{"nbformat":4,"nbformat_minor":0,"metadata":{"colab":{"provenance":[],"authorship_tag":"ABX9TyMfWViYOFECvs7d148+pMyT"},"kernelspec":{"name":"python3","display_name":"Python 3"},"language_info":{"name":"python"}},"cells":[{"cell_type":"markdown","source":["# Variance reduction"],"metadata":{"id":"856tFswmENXK"}},{"cell_type":"code","execution_count":66,"metadata":{"id":"V0_GOOUdY9X_","executionInfo":{"status":"ok","timestamp":1698055014098,"user_tz":-120,"elapsed":627,"user":{"displayName":"Julien Reygner","userId":"06465841509771183939"}}},"outputs":[],"source":["import numpy as np\n","import matplotlib.pyplot as plt"]},{"cell_type":"markdown","source":["## 1. Control variate\n","\n","For $t>0$ and $x \\in R$, we set $$f_t(x) = \\frac{1}{1+t x^2}, \\qquad g_t(x) = 1-t x^2.$$"],"metadata":{"id":"e9XL05DXL_CB"}},{"cell_type":"code","source":["def f(t,x):\n"," return 1/(1+t*x**2)\n","\n","def g(t,x):\n"," return 1-t*x**2"],"metadata":{"id":"Mklav__ovfcr","executionInfo":{"status":"ok","timestamp":1698047197420,"user_tz":-120,"elapsed":233,"user":{"displayName":"Julien Reygner","userId":"06465841509771183939"}}},"execution_count":8,"outputs":[]},{"cell_type":"markdown","source":["For $X \\sim \\mathcal{N}(0,1)$, our aim is to estimate the value of $\\mathcal{I}:=E[f_t(X)]$, using $Y=g_t(X)$ as a control variate. Note that $E[Y] = 1-t$."],"metadata":{"id":"aq08HibqKlOW"}},{"cell_type":"markdown","source":["With the notation of the lecture notes, complete the following code so that the function `MCandCV(t,X)` takes in input a number $t>0$ and an array $X$ which contains $n$ realisations of independent standard Gaussian variables, and returns the list $(\\hat{\\mathcal{I}}_n, \\hat{\\sigma}_n, \\hat{\\mathcal{I}}^\\mathsf{CV}_n, \\hat{\\sigma}^\\mathsf{CV}_n)$."],"metadata":{"id":"0SEbykCDKWh3"}},{"cell_type":"code","source":["def MCandCV(t,X):\n"," n = len(X)\n"," fX = f(t,X)\n"," Y = g(t,X)\n","\n"," mean_fX = np.mean(fX)\n"," mean_Y = np.mean(Y)\n","\n"," var_fX = np.mean((fX-mean_fX)**2)\n"," var_Y = np.mean((Y-mean_Y)**2)\n"," cov_fX_Y = np.mean((fX-mean_fX)*(Y-mean_Y))\n","\n"," beta = cov_fX_Y/var_Y\n","\n"," estim_CV = np.mean(fX-beta*Y)+beta*(1-t)\n"," estim_var_CV = var_fX - cov_fX_Y**2/var_Y\n","\n"," return [mean_fX,var_fX**.5,estim_CV,estim_var_CV**.5]"],"metadata":{"id":"hfv2ACrNwcgN","executionInfo":{"status":"ok","timestamp":1698048832693,"user_tz":-120,"elapsed":246,"user":{"displayName":"Julien Reygner","userId":"06465841509771183939"}}},"execution_count":19,"outputs":[]},{"cell_type":"markdown","source":["We use this function to plot the estimated values of $\\mathcal{I}$ as a function of $t$ obtained with both the naive MC estimator and the CV estimator, together with respective $95\\%$ confidence intervals."],"metadata":{"id":"RwL3Fk3gnzgU"}},{"cell_type":"code","source":["# Sample\n","nX = 1000\n","X = np.random.normal(0,1,nX)\n","\n","# Values of t\n","nt = 100\n","t = np.linspace(.1,10,nt)\n","\n","# Estimators and standard deviations\n","estim_MC = np.zeros(nt)\n","sigma_MC = np.zeros(nt)\n","estim_CV = np.zeros(nt)\n","sigma_CV = np.zeros(nt)\n","\n","for i in range(nt):\n"," [estim_MC[i],sigma_MC[i],estim_CV[i],sigma_CV[i]] = MCandCV(t[i],X)\n","\n","# Plot of estimators and 95% confidence intervals\n","plt.plot(t,estim_MC,label='Naive MC',color='blue')\n","plt.fill_between(t, estim_MC-1.96*sigma_MC/nX**.5, estim_MC+1.96*sigma_MC/nX**.5, color='blue', alpha=.1)\n","\n","plt.plot(t,estim_CV,label='Control Variate',color='orange')\n","plt.fill_between(t, estim_CV-1.96*sigma_CV/nX**.5, estim_CV+1.96*sigma_CV/nX**.5, color='orange', alpha=.1)\n","\n","plt.legend()\n","plt.show()"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":430},"id":"QiSil_xyNGKX","executionInfo":{"status":"ok","timestamp":1698049633189,"user_tz":-120,"elapsed":574,"user":{"displayName":"Julien Reygner","userId":"06465841509771183939"}},"outputId":"239cde84-6068-486b-9a20-f2d52a60d258"},"execution_count":27,"outputs":[{"output_type":"display_data","data":{"text/plain":["
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M2zYsD0+b9GiRaxcuZJNmzbVBoi///3vjBw5ki+++IKjjjoKcELL008/TWZmJgAXXXQRixYt4vbbbycrKwvDMPD5fE0ucdx6662cdNJJAASDQR555BGefvppTj31VAAef/xxFi5cyBNPPMENN9zQol/DpZdeyvXXX88DDzxARkYG1dXVvPzyyzzwwAMAnHjiiQ2e/9hjj5Gdnc3ixYv50Y9+VHv/+eefX1urCbBx48YGr+vbty/XX3997Z+vuuoq3nnnHV566SUmTJjQ7Gd/6KGHGDt2LHfccUftfU8++ST5+fmsW7eOQw89tEWfs7VkZiSpJO78xY+XfSdn1AghRJrYLZxBWb16Nfn5+Q1mMkaMGEF2djarV9edwj5w4MDaIALQu3fvZs9S292RRx5Z+88bNmwgHo9zzDHH1N6n6zoTJkxo8H57c95552GaJi+99BLgnPemqmrtjE5hYSGXXXYZQ4cOJSsri0AgQE1NDVu3bm12bE0xTZPbbruNUaNGkZubS0ZGBu+8806j6+xuxYoVvP/++2RkZNTeUsFww4YNLf6crSUzI0lRt/PL9sSdA/P2o05JCCEOTprPmaFI13u3wNChQ1EUpc2KVHcvCFUUBcuyWvTappaJ9lcgEOBnP/sZTz31FJdccglPPfUU5557LhkZzozVxRdfTGlpKffffz8DBgzA7XYzadIkYrFYq8Z29913c//993PfffcxatQo/H4/1157baPr7K6mpoYzzjiDu+66q9FjvXv3buWnbTn5yk3SkwfmdTPWk4glwCO/GiFEJ6MoLVoqSafc3FymTJnC3Llzufrqqxt96VZUVJCdnc3w4cPZtm0b27Ztq50d+fbbb6moqGDEiBEtfj/DMDBb0GBqyJAhGIbBxx9/zIABTo1hPB7niy++aPX22EsvvZQTTjiBN998k08++YS777679rGPP/6Yhx9+mNNOOw2Abdu2UVJS0qrrp65z5plncuGFFwLOctW6desa/G6a+uzjxo3jlVdeYeDAgfu1e6i1ZJkmqftgJ4z0ytxEPBJM82iEEKLrmjt3LqZpMmHCBF555RW+++47Vq9ezQMPPFDbYHPy5MmMGjWKCy64gGXLlrFkyRKmTZvG8ccfv9cljPoGDhzI559/zubNmykpKWl21sTv93PFFVdwww03sGDBAr799lsuu+wyQqEQl156aas+33HHHcchhxzCtGnTGDZsGEcffXTtY0OHDuXZZ59l9erVfP7551xwwQV4vd5WXT91nYULF/LJJ5+wevVq/t//+3+Nuqc39dlnzJhBWVkZ5513Hl988QUbNmzgnXfeYfr06S0KbftKwkjSkBF5VASz0FSLWMnadA9HCCG6rMGDB7Ns2TJ+8IMf8Jvf/IbDDz+ck046iUWLFvHII48AznLL66+/Tk5ODscddxyTJ09m8ODBzJ8/v1Xvdf3116NpGiNGjKBHjx57rKm48847+elPf8pFF13EuHHjWL9+Pe+88w45OTmtek9FUbjkkksoLy9vtBPmiSeeoLy8nHHjxnHRRRdx9dVX07Nnz1ZdH+Cmm25i3LhxTJkyhRNOOIFevXpx1llnNXhOU5+9T58+fPzxx5imycknn8yoUaO49tpryc7O3uOZcvtLsVtaLZRGVVVVZGVlUVlZ2W6t4S0Lvrz9e0wY8jnfBp5ixI9+0S7vI4QQB0okEmHTpk0MGjQIj8eT7uGITmpPf89a+v0tMyNJqgo7g85STahwneyoEUIIIQ4QCSP11ChOGFFD69lLwbEQQggh2oiEkXrsgBNGspTviEdlakQIIYQ4ECSM1JPRdyQAfTPXEAmG0jwaIYQQomuQMFJP/xFDqAhm4dEjJEq+bulRCkIIIYTYDxJG6hl1hMpXW8YDUL35C+LxNA9ICCHaQAfYNCk6sLb4+yVhpB7DgC3V4wCIFa4gHpP/AwshOq5UK/RQSJadRftJ/f3avfV+a0jP892EvM5Jj5nW18SiCfwZ+/7LFUKIdNI0jezs7NqD4Xw+H4qipHlUorOwbZtQKERRURHZ2dlomrbP15Iwsht/vtNGuK//G0prgtAtO70DEkKI/ZA6Hr6lJ9UK0VrZ2dm1f8/2lYSR3QwdM4iyb3LIzSjHLF2BlX887dgBVwgh2pWiKPTu3ZuePXsSl0I40cZ0Xd+vGZEUCSO7OWK0wmdvjWfy4f+lessS4iOPx+1O96iEEGL/aJrWJl8aQrQH+W/+3fj9sLna2VGTKP5ailiFEEKIdiZhZDeKAiGPU8QaML8mFpVpTSGEEKI9SRhpQkZ/p4i1T8ZqwtVVaR6NEEII0blJGGnCYWP7U1LdDV2LQ9kKTDPdIxJCCCE6LwkjTThitMLSzc7sSNUW6cQqhBBCtCcJI03w+2FLlVPEapasJBa10jwiIYQQovOSMNIEVa0rYs2ypIhVCCGEaE8SRpqROdCZGemdsZpwZWWaRyOEEEJ0XhJGmnHYmH4UVvbEpZpQ/hWJRLpHJIQQQnROEkaaUb+INbhtKbFYmgckhBBCdFISRprh98PW6nEAmKVfE4/J/l4hhBCiPUgYaYam1RWxZttfE6qRIlYhhBCiPUgY2YOMAc4yTS//WiJV5dL8TAghhGgHEkb2YPi4Puws742mWqgVUjcihBBCtAcJI3twxBGwLFnEGtn5OdGIND8TQggh2pqEkT3w+2Fd+TEAuCq+IFgjUyNCCCFEW5MwsgcuF1R5jwcgz/U5kWBE6kaEEEKINiZhZC/6HTGemoifDKMCrXIZ0Wi6RySEEEJ0LhJG9uK443U+Wnus84eixVI3IoQQQrQxCSN70bcvrCw4DgC7eAmhoNSNCCGEEG1Jwshe6DpUeU8AoIe6hGhNWM6pEUIIIdqQhJG90HXoMewoghEfGUYZSuVXUjcihBBCtCEJI3uhKHDscToff+ds8dVKPyQWlS01QgghRFuRMNICgwfD8p1O3YhZtIRgldSNCCGEEG1ln8LI3LlzGThwIB6Ph4kTJ7JkyZI9Pv++++7jsMMOw+v1kp+fz3XXXUckEtmnAaeD2w2VhhNGuiufEw1FpG5ECCGEaCOtDiPz589n5syZzJ49m2XLljF69GimTJlCUVFRk89//vnnufHGG5k9ezarV6/miSeeYP78+fzud7/b78EfKIYB3Q6dSCjqJdMoRa1eLnUjQgghRBtpdRi59957ueyyy5g+fTojRoxg3rx5+Hw+nnzyySaf/8knn3DMMcdw/vnnM3DgQE4++WTOO++8vc6mHEw0DY7+vptPv5vk/Ln0Q6IRqRsRQggh2kKrwkgsFmPp0qVMnjy57gKqyuTJk/n000+bfM3RRx/N0qVLa8PHxo0befvttznttNOafZ9oNEpVVVWDW7oddhgs2+G0hreKlhCskqkRIYQQoi24WvPkkpISTNMkLy+vwf15eXmsWbOmydecf/75lJSUcOyxx2LbNolEgssvv3yPyzRz5szhj3/8Y2uG1u4MA8p0p24klyUUhSLEYj4MI80DE0IIITq4dt9N88EHH3DHHXfw8MMPs2zZMl599VXeeustbrvttmZfM2vWLCorK2tv27Zta+9h7pVhQM6Q7xGOeQgYxShVK+lANbhCCCHEQatVMyPdu3dH0zQKCwsb3F9YWEivXr2afM3NN9/MRRddxC9/+UsARo0aRTAY5Fe/+hW///3vUdXGecjtduN2u1sztHan6zDxaA+fLfwePxjxAZ6qDwlWH0Mg0KpfoRBCCCF206qZEcMwGD9+PIsWLaq9z7IsFi1axKRJk5p8TSgUahQ4NE0DwLbt1o43rUaMgC+3Oks1icLPCQcjmFLHKoQQQuyXVv9n/cyZM7n44os58sgjmTBhAvfddx/BYJDp06cDMG3aNPr27cucOXMAOOOMM7j33nsZO3YsEydOZP369dx8882cccYZtaGko/B6oURzilhzrCXsCoWIRDLw+9M8MCGEEKIDa3UYmTp1KsXFxdxyyy0UFBQwZswYFixYUFvUunXr1gYzITfddBOKonDTTTexY8cOevTowRlnnMHtt9/edp/iADEMyBx0NMGIj4CniPLQ54RDp+P3SyNbIYQQYl8pdgdYK6mqqiIrK4vKykoCgUDaxmHbsGgRhP7zY348/t8U515NzZA5DBjio4nSFyGEEKJLa+n3t3yFtoKiwKhR8NGmUwGwiz4iFolIN1YhhBBiP0gYaSWvF4KZThjprn2FFtpCJHzQTy4JIYQQBy0JI61kGDBq0gBWbDkCVbEJRP5LtZziK4QQQuwzCSOtZBhw7LEK76x02tnbhYuJhSLEJI8IIYQQ+0TCSCupKnTrBpvjThgJRD4iHq6QbqxCCCHEPpIwsg98PsgZ+j1KqrvhdVWSGfmYYHU83cMSQgghOiQJI/vAMOD4E3QWrDgFAHf5e4RqIiQSaR6YEEII0QFJGNkHhgFDhsDSXc6uGq3sQ+LhkCzVCCGEEPtAwsg+0DTIyIB4t5MwLZVc1zrcsbWEgnJQjRBCCNFaEkb2kdcL4yb14JN1RwMQCC6kplIOzhNCCCFaS8LIPnK7YcJEhYXfnA6AWfAhsVCYUCjNAxNCCCE6GAkj+8gwnKWaQpdTxJpjfo7LLKCm2krzyIQQQoiORcLIPlIUyMyEfoePYktJf3Q1QlZ0MaHqMHHZ5SuEEEK0mISR/eDxwHHHabz51Y8A0Eo+IBYKEg6neWBCCCFEByJhZD94PNC3LywrOgsAX+V/0SmlukqqWIUQQoiWkjCyHzQN/H7wDzyOgoo8PFoF2ZEPCFWH5awaIYQQooUkjOwnnw9OnGzw8pKfAaCXvkMiUi1LNUIIIUQLSRjZTx4PDBuu8Ml2J4x4Kt7DTSlVFVLFKoQQQrSEhJH9ZBhOIMkb8T22l/XFrVaTFXmPSDBMNJru0QkhhBAHPwkj+0lRnH4jPzzJ4J+fnwOAq3ihLNUIIYQQLSRhpA14vTBosMqSgp86f656H49aSmV5DNtO8+CEEEKIg5yEkTbgdju3/NFHsbl4AIYaJCu8iEgwLCf5CiGEEHshYaQNuFzO7Mjkk3Re+vxcALSihdixKoI1MjUihBBC7ImEkTaSkQF9+qosK3aWanzVH+BXi6iqiJJIpHdsQgghxMFMwkgb8XhA12HwuDGsLxiCrobJjLxHtCYoJ/kKIYQQeyBhpI243U4DtB+erDP/s6kAKAX/RVcrqKwwpZBVCCGEaIaEkTaUmQl5eSpfl58NgL9mMX61kHB1SApZhRBCiGZIGGlDXq+zVHPohJF8vXUULjWKv2oBVqyKYDDdoxNCCCEOThJG2pBh1O2qeXLxJQBoO1/D66qhqjyCKYf5CiGEEI1IGGlDigKBAHTrrrEuci6xhI4/vpIMcyWxYFBmR4QQQogmSBhpYx6P03fk5DO68dqXZwHgLnkDl1JBVaUUsgohhBC7kzDSxtxuZ6nmBye6eOHzXwBgFP4br1ZBqCokh+cJIYQQu5Ew0sZSSzUuXSNr6HFsLcnHUCrw13yAFauipibdIxRCCCEOLhJG2oHXC5oGP/uZi6f+Nx0AvbCukFU6sgohhBB1JIy0A7fbqR05dLibD7ddgGUp+IMf4WMz0ZoamR0RQggh6pEw0g4UBbKyIBZXOG5KPou++SEA3tI38WiVlJclsKw0D1IIIYQ4SEgYaScej7NU8+OzNJ75yOk54tr1L7x6DZGaGtnmK4QQQiRJGGknHo9TO+LLMAhlnUJZTQ4eewfu4FJ0u4KKcku2+QohhBBIGGk3igLZ2RCLwc/OdfOPjy8EwF34T3xGkFBVkHA4vWMUQgghDgYSRtqRz+e0iJ94tJtXV/wSAE/Fu+hmASQqqKqUqREhhBBCwkg7MgzIyIBYXGPs9w/h3ZUnoSoWvuL5ZBg1VFeEpQmaEEKILk/CSDvLzATThPMvUHno3asBMArmoys1JCKVVFeneYBCCCFEmkkYaWc+n1PImtvdjZJ3POt2DUW3q3CXL8DnqqKyLEI8nu5RCiGEEOmzT2Fk7ty5DBw4EI/Hw8SJE1myZEmzzz3hhBNQFKXR7fTTT9/nQXckmua0h49EFS65FB589yoAPDuewW3EiAWlCZoQQoiurdVhZP78+cycOZPZs2ezbNkyRo8ezZQpUygqKmry+a+++iq7du2qva1atQpN0zjnnHP2e/Adhd/v7K4ZO97Dl8U/pzIUwB3fgFH9BV5XBWUlcWkRL4QQostqdRi59957ueyyy5g+fTojRoxg3rx5+Hw+nnzyySafn5ubS69evWpvCxcuxOfzdakw4vU6yzWRmM55F7h54oNLnfsLnsVrRIjVVEntiBBCiC6rVWEkFouxdOlSJk+eXHcBVWXy5Ml8+umnLbrGE088wc9//nP8fn+zz4lGo1RVVTW4dWS17eFjcMaZOi98eTmWpeCt+gA1tlNmR4QQQnRprQojJSUlmKZJXl5eg/vz8vIoKCjY6+uXLFnCqlWr+OUvf7nH582ZM4esrKzaW35+fmuGeVBK9RxRNA/HT8njjWU/BsBf9BxeI0w0WC21I0IIIbqkA7qb5oknnmDUqFFMmDBhj8+bNWsWlZWVtbdt27YdoBG2H7fbqR0JRxQuukhj7n+dQlaj8GWwo/i0ckqLZXZECCFE19OqMNK9e3c0TaOwsLDB/YWFhfTq1WuPrw0Gg7z44otceumle30ft9tNIBBocOsMAgGn50i3PC9Zg49i+ZbRuAjhK3o+OTsiO2uEEEJ0Pa0KI4ZhMH78eBYtWlR7n2VZLFq0iEmTJu3xtf/85z+JRqNceOGF+zbSTiDVcyQS1bh0usWcN2YB4N35JGDi08ooLUlgmukdpxBCCHEgtXqZZubMmTz++OM888wzrF69miuuuIJgMMj06dMBmDZtGrNmzWr0uieeeIKzzjqLbt267f+oOyhNg9xciERg9HgvW+Ons3bnobisCrzF/3RmR2qqZWeNEEKILsXV2hdMnTqV4uJibrnlFgoKChgzZgwLFiyoLWrdunUrqtow46xdu5aPPvqId999t21G3YH5/U4hazTu5spfB5nzzCye/n/T8e14jFDP8/CqZZSWZJKR4cLV6v91hBBCiI5HsW37oD86tqqqiqysLCorKztF/UhBAZSVQXZGkB+dbvLqL49gYI8tVA28jXC3sygN9yUvP4cuPIkkhBCiE2jp97ecTZMGmZnOTxMfv748yl3//i0A3u0PYysaflcZ5aVxYrE0DlIIIYQ4QCSMpIHP5yzXhMIKp57u5sNt57GzvDd6YhfesrfxGmHiwUoqK9M9UiGEEKL9SRhJA0WB7Gyc03pdGfzqsjh/eet6AHzbH8JSDTLcZVSURolE0jpUIYQQot1JGEkTvz+1zVflrJ8YvLV6OiXV3dBjm/GULcRwxTAj5VRUpHukQgghRPuSMJImLpdzXk0kAi5PBpdMT/DX/1wHgH/bX7FVD5nuCirLQoRCaR6sEEII0Y4kjKRRRoYTSmJxjXOnupi/9HJndiS6AW/pG7hcFsTKKS+zOfj3PAkhhBD7RsJIGnk8Tov4UAjc/gwuusjmjtd/B4B/61+wFRcBTyVV5UFpEy+EEKLTkjCSZqlt1wlL58KL4JXll7GttB+u+C58hc+haiq6XUppiSVt4oUQQnRKEkbSzOdzAkkwCN7MADMuDzH7lT8C4N/+INgWmZ4aQlXVUswqhBCiU5Iwkmapbb62DaZtcN55Cv/bMpXVO4ahmeX4dz0Oqhu/VkJZSZxoNN0jFkIIIdqWhJGDQG0TtBBongDXX13J71+6HQDvzsdQzCAeI0wiVE5ZGVLMKoQQolORMHIQUFXIyXGaoFmKhx+f6WJdzSks2XAUmh3Cv+MBbC2DgKeMitIQwWC6RyyEEEK0HQkjB4mMDGd2JBwGjCx+95sybnzxTgC8Bc+ixXbhclm4LClmFUII0blIGDlIqCrk5kI0Crbi5gc/dBP2TeTdlSehEidzy23YWoBMdyXBymqqqtI9YiGEEKJtSBg5iKRaxIfDYOvO7MjMf9xLwtTwlP0Ho+pjFM2DXyuhtFiKWYUQQnQOEkYOIi6XMzsSiYCtGBz5PS/9Du3PI/+9AoDMTX/AVg28yWLW0lIpZhVCCNHxSRg5yGRmJg/Qi4DtCnDLjeX86Y0/UFqdix5eg6/gWSwtg0xPGRUlQaqr0z1iIYQQYv9IGDnIpGZHwmFndmTQ0Ax+drbFzS/fBoB/219QzBpcGnjUEoqLTOLxNA9aCCGE2A8SRg5CqdmRcNiZHbnuympeXXEJX28dhWZWkLHtL9haJhlGNbFq6T0ihBCiY5MwchByuaBbt1TtiI4/N4cbry3hmmfvB8BX+Cyu0BoszUfAU0pZcUgO0hNCCNFhSRg5SGVm1nVltV0Bzj0nQZk6iZeX/BQFi8Dmm0Fx43KZ6HYpJcUWiUS6Ry2EEEK0noSRg5Sm1fUdsWwNxZ3Ln36/k+uf+wuhqBej6lO8xS85vUeMSsJVFZSVpXvUQgghROtJGDmIZWQ4t1AIbC2ToyYYjJ+Uwx9e/QMAmVtuRUmUY2teAu4SSosisrtGCCFEhyNh5CCWmh2JxcCyFWw9l5tuKObRxdewYssRqIkKApv/AJoXXUtg2MUUFVqyu0YIIUSHImHkIJeR4dSPBINgaz5698vk+ivLuOxvj2NZCt6SVzEq/oftCpBhVBKrLqOkRHbXCCGE6DgkjBzkUmfWmCYkEmDp2Vw6rYao7wgeWnglAIGNN4IVxXb5CHhLKC8Oytk1QgghOgwJIx1ARgZkZeFs31XdKJ5c7rl1O7e8fBvby/riim4hY/t9oHpwaeBViykuSsjZNUIIIToECSMdgKI4syOaljzV15XF4UdonHeeyYyn5wLg3zkPV/AbbC0Tv1FDIlhKSbGNZaV58EIIIcReSBjpILxeyMlxakdQXdh6d264qpBlBafzypKzUUiQtf5asONYWiYBTymVpTWUl6d75EIIIcSeSRjpQLKzweNJtonXMvAFsrhz9g5mPD2Xkupu6KFvk8s1OprLRYariOLCmHRnFUIIcVCTMNKBGIbTJj4cBhsFy5XL5BNifO9YP5c/OQ8A/46H0Ku/wtYycOsRXGYxhQUWsViaBy+EEEI0Q8JIBxMIOG3ig0FA82Dpudxx03Y+WP8Tnv/kPBRMsjZcC2YY25VFplFeu91X6keEEEIcjCSMdDCa5syOxOPOdl9bzya3h5e/3LaLK59+iJ3lvXGF15O57S5QVGyXnyxfMRXF1VI/IoQQ4qAkYaQDysx06keqqwFFw9a7c+rkKiafbPDLx/8GgG/X39ArPwXVjaZpZLiKKCmMSv2IEEKIg46EkQ5IUZzZEZcLIhGwNT+23o0//W4rywun8Nh7l6Fgk7XhOpREZW39iJasH5H+I0IIIQ4mEkY6KI/HCSShULKYVc8lkO3mr3fs4jfP3cOGwsG4otsIbPwt2Da2K4uAu4J4TSlFhTamme5PIIQQQjgkjHRg2dlOd1anM6uObfTghGOrOPtsOO+hF4ibLryl/8Zb9AIoKpaWQZa3iOryKjm/RgghxEFDwkgHpmnQvbtTyBqPg61lYrtymX39NorNcfxu/h0ABDbfjBZaB6qBqrnJMgopLQpRWZnmDyCEEEIgYaTD8/udVvE1NYDiLNf4M93M++tOHlg4kwUrpqBYEbLXXeFs99V8uFwmfrWQwl0xZ4uwEEIIkUYSRjq41Lk1brdTP4JqYBk9GHN4DbOur+Liec9QUJmHHl5DYMsfAbC1AB49iJYopmCXKQWtQggh0krCSCdgGM5yTSSS7D2iZWK7cvh/F+1k1LgsLnr4WQB8hc/iLv03KAq2K5uAu7y2oDWRSPOHEEII0WVJGOkkAgHnIL2qKpLLNd1QXB7uv3MHK4tOZM4bNwKQtf43aKHvagtas73FVJdVUFQkHVqFEEKkh4SRTkJVndkRt9s5uwbVwDZ60j0nykP3FHPLy7fy3jc/QLWC5Ky9FCVRDaoBmodsbyEVxdWUlsoOGyGEEAfePoWRuXPnMnDgQDweDxMnTmTJkiV7fH5FRQUzZsygd+/euN1uDj30UN5+++19GrBontvtBJJwOLlc48rENrpz3IQSrrw8zM8fepHtZf1wRTaQtWGmkzw0L5qmETAKKCkMUVGR7k8hhBCiq2l1GJk/fz4zZ85k9uzZLFu2jNGjRzNlyhSKioqafH4sFuOkk05i8+bNvPzyy6xdu5bHH3+cvn377vfgRWMNlmsAy5WLpWVyw5U7GD4qi5/e9zKxhI6n7G38Ox8BnA6uuiuBT3F22FRXp/EDCCGE6HIU227dxPzEiRM56qijeOihhwCwLIv8/HyuuuoqbrzxxkbPnzdvHnfffTdr1qxB1/V9GmRVVRVZWVlUVlYSCAT26RpdSTQK27Y5O228XlDMEGpkOyXlOlN+ks+Phv+NeZdcgY1K+YjniWV9H2wbJVFBMBbANHrTt58Lny/dn0QIIURH1tLv71bNjMRiMZYuXcrkyZPrLqCqTJ48mU8//bTJ17zxxhtMmjSJGTNmkJeXx+GHH84dd9yBuYd+5NFolKqqqgY30XKNlms0H5bRgx7ZIf42t5SnPvwVTy3+BQoW2et+jRbZmtxhk0WGUQmRInbtNIlE0v1JhBBCdAWtCiMlJSWYpkleXl6D+/Py8igoKGjyNRs3buTll1/GNE3efvttbr75Zu655x7+9Kc/Nfs+c+bMISsrq/aWn5/fmmEKnOWa3Fxqu6zariwsVzZjRxRzxx8q+fVTD/PlxvGoiTKy1/zCKWhVVCxXFlmeMsxQCQW7LGKx9H4OIYQQnV+776axLIuePXvy2GOPMX78eKZOncrvf/975s2b1+xrZs2aRWVlZe1t27Zt7T3MTie1u8brTXVnVbGMHtiqlwt/VsTZZ1ucee/r7KrojR5eS9Z3vwbbBEXDcgXIchcTqSylYJf0IBFCCNG+WhVGunfvjqZpFBYWNri/sLCQXr16Nfma3r17c+ihh6JpWu19w4cPp6CggFgz/9ntdrsJBAINbqL1DAN69oREAmeGQzWwjTzA5vZbiuiZ34Mz/vIGkbgHT8V7ZG5JzlYpLmxXBtneIoJlZRQWIqf8CiGEaDetCiOGYTB+/HgWLVpUe59lWSxatIhJkyY1+ZpjjjmG9evXY9XrqLVu3Tp69+6NYRj7OGzRUpmZ0KOHMztiWWC7MrD0HvhcIZ6aV8zO8FimPfIMAP5dj+EtfN55oWqAy0eWt5DKEqcpmgQSIYQQ7aHVyzQzZ87k8ccf55lnnmH16tVcccUVBINBpk+fDsC0adOYNWtW7fOvuOIKysrKuOaaa1i3bh1vvfUWd9xxBzNmzGi7TyH2KDfXqSFJ1QHbeg6WnkvvbuU8/Wgp/17xM2552Tm3JrBpFkblx84TVTeqy022p4Dyoirp0iqEEKJduFr7gqlTp1JcXMwtt9xCQUEBY8aMYcGCBbVFrVu3bkVV6zJOfn4+77zzDtdddx1HHHEEffv25ZprruG3v/1t230KsUea5syORCLODhuvV8XSu6NaUUYPL+XBe1xcNuNmhvdZzXlHv0j22l9Sdvi/SPiGOU3RsMly76K8CFQ1QI8eTk2KEEII0RZa3WckHaTPSNuoqIAdOyAjA3S9rv+IrWjcN68H9z+gs3DWyRx72EeYRm9KD38Dy90HAMUMkkhYVEb70L13Jj16OH1MhBBCiOa0S58R0bFlZTk7bKqrnU7wTv+Rnih2lGsuL+PUU21+fM/rrNk5HC22i5zVF6EknL3BtubH5VLJMnZRWlBDSYmcYyOEEKJtSBjpQhQFunVzilrr9x+x9R5oVg333lnKYSP9nHznAnZV9kEPryFn7aVgRZ3nan5cOmTquyjaFZRAIoQQok1IGOliXC7Iy3O2/QaDgKJg6d2wXDl4tUqemleKr1tvpsz5D9WRAEbVp2Stv8bpQQLYWga6yyJL31kbSKSoVQghxP6QMNIFeTxO/5F4PNl/RFGxjJ7YWgY5GRU892QpJfGRnHnPv4ibOt7SfxPY9LvaaRDblYnussg2dlKyq0YCiRBCiP0iYaSL2r3/CKqOZeQBOv3yqnnuyVK+3HoCF8z9B5at4Cv8h9MUrV4gcblsAvpOSnZVU1wsgUQIIcS+kTDSRSmK038kJ8fZZWPbgObFcueBlWD4oTU89WgZr391Dpc9/jgA/l3z8O+4v/YatpaBSydZ1CqBRAghxL6RMNKFpfqP+P31GqK5AljuXihWmGMm1jD3r+U8/eElXPfsvQBkbrsb364naq/hBBKFLPdOSguqpHW8EEKIVpMw0sUZBvTq5RS2BoPOfbYrG1vviWrW8KNTarj3zgruW3Ads1/5AwCBzbfgLXqx9hrOtl+NbM9OyosqKChADtcTQgjRYq3uwCo6H6/X2WGzfTtEo+B2OztssBOo8VKm/lSlukbl5ltvIctbyczT/kpgw/WAQrjnVMDpWaISJtu9k8oiE8vMpVdvBV1P72cTQghx8JMwIgDn7Jq8PCgocJZvXC4VS+8BdgIlUcEvL1aoCSr85p57cOtRZpz0MIENv8FGIdLzXOcimhdVUchRCigvtdlpdaNXbwW3O72fTQghxMFNwoiolZvrbPUtLYXsbFBVF5aRh2qbKIlKrvm1Qk21wpWPPYSCza9PeoSsDTMB6gKJ6gGXSq6/gPIKk51WD3r1VvF60/e5hBBCHNwkjIhaiuL0H0kknB02OTmgqAaW0Qs1ugPVrOT3v4VoTOHKZx4CqAskikKkxznOhVQDm0xyvMVUVlvsMHvQq7eLjIz0fTYhhBAHLwkjogFNc5ZrTNNpGZ+dDWgebHdvlOhOVLOaW28GG7jymYdQFYvLJz9K1vrrUKwE4bzznAupOraeRbZSSlUkwY5tefTqY5CVlcYPJ4QQ4qAkYUQ0ktphs2OHc6heZmbqUL1eqNGdqFaI2252nvvrpx/GRuGKyfPI2ng9ihUk1PuXzoOKhuXKJpMKQjGTXdt7kUh4yM2VE3+FEELUkTAimuTx1AWSUAh8PrBdGVg4gUSxwk4gseHXTz1MKOrjN6ffS2DzbBQzSLDv1U7iUFRsVw4+KtHiOyjc0YtEwk/37s4sjBBCCCFhRDTL768LJOGwswXYdgWwbAs1uhOA224BFLj+6b9QFQnwx5/+gcxtf0Yxg9T0n5UMJAq2no1bqUZL7KC0II9EPEDPPNn6K4QQQsKI2ItAwKkf2bULVBXcbrD1bCxs1OguUBRuuxl8XptbH5lNTSSDey64noydc1HMGqoH3QaKMwViuzLRlDDZyg4qi+MkErnk9VLxeNL8IYUQQqSVhBGxV9nZzpkzBQXORIdhgK3nYNk2arwAVPjdDZDht5nzl99QHc5k3qWX4y98Bi1eQsXQB5wtv5DsRaKSqxRQURFnR6IHeb1kp40QQnRlEkbEXqUO1bMsKCx0Clp1PRlIcAKJjcLVvwa/3+amP/6KilA2z824CE/ZW+SsLqPisCexXQHngqobW9fIUUqpDifYsbUnPXu7yc6WwlYhhOiK5Gwa0SKKAt26OQfrVVcnD8NTFGw9F0vvhWKFwIpy6cVB/npXOa98cQ5T7vwPwVgm7qpPyf3mbNRYQb0LurBc2WS4q/DY2ynYVkNRkRyyJ4QQXZGEEdFiquqEkW7dnB4kDQNJXm0g+fk5YR6fW84n63/AsX9YTGkwDz20mm6rzkQLr6+7oKJi6zl4jAQBfTulu8rZucMmGk3bRxRCCJEGEkZEq6iq06U1N3f3QNItOUMSBivCaVMivPj3UjaVj+ao33/C5tJD0KLb6bbyxxiVHze4pu3KRNN1cr07CZYVsX2bSU1Nej6fEEKIA0/CiGi1VJfWnJzmZkgiYEX43oQYr80vIaIN4Kjff8KXm7+HalaSs/p8vEXzd7uoF1wZ5PqKscM72bE1QmmpU6cihBCic5MwIvbJ7oHEsqgLJEZvFDMCZphhhyV4458lZPfM5tg/vM/LX5yLYifI2jCTjK1/Btuuu6iqOx1b3dV47e0U7aiisBDi8bR9TCGEEAeAhBGxz1yuukBSUVE/kORguXujWFEww/Tra/LGP0sYO1bh3PtfYM4bswDI2HE/Wd/NADNcd1FFdRqkuW0CxnYqCkrYsd0iFErLRxRCCHEASBgR+yUVSLKznUCS2g1j6zlYnr4odhzFDJKTbfPiM6X89KwIv5t/B5c89gSm5cJb+jrdvjm7tqNriq350XQfub4CYlU72b4lSnl5w4kUIYQQnYOEEbHfXC6nbXyDGhLAdmVhufuCbaMkqnG74YG/VHDDtVU8tfgSfnjHQioj3dCDX9Nt5eno1UsbXlg1sPVssryVuM3t7NpaTUGBLNsIIURnI2FEtIlUIEntskkknPttVyaWpy8oGkqiEkWBmVfXMPev5Xy64XjG3PgFawsPR4sXkfvNz/AUvdTwwqntvx6TLPc2KgqK2b7NJBg88J9RCCFE+5AwItpMqqi1WzeoqqoXSDQ/prsvtmKgxCvAtjn7zDAvP1dC0O7P+Fmf8uaKs1DsGNkbriNz0y1gNZz+sLWM5LJNIYmanWzfEqGkRHbbCCFEZyBhRLSpVCDp3t0JJLVLKpoXy90XW/OhJCrAtjhqfJz/vFbMkKEGP777FW577WYA/AVPkPvtuaixwoYXTy7bBDzV+NhG0Y4Kdmy3iUQO6EcUQgjRxiSMiDaXaozWs6fTOr62o6rmcQKJKws1UQF2gr59LP41v5QzT49wyz9v5cf3vE4oHsCoXkK3r09Br1rS8OLJ3Ta6oZLr2UGorIBtW2JUVkpxqxBCdFQSRkS7SLWO79ULwmHn5jygY7l7Y+ndUBNVYMXweW0evr+C391QxZtfncHo337Jd0UjnTqSb8/Bt+tvjZOG5gU9k2xfGXp8Gzu3VFOwyyYWO+AfVQghxH6SMCLaTepwvd69IRajrleIomEZeVhGHooVBDOMosBVV9Tw/NNllEaHMObGz3nly6kodoLA5tlkr/sVSqJytzdwYes5eD0mAWMblUVFbNuaoKpKZkmEEKIjkTAi2pWiOFt++/RxtvzWnjmjqFh6dyyjT7IXifPACd+P8s4bxRwyVOdnf32Ba569j4Sl4yl7m25fn4KrZkWj93CKW/3k+opRI9vYsaVGOrcKIUQHImFEHBBZWdC3r7N8U1vfUduttS+gJgtbbfL7mbz+zxLOPzfEAwuuYdLsj9lVNQBXdCvdVp2Jb9eTjac+VB3LlYPPHSPg2kb5riK2bklQXS2zJEIIcbCTMCIOmIwMJ5B4PPXax+P0IjHdfbHVup02Hjfcc2cl98ypYNWOIxn+m+W8/fVZKHacwOabyV57KUq8rOEbKIpzArDhJddfhBLezvZNziyJ1JIIIcTBS8KIOKC8XieQBAIN28fXbv11ZTs7bZJ9Rs6fGuLtfxXTo7ef0+96tW7Zpvwduq+YjFH5YeM3UQ1sVw5+T5SAvpXyXYVs2xKXWhIhhDhISRgRB5xhOEWtqW6ttbMWqo7l7oVl9KwtbAUYPizBgtdKOOfsMA8suIajbvqcLWWHocULyfn2PDK23A7WblMftbMkPrr5i1EjW9mxuZpdO+26rcZCCCEOChJGRFqk2sfn5UEwWG/rr6Jh6T2Sha0mSsKZzvD7bR74SwX3/bmctUWjGf6bZTz10a9QsMnY+TDdVp2JFl7f+I1Uw6kl8Zpk6duoKi5g22bn0D3p3iqEEAcHCSMibVTV6dTat6/TOr66OvmAomDr2VievtiqGyVRDraTHKb+LMy7/y7mkENdXPLIo/zkr69SE8tFD35N9xVTkj1JdksZioKtZaC6M8jxlaLHt7FrawU7tlt1242FEEKkjYQRkVaKAtnZTiBxuZw6klRdh635d6sjcZZiDhls8ubLJfz6VzW89uVPOGzm1/zvu5NR7AiBzbPJ+fbnqNEdTbyZC1vPxe1RyHVvJ1S2g22bQxQXyzZgIYRIJwkj4qCQ2mnj90N5ed0he6gGlrsPptELxQyhmM5xvYYBN99Yxfy/l2DqvTj+Dwu48pm5xEwf7qqP6b7ih3iL5jddsap5wcgm21+Nz95KyfZitm6WAlchhEiXfQojc+fOZeDAgXg8HiZOnMiSJUuafe7TTz+NoigNbh6PZ58HLDovj8cJJKlTf2sPwFNUbKM7lqcfoCRP/nWWYo47NsZ7/ynijNMizH3314y8YTlf75iIalaTtWEmOWumoUZ3Nn4zRcV2ZaN73OT6C5MFrpXs3GHV1a8IIYQ4IFodRubPn8/MmTOZPXs2y5YtY/To0UyZMoWioqJmXxMIBNi1a1ftbcuWLfs1aNF5uVxOUWuqhXxtx1bAdgUw3f2wXZkNlm1yc2wefbCch+8rpzg8hHE3fsRN/5xDwjJwV7xH9xUn4i16selpD9WN7crB57XINrZRU7JTlm6EEOIAa3UYuffee7nsssuYPn06I0aMYN68efh8Pp588slmX6MoCr169aq95eXl7degReemqs7sSN++oGkNG6Q5J//2cbb/1lu2URT4yY/DvL+giGOPSXD7azcy6rfLWblrQnKW5DfkrL4QLbKt8RsqCrbmRzGyyPZX4bW2UrK9kK2bndOAZdeNEEK0r1aFkVgsxtKlS5k8eXLdBVSVyZMn8+mnnzb7upqaGgYMGEB+fj5nnnkm33zzzb6PWHQZmZnQr5/zs6KiXj+S1PZfTz9ARYnX7bbp3cvihafLuOtPFWyrPIwxN3zCrPl/Jm55cFd+QLcVP8C38zGwzcZvqGjJpRsPuf5i1PBWdmwqZ8d2k2BQ6kmEEKK9tCqMlJSUYJpmo5mNvLw8CgoKmnzNYYcdxpNPPsnrr7/OP/7xDyzL4uijj2b79u3Nvk80GqWqqqrBTXRNbrdzyF7Pns6pv3Un/yrOso2nH7YrK7lsE009xLTzQ3ywoJjvHxvnzjduYOQNK/hy63GoVpjAlj/SbeUZuIKrmn5T1cDWc/H6FHI9O4iUbWP7pmoKdtl1dSxCCCHaTLvvppk0aRLTpk1jzJgxHH/88bz66qv06NGDRx99tNnXzJkzh6ysrNpbfn5+ew9THMQ0DXr0cJZtbHu3ZRvVXbfbxorWNkkD6NfX5IWny7j3znIKag5hwu/e5/InHyWcyEIPrqDb16eRseVPKGYzzUaSu24C/igZ6jYqC3eybVOIkhKpJxFCiLbUqjDSvXt3NE2jsLCwwf2FhYX06tWrRdfQdZ2xY8eyfn0T3TKTZs2aRWVlZe1t27Ym1vlFl6Iozsm/+fnONuCGyzap3Tb52KoHNVFee7aNosB554ZZ/G4RU06K8uiiXzH4mtX8Z9VPUTDJ2PkI3ZefgLvsnWbeWHXaynsyyMmowm1upXhbIVs3RxuerSOEEGKftSqMGIbB+PHjWbRoUe19lmWxaNEiJk2a1KJrmKbJypUr6d27d7PPcbvdBAKBBjchoG77b2rZJhise8xpktYPS++BYgVri1vBqSV5al45f3u4DNvoyWlzXuZHf/k3xaH+aLEd5Ky9hOw1v2i6wBWchmmuLAyvhxx/CVpkK7s2ldQewCdFrkIIse9avUwzc+ZMHn/8cZ555hlWr17NFVdcQTAYZPr06QBMmzaNWbNm1T7/1ltv5d1332Xjxo0sW7aMCy+8kC1btvDLX/6y7T6F6FJSyzb9+jk7b8rL681QqC4soyeWu35xa6L2taefEuF/7xYx7fwgb331IwbMWM2979yIabvwlC+k+4oT8G+/H6xmikNUA/QcPF6VXH8BVs1WdmyuYMd2k5oaKXIVQoh90eowMnXqVP7yl79wyy23MGbMGJYvX86CBQtqi1q3bt3Krl27ap9fXl7OZZddxvDhwznttNOoqqrik08+YcSIEW33KUSXoygQCDiBJDvbOf23rklaqrg1H1vPRTGrG9SFBAI2d/2pkjf+WcygwTq/+fscRv12BUu3HYdiRcjc9me6Lz8Rd/nC5gegebFdOfh9Ntn6diJlW9m+qYqdO+S8GyGEaC3Ftg/+/5arqqoiKyuLyspKWbIRjViWMztSUuL8cyDghBUAbBvFrEKNFYMVwXYFQHHVvjaRgCee8XP3fZkEgwoXHPs8D06/gRyPE6gj2SdSPfCPmN7BzQ/AtlGsIGY8Rk0kE1vPJZDrJztHxettxw8uhBAHuZZ+f0sYEZ1GKATFxc7pvxkZzvk1tawoarzUOQFYMbA1f4PX7tylMvtPWbz5Hy8Znmpu//mtzJh8P5oSx1Z0Qr0upabfNU6YaY5toZhBEvEENdEAijuHrFw/WdkKcgKCEKIrkjAiuiTThLIyZ5ZEVZ1Q0niWpCQ5S5LZYJYE4H8fGdx0axbfrdc5tPdanvz11Rwz+F3n2q5u1PS/gXDP80HRmh+EbaKYQeIxi2AsszaUZOcouN3t9MGFEOIgJGFEdGk1Nc4sSTDY1CxJrN4siQtb9ddLLE4PkSee8XPPA5nU1KicMvo/PH75dfQLrHUe9w2nesAtxLKP2/MgbBPFrCEes6mJBtC8OWTl+sjKllAihOgaJIyILi+RgNJS56ZpTc2S1KDEilGtEJaW4eyUqaeoWOWOPweY/4oPlxbnqimP8Keps/G5KgCIZp1A9YDfk/DvpRi7XigJxgKonmwCObJ8I4To/CSMCJG051mSBGqiHCVeCtjYWiYoDTeZrVipM/tPAT7/wk1uRilzzr+VS497xKknQSHc41xq8m/AcjffOwcAO5FcvrGpiWaieXMklAghOjUJI0LUk0g4tSSlpc7sSGZmg5UZ5wTgeAlqogpL9Tqt4OuxbXjrPx5uvTPAtu0uBvfcwNzLfsspI15xHlc9BHtdSrDvr7Fd2XseTKqmJGoRjGeiuFMzJbL7RgjRuUgYEaIJwaAzS1JTAz4fDWckbAslUYkaL3UKXLVMUPUGr49E4cln/Nw/N5OqapWJh3zG45fPZFRv59RqS8si2HcGwV6XNAo0jdQWupqEYhlg5JCR7ScrW8PnaxiWhBCiI5IwIkQzTNPpS1Ja6vxzIODsvKllRVHjZU73VkXF1jIaLd2UlSvc91AmT//DTzwOZ4z7Nw/98kb6Z6123kPvRU2/awj3/HmjWpRGarcExwlF/ZiuXDKy/WTnuPD5dhubEEJ0IBJGhNiLcNjZAlxVBW63M1NSy7ZRzKCzdGPWNLl0A7B5i8acvwR44y0vqmLyi+Of5c8X3kI3r3PGTcLdn5p+1xHp8dM9bwcGJ5RYYcx4lFDUS0LNxZuZQU43Hb/fKcIVQoiORMKIEC1gWU4YKS11wklmJuj1V2ZsEyVRlVy6iTrN0pqY6VixUmfOXzJZ/KEHwxXlqlPmcctP7yBgFAGQ8B5CTb+ZRLqd0WiWpRHbRrHCWIkIoaibmJWDN5BBdjcPfv9u4xNCiIOYhBEhWiEWq1u6STVLa7h0E0ONl6PEy0AhuXTTeKrio08M7rg7wFcrDHzuIL/50UP89sd34XeVAxD3Hkaw37VEuv1o76EEwAyDFSEScRE2Axi+LLJyvWQGpFeJEOLgJ2FEiFaybafAtbTUaSnv9dJod4uzdFOGYlaBojdqmJa6zoKFHu7+ayar1+pkequYddZfufbUv+LVKoF9CCVWDMwQsahCKJ6B5s0mM9tPIMvZgSPFrkKIg5GEESH2kWnWLd1Eo84sScOlG8tpmBYvRTWDzdaTWBb8+20Pd9+XyYaNOlm+Cm78yV+5Zsr9taEk4R1KTd+riHQ/s1Fr+ibZCRQzRDxqEo77sFw5+AJ+snN1fD6pKxFCHFwkjAixn1JLN2Vlzp8zMnb7srcSTj1Joiy5FdgPauO1k0QCXn3dy18fymTzFlfTocQ9kGDfKwn3+Oned9/AbsWuHuJ2Fp7MDLJzPfgzlIaN3YQQIk0kjAjRRoJBJ5BUVtbtummwLGLFUBMVzlZgO+HUk6iNq0wTCXj1DS/3PZjJpi0uAt5Krv/xQ1x36l/J0EsBMI0+BPtcTrjnediar9E1mmRFIB4iGnMRSmSi+wJkZvvJDMgSjhAivSSMCNGGLMupIykpcXbdNFVPghlOhpKKPRa5JhLwr387oWTjZhc+d5CrT53HjWfeTZZR6LyfK4dg70sJ5f0CW89p4SDjKFaIWNQiHPNh69n4An6ycgzZGiyESAsJI0K0g0TCmSEpK3OWcfz+3c66sW0UK5Qscq0GNGf5pokiVdN0akrun5vJmnU6bj3CL098ilt+djc9fZsAsFQf4bwLCPa+DMvdt2WDrLeEE465idsBDF8mWble/BlyDo4Q4sCRMCJEO4pGoaLCqSmxbSeUuOrXn6ZOBY6XoZrV2KqBrfqaDCWWBe/818P9czNYsdJAUxNMnfRPbj9/DgOzVzqXU1xEuv2YYJ/LSfhHtnygVhTMMLEohGN+VG82vkwfgWwpeBVCtD8JI0IcAKGQE0gqK5vpT1K786YM1azBVt3JUNK4kMO24cOPDR6al8mHn7gBmylHvMOci+5ibJ8Pap8XzTqOYO9fEcs+oeUFIbaJYoVJxGKEox4SSgDDl0F2N2e2RHqWCCHag4QRIQ6QVH+SsjJnS7BhODMlDXKCbdYLJcE9hhKA5St0Hno0g7ff8WDbCuMGLuX2C/7MycNfQVVMAOLeQwn1/iXh7mfv/VC++oO1o9jxMLGYSiTuR/VmyWyJEKJdSBgR4gCzLOc04LIy56fHQ+PdLLaJkqh2epRYYWzVg602v+Vlw0aNR5/I4KVXfERjCgO6b+amn93HRUc/iVurdt7XlUso7yJCvaZhGb1aPuDkbEk8GiMSc9fOlji1JSput+zEEULsHwkjQqSJaTo7b1Ln3TQZSqxEXeM0K7zXmZKSEpWnnvXz1D98lJdrZHqrmHHy37j+jPvp5t0KJOtKcn9EqPelxDPHtW7QVgTMCNEIRBM+MLLwZvrIynHj8+1WDyOEEC0kYUSINEsknFBSVraH7cCtDCWhsMI/X/Xy+FN+NmzU0dQEZx/1L26Zej+H531c+7xYxlhCvX7htJtXW7F9xjZRrEhyJ45BzPRj+ANkZPnIyNTweneriRFCiD2QMCLEQSIed2pJysshEmkmlKSWbxLlyZqS5nffgLMk9P5iN489mcH/PnaqT8cOXMYt597Pj454EZcSc57nyiWUdz6hvGkt3xpc+yYxMMPEYyaRuBdTCeDJ8JOV68XrU2QZRwixVxJGhDjItDiUmDUo8XJn942iN9unJGXNWhd/e8bPK695iURUegSKuOa0R/n1yY+S497hXBaVaM5JhPIuIpZ9fMsO56sdk+20u09EiMYUonEfittZxglkO8s4euOGs0IIIWFEiINVU6HE49m90NWqDSWKFQRUpz38Hg7TKytXeOElP08962PHTheamuCsI1/n9+c81GBrcMLdn3DehYR7TsXSu7du8MllnEQsSjSuEzN96N4AvoCPzCwdr1d24wgh6kgYEeIg1/JQEkJJVKAkqpw286q/ybNvUhIJWPieh6ef9dcu4Qzrs5r/+8nD/Hzi3/FqVc6lFZ1I7qmE8y4gFji6dbMlkGw/HyYWSxCJujHVDAxfBplZXnwZLqkvEUJIGBGio4jH67YEN7v7JtVmPlGFkqgE28TWvHstTv1ug4tn/uHjpVd8VNeoeI0QF37/BW448xGGdlta+7yEeyDhvPMJ9zgXy+jR+g9R2+nVIhL3YmuZGD4/gRwPPr/WOGQJIboECSNCdDCJhBNKUluCDcM5IbjR7IIZRjWrUeKVKHYUS02Gkj182weDCq+96eWZ53ysXOUcpjN24DJuOOtRzh73Qm3PEltxEc05iXDPnxPNPmGPy0JNStWXmFFiMYjG3eDKwu33Ecjx4PVJ/xIhuhIJI0J0UKbphJLycqezq6bR9Km7VgwlUe2cFGyF97oDJ2X51zrPPu/jX//2Eg6r+N01nH/sfK7/8aMc2v2LunHovQj3PIdwz59jega2/oPYFljReoWvXhQjgMfvJTNbgokQXYGEESE6OMtywkhFhdOvBJxQ0mjnipVAsYLJYtcQoDihZA91JQBVVQr/+reXf7zoY9U3zmzJyH6ruO6Mx/n59/6B31VW+9xY5kTCPacS6XY6tpbR+g9jW8n+JRGiMZVYwgkm3gwfmdluPF4JJkJ0RhJGhOgkbNs5kK+y0il4NU1n+abR4Xa1xa6VKGY12Amn1fxelnAAVqzU+ceLPl77t5eaGhXDFeWsI1/nN2c+wZH5/0VVLAAs1Ue022mEe5yzb0WvkNyREyURixKLq8QSHlR33YyJBBMhOg8JI0J0QuGwE0gqKyEWa2YHDiTrSmqcYGJFWryEEwopvPkfDy/808dnS5y00ydnB7+a/Hf+30lP0cv/Xd1bGH0I9/gp4R4/w/Qesm8fqEEwUYglPMmlHJ8s5QjRCUgYEaITi8WcpZuKirpi1yZ7fFhxFDPobA22QgDJJRxjr++xcZPGiy/7ePlfPnYVaIDN9w75jGvOeIozx7yE11VZNx7/GCI9zibS7cf7thsHaoOJGY8SjSnEEm5UI4DhSwUTDbdbtgsL0ZFIGBGiC0gknLqS8nJnKQecJRxj96yRWsIxq1AS1WDHW7yEY5rw4SduXnrZy3/e9RKJKrj1CGeM/TfXnfk03xvwDqpiOm+DRiz7OMLdf0I09xSne+y+SBa/WvFIcsbEDa4MdK+PzCwvXr8Lj0carAlxsJMwIkQXYllOGKmqcmZMEgln+abJJRwripKoQUlUOIfzKa5kMNl7T/fKKoU3/+Pln696+fwLZxmnR6CIC78/n8unPMuh3ep249iqh0jOyUS6/8TZJtyC2ZgmpYJJIko8ZhGNe7A0P4bXjz/gwZ9p4PHIycJCHIwkjAjRRUUiztbgysq9LOHYZr2C1xqnkZrqadFsCcCWrRqvvObln//ysXmLkwQOyfuOX530d6Yd9wJ5/g21z7W0bCLdTiPS/cfEApNa37+kdsw22FHsRJR4zCQWN0jgxfBl4s3w4Mtw4/EqGIbUmQhxMJAwIkQXl1rCqahwfoITShrvwrGd2RKzBjVV8KpoLZ4tsW34aoXOK695ef0tL6WlTn3J+EFLuXzKPzhnwnyy3AW1zzf17kRyf0Sk+4+JZx61bztyUqwYmFHisTixhIuE6UZ1Z+Lxe8nI8uDxSp2JEOkkYUQIAdRtDa6udpZxYjEnkDR5dkztbElVcrYkDqrHmTFpQWiIx+F/H7l59Q0vCxZ6CIVUVMXkuOH/4/JTnudHR7yKX6/rX2LqvYh0O51Itx8Rzzxy/4JJgwJYnJ05uh/d4yMjy4PXL8s5QhxoEkaEEI3EYnWzJeGwc1+TBa8AZqRuJ44dwWmm5gV196mVpoXCCv99zwkm733gIR5XcGlxfjhyEZef8gJTDn8dr1a3I6cumJyWnDHZj+rU2jqTGIm4s5xjKl50bwZun4eMgBu3R7YNC9HeJIwIIZqVKnitrnZue58tCaOY1SiJahQ7luxb4m1xYKioVFjwrofX3/Ly4cduTFPBcEU5adRCfjVlPiePeANP8jRhAFPvQTT3FCLdTt+/GpPaDxwDK0YiGiMaV0lYblQjE8PnxZ/pwevXcbtl1kSItiZhRAjRItGoE0zqz5Y0WVsCznk4tUWvIcBs8RbhlNIylbff8fDvt7x8/JmBZdUFk8tOfomTR7zRoIeJ5comkjOFaO4pRLOP2+tJxXtlm2DFnG3DMZt4wsDSvOiejNoiWJk1EaJtSBgRQrRKarakpsaZLYlG97QTxzmd11nGqdynZRyAkhKVt9/18ObbdcFE12KcOPI9Ljnpn5w+6rUGNSaW6iea8wOiuacSzf4Btitr/z60bTt1MWaUeDROLJGcNdEz0L2pIlgdt0dpfCaQEGKv2jWMzJ07l7vvvpuCggJGjx7Ngw8+yIQJE/b6uhdffJHzzjuPM888k9dee63F7ydhRIgDKxarOw8nFHKCSrN9Sxos49Sg2FFsRU9uE275N3hJqco7//Xw5n88fPSJm0RCQVMTHHvYR1x84sucOe41cj076t5WcRELHE0kdwrRnJOx3H32/4PXNluLEo9bxOI6tuJB89SrNfG6ZIeOEC3UbmFk/vz5TJs2jXnz5jFx4kTuu+8+/vnPf7J27Vp69uzZ7Os2b97Msccey+DBg8nNzZUwIkQHYNtO35Jg0AkmkYjzJezxNLeME0exUrtxQih2vNX1JeDUmLy7yMPbCzws/tBDJKoANkcO/pLzv/8q50x6jX6Zaxq8Ju4fRSTnZKK5J5PwjWybNRYr7tSaxGLE4gpx0wDNi+H1481w1y7pSF8TIZrWbmFk4sSJHHXUUTz00EMAWJZFfn4+V111FTfeeGOTrzFNk+OOO45LLrmEDz/8kIqKCgkjQnQwpllX9FpT48yeGAbNb5e1ovXqS8KABaq7xduEU4JBhff/5+btdzz89z0P1TXOa4f2Wsc5k17jguP+xfAen6Modf8qM40+RHMmE8k5iVjW0ftfZwL1Gq7FiMcSxOIapm2gGc6Sjj/gxu0xcHuUpncnCdEFtUsYicVi+Hw+Xn75Zc4666za+y+++GIqKip4/fXXm3zd7Nmz+frrr/nXv/7FL37xi72GkWg0SjQabfBh8vPzJYwIcZBILeNUVTk/TdOZKfF4mtqNk6ovSZ6NY4YBO7mM425VMInF4JPP3byz0MM7//UkD/BzWtKfMe5Npv3gdSYN/i+GGqp9jaV6iWUdRzRnMtGcE7GMXm3wG6BeIWyUeNwmntAwbbezpOP1JMOJU28iu3REV9XSMNKq/4uUlJRgmiZ5eXkN7s/Ly2PNmjVNvuajjz7iiSeeYPny5S1+nzlz5vDHP/6xNUMTQhxAhuHcsrKcpZtUfUlVlZM9GtSXKApoXmzNi23noFgRSM2YJCpxCl/dLQomhgEnfD/KCd+PcvsfKvl6pc6C/3p49785PPnBJTz5wSV49DA/GPE+Fxz/OqeOfotczw485e/gKX8HgLj/cKLZJxLN+SHxjLH73s9E0UDzompe3B5w2wkUK0YiVkSsCkrLdWzFqK038WcaGB5DthAL0YR2/b9EdXU1F110EY8//jjdu3dv8etmzZrFzJkza/+cmhkRQhxcFMXZbeP1Qk6OszU4GHSWcioqnMdT9SVOMFGxNR9oPmzdCSaKGYJEFYpZBTYtDiaqCmNGxxkzOs6Nv6lm23aNdxd5eGehh4VLTuU/K04DbEYPWMHPvvcG5xz9Jod2+xI9uAo9uIqMHQ9guXKIZp9ANPsHRLNPwNa77ccvw4WtudC8Prxe6tWbFBKrgHCpjq24cXkz8PjdeP1u3F4dw5BwIkS7LtMsX76csWPHotXbF2hZFgCqqrJ27VqGDBmy1/eVmhEhOhbTbBhMotG9FL7aZjKYBFES1WBHWxVMdldVpbD4IzcL3/Ow6AM3ZWXOv4O6ZxZz2pi3ueCEN/n+IQsb9DOxUYj7RxPLPoFozgnJWZM2SgmpLcRWjHg0TjwBiYSBrbnRPX4JJ6LTatcC1gkTJvDggw8CTrjo378/V155ZaMC1kgkwvr16xvcd9NNN1FdXc3999/PoYceitGCSi8JI0J0XIlEXf+SYNAJJi6XE0ya/L9/vWBCotpZ1oF9qjEBJxgtW66z6AMPi973sOpbZ7uxpib43iGf8dNJb3LWUf9hUM7XDV5naVlEs79PLOsEotnHYbn77tPnb1IynNhmjEQsFU500Ny4PH7cPg++DAPDLTUnomNr1629F198MY8++igTJkzgvvvu46WXXmLNmjXk5eUxbdo0+vbty5w5c5p8fUsKWPf1wwghDm7xuDNjUl3tBJNYrGXBBDPs1JhYUZxdOR5n1mQf6j12Fai8t9jDovfdfPiJm5rk7pze2Ts5ZfQCfn7cfzj2kP/ic1U0eF3CO5Ro1nFEs48nHpjkLDe1ldpwEicRi5FI2CRMo67mxOvGl2lguA1pwCY6lHYpYAWYOnUqxcXF3HLLLRQUFDBmzBgWLFhQW9S6detWVOkGJIRogq47t0DACSLhcN2MSTDodHp1u+st5SgatuYHzY+t59YVvyYbrIEJqpHcLtyyf5317mVxwdQQF0wNEYvBl8sM3vvAzXuLe/DU4kt4avElaGqCowZ/wY+PWsBZE97hsB5f4Ap/hyv8Hf6CJ7AVnXjmkUSzvk8s6zjiGUfs38F+igKKgaIa6LofHRrUnCQqbUrLdGxFRzP86F4vvgwDt9fAcEufE9HxSTt4IUTa1Q8moZCzlKNpdTMmjbu+JrcLWxGnwZoVRrETyc6vblD3rdHHrgKVxR+6+eBDD4s/dFNR6fyHVbavnBNHvsc5x/yHH478Lz18Wxq8ztKyiGUdQzTrGGJZ38f0DG77dGAnwIyRiMcw4xbxhIalGKi6D5fbiy/TwOM1MDwuDEM6xIqDg5xNI4TokGIxZ7twasYkFmtiV059tu00WLMizoyJGXZOFla05FLOvp14Z5qwYqXOBx+6Wfyhm6VfGZim0wl2SN4GphzxLj875l2+N+iDBoWw4DRdqw0ngWPaplX97pJ9TsxYnEQiQTyuYNkGuJJFsT43Hr9Td2K4ZWlHpIeEESFEhxePNwwm0aiTK1JLOU3+178Vq9uZY4aSO3PsZPfXfaszAWeHzsefufnfR24+/NjNhk3OspCmJhg/aCmnjP0vZ01YyKjen+JSYw1em/AMJpZ1NLHAMcSyjsbSW97qoMVsG+wYViJOIhZP1p3ooOq43H5cHk+9pR0NXZfZE9H+JIwIITqVRKKuwVpqu7Bl1XV+bXSyMOxWAFuFYkVRMPd7OQdg+w6NDz82nHDyqZvSUmcAXiPEMYd+zGnj/8vp4xZxSLevUBWrwWvj3sOS4eRoYoHvYeu5+zyOPbLiYMWJx2JYieTSDjqK7sHl9uH1GQ1mT2TXjmhrEkaEEJ2WaTphJLUzJxx27tN1J5w0vTPHSi7nROst58SxFQVb2bd+JimWBWvWuvjwE2fW5LMvDIJB51pZvgqOH76YHx35Hicd8R4Ds1c1en3cN5xYYBKxwPeIBya2z8wJOL8DO4YVj5OIx5NLOy5sZbfZE4+O7pbaE7H/JIwIIboE23aCSSRSF0xisYY7c5osGbFiyWASArMmuW3YBtVp474/sybxuFNv8tGnbj7+xM0XSw2iMWcQ3TOLOX74Yk4/8n1+ePj79M9a3ej1Ce9QYoHvJQPKxLY7T6cptbMnccy4STyhOoWxLgOX4cPtc+P16xgeA91Q0XXZuSNaTsKIEKJLShXAprYLx2JOYEnNmDS5FGFbznJO7axJJFkEq+73rAlAJApLlxl8/Jmbjz81+GqFQTzufKP3CBRx3LD/cfr49znx8A8YkP1to9cn3AOJBSYSD0wkljkB0zOw/RKBbYEdT9aeJEjELczk7IlmeNAMHz6/jttnoBuyvCP2TMKIEKLLM00nmKSWc6JRp/ZE1+sO+2v5rIkFio6tGqDsX2OPUFhh6TKdTz5388lnDcNJbkYp3z/sQ04Z+z4njV7MoJyVjWpOTL0n8cyjiCXDScI/vO1a1zfFNrGtGFY8QSKeIJEA03KhaDqa4cPl9uDN0HF7DHTDheFWmq7hEV2OhBEhhKin/nJOMOgUwqa2DRuGM3PSdBGsVW/rcA2KGQFiYCvJhmvGfgeBUFhh2Vc6ny1x8+kSg2VfGUSiTjgJeCs5+tBPOOmIxZw8ZjHDei7FpcYbvN5S/cQzxxHLnEA880jimeOdZnHtyYpjWwkSsRimaZGIg2XrKC4noOgej7O849Yx3C50QwJKVyRhRAgh9iAeb1gEG406MykuV92STtOzJnFnpiQVTqxIsuGalqw12b8lHXDGsvxrg8+WGHz+pcEXS43atvUePcxRQ77g+OEfcur4/zE2/zO8rqoGr7dRSfhHEMs8injmkcQyj8Iy+rRvsUdtS/sE8VgcK2GSSCjO7h2XgWb4MLxuPD7dKZA1JKB0BRJGhBCihSyrYa1JOOwEAqibNWm61sTp7VG3pBNEsWLJ7cOu5KzJvjVdqy+RgG/X6Hy2xGDJlwaff2FQktxKrComh+ev4phDP+bU8f9j0iGf0N23rdE1TL0X8czxxJIzJ3H/4U5wak/JgGIlEpjxGGbCImEqyRkUA83wonvceH2pAllZ4ulsJIwIIcQ+SvU0STVcS9WaqGpdOGlyy2uD7cNBFCsMVjLVqM7ZMvtbbwLOd/ymzRpLvnTCyZKlBhs21rVY7Zu7naOHfsIPj/iIH4z8mCHdV6ApZsNrKG7i/sOTyzvjiWeMa9uTifc0+GRAScTjmKaFGQdb0VE0F2pqicenY3h0dENHN6RItqOSMCKEEG3Atp1Zk2jUqTNJdYK1bWe2ZM+FsAkUO+oUxDaoN6FuC7HSNntlS0pVvlzmLOl8sVTn65V124l97iBHDvqSY4d9zEljPmH8gE/JNMoaXcOZPRlHPGMsscxxJPxHtO3pxM2pt8TjBBSTRLIHiqK50AwvLrcXj9flBBS3jm5oss24A5AwIoQQ7cCynDASjdYt6cRizv2pYNLsl6QVR7FjYKbqTaJAPBlOjOTMSdt8w0ajsOpbPRlODJZ+ZVBYlFr/cM7XmTT0U35w+KccN+ITBuWuajx7gkbCdxjxjLHJkDKGhHfo/p1Q3FK2DXYC20xgmgnM5C4ey9acbca6gaZ78fgN3N7UDIqGS5dlnoOJhBEhhDgAEom6cFJT4yztxGLOMk5qC3Gzh9Q1KIYNtms4sW3YvlNj6TKdL79yduys+lav3VLscwcZN3AZk4Z+yg9Hf8ZRgz4j17ur8ZBVP4mMI4hljCGeMZp4xlgso++Bm6KwTWwrjhVPYCbiJOI2pqVio6Pqem0diseb3MXj1nHpKi6XzKKkg4QRIYRIg1is4cnDqXoTcGpNdH1P4SSGYsWSzdd2nzlpu5qTlHAEVn2js/QrZ+Zk2XKdnbvqijP65Oxg4pDPOWbY55ww6jMO770UtxZsdB3T1Y14xmgStQFlTPu1tG+KbYGdqFeH4izz2CSXeXQ3mu5xzuGp3cnjklmUA0DCiBBCpJlt120hToWTWMy5L1UM22xXWKg3cxJFsYIoZiqc2KC4sNVUOGm7A2QKi1SWLXeCyVcrDJZ/rdees6MqJsP6rGHCkCUcN/Jzjhm2hCG5K9HURKPrmEYfJ5j4j0j+HNV+BwI2x05gW3HMuLPMYybqz6JouAwvmtspltWN1CyK1KK0JQkjQghxkEkVw8ZiTq1Jql19PO58+e195iRZc2LFkss6EbBioNiAU0uBarRpTYdpwvoNLr5aobNsucHylTqr1+gkEs63tVuPMGbAco4a/AXHH/45Ew/5kr6BdahK468W090vGU6OcH6mJaAkZ1HiCRJmAjNhYiZsbFzYuNAMHdXlxe0z8HidkOIydHRDRdMkpLSWhBEhhDjI1Q8nqR4nqWUdRakLJs0XxCbqwokVBjOEYseTTdhUUJJ1J2pz6WbfhCPw7eq6mZPlX+sNthZneKoZO/Arjhz0JceP+oKjBn9Bn8wNTV7LNPoS948injGKuH8UCf/hWEZem463RWprUUynFsW0MRM43XVVV23BrNun4/bquPTUUo8q2473QMKIEEJ0MPWXdVIFsamZE6gLJs1uJbYtJ5jYMWfWJNWEzY5jo9SrO9HbdGkHoKpKYeU3Osu/Nvh6lc6KlTpbttZ9S2f5Khg78CvGD1zKcYd/ybhBy+gX+K7Ja5l6T+L+w0n4D0/+HInpHnDgpyVsGzCxTKejrBmPY5o4IUVtWI/i9hkYnlRA0XHp0hsFJIwIIUSHlwonqdmTVEFs6iRil8sJJ803YXP6d9QWxVqhZA1KDAWrXgt7vV0O2iuvUFi5SmfFSiegfL1KZ+u2uvfJ9FYxZsByxg9cyvdHfsH4QcvJz17b6GBAAEvLJOEbSdw/koQ/+dN7qLMsdaAltx2bplM0ayWcbce2BRY6qkurF1L02qJZl971ZlIkjAghRCeUSDRswpbaSmyaLS2Krb+0k5w9sRPO0g606+wJ1AWUld8YrPxGZ+UqnY2b6wbrNUIc0f9rxg1cxvcOW8aEQ75iSPdV6Gqs0bVsRSfhHUrCP4K4b2Ty54gDX4dSO6DmQ4qtODMpqsvAZXgxvDpuTzKgGC5cutYptx9LGBFCiC7ANBvXncTjdduJW7S002j2JNZg9gQlFVDa59uyulph5bc6q77RWZX8uW69C9N03sulxRneZ7VThzLkKyYdtpzhvVbgNyqa/p0YvYj7RpDwjyDhG0HcPxLTM+jANGtrSjKkWFYCM246Z/WYFqapYKOhunRUl8spnPUauL2pWZRkSNGVpme+OgAJI0II0QWlimJTtSepoth4ckdwavZE11swe2LHnRb2ydmTum3FOrbiavOdO/WFI7B2rc6q1XUh5ds1LsLh1LeyTf/uWxk74CvGDFjOpGErGN1/Ob0yNzd5PVvxkPAdStw3PBlShhH3DcfWu7XL+FvMTmCZCayEE1ISppXc3aOhKC5UXUPTvehuN26vC91dP6ioB32fFAkjQgghgLqlndSW4lDIuW/32RNdb0HtiR1HMUPOEk9q505tcayrzfue1GeasGmLxjffOuHkm9U633yrU1Rc940c8FYyKn8lowes4MjBKzhq6AqG9liF2xVq+pp6TxK+YbW3uG+40/Je87bLZ2gx28S2Elhxk0QyrJgW2JYCigvFpaG5DKcuxatjeOqFFMN10Cz5SBgRQgjRJMtqWBib6neSKoxVlIazJ3veuRNPLu+Ek8WxcRTMZEAxkgGlfepPUkpKVL5Z7cycfLNa59vVOt9tcNX2QlEUi8E9N3JE/tcc0f9rJg1bzqj8lfQJbGzyejYqpmcACd9w52yeZFAxPQPbpdC3VRrUpaSWfGxME2eWStHQXBqq7kF3e5wlH6P+ko92QPulSBgRQgjRYolEXUCJRJzZk3i8bluxptXNnjS7vGOb9QJKqjFbcjYFG1tR6pZ42rCtfVNiMfhug4tv1zjhZPVaF6vXNJxF8btrGNnvG47o/zVjB37NkYesYljvVQTcJU1/PMUg4R3izKJ4D03OphyK6e7frmGrxZqaTTFtsBUsXMm6FBeqy43b6643m6Lh0l24PW3/GSSMCCGE2GepbcWpgBIOO7d43FkugbqtxbpO87ULViK5nBNzZk7MEIoVbyagtO8MCkBJqcqatS5Wr9VZs9YJK2u/a1iL0jNQxKj8lRyev4pxg1Yydsgqhvb4Bo+r8bk8ALbqcXb1eIeS8B2WDCoHU0hx+qWYiXqzKZbpFAjbzhk+hlcnb0AfvBnuNn1rCSNCCCHaVP3lnXjcmT1JFccmEk69ictVF1L2FFBqC2QbBRQr2T32wAUUy4ItWzXWrHNmUNYmf27cVLejR1EsBnTfwsh+33B4v1WMG7KKsQNXMjB3LboWbfK6Tkg5xAkn3qEkfM5P0zMg/cs9KbaNbSUIVtTQ59BB+AK+Nr28hBEhhBDtzjTrZlCiUWf2JBVQrGTvstTSTstmUOot8djxZA2KlaxBcdULKO2/jSQahQ0bXaxZ58yerFnr/Ny6TcO2nZCiKiaDe25kZL9vGNnvG8YNXsURA77dc0hRDBKeQSR8QzFTMyreQ0h4BqelcNa2bILlFRJG9kbCiBBCdBypgJIqig2F6mZTLMspFWnpDEqDgGKF62pQbBOb5OnFtT1QDswWklBY4bv1Ltauc7HuOxdrv3P6otTvLpsKKSP6fcvIvt8weuC3HDFgNYO6r8HTzM4eGwXTnU/Cewim95C6kOI9pF0buUkYaSEJI0II0bHtHlDC4bqAUr8GJRVQ9lgkayfqlnXMMFhhFNsEEoANqMmQcmCWeVJCIYV16118t8HFunUu1q13QsqWrXUzKYpi0b/bVob3Xc3wPqs5YsA3jB60mkN6rCHTXd7stS1XbjKYDHHCimdw8mf//V7ykTDSQhJGhBCi86m/xBOPOwElEqmrQVGUul08qaDS9DZju94MilMsW1eHkkgu81AXUFQdOHD7W8MR2LjJxbrvnC3H675zAsumzS7i8dQYbHoEihneZzXD+65mRN9vGT1oLcP6rCYvY1uz17YV3dmG7BlCwjsE0zuk9p9tV06LPqOEkRaSMCKEEF1Dqkg2dYtEnJCS2nqc6iJbvw6l2Vbp9Zd57ERyFiWS7Cab7PhG/d08rgPaMj6RcApnv9ug8916F+s3uli/wcV3611U19R9KJ87yKG91jGszxoO672WIwasZmT/tQzMXYfbFW72+paWTcI7GNM7mIRnsBNWPINJeAY1qE2RMNJCEkaEEKLrqr/NOJGoOyQw9edUoWzLlnksJ5jsPotiJ+oVy9avRdGSSz0Hrp2pbUNhkcqGVDhJ/ly/wcWOnXUfTFEs+uVu57Deazms91qG913N6IFrGdprHXmZzc+mAJhGHxKeQU5QcQ+iJtGT7MPPxNe9X5t+FgkjQgghOrXUbEn9ZZ5otGGr+923G+95FiVRN4tiRcAMOyGldqlHBUWrN4ty4Huuh8IKmzZrrN/gcsLKRufnhk0ugsG6D+c1QhySt57Deq/l0N7rGN5vLYf3X8eQHuuarU2JfO8tPINPa9PxtvT7+yDZ6CyEEEK0TipkeOvthk0t89TvKJs6iycScepUmqxFUV2ACxsP4JTBplqvN5hJsSLOUo8VBbsGJfVcVcdGa/eQ4vPajByeYOTwRIP768+mbNjk9EjZuOlQlm0cwb+WarX9UgByM0oZ2us7Du21jkN7r2NEv3WM6LeW8g2HMmlwuwx7r2RmRAghRKeXmi2pX4sSidTdl5Laapz6ueeCWacmpS6kROvNpBzYkLIn8Ths3aaxaXNdUHH+WWPnrro5iY//F+bo77dtnxOZGRFCCCGSUjMgHk/dfbZdF0ZSJxtHInXN2xKJuoLZ1OudmwKKgY1Rdy1oWI+SqkFJhRQr0jCk1G49bv+QouswZLDJkMEmk2nYiC0UVti8WWX1qiiHHdq93cawNxJGhBBCdEmKUne2Tn2W1TikpM7lCYXq+qKkXq9pqZCi7iWkJJIhJVYvpETBDtZ1mVW03YJK+/ZI8Xlthh+WoH/ParxeCSNCCCHEQUFVwTCcW32m2XC5JzWT0lRIqT+TomkqitpUSLEbzqTYCRQzVZMSBztcr0dKqpFbvSWfTqRzfRohhBCinWiac3PvdrBt/ZCy+0xK/eWe3UOKy6WgqDo2dVMztk4ypJj16lISTht8K7m7xwoBJoptJ2dTUjMp2gGZTWkPEkaEEEKI/dBcSNl9uSdVOBuNOrdgsO6sntRSj1M8q6DW7u5x1O40SbXDr13yqV+XEgM7VG82RWs4m3IAu862loQRIYQQoh00t9yTauJWfzal/u6eSKSukVuqeLYurGgoqoZNXfJpvA25/mxKsuOsFd6tgLZeUOHAdZ1tzj7N5cydO5eBAwfi8XiYOHEiS5Ysafa5r776KkceeSTZ2dn4/X7GjBnDs88+u88DFkIIIToyRXECis8HgQDk5kKfPjBoUN1twADo2xe6dXNmXGzbWfKpqICyMigvh+pqp1YlGgXTUpwzdzQvtisTW8/BcudheQdgegdiegdheQdhevKx3L2xtQC2oqHYJooZwlaMtM6atHpmZP78+cycOZN58+YxceJE7rvvPqZMmcLatWvp2bNno+fn5uby+9//nmHDhmEYBm+++SbTp0+nZ8+eTJkypU0+hBBCCNHR1a8p2V1qyWdPsympAtr6synOjIqGqmoNC2hTZSpWAts2iUdNUN2N3/gAaXXTs4kTJ3LUUUfx0EMPAWBZFvn5+Vx11VXceOONLbrGuHHjOP3007ntttta9HxpeiaEEEI0ZtuNC2jj8bq6lNRjqW/6VFCp3zsFnBmXQYOc2Zq21C5Nz2KxGEuXLmXWrFm196mqyuTJk/n000/3+nrbtnnvvfdYu3Ytd911V7PPi0ajRKN1jVmqqqpaM0whhBCiS9jTbEqqqdvuQSW1Hbl+Ea2W5rKRVoWRkpISTNMkLy+vwf15eXmsWbOm2ddVVlbSt29fotEomqbx8MMPc9JJJzX7/Dlz5vDHP/6xNUMTQgghRD3NNXWDumWf1MyJaTbeDXQgHZDdNJmZmSxfvpyamhoWLVrEzJkzGTx4MCeccEKTz581axYzZ86s/XNVVRX5+fkHYqhCCCFEp5fa6XOwaFUY6d69O5qmUVhY2OD+wsJCevXq1ezrVFXlkEMOAWDMmDGsXr2aOXPmNBtG3G437nRGNCGEEEIcMK3a2msYBuPHj2fRokW191mWxaJFi5g0aVKLr2NZVoOaECGEEEJ0Xa1eppk5cyYXX3wxRx55JBMmTOC+++4jGAwyffp0AKZNm0bfvn2ZM2cO4NR/HHnkkQwZMoRoNMrbb7/Ns88+yyOPPNK2n0QIIYQQHVKrw8jUqVMpLi7mlltuoaCggDFjxrBgwYLaotatW7eiqnUTLsFgkF//+tds374dr9fLsGHD+Mc//sHUqVPb7lMIIYQQosNqdZ+RdJA+I0IIIUTH09Lv7453tJ8QQgghOhUJI0IIIYRIKwkjQgghhEgrCSNCCCGESCsJI0IIIYRIKwkjQgghhEgrCSNCCCGESCsJI0IIIYRIqwNyau/+SvVlq6qqSvNIhBBCCNFSqe/tvfVX7RBhpLq6GoD8/Pw0j0QIIYQQrVVdXU1WVlazj3eIdvCWZbFz504yMzNRFGWfr1NVVUV+fj7btm2TtvLtTH7XB478rg8c+V0fOPK7PnDa83dt2zbV1dX06dOnwbl1u+sQMyOqqtKvX782u14gEJC/3AeI/K4PHPldHzjyuz5w5Hd94LTX73pPMyIpUsAqhBBCiLSSMCKEEEKItOpSYcTtdjN79mzcbne6h9Lpye/6wJHf9YEjv+sDR37XB87B8LvuEAWsQgghhOi8utTMiBBCCCEOPhJGhBBCCJFWEkaEEEIIkVYSRoQQQgiRVl0mjMydO5eBAwfi8XiYOHEiS5YsSfeQOp05c+Zw1FFHkZmZSc+ePTnrrLNYu3ZtuofVJdx5550oisK1116b7qF0Wjt27ODCCy+kW7dueL1eRo0axZdffpnuYXU6pmly8803M2jQILxeL0OGDOG2227b69kmYu/+97//ccYZZ9CnTx8UReG1115r8Lht29xyyy307t0br9fL5MmT+e677w7I2LpEGJk/fz4zZ85k9uzZLFu2jNGjRzNlyhSKiorSPbROZfHixcyYMYPPPvuMhQsXEo/HOfnkkwkGg+keWqf2xRdf8Oijj3LEEUekeyidVnl5Occccwy6rvOf//yHb7/9lnvuuYecnJx0D63Tueuuu3jkkUd46KGHWL16NXfddRd//vOfefDBB9M9tA4vGAwyevRo5s6d2+Tjf/7zn3nggQeYN28en3/+OX6/nylTphCJRNp/cHYXMGHCBHvGjBm1fzZN0+7Tp489Z86cNI6q8ysqKrIBe/HixekeSqdVXV1tDx061F64cKF9/PHH29dcc026h9Qp/fa3v7WPPfbYdA+jSzj99NPtSy65pMF9Z599tn3BBRekaUSdE2D/61//qv2zZVl2r1697Lvvvrv2voqKCtvtdtsvvPBCu4+n08+MxGIxli5dyuTJk2vvU1WVyZMn8+mnn6ZxZJ1fZWUlALm5uWkeSec1Y8YMTj/99AZ/v0Xbe+ONNzjyyCM555xz6NmzJ2PHjuXxxx9P97A6paOPPppFixaxbt06AFasWMFHH33EqaeemuaRdW6bNm2ioKCgwb9LsrKymDhx4gH5ruwQB+Xtj5KSEkzTJC8vr8H9eXl5rFmzJk2j6vwsy+Laa6/lmGOO4fDDD0/3cDqlF198kWXLlvHFF1+keyid3saNG3nkkUeYOXMmv/vd7/jiiy+4+uqrMQyDiy++ON3D61RuvPFGqqqqGDZsGJqmYZomt99+OxdccEG6h9apFRQUADT5XZl6rD11+jAi0mPGjBmsWrWKjz76KN1D6ZS2bdvGNddcw8KFC/F4POkeTqdnWRZHHnkkd9xxBwBjx45l1apVzJs3T8JIG3vppZd47rnneP755xk5ciTLly/n2muvpU+fPvK77sQ6/TJN9+7d0TSNwsLCBvcXFhbSq1evNI2qc7vyyit58803ef/99+nXr1+6h9MpLV26lKKiIsaNG4fL5cLlcrF48WIeeOABXC4Xpmmme4idSu/evRkxYkSD+4YPH87WrVvTNKLO64YbbuDGG2/k5z//OaNGjeKiiy7iuuuuY86cOekeWqeW+j5M13dlpw8jhmEwfvx4Fi1aVHufZVksWrSISZMmpXFknY9t21x55ZX861//4r333mPQoEHpHlKn9cMf/pCVK1eyfPny2tuRRx7JBRdcwPLly9E0Ld1D7FSOOeaYRtvU161bx4ABA9I0os4rFAqhqg2/mjRNw7KsNI2oaxg0aBC9evVq8F1ZVVXF559/fkC+K7vEMs3MmTO5+OKLOfLII5kwYQL33XcfwWCQ6dOnp3toncqMGTN4/vnnef3118nMzKxdZ8zKysLr9aZ5dJ1LZmZmo1ocv99Pt27dpEanHVx33XUcffTR3HHHHZx77rksWbKExx57jMceeyzdQ+t0zjjjDG6//Xb69+/PyJEj+eqrr7j33nu55JJL0j20Dq+mpob169fX/nnTpk0sX76c3Nxc+vfvz7XXXsuf/vQnhg4dyqBBg7j55pvp06cPZ511VvsPrt336xwkHnzwQbt///62YRj2hAkT7M8++yzdQ+p0gCZvTz31VLqH1iXI1t729e9//9s+/PDDbbfbbQ8bNsx+7LHH0j2kTqmqqsq+5ppr7P79+9sej8cePHiw/fvf/96ORqPpHlqH9/777zf57+iLL77Ytm1ne+/NN99s5+Xl2W632/7hD39or1279oCMTbFtaWsnhBBCiPTp9DUjQgghhDi4SRgRQgghRFpJGBFCCCFEWkkYEUIIIURaSRgRQgghRFpJGBFCCCFEWkkYEUIIIURaSRgRQgghRFpJGBFCCCFEWkkYEUIIIURaSRgRQgghRFpJGBFCCCFEWv1/ZSk0NGKtLiQAAAAASUVORK5CYII=\n"},"metadata":{}}]},{"cell_type":"markdown","source":["Using the quantities computed above, plot as a function of $t$ the ratio $\\hat{\\sigma}^\\mathsf{CV}_n/\\hat{\\sigma}_n$, which quantifies the variance reduction due to the use of the control variate. What do you conclude?"],"metadata":{"id":"vpE-9zbFxjH8"}},{"cell_type":"code","source":["# Second plot\n","plt.plot(t, sigma_CV/sigma_MC)\n","plt.show()"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":430},"id":"C9FKpNdbx4To","executionInfo":{"status":"ok","timestamp":1698049751533,"user_tz":-120,"elapsed":281,"user":{"displayName":"Julien Reygner","userId":"06465841509771183939"}},"outputId":"4d60186d-207d-4341-93e4-1b4357c2d0bf"},"execution_count":28,"outputs":[{"output_type":"display_data","data":{"text/plain":["
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\n"},"metadata":{}}]},{"cell_type":"markdown","source":["## 2. Importance Sampling\n","\n","For $p \\in [0,1]$, we let $(X_n)_{n \\geq 1}$ be a sequence of iid random variables such that $P(X_1=1)=p$, $P(X_1=-1)=1-p$. We next define $S_0 = 0$ and $S_{n+1} = S_n + X_{n+1}$, and for $N \\geq 1$ we let $$T^\\pm_N = \\inf\\{n \\geq 1: S_n = \\pm N\\}.$$\n","\n","We are interested in the evaluation of the probability of the event $\\{T^+_N < T^-_N\\}$, which is *small* when $p$ is close to $0$. In fact, it can be proved that we have\n","$$\\mathcal{I} := P(T^+_N < T^-_N) = \\begin{cases}\n"," 1/2 & \\text{if $p=1/2$;}\\\\\n"," \\frac{1-\\alpha^{-N}}{\\alpha^N-\\alpha^{-N}} & \\text{otherwise;}\n","\\end{cases} \\qquad \\alpha = \\frac{1-p}{p}.$$\n","So when $p$ is small, this probability is equivalent to $p^N$.\n","\n","We first fix values for $N$ and $p$."],"metadata":{"id":"zpKIMSDyPWoK"}},{"cell_type":"code","source":["p = .25 # Must be lower than .5\n","N = 5\n","\n","proba_to_estim = (1-(1/p-1)**(-N))/((1/p-1)**N-(1/p-1)**(-N))\n","\n","print(\"Probability to estimate:\",proba_to_estim)"],"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"BjRoZ-uEzdK9","executionInfo":{"status":"ok","timestamp":1698055778534,"user_tz":-120,"elapsed":211,"user":{"displayName":"Julien Reygner","userId":"06465841509771183939"}},"outputId":"504180d1-c534-4b3f-b1ae-3a1c3aa0c2dd"},"execution_count":82,"outputs":[{"output_type":"stream","name":"stdout","text":["Probability to estimate: 0.004098360655737705\n"]}]},{"cell_type":"markdown","source":["Our purpose is to fix $p$ small and to use importance sampling with random variables $(\\tilde{X}_n)_{n \\geq 1}$ which are such that $P(\\tilde{X}_1=1)=q$, $P(\\tilde{X}_1=-1)=1-q$. We denote by $\\tilde{T}^\\pm_N$ the associated hitting times of $\\pm N$. We want to study variance reduction as a function of $q$.\n","\n","Complete the following code so that the function `OneTraj(q)` returns a realisation of the pair $(\\tilde{T}^+_N \\wedge \\tilde{T}^-_N, 1_{\\tilde{T}^+_N < \\tilde{T}^-_N})$."],"metadata":{"id":"vhIUlwlWtV-d"}},{"cell_type":"code","source":["def OneTraj(q):\n"," S = 0\n"," T = 0\n"," while (abs(S)0]"],"metadata":{"id":"EoF3c8Jo6-O7","executionInfo":{"status":"ok","timestamp":1698055786818,"user_tz":-120,"elapsed":239,"user":{"displayName":"Julien Reygner","userId":"06465841509771183939"}}},"execution_count":83,"outputs":[]},{"cell_type":"markdown","source":["We now claim that for any $k \\geq N$,\n","$$P(\\tilde{T}^+_N \\wedge \\tilde{T}^-_N = k, \\tilde{T}^+_N < \\tilde{T}^-_N) = \\left(\\frac{q}{p}\\right)^{(k+N)/2}\\left(\\frac{1-q}{1-p}\\right)^{(k-N)/2}P(T^+_N \\wedge T^-_N = k, T^+_N < T^-_N),$$\n","therefore the IS estimator of $\\mathcal{I}$ writes\n","$$\\hat{\\mathcal{I}}^{\\mathsf{IS},q}_n = \\frac{1}{n}\\sum_{i=1}^n w_q(\\tilde{T}^{+,i}_N) 1_{\\tilde{T}^{+,i}_N < \\tilde{T}^{-,i}_N},$$\n","with\n","$$w_q(k) = \\left(\\frac{p}{q}\\right)^{(k+N)/2}\\left(\\frac{1-p}{1-q}\\right)^{(k-N)/2}.$$\n","\n","Complete the following so that the function `IS(q,n)` returns an array of $n$ realisations $w_q(\\tilde{T}^{+,i}_N) 1_{\\tilde{T}^{+,i}_N < \\tilde{T}^{-,i}_N}$."],"metadata":{"id":"R_RHpB327eOr"}},{"cell_type":"code","source":["def IS(q,n):\n"," vect_of_I = np.zeros(n)\n"," for i in range(n):\n"," [T,B] = OneTraj(q)\n"," if B:\n"," vect_of_I[i] = (p/q)**((T+N)/2)*((1-p)/(1-q))**((T-N)/2)\n"," return vect_of_I"],"metadata":{"id":"fizYUEu1_OUq","executionInfo":{"status":"ok","timestamp":1698055789944,"user_tz":-120,"elapsed":408,"user":{"displayName":"Julien Reygner","userId":"06465841509771183939"}}},"execution_count":84,"outputs":[]},{"cell_type":"markdown","source":["To check that the IS estimator is unbiased, we plot its value as a function of $q$, together with the true value of $\\mathcal{I}$."],"metadata":{"id":"Jk_cIltG_Gms"}},{"cell_type":"code","source":["# Values of q\n","nq = 20\n","q = np.linspace(.05,.95,nq)\n","\n","estim_IS = np.zeros(nq)\n","\n","for i in range(nq):\n"," isq = IS(q[i],10000)\n"," estim_IS[i] = np.mean(isq)\n","\n","plt.plot(q,estim_IS,label='IS estimator')\n","plt.axhline(y=proba_to_estim, color='black',label='True probability')\n","plt.legend()\n","plt.show()"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":430},"id":"8II-RIbA_WXg","executionInfo":{"status":"ok","timestamp":1698055797183,"user_tz":-120,"elapsed":4880,"user":{"displayName":"Julien Reygner","userId":"06465841509771183939"}},"outputId":"0d0b6a92-7457-40cc-8cbe-1b1775006015"},"execution_count":85,"outputs":[{"output_type":"display_data","data":{"text/plain":["
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\n"},"metadata":{}}]},{"cell_type":"markdown","source":["Using a similar code, plot the empirical variance of $\\hat{\\mathcal{I}}^{\\mathsf{IS},q}_n$ as a function of $q$. Which value of $q$ seems to be optimal?"],"metadata":{"id":"CZrywyom-y4I"}},{"cell_type":"code","source":["sigma_IS = np.zeros(nq)\n","\n","for i in range(nq):\n"," isq = IS(q[i],10000)\n"," sigma_IS[i] = np.std(isq)\n","\n","plt.plot(q,sigma_IS)\n","plt.axvline(x=p,color='black')\n","plt.show()"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":430},"id":"DNLAkfxl2Ixr","executionInfo":{"status":"ok","timestamp":1698055807434,"user_tz":-120,"elapsed":5053,"user":{"displayName":"Julien Reygner","userId":"06465841509771183939"}},"outputId":"6712a114-67f5-42ac-878d-47eafa513e86"},"execution_count":86,"outputs":[{"output_type":"display_data","data":{"text/plain":["
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gcXhzGXNlo2wFLePsLBSCAAwmAgsDmflsmZJmlzUPsJyJNysOrboBwAMEgKLw3UefGhNYAllpqs0FJQkvc8oCwBgkBBYHC6cXCVkTQ+L1LmBHCuFAACDhcDicBGLVwlJnSuFdh1jpRAAYHAQWBzMMIzOERaLVglJnSuFdjPCAgAYJAQWB2uIxhRPtB/VbOkIS7G5F0u9DPPoaAAABhCBxcHMFUKBNK+C6UN7UnNX40dmKd3nUX00piPhZsvqAAC4F4HFwazclr+rdJ9XE0a2b9HPBnIAgMFAYHGwzj1YrOtfMSX7WKoJLACAgUdgcTBzl9uQxSMsUudKoZ2sFAIADAICi4NZvcttV5NZKQQAGEQEFgezSw+LJE3tWCm0r6ZR0Vjc4moAAG5DYHGwzm35re9hKcoNKC8zXfGEoT3HG6wuBwDgMgQWB6vtmBIK2WBKyOPxJA9CZKUQAGCgEVgczOxhybfBCIskTe1ovGWlEABgoPUpsDz88MMqLy9XMBjU3Llz9frrr5/x2nfffVef+9znVF5eLo/HozVr1nzomvvuu08ej6fbY8qUKX0pbViJNFt7UvPpkocgslIIADDAUg4szzzzjJYuXaoVK1Zo+/btmjFjhhYuXKjjx4/3eH1TU5PGjx+vVatWqbi4+Iz3veCCC3Ts2LHk45VXXkm1tGHHTquEJM4UAgAMnpQDy0MPPaTbbrtNS5Ys0bRp07R27VplZmbq8ccf7/H6Sy65RA8++KCuvfZaBQKBM943LS1NxcXFyUdhYWGqpQ075sGHdtiHRZImdfSwHK+P6mRD1OJqAABuklJgaW1t1bZt21RRUdF5A69XFRUV2rJlS78K+eCDD1RaWqrx48frhhtuUGVl5RmvjUajqqur6/YYbgzDUMQcYbFJD0tWIE3njciUxCgLAGBgpRRYampqFI/HVVRU1O35oqIiVVVV9bmIuXPn6oknntCGDRv0yCOPaP/+/brssstUX9/zD72VK1cqFAolH2VlZX3+2k7V3BZXazwhyT5TQpI6VwoRWAAAA8gWq4Q+9alP6Qtf+IKmT5+uhQsX6vnnn1c4HNavfvWrHq9ftmyZIpFI8nHo0KEhrth6Zv9Kus+jTL91JzWfztyif1fV8Bv1AgAMnrRULi4sLJTP51N1dXW356urq8/aUJuqvLw8TZo0SXv27Onx84FA4Kz9MMNBOLkHi18ej8fiajrReAsAGAwpjbD4/X7NmjVLGzduTD6XSCS0ceNGzZs3b8CKamho0N69e1VSUjJg93SbsM2WNJu6ntocTxgWVwMAcIuUp4SWLl2qdevW6cknn9TOnTt1xx13qLGxUUuWLJEk3XTTTVq2bFny+tbWVu3YsUM7duxQa2urjhw5oh07dnQbPfn617+ul19+WQcOHNDmzZv113/91/L5fLruuusG4Ft0p4jNljSbzhuRpWC6Vy1tCVWearK6HACAS6Q0JSRJixcv1okTJ7R8+XJVVVVp5syZ2rBhQ7IRt7KyUl5vZw46evSoLr744uSfV69erdWrV+vyyy/Xpk2bJEmHDx/Wddddp5MnT2rkyJG69NJL9dprr2nkyJH9/PbcK2yjgw+78nk9mlSUo7cOR7TrWJ3GFWZZXRIAwAVSDiySdNddd+muu+7q8XNmCDGVl5fLMM4+NbB+/fq+lDGs1XYcfBjKsMeS5q4mm4Glql6fuohpPQBA/9lilRBSF0meI2SvERaJlUIAgIFHYHGo5Lb8dgwsrBQCAAwwAotDmauEQjbZ5bYrM7AcPNWkptaYxdUAANyAwOJQdjv4sKsR2QEVZgdkGNL71Q1WlwMAcAECi0NFbLpKyDS1pGOL/mP0sQAA+o/A4lCdIyz2mxKSOqeFOFMIADAQCCwOZdedbk2Ti1kpBAAYOAQWB2ppi6ulrf2k5pBNA0vXlULn2ocHAIBzIbA4kNm/4vN6lBPo095/g+78UdnyeT2qbWrT8fqo1eUAAByOwOJAnSc1p9vqpOaugum+5Lb8O2m8BQD0E4HFgcId2/LbcUlzV5PZQA4AMEAILA5Ua46w2LR/xTSVlUIAgAFCYHGgSMcKoXwb7nLbVedKIQILAKB/CCwOZOddbrsyVwrtOV6vtnjC4moAAE5GYHGgcLMzpoTG5GcoO5Cmtrih/TWNVpcDAHAwAosD2X2XW5PH40k23rJSCADQHwQWB4rYfJfbrlgpBAAYCAQWB0qOsDggsLBSCAAwEAgsDtR14zi7M1cKMcICAOgPAosDmVvz59l8WbPUOSV0JNycrBsAgFQRWBzIKTvdSu2jQKWhoCTp/WpGWQAAfUNgcZjWWEKNrXFJzuhhkaQpJR0byLFSCADQRwQWhzGnVTweKSfojMAymcZbAEA/EVgcxpwOyg2my+e150nNp5tCYAEA9BOBxWHCzc5Z0myaWtK5UsgwDIurAQA4EYHFYTr3YLH/CiHTuMIspfs8aojGdLi22epyAAAORGBxGCetEDKl+7w6fxTTQgCAviOwOEzEgVNCUmcfy+4qVgoBAFJHYHGYzoMPnRlYdjLCAgDoAwKLw4Q7Dj4MOaiHReIQRABA/xBYHMapIyzmSqH9NY1qaYtbXA0AwGkILA7j1B6WUTkB5WWmK54wtOd4g9XlAAAchsDiMJ3Lmp0VWDweDxvIAQD6jMDiMMkelgxn9bBI0pRicwM5VgoBAFJDYHEYp46wSGzRDwDoOwKLg8TiCdW3xCQ5r+lW4hBEAEDfEVgcxGy4laSQAwPLpKIceTzSifqoTjZErS4HAOAgBBYHMQ8+zAmmKc3nvLcuK5CmsQWZktiPBQCQGuf91BvGnNy/YmLHWwBAXxBYHCTSbB586LwVQqbJrBQCAPQBgcVB3DDCMpXGWwBAHxBYHMQMLE5suDVNKTFHWOoVTxgWVwMAcAoCi4OEHbotf1djCzIVTPcqGkvo4MlGq8sBADgEgcVBIk3O72HxeT2aXMS0EAAgNQQWB3HDCIvEBnIAgNQRWBzEDT0sUueZQruOsVIIANA7BBYH6Rxhce6UkNS5F8vuakZYAAC9Q2BxkGQPi0umhA6ebFJjNGZxNQAAJyCwOEituQ+Lw6eERmQHNDInIEl6n1EWAEAvEFgcIp4wVNfijikhqXNaiMZbAEBvEFgcor6lTUbHPmtOb7qVuvSxEFgAAL1AYHEIc4VQlt8nf5rz3zZzpdBOVgoBAHrB+T/5hgm3rBAyTe6yUsgw2KIfAHB2BBaHCHesEHLDdJAknT8qWz6vR+GmNlXXRa0uBwBgcwQWh4i4ZJdbUzDdp3GFWZKkXVVMCwEAzo7A4hBmD4tbAovESiEAQO8RWByic1t+d/SwSF0CC423AIBzILA4RLjZHbvcdpU8U4gRFgDAORBYHCLikl1uuzJXCu090aC2eMLiagAAdkZgcYiwy5puJWlMfoayA2lqixvad6LR6nIAADZGYHGI2uSyZvf0sHg8ni6Nt/SxAADOrE+B5eGHH1Z5ebmCwaDmzp2r119//YzXvvvuu/rc5z6n8vJyeTwerVmzpt/3HI4iLlwlJHVOC9HHAgA4m5QDyzPPPKOlS5dqxYoV2r59u2bMmKGFCxfq+PHjPV7f1NSk8ePHa9WqVSouLh6Qew5H5pRQvkt2ujVNKelovGWlEADgLFIOLA899JBuu+02LVmyRNOmTdPatWuVmZmpxx9/vMfrL7nkEj344IO69tprFQgEBuSew00iYSR3unXbCAuHIAIAeiOlwNLa2qpt27apoqKi8wZeryoqKrRly5Y+FdCXe0ajUdXV1XV7uFlDa0wJF53U3JU5JXQ00pKc9gIA4HQpBZaamhrF43EVFRV1e76oqEhVVVV9KqAv91y5cqVCoVDyUVZW1qev7RTmD/JgulfBdJ/F1Qys3GC6RudlSGo/CBEAgJ44cpXQsmXLFIlEko9Dhw5ZXdKgSm7L76IVQl2xUggAcC5pqVxcWFgon8+n6urqbs9XV1efsaF2MO4ZCATO2A/jRm7c5barycU52rjrOCuFAABnlNIIi9/v16xZs7Rx48bkc4lEQhs3btS8efP6VMBg3NNtOs8RcmdgYaUQAOBcUhphkaSlS5fq5ptv1uzZszVnzhytWbNGjY2NWrJkiSTppptu0ujRo7Vy5UpJ7U217733XvLjI0eOaMeOHcrOztb555/fq3sOd27c5bYrc0ro/eoGJRKGvF6PxRUBAOwm5cCyePFinThxQsuXL1dVVZVmzpypDRs2JJtmKysr5fV2DtwcPXpUF198cfLPq1ev1urVq3X55Zdr06ZNvbrncBcxlzS7tIdlXGGW/D6vGqIxHQk3q6wg0+qSAAA2k3JgkaS77rpLd911V4+fM0OIqby8XIZh9Ouew13YpbvcmtJ9Xk0Yla2dx+q0q6qewAIA+BBHrhIabmrNHhaXBhZJmmquFKKPBQDQAwKLA0Sa3T0lJHGmEADg7AgsDmBOCeW7eIQluVKIvVgAAD0gsDiAuUrIzVNC5kqh/TWNammLW1wNAMBuCCwO4PadbiVpVE5A+ZnpShjSnuMNVpcDALAZAovNGYbR2cPi4hEWj8ejKcXt00I7abwFAJyGwGJzTa1xtcXbl4W7ObBInY23u2m8BQCchsBic2b/it/nVYbLTmo+3dQSVgoBAHpGYLG5cMcut6HMdHk87t6yfnKxuVKIwAIA6I7AYnORZMOtu6eDJGlSUbY8HqmmIaqahqjV5QAAbITAYnNuP/iwq0x/ms7r2JafPhYAQFcEFpszlzSHXLykuStWCgEAekJgsbnaJvcvae6KlUIAgJ4QWGwu0jx8elgkVgoBAHpGYLE5c5VQftbwmBIyVwq9X12veMKwuBoAgF0QWGyus4dleIywjC3IVEa6T9FYQgdONlpdDgDAJggsNjecVglJks/r0aSibEnSrmNMCwEA2hFYbC4yDA4+PJ25Umh3FSuFAADtCCw2Fx4GBx+ezlwptJPGWwBABwKLzQ23HhZJmlLC0mYAQHcEFhtraYsrGktIGl4jLOaUUOWpJjVEYxZXAwCwAwKLjZmjKz6vR9mBNIurGToFWX6NyglIal/eDAAAgcXGkv0rGe4/qfl0Zh8LK4UAABKBxdaS/SvDaDrINLWElUIAgE4EFhsLNw2vbfm7msJKIQBAFwQWGwsnDz4cPnuwmJJLm4/VqS2esLgaAIDVCCw2Ntx2ue1qUlGOCrMDqm+J6T92HLW6HACAxQgsNhYehrvcmtJ9Xn3x0nJJ0tqX9yrBQYgAMKwRWGwsMgx3ue3qhrnnKTuQpg+ON+gPu45bXQ4AwEIEFhtLjrAM08ASykjXDXPHSmofZQEADF8EFhsbjtvyn+6Ll46T3+fVGwdrtf3gKavLAQBYhMBiY51Nt8Ovh8VUlBvUX188WpL02Cv7La4GAGAVAouNRZo6d7odzm6/fLw8HumPu05YXQoAwCIEFhsbzsuau5owMlsLpxVbXQYAwEIEFpuKxuJqao1LGp7Lmk/35Y9PsLoEAICFCCw2FekYXfF4pJzg8Dmp+UxmluVpzrh8q8sAAFiEwGJTkS4rhLze4XVS85nceum45MfmsQUAgOGBwGJTtcP44MMzuWziyOTHT2+ttLASAMBQI7DYlDmCEBrGS5pP5/F0jjT929aDau7o8QEAuB+BxabMFUL5w3yF0JnUNrbpV28csroMAMAQIbDYVIQpoXNa9+d9isUTVpcBABgCBBabCicPPmRKqCf5Wek6XNus/3r7mNWlAACGAIHFpjhH6OxunHueJGnty/tkGIbF1QAABhuBxabY5fbsrps7Vpl+n3Yeq9PL77NlPwC4HYHFppI9LASWHuVl+nXdnLGSpEc27bW4GgDAYCOw2FSyh4Vt+c/o1kvHKc3r0db9p/RmZa3V5QAABhGBxaaSPSyMsJxRaV6GFl08WpK09mVGWQDAzQgsNsWy5t758uXjJUkvvletvScaLK4GADBYCCw21BZPqD4ak8Sy5nM5f1SOKqYWyTCkR1/eZ3U5AIBBQmCxIfOkZknK5aTmc7rj4+2jLL9987CqIi0WVwMAGAwEFhsy+1dygmlK8/EWncus8wp0SXm+2uKGHn91v9XlAAAGAT8NbSjSsUIon+mgXrvj4xMktZ/i3HWECgDgDgQWGwqzB0vKPjF5lCYX5aghGtO/vXbQ6nIAAAOMwGJDbMufOo/Ho7/rWDH0r68eUEtb3OKKAAADicBiQ53b8jMllIrPzijV6LwM1TRE9e/bD1tdDgBgABFYbCjSZO5yywhLKtJ9Xt166ThJ0qN/2qd4gkMRAcAtCCw2xMGHfXftnDLlZabr4Mkm/fc7x6wuBwAwQAgsNkQPS99l+tN087xySe3b9RsGoywA4AYEFhuih6V/bp5frmC6V+8cqdOre05aXQ4AYAAQWGyIHpb+Kcjy69pLxkriUEQAcAsCiw3Rw9J/t146Tj6vR6/sqdHbhyNWlwMA6Kc+BZaHH35Y5eXlCgaDmjt3rl5//fWzXv/rX/9aU6ZMUTAY1EUXXaTnn3++2+dvueUWeTyebo8rr7yyL6W5Qm1jxwgLgaXPygoydfWMUknS2j8xygIATpdyYHnmmWe0dOlSrVixQtu3b9eMGTO0cOFCHT9+vMfrN2/erOuuu0633nqr3nzzTS1atEiLFi3SO++80+26K6+8UseOHUs+fvnLX/btO3K4eMJQXUv7Sc2hDHpY+sPcSO6/3z6mAzWNFlcDAOiPlAPLQw89pNtuu01LlizRtGnTtHbtWmVmZurxxx/v8fof//jHuvLKK/WNb3xDU6dO1f3336+PfOQj+ulPf9rtukAgoOLi4uQjPz+/b9+Rw9V1OQeHEZb+mVKcq09MHqmEIT36531WlwMA6IeUAktra6u2bdumioqKzht4vaqoqNCWLVt6fM2WLVu6XS9JCxcu/ND1mzZt0qhRozR58mTdcccdOnnyzKs7otGo6urquj3cwuxfyQ6kKZ2Tmvvty5e3H4r4m22Hdby+xeJqAAB9ldJPxJqaGsXjcRUVFXV7vqioSFVVVT2+pqqq6pzXX3nllfrFL36hjRs36oc//KFefvllfepTn1I83vN5MCtXrlQoFEo+ysrKUvk2bC3csUKIPVgGxpxxBfrI2Dy1xhJ64tUDVpcDAOgjW/wKf+211+rqq6/WRRddpEWLFum5557TX/7yF23atKnH65ctW6ZIJJJ8HDp0aGgLHkSsEBpYHo8nOcryf187qPqWtnO8AgBgRykFlsLCQvl8PlVXV3d7vrq6WsXFxT2+pri4OKXrJWn8+PEqLCzUnj17evx8IBBQbm5ut4dbRJoILAOtYmqRzh+VrfqWmJ7eWml1OQCAPkgpsPj9fs2aNUsbN25MPpdIJLRx40bNmzevx9fMmzev2/WS9NJLL53xekk6fPiwTp48qZKSklTKc4VwctM4VggNFK/Xo9s/1r5i6Oev7Fc01vNUIwDAvlKeElq6dKnWrVunJ598Ujt37tQdd9yhxsZGLVmyRJJ00003admyZcnrv/rVr2rDhg360Y9+pF27dum+++7TG2+8obvuukuS1NDQoG984xt67bXXdODAAW3cuFHXXHONzj//fC1cuHCAvk3nMKeEQoywDKhFM0erODeo4/VRPfvmEavLAQCkKOXAsnjxYq1evVrLly/XzJkztWPHDm3YsCHZWFtZWaljxzpPyZ0/f76efvppPfroo5oxY4Z+85vf6Nlnn9WFF14oSfL5fHrrrbd09dVXa9KkSbr11ls1a9Ys/fnPf1YgEBigb9M5zIMP2ZZ/YPnTvPrSZeMkSf/yp31KJDgUEQCcxGO44Djburo6hUIhRSIRx/ez3P3MDv3uzSP6zlVTdPvHJlhdjq00NjYqOztbUvvIXFZWVkqvb4jGNH/lRtW1xLT2b2fpygvP3EcFABh8qfz8tsUqIXSih2XwZAfSdNO8cknSIy/vlQuyOgAMGwQWm6ltoodlMN2yoFyBNK/+51BYr+07ZXU5AIBeIrDYTKSZHpbBVJgd0Bdmj5EkrX2ZQxEBwCkILDaTnBLKZEposNx+2QR5PdLL75/Qe0fdc6wDALgZgcVGEgkjOcKSz5TQoBk7IlOfnl4qSfqn595VSxv7sgCA3RFYbKQ+GpO52jaXKaFB9b8/eb4y/T69tu+U7np6u1pjCatLAgCcBYHFRsxt+TPSfQqm+yyuxt0mFuXo5zdfokCaV7/feVxfe+ZNxeKEFgCwKwKLjYSbzf4VRleGwrwJI/ToTbPl93n1/NtV+sZv3lKcDeUAwJYILDZi7nIbYjpoyFw+aaR+ev3FSvN69Ls3j+i7v3ub/VkAwIYILDZiniPECMvQuuKCYq25dqa8Hmn9Xw7pH//zPUILANgMgcVGIuxya5nPTC/Vg5+fIY9HemLzAa3asIvQAgA2QmCxkeTBh4ywWOJzs8bo+4sukiT9y8v7tOb3H1hcEQDARGCxEXNKiG35rXP93LFa/plpkqQfb/xAj2xiN1wAsAMCi40kR1iYErLUFy8dp29eOVmS9MMNu/T4K/strggAQGCxkc5t+RlhsdpXPn6+/vdfTZQk/dNz7+nprZUWVwQAwxuBxUbCHHxoK3dXTNTffWy8JOm7z76t324/bHFFADB8EVhshIMP7cXj8ejbn5qim+edJ8OQvv7r/9Fzbx21uiwAGJYILDYSYR8W2/F4PFrx2Qt07SVlShjS19bv0IvvVlldFgAMOwQWmzAMg2XNNuX1evT9v75Ii2aWKpYwdNfTb+rl909YXRYADCsEFptobI0r1nGODauE7Mfn9Wj1F2boqouK1RpP6PZfvKHNe2usLgsAhg0Ci02Y/Sv+NK+C6bwtdpTm82rN4otVMXWUorGEvvTkG9p28JTVZQHAsMBPRpvo3IMlXR6Px+JqcCb+NK9+ev1HdNnEQjW1xnXL43/RW4fDVpcFAK5HYLEJGm6dI5ju06M3ztaccQWqj8Z0489f185jdVaXBQCuRmCxCXa5dZYMv0+P33KJLh6bp0hzm/72sa3ac7ze6rIAwLUILDYRbm7vYeEcIefIDqTpiSVzdOHoXJ1sbNX167bqQE2j1WUBgCsRWGyiaw8LnCOUka7/+8W5mlyUo+P1Ud3w2FYdrm2yuiwAcB0Ci01wjpBz5Wf59W9fmqvxhVk6Em7W9eu2qirSYnVZAOAqBBab6Nw0jh4WJxqZE9BTt81VWUGGKk816YbHXlNNQ9TqsgDANQgsNhFmlZDjlYQy9PSXPqrSUFB7TzTSiAsAA4jAYhMRVgm5QllBpp667aMalRPQrqp6VTz0J93yr6/rlQ9qZBiG1eUBgGMRWGzCXCXECIvzjSvM0jN/N08LLyiSxyNt2n1Cf/vzrfrUj/+sX71xSNFY3OoSAcBxCCw2YfawhFgl5ArjCrP0LzfO1qavf1y3zC9Xpt+nXVX1+uZv3tKCVX/UTzZ+oJP0uABArxFYbMAwDHpYXOq8EVm67+oLtGXZX2nZp6aoJBRUTUNUD730vuav+oOW/fZt+lwAoBcILDbQ0pZQaywhiVVCbhXKSNffXT5Bf/rmJ/Tja2dq+piQorGEfvl6JX0uANALaVYXgM7+lTSvR1l+n8XVYDCl+7y6ZuZoXT2jVG8crNVjf96nF9+r1qbdJ7Rp9wlNKc7RFy8dp2tmliqQxn8LAGAisNhA5x4snNQ8XHg8Hl1SXqBLygt08GSj/vXVA/rVG4eSfS4PbNitm+adpxvmjtWI7IDV5QKA5ZgSsgEaboc3+lwA4NwILDYQSS5ppn9lOKPPBQDOjCkhG6jl4EN0QZ8LAHwYgcUGklNCLGlGF+fqc1n5/E5VTC3SwguKdenEQgXTCS8A3IvAYgPmKqF8poRwBmafy93/a5LWv16pJzYf0LFIi3697bB+ve2wMv0+XT5ppBZeUKxPTBlFPxQA1yGw2ECEKSH0ktnncuul4/T6gVN68d1qvfhulY5GWvTf71Tpv9+pUprXo3kTRuiKaUW64oJiFeUGrS4bgINFmtu07k/71BCN6b6rL7CsDgKLDXRd1gz0RprPq/kTCjV/QqFWfHaa3jlSpxferdKL71Xp/eoG/fmDGv35gxrd+//e1cyyPF1xQfvU0YSR2VaXDsAhGqMxPbH5gP7l5b2qa4nJ5/XoiwvGaeyITEvqIbDYgDklFGJKCH3g8Xh00ZiQLhoT0tcXTtb+msb28PJulbZXhrXjUPvjgQ27df6obF0xrT28TB8TYt8fAB/S0hbX01sr9bNNe1TT0P7zaVJRtpb+r8kqK8iwrC4Ciw2EmRLCABpXmKUvXz5BX758go7XtejF96r14nvV2rK3RnuON2jP8Qb9bNNeFecGkyMvc8YVKN3HLgfAcNYWT+jftx3WTzZ+oKORFknSeSMydXfFJH12Rql8Xmt/wSGw2ECEgw8xSEblBvW3Hz1Pf/vR8xRpbtOm3cf14rvV+uPu46qqa9EvthzUL7YcVCgjXX81ZZSuuKBYH5tUqEw//2sAhotEwtB/vnVU/+el93XgZJMkqTg3qK9WTNTnZ42xzS8z/F/JBjpHWJgSwuAJZaTrmpmjdc3M0Wppi2vz3hq98E61fr+zWicbW/XbN4/ot28eUTDdq8smjtTHJhZq+pg8TSnJYb8XwIUMw9BL71XrRy++r93V7btpj8jy6yufOF83zB1ru60SCCwWa2mLq7ktLol9WDB0guk+fXJKkT45pUjxhKFtB2v1wrtVeuHdKh2ubdZL71XrpfeqJUnpPo8mF+do+pg8TR8d0vQxeZpYlG2b37oApMYwDL2yp0arX3xf/3MoLEnKCabp7z42XksWjFNWwJ7RwJ5VDSN1HdNBXo+UY9P/SOBuPq9Hc8YVaM64At3z6anaeaxeL71Xre2VtXrrcFi1TW1650id3jlSp6c7XhNI82paaa5mjMnTRaNDmj4mpPEjsy2f4wZwdtsOntKDL+zWa/tOSZIy/T4tWVCu2y+bYPtfmvkJabFwc+fBh17+Zw+LeTweTSvN1bTSXEntv4kdrm3WW4cjeutIWG8fjujtwxHVR2N6szKsNyvDyddm+X26YHSofRSmrH005rwRmaxEAmzgnSMR/ejF3frj7hOSJL/Pqxs+OlZf+fj5GpnjjBPhCSwWq23k4EPYl8fjUVlBpsoKMvXp6SWS2hv0Dpxs1NtHIvqfQxG9fSSsd47UqbE1rtf3n9Lr+08lX58bTNP0MXm6aExnkCkNBQkxwBDZc7xeD730vp5/u0pS+4jq38weo7//5ESV5lm3RLkvCCwW6zrCAjiB1+vR+JHZGj8yW9fMHC1JiicM7TneoLcOhztGYyLaebROdS0xvbKnRq/sqUm+fkSWXxeNCWlSUY7KR2RpXGGWxo/M0qicAEEGGCCHTjVpze8/0O/ePKyEIXk80tUzSvW1ikkaV5hldXl9QmCxWIRdbuECPm97Y+7k4hx9YXaZJKk1ltD71fV663D7KMz/HIro/ep6nWxsTZ483VWm36dxhR0BpjBL40ZmqXxElsYXZtt+bh2wi+q6Fv30D3u0/i+VaosbkqQrphVp6RWTNKU41+Lq+ofAYjEOPoRb+dO8unB0SBeODkkaK6l9VdzOY3V650hE+2oatb/jcehUk5pa43r3aJ3ePVr3oXsVZPmTYcYMNOWF7YEmw2+vpZfAUGppi+tkY6tONbTqP986qic3H1A0lpAkXTaxUP9wxWTNLMuztsgBQmCxmLkHC1NCGA6C6T5dPDZfF4/N7/Z8ayyhylNN2l/TqAM1jR1hpkH7axpVXRfVqcZWnWps1baDtR+6Z2koqHEjzTCTrfGFWTpvRKZKQhmEGThKImGorqVNJxtbVdvY2u2fp7o+19Sqkw3tz5nbYnQ167x8ff2KyZo3YYQF38XgIbBYLMwut4D8aV6dPypb54/68OGMjdFYciTmQMc/99U0at+JBtW1xHQ00qKjkRa9uufkh16bn5mu4lCGSkJBFYeCKslt/2dpXkb7n0NBdvXFoDMMQwdONundoxHV1HcE8KbWZBA3H7VNbYonjJTvn+7zqCDLr/IR7cdyfHzySFf2g/E31WIRzhECziorkNZlaqmTYRiqbWrrGIlpSo7I7DvRqIMnm9TcFldtU5tqm9q089iHp5lMucE0lYQyVJLXHmCKczsDTmleUMWhDGWzRxJS0NIW1ztHItp2sFZvHKzV9oO1OtmxIrQ3cgJpKsj2Kz/TrxFZfhV0eeRnffi57ECaKwPK6fhbaDGzh4VlzUBqPB5Px/+wCzTrvIJunzMMQ3UtMVVFWnQ00qyqSIuORVpUFWnWsY6Pj4Wb1dgaV11LTHUt9cmtyXuSE0hTsTlKEwqqKDeY/GFRmB1QQccPkfwsPzsAD0M1DVFtO1jbHlAOnNI7R+rUGk90u8af5tUFpbkqCZn/7QRUkJmuguyACjLb/1sake1XXmY6R2GcAYHFYskeFqaEgAHj8XgUykhXKCNdk4tzznhdfUtbMswc6wgzneGmPezUt8RUH42p/niDPjjecM6vnRtM04jsQPK34BHZ/uQPqBFd/jwiqz3o+NMIOE6SSBj64HhDx+jJKW0/WJs8MLCrwmy/Zp2X3/Eo0IWjcwki/URgsViYKSHAMjnBdOUE0zWx6MyhpiHaPlJT1SXUnOjoQ6hpiHbpP2hVwlDHiE17302vaugY/h/RJdTkZ/lVkJWuvEy/CjL9yu/ycW5GOkcgDKGm1ph2HApr24Fabatsn96pa4l1u8bjkSaNytFHzsvX7I6Qwi7PA4/AYrFIsumWKSHAjrIDaWdsCO4qnjAUaW7TqcaoajpWcJjLTU81RlWT/LhzpUc8YbSP3kRjOtjDb+k98Xjaf8HJz2wPNvmZHWEmq306oSDTn/yz+bm8zHSmqnrpWKRZbxyoTU7xvHes7kONsBnpPs0sy9Ps8nx95Lx8fWRsPis9h0CfAsvDDz+sBx98UFVVVZoxY4b++Z//WXPmzDnj9b/+9a9177336sCBA5o4caJ++MMf6qqrrkp+3jAMrVixQuvWrVM4HNaCBQv0yCOPaOLEiX0pzzHa4gk1RNuTOiMsgLP5vJ5kX8v5o859vbmE1Qw3pxqjnQGnqVXhpjadamxVuKnjz41tqo/GZBhKNhOrl6M4UvtpvO2hpiPIZLSHmVBGuvIyOx4Z/o6P/crLSHfdaI7Z23Q03Kyj4WYd6XgcDbfoaLhZh0416Xh99EOvK84NalZ5++jJ7PMKNKUkhwBogZQDyzPPPKOlS5dq7dq1mjt3rtasWaOFCxdq9+7dGjXqw39LN2/erOuuu04rV67UZz7zGT399NNatGiRtm/frgsvvFCS9MADD+gnP/mJnnzySY0bN0733nuvFi5cqPfee0/BYLD/36VNmdNBkpRLYAGGFa/X0zH60fvR1dZYQuHm08JMY5tqmzo/TgacjmvMUdz6lpjqW3o/kmPKDaYlR2lCHSM7eR2BJ9QRbMzAE8poD0CZfp+C6b4hDztt8UR771G4WUcj7UHkSEc4OdoRTMxfEs/E65GmluS2T+2UF2jWefka7bAzd9zKYxhGSou+586dq0suuUQ//elPJUmJREJlZWX6+7//e33729/+0PWLFy9WY2OjnnvuueRzH/3oRzVz5kytXbtWhmGotLRU//AP/6Cvf/3rkqRIJKKioiI98cQTuvbaa89ZU11dnUKhkCKRiHJznbP18J7j9ap46E/KDabprfsWWl2O7TU2Nio7u31YvqGhQVlZzjwPAxhKnVNVZqhpVbi5TZGmtmT46frn2sY2RZrbzvmDvTcCaV5l+H3KTPcpw+/r+DhNwQ891/lxRrovGXgy/WldPm7/XFNrPBlIuo6OHA03q7quRb3ZxqQgy6/SvKBG52WoNC8j+c+SUFCTinKUxTL2IZPKz++U3pXW1lZt27ZNy5YtSz7n9XpVUVGhLVu29PiaLVu2aOnSpd2eW7hwoZ599llJ0v79+1VVVaWKiork50OhkObOnastW7b0GFii0aii0c5hu7q6M++x0B+xeELff37noNxbkk50DD3SvwJgsHSdqkpFWzyhSHObwk1tinQEm9qm9hEc8/lwc+efaztGdeq7NKRGYwlFYwmF1XaWrzSw/D6vSvPaNwcsTQaSLn9mB2THSimw1NTUKB6Pq6ioqNvzRUVF2rVrV4+vqaqq6vH6qqqq5OfN5850zelWrlypf/zHf0yl9D5JGNK/vnpg0L9Occi9014AnCnd51VhdkCF2YGUXpdIGIrGEmpqjam5La7m1riaWuPJj5vbuv45pubWhJraYmo5w3Utbd2f96d5O0ZEgt1GR0o7nivMCsjror4bdHLkuNeyZcu6jdrU1dWprKxswL+O1yPd+YkJA37frnwejz4zo3RQvwYADBWv15Oc3gEGUkqBpbCwUD6fT9XV1d2er66uVnFxcY+vKS4uPuv15j+rq6tVUlLS7ZqZM2f2eM9AIKBAILXU3xdpPq++sXDKoH8dAABwdimty/L7/Zo1a5Y2btyYfC6RSGjjxo2aN29ej6+ZN29et+sl6aWXXkpeP27cOBUXF3e7pq6uTlu3bj3jPQEAwPCS8pTQ0qVLdfPNN2v27NmaM2eO1qxZo8bGRi1ZskSSdNNNN2n06NFauXKlJOmrX/2qLr/8cv3oRz/Spz/9aa1fv15vvPGGHn30UUntW2h/7Wtf0/e+9z1NnDgxuay5tLRUixYtGrjvFAAAOFbKgWXx4sU6ceKEli9frqqqKs2cOVMbNmxINs1WVlbK6+0cuJk/f76efvpp3XPPPfrOd76jiRMn6tlnn03uwSJJ3/zmN9XY2Kjbb79d4XBYl156qTZs2ODqPVgAAEDvpbwPix05dR8WpIZ9WADAXVL5+c3ewgAAwPYILAAAwPYILAAAwPYILAAAwPYILAAAwPYILAAAwPYILAAAwPYILAAAwPYILAAAwPZS3prfjszNeuvq6iyuBIOpsbEx+XFdXZ3i8biF1QAA+sv8ud2bTfddEVjq6+slSWVlZRZXgqFSWlpqdQkAgAFSX1+vUCh01mtccZZQIpHQ0aNHlZOTI4/HY3U5w1ZdXZ3Kysp06NAhznSyAd4P++E9sRfeD+sZhqH6+nqVlpZ2Ozi5J64YYfF6vRozZozVZaBDbm4uf/lthPfDfnhP7IX3w1rnGlkx0XQLAABsj8ACAABsj8CCARMIBLRixQoFAgGrS4F4P+yI98ReeD+cxRVNtwAAwN0YYQEAALZHYAEAALZHYAEAALZHYAEAALZHYEFKHn74YZWXlysYDGru3Ll6/fXXz3jtunXrdNlllyk/P1/5+fmqqKg46/VIXSrvR1fr16+Xx+PRokWLBrfAYSbV9yMcDuvOO+9USUmJAoGAJk2apOeff36Iqh0eUn1P1qxZo8mTJysjI0NlZWW6++671dLSMkTV4qwMoJfWr19v+P1+4/HHHzfeffdd47bbbjPy8vKM6urqHq+//vrrjYcffth48803jZ07dxq33HKLEQqFjMOHDw9x5e6U6vth2r9/vzF69GjjsssuM6655pqhKXYYSPX9iEajxuzZs42rrrrKeOWVV4z9+/cbmzZtMnbs2DHElbtXqu/JU089ZQQCAeOpp54y9u/fb7zwwgtGSUmJcffddw9x5egJgQW9NmfOHOPOO+9M/jkejxulpaXGypUre/X6WCxm5OTkGE8++eRglTis9OX9iMVixvz5843HHnvMuPnmmwksAyjV9+ORRx4xxo8fb7S2tg5VicNOqu/JnXfeaXzyk5/s9tzSpUuNBQsWDGqd6B2mhNArra2t2rZtmyoqKpLPeb1eVVRUaMuWLb26R1NTk9ra2lRQUDBYZQ4bfX0//umf/kmjRo3SrbfeOhRlDht9eT/+4z/+Q/PmzdOdd96poqIiXXjhhfrBD36geDw+VGW7Wl/ek/nz52vbtm3JaaN9+/bp+eef11VXXTUkNePsXHH4IQZfTU2N4vG4ioqKuj1fVFSkXbt29eoe3/rWt1RaWtrtfyDom768H6+88op+/vOfa8eOHUNQ4fDSl/dj3759+sMf/qAbbrhBzz//vPbs2aOvfOUramtr04oVK4aibFfry3ty/fXXq6amRpdeeqkMw1AsFtOXv/xlfec73xmKknEOjLBgSKxatUrr16/X7373OwWDQavLGXbq6+t14403at26dSosLLS6HEhKJBIaNWqUHn30Uc2aNUuLFy/Wd7/7Xa1du9bq0oatTZs26Qc/+IF+9rOfafv27frtb3+r//qv/9L9999vdWkQIyzopcLCQvl8PlVXV3d7vrq6WsXFxWd97erVq7Vq1Sr9/ve/1/Tp0wezzGEj1fdj7969OnDggD772c8mn0skEpKktLQ07d69WxMmTBjcol2sL38/SkpKlJ6eLp/Pl3xu6tSpqqqqUmtrq/x+/6DW7HZ9eU/uvfde3XjjjfrSl74kSbrooovU2Nio22+/Xd/97nfl9fI7vpX4t49e8fv9mjVrljZu3Jh8LpFIaOPGjZo3b94ZX/fAAw/o/vvv14YNGzR79uyhKHVYSPX9mDJlit5++23t2LEj+bj66qv1iU98Qjt27FBZWdlQlu86ffn7sWDBAu3ZsycZHCXp/fffV0lJCWFlAPTlPWlqavpQKDEDpcGxe9azuusXzrF+/XojEAgYTzzxhPHee+8Zt99+u5GXl2dUVVUZhmEYN954o/Htb387ef2qVasMv99v/OY3vzGOHTuWfNTX11v1LbhKqu/H6VglNLBSfT8qKyuNnJwc46677jJ2795tPPfcc8aoUaOM733ve1Z9C66T6nuyYsUKIycnx/jlL39p7Nu3z3jxxReNCRMmGH/zN39j1beALpgSQq8tXrxYJ06c0PLly1VVVaWZM2dqw4YNyaa2ysrKbr+dPPLII2ptbdXnP//5bvdZsWKF7rvvvqEs3ZVSfT8wuFJ9P8rKyvTCCy/o7rvv1vTp0zV69Gh99atf1be+9S2rvgXXSfU9ueeee+TxeHTPPffoyJEjGjlypD772c/q+9//vlXfArrwGAbjXAAAwN749QsAANgegQUAANgegQUAANgegQUAANgegQUAANgegQUAANgegQUAANgegQUAANgegQUAANgegQUAANgegQUAANgegQUAANje/weSXhdeGubxwgAAAABJRU5ErkJggg==\n"},"metadata":{}}]},{"cell_type":"markdown","source":["You may now try to play with the values of $p$ and $N$ to make the event $\\{T^+_N < T^-_N\\}$ even more unlikely, and see how IS performs."],"metadata":{"id":"dv5q938lAVzo"}}]}