{ "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\n", "css_file = './custom.css'\n", "HTML(open(css_file, \"r\").read())" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Python version 3.8.5 (default, Jul 28 2020, 12:59:40) \n", "[GCC 9.3.0]\n" ] } ], "source": [ "import sys #only needed to determine Python version number\n", "print('Python version ' + sys.version)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# M62_CM1 Introduction à l'approximation numérique d'EDO \n", "\n", "On ne peut expliciter des solutions analytiques que pour des équations différentielles ordinaires très particulières. \n", "Par exemple : \n", "- dans certains cas, on ne peut exprimer la solution que sous forme implicite. \n", " C'est le cas par exemple de l'EDO $y'(t)=\\dfrac{y(t)-t}{y(t)+t}$ dont les solutions vérifient la relation implicite \n", "$$\n", "\\frac{1}{2}\\ln(t^2+y^2(t))+\\arctan\\left( \\frac{y(t)}{t} \\right)=C,\n", "$$\n", " où $C$ est une constante arbitraire.\n", "- dans d'autres cas, on ne parvient même pas à représenter la solution sous forme implicite. \n", " C'est le cas par exemple de l'EDO $y'(t)=e^{-t^2}$ dont les solutions ne peuvent pas s'écrire comme composition de fonctions élémentaires.\n", "\n", "Pour ces raisons, on cherche des méthodes numériques capables d'approcher la solution de toutes les équations différentielles qui admettent une et une seule solution." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Position du problème\n", "\n", "Considérons le problème de Cauchy:\n", "\n", "
\n", "trouver une fonction $y \\colon I\\subset \\mathbb{R} \\to \\mathbb{R}$ définie sur un intervalle $I$ telle que\n", "$$\n", "\\begin{cases}\n", "y'(t) = \\varphi(t,y(t)), &\\forall t \\in I=]t_0,T[,\\\\\n", "y(t_0) = y_0,\n", "\\end{cases}\n", "$$\n", "avec $y_0$ une valeur donnée et supposons que l'on ait montré l'existence et l'unicité d'une solution $y$ pour $t\\in I$.\n", "
\n", "\n", "Pour $h>0$ soit $t_n\\equiv t_0+nh$ avec $n=0,1,2,\\dots,N$ une suite de $N+1$ nœuds de $I$ induisant une discrétisation de $I$ en $N$ sous-intervalles $I_n=[t_n;t_{n+1}]$ chacun de longueur $h=\\frac{T-t_0}{N}>0$ (appelé le *pas de discrétisation*).\n", "\n", "Pour chaque nœud $t_n$, on cherche la valeur inconnue $u_n$ qui approche la valeur exacte $y_n\\equiv y(t_n)$. \n", "- L'ensemble de $N+1$ valeurs $\\{t_0, t_1=t_0+h,\\dots , t_{N}=T \\}$ représente les points de la *discrétisation*. \n", "- L'ensemble de $N+1$ valeurs $\\{y_0, y_1,\\dots , y_{N} \\}$ représente la *solution exacte discrète*. \n", "- L'ensemble de $N+1$ valeurs $\\{u_0 = y_0, u_1,\\dots , u_{N} \\}$ représente la *solution numérique*. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Construction élémentaire des méthodes d'Euler explicite et implicite" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Une méthode classique, la **méthode d'Euler explicite** (ou *progressive*, de l'anglais *forward*), est obtenue en considérant l'équation différentielle en chaque nœud $t_n$ et en remplaçant la dérivée exacte $y'(t_n)$ par le taux d'accroissement\n", "$$\n", "\\varphi(t_n,y(t_n))=y'(t_n)=\\lim_{h\\to0}\\frac{y(t_n+h)-y(t_n)}{h}\\simeq\\frac{y(t_{n+1})-y(t_n)}{h}.\n", "$$\n", "Cela permet de construire une solution numérique par une suite récurrente:\n", "
\n", "$$\\begin{cases}\n", "u_0=y(t_0)=y_0,\\\\\n", "u_{n+1}=u_n+h \\varphi(t_n,u_n),& n=0,1,2,\\dots N-1.\n", "\\end{cases}$$\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "De même, en utilisant le taux d'accroissement\n", "$$\n", "\\varphi(t_{n+1},y(t_{n+1}))=y'(t_{n+1})\\simeq\\frac{y(t_{n+1})-y(t_n)}{h}\n", "$$\n", "pour approcher $y'(t_{n+1})$, on obtient la **méthode d'Euler implicite** (ou *rétrograde*, de l'anglais *backward*)\n", "
\n", "$$\\begin{cases}\n", "u_0=y(t_0)=y_0,\\\\\n", "u_{n+1}-h \\varphi(t_{n+1},u_{n+1})=u_n,& n=0,1,2,\\dots N-1.\n", "\\end{cases}$$\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Ces deux méthodes sont dites *à un pas*: pour calculer la solution numérique $u_{n+1}$ au nœud $t_{n+1}$, on a seulement besoin des informations\n", "disponibles au nœud précédent $t_n$.\n", "Plus précisément, pour la méthode d'Euler progressive, $u_{n+1}$ ne dépend que de la valeur $u_n$ calculée précédemment, tandis que pour la\n", "méthode d'Euler rétrograde, $u_{n+1}$ dépend aussi \"de lui-même\" à travers la valeur de $\\varphi(t_{n+1},u_{n+1})$. \n", "C'est pour cette raison que la méthode d'Euler progressive est dite *explicite* tandis que la méthode d'Euler rétrograde est dite *implicite*.\n", "Les méthodes implicites sont plus coûteuses que les méthodes explicites car, si la fonction $\\varphi$ est non linéaire, un problème non linéaire doit être résolu à chaque temps $t_{n+1}$ pour calculer $u_{n+1}$. \n", "Néanmoins, nous verrons que les méthodes implicites jouissent de meilleures propriétés de stabilité que les méthodes explicites." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Implémentation des schémas d'Euler explicite et implicite\n", "\n", "Voyons un exemple complet: considérons le problème de Cauchy\n", ">trouver la fonction $y \\colon I\\subset \\mathbb{R} \\to \\mathbb{R}$ définie sur l'intervalle $I=[0,1]$ telle que\n", ">$$\n", "\\begin{cases}\n", "y'(t) = 2ty(t), &\\forall t \\in I=[0,1],\\\\\n", "y(0) = 1.\n", "\\end{cases}\n", "$$\n", "(Sachant que la solution est $y(t)=e^{t^2}$, on pourra éstimer la qualité du schéma) " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "On commence par importer \n", "- le module `matplotlib.pylab` \n", "- la fonction `fsolve` du module `scipy.optimize` pour résoudre les équations implicites présentes dans le schéma implicite.\n", "\n", "Rappel: avec `pylab` on importe aussi le module `numpy` sans alias. Ainsi, non seulement on pourra utiliser ses fonctions spécifiques comme `linspace` mais, de plus, `numpy` rédéfinit toutes les fonctions mathématiques du module `math` (donc il est inutile de l'importer) et ces fonctions sont vectorisées (e.g. on pourra écrire directement `yy=sin(xx)` avec `xx` une liste au lieu d'écrire `yy=[sin(x) for x in xx]`)." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "%reset -f\n", "%matplotlib inline\n", "\n", "from matplotlib.pylab import *\n", "from scipy.optimize import fsolve" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "On initialise le problème de Cauchy" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "t0 = 0\n", "tfinal = 1\n", "y0 = 1" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "On définit l'équation différentielle : `phi` est une fonction python qui contient la fonction mathématique $\\varphi(t, y)=2ty$ dépendant des variables $t$ et $y$." ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "phi = lambda t,y : 2*y*t" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "On introduit la discrétisation: les nœuds d'intégration $[t_0,t_1,\\dots,t_{N}]$ sont contenus dans le vecteur `tt`. \n", "On a $N+1$ points espacé de $h=\\frac{t_N-t_0}{N}$." ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [], "source": [ "N = 8 \n", "tt = linspace(t0,tfinal,N+1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "On écrit les schémas numériques : les valeurs $[u_0,u_1,\\dots,u_{N}]$ pour chaque méthode sont contenues dans le vecteur `uu`." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Schéma d'Euler progressif :**\n", "$$\\begin{cases}\n", "u_0=y_0,\\\\\n", "u_{n+1}=u_n+h\\varphi(t_n,u_n)& n=0,1,2,\\dots N-1\n", "\\end{cases}$$" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [], "source": [ "# ici y0 est une variable globale\n", "def euler_progressif(phi,tt):\n", " h = tt[1]-tt[0]\n", " uu = [y0]\n", " for i in range(len(tt)-1): \n", " uu.append( uu[i]+h*phi(tt[i],uu[i]) )\n", " return uu\n", "\n", "def euler_progressif(phi,tt):\n", " h = tt[1]-tt[0]\n", " uu = [y0]\n", " N = len(tt)-1 # car tt contient N+1 points\n", " for i in range(N): # = 0,1,...,N-1 \n", " uu.append( uu[i]+h*phi(tt[i],uu[i]) )\n", " return uu" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Rappels: \n", "- `len(tt)` = nombre d'éléments de la liste `tt` = $N+1$ \n", "- les indices des éléments de `tt` vont de $0$ à $N$\n", "- `range(M)` produit les nombres entiers de $0$ à $M-1$\n", "\n", "Conclusion : `range(len(tt)-1)` donne $0,1,2,\\dots,N-1$ comme souhaité" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Schéma d'Euler régressif :**\n", "$$\\begin{cases}\n", "u_0=y_0,\\\\\n", "u_{n+1}=u_n+h\\varphi(t_{n+1},u_{n+1})& n=0,1,2,\\dots N-1\n", "\\end{cases}$$\n", "\n", "Attention : \n", "- $u_{n+1}$ est solution de l'équation $x=u_n+h\\varphi(t_{n+1},x)$, c'est-à-dire un zéro de la fonction (en générale non linéaire) $$x\\mapsto -x+u_n+h\\varphi(t_{n+1},x)$$\n", "- la fonction `fsolve` du module `scipy.optimize` requiert deux paramètres : une fonction et un point de départ. Elle renvoie deux valeurs: le zéro approché et l'estimation d'erreur." ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [], "source": [ "# ici y0 est une variable globale\n", "def euler_regressif(phi,tt):\n", " h = tt[1]-tt[0]\n", " uu = [y0]\n", " for i in range(len(tt)-1):\n", " temp = fsolve( lambda x: -x+uu[i]+h*phi(tt[i+1],x) , uu[i] )\n", " uu.append(temp[0])\n", " return uu" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "On calcule les solutions approchées:" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [], "source": [ "uu_ep = euler_progressif(phi,tt)\n", "uu_er = euler_regressif(phi,tt)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Comme on la connait, on définit la solution exacte pour calculer les erreurs:" ] }, { "cell_type": "code", "execution_count": 44, "metadata": {}, "outputs": [], "source": [ "sol_exacte = lambda t : y0*exp(t**2)\n", "yy = [sol_exacte(t) for t in tt]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "On compare les graphes des solutions exacte (en bleu) et approchées (en rouge) et on affiche le maximum de l'erreur:" ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [ { "data": { "image/png": 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uXsPG5zff1w96Wfwlu7g+cpyFbhA/R699PLH6iLvPIvwfb82GrQ/KqosUn9uzsfOaTvgy1iK2TXnqI2XVacpSVfWR4t/T1UcuJ7S+fD91RZSguo+QoFpT0otF2z1GaFWy0fgmFrrrvBId5+3i5e4+zd2/L058EcYdKFcCy93/F+2/NvoC/8d0+0YJqk8Jn687YstLq2/E9/8voR4+ZlPOOXIuG3ZhKPWcLXTXbuTuT5Z27hZavZ6aemxC0u7/ET4zWxP+7h9Hyk0JhupvGRveedzoghUzB7gspXJQL/rDLFba3cI5wGMp+zeIMpvF2dsHCE2Yells1PvIutYKFvqRb0vIdpZmDukHuUuVGvd8wj/BdrFYG3sYzGXjnd3b+foBX95Ot80m+kcU0z4e7uaeTaxZenT+AwiD59xuUZ/FyBTC3d+SlPo+FJ9Smv3iy+YQ7hI3ix1ja3cvbrJZns9Vee4s702seaFtOGpv6uPmaLOpwD5RK5Ri+1ByU0HM7A5Ci4yj3X3JBkG6P+3u7d29KeGu+C7AxGjdCne/wt1buvuuhEz1J755o5MXl4elLD8PeCYlOz6V0B80XnnrwPpzPBLoGzU5LG5K+b6ZnRnFPdvdu7t7c3c/kDBQ5EdlxGbRvhU559TuGmXZgtL/ftfFRTj/F939/0UVnVcJSaFDNvE1RaTqVWVdJBPirSd3puy6yHxgJZtXH5kDvJlyvg3dfaNm7O7+dqwuUuE+3xZmPBhH6N7RwkMX0vFsWB85njBu0X/YcPaqr8Jqiyd+Us0Bjks5t7q+4SDV5amPlFanqXB9JLqjvhtRfcQ2nqEg9VHcRWIqsW6e0e8/prmBBuG6fXLsun0I0N/COBxbE7pOPBmtmxjtM7f4plZU53mIkJz5vae0DI1a+LxOaElb1sCb8Wvrpipr3025rpe6b1nnHG1zKCEB+HTqulJe90igc+y9OA242syeT9nnFEKCsyBleQfgAQ+tYgoJrTE3pUVNzlOCofqbDBxuYR7mxoSRa0syFLjJon5/Fgb5O3UTXutxQn+nY8ysdtQ8u6uZ7RStL/6CeCHhYvaobTgC/PEW+mJtSRiL4UN3Lyvr/BBwgZkdaWEwoZZmtlea7X4k9CPfEtY1+xtO6Gu4XXS+LS2MCFwVGhEy34uii/MNKesHEr7UXUzoexjvrzeeWHeKNMp6H8rk7j8Qmkn2N7Oto7L9P1s/hsSmfK5K04VYvzXfcLTf1Mffo80KCHdArrIwUGdxRjvt4E1mdhNwJvCbdBd9M+sUlVNzQgLsRY+a4EWfiR0tOIgwivFfYvtuYWZ1gdqE8QHq2vrBsA40sz2jsmtKaI5X4LFWB7Hs+Mh4TO7+/whl/JfomCcTkijFfQD3IFzgOkYPCH1un42Ou7eZNTKzLc3sbEILgnuide0sTPNZO0pk9Qf+S7izVKFzBt4CviP8H9kiuvB3JeqSYmHQ1eK/t7aEz81/ouc7m9mhUcx1zewGwl20d6NjTwROMLNdo9h+E5XDF6nvqYhknaqsi2TCDRb6V7ci3HUt9U5nVKd4GLgn+v9Z28wOtvRTFv7IhgMfvwTsYWbnmFmd6LG/xbqWVaItga2AecAaC9MAxgeAbEaoZ11MSIafaCHhQPRl73VKr48MBe6KvvxioevdbzcxxrLqNJOB06Ny68zmdS05gNBNZTZsPENBmkdxF4lHgYvMrK2Fvvh/JuV6HnM+4aZK8XX7Y8Kd/ltY362weF3xl9VOrJ/h4P5o/xM9patHVI98Axji7huN72BhMPRtomvnAYQuL8/H1teJruu1gC2iMi4e1Ltr9Hdr0d/D3cX7RvWby1KO3Zv11/Wy6htnxY69C6EF5X9ioZd4zjHnAeN8wzGryjrnW1k/BltHQvef4YQxSFKPnW7gyInAxRYGca1HGG/sM6T83F2PLH8Q+iutIHxhLX48G1s/hNDs62tCPyQHtojWFQAXx7Y9hzBa/RJC1vjh2DoHdisjlgMJIwr/TLhgvUzI/nciNKfbLdquNuGLwy3R85GEC9FrUfxvAW3SvXa07Z2xdScT7pwujc7xmNRzI1xEX47imh8tq0vo2/ltdL7Tgas28z1oHS/XaNlcoGvs+ePAn6Pf2wGfROc6mTCS7dxo3W8J/4C3jZ43jM7rrOh5HcKXuB1jxy5Iea2070O697yUZY0J/9znEi5+nwKnb87nKlp2PmFchuLn+wOfbmZ57xuV3wpCX7h9Y+vOIsw2EP/srGLDv4+bY+vfiT47PxMSDA1i64oHDV1OGFjzrJQ4bo+OH3/cHq07gzDi8TLCnfZHge1T9j+D0PzWSvhMFUTnOAM4qpTy2OBvE7g6et+XRefXObbuiOh4ywjdHp4Dds/EOcc+2+9Hx58GnBxbN4JQuV4WvUZfoG5svynRugWESkY8biMMxvRd9H5NB87ZnM+PHnrokfkHWVIXIeV6nObYd7Lhtego4OuU419FqBssIHwpqh2tOx94J10sQD1Cy8P/Eq6Zb0XLUuM5mNC8eiFhxh6APQnX6XnRa74BdNzM9+F24PGU85sVe75FFM9O0fPe0f/lRYSm6GOI6ljAM8DQ2L7HEVpzNI2enwC8klL28deqBVxLuJYsJTQl/3u696mkZdHy0uo0uxK+hBdGywcVn38pxytgwzrTEDa//ndtVH5LCNe4rWLrXiFW30gTw8UlrEv9zOwSPV/Jhn9fxfXCv0Tr4+sKY8cbHX2uCgljKlyV8noj2fi6fn7s/P5LqBPMIYwR0Sj2/r4avS+FhM/1zUR1Gsqub9xFqGMui34Oi322Sj3naJu6hM/tkWnKsNRzTnP+d6Ysa0k00Hma7dsQpt1eEJ37q/Hz0qPsR/EHRKRSmdlIwhfsPycdS3VgZpcCbd396uh5AeFLXkGScZXGzM4nXNDPj56PI0zvND7JuERERIqZmRO+LHyddCzVgZm9Qxi48FMLMwQUuHvrRIMqQ7zOFLWqe5Nwo2JlspGJ5Ia08xOLSLLcfVjSMVSUu/++7K1EREQkW7n7r5OOoSI8zJJVFd1RRCSiBINI9TCSNFP7ZJnJbPoIzSIiIlI9LCJ0E8l2I8n+OpNIjaUuEiIiIiIiIiJSYZpFQkREREREREQqLCu7SDRr1sxbt26dseMtW7aMBg0aZOx4uUxlmTkqy8xRWWaOyjJzKqMsP/nkk/nu3jyjB5W0Ml0XAf19ZZLKMnNUlpmjsswclWXmVHV9JCsTDK1bt+bjjz/O2PEKCgro2rVrxo6Xy1SWmaOyzByVZeaoLDOnMsrSzGZn9IBSokzXRUB/X5mksswclWXmqCwzR2WZOVVdH1EXCRERERERERGpMCUYRERERERERKTClGAQERERERERkQrLyjEY0lm9ejVz585l5cqVm7xv48aNmT59eiVEVXPUrVuXnXbaiTp16iQdioiISFaqSF0EVB8pD9VHRESqt2qTYJg7dy6NGjWidevWmNkm7bt06VIaNWpUSZFVf+7OggULmDt3Lm3atEk6HBERkaxUkboIqD5SFtVHRESqv2rTRWLlypU0bdp0sy7oUjozo2nTppt9R0ZERCQXqC5SuVQfERGp/spMMJhZKzPLN7PpZjbVzPqk2eYGM5scPb4wsyIz2zZaN8vMPo/WVWi+J13QK4/KVkREqiMzq2tmH5nZZ1E95Y4023Q1s8WxusptFXi9igUspVL5iohUb+XpIrEGuM7dJ5lZI+ATM3vN3acVb+DufYG+AGZ2InCNu/8cO0Y3d5+fycBFREREgFXAEe5eaGZ1gHfM7BV3/yBlu7fdvXsC8YmIiOSMMlswuPsP7j4p+n0pMB1oWcouZwCjMxNedqlduzYdO3Zc97j77rszduzJkyczfvz4jB1PREQkF3hQGD2tEz08wZAqleoiIiKSzcy9/NdgM2sNvAW0d/cladbXB+YCuxW3YDCzmcBCwsX+AXcfVsKxLwUuBWjRokWnMWPGbLC+cePG7LbbbuWONa6oqIjatWtv1r5xO+ywAz/88EOFj5POqFGjmDRpEv3796+U45fH119/zeLFi0vdprCwkIYNG1ZRRDWbyjJzVJaZo7LMnMooy27dun3i7p0zetAawMxqA58AuwFD3P1PKeu7AuMIdZTvgevdfWqa41RaXQQyUx+p6XURUH2kqqksM0dlmTkqy8yp8vqIu5frATQkXLxPKWWb04AXU5btGP3cDvgMOLys1+rUqZOnmjZt2kbLymvJkiWbvW9cgwYNNlq2aNEi32OPPfzLL790d/fTTz/dhw0b5u7uPXv29E6dOnnbtm39tttuW7fPRx995AcffLDvs88+vv/++/uiRYu8VatW3qxZM+/QoYOPGTPGCwsL/YILLvDOnTt7x44d/bnnnsvIOZSmPGWcn59f6XHkCpVl5qgsM0dlmTmVUZbAx17O63YuPoAmQD7hRkh8+dZAw+j344GvyjpWpusi7pmpj9T0uoi76iNVTWWZOSrLzFFZZk5V10fKNU1l1KdxHDDK3Z8pZdPTSeke4e7fRz9/MrNngQMIrSA229VXw+TJ5d++qKgeZd0w6NgRBgwofZsVK1bQsWPHdc9vuukmTjvtNAYPHsz5559Pnz59WLhwIZdccgkAd911F9tuuy1FRUUceeSRTJkyhb322ovTTjuNJ598kv33358lS5ZQv359/vrXv/Lxxx8zePBgAG6++WaOOOIIHn74YRYtWsQBBxzAUUcdRYMGDcp/4iIiIjnE3ReZWQFwLPBFbPmS2O/jzew+M2vmFRgfalPrIlB2fUR1ERERqe7KTDBYGM73IWC6u99TynaNgS7A2bFlDYBa7r40+v1o4K8Vjjoh9erVY3Ka2sRvfvMbnnrqKXr37s1nn322bvnYsWMZNmwYa9as4YcffmDatGmYGTvssAP7778/AFtvvXXa15owYQIvvPAC/fr1A8LUWN999x177713JZyZiIhI9WRmzYHVUXKhHnAU8M+UbbYHfnR3N7MDCGNQLaj6aCtOdRERESm3/HwOOuMMGD0aunWrkpcsTwuGQ4FzgM/NrPiKdjOwM4C7D42WnQxMcPdlsX1bAM9GUw5tATzh7q9WNOiysvupli5dQaNGjSr6siVau3Yt06dPp169evz888/stNNOzJw5k379+jFx4kS22WYbzj//fFauXIm7l2sKJndn3Lhx7LnnnpUWt4iISA2wA/BINA5DLWCsu79kZj1hXT2lB9DLzNYAK4DToyaem21T6yJQufUR1UVERGQD+fnQvTt1ly+H7t3hpZeqJMlQnlkk3nF3c/d93L1j9Bjv7kNjyQXcfaS7n56y77fu3iF6tHP3uyrjJJJ27733svfeezN69GguvPBCVq9ezZIlS2jQoAGNGzfmxx9/5JVXXgFgr7324vvvv2fixIkALF26lDVr1tCoUSOWLl267pjHHHMMeXl5xX1H+fTTT6v+xEREZLMsWwZffaXBqaqCu09x932jekp7d/9rtHxdPcXdB0f1kA7ufpC7v5ds1JmnuoiIiKwTJRdYvjw8L04y5OdX+kuXawwGCVL7PR577LFceOGFPPjgg3z00Uc0atSIww8/nDvvvJM77riDfffdl3bt2rHrrrty6KGHArDlllvy5JNPcuWVV7JixQrq1avH66+/Trdu3bj77rvp2LEjN910E7feeitXX301++yzD+5O69ateemll5I6dRER2QQjRsCVV3bm4IOhffuko5GaRHUREREpVWpyoVgVtWRQgmETFBUVpV0+ffr0db/fc8/6YSpGjhyZdvv999+fDz74YKPlxXcSij3wwAObEaWIiCSpqAjuuQfatVtM+/aNkw5HahjVRUREpFQXXLBxcqHY8uVh/axZlfbyZXaREBERkfJ75hmYORNOO21O0qGIiIhIrhkxAurXT7+ufv2wvhIpwSAiIpIh7tC3L+y2GxxyyGbPgCgiIiKyebp1gxdfhNTBfOvXr5KBHtVFQkREJEPeeQcmToT774fatZOORkRERHLS2rXhrseWW8Ivv1RZcgHUgkFERCRj+vWDZs3g3HOTjkRERERy1qBBsN128MILrGzRosqSC6AEg4iISEbMmAEvvAC9e5fc9VFERESkUn37bUgoXHopHHMMH4wZU8El9cgAACAASURBVGXJBVCCQUREJCPuuQfq1oXLL086EhEREclZ990X+mn27JnIy9fsBEN+PrRuTe233ko6kiq1dOlS7r//ftw96VBERHLCTz/BI4/AeeeFFoki60R1EfLzk46kyqk+IiJSxZYtg4ceglNOgZYtEwmh5iYY8vOhe3eYPZt6p56asQv7s88+i5nx5ZdfZuR4m2rWrFm0b9++xPW//PILl19+OV26dMFSRw4VEZFKMXhwGEPpmmuSjkSySqwuQvfuGb3hofqIiIhsZNQoWLQIrrwysRBqZoKh+IK+fDkAtmJFeJ6BJMPo0aP59a9/zZgxYyp8LIA1a9Zk5DjFttxySx577DHatm2b0eOKiEh6y5fDkCFw0kmw555JRyNZI6UuwvLlGb3hofqIiIhswB3y8mDffeHQQxMLo3pOU3n11TB5cvp1CxfCF1+EqTnili+Ho46C9u1hm2023q9jRxgwoNSXLSws5N133yU/P5+TTjqJ22+/nYKCAm677TaaNm3KjBkzOPzww7nvvvuoVasWDRs25LLLLiM/P59tttmGMWPG0Lx5c7p27cohhxzCu+++y0knnUSPHj248MILmTdvHs2bN2fEiBHsvPPO/Pjjj/Ts2ZNvv/0WgPvvv58dd9yRoqIiLrnkEt577z1atmzJ888/T7169fjmm2/o3bs38+bNo379+gwfPpy99tqLefPm0bNnT7777jsABgwYwKEJfuhERGqSkSPh55/hhhuSjkSq1GbURWzFigrXRUD1ERERSePNN8O15+GHIcGWYzWvBcOMGRsnF4qtXRvWb6bnnnuOY489lj322INtt92WSZMmAfDRRx/Rv39/Pv/8c7755hueeeYZAJYtW8Z+++3HpEmT6NKlC3fccce6Yy1atIg333yT6667jiuuuIJzzz2XKVOmcNZZZ3HVVVcBcNVVV9GlSxc+++wzJk2aRLt27QD46quv6N27N1OnTqVJkyaMGzcOgEsvvZS8vDw++eQT+vXrx+XRSGN9+vThmmuuYeLEiYwbN46LL754s8tARETWKyoKgzsedBAcckjS0UjWqMS6CKg+IiIiaQwaBE2bwumnJxpG9WzBUFp2P7VJYlz9+hWaA3T06NFcffXVAJx++umMHj2aE044gQMOOIBdd90VgDPOOIN33nmHHj16UKtWLU477TQAzj77bE455ZR1xypeDvD++++vqwScc845/PGPfwTgjTfe4NFHHwWgdu3aNG7cmIULF9KmTRs6duwIQKdOnZg1axaFhYW89957nHrqqeuOu2rVKgBef/11pk2btm75kiVLWLp0KY0aNdqschARkeD55+Gbb+Cf/0z0ZoEkIaG6CKg+IiIiKWbPDpWSP/4R6tVLNJTqmWAoTbdu4cKdemGv4AV9wYIFvPHGG3zxxReYGUVFRZgZxx9//EaDF5U0mFF8eYMGDUp8rbIGQ9pqq63W/V67dm1WrFjB2rVradKkCZPTNNdcu3Yt77//PvUS/rCJiNQ0/frBrrvC736XdCSSVUqoi3i9elgFkwuqj4iIyEbuvz/87NUr2TioiV0kYP2FvX59IFzQK3q34Omnn+bcc89l9uzZzJo1izlz5tCmTRveeecdPvroI2bOnMnatWt58skn+fWvfw2EC+nTTz8NwBNPPLFueapDDjlk3SBNo0aNWrfdkUceyf3Rh6WoqIglS5aUGN/WW29NmzZteOqpp8I5u/PZZ58BcPTRRzN48OB126a76IuIyKZ59114/3249tow3bTIBlLqItSvz4qnnqpQXQRUHxERkRQrVsDw4eFux847Jx1NDU0wwPoL+y67ZOSCPnr0aE4++eQNlv3+97/niSee4OCDD+bGG2+kffv2tGnTZt12DRo0YOrUqXTq1Ik33niD2267Le2xBw0axIgRI9hnn3147LHHGDhwIAADBw4kPz+fX/3qV3Tq1ImpU6eWGuOoUaN46KGH6NChA+3ateP5559fd/yPP/6YffbZh7Zt2zJ06NAKlYWIiITWC9tuC+efn3QkkrVidRFeeomiww+v8CFVHxERkQ2MHh1Gm05wasoNuHvWPTp16uSppk2bttGy8lqyZMlm71uW/Px8P+GEE9Kua9CgQaW9bmUoTxnn5+dXfiA5QmWZOSrLzFFZls+MGe5m7n/+c8nbVEZZAh97Flync+GR6bqIu+oj5aX6SNVSWWaOyjJzVJbltHate8eO7r/6Vfg9jaquj9S8MRhEREQq2b33wpZbwhVXJB2JiIiI5Kx33w1TJg8bljWjTSvBUEFdu3ala9euadcVFhZWbTAiIlLp5s2DkSPh3HOhRYukoxEJVB8REclBgwbBNtvAWWclHck61WoMhtAaQyqDylZEpHzuuw9WrgyDO0ru0fWycql8RUTKae5ceOYZuOii9QMKZ4Fqk2CoW7cuCxYs0IWnErg7CxYsoG7dukmHIiKS1ZYvh8GD4cQTYa+9ko5GqprqIpVL9RERkU0wdCisXQuXX550JBuoNl0kdtppJ+bOncu8efM2ed+VK1fqYlWGunXrstNOOyUdhohIVnv0UZg/H66/PulIJAkVqYuA6iPlofqIiEg5rFwZxl048URo0ybpaDZQbRIMderUoc1mFl5BQQH77rtvhiMSEZFcUlQE99wD++8Phx2WdDSShIrURUD1ERERyZCxY8OgUNkyNWVMtUkwiIiIJOnFF+Grr8I1PUsGahYREZFc4x4Gd9x7bzjyyKSj2YgSDCIiIuXQr19ohXjyyUlHIiIiIjnrgw/gk0/CqNNZeMdDCQYREZEyvP9+mGp60CDYQldOERERSUpeHjRuDOeck3QkaVWbWSRERESS0r9/mGb6gguSjkRERERy1g8/wFNPhQpJw4ZJR5OWEgwiIiKl+PrrMM10r15Zey0XERGRXPDAA2HU6d69k46kREowiIiIlOLee6FOHbjiiqQjERERkZz1yy8hwXDccbDbbklHUyIlGEREREowfz6MGAFnnw077JB0NCIiIpKznn4a/ve/rJyaMk4JBhERkRLcfz+sWAHXXZd0JCIiIpLT8vJgjz3g6KOTjqRUSjCIiIiksXJluJafcAK0bZt0NCIiIpKzJk4M01NecQXUyu6v8NkdnYiISEIeewzmzYPrr086EhEREclpeXlhpOnzzks6kjIpwSAiIpJi7dowNWWnTtClS9LRiIiISM766Sd48kk4/3zYeuukoynTFkkHICIikm1eeglmzIDRo8Es6WhEREQkZw0bFmaQqCbTWakFg4iISIp+/WDnnaFHj6QjERERkZy1enUYcfroo2HPPZOOplzUgkFERCTmww/h7bfh3nthC10lRUREJCnPPgvffx9aMVQTasEgIiIS078/NGkCF12UdCQiIiKS0/Ly4P/+D447LulIyk0JBhERkci338K4cdCzJzRqlHQ0IiIikrM+/RTeeQd69876qSnjqk+kIiIilezee6F2bbjyyqQjERERkZyWlwf168MFFyQdySYpM8FgZq3MLN/MppvZVDPrk2abrma22MwmR4/bYuuONbMZZva1md2Y6RMQERHJhAUL4OGH4ayzYMcdk45GREREctb8+fDEE3DuuaHfZjVSnuGr1gDXufskM2sEfGJmr7n7tJTt3nb37vEFZlYbGAL8BpgLTDSzF9LsKyIikqihQ2H5crjuuqQjERERkZz24IOwalW1mZoyrswWDO7+g7tPin5fCkwHWpbz+AcAX7v7t+7+CzAG+O3mBisiIlIZVq4MLRGPPRbat086GhEREclZa9bAfffBEUdAu3ZJR7PJNmkMBjNrDewLfJhm9cFm9pmZvWJmxSXREpgT22Yu5U9OiIiIVIlRo+DHH+GGG5KORERERHLaCy/AnDlw1VVJR7JZyj3Dt5k1BMYBV7v7kpTVk4Bd3L3QzI4HngN2ByzNobyE418KXArQokULCgoKyhtamQoLCzN6vFymsswclWXmqCwzJxfLcu1a+Nvf9mf33ddi9gmZOv1cLEsRERGpoLw8aN0auncvc9NsVK4Eg5nVISQXRrn7M6nr4wkHdx9vZveZWTNCi4VWsU13Ar5P9xruPgwYBtC5c2fv2rVrec+hTAUFBWTyeLlMZZk5KsvMUVlmTi6W5csvw+zZoRVDt25dM3bcXCxLERERqYApU6CgAP71rzCtVTVUnlkkDHgImO7u95SwzfbRdpjZAdFxFwATgd3NrI2ZbQmcDryQqeBFREQqqm9faNUKTj016UhEREQkpw0eDPXqwUUXJR3JZitPC4ZDgXOAz81scrTsZmBnAHcfCvQAepnZGmAFcLq7O7DGzK4A/g3UBh5296kZPgcREZHNMnEivPkm9O8PdeokHY2IiIjkrJ9/hscfD/Nlb7tt0tFstjITDO7+DunHUohvMxgYXMK68cD4zYpORESkEvXvD1tvDRdfnHQkIiIiktMefhhWrIArr0w6kgrZpFkkREREaopZs+Cpp+Cyy0KSQURERCQRRUUwZAgcfjjss0/S0VSIEgwiIpKTBgyAWrWgT5+kIxEREZGc9vLL4c5HNZ2aMk4JBhERyTkLF8KDD8KZZ0LLlklHIyIiIjktLy+MOP3b3yYdSYUpwSAiIjln6FBYtgyuuy7pSERERCSnTZsGr78OvXrBFuWZgyG7KcEgIiI5ZdUqGDQIjj662ndzFBERkepu8GDYaiu45JKkI8mI6p8iERER2QRPPAH/+x88+mjSkYiIiEhOW7w4VEjOOAOaNUs6moxQCwYREckZ7tCvX2i5cNRRSUcjIiIiOW3EiNBns5pPTRmnFgwiIpIzXn01dHV89FEwSzoaERERyVlr14apKQ85BPbbL+loMkYtGEREJGf06xdmjTj99KQjERERkZz26qvw9dc1YmrKOCUYREQkJ0yaBG+8AVdfDXXqJB2NiIiI5LS8PNhxRzjllKQjySglGEREJCf06weNGtWYQZpFRESkupoxI7Rg6Nmzxt31UIJBRERqvNmzYexYuPRSaNw46WhEREQkpw0ZAltuGSomNYwSDCIiUuMNHBgGdezTJ+lIJNPMrK6ZfWRmn5nZVDO7I802ZmaDzOxrM5tiZjVnNC0REaleli6FkSPhD3+AFi2SjibjNIuEiIjUaIsWwfDhcNpp0KpV0tFIJVgFHOHuhWZWB3jHzF5x9w9i2xwH7B49DgTuj36KiIhUrUceCUmGGjQ1ZZxaMIiISI02bBgUFsL11ycdiVQGDwqjp3Wih6ds9lvg0WjbD4AmZrZDVcYpIiLC2rUweDAceCAccEDS0VQKtWAQEZEa65dfQveIo46Cjh2TjkYqi5nVBj4BdgOGuPuHKZu0BObEns+Nlv2QcpxLgUsBWrRoQUFBQUbjLCwszPgxc5XKMnNUlpmjssycmlqW20ycSIcZM5h28838VEXnV9VlqQSDiIjUWKNHw/ffw8MPJx2JVCZ3LwI6mlkT4Fkza+/uX8Q2sXS7pTnOMGAYQOfOnb1r164ZjbOgoIBMHzNXqSwzR2WZOSrLzKmxZdmvH7RoQdu//IW2W25ZJS9Z1WWpLhIiIlIjuYfrePv2cPTRSUcjVcHdFwEFwLEpq+YC8RE4dgK+r6KwRERE4JtvYPx4uOyyMINEDaUEg4iI1EgTJsAXX4SxFyzd/WupEcysedRyATOrBxwFfJmy2QvAudFsEgcBi939B0RERKrKkCFQu3ZIMNRg6iIhIiI1Ur9+sOOOcMYZSUcilWwH4JFoHIZawFh3f8nMegK4+1BgPHA88DWwHLggqWBFRCQHFRaG/po9eoTKSQ2mBIOIiNQ4kyfD66/D3XfX6FaIArj7FGDfNMuHxn53oHdVxiUiIrLO44/D4sU1dmrKOHWREBGRGqd/f2jYsMa3QhQREZFs5x6mpuzUCQ4+OOloKp1aMIiISI0yZw6MGRNuEjRpknQ0IiIiktPy82HqVBg5MicGhVILBhERqVEGDgw3C/r0SToSERERyXmDBkGzZnDaaUlHUiWUYBARkRpj8WIYNgz+8AfYZZekoxEREZGcNmsWvPgiXHop1K2bdDRVQgkGERGpMYYPh6VL4brrko5EREREct5994VuEb16JR1JlVGCQUREaoRffoEBA6BbtzCOkoiIiEhili+HBx+Ek0+GnXZKOpoqo0EeRUSkRhg7Fv7739BFQkRERCRRTzwBCxfmxNSUcWrBICIi1Z479O0LbdvCccclHY2IiIjkNHfIy4MOHeCww5KOpkqpBYOIiFR7r78OU6bAww/nxAxQIiIiks3efjtUTB58MOcqJmrBICIi1V6/frD99nDmmUlHIiIiIjlv0CDYdtucrJgowSAiItXalCkwYQJcdRVstVXS0YiIiEhOmzMHnnsOLr4Y6tVLOpoqpwSDiIhUa/37Q4MGcNllSUciIiIiOe/++8MYDJdfnnQkiVCCQUREqq25c8MgzRddFFoiioiIiCRm5UoYPhxOOgl22SXpaBKhBIOIiFRbeXmwdi1cc03SkYiIiEjOGzMG5s/Puakp45RgEBGRamnJEhg6FE49FVq3TjoaERERyWnFU1O2awfduiUdTWI0TaWIiFRLDz4YkgzXX590JCIiIpLz3n8fJk0Kdz9ybGrKOLVgEBGRamf1ahgwALp0gc6dk45GREREct6gQdCkCZx9dtKRJEoJBhERqXaeeirMAqXWCyIiIpK477+HcePgwgvD1FY5TAkGERGpVtyhXz/Yay84/vikoxEREZGcN3QoFBVB795JR5I4JRhERKRayc+HTz+F666DWtl6FcvP56DTTw/BioiISM21ahU88ACccALsumvS0SQuW6tmIiIiafXtCy1aZHEXx/x86N6duj/+CN27K8kgIiJSkz31FPz0E1x1VdKRZIUyEwxm1srM8s1suplNNbM+abY5y8ymRI/3zKxDbN0sM/vczCab2ceZPgEREckdX3wBr74appeuWzfpaNKIkgssXx6eL1+uJIOIiEhNlpcX+m0edVTSkWSF8rRgWANc5+57AwcBvc2sbco2M4Eu7r4P8DdgWMr6bu7e0d011reIiGy2/v2hfn3o2TPpSNJITS4UU5JBRESkZvrwQ/joI7jiipyemjKuzASDu//g7pOi35cC04GWKdu85+4Lo6cfADtlOlAREclt338Po0aFAZqbNk06mjQuuGDj5EKx5cvDehEREak58vKgUSM499ykI8kaW2zKxmbWGtgX+LCUzS4CXok9d2CCmTnwgLuntm4oPvalwKUALVq0oKCgYFNCK1VhYWFGj5fLVJaZo7LMHJVl5mRzWQ4f3oaiop056KAPKShYmXQ4G2nSpw+/uukmaq9atdG6oq224vM+fViUpWUrIiIim+h//4OxY6FXr5BkEGATEgxm1hAYB1zt7ktK2KYbIcHw69jiQ939ezPbDnjNzL5097dS940SD8MAOnfu7F27di3/WZShoKCATB4vl6ksM0dlmTkqy8zJ1rJcuhROPhlOOQXOOuugpMNJr2tXmDcP/vGPDZfXr0/tl16iY7duiYQlIiIilWDYMFi9WlNTpijXLBJmVoeQXBjl7s+UsM0+wIPAb919QfFyd/8++vkT8CxwQEWDFhGR3PLQQ7BoEVx/fdKRlGL+fBg+HPbeOwwUAeHnSy+BkgsiIiI1xy+/wNChcOyxsMceSUeTVcozi4QBDwHT3f2eErbZGXgGOMfd/19seQMza1T8O3A08EUmAhcRkdywZg3cey8cdhgceGDS0ZTi2mth8eLQXPKll1jZooWSCyIiIjXRM8/ADz9oaso0ytNF4lDgHOBzM5scLbsZ2BnA3YcCtwFNgftCPoI10YwRLYBno2VbAE+4+6sZPQMREanRnn4avvsujKOUtSZMgMcegz//Gdq3B+CDMWOysruJiIiIVFBeHuy+OxxzTNKRZJ0yEwzu/g5Q6pwb7n4xcHGa5d8CHTY7OhERyWnu0LdvaH3YvXvS0ZRg2bIwb+aee8IttyQdjYiIiFSmTz6B996DAQOgVrlGHMgpmzSLhIiISFV6802YNAkeeCCLr+G33w4zZ4Zg69ZNOhoRERGpTHl50KABnH9+0pFkpWytromIiNCvHzRvDueck3QkJZg0Ce65By65BA4/POloREREpDLNmwdjxsB550HjxklHk5WUYBARkaw0bRq8/DJccQXUq5d0NGmsWQMXXwzbbQf/+lfS0YiIiEhlGz4cVq0KlZNqYP58+PbbBlX6mkowiIhIVurfPyQWLr886UhKMHAgfPopDBoETZokHY2IiIhUpjVr4P774aijwpTU1cA110Dv3vsxf37VvaYSDCIiknV++AEefxwuuACaNUs6mjRmzoTbboMTT4QePZKORkRERCrbc8/B3LnVZmrK8eNDXerUU+dUaV1KCQYREck6gwfD6tUh85513MOsEbVqwZAhYKVOtCQiIiI1QV4etGkDxx+fdCRlWrIkVFX23hvOPnt2lb62ZpEQEZGsUlgYWiCefDLstlvS0aQxahRMmBAqGq1aJR2NiIiIVLbPPoO33gqjT9eunXQ0ZbrpptDY4t13YdUqr9LXVgsGERHJKiNGwMKFcP31SUeSxvz5oVnFQQdBr15JRyMiIiJVIS8P6teHCy9MOpIyvf023Hdf6Mlx8MFV//pKMIiISNZYswbuvRcOOSSZi2KZrr0WFi8Oo0hXgzsYIiIiUkELFoTWi2efDdtsk3Q0pVqxAi66CFq3hjvvTCYGdZEQEZGs8cwzYfzE/v2TjiSNCRPgscfgz3+G9u2TjkZERESqwkMPwcqV1WJqyr/+Fb76Cl57DRo2TCYGtWAQEZGs4B66Nu6+O5x0UtLRpFi2LIyWtOeecMstSUcjIiIiVaGoKPQ36NoVfvWrpKMp1aRJ0Ldv6MVx1FHJxaEWDCIikhXefhsmTgwDPGZd74Pbbw9NK958E+rWTToaERERqQovvgizZ4f+m1ls9erQNaJ583CzJklKMIiISFbo1w+aNYNzz006khSTJsE998All8DhhycdjYiIiFSVvDzYeWc48cSkIylVv34weXLoapr0MBHqIiEiIon7/PNwk6B37zBIc9ZYsyYkFpo3h3/+M+loREREpKp88QW88QZcfjlskb335b/8Eu64A3r0CFN8Jy17S0pERHJCUVH4Dt+0aRaOnzRwYGjBMHZs8rcEREREpOoMHhy6RV58cdKRlGjt2hBe/fqhsUU2UIJBREQSNXAgfPhhmAGqWbOko4mZORNuuy00i+zRI+loREREpKosXBhmjjrzzHAHJEvddx+8+y6MHAnbb590NIG6SIiISGK++SbM+ti9O5xxRtLRxLiHWSNq1YIhQ8As6YhERESkqowYAcuXw5VXJh1JiWbPhhtvhKOPzq7xq9SCQUREEuEeukbUqQNDh2bZd/hRo2DChNDesFWrpKMRERGRqpCfD+efH6ZlOOww6Ngx6YjScofLLgu/P/BAdtWhlGAQEZFEDB8eruPDhkHLlklHEzN/PlxzDRx0EPTqlXQ0IiIiUhXy80OTyuXLw/NLLkk2nlI89hj8+98waBC0bp10NBtSFwkREalyc+fC9dfDEUdk4dhJ114LixeHDEjt2klHIyIiIpUtNbkAYe7H/PzkYirBjz/C1VfDIYeECS6yjRIMIiJSpYqHNygqCt/hs6lZHxMmhNsCf/oTtG+fdDQiIiJS2dIlFyA8794965IMV14Jy5bBgw9m530QJRhERKRKPfEEvPwy3HUX7Lpr0tHELFsWMh977gm33JJ0NCIiIlIVLrhg4+RCseXLw/os8eyz8NRT8Je/wN57Jx1NekowiIhIlfnpJ+jTJwxvkHUDM99+e5ia8oEHwrzXIiIiUvONGAH166dfV79+WJ8FFi4MXSI6dIAbbkg6mpIpwSAiIlXmyith6VJ46KEsa9Y3aRLcc08YEKJLl6SjERERkarSrVvoHpmqfn146aWwPgvccAPMmwcPPxxm4MpWSjCIiEiVeO45GDsWbr0V2rZNOpqYNWvCSNHNm8O//pV0NCIiIlLVxo6FLbZY34Ixy5IL//lPuDlz/fWw335JR1M6JRhERKTSLVwYZnzs0CGMn5hVBg4MLRjy8mCbbZKORkRERKrS88/Dk0+GrpLjx8Muu2RVcmHZsnAfZPfdw9gL2W6LpAMQEZGa7/rrQ7O+l1/OsmZ9M2fCbbfBiSdCjx5JRyMiIiJVadGicAdkn33gj38MlZRZs5KOagO33hqqK2++CfXqJR1N2ZRgEBGRSvXaa6G/4I03ZlmzvuL5MmvVgiFDsmy+TBEREal0N9wQRqB+8cUsuwMSfPABDBgQciCHH550NOWjBIOIiFSawsLQrG/PPbOwWd+oUTBhQuga0apV0tGIiIhIVfrPf+DBB0PfzU6dko5mI6tWwUUXQcuWcPfdSUdTfkowiIhIpbn5ZvjuO3j77Syb+XH+fLjmmjBfZq9eSUcjIiIiVakaDGzw97/DtGlhOIitt046mvJTgkFERCrFu+/C4MFwxRVw6KFJR5Pi2mth8WIYPjzL5ssUERGRSpflAxtMmRISDGedBSeckHQ0m0azSIiISMatXBma9e28c7hAZpUJE8J813/6E7Rvn3Q0UkFm1srM8s1suplNNbM+abbpamaLzWxy9LgtiVhFRCQLFA9scPnlWTmwwZo1oQ61zTYhzOpGLRhERCTj7rgDZsyAf/8bGjZMOpqYZcvCwI577AG33JJ0NJIZa4Dr3H2SmTUCPjGz19x9Wsp2b7t79wTiExGRbFE8sMFOO8E//pF0NGkNHAgffwxjxkCzZklHs+mUYBARkYyaNAn69oULLoCjj046mhS33x6aRBYUZNmgELK53P0H4Ifo96VmNh1oCaQmGEREJNcVD2wwfnxWDmzw9deh98ZJJ8Ef/pB0NJtHCQYREcmY1avhwguheXPo3z/paFJMmgT33AMXXwxduiQdjVQCM2sN+XDzfQAAIABJREFU7At8mGb1wWb2GfA9cL27T02z/6XApQAtWrSgoKAgo/EVFhZm/Ji5SmWZOSrLzFFZZk5llGWDb76h01138dNvfsOX9eqFmw1ZxB2uvbYDtWo14uyzP+LNN3/JyHGr+nOpBIOIiGTMv/4Fn30Gzz4b+g5mjTVrwmjRzZuHIKXGMbOGwDjgandfkrJ6ErCLuxea2fHAc8Duqcdw92HAMIDOnTt7165dMxpjQUEBmT5mrlJZZo7KMnNUlpmT8bJcswZuuAG23ZbtR49m+6ZNM3fsDBk+HCZPhmHD4NRTD8nYcav6c6lBHkVEJCOmTYO//jU06fvd75KOJsXAgaEFQ15elmU+JBPMrA4huTDK3Z9JXe/uS9y9MPp9PFDHzKphz1YREdksAwaEgQ0GD4YsTC78979w/fXQrVtoaFmdKcEgIiIVVlQUxkxq1Ch8h88qM2fCbbfBiSdCjx5JRyMZZmYGPARMd/d7Sthm+2g7zOwAQv1nQdVFKSIiifnqqzCwwe9+B6eemnQ0G3GHXr1CN9NhwyBcraovdZEQEZEKy8sLsz49/jhst13S0cS4h1kjatWCIUOq/1Vb0jkUOAf43MwmR8tuBnYGcPehQA+gl5mtAVYAp7u7JxGsiIhUobVrQxfJrbbK2nrA2LHw4ovQrx/stlvS0VScEgwiIlIh334bZnw84QQ488yko0kxahRMmBAyIK1aJR2NVAJ3fwcotcbo7oOBwVUTkYiIZI3hw+HNN+HBB2HHHZOOZiPz58OVV8L++0OfPklHkxlKMIiIyGZzDzcGateG++/PshsD8+fDNdfAQQeFtociIiKSO+bODQM7HnlkmOIqC11zDSxcCP/5D2xRQ76ZlzkGg5m1MrN8M5tuZlPNbKPcigWDzOxrM5tiZvvF1h1rZjOidTdm+gRERCQ5Dz4Ib7wBfftmYQOBa6+FxYvD3YvatZOORkRERKpKcRfJoqKsHdhg/PjQtfTmm+FXv0o6mswpzyCPa4Dr3H1v4CCgt5m1TdnmOMJ0T7sT5o++H8DMagNDovVtgTPS7CsiItXQ3LlhxOOuXUMrhqwyYQI89hj86U/Qvn3S0YiIiEhVGj0aXn4Z7roLdt016Wg2smQJXHYZtG0bEgw1SZkJBnf/wd0nRb8vBaYDLVM2+y3wqAcfAE3MbAfgAOBrd//W/z97dx5nY/3+cfx126qxlj1ZUpRQCG0qI8maUkq2siQUoVRfSkipSKQiikQpikj2mrGTsq9lzZLswxjbzNy/Py7zmzEzsp1z7jNz3s/HYx4zzjnOfbmbR/d9rs/1uS7XPQV8e+a1IiKShiXtePz559ZDMWgcO2arFiVLWnMIERERCR379kHHjrZFskMHr6NJ1Wuv2WjKL76w/pPpyUXt9HAcpxhQHliS7KlCwI4kf9555rHUHr/jHO/dBqt+IH/+/ERGRl5MaP8pOjrap+8XynQufUfn0nd0Ln3nQs/lL7/kY8qUW2jXbhM7duxkx47z/pWAKT50KEW2bmX5hx8StXixZ3Ho91JERMQDHTvC0aP26T0It0jOnWt9qxLaRKU3F5xgcBwnG/AD0Ml13SPJn07lr7j/8XjKB113GDAMoGLFim7VqlUvNLTzioyMxJfvF8p0Ln1H59J3dC5950LO5b59Nkb6jjtg8OAbyZgxiGYqLVsG48dD69aU79TJ01D0eykiIhJgkyfDt99C7962/yDIHD8OrVvD9dfDW295HY1/XFCCwXGczFhy4WvXdSek8pKdQNL2XtcBu4Es53hcRETSqI4drXdi0C0MxMZaM4i8eeH9972ORkRERAIpKsr2b956q/VgCkK9esFff8Hs2ZA1q9fR+MeFTJFwgC+A9a7rDjjHyyYDzc9Mk7gTiHJd9x9gKVDCcZzrHcfJAjQ681oREUmDEhYG3ngDSpf2OppkBg2yCobBg+Hqq72ORkRERAKpa1fYs8dWQLJk8TqaFP74A/r3h1atbHJmenUhFQz3AM2A1Y7jrDjzWDegCIDrukOBqUBtYBMQA7Q481ys4zgvADOAjMAI13XX+vRfICIiAXH4sPVOLFs2CBcGtm6FHj2gXj14/HGvoxEREZFA+vVXG0vdtStUrOh1NCmcPm2JhXz5LMmQnp03weC67nxS76WQ9DUu8Pw5npuKJSBERCQNe/ll+Pdfq2IIqoWBhFnXGTLAJ58E5axrERER8ZOYGNsieeONtgchCPXrBytXwsSJkCuX19H410VNkRARkdA0e7ZVHL7yShAuDHz9NcycaVsjChc+/+tFREQk/XjjDdiyBSIj4aqrvI4mhQ0bLO/RsCE88ojX0fhfME0uFxGRIBQdbQsDJUpAz55eR5PM/v2Jc57atfM6GhEREQmkJUtg4ECrZLz/fq+jSSE+3rZGZM1q6yChQBUMIiLyn7p3h23bbG5z0C0MdOlizSGGDw+ykRYiIiLiV6dO2af3a6+F997zOppUffopLFwIo0ZB/vxeRxMYSjCIiMg5LVxoGffnn4d77/U6mmRmzoTRoy0DUqaM19GIiIhIIL3zDqxdC1OmQI4cXkeTwvbt8Npr8NBD0KyZ19EEjrZIiIhIqk6csIWBwoWhb1+vo0nm2DErhyxZEl5/3etoREREJJBWr7YEQ5MmUKeO19Gk4Lrw3HP282efhVb/aVUwiIhIqt56yxoTzZgB2bN7HU0yvXrZaMrISLjySq+jERERkUCJi7MVkFy5rP9CEPrqK7t/GjwYihb1OprAUoJBRERSWL7ctjM+8wzUqOF1NMksXw4DBkDr1kHZ0ElERET8aOBAWLoUvv0W8uTxOpoU9uyx/tP33APt23sdTeBpi4SIiJzl9Glo2dKu2R984HU0ycTGWmIhTx54/32voxEREZFA2rTJxlI+/DA88YTX0aSqQweIibHx3hlC8NO2KhhEROQs/frBihXwww9wzTVeR5PMoEGwbBmMGwdXX+11NCIiIhIormtzszNntvEMQdjYYMIE+P57aw9x001eR+MNJRhEROT/bd8eRq9e8Pjj0KCB19Eks3Ur9OgB9epZgCIiIhI6hg+33kvDh0OhQl5Hk8KhQzZ1q1w5ePllr6PxjhIMIiICWM+kfv1uIls2+Phjr6NJxnVtakSGDPDJJ0G5aiEiIiJ+snMndO0K4eHW4DEIvfwy7NsHP/9sRRahSgkGEREBLKmwdm1OvvoK8uf3Oppkvv4aZs60dsyFC3sdjYiIiASK60K7dtYkavjwoFxkmD0bRoyA116DChW8jsZbIdh2QkREktuyBbp1gzvuOEDTpl5Hk8z+/daO+c477QZDREREQsd338GUKfD223DDDV5Hk8KxY9YaomRJ28kZ6lTBICIS4lwX2rSBjBmhS5c/cZy7vA7pbF26wOHDMGyYBSkiIiKhYd8+G8tQuTJ07Oh1NKl6/XXYtg3mzoWrrvI6Gu8pwSAiEuJGjIBffoEhQyBfvpNeh3O2mTNh9Gjo3h3KlvU6GhEREQmkTp0gKspuVoJwkWHxYhtw1b493Huv19EEB22REBEJYbt3w0svwX33WRVDUDl2zBo7lixpywMiIiISOqZMgW++sXuA0qW9jiaFkyet3+R110Hfvl5HEzxUwSAiEqISeiadPAmff24DGoJKr142mjIyEq680utoREREJEAyRkfbIkOZMtY5MQi98w6sW2dTI3Lk8Dqa4KEEg4hIiPruO5g8Gfr1gxIlvI4mmeXLYcAAaN0a7r/f62hEREQkgG4YNgz++QcmToQsWbwOJ4VVqyzB0LQp1K7tdTTBJdjWq0REJAASeiZVqmTbG4NKbKwlFvLkgfff9zoaERERCaTISK796Sdr8lypktfRpBAba1sjrr4aBg70OprgowoGEZEQ9OKLiT2TMgXblWDQIFi2DMaNs6u3iIiIhIaYGGjdmuPXXstVvXp5HU2qBg6E33+3StDcub2OJviogkFEJMT89BOMHWuDGcqU8TqaMyIioFgxa+bUowfUqwePP+51VCIiIhJIb74JmzezsWtXCAvzOpoUNm2CN96A+vWhYUOvowlOwbZuJSIifnT4sPVMKlsW/vc/r6M5IyIC6ta1VYtmzWyv5SefgON4HZmIiIgEytKl1n/puec4XK6c19GkEB9vOzivuAI+/VS3KeeiCgYRkRDyyiuwZw988UWQ9ExKmlwAu3rHxdkSgYiIiISGU6essUHBgvDee15Hk6rPP4c5c6B/f7j2Wq+jCV5KMIiIhIhff4Xhw+Gll4KkZ1Ly5EKC06ft8YgIb+ISERGRwHr3XVi9GoYOhZw5vY4mhZ07oWtXCA+3PIicmxIMIiIh4NgxK+u78Ubo2dPraM5o0SJlciFBTIw9LyIiIunb2rXQpw80bmwLDEHGdaFdO1v/GD5cWyPORwkGEZEQ8PrrsHWrbY0Imp5JI0eeO5iwMHteRERE0q+4OCsJyJkzaGc+fvcdTJliOZAbbvA6muCnBIOISDq3aJFNfmzXDu67z+tokqhaFapVS/l4WJhdycPDAx6SiIiIBNBHH8GSJfY9b16vo0lh/37o0AEqV7YR33J+SjCIiKRjJ0/awsB119n2xqDy7ruWSGjcOLGSQckFERGR0LBli83MrlcPGjXyOppUdeoEUVFWAZoxo9fRpA1KMIiIpGNvvQXr18OwYZAjh9fRJDF6NHTrBk2awJgxllQoWlTJBRERkVDguvDss5A5c9DOfPz5Z/j6a7tdKVPG62jSjkxeByAiIv6xYoUVCTRvDjVreh1NEr/8Ai1b2vaIESPspiI8HLZt8zoyERERCYQvvrDxVp99ZmWWQebIEWjbFkqXtgSDXDglGERE0qHTp+0zfJ488OGHXkeTxKpV0KABlCoFEyZAlixeRyQiIiKBtGuXzcwOD7cqhiD02muwezf88INuVS6WEgwiIunQBx/A8uXw/fdwzTVeR3PGjh1QuzZkzw5TpwblnGsRERHxI9eF9u1tJWTYsKDcGjF3LgwZAl26WHNHuThKMIiIpDMbN0LPnvDYY/YVFA4ftuTC0aMwf35QlkOKiIiIn40bB5MnQ//+cOONXkeTwvHj0Lo1FC8OvXt7HU3apASDiEg6Eh9vUyPCwuDjj72O5oxTp2xbxMaNMH06lC3rdUQiIiISaElnPnbq5HU0KcTF2fbSv/6C2bMha1avI0qblGAQEUlHPvkEFiyAL7+EAgW8jgbLeLRsCRERNjmiWjWvIxIREREvdOpkFY1BOPPRdeGFF+Dbb61B9gMPeB1R2qUxlSIi6cS2bfC//9nEiObNvY7mjO7dbcbTO+9A06ZeRyMiIiJeCPKZj6+/DkOHwquv2pdcOiUYRETSgYRx0o5jE5+ComfSkCG2DNC2rbVjFhERkdCTMPOxTJmgnPnYv7+tg7RpA337eh1N2qctEiIi6cDIkbZf8JNPoEgRr6PBGji98ALUrQuDBwdJxkNEREQCLohnPn7+OXTtCk8+CZ9+qtsVX1AFg4hIGrd7t41Suu8+WyDw3JIl0KgR3H67bWbMpFy2iIhISJozxyoaO3UKupmP48db1ULNmvDVV0HXFiLNUoJBRCQNSxgnffKkZeEzeP1/9U2brGqhYEGYMkUtmEVEREJV0pmPb73ldTRnmTEDmjSBu+8OysKKNE3LSiIiadi4cTBpErz/PpQo4XEw+/ZBrVqW9Zg+HfLl8zggERER8UzPnrbw8OuvNj87SCxcaNOzS5e2tZAgCi1dOG+CwXGcEUBdYK/ruilafjqO0xVokuT9SgF5Xdc96DjONuAoEAfEuq5b0VeBi4iEuoRx0rffDp07exxMTAzUqwc7d9qNhOfZDhEREfHM779b98Rnn4XwcK+j+X8rV0Lt2lCokK2F5MrldUTpz4VUMHwJfAx8ldqTruv2A/oBOI5TD+jsuu7BJC8Jd113/2XGKSIiyXTqBIcOWXNHT9scxMXBU0/Bb7/BhAlw110eBiMiIiKeOnUKWraEAgWgXz+vo/l/f/0FDz0E2bPDrFmQP7/XEaVP570ldV13ruM4xS7w/Z4Cxl5OQCIicn6DB9s46R494NZbPQzEdaFjR5saMXgwPPKIh8GIiIiI595/H1avtnuDnDm9jgaAXbvgwQdtTSQyEooW9Tqi9Mtn7cAcxwkDagI/JHnYBWY6jvOH4zhtfHUsEZFQNmaMfaZ/9FF44w2Pg+nXz+Y6de1qYylFREQkdK1bZw0dGzWyrZNBYP9+Sy4cPGjbIm6+2euI0jdfFtXWAxYk2x5xj+u6ux3HyQfMchxng+u6c1P7y2cSEG0A8ufPT2RkpM8Ci46O9un7hTKdS9/RufSdUDqXCxfm5o03ylChwmHatl3N/PnxPn3/izmX+WbP5pa33+bfatVYX7OmLQnI/wul30sRERHi4qBVK9uD8NFHXkcDwJEj1n9661ZLLtx+u9cRpX++TDA0Itn2CNd1d5/5vtdxnIlAZSDVBIPrusOAYQAVK1Z0q1at6rPAIiMj8eX7hTKdS9/RufSdUDmXkZHQuzdUrAizZ19N9uz3+eEYF3guIyKsBPL++8k/dSr5r7jC57GkdaHyeykiIgLAxx/D4sW2hzNvXq+j4cQJqF8fli+HH3+E++/3OqLQ4JMtEo7j5ATuByYleSyr4zjZE34GagBrfHE8EZFQ88cf8PDDcMMNMHWqLQ54Zs0a259RooRdsZVcEBERCU0REVCsmCUVunWDOnWs8bPHTp+GJ5+EOXNg1CioW9friELHhYypHAtUBfI4jrMTeBPIDOC67tAzL3sUmOm67rEkfzU/MNFxnITjfOO67nTfhS4iEho2bICaNSF3bpg50757ZtcuqzUMC4Np0zTfSUREJFRFRNgn95gYaN7cFhyGDAH7/OeZ+HjbqTF5MnzyCTRp4mk4IedCpkicNwXluu6X2DjLpI9tAW671MBERAT+/tsaE2XMaCOVChXyMJgjR2x4dFQUzJ0LRYp4GIyIiIh4JmlyAexTfVwcbNoEhQt7Fpbr2hjv0aOhTx9o396zUEKWz6ZIiIiIb+3da8mF6GirXLjxRg+DOXUKHnvMukP/8AOUK+dhMCIiIuKZ5MmFBKdO2eMREd7EBfTsaVOzu3SxHRsSeEowiIgEoago2xaxYwf8/DPcequHwbgutG4Ns2fD559b1kNERERCU4sWKZMLCWJi7HkPDBxozbBbtoT+/T3fqRGylGAQEQkyMTE2OnrNGpg4Ee6+2+OA3njDag3fegueftrjYETO5jhOYcdxIhzHWe84zlrHcV5M5TWO4zgfOY6zyXGcVY7jVPAiVhGRdGHkSLjqqtSfCwuz5wPsyy+hc2crthw2TMkFLynBICISRE6fhoYNYf58GDMGHnrI44CGDYO337YKhu7dPQ5GJFWxwEuu65YC7gSedxznlmSvqQWUOPPVBhgS2BBFRNKRsmUhf/6Uj4eFwZQpEB4e0HAmTrSmjg8+aMMsMmYM6OElGSUYRESCRHy8FQhMnQpDh8ITT3gc0JQp0K6dNXYMgq7QIqlxXfcf13WXnfn5KLAeSN4OtT7wlWsWA7kcxykY4FBFRNK+Q4egRg3Ys8f2IYSF2eMeJRd++QUaNYLKlWHCBE3ODgbnnSIhIiL+57rQoQOMHQvvvgtt2ngc0NKlNkC6fHn47jvIpMuFBD/HcYoB5YElyZ4qBOxI8uedZx77J9nfb4NVOJA/f34iIyN9Gl90dLTP3zNU6Vz6js6l76T3c5nx2DFue/llsm3ezJo+fTh4++3k6tOHm997jw2vvsphxwEf/fsv5FyuW5edl14qR6FCx/nf/1bw+++xPjl2ehPo30vdMYqIBIEePeDTT+GVV+DVVz0OZssW6wKdL5+tRmTL5nFAIufnOE424Aegk+u6R5I/ncpfcVM84LrDgGEAFStWdKtWrerTGCMjI/H1e4YqnUvf0bn0nXR9LqOjrfv0pk0wYQK31qtnj1etCp074+vZUuc7l2vWwOuv2/juefOyUbBgFR9HkH4E+vdSCQYREY8NGGCzmlu3tuoFL2VOGF8RGwvTp0OBAt4GJHIBHMfJjCUXvnZdd0IqL9kJJB3Mfh2wOxCxiYikeQndpxcvtqrGhOSCR7ZssV0aV14Js2ZBQW14CyrqwSAi4qGRI+Gll6yx49ChHrc5OH6cMt27w99/w+TJcNNNHgYjcmEcx3GAL4D1rusOOMfLJgPNz0yTuBOIcl33n3O8VkREEpw4AY8+CnPmwFdf2ZgGD/3zjzVzPHnSkgvXX+9pOJIKVTCIiHhk4kSrWqhRw6ZAetr1OC4OmjQhx7p1MH483HOPh8GIXJR7gGbAasdxVpx5rBtQBMB13aHAVKA2sAmIAbwZ0i4ikpacOmUrIDNnwogR0Lixp+EcPGj3THv3WnPH0qU9DUfOQQkGEREPJHQ9vuOOIOh67Lo2PHriRDY9/zwlPF6dELkYruvOJ/UeC0lf4wLPByYiEZF04PRpeOop68U0dCi08DYvGx1tQ63+/BOmTbOpERKctEVCRCTAliyB+vVtB8LPP0PWrB4HNGAADB4MXbqw6/HHPQ5GREREPBUXB82b2wrIwIHw3HOehnPypO3S+P13awFRrZqn4ch5KMEgIhJAa9daBj5/fpgxA66+2uOAvv0WXn7ZSiD79fM4GBEREfFUfDy0amX3B++9By++6Gk4sbG2M2P2bPjiC3jkEU/DkQugBIOISIBs3Wp7B6+4Iki6Hs+ZA08/DVWqWOOmDLokiIiIhCzXhXbtYNQo6N3bZmd7KD4e2rRJLKR4+mlPw5ELpB4MIiIBsGePdT0+fhzmzoXixT0OaN06WwYoXhwmTbJZTyIiIhKaXNeqFYYNg27d4PXXPQ/n5Zdt2tabb3peSCEXQQkGERE/O3QIHnrIkgyzZ0OZMh4HtHs31KplSYVp0+CaazwOSERERDzjulatMHiwzc7u08fjudkwZkxRRoyAjh0twSBphxIMIiJ+dOwY1K0LGzZYQ8c77/Q4oKNHoU4dOHDASimKFfM4IBEREfFUjx7Qvz88/7z1Y/I4ufDxxzBixPU0bw4ffuh5OHKRlGAQEfGTU6fgscdg8WIYPx6qV/c4oNOn4fHHYfVqGztVoYLHAYmIiIin+vSxr9at4aOPPP80//XX0KED3HPPfr74Io/aQ6VB+k8mIuIHcXHQrJlNihg+HBo08Dgg17VOSTNn2v7KmjU9DkhEREQ81b8/vPGG3bB89pnnzZ5/+skaOYaHQ48e68ikpfA0SQkGEREfc11o3x7GjbNrd8uWXkcE9OwJX35pGxmDIiARERHxzODB0LUrPPkkjBjheXIhMtImZleoYL2ns2SJ9zQeuXRKMIiI+Fi3bolNmF96yetogM8/t3FTLVqoU5KIiEioGzbMuic++iiMHo3XpQK//w4PPww33GC9p7Nn9zQcuUxKMIiI+ND778O770Lbtral0XPTplkwDz1k5Y/qlCQiIhK6Ro2y+4I6deDbbyFzZk/DWb/edm3mzm27OHPn9jQc8QElGEREfGT4cHj1VWjUyDoge/5Z/o8/rN7w1luty6THNxEiIiLiobFjbZvkAw/A999DliyehrN9Ozz4oBVQzJoFhQp5Go74iFpniIj4wPjx8NxzUKuWLQ5kzOhxQFu32upE7tw2H1P1hiIiIqFrwgRr5njvvdbk4MorPQ3n339tutaxYzBnDtx4o6fhiA8pwSAicplmzIAmTeDuu4NiQQAOHrRMx8mTEBEBBQt6HJCIiIh4ZsoUK6+84w77OSzM03AOH7adm7t3w+zZVmgp6YcSDCIil2HhQhtBecstQXHNhhMnrFPS1q121S5VyuOARERExDMzZ8Jjj0G5cjB1KmTL5mk4MTFQty6sW2cFlnfd5Wk44gdKMIiIXKJVq2wXwrXXWhVDrlweBxQfb+WPCxbAd99ZGaSIiIiEpogIqF/fVkFmzICcOT0N59Qpy3UsWmS3KQ8+6Gk44idKMIiIXIJNm6BGDcia1RoT5c/vdUTAyy/bHo0PPoAnnvA6GhEREfHK/PlWKnDDDVbFcPXVnoYTF2drINOn2/Tsxx/3NBzxIyUYREQu0q5dlnWPjYVff4VixbyOCPjwQ/vq2BE6d/Y6GhEREfHKb79B7dpw3XW2XTJvXk/DcV1o1w7GjYP+/aFVK0/DET/TmEoRkYtw4IBVLuzfD9OmWdWhZyIiLLvx5pvw0kvWDGLAgCCYjykiIiKeWLbMOijmzWurIAUKeB0Rr71mo7y7dbPbFUnfVMEgInKBoqNtQWDzZksuVKrkYTAREVb6GBMDvXtD6dIwZkwQzMcUERERT6xebasgOXJYcqFQIa8j4r334P33oX176NPH62gkEFTBICJyAU6ehEcegT/+sMZE4eEeBpM0uZBgyxZYvNi7mERERMQ7GzZA9epw5ZWWXCha1OuI+Owzq15o3BgGD1aBZahQgkFE5DxiY+3i+MsvMGKENWT2TGrJBYDjx+3xiAhv4hIRERFvbNoE1arZJ/hffrHGjh777jvru1CnDnz5JWTQp86Qof/UIiL/wXWhTRuYMAEGDoTmzT0OqEWLlMmFBDEx9ryIiIiEhm3bLLlw+rQlF266yeuImDYNmja1adnjx0PmzF5HJIGkBIOIyDm4rk1+HDnS+ii++GIQBFS79rmfDwuzYEVERCT927HDkgvR0TYtonRpryNi3jx47DG49VaYPBmuusrriCTQlGAQETmHvn1tKEOHDpZg8NSJE1adMGSILQkkv2KHhcGUKR43hxAREZGA+OcfeOABG281YwbcdpvXEbF8ue3WLFIEpk+HnDm9jki8oASDiEgqhgyB7t2txG/gQI8bE+3eDVWrwqhR0KsXREbCzz9bUgGUXBAREQkle/dacmH37iAYa2X+/NOmY+bMCbNm2ZRMCU1KMIiIJDN2LDz/PNSrZ00dPW1MtHSp3TisWWONIHr0sIDCwy2pULSokgvEaozWAAAgAElEQVQiIiKh4sABePBB673w889w991eR8SOHRYS2E6NwoW9jUe8lcnrAEREgsnUqdbI8b77rAOyp42JxoyB1q2hYEFYtAjKlj37+fBwu8EQERGR9O/wYSsT2LjRFhfuv9/riNi5E2rUsNAiI6FkSa8jEq+pgkFE5IygaUwUFwevvALNmsFdd1kVQ/LkgoiIiISOo0ehVi1YtcoqGqtX9zoipk2DcuUsyTBlCpQv73VEEgyUYBARIbExUdGi1pgoRw6PAjl82PZm9OsH7dvDzJmQJ49HwYiIiIjnjh2DOnVswWHcuP+eKBUAsbHQrZuFUagQ/P679Z8WAW2REBEJnsZEf/4JDz8MmzfDZ59BmzYeBSIiIiJB4fhxuzdYsMCaRD3yiKfh7NoFTz1lVZ/PPguDBmkUpZztvBUMjuOMcBxnr+M4a87xfFXHcaIcx1lx5qtHkudqOo6z0XGcTY7jvObLwEVEfGHnzsTGRLNmediYaPp0qFwZDh6EX39VckFERCTUnTwJDRpARIRNknriCU/DmTHDtkQsW2ZtooYNU3JBUrqQLRJfAjXP85p5ruuWO/PVG8BxnIzAJ0At4BbgKcdxbrmcYEVEfGn/fksuHD5sn+9vusmDIFwX+ve30sfrr7fyR9UZioiIhLbTpy2hMH26fZJv2tSzUGJj4fXXrQVEgQK2JaJJE8/CkSB33gSD67pzgYOX8N6VgU2u625xXfcU8C1Q/xLeR0TE544csQvltm3w009QoYIHQZw4YSMruna17pLz51sTCBEREQldsbH2CX7yZPj4Y5so5ZHdu62f5NtvQ8uWsGQJ3HyzZ+FIGuCrHgx3OY6zEtgNvOy67lqgELAjyWt2Anec6w0cx2kDtAHInz8/kZGRPgoNoqOjffp+oUzn0nd0Ln3nYs/lyZMZeO21sqxenZO33lpDfPxBAv2fIsv+/ZR54w1ybNjA1pYt2d60qVUveEy/l76jcykiIhctLg6eeQbGj4cBA+D55z0LZdYsy3McO2Y7NJo39ywUSUN8kWBYBhR1XTfacZzawI9ACcBJ5bXuud7Edd1hwDCAihUrulWrVvVBaCYyMhJfvl8o07n0HZ1L37mYc3n6tBULrFxp+wcbN77Vv8GlZskS6NjRRk5NnMj1jzzC9YGPIlX6vfQdnUsREbko8fHWg+nrr+Gdd6BzZ0/CiIuDXr2gTx8oVQoiI+EWbXSXC3TZYypd1z3ium70mZ+nApkdx8mDVSwkbZd2HVbhICLiifh4aNXKtkQMHgyNG3sQxFdfwf33w5VXwqJFnneDFhERkSDguvDCCzBiBPToAf/7nydh/POPbYl46y14+mn47TclF+TiXHaCwXGcAo7jOGd+rnzmPQ8AS4ESjuNc7zhOFqARMPlyjycicini4+HFF2H0aLtoBrziMC4OXn7Zrtb33GPbIcqUCXAQIiIiEnRcF7p0gSFD4NVXoWdPT8L45RcoX94KLUeOtK+sWT0JRdKw826RcBxnLFAVyOM4zk7gTSAzgOu6Q4HHgXaO48QCx4FGruu6QKzjOC8AM4CMwIgzvRlERAJq3z5rvjxzpl2/u3cPcACHDtnQ6BkzoEMH+OADyJw5wEGIiIhI0HFdq1YYONBWQvr2BSe1neb+Exdn2yF69bIGjr/8AqVLBzQESUfOm2BwXfep8zz/MfDxOZ6bCky9tNBERC7f3Ln22f7AAfjsM3j22QBftzdsgIcftnEVw4d72glaREREgkzv3vDee9C2LXz4YcCTC3v2WCPHX3+1Jo6ffqqqBbk8l71FQkQkGMXH2yJAeDiEhcHixdY3KaDX7alT4Y47ICrKrtxKLoiIiEhEBBQrZqsePXtCixbwyScBTy5ERNiWiIUL4Ysv4MsvlVyQy6cEg4ikO/v3Q5060K0bNGwIf/wB5coFMADXhfffh7p14YYbrN9ClSoBDEBERESCUkSE3R9s3w6ffw4PPGAVjhkC97EsLs76UVWvDrlyWSPHli0Dnt+QdEoJBhFJVxYssGTCr79amd/YsZAjRwADOH4cmjWzJk0NG8L8+VCkSAADEBERkaCUkFyIiUl8bNEi288ZIHv3Qs2aNqjiqadsDaRs2YAdXkKAEgwiki7Ex1vRQMIEyMWLoV27AGfjd+2C++6Db76Bt9+Gb7+1/RkiIiIS2iIirLwyaXIB7M9169rzfjZnji3CzJ9vRROjR0O2bH4/rIQYJRhEJM2LispEvXpWNPDoo7Ylonz5AAexeDFUrGhNHSdNsv0ZqjUUERERgMaNrcoxNTEx1ofBT+Ljbd2jWjXInt3GULZurdsU8Y/zTpEQEQlmCxfCs89WJCoKPv4Y2rf34II5apR1kCxcGGbP1mwnERERMbGxNiVi7167QXHdlK8JC4ORI/1y+KSjup96yiZqZc/ul0OJAKpgEJE0ynWhf3/bEpEpk8vChfD88wFOLsTGQpcu8MwzcO+91iVJyQUREREB2LzZblRefx2eeAJ+/DHl1smwMJgyxcZe+di8ebYlYs4cSyx8/bWSC+J/SjCISJpz8CDUrw9du8LDD8OwYX9w++0BDuLQIahd22ZWd+wI06fDNdcEOAgREREJOq5rcx/LlYO1a+2T/dixdtMyZUpiksFPyYWko7qzZvVoVLeELCUYRCRNWbzY+itMnw4ffQTffw/ZssUGNoj166FyZYiMtBuIQYMgk3aciYiIhLy9e60hVOvWUKkSrF5t/RcShIdbUqFoUb8kF5KO6n78cfj99wCP6paQpztiEUkTXNeKBV59Fa67zsZRVqrkQSA//2ybGK+6yjo+33OPB0GIiIhIsMm9aBE8+SRERcGAAfDii5AhlfXc8HDYts3nx1+wwA6/b5+N6m7bVlULEniqYBCRoHfokC0GvPSSTXJatsyD5ILrWpOmevWgRAlbElByQURERI4dg+eeo2y3blCggN0jdO6cenLBD4JiVLfIGapgEJGg9ttv1hdp924YONDaHQT8gnn8uJU6fvMNNGpk2yKSN2kSERGR0LNkiY1p2LyZv598kiKjRsEVVwTs8AcOQPPmMHUqNGwIn38OOXIE7PAiKaiCQUSCkutaa4MqVezP8+dbpWHAkws7d9qEiLFjrWPSN98ouSAiIhLqTp+Gnj2tmvHUKYiIYEvbtgFNLixcaP0VZs+2Ud3ffafkgnhPCQYRCTqHD8Njj0GnTlCrlm2JqFzZg0AWLoSKFeHPP2HyZHjtNdUbioiIhLo//7TEQq9e1sBx1SrbnxAg8fGJo7qzZMGbUd0i56AEg4gEld9/hwoV4Kef4IMPbGS0J9MfR4ywJkzZs9tmxrp1PQhCREREgobrwpAhVjaweTOMGwdffQU5cwYshKSjuuvXt0WYgI/qFvkPSjCISFBwXRg8GO6+G2JjYe5c6NLFg2x8bKyVTrRqBffdZ3srb7klwEGIiIhIUNmzxxYb2re3rZOrV1vTgwBKGNU9Y4aN6h4/PqC5DZELogSDiHguKsoaOXbsCA89BCtWwF13eRDIwYO2J2PQIEsyTJvmUfmEiIiIBI0ff4SyZeHXX201ZPp0uPbagB3edW3q5b33QsaMNo6yQwdtiZDgpCkSIuKpZctsAWD7dhux9NJLAZvqdLZ16+Dhh2HHDhg5Ep55xoMgREREJGgcPWodpkeOtP2bY8ZAqVIBDeHQIbslmTzZRnaPGAG5cgU0BJGLogoGEfGE68Knn1qlwqlTtiWia1ePkgs//QR33mlzrCMjlVwQEREJdfPnw223wahR0L07LFoU8OTCb7/Zlohp02xU9w8/KLkgwU8JBhEJuCNHoFEj63hcvbptibj7bg8CcV0bPVm/PpQsCUuXerQ3Q0RERILCqVPQrZuNaHAcWwHp08fGNQSI61pCwfNR3SKXQAkGEQmoFSus2/EPP8C771rxQO7cATp4RAQUK2bfY2JstFS3bvDUUzBvHlx3XYACERERkaCzbp1VNPbtCy1a2E3LPfcENIRDh6BBA+jcGWrXhuXLPRrVLXKJlGAQkYBwXfjsM7tuHz9uOxFefTWAWyIiIqz78/btUKeOjZj67jt47z3bU3nVVQEKRER8zXGcEY7j7HUcZ805nq/qOE6U4zgrznz1CHSMIhLE4uNtLMPtt1svpokT4fPPbVR1AC1daq0epkyxUd0TJ8LVVwc0BJHLpgSDiPjd0aNWLNC2LYSHWzY+oewvIBKSCzEx9ufjx+Gvv+Dtt+GVV1RzKJL2fQnUPM9r5rmuW+7MV+8AxCQiacGuXVCzpu1BeOABWLMGHnkkoCEkjOq+5x7Ldcyb59GobhEfUIJBRPxq5UpbEBg3Dt55B37+GfLmDWAAyZMLSfXpY8+LSJrmuu5c4KDXcYhIGjNunI2fXLAAhg61fZv58wc0hMOH4fHHbVR3zZq2CHPnnQENQcSnNKZSRPzCda26sGNHK++LiID77vMgkBYtUk8ugD3eogVs2xbQkETEE3c5jrMS2A287Lru2uQvcBynDdAGIH/+/ERGRvo0gOjoaJ+/Z6jSufSdUDyXmaKjufGjjygwaxZHbr6Z9d26cbxwYZgz57Le92LP5caN2ejduzR79lxJ27ZbeOKJHaxadVkhpBuh+HvpL4E+l0owiIjPRUfbdoivv4YaNWD0aMiXz4NA9u2zlYnt21N/PizMZluLSHq3DCjqum604zi1gR+BEslf5LruMGAYQMWKFd2qVav6NIjIyEh8/Z6hSufSd0LuXEZGQvv2sHs39OxJju7duSOTbz4SXei5TBjV3aWLFUzMmwd3330DcINP4kgPQu730o8CfS61RUJEfGr1aqhYEcaOtR0I06Z5kFw4fRoGDbLRk9Onw2OPpWziGBZmXZTCwwMcnIgEmuu6R1zXjT7z81Qgs+M4eTwOS0QC6eRJePllqFYNrrwSFi6EN98EHyUXLtSuXfDEE/DCCzaqe/lyj0Z1i/iJEgwi4hOuC198YaOUoqLgl1+ge/cATolIMGuWTYjo1MmCWbUKvv/emj+EhdlrlFwQCSmO4xRwHGuX5jhOZez+54C3UYlIwKxeDZUq2WiG557zZPbjgQPQtSvceCNMmmRDrAI6qlskQJRgEJHLduwYPP00tG5tHZBXrICAV7Vt3gz169uejJMn7eo9fTqUKmXPh4dbUqFoUSUXRNIZx3HGAouAmxzH2ek4TivHcdo6jtP2zEseB9ac6cHwEdDIdV3Xq3hFJEDi4y2pULEi7N1r1/8hQyBr1oCFcPQovPUWFC9uoTzxBGzcaEOsAr4IIxIA6sEgIpdl7Vpo2BA2bIBevaxqIWPGAAYQHc31w4dblUKWLPDuu1a9cMUVKV8bHq6GjiLpkOu6T53n+Y+BjwMUjogEg7//ttWPyEgbOzlsWEDHWJ04YYMp3nnHWkI9+qglGkqXDlgIIp5Q3kxELtmXX1rF4cGDMHs29OgRwORCfLx1jyxZkqLffAONGtmSwKuvpp5cEBERkfTPda3L9K23wu+/w4gRMGFCwJILsbG2ZbRkSejcGW67DZYssRCUXJBQoASDiFy0Y8dsumOLFjarecUK65kUMEuX2l6M5s2hcGGWffIJjBoF114bwCBEREQkqBw8aAsOTZtCmTKwcqXdrFgLFr+Kj4fIyLyUKWNbRgsWtH5Us2YFvN2DiKeUYBCRi7JunV0oR42y5suzZkGBAgE6+J49dqNQubJtdfjyS1i0iCO33BKgAERERCQozZplo6knTIC334Y5c6zxgZ+5rrV8qlQJevUqTaZMMHEiLF4c4MUXkSChBIOIXLDRo+0Cun8/zJwJPXsGaEvEqVPQr5/VG379tXVG2rjR9laqQ5KIiEjoOn4cXnzRmjznzGn7Ebp1C8gNysKF1tS6Vi0rnvjf/9azcqW1fAhA0YRIUNKduYicV0wMtGplOxIqV7YtEdWrB+DArmsdn8uUsaTC/fdbV8n33oMcOQIQgIiIiASt5cttQsRHH0GHDvDHH1Chgt8Pu3Il1K1ruzX//BM++cTWPWrU+Dewja5FgpASDCLynzZsgDvugJEj4Y03rAKxYMEAHbh2bahXz6oUpk2zgdElSgTg4CIiIhK04uKgb1+7QTl8GGbMsCTDVVf59bB//QWNG0O5crBggYWwaRO0b2+DrERECQYROYf4eEsqVKwI//5r+wt794ZM/h5uGxUFL71k+ygXLoQBA2D1aqhZ088HFhERkaAUEQHFitn3rVutorFbN9uLsHq1bY/wo1274LnnoFQpmDTJDr1lC7z2GmTN6tdDi6Q5/v6oICJpjOvC5MlWrbB6Ndx3H4wdG4ABDXFx1rSxWzcbGN2qlTVpypfPzwcWERGRoBURYfsRYmJssSFDBisXGD0amjTxa7OD/fvh3XdtC0RcHLRrB927B7C5tUgapASDiACWWJg5E15/3cZGlyxpiYUnnghAH8UFC6BjR1i2zDY0TpsWkD2UIiIiEsSSJhfAmj5nyAAjRsBTT/ntsEePwocfQv/+Npq7WTObnHX99X47pEi6oS0SIsLcuVZtWLOmZetHjrReio0a+Tm5sHOnrT5UqWL7ML75BubNU3JBREQk1EVEQJ06icmFBPHx0Lq1Pe9jJ05YYqF4cUsoVK9u1ZxffqnkgsiFUoJBJIQtWWLbFu+/HzZvhiFDrAvyM8/4udfCiRPQpw/cdBP88IOVTWzcaKsRmuskIiIS2vbssf4Kx4+n/nxMDLRo4bPDxcbCF19Y9WaXLtbE8bffYMIEuOUWnx1GJCScN8HgOM4Ix3H2Oo6z5hzPN3EcZ9WZr4WO49yW5LltjuOsdhxnheM4v/sycBG5dCtWwMMPw5132s8DBlgX5LZt/dwF2XXtal2qlDV5qFUL1q+Ht95SlyQREZFQt2WLjWQoVsz2KZxr5mNYmJVbXqb4eBg3DkqXtqKIggXhl19sYlalSpf99iIh6UIqGL4E/qt9+1bgftd1bwXeAoYlez7cdd1yrutWvLQQRcRX1q+3ngrly9tOhHfesWt5585+n+xkNYbVq8Njj0G2bHYF//571RyKiIiEutWrbctkyZJWStC8uVU2zpplyYSkwsJgyhQID7/kw7mutXuqWBGefBIyZ4Yff4TFi6Fatcv8t4iEuPMmGFzXnQsc/I/nF7que+jMHxcD1/koNhHxkc2b7VpdpoxdUN94w6Y8/e9/9lnfrw4ehBdesHrD5cvh44/tu67gIiIioW3BAmvieOutNsKqc2e7QRk2DEqUsCTClCmJSQYfJBcWLLCtobVrw+HD8NVXsHIl1K+vXZoivuDrXdatgGlJ/uwCMx3HcYHPXNdNXt3w/xzHaQO0AcifPz+RkZE+Cyo6Otqn7xfKdC59JxDncu/eKxg9uijTphUgUyaXhg138dRTO8iZ8zQrVvj10DhxcRT86SeuHzmSTNHR7H74YbY+8wyxOXPC/Pk+PZZ+L31H59J3dC5FRFKRUD7Qt6/dD+TJY1sln38err465esTkgwtWti2iEtMLqxYYSMmp061MZOffGLbIvy6NVQkBPksweA4TjiWYKiS5OF7XNfd7ThOPmCW4zgbzlREpHAm+TAMoGLFim7VqlV9FRqRkZH48v1Cmc6l7/jzXO7ZY9ftoUPtz+3bW7VCwYJFgCJ+OeZZIiNtFWL1arsRGDSIQmXLUshvh9Pvpa/oXPqOzqWISBKxsTB+PLz7LqxaBUWKwEcfQcuW5+/DFB4O27Zd0mH/+gt69IBvv4VcuezwL7yg1k8i/uKTBIPjOLcCnwO1XNc9kPC467q7z3zf6zjORKAykGqCQUQu34ED8P77MHiwjYpu2dIGNBQJQE4BsIt/167WW6FoUfveoIFqDkVERELViRM257FfP2v8VKqU/blxY2t+4Cc7d0Lv3jBiBFxxBXTrZrcouXL57ZAigg8SDI7jFAEmAM1c1/0zyeNZgQyu6x4983MNoPflHk9EUoqKsrnNAwZAdLT1SXrzTbjxxgAFcOwYvPee3TxkyGClji+9FIDOkSIiIhKUjhyx+dcffgj//guVK8MHH9gYqwwX0mf+0uzfb1UKH39sUyLat7fkQoECfjukiCRx3gSD4zhjgapAHsdxdgJvApkBXNcdCvQAcgOfOrZKGXtmYkR+YOKZxzIB37iuO90P/waRkHXsmFUrvP8+HDoEjz8OvXoFcGaz68J339mSwM6d8NRTlmgoXDhAAYiIiEhQ2bsXBg2yJgdRUfDgg7ZPs2pVv1Y0Hj1qCy0ffGD3R82aQc+eNvFSRALnvAkG13WfOs/zrYHWqTy+Bbjt0kMTkXM5ccL6K/Tta9fxOnWsDLBChQAGsXw5dOxoDZrKl4exY6FKlfP/PREREUl/tm2D/v1tzOTJkzaW+rXX4Pbb/XrYEyesUOKdd6x6oUEDK6QM2GKLiJzFf/VJIuJzp07BZ5/Z1ofOnaFsWVi40JorByy5sG8ftGljNwwbNtgoqaVLlVwQEREJRWvWWLnAjTfaPUGTJnZ/MH68X5MLsbHw+ec2zbJLF1vr+O03+OEHJRdEvKQEg0gaEBsLo0bBzTdD27bWP/HXX2H2bLjrLj8eOCLCagsjIuD0adtHWaKEjYnq1MlaMz/7LGTM6McgREREJOgsWmT9FMqWhYkT4cUXYetW+9RfsqTfDhsfD+PGQenSdgtSqJDdE82cCZUq+e2wInKBfDamUkR8Lz7eFgDefBM2brSFgE8/hYceCsBghogIqFsXYmKgVi3Ilw927LCDf/ihdYEWERGR0OG6MGOG7dGcOxdy57bmT88/bz/7+dDTp0P37rZLs3Rp+PFHy3FoWJVI8FAFg0gQcl2YPNnK/Ro1gkyZYMIE24lQs2aAkwtgeyl37oS334Zp05RcEBERCSVxcdbUuUIFW3TYsgUGDoTt26FHD78mF1wX5syB+++H2rXh8GEYPRpWroT69ZVcEAk2SjCIBBHXtRK/O+6wi+bx4/DNN3YRffTRAF1EZ82ym4eE5ELS4N5+GyIjAxCEiIiIeO7kSeurcNNNtuJx4oRtk9y82bZEZM3qt0Nv3mwNrG+6yQZQ/PWXVXFu2ABNm2p3pkiw0hYJkSAxdy68/jrMmwdFilgT5ubNrXohILZssX2T771nezNSExMDLVpYp2gRERFJn44etXFVH34I//wDFSta98RHHoEM/lufPHDA+iuMHm0tHhzHkguvvWb5jbAwvx1aRHxECQYRjy1ZAm+8YYUDBQva2OhWreCKKwJw8NOnbS/GZ59ZABkyWPnEsmW2apFcWJitXIiIiEj6s28fDBpkNyOHD0P16vZpv1o1v5VRnjhh07DGjIGpU+3WpHRpePddaNwYChf2y2FFxE+UYBDxyKZNWRkwAH76CfLkgQ8+gHbt4KqrAnDwzZutWmHkSPj3X7t69+oFLVvCddel7MEAllyYMgXCwwMQoIiIiATM9u3Qv7+VT544AQ0awKuv+m0sQ3w8zJ9vuYvx4yEqyhZZOna07Q+33abeCiJplRIMIgG2fj307AnjxlUiVy5ra9ChA2TP7ucDnzoFkybZXsrZs61aoW5daNPGOkcm3cwYHm7JhIQkg5ILIiIi6c/atdzct6/NeQRo1gxeecXmYvvB+vWWVPj6a/j7b2vh8NhjllSoVk19FUTSAyUYRAJkyxYrEhgzxj6vN2u2jY8+KkauXH4+8F9/JVYr7NtnDR5697ZqhUKFzv33EpIMLVrY31VyQUREJH1YvNj2IEyaRN4rr4QXXoAuXfyyH2HPHvj2W0ssLFtm6xs1atiky/r1/donUkQ8oASDiJ/t3Al9+ljVYaZMdv1+5RVYu3YbuXIV889BT5604dDDhtmqRMaMUK+eVSvUqHHhSwTh4WroKCIiktZERKRcIHBd67fUt69NhLr6anjzTRaVL0+V+vV9evhjx+w2ZMwYm44VHw+33249Ixs1ggIFfHo4EQkiSjCI+Mm//9o1fOhQu7A+9xx07257DP3mzz9h+HD48kvYvx+KFrXsRosWcO21fjywiIiIBIWkfZTq1rXtkYcOWcXCsmVWvThgADz7LGTLRqyPxk/HxcEvv1hSYcIESzIUKWITIJo2hVKlfHIYEQlySjCI+NiBA9CvHwwebIUEzzxjUyKKFvXTAU+etCv5sGG2IpExo9UctmkDDz7o13FSIiIiEkSSN2mOibHKRdeFkiWtnLJJE5+NqnJdWLnStj+MHWsTLXPmtOkPTZtClSq6DREJNUowiPiA68Ly5bbH8LPPbHx048bw5ptQooSfDrphg1UrjBplWY3rr4d33rGMhl/LJERERCTopDYBCuwmJUsWGz1ZvbpPDrVjhzVqHDMG1q6FzJmhdm3rEVmnDlx5pU8OIyJpkBIMIpcoIWs/bpx9bd5sPRbq17cpEWXK+OGgJ07ADz9YtcLcuXbARx6xaoUHHtAygYiISKiJj4clS+x+IHlyIcGpU9C69WX1VYqKsluQ0aNhzhy7D7r7bhgyBBo2hNy5L/mtRSQdUYJB5CK4LqxenZhU+Osv25HwwAPwv//Ztd0vF9j16xOrFQ4ehOLFbS/lM89A/vx+OKCIiIgErRMnrOHBpEnw0082qiFDBvuKj0/5+rAwa/h4kU6fhunTrVJh8mQ7bIkStpDSpAnccMPl/1NEJH1RgkHkPFwX1qyB8eMtqbBxo12/q1WDrl3h0UchTx4/HPj48cRqhXnzrP7w0UetWiE8XNUKIiIioeTAAfj5Z0sqzJhhXRSzZbO9CfXrQ61asGJFym0SYWE2dvoCx027rhVEjBljWz8PHLD7nNatra9C5crgOH76N4pImqcEg8g5rF2bWKmwYYN9ng8Ph86doUEDyJvXjwcePhy++sq6Pt94I7z3nlUr5Mvnp4OKiIhI0Nm61RIKkybZYkNcnE2FatbMkgrh4Wc3bAwPt2RCQpLhIpILmzdbUmHMGNi0yfoo1K9vSYWHHrJ1DhGR81GCQSSJ9esTkwrr1uLE9+MAABYoSURBVFmGvmpVePFFSyr47fP98eNWIjFsGCxYYFfxBg2sWqFqVVUriIiIhALXhT/+SEwqrF5tj5cpY/Me69eH22//7/uChCRDixa2LeI/kgsHDsCkSdfSrRssWpR439Otm92G5Mzp23+eiKR/SjBIyNu4MTGpsGaNXVzvu8+aLTdoAAUK+PHga9ZYUmH0aDh82DY29usHTz/txxIJERERCRqnTtmY6UmTrNHBzp2WQKhSBT74wJIKF9vsIDz8nA0dT5yw/MPo0TBtGpw+XZLSpa21U+PGULjwZf+LRCSEKcEgIenPPxN7KqxaZUmFKlVg8GB47DE/T3mMiYFx4yjfv79th8iSxQ7apg3cf782NoqIiKR3UVEwdaolFaZNgyNHbDvDQw9Bnz4269GHDZ7i422HxZgxdv8TFWX3Oh07ws03L6VVq0q6/RARn1CCQULGpk2JSYUVK+yxKlVg0CD7fF+okA8OEhFx7pLEVausWmHMGIiKInPhwrYy0by5n7pEioiISNDYscMqFCZNsvuF2Fjbe9mwoVUpVK8OV13l00OuW2e3HV9/DX//DVmz2j1P06bWrDpjRoiMPKbkgoj4jBIMkq5t3pyYVFi+3B67+24YONAusNdd58ODRUQkNlWqW9fqDytXhu++s8TCkiXWiOnxx6FNG36Li6PqBXZ0FhERkTTGdW1xIaGfwrJl9vhNN0GXLpZUuOMO+5TvI6dO2WHmzLF7n2XL7O1r1IC+fe2QWbP67HAiIikowSDpztatiUmFP/6wx+68EwYMsM/2ftlbmDS5APa9Rg3b/hATA6VKwYcfWtfn3LntNZGRfghEREREPBMba3sREpIK27bZ1sc777QmB/Xrw803++xwUVHWnHH+fPtassR6LID1ghw4EBo1gvz5fXZIEZH/pASDpAvbtycmFZYutccqV4b+/S2pULSoHw+ePLmQIDbWVi8GDYIOHdRbQUREJD06ehRmzLCEws8/24jpK66ABx+E7t2hXj2ffcLftSsxmTB/vhVIxMdblUKFCtCunW3/vOceJRVExBtKMEia9fff8P33llRYssQeq1gR3n/ftjMWK+bHg8fH28bGyEh49dWUyYUEcXFWOtGxox+DERERkYD65x/46SdLKsyebXsTrrnGkgn161sVY7Zsl3WI+Hgbn500oZAwGCJrVrjrLujRA+691xZVLvNwIiI+oQSDpCk7diQmFRYvtscqVLCqw4YNoXhxPx04aUIhMtI2N+7fb8/lywcnT1oyIbmwMGv4KCIiIsHrv5o0g1Ukrl+fuPUhYWWjeHF4/nlLKtxzD2S69Fvrkydta+f8+bbLYsECK4YAq0a4917o1MkqFG677bIOJSLiN/pfkwS9XbsSkwoLF9pj5ctbs6KGDS9+NPQF+a+EQpEiNj6qalX7KlYs9W0SYWHW6FGNHEVERIJXak2aw8Nt4WDRIkso/PijjaMCK5d86y145BEoXfqSt0AePmz3NQnVCb/9ZkkGsD6QDRpYUqFKFctjaKeliKQFSjBIUNq9G374wZIK8+fbY7fdBm+/bUmFEiV8fMCLTSgkFx5uNyQJNyhKLoiIiAS/1Jo016pl1+8//oB9+yBzZpvp2KULPPzwJc+13rHDKhMSEgpr1lhhRKZM1pDxhRcS+yfkzevDf6OISAApwSBBY8+exKTCvHl20S1b1hYJGja0bL7PXG5CITUJSYb/KrEUERGR4BARYdf748fPfvzkSZg+3ZIKzz0HNWtCjhwX9dbx8bB27dn9E/7+257Lnt1GZj/xhCUUKle2dQkRkfRACQbxjOvCli0wc6YlFebMscdKl4aePS2pUKqUjw52voRC7dpnJxQutQ4xPDyxA5OIiIgEB9e1xowrViR+TZxoE5/OZfNmywJcgBMnbIpVQjJh4ULbAgFQsKBtdXj5ZUsolC2r/gkikn7pf28SEKdO2ef75cvtmr58OaxcCUeO2POlSsGbb1pS4ZZbfHBA1z07oRAZmZhQKFzYdwkFERERCS6xsbBx49nJhBUrEu8DwBo43XWXdYw+fTrle5ynSfPBg2f3T1i61O51wO5jEqoTqlTRbYaIhBYlGMTnoqIseZA0mbBuXeL1OyzM+ik0bQrlylmZYOnSl3lQJRRERERCz5EjsGrV2YmENWsSuyVecQWUKWNTHsqVs69bb03c8nABTZpdF7ZvP3u7w9q19tLMma3n44svJvZPyJ07gP9+EZEgowSDXDLXtQkPCUmEhO9btya+Jl8+m/hQq5Zd08uXt0WDjBl9cHAlFEREREKD68LOnf+fRCg9a5bdhGzZkviaPHnsZqNDh8Rkwk03/fd+hFSaNMdNmsKaa8KZ93FiQmHXLnt5jhyWRGjc2BIKlSrBVVf5958uIpKWKMEgFyQ2Fv78E2bPzsfPP6debViihGXxn3028bpesOB53vh8c6cTJE8ozJljnZ1BCQUREZH05NQp2LAh5RaHQ4fseccha6FCtsWhVauzbzou8vp/6hRsvy6cQz2mUPLdFvQtMZKhj4X//xbOQoWsf0LCuMjSpX2wSCIiko4pwSApxMQkVhsmVCasXp3QZPkWsmSxBkUJ1Ybly1u1YfbsF3mgc82dhvMnFGrVUkJBREQkrTt82PZVJk0krF2buK/yqqvspqNhw8REQtmy/Pb771StWvW8b++6dvuwZUvqXzt32msgHNhGmeOJ1QlVqlgfaN1iiIhcOCUYQty+fWdvb1ixwioV4uPt+Vy5LIHQtq1d02Njl9KsWSUyZ77MA6c2d7p2bWjTBnbvVkJBREQkmF1oBWKChEYGyasStm9PfE3+/Haz8dBDicmEEiXOWzJw/LgNcEotgbB1Kxw7dvbrCxaE4sXtluL66+3n4sWtOuGaay76TIiISBJKMISI+Hi7yCZPJuzenfiaIkXsWv7kk4mVCckz95GRxy4vuRAXB99/D08/ndiAKcGJE/DRR5A3rxIKIiIiweq/KhDBru/r1p2dSFi50rpAA2TIACVL2haHdu3spuO226BAgVQPFx8Pe/acnThYuPBmXn/d7m2S3suA9WhMSBpUr352EqFYMXteRET8QwmGdCjhup6QREj4OnrUns+Y0cZCVqtmSYSERQKfZO1PnoS//7YVidS+du7875nTYFf+UaN8EIyIiIj4VGoViLVqQcuWEB1tNxzr1yde67NmtX2UjRsn3nCUKZPiU350NGxdnXoFwtattgaRwHEgb95clCplxQ7Fi5+dRMiXT+sSIiJeUYIhjUvYupg0mbB27dnX9dtug2bNEpMJpUtfesfjjDExNv7pXAmEf/45+y84jnVIKlrU5lEWLWp3CZ9+mrKCAc47d1pEREQC4Ngx+Pdf2LvXvv79FxYvhq++SrlQcPIkDBli8xnvuAPq1UusSjgzOiouLnHow5ZvUyYR9u49+y2zZ7e/WqoU1KlzdgKhaFFYtGjxBfVgEBGRwFKCIUjFxlolYVSUJREOH078eceOxGRC0pGQ+fMnjoRMSCZc1EhI17WxEOdKHmzfzr0JHZwTZMliPRKKFoWaNe17sWL2vWhRuO46Ut1TUa/eeedOi4iIiI/Ex8OBA4nJgqSJg9QeS3p9vtBDhGVjZZ+fE5MHvyYmELZtS+zbCHZvUqSIJQzq109ZhXDNNapCEBFJi86bYHAcZwRQF9jrum6ZVJ53gEFAbSAGeMZ13WVnnqt55rmMwOeu6/5fe/cfG/dd33H8+W7c1Dm7rU3Tc2kSaBtlQP+Apgtr2NhWb6LQEqnixx9kE50qJFSxIthfVJPY/ug/RZPQNLENdYhG3SRnaEWjizoQEnWLBNkKohRKxAgtS7K0SdOUtLbjpnY+++N7xr/O9tn38fd8d8+H9JW/971vzh+/eqfPu+/7/ngg49gb8/jj7N2/H0ZGSvsf15SKL+nrNQcaXV94QaIZt/I4D3E3v9j2EO/+veHf3BJy9+4lT12cNT1dHGEwt2nwq1/Nrh87trig6O+fbRa85z38cnqancPDs9uuuaY4l3K16tx32uaCJGktmqlVSrMe9cjk5PJNgrk/X3pp9grOc23aVJxTUK3C0BAXd+7ijcEqrw8MMXF5lfG+IV7bUuVc7xA9P3+WW77wYXouLG4+TFDhg8cfYvTm2W1velPRLNi9Gz7ykfkNhB076n//IElqb40cwXAA+BLw8BLP3w7sqi23AP8I3BIRm4C/B94HnACeiohHU0o/a3bQDaudJ9i71EWIlnDxYnEu4FqaAzM/L1xY/nf09MCVVxZ3aRgYKNavuWbxtrnr23/xODd8Zh+XnJ/gn1/ZB/fUuajS8eNLH4Fw/PjiwxqvuqpoFNx4Y3How0zjYGYZHJz3FcLx0VF25jokcabJsJqrUEuStNgB1lCrlDIyaLweSQleeWXZowvSqVOkU8W2S8Zeq/vr3ujtZ6K/ylhliFe33MC56l7Obhvi5U1VTscQp1KVk9ND/N8bVV6YHGT8/CWMH4PxIyvVL9u5lUMcYh99zDYZJi+p8OC+Q9zx3mHuvWH2aISBgeZikyS1nxUbDCmlJyPiumV2uRN4OKWUgMMRMRARbwauA46mlJ4DiIiDtX3LaTDUuQjR1G23890PPsDzV+5mbKxoIoyPM2/9tTEYH4O0wstftrn4cr+/H66u/ezfAX1vn93e1wf9l0N/34Jt/dDbu8pD/370I7jvvtnrFkxMwG23FTdpnpwsGggvvjhzM+dCBFx7bdEo2Lu3uD3EwgZCX99qUs1veLg4ikKSpDVaa62SUnphmX+TR516ZPp97+f53R/iApdx2bnT9L56mr6xU/RPnKYnLb4Q8kWCs7GVUzHEixernGYPpxjiNNXf/Jy7fn6yAnMuilipFNP9zFKpQN8Vtes09c1/buVlmPNPH6LyZ/uI88URiL2HDvFZvySQJJHnGgzbgONzHp+obau3vZxvCxZO5jU9U68z/I2/IMsUeAE4W1taZWoKnniiOEdi5voHc5ft24trJEiS1N2WqlXWt8GwRD2yafoNdv7ga5yiyjHeyktcy9lLd/PrSpVXtwwxVqky0V/l/BVDXBioMj24lcrlmxY1Cnb2wTtXaAhs2bK2MxmXtWsYtnoEoiRpsRwNhnrfw6dlttd/kYhPAp8EGBoaYnR0dM0D2rt/f3EY4hJeHxzkyOc/v+bXL9uN99/P5oUXV5yREpMnT3L4i1+cv/3YsWLJbGxsrKn/NppllvmYZT5mmY9ZbigN1SQ5axFYvh4J4Iqrg1//y99QuTTRF7Bj0V4v15bGnD9fLGfOrHHAqxEBBw4U6y14n/v5yscs8zHLfMwyn7KzzNFgOMH8OXE7cBLYvMT2ulJKDwIPAuzZsyc1deuhkZG63xgAUKlw2SOPcFM7ddtvumnZv6d3ZKS0WzWNjo56W6hMzDIfs8zHLPMxyw1lqVplnqy1CKxYj1T+dYTbhv+wud/Rpfx85WOW+ZhlPmaZT9lZ5jho7lHgrijsBc7Vzml8CtgVEddHxGbgY7V919/MxQMrlfnb2/VOBZ3290iSVK6lapX15fwtSeoyKzYYImIE+D7wtog4ERGfiIh7IuKe2i6PAc8BR4F/Aj4FkFKaAu4FvgUcAb6WUnp2Hf6G+hZO6u0+mXfa3yNJUiZrrVVK4fwtSeoijdxFYv8Kzyfgz5d47jGKSb01apP65P799Oa873SreFtHSZIWaaZWKUWn1SOSJC0hxzUYNrbhYQ4fPNg55/B4W0dJktpPp9UjkiTVkfvGRZIkSZIkqQvZYJAkSZIkSU2zwSBJkiRJkppmg0GSJEmSJDXNBoMkSZIkSWqaDQZJkiRJktQ0GwySJEmSJKlpNhgkSZIkSVLTbDBIkiRJkqSm2WCQJEmSJElNs8EgSZIkSZKaFimlVo9hkYh4CfjfjC+5FTiT8fW6mVnmY5b5mGU+ZpnPemT51pTS1ZlfU3WsQy0Cfr5yMst8zDIfs8zHLPMptR7ZkA2G3CLiBymlPa0eRycwy3zMMh+zzMcs8zFLLeR7Ih+zzMcs8zHLfMwyn7Kz9BQJSZIkSZLUNBsMkiRJkiSpad3SYHiw1QPoIGaZj1nmY5b5mGU+ZqmFfE/kY5b5mGU+ZpmPWeZTapZdcQ0GSZIkSZK0vrrlCAZJkiRJkrSObDBIkiRJkqSmdVSDISI+EBE/j4ijEXFfnecjIv6u9vwzEXFzK8bZDhrI8k9rGT4TEd+LiHe1YpztYKUs5+z37oiYjoiPljm+dtJIlhFxa0Q8HRHPRsQTZY+xXTTwGb8yIv4jIn5cy/LuVoxzo4uIr0bE6Yj46RLPO+90IeuRfKxH8rEeycd6JB/rkTw2VD2SUuqIBdgE/BK4AdgM/Bi4ccE+dwD/CQSwF/ivVo97Iy4NZvm7wGBt/XazXHuWc/b7DvAY8NFWj3sjLg2+LweAnwFvqT2utnrcG3FpMMu/BL5QW78aOAtsbvXYN9oC/AFwM/DTJZ533umyxXqk9CytRzJlOWc/65Ems7QeyZql9UhjWW6YeqSTjmD4HeBoSum5lNIF4CBw54J97gQeToXDwEBEvLnsgbaBFbNMKX0vpfRK7eFhYHvJY2wXjbwvAT4NPAKcLnNwbaaRLP8E+HpK6RhASsk862skywRcHhEB9FNM6FPlDnPjSyk9SZHNUpx3uo/1SD7WI/lYj+RjPZKP9UgmG6ke6aQGwzbg+JzHJ2rbVruPVp/TJyg6YlpsxSwjYhvwIeDLJY6rHTXyvvwtYDAiRiPihxFxV2mjay+NZPkl4B3ASeAnwGdSShfLGV5Hcd7pPtYj+ViP5GM9ko/1SD7WI+Upbd7pWY8XbZGos23hPTgb2UeryCkihikm9Peu64jaVyNZ/i3wuZTSdNGc1RIaybIH+G3gj4EtwPcj4nBK6X/We3BtppEs3w88DfwRsBP4dkR8N6X06noPrsM473Qf65F8rEfysR7Jx3okH+uR8pQ273RSg+EEsGPO4+0Una7V7qMGc4qIdwJfAW5PKb1c0tjaTSNZ7gEO1ibzrcAdETGVUvr3cobYNhr9jJ9JKY0D4xHxJPAuwAl9vkayvBt4IBUn7h2NiOeBtwP/Xc4QO4bzTvexHsnHeiQf65F8rEfysR4pT2nzTiedIvEUsCsiro+IzcDHgEcX7PMocFftKpp7gXMppRfKHmgbWDHLiHgL8HXg43Zjl7Viliml61NK16WUrgP+DfiUk3ldjXzGvwH8fkT0REQFuAU4UvI420EjWR6j+OaFiBgC3gY8V+ooO4PzTvexHsnHeiQf65F8rEfysR4pT2nzTsccwZBSmoqIe4FvUVyR9KsppWcj4p7a81+muCLuHcBRYIKiI6YFGszyr4CrgH+odbqnUkp7WjXmjarBLNWARrJMKR2JiG8CzwAXga+klOrerqebNfi+vB84EBE/oTis7nMppTMtG/QGFREjwK3A1og4Afw1cCk473Qr65F8rEfysR7Jx3okH+uRfDZSPRLF0SaSJEmSJElr10mnSEiSJEmSpBaxwSBJkiRJkppmg0GSJEmSJDXNBoMkSZIkSWqaDQZJkiRJktQ0GwySJEmSJKlpNhgkSZIkSVLT/h93Yh5S8CITDQAAAABJRU5ErkJggg==\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "figure(1, figsize=(18, 7))\n", "\n", "subplot(1,2,1)\n", "plot(tt,yy,'b-',label='Exacte')\n", "plot(tt,uu_ep,'r-D',label='Approchée')\n", "erreur=[abs(uu_ep[i]-yy[i]) for i in range(N)]\n", "# title(f'Euler explicite - max(|erreur|)={max(erreur):g}') # synatxe pour python >= 3.6\n", "title('Euler explicite - max(|erreur|)='+str(max(erreur))) # synatxe \"OLD\"\n", "grid()\n", "legend();\n", "\n", "subplot(1,2,2)\n", "plot(tt,yy,'b-',label='Exacte')\n", "plot(tt,uu_er,'r-D',label='Approchée')\n", "erreur=[abs(uu_er[i]-yy[i]) for i in range(N)]\n", "# title(f'Euler implicite - max(|erreur|)={max(erreur):g}') # synatxe pour python >= 3.6\n", "title(f'Euler implicite - max(|erreur|)='+str(max(erreur))) # synatxe \"OLD\"\n", "grid()\n", "legend();" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Convergence des schémas d'Euler\n", "\n", "Une méthode numérique est *convergente* si\n", "$$\n", "|y_n-u_n|\\le C(h)\\xrightarrow[h\\to0]{}0 \\qquad\\forall n=0,\\dots,N\n", "$$ \n", "Si $C(h) = \\mathcal{O}(h^p)$ pour $p > 0$, on dit que la convergence de la méthode est d'ordre $p$. \n", "\n", "Remarque: $N\\to+\\infty$ lorsque $h\\to0$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ ">Soit $u_{n+1}^*$ la solution numérique au temps $t_{n+1}$ qu'on obtiendrait en insérant de la solution exacte dans le schéma (par exemple, pour la méthode d'Euler explicite on a $u_{n+1}^*\\equiv y_{n} + h\\varphi(t_{n}, y_{n})$).\n", "Pour vérifier qu'une méthode converge, on écrit l'erreur ainsi\n", "\t\\begin{equation}\\label{quart7.9}\n", "\te_n \\equiv y_n - u_n = (y_n - u_n^*) + (u_n^* - u_n).\n", "\t\\end{equation}\n", "Si les deux termes $(y_n - u_n^*)$ et $(u_n^* - u_n)$ tendent vers zéro quand $h \\to 0$ alors la méthode converge.\n", "\n", "\"erreurEuler\"" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- La quantité\n", "$$\n", "\\tau_{n+1}(h)\\equiv\\frac{y_{n+1}-u_{n+1}^*}{h}\n", "$$\n", "est appelée **erreur de troncature locale**. \n", "Elle représente (à un facteur $1/h$ près) l'erreur qu'on obtient en insérant la solution exacte dans le schéma numérique. \n", "\n", "- L'**erreur de troncature globale** (ou plus simplement l'erreur de troncature) est définie par\n", "$$\n", "\\tau(h)=\\max_{n=0,\\dots,N}|\\tau_n(h)|.\n", "$$\n", "\n", "Si $\\lim_{h\\to0} \\tau (h) = 0$ on dit que **la méthode est consistante**. \n", "On dit qu'elle est consistante d'ordre $p$ si $\\tau (h) = \\mathcal{O}(h^p)$ pour un certain $p\\ge1$.\n", "\n", "Remarque: la propriété de consistance est nécessaire pour avoir la convergence. \n", "En effet, si elle n'était pas consistante, la méthode engendrerait à chaque itération une erreur qui ne tendrait pas vers zéro avec $h$. \n", "L'accumulation de ces erreurs empêcherait l'erreur globale de tendre vers zéro quand $h \\to 0$. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "Proposition. \n", "Si la fonction $\\varphi$ est lipschitzienne par rapport à sa deuxième variable, la méthode d'Euler explicite appliquée au problème de Cauchy $y'(t)=\\varphi(t,y(y))$ pour $t\\in[t_0;T]$ et $y(t_0)=y_0$ est convergente d'ordre 1.\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Preuve** \n", ">On étudie séparément l'erreur de consistance et l'accumulation de ces erreurs.\n", ">+ **Terme $y_n - u_n^*$.** \n", " Il représente l'erreur engendrée par une seule itération de la méthode d'Euler explicite. \n", " En supposant que la dérivée seconde de $y$ existe et est continue, on écrit le développement de Taylor de $y$ au voisinage de $t_n$:\n", "\t$$\n", "\ty(t_{n+1})=y(t_n)+hy'(t_n)+\\frac{h^2}{2}y''(\\eta_n)\n", "\t$$\n", "\toù $\\eta_n$ est un point de l'intervalle $]t_n;t_{n+1}[$. \n", "\tDonc il existe $\\eta_n \\in ]t_n, t_{n+1}[$ tel que\n", "\t$$\n", "\ty_{n+1}-u_{n+1}^*=y_{n+1}-\\Big(y_{n} + h\\varphi(t_{n}, y_{n})\\Big)=y_{n+1}-y_{n} - hy'(t_n)=\\frac{h^2}{2}y''(\\eta_n).\n", "\t$$\n", " L'erreur de troncature de la méthode d'Euler explicite est donc de la forme\n", "\t$$\n", "\t\\tau(h)=M\\frac{h}{2},\n", "\t\\qquad\n", "\tM\\equiv\\max_{t\\in [t_0,T]}|y''(t)|.\n", "\t$$\n", "\tOn en déduit que $\\lim_{h\\to0} \\tau (h) = 0$: la méthode est consistante.\n", " \n", ">+ **Terme $u_{n+1}^* - u_{n+1}$.** \n", " Il représente la propagation de $t_{n}$ à $t_{n+1}$ de l'erreur accumulée au temps précédent $t_{n}$.\n", "\tOn a\n", "\t$$\n", "\tu_{n+1}^* - u_{n+1} \n", "\t=\n", "\t\\left(y_{n} + h\\varphi(t_{n}, y_{n})\\right)\n", "\t-\n", "\t\\left( u_{n} + h\\varphi(t_{n},u_{n})\\right)\n", "\t=\n", "\te_{n}+h\\left( \\varphi(t_{n},y_{n})-\\varphi(t_{n},u_{n}) \\right).\n", "\t$$\n", "\tComme $\\varphi$ est lipschitzienne par rapport à sa deuxième variable, on a\n", "\t$$\n", "\t|u_{n+1}^* - u_{n+1}|\\le (1 + hL)|e_{n}|.\n", "\t$$\n", " \n", ">+ **Convergence.** \n", " Comme $e_0 = 0$, les relations précédentes donnent\n", "\t\\begin{align}\n", "\t|e_n| \n", "\t&\\le\n", "\t|y_n - u_n^*| + |u_n^* - u_n|\n", "\t\\\\\n", "\t&\\le\n", "\th|\\tau_n(h)| + (1+hL)|e_{n-1}|\n", "\t\\\\\n", "\t&\\le\n", "\th|\\tau_n(h)| + (1+hL)\\left( h|\\tau_{n-1}(h)| + (1+hL)|e_{n-2}| \\right)|\n", "\t\\\\\n", "\t&\\le\n", "\t\\left( 1+(1+hL)+\\dots+(1+hL)^{n-1} \\right)h\\tau(h)\n", "\t\\\\\n", "\t&=\\left(\\sum_{i=0}^{n-1}(1+hL)^i\\right)h\\tau(h)\n", "\t\\\\\n", "\t&=\\frac{(1+hL)^n-1}{hL}h\\tau(h)\n", "\t\\\\\n", "\t&\\le\\frac{(e^{hL})^n-1}{hL}h\\tau(h) \\qquad\\text{car }(1+x)\\le e^x\n", "\t\\\\\n", "\t&=\\frac{(e^{hL})^{(t_n-t_0)/h}-1}{L}\\tau(h)\\qquad\\text{car } t_n-t_0=nh\n", "\t\\\\\n", "\t&=\\frac{e^{L(t_n-t_0)}-1}{L}\\tau(h)\n", "\t\\\\\n", "\t&=\\frac{e^{L(t_n-t_0)}-1}{L}\\frac{M}{2}h=\\mathcal{O}(h)\n", "\t\\end{align}\n", "\tOn peut conclure que la méthode d'Euler explicite est convergente d'ordre 1. \n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "On remarque que l'ordre de cette méthode coïncide avec l'ordre de son erreur de troncature. \n", "On retrouve cette propriété dans de nombreuses méthodes de résolution numérique d'équations différentielles ordinaires.\n", " \n", "\n", "Remarque: l'estimation de convergence est obtenue en supposant seulement $\\varphi$ lipschitzienne. \n", "\tOn peut établir une meilleure estimation si $\\partial_y\\varphi$ existe et est non négative pour tout $t \\in [t_0;T]$ et tout $y\\in\\mathbb{R}$. \n", "\tEn effet dans ce cas \n", "\t\\begin{align*}\n", "\tu_n^* - u_n \n", "\t&=\n", "\t\\left(y_{n-1} + h\\varphi(t_{n-1}, y_{n-1})\\right)\n", "\t-\n", "\t\\left( u_{n-1} + h\\varphi(t_{n-1},u_{n-1})\\right)\n", "\t\\\\\n", "\t&=\n", "\te_{n-1}+h\\left( \\varphi(t_{n-1},y_{n-1})-\\varphi(t_{n-1},u_{n-1}) \\right)\n", "\t\\\\\n", "\t&=\n", "\te_{n-1}+h\\left( e_{n-1}\\partial_y\\varphi(t_{n-1},\\eta_n) \\right)\n", "\t\\\\\n", "\t&=\n", "\t\\left( 1+h\\partial_y\\varphi(t_{n-1},\\eta_n) \\right)e_{n-1}\n", "\t\\end{align*}\n", "\toù $\\eta_n$ appartient à l'intervalle dont les extrémités sont $y_{n-1}$ et $u_{n-1}$. \n", "\tAinsi, si\n", "\t$$\n", "\t0" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "figure(figsize=(10, 7))\n", "loglog(H,err_ep, 'r-o',label='Euler Explicite')\n", "loglog(H,err_er, 'g-+',label='Euler Implicite')\n", "xlabel('$h$')\n", "ylabel('erreur')\n", "legend(bbox_to_anchor=(1.04,1),loc='upper left')\n", "grid(True);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Pour estimer l'ordre de convergence on doit estimer la pente de la droite qui relie l'erreur au pas $k$ à l'erreur au pas $k+1$ en echelle logarithmique.\n", "Pour estimer la pente globale de cette droite (par des moindres carrés) on peut utiliser la fonction `polyfit` (du module `numpy` que nous avons déjà importé avec `matplotlib.pylab`). " ] }, { "cell_type": "code", "execution_count": 49, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Euler progressif 0.96\n", "Euler regressif 1.05\n" ] } ], "source": [ "# ln(e) = a ln(h) + b\n", "a_ep, b_ep= polyfit(log(H),log(err_ep), 1) # polyfit ( [liste des abscisses], [liste des ordonnées], degré du polynome)\n", "print (f'Euler progressif {a_ep :1.2f}') # syntaxe print pour python 3.6 ou superieur\n", "a_er, b_er= polyfit(log(H),log(err_er), 1)\n", "print ('Euler regressif %1.2f' %a_er) # syntaxe \"OLD\"" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "On peut bien sur afficher la droite obtenue par régression linéaire en même temps que les points.\n", "\n", "Soit on affiche les logarithmes des données et l'équation de la droite avec la commande ``plot``:" ] }, { "cell_type": "code", "execution_count": 50, "metadata": {}, "outputs": [ { "data": { "image/png": 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q9ozoF1qudqr4OR3AGHOjMWaNMcZljAn7i+0uN8ZsMMZsMsY86s6MIiIix5ubkMotU3+lSmAFFozrp6JCREREPMqSzfu45q2f2Z2Rx/Q7e3FH//J39KcnHFmRDFwPvHuyDYwx/sBbwKVAKrDcGPOptXateyKKiIhAkcvyjy/XM/XHLfRrUYcp4d2pGVTR6VgiIiIiAFhriVm6jWf/s5bmdavw3vAwQutWcTrWWXG8rLDWrgNO1fL0AjZZa7cc2XYWcC2gskJERNwiM7eAe2eu4PsNexnetylPXtWeAH/HD1AUERERASC/0MXTnyYzc9l2BrWrz79u6kq1SgFOxzprjpcVp6kRsP2426lA75NtbIyJBCIBgoODiYuLK9NwpSkrK6tc5fV2Gg/PovHwLL40HruzXbyRmMvuHMvw9hW5uMY+Fv/0o9OxjvGlsRAREZE/25eVx9iPEli+9QDjLmrBQ5e2wc+vfJ328UduKSuMMd8ADUp4KMpau/B0XqKE++zJNrbWTgWmAoSFhdmBAweeTkyPEBcXR3nK6+00Hp5F4+FZfGU8lmzex0uxiUAFYkZ1p1+Luk5H+hNfGQsRERH5szVph4iMTmBfVh6TbunGNV0aOh2pVLilrLDWDjrHl0gFmhx3uzGQdo6vKSIi8pdilm7j2U/X0KxuFaaNCKNpnfJ5zqeIiIh4p89W7eThj1dSMyiAuWP60alxDacjlZrychrIcqCVMaYZsAO4GbjV2UgiIuKtCopcPPeftcQs3cbFbevzxs3l+5xPERER8S4ul+X/vtnIpO820aNpLd6+rTv1q1VyOlapcrysMMYMBd4E6gGfGWOSrLWXGWMaAtOstUOstYXGmPHAV4A/8IG1do2DsUVExEsdyM7n7thEftmyn9EXNOfvl7fFv5yf8ykiIiLeIzuvkAdmJ7Fo7W7+FtaY56/rSGAFf6djlTrHywpr7QJgQQn3pwFDjrv9OfC5G6OJiIiP2bQnk7umx7PzYC7/vLELN/Ro7HQkERERkWO2p+cQER3Pxt2ZPH11e+7oF3qqK2uWW46XFSIiIp7g+/V7uHfmCgID/JkZ2YceTWs5HUlERETkmCWb9zEuNhGXhekje3F+q3pORypTKitERMSnWWt576ctvPTFetqfV533hofRsGZlp2OJiIiIAMVzlY+WbuOZ/6wtXvR7eBihdb1/0W+VFSIi4rPyCot4fH4y8xJTGdKpAa/d2IWgivpoFBEREc+QX+jimf+sYcavKT636LdmZCIi4pP2ZuYx5qMEErYd4P5Brbj34lb4aSFNERER8RD7s/IY+1Eiy7amM3ZgCx4e3ManFv1WWSEiIj4necchIqPjSc/JZ0p4d4Z0Os/pSCIiIiLHrE3LICI6nn1Zebxxc1eu7drI6Uhup7JCRER8yherd/LgnJXUDApg7ph+dGxUw+lIIiIiIsd8vnonD81ZSY3KAXw8pi+dG9d0OpIjVFaIiIhPsNYy6dtN/OubjXQLqcm7t/egfrVKTscSERERAcDlsvzft78x6dvfNFdBZYWIiPiAw/lFPDx3JZ+t2sn13Rvx4tBOVArwdzqWiIiICADZeYU8OCeJr9bsZliPxkwc2pHACr49V1FZISIiXm3nocNERMezJi2Dx4e0JeL85hjjO4tTiYiIiGfbnp5DRHQ8G3dn8uRV7RnZP1RzFVRWiIiIF0tMOcDomAQO5xcxbXgYl7QLdjqSiIiIyDG/bN7P3bEJFLksH97Ziwta13M6ksdQWSEiIl5pfmIqj85fTYPqlYgd1ZvWwdWcjiQiIiJyTMzSbTz76Rqa1gli2oieNKtbxelIHkVlhYiIeJUil+XVrzbwzg+b6dO8Nm+H96BWlYpOxxIREREBIL/QxbP/WUPsrylc3LY+/3dzV6pXCnA6lsdRWSEiIl4jM7eA+2cl8e36PYT3DuGZazoQ4O/ndCwRERERAPZn5TE2NpFlv6cz5sIWPHJZG/z9tD5FSVRWiIiIV0jZn8Oo6OVs3pvN89d24Pa+oU5HEhERETlmbVoGEdHx7MvK442bu3Jt10ZOR/JoKitERKTcO7o4lctC9Mhe9G9Z1+lIIiIiIsd8sXonD85ZSY3KAXw8pi+dG9d0OpLHU1khIiLl2oxfU3hqYTJN6wTx/oiehGpxKhEREfEQLpfljW9/441vf6NbSE3eva0H9atXcjpWuaCyQkREyqXCIhfP/3ct03/ZxsA29Zh0SzctTiUiIiIeIzuvkIfmrOTLNbsY1qMxL1zXkUoB/k7HKjdUVoiISLlzMCefcTMSWbxpPxHnN+PRK9ppcSoRERHxGNvTc4iIjmfj7kyeuLIddw1ohjGaq5wJlRUiIlKubNqTxajpy0k7mMurwzpzY1gTpyOJiIiIHLN0y37ujk2ksMjFh3f24oLW9ZyOVC6prBARkXIjbsMe7pm5gsAKfsyM7E2PprWdjiQiIiJyzEdLt/HMp2sIqRPEtOFhNK9X1elI5ZYuPi8iIh7PWsu0n7Yw8sPlNK4VxMLxA1RUiCOMMTcaY9YYY1zGmDCn84iIiGcoKHLxxCereeKTZM5vVZdPxvVXUXGOdGSFiIh4tLzCIp5YkMzHCalc3qEBr9/UhaCK+vgSxyQD1wPvOh1EREQ8Q3p2PmM/SuDX39MZc2ELHrmsjdbSKgWa7YmIiMfal5XHmJgE4rcd4N5LWnH/Ja3w04e/OMhauw7QImkiIgLAup0ZRETHsyczj/+7qSvXdWvkdCSvodNARETEM8TGQmgo+PlBaChrp83k2smLSU47xORbu/Hgpa1VVIiIiIjH+DJ5Jze8vYSCIhcfj+6roqKU6cgKERFxXmwsREZCTg4AX1ZsyINrK1I9KJuPx1xAp8Y1HA4ovsQY8w3QoISHoqy1C0/zNSKBSIDg4GDi4uJKL6AbZGVllbvM3kzj4Vk0Hp7DqbFwWcunmwv4ZFMBzWv4cW83Pw5sTiJus9ujeJTSHg+VFSIi4ryoKMjJwQJv9f0br10wnK5p65m6bDr1X1jldDrxMdbaQaXwGlOBqQBhYWF24MCB5/qSbhUXF0d5y+zNNB6eRePhOZwYi5z8Qh6as5IvNu3ihu6NmTi0I5UC/N2awVOV9niorBAREeelpJBboSKPXHEf/2l/IUOTv+OlL9+kkqvQ6WQiIiIiAKQeyCEiOoENuzJ44sp23DWgmdYwKkMqK0RExHG72nQistcdrG7Qkglx/2bMr/MwAE2bOh1N5ATGmKHAm0A94DNjTJK19jKHY4mISBn7dct+xsYmUlDk4t939uLC1vWcjuT1VFaIiIijkrYfJPKGF8jOyeO9eS8waPOy4geCgmDiRGfDifyBtXYBsMDpHCIi4j6xv27j6YVrCKkTxLThYTSvV9XpSD5BVwMRERHHLEzawd/e/YXA6lWY36GAQYW7wZjiIyqmToXwcKcjioiIiI8qKHLx5CfJRC1IZkCrunwyrr+KCjfSkRUiIuJ2LpfltUUbmBK3md7NavP2bT2oXaUi3HWL09FERERESM/O5+7YBJZuSWf0hc35+2Vt8dcl1N1KZYWIiLhVVl4hD8xO4uu1u7mlVwjPXtOBihV0oJ+IiIh4gNhY1r/yFqP6jmJPtdr8q2k+Q69o53Qqn6SyQkRE3GZ7eg6jpsezaW8Wz17TgeF9m2oVbREREfEMsbF8+fJ7PHjpg1TNy2FO7AS6HkqFWgU6NdUB2pUlIiJu8euW/Vz71mJ2HjrM9Dt7MaJfqIoKERER8QjWWibN+IkxVz5Cq30p/Cf6Abru3Ag5ORAV5XQ8n+R4WWGMudEYs8YY4zLGhP3FdluNMauNMUnGmHh3ZhQRkXMza1kK4dN+pWZQAJ+M68+AVnWdjiQiIiICQE5+IeNmJPJ6p6u5Pvk7Zs94lOCs9P9tkJLiXDgf5gmngSQD1wPvnsa2F1lr95VxHhERKSWFRS4mfr6Ofy/eygWt6/HmLd2oUTnA6VgiIiIiAKQeyCEiOoENuzKIWjGfUYs+4E/HfYaEOBHN5zleVlhr1wE6FFhExMscyilg/MxEfvptH3cNaMZjV7Slgr/jB/SJiIiIALDs93TGfpRAfpGLD+7oycCOB+HnWcWnfhwVFAQTJzoX0oc5XlacAQssMsZY4F1r7dSTbWiMiQQiAYKDg4mLi3NPwlKQlZVVrvJ6O42HZ9F4eJa/Go+dWS7eSMxl72HLyI4VOb/qHn7+aY97A/oQ/W6IiIicmZnLUnjyk2RC6gTx3vAwWtSrCm2OLKIZFVV86kdISHFRocU1HeGWssIY8w3QoISHoqy1C0/zZfpba9OMMfWBr40x6621P5a04ZEiYypAWFiYHThw4NnEdkRcXBzlKa+303h4Fo2HZznZePy4cS8vzkikon8As0b3oGdobfeH8zH63RARETk9BUUunv/vWqJ/2caFresx6Y+nqIaHq5zwEG4pK6y1g0rhNdKOfN1jjFkA9AJKLCtERMT9rLX8e/FWXvhsLa2DqzFtRBiNawU5HUtEREQEgPTsfMbFJvLLlv1EXtCcCZe3xd9PyxF4qnJxGogxpgrgZ63NPPL9YOA5h2OJiMgR+YUunlqYzKzl2xncPph/3dSVKoHl4iNGREREfMD6XRlERMezOyOP1//Wheu7N3Y6kpyC4zNJY8xQ4E2gHvCZMSbJWnuZMaYhMM1aOwQIBhYcWYSzAjDDWvulY6FFROSY/Vl5jP0okWVb07nn4pY8MKg1ftpLISIiIh7iqzW7eGB2ElUDKzBndF+6NqnpdCQ5DY6XFdbaBcCCEu5PA4Yc+X4L0MXN0URE5BTW7cxg1PR49mXl8cbNXbm2ayOnI4mIiIgAxaeoTv5uE//8eiNdGtdg6vAwgqtXcjqWnCbHywoRESmfEncXMu27JVSrVLyXoov2UoiIiIiHyMkv5JGPV/HZ6p0M7daIl67vRKUAf6djyRlQWSEiImfEWsuUuM28uSKPztpLISIiIh4m9UAOkdEJrN+VweND2hJxfnOOLCkg5YjKChEROW25BUVMmLeKhUlp9DnPnw9H99VeChEREfEYy7emMyYmgfwiF+/f0ZOL2tR3OpKcJZUVIiJyWnZn5BIZHc/K1EM8clkb2rNdRYWIiIh4jJnLUnhqYTJNagXx3ogwWtSr6nQkOQcqK0RE5JRWbj9IZEw8mbmFTL29B4M7NCAuLtXpWCIiIiIUFLl44b9rmf7LNi5oXY83b+lGjcoBTseSc6SyQkRE/tKnK9N45OOV1KsWyPy7+9G2QXWnI4mIiIgAcCA7n7tjE/lly34izm/Go1e0w1+XUPcKKitERKRELpfl9a83Mvn7TfQKrc3bt3WnTtVAp2OJiIiIAJCa6eLJt35md0Ye/7yxCzf0aOx0JClFKitERORPsvMKeWB2EovW7ubmnk147tqOVKzg53QsEREREQAWrdnFC0sPUz0okNmRfegWUsvpSFLKVFaIiMgJtqfnEBEdz8bdmTx9dXvu6Beqy32JiIiIR7DW8tb3m3ht0Uaa1fBj5t0DaFBDl1D3RiorRETkmOVb0xkdk0BBkYsP7+zFBa3rOR1JREREBICc/EIembuKz1btZGi3RlxR94CKCi+mY3pFRASA2ctTuPW9pdSsHMAn4/qrqBARERGPsePgYW585xc+X72Tx65oy+t/60JFfx356c10ZIWIiI8rLHLx4ufr+WDx75zfqi6Tb+lOjSBd7ktEREQ8w/Kt6Yz9KIG8AhcfjOjJRW3rOx1J3EBlhYiIDzt0uIB7Zq7gx417ubN/KFFD2lHBXwfdiYiIiGeYtSyFJxcm07hWELMiw2hZv6rTkcRNVFaIiPioLXuzGBUdz/b0HF6+vhM39wpxOpKIiIgIAAVFLiZ+to4Pl2zlgtb1ePPmbjry08eorBAR8UE//baXcbGJVPD346O7etO7eR2nI4mIiIgAcCA7n3EzElmyeT+jBjTj0Sva6shPH6SyQkTEh1hrmb5kK89/to6W9aoybUQYTWoHOR1LREREBICNuzMZNT2eXYdyee3GLgzr0djpSOIQlRUiIj4iv9DF058mM3PZdga1C+b/bu5K1UB9DIiIiIhn+Hrtbu6ftYKgwArMGt2H7iG1nI4kDtIsVUTEB6Rn5zPmo8ge1FwAACAASURBVASW/Z7O3QNb8PDgNvj56XJfIiIi4jxrLW99v4l/fr2RTo1qMPX2MBrUqOR0LHGYygoRES+3YVcmo6KXszsjjzdu7sq1XRs5HUlEREQEgMP5RTwydyX/XbWTa7s25B83dKZSgL/TscQDqKwQEfFiRw+nrBJYgTmj+9K1SU2nI4mIiIgAkHbwMJEx8axJy+DRK9oy+oLmGKMjP6WYygoRES9kreXtHzbz6lcbdDiliIiIeJz4remM+SiBvAIX748I4+K2wU5HEg+jskJExMvkFhTx6LxVfJKUxtVdGvLqMB1OKSIiIp5j9vIUnvgkmUY1KzMrMoyW9as5HUk8kMoKEREvsicjl4iYBFZuP8jDg1sz7qKWOpxSREREPEJhkYsXPlvHh0u2cn6ruky+pTs1ggKcjiUeSmWFiIiXWJ16iIjoeDJyC3jnth5c3rGB05FEREREADiYk8/4GSv4edM+7hrQjMeuaEsFfz+nY4kHU1khIuIF/rMyjUfmrqROlUDmjulH+4bVnY4kIiIiAsDG3ZlERMez82Aurw7rzI1hTZyOJOWAygoRkXLM5bL83zcbmfTdJnqG1uLt23pQt2qg07FEREREAPhm7W7um7WCoMAKzIzsQ4+mtZyOJOWEygoRkXIqJ7+QB2ev5Ms1u7ixR2NeGNqRwApaSFNEREScZ61lStxmXlu0gY4NazB1eA/Oq1HZ6VhSjqisEBEph1IP5BARncCGXRk8cWU77hrQTAtpioiIiEc4nF/E3+et4j8r07imS0Ne0ZXJ5CyorBARKWeOvy75B3f0ZGCb+k5HEhEREQEg7eBhImPiWZOWwYTL2zLmwubaoSJnRWWFiEg5Mid+O1ELVh+5LnlPWtav6nQkEREREQAStqUzOiaR3IIipg0P45J2wU5HknJMZYWISDlQ5LK89Pk6pv38OwNa1mXyrd2oGVTR6VgiIiIiAMxZvp0nPkmmYc1KzIrsTcv61ZyOJOWcygoREQ+XkVvAvTNXELdhL3f0C+WJK9vpuuQiIiLiEQqLXEz8fB3/XrxVO1SkVKmsEBHxYL/vy2bU9OVs25/DxKEdCe/d1OlIIiIiIgAczMln/IwV/LxpHyP7N+PxIW21Q0VKjcoKEREPtXjTPu6OTcTPQMxdvenboo7TkUREREQA+G13JqOi49l5MJdXhnXmb2FNnI4kXsbx2ssY86oxZr0xZpUxZoExpuZJtrvcGLPBGLPJGPOou3OKiLiLtZboX7Yy/INlBFcPZOG4ASoqRERExGN8u243Q6csITuviJmRvVVUSJlwvKwAvgY6Wms7AxuBx/64gTHGH3gLuAJoD9xijGnv1pQiIqUpNhZCQ8HPr/hrbCwABUUunvgkmacWrmFg63rMG9uPkDpBjkYVERERgeIdKlPiNjEqOp5mdavw6fj+9Gha2+lY4qUcPw3EWrvouJtLgWElbNYL2GSt3QJgjJkFXAusLfuEIiKlLDYWIiMhJ6f49rZtEBnJgULD2LxmLN2SzpgLW/DIZW3w99N1yUVERMR5h/OLmDBvFZ+uTOPqLg155YbOVK7o73Qs8WKOlxV/MBKYXcL9jYDtx91OBXqf7EWMMZFAJEBwcDBxcXGlGLFsZWVllau83k7j4Vm8ZTz6PPQQlY4WFUdsDKrLXb/msbNmOpGdA+lTeRc//bjLoYSnx1vGwxtoLEREpCztPHSYyOgEktMO8ffL2zD2whYYox0qUrbcUlYYY74BGpTwUJS1duGRbaKAQiC2pJco4T57svez1k4FpgKEhYXZgQMHnmlkx8TFxVGe8no7jYdn8Zrx2LPnhJvftujJfVc/QuWCXD4e049uIbUcCnZmvGY8vIDGQkREykrCtgOMjkkgt6CIacPDuKRdsNORxEe4payw1g76q8eNMSOAq4BLrLUllRCpwPGrtjQG0kovoYiIG4WEwLZtWGBqr+t5eeAddNi9hfd+/TfnTb7d6XQiIiIiAMyJ384TC5I5r2YlZkb0plVwNacjiQ9x/DQQY8zlwATgQmttzkk2Ww60MsY0A3YANwO3uimiiEjpmjiR3LHjePz8kczvdAlXrvuR1+KmUnnKZKeTiYiIiFBY5OLFz9fzweLfGdCyLpNv7UbNoIpOxxIf43hZAUwGAoGvj5z3tNRaO8YY0xCYZq0dYq0tNMaMB74C/IEPrLVrnIssInL29lxzA6M3VWXF4Qo88HMs96YuwUyZDOHhTkcTERERH3cop4DxMxP56bd93Nk/lKgh7ajg7wkXkRRf43hZYa1teZL704Ahx93+HPjcXblERMpC8o5DRETHc7AwkLfDu3DFy1c6HUlEREQEgE17Mhk1PZ4dBw/zyg2d+VvPJqd+kkgZcbysEBHxFZ+t2slDHydRO6gic8f2pUPDGk5HEhEREQHgu/W7uXdmEpUC/JgZ0Yew0NpORxIfp7JCRKSMuVyWN779jTe+/Y3uITV59/Yw6lULdDqWiIiICNZa3vlhC698tZ4ODasz9fYwGtas7HQsEZUVIiJlKSe/kIfmrOSL5F0M69GYiUM7EljB3+lYIiIiIuQWFDFh3ioWJqVxdZeGvHJDZypX1DxFPIPKChGRMrLj4GEipsezflcGUUPaMer8ZhxZSFhERETEUTsPHWZ0TAKrdxzikcvacPfAFpqniEfRsq4iImUgYdsBrp28mO3pObw/oicRFzTXBEBEREQ8QmLKAa6ZvJjNaQd574e3GXdJa0yzZhAb63Q0kWN0ZIWISCmbm5DK4/NXc17NSsyM6E2r4GpORxIREREB4OP47UQtSOY8/wJiYx6hderG4ge2bYPIyOLvdTl18QA6skJEpJQUuSwvfr6Ohz9eSVhoLT65u7+KChEREfEIhUUunv/vWh6Zu4qezWqxcNaj/ysqjsrJgagoZwKK/IGOrBARKQWZuQXcO3MF32/Yy/C+TXnyqvYE+KsPFhEREecdyilg/MxEfvptH3f0C+WJK9tRIXJ9yRunpLg3nMhJqKwQETlH2/Znc9f0eLbuy+aF6zpyW5+mTkcSERERAWDTnkwiohNIPZDDP27oxE09Q4ofCAkpPvXjj0JC3BtQ5CS0209E5Bws2byPa99azL6sPKLv6qWiQkRERDzG9+v3MPStJWTmFjAzos//igqAiRMhKOjEJwQFFd8v4gFUVoiInKWYpdsY/v4y6lYNZOG4/vRrUdfpSCIiIiJYa3k7bjMjpy8npE4QC8cPICy09okbhYfD1KnQtCkYU/x16lQtrikeQ6eBiIicoYIiF8/9Zy0xS7dxcdv6vHFzV6pVCnA6loiIiAi5BUVMmLeKhUlpXNn5PF4b1oXKFf1L3jg8XOWEeCyVFSIiZ+BAdj53xybyy5b9jL6gOX+/vC3+fsbpWCLiJsaYV4GrgXxgM3Cntfags6lERIrtOpRLZEw8q1IP8chlbbh7YAuM0TxFyieVFSIip2nTnkzumh7PzoO5/PPGLtzQo7HTkUTE/b4GHrPWFhpj/gE8BkxwOJOICIkpBxgdk0BOXiHvDQ/j0vbBTkcSOScqK0RETsP36/dw78wVBAb4MzOyDz2a1nI6kog4wFq76LibS4FhTmURETlqbkIqj89fTYMalYgd1ZvWwdWcjiRyzlRWiIj8BWst7/20hZe+WE/786rz3vAwGtas7HQsEfEMI4HZJT1gjIkEIgGCg4OJi4tzY6xzl5WVVe4yezONh2fxpPEoclnmbMznq62FtKvtx7iukLYugbR1TidzD08aCyn98VBZISJyEnmFRTw+P5l5iakM6dSA127sQlBF/bUp4u2MMd8ADUp4KMpau/DINlFAIRBb0mtYa6cCUwHCwsLswIEDyyZsGYmLi6O8ZfZmGg/P4injcSingHtmreDHrTnc0S+UqCvbEeDvWxd79JSxkGKlPR6adYuIlGBvZh6jY+JJTDnI/YNace/FrfDTQpoiPsFaO+ivHjfGjACuAi6x1lr3pBIR+Z9Ne7KIiI4n9UAOL1/fiZt7hTgdSaTUqawQEfmD5B2HiIyOJz0nn7du7c6Vnc9zOpKIeAhjzOUUL6h5obU2x+k8IuJ7/reOlh8zIvrQM7S205FEyoTKChGR43yxeicPzllJzaAA5o7pR8dGNZyOJCKeZTIQCHx95HKAS621Y5yNJCK+wFrL1B+38PKXxetoTR0eRiOtoyVeTGWFiAjFE4BJ327iX99spFtITd69vQf1q1VyOpaIeBhrbUunM4iI78ktKOKx+atZsGIHV3Y+j9eGdaFyRX+nY4mUKZUVIuLzDucX8fDclXy2aifXd2/Ei0M7USlAEwARERFx3q5DuYyOiWdl6iEeHtyacRe15MiRXSJeTWWFiPi0nYcOExEdz5q0DB4f0paI85trAiAiIiIeYUXKAUbHJJCdV8jU23swuENJFyoS8U4qK0TEZyUemQAczi9i2vAwLmkX7HQkEREREQDmJaTy2ILVNKheiZi7etOmQTWnI4m4lcoKEfFJ8xNTeXR+8QQgdlRvWgdrAiAiIiLOK3JZXv5iHe/99Dt9m9dhSnh3alWp6HQsEbdTWSEiPqXIZXn1qw2888Nm+jSvzdvhPTQBEBEREY9w6HAB985cwQ8b9zKib1OeuKo9Af5+TscScYTKChHxGZm5Bdw/K4lv1+8hvHcIz1zTQRMAERER8Qib92YRMT2e7QdyeOn6TtzSK8TpSCKOUlkhIj4hZX8Oo6KXs3lvNs9f24Hb+4Y6HUlEREQEgO837OHeGSuoWMGPGRF96Bla2+lIIo5TWSEiXu+Xzfu5OzYBl4Xokb3o37Ku05FEREREsNby3k9beOmL9bRrUJ33RoTRqGZlp2OJeASVFSLi1Wb8msJTC5NpWieI90f0JLRuFacjiYiIiJBbUMRj81ezYMUOrux0Hq/e2JmgivrnmchR+m0QEa9UWOTi+f+uZfov2xjYph6TbulG9UoBTscSERERYXdGLpExCazcfpCHLm3N+ItbYoxxOpaIR1FZISJe52BOPuNmJLJ4034izm/Go1e0w99PEwARERFx3oqUA4yOSSA7r5B3b+/BZR0aOB1JxCOprBARr7JpTxajpi8n7WAurw7rzI1hTZyOJCIiIgLA/MRUHp2/muDqgUTf1Y+2Dao7HUnEY6msEBGvEbdhD/fMWEFggB8zI3vTo6lW0hYRERHnFbks//hyPVN/3ELf5nV4K7w7tatUdDqWiEdzvKwwxrwKXA3kA5uBO621B0vYbiuQCRQBhdbaMHfmFBHPZa3l/Z9/58XP19GmQXWmaSVtERER8RCHDhdw78wV/LBxL8P7NuXJq9oT4O/ndCwRj+d4WQF8DTxmrS00xvwDeAyYcJJtL7LW7nNfNBHxdHmFRTyxIJmPE1K5vEMDXr+pi1bSFhEREY+weW8WEdPjSUnP4cWhnbi1d4jTkUTKDcdn9NbaRcfdXAoMcyqLiJQv+7LyGBOTQPy2A9x7SSvuv6QVflpIU0RERDxA3IY93DNzBRX9/ZgR0YdezXR6qsiZcLys+IORwOyTPGaBRcYYC7xrrZ16shcxxkQCkQDBwcHExcWVds4yk5WVVa7yejuNh2c5fjxSMop4IzGPzHzL3V0C6R6Qxo8/pjkb0Mfo98NzaCxERDyHtZZpP/3OS18Un5763vAeNK4V5HQskXLHLWWFMeYboKRr8kRZaxce2SYKKARiT/Iy/a21acaY+sDXxpj11tofS9rwSJExFSAsLMwOHDjwXH8Et4mLi6M85fV2Gg/PcnQ8vkzexcvfJVG9UiDzRoXRqXENp6P5JP1+eA6NhYiIZ8gtKOLx+auZv2IHQzo14LUbdXqqyNlyy2+OtXbQXz1ujBkBXAVcYq21J3mNtCNf9xhjFgC9gBLLChHxTtZa3vz2N/759Ua6NqnJ1Nt7UL96JadjiYiIiLA7I5fImARWbj/Ig5e25p6LW2KMTk8VOVuO13zGmMspXlDzQmttzkm2qQL4WWszj3w/GHjOjTFFxGGH84t4Z2Uev+7ayNBujXjp+k5UCvB3OpaIiIgISdsPEhkdT1ZeIe/c1oPLO5Z0ULmInAnHywpgMhBI8akdAEuttWOMMQ2BadbaIUAwsODI4xWAGdbaL50KLCLutetQLhHR8STvKmLC5W0Zc2Fz7akQERERj7BgRSoT5q2mfrVA5t/Vj7YNqjsdScQrOF5WWGtbnuT+NGDIke+3AF3cmUtEPMPRPRXZeYXc2z2QsQNbOB1JREREhCKX5ZUv1/Puj1vo07w2U8J7ULtKRadjiXgNx8sKEZGTWZi0g0fmriK4eiAxd/Vn5/oEpyOJiIiIcOhwAffNWkHchr3c3qcpT13dngB/P6djiXgVlRUi4nFcLsurizbwdtxmejerzdu3Fe+p2Lne6WQiIiLi63Zluxg6ZTEp+3N4cWgnbu0d4nQkEa+kskJEPEpWXiH3z0rim3W7uaVXCM9e04GKFbSnQkRERJz3w8a9PPvLYSoHViR2VG96N6/jdCQRr6WyQkQ8xvb0HEZNj2fT3iyevaYDw/s21UKaIiIi4jhrLe///Dsvfr6ORlX9mDG2P01qBzkdS8SrnXFZceTSobnW2qIyyCMiPurXLfsZG5tIYZGL6Xf2YkCruk5HEhEvoHmLiJyr3IIiHl+wmvmJO7iiYwOua5ChokLEDU55bLUxxs8Yc6sx5jNjzB5gPbDTGLPGGPOqMaZV2ccUEW82c1kK4dN+pWZQAAvHD1BRISJnTfMWESlNuzNyuXnqUuYn7uCBQa1569buBFbQUZ8i7nA6J4J/D7QAHgMaWGubWGvrA+cDS4GXjTG3lWFGESnvYmMhNBT8/Iq/xsYCUFjk4plP1/DY/NX0a1mXBXf3p1ndKo5GFZFyT/MWESkVK7cf5JrJP7Nxdybv3NaD+wa1ws9PRYWIu5zOaSCDrLUFf7zTWpsOzAPmGWMCSj2ZiHiH2FiIjIScnOLb27ZBZCSHigzjC1rw02/7uGtAMx67oi0VdMkvETl3mreIyDn7ZMUO/j5vFfWrBTJvbD/anVfd6UgiPueUZcXRD/wjh00+Bhy21o4raRsRkT+JivpfUXHE5kq1iFiay/Za+3nlhs78rWcTh8KJiLfRvEVEzkWRy/LKV+t594ctJ1w+XUTc70x2Y8YAH1N8GCXGmI7GmOgySSUi3iMl5YSbP4Z247rb/8kh/0BmRPRRUSEiZUXzFhE5Ixm5BYyavpx3f9jC7X2a8tGo3ioqRBx0JmWFn7X2C6AIwFqbDHQsk1Qi4j1CQgCwwAc9ruGOG5+hUcZeFn77T3qG1nY2m4h4M81bROS0bdmbxXVvLean3/YxcWhHnr+uIwE6PVXEUWdy6dI0Y0wziv/NgTHGAJXLJJWIeI+JE8kfM5anBtzBrC6XMXjjL/zruylUeetNp5OJiHfTvEVETsuPG/cyfkYiFfz9iB3Vm97N6zgdSUQ4s7LifmAa0MAYcydwOZBcJqlExGvsv3YYYzdXZVlOBcYvmc2DKT/h99abEB7udDQR8W6at4jIX7LW8v7Pv/Pi5+toHVyN94aH0aR2kNOxROSI0y4rrLVbjTGXA9cBXYAfgA/KKpiIlH/rdmYwano8+/IrMumWLlzz8pVORxIRH6F5i4j8ldyCIqIWJDMvMZUrOjbgtRu7UCXwTPbjikhZO+VvpDHGWGstgLW2EJh75L8StxERAVi0Zhf3z06iWqUKzBndly5NajodSUR8gOYtInIqezJyGf1RAitSDnL/oFbce3Er/PyM07FE5A9OZ9WY740x9xhjQo6/0xhT0RhzsTFmOjCibOKJSHljreWt7zcx+qMEWtWvyqfjB6ioEBF30rxFRE5q5faDXDN5Met3ZvLObd25f1BrFRUiHup0jnW6HBgJzDyyUNVBiheo8gMWAf+y1iaVXUQRKS9yC4qYMG8VC5PSuLZrQ/5xQ2cqBfg7HUtEfIvmLSJSooVJO/j73FXUrRrIvLH9aN+wutORROQvnLKssNbmAlOAKcaYAKAucNhae7Csw4lI+bE7I5fI6HhWph7ikcvacPfAFhQvvi8i4j6at4jIHxW5LK98tZ53f9hC7z0bmTL7Weq8UxMmTtSC3yIe7IxWkbHWFgA7yyiLiJRTK7cfJDImnszcQqbe3oPBHRo4HUlERPMWESEjt4D7Zq7g+w17uW3VVzz91RQCXEWw7RBERhZvpMJCxCOddllhjAkEbgBCj3+etfa50o8lIuXFpyvTeOTjldSrFsj8u/vRtoEOqRQR52neIiK/78tm1PTlbNufw/Pxs7j9249O3CAnB6KiVFaIeKgzObJiIXAISADyyiaOiJQXLpfl9a83Mvn7TfQKrc3bt3WnTtVAp2OJiByleYuID/tx417Gz0jE38/w0aje9Hn5qpI3TElxbzAROW1nUlY0ttZeXmZJRKTcyM4r5IHZSSxau5ubezbhuWs7UrHC6VxcSETEbTRvEfFB1lre//l3Xvx8Ha2Dq/He8DCa1A6CkBDYtu3PTwgJ+fN9IuIRzuRfF0uMMZ3KLImIlAupB3K44e0lfLNuN09f3Z6Xru+kokJEPJHmLSI+Jq+wiEfmruKFz9YxuH0D5o3tV1xUQPFimkFBJz4hKKj4fhHxSGdyZMUA4E5jzBaKD6c0gLXWdi6TZCLicZZvTWdMTAL5RS4+vLMXF7Su53QkEZGT0bxFxIfsycxlTEwCiSkHue+SVtx3SSv8/I67KtnRdSmioopP/QgJ0dVARDzcmZQVl3Pkg76MsoiIB5u9PIUnPkmmSa0g3hsRRot6VZ2OJCLyVzRvEfERq1IPEhmdwKHDBbwd3p0rOp1X8obh4SonRMqRU5YVxphMSv6gPzoB0NL/Il6ssMjFi5+v54PFv3N+q7pMvqU7NYICnI4lIlIizVtEfMvCpB38fe4q6lYNZN7YfrRvqF9xEW9xyrLCWlvNHUFExPMcOlzAPTNX8OPGvdzZP5SoIe2o4K/1KUTEc2neIuIbilyW1xZt4O24zfRqVpu3w3VVMhFvcyangYiID/l9XzZ3TV/O9vQcXr6+Ezf30mrZIiIi4rzM3ALum5XEd+v3cGvvEJ65uoMW+xbxQiorRORPfvptL+NiE6ng78dHd/Wmd/M6TkcSERER4fd92UREx7N1XzbPX9eR2/s0dTqSiJQRlRUicoy1lulLtvL8Z+toWa8q00aE/e+SXyIiIiIOOrozxd/PEHNXb/q20M4UEW+mskJEAMgvdPH0p2uYuSyFQe2C+b+bu1I1UH9FiIiIiLOstfx78VZe+GwtrYOr8d5w7UwR8QX6l4iIkJ6dz9iPEvj193TuHtiChwe3OfHa5CIiIiIOyCss4okFyXyckMplHYJ5/W9dqaKdKSI+Qb/pIj5uw65MRkUvZ3dGHm/c3JVruzZyOpKIiIgIezJzGROTQGLKQe69pBX3X9JKO1NEfIjKChEf9vXa3dw/awVVAiswZ3Rfujap6XQkEREREVanHiIyJp6DOQVMCe/OkE7nOR1JRNzM8Wv8GGOeN8asMsYkGWMWGWManmS7y40xG4wxm4wxj7o7p4g3sdbydtxmImPiaVG/Kp+OH6CiQkRERDzCpyvTGPbOEvyMYe7YvioqRHyU42UF8Kq1trO1tivwX+CpP25gjPEH3gKuANoDtxhj2rs3poh3yC0o4sE5K/nHl+u5qnND5ozuS4MalZyOJSIiIj7O5bK88uV67p25gi6Na7JwfH86NKzhdCwRcYjjp4FYazOOu1kFsCVs1gvYZK3dAmCMmQVcC6wt+4Qi3mNPRi6RMQkkbT/Iw4NbM+6ilhijcz9FRETEWZm5Bdw/K4lv1+/hll4hPHtNBypW8IT9qiLiFMfLCgBjzERgOHAIuKiETRoB24+7nQr0/ovXiwQiAYKDg4mLiyu1rGUtKyurXOX1dt40HlsPFfFGYh45hZZ7ugXS0W8HP/yww+lYZ8SbxsMbaDw8h8ZCRMqzrfuyGRUdz+/7snn+2g7c1qepdqaIiHvKCmPMN0CDEh6Ksvb/27vv+Cqrw4/j35NBAggEGWETNiIQRthawdWKigNrVQoOSBhSrbPVWOuita5q1aqBWkkIIKIWFQcixo2SBAJhDyFAGIYRAiHznt8fCS36A0UN9zz33s/79fJlnnsf47c9gCffe55z7DxrbbKkZGPMXZImS/rzd7/FMf7ZY63AqHrD2hRJKZKUkJBghw4d+pNyu5CRkaFAyhvsgmU83lqer4c/yFGjutFKH5Ogbi3qu470kwTLeAQLxsM7GAsAgerT9QW6cWa2jJHSxvbX4A6NXUcC4BF+KSusteee4K0zJc3X/y8rtklqfdR1K0n5NRANCGo+n9WTC9fpH4s2qF9cQz33275qfEqU61gAACDEWWv10ueb9dD81erY5BRNHZOgNo3quI4FwEOcPwZijOlkrV1ffTlC0ppj3LZEUidjTDtJ2yVdJekaP0UEAlJxWYVufTlH767cqV/3baWHLuuuqIhw17EAAECIK62o1J/+k6s5mdt0frdYPfGbXjolyvmPJQA8xgt/KjxsjOkiySdpi6QJklR9hOk0a+1wa22FMWaypPckhUt60Vq70lliwOO27StWYmqW1u48oHsuPE1jz2jHs58AAMC53UUlmjgjW1lb9ummszvq9+d2VlgYcxQA/5/zssJaO/I4r+dLGn7U9duS3vZXLiBQZW3Zq/FpWSot9+nF6/ppaJemriMBAAAod3uhElMzta+4TM9e00cX9mzuOhIAD3NeVgCoOa9kblXy67lqEROt2Un91LHpKa4jAQAA6M2cfN0xN0en1qmluRMGq3vLBq4jAfA4ygogCFT6rB5+Z7WmfvK1hnRspGev6aOYOrVcxwIAACHO57N6/P21evbDjWz2DeBHoawAAtyBknLdNGupMtZ+o2sHtdU9F3VTZHiY61gAACDEFZWU65aXl2nh6t26un9r3T+iRAzvmgAAIABJREFUu2pFMEcBcGIoK4AAtrngkMZOX6Ite4o15bLuGjWgretIABDUjDEPSrpEVRuD75Z0XfU+WwCOsmXPIY2bnqlNBYf0wCWna/TAtmz2DeBHoawAAtTnGwo0MT1bxkhpYwdoUIdGriMBQCh41Fr7J0kyxtwk6V5Vn2QGoMpnGwo06cgc5Yb+GtyxsetIAAIQZQUQgNK+2Kz73lylDk3qatqYfmrTqI7rSAAQEqy1B466rCvJusoCeI21Vi99vlkPzV/NHAXAz0ZZAQSQ8kqf7ntjpdK/zNM5XZvqyat6qV50pOtYABBSjDFTJI2RVChp2HHuSZKUJEmxsbHKyMjwW76acPDgwYDLHMwCYTzKfVZpq8r08bYK9W4arqQePm1a8ZU2uQ52EgTCeIQKxsJbano8KCuAALHvUJkmpmdp8aa9mnBWB93xyy4KD+PZTwCoacaYhZKaHeOtZGvtPGttsqRkY8xdkiZL+vN3b7TWpkhKkaSEhAQ7dOjQk5i45mVkZCjQMgczr4/HN0WlmjgjS5nbivW7szvqlnM7KyyI5yheH49Qwlh4S02PB2UFEADW7yrS2OmZ2llYoieujNflfVq5jgQAQctae+4J3jpT0nwdo6wAQkXu9kIlpWZqb3GZnrmmty7q2cJ1JABBgrIC8LhFa3bpplnLFB0ZrtnjB6pPm4auIwFAyDLGdLLWrq++HCFpjcs8gEtv5uTrjrk5OrVOLc2dMFjdWzZwHQlAEKGsADzKWquUjzfp4XfX6PQW9ZUyOkEtYmq7jgUAoe5hY0wXVR1dukWcBIIQ5PNZPfH+Oj3z4QYltG2o537bV03qRbmOBSDIUFYAHlRSXqm7X1+h17K368KezfXYFfGqXSvcdSwACHnW2pGuMwAuHSyt0O9nL9PC1bv0m4TWevDS7qoVEeY6FoAgRFkBeMzuohKNT8vS0rz9uvW8zvrd2R1lTPBuUgUAAALDlj2HlJiaqY3fHNJ9F3fTtYPjmKMAOGkoKwAPyd1eqMTUTO0vLtdzo/rogh7NXUcCAADQZxsKdOPMbElS6g39NaRjY8eJAAQ7ygrAI+Yv36HbXllWtUnVxEE6vQWbVAEAALestUr9YoseeGuVOjSpq6ljEtS2UV3XsQCEAMoKwDGfz+ofi9bryYXr1adNjF4YncAmVQAAwLmyCp/unZer2Uu26tzTYvXkVb10ShQ/PgDwD/60ARwqLqvQ7a/k6O0VOzWyTyv95fLuiopgI00AAOBWwcFSTZyRpSWb92nysI669bzOCgtjfwoA/kNZATiSv/+wElMztWrHASUPP03jzmzHJlUAAMC53O2FSkrN1N7iMj19dW9dHN/CdSQAIYiyAnAga8s+jU/LUml5pV68tp+GdW3qOhIAAMC399CaMFjdW7KHFgA3KCsAP3s1a5vuem2FmsdEa1biAHWKrec6EgAACHE+n9XfF67T04s2qG/bhnr+t33ZQwuAU5QVgJ9U+qweeXeNXvh4kwa1b6R/juqjhnVruY4FAABC3MHSCt3y8jK9v2qXfpPQWg9cejp7aAFwjrIC8IOiknLdPHuZFq3ZrdED2+rei7spMjzMdSwAABDi8vYUa1zqEm385pDuu7ibrh0cxx5aADyBsgKoaenpUnKylJcntWmjLff+VeP2t9CmgkN68NLuGj2wreuEAAAA+nxjgSalZ8taKfWG/hrSsbHrSADwX5QVQE1KT5eSkqTiYknS57a+JuUY2VMOKO2GQRrMJAAAADhmrVXa4i26/81Vat+4rqZdm6C2jeq6jgUA38I6dKAmJSf/t6iY0esCjbnyQTU+tE/z5j1AUQEAAJwrq/Dp7tdX6N55KzWsSxO9NmkwRQUAT2JlBVCT8vJUHhauB89OVGrfizRs4xI99cajql9+2HUyAAAQ4goOlmrijCwt2bxPNw7roNvO66KwMPanAOBNlBVADdrfsasm9fmtPo+LV+JXr+mPGS8p3PqktuxTAQAA3FmZX6ik1CztOVSqf1zdWyPiW7iOBADfi7ICqCEbdhdp7FUPa8dhnx6b/4SuyF1U9UadOtKUKW7DAQCAkDV/+Q7d/kqOYupE6pXxg9WjVQPXkQDgB7FnBVADPly7W5c9+7kO1aqtWZ1KdEXRRsmYqhUVKSnSqFGuIwIAgBDj81k9sWCtbpyZrW4t6mve5CEUFQACBisrgJ/BWqtpn3ytv76zWl2b1dfUaxPUMqa2lHSV62gAACCEHUxN160LNmtBq166ctPnevD0boqqN9h1LAA4YZQVwE9UWlGp5NdzNTdrmy7o3kyPXxmvOrX4LQUAANza+u9ZGvfZAW1o0UP3LkzR9VlvyMyvU7WmmtWeAAIEP1kBP8E3RaWaMCNLWVv26eZzOunmczqxmzYAAHDu840FunF5mHx1G2r6nHt1xpacqjeKi6uOWKesABAgKCuAH2llfqESp2dqb3GZnr2mjy7s2dx1JAAAEOKstZqxeIvue3OV2hXt1bRXH1Tc/h3fvikvz004APgJKCuAH+Hd3B265eWq3bTnThis7i3ZpAoAALhVVuHTn99YqVlf5emcrk315JxbVe+7RYUktWnj/3AA8BM5LyuMMQ9KukSST9JuSddZa/OPcd9mSUWSKiVVWGsT/JkToc1aq6cXbdAT769T7zYxemF0XzWtF+06FgAACHF7DpZq4oxsfbV5ryYN7aDbzu+i8Mg/SUlJVY9+HMFR6gACjPOyQtKj1to/SZIx5iZJ90qacJx7h1lrC/yWDJB0uKxSt8/N0fzlO3R575b6y+U9FB0Z7joWAAAIcavyDygxNVMFB0v11FW9dEmvllVvHNmXIjm56tGPNm2qigr2qwAQQJyXFdbaA0dd1pVkXWUBvmtviU9XvvCFcvML9ccLumr8L9rLGDbSBAAAbr29Yodum5OjBrUj9cqEQerZKubbN4waRTkBIKA5LyskyRgzRdIYSYWShh3nNitpgTHGSnrBWpvyPd8vSVKSJMXGxiojI6NmA59EBw8eDKi8wWzj/ko9lXVYZb7Durl3lLrarfroo62uY4U0fn94C+PhHYwFEDp8PqsnP1ivf3ywXn3axOh5Hk0FEKT8UlYYYxZKanaMt5KttfOstcmSko0xd0maLOnPx7h3iLU23xjTVNL7xpg11tqPj/Xvqy4yUiQpISHBDh06tEb+d/hDRkaGAilvsHp96Tb9beEKNYgM09wJZ6pzbD3XkSB+f3gN4+EdjAUQGg6VVujWOcv03spd+nXfVnrosu6KiuDRVADByS9lhbX23BO8daak+TpGWXFk001r7W5jzOuS+ks6ZlkB/FQ+n9Uj763V8x9t1MD2p2pUXAlFBQAAcG7r3mIlpmZq3a4i3XtRN10/JI5HUwEEtTDXAYwxnY66HCFpzTHuqWuMqXfka0nnS8r1T0KEioOlFUpKy9TzH23UNQPaKG3sANWrxSQAAAC4tXpPpUY886ny9x/WS9f31w1ntKOoABD0vLBnxcPGmC6qOrp0i6pPAjHGtJA0zVo7XFKspNer/1COkDTTWvuuo7wIQnl7ijUudYk2fnNID1xyukYPbMskAAAAOJe2eIseyyxRuyanaOqYBLVrXNd1JADwC+dlhbV25HFez5c0vPrrTZLi/ZkLoWPxpj2aOCNLPiul3tBfQzo2dh0JAACEuLIKn+57c6Vmfpmn+CbhmjFpsOpFR7qOBQB+47ysAFya+WWe7p2Xq7aN6mjatf34tAIAADi352CpJqZn66uv92ri0A7qF7WDogJAyKGsQEiqqPTpofmr9dLnm3VW5yZ6+preqs8kAAAAOLYq/4ASUzNVcLBUT13VS5f0aqmMjJ2uYwGA31FWIOQUFpfrxpnZ+nRDgcad0U53DT9N4WHsTwEAANx6Z8UO3TonRw1qR+qVCYPUs1WM60gA4AxlBULKht0HlZiaqW37ivXIFT11ZUJr15EAAECI8/msnvpgvZ76YL16t4nRC7/tq6b1o13HAgCnKCsQMjLW7tbvZi1VVESYZiUOVELcqa4jAQCAEHeotEK3zcnRuyt36oq+rTTlsu6Kigh3HQsAnKOsQNCz1urFzzZryvxV6tKsvqaO6atWDeu4jgUAAELc1r3FSkzN1LpdRbrnwtM09ox2HJ0OANUoKxDUSisq9af/5GpO5jb98vRYPXFlL9WN4pc9AABwa/GmPZqUnq2KSp9eur6/ftG5ietIAOAp/NSGoFVwsFQTZ2RpyeZ9uunsjvr9uZ0VxkaaAADAsbTFW3T/Gys5Oh0AvgdlBYLS0cd+PX11b10c38J1JAAAEOLKK326742VSv8yT8O6NNFTV3N0OgAcD2UFgs57K3fqlpeXqX50pOZOGKwerRq4jgQAAELcnoOlmpSerS+/3qsJZ3XQHb/swtHpAPA9KCsQNKy1evbDDXpswTrFt47R1NEc+wUAANxbveOAxk2vWvH55G966dLeLV1HAgDPo6xAUCgpr9Qdc5frzZx8XdqrhR4e2VPRkRz7BQAA3Ho3d4dunZOjetERmjN+kOJbx7iOBAABgbICAW9nYYmS0jK1Ynuh7vxVF008qwPHfgEAAKd8PqunPlivpz5Yr16tY5TCik8A+FEoKxDQlm3dr6TUTB0qrdDU0Qk6t1us60gAACDEHSqt0G1zcvTuyp0a2aeVplzWnRWfAPAjUVYgYM1btl13zF2upvWilDZ2iLo0q+c6EgAACHFb9xYrMTVT63YV6Z4LT9PYM9qx4hMAfgLKCgQcn8/qsQVr9c+Mjerf7lQ9N6qPGp0S5ToWAAAIcYs37dGk9GyVV/r07+v766zOTVxHAoCARVmBgHKwtEK3vLxM76/apav7t9b9I7qrVkSY61gAACDEzVi8Rfe9sVJtGtXRtDEJat/kFNeRACCgUVYgYBxZVrl+90Hdd3E3XTs4jmWVAADAqfJKn+5/c6VmLM7T0C5N9I+re6t+dKTrWAAQ8CgrEBC+3LRHE9OzVVHp00vX99OZnVhWCQAA3Np7qEyT0rO0eNNejT+rve78ZVeFh/FBCgDUBMoKeN7sr/L0p3m5an0qyyoBAIA3rN5xQImpmdpdVKq//yZel/Vu5ToSAAQVygp4VkWlT1PeXq1/f7ZZv+jcRE9f3VsNarOsEgAAuPVu7g7dOidH9aIjNGf8IPVqHeM6EgAEHcoKeFJhcbkmz8rWJ+sLdMOQdrp7eFdFhLORJgAAcMfns3p60Qb9feE6xbeOUcrovoqtH+06FgAEJcoKeM6mbw5q3PRMbd1XrEdG9tSV/Vq7jgQAAEJccVmFbpuTo3dyd+ryPi31l8t6KDoy3HUsAAhalBXwlI/XfaPJM7MVER6mmYkD1S/uVNeRAABAiNu2r1iJqVlau/OA7rnwNI09ox0nkgHASUZZAU+w1urfn23WQ/NXqXNsPU0dk6DWp9ZxHQsAAIS4IyeSlVf69OJ1/TS0S1PXkQAgJFBWwLmyCp/unZer2Uu26rxusXryN71UN4pfmgAAwK2ZX+bp3nm5atOojqaOSVAHTiQDAL/hJ0I4tedgqSamZ+urr/dq8rCOuvW8zgrjfHIAAOBQeaVPD7y5SmmLt+iszk30D04kAwC/o6yAM2t2HtC46Zn6pqhUT13VS5f0auk6EgAACHF7D5VpUnqWFm/aq/G/aK87f9VV4XyQAgB+R1kBJxas3KlbXl6mulFV55PHcz45AABw7MgHKbuLSvXElfG6vE8r15EAIGRRVsCvrLX6Z8ZGPbZgrXq2bKCUMQmcTw4AAJx7N3enbp2zTKdUf5DSiw9SAMApygr4TUl5pf7w6nLNW5avEfEt9MgVPTmfHAAAOGWt1dOLNuiJ99cpvnWMUkb35YMUAPAAygr4xa4DJUpKzVTOtkLd8csumjS0A+eTAwAAp4rLKnT7Kzl6e8VOXd67pf5yeQ8+SAEAj6CswEm3fNt+JaZmqqikQimj++r805u5jgQAAELctn3FSkzN0tqdB3T38K5KPLM9H6QAgIdQVuCkeiMnX3e8kqPGp0Tp1YmDdVrz+q4jAQCAEPfV13s1cUaWyip9+td1/TSsS1PXkQAA3xHmOsARxpjbjTHWGNP4OO//yhiz1hizwRjzR3/nw4/j81k9vmCtbpq1VPGtYvTG5CEUFQAAwLlZX+XpmqmL1aB2pP5z4xCKCgDwKE+srDDGtJZ0nqS847wfLunZ6nu2SVpijHnDWrvKfylxog6VVujWOcv03spduqpfaz1wSXfVivBMLwYAAEJQeaVPD761SqlfbNEvOjfR01f3VoPaka5jAQCOwxNlhaS/S7pT0rzjvN9f0gZr7SZJMsbMlnSJJMoKj9m2r1jjpmdq3a4i3XtRN10/JI7nPwEAgFN7D5XpxvRsfbFpjxLPbKc/XnCawsOYnwCAlzkvK4wxIyRtt9bmfM8PtS0lbT3qepukAd/zPZMkJUlSbGysMjIyaiasHxw8eDCg8h5t3b5KPb20RBU+6ZY+UWpfsUUffbTFdayfJZDHIxgxHt7CeHgHYwEc35qdB5SYmqldB0r1+K/jNbJvK9eRAAAnwC9lhTFmoaRjHQGRLOluSef/0Lc4xmv2eDdba1MkpUhSQkKCHTp06IkF9YCMjAwFUl6lp0vJyZrToLMePf9GtaptNO3ms9WhySmuk9WIgBuPIMd4eAvj4R2Mhf8ZY26X9KikJtbaAtd5cGzvrdypW15eplOiIvRy0kD1btPQdSQAwAnyS1lhrT33WK8bY3pIaifpyKqKVpKyjTH9rbU7j7p1m6TWR123kpR/kuLiRKWnq2L8BP11wFX6V79LdebX2Xrm/afVoMvfpVGjXKcDAOCk+KG9tuCetVbPLNqgx99fp/hWDfTC6AQ1axDtOhYA4EdwuuuhtXaFtbaptTbOWhunqlKiz3eKCklaIqmTMaadMaaWpKskveHnuPiOwvse0g3D79C/+l2q6zLf0L9fuU8N9n0jJSe7jgYAwMl0ZK+t467yhB+lp0txcVJYmBQXp+K0dE2euVSPv79Ol/VuqZfHD6KoAIAA5HzPiuMxxrSQNM1aO9xaW2GMmSzpPUnhkl601q50mzC0fV1wSGOH/l55Mc3013ef1tU57/3vzTw+aAIABKcT3GsroPfPkgJnH5SmCxeqy2OPKby0VFLVRt9JH+7V6qb5urJLLV3QdJ8Wf/aJ45Q/X6CMR6hgPLyDsfCWmh4PT5UV1asrjnydL2n4UddvS3rbQSx8x6frC3TjzGyF122g9NnJGrDtO71RmzZuggEAUANqYK+tgN4/SwqgfVCuu06qLiqWtOymCZfdrbKISL340XMa9vB8t9lqUMCMR4hgPLyDsfCWmh4PT5UV8DZrrVK/2KIH3lqljk1O0bQ2Vq2nfv3tm+rUkaZMcRMQAIAaUAN7bcFfqldzzu55vv50/kS1KtytqTMfVMd92x0HAwD8XJQVOCFlFT79+Y2VmvVVns49LVZPXtVLp0RFSLV8VXtU5OVVraiYMoXNNQEAQclau0JS0yPXxpjNkhI4DcSd8rZxeqjj+Zre9+Kqjb7n/U0NSg9Jbdu6jgYA+JkoK/CD9h4q08QZWfry672aNLSDbj+/i8LCqp/THTWKcgIAAPjdvkNlmnTDE/riUKQSv3pNf8h4SRHWxypPAAgSlBX4Xut2FWns9CXadaBUT/6mly7t3dJ1JAAAPOHovbbgX2t3Fmlc6hLtKonS462KNHL2B5Js1YoKVnkCQFCgrMBxLVy1SzfPXqq6URGaM36QerWOcR0JAACEuAUrd+qWl5epTlSEZo8fqD5tGkqTr3QdCwBQwygr8P9Ya/X8R5v0yHtr1L1FA00dk8D55AAAwClrrZ79cIMeW7BOPVs1UMpo5icAEMwoK/AtJeWVuuu1FXp96XZd1LO5Hr0iXrVrhbuOBQAAQlhxWYXumLtc85fv0KW9WujhkT0VHcn8BACCGWUF/mv3gRIlpWVp2db9uu28zpp8dkdVH80GAADgxPb9h5WUmqlVOw7ojxd01fhftGd+AgAhgLICkqQV2wqVmJqpAyXlev63ffWr7s1cRwIAACFuyea9mjgjS6XlPr14bT8N69r0h/8hAEBQoKyA3lqer9tfyVGjulGaO2GwurWo7zoSAAAIcbO/ytOf5uWqZUxtzU5KUMem9VxHAgD4EWVFCPP5rJ78YL3+8cF6JbRtqOdH91XjU6JcxwIAACGsvNKnKfNX66XPN+vMTo31zNV91KBOpOtYAAA/o6wIUcVlFbptTo7eyd2pX/dtpYcu666oCDaqAgAA7uw7VKYbZ2br8417NO6MdvrjBV0VER7mOhYAwAHKihC0ff9hJU7P1JqdB3TPhadp7Bnt2KgKAAA4tW5XkcZNz9TOwhI9ekVP/TqhtetIAACHKCtCTNaWvRqfVr1R1XX9NLQLG1UBAAC33l+1S7+fvVR1oiI0K2mg+rZt6DoSAMAxyooQ8krmViW/nqsWMdFsVAUAAJyz1urZDzfo8ffXqXuLBkoZ01fNG9R2HQsA4AGUFSGg0mf18DurNfWTrzWkYyM9e00fxdSp5ToWAAAIYYfLKnXH3By9tXyHLunVQn8b2VPRkeyfBQCoQlkR5A6UlOvmWUv14dpvdO2gtrrnom6KZKMqAADgUP7+w0pKy9TK/AP6w6+6asJZ7dk/CwDwLZQVQWxzwSGNS83U5oJDmnJZd40a0NZ1JAAAEOIyN+/VhBlZKin3adqYBJ1zWqzrSAAAD6KsCFKfbyjQxPRsGSOljR2gQR0auY4EAABC3MtL8nTPf3LVMqY2+2cBAL4XZUUQSvtis+57c5U6NKmraWP6qU2jOq4jAQCAEFZR6dND81frpc8368xOjfXM1X3UoE6k61gAAA+jrAgi5ZU+3f/mSs1YnKdzujbVk1f1Ur1oJgIAAMCd/cVlmjxzqT7dUKAbhrTT3cO7KoL9swAAP4CyIkjsO1SmSenZ+mLTHk04q4Pu+GUXhYexURUAAHBn3a4iJaZmasf+Ej1yRU9dmdDadSQAQICgrAgC63cVaez0TO0sLNETV8br8j6tXEcCAAAhbuGqXbp59lLVrhWhWUkD1bdtQ9eRAAABhLIiwC1as0s3zVqm6MhwzR4/UH3aMBEAAADuWGv1z4yNemzBWnVv0UApY/qqeYParmMBAAIMZUWAstZq6ieb9Nd31qhb8/qaOiZBLWKYCAAAAHcOl1XqzleX682cfF0c30KPjOyp2rXCXccCAAQgyooAVFJeqbtfX6HXsrfrwh7N9eive6pOLYYSAAC4k7//sJLSMrUy/4Du/FUXTTyrg4xh/ywAwE/DT7gBZndRicanZWlp3n7dcm5n3XRORyYCAADAqawtezU+LVsl5ZWaNiZB55wW6zoSACDAUVYEkNzthUpMzdT+4nI9N6qPLujR3HUkAAAQ4uYs2ark/6xQi5jampU4QJ1i67mOBAAIApQVAeLtFTt025wcNawTqVcmDFL3lg1cRwIAACGsotKnKW+v1r8/26wzOjbWM9f0VkydWq5jAQCCBGWFx/l8Vv9YtF5PLlyvPm1i9PzovmpaL9p1LAAAEML2F5dp8syl+nRDga4fEqfk4acpIjzMdSwAQBChrPCw4rIK3f5Kjt5esVMj+7TSXy7vrqgIdtQGAADurN9VpHGpmcrff1iPjOypK/u1dh0JABCEKCs8Kn//YSWmZmrVjgNKHn6axp3Zjo00AQCAUx+s3qWbZy9TdGS4ZicNVN+2p7qOBAAIUpQVHpSdt09JqVkqKa/Ui9f207CuTV1HAgAAIcxaq+c+2qhH31ur01vUV8roBLWIqe06FgAgiFFWeMxn28s1/f3Fah4TzY7aAADAucNllfrDq8v1Rk6+Lo5voUdG9lTtWjyWCgA4uSgrPKLSZ/XIu2s0dUWZBrVvpH+O6qOGddlRGwAAuLOj8LCSUrOUm1+oO37ZRZOGduCxVACAX3imrDDG3C7pUUlNrLUFx3h/s6QiSZWSKqy1Cf5NePIUlZTr5tnLtGjNbp3dJkIvjO2vSHbUBgAADmVt2afxaVk6XFahqaMTdG63WNeRAAAhxBNlhTGmtaTzJOX9wK3DjlVkBLK8PcUaO32JNhUc0oOXnK7WpZspKgAAgFOfbCtXWvVjqTMTB6gzj6UCAPzMKz8V/13SnZKs6yD+9PnGAo149lPtLipV2g39NXpQnOtIAAAghFVU+vTAm6v0r9wy9WvXUPNuHEJRAQBwwvnKCmPMCEnbrbU5P/AMpJW0wBhjJb1grU35nu+ZJClJkmJjY5WRkVGDiWvGorxypa8uU2wdo5v7RqtsW64ytkkHDx70ZN5QxXh4C+PhLYyHdzAWqAmFxeWaPCtbn6wv0HltI/Tc9f0VwWpPAIAjfikrjDELJTU7xlvJku6WdP4JfJsh1tp8Y0xTSe8bY9ZYaz8+1o3VRUaKJCUkJNihQ4f+tOAnQXmlTw++tUqpq7ZoWJcmeurq3qofHfnf9zMyMuSlvKGO8fAWxsNbGA/vYCzwc23YXaRx0zO1ff9h/W1kD8Ue2kRRAQBwyi9lhbX23GO9bozpIamdpCOrKlpJyjbG9LfW7vzO98iv/vtuY8zrkvpLOmZZ4VX7i8t048xsfbZhj5J+0V5/+FVXhYexozYAAHBn0ZpdumnWMkVHhmlW4kAlxJ2qjIxNrmMBAEKc08dArLUrJDU9cl194kfCdzfRNMbUlRRmrS2q/vp8SQ/4M+vPdeQTi/z9JXrs1/G6om8r15EAAEAIs9bq+Y826ZH31qhb8/pKGZOgljG1XccCAECSB/asOB5jTAtJ06y1wyXFSnq9evVFhKSZ1tp3Xeb7MT5cu1s3zVyqqMgwzUoaoL5tT3UdCQAAhLCS8kr94dXlmrcsXxf1bK5Hr4hX7VrhrmMBAPBfniorrLVxR32dL2l49debJMU7ivWTWWv1r0+/1l/eXq2uzepr6rV8YgGp82MFAAAKKElEQVQAANzaUXhY49OytHxboe74ZRdNGtpBP7DJOQAAfuepsiKYlFZUKvn1XM3N2qYLujfT41fGq04t/u8GAADuZOft0/i0LBWXVmjqmASd1y3WdSQAAI6Jn55Pgm+KSjVhRpaytuzTTed00u/P6aQwNtIEAAAOvZK5Vcmv56pZg2iljxugzrH1XEcCAOC4KCtq2Mr8QiVOz9Te4jI9e00fXdizuetIAAAghFVU+vTXd9boX59+rcEdGunZa/qoYd1armMBAPC9KCtq0Lu5O3TLyzlqUDtScycMVveWDVxHAgAAIaywuFyTZ2Xrk/UFum5wnJIvPE2R4WGuYwEA8IMoK2qAtVZPL9qgJ95fp16tY5Qyuq+a1o92HQsAAISwDbuLlJiapW37ivXw5T10Vf82riMBAHDCKCt+psNllbpjbo7eWr5Dl/duqb9c3kPRkRz9BQAA3PlwzW7dNKvq2PSZiQPVL45j0wEAgYWy4mfYUXhYSalZys0v1B8v6Krxv2jP0V8AAMAZa61e+HiT/vbuGnVrXl8pYzg2HQAQmHho8SdamrdPI575TJu+OahpYxI04SzOKAcAACdZeroUFyeFhVX9PT39v2+VlFfqlpeX6eF31mh4j+aaO2EwRQUAIGCxsuIneH3pNv3h1RWKrR+lGWOHqEszjv4CAAAnWXq6lJQkFRdXXW/ZUnUtaedFI5WUlqnl2wp1+/mddeOwjnyIAgAIaJQVP4LPZ/XogrV6LmOjBrQ7Vc/9tq9O5egvAADgD8nJ/ysqjiguVvYTUzV+c6yKSyuUMrqvzj+9mZt8AADUIMqKH+GfGRv0XMZGXTOgje67+HTViuApGgAA4Cd5ef/vpbndz9bdw36nZpHhmjF2AKs9AQBBg7LiRKWna8x9D6pZdCuNnL1BpniKNGqU61QAACBUtGlT9eiHpAoTpoeHXq9p/S/ToF1r9c/7J6shqz0BAEGEsuJEVD8jWr+4WFdobdVr1c+IUlgAAAC/mDJFSkpSYaXR70bcqY/b99W1Oe/onlGDFElRAQAIMpQVJ+I4z4gqOZmyAgAA+Ef1nOP+N9bqizY99dev0nX12AuZiwAAghJlxYk4xjOi3/s6AADAyTBqlO4aUapr9hxSQtylrtMAAHDSsEPkiWjT5se9DgAAcJI0qRelhLhTXccAAOCkoqw4EVOmSHXqfPu1OnWqXgcAAAAAADWKsuJEjBolpaRIbdtKxlT9PSWFZ0QBAAAAADgJ2LPiRI0aRTkBAAAAAIAfsLICAAAAAAB4CmUFAAAAAADwFMoKAAAAAADgKZQVAAAAAADAUygrAAAAAACAp1BWAAAAAAAAT6GsAAAAAAAAnkJZAQAAAAAAPIWyAgAAAAAAeAplBQAAAAAA8BTKCgAAAAAA4CmUFQAAAAAAwFOMtdZ1hpPKGPONpC2uc/wIjSUVuA6B/2I8vIXx8BbGwzt+yli0tdY2ORlh8G0BOBeR+P3tNYyHtzAe3sFYeEuNzkeCvqwINMaYTGttguscqMJ4eAvj4S2Mh3cwFqhp/JryFsbDWxgP72AsvKWmx4PHQAAAAAAAgKdQVgAAAAAAAE+hrPCeFNcB8C2Mh7cwHt7CeHgHY4Gaxq8pb2E8vIXx8A7GwltqdDzYswIAAAAAAHgKKysAAAAAAICnUFYAAAAAAABPoazwKGPM74wxa40xK40xj7jOE8qMMfcZY7YbY5ZV/zXcdaZQZ4y53RhjjTGNXWcJZcaYB40xy6t/XywwxrRwnSmUGWMeNcasqR6T140xMa4zIfAxH/EG5iLexHzEG5iPeEtNzkcoKzzIGDNM0iWSelprT5f0mONIkP5ure1V/dfbrsOEMmNMa0nnScpznQV61Frb01rbS9Jbku51HSjEvS+pu7W2p6R1ku5ynAcBjvmI5zAX8RDmI57CfMRbamw+QlnhTRMlPWytLZUka+1ux3kAL/m7pDslsTuwY9baA0dd1hVj4pS1doG1tqL6crGkVi7zICgwHwGOj/mIRzAf8ZaanI9QVnhTZ0lnGmO+NMZ8ZIzp5zoQNLl6KdOLxpiGrsOEKmPMCEnbrbU5rrOgijFmijFmq6RR4pMML7lB0juuQyDgMR/xFuYiHsF8xHuYj3jWz5qPcHSpI8aYhZKaHeOtZElTJC2SdLOkfpJeltTeMlgnzQ+Mx2JJBapqaR+U1Nxae4Mf44WUHxiLuyWdb60tNMZslpRgrS3wZ75Q833jYa2dd9R9d0mKttb+2W/hQtCJjIcxJllSgqTL+e8GfgjzEe9gLuItzEe8hfmIt/hrPkJZ4UHGmHdVtewyo/p6o6SB1tpvnAaDjDFxkt6y1nZ3HCXkGGN6SPpAUnH1S60k5Uvqb63d6SwYJEnGmLaS5vN7wy1jzLWSJkg6x1pb/EP3A9+H+Yg3MRdxi/mItzEf8Yaamo/wGIg3/UfS2ZJkjOksqZaq2nQ4YIxpftTlZZJyXWUJZdbaFdbaptbaOGttnKRtkvowMXDHGNPpqMsRkta4ygLJGPMrSX+QNIKiAjWE+YhHMBfxDuYj3sN8xFtqcj7CygoPMsbUkvSipF6SyiTdbq1d5DZV6DLGpKlqLKykzZLGW2t3OA0FsezSPWPMq5K6SPJJ2iJpgrV2u9tUocsYs0FSlKQ91S8tttZOcBgJAY75iHcwF/Eu5iPuMR/xlpqcj1BWAAAAAAAAT+ExEAAAAAAA4CmUFQAAAAAAwFMoKwAAAAAAgKdQVgAAAAAAAE+hrAAAAAAAAJ5CWQEAAAAAADyFsgIAAAAAAHgKZQWAn80Yc/AE7qltjPnIGBNefX2uMSbtO/fUMsZ8bIyJOFlZAQBA8GEuAgQfygoA/nKDpNestZXV1/GSlh59g7W2TNIHkn7j52wAACD4MRcBAghlBYAaYYyJM8asNsZMNcasNMYsMMbUPuqWUZLmHXUdL6mZMeYTY8xOY8y51a//p/peAACAE8ZcBAgulBUAalInSc9aa0+XtF/SSKlqSaWk9tbazUfdGy+pwFp7pqRJ+t+kIFdSP78lBgAAwYS5CBAkKCsA1KSvrbXLqr/OkhRX/XVjVU0YJEnGmEhJp0p6rPqliCPvVy/NLDPG1PNHYAAAEFSYiwBBgrICQE0qPerrSlX9h1+SDkuKPuq9bpJyrLW+6uueqvoU44goSSUnKyQAAAhazEWAIEFZAeCks9bukxRujDkySYiXlHPULT0lLZckY0wjSd9Ya8v9mxIAAAQr5iJA4KGsAOAvCySdUf11vKonBNW663+fZgyT9LYfcwEAgNDAXAQIIMZa6zoDgBBgjOkt6VZr7egfuO81SXdZa9f6JxkAAAgFzEWAwMLKCgB+Ya1dKulDY0z48e6p3qn7P0wOAABATWMuAgQWVlYAAAAAAABPYWUFAAAAAADwFMoKAAAAAADgKZQVAAAAAADAUygrAAAAAACAp1BWAAAAAAAAT6GsAAAAAAAAnvJ/TKV3laMJisUAAAAASUVORK5CYII=\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "figure(figsize=(18, 7))\n", "\n", "subplot(1,2,1)\n", "plot(log(H),log(err_ep), 'ro',label='Euler Explicite')\n", "plot(log(H),[a_ep*log(h)+b_ep for h in H])\n", "xlabel('$\\ln(h)$')\n", "ylabel('$\\ln(e)$')\n", "legend(loc='upper left')\n", "grid(True);\n", "\n", "subplot(1,2,2)\n", "plot(log(H),log(err_er), 'ro',label='Euler Implicite')\n", "plot(log(H),[a_er*log(h)+b_er for h in H])\n", "xlabel('$\\ln(h)$')\n", "ylabel('$\\ln(e)$')\n", "legend(loc='upper left')\n", "grid(True);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Soit on affiche les données en échelle logarithmique et l'équation de l'erreur $Ch^p$ avec $C=\\ln(b)$ et $p=a$ avec l'instruction ``loglog``:" ] }, { "cell_type": "code", "execution_count": 51, "metadata": {}, "outputs": [ { "data": { "image/png": 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