9+vX06tXLjdhSz3Q6djC/thnLoN1vQ+YOt+OIiEgNefwzIMLCNAOiXhrSsRltmkTz7rJtv9mekpDCNX2v4dNfP2X+lvkupRMRkbpq+PDh7Nixg+OPP56kpCSioqIYPnw4AKeddhppaWkAJCYmMmzYMHr16sWUKVMAmD9/PmPHlrNwpkg1dDz3ATzeIljwmNtRRESkhnL9j+HcmRNaq0CE1nwNF3k8hvH9W/P3r37hYG4RCY0jDu27/NjL+eiXj3jw2wcZkjyExhGNXUwqIiJ1yahRoygqKjr0ev369Yc+v/TSS7nuuuu46qqrAPj3v//9m3Nnz57NBx98UDtBpf5r1pGDPS8iZslL7O01iVYdurudSEREqqmgqJjGwJ680CpAaAZEAC4f1pGvbxv5m+ID+J6KcdfQu9iVu4t//PgPl9KJiEioadWqFZMmTSIzM/OIfXv27GHy5Mk0barHJopzcodOpsiGsf29u92OIiIiNVL6FIzQ+pU+tNK6rHlsI1rGR5W777ik4xjbaSyzVs1ia+bWWk4mIiKh6vzzzyc+Pv6I7S1atGD8+PEuJJL6LLltR5a1Oo9+++eRvm6Z23FERKS6rH8R69BaAkIFiEBt2J3NxBe/5eddWUfsmzxgMuGecB5Z+ogLyURERESOrtv/u4scosj4711uRxERkWornQERWhUIFSACFB8Vztcb9vLhj0euIN2ycUuu6XMNaVvT+Hrb1y6kExEREalc85bJrGg/kb7ZX7Np+ZduxxERkeo4tPSDChD1Wsv4KAalNGNuOQUIgIk9J9IhvgMPf/cwRSVF5R4jIiIi4qbev7ud3IimtF/+uNtRRESkGppEhwHQq3mYy0kCowJENYztncz6Xdls2H3kbRiRYZFMHTSVzZmbeX3d6y6kExGRum7Hjh2ceeaZACxfvpy5c+ce2jdnzhzuuecet6JJA5HQtBmNT55K+OYvYZNmQYiIhBpTugaEJ7QebKkCRDWc3qsVxsCHK3eWu394m+EMTR7KCytfILPwyJXNRUSkYXviiSeYNGkScGQBYuzYscyePZvc3Fy34klDMfAK8hsns+Wt27Ber9tpREQkALn5hQBszXY5SIBUgKiGlvFRXDi4PW2aRpe73xjDrQNvJbMgk3+s1GM5RUQaqrvuuounnnrq0Os777yT6dOn88477zBmzBgKCwu5++67eeONN+jXrx9vvPEGxhhSU1OZM2eOi8mlQYiIYlmnP9A+bw0/fvaq22lERCQAhUW+AsTu/NBaAyK05mvUIX85p3el+7s36864zuN4bc1rTOg+gdaxrWspmYiIlOfh7x5m7b61jrbZvVl3bht8W4X7r7zySn73u99x88034/V6ef3113nzzTd5+eWXadSoEYWFhdx///0sXbqUZ5555tB5AwcOZMGCBZx//vmO5hU53KCzr+OX1f+g6eKHKBk5gbCISLcjiYhIFRhvMQBetAZEg5FdUMyG3RXPebmx/40YY5j+w/RaTCUiInVFSkoKiYmJ/PDDD8ybN4/+/ftTUFBAixYtKj2vZcuWbN++vZZSSkMWERHB3iHTaOfdxorZT7sdR0REqqq0AGFCqwChGRA1cOWsJeQUFjPnxuHl7m8V04rf9/g9L/70IhN7TuTYxGNrOaGIiJSqbKZCMF111VXMmjWLnTt3csUVVxAdHU1+fn6l5+Tn5xMdXf5tfiJOGzj6QlYteZYOP04nf8xVRMUkuB1JRESOxlvi+yPE5hSEVto6ZmT3lvy0LZNtB/IqPObK3lfStFFTnlj6BNbaCo8TEZH66ZxzzuHjjz9myZIlnHbaaRxzzDFs3rz50P64uDiysn77VKX169fTq1evWk4qDZXxePCceh+JHCByyQtuxxERkSowNjRnQKgAUQOjeyYB8NnqXRUeExcZxzV9r+G7nd+xYNuC2oomIiJ1RGRkJCNHjuT8888nLCyMmJgYOnfuzIYNGwAYOXIkq1evPrQIJcD8+fMZO3asm7GlgekxeDR0PxPPoumQs9ftOCIichTxkb7FJwcmN3I5SWBUgKiBzi1i6dwihnmry38cZ6nzu51P29i2PP3D03itHnMlItKQeL1eFi9ezJVXXnlo2w033MCsWbMAaNasGUuWLGH58uVccMEF7Nq1i7y8PHr3rnyxYxGn2VF34y3MYeV//uR2FBERORr/GhBWMyAaltE9W/Htpn0czCuq8JgITwTX97+etfvWMu/XebWYTkRE3LR69Wq6dOnCqFGj6Nq166Ht55xzDikpKeWes2XLFh5//PFaSijyP6ZFN75JOIPuW99k79Z1bscREZFK5BUUALDhoMtBAqQCRA1NPL4DH940nPioytfzPD3ldLo06cKzPzxLsb9aJSIi9VvPnj3ZtGlTuQWFq666qtxzBg0aRL9+/YIdTaRcHX53HyWEkf7OnW5HERGRShQX+d4A35lnXE4SGBUgaqhNk2i6tYrDmMq/8WGeMG7sfyObMzcze+PsWkonIiIiUnVtO3RhSasL6HfgU9JXL3Y7joiIVOTQUzB0C0aD89O2g0x7ZyUFxSWVHjey3Uj6NO/D8yuep6CkoJbSiYiIiFRdz/Pu4YCNJXOOZkGIiNRZh56CEVq/0odW2jpqT3YBry/ZyqKNGZUeZ4zhpuNuYmfOTt5c92YtpRMRERGpuubNW7D52OvombsUNs53O46IiJTD+G/r1wyIBuiEzonERIYxb1XFj+MsNSR5CEOSh/CPH/9BTlFOLaQTERERCUy/3/0fJLSHz+4Fr57gJSJS1xj/LRieMBUgGpxG4WGkdmvJZ2t24fXaox5/U/+b2Je/j1dXv1oL6UREREQCFN6I/OHTYMdy1n7+sttpRETkMDERvj+Pb9PI3SABUgHCIaN7JrEnq4Dl6QeOemyfFn04ud3JzFo1i4MFIfbcFBERqRFrLSeffDKZmZkcOHCA55577tC+PXv2MGbMGBfTifyPp8/5bDApJHzzEN4irV0lIlKn+G/BsEYzIBqkkd1a0iGxMfuyC6t0/A39byCnKIcXf3oxyMlERKQumTt3Ln379iU+Pp6DBw/+pgDRokULkpOTWbhwoYsJRXwiIyPYPeR2kr07+em/T7kdR0REysjJ9xWG1+xzOUiAwt0OUF8kNI7gyykjq3x816ZdGdtpLP9e828m9phIi8YtgphORER2/uUvFKxZ62ibjXp0p9Udd1S4f8aMGcyYMQOAgwcPkpKSQnJyMldffTUA99xzDxs3bqRfv36MHj2aRx99lPHjx/Paa68xbNgwR7OKVMfQ0eezfOmzdFj5DIWnXUNkTILbkUREBCgq9L3xnRFiE9Q0A8JhJV5LflHlj+MsdV2/6yjxljBz5cwgpxIRETdce+21LF++nCVLltC2bVsmT57MwoULGTBgAAD33XcfnTt3Zvny5Tz66KMADBw4kAULFrgZW+QQT5gH76h7acpBVr/9Z7fjiIiIn7fEdwuGJ8RuwXBkBoQxZgzwFBAG/MNa+9Bh+6cAF5e5Zg+ghbV2nzFmM5AFlADF1tqBTmRyw8G8IkY9nsbVIzpx9YjORz2+XVw7zul6Dm///DaX9bqMNrFtaiGliEjDVNlMhWC7+eabOfnkkxk3bhwXXXQRcXFxFR7bsmVLtm/fXovpRCrXf+jJrF82mj5bX4HMyRDf2u1IIiINnrekiCIbRliYcTtKQGo8A8IYEwY8C5wO9AQuNMb0LHuMtfZRa20/a20/4HbgS2tt2btVRvr3h2zxASAhOoIWcVF8tnp3lc+5ps81ePDw/PLng5hMRETcMmvWLH799VfuueceAMLDw/FW8ljD/Px8oqOjayueyFEZYzjmosfwWC/Mf9DtOCIiAtiSIooJI8TqD47cgjEY2GCt3WStLQReB86u5PgLgf84cN06aXTPJJb+uo99OVVbjDIpJokJ3Sfw303/ZdPBTUFOJyIiten777/nscce49VXX8Xj8f3I7datG5s2+f6/j42NJSsr6zfnrF+/nl69es5S56kAACAASURBVNV6VpFKNU0hs88VeH94jV3rl7qdRkSkwQvzFlJkIogKD60KhBO3YLQBtpZ5nQ4MKe9AY0xjYAxwQ5nNFphnjLHAC9bachdEMMZcDVwNkJSURFpaWs2Tl5Gdne1Im81yS/BaePa9LxneNqJK53Qv6U4EEdw7716uaHFFjTO4zam+FPWlk9SXzgmlvkxISDjiF/za9Le//Y2MjAxOOukkAPr3788pp5zCxx9/zKWXXkqTJk0YPHgwPXv2ZPTo0TzwwAN8/PHHjBo16qi58/PzQ+b7IPVDzpA/4l32Cvs/mEbSlM/cjiMi0qA1awTENKZnYsNbA6K8kout4NhxwMLDbr8YZq3dboxpCXxqjFlrrf3qiAZ9hYmZAAMHDrSpqak1jP1baWlpONGmtZYXVn9Buk0gNbXqd5Rs/mEzM1fOJKl3Ej0Se9Q4h5uc6ktRXzpJfemcUOrLNWvWVLreQrC9+uqrR2zbsWMHl1xyCTfccANZWVm89dZbv9n/ySef8MEHHxw1d1RUFP3793c0r0hlklsl83nKJEb9+iS/LJ5Nx6FnuR1JRKThKimEsEZupwiYE7dgpAPtyrxuC1S0etYEDrv9wlq73f/nbuA9fLd0hCxjDHec0YNLT0gJ6LxLj72U+Mh4nln+THCCiYhInZCcnMykSZPIzMw8Yt+ePXuYPHkyTZs2dSGZyNENOn8qW2lF2Gd3Yf0rsIuISO07kJXNzlwv27IqXleqLnKiALEE6GqM6WiMicRXZJh9+EHGmATgJOCDMttijDFxpZ8DpwI/OZDJVeP6tuaEzs0DOic+Mp7Le13OV+lfsXz38iAlExFpeKytaFKee84//3zi4+OP2N6iRQvGjx9/1PPr4tckDUN8TAwb+txK++LNrP14httxREQarMKCPA4WesgrCa0xQY0LENbaYnxrOnwCrAHetNauMsZca4y5tsyh5wDzrLU5ZbYlAV8bY1YA3wEfWms/rmmmumBl+gE+/mlnQOdc1P0iEqMSmf7DdA0uRUQcEBUVRUZGRr36P9VaS0ZGBlFRUW5HkQZq2Lgr2RHfh25rpkNhztFPEBERx5mSQgoJJ8TWoHRkDQistXOBuYdtm3HY61nArMO2bQL6OpGhrnnhy018t3kfp/ZMwuOp2t+KxhGNmdRnEg999xCLdyzm+NbHBzmliEj91rZtW9LT09mzZ4/bUcqVn59frUJCVFQUbdu2DUIikaOLjAgj+dzH4J+nwqJnIPU2tyOJiDQ4pqSQIsKp4q+adYYjBQg50uieSXz44w5WpB+gf/uq38t73jHnMWvVLJ7+4WmGJg/FmBD7GyUiUodERETQsWNHt2NUKC0tTQtJSmhqP4T9KWcQ/eUTePtcTONmKoiJiNQm3wyICMJCrALhxBoQUo7Ubi0I8xg+Xb0roPMiwyL5Q98/8OPeH0nbmhaccCIiIiI1tG3AFDzeIja+8Se3o4iINDjhFBEeGUVkiP1GH2JxQ0eTxpEMTmnGZ2sCK0AAnNX5LDrEd+Dp5U/jtaG1qqmIiIg0DL16H8eCJuPpufN9Mn75we04IiINSkKEl4GdWtGicWj9Sh9aaUPM6J5JbNufR0Z2QUDnhXvCua7vdfy8/2c+/qVerMkpIiIi9VCXc+8jm8bsfXea21FERBqW4kIIj3Q7RcBUgAiiCYPbsezu0STGNgr43DEdx9C1aVeeXf4sRd6iIKQTERERqZkO7dqxuO3ldMtazLbvP3Q7johIg5GTl8eXGw+SXRhaT/pSASKIGkeG0yg8rFrneoyHG/vdyJasLczeMNvhZCIiIiLOGHT+NLKjW9P62wfBW+J2HBGRhqG4gF05oVV8ABUggu7rn/dy+lML2J9TGPC5qe1S6d28NzNWzqCwJPDzRURERIKtWUIcsWf8GbN7Fax43e04IiINgsdbSCHhhIXYb/QhFjf0xEeHs2ZHJl+s3R3wucYYbux/IztzdvLW+reCkE5ERETEAb3+H/ub9mHfnLvxFuS4nUZEpN7zeIsoJIIQewqnChDB1qt1Aknxjar1NAyAoclDGdxqMDNXziS3KNfhdCIiIiIOMIYVPf6PZiV7WfveX91OIyJS74V5CykkgjAVIKQsj8dwSo8kvly/h/yiwO+LLJ0FsS9/H/9e++8gJBQRERGpuRGnnMXCyBNIWft38jLS3Y4jIlJ/WUuYLaJZfIwKEHKkU3omkVtYwjcbM6p1fr+W/RjRdgT//OmfZBZmOpxOREREpOY8HkPM2AcJs8X88qYeyykiEjTeYgyW84d0wZjQqkCoAFELTuicyJl9komPjqh2Gzf2v5GswixeXvWyg8lEREREnNOv73F82fT/0X3nHA5uXOJ2HBGR+qm4wPdneKS7OapBBYha0Cg8jGcuOo4BHZpWu43uzbpzWsppvLL6FTLyqjeTQkRERCTYup9/H8VRTUn48m6wofeIOBGROs9fgPj7om0uBwmcChC1aOu+XHZl5lf7/Ov6XUdBSQEv/vSig6lEREREnNO+dTKRo++CLYuwqz9wO46ISP1TnAfAtpzQuv0CVICoNZn5RaQ+lsYr3/xa7TY6JXRiXKdxvLH2DXbm7HQwnYiIiIiD+l9CRkwX9rx3G7Yoz+00IiL1i///1SJPI5eDBE4FiFoSHxXBwA5Nq/04zlJ/6PcHvHh5fsXzDiUTERERcVhYOMu6/x8ti3fy8+zH3E4jIlK/FOX6/vBEuRwkcCpA1KLRPZNYuzOLrftyq91Gm9g2TOg2gfc3vM+G/RscTCciIiLinNQzLuCbsEG0+fFZCg9o5qaIiGNKZ0AYFSCkEqN7JgEwb3XNZkFc0+caYsJjeHLZk07EEhEREXFcRJgHc9oDRNpCNr11h9txRETqD/8MiM5tWrgcJHAqQNSiDokxHJMUy+c1vA2jSVQTrux9JV+mf8mSnXrElYiIiNRNQwYN4fO4s+i67T0K0le6HUdEpH4o8j3Y4IZTe7scJHAqQNSyx8/rx9MX9q9xOxf3uJikxkk8vvRxvNbrQDIRERERZxlj6H7Bn6FRPI0+/5Meyyki4gT/DAgiGruboxpUgKhlvdsmkBhb89VKo8KjuLH/jazKWMW8zfMcSCYiIiLivJR27Qg7+Q745UuK13zodhwRkdDnXwNiygfrXQ4SOBUgXDBn5XYe+mhtjds5s9OZHNP0GJ5c9iSFJYUOJBMREREJgoFXsLtRB/a+OxWKNWYREakRfwFiZ17o/TofeonrgdXbM/n7gk3sz6nZD+AwTxi3DriVbdnbeGPdGw6lExERqT+MMc2MMZ8aY372/9m0nGPaGWPmG2PWGGNWGWNudiNrvRYWwZJjbqVV8TZ++UiLaIuI1Ij/FgxPRLTLQQKnAoQLzuidTInXMm91zR9JdUKbEzg++XheWPkCmYWZDqQTERGpV6YBn1truwKf+18frhi41VrbAxgKXG+M6VmLGRuEUeMu5ltPf5ove5KS7L1uxxERCV3+GRBhkSpASBUc2zqeds2imfujM8/EvmXALWQWZPLijy860p6IiEg9cjbwsv/zl4Hxhx9grd1hrV3m/zwLWAO0qbWEDURUZDh5J99HtDePTW/d6XYcEZHQVZxHAZFENYpwO0nAwt0O0BAZYzijVzIvfv0LB3OLSGhcs784PRJ7cGanM3l19atM6DaB5Nhkh5KKiIiEvCRr7Q7wFRqMMS0rO9gYkwL0B76tYP/VwNUASUlJpKWlOZm1xrKzs+tcprKstcwNH8UZv77J4jnDyI9t73akOt9ndY36KzDqr8Cov6qmy+YNNDGRJBTtIzu7IKT6TAUIl5zRO5nvNu9jV1Z+jQsQADf2v5F5v87jb8v+xiMjHnEgoYiISGgwxnwGtCpnV0BvsxtjYoF3gD9aa8u9r9FaOxOYCTBw4ECbmpoaWNggS0tLo65lOtzmzp3wvHYiQ/e/D2PfAWNczRMKfVaXqL8Co/4KjPqrig6+BZnx/PWy0SHXZ7oFwyV92zXhveuGcUxSnCPtJccmc9mxl/HRLx/xw+4fHGlTREQkFFhrT7HW9irn4wNglzEmGcD/5+7y2jDGROArPrxmrX239tI3PCntO+A56TbY+Dm5q/RYThGRgBXlQQguQAkqQLguM7+I/KISR9q6otcVtGzckoe+ewiv9TrSpoiISIibDVzq//xS4IPDDzDGGOBFYI219olazNZg2cGT2B7eluwPpkJxgdtxRERCii3KZe2+Yp6dv8HtKAFTAcJFG/dkM/DPn/HRTzscaa9xRGNuGXALqzNW88GGI8ZXIiIiDdFDwGhjzM/AaP9rjDGtjTFz/ccMAyYCJxtjlvs/znAnbsNgwhuxpNtUWhZtY/Pcx9yOIyISUmxhHjneSKy1bkcJmAoQLuqYGENibKRjT8MAGNtxLH1a9GH6D9PJKcpxrF0REZFQZK3NsNaOstZ29f+5z799u7X2DP/nX1trjbW2j7W2n/9jbuUtS02dNv73fO0ZRMtlT1N8YLvbcUREQoa3MI88G0lURJjbUQKmAoSLPB7DmF6t+HL9HjLzixxp0xjDtEHT2Ju3l7+v/LsjbYqIiIg4LSoijJLRDxBmi/j1zdvcjiMiEjJsYQ75RBIdqQKEBGhc39YUFnv55CfnZkH0btGbszqfxb9W/4utmVsda1dERETESSOGDuHjuN/Refts7NYlbscREQkNhdnkEE1UuAoQEqD+7ZrQrlk0s1c4O/Xw5uNuJtwTzuPfP+5ouyIiIiJOMcZw/GV/xca2wnw0FbxaRFtE5GjCirJp1aI5nVrEuB0lYCpAuMwYw0O/68O9Zx3raLstG7dkUu9JfL7lc77Z/o2jbYuIiIg4pWXz5pjR98H2ZWR++y+344iI1HmewhyGdGtP//ZN3Y4SMBUg6oBhXZrTuUWs4+1ecuwltItrx1++/QuFJYWOty8iIiLiBG+v81gT1g3vp/di8w+6HUdEpO4qKYbiPGyk878/1gZHChDGmDHGmHXGmA3GmGnl7E81xhws82iru6t6bkPxzcYMHvtknaNtNgprxB1D7mBz5mZmrZrlaNsiIiIiTvGEhfHzcXfRxLufLe/f73YcEZG6qzAbgAc/38ryrQdcDhO4GhcgjDFhwLPA6UBP4EJjTM9yDl1Q5tFW9wd4br23Iv0Az8zfwJaMXEfbPbHNiYzuMJqZK2eSnpXuaNsiIiIiTjl9zFg+ijiF1mtnUbhrvdtxRETqJn8BIttGExURejc0OJF4MLDBWrvJWlsIvA6cXQvn1ivj+rYGYPaKbY63PXXQVDzGw8PfPex42yIiIiJOiAjzkHDmA+TZCHa+eYvbcURE6qYCXwEix0YRHRF6T8EId6CNNkDZZz2mA0PKOe54Y8wKYDvwf9baVQGcizHmauBqgKSkJNLS0mqevIzs7GzH2wzUMU09/Hvhzxxr0jHGONr2abGn8X76+zw992l6N+7taNuHqwt9WV+oL52jvnSO+tI56kuR3zqhbw/eSLuECzJewK7/BHPMaW5HEhGpW0pnQBDdYAsQ5f2mbA97vQzoYK3NNsacAbwPdK3iub6N1s4EZgIMHDjQpqamVjtwedLS0nC6zUClR/3Kn97/iZbdjuPY1gmOtj3MO4yf/vsTc3LncOXoK2kc0djR9suqC31ZX6gvnaO+dI760jnqS5EjnXHVPdh/foH55A7oNBLCI92OJCJSdxRkAb4ZEFGRoVeAcOIWjHSgXZnXbfHNcjjEWptprc32fz4XiDDGNK/KuQ3J2N7JdGwew56sAsfbjvBE8Kehf2J7znb+/uPfHW9fRERExAlxMTGYMQ9BxgYyvpjudhwRkbrFPwNiVL9OITkDwokCxBKgqzGmozEmEpgAzC57gDGmlfHfU2CMGey/bkZVzm1ImsZE8sWtJ5HarWVQ2h+QNICzOp/FrFWz2HhgY1CuISIiIlJT3s6jWBQ2kOhvHsNm7XQ7johI3eFfA+KaU/oSEdYAF6G01hYDNwCfAGuAN621q4wx1xpjrvUfdi7wk38NiOnABOtT7rk1zRTKjDEUlXiDMgsCYPKAycRExHDvonvxWm9QriEiIiJSEx6P4cDw+wj3FrL1rdvcjiMiUnf4Z0Dke4J3S30wOVIysdbOtdYeY63tbK190L9thrV2hv/zZ6y1x1pr+1prh1prF1V2bkNmreWsZxZyx3s/BqX9xOhEpg6ayvI9y3l97etBuYaIiIhITY0ZMYzZ0efQfsv75G1a7HYcEZG6wb8GxKhnvnc5SPWE3pyNes4Yw4iuzZm/djd7s4MzC2Jcp3Gc0PoEnlr2FDuydwTlGiIiIiI14fEYOp97DzttUw6880fwlrgdSUTEfYXZePFgIqLdTlItKkDUQecOaEux1/L+D9uC0r4xhruPvxuL5f7F92NtuQ8eEREREXFV/y7t+aztjSTnrMEu+5fbcURE3FeQTb4nmuhIJx5oWftUgKiDuibF0bddE97+Pj1oxYE2sW24qf9NfL3taz785cOgXENERESkpi684hbocCLm8/sgd5/bcURE3FWYTb5pTHQIPoITVICos84d0Ja1O7NYvSMzaNe4sPuF9Gneh4e/e5h9+fqBLiIiInVPWJgHzngUm5/J7vfvcDuOiIi7CrLINdFEhasAIQ46q29r/jNpKD1axQftGmGeMO474T6yi7J5YPEDuhVDRERE6iTbsgfvRZ5J8/WvU/DrUrfjiIi4pzCb6NgEzhvY1u0k1aICRB2VEB3B8Z0T8XhMUK/TpWkXru93PZ/++ikf/fJRUK8lIiIiUh3GGFqffR97bQL7374JvHqUuIg0UPkHSUxsyXkD27mdpFpUgKjD8otKuHf2Kj5YHpzFKEtdduxl9GnRhwe/fZDdubuDei0RERGR6hjasyNzW11Hq6xV7Fv4ottxRETckX+Q/PBYsvKL3E5SLSpA1GGNwj0s2riXF7/+JajXCfeE85cT/0JhSSF3L7pbt2KIiIhInTR6wo0std2JnH+/FqQUkYYp/yBzf87jwQ/XuJ2kWlSAqMOMMVw8pAMr0w+yMv1AUK/VIb4Dtwy4hYXbFvLOz+8E9VoiIiIi1dGmaWM2D7mfxt4c7BcPuB1HRKR2WQv5B9lX0pioCC1CKUFwznFtiI4I49/fbgn6tSZ0n8CQ5CE8uuRRtmZtDfr1RERERAJ17hmn4RlyNWbpP2H7D27HERGpPUV5UFLI3pIo4qLC3U5TLSpA1HHxURGc1bc1HyzfTmaQ7/PxGA9/PuHPeIyH2xfcTrG3OKjXExEREamWkbdTEJVIxptakFJEGpD8gwBk2hjioyJcDlM9KkCEgInHd2Bc32TyCkuCfq3k2GTuGnoXK/asYMaKGUG/noiIiEjAohJ4JfYKEg+sJHPxLLfTiIjUjkMFiMaaASHB06tNAo+c25ek+Khaud4Znc7g7M5nM3PlTJbsXFIr1xQREREJxCkX3Mz33m54Pr8X8va7HUdEJPjyfesCnjmkBwM6NHU5TPWoABFCVqYfYPX2zFq51h1D7qB9fHtuX3A7BwsO1so1RURERKoqpUUsP/W7i+jiTHZ9cJfbcUREgs8/A2LMgO50TYpzOUz1qAARIopKvFz58lIe+WRtrVyvcURjHhnxCBn5Gdyz6B49mlNERETqnPPOPJ13w0+n+drXKNm23O04IiLB5S9A/JobTmFxaK5/owJEiIgI8zBxaAfS1u1hw+6sWrlmz8Se/PG4P/L5ls95c92btXJNERERkapqHBlOi3H3UxTZBM9HU7QgpYjUb/4CxDn/XE36/lyXw1SPChAh5OIh7WkU7uHFrzfX2jUn9pzIiW1O5OElD7Nq76pau66IiIhIVaT260rU6Q9g0r+DFf9xO46ISPDk+daAyKIxcXoKhgRbYmwjfndcG95dls6+nMJauabHePjriX+leXRzJqdN1noQIiIiUvf0vZDdTfqR8+GdkLvP7TQiIsGRf4AiTxRFhOspGFI7rhjWkUbhnlpbjBKgSVQTnkh9gj15e5i2YBpeq+mNIiIiUod4PMxueyuNig6y54M73U4jIhIc+QfJD4sjMtxDVESY22mqRQWIENM1KY7v7jyFE7s2r9Xr9mrei2mDp/H1tq+ZuXJmrV5bRERE5GguHHcGb4WNJXHdfyje8p3bcUREnJd/gNywWOJDdPYDqAARkqIiwvB6LTsO5tXqdc875jzGdRrHc8ufY9G2RbV6bREREZHKxDQKp9mZ97DLNuXgWzdASbHbkUREnJV/kJj4RO4ed6zbSapNBYgQdetbK5gwczHFJbV3O4QxhruOv4suTbsw5aspbM3cWmvXFhERETmaU/t34e0W15OYtY78RTPcjiMi4qz8g8QmJHJW39ZuJ6k2FSBC1Jherfg1I5c5K3fU6nWjw6N5KvUpjDHc8MUNZBdm1+r1RURERCpijOGsCdeyv/UIohY8BJm1O04SEQmq3P3sJ5ZNe0L3dzAVIELU6B5JdG8VxzPzN+D12lq9drv4djxx0hNsydzCbQtuo8RbUqvXFxEREalIh+axND33KfAWUfLx7W7HERFxTm4Gn/1azGPz1rmdpNpUgAhRHo/h+pFd2LA7m49X7az16w9OHsy0wdP4Kv0rnlr2VK1fX0RERKRCzTqxpN3lhK1+j6J1n7mdRkSk5oryoCiHPSWxxDWKcDtNtakAEcLO6J1Mp+YxvPN9uivXv6D7BVzQ7QJeWvUSszfOdiWDiIiISHlyBl3PRm8yOe//EYry3Y4jIlIzufsA2FkUQ5yegiFuCPMY/nnZIGZMHOBahtsG38aQVkO4d9G9LNm5xLUcIiIiImWl9mzH7DaTaZK3lczPHnE7johIzeRmALCrOIb4aM2AEJekNI8hIsxDXmFJra8FARDhieDx1MdpF9eOm7+4mQ37N9R6BhEREZHynH/BROZ4hxH97XTI2Oh2HBGR6vMXIPbZOM2AEHdt3pvDSY/OZ86P7qz0nNAogRmnzCAqPIprP7uWA8UHXMkhIiIiUlabJtFknHgP+Tac3Pf/CLb236wREXGEvwAx+ayhnNy9pcthqk8FiHqgXbPGNIuJ5PF56ygq8bqSITk2medOeY7somye3/08WYVZruQQERERKevCkwdRcNKdNN76Fax61+04IiLV418D4vjex9AhMcblMNWnAkQ9EOYxTB3TjV8zcnl9yVbXcnRv1p0nUp9gZ9FObpl/C4Ulha5lEREREQGIDPfQPPU6SO5H8dzbIT/T7UgiIoHzz4CYv6WQzPwil8NUnwoQ9cTIbi0ZnNKM6Z//TG5hsWs5Tmh9AhcnXsy3O79lypdTKPKG7j8OERERqSc8YcxuNwVPzm4OzL3X7TQiIoHLzaAwIoHL/7Wcrfty3U5TbSpA1BPGGG47vRt7sgqYs9KdtSBKDY4dzO2Db+eLrV/wp6//RIm3xNU8IiIiIkOHn8KbZjTxK1/Cbv/B7TgiIoHJ20dBZBMA4qP0FAypAwZ0aMYH1w/jvAFt3Y7CRT0u4ubjbmbuL3P58+I/Y7Xok4iIiLioZVwU3pF3kWHjOPjmDaA3SEQklORmkBueAKCnYEjd0bddE4wxHMh1f/2Fq3pfxaTek3jn53d4dOmjKkKIiIiIqy4Y3pt/xl5NkwM/kb/webfjiIhUXW4G2WG+AkRsIxUgpA75+ue9HP/XL1i2Zb/bUbix/41c3ONiXln9Cn/7/m8qQoiIiIhrwjyGMRdcz0LTj4gv/wIH092OJCJSNbn7yDLxxESGER4Wur/Gh25yqVC/9k2Iiwrn3tmr8Hrd/YXfGMPUQVOZ0G0CL616iUeWPKIihIiIiLimb/umDLzuJcKMhblTQOMSEanrrIXcDDq1b88/LxvkdpoacaQAYYwZY4xZZ4zZYIyZVs7+i40xK/0fi4wxfcvs22yM+dEYs9wYs9SJPA1dbKNw7jijByvTD/LW9+49lrOUx3i4Y8gd/L7H73l1zas8+O2DeK3X7VgiIiLSQDVq0QnvSbfDurk02/ON23FERCpXlAvF+cQnJjGkU6LbaWqkxgUIY0wY8CxwOtATuNAY0/Oww34BTrLW9gH+DMw8bP9Ia20/a+3AmuYRn7P7tWZQSlMe+XgdB3PdfxRm6UyIy3tdzhvr3uD+b+5XEUJERERcsyDxXFZ5O9Bh3d8h/6DbcUREKpazB4BVByL57pd9LoepGSdmQAwGNlhrN1lrC4HXgbPLHmCtXWStLV2QYDHg/mMa6jljDPed1Yus/GK++nmP23EAX6ZbjruFq/tczTs/v8O0r6ZRVOJ+cUREREQanhHdk3mnzRRii/eTPfdut+OIiFQsezcAr67K46WFv7gcpmacWD6zDVB2nn86MKSS468EPirz2gLzjDEWeMFae/jsCACMMVcDVwMkJSWRlpZWk8xHyM7OdrzNuuDh4Y2I27+etLT1tXbNo/Vlb3pzVpOzmL15Npt2buKqFlcR5YmqtXyhpL7+vXSD+tI56kvnqC9F3GOM4fLz/h+v/u2/TFz5Mgy6GNoNdjuWiMiR/AWIzfmxdI6NdDlMzThRgDDlbCt3NR9jzEh8BYgTy2weZq3dboxpCXxqjFlrrf3qiAZ9hYmZAAMHDrSpqak1Dl5WWloaTrdZlyzdvI/uyfG18siWqvRlKqkM2jCIexfdy0u5L/HcqOdIjA7t+5mCob7/vaxN6kvnqC+do74UcVe7Zo3ZkHIxO7csIe6t64m7eRGERbgdS0Tkt7J3AbApL4ZBMY1cDlMzTtyCkQ60K/O6LbD98IOMMX2AfwBnW2szSrdba7f7/9wNvIfvlg5x0NZ9uVwwczGPfrzW7Si/Mb7LeKafPJ1NBzZxyUeXsDXL/QUzRUREpGE5qWM8HyTfQlzmelj0tNtxRESO5F8DIoN4mof4DAgnChBLgK7GmI7GmEhgAjC77AHGmPbAu8BEa+36MttjjDFxpZ8Dp/L/2bvv8KjKtI/j3+dMS5v0G3L9MwAAIABJREFUXoBAIFTpRRBEAQugiNj7WkFR3l173V23Ka5t17X3snZELCsgEhVQQJTeQi8hCSFtkkySKc/7xwwIiAqmnGRyf67rXGfmlJnfHJIwc89TYHUjZBIHyYqP4NIh7Xnlm+0tbtCSEZkjeO6U56ior+DiTy5mWdEysyMJIYQQog2xGoopk2+CbmfAlw9C6RazIwkhxKGqivCGxePFSnxkGy9AaK29wFRgNrAOeEdrvUYpNVkpNTl42H1AAvDkYdNtpgALlFIrgCXAJ1rrzxqaSfzUrafmkhUfzu/fXk6Fu2UN/NgnuQ+vn/46MY4Yrp5zNTM3zTQ7khBCCCHaGPfov1PnN3C9fxPoI/YmFkIIc1QVYziTmfP7EQzPSTI7TYM0RgsItNafaq27aK07aa3/Ftz2tNb66eDtq7XWccGpNg9MtxmcOaN3cOmx/1zR+CIdVh6/oC+FlbXc/cEqdAv7j7VDTAdeH/s6/VP6c+/Ce3lk2SP4/D6zYwkhhBCijaiPSONxLsK5+2v8K98xO44QQvyoqhgjKoUuKU5iIlr3ODWNUoAQrUO/dnHcflou/drFmR3liGIcMTw1+inOzz2fl1a/xLT506iok3m5hRBCCNH0YsJtdDvz//jBn0Pdx7dDTcvqtiqEaMOqiihVsby8cCv1Xr/ZaRpEChBtzLUjOnHlCdkopVpcKwgAm2Hj7sF3c/fgu1lYsJALPr6A9aUta/BMIYQQQoSm8b0zeT/9Vqz1ldR8cqfZcYQQItAlrHovW9yR3P/xWqzGkSahbD2kANFGzVlTyIXPfYu7vuV1c1BKcUHXC3jp1Jeo99dzyaeX8EH+B2bHEkIIIUSIU0px7bln8pIeR8Sat2DrT2aGF0KI5lVfBZ4ainUs8ZF2DClAiNbIZjVYvLWUW99b0SJbQkBgcMp3z3iXPsl9uG/Rfdy38D5qPDVmxxJCCCFECGuXEEHmhD9RH90BZt0E9fLeQwhhoqpiAIr8zlY/AwZIAaLNOik3mVtPzeXjlXt4+suWO91UfFg8z4x+hmt6XcPMTTM5/+PzWbNvjdmxhBBCCBHCxvbvhH3iv6FsK/75fzc7jhCiLQsWIHbVO0mIdJgcpuGkANGGTTmxE+OPS2P67PXMX19sdpyfZTEs3NTvJp4/5XlqvDVc8uklvLj6Rfy6dQ/AIoQQQoiWy99+OAuix8E3/4Hd35sdRwjRVlUHPqdtr4siPkpaQIhWTCnF9HOOo2tqNAs2lZgd51cNShvEjDNncFLWSTy67FGunXMtu6t2mx1LCCGEECHIMBQLs6exV0dT894U8HnMjiSEaIuCLSAeueoU7j+zh8lhGk4KEG1chN3Ku5OP555x3cyOclRiHDE8fOLD/Hnon1lVsoqJH07kzfVvSmsIIYQQR6SUildKzVVK5QfXPzsXtVLKopT6QSn1cXNmFC3X1LH9edQ+mYiy9Xi/fszsOEKItqiqCJRBdHwqCVHSBUOEgCiHFaUUm4pdXP7iEspr6s2O9IuUUpzd+WxmTphJ3+S+/H3x37ly9pXsqNxhdjQhhBAtzx3APK11Z2Be8P7PmQasa5ZUolWIdFg57Zyr+MQ3CL6cDns3mh1JCNHWVBbgj0zhgdn5rC2oNDtNg0kBQhxQUlXPN5v3cdUr37XI6TkPlxaVxtOjn+b+ofezsXQjk2ZN4pkVz1DnqzM7mhBCiJZjAvBK8PYrwFlHOkgplQmMA55vplyilRiZm8zirndQre34PrwR/NLqUgjRjCoLqI1I5ekvN7OjtNrsNA1mNTuAaDmGdEzgsQv6cMN/v+fa177j2UsHEG63mB3rFymlmNh5IkPTh/Lg0gd5YvkTzNo8izsH38kJGSeYHU8IIYT5UrTWewC01nuUUsk/c9xjwG2A85ceTCl1LXAtQEpKCnl5eY0YteGqqqpaXKaW7miu2aAkgy36d/Td9G82vnkHBRljmydcCyQ/Y8dGrtexkev1UwML8ym0ZgKwfeNa8ko2HLK/tV0zKUCIQ4ztlcb0Scdx2/srueqVpbxw+cAWX4QASIlM4ZGRj7Bo9yL+seQfTPl8Cidnncxtg24jIyrD7HhCCCGakFLqcyD1CLvuPsrzxwPFWutlSqmRv3Ss1vpZ4FmAAQMG6JEjf/HwZpeXl0dLy9TSHfU10yPRr62m07bX6TL+JojJbPJsLZH8jB0buV7HRq7XEXxTCak5UAyjhw+mU1LUIbtb2zWTLhjiJ84dkMU/z+mNX2v8Wpsd55gMzRjK+2e+z7R+0/hmzzdMmDmBx79/HFe9y+xoQgghmojWerTWuucRlg+BIqVUGkBwfaR5p4cBZyqltgFvAScrpV5vthcgWgel+E/kVOrqvdTOnAat7D2SEKIVqq2Eukr2GYkAJETKNJwiRE3qn8l/rx5CpMNKhdvT4gemPJjdYufqXlfz4YQPGdVuFM+vep6xM8by2trXqPe1ntchhBCiUcwCLg/evhz48PADtNZ3aq0ztdYdgAuAL7TWlzRfRNFaTDhpKP/S5xO29XP0qvfMjiOECHWuPQCUGAlYDUV0mM3kQA0nBQjxswxDobXmpjd/4Jynv2FXWY3ZkY5JWlQaD454kLfHv01ufC7Tl07nzJln8smWT/D5W/4gm0IIIRrFA8AYpVQ+MCZ4H6VUulLqU1OTiVYnKz6CpNE3sdzfifqPb4XqfWZHEkKEssoCAMYO7c+qP52KYSiTAzWcFCDEL1JKcf3IThRX1jLxyUWs3l1hdqRj1j2hO8+NeY5nRj9DlC2KO76+g4mzJvLxlo/x+r1mxxNCCNGEtNb7tNajtNadg+vS4PYCrfVPRhLUWudprcc3f1LRWlxxQg4vxP8Bo95F3ce3mR1HCBHKggUIotNbxbh8R0MKEOJXDe6YwPtThmK3GJz3zDd8snKP2ZGOmVKKoRlDeeeMd3joxIewGlbu/PpOJsycwMxNM/H4PWZHFEIIIUQrYDEUU847k7cc5+BY9x5snG12JCFEqHIFChD//q6KNxZvNzlM45AChDgqnVOcfHD9UHJTnfz903XUelpnFwZDGZzW4TTeO+M9Hhv5GJG2SO5deC9nfHAGr619jar6KrMjCiGEEKKF654ezcU3Pw5J3eCjaeAuNzuSECIUVRZARALvrShh8ZZSs9M0CilAiKOWHB3GW9cO4b/XDCbMZqHe66ekqs7sWL+JoQxGtR/F2+Pf5j+j/kNyRDLTl05nzHtjmL50OrurdpsdUQghhBAtmGEPo3b8v/G7ivF+dpfZcYQQoaiyAO1Mo8RVR0JU658BA6QAIY6Rw2qhfUIkANM/W88pj37F7DWFJqf67ZRSjMgcwaunv8qb495keOZw3lz3JmNnjOUPeX/g2z3f4td+s2MKIYQQogVa6e/EU95xWFe8Afmfmx1HCBFqKgvwRqVRXe8jPSbc7DSNQgoQ4jc7f2AWaTFhXPfaMm57bwVVda17QMeeiT2ZPmI6/5v0P67ocQVLCpdwzZxrGDdjHM+vep4Sd4nZEYUQQgjRggzKjqewzzTy/RnUz5wKta1vsG4hRAtWWUC1IwWA1Jgwk8M0DilAiN8sMC7EMKaelMN7y3Zx6qNfsWRr6++blBqZyu/7/555587jgeEPkBaVxuPfP86Yd8fwf/P/j3nb51Hvqzc7phBCCCFagFvH9+YBx41YqovwfXa32XGEEKHCWwc1JVQ5knCGWUkLkQKE1ewAonWzWw1uOTWXk7om86dZa4iPtFFjdqhG4rA4GNdxHOM6jmNrxVZm5M9g1uZZzNsxD6fdySntT2Fs9lgGpA7AUFLLE0IIIdqi6DAbl0yaxHOvL2Ty8teg50TIGWV2LCFEa1cZGJMus11nVk04Fa21yYEah3xqEo2if/s4Zk0dRk6yE4Bb3l3BI3M2tPpuGftlx2Rz84CbmXfuPJ4e/TQjM0fyv63/46o5VzHmvTE8tPQhvi/6Hp+/dc4OIoQQQojf7qSuyVhOvov6uJzArBi1lWZHEkK0duU7Auu49kBg7LpQIC0gRKPZ/0vh8WtqPT7+9cUm3li8g2mjO3PBwHbYra2/3mU1rAzLGMawjGG4vW6+3Pkln2z5hDfXv8mra18lPiyekVkjOTnrZAanDSbMGhpNpYQQQgjxy645uTt0fhr94ikw9z7UGY+ZHUkI0ZoFCxCvr9NsXLGa+yf0NDlQ45AChGh0NkPxxEX9uGZ4OX//dB33fbiGp/M288TF/ejXLs7seI0m3BrOadmncVr2aVTVV7GgYAFf7PiCOdvmMCN/BuHWcIakDWFY+jCGZgwly5lldmQhhBBCNKG6tH58FT2JMctegh5nQceRZkcSQrRW5TtAWZi720J5XegMcCsFCNFkemfF8ta1Q/gqv4Tnv95CdnD6zk3FVSRFOYiJsJmcsPFE2aM4rcNpnNbhNDw+D0sKl/DFji9YsHsB83fOByDLmcXQ9KEMSx/GgNQBOO1Ok1MLIUTLoZQ6HrgEGA6kAW5gNfAJ8LrWOnTefYmQ5bBa+DTpKjqVLyDrgxuwTf0WHPL/vRDiNyjfAdEZ7K70kJMUZXaaRiMFCNGklFKc2CWJE7skAaC15g/vLGdjkYszjkvn4iHt6Z0ZEzJ9mgBsFtuBbhpaa7ZVbmNRwSIWFSxi1uZZvL3hbRSK3Phc+qf0p39Kf/ol9yMhPMHs6EIIYQql1P+AAuBD4G9AMRAGdAFOAj5USj2itZ5lXkohjs49Z/Xjlkdu4AXXvfjn/glj/MNmRxJCtEblO9CxWezZ6mZ450Sz0zQaKUCIZqWU4oGzj+P1xduZ+cNu3l22i54Z0Uwb1YUx3VPMjtfolFJkx2STHZPNxd0upt5Xz4q9K/iu8DuWFS3j/Y3v88a6NwDoEN2B/in96ZnYk56JPcmJzTE5vRBCNJtLtdYlh22rAr4PLg8rpULn3ZcIaQlRDiZOmMSL7y7k6u+ehx4TIHuE2bGEEK1N+Q487YZTXe8LmSk4QQoQwgTd06P5+8Re3Hl6V2b+sJs3Fu+gqLIWgLLqeuZvKOaUHqlEOULvx9NusTMwdSADUwcC4PF5WFu6lu+LvmdZ0TLmbp/L+/nvAxBmCSPNmsbiJYvpkdiD7gndaedsh9UIvesihGjb9hcflFIPaq1vP3jf/m1HKFAI0WKdcVwaU3+Yys7ty8n8cCpqyiJwhE4TaiFEE/PWQ2UBtZEZ5KY4aRcfaXaiRiOfZIRpnGE2Lj2+A5cMaY8/OK3t5+uKuPW9ldgtqxjcMZ7R3VI4uWsyWfER5oZtIjaLjd5Jvemd1Jvf9fwdWmt2uXaxqmQVq/etZuHmhbyf/z6vr3sdALthp1NsJzrHdaZLXJcD64SwhJDqxiKEaLPGALcftu30I2wTokVTSvGnSQOx734G9dYEmHc/jJ1udiwhRGtRuQvQRKd2YvbpodWCSgoQwnRKKSzBz86T+mWSnRjJ7DWFzFtfzB9nreGPs9aw5K5RJEeHsX1fNVEOKwlRDnNDNxGlFFnRWWRFZzG241jyqvM4YcQJbC7fzIayDeSX5bOxbCPfFHzDrM0/doWOc8SRHZNN++j2tI9uT4foDrSPbk+76HbYLXYTX5EQQvw6pdQU4Hqgo1Jq5UG7nMBCc1IJ0TBJTgd0PRH/oGsxljwD3c6A7OFmxxJCtAbBKTiJDb1Z9KQAIVoUw1AM6BDPgA7x3D2uO1tLqlm2vYzk6EC/p799so45a4vITXEyKDue3lmx9MmKJSc5dJs1Wg0rufG55MbnHrK9rLaM/LJ88svzyS/LZ1vlNr7e/TUfbPrgwDEKRXpUOh2iO5DpzCQtMo30qHTSItPIiMogITwBQxnN/ZKEEOJw/wX+B/wDuOOg7S6tdak5kYRoHP/0ns/5+kMyPpiC9YZvZFYMIcSvK98JwLubDd6d8w1vXTMEwwiN1s5SgBAtWnZiJNmJP/Z5mnpyDr2zYvl2yz5mfL+L177dTt92sXxw/TAAXlywlagwK11SnHROjiIyBMeR2C8uLI5BaYMYlDbokO1V9VVsd21ne8V2tlduZ1vlNrZVbmP1vtVUHDaHsM2wHShKpEelkxKRQmJ4IknhSSRHJJMYnkhCeIKMOyGEaGo+rfU24MKfO0ApFaW1rmq+SEI0jotHdOf2ZTfwauUf0Z/dhZrwb7MjCSFauvIdoAyWlYWzraQsZIoPIAUI0coclxnLcZmx3HBSDj6/ZsveKqrrfQD4/ZrH5+VT4fYcOD4jNpyLh7Tj+pE5aK35Kr+EzLhwMmLDCbNZzHoZTSrKHkWPhB70SOjxk301nhoKqgooqC44sN5TtYeC6gK+2vUV+9z70OhDzlEo4sLiSApPIjEiUJxICEsgLiyOWEcscWFxxDhiiHPEERsWi9PmlPEohBDH6kOl1HIC03Au01pXAyilOhKYhvM84DngPfMiCvHbZMSGc8b4s3j2wyVM/uHVQFeMLqeYHUsI0ZKV74DoDHZXekmLDTc7TaOSAoRotSyGonPKj80YDUPx/b1j2FFaw8YiF/lFLjYWVREbHhgDocLt4fIXlxw4PjU6jKz4cC4f2oHxx6VTXefl83VFJDvDSIl2kBwdFnIzcUTYIsiJyyEn7shTfHr8HkrdpZS4SyiuKWavey8l7pLAuiawzi/Np7SuFK/fe8THsCproCARLEw47U6i7dFE2aKIskfhtDmJsh9622lz4rQHbodZwqSAIUQbo7UepZQaC1wHDFNKxQMeYAPwCXC51rrQzIxCNMR5A7K4euVkTt6xnI4zp2Kd+i1ExJsdSwjRUpVvh5gs9pTXkpMUWl3NG+XTlVLqNOBxwAI8r7V+4LD9Krh/LFADXKG1/v5ozhXiWFgMdaDbxqk9Ug/ZF2G38v6U49lZ6mZHac2BxRecgmNHaQ3T3lp+yDnhNgsPTOrFhD4ZbN5bxSNzNhIbYSM2wkZchJ2YcBsndE4kLSYcV62HPRW1RDqsRDmsRNotWC2ta3wFm2EjJTKFlMgUevDTFhT7aa2p9lRTVldGRV0FZbVllNeV/7iuK6O8NrAuqCpgY/1GXB4XVfVVP2lhcTiLshBuDT9kibBF/HSbNYJw26HbHBYHdsOO3WIP3Lb8eLvEU0JRddEh26VriRAth9b6U6XULcANWutP9m9XSj0rxQfR2iml+Os5/Xnwpdt4tOL38OktcM6LZscSQrRUpVshZxSF22sZ3jnR7DSNqsHvvpVSFuA/BKbO2gUsVUrN0lqvPeiw04HOwWUw8BQw+CjPFaJR2K0G/dvH07/9kfd3Sori8z+cSHFlLUWuWoor69jrqqNjYqDqWOn2sK6wkooaD+Vuz4HCxUtXDCQtJpxvNu/j2teWHfKYYTaD168azIAO8XyxvohH5+YTZjNwWC0H1nec3pWs+AiWbS9l7tpiHFaDMJsFm0WxdbuHfrUeosNsbCxysW5PJVbDwGpR2CwKi2FwfMcE7FaDgnI3JVV1B/ZbDYWhFO3iIzAMRUWNB7fHh6ECb4QMBYZSxEUGWojUenxoDSq4ff/+I/U5U0odaMWQ5Tz60Xn92k+Np4YqTxWueteP6/rA2uVxUeOpwe11U+Otwe1x4/YGlqr6Koprig/c378ck8Mab1uU5cciheE4UJSwGlZshu3AbathxaIsh+5T1kP2H7Icvk9ZMZSBoQwsyoJhGBgE7xuWwD4MDCO4/5f2BR9n/2MFZpGx/OS+QqGUOnR9pG2HrfcPSnrwOYdv82gPHp8HFAf2738MIRqoA3CbUqq/1vr+4LYBJuYRotGkxYTz6LRLUV8Vwvy/Qdfx0PNss2MJIVqa+mqoKsQbm82ADnH0SI8xO1Gjaoyv/wYBm7TWWwCUUm8BE4CDiwgTgFe11hr4VikVq5RKI/BG49fObVK+igq2X3opEYMGwMiRzfW0ogWyWw1ykqN+dkaNvu3i+OLmkUBgvAlXnZeKGg+JzsAH+OMyY/n3hX2prvNSVeelus5Hdb2X1JjADB4Oq4XEKDu1Hj819V5Kq/3UeX14g4WMtXtcvLhgK/U+/yHPO7kmUICYs6aQf87Z+JNcP9wzGnuknde+3c4reWtIVBXY8WDDhxUfM64biJHRm4fnbmb+t0vIVbuw4sOGl3DDz/Szu0HPc7hr5ka2//AFA40NGPgx0ETaFVNGZMPwm7nh7TV41n3KYGMdFjQW5SfaYWFC71QY9zDXvvodSVveZyBrsQTPjwmzcEJuOkx8istfXMJxO1+nn16LQmMoiA6zMqJzBzj7GS549htOLnyJ/nojClBoYiNsdOucC2c9z8QnF3LuvqfprEtQaBQ2YiLCSe/SC/eYP3PR819zXtWLJFGIV2m8ChxhBtFZXajrdzH3zVjKBDWXMFx4lcajwGIzsCWl4c4cyKyVO+jt+x4L9XgV+ADDamA446l1ZrJmz17a+XfgV368gE+BNhR+ezj11jAq3LU4qMEXPNff2J/FtcbQYPGB1Q9WH1j8P963BBfDD4Y+aH3gtv6Ffb90XmCt9KHbF7wZ2KZ04N+K4D446FhAYYAGpTUWvw7822pQBI5Rhg2/X6P8fiz4f3zM/edbbPh8GsPvw6I16uDnAJRhx+/zo/SP+/dfehU83+vTGNqHofWh+1Bg2PD6/UfcDwplseHx+bH6fRgHtd45cL7Fhsfrx4oXpfVB54KB8eN+7Q3+3P6YXxkWiof2l/97DlUOjAL+pZT6CLjE5DxCNCqlFDWDb6Js8QxSP/4DlvbDwJlidiwhREtSugUAa2InXh456FcObn0aowCRAew86P4uAq0cfu2YjKM8FwCl1LXAtQApKSnk5eU1KPSBx/XUkrwxH2d8baM9ZltXVVXVpq7lloNuO4MLEPjtssKmFUVsCm66PBuU34fV68LmcWHx1VD51XIWRnUgyx7La8PKSSr6EuWtweKtgdpKwl//J0s7XkYHewde6bmGvjtfxOKvw+Kvx+qvx/JPD8v6PUSW0Yl/ZS9m9J5nDg34MiwZ+B+y/Gnck76cU/e9cuj+WfBNcQTtVSynJa/jlIp3ftyngS/ha19vMpSD/jH5jKyZi0YFSgweRf1yC4sixpGCj+MjChhcuxo/Cr8yMLwGrs3bWZaXR6ry0C28nJy6YjSglcLqU5TurGVlXh4pqp529irSvJXs/xhn8yiKd29lbV4eKUY9idY6Ynx1gfNRWDw+yvcUsnHxKtKMcNprK2l+S/BjniIchb0UNm8Lo53uy/G1y4j3awh+FIyyK2zhqWxyD2OVewCXVhcQFRj7DoAYhwJ7DzZFj+etvS6mVD6FzVOH8mrwaqIs4I/uyO6I3ny73c24mk/Apw8sDgXesGRKLSns3ucmt249+ED5NPg1VsBviaBOh+Gu9RDrq0D5NfhB+QMFB/wG+DXGYYWplijw7wqowG0UaKXxB28Htx44BqXwG358Wu2/++N5wft+q8LjV4GiFb7gPhVofaHAZ7VQ77cEijPKd8j5FkPhtdip82msuh7LgfODT28ofBYbtX6w+esC+4P5AvsNfBY7bh/YdS0G/h9fhwq0/PAaDrTWOHRtIN/+EoMKFBh8Fgd1fo2BO1CoCVIKlMWCgb1N/b08Ckpr7QWuV0pdASwA4syNJETjqvLADe7reFvfhpp1I8ZFbwf+KAghBBwoQJDQydwcTaQxChBH+ot5eCfvnzvmaM4NbNT6WeBZgAEDBuiRjfiN0Ub7NFRdPY35mG1ZXl5e272WtZWw+QuoLICqIqjZF1gGXgU5o2HHYnjxVH7yY37uK9BjJGyeD0veBFskOJzU+CxExCaR2KcXZA6AnVGweAPYwsAafmDdv//p9I/JhJJ2sKsvWO1g2MASWAZlDWGQIwpcPcH1u0P2Ydg43pnG8RYreIeD/3EwLKAsoAwwDIYDwwEYfcSXPTK4BHpT/cz+kUez/8ijgif/yv70A/tPBQJjVGi3G5+rCr+rkgSXiz84VtMj9xr8VS58lS78Lhe+Khd6fQ1dv5/BP2vc+Gui8btt+Gtq8Ltr8FXX4HfPo73+nNuP8LyBYTiXkcYyJh4pmN2OzVFCqsNFut2OcqSjwh0ohx3DZkfZbWC1oqw2lNUaWGw2sAVv799us6Fs1kOPDW5T1oO3W8AwUJaD1xaUxfj5tcWCMo68PuSxDtr31cKFnDhyJBhG8JO0dL/4rdr038sje3r/Da31y0qpVcANJuYRotElO8O48qxTmf7Od9yb/zosfwP6SmMfIURQsAAxc0cYD73yBR/cMJRkZ5jJoRpPYxQgdgEHdwLPBAqO8hj7UZzb5Cxh4K+ua+6nFa2Rxw0r3w78YSjdAhW7A8WGoTfC0KmBYsO7lweONWwQmQgRCVAXnLo+rj2MvCOwLTwOwmLA4YTELoH92SPg3n1gCfxqLjn8w0nWwMDycxJzAsvPcab8clNPq+PXr0Ez034/vooKfGXl+MrL8JWW4i0rC9wvLcVXVoa3vAxfaRm+igr8lZX4qqrAe+gsHXEc9sfFZsMSFYURGYkRHo4REYGKCMcWH3/gvhEejhEZgTpwPyKwjghHOcIwHHaUwxFY7I5D79tsgQ/tochmCxRKhGhkWutnDru/DLjSpDhCNJkzjktj9qorWLxxGQM+vR1L9giIbWd2LCFES7BvM0Qms7VSUVDhJi7CbnaiRtUYBYilQGelVDawG7gAuOiwY2YBU4NjPAwGKrTWe5RSe4/i3CZniTDw19Q399OKlkjrwLQ3RWsCS+EqKMmHLqfCmD8HWgR89H+BlgNxHSAmE1K6Q1Ju4PyYLJi8EKLTAwWGw78VdqYGChA/x7A02UtrabTPh6+0FE9xMd69e/EWF+MtDq4P3C/Gu28f+I/c/UCFh2ONi8MSXOzt2mGJicaIcmKJdv64dkazIn8jA0aOxIiKwhIdHSgSyLf2QgghTKCU4v6zjuOKR6byjvcWwmdeD5fNCrQsE0K0baVbIb4jBeVukqIc2FrZrHq/psEFCK21VynoP3MaAAAgAElEQVQ1FZhNYCrNF7XWa5RSk4P7nwY+JTAF5yYC03D+7pfObWimY2UJt1Jf6WnupxVm0xoqdsGuJeD3wXHnBQoGz4+G6r2BY+I7QlJXSAi2KrA64PerwZl25GKBxQqpPZvvNbRg2uvFW1RE/e7deHYX4Nm9+9ClsBB8vp+cZ0lIwJqUhDU5CUe3rlgTE7HGx2OJiw8WGmID92NjMcLDjzqPx+fF0bFjY75EIYQQ4jdLiHJw58WnUb2rhvD5t8KSZ2HIZLNjCSHMVroFOo5ke1ENHRIizU7T6BqjBQRa608JFBkO3nZwP07Nz/ThPNK5zc3aoQfepT+dXUCEqBVvw/qPYddScO0JbEvpGShAAJz5RKCLRHI3cBxhRoyYzObL2sJpvx/vnj3Ubd1G/datgWXbVup37AwUGA7uBqEU1uRkbBkZhPfrR3RaGtaU5MC25MDampCAsodWMzMhhBDi5wztlAgdr0HvnAOf/xGVMwoSO5sdSwhhlvoacBVAQke2ra3mxC5JZidqdI1SgGjtLDmD4MtVaK2lSXaoqSmFTfMCrRxOnx5o4bDta9izHDqcAJmDAmMqpBzUaiH3NPPytlDa78ezcye1GzZQt34DdVu2BIoN27eja2sPHGdERWHPzia8Tx+iMzOwZWRgzwisrWlpGFJcEEIIIQ6lFA/ab+B637dEvX8NxtVzA109hRBtT9lWAPxxHRnTPYVB2fEmB2p8UoAALFEOlN+P3+XCEh1tdhzRUFV7Ye1MWPcRbFsA2gdhsTBsWqD1wriHW+Rgiy2Fv6aG2nXrfiw2bNhAbX4+uqYmcIBhYMvMxJ7dgcjjj8eenY09uwOO7GwsiYlSxBNCCCGOUf+e3blt+VU8vecx+PJBOPkesyMJIcwQnAHDSOjE33r1MjlM05ACBGDZORsAX1mZFCBaqzoX+L2BgR93LYVPb4GEzoGiQ9dxkN73xzEbpPhwgPb5qNu0CffKldSuXIV75Urq8vMPDPpoREcTlptL7KRJhOV2wZGbiyMn55jGXhBCCCHELxvTPYXZfc/m3ZU/cM7XD6NyRkO7IWbHEkI0t32bAaiNbo/Nr7EYoffFnhQgAGuMEwBvaSn29u1NTiOOmtaBFg4/vAZrZ8Gwm+Cku6DTyXD9YkjuanbCFsfvduNesYKaJUup+e473KtXH2jZYMTEEN6rF85RJxPWsxdh3bpiTU2VFg1CCCFEM/jjGd05e9N1DK1fT9qMazEmL4Aw+WJMiDaldDNEJPLyslIenvMNy+87hUhHaH1kD61X8xtZYgJ/3H1l5SYnEUdt+ZvwzRNQtBocMdD7Asg9PbDPFibFhyB/TQ01y76nZulSapYuxb16NXg8YBiEdetG7KRJhB/Xi/BevbC1by/FBiGEEMIkzjAbfz1/CPe+cRMvVNwLn90BZz1pdiwhRHMqyYfELmzfV010mC3kig8gBQgALPEJAPj2lZicRPwijxtswab/Gz4JtIA48wnodc6P29s4rTV1+flUf72AqgVf4/5uGdrjAauV8B49SLjiciIGDiS8Xz8sUUeY4UMIIYQQphncMYHed0xBfV0BX02HzqdAj7PMjiWEaC57N0D3M9m2p4b2CRFmp2kSUoAArPGB0UV9+4pNTiKOqKYUvn0KljwDV30OSV1gwpPgcAZmtWjj/LW1VC9ciGv+fKq/XoC3qAgAR+fOxF16KZFDhxLRry9GRGj+ERNCCCFCSZjNgm/4rexb/gmJH/0fRtYgiE43O5YQoqlVl4C7FBJz2b66miEdE8xO1CSkAAGobqPB+gLe8gqzo4iDeWrh2//AgsegrhK6TwBL8Ee2jfeJ9LlcVOV9iWvuXKq+/hrtdmM4nUQOG0bU8BOIHDYMW2qq2TGFEEII8RtsK69nyr6r+Mh+F/aZ16MumQGGYXYsIURT2rsegPq4HAoqaumQGGlyoKYhBQhApR2HzxmDr6LK7ChiP58Hnj4B9uVD7tjAdFQpPcxOZSp/dTWuzz+n4uNPqP72W/B4sCYlEXPWBKLHjCFi4ECUTeYNF0IIIVq7TklRnHPqSP40+2L+seWFQCvQIVPMjiWEaEp7NwDgScjlllPqGZqTaHKgpiEFCAhM4Rhhw1ciXTBM5yoEZypYbDBkcmAqzY4nmp3KPD4fVV99RcWsj3DNm4d2u7FlZBB/2aU4R48mvHdvlHwjIoQQQoScq07oyIVrz2H+nuWMnPtHVPaJkNLd7FhCiKZSshHsUUQmtmPqyaHbzVwKEAC7lxHt3YF3rwzKZxpvPSx8HL56CM5/DbqcCgOvNjuVaeo2baL83XdJnPEBO10ujJgYYiacScyZZxLet6/MViGEEEKEOIuhePi8Plz8+GQ+0rcSPeNq1DXzweowO5oQoinsXQ+JnSly1aGAJKcjJN/zSwECwO7EEuanXsaAMEfRWnj/aiheAz3OhvS+Zicyhb+uDtecOZS99TbuZcvAZsPTsyftr7qSyBEjMOx2syMKIYQQohllxUdw17nDKS5/hJjPr4Qv/gKn/NXsWEKIprB3I2SP4F/z8vl45R5W/PEUsxM1CSlAANgjsTj8+PbKGBDNbtnL8OltEBYDF74FuaebnajZeXbvpvSN/1IxYwa+8nJs7duRfOstxEycyIKVK3GOHGl2RCGEEEKY5LSeacAkKF+IXvQEKmdM2+6eKkQoqq0EVwEkdWH7hpqQHYASpAARYI/E6vDjr6nDX18v3zQ3J4s98J/ohCchKsnsNM3KvXIlpS+/TOXsOQA4R48m7oLziRg8WMZ1EEIIIcQh/mP7HWcas8n8YDJqykKIiDc7khCisZTkB9aJuWxbVE3/9nHm5mlC8ikHDrSAAPCVlZkcpg0oXA1rPwzc7n0hXPROmyk+aL8f1+efs+3iS9h23vlUfb2A+CsuJ+fzuWQ+/hiRxx8vxQchhBBC/ES39slMcU/GX1UMH90EWpsdSQjRWPZPwRnfmYJyN+0TQrcFhHzSAXBEU9hpHAC+fftMDhPi1nwAL4yBuX8MDDypVGAJcdrvp/Kzz9h61kR2Tb0R7549pNx5Bznz55Ny663Y0tLMjiiEEEKIFuzkrin0GjiSBzznw7qPYNlLZkcSQjSWkg1gsbOTFPwaOiREmJ2oyUgXDACLldKs44lnIV4pQDQNreHLByHvH5A1GM57Dayh39VF+3y4Zs+m5KmnqMvfhD07m/TpDxI9dizKKr9+QgghhDh694zrxvhNkxjtXsOgz+5EtTsekruZHUsI0VDF6yAhhwRnBA+f25vBHRPMTtRkpAVEUIQqAcBbIgWIRuf3w8f/Fyg+9LkYLv8InClmp2pSWmtc8+ez9ayz2P2Hm9E+P+kPPUTHjz8i5swzpfgghBBCiGMW6bDy2IX9ud1/PR5rJLx3FXjcZscSQjRU4WpI6UlshJ1J/TPJiA03O1GTkU9BQbm7n2M7NnylUoBodIYRmOVi+C1w8j0h3+XCvWoVxdMfombpUuzt25P+8D+JPu00lMVidjQhhBBCtHK9s2KZdefZ2HcmwBuTYM69MO6fZscSQvxWNaWBGTBSe7JyVzmGUvTMiDE7VZORAkSQ3+FA2bS0gGhMdVXgKoTEHBj955AvPNTv3MneRx+j8tNPscTHk3LvPcSddx7KZjM7mhBCCCFCSHSYDZ0zik0dL6fz0ueg00nQdZzZsYQQv0XR6sA6pQcPzd5AeY2Hj248wdxMTUi6YAT5rRFYI6x495WYHSU01FfDf8+Dl8cFbodw8cFfW8vef/2bLePG4/riCxKmTKbTnNnEX3yxFB+EEEII0SQq3B4u2noqmyyd0B/eABW7zY4khPgtitYE1im9WF/ookuK09w8TUwKEEE+SxiWCIVPWkA0mOGrg/+eDzu+gdP+DvbQnUbGlZfHlvFnUPLkkzjHjKHT7M9InjYNS1SU2dGEEEIIEcJiI+z89Zz+XFNzPZ66WvjgOvD7zI4lhDhWRashIpFSFcteVx1dU6UA0Sb4LOFYw7TMgtFQHje9Vv0Vti+Eic9Cz0lmJ2oSnt272XnDVHZNnoKy22n38ktkPPxPbCmhPbimEEIIIVqOU3ukMnTQYO6quwy2fQ0LHjE7khDiWBWuhtSebCiqAiBXChBtw/b252LNHYJXBqFsmAWPEle+EiY8Cceda3aaRqf9fkpff4PN48+getEikm+5mY4zPyByyBCzowkhhBCiDbpnXHeWx49lthqGnv8P2LnE7EhCiKPl88Le9ZDSkw2FlUDoFyBkEMogV3RnLB2q8c1dhPb7UYbUZn6TE37Pqn1WevW50Owkja5+50723HU3NUuXEjl8OGl//hO29HSzYwkhhBCiDQu3W3j8wr7sKXoc9eXZgak5J38N4bFmRxNC/JrSzeCthZSeTOySSZcUJ8lOh9mpmpR8yg4Kr9mF1bMHfD585eVmx2l91nwA7nKwhbMvcZDZaRrV/lYPW86cQO26daT97a9kPfuMFB+EEEII0SL0SI9hdN/OMOlFdOVumHUjaG12LCHErzloBoyYCBtDcxJRITx4P0gB4oDk4gVY170MgLdEZsI4Jms+gHevCMl+h57CQnb87kqK/vpXIvr3p+NHs4idNCnk/zAIIYQQovX5rCKLR/SFsG4WLHnO7DhCiF9TuBoMKzqxC0/lbWbdnkqzEzU5KUAE+SxhWBz+wO3SUpPTtCJFa2Dm9ZA5CEbeZXaaRuWaN4+tE87CvWoVqX+5n6znnsWWlmZ2LCGEEEKII+rfPo43jTNZYh2AnnM3FCw3O5IQ4pcUrYbELuxy+Xnws/X8sCP0W+JLASIoMAtGoADhlak4j467DN66CBzRcP5rYAszO1Gj8NfVUfiXv7LrhqlYM9LJfv894s49V1o9CCGEEKJFS3I6eOi8vlxXdTWVRmyghWpthdmxhBBHonWgSJjelw2FLgByU6NMDtX0pAARFChABOZO9u2TLhhH5bO7oGI3nPcqOFPNTtMo6rdvZ9t551P2xhvEX3EFHd56C0d2ttmxhBBCCCGOykldk5kw9DiurJqCv3wHzLpJxoMQoiWqLIDq4kABoihQgOiSEtozYIAUIA7wWiMw7BosFmkBcbRG3QfnvgTtBpudpFG45s9n6znn4i0sJOuZp0m543YMu93sWEIIIYQQx+SO07viTh3I0o43wNqZ8N0LZkcSQhyu4IfAOr0v6wtdZMSG4wyzmZupGUgBIqgyOhd11Wys8fF4S6UA8YvKtoPfD9Fp0O0Ms9M0mPb52Puvf7FryvXYs7Lo8P77RJ14otmxhBBCCCF+kzCbhZk3DGPwxX+GnDHw2Z2wZ4XZsYQQByv4AQwrpPRgy94qclNDv/UDSAHiAK/NCe2GYElKxCctIH5eTSm8eBp8PM3sJI3CV1nJzilTKHnyKWImTqT9f9/AnplhdiwhhBBCiAaxWw0wDBb3+TvV1v3jQYT+CPtCtBoFP0ByN7CF8+ENw3j43N5mJ2oWUoAIMnx1sPxNrM4wmYbz52gNH02D6r0w4Eqz0zRY/c6dbLvwIqq/+ZbUP/2RtL//DSMsNAbSFEIIIYQAeHtNDb9zTUaXbQ+8j5PxIIQwn9aBAkR6XwCsFoO4yLbR9VsKEEGG3wMzJ2O1ufHu3Wt2nJbph9cC80qffM+BX5bWqub779l23vl4S0po98LzxF1wgcxyIYQQQoiQc/9ZPdkb35+njAtgzQxY9pLZkYQQ5TvAXQrpfVm0qYR7Z66mosZjdqpmIQWIIK81HABrlBXvvn1on8/kRC1M6Vb43+2QPQKG3mR2mgap+Ogjdlx+BZboaDq89SaRgwaZHUkIIYQQoklEOaw8cVFfHq8bx6qwAej/3QGFq8yOJUTbdtAAlAs3l/Dmkh2E2y3mZmomDSpAKKXilVJzlVL5wXXcEY7JUkrNV0qtU0qtUUpNO2jfn5RSu5VSy4PL2IbkaRBlAbsTa6QBPh++sjLTorRI7lJIyIGzngKjddattNaUPPMsBbfeRnifPnR4W6bYFEIIIUTo65Eew93jenB5+dXU2WLgnctlPAghzFTwA1jskNydDYUuOiZFBsZtaQMa+irvAOZprTsD84L3D+cFbtZadwOGADcopboftP9RrXWf4PJpA/M0jMOJNdwPgLe42NQoLU5Gf7juK4jJNDvJb6L9foofeIC9jz5K9PjxtHvheSyxsWbHEkIIIYRoFpcOac+Dl52M44KXoWwbfHiDjAchhFkKfoCUHmB1sL7QRZeUtjEDBjS8ADEBeCV4+xXgrMMP0Frv0Vp/H7ztAtYBLXOagbBorGGBvjcyDkSQqwi++Ct43NBKx0jQHg8Ft99B6SuvEnfZpaRPfxBlbxuDvAghRFt3NK01g8fFKqXeU0qtD7baPL65swrRlJRSjOmeguowjPJh9wTG9frmCbNjCdH2+P1QsBzS+1JV52VXmZuubWQKTgBrA89P0VrvgUChQSmV/EsHK6U6AH2BxQdtnqqUugz4jkBLiSP2fVBKXQtcC5CSkkJeXl4Dox+qqqqKxdk3oitqiWEDqxcsoFaqwnRf8yCJJUtZWpuNO+Lo6kZVVVWN/u/zm9XVEfvsczjWrMF11gSKjj+e9V99ZXaqo9airmUrJ9ey8ci1bDxyLZvF/taaDyil7gjev/0Ixz0OfKa1PkcpZQcimjOkEM2lus7LqEU9eS5yBH3n/hGV3g86DDM7lhBtx971UFcBWYMprHCTHhNGt7Ros1M1m18tQCilPgdSj7Dr7mN5IqVUFPA+8H9a6/2dzp4C/gLo4Pph4IjzO2qtnwWeBRgwYIAeOXLksTz9r8rLy2PwyPH46+vZ8OeHyYmPJ7GRn6PVWf8J5C2CUfcxePjFR31aXl4ejf3v81v4qqrZOfk63OvWkfqX++l27rlmRzpmLeVahgK5lo1HrmXjkWvZLCYAI4O3XwHyOKwAoZSKBkYAVwBoreuB+uYKKERzinRYmTwyh8s+vYyF8duJee93gW62ziO93RdCNLqdwe/iswaRE+9k0Z2j0G3oi+9fLUBorUf/3D6lVJFSKi3Y+iENOOLACUopG4Hiwxta6xkHPXbRQcc8B3x8LOEb3eYvMMp3YImJwdPWx4DwuOGzOyC5e6uc9cJXVcXOa67FvXIlGf98iOix5o1vKoQQwlRH01qzI7AXeEkp1RtYBkzTWlcffmBTt8hsKGlVc+za4jXrpDWdkpycv+8GPgq7l+rnJ7Ki91/Qxq83jm6L16sh5Hodm7Zwvbqum0W8LYZFK7aD2tHgx2tt16yhXTBmAZcDDwTXHx5+gFJKAS8A67TWjxy2L23/mwJgIrC6gXkaZs0HsHE21uSuMgbEon8H5qe9/COw2MxOc0x8lZXsuOYaatesJeORR4g+9RSzIwkhhGhCjdBa0wr0A27UWi9WSj1OoKvGvYcf2NQtMhtKWtUcu7Z6zfoNrmfcvxbwN9/1/KniUU70zIdT//ar57XV6/VbyfU6Nm3ieq38A3QazsiTTmLSU4sY2yuNq0747TPztbZr1tACxAPAO0qpq4AdwLkASql04Hmt9VhgGHApsEoptTx43l3BGS+mK6X6EOiCsQ24roF5GsYRDbWVWJOSpADRdXxg0MnsEWYnOSa+igp2XHU1tRs2kPn4YzhHjTI7khBCiCbWCK01dwG7tNb7x6h6jyPP7CVEyIiNsPPkxf14Y3ECHocL2zdPQOZA6PGTMeWFEI2lai+Ubob+l1NcWcuy7WWc3rNtdX9qUAFCa70P+MknPK11ATA2eHsBcMTpE7TWlzbk+RudIxq8bqyJCdRt3Wp2GnOldA8srYivqoodV11N3YYNZP7rcZwnnWR2JCGEEOb71daaWutCpdROpVSu1noDgfc2a5s3phDNr3dWLL2zYsH7D3ThctSHUwNTAyZ2NjuaEKFp15LAOmswy3eWA9AnK9bEQM2vodNwhpawwOij1vgYvCUlbWowkAO2LYT3roTqfWYnOSZ+t5udkydTu349GVJ8EEII8aMHgDFKqXxgTPA+Sql0pdSnBx13I/CGUmol0Af4e7MnFcIkOyu9XOq6Ho+ywduXQF2V2ZGECE07F4Nhg7Q+rNhVjsVQ9MyIMTtVs5ICxMEcwQJEbCR4PPjKy00O1Mz8/sDAkzsWgy3c7DRHzV9fz64bb8L9/Q9kTH9Qig9CCCEO0Frv01qP0lp3Dq5Lg9sLgl1F9x+3XGs9QGt9nNb6rJ+bFlyIUBQXaadAJ3CT50Z0yUb4aBq0xS/ihGhqO5dAeh+whbF8ZzldU52E2Sxmp2pWUoA4WPcz4Q/rsbbrAoC3uI2NA7FmBhSuhFH3gb11TH+uvV4Kbr6F6gULSPvL/TLbhRBCCCHEMYpyWHn6kv7kebrzVtRlsPo9WPKc2bGECC3eetj9PWQNBqBXRixn9E43OVTza+gglKHF4QSHE2tKAQDe4mLI7WJyqGbi88AXf4WUntDrXLPTHBWtNXvuuRfX3Lmk3HUnsZMmmR1JCCGEEKJV6pLi5G8Te3LzOx4Gpm0mZ/ZdkNYb2g02O5oQoWHPcvDVBQZ7Be44vavJgcwhLSAOVlUMeQ9gNSoB2tZMGN+/CmVb4eR7wWgdPxZ7H3ucipkzSbxxKvGXXWZ2HCGEEEKIVu3sfplcMKg996kb0DGZ8M6lULnH7FhChIZtXwfW7YfhqvXg87fNbk6t45Nmc6mthLx/YPXuBsBbXGRyoGbUdTyM+Qt0OdXsJEel7K232PfMM8Sedx6J119vdhwhhBBCiJDwxzN68NL1p6Au+G9gMMp3LgVvndmxhGj9ti2ApG4QlcT0zzZw/D/mtclJD6QAcbDgLBiGdmOJjcVT1IYKEM4UGHYTqCPOmNqiuL74gsL7/0LUyJGk3ncvqhVkFkIIIYRoDcJsFhxWCxXRnXmv3d2wayl8eosMSilEQ3jrYce3kD0cgBW7yslJjmqTn2OkAHEwhzOwrqvEmpqKd0+huXmag7sc3jgXCleZneSouJcvZ/cfbiasZ08yHnkYZZVhTIQQQgghGtvyneXcsqYDcxMvDXTV/e5FsyMJ0XoV/ACeGuhwArUeH+v2VNI7K9bsVKaQAsTBrGGBeVlrK7GlpLSNFhBLnoX8OaD9Zif5VfW7drFzyvVYk5PJeupJjIjWMVOHEEIIIURrc2KXJKaN6sx1u05lV+Jw+N9tsP0bs2MJ0Tpt+yqwbn8Cawoq8fg0faQAIVAq0A1jfwuIwhBvAVHngm+fhC6nBUY5bsF8VVXsmjIF7fOR9czTWBMSzI4khBBCCBHSpo3qzIm5KZxRcDm1UZnwzmU4akvMjiVE67NtAST3gMgEVuwsB5AChAi6aTmM/Se2tFR8ZWX460J40J3vXgR3GYy41ewkv0j7fBTccit1W7aS+dijOLKzzY4khBBCCBHyDEPx2Pl9iY5L5PfqNrSnhh5rHgBPrdnRhGg9vHWwY/GB8R8GZcdz5+ldSYkOMzmYOaQD/eGCA1FaU1IB8BYVYW/XzsxETcPjhkX/ho4nQeYAs9P8ouJHHqEqL4+U++4lcuhQs+MIIYQQQrQZMRE2Xrh8AM4wG2p3DNFvXwyf3AwTnmgVg5cLYbrd34PXDR0CBYieGTH0zIgxOZR5pABxuO9egjoXttSBAHgKC0OzAAEwbBpkDjI7xS8q/2AmpS+8SNxFFxJ/0UVmxxFCCCGEaHNykgMDtfuixrE08RyGLH8d0o6DwdeZnEyIVmDb14CC9kMpra5nTUEFAzvEE2azmJ3MFNIF43CbPocVbx3SAiIk2cJh6I3QbrDZSX6We+VKCu+7j4jjh5By551mxxFCCCGEaNPeXLKDC3edxf+zd99xVZb/H8df9zkc9t4IoqAo7q2J5dbMSs3MUTmytGFlO233zdavXZZpllqZmqZpZcNF5ihx74kLVGTJPnA45/79cROpgKgcuAE/z8eDB4f7vs91vc8dmudzrnE6uAf8NhmOrNY7khDVX3wsBLcAV19iD5xl5JebiE/O0TuVbqQAcTEXb8hLxxQUCGgjIGqdvctg+3dgq747XxSmpZHw6EQcAgMJ++ADFJNJ70hCCCGEENe0YR3q0sjHgVsSR2H2iYKFYyDlsN6xhKi+zJlw8h9o2BuADUdS8XE1ER3soXMw/UgB4mIuPpCXjsHNDYOnJ4Wna1kBwmaD1a/BP59X23l7amEhiU88iTUtjdCPP8LofW2uECuEEEIIUZ2YjAYeau2MycWLkbmPY1McYN4wbVFzIURJR9eCrRAa9kJVVTYeSeW6SD8Mhur5PqwqSAHiYs7e2iIhFjOmoCAstW0KxuGVkHIQOj9SbQsQyR99TO7ffxP88su4NGumdxwhhBBCCFHEy0lh2t1t2ZHlxf95vwjpx7WRENZCvaMJUf0cXgmOHhDWkZNpeSSeyyOmgZ/eqXQlBYiLufiA0QnyM3EICaawtk3B2DgVPOpAs0F6JylV5ooVpH7xBd5Dh+J9+2C94wghhBBCiIu0CffhnTta0q//bXDLB9oc99+f0zuWENWLqsLhVRDZDRwc2RifAkDna7wAIbtgXKzdPdDhXgBMQcGY9+7TOZAdndkFR/+E3q+CsfqtqVBw7BinJ03GuUULgl54Xu84QgghhBCiDANbh2oP6o4k6+QuPDZNh8BoaD9W32BCVBephyHjBFz/GAC3tw2jSYgnDQLcdQ6mLxkBcTHDf7fEITgIa0oKakGBjoHsKD8b6l4H7cbonaQEW0EBiU88CQ4OhH34AQZHR70jCSGEEEKIciyIO0GHuG5khHWH5U9rc96FENr0C4CGvQBwMBpoGeaNUk2nwVcVKUBcLCMRljwACZsxBYcA1J51IOp1hnt/13b6qGaS33sf89691HnjdUyhoXrHEUIIIYQQl6Fv02ACPV0ZeGYsFu8I+H4UpMXrHUsI/R1eCX5R4FOfoyk5vLx0NyfTcvVOpTspQFzMWgA75kHyAUyhdQCwJJ7SOZQdHAYqh1YAACAASURBVN8I5gy9U5QqKzaWtDlz8LnrLjx69dI7jhBCCCGEuEw+bo7MHN2e5AInHrI9iwrw3XDIO6d3NCH0YzHDsfXFox/+OpTMnI3HUVWdc1UDUoC4mIuP9t18rviTeEtioo6B7MBihvl3wk+P6Z2kBEtSEqcnP4dTdDSBzzytdxwhhBBCCHGFGgV58OHwNqxMcuUT/5dQ0+Lh+5FQWEumMQtxpY7+qe2s2LAPABuPpBLq7UJdXxedg+lPChAXc/IExQB56ZiCgkBRsJyq4SMg9i6FvDRoN1rvJBdQrVZOPf0MNrOZ0Pffx+DkpHckIYQQQghxFfo0DWJSv2jqtu2LMuATbS2Inx9HPvIV16T9v2jbb0bcgM2msjE+lc4N/K759R9AdsEoyWAAZ2/IS0dxdMQhMLDmj4CImwl+DSGim95JLpA6Ywa5mzYR8sYbOEVG6B1HCCGEEEJUwP3dGhQ9GoEl5Qimde+AbwR0fUrXXEJUKZsNDvwKUb3BwYn9pzI5l2uhc+S1vf3mv2QERGl86oFBq82YQkNr9giI0zsgYRO0vxeqUcUtd9s2kj+Ziuett+J12yC94wghhBBCCDtZtS+Jzhs7ktloMKx+DXYt0juSEFUncTPknIXGNwOQlGkmwMOJzg2kAAEyAqJ042OLH5rq1CFv2zbdolRYfCyYXKH1CL2TFLPl5HDq2UmYQkIIfvklGYokhBBCCFGLNAnRpjQPThjBb2GncfjxQfAM1XZkE6K22/+L9mF2lLb+Q4/oQDY910ve8xSRERDlMIWGYklKQi0s1DvK1ekyESbu+G9xzWog6d13sZw8Scibb2B0d9c7jhBCCCGEsKM63i5MH9mWExlWxlueQPUK1xZETz2idzQhKt+B5VCvC7h4Y7OpqKoqxYfzSAGiNH9/DkseALQREBQWUnj2rM6hroK1qGjiHqhvjvNk/7WOc/Pm4ztmDG4dO+odRwghhBBCVIJ29Xz5vyEtWX3cwpu+/9O255x7B+Sm6R1NiMqTcghSDkK0Nv1iY3wqXd5azb7TmToHqz6kAFGa1ENw8DeA/7birGnrQKgqzOgOa97UO0kx67lznH7+eRwbNiDgsYl6xxFCCCGEEJVoUJtQHusdheIbiTrsO8hIgPl3QWG+3tGEqBz7f9G+N+4PwIq9SaTmFFDPz1XHUNWLFCBK4+ID5gyw2bQREFDzdsI4sRGSdoF3Xb2TFDvz2hQK09Ko8/bbsuWmEEIIIcQ1YGKvKCb3b4KhfmcKB3wKJzbAjw9pOwUIUdvs+wmCW4J3XVRVZcXeJG6I8sfVUZZe/JcUIErj7A2qDfIzMdUJAWrgCIit32h7zza7Te8kAGQuX07mL78QMOEhXJo10zuOEEIIIYSoAv/Ofd9/JpMev/uR0O4Z2L0IVryoczIh7Cz9mLYDRtH7r/1nskg8l0fvJkH65qpmpABRmn8XbDSfw+DsjNHfn4KaNALCnAl7f4Tmg8HRTe80WM6e5cyr/8O5ZUv8xo3TO44QQgghhKhigR7OGBSFgds6kNXyXtg4FTZ8oncsIexnzxLte/PBgDb9QlGglxQgLiAFiNJ4BENANFgtAJhC69SsKRh7FoMlF9qO0jsJqqqS9Npr2Mxm6rz1FoqDDD8SQgghhLjW+Lo58tWYDhSqcPvRWyhoPBD+eAF2LtQ7mhD2sfsHCG0PPvUB6BThyxO9GxHgIVPPzycFiNI07AUT/gH/KAAcw+piOZmgc6grENkDbnwDQtvpnYSs3/8ga8VKAh55GKfICL3jCCGEEEIInTQIcOfzu9txNM3M+Oz7sNW7Hn58EI6s1juaEBWTcgjO7Coe/QDQKdKPR3pF6RiqepICxGVwrBeO5dQp1IICvaNcHp960HkC6LzfbGF6Omdeew3nZs3wHTNG1yxCCCGEEEJ/nRv48ebgluDgTN7gbyCgMSwYCae26R1NiKu3ezGgFK//sCshg72nMlFVVd9c1ZAUIEqTlw5f9Sv6RQJT3XCw2WrGQpTb5sLBP/ROAcDZt97GmpFByOtTZOqFEEIIIYQAYEi7MGaN6YCbly/WOxdq66/NvQPS4vWOJsSVU1Vt+kV4Z/DUdlB8b8UBHpy7Redg1VOFChCKovgqirJCUZRDRd99yrjumKIouxRF2a4oyuYrfX6VM7lq21imHgHAMVzbyrLgxAk9U5WvsEBbUXjbN3onIXvtWjKWLsVv3H04R0frHUcIIYQQQlQjiqKQmp3P4G+PsqbD52ArhG8GQ3ay3tGEuDJn90LKgeLpFzn5hWw4nEqfJkHFu8CI/1R0BMQkYJWqqlHAqqKfy9JDVdXWqqq2v8rnVx0HJ3DygtwUABzDwwEoOHFSz1TlO/gr5KbqvvikYjZz+uVXcGzQAP8HH9Q1ixBCCCGEqJ7cnBxwMhq4/9csdnadAVlnYO4QyM/WO5oQl2/HfFCM0HQQAGsPJlNgtdG7qex+UZqKFiAGAnOKHs8BBlXx8yuPmx/kaBVYo78/iqsrBSeO6xyqHDsWgHsQNOipawz3JUsoPHOGkCmvYXB01DWLEEIIIYSonpxNRr4Y1Z76/q7c+ZvKiV6faQv5LbgbCvP1jidE+ayFsHMBNLoR3AMAWLEvCW9XE+3rVY/B/dVNRSfmB6mqehpAVdXTiqIElnGdCvyhKIoKTFdVdcYVPh9FUcYD4wGCgoKIjY2tYPQLZWdnX9Bmm0JHbAmH2FF0zNfHhzPbtrPfzv3ai4Mlk5iDv5MYejNH1v6lWw7T4SP4/rmWnJ492JSRAdX0ftUUF/9eiqsn99J+5F7aj9xLIcS1zsvVxOx7OjL4sw0MWe3Jr73ew2/lY7BoLNwxB4yyjpioxo6sguwkaH0nAKqqsuFwKj0bB+JglOUWS1Pun2hFUVYCwaWcev4K+umiquqpogLDCkVR9ququvYKnk9R0WIGQPv27dXu3btfydPLFRsbywVtmnuBJaf4WMIPP5Aff5TWdu7XbhK3wJ5Q6t78FHVDWuoSQbVYOPre++T6+ND2nXcwuLnpkqM2KfF7Ka6a3Ev7kXtpP3IvhRAC6ni7MGdsR579YSe5TYfh52CG3ybBsodh4GdgkDdyopra9i24+kHUjYC2tsmqJ7uRnV+oc7Dqq9wChKqqvcs6pyhKkqIoIUWjF0KAs2W0caro+1lFUZYAHYG1wGU9Xxf93rjgR1N4ONl/rkW12VCq41+Coe1g4g5dI6TNmUP+oUNkPfCAFB+EEEIIIcRlaxzswZKHYlAUBVvHB7DmZmBa+yY4ukP/d3TfXl6IEnJS4cCv0HEcOPw37dzNyQE3Jxm5U5aKvpNeBowuejwaWHrxBYqiuCmK4vHvY6AvsPtyn19dONYNRy0ooDApSe8oJeVnaTtgKIpufzkXJCSSPPVT3Hv1Ir91K10yCCGEEEKImktRFFRV5alFO3jwRE9s1z0McV/A6tf0jiZESbsWgs0Cre8CwGyxMvTzjaw5UH0+U6+OKlqAeAvooyjKIaBP0c8oilJHUZTlRdcEAesURdkBbAJ+UVX1t0s9v1rYtQg+agW5acB5W3Eer4ZbcW78DN5vohUidKCqKklTpoCiEPz8c7pkEEIIIYQQNZ+iKLQJ92Hl/mSezxmK2nY0/PUerPtA72hCXGj7txDSCoKbA7Bq31k2HUvDwSCjdS6lQmNDVFVNBXqVcvwU0L/ocTxQ6kfiZT2/WlBVSD8GOSng6ospvB4ABSeO43ZdJ32znU9VYed8CGwCTh66RMhetYrs2FgCn34aU506cPCgLjmEEEIIIUTNN/K6eiRlmJm65jBBPR/ksebZsPIV7d+6He7TO54QcGq7tmPLTe8UH1qyLYEgTydiGvjrGKz6q4aLGVQTbn7a96KtOE0hwShOThQcq2ZbcSZugbR4aDlMl+5tOTmcmfI6To0a4TtqpC4ZhBBCCCFE7fJk30YMbR/Gh6vjmRf6PDS6CX55Stt2Xgi9xX0BJldoORSA1Ox8Yg8kM6h1KEYZAXFJsjpGWdy0fVzJTQFAMRpxrFePgvh4HUOVYsd8cHCGpgN16T556qcUnjlD6Afvo5hMumQQQgghhBC1i6IovHFbC8wWG/UCvaD9bJg7BH58EJzcIfpmvSOKa1VumjZdv9VwcPEG4Kcdpyi0qdzWNlTncNWfjIAoi2vR0JmiERAAjpGR5B+tRgUIqwV2/wCN+4OzZ5V3b96/n7Svv8Z76FBc27Sp8v6FEEIIIUTt5WA08PGINtqQdpMzybfMhjptYOEYOLRS73jiWrV9LhSaocO44kP1/N24s1M40cFV/56sppECRFnc/CGqL3iEFB9yiozAcjIBW0GBjsHOoxhh+Fy4/vEq71q12Tjz8isYvbwIfKLq+xdCCCGEENeOJdsS6PrRZjZf/wUERMP8O+HwKr1jiWuNzQZxX0J45+LFJwF6NA7kjdta6Bis5pACRFmMJrhr4QXDuxwjIsFmw3K8mqwDYTBAvRgIaVnlXWf8uJS8HTsIfPppjN7eVd6/EEIIIYS4dtwQFUAdb2fGzD/Ent5fg38jrQhxZI3e0cS15MhqSD96wWKom4+lcTbTrGOomkUKEFfAMTICgPz4ozonAQpy4bfJkHKoyru2ZmVx9v33cWndGq+BA6q8fyGEEEIIcW3xd3fi2/s64e1q4q7vDnG431zwbQDzRkD8n3rHE9eKuC/ALRCaaO+BbDaVifO38/SinToHqzmkAHEp8++Cb28v/tGpfn0ACo5WgwLEoT/g788g81SVd53y2TSsqakEPf88ikF+hYQQQgghROUL8XLhu/uuw9FoYMR3h8gc9gP41Id5w+HYOr3jidou5RAc/B3ajQEHRwD+OZpG4rk8Bsvik5dN3j2WJyOx+KHBzQ2H4GAKqsNClHsWazt11L++SrvNj48n7Ztv8B5yOy4tmpf/BCGEEEIIIewk3M+Vufd14rHeUXj6hcDoZeBVF+YOheMb9I4narMNn4CDE3QcX3xoybYE3ByN9G0arGOwmkUKEJfi5l+8Dee/nCIj9J+CkZ8NB//Qtt40GKusW1VVSXr9DQwuLgQ89liV9SuEEEIIIcS/ooI8uKtTPQB2pDuSMmQReNaBb4fA0b90Tidqpawk2DEPWt8J7gEAZJot/LLzNDe1CMHFserek9V0UoC4FLcAyE3VVjst4hgRSUF8PKqq6pfr4G9QmAfNBldpt9lr1pCzfj0BjzyMg59flfYthBBCCCHE+cwWK/fO2czd84+RMWwxeNeFuXdoCwUKYU//fA5WC3R+uPjQ30dSybVYGd25vn65aiApQFyKWwCoNshLKz7k2CASW04OhUlJ+uXKS4eAJhB+XZV1acvPJ+nNt3Bs2ACfESOqrF8hhBBCCCFK42wy8tHw1sQn5zBq4Umy71wKfg3gu+HaaGEh7CE/S9t6s+kA7ferSN9mwWyc1IsWYV46hqt5pABxKSGttEVGzhvt4BwVBUD+wYM6hQI6joOHNlbp9Iu0WbOxnDxJ8HPPoZhMVdavEEIIIYQQZenS0J9P72rL7sQMxn5/lNwRSyAwWtuic9/PescTtcGWOZCfAV0mFh+yWLUR8sFeznqlqrGkAHEp4dfBrR8Vz/MBcGrUCNCxAGHO1AoiilJlXVrOnCFl+nQ8+vTBLSamyvoVQgghhBCiPH2aBvHBsNZsPpbGrG2ZMGqZ9kHiwtGwZ4ne8URNZsmDjVOh3vUQ2q748NjZcUxevEvHYDWXg94Bqj2bDWwWbcVTwOjlhUNwMGa9ChCLx4MlB0b/VGVdnn3nXbDZCHz22SrrUwghhBBCiMs1oFUdgjycaFfPB4wGGLkEvhsKi8ZCYT60Gq53RFETbZ4FWadh8BfFhw6cyeKvQyk82y9ax2A1l4yAuBSrBaYEwroPLzjs1CiK/AM6FCDyzsHhlRDcssq6zN28mcxffsHv3ntxDJP9bYUQQgghRPXUKdIPB6OBs1lmnvv1OHlDv9e2rF9yP/wzXe94oqYpyIF170P9GyDihuLDszccxdlkYHiHujqGq7mkAHEpRhM4e0HWqQsOOzdqRH58PKrFUrV5DizXRmM0u61KulOtVs5MeR2HkBD8xt1XJX0KIYQQQghREVuPpzNv0wnGzd+Heeh8iL4Ffn0GYt++YG03IS4p7kvISYYezxcfSs8pYPHWRG5rE4qPm6OO4WouKUCUxzMEss5ccMipcWOwWCg4dqxqs+xeDF7hF8w/qkznFi4kf/9+gp59BoOLS5X0KYQQQgghREX0ax7CO0Nasf5ICvfN3U3eoK+g9V0Q+wb8NkmbYi3EpeRnwfoPoUFPqNe5+PC8uBPkF9oYExOhY7iaTdaAKI9HiDbv5zz/LkRpPngQp6JdMSpdbhrEr4HrHqqSBSit586R/MGHuHbsiMeNN1Z6f0IIIYQQQtjLkHZhADy9aAf3fL2VL0d9iJuzF/z9GZgzYMBUMMpbIVGGf6ZDbuoFox8AhravS5CHM42DPXQKVvPJn7ryeATDqe0XHHKKiAAHB20diJtvrpocJle4fSYEtaiS7pI//gRrVhZBzz+PUoU7bgghhBBCCGEPQ9qFYTIqTIs9grlQxe3GN8DFF9ZM0YoQQ2aBSbZRFBfJSYH1H0HUjRDW/oJT/u5O3F5U3BJXRwoQ5Wl8M3iHX7D1peLoiFNEBPkHDlRdDpNzla39YD5wgPT58/EZMQLnxo2qpE8hhBBCCCHsbWDrUG5uEYKD0UBBoY28jo/h5eINy5+Cb26D4XPB1VfvmKI6iX1LW4Cyz/+KD6mqypMLd3B72zC6NPTXMVzNJ2tAlKdxP+j6dIlpD85Nm2Deu7dqMuSkwtp3ISup0rtSVZWkKa9j9PQk4JGHK70/IYQQQgghKpODUXvL88yiHYyY8TdpzUZrox8SN8NX/eDcCZ0Timrj7H7Y/BW0vwcC/9tm8/c9SSzemsjpDLOO4WoHKUCUx2aFzNOQn33BYedmzSlMTsaSdLbyM+xbBqtfg5zK7yvrt9/IjYsj4LHHMHp7V3p/QgghhBBCVIXb2oZxJDmb4TM2khTeH0YugewzMLM3nN6hdzxRHax4ERzdoPvk4kNWm8r7Kw4QGeDGoNZ1dAxXO0gBojxJe+D9aDiy6oLDzs2bAWDes6fyM+xZAn4NIah5pXZjy80l6f/ewalJE7zvGFKpfQkhhBBCCFGVujUKYNY9HUhMz2PI5xs47tEGxv4OBhPM6g+HV5XfiKi9Dq+CQ39A16fA7b9pFj/vPMXBpGwe792oeDSNuHpyB8vjWVTlumgrTufoaDAYMO/eXbn9Z5+FY39Bs8GVvvtF6syZFJ4+TfALz6MYjZXalxBCCFEVFEXxVRRlhaIoh4q++5Rx3eOKouxRFGW3oijzFEWRlemEqIViGvgzd9x1ZJkLuXfOZqz+0XDfCvCpD98Nhe3f6R1R6KGwAH5/DrzrQacH/jtstfHhykNEB3twc4sQHQPWHlKAKI+Lr1YVvWgrToOrK04NGlT+CIh9y0C1VfoClAUJCaTO/BLPW27BtV27Su1LCCGEqEKTgFWqqkYBq4p+voCiKKHAo0B7VVWbA0ZgeJWmFEJUmdZ1vVl4f2fevr0FRoOifeB4z69Qrwv8+CCseVNbgF5cOzZ8DMn74ab/AwenC06N7xrJc/2bYDDIzoD2IAWI8hgM4BGirQNxEedmzcjbswe1Mv+CykjUtt4Malp5fQBn334bHBwIfPqpSu1HCCGEqGIDgTlFj+cAg8q4zgFwURTFAXAFTlVBNiGETqKCPGhXT9v9YuZf8aw7WQB3LYLWd8Gfb8Gie8CSp3NKUSVSj8Dad6DpQG0DgvM4GA2M6BhO10YBOoWrfaQAcTk8giEzscRh5+bNsaakUJhUibtT9H4ZxsdWXvtA9vr1ZK1Yif/992MKCqrUvoQQQogqFqSq6mmAou+BF1+gqmoi8C5wAjgNZKiq+keVphRC6MJssbJoSwL3zN7Esj0pMPBTbfvFPT/CrJtK/RBS1CKqCr88qY147/f2BaeWbk/k643HsNpkNIw9OegdoEaIeRiUkmsiODfTRiWYd+/GFBxs/34LC8DBEYyV959JtVhIeuNNTOHh+I4ZXWn9CCGEEJVFUZSVQGn/I37+Mp/vgzZSIgI4ByxUFOVuVVW/LeXa8cB4gKCgIGJjY682dqXIzs6udpmqO7lnV6Y23q9Hmql8vFXh0Xnb2LhtDzfWb4Vf88k03fs+hVNj2NXiebI9Gl5V27XxflWmqr5fgUlraRq/hkMNx5O49QBwAIACq8ora/Pwc1Goaz6KUslr8VVETfsdkwLE5Wg6sNTDzk2agMlE3o4dePTubf9+vx4Ifg1g4FT7t10kbe5cCo4cIeyzTzE4OZX/BCGEEKKaUVW1zP8JK4qSpChKiKqqpxVFCQFK29O6N3BUVdXkoucsBmKAEgUIVVVnADMA2rdvr3bv3t0Or8B+YmNjqW6Zqju5Z1emtt6vXt2tPL5gO/N2n8EzKIzJQ56FMzdjnDec9jtegNs+h2ZlzeAqW229X5WlSu9X9ln4bCzUaUvUnW8RZfjvA+eZf8WTnr+PT0d1JKaB/yUa0V9N+x2TKRiXoyAHEjZDftYFhw3Ozrg0bUrulq327zMjEU5s0FZirSSFycmkfDIVt6434N6jR6X1I4QQQuhoGfDvEL/RwNJSrjkBXKcoiquifczVC9hXRfmEENWAs8nI1DvbMiamPvV83bSDwc1h3GoIbgELR2uLU9ps+gYV9qGqsOwR7X3eoGlwXvHhbJaZj1Ye4oYo/2pffKiJpABxORI2w8xekFiy0ODSti3m3buxFRTYt8+9Rf8+qsTdL86+9z62ggKCJk+u1sOKhBBCiAp4C+ijKMohoE/RzyiKUkdRlOUAqqr+AywCtgK70P59NEOfuEIIvRgNCq8MaMadncIBiDuWRobRB0b/BK3u1BannDcM8tJ1TioqbOvXcPA36P0KBEZfcOr1X/aRX2jj1QHNdIlW20kB4nL41Ne+px8rccq1XVvUggLMu+28HeeexVq11f/q5puVJ3frNjJ+/BG/MWNwioiolD6EEEIIvamqmqqqai9VVaOKvqcVHT+lqmr/8657WVXVaFVVm6uqOlJV1Xz9Ugsh9JaRZ2Hs7Dhun7aBk1k2GPQZ3PweHFkDM7rD6Z16RxRXKy0efpsMEd2g4/0lTg/rUJcXb21KZIC7DuFqPylAXA7PUG0RynPHS5xyadMGgLytW+zXX/oxSIiD5rfbr83zqFYrSVOm4BAUhP8DJf/QCSGEEEIIcS3zcjExY2R7krPyGfTperacOAcd7oN7lkNhPnzZB3bM1zumuFJWCyy+HwwOWlHJUPLtcEwDf0ZeV3nT4K91UoC4HEYH8K5b6ggIBz8/HOvVI3frNvv15+wFN/0fNB9ivzbPc27hIsx79xL4zNMY3NwqpQ8hhBBCCCFqss4N/FjyUAwezg6M+OJvlu04BXU7wv1rIawDLLlf28Kx0M5TsUXlWfEyJGyCWz8Ar7ALTn265jCv/7IXm2y7WamkAHG5vOtBeskREAAu7dqRt3UrqmqnX1YXH+h0v1b0sLPC9HSSP/gA1w4d8Ozfv/wnCCGEEEIIcY2KDHBnyUNdaF3Xm7ijadpB90AY+SPEPAJxM2HWTaV+UCmqmT0/wt+fatMuLhppfiwlh49WHeJUhhmDQdbGq0xSgLhcPZ6DG98o9ZRru3ZYz50j/+ChiveTdhS2fQv52RVvqxTJH3+MNTuboBdekIUnhRBCCCGEKIePmyPf3NuRl29tCsDRlBxyrUDfKTD0a0g5BJ/fALsW6RtUlC3lECydoI1c6TvlglOqqvLSsj04Gg28dEtTnQJeOypUgFAUxVdRlBWKohwq+u5TyjWNFUXZft5XpqIojxWde0VRlMTzzlXfj+TDr4PwTqWecovpDEDOhg0V72fnAlj6cIktP+3BvHcv5xZ8j8+dd+LcuJHd2xdCCCGEEKI2cnIw4mA0kF9oZdRX/zBk2kZOpuVC04HwwF8QEA0/3As/TtC2dhTVhzkTFtwNDk5wxxxwcLzg9PJdZ1h7MJkn+zYiyNNZp5DXjoqOgJgErFJVNQpYVfTzBVRVPaCqamtVVVsD7YBcYMl5l3zw73lVVZdXME/lyU3Thu1kJ5c4ZQoJwTEiouIFCFWF3T9A/evBM6RibV3ctM3GmSmvY/T2JuCRh+3athBCCCGEENcCJwcj/xvYnJPpuQyYuo4NR1LApx7c8yvc8BRsnwvTu8HpHXpHFaAtOrlwNKQehiGzwCv0gtMWq403lu+jWR1PWXiyilS0ADEQmFP0eA4wqJzrewFHVFUtfTGF6iztqPbLmxBX6mm3mBhy4+KwFVRgEZqk3ZByEJoPvvo2ypCxeDF5W7cS+OSTGD097d6+EEIIIYQQ14IejQNZ9vD1+Lk7MfLLTcxafxTVYIReL8LoZVCQDTN7w8bPQLXpHffapaqw/Ck4shpu+RAiu5W4xGQ0MOueDnw0vA0ORlmdoCo4VPD5QaqqngZQVfW0oiiB5Vw/HJh30bGHFUUZBWwGnlRVNb20JyqKMh4YDxAUFERsbGyFgl8sOzv7km06WDK5Hjgc9wcJZ1xLnHfy9MDbbGbjrNlYrnJ6Q0T814RjYEOaPxY7vj4lMxP/N96kMCqK7b4+YOd7d7Hy7qW4fHIv7Ufupf3IvbQfuZdCCFEzRfi7seShGB5fsIM/9iQxqnN9jAoQ0RUeWK+tN/D7ZFp5N4dWEeAboXfka8/6j2DLbLjhSWg7ssTp/WcyiQ72pFGQR9Vnu4aVW4BQFGUlEFzKqeevpCNFURyBAcDk8w5PA14D1KLv7wFjS3u+qqozgBkA7du3V7t3734l3ZcrNjaWctvcHkBDTwsNS7nO2r49B2d8QVRuLoFXm+3MF9CwJ136Dri655ch8ZlnyLRYiP7wA5waNLBr26W5rHspLovcS/uRe2k/c5mhVQAAIABJREFUci/tR+6lEELUXB7OJmaMbEeuxYrRoHA2y0xmXiENA/1gxDzY+jUey5+FaV2gz6vQ/l4wyKfsVWL7d7DyZWg2GHq8UOL0pqNpDJ+xkdcGNeeuTjL1oiqV+ydAVdXeqqo2L+VrKZCkKEoIQNH3s5do6iZgq6qqSee1naSqqlVVVRvwBdCxYi+nkgVEw9n9pZ4yurvj0ro12WvXXn37w+fCsG+v/vmlyNm4kcxlP+F3371VUnwQQgghhBDiWmEwKLg7aZ/pvrJsDwOmrmPp9kRQFGg3mrgOn0DdjtpUgK8HyHadVWHXIm0ESmR3GDStRNEnPaeAifO3Ee7rysDWoaU2ISpPRUtwy4DRRY9HA0svce0ILpp+8W/xoshtwO4K5qlcgU0geb82n6gUHj17kr9vHwUJiVfets2qfTe5VCDgRU3m53P6lVcw1QvH//777dauEEIIIYQQ4kIv3dKMpiGeTJy/nRd+3IXZYiXfOQBGLoFbP4JT2+GzGIibCTZZG6JS7F0Gi8dDeGcYPg9MF+5qoaoqTy3cQWp2AVPvbFtcPBJVp6IFiLeAPoqiHAL6FP2Moih1FEUp3tFCURTXovOLL3r+/ymKsktRlJ1AD+DxCuapXDGPwIPryzzt0bsXANmrVl5Zu9ZCmNoeNnxSkXQlpE6fjuX4CUJefhmDs2wpI4QQQgghRGUJ9nJm3vjrGN81km//PsGQzzeQkmcrGg0xBh7aqI2G+OVJmN0fzlTvz15rnL3LYNFYCG0Hdy4Ax5Lr9n21/hir9p9lcv9omod66RBSVKgAoapqqqqqvVRVjSr6nlZ0/JSqqv3Puy5XVVU/VVUzLnr+SFVVW6iq2lJV1QH/LmhZbXmHg0997S+RUjjWq4dTVBRZK1ddWbtHYyEtXmvfTswHD5LyxUw8b70Vt5gYu7UrhBBCCCGEKJ3JaOC5/k2YMbIdqgouDue9b/Cuq42GGDBV2/luelf4bTKYM8puUFyeLbO1HQvrtIa7F4FT6QtLhnq7cHvbMMbE1K/SeOI/sgrKlbDZtO10Dpc9wsGjT29yt2yhMC3t8tvdMR+cvaFRPzuEBNVi4fTk5zB6eBA0eZJd2hRCCCGEEEJcnr7Ngvn5ketxMynkF1p5Y/k+0nIKtA8y246EhzdDu9Hw9zSY2gF2fl/mNG9xCaoKf70HP02EBj1h1FJwLjmyISe/EIB+zYN5b2grlDI+UBaVTya9XAmDQdvOpUFPaNi71Es8evcm5bNpZK1cic/QoeW3ac6EfT9D6zvBwckuMVO//BLznj2EfvghDr6+dmlTCCGEEOWzWCwkJCRgNpt16d/Ly4t9+/bp0ndVcHZ2JiwsDJPJpHcUIcr175vc7SfOMXv9MX7clsh7Q1txQ1QAuPrCLR9Am5HalIzF47RP8fu/C0FN9Q1eU1gt2giSuC+gxVAY9BkYS/7dcCI1lzumb2DSTdHc1iZMh6DifFKAuFKB0ZBc9v/YnZo0wTEykoylyy6vALF3KRTmaQUIOzAfOEDyp5/h2f8mPPvdaJc2hRBCCHF5EhIS8PDwoH79+rp8wpaVlYWHR+3c015VVVJTU0lISCAiIkLvOEJctk6RfiyZEMPE+dsZ+eUmxsTU55l+jXF1dIDQtnDfKtg6B1a9Cp93gdZ3QfdJ4CVvlsuUkwLfj4bj67R1+nr/r9QtTlOz8xn11T/kF9poEeqtQ1BxMZmCcaUCmkDygTJXrlUUBa/bBpG3ZQsFx4+X3169GOj1srZYSgWpFgunJk/G6OlJ0IsvVrg9IYQQQlwZs9mMn5+fDO+tBIqi4Ofnp9voEiEqolkdL356+HrGxNRn9oZjPL5g+38nDQZofw88vAU6PQA7F8DHbeGPFyD3CqZ1XytObYcZ3SFxMwz+AvpOKbX4kFtQyNjZcZzOMPPl6PY0DHSv+qyiBClAXKmgZmDJ1RaNLIPXgAFgMHDuxx/Lb8+vAdzwRJkLW16JlGnTyN+7j+CXX8LBx6fC7QkhhBDiyknxofLIvRU1mYujkVcGNGPhA515vE8jADLNFjLNFu0CNz/o9yY8sgWa3w4bpsJHrWDtu1CQo2PyasJm0+7Jl320tR/G/g4tSx9xbrWpTJi7lV2JGUy9sy3t6sm09OpCChBXKrQtKAZt5doymIKCcIuJIWPpUlSrtey2di2CI2vsEivnn02kTPscr0GD8Ozb1y5tCiGEEEIIIeyrQ31fooM9AXjtp73c+MFa1uw/+98F3uFw2zR4cAPUvx5WvwYft9EWrCzI1Sm1zjIS4JuB8MfzENUX7l+r7XhRBoMC7ev7MmVQC/o0DarCoKI8UoC4UgFNYNJJiO5/ycu8hwyh8NRpslavLv0CqwV+fw7++bzCkQrT0zn1zDM4hocT/OILFW5PCCGEEKKyZWVlMW3aNFRZ+V9cw+7sFI67kwP3zI7jgW+2kHgu77+TQU1hxDwY+wf4RcFvk+DD5tqIiLxz+oWuSjYrxH0J02IgYQsM+ASGfauNFilFRq6F3YkZKIrChB4NubNTeBUHFuWRAsSVMhjAqfz5Qx69e2EKCyNt1uzSL9j/C2QnQfuxFYqj2mycfv4FCtPSqPP+exjc3CrUnhBCCCFqviVLlqAoCvv379el/2PHjtG8efMyzxcUFPDQQw/RrVs3mVYhrmltwn34+dHrefrGxsQePEuv92L5Y8+ZCy8K7wT3/AL3/AZ12mojIj5oDr9OuuS08BovYQt80RN+eQKCW8KD66DtqDKnrp/JMDN0+kbumR1HXsElRqELXUkB4mocWQ1fDwJLXpmXKA4O+I4aRd7WreRt317ygs1fgld4mdt5Xq7U6dPJXr2aoKefwqVZswq1JYQQQojaYd68eVx//fXMnz/fLu0VFhbapZ1/OTo68s0339C0qWw3KISTg5EJPRqy8olu9G0aTIswLwDMloveRNfrDHcv0qYfRPeHuJnaYpXzRsChldpogdogLR4W3w8ze0HWGbj9Sxj9E/hGlvmUw2ezuX3aBhLSc/lwWGtcHI1VGFhcCdmG82oUFkD8Gkjcos3LKoP37YNJnjqVlBlfUPezT/87kXIIjq6Fni+C4er/cGTFxpL88Sd43norPiNHXnU7QgghhKgcw6ZvLHHslpYhjOxcn7wCK2NmbSpxfki7MO5oX5e0nAIe/HbLBecW3N+53D6zs7NZv349a9asYcCAAbzyyivExsby0ksv4efnx4EDB+jatSufffYZBoMBd3d37r//ftasWYOPjw/z588nICCA7t27ExMTw/r16xkwYABDhgxh7NixJCcnExAQwKxZswgPDycpKYkHHniA+Hjtk9hp06ZRp04drFYr48aNY8OGDYSGhrJ06VJcXFw4cuQIEyZMIDk5GVdXV7744guio6NJTk7mgQce4MSJEwB8+OGHdOnS5WpuuxA1UpiPKx+PaANo286OmbUJdycTz/ZrTFTQedvrhrSCwTOg96vah5qbZ8GB5dqHm21HQqsR4F1Xp1dRAedO0OjAVPhzNRgdte01uz4Nzp6XfNr2k+e4Z9YmjAaFBfd3pnmoVxUFFldDRkBcjfBOgALH1l/yMoObG35j7yF79Wpy4+L+O3HuuLa4TNtRVx0hPz6eU089jVOTaEL+96oMXxRCCCEEAD/++CP9+vWjUaNG+Pr6snXrVgA2bdrEe++9x65duzhy5AiLFy8GICcnh7Zt27J161a6devGq6++WtzWuXPn+PPPP3nyySd5+OGHGTVqFDt37uSuu+7i0UcfBeDRRx+lW7du7Nixg61bt9KsaETmoUOHmDBhAnv27MHb25sffvgBgPHjx/PJJ5+wZcsW3n33XR566CEAJk6cyOOPP05cXBw//PAD9913X5XdMyGqG6tN5YaoAP6JT+XGD9fy7KKdnMm4aAtazxDo+QI8sQ/umA1+kbDmdW2diK9ugs1fVf9tPFUVjq2DBXfDR60IPrMGOtwHE7dD39fKLT4AfPfPcTycTfzwYIwUH2oAGQFxNVx8ILg5HPsLePaSl/qOHk36/AUkvf1/1P9+AYrBoE27mLjzqrfetJw5w4n77kNxdKTuJ59gcHG5qnaEEEIIUbkuNWLBxdF4yfO+bo6XNeLhYvPmzeOxxx4DYPjw4cybN4+bb76Zjh07EhmpDWEeMWIE69atY8iQIRgMBoYNGwbA3XffzeDBg4vb+vc4wMaNG4uLFiNHjuSZZ54BYPXq1Xz99dcAGI1GvLy8SE9PJyIigtattVXq27Vrx7Fjx8jOzmbDhg3ccccdxe3m5+cDsHLlSvbu3Vt8PDMzk6ysLDw8zvvkV4hrhIPRwIQeDRnRMZypqw/zzd/H+HF7IrPGdCCmof9FFztCs9u0r/RjsGsh7Pwefn4cfnkK6sVA9M3QuD/41NPl9ZRw7iTsWQw7FsDZPdr7q5hH+Mfags797ij36anZ+WSaC4nwd+OlW5uRb7Hi5+5UBcFFRUkB4mo16AUbp4I585KVOYOLCwGPTeT0pMmcW7QInx6twLcBGK/u1hemp3PivvuwZWQS/vUcTKGhV/sKhBBCCFHLpKamsnr1anbv3o2iKFitVhRFoX///iVGS5Y1evL8426XWNy6vNGXTk7/vRkwGo3k5eVhs9nw9vZmeynrY9lsNjZu3IiLfLAiRDFfN0deurUp93Spz/S1R2gd7g3A7sQMwnxc8HZ1vPAJPvW1aQs3PAWnd8C+n7TF73+bpH35NoDIbhDRTStMuAdWzQux2SBptzaNff9yOPm3djy0nbazRYs7wORCfmxsuU2tPZjMkwt3EOLlzNIJXXB3csDdSd7W1hQyBeNqNb4J6nWBnORyL/UaMADXTp1IevMtCj6+FZY+dFVdWjMyOHn/A1iOnyDs009l0UkhhBBCXGDp0qWMGjWK48ePc+zYMU6ePElERATr1q1j06ZNHD16FJvNxoIFC7j+em0dK5vNxqJFiwD47rvvio9fLCYmpnhRy7lz5xZf16tXL6ZNmwaA1WolMzOzzHyenp5ERESwcOFCQJvnvmPHDgD69u3L1KlTi68trUghxLWqrq8rUwa1wNXRAVVVmTh/G13eWs0by/dxNtNc8gmKAnVaQ68XYcLf8MhWuPFN8I/SRkcsHA3vRsGHLWDhPbD+Yzi0QhuZYI+tcfPSIT4W1n8EC8fAe41g+g2w4iXIz9Kmjjy6Dcat1qalm8ovPOYXWnn9l72M+moT3i4m3r69pUxDr4GkVHS1wq+D0csu61LFYKDOW28Sf/NNJK6yUm/E7Vdc+SlMTubEfeMoiI8n9MMPcLuu05VnFkIIIUSttmjRIp5//vkLjt1+++1MmzaNzp07M2nSJHbt2kXXrl257bbbAG2Uw549e2jXrh1eXl4sWLCg1LY//vhjxo4dyzvvvFO8CCXARx99xPjx4/nyyy8xGo1MmzaNkJCQMjPOnTuXBx98kClTpmCxWBg+fDitWrXi448/ZsKECbRs2ZLCwkK6du3K559/bqc7I0TtoSgKn93Vjmmxh5n5Vzyz1x9jSPswHuzWgLq+rqU/ya8BdH5I+7JaIHErJGyChM1wcpM2HeJfJjfwCgPPOuAVCq7+4OQBTp7geF77qqoVE8znwJyh7Vhx7jikH4e889ae8AqHyB7QoAdEdtfavULHUnIY+dU/nEzL4+7rwnnh5qY4m2Sni5pIChAVlZ2s/YE0OV/yMpOfJyExeSSudCTxoyWETe2JYjJdVhd5O3eS8MijWDMzqTv9c9xiYuyRXAghhBC1zPLly0usmfDoo4/SsmVL3n333TKLC6+99hqvvfbaBcdiLxoKXb9+fVavXl3iuUFBQSxdurTE8d27dxc/fuqpp4ofR0RE8Ntvv5W43t/fv8x8QogLNQ724MPhbXiiT2Omrz3Cws0JxDTwo66vK3kFVkxGBQdjGR95Gk3aovrh532gmZsGyQcgeT+kHISMBMhMhEP7IDcVbJZLB3LyBLcAbY2JkNbaVJCQlhDcCtz8rvp1pucU4OPmSKiPC83rePHGbS24ISrgqtsT+pMCREWc3gEzemjb4LQYculrN36Kp/8ZrI9O4szHX3Py4YcJffttjN7eZT5FtVhI/fIrUj79FIfAQOp/NxfnJk3s/CKEEEIIIYQQNVG4nyuv39aCib2iiteD+HJdPN/8fZxhHcIZ3qEudbwvY10VV1+o11n7Kk1hvjbaoSDnv2OKAo7u4OwFBvuORjiaksPbv+5n8/F0Yp/ujruTA9PubmfXPoQ+pABREUHNtT12N31x6QKEqsKRNRB9Cz7DJ4NvBGdef4P42wYTMPFRvPr3R3H8bwEZW14emct/JXXmTAqOHsXjxhsJfuVlHHx8quBFCSGEEKK26d69O927dy/1XHZ2dtWGEULYXaDnf6Ox24T7sPl4Op+sPsTU1Yfo0tCfAa3qcEf7ulffgYOT9uXmX/61V8lqU9l+tpBv58Sxev9ZnE1GHujWAKOs81CrSAGiIgxG6Hg//D4ZTvxz4TCm8ykKjPkF8rVFmXyGD8e5WXNOP/ccpydNJunNt3Bu2gSjuweF6WmYd+5CLSjAqXFjwj77FI+ePavwRQkhhBBCCCFqqi4N/enS0J+TabksiDvJzztP8fueM8UFiOW7TtMxwhf/arJtpaqqKIrC7sQMPtyaj797Bg90a8CYLvUJ9Lj0NHdR80gBoqLajtJWd/3jBRj7Oxgummt18HcIba/NfXL5b7qFS4vmRCxbSs5ff5H5xx/k79tPYXIyRg9PfO68E/fu3XDt1ElWdhVCCCGEEEJcsbq+rjx1Y2Oe7NuI7PxCAM5kmHlo7lYUBZqGeNI50o/rIv3oGOmLp/PlrU9XUaqqcjApm78OJfPXoRTCfV15bVBzWoZ58UQ7Jx4c3BNTWetXiBpPChAV5eQOvV+GnybCmR1Qp81/5/Yu07adaTUCBn1a4qmKouDetSvuXbtWXV4hhBBCCCHENUNRFDyKigtBnk78OvEG/tiTxN/xqXz993FmrjvK/93ekqEd6nIyLZd1h1OICnSnYaB78boSFVFQaMPRQSso/N9v+1m4JYHkrHwAIv3daBPuXZyzZYCDFB9qOSlA2EOrEdq2nL6RYC2EwyvhyCptbYjQdnDTW3onFEIIIYQQQlzjFEWhSYgnTUI8mUgUZouV7SfPERXoDsA/R9OYvHhX8fUBHk40DHDn3aGtCPV2YXdiBttPnsPb1YSXiwlVBbPFSvfGgTg6GNhyPJ3Nx9I4npbLsZQcjqXkkJFnYferN6IoCgZF4bpIP25o6E+XKH9CL2eBTFGrSAHCHhRFKz4A7P4BlowHxQjtx0LfKRfulyuEEEIIUYmMRiMtWrQo/nn48OFMmjTJLm1v376dU6dO0b9/f7u0J4TQl7PJyHWR/22TObhNKJ0ifDl8NptDZ7M4lJTNwbPZGIpmhf95MJl3fj9Qop3NL/TG392J2ANn+WT1YbxdTdT3c6NTpB/1/dwosNpwcjDy1I2Nq+qliWpKChD21qAn3LsC/Bpq29kIIYQQQlQhFxcXtm/fXiltb9++nc2bN0sBQohaymBQqOvrSl1fV3pEB5Y4P+6GSO5oF0ZGnoVzeRYUtCKGl4s2xeOBbg0Y3zWyeMqHEBeTAoS9uQdoX0IIIYS4tv06Cc7sKv+6KxHc4qqmdmZkZNCxY0eWLVtG48aNGTFiBD179mTcuHE8+OCDxMXFkZeXx5AhQ3j11VcBiIuLY+LEieTk5ODk5MSKFSt46aWXyMvLY926dUyePJlbbrmFRx55hF27dlFYWMgrr7zCwIED7fuahRDVhqODgUBP5wu2/Tyfm5O8vRSXJr8hQgghhBC1SF5eHq1bty7+efLkyQwbNoypU6cyZswYJk6cSHp6OuPGjQPg9ddfx9fXF6vVSq9evdi5cyfR0dEMGzaMBQsW0KFDBzIzM3F1deV///sfmzdvZurUqQA899xz9OzZk6+++opz587RsWNHevfujZubmy6vXQghRPUmBQghhBBCiMqg0yLUZU3B6NOnDwsXLmTChAns2LGj+Pj333/PjBkzKCws5PTp0+zduxdFUQgJCaFDhw4AeHp6ltrXH3/8wbJly3j33XcBMJvNnDhxgiZNmlTCKxNCCFHTSQFCCCGEEOIaYLPZ2LdvHy4uLqSlpREWFsbRo0d59913iYuLw8fHhzFjxmA2m1FVFUVRym1TVVV++OEHGjeWheWEEEKUTzZZFUIIIYS4BnzwwQc0adKEefPmMXbsWCwWC5mZmbi5ueHl5UVSUhK//vorANHR0Zw6dYq4uDgAsrKyKCwsxMPDg6ysrOI2b7zxRj755BNUVQVg27ZtVf/ChBBC1BgyAkIIIYQQoha5eA2Ifv36MXbsWGbOnMmmTZvw8PCga9euTJkyhVdffZU2bdrQrFkzIiMj6dKlCwCOjo4sWLCARx55hLy8PFxcXFi5ciU9evTgrbfeonXr1kyePJkXX3yRxx57jJYtW6KqKvXr1+fnn3/W66ULIYSo5qQAIYQQQghRi1it1lKP79u3r/jx+++/X/x49uzZpV7foUMH/v777xLH/x0V8a/p06dfRUohhBDXIpmCIYQQQgghhBBCiEonBQghhBBCCCGEEEJUOilACCGEEELY0b8LMgr7k3srhBA1mxQghBBCCCHsxNnZmdTUVHmjXAlUVSU1NRVnZ2e9owghhLhKsgilEEIIIYSdhIWFkZCQQHJysi79m83mWv0G3dnZmbCwML1jCCGEuEoVKkAoinIH8ArQBOioqurmMq7rB3wEGIGZqqq+VXTcF1gA1AeOAUNVVU2vSCYhhBBCCL2YTCYiIiJ06z82NpY2bdro1r8QQghxKRWdgrEbGAysLesCRVGMwKfATUBTYISiKE2LTk8CVqmqGgWsKvpZCCGEEEIIIYQQtUyFChCqqu5TVfVAOZd1BA6rqhqvqmoBMB8YWHRuIDCn6PEcYFBF8gghhBBCCCGEEKJ6qoo1IEKBk+f9nAB0KnocpKrqaQBVVU8rihJYViOKoowHxgMEBQURGxtr15DZ2dl2b/NaJffSfuRe2o/cS/uRe2k/ci+FEEIIcS0ptwChKMpKILiUU8+rqrr0MvpQSjl2xUtDq6o6A5hRlCm5R48ex6+0jXL4Ayl2bvNaJffSfuRe2o/cS/uRe2k/9r6X9ezYlijHli1bUhRFsfe/Ryrq/9u795DJ6jqO4+9PXtJSMspsc5UVs4tJbmllWOQly0w06aJp3ggi0jISuhGFSWBEIVEWpmLRDSm3Nt3WojQjs5ual7TcTGrZaMtuZpGsffvjHPFBd33O7PPMnDnPvF/wsHNmzsx++DJz5ne+55zf+PkcnTUbjfUajfUajfUa3TTWbIvjkXkbEFX18gX+5+uBPeYsLwc2tLf/lGRZe/bDMmBjlxesql0XmOkRkvy8qg5c7NedRdZy8VjLxWMtF4+1XDzWctjGMR5ZKN9To7Nmo7Feo7Feo7FeoxtazRY6CWUXPwP2SbJXku2BE4DV7WOrgVPb26cCXc6okCRJkiRJA7OgBkSS45KsB14MXJnkqvb+pyVZA1BVm4AzgauA24HLquq29iXOA45IcidwRLssSZIkSZKWmAVNQllVq4BVm7l/A3DUnOU1wJrNrHcPcPhCMiyiC/sOsIRYy8VjLRePtVw81nLxWEstNt9To7Nmo7Feo7Feo7FeoxtUzVI18nyQkiRJkiRJI5nEHBCSJEmSJGnG2YCQJEmSJEljN/MNiCRHJvl1knVJ3tt3niFLckmSjUlu7TvLkCXZI8nVSW5PcluSs/rONFRJdkjy0yS/bGt5Tt+Zhi7JNkluTHJF31mGLMndSW5JclOSn/edR8Mz3/gljU+2j9+c5Pl95JwWHep1Ulunm5Ncl2T/PnJOk65j5CQvSPJAktdNMt+06VKvJIe02/3bkvxg0hmnSYfP5BOSfGvOGO70PnJOi/n2s4a0zZ/pBkSSbYBPA68C9gXemGTfflMN2qXAkX2HWAI2AWdX1bOBg4AzfF9utf8Ch1XV/sBK4MgkB/WcaejOovlFIy3coVW1cki/3a3p0HH88ipgn/bvLcBnJhpyinSs1++Al1XVc4FzGdikbout6xi5Xe+jNL92N7O61CvJLsAFwDFV9Rzg9RMPOiU6vr/OAH7VjuEOAT6eZPuJBp0ul/Lo+1mD2ebPdAMCeCGwrqruqqr7ga8Cx/acabCq6lrgr33nGLqq+mNV3dDevpdmZ2/3flMNUzX+1S5u1/458+5WSrIceDVwUd9ZpBnXZfxyLPCFdjt4PbBLkmWTDjol5q1XVV1XVX9rF68Hlk8447TpOkZ+O/B1YOMkw02hLvU6Ebi8qn4PUFWzXLMu9Spg5yQBdqLZx9g02ZjTo8N+1mC2+bPegNgd+MOc5fW4o6cpkmQF8DzgJ/0mGa72koGbaAZH360qa7n1zgfeDfyv7yBLQAHfSfKLJG/pO4wGp8v4xTHOQ0atxZuBb4810fSbt2ZJdgeOAz47wVzTqst77BnAE5Nc0277T5lYuunTpV6fAp4NbABuAc6qKscfWzaYbf62fQfoWTZzn0dHNRWS7ERzVOGdVfXPvvMMVVU9AKxsT31clWS/qnKekhElORrYWFW/SHJI33mWgIOrakOSpwDfTXJHe3RD6qLL+MUxzkM61yLJoTQNiJeMNdH061Kz84H3VNUDzUHqmdalXtsCBwCHAzsCP05yfVX9ZtzhplCXer0SuAk4DNib5rvyh46Jt2gw2/xZPwNiPbDHnOXlNF02qVdJtqNpPnypqi7vO89SUFV/B67BeUq21sHAMUnupjlV8rAkX+w30nBV1Yb2343AKprTUaWuuoxfHOM8pFMtkjyX5hKzY6vqngllm1ZdanYg8NX2e+F1wAVJXjOZeFOn62dybVXdV1V/Aa4FZnWy0y71Op3mkpWqqnU087Q8a0L5hmgw2/xZb0D8DNgnyV7FVnHfAAADdElEQVTtpCYnAKt7zqQZ117rdjFwe1V9ou88Q5Zk1/bMB5LsCLwcuKPfVMNUVe+rquVVtYJmW/n9qnpTz7EGKcnjk+z84G3gFYBn5WgUXcYvq4FT2pnRDwL+UVV/nHTQKTFvvZLsCVwOnDyjR6Qfbt6aVdVeVbWi/V74GvC2qvrG5KNOhS6fyW8CL02ybZLHAS9idid17lKv39OcLUKS3YBnAndNNOWwDGabP9OXYFTVpiRn0szcuw1wSVXd1nOswUryFZpZap+cZD3woaq6uN9Ug3QwcDJwSzt3AcD7q2pNj5mGahnw+Xa25ccAl1WVPx+pvu1GczkQNN/DX66qtf1G0pBsafyS5K3t458F1gBHAeuAf9McTZxJHev1QeBJNEfxATbN8i/UdKyZWl3qVVW3J1kL3Ewzl9JFs3pJaMf317nApUluobm84D3tmSMzaXP7WTSTqw9um5+qqbw0RJIkSZIkLSGzfgmGJEmSJEmaABsQkiRJkiRp7GxASJIkSZKksbMBIUmSJEmSxs4GhCRJkiRJGjsbEJK2WpJdkryt7xySJGk2JVmW5Ir29sokR8157Ogk5/SXTtLD2YCQtBC7ADYgJElSX94FfK69vRI4as5jVwLHJHncxFNJ2iwbEJIW4jxg7yQ3JflY32EkSdLSlOTcJGfNWf5IkncArwXWJtke+DBwfDsuOb6qCrgGOLqX0JIeIc3nUpJGl2QFcEVV7ddzFEmStIS1Y47Lq+r5SR4D3Am8Abiwqg5o1zkNOLCqzpzzvJOAg6rq7RMPLekRtu07gCRJkiQ9mqq6O8k9SZ4H7AbcCDwW+PM8T90IPG3c+SR1YwNCkiRJ0hBcBJwGPBW4BPgPsMM8z9mhXU/SFLABIWkh7gV27juEJEmaCato5nnYDjiRprmwYs7jmxuXPAO4dRLhJM3PSSglbbWqugf4UZJbnYRSkiSNU1XdD1wNXFZVD1TVfcBvkzy9XeVqYN8HJ6Fs7zuU5tcwJE0BJ6GUJEmSNPXaySdvAF5fVXe29x0HHFBVH9jM+rsBX66qwyebVNKWeAaEJEmSpKmWZF9gHfC9B5sPAFW1Crh7C0/bEzh7/OkkdeUZEJIkSZIkaew8A0KSJEmSJI2dDQhJkiRJkjR2NiAkSZIkSdLY2YCQJEmSJEljZwNCkiRJkiSN3f8BQB0J9CtdJ3oAAAAASUVORK5CYII=\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# x0, y0, h variables globales\n", "def euler_progressif(phi1,phi2,tt):\n", "\tuu = [y0]\n", "\tww = [z0]\n", "\tfor i in range(len(tt)-1):\n", "\t\tuu.append(uu[i]+h*phi1(tt[i],uu[i],ww[i]))\n", "\t\tww.append(ww[i]+h*phi2(tt[i],uu[i],ww[i]))\n", "\treturn [uu,ww]\n", "\n", "[uu_ep, ww_ep] = euler_progressif(phi1,phi2,tt)\n", "\n", "\n", "figure(figsize=(18,7))\n", "\n", "subplot(1,2,1)\n", "plot(tt,uu_ep,'--',tt,ww_ep,'--')\n", "plot(tt,yy,tt,zz)\n", "xlabel('t')\n", "legend(['u(t)','w(t)','y(t)','z(t)'])\n", "title('Euler progressif - y(t) et z(t)') \n", "grid()\n", "\n", "subplot(1,2,2)\n", "plot(uu_ep,ww_ep,'--')\n", "plot(yy,zz)\n", "xlabel('y(t)')\n", "ylabel('z(t)')\n", "legend(['Approchée','Exacte'])\n", "title('Euler progressif - z(y)')\n", "grid()\n", "axis('equal');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ ">**Q3 [2 points]** \n", "Calculer la solution approchée obtenue par la méthode d'Euler Regressif avec $301$ points. Afficher $t\\mapsto x(t)$, $t\\mapsto x'(t)$ et $x\\mapsto x'(x)$ en comparant solution exacte et solution approchée. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Euler implicite**\n", "$$\n", "\\begin{cases}\n", "u_0=1,\\\\\n", "w_0=0,\\\\\n", "u_{n+1}=u_n+h\\varphi_1(t_{n+1},u_{n+1},w_{n+1}),\\\\\n", "w_{n+1}=w_n+h\\varphi_2(t_{n+1},u_{n+1},w_{n+1}),\n", "\\end{cases}\n", "$$\n", "qu'on peut rendre explicite par un calcul élementaire:\n", "$$\n", "w_{n+1} \n", "= w_n + h(-6u_{n+1}-5w_{n+1})\n", "= w_n + h(-6(u_{n}+hw_{n+1})-5w_{n+1})\n", "\\quad\\implies\\quad\n", "(1+5h+6h^2)w_{n+1} \n", "= w_n-6hu_{n}$$\n", "ainsi\n", "$$\n", "\\begin{cases}\n", "u_0=1,\\\\\n", "w_0=0,\\\\\n", "u_{n+1}=u_n+h\\dfrac{w_n-6hu_{n}}{1+5h+6h^2},\\\\\n", "w_{n+1}=\\dfrac{w_n-6hu_{n}}{1+5h+6h^2}.\n", "\\end{cases}\n", "$$" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# VERSION AVEC fsolve (il est bien sur possible d'implementer la version explicitée)\n", "\n", "from scipy.optimize import fsolve\n", "\n", "# x0,y0, h variables globales\n", "def euler_regressif(phi1,phi2,tt):\n", "\tuu = [y0]\n", "\tww = [z0]\n", "\tfor i in range(len(tt)-1):\n", "\t\tsys = lambda z : [ -z[0]+uu[i]+h*phi1(tt[i+1],z[0],z[1]) , -z[1]+ww[i]+h*phi2(tt[i+1],z[0],z[1]) ]\n", "\t\tutemp,wtemp = fsolve( sys , (uu[i],ww[i]) ) \n", "\t\tuu.append(utemp)\n", "\t\tww.append(wtemp)\n", "\treturn [uu,ww]\n", "\n", "[uu_er, ww_er] = euler_regressif(phi1,phi2,tt)\n", "\n", "figure(figsize=(18,7))\n", "\n", "subplot(1,2,1)\n", "plot(tt,uu_er,'--',tt,ww_er,'--')\n", "plot(tt,yy,tt,zz)\n", "xlabel('t')\n", "legend(['u(t)','w(t)','y(t)','z(t)'])\n", "title('Euler regressif - y(t) et z(t)') \n", "grid()\n", "\n", "subplot(1,2,2)\n", "plot(uu_er,ww_er,'--')\n", "plot(yy,zz)\n", "xlabel('y(t)')\n", "ylabel('z(t)')\n", "legend(['Approchée','Exacte'])\n", "title('Euler regressif - z(y)')\n", "grid()\n", "axis('equal');\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ ">**Q4 [1 point]** \n", "Calculer la solution approchée obtenue par la fonction `odeint` du module `scipy`. Afficher $t\\mapsto x(t)$, $t\\mapsto x'(t)$ et $x\\mapsto x'(x)$ en comparant solution exacte et solution approchée. " ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "from scipy.integrate import odeint\n", "\n", "pphi = lambda yy,t : [ phi1(t,yy[0],yy[1]), phi2(t,yy[0],yy[1]) ]\n", "\n", "yy0 = [y0,z0]\n", "sol = odeint(pphi,yy0,tt)\n", "\n", "figure(figsize=(18,7))\n", "\n", "subplot(1,2,1)\n", "plot(tt,sol[:,0],'--',tt,sol[:,1],'--')\n", "plot(tt,yy,tt,zz)\n", "xlabel('t')\n", "legend(['u(t)','w(t)','y(t)','z(t)'])\n", "xlabel('t')\n", "grid()\n", "\n", "subplot(1,2,2)\n", "plot(sol[:,0],sol[:,1],'--')\n", "plot(yy,zz)\n", "xlabel('y(t)')\n", "ylabel('z(t)')\n", "legend(['Approchée','Exacte'])\n", "grid()\n", "axis('equal');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exercice : étude d'un schéma RK\n", "\n", "Soit le schéma de Runge-Kutta dont la matrice de Butcher est\n", "$$\n", "\\begin{array}{c|cc}\n", "\\dfrac{1}{3} & \\dfrac{1}{3} & 0\\\\\n", "1 & 1 & 0\\\\\n", "\\hline\n", " & \\dfrac{3}{4} & \\dfrac{1}{4}\n", "\\end{array}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ ">**Q5 [1 point]** Écrire le schéma sous la forme d'une suite définie par récurrence." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Le schéma associé à cette matrice est semi-implicite ($K_1$ dépend de lui même) et permet de calculer $u_{n+1}$ à partir de $u_n$ par la formule de récurrence\n", "$$\\begin{cases}\n", "u_0\t = y_0 \\\\\n", "K_1 = \\varphi\\left(t_n+\\frac{h}{3},u_n+\\frac{h}{3}K_1\\right)\\\\\n", "K_2 = \\varphi\\left(t_{n+1},u_n+hK_1\\right)\\\\\n", "u_{n+1} = u_n + \\frac{h}{4}\\left(3K_1+K_2\\right) & n=0,1,\\dots N-1\n", "\\end{cases}$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ ">**Q6 [2 points]** Étudier théoriquement l'ordre du schéma." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Soit $\\omega$ l'ordre de la méthode.\n", "\n", "+ C'est un schéma semi-implicite à $2$ étages ($s=2$) donc $\\omega\\le2s=4$\n", "\n", "+ Si \n", "$\\begin{cases}\n", "\\displaystyle\\sum_{j=1}^s b_{i}=1&\n", "\\\\\n", "\\displaystyle c_i=\\sum_{j=1}^s a_{ij}&i=1,\\dots,s\n", "\\end{cases}$\n", "alors $\\omega\\ge1$\n", "\n", "+ Si $\\displaystyle\\sum_{j=1}^s b_j c_j=\\frac{1}{2}$\n", "alors $\\omega\\ge2$\n", "\n", "\n", "+ Si \n", "$\\begin{cases}\n", "\\displaystyle\\sum_{j=1}^s b_j c_j^2=\\frac{1}{3}\\\\\n", "\\displaystyle\\sum_{i=1}^s\\sum_{j=1}^s b_i a_{ij} c_j=\\frac{1}{6}\n", "\\end{cases}$\n", "alors $\\omega\\ge3$\n", "\n", "\n", "+ Si \n", "$\\begin{cases}\n", "\\displaystyle\\sum_{j=1}^s b_j c_j^3=\\frac{1}{4}&\n", "\\\\\n", "\\displaystyle\\sum_{i=1}^s\\sum_{j=1}^s b_i c_i a_{ij} c_j=\\frac{1}{8}\n", "\\\\\n", "\\displaystyle\\sum_{i=1}^s\\sum_{j=1}^s b_i a_{ij} c_j^2=\\frac{1}{12}\n", "\\\\\n", "\\displaystyle\\sum_{i=1}^s\\sum_{j=1}^s\\sum_{k=1}^s b_i a_{ij}a_{jk} c_k=\\frac{1}{24}\n", "\\end{cases}$\n", "alors $\\omega\\ge4$\n", "\n", "Calculons donc toutes ces sommes:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Consistance (ordre 1)\n" ] }, { "data": { "text/latex": [ "$\\displaystyle \\sum_{j=1}^s b_j=1$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle i={}0\\quad \\sum_{j=1}^s a_{ij}-c_i=0$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle i={}1\\quad \\sum_{j=1}^s a_{ij}-c_i=0$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Ordre 2 (si = 1/2)\n" ] }, { "data": { "text/latex": [ "$\\displaystyle \\sum_{j=1}^s b_j c_j=\\frac{1}{2}$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Ordre 3 (si = 1/3 et 1/6)\n" ] }, { "data": { "text/latex": [ "$\\displaystyle \\sum_{j=1}^s b_j c_j^2=\\frac{1}{3}$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\sum_{i,j=1}^s b_i a_{ij} c_j=\\frac{1}{6}$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Ordre 4 (si = 1/4, 1/8, 1/12 et 1/24)\n" ] }, { "data": { "text/latex": [ "$\\displaystyle \\sum_{j=1}^s b_j c_j^3=\\frac{5}{18}$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\sum_{i,j=1}^s b_i c_i a_{ij} c_j=\\frac{1}{9}$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\sum_{i,j=1}^s b_i a_{ij} c_j^2=\\frac{1}{18}$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\sum_{i,j,k=1}^s b_i a_{ij}a_{jk} c_k=\\frac{1}{18}$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import sympy as sym\n", "sym.init_printing()\n", "\n", "from IPython.display import display, Math\n", "\n", "c=[sym.Rational(1,3),1]\n", "b=[sym.Rational(3,4),sym.Rational(1,4)]\n", "A=[[sym.Rational(1,3),0],[1,0]]\n", "s=len(c)\n", "\n", "print('Consistance (ordre 1)')\n", "display(Math(r'\\sum_{j=1}^s b_j='+sym.latex(sum(b)) ))\n", "for i in range(s):\n", " display(Math(r'i={}'+str(i)+'\\quad \\sum_{j=1}^s a_{ij}-c_i='+sym.latex(sum(A[i])-c[i]) ))\n", "\n", "print('Ordre 2 (si = 1/2)')\n", "display(Math(r'\\sum_{j=1}^s b_j c_j='+sym.latex(sum([b[i]*c[i] for i in range(s)])) ))\n", "\n", "print('Ordre 3 (si = 1/3 et 1/6)')\n", "display(Math(r'\\sum_{j=1}^s b_j c_j^2='+sym.latex(sum([b[i]*c[i]**2 for i in range(s)]).simplify()) ))\n", "display(Math(r'\\sum_{i,j=1}^s b_i a_{ij} c_j='+sym.latex(sum([b[i]*A[i][j]*c[j] for i in range(s) for j in range(s)])) ))\n", "\n", "print('Ordre 4 (si = 1/4, 1/8, 1/12 et 1/24)')\n", "display(Math(r'\\sum_{j=1}^s b_j c_j^3='+sym.latex(sum([b[i]*c[i]**3 for i in range(s)]).simplify()) ))\n", "display(Math(r'\\sum_{i,j=1}^s b_i c_i a_{ij} c_j='+sym.latex(sum([b[i]*c[i]*A[i][j]*c[j] for i in range(s) for j in range(s)])) ))\n", "display(Math(r'\\sum_{i,j=1}^s b_i a_{ij} c_j^2='+sym.latex(sum([b[i]*A[i][j]*c[j]**2 for i in range(s) for j in range(s)])) ))\n", "display(Math(r'\\sum_{i,j,k=1}^s b_i a_{ij}a_{jk} c_k='+sym.latex(sum([b[i]*A[i][j]*A[j][k]*c[k] for i in range(s) for j in range(s) for k in range(s)])) ))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "D'après ces résultats, le schéma est d'ordre $3$.\n", "\n", "\n", "Sinon, en utilisant les **Théorème de Butcher**: si les coefficients de la méthode de RK vérifient les conditions $B_p$, $C_\\eta$ et $D_\\zeta$ pour $p\\le\\eta + \\zeta + 1$ et $p \\le 2\\eta + 2$ alors la méthode est d’ordre $p$ où:\n", "+ $B_p$: $\\sum_{i=1}^s b_i c_i^{q-1}=\\frac{1}{q}$ pour $q=1,\\dots,p$\n", "+ $C_\\eta$: $\\sum_{j=1}^s a_{ij} c_j^{q-1}=\\frac{c_i^q}{q}$ pour $i=1,\\dots,s$ et $q=1,\\dots,\\eta$\n", "+ $D_\\zeta$: $\\sum_{i=1}^s b_i c_i^{q-1}a_{ij} =\\frac{b_j}{q}(1-c_j^q)$ pour $j=1,\\dots,s$ et $q=1,\\dots,\\zeta$" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\omega = 2$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import sympy as sym\n", "sym.init_printing()\n", "\n", "from IPython.display import display, Math\n", "\n", "c=[sym.Rational(1,3),1]\n", "b=[sym.Rational(3,4),sym.Rational(1,4)]\n", "A=[[sym.Rational(1,3),0],[1,0]]\n", "s=len(c)\n", "\n", "B = lambda p : sum([sum([b[i]*c[i]**(q-1) for i in range(s)]) != 1/q for q in range(1, p+1)])==0\n", "C = lambda e : sum([sum([A[i][j]*c[j]**(q-1) for j in range(s)]) != c[i]**q/q for q in range(1, e+1) for i in range(s)])==0\n", "D = lambda z : sum([sum([b[i]*c[i]**(q-1)*A[i][j] for i in range(s)]) != (b[j]/q) * (1-c[j]**q) for q in range(1, z+1) for j in range(s)])==0\n", "\n", "omega=0\n", "for z in range(1, 2*s):\n", " for e in range(1, 2*s):\n", " if C(e) and D(z):\n", " p=1\n", " while B(p) and p<=min(z + e + 1,2*e + 2):\n", " p+=1\n", " omega=max(omega,p-1)\n", "display(Math(f\"\\omega = {omega}\")) " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ ">**Q7 [2 points]** Étudier théoriquement la A-stabilité du schéma." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Soit $\\beta>0$ un nombre réel positif et considérons le problème de Cauchy\n", "$$\\begin{cases}\n", "y'(t)=-\\beta y(t), &\\text{pour }t>0,\\\\\n", "y(0)=1.\n", "\\end{cases}$$\n", "Sa solution est $y(t)=e^{-\\beta t}$ donc $$\\lim_{t\\to+\\infty}|y(t)|=0.$$\n", "\n", "Le schéma appliqué à ce problème de Cauchy s'écrit\n", "$$\\begin{cases}\n", "u_0\t = y_0 \\\\\n", "K_1 = -\\beta\\left(u_n+\\frac{h}{3}K_1\\right)\\\\\n", "K_2 = -\\beta\\left(u_n+hK_1\\right)\\\\\n", "u_{n+1} = u_n + \\frac{h}{4}\\left(3K_1+K_2\\right) & n=0,1,\\dots N-1\n", "\\end{cases}$$\n", "\n", "L'équation $K_1 = -\\beta\\left(u_n+\\frac{h}{3}K_1\\right)$ \n", "donne \n", "$K_1=\\frac{-3\\beta}{\\beta h+3}u_n$ \n", "ainsi\n", "$K_2=\\beta\\frac{2\\beta h-3}{\\beta h+3}u_n$\n", "et enfin\n", "$$\n", "u_{n+1} \n", "= \n", "u_n + \\frac{h}{4}\\left( \\frac{-9\\beta}{\\beta h+3}+\\beta\\frac{2\\beta h-3}{\\beta h+3}\\right)u_n\n", "= \n", "\\frac{(\\beta h)^2-4\\beta h+6}{2(\\beta h+3)}u_n\n", "$$\n", "\n", "Vérifions ce calcul:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "scrolled": true }, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle K_1=- \\frac{3 \\beta u_{n}}{\\beta dt + 3}$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle K_2=\\frac{\\beta u_{n} \\left(2 \\beta dt - 3\\right)}{\\beta dt + 3}$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle u_{n+1}=\\frac{u_{n} \\left(\\beta^{2} dt^{2} - 4 \\beta dt + 6\\right)}{2 \\left(\\beta dt + 3\\right)}$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import sympy as sym\n", "sym.init_printing()\n", "from IPython.display import display, Math\n", "\n", "# pour ne pas effacer l'affectation de \"h\", ici je vais l'appeler \"dt\"\n", "\n", "sym.var('u_n,u_np1,beta,dt,x')\n", "K1 = sym.solve(-x-beta*(u_n+dt/3*x),x)[0]\n", "display(Math('K_1='+sym.latex(K1)))\n", "K2 = -beta*(u_n+dt*K1).factor()\n", "display(Math('K_2='+sym.latex(K2)))\n", "u_np1 = (u_n+dt/4*(3*K1+K2)).factor()\n", "display(Math('u_{n+1}='+sym.latex(u_np1)))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "On note $x=\\beta h$ et on étudie la fonction $q\\colon \\mathbb{R}^+\\to\\mathbb{R}$ définie par $q(x)=\\dfrac{x^2-4x+6}{2(x+3)}=\\dfrac{1}{2}\\left(x-7+\\dfrac{27}{x+3}\\right)$.\n", "Le schéma est A-stable ssi $|q(x)|<1$. \n", "D'après les calculs ci-dessous on conclut que le schéma est A-stable ssi $h<\\dfrac{6}{\\beta}$." ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle q(x)=\\frac{x}{2} - \\frac{7}{2} + \\frac{27}{2 \\left(x + 3\\right)}$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle q(0)=1$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\lim_{x \\to \\infty}\\left(\\frac{x}{2} - \\frac{7}{2} + \\frac{27}{2 \\left(x + 3\\right)}\\right)=\\infty$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "On cherche les points stationnaires dans R^+\n" ] }, { "data": { "text/latex": [ "$\\displaystyle q'(x)=\\frac{x^{2} + 6 x - 18}{2 \\left(x + 3\\right)^{2}}$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Le signe de q' est le signe du numérateur qui est une parabole convexe dont on cherche le sommet\n" ] }, { "data": { "text/latex": [ "$\\displaystyle q'(x)=0 \\iff x\\in\\left[ -3 + 3 \\sqrt{3}\\right]$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle x=-3 + 3 \\sqrt{3}\\text{ est un minimum et}$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle q(-3 + 3 \\sqrt{3})=-5 + 3 \\sqrt{3}\\approx0.196152422706632$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Conclusion: q(x)>0 pour tout x>0. Vérifions ce calcul:\n", "q(x)>0\n" ] }, { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAACoAAAAOCAYAAABZ/o57AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAB0UlEQVRIDb2W7U0CQRCGD0IBqB1oBxAqEDuA2IF0oH/5Z6ADtAKFDqAEpAOxA0MH+Dzn7uVyhFNAb5JhZudj592vOWrb7TYZDodvSZI04Q38GWQLKWlbw/rPg1yTc4NeGTVCJUH1KT6LldHv0UfwAP0pZ++hP8dxVbIOCHdqlgdZVjzELcti/sNXZ1KPc37g5KuwwAPTjg8XqDvqHTyE3gl2gZVRg51ZHVHNhU3JvUQ+wo47cBfuwy7ee6x/QZy2BHmHGMBpHuOx9kiMtT/AbsQF3MRmfFL351Ai2eLtkHcV5ATpo+zi82rot4MIOiVsPsprOLN9e9JFmGv3GRE3hlPASG3HATUxkEAE5mN0V8+QWYdgvHOl8G9CblFMMbyGeVIfujveQvZieyom/XZs0QxMCYjS+chzNz32dPcKwV7NzqlAndNdPZUEKbUB7T3Ok1dq+RdA9x1lvthPejyVOUCzj04+qZ4fVKEDJO5eVg6bx+uCbzNjTsHfPRVok/nkfeSHpAjMFiYo20+e7AY+TP0ZMfYzvq75pyRSCLLn2cxNEISTLmDv4oQYW48++5vf/eh/wb5zbNi8Y5K90dhX+AOWlvizPzfoLsq2JMVeas31Fx2vogF49TMKAAAAAElFTkSuQmCC\n", "text/latex": [ "$\\displaystyle \\text{True}$" ], "text/plain": [ "True" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "On resout maintenant q(x)=1\n" ] }, { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle \\left[ 0, \\ 6\\right]$" ], "text/plain": [ "[0, 6]" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "On a trouvé q(0)=q(6)=1\n", "Conclusion: q(x)<1 pour x<6. Vérifions ce calcul:\n", "q(x)<1\n" ] }, { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle 0 < x \\wedge x < 6$" ], "text/plain": [ "0 < x ∧ x < 6" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import sympy as sym\n", "sym.init_printing()\n", "from IPython.display import display, Math\n", "sym.var('x',nonnegative=True)\n", "\n", "q=((x**2-4*x+6)/(2*x+6)).apart()\n", "display(Math('q(x)='+sym.latex(q)))\n", "\n", "display(Math('q(0)='+sym.latex(q.subs(x,0))))\n", "lim=sym.Limit(q,x,sym.oo)\n", "display(Math(sym.latex(lim)+'='+sym.latex(lim.doit())))\n", "\n", "print('On cherche les points stationnaires dans R^+')\n", "dq=(q.diff(x)).factor()\n", "display(Math(\"q'(x)=\"+sym.latex(dq)))\n", "print(\"Le signe de q' est le signe du numérateur qui est une parabole convexe dont on cherche le sommet\")\n", "sol=sym.solve(dq)\n", "display(Math(\"q'(x)=0 \\iff x\\in\"+sym.latex(sol)))\n", "minimum=sol[0]\n", "display(Math(\"x=\"+sym.latex(minimum)+\"\\\\text{ est un minimum et}\"))\n", "qmin=q.subs(x,sol[0])\n", "display(Math(\"q(\"+sym.latex(minimum)+\")=\"+sym.latex(qmin)+\"\\\\approx\"+sym.latex(sym.N(qmin))))\n", "sym.plot(q,1,-1,xlim=[-1,15],ylim=[-1.5,1.5]);\n", "\n", "print(\"Conclusion: q(x)>0 pour tout x>0. Vérifions ce calcul:\")\n", "print(\"q(x)>0\")\n", "display(sym.solve(q>0))\n", "print(\"On resout maintenant q(x)=1\")\n", "display(sym.solve(q-1))\n", "print('On a trouvé q(0)=q(6)=1')\n", "print(\"Conclusion: q(x)<1 pour x<6. Vérifions ce calcul:\")\n", "print(\"q(x)<1\")\n", "display(sym.solve(q<1))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ ">**Q8 [2 points]** Implémenter le schéma et approcher la solution du problème de Cauchy suivant avec $N+1$ points et $N=8$. \n", "$$\n", " \\begin{cases}\n", " y'(t)=-6y(t), &t\\in[0;1]\\\\\n", " y(0)=1\n", " \\end{cases}\n", "$$ " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Dans chaque point $t_i$, il faut approcher $(K_1)_i$ en resolvant une équation. Si on utilise la fonction `fsolve` du module `scipy.optimize`, il faut initialiser `fsolve` avec une approximation de $(K_1)_i$. On choisira donc $\\varphi(t_i,u_i)$:" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "A-stable ssi h < 1.0\n" ] }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "%reset -f\n", "\n", "from matplotlib.pylab import *\n", "from scipy.optimize import fsolve\n", "\n", "def RK(phi,tt):\n", " uu = [y0]\n", " for i in range(len(tt)-1):\n", " k1 = fsolve( lambda x : x - phi( tt[i]+h/3, uu[i]+h/3*x ) , phi(tt[i],uu[i]) )[0]\n", " k2 = phi( tt[i+1], uu[i]+h*k1 )\n", " uu.append( uu[i] + h/4*(3*k1+k2 ))\n", " return uu\n", "\n", "t0, y0, tfinal = 0, 1, 3\n", "\n", "sol_exacte = lambda t : exp(-6*t)\n", "phi = lambda t,y : -6*y\n", "\n", "print('A-stable ssi h <',6/6)\n", "figure(figsize=(10,5))\n", "N=8\n", "tt = linspace(t0,tfinal,N+1)\n", "h = tt[1]-tt[0]\n", "yy = [sol_exacte(t) for t in tt] \n", "uu = RK(phi,tt)\n", "plot(tt,yy,'b-',label=(\"Exacte\"))\n", "plot(tt,uu,'ro',label=(\"RK\"))\n", "title(r' $N$={}, $h$={}'.format(N,h))\n", "xlabel('$t$')\n", "ylabel('$u$')\n", "legend() \n", "grid(True)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exercice : étude d'un schéma Predictor-Corrector multipas\n", "\n", "On notera $\\varphi_k\\stackrel{\\text{déf}}{=}\\varphi(t_k,u_k)$.\n", "Soit les méthodes multipas\n", "\\begin{align*}\n", "u_{n+1}&=u_{n-3}+\\frac{4h}{3}\\left(2\\varphi_{n}-\\varphi_{n-1}+2\\varphi_{n-2}\\right)&\\text{[P]}\n", "\\\\\n", "u_{n+1}&=u_{n-1}+\\frac{h}{3}\\left(\\varphi_{n+1}+4\\varphi_{n}+\\varphi_{n-1}\\right)&\\text{[C]}\n", "\\end{align*}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ ">**Q9 [3 point]** \n", "Étudier consistance, ordre et zéro-stabilité de la méthode P." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "P est une méthode à $q=4$ pas explicite : \n", "- $p=3$\n", "- $b_{-1}=0$\n", "- $a_0=a_1=a_2=0$ et $a_3=1$\n", "- $b_0=\\frac{8}{3}$, $b_1=\\frac{-4}{3}$, $b_2=\\frac{8}{3}$ et $b_3=0$\n", "- La méthode est consistante d'ordre $\\omega=4$ car\n", "$$\n", "\\begin{cases}\n", "\\displaystyle\\sum_{j=0}^p a_j=1,\n", "\\\\\n", "\\displaystyle\\sum_{j=0}^p (-j)^{1}a_j+1\\sum_{j=-1}^p (-j)^{0}b_j=1\n", "\\\\\n", "\\displaystyle\\sum_{j=0}^p (-j)^{2}a_j+2\\sum_{j=-1}^p (-j)^{1}b_j=1\n", "\\\\\n", "\\displaystyle\\sum_{j=0}^p (-j)^{3}a_j+3\\sum_{j=-1}^p (-j)^{2}b_j=1\n", "\\\\\n", "\\displaystyle\\sum_{j=0}^p (-j)^{4}a_j+4\\sum_{j=-1}^p (-j)^{3}b_j=1\n", "\\\\\n", "\\displaystyle\\sum_{j=0}^p (-j)^{5}a_j+5\\sum_{j=-1}^p (-j)^{4}b_j\\neq1\n", "\\end{cases}\n", "$$\n", "Vérifions ces calculs ci-dessous:" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\omega\\ge1$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\omega\\ge2$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\omega\\ge3$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\omega\\ge4$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\omega<5$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import sympy as sym\n", "sym.init_printing()\n", "from IPython.display import display, Math\n", "\n", "p=3\n", "bm1=0\n", "a0=0\n", "a1=0\n", "a2=0\n", "a3=1\n", "b0=sym.Rational(8,3)\n", "b1=sym.Rational(-4,3)\n", "b2=sym.Rational(8,3)\n", "b3=0\n", "\n", "aa=[a0,a1,a2,a3]\n", "bb=[b0,b1,b2,b3]\n", "\n", "cond=[]\n", "cond.append(sum(aa)) \n", "cond.append(sum([-j*aa[j]+bb[j] for j in range(len(aa))])+bm1)\n", "if cond==[1,1]:\n", " display(Math(\"\\omega\\ge1\"))\n", " for n in range(2,9):\n", " cond.append(sum( [ (-j)**n*aa[j]+n*(-j)**(n-1)*bb[j] for j in range(len(aa)) ])+n*bm1)\n", " if cond[n]==1:\n", " display(Math(\"\\omega\\ge\"+str(n)))\n", " else:\n", " display(Math(\"\\omega<\"+str(n)))\n", " break\n", "#print(cond) " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- Le premier polynôme caractéristique est\n", "$$\n", "\\varrho(r)=r^{p+1}-\\sum_{j=0}^p a_jr^{p-j}=r^4-a_0r^3-a_1r^2-a_2r-a_0\n", "=\n", "r^4-1\n", "$$\n", " dont les racines sont \n", "$$\n", "r_0=1,\\ r_1=-1,\\ r_2=i,\\ r_3=-i\n", "$$\n", " La méthode est zéro-stable ssi\n", "$$\n", "\\begin{cases}\n", "|r_j|\\le1 \\quad\\text{pour tout }j=0,\\dots,p\n", "\\\\\n", "\\varrho'(r_j)\\neq0 \\text{ si } |r_j|=1,\n", "\\end{cases}\n", "$$\n", " ce qui est bien vérifié.\n", "\n", "- La méthode est convergente car consistante et zéro-stable." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ ">**Q10 [3 point]** \n", "Étudier consistance, ordre et zéro-stabilité de la méthode C." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "C est une méthode à $q=2$ pas implicite : \n", "- $p=1$\n", "- $b_{-1}=\\frac{1}{3}$\n", "- $a_0=0$ et $a_1=0$\n", "- $b_0=\\frac{4}{3}$ et $b_1=\\frac{1}{3}$\n", "- La méthode est consistante d'ordre 4 car\n", "$$\n", "\\begin{cases}\n", "\\displaystyle\\sum_{j=0}^p a_j=1,\n", "\\\\\n", "\\displaystyle\\sum_{j=0}^p -ja_j+\\sum_{j=-1}^p b_j=1\n", "\\\\\n", "\\displaystyle\\sum_{j=0}^p (-j)^{2}a_j+2\\sum_{j=-1}^p (-j)^{1}b_j=1\n", "\\\\\n", "\\displaystyle\\sum_{j=0}^p (-j)^{3}a_j+3\\sum_{j=-1}^p (-j)^{2}b_j=1\n", "\\\\\n", "\\displaystyle\\sum_{j=0}^p (-j)^{4}a_j+4\\sum_{j=-1}^p (-j)^{3}b_j=1\n", "\\\\\n", "\\displaystyle\\sum_{j=0}^p (-j)^{5}a_j+5\\sum_{j=-1}^p (-j)^{4}b_j\\neq1\n", "\\end{cases}\n", "$$\n", "Vérifions ces calculs ci-dessous:" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\omega\\ge1$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\omega\\ge2$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\omega\\ge3$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\omega\\ge4$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\omega<5$" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import sympy as sym\n", "sym.init_printing()\n", "from IPython.display import display, Math\n", "\n", "p=1\n", "bm1=sym.Rational(1,3)\n", "a0=0\n", "a1=1\n", "b0=sym.Rational(4,3)\n", "b1=sym.Rational(1,3)\n", "\n", "aa=[a0,a1]\n", "bb=[b0,b1]\n", "\n", "cond=[]\n", "cond.append(sum(aa)) \n", "cond.append(sum([-j*aa[j]+bb[j] for j in range(len(aa))])+bm1)\n", "if cond==[1,1]:\n", " display(Math(\"\\omega\\ge1\"))\n", " for n in range(2,9):\n", " cond.append(sum( [ (-j)**n*aa[j]+n*(-j)**(n-1)*bb[j] for j in range(len(aa)) ])+n*bm1)\n", " if cond[n]==1:\n", " display(Math(\"\\omega\\ge\"+str(n)))\n", " else:\n", " display(Math(\"\\omega<\"+str(n)))\n", " break\n", "#print(cond) " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- Le premier polynôme caractéristique est\n", "$$\n", "\\varrho(r)=r^{p+1}-\\sum_{j=0}^p a_jr^{p-j}=r^2-a_0r-a_1\n", "=\n", "r^2-1\n", "$$\n", " dont les racines sont \n", "$$\n", "r_0=1,\\ r_1=-1\n", "$$\n", " La méthode est zéro-stable ssi\n", "$$\n", "\\begin{cases}\n", "|r_j|\\le1 \\quad\\text{pour tout }j=0,\\dots,p\n", "\\\\\n", "\\varrho'(r_j)\\neq0 \\text{ si } |r_j|=1,\n", "\\end{cases}\n", "$$\n", " ce qui est bien vérifié.\n", "\n", "- La méthode est convergente car consistante et zéro-stable." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ ">**Q11 [2 point]** \n", "Calculer empiriquement l'ordre de convergence des méthodes P et C ainsi que de la méthode Predictor-Corrector associée aux deux méthodes P et C sur le problème de Cauchy\n", "$$\n", "\\begin{cases}\n", "y'(t) = 1+\\big(t-y(t)\\big)^2, &\\forall t \\in I=[2,3],\\\\\n", "y(2) = 1,\n", "\\end{cases}\n", "$$\n", " dont la solution exacte est $y(t)=t+\\dfrac{1}{1-t}$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Il est d'ordre $4$ car le prédictor est d'ordre $4$ et le corrector d'ordre $4$.\n", "\n", "On définit la solution exacte pour estimer les erreurs et on calcule la solution approchée pour différentes valeurs de $N$ pour les trois schémas:" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Multipas P ordre=4.26\n", "Multipas C ordre=4.04\n", "Multipas PC ordre=4.12\n" ] }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "%reset -f\n", "%matplotlib inline\n", "\n", "from matplotlib.pylab import *\n", "\n", "# variables globales\n", "t0 = 2\n", "tfinal = 3\n", "y0 = 1\n", "\n", "phi = lambda t,y : 1+(t-y)**2\n", "\n", "sol_exacte = lambda t : t+1/(1-t)\n", "\n", "from scipy.optimize import fsolve\n", "\n", "\n", "def multipas_P(phi, tt):\n", " h = tt[1] - tt[0]\n", " uu = [y0]\n", " uu.append(sol_exacte(tt[1]))\n", " uu.append(sol_exacte(tt[2]))\n", " uu.append(sol_exacte(tt[3]))\n", " for i in range(3,len(tt) - 1):\n", " uu.append( uu[i-3]+4*h/3* ( 2*phi(tt[i],uu[i])-phi(tt[i-1],uu[i-1])+2*phi(tt[i-2],uu[i-2]) ) )\n", " return uu\n", "\n", "def multipas_C(phi, tt):\n", " h = tt[1] - tt[0]\n", " uu = [y0]\n", " uu.append(sol_exacte(tt[1]))\n", " for i in range(1,len(tt) - 1):\n", " temp = fsolve ( lambda x : -x+uu[i-1]+h/3* ( phi(tt[i+1],x) + 4*phi(tt[i],uu[i])+phi(tt[i-1],uu[i-1]) ), uu[i])\n", " uu.append(temp)\n", " return uu\n", "\n", "def multipas_PC(phi, tt):\n", " h = tt[1] - tt[0]\n", " uu = [y0]\n", " uu.append(sol_exacte(tt[1]))\n", " uu.append(sol_exacte(tt[2]))\n", " uu.append(sol_exacte(tt[3]))\n", " for i in range(3,len(tt) - 1):\n", " u_tilde = uu[i-3]+4*h/3* ( 2*phi(tt[i],uu[i])-phi(tt[i-1],uu[i-1])+2*phi(tt[i-2],uu[i-2]) )\n", " uu.append( uu[i-1]+h/3* ( phi(tt[i+1],u_tilde) + 4*phi(tt[i],uu[i])+phi(tt[i-1],uu[i-1]) ) )\n", " return uu\n", "\n", "\n", "H = []\n", "err_p = []\n", "err_c = []\n", "err_pc = []\n", "N = 10\n", "for k in range(7):\n", " N+=20\n", " tt = linspace(t0, tfinal, N + 1)\n", " h = tt[1] - tt[0]\n", " yy = [sol_exacte(t) for t in tt]\n", " uu_p = multipas_P(phi, tt)\n", " uu_c = multipas_C(phi, tt)\n", " uu_pc = multipas_PC(phi, tt)\n", " H.append(h)\n", " err_p.append(max([abs(uu_p[i] - yy[i]) for i in range(len(yy))]))\n", " err_c.append(max([abs(uu_c[i] - yy[i]) for i in range(len(yy))]))\n", " err_pc.append(max([abs(uu_pc[i] - yy[i]) for i in range(len(yy))]))\n", " \n", "print ('Multipas P ordre=%1.2f' %(polyfit(log(H),log(err_p), 1)[0]))\n", "print ('Multipas C ordre=%1.2f' %(polyfit(log(H),log(err_c), 1)[0]))\n", "print ('Multipas PC ordre=%1.2f' %(polyfit(log(H),log(err_pc), 1)[0]))\n", "\n", "figure(figsize=(8,5))\n", "plot(log(H), log(err_p), 'r-o', label='Multipas P')\n", "plot(log(H), log(err_c), 'b-o', label='Multipas C')\n", "plot(log(H), log(err_pc), 'c-o', label='Multipas PC')\n", "xlabel('$\\ln(h)$')\n", "ylabel('$\\ln(e)$')\n", "legend(bbox_to_anchor=(1.04, 1), loc='upper left')\n", "grid(True);" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.5" }, "toc": { "base_numbering": 1, "nav_menu": {}, "number_sections": true, "sideBar": true, "skip_h1_title": true, "title_cell": "Table of Contents", "title_sidebar": "Contents", "toc_cell": true, "toc_position": {}, "toc_section_display": true, "toc_window_display": false } }, "nbformat": 4, "nbformat_minor": 4 }