{ "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_CM3 : problèmes bien posés (mathématiquement et numériquement), schémas A-stables\n", "\n", "On comprend bien que, en générale, il ne suffit pas qu'un schéma numérique soit convergent pour qu'il donne des bons résultats sur n'importe quelle équation différentielle. \n", "Encore faut-il que le problème soit \n", "- mathématiquement bien posé (existence et unicité de la solution), \n", "- numériquement bien posé (continuité suffisamment bonne par rapport aux conditions initiales)\n", "- bien conditionné (temps de calcul raisonnable)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Propriétés du problème de Cauchy \n", "\n", "- **Problème mathématiquement bien posé** \n", " Un problème de Cauchy est un problème bien posé si sa solution est unique.\n", "\n", "- **Problème numériquement bien posé** \n", " Un problème de Cauchy est numériquement bien posé si sa solution n’est pas trop perturbée par une erreur initiale ou des perturbations du second membre.\n", " \n", "- **Problème bien conditionné** \n", " Un problème est bien conditionné si les méthodes numériques usuelles peuvent donner sa solution en un nombre raisonnable d’opérations. Sinon on parle de problème raide.\n", " \n", "Cf. page 235-236 Jean-Pierre DEMAILLY, *Analyse Numérique et équations différentielles*" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Problème mathématiquement bien posé\n", "\n", "
\n", "On dit qu'un problème de Cauchy est mathématiquement bien posé s'il existe une et une seule solution. \n", "
\n", "\n", "Si ce n'est pas le cas, on dit qu'il est **mathématiquement mal posé**." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Exemple de problème mathématiquement mal posé\n", "Par exemple, on cherche une fonction $y\\colon t\\in\\mathbb{R}^+\\mapsto y(t)\\in\\mathbb{R}$ qui satisfait\n", "$$\n", "\\begin{cases}\n", "y'(t) = \\sqrt[3]{y(t)}, &\\forall t>0,\\\\\n", "y(0) = 0.\n", "\\end{cases}\n", "$$\n", "Il est simple de vérifier que, pour tout $t\\ge0$, les trois fonctions \n", "- $y_1(t)=0$ et \n", "- $y_{2,3}(t)=\\pm\\sqrt{8\\frac{t^3}{27}}$ \n", "sont solution du problème de Cauchy donné.\n", "\n", "Dans ce cas, lorsqu'on utilise une méthode numérique, on ne sait pas quelle solution ce schéma approche et différents schémas peuvent approcher différentes solutions." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Exercice \n", ">1. Montrer que le problème de Cauchy\n", "$$\n", "\\begin{cases}\n", "y'(t)=2\\sqrt{|y(t)|}, & t>0\\\\\n", "y(0)=0,\n", "\\end{cases}\n", "$$\n", "admet une infinité de solutions de classe $\\mathcal{C}^1(\\mathbb{R})$.\n", ">1. Parmi ces solutions, quelle solution approche-t-on si on utilise la méthode d'Euler explicite? et la méthode d'Euler implicite? \n", ">1. Que se passe-t-il si la donnée initiale est $y(0)=y_0>0$?" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Correction** \n", "Sur $\\mathbb{R}^+$ la fonction $\\varphi(t,y)=2\\sqrt{y}$ n'est pas lipschitzienne par rapport à $y$ au voisinage de $(t,0)$ (car $\\partial_y\\varphi=1/\\sqrt{y}\\to\\infty$ lorsque $y\\to0$), donc le théorème d'existence et unicité locale n'est pas valable au voisinage de $(0,0)$. Même raisonnement sur $\\mathbb{R}^-$. \n", "\n", "L'EDO est à variables séparables, on peut donc calculer explicitement toutes les solutions du problème de Cauchy: \n", "- elle admet une seule solution constante, la fonction $y(t)=0$ pour tout $t\\in\\mathbb{R}^+$, \n", "- et des solutions de la forme $y(t)=(t+c)^{2}$ pour tout $t\\ge c$. \n", "\n", "En imposant la CI on trouve que, pour tout $b\\in\\mathbb{R}^+$, les fonctions\n", "$$\n", "y_b(t)=\n", "\\begin{cases}\n", "0 , &\\text{si }0\\le t\\le b,\n", "\\\\\n", "(t-b)^{2} , &\\text{si }t\\ge b,\n", "\\end{cases}\n", "$$\n", "sont de classe $\\mathcal{C}^1(\\mathbb{R}^+)$ et sont solution du problème de Cauchy donné.\n", " \n", "Traçons quelques curbes:" ] }, { "cell_type": "code", "execution_count": 94, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "%matplotlib inline\n", "from matplotlib.pylab import *\n", "\n", "xx=linspace(0,2,101)\n", "yyr=[0 for x in xx]\n", "\n", "f = lambda x,b : (x<=b)*0 + (x>b)*((x-b)**2)\n", "\n", "yy1=[f(x,1) for x in xx]\n", "yy75=[f(x,0.75) for x in xx]\n", "yy0=[f(x,0) for x in xx]\n", "\n", "figure(figsize=(10,7))\n", "plot(xx,yyr,'r-',xx,yy1,'b:',xx,yy75,'g*',xx,yy0,'yd');\n", "legend([r'$y(t)=0$ $\\forall$ $t$', '$y_{b=1}(t)$', '$y_{b=0.75}(t)$', '$y_{b=0}(t)$']);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "La méthode d'Euler explicite construit la suite \n", "$$\\begin{cases}\n", "u_0=y_0=0,\\\\\n", "u_{n+1}=u_n+h \\varphi(t_n,u_n)=u_n+hu_n^{1/2},& n=0,1,2,\\dots N_h-1\n", "\\end{cases}$$\n", "par conséquent $u_n=0$ pour tout $n$: la méthode d'Euler explicite approche la solution constante $y(t)=0$ pour tout $t\\in\\mathbb{R}^+$.\n", "\n", "\n", "La méthode d'Euler implicite construit la suite \n", "$$\\begin{cases}\n", "u_0=y_0=0,\\\\\n", "u_{n+1}=u_n+h \\varphi(t_{n+1},u_{n+1})=u_n+hu_{n+1}^{1/2},& n=0,1,2,\\dots N_h-1.\n", "\\end{cases}$$\n", "par conséquent $u_0=0$ mais $u_1$ dépend de la méthode de résolution de l'équation implicite $x=0+h\\sqrt{x}$. Bien sur $x=0$ est une solution mais $x=h^{2}$ est aussi solution.\n", "Si le schéma choisit $u_1=h^{2}$, alors $u_n>0$ pour tout $n\\in\\mathbb{N}^*$.\n", "\n", "Notons que le problème de Cauchy avec une CI $y(0)=y_0>0$ admet une et une seule solution, la fonction $y(t)=\\frac{1}{4}(t+2\\sqrt{y_0})^{2}$.\n", "Dans ce cas, les deux schémas approchent forcement la même solution." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Problème numériquement bien posé" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Une fois calculée la solution numérique $\\{u_n\\}_{n=1}^{N}$ d'un problème de Cauchy mathématiquement bien posé, il est légitime de chercher à savoir dans quelle mesure l'erreur $|y(t_n)-u_n|$ est petite. \n", "Cela **dépend** bien sur du schéma choisi mais aussi **du problème en question**. \n", "\n", "Étant donné que les erreurs d’arrondi amènent toujours à résoudre un problème perturbé, il est important de savoir si **la solution d'un problème perturbé est proche de la solution du problème non perturbé**.\n", "\n", "
\n", "On dit qu'un problème de Cauchy est numériquement bien posé si la solution d'un problème faiblement perturbé (second membre ou condition initiale) possède une solution proche de celle du problème original. \n", "
\n", "\n", "\n", "### Exemple de problème numériquement mal posé\n", "On se donne $\\varphi(t,y)=3t-3y$ et $y(0)=\\alpha$ (un nombre quelconque). \n", "On cherche une fonction $y\\colon t\\in\\mathbb{R}\\mapsto y(t)\\in\\mathbb{R}$ qui satisfait\n", "$$\n", "\\begin{cases}\n", "y'(t) = 3y(t)-3t, &\\forall t\\in\\mathbb{R},\\\\\n", "y(0) = \\alpha.\n", "\\end{cases}\n", "$$\n", "Sa solution, définie sur $\\mathbb{R}$, est donnée par \n", "$$\n", "y(t)=\\left(\\alpha-\\frac{1}{3}\\right)e^{3t}+t+\\frac{1}{3}.\n", "$$\n", "Calculons $y$ en $t=10$:\n", "- si $\\alpha=1/3$ alors $y(10)=\\frac{31}{3}$,\n", "- si $\\alpha=0.333333$ alors $y(10)=(0.333333-1/3)e^{30}+10+1/3=-e^{30}/3000000+31/3\\approx31/3+10^7/3$\n", "\n", "Vérifions-le: " ] }, { "cell_type": "code", "execution_count": 55, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$\\displaystyle y{\\left(t \\right)} = t + \\left(\\alpha - \\frac{1}{3}\\right) e^{3 t} + \\frac{1}{3}$" ], "text/plain": [ " 3⋅t 1\n", "y(t) = t + (α - 1/3)⋅ℯ + ─\n", " 3" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGMAAAArCAYAAABsMDaUAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAFoElEQVR4Ae2a7XEUMQyGL8wVEKCD0EEgFRA6IOkA0kEY/uUfAx1AKsiEDggVENIB6YCQDsL7GMvn9Xr39uuYuxtrxmdblmVbkiXbtzsPDw+zAsMlcHZ2tqfeJ57DrnLqH4W/8rhKJvy+EJdKz1W+jxvncaWU+0lAwkT475SbMmYqvxbum/Ijpa9wVA7dudKd0gslFFaDRzVMQfSRwFsRv5WwUYCB7Yj3hlD7vRLKQWkXhk/zooxUIv3qNyLH1QR3g+D7sVhQFze1kEXvkgTPLngcdxTOdsnnGN+lXHZGFyl1pJEiDkX6UelE5S8duwWysjOCKIYXJHhOSCjiQAnXda3UG4oyeous3kHKQAGkmcq4qZ/Kw2kKfBcobqqLlHrQSAkcZwnilyrv9ug6K8roI62EVsLeJyVoquamcF2doSijs6iyhD+FxSX12gFZTkKWmNEkmW543NGVlEEeA7dswC6A/2pLfifZGZrM3pJxBjevkvfgSS06vlPx26IaAjg7heNtqiRIn3r6Jz4P2c7Yh0INeCput8rdO0zgPFFBfPG7+OZPE7GclI2f31HEFMOsPRSKjsdBgPWgLE5ft0q8Y7k7yShliAnHuAPlWEgNhG98oTRi0XBJAn4rPVNiIUwygOoonPed3hepwGQDCoOVIcGg3e/Kn8fr9Pj4hRKFPBa+tmWFIwB+UO52le8L7pXKqULAvxS+xkf4rYAxMQOLrr2/ICylpS+UouHFc1d5cG/0FY56ja/H2S5SdftgjDKOJbwxbgM/626tiVh/qH4o3uy8AH4sxqzgA8EWFAYpQwIhVlTcyABZEMjuMv2ML+0p0HacIrelPmch3tpwGwRQLjHB4n3bufL4xPBKdL3O0KIP4HmGekOhdvQTHWMydphfQ9+NRDtlaObvJSD+PsTiCb7xYrFE8DG8UCXn12OatrIJui0Y59zRLzHN7ZjsWFoPc+xM75ncqF9seFneq0DONTCnHfw0gNWlriO3CxBUSkf/KcEuRzFPxtyLEW1lrS38N91Gty5txIz4wsYuSC0ey6rcMlXHstusWs2t0KZI2zXcO1IgZuR2TEq3kXV2hhOqcoTOQoOL8rsG3OD4kJMKYyrRlBOs4SyQxyzGGkHMa1RZ85/8G6d5NCN31ERQEQ4FIbibCEcRyzahJU2dqyg453JsZ+QMgDHbdlVlcM17ZTFDvHcqg01QiZWBYFJrzMULhoUuJ0jaugJvNblLHDd6gmhsFMYTRaVztLZaLh4bFzNsEZVFaiHsClIaL6BnpxxQWAIWhM3aA7n44w7vlIeTmspYPnHrTSCsFpyiqqjtqcU7g8c+7hNYK8GTOweQcxcXwtsrpCOKf8TD2lAmwF+QKDu8UDqsPnFUzsMgimVMct6fUrcotAP4bZS1+3l3yhofCiUQlMKzBAKrgfCc+XmDahJcrc8YhMbBLaJMM5Ix7Nayr9sZWmBF8KrjLriRN7kLFkMfrPR/WSo7lzHXHrzhmFyQJYZU+48jXcgjj8BvX0SN7hYupuFFNWpzRbXh8/f8wGnzpHU/BmOFY/ekA0zITHNE+LxmWEIpnOrY1SE+5oY0ZWB1HGFPlej4GWa5DgmO43B6SUxIJqkyhlnaJAxXyKTTx9C58RtjRo44h5PS2IKvla/kb1HxPRX/r8orp73cXNYBp3lyyOAAQzwNhx+VuSRyZM/GYOY+WhkwKdAuASkA94SCWr/BNTfVzq20DpaAFMFOcYcdlVtjnjtNDR6pdGyUgAS/r0YUwd2J4/+1UisUN9UqnmkaIzdFHGk8oRZlTCPvpVykhD8i4tib/VIGBiVmIIUJAfdEyrA0N4XrykJRRlYso5B83zXoY+gSwEfJPdv5XthBH0MXZWTlOQpZe7mQ2+KeQbxo+hjaDVgC+Ci55ztL+MQFnooMeKVY+lD4F27H15aoH6ijAAAAAElFTkSuQmCC\n", "text/latex": [ "$\\displaystyle y{\\left(10 \\right)} = \\frac{31}{3}$" ], "text/plain": [ "y(10) = 31/3" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle y{\\left(10 \\right)} = \\frac{31}{3} - 3.33333333324415 \\cdot 10^{-7} e^{30}$" ], "text/plain": [ " 31 30\n", "y(10) = ── - 3.33333333324415e-7⋅ℯ \n", " 3 " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Difference:\n" ] }, { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle 3562158.19374618$" ], "text/plain": [ "3562158.19374618" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import sympy as sym\n", "sym.init_printing()\n", "\n", "t=sym.Symbol('t')\n", "y=sym.Function('y')\n", "\n", "edo=sym.Eq( sym.diff(y(t),t) , 3*y(t)-3*t )\n", "#display(edo)\n", "\n", "solgen=sym.dsolve(edo,y(t))\n", "#display(solgen)\n", "\n", "alpha=sym.Symbol('alpha')\n", "t0,y0=0,alpha\n", "consts= sym.solve( [solgen.rhs.subs(t,t0)-y0 ], dict=True)[0]\n", "solpar=solgen.subs(consts).simplify()\n", "display(solpar)\n", "\n", "sol1=solpar.subs(t,10).subs(alpha,sym.Rational(1,3))\n", "sol2=solpar.subs(t,10).subs(alpha,0.333333)\n", "display(sol1)\n", "display(sol2)\n", "print('Difference:')\n", "display((sol1.rhs-sol2.rhs).evalf())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Si nous cherchons à résoudre le problème de Cauchy jusqu'à $t=10$ avec $\\alpha=1/3$, nous obtenons $y(10)=31/3$. \n", "Par contre, si nous faisons le calcul avec l'approximation $\\alpha=0.333333$ au lieu de $1/3$, nous avons $y(10)=31/3-e^{30}/3000000$ ce qui représente une différence avec la précédente valeur de $e^{30}/3000000\\approx10^7/3$. \n", "\n", "Cet exemple nous apprend qu'**une petite erreur sur la condition initiale (erreur relative d'ordre $10^{-6}$) peut provoquer une très grande erreur sur $y(10)$ (erreur relative d'ordre $10^{6}$)**. Ainsi, si le calculateur mis à notre disposition ne calcul qu'avec $6$ chiffres significatifs (en virgule flottante), alors $\\alpha=1/3$ devient $\\alpha=0.333333$ et il est inutile d'essayer d'inventer une méthode numérique pour calculer $y(10)$. En effet, la seule erreur sur la condition initiale provoque déjà une erreur inadmissible sur la solution. Nous sommes en présence ici d'un problème **numériquement mal posé**" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Exemple de problème numériquement bien posé\n", "Considérons le problème de Cauchy\n", "$$\\begin{cases}\n", "y'(t) = -y(t), &\\forall t>0,\\\\\n", "y(0) = y_0+\\varepsilon.\n", "\\end{cases}$$\n", "La solution est $y(t)=y_0e^{-t}+\\varepsilon e^{-t}$: l'effet de la perturbation $\\varepsilon$ s’atténue lorsque $t\\to+\\infty$ puisque $\\varepsilon e^{-t}\\xrightarrow[t\\to+\\infty]{}0$. Cela suggère que si une erreur est faite dans une étape d'une méthode d’itération, l'effet de cette erreur s’atténue au cours du temps: **le problème est numériquement bien posé.**" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "%matplotlib inline\n", "from matplotlib.pylab import *\n", "\n", "y = lambda t,eps : (1+eps)*exp(-t)\n", "\n", "tt = linspace(0,5,101)\n", "yye = [y(t,0.1) for t in tt]\n", "yy0 = [y(t,0) for t in tt]\n", "#xkcd()\n", "figure(figsize=(10,7))\n", "plot(tt,yye,tt,yy0);\n", "legend([\"$y_0=1.1$\",\"$y_0=1$\"])\n", "axis([0,5,0,1.1]);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Exemple de problème numériquement mal posé\n", "\n", "Considérons maintenant le problème de Cauchy\n", "$$\\begin{cases}\n", "y'(t) = y(t), &\\forall t>0,\\\\\n", "y(0) = y_0+\\varepsilon.\n", "\\end{cases}$$\n", "La solution est $y(t)=y_0e^{t}+\\varepsilon e^{t}$: l'effet de la perturbation $\\varepsilon$ s’amplifie lorsque $t\\to+\\infty$ puisque $\\varepsilon e^{t}\\xrightarrow[t\\to+\\infty]{}+\\infty$. Cela suggère que si une erreur est faite dans une étape d'une méthode d’itération, l'effet de cette erreur s’amplifie au cours du temps: **le problème est numériquement mal posé.**" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "%matplotlib inline\n", "from matplotlib.pylab import *\n", "\n", "y = lambda t,eps : (0+eps)*exp(t)\n", "\n", "tt = linspace(0,5,101)\n", "yye = [y(t,0.01) for t in tt]\n", "yy0 = [y(t,0) for t in tt]\n", "#xkcd()\n", "figure(figsize=(10,7)) \n", "plot(tt,yye,lw='2')\n", "plot(tt,yy0,lw='5');\n", "legend([\"$y_0=0.01$\",\"$y_0=0$\"])\n", "axis([0,5,0,1.1]);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Exemple de problème numériquement bien posé\n", "\n", "Considérons le problème de Cauchy\n", "$$\\begin{cases}\n", "y'(t) = y(t)\\sin(t), &\\forall t>0,\\\\\n", "y(0) = -1+\\varepsilon.\n", "\\end{cases}$$\n", "La solution est $y(t)=-e^{1-\\cos(t)}+\\varepsilon e^{1-\\cos(t)}$: l'effet de la perturbation $\\varepsilon$ reste borné lorsque $t\\to+\\infty$ puisque $|\\varepsilon e^{1-\\cos(t)}|\\le \\varepsilon e^2$. Cela suggère que si une erreur est faite dans une étape d'une méthode d’itération, l'effet de cette erreur reste borné au cours du temps: **le problème est numériquement bien posé.** " ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "image/png": 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d+Ovdk3h4TjR/33Wa+9/ZR01Tq45BCyGu1Jvbsnn0vYMkBrvz+RMzGRt23im7vN3wf7O1vnQ3/gHu/hT8YvUJdjB4RcDdn2jbe8VH4C8ztRW4LiJ8XVj16FWMD/fiux8c4i9bstCjpET0n9TUVGbOnGn99cGDB1mwYIFu8cyZMwcfHx/dfr5UAPankPHw4Ffa7GzrC1B4AG55XbvzCjAaFJ65PokYf1ee/eQYd76+m/cemibXywgxDLyyKYMXN6RzQ0ow/7tinO1JXIsFdv1Zqx3yioCHvoLgcfoFO5gURTvFHLsQ/nMf/PsemP0UzH9WW7kCvFzM/OOhqfzoP0f57fqT5FU18MtlyXITRF+t+7F2O0d/CkqB61646EuSk5PJysqivb0do9HIk08+yUsvvXTR98yePZva2tpu4y+++CILFw7vLW9JpPqb2QVueg3Cp8Da/4bX58KKf9p8qK6YEkGghxMP/+MAK9/cI8mUEEOYqqr84at0/rQpk1smhPK75WNtE4DGavjkEUhfB6OXwdI/X/CKFbvmGQr3rda2NLe9qP0Ff+sb1j8LRwcjf1wxnjBvZ17bkkVzazu/v21cz93exZBmMBhITk7m+PHjZGRkEBERwcSJE6mvr+exxx7DbDYzb948Vq5caX3Ptm3brvjnLFy4kJKSkm7jzz//PMuWLevTP0N/kkRqICiKdr1C0FhtdvbWIlj2KqQst75kXkIAr989SZIpIYYwVVV5Yf1J/ro1m9snh/GbW8ba/sVfUwz/vFW7s+6638HUh4dXS4P+5uCobfMFj4N1T8MbC+COD8A/HgCDQeG/r03ExWzkxQ3pOBgVXrhlbPeeW+LyXGLlaCBNnz6dHTt28Nprr1nrk1atWsXy5ctZsmQJK1assEmkerMitXHjxoEJvp9JIjWQwibDd76Bf98LHz+k3cI+9dvWx5JMCTF0qarKL1an8faOXL41PYJfLB1j+xd+RSb842ZorIKV/9Eu+xVaIjnlIfBP0iaSbyyAlf+GUVdZX/LEgjha2iz8aVMmJqOBX9005sIXOYshafr06dx33308/vjjhIZqBy4KCgpISUkBwGi0bULdmxWp4UI2qAeaqx9862Pt4uO1T2nNPLs4l0xllNWx8s09VDe06BSoEKKrX689wds7crl/ZiS/XHZeElV4AP62CFob4N4vJInqSeRMeHgLeARrq3ZZm20e/9c18TwyN4b39uTx8y/SpAB9mElMTMTR0ZGnn37aOhYWFkZBQQEAFsvgnUy/8847mTFjBqdOnSIsLIy33npr0H42SCI1OExOsOIfkHI7fP0L+Oo5rbFdh67J1L1/20tjS7uOwQoh3t6Rwxvbcrhnxih+euNo29WSrE3wzhLttO6DGyB0on6BDnVe4XDfGq0Z6fsrIP1L6yNFUXj6Wu06nXd25vKbdSclmRpGXn75ZX7zm9/g6upqHbvlllv4+OOPefTRR1myZMmgxfLBBx9QXFxMa2srBQUFPPjgg4P2s0ESqcFjNMHNf4XJD8KOP8Lq/9IuRe4wLyGAV+6cwNHCs/zw34exWOQDRQg9rD9Wwi9Wp7FodCDPLUm2TaJOrIb3bgfvSHhgA/jG6BbnsOEWoBWhByTChyvhxBfWR4qi8JMbk7h7+ihe/yab17/J1jFQcTmysrJITEyksbGRe++91+aZq6srb7/9Nn/5y19s6qPsnSRSg8lg0LoBz/ohHHgbPv+uzcrUouQgnr0+iXXHSvjtlyd1DFSIkelg3hm+/+EhxoV58fIdE2wLy7O3wkf3a4XU96/VtqzE5XHxgXs+h5AJWs1o6kfWR4qi8POlydyQEswL60+yMa1Ux0DFpcTExHDy5MlB3z4byiSRGmyKAgufg7k/hsPvwcaf2Tx+cFYU35oewV+3ZvPB3jx9YhRiBMqtqOehd/cT5OnEW/dOxtncpVi28AB8eBf4xGiF5c5eF/6NRM+cveDuVRAxXTt8k/a59ZHBoPDibeMYE+LJ9z88xMmSGh0DFeLKSCKll3k/1lok7Pgj7P6LdVhRFH62JJm58f785NNjbM+o0DFIIUaGyrpm7nt7LwDv3D8VXzfHzoflp+Cfy7VVlbs/sTbYFb3g6A4rP9JONK/6NuTvtT5yNht5457JuDo68NC7+6msa9YxUCEunyRSelEUuP5FSFoC6/8Hjn1sfeRgNPDKXROI8Xfj0fcOkFHavfeGEKJ/tFtUHn//IMVnm3jjnslE+XUWz1Kdr7U4MDho173Idl7fmV3gzg/BI0QrQK/Msj4K8nTijXsmU17bzCP/PEBzmxy86YkU5Q+c3vzZSiKlJ4MRbnkTImbAqu9A9hbrI3cnE2/dNxlHByPf/vt+auVePiEGxB83prM7u4rnb05h0ijvzgf1FVoS1VynbUlJYXn/cfXTVqYURWuNUN+58j4u3IuXbh/HvtwzPPvJMUkazuPk5ERlZaX8uQwAVVWprKzEycnpit6n6PEvY/Lkyer+/fsH/ecOWY3V8PZ12uz3/jU218nsy61ixV93sWRcCH9cMV6a1gnRj7aml3Pf23tZPjGM39/W5W689lZ4dykUHdRWokbN0C9Ie5a/D969EQLHaP24zC7WR3/4Kp2Xv87gF8uSuWdGpH4xDjHnjvg3NTXpHYpdcnJyIiwsDJPJZDOuKMoBVVUn9/QeSaSGirOF2lUyqFo3dFc/66M/f53BS1+l87vlY7l9crh+MQphR4rPNnLDn7bj7+bIp4/PtC0uX/c07Pk/bcV47G36BTkSnPgC/nU3JN4At/9dW6lHWx148N39bM+s4IsnZpEQ5K5zoGIku1giJVt7Q4VnKNzxnrbE/fFDNj2mHpsfy1Uxvjz32XEyy6ReSoi+amu38L0PDtHU2s6rKyfaJlFH/qUlUdMfkyRqMCQtgWtfgJOrYUvn3XGKovC75WPxcDJZ/10JMRRJIjWUhIyH638H2Zth62+tw0aDwh9XjMfFbOTx9+QDRYi+enFDOvtyz/CbW1KIDXDrfFB8FL74PoyaBdf8Qr8AR5rpj8D4lfDN722ukvFzc+Sl28dxqrSW36w9oWOAQlyYJFJDzcR7YdxdsPV3kNF583WAh5P1A+WXq9N0DFCI4W3zyTL+b2sWd06NYNn40M4HDVXwr2+Bszfc9rZ2G4EYPNf/HvwTtLYItSXW4bnx/jw0K4p3d53m6xPSrFMMPZJIDTWKonU/DxgNqx6C6s6mnPMSAvjO3Gje25PHmqPFOgYpxPBU3dDCjz46SmKQO88tGd35wNKubanXFmv3YroF6BfkSGV2hdvehZb6buUNP7o2geQQD3700VHKaqTIWgwtkkgNRWYXreiyvU27TqGtszHdU4sSmBDhxf+sOkpZrXygCHElfrE6jeqGFl66fRxOpi51Ud/8HrK+hut+pzWLFPoISNQmkrnbbMobHB2M/OnOCTS2tPPDfx+Ru0jFkCKJ1FDlFws3vaYdv97w/6zDJqOBF28bR1ObhZ9/IVt8QlyuzSfLWHWwkEfnxZAc4tn5oPCgtpWecjtMuk+3+ESH8Xdp9VJbf2dTLxXj78ZzS0azPbOCd3bm6hefEOeRRGooG70Upj0Ke1/XLkztEOPvxvevjmPN0WK+kgs+hbik2qZWnvkklbgAN55YENv5oLUJPnkE3AK1gx7Sp21ouEC91Iop4cxP8OelDacoqm7UMUAhOkkiNdRd/VPtotTPn9A6LHd4eE40iUHu/OTTY9L1XIhL+M26k5TWNPG75WNxdOiypbf5V1BxCpb9WSsyF0PDuXqp5lpY8yR09DtUFIVfLBuDRYXnPj+uc5BCaCSRGurMLrDsVa3r+cafWYdNRgMv3DqWstomfrf+lH7xCTHE7cys4P09eTw4K4oJEV2SpdO7YOcrMOl+iF2oX4CiZwGJMP8Zrb9U2mfW4XAfF/7rmji+Sivly+MlF/kNhBgckkgNB6NmwLRHYN8bkLvdOjw+3Iv7Z0bxj92n2Z9bpWOAQgxNDS1tPL3qKJG+LvzwmoTOB8118Okj4BUBi36pX4Di4qY/DsHjYe1TWnuKDvfPjCIp2IPnPjsuK/JCd5JIDRdX/wS8o+Czx7XjwR2eXBRPmLczT398VG5KF+I8L21IJ7+qkd/eOta2e/nG5+DMae1Ah6NcPTJkGR1g2SvQeAa+fNY6bDIa+M0tKZTWNvHShnQdAxRCEqnhw+yqbfGdyYWvOzsuu5gdeP7mFLLK63l1c5Z+8QkxxGSU1vLOzlzunBrBtGjfzgdZm2Hfm9oVMJGz9AtQXJ6gFJj5AzjyPmR+bR0eH+7FPdNH8e6uXI7kV+sXnxjxJJEaTiJnwtTvwJ6/wumd1uG58f7cPCGUv2zJ5HRl/UV+AyFGjl+tOYGL2chTi+I7B1ubYPV/gW+stsorhoc5PwK/ePjiBzaHbp5cnECAuyPPfJJKW7tFxwDFSCaJ1HCz8DnwHgWffxfaWqzDP74uEZPRwG/Xn9QxOCGGhs0ny9iaXs73r47D182x88GuV+BMjna83uSsX4DiypicYOmf4Ww+bOqsafNwMvGzJckcL6rh77tO6xigGMkkkRpuzK5w/UtQmQl7/2odDvRw4pG5MaxNLZHCczGitbZb+OWaNKL9XLlnRmTng7MFsO0lSLwRYhboFp/opYjpMPXb2op8/l7r8LVjgpgd58fLX2dwtkEKz8Xgk0RqOIpbCHGLtc6/deXW4W/PjibIw4lfrjkhVyiIEevvu06TXV7PszckYXbo8hG34SegWmDxr/ULTvTN1T8F92BY9zRYtK08RVF45vokappaeW1Lps4BipFIEqnhavHz0Npgs8ztbDbyo8UJHMmv5oujRToGJ4Q+qupbeHljOrPj/FiQ2OXi4ZxtcHyVVrTsPUq/AEXfOLpr5Q1FB+HYR9bhpGAPbp0Yxts7cyk406BjgGIkkkRquPKL03pLHfw7FB+xDt88IZQxoR78dt1JmlqlHYIYWf73q1PUt7Tz0xtHo5y77qW9TVvB8IyAWT/QN0DRdym3a72lNv4MWjqTpicXxaMAL34pDYrF4JJEajib8yNw8YF1P7ZeoWAwKPy/G0ZTdLaJt7bn6BygEIPnZEkN7+/J41vTIogL7NIbav9bUHZcW8WVAvPhz2DQtmdrCmHXq9bhYE9nHpwVxaeHi0gtOKtjgGKkkURqOHP2ggU/gbydcPwT6/D0aF8WJwfy2uZMymubdQxQiMHz/JoTuDuZ+MHCLu0O6itg8/MQPR+SlugXnOhfkTO1f5/b/2BzqfEj82LwcTXz67UnUFWpExWDQxKp4W7iPRCYAl/9FFo7b0P/8XVJNLdZ+N+vpOuvsH97c6rYllHBE/Nj8XY1dz7Y9EvtJoDrfgfntvqEfbjmF9DeApt+ZR3ycDLx/avj2JVdyZZT5Rd5sxD9RxKp4c5ghOte0Pqr7PyzdTjKz5W7Z4ziX/vyyC6vu8hvIMTw94ev0vFzc+Rb07sUkldmwcF/wJSHwD/+wm8Ww5NPNEz7Dhz6J5SkWofvnBpBpK8Lv1l3Qpp0ikEhiZQ9iJwFSUu1Ze4u7RAemxeL2cHAK5vlSLCwX7uyKtmVXclj82Js79Pb8gI4OMKsH+oXnBhYc34Ezt7w5TPWOlGzg4Gnr00kvbSOjw8W6BygGAkkkbIXV/8U2ppg58vWIX93R741bRSfHS4it0KujhH2R1VV/rAxnQB3R+6aFtH5oDQNUv8DUx8G90D9AhQDy9kL5v0P5HwD6eutw9eOCWJ8uBd/3pRJq6xKiQEmiZS98IuDlNtg75tQW2odfnhuNA4GRValhF3amVXJ3pwqHp8fi5Op62rUr7WeQzO/r19wYnBMvh98YrRDBR2rUoqi8L2rYyk408inhwp1DlDYuz4lUoqi/F5RlJOKohxVFOUTRVG8+isw0Qtzn9aKL3d0rkoFuDuxctooPjlUKBcaC7uiqip/+CqdIA8nVkwJ73xQdAhOfAEzHtfagwj7ZjRpW3wlqXBqnXV4fkIAySEevLYli3a56UEMoL6uSH0FjFFVdSyQDvxP30MSveYbA2NXaH1zuh4J7liVelVWpYQd2ZZRwf7TZ3h8wXmrUZt/rdXNTH9Uv+DE4Eq5DbyjYOsLNqtS310QS05FPQahzbAAACAASURBVKvlpgcxgPqUSKmqukFV1baOX+4GwvoekuiTuT+C9lat8LxDgIcTd06NYNXBQvKr5PoEMfydq40K8XTi9sldPnby9kDGBm1Lz8lTvwDF4DI6wJyntFseMjZYhxeNDiIuwI1XN2fK/aNiwPRnjdQDwLoLPVQU5WFFUfYrirK/vFz6ewwYn2gYfyfsfxtqOmdhj8yNwaDIqpSwD1vSyzmUV80TC+JwdOiyGrXpl+DqrxWZi5Fl7ArwGqWd1uxy08MTC2JJL61jQ1rpJX4DIXrnkomUoigbFUU51sPXsi6veRZoA9670O+jqurrqqpOVlV1sr+/f/9EL3o250egttusSgV5OnHH1HA+OlAgq1JiWFNVlT9uzCDUy5nlk7qsRmVvhdxtMPtJMLvqF6DQh9Gk/bsvOgiZG63DN44NIcrPlT9vypBu52JAXDKRUlV1oaqqY3r4+gxAUZR7gRuBlar8Vzo0eEfC+JVw4B0429lH5dF52qrUa1uydAtNiL7am1PFkfxqHpkXg9mhy0fY5l+DRyhMul+/4IS+xt2pXU7dZVXKaFB4dF4Mx4tqpNu5GBB9PbV3LfA0sFRVVVnmGErmPKV9kGz7X+tQsKczt08J46MD+ZScbdIxOCF6741t2Xi7mFg+8bzaqPzdcNX3wOSkX3BCXw5mmP1fULgfsjZZh2+eEEqolzN/klUpMQD6WiP1CuAOfKUoymFFUf6vH2IS/cErAiZ8Cw7+3eYE33fmxNBuUXl3V65uoQnRW1nldWw8Ucbd00fZdjHf+Sdw8tL+mxcj2/iV4BEGW39rXZUyGQ08Oi+GQ3nV7Myq1DlAYW/6emovVlXVcFVVx3d8PdJfgYl+MPN7YGmDva9bh8J9XFg0Ooj39+TR0NJ2kTcLMfS8tT0Hs4OBu2dEdg5WZMLJNdqdeo5uusUmhggHR5j1A8jfAzlbrcPLJ4UR6OHIa1vkwI3oX9LZ3J75REPiDbD/b9DS2YzzwdlRnG1s5eOD0vFXDB+Vdc18fKCAWyaE4u/u2Plg1ytaofG07+gXnBhaJt4D7sE2B26cTEbumRHJjsxKTpXU6hicsDeSSNm7GU9A4xk48oF1aPIob8aGefL29hzprSKGjX/sPk1zm4WHZkd1DtaVa/9tj7sD3AL0C04MLQ6OMPXbkL0Fyk5Yh++aGoGjg4F3duboF5uwO5JI2buI6RA6CXa9Bhbt8k5FUXhwVhTZFfVsSS/TOUAhLq2ptZ1/7DrN/AR/YgPcOx/se0O7rHvGd/ULTgxNk+4HByfY/RfrkLermZsnhLLqYCFn6lt0DE7YE0mk7J2iaHeOVWXZ3I5+fUowQR5OvLVdZmZi6PvkUCGV9S18e05052BLA+x9A+KvA/94/YITQ5OLj7ZSefRfUN9ZYH7fzEia2yx8sC9Px+CEPZFEaiRIWgae4bDrVeuQyWjgnqtGsSOzkhPFNToGJ8TFWSwqb27LJjnEgxnRvp0PDr8HjVXaoQohejLtUW3F8sDb1qHEIA+uivHlH7tO09pu0TE4YS8kkRoJjA5aIe7p7VB0yDp819QInE1G/iarUmII23yqjKzyer49OxpFUbRBS7s2MQidDBEz9A1QDF0BiRCzAPa9CW2dW3n3z4yi+GwTXx4vucibhbg8kkiNFBPvAbO7zaqUl4uZWyeF8tnhIsprm3UMTogLe2NbNsGeTtwwNrhz8ORqOJMDV31X274W4kKmPwa1xZD2mXVoQWIAET4uvL0jV7+4hN2QRGqkcPLUkqnjn9hcG3P/zCha2i38Y/dpHYMTomfppbXszq7inhmRmIxdPq52vqJdhZS0RLfYxDARczX4xsHuV22ujbn3qkgOnD7D0YJqnQMUw50kUiPJtO+AarFp0Bnj78aCxADe232aptZ2HYMTorv39+RhNhq4fXKX62CKj0DBXpj6HTAYL/xmIQAMBpj+iFbWkL/XOnzb5DBczUZZlRJ9JonUSOI9CpKWwv53oLnOOvzAzCgq61tYf0zqBcTQ0djSzscHC1g8Jghfty4NOPe/rR1rH3+nfsGJ4WXcndqq/O7XrEMeTiZumxzO6qNFlNXI3aOi9ySRGmmmPQLNZ7Utvg5XxfgyyteF9/fKcWAxdKw+WkRtUxsrp0V0DjbXQup/YMyt4OytX3BieDG7wqT74MTnUN35OXfvVZG0WVT+uUc++0TvSSI10kRMB78EOPCOdchgUFgxJZy9OVVklddd+L1CDKL39uQR4+/KtCifzsHU/0BLHUx+QL/AxPA05duAovUe6xDl58q8eH8+3JtHm7RCEL0kidRIoyjazKxwP5Qcsw4vnxSGg0HhQ1mVEkNAWlENh/OruWvaqM6WB6oK+/4GgSlat34hroRXOCRer/Uf69IK4Y6pEZTVNrPlVLmOwYnhTBKpkWjcHWB0hIPvWocC3J24ZnQgHx0ooLlNis6Fvt7fexqzg4FbJ4Z2DhYegNJUmHy/tDwQvTPxPmiohFNrrUMLEgPwc3Pkw335+sUlhjVJpEYiFx8YvQyO/Eu7ZqPDnVMjONPQyobjpToGJ0a6+uY2Pj1UxI0pwXi5mDsf7H8bzG4w9nb9ghPDW8x88AiDg3+3DpmMBpZPCmPzqTJKpehc9IIkUiPVpHu1ovO0T61Ds2L9CPN25gPZ3hM6+vxIEXXNbayc3qXIvPEMHPsYUm4DR/cLv1mIizEYYcK3IGuTTdH5iinhtFtUPjpQcJE3C9EzSaRGqlEzwTe2W9H5HVPC2ZlVSW5FvX6xiRHtvT2nSQh0Z2JEl1N5R/4FbY3atp4QfTFhpfb90HvWoSg/7VDDv/fnY7GoOgUmhitJpEaqc0Xn+Xug7IR1+LbJ4RgNitQLCF0cLajmWGENK6dH2BaZH3hbKzAPHqdvgGL484rQtvgO/VO7s7HDHVPDOV3ZwJ6cKh2DE8ORJFIj2bi7wGiGA51F54EeTixIDOCjA/m0tMlxYDG43t+Th7PJyE0TuhSZ5+2C8pPS8kD0n4n3QE0BZG22Dl03Jhh3Jwf+tU9KG8SVkURqJHP11e4qO/IBtHYWWd41NYKKuhY2npCiczF46prb+OxwEUvGBePhZOp8sP9tcPSE5Fv0C07Yl4TrwcXX5uSyk8nIzRNCWXushLMNrToGJ4YbSaRGuon3QlO11vG3w5x4f0I8naToXAyq9cdKaGxtZ8WU8M7BxjOQ9hmMWwFmF/2CE/bFwVG7NubUWqjr7B+1Yko4LW0WPj1cqGNwYriRRGqki5wNPtE2RedGg8KKKRFsy6ggv6rhwu8Voh+tOljAKF8X2yLz459CezOMv0u/wIR9mnA3WNq0FfkOySGepIR68sHePFRVis7F5ZFEaqQzGLRVqdM7oCLTOrx8chgAnx6SmZkYeEXVjezKruTmCaGdReYAR/+lXWkUPF6/4IR9CkiE8GlaT6kuSdOKKeGcLKkltfCsjsGJ4UQSKQFjV4BigNR/W4dCvZyZHu3DJ4cKZWYmBtynhwtRVbhlQljn4JlcrdB83ArpZC4GxsR7oDID8nZbh5aOD8HJZOCDvXJyWVweSaQEeARD1Bxt9t8labp5QijZFfUcLZCZmRg4qqqy6mAhUyK9ifDtUgd1tCOxT5FO5mKAjL4JzO42nc49nExcPyaY1UeLaGqV67LEpUkiJTQpt2srAAX7rUPXjgnG7GDgE9neEwMotfAsmWV13Nx1NUpV4ciHWg2fV/iF3yxEXzi6QfIy7bBNa6N1+KYJodQ2tbHlVJmOwYnhQhIpoUlaAg5O2qpUB09nEwuTAvjiSBGt7dJTSgyMVQcLMTsYuCEluHOw8ABUZWnbzkIMpJTboaUO0tdbh66K8cXPzcynh4p0DEwMF5JICY2TByRcB8dXQXtnD5WbxodSWd/C9owKHYMT9qq13cLnR4q4JikQT5cuvaOOfKgl9qOX6RecGBkiZ4FbEKR+ZB1yMBq4cWwIm06WcbZRekqJi5NESnQauwIaKrULPTvMSwjAy8Uk23tiQGw9VU5VfQu3TOzSybytRbugOOF6LcEXYiAZjDDmFsjYoPUt63DThFBa2i2sP1asY3BiOJBESnSKuRqcvTuLfAGzg4EbxwazIa2EuuY2HYMT9uiTQ4X4upqZE+/fOZi5ERqrYNwd+gUmRpaU5dDeAie+sA6NC/Mk0tdFtvfEJUkiJTo5mLVrOE6ugeZa6/DNE0JparXw5bESHYMT9uZsQytfnShlybgQTMYuH0VHPgAXP4hZoF9wYmQJmag1Jk79j3VIURSWjQ9ld04lJWebLvJmMdJJIiVsjb0d2hrhxGrr0MQIb8J9nOXaBNGv1qQW09Jm4daJXU7rNZ7Rin5TloPRdOE3C9GfFAVSboOcbVDTuZV304RQVBW+OCKrUuLCJJEStsKngVeETXNORVG4eXwoOzIrKK2RmZnoH58cKiAuwI0xoV3qoI5/qm2xyGk9MdjGLAdU7cBNhyg/V8aFecokUlyUJFLClqJox4Gzt0BtqXX4pgmhWFT4/LDMzETf5Vc1sC/3DDdP7OlKmHgImaBfcGJk8o+H4HE223sAy8aHcryohsyy2gu8UYx0kkiJ7sbeDqpFOznVIdrfjXHhXnJ6T/SLtana9smSsSGdg2cLtCthxt4uV8IIfaTcBkWHoDLLOnTjuGAMClJ0Li5IEinRnX+CNjPr0pwT4ObxIaQV13CqRGZmom/WpBYzLsyTcJ8uV8KkfaZ9T75Fn6CESL4FUGx6SgW4OzEz1o/Pjsi9o6JnkkiJnqXcDsWHoSLDOnTjuBAMCqw5KjMz0Xt5lQ0cLTjLDWODbR+kfQaBKeAbo09gQniGag06U/9jc+/osvGh5Fc1cjDvzEXeLEYqSaREz8Z0rAqkfWod8nNzZFqUL2tSi2VmJnptTce23vVdr4SpKYL8PdLJXOgvZTlUZkDxEevQ4uRAHB0Msr0neiSJlOiZRwiETYG0z22Gr08JIqu8noyyOp0CE8PdmtQixod7EebddVuv47+z5Jv0CUqIc5KWgsFkU3Tu7mRi4ehA1qQW0yb3jorzSCIlLixpKZQchaoc69DiMUEoCqw5KtcmiCuXW1HPscIabuxpWy9gNPjF6ROYEOe4+EDsQjj+ic323g0pwVTVt7A3t0rH4MRQJImUuLDRS7XvXa5NCHB3Ymqkj/XUlRBX4ty23nVdt/VqS7TTerKtJ4aK0cugphCKDlqH5iX442QysC5VbngQtiSREhfmHQlBY+HE+dt7wWSU1ZFRKqf3xJVZm1rMhAgvQr2cOwdPfAGoMFq29cQQEb8YFKPNJNLF7MD8hADWHy/BYpEaUdFJEilxcaOXQsE+ONvZP+raju29tTIzE1cgp6Ke40U13JDSw7aeXwIEJOoTmBDnc/GBqNlaItVle+/aMUGU1zZzQE7viS4kkRIXl9Sx3XKy8+69QA8nJo/ylu09cUXW9nRar64MTu+QbT0x9CQtgcpMKD9lHVqQGIDZQbb3hK0+JVKKovxSUZSjiqIcVhRlg6IoIZd+lxhW/OPBP7GH03vBnCqtJVNO74nLtPpoMZNGeRPSdVvv5Gqti76c1hNDTeKNgGKzvefuZGJOnB/rj0kLGNGprytSv1dVdayqquOB1cBP+yEmMdQkLYW8nVBXbh26boy2qrBOVqXEZcgqr+NEcY3tahRolxT7xmon9oQYStyDtBYw59WIXjcmmKKzTRwpOKtTYGKo6VMipapqTZdfugKSotuj0Uu1VYMu23tBnk5MGuVtPYUlxMWsPXpuWy+oc7C+AnK3a9t6creeGIqSlmgtYM6ctg4tTArEwaDIJFJY9blGSlGU5xVFyQdWIitS9ilwDHhH9Xh672RJLdnlsr0nLm5NajGTR3kT7Nl1W28NqO1yWk8MXUk3at+7TCI9XUzMjPVjrWzviQ6XTKQURdmoKMqxHr6WAaiq+qyqquHAe8ATF/l9HlYUZb+iKPvLy8sv9DIxFCmKtiqV8w00dp5WuW6Mtrqw7pgUXooLyyqv42RJbfdtvbRPtQQ9KEWfwIS4FJ9o7f7HLnVSoH325Vc1cryo5gJvFCPJJRMpVVUXqqo6poevz8576fvArRf5fV5XVXWyqqqT/f39+xq3GGxJy8DSBqfWWYdCvJyZEOElXc7FRW04XgpoR8etGqq0xFy29cRQl7QE8nZDbal1aFFyEEaDwrpj8tkn+n5qr+t9DkuBk30LRwxZoRPBI6zb6b0bUoJJK64ht6Jep8DEUPdVWgljQj1sT+tlbtQS86Ql+gUmxOVIuhFQ4dQa65CPq5lpUT6sSy2R7T3R5xqpFzq2+Y4Ci4Dv90NMYihSFO0vvaxN0NzZ0fzcKsP647K9J7orq23iUH41i0YH2T44tQ5cAyBkoj6BCXG5AkZrW3wnVtsMX5cSTHZFPemlUiM60vX11N6tHdt8Y1VVXaKqauGl3yWGrdFLob0ZMjZYh8K8XUgO8WBjWulF3ihGqq9PlKGqcM3owM7B9lbI/BriF4FBegKLIe7cJDJnKzRWW4cXJweiKMj2npDO5uIKhE8DZx9I/9JmeGFSIAfyzlBR16xTYGKo2nC8hHAfZxKD3DsHT++E5rOQcL1+gQlxJZKWalvRXT77AtydmDLKR7qcC0mkxBUwGCFukbYiZWm3Dl8zOhBVhU0ny3QMTgw1dc1t7MiqZNHoIJSuBeWn1oHREaLn6RWaEFcmZCK4B3drAbN4TBCnSms5XSk1oiOZJFLiysQv1logFOyzDiWHeBDi6cRXsr0nuvgmvZyWNovttp6qQvo6iJ4LZlf9ghPiShgMkHiDViPa2mQdXpgUAMDGEzKJHMkkkRJXJmYBGBwgfb11SFEUFo4OZFtGOY0t7Rd5sxhJvkorxdvFxORR3p2D5afgTC4kXKdbXEL0Svy10NqgdePvMMrXlbgAN6kRHeEkkRJXxtkLImZ0q5O6ZnQgTa0WdmRW6BSYGEpa2y18faKUBYmBOBi7fMykd/Qhi79Wn8CE6K3IWeDgDBnn1YiODmRvbhVnG1p1CkzoTRIpceUSroOyNJv7p6ZF+eLu6CDbewKAfTlV1DS1sSg50PbBqXUQPA48QvQJTIjeMjlrW9LpX2pb1B0WJgXSblHZki7beyOVJFLiyp1bTeiyKmV2MDA3wZ+vT5bSbpEGdSPdhrRSHB0MzI7z6xysr4D8vXJaTwxf8Yuh+rS2Rd1hfLgXfm5mqZMawSSRElfONwZ8Y23qpEDb3quoa+FwfvUF3ihGAlVV+SqtlNlx/riYHTofZGwAVNnWE8NX3CLte5ftPaNBYUFiAFtOldHabtEpMKEnSaRE78RfC7nboLmzq++8hAAcDIps741wx4tqKKxu7Hlbzz1E29oTYjjyDIPAMZC+wWb46qRAapva2JdTpVNgQk+SSIneib8W2lsge4t1yNPZxLRoH75KkwZ1I9lXaaUYFLg6MaBzsK1ZOzoev1guKRbDW9wiyNtl0+V8dpwfZgcDX52QSeRIJImU6J2I6eDo2XkKq8M1SYFkldeTXS73T41UG9JKmTzKB183x87B3G3QUif1UWL4i18Mars2MejgYnZgVqwfG0+UyiXGI5AkUqJ3jCaIvVpb4rZ01gUs7Gi+uFFmZiNSflUDJ4prbJtwApxaDyYXiJqjT2BC9JewKeDsbXPnKGin9/KrGuUS4xFIEinRe/HXQn0ZFB+yDoV5u5AU7CF1UiPU5lPayaWF3bqZr4fo+WBy0ikyIfqJwQix13S7Kutqa5dz+ewbaSSREr0Xdw0ohh6bcx44fYZKucR4xNl0sowoP1ei/Lpc/1KWBmfzIUFO6wk7Eb8YGiqh8KB1KNDDibFhnpJIjUCSSInec/GB8Gnd2iAsGh2IRS4xHnEaW9rZlVXJ/IQA2weZG7XvsdcMflBCDISYBdok8vwu50mBHM6vpqy26QJvFPZIEinRN/GLofgI1BRZh5JDPAj0cGTLqXIdAxODbWdWBc1tFuYn+ts+yNwIAcngEaxPYEL0NxcfCJ/ebTV+YVIgqgqbZRI5okgiJfrmXHPFLoWXiqIwPyGAbzLKpUHdCLL5VBkuZiNTo3w6B5vrIG+3djBBCHsSvwhKjtpMIpOC3Qn1cpYu5yOMJFKib/wTwSMUMr+2GZ6XEEBtUxsHT5/RKTAxmFRVZfPJcmbF+uHoYOx8kLtd6zcmiZSwN3GLte/nTSIXJgWwLaOcptb2C7xR2BtJpETfKIpWL5C9FdrbrMMzY30xGRU2y/beiJBeWkdhdSPzE3uojzK5QMQMfQITYqAEJIFneLcu5/MTA2hqtbA7u1KnwMRgk0RK9F3MAmg+C0WdJ1jcnUxMifRhyylZ4h4Jzh0s6FZonvW11jvKwbGHdwkxjCmK1uU8e4vWub/D9GhfHB0MUiM6gkgiJfouep52guW87b35CQGcLKmlqLpRl7DE4Nl8sozRwR4EeXbpE1WZBVXZECPbesJOxS6E1nrI32sdcjIZuSrGl63pkkiNFJJIib5z8YGQidrqQxfnTm/JzMy+nW1o5UDeme6n9c5doSH1UcJeRc4Cg4PNdTGg1YjmVNSTW1GvU2BiMEkiJfpH7NVQeAAaO4vLY/zdCPN2tna7Fvbpm4xy2i0qC7rVR30N3pHgG6NLXEIMOCcPCJvaQyJ1bhIpn30jgSRSon/EXA2qRSs673CuDcKOzAqa2+QEi73afLIMLxcT48O9OwfbWiDnG23rQwh7FrNA66VXX2EdGuXrSrSfK1tke29EkERK9I/QSeDo2eP2XkNLO/typA2CPWq3qGxJL2duvD9Gg9L5IH+3Vjsi9VHC3sUsAFSt6LyLuQn+7MqqlDYII4AkUqJ/GB0geg5kbtIuqe0wI9oPs4NBtvfs1NGCaqrqW3rY1tsIBhNEzdYnMCEGS8h4cPKCrM02w/MTAmhus7BL2iDYPUmkRP+JuRpqCqAi3TrkbDYyI9pXEik7tflkGQYF5sSdfy3M1xAxHRzd9QlMiMFiMGonl7NsJ5FTo3xwNhnZItfF2D1JpET/OXc667zCy/kJ/mSX13O6Uk6w2JtNp8qYEOGNt6u5c7CmGEqPyWk9MXLELIDaIig/ZR1yMhmZEeMrdVIjgCRSov94RYBvXI/XxYC0QbA3ZTVNHCus6b6tdy6RlvooMVLEzNe+9zCJPF3ZQI60QbBrkkiJ/hWzQLtfrbXJOhTpp51gke09+3Jupn3uqLdV1tfgFghBKTpEJYQOzk0ie+gnBdoWuLBfkkiJ/hV7NbQ1Qt4um+F5CQHsyqqksUVOsNiLb9LL8Xd3ZHSwR+egpV37yyTmau0KDSFGipj52iSyy3Ux4T4uxPhLGwR7J4mU6F+Rs8Bo7r7EnehPc5tc5Gkv2i0q2zIqmB3nh9I1YSo6rDVllfooMdLELOiYRO62GZ6XEMDubJlE2jNJpET/Mrtqp7XOS6SmRvngYjbK9p6dOFpQzdnGVubGn7etl91xBDx63mCHJIS+LnhdjD8tbRZ2ZVdc4I1iuJNESvS/mAXaqa3aEuuQo4OR6dG+fCNL3Hbhm/QKFAVmn9/2IGcrBI4BVz99AhNCL47uED6tx0mks8koh23smCRSov/F9NwGYU6cH7mVDeRVNugQlOhP32SUkxLqiU/XtgetjZC3B6Lm6heYEHqKmQ8lR6GuM2lydDAyM1brpad26TMl7IckUqL/BaWAi5/NvXsAczq2gb7JkJnZcHa2oZVDeWe6b+vl7Yb2ZtnWEyNXzALte7frYgLIr2okW9og2CVJpET/UxSImqNt83SZgUX5uRLq5Szbe8PcjqwKLGpnYmyVs1WrERl1lT6BCaG34PHg7N1tNX5uxxb49gypk7JHkkiJgRE9F2qLoTLTOqQoCnPi/dmZVUlru0XH4ERffJNejrujA+PDvWwfZG+F0Mng6KZPYELo7QLXxUT4uhDp6yKTSDsliZQYGFFztO/nL3HH+1HX3MahvOrBj0n0maqqfJNezsxYP0zGLh8fjWeg+LCWQAsxkkXPh7oSm+tiQDuYsSu7kpY2mUTaG0mkxMDwjgLPCG27p4urYv0wGhSZmQ1TWeV1FJ1t6r6tl7sdVIvURwlxbhKZ843N8Ow4Pxpa2jlw+owOQYmBJImUGBjWOqltYOmcgXk4mZgQ7iUF58PUuSPcc+LPa2+QvRVMLtrWnhAjmXdkj5PIGTG+OBgUtslnn92RREoMnOi50FStHQfuYnacP6mFZ6mqb9EpMNFb32RUEO3vSpi3i+2DnK1akbmDuec3CjFSnJtE5m7Xrkzq4O5kYmKEt0wi7ZAkUmLgWJe4z2+D4IeqwvZMOcEynDS1trMnu7J724OaIqhIl/5RQpwTNadjEplqMzwn3o9jhTVU1jVf4I1iOJJESgwc9yDwS+hWKzA2zAsvF5PUSQ0ze3OqaG6zdK+POtcvLHreYIckxNB0wTqpjjYIMom0K5JIiYEVPRdO74S2zm08o0FhZqwf2zLKpdPvMPJNejlmBwPTo3xtH+RsBRdf7WoYIQR4BINffLdEakyoZ8ckUhIpe9IviZSiKE8piqIqiiIXbAlbUXOhtQEK99sMz4nzo7SmmVOltToFJq7U1vRypkX54Gw2dg6qqrYiFTkbDDIvE8Iqao42iWxvtQ4ZDQqzZBJpd/r8yacoSjhwDZDX93CE3YmcCYqh28zMel2MbO8NC0XVjWSU1THn/EuKKzOhtkj6Rwlxvqg50FoPhQdthufE+VNWK5NIe9IfU8g/AP8NSHotunP2huBx3e7dC/Z0Ji7AjW1yZcKwcO7Idvf6qC3a9+h5gxmOEENf5Gzt+/l1Uh2tQ7bJ9p7d6FMipSjKUqBQVdUjl/HahxVF2a8oyv7yclmFGFGi5kLBPmixvbBzTrw/e3KqaGxpv8AbxVCxLaOCQA9H4gPPu/4le4vWM8c7Spe4hBiyXHy0C9xzep5EShsE+3HJREpRlI2Kohzr4WsZrrkWTQAAIABJREFU8Czw08v5Qaqqvq6q6mRVVSf7+/tf+g3CfkTNAUsr5O2yGZ4d50dLm4U9OZU6BSYuh8WisjOrkpmxfiiK0uVBO+Rug+g5Wu8cIYStqLmQvxdaG22GZ8dpk8imVplE2oNLJlKqqi5UVXXM+V9ANhAFHFEUJRcIAw4qihI0sCGLYSdiBhhM3bb3pkf7YnYwyAmWIS6tuIaq+hZmx513lqT4CDSdhah5usQlxJAXNQfam7VkqovZ8dokcm9OlU6Bif7U6609VVVTVVUNUFU1UlXVSKAAmKiqakm/RSfsg9kFwqd2W+J2MhmZGunD9kxZ4h7KztWxzYw9L5HK3aZ9P9czRwhhK2IGKMZudVLTo3wxGw1y2MZOyHllMTii5kLxUWiwnYHNivMjvbSO0pomnQITl7I9s5zEIHcC3J1sH+Ru13rluAfqE5gQQ52TB4RO7JZIOZuNTInylsM2dqLfEqmOlSn5r0L0LHouoGp/+XYxq2OVY4d0+h2SGlva2Zdzxvrvyaq9DU7vgshZ+gQmxHARNQcKD0CzbbuD2XH+nCqtpeSsTCKHO1mREoMjdBKYXLvNzEYHe+Djama7zMyGpL25VbS0W5h1fn1UyRFoqZVESohLiZoDars28ejiXM2hXBcz/EkiJQaH0QQR0+D0Dpthg0HhqhhftmdWSKffIWh7Rjlmo4Fp518Lc25lcZQkUkJcVPg0MJq71YgmBXng62qW1Xg7IImUGDyRs6AsDeptPzhmx/lRVttMemmdToGJC9mWUcGkUd6218KA1EcJcblMzloydd5qvMGgcFWsn0wi7YAkUmLwnOv0e96q1KyOa0e2SYO6IaW8tpmTJbXdt/Ws9VGz9QlMiOEmag6UpHY7bDM71o9yuS5m2JNESgyekAlgcoFc20Qq1MuZaD9XqRUYYs5tOXS7X69Y6qOEuCKRswBVu8S4i5nn6qSkRnRYk0RKDB6jSVviPu/kHmhtEPZkV9HSZtEhMNGTbRkVeLuYSA7xsH1wrn+UJFJCXJ7QSeDg1O2zTyaR9kESKTG4ImdB2XGot70WZmasH42t7RzMO6NTYKIrVVXZnlnOVbF+GAznXf+Sux38EsAtQJ/ghBhuHBwhbAqclkmkPZJESgyuc6sYebZL3DNifDEaFFniHiIyy+oorWlmdk/9o/Kkf5QQVyxyFpQcg0bbyaJMIoc/SaTE4AqZCA7O3Za4PZxMjAvzZJsscQ8J5zoudys0Lz4CLXWSSAlxpax1Urb9pGQSOfxJIiUGl4NZ6yfVY52UP6kF1ZxtaNUhMNHV9swKovxcCfN2sX0g9VFC9E7oZDA6dju1fG4SKXVSw5ckUmLwRc6C0mPdjwLH+WFRYVe2fKDoqaXNwu7syu7XwoDURwnRWyYnrU6qp0lkrB9HZRI5bEkiJQbfuW7Y5x0FHh/uhavZKBd56uxg3hkaWtp77h8l9VFC9F7kTCg5Ck1nbYZnxfnLJHIYk0RKDL7QnuukTEYD06N9ZYlbZzsyKzAaFGbEnHctTPFhqY8Soi9GzQTVAnm7bYYnRGiTSPnsG54kkRKDz8ERwqdesJ/U6coG8qsadAhMgFYfNS7MEw8nk+0Da32UdDQXolfCpmj37vUwiZwW7SsF58OUJFJCH/+/vTuPj/Oq7z3+OaN9sWTJWixZtiRb8r4mXmXLcewsxEkIIQQCoUChDS2F9PbS9kLbW9q+SuFSoKUtOw1JW5ZACGvi7I73eEu874sW27K12JJl7dKc+8cjObE1kkbLzDPL9/166SWi55mZXyz85HfO+Z3fGaROCtDynkua2rrYX904cH1U9kxIze5/TUSGFp/sNOe8qeAcnDqpCg0iw5ISKXFH31bgqhu3Ak/LTmViWiJbT+ncPTe8caYBr3V629ygp8tZjtCynsjoFK6EC/ug48bz9foGkdu0vBd2lEiJOwY4MsEYw6rSLLafbsDr1YnowbbtVD3J8TEsmpJx4wX1jxIZG0WrwPZA1c4bflySk0puWoJ66YUhJVLijr4jE/rqbt5hVUkWja1dHL5w1YXAotvWk/UsK84kPvamR0Nfwlu4MvhBiUSSyUvBE9vvuBhjDCtLsth+ql6DyDCjRErcU1Tu88iEshJnt5h2sATX+cY2ztS39F/WA6emI2u6+keJjFZ8inPCwwD9pK60dnGkRoPIcKJEStwzwJEJOeMSmTlxnOqkgqyvNqO89KZicm+PUx9VWOZCVCIRqGgVXHgLOltu+HHfJg8NIsOLEilxz6RbnSMTfIzMVpZksbviCu1dPS4EFp22naonKzWB6bmpN164dAg6rmpZT2SsFK0EbzdU31gnlZOWyPTcVLVBCDNKpMQ9fUcmDLAVuLPby54KnYgeDF6vZdupelaVTMAYc+PFvg70mpESGRuTl4OJGWB5L5tdFZc1iAwjSqTEXYVlvUcm3FgTsLQ4k7gYwxYt7wXF8UvN1F/rZNXNy3rgJLrjCyG9IPiBiUSihFTIXwQVPgaRpRPo7Payt1KDyHChRErcVVjmHJlQveuGH6ckxLJoSoZ6qgRJ35/zypKbjoWx1pmR0rKeyNgqWgnn90LnjQ04lxVPINZj1JQ4jCiREndd3wrse3nv8IWrXG7pdCGw6LLlZD3TslPIS0+68ULdcWht0LKeyFgrXAXeLjjXfxB5iwaRYUWJlLgrPgXyFr5dh/MOq0qzsBa2n9YDJZA6unvYdfZy/9168HaCW6QZKZExNWUZYPrtWgZns82hC01c0SAyLCiREvcVljlT3F1tN/x4/qR0xiXEamQWYG9VNdLW1TNA/6jtMC4PMoqDH5hIJEtMh4nzoGqwQWSDC4HJcCmREvcVruyd4t5zw49jYzwsnzZBPVUCbNupemI8hmVTM2+8cL0+qgxu3sknIqNXuBKqd0P3jTNPCwqcQaSefeFBiZS47/oUd/+RWXlpFtWX26hsaOn/OhkTW07Ws3DyeNIS4268cKUCmi+oPkokUArLoLsNavbd8OO3B5HatRwOlEiJ+5IyIHeuzynuler0G1BNbV0cONc48LEwoB17IoHSN0gZYLNN9eU2qhpa+12T0KJESkJDYZnTAqGn64YfT81KIS89UXVSAfLGmQa89u2jKW5QuR2SJ0D2zOAHJhINUrKcMywHKDgH1EsvDCiRktBQWAZdrVCz/4YfG2NYVZLFtlMN9OhE9DG39WQ9yfExLJw8vv/Fym0wZYXqo0QCqbDMOcvSe2Mnc6cdiQaR4UCJlISGwaa4S7Noauvi8IWmIAcV+baeqmf51AnEx970KGg679RIaVlPJLAKV0JHE1w6fMOPjTGsLMli+2kNIkOdEikJDak5MKHUZ8F52TTVSQXCuSutnK1v8b2sV9W71KBCc5HAuj6I9L3ZprFVg8hQp0RKQkdhmVMrcNMUd/a4BGZOHKcT0cdY359neamPRKpiKySkOX1uRCRw0gsgfYrP2XgNIsODEikJHX1T3LVH+l0qL81iT8UV2jp1IvpY2XKynty0BEpyUvtfrNwOU5aDJyb4gYlEm8IyZxbY3riEp0FkeFAiJaFjkCnuVaXZdPZ42XlWnX7HQo/Xsu10PatKsjE3F5Nfq4P641rWEwmWwjJoqYOGU/0urSrRIDLUKZGS0DF+8oBT3EuLMomP8WhkNkYOX2iisbXL97Le9fooFZqLBEXf3zUfz76VpVl09njZXXE5yEGJv5RISWgpLHNmpG6a4k6Kj2FJcYZqBcbIlt6EdMDz9WKTnMOkRSTwJkyDlByfs/HLinsHkXr2hSwlUhJaBp3izubYxWZqr7a7EFhk2XqynpkTx5E9LqH/xcptULAYYuODH5hINDIGClf4bMyZHB/LrYUZbD6hxpyhSomUhJZBprj7lqE0Mhudts4e9lZe8b2s194Elw5pWU8k2ApXQlMVNFb1u1Q+PcsZRDZrEBmKlEhJaOmb4q7on0jNzktjQkq86qRGaefZBjp7vKwqze5/sXoXWK8zOhaR4Lm+2ab/rNTq3r+r6nIempRISWjpm+Ku6v8w8XgMZSVZbDlVj7Xq9DtSW0/WEx/jYWlRZv+LldvBEwsFS4IfmEg0y5kNiek+Z+Nn56WRmRLPlhNKpEKREikJPVPKoKna9xR3aRZ1zR0cv9TsQmCRYeupehYXZZAU76NHVOV2p8g8PiX4gYlEM08MTF4+4CBypQaRIWtUiZQx5u+MMeeNMft6v9aPVWASxQaZ4r5eJ6XlvRGpbW7n2MVmVvmqj+pqgwtvqn+UiFsKy6D+hNPL7SYaRIausZiR+hdr7cLer+fH4P0k2uXOcY4nqeq/FTgvPYlp2SlsViI1In01FuUlPuqjzu+Fnk4lUiJu8WOzjZb3Qo+W9iT0eGKc40l89FQBKC/NZtfZBtq71Ol3uLacqCcjOY45+Wn9L1buAIzzZy8iwZe3wOnh5mN5Ly89iZKcVDafVBuEUDMWidSnjTEHjDFPGGMyBrrJGPOYMWaPMWZPXZ3+jyBDmLJi0Cnu9i4veyuvuBBY+LLWsvVUPWUlWXg8pv8NlducgtekAf8ai0ggxcbD5CWDDCKz2HX2sgaRIWbIRMoY84ox5pCPrweAbwPTgIVADfC1gd7HWvs9a+1ia+3i7Gwfywoi79Q3xe1jZLZs6gRiPeZ6d27xz4lL16ht7qDcVzfznm6n9YHaHoi4q3AlXDzo9HS7yerSbDq6veyp0CAylAyZSFlr77DWzvXx9Wtr7SVrbY+11gt8H1ga+JAlKuQvgthEn4lUakIstxRmsPWUZjaHY0vvkoDPQvOL+6GrRfVRIm6bsgKwzsDmJsumZhIXY67/XZbQMNpde3nv+McHgUOjC0ekV2y808vIR9ElQHlJFofOX6XhWkeQAwtfW07WMzUrhYKM5P4X+3ZITlEiJeKqgiVOLzcfz76+42I0Gx9aRlsj9RVjzEFjzAHgduDPxiAmEceUFb1T3Ff7XeqbVdl2uiHYUYWl9q4e3jjTwOrpAyyrV26HjGJIy/N9XUSCIz7ZmZH30f4FnM02R2quUtesQWSoGFUiZa39PWvtPGvtfGvtu621NWMVmAiFZc5xJT6muOcXjCctMZatmuL2y66zl+no9nLbDB+JlNfrLKFqWU8kNBSWOe1Iutr6Xeprg6DjYkKH2h9I6CpYAibGZz+pGI9hVWkWm0+o068/Np2oIz7Ww/LiCf0v1h+HtstKpERCxZQy8HY5ydRN5uSnk5Ecp+W9EKJESkJXQirkLxxwivu26dlcvNrOiUvXghxY+Nl0oo5lxZkDHwsDvUWuIuK6KcsA47MNQkzfcTEn6zSIDBFKpCS0TVkB5/dAV3u/S331PptO1AY7qrByvrGNU7XXuG2w+qjUiZA5NbiBiYhvSRnOCQ8DbLZZXZpNbXOHBpEhQomUhLbClc6xJRfe7HcpLz2JGbnjeP246qQGs/mE8+fjs9DcWieRKlwBxkeTThFxR2EZVO+Gnq5+l/o22/T93RZ3KZGS0NZ3XMkAI7M1M7LZXXGZlo7uIAYVXjYdryMvPZHSnNT+FxsrofmC2h6IhJrCMqe3W82BfpfyxycxPTeV1zUbHxKUSEloS850ji0ZpE6qq8eyQ20QfOrq8bLtVD23Tc/G+Jpx6vtzVaG5SGjpG9z42GwDsGZGDrvPXtEgMgQokZLQN2UFVO90jjG5ya1FGSTHx2hkNoB91Y00d3QPUh+1DRLTnWRVRELHuFzInDbguXtrpmfT2eNluwaRrlMiJaGvsAw6r8Glg/0uJcTGUDYti9ePaweLL5uO1xHjMZT5Ol8PnIf0lDLw6FEgEnIKVzg93rzefpcWF2WSEh/D68c1iHSbnp4S+vqWnQYYmd02I5tzV9o4W98SxKDCw+aTdSyaPJ70pLj+F5svwuXTWtYTCVWFK6HtCtQd63cpPtZDWYkGkaFAiZSEvrR8yCgadIob0O69m9Rf6+DAuabB2x6A87AWkdDT19ttkM025xvbOF2nNghuUiIl4aFwpfMffh8jr8mZyUzNTmGTtgLfYGtv5+NBz9eLS4G8+UGMSkT8llEE4/Kd5T0f1szIATSIdJsSKQkPhWXOMSY+prjB2b33xpkG2rt6ghxY6Np8oo7MlHjmTUr3fUPldpi8FGJ8LPuJiPuMceqkBhhEThqfRGlOKhtVJ+UqJVISHvqWnwac4s6ho9vLG2e0gwXA67VsPllHeWkWHo+Ptgetl6H2MBRpWU8kpBWWQXMNXKnweXnNjGy1QXCZEikJD31T3BW+E6llxZkkxHq0vNfrSM1V6q91Dlwf1bdUoPookdA2xCDy9hk5aoPgMiVSEh6McWZPBpjiToyLYfnUCWxSrQDA9YSyvHSQ+qiYBMi/JYhRiciwZc2ApMwBN9uoDYL7lEhJ+Cgsg2sX4fIZn5fXzMjmTH0LVQ2tQQ4s9Lx+vJY5+Wlkj0vwfUPlNihYDHGJwQ1MRIbH43GefRVbfV5WGwT3KZGS8FG4yvk+wBR33zLWppPRPSt1paWTvZVXWDczx/cNHc1Qs1/9o0TCRdEq51zMpnM+L6sNgruUSEn4yCqF5KwBp7iLs1KYnJnEpiif4t54vBavhXWzcn3fUL0TrFeJlEi4GKIpsdoguEuJlIQPY3qnuH3PSBljWDM9h22norsNwqtHa8kelzB42wNPLExeFtzARGRkcudCQvqAy3t9bRCUSLlDiZSEl6JV0FQFjVU+L98xO5e2rh52ROkOls5uL5tP1LFuZo7vtgfgJFJ5CyE+JbjBicjIeGJ6+0n5HkSCs7y36+xltUFwgRIpCS/Xp7h9d/pdPtXZwfLy0UtBDCp07K64THNHN2sHqo/qaoPze7WsJxJuCldCwynnjEwf1vS2QYjWQaSblEhJeMmZA4npUOl7ijshNobbZmTz6tFLeL3Rt4PllaOXiI/1sKo0y/cN5/ZAT6f6R4mEm+v9pHzXSS0pyiQ1IZZXj0XnINJNSqQkvHg8MKVswIcJwB2zcrl0tYNDF5qCGJj7rLW8erSWldMmkBwf6/umyu2AgSmqjxIJK3kLID51wOW9+FgPt03P5pWjtVE5iHSTEikJP0WDT3HfPiMHj4FXjkTXyOx03TWqLrcOvFsPnIdw7lxIygheYCIyejG9G0QG2GwDcOfsXOqaO9h3rjGIgYkSKQk/Q2wFzkiJZ3FRJi8fja42CK/0/vsOWB/V3QnVu3S+nki4KloJdUehxXcd1O0zcoj1GF6OskGk25RISfiZOPgUN8Ads3I4WnOVc1eip8v5q0cvMTsvjfzxSb5vqNkP3W0qNBcJV311UlW+B5HpyXEsm5qpRCrIlEhJ+Omb4h6iTgrgtWPRMSvV1838jlkDzEbB24nnFCVSImEp/xaITRp8eW9WLqdqr3FGXc6DRomUhKfCMqg9Aq2XfV6emp3K1OyUqBmZvX5iiG7m4DTzy5oOqQMcZCwioS02HiYvGXDXMji99ICoefaFAiVSEp6K+s7dG3hW6s5ZubxxpoHm9q4gBeWeV4bqZt7TBVU7oKg8uIGJyNgqXAUXD0Gb74LygoxkZuelRUUi5exUvuT6SRZKpCQ85S+C2MTBl/dm59LVY9l8oj6IgQVfZ7eXzcfrWDtjkG7mNfuh89rbCaiIhKfCMsBC1RsD3nLXnFz2Vl2h/lpH8OJywcHzTXziqT38dv8FV+NQIiXhKTYBCgaf4r5lSgYZyXG8EuFdzvu6ma8brD7q7Gbnu2akRMJbwWKIiR/02Xfn7FyshdcifOfy8wcvEusx3Dl7kJKGIFAiJeGrqBxqDkDbFZ+XYzyGtTNzee1YLd093iAHFzyvHq0dvJs5OPVR2TNVHyUS7uKSYNLiQQvOZ+elMWl8Ei8d8d1rLxJYa9lwqIYV0yYwPjne1ViUSEn4Ki4H7OB1UrNzaGrrYk+l72Qr3FlrefHwRVaVZA3czbyny1kG0GyUSGQoWuks13c0+7xsjDNLs+VkPa2dkXmI8ZGaq1Q2tLJ+Xp7boSiRkjA26VZnK/DZLQPeUl6aTXyMJ2K7nO8/18T5xjbumTtx4JsuvAVdLaqPEokUhWVge6B654C33Dk7l45uL1tORmaN6AuHLuIxcJfLy3qgRErCWWyCc2ZcxcCJVEpCLGUlE3j56CWsjbzzp54/WENcjOGu2YMkUtfro5RIiUSEycvAE+cs2Q9gaXEmaYmxEbl7z1rLcwdrWFY8gQmpCW6Ho0RKwlxROVw6BC0Dj7runjORyoZWjtRcDWJggWet5bkDNawqySI9OW7gGyu2Qs5sSBmkhkpEwkd8ilN03jdI8iEuxsPamTm8evRSxNWInqy9xpm6FtbPG2QAGURKpCS8Fa92vg8yMnvXnInEegy/3V8TpKCCo29Z7975+QPf1N3pTP+rPkokshSvdpbt25sGvOXO2RO50trF3girEd1w8CLGOIPkUKBESsJb/iKISxl0eS8jJZ6VJVn87sCFiFree+7ABeJihtj6e+FN6GrVsp5IpCkqB+sddLPNbTOcGtEXDkfW7r0Nh2pYXJhBTlqi26EASqQk3MXEQeGKQQvOAe5fkM+5K23sq/bdDTjcWGt5/uBFykuzSU8aZFmv789FiZRIZClY4jQlHmR5LzUhljUzsnnuQA093sgYRJ6pu8axi83cM9f93Xp9lEhJ+Csqh/rj0DxwUeVdc3KJj/HwuwORsby3r7qR841tQ2/9rdgCuXMhOTM4gYlIcMQlOkXnQwwi370wn9rmDnaeaQhSYIG14ZAzu/auwXYqB5kSKQl/xb31P4Ms76UlxnFb78jMGwEjs77deoMu63V3qD5KJJIVr4ZLB6Fl4CRp3cxcUuJj+I3Lx6iMlQ2Halg4eTz545PcDuU6JVIS/iYugIS0QRMpgPvm53HxanvYN+f0e1nv/F7obteynkikur7ZZuBnX1J8DHfPmcjzB2vo6Hb3cN/Rqmpo5dD5qyGzW6+PEikJfzGxULhyyCnuO2blkhjncf2Ay9HqW9a7d6hlvbNbANN7yKmIRJz8RRCfOmidFMD9C/O52t4d9ge4v3DYKc0IpfooUCIlkaK4HC6fhqsDJ0kpCbGsm5nLhkM1Yd1X5bkDzrLeHUN19K3YAhNVHyUSsWLinIHSELPxq0qyyEyJ59f7zgcpsMB4/uBF5k5KY3Jmstuh3GDUiZQx5jPGmOPGmMPGmK+MRVAiw9ZXBzTk7r086q918saZy0EIauw5B3VeZPVQy3pd7VC9C4pWBy84EQm+4tVQfwKuDryRJi7Gw/p5E3nl6CVaOsLz7L3zjc6u61CbjYJRJlLGmNuBB4D51to5wFfHJCqR4cqdC0kZQ05xr5mRQ0p8DL87EJ7Le37v1ju/B3o6VB8lEumKht5sA/DAwkm0d3nD9siYX73lzKYNWdLggtHOSP0x8GVrbQeAtbZ29CGJjIDH49RJVQyeSCXGxXDn7Fw2HLpIZ3f4Le89d6CG+BjP0Mt6qo8SiQ4T50HieDi7adDbbp2SwaTxSWG5vGet5ed7qllalElRVorb4fQz2kRqOlBujNlpjNlkjFkyFkGJjEjxamisgiuVg952/4J8mtq62HYqvAovu3u8/O5ADaunZw2+rAfOzFzefEgaH5zgRMQdnhhn5nmIsgaPx3Dfgjy2nKzncktnkIIbG3sqr1DR0MrDiwvcDsWnIRMpY8wrxphDPr4eAGKBDGA58BfAz4wxZoD3ecwYs8cYs6eurm5M/yVEAL+nuMtLs0lLjA273XubT9Zx8Wo777t18uA3djTDuV0w9fbgBCYi7iq+DRor4UrFoLc9sGAS3V7L8wfDqzHxz/dUkxwfM3RJg0uGTKSstXdYa+f6+Po1cA541jp2AV7A5xHz1trvWWsXW2sXZ2dnj+2/hQhAzixIzhpyZBYf6+Fdcyfy0pFLtHWGT1+Vn+yqJis1nnWzcga/sWIbeLthmhIpkahQ7N9mm1l54yjJSQ2r5pytnd08d6CGe+flkZIQ63Y4Po12ae9XwFoAY8x0IB4Ir/USiRzG9E5xb4YhDid+6JYCrnV0h03Ree3Vdl47VstDtxYQFzPEX9szG50zuCYvD05wIuKu7JmQkj3kZhtjDA8syGfX2ctcaGwLUnCj8/zBi7R09vDw4iFm4l002kTqCWCqMeYQ8FPgo9YO8V8wkUCadjs0X4C644PetrQ4k5KcVH60sypIgY3OM2+eo8dr+YA/D5PTG50i87jQOBldRALMGKdGtGLLkIPIdy/MBwib0oaf7ammaEIyS4oy3A5lQKNKpKy1ndbaD/cu9d1irX1trAITGZFpa53vpwf/v6Ixhg8tncK+6kYOX2gKQmAjZ63l6d3VLCvOZGp26uA3N513DnDu+3MQkehQvBqaa6Dh1KC3FU5I4dbCDH66uzrkzx2tbGhh19nLvO/WAgYovw4J6mwukWX8FJhQOmQiBc7yXkKshx+H+KzUjjMNVDa08shSP2ajzmx0vqvQXCS69G22OfP6kLd+ZEUhZ+tb2BriO5ef2XsOY+ChW0Nzt14fJVISeaathYqt0N0x6G3pyXHcNz+fX711nmsh3O336d3VjEuM9a+j7+mNkJIDuXMCH5iIhI7Mqc5A8vTGIW9919yJZKXG8187KgIe1kj1eC2/2HuO8tJs8tKT3A5nUEqkJPJMWwvdbVD1xpC3Prp8Ci2dPfxmX2jWCzS2drLh0EUeXDSJxLiYwW/2ep3R6NQ1Ts2EiEQPY2DaOqfgvKdr0FsTYmN4ZMkUXj1WS/Xl1iAFODzbT9dzoamdh0N8NgqUSEkkKloFnji/lvcWTR7PrLw0frSzklDcJ/Grt87T2e3lA0v8WNa7dAha69X2QCRalayDzmbnnM0hfGjZFAyE7Iabn+85R1piLHcOdYpDCFAiJZEnIRUmL4PTrw55qzGGDy2bwuELV9l/LrSKzq21/HR3NfML0pmTnz70C67XR60JZFgiEqqKV4OJgVOvDHnD+XFCAAAY5UlEQVRr/vgk7po9kad3V9HeFVr99Jpau3jh8EUeWOjHTHwIUCIlkWna7XDxIFwb+vjH9yzMJzk+hh/vHPxomWDbf66JYxeb/ZuNAqc2InsmpOUHNjARCU2J6X4PIsEpOr/S2sXvDoRWp/Of7amms9vL+0O4d9Q7KZGSyFSyzvnuxw6WcYlxPLBwEr/Zf4GmtsFrC4Lp6d1VJMXF8O4FfiRGXe1QtUO79USiXclaqNkP14Y+im3FtAmU5KSGVNF5R3cPP9h6huVTM5lX4MdMfAhQIiWRaeICSMr0q04K4NFlU2jv8vLLN88FODD/NFzr4FdvXeC++XmMSxzigGJwkqjudtVHiUS7aX2DyKF37xlj+MiKQg6ca2JfdWOAA/PPs2+e59LVDj61psTtUPymREoik8fjJBWnXxuy0y/A3EnpLChI58e7qkKi6Pz7W87S3t3DJ2+b5t8Lzmx0CuwLVwY2MBEJbXkLIXmCX3VSAA8umkRKfExIzEr1eC3f3XSauZPSKC/1eWxvSFIiJZFr2lq4dglqj/h1+6PLCzlx6Robjw9dVxVIl1s6+a8dFdw3P5+SnCE6mfc5vREmL3UK7UUkenk8zrPv9GtOS5QhjEuM4723FPC7/TU0XBu8916gbThUQ0VDK59aU+JfJ/P2JnjyPqdvoIuUSEnk6qsXOuVf4eWDiyYxJTOZr754wtWjE36w5QxtXT08vtbPqe2Werh4QMt6IuKYtg5a6uDSQb9u/8iKQjp7vPx0d3WAAxuYtZZvbTzN1KwU7p4z0b8XnX7NOV/QuLuzT4mURK70Sc4uNj/rpOJiPPzZnaUcqbnKhkMXAxycb42tnTy1vYL18/IozR3n34v6Cuqn6nw9EeHtszb9XN4rzR1HeWkW/7n1LFfb3dlws+lEHUdqrvJHt00jxuNnQ+ETL0HieChYEtjghqBESiLbtLVQuR262vy6/d0LJlGak8rXXz5Od8/Q0+Jj7T+3nqWls4fH15b6/6IzG52HSf7CwAUmIuFjXC5MnAen/BtEAvzl3TO53NLJd14/HcDABvat10+Tl57IexZN8u8FXi+cfAlK74SY2MAGNwQlUhLZpq2Dng4nmfJDjMfw2bumc7quhV++dT7Awd2oqbWLJ7dVcM/cicyY6OdslNcLJ1+BqbeBJ/Qb14lIkExbB9VvQEezX7fPK0jnPQvz+c+tZ6lp8m/gOVb2Vl5m19nL/EH5VOJj/UxLLrzpnORQendgg/ODEimJbIVlEBPv9/IewN1zJjJvUjrfePUknd3Bm5V6YttZmju6eXzdMGajavbBtYsw/Z7ABSYi4adkHXi74ewWv1/y2btmYC18/aUTAQysv29tPE1GchwfXDqMBpwnXgTjebtnoIuUSElki0+GKSuGlUgZ48xKnbvSxtO7g3MOVVNbF09sO8vdc3KZlZfm/wtPvOA8TErvClxwIhJ+Ji+HuBS/66QAJmcm87GVRTzz5jmO1lwNYHBvO3LhKq8eq+VjZcUkxw9jie7ki1CwFJIzAxecn5RISeQrucNpgdDof1J02/RslhRl8O+vnaKtM/DnUD21vYLm9m4+M5zaKIDjG5yHScqEwAQmIuEpNt45e8/P42L6/MmaEtIS4/jyhmMBCuxtPV7L3/zqIOlJcXy0rND/F16tcbq3T3d/WQ+USEk0mLHe+X58g98vMcbw53fNoLa5g/9+oyIwcfWqaWrj+1vOcMesXOZOGsaRCE3nnbYHM94VuOBEJHyVrIMrFdDgfwF5enIcn769hE0n6th6sj5wsQE/3HaWN6sa+bt3z2Z8crz/Lzz5kvNdiZRIkGSVQNZ0OPbcsF62bOoEykuz+Pbrp2lqDcyWYK/X8tmf7afHa/nre2cN78UnXnC+qz5KRHy53gZheLNSHykrpCAjiS9tOBqwnnoV9S189aXjrJ2Zw3sW+rlTr8/JlyCtAHJmByS24VIiJdFhxnqo3AZtwztP6nP3zORaRzef/fm+gDxQfrD1DNtPN/CF+2dTnJUyvBefeAEyiiB7xpjHJSIRYMI0mFACJ/yfjQdIiI3hL+6eweELV/n1/rHfvez1Wv7PLw4Q5/HwTw/O86+LeZ/uDuckh+l3w3BeF0BKpCQ6zFjv7GAZRuElwJz8dP56/SxeOVrLdzefGdOQDl9o4p9fPM7dc3J5/+Jh7FYB6GyBM5uc2agQeZiISAiaeS+c3TzsQeT98/OZX5DOP/z2CJUNLWMa0o92VrLz7GX+5r5ZTExPHN6LK7ZCV0vILOuBEimJFgWLISV72Mt7AB8tK+Le+Xn884vH2HG6YUzCaevs4U9/uo+M5Hi+/N75wxuRgdPNvKdD9VEiMriZ9zmDyL66Ij95PIZ/e2QRFvjEU3vGrON59eVWvrThGOWlWcMfQILz7xGbCEXlYxLPWFAiJdHBEwPT3+XMSHV3Duulxhj+30PzKcpK4TM/eYvaq+2jDudLG45yqvYaX3v/AjJShlFk2ef4BkhIgyllo45FRCLYpMWQOhGO/W7YLy3KSuHbj95KRX0Ln/7xW6M+7cFay1/98iAG+NJ7h7mk57yBU9JQvNppbRMilEhJ9JixHjquQuXwTwpPTYjl24/eyrWOLj7zk9E9UDYeq+W/dlTyiVXFlJdmD/8NvF6nGV3JOmeLs4jIQDwemLneOQHBz6Oy3mnFtAn843vmsvlEHV98/uiIw7DW8o1XT7LlZD2fWz+LgowRJEL1J51diCG0rAdKpCSaTF0DsUlw7PkRvXzGxHH804Pz2Hn2Ml97eWSdf3edvcxnf76fmRPH8Rd3j7BI/MJb0FKr3Xoi4p+Z9zp1RWc2jejljyydwh+sKuaH2yr40c7KYb++q8fL535xkH995SQPLprEo0unjCgOTr7ofA+BY2HeSYmURI/4ZGc78PENzhTxCLz3lgI+uHQK3379NH//28O0dHT79Tqv1/LNjad45Hs7SE+K45uP3kJi3AjPxjuxobeb+Z0je72IRJei1ZCQDsd+O+K3+Pz6Wdw+I5sv/PrwsPpLXW3v4uNP7ubpPdU8vraEr79/AR7PCDfInHjRaXkwfgS1VQGkREqiy8z1cPWc08hyhL5w/2w+sqKQJ7dXcNe/bGbjsdpB72+41sHHntzNP794nPXz8vjNp1cyLTt1xJ/P8Rec4x9C4GgEEQkDsfEw/S5nENnj3+DvZjEew799cBHTslP5yBM7+d9P76OqoXXQ15xvbOPhb+9gx+kGvvK++fzvu2YMvy6qT3sTVO0IyeOwlEhJdCm9GzAjXt4DSIyL4R8emMszf7SC5PgYfv/J3Xz6x29S19xx/Z7uHi9XWjrZcrKO9f+2hTfONPDFB+fy7x9cxLjEuJHH31gNlw5qt56IDM/Me6G1Aap3jvgtxiXG8fQnl/OH5VN57mANa7/2On/1y4PUNL1de9XU2sXOMw08tb2C93xzGxca23jq40tHtkPvnU686Ow+nBF6JQ3DOCFQJAKkZsPkZXD8ebj986N6q1sLM3nu8XK+s+k0//HaKTYeqyU1MZbm9m5a33E+X9GEZJ741BLm5A/j+JeBqJu5iIxEyR0Qk+Ds3itaOeK3GZ8cz+fXz+Ljq4r55sZT/GRXFc/sPcfiwgzO1LVw8R27mouzUvjRHyxjeu640cd/6FkYl++cLRpilEhJ9Jm5Hl7+W2d2Z5Rr7fGxHh5fV8r6eXn8YMsZrIVxibGMS4xjXGIsGSlx3DErd3SzUO904gXInApZwzzcWESiW8I4Z8PN0d/B3f806ka+uWmJ/MMDc/nD8qn8+2snOVJzlRXTJjBj4jhmThzHzIlp5KYljHwp753arjita5Z90tmFGGKUSEn0mdGbSB3fAMseG5O3LMlJ5csPzR+T9xpQ62WnEefyP1Y3cxEZvln3OTvfLh6EvLF5Xk3OTOYr71swJu81oGPPgbcL5rw3sJ8zQqGX2okEWlYpTCh1lvfCyeFfOjUC897vdiQiEo5mrHd2/I6gOaerDj0L4wth0i1uR+KTEimJTjPXQ8UWZ5YnXBx8BrJmwMR5bkciIuEoJcvZ8Xs0jBKplnpnJn7ue0N2Jl6JlESnue9zZncOP+t2JP5prIaq7TD/4ZB9mIhIGJh1H9Qehstjewh7wBz9DdgemPuQ25EMSImURKeJ85zGbvufdjsS/xz6hfN97vvcjUNEwtvMe53vIzjA3RWHnnVKMXLnuh3JgJRISXQyBuZ/AM7tgobTbkcztIM/h4IlkFnsdiQiEs4yiiBvARz4mduRDK35IlRsdWajQngmXomURK95DwMGDoT4rNSlI3DpUG+8IiKjtPBR53SHiwfdjmRwh38FWKc+KoQpkZLolT4Jpt7mJFIjPHsvKA49AyYG5jzodiQiEgnmPQyeONj3Y7cjGdzhZ50lvewRHvAeJEqkJLrNfwSuVIzq2ISAstZZ1pu6BlJz3I5GRCJBcqZz1MqBp6G70+1ofGusdp7LYTCAVCIl0W3W/RCXDPt/6nYkvlXvgsYqLeuJyNha9GHn7L2TL7kdiW+Hf+l8D/FlPVAiJdEuIRVm3udMIXe1D31/sB38OcQmOluWRUTGyrR1kJoL+37kdiS+HfoF5C9yjsQKcUqkRBY8Au1NztEJoaSnyxmVzbjHOSdLRGSsxMQ6O5dPvAjXat2O5kYNp6FmX0j3jnonJVIiU9dA6sTQ6yl1ZhO01mtZT0QCY9GHnWaXodYK4c3/co6yCdGz9W6mRErEEwPz3ufMSLU0uB3N2w7+DBLToeQOtyMRkUiUPQMmLXaW90Jl53JXG7z5lNM4NH2S29H4RYmUCMCCD4bWkTGtl+Hob50dK7EJbkcjIpFq4Yeg9ghceMvtSBwHn4G2K7D0k25H4rdRJVLGmKeNMft6vyqMMfvGKjCRoJo41+lXEiq79/b+ELpaYeljbkciIpFs7kPOhpZQKDq3FnZ91zm+q2iV29H4bVSJlLX2A9bahdbahcAvgBAZzouMwIJH4Pwe97v9dnfCzu/B1Nshd467sYhIZEsa7+xcPviM+zuXq3c6z9+lfxjSR8LcbEyW9owxBng/8JOxeD8RVyz6MMSnwrZvuBvH4V/CtYuw4tPuxiEi0WHRo9DeCMefdzeOnd916kLnf8DdOIZprGqkyoFL1tqTA91gjHnMGLPHGLOnrq5ujD5WZAwlZcCtH3NOG79S4U4M1sKO/4DsmVCyzp0YRCS6FN8GaQWw5wn3YrhaA0d/A4t+D+JT3ItjBIZMpIwxrxhjDvn4euAdt32QIWajrLXfs9YuttYuzs7OHm3cIoGx4k+cbbfb/8Odz6/Y6hwmuvxTYTW1LSJhzBMDKz4FFVugcrs7Mex5Arw9sOQP3Pn8URgykbLW3mGtnevj69cAxphY4L1AiDXhERmBtHynVuqt/4ZrLsyc7vgmJGfB/PcH/7NFJHrd+vuQkg2bvhL8z+7ucDbYTL8bMouD//mjNBZLe3cAx6y158bgvUTct/JPnb/Yu74b3M+tPwUnNjgjsrik4H62iES3+GQoexzObHTO+AymI7+GljqnyDwMjUUi9QgqMpdIklXqnG2363vQ0Ry8z33jWxCTAEs+EbzPFBHps/jjkDwBNv2/4H7uzu/ChBKYuja4nztGRp1IWWs/Zq39zlgEIxIyVv6Zc/7e3ieD83mtl2Hfj50lvdSc4HymiMg7JaQ6u4VPvQLn9gbnM8/tddrOLH0MPOHZIzw8oxYJtIJboXi1U7PU3RH4z9vzBHS3OUXmIiJuWfqHzg7mzUGolbIWXvobZxZswQcD/3kBokRKZCAr/xc01wT+QM/2q87U9rS1kDs7sJ8lIjKYhHHO7uUTL8CFAB9WcvAZqNoO674AiWmB/awAUiIlMpBpa2HifNj2r8623EB59R+cQsvb/yZwnyEi4q+ljzmNMQO5g6+j2ZmNyl/k9I4KY0qkRAZiDJR/FhpOOTNGgVC9C3b/AJZ90llOFBFxW2I6LP8TOP4c1BwIzGds/mfnBIf1Xw3b2qg+4R29SKDNfgCm3wOv/B3UHhvb9+7uhN88DmmTYK1mo0QkhCz7JCSkOTPm1o7te9efhB3fgoUfhoLFY/veLlAiJTIYY+Dd/+bsZvnlY07yM1a2/SvUHYX7vu7UJYiIhIqk8XD7X8Opl53WLGPFWtjwlxCXDHd8Yeze10VKpESGkpoD938DavY709Fjoe6E815z3ut08xURCTXLPgkz74OX/xaqd4/Nex57Dk6/Brd/PmJavSiREvHHrPthwYdgy9fg3J7RvZfXC7993BmR3RPkxnciIv4yBh74pnN01jO/7/S7G42uNnjx85AzG5aEZxdzX5RIifjrni87D5RnH4POlpG/z5tPQtUOuOsfI2ZEJiIRKmk8PPwkNF+EX/2xMxAciZ5upya0sQru+QrExI5pmG5SIiXir8R0eM+34PJpZ6p7JE5vhJf+ForKYdGHxzY+EZFAmHQr3P1Fp7fUjn8f/uu7O+EXH4eDP4O1/xeKy8c+RhcpkRIZjuLVzrbg3T+A1/4Rerr8f+2u78P/PATjJ8OD33GmzUVEwsHSx2DWu+GVv4eqN/x/XVc7/Oz3nIOJ7/4SrP7zwMXoEiVSIsO17m9h4aNOsfgTd0PD6cHv7+mG5/4cnv9zKLkDPv4ipBcEJ1YRkbFgDDzwH85A8L/f69SLdrUP/prOFvjJB5yZrPv+BVZE5hFYSqREhisu0Vnie/gpJ4n6Tjnsfcp3r5W2K/Cjh2D396HsM/DBn4T1UQgiEsUS0+Gjv4Vptzv9pb65FI7+tv+zz1qnFup/3gdnN8N7vgOLP+5OzEFg7Fg32vLD4sWL7Z49o9z5JBIKms7Dr/7IeVhMfxdkz4T2JmhvhLZGqDvuHP9y/zdg0aNuRysiMjZOb4QXPu/0wite7czS1x1z2sTU7IfWBvDEwkM/gDkPuh3tqBlj9lprfXYPVSIlMlpeL7zxTXjti+Dtdna5JI53vidPcA4/LlzhdpQiImOrpxv2/hA2ftGZfffEQc4syFvgfBWVQ85Mt6McE0qkRILB2wPGoyJyEYkubY3QVA1Z0yE2we1oAmKwRCpyGjmIuM0T43YEIiLBl9Q7Ax+lVGwuIiIiMkJKpERERERGSImUiIiIyAgpkRIREREZISVSIiIiIiOkREpERERkhJRIiYiIiIyQEikRERGREVIiJSIiIjJCSqRERERERkiJlIiIiMgIKZESERERGSElUiIiIiIjpERKREREZISUSImIiIiMkLHWBv9DjakDKgP8MVlAfYA/Q4ZPv5fQo99JaNLvJfTodxKagvF7KbTWZvu64EoiFQzGmD3W2sVuxyE30u8l9Oh3Epr0ewk9+p2EJrd/L1raExERERkhJVIiIiIiIxTJidT33A5AfNLvJfTodxKa9HsJPfqdhCZXfy8RWyMlIiIiEmiRPCMlIiIiElBKpERERERGKCITKWPMu4wxx40xp4wxn3M7nmhnjJlsjNlojDlqjDlsjPlTt2MShzEmxhjzljHmd27HIg5jzHhjzDPGmGO9f2dWuB2TgDHmz3qfX4eMMT8xxiS6HVO0McY8YYypNcYcesfPMo0xLxtjTvZ+zwh2XBGXSBljYoBvAvcAs4EPGmNmuxtV1OsGPmutnQUsB/5Ev5OQ8afAUbeDkBt8A3jBWjsTWIB+P64zxkwCHgcWW2vnAjHAI+5GFZWeBN51088+B7xqrS0FXu3956CKuEQKWAqcstaesdZ2Aj8FHnA5pqhmra2x1r7Z+7+bcf7DMMndqMQYUwDcC/zA7VjEYYxJA1YD/wlgre201ja6G5X0igWSjDGxQDJwweV4oo61djNw+aYfPwA81fu/nwLeE9SgiMxEahJQ/Y5/Pof+ox0yjDFFwCJgp7uRCPCvwF8CXrcDkeumAnXAD3uXXH9gjElxO6hoZ609D3wVqAJqgCZr7UvuRiW9cq21NeAM2oGcYAcQiYmU8fEz9XgIAcaYVOAXwP+y1l51O55oZoy5D6i11u51Oxa5QSxwC/Bta+0ioAUXlirkRr11Nw8AxUA+kGKM+bC7UUmoiMRE6hww+R3/XICmYF1njInDSaJ+ZK191u14hJXAu40xFTjL32uNMf/jbkiC8/w6Z63tm7F9BiexEnfdAZy11tZZa7uAZ4Eyl2MSxyVjTB5A7/faYAcQiYnUbqDUGFNsjInHKQj8jcsxRTVjjMGp+Thqrf262/EIWGs/b60tsNYW4fwdec1aqxG2y6y1F4FqY8yM3h+tA464GJI4qoDlxpjk3ufZOrQJIFT8Bvho7//+KPDrYAcQG+wPDDRrbbcx5tPAizg7K56w1h52OaxotxL4PeCgMWZf78/+ylr7vIsxiYSqzwA/6h0IngF+3+V4op61dqcx5hngTZxdyG+h42KCzhjzE2ANkGWMOQd8Afgy8DNjzCdwEt6Hgx6XjogRERERGZlIXNoTERERCQolUiIiIiIjpERKREREZISUSImIiIiMkBIpERERkRFSIiUiIiIyQkqkREREREbo/wOF2iAHRqq8NwAAAABJRU5ErkJggg==\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "%matplotlib inline\n", "from matplotlib.pylab import *\n", "\n", "y = lambda t,eps : (-1+eps)*exp(1-cos(t))\n", "\n", "tt=linspace(0,10,101)\n", "yye = [y(t,0.1) for t in tt]\n", "yy0 = [y(t,0) for t in tt]\n", "#xkcd()\n", "figure(figsize=(10,7)) \n", "plot(tt,yye,tt,yy0);\n", "legend([\"$y_0=-0.9$\",\"$y_0=-1$\"]);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Exercice\n", "\n", "L'objectif de cet exercice est de bien mettre en évidence l'influence d'une petite perturbation de la condition initiale sur la solution d'un problème numériquement mal posé et d'appliquer la méthode d'Euler explicite pour en approcher la solution exacte.\n", "\n", ">Considérons le problème de Cauchy\n", "$$\\begin{cases}\n", "y'(t)=3\\dfrac{y(t)}{t}-\\dfrac{5}{t^3},\\\\\n", "y(1)=a\n", "\\end{cases}$$\n", ">1. Calculer l'unique solution exacte à ce problème de Cauchy en fonction de $a$ pour $t>0$. Remarquer qu'avec le changement de variable $u(t)=\\frac{y(t)}{t}$ on obtient une EDO linéaire d'ordre 1\n", ">2. Tracer la solution exacte sur l'intervalle $[1;5]$ pour $a=1$, $a=1-\\frac{1}{1000}$ et $a=1-\\frac{1}{100}$.\n", ">3. Pour $a=1$, sur un autre graphique tracer la solution exacte et les solutions obtenues par la méthode d'Euler explicite avec $N=20$, $N=100$, $N=300$, $N=5000$. \n", "Comment expliquez-vous ce comportement du schéma?" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Correction**\n", "\n", "Calculons les solutions de l'edo pour $t>0$.\n", "On pose $u(t)=\\frac{y(t)}{t}$ ainsi $y(t)=tu(t)$ et $y'(t)=u(t)+tu'(t)$. \n", "On obtient l'edo linéaire d'ordre 1 suivante:\n", "$$\n", "tu'(t)-2u(t)=-5t^{-3}. \n", "$$\n", "On a $a(t)=t$, $b(t)=-2$ et $g(t)=-5t^{-3}$ donc \n", "+ $A(t)=\\int \\frac{b(t)}{a(t)}\\mathrm{d}t=\\int -2/t\\mathrm{d}t= -2\\ln(t)=\\ln(t^{-2})$,\n", "+ $B(t)=\\int \\frac{g(t)}{a(t)}e^{A(t)}\\mathrm{d}t=\\int -5t^{-4} e^{A(t)}\\mathrm{d}t=\\int -5t^{-6} \\mathrm{d}t = t^{-5}$. \n", "\n", "On conclut que $u(t)=ce^{-A(t)}+B(t)e^{-A(t)}=ct^2+t^{-3}$ et $y(t)=tu(t)=ct^3+t^{-2}$.\n", "\n", "En imposant la condition $a=y(1)$ on trouve l'unique solution du problème de Cauchy donné:\n", "$$\n", "y(t)=(a-1)t^3+\\dfrac{1}{t^2}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Vérifions ce calcul avec `sympy`:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$\\displaystyle \\frac{d}{d t} y{\\left(t \\right)} = \\frac{3 y{\\left(t \\right)}}{t} - \\frac{5}{t^{3}}$" ], "text/plain": [ "d 3⋅y(t) 5 \n", "──(y(t)) = ────── - ──\n", "dt t 3\n", " t " ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": 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"text/latex": [ "$\\displaystyle y{\\left(t \\right)} = C_{1} t^{3} + \\frac{1}{t^{2}}$" ], "text/plain": [ " 3 1 \n", "y(t) = C₁⋅t + ──\n", " 2\n", " t " ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": 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"text/latex": [ "$\\displaystyle y{\\left(t \\right)} = t^{3} \\left(a - 1\\right) + \\frac{1}{t^{2}}$" ], "text/plain": [ " 3 1 \n", "y(t) = t ⋅(a - 1) + ──\n", " 2\n", " t " ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import sympy as sym\n", "sym.init_printing()\n", "\n", "t=sym.Symbol('t')\n", "y=sym.Function('y')\n", "\n", "edo=sym.Eq( sym.diff(y(t),t) , 3*y(t)/t-5/t**3 )\n", "display(edo)\n", "\n", "solgen=sym.dsolve(edo,y(t))\n", "display(solgen)\n", "\n", "a=sym.Symbol('a')\n", "t0,y0=1,a\n", "consts= sym.solve( [solgen.rhs.subs(t,t0)-y0 ], dict=True)[0]\n", "solpar=solgen.subs(consts)\n", "display(solpar)" ] }, { "cell_type": "code", "execution_count": 57, "metadata": {}, "outputs": [ { "data": { "image/png": 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diQlwt3ZY4gqqXaJm52BDdMdAUr2bEx7vQurBJew9dd7aYQkhhBBWtepgCv0mrGPPqfN8dGcTPh7WFGdp6qz0ql2iBtAotg4oAxleHemXNo8JK45YOyQhhBDCKgqKS3jzt/2M/moLPq72LHysI0NaBFo7LFFO1TJRc/d2IqSRF4mBnai/r5D9+7dIrZoQQoga52hqDndM2cj09Qnc2y6EXx/tQD2ZwLZKqZaJGkDjroGU2Lii7JoxLOsbqVUTQghRY2itmbMlkQET1pOceYFp97bk9dsa4mArE9hWNdW2cTqoQW1q+TqScKErUbu28J3bTvYlR0inSSGEENXa+QtF/HveHhbtPk37ep58fFdT/NwdrB2WuE7VtkZNGRRNewRzwTEEx5JIRuTMlFo1IYQQ1dqW4xn0G7+OJXvP8HyfSL79VxtJ0qq4apuoATRo64+Tux3x9XoTvTuDbfv2sC9Z+qoJIYSoXopKTHyw5CDDvvgTo0Ex9+H2PBIbjtEgc6NVddU6UTPaGmjWM5gc10jc8kMZkSe1akIIIaqXY6k5DP1sI5NXHWVI80AWP9GJpkG1rB2WqCDVOlEDiO4YgL2TDfF1exO1J5XNew9IrZoQQogqT2vN7LiT9J+wnuPpeXw2qjkf3NlEHgNVzVT7RM3OwYbG3YI4X6sx3ln+jCycIbVqQgghqrS0nALGfruNl37ZQ/OQWix5sjN9G/lbOyxxE1T7RA3MU3XY2BmIr9uLqD1n2LjvsNSqCSGEqJKW7T9Ln0/XsuZQKi/3j+LbB2TAQHVWIxI1B2dbGnYJJMOzFXXSPBlV9BXv/XHI2mEJIYQQ5ZZTUMzzc3fx4Ddb8XZ1YOFjHRnTqS4GGTBQrdWIRA2gaY8gDEZzrVqDvYlsP5zAmsOp1g5LCCGEuKq4hAz6jl/L3G1JPBJbj/mPdiDSz9XaYYlboMYkas7u9kR1CCDNpx2hp925T3/F24sOUGLS1g5NCCGEKFN+UQnvLD7AsKl/olDM+X/teL5PA+xsaszXd41Xo+50s57BYDByNKw7kTuOk3rmOHO2Jlo7LCGEEOISu5MyGThxPV+sPcbwVsEsfqITLUNrWzsscYvVqETN3duRiFa+pAR0ITTZmX+VTOejpYfJKSi2dmhCCCEEAIXFJj5eeojBUzaSnV/MV6Nb8c4djWTajRqqRiVqAM17h2DSNhyO6E7E5lSc8w7wxZqj1g5LCCGE4OCZLG6fvIEJK+O5rWkAS57qTGykj7XDElZU4xI1zwAXwlv6kOrfA99MV+7J/oapa4+RnHnB2qEJIYSooYpKTExYcYSBE9eTkp3P1Hta8PFdTXF3tLV2aMLKalyiBtBmYF00Rg5E9ydycw6N2MaHS2S6DiGEELfe/mRzLdrHyw7Tp6E/S5/qQq8YP2uHJSqJGpmo1fJ1Iqp9ABm1O+JcWJsBKT/xy44k9iTJJLhCCCFujcJiE58uP8ygSes5m1XA53e3YOKIZtR2trN2aKISqZBETSnVRyl1SCkVr5R6sYz1zymldlpee5VSJUqp2pZ1x5VSeyzrtlZEPOXRqn8oBqOR/Y0HEx1XyACHVby5aD9ay3QdQgghbq69p85z2+QNfLr8CAMa+7Psqc70aSi1aOJSN5yoKaWMwGSgLxANjFBKRZfeRmv9gda6qda6KfASsEZrnVFqk66W9S1vNJ7ycvFwoFHXQM47NQeDP22O/sHWhFTm70y+VSEIIYSoYfKLSnjvj4PcNnkDaTkFTL2nBZ8Ob4aH1KKJy6iIGrXWQLzW+pjWuhD4AbjtCtuPAGZXwHlvWIveIdg5GDnY/E4a7zDxgMuvvLloP5l5hdYOTQghRDUTl5BBv/Hr+Gz1UYY0r8Ny6YsmykHdaFOfUmoo0EdrPcby/h6gjdZ6XBnbOgFJQPhfNWpKqQTgHKCBL7TWUy9znrHAWABvb+8Wc+bMuaG4/5KyV5O6V9Nk5wecrHOct+u/QYs6roxuaF8hx7+VcnJycHFxsXYYlYqUSdmkXMom5VI2KZdLXUuZXCjW/HSokJWJxXg5KkbH2BPjZbzJEVqHfFbK1rVr123X22pYEbPnlfU02MtlfwOBDRc1e3bQWicrpXyAZUqpg1rrtZcc0JzATQWIjIzUsbGxNxi2WWHbYr575U+ONhtBy9Xv8GSzmbyZ9AiPDWhV5WaAXr16NRVVLtWFlEnZpFzKJuVSNimXS5W3TFYcOMsbv+7ldFYxD3QI49ne9XGyq74T18pnpeJVRNNnEhBU6n0gcLmOXsO5qNlTa51s+TcFmIe5KfWWsXOwoUWfUHJ0IGd8G+C38hjt3BL597w9FBabbmUoQgghqomUrHwe/X47//p6Ky4ONsx9qD3/HRhdrZM0cXNURKK2BYhQSoUppewwJ2MLLt5IKeUOdAHml1rmrJRy/etnoBewtwJiuiYNO9fBpbY9p5o/QN1kRZ+zX3D4bDbT1h271aEIIYSowkwmzey4k3T/eA3LDpzl2V71+e2xTrQI8bB2aKKKuuFETWtdDIwDlgAHgDla631KqYeUUg+V2nQwsFRrnVtqmS+wXim1C4gDFmmt/7jRmK6V0dZAm4F1yc5zJj6mI5Hr8nnMbyUTVhzhZHrerQ5HCCFEFRSfksPwaZt46Zc9xAS48ccTnRjXLQI7mxo5ZamoIBVSB6u1XgwsvmjZ5xe9/wr46qJlx4AmFRHDjYps48ee1UmkOo0g5FAcPpt+p1ZkR16ev5evR7dCqbK64gkhhKjp8otKmLwqns/XHMXJzob3hzTmzpaB8r0hKoSk+RbKoOg0rD75eZrDne6m+S4Y5zidtYdTWbBL5lYTQghxqbWHU+n96VomroxnQOMAlj/dhbtaBUmSJiqMJGql+NV1J7KNH2mGFpyr5Y3H0gT6+ifzvwX7SMnKt3Z4QgghKonMfBPjZm3n3hlxGJVi1pg2fDKsKd6uVW9qJ1G5SaJ2kXaD62GwMXCqy5PUS4Z2CVMoKCri+Z93y+OlhBCihisuMfH1xuO8tP4CS/ef5ake9fn9yU60D/eydmiimpJE7SLOtexp2TeEtMxaHG/QjPpr83mt3hpWH0rl+80nrR2eEEIIK9l24hyDJm3gfwv2UdfdwJInO/NEjwjsbarn5LWicpBErQxNuwfj5u1IWtSDOOcbKFi8kP51Tby16AAJablXP4AQQohqIyO3kOfn7mLIZxvJyC1k8sjmPNvSgTAvZ2uHJmoASdTKYLQ10HFoOFnnNAk97qHlbuh4+h3sjIqnftxJcYlMhCuEENVdiUnz/eYTdP1wNb9sP8XYznVZ/kwX+jf2l8EC4paRRO0yQht7ERRdm7OGdqT51CJg8TnearCRnYmZTFl91NrhCSGEuIm2nTjHbZPX8595e2ng58riJzrx735RuNjLkwXErSWJ2mUopeh4ZwTFhSbOD3wTr/OQ/uscRkYbmbDiCLuTMq0dohBCiAqWkp3P03N2MuSzjaRlFzJhRDN+GNuW+r6u1g5N1FCSqF1BbX9nmvYM4sRRRXKvAbTeCU1OvIGXsx1P/biTC4Ul1g5RCCFEBSgqMTF93TG6fbiGhbuSeTi2Hiue6cKgJgHSzCmsShK1q2jVPww3b0fOuN5OpqcTvosyeCtqI8fScvnPr3tkyg4hhKjiVh1Koc+na3lz0QFahXqw9KkuvNCnAc7SzCkqAUnUrsLGzkjXUZFkpReQM+w9fDPh5NzZvNDelV+2n2J2XKK1QxRCCHEdjqbmMHpmHKNnbsGk4cv7WjJzdGsZzSkqFfnvQjkENqhNVHt/Dm46g03PHrRetpx9Mf+mU/iHvLpgH43quNMo0N3aYQohhCiH8xeKmLDiCF9vPI6jrZGX+0dxb7tQeXi6qJTkU1lO7YeE4+BsQ6rvKHI87PD+LZ1n6yzBy8WOh7/fRmZeobVDFEIIcQXFJSa+/fM4XT9czYwNCdzZMpBVz8UyplNdSdJEpSWfzHJycLal07D6pCblUXDfh/icg50zfmByLwNns/J5es4uTCbpryaEEJWN1ppVB1PoM34dr8zfR31fFxaO68g7dzTGy0WezSkqN0nUrkF4Cx9CG3lyYJ8tWYMH0noPbJ/9OK/2DmXlwRQ+WyPzqwkhRGVy4HQW986IY/RXWygxaabe04LZD7alYR3priKqBknUroFSis4jIlEGRYrvcM6FutNkWTE+B15iYJMAPlp6iA3xadYOUwgharyzWfm8MHc3/SesY3fSef43MJolT3amV4yfTLchqhRJ1K6Ra20H2g2uR9LBTBwf/gIjivwfD/N40AbqerswbtZ2jsvzQIUQwiqy84v4aOkhunywil92JDG6QxhrnotldIcw6YcmqiT51F6Hhp3rENjAg7gV57B75mXCk2H1tMlMH+CGBh74agvn84qsHaYQQtQYRZaBArEfrGbiynh6Rvux4ulYXhkQTS0nO2uHJ8R1k0TtOiiDovt9URhtDBw4XZ/sbs1pvxlWf/cA00Y2IvFcHg99t43CYnl4uxBC3ExaaxbvOU3vT9byyvx9hPu4MP/RDkwc0YxgTydrhyfEDZNE7Tq5eDjQZUQkZxOy0D3/R46PIxGL8yna9BzvDWnMn8fS+c88eXKBEELcLBvi07ht8gYe+X47RoPiy/ta8sPYtjQJqmXt0ISoMJKo3YCIVr5EtPJl27JTeP13Kq4X4NT322hdOJ/Hu4Xz07YkGQkqhBAVbE/See75cjOjpm8mLbuAD4Y25o8nO9M9ylcGCohqR55McIO6jKjP6fhM/lxbRMsnx9Hwo0n89tlEHnq9KcfTA3j/j0OEejrTr5G/tUMVQogq7WhqDh8vO8yi3afxcLLl5f5R3N02BAdbo7VDE+KmkUTtBtk72dL9/mjmf7qD05Hdce+1hS5LNzNz4oO899JSks7l8dSPO/F1s6dFSG1rhyuEEFVOYkYe41cc4ZftSTjYGnmsWzgPdq6Lm4OttUMT4qaTps8KEBjpQdMewexfl0yt+98lq54X7Zdo5k8fwrSRjfB3d+D+mVvYl3ze2qEKIUSVkZKVzyu/7qXbR6tZsCuZ0R3CWPt8V57pFSlJmqgxJFGrIG0H1cWzjgurZh0h4pPZmBxs8J2XxaGFY/juX61xtbfh3i/jOJqaY+1QhRCiUkvLKeCtRfvp/MEqZsed5M6WQax5zjzVhjzySdQ0kqhVEKOtgd4PxlBSZGLVr2cJmfQFXlmQNGsPJXsn8d2YNigFd0/fTNK5PGuHK4QQlU5GbiHv/H6ATu+t4sv1CfRr6M+KZ7rw9uBG+Ls7Wjs8IaxCErUK5OHnTNd7GnDm2HkOJnrj/NzjNEqAxdNn4ntuPd880IbcgmLunr6ZlOx8a4crhBCVQmZeIR8sOUin91Yyde0xesX4suzpLnw8rCkhns7WDk8Iq5JErYJFtPSlUZc67FqeiG46hIK+7YmNU0z97Enqk8DM0a1JyS7gnulxZOYVWjtcIYSwmoxcc4LW8b1VTFl9lK4NfFj6ZGfGD29GPW8Xa4cnRKUgidpN0GFoBD4hrqz8+gAhz35MblQdui9VfPHZCJq7ZTPt3pYkpOVy74w4edSUEKLGScsp4J3fD9DxvZVMWX2ULpHe/PFEZyaNbE6Er6u1wxOiUpFE7SYw2hroPbYhyqBY+tUhGk2dQ6G3K60Xar79fBAd6hiZMqo5B09nM2LaJtJzCqwdshBC3HQpWfm8+dt+Or63kmlrj9Er2pdlT3Vm8sjmRPpJgiZEWSRRu0ncPB3pMTqatMQcNi5JodF3v2C0tSXk1wIWTh9Ijwh3pt3XkqOpOQybuomULOmzJoSonhIz8nj51z10fH8VMzcep18jf5Y/3YVPhzcj3EcSNCGuRBK1myi0kRfN+4Swf30yR08YiPjyWzzyFMa56WycNZwu4Z58Nbo1yZkXuOuLPzmVecHaIQshRIWJT8nm6Tk7if1wNXO2JDGkeSArn+nCx3c1pa70QROiXCRRu8naDAwjKMqDNbMOke0Ugv8nHxOSAifmHOLAgkdoV7c23/6rDem5hdz1+Z+cSM+1dshCCHFDdidl8vB32+j5yVp+33OG+9uHsvb5rrxzRyMZxSnENZJE7SYzGA30GtMQNy9H/vhiD/ZNu+Dy0lM0PWsGO4MAACAASURBVArrZq3h+JIXaRHiwewH25JXWMydn//JkbPZ1g5bCCGuidaaNYdTGTF1E4MmbWB9fBqPxoaz/oWuvDIgGj93B2uHKESVJInaLeDgbEv/RxpjKtEs/mwP/nc9gOHeoXTaDQu/X0DS8v/SsI47P4xthwaGfLaRP4+mWztsIYS4quISEwt2JdN/wnrumxHHsbQc/t2vARtf7MazvSPxlCcJCHFDJFG7RWr5OtFrTAwZyTksn7mfiBdeg8G96bEFfpo1h7Nr3yXSz5VfHm6Pj5sD987YzLwdSdYOWwghypRTUMyM9QnEfriax2fvIL+4hPeHNGbt810Z27kervIsTiEqhI21A6hJgqM96XBnBOvnHCHutwTavPUx+y88Qp8/1vCN7VeMNjoQ1OFJfn6oPf/vu6089eMukjIuMK5bOEopa4cvhBCcOZ/PVxuPM2vzCbLyi2kZ4sErA6LpGeWLwSB/p4SoaJKo3WKNuwaSkZzLtt9PUNvfmeiPJrM3/wEGrI5jus1nPGS0o1bbR/j6gda8+PMePlp2mKRzF3hzcENsjVIBKoSwjn3J55mx/jgLdp2ixKTp09CPMZ3q0jzYw9qhCVGtVUiippTqA4wHjMB0rfW7F62PBeYDCZZFv2itXy/PvtWNUorOw+uTeTaPld8cxMXDnoYTp7Nn7N3ctmI3U2zGM85og1ursXx8VxMCPRyZuDKe5PMXmDSiOe5O0pwghLg1SkyaFQfO8nHcBQ7+sR5HWyOj2oTwQIcwgj2drB2eEDXCDVfRKKWMwGSgLxANjFBKRZex6TqtdVPL6/Vr3LdaMdoY6PtQI9y8HFg0ZQ8ZKQU0/PwbCptFcttSA+Nnf0Tmhk9QSvFMr0jeH9KYTcfSGThpPQdOZ1k7fCFENZedX8SM9Ql0/XA1Y7/dRmqe5t/9GrDppe68OihGkjQhbqGKaEtrDcRrrY9prQuBH4DbbsG+VZqDsy0DHmuCrZ2BhRN3kZurafTlLIobhnPH7wYmzvmctBWvg9bc1SqIH8a2o6C4hMFTNjB/5ylrhy+EqIaOpubwv/l7affOSl7/bT8+rvZMGdWc9zs7MrZzPanRF8IKlNb6xg6g1FCgj9Z6jOX9PUAbrfW4UtvEAj8DSUAy8KzWel959i11jLHAWABvb+8Wc+bMuaG4K4v8TE3CCo2tI4R2V9ioIuwmfYTH4ZP83EPTp1kzcuqOBaXILDAxZWcBh8+Z6BViw12RdtiU6rybk5ODi4vM9l2alEnZpFzKVhPLxaQ1u1JLWH6iiH3pJmwUtPa3oUewDXVrGYGaWS5XI2VSNimXsnXt2nWb1rrl9exbEX3Uyhrmc3H2tx0I0VrnKKX6Ab8CEeXc17xQ66nAVIDIyEgdGxt73QFXNqeizrFg4k6y9rgx6PGmGLp0Yf/DDzBk+XbmFe/kbs/ZBA/6DAxG+vcw8fbiA8zccJxMgxuTRzbH29U8T9Hq1aupTuVSEaRMyiblUraaVC7pOQX8tC2J7+NOkJhRgJ+bA8/2CmZ462C8Lpr7rCaVS3lJmZRNyqXiVUTTZxIQVOp9IOZas79prbO01jmWnxcDtkopr/LsWxPUifSgx/3RnD56nqVf7gNbO2I+nwld2zF4Ncz+bR3xc++B4kJsjQb+NzCGT4Y1YXdSJn3Hr2P1oRRrX4IQogrQWrP1eAZP/rCDdu+s5N3fDxLg7siUUc1Z90JXxnWLuCRJE0JYV0XUqG0BIpRSYcApYDgwsvQGSik/4KzWWiulWmNOENOBzKvtW1NEtPQlL6uQ9XOOsPr7g3Qd1YAGE6dy+LnHGfT7KuYX76B7Xj+aDpsLjrUY3CyQKH83npi9k/tnbuGBDmG0c76xZmwhRPWUlV/E/J3JfL/pBAfPZONqb8PINsGMahNMhK+rtcMTQlzBDSdqWutipdQ4YAnmKTZmWPqfPWRZ/zkwFHhYKVUMXACGa3PnuDL3vdGYqqom3YLIzyli6+Lj2Nga6TQsgvofTSLe4Tn6zVvM6rxk0nO70X3Ur1ArmAZ+bswf14F3Fh9gxoYElrkaCI3Jlj+8Qgi01uxIzGT25pP8tvs0F4pKiAlw4507GnFb0wCc7GQaTSGqggr5TbU0Zy6+aNnnpX6eBEwq7741WeuBYRQXlrBzeSJGWwPt76hH+NsfkujtQ+zUr9iRc4HZhb0Zcdf3UKc5DrZGXrutIZ3re/PkrK0MmLielwdEc3ebYHmagRA10Pm8IubtSGJ2XCKHzmbjZGfk9mYBjGgdTKM67vJ3QYgqRv5LVckopWg/JJySIhM7l53ExtZAm0F1CX76BVL869Dk9bdImA+fFt3F44PGY2jQH4DuUb680dGRn5OceeXXvSzdd4a3BzciqLbMdyREdWcyaTYeTWfO1kT+2HeGwmITjQPdeeeORgxsEoCLvfypF6Kqkt/eSkgpRadh9SkuNpmbQe0MtOgTis+Iu7H39cf05BO4/AJvFD3OC7FHcWj/GChFLXsDX49uzfdxJ3l38QF6f7qW53pHcm+7UIzyDD4hqp3EjDzmbkti7rYkTmVewN3RlhGtgrizZRAN67hbOzwhRAWQRK2SUgZF7KgGFBea2PTrMYw2Bpr2CMa9W3fCv5uFHnM/febCK4WTeO7MDnwGTQHAYFDc0zaEbg18+M+8Pby2cD+/7T7Ne0MaEe4jfdeEqOpyCor5fc9pft6exKZjGSgFHcO9eLFvA3pG++Jga7R2iEKICiSJWiVmMCh63B+FqdjEhrnxADTtEYxj48Y0+Gkeh0bfwz1zU/k4axPD07tjH/zU3/vWqeXIzPtb8evOU7y2cD/9xq9nXLdwxnauK3/IhahiSkyajUfT+GX7Kf7Ye4YLRSWEejrxdM/6DGkRSJ1ajtYOUQhxk0iiVskZjAZ6jomBL/ezYW48xYUltOwXhl1ICFE/zyd+3EPc9/tuFqdlEtT0RdpF+kFYJ8DchDq4WSCdIrx5dcE+Pl52mLnbknhlQDQ9onykU7EQlZjWmn3JWczfeYoFu5I5m1WAq4MNg5vXYUjzOjQP9pDfYSFqAEnUqgCj0UCvf0Wz0tbA5gUJFBeaaHNbXWw8PIj8+nsS336DfrPmsCvVng9M9/Jkx2ewbfsoWP6Ie7nYM2lkc0a0TuN/C/bx4Ddb6VLfm/8NjKautzzqQ4jKJDEjjwW7kvl1xymOpORgY1DERnrzyoA69IiSpk0hahpJ1KoIg9FA9/uiMNoa2PbHCYqLTHQYGo6ysSH4v6+RHhVDw1dfxecXB14sGs9LJzfiNWgKONb6+xgdwr34/YlOfL3xOOOXH6H3p2t5oGMY47qG4+ogD1sWwlpSsvJZtOc0C3cls/1kJgCtQj148/aG9G/kj4eznZUjFEJYiyRqVYh5gEEkNrYGdq1IpLjIRJfh9VEGheedd3HwfDbe0z9n1A85vNV3B8NTO9Hm9pkQ+H/PgbU1GhjTqS6Dmgbw/h+H+GLNMX7amsSjXcO5u20w9jbyv3UhboWM3EJ+33ua33adZlNCOlpDAz9XnusdyaAmATK1jhACkEStylFK0fGuCGzsDGxfcpLighK63tsAo9FAUXg9GsxbQPyjDzF2/mEWtyxhe+FQxrZ+AmP7J8Dwf4929XF14MM7m3BvuxDe++Mgb/y2nxnrE3i6Z31ub1ZHpvMQ4iZIyylgyb4z/L7nDH8eS6fEpKnr7czj3SIY2MRfRmYLIS4hiVoVpJSi7e31sLW3YfOCY1zIKaT3gw0BsPX3p8EPP3Hq/Xfp991sjpxy5NniKfwnYQ1ed0wHZ69/HKtxYC2+H9OWdUdSee+Pgzzz0y6mrj3Gs70jZcCBEBXgbFY+S/edYdGe08QlZGDSEOblzENd6tKvkT/R/m7yeyaEuCxJ1KoopRQt+4Xi5GbH6u8PMv+THXg0Mz+UXdnZEfjyf8lq1ZaSl57Hfxa82u8QI6d1oH3fTyGy7yXH6xThTYd6Xizee5oPlxziwW+2EuXvxmPdwukT44dBatiEKLejqTks3XeWJfvOsDPR3OcswseFcd0i6NfIj0hfV0nOhBDlIolaFRfdMQBHV1uWTN9HZoYmq+UF3LzMcyq59e5F/agGHBv3CI/+cpT5bYys0w/zxP4+OPR9Dxz+OXO5waAY0DiA3jF+zN+ZzJRV8Tzy/XbCfVx4JLYeg5oEYGM0lBWGEDWayaTZmZTJigNnWbLvLPEpOQA0DnTnud6R9Ir2JcJXmjWFENdOErVqIKyJN7c90ZT5E7bz8/vbGPBYE7yDzF8KdsHB1P/pZ5LfepPb5swl4bgj4/qu5qnEDsQMmAx1u1xyPFujgaEtAhncrA6L95xm8qp4np6zi0+XH+HBTmEMaRGIk518dETNlldYzLojaaw4cJaVB1NIyynEaFC0CavNPW1D6BntS4BMRCuEuEHybVtN+IfXIrS74uwmxbyPttNnbEOCoz0BMNjbE/j6G2THdsX07xd57LtsvonVhM6/mwfrD8Om52tg53zJMY0GxcAmAfRv5M+KgylMWhXPK/P38cGSQ4xoE8y97UJlRnRRo5xIz2XVwRRWH05l49F0CotNuDrYEBvpQ48oH2Lr++DuJFPdCCEqjiRq1YiDu2LI8y34bdJufpu0m87DImjYJfDv9a7duhH522JOvvQi9y/fwM6jrjxiWsCL8Uuo2+9jiOhZ5nENBkXPaF96RPmw/eQ5Zqw/zrS1x5i+LoE+MX6M7hBKixCZJV1UP/lFJWxOyGD1oRRWH0olIS0XgLpeztzdJoQeUT60CquNrXQJEELcJJKoVTMuHg7c8Vxzln25jzWzD5NxJo+OQ8MxWL5IbLy8CJs6jcwff6TxO28T/h182NNEk98e4IHA7tj2fR9cfMo8tlKKFiG1aRFSm6RzeXz75wlmx51k0Z7TRPq6Mrx1EIOb1aGWk0zOKaomrTUHz2Sx7nAaa4+kEpeQQUGxCXsbA+3qeXJ/+1BiI70J8by0BloIIW4GSdSqITsHG/o+3JiNP8eza0Ui51Mu0HtMDHaO5tutlMJj+HCcWrfh5PPP8NiCA2yJcOVf3Tbx7BdtaBz7KjS75x/zrl0s0MOJl/pF8USPCObvTOaHuJO8tnA/7/x+kL4N/RjeKpi2dWtLLZuo9JIzL7DxaDob49NYuf8CmUvWAeZRmqPahNApwou2dT1xtJPJoIUQt54katWUwaDoeGcEHn5OrJ19mJ8/2Eb/Rxr/PSIUwL5uGOE//kTG19/Q8tNPiPm6mC+7OhGw/r88tmsWTv0+BL9GVzyPk50NI1oHM6J1MPuTs/hhy0nm7TjF/J3JBNd24ramAdzWtA7hPvJMUVE5pOUUsPlYBhuPprHxaPrfzZm1ne2o72FgaMcYOkV44e8u/S+FENYniVo1F9OpDm7ejiyZupef3t1K7zExBDao/fd6ZTTi+cBoXLt3I+k//+ah37ezO9SZB3ol8uC3vegWNQLV7WVwqn2Fs5hFB7jx+m0NealvFIv3nGbejlNMXhXPxJXxNKzjxm1N6jCwSQB+7g4385KF+IeU7Hw2H8tgc0I6m49lcMQydYaznZE2dT0Z1SaY9vW8aODnytq1a4htGWTliIUQ4v9IolYDBDWozdAXWrL4s90sGL+T9kPCadI96B/NknYhIYR98y2Zc+ag33+fBjPz+alDbX52WsjzU+YR2uU/0GI0GK7e/ONoZ2RIi0CGtAgkJSufhbtPM3/nKd5afIC3fz9Ai2APesf40TvGj2BPeZ6hqDhaa46l5bLt+Dm2nshg6/FzHLPUmDnbGWkZWpvBzevQJsyTxoHuMghACFHpSaJWQ9TydWLoiy1Z8fUBNsyNJ+V4Fl3vicLW/v8SL2Uw4DF8OC5dunD6rbcYtXwFyXvs+E8vO1ptfJ2xW2fi1OcdCOtc7vP6uDnwr45h/KtjGMdSc1i46zRL9p3hrcUHeGvxAaL83egd40vPaF95lI64ZnmFxexJOs+OxEy2nTjH9hPnSM8tBMDDyZYWIR4Mbx1EmzBPYgLcZMJmIUSVI4laDWLnYEOfsQ3ZvuQEm+YfI+N0Ln0faoS79z9rtWz9/QmeNIns1asxvPE6L88+zbpoF+6OzWTM3GH09WuP6vU6+MZc0/nrervwRI8InugRwcn0PJbuP8OSfWcYv+IIny4/go+rPZ3rexMb6U3HcC8ZPSr+ocSkOZqaw67ETHYmZrLjZCaHzmZTYjI/Oi3My5muDXxoGeJBy9Da1PN2lsRfCFHlSaJWwyilaNEnFO8gV5Z+uY+f3tlKj/ujCW3sdcm2rrGxRLRtS/rUaXScNpVWx0z82KE2s1z389SMrrSIGgpd/w3ugWWc6cqCPZ0Y06kuYzrVJTW7gNWHUlhzOJVl+88yd1sSBgVNg2rRMdw84q5ZsIeMuqtBSkyahLRc9iWfZ1fiefacymRfchZ5hSUAuNrb0DS4Fo9G1aNZsAdNgmpR21kSeyFE9SOJWg0VHOPJnS+14o+pe1g0ZTfNegbT5va6GC9qGjI4OOD9+GO4DxrI6Tff5L4VG0jZYWRSV2/ck5fx5Ge/EtpiDHR4slwDDsri7WrPnS2DuLNlEMUlJnYlnWfN4VTWHE5l0qp4JqyMx9aoaBpUi7Z1PWkT5kmTIHdcHWQG+Oogr7CYw2dz2J+cxb7k8+w/ncXB09lcKDInZQ62BmIC3LmrZRCNA91pHOhOXS8XDAapLRNCVH+SqNVg7t6ODHm+BRt+imfHspOcPppJrzENca196ahMu9BQgqdNI3ftWmzffZfnfz7O/hB7nujuQJsD3zB220y8Wo2Fdo9ed8IGYGM00CLEgxYhHjzdsz7Z+UVsPXGOTcfS2XQs4+9RpEpBfR9XmofUolmQB82Ca1HPW768K7PCYhMn0nM5dDabw2eyOXgmm0NnszmZkYc2t17i6mBDtL8bw1sHERPgTkyAGxE+LtK3TAhRY0miVsPZ2BrpMjKSgPq1WPXdQX58M+6yTaFKKVy6dKFe+/ac+3EOauIE3pmZxaomzozsaKTf3mmM3jIN99b/D9o9Ao4eNxyfq4MtXSN96BppflpCdn4R209msvNkJttPnmPR7tPMjksEzKP6ovzdiA5wIybAjZgAd4os/ZfErZOZV0hCWi4JabnEp+SYX6k5nEjP+7s/mUFBqJczMQFu3NEskEg/F2IC3An0cJR+ZUIIUYokagKAiJa+eAe5smT6XhZN2U3THkG0vb0eRptLazKUrS217x6F+8ABpE2ZQrfvZ9F5XwmLWrgypK2Robs/5564L3Bu8xC0eeiGatgu5upgS5f63nSp7w2AyaRJSM9l+4lz7LM0nf28LYlv/jQ3mxkVhO1cQ4SPCxE+LoT7uhLh40KYlzMOttLn7XqYTJqz2fkkZlwgMSOPxHN5nEjPIyEtl+PpuWTmFf29rY1BEerlTH0fV/o19KeejzMRPq6E+7hI+QshRDlIoib+VsvX6e+m0J3LE0k6dI6eD8RQ27/s5xoa3d3xfeklPO6+m9SJE7lt4W/02aX5pbUbg1raMGrnFIb9OQnn5veZm0SvY9DB1RgMinreLtTzduFOyzKTSXMiI499yef5Y9NeCh2cOXQmmyX7zvBXBZtS4OfmQIinEyG1nQnxMv8bUMuBgFqOeLvY18hmVK01WfnFpGbnk5yZz+nzF0jOzCc58wKnz+dzKvMCp85doLDE9Pc+SkGAuyOhXk70b+RPmJczoZ7OhHo5E+LpJHOVCSHEDZBETfzDX02hQdG1WfXtQX56ewsdhoYT07nOZZuk7IKCqPP++3j+awypn37KiFWrGLRd80Nbd/o3s2XYwVmM3Dod94Z3QccnwSvipl6DwaAI83ImzMsZl4zDxMa2BCC/qITj6bkcPpvDcUvtz4n0PFYcTCEtp+Afx7A1KnzdHAhwd8TP3QEvF3u8XO3wdrHHy9Uebxd7PJztqOVoi5OdsdI212mtySss4fyFIs5fKOJcbiHpuYVsPlHEjmWHycgtJC2ngJTsAlKy80nJKqCg2PSPYygF3i72+NdyJNrfjV4xvgR5OBFU24kgD0fqeDhibyO1Y0IIcTNIoibKVLepN75hbqz4+gBrZh/mxN50ut4ThZPb5adAcIisT9BnU8jbvp3Ujz/hX8u2MiwOfmrlwoBmtRia+Af3TJlN7fBe5j5soZ3MWcAt4mBrpIGfGw383C5Zl1NQzMn0PEvN0QWSz+dzOtP8766kTNKyC8i1TA1xMRuDopaTLe6Otrg52uJib4OTnRFnOxuc7W1wsjfiYGPEzsaAvY0BOxsDdkYDtkYDRoNCKTAoZXmZj1miNSUmjUlrSkzmWsLCEhMFxSYKi00UFJdQWGwiv8hEXmExuYUl5BUUk1tYTF5hCVkXisjKLybrQhHFl+mnpw4eoZajLZ4u9vi42tMi2AMfNwd8XO3xdrXH390Rf3cHfN0csCujCVwIIcTNJ4mauCxnd3sGjmvC7lVJ/DnvKD+8GUe3exoQ2ujSgQalOTVvTvC335C3eTNpkyYzevlWhm1R/NTSgQHNg+h/fif3fj+YIM8oaPswNBwCttZ9/qeLvQ3RAeaBCJeTV1hMWnYhqTkFpGYXkJlXyPkLRWRaaqvO5xWRlV9EbkExKVkFfydNOQXFFF5US1URbAwKexsDzvY2lpcRJzsbajvbEerpjJujjTl5dDAnkO6OttR2tqO2sx0Hd26lX48uMppSCCEqOUnUxBUpg6JJ9yACG3iwbMY+Fk3eTYP2/nS8MwJ7x8t/fJRSOLdti3PbtuRujiNt8mTuWxHHXVsNzGthy9Cm/nQw5nLfH0/SZPn/oOUD0Pw+cK9zC6/u2jjZ2RDsaXNdzyf9q0assMRcI/bXSwMmrdFaY9J//QxGg7mGzWhQGJXCYAA7owF7S82cnY25Nu56JdsrSdKEEKIKkERNlItnHRfufLEVcYsS2LHkBEkHMuh6TwOCoz2vuq9zm9Y4t2lN3pYtpH32GSNX/smdm+xY2gwebe5LXRcn7t0ykdi1H2IT2dectNXtCobqk0gYDAoHg1FGOgohhLgm1eebUNx0RlsD7W6vx5DnW2Jrb2ThhF2s/v4ghfnF5drfqVUrgmfMIHTuXGp37kb/Pwv54nPoslzznq0n/eqFMz1tK+dmDYWJzWHDeMhNu8lXJYQQQlRekqiJa+Yb5sZd/2lF057B7FufzA+vx3Fyf3q593dsGEPgp59Q7/fF1L5jKB12FTJ+qonHfrVleYYNPUKCednNnv1r3oCPIuHHu+HwEigpX0IohBBCVBeSqInrYmNrpMOQcO54tgVGWwMLJ+xi+Vf7yc8puvrOFnYhIfi/9irhK5bj/fBDRJ7S/G+WiUnfOXFhfyF3+/oxMjyaX87GkTd7GHwSA8tfhbT4m3dhQgghRCUiiZq4If713Bn2cita9gvlSNxZZr22iSNbzqJ1+R/dZOPtjffjjxO+aiX+b72Jr6MPYxbm89UX9nRYZ2ICdnSrG84bvr4c2DIFJrWAad1h8xeQk3oTr04IIYSwLknUxA2zsTXSZlBd7vpPK1w9HVn65T4WTdlNdkb+NR3H4OBArSFDCJv/K8EzZ+DRog3dVp9jyucmXl/kxrFjWQzz92FYg+bMIovMJS+am0a/Gwq7f4LC3Jt0hUIIIYR1yKhPUWE867gw5PkW7FmVxKb5R5n12mZa9w+jcfdAjNcwFYRSCud27XBu147CpFNkzpmDzdy5PL+zgAI/D1a30Eyud4EPQkPoau/HoJQDdPhlDLY2jlC/F8QMhoheYFf2o6+EEEKIqqJCEjWlVB9gPGAEpmut371o/SjgBcvbHOBhrfUuy7rjQDZQAhRrrVtWREzCOgyWedfCmnix7sfDbPwlnoObThM7MhL/8FrXfDy7wDr4PP0UXuMeJXvpMs7Nnk3vRdvobTSS3NyPuZGZPF5H4eEZQz+jB/1O/UnD/fNRtk5QvzfeOgIKW0nSJoQQokq64URNKWUEJgM9gSRgi1JqgdZ6f6nNEoAuWutzSqm+wFSgTan1XbXWMg9DNeLm5Ui/RxqTsCuNdT8e5pcPtxPV3p92d9TD0eXyj6G6HIOdHe4D+uM+oD8F8fFkzv0Z4/z5PL7l/P9n7z7D47rue99/1/TBoPdB7wABAixgFymxSKJESRZVLFt23B3HN3ae3MTOsXOTk+LEse89ThznJDruJYklW5JVTYliBTspVgAEKwoBDDpA9DYDzL4v9gAYkGAFSLT/53n2s/tgzX6AwW/W2nstvhIVxpnl4fxXqov/DreRGLeULSqILTUHyet+Ey79m94v24InIetxcNy67zchhBBiJpiKGrUVQLmmaZUASqnfAE8Do0FN07TDfscfBRKm4OeKGU4pRdriKBJywjix7QrFu2upLG5h1dPp5K6Nw3CXPetbMzKI+eY3iP7zP6N7z146Xn+dwvcPUahp9OWlcLDAxH87r/CjSDvJ0UvYao9gs6uUxEvvgzJA0hrI2QJZj0FE+hS/ayGEEGLqqDt5Om/CF1DqeeAxTdO+6Fv/FLBS07Sv3uD4rwM5fsdXAe2ABvxI07Qf3+C8LwFfAoiKiip89dVXJ1Xuuainp4fAwMDpLsYNDXRoNJzU6GsBWxg4CxUBkVMzKLvh6lVsx45hP3oMU1MTXrOJ2tw4diwYZE9SK8NGRZIxioc8Np66WsuC7hoA+uxxtEUsoy1iGZ0huWgG85SUZ6ab6b8r00Wuy8TkulxPrsnE5LpMbMOGDSfv9tauqQhqHwU2XxPUVmia9icTHLsBeAlYq2lam29bnKZp9UqpaGAn8Ceapu2/2c/Mzs7WLl68OKlyz0VFRUWsX79+uotxU5qmUX6imUO/K6e3Y5CslTGseTYDR4h1yl5/oLSUzrfepmvbNoY7O1GhITStSGN7Zi/bgirQlCIrOIWN5ig2Xm0kp/o4atgNliBIewjSN0LGJghLmZIyzUSz4XdlOsh1mZhc6xI9RwAAIABJREFUl+vJNZmYXJeJKaXuOqhNRdOnC0j0W08A6q89SClVAPwUeHwkpAFomlbvmzcrpd5Eb0q9aVATs5dSiszlMSTnR3BqezWnd9VQdaaVZVtSWLQxEaN5cj3GKKWwFxRgLygg5pvf4OhL/4ekmmrYs5dP7xjgMzFRNKxKY3tGLz+2nOCHaDhzFrMhKI2NvX0srTmN+cLv9RcLTx8LbckPgC14Cq6AEEIIcfumIqgdBzKVUqlAHfBx4BP+ByilkoA3gE9pmnbJb7sDMGia1u1bfhT41hSUScxwFpuJVVvTWfCAk4OvlXPkzQrKDtSx+pkM0pdGodTkm0SVxcLg4kXE/99/ire3l+49e+nato3YbQf57NAQX4iPo3llOnszBvjdwAle9roJiglmdd7HeVCzs7alhogzv4bjPwFlhLgleo1b6oOQuBLM9im4EkIIIcSNTTqoaZo2pJT6KvABevccP9c0rUwp9WXf/h8CfwNEAC/5/gGPdMMRA7zp22YCXtY0bftkyyRmj5CoAJ744wJqz13l0O8u88FPzuLMCGHtRzOJTp66GiyDw0HIU08S8tSTDHd00L17N10ffEDEu0d53uPhY85YOtfkcjRT462r59kx2IpCkb9oPWsDU1jbP0hu/VmMh34AB/4ZjFZIWKbXtCWvgcQV0gWIEEKIKTcl/ahpmvYe8N41237ot/xF4IsTnFcJLJqKMojZLTE3nBdyVnD+UD3H3qnkte+cIHtlLKu2phEYZpvSn2UMDSX0uecIfe45hjs76d67l+7tH+B95wAPezxsDg9naM1DnM0N4F1PNS9V/56XgBBHCGvWfZo15gjWdLUT7ToJB74H+71gMOk1bslrIHGVHtwckVNabiGEEPOPjEwgZgyDQZG3Lp7MZTGc3F5N8e5ayk81s2hjIks3J2ENmPonMo0hIYRu3Uro1q0M9/TQu38/3bt207N3H/m/76UgIADL6geozY9hf2QPu5tP8v6AfotlRkwGqxb+GauMIRR2thBY+yEceQkO/UB/8YiMsdCWuAIis8Ego7YJIYS4fRLUxIxjsZtY/Uw6eeviOPZuJac+qKbsYB3Lt6Sy8MH4ST9wcCPGwECCt2wheMsWvG43fcc+pHvXLnr27iV69z6eV4o/yM9ncOUmTmea2Gut5LXKd/jv4UGMyshC50JWLvorVphCWNTVhs11Ei6+B2f+W/8B1mC91i1hGcQv0+eB0ffkvQghhJgbJKiJGSs40s4jn8tj8aYkjrxZzsHXLlOyt5aVT6eRWRiDussOc2+HwWIhcN1aAtetRfu7v2Xw/Hm6i4roKdqH9pOXWQwsi47GvvYRmvPjORrfz6HuM/z03C/5sebFbDCTH5nP8gVfY7nNSUF/D/b6M+A6AQf/FbRh35tMgLjFeoAbmQLC79n7EkIIMbtIUBMzXlRSEB/50yXUnGvj8BsV7PzZOU7vqGHlR9JIXhgxJU+I3oxSCltuLrbcXKL++I8ZammhZ/8Beg4coHfXHuxvdLHBYGDLokWY13yO6pwwjoa0cLz5JD8p/Qk/0ryYDCZyI3JZumwrS8P/iiVeI6HNF6D+DNSfhpEuQQBCksBZALEFvnk+BMfDPX6fQgghZh4JamLWSMqNIDEnnMsnmjj2TiXb/qMEZ0YIq7amE3cXA77fLVNUFKHPPUvoc8+iDQ3RX1Kih7b9B+j6jx8RBjwRHMwLK1diWvnnlGcFcsJYy+nm0/z6/K/5pdcDQFpIGovjFrN40ZMsCk4lpacdQ0MxNBRDYwlc2IY+YAcQEAExeRCz0DfPg6gc6SJECCHmOAlqYlZRBkXWiljSl0Zz/lA9x7dd4c3vnSJ5YQQrn04jKjHo/pbHZCJg6VICli6FP/1Thq5epffIEXoPH6b38BGGdu4kAngyPp4XVq3EsuJpajJCOOWt4lTTKXZV7+KNy28AEGQJoiCqgEVZK8hf8wXyg1IJ6aiFxlI9vDWVwYlfwFD/yMXQO+WNzoGoBRDtm8LTwXTnA98LIYSYeSSoiVnJaDKw8KEEslc7Kd3r4tQH1bz67eOkL41i+ZOpRMRNz1hzpvBwQp54gpAnnkDTNNxVV+g9fJi+Y0fp3rUb7+/ewA5sSEvjyVUrsS17kracGEqGayhuKeZM8xkO1x1G89WkJQcnszByIflZq1m45g/JDsnA1tUAzWV6cGsqg+bzeu2b5tULYTDpYS0yE6KyITLLN2WC9f4GWSGEEJMjQU3MamaLkaWbk8lbF8eZ3bUU766l4nQLmctiWPFk6rSWTSmFNS0Va1oq4X/wSbThYQYvXqT36DF6jx2l8623aX/5FQAKUlNZtXw5Acu/iHdDDpfMbZS2llLaUsqHDR+yrXIbAEZlJD00nbyIPPKceeQufIrMsExsGtB6CVou6MGt9RK0XISL7489uAAQ5ISIDLIGA8BSChGZEJEOoUlgmprxVoUQQkwdCWpiTrAGmFn5VBqLNiRyemc1JXtdlJ9oIiQZOnL7CI0OmO4ioozG0YcSIj7/ObShIQbOnaPv+HH6PjxO13vv0fHqqwBExMfzeOFSnlu6BvvSr9DuDORc+3nKWss4d/Uce2v38mb5m4Ae3lJDUlkQvoAFEQvIWbiF7PA/I9gSDENuaK8aC25XK6H1MlEtZ2DHB36FM0BIAoSnQViqb54yNsk4p0IIMS0kqIk5xRZoZvUzGSzalMSpD6opKarl5b87RvaKGAq3pMyIwDZCmUyjA8hHfOELaMPDDFy4QP/Jk/SdPEXv4SN0vfMuAIaQELIWL2bRksXYF38O2+qFNGmdlLWVcb7tPBeuXuBow1HerXx39PXjHHFkhWeRFZZFdlg22Ys/SkJgAkaDkUNFRaxfUQBt5Xp4u1oJV6v0+bm3of/q+MLaw/TAFpqs1775TyGJYJ2epmYhhJgRhtz652Zf29jU2wp9vm2TIEFNzEkBwRbWfjST/kAX9u4Ezh6o4+KHTWStiGHZ4ymExsycwDZCGY3Y8/Kw5+UR/ulPo2kantpa+k6eou/kCfrPnKFn3z79YKMRW3Y2+YsXs2LxIuwFH8GcnEzbQBvn285zsf0il65e4mL7Rfa79uP13b9mM9pID00ncCCQ6qp1ZIZlkpG5kSj7x8Z3c9LfAR3V0H5l/NRYqjenDg+OL7w9TO8TLuSaKThOn4Kc0rQqhJj5NA3cvdDf7gteV/3m7ePX+9rGlge7bvyatpBJFUmCmpjTzHbF2sczWbI5idM7aijbX8elY41krYil8PFkwmJn7kDqSiksSUlYkpIIfWYrAMOdnfQXF9N3+jT9p8/Q+dZbtL/8MqCPYWoryCenYBFLFhVgK3gOU1gYA0MDVHRUcKn9EpfaL1HeUU5ZexnHThwb/VnBlmDSQ9P1KUSfp4WkER1bcH0/dV4v9DZDR60e5jpqoKsOOl3QWQs1R2Cg4/o35IjSA1twHATF6sv+88BYfXxUg/GeXVMhxDzh9cJgp/6ls79d/0zqbx9bv265fSycDbtv/LqWIAgIA3u43m1SRIY+D4jQOysPCIeASH3dEal/iTWa4S/vvh9MCWpiXnCEWFn70UyWPJrE6Z16YLv4YSMZS6MpfDyFyITZ0XRnDAkh8MEHCXzwQQD9AYXyCvqLz9BfUsJAcTGtBw7q3woBc0IC9oJ8Yhfmk5q/kI/kPoox0EFRUREFqwq43H6Z8o5yKjoqqOioYGf1Tl4ffH305znMDlKDU0kLTSM1JJXU4FRSQlJIDErEEhQLicsnLuhgN3TWQXc9dI1Mdfq2rjqoOwm9Ldefpwz6h1xQDAT6JkeUPgVGj81HPgiN8hEmxJykaeDpg4FO39Tlt9zhm3zr/R1j2/tH9nUx2g/lREx2PVTZw/QpMgNsob5t4X77fPOACH0+DV0fyaecmFccIVbWPp9J4eZkzuyupbTIRfnJZlIKIlm2JYWYlNl107wyGrFlZ2HLziLshRcAGO7pYeBsGQNnS+kvPUv/mWK63nvfd4LCkpJCcGQk2pWN5OflsWzBVowL9JpFTdNoG2ijoqOCys5KqjqrqOys5Gj9Ud6peGf05xqUgfjAeFKCU0gJSSElOIWk4CSSg5KJccRgsAbp/btF59y48ENuvWauu1EPcj1N0NMMPY36vLsRms7pgc7XSfB17GF6aHNEgSPC75utbxr3gRumN0FIjZ0Q99awR/+y5u7RA9Ngt2/qGpsPdI2tjywPdI7NB7rGP7E+EZMd7KH637UtRK+Vj8rRA5c99Jq57zNgZN1suz/XYgpIUBPzkj3Iwuqt6Sx5JInSIhfFu2t5/bsnSFwQRuFjKcRlhd7zoanuFWNgII5VK3GsWjm6baitjYGzZ+kvK2PgbBmWU6doPn58dL8lORlr7gL9qdQFuRTmLmBlzspxr9vj7uFK1xWqOqu40nWFK51XqO6q5njjcQaGB0aPsxqtJAYlkhSURFJwEolBiaOT0+HEOBKUTJaxe9luRtP0b8g9LXqw62n2u1G3VQ9yvW3Qehn6jun7bvgBr6754Pb7MLeHkVjXCievjH3wW31zW7DeB53JJkN5iblF02BoUK+9cvfo92e5R5ZH1ntg0H+5e2zbYDe4u337e1jX3wlFN2k6HKEMYA3WJ5tvHhwH1hx93Rbi2xfiN4WO/T3aQubNfa8S1MS8ZnOYWf5EKos2JXJ2Xx1ndtfy1vdPE5sWzNLHUkjJv/djid4PpogIAh96iMCHHgKgoqiItQsXMlBWxsC5c/pUUkr3+9vHzomJwZqTjS1nAbacbKw5OeQl57IwcuG41/ZqXpr7mqnuqqa6q5qarhqqu6u50nWFg3UHcXvHPrRNBhPxgfEkBCbo86AEfQpMID4oXu9S5FpKjX0bjsq69Zv1evVv5SNPXo3cezJyM3D/1bH7UwY6oL16tMkkXRuGyl/d+LUNZj2wjQQ3S5A+twb61kfmDt8U6JsCwOzQ5xbH2LLJDgbDrd+TmJ9GQtRQP3gGxuaeft+y/9Q3wbxPD13jlkeCWK9vW++ta678mR3677sl0DcP0sci9v3+1zV3kJSRO/b3MPK3Yg3x+1sJ1v8O5sBn6/0gQU0IwGIzsXRzMgUbEjh/uIHTO2t476USIuIdLN2cTEZhNAbj3PqHaoqMHBfeAIY7Ohi4cIGBc+cZuHCewQsXaTt4CIb1D3Jlt2PNzMSWnY01OxtbdhbWrCxiQ2KJdcSy0jm+Fm4kxNV211LbXUtNVw2uHheubhdn287SOdg57vggcxDxQfHEOeKIC4wjPjCeuEB92elwEmwJvnVwNhh8NWSheme+t0vTOLB7O+uWF1zTBNPp11TTPb65xt2jN9m2lY/VNHj6bv9ngh7WzHYwB/jmNt+2ieY2vRbBf2606MtGq15L6T83WvQbmUeOMZj0dYPZt923bDDpTcLz8R+n1wveId/kgeGRuWds+7BHv8HcO6TPh92Et52CC336E9Aj+4fdepP+8ODYfGTb0IBvefD65aGBsWXPgN+2gVuXfyLKMPZlwOybLL7fL3vY9V8YRr5UmP2WLQHXf/EwO275xaKyqIikh9bfXbnFhCSoCeHHZDGSvz6B3HVxlB9v4uT2anb+/BzH3qlk8cNJ5KxxYrbM3XucjKGhOFatwrFq1eg27+Agg+XlDF64yMCFCwxeukT3jh10vPba6DGm2FismZlYszL1IJeVhSU9HYPVSqxDD3HLY69/8KDL3UVddx2uHhf1PfW4ul3U99ZT3VXNkYYj9I+Ma+rjMDtwOpzEBcYRGxCLM9BJrCMWp0OfRwdEYzaY7+7NK8WwyQ4h8fp0t7zDYzUV7l5f01Dv+G0jy+NqQPxqQTz9+j/p/o5r/nmP/JMfvHU57obB5JvM+j9kgwmUkdWeITjt0AOAwQjKeM2y0tdHJ986yhf+lF8IVDcOhNrIzd+a37qmD482ujyy7ptGtnmH9Zoh77Bv37Aewka2eYf8lofHQtjNbji/iQKA0ts40Gj1hWrr+OWR9ZHw5B+8zXZ9eWRusulB3Rwwtt1s9wv5/kE/QA/g8zF0z1ES1ISYgNFoIHuVk6wVsVSVtHJ6RzX7f3OJD39fRcGGBPLXJ2Bz3GUgmGUMVuto/24jNE1jqLmFwUu+8Hb5MoOXLtN39Ciax3fjv8GAJTERS2aGHuIyMrBmZGJJTcFg0Z+cCrYEExwRzIKIBdf9XE3TaB9sp6Gngfreeup76mnobaCup46m3iZKWkroGBzfDYhCEWGPIDYgdjQgRgdEExMQo88d+txqvIf3thiMviaeeziuqtfrq5Hp99XcuMdqaEZrc0Ymz/hl78j60Niyf3AZrUkaCT5DXK1z4YyN8dvuF4A0TV++NkD5h6gR/mHLn6aND3IjYU5xTdAb2X6jcGj0hUfD+OWRGkNf8NRDqNkvmBrHlo3m8XP/2kejxbfPwqniUpYuX+WrtbT4ajH9ppEaTAlMYpIkqAlxE8qgSFscReqiSBrKOzn1QTUfvlvFqR015D0QR8GmBIIj7NNdzPtOKYU5JhpzTDSB69aNbteGhnBXV/uC2yUGL5czWF5Oz96i0eZTjEa9f7j0NKzpGVgz0rGkpWFNTcUQEDDuZ4Tbwgm3hZMXmcdE+jx9NPY10tjTSENvA019TTT2NtLY20hlZyWH6w/TN3R9U2SoNZTogGiiAqKICYghyh5FdEA0TX1NRLVGEWmPJMIegckwQz8iDQYw2O7bk2sXi4pwrl9/X37WbNF1xQNxi6e7GGIemKGfQkLMLEop4jJDicsMpa2uh1M7qiktclFS5CKjMJoljyYRlXgPa1BmCWUyYU1Px5qeDo89Nrrd63bjrqpi8HI57soKBssrGKyooKdoHwwNjR5njovTQ1t6GpbUkXkqxoiJH+oIMAeQFpJGWkjahOXRNI0eTw/Nfc009TXR1Ns0utzS30JzXzOXrl6ibaBtdPSGH2/7sf5e0INiVIAe3KLsYwHOfznCFoHD7JgTD50IIWYeCWpC3KGI+EAe+Vweq55Op3hPLecO1HP5eBMJOWEseTSJxAXh8k/7GgaLBVt2Nrbs7HHbNbcbd02NHtwqK3BXVjFYWUHfiRNoA2M3UhuCgrCkpWJNScWSloYlJQVLagqW5GQM1hs3YyqlCLIEEWQJIj30xg8WDHmHaOtvY/vB7SQtSKKlv0Wf+vR5a38rl9ov0dbfxvAET8jZjLbR0BZuD9fntvDRbRF2fT3MFkaIJWSsixIhhLgFCWpC3KWgcBtrn89k+ZYUzu6vo2SPi3f/rZiI+EAWP5JI5rIYjKa59aToVFMWi+/etYxx2zWvF099A+6qKtxVlQxWVeGuukLvkSN0vv223wsovRYuNVUPb8nJWFKSsaSkYI6LQxlvLxCZDCZiHDEkW5NZn7T+hsd5NS8dgx209LXQ1t9G20Abrf2t45breuoobSmlfbB9tJbOn0EZCLWGjga3MGsYYbawsXVbGOHWcEJtoYRZwwi1hd79AxJCiFlPgpoQk2QNMFP4WAqLNyVx6XgjZ3bVsvuX5zn6ZgUFGxPJXRs3bx48mCrKYMCSEI8lIR7WrR23b7inF3f1FdxVV3Bf8U1VVXSePo23t3fsQLMZS0KCHt6SkzEnJ/mWUzA7Y287xPkzKMPofXO3MuwdptPdORri2gfauTpwdWzqv0r7YDuX2i9xdeAqXe4bD+ocZA4i1BZKqDWUEGsIYdYwQqwhhFp922whhFj81q0h2E12qdkVYg6QoCbEFDGaDSxYE0fOaic1565yZmcNR96s4Ph7V8hd46RgYyIhUfPvwYOpZgx0XPcUKuj3ow23tenBrboa95Xq0eXeo0fHNaUqsxlzQgKWpCTMvoHvLclJGJua0NxulGXy4/kZDcbRUJdJ5i2P93g9dA52cnXgKp2DnbQPtNMx2EH7QDvtg/pyx4C+XtVZRftA+4QPSowwG8yEWPUAF2INIdgaTIhl4nmwxTdZgwmyBEkNnhAziAQ1IaaYUorkvAiS8yJoqe3mzK4azu6ro7TIRdriKBY9nERs2m103CruiFIKU2QkpshIApYtG7dP706kWQ9w1dV4qqtx19Tirqmh9/hxtD498EQCF/7+W5hjY/UAl5iIOSlRnyckYklKxBh8b8aDNRvMRNojibRH3vY57mE3nYOddA520jHYoS+7/ZYHO+lyd9E52El9Tz0X3BfoGuy6acADsJvsY8HNHIS7282OAzsItgYTaA4kyBJEsCWYQIu+HGQOItASSKA5kGBLMGajBD0hpooENSHuoajEIB75XB6rt2ZQus9F2f46Kk63EJMazKJNiaQviZpzIx7MRHp3IjGYY2JwrFgxbt9oTVx1NSU7dpDucOghrraG7l27GG5vH3e8ISQES0IC5sRELIkJmOMTMCcm6NuczimpjbtdFqOFqIAoogKi7ug8z7BHD3DuTroGu+hy+ya/5W53N93ubrrcXXQMdXCq+RRdg130eHrQbtFJrNVoHQ10geZAHBbHuDA3OvftG102O0a3OUwOeehCCCSoCXFfBIZZWb01nWWPp3DhSAPFu2vZ8dMyAsOtFKxPJHetE2uA1EJMB/+auIHubqKu6S9suKcHj8uFu7YWT60Lj6sWd62LwfPn6dm9e6yDXwCDAVNsDJa4eMwJCb4pXg9x8fGYoqPv6t64qWY2mvUnUu0Rt3V8UVER633Xxat56fX0jga5kanH00OXu4sedw89np7RbSPrrX2tdHu66XH33LJGb4TdZCfAFECgJZAAUwAOswOH2UGA2bdsumbd7CDAFECAWZ8cprF9dpMdg5IvRWL2kaAmxH1ktupDVOU9GE91aSvFu2s5/EY5H26rYsFqJwUbEwiNDrj1C4n7xhgYiDEnB1tOznX7NK+XoeZmPLW1uF11eGpr8dS5cLvq6D1yhKHm5vG98JvNmJ1OzPFxmOPjscTHY/abTFFRMyLI3YxBGUa7PLlbXs1Ln6dvXJDr9fSOTj0e3zZ3L71DvaPzHncPTX1No8f1efoYGL798TDtJvto+AswB1w3998/uuy33X/qGOqgy92F3WSXe/rEPSVBTYhpYDAoUhdFkbooipaabor31FJ2oI7SfS5S8iNZtCmR+KxQuY9thlMGg34/W2wsAcuvH8vU63bjqavD46rDU1+Px+XS1+vq6Cnax3Br6/gTzGb99eLjMcfFjZ/i4zDHxNzXptV7xaAMevOnJRAck3utIe8QfUN99Hn0qdfTS9/Q2Hxk++iyb1//UD99Q330uPUOkfuH+unz9NE/1H/b4e9/vvI/Ab17F7vRF+LMdmxGG3aTHZvJhs1kGw13/ttH1v2PsRltWE3Wse1++y0Gi3wezFMS1ISYZlFJQTz82VxWP5PO2X11nN1fx5WSViLiAynYmEDWihhM5pldyyImZrBYsKamYk1NnXC/d2AAT33DaHgbnerr6T14UK+R86cUpujoa0KcE3NcHCanE3NcPMbASSafWcZkMI0+tTpVhr3DDAwPjAtv/st9Q30UnysmMS2RgaGB0f39Q/36+vDYevtA+7jtA0MDeLyeWxfiGgqFzWTDarRiNVqxm+z6si/YWY3WcfttJhsWo2XcPv91/3MtRsvYNt92q9Eq4XCGkKAmxAzhCLGy8iNpFD6WzKXjTZTsqWXvf13g6FsV5K2LZ+FD8ThC7uFg4uK+M9hsWNNSsabdIMi53Qw1NOi1cfX1eOrq8fjW+0tK6NqxAzzj/+kbgoP1AOd06lOcUw9xTj3UzYbm1elmNBhxGPR73rhBjzpBtUGsz1t/V6/v8XoYHBocrb0bGBpgcHhwNNwNDg8yMDTAwPAAg0ODo6FxZPna/YPDg/R4esbtH1m+m1Doz2KwjAY3q9GKxWjRt/mWR7f5llvbWjly7MjoNv9jx02G8XOz0YzVMHac2WAed+x8DowS1ISYYUwWI7kPxLFgjZO6i+0U73Fx4v0rnPqgmoxl0SzamEh08r3pIkLMLAaLZbTD3olow8MMtbbhqa/zC3S+eUMDfadO4e3sHH+S0YgpJloPbrGxeniLjdXXnbGYYmMxhkqz+71kNpgxW8x60+895tW844Kbe9g9NvcFxMHhQdzD7tHla7f577t2W7e7W1/3unEPu+nu7+Zs5dnRY271hPDtMhlM48Kd2WgeC3Qj6775aMjzWzcbzGPH+Jb9z/U/ZnT5Fusmg2l0+V4+oSxBTYgZSilFQk44CTnhdDT3UbrXxfkjDVw61kRsWggFGxOke495ThmNmGOiMcdEw5IlEx4z3NPLUIOvJq6hEU9DvS/UNdBfXDxhrZyy2/UQ54zFFOvEMThAR2srplgnZqd+T57BMb+aWGcrgzKM3iN3P/g/IaxpGkPaEO5h97iA5x52jwY7j9czts/rxjPsmXD/tds8w57R9dHzvG56Pb36uu88z7BHP943TWV49GdQBkzKNBrk/EPcZB82kaAmxCwQGh3Auo9lsfIjaZw/3EBJkUvv3iPMysKH4slbG48tUJ48E9czBjowZmZizZx4dATN62WotZWhxka9Nq6xQQ9yDY14GhsZPHAAR2srDdveG3eeITgYc0wMJmcsZl+AM8XEjtbKmWNjMdhlJI75TCmFWelBxWGeGcFe0zSGteGxwOf1MOQdC5ND2tCE4W7IO3TDbeP2DY9fH1meDAlqQswiFruJRZsSyd+QQHVpKyV7XRx9q5Lj266QvSKGgo2JRMTf++YUMXcogwFzdDTm6GjsBQUTHlO0axdrchYw1KgHuKGmxtEgN9TQwMDZMoavXr3uPGNIyGho0+cxeq1cbIwe6mJjMARIdzTi/lFKYVImTIb7G3++z/fv+lwJakLMQv7de7TV9VCy18XFY42cO9RAfHYoBRsSSSmIxGCQ+4zEFDCZsCTEY0mIv+Eh3sFBhpqaxgW50UDX1Eh/Scl1ozyAPtLDaM1cTCym2JixeaxeSzffnmQVwp8ENSFmuYj4QDb8QQ6rt6bxkrmwAAAgAElEQVRz7lA9pUUu3v9hKUERNvLXJ5D7gIx6IO49g9WqD26flHTDY7wDA3qYa2y6LswNNTXpNXNtbde/dmDg+AB3TZAzx8ZgCJbxc8XcNCVBTSn1GPADwAj8VNO0716zX/n2bwH6gM9qmnbqds4VQtweW6CZpZuTWfxwIpVnWinZW8vh35Xz4buV5Kxykr8hgXCn1EyI6WOw2W76FCv4auaam/V75hobxwe7xiYGLl1kuLVt/IgP+B6AiI4ea2IdDXVjzazG8HCUQR6+EbPLpIOaUsoI/AfwCOACjiul3tE07ZzfYY8Dmb5pJfB/gJW3ea4Q4g4YjAYyCqPJKIympaabkr21nDtcz9n9dSTmhlOwIQFNm/qnnoSYCgarFUtiIpbExBseo3k8DLW0jAW4piaGGhrxNDcx1NhE7/HjDDW3wNDQ+BPNZj3MxcSMu0/OFBPj2xar9zNnksYmMXNMxW/jCqBc07RKAKXUb4CnAf+w9TTwn5r+3+GoUipUKeUEUm7jXCHEXYpKCmLTZ3JZ/UwG5w7WcXZfHdv+owRLIIRrtSxY7cRil39KYnZRZvPoyAw3onm9DLe1XVMr14SnqZGhhkb6y8oY2rMXbeCa4aIMBkwREZhiY/X+5mJixwe7GD3oCXG/TMUndDxQ67fuQq81u9Ux8bd5rhBikgKCLSzbksqSzclUnm7hwFtlHHz1MsferiRnjZOC9QmExsjTd2LuUAYDpqgoTFFRkJ8/4TGapuHt7NRr5JqafKGuWQ9zjU14qqvpO/Yh3u7u686NcjiojI+/5t658bV0hqAguW9OTNpUBLWJfguvbVe50TG3c67+Akp9CfgSQFRUFEVFRXdQxPmhp6dHrss15JpMLHpVH0a3g7ZLw5zd56J0r4tAJ4RnKQJjmbf/XOT3ZWLz5rpER+tT/sJxm9XAAIaODowdHRg6OjC0t+NtbsHd24vxyhUMp89gnCDMea1WvKEhDIeG4Q0L1eehofpyWBjDoaFogYEwh+6bmze/K/fRVAQ1F+B/M0ECUH+bx1hu41wANE37MfBjgOzsbG2k52Mxxr9HaKGTazIx/+vS1+Wm7IDeLFqzz01oTAD56xPIWR2LxTa/mkXl92Vicl2uV1RUxCq/a+J1u/WHIPxq5vT755r1DoSraxg6fgKGh8e9jjKbMY08BBETM+5hCL3fuVhMkZGzZnxW+V2ZelPxKXwcyFRKpQJ1wMeBT1xzzDvAV333oK0EOjVNa1BKtdzGuUKIeygg2MLyJ1JZujmZilPNFO9xceC3lzj2dgUL1sSRvyGekChpFhXiZgwWC5aEBCwJCTc8RhseZqitbSzMNTYx1Oy7f66xkf6zZxnavRttcHD8iUYjpsjIm3dREh2Fslju8bsU02HSQU3TtCGl1FeBD9C72Pi5pmllSqkv+/b/EHgPvWuOcvTuOT53s3MnWyYhxJ0zmgxkrYgla0UsjVWdlOxxUVrkonhvLSn5kRRsTCAhO2zeNosKMVnKaBwbBeIm980Nd3SMC3MjD0AMNTcxWF5O78GDePv6rjvXGBk5Vit3be2cU38owmC13uu3KabYlLRraJr2HnoY89/2Q79lDfjK7Z4rhJhesakhxH4hhN7nMji7v46yA3W886+thMc5KNiQQNbKWMyW2dEUI8RsopTCFBaGKSwMW07ODY8b7unRm1RHmlj9Qp2npoa+48fxdnVdd54xLMxvWC+9Zm7cOK0xMRhstnv5FsUdml83oAgh7ogj1MrKj6RR+Hgyl483U7K3lqJfX+TImxXkro0jf30CQeHyoS7E/WYMDMSYmYk1M/OGx3h7e/2CXJM+VquvmdVTX0//qVMMd3Ze/9q3CnOxsVIzdx9JUBNC3JLJbGTBGic5q2NpqOikZE8tZ3bWcGZXLWmLIynYmIgzPUSaRYWYQQwOB9a0VKxpqTc8xtvf79fXnK9WrrFB70C4oeHmYc4ZiznW6XvowYnZGYu5sRF3Rgbm6Gi5Z26KSFATQtw2pRRxGaHEZYTSfXWA0iIX5w7WU3GqhaikIAo2JpBZGIPRPHe6GxBiLjPY7VhTU7Gm3kaYa/SrmWto1JtZXS76TpwYbWYNByr++V8A3z1zsSO1cE59Oc6p19Y5nfooELPkadbpJEFNCHFXgsJtrHk2g+VPpHLxWCMle13s/uV5Dr9RwcJ1ceQ9GI8jRJpHhJjtbivM9fbiaWzk5Ac7yI2J1oOcr2ZusLKK3kOHr38AwmTCFB2F2Rk3FuicTsy+yRQbizE0dN7X1EtQE0JMitlqZOGD8eSti8N1vp3ivbUc33aFk9uryVwew6KNiUQlBU13MYUQ95DB4cCano47dwGhE/Sjpmka3u5uPA0NeBoa9Nq5hsbR2rn+khK6d+xA83jGnafsdl+Ic+pNrc44X5AbC3Vz/eEHCWpCiCmhlCIxN5zE3HA6mvoo2evi/JEGLh5tJC4zlEUbE0lZFInBML+/HQsxHymlMAYHYwwOxpadPeExmtfL8NWrY2GuwdfE6lsf3H+AoZaW684zhofr4S3O6QtvvjAXr8+NERGzulZOgpoQYsqFxgTw4MezWPmRVM4fbqBkj4v3f1RKcKSN/PUJ5D4QJ4PBCyHGUQaD3rFvZOQN+5nzut3X1MY14Kn3BbmqiZtYlcWi18bFxY2FuDinvh4Xh8npxDCDH3yQT0ohxD1jDTCz+OEkCjYkUFXSSvHuWg69Xs6Hv69iwRonBRsSCYmyT3cxhRCzhMFiwZKUhCUpacL9mqbh7eoaC3D19Xga6vXaufoGeg8dYqi5GbTxw4oboyLHglyc/+TEHB+PMWj6bt+QoCaEuOcMRgPpS6JJXxJNc3UXxXtqOVtUR8leF6kFkSzalEhcptw0LISYHKUUxpAQjCEhN+wwWHO78TQ346mrHw1ynvp6hurrGTx/np49e9Dc7nHnGAIDxwe4+DjM8fGj6/eyeVWCmhDivopODuaRz+Wx5pkMSve5KNtfT1VxK1FJQSzalEhGYTRGk3TvIYS4N9QtxmXVNI3htjY9xNXXjwU639R38iTe7u7xr2mz+e6L84U3/3l8/KTKK0FNCDEtHKFWVj2dTuHjKVw61kjx7lp2/eIcR94oJ39DAnnr4rE5zNNdTCHEPKOUGrtXrqBgwmOGu7vHQlxdnV+oq2Pg3DmGr16dsvJIUBNCTCuzxUjeunhyH4ij5vxVinfVcPStSk5su0LOaieLNiUSGhMw3cUUQohRxqAgjNnZN3yC1dvXNy688YlP3PXPkqAmhJgRlEGRnBdBcl4EbXU9FO+u5dzhes4eqCMlP5LFD8t9bEKI2cEQEIA1IwNrRoa+QYKaEGIuiYgPZOOnF7BqazqlRS7O7qvjrRL9PrbFDyeSXhiN0Sj3sQkh5j4JakKIGSsg2MLKj6RR+FgyF481cmZXLTt/fo4jb1ZQsDGRvLXSH5sQYm6TTzghxIxn8ruPrfpsG2d21XD4d+Wc2FZF7to4CjYmEhQ+t4eREULMTxLUhBCzhjIoUgoiSSmIpLm6izO7aine46Jkj4v0wmiWPJIk44oKIeYUCWpCiFkpOjmYR7+Qx6qtaZTscXHuYD2XjzeRkBPGkkeSSMwNlwcPhBCzngQ1IcSsFhxhZ+1HM1n+RAplB+op3lPLu/+7mIj4QJY8kkjG8hh58EAIMWtJUBNCzAnWADNLNyezaGMil443cWZXDbt+eZ6jb1eyaFMiuWvjsNjkI08IMbvIp5YQYk4xmg0sWOMkZ3Ws/uDBzhoOvV7OifeusPDBeAo2JhIQbJnuYgohxG2RoCaEmJOUUqTkR5KSH0lTVRend1Zz8oNqzuyqJXt1LEseTpIRD4QQM54ENSHEnBeTGsxjX8qno6mPM7truXC4gXMH60lfEsXSzclEJwdPdxGFEGJCEtSEEPNGaEwA6z+RzYonUyneU8vZfXVUnGohISeMpY8mo2nadBdRCCHGkaAmhJh3AoItrN6aTuHmZMoO1HNmdw3v/NsZbGGQGNxM2pIoDAbp2kMIMf0kqAkh5i2L3cSSR5Mo2JDAxWONHHr7Ah/85CyhMQEseTSJ7JWxGE3StYcQYvpIUBNCzHtGs4HctXE0uS+SGJLHye1X2PtfF/jw3SoWP5xI3rp4zFbjdBdTCDEPSVATQggfZVBkFEaTvjSK2nNXObm9mkOvl3Py/WoKNiaQvz4Bm8M83cUUQswjEtSEEOIaSimS8iJIyougsbKTk+9f4cN3qzi9s4b8h+JZtClJ+mITQtwXEtSEEOImYtNCeOIri2h1dXNyezWndtRQvMdF7to4ljySRFC4bbqLKISYwySoCSHEbYhMCGLzFxey8qk+Tn5QTdm+Osr215Gz2snSzcmERNmnu4hCiDlIgpoQQtyB0JgANn16AcufSOH0jhrOH2rg/OEGspbHsPSxZMKdjukuohBiDpGgJoQQdyE4ws5DL2az7PEUTu+qoWx/HRc/bCRjaTTLtqQQER843UUUQswBEtSEEGISHKFW1j6fSeHmZM7srqW0yEX5Sb3T3GVbUohKDJruIgohZjEJakIIMQXsQfpoB0seSaJ4Ty0le1xUnm4hpSCS5U+kyHiiQoi7IkFNCCGmkM1hZuVTaSzelEjJXhfFu2t57TsnSM6PYPkTqcSkSGATQtw+CWpCCHEPWAPMLH8ilUUbEynd5+L0zhpe/+4Jkhf6AluqBDYhxK1JUBNCiHvIYjdR+FgK+esTKC1ycWZnLa//vydIyotgxZMS2IQQNydBTQgh7gOLbeLAlpyvBza5h00IMRHDZE5WSoUrpXYqpS775mETHJOolNqrlDqvlCpTSv2p376/U0rVKaXO+KYtkymPEELMdCOB7VPfXs2qrWk0VnTy2ndOsO2lElpquqe7eEKIGWZSQQ34JrBb07RMYLdv/VpDwNc0TVsArAK+opTK9dv/fU3TFvum9yZZHiGEmBVGAtunv72GlR9JpaG8g1f/6Tjv/7CUtrqe6S6eEGKGmGzT59PAet/yr4Ai4Bv+B2ia1gA0+Ja7lVLngXjg3CR/thBCzHoWu4llW1LJX59A8R4XxbtqqCxuIbMwmuVPphIWKyMdCDGfKU3T7v5kpTo0TQv1W2/XNO265k+//SnAfmChpmldSqm/Az4LdAEn0Gve2m9w7peALwFERUUVvvrqq3dd7rmqp6eHwEDpDd2fXJOJyXWZ2Ey4LkODGm0XNa5eAu8whCZD1EKFJVBNW5lmwnWZaeSaTEyuy8Q2bNhwUtO0ZXdz7i2DmlJqFxA7wa6/An51u0FNKRUI7AO+rWnaG75tMUAroAH/ADg1Tfv8rQqdnZ2tXbx48VaHzTtFRUWsX79+uosxo8g1mZhcl4nNpOvS1+Xm1I5qzu6rQxvWyHnAyfItKQSG2e57WWbSdZkp5JpMTK7LxJRSdx3Ubtn0qWnawzf5wU1KKaemaQ1KKSfQfIPjzMDvgF+PhDTfazf5HfMT4Pd3UnghhJirAoItrH0+kyUPJ3Hy/SuUHazn4pFGFj4UT+FjydiDLNNdRCHEfTDZhwneAT7jW/4M8Pa1ByilFPAz4Lymaf9yzT6n3+ozwNlJlkcIIeYUR6iVB1/M5pN/v4rMFTGU7KnlP//6CMfeqWSwzzPdxRNC3GOTDWrfBR5RSl0GHvGto5SKU0qNPMH5APApYOME3XD8f0qpUqVUCbAB+LNJlkcIIeak4Eg7mz69gBf/diUpCyM48d4V/uuvj3Dqg2o87uHpLp4Q4h6Z1FOfmqa1AZsm2F4PbPEtHwQmvAtW07RPTebnCyHEfBMW62DzHy5k6WPdHHu7kiNvVlC8p5blT6Sy4AEnRuNkv38LIWYSGZlACCFmoajEIJ786iLqyzs4+mYF+16+yOmdNaz8SCqZhTEow/Q9JSqEmDry1UsIIWaxuIxQnvn6Up74SgFmi5GdPzvHb//pONVlbUym+yUhxMwgNWpCCDHLKaVIyY8kOS+CyyeaOPZOJb//38XEZ4ey+pkMYlJkHFEhZisJakIIMUcogyJrRSzpS6MpO1DHifeu8Pp3T5C+NJpVT6cRGhMw3UUUQtwhCWpCCDHHGE0GCjYkkrPayZmdNZzeVUvlmRby1sax/MlUAoKlDzYhZgsJakIIMUdZbCZWPJXGwocSOL6tinMH6rl4rJEljyaxaFMiFpv8CxBippO/UiGEmOMCgi089GI2izYmcvStCj58t4qz++pY8VQqC9Y4MUiXHkLMWPLXKYQQ80RoTACP/VE+z/5FISFRdop+fZHf/MOHVJW0yhOiQsxQEtSEEGKecaaH8MzXl/L4H+WjafDeSyW8/a+naanpnu6iCSGuIU2fQggxDymlSFsSRXJBBGX76zn++ype/afjZK+MZeXTaQSF26a7iEIIJKgJIcS8ZjQaKNiQQPaqWE5tv0Lxbhflp5pZ/HAiSzcnywMHQkwz+QsUQgiB1W5i9TMZ5D0Yz9G3Kjn5fjXnDzWw8uk0NK/cvybEdJF71IQQQowKjrDz6BfyeO4bhQRH2tj7Xxeo3KHhunB1uosmxLwkQU0IIcR1YlNDePYvCnn0i3kMu+Htfz3DtpdK6Gjqm+6iCTGvSFATQggxIaUUmctiyHhCsWprGnUX23nlW8c49PplBvs80108IeYFuUdNCCHETRmMisJNKeSsdnLs7UrO7K7l4rFGVjyVRu7aOAwGNd1FFGLOkho1IYQQt8URYmXjpxfwwl8uJzQmgH0vX+TVbx/HdbF9uosmxJwlQU0IIcQdiUoK4pmvLWXzHy7E3T/E298/zfYfl9LV1j/dRRNizpGmTyGEEHdMKUVGYTQp+RGc3lnDqe3VXCltY+mjSSzZnIzZYpzuIgoxJ0hQE0IIcddMFiPLn0glZ7WTw2+Uc3zbFc4faeCB5zJJXxqFUnL/mhCTIU2fQgghJi0o3MbmLy5k658vwRpg5oOfnOXtfz1NW33PdBdNiFlNgpoQQogpE58Vxgv/z3Ie/HgWrbU9/PYfj3PwtcsM9g9Nd9GEmJWk6VMIIcSUMhgU+esTyFgWzdG3KyneU8ul402seTad7BWxKOnOQ4jbJjVqQggh7gl7oIUNn8zho99cRnCEjd2/PM8b3ztFS233dBdNiFlDgpoQQoh7Kjo5mOf+opCNn86hs6WP1/7pOPt/e0maQ4W4DdL0KYQQ4p5TBsWCNXGkLori2DuVlBa5KD/ZzAPPZZC1IkaeDhXiBqRGTQghxH1jc5h56MVsPvrNZQSF29j1i3O89S/ydKgQNyJBTQghxH0XnRzM8/+jkA1/kENbfQ+v/uNxDr9RjmdweLqLJsSMIk2fQgghpoUyKHLXxpG6OJIjb1ZwekcNl080se6FLNIWR0138YSYEaRGTQghxLSyB1rY+KkFPPv1pVhsJt7/YSnbXiqRsUOFQIKaEEKIGcKZEcoLf7WcNc9m4LrYzit/f4xTO6oZHvZOd9GEmDbS9CmEEGLGMBoNLHk0iYxl0Rz47SWOvFHBpWNNrP9kNrFpIdNdPCHuO6lRE0IIMeMEhdvY8n8V8PiX8xns8/C7/3WSfa9clL7XxLwjNWpCCCFmrLTFUSTkhOl9r+11UXm6hbUvZJJRGC19r4l5Yc4ENY/Hg8vlYmBgYLqLMm1CQkI4f/78PXltm81GQkICZrP5nry+EELciMVmYt0LWWSvjKXo1xfZ8dMyLh1r5MEXswkKt0138YS4p+ZMUHO5XAQFBZGSkjJvv2V1d3cTFBQ05a+raRptbW24XC5SU1On/PWFEOJ2RCcH8/w3CinZ6+LYO5W88vfHWLU1jYUPJWCQgd7FHDVn7lEbGBggIiJi3oa0e0kpRURExLyurRRCzAwGo4HFDyfx4t+sxJkewoHfXuaN/3WStjoZ2UDMTXMmqAES0u4hubZCiJkkONLOk3+yiEc+n0tXaz+vfvs4R9+uYMgjIxuIuWXONH0KIYSYX5RSZK2IJSk3goOvX+bk+9VUnm5h/R/kEJcROt3FE2JKTKpGTSkVrpTaqZS67JuH3eC4K0qpUqXUGaXUiTs9XwghhLgRW6CZhz+by1N/soght5c3v3eKfa9cxC1deYg5YLJNn98Edmualgns9q3fyAZN0xZrmrbsLs8XQgghbigpL4KP/80KCjYmcHZ/Ha986xhXSlunu1hCTMpkg9rTwK98y78Ctt7n8+eVz3/+80RHR7Nw4cIbHrN9+3ays7PJyMjgu9/97h2fL4QQs9lIVx7P/UUhZpuJbf9Rwq5fnmOg1zPdRRPirihN0+7+ZKU6NE0L9Vtv1zTtuuZLpVQV0A5owI80TfvxnZzv2/cl4EsAUVFRha+++uq4/SEhIWRkZNz1e5kNDh06hMPh4I/+6I84duzYdfvdbjfLli3j7bffJj4+nvXr1/Pzn/+cnJyc2zr/VsrLy+ns7Jz0+7ifenp6CAwMnO5izDhyXSYm12Vis/W6eIc1Ws9ptJwDkxWcyxTBCVPzYNRsvSb3mlyXiW3YsOHkNS2Kt+2WDxMopXYBsRPs+qs7+DkPaJpWr5SKBnYqpS5omrb/Ds7HF+5+DJCdna2tX79+3P7z58+P9iH29++Wca6+605e/pZy44L526fybnpMaWkpX/7ylzl06BAAp06d4utf/zp79uyZkjI89thjXLlyBYPBMGF/abt27SIrK4uCggIAPvGJT7Br1y6WL19+W+ffis1mY8mSJZN7E/dZUVER1/6uCLkuNyLXZWKz+rpsgpbabvb853lqD/aQURjFuo9lERBsmdTLzuprcg/JdZl6twxqmqY9fKN9SqkmpZRT07QGpZQTaL7Ba9T75s1KqTeBFcB+4LbOny3y8vKoqKhgeHgYo9HI1772Nf75n//5puesW7eO7u7u67Z/73vf4+GHb3jpJ9TQ0EBiYuLoekJCwl3VnAkhxFwSlRjE899cxukPajj+XhWui+08+LEsMpbJMFRi5pts9xzvAJ8Bvuubv33tAUopB2DQNK3bt/wo8K3bPf9u3Krm614xGAzk5eVRVlbG5cuXSUpKYunSpfT29vLHf/zHWCwW1q9fzyc/+cnRcw4cODBlP3+iZmz5EBJCCDAaDSzbkkLq4kj2/OcFdvysjPJTzTz0Yvaka9eEuJcmG9S+C7yqlPoCUAN8FEApFQf8VNO0LUAM8KYvMJiAlzVN236z82ezVatWcejQIV566SW2b9ff5htvvMHzzz/PU089xcc+9rFxQW0qa9Ti4uKora0dXXe5XMTFxd3lOxFCiLknIi6Q5/5iKWd21XLs3UrqL3Xw4ItZZC6Lme6iCTGhSQU1TdPagE0TbK8HtviWK4FFd3L+bLZq1So++9nP8pWvfIX4+HhAD0z5+fkAGI3GccdPZY1aYWEhly9fpqqqivj4eH7zm9/w8ssvT9nrCyHEXGAwGli6OZmU/Eh2/+ocO35aRsWpZh78uNSuiZlnTg0hNRPk5ORgtVr5xje+MbotISEBl8sFgNfrvevXfvHFF1m9ejUXL14kISGBn/3sZwBs2bKF+vp6TCYT//7v/87mzZtZsGABL7zwAnl5ebc8Xwgh5qPwOAfP/Y9CVm1No6qklVe+dYyKU7P6VmkxB8kQUlPsBz/4Ad/5zndwOByj25599lm++tWvsm3bNp566qm7fu1XXnllwu3vvfceAN3d3WzZsoUtW7bc0flCCDFfGYwGCh9LIaUgkt2/PM/2H58lc3kMD348C5vDPN3FE0KC2lSpqKjgiSee4IEHHuAzn/nMuH0Oh4Nf/OIX01QyIYQQtxIRF8hz3yjk5PvVnHzvCvWX2tnwqQUkL4yY7qKJeU6C2hRJT0/nwoUL010MIYQQd8loNLDiyVRSCyLZ9ctz/P7fi8l9wMkDH83EYpN/l2J6yD1qQgghhJ+opCBe+MvlLHk0iXOHG/jNP3xI/eWO6S6WmKckqAkhhBDXMJoNrHk2g2e/thSl4M1/OcXh35Uz7Ln7B8KEuBsS1IQQQogbcGaE8rG/XkHu2jhO76zhte8ep9XVM93FEvOIBDUhhBDiJiw2Exs+mcMTXymgr9vDa985Tst5Da/3+tFghJhqEtSEEEKI25CSH8mLf7OC1IJImos13v7+abra+qe7WGKOk6AmhBBC3CZ7oIXNX1pI/EpFS203v/2HD7l4rHHCsZaFmAoS1IQQQog7oJQiNFXx8b9eQUR8ILt+cY4dPytjoNcz3UUTc5AENSGEEOIuBEfa2fq1pax8Oo3KUy389h8/xHXh6nQXS8wxEtRmkc///+3de3hV1bnv8e/Iyg3CXQiXrCCXYC4rEQgXYykSW5BNMF4CKpbuTYptNyXsXjaU1HY/53iesw9Qi27TjbSnaKm2B6jWIEgCrUgt7JQaIAqBBIwpCAEEjSIBxEsY548ssglZuZCsZCZZv8/zrIc15xhzrjcvQ3kz1pxzzJ9PZGQkiYmJDfbZtm0bsbGxxMTEsGLFima1Nee8IiJSX1CQYfyMYczKHkdwqItNOW/x19x3qP5Cj/EQ/1Ch1olkZmaybdu2Bturq6vJyspi69atlJSUsH79ekpKSppsa+q8IiLSuMibe/HgTybg+fIQ3vzTcV56fB/nzlxyOizpArrmmhhbfwTvFfv3nIOSYMaKRrsUFxezYMECCgoKACgqKmLJkiXs2LHDLyHccccdHDt2rMH2vXv3EhMTw4gRIwCYM2cOmzZtIiEhgcLCwgbbmjqviIg0LSTURercOIYm3MSO35Xy+/9TyOSHbiH+S4MxxjgdnnRSXbNQc4jH46G8vJzq6mpcLheLFy/miSeeaPSYyZMnU1VVVW//ypUrmTp16g19/unTp4mOjq7ddrvdvPHGGwCcPHmywTYREfGfEWMHEDmsF9t/U8Kff3uY4wcrSf16HOERIU6HJp1Q1yzUmpj5aitBQUF4PB4OHTpEWVkZQ4cOJTk5mYsXL7Jw4UJCQ0NJTU1l7ty5tcfs2rXLb5/v62xYb+cAABilSURBVPbwq7/FNdYmIiL+1aNvGPd+bwxvbj/OG5v+zpl/L2TafA9DRvVxOjTpZLpmoeaglJQUCgoKWL16de11X7m5ucyePZv09HQeeuihOoWaP2fUhgwZwokTJ2q3KyoqGDJkCFAzg9ZQm4iI+J8JMiTfdTPu2L786ZlDvPxkERPuHs64GcMICtIvytI8KtT8LCUlhczMTLKysoiKigJqiqKkpCQAXC5Xnf7+nFEbN24cZWVlHD16lKioKDZs2MC6desAmDBhQoNtIiLSdq7eaLBz/dsUvnKUisMfMfUbCfTsF+50aNIJ6K5PP4uLiyMsLIzs7OzafW63m4qKCgCuXGn5LdsPP/wwt99+O0eOHMHtdvPss88CkJaWxqlTpwgODmbVqlVMnz6d+Ph4HnzwQTweD0CjbQ2dV0RE/CM0PJip30hgamY87x+v4vf/Xsjf33zf6bCkE9CMmp/l5OSwfPlyIiIiavdlZGSwaNEi8vLySE9Pb/G5169f73N/fn4+AFVVVaSlpZGWluazX0NtDZ1XRET8KzZlMANH9ObVZw+x9f8Wk5Tq5kuzRhIc4mr6YAlIKtT8pLy8nJkzZzJp0iTmzZtXpy0iIoK1a9c6FJmIiHQkfSK7k/HDcex+uZz9209wuvwc07+ZSJ+B3Z0OTTogFWp+MnLkSA4fPux0GCIi0gm4goP48uxRuG/py/bnSvj9sj2kfi2W2NsGOR2adDC6Rk1ERMQhw27tz5x/m8iA6JrF3V97roTPP612OizpQFSoiYiIOKhH33Du+8FYxqcN4/Df3uPF5XuoPHXB6bCkg1ChJiIi4rAgVxC33TOCe747hssXP+cPK/ZyePdpp8OSDkCFmoiISAcRHd+Ph/5tIgOH9eK150p57flSPv9MX4UGMhVqIiIiHUhE7zDu+d6Ymq9Cd5/mDyv28uHpi06HJQ5RoSYiItLBXP0qNP1fRvNJ1We8uHwPbxe+53RY4gAVaiIiIh3U0ISbeOgnExkwtCev/rqEv6w7QvXnLV/hRjofFWoiIiIdWESfMO79wVjGThvKwZ0nyV25j/MffOJ0WNJOVKh1IvPnzycyMpLExESnQxERkXbkcgXxpVkxzFiQxLmzn/DCsj0cO/CB02FJO1Ch1olkZmaybds2p8MQERGHjBgzgAd/PIGeN4WTt/oAu18u58oV63RY0oa65BJSPy38KYc/9O9yTnH94siemN1on+LiYhYsWEBBQQEARUVFLFmyhB07dvglhjvuuINjx4755VwiItI59R7QjVlLx7Hr92UUbXuXs8fOc9cjHrr1DHU6NGkDXbJQc4rH46G8vJzq6mpcLheLFy/miSeeaPSYyZMnU1VVVW//ypUrmTp1aluFKiIinVhwiIs7vx7HoBG9+Mu6t3lh2R6mfzuRQcN7Ox2a+FmXLNSamvlqK0FBQXg8Hg4dOkRZWRlDhw4lOTmZixcvsnDhQkJDQ0lNTWXu3Lm1x+zatcuRWEVEpPOL/9IQ+rt7su1XxWxcWcSXHxhF4pQojDFOhyZ+0iULNSelpKRQUFDA6tWra68ny83NZfbs2aSnp/PQQw/VKdQ0oyYiIq0xYGhPHnh0AtvXlrBzw9ucOXqeKXNjCQl1OR2a+IEKNT9LSUkhMzOTrKwsoqKiAKioqCApKQkAl6vufziaURMRkdYKjwhh5sJb2bv1GIVbjvLByQvM+Ockeg/o5nRo0kqtuuvTGNPPGPOqMabM+2dfH31ijTFvXfM6b4z5vrftMWPMyWva0loTT0cQFxdHWFgY2dn//fWr2+2moqICgCtXWv6gwocffpjbb7+dI0eO4Ha7efbZZ1sdr4iIdA0myDBh5nDuzhrNhQ8v8+LyPbx7qNLpsKSVWjuj9iPgNWvtCmPMj7zbdS4Qs9YeAcYAGGNcwElg4zVd/sNau7KVcXQYOTk5LF++nIiIiNp9GRkZLFq0iLy8PNLT01t87vXr1/sjRBER6cJuTryJBx4dz9ZfHmTLqv3cds8Ixk2/GROk69Y6o9YWavcCqd73zwGvc12hdp2vAuXW2ndb+bkdTnl5OTNnzmTSpEnMmzevTltERARr1651KDIREQk0vQd0Z9bScfz5d4d5Y9PfOXvsPFMzEwjtpiueOhtjbcsflGeMOWet7XPN9kfW2npff17T/mugyFq7yrv9GJAJnAf2AouttR81cOy3gW8DDBgwYNwLL7xQp713797ExMS0+GfpCq4+FqStvPPOO3z88cdtdv62cOHCBXr06OF0GB2O8uKb8uKb8lJfZ8mJtZYP34b33rKE9oChkw1hvdpuZq2z5KW93XnnnfusteNbcmyThZoxZjswyEfTT4DnmluoGWNCgVOAx1p7xrtvIPABYIH/DQy21s5vKujY2Fh75MiROvtKS0uJj49v6tAuraqqip49e7bZ+Ttjjl9//XVSU1OdDqPDUV58U158U17q62w5Ofn2R/xxzUGqP7/CtPkeht3av00+p7Plpb0YY1pcqDV5M4G1dqq1NtHHaxNwxhgz2BvEYOBsI6eaQc1s2plrzn3GWlttrb0CrAEmtuSHEBERkYZF3dKXBx6dQO/I7uT94gB7849itfRUp9DatT43A1cvyJoHbGqk78NAnavhrxZ5XvcDB1sZj4iIiPjQs184GUuSuWXiQN7YfJRtaw7y2eUvnA5LmtDaQm0FMM0YUwZM825jjBlijMm/2skY093bnnvd8Y8bY4qNMQeAO4EftDIeERERaUBwqIupmQl8+YFRHN3/AS89vo9zZy85HZY0olW3f1hrK6m5k/P6/aeAtGu2LwE3+ej3j635fBEREbkxxhhGfzWaflER/GnNIf6wYi/Tv5VIdHw/p0MTH1o7oyYiIiKdUHRcPx54dDwRfcJ45edvsf+1E7TmSRDSNlSoiYiIBKhe/bsxa+k4ht3an/96sYwdvz1M9ectX0FH/E+FmoiISAALDQ9mxj8nMX7mMA7/9TQv/0cRFz/+1OmwxEuFWieybds2YmNjiYmJYcWKFT775OTkkJiYiMfj4amnnmpWW2PHiIhI12eCDLelj2D6txL5oOICLy7fy/vHq5wOS1Ch1mlUV1eTlZXF1q1bKSkpYf369ZSUlNTpU1JSwpo1aygsLGT//v1s2bKFsrIyAA4ePOizraH9IiISeGLGRZLxw3EYA7k/28c7+xp7PKq0hy656Nd7y5bxaelhv54zLD6OQT/+caN9iouLWbBgAQUFBQAUFRWxZMkSduzY0erPLywsJCYmhhEjRgAwZ84cNm3aREJCQm2fI0eOkJKSQvfu3QGYMmUKGzduZOnSpZSWlvpsGz58eIPHiIhI4BkQ3ZMHHp3A1l8e4I9rDvLRe8MZnzYMY7SouxM0o+ZHHo+H8vJyqqurAVi8eDErV65s9JjJkyczZsyYeq/t27fX6Xfy5Emio6Nrt91uNydPnqzTJyEhgZ07d1JZWcmlS5fIz8/nxIkTACQmJvpsa2i/iIgEru69QrnvB8nEpgyi8JWj/OmZQ3z+WbXTYQWkLjmj1tTMV1sJCgrC4/Fw6NAhysrKGDp0KMnJyVy8eJGFCxcSGhpKamoqc+fOrT1m165dzTq3r1umr//tJjY2luzsbKZNm0aPHj0YPXo0wcE1f8Xx8fE+2xraLyIigc0VEsRX58XTb3AEu18u5+P3PyHtO7fSo2+Y06EFFM2o+VlKSgoFBQU89thjLFu2DIDc3Fxmz57NmjVr2Lx5c53+zZ1Rc7vddWa6KioqGDJkSL3Pf+SRRygqKmLnzp3069ePUaNGNdnW2DEiIhK4jDEkT7+ZtAVJnDtziT+s2MPZd887HVZA0dSJn6WkpJCZmUlWVhZRUVFATVGVlJQEgMvlqtO/uTNqEyZMoKysjKNHjxIVFcWGDRtYt25dvX5nz54lMjKS48ePk5uby+7du5tsa+wYERGR4aMHkPHDceSt3s/GJ4qY9g0PI8YOcDqsgKBCzc/i4uIICwsjOzu7dp/b7aaiooIxY8Zw5UrLHiQYHBzMqlWrmD59OtXV1cyfPx+PxwNAWloazzzzDD179mTWrFlUVlYSEhLC008/Td++fWvP0VBbY8eIiIgA9Hf3YHb2eLb+spitvyrm9vtGMvauobrJoI2pUPOznJwcli9fTkRERO2+jIwMFi1aRF5eHunp6S0+d1paGmlpafX25+fnA1BVVdXoDF1Dbc2d1RMRkcAW0TuM+34wlteeL2X3xnI+OnOJ1K/F4grWlVRtRYWan5SXlzNz5kwmTZrEvHnz6rRFRESwdu1ahyITERHxn+BQF3fN99BnYHf25h3j/PufMGNBEuERIU6H1iWpUPOTkSNHcviwf5/dJiIi0hFdXcmg78DuvPZ8KS89vo+ZWbc6HVaXpLlKERERaZFbJg7i3u+P5fKFz3npp/u4+H79R0lJ66hQExERkRYbEtOHWdnjCIsI5t0/W97e857TIXUpKtRERESkVfpEdmf20vF0uwlefbaEvflHfT6oXW6crlETERGRVgvvEcLNqYYrxwfyxuajfHz2E1K/Hqc7QltJhZqIiIj4RZDLcGdmPL0GdGPPlqNUffQpMxYkEdZN5UZLqcwVERERvzHGMPHu4Xx1Xjyny86R+7N9VH142emwOi0VaiIiIuJ3cbcP5u7vjubCh5f5w0/38v7xKqdD6pRUqImIiEibiI7rR8YPxxEUZNj4RBHvHqx0OqROR4VaJ7Jt2zZiY2OJiYlhxYoVPvvk5OSQmJiIx+PhqaeeanabiIhIW7gpqmaN0N6R3chbfYCSglNOh9SpqFDrJKqrq8nKymLr1q2UlJSwfv16SkpK6vQpKSlhzZo1FBYWsn//frZs2UJZWRkABw8ebLBNRESkLUX0CeP+xclEx/Xlz789TOErf9fjO5qpS96GseuFt/ngxAW/nrN/dA8mP3hLo32Ki4tZsGABBQUFABQVFbFkyRJ27NjR6s8vLCwkJiaGESNGADBnzhw2bdpEQkJCbZ8jR46QkpJC9+7dAZgyZQobN25k6dKllJaWNtgmIiLS1kLDg0nLupXXf3eYPXnHuHDuU6Z8LRaXS3NGjVF2/Mjj8VBeXk51dTUAixcvZuXKlY0eM3nyZMaMGVPvtX379jr9Tp48SXR0dO222+3m5MmTdfokJCSwc+dOKisruXTpEvn5+Zw4cQKAxMTEBttERETag8sVxFf+KZ7xacMoLThN/upiPrv8hdNhdWhdckatqZmvthIUFITH4+HQoUOUlZUxdOhQkpOTuXjxIgsXLiQ0NJTU1FTmzp1be8yuXbuadW5fU8TGmDrbsbGxZGdnM23aNHr06MHo0aMJDq75K46Pj2+wTUREpL0YY7jtnhH06BvGX9Yd4eUn3+TuRaPp3ivU6dA6JM2o+VlKSgoFBQU89thjLFu2DIDc3Fxmz57NmjVr2Lx5c53+zZ1Rc7vddWbAKioqGDJkSL3Pf+SRRygqKmLnzp3069ePUaNGNatNRESkPXkmRzHjO7fy0emLvPT4Xs6dueR0SB2SplT8LCUlhczMTLKysoiKigJqiqqkpCQAXC5Xnf7NnVGbMGECZWVlHD16lKioKDZs2MC6devq9Tt79iyRkZEcP36c3Nxcdu/e3aw2ERGR9jb81v7c+69jyVt1gNyV+7h70Wgib+7ldFgdimbU/CwuLo6wsDCys7Nr97ndbioqKgC4cuVKi84bHBzMqlWrmD59OvHx8Tz44IN4PB4A0tLSOHWq5nbnWbNmkZCQQHp6Ok8//TR9+/atPUdjbSIiIk4YNLw3GT9MJjjUxcYn3+R4iZ61di3NqPlZTk4Oy5cvJyIionZfRkYGixYtIi8vj/T09BafOy0tjbS0tHr78/PzAaiqqmp0hq65s3ciIiLtqe+gCGYtHccr/7mfvFUH+Mq8eGJvG+R0WB2CCjU/KS8vZ+bMmUyaNIl58+bVaYuIiGDt2rUORSYiItLxRfSuedba1l8cYPvaEi6d/4yx04Y6HZbjVKj5yciRIzl8+LDTYYiIiHRaYd2CSf+XMby6toS/vvQOlz7+lC9lxGCCTNMHd1G6Rk1EREQ6DFdIEHd900PSlCje2n6CHc+XUl3dsuu7uwLNqImIiEiHEhRkmDznFrr1CqXwlaNcvvg5d30rkZBQV9MHdzFdakZN64a1HeVWRETakzGGCTOHM+XhWzh2sJJXct7i8sXPnQ6r3XWZQi08PJzKykoVFG3AWktlZSXh4eFOhyIiIgEmcYqb6d9M5Myx87z8ZBEXz33qdEjtqst89Xn1WWXvv/++06E45vLly21WTIWHh+N2u9vk3CIiIo2JGRdJWPdg8n9ZzEs/28c93xtDn8juTofVLrpMoRYSEsLw4cOdDsNRr7/+OmPHjnU6DBEREb+Lju/H/f86lld+vp/clUXc890x9Hf3cDqsNteqrz6NMQ8YYw4ZY64YY8Y30u8fjDFHjDHvGGN+dM3+fsaYV40xZd4/9ah8ERER8Sny5l7cvyQZl8vw8pNFnC7/2OmQ2lxrr1E7CGQAOxvqYIxxAU8DM4AE4GFjTIK3+UfAa9baUcBr3m0RERERn/oNjuD+JcmE9whhc86bHD/UtZecalWhZq0ttdYeaaLbROAda+3frbWfARuAe71t9wLPed8/B9zXmnhERESk6+t1Uzcyloyjz8Du5K0+wDv7zjodUptpj2vUooAT12xXALd53w+01p4GsNaeNsZENnQSY8y3gW97Nz81xhxsi2A7uf7AB04H0cEoJ74pL74pL74pL/UpJ745k5fV7f6JNyq2pQc2WagZY7YDvlZG/Ym1dlMzPsPXug83/AwNa+2vgF95Y9prrW3wmrhApbzUp5z4prz4prz4przUp5z4prz4ZozZ29JjmyzUrLVTW3pyrwog+pptN3DK+/6MMWawdzZtMNB15y5FREREblB7PPB2DzDKGDPcGBMKzAE2e9s2A/O87+cBzZmhExEREQkIrX08x/3GmArgdiDPGPNH7/4hxph8AGvtF8Ai4I9AKfCCtfaQ9xQrgGnGmDJgmne7OX7Vmri7MOWlPuXEN+XFN+XFN+WlPuXEN+XFtxbnxWjJJREREZGOqcus9SkiIiLS1ahQExEREemgOmyhZoz5tTHmbEPPSzM1fu5dluqAMSa5vWN0QjPykmqM+dgY85b39T/aO8b2ZoyJNsb82RhT6l3S7Hs++gTceGlmXgJxvIQbYwqNMfu9eflfPvoE1HhpZk4CbqxcZYxxGWPeNMZs8dEWUGPlWk3kJSDHizHmmDGm2Psz13skR0vGS0delP03wCrg+QbaZwCjvK/bgF/w3w/S7cp+Q+N5Adhlrb27fcLpEL4AFltri4wxPYF9xphXrbUl1/QJxPHSnLxA4I2XT4GvWGsvGGNCgP8yxmy11v7tmj6BNl6akxMIvLFy1feouRmul4+2QBsr12osLxC44+VOa21DD/294fHSYWfUrLU7gQ8b6XIv8Lyt8Tegj/dZbF1aM/IScKy1p621Rd73VdT8jyPqum4BN16amZeA4x0DF7ybId7X9XdVBdR4aWZOApIxxg3MBJ5poEtAjZWrmpEX8e2Gx0uHLdSawdfSVAH/j5DX7d6vMLYaYzxOB9OejDHDgLHAG9c1BfR4aSQvEIDjxfuVzVvUPGT7VWttwI+XZuQEAnCsAE8BS4ErDbQH3FjxaiovEJjjxQJ/MsbsMzVLX17vhsdLZy7U/LI0VRdUBNxsrR0N/CfwssPxtBtjTA/gJeD71trz1zf7OCQgxksTeQnI8WKtrbbWjqFmpZSJxpjE67oE3HhpRk4CbqwYY+4Gzlpr9zXWzce+Lj1WmpmXgBsvXpOstcnUfMWZZYy547r2Gx4vnblQa2xpqoBlrT1/9SsMa20+EGKM6e9wWG3Oe13NS8D/s9bm+ugSkOOlqbwE6ni5ylp7Dngd+IfrmgJyvEDDOQnQsTIJuMcYcwzYAHzFGPO76/oE4lhpMi8BOl6w1p7y/nkW2AhMvK7LDY+XzlyobQb+yXsHRQrwsbX2tNNBOc0YM8gYY7zvJ1Lzd1zpbFRty/vzPguUWmufbKBbwI2X5uQlQMfLAGNMH+/7bsBU4PB13QJqvDQnJ4E4Vqy1j1pr3dbaYdQsf7jDWvv167oF1FiB5uUlEMeLMSbCe+MWxpgI4C7g+ic03PB46bB3fRpj1gOpQH9Ts0zV/6TmAlestb8E8oE04B3gEvANZyJtX83Iy2zgO8aYL4BPgDm26y8/MQn4R6DYe40NwI+BoRDQ46U5eQnE8TIYeM4Y46LmH48XrLVbjDELIGDHS3NyEohjxacAHysN0nhhILDRW58GA+ustdtaO160hJSIiIhIB9WZv/oUERER6dJUqImIiIh0UCrURERERDooFWoiIiIiHZQKNREREZEOSoWaiIiISAelQk1ERESkg/r/hj9l8I0p1ocAAAAASUVORK5CYII=\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "%matplotlib inline\n", "from matplotlib.pylab import *\n", "\n", "sol_exacte = lambda t,y0 : (y0-1)*t**3+1./t**2\n", "\n", "# INITIALISATION\n", "t0 = 1\n", "tfinal = 5\n", "c = 1\n", "\n", "tt=linspace(t0,tfinal,101)\n", "Y0=[c+1.e-2,c+1.e-3,c,c-1.e-3,c-1.e-2]\n", "figure(figsize=(10,7))\n", "for y0 in Y0:\n", " yy=[sol_exacte(t,y0) for t in tt]\n", " plot(tt,yy,label=('$y_0=$'+str(y0)))\n", "legend()\n", "grid()\n", "title('Solution exacte - differents valeurs de $y_0$')\n", "axis([t0,tfinal,-c,c]);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "On constate qu'une petite perturbation de la condition initiale entraîne une grosse perturbation de la solution, en effet\n", "$$\n", "\\lim_{t\\to+\\infty}y(t)=\n", "\\begin{cases}\n", "+\\infty &\\text{si } a>1,\\\\\n", "0^+ \t&\\text{si } a=1,\\\\\n", "-\\infty\t&\\text{si } a<1.\n", "\\end{cases}\n", "$$ \n", "Si $a=1$, la solution exacte est $y(t)=t^{-2}$ mais **le problème est numériquement mal posé** car toute (petite) erreur de calcul a le même effet qu'une perturbation de la condition initiale: on \"réveil\" le terme dormant $t^{3}$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Lorsqu'on essaye d'approcher la solution avec les méthodes d'Euler explicite et implicite, bien sûr lorsque $N\\to+\\infty$ on converge vers la solution exacte, cependant en pratique $N$ est fini et on voit qu'inevitablement nous allons approcher une autre solution que celle corresponsante à $a=1$." ] }, { "cell_type": "code", "execution_count": 62, "metadata": {}, "outputs": [ { "data": { "image/png": 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n6T87grR0Gx4WEx/3a0ST6r4Ouf7o0aNp3LgxQ4YMyWrLywiKixcv8thjjzFx4kRCQ0MBKF++PElJSaSnp2OxWIiPj8fPz88hcQohhBCuoMILIwl9/wMs6elYAwNJKl4cnytXAKhX1YeXHn6AD3+JYdm24zzRxN+h1z53OZVxy3ZRu0ppRnWs6dBzi0IgLgrmP25M6TB7QsmKULurs6MS+SAzOaEdUHjBbUdQ+PbtS7HUVJrFxqJNJjbUrYMCzmer3i+EyB9NqvuyaFgoLz8SzKJhoQ5LTgCULVuWPn365ChmeacRFKmpqfTs2ZOBAwfSu3fvrOOUUrRv3z5rv/nz59O9e3eHxSqEEEI4m2/fvhS/do0mBw5gM5nYWMfoE1/Zvh2A59rcT/P7yvL2D3uIO3fFodd+a8VuLlxNZVqfhnha3PZrhcgvW+cbyQkAW4bUnSikHJmcADdOUACYvb1pbR/Stq5ePQASpt19gT0hxN1rUt2Xke2DHJqcyDRmzJi7Ws1jyZIlrFu3jnnz5hESEkJISAhWqxWAyZMnM23aNIKCgkhMTGTo0KEOj1cIIYRwJpPFkmM1D4CDr7wMgNmk+KhPQwDGLNlBhs0xXyJ+3HGCn3eeZHSnB6hdpbRDzikKkeQE2PczoECZpe5EIZVbcsKEifRz6X97iU23neIB4Pt0f9r88COT+/Rmfd26pJlM6AsXnB2WEOJvSE5OznpfqVIlrlzJ+195nn76aZ5++ulctwUGBhIVFXXP8QkhhBCuqlSXzrTevIUpvXqxrl490k0mTMdPZW0PKFucCd3q8kr4DmauO8SIdvff0/V+jz7Nq0t38EClkjzXJvBewxeFzdEIWD4C0i5Dzy/gYryRnJC6E4XKrZIT40PH0+din7z/pfEGbj2CotIrr1A9IYHAkye5WKIEW2vK3DchhBBCCFG0+E+ZQo1Tp6h++jRJJUuyLeh+FDajBoDdE42rEla/MtN+i2H38bz/QS/DpjmQkMwPO04wadU+un/yF8MWbCElzcbRxCvsiJc/Dops4qJg/mNw/hBoDWVrQOsxkpwoZG6XnOgd3Ps2R96ZWycoAFCKjtYdAPzeKAQFHOrdx7kxCSGEEEIIUYDMStHRPr3xj5AQFCZ2vTgma7tSivd61Me3uCcvLbaSknZzpf2UtAx2xCXxdeQxxi/fRc/PNlDv7V/pNO1/jPpmO3P+OsTJCylZ+6dn2Ig4lJj/Nyfcx/aF1+tOaJvUnSiE8jM5AW4+xQOgZMuWdLRamdWlM2saNuSNxUu4umuXs8MSQgghhBCiwHjVq0cn6w6+fPRR/mjYkHFLwlH7co6y9i3hydTeDRn4ZRT9Zm6iR0hV0jI0e05cIPrkRQ6euZxVo6KUl4XafqXp1zyAOlVKU9fPh6CKJdl1/EKOlbxCA8s543aFK7p0GqJ/xKg7YZK6E4VQficnoBAkKKrPmc2lWrWpfO4cp8qWZXf1atQ/8rdrcgghhBBCCOF2AsOXcLl2HSokJXGyXDmiq1WjzrFjN+1XwsuC2aSwxl3AGmdMz6ji402dKqXpXLcydfxKU6eKDwFli6GUuun4zJW8Ig4lEhpYLl+KZQs3dHQTfP+81J0oxAoiOWGcsxBQQIcdxjQPY0gbHB06zKkxCSGEEEIIUZAsWmf1iX+394l3vZizTxxxKBGtjS8YJgUj29/Pptc7MmdwM15+JJjO9apQrVzxXJMTmfJzJS/hhuKiYH5XSDoidScKqYJKThjnLQS8/CrlmHMHkLxhgzNDEkIIIYQQokCZK1WiU1af2FhalN9z9olDA8vhaTFhVuBpMdGhVqWCDlMUNlGzpO5EIVaQyQnj3IWA30fTaRJ7AJ/kZA5XrszBypWdHZIQ4i4ppRgz5noxr6lTpzJhwoQ8H9+5c2fKlClD165dc7QfPnyYFi1aULNmTfr27UtqaioAWmtGjRpFUFAQDRo0YNu2bQ65DyGEEMJZ/P8znWYx+yl9+QoH/fw4UrEiJp1zn8wpGi8/EsyiYaEyCkLcm7OxsPcnjLoTZqk7UchYE6xMjJxYYMkJ4/yFQPFGjbDYbLTfuROAXxs3lmkeQrgZLy8vli1bxtmzf2/Z5FdffZWFCxfe1D527FheeuklYmNj8fX1Zc6cOQCsWrWK2NhYYmNjmTlzJiNGjLin+IUQQghnK96oEZ42G+122fvETYw+8a4BOVe4kykawiEOroW5XcBsgT4LocObMOgHmdpRiHy4+UNs2pb1WaHyNTkBhSRBAWCuWInOW7YCsKpZUzQyzUOIfBUXBes/yrHG+r2wWCwMHz6c6dOn/63jO3bsSKlSpXK0aa1Zs2YNvXr1AmDQoEEsX74cgBUrVjBw4ECUUoSGhpKUlMTJkyfv7SaEEEIIJ/OoXj2rT7yyaTMA1GZZ4U442LFI+KonXD4D6SlQqpLUnShkxv81nl1ncz472ge0z9fkBBSCVTwyVft4Oi36P43vpUscqlKFff7+1IqPd3ZYQrifVePg1B06MtcuwundxjxDZYJK9cCr9K33r1wfuky646VHjhxJgwYNeO2113K0L1q0iClTpty0f1BQEEuXLr3l+RITEylTpgwWi/Go8/f35/jx4wAcP36cgICArH0zt1WpUuWOcQohhBCuym/SBzz49AB8kpM5UNWP/X5+1DwRb/xBQb48CkdZ+4HRDwSwZRh1J+S/r0LBmmDlnY3vEHshNke7GTND6g3J9+sXmhEUxRs1wsNm4xH7PPKVzZrKNA8h8kvKheu/lLTN+OwApUuXZuDAgcyYMSNHe//+/bFarTe9bpecALKqlGeXWZX8dtuEEEIId1W8USO8MjJ4ZNt2AFY2a4bCxK4Xx9zhSCHyIC4Klj0Hh/40ak5I3YlCJTwmnAGrBtyUnDBh4s3QNwmpGJLvMRSaERQA6b4VeGzzFha3bcuqpk15afkKmeYhxN3Kw0gHYzmpbpCRavxSemK2w7Lmo0ePpnHjxgwZcj1D+3dHUJQvX56kpCTS09OxWCzEx8fj5+cHGCMm4uLisvbNvk0IIYRwZ1716xO2ZTPhbVqzqmlT/rViBWrf36vxJESWzOVE068BCjpPgtRLRnJCRk+4vSUxS3g34t2b2vO7KObN1ytEyk2ZQsjBg1Q+d46T5cphDazh7JCEKJwCmhtFkPKhGFLZsmXp06dPVjFL+PsjKJRStG/fPmu/+fPn0717dwC6devGggUL0FoTERGBj4+PTO8QQghRKASGL6Fx7AEqJiURX6E8O2vUQMYIinsWu9qenMCY4pt6SepOFBLf7vs21+SEWZkLNDkBhSxB4f9QC0xa0yWzMFCzZijgUO8+tz9QCHH3Aprn2y+lMWPG3PVqHq1bt6Z379788ccf+Pv78+uvvwIwefJkpk2bRlBQEImJiQwdOhSAsLAwAgMDCQoK4tlnn+Wzzz5z+H0IIYQQzuKhNZ23bAGuT33e9WhH5wYl3Fd6KuxfbbxXJpnWUYgsjF7Ie5Hv5WhTKDoEdGBe53kFmpyAQjbFA+BaJR/CNkcx95GH+bVxE8YuCUfvksrFQri65OTkrPeVKlXiypUrd3X8+vXrc20PDAwkKurmlUaUUnz66ad3F6QQQgjhJrzq1yds8xYWdOrEqiZNeHXpd5iPnnB2WMIdHYuEX8bBqR3QZix4eMm0jkLiy11fMn1bzhX0CnpKx40cMoJCKdVZKRWjlDqglBqXy/ZXlVJW+2u3UipDKVXWvu2IUmqXfduWe40l+e23qBV3jPtPnOBc6VKsr18PBVzZvv1eTy2EEEIIIcQtuVKfODB8CXWPHqXGqVMk+vjwV906KGDn9Ffu9dSiKImLgnmPwYltYLJAzU4yraOQ+HT7py6XnDBiuEdKKTPwKdAFqAM8qZSqk30frfUUrXWI1joEeB34n9b6XLZd2tu3N73XeB5o0hGFiX9s3AjAspYtAYgb+cK9nloIIYQQQohcuVqfGMBsNtPT3if+3t4nZs4PxpdOIfIiahbY0oz3WhvLiQq3N2XzFD7f+XmONldIThhx3LvmwAGt9SGtdSrwLdD9Nvs/CXzjgOvmqrKPN9F+tekaGYUlI4N19epxpnRp0s+du/PBQgghhBBC/D0u1ScGKBXWhW4RkZgzMljboAGJpUphTlfyJVPcWVwU/PQy7FkGSslyooWENcFKv5/6sSB6QY52V0lOGLHcu6pAXLbP8fa2myiligOdge+yNWtgtVJqq1JquAPiYevICZS7dIm2u3aRYTbzY4sWKCD+1VcdcXohhBBCCCFu5HJ9Yv8pU6h48SKtd+8h3Wzmx+bNUZjYtSDSEacXhVXmcqJb5oAtAzq9my8rt4mCFR4TzsBVA9mTuCdHuyslJ8AxCYrcVi3St9j3cWDDDUPZWmmtG2MMhxuplGqT60WUGq6U2qKU2nLmzJnbBhQaWI40k4keGzcB8H3LB9HApZ9X3uFWhBBCCCGE+Ftcrk8MoLy96bnp+jQPDbBeCsiL29j/a87lRG2pUnfCzS3et5h3It5B3/BIcrXkBDgmQREPBGT77A/cqkRwP24Yyqa1PmH/3wTge4zhcTfRWs/UWjfVWjetUKHCbQNqEViW5fe3ofXu3ZS7cIFDVaqws0YNbDZb3u5ICCGEEEKIu+NyfWKASq+Po82u3ZS9eIkDVf3YXb06ZmDHmiV3PFYUQamXYe+PxntZTrRQWBi9kImRE29qNyuzyyUnwDEJis1ATaVUDaWUJ8YD94cbd1JK+QBtgRXZ2koopUplvgceAXbfa0BVfIrxZ5s+WGw2ukUaQ9iWPtQKBRwIe+xeTy+EyAdKKcaMGZP1eerUqUyYMCFPxx49epQmTZoQEhJC3bp1+fzz60V/Dh8+TIsWLahZsyZ9+/YlNTUVAK01o0aNIigoiAYNGrBt2zaH3o8QQogix+X6xAC+ffvilZHB4/Y+8XcPtQIg45X/k2KZIqejG+GLNnB2P3ScAB3Gy7QONzdrxyw+3PxhjjaFokNAB+Z1nudyyQlwQIJCa50OvAD8CuwFlmit9yilnldKPZ9t157Aaq315WxtlYC/lFI7gCjgZ631L/caE0BojXKcLl2S3n/9BcDPzZqRVLw41w4dcsTphRAO5uXlxbJlyzh79uxdH1ulShU2btyI1WolMjKSSZMmceKE8UersWPH8tJLLxEbG4uvry9z5swBYNWqVcTGxhIbG8vMmTMZMWKEQ+9HCCFE0eKqfWIAr/r16WXvE//UvDkXixXD84oN6zf/kCSFMByNgHldIfGAsZzofa1kWocbsyZYGbxqMDOsM3K0mzDxVuhbfNzhY0IqhjgputtzxAgKtNYrtdYPaK3v11q/Z2/7XGv9ebZ95mmt+91w3CGtdUP7q27msY4Qen9ZtjzciGoJp2m5J5prnp4sb/mgFMsUwkGsCVZm75qNNcHqkPNZLBaGDx/O9OnT77zzDTw9PfHy8gLg2rVrWdO5tNasWbOGXr16ATBo0CCWL18OwIoVKxg4cCBKKUJDQ0lKSuLkyZMOuRchhBBFkyv2iQECw5dw/+nThO7dx1UvL5Y/+CAm4PLqYrKihzCWD/3t36Az7J9t8t+FG8sshrk1YWuOdlesN5Ebi7MDyC8tapRjoa02j6n/8dTatWysW4dv27Zl4B9ruPTjTzBlirNDFMIlTY6azL5z+267T3JqMjHnY9BoFIpg32BKepa85f61ytZibPOxd7z2yJEjadCgAa+99lqO9kWLFjEll3+zQUFBLF26FIC4uDgee+wxDhw4wJQpU/Dz8+Ps2bOUKVMGi8V41Pn7+3P8+HEAjh8/TkDA9anCmduqVKlyxziFEEIId6O8vXlq7Voiatfi27ZtePrPP/GJN0l9gaIuLgr+fA/iI8FkNsq6St0Jt7UkZgkTIya6RTHMWym0CQq/MsU46xtCbJMHabM1Cr/EROIqVOCvunVos3sPV7Zvp3ijRs4OUwi3dCntUtaDT6O5lHbptgmKvCpdujQDBw5kxowZFCtWLKu9f//+9O/f/7bHBgQEsHPnTk6cOEGPHj3o1asXJtPNg8SUMoqsa31zYfXMbUIIIURhU+n1cbT7v3eofO4cRytVYmPtWjwUvZedv66nwTAZxl8kxUXB3DCwpYEyQ6k02D0AACAASURBVJepkHLeSE7I1A63syh6EZM2T7qp3azMvNniTbdITkAhTlAAhAaW5e2rT/H1lkj6/m8d0//Rk2/atqXN7j3EjXyB4I0bnB2iEC4nLyMdrAlWnl39LGm2NDxMHkxqPclh89hGjx5N48aNGTJkSFZbXkZQZPLz86Nu3bqsX7+eJ554gqSkJNLT07FYLMTHx+Pn5wcYIybi4q4vV599mxBCCFHY+Pbty6m3J9B33Xo+7tGdb9u246HoveiP58CwMXc+gSh8NnxsJCcypZw36k4ItzNzx0z+a/1vjjaFon1Ae4bUG+Ky9SZy45AaFK4qNLAcF66mcS2wGk9s2IBnWhrr69blWPnypJ87d+cTCCFyFVIxhFmPzOKFRi8w65FZDn3olS1blj59+mQVswRjBIXVar3plZmciI+P5+rVqwCcP3+eDRs2EBwcjFKK9u3bZ+03f/58unfvDkC3bt1YsGABWmsiIiLw8fGR6R1CCCEKteKtWvHEhg14pKWxtn494suVw5Kmsa4eK8Uyi5K4KFgyCPb9ZCwlqswyrcNNWROsDPh5wE3JCXcohnkrhTpB0SKwHAA73vycMpcvE7Z5M9pkYkGnjijgUO8+zg1QCDcWUjGEYfWH5ctDb8yYMXe1msfevXtp0aIFDRs2pG3btrzyyivUr18fgMmTJzNt2jSCgoJITExk6NChAISFhREYGEhQUBDPPvssn332mcPvQwghhHAl1efMpvylS3TZuhVtMrGwYwcUcPXtZTC/myQpioK4KJgXBtHLjeRE58nQ4U1ZTtQNZRbDtJ7NWbDenepN5KZQT/GoWqYYAWWLEXEokWbFLAz+/Q+Wt2zJ9y1bMvLHnyiza5ezQxRC2CUnJ2e9r1SpEleuXMnzsQ8//DA7d+7MdVtgYCBRUTd3uJRSfPrpp3cfqBBCCOHGzH5+DPrtd34IDeW7Vq0Y8fNKSp+/jNUCIUfWy5fUwi7yc8jInNahIPWSTOtwQ1/v/ZoPoj64qd3dkxNQyEdQAITWKEfk4XPUeLEzNU8cp82uXaR4evJNu3ay5KgQQgghhChSHljzB7WPH6fVnj1c9fLi2zZtMAGXVpaSIf6FWVwUhA+B3d/JtA43Zk2wMnz18FyTE2ZldvvkBBSFBEVgOZKupHGmURhaaYb89hsAX7dry1UPD2PJUSGEEEIIIYoKT0+eWW3vE7dvR4qHB2USLFhVupMDE/kic1rHnmVGcuLRD2RahxvKnNKx6eSmHO0KRYeADszrPM/tkxNQBBIULQLLArD2yn2U796BZvv3U/fIEc6XKsXylg+ikVEUQgghhBCi6Kj85huExsRQ+9gxEkuX5ofQFpiBc8+NdHZoIj9snJFzWkfaZWNahyQn3MaSfUt4J+IdNDpHuzsXw7yVQp+g8Pctjr9vMSIOnaPSpP8HKIbaM8bzO3Ui3WSSURRCCCGEEKLI8O3bF6VU1iiKuQ8/TLrJRIWDKVh/HCHFMguLuChY1Bf2/ijTOtyUNcHKiN9G8G7kuzdtKyxTOm5U6BMUYEzziDyciM2msbWqTaft26mWkEBchQr81Ly5jKIQQgghhBBFStmhz/Dotm0EnDnDsYoVWdW0KWbg5KzfZUWPwuBYJMztArG/GMmJLh/KtA43Ex4TzqBVg/jrxF852gvblI4bFZkExfkraexPuET9WUsxac3zP68E4POwLjKKQgghhBBCFCmVXnkFi83GcytXAfB5WBjpJhP++y1YzRqOrHdyhOJvs9ngt7fAlllTRMG1izKtw42Ex4TzbsS72LDlaC+MUzpuVCQSFC1qGHUoIg4mgsmErVYFHtu8meqnTxNXsSI/tjBGUZyeOtW5gQpRhCmlGDPm+jJXU6dOZcKECXk+3mw2ExISQkhICN26dctqP3z4MC1atKBmzZr07duX1NRUALTWjBo1iqCgIBo0aMC2bduyjvnll18IDg4mKCiISZMm3fvNCSGEEC6o1ONdeTwykoAzZzhSuRIrm9lHUWwqDZXqOTs88Xcc3Qgz20BcJJjMMq3DzVgTrPzzt3/mWm+isE7puFGRSFAElC1O1TLFiDx8DoD6U/6N2ZbO8yuNURRfdDFGUZybPceZYQpRpHl5ebFs2TLOnj37t44vVqwYVqsVq9XKDz/8kNU+duxYXnrpJWJjY/H19WXOHOPf+apVq4iNjSU2NpaZM2cyYsQIADIyMhg5ciSrVq0iOjqab775hujo6Hu/QSGEEMLF+E+ZgofNxnMrM0cWZxtFsWkarP9Ipnq4k4NrYd5jcGoXmCwQ9pFM63AjmVM61p/IOXqpsE/puFGRSFBAZh2Kc9hsGk7twFYljbDNW24aRSG1KITImyvbt3P2i5lc2b7dIeezWCwMHz6c6dOnO+R8YIySWLNmDb169QJg0KBBLF++HIAVK1YwcOBAlFKEhoaSlJTEyZMniYqKIigoiMDAQDw9PenXrx8rVqxwWExCCCGEKyneqhXdIqMISEjgaKXroyguzTkEf0yUehTuIC4Kfv8/WDoEtH1KgNZw9ZxM63ATi/ctLrJTOm5kcXYABSU0sCzfbYsnNiGZ4PtaU7/jZPZ85cGIn1cy7pkhfNa1K2Gbtxi1KKZMcXa4QjjNqfff59refbfdJyM5mWv79hm//JTCq1YtzCVL3nJ/r9q1qPzGG3e89siRI2nQoAGvvfZajvZFixYxJZd/l0FBQSxduhSAlJQUmjZtisViYdy4cfTo0YPExETKlCmDxWI86vz9/Tl+/DgAx48fJyAgIOtcmdtya4+MjLxj7EIIIYQ7qj5nNntr1eb5lat4c/AgPu36OJ23bqNMAoSXLEbvyylGPQr5kuua4qJgfldIv2Z8NnkYSQqZ1uEWrAlWPt/xORtObLhpm1mZebPFm0Vi1ER2RWoEBUDEoUTjATvoJ2wBZsI2b+aB+HhOlCvHN+3aooGjQ4c5N1ghXJzt4kUjOQGgtfHZAUqXLs3AgQOZMWNGjvb+/ftnTd/I/spMTgAcO3aMLVu28PXXXzN69GgOHjyI1vrGS6CUsoed+7bbHSOEEEIURmWHDeXxqCjuP3GC+Arl+bZNG8xAqWU+WL08oForZ4cobmXrvOvJCWWCxk/LtA43ER4TzsBVA29KThS1KR03KjIjKK7XoUhkUMv7IKA59b/4jD1hz/Hysu95ftSLfNEljB4bN6E23JzBEqKoyMtIhyvbt3NsyDPotDSUhwd+U6dQvFEjh1x/9OjRNG7cmCFDhmS15WUEhZ+fHwCBgYG0a9eO7du388QTT5CUlER6ejoWi4X4+Pis/fz9/YmLi8s6V+a21NTUXNuFEEKIwqrSK69w7su5jFn2Pf98YSSfh3Whx6ZN+J+7yvclihGyeykc22D8RV6+9LqGuChYPw32rwKUkZwwe0LDp+T/IxdnTbAya+cs1h1fd9M2E6YiUQjzdorMCAqAoIol+HPfGbYeMYplUqMttlqePBQdTejefVwsUZxZXTqjgUO9+zg1ViFcWfFGjag290sqjBpFtblfOiw5AVC2bFn69OmTVcwS7jyC4vz581y7Zvz14OzZs2zYsIE6deqglKJ9+/ZZ+82fP5/u3bsD0K1bNxYsWIDWmoiICHx8fKhSpQrNmjUjNjaWw4cPk5qayrfffptjVRAhhBCiMKr89r9pu3s3zWJiuFCyJLM6P4oJaD+3GNad82GN1KNwGUc2wJedjeSEMkHnSTJqwk1kFsLMLTlRVFbpuJMik6DYevQ8Gw8mcjUtg6dmR7L16HlQivqjngRsjFm2DIBF7doRX64c13btcm7AQri44o0aUf654Q5NTmQaM2bMXa3msXfvXpo2bUrDhg1p374948aNo06dOgBMnjyZadOmERQURGJiIkOHDgUgLCyMwMBAgoKCePbZZ/nss88Ao1jnJ598wqOPPkrt2rXp06cPdevWdfg9CiGEEK7Et29flIcHr35n9Im/6tCBE2XLUvYKbEguZdQ1yEg16lEI58gshhk+GHSGvVFB2mUphunirAlWXvzjRd6JeOemQphFfUrHjYrMFI+IQ4lk2Iy55anpNiIOJdKkui+YPbDVvUqdPXE8HhHJj6EtmN6zBx/NnsPeJk2pvXWLkyMXomhITk7Oel+pUiWuXLmS52NbtmzJrlskFQMDA4mKuvmvPUopPv3001yPCQsLIywsLM/XF0IIIQqD6gvmw5NPERa1mZXNm/GfHj348MsvafOdF+HDStD78lUICHV2mEVTXBTM6woZmcUwLUY9MCmG6fLCY8KZGDHxpsQEFN1CmLdTZEZQhAaWw9Ny/XZDa5Q13gS2pX7DK9hUBqNWrMA7NZVfmjYlIjgYLl/m/OLFTopYCCGEEEKIglO8USOUjw8vrViOV2oqPzdvRtQDNTED5zb6YPW0QMRnsP4jmepRUOKiYN1H8Oub15MTygSNB8i0DhdnTbAy6o9RMmriLhWZBEWT6r4sGhZKWL3KaIDMovwBzWHA92Q8lorf+fM8t3IVAO/168s1s5lTb09wUsRCCCGEEEIUrFqREVRNPMezv/wCwHt9+5FmMtEyGn6s1gD2/QR/vCv1KApCXBTMfxzWvAPxUUZiQpnB7GUUw5RpHS4rs9bEn/F/3rTNrMy8FfoWH3f4mJCKIU6IzrUVmQQFGEmKKb0bUszDzPfbj1/fUKM1DXu+zJXi6Qz+/Xeqnz7NoSpV+KpDB1l2VAghhBBCFCle9eszdPVvBCQkcKCqH4vat8cMtPrPCcJLlgC0sbSl1KPIXzu+gfQU+wcFjQfJqAkXJ7Um7l2RSlAAlPCy8EjdSvy08ySp6dn+o2n6DJanbXikp/P64iUAfPZYGKfKlOGKLDsqhBBCCCGKiMDwJXilp/PGknAAPn28Kwk+PlS+BIuv+dqTFDZIPiPTPfLD0U2woAdsmUvWEqIWbwiRUROuLHPUxNr4tTdtk1ETeVfkEhQAPRpVJelKGmtjEq43epUkpGZXkv1SaR0dTadt27nq7c0HfftgA/aGOH6lAiGEEEIIIVxR2WFDabt7Nx2sVq54e/NBH+Mvvm8vtPF+hfJYff0g8v/J8qOOtud7mBcGh/60LyH6PnQYL6MmXJiMmnCsIpmgaB1UnnIlPFluPZ5zQ8mKtGhzFo2NceHhFE9J4fdGjVjduDGkpHB66lTnBCyEEEIIIUQBqvTKK+DpyetLwimWksLqJk1Y3agRZuDtuWn86FfT2FHbjOKNMt3j3hyLhG+ehKXDjJ9pprSrMmrChcmoCccrkgkKi9nE4w39+H1vAheupl3fEPQwmCxkhF6myvnzvGJfB3piv36cK1GCc7PnOCliIQo/s9lMSEhI1mvSpEkOO7fVamXlypUOO58QQghRFNTeuYOq584x5vvlAEzs15ekEiW4/xR8d+Uo4T5ljB21Da6ck+kef9ee5TC3M8SsBJ1hLB2qzLKEqAuzJlgZ+ftIGTWRDxySoFBKdVZKxSilDiilxuWyvZ1S6oJSymp//Tuvx+aXno2qkppu45fdJ683BjSHp5fRoKaNFC/o/ddfNIuJ4VzpUnzQtw8a2NdC1n4WIj8UK1YMq9Wa9Ro3znGPA0lQCCGEKAju2Ce+k+KtWtFv3TqaxMaS6OPD5N69MAMLP0hlYlkfwkN6QIlKsOkT+EOme9yVo5vgq16w9JnroyaUCRr1l2KYLsqaYOVfa/7FwFUDWXd83U3bZdTEvbvnBIVSygx8CnQB6gBPKqXq5LLreq11iP31zl0e63AN/H0ILF8i52oeAIFtodMETP3OorTmnYVf4Z2ays/Nm7OmYQP0hQucX7y4IEIUosi7cOECwcHBxMTEAPDkk08ya9YsAEaMGEHTpk2pW7cub7/9dtYxmzdvpmXLljRs2JDmzZtz4cIF/v3vf7N48WJCQkJYvHgxly9f5plnnqFZs2Y0atSIFStWOOX+hBBCFB7u2ie+k+pzZmNSincXLsQrNZUfQkNZW78eZmDyF2m8f3En1rpd7HvbjFUnZLrHrcVFGSNNfnsb5naBA78ZyYnsoyZkCVGXlDmdY03cGjQ6xzYZNeE4FgecozlwQGt9CEAp9S3QHYjO52PviVKKHo2qMu23/RxPukrVMsWub2wymJBN/+V/TWxU23qWfy1fweQ+vfn3009T//BEeHsCvn375neIQjiFWrs2X86r27W77farV68SEnI90/z666/Tt29fPvnkEwYPHsy//vUvzp8/z7PPPgvAe++9R9myZcnIyKBjx47s3LmTWrVq0bdvXxYvXkyzZs24ePEixYsX55133mHLli188sknALzxxht06NCBL7/8kqSkJJo3b06nTp0oUaJEvty7EEKIIsEt+8R5UTt6D9SqzagffmBKr178e8AAlr07kWrnLhEYl8aPVTIIsXjbl8TUcGgdaA012siX7OziomD+49mWDrXLHDXhE2BM6ZCfmUuxJliZs2tOrnUmwBg18WaLNyUx4SCOmOJRFYjL9jne3najB5VSO5RSq5RSde/y2HzRI8S41A/WEzk3WDyhQT/a1ownzZzO03/+SejefZwvVYo3Bg3CphR769YrqDCFKBJunOLR154EfPjhh6lfvz4jR45k9uzZWfsvWbKExo0b06hRI/bs2UN0dDQxMTFUqVKFZs2aAVC6dGkslpvzsKtXr2bSpEmEhITQrl07UlJSOHbsWMHcqBBCiMLKbfvEeeFVvz4D/1hD85gYEkuXZvyggWhgwsIMvjv5F+GdxkCHt+C+NnB4Lax51/gyLtM9jJ/B2snwy+s5kxMmi4yacGHZp3PklpwwYZJRE/nAESMoVC5t+obP24DqWutkpVQYsByomcdjjYsoNRwYDlCtWrW/H2021coVp2l1X77fHs/zbQNRKls4Fi8jmKeTUPPL8/78efxj/Hg21q3Dgg7tGfzHGg717kNg+BKHxCKEq7jTSIeCZrPZ2Lt3L8WKFePcuXP4+/tz+PBhpk6dyubNm/H19WXw4MGkpKSgtc757/gWtNZ89913BAcHF8AdCCGEKCLctk+cF4HhS9hbuw6T5s6jx1vjWV+vHovat+fpP/9k9uRrDH1tEYSOp7dqZ5/ioY0v4+unQUCzojsy4GgELOgKGfbC/MoEKCMp0XkSXE0suj8bF2VNsDJ391z+jPvzpqkcYEznaB/QniH1hkidiXzgiBEU8UBAts/+QI4hCVrri1rrZPv7lYCHUqp8Xo7Ndo6ZWuumWuumFSpUcEDYhh6NqrL/dDLRJy/m3FCjDZg9CbmWytZ2qVRKusC7CxcCML1HD6L9/bm2axdXtm93WCxCiJtNnz6d2rVr88033/DMM8+QlpbGxYsXKVGiBD4+Ppw+fZpVq1YBUKtWLU6cOMHmzZsBuHTpEunp6ZQqVYpLly5lnfPRRx/lv//9L1obv3S2y79jIYQQ986t+8R5UXtvNJWTknh34VcAfPSPnsRUrUrxDBj7TSoTIyYS7pEBFm/7F3Fg/yr4owiOpjgWCcuGwzf9ciYnmgy6XgCz6WAZNeFibldnAqQIZkFwRIJiM1BTKVVDKeUJ9AN+yL6DUqqysv9ZUynV3H7dxLwcm98eq18FD7Ni+Y3FMgOaw+CfoVwQDe5PJbE4dNixk77/W0eahwdjnh3GJW9vjj75VEGGK0ShlVmDIvM1btw49u/fz+zZs/noo49o3bo1bdq0YeLEiTRs2JBGjRpRt25dnnnmGVq1agWAp6cnixcv5sUXX6Rhw4Y8/PDDpKSk0L59e6Kjo7OKZL711lukpaXRoEED6tWrx1tvveXkuxdCCFEIuHWfOK9KPd6Vh61Weq9fT6q9T5zs7U3IEbg/PoP3Yr7G2mMadBgPIQPsR9lHU2z4uHAvRRoXBeumwo+j4ctHYediSDmfbSqHl0zlcFHWBCuj/hiV67KhINM5CpLK/AviPZ3EGKL2H8AMfKm1fk8p9TyA1vpzpdQLwAggHbgKvKy13nirY+90vaZNm+otW7bcc9yZnl2whR1xSWx6vSNm0w0j7M7EwGcPEn5/c2pNPEqqhwdPjX2N/f7+dLBamfHFTEwmE7X37HZYPEIUtL1791K7dm1nh+FWcvuZKaW2aq2bOikkIYQQTubufeK82tugIVeAJ8e+RmzVqjy8bTvTZ87EBjz1uoUOAR34uMPH9qKQ3SDj2vVlNFHGVOpBPxaOL+lxUcaUFq/SRo0JW1rO7coMTQZKAUwXdafpHCZMtAtoJ9M57tK99IkdkqAoaI5+GK/cdZJ/LtrGV0Nb8FDN8jfvsHgg7F3BO+XK8sSn3sSVL0+fN17nUvHijP5+OcN//RWznx8PrPnDYTEJUZAkQXH3JEEhhBDC2ZyVoADYW6s2RypWoM/rr5NcrBgvL1vG0NW/kQH0f92Dt0LfMv7SnPkFPvEQWBeRVVqjalOo+Qjc3959v7RnrcpxjZwlQxSYzMZKJmZPYzqHu95jISV1JvLXvfSJHTHFw+11qFWRUt4Wvr9xmkemCg8A0C05mb/qQLWzZ5k0dy4AM7p3Y0PtWmScOMH5xYsLKmQhhBBCCCGcpva+vdyXcIZJc+cB8J8ePYgIDsYMzJ6aZtSjiAk3vpi3HmPUXrB4GyMKUHB8C6x9H+aGQeQX7jX14+gmWP5P+G7Y9aVVwbg3ZTbuM+yj67UmJDnhMrKvzCF1JlyTI1bxcHveHmYeq1+FH3ecYGKPehTzNOfcoeYj8Nd/CLmWSuxDyVyJKUm7XbsZ8dPP/L+ujzFm2DC+nfwhvD0BX/vSiEK4m7yugCHAHUeeCSGEEI5WdthQOsyew3MrV/FFWBfGPDuMryd/SPUzZ4yimUwEMEZSBDQ3vqwfWQ8X4mHLPMBmTIlY9RrG1A9P6Pyha65scSwSdi+Fs/vh0Nrr7cr+vUFW5XBpdxoxATKdw1XIFA+7iEOJ9JsZwcf9Qugeksuy00c3wdIhkH6N8K7vUOuJ91FK8eLzz7O2YQOqJSTwzeQP8b18mdr79jo0NiHy2+HDhylVqhTlypWTJMUdaK1JTEzk0qVL1KhRI8c2meIhhBCiIDlzikemfS1CSb94kZH/HMG6+vW579RpFn34IWWuXGH8ADMH/c2MDx2fs7BgVm2KVOOzzsh2Rns/xOIFnScX/Bf+zCkp97UGWwbs/BbOH4FD/+OmlV+lvoTLk8SEc0gNCgew2TQPTV5DcOVSzB1yi4fL8a0wqyPU7ckE8wV6vbGXq15eDBrzMnurVaNx7AG+/PhjPNPTJUkh3EpaWhrx8fGkpKQ4OxS34O3tjb+/Px4eHjnaJUEhhBCiILlCggJgb+06XPb05OlXxhATEECz/fuZ+fEMzBkZPPW6BROm3JMUR9ZDsXLwyzh7skJnK6YJ15MV+TyyIjMW7zJGocvMxEmuX2hNYDJJfQk3MG3LNObtmSeJCSeQBIWDTP5lHzPXHSLyjY6UL+mV+07fPAUxP2P19ua99PL831c2Tpfx4cmxYznt60vXyEgmz52HMptlZQ8hihhJUAghhChIrpKgAKNo5qkyZeg3biwJZcrQbVME78+fn7WyR65Jikw3JSvsRSdz+55i8oAWzxkjLGo+CkpdH/EAeX/v1wgunoAd38LRjTeM4rjpojmTEjKVw6VZE6x8tPkjrGetuW6XxET+kwSFg+w/fYlHpq9jwuN1GNyqRu47rZkI66YAEF6qFGm/lSLkCOz192fAK2O46u3N4N9+49XvlqG8valt3e7wOIUQrkkSFEIIIQqSKyUowEhSRAcEGH1iLy+G/vIrLy9fTgZGkkKhrq/ucSt5GlmRnTK2YzKSFdoGyr4OQNb7zOMzp7He5vtPZk0Jk72Ypy1dkhJuwppgZdbOWaw7vi7X7ZKYKDiSoHCgsI/X42ExsWJkq9x3iIuCeWGQkQYmD6a1HsKDo3/C5xqsr1OHF/45gnSLhVErVvD8ql8wlS9P8F/r8yVWIYRrkQSFEEKIguRqCYor27dz9Mmn+F+9urw4YgTpZjOjv/+eZ39dfXdJiky5JSuUMmpD3C7J8LcoY/WN7IkIuD7yQpISLsuaYGXmzpmsP37r71xNKjZhdJPRkpgoILLMqAP1bFSVHXFJHDqTnPsOAc1h0M9QriYoReniFXjuZQ/SgNbR0Xz45VxMNhszundnUbu22M6e5UDYYwV6D0IIIYQQQhS04o0aUerxrrTdvYdJc+ehbDb+07Mn37Zpgxn4+oN0NJp3I941liC9k8wlSpsONmo9dHjTWL4zc7lSsyeYvRzzvukQ4xpNBxvXDGh+/fqSnHBJ1gQrz//2PANWDbhlckKhGFJ3CPO6zJPkhJuQERQ3OH0xhQc/+IMX2gfx8iPBt97xQjx8Goq1fDWGeCWTjubrD9IxA9+1bMm/Bw4A4L358+m5KQKPwECCVv6cLzELIVyDjKAQQghRkFxtBEWmA2GPkXboEEseeogJT/dH2Wy8P38+3SKjskZSAHQI6PD3httnX2kDHPNekhBuw5pg5ZPtnxB5KvKW+8h0DueSKR4ONmBOJEcSL7Pu1fa3X3Jx9VuwcQbhJUsysbwvNqWykhTzOnVkSq9eKJuNd75axBMbN2L28+OBNX/kW9xCCOeSBIUQQoiC5KoJCoCYh1pjO3uWOY88zEf/+AfKZuPdhQvpuSkiR5LCpEyMb3GL4plC2FkTrPxw8AesCVZik2JvuZ8kJlyDTPFwsB4hVYk7d5Vtx87ffkdvHwB6JyczPjEJE8bDNgMY/PsfvLTse7TJxFsDB/BtmzZknDhBzEOt8z1+IYQQQgghnCn4r/VQogRDV//Gv5avQJtMjB80iCUPPZQ13QPApm28E/EO/1rzL6wJua+6IIoua4KVF/94kYGrBhK+P/yWyQkTJjoEdGB+l/l83OFjSU64MUlQ5OLRepXx9jDx/fbjt9+xRhtjDhzQO/ky4x8YgAlTVpJi2OrVvBZuzK9756knWdihPbazZ9nbRP7AKoQQQgghHTk8rAAAIABJREFUCrfa/5+9Ow+Pqrz7P/4+M9lYQkICAQIhIYDsi4AQ0CBLbRFB26eitiiBqrQ+dlHRx1q1pSKtPqKt7U8fpaCgxQ21dWlRKbtCjCxBwCB7SNgCgYQ1y2TO74/JhCF7yGTOzOTzuq5cmTlzzuQ7COM9n9z39960ESIi+Oknn/DQu+8C8Pvbp7JkzLWXhBQAK3NWMv2T6fXrTSFBLTMvkzkb5vDjf/2YO5bdwerc1Zg1NEVVMBF8FFBUo3V4CN/t25GPvz5CiaOmLY0ob5j5EfSZDKaTKQX5PNZuBHaPkCJtxUoeffMtAP54yy38ddIkzHPnyBo4yDcvRkRERETEIn0yt0BEBDP+s4LfvPU2AH+47TZenDgRG5eGFGVmmWZTNGOZeZn8auWvmLZsGu/seodtJ7bVeK6CieClHhQ1WLUzjxmLvuJv04ZxXd8OtZ9smvDKBMhJB8PGE7GxLI1sAXBJ48zZt0/FabPxw8+/4HdvvEEI0OebHU36OkTEd9SDQkREfMmfe1BUljX4SigqYuk1V/P7H/8Yp83GzevW8fibb2F3OvlghMGb4+wV59sNO4+OeFS9KYJcZl4mH+39iF0nd5F5ovZQyoaNK+OuJDk6mRu736hQwo+pSWYTKC1zkvKHFaQkx/LC1CF1X7DiCVj3LACZES2Y0SkOR/lUJHdIsWrAAB68+y6KwsIYvW0bz/1tAS1LSuizM6sJX4mI+IoCChER8aVACigAdo5IwSwsZMWggTx4550Uh4UxZuvXzFuwgBalpeS3hP/+Vcgl11z2Th/i1zLzMnll2yu1Lt9wU+PLwKMmmU0g1G5j8qB4lmcdo/BCad0XXDHBtYcyMLikhN/0uh1b+R+ve7nH2G3beOVPfyb67FnWDhhA2qwHOBodTVbvPk34SkRERERErNf7y3TCBwxg/NaveeXPzxN19iyrBw1kxqwHOBYdRex5WOKx5ANcvSmmLZumZR9BwN1b4taPbuWOZXewKndVrb0lhsYNZcoVU7SMo5nRDIpabM0p4KYXvuDpHw7g1qu61n1BTgZ8+hjkfgnXPcHSgh08eSIdZ6WZFPs7dOBnP/85ue3bEVtYyF9ens+V+/bR8fezaXvrrU37okSkyWgGhYiI+FKgzaBwO/X22xz93Wz2duzIz35+L4fataN9QQHPvzyfQfv3X7INqSf9Jj3wuJdw7Dy5k69PfF3n+fpvHBy0xKOJmKbJ+GfXENcmnLdmjqzfRY5ieDEFTu4Dw8bSNm14MiaqSkhxqlUrZt19N1/27kVoaSm/ffNNfrh+A/b4eK5YuaLpXpSINBkFFCIi4kuBGlC4ZfXuw8nWrbl/5t18dcUVhJaW8vslb3BTejomMH+CjZVXVp3wrQ+x/q0hfSVAvSWCkQKKJvSXFbt5bvkuvvj1ODpHt6jfRZ88Aukvum4bdpYOu5knT3yJE9eOIIufdhDhhFKbjWduvpkl48YC8KPVa/ifd98lvKyMPlnfNMXLEZEmpIBCRER8KdADCnCFFKU2G0/fMoU3xowB4MerVvHQe+8T5nBQGA4zH6g6mwIUVPgTz1Bi64mtdfaVAP33C2YKKJrQwfzzjH5mFQ9P6M09Y7rX76KcDFg0EcpKwRYCM5ax9Hw2T6Y/WRFSPPOyg64nXae/P2oUT/zoNkpDQ+lz8CDPLlhAUt5xNc8UCTAKKERExJeCIaAAKvqxLb3maubcdhuOkBD6ZrvGxF2PH6cMuP2RsIpxdGX6oOt77kDixIUTHL9wvNYtQT1ptkTzoICiif3w/9ZzpqiUT+8bjWEY9bsoJwOWPQyHN8N1T4DTwdLQMp78dknFm+u4LU7u+sSJHfima1ceuPsuctq3p2VREbOXvMGkr74iNDmZHv/+V9O9OBHxGgUUIiLiS8ESUMDFHT62JSYy6667yG3fjlYXLvD7JW8wceNGnMCxxEju/3FxjUGFgcGQuCH68NtE3KHE3oK9bM7bXK9ZEqBQojlSQNHEXk/P5vF/buffv0ylb3yb+l9YWgT/NwpO7gXDBvZwln7ngUtCCrjYl+JsRAS/u30qnwxz/bf8/oYNPPL2O0QWFWk2hUgAUEAhIiK+FEwBBcCxefM4uWAhZyIiePyO2/ls6FAAfvDFen69dCmRRUU4gAUvfpfVOatrDCpAH4q9wXOWxIkLJ9h2YptCCakXBRRN7NS5Eob/4T9MH5XEozf0bdjFnz4GG/7qum3YYdyjLI3reslyD7gYUpjA0tRreGrKFIrDwuh48iRzXv87V2dl0fLqq0lcuMBrr0tEvEsBhYiI+FKwBRRuWb37YAJvjx7NU1NupiQ0lI4nT/Lka68zaudOABx9klnwix51BhWgD8v15RlInLxwst69JNz05yxuCih84O7XNrI1p4ANj4zHbqvnMg8o70dxA5SVuAKKGcug6wiWfru0Skgx51UH3Y9SsRXpI9PT2NatGwC3rlnLg++/T6viYs2mEPFTCihERMSXgjWgAMgafCUUFbGnU0ceTUtjW1IS4Goq/8A//kGr4mIAite+yavbX61XUAEXl4FEhUcR2yK22X6Q9gwjAI5fOM72E9svK5Bo7n+WUpUCCh/497Yj/PeSzdwyrAu3XtWVoYlt639xTgas+gPsWwXDZ0JkR0hKrdI40809m6LMZuOV667j/02ehCMkhE75+Tz21tuM3baN8AEDSF76jndfpIg0igIKERHxpWAOKODikg+HzcaC732XF2+4AUdICJ1PnOCxt97m2u3bATCioij66KWK/ghb8rbUK6yA5vEhu3IYcaroFJnHMxsURrhploTUhwIKH9iw9wQ/+tuXAESE2lhyV0rDQgrThMWT4cA6wAYh4ZD2IUvPZzM3fS5llF1yunsrUgPYFR/Pb9LS+CaxKwDjMzN55O13iD91SrMpRPyIAgoREfGlYA8o3LL6DwCHg52dO/ObtDR2dk0A4DtbtvDIO0vpdOoUQMVyaM9mjg0JK+DSGRZAQIUWlYMIoMG9IyprDgGOeJ8CCh94YdUenvn0WwDsBjzw3V7cO7ZHw55k1R9hzVOu2+X9KEidRWZeJq9uf5VVOasuefP48coyJn1pEgI4bDbeHHMtf5k8mXMtWtCiuJj//vhf3LFyJWEREfTZFPz/cxLxdwooRETEl5pLQAGXzqZ4Y8wY/nLjZM5HRNCiqIiff/wxU1etJqzM9Qu/xDffoOWVVwI0KqxwqxxaePLVh/bqwgc3p+nk+PnjfHPym8sOItwUSIg3KKDwgU3Zp/jx39IpdjgJsRm8/dORDZtBAa6lHosng6MIMGDqu9DzOxUPV9eXAi4u+TCAY9FRPDVlCp+WdzXumpfHrPf/wXcyM4nQsg8RSymgEBERX2pOAYWbuzfF0ehonrplCp8NGQJA4rFjzHr/H4zfuhUDwDDok/XNJdd6fsgvLC687MCiMgODQe0HERMR0+jnKiwu5FTxKdqGtyUqPAqH08GJCyfIOpnV6PChMs8wAgJrtoj4N8sDCsMwJgDP4+rvuMA0zacqPT4VeLj87lngHtM0t5Y/dgA4A5QBjvq8EKvejDdln2LWO5kUXijly998h7AQW8OfJCcDMt+ALa9DXD/oMxmSr4WE4UDNIcXPPyxj1A7XbAqAdX378vSUm9nXqRMAQ3fv5n/efY8B2dlETp5El2eeacxLFZHLoIBCRKR5ay5jYqud37KF7B/9GIA1/fvx9M1TONCxAwDDdu3i4Xffo9/Bg66T7Xb67Nhe7fM0VWDhjxRGiC9ZGlAYhmEHdgHXAbnAV8CPTNP8xuOcUUCWaZqnDMO4HphtmuaI8scOAMNM06w6X6kGVr4Zr/o2jxmvfsWfbx3M96/sfPlPtHIurP1f1+2QFpD2YZ0hhQ0bf3/WSUiJA4BSm42lqam8MOkGTkVGAnBDxlf897/+Rbdjxy6Z3iYiTU8BhYhI89XcxsT+YM/EGyjdt49Sm413RqfywqRJFLRuDcANGRnc+/G/SMrLc51ss9Hnmx21Pl/lZRSBGFpUDiJAYYT4ntUBxUhcb67fK7//CIBpmn+s4fy2wHbTNDuX3z9AAL0ZO50m1/1pDa3CQ/jg3qsxjAZsOepp3bOwYg5gAgaMfxxSZ1U8vPTbpdU2zzQw+EN0Gt0fXlBx7ExEBPOvn8Br48ZRGhqKzelkUkYGP/v3v0nKO65GmiI+ooBCRKT5am5jYn/iXvZxukULXr7+ev4+dkzFmHjyl64xceLx466Tq1n6UZvaej/4OsCoLnzwpCBC/IXVAcXNwATTNO8qv38HMMI0zZ/XcP6DQG+P8/cDp3B9Un/ZNM35df1Mq9+M/56ezWP/3M7Sn43kqqTLXGuWkwGLbyzvR2HC4NshNhmSUitmUtTUPBNgXMI47vrrHkKy9lUcO9y2LfOvn8D7V1+Nw27HXlbG5C+/5KfLPiHx+HHNqBBpYgooRESar+Y4JvY3WX36gmlyKCaG+ddP4B+jRlWMiW9M/5KfLltG1xMXg4aYu+6kw4MPNupn1hZgXI7KPSjcFD5IILE6oJgCfK/Sm/Fw0zR/Uc25Y4EXgWtM08wvPxZvmuZhwzDigOXAL0zTXFvNtTOBmQBdu3Ydmp2d3ai6G+N8iYORf1zJqO6x/N/tQy//iXIyYN9q2Po2nNyD5/aj7pACal7yYTfsPDriUQZMexazsLDi+KHYGF66fiL/HJlCmd2O4XRyXWYmMz5bzqADB+j4+9m0vfXWy69bRKqlgEJEpPlqjmNif+TZnyI3NpaXJl7PBymuMbHN6eS6LVv4yWfLGeDx52aPj+eKlSusKlkk6FgdUNRrOpthGAOBfwDXm6a5q4bnmg2cNU1zXm0/0x/S4qc/2cnLa/ay5qGxJMS0bNyTefajMGww7rFLlntAzSEFuGZTzOg/g/DvpEFJScXxnHbtePn6CXw4YgSOEFd7zWG7dvGT5csZvX0HrUeNInHhgirPJyKXRwGFiEjz1VzHxP7q1Ntvc/R3swHIbt+e+RMm8NGI4RVj4qt27eInny0ndccObO7PQ7U01BSR+rM6oAjB1RBoPHAIV0OgH5umucPjnK7ASmCaaZrrPY63AmymaZ4pv70ceMI0zU9q+5n+8GZ8pPAC1zy9ihmjknhsUt/GPdkl248C4x4Hw7hkuQfU3JcCLs6mmNJrCll9+4HzYpCRFxXF38eO5e3RqZxp6QpTuh09ym1r1nJTejptbDb6bNL/3EQaSwGFiEjz1VzHxP7Oc0bFsWj3mHg0Z1u0ACD5yJGKMXFkUVHFdZpVIXL5/GGb0YnAn3FtqfSKaZpzDcP4GYBpmi8ZhrEA+CHgnkvlME1zmGEYybgSZIAQ4A3TNOfW9fP85c34F29uYfXOPDb8Zjytw0PqvqA2ORmw4x+wcRE4LrgCCnvV5R619aUwMBibMJYZ/WcwOG5wxTo8t7MRESy95mpeHzeOozGu3hktiou54auvuG3NGvrmHmpQ0yARuZQCChGR5q25jokDhXtsfCYigqWp1/D6uHEca9sWcI2JJ32ZwW1r19InN/eS67Q8WqRhLA8ofM1f3owzcwr4/gtf8LvJfZlxdTfvPOnH98PGV1y3DTuMe7TKcg+o/2wKoMqMilKbjVWDBvHmtaP5snfviuMD9+3nh+u/YMLGTUTHxCg1FmkgBRQiIuJL/jImDjRZ/QeAw0GpzcbKwYN4a/S1fNm7V8Xjg/fu5YdfrOd7mzfT2mNWRX22KhURBRSW+q8XvyD/XAkrZ43BbrvMLUc95WTAohugrMTVj2LC01BypspyD2jYbAqArH79oezSQGNfhw68PXo0/xyZUrH8I7ykhO9kbuWm9A2MzNpJ8htLtPuHSD0ooBAREV/ypzFxIPr2mlSc5bt67O3YkbdHp/LPkSMrln9ElJQwPjOT729IJ2XnTuyen5vUr0KkRgooLPTx14f5+Rtb+Nu0YVzXt4N3njQnA758Cba/5wopMMAeVmW5h1tDZlPAxb2iPZ0PC+PToUP4ICWFjF4XE+T2BQVMzshg4lcb6XPkCH23b/POaxQJQgooRETEl/xpTBzIPBtqng8L45OhQ/lgZApfXXFFxTkdTp1i8pcZTNi0kT45uVT+taSWgYhcpIDCQo4yJ6P/dxVdY1vy1syR3n3yt6bCzo9dt2tZ7gENn00BsG/KLRRvqxo4HIqN4cMRI/ggJYWDcXEVxxOOH+d7mzYzYdMm+hw/Tt/MLV54kSLBQwGFiIj4kj+NiYOF56yK3NjY8jHxCHI8x8R5eXxvc/mYuJqwQg02pblTQGGxl9fs5Y/LdvKvX15Dv/go7z1xTga8OhGcpa6AYvq/ITGl1ksaOpvCrXJDTQATyExO5uPhw/nsysHkR118be6wYvzWrQw4epR+WzZf3msUCSIKKERExJf8bUwcbNzLo01gc/fu/Gv4VSy/8kry27SpOMcdVozP3MqA7OyLW5a6GYaa0Euzo4DCYoXnS0n54wpuGNiJeVMGeffJczJg1R9g3yrodQN0Hgrdqvaj8FTbbAqAcQnjqsymcPNMjT2VGQabevbg0yFDq4QVsadPk7p9O2O2bWfU7t1ctfGry3yxIoFNAYWIiPiSv42Jg5XnVqVlhsHGnj35dOiQKmFF7OnTjN62nTHbtjEqK4tWxcVVnywkhD5aMi1BTgGFH/jtB9t5KyOHz389lrjICO8+uWnCP34GX78FGBASUWM/Ck+1zaawYWNMwpgagwqouvuHmzus+OzKK1k9cCCHY2MrHgtxOBi+axfXbt/OqG+yGHXPz4jRejxpJhRQiIiIL/njmDjYefarcIcVy4dcyaoBAzjiMSYOLS1l+K7djN6+nVFZWSQfPVplKQignUEkKCmg8AP7jp9l3LNr+NX4ntx/3RV1X9BQa5+FlXNwLbwwYNzjMLr6fhSe6ppNUduyD7dj8+ZxcsHCah8zgT3x8awe0J81AwaQmZyM02areLx9QQEjd+5kZNZOUg4eZMyqlXXWLBKoFFCIiIgv+eOYuLnxXAayOz6e1QMGsGbgADK7dcP0GBPHFRQwMmsnI3dmkbLzW+IKC2t8zvABA0he+o4PqhdpGgoo/MRPFn3F17kFfP7wOCJC7d598pwMWHwjOIoAE5LHQdI1dS73cKttNkVNTTSrk33nXZz/4osaHz/VqhXr+vVjXf9+pPfqdclSEIDkI0dI2bmTq3bvJiU0jJTFi+qsXSRQKKAQERFf8tcxcXPluVT6ZOvWfN6vL+v69Se9d69LloKAa0w8Mmsnw/bsZsievbQ/fbrW54656046PPhgk9Uu4k0KKPzEF3tOMHXBl/zvzQO5ZViC939ATgbsXwd7V0D2FzRkuQdcnE2xOmc1Tqou3ajPsg9PdYUV7tkVG3r3Jr13LzKuuILzEZcuf0nIy2Po3r0M2bOX1IQuXPu732EY1U6AE/F7CihERMSX/HVMLC7uJvTu2RUb+vQmvVdvMq7oyYVaxsRD9uyh27Fj1S8J8aDdQsRfKaDwE6ZpMuHP6zAMWPar1Kb7oL12Hqx8koYu93Cra9lHQ4MKqH0ZiFupzca2bkl82asXm7v3YEv35CqBRfTZs1y5dy8DDmQzuLiYm57/MzGhofV+bSJWUkAhIiK+5K9jYqnKc6xcYrezLal8TNyjO5nJVcfEbc+cYfC+fQw4cIABB7Lpl51N9Pnz9fpZkZMn0eWZZ7z+GkTqSwGFH3n7q4M8/N423rh7BKO6t2uaH1JluccYSBpd7+UebrUt+4D69aeoiXs9Xm0cNhu7OsezuXsPNvfozqYePTgeHV3lvIS8PAZkZ9P/0CEmzZ7NkMhIWtm9vIRGxAsUUIiIiC/585hYaue5M4hrTNyZTT0ujolPVFomDdA1L48BB7Lpn32A/gey6ZOTQ8uSkvr/0IgI+mRu8dZLEKmRAgo/UlRaxqinVjKkazQL0q5quh9UsdxjFWSvo6HLPdzqWvbRkP4UNdk35RaKt9W9nZIJHIqNZUv3ZLYnJrE9KZFvEhIoDgu75Dyb00nSsWP0yj1EP4eDMTPvZlCrVnQOD9fyELGUAgoREfElfx4TS8NlDR0G585hArnt2rElOZntSYlsT0wkq5Yxce/cXK7IPUSvQ7n0zs0lrqCwzuUhVdjt9Nmx3VsvRZo5BRR+5rnPvuWvq/awctYYurVr1bQ/rPLuHkOnQ3QCJDVsNoW3+1PUxv3mWx8Om409nTqxrVsS2xMT2ZaYxO7O8ZRVM4Mi6uxZeh06RD+nk2u+/336t2pF75YtaRMS0qh6RepLAYWIiPiSv4+JpfGy+g8Ah4NSm4098fEVgUVdY+LeuYe44tAheuXm0uPIYZKPHqN1UVGjalGjTqkvBRR+Ju90EVc/vZKpIxKZfWO/pv1hlZd7GDbAAHtYg2dTQNP0p6hL1sBB0IDpacUhIezt1ImdXbrwbZcu7OrcmZ1dOlPYunW158cVFNA9P5+ho0fTu2VL+pR/dQwL04wL8SoFFCIi4kv+PiaWpuHeLaQ4JIQ98fF827kz33bpwrddOrOzSxdOt6r+F6QdTp2i29FjJB89QvejR0k+cpTko0dpd/p0w2dc1EQzMQQFFH7pgXcy+WT7UTY8Mp6oFk3c5NG93CPrAziy1XXMsMO4RyG1/s0zPdXVn6Ipggq3nSNSMGvZG7o6JnAsOppvu3Tm286u4GJfp47s79CBkhqabEaeP0/vyEh6t2tH9xYt6NGiBd0jIujeogXtQkMVXkiDKaAQERFfCoQxsfhO1sBBmCUl5WPiLuws/0Xe3k4dOVDHmDj56FES8/LomnechBPH6Xr8BF3z8og+d8574UV11BcjKCmg8EPbDxUy6a+f8+jEPtw9Otk3PzQnA169HpwO10yK65+B4sIGL/dwq2vZBzRtUOFW13amtSkzDA7FxrK/Y0f2durIvo7ur06cbtWyxuva2O10b9GiSnCRFBFBl/BwQm22y305EsQUUIiIiC8FwphYrJd9512cWb+eQ7Gx7OvUkb0dO7G/Y4d6jYlbX7hAwvHjdD1+vPz7Cboez6Nzfj5xpwoIdVb/GcFXEt98g5ZXXmlpDVKVAgo/dcvLGzh06gJrHhpDiN1HH2hzMlzbkO7+tNHLPdz8Jajw1JA+FtUxgfzISPZ16sjB9nEcbN+OnPbtOdi+PTnt23O2RYsar7UB8eHhdA0PJzEi4uL3iAgSy29Hqu9Fs6SAQkREfClQxsTiv06+/Tbbn32O/R07VIyJ3ePhg+3bc662MbHTSVxBAfEnT9Lp5Ck6ncwn3uN7fH4+rYqLffhqvCwkhD7b627039xVt1x/yoEDbC+6cFmTbxRQNKFPdxzlp69v4sWpQ5g4oJNvf/hbU2Hnx67bjVzu4eaPQYWnrMFXQiOb/4ArvCho1YqDceXBRbv2HIxzvVEfjokhLyoKs44ZFNEhISSGh9M5PJz48HDiw8Jct8PCKu63DwvDrmUkQUUBhYiI+FKgjIklMJmmyYnSUlb81w85GNOWg+3jyCn/pV59x8Rtzp2n08l8Op4qIK6wgPaFhcQVFBJXWEBcQSHtCwuJOXMGewB+JpWaKaDwU2VOkzHzVtEhMoJ37xnl2x+ekwGLJkJZKRgGXDcXyooue7mHJ38PKjztGjeessOHvfqcpTYbx9q25XBsDEfaxri+x8RwOObi98rbQFXHDnQsDyw8w4uOYWHEhYbSofx7XFgYLavp0Cz+RwGFiIj4UqCMiSU4lTidHCouJruoiKy1a8las4bDMbEciWlb8b1eY+KyMtoXusKKuPIAo31hAe0LTxN75jQxZ84Qe+YMMafP0KK01AevTBpLAYUfW/j5fuZ8/A0f3Hs1gxKiffvDczLgqwWw7V0wna6gwh7eqOUengIpqKgsq19/KKu+AWhjmcCp1q1dyXJ0NHlRURyPjuJYdDR5UdEcj44iLyqKU5GR9X7OVjYbcWFhl4QW7u8dPO63Cw0lJjSUMPXIsIQCChER8aVAGhNL82OaJsdLSzlYVMThkhIOFRdzuKSEw8XF7N20iaOm2eAxcYuiItqdOUNM+Ves+/tpj9tnzhB99ixR584R1kTjfamdAgo/dqaolJF/XMn4PnE8f5tFDVw++Dlsed1120vLPTwFclBRWVafvuCjfxMlISEcb9OG41FR5EVHuwKM6Cjy27QhPzKSk5GR5Jd/ldbQdbkmre12YkNCiAkNJTY0lNiQEGLLw4uabkeHhGjJSSMpoBAREV8KpDGxSE2KnU6OeIQX7jDjWEkJeaWl5JV/P1ZSQkkDx+kti4qIPneO6LPniDrv+h597hxR51zfo8+dLb99nqhzZ4k+e442Fy5gC8DPyP5EAYWfe+Kjb3htwwE+f3gcHaMifF9ATgYsmgRl5U1qxjwC9lCvLPfwFExBRWXe6m9xOUzgbEQE+W1cocXJyEhORLZx3W7jCjAKOnehMLkb+aWlnCwtrWFz2NoZuHpneH5Fub/b7Zfcv+Qxj3NCmvnMDQUUIiLiS4E2JhZpDNM0OV1WdklgkVdNiHGspIR8h4OTpaU1fCKpnWGatDl3jsgLRUReOE+b8xdoXXSByPPnLz124QJtLpyn9YUi2pw/T+SFC0RecB0PsXh3EytETp5El2eeAbSLh987mH+ea+et4p5ru/M/E3pbU0ROBmx/D7YsgZIzrh0+vLjcw1MwBxXV8eWsi/pwGgZnIyIobNWKgvKvwlatKEoZQdnEiZx0OMgvLXWFGR63C70wBa6VzVYRWkSFhBBpt9PabifSbiey/L77q3U1xyJDQirOD8RlKgooRETElwJtTCziS07T5LTDURFW5JeWXnLbF2Pi6JAQ2niMd+sa/2pMrIDCZ376+ka+2HOCu1OTuaZne4YmtrWmkE8fhQ3/z3XbsMG4x7y63MNTcwsqqpPVfwA4HFaXUS8Om43T7drR4ZNlFJaVUeBwUOBwUFjD9wKHo+K8wvIvb2bFYYZR7Zt2S7vbkq6VAAAgAElEQVSdVjab67vdTkubreJ7dcdaVbqmpc1GuM2G0QTLWRRQiIiILwXimFjE3zmcTk55jn09xrtVxsI+HhN7/vKvPuPf2sbErex2wgzD78bECih85PUNB3j8gx0YQHiojSV3pVgTUuRkwOLJ4ChfrjDsTojq7PXlHp4UVFQv96GHOPPRx1aXcVns8fFcsXLFJcdM0+SsR7BxuqyMMw4HZ8rKOFNWxtny757HPO+frXSsKVsa2aDaoKOl3U4Lm+3iV/n9iHoeuzo6WgGFiIj4TCCOiUWCnbN8TFxYw5i4Ytxbw5i48ri5uY2JFVD4yAurdvPMp7sAsBvwwHd7ce/YHtYUk5MBuz+Dre9AYTZgQEhEkyz38FSfoMLAYHq/6Tww7IEmqyNQHJs3j5MLFlpdRqPY2rWj1+frGnydaZoUOZ3Vvomfdzo5X1bGOc/blY6dq+Y8z++lTfW+N3asAgoREfGZQBwTi0j9VR4Tn3E4KsKLmsa6NY1/zwfImFgBhY9syj7FbfM3UFpmEmo3eGvmSOuWebit+iOseeri/e7jXA00mzCkgPoFFUPjhnLf0PuazWyKy2Vl806vMgz6ZH3jsx9X6nRyoZo37XNOJxfKyigqf7ziq5pjReXHPY+lDx2qgEJERHwmEMfEIuI/Sp3OilCjujFxxZjXh2NiBRQ+tPHASWa+vpHI8FBWPzSmSdb7NEjl5R4AIS2afCaFW11BhYHB2ISxzWrZR1PIGjgISkqsLsN7WrWizyb//PevHhQiIuJLgTomFpHgZnkPCsMwJgDPA3ZggWmaT1V63Ch/fCJwHphumubm+lxbnUB+M166MYeH3v2aV2dcxdhecVaX4wopVv0B9q0qP2C4GmeOftBnJWTmZfLnTX9mU96mah9XUNH09k25heJt26wuw+uq65XRlBRQiIg0bxoTi4hYHFAYhmEHdgHXAbnAV8CPTNP8xuOcicAvcL0ZjwCeN01zRH2urU4gvxmXOJxc+8wqEmJa8s5PR1pdjktOBiy+sXwmhQkdB0KfyZA8xiczKdye2/gci3YswqT6v5MKKqz37TWpOE+csLqMJhE+YADJS99p1HMooBARab40JhYRcWnMmDjECz9/OLDHNM195cW8BdwEeL6h3gS8ZrrSkHTDMKINw+gEJNXj2qASFmLjrtRk5nz8DZuyTzI0McbqklwhRNqHcGAdHPkavvknHP0a1j0LaR/5LKR4YNgDjOs6rsZlHyYmK3NWsipnlYIKi9S34eWuceMpO3y4iavxruJt28jq3ade50ZOnkSXZ55p4opERCTAaEwsItJI3ggoOgM5HvdzcSXCdZ3TuZ7XBp3brkrgryt383+r97EgzQ8CCnCFEAnDXaHENx8ApmtGxa5PfDqLYnDcYJ4f93yt/SkUVPi/hiyryOrTFwKsF86Zjz4mq5otYvuFRwy1oBwREfEPGhOLiDSSNwKK6jo9Vv60UdM59bnW9QSGMROYCdC1a9eG1Od3WoWHMH1UEn/+z252HTvDFR0irS7poqRU15ajZcVgOmHTYihzQJ9JCiqkSdR3945g7ZMhIiJBQ2NiEZFG8kZAkQskeNzvAlSe213TOWH1uBYA0zTnA/PBtd6ucSVbL21kEi+v2cdLa/by3C1+9KHac7lHUSF88Tysfx4yXoK0j30aUoCCCrmoIf0hgrlXhoiI+C2NiUVEGskbAcVXQE/DMLoBh4DbgB9XOudD4Ofl6+lGAIWmaR4xDON4Pa4NSm1bhfGj4V15bcMBHrjuCrq0bWl1SRd5LvcwbK6ZFI5i2PqmzwMKNwUV0hD17ZUBkDV0GJw714TViIhIM6ExsYhIIzU6oDBN02EYxs+BT3Fti/SKaZo7DMP4WfnjLwH/xtWteA+uLZVm1HZtY2sKFHelduO1DQdYsG4/s2/sZ3U5VSWlgj0cykpcIcXm16G0GIZNV1AhQaPPpvp3P88aOAhKSpqwGhERCVQaE4uINF6jtxm1QjBtqfTg0q18/PVh1v96PDGtwqwup6qcDNdyD1sYLH8cMMEeCtP/bVlI4am2oMJN25OKr2ibURER8aVgGhOLSPBozJjY5u1ipGF+dm0yRaVOFn2x3+pSqpcwHFJngbPEtdwDoKwUPnvctQQkJ8PS8twzKhZfv5hxCeOwVfNX2j2jYtqyaUxfNp0nNjxBZl6mBdWKiIiIiIhITRRQWKxHXCTf7duBxRuyOVvssLqcmiWlgj0MDDtgQE46rJgDi2+0PKSA+gcVm/I2sXTXUtKWpfGrlb9SUCEiIiIiIuInFFD4gZ+N6U7hhVLeyjhodSk1c+/uMe5RGJpWftAERxHsX2tpaZ7qE1QAOHGyMmelggoRERERERE/oYDCDwzp2paU5Bj+tm4fxY4yq8upmXu5x+CpENIC15bdJuz8N6x5xi9mUrgpqBAREREREQksapLpJ9buOs60VzL43x8O5JarEuq+wGo5GbB/HRzaBN/+y3UsJALSPvKL5pmVZeZl8tHej9hbsJcteVtqbKhpw8aYhDFqqCmXRU0yRUTEl4JxTCwiga8xY+JGbzMq3pHasx394tvw0tq9/HBoF+w2w+qSapcw3PW17ln49t9ULPfYs9wvA4rBcYMrAofadv5wz6hYnbNaQYWIiIiIiIgPaYmHnzAMg3vGdGff8XMs/+ao1eXUX1Kqa+aE+6/S5r/Diif8arlHZfVZ/qGlHyIiIiIiIr6lJR5+pMxpMu7Z1US3COWf916NYfj5LAq3nAw4sA7O5EHGS65jfrzco7LaZlS4aemH1IeWeIiIiC8F65hYRAJbY8bEmkHhR+w2g5+O7s7W3EI27M23upz6czfPjIwDo/yvlKMIdvzT2rrqSTMqRERERERErKeAws/815DOtI8M5//W7LW6lIZLSgV7+MWQYtMi+PdDfr3cw5OCChEREREREesooPAzEaF27rymG+t2n2BbbqHV5TRMwnBI+xDGPQZjHoXS85AxHxZNCpiQAhoWVEz/ZDpLv11qQZUiIiIiIiLBRQGFH5o6oiuRESH835o9VpfScO7lHnY7uHtolBXDpsXW1nUZ6hNUlJllPJH+hGZTiIiIiIiINJICCj8UGRHKHSmJLNt+lH3Hz1pdzuWpWO5hBwzY+ia8d3dAzaRwq09QsTJnJdOWTVNQISIiIiIicpkUUPipGVd3I8xuY/7afVaXcnkqlns8Ct+bCzhh2zuwaGJAhhRQNagwuHSXFRNTQYWIiIiIiMhlUkDhp9pHhnPLsATe25zL0cIiq8u5PO7lHo4iKv6qlZXCf34H654N+KDi8ZTHsWOv8rg7qFAjTRERERERkfpTQOHHZo5OxmnCK1/st7qUxklKBXvYxeUe2ethxRxYfGPAhhQAU3pNYdH1i7Tjh4iIiIiIiBcooPBjCTEtmTSwE0vSsyk8X2p1OZfPc7nHkGnlB03XzIoD6ywtrbG0NamIiIiIiIh3KKDwcz+7tjvnSsp4Pf2A1aU0jnu5x5W3Q0gLwABMyN4AawN3uYebggoREREREZHGUUDh5/p0asPYXu159YsDXCgps7qcxquYTfEYJKTAnuWwMvCXe7g1JKiY/sl0ln671IIqRURERERE/I8CigBwz5ge5J8r4RdvbmZT9imry2m8hOEw+kHo+V0qZlI4imD/Wqsr85r6BBVlZhlz0udoNoWIiIiIiAgKKAKC3QDDgP9k5TF1QXpwhBQA3VIhJIKKkGL3clg7LyhmUrjVFVRoxw8REREREREXBRQBIH3/STBdt0scTtL35VtbkLdULPd4HHpeBznpQbXcw1PloMLAuORx9acQEREREZHmTgFFAEhJjiU81PWfyjQhpVuMxRV5UcJwGD0Luo4E94d2RxHsW2NpWU3FHVQ8nvI4duxVHldQISIiIiIizZUCigAwNLEtS+5KYUK/DpjAhVKn1SV5X1Kl5R7f/gvWPBN0MyncpvSawqLrF6mRpoiIiIiISDnDNE2ra2iwYcOGmRs3brS6DJ8rdpRxzdOr6N0xktfvHGF1Od6XkwEH1sGhzbDzY9exkAhI+8g10yJIZeZl8ur2V1mdsxonVcMnA4OxCWOZ0X8Gg+MGW1Ch1JdhGJtM0xxmdR0iItI8NNcxsYj4t8aMiTWDIoCEh9iZcXUS63afYMfhQqvL8b6E4ZA6CzoPoTks93BTI00REREREREFFAFn6ohEWoXZmb92n9WlNJ1LlnsAWR8H9XIPNzXSFBERERGR5kwBRYCJahHKj4Z35eOvj5B76rzV5TQN9+4e4x+HPjfB0UxY9WRQ7u5Rnfo20lR/ChERERERCSYKKALQT67phgEs/Hy/1aU0Hfdyj/hBXFzucSHol3t4qquRZplZxpz0OZpNISIiIiIiQaFRAYVhGDGGYSw3DGN3+fe21ZyTYBjGKsMwsgzD2GEYxq88HpttGMYhwzAyy78mNqae5iI+ugU3Dorn7a9yKDhfYnU5Tavyco9vl8Haec1iJgWoP4WIiEgg0JhYRMQ7GjuD4tfACtM0ewIryu9X5gBmmabZB0gB7jUMo6/H438yTXNw+de/G1lPszHz2mTOl5Tx9/Rsq0tpWp7LPXrdAIc3wco5zWa5h1t9+1No2YeIiIglNCYWEfGCxgYUNwGLy28vBr5f+QTTNI+Yprm5/PYZIAvo3Mif2+z17tiGa69oz6L1BygqLbO6nKblXu7RZSiX7O6xf62lZVmhrv4UWvYhIiJiCY2JRUS8oLEBRQfTNI+A600XiKvtZMMwkoArgS89Dv/cMIyvDcN4pbrpcB7XzjQMY6NhGBuPHz/eyLKDw0+vTebE2RLe33zI6lJ845LlHibsXQVrn21WMyncautP4V72MW3ZNJ7b+JxFFYqIiDQrGhOLiHiBYZpm7ScYxn+AjtU89Ciw2DTNaI9zT5mmWe0bqmEYrYE1wFzTNN8vP9YBOAGYwBygk2maP6mr6GHDhpkbN26s67SgZ5omN73wBWeLHCx/4FrsNqPuiwJdTgbsXwcH1sK+1YDhCi3SPnTNtGiGMvMyeXX7q6zKWYVJ1X/PM/rN4IFhD1hQWfNjGMYm0zSHWV2HiIh4n8bEIiL105gxcZ0zKEzT/I5pmv2r+foAOGYYRqfyIjoBeTUUGAq8ByxxvxGXP/cx0zTLTNN0An8DmucnzMtkGAYzRyez78Q5ln9zzOpyfCNhOIyeBUmjqZhJ4SiCA+usrswydS37eHXHq1ryISIi0kgaE4uINL3GLvH4EEgrv50GfFD5BMMwDGAhkGWa5nOVHuvkcfcHwPZG1tPsTOjXka4xLXl57V7qmg0TVLpVWu5xaDOsa57LPdzcyz6Gxg2t8tjKnJWkfZKmBpoiIiJNQ2NiEREvaGxA8RRwnWEYu4Hryu9jGEa8YRju7sNXA3cA46rZOul/DcPYZhjG18BY4P5G1tPshNht3JXajS0HC9iYfcrqcnzHvbvHuMcgfgjs/BhWNL/dPSobHDeYRdcvYka/GVV3+jCdPJH+hGZTiIiIeJ/GxCIiXlBnDwp/pPV2l7pQUsaop1YwNLEtC9Kusroc31vzDKx6svyODcY/5tr1o5lb+u1S5qbPpYyqu7wYGIxNGMuM/jMYHDfYguqCk3pQiIiIL2lMLCL+qEl7UIj/axFmZ9rIJP6TlceevDNWl+N7ydeWL/cAcFJNn8hmyXOnj8qzKdw7fUz/ZLqWfYiIiIiIiF9QQBEkpo1MJCLUxvy1+6wuxfcShkPaRzDmEWjfB1bNhX/+d7Ne6uFWVwPNMrNMyz5ERERERMQvKKAIErGtw5kyNIF/bjnMsdNFVpfjewnDYcyv4XtzwXRC5hJYNEkhRTnP2RS2av7ZazaFiIiIiIhYTQFFELkrtRsOp5NXvzhgdSnWOZIJRvlyhrJi2P6etfX4EfdsisXXL6522YdmU4iIiIiIiJUUUASRxNhWXD+gE0vSszlTVGp1OdZISgV7OBjlyxm2LIFPH9NMCg91LfvQbAoREREREbGCAoog89PRyZwpdvBWRo7VpVijYvvRR+HaR6DkDGz4KyyerJCiktqaaGo2hYiIiIiI+JoCiiAzsEs0I5NjWfj5fkocTqvLsUbCcNc2oyGhYJT/FXcUwZ7l1tblhzSbQkRERERE/IUCiiA089pkjp4u4qOth60uxVoVyz3K/5pvex9WP62ZFNXQbAoREREREbGaAoogNOaK9vTqEMn8tfswTdPqcqxTsdzjMRh8B5zcA6v/AItvVEhRDc2mEBERERERKymgCEKGYTBzdDLfHjvD6l3HrS7HWu7lHrHdwD0zwFEE+9dZWpY/02wKERERERGxggKKIDV5UDydoiJ4ec1eq0vxD0mpEBKBK6Qw4fAmWPesZlLUQLMpRERERETE1xRQBKmwEBs/ubob6ftOsjWnwOpyrOe53CN+KOz8F6yYo+UedahrNsWc9DmaTSEiIiIiIl6hgCKI3TY8gciIEOav3Wd1Kf4hYTiMfhB6TSg/YEJZMRzQco/a1DabwsTUbAoREREREfEKBRRBLDIilNtTElm2/QjZ+eesLsd/JI8pX+4BmE6IjLeymoBR12yKJ9OfVEghIiIiIiKXTQFFkJsxKokQm40F6/ZbXYr/SBgOaR/B1fdBy3bw2aOw/Hda6lEPtc2mcOJUA00REREREblsCiiCXFybCH5wZWfe+uogz3y6k03Zp6wuyT8kDIfrfg/fmwvn8+GLP8PiyQop6qm22RRa8iEiIiIiIpdDAUUzcHWPWErLTF5ctZepC9IVUng6fYiKfwaOIti3xtJyAonnbApbpbcSNdAUEREREZGGUkDRDOScugCACZQ6nKTvy7e2IH+SlAoh4VT8U9izHNZq+9GGmNJrCo+lPKYGmiIiIiIi0ighVhcgTS8lOZYwu0FJmYnNZpCSHGt1Sf7Dvf3ogXVwcAPsXu4KJ0IiXMcThltdYUCY0msKPdv25NXtr7IqZxUmZsVj7gaa7vNERERERESqoxkUzcDQxLa8cXcK0S1D6dauFUMT21pdkn9JGA6psyBhZPkBExzafrSh1EBTREREREQaQwFFMzEsKYZfjuvJrmNn2ZZbaHU5/qlb6sXtR3FCy/aWlhOo1EBTREREREQuhwKKZuTmYV1oGWZn0foDVpfin9zbj6Y+CK07wvLfwme/VT+Ky1BXA80n059USCEiIiIiIpdQD4pmpE1EKD8c0oW3v8rhkYm9adc63OqS/E/CcNdX+97w/l2w/nnIeNkVXKgfRYO5e07MTZ9LGWUVx91LPj4/9Dkz+s9gcNxgq0oUEREJKqWlpeTm5lJUVGR1KQEhIiKCLl26EBoaanUpIoICimYnbVQir6dn81bGQX4+rqfV5fivwoO4Jhg5XduP7l+rgOIy1dZAc2XOStbkruHREY+qgaaIiIgX5ObmEhkZSVJSEoZh1H1BM2aaJvn5+eTm5tKtWzeryxERtMSj2ekRF0lqz3b8Pf0gpWVOq8vxXxXbj5b/jz17PazT9qOXS0s+REREfKOoqIjY2FiFE/VgGAaxsbGabSLiRxRQNENpI5M4erqIz3Ycs7oU/+XefnTc49B1JOxdASvmwOIbFVI0wpReU3gs5bFqd/mYkz5Hu3yIiIh4gcKJ+tOflYh/UUDRDI3tHUdCTAsWq1lm7RKGw+hZ0H18+QETyrT9aGPVtMuHialdPkREREREmjEFFM2Q3WYwLSWJjAMn2XFYW47WKfnai9uPmk5o38faeoKAlnyIiIgEL8MwmDVrVsX9efPmMXv27Hpdm5mZyciRI+nXrx8DBw7k7bffrnhs//79jBgxgp49e3LrrbdSUlLi7dJFxGKNCigMw4gxDGO5YRi7y7+3reG8A4ZhbDMMI9MwjI0NvV6875ZhCbQItWsWRX24tx8d8TOwR8B/ZsPqp7XUwwtqW/KhkEJERAJFoI+JN2Wf4oVVe9iUfcorzxceHs7777/PiRMnGnxty5Ytee2119ixYweffPIJ9913HwUFBQA8/PDD3H///ezevZu2bduycOFCr9QrIv6jsbt4/BpYYZrmU4Zh/Lr8/sM1nDvWNM3K71INuV68KKplKD8Y0pn3NuXyyPV9aNsqzOqS/Jt7+9HWHWDF72H1H+DzP7n6VGh3j0apaZcPbUUqIiIBxC/HxL//aAffHD5d6zlnikrZefQMThNsBvTuGElkRM1bbvaNb8PvJver9TlDQkKYOXMmf/rTn5g7d26Dar7iiisqbsfHxxMXF8fx48eJiopi5cqVvPHGGwCkpaUxe/Zs7rnnngY9v4j4t8Yu8bgJWFx+ezHwfR9fL42QNjKJYoeTt77KsbqUAGJSsbOHo0j9KLyktiUf6kshIiIBIGDHxKeLHDjLdwB3mq773nDvvfeyZMkSCgsvXU68ZMkSBg8eXOXr5ptvrvIcGRkZlJSU0L17d/Lz84mOjiYkxPX71S5dunDo0CGv1Coi/qOxMyg6mKZ5BMA0zSOGYcTVcJ4JfGYYhgm8bJrm/AZej2EYM4GZAF27dm1k2QLQq2MkI5NjeX3DAe5O7UaIXS1J6pSU6upH4bgAmBAWaXVFQWVKrykAPJn+JE4uboPr7kvheY6IiIgf8csxcV0zHcC1vGPqgnRKHU5CQ2w8f9uVDE1s/AqTNm3aMG3aNP7yl7/QokWLiuNTp05l6tSpdV5/5MgR7rjjDhYvXozNZsM0zSrnaAcOkeBTZ0BhGMZ/gI7VPPRoA37O1aZpHi5/s11uGMZO0zTXNuB6yt/A5wMMGzas6juUXJbpVyfx09c38Z+sY0zo38nqcvyfe/vRPf+BzDdh5RwozIU+k7TUw0vcAcTc9LmUUVZx3N2XwvMcERERXwnWMfHQxLYsuSuF9H35pCTHeiWccLvvvvsYMmQIM2bMqDi2ZMkSnnnmmSrn9ujRg3fffReA06dPc8MNN/Dkk0+SkpICQLt27SgoKMDhcBASEkJubi7x8fFeq1VE/EOdAYVpmt+p6THDMI4ZhtGpPOntBOTV8ByHy7/nGYbxD2A4sBao1/XSdL7TpwOdo1uwaP0BBRT15e5H0a43vDcD1j8PGS+7GmkqpPCK2vpSzEmfo74UIiLic8E8Jh6a2NarwYRbTEwMt9xyCwsXLuQnP/kJUPcMipKSEn7wgx8wbdo0pky5+AsJwzAYO3Ys7777LrfddhuLFy/mpptu8nrNImKtxs7p/xBIK7+dBnxQ+QTDMFoZhhHpvg18F9he3+uladltBneMTCR930l2Hq29iZJUUrCfin9CjmLYr34U3lRTXwoTU30pRETE32hMXINZs2Y1aDePd955h7Vr17Jo0aKK/hSZmZkAPP300zz33HP06NGD/Px87rzzzqYqW0Qs0tgeFE8B7xiGcSdwEJgCYBhGPLDANM2JQAfgH+VrxEKAN0zT/KS268W3bh2WwJ+W72Lx+gP88b8GWl1O4EhKhZBwV7NMTCjyztZccin1pRARkQCgMbGHs2fPVtzu0KED58+fr/e1t99+O7fffnu1jyUnJ5ORoW3eRYJZowIK0zTzgfHVHD8MTCy/vQ8Y1JDrxbfatgrj+4M7848th3h4Qm+iW2rL0Xpx96PYvxZ2fQobXoSSczDoR1rq4WXqSyEiIv5MY2IREe/Qtg0CQNqoJIpKnbyzUVuONkjCcBj9IIx5GEwnbHwFFk+GHKX73jal1xQWXb+IcQnjMLjYtdvdl+JXK39FZl6mhRWKiIiIiEhjKKAQAPrGt2F4txhe25BNmVObpDTYka3g3urKUQT71lhbT5BSXwoRERERkeClgEIqTB+VRO6pC6zcqc1UGiwpFezhVPyTOpgO657VTIomMqXXFB5LeeySkAIu9qVQSCEiIiIiEnga2yRTgsh3+3agU1QEi9bv57q+HawuJ7C4+1EcWAe7PoO9/4G9K11NNNM+VE+KJqC+FCIiIiIiwUUzKKRCiN3G7SmJfLEnn93HzlhdTuBJGA6ps6D7uPIDTigrcYUW0iRq60uhmRQiIiIiIoFFAYVc4kfDuxIWYmPxhgNWlxK4uo91zZwAV+PMhBHW1hPkPPtSKKQQERGxnmEYzJo1q+L+vHnzmD17dr2vnzBhAtHR0UyaNOmS4/v372fEiBH07NmTW2+9lZKSEgBM0+SXv/wlPXr0YODAgWzevNkrr0NEfE8BhVwiplUYNw6K5/3Nhyi8UGp1OYEpYTikfQwDbgFM+HK++lH4wJReU6o0z9QOHyIiIvWQk+HVsUp4eDjvv/8+J06cuKzrH3roIV5//fUqxx9++GHuv/9+du/eTdu2bVm4cCEAy5YtY/fu3ezevZv58+dzzz33NKp+EbGOelBIFdNHJfHuplyWbszhrtRkq8sJTAnDXV/OMtjxHmR9CCER6kfRxNw9J55MfxInTuDiDh9rctfw6IhH1ZdCRESaj2W/hqPbaj+n+DQc2+6a9WnYoEN/CG9T8/kdB8D1T9X6lCEhIcycOZM//elPzJ07t8Fljx8/ntWrV19yzDRNVq5cyRtvvAFAWloas2fP5p577uGDDz5g2rRpGIZBSkoKBQUFHDlyhE6dOjX4Z4uItTSDQqro3zmKYYlteT09G6e2HG2cuN7lN0z1o/AR7fAhIiLSAEWFrnACXN+LCr3ytPfeey9LliyhsPDS51uyZAmDBw+u8nXzzTfX+nz5+flER0cTEuL6/WqXLl04dOgQAIcOHSIhIaHiXM/HRCSwaAaFVCttVBK/eHMLq3flMa63dvS4bMljXFMmHUXqR+FD2uFDRESEOmc6AK5lHYtvdP0ixR4GP1zgldmebdq0Ydq0afzlL3+hRYsWFcenTp3K1KlTG/x8pln1l2aGYdT5mIgEFs2gkGpN6N+RDm3CWbQ+2+pSAlvCcEj7CAbeCpiQ8Tf1o/AR7fAhIiJSD+6t0sc96vWlqPfddx8LFy7k3LlzFccudwZFu3btKCgowOFwAJCbm0t8fDzgmuuP/kAAACAASURBVDGRk5NTca7nYyISWBRQSLVC7Tamjkhk7a7j7D1+1upyAlvCcPiv+dD3B/DNP2HFk67fVCikaHLa4UNERKQe3Fule7lPVkxMDLfccktFM0twzaDIzMys8vXuu+/W+lyGYTB27NiK8xYvXsxNN90EwI033shrr72GaZqkp6cTFRWl/hMiAUoBhdToR8O7Ema38dr6A1aXEhw69C2/4VQ/Ch9z7/ChkEJERMS3Zs2a1eDdPFJTU5kyZQorVqygS5cufPrppwA8/fTTPPfcc/To0YP8/HzuvPNOACZOnEhycjI9evTg7rvv5sUXX/T66xAR31APCqlR+8hwJg3sxLubcnnwe72IjAi1uqTAljwG1s0DRzFgQtdRFhfUvFS3w4d7G9LPD33OjP4zGBw32MoSRUREgsLZsxdn33bo0IHz58836Pp166r/JU5ycjIZGVVnoBqGwQsvvNCwIkXEL2kGhdQqbVQS50rKuO+tTDZln7K6nMCWMBzSPoZ+P3A1zNzyd/Wj8LHqdvhwb0M6/ZPpmk0hIiIiImIhzaCQWjmcJoYBK3bm8cXeEyy5K4WhiW2tLitwJQx3fRWfhcy/AzYICfd6UyqpWXUzKeDiNqSe54iIiIiIiO9oBoXUKn1fPpTv3FTicLruS+N1Hlp+Q/0orOCeSWHHfslx9aUQEREREbGOAgqpVUpyLOEhrr8mBgYpybEWVxQkeowHe3j5HRMSr7G0nObIcxtSTwopRERERESsoYBCajU0sS1L7k5hUJco7Dbo2aG11SUFh4ThMP1j6PEdVz+KE99aXVGz5N6G9Lcpv9UOHyIiIiIiFlNAIXUamtiWJ27qT0mZyUdbD1tdTvBIGA4/XgrdroV/PQif/EYNMy2ibUhFRERERKyngELqZWCXKHp3jOSdr3KsLiW42Gww4qdQVgzpL8DiyQopLKKQQkRExDsMw2DWrFkV9+fNm8fs2bPrdW12djZDhw5l8ODB9OvXj5deeqnisf379zNixAh69uzJrbfeSklJCQCmafLLX/6SHj16MHDgQDZv3uzV1yMivqOAQurFMAymDEtga24hO4+etrqc4HJ8JxX/FB3FaphpoZpCirnpc8nMy7SwMhERkaaTmZfJgm0LvPb/uvDwcN5//31OnDjR4Gs7derE+vXryczM5Msvv+Spp57i8GHXDN6HH36Y+++/n927d9O2bVsWLlwIwLJly9i9eze7d+9m/vz53HPPPV55HSLie9pmVOrtB1d25qllWbzzVS6/ndzX6nKCR1Kqa6tRRxHw/9m78/iqqnP/458nCYQxEECQmIiGME9hHpTREVTQK4MWCyLCLcVaLCjaQWmv9oeF4tVWb4uiYouIUgSq4khVRJExMmMQRBIQISTMISRZvz/OSQyQkIScZJ8k3/frdV45Z++19372NjxZPmfttR1Uq+t1RJVaziNG/7DqD7nLssjipc0v8fSAp70KS0REpNieXP0k2w9vv2Cb4xnH2ZG6A4fDMFpEtqBW1YLnHGtZryVTu0294D7DwsIYP348Tz31FE888USxYq5atWru+9OnT5Od7XskuHOO5cuX8+qrrwIwevRopk2bxoQJE1iyZAmjRo3CzOjRowdpaWns37+fxo0bF+vYIuI9jaCQIqtXsyrXt76UNzckcTozy+twKo6YbjB6KfSdCrUvhU+ehP/8Ubd6eGhYi2HnPd3jP3v/o1s9RESkwjl25hjO/0x5h+PYmWMB2e/EiROZN28eR44cOWv5vHnziI+PP+81dOjQ3DZ79+6lffv2xMTEMHXqVKKiokhJSaFu3bqEhfm+X42OjiY5ORmA5ORkYmJicrfPu05EyheNoJBiGd41hrc37efDrT9wU3tVpQMmppvvVbsxvPVLX5Fi5TO+wkVMN6+jq5TGtB3DJ0mfkOV8xTiH439W/Q/w4ygLERGRYFbYSAfw3d4x7v1xnMk+Q5WQKkzvPZ34hvElPnZERASjRo3imWeeoXr16rnLR44cyciRIy+4bUxMDBs3bmTfvn3ceuutDB06lJCQ879XNfPdkumcK3CdiJQvGkEhxXJ1XAOi6lRjwVpNllkqTqVAzvwHWZqPwkvxDeP5TfffEJInTeYUKTSSQkREKor4hvE8f/3z3NfxPp6//vmAFCdyTJo0iTlz5nDixIncZUUZQZEjKiqKNm3asGLFCho0aEBaWhqZmZkAJCUlERUVBfhGTOzd+2PfNO86ESlfVKCQYgkNMYZ2jmZF4kH2pZ3yOpyKJ2c+CgCXDY3aehtPJTesxTB+2+O3KlKIiEiFFt8wnnvb3RvQ4gRAvXr1GD58eO5kluAbQZGQkHDea+HChYCvuHDqlK+PmZqaysqVK2nRogVmRv/+/XPbzZ07lyFDhgAwePBgXnnlFZxzrFq1ijp16mj+CZFySgUKKbZhXWJwDhauS/I6lIonphuM/jd0nwAhVeCzp+DTP2s+Cg+pSCEiInLxJk+eXKyneWzbto3u3bvToUMH+vbty5QpU2jXrh0ATz75JLNmzSIuLo6UlBTGjh0LwKBBg4iNjSUuLo5x48bx3HPPlcq5iEjps/zu2Sryxmb1gAXAFcC3wHDnXOo5bVr42+SIBR51zv2vmU0DxgEH/et+7Zx7p7DjdunSxa1du/ai45aS+8nzq/ju8Ek+fbA/ISG6x69UvPMQrP47YBBWTfNReOyNHW/w+KrHySb7rOUDYgYwpu0Y4hvGY2brnHNdPApRREQ8Ekx94m3bttGqVauLO5FKStdMJLBK0icu6QiKh4GPnHPNgI/8n8/inNvhnIt3zsUDnYGTwJt5mjyVs74oiViCw4iuMSSlnuKLXSleh1Jx1Wrof+MgK0PzUXgsv5EUAMv3Lufud+/WaAoRkcpNfWIRkQAoaYFiCDDX/34ucGsh7a8BvnHO7SnhccVjN7S5lIhqYbyuyTJLz5V9fpyPAgdNrvY0HCm4SJHlsnh81eOERYQ18Cg0ERHxlvrEIiIBUNICRSPn3H4A/8+GhbS/A5h/zrL7zGyjmb1oZpEFbWhm481srZmtPXjwYEHNpIxUqxLKrR0vY9nm7zly8ozX4VRMMd1g9FvQ8ibfhJkHNnkdkfBjkSKU0LOWZ5NNWL2wJh6FJSIi3lKfWEQkAAotUJjZh2a2OZ/XkOIcyMyqAoOBvOOg/w9oCsQD+4E/F7S9c262c66Lc67LJZdcUpxDSykZ3iWGjMxslnyV7HUoFVdMNxgxD5oOgHd/A+/9VhNmBoFhLYbx8sCXGRAzAENzsIiIVAbqE4uIlL5CCxTOuWudc23zeS0BDphZYwD/zx8usKuBwHrn3IE8+z7gnMtyzmUDzwOaAbAcaXtZHVo3jtBtHqXNDLqMhax0+OIvMHewihRBIL5hPE8PeJrf9fjdebd8iIhIxaM+sYhI6Stpr3opMNr/fjSw5AJt7+ScoWw5idzvNmBzCeORMjaiawybk4+yZd8Rr0Op2A7tgJxv6jPTNWFmEMm55SPMwrwORUREvKM+sYhIAJS0QDEduM7MEoHr/J8xsygzy5192Mxq+NcvOmf7P5nZJjPbCPQHHihhPFLGhsRHUTUshNfXaBRFqbqit+9RowA4qNfU03DkbMNaDOOlG18i80im7ncSEamc1CfOw8yYPHly7ueZM2cybdq0Im8fGhpKfHw88fHxDB48OHf57t276d69O82aNWPEiBFkZGQA4Jzj/vvvJy4ujvbt27N+/frcbd59911atGhBXFwc06dPL/nJiUipKlGBwjmX4py7xjnXzP/zsH/5PufcoDztTjrn6jvnjpyz/U+dc+2cc+2dc4NzJheS8qNujarc0OZSFifsI/1MltfhVFwx3WD0UrhqElSpAStmwaczdatHEIlvGE9maub3XschIiJlr7z3iU9u2MChv8/m5IYNAdlfeHg4ixYt4tChQxe1ffXq1UlISCAhIYGlS5fmLp86dSoPPPAAiYmJREZGMmfOHACWLVtGYmIiiYmJzJ49mwkTJgCQlZXFxIkTWbZsGVu3bmX+/Pls3bq15CcoIqVGY5KlxEZ0ieHfX+3j/a0HGNwhyutwKq6Ybr6XhcBns+D7jb5RFaOX+paLiIiI5PH9H//I6W3bL9gm6/hxTm/fDs6BGeEtWxJaq1aB7cNbteTSX//6gvsMCwtj/PjxPPXUUzzxxBMXFfu5nHMsX76cV199FYDRo0czbdo0JkyYwJIlSxg1ahRmRo8ePUhLS2P//v18++23xMXFERsbC8Add9zBkiVLaN26dUBiEpHA08xuUmK9mtbnsrrVdZtHWama02lwkJWh+ShERETkomUfPeorTgA45/scABMnTmTevHkcOXL2PGXz5s3LvX0j72vo0KG5bdLT0+nSpQs9evRg8eLFAKSkpFC3bl3Cwnzfr0ZHR5Oc7LuzMjk5mZiYmNztc9YVtFxEgpdGUEiJhYQYw7pE878fJrL38Eli6tXwOqSK7Ur/fBSZ6YCDJld7HZGIiIgEocJGOoDv9o7vxtyDO3MGq1KFqJkzqNGxY4mPHRERwahRo3jmmWeoXr167vKRI0cycuTIC2773XffERUVxa5duxgwYADt2rUjIiLivHZmvgnEXU6B5Zx1BS0XkeClERQSEMO6xGAGb6xL8jqUii+mG4z+NzQfCC4bDn/jdUQiIiJSTtXo2JHLX3qRS+6/n8tfejEgxYkckyZNYs6cOZw4cSJ3WVFGUERF+W4Zjo2NpV+/fmzYsIEGDRqQlpZGZmYmAElJSbntoqOj2bv3x5G8OesKWi4iwUsFCgmIy+pW5+q4Bixcu5es7POr1RJgMd3gjlchpge88yB8+HtNmCkiIiIXpUbHjjT47/EBLU4A1KtXj+HDh+dOZgm+ERQ5E2DmfS1cuBCA1NRUTp8+DcChQ4dYuXIlrVu3xszo379/bru5c+cyZMgQAAYPHswrr7yCc45Vq1ZRp04dGjduTNeuXUlMTGT37t1kZGTw2muvnfVUEBEJPipQSMAM7xLDviPprNx5cTM2SzGFhEC38ZBx3Ddp5tzBKlKIiIhIUJk8eXKxnuaxbds2unTpQocOHejfvz8PP/xw7qSWTz75JLNmzSIuLo6UlBTGjh0LwKBBg4iNjSUuLo5x48bx3HPPAb7JOv/6179yww030KpVK4YPH06bNm0Cf5IiEjCag0IC5vo2jahbowoL1u6lT/NLvA6nckj7FjB8E2ae9k2YqSd6iIiIiIeOHz+e+75Ro0acPHmyyNv26tWLTZs25bsuNjaW1avP/zLGzHj22Wfz3WbQoEEMGjQo33UiEnw0gkICJjwslFvjL+ODLQdIPZHhdTiVwxW9ISzc995lQ1Qnb+MRERERERG5SCpQSEAN7xJDRlY2ixP0CKcykTNhZpd7fZ/XzoEVf9atHiIiIiIiUu7oFg8JqNZREbS7rA4L1uzl7l5X6FFOZSGmm+916jBsWQTb34bQcBi9VLd7iIiIiIhIuaERFBJww7vGsP37Y2xKPuJ1KJVLg+a+ny4bsjJ881GIiIiIiIiUEypQSMAN7hBFeFgIr6/dW3hjCZy4ayC0qu+9mW9+ChERERERkXJCBQoJuDrVqzCw7aUsSdhH+pksr8OpPGK6wd1vQ8PWgEGN+l5HJCIiIiIiUmQqUEipGN41hmPpmSzbvN/rUCqXmG5w1yKoUh0W3gOfasJMERERKVuhoaHEx8fnvqZPnx6wfSckJPDOO+8EbH8iElw0SaaUih5X1ufyejVYsGYvt3WM9jqcyiWiMXQeDZ//BfZ/BWHVNGGmiIiIlJnq1auTkJBQKvtOSEhg7dq1DBo0qFT2LyLeUoFCSkVIiDG8SzQz3/+aPSknaFK/ptchVS7V6vrfuB8nzFSBQkREpFKxjz8ulf26fv2Kvc2RI0fo1q0bS5cupUWLFtx5550MGDCAcePGMWHCBNasWcOpU6cYOnQov//97wFYs2YNv/zlLzlx4gTh4eF88MEHPProo5w6dYrPPvuMRx55hJtvvplf/OIXbNq0iczMTKZNm8aQIUMCfMYiUlZ0i4eUmts7RxNi8MbaJK9DqXyu7ON71GgOTZgpIiIiZeTUqVNn3eKxYMEC6tSpw1//+lfuvvtuXnvtNVJTUxk3bhwATzzxBGvXrmXjxo188sknbNy4kYyMDEaMGMHTTz/NV199xYcffkjNmjX5wx/+wIgRI0hISGDEiBE88cQTDBgwgDVr1vCf//yHBx98kBMnTnh8BUTkYmkEhZSaxnWq06f5Jbz65XdUqxJCz6YN6Nwk0uuwKoeYbnD3W/D+72DvKsC8jkhERETK2MWMdAiEgm7xuO6663jjjTeYOHEiX331Ve7y119/ndmzZ5OZmcn+/fvZunUrZkbjxo3p2rUrABEREfke6/3332fp0qXMnDkTgPT0dL777jtatWpVCmcmIqVNIyikVHVtEsnhkxn8+YOvGfnCKtbtSfU6pMojphvctRBqR8Gi8fDJDE2YKSIiIp7Jzs5m27ZtVK9encOHDwOwe/duZs6cyUcffcTGjRu56aabSE9PxzmHWeFfsDjn+Ne//kVCQgIJCQkqToiUcypQSKnKcr6fzsGZzGxW7UrxNqDKJrw2dB0LqbvgP0/A3MEqUoiIiIgnnnrqKVq1asX8+fO55557OHPmDEePHqVmzZrUqVOHAwcOsGzZMgBatmzJvn37WLNmDQDHjh0jMzOT2rVrc+zYsdx93nDDDfzlL3/BOV+nc8OGDWV/YiISMCpQSKm6Kq4BoSG+6neV0BB6xNb3OKLKKOfbhzwTZoqIiIiUknPnoHj44Yf5+uuveeGFF/jzn/9M79696dOnD48//jgdOnSgY8eOtGnThnvuuYerrroKgKpVq7JgwQJ+8Ytf0KFDB6677jrS09Pp378/W7duzZ3b4ne/+x1nzpyhffv2tG3blt/97ncen72IlITmoJBS1blJJI/d0ppHl2zh5/3jNAeFF67sDWHhkHna91kTZoqIiEgpysrKynf5tm3bct/PmjUr9/3LL7+cb/uuXbuyatWq85bnjKrI8fe///0iohSRYKQRFFLqRnZvQqOIcDYnH/E6lMopphuMfsv3ZA+XBZnpXkckIiIiIiJyHhUopNSFhhg3tYvi4x0HOZp+xutwKqeYbvCT16FuE1g8URNmioiIiIhI0FGBQsrELR0ak5GVzftbDngdSuVVpTp0vReOfKcJM0VERCqwnAkjpXC6ViLBRQUKKRPxMXWJjqzOv7/a53UolVt2zggWTZgpIiJSEVWrVo2UlBT9j3cROOdISUmhWrVqXociIn6aJFPKhJlxS4coZn+6i8MnMqhXs6rXIVVOV2jCTBERkYosOjqapKQkDh486HUo5UK1atWIjo72OgwR8VOBQsrMLe2j+L+Pv2HZ5v2M7N7E63Aqp5wJMz/6H/j2U8jSnCAiIiIVSZUqVbjyyiu9DkNE5KKU6BYPMxtmZlvMLNvMulyg3Y1mtsPMdprZw3mW1zOzD8ws0f9Tz6CswFo1rk3TS2rqNg+vxXSDnyyAOjGw5D5NmCkiIlJC6hOLiARGSeeg2Az8F/BpQQ3MLBR4FhgItAbuNLPW/tUPAx8555oBH/k/SwWVc5vHl7sPc+CoHnXpqao1oPPdkLpLE2aKiIiUnPrEIiIBUKIChXNum3NuRyHNugE7nXO7nHMZwGvAEP+6IcBc//u5wK0liUeC3y0donAO3tq43+tQBPP/1ISZIiIiJaE+sYhIYJTFHBSXAXvzfE4CuvvfN3LO7Qdwzu03s4YF7cTMxgPj/R9Pm9nm0gi2BBoAh7wO4hxBG9O9T8K9Xkfyo6C9TqV5gNpVqdmsXkhzDMPhEv845etjGVNOeBlTMQVbPAAtvA5ARESClvrE3lFMRaOYChds8UBwxnTRfeJCCxRm9iFwaT6rfuOcW1KEY1g+y4r93CPn3Gxgtj+mtc65Au/v84JiKhrFVDSKqXDBFg/4YvI6BhERKR3qExeNYioaxVQ0wRZTsMUDwRvTxW5baIHCOXftxe7cLwmIyfM5GsiZJfGAmTX2V4obAz+U8FgiIiIiIgGnPrGISOkr6SSZRbEGaGZmV5pZVeAOYKl/3VJgtP/9aKAo1WcRERERkfJGfWIRkUKU9DGjt5lZEtATeNvM3vMvjzKzdwCcc5nAfcB7wDbgdefcFv8upgPXmVkicJ3/c1HMLkncpUQxFY1iKhrFVLhgiweCMyYRESll6hOfRTEVjWIqmmCLKdjigQoWkzlX7FvfREREREREREQCqixu8RARERERERERuSAVKERERERERETEc0FboDCzF83sh4Ke7Ww+z5jZTjPbaGadgiCmfmZ2xMwS/K9HyyCmGDP7j5ltM7MtZvbLfNqU2bUqYjxlep3MrJqZrTazr/wx/T6fNmX6+1TEmMr898l/3FAz22Bmb+Wzrsz/3RUhJi/+3X1rZpv8xzvvMUpeXScREal41CcuckzqExcek/rExYtNfeLC4wl4n7jQx4x66GXgr8ArBawfCDTzv7oD/+f/6WVMACucczeXchx5ZQKTnXPrzaw2sM7MPnDObc3TpiyvVVHigbK9TqeBAc6542ZWBfjMzJY551blaVPWv09FiQnK/vcJ4Jf4Ju+KyGedF//uCosJvLlO/Z1zhwpY59V1EhGRiudl1CcuCvWJC6c+cfGoT1w0Ae0TB+0ICufcp8DhCzQZArzifFYBdc333GgvYypzzrn9zrn1/vfH8P3CXnZOszK7VkWMp0z5z/u4/2MV/+vc2WHL9PepiDGVOTOLBm4CXiigSZn/uytCTMGozK+TiIhUTOoTF436xEWKSX3iIlKfOGCKfZ2CtkBRBJcBe/N8TsLjf/R+Pf1DlJaZWZuyPLCZXQF0BL48Z5Un1+oC8UAZXyf/cKgE4AfgA+ec59eoCDFB2f8+/S/wEJBdwHovfpcKiwnK/jo54H0zW2dm4/NZH6z5SUREKp5g/ZujPnHh8YD6xOoTBy4mqAB94vJcoLB8lnldbVsPNHHOdQD+AiwuqwObWS3gX8Ak59zRc1fns0mpXqtC4inz6+Scy3LOxQPRQDcza3tuyPlt5nFMZXqdzOxm4Afn3LoLNctnWaldpyLG5MW/u6ucc53wDVubaGZ9zlkfjPlJREQqpmD8m6M+cdHiUZ+4aDGpT1yJ+sTluUCRBMTk+RwN7PMoFgCcc0dzhig5594BqphZg9I+rv9+rX8B85xzi/JpUqbXqrB4vLpO/uOlAR8DN56zyrPfp4Ji8uA6XQUMNrNvgdeAAWb2z3PalPV1KjQmL36fnHP7/D9/AN4Eup3TJOjyk4iIVFhB9zdHfeKixaM+cdFiUp+4aDFVlD5xeS5QLAVG+WcG7QEccc7t9zIgM7vUzMz/vhu+65tSysc0YA6wzTk3q4BmZXatihJPWV8nM7vEzOr631cHrgW2n9OsTH+fihJTWV8n59wjzrlo59wVwB3AcufcXec0K9PrVJSYPPh9qmm+ya4ws5rA9cC5s5gHXX4SEZEKK+j+5qhPrD5xIGNSn7hy9YmD9ikeZjYf6Ac0MLMk4DF8k6bgnPsb8A4wCNgJnATGBEFMQ4EJZpYJnALucM6V9hC7q4CfApvMd+8WwK+By/PEVZbXqijxlPV1agzMNbNQfP9QX3fOvWVmP8sTU1n/PhUlJi9+n87j8XUqSkxlfZ0aAW/6838Y8Kpz7t1gvE4iIlL+qU9cZOoTF0594hIIxr5eRewTmwf/bUVEREREREREzlKeb/EQERERERERkQpCBQoRERERERER8ZwKFCIiIiIiIiLiORUoRERERERERMRzKlCIiIiIiIiIiOdUoBARERERERERz6lAISIiIiIiIiKeU4FCRERERERERDynAoWIiIiIiIiIeE4FChERERERERHxnAoUIiIiIiIiIuI5FShERERERERExHMqUIiIiIiIiIiI51SgEBERERERERHPqUAhIiIiIiIiIp5TgUJEREREREREPKcChYiIiIiIiIh4TgUKEREREREREfGcChQiIiIiIiIi4jkVKERERERERETEcypQiIiIiIiIiIjnVKAQEREREREREc+pQCEiIiIiIiIinlOBQkREREREREQ8pwKFiIiIiIiIiHhOBQoRERERERER8ZwKFCIiIiIiIiLiORUoRERERERERMRzKlCIJ8zsPjNba2anzezlAOzvWzM7ZWbH87z+GoBQCzvmtaV5jCLGkO95+9dlmFmDc7ZJMDNnZld4EbOIBA/l4tKLwczuNrPPzmlTptdGRIKf8nBA4/iJ/1oeN7P9ZrbMzK7OE+MBM6uZp/29ZvaxZwFLvsK8DkAqrX3A48ANQPUA7fMW59yHAdpXeXKh894N3An8BcDM2hG46y0i5Z9ycdnStRGRcykPB4CZ/Qp4GPgZ8B6QAdwIDAFyisVhwC+BP3oRoxSNRlAIAGZWy8yyzKxxnmVt/dXH2oE+nnNukXNuMZAS6H3nZWZNzeywmXXyf44ys0Nm1i9Pm4fN7BszO2ZmW83stjzrYsxskZkdNLOUPKMT/gFcDvzbX6V9yL/vf/nb7jaz+wuI6WEzW3jOsqfN7Jk8n6eaWbI/ph1mds1FXoJ/AKPyfB4NvHKR+xKRUqZcXGFzsYiUE8rD5S8Pm1kd4A/ARP/1POGcO+Oc+7dz7sE8TWcAU8ys7kVcQikjKlAIAM6548B2oFOexdOBPzrnjhW2vZm9ZWZpBbzeKq24C+Oc+waYCswzsxrAS8DLzrmP8zT7BugN1AF+D/zTzBqbWSjwFrAHuAK4DHjNv9+fAt/hq1DXAmYC/wa+8re7BphkZjfkE9Z8YJCZRQD4jzMceNX/uQVwH9DVOVcbX0X924u8BKuACDNr5T/OCOCfF7kvESllysUVNheLSDmhPFwu83BPoBrwZiGXYS3wMTClkHbiIRUoJK81+JOxmfUBWgN/nGHVjQAAIABJREFU939+wsxWmNlCf1I7i3PuZudc3QJeN5dR/IvP+SMwzh/b80Ai8CXQGPjNObG/4Zzb55zLds4t8Lft5n9FAQ/6K7HpzrnPyF9X4BLn3B+ccxnOuV3A88Ad5zZ0zu0B1gO3+hcNAE4651b5P2cB4UBrM6vinPvW/0elWOedR84oiuvw/cFNvsC+RMR7ysUVIBcDzxX12ohI0FEeLl95uD5wyDmXWYRr8yjwCzO7pAhtxQMqUEheuckY+BPwO+dchpm1BZo653oDHwL3lGVQZjbSfpzkZ9kFmt56zh+B5/Osex5oC/zFOXf6nP2PMt/EkTmdyrZAAyAG2FPEZNcEiDqnc/proFEB7V/FNzcEwE/8nwFwzu0EJgHTgB/M7DUzi7rI8wZfgeInwN3o9g6R8kC5uALkYuDnhbXJJ1+LSHBQHi5feTgFaGBmhc6v6JzbjG80yMNFOBfxgAoUktcaoJOZ3Y5vkp75/uW9gZwkuAy4+twNzTdL7vECXhdKoIVyzs1zztXyvwYWd3szqwX8LzAHmGZm9fKsa4IvUd8H1Pd3KjcDBuwFLr9AsnN53u8Fdp/zx6C2c25QAdu+AfQzs2jgNvIkY/85v+qcuxpfknfAk8U767P2tQffZJmDgEUXux8RKTPKxRUwF4tIuaI8XL7y8BdAOj+OxCjMY8A4fLegSJBRgULy+gq4FPgz8LBzLtu/PBI44n9/BKh37obOuYF5Eua5r/MSqJmFmVk1IBQINbNqRal6XqSngXXOuXuBt4G/5VlXE1+yO+iPawy+ajHAamA/MN3MavpjvCrPtgeA2Dxtj5pvIp/qZhZqvgmVuuYXkHPuIL574F7Cl8S35awzsxZmNsDMwvEl21P4hriVxFhggHPuRAn3IyKlT7m44uZiESkflIfLUR52zh3Bd+vGs2Z2q5nVMLMqZjbQzP6UT/udwAIg38k7xVsqUEgu/zCvTcC3zrm8Fd5UfJPl4P95OACH+y2+JPMwcJf//W9LuM+c2YNzXm+a2RB8jxj6mb/Nr/BVxEcCOOe24vvj8wW+5NoOWOlflwXcAsThm/wnCd8kkzn+H/Bb/9C1B/xt4/GNVjgEvMCP1y0/rwLXck6lGN+9dtP9+/geaIhvaFyRz/vcBs65b5xzay+wDxEJEsrF5TYXF0Wh+VpEvKc8XP7ysHNulv+cfouvyLIX32iQxQUc8w/4ijISZMw5V3grqRTMrCqwExjufpycBjNrBzzinPuJmY0Hwp1zf/EqThGRiky5WETEW8rDIt4preFDUj49BqzMm4gBnHObzGyPma0AfsD3RAgRESkdysUiIt5SHhbxSEBu8TCzF83sBzPbXMB6M7NnzGynmW00s0551t1oZjv86zSbqgfMrJOZHQH6AL/Ir41z7hHnXG/n3O2ax0Ak+CgPl3/KxSLln3Jx+aY8LOK9gNziYb7nAx8HXnHOtc1n/SB8/8gHAd2Bp51z3c0sFPgauA7fvUxrgDv990CJiEgRKQ+LiHhPuVhEpGQCMoLCOfcpF54kZgi+RO38Q6XqmlljoBuw0zm3yzmXAbzmbysiIsWgPCwi4j3lYhGRkimrOSguwzeTao4k/7L8lnfPbwf+iWjGAxAR0bnlmTMQAjVbti6VgEVEimvdunWHnHOXeB1HAQKehxvUqEGj1DRCatWi6hVNSiVoEZHiqmy5WH1iEQk2JcnDZVWgsHyWuQssP3+hc7OB2QDWooX7Rfv29N20mTZr9eREEQkOZrbH6xguIOB5+NJHHmHB9CexkBBaKReLSJCobLl4Utt2XL1li/rEIhI0SpKHA3KLRxEkATF5PkcD+y6wvFCrWrbEgKQHHwxUjCIiFVlg87BzbLn8co7UqAHZ2YGMU0SkIiuFPnEL9YlFpMIoqwLFUmCUf+biHsAR59x+fBMANTOzK/3PG77D37ZQX7ZoAcCRt98ppZBFRCqUgObhGqdP40JCWNO8GQ44uWFDqQYvIlJBBLxPvKpFS0B9YhGpGAJyi4eZzQf6AQ3MLAnfs4OrADjn/ga8g2+24p3ASWCMf12mmd0HvAeEAi8657YUejzn+Do6mpTatal37FggTkFEpFwr6zxcM/00J4EvW7bk2oSv2P/7P9B08ZuBPzERkXLEiz7x9stjSK1Zk7on9MRLESn/AlKgcM7dWch6B0wsYN07+JJ1kdU4fZoT+EZRDFq7lpMbNlCjY8fi7EJE8jhz5gxJSUmkp6d7HUq5UK1aNaKjo6lSpYrXoeQq6zxcK/0UB4FV/tFsGV9/XZzNReQcysPFp1z8Y594dYsW3LB+vfrEIiWkXFw8pZGHy2qSzICqmZ7uK1C09BUo9M2dSMkkJSVRu3ZtrrjiCszym6dLcjjnSElJISkpiSuvvNLrcDxTLSODGunp7GrcmAN163Bp2hGvQxIp15SHi0e52CenT7yqpa9AoT6xSMkoFxddaeXhspqDIqBqnfJVtHLuuUv/eoeX4YiUe+np6dSvX1+JuAjMjPr161f6yro56JKYCPhGszkgdcECb4MSKceUh4tHudinpv/8V7VUn1gkEJSLi6608nC5LFBUO5NBxImTJF3SgKT69SE736cwiUgxKBEXna4VEGJ03+HrCH/p7xgf/MtfvYxIpNxTbikeXS+onpFBrVOn+K5hQ/ZFRqpPLBIAyi1FVxrXqlwWKHDQzV8h/qKV73Gj+uZORKQMRUbQfbs/D7dsiQOyUlK8jUlEpLJx0NU/B9Aq9YlFpAIolwUKCw3hqi1bAfisTRsAfpj1lJchiUgJmRmTJ0/O/Txz5kymTZtWpG0TEhLo2bMnbdq0oX379izI0znbvXs33bt3p1mzZowYMYKMjIxAh14pVW8cTfPkZBocOcKByEgSo6LA6Zs7kfJMebj8sdAQrvb3iVeoTyxSIVT2XFwuCxSh9erRe4vvyUuft2pFRmgomUc0QZtIeRYeHs6iRYs4dOhQsbetUaMGr7zyClu2bOHdd99l0qRJpKWlATB16lQeeOABEhMTiYyMZM6cOYEOvdIKcY7em325+NO2bXHAgZkzvQ1KRC6a8nD54+sTbwbg89atORMSoj6xSDlX2XNxuSxQVGnUiEtTU2mWnMzJatVYHxeHvrcTKVvr9qTy7H92sm5PakD2FxYWxvjx43nqqeJ/89O8eXOaNWsGQFRUFA0bNuTgwYM451i+fDlDhw4FYPTo0SxevDgg8QqkN6iaWyxe0db3zd3hf/zTy5BEKhXlYanSqBFRKYeJ3b+f49Wrk9C0qfrEImVMuTiwyuVjRgEw6L15M4mXXcaKtm3osWMHSQ8+SPSMGV5HJlKu/f7fW9i67+gF2xxLP8P274+R7SDEoOWltaldreDnH7eOiuCxW9oUeuyJEyfSvn17HnroobOWz5s3jxn5/NuOi4tj4cKFZy1bvXo1GRkZNG3alJSUFOrWrUtYmC/VRUdHk5ycXGgcUjRhD4ykx+OvEZqVxYamTTlWrRoRlXxGfZFAUB6WYjHos3kzuxo35tO2beiamKg+sUgAKBd7o1yOoAAIv7wJffxDi3PuuTvy9jtehiRSaRxNz8ydKDzb+T4HQkREBKNGjeKZZ545a/nIkSNJSEg473VuIt6/fz8//elPeemllwgJCcHlMyeCZmYOnA63P0TEqVN0/OYbskJD+bx1KxxwcsMGr0MTqfCUhyVH+OVNcm+3+0x9YpEypVwceOV2BEXU9P9H/Mi7qHXqFN9ERZFcvx5RKYe9Dkuk3CtKVXfdnlRGvrCKM5nZVAkL4ek7OtK5SWRAjj9p0iQ6derEmDFjcpcVpVp89OhRbrrpJh5//HF69OgBQIMGDUhLSyMzM5OwsDCSkpKIiooKSJzi40JDuHrLFtY2b86KNm25Yf0Gkh9+hGbvvet1aCLllvKwFEfU9P9Hp7t+So30dL6OjmZ/ZCSXpgZmqLlIZaZc7I1yO4KiRseOhGVn03PbNgBWtGmLoQnaRMpC5yaRzLu3B7+6vgXz7u0RsEQMUK9ePYYPH37WxD2FVYszMjK47bbbGDVqFMOGDcvdzszo379/bru5c+cyZMiQgMUq4Ab0/HE0W9s2OCBzzx5vgxKpBJSHJUeNjh2pkpVFz+3bAd/IYj1uVKRsKBcHXrktUACE1a1Ln82+mYs/bdsWgEOvvOJlSCKVRucmkUzsHxfQRJxj8uTJxZq5+PXXX+fTTz/l5ZdfJj4+nvj4eBISEgB48sknmTVrFnFxcaSkpDB27NiAx1uZtfvLCzRLTqZRaiqH6tRhW0wM2V4HJVJJKA9LjtBq1eid0ydu5+sTJ8/UHBQiZUG5OLDK7S0eAA0fmETvP88CYFXLFpysWpXqQfo8VxG5sOPHj+e+b9SoESdPnizytnfddRd33XVXvutiY2NZvXp1ieOTghm+SYsX9u7Nx+3b03rvXk3QJlIOKQ+XX5F3jaT3wn8BsKplS9KrVKHqsRMeRyUiF6Oy5+JyPYIicsQIGhw9SvtduzldtSorW7fWkDYRkTLmLmvENV99BcCHHeMBOKYJ2kREykyjKVNolJZGm2+/5VR4OJ+3bkUImrRYRMqfcl2gAAgJC+Ma/7CVj+J9HeMfZhX/mbEiInJxYmc+RY/tO6h56hQ7oqPZ26ABZOtGDxGRshQSEsK1Cb5icU6fOPnhR7wMSUSk2Mp9gaL2wBu51l+g+KRdOzJCQ8k8csTjqEREKo8aHTtSJTMzd06gj+I74NCkxSIiZalGz565X9r9p107zoSEkKFJi0WknCn3BYroGTNo8sMPNN23j6M1a7CmeXMMDWkTESlLmTVCc4vFOd/cHZ6rSYtFRMpKkzkvEPv991z5/fccqVWLdc2aeR2SiEixlfsCBQBmXKMhbSIinrGf3kjvzVuoeuYMG2JjOVS7Npw543VYIiKVTs4oig87xmPAnrH3ehuQiEgxVIgCRa1evXK/uVveoT3ZZhrSJiJShto/MJMap0/Tc9t2XEgIyzvoNg8RkbJWvV273ALF8g4dyDbj+MqVHkclIlJ0FaJA0WTOC7T67jsap6RwsG5dNl5xhdchiUgxmRmTJ0/O/Txz5kymTZtW5O1vvPFG6taty80333zW8t27d9O9e3eaNWvGiBEjyPA/itg5x/33309cXBzt27dn/fr1ATmPyiy7ap5Ji/1P8zj8j396GZKIFIPycPkX+8brtNnzHY1SUzkQGcnmJpd7HZKIFFNlz8UVokABvv+QObd5vNulMwbsGjbc26BEpMjCw8NZtGgRhw4duqjtH3zwQf7xj3+ct3zq1Kk88MADJCYmEhkZyZw5cwBYtmwZiYmJJCYmMnv2bCZMmFCi+AUYOpD+GzcSmpXFqpYtSatZE06f9joqESki5eGKIYQfb/N4r7P6xCLlTWXPxRWmQFGrVy8GrV0LwLudO5NlxqlNmzyOSqQC27saVvzZ9zMAwsLCGD9+PE89dXGPCb7mmmuoXbv2WcuccyxfvpyhQ4cCMHr0aBYvXgzAkiVLGDVqFGZGjx49SEtLY//+/SU7iUqu3aN/IvL4cbpv30FmaCjvd+qo2zxESpPysOSjVq9eDMzTJ85Wn1ikdCkXB1SYZ0cOsCZzXuBYy1ZEHzxE0iUNWNusGd2+/trrsETKn2UPw/eFdGROH4UDm8Flg4VAo7YQHlFw+0vbwcDphR564sSJtG/fnoceeuis5fPmzWPGjBnntY+Li2PhwoUF7i8lJYW6desSFuZLddHR0SQnJwOQnJxMTExMbtucdY0bNy40TimYCwvlpjVr+LxNa97p2pXhKz7j8D/+SaMpU7wOTaT8UB4uNE4pWJM5L3C0VWuiUlLYV78+65s2pfPOnV6HJVL+KBcXGmdpqDAFCvDd5jFo7RpmDxzIO1270P3rr9l02020e/Ntr0MTqVjSj/gSMfh+ph+5cDIuooiICEaNGsUzzzxD9erVc5ePHDmSkSNHFnt/zrnzlplZoevk4rnhg7h20Qf8/sxPWBsXx4G6dWiYdsTrsEQqHuVhuYAQYOCatcy58Qbe6dqVLjt3qk8sUhqUiwOuQhUoavXqxaA1a5k9cCDvd+rEb15bQJVtu7wOS6R8KUJVl72rYe5gyMqA0Kpw+wsQ0y0gh580aRKdOnVizJgxucsutlrcoEED0tLSyMzMJCwsjKSkJKKiogBfdXjv3r25bfOuk4vX7tE/seXVVvTZtJkPO3VkWecu3P3RRxyYOVOjKESKSnlYSsjXJ17DnBtv4L3OnXhkwQLC1CcWKR7lYk9UmDkowDekLW7fPpolJ3O0Zk0+b90KA05u2OB1aCIVS0w3GL0UBvzG9zNAiRigXr16DB8+PHfiHvBVixMSEs57XSgRg6/6279//9x2c+fOZciQIQAMHjyYV155Beccq1atok6dOhpWHChhvtFsAMu6dgHg8NxXvIxIpOJRHpYLaDLnBZonJxO7fz9ptWqxqpX6xCKlQrk44CpUgQIgxIxBa3wd43e6dgXgm1/c52VIIhVTTDfoPTmgiTjH5MmTiz1zce/evRk2bBgfffQR0dHRvPfeewA8+eSTzJo1i7i4OFJSUhg7diwAgwYNIjY2lri4OMaNG8dzzz0X8POorBr81yD6btpEzVOn2HzFFexpeAnZZ854HZZIxaM8LBcQelaf2FcsVp9YpBQoFwdUhbrFA6BGr14MXLuOp2+9leUdOnAiPJwahw57HZaIFOL48eO57xs1asTJkyeLtf2KFSvyXR4bG8vq1efPqmxmPPvss8ULUoqk0R9mcuj1t7nmq69Y2qMHb3XrxsS33mbP2HtpMucFr8MTkQIoD1csNfy3Pv918GA+jI/nd1XnU119YpGgV9lzcUBGUJjZjWa2w8x2mtnD+ax/0MwS/K/NZpZlZvX86741s03+dWtLGkuTOS8QfegQnRJ3cio8nPc6d8LQY+5EpOILplxMzWoMXvUlAIt79iTbjJMrV5Z4tyIiwSyY8nCTOS9w+cGDxH/zDSerVeP9TuoTi0jwK3GBwsxCgWeBgUBr4E4za523jXNuhnMu3jkXDzwCfOKcy1vC7e9f36Wk8QCEhIXxX59/DsCbPXsBcOjllwOxaxGRoBRsuTjqoYfpvmMHjf2PuVvdvDnZJd2piEgQC7Y8DL4+8W2ffwHA4l49AfWJRSS4BWIERTdgp3Nul3MuA3gNGHKB9ncC8wNw3ALVu3s0169fT/X0dNY3i+Pbhg0hM6s0Dyki4rWgysWRI0ZgznHbF76O8Zu9emLAzkE3ldYhRUS8FlR5GHx94hvXraP66dOsad6cPZdcoj6xiAS1QBQoLgP25vmc5F92HjOrAdwI/CvPYge8b2brzGx8QQcxs/FmttbM1h48ePCCATWaMoUap08zcN064MeO8aafDi/K+YiIlEelnouLk4cBslvFMuSLVQB80LEjR6tX5/QuPeZORCqsoOwT10xP5/r16wHfLXfqE4tIMAtEgcLyWeYKaHsLsPKcoWxXOec64RsON9HM+uS3oXNutnOui3OuyyWXXFJoUGGRkdzmv81jSY8eZIaEYGs2FbqdiEg5Veq5uLh5uN2bb3NZSgrdt2/ndNWqLOvSRfc/i0hFFrR94v9a6e8T9+xBlpn6xCIStAJRoEgCYvJ8jgb2FdD2Ds4Zyuac2+f/+QPwJr7hcSUW89yzdPxmF1d8f4CDdeuysnVrQoDUBQsCsXsRkWATlLmYEH6cE+gq35xAh198KSC7FhEJMkGZh2Oee5bOO3dy+Q8/cCAyks9btVKfWESCViAKFGuAZmZ2pZlVxZdwl57byMzqAH2BJXmW1TSz2jnvgeuBzQGIiRodOxIC3Pa5b9b4N3pfDUDyH/8YiN2LSICZGZMnT879PHPmTKZNm1akbffs2UPnzp2Jj4+nTZs2/O1vf8tdt3v3brp3706zZs0YMWIEGRkZADjnuP/++4mLi6N9+/as9w9/LceCMhfX6deLazdsoPbJk2y64gq2RUfjsrM5uWFDIHYvIgGkPFxiQZmHf+wT+4rFC9UnFglqlT0Xl7hA4ZzLBO4D3gO2Aa8757aY2c/M7Gd5mt4GvO+cO5FnWSPgMzP7ClgNvO2ce7ekMeWocdVV3PrFKsIyM/mkXTv21auHnc4I1O5FJIDCw8NZtGgRhw4dKva2jRs35vPPPychIYEvv/yS6dOns2+f70urqVOn8sADD5CYmEhkZCRz5swBYNmyZSQmJpKYmMjs2bOZMGFCQM+nrAVrLo5+bg7hZzJzHzn6Wr++ACQ98KtA7F5EAkh5uGSCNQ+Dr0982+dfEJaVxfIOHdgfGak+sUiQquy5OBAjKHDOveOca+6ca+qce8K/7G/Oub/lafOyc+6Oc7bb5Zzr4H+1ydk2UJrMeYH6x45x47p1ZIeEsKBPb00MJBIgCT8k8MKmF0j4ISEg+wsLC2P8+PE89dRTxd62atWqhIeHA3D69Gmys30PtHTOsXz5coYOHQrA6NGjWbx4MQBLlixh1KhRmBk9evQgLS2N/fv3B+RcvBKsuTi7VSx3fPIJAG9168aRGjU48/33gTyESKWkPBx8gjUPN5nzAg2OHuW69evJDgnh9d7qE4sEinJxYIV5duQyElqrFnd+/Alvde/Ov666ip+/9TZVNDGQSIGeXP0k2w9vv2Cb4xnH2ZG6A4fDMFpEtqBW1VoFtm9ZryVTu00t9NgTJ06kffv2PPTQQ2ctnzdvHjNmzDivfVxcHAsXLgRg79693HTTTezcuZMZM2YQFRXFoUOHqFu3LmFhvlQXHR1NcnIyAMnJycTE/HircM66xo0bFxqnFE+7N98mpGUrem7bxhetWrG4Zw9Gf7ScPWPvpcmcF7wOTyToKA8rD5eGnD7xsq5d+dfVVzHhnXcIU59YpEDKxd7k4oCMoAhmjR6cQofdu2n13Xek1q7Nu507E4JmkRcpiWNnjuH8E5M7HMfOHAvIfiMiIhg1ahTPPPPMWctHjhxJQkLCea+cRAwQExPDxo0b2blzJ3PnzuXAgQM4d/7k6Wa+SdYvtE4Cz4WGcufHvlEUr/XtS7YZJ1eu9DgqkfJLeViKq9GDU+j0zTc0T0oiJSKC9zt1VJ9YpISUiwOvwo+giBwxgu8fm8adH3/Co6N+yvx+fRny5ZccmjOHRlOmeB2eSNApSlU34YcExr0/jjPZZ6gSUoXpvacT3zA+IMefNGkSnTp1YsyYMbnLilItzhEVFUWbNm1YsWIFt99+O2lpaWRmZhIWFkZSUhJRUVGArzq8d++Pj6vPu04Cr86oUfSdO5dLDx/mu4YN+bxVS67euo3UBQuIHDHC6/BEgorysJSGvH3i3981kvl9+3Lz6jXqE4sUQLnYGxV+BAVA7VtuZtCaNUScOMGmK69k4xVNMIdmkRe5SPEN43n++ue5r+N9PH/98wFLxAD16tVj+PDhuRP3QOHV4qSkJE6dOgVAamoqK1eupEWLFpgZ/fv3z203d+5chgwZAsDgwYN55ZVXcM6xatUq6tSpo2HFpShm6kOEZmdzxyefAvBqv/4AfP+EZpEXuRjKw3Ixat9yMzevXk3tkydJaNqULZdfrj6xSAkoFwdepShQRM+YQbUzZxj62WcAvHzddQB887P/9jIskXItvmE897a7N6CJOMfkyZOLNXPxtm3b6N69Ox06dKBv375MmTKFdu3aAfDkk08ya9Ys4uLiSElJYezYsQAMGjSI2NhY4uLiGDduHM8991zAz0POltIsmttXriQ8I4NP2rdjZ+NLyc7QLPIiF0t5WIoresYMqmdkcLv/FruXr7sWUJ9YpCSUiwPL8rvnJNh16dLFrV27tljbbLr+RlKOpHHdE4+THRLC248+RsyhQ7TZvq2UohQpP7Zt20arVq28DqNcye+amdk651wXj0IqUxeTh3cdPM6p3l35nzvvYEHfvty28nMe/8c/CIuKovnyj0opUpHyQXn44igXFz8X7xx0E3tSU7nx8f8h24xlv3uUy1JS1CcWQbn4YgQ6D1eKERQAnz7yVy45coSbV68mOySEudde63u80g3XeB2aiEilEHtJLfbVrcfoDz8iJDubf3fvxg916nDG/3xuEREpfUf+759cmprKoDVryA4J4ZVrrlGfWESCRqUpUPSIrc/xKtUY88GHACzu1ZPDtWoRskcdYxGRsvLFb57n8oMHuXbDBjLDwvjHgP4YsGvYcK9DExGpFFbtSuF4WDXu9veJF13Vi7SaNdUnFpGgUGkKFJ2bRFL1T08Tt38/fTduIr1qVeb36+urGN92k9fhiYhUCte0asjxKlW55/0PAHi9Tx+OV6vGqU2bPI5MRKRy6BFbnz/2GU/z5GSu3ryFU+HhzO/bx9cn/qmKxSLirUpToADoOLAPVrUqYz7wdYxf7dePE+Hh2LZdHkcmIlI5hIUYj/UcT7s9e+j69dccr16d1/r00SgKEZEy0rlJJI9OvQMXEpKnT9yfk1WrwhoVi0XEW5WqQAFw6W9+TZfEROK/+Ya0WrWY37cvIWgUhYhIWVi7J5Xt9a4gG7j33XcB3yzyJ8LDNYpCRKSMdG4SySX3jKH7jh20272bwxG1WdC3D6FoFIWIeKvSFSgiR4wgJCSEn7/1FgAvXX+dRlGIiJSRHrH1qRoWwmfNOnDV1q102LWL1Nq1ec0/vHjP2Hu9DlFEpFJoNGUKIcBEf5/4xeuu1ygKEfFcpStQAFz62KP02rY9dxTFq/00ikLEa2bG5MmTcz/PnDmTadOmFXn70NBQ4uPjiY+PZ/DgwbnLd+/eTffu3WnWrBkjRowgIyMDAOcc999/P3FxcbRv357169fnbvPuu+/SokUL4uLimD59eslPTnJ1bhLJ/HE9eK3Pf+OwH4vF1/mKxcdXrvQ4QpHKS3m48ql371iu3rKV9rt8oyhe69tXoyhEPFbZc3GlLFBEjhhBiFluxfjl6zSKQsRr4eHhLFq0iEOHDl3U9tWrVydC33VyAAAgAElEQVQhIYGEhASWLl2au3zq1Kk88MADJCYmEhkZyZw5cwBYtmwZiYmJJCYmMnv2bCZMmABAVlYWEydOZNmyZWzdupX58+ezdevWkp+g5OrcJJInbmvHwmb9uGrrNuK/+YbU2rWZ37ev5qIQ8ZDycOWTM4oip1j84vXXaRSFiMcqey6ulAUKgHpj76Hntu103LnTP4qin0ZRiBTDyQ0bOPT32ZzcsCEg+wsLC2P8+PE89dRTAdkf+CrCy5cvZ+jQoQCMHj2axYsXA7BkyRJGjRqFmdGjRw/S0tLYv38/q1evJi4ujtjYWKpWrcodd9zBkiVLAhaT+PRqWp91148E4OdvvQ38eMud5qIQKRrlYQmE2rfczNV5brmb388/ikJ9YpEiUS4OrLBSP0KQajRlCodffImJb73NvZN+yZwbrmfYZ58RoVEUUsl9/8c/cnrb9gu2yTp+nNPbt4NzYEZ4y5aE1qpVYPvwVi259Ne/LvTYEydOpH379jz00ENnLZ83bx4zZsw4r31cXBwLFy4EID09nS5duhAWFsbDDz/MrbfeSkpKCnXr1iUszJfqoqOjSU5OBiA5OZmYmJjcfeWsy2/5l19+WWjsUjxmxn/3jeWN9/sxbNvHxH/zDQlNm/Lyddcy8a232TVsOLFvvO51mCKeUB5WHi5L0TNmcOytt/n5W2/x3/ffzws33MDQz1ZSW31iqeSUi73JxZW2QAG+uSh6PjaNXlu38nnr1vx94I1MXfgvNt1wDe3e+8jr8ESCVvbRo75EDOAc2UePXjAZF1VERASjRo3imWeeoXr16rnLR44cyciRIy+47XfffUdUVBS7du1iwIABtGvXjoiIiPPamZk/bJfvuoKWS+ANbNuYGX2GMyzxY3715mJGTZnMS9dey/BPV9BAoyhELkh5WALp0mmPcdVj0+i+fTtftmzJ8zfeyJRFi9QnFimEcnHgVeoCReSIEXz/+z/wq0WL+KJlS17t14+R//mYy/bs8zo0Ec8Upap7csMGvhtzD+7MGaxKFaJmzqBGx44BOf6kSZPo1KkTY8aMyV1WlGpxVFQUALGxsfTr148NGzZw++23k5aWRmZmJmFhYSQlJeW2i46OZu/evbn7ylmXkZGR73IJvNAQY1zvWFZ82oE+iQkMSEhgeXw8z918E4+9Op+vB1xD8+XqGEvlozysPFzWcvrEkxctYvivf80/+/fjzk8+Jkp9YqnElIu9ycWVdg6KHJc+9iitkpK55cvVZIaF8b+3DsGATZ0C84slUhHV6NiRy196kUvuv5/LX3oxYIkYoF69egwfPjx34h7wVYtzJvvJ+8pJxKmpqZw+fRqAQ4cOsXLlSlq3bo2Z0b9////P3p3HRVW2fxz/nFlg2DdFUBQFVBBRUdyXTLNsMcssNevJXNr3vX6tT/uTtmm22WKLpraplaVlZVZuKIqKC6ggooLs2zAwc35/zAwCgoIOjMD1fr14ATPnzNzH6tvlNfd9n8rjFi5cyPjx4wG48sor+eyzz1BVlQ0bNuDj40NwcDD9+/dn//79HDx4EJPJxFdffVVtB2ThWBP7hbBg4AwsKNz/3fdozWa+GTqUlKAgyjOkMBaiLpLDwtGCnnma6LTDXLFxI+V6PW9feaXUxEKcgWSx47X6BoXfpEkoBgN3r1iBS3k5q/r3JzE0FG2JkdwlS5w9PCHOW+6xsbS59RaHBrHdgw8+2KCdi5OSkoiLi6N3795ceOGFPPbYY/To0QOAV199lddff52IiAiys7OZMWMGAJdddhlhYWFEREQwa9Ys5s+fD1g3Jpo3bx6XXHIJUVFRXHfddURHRzv8GoWVQa9l2pDOzO1zDWHHjzNx/d+YtVreuPoqFCCpV29nD1GI85bksHAke018z4oV6MvL+WHgQHZ37Cg1sRBnIFnsWEpta0vOd3FxceqWLVsc+pq7IqN446qr+GjsJcSmpPDZ7DmoqkrMniSHvo8Q56OkpCSioqKcPYxmpbY/M0VR4lVVjXPSkJqUI3M4v6Scwa/8xhffP0qRqytjn/8vpQYDC958i8F79hAwcwbtHnrIIe8lxPlKcvjsSBY3Tk08Z8IEPrl4DP3272fhnNexgNTEolWQLG44R+dwq59BYadv355ZP/9MQH4+28LDWTFwoPUWSzde5+yhCSFEi+bjrmfKgE5ce8XLBBQWcuuqVQC8OHkSJq2WEws+OsMrCCGEcBR9+/bcsmoVAQUFxHftyg8DBkhNLIRoMtKgsOm29jc8jUYe+vZbAF6fcDUFbm6wWXaSF0KIxjZjWBcUoKCNDzf9+htdjh3jYFAQn48ehQIkX3a5s4cohBCtQre1v+FVWsr9330HwOxrJlBoMEhNLIRoEtKgqMJz6FDGbdxE3/3JZHt7M2/cOGvHuH+rmCUohBBO097XjSv7tGf6hc+iN5t5wrbe+d3LLuOonx9lBw44eYRCCNF6eA4dyvgNG+mTksIJHx/eGXeF1MRCiCYhDYoqQj9agAZ48quv0JrNLB55AUkhIWgLi2VzICGEaGS3XRBOabmZ1DHXMCRpDxfHx1NqMPC/iddYN8yM7unsIQohRKsQ+tECNKrKk4u/QmOxsGjkSPZ26CA1sRCi0UmDooag556l+5EjTPnjTywaDc/dMJUKjYb0Z5519tCEEKJF69bOi9GRgTwZeAFmvY5Hvv4Gt7IyVvfrxx8xPVHNZtIfftjZwxRCiFYh+LlniUpPZ/Kff2LWanl26lTMiiI1sRCiUUmDoga/SZNQPDy4e+VKgnJySOzcmc9Gj7ZOa7ta1kALIURjuvWCcHKKTWz7YCVBubncvWIFAM9NnUqBmxv5K39w8giFEKJ18Js0CcXdnXuWr6Bdbi47wrrw+ahRUhMLIRqVNChqERW/BQ+jkWe/XATA3CvHcbBdO5QkWQMtRGPRarX06dOn8uuVV15x2GsnJCTw008/Oez1ROPp39mPvp18+fCvA1gi2nPD2t/pfeAAmb6+zL5mgnWpR6/ezh6mEC2S5LCoKWprPJ5GI08vstbEb4+/ktTAtlITC9GIWnsWO6RBoSjKWEVR9iqKkqwoymO1PD9SUZR8RVESbF9P1/dcZ/EcOpThu3Yx/t9/Men1PHXjDaAoJEbKfXGFaAxubm4kJCRUfj32mOPioDmEsSO0hCxWFIVbLwjncE4pB1/8HI2q8vxnn6MvL+ebYcP4JyoS1WSSpR5CNALJ4XPXEnK4Ju2AQYxM3Mm4DRspc3Hh6RukJhaiMbX2LD7nBoWiKFrgHeBSoAcwRVGUHrUc+peqqn1sX/9t4LlNLvSjBSgaDY8s+5o2+flsi4jgiwsvRAskDJMdjIVoCvn5+XTv3p29e/cCMGXKFD788EMAbr/9duLi4oiOjuaZZ56pPGfz5s0MGTKE3r17M2DAAPLz83n66adZsmQJffr0YcmSJRQXFzN9+nT69+9PbGwsy5cvd8r1OVJLyuIxUe0Ib+vBe3+k0P7ZZwk/dow7f/gRgGduuIEig0GWegjRRCSH668l5XBVkZ99ghmFx5YtI6CggC3durH4ggvQAlsvGu7s4QnRKrSmLNY54DUGAMmqqh4AUBTlK2A8sLuRz210Ubt3sSsyiqcXLeae22/jjauvYtCePXTLyGDHgjn0mvmgs4cohMMpf/zRKK+rjhx52udLS0vp06dP5e+PP/44kyZNYt68eUybNo17772X3NxcZs2aBcCLL76Iv78/ZrOZ0aNHs2PHDiIjI5k0aRJLliyhf//+FBQU4O7uzn//+1+2bNnCvHnzAHjiiScYNWoUH3/8MXl5eQwYMICLLroIDw+PRrn2JtJislijUbh1RDiPfLODnZeNwtfrNaatWcOavrHsCg3lhcmTeeXTT0mK7knUrp3OGKIQjUpyuNlqMTlcU8CGeDSD+vLUosXcd9utzJlwNQP37iUi/Sjb1y6l96jrnD1EIRxOstg5HLHEowNwuMrv6bbHahqsKMp2RVFWKYoS3cBzURTlFkVRtiiKsiUrK8sBw64fz6FDGb19O9esX49Jr+fhGTMw6vUosxc02RiEaA1qTmebNGkSAGPGjCEmJoY777yTBQtO/ne3dOlS+vbtS2xsLLt27WL37t3s3buX4OBg+vfvD4C3tzc63al92NWrV/PKK6/Qp08fRo4cidFoJC0trWkutPE0ehY3ZQ6Pj21PO29X3vszhZjNW9BZLLz68Se4lZWxctBAfhjQH9Vs5sC1UhQL4SiSw+esxdbE7X3dONatN2MSErj6738oc3Hh4RnTKdPp0NzxzJlfQAhRb609ix0xg0Kp5TG1xu9bgVBVVYsURbkM+B7oWs9zrQ+q6gfABwBxcXG1HtMYQj9aQGKPaB5buoz4iK4kd2jP7Gsm8ORXS0js0YOY3edFY1sIhzlTV7epWSwWkpKScHNzIycnh5CQEA4ePMjs2bPZvHkzfn5+TJs2DaPRiKqqKEptsVKdqqp88803dO/evQmuoMk0ehY3ZQ676rRMH9qFl1ftITE9n5D7bkZ582MeXbaMZ2+4gf9OmULvAwfpmJhIybZtuMfGNuZwhGhSksPNVouuiWO+/JTj/WN5fOkS4rtGsC8khNcnXM3jS5eR2LMnMTtlRptoWSSLncMRMyjSgY5Vfg8BMqoeoKpqgaqqRbaffwL0iqK0qc+554Oeu3biZjLxv48/RldRweKRI/k9JgatRZXbLAnRyN544w2ioqJYvHgx06dPp7y8nIKCAjw8PPDx8eH48eOsWrUKgMjISDIyMti8eTMAhYWFVFRU4OXlRWFhYeVrXnLJJcydOxdVtdZ127Zta/oLc7wWl8VTBnbCy1XHe+tS8LvtEcw+nkxc/zcXbd1GsZsbj06/mXKNhpQp1zt7qEK0aJLD9dbicriqQC8DP77xHe5lJv730cfozGa+GDWKdT2j0VaYSbxRZrQJ0ZhaSxY7okGxGeiqKEoXRVFcgMnAiqoHKIoSpNhaOIqiDLC9b3Z9zj0fKIrC0bETiU5L497l1uE9Me0m0gMC0CYdoOQ8+AcpRHNnX29n/3rsscfYt28fCxYsYM6cOQwfPpwRI0bwwgsv0Lt3b2JjY4mOjmb69OkMHToUABcXF5YsWcLdd99N7969GTNmDEajkQsvvJDdu3dXbgj01FNPUV5eTq9evejZsydPPfWUk6/eIVpcFnsb9EwdFMqqxKOkZhcTs3ELFuC5L76gXW4u28PCeOPqq9ECST1jnD1cIZo9yeFz1uJyuKZbR4TxV2g/YlJTuWuFdXiPT5tGhr8/2s2JUhML4QCtPYsVe7fknF7EOkXtTUALfKyq6ouKotwGoKrqe4qi3AXcDlQApcADqqr+U9e5Z3q/uLg4dcuWLec87oZ45/dkYu+ehLe5lLtuv50/e8UQlZbGF6/NRl9eTsyepCYdjxCOlJSURFSU3C6sIWr7M1MUJV5VVafd5qcps7ipcjizwMiwV3/nuv4hvHCVtQmxKzKKreHhTH/gfiq0WuZ8+CFj47diiIkhbNnSRh+TEI1BcvjsnG9Z3Bpq4jfW7GP4vRMxqBXccecd/NWzJ9GpqXz+2mx0FRVSE4tmTbK44Rydw46YQYGqqj+pqtpNVdVwe5iqqvqeqqrv2X6ep6pqtKqqvVVVHWQP4rrOPR8NCgvg5qteQFVVXv70EzpmZZHUqRPPT5mMBuRe0EIIp2uJWRzobWBC3w4s25LOiaIyAHxGDaFfSjIPf/0NAE/eeCPJwcEYE+XTOyGEc7XEHK4prrMfE698GVVVeeXjjwnJOsGu0FBenDwZLVITCyHOjUMaFK1Bv1A/Fs0axKKXluJdUspb772PwWTi+yFDWDp8uDWQ+8gUYyGEcLRbRoRhMlv49O9DAITM/wizi56pv//O5Zs2UWowcO+tt1JoMMh+FEII0ch2pOejAOOumo1PSSlvfvA+riYT3wwbyrJhQ601cV/ZuFgIcXakQdEA/UL9eHlCDEfHTqT7kSM8+8WXALw0eRIbundHa6yQTTNFs+WI5V6thfxZNa2wtp5c0iOIz/49RHFZBQAxOxKxAM9+8SXd0tM5FNSOh2bORNVo5NM70WxJtjSM/Hk5x6CwAFz11r9CLOs6kqjD6Tzz5SIAXpg8mU3duqItMcptoEWzJdlSf43xZyUNirMw+o3/UmLwYNymTUxbs4YKrZb7b7mFA+3aoU06QO6SJc4eohANYjAYyM7OlkCuB1VVyc7OxmAwOHsorcqtF4RRYKxg8aaT9+aO2ZOEm8nE2++9j19hIet7RvPSpOtk2Z1oliSHG0ay2Hn6hfrx5cxB3De6K2uGTqRQ7874jRu58bffqNDpuPfWWznYrh3GxESpiUWzI1lcf42Vww7ZJLOpOWNDoJpKTWaSe/UEReH+W2/htz59CMk6weJXXyWgqIgo2SBINCPl5eWkp6djNBqdPZRmwWAwEBISgl6vr/a4szfJbErOyOFJ7/9LWk4J6x65EL3W2l/Pfe9/HH3zY7aFhTP9/vsw6fU8smwZN/22FkWnI2pnYpOOUYizJTnccJLFzq+J9x4rZPw761m27AFQFO697VZ+792bjpmZLH71f/gXF0tNLJoVyeKGaYwclgbFOTicU0LekH6Uubgw7YH72dW5M31SUvjozTdxK6+QQBailZGiuHH9vieTmz/dzJxre3NNv5DKx5MvHo0pLYOf4uJ4eOYMFIuFNz/4gIsStqN4eBAV7/z/Xwghmo5kcdNauuUwj3y9g5XfP4TR1ZVpDzzA7tBOxCYns+Ctt6QmFqIVcvpdPFqrjv7u5K5Yh8Fk4p357xKUk0NCeDj333ILZVotSTLFWAghHGZk97ZEBnnx5q/7mLd2P/GpuQBErP4NxcODy7Zs4Z7ly1E1Gh6eMYON3bqhFhezb9RoJ49cCCFaruviOjKxXwhXXj0bt7Iy3pk/n6CcHLZFRPDgzFmYNBqpiYUQ9SYNinM0oltbfrvvTtoWFPDB23PxLSpiXUwMT0y7iQpFkUAWQggHURSFS6LbcTi3lDmr9zF1wYbKJkVU/BbMCtyy6mcm/bkOk17PXXfcTmJoKBUZGaTOmOnk0QshRMv1/PiedA305L+XTyYwP5/3587Dp6iIP3r34v+m3YRZamIhRD1Jg8IB7rn1TrZFhxF+7Bjvvz0Xj9JSVvXvzwtTJmNBlUAWQggH0dn2nlCB8goLGw5kVz4Xs3ohFip48quvuHzTJkoMBm69+y6Sg4Mp/vtv0h9+2EmjFkKIls3NRcv8qX3Z6TGQTVFhRBw9ynvz5uFuNPLjgAG8OGmS1MRCiHqRBoUDKIrC+EXLOeLnRc+0NN6Z/y6uJhNLR4zgtWsmYgEJZCGEcIAh4W1w0dn+16UoDAoLOPlkxwHErPkSVVV58dOFXLBjB/mensy89x4OtmtHwcofpEkhhBCNJCLQi5eujuHZ7ndwoq0vvQ6lMm/+u7iUl/PVyAuYM2GC1MRCiDOSBoWDeLrqCFuxlnwXPf337+eNDz5EV1HBwjEX8cp110ogCyGEA/QL9WPxrEHEhfphtqhYam703HEAMXuS0FgsvP7hAvrv3UuWry/THrif5OAgaVIIIUQjuiq2A1MGdOI/Q5/E6O7KwH37eP2DD9GZzXxy8cW8eq18cCeEOD1pUDhQeFtPChevpkSr4YKdO3nz/Q/Ql5fzxahRPD9lCmZFAlkIIc5Vv1A/Fk4fQAdfNx7/NpGyCvMpx8SsWYi+vJT578xn4J49nPDx4eb7H2Bvhw4UrPxB9qQQQohG8sy4HkQFezP9qv9hdtFxYWIib3zwAbqKCj4fPZoXJk+WmlgIUSdpUDjYxdFB/PnGt5RoNVyYmMjcd9/D1WRiyQUjeOaGG2TjTCGEcAAPVx0vXNWT5Mwi3v/zwKkH2JZ7uJpMzH9nPkN37SLH24ub77+P3R07Uvz333J3DyGEaAQGvXU/igqzytO3vYdZp2XU9h3Mfe+9yuUez11/vdTEQohaSYOiEdx3UTfmPfQJJVoNw3fvZv478zGYTHw7dCgP3DKLUr2eRAlkIYQ4JxdGBnJFr2DmrU0mJavo1ANsyz305eXMffe9yj0ppj1wPxu6d6ciI4OkPrFNP3AhhGjhurTx4NVrerEtLY/lryzFrNUwYucu3pk/H1eTia+HD+fBWTMx6nRSEwshqpEGRSPQahTemhTLg9PmUaLVMGjvXj546228i0v4NTaWGffdS6G7OzsjoyjZts3ZwxVCiGbr6XE9MOg1PPFtImrN/ShsYtYsRFdh5M33P+DSzZspdnPj1rvv4ocB/VGNRhKjejTxqIUQouW7vFcwNw0OZcH6g2Qs/Q2zVsOQpD188PZcvEpKWNO3L7PuvYciqYmFEFVIg6KR+Ljree+Gftw4YQ4lei39UlL4fPZsgnJySAgP58aHH+KYvz+Hplwva6GFEOIsBXoZePyyKDYezGHZlvTaD7It99Cazfzv40+YtmYNFTodj06fzsdjxqBRVfkETwghGsETl0fRK8SHh5Ztx2fdZswuOuKSk/l89hza5eYS37UrNz70IMf8/KQmFkIA0qBoVFHB3rw6sRcTL3+VXB9PIo4e5cv/vUbXI0c4EBzM5EcfYVtYGMV//y3TjIUQ4ixNiuvIgM7+vPhTEieKymo/yLbcQ1VVHv7mWx5ZtgyAOddM4Nmp12PWaqVJIYQQDuaq0/LO9X0BuHPRVrptTcDU1oeuGRl8+b/XiDiSQUr79kx57FG2d+kiNbEQQhoUje3K3u2ZOawLUy98lpTYPgTl5fHZ7DkMStpDto8P0x64n2+GDEE1GmWjICGEOAsajcJLE3pSYqrg+R92n/bYmDULMVPBTb+tZc6HH1auhZ5+/33kenmxKzJKbkMqhBAO1NHfndnX9ibxSD4v/ZhEn782YB4zlODcXD6bM4cBe/ZywseHmx64n+8GD5KaWIhWThoUTeCxSyOJbu/F3aE38NCIu/AoLeX9uXO5Ye1aKnQ6nv7Pjbx03XWYNBr5BE8IIc5CRKAXd4yMYHlCBn/szaz7QNtyDzMwNn4rn8+eQ1BODtsiIpj0+GPs7tSJgpU/kBgT02RjF0KIlu6S6CBmDuvCwn9T+WFHBjFzF9B58SI8S0r44O23uf733ynX63nyppt4+dprKZeaWIhWSxoUTUCn1TCyeyAAu/07M/7q2SgWC48vXcbzn32GvrycL0ddyIz77yPL15ddkVGyBk8IIRrojgvDCWvrwZPf76TEVFH3gbblHoqrC9FpqXz1yqv0SUnhmL8/Nz70IMuGDUVTXiHFsRBCONCjl0YS28mXx75J5IcdGXyS54Xpl3/QWCz835KlPPf5F+gqKvhi9Chm3ncv2T4+UhML0QpJg6KJjIpsh6vO+sdtUWHtOyswAxP++ZdPX3+Dtnl5xHftyjVP/h9/9Yy2rsGT4lgIIerNVafl5atjSM8t5a1f95/x+Kjt2/EeNRT/ggI+eeNNJv71F2UuLjx7ww08PGM6pQYDuyKjOD57dhOMXgghWja9VsO86/sCKncv2sac1XuZumADpl/+wQxM/PtvPnnjDdrk57O5WzcmPPl/rO/Rw1oTy92WhGg1pEHRRPqF+rFo1iDuGR1BVLAXb/66j4RPf8as1dDn4EG+efElhu7aRZ6nJ7ffdRezJ0zApFHYFRlF7pIlzh6+EEI0CwPDApjcvyML1h9kV0b+GY8Pmf8RMXuS0FZU8NyXi3jl409wMxpZ1b8/1z7xOEkdO5K94CNZ8iGEEA7QwdeNS6KDULF+YFdeYWHDgWxi9iRh1mrom3KAb154kSG7d5Pr5cVt99zN61ddhUlBamIhWglpUDShfqF+PDCmO9/dMZTRkYE89f1O/nj7e8z9Y/AtLOS9ee9w33ffoTWb+eTiMUx99FFSgoM5+syzJPXu7ezhCyFEs/D4pVH4ubvw+LeJmC1qvc6xb545btNGvn7pZbofPkxaYCDXP/IwH15yMarZws7IKEq2bWvk0QshRMt2/cBQ9FoFABXo28kXgJhduzD3j8GvsJD3587j3u+Xo7FY+GjsJdzwyCMcCAqy1sRylw8hWjRpUDiBQa/l3Rv6MSG2A3PW7OO7654levduVFVl1i+r+fT11wnOzmZXaCgTn3icj8eMocJkks6xEELUg4+7nqfH9WBHej4L/zlUv5Nsm2cqrno6Z2ay+NX/MfmPPynX63nz6qu58aGHSG3XjkNTriexrxTHQghxtvqF+vHVLYMZG90Oiwrv/nkAY7kZgJjPl1beEvqWn3/m09ffICgnh52dOzPx/57g04tGU1FWJjWxEC2Yoqr1+3TpfBIXF6du2bLF2cM4ZxaLygs/JvHx3weZ0LcDr17Tiz09o9GqUGQw8L+JE/lm2FAA+qSk8NLChYRmZqHodUQlJjp59EKImhRFiVdVNc7Z42gK53sOq6rKzZ9uZtPBHNY8cAEdfN3qfe7xpx8ic+kKtGhZ3yOKp2+8keN+friaTNz3/XKm/v47GlWl/XPP4jdpUiNehRDibEgWNx9LNqfx6DeJjOjWlg9u7IdBr618LjEyCi1QaDDw6rXX8t3QIQDEJifz4sLPCM3KQnFxIWrHdieNXghRl3PJYWlQOJmqqsxbm8ycNfu4KCqQedf3Zf/0KbA5ES2wrmc0z9xwA5m+vriUl3PLqlXMWL0afYUZz6FDCf1ogbMvQQhhI0Xx+eVwTgkXv7GOIeEBLLgpDkVRGnDyJhLH3IQWKHBz45XrrmX54MEAxBw8yDOLFhF1OB2zRkPM7l2NcwFCiLMiWdy8LNmcxmPfJjK8ay1NiqsvR0k6gAb4I6Ynz9xwAyd8fHA1mbh11SpuXvMr+ooKfMZdQchrrznvIoQQ1UiDogX4fEMqTy/fSf/O/iy4KQ5vg76yc5zv7s6r106sLI47HzvOU4sXM2jvXsxAzJ4kp45dCGElRfH558N1B3jxpyTmT+3LZTHBDT4/+chhSnYAACAASURBVOLRlKWlo6Bhba9ePH/9FDJ9fdFYLFz/+x/cvXIlHkYjlqA2xPzxVyNcgRCioSSLm5+lmw/z6Lc7am1SwMnZFHnu7rx63bWsGDQIgC7HjvHUosUM3LdPamIhziPSoGghlicc4cGl2+ke5MXC6QNo4+lK4iWj0aRmoACbunXl+SlTOBBsLbIv27SZB7/9lnZ5eVjcDcRslc3bhHAmKYrPPxVmC+Pf+ZvMwjJ+feACfNz0DX+RKrMpil1dmTfuCr4YNQqLRkNgXh4Pff0Nl9n+LNrMnEG7hx5y7EUIIRpEsrh5OmOTYvRwNEdOoAAbunfnhSmTORgUBMAVGzfywHffEZiXj8bDg6j4lvFnIkRzJQ2KFuSPvZnc9kU8wT5ufD5jACF+7sDJzrFJq+XTMWN477JLKXNxwWAyMW3NGqavXoN7WRm6Nm3ovl4+xRPCGaQoPj8lpucz/p31TBnQiRevPvvbhVadTZEUEsJ/r7+eHWFdAOh14CCPfP01sQcOyKd4QjiZZHHzdaYmBVSpiXU6PhlzEe9faq2J3crKuHnNGm5evQY3kwl9+/Z0W/tb01+EEML5DQpFUcYCbwFaYIGqqq/UeH4q8Kjt1yLgdlVVt9ueOwQUAmagoj4X0tLCuKb41Bxu/mQz7i46Pp8xgK7tvIDq6/DSAwJ4fcLV/NKvHwBt8vO5e8VKrv7nHzSqimtYGBE//ejEqxCi9XF2UdyUWdzccvj5H3bz0fqDLLttMP07+5/9C1WZTWFRFL4bPJi3x1/JCR8fAC6Oj+f+776n44kTWGR/CiGcwplZLDXxuatXk+KK0SjJGWiAw23a8PqEq1ndty8AbfPyuGfFCsb/u0FqYiGcxKkNCkVRtMA+YAyQDmwGpqiqurvKMUOAJFVVcxVFuRR4VlXVgbbnDgFxqqqeqO97tsQwrinpaAH/+XgT5WYLn948gD4dfSufS4yJQVteAcDW8DBeu2Zi5ad44RkZ3PnDj4zZtg1FVXGLiSFs2VKnXIMQrY2Ti+ImzeLmlsPFZRVc/MY6FAWui+vI0Ig29Av1O+vXS79jBvlr16OgUOxq4OOLx/DpmDEYXVzQVVQwcf3fzPr5Z+sSPK2WmF07HXg1QojTcVYWS03sOPVpUgAkRvdEa7beojQ+PJzXJl5DYhdrTdz1yBHu+OFHLkpIkJpYiCbm7AbFYKzheont98cBVFV9uY7j/YCdqqp2sP1+CAnjWqVll3DDRxs5UVTGwxd3p6TczKCwAPqF+lGybRspU65HC6jAqrg43rxqPEfatAGgW3o6d/zwI6O3b5dQFqKJOLlB0aRZ3Bxz2L5hpgK46jV8OXPQOTUpOLyJvRNupaLQiAIc8/Xl7fFXsmLgQFSNBpfycq79az2zfv6ZNgUF0qgQook4sUEhNbED2ZsUwyLa8OF/4upsUlStiS2Kwqp+/Xjz6qvICAgArDXxXSt/YNR26+1IpSYWovE5u0ExERirqupM2+83AgNVVb2rjuMfAiKrHH8QyMX69+z3VVX9oI7zbgFuAejUqVO/1NTUcxp3c5FZYOTa9/4lNaek1qL6wLXXUZqYiIJ1f4rvhgzhg0vHcszfOoU5Mu0wt/30E6O2b0ejqrIeT4hG5OQGRaNncXPP4Xd+T+a1X/YCoFHgwYu7c+eFEef+woc3kXjxVDSqBgUNycHBvHv5ZfwcZ/1XwdVk4rq//mLaml+tMyqQPSqEaExObFBITexg9W1SQPWl0Cadjm9tNfFxP2vNHJWWZquJd6BITSxEo3J2g+Ja4JIaYTxAVdW7azn2QmA+MExV1WzbY+1VVc1QFCUQWAPcrarqutO9Z0vuFtfm9dV7eXttMgAKcP+Ybtwzumu1YxJHDkdzzLqzsUmn4+uhQ/nw0rFk+lqXhoQeP860X3/lyg0bcS0vx+LqQoytkyyEcAwnNyiaNIubYw7Hp+Zy/YcbKKuwAPDJzf25sHugY1788CY49BeJsxagRQUU9nbowLuXX86avrEA6MxmLtu0mZvXrKFrhnXttNe4Kwh57TXHjEEIATi1QSE1cSNYuuUwj35TvyYFQOKQQWhy8lGAMp2Or4cN48Oxl5Blq4k7HzvOTb/+yvgNG3CpqJCaWIhG4OwGRb2msymK0gv4DrhUVdV9dbzWs0CRqqqzT/eerSGMq4pPzWWqrahWgbA2Hnxyc39CAzxOOTaxXyyaYmO1UF540ejKpR/+BYVM/f13Jq9bh09xsfX1Fi/CPTa2Sa9JiJaoOSzxcFQWN9ccjk/NZfm2IyzalMawrm34+Kb+aDSK497g8CZyn5zIkY2uaNACCkkdQ1hwySWs7tsXi0YDwPCdO7l59RoG7LP+I5A7MAnhOOf7Eg+piRuuoU0KqF4TG/X6yprYvvQjID+fqX/8waQ/1+FTUiKz24RwIGc3KHRYNwQaDRzBuiHQ9aqq7qpyTCdgLfAfVVX/qfK4B6BRVbXQ9vMa4L+qqv58uvdsLWFcVXxqLhsOZKMo8P6fBzBbVF6aEMOVvdvXenxi31g0JdZQrtBoWN23L5+MGcPu0E6AdcrxZZu3MPnPP+mZloYKMtVNiHPk5AZFk2Zxc8/hzzek8tT3O3n4Egct86jKNpvi+Je/cuKPZBRbo+JwmzZ8NnoU3w4ditHFBbCujZ607i/GbdyIe1kZGo2GKLnzhxDnxIkNCqmJG5G9SdGrgw+jIgMZ1rVtvfYRqlkT/9K3L59cPIakTtaa2GAycdnmzUz+cx090tIAqYmFOFfnw21GLwPexHpLpY9VVX1RUZTbAFRVfU9RlAXANYB9kVyFqqpxiqKEYe0gA+iARaqqvnim92tNYVybI3ml3LN4G/GpuUyK68gzV/bA3UVX67FVQ1kFNnbvzqdjLuKvnj0rj4k+dIgpf65j7JYtGMrLpYMsxFk6D24z2mRZ3NxzWFVV7l68jZ8Sj7Jo1iAGhQU4/k0Ob4KFV3I83oXMJAMatChAnocHiy8YweKRI8n29gbA3Wjkyg0bmbxuHRG25R9aKZCFOCtObhZLTdyIXvtlD+/8ngKAq07Doln13+y4tpr4k4vHsD46uvKYmIMHmbRuHZduibcuiUZqYiHOhtMbFE2ttYVxbSrMFt74dR/z/0ghvK0n866PJTLIu87jEwfGockvxj6RObVtW5aOGM63Q4ZQ4GFdKuJdXMxlm7cwfsO/9DyUigJY3A3EbN3W+BckRAvg7AZFU2oJOVxUVsGVc9dTVFbBj/cMp62Xq+PfxDabArcAEtY+ifZjNzToKjc2/jU2liUjhrOlW7fKU2KTkxm/YQNjt8TjaTSiAfxnzqDdQw85fnxCtECSxS3XO78nM/uXvdj/9jJzWBeevKJHg17jlJo4sC1Lho/guyGDq9XEl2/ezPh/NxCdKjWxEA0lDYpWbP3+E9y3JIFCYzlPj+vB9QM6oSh1r6dOnTGTor//Bqhck7cqrh9LRoyovG80QNjRo4z/dwPjNm0kMC8fANewMCJ++rFRr0eI5kyK4uYn6WgBV73zN3Gd/fhs+kC0jtyPoiZbsyLhxE40/7cercV61w+A/e3bs2TECJYPGkiJwQBYl+KNTtjO+A3/MjhpDxpVlSUgQtSDZHHLFZ+ay9QFGzBVWLCo4O6i5YMb4xjWtU2DX6tmTVyq1/NzXBxfXTCCnZ07Vx4XnpHB+A0bGLdxE23zpSYWoj6kQdHKZRWW8cDSBP7af4LLY4J5aUIMPm76M56XGNUDjapWdpCTQkJYPngQP/YfQI63FwAai4VBe/Zw6ZYtjErYjk9JCSDBLERtpChunpZuPswj3+zg3tFduX9MtzOfcK5sSz8wm0hcEYCm+GSjotjVldV9Y1kxaBCbunevPCUwL4/LNm/mkvit9Dx0CAXQaLVE7drZ+OMVopmRLG7Z7PuyhbXx4M1f95OcVcQz43pw46DQ035IV6fDm0gcMxUNJ7M4qWMI3w8azE/9+1eriQcnJXHplnhGbd+Ot60mdouJIWzZUoddnxAtgTQoBBaLygd/HWD2L3sJ8jEwd0ossZ3quSZvYBya/EIAFDSUazSsj45mxeBB/B4TQ7ne2uzQmc0M3LOHsfHx1ZoVFn8fYv7Z0DgXJkQzIkVx86SqKg8t28G329L5bPoAhndt2/hvWmXpBz8/xvYcLZqf3CuXfwAcCfBnxcCBLB80iMOBJ2+HGpydzcVbtzE2Pv5ks0JRiEra3fjjFqIZkCxuPYrKKrjvq238mpTJ1IGdePbKaPRaTcNfyH6r6Ac+R1NYCpysif/q2ZMVgwbxe68YKnTWPd90ZjODkqw18YXbT9bEckcmIaykQSEqbU3L5Z7F2ziWb+ThS7oza3hY/W6hZw/mWQvQYEFBARTy3N1Z0zeW1X37srF7d8xa622d7M2K0QnbGbFzJ0G5uQBY9DpiEhMb8QqFOH9JUdx8lZgqGD/vb3KKTfx073DaeRua7s1r7FNR8ZUbbiUaQFO5mVtCWBi/9OvH6r6xHPc72XwOzs5mzLZtjNyRSGxyMnqLBQ3gNe4KQl57remuQYjziGRx62K2qMxevZd3/0hhUJg/707th5+Hy9m/YF01sYcHq2NjWd2vL5u6datWEw9K2sPo7QmMSNxJu7w8QGpi0bpJg0JUk19azmPf7GDVzmNc0K0tc67rTRvPBmz+dngTB2Y9SOmhE4ClcrpbrocHv/Xpwy/9qjcrACLTDnNBYiIjExOJTk1FY/v3ykeKZNGKSFHcvCVnFjJu7t/EdPBh0ayB6M7mU7hzZW9W5KeTsHMR2oXeaNST044tisL2Ll1qbVZ4lpYybNcuLkhMZPjOXfgWF1vPcnEhasf2pr8WIZxEsrh1+nZrOo99k0iQj4GPboqjazuvc3/ROmriHE9PfuvTm1/69avWrABrTTwycQcXJCYSnZomNbFolaRBIU6hqipfbkzjvz/sxsdNz50jwyk2mRkUFlDv2zEB1nV542egKamgZrPi9969+DMmhr+joig1nPy0MaCggGG7djFkdxID9+6lTUEBgNyqSbR4UhQ3f99tS+f+Jdu5fWQ4j46NdN5AquxTgaKwPdcVzSqPamuk7c2KtbYsTmnfvvJ0jcVCr4MHGb5zF4P37KFHaio62+wKuX2paOkki1uvrWm53PJZPMZyM3OnxHJhZOCZT6qv09TEa3v35s9eMfwTFUWp68kPBQMKChi+cyeDk/YwaM8eAgqtS6otikKMLMsTLZg0KESdko4WMHPhZo7kWe/77KrT8GUD7hldzSnBbJ3yVqbTsblbV/6MieGPXr3ICAiodlrXI0cYtGcPg5P20G//fjzKygAJZ9HySFHcMjz+bSKLN6Xx8bQ4RkW2c95AauxTYW1WwI4D7igb3G0LQDSVe1YcbtOGdT178mdMDBu7d6tcKw3W2RUD9u5l0J69DNqzhy7HjtnOBlfZ4E20MJLFrVtGXimzPtvC7qMFPHFpFDOHdzm7zTNP5/AmEq+YjqbMTNVmRZlOx6Zu3VjXK4bfY2I4WqMm7paebq2J9+yh3/5k3KUmFi2UNCjEab356z7e/HV/5e/X9O3AnOv6nNuLHt7Evkl3UJ5TTNVmhQqkBAfzV89oNnSPZEu3rhhdTq4D1JnNxBw8SP/9++mbnEKvAwfwLi2tfL79c8/iN2nSuY1NCCeRorhlMJabmTD/H47klfLjPcMI8XN39pBqb1YACS5atAt90XByzwqw3g3k36hI/o2K4t/ISFLbVW+0BOblMWDvXvolJ9M3OYUux45Zb2MK4OFBVHzL/GcrWgfJYlFiquDBpdtZtfMYE/uF8OLVPXHVac984tk4vInESbeiyTFSsyZObt+ev6Kj2RjZnS1dT62Jex04SFyytSbunZKCl9FY+bzUxKI5kwaFOK2q94xWVeuGa5f3CuaZcT0I9HLARnCHN5H+5HPkbzyAdSEHlZ1kk07H9i5d+Dcykg1RkSR27oxFc3Jdt2Kx0DUjg77JKfRNTqZvSkrlhpsAGimURTMiRXHLcehEMVfMXU9EoCdLbx2Mi84J+1HUpdZmhUqCiw7lC1905urNCoAMPz82Rkbyb1QkGyIjyfb2rvaS3sXFxKakVDYseqSl4VJRgf2qg6RQFs2IZLEA6x3u3vptP2/9tp9+oX68f2O/hu3JdjbOUBMnhHVhQ6S1cbyzc+gpNXG3I0fom5JSWRfbN9zUAIqPD5Eb5a55onmQBoU4I/s9o+NC/dh0MIe5a5Mx6DU8dmkUk/t3rN+dPurDvvPxbR+hMZsrH7aHc6HBwJauXdkWEc7WiAgSQ0OrTUMG6670PVNT6XkolZ6pqfRITa3WUXYNCyPipx8dM14hHEiK4pblp8Sj3PHlVqYP7cLT43o4ezi1O02z4ui/voTs09jS92TDQgX2t29fmcXxERHVNtsEcCkvp0daWrUs7pSZeXKWhVZL1K6dTXihQtSfZLGo6ocdGTy0bDsBHq58+J84erT3PvNJjlBLTWyfXQFQ4OZ2siYOD2dnHTVxzKFDlVncIy0NT1tNrAH0UhOL85Q0KESDHcgq4onvEtlwIIf+nf14eUIMEYEO2O24Knvh3Hk4iWNuqvwkzjrpzRrORr2eXaGhbA0PZ2tEONvCwyl0P3U6dZdjx+h5KLUypLump+NWXl75vMy0EOcDKYpbnmdX7OLTfw7x3g39GNszyNnDOb06mhWoFhJcXahY5mu7fSnUbFhkBPizNTyCrRHhbA2PILlD+1Ne3qukxNq0sDUsolNTCc7JqdzLAmSmhTg/SBaLmhLT85n12RYKjOXcNSoCVaXhG8efC1s+l6QbSXnuU+yt46oNC6Nez87QULZGRFTWxEVubtVeRrFY6HL8eLXmcdWaWJbpifOFNCjEWVFVlWXx6bz4YxIlpgruGBnBHReGN94aPZuSXxaRcu/z9n3ogZMzLCyKwoGgIBI7h7IrtDM7O4eyp0MHyvX6aq+hWCyEZmbR/Ug63dOtX93SjxCUm1tZdGsA96FDCf1oQaNejxB2UhS3PGUVZq57718OnCjmh7uHERrg4ewh1c9pmhUACa4usMgXffmpDQuAPHd3doWGsrNzKDtDQ9nZuTOZvr6nvI13cQndjqTTPf0I3Y4coXt6OhEZGRjKyyubFnKbU9HUJItFbTILjFy/YCPJmUXWjeP1Gr6ceZYbx5+LKvlc8vn/kfKLe2XDAk7WxGZF4UBwsC2DrVm8JyTklFkWGouF0MxMay1sy2GpiYWzSYNCnJMTRWU8/8NulidkENbWg5eujmFQWMCZT3QE+/S3WxagsRXOcDKcwbpmb1/79iR26czO0FB2hYZyICio2j2n7byLi+l+5Ajd0o8QkZFB2LFjdDl2DP+iospjZNd60VikKG6ZDueUcPnbf9EpwJ2vbxuCQd+4TVyHq2ODTdSTy/ASXF0oXOWLX6amWv5WbVpk+vjYiuTOJHYOJaljR3K9Tp15p7FY6Hz8ON2OWJsWYUetOdwpMxNXy8mcl2UiorFIFou6vPXrPt6osnH8eXFLadtsY6CWmrh689ik07G3Q4eTzePQzhwManfamrh7ejrhGUcJP3aMLkeP4ldcXHmM1MSisUiDQjjEn/uy+L/vEknPLWVy/448fmkUPu76M5/oKFWXhNz+OEpyhi2Uq8+yAGtApwQHsbdDCPtCOrC3Qwh7QzrUWiwD+BYVWYvk48cIO3rMGtLHjhKck4umyn8DGmSKsjh7UhS3XGt2H2fWZ1u4JLodvUJ8m3ZqsCPVo1kB1obFSk9PrppvQFOlTKhaKKtAlo8Pezt0YG/IySyuq1jWmc10zMoi7Jg1h8OOHaXzseOEHTuGR1kZVbch1bRpQ/f1fznsskXrIlks6mLfOL7MtnG8l0HH3CmxjOwe6OyhVWfL6gS1jNJHP8E7t1pCVsviMp2OA8FB7AkJYX+HDuwJCWFvSAh5np61vrS9Jg47dtSax8eOS00sHE4aFMJhSkwVvPXrfhasP4ifu56nx0Uzrlew4+8f3RD2WRb3LkRTUlbtqapNi5rF8sHgIFKCgjgQFERxjTV8dgaTiU6ZmXQ8cYKOWVl0ysyi44ksOmZmEZSbi67Kp32yGZE4EymKW7Z7v9rG8oQM504NdqTamhWKAhYz1kStLsHgRuIhD/r8oas2HRmos1hObt+eA0FBHAwKIj0gAFVT/Ty7drm5dMrKomNWFh2zTtApK5OQLGsu+1a5FTUgMy/EGUkWi9Oxbxzf0c+N+X+ksOdYIbePDOfBMd3Qac+jOzbZ1cjqBB1YFnvhWlZ306KyJg4JYW+HDg2qiTvZcrhjVhadsrIIyZKaWDScNCiEw+3KyOfxbxPZkZ5PbCdf+nf255LoIOcX4zWmwiXOvAclNR+FKtOG6wjpA0FBpAQHcTAomANB7TgQFERWLWuq7XRmM+2zs082LbKy6JCdTXB2Du1zcvApLq72PhrkU7/WTorilm3u2v3MWb0PAI0CD17cnTsvjHDyqBykgc0Ke0siwc2DlVEXcNWj26rNtDh5xElGvZ5DgYEcCA62NS2sOXyoXTtM+rpn6/kWFVUWyh2zsgg5cYL22TkE5+QQnJuLi7n67A8Uhaik3Q3+IxAth2SxqC9juZnnVu5m8aY04kL9eHtKLO19a/8L/HmhliUhuAWQsPZJCn9wsy3Tq+rUmjjT14cDtlrYWhNb6+MTPj51vq3ObKbDiRPWxsUJ6wd6HbKzCco5tSau3NlIauJWTRoUolGYLSov/LibT/4+BIBOo/DZ9AEMiWjj3IHVpkpgJ+Qno0x+Hl0FcJrGBVhve5oWGMjhtm1Ia2v/3pbDbduectu9mtyMRoJzcwnOybEWy7k5J4vmnBwC8/KqdZutI5D7WLdkUhS3bPGpuVz/oXVqMMCC/8RxUY92Th5VI6itWaGxLdmwVGDdbLOW2kGjh6H3kWAuYKVSQswn2wjfmkPNzyNr5rBZUcgICOBw2zYcbtOWtEBrBtuz2OjiUudQFYuFtgUFlTkcVCWHO+Tk0D47u9ptqk+eKE2MlkyyWDTUiu0ZPP7NDvQ6DXOu7c3oqGaW7adpXCgL3dCZaybxqTVxgZtbtew9m5rYnr/tc07Ww7XVxJV39pOauMWSBoVoNO/8nsyc1Xux2P418TboeGlCDJfHOHnZR31VKbQT1j4Jn7tV2bW+qlOD2qjXk97GGs5pbduS3rYtGf7+HPX3IyMg4JRbP53yihYLbQoKaJuXR2B+PoF5eQTm5ROYn0fbvHza2h7zrTETwzoawGAgKmHb2V+7aHJSFLd88am5rEjIYPGmNHq09+arWwY1v00zG6KOore2O4NUp4BGY21kaF1JGHEPK0/Ec8LVnaj4YoYuSjqlcWE7qxoVOOHtTVpgIGlt23C4bSDpbQI46udPRoA/mb6+WOpYNmLnUVpaLYPb5tsz2ZrHgbY8dq2oqPV8/5kzaPfQQ2f4gxLnE8licTYOnijmzi+3svtoATOHdeGRsZG46M7DJR8NUUfDOcHFBb5wr3dNXKrXc6RNG9IC25LW5mRNnBHgT4a/f53LRuy0ZrO1JrZlcdsqGWyvidvl5dU6OxmQW6c2Q9KgEI3GvplQeYUFrUZDsK+B1OwSBnTx55lxPYhuX/d0sPPOaQrtBB1oP/U6ZV211alBDdbZFxkBAdaGhX8Ax2zfj/r7cdTfn0wfnzrXW1elLy8nMP9kw6JNQQH+hYUEFBTiX2j9CrB99zAaa29mIBsZnQ+kKG49ViUe5fYvt3JN3xBmX9ureTRsHeksl4Sg1cPFL4KpsDKLE/Z8w0qKOaHVMvnJP/HOKavl/FObFwAVGg2Zvr4ctTeP/QNqfPen1GCo1yX5FhVVFs5t8/MJKCgkoLCAAHsGF1i/+xUVnTI7rnKM8mngeUGyWJwtY7mZl35K4rN/U+nT0Ze5U2Lp6O/u7GE5xpkazhotoJCg1zi0Js4ICCDL2/usa+Kq9bC9Jg4oKMC9rOyUZSV2UhM7nzQoRKOybyY0KCyAPh19+WpzGrN/2Ut+aTmTB3TiwTHdCPB0dfYwz15tgV3l5wS1DMvdn9TYjKiq2sPapNWS7e1Nlo8Px319yfT1IcvHlyxfHzJtj2X5+FLgUf//8bmaTKc0LezBHVBYiG9RET4lxfgUFeNbXIyn0XjKjsxVyT2xHUuK4tbljTX7eOu3/Tx5eRQzh4c5ezjOc7olIWbbkpDamhaK1trUUC2gdYWxr0BptjV/Ow4gITOBlSkrOVF6AoCo9YfrnHlR+ZI1fleBAnd3sny8yfT1JdPHlywfn8o8tmfzCR8fKmq580hdfIuKauRwQeXP/oVF+BZbM9iaySW41DE7AwAXF6J2bK/3e4szkywW5+qnxKM8+vUOFAX+N7E3Y3sGOXtIjaeejQssFSS4utSyQaed9bHT1cTWHD6ZwZm2DM708TnnmrjmB3u+RUX4FBfjY8tjr9JSFE6the2kJnYsaVCIJpdfUs6bv+3js39TcXfRct9F3fjP4FD05+Pux45whtkXpm+88Cg83bXX3sQA67Q5a8FsLZZzvL3I9vImx8uLHC8vsiu/e9b7k0A7rdmMd0kJvraA9qksmovxKbF9r/K4jy3Aa87UqHUqtnxSeAopilsXi0Xl9i/jWbP7OJ/ePIAR3do6e0jOd8YlIZxyS9OT7LMsXGDsy2DMO7VxXEvjoqqZd6w+bQOjyrtUY1EUcj09bYWytWC2Z3DVHM7x8iLH07NenwRW5WY02jK2xNa8KKrl5yJbNpfgU1yMZ2npqRuAnnIxspdGbSSLhSOkZZdw1+Kt7EjPZ9qQzjx+WSSuuha8pK+mBjQuUBQS9FpWerpz0Wd6vOusietuYoC1Js709a38cC/b2/uULM729iLb2/u0+xPVpr41cdXnvUtKKmdqnLbKlw1BTyENCuE0+48X8t8fdvPX/hOEt/Xg6XHRXNCaivQGTJcrXe5V4z7Wtam7kQFQ4uJSe8FspUA7KQAAIABJREFUa2rkeXiQ5+lh/e7hccY1gXWOwmLBs7QUr9JSvEpK8TKW4lVSgmepEe/SErxKSm3Pl+BVasSrpMR6bKn9uFL0tWwQWpuW1OiQorj1KS6r4Jp3/yEjr5Tldw2jSxsPZw/p/FTXLAtVtW28CWczy6I2p2tg5Jfl8+D9G8/YxIC6C2izopDn6XlKDmd7n2xg5Hl4kufpQb67O/meng2anVGVwWTCs7QU72oZa81fexZb89n2nP0x27FVp0CflkZD1O5dZzXG85FksXAUU4WFV1bt4eO/DxLTwYfbLgjjUHYJg8ICnH9nO2dpSOOi2gbLkOCiY6WnByMX6fE/bU18+kYG1F4T2xsa2d5e5Hl4kO/ReDVxzUyumb9Va2LXOpYF1iQ1se1caVCIc6WqKr8lZfL8j7tJzS7hoqhA/u/yHq27UD/DspE6p84ZDOT86kHbw7Wv/KvuzOFdrtFYw9nTg3x3D/Irmxee5Hu424Lb+rO1ueFJocHQ4JkatTGYTHgYjbiXleFuNOJhLMOjzIiH7We3sjLrz2VG3I01fq76mO01tKdZqlIb15gYwpYtPefraAgpilunwzklXDlvPf4eLnx351C8DXXfMlNwDhtvQrVZFhe/UG0vi6qzLE7ndA0Mu/yyfO57YCP6epZIp8thFSg2GE4plvPtv9vyuWpzudDdnUI3N8xn2diw05rNuNuy1v7dw1iGe9mpmWzP3crnbPnrUVZWea5reXn9Gh52TtrYTrJYONrqXce4b0kCJSYzCuCq1/DlzEGtt0lRm7OsfQFb5p8M3ARXF1Z6ejD0az2Bx0/34V31ivB0+WTSailwd7dmbtXa18PTlsfu1ZrLjVITV2Zx9fx1r63+LavxnNGIVy01cX00t5pYGhTCYcoqzHzy9yHm/rYfk9nC9KFduGtUBF5SrNfubIPc1o1O0GtY6eHO+Pk6tOqZZ2ZUdaYCs1yjodjNjQI3N4rc3Chwd6fIzUChm7VoLnR3s/1soNDdnSKDGwXutmPd3Ch0dz/jzvoNZQ93g8mEW5kJQ7kJt7Iy3Ewm3EwmDLYv9zLrdzdT2cljbcfYj3Mrsz7nXuU8ncVSr8ZHbezrFqUobr3+Tcnmxo82MqJbWz78TxxaTSvbNNMRznaWBRrbHUPqP8uiPurbyNiWuY1X3zcRknPmbK3qTMeqQKmLiy1vbZnr7kahwc3WwDBQVCWTC9ysOW3Na+s5pa6O3R9KazZbG8ymsspsdbdlbdWcdauSwydz1/58GQZTOe62HK6azQ1ugFQdW/v2dFv7GyANCtE4Xv4piffXHQCs//3eO7or943p5txBNTcNnn2h2jZfptbGtb2RccK2xHzm29paNvdUOF3inktNbK99a6uJqz7m6JrYzdY4tmdtbTWxPXNrr4nLcDOV416lJj6Z3w1vgFTlNe4KQl57TRoU4vySWWjktZ/3siw+nTaerkzqH4KbXsvg8DbSaW6oc2xi2EPdOqXOvTLAb3pXi2utt5aq6dRj6lM82gvrYoOBEoMrJa6uFLsaKDbYv6y/l7q6nvzdYKDE1WA91uB68lhX6/kNXfPdUPryctxM5biWmzCUl+Nq+9m1vAJDubVwNpjKcamwfrcf51JeXnn8dwsXsstY2ir+Zio5fKrP/z3EU8t3cfvIcB4dG+ns4TRvDtnLQgcjHrEuE+nSsFkWDVGfRgacbGZYsBbZdy6vYOju+s0Iq019gqZco6HEnqM18rfElr8l9rytkr8lVfLXms3W7yZ9437goFgstkL51Iw1VMnkqjltKDfhYstp+2PLvviCnZLFwsHsd7YrK7egAsE+Bt6/sR+9QnydPbTm72zrXQWwVJ99UZeazQx7it78jga9WcPZpPHZ1MRV8/dMNXGxvYZuhjXx9wsXnnUOS4NCNJrth/N4+Ovt7DteBICLVsPiW2Q6nMOdUxOjys+qerLgt3WpTw1zmPi59gzrBmuq/diGJFbVcDe6uGB0caHU9mV0caHU1f67a+Xv9uNKKo+3PldS23Ourg7pbkePvVSK4lZMVVWe+G4nizel8dbkPozv08HZQ2p5znTHEIU6locooGisOafVQZ+p0Od661ON0LSoS32bGXBqQ6OmRS9XnHVzw66hYVWu0VBssBbS1hzVU1Ijd+2/V33Mntelrq4Y9fafT33OUQ0QyWLRWOx3tvMy6Hj3jxSyCst44OJu3DoiXGbONbYGz76oewnJmSm24zUkGFxZ6WHghFYHbSKsT3tY99ub+PI2/E+cYTPjer5bfVkUBaNeX2tNXD1XXWvksCulLvpaa+Jqz50HNbE0KESjmrd2P3NW76uMhK6Bnnzwn7jWvT+FM5ypiXEWszLqG/a1NTmgPo2OswvHs0lCFSjX6ShxcaHMRU+ZTk+ZiwtGvZ4yvR6jix6TXo9R71L5u/UYPWV623EuehJffkWK4lbOVGHhhgUb2Z6ex7LbBssna42pXrMsTpdTVZoWOhfb0pCcJmtW1EdDGhpw5qZGTV3TVZ773HzOjY6azjYEzYpSWVxbM9earfbsrcxfWx5XPcb+WJmLnqQXXpQsFo0uv6T8/9u77/Aoy7SNw79nUgklpNADgdB7CUIoVoqIBXUtgCIWRMUu7rer67quuuquq6wFC6AuasSyNlQUAStKC01C7zV0CJ2Ueb4/ZsCIgUxIMu+U6zyOmJnMO5krY7jnzT1P4cGPF/HFohy6pSXx7NXtqRN/egsxShmU02jjsjQxfl3MOQoy7gBjoen5LNi/ls9WfMjOGO/2qYd2QVwSVEr8zXe64tHpJG49dHo/fwkJS+vEc+Ki9ffXGhz9m9pbtP4eu5z95JNqUEhgOjYcLr/AjTGGCJfBbS1DMhpyV68mVI8r3RZB4iflNSoDvN1rcxqF//cWxMR4mx0nzic0nq62tQx9YiMxR/1T106suletW6eTYmHngaMMePEnCt2WiXf2oGbVsi+wJaVwylEW+ZRYh1yR0GkoxFSF5v09J55+HGVRVqUdpZFzMIecgznYMtbnoh57o4DGW09/+kppFFdwVYvFX6y1fDB3E49MXExUhIt//qEt/drUcTqWnMixJsYxxtvEsJ7XmDNu8nytaR+IrgLrp5/yTcQFMdGlalYfM/TeqcQc9a1hXVbleU5cLg0KY0w/4DkgAhhnrX3qhNuN9/b+wCHgemvtPF/uWxwV4+BybDhcRloS9RMr8ezXK3g/ayNVY6O4u1dTrs1IJTrSH6cxUq7KbVRGxTU0fn1BcP963RUJ6Td4prM0OhNi41mwejKfmYPszMv9tbsN5O7fwp6ICBKiqhCff+R3Xe8b7p5CVP6vGa9at87RNSj8WYtVh09t8ZZcrnh5Bi3rVGXC8AxiIsu2G4OcphPr1MJ3YP47xW5/d9KdQ45PDYmCcx70HFeBa1o4obQjNY4p7YiNivDScwUknvDGo5O1WOfE4WntzoPc/e58ftmUy8Az6vPwxa2Ii450OpaURnm8OQfl08Qw3lEZbjdERkGfxyDvwKnPuUvxOnQ6o/P2HN1DQkwC8THxxR5z451fE1lktktZ6nCZGxTGmAhgBdAH2ATMAQZZa5cUOaY/cCeeYtwVeM5a29WX+xZHxTj4Lc3Zxz++WMr0VTtplFyZBy5oQZ9WtfC8bktICpaGBpaTvkhERMHZf/Lkanzeb99ZBVJaZ2zetM+dUsYQp5fcz7VYdbhkX/ySw+3vzOPK9BT+dUU71bdAUdLUEGO875adqp4E/vQQfznd5sYxFdHkWPXIKg6v9X+DQufE4S2vwM2oqSt45fvVNEqqzPODOtKmXvF/zEmQKo8mRikW9iye91zVRPz6ehURDWf/n+dy43OLz+bntZYeu+SxHXk78mqezvcojwZFN+ARa+353usPAFhrnyxyzKvAd9baCd7ry4FzgIYl3bc4KsahwVrLd8t38I9JS1m1/QBdGyXy14taqZiLR3k2NArzAfdvmw3HGhHlMqzZ09Do/Oo+srYUOvWunV9rseqwb579ejnPf7OKhy9qxY09GzkdR07lpFNDSvHOmCsSOg7xDNlteXHQTQ9xSnk3ORxsUOicWPh59U7ue28huw4e5f6+zbn5zDRcWkAzfJTbNOmiW6xayrw+hisCmp0P7a72jAbeNLvcRmQUm6AM24yWx9ijesDGItc34ekIl3RMPR/vKyHKGMO5LWpyZtNkJszZyKgpK7j4xelc3jGFP57fnNrxmrcd1up3+W1xLOlyrVYlvyAc3lWKF4bSjNSw3hcPR6kWB6B7ejdj2db9PP7FEprWqsKZTWs4HUlOpmjNKa6eFLemxYnTQ9wFMPcNz+UZL/D7LU9d0OisX7+nGhcAdKjZgQ41O5TpexRtciw/sHxHOUUrLdVhoXvjZL68+0we+GgRT365jB9W7uCZKzvovDZc+HL+WtI5a6nWxyju/PTYZ3eRiwWw7AvPR1HHRwxH8LsRw7YQ0s45ecYKev0qjwZFcS3BE8/UT3aML/f1fANjhgPDARo0aFCafBLgIiNcDMlIZUCHuoz+dhVvTF/HpEU5DD8rja6NEpm/cS8ZaUnanlROzdeGBpTuhaEUXW/LASe7FBVei1WHS8/lMjx7dQcuf+knbn1rLoO7NqBfmzqqZ4HuZPWkpMbF76aHeD8X5sO3//B+7dj0ELdnesj5T8GRPWpWlFHRJsfzO5/f4FAMnRMLAAmVo3n52k68N2cjf/9sCRc89wP//EM7+rau7XQ0CQQV8SbcKUcS5/FrOTlh9PCxN9hs4a9fLjwK3zzqufztP4rc54QdS7rd4fl60/M9j1UkV72q5rR/2cujQbEJqF/kegqwxcdjon24LwDW2jHAGPAMZytbZAlE1WKjeOCCllzbNZWnvlrGc9NWAp5/EjFRLjKHZeikXspHaV8Yil4+xYvEtqcziq1fflLhtVh1+PRUiYnknl7NGPHOPMb+uJa3Zqwn82bVs6BUUuPCp+kh1nMiCFBwFL6413PZFQntBkJsNWh5yW9P9tS4CBY6J5bjjDEM7NKAMxolcve78xn+1lz6tKpFm7rV6Nm0hl4DxHdlHZXxuwWiSztiuJgRGYV5MP1Zz+Xpo4qE9Ux7rl3F1DvdH7c8GhRzgKbGmEbAZmAgMPiEYyYCdxhj3sUzXC3XWptjjNnhw30lzNRPjGP04E5Ur7SIzFkbsMCRfDcfZG1UMRfnneJFYvN+u9WBRMeoFgewtbsO4jLgtnCkwM2M1TtVz0JJqaeHGHDne24vOj1kwdueyzNf4jfTQ86839PA0PSQQKc6LL/TuEYVPrqtB3/830I+XbCFKUu2Mfq71UxQo1rKU0lNjPaDy3exeqiwac9lblBYawuMMXcAk/Fsi/S6tXaxMeZW7+2vAJPwrFa8Cs+WSjec6r5lzSSh4fJOKXw4bxN5BW7cFt6ds5G9h/J5oH8LUpMqOx1PJKCoFge2jLQkoiNdHM13Y/FsSSchqiKmh3x3bJ1ETQ8JZKrDcjLRkS6a1ap6vFGdV+Bm9LereG1oZ+3wJP7h52klZZn2XOZdPJygFYvDx9z1e5i5ZhedGlQna90eXv5+NQWFlut7NOSO85pQLTbK6Ygix5VlxeJgozpcep56tpP5G/Yydel2XhjUkYvb13U6ljitvHYP6XANxFTT7iGoFktgmrt+D9eMm0l+gadR7bZwXouaPH1FO5KqxDgdT6T0TrFjSUrrjM2b9rlTTufbqkEhQWXbviP8e/Jy/jdvE4lx0dzbpxkDz6hPZITL6WgiOikWn+QVuBk8dibZW3L5363dtbWy/Kq4kz1fpoecyBhPPyMiCs590LMqeyP/rL4eCFSLJVAde+Mto1Eiizbn8sSXy4ivFMWzV7XXLk8SUspSh9WgkKCUvTmXRz9fwuy1u2lWqwoPXdiKs5qpsIuzdFIsvtqx/ygDXpwOwKd39KRGVb17JqdQUuPid9NDinNseoj1TA/p90/P1ssh2KxQLZZgsTRnH3dOmM+q7Qe45aw0RvZtTnSk3nST4KcGhYQlay2TF2/liUnL2LD7EOe1qMmD/VvSpGYVp6NJmNJJsZRG9uZcrnjlZ9rUjeedmzN0UiqlVy7TQ6Kg2wiIioPG53m+FuSjLFSLJZgczivk0c+XMGH2BtqlxPP8wI40TNZaaxLc1KCQsHa0oJDxP6/jhWmrOJxfyLUZqdzdqykJlaOdjiZhRifFUlqfLdzCnRPmM6hLfZ64rK0WS5PT59P0EE5oXBSj6CiL85+EI3uDrlmhWizB6KvsHP704SIKCt08OqANl3eqp9cECVplqcPlsc2oiKNiIiMYflZjLu+UwqgpK3hzxjo+nr+Zu3s1pU29asxZt4eMtCRt5SQiAefi9nVZmrOPl75bTcs61biuW0OnI0mwKq/dQ441LgqOwhf3eS67IqHTdRBbHZpf8Ov3CbLGhUgg69emDu1SqnPPewsY+cFCfli5g8cvbUNVLQgvYUYjKCTkLN+6n8e/WMKPK3ce20WemCgXmcO037RULL1rJ6fD7bbc/GYW363YwVs3daF742SnI0k4KHF6iC15lEVENPR9HPL2B1SzQrVYglmh2zL621U8N20ldavH8tzAjnRqoPNXCS6a4iFyAmst93+wkA/nbT7+tZvPbMRfLmzlYCoJdTopltO1/0g+l730M7sOHGXiHT2pnxjndCQJJ+WxCKcrAtoPhkoJ0OIiR7c6VS2WUDB3/W7umrCArfuOcF+fZtx6dmMiXJryIcFBDQqRYhzbb/povme/6UiXYfhZadxxXhPiojW7ScqfToqlLNbuPMiAF6dTt3olPrytO5VjVKfEYT4twllY/H0dHGWhWiyhIvdwPn/5eBGf/5JDm7rVOLNpDXq3qqURwRLw1KAQOYlj+003r1WVSYty+Gj+ZurEx/Jg/5Zc1K6OFh+ScqWTYimrH1bs4Po3ZtOnVS1eviYdl94tk0BRLqMsIqHjEM8oiwpcy0K1WEKJtZZ/T17O6O9WAxAdYZgwvJuaFBLQtEimyEmkpyYcL+C9W9XimowGPPzpYu6cMJ/MWet55JLWtKhdzeGUIiIeZzWrwYP9W/L4F0t5btpK7u3TzOlIIh4lLcLpyygLdwHMfcNzefqzJ+wY8hQc2RNQa1mIBAJjDHExkbgMuC3kFVr+9dUyMod1JTJC21NL6FGDQsJKemoiE+/oybtzNvD05OVc+Px0hmSkcm/vZsTHaZVkEXHeTT0bsTRnP89NW0mL2lW5oG0dpyOJnFzRxkWZdgy513PZFQkZIyC6CjQ+99fvo8aFhLGMtCSiI13kF7gBw6y1uxk8bhYvDOpIrWqxTscTKVea4iFha++hPJ75egWZs9ZTPS6a/zu/OVd1rq8h1XLaNKxYysuR/EIGjpnJ8q37+WhEd1rW0UgvCWKnvWOI+e0oi37/hMO7SmxWqBZLKDo2bTkjLYn1uw7yl4+ziYuO4LmBHenZVLs/SWDRGhQiZbB4Sy6PTFzMnHV7aJ8SzyOXtKajtnOS06CTYilP2/cd4eIXpxMV4WLiHT1JrBztdCSRsivzWhYGIqLgvIfBnVdss0K1WMLBym37GZE5j1U7DnDXeU25q1dT7fIhAUMNCpEystby6YItPDFpKdv3H+XK9BT6tanNsq37yUhL0kJE4hOdFEt5W7hxL1e+OoNODarz1k1didJ8YwlVJx1lQZG1LIoZZWEioP0gqJwMLS4EIKV1xuZN+9wpfkruKNXi8HYor4CHPs7mo/mb6dEkif9c3ZEaVWOcjiWiBoVIeTlwtIAXvlnJuB/XUOgGA8REucgclqEmhZRIDQqpCB/P38S97y2kX+vatE2JV9NUQl9JoyzA26wo5hzWuOj86j6ythSGxVvJqsVireX9rI08/Oli4itF8cKgjnRNS3I6loQ57eIhUk6qxETywAUtKSi0vDZ9LRY4ku/mk/mb9QeBiDjiso4pfLd8B58u2MLkxVvVNJXQV5odQ0628KZImDDGcPUZDWiXUp0RmfMYNHYmI/s257azG2tdNQlKGisqUoz+besQG+XiWF1/e+Z6Hpm4mANHC5wNJiJhqWnNKoDnT7C8Ajcz1+xyNpCIE+p3gTNHQufrYehEOO8v0P8ZiIz1TPWIiIaIGOzJF7AQCVkt61Tjszt7cmG7ujw9eTk3jp/DnoN5TscSKTWNoBApRnpqApnDMpi5ZhftUuKZsmQb42es46vsrfx9QGvOb13b6YgiEka6NU4mJnIVRwvcWAut62pXDwlzp9jedNvTGVscTCbimCoxkTw/sANdGiXy2GdL6P/8j7w4uCPpqYlORxPxmUZQiJxEemoCt5/bhDOb1uDRAW348LbuVI+L4pa35nLzm1ls2XvY6YgiEibSUxN45+YMhmSkEuEyvPHTOtxuvUksAvw6ssLbtNi83251OpKIU4wxDMlI5cPbuhMV4eLqV2cy7sc1BOO6gxKe1KAQ8VGnBgl8dmdP/nxBC35cuYM+z37P69PXUqg/EkTED9JTE3js0jY8cklrvl+xg+e/Wel0JBERCVBtU+L57M6e9GpZk8e/WMpVr87g2a+XM3f9HqejiZySGhQipRAV4eLWsxsz5d6z6dwwkUc/X8JlL/1E9uZcp6OJSJi4pmsDLu9Uj+emreS75dudjiMiIgEqvlIUr1ybzg09GjJn3R6e/2YVg8fOVJNCApoaFCKnoX5iHP+94QxeGNSRLXuPcMmL03n88yUc1CKaIlLBjDH849K2NK9VlXveW8DG3YecjiQiIgHKGENylZjjC78fLXDz1ox1TkYSOSU1KEROkzGGi9vXZdp9ZzOwSwPGTV9L31E/MG3pNqejiUiIqxQdwSvXplNYaBmROY8j+YVORxIRkQCVkZZEdKSLCOPZmfeTBVsYNWWF1jKSgKQGhUgZxcdF8cRlbfnfrd2Ii47gpvFZjMicy9Ql2xj97SoNoxORCtEwuTLPXNWeRZtzefTzJU7HERGRAHVsd7r7+jbnnWFd+UOnFJ6btpJb357LAY3+lQCjbUZFyknnhol8cdeZjP1xDf+ZuoJJi7ZigJgoF5nDMkhPTXA6ooiEmL6ta3PbOY15+bvVdGqQwBXpKU5HEhGRAJSemnD8XDQjLYnWdavxj0lLuWz0T4y9rjMNkys7nFDEQyMoRMpRdKSL289twvXdGwJggSP5br5eoh3PRKRijOzTjG5pSfzl40Us2bLP6TgiIhLgjDHc2LMRb97YhR0HjnLJi9P5fsUOp2OJAGpQiFSIfm3qEBvlwrseEW/+vJ735mzQHtQiUu4iI1w8P6gj1eOiuC1zLrmH852OJCIiQaBHk2Qm3t6TutUrccMbsxnzw2qdq4rj1KAQqQDH5vrdf35zXr6mE+3rx/OnDxdxw3/nsDX3iNPxRCTE1Kgaw0vXdGLznsOMfH+BFj4TERGfNEiK48PbutOvTW2emLSMe95boIWXxVFlalAYYxKNMVOMMSu9n383yd4YU98Y860xZqkxZrEx5u4itz1ijNlsjFng/ehfljwigSQ9NYHbz23CBW3r8M6wDP5+SWtmrdlNn1Hf8+HcTepQS7lRLRaA9NRE/nJhS6Yu3c7L3692Oo5IWFEdlmBWOSaS0YM7cX/fZkxcuIUrXvmZzXsPOx1LwlRZR1D8GZhmrW0KTPNeP1EBMNJa2xLIAG43xrQqcvsoa20H78ekMuYRCUgul2Fo94Z8efeZNK9VlZEfLOTmN+eyfb9GU0i5UC0WAK7v3pCL29flma+X89OqnU7HEQknqsMS1Iwx3HFeU8Zd15l1Ow9xyQvTmb12t9OxJAyVtUExABjvvTweuPTEA6y1Odbaed7L+4GlQL0yPq5IUGqYXJn3bunGQxe25IeVO+g76gcmLtyi0RRSVqrFAnhOMJ+6vC2Na1ThrgnzycnVO2AifqI6LCGhV8tafHJ7D+IrRTF47Ezemrle56niV2VtUNSy1uaAp+gCNU91sDGmIdARmFXky3cYY34xxrxe3HA4kVAT4TIMOzONSXedSWpSZe6aMJ8RmfPYdeCo09EkeKkWy3GVYyJ5+dp0juQXMiJzHnkFbqcjiYQD1WEJGU1qVuHj23twZtNk/vpJNsPfzOL5aSuZu36P09EkDJTYoDDGTDXGZBfzMaA0D2SMqQJ8CNxjrT22D9rLQGOgA5ADPHOK+w83xmQZY7J27NA2OBL8mtSswoe3duP/+jVn2tLt9B31A18uynE6lgSoQKjFqsPBo0nNKjx9ZXvmb9jLE5OWOh1HJCQEQh323l+1WCpcfKUoxg09g8s71mPK0u08O2UFg8fOVJNCKlxkSQdYa3uf7DZjzDZjTB1rbY4xpg6w/STHReEpxJnW2o+KfO9tRY4ZC3x+ihxjgDEAnTt31jgjCQmRES5GnNOEXi1qMfKDBdyWOY9L2tfl75e0JqFytNPxJIAEQi1WHQ4u/dvWYVjPRoybvpaODaozoINGkouURSDUYe+xqsXiFxEuQ+OaVTCABY4WuPkyO4f0VA3wkYpT1ikeE4Gh3stDgU9PPMAYY4DXgKXW2mdPuK1OkauXAdllzCMSlJrXrsrHI3pwb+9mTFqUQ59RP/DK96sZ/e0qdarFF6rFUqw/XdCCMxom8McPfuHvny1WPRGpOKrDEpIy0pKIiXLhMp7r783eoNcSqVCmLIueGGOSgPeBBsAG4Epr7W5jTF1gnLW2vzGmJ/AjsAg4NhH2QWvtJGPMW3iGsllgHXDLsfl7p9K5c2eblZV12rlFAtniLbmMeHse63cfAiA20kXmzRnqVgcBY8xca21nBx7X77VYdTh4TF2yjWFvev5fqZ5IOHCiFuucWELZ3PV7mLlmFw2T4vjX5OVszT3CcwM70q9NbaejSYAqSx0ucYrHqVhrdwG9ivn6FqC/9/J0wJzk/kPK8vgioah13Xj+kJ7CqCkrsMCRAjefL9yiPyjkpFSL5VSWb9uPy4DbeobnzlxjN7+AAAAgAElEQVSzS/VEpJypDksoS09NOP66kZGWxE3js7gtcy4PX9SKG3o0cjidhJqyTvEQkQrQo0nyb4bTvT1rPW9rmycROQ0ZaUlER7qOzyFGdURERE5TUpUYJtycQZ+Wtfj7Z0t47PMluN16XZHyowaFSABKT00gc1gGI/s257WhnclIS+KhT7IZkTmP3EP5TscTkSByrJ7c17cZTWpUZsyPa8nJPex0LBERCVKVoiN4+dp0ru/ekNemr+WOCfM4kl/odCwJEWpQiASo9NQEbj+3Cb1a1mL8DV144IIWTFmyjf7P/0jWut1OxxORIJKemsCd5zVl7NAzyC90c997CynUO14iInKaIlyGv13ciocubMmkRVu5dtws9hzMczqWhAA1KESCgMtluOXsxnx4W3ciXIarx8zkhWkr9QeGiJRKo+TKPHJxa2as2cXYH9c4HUdERIKYMYZhZ6YxenAnftmcyx9e/pkNuw45HUuCnBoUIkGkff3qfHFXTy5qV4dnpqzgmnEz2Zp7xOlYIhJEruycQv+2tfn35OUs2pTrdBwREQlyF7arQ+awruw+lMdlL/3Ego17nY4kQUwNCpEgUzU2iv9c3YGnr2jHwo25XPDcD0xbus3pWCISJIwxPHFZW2pUjeHud+dzKK/A6UgiIhLkzmiYyIe3dScuJoKBY2YwdYnOTeX0qEEhEoSMMVzZuT6f39WTOvGVuGl8Fo9MXMzRAi1QJCIlqx4XzbNXdWDtroM89vkSp+OIiEgIaFyjCh/d1oNmtaoy/K0s3pq53ulIEoTUoBAJYo1rVOHj27tzQ4+G/PfndVw2+mdW7zjgdCwRCQLdGidx69mNmTB7I19lb3U6joiIhIAaVWN4d3gG5zavyV8/yebJL5dqG1IplUinA4hI2cRERvC3i1vTs0ky93+wkItfmM4NPRoSFx1BRloy6akJTkcUkQB1b+9mTF+5kz9/9Asd6lendnys05FERCTIxUVH8uqQdB75bDGvfr+G7M376NoogR5Naui8VEqkERQiIaJXy1p8efdZNEqOY/S3q3l68gquGTuTuev3OB1NRAJUdKSL5wZ24Gi+m5EfLNC7XCIiUi4iI1w8NqANQzJS+WnVTp6dslLnpeITNShEQkjt+FguaFMH471+pMDN5MUaui0iJ5dWowp/u7gVP63S1qMiIlJ+jDHUjo/9zXnp98u3O5pJAp8aFCIhplvjZGKiXLi8rwaZM9fzw4odzoYSkYB29Rn16de6Nv/+ejnZm7X1qIiIlI+MtKTfnJd+/ssWdh/MczaUBDQ1KERCTHpqApnDMhjZtzkvDe5E/cQ4hr4xm5e+W4W1Gr4tIr9njOHJy9uSVDmGu7T1qIiIlJOi56V/6d+SzXuPMHDMDLbvP+J0NAlQalCIhKD01ARuP7cJ/dvV4aMR3bmoXV3+9dVyRmTO48BR/eEhIr+XUDmaZ69qz9qdB3n8i6VOxxERkRBx7Lz05rPSeOOGM9i05zBXvzqTLXsPOx1NApAaFCIhLi46kucHduChC1vy9ZJtXDb6J9ZoK1IRKUb3JskMPzONd2Zt0Po1IiJS7ro3Tuatm7qwc/9RrnxlBht2HXI6kgQYNShEwoAxhmFnpvHWTV3YdTCPAS/+xNQl25yOJSIBaGTf5rSpV40/f/gL2/ZpCK6IiJSv9NRE3rk5g4N5BVz56s+s2q43zuRXalCIhJHujZP57M6eNEyuzLA3sxg1ZYW2FRSR3/BsPdqRw/mFjHx/oWqEiIiUu7Yp8bw3vBuFbrj61Rks2bLP6UgSINSgEAkz9apX4oNbu3FFegrPTVvJzW9mkXs43+lYIhJAGteowsMXtWb6qp28Nn2t03FERCQENa9dlfdvySA60sWgsTNZsHGv05EkAKhBIRKGYqMiePqKdjw2oDXfr9jBpaN/YsW2/U7HEpEAMqhLffq2qsW/Ji9j8RZtPSoiIuUvrUYV3r+lG/GVorh23Cxmr93tdCRxmBoUImHKGMOQbg2ZMDyDA0cLuHT0T0xalON0LBEJEMYYnvpDOxLiohn+ZhbPTV3B3PV7nI4lIiIhpn5iHO/f0o1a1WK47vVZTF+50+lI4iA1KETC3BkNE/n8zp60qF2VEZnzeOrLZcxZt5vR367SHyMiYS6xcjS3nd2YzXuPMGrqSq4ZN1N1QUREyl3t+Fjeu6UbDZMqc+P4OVrMPYypQSEi1KoWy7vDu3FN1wa88v1qrn51Bs98vVx/jIgIh/ILj1/OK3Azc80uB9OIiEioSq4Sw7vDM2hZuyq3vj2Xz3/Z4nQkcYAaFCICeFbu/8dlbenXujZuC26rP0ZEBDLSkoiN9JwuWAudGlR3OJGIiISq6nHRvD2sKx0bVOeuCfP539xNTkcSP1ODQkR+4+az0oiO8JQGt4WEuGiHE4mIk9JTE8i8OYMr0lOwwM+r1bQUEZGKUzU2ivE3dqF742Tu/2AhT0xaqqnHYSTS6QAiEljSUxOYMDyDL7Nz+HzhFv7+2WKqx0XRv20dp6OJiEPSUxNIT03A7ba8/N1q+retQ8s61ZyOJSIiISouOpJxQztz7bhZjPlhDQaIiXKROSyD9NQEp+NJBdIIChH5nfTUBB66sBWT7j6LNvXiGZE5j1e/X4211uloIuKgv17UivhKUfzpw18oKHQ7HUdEREJYbFQEZzWrAYAFjuZr6nE4UINCRE4qsXI0mcO6clG7Ojz55TIe+iRbf5SIhLGEytE8cklrftmUyxs/rXM6joiIhLgeTZJ/XQcJPIshSUhTg0JETik2KoLnB3bktnMakzlrA8PezOLA0QKnY4mIQy5qV4feLWvyzJTlrN910Ok4IiISwo6tg3RP76a0qF2V/0xbybfLtjsdSypQmRoUxphEY8wUY8xK7+diJwQZY9YZYxYZYxYYY7JKe38RcZbLZfhTvxY8eXlbfly5kytfmcHW3CNOxxIv1WLxJ2MMj13ahkiXiwc+WqSpXyKoDotUpPTUBO7p3Yz3b+1Gc+8WpDO0YHPIKusIij8D06y1TYFp3usnc661toO1tvNp3l9EHDaoSwNev/4MNu4+xKWjf2LJln1ORxIP1WLxqzrxlfjzBS34efUu3s/a6HQckUCgOixSwarFRvHmjV1pkBjHTePnMG+DdvUIRWVtUAwAxnsvjwcu9fP9RcTPzm5Wgw9u7YYxcOUrP/Ptcg2zCwCqxeJ3g7s0oEujRB7/Yinb92lElYQ91WERPzi2PlqNqjFc//psFm/JdTqSlLOyNihqWWtzALyfa57kOAt8bYyZa4wZfhr3F5EA0rJONT65vQcNkyszbHwWmbPWOx0p3KkWi9+5XIanLm/L0QI3D3+62Ok4Ik5THRbxk5rVYskc1pUqMZFc99psVm0/4HQkKUclNiiMMVONMdnFfAwoxeP0sNZ2Ai4AbjfGnFXaoMaY4caYLGNM1o4dO0p7dxEpZ7WqxfL+Ld04u1kN/vJxNk9+uRS3W3PRK0og1GLVYTlRWo0q3NO7KV8t3sqXi3KcjiNSoQKhDntzqBZL2EtJiOPtYV0xxnDtuFls3H3I6UhSTkpsUFhre1tr2xTz8SmwzRhTB8D7udix3tbaLd7P24GPgS7em3y6v/e+Y6y1na21nWvUqFGan1FEKkjlmEjGDElnSEYqr36/hjsnzOdIfqHTsUJSINRi1WEpzs1nptG6bjUenriY3EP5TscRqTCBUIe991UtFsHTJH97WBeOFBQyeNxMLeAeIso6xWMiMNR7eSjw6YkHGGMqG2OqHrsM9AWyfb2/iAS2yAgXjw5ozUMXtmRSdg6XvDidpycvY+56LVzkR6rF4pioCBf//EM7dh/M4x+TljgdR8QpqsMiDmhRuxrjb+jCnoP5XDNuJjsPHHU6kpRRWRsUTwF9jDErgT7e6xhj6hpjJnmPqQVMN8YsBGYDX1hrvzrV/UUkuBhjGHZmGn/s24wV2w4w+tvVDB47U00K/1EtFke1qRfPzWem8X7WJn5atdPpOCJOUB0WcUj7+tV5/foz2Lz3MENem63RfEEusix3ttbuAnoV8/UtQH/v5TVA+9LcX0SCk8XgMuC2cLTAzReLckhP1VbuFU21WALBPb2bMnnxVv780S9Mvucs4qLLdIohElRUh0Wc1aVRImOGdGbY+Cyu/+9s3rrJs4imBJ+yjqAQETkuIy2J6EgXLuO5/kHWRrI3a/snkXAQGxXBk5e3ZePuwzz79Qqn44iISJg5q1kNXhjckV825TJs/Bytixak1KAQkXKTnppA5rAMRvZtzouDO1ItNopBY2YyZ91up6OJiB9kpCUxuGsDXv9pLQs27nU6joiIhJnzW9fm2avaM2vtbm57ey55BW6nI0kpqUEhIuUqPTWB289twkXt6vLBrd2oUTWGIa/N4ocV2gpNJBz8+YIW1Kgaw5/+94tODEVExO8GdKjHE5e15dvlOxj6+ixe+Gal1kULImpQiEiFqVu9Eu/f2o1GyVUYNj6Lr7JznI4kIhWsWmwUj1/aluXb9vPK96udjiMiImFoUJcG3NCjITPW7OaZr1dwzTgt3h4s1KAQkQqVXCWGd2/OoE29aozInMf/5m5yOpKIVLA+rWpxUbs6vPjNKlZt3+90HBERCUPJVWLwLovG0Xw3M9fscjSP+EYNChGpcPFxUbx1U1e6N07m/g8WMv7ndU5HEpEK9sglrYmLieD//vcLhW7rdBwREQkzGWlJxER5/ty1wGEtmhkU1KAQEb+oHBPJuKGd6duqFn+buJgXv1mJtfqjRSRUJVeJ4a8XtmLehr0MfzNLQ2tFRMSvji3efl+fZrStV41XvlvNz6t2Oh1LSqAGhYj4TWxUBC9d04nLO9bj31+v4Kkvl6lJIRLCGibF4TIwbdl2Bo/V/F8REfGv9NQE7urVlLeHZZBWozK3vD2Xlds09TCQqUEhIn4VGeHi31e2Z0hGKq/+sIa/fJKt4d8iIWrm2l+3GD5aoPm/IiLijPhKUbx+/RnERkVw/Rtz2L7/iNOR5CTUoBARv3O5DI8OaM1t5zTmnVkbuO/9BeQXajtCkVCTkZZEdKTr+CJl1StFOZpHRETCV0pCHK8PPYPdB/MYNj6LQ3kFTkeSYqhBISKOMMbwp34t+L9+zfl0wRZue3suR7R4kUhIOTb/9+7eTUmMi+bdORtxa8SUiIg4pG1KPC8M6kj25lzumjBfo3gDkBoUIuKoEec04bFL2zBt2XZueGMO01ftZPS3qzRXXSREpKcmcE/vZjx8cSsWbc7VVsMiIuKo3q1q8cglrZm6dDuPfrZY66EFmEinA4iIDMlIpUpMBCPfX8jM13ZhgOhIF5nDMkhPTXA6noiUgwEd6vLWzPX8a/Iy+rWtTbVYTfcQERFnXNetIRt2HWLc9LU0SKrMTT0bOR1JvDSCQkQCwmUdUxjQvi7WgttCvhbUEwkpxhj+dnErdh3M48VvVjkdR0REwtyD/VvSr3VtHv9iCV9lb3U6jnipQSEiAePabg2JivAsp+cG2qXEOxtIRMpVu5TqXJmewhs/rWXNjgNOxxERkTDmchn+M7ADHepX55735jN/g6YXBwI1KEQkYKSnJvDu8G5c1qEuLuD5aSs5eFQrLIuEkvvPb05MZASPf7HU6SgiIhLmYqMiGHtdZ2pWjWXY+Cw27DrkdKSwpwaFiASU9NQERg3syAuDOzFvw15uGj+Hw3na3UMkVNSsGstdvZrwzbLtfLt8u9NxREQkzCVXieGNG86gwG25/r+z2Xsoz+lIYU0NChEJSP3b1uHZq9oze+1ubn4zS1uQioSQ67s3olFyZR77fAl5BW6n44iISJhrXKMKY4aks2n3YW55ay5HC3Te6RQ1KEQkYA3oUI9/XdGen1bv5Na39WIhEiqiI1389aKWrNlxkDdnrHM6joiICF3Tknj6ynbMWrubP/3vF20/6hA1KEQkoF2RnsITl7Xlu+U7uD1znt5tFQkR5zavydnNavDc1JXsPHDU6TgiIiIM6FCPP57fnE8WbGHUlBVOxwlLalCISMAb1KUBjw1ozdSl27lrwnzyC9WkEAl2xhj+elErDucX8u/Jy52OIyIiAsCIcxpzdef6PP/NKp6evIzR365i7nrt8OEvalCISFAY0q0hf72oFV8t3sp97y+k0K1hdyLBrknNKgzt3pD3sjaSvTnX6TgiIiIYY3j8sja0T4ln9Lereebr5VwzbqaaFH6iBoWIBI2bejbigQta8NnCLfzxAzUpRELBXb2akhgXzSMTF2u+r4iIBISoCBfnNK8BgNtCXoGbmWt2OZwqPKhBISJB5ZazG3N/32Z8NH8zD360CLeaFCJBLb5SFPef35ys9Xv47Jccp+OIiIgAcFazmsREev5ctkCnBtWdDRQm1KAQkaBzx3lNueu8JryXtZG/fpqtd11FgtxVnevTum41npy0lEN5BU7HERERIT01gXduzuCyDnWxFj6ev1nnnH6gBoWIBKV7+zTj1rMbkzlrA3//bIleMESCWITL8MglrcnJPcIr369xOo6IiAjgaVKMGtiRO89rwvtZm3hn9ganI4U8NShEJCgZY/hTv+bc1LMR//15HU9MWqomhUgQO6NhIhe3r8ur369m055DTscRERE57p7ezTi7WQ0embhYi2VWMDUoRCRoGWN46MKWXNctlbE/rmXk+wsZ/e1KvXCIBKkHLmiBMfDkpGVORxERETkuwmV4bmAH6sRXYkTmXLbvP+J0pJBVpgaFMSbRGDPFGLPS+zmhmGOaG2MWFPnYZ4y5x3vbI8aYzUVu61+WPCISfowxPHJxa/q0rMVH8zfz78krwm4rKNViCRV1q1fitrOb8MWiHGas1mrpEjxUh0VCX/W4aF65Np3cw/nckTmf/EK305FCUllHUPwZmGatbQpM817/DWvtcmttB2ttByAdOAR8XOSQUcdut9ZOKmMeEQlDLpehff14wLPK8tH8sNsKSrVYQsYtZ6dRr3ol/v7ZYm0lLMFEdVgkDLSqW41//qEds9ft5h9fLHU6Tkgqa4NiADDee3k8cGkJx/cCVltr15fxcUVEfqNb42Rio37dCirM1qNQLZaQERsVwYP9W7Js634maDEyCR6qwyJhYkCHetzYw7MG2kfzNjkdJ+SUtUFRy1qbA+D9XLOE4wcCE0742h3GmF+MMa8XNxxORMQX6akJZA7L4N7eTWleuyrPT1sVTkPEVYslpPRvW5uujRJ55uvl5B7KdzqOiC9Uh0XCyAP9W9C1USIPfLSI7M25TscJKSU2KIwxU40x2cV8DCjNAxljooFLgA+KfPlloDHQAcgBnjnF/YcbY7KMMVk7duwozUOLSJhIT03g7t7NeG94Bg2S4hj+ZhZLc/Y5HatcBEItVh0WfzHG8LeLW5N7OJ9RU1c4HUcECIw67L2/arGIw6IiXLw4uBMJcdHc+vZc9hzMczpSyCixQWGt7W2tbVPMx6fANmNMHQDv5+2n+FYXAPOstduKfO9t1tpCa60bGAt0OUWOMdbaztbazjVq1PD15xORMFQ9LprxN3ahckwk178xOyS2LAyEWqw6LP7Uqm41BnVpwJsz1vHoZ9rWTZwXCHXYe6xqsUgAqFE1hpev7cT2fUe56935WjepnJR1isdEYKj38lDg01McO4gThrIdK+RelwHZZcwjIgJAveqVGH9jFw7lFTL09dmh3tlWLZaQ1LtVLdwWXv9pXdjtziNBR3VYJAx1bJDA3we05seVO3nm6+VOxwkJZW1QPAX0McasBPp4r2OMqWuMOb76sDEmznv7Ryfc/1/GmEXGmF+Ac4F7y5hHROS45rWrMva6zmzcc5ibxs/hcF6h05EqimqxhKQlW/ZhvJfzCsJudx4JLqrDImFqUJcGDOpSn5e+W81X2TlOxwl6kWW5s7V2F55ViE/8+hagf5Hrh4CkYo4bUpbHFxEpSUZaEs9d3YER78zjzgnzeOXadCIjytqbDSyqxRKqMtKSiIl0caTAs9d8RqNEhxOJFE91WCS8PXJJa5bk7Gfk+wtpUrMKTWpWdTpS0Aqts3QRkWJc0LYOj17SmqlLt/PQJ9nhtgWpSNBKT00g8+YMerWoidvCofyQHQUlIiJBLCYygleu7USl6AiGvzWX/Ue0A9XpUoNCRMLCkG4NuePcJrw7ZyOjpq50Oo6I+Cg9NYGXru1EveqV+Pfk5WowiohIQKoTX4kXB3di/a5D3Pf+QtxaNPO0qEEhImFjZN9mXNU5heenreTtmeudjiMiPoqJjOCuXk1YuCmXKUu2lXwHERERB2SkJfFg/5ZMWbKNl75b5XScoKQGhYiEDWMMT1zWlvNa1OThT7P5Knur05FExEd/6JRCo+TKPPP1Cr0rJSIiAevGHg0Z0KEu//56BSM/WKAdqEpJDQoRCSuRES5eHNyRdinVuevd+cxZt9vpSCLig8gIF/f2acbybfv57JctTscREREpljGGqzvXxwAfzt3M4LHaJrs01KAQkbATFx3J69efQUr1Stz03zms2Lbf6Ugi4oOL2tahRe2qjJqygvxCt9NxREREijV/416Md5/sowVuZqze6WygIKIGhYiEpcTK0Yy/sQsxUREMfX02W/YedjqSiJTA5TKM7NucdbsO8eHcTU7HERERKVZGWhLRkS68PQqdZ5aCGhQiErbqJ8Yx/oYuHDhSwNDXZ5N7SFtCiQS63i1r0r5+dZ6ftpKjBdp2VEREAk96agKZwzK4//xm9GiSzHtZmzTNw0dqUIhIWGtVtxqvXpfO+l2HGDhmBs9NXaEXEJEAZozhj32bsyX3CO/M2uB0HBERkWKlpyZw+7lNefnaTtSJj+Xud+ez74jeDCuJGhQiEva6N07mjvOasHTrfkZNXck147SYkUgg69EkiYy0REZ/u4pDeQVOxxERETmparFRPDewIzm5R3jo42ys1U5Up6IGhYgIEOEyx+cJHs13M3PNLkfziMjJGWP44/nN2Xkgj//+vM7pOCIiIqeUnprAPb2aMnHhFj6ct9npOAFNDQoRETyLGcVEeUqiBdxudbdFAll6aiLnNq/Bq9+vIfewhsyKiEhgG3FuE7o2SuThT7NZu/Og03EClhoUIiL8upjRfX2a0qxWFUZ/t4rszblOxxKRUxjZtzm5h/N57cc1TkcRERE5pQiXYdTVHYiKcHHXhPnkFWi77OKoQSEi4pWemsBdvZqROSyDxLhobn4zi+37jjgdS0ROok29ePq3rc1r09ey68BRp+OIiIicUt3qlfjnH9qxaHMuz3y93Ok4AUkNChGRE9SoGsPYoZ3Zeyif4W/N5Ui+tjIUCVT39WnG4fxCXvl+tdNRREREStSvTW0Gd23Aqz+s4ceVO5yOE3DUoBARKUbruvGMuro9Czbu5YGPFmnFZZEA1aRmVS7tWI83Z6xnm0Y8iYhIEPjrha1oWrMK972/UCMAT6AGhYjISfRrU4eRfZrx8fzNvPK95riLBKp7ejWj0G154ZuVTkcREREpUaXoCJ4f1JHcw/n88X+/6I2wItSgEBE5hTvOa8LF7evyr8nLmLJkm9NxRKQYDZLiGNilPu/O3sjG3YecjiMiIlKilnWq8eAFLfhm2XZtmV2EGhQiIqdgjOHpK9rRtl4897w7n2Vb9zkdSUSKced5TYlwGf4zVaMoREQkOAzt3pBeLWry5KRlLNmic0xQg0JEpESxURGMGdKZyjGRDBufpbmCIgGoVrVYruuWysfzN7Fq+36n44iIiJTIGMO/rmhH9bgo7pwwj8N5WphdDQoRER/Ujo9lzHWd2bH/KLe9PU97V4sEoNvOaUKlqAhGTdEoChERCQ5JVWJ49qoOrNl5kEc/X+J0HMepQSEi4qMO9avzryvaMXvdbv76SbYWNBIJMImVo7mpZyO+WJRD9uZcp+OIiIj4pGfTZIaflcaE2Rv4KjvH6TiOUoNCRKQUBnSox53nNeG9rI288dM6p+OIyAmGnZVGfKUonp2ywukoIiIiPhvZpzntUuL504eL2LL3sNNxHKMGhYhIKd3buxnnt67F418s4bvl252OIyJFVIuN4paz0/hm2Xbmrt/jdBwRERGfREe6eH5gRwoK3dzz3gIK3eE5UlcNChGRUnK5DM9e1YHmtatx5zvzWbX9gNORRKSI67s3JLlKDP+evNzpKCIiIj5rmFyZRwe0Yfba3fzl40WM/nZV2DXb1aAQETkNlWMiGXtdOjFRLoaNn8PeQ3lORxIRr7joSG4/tzEz1uzip1U7nY4jIiLis8s71ePMpsm8O2cjz3y9nGvGzQyrJoUaFCIipyklIY5Xh6SzZe8Rhrw2i4gqSbWdziQiHoO7NiC5cjT3f7CQuet2Ox1HRETEJ8YYOjWoDoDbQn6Bm5lrdjmcyn/UoBARKYP01ERuPqsRizbvI6JyQj2n84iIR/bmfeQeyScn9wiDxobXu08iIhLczmpWk+gI75/qxpCRluRsID9Sg0JEpIzioiMxgOc/IhIIZq7ZdXyBsbxCG1bvPomISHBLT01gwvAM2qXE43ZbTBidY5apQWGMudIYs9gY4zbGdD7Fcf2MMcuNMauMMX8u8vVEY8wUY8xK7+eEsuQREXFCRloSMVEusNaR5ZZVi0V+LyMtiehI1/G+YWLlaEfzSGhTHRaR8paemkDmsK7UiY/ljx8s5Eh+odOR/KKsIyiygcuBH052gDEmAhgNXAC0AgYZY1p5b/4zMM1a2xSY5r0uIhJUPC8gGRQe3LvFoQiqxSInOPbv8u5eTakcHcG3y7QlsFQo1WERKXdVY6N48g/tWL3jIM9NW+l0HL8oU4PCWrvUWlvSHl5dgFXW2jXW2jzgXWCA97YBwHjv5fHApWXJIyLilPTUBAoP7NrqxGOrFosULz01gXv6NOOmM9P4esk2Vm3f73QkCVGqwyJSUc5uVoOrOqfw6verWbhxr9NxKlykHx6jHrCxyPVNQFfv5SADEJwAAAbVSURBVFrW2hwAa22OMabmyb6JMWY4MNx79agxJrsiwpZBMhBoe5kpk2+UyTeBlinQ8gA0dzrAKZS5FqsOnxZl8o1fMjX9Z6kOD9vnqZQCMVOg1mKdEztHmXyjTCVzNE+Hp4r9cqA9R1CGOlxig8IYMxUobuu8v1hrP/XhMYpb0qPU87SttWOAMd5MWdbak87vc4Iy+UaZfKNMJQu0PODJVIHf2/FarDpcesrkG2XyjTL5pqJqcSDUYVAtPh3K5BtlKlmg5YHAzXS69y2xQWGt7X2639xrE1C/yPUU4Ng87W3GmDreTnEdQBNERUSKoVosIuIs1WERkYrnj21G5wBNjTGNjDHRwEBgove2icBQ7+WhgC/dZxERKT3VYhERZ6kOi4iUoKzbjF5mjNkEdAO+MMZM9n69rjFmEoC1tgC4A5gMLAXet9Yu9n6Lp4A+xpiVQB/vdV+MKUvuCqJMvlEm3yhTyQItDziUyaFarOffN8rkG2XyjTL5xu+ZdE78G8rkG2XyTaBlCrQ8EGKZjLWlnvomIiIiIiIiIlKu/DHFQ0RERERERETklNSgEBERERERERHHBWyDwhjzujFm+8n2djYezxtjVhljfjHGdAqATOcYY3KNMQu8Hw/7IVN9Y8y3xpilxpjFxpi7iznGb8+Vj3n8+jwZY2KNMbONMQu9mf5ezDF+/X3yMZPff5+8jxthjJlvjPm8mNv8/u/Oh0xO/LtbZ4xZ5H28322j5NTzVBECrRarDpdrJtXiAK3FqsM+ZwqLWhxoddjHTGFfi1WHyzWTzol9yxQa58TW2oD8AM4COgHZJ7m9P/Alnj2lM4BZAZDpHOBzPz9PdYBO3stVgRVAK6eeKx/z+PV58v7cVbyXo4BZQIaTv08+ZvL775P3ce8D3inusZ34d+dDJif+3a0Dkk9xuyPPUwX9rAFVi1WHyzWTanGA1mLVYZ8zhUUtDrQ67GOmsK/FqsPlmsmpGqNaXHKecq/DATuCwlr7A7D7FIcMAN60HjOB6sazb7STmfzOWptjrZ3nvbwfz6rQ9U44zG/PlY95/Mr7cx/wXo3yfpy4Oqxff598zOR3xpgU4EJg3EkO8fu/Ox8yBSK/P08VJdBqsepwuWbyK9Vi36gOl6uQqMWBVod9zOR3gVaLVYfLNZPfqRaXm1I/TwHboPBBPWBjkeubcPgfvVc37xClL40xrf35wMaYhkBHPJ3Hohx5rk6RB/z8PHmHQy0AtgNTrLWOP0c+ZAL//z79B/g/wH2S2534XSopE/j/ebLA18aYucaY4cXcHqj1qSIE4s+qOuxbJlAtDsRarDrsO9Vij0D9OVWLS84DqsOBWIdBtdhX5V6Hg7lBYYr5mtPdtnlAqrW2PfAC8Im/HtgYUwX4ELjHWrvvxJuLuUuFPlcl5PH782StLbTWdgBSgC7GmDYnRi7ubg5n8uvzZIy5CNhurZ17qsOK+VqFPU8+ZnLi310Pa20n4ALgdmPMWSfcHoj1qaIE2s+qOux7JtVi3zL57XlSHS411WKPQPw5VYt9y6M67FsmnRMHbi0u9zoczA2KTUD9ItdTgC0OZQHAWrvv2BAla+0kIMoYk1zRj2uMicJT+DKttR8Vc4hfn6uS8jj1PHkfby/wHdDvhJsc+306WSYHnqcewCXGmHXAu8B5xpi3TzjG389TiZmc+H2y1m7xft4OfAx0OeGQgKtPFSigflbVYd8zqRb7lsnPz5PqcCmoFh8XcD+narFveVSHfcukc2LfMoXKOXEwNygmAtcZjwwg11qb42QgY0xtY4zxXu6C5/ndVcGPaYDXgKXW2mdPcpjfnitf8vj7eTLG1DDGVPdergT0BpadcJhff598yeTv58la+4C1NsVa2xAYCHxjrb32hMP8+jz5ksmB36fKxpiqxy4DfYETVzEPuPpUgQLqZ1Ud9j2TanHg1WLVYd+pFv9GwP2cqsWqw+WZSefEgVmLK6oOR1ZI2nJgjJmAZyXSZGPMJuBveBZNwVr7CjAJz6qgq4BDwA0BkOkK4DZjTAFwGBhora3oIXY9gCHAIuOZuwXwINCgSC5/Ple+5PH381QHGG+MicDzD/V9a+3nxphbi2Ty9++TL5mc+H36HYefJ18y+ft5qgV87K3/kcA71tqvAvF5Kg+BVotVh8s1k2pxkNTiQKwvAfAchU0tDrQ67GMm1WLV4fLM5HgdBtXiYlRIHTYO/L8VEREREREREfmNYJ7iISIiIiIiIiIhQg0KEREREREREXGcGhQiIiIiIiIi4jg1KERERERERETEcWpQiIiIiIiIiIjj1KAQEREREREREcepQSEiIiIiIiIijvt/BQW1MWcKHX4AAAAASUVORK5CYII=\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "from scipy.optimize import fsolve\n", "\n", "phi = lambda t,y : 3*y/t-5/t**3\n", " \n", "t0 = 1\n", "tfinal = 5\n", "y0 = 1\n", "\n", "\n", "def EE(phi,tt,y0):\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 EI(phi,tt,y0):\n", " h = tt[1]-tt[0]\n", " uu = [y0]\n", " for i in range(len(tt)-1):\n", " uu.append(fsolve(lambda x : -x +uu[i]+h*phi(tt[i+1],x),uu[i]))\n", " return uu\n", "\n", "def EM(phi,tt,y0):\n", " h = tt[1]-tt[0]\n", " uu = [y0]\n", " for i in range(len(tt)-1):\n", " utilde=uu[i]+h/2*phi(tt[i],uu[i])\n", " uu.append(uu[i]+h*phi(tt[i]+h/2,utilde))\n", " return uu\n", "\n", "def CN(phi,tt,y0):\n", " h = tt[1]-tt[0]\n", " uu = [y0]\n", " for i in range(len(tt)-1):\n", " uu.append(fsolve(lambda x : -x +uu[i]+h/2*(phi(tt[i],uu[i])+phi(tt[i+1],x)),uu[i]))\n", " return uu\n", "\n", "def HE(phi,tt,y0):\n", " h = tt[1]-tt[0]\n", " uu = [y0]\n", " for i in range(len(tt)-1):\n", " utilde=uu[i]+h*phi(tt[i],uu[i])\n", " uu.append(uu[i]+h/2*(phi(tt[i],uu[i])+phi(tt[i+1],utilde)))\n", " return uu\n", "\n", "NN=[20,100,300,5000]\n", "\n", "figure(1, figsize=(18, 12))\n", "\n", "subplot(2,3,1)\n", "for N in NN:\n", "\ttt=linspace(t0,tfinal,N)\n", "\tuu=EE(phi,tt,y0)\n", "\tplot(tt,uu,'.-')\n", "plot(tt,[sol_exacte(t,y0) for t in tt],'c-',lw=2)\n", "legend(['N=20','N=100','N=300','N=5000','Exacte'])\n", "title('$y_0=1$ - Exacte vs EE')\n", "axis([t0,tfinal,-c,c]);\n", "\n", "subplot(2,3,3)\n", "for N in NN:\n", "\ttt=linspace(t0,tfinal,N)\n", "\tuu=EI(phi,tt,y0)\n", "\tplot(tt,uu,'.-')\n", "plot(tt,[sol_exacte(t,y0) for t in tt],'c-',lw=2)\n", "legend(['N=20','N=100','N=300','N=5000','Exacte'])\n", "title('$y_0=1$ - Exacte vs EI')\n", "axis([t0,tfinal,-c,c]);\n", "\n", "subplot(2,3,4)\n", "for N in NN:\n", "\ttt=linspace(t0,tfinal,N)\n", "\tuu=EM(phi,tt,y0)\n", "\tplot(tt,uu,'.-')\n", "plot(tt,[sol_exacte(t,y0) for t in tt],'c-',lw=2)\n", "legend(['N=20','N=100','N=300','N=5000','Exacte'])\n", "title('$y_0=1$ - Exacte vs EM')\n", "axis([t0,tfinal,-c,c]);\n", "\n", "subplot(2,3,5)\n", "for N in NN:\n", "\ttt=linspace(t0,tfinal,N)\n", "\tuu=HE(phi,tt,y0)\n", "\tplot(tt,uu,'.-')\n", "plot(tt,[sol_exacte(t,y0) for t in tt],'c-',lw=2)\n", "legend(['N=20','N=100','N=300','N=5000','Exacte'])\n", "title('$y_0=1$ - Exacte vs HE')\n", "axis([t0,tfinal,-c,c]);\n", "\n", "subplot(2,3,6)\n", "for N in NN:\n", "\ttt=linspace(t0,tfinal,N)\n", "\tuu=CN(phi,tt,y0)\n", "\tplot(tt,uu,'.-')\n", "plot(tt,[sol_exacte(t,y0) for t in tt],'c-',lw=2)\n", "legend(['N=20','N=100','N=300','N=5000','Exacte'])\n", "title('$y_0=1$ - Exacte vs CN')\n", "axis([t0,tfinal,-c,c]);\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Problème bien conditionné" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "On dit qu'un problème de Cauchy est bien conditionné si les méthodes numériques usuelles peuvent donner sa solution en un nombre raisonnable d’opérations. \n", "
\n", "\n", "Dans le cas contraire on parle de **problème raide**. \n", "\n", "\n", "### Exemple de problème mal conditionné (= raide = *stiff*)\n", "On se donne $\\varphi(t,y)=-\\beta y$ et $y(0)=1$. \n", "On cherche une fonction $y\\colon t\\in\\mathbb{R}\\mapsto y(t)\\in\\mathbb{R}$ qui satisfait\n", "$$\n", "\\begin{cases}\n", "y'(t) = -\\beta y(t), &\\forall t\\in\\mathbb{R},\\\\\n", "y(0) = 1.\n", "\\end{cases}\n", "$$\n", "Ce problème est \n", "- **mathématiquement bien posé** pour tout $\\beta\\in\\mathbb{R}$ : il existe une et une seule solution, elle définie sur $\\mathbb{R}$ et est donnée par $y(t)=e^{-\\beta t}$;\n", "- **numériquement bien posé** pour tout $\\beta>0$ : l'erreur sur la solution sera au pire de l'ordre de l'erreur sur la condition initiale;\n", "- **mal conditionné** ou **stiff**: si nous cherchons à résoudre le problème de Cauchy lorsque $\\beta$ est très grand, il faut prendre un pas $h$ très petit et ce quel qu'il soit le schéma choisi car la solution devient de plus en plus raide (*stiff*).\n", "\n", "Voici ci-dessous un exemple avec des valeurs de $\\beta$ de $1$ à $100$:" ] }, { "cell_type": "code", "execution_count": 93, "metadata": {}, "outputs": [ { "data": { "image/png": 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ImxDidx9V8Lm7i7hieJDZeQEa/5+JrsrKSk+9Rl7LA97M1EkRqzM4dq2VH27cdPGyD6jbHhfZ9B3gtffMa3nAe5m8lqeLHNmnBfsOZn3JRoqOemj3ePE981om5XGUY3X28foNFNV84kTmTvHa+6Y87fNips4Kp5E7VgfT9uuQzwErgdOBQcDLxpil1tr9R/0lax8BHgEYNmyYLSws7HBgJxUVFVFYWEghcMU5h/nOM8X8fnU52xp6cfvFY8hMDrmSxyu8lge8mamTIlZncOxaq2+w/Ocbi4jPyqewcHiE44fPa++Z1/KA9zJ5LU8XObJPC/UbYnPy+7taW6158T3zWiblcZRjdVZQMIjC2QUORO4cr71vytM+L2bqrHBOrSwF8lvdzqPx25PWrgH+ahuVAJsBb+zNOqlfWgJPfG0qPzxvBK9v2M3n7lmqiVDESY7Xmd9nyOuVwJZynTIsPZojtWbQqZUirTi2T9M1ciItwmnk3gGGGGMGNl2EejmNQ+GtbQXOADDG9AGGAZsiGdQNPp/h2lkFPH/DDLKSg1zz+Dv88LmPOFxT73Y06X6iUmcDs5LYrGs/pWdzpNYMWn5ApBXH9mkNmrVS5Ih2GzlrbR1wA/AisBZ42lq72hgz3xgzv2mz/wFOMcZ8BLwC3Gyt3eNU6Ggb3jeVBTfMYN7sAv60fCvn3beUD0v3uR1LupFo1VlBdmMjp4vFpadyqtaM0ayVIs2c3KfVaeRb5IhwrpHDWrsQWNjmZw+3+nMZcHZko3lLKODnB3NGUDgsm28/XcxFD77FTWcM4ZuFg4jz99h11SWColFnA7OSqa5roKziMHm9ErvyUCIxy6la0zpyIi2cqjOtIyfSQh1IB50yKItF35rNnDH9uOvl9Vz04Fus+/SA27FEwlKQnQSg0ytFIswYnVopEg31DaozkWZq5DohLTGOe+dO4KErJ1K27zDn3/cGDywpoU7fxorHFWQ1NnKbdquRE4mkxslOtA8QcZIBVGYiLdTIdcG5Y/rx0v+bzVmj+nDHi+u46KG32LBTo3PiXdkpIZJDATbtrnQ7iki3oslORKJDI3IiLdTIdVFmcogHrpjIA1dMpHTvYc679w0eKtqo0TnxJGMMBdlJbNKplSIRZQzU6P99EcepzERaqJGLkPPGNo7OnTGiN7cv+piLH36bkl0anRPvGZiVpFMrRSLMANW1OsIUcZIxGpETaU2NXARlJYd48MqJ3Dd3AlvLDzLn3je4/9UNOt1GPKUgK5myisNU1Wo9RJFIMcZQVaeaEnGaZq0UaaFGLsKMMZw/LoeX/t+pnDWyD3e+tJ4v3P8GH2zd63Y0EQAGZidhLXxSrlE5kUgxoC9HRBzWONmJGjmRZmrkHJKd0njt3G+/Ool9h2q56KG3+MnfV3Owus7taNLDNc9cuVmnV4pEjBYEF4kONXIiLdTIOeyskX14+T9m8+WpJ/H4W59w9t2vs2TdLrdjSQ82sHkJAk14IhIxGpETcZ4B6tTIiRyhRi4KUuLj+J8vjuaZ66aTEPRzze/f4VtPfUB5ZbXb0aQHSgoF6Jsaz0YtQSASMcZAlSY7EXFcgxo5kSPUyEXRpAEZ/OOmmXzrjCEs/GgHZ971Gn95rxSrC3clygqyNXOlSCT50IiciNOMgXodMokcoUYuykIBP//vrKH846ZZFGQn851nirnskWWs10LiEkWDspPZuKtSXyKIREjzNXKqKRFnafkBkRZq5FwytE8Kz1w3ndsuGsP6nQeY8+ul/GLhWk2GIlExtE8yB6rr+HR/ldtRRLoF0/S7JjwRcZYmOxFpoUbORT6f4fIp/Xn124VcPDGP37y+iTPveo1Fq3boW11x1NA+KQCs36nr5EQiwTR1cjq9UsQ5Wn5A5Ghq5DwgIynI7ZeM5dlvTictIY75f3yfax5/h12H9M2uOONII/epTukViYTmETlNeCLiLDVyIi3UyHnIySdl8MKNM7nl8yN5Z/Nn/OCNw/x68QZ9wysR1yspSFZySNdmikSIRuREnGeMlh8QaU2NnMcE/D6+PnMgr3y7kIm9/dy9eD1n3/06L67+VKdbSkQN65usRk4kQo6MyNWpkRNxUp2mrRQ5Qo2cR/VNi+ffxsfzp2unEh/n47on3uMrv1uhA2+JmCG9U9iwq1Jr8ohEQMuInE6tFHFK44LgqjGRZmrkPG7G4CwW3jSLn3xhFB9tr+DcXy/lx8+vpuJQrdvRJMYN65vCoZp6tu877HYUkZjnaxqTq9aplSKOMcZQoxE5kSPUyMWAgN/HVacMYMl3Cpk7JZ8/vP0JhXcu4YllW6ir1zdT0jlD+yQDaJRXJAKOjMhp+QERR+m4R6SFGrkYkpEU5KdfHMM/bprFsL4p3PLcKj5/3xu8vbHc7WgSgwb31hIEIpHSMmulRuREnGLQNXIiramRi0Ej+qXy5Dem8dCVEzlQVcfc3y5j/hPvsXnPQbejSQxJS4ijX1q8RuREIkCzVoo4zwC1ukZO5IiA2wGkc4wxnDumH6cN781vX9/Ew69tZPHanVw5tT83nTGEzOSQ2xElBgzpk6JGTiQCmkfkqjXZiYhzjEbkRFoLa0TOGHOOMWadMabEGPP942xTaIxZaYxZbYx5LbIx5Xji4/zceMYQir57GpdNzuePy7dSeEcRDxaV6JvhGONGnQ3tncyGXZVaYFV6FCdqreUaOf2/KwIO1Rm6Rk6ktXYbOWOMH3gAOBcYCcw1xoxss0068CDwBWvtKOBLDmSVE8hOCfGzC8fw4r/PYmpBBr9ctI7T7yzi2fdKNb18DHCrzob3S6WmroFNu3WdnPQMTtWarpETaeFknWnWSpEW4YzITQFKrLWbrLU1wFPABW22uQL4q7V2K4C1dldkY0q4BvdO4dGrJvPkN6aRmRzi288Uc/79b/BmyR63o8mJuVJno3JSAVizY39XH0okVjhSa1pHTuQozuzTjNaRE2ktnEYuF9jW6nZp089aGwr0MsYUGWPeM8Z8NVIBpXOmD8pkwfUz+PXl49l3qJYrH13OVY+tYNX2CrejybG5UmeDeycTDPhYXaZGTnoMR2rNAH6f0YicSCPH6kzXyIm0CGeyE3OMn7WtogBwMnAGkAC8bYxZZq1df9QDGTMPmAeQnZ1NUVFRhwM7qbKy0lOZIpEnDbh1suGVrUFe2LSbz9+3m8l9/Vw4OEhOcscmLfXa6wPezNRJEasz6Fit5STCG6s/oShxZyejd4zX3jOv5QHvZfJani5ybJ+WYSwlm7dQVPSpA7E7xovvmdcyKY+jHKmzlN75VNfWeep18tr7pjzt82KmzgqnkSsF8lvdzgPKjrHNHmvtQeCgMeZ1YBxwVDFaax8BHgEYNmyYLSws7GRsZxQVFeGlTJHMczbwX1W1PLp0M79buon33jzMxRPz+NaZQ8jrlRj1PJHixUydFLE6g47V2rTyD1m0+lNOPfVUjDnWvjeyvPaeeS0PeC+T1/J0kWP7tKT4IFl9+1JYOMax8OHy4nvmtUzK4yhH6qz3SUNtA8ZTr5PX3jflaZ8XM3VWOEMy7wBDjDEDjTFB4HLg+TbbLABmGWMCxphEYCqwNrJRpatS4+P4j7OG8vr3TuOaGQNZUFzG6Xe+xo+fX83uA9Vux+vpXKuzUTmp7DtUS1lFVVcfSiQWOFZr8XF+nVop0siROjMG6hss1ur0ShEIY0TOWltnjLkBeBHwA49Za1cbY+Y33f+wtXatMWYR8CHQADxqrV3lZHDpvMzkELd8fiRfnzmQ+17dwBPLtvB/72zjazMHMG/2INIS4tyO2OO4WWcjc9IAWL29gtz0hK4+nIinOVlroTgf1XWaiEHE6X1abb0lGHD+DBIRrwtrQXBr7UJgYZufPdzm9h3AHZGLJk7LSU/gFxeN5RuzCrh78QYeWLKRJ97ewrWzCrh6xgBS49XQRZNbdTaiXwrGwOqy/Zw9qm8kH1rEk5yqtfiAn2qNyIkAztRZc+tWW99AMNCx6/xFuiNVgVCQncx9cyfwj5tmMmVgJne9vJ6Zt73KrxdvoOJwrdvxxGGJwQAFWUmauVKki+LjfFp+QMRBzZdxa+ZKkUZq5OSIUTlpPHrVJF64cSZTCzK5e/F6Zt7+KvcsXq+GrpsblZPGmjItTSHSFbpGTiQ6arWWnAigRk6OYXRuGr/9amNDN70gk3sWb2Dm7a/ytw01aui6qVE5qZRVVLH3YI3bUURiVnycn6o6NXIiTmk+tVIjciKN1MjJcY3OTeORr07iHzfN5JRBmSzYWMvM217lrpfXU3FIDV13Mia3ccKTD7VgvEinJcT5OVyjRk7EKc2nVtbWa0ROBNTISRhG5aTxm69M4r9PiWfG4CzufWUDp9z2Cr9YuJZd+zVlfXcwJi8NY2Dl1n1uRxGJWYlBNXIi0VDXoBE5EQhz1koRgP6pfr76hZNZu2M/DxVt5LdLN/H7Nz/h4pPzmH9qASdlJrkdUTopJT6OwdnJrNy21+0oIjErKRTgoBo5Ece0nrVSRDQiJ50wol8q986dQNF3TuNLk/J49v1STruziBuf/IA1mvkwZk3on87Kbfu00KpIJyUG/RyqqXM7hki3pVMrRY6mRk46rX9mIj+7cAxvfO80vjG7gCUf72LOvUu55vcreOeTz9yOJx00Pr8Xew/VsvWzQ25HEYlJSaEAtfWWGi0KLuIoTXYi0kiNnHRZ79R4/vPcEbz5/dP5ztlDKS6t4EsPv82XHn6LxWt20qBz2WPC+Px0AFZu03VyIp2RGPQDaFROxCFHZq3U8gMigBo5iaC0hDhuOH0Ib958Oj/5wijK9lVx7R/e5cy7XuOPy7ZoEgCPG9onmYQ4Px9owhORTkkKNl52ruvkRJxhmlq5Wo3IiQBq5MQBCUE/V50ygNe+W8h9cyeQEh/gh8+t4pTbXuFXL61j1wHNdOlFAb+PMXlpfKAROZFOSQw1jsgdrNaInIgjmobkdGqlSCM1cuKYgN/H+eNyeO76GTwzfzqTB2Rw/5ISZt62hO8+U8zHn2piFK+Z0D+dtWX7qdaixiIddmRETo2ciCM0a6XI0bT8gDjOGMPkARlMHpDBJ3sO8tibm3nm3VKeea+UWUOyuHZWAbOHZGGap6MS10zIT+c39Q2sKdvPhP693I4jElNarpHTFyEiTjCARY2cSDONyElUDchK4r8vGM3b/3k63/3cMNZ9eoCrHlvBGXe9xv++9QmV+ibbVc3N23tbtJ6cSEclhTQiJ+Ko5lMrNYmaCKBGTlySnhjk+tMG88bNp3P3ZeNIiY/j1udXM+3nr3DrglWU7Kp0O2KP1Cc1nv4ZiVo+QqQTNCIn4iydWilyNJ1aKa4KBnxcOCGPCyfksXLbPv7w1ic8uWIb//v2FmYNyeKr0wdw+vDe+H067TJapgzM4JW1jctG+PS6i4TtyIiclh8QccSR5Qc02YkIoBE58ZDx+encddl43mo67bJkVyXf+MO7nHrHEn7z2kb2HqxxO2KPMHVgBnsP1VKyW6OiIh1xZESuWiNyIk7SOnIijdTIiedkJYe4/rTBLP3eaTx05URy0xP4xT8/ZtovXuE7zxTz3pa9WKtv45wydWAmAMs36/RKkY5IDGpETsRJzXOi1WhETgTQqZXiYQG/j3PH9OPcMf34+NP9/OHtLSz4YDt/ea+U4X1TmDulP1m1+s880vIzEuibGs/yTeV8ZdpJbscRiRl+nyE+zqdr5EQc0nJqpUbkRECNnMSI4X1T+fmFY/jBnBH8vbiMJ1ds5dbnVxP0wZJ9xcyd0p+J/dO1hEEEGGOYMjCDZZvKsdbqNRXpgORQQLNWijhM18iJNFIjJzElORRg7pT+zJ3Sn1XbK7jzuWX886MdR43SfXFCLmkJcW5HjWlTCzJ4vriMLeWHGJCV5HYckZiRGAxoRE7EIS2nVmpETgR0jZzEsNG5aVw9KsTy/zqTX1w0hmDAx63Pr2bqzxfz7aeLWd40oiQdN3VgBgDLN5e7nEQktiQG/RqRE3FI8/kh1bX6skQENCIn3UDbUbo/r9jK8yvLePb9UvpnJHLJyXlcfHIeuekJbkeNGYOyk8lMCvL2xnIum9zf7TgiMSMppBE5ESeFAj6q6zQiJwJq5KSbGZ2bxngsDfsAABpNSURBVM8vHMMPzxvBolWf8pf3Srnr5fXcvXg9pwzK5Esn5/O5UX1JaJomXI7NGMPMIVm8UVKu9eREOiAx6KdSI3IijlEjJ9IirFMrjTHnGGPWGWNKjDHfP8F2k40x9caYSyIXUaTjEoMBLpqYx5+/MY2l3zuNb50xhC3lh/j3/1vJlJ8t5j//+qHnljHwWp3NGpLNnspqPv70gJNPIxJ1TtZaUjCgdeREcK7OggG/GjmRJu2OyBlj/MADwFlAKfCOMeZ5a+2aY2x3O/CiE0FFOis/I5F/P3MoN50+hOWbP+OZ97bx3AdlPLliGwXZSVw8MY8vjMshPyPRtYxerLNZQ7IAeH3DbkbmpDr9dCJR4XStJYb8WkdOejwn66xxRE5flohAeCNyU4ASa+0ma20N8BRwwTG2uxF4FtgVwXwiEePzGaYPyuSuS8fzzg/P5JcXjyUrKcQdL65j1i+XcMlDb/HEsi3sPVjjRjzP1Vmf1HiG9Ulh6YbdTj+VSDQ5WmtJQS0/IIKDdRaK06mVIs3CaeRygW2tbpc2/ewIY0wucCHwcOSiiTgnORTg0sn5PD1/Oku/dxrf/dwwKg7Xcstzq5j8s8V8/fF3+HtxGYejN2mBJ+ts9tAs3tm8N5qvg4jTHK211IQA+6vqPHXatogLHKuzUMBPjRo5ESC8yU6ONctB2z3UPcDN1tr6Ey0ebIyZB8wDyM7OpqioKMyY0VFZWempTMrTvkhlGmVg5ATL1gPxvF1Wz/LNu3nl413E++HkPgGm5/gZkeHH79ykHxGrM4hcraUdrqOmvoHfLljC2OzIzI3ktc+R1/KA9zJ5LU8XObpP211WQ32D5cVXiogPuDdJkBffM69lUh5HOVZn1Ycq2VF70DOvldfeN+VpnxczdVY4R2alQH6r23lAWZttJgFPNRViFjDHGFNnrX2u9UbW2keARwCGDRtmCwsLOxnbGUVFRXgpk/K0z4lMVwH1DZblm8tZ8EEZC1ft4M2yarKSQ3x+bD/OG9uPk/v3ivRMjhGrM4hcrU2rree+lS9RkZBDYeHITj1GW177HHktD3gvk9fydJGj+7RPE7fy9LqPGDd5Gv3S3FvyxIvvmdcyKY+jHKuzrIx0DFBYON2p7B3itfdNedrnxUydFU4j9w4wxBgzENgOXA5c0XoDa+3A5j8bYx4HXjjWwaVIrPD7DKcMyuKUQVn85IJRFK3b1TRBylYef+sT+qbGM2dMY1M3sX867Y2QhcGTdRYf52dqQSZL1u3ils9HppETcZmjtZaaEAdAxeFaVxs5EZc5VmehgE9LfIg0abeRs9bWGWNuoHFGIT/wmLV2tTFmftP9ui5OurX4OD/njO7HOaP7UVldxytrd/L34h38cdkWHntzM7npCcwZ05fzxuZ0+jm8XGdnjujNjxasZuPuSgZlJ7sVQyQinK611PjGRm7/YR1oSs/lZJ2FAj7KK3WNnAiEuSC4tXYhsLDNz45ZhNbaq7seS8SbkkMBLhifywXjc9lfVcviNTt54cMdPP7WJ/x26eYuPbZX6+zMEX340YLVvLxmJ4NOVSMnsc/JWktNaNyt7j9c28l0It2DU3UWCvi1/IBIk8jMXiDSA6XGx3HRxDwumphHxaFaXlzzKZfd7naqyMtJT2B0biovr9nJ/FMHuR1HxNPSWp1aKSKRFwr4qKnXiJwIhLf8gIi0Iy0xjksn5be/YYw6e2Rf3t+6l90Hqt2OIuJpR06trFIjJ+KEYMBHda0aORFQIyciYThrZB+shVfW7nQ7ioinpcQ3n1qpa+REnBAKaEFwkWZq5ESkXcP7ppDXK4GX16iREzmRgN9HciigETkRh4TidI2cSDM1ciLSLmMMZ4/sy9KSPRzQAarICaXGB3SNnIhDQgEfNXUNWNt2fXGRnkeNnIiE5fPj+lFT18BLqzUqJ3IiqQlxmrVSxCFBv48GC3UNauRE1MiJSFgm5KeTn5HAguIyt6OIeFpqfJxOrRRxSCiu8dBV18mJqJETkTAZYzh/bA5vluyhvFKzV4ocT2pCHBWa7ETEEaGAH4AaNXIiauREJHxfGJ9DfYNl4Uc73I4i4lmpCQGdWinikGCgeUROE56IqJETkbAN75vKsD4pLFip0ytFjkenVoo4J9TcyGktORE1ciLSMV8Yn8O7W/ay7bNDbkcR8aTUhDgOVNVRr8kYRCKu+dRKXSMnokZORDroixNyMQaeeXeb21FEPCkzKQjA3kM1LicR6X4Sgo2HrodrdWqliBo5EemQ3PQETh2azdPvllJXr29ERdrKTG5s5Mor1ciJRFpiMADAoWpNKCSiRk5EOuzyyf35dH8Vr63f7XYUEc/JTAoBaHZXEQckBhtPrTxUoxE5ETVyItJhZ4zoTVZyiKfe0emVIm1lNY3I7TmoETmRSGsekTtYoxE5ETVyItJhcX4fl5ycx6sf72Ln/iq344h4SmayRuREnJIUahyRO6wRORE1ciLSOZdPzqfBWv60fKvbUUQ8JT0hDp+BzzQiJxJxiXHNI3Jq5ETUyIlIpwzISuKM4b3507ItVGn2MJEjfD5DRlKIPZrsRCTiEoLNI3I6tVJEjZyIdNrXZg6k/GANz32w3e0oIp6SlRzUqZUiDggGfMT5jUbkRFAjJyJdML0gk5H9UvndG5uxVosfizTLTA5SrlMrRRyRGAzoGjkR1MiJSBcYY/j6zIFs2FXJ6xv2uB1HxDMyk0IakRNxSGLQz0GtIyeiRk5Euub8cTn0SQ3xwJISjcqJNMlMDmpBcBGHJAb9HNK12SJq5ESka4IBH/9WOJgVmz/j7U3lbscR8YSs5BAHqus0EZCIAxKDAQ5pRE5EjZyIdN1lk/PpkxrinsUbNConQqtFwXV6pUjEJQb9HNI1ciLhNXLGmHOMMeuMMSXGmO8f4/4rjTEfNv16yxgzLvJRRbq3WK6z+Di/RuUkZkSj1vqmJQCwo6IqAolFYo+TdaZGTqRRu42cMcYPPACcC4wE5hpjRrbZbDNwqrV2LPA/wCORDirSnXWHOmselbvzxXUalRPPilat5abHA1C273CX8orEIqfrLDEU4JDWkRMJa0RuClBird1kra0BngIuaL2BtfYta+3eppvLgLzIxhTp9mK+zuLj/PzHWUN5f+s+Xvhwh9txRI4nKrXWr2lErmyfRuSkR3K0zhLj/Bys1oicSCCMbXKBba1ulwJTT7D914F/HusOY8w8YB5AdnY2RUVF4aWMksrKSk9lUp72eTFTJ0WszsC9Wsu2lvwUHz/+20pCe9YR9Jt/2cZr75nX8oD3MnktTxdFbZ+WFAfvrClhxFFPFx1efM+8lkl5HOVonVXsqWbfwTpPvF5ee9+Up31ezNRZ4TRy/3okBsc8b8oYcxqNxTjzWPdbax+haeh82LBhtrCwMLyUUVJUVISXMilP+7yYqZMiVmfgbq2F8vdw5aPLKfHn82+Fg//lfq+9Z17LA97L5LU8XRS1fVr/4qX4kuIpLJzclbyd4sX3zGuZlMdRjtZZcd0GXtqynhmzZhPnd3fePq+9b8rTPi9m6qxwPv2lQH6r23lAWduNjDFjgUeBC6y1mu1ApGO6TZ3NGJzFmSN6c/+rJWzX9UHiPVGrtZy0eLbr1ErpmRyts7SExnGI/Ydru5ZSJMaF08i9Awwxxgw0xgSBy4HnW29gjOkP/BX4irV2feRjinR73arObj1/FNbCLc+t0sQn4jVRq7Wc9AR2VOjLDOmRHK2z9MTG5T0q1MhJD9duI2etrQNuAF4E1gJPW2tXG2PmG2PmN232IyATeNAYs9IY865jiUW6oe5WZ/kZiXz77KG8+vEu/vGRJj4R74hmreWkJ7DvUC0HtXCx9DBO11laQhygRk4knGvksNYuBBa2+dnDrf58LXBtZKOJ9Czdrc6uPmUAC1aW8ePn1zC9IJPM5JDbkUSA6NVaTqslCIb0Senqw4nEFCfrLFWNnAgQ5oLgIiIdFfD7+OUlY9l/uJabn/1Qp1hKj3NSZhIAn5QfcjmJSPeiETmRRmrkRMQxI/qlcvO5w1m8dhd/XL7V7TgiUVWQ3djIbdxd6XISke5FjZxIIzVyIuKoa04ZwOyh2fz0hTWs3bHf7TgiUZMaH0fvlBAbd6mRE4mkI43cITVy0rOpkRMRR/l8hju/NJb0xDjmPfEulTU6xVJ6jkHZyRqRE4mwYMBHYtCvETnp8dTIiYjjeqfE8/CXT2ZnRTUPFldRV9/gdiSRqBjUO4mNuw/qGlGRCEtLiGOfGjnp4dTIiUhUTOjfi59eOJo15Q3csmC1DmylRxiUnUzF4Vr2VNa4HUWkW8lKDrGnstrtGCKuUiMnIlFz6aR8Pl8Qx5MrtvKrlzy9prlIRAzKTgagRNfJiURUn9QQO/erkZOeTY2ciETVxUPimDsln/uXlPDo0k1uxxFx1PB+jevHrS6rcDmJSPfSOzWe3Qeq3I4h4qqwFgQXEYkUYww//eIY9h2q5af/WEuDtcybPcjtWCKO6J0ST256Aiu37XM7iki30jslxJ7KGmrrG4jza1xCeiZ98kUk6vw+w68vn8B5Y/vx84Ufc9dL63TNnHRb4/LTKC5VIycSSX1S4wHYfUCnV0rPpUZORFwRDPi49/IJXDYpn3tfLeG/nltFrWazlG5oXF462z47TLkmZhCJmN4pIQB2qZGTHkyNnIi4xu8z3HbxGL5ZOIg/L9/Klx9dzmcHNbufdC/j8tMBNConEkHNI3KfVug6Oem51MiJiKuMMdx8znDuuWw8H2zbxxfuf4P3tux1O5ZIxIzNSyPo9/FWSbnbUUS6jdz0BABK9x5yOYmIe9TIiYgnfHFCLs9cNx2AS3/zNr9evEELh0u3kBgMMLUgg6L1u92OItJt9EoKkp4Yx+Y9B92OIuIaNXIi4hnj8tNZ+K1ZnD+2H3cvXs9FD73FR6Watl1i36lDsynZVanRA5EIGpiVpEZOejQ1ciLiKanxcdxz+QTumzuBsn1VXPDAG9y6YBUVh2vdjibSaYXDsgFY8vEul5OIdB9q5KSnUyMnIp50/rgcXvn2qXx52kn8YdkWZv9yCQ8VbeRwTb3b0UQ6bFB2MkN6J/PXD7a7HUWk2xiUncyOiioOVte5HUXEFWrkRMSz0hLi+O8LRvPCjTOZ2D+d2xd9zOw7lvDI6xs1QicxxRjDpZPy+WDrPjbsPOB2HJFuYXjfFABWbdcp+NIzqZETEc8blZPG76+ZwjPzpzM4O5mfL/yY6b94hR8/v5pNuyvdjicSlgsn5hLwGZ5YtsXtKCLdwoT+vQB4f6uW9pCeSY2ciMSMyQMyeHLeNF64cSbnjO7Ln5Zv4fRfvcaFD77JH5dtYd8hrUEn3pWVHOLiiXk8tWIbZfsOux1HJOZlJAUZmJXE+1u1ZI30TGrkRCTmjM5N465Lx/Pm90/nB3OGc6i6nh8+t4pJP13MlY8u4/dvbmbbZ5odULznpjOHAHDHi+tcTiLSPUwdmMGyjeVU1er6ael51MiJSMzqnRLPvNmDWPTvs3jhxplcO6uAnfur+cnf1zDrl0s4/VdF/OBvH7Fg5XZ27q9yO64IuekJXHdqAX/7YDv//GiH23FEYt45o/tyoLqOpRv2uB1FJOoCbgcQEekqYwyjc9MYnZvG988dzuY9B3ll7U7eLNnD31eW8eflW4HGg+jRuamMyU1jVG4ao3JSsda6nF56mpvOGMLr63fzH08X0zs1xMknZbgdSSRmzRicRUZSkD8v38JZI/u4HUckqsJq5Iwx5wC/BvzAo9ba29rcb5runwMcAq621r4f4awi3ZrqLHIGZiVx7awCrp1VQH2DZU3ZfpZtKqe4dB+ry/bz4uqdR7ZNCMDQ1W9QkJ1MQVYSA7KSyElPICc9nt4p8fh9xsV/iTjB7VqL8/v47VWTuPTht7ny0eX89ItjuHhiLo1PK9I9RKvO4vw+vj5zIHe8uI5lm8qZVpAZgfQisaHdRs4Y4wceAM4CSoF3jDHPW2vXtNrsXGBI06+pwENNv4tIGFRnzvH7DGPy0hiTl3bkZ/urallTtp+Pd+xnafF6qoNxLN9Uzt/arPHl9xl6p4TolxZP37R4eiUGyUwK0ispSEZSkF6Jjb+nJcSREh8gKRQgzq8z1r3MK7XWOyWep+dP54Y/fcB3ninmz8u38I1ZBZw+ojehgD+STyUSddGus6tPGcAz727jpic/4I/XTmVon5Su/QNEYkQ4I3JTgBJr7SYAY8xTwAVA62K8APiDbTxHaZkxJt0Y089aqwsARMKjOoui1Pg4phVkMq0gkwG1WygsbDx2OFRTx9bPDrFjXxVlFYfZsa+KHRVV7Kg4zLpPD7D3UC17D9VworMxQwHfkaYuOdTye3ycj6DfRyjgJxTnIxRo+nPA13S78c/BgI/1ZXUc/HAHfp8h4DP4/U2/G9P4M7/B7/M1/qx5G58h4PNhDE2/DAbwGdN4m6afmaaf0Wq7pvtbtjUtj4OhvsFS32CP+jsxzDO11jslnifnTeMv723j7pc38M0/vU9CnJ8J/dMZlZNK/4xEcnslNH1R0PhlQWJcoOXz0PTex/j7Id1TVOssKRTgN1+ZxJWPLue8e5dy0YQ8Th2WTUF2EtnJIRKDAUIBHz6dYSHdTDiNXC6wrdXtUv71G5NjbZML6ABTJDyqMw9IDAYY3jeV4X1Tj7tNfYOl4nAtnx2sOfKr4nANldX1HKyu42B1HQeafj9YXceBqjp2H6imuq6e6roGqmsbjvy5qraehuM1hR967KzZlxYe967WfYQ56udHHzQdfV/rn7c5uDru47WbMhyeqjW/z3DZ5P5cPDGPN0r2ULRuN+988hl/eHsL1XUNYT2Gz0DA5zvqxWpoaMC3+J9tX1ng2K/jv7wHx92u8+rr6/G/uqgLjxBZyuOoqNfZsL4pLPzWTO5+eQMLVm7n/97d9i/bBJoaueYvqTjyJVerL7A6+Lxee9+Up31ezNRZ4TRyx/pMtz30CGcbjDHzgHlNN6uNMavCeP5oygK8NO2R8rTPa5mGdfLvRazOwPO15rX3zGt5wHuZvJYHPFBrqrMO81om5Wmf6qx9XnvflKd9XsvU2ToLq5ErBfJb3c4DyjqxDdbaR4BHAIwx71prJ3UorcO8lkl52ue1TMaYdzv5VyNWZ+DtWlOe9nktk9fygDdqTXXWMV7LpDztU521z2uZlKd9XsvUhToLax25d4AhxpiBxpggcDnwfJttnge+ahpNAyp03Y5Ih6jORKJDtSbiPNWZSBS0OyJnra0zxtwAvEjjFLKPWWtXG2PmN93/MLCQxuljS2icQvYa5yKLdD+qM5HoUK2JOE91JhIdYa0jZ61dSGPBtf7Zw63+bIHrO/jcj3Rw+2jwWiblaZ/XMnU6j0N11qVMDlGe9nktk9fygPdqzWuvkdfygPcyKU/7VGft81om5Wmf1zJ1Oo+xJ5pHW0RERERERDxHK9eKiIiIiIjEGMcbOWPMOcaYdcaYEmPM949xvzHG3Nt0/4fGmIku57myKceHxpi3jDHjnMwTTqZW2002xtQbYy5xO48xptAYs9IYs9oY85qbeYwxacaYvxtjipvyOHqevTHmMWPMruNNgRztz3TTc3qqzsLMFNVaU511PZNqzXu1pjqLTCbt01RnXcyjY0fVWXt5nKkza61jv2i8wHUjUAAEgWJgZJtt5gD/pHE9kWnAcpfznAL0avrzuU7mCTdTq+1epfF880tcfo3SgTVA/6bbvV3O8wPg9qY/ZwOfAUEHM80GJgKrjnN/1D7THXiNvJgparWmOotYJtWah2pNdRax10j7NNVZV/Po2FF11l4mR+rM6RG5KUCJtXaTtbYGeAq4oM02FwB/sI2WAenGmH5u5bHWvmWt3dt0cxmN65o4KZzXCOBG4FlglwfyXAH81Vq7FcBa62SmcPJYIMUYY4BkGouxzqlA1trXm57jeKL5mQbv1VlYmaJca6qzyGRSrXmr1lRnkcmkfZrqrEt5dOyoOmuPU3XmdCOXC2xrdbu06Wcd3SaaeVr7Oo3dsZPazWSMyQUuBB7GeeG8RkOBXsaYImPMe8aYr7qc535gBI0LiX4EfMta2+BgpvZE8zMd7vN5MVNrTtea6iwymVRr3qo11VkEMqF9WntUZ96qM/BeranOuq5Tn+mwlh/oAnOMn7WdJjOcbSIl7OcyxpxGYzHOdCjLkac6xs/aZroHuNlaW9/4xYHreQLAycAZQALwtjFmmbV2vUt5PgesBE4HBgEvG2OWWmv3O5AnHNH8TIf7fF7M1LhhdGpNdRaZTKq19p+vJ+/TvFZn4L1aU51F5vl6cp2B92pNddZ1nfpMO93IlQL5rW7n0dj5dnSbaObBGDMWeBQ411pb7lCWjmSaBDzVVIhZwBxjTJ219jmX8pQCe6y1B4GDxpjXgXGAE8UYTp5rgNustRYoMcZsBoYDKxzIE45ofqbDfT4vZopmranOIpNJteatWlOdRSaT9mknpjrzVp2Fm6kn79N6Tp1ZZy/GDACbgIG0XGw4qs0253H0xX0rXM7THygBTnHytelIpjbbP46zF6yG8xqNAF5p2jYRWAWMdjHPQ8CPm/7cB9gOZDn8vg3g+BesRu0z3YHXyIuZolZrqrOIZVKteajWVGcRe420T1OddTWPjh1VZ+HkinidOToiZ62tM8bcALxI4wwyj1lrVxtj5jfd/zCNM+nMobEADtHYIbuZ50dAJvBg07cYddbaSS5nippw8lhr1xpjFgEfAg3Ao9baY06nGo08wP8AjxtjPqKxAG621u5xIg+AMeZJoBDIMsaUArcCca3yRO0z3fScnqqzDmSKWq2pziKTCdWap2pNdRaZTNqnqc4ikEfHjqqzE3KqzkxTFygiIiIiIiIxwvEFwUVERERERCSy1MiJiIiIiIjEGDVyIiIiIiIiMUaNnIiIiIiISIxRIyciIiIiIhJj1MiJiIiIiIjEGDVyIiIiIiIiMUaNnIiIiIiISIz5/6vKXDi57d9JAAAAAElFTkSuQmCC\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "%reset -f\n", "%matplotlib inline\n", "from matplotlib.pylab import *\n", "\n", "exacte = lambda t,b : exp(-b*t) \n", "\n", "t0 = 0\n", "tfinal = 1\n", "tt=linspace(t0,tfinal,501)\n", "\n", "figure(1, figsize=(15, 3))\n", "for i,b in enumerate([1,10,50,100]):\n", " subplot(1,4,i+1)\n", " yy=[exacte(t,b) for t in tt]\n", " plot(tt,yy,label=('$b=$'+str(b)));\n", " axis([0,1,0,1])\n", " legend()\n", " grid()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Exemple de problème raide et approximation numérique\n", "\n", ">Soient $b,g>0$ et considérons le problème de Cauchy\n", "$$\\begin{cases}\n", "y'(t)=-by(t)+g,& t\\in[0;1]\\\\\n", "y(0)=y_0.\n", "\\end{cases}$$\n", ">1. **Solution exacte**\n", "\t1. Donner l'unique solution exacte à ce problème de Cauchy et tracer le graphe $t\\mapsto y(t)$ pour $t\\ge0$ en fonction de $y_0$.\n", "\t1. Soient $b=g=10$. Tracer avec python les solutions exactes sur l'intervalle $[0;1]$ pour $y_0=\\frac{g}{b}$ et $y(0)=\\frac{g}{b}+10^{-8}$ pour $b=g=10$. \n", "\t1. Sur un graphe à coté tracer les mêmes courbes avec $b=g=40$. Que peut-on dire lorsque $b$ devient de plus en plus grand?\n", ">1. Méthode d'**Euler explicite**\n", "\t1. Vérifier que la méthode génère une suite arithmético-géométrique et calculer analytiquement pour quelles valeurs de $h$ la suite ainsi construite converge et pour quelles valeurs la convergence est monotone.\n", "\t1. Soient $b=g=40$ et $y_0=\\frac{g}{b}$. Tracer les solutions approchées obtenues avec $N=19$, $N=23$, $N=45$, $N=100$. \n", "\t1. Répéter pour $y_0=\\frac{g}{b}+10^{-8}$ et expliquer chaque résultat. \n", ">1. Méthode d'**Euler implicite**\n", "\t1. Répéter le même exercice pour la méthode d'Euler implicite. Noter que, étant $\\varphi$ linéaire, la méthode peut être rendu explicite par un calcul élémentaire.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Correction**\n", "\n", "**Solution exacte** \n", "Il s'agit d'une edo linéaire d'ordre 1 dont la solution est \n", "$$\n", "y(t)=\\left( y_0-\\frac{g}{b} \\right)e^{-bt}+\\frac{g}{b}\n", "$$ \n", "et \n", "$$\n", "\\lim_{t\\to+\\infty}y(t)=\\frac{g}{b} \\text{ pour tout } y_0\\in\\mathbb{R}\n", "$$ \n", "le problème est numériquement bien posé, c'est-à-dire que l'erreur sur la solution sera au pire de l'ordre de l'erreur sur la condition initiale.\n", "\n", "Cependant, lorsque $b$ devient de plus en plus grand, la solution devient de plus en plus raide:" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "%reset -f\n", "%matplotlib inline\n", "from matplotlib.pylab import *\n", "\n", "exacte = lambda t,b,g,y0 : (y0-g/b)*exp(-b*t)+g/b \n", "\n", "t0 = 0.\n", "tfinal = 1.0\n", "tt=linspace(t0,tfinal,10001)\n", "\n", "figure(1, figsize=(15, 5))\n", "subplot(1,2,1)\n", "g=10.\n", "b=10.\n", "Y0=[g/b,g/b+1.e-8]\n", "for y0 in Y0:\n", " yy=[exacte(t,b,g,y0) for t in tt]\n", " plot(tt,yy,label=('$y_0=$'+str(y0)));\n", "title(\"$g=b=$\"+str(g))\n", "axis([0,1,g/b-1.e-8,g/b+1.e-8])\n", "legend()\n", "grid()\n", "\n", "subplot(1,2,2)\n", "g=40.\n", "b=40.\n", "Y0=[g/b,g/b+1.e-8]\n", "for y0 in Y0:\n", " yy=[exacte(t,b,g,y0) for t in tt]\n", " plot(tt,yy,label=(r'$y_0=$'+str(y0)));\n", "title(\"$g=b=$\"+str(g))\n", "axis([0,1,g/b-1.e-8,g/b+1.e-8])\n", "legend()\n", "grid()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Méthode d'Euler explicite** \n", "1. La méthode d'Euler explicite donne la suite définie par récurrence $u_{n+1}=Au_n+B$ et $u_0=y_0$ avec $A=1-bh$ et $B=gh$. Pour calculer $u_n$ sans passer par la récurrence, on introduit une suite auxiliaire définie par $v_n=u_n-\\alpha$ ainsi $u_n=v_n+\\alpha$ pour tout $n\\in\\mathbb{N}$. On a $v_{n+1}+\\alpha=A(v_n+\\alpha)+B=Av_n+A\\alpha+B$. En posant $\\alpha=\\frac{B}{1-A}\\ (A\\neq1)$, la suite $(v_n)_{n\\in \\mathbb{N}}$ est géométrique de raison $A$ donc $v_{n}=v_0A^n$. En remplaçant $v_n$ par $u_n-\\alpha$ et $v_0$ par $u_0-\\alpha$ on en déduit que $u_n=A^n(u_0-\\alpha)+\\alpha,\\quad \\alpha=\\frac{B}{1-A}$ soit encore\n", "$$\n", "u_n-\\frac{g}{b}=(1-hb)^n\\left( y_0 -\\frac{g}{b}\\right)\n", "$$\n", " \n", "1. Comparons la solution exacte et la solution approchée:\n", " \\begin{align*}\n", " y_n&=\\frac{g}{b}+\\left( y_0 -\\frac{g}{b}\\right)e^{-bt_n}\\\\\n", " u_n&=\\frac{g}{b}+\\left( y_0 -\\frac{g}{b}\\right)(1-hb)^n\n", " \\end{align*}\n", " Pour que $u_n\\simeq y_n$ il faut que $(1-hb)^n\\simeq e^{-bt_n} \\in]0,1]$, donc il faut que $0\\le(1-hb)<1$, soit encore $h<\\frac{1}{b}$. Comme $h=\\frac{T-t-0}{N}=\\frac{1}{N}$, cela implique $N\\ge b$. \n", "\tDans cet exemple, pour obtenir une bonne convergence il est nécessaire de prendre un pas $h$ de pus en plus petit que $b$ est grand, c'est-à-dire un coût en calcul plus élevé. \n", "\tOn parle de **problème raide** ou mal conditionné." ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "image/png": 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jlHBI4cefOa90P88Af5pzz74KOKm1PoxlQnWpUmpirjfw53LHCoIgCIIguI2r8YgaXvkJOlK5BEW8W4IiFslvDypdKWvRXhULD3hMZTRMVzo4i3ubZNagMjrwvLtTEQ2T8nlh35101iRebILC8Pd7kTFMokXq6u3EQNbw5z+4fd1SFsrhkPIlwWInRdxY5IdDyvMki2FqQi4t8O1xTI/uQWuNqSHkYl2SUt7NvzuuKChyLbbmAR/Ktbj6OhAF0Fr/ENiM5Zj9NtAF/K/cvqxS6s+AF4Ew8E9a691uzEkQBEEQhLMLr+OR4ZagaE/1p6AI5sK+O/b8Bk1QxMKBUh/YJNImFQ4TFPGI9dY7Y5glvT13i1TWZERFYUuFWG7efida0lmTWLi4hVo0d17aJwVFNnfdcAkLzWhY+ZJgyZqlz90mrERBUQj2o3JfQTFMExRa6xuH2K+Brw6wbzNWwCAIgiCUkUwmQ0tLC8lk0tPrnnPOOezdu9fTa57pVFRUMG7cOKLR4uqTz1S8jkdCwXnZ7Qi7xGNE/PTPTdUwUFAkMoOXeICVoAhkiUfWIB51lmywExmpbFASFEbBJR7xgJR4ZAzT8XPvTdT3Eg9NNKwclQUNRCTsU4mHQ/WHk3jkgT86j1gEV+OHoeKR6ybC4ovOc+WaF5gGjy25kPSxg+w9Uf7/z1prHltyIedUJlx7Zt+YM4poODToeOWIR7zyoBAEQRB8pqWlhREjRjBhwoSSAp9CaW9vZ8SIEZ5d70xHa01bWxstLS1MnFiaCZlQGqFhpqDoSGUIhxQV3RZuNfEInaksWmtPfy8UQucwLvFIZZyXeNgL6lTG6FGG4xdWiYezudsExYMiY5g9lEKF4H+Jh0mkxM5XkZDKe1l4ia14iAyRYHMSj4Teb6cyGubi0VWuzW+oeOTd4110pbNMumBkydc6lcgQaevkkjE1gyZX3cI0NdlDJ7lgZAVjRrpjUBt6v52KaIjxo/s3+SxXPOJ/elYQBEHwhGQyyejRowO7CBGcoZRi9OjRnithhL4oDUY22OqD7nQks9TEIz1+B1THI5jaf0n+YNiJh8rhWOKRM8l0gq0+SAbke1GSSaZP6gObtFG8SabfJR4ZQxMpsjzFJhIKobX3/gFOPSicxiO6/2ZK5UNr1JANgYNJ/km5OP2hhipXPCIJCkEQhLMISU6cGcj3MRgoDW1Hhk9n1/ZUts+b+eq4tXgOcpmH3cWjerASj2iYRAAVFMmM2UOxMhj5Eo+AJFqKMskMB0NBkc4a+bkUSr7Ew6d7yJqll/jYCQ6vVRSFeFAM9XfMjz9zA/WcLgZ7/t5ZOFgXcjXB4mCocsQjkqAQBEEQBEEoktaDb/k9BcfYCoru2It+u1NGEHFkkhkNqAdFISUetoIiExwFRbElHn75N9hkDF2w+sMmX+LhUxvhrKHzKo5isRUMXpepuNnFQ+Hl4t5Cu5mh8JhyPSsfPDIlQSEIgiAIglAsHxw56PcUHNOZzvapyx8OCgpbGVE1iC9DRSyYCopExiDu2IPCNskMxn0UY5IZDUgXD6sTyvDs4pF2w4PCpySLfT03unig8LrAw76sq+N4fQ9uChr8ytVIgkIQBEEQBKFIOo8d8nsKjulXQZH72i6jCCKdubkNpkSoCqiCIpUxqXCoQgiSgsI0NRlDF1ziEaQuHsV7UPhc4uGqgsLjEo98F4/Sl5gKhfb49b3GxQW+xzUeZbmKT0kiSVAIgiAInvLoo4+ilOrRtqq2tpbm5uaCx/riF7/ImDFjqKur67Pvu9/9LnV1dUyZMoWHH364lCm7xtq1a1m3bp3f0xBcJHGi1e8pOKY91VdBYbvLB11BEQ0N/lY2sG1GMwaVsULbjPp/H7Z6oNBWnUExySylVavvJR6mOWQXjKGwPSi8V1CYPa5fCl68ve8dj2itWfTJj7kSj3RXUHgZjzh9bk7iEb8MQyVBIQiCIHjKzp07aWho4PnnnwcglUpx5MgRxo8fX/BYt9xyCy+88EKf7bt27eKxxx7j1Vdf5Y033uC5555j//79Jc9dEHqTOXXC7yk4piOZZUQvBYWtqOgMuAdFxRAihKC2GU1kjIIVFH6XR4Cl/AAKNpoMiklmxijc4NPGVi/45aORMXTJHg7RkD9JFjc9KLx4e99fPNJ29Kir8cju3d7EI3mhhss5Ba9VLCAJCkEQBMFjmpqauPvuu/MBwe7du6mtrS3KCXrOnDmcd955fbbv3buXq666iqqqKiKRCHPnzmXTpk09jmlubqa2tpYvfelLTJkyhYULF5JIJAB46KGHqKuro66urs/bjscff5yZM2fS0NDA7bffjmEYNDc391BxrFu3jrVr1wLwwAMPcPnllzN//nz27dsHQGdnJ9dccw1Tp06lrq6OJ554ouB7F4KB0XHK7yk4pmOQLh6dAS/xiA3xRrYyFiadNfMLpCCgtSZZUJtR67ggtEtNGdYcnPpn2MQCU+JRSptRf+8hW0J5ik3YrxIPFz0oFJQ9Q9E7Hnlr314+eullrsQj9hD73nzTk3jkjv/9FQzD4OCBA67FI355UAzsNiQIgiCcsXzj2d3sOeTuwmry2JF8/TNThjxuz549LFmyhPvuu4+TJ0/S1NREfX19j2Nmz55Ne3t7n3PXrVvH/Pnzh7xGXV0dq1evpq2tjcrKSjZv3szHPvaxPsft37+fDRs28Nhjj3HDDTfw85//nNraWn70ox/xyiuvoLVm1qxZzJ07l2nTprF3716eeOIJtm7dSjQa5Y477mD9+vXMmTOn33ls27aNjRs3sn37drLZLNOnT2fGjBm88MILjB07Nh8UnTx5csh7EoKJ7ur0ewqOMExNV9roa5IZsxUUwU1QJNIGg/hjAqf9KZIZI++r4TcZQ2NqCmgzGjwFRbxQBUUAEhSGqTHM0hMU/pV46JJLJPwq8TCK8KAYKB5JZgw0g3vPDESx8chbe3ZzeW3P84qNR+zvYO3kKTzwja+XPR65/Sv/m82bfsY1C/+43/kUFY/45EERjN/ggiAIwlnBwYMHGT16NJWVlSxYsIAXX3yRnTt3csUVV/Q4rrGxsaTr1NbWctddd7FgwQJqamqYOnUqkUjfP3kTJ06koaEBgBkzZtDc3ExbWxtLly6luroagOuvv57GxkamTZvGyy+/zLZt27jyyisBSCQSjBkzZsAERWNjI0uXLqWqqgqAJUuWAFBfX8+qVau46667uPbaa5k9e3ZJ9yv4g1agEim/p+EI22NiYJNM/9/aD0Rn2qDCgYICrJKKoCQokjkviUIVFKkAKCiK9aCIhBRK+etBYZdmRCPFLfIjvpd4mPkSjWKxEwRetxm1PShc6eJRZvqLR97cs4tJk3t6WhUfj1jP4PJJkzyJR7q6Elxddc6Aqodi4hE/Wr2CJCgEQRDOSpy8WSgHO3fuzKslFi9ezPr16zl8+DDXXXddj+NKVVAA3Hrrrdx6660A3HvvvYwbN67PMfF4PP95OBwmkUgMWm+ptebmm2/mwQcf7LG9paUF0zwdzCaTyfzn/UlFL7vsMrZt28bmzZu55557WLhwIWvWrHF0X0JwMBWEk8MrQTGil4IiFgkRDauAm2RmiQ+xxreTAEFqNZpMF5agCKSCokAfB6UUsXDIVwWFnRwp1D/DJt/Fw+PFvU3W0AW3d+2NX0kWW7FRSBeSgeKR5mOdpA2Ty84f4crcetNfPPK7AwdZuPgzPY4rWkHRrYmHF/FIMmPw1pF2wonjrsYj0sVDEARBOKPpXs4xd+5cGhsbewQJNo2NjezYsaPPh9PkBEBrq9Vd4d133+UXv/gFN954o6Pz5syZw1NPPUVXVxednZ1s2rQp/0bh6quv5sknn8yPffz4cQ4cOMD5559Pa2srbW1tpFIpnnvuufxYmzZtIpFI0N7ezrPPPgvAoUOHqKqq4qabbmLVqlW8/vrrju9LCA46BOFkcBf23elI2gqKaJ991fEIXQFOUHSlDeJDLHiquikogoLdLnRYelDk1B/FLJRjkZCvSRa7PWixi/xYPkHhk4KihPIUGztB4LUni+GmB0WZRRj9xSP79uymdnLPhMlwikcOtbzLmDHDPx4RBYUgCILgGU1NTSxbtgyw3hbU19ezfft2Ro0aVdR4N954I1u2bOHYsWOMGzeOb3zjG/m3FMuWLaOtrY1oNMojjzzCueee62jM6dOnc8sttzBz5kwAbrvtNqZNmwbA5MmTuf/++1m4cCGmaebHHj9+PGvWrGHWrFlMnDiRSZMm5cdavnw5DQ0NjB8/Ph9YNDU1ceeddxIKhYhGo/zgBz8o6v4FfzEVRNP+v+12gq2QqO5HilAdi9AR8C4eYxx6UARKQZFb5DutoY+GFSEVDAWFrYCIO+xA0p14JOTb4h5OKx+KXeTn1Qc+fR8yWbMgBUJ/hPNdPDxWUBThQTEY5Swv6C8eeW3b64w8x5145Gtrvs7HF/8JGm/ikXAkyl+u/RaxKya5Fo8opUB7//9AEhSCIAiCZ6xfv77H108//XRJ423YsGHAfUPVjU6YMIFdu3blv161alX+85UrV7Jy5cp+z1u+fDnLly/vs33FihWsWLGiz/bVq1ezevXqPtsXLVo06PyE4KMVRFP+LyadMFCJB1hJiyCbZHals8QrhvCgyCUBguSlkciXeDhbrCmliEfCgUhQpLLFlXiAlRjwtcQjd+2iExS5t/8Z30wyzZIX+FH7HjwuU8krKEpMsIBdjlC++fcXj7z1fnvRyo3e8Ugqa7Dv/XbQ3sQjibTB/tZ2UMq1eMQvDwop8RAEQRAEQSgCrSCeCk5by8EYqsQjyG1GrRKPwY+xTTKDUB5hY8/FaYkHWKaUQbiHVAllErFIyFeTzLwHRZElHraPhl8qkKzhRhcP6969LvGwPSgiLrUZ9XpxrHGvtaY9jne3MDz+FjlBEhSCIAiCIAhFoEOKeNrvWTijI5UB6NNmFKwSj6AqKLTWjjwoKoPoQZEtzIMCoCISzhtU+kkpJR5+m2Rm8iaZxS81I2HlX4mHaZbsQREO+WWS6W4XD6+X3BrtovmFOj2mB9hXcdu6Q0wyBUEQBEEQhglaKSqGRxMP2pP9txkFq8QjSKUR3UkbJoaph1ZQBNCDotASD8gpKLL+30OpJplBSFCUssiPhkN5NYDXZA1dsgLBL5NM24Oi1DapUH6TzH7RLiooPJZQ6DJkKHz5HiAJCkEQBEEQhOIIKSrTkEp0+T2TIbE9KPpNUMQigW0z2pUz76wYSkFhe1AESEFhL/KHt4Ji+JV4uJWg8OseMobOl2gUS/hM8KDAnxKP4TDmYLiuoBAPCkEQBEEQhOGBzr1eOtKy3+eZDE1HMktVLNyv9Lo6HgmsgsJOOMSG6uJhe1AE6D5sLwmnXTwgSAqKEhIUYX/bjKZKNMm0zvWvxCNrlt7Fw753z7t4uOlBoZRn5RE9r+vSOO4M4+t1lU93IQkKQRAEQRCEYsjJmNta3vZ5IkPTkcr2q54AqIqHA6ugSOTMO4dSUNgqhSB5UJwu8SggQREJBUJBMbxLPHR+HsXif4lHaUs0O0HgdYmH4bIHhdf5Ce1iiYc9kFcKhLJcR3nnodEdSVAIgiAIgiAUg7LCqJOtB32eyNC0p7L9GmQC1MQipLOmb10LBqMzV+IxlAdFNBwiGlaBUoKcNsl0Hm5XRMP55ICflGKSGfc7QWF3IClRQeFfiYdJNFJiF49cgsPrEg/7eu4oKIa3SabyvI9H7jouGkf4pQKRBIUgCIIgCEIxhKzFW+fxIz5PZGg6B1FQVOe2234PQcJOOAzVxQOsUoogtOi0ybcZLWCRH4+ESAZCQWGiFEWVGsQi/rXohG4eFCUs8qPhkH9dPAyzZJPJSN4k09t7MExNOKRQLi2S/dCwDNc2o+Xq4uHHN0ESFIIgCIKnPProoyil2Lt3b35bbW0tzc3NBY1z8OBB/uiP/oja2lqmTJnCd7/73fy+ZDLJzJkzmTp1KlOmTOHrX/+6W9MvibVr17Ju3Tq/pyG4RdhaeKY+OOrzRIamIzlYgsK6j4508Mo8EhlrTgNMvQeVsXCwunhkDGKREKEC3ibHA6SgiIVDRS00/TSYBPLXHo5dPExTY+rTCYZiifhkkpnNJSAZMnoAACAASURBVCjcQKHKXh/ROx7RGubOmuZOPJJ7DMmEN/FIoU/KSTyiihjXDRz8uhcEQRAE99i5cycNDQ08//zz1NbWkkqlOHLkCOPHjy9onEgkwne+8x2mT59Oe3s7M2bMYMGCBUyePJl4PM6//uu/UlNTQyaT4ZOf/CSf/vSnueqqq8p0V8LZSChkhVHZ9g98nsnQdKSyXFxd1e++0wqK4CUoTpd4OFNQBMmDIpUxqSjQByFICopiDDLBKq3ws8Qj7UKJRySsfFGBZMzSkytAvgtI1uN7MEzTlfIOOF3iobV2TZHRm97xSDqV4lhrqyvxyPz58+GcPyDmVTzifoWHbzUeoqAQBEEQPKWpqYm7776b559/HoDdu3dTW1tbcABy4YUXMn36dABGjBhBbW0t7733HmC5f9fU1ACQyWTIZDJ9xm9ubqa2tpYvfelLTJkyhYULF5JIJAB46KGHqKuro66ujocffrjHeY8//jgzZ86koaGB22+/HcMwaG5upq6uLn/MunXrWLt2LQAPPPAAl19+OfPnz2ffvn0AdHZ2cs011zB16lTq6up44oknCrp3IRiocBQAs6Pd55kMTXtyYA+K6lyLjCAaZSbSzjwoACpjwepGkswYBRlkgu1BEYQEhUGsCP8JOHNMMv24h6xLHg62AsNrFYi7CgqLct5B73hk/769XHr55a7GI3gUj/zFn92BYRgccDke8aPNqCgoBEEQzkZ+eTe83+TumBfUw6e/NeRhe/bsYcmSJdx3332cPHmSpqYm6uvrexwze/Zs2tv7LvrWrVtnvZXoRXNzM9u3b2fWrFn5bYZhMGPGDN5++22++tWv9thns3//fjZs2MBjjz3GDTfcwM9//nNqa2v50Y9+xCuvvILWmlmzZjF37lymTZvG3r17eeKJJ9i6dSvRaJQ77riD9evXM2fOnH7vddu2bWzcuJHt27eTzWaZPn06M2bM4IUXXmDs2LH5oOjkyZNDPjcheIQiVoKCri5/J+KAjlSWEUN5UARocW/T6bCLB0BlNBQ4Dwq7/alTrC4e/t9DSQoK3xMUpasQYuGQL2qcfIKiVAVFyJ8EhWHqwpMrA8QjowyTqqyJiocp+FV+kfHI/jf3MGlyXY9jio1HrrrqKg6cMkF7E4/c9uWvsHnTz1i2uO+coLh4xK82o5KgEARBEDzj4MGDjB49msrKShYsWMCLL77Izp07ueKKK3oc19jY6HjMjo4Oli1bxsMPP8zIkSPz28PhMDt27OCDDz5g6dKl7Nq1q8dbBYCJEyfS0NAAwIwZM2hubqatrY2lS5dSXV0NwPXXX09jYyPTpk3j5ZdfZtu2bVx55ZUAJBIJxowZM2CCorGxkaVLl1JVZUnrlyxZAkB9fT2rVq3irrvu4tprr2X27NmO71cIDkqFSEVBJVN+T2VQtNZWm9EBFBRVuUV0EBUUdtLEyTq/MhasEo9ExijIIBOsrhnBUFCYxAvoPtKdWCREyk8PiqydoCh+cRUJKzJJP0s83Oni4XWJR8bQhEs0+PSK3vHICy+8wFt7dzN5Ss84oaR4pP0kGu1JPNLZ1UW0ZtSANR5FxSM+tRmVBIUgCMLZiIM3C+Vg586debXE4sWLWb9+PYcPH+a6667rcZzTNxaZTIZly5bxhS98geuvv77fa44aNYp58+bxwgsv9AkI4vF4/vNwOEwikUAPomfUWnPzzTfz4IMP9tje0tKC2c0tPZlM5j/vTyp62WWXsW3bNjZv3sw999zDwoULWbNmzYDXFYJLIgbhZMbvaQxKMmNimJqaeLTf/bZ5ZmcAExSJtEE4pHCyVq6MhjneGZzvRTJjFtRiFKyWpGnDxDR1QeaabpPKmEV7OMRz5RHl9A4YDNsks9QSD68NJqF7icdwVVCYhSdXBohHTranOHwyweSxI0t+Hv3ROx55fP169v/+IMuudy8e6f0kyhmPfNCV5t3jXUQyJ12LR6TNqCAIgnDG072cY+7cuTQ2NvYIEmwaGxvZsWNHn4/uwYDWmltvvZXa2lpWrlzZ4/yjR4/ywQeWcWEikeDXv/41kyZNcjTHOXPm8NRTT9HV1UVnZyebNm3Kv1G4+uqrefLJJ2ltbQXg+PHjHDhwgPPPP5/W1lba2tpIpVI899xz+bE2bdpEIpGgvb2dZ599FoBDhw5RVVXFTTfdxKpVq3j99dcLfZRCQEjGIRLAhX132lPWon1ADwo7QRHQEo+qaNjRQrcyFglciUehHhTxnOLCbxVF2jCJFzh3Gzsx4McC37puToVQwqI2FvanVap9zVK7eIRCipA6nfDwClc9KMpsQtE7HvnPxkb2v9m/gqKUeOSYR/HIiePHOdTyruvxiHhQCIIgCGc0TU1NLFu2DLDeFtTX17N9+3ZGjRpV8Fhbt27lJz/5CfX19XlZ5De/+U0WL17M4cOHufnmmzEMA9M0ueGGG7j22msdjTt9+nRuueUWZs6cCcBtt93GtGnTAJg8eTL3338/CxcuxDRNotEojzzyCOPHj2fNmjXMmjWLiRMn5oOP6dOns3z5choaGhg/fnw+sGhqauLOO+8kFAoRjUb5wQ9+UPD9C8EgHVdEUv5L8gejI2klUAb2oLAWokFVUDj1caiMhgLVZjSZMRhVFSvoHFtxkcoW7l/hJqmMUZIHBVhJjlJUDMWSMaxOEqUoUPzq4mErHkot8QDLx2JYeFAMQLlNMnvHI3V19by2bZur8cj4qZ/g/fff50//5LNlj0dC4QirvvG3xKJTXItHfBBAAZKgEARBEDxk/fr1Pb5++umnix7rk5/85IDyxyuuuILt27cPev6ECRPYtWtX/utVq1blP1+5cmWftyA2y5cvZ/ny5X22r1ixghUrVvTZvnr1alavXt1n+6JFiwadnzA8yMQUsaQ/b4qdYntL1AyQoKiMhlEqmG1Gu9JGXuExFEFrM1pMiYetoPC71WjaMAf8eRkKuzQknTUhPsTBZSCdNUtu0xkNhzxXH8Bpz4hS5w9WmYfXHhRuKijKnaHoHY9semoTuw+dopjChoHikd2HTlJbV+9JPHK8M03LiS5Q7sYjfvx1kxIPQRAEQRCEIslUhImnh0eCYqCFvlKK6liEjlRwFvc2XekslQ5LDSpjkWApKLKG47nbdFdQ+EkqU0oXD+ue/VAgWNfVJSs3ouFQ3svCS+xruuG5EAkp79uMGqZrfhF2BwmvTBrt/IKbqgGF8rxGwk3Rg3hQCIIgCIIgDDOysQjxYDfxOF3iMYAHBVhlHl3pYCooqhyXeIRJG6bnb40HIpEu3oPCbwVFKmsUvci3yxP8ajWaNtxQUPhU4mG4V+IRDYfImt7eg2Hqkv0zbMpd4tGbcl3Hu/nbV3I5ReGDCYUkKARBEARBEIrErIhSGfQExRAlHkBOQRG8BEVn2qDKaYlHzAprkwFo0wnFmmQGQ0GRNsx8sqRQYvl78ElBkTWJlbhI9q3Ew7RNMktfooVDyheTTNc8KHLDeL0+dnV5r/A8w+Kqb4SSEg9BEARBEIRhhVlZQcyAUyda/Z7KgOQTFIMqKCIBNcm0ung4wS6nCIoSJJk1C05Q2Mf73cWjlBIP+zxfFRTDtMTD7nwSdWGR70erVMNNDwqPKU+Jhz8LfLfIq1g8zhJJgkIQBEEQBKFIVFUVAEfe3e/zTAamPelAQREPB7LNaFfaoCru3IMCIJn2X0FhmJp0tgiTzNzxfrdLTWWL78DRvYuHH2QMM2/UWSx+l3i4oaCIhBWGxyUeWUO750HhuYKiDCUSHuZqylHg0Xtsr5AEhSAIgiAIQpGEq0cCcPzw73yeycB0pLJEw2rQN+LVsWAqKAr1oAAC0cnDLtEoWEGRK6tI+d3FI1uCSWY4nB/DD9JZ7UoXD62tRJOXZPIlHqUvM8MhRcaHNqNuKSiUxy4UeQWFi2MqvEuwlOM6YpIpCIIgCIIwzIiccy4A7UcP+TyTgelIZqmJR1CDaJer4xG6AqmgyFIVK8yDIggJCtvkstAuHnkFhd9dPEowyYz5XOKRcaHEw04QeK2iyJtkuqBCiIZCGB6XeGRM0zWTTBuvTTJd9XBAedaFJH/FcrTx8FhC4UqCQin1KaXUPqXU20qpu/vZf6dSakfuY5dSylBKnZfb16yUasrt+60b8xEEQRAE4ezDj3ikYtRoABInjrh2H27TkcoO6j8BVolH0EwyDVOTzJgFKCisewyCB4WdJCm0xCMICgrD1Jiakk0y04Y/SRarxKO0VZpdIuJ1mYrdgcYtBYUvXTyGuUmmm3irQHC/yMOn/ETpCQqlVBh4BPg0MBm4USk1ufsxWutva60btNYNwD3Av2utj3c75I9y+z9W6nwEQRCEYPPoo4+ilGLv3r35bbW1tTQ3Nxc1nmEYTJs2jWuvvbbH9gkTJlBfX09DQwMf+1gw/rysXbuWdevW+T2NMxK/4pGa0WMBSH/QVuotlI32ZJaaeHTQY6pjEboClqCwF/mOExQxu0VnEBQUxZV42AoKP00y7dxI8SUePptkZt1oM2qd73UXDLsko9T5W2Moz00ys4Ym7JYHhSujDE73eMROhMyafoV78UiuS6cX8Uih6Qln8Yg/KQo3foJmAm9rrd/RWqeBjcBnBzn+RmCDC9cVBEEQhiE7d+6koaGB559/HoBUKsWRI0cYP358UeN997vfpba2tt99//Zv/8aOHTv47W9FoHcW4Es8cs4FFwOQ7ThV6lBlozOVpWYIo8mqeITOtIHpcc36YNhKiEqnJR62B0UATDLtBEWhKgQ7KeBnksVOUJRukunPz1LGKN7g08avEo9MLqkTdUFBEQmHPPfQcFNBYUsoytlBomc8okmnUrS2troWj3R/EmWPR8ro8em1isXZb/zB+QPgYLevW4BZ/R2olKoCPgX8WbfNGnhJKaWBR7XWfz/AuV8Gvgzw4Q9/mC1btpQ+c2FIOjo65Fl7iDxv7zgbn/U555xDe3u759c1DKPHdXfs2MFf/MVf8E//9E/cfvvt7Nixg8suu4yOjo6Cx37vvfd45plnWLVqFd///vd7XEdrTUdHB/F4vN9zDxw4wLJly/jDP/xDXnnlFS688EI2btxIZWUl3//+9/nJT34CwJ/+6Z/y1a9+NX/exo0b+eEPf0gmk+FjH/sYDz30EC0tLdxwww288sorAHzve9+jo6ODe++9l29/+9ts2LCBcePGMXr0aKZNm8b777/PzTffzKFDhzAMg7/6q79i2bJlBd17Mpk8636Gh8CXeOTA4ZNcAqROHA/s9+PwsQQj42rQ+R1pyQDw4r9uoTISjDaBRzqtxdq7v3uLc89JDfl8W7us41/fuYvq4/vKPb1BefuElWDYv3cXFcfedHyevajf+9Z+tmQPlGVuQ3GyvRNQHHjnbbZkCp+D/X3YuWs3I0+85fLshub4yQSh9OA/70PxTu7/Q+PW/+JDleW17Osej+zKXfe3r77C70u8bsepBB3g6e+lUx1dHFddQ17TSTySzFr/F7q6EuhM0pX5DRaP3HLbl/ndW29y6aXuxSOmaZLRpifxSH3DDFb9zTp279rF8uXLXYlHUinre9DR0TGo+anb8YgbCYr+ZjtQnuUzwNZecspPaK0PKaXGAL9SSr2ptf6PPgNagcLfA1x++eV63rx5JU5bcMKWLVuQZ+0d8ry942x81nv37mXEiBEA/O2rf8ubx50HzU6YdN4k7pp5V5/t7e3t+esC7Nu3j+XLl/Ptb38b0zR55513aGho6HHM7Nmz+w1e1q1bx/z58/Nff+1rX+M73/kO7e3tRCKRHmOEQiGuv/56lFLcfvvtfPnLX+4xVk1NDb/73e944oknaGho4IYbbuCll16itraWn/70p7z22mtorZk1axaLFi1i2rRp7N27l2eeeYbf/OY3RKNR7rjjDp555hnmzJlDKBTKXz8ej5PJZHjrrbfYtGkTb7zxBtlslunTp3PVVVexdetWLr74Yl588UUATp482WPuTqioqGDatGkFnXOG40s88qklf8JbX3uAWNYI7O8U9dstjB97DvPmDfzz8l7lAZ7Yt4sZM/+QMSMrPJzdwOw5dAoaG5kxtY6KY/uGfL5H21PwH79mwkcvZd4fTvBkjgMRffsYvPIKM2dMY9ZHRjs+zzQ1/GozYy+awLx5l5VxhgPzL5v/FUhQP7mWeTPGFXz++yeT8B8v85FLLmferIvdn+AQxF//d8aeP4J586YXPcYH29+DXTuYceUsJn6o2sXZ9aV7PNLymwOwaxezP/Hxkv8fPvb2b0hlTObN+7gLs3RG7LV/48ILRg36uwacxSOmqUlkDCqi4YI7gxQTj6TTGfa/uYf6K65wLR4JJ6yFvRfxyC233c7mTT/j859d5Fo8klYpSCaorqkZtOzI7XjEjQRFC3BRt6/HAQNZWX+OXnJKrfWh3L+tSqlNWBLNPgGBIAiCMPw5ePAgo0ePprKykgULFvDiiy+yc+dOrrjiih7HNTY2DjnWc889x5gxY5gxY0a/mfutW7cyduxYWltbWbBgAZMmTWLOnDk9jpk4cSINDQ0AzJgxg+bmZtra2li6dCnV1VZQev3119PY2Mi0adN4+eWX2bZtG1deeSUAiUSCMWPG9Bm3+30sXbqUqqoqAJYsWQJAfX09q1at4q677uLaa69l9uzZQ96vMCS+xCPRWJxkHELJVFGT9oL2XBePwajOlVF0pLKM8WJSDkhkrBKPqlgEJ0J724MiGF08ivOgCIUUsUgoEB4UpXfx8Of7YHlQlKYC8q+Lh22SWbpqIxwKkTG9/R5kDU3EJQ+KcptQ9I5Hfv2rl3hr724a6up7HFdqPKK19iQe6ejsonLkuQN2axpO8YgbCYrXgEuVUhOB97D+6H++90FKqXOAucBN3bZVAyGtdXvu84XAfS7MSRAEQRiE/t4seMHOnTupr7f++C9evJj169dz+PBhrrvuuh7HOXljsXXrVp555hk2b95MMpnk1KlT3HTTTTz++OMAjB1rmReOGTOGpUuX8uqrr/YJCLrLLcPhMIlEYtB6V601N998Mw8++GCP7S0tLZjd3NKTydNy1P6Chcsuu4xt27axefNm7rnnHhYuXMiaNWsGvK7gCN/ikWQMwgEzmOxORyrDiCG7eNgdMPxf3Nt0pk6bZDoRXNseFEG4h3ybUYcGn92JR0L+elDkykyKNsnMe1D412a0VA8K+22x5wmKnGeEG108oiGVT3h4RTEeFAPFI4l0lv2tHYwfXc05lYOb/BZD73jkx//8E37/7nt84Yae5ZalxCPfeOiHgDfxyPsnExxtTxOh3b14ZLi2GdVaZ7FqOF8E9gL/orXerZT6ilLqK90OXQq8pLXu7LbtfOA/lVJvAK8Cz2utXyh1ToIgCEIwaWpqygcEc+fOpbGxsUeQYNPY2MiOHTv6fHSXUz744IO0tLTQ3NzMxo0b+eM//uN8cqKzszMfUHR2dvLSSy9RV1fnaI5z5szhqaeeoquri87OTjZt2pR/o3D11Vfz5JNP0traCsDx48c5cOAA559/Pq2trbS1tZFKpXjuuefyY23atIlEIkF7ezvPPvssAIcOHaKqqoqbbrqJVatW8frrrxf7SIUcfsYjyQqIBDRBkTFMkhnTgYLCWkgHqdWonWhwusgP59QHQVBQ5NuMFtGqMx4JD28Fhc9dPDJG6V08YvkEhcddPHLXi7qgQoiElecmmVlTE3YhuQLdFtNlcmjsHY/899at7H9zN1PcjEcUdHZ4FI+cOM6h9951NR7xq82oGwoKtNabgc29tv2w19c/Bn7ca9s7wFQ35iAIgiAEn6amprz5Ujwep76+nu3btzNq1ChXr3PkyBGWLl0KQDab5fOf/zyf+tSnHJ07ffp0brnlFmbOnAnAbbfdlq+tnDx5Mvfffz8LFy7ENE2i0SiPPPII48ePZ82aNcyaNYuJEycyadKk/FjLly+noaGB8ePH5wOLpqYm7rzzTkKhENFolB/84Aeu3v/Zil/xSDqmiKaC0/2iO525hMOQCYq8giI4CQq7xKM6FuGow3Mqo2GSgVBQ2CUehS80K6IhUj6VRwDYeYVCO5DY2OUVw7nNqP8lHi508QiFPJ+/YZrudfHIUa7frL3jkcl1U3j99e2c62I8ooCjR1u56TpLsFfOeIRQmLvu+zbRaL2L8Yg/KQpXEhSCIAiC4IT169f3+Prpp592Zdx58+b1MND7yEc+whtvvDHoORMmTGDXrl35r1etWpX/fOXKlaxcubLf85YvX87y5cv7bF+xYgUrVqzos3316tWsXr26z/ZFixYNOj9h+JCJhajq8H9R3B/tyVyCYsgSD1tBEZz76F7i4ZTKaDgQCop8gqLIEo9Uxk8FRa7Eo4jkClhvvmORECmfSjzSw7jEw372bizyI2GVLxnxiqypCza0HIhyL417xyMbfvYLmo91lmx90TseuXjCRE/ikUMfJDjRmQbci0f8UlCUt2+OIAiCIAjCGU62Ikw87fcs+scu2RjhUEHRGaASj0SBJR5gJTOC4UFRfIlHRTTsq4IiX+JRggohHg75WOKhS5o7dE9QeLzANyyDz4GMDgshEgqR9Xz+hXtQDESZKzz6Yl/HRQGIUgrt5fLebWPR4epBIQiCIAiCcDaTjUeoCGgTj3yJh0OTzCAlKDrTp7t4OKUiGvbVYNImmTEJKYrqJmGZZPrvQVFMeYpNLOJPgsIwNYapSy7xsL9vGY/vIWOYrnXBiIQUWdMHk0wXOpBYeLs6LkN+whrLo8V9OS4jCgpBEARBEIRhiFkRpzIFRjY4i3ub9lzCoXoIBUVVrgNGZ4BKPBJpg3gkVJBkvDIWnBKPymi4qDfhviso8l08ivOgACtB4XV5BJwuyYhGSltm2gkOrxf4GUO74j8BfplkuudB4b2CQve8sFvDujraYBfSqHL3ZvUISVAIgiAIgiCUQlUFYQ1tRw74PZM+dCSdlXhEwiEqoqFAmWR2pY2C/CfA8qAIQolHImNQES1uge+3giJbYhcPsBb4figo7NampZd45Iw+vS6RMEs3+LSJhJSnJSqmqTE1w8aDojdlUVB4mC/Q4HqJhygoBEEQBEEQhiNV1QAcPbjf54n0pcNhiQdY3TKC1Ga0M50tqLwDcgqKACQokhmzhARFMDwo4iUkKGKRUD5Z4CV2SYZrJpkeJ1nc9HCIhEP5riBeYGj3DD574PHq2O2cgpceGq7nQzyXsVhIgkIQBEEQBKEEIjXnAHDiyLs+z6QvtoJiqDajYJWBBEF9YJMoUkERCA+KrFG0h4PVZtT/Lh6lLPJjPikobMVA6R4U/pV4uKag8LiLh23IGXbJQyO/NnZltKHR5TDJ9LDkQlM+j0yvkQSFIAiCIAhCCcTOOQ+AzmOHfJ5JX/IeFA6UCFWxcKAUFF1pgyoHiZXuBKbNaLqUEg9/kyxZF7p4xCL+JFnyHhQlLvIjPpZ4uOZBEfI4QWHaz96tEg9rHK+6YJSrxMPTLh5lQko8BEEQBEEQhhEV534YgMSJVp9n0peOZJaaeISQA9l1TTwSqC4eXels3rzTKZVBaTOaLT5B4b+CwlrcltKNwa8uHvZzK3WRHDsTSjxCIQxToz2S59uGnG55UHhtgKDLkqLA2y4e0mZUEARBEArn0UcfRSnF3r1789tqa2tpbm4ueKwvfvGLjBkzhrq6uj77XnjhBS6//HIuueQSvvWtbw253U/Wrl3LunXr/J6GUCQjPjwOgMypEz7PpC8dqYyj8g6wSjw6A7C4tynKJDMWkBKPjFl0iUfc5zKVjKFL9nCI++VBkbtmKf4Z4GeJh3smmXaSxisVhX0d17p45P4t5+x7xCO5C11RP8W1eERhDetJPOJSF4/u8YiYZAqCIAhnBTt37qShoYHnn38egFQqxZEjRxg/fnzBY91yyy288MILfbYbhsFXv/pVfvnLX7Jnzx42bNjAnj17BtwuCKVw7gXWz67RccrnmfSlI5V1ZJAJUB0PB0xBUVyJR8bQvrS47I7dZrQYKnLlEV69+e5NRpe+wPfPg8LdEg8vu2CAtch3q8TD9oLIenQPpxUUbnlQ5Eo8yjj97vGIBtKpFK1uxiMKsh7FI8O/kOQ0kqAQBEEQPKWpqYm77747n6DYvXs3tbW1+WCkEObMmcN5553XZ/urr77KJZdcwkc+8hFisRif+9znePrppwfc3pvm5mZqa2v50pe+xJQpU1i4cCGJRAKAhx56iLq6Ourq6nj44Yfz5zz++OPMnDmThoYGbr/9dgzDoLm5ucfblHXr1rF27VoAHnjgAS6//HLmz5/Pvn378sd0dnZyzTXXMHXqVOrq6njiiScKfi6Ct4y56DIAdFenzzPpS3uuxMMJ1bEIXYFKUBRR4pE73m8fikTGIF6sB0U0jNb4okAAyBiWD0Yp+FXi4VaCIppbZHud6CqPgsKbe3BbQQG2D0X5lt494hENv3vrTSZNci8eUUDT69s8iUf+8YeP5I1F3YpHujo7+bObb+CqK6d7Go8UlpYWBEEQzgje/+Y3Se1909Ux47WTuODee4c8bs+ePSxZsoT77ruPkydP0tTURH19fY9jZs+eTXt7e59z161bx/z584e8xnvvvcdFF12U/3rcuHG88sorA27vj/3797NhwwYee+wxbrjhBn7+859TW1vLj370I1555RW01syaNYu5c+dSUVHBE088wdatW4lGo9xxxx2sX7+eOXPm9Dv2tm3b2LhxI9u3byebzTJ9+nRmzJgBWJLPsWPH5hM4J0+eHPJ+BX8557wLeC8EKpHyeyp96EwVkKCIB6vNaFfaoLKIEg+wTCpHVkTLMS1HpDImFUUu8m31QiprlpwoKIasWXqJRywS8kXFctqDorT5h0KKSEj5k6BwSYFge0F4paCwW5oW6kExWDwSTmc5FQ6RKvD7WWw8sv/NPdTX9ywZLSkeUYoj7x/yJB6Z/rErmT7r4xjHfN+i1wAAIABJREFUR7sWj/z6Vy/x4fMvZPPzz1NTEfUsHpEEhSAIguAZBw8eZPTo0VRWVrJgwQJefPFFdu7cyRVXXNHjuMbGxpKu0580Wik14Pb+mDhxIg0NDQDMmDGD5uZm2traWLp0KdXV1QBcf/31NDY2EgqF2LZtG1deeSUAiUSCMWPGDBgQNDY2snTpUqqqqgBYsmRJfl99fT2rVq3irrvu4tprr2X27NkF3LngB+FIhEQcwsngJSg6UlnGjKhwdGx13DKY1FoX9QbRTbTWdKUNquPDU0GRzBhUxor3oAAryYGzb52rZEx3PBz8bDNaaoIFrDIPz0s8XPD/sLFNTjNeKyhcKlHJU6ZvQe945F9ffom39u5m5lT34hGFd/HIomuWsO2V/6LlnCrX4pEpdXX81V/dyV+vvofrr/usZ/GIJCgEQRDOQpy8WSgHO3fuzKslFi9ezPr16zl8+DDXXXddj+NKVVCMGzeOgwcP5r9uaWlh7NixA27vj3g8nv88HA6TSCQGrAnXWnPzzTfz4IMP9tje0tKC2S04TCaT+c8HCkQuu+wytm3bxubNm7nnnntYuHAha9asGeRuhSCQjEM4GRz1gU1H0rkHRVUsQtbUpLJm0R0o3CJtmBimpspBe9Tu2AoKvxMUiYxRsoLCL6PMjAmxIg0+bWJ+mWTmkiKltEi1iYa9V4FkTE2VWyUeOSWD4ZFJppEv8Shs/oPFI7sPnaSmKsbYUZUlza0/escj//jjf+bdlkPcetMNPY4rNR4Zc+FYXhym8cill17Gxue38OZv/93TeEQ8KARBEATP6F7OMXfuXBobG3sECTaNjY3s2LGjz4eTYADgyiuvZP/+/fz+978nnU6zceNGlixZMuB2p8yZM4ennnqKrq4uOjs72bRpE7Nnz+bqq6/mySefpLXVajN5/PhxDhw4wPnnn09rayttbW2kUimee+65/DibNm0ikUjQ3t7Os88+m7/GoUOHqKqq4qabbmLVqlW8/vrrjucn+Ecqroim/TVm7I/2Ako87OOCYJSZyHUTKaaLB+Brq1GtNclMKW1GcwoKn1qNZk3tikmmH/PPe1BESn+LH/MhQZE1zHxioVS8L/Fwuc0olNWBonc88pv/2sr+N3dzhYvxiFJQN3W6J/HIi5uf5WNXfcLVeOTwoUNUVFay/MYveBqPiIJCEARB8IympiaWLVsGWG8E6uvr2b59O6NGjSpqvBtvvJEtW7Zw7Ngxxo0bxze+8Q1uvfVWIpEI3//+91m0aBGGYfDFL36RKVOmAAy43QnTp0/nlltuYebMmQDcdtttTJs2DYD777+fhQsXYpom0WiURx55hPHjx7NmzRpmzZrFxIkTmTRpUn6c5cuX09DQwPjx43vIJpuamrjzzjsJhUJEo1F+8IMfFPVsBG9JxxTRVLASFFprOlJZRjju4mEd15U2GF3OiTmgs9gERfS0B4VfZAyNqSnYP8MmCAqKkSV6X8R9MslMu2SSCbkSj6z3JR5ulUicbpXqtYLCzRKP/ksh3KB3PFI7pY43drgbjyz+H58n7FE8svwLNzOlfiqXjKlxLR7ZvXsXd955JxXRCPF4zLN4RBIUgiAIgmesX7++x9f9OVYXwoYNGwbct3jxYhYvXux4e3cmTJjArl278l+vWrUq//nKlStZuXJln3OWL1/O8uXL+2xfsWIFK1as6LN99erVrF69us/2RYsWsWjRokHnJwSPbDxE1Qn/lQfdsfwkKKCLh7UoDYJRZiJtzaGy0BKPAHhQ2NcuVoXgt4IiY5bu4WCXeHjtZ5J2u8TDI/8Gm4xp5r0jSuW0gsKbe7CfVdhFDwpVRglF73jkxz/9GUfbk0X/vPYXj7x/MgHam3jknaMd+ZasbsUjCxcuZGLDx5kwupqRld6ZDkuJhyAIgiAIQolk4mHiab9n0RM70eDUg+K0gsL/BEVnylrkVw/DEo9ULkFRbIlHvouHXwoKw50SD629e3tv46ZJpuVB4b2Cwq0SD7vNqFf3YCso3OpCAuUt8eiNRoPbyTSl0OiyqUC6o8F6YOUa20MkQSEIgiAIglAiZkWUioA18WjPmXY6bzNqKyj8NZiE0wmGgtuMBkBBkcyYPeZSKN3bjPpBxjzdSaRY7ASB12UeGRdLPKJhlTfd9Iqs4Z6Cwjar9MoksxweFChvF8dur+897YWky3A9O2HjQYKlO5KgEARBOIvwIosvlB/5PgYPsyJOZRpSiS6/p5LHVlAU7EERhBKPjDWHYrt4+OXfAKeTI6WbZPpzD1mz9BIJ/xMUpS/VouEQWc9LPLQryRU4XWrhVZmKUWCbUSd/x5SHGYpy/Fm1n4QXt1COaziZfzniEUlQCIIgnCVUVFTQ1tYmi9thjtaatrY2Kioq/J6K0J1cD/kjLft9nshp7G4cNXFntcPVuWRAEDwoii7xsBUUPpZ4JPMJiuLC7NMmmX4pKDRxF9qMAp63GrVVJ+6YZIZIe1zikTFMV5IrcLrUwrMuHrYHhQMFhdN4xCrx8O574LpdipcZCrz1e4HyxSNikikIgnCWMG7cOFpaWjh69Kin100mk7KYdpmKigrGjRvn9zSEbqiaEQC0HdzPxZdO9Xk2FnaJh126MRTVAWwzWmyJh58eFHaCotgSD78VFBk3FBRhfxUUbphkxnwp8dD50oxSsZUMXqlACuni4TQeOXIqSSSk6GqNuzLHweKRD7rSJNIGoZOVrlwLoD2Z4WQiS/hUBaEyJw9aTyUJhxTJo+48K7BKjo6cSpFpiw6oZitHPCIJCkEQhLOEaDTKxIkTPb/uli1b8q04BeFMRZ17HgCnWt7xeSanyZd4OFVQ5BIZnT4u7m1so87qAks8QiFFPBIKRIlHsT4O/isoGLYKioxhEgkpQi74IFgmmd7P3y0FRSTfxcMbBUKmAA8Kp/HInf+nkfNHVPCPtzSUPD8YPB65d1MTL+1u5bdfW+DKtQD+ofEd7n9+LzvXLmRkRXm7YPzldxsZO6qCf7jZnWcFcKCtk888voWHbpjK9bXevRSREg9BEARBEIQSiY25AIDk+4d8nslpOpIZwHkXj1g4RCSkAqGg6CxSQWGfEwSTzKJLPHxUUGityZoQj5RokumbgsI9Dwc/Sjyypnbs4TAUttmmVyaZ+S4eLj1/gLBSnnWCMU3trsEn5FUTpgf3YGrtukrDHs+rn6H8dT29miAIgiAIwhlI1diLAcgeO+LzTE5jKyiclngopaiOR3wtj7BJpA1CiqLaXVZFw/62Gc2WaJLpo4LCVjyU3GbUJ5PMdNY9BYLXJR5aawzTxRKPkN1m1Jt7KMSDwinhkML0yDcra2rCLi/w7WfhxQLfKEeCJTee19ZlkqAQBEEQBEEokXPHTiQTBnX8A7+nkqc9lSUWCRX0Nrw6Fg6GSWY6S3UsUpTpW4XvCorSPCgi4RDhkPJFQWGbTLqWoPC4RCJtmMRKVH/YeN3Fwy6RcK3EI+9B4a2CwokHhVMioZBnJSqmqfOdT9zCXuAbHqzwDa1dKW3qjj2cF/PvcV1PryYIgiAIgnAG8qFzx3GiBkInO/yeSp6OZJYR8cI8HCwFhf8JikTaKKq8A6zEQNJHBYVt8FmsggIsFUXKBwWFfc2SExR+lXhkTWIulkhkPCzxsJMhEbdKVOwuHh4lKOzruK2g8Kq8oCwKinyJh6vD9ovWlG/+kqAQBEEQBEEYXpw/+iI+qIbYqS6/p5KnI5V17D9hUxWP0JHyv8SjK21QVUKCwlcFRbY0DwqwfCiSPigobMVDbJiWeGQMk2iJc7eJhpWnJpl2MsQtBcJpk0yPSjzy83fRgyKkvOtCot0vkYh4qaAwNS5PP69g88JDozuSoBAEQRAEQSiRqopqTtZARUfG76nk6UhmqSlQQVETDwfCJLMrnR2wrd1QVMb89aCwSzwqSig18E9BketAUqpJZi5JkPLag8IwXTNpjHncxSPrUnLIJl/i4ZEKxMgrQFxWUHi0NjaM8nk4GB7chGG6X+JhPw+P8xOSoBAEQRAEQXCDjmqo7PB/cW/TkSo8QVEdiwQkQVGagsLvNqOxSKikxYKloPAhQZF1Z5Ec98uDIqvz5SWlEgkrT0s8Mi4rEOxEjdclHu56UKh84qPcGGXogmH/KHqhoDC1+yUqeQ8KUVAIgiAIgiAMPxLVioqUxkwm/Z4KpDqKS1DEI3QGwIOiK21QVeDcbfxuM5rKmPlOHMUSj4TyagYvSbtlkhm2kktedsEAt0s8vFVQ2NdyS4Fgv/32rESiTB4UXppkuqn+AG/bdJazi4d4UAiCIAiCIAxDEtW5N5ZHj/o7kda98K2LGZN4u2APiup4mK5AeFBkqSrSZLIqACUexRp82sSjYc/LI6B7F4/S5h+NWAsbrxUUGcM9k0yvExS2AsGtLh5R2yTTowX+aQWFe8vLSHiYm2R6uMA3Ne6XeIhJpiAIgiAIwvAlVWUlA7Ktrf5O5NhboA3+IPlOUSUeQWgzWkqJR4XfXTwyRkkdPMBSMPhRppJ2qcTDry4e6ax7HhRRj0s8bA8Ktxb44fDwV1CElHcJCrMMJplhDxUUpnbfJPO0AsTdcYe8rreXEwRBEARBODNJ18SAACgoOo8BcE72aBEKigiprOmZ8/9AWCUew7SLR8YoySATrCSLPwoK2yRz+HbxcMtkMhoOYZjasw4GdjLELQWF7QXhVZIlX6LitgeFR2/vs+U0yfSqxMNtD4rcfyVRUAiCIAiCIAxDsjUV1r9+Kyi6jgPwYd3GiAIVFLZqodNHBQKU2MUjGiZrak/l+d1JZkwqSi3x8ElB4ZZJZswvk0xDu6igsMbJeKRAsJUObptkeqVAsNtcullmEA6FvOtCUkYFhSclHmXo4hGSNqOCIAiCIAjDF1VVRSYchASFpaAYq9qKaDNqHe9nJw/D1CQzJpVFlknY/g9++VAkMkbJJpkV0bDn6gNw0yTTnzajlgeFeyUe1pjeKijcMmq016peqaGypnbVfwLsLh7eJVhcT1B4qaAoQxeP0wkWV4cdEklQCIIgCIIguEBldAQnaiBx+D1/J5Ir8bhAHaemIlrQqdW5BEWXj5087PKM6mJLPHIJCr9ajaaGsQdFvsSjxPkrpYiFQ76UeLhmMmm36fRqgZ+7jlsKEKUU0bDyrM1oubpIeDl/t9uMetkFoxweGvbj8KrMxsaV/wFKqU8ppfYppd5WSt3dz/55SqmTSqkduY81Ts8VBEEQBEFwgt/xSFW0hg+qIXHI5wRFTkFxoTpeRJtRa2Ha4WMnDzs5UllCiQdAwicFRSnqD5t4JOSLB0XeJNOFRXIs4n2Cwk2TzEjY2zKV01083Ht/HPZwgZ81tKv+E2ApKLzyPzB1eeYP3phMmqaVlHITpRQhBdrjBEVxv/m7oZQKA48AC4AW4DWl1DNa6z29Dm3UWl9b5LmCIAiCIAgDEoR4pCY2ihMjFNmjPpd4dLYB8GF1kpHRwiLj6lxSoMvHEg87sVBdpI9DVRBKPKKll3j42ma0xPlDLkFhePs9cNMkM+ZxiYedCHGrxAOsVqNeebFkTTPfOcQtwiHloYKlDAoQ5V0nFctDw/1xveykkr+mC2PMBN7WWr+jtU4DG4HPenCuIAiCIAiCje/xyP/P3nvHSXKV5/7fU1UdpnvC7s7mVVrtrsIq7EpCASWPBMggg2UM2BI43B/JAgOG6/u74BtwNmBsgzHJYJJtEMZEGQQSSBplCaXNCpu0q807s2Gmu6dT1bl/VFfP7MTu6TpnNKv3+/ns9kx39Tmna1Kdt5/3edrb5nGkHThyvNmnxkuhn8AJWzvm+P1NPTVq8ZjJqNF8Tb3RSswoMGNJHsVZ3eIRjwcFMCMtHvHGjNpu8agpKGL0cfBce5vL2e5BYSRmNGrxsPAtZCLFA8I2FdseFC0rKIBlwIsjPt8DXD7Oca9USq0H9gH/Q2u9uYnnopR6N/BugAULFtDb29v6yoUpyeVycq4tIufbHnKu7SHnWrDEjF+PDBwZ4li7wi0U6b3rLkgmW3g500Rrrs0d4nDyVBaVdrJ3/X0cOtR4keJAPrySfmL9JpKHnzW1yknZejTcmG99ZjPpvueA5n6PPHckfP5jTzzF4M7WCgXTITdUou/gfnp7j0x7jP17ylQDzd333Bv7pmkynt9WBuDhB+5vWS7uV0q8uG8/vb1H41haQxQrVQ7s20tvb+tRv8/vD4t0Dz3yGDvbzdn2Rd/b6w6E8617+kn6t8Uzn1+tsvvFvfT29sUy3mTs2VvCr/ix/r3ft7dMuRrfmJP9HhkYLNCvC7GuP/pd9vS6dVT3xrHtHp+oDWb37l309u6PefCAF3btprf3QLzjTkIcZ2q8316j6yxPAadrrXNKqRuBHwKrGnxueKfWXwK+BHD22Wfrnp6eaS9YaJze3l7kXNtDzrc95FzbQ861YIkZvx4JHu/n++t/BMCV555L8tRTm3wJMVAcgPuq5OavYdHenVy6ops5l/VM+bSIQwNFeOBuTjtzFT1XnG5unZPgPH8YHvslr7z0Yl5xxjygud8j8/Ycg18+xKpzz6dn9SKDKx2f6i9+ysrlp9PTc860x3hObYdtz3LFVdfUVS02eGToGbydO7juuutaHqvzyV7mdXfS03NxDCtrDP+uO1ix/LSWzn1EcdN+WP8UF13yCs5d0hnD6sYn+t4eWL8P1j3NKy+/lJULO2IZO/vI3SxYNJ+enjWxjDcZ/3VoPZlcf6x/739ZfBa9a0dsY072eyT9RC+LF8X7/dq1+yg89jDnX3AhPecsjG3c0VT9AO78KSuWL6enZ1WsYyfu+RmnnHIKPT2rYx13MuIoz+0BRv4FPoXwXYk6WusBrXWu9vEdQEIpNb+R5wqCIAiCIDTAjF+PzO9awrH28OMZixqtGWTuz5wFQFvxYFNPz74EYkYj74i2Fj0oZqLFww805WrQsgdF1GJh24eiVAmIwX4CgKRn10fDDzR+oEm68ahmohYPax4OkQfFLG3x8IMgVv8MqLV4WDJo9A2YZNqKGY3OkWNAbeUoe1+D+pwxjPE4sEoptVwplQRuBm4feYBSarGq6cSUUpfV5u1v5LmCIAiCIAgNMOPXI/PnLuNIe80U7XDrEvNpUQjbCg65ixnUbSTzzcl92xIuSkF+hgwmAYYqYXEkM80Uj8j/oTgDryGK6WzVg6L+GiwXWcp+nAUKx1oCBgwXEhJePJs0z3qBQtfmjW+T6TkOFYseFPF7ODhoDYGF11D1dewb/Mgk0/QGP/K4iDsmFWoeFJZNKFrWjGmtq0qp9wF3Ai7wVa31ZqXUrbXHvwi8GXiPUqoKDAE36zCvZNzntromQRAEQRBeXrwUrkcWzl3KsfbwQm7GFBT5UEHRrzs5pObRMdBc5KnjKDIJd0YVFJFJ5nRTPNpm0CSzWAlOWMN0iVI0ZkZBEc8mJ+U6lKv2vgZRISGOiFSAhOUUj0ptlxlnzGhoMmmvwBJ7TKcbpWBokoa9WEzEjA6bZNpRUJhI8XBnqUlmJJO8Y9R9Xxzx8WeBzzb6XEEQBEEQhGaZ6esRz0ugkxrfnfkWj8NBO/1qPisGmu+czaY8CuWZjxmdfotHLSp1BhQUUVGk5ZhRL3ztJYsbfAgVFDGldJL0HKvfR1FiSNwpHtYVFDFukl1HWSuwmEjxqCsQbCgojChA7CgoovNjREGhzK9/zJxWZxMEQRAEQTiJyWpFPuvMXItHTUFxsNrBEW8BTLNAkSvNXItHVFiYbotH5N8wMwqKeFo8IgVFpMiwRaniz+IWD12fNw5sFyiiebwY3wZPuI61mFRTHhRgZ4McGChQ2Cqw6LqCwowHhZYChSAIgiAIwuykTTsMZqEykwoKL01/2eN4YiEMHgC/0tQQ2ZRLYUZNMqukPGfaF9uOo0gnHOv+DRBfgaKuoLD8GkrVoN7a0CpJ16mrGmxQ96CYpS0e1domNq4WFQhbJKqz2oOitsG38DWoBho3ZgVCVGAJZrWCwp7Ran1Oq7MJgiAIgiCcxKS1y/HsDJtkZrrJlX3y6UWAhlxzSR6ZpEduhlM8MtNs74jIJGemTSV2BYVlD4pyNWaTTIvrL9cLFPFs0qwrKKqRgiJOk0xVbx0xjR+Y9KAw/zUIgvhNMqMCi+mvgckUj5nwoJAChSAIgiAIQky0aY9j7ZrqoRls8ch0kytVKaQXhvc12ebRnvLIz6AHRb5cnXZ7R0RbwmWobHdzD8MtGekW2wxSM6agiLfFw5b6AIY9KOIzybRcoAji96DwHMfK5h7CTbgxBYWFHbKJmFHHkoIi+hLHrQABUMpOispIpEAhCIIgCIIQE2mVor8dgoEBgmLR/gIKfZCdT65YpZxZHN7XZJJHNuVRmEEPiqEYFBQz3eIxXYPPiPQMKShK1SC2TVrCdaymkNRTPGLzoLDc4uGH517FuMm02+IRxJpAAsMbbhuvoWpCQVH3oIh12DGYT/GQAoUgCIIgCMKsJK1SHO4IP56RNo98H2TmkytVqWaXhvc1qaDIJt1Z3+LRlnRnxCRzKK4WjxlSUMTZ4pHyZiZmdNameATaiMmkzRaP2aygCAy0qEShJqZNPgPTHhTS4iEIgiAIgjA7aXMzHOyoves3E0aZhSPozDxypSpedi4kMs0XKFLejER0RhRiaPHIJGbKgyJq8YjHg8KmAiGab7ameJSr4S4q9gKFpa9BxQ9IxBzT6bmOVZNMUx4U1mJGY97gR+OZbpGIFA6mYkZFQSEIgiAIgjBLyXjt9EUFCtsKimoJyoOUU/MA6EgnoHMpHN/T1DDZpEu+XLUeLRcRh4IinXQZshzRCSNMMpPxeFDYblMpx9jiYTvFo1xv8YirRcVeewGEHg5mFBS2YkZNKCjCnyPTX4OogODGXCCypQDx6+s3o6AQDwpBEARBEIRZSluik6Pt4cfWFRT5PgCGEnMBaE97YYFiGgoKrZmRFgkIPSha9XBoSzgUZ0AFElvM6IwpKHxaXHqdpOcQaKxtkCt1k8x4XkCkoLClAqkGAV7MJgL2FRQxr9/SBr8amPFwsGaSaTzFQwoUgiAIgiAIs5L21FxybaA9136BohAWKArenNpaPOg8pekCRSYVtlfMlA9Fvlwl22qLR9KjUJnBmNEWWzyiJAr7KR7xtniAvQ1+3YMiNgVF1OJhZ3NW8TWJmDeYCUdZTPEIYn8H36mbZJp9DaY2+MMmmaYVFCfOFydKKeMmn6ORAoUgCIIgCEJMdLbNA6Wodmbtt3jUFBSDbq1AESkoBvdD0PhGtz0Vbq7zM5TkUYhBQZGewZhRRw23B0wXpRQpz24KBkQmmfG1eERj2qAcs0mm6ygcZX5zHFH1AxIxJZBEuBZNMk2ZfMJwjKYpfAMRrzCixcOSgsJMigfW2/2kQCEIgiAIghATndn5AJQ6klSsKyiOADDodALQkaoVKLQPucbXEqkX8jOgoNBaUyj7ZFOttni4MxYz2pZwY4mKTFt+DX6gqQY6fgWFNZPJcBOVjHGX5rn2jD4rvgmTSXstHr4Bk0zXtaOgqBpKwXAsmWRGBZY4I2ojwhQPKVAIgiAIgiDMSuZ1Lgag0O7ZV1DUWjyOERYoQgXFsvCxJto8sqmZK1CU/QA/0C2neLQlHYYqvvV3/oYqfsv+ExG2FRRRISGuN/GjAoWt1xCtPy4FBYTFDlsKhIofxLp2sGuSWQ107CaTtjwoAkMKiuH1xzrsGOoKClMFCjHJFARBEARBmJ0s6FoKQC6jqB6agRYP5XJEZ4GaEqIzXA8DexseJipQzETU6FBtzrYWN/mZpIcfaKsxlxC2eMRVoEgnXKsFilI1PPdxtXikagWKimUPimSMbRKeq6yt30iLhGuvxcOIgqLuQWHLJDNmBUW9QGH2e8hkiofrKGwHOkmBQhAEQRAEISYWdp8KQC4TEAwMEBSL9iYv9EFmHrlSeDHcMV0FRc3/YSZMMvO1AkWrLR5RkaBo2YeiWPXrCRytkvIcqy0eUTEkrhYP2ykYdZPMGDf5CdexWmCJOwUjYbHFo2LAJNN17LRIDHs4xL81dh3zLRImUzwcZV7BMmZOq7MJgiAIgiCcxHRl5+JpzbFMuKmx2uZR6IdMd72wkE15kJkHbmqaCgr7BYqh2pxtrbZ41AoUtqNSi+X4WjxsKyhib/GY5SaZEL6Gii2TSV/HWlyBmkmmJZNPP4h//Z5rW0ER/9iuhRSM6PSYavGQmFFBEARBEIRZiuO6ZAPN8Uy40bYaNZrvh8x8cqUq6YQTbtSUCts8milQJKOYUfstHlFbSablFg+3Np7dIkuooIjPg8KugqLW4hHTJtO2SWY0T7wmmTZbPAwoKBxFxddWvFhMeFBE49nyoIjbJBPAcTC+wffr649/bClQCIIgCIIgzHKygeJoJtzs2VVQ9EG2m8FilfZUYvj+zmVNmmRGMaMz0OJRK4pkYmrxsK6gqAQt+2dEpBJ2TTKLlXhbPOyneAR4jopV5m63xcOEB0X4NbCh0DfhQRGNZ1pBUY8Zjfn8Q6SgsFRgMeRBYbnDQwoUgiAIgiAIcdIWOPRlZ0JB0VdXUHSkR7RIdC1rSkHhuQ4pzyE/Ey0elXDO1lM8ah4UlgsUQ+X4PCjSnt2Y0XqLRNwpHhY3+HGnYCRstngE8ad4RB4OpossWmv8QMdvMqnsmEyaihmFsGhgukDha3MmmUo8KARBEARBEGY3aVyOtlVQiYS9AkXgw9BRyM4nV6zQnhqxwe9cCgP7oYmL/GzKm1EFRWTUOV3qHhQzYJKZmqUKilJdQRFTi4dtD4pqELsHQsJmi4cBD4poPOMbZFMxnfX1xzrsGKIWhrhbbCBSINhq8TCV4iEFCkEQBEEQhFlLm04w5Gi8BQuo2CpQDB0FNGS6yZf8UQWKZRBUwhaQBsmmXAoz4EFRjxltsUAxUx4UpRhbPNKeW/eFsEGkoIjLJDNl24PCD2KNGAW7LR5lP6hRo/T2AAAgAElEQVS3ZMRF5OFgOmq0bjJpwOQzHN+wgsI3bZJp9vxH9QMTCgpHmU8hGTOn1dkEQRAEQRBOclIqxZATkFi2jMqLe+xMmq8VHzLdDJaqtKdHKSigaaPMmYgZjQoKrbZ4zJQHxVAlxpjRhFP3hbBBqXauZq0HRTWI1SATIgWFxRSPmDeYkYKiYqlFIhGzAiFSZBj3cNBmWzxmvUmmXSGaFCgEQRAEQRDipM1pI+9AcuUKStu22ZHHRuqI7HxypXFaPKBJo0yvnqhhk3yU4tFqi8cMeVAUKz5pL64UD7deNLBB1E4Sd4qHPZPJgMQsVlBUDSgoPEspGL5vxgMhKhjMZpNMz6IHhZECizKfQjJmTquzCYIgCIIgnOS0OVkKjoO3/AyCwUE7PhR1BcV8csXq2BYPaKpAkUm6M6KgGCr7OGq4PWC6RDGlNossWmuKFb/l9pSIVMKhaNGDIlI6xKWgiAwfy7PeJNPS+oP4PSg8SyaZUQtG/Ckk4XiB4Q2+UZNMpYwXWKLzY6LFw4aHxmikQCEIgiAIghAjmUQ7ALkFnQCUtm0zP2lNQaEz88iNbvHIzAcn0VSLR/sMmWQWyj6ZpIdqcaMQFQlstniU/YBAD7eXtErKcylXA2sGdaWYCxS2WzxK1fhTMBKuMu7fEFH1g9hNGqMNvunX4BvaILuWYkaNm2TO4hQPx4KHxpg5rc4mCIIgCIJwkpNJdgFwdG4SgLKVAsURAErJuVR8faKCwnGgc8msaPEolKstt3dAqMBQCooWX0PkF9Gq+iMi8rKwleQRGXLGlcQQ+UHYWn/FgEmm5zrWFCBVXxtQINRMMi0pEGJP8bDUohIVcAzUJ3AdhekaV3R6zHloxD7s5HPanU4QBEEQBOHkpj09B4B+crhz59pRUOT7INVFrhpe2nWkR5lMdi6D482YZM5Mi0eooGi9QKGUoi3hWlVQRH4RcSoownHtxXRCjAoKyzGjFT8gGfMGP2m1xSN+BYhnPQUj7hSS2a+gcJT5FpXAoEmmKx4UgiAIgiAIs5vOtm4AjuUOk1qxgtK27eYnLfRBtptcMSwqnKCggLBA0UyKR8qzHtEJkYKitQSPiLaEa1UFEikoYosZrSso7LyGuFs8HEeRcJVFDwozG3x7LR7agALBTotHVACJ20PDrad42EkhMRIzasMk06AHhbR4CIIgCIIgzHLmtC8EYKBwmOSqlXaSPPJ9oUFmaaICxdKwxaPBdWRTHhVfW9scR8SloIBQyWBTQTFkSEFhK2q0XA1wVLybnKTrWFNQlE14UHh2FBRaa6qBjj/Fw7WjQDC1QR6OGY112DEEhk0yfdMxo4ZjUi0LKKRAIQiCIAiCECfzOhcDMFg8SmrlSjtJHoUjkOlmMFJQjNfi4ZfqXhVTka0VCQol+wWKuFIw2pKu1ZjRYr1AEc/ldeRlYU9B4deLInGR9CwWKHwduwdF2OJhfncWFRDiblGJWhaqxlM8zHhQ2FJQ1GNGZ6lJpskUD0eZ9wAZM6fV2QRBEARBEE5yFswNYz3z5QFSK1YCFpI8ai0eUfJGRypx4uOdS8PbBts8MjUFhm0fikK5SjamFo9M0naLRzhXfC0edhUUpWpAKq7+jho2CxShB0X8LR42FBTRHLNfQRGzB4Wys/56zKgxk8zZm+IhMaOCIAiCIAiznEXdpwIwVBkktSosUBhN8tB6bIvHeAoKaDjJI2oRyVv2oYi9xcNigSJq8UjF1uJhV0FRrsa/wU969lIwQg+KeDdotlo8IpWGqRQM0x4U9QJLzOt3HGXlHXyzJpnmPRxMpngoJQUKQRAEQRCEWU0mnSUdaIb8HF53t/kkj9IABBXIzmewVqDIpkZtkptUUGSjAoWJFo8nvw5fuWFcP4yhOFs8ErZbPMJNWlwtHpGCwl7MqAEFhcWYTiMeFLUWD9MeMlELRuwmn7WCTcVWi0TMBSII38G3paAwYZLpWVAgmE3xkJhRQRAEQRCEWU820AwFRQDzSR75vvA2M5ziMabFo30hKLdhBUXkQZE30eKx/R548TE4vmfMQ/lytV4caZVM0nLMaDXeFo9IQWGryFKq+rErKBIWTTIrfhC7B0XCUsxl1dAGP1FTBPjGUzwMtxhY83AwoKCwkARjNsVDPCgEQRAEQRBmPZlAUdRhgcJ4kkdkfJmZT65UwXXU2HfxHRc6ljReoKgVCYxEjfbXijX7151wtx9oipUgtg2+7ZjRqJ0krhQP2wqKcjWI3SQzZdMk01CKB2C8zSMaPxG3h0O9wDJ7TSY9x7GnoDDQIuFaaJGIxndMFCjEg0IQBEEQBGH206ZdilQAzCd5FGoKimyooGhPeajxLrQ7l8LAWNXCeERGlbm4WzyCYLhAse/pEx6K1A5j2lOmSXrGUjxmq4Jitptkxp/iERU8TCd5RO+wx66gsGSSaVpBYdzDwbRJpi2TUkMxqaYVLGPmtDqbIAiCIAjCy4C09iiqUH1gPMmj3uIRelC0T9Qi0bWsCQVFLWY0bgXF4D6oDoUf7ztRQRHN1RZTikebZZPMYm0jHluLRyIyybTnQSEmmScSjWdaQREpHOJO8agrKIy3GJgxyYTIg8KwAsSkSaajMJ1UG9UPzKV4xD7spEiBQhAEQRAEIWbSKknBCS+qjSd51BUU88kVq3SMTvCI6KwVKBqQ62ZNxYz2187B3DPCFo8Ra4mKCZmYNviZpEuh4hs3OIyI1p+K6V38tBfFjNpUUMTb4pG05EERBJpqoI2YZIKNFo/wezQR8wbT9vrNKShiH/YEjMaMKsx7aNR+xxkQUKAUxmNSRyMFCkEQBEEQhJhJk6Kgau/KmU7yKPSDl4ZEhtxkCorOpVApQPHYlEOmPAfXUfGbZEYFivPfHK77+Iv1h6LEkNhaPBIuWttTIBSrPknPia0P3LqCouLHVlyJsNXiEak0TLV4mFYgROObSvGw1WIQ9/ohVGX4hhUUgUEPjdne4uEqZa3IGxHLV0Ep9Vql1HNKqW1KqY+M8/jblFIbav8eVkqtGfHYC0qpjUqpdUqpJ+JYjyAIgiAILz9eStcjbU4bhREbVaNJHvl+yMwHFRYU2idUUERRo1O3eSilyCbd+GNG+7dDIgNn3xh+PqLNY6gSf4sHWFQgxGjwCdQNK0sVSyaTBlIwkp5rpcUjUgjEn0IS/gybfg3R+HF7UESKhsos96CYzSaZjgWTTLMpHuYLLGPmbHUApZQLfA54HbAauEUptXrUYTuBX9FaXwj8JfClUY9fp7Veq7V+RavrEQRBEATh5cdL7XqkzctSchSD+VCtYDTJo9AH2W6AyT0oOpeFt00keRhRUMxbAYvOA8c7wSgzStzIJONr8QCsRY0Olf2x6Skt4DqKhKsoVu0VWGJXUFhq8agYUiDYapGoRikeca+/HjNqOsXDnAeFZ9Ek0425QASWTD61RinGN0duEafmQWFTRRHHT8FlwDat9Q6tdRn4NnDTyAO01g9rrY/WPn0UOCWGeQVBEARBECJeUtcjmUQHAAf7w9QMo0ke+b5QQQH1FI9xqSso9jY0bDblxR/T2b8NuldAIg0LV58QNRqpNeIqULQlI6NPOxv8YtWPLcEjIu25VhUURlo8LCgooiJI7C0Slkwmq/UWg5gVFLZSPAx6UDizXUHhKOMeDoHWRtYOw+fEZpdHHBq6ZcCLIz7fA1w+yfHvAH464nMN3KWU0sA/a61Hv5sBgFLq3cC7ARYsWEBvb28raxYaJJfLybm2iJxve8i5toeca8ESL6nrkXJBQxIeeuxu9uzoI5HLMQ94/Hvfp7z63EZfU0NcfmQPA50dPNPby/FCiaOH99Pbe2Ts2oMq1+Kwa8PDvDB4xpTj+qUhdu8fiu3nVwUVrj3yArvbL2Fnby9nqUUs2PUoD917LyjFU/tCtcbGp5/k8PMnbjSn83tk28FwvAcfeYwXO+MtHIzHnv1F/FIQ6+87pavs2P0ivb2GImpHkB8qcfjAfnLJSmyv4dCBEoVi1fjfgMOFsECxY9tz9A7tiG3cZw+H30OPPv4EfVvNfA/lcjl2PB0W6jauX8fQ7vjmKdUKB89t3Uavvzu2cUez+cUwUvnxXz7K9nS8RaLSUIGDB4uxfA9N9Htk2/YyAA8+eH/sRqX9h4vk8/H+XhjNzhfKoLWROXbtCs/NPb29RhQy4xFHgWK8lY5bY1FKXUd4QXD1iLuv0lrvU0otBH6ulHpWa33/mAHDC4UvAZx99tm6p6en5YULU9Pb24uca3vI+baHnGt7yLkWLPGSuh7ZOfgAHHmSBUvm0nN1D9ULLmDrpz7NOdkM8+L+eXi4QNuZ5zH/2l+h9LM7OHflcnp6zhr/2KcXcca8BGc0sIYvbX2Uih/Q03NlPOvs2wr3B5x+8as4fW0PtO+AH/+cnrVnwtzT2fPoLtiwieuvuZKFnekTnjqd3yPO84fh6V9y3oUX8Yoz5sXzGibhazt+STVZoafnqtjG7Hj0HroXdNPTs2bqg1vE/8VPWXHGabRnDsb2O/uh/BYe3r/b+N+A7YdzcP99XHDeanrWLott3MS2PnjyMS5ccxGXLTfzPdTb28vqM8+FJ5/gsksv4cJT5sQ2dsUP4Oc/5fQzltPTsyq2cUfz4qO7YPMmrrnqKhZ0pGIdu3P9A8yd20ZPT+tOABP9Htngb4Wtz3N9T0/sKpDbD65jb+mI0Z+BhwvP4O15wcgcm/U22Poc11x7bd0XxzRxlLj2AKeO+PwUYExzo1LqQuBfgJu01v3R/VrrfbXbQ8APCCWagiAIgiAIzfCSuh7prHlCHMsfBgwmeVSKUM5BprseCTphiweEbR7HG2vxyCQ9cnGaZEYJHt1h7CpL1oa3NR+KKKazLWYPClstHkMVn3TMLRKphGPFg0Jrbcgk006LhymTzOgdY+MeFHUPB1PrN2zS6M9uD4p6zKgBgYDjKHzT5z8w1+Lh1MY1HKRy4pwxjPE4sEoptVwplQRuBm4feYBS6jTg+8Dvaq2fH3F/VinVEX0M3ABsimFNgiAIgiC8vHhJXY/M7VgEwGChr36fkSSPaPzs/HqBomOiFA8IjTIbNMlsT7nxmmTWCxQrwttF54GTqPtQDJtkxpPiEflB2DLJLFXi96BIWfKgqPgarTFgkuniB9r4BtOUB0Widj5MF1mGTT7j3WQqpWopGKYLLOZMJm14UASBxnWUEZNJV5n3oPADHVu88WiiHynTSSQjafkvgNa6qpR6H3An4AJf1VpvVkrdWnv8i8BHgW7g87UvfLXmkL0I+EHtPg/4ltb6Z62uSRAEQRCElxcvteuR+V1LAMiVjtXvS65aycCPf4LWOr4L4XytQJGZT64YKSgSEx/fuQy239vQ0KFJZswFirZ5kKlJ5b0ULFpdV1AUylVSnhObxDpSYtiKGS3GHDMKkE44lCwoKKI5Up4LMe5lI0VGuRrEpowZj7qCwkAKCdgwyYxiRuNdP4QKBNMbfN+QyWc0ZmBBQWFMgeAoTIuItNZGDEphWEFhusgyklhK1FrrO4A7Rt33xREfvxN45zjP2wGYb6oTBEEQBOGk56V0PTJ/btgHn68M1O8bmeSRWLQonokKtS6VTDe5UmhU1z6pgmIplAehOADpzkmHzqa8uiojFvq3w/xRffBLL4LNPwStKZT92BI8YIZaPGKMGYVQ0WBDQREpEJKeA+X4xrVVoChXzcSMeq6dFg9TCgqoFSgspZCY2CTbUIAEWhNzd00d1zGvPvC1rhcS4iYaV8+yFg9BEARBEARhBAvnLkVpzVAlV78vtSrcnBc3b4lvoqhAkZ1f94uY0oMCGmrzyCY9ipUgPnl+39Zh/4mIJWuheAyOvlArUMTT3gHU1QxDtmJGTbV4WFFQBLX54vegACj5Zl/DsIIi3k1aVPAwX6Aw06ICoSrDuIeDHykozChATK/fD7SRtUOtxcP4+jFYoKjNYVFBIQUKQRAEQRCEmPG8BNlAM+Tn6/e1rVmDSqfJP/RQfBPVWzy66y0eU3pQAAxMbZSZTYWb7XwcbR6lQcgdGPafiFhaM8rcv45CuRqrgsK2B4WJAkU64VC0raCIkZQ7rKAwiSkPimS9QGFrg29GQWG6wOIHAUqZVFBY8HAwlKDpWGhRCT00zIwdfU1telBIgUIQBEEQBMEA2QCKulj/3EmlyFx2KfkHH4xvkkIfKBfSc4ZbPOJSUNTGKcSR5NFfMwcdraBYuBrcJOxbF3uLR8pzUMquB8XsV1DEu/6RLR4mMeVBYa/Fw6AHhWunxcNEcQXCDbIVBYWhHb4Vk0xtzkND1VM8pEAhCIIgCIIwq2nTDkO6dMJ97VdfTXnXLsp79sQzSb4vNJ10HAZrCorsZAWKjtC8s5ECRVQsiMWHYnTEaISXCosU+56uKSjia/FQSpFJuFY8KPwgjOmM24PCloJi2CTTTIuHaQVC2VCLhK0Wj0ghYMaDwrGiQDBl0mgrZtRUi4SNAkvooWFu/eEcRoYfFylQCIIgCIIgGCCtXYrqxM199uqrAeJTURT6ITMfGC4kTKqg8JKQXdhQi0c0TixRo5GCYt6ZYx9belHY4lGKt8UDwiQPGy0e0QY/7hQPWwoKUy0eCUstHlEBJGmsQGG6xaOmoDDh4eDaiRk15uFgY4NvUAHiOMp4e0RgsMAiHhSCIAiCIAgnCW3aG1OgSC5fjrd0Cbk4CxTZWoGiGG7wp3wns3Npgx4UtQJFHB4U/dug61RItI19bOlaKB5nbmlv7EkP6YRL0YKCIjLiNOFBUTK8uQfzJpllwyaZpjwoEpLi0RBVPzCooDCvAKnOcgWIr834f8Cw+aa0eAiCIAiCIMxy0ipFQZ14UaeUov2qqyk88ii6Uml9knwfZLqBUEExqXoiouuUhlM8APKxeFBsG2uQGbH0IgBOKz1fnzMuMkk7LR7F2gY5/phRl2LFRxt+97Le4hFzgSVSNJguspjyoKgrKAyvvxqEG3xl4F3whOtYUVCYKK6AzRYJM2M7ShFojP4MByZNPpWYZAqCIAiCIJwUpFWKgjP2oi57zdUE+TxD69e3Pkmhr66gGCxVaZ8swSOiYQVFuFkttKqg0Dps8RjtPxGx4Fxwk6z0t8WuoGhL2GnxiIw44zfJdAg0xt9Brrd4xJ2CYdkkM+5NciT7r1iI6TRpMmlaQWHSgyJM8TCdQmK2RSWawxSmz380hy2kQCEIgiAIgmCANjdD3lEEo+Tt2Ve+Ely39TYPvwpDx4YVFMUqHY0oKDqXQvE4lHKTHha1eLRskpnvg9LxiQsUXhK96HzOCbbH7kGRtlSgMNfiEY5nOomk3uIRuwLEUsyoIZNMpRQJ13xMZ8XXsReHIjzXTouEyQ2+4fqE0ZjR+gbfpIJCG/SgEJNMQRAEQRCEk4M2rx1fKfoHDp5wv9vRQduaNeQfaLFAMXQU0HWTzHzDCopl4e3g/kkPiy1mdKIEjxEEi9dwvtpJNuYCRSbpNr+5z/dBabCpp0QtErErKBJ2WiRKphUUpjf41cjDIf6tTcJ1jLd4VPygHmkaN54lBYJJD4fZrKAY9nAwMnw4tskChRqewxZSoBAEQRAEQTBAJtEJwKEjY/0esldfRXHLFqpHjkx/gkJfbbAmPSg6l4a3U7R5ZBIxxYzWCxQTeFAApQUX0KkKLKxO7Y3RDG3NelBoDV99LfzwPU3NE0WBpmP2QEh7s1tBkbSU4lH2fVxHGdkkhxtk0wqEAM+UgsJR5lNIDKZg2PCgqAYmYzrDW5MKCqMtHuJBIQiCIAiCcHLQkZoDQP/xsYWA9quvBq3JP/Tw9CfI1woUNQXFYLFKeyox9fPqBYrJiwGOo8gk3dZjRvu3gZOArtMmPCTXfSEAy4aebW2uUaQTbr39oiGO7ID+rfDcT4fPbwNEBYS4PTSsKShq6095MZtkWvOgMNcikfQc8woQX5MwtMFMuI75Db7BFA/XQoEo0AZjRpUFDwqNsQKLsrD+0UiBQhAEQRAEwQAdbaGy4eioFg+A9Hnn4XZ1kW/FhyJSUJyQ4tHABrOjMQUFhG0e+VZTMPq3wbzl4E6s7hjoWEFJJ1g4GG+Boi3RZIvH9nvC26AKG7/b8NOGjJlkhuOVKnY8HEzFjJr2cChXA2MpEgnXoWp4/VXfnIIiNMk0n+Jhcv2zW0FhPqYzCDSGvv3r67cooJAChSAIgiAIggk6a+kaxwtj34lXrkv2qivJPfzQ9OPnCv3hbXY+WuuwQNGIB0UiHRY1GooajUNBMUmCR42C7/CMPo15xze3Ntcomo4Z3X4vzDkNlqyB9bc1/LThFg8zCopi1XCLR8WMB0XCYsxo3BGjEZ5rvkWiEmhjHhQJ17wCwTfY4uHZiBm1sME3bZJpSsESDSsKCkEQBEEQhFnOvM7FAAwOje8zkb3qavzDfZSee256E+RrBYpMN8VKgB/oxlo8IGzzGKf1ZMwaU15rMaOBH7ZNTOI/AZAv+WwMltN5bHOsbnJRzGhDRSC/AjvvhxXXw5q3wv51cOiZhuapx4wmzaRg2FBQJF0n9neRU5ZMMkMFhZltTcI13+JR9QMSBlMwTMeMVo3GXJpPIbFjkmnWg0IZT/GQAoUgCIIgCMKsZkFX2EqRKx0b9/Hs1VcDTL/No9AH6S5wEwyWKgCNKSgAOk9pUEHhtWaSeXwP+KUpFRRDlSob9Jl4lVxY0IiJdM0ToqF38Pc+CeXBsEBx/pvA8WD9txuap2ioxaMeM2pBQWFCgWDLJNOkgiJppcXDnILCcx0qxlMwglmtoPADjaH6RP28GFdQGEvxkAKFIAiCIAjCScHC7lMBKFTGj6xMLFpI6qyzyPXeN70J8n11g8xcMSwidDSS4gGhgqIhD4omWyRG00DEKECh7LMpWB5+sn/d9OcbRZRE0pBR5vZ7QDmw/FpoXwArXwMb/iNUgUxBvUARd4uHJQVFqerH7j8B4buvnqOsmGSaUlDYa/EwpACxsMGv+AZbDGrrn3YrXAP42qCConZeTKpY7KR4GBl+XKRAIQiCIAiCYICu7Fw8rRmq5iY8pvPGGyk88QTFZ6dhDlnoP8EgE2gsZhTCAsXQEagMTXpYJtWigqJ/e3g7VYGi5LNVL0O7adj39PTnG0WUqlFoxChz+72w9GJomxt+vuZmGNwPO6cuIBUrYYpB3EaNkYKiZFhBUa4aVCB4joWYUbMtHqZNPsMWD4MtEoYLLH5gsEDkmN8gGzXJtKBAMJniIR4UgiAIgiAIJwmO65INNENBYcJj5t5yM04mQ/9Xvtr8BIV+yJ6ooGi8xWNZeDtFm0d70mvNJLN/GyTboX3RpIcVylWqePgLz4N98Sko0o0qKIaOwd4nwvaOiLNeG7bQNNDmMVTxSXtO7H3g9hQUgREFBdiJ6SxXA5IGUzzMFyjMbfATrjK/fqMeFDUFgsE2FSsmmQY3+FqbW794UAiCIAiCIJxEZANFUZcmfNzt6mLOb/0WA3fcQXnP1C0XJ5Dva01BAVMWKLIpj0KpxRaP7hUwxca9HmW6dC3sXx+bUWZb5OEwlYJi5/2gA1hx3fB9iTSc95vwzH9Bafw2nYhixY/dfwJGxIya9qCo+vW54ibpmldQmPSgSFho8Sj7gUEPChseDmY9KMI5TLdImG3xMKqgCHTdKyJuhk0+jQw//pz2phIEQRAEQXh5Mcd3eMEtUA0mViHM+2+/D0px5BvfaHxgrU9UUNQKFB0xKyiyKZd8uTr9/u/+bVO2d0CocHAUuMsuCo0qj2yf3nyjyCTD8zGlj8aOe0OlxymXnnj/2rdCpRAWKSahWAmMFCjSUcyo6RSPWd7iUZntLR6BufV7jh0FiHkFhekChZmxoxYPk18C32SLSu28mDT5HI0UKARBEARBEAzx2kKWA57PD7f9cMJjEosX0/X613Psu9+levRoYwMXj0NQaUFBsSS8ncIoM5vyCPQ0N8jVEhzb3VCBolD2ySQ91NKLwjtiavNoq8V+Dk2loNh+D5xxDbijYlpPuRTmnQnrb5v06aGCIv7LansKCrMtHiXTLR4mTTIdx7iCourrWZ+CYUoBEhUojMZ0GjTJrG/wDa5fUjwEQRAEQRCEhlhTncs5Rfj8us8zVJ3YkLL7HW9HDw1x9FvfamzgQn94W0vxGKx5UGQbLVAks5CeM7WComYyOS2jzCM7Ad1ggaJKJunCgnPAi88osyEPiiM74OgLJ/pPRCgFa26BnQ/AsRcnHMJUi0fCVThqlisoXIeKaZPMqjkFQtIz7+FgUgHiuoqK4QJF1WCLhGdJQWHOZNJOi4exFBVlvkA0Zk5rMwmCIAiCILzMqCa6+IMjQxweOsw3n/nmhMelVq2ivaeHo//+TYKhyZM1gOECxYgWj4SrmnsXvHNZQwoKCAsITVOPGF0x5aGhgsIF14PFF8QWNRq1eEzqQbH93vB2vAIFwIW/BegwcnQCilW/7ncRJ0opUp57ooKiNHEqzHSZ7SaZFd/c+u20eJhTICQch6qNFhWDKSRg3oPClILFjkmmuRQP10KKymikQCEIgiAIgmCIINnBFcUcPaf28JWNX+FY8diEx3a/6534R49y7Pvfn3rgfF94G7V4FKu0p7zmUiQ6l05ZoIg2+NNSUEQFinmNFSjaanOx9KLYjDKjosGkHhTb74GuUycupMw9A06/KkzzmOBd0KGyGQUFhD4UpUiBsG8dfHIFPPqFWOc4GUwy4454jfAsxHSGLR6GFAiuItCGWyQMelBYU1CYapGIChRGY0Y1hk5/3d9YYkYFQRAEQRBOAnSqk3Y1xPsufB+FaoEvbfzShMe2XXwxbWvXcuRrX0dXpygIFGoFihEKioYjRiO6lk0dM1pTUOSnk+TRvw2yC6BtzpSHFsrVejsJS9ZCOTdc4GiBqEAxoQeFXw3bN1ZcN3nSyJpboH8r7H1q3IdDk0wzl9Upzx1WgNz3CagW4ed/Coefj22OWW+SabjFw4YCxFyBxQ8RTRkAACAASURBVPwG36QCpL7BN1gksmGSabRAFJjzoIgKT9M2Sp4GUqAQBEEQBEEwhGrrAmCxM4+bVtzEt5/9Nntz46sWlFJ0v+udVPbs4dh//ufkA49SUAwWq7SnEpM8YRw6l0H+cGhmOQHZVLjBz0+rxWN7Q/4TECkoagWKulFm6z4U6WSUgjFBgWLfU1A6PnF7R8Tqm0JvjAnMMotVn5QhBUUqUlDsXw/P3QGXvguSGfjhrWGBJQZme4tH2Q9IGGzxMN8iYW6D79V23lWDOZEmPRA8SwoE0x4aRk0yLXhoSIqHIAiCIAjCSYBTK1DkB47w3rXvxVEOn3v6cxMe337ddWSvvJKDH/s4Q5s3TzxwoR+8ttDsEsiVKnQ0apAZ0bk0vB3cP+Eh2bqCYpotHg34T8AIk0yA+WeFry0GH4qk6+A6amIPje33AAqW/8rkA6U74ZzXw6bvjlvQKVUCIx4UAOlIQdH7CUh3wav+L9z4d7D3SXjkn2KZo1QNSBlSgNho8ShXA5KzOMXDpEmmDQVFxQ8MpmBEG3zTBRYzYzsWChS+lRQPI8OPP6e9qQRBEARBEF5eeJmwvWFooJ/F2cW87dy38eMdP+a5I8+Ne7xyHJb+/d/hzpvH3g/8Ef7x4+MPXOivt3dA2ILRdItHVKCYpM2jbpLZbItH8TjkDzWloMhGHhSRUWYMCgqlFG0Jl6HyBJub7feGio3MvKkHW3MLDB2FrXeNeWjIUMwohAqKRYXn4bmfwBV/GBYpzn8TnPvrcO/fwMEtLc8RbvANeVDYaPHwtbEWlYS1Fg/DBQrDLRImY1LBvAeF8QKLQQVCYNAkMxpWUjwEQRAEQRBOApLZsEBRzIXmmG8//+10JDv41FOfmvA53ty5nPLpT1E5dIh9H/4Ierx3DvN99fYOqHlQNK2gWBbeTlagmG7MaP/28LbBAsXQyBYPqBllboBgGt4Xo0gn3PE9KIrHYc/jof9EI5zZA+2LQrPM0UNVfNKGTCbTnsvrj/4bpLrg8j8I71QKfu0fINUBP3wP+JWW5ihVfXMKCm+EyachTHo4JG20ePgGN/gWWjyqgcY1dP5dSwUWYyaZylKLhyGTzOEUDylQCIIgCIIgzHpS7XMBqOTDAkVXqot3XfAuHtr7EI/tf2zC57WtXcuiD3+YXG8v/V/68tgDCn0nKCgGi9MwyawrKCZO8ohSPJqOGa0XKFY1dHi+XK2rNQBYuhYq+ViMMjNJd3wPihceBO1P7T8R4Xph5Ojzd0K+v3631ppiZVSBJUZW6Z1cVnwYXvneEw1H2xfA6z8VtsI8OHHBayqCQIcKBEPv4KcMe1AEgaYaaIMKBIdAm9tgaq1rHhSioBgPGzGdvkEPEBsb/NBDY/YWWMbMaW0mQRAEQRCElxltHWHrQCV/tH7fLefewuLsYj715KcmdUaf+7a30nnjjRz+zGfIP/LIiQ8W+kcpKCrNKyhSHZDqnFRBkfQckq5DrtkWj/6tgIJ5y6c8NAg0xdEeDjEaZbYl3PELLNvvgUQWTrms8cHW3AJBBTYPR8GW/YBAYyxm9E2D3ySnsnD5rWMfXH1T2O5x3ydCxck0iIoHsSgoBg/Ahv88QfmSdB0qBgsU0fpNFSgSXrhBM/UaorpBwrSCwlCBIiqwmDKZtNEiYVJB4dY3+EaGr41tPibVooBCChSCIAiCIAimyHaGBQp/aNhLIuWm+MO1f8jm/s3ctWusn0GEUoolf/kXJJcvZ88ffZDc/fcPP5jvh0yooKj4AcVK0HyBAkIVxSQKCgiTPJo2yezfBnNOAy815aFR+0VmpAJh/lmQyMC+1o0y00mXoco4u4Pt98AZV4OXbHywReeF/hjrvlW/q1gb20gKxoFNXFx4kO8lXj9xXOuNfwdt8+CH74VquekpSvX1t1hgKRyBb7wBvv9OuO2WsIWGsHBg0oMiKhwYSyGpbfBNFyhMKSii1hdTLR7RG+vmPCjC82LcZNKYSWZtDsMtHuYUFOGtpHgIgiAIgiCcBLR3hQUKXTzR7PINZ76BlXNW8pmnPkMlmNg/wMlmOfWf/5nE0qW8+Ae3cvifPosu5cP2h2yooIiKB9MuUByfqkDhNR8z2r+tYf+JaOzMyPU7Liy+MBYFRSbhUiyPUoAcfQGO7Gi8vWMka24J40kPh0anpVqBxUiLx32fYMjJcpt6/cTHZObBGz4NBzfC/Z9seoqSH66/JZPJch6++RY4thuufD9svxu+/Cro22rcJDNK2DBtMmkqySOqe5jy0HANm0xGhRtTG2TTHhRa61qKh1kFiMkWj0AbPP9KPCgEQRAEQRBOGhLJFAWdQo0qULiOy4cu+RC7B3fzvee/N+kYyVOWccZt36Lrppvo+9znePEPbsUvqbqCYrBYK1A060EBoVHmJC0eANmk15yCQuvQg6IJg0wICwknsHQtHGjdKLMt6VKojFr/9nvD2+kUKC54Cyi3bpYZKUBiN8k8sAmeuZ1HFryFPr9t8mPP+TW48GZ44O+bLuqUWlWA+BX4zu+FRZs3fw1u+Cv4vdvDxJMvX89ZA49QDbSxFICK8RYPSwoKwwoEUxv8SBlgusBiSoEQDWsqptO14OHga42h5aOiAoV4UAiCIAiCIJwc5FUGpzw45v5rll3DJYsu4Qvrv0ChUph0DKetjSUf+xsW/9mfUXhyHTvvXEDu+T601vWEjY5pKSiWQe7gpCkQ2ZRLYbQCYTJyB6Gca1xBURqnxQNCH4pKAfqeb3zucQhjRketf/s94Wuf35iJ5wm0L4SVr4YN/wFBUG/xiN2D4v6/hVQnjy++uV5EmJTXfTxc2w/eA9VSw9OUW2mRCIKwtWTbL+AN/wjn3Bjef8ZV8O5emHsGN235EO9xb6dcbT2RZTwidUZDG+TcoSkVQ6NJOIYLFLWNn3GTTEMtHpEyY7Z6UAyff7MeDkYVFIE2V2CxYFI6GilQCIIgCIIgGKTgZHErYwsUSik+dMmHOFI8wje2fGPKcZRSzL35tzn9438EwIt//i/sfNObKP78ThwdTFNBsRTQobnhBGRTXnMxo1HyRveKhg4fqozT4gGwZG1426IPRTrh1osIQKjI2HlfGC863Yv6NTeH3h0vPFBPCGlLxnhZfXAzbPlRGCvaNpdiI5v7trnw6/8Eh5+B3o81PNW0FRRaw13/GzZ+B171Ubj49058fM6p8PY72bHoBj6c+DbO998B5ckLcdMhKrBM2qJSGYLej8OnL4BPnw//8Tuw65GGnP+GTTINbZBrw5pKUYk23uZaVGobfGMKkGiDbLZAZMpk0rORQmIwxWO4xcPI8OMiBQpBEARBEASDDDntJMcpUACsWbCGV5/2ar6+6ev0D/WPe8xo2k7p4MxfO8SSj7wfPVQk/Tcf5cu/+ARdd/6IyoGJCw3j0rksvJ2kzSOb9Cg0k+JRL1A0pqCI1BljFBTzV4UpGy36UGSSbr0NAwjHKx6HM6+b/qBnvw5SXbD+NjMtHvf9LSQ74Ir3kvIcKr5ubIOz6jVw0e/CQ/8Ie55oaKpSrfjRtEnmg5+CRz8PV7wXrv7v4x+TzPDwmk/w8crNJJ79EXz1htCnIkYiZcOEG/znfgafvyIs2px9I1z1wTBi9muvhS9fDxu/O6mCKGHYJDOy5zD1Dn60flMb5EiZMVs9KCJlhqkCS1T4MOUBorVGa3MFFlX7sZp1HhRKqdcqpZ5TSm1TSn1knMeVUuoztcc3KKUubvS5giAIgiAIjfBSvR4pe+2k/NyEj3/g4g9Q8kv82cN/xv177udI8cjkAxb6cFyY89u3cOZPfsz+P/5zBpIZEp/9e7b1XMeON7yBg5/4W3IPPURQmOId686l4e0kSR6ZlNu8gsJNQdcpDR0+YYuH48KSC2F/awqKtuSomNHt9wKqtQJFog3O+w3YcjuVwgAAqbhaPA5ugS0/DNUTmXn11pGGjSZ/9a+hYyn84NZQOTAF0bhNmWQ+9W9w95/DBb8FN/z1pEqUZMLli/6v03/Tv8HR3fClnrBAEBOV6gQmmUd2wrduhtt+O/x+/L3b4S1fg1f/KXxoC/zaP0BpAL73DvjHNfDgp0PfjFF4hls8tF/heucpXrHu/8AnV8EXr4GffgS23A75vpbHH97gGyqw+IYVFK5ZBYJfW79juMBiysOh7qFxEplkTkMLeCJKKRf4HPAaYA/wuFLqdq31lhGHvQ5YVft3OfAF4PIGnysIgiAIgjApL+XrkYrXztzSxAqF5V3LeccF7+DLG75M755eAJZml3Le/PM4r/s8zp9/Pqu7V9OR7AifkO8LTRrTc1BKcXDNFfzJtRke+u0zSK97nPyDD3D03/+dI1/7GgCJZctIrVxJatVKkmeuILFkMd7ChXgLFuB0LEHBpAqK9mZTPPq3w7wzwwJDA9RbPJLjXJYuWQtPfh21fPr+BVGLRxDocBOy/R5YsqaegjJt1twCT32Drl13AqfSFleB4v6/hWQ7vPIPgeHWi2LFbywpJN0FN30W/u034J6/CgsWk1CqNtni8ewd8F8fgBWvgps+N5yjOAFR4WPw1OuZ/6574Lab4V9vgtd+HC595/TbbGqUaykkkZkllaFQQfLAP4DjwWv+Ei6/9cQ42WQGLn0HXPL/wda74NHPwS/+NFSuXPS28Phai1LSRItHZQi23Q1bfsRvbv4xNycLVPZ3wNk3QP4wPPl1eOwL4bHzz4bTr4TTrwpvu5Y1NVXkzVGZzga5nIdDz8DBTWHh7PiL0LEE5p5R/xewCDDnoeE2qkAIAkA3/HsnIlJQGBKwWPPQMFSfqCszDNW3xqXlAgVwGbBNa70DQCn1beAmYOQf9ZuAf9Vaa+BRpdQcpdQS4IwGnisIgiAIgjAVL9nrkWqyk7bc5EqG91/0ft5+/tvZ0r+FLf1b2NS3iU19m/j5rp/Xjzmj8wzOm38eK/q3kOxeSOLZ20i4CX55+Bhe1yGe7HRpf/XpJG5YSaL8+6Q2bMXZuovqzj2UXtiBevgh1Kg0C51KQmIx3Pd19MK7oT0L2Qw6nYRUClJJzjo8xA2HCmz6wtPgeuC5kPDAdcPNqeOAW/unHHj6GWhfAD/9JqDCDagClEIz/DHhyaZ/Zz/nDe5j/2MD5NoS9fsBOKyhXzGw4SdsdI6NOmvjXJGPs9lNvHCQc/IHWNc7REJXYPOG0OTyvh9M+jWZGg3+qeR++e+ck381B54ewN+abm3IwX3wyF1w1g3w+H0AVHf0c05+D5sfzNOVTjQ+1rwb4M6vQX7OpO02+/Yd55z8Cxxed4yNuzIcfv4ZNqqxSgIA+rfCI/8EnWfD0rfAQz+ZchmFPcc4J7+LbQ8PkOtKw+oPwRNfh29/FB65Ey787bCQME1ePJzjnPx2jm44wsZ1O2HDd6DQD8sug/N/E6pzp17nGW+Hua+BHffC3bfBL74Jiy+AlddzrLqYc/I72f34Edwd7dNeJ34RDmwJ004Obga/BMkMBxOr+fHx03nlFVeyekE3LABO+z04vhv6toaKpAd+BPf+RzhOpjtsf5q3MiyitC9k3J+FGgeP5jknv42DTx9m4/6u8Q/SQVgYGdgXqqkG94Vmovk+IIoZSUHbPCg+NkaZ821S8IP5bLx3EWQXhMW/bDdkFoTeKGr6xbvDA3kuy6+n+NhTbHxOw9CxUPlSPD78rzQAxUHQPnhtkMqE7WHJdkhkIJWFRJbq0Rwb9/wMktnwXyJDTqc5N7+D8vP72Fid3/jCtA/VYvh1rBTDjyvF4Y/98LY8lOd/Dx1k6c9cNj6cDtU8bgLc5MT/vHEe95LgJId7LmpUfZ9z8pvwt77IRrUOgioElfC2Wg7X6VeG7/dHPB6MfKwaju244c+jCm99pXhTYS/tT3azsa/7xMeVG/5NUE54ixv+XVCtlRiUbrGao5R6M/BarfU7a5//LnC51vp9I475MfBxrfWDtc/vBj5MeEEw6XPHY1Vnm/7Hy5a3tG5BEARBONn4tbufeVJr/YqZXsdMMBPXI2effbZ+7rnnplzbo198L2v3f4f0nzcv1z5eOs7mvs1s6t9Uvz1UONT0OABOoFl4DObmYN6gZm4O5uY0nQXIlCBT0mRKkC1CsgLJKqQq4Fo0RxMEQRBmP6ufe3ba1yNxKCjGK9mN/lM20TGNPDccQKl3A+8GWNnRYnVaEARBEISTDevXIwsWLKC3t3fKheWHAtKqwi9+cRfeSJl5E5zFWZzlnMUbF7yR1U99mLLjse6CP8HXPt/bVuCXB0p89Mokvvbx8anqKr72ay9k+KXoxSM+rt1/+vZ/RekK21e+fdy51+8r8/CuEu9a7ZFRAcoPcPwAggAVaFQQoLRG+QGJ8gCn7/oOhxZcQ659JWhQ6PBsah2e6OjNMR3+90y/z5Z+n99clcBBccKp15rTdv0nA5nTObbg8knPkZrgTbcXBnyePhjwq2d4LBt4ko7BHew+/U0tvasb4VVynLLnv3ggOJ+Fp59Pypu+zjpRPs6yvXdwrGs1x+atqd+/dzDglwd8rj/NoyvV3PipoYMsOXAPxzvP4mj3JeMes3vA58mDAa853aM9qSiVy6SSJ36fepU8i/ffBcph/5LX4HuZhtdwIB/wyD6fa0/x6G47cf2Z/G7mH36UwElxaNE1lFPzmnp9BD7e4c0sLjyLUorjc85noOvsMe8yT4vApz2/k+yxZ2mrDlJ228h3nsVgx0oCd+KfY8cv01bYR7awm7ahAyjtU3XTFLKnks+cSim98ASlz/6BMo8eVFy11GNhtsGvr9YkKgOkSodJFw+RHjqM54cqLd9NUkotoJheSCk1j0qxwMH+I6xMD9ARHMerDqu5fDdJOTGHcnIuleQcysk5lBNdTbVJHC9p7tld5bLFLss6HNAa1y/gVXIkqjm8Sh7Pz9U+z+P6xUlGU1TdNL6Xxncz+G6aokqz4ViSRZ0Z5nVm8d02fDfV/M+v1lRLOdIOuEEJJyjj+GX8apEXjhQ5NVOhy6vg+iWc2uMAWiUInASB453wceAkRnyemOAxj3Kg+MmOKucvcFg1xz1hPRDgBD7oKk7go6jWPvdxgiqK8Db63NFVlPZRteO1cvBRbD2u6M54zG3z0Dhox0MrB42LdpzafS5auYBDoFxQDlp5BEqB8tBKhb9DdRCuq/6xz327y5zZCWd0hut2dAA6QBGgRnxcvy8I4K+ebe7rM4I4ChR7gFNHfH4KMLqRcaJjkg08FwCt9ZeAL0H4jsWNv5AuEBv09vbS09Mz08t42SDn2x5yru0h59oihly8Zwkzcj3SyPf2YwcfheNw0QWr6V7UmHHkpGyswuJzOeVVrwfgoWPrmXOkj7fe8KrpjffDTbCjF173h+M+/J3HX+SurRt4069exylzp9iYbv05fPNrcPMH4fRXNjT9x+54ho0Pv8D3PvS68Q/46iaOH+un67//bUPjjeb29fv44m1P89dvv5aV/3EdrL6UK3/nH6Y11njs/9QWuo/u5pQP/pi20VGpzfDdt4d99B+87QR/jLufOcg/feMJ/tfvXsWaU+c0P+4d/xN++c/wm38Oy68Z8/C3f7mbz39/I59+7/Us6Wob+zs7dxi++quwpAJvvxMWntPU9A9v7+PTX36MD95yBa9cMY7vx/4N8O23Qe6/4NWfCSNcG+G5n8JPPwyZXdyeeiXn/v4/ctXKs5taWyNsePEI//CFL/D3pz5E98EHofIUrL4lTC+ZX2udKRyBZ38Cz9xeM2GtwOJlsPqtcO6vw6mXT+jV8fnv3c1jjxf50O9cwRVnTtMXRWs4tgt2PQy7Hgpvj9wPFcCF8gKXoa6VdJ3RA4vOq/07H9oXtfx3Y9Pe43zsnx7kHW+5hBvOWzz1E0qDcHQXHH0BiscguxA6Fof/Mt1jiiMHjhd5/8fu5m/eeAG/eflpLa11vOuR3f0FPvjJe/m7t6zh1ZfE8Pt5FIPFCh/9s7t49WvO5eZrz4x9/GOFMrf+xc/56OtX87qr4+8w0FrzP/7kDj7wqlW88TVnNf7Ev5r+91UcBYrHgVVKqeXAXuBm4K2jjrkdeF+tp/Ny4LjWer9S6nADzxUEQRAEQZiKl+z1iNsW9n0XBo/GU6DI94UX8jVypSrt6RYu6TqXwuCBsDfZHTtOtrbpzjcSNdpkxCiEMaPZycwfl15E+54vT7i+qcjUzCsr/btCD4VXjK8UmS6b5t/Ia47/DcHhDXDKxVM/YTwOPwebvg9X/dEY884o/rPUaIrHaF79p7Dt5/Cj98J7HoHUiT4K0bjjxnSWBuGbbw69CX7/9qaLEzBsvlmeyGVvyYXw7nvhP/8b/OAP4MBGePWfT/y1PrITfvYReP5nMP9sHrrqa3zg7hR3N2ke2Sie69EbrOXxa97OaxccCaNVn/43eOIrsOpXwS/DzvvDXv85p8EVt8Lq34ClF09pIArDJoeJVlwalRo2rlxb+9U1sB8ObGAf87n2q3v4xK9cwpsMbMAj88qGY1JTHbD4/PBfAxg3mYxMMs14fBpfv+kUD6UUjgoLFbZo+Uuhta4C7wPuBJ4BvqO13qyUulUpdWvtsDuAHcA24MvAeyd7bqtrEgRBEATh5cVL+XrEy8wFYGhgivjQRvCr4buOmWEzt8FilfZW3rnvXBpurvLje1tkU+EGuaEkj/5tkOqCbONmc4WyP36CR8TStbhBGfqm9vsYjyj5IrX7/vCOFS3Ei47DxjnXUdIJnA3fnv4g9/1taOZ35fvHPJRKhJfrpeo0k0ySWfiNL8CxF+Hn/3fMw9G4Y2JSq6VQ2XBgI/zWv8Kpl01vereBmNTsfPjdH8Bl74ZHPhsWRQqjfl4qQ3Dvx+Bzl4cxpTf8FbznIfbPu6w2j5kdZpTiUfZ1qDy46XPwoc3wKx+BfU+HyoWrPgDv7oU/2hCu65RXNFScAIjCQbwGj2+YziVw1q+iF66mimcupjMIv66xr79GFF/qG4qRiM6La2j9wykYszPFIxxbGVv/eMShoEBrfQfhH/2R931xxMcaGFc3ON5zBUEQBEEQmuWlej2SbA9l+aXc6BSKaTBU27SNKADkSlU6WlJQ1N55HtgXFitGMaygaLBA0b2iKdl4oVwlM4WCAgg3g4vOa3jciHRt492+94EwInFB8yqAyRjQWe5Vl/LaTd8NN6fN+owcfh42fS/c5I5T2EnXFBTFSgsbtNOuCGNLH/ksnPsGWHF9/aHyeAqKwIfvvxt23gdv/OcwVWSaRDGjkxYoIEw2uPGTYevBT/4Yvnw93HIbLDx3uJ3j2C44/81ww1/Wv1crtY1rwlCBIhq3OnKD3L4QrvuT8F+LRAUKU+uPNviVwMwGv1p7AZ6hHbLrNhgzOk3qBQpDLZLReQkMrT+oKRscgxUKx1FYrE+0rqAQBEEQBEEQJibdHiooKoUJohubIV9LAhnR4pEvxaCggDBecByyyWZaPLY31d4BkYJikgLFvBVU3TbYt66pcSMySReHgDkHHgo35jFvRIbKPnd6PWG05bZfND/A/Z+ERBtc+YFxH25ZQRFx/f+B7lXwo/dDcaB+d6kaoNSIFgOt4af/E7b8MCy4NOoJMQH1AoXf4Pov+X34bz+Bch7+5dXwjV+H224Oz9Hv/xe8+SsnFNKiwkdLLRKT4NUKBxVD7+BHdRtj669tXKNCQtwMKxDMrt+0AsHU+qNxZ2uBBUJ1RjCbWjwEQRAEQRCEiWnrCAsU1XwMCopCrUAxUkHRcovHCAXFONRbPKZSUFSG4PiLTRcohsp+vQ1jXByHXPuZoYJiGrQlXM5TL5AsH4cz423vAChWfdYnL4HsAlj/reae3LcVNn0XLn3nhG0xsSgoINzgv/GLMLgP7vxf9btL1YCU56BqG5zTd/0HPP4vYcFknJaTZmlYQTGS0y4PWybmnwV7nwwLJbc+CMuvHXNoVDiI5ombqHBQNrXBj1o8TCkoIgWIoQ1y0x4UTRK1SBjf4Bv0cFAGN/j1Fg+DCgrXcouHFCgEQRAEQRAMku0K1Q5B8XjrgxX6w9sRCorBVk0y2+aCl55YQVErfhSm8qA4siO87V7R1PT5crWu0piIwY6VcHAT+JWmxobQg+IaZ0P4yZk9TT9/KooVn2QyCRf8Fjz3s7HeCZNx/yfDcz+BegJiVFBA6I1w1R+FJo9bfw6EhYN6e8fjX2H5C7fB2rfBa/6i9fkYbh1pqkAB0LUM3vkL+ONnw0KJmxj3sLLpFg9nnBaPGIk2fqZaJIYVFKYUIOG4pjwcTLdIROaVps4/mN3gR3UPowoKR4mCQhAEQRAE4WShvWMOgVbooYGpD56KeotH+G671jr0oGhFQaFUKJmfSEFRKx7kpmrxmEaCBzSgoAAGO1ZAtQiHn21qbAg9KK51N9LXfja0L2j6+VMxVAlCg8k1N0NQgc3fb+yJfdtg43/Cpe+YdF2xKSgiev4EFpwLt78fho5Sqvrh+jf/EH7yx/R1Xwpv+ExsrTCRsmFaKSSOG6Y+TEKlGqVgGCpQeGZbPIx7UBj2cDDuQWG8RSL8upr2cDCdQmKoPhSOrZSxAtG481mbSRAEQRAE4WWI47rkaEOV4lRQzAs/LftoPaxymDady+D4+AqKdMLBUQ20eNQLFM0rKCb1oKCmoIBp+VBkGOJi9Ty75lze9HMboVjxSXtOGJe56HxY32Cax/2fBDc1qXoCYlZQAHgpeOMXIHcIfvYnlKoBr1Sb4PvvglMvZ8vq/39aca4Tkapv8M1scMq+j+soYxL9qMXD1Pqjuo2pFgmvrgAx22Jgav1KhV9bcx4U4a1JBYJrcIM/nOJhcP1ikikIgiAIgnByUVBZ3PJg6wPl+yA9py53z9WKBi21eEBYoJhAQaGUIpvypo4Z7d8O7YunfMd7NFPGjAJDbUsg2TEtH4rEi4+QVD5bO6YXkzkVpcoIBciam2HP46E6YjL6tsHG79TUEwsnPTRqkSjFpaCAMBnlmj+G9bfRc/Bff/jHLgAAIABJREFU+Vj54zBvBbz12wRuKr55GFYGNN3i0SAVXxuLGIXhFg/jCgpDb4G7TuiBUDWV4mG4RQX+H3tvHiNJdt/5fV9E5FVZVT0H5+oRrxlRI5IiOTwkkaIkFyVRkGhgKS12YQmGlpZg8B8LWC3WNmgIMGTs/iEsVivAxmptCmub8LG2DEsWtaQlkjNsDilSFGmKHM7B4XTPfbOrr6o843j+48WLzKrKrMrMeO8XkVXfDzDI7qzMeJFR1TX5fvk9zAZ/XTMo7LE9/fgU1guf5x8oeFOAzFxPbCVCCCGEkDPKIOgiih1YPPqXD+ZPDPMBRWkFxXkTnjhnE9NtRospKJa0d2itMTipxQMAVACcvx94eYUmj0sPYqgbuNhevqJ0EQZxWtgw8I5/aM71O//u+Cd9+V8a9cQH//GJxw8ChWYUYOhKQWH52f8CuOMd+Hu7/xb7wSbwG39q8kgcY9UNC7d4LMk4ybw1YADm+oeB8jig8KtAAMzww/8G39+2MvSYgSAxoPDZgiHT4qGgOaAghBBCCDk9DMJNNJP98gfq7x5s8MiHBlulFRTngSwBej+Y+eVuK0RvvEAGxZL2jnGaIcn0YhaVu94FvLJCUOalL+Lvgrejl7qzLUwzjDO0cxsGtu40VaYP/59zhz3YvQQ8/CfA+37rRPWEpRUFbhUUABA1gb//Sfxd+/34b8798wPVna5phoFHBUXmrcHD0giVN4tH6tniAZjNt7+QTP8KiihQ/iwqAgoEnxaVrMig8DugYIsHIYQQQsgpYhxtopX2yh+ot1sEZAKmYhQANluzGw4Wpqgand/kcayCon/FDE9WCMgETBXoiZx/N5COgNceX3yB6y8Cl5/AtxrvRv+kAcuKDONDIZ/v+nVTt/rsX89+wkP/0lh0FlBPWNqN0F0GxTR3vA2/f/Pv4XL7je6PPUUz8jug8BUwaWkEwdpaPAAz/PAXkmlbPDxu8ENVhFm6Js0Ezt9jSKa9LMygIIQQQgghCxM3trCRuVBQXAa6E4tHkUHhwuIBHNvk0T+uxaOoGF1uQGGHBidaPAAzoACWy6F46osAgO+23lMMQ1wziFO0oqnz/9H/EGhtzw7L3L1k1BXv+y1g646F1/CioMgZp1kRZOmLZhQUdaCuMRYPzwOKyN+AIsnMBtDnJ+A+FQiJ55BMID9/zyGZXjM0Ao8hmYUCxMvhAZhCH7Z4EEIIIYScIrLmFjZ0SQWF1kalsOHD4mEVFHMGFK2wWGsmK1aM9vPgzY1FBiw3v9ls/JfJobj0ILB5B15t34NB7GdAMYoztKcVII0O8LaPAo/9P8C4f/DBX/6DpdUTgFFQOM+gyBnFAgOKMFitZnQB4lTLWDwSXxtMv5tjAIjCwFtIZtHi4VEB4tNiUNSMem7x8J0B4l9BwQEFIYQQQsipIWttY1P3octsEobXTU7EVEjm/tDkMZRWUGzcCoTN4y0ex7V47F404ZA3v2mpZQsFxSIWjyAwORSLVo1mGfDUBeCeD6HTjLwoKNJMY5xOZVBY3vXrwHgf+N5nJvddecqoKt77myarYgl8KihGySEFiAdaHi0eYwGLRxQEiD1u8P1bVPwrKHxaJCKJmlGfGQ4eFRQyLR4Knn58Zq8ntxQhhBBCyBmlfQ6RytDvlWjy6O+a2xkhmQuFTB5HEABbdx2joIjQO87isXsRuOmNJnhxCewxF7J4AKbJ49VHgGR88mNfedhcs3t/Dp1G6EVBYXMhjmRovOEDwE1vAL7zv0/ue+gPgCBaWj0B5AMKjxt8EYuHz5BMj/YCwJy/t5BM7dceAZgMB3+f4Jvvq1eLROhxQCEVkuktg0KixcNfC8nM9cRWIoQQQgg5owTtcwCA3o2rqx+kd9ncTlk89kYJmlHgRuK+ffd8BUUzPD4kc4WKUQAYxEtYPIA8KHMM/GCBoMxLD5rbe3bQbvoZUFhVRvvwgCIIgHf+mlFw3HgJuPK0qR59328C23ctvU67EWLo0aLi2yLR9JjhIJJBEfprwUgzv/YIwG/Ipx3c+FVQ+KxJFQjJ9GlRyQcHynPNKDMoCCGEEEJOEdGGGVAMblxZ/SD9fEAxHZI5TLBVVj1hOXfMgKIVYRCns99ka23CH1/3lqWXXCokEwDuut/cLhKU+dQXgTt+DNi6AxuNEEMPFo9hrgo4YvEAgHf9GqAz4Lv/l8meCCLgg7+z0jo+FRSjxL+CohH6C8mUqBmNPLd4NDwrKCKfCoT8uD6HRD5rOiVCMgOPGQ6ZgEWFGRSEEEIIIaeMqHszAGCwV2JAUSgoDrZ4bJYNyLRsnzef9s94I9ptmjX6s3Io9l4G4j5w671LL9lf1uJxyz1A69zJORTjHvDc3wD3fggA0GmG6HtQIFhVwxEFBWCux+t/EvjbPzbqiff+JyupJ+zxfSkoxomAgiL0mUEhkOEQBRj7ynDQ2r/FI/BnUUmKkEYvhwdgQyb91oz6bFGRUFD4bfFQ8DSfmwkHFIQQQgghnmlt3gQAGPWurX4Qm0Ex3eIxTMoHZFq27zb2CbvOFDbjoj9LhXD5SXO7gsWjaPFoLvgalALOv+tkBcWzXzWv5Z58QNEIvYRkzrV4WN71a8D1502A6E+vpp4AfCso/Idkes2gkLB4BH4tHg3fFo9QFRtx16RZhihQXi0GRkHh59hSCgpf55+JtHgwg4IQQggh5FTR2TQKirjsgKKxATQ3irv2Ri4HFOfN7QybR7dlNrAzq0ZXrBgFUKgaFlZQACaH4tVHgWQ0/zGXvgiELeCNPwXAKChGSebcR21DMucOKN7+q0BzE3jfb02u7wq0G2GxlkuSNEOmIRKS6TPksxn5VSA0Qt8WD7/XPwz8hWQmmfZqLwCsRcWzgmJNN/gSLR6hosWDEEIIIeRUsbFtbBlpv8SAonf5gHoCAHqjBFsuLR7AzCYPa/GYGZS5ewmIOsDW8hvw/ihFoJbcIN91P5DFwGuPzX/MpQfNcKLRATBp2XAdlDnMqz+PtHhYOjcDv/0N4Bf/eal1WlFQrOUSOzSQCMn0mkGxzhaPzH+Lh8+QzDTVXtUHgN8BSypQk+rV4iGgoFAez38WHFAQQgghhHhm89wtAIBsWKZm9PKBgEwgz6BwafEA5igo7IBixgZ/96LJW1hBpt4fp9hoRsvJw8/boMw5ORQ3XjItH3n+BGAUFID7AcXE4nHMa98+D4SNUuu0PCko7IDCt4Ki5TGDYu0tHhqIPJ+/z5BMCQWF3wyHfI11DcnUEhYPNSuayBscUBBCCCGEeKbd6SLWITAsq6A4NKAYOgzJ7N5mmiZmKShyi8dsBcXFlQIyAZNB0VnG3gEAN78ZaJ+bn0Px1AVze+/PFXdZC4brHIrhSRYPR7RzBYV2vEsYFwqK9c2gGKcaDYEWEm8KhEyjIaBA8BeS6X9A5LfFw3/NaCTQQuJ1wKJABQUhhBBCyGlCBQH21QaCURkFxZUjFg+TQVHu0/mCIAS27pozoMgVFIdbPNIYuPrMSvkTgFFQdJcdUChlbB4vz1FQXHoQ6N4O3P724i6bceG6CeNEi4cjWvnxXdskrCpDIoPCl8VjnKQiFg9fG3yjoPCfoeGvBUMqg2KNa0aVR4uKQItHwAwKQgghhJDTR091Ecb7qx+gfxnoTgYUoyTFOMncZVAAxo5w/YUjd08yKA5t8K8+C+i01ICis2iDxzTn3w28+tjRoMwsMwqKe3YOWE7sAGFmC0kJrGWkdZzFwwF2gOA6h6KweHg+f581o3GqvWdoNALlUUHhPyQzChQSXwoKkQyKwGMGhURIpnIe0GvRAhYPDigIIYQQQk4hg2ATjXhFBcW4D8T9AxYPOyxYWoFwHNt3H2vx6B9WUJRo8LDHW6rBw3I+D8p89dGD97/6CND7wQF7B+AvJHMUy1g8rILCdQ5FYfHwvEFuRv4sEnGaoSGgQPDZ4uF7gx+FfkMmQ8/Xf90tEmGgCqWDayRCMsNAQdDhwQEFIYQQQogEo2gTzWRFBUX/srmdGlDsD82wYLPtyOIBGAXFjZdwOBFtI1c5HKkZLT2gSFccULzb3B62eVx60Nzes3Pgbl8hmdYy4t3ikSsERs4VFFYB4vf8zQZfO/8UOcs0kkz7D8mM/GU4iIRkBoG3DX6caUQrBOQug4RFwueMKFD+FBQSLSSKGRSEEEIIIaePONpEO+2t9uT+rrmdsnjsjWIAcNfiARgFRTIABlcP3B0GCp1GeDQkc/eiqdLcuGWl5QarDihueiPQvuloUOZTXwRufxuwfdeBu4sBhQeLRxgo7xvkticFhVSLh7VguM6hsMfzb5HwHJIpoEDwd/6ZfwWIR4tEmmUIA7Vck9CS+FRQFC0enhUgrgN6j4MDCkIIIYQQAeLGNjayFRUUvXxAMRWSaRUUzjMogLlBmb3DG/zdiyurJwATutldJYNCKaOimK4ajQfAs187Yu8Apiwerls84gxtz5t7wH8Ghe8Mh5anAYXddEsMWPxaPPzXjPrMoPBeMxoqjyGfftUHgFFQePrxmVhUPGdQ+BqwzFxPbCVCCCGEkDNM1txCV/dXe7K1eEwpKKzdwrmCApibQ3FUQXGp1IBiME6Xrxm1nL8feO0xIB6avz/7VSAdAfd86MhDfWVQDOPUe/4E4FFBEQsrKBwHZVrbhXeLh88NvkRIZugzZFJ7byHxm0GRed3cA6Zhw5cCZKKg8HJ4c2yl4Gk+NHs9uaUIIYQQQs4uurWNTTVAmiQnP/gwvRkZFHZA4UVB8eKRL3Wb0cEWj9E+sPcScOu9Ky/XWzUkEzBVo1kCvJYHZV56EAibwBt/6shDOx5rRiUGFL4yKMZSCoTQ14BCzuKRZO4zNACjoJCwePhSICSZRuhZARL6zKDI/IeUilg8vCoowBYPQgghhJBTR/scAGD/xtUTHjiD/mUgiIpjAJMBxZZLBcXmHYAKFlNQXHnK3K6ooMgyjWGcFQGcS2ODMm0OxaUvAm/4ANDcOPLQtqeaUaOg8P922p7/0LmCIg/JjPwOWXwpKOzxfG/w7fnHHjb5MgqEAKmvkM9MombUbwaFz/wGQCgk03eGBkMyCSGEEEJOF+GGGS70blxZ/sn9XaOemHoTOmnxcDigCCNg8865GRQHakZLNnhYu8XKCoqb3mACOl/6NrD3ilFS3HvU3gGYT9gboVpbi8faKyg8h2T6ztCwG3AfNo9EKIPCx3AFMCoW3xkOXmtStf8MjUiiZtTjawgCRQUFIYQQQshpo5EPKPp7KwwoersHAjIBo6AIlIeKy+3zwI0XjtzdbUYHa0Z3L5nbW+5ZaZlePuxYeUAxHZT51AVz34yATEunEboPyUxS7xWjwNSAwrECwQ48fG/wfVs8mt4zKIID67lEzOLhUUHh+/x9foIvEpLp8frbwYHP1xAoBUEBBQcUhBBCCCESNLo3AwBGeytaPLq3Hrhrb5ig24rc1+Odu/sYi8fUBn/3ognVbHZXWsYOC1a2eAAmh+IHjwNPfNYMcO54x9yHdpruBxSDsWxIpusMjUnNqJDFw3WLRyIUkunp/IE8A0EoJNNHVaREBoXNAPGBSEim8qdAsJfFq8WDGRSEEEIIIaeP1qYZUMS9a8s/uXf5QEAmYBQUTvMnLNt3A9dfBA69ITU1o4csHiUCMvvjkhYPwCgosgR4/C+Ae3aOjbLvNEIPFo9MJIPCl4JiLFQz6ktBYQcGDc/n3/Bk8dBaGwWFgMUAgBcVgkQGhanpXF8FhV8FiDmuzxmLz+s/cz2xlQghhBBCzjCdYkCxioJihsVjmLjNn7BsnwfiHjC6ceBu0+KRmE9htQZ2nyxVMWrzLDbKDFnO329udXasvQMAOs3I/YAiWXcFRYooUN43aOsekunL4mE3ff4VFPmAxcMm0ygo/GdQeK0ZFbB4eFNQZAIWj0Adnld7hQMKQgghhBABNrZvAQCkg+vLPTGNgeE1oHs0g2LTi4LCVo0etHl0WxEynX+K378CDK+XHFA4UFCce/1EWTInINPSaQTua0aFLB7eMiiSzHtAJuBvQBELhXxahYbrAUVSDCg8D1hyZZGXAUWaydR0egvJFKgZ9akAsRkUnmtGqaAghBBCCDllbJ4zG2k9XHJA0c9DNQ9ZPPZGCTbbDRendpDtu83tjRcP3N1tmY34/igp3eABoMizKBUyqRTw+vcDd75jMliZQ6cZuq8ZTWQsHlEYIAqU8wHLOMm82zsAjxkU1uLhO4Mi38DGji0eUiGfYWFR8VWT6r9FJfHUQiJRM+pzwJIJtHiEwi0eHsbuhBBCCCHkMM1WC33dglp6QHHZ3B5SUPRGCX7opo6js5tinoIiD7Psj1InA4pBbCwe3bIqkF/510CanPiwTiPE1V5cbq1DDGOZFg/AqATcKyhS7wGZgP8WD+8DCk8WDzvw8P0JfsOzxUNCQZFpsxl3vRFPM+09JNNnC0am/WdoKI8hn7PggIIQQgghRIie2kAw3lvySfmA4nBI5tCTxWPzTgBqhsXjkIIiiICb3rDyMk4sHgDQuXmxhzUjpwoErTUGsYzFAwBajRCjxL2CoiWgAPFl8RglQgMKXxaP/Hi+FQi2ZcNH1WUqkEFhBwip1gjgekAhEZLpzyKRag3Pp5+3kPhdY5pS/xqUUrcopT6vlHoyvz3yfwil1OuVUl9USj2ulHpUKfWPp772e0qpF5VS387/+0iZ8yGEEELI2WOd3o/0gy6ieMkBRX/X3B4OyRx5CsmMmsDm7TMsHmat3jgfUNz8JiBc3WLStxaPsgOKBek0AqcWj3GaQWuIDSjaUYBh7H6D79teAEwGFCNPCgTvGRS+LB6ZrUn1HzIJuB+wAECSCWRQhD5bSGRCMlOPIZmBdwXIemVQfALAA1rrtwB4IP/7YRIA/1Rr/VYA7wfwnyml3jb19T/UWt+f//fZkudDCCGEkLPH2rwfGQSbaKw6oJiyeGSZ9heSCRibx4yQTMBYS7B7qZS9A5hSUAht8F3XjNphgayCwkNIpkRNamiu0dpaPHwrKI6px3VBw+sGX6DFw2dNqoBFIgpUkRXhGonr77OFZOZ6JZ//UQCfyv/8KQC/cvgBWuuXtdbfyv+8B+BxAHeXXJcQQgghxLI270fG0SZa6f5yT7IWj84tk7vyis4tHwoKwARlXj+koMgzKHrDGLjiYEARJ2hGgXd5u8V1zai1i0iEZAJGJeAjJFMkg8Jzi8e61owWGRSez7+weHgImoxTiQwKfy0kEgqKUCkv5w4Yi4dIhoaggqLs/9Xu0Fq/DJj/8Sulbj/uwUqpNwF4N4CvT93920qpfwTgmzCfbMwsB1dKfRzAxwHgtttuw4ULF0qeOlmE/f19XmtBeL3l4LWWg9eaCLA270eaaQPbyXL/Jt7y5Hdwe7SJv/7yV4r7rgzNRuOFZy7hQvrcUuewCD+8p3HnlefwlanzvDwwa1781peAZIgndlO8XOLf9pNPj9BU2ULXwsXvkVdfHGOcZHjwi190Iol+rW+uxzMXv48Lg6dLH+8kxoMBXn6t7/T36Wu7A4QKB47p43e2/fT1+xcv4QKed3bcx542oadf/9pXsdHwt0l79oYZDP3dd76L4JXHnR33+T3zM/T97z2OC9eedHbcwzzxihlofu3r38ALW24HaqNxjFdeehEXLlx2etxpnnrWfJ8f+vJXsNVc/fs862d798oAmYbX9ynPPTcGAGe/e6Z5/vkRsizxev4vvjBGnKZi7+VOHFAopb4A4M4ZX/rdZRZSSm0C+L8B/I7W+kZ+978B8M8A6Pz2DwD81qzna60/CeCTAHDffffpnZ2dZZYnK3LhwgXwWsvB6y0Hr7UcvNbEBafl/cjXH/632Lzy+HL/Jn7wPwOjOw8858lX94ALD+G973w7dt51fL3mSkTfBl78DHY+8F6gtQUAuNobA1/6PO67VQHPAvd94CO4780/u/ISf/Had3Duxu5C18LF75HvB5fwZxe/h5/4qZ9xYo154pU94KGHcP87fgw777yr9PFO4o+e+BoAYGfnA86O+a8e+Qpu6Taxs/MTxX2+fmcHn/sM7n79G7Gzc5+zYz6Gi8ATT+Dndn7Wq9Xm+6/uAV99CPe99W3Yeae7f2+PvHgd+Ouv4F3v+DHsvH3Wrzc3xI+9Cnz7m3j3e96LH7v7nNuDP/D/4k1vfAN2dt7q9rhTPP83zwKPP4L3f+CncNtWa+XjzPrZ/qPvfQ1hoLCz8/6SZzmf76ZPAhe/j5/52f/AuR3pgWuPoHn5Ja/vs742eBx44Rmx93In/nbWWv/CvK8ppV5VSt2Vf1pxF4DX5jyuAfNm4H/TWv/p1LFfnXrMHwP498ucPCGEEELOBqfl/YhubmFT95Z7Uu/ykYDMvZH5RNRLSCZgLB4AcONl4DYzoLAZFJ0buVqgdAZFIhaQCaCoAx2MUycDCmu36DRlLB7tRojrA7c1qWOhkEzA2DzGri0SiQ2Z9PsarIXBdQuGVIZGFPgLyVz3DIoky9CM/BZbBlPn73qOlmqZFhVJi0fZfw2fBvCx/M8fA/Dnhx+glFIA/i2Ax7XW/+rQ16bHzb8K4JGS50MIIYSQs8favB/R7W20VYzRsL/4k/q7BwIyAVMxCgBbPkMygQNNHs0oQCNU6O4/AzS6wFY51UB/nKIrOaDIMzRc5TjYPIu2QIYDYDIoRo4zKExIpsz5N8PASwZFGCjvGzQ7QHA9YEkyoQGLx5DMJJPIoMgHRB4yNCRCMu3xfQRNSrR4hGsWkvn7AD6slHoSwIfzv0MpdV4pZROwPwjgNwD83Iz6rn+hlPquUuphAB8C8E9Kng8hhBBCzh5r835EtW8CAOxfv7L4k3qXgY1bD97lXUFhBxRHmzy2+88Ct94LlHxTPBinlSgoXFWN2kGH1Aa/7aHFw4RkSikoPJx/mnkPyAQmIZ/OQzLz6+E/JNNPTWqaaWg9CbH0hdcWD6GQTLPWeraoKKWQaUALDSlK/V9Na70L4Odn3P8SgI/kf/4KgJlXTWv9G2XWJ4QQQghZp/cj0Ybxf/f3ruLWO37o5CdkmVFQHBpQFBYPXwoKq444PKBoRrhl8Bzw+vJ+7d44wR3b7dLHWRRrxXDV5GFrRjtSNaNeFBRpsfn2TStyr6CQsqh4s3gUCgoZBYhrBYI9ntSAxU+Lh38FhbV4eBCAINPwr6DIj6916bn0Qsj8RiKEEEIIIYg2jIJicGNBBcXoOqDTYyweDafnV9Bom9yLGwerRm9qZrglfqV0/gRgFBQbogqKqFjXBdI1o+1GgKHjDf4ollRQeMigSDORAUvDk4IiyY8XCSkQXG/wrSJAyuLhIwchzTLvNZ12fpP6sHhoDc8/Pgg8nv/M9URWIYQQQgghaHbNgGK0f22xJ/R2ze2hkMz9XEHRbXnc4G+fP6KguCf6AQJkTgYUfekBRb7WIE6cHG8yoJBSUITuFRRphpZQhobJoHB7/nGaec9vAFCoNNwPWPINvmcFgh2AuFaA2IGHVEimHwWFQMikV4uK9j5gCTxmaMxcT2QVQgghhBCC9tYtAIC4f3WxJ/Qvm9vuQYvH/ihBpxEi8rk5O/dDRxQUb1avmD84GFD0xgk2mn7T86eZtHi42WQWLR6CFg+XCgqttbFISCoonIdkapEBhS+Lh7VIyIVkur3+aSqloDDXx8cGPxMIyfS5wU+1Lo7vC2sh8WFRmbmezDKEEEIIIaSzdTMAIOktqqDIBxSHMyiGSVH76Y3t80cGFG/UuaLi1ntKHVprXYHFwyooXLV4mHfrUgqKdiNEmunCFlAWqwaQtHi4DmkcJzIhmWGgoJQPi4fMBj/yFJJZKCg8D1js4X0oKBKBkEyfIZ+ZgILCXn8qKAghhBBCThndc2bQkA2vL/aE/nyLx5avBg/L9nlgcBUYTypR785exFVsA52bSx16nGZIMl2RxcNxi4dgyCQAZ00Y9jhS598IlfuQzDRDU8CiopRCI/SToQFIKCg8h2Sus4JCIiTTY4tHpv1bVIrz54CCEEIIIeR0sbl1EzKtoIc3FntCYfE4HJIZ+2vwsGzfbW73Xi7uuiN+Ec/grtKHtkGVohYPO6AYO8qgSFK0osC7vNpilRpDRwOWUSytoAgx8hGSKaCgAIBGoLxlOPjPoPBkURFWgPjY4CcSIZleMyjMAM0ndkChafEghBBCCDldBGGIfXSgFlVQ9HaBRhdodA7cvT9KBAYU583tlM3jtvHzuJSVH1D0iwFFBRYPVxkU41TM3gG4V1BMLB6SIZnuBxQSGRSAafLw1eIhlUHhrcVDrGbU/Q45zYBQ6Px9tXj4/ifAFg9CCCGEkFNMX3URjvcWfPDlIwGZALA/SrHp3eKRKyhsk8fwBjbjXVxK7yy9UevnKoaO4IAiDBSaUeDQ4pGJVYwCPhQU5jhSIZmtyH2Lh8mgkLKouB9QjHMFQsN7zaht8XBt8bAtHjI1qX4UCP4VFJOQyfVs8QjZ4kEIIYQQcnoZBF1E8YIDit7lIwGZALA/irHlW0GxlSslrILiyiUAwFP6TvRH5TaaVkHRFbR4AEZF4dLiIdXgAax/BkUzcp/hME612IClESj3IZP59fCtQGj4VlAItWCsfc2oJwWFb5uZ8jhgmQUHFIQQQgghggzCTTSTRRUUu0cCMgFgf5j4V1A0N0wYplVQ7JoBxdP6LuyX3ORXYfEA8gGFqxYPaYtHrtZwpaCwdguxmlEfFg9JBYUPi4e0RcJTTapUC4aPDbJIzajvkEypDA0qKAghhBBCTh/jaBOttLfYg/u7RwIytdYyGRSAsXkUA4qL0FB4Vt+B/qjsgMI8f0PiNUyx0QyLetCyDJMMLcEBRTvPinCvoBDKoIj8ZFA0I5mQzChQzgcURYuHZ4tEo2jxWM+QzNCjgkKiZrSwSHgImUwzXQxAfGEHIEICCg4oCCGEEEIkSRqb6GT7iz14hsVjlGSIU+1fQQGYAcX1F8yfdy9iuHEeIzQbh8NFAAAgAElEQVSxX3pAUY2Cot0IiwaRsgzjFG0h9QEwUVA4C8m0AwqhHA0fA4qxZEhmGHiweGgowLtEf6Kg8KUAkcnQWNeaUXt5vFg8MsDzfAt2/kGLByGEEELIKSRpbqOrF1BQjHtAMjgyoLDDAe8ZFIBp8phSUIzPvRkA0CubQZE/XzLDATChnIPYTQbFKE5FQz6t0sFZSGYeWNkU2uD7yKCIk0z0/J0rKLLMewMDMFE4xGuaQeFdQSEUkukl5FPLZWgwJJMQQggh5BSiW9vY1H3ok/S+/V1ze8jisT80G+yulMWjfxmIh8DuJSQ33wsA6JXOoBB8DVN0HCooBnFa2C4kaDtWUIykFRS5AkE73OSMU42GkIolCpT7DIdUQ8KhopRCFCikjj0GUhkURQaC4/PXWotkUFgFiI8NvoTFw+eAZeZ6IqsQQgghhBAAgGptI1IZ+r0bxz+wd9ncHgrJtAoKmQyK8+b2lYeB0Q3g1nxAUdbiEVcUkukyg0K4ZtS1gmJcQQYFAKcqijiVU1A0QvcKkCSVUVAAZhPuesAipaCY1Iy6PW5a1KT6biExt66vP5CHZAq1qDCDghBCCCHkFKI6NwEAejeuHv/AOQqKvVxBIZNBkQ8onv4SACC87UcAAL2SKoT+KIVSchWXFqc1o9IWD+cKitziIdjiAcBpDsU4yYoKTd/4sXhoCJ2+twwNYH0VFDYTwvv5K38WCYkWD3t5aPEghBBCCDmFRBvnAACDG1eOf2ChoJiXQdFwfm5H2L7b3D79EACgdcdbzKk5CMnsNiMoz2+sD+O0ZjROxdQHwETpMHKWQWEVFHIZDoDbAYVp8Vhni0fmXX1giUIfFg9zPXwHlUaeMijs5ZAbsPiweMD771GfA5ZZcEBBCCGEECJI1L0ZADDYO2FA0Z89oLDDARkFxV3m9rmvA0ED7de9CUqhdM3oIE5E1QeWTtNdBsUoztCWrBn11OIhpqBwbPHIMo0k08ItHq5rRuUUFFGgPIRkymRQBJ42+EWGhm8Fgj1/Ly0e2rtNSDGDghBCCCHk9NLqGovHqHft+Af2d4GgAbTPHbh7TzKDorUFtM4B6Qi45R6oMEK3GWG/ZItHb5SiW9GAYugggyLNNMZpJtpC0gwDKLXGCgrHFo8431yKDSh8tJCkmeCAIkDqWgEinkGxpgoKq0BY8xYPIQEFBxSEEEIIIZJ0toyCIu6fMKDoXTbqiUOf7tkWjy0JBQUwyaG49YcBAN1W6MTi0WnKNngAxuIxTjMkJTeaNqhSMiRTKYVWFGDouMVDsqYTcDegGAuff8NTi4dkSGbsOsNBKGTSl0VCvoXEj4LCf4uHuaWCghBCCCHkFLKxbSwb6SIKikMBmQCwP4oRBUouYPJcnkORN3h0W5GTmlHpBg8AheKhbA7FZEAh+xpaUehQQZGiGQViOSB2QOHKomIDH6UsKj4sHkmWebcXWBqhnwELMKnR9IU9vusMCmu5CHxbVHyHZIq1eHBAQQghhBBy6tg8dwsAIBsuUDO6ccuRu/eHCTbbggGThxUUzciJgqKSAUXT0YAi32RLWjwAY8dwYVEBTIaGZIuK6wwKOyyQtHh4yaAQ+hZEYeBPgeDZp+JLgWAFJb4tKpPzd3/sVEsoKDigIIQQQgg5tbQ7XcQ6BIbXj39g/zKwcVRBsTdK0JW0R9gmj2mLR8mgyUFVAwqroHBw/sCk+lOKdiMs6kHLMk4z2RaSfCceO7Z4SNWMNgLlvqYzk8ygUB4UIHmLh9AG37kCRCgk0w6h/IRkwvuAYtLi4XWZAg4oCCGEEEIEUUGAfbWBYHTSgGKOxWOYyOVPAMCd7wSaW8DtbwXgRkHRGyfYqCKDwpWCojKLR+DMIiGtoGg4VlDY46yzxUO0xSNU7i0SQhkURQaC4w2+XM1okK/no2bUvwpHOoNC/v8MhBBCCCFnnJ7qIoz35z8gjY3CYoaCYn+UyDR4WO77ZeC/fAqImgBMBkV/XRUUTTcKCqtikLZ4tBthMRwpyyhJZS0erls8UtmQzMhHBkWaQepbEAWB8wGFVAaFUgpRoIpaU1eIhWTmCgTX1x+QafEoMigYkkkIIYQQcjoZBl1E8d78B/R3zW331iNf2h+ZDAoxlCqGE4CxeOyvawaFo5DMwdhsbNZZQTFOMjH1AeC+xSNOcnuBVAtJaCwe2uGn+Enmf3NpiQJVur3mMIWCQkAGEgbuFSA2U8H/Bj9fz8MGXwtkUIQBLR6EEEIIIaeaYbiFVnLMgKJ32dxuzBlQSCooDlHW4pFlGoM4rcbi4SiDooqaUbOeSwXFeodk2uM0BC0egNtPwceJYAaFB4uHrS31HTIJmE1y6jyDQrgm1UMGRSpZM8qQTEIIIYSQ00nc2EQ77c1/gFVQzLJ4SGdQHMJaPFb9NNCqF9a7xWP9MyjGiWxIprViuKsZlQ3JjOyAwuEmOckEMyiCwL2CorB4CA0oHG+QrQJEKmTSR4ZDKqDCYYsHIYQQQsgpJ25sYyM7JoOinysoZoVkVq2gaJlNbX/FTX5vbNQX69ziYas+xWtGG4HTDApJi0fLscXDHkcqg8IOQlwpQADhDAoPCgopBQKAPINiPWtGiwwHHy0e2v+ApRhQMIOCEEIIIeR0kjW30NX9+Q/ozVZQpJlGf5xis9XweHbH082HI/0VbR52OLDOLR72+eI1o1HorsWjKouH65BModdg13EZlGlaPKQyKALnNZ3203sl8BpCHyGfwiGZ/hQUzg97AGZQEEIIIYSccnRrG5tqgDSZs8m3CorOzQfutuGUoiGZh+jmg4VVgzL74wotHo4UFKOqakYbji0eggMW1xkUE4uHUItH4MPikXnfXFqiQBUbcldIh3y6zqCQC8n0N6DItC6O7ws7f5KqGeWAghBCCCFEmvY5AMD+jauzv967bIYT4cFBhB0KbFVq8cgVFCtu8vu5xaOzxi0e1mYhbvGI3IZkStkjAPc1o+NUtsXDWjxcKiiSVDCDwoPFI80ykfwJwE+Lhx02SYVk+rF4yLV4uGywOQ4OKAghhBBChAk7ZkDRu3Fl9gP6u3MDMoHJkKAKuvlgoayCoorXEAQKrShwYvEIAyW2ObY4V1AIhmRGYYBAudvgS2dQuFaAAOZayIVkKucWjziVU1CEgXK+wU+FFBQTi4f7Y6eZf5uQHYCwxYMQQggh5JTS6JoBxWBvjoKivzsnIDMGULHFIx8srFo1agcU0uoDS6cZOgnJbAvmN1jaUYhxkjkJq5MOyQSM2mFdMyj8WDz85wdYotBDi0emBS02PhQgUhYPc+t6wKK1NiGZYi0eXpeZrCezDCGEEEIIsTS6JltiuDdHQdG7DGzceuTuvVxBUW2LRz6gKGnxqCKDAjCDkfIDirQSi4rNjHDxKb50SCZghgnrWjPqy+IRCYVkNjy1eEgqKFLHGRpSNaN2uOW8hSQ/nH8FRb4eMygIIYQQQk4nrU0zoIh712Y/oD97QNEbmY31VqUKijA/l/WzeAC5gsKBxUPSHmGxa7rIoZAOyQRM1agri4RVYjSEhiwNxxYPrTXGqVxIpo8MB/EMCk8hmd5rRvPDu77+k5BPp4c9gs8MjVlwQEEIIYQQIkzHDij6MwYUWQb0rxxv8aiDgmLVAUU+ZKlCgQC4UVCM4gxt4c09gGLNsiqEJM2QZBrNUPZ70HRo8bCDAqkMioZji0dhLxDLoAicqj+AKhQU6xmSqZRCoNwrEOz18F3zGnisSZ25nsgqhBBCCCGkYGP7FgBAOmtAMbwG6HRmSGZh8ahQQbHRsAqKVS0e6YHjSNNplFdQVGbxyBUUo7jcRtNu7qUVFM3IYQZFst4tHkkm8+m3pRH62eBLKSiiQDkPaZSqGbVrrOv5B0WLh9dlJuvJLEMIIYQQQiyb54x9Qw+vH/1if9fczlRQ5C0ezeoGFFEYoN0IiiyJZenHCZpRgEi4AcPiyuLRrsDiYRUUw6Tc+dshQRUZFC5DMsNAiX2Cb39eXVk87KDDd36AJQwC5xaJNNNi/469KCiEQjIBo0LwpaCQyqBYixYPpdQtSqnPK6WezG9vnvO4Z5RS31VKfVsp9c1ln08IIYQQMo91fD/SbLUw0E1gdOPoF3uXze3GLUe+tD9M0G2GYpuyeXSb0eo1o6O0soBMwF1IZrsCBYgrBYW1iEi3eDQdZlDEaSYWkAlMrCSuNvn2OJHQSzAhma4tHnIZFJGnAQsgqKBwnUGRfzt9t3iEa2bx+ASAB7TWbwHwQP73eXxIa32/1vp9Kz6fEEIIIWQWa/l+ZF91Ec4aUPTtgGK2gqJKe4el24pKhWRWqQBxoaAYxlklAwr3Cor1zaAYJZmYvQMAGpFbi0ec7y4lQzIz7TYHIRXMoAgC9xtkKQWCXcObxcPz6U8sHusxoPgogE/lf/4UgF8Rfj4hhBBCyFq+H+kHXUTx3owvzLd47I2Sytovpum2opVrRgdxUllAJmDqTd0oKOQtKu4UFOb1r7uCQtKiYqsinWVQpDKbS4sd5sQOVRRJJplBETjf4EsqKILAg8Ujvx6+FRTSIZll/w93h9b6ZQDQWr+slLp9zuM0gM8ppTSA/0Fr/cklnw+l1McBfBwAbrvtNly4cKHkqZNF2N/f57UWhNdbDl5rOXitiQBr+X7kNt0B+leOHOMNz34D9wB46JuPIQsvHvjacy8NoWNd+b+pZDDAC6PV/m0///IQ6Xi51+Dy98jlV0fYGySljndtv49ruyPx78NT181g4Rt/920kL67+Nv7ZG+Y4T37vMVy4+v0DX/P5O3v/+hD9xM3P73MvjJAlqdj34LW+2dh/99HHcO7ak86OF49lfo6efWYMALhw4SG0HPlKfnB5gFEKkfO/fm2IXsnffYd/th97wbQi/e3X/waXOn6HXVkS4/kXXsSFC5edHfPa0PwMXXzySVwYPePsuIfpxWYw8f0nL+JC/Ky3dSwn/mZTSn0BwJ0zvvS7S6zzQa31S/n/8D+vlPqe1vqhJZ6P/E3EJwHgvvvu0zs7O8s8nazIhQsXwGstB6+3HLzWcvBaExecxvcjD//NOWwmN/Cew8f4y78CXtzEz/78Lx55zn/3+FdxvhFgZ+f9K6/rgv/pqb/FtUGMnZ0PLv3cP/re19ANgJ2dDyz8HJe/R/52+D188fmnyh3vy5/Hm15/J3Z23uHknBblrlf2gK89hB/50bdj5513rXycbz13FfjqV/He+9+JnfsOzuN8/s7+X5/9Bl66NsTOzs+UPtafv/ptbA6uiv3/5eXrA+ChB3HvW+7Dzk+8ofTxLv1gH3joS+h22iKv4WL4FPDE43j/B38a5zoNJ8f877//NXSz5f4tr8r/8sw38OpeuZ+dwz/bL379WeCRR/DTH/wp3LHddnCW82l95Qu44647nP7OePn6ALjwIN76o25+JudxYxgDD3wO99x7L3Z+5h5v61hOHFBorX9h3teUUq8qpe7KP224C8Brc47xUn77mlLqzwD8BICHACz0fEIIIYScbU7j+5E42sRN45ePfqF3eWZAJmBCMl+3ueH5zE6m2wrx4rXBSs/tjRPvm4Hj6DRCJJnOQxZX+9R0GKfoVBKSac53VDKDwlpE1tniMU5kQzKtxSNxbPGQ+hY0QrfnDxjJv1QOSBgo5yGZmaDFI/Jh8RDK0LDHz9Ykg+LTAD6W//ljAP788AOUUl2l1Jb9M4BfBPDIos8nhBBCCDmBtXw/kjS3sZH1jn6hvzszIBPIQzJbbj79LEO3GaG/YkjmYFxxi0e+9qpBmVprUzNaSUimWXPoKINinUMyx2mGpuD5N4uaUTebtEnNqJPDnUgUus8RSARDMr3WjEqEZAaqWM8Vdl4gl0HhdZnJeiWf//sAPqyUehLAh/O/Qyl1Xin12fwxdwD4ilLqOwD+FsBntNZ/edzzCSGEEEKWYC3fj2TNLWzqWQOKyzMDMgEzoNiqSYvHyjWjdRlQrBiUOU4zaI2KakbdKCgmLR4VKCgcDSjiNENTUEHhvMUjlW3xsGGWseMWD6mQzDBw34JhBx6+N/hmDfcKhOL8vbd4mFspBUWp/8NprXcB/PyM+18C8JH8z08BeNcyzyeEEEIIWZS1fT/S3kZbxRgN+2i1p2wbvV3g9rcfebjWOldQ1GFAEaI3TqG1hlry08feOMFGlTWjjXIDCqteqGRAYWtGSysoKhxQONzgS9aMOrd4CH56D0zOP3Vok0hSjTCQ+R5EHhQUdsMtMWQJlfvztwMb3yoWq6BwbVGZu57IKoQQQggh5ACqfRMAYP/6lcmdWhsFxYwMimGcIc00NmuioEgzXWx0l6Fqi8dGSYvHMH9epTWjzhQU0haP0J2CIpHLPwBQ5F2su8XDbc1oJqigCJxnUCTCNaOuFSBZoaCQyqDwukwBBxSEEEIIIRUQbZwDAPT3rk7uHPeAZDjT4rE3MpV4tVBQ5AqI3pI2j3GSIcl0pQMKq3woPaAQ3twDZiPVCJUzBUUlIZmOBhSjNEND8PyVMtfelcVDOiRzogBxm0ERCU1YwsBtfgYgG5IZKg8hmUIKCjv/cD1gmQcHFIQQQgghFRBtGAXFcHpA0d81tzNCMveHZhhQlwwKwORJLEN/bF5D5xRYPDoVDVnaUVi+xaMIyazG4qEdbHTiJCuCK6WIgsChxaMaBUXiUEEhm0ERON8gS4dkuh+wmFvfCgqlFAIFJ/9uF4EDCkIIIYSQCmh2Zw0oLpvbGQoKG0rZrXBzb+nmm/NlgzLtQKO7xiGZgwotHoDJoVjFWjNNYfEQfg12IBI7+BQ/TjM0I7mQTAC5gsKVxcN++u3kcCdiBwlOFRTrnkGRaSglFJKplPOQSXs8iRlR4CFDY+5aIqsQQgghhJADtLdMzkTcnxpQ9E5WUNQlgwKYKCIWZaKgOAUZFBVYPACTGzFc8dwthcVDWIEwyXEo/ym+dEgmADRCdyGfdlAgFpKZXyuXVZfSLR6u1CuWJNNi19+HgiJd8wyNuWuJrEIIIYQQQg7Q2TIKiqR/fXKnVVDMCMncy9UKtcigaFkFxbIWD6ugqO41tEtbPPIBRUVDFlcKijBQxaZVCjsQcZFDMU6qGVC4tnjIZVBYBYXLkEyNUMij4kNBkWotop4A7Abf7THtwEDiNYRKQWg+wQEFIYQQQkgVbJwzKolscG1yp82gmGHxsIGUdcqgWDYk0w4oqgzJ7KxxSCZgFBSj0gqKVFw9AQDN/Jo5GVCkWjzksxG5s3jYayCWQWEHFA43+bItHsrpuQPG4iF1/lHgPiQzE8zQCJT7kNK5a4msQgghhBBCDrC1dRMyraCHNyZ39i4DQQNobR95/H6dFBQrtnjUw+Jhzn31AYXZWFaVQdF2oKAYJZl4/gQwaQ1xMaCIU/mQzEbg0OKRCWdQ+LB4pLpoB/FNGLjPcBC1eCjlNKAUmNR+Slk8XF//uWuJrEIIIYQQQg4QhCH20YEaHrJ4dF836XWbYq+GGRSrKii6FQ5ZbFBjWYtHVUOWVhRgVLJmdJxk4g0ewNSAIi2nAAFsBoV0SKZDi0dqFRRyn+BPr+sCyZrRyJOCQsqiEgST1g1XWEWDxI9Q4KEmde5aIqsQQgghhJAj9NUGwvHe5I7eLrBx68zH7o8SNMMArYqsBdPYDIre0jWj+ea+Ud1rCAKFdiNYWUExqNji0W6EGJauGc3E7RHAJIOirAIEqCiDwqHFo2jxEK4ZdXX+gNkgS3x6D5iaUa3hdJMsHpLpqcVDqiZVaD7BAQUhhBBCSFUMgk1E8dSAon95/oBimNRCPQGYjWYUqOUVFPnjq8ygAMyAZHUFhbV4rK+CYpSklQy6Wo4sHlmmkWTyGRRRECB2HJIpNWOxwxyXOQKyGRR2TXfnn2m5AYuPmk7RFg8FtngQQgghhJx2BuEmmsn0gGJ3ZkAmYBQUdcifAAClFLqtqFBELEo/rt7iAZgcirIhmVVYJAA3CorKLR4lBxRxvrmXVlA0Q3cDCmkFRViEZLo5/yzTyLTM5hgwCgoATnMQklRSAeI+w0GyxSNQCpoDCkIIIYSQ08042kQr7U3u6O0CG7MHFHvDpPKN/TTdZlgEdy5Kf5RCqeo295Z2I1hdQZGkaEWBWD3hYdwoKCqyeORrlrUZ2OeLh2Q6tHgk+XGkvg2NfIOfuDr//NN7qSGRjxaSVGsEgiGZrhUUWtDi4UMBMnctkVUIIYQQQsgRksYmOtl+/pcxMLp+jIIixladBhStaKWQzG4zghLaFMyj0wxXV1CM08rsHYCpGXWRQVHFkMhuZsuGZFoFhnRIZhQ4DMnMMigFuQ1y6FZBIWkvmF4ndZihkQmGfAaBD4tHfmxmUBBCCCGEEBckzW10da6g6O+a241bZj52f1SfDAogH1AsqUIYxEmlFaOWjUZUKoOiqopRIK8ZdaKgkP8+WMVDaYtHvjOTfg2NMMDYYUimpEWlEbgNybSDDqkMisjxgMUcS7Zm1LnFI7MWD6eHnYlSbgNKj4MDCkIIIYSQitCtbWzqPnSWmYBMYK7FozdKa5NBAZgmj2UVFL1RWnlAJgC0yygokrTSFpJWFGKUpKX84KM4rTSDomyLR1UKimakHGZQZMXQQILIcUimtILCqgRcqhAkQzJDDwqKosVD4DX4yNCYBwcUhBBCCCEVoVrbiFSGQX9voqCYY/HYq1GLBwB0m6tZPDaa1b+GTokMikHFFo92I0Cmy30SPk6rsXi4avGYKCjkWzycWTzSrBgaSBAWCgpXFhWboSGkoLAWj7UOyXR7zGJIJJVBQYsHIYQQQsjpRnVuAgDsX78C9I5XUNQyg2K83IBiECe1UFCUavFIMrQqVlAApip0VUZxtSGZ45KbZPt86RaPRhg4s0jEmRZVgDRCtyGTNmwzlPAXYKqFxGUGhWRIZqCc2lOAiYJCpsXDbYPKsWuJrEIIIYQQQo4QbZwDAAxuXJnKoLj1yOPiNMMwzmpn8eiPltsk18bi0Shh8YhTdCrMoGjlaw9L5FCYkMw1zqBIKmrxCFXp4YolSTNEQpt7AMVarmwGRQaF0JDFruPSJpFKhmQqBcfzicmAQkhBwQwKQgghhJBTTmPDKCgG+1dzBYWaGZJprRR1s3gsWzM6GNdjQNFphCVCMiu2eDhQUIyTajMoyg4oCgWF8GtohC4tHnKbY2BikXBl8UiFLR5WqeGyZjTJJBUUbocrwKTFQ8Li4SNDYx4cUBBCCCGEVERz82YAwGj/qlFQdG4GgqOb371hPqColYIiwijJltqw9cZJPTIomgEG8WpBk8M4LYYEVeBOQbG+A4o4rSYk073FQ+57EAQKgXKpoBCuGfUUkik3YFFO8zOASauGhBAnUKwZJYQQQgg59XS2zIAi7l8zLR5zAjKtUmGrTgqKfFiyTNVoXRQUG80IaaZX2mwO46zSqtSyGRRa68pCMqNAQanyGRRFSObaWzxkByxR4G7AMlFQyGZQuBxQJKkWyW8A/FgkUsEWjyBgBgUhhBBCyKlnY9vkTaS9a0Bv95iATDOg6NZJQZFv0pdp8ujXZEBhLRqr5FAM4hTtGmRQrFrVGacaWss3YACAUgrNMCgfkplUF5LpyuIRp1q0xQMwOQ7uzt8cR0pBEXkYUKy9gsIOKCQsHoo1o4QQQgghp57uOaOgyIY3jIJiRv4EAOzX1OIBAP0FmzyyTGMQp+jUweJhBxQr5FAM47SSgEmLtZcMVwz5tMqLql5DMwzWtma0EZqKVxeb5CTL0BS2qESBcpbhYK+BlM0mLFpI3CVNJplczWig3Gc4WEWGEhhQKA/nPw8OKAghhBBCKqLT2USsQ2B43YRkrpXFw2xw9xds8rBqhW4NFBRWxbGKgmJUtcWjpILCDgdaFalAmlH5AcU4tZtjeQUC4CZoMk6zChQUgbMN/qnIoBANyfRg8RD8HoSBgpCAggMKQgghhJCqUEGAfbWBcHQNGFw50eKx2WpInt6xdHMlRH9Bi0c/VyvUyuKxpIIizUx+Q6UhmblqYLSygqKa/AaLiwFFXNFrsOu5GVDI2QsskcMmBukMCh8Wj3TNLR42TkTC4uEyYPXEtURWIYQQQgghM+mpLjaHLwM6m6+gGNawZjS3eCxaNWqtILWweBQKiuVqUq2tosoMCjtcWWsFRdkMiqJmVLrFwyooHFg80kxeARIoZyGZSSqsoDgVIZlujynf4sEBBSGEEELIqWcYdHHr6AXzl41bZz5mb5RAKWCjUb36wDJp8VhOQVEHi8ckg2K5HcNkQFG9gmL1DIp8QHEaMigqsEgAcBI0mWS6sIxIETkM+SwUFEKvISoyKNYzJDPyoqAQbPHggIIQQggh5GwwDLfwuuQV85c5A4r9YYLNZiT2ad8i2AyK3oIZFBMFRfUDilUzKIb5xrpT4YCirILChmSus8WjaPEQDsm018xF1aixeFTQ4uFogx9nsi0eYX6tnCooMkEFRW6v0Q43+XZgIJGjEQYKQg4PDigIIYQQQqpkHG0iRL7hmRuSGdfK3gFMMigWrRktFBQ1aCJZtWbUZlZUZY8ApjMozq7Fw9oUpIcs1lLizuJRQYuHI4tHaoNKhYYsNmfBqYIiE8ygyM/f5Sa/sHiItHgwg4IQQggh5EyQNrcmfzkmJLNOFaOAUREoBfQWDJq0A4oq1QeWIoNiQXuKpQ4WD7t2WYtHZQqKMFhZ/WEpFBTiGQ6uLR7y57+2LR4+MigyLRIwCQD2W+005DOzx5Zq8eCAghBCCCHk1JM2tyd/mZdBMUxqoTyYJggUNhrhEgoK87g6tHhsrNjiYe0RVQ5ZokAhUOUtHq2KXkMzCkq3YMRphjBQYptjS8OpxSNDQ/z83Vk8qsqgcF4zKmjxAOA0xyEtLB7ODjmXQLnP0Ji7lsgqhBBCCCFkJrplBhS6uQk02jMfsz9KsFUziwdg7Br9ZUMyazBombR4LLfRtKGaVSoolFJoReHKCorC4iGc32BxFZIpbSiBzoIAACAASURBVI8AXLd4yIdkhg4tHol4BoW1eLirwhCtGVWeBizK/E7wTaAUHOWrnryWzDKEEEIIIWQWqp0PKDqz1RNAHpJZg439YbqtCPsLhmRatUIdQjLt5nx1i0e1b6HbjdVtEoXFo6oBhYuQzAoqOoGJgsKNxaOCmtHQocUjH3Ss8wY/FVRQFBYVxyGZcgMi0OJBCCGEEHIWCDo3AQDi9i1zH9OrYQYFYJo8FrV42LaPOlSlKqXQaYQrtHhUb/EAUEpBMapaQeEgJHOcZJVkaLi0eIyTCgYULkMyC4uHUEimhwyKVDAk0wZZZi7PX2sR9QRgFRQcUBBCCCGEnHoa3XMAgHFr/oBib5TUrsUDME0eC2dQxAmaUSAeDDiPjebyAwqrAqnS4mHWL6+gaEUVZVA4snhUoQBxavEQ3BxbojBA7GiTabMspF6DjwyKJNMiDRiAnwFLJhjyGQTKaX7GsWuJrEIIIYQQQmbS6N4MABg2bpr5da21yaCopYIiQm/RDIpRWouATEu7ERaZEosyrLii09KKwiLscllG+VBmnS0ecarX3+KRyrd4NAKF1JHFI60sg2I9a0YDDxaPNJO7/oFSTitSj11LZhlCCCGEEDKL1qYZUPSi2QOK/jiF1qingqIVob9gBkV/nNbC3mHpNEMM4uUyKOzmvmqLR7sRYLhkwKfF2hMqtXg4yaCoIiTTXLOyLSQAEGfyr8FtSKawgiKveHVeMyp2/tbi4e6YmdYiDR4AECq3DSTHUeo3k1LqFqXU55VST+a3N894zH1KqW9P/XdDKfU7+dd+Tyn14tTXPlLmfAghhBBy9lj39yOdfECxF5yb+fX93EKx2WqIndOidJthcX4nMYgTbNRIBbLRDJeuGa2LxaOcgiIPyazIatOMAowcZFBUo6Awu8FxyU1+mmloPdl0S9EIy1e8WuygQExBoTwoKLRgSGZx/u4mFJIhmeuUQfEJAA9ord8C4IH87wfQWj+htb5fa30/gPcC6AP4s6mH/KH9utb6syXPhxBCCCFnj7V+P9J53evx9exH8dzm/TO/vjfMBxR1VVAsuMnv1dHisUJIZhioSjbH07RKKChGecCk1MbsMK08g6JMI0CcZpUoQFxZPOyQQLpmNArdbTLjosVDKCQz9BAyWYHFw6WCIhXM0AgCBSEBRekBxUcBfCr/86cA/MoJj/95AJe01s+WXJcQQgghxLLW70e2NrfwH43/azzVftvMr1uFQi0zKJoheuNkoc3mYFyvAUWnsbyCYhhnlds7AKugWNHikVSzubfY7IsyQZNxVTWjkRuLh1UBVGHxcBHwCUwyKKSGLJHjDAqtNTINwZBMc+u6ZlRq0Bgot/aa4yj7f7o7tNYvA4DW+mWl1O0nPP7XAPy7Q/f9tlLqHwH4JoB/qrW+OuuJSqmPA/g4ANx22224cOFCqRMni7G/v89rLQivtxy81nLwWhMB1v79SDMAHn3yaVwIXzzytUcvm0309x/7LtQr1W+Op3nlxTG0Bv7qgQtoR8e/UX71ygA3tdRK18zH75He9SEu97Kljnvp2RECnVT+O+3G1SGu3lju3C1PPzcCjnkNvn9nP/9sDAB48MKXTvyZmccPdgcIFcS/D9eGZlP+6ONP4ELvqZWPsz82G71nnnoK5183Ensdl18boT9Inax38akxAODLD31JZJNvBxNPXrqEC3h+pWNM/2zbzfbzzz2DCxdecnKOx/HES2bQ/LW/+Tqe3XQzXHvhxRGSsZvv50m8+soIw5HMWicOKJRSXwBw54wv/e4yCymlmgD+HoD/aurufwPgnwHQ+e0fAPitWc/XWn8SwCcB4L777tM7OzvLLE9W5MKFC+C1loPXWw5eazl4rYkLTvv7kXN//QXcdNvt2Nl555GvDR95Gfjmt/Az7/9xvO38tpP1XPFC+1n8yROP4D0/+QHcvtU+9rHhNy/gh+7axs7Oe5Zex8fvkU+/+m28+uyVpY776de+ja295Z7jg7947Tt4cbi70nl85gffwdaNy3Of6/t39tONp4EnHsNPfuCDuLnbXOkYf/joX+Ncp4GdnZ9wfHbHc7U3Bi58Hm++94ex88E3r3ycH+yNgAe/gLf+6I9gc/i02M/T565+F49de8XJev/f+AkEly7i5z70ofIntgBZpoHPfRZveOObsLPzIysdY/pnexinwOf+Evfecw92dn7Y4ZnOZv/hl4CH/w7v+/Efx4/cseXkmH/x2new0Vvt98CyfO7qd/Goo5+dkzhxQKG1/oV5X1NKvaqUuiv/tOIuAK8dc6hfBvAtrfWrU8cu/qyU+mMA/36x0yaEEELIWeK0vx/ZakdF1sRh9vOWjK1aZlAYRUdvlAInvOfuj1N0a2TxaK8QkjmKM7QrrhgFTAbFyiGZSYZWhTYVa/EYl7BJxHmOhjSuLB72+Q3hHJCGQ4tHkmnRkM8gUFAObQa2kUI65NOlTUJrDalvQbhGNaOfBvCx/M8fA/Dnxzz213FITpm/ibD8KoBHSp4PIYQQQs4ea/9+ZLvdwI1hPPNr+/n9m7XMoDDn1FugyaM3TrDRrM9rWC2DIkWnBkOWVlQmJDOtrMEDmLSHlKkajdMMzRXtIWWwOQhlN/m26jMS/j5EYeBsg5ykmdjm3hIF7kI+U+GaVJsV4XJAkWrBkEzBDIqy/yp+H8CHlVJPAvhw/ncopc4rpYoEbKXURv71Pz30/H+hlPquUuphAB8C8E9Kng8hhBBCzh5r/35kqx3hxlwFhbm/W8cBRWvxAcVgXI/NvWWjaVo8lmmTGMQp2lH1r6HdWL1mdJxkaFWoArEKilVDPoEKQzJDRwqKPGBSOiQzCpSzmtEk0+ItJKGHAYVYSGa+TuYwJDPNdHFc3wSBcnrux1Hq/3Ra612YJOzD978E4CNTf+8DuHXG436jzPqEEEIIIafh/ch2p4EXrw1mfm1vlKAVBcXGrk7YAcVJVaPjJEOS6XpZPBohMm2sBq0Fhw7DOK2FCqQVBYhTbTYoS34CPKrIHmGxDSLlFBS6kgFFGCgEykGLh3BFp8VlzahkRaclCgJnLR6FgkJoyBJ6UFDItngopxWvx64lsgohhBBCCJnLdjvCjcEcBcUwqWX+BIBi4LB/goKiPzZf79Rgc2+xdaHL2DyGcYZ2DWpG7TmsoqKoi4KiTAbFKKlGQQEYFUVS0uJhBxzyCgSzwV9GNTSPJNMIhQcsLm0G4gqKYL0VFGGgnFakHgcHFIQQQgghFbPVbmBvXgbFKKll/gSwuMXDKizqpKDYyM9lEC8zoEjrEZJpbRIr5FCMksUVIz5ohmbtshkUrYoURc0wKDVcASaVmdIWDxvK6UKFkKYVKCgcZmjYzbbUa7ADirLDrWkyDVkFxZqEZBJCCCGEkJJstyOMkmzmJ+L7wwSbdVVQ2AHFCSoEO6CoUwaFPZflFBRprRQUwxUUFJWHZDqxeGTim3tLFJbPcUisgkLc4mHWc7HJT1awF5UlDJQzi4cdFEhu8AE4VSFkmYbUtyBQoMWDEEIIIeSssNVuAMDMqtG9UVK0ZdQNq4g4WUFhvl6H/AaL3eSflJ8xzTDJCmtIlZRRUNTF4lFmk19VSCbgyuIhm39gmbSQlA/KTLKskvNPMzchn0XNqLTFw83pAzDDDrGaVMGQTA4oCCGEEEIqxmZMzBpQ1DmDIgoDtKIAvfH6WTzsoGG4hMVjMK6LxaOMgqLakEyrfFi1xSPLdGUhmYAZUJS3eJjnS38f7EDBhc0gqSAk06WCQj4kM1/XcQaFVIaGyi0eLvJLTqL637CEEEIIIWec7UJBcTSHojeubwYFYGweiyoo6mTxWDaDQmuNYVIXi0fJDIoKhyytkiGZtqKzqlabRqgKBcSqTEIyq7F4uMugkD3/da4Zteu4tElkkgqK/PwlRBQcUBBCCCGEVIxVSMxq8qhzBgUAdFsh+qPFMijqaPFYNINinGbQGrUYUBQKiiXUH5bxmodk2uFAVSoQY/Eop6AoLB7iNZ02JNOFxaOaDIp1D8l0WjOayVlU7GWSaPLggIIQQgghpGK2jlFQ7I0SbLYa0qe0MN1mtEDNqB1QVL+5t3SWVFAMx2ZTV4cBRaGgWGGTP0rSytQHQPmQzDh/XlUhmY0wcBCSaVs8hBUUDpsk0soyKBiSaUm1htB8orhOEjkUHFAQQgghhFTMdmd2BsUoSTFOstpmUAC5xeOkDIqRDcmsfnNv2ViyxcPmPdQpg2LZAYXNb6iqohOYHlAsr/4AJtaQRoUWj3HJDb5VMIhv8G0Gxdq2eATOMiiyihQUTi0egt+DiUVFYC3/SxBCCCGEkOOwCoobhxQUvdw6Uf8MihMsHrlKoVuj12FDMhdWUOSPq0OLhx2SLGvxsJv7Si0eJTMorPLiNFg8GtI1o/l6Zc/fHEOLn38YuLNI2EGHlIKisHg4VlDItXiYWyooCCGEEELOAHYAceOQgmI//3utBxTN8MSQzME4hVKo9JP7wyxbM2oHGXWweKyqoLChmpVaPMKSFo+06pBMFxaPahQUDYcKirQiBYWrAYVVMshlOPjIoJBr8fBhUZm7lvcVCCGEEELIsYSBwlYrOpJBsTcyf693SGZ04ia/N0qx0QihpAzTC9CKAii1uAphGNsMiurfPrdWVFCMUvP4KgdFdpNcNiSzqprRyEWLh3DFpSUsFBQuLB7rnUFR1IwKDVkiDxkOmYa4xUPT4kEIIYQQcjbYakdHWjysgmKr5gqKk0IyB3GCjZq9BqUUNhrh4hkUVkFRoT3C0l5jBYVSCs0owKikxaOqAUXToYJC3OKRDxRiB0EC1SgolJMGEmCqZlTY4uFiOGRJMw2pbwFbPAghhBBCzhjbncYRBYXd+Ncpu+EwRkFx/ICiN0prFZBp6TTDpTMo2jV4HSsrKBKbQVHtFqAVBoiT1TY644otHkZB4abFQ9zikQ9EXKgQkkxXUpO6rjWjPlowMi1n8fBRkzoPDigIIYQQQmrAVjs6EpJpBxR1t3jEqcbomFaG/jitRbjkYdprqqCwA4ZlFRTjpPqQTMA0cIzT1Vo87HCgyprRsp+CWwWDtArEbjLLDlgAM2SpRkGxpiGZRQaFu2NKqljsddJUUBBCCCGEnA222o0jNaN2QFF3iweAY5s8BnFSSxXIxlIKivpkUBQ2iSWrOu3jq1ZQNMOgfEhmhRaPVRtILFY9Ij2gsEMdNwqKDFEFAxZXNZ2ZcAaFdfO4bvGQGrAwJJMQQggh5Iyx3Y6ODiiG66GgAHBsk0dtLR6N5S0enZq8jnYUFJkSizKuicWjGa0+oKg6g8KJxSPLoJRcwKHFZQ5CWpHFw7mCQsoika/jasACAFrLtZAU5+9/PsEBBSGEEEJIHdhqN2ZaPAKFWtojLMWA4pgcisG4ngOKdiNcvma0BhYPAGg1whUUFPmAomIVSDNaXYVQh5rR0haPVIsHZAKToY4Ti0dFIZnOa0aFQzJdZjhIhmTaOYjLAcs8OKAghBBCCKkBW7mCYtrjuzdMsNmKalXPeZiJguKYDIo4wUazfiqQjWa4Qs1oTQYUUVCc06LYAUUzrPY1lLF4jCuuGW04sHgkqXxFJzAJ5XSxSa5GQRE4U1CchpDMNJOzeIQezn8eHFAQQgghhNSA7U4DaaYPWA72Rwm22o0Kz+pkJhkU8xUU/VFaG2vENJ3m8iGZVdsjLO0VFBTjGikolg34tMRJtRkUDScWD/nNPTDZjMcONvlxqhEKq0ACly0elYVkum3xkLJ4BB7Of+5a3lcghBBCCCEnspXnTNwYTDb6+7mCos5YBcVxVaP9cVoMMupEe8kMilYUiG1oTmI1BYV5rVVt7i1lMiiKFo9ojVs80qwSBUhU1IyWt3ikWSbepOK0ZlQ4JLOweDhWUEi3eDCDghBCCCHkjGCVEntTORT7o6TWAZkA0M2tG/tzLB5Zrgrp1NDi0VmyZrQu9g5gvRUUrRIZFOO02pDMRmhsBmW8+EmqK7F4TGpGXbR4rHcGhXhIpt3gO1VQyClA7DK0eBBCCCGEnBG2rYJiqsljb1TPes5puq3jLR5WoVDHkMxla0brFFbaWqHFowjJrDjosxkGK9sk7JClupBMa5NYXYUQZ1mhZpDEDnXWu8WjvPoDqCAkMx+EuMrQAMywQOpbMGnx4ICCEEIIIeRMYBUU000e+8MYW7UfUBzf4mFbMupo8bA1o3qBN92DOEW7YuXBNK0owHDpFo/c4lFxjkajREim/fS/ugyK8pv8JNWVfA8mIZmuWjzWOIOiqpBMxy0eUhkUihkUhBBCCCFnC6ug2JtSUPRGae0zKFpRgDBQcxUUNpuijhaPdjOE1lgosLGWFo8lFRSFxaPiAYWTDIqKBxRlgjLjNKs2JHONFRTrGpIJ5BYVhwqErIIWDwEBBQcUhBBCCCF1YLuznhkUSil0m+HcmtE6Kyg28oHDIjkUwySr1YBiNQVFhkDJfWo8jzIDinH+GqTzDyzW4lGmajRONaIqQjKt+qOkgkJrLRrQaAkD5a5mVDgkEzA2iZIFMAdIRVs88jWpoCCEEEIIORscbvHIMm0GFDVXUADG5jFfQWE20XWtGQWwUA7FcFwvi8cqCopRkqEZBYVcuyqaJUIy4zSr1KLixOJRQQMGMNmMl93k2+efhhYPSQVFELjNcMh0FS0eHFAQQgghhJwJOo0QUaAKBYXNdNiquYICMAOK/hwVgrV4bNTR4tFYYkCR1MviYWpGl2/xqDogEzD5EYvYamYxrqii0xI5sHgkqbw9ApgaUJS0eKRFwKTs9yEMglOgoHBp8YDYsDFgSCYhhBBCyNlCKYWtdlSEZO7nioS1UFA0w+J8D2MHF3Vs8egsY/GIU7RrsLm3mJrRZRUUaeUBmUBeM1oig6KqgExgqsWjxCY/TrNKLB6hYwWF9JAlDNxZDKRrRgG3IZ9AbvEQ+jEKi5BM/2tV/xuKEEIIIYQAME0eNiRzP7+tewYFcJLFwyoo6rO5t1hVx0IKijirlU2lFRkVwiINJJZRklUekAlMLB7LnLtlnFSroGi6UFBkuhKLh1LKVHWW3GWmqWxFpyUMAqSZXunn5jDSNaN2LZcKBMkWDyuWoYKCEEIIIeQMsd2JigHF3jopKFoRenMtHlZBUb/X0Wmat8KLKChqVzOaqz+WUVHUZkARBtB6tU/y44oqOi1uLB4ZImF7hCUKywdNxnnIZlRBBgUAuBAh2DYNqQ2+XcuVgiITztAoLB4MySSEEEIIOTtstRq4McgtHsM1yqBohnMVFHbzv9Gqj/rAYjMl5uVnTDOM01rkN1jsoGGpAUWcoVmD12AHDKvYPEwGRXUhn24sHtUoKACgEQTOMiikhywTi0p5n0GaaSglXzPqSoGQCQ9YQofDoZPggIIQQgghpCZstScKCpvp0F0TBYW1chzG1o9u1Chg0mJVHYuETY7qZvGwCoolgjLHaU0UFNHqKoT4VFg8qnsNYahKb/Cry6CwOQjld8lpJh9UGgaq9HDIYhUgcgqKg+t6Xcv7CoQQQgghZCG2O42ixWOtQjJb0fyQzDhBMwoqCQU8ic6CLR5ppjFOs3qFZK6koEhrMaCwm/NVFRTrbvGIU13Zv4coCEqpP4DqMigixwMKyYBMwNgkXG3w7YxJ6jWwxYMQQggh5AxiWjwOhmRutRpVntJCdJsRhnE2c+MwGKe1DMgEFm/xsAqLOmZQLFM1Okqq3dxbmisMVyx1afEo80l4nGZoVFAzCpjzT0srKKrJoDgNCgpXGQ5FhobQPwVmUBBCCCGEnEG22g3sjxKkmZ6yeNRzcz+NPcfeDJtHb5TW0t4BAG0bknnCJt8OAepk8VhFQTFOslrkaFgVx3gli4eu1OJRqD9KhWRq8c29xYXNIK2gAQOYKCjKhnzaY0jmTwDmejlyeBTfAykFBTMoCCGEEELOINt5IOb+KMH+KEGnEdbSGnEYm5MxKyhzECfYqKlNpRkGCAN1ooLCDjDqZPGYtHgso6Coh8WjWdLi0ajwNTQcZVBU9e+6EQblWzzSqjIozDVzoaDItBYfsATKnQJBa9khkZ2DuGohOY7qf0MRQgghhBAAwHbb2DluDGLsDRNsrkGDBzA9oDi6We6N6mvxUEqh0wgXUFCYzWirThaPfJNuz20R6haSudKAIsnQrEGLRzmLh67M4hEG5UMyq2vxMLcuFBRVWTxcbfCrUlBoZlAQQgghhJwdbKXo3tAoKLZqqjw4TDcfQMxUUIzTIuuhjrQb4Yk1o4XFo0avo72KgiLOajFkaZaxeFQckunG4lGdgiIKVOmQTDvgCMUzKMw1c6FCWPeQTPkWD3VgXa9rlXmyUuofKqUeVUplSqn3HfO4X1JKPaGUuqiU+sTU/bcopT6vlHoyv725zPkQQggh5Oxxmt6PbHeMgmJvGGN/GK+fgmJGBkU/TmpdlbrRDE8MmpyEZNZnQLGKgmKUVBswaSlj8YjTamtGXVg84qy6DIpGGJT+FD+tqGbUZQbFuodkWhFMKN7iIbBWyec/AuDvA3ho3gOUUiGAfw3glwG8DcCvK6Xeln/5EwAe0Fq/BcAD+d8JIYQQQpbh1LwfsQqKG7mCYh0qRgHT4gHMtnj0R2mtwiUP02mEC7R4mN1AnQYUqygoxklWZFdUSRmLR5xWHZJZ3uKRpBkawvYISxioUsMVYDIgkM5wmLR4lDt/c4yqQjLXtcXD3Na+xUNr/bjW+okTHvYTAC5qrZ/SWo8B/B8APpp/7aMAPpX/+VMAfqXM+RBCCCHk7HGa3o/YDIq9ocmgqLPyYJqixWOGxaM/rm+LBwC0myH6CysoqlcfWFZTUKT1UFCUqBkdV62giMopKLJMI9PyFZ2WRuiuxUM6g8KpgqKCkEyXGRRZRRkUEiGZykXQhVLqAoD/XGv9zRlf+wcAfklr/Z/mf/8NAD+ptf5tpdQ1rfVNU4+9qrWeKatUSn0cwMfzv/4YzKclxD+vA3C56pM4Q/B6y8FrLQevtRz3aa23qj6JquD7kVMNf4/IwWstC6+3HLzWcqz8fuTEsbxS6gsA7pzxpd/VWv/5AmvMGussPRXRWn8SwCfzc/qm1nqux5S4g9daFl5vOXit5eC1lkMpdWRjflrg+5GzDa+1HLzWsvB6y8FrLUeZ9yMnDii01r+w6sFzXgDw+qm//xCAl/I/v6qUuktr/bJS6i4Ar5VcixBCCCGnEL4fIYQQQk4/EsahbwB4i1LqzUqpJoBfA/Dp/GufBvCx/M8fA7DIJyCEEEIIIcvC9yOEEEJIzSlbM/qrSqkXAHwAwGeUUn+V339eKfVZANBaJwB+G8BfAXgcwJ9orR/ND/H7AD6slHoSwIfzvy/CJ8ucN1kKXmtZeL3l4LWWg9dajjN5rfl+5EzAay0Hr7UsvN5y8FrLsfK1dhKSSQghhBBCCCGEEFKG6nuGCCGEEEIIIYQQcubhgIIQQgghhBBCCCGVU+sBhVLql5RSTyilLiqlPjHj60op9d/mX39YKfWeKs7zNLDAtf6P82v8sFLqq0qpd1VxnqeBk6711ON+XCmVKqX+geT5nTYWud5KqR2l1LeVUo8qpb4kfY6nhQV+j5xTSv2FUuo7+bX+zSrO8zSglPoflVKvKaUemfN1/v/REXwvIgvfj8jB9yNy8L2ILHw/IoO39yJa61r+ByD8/9u7exC5qjCM4/8X1gVFIcGoRTQkip+FggYNouJHIdkmCBYiGBAbEcUyYKFFmthZiKQIIjam0KAp/EAQiaCrIqhRA7JGiIuCREUhKWTNa3GnWJYsc2Z25tw7M/8fXNiZucXLw2XOw9mZucBPwNXAPPANcNOacxaA92jubb4L+LztuSfxKMz6TmBz7+/dZj2+rFed9xHwLvBw23NP6lF4bW8CfgC29R5f3vbck3gUZv0c8GLv78uAP4H5tmefxAO4B7gV+G6d110fR5OzXaR7edtHKmW96jz7yJiztotUz9s+Mpqsx9JFuvwJituBpcw8mZn/AoeBPWvO2QO8no1FYFM09y/XYPpmnZmfZuZfvYeLNPeP1+BKrmuAZ4C3gN9rDjeFSvJ+FDiSmacAMtPMh1OSdQKXREQAF9MUgpW6Y06HzDxGk996XB9Hwy5Sl32kHvtIPXaRuuwjlYyri3R5g2Ir8Muqx8u95wY9R/0NmuMTNLthGlzfrCNiK/AQcLDiXNOq5Nq+DtgcER9HxFcRsbfadNOlJOuXgRuBX4HjwLOZea7OeDPH9XE07CJ12UfqsY/UYxepyz7SHUOtj3NjG2fj4jzPrb0nask56q84x4i4j6YQ3DXWiaZXSdYvAfsy879mY1cbUJL3HHAb8ABwIfBZRCxm5o/jHm7KlGT9IPA1cD9wDfBhRHySmf+Me7gZ5Po4GnaRuuwj9dhH6rGL1GUf6Y6h1scub1AsA1etenwlzS7XoOeov6IcI+Jm4BCwOzP/qDTbtCnJeidwuFcGtgALEbGSmW/XGXGqlL6PnM7MM8CZiDgG3AJYCgZTkvXjwIFsvpi4FBE/AzcAX9QZcaa4Po6GXaQu+0g99pF67CJ12Ue6Y6j1sctf8fgSuDYidkTEPPAIcHTNOUeBvb1fCN0F/J2Zv9UedAr0zToitgFHgMfczd2Qvlln5o7M3J6Z24E3gacsA0MreR95B7g7IuYi4iLgDuBE5TmnQUnWp2j+O0REXAFcD5ysOuXscH0cDbtIXfaReuwj9dhF6rKPdMdQ62NnP0GRmSsR8TTwAc2vsb6amd9HxJO91w/S/KLwArAEnKXZDdOACrN+HrgUeKW3k76SmTvbmnlSFWatESnJOzNPRMT7wLfAOeBQZp73dklaX+G1vR94LSKO03zsb19mnm5t6AkWEW8A9wJbImIZeAG4AFwfR8kuUpd9pB77SD12kbrsI/WMq4tE88kWSZIkSZKk9nT5Kx6SJEmSJGlGuEEhSZIkSZJa5waFJEmSJElqnRsUwibsKwAAAChJREFUkiRJkiSpdW5QSJIkSZKk1rlBIUmSJEmSWucGhSRJkiRJat3/FMd5QXU/S34AAAAASUVORK5CYII=\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "%reset -f\n", "%matplotlib inline\n", "from matplotlib.pylab import *\n", "\n", "phi = lambda t,y,b,g : -b*y+g\n", "\n", "def EE(phi,tt,y0,b,g):\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],b,g))\n", " return uu\n", "\n", "t0 = 0.\n", "tfinal = 1.0\n", "\n", "g=40.\n", "b=40.\n", "Y0=[g/b,g/b+1.e-8]\n", "NN=[19,23,45,100]\n", "\n", "figure(figsize=(18, 7))\n", "subplot(1,2,1)\n", "y0=Y0[0]\n", "for N in NN:\n", " tt=linspace(t0,tfinal,N+1)\n", " yy=EE(phi,tt,y0,b,g)\n", " plot(tt,yy,label=('$N=$'+str(N)+' noeuds'));\n", "title(\"Euler explicite, $g=b=$\"+str(g)+ \", $y_0$=\"+str(y0))\n", "axis([0,1,g/b-1.e-8,g/b+1.e-8])\n", "legend()\n", "grid()\n", "\n", "subplot(1,2,2)\n", "y0=Y0[1]\n", "for N in NN:\n", " tt=linspace(t0,tfinal,N+1)\n", " yy=EE(phi,tt,y0,b,g)\n", " plot(tt,yy,label=('$N=$'+str(N)+' noeuds'));\n", "title(\"Euler explicite, $g=b=$\"+str(g)+ \", $y_0$=\"+str(y0))\n", "axis([0,1,g/b-1.e-8,g/b+1.e-8])\n", "legend()\n", "grid()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Méthode d'Euler implicite** \n", "1. La méthode d'Euler implicite donne la suite définie par récurrence $u_{n+1}=Au_n+B$ et $u_0=y_0$ avec $A=\\frac{1}{1+bh}$ et $B=\\frac{gh}{1+bh}$. \n", "\tOn introduit une suite auxiliaire définie par $v_n=u_n-\\alpha$ ainsi $u_n=v_n+\\alpha$ pour tout $n\\in\\mathbb{N}$.\n", "\tOn a $v_{n+1}+\\alpha=A(v_n+\\alpha)+B=Av_n+A\\alpha+B$.\n", "\tEn posant $\\alpha=\\frac{B}{1-A}\\ (A\\neq1)$, la suite $(v_n)_{n\\in \\mathbb{N}}$ est géométrique de raison $A$ donc $v_{n}=v_0A^n$. \n", "\tEn remplaçant $v_n$ par $u_n-\\alpha$ et $v_0$ par $u_0-\\alpha$ on en déduit que\n", "\t$u_n=A^n(u_0-\\alpha)+\\alpha,\\quad \\alpha=\\frac{B}{1-A}$ soit encore\n", "\t$$\n", "\tu_n-\\frac{g}{b}=\\frac{1}{(1+hb)^n}\\left( y_0 -\\frac{g}{b}\\right)\n", "\t$$\n", " \n", "1. Comparons la solution exacte et la solution approchée:\n", " \\begin{align*}\n", " y_n&=\\frac{g}{b}+\\left( y_0 -\\frac{g}{b}\\right)e^{-bt_n}\\\\\n", " u_n&=\\frac{g}{b}+\\left( y_0 -\\frac{g}{b}\\right)\\frac{1}{(1+hb)^n}\n", " \\end{align*}\n", " Pour que $u_n\\simeq y_n$ il faut que $\\frac{1}{(1+hb)^n}\\simeq e^{-bt_n} \\in]0,1]$, donc il faut que $0\\le\\frac{1}{1+hb}<1$ c'est-à-dire pour n'importe quelle valeur de $h$.\n", "\tDans cet exemple, pour obtenir la convergence il n'est pas nécessaire de limiter le pas $h$. \n", "\tCependant, pour une edo quelconque, à chaque pas de temps on doit résoudre une équation non-linéaire. " ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "%reset -f\n", "%matplotlib inline\n", "from matplotlib.pylab import *\n", "from scipy.optimize import fsolve\n", "\n", "phi = lambda t,y,b,g : -b*y+g\n", "\n", "def EI(phi,tt,y0,b,g):\n", " h=tt[1]-tt[0]\n", " uu = [y0]\n", " for i in range(len(tt)-1):\n", " uu.append(g/b + (y0-g/b)/(1.+h*b)**i)\n", " #uu.append(fsolve(lambda x: x-uu[i]-h*phi(tt[i+1],x,b,g), uu[i]))\n", " return uu\n", "\n", "t0 = 0.\n", "tfinal = 1.0\n", "\n", "g=40.\n", "b=40.\n", "Y0=[g/b,g/b+1.e-8]\n", "NN=[19,23,45,100]\n", "\n", "figure(figsize=(18,7))\n", "subplot(1,2,1)\n", "y0=Y0[0]\n", "for N in NN:\n", " tt=linspace(t0,tfinal,N+1)\n", " yy=EI(phi,tt,y0,b,g)\n", " plot(tt,yy,label=('$N=$'+str(N)+' noeuds'));\n", "title(\"Euler implicite, $g=b=$\"+str(g)+ \", $y_0$=\"+str(y0))\n", "axis([0,1,g/b-1.e-8,g/b+1.e-8])\n", "legend()\n", "grid()\n", "\n", "subplot(1,2,2)\n", "y0=Y0[1]\n", "for N in NN:\n", " tt=linspace(t0,tfinal,N+1)\n", " yy=EI(phi,tt,y0,b,g)\n", " plot(tt,yy,'-*',label=('$N=$'+str(N)+' noeuds'));\n", "title(\"Euler implicite, $g=b=$\"+str(g)+ \", $y_0$=\"+str(y0))\n", "axis([0,1,g/b-1.e-8,g/b+1.e-8])\n", "legend()\n", "grid()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Propriétés d'un schéma d'approximation: consistance, stabilité, convergence\n", "\n", "Le héorème d’équivalence de Lax-Ritchmyer affirme qu'un schéma consistant est convergent ssi il est zéro-stable:\n", "
\n", "Consistance + zéro-stabilité = convergence \n", "
\n", "\n", "Rappelons ci-dessous la définition de schéma convergent et de schéma consistant, avant d'introduire la (les) notion(s) de stabilité d'un schéma." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Convergence\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", "- $N\\to+\\infty$ lorsque $h\\to0$ pour $T$ fixé (notion liée à la zéro-stabilité)\n", "- $N\\to+\\infty$ lorsque $T\\to+\\infty$ pour $h$ fixé (notion liée à la A-stabilité).\n", "\n", ">Soit $u_{n+1}^*$ la solution numérique au temps $t_{n+1}$ qu'on obtiendrait en insérant de la solution exacte dans le schéma.\n", "Pour vérifier qu'une méthode converge, on écrit l'erreur ainsi\n", "\\begin{equation}\\label{quart7.9}\n", "e_n \\equiv y_n - u_n = (y_n - u_n^*) + (u_n^* - u_n).\n", "\\end{equation}\n", "- Le terme $y_n - u_n^*$ représente l'erreur engendrée par une seule itération du schéma. \n", "- Le terme $u_{n+1}^* - u_{n+1}$ représente la propagation de $t_{n}$ à $t_{n+1}$ de l'erreur accumulée au temps précédent $t_{n}$. \n", ">\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", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ " \n", "### Consistance\n", "On appelle *erreur de troncature locale* la quantité\n", "$$\n", "\\tau_{n+1}(h)\\equiv\\frac{y_{n+1}-u_{n+1}^*}{h}\n", "$$\n", "et *erreur de troncature* \n", "$$\n", "\\tau(h)=\\max_{n=0,\\dots,N}|\\tau_n(h)|.\n", "$$\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 $p\\ge1$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Stabilité : zéro-stabilité et A-stabilité\n", "De manière générale, un schéma numérique est dit stable s’il permet de contrôler la solution quand on perturbe les données. Il existe de nombreuses notions de stabilité.\n", "\n", "Considérons le problème de Cauchy:\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", "\n", "avec $y_0$ une valeur donnée et supposons que l'on ait montré l'existence et l'unicité d'une solution $y$ pour $t\\in I$ (donc il est mathématiquement bien posé) et qu'il soit numériquement bien posé. \n", "\n", "Deux questions naturelles se posent: \n", "- que se passe-t-il lorsqu'on fixe le temps final $T$ et on fait tendre le pas $h$ vers $0$? \n", "- que se passe-t-il lorsqu'on fixe le pas $h>0$ mais on fait tendre $T$ vers l'infini? \n", "\n", "Dans les deux cas le nombre de nœuds tend vers l'infini mais dans le premier cas on s'intéresse à l'erreur en chaque point, dans le deuxième cas il s'agit du comportement asymptotique de la solution et de son approximation. \n", "\n", "Ces deux questions font intervenir deux notions de stabilité:\n", "- **Zéro-stabilité:** \n", " la propriété de zéro-stabilité assure que la méthode numérique est peu sensible aux petites perturbations des données (encore faut-il que le problème soit **numériquement bien posé**). L'exigence d'avoir une méthode numérique stable provient avant tout de la nécessité de contrôler les (inévitables) erreurs introduites par l'arithmétique finie des ordinateurs. En effet, si la méthode numérique n'était pas zéro-stable, les erreurs d'arrondi commises sur $y_0$ et sur le calcul de $\\varphi(t_n, y_n)$ rendraient la solution calculée inutilisable.\n", "\n", "- **A-stabilité:** \n", " la zéro-stabilité s'intéresse à la résolution du problème de Cauchy sur des intervalles bornés. Dans ce cadre, le nombre $N_h$ de sous-intervalles ne tend vers l'infini que quand $h$ tend vers zéro. Il existe cependant de nombreuses situations dans lesquelles le problème de Cauchy doit être intégré sur des intervalles en temps très grands ou même infini. Dans ce cas, même pour $h$ fixé, $N_h$ tend vers l'infini. On s'intéresse donc à des méthodes capables d'approcher la solution pour des intervalles en temps arbitrairement grands, même pour des pas de temps $h$ \"assez grands\" (si on peut choisir $h>0$ quelconque, on dira que la méthode est inconditionnellement A-stable, sinon on aura une condition de stabilité sur $h$ qui détérminera le temps de calcul). Considérons un problème de Cauchy dont la solution exacte vérifie la propriété $y(t)\\xrightarrow[t\\to+\\infty]{}0$. Soit $h>0$ fixé et considérons la limite $T\\to+\\infty$ (ainsi $N_h\\to+\\infty$). On dit que la méthode est A-stable si $u_n^{(h)}\\xrightarrow[n\\to+\\infty]{}0$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Étude de la zéro-stabilité pour les schémas à un pas\n", "\n", "La zéro-stabilité garantit que, sur un intervalle borné, des petites perturbations des données entraînent des perturbations bornées de la solution numérique quand $h\\to0$.\n", "\n", "Plus précisément, une méthode numérique pour approcher le problème de Cauchy sur l'intervalle fini $I=[t_0,T]$ è **zéro-stable** si\n", "$$\n", "\\exists h_0>0,\\ \n", "\\exists C > 0,\\ \n", "\\exists \\varepsilon_0> 0 \n", "\\text{ tel que } \n", "\\forall h \\in ]0,h_0],\\\n", "\\forall \\varepsilon \\in ]0,\\varepsilon_0],\\\n", "si |\\varrho_n| \\le \\varepsilon,\\\n", "0 \\le n \\le N,\n", "$$\n", "alors\n", "$$\n", "|z_n - u_n | \\le C\\varepsilon,\\\n", "0 \\le n \\le N\n", "$$\n", "où\n", "- $C$ est une costante qui ne dépend pas de $h$,\n", "- $u_n$ est une solution obtenue par une schéma,\n", "- $z_n$ est la solution obtenue par le même schéma sur le problème perturbé,\n", "- $\\varrho_n$ est la perturbation au $n$-ième pas,\n", "- $\\varepsilon_0$ est la perturbation maximale.\n", "\n", "Naturellement, $\\varepsilon_0$ doit être assez petit pour que le problème perturbé ait encore une unique solution sur l’intervalle d’intégration donné. \n", "\n", "\n", "Pour une méthode consistante à un pas, on peut prouver que la zéro-stabilité est une conséquence du fait que $\\varphi$ est lipschitzienne par rapport à sa deuxième variable. Dans ce cas, la constante $C$ qui apparaît dépend de $\\exp((T − t_0 )L)$, où $L$ est la constante de Lipschitz. Cependant, ceci n’est pas toujours vrai pour les autres familles de méthodes.\n", "\n", "\n", "### Exemple: Zéro-stabilité de la méthode d'Euler explicite \n", "Soit $u_n$ l'approximation obtenue avec la méthode d'Euler explicite du problème donné:\n", "$$\n", "\\begin{cases}\n", "u_0=y_0,\\\\\n", "u_{n+1}=u_n+h\\varphi(t_n,u_n),&n=0,\\dots,N-1\n", "\\end{cases}\n", "$$\n", "et $z_n$ l'approximation obtenue avec la méthode d'Euler explicite du problème perturbé:\n", "$$\n", "\\begin{cases}\n", "z_0=y_0+\\varrho_0,\\\\\n", "z_{n+1}=z_n+h\\left(\\varphi(t_n,z_n)+\\varrho_n\\right),&n=0,\\dots,N-1\n", "\\end{cases}\n", "$$\n", "On a\n", "\\begin{align}\n", "|u_n - z_n|\n", "&\\le\n", "|u_{n-1}-z_{n-1}| + h|\\varphi(t_{n-1},u_{n-1})-\\varphi(t_{n-1},z_{n-1})|+h|\\varrho_n|\n", "\\\\\n", "&\\le\n", "(1+hL)|u_{n-1}-z_{n-1}| + h\\varepsilon\n", "\\\\\n", "&\\le\n", "(1+hL)^2|u_{n-2}-z_{n-2}| + (1+(1+hL))h\\varepsilon\n", "\\\\\n", "&\\dots\n", "\\\\\n", "&\\le\n", "(1+hL)^n|\\varrho_0|+\n", "\\left( 1+(1+hL)+\\dots+(1+hL)^{n-1} \\right)h\\varepsilon\n", "\\\\\n", "&=(1+hL)^n|\\varrho_0|+\n", "\\left(\\sum_{i=0}^{n-1}(1+hL)^i\\right)h\\varepsilon\n", "\\\\\n", "&=(1+hL)^n|\\varrho_0|+\n", "\\frac{(1+hL)^n-1}{L}\\varepsilon\n", "\\\\\n", "&\\le\n", "(e^{hL})^n|\\varrho_0|+\\frac{(e^{hL})^n-1}{hL}h\\varepsilon \\qquad\\text{car }(1+x)\\le e^x\n", "\\\\\n", "&=\n", "e^{nhL}|\\varrho_0|\n", "+\\frac{(e^{nhL})-1}{L}\\varepsilon \n", "\\\\\n", "&\\le\n", "\\left(e^{nhL}+\\frac{(e^{nhL})-1}{L}\\right)\\varepsilon \n", "\\\\\n", "&=\n", "\\frac{(1+L)e^{L(T-t_0)}-1}{L}\\varepsilon \\qquad\\text{car } t_n-t_0=nh\n", "\\end{align}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Étude de la A-stabilité pour les schémas à un pas\n", "\n", "Dans la section précédente, on a considéré la résolution du problème de Cauchy sur des intervalles bornés. Dans ce cadre, le nombre $N$ de sous-intervalles ne tend vers l’infini que quand $h$ tend vers zéro.\n", "Il existe cependant de nombreuses situations dans lesquelles le problème de Cauchy doit être intégré sur des intervalles en temps très grands ou même infini. Dans ce cas, même pour $h$ fixé, $N$ tend vers l’infini, et\n", "un résultat comme la zéro-stabilité n’a plus de sens puisque le membre de droite contient une quantité non bornée ($T-t_0$). On s’intéresse donc à des **méthodes capables d’approcher la solution pour des intervalles en temps arbitrairement grands, même pour des pas de temps $h$ \"assez grands\"**.\n", "\n", "
\n", "Soit $\\beta>0$ un nombre réel positif et considérons le problème de Cauchy\n", "$$\\begin{cases}\n", "y'(t)=-\\beta y(t), &\\text{pour }t>0,\\\\\n", "y(0)=y_0\n", "\\end{cases}$$\n", "où $y_0\\neq0$ est une valeur donnée. Sa solution est $y(t)=y_0e^{-\\beta t}$ donc $$\\lim_{t\\to+\\infty}y(t)=0.$$\n", "\n", "Soit $h>0$ un pas de temps donné, $t_n=nh$ pour $n\\in\\mathbb{N}$ et notons $u_n\\approx y(t_n)$ une approximation de la solution $y$ au temps $t_n$.\n", "\n", "Si, sous d'éventuelles conditions sur $h$, on a\n", "$$\n", "\\lim_{n\\to+\\infty} u_n =0,\n", "$$\n", " alors on dit que le schéma est A-stable.\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "On peut tirer des conclusions analogues quand $\\beta$ est un complexe ou une fonction positive de $t$. \n", "D'autre part, en général il n'y a aucune raison d'exiger qu'une méthode numérique soit absolument stable quand on l'applique à un autre problème. \n", "Cependant, on peut montrer que quand une méthode absolument stable sur le problème modèle est utilisée pour un problème modèle généralisé, l'erreur de perturbation (qui est la valeur absolue de la différence entre la solution perturbée et la solution non perturbée) est bornée uniformément (par rapport à $h$). \n", "En bref, on peut dire que les méthodes absolument stables permettent de contrôler les perturbations. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Remarque:** ce résultat peut être généralisé à un système de $n$ EDO de la forme\n", "$$\n", "\\mathbf{y}'(t)=-\\mathbb{A}\\mathbf{y}(t)\n", "$$\n", "où $\\mathbb{A}$ est une matrice constante ayant $n$ valeurs propres positives $\\lambda_i$, $i=1,2,...,n$. \n", "La solution s'écrit \n", "$$\n", "\\mathbf{y}(t)=\\sum_i c_i \\mathbf{v}_i e^{-\\lambda_i t}\n", "$$\n", "où $\\mathbf{v}_i$ est le $i$-ème vecteur propre associé à la valeure propre $\\lambda_i$ donc \n", "$$\n", "\\lim_{t\\to+\\infty}y_i(t)=0 \\quad \\forall i=1,2,...,n.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### A-stabilité du schéma d'Euler explicite \n", "Le schéma d'**Euler progressif** devient\n", "$$\n", "u_{n+1}=(1-\\beta h)u_n, \\qquad n=0,1,2,\\dots\n", "$$\n", "et par suite\n", "$$\n", "u_{n}=(1-\\beta h)^nu_0, \\qquad n=0,1,2,\\dots\n", "$$\n", "Par conséquente, $\\lim\\limits_{n\\to+\\infty}u_n=0$ si et seulement si \n", "$$\n", "\\left|1-\\beta h\\right|<1,\n", "$$\n", "ce qui a pour effet de limiter $h$ à\n", "$$\n", "\\boxed{h<\\frac{2}{\\beta}.}\n", "$$\n", "Cette condition de A-stabilité limite le pas $h$ d'avance en $t$ lorsqu'on utilise le schéma d'Euler progressif. \n", "De plus, cette convergence est *monotone* ssi $1-\\beta h>0$, i.e. ssi $h<\\frac{1}{\\beta}$.\n", "\n", "Notons que si $1-\\beta h>1$ alors $u_n$ tend vers $+\\infty$ lorsque $t$ tend vers l'infini et si $1-\\beta h<-1$ alors $u_n$ tend vers l'infini en alternant de signe lorsque $t$ tend vers l'infini. \n", "Nous dirons dans ces cas que le schéma d'Euler progressif est instable. \n", "\n", "\n", "**Remarque:** dans le cas du système, le schéma d'Euler progressif est A-stable ssi \n", "$$\n", "h<\\frac{2}{\\lambda_\\text{max}}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### A-stabilité du schéma d'Euler implicite \n", "Le schéma d'**Euler rétrograde** devient dans le cadre de notre exemple\n", "$$\n", "(1+\\beta h)u_{n+1}=u_n, \\qquad n=0,1,2,\\dots\n", "$$\n", "et par suite\n", "$$\n", "u_{n}=\\frac{1}{(1+\\beta h)^{n}}u_0, \\qquad n=0,1,2,\\dots\n", "$$\n", "Dans ce cas nous voyons que **pour tout $h>0$** nous avons $\\lim_{n\\to\\infty}u_n=0$, le schéma d'Euler rétrograde est donc toujours stable, sans limitations sur $h$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### A-stabilité du schéma d'Euler modifié\n", "Le schéma d'**Euler modifié** pour notre exemple devient\n", "$$\n", "u_{n+1}=u_n+h\\left( -\\beta\\left( u_n+\\frac{h}{2}(-\\beta u_n) \\right) \\right) =\\left(1-\\beta h +\\frac{(\\beta h )^2}{2} \\right) u_{n}\n", "$$\n", "Par induction on obtient\n", "$$\n", "u_{n}=\\left(1-\\beta h +\\frac{(\\beta h )^2}{2} \\right)^nu_0.\n", "$$\n", "Par conséquent, $\\lim\\limits_{n\\to+\\infty}u_n=0$ si et seulement si \n", "$$\n", "\\left| 1 - \\beta h +\\frac{1}{2}(\\beta h)^2 \\right|<1.\n", "$$\n", "Notons $x$ le produit $\\beta h$ et $q$ le polynôme $q(x)=\\frac{1}{2}x^2-x+1$ dont le graphe est représenté en figure:" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "%matplotlib inline\n", "from matplotlib.pylab import *\n", "\n", "xx=linspace(0,3,101)\n", "yy=[1-x+0.5*x**2 for x in xx]\n", "#xkcd()\n", "plot(xx,yy,[0,2,2],[1,1,0],'m:')\n", "axis([0,3,0,3])\n", "xlabel(r'$h\\beta$')\n", "ylabel(r'$q(h\\beta)$');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Nous avons $|q(x)|<1$ si et seulement si $00$ pour tout $x$, donc **cette convergence est monotone** (tandis qu'avec le schéma d'Euler explicite la convergence n'est monotone que si $h<\\frac{1}{\\beta}$). " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### A-stabilité du schéma de Cranck-Nicolson \n", "Le schéma de **Crank-Nicolson** appliqué à notre exemple s'écrit\n", "$$\n", "\\left(1+\\beta\\frac{h}{2}\\right)u_{n+1} = \\left(1-\\beta\\frac{h}{2}\\right) u_{n}\n", "$$\n", "et par suite\n", "$$\n", "u_{n}=\\left( \\frac{2-\\beta h}{2+\\beta h} \\right)^{n}u_0, \\qquad n=0,1,2,\\dots\n", "$$\n", "Par conséquent, $\\lim\\limits_{n\\to+\\infty}u_n=0$ si et seulement si \n", "$$\n", "\\left|\\frac{2-\\beta h}{2+\\beta h}\\right|<1.\n", "$$\n", "Notons $x$ le produit $\\beta h>0$ et $q$ la fonction $q(x)=\\frac{2-x}{2+x}=1-2\\frac{x}{2+x}$. Nous avons $0<\\frac{x}{2+x}<1$ pour tout $x\\in\\mathbb{R}_+^*$, donc $|q(x)|<1$ pour tout $x\\in\\mathbb{R}_+^*$. \n", "La relation $\\lim\\limits_{n\\to+\\infty}u_n=0$ est donc satisfaite pour tout $h>0$: le schéma de Crank-Nicolson est donc toujours stable, **sans limitations sur $h$.** \n", "Cependant, si on cherche $00$)\n", "$$\n", "1-2\\frac{x}{2+x}>0\n", "\\iff\n", "x<2.\n", "$$" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "%matplotlib inline\n", "from matplotlib.pylab import *\n", "xx=linspace(0,10,101)\n", "yy=[1-2*x/(2+x) for x in xx]\n", "plot(xx,yy)\n", "plot([0,10],[1,1],'r--',[0,10],[-1,-1],'r--')\n", "plot([0,10],[0,0],'m:',[2,2],[-1.5,1.5],'m:')\n", "axis([0,10,-1.5,1.5])\n", "xlabel(r'$h\\beta$')\n", "ylabel(r'$q(h\\beta)$');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### A-stabilité du schéma de Heun \n", "Le schéma de **Heun** pour notre exemple devient\n", "$$\n", "u_{n+1} =u_n+\\frac{h}{2}\\left( -\\beta u_n-\\beta\\left( u_n+h(-\\beta u_n) \\right) \\right) =\\left(1-\\beta h +\\frac{(\\beta h )^2}{2} \\right) u_{n}\n", "$$\n", "ce qui coïncide avec la méthode d'Euler modifiée. Par conséquent, $\\lim\\limits_{n\\to+\\infty}u_n=0$ si et seulement si \n", "$$\n", "\\boxed{ h <\\frac{2}{\\beta} }\n", "$$\n", "et cette **convergence est monotone**. \n", "Cette condition de stabilité limite le pas $h$ d'avance en $t$ lorsqu'on utilise le schéma de Heun." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### A-stabilité du schéma de Simpson implicite \n", "Le schéma de Simpson implicite appliqué à notre exemple s'écrit\n", "$$\n", "\\left(1+\\beta\\frac{h}{6}\\right)u_{n+1} = \\left(1-\\frac{5}{6}\\beta h+\\frac{1}{3}(\\beta h)^2\\right) u_{n}\n", "$$\n", "et par suite\n", "$$\n", "u_{n}=\\left( \\frac{ 1-\\frac{5}{6}\\beta h+\\frac{1}{3}(\\beta h)^2 }{ 1+\\beta\\frac{h}{6} } \\right)^{n}u_0, \\qquad n=0,1,2,\\dots\n", "$$\n", "Par conséquent, $\\lim\\limits_{n\\to+\\infty}u_n=0$ si et seulement si \n", "$$\n", "\\left|\\frac{ 6-5\\beta h+2(\\beta h)^2 }{ 6+\\beta h }\\right|<1.\n", "$$\n", "Notons $x$ le produit $\\beta h>0$ et $q$ la fonction $q(x)=\\frac{6-5x+2x^2}{6+x}$ dont le graphe est représenté en figure." ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "%matplotlib inline\n", "from matplotlib.pylab import *\n", "xx=linspace(0,4,101)\n", "yy=[(6-5*x+2*x**2)/(6+x) for x in xx]\n", "#xkcd()\n", "plot(xx,yy,[0,3,3],[1,1,0],'m:')\n", "axis([0,4,0,3])\n", "xlabel(r'$h\\beta$')\n", "ylabel(r'$q(h\\beta)$');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Nous avons $q(x)>0$ pour tout $x\\in\\mathbb{R}_+^*$ et $q(x)<1$ si et seulement si $x<3$, donc $|q(x)|<1$ pour tout $x\\in]0;3[$. Par conséquent, $\\lim\\limits_{n\\to+\\infty}u_n=0$ si et seulement si \n", "$$\n", " h <\\frac{3}{\\beta}.\n", "$$\n", "Cette condition de stabilité limite le pas $h$ d'avance en $t$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### A-stabilité du schéma de Simpson explicite \n", "Le schéma de Simpson explicite pour notre exemple devient\n", "$$\n", "u_{n+1} = u_n+\\frac{h}{6}\\left( -\\beta u_n -4 \\beta\\left( u_n+\\frac{h}{2}(-\\beta u_n) \\right) -\\beta\\left( u_n+h(-\\beta u_n) \\right) \\right) =\\left(1-\\beta h +\\frac{3}{2}(\\beta h )^2 \\right) u_{n}\n", "$$\n", "Par induction on obtient\n", "$$\n", "u_{n}=\\left(1-\\beta h +\\frac{3}{2}(\\beta h )^2 \\right)^nu_0.\n", "$$\n", "Par conséquent, $\\lim\\limits_{n\\to+\\infty}u_n=0$ si et seulement si \n", "$$\n", "\\left|1-\\beta h +\\frac{3}{2}(\\beta h )^2\\right|<1.\n", "$$\n", "Notons $x$ le produit $\\beta h$ et $q$ le polynôme $q(x)=\\frac{3}{2}x^2-x+1$ dont le graphe est représenté en figure." ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "%matplotlib inline\n", "from matplotlib.pylab import *\n", "xx=linspace(0,3,101)\n", "yy=[1-x+1.5*x**2 for x in xx]\n", "#xkcd()\n", "plot(xx,yy,[0,2/3,2/3],[1,1,0],'m:')\n", "axis([0,3,0,3])\n", "xlabel(r'$h\\beta$')\n", "ylabel(r'$q(h\\beta)$');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Nous avons $|q(x)|<1$ si et seulement si $00$ \t| $\\forall h>0$ \t|\n", "| EM \t| $u_{n+1}=u_n+h\\varphi\\left(t_{n}+\\frac{h}{2},u_n+\\frac{h}{2}\\varphi(t_n,u_n)\\right)$ \t| Explicite \t| 2 \t| $00$ \t| $0On considère le problème de Cauchy\n", "$$\n", "\\begin{cases}\n", "y'(t)=-y(t),\\\\\n", "y(0)=1,\n", "\\end{cases}\n", "$$\n", "sur l'intervalle $[0;12]$. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Solution exacte\n", "Il s'agit d'une EDO à variables séparables. \n", "L'unique solution constante de l'EDO est la fonction $y(t)\\equiv0$, toutes les autres solutions sont du type $y(t)=C e^{-t}$. \n", "Donc l'unique solution du problème de Cauchy est la fonction $y(t)=e^{-t}$ définie pour tout $t$ $\\in$ $\\mathbb{R}$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Solution approchée par la méthode d'Euler explicite \n", "La méthode d'Euler explicite pour cette EDO s'écrit\n", "$$\n", "u_{n+1}=(1-h)u_n.\n", "$$\n", "En procédant par récurrence sur $n$, on obtient\n", "$$\n", "u_{n+1}=(1-h)^{n+1}.\n", "$$\n", "De la formule $u_{n+1}=(1-h)^{n+1}$ on déduit que\n", "+ si $02$ alors la solution numérique oscille et diverge.\n", "En effet, la méthode est A-stable si et seulement si $|1 - h| < 1$.\n", "\n", "Remarque: la suite obtenue est une suite géométrique de raison $q=1-h$. On sait qu'une telle suite \n", "+ diverge si $|q|>1$ ou $q=-1$, \n", "+ est stationnaire si $q=1$,\n", "+ converge vers $0$ si $|q|<1$.\n", "+ converge de façon monotone vers $0$ si $00$:\n", "* si $h=4$ alors $t_n=4n$ et $u_{n}=\\left(-4\\right)^{n}$ tandis que $y(t_{n})=e^{-4n}$,\n", "* si $h=2$ alors $t_n=2n$ et $u_{n}=\\left(-1\\right)^{n}$ tandis que $y(t_{n})=e^{-2n}$,\n", "* si $h=1$ alors $t_n=n$ et $u_{n}=0$ tandis que $y(t_{n})=e^{-n}$,\n", "* si $h=\\frac{1}{2}$ alors $t_n=\\frac{n}{2}$ et $u_{n}=\\left(\\dfrac{1}{2}\\right)^{n}$ tandis que $y(t_{n})=e^{-n/2}$,\n", "* si $h=\\frac{1}{4}$ alors $t_n=\\frac{n}{4}$ et $u_{n}=\\left(\\dfrac{3}{4}\\right)^{n}$ tandis que $y(t_{n})=e^{-n/4}$.\n", "Ci-dessous sont tracées sur l'intervalle $[0;10]$, les courbes représentatives de la solution exacte et de la solution calculée par la méthode d'Euler explicite. En faisant varier le pas $h$ nous pouvons constater que si $h>1$ l'erreur commise entre la solution exacte et la solution calculée est amplifiée d'un pas à l'autre. \n", "(Remarque: la première itérée a la même pente quelque soit le pas $h$ (se rappeler de la construction géométrique de la méthode d'Euler))." ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "%matplotlib inline\n", "from matplotlib.pylab import *\n", "\n", "# SCHEMA\n", "def EE(phi,tt,y0):\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", "phi = lambda t,y : -y\n", "sol_exacte = lambda t : exp(-t)\n", "\n", "# INITIALISATION\n", "t0, y0, tfinal = 0 , 1 , 12\n", "\n", "# CALCUL\n", "H = [ 2**(k-2) for k in range(5) ]\n", "tt = [] # liste de liste\n", "uu = [] # liste de liste\n", "\n", "for h in H:\n", "\tNh = int((tfinal-t0)/h)\n", "\ttt.append( [ t0+i*h for i in range(Nh+1) ] )\n", "\tuu.append(EE(phi,tt[-1],y0))\n", "\n", "# AFFICHAGE\n", "figure(1, figsize=(15, 10))\n", "yy = [sol_exacte(t) for t in tt[0]] # affichage de la sol exacte sur la grille la plus fine\n", "axis([t0, tfinal, -8, 8])\n", "\n", "plot(tt[0],yy,'-',label=('$y(t)=e^{-t}$'))\n", "for k in range(5):\n", " plot(tt[k],uu[k],'-o',label=('h='+str(H[k])))\n", "legend()\n", "grid();\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Solution approchée par la méthode d'Euler implicite \n", "La méthode d'Euler implicite pour cette EDO s'écrit \n", "$$\n", "u_{n+1}=\\frac{1}{1+h}u_n.\n", "$$\n", "En procédant par récurrence sur $n$, on obtient\n", "$$\n", "u_{n+1}=\\frac{1}{(1+h)^{n+1}}.\n", "$$\n", "De la formule $u_{n+1}=(1+h)^{-(n+1)}$ on déduit que la solution numérique est stable et convergente pour tout $h$.\n", "En effet, la méthode est inconditionnellement A-stable.\n", "\n", "Remarque: la suite obtenue est une suite géométrique de raison $q=\\frac{1}{1+h}\\in]0;1[$. " ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "data": { "image/png": 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "%matplotlib inline\n", "from matplotlib.pylab import *\n", "from scipy.optimize import fsolve\n", "\n", "# SCHEMA\n", "def EI(phi,tt,y0):\n", " h=tt[1]-tt[0]\n", " uu = [y0]\n", " for i in range(len(tt)-1):\n", " uu.append(fsolve(lambda x:-x+uu[i]+h*phi(tt[i+1],x),uu[i]))\n", " return uu\n", "\n", "phi = lambda t,y : -y\n", "sol_exacte = lambda t : exp(-t)\n", "\n", "# INITIALISATION\n", "t0, y0, tfinal = 0 , 1 , 12\n", "\n", "# CALCUL\n", "H = [ 2**(k-2) for k in range(5) ]\n", "tt = [] # liste de liste\n", "uu = [] # liste de liste\n", "\n", "for h in H:\n", "\tNh = int((tfinal-t0)/h)\n", "\ttt.append( [ t0+i*h for i in range(Nh+1) ] )\n", "\tuu.append(EI(phi,tt[-1],y0))\n", "\n", "# AFFICHAGE\n", "figure(1, figsize=(15, 10))\n", "yy = [sol_exacte(t) for t in tt[0]] # affichage de la sol exacte sur la grille la plus fine\n", "axis([t0, tfinal, -1, 1])\n", "\n", "plot(tt[0],yy,'-',label=('$y(t)=e^{-t}$'))\n", "for k in range(5):\n", " plot(tt[k],uu[k],'-o',label=('h='+str(H[k])))\n", "legend()\n", "grid();\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Solution approchée par la méthode d'Euler modifié\n", "La méthode d'Euler modifié pour cette EDO s'écrit \n", "$$\n", "u_{n+1}=(1-h+\\frac{1}{2}h^2)u_n.\n", "$$\n", "En procédant par récurrence sur $n$, on obtient\n", "$$\n", "u_{n+1}=(1-h+\\frac{1}{2}h^2)^{n+1}.\n", "$$\n", "La suite obtenue est une suite géométrique de raison $q=1-h+\\frac{1}{2}h^2$:\n", "+ si $02$ alors la solution numérique diverge vers $+\\infty$.\n", "\n", "En effet, la méthode est A-stable si et seulement si $|1-h+\\frac{1}{2}h^2| < 1$. De plus, comme $1-h+\\frac{1}{2}h^2>0$ pour tout $h>0$, la suite est positive." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "%matplotlib inline\n", "from matplotlib.pylab import *\n", "\n", "# SCHEMA\n", "def EM(phi,tt,y0):\n", " h=tt[1]-tt[0]\n", " uu = [y0]\n", " for i in range(len(tt)-1):\n", " utilde=uu[i]+h/2*phi(tt[i],uu[i])\n", " uu.append(uu[i]+h*phi(tt[i]+h/2,utilde))\n", " return uu\n", "\n", "phi = lambda t,y : -y\n", "sol_exacte = lambda t : exp(-t)\n", "\n", "# INITIALISATION\n", "t0, y0, tfinal = 0 , 1 , 12\n", "\n", "# CALCUL\n", "H = [ 2**(k-2) for k in range(5) ]\n", "tt = [] # liste de liste\n", "uu = [] # liste de liste\n", "\n", "for h in H:\n", "\tNh = int((tfinal-t0)/h)\n", "\ttt.append( [ t0+i*h for i in range(Nh+1) ] )\n", "\tuu.append(EM(phi,tt[-1],y0))\n", "\n", "# AFFICHAGE\n", "figure(1, figsize=(15, 10))\n", "yy = [sol_exacte(t) for t in tt[0]] # affichage de la sol exacte sur la grille la plus fine\n", "axis([t0, tfinal, -1, 8])\n", "\n", "plot(tt[0],yy,'-',label=('$y(t)=e^{-t}$'))\n", "for k in range(5):\n", " plot(tt[k],uu[k],'-o',label=('h='+str(H[k])))\n", "legend()\n", "grid();\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Solution approchée par la méthode de Crank-Nicolson\n", "La méthode de Crank-Nicolson pour cette EDO s'écrit \n", "$$\n", "u_{n+1}=\\frac{2-h}{2+h}u_n.\n", "$$\n", "En procédant par récurrence sur $n$, on obtient\n", "$$\n", "u_{n+1}=\\left(\\frac{2-h}{2+h}\\right)^{n+1}.\n", "$$\n", "La suite obtenue est une suite géométrique de raison $q=\\frac{2-h}{2+h}\\in]-1;1[$. \n", "On déduit que la solution numérique est stable et convergente pour tout $h$.\n", "En effet, la méthode est inconditionnellement A-stable. \n", "\n", "De plus, la suite est positive ssi $0" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "%matplotlib inline\n", "from matplotlib.pylab import *\n", "from scipy.optimize import fsolve\n", "\n", "# SCHEMA\n", "def CN(phi,tt,y0):\n", " h=tt[1]-tt[0]\n", " uu = [y0]\n", " for i in range(len(tt)-1):\n", " uu.append(fsolve(lambda x:-x+uu[i]+h/2*phi(tt[i],uu[i])+h/2*phi(tt[i+1],x),uu[i]))\n", " return uu\n", "\n", "phi = lambda t,y : -y\n", "sol_exacte = lambda t : exp(-t)\n", "\n", "# INITIALISATION\n", "t0, y0, tfinal = 0 , 1 , 12\n", "\n", "# CALCUL\n", "H = [ 2**(k-2) for k in range(5) ]\n", "tt = [] # liste de liste\n", "uu = [] # liste de liste\n", "\n", "for h in H:\n", "\tNh = int((tfinal-t0)/h)\n", "\ttt.append( [ t0+i*h for i in range(Nh+1) ] )\n", "\tuu.append(CN(phi,tt[-1],y0))\n", "\n", "# AFFICHAGE\n", "figure(1, figsize=(15, 10))\n", "yy = [sol_exacte(t) for t in tt[0]] # affichage de la sol exacte sur la grille la plus fine\n", "axis([t0, tfinal, -1, 1])\n", "\n", "plot(tt[0],yy,'-',label=('$y(t)=e^{-t}$'))\n", "for k in range(5):\n", " plot(tt[k],uu[k],'-o',label=('h='+str(H[k])))\n", "legend()\n", "grid();\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Solution approchée par la méthode de Heun\n", "La méthode de Heun pour cette EDO s'écrit \n", "$$\n", "u_{n+1}=(1-h+\\frac{1}{2}h^2)u_n.\n", "$$\n", "On retrouve le schéma d'Euler modifié." ] } ], "metadata": { "hide_input": false, "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.5" }, "toc": { "base_numbering": 1, "nav_menu": {}, "number_sections": true, "sideBar": true, "skip_h1_title": true, "title_cell": "Table of Contents", "title_sidebar": "Contents", "toc_cell": false, "toc_position": { "height": "calc(100% - 180px)", "left": "10px", "top": "150px", "width": "384px" }, "toc_section_display": true, "toc_window_display": false } }, "nbformat": 4, "nbformat_minor": 4 }