03_causality.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import torch\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "import math\n",
    "from torch.nn.functional import conv1d\n",
    "from torch.nn.functional import pad\n",
    "from torch.autograd import grad\n",
    "\n",
    "plt.rcParams.update({'font.size': 16})\n",
    "plt.rcParams['font.family'] = 'serif'\n",
    "plt.rcParams['figure.figsize'] = [4, 2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# define signal and kernel\n",
    "L = 100\n",
    "x = torch.linspace(0, 2*torch.pi, L)\n",
    "freq = 5\n",
    "u = torch.sin(freq * x)\n",
    "\n",
    "M = 30\n",
    "x = torch.linspace(-2, 2, M) \n",
    "h = torch.exp(- x.pow(2))\n",
    "\n",
    "# toeplitz utilities\n",
    "def centered_toeplitz(h, L, M):\n",
    "    T = torch.zeros(L - M + 1, L)\n",
    "\n",
    "    T[0, :M] = h\n",
    "    for i in range(1, L - M + 1): T[i] = torch.roll(T[i-1], 1)\n",
    "    return T\n",
    "\n",
    "def full_toeplitz(h, L, M):\n",
    "    T = torch.zeros(L + M - 1, L)\n",
    "    for i in range(M): T[i, :i+1] = h[-i-1:]\n",
    "    for i in range(M, L): T[i] = torch.roll(T[i-1], 1)\n",
    "    for i in range(L, L + M - 1):\n",
    "        T[i, -M+i-L+1:] = h[:-i-1+L]\n",
    "    return T\n",
    "    \n",
    "def circular_toeplitz(h, L, M):\n",
    "    T = torch.zeros(L, L)\n",
    "\n",
    "    T[0, :1] = h[0]\n",
    "    T[0, -M+1:] = h[1:]\n",
    "    for i in range(1, L): T[i] = torch.roll(T[i-1], 1)\n",
    "    return T\n",
    "\n",
    "def fft_conv(u, h):\n",
    "    L, M = u.shape[0], h.shape[0]\n",
    "    T = circular_toeplitz(h, L + M - 1, M)\n",
    "    u_pad = pad(u, (0, M - 1))\n",
    "    return T @ u_pad\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Causality"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f7612321330>]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 400x200 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "u.requires_grad = True\n",
    "y = fft_conv(u, h)\n",
    "plt.plot(y.detach())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Is this a causal convolution? In other words, does $y[k]$ depend only on values $u[n]$ with $n \\leq k$? One way to check\n",
    "is by computing the gradient $\\frac{d y[k]}{d u[n]}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "dydu = torch.zeros(L, L)\n",
    "for n in range(L):\n",
    "    dydu[n] = grad(y[n], u, retain_graph=True)[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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J6ZdfpDNnbGZpsH/1/HwpFLIleIkRCllYKimRxo2zkZ8/hj8EAAAAkKKMOLx1z549isfj8vv9mjNnzoDPqa6uVnNzs3bv3j2sUISr5edL06fbSOjpkX791QLS2bM2WlulSMRC1PnzNoYSCCQDUiAw8CgsHDo8+f0WtEpKbOaqpMQOtQUAAADGkrO3oMeOHZMklZWVDdphburUqf2ee6VoNKpoNNp3PxKJjHCV2amgQCovt3G5eNyCUXu7FA4n9yaFw9LFizY6O+150aiNa4WnVBUWWjhiRR0AAEDmevxx2/+eKZyFokSAGWqvUElJiSQpHA4P+PjatWu1Zs2akS8uR+XlJZfMDcbzpEuXkgHp4sVkQEqM7u7k7Xj86tfo6Ynpyy//Xj090sKFKxSN+tXVZc/t7rYBAACAzJVp7+cyerFSbW2tVq1a1Xc/EomosrLSYUXZz+ezpW7D6HsxqO5uT8ePt0iSnnvOU2Ghha1o1IJWZ6ct8QMAAEBmmjzZdQWpcRaKEpudurq6Bn1OZ2enJCkUCg34eCAQUCAQGPniMOZ8PqmoyMYNN7iuBgAAALlk4M08YyDRXKG1tVXxgdZYSTp16lS/5wIAAADASHMWihYuXKi8vDzFYrFBGykcPnxYkrRo0aKxLA0AAABADnEWiiZPnqxly5ZJkj777LOrHq+vr1dzc7OCwaAeffTRsS4PAAAAQI5wFook6bXXXpMkvf322+q5Ymf9unXrJEkvvfRSXxc6AAAAABhpTkPRsmXL9MYbb+jgwYN64okntG/fPjU0NGj16tXasGGDHnvsMdXW1rosEaOkpKSEsAsAAIC04PM8z3NdxLZt27Ru3Trt2bNHsVhM8+bN04oVK7Ry5Url5+cP+3UikYhCoZDC4XBfdzsAAAAAuSeVbJAWoWikEIoAAAAASKllA6fL5wAAAADANWeHtyJ3xWIxbd68WZL01FNPye/3O64IAAAAuYxQhDHneZ4aGxv7bgMAAAAusXwOAAAAQE4jFAEAAADIaYQiAAAAADmNUAQAAAAgpxGKAAAAAOQ0us/BCdpwAwAAIF34vCzqiZzKqbUAAAAAslcq2YDlcwAAAAByGqEIAAAAQE5zHop6e3tVV1enoqIiPfTQQ67LwRjo6enR5s2btXnzZvX09LguBwAAADnOaaOFQ4cOacWKFdq1a5fLMjDG4vG4jhw50ncbAAAAcMnZTNHGjRu1YMECTZs2TW+99ZarMgAAAADkOGehaOfOnfrwww+1detWTZkyxVUZAAAAAHKcs+Vz7777rvLz8119ewAAAACQ5HCmiEAEAAAAIB047z4HAAAAAC457T53vaLRqKLRaN/9SCTisBoAAAAAmSijQ9HatWu1Zs0a12UgRYWFhXr99dddlwEAAABI+h2h6OWXX9ZHH32U8jfauHGjlixZkvKfG0ptba1WrVrVdz8SiaiysnJEvwcAAACA7JZyKGppaVF9fX3K3+jChQsp/5lrCQQCCgQCffc9z5PEMjoAAAAg1yUyQSIjDCXlULRp0yZt2rQp9arGQEdHhyQxWwQAAABAkmWEUCg05HMyek/RlSoqKtTU1KRgMCifz+e6HAAAAACOeJ6njo4OVVRUXPO5WRWK8vLyNH36dNdlAAAAAEgD15ohSuCcIgAAAAA5zdlMUXd3t86fPy9JCofDfddOnz4tSSouLh52sgMAAACA38vnDacdwyj44osv9PDDDw/6+NNPP60PPvhg7AoCAAAAkJOchSIAAAAASAfsKQIAAACQ09ImFO3YsUNVVVXy+XxqbGx0XQ4AAACAHOE8FHV1denFF1/U0qVL1dDQ4LocAAAAADnGaSg6duyYampqtGXLFm3bts1lKQAAAABylNNQdPDgQS1fvlwHDhzQI4884rIUAAAAADnKafe53t5e5efnJ4vx+SRJx48f16xZs1J+vXg8rpaWFgWDwb7XAgAAAJB7PM9TR0eHKioqlJc39FyQs8NbJfULRCOhpaVFlZWVI/qaAAAAADJXU1OTpk+fPuRznIaikRYMBiXZD15aWuq4mrHneVIsNvjjeXlSQVb9iwMAAAADi0Qiqqys7MsIQ8not8jRaFTRaLTvfkdHhySptLQ0q0NRLCadOyf9+quNy28PFYp8PikUkiZNsnHjjcnb48fb4wAAAEA2Gc62mowORWvXrtWaNWtclzHqenulpibpyBEbZ8/+vtfxPKm93cbRo/0fKyqSZs+W5s2Tbr7ZQhLGXnd3t958801J0iuvvKLCwkLHFQEAAGS/jA5FtbW1WrVqVd/9xBRZNrh40YJLQ4N07Jh06VL/x4uLk7M8l49gcPAZn+7u/rNKidHWZq9/6JANSaqosIA0d67dZhYJAAAA2SqjQ1EgEFAgEHBdxoiJxaSDB6V9+2xm6PK+gCUlNoMzb540a5Y0blzqQcXvtz83Y0b/6z090unTyZmolpbk+OILmzW6/XZp4UILXgAAAEA2yehQlC3OnJH27pV++KH/jNDUqTZTM2+ezdZco5Pg71ZQIE2fbuPhh6WOjuQs1c8/SxcuSN98Y2PmTAtH8+fTtGE0+Hw+VVRU9N0GAADA6HN6TtGVrvecokgkolAopHA4nPaNFrq7bVZo716puTl5feJEacEC6c47pXT4EXp7bfne3r0WkhK/LcXFUk2N1VpW5rREAAAA4CqpZAM+6x9jFy5IO3dKu3cnZ4Xy8qTqapuBmT07vfbv5OfbTNW8eVIkIn33nS3vC4eTs0ezZ0tLlkg33ZRetQMAAADDQSgaI21t0o4dFip6euzaxIkWhGpqMqPbW2mptGyZ9OCDydmj+npbYvfzz7bEb8kSC3ijtdQPAAAAGGnOQ1Fra6t6e3uvulZUVCRJKi8vd1HWiDl1Svr6a1sql1h6Nm1aMjxk4sxKXp7tdZo719p7J8JeS4v0j/9o5x/df78tAWTfUWpisZjWr18vSVq5cqX8fr/jigAAALKf87esixYt0okTJ/pdW7x4cd/tNNrylJLmZunf/s1mVBJuvtnC0MyZmRmGBjJhgvTYYzaDtGuXjXPnpH/+Z/v577tPWrRI4rid4fE8T+3t7X23AQAAMPqch6LGxkbXJYyK1lYLRD6fdNtt0gMPSBk+6TWkceOsc90DD9ieox07bA/Sp5/aTNl990mLF0tZ1EEdAAAAWcJ5KMpWd9xhMyYLF9reoVxRWCjde6/NDv3wg7R9u3T+vPTZZxaU7r1Xuuce6bfVkQAAAIBzhKJRkp8v/fGPrqtwJz9fuusu21d04ID05ZfSr7/akrodOywY3XuvHUoLAAAAuEQowqjKy7NZs9tukw4dsnB09qx9/fZb6e67bWldMOi6UgAAAOQqQhHGRF6eBaNbb5V++slC0enTNmu0a5fNKj3wgDVuAAAAAMYSoQhjyueT5s+XbrlFOnrUwlFTkx1mu3evdPvt1qGvrMx1pW74fD6V/fbD+7KlRSEAAECa83lZ1Pc3EokoFAopHA6rtLTUdTkYBs+TTpywhgyJ9uU+n53hdO+90owZ2dO+HAAAAGMnlWzATBGc8vmkWbNs/PKLhaPDh22J3U8/SVOnWji69VYOggUAAMDoYKYIaefsWWnnTun776WeHrs2fry1+b77bjsTCQAAABhKKtmAUIS01dlp+4x27ZI6Ouxafr41bLjrLmnmzOxbWheLxfTee+9Jkp599ln5/X7HFQEAAGQmls8hK5SUSA8+KN1/v7Xz/vZbW2L3/fc2Jk6UamrsLKRs6VrneZ5aW1v7bgMAAGD0EYqQ9vLzrSvd7bdLzc3Svn3SwYNSW5sdBvtv/ybddJMFpFtukQoLXVcMAACATEIoQkaZPt3Gn/5kjRj275eOH0+O//f/lebMkaqqpHnz2H8EAACAa0uLUPTxxx9r3bp12rt3r6LRqKqqqvTMM8/o+eefV15enuvykIb8fumOO2yEw7acbv9+6fx56153+LDtN6qstIBUVSXdeGP27UECAADA9XMeiurq6rR69Wo9/vjj+vTTTxUMBvX+++/rhRde0CeffKKtW7eqgF7MGEIoJC1davuPzpyxQFRfL506JZ08aePTT6UbbrDW3zNmWJOGCRMISQAAAHAcirZv367Vq1dr/vz52rp1a1+nrbq6OrW1tWnDhg2qq6vTq6++6rJMZAifTyovt/HQQzaD1NBgAen4cZtFOn/e9iRJUjBoASkxJk+2/UsAAADILU5bci9fvlyff/653nnnHT333HP9Hquvr1d1dbVKS0t1+vRpFRcXX/P1aMmNwUSjUmNjcuaopUXq7e3/nLw8adIkC0dTptjXyZPHdkYpFotp/fr1kqSVK1fSkhsAAOB3yohzilpbW1VeXq54PK6GhgbNnTv3qudUVlaqublZW7Zs0ZNPPnnN1yQUYbhiMWvvnQhJzc3SpUsDP7ew0Np/T5iQHKFQ8nZxMcvwAAAA0k1GnFO0Z88exeNx+f1+zZkzZ8DnVFdXq7m5Wbt37x5WKAKGy++3/UWzZtl9z5MiEensWduXdPasjdZWqbvbrp05M/Br5eVZl7uSEvt6+e2iIikQuHoUFkpDbZUrKLAaAQAAMPqchaJjx45JksrKygbtMDd16tR+zwVGi89nsz+hkHT5pGVvr52H1N4+8LhwQYrHpY4OGyOpoMDCVWIUF9vXQGBkvw8AAMBIq6mxbQmZwlkoikQikjTkXqGSkhJJUjgcHvDxaDSqaDR61WsCIyU/3/4HPdj/qHt6pIsXpc7O/l8Tty9dsv1M0ajNOCVuR6M2O3Wl3t6Y9u//e0lSTc0K9fT4xa81AADINDNnEorGzNq1a7VmzRrXZSCHFRQkZ5hS4XkDh6Lubk9r17aot1d64QVPPT0Wri4f3d0jUzsAAMBomTDBdQWpcRaKEpudurq6Bn1OZ2enJCk0yDvO2tparVq1qu9+JBJRZWXlCFYJjA6fb+DmDHl5dr2gwP5jUlg45qUBAADkHGehKNFcobW1VfF4fMB9RadOner33CsFAgEF2GABAAAA4DoM3OFgDCxcuFB5eXmKxWKDNlI4fPiwJGnRokVjWRoAAACAHOIsFE2ePFnLli2TJH322WdXPV5fX6/m5mYFg0E9+uijY10eAAAAgBzhLBRJ0muvvSZJevvtt9XT09PvsXXr1kmSXnrppb4udAAAAAAw0pyGomXLlumNN97QwYMH9cQTT2jfvn1qaGjQ6tWrtWHDBj322GOqra11WSIw5kpKSvggAAAAYAz5PG+gxsBja9u2bVq3bp327NmjWCymefPmacWKFVq5cqXy8/OH/TqRSEShUEjhcLivux0AAACA3JNKNkiLUDRSCEUAAAAApNSygdPlcwAAAADgmrNzigBcLRaLafPmzZKkp556Sn6/33FFAAAA2Y9QBKQRz/PU2NjYdxsAAACjj+VzAAAAAHIaoQgAAABATiMUAQAAAMhphCIAAAAAOY1QBAAAACCn0X0OSDO04QYAABhbPi+L+v6mcmotAAAAgOyVSjZg+RwAAACAnEYoAgAAAJDTnIei3t5e1dXVqaioSA899JDrcgCnenp6tHnzZm3evFk9PT2uywEAAMgJThstHDp0SCtWrNCuXbtclgGkjXg8riNHjvTdBgAAwOhzNlO0ceNGLViwQNOmTdNbb73lqgwAAAAAOc5ZKNq5c6c+/PBDbd26VVOmTHFVBgAAAIAc52z53Lvvvqv8/HxX3x4AAAAAJDmcKSIQAQAAAEgHzrvPAQAAAIBLTrvPXa9oNKpoNNp3PxKJOKwGAAAAQCbK6FC0du1arVmzxnUZwIgpLCzU66+/7roMAACAnJJyKHr55Zf10UcfpfyNNm7cqCVLlqT854ZSW1urVatW9d2PRCKqrKwc0e8BAAAAILulHIpaWlpUX1+f8je6cOFCyn/mWgKBgAKBQN99z/MksYwOAAAAyHWJTJDICENJORRt2rRJmzZtSr2qMdDR0SFJzBYBAAAAkGQZIRQKDfmcjN5TdKWKigo1NTUpGAzK5/O5LgcAAACAI57nqaOjQxUVFdd8blaFory8PE2fPt11GQAAAADSwLVmiBI4pwgAAABATnM2U9Td3a3z589LksLhcN+106dPS5KKi4uHnewAAAAA4PfyecNpxzAKvvjiCz388MODPv7000/rgw8+GLuCAAAAAOQkZ6EIAAAAANIBe4oAAAAA5LS0CUU7duxQVVWVfD6fGhsbXZcDAAAAIEc4D0VdXV168cUXtXTpUjU0NLguBwAAAECOcRqKjh07ppqaGm3ZskXbtm1zWQoAAACAHOU0FB08eFDLly/XgQMH9Mgjj7gsBQAAAECOctp9rre3V/n5+clifD5J0vHjxzVr1qyUXy8ej6ulpUXBYLDvtQAAAADkHs/z1NHRoYqKCuXlDT0X5OzwVkn9AtFIaGlpUWVl5Yi+JgAAAIDM1dTUpOnTpw/5HKehaKQFg0FJ9oOXlpY6rgYAAACAK5FIRJWVlX0ZYSgZHYqi0aii0Wjf/Y6ODklSaWkpoSjH9fRIbW3Sr78mRyQy+PPz8qSJE6Ubb5QmTbIRCtl1AAAAZK7hbKvJ6FC0du1arVmzxnUZcMzzpDNnpCNHpKYmC0BtbXb9ehQUJEPSrFnS3LnShAkjUTF+j+7ubr355puSpFdeeUWFhYWOKwIAANkio0NRbW2tVq1a1Xc/MUWG7NfdLf38swWhI0cGngUKBCzQJILNhAmDz/z09Ejnzydnlc6ds2tnztg4eNCeN3myhaN586TKSmaSAAAAskFGh6JAIKBAIOC6DIyRaFQ6cEA6dEhqbJR6e5OP+f3S7Nk2Jk+2EDR+vPR7mxDG41J7u4Wj06eTs1Bnz9r4+mupqEi6+WbpttssKI1w3xAAAACMkYwORch+nie1tEh791og6u5OPjZxYnLWZtYsW+42UvLypBtusDF3rvTgg1JXl3TsmNTQIB09KnV2Wk0HDkjBoHTXXdKCBSyxGy0+n08VFRV9twEAAEaK03OKrnS95xRFIhGFQiGFw2EaLWS4aFT68UcLQ6dOJa9PmmTho6rKlsW5em8cj0u//CIdPizt3y9dvGjXfT5pzhxp4UILa8weAQAAuJFKNmCmCGnl3Dlpxw4LRIlZoYICaf58CxozZrgLQpfLy7M9RZWV0h/+INXXS3v22D6no0dtjB8v3X23tHixVFLiumIAAAAMhlCEtPDLL7ZP56efkl3jJk2yUHHHHekdKvLzLbTNn2/NGvbtk777TrpwQfriC/u5FiyQ7ruPpXUAAADpyHkoam1tVe/lO+Z/u1ZUVCRJKi8vd1EWxoDn2czKV19Jx48nr8+bJ91/vzRzZnrMCqXihhukP/5RevhhC3hff23L/3bulHbvtqYMDzwgTZniutLME4vFtH79eknSypUr5ff7HVcEAACyhfNQtGjRIp04caLftcWLF/fdTqMtTxgh8bgFhq++Su4XysuTbr/dAsPkyW7rGwn5+RaAbr3VAt9XX1kA/OEHG/PmSUuW2HJADI/neWpvb++7DQAAMFKch6LGxkbXJWCMxOPWqe3LL+0sIMlaaS9caEvLQiG39Y0Gny/ZKrylxWaODh2yDnYNDdJNN0nLlmXmrBgAAEC2cB6KkP16e212ZPt223MjScXF0j335FYTgooK6T/9p2Qzif37bRbp+HELRUuXWngiHAEAAIwtQhFGTW+vvfHfvt0OQpUsAN13n4WhXD1398YbpX//7y0EffWVNWY4cUL6H/9Dmj7dZo5uvplwBAAAMFYIRRhxsZi90d+xQwqH7dq4cbZf6O67pcJCt/Wli1BI+nf/zg6G3bHDWno3N0ubN9us0oMPStXVhCMAAIDRRijCiIlGrcPaN98kDzMNBi0MLVxo+4dwtdJS6dFHrfHCjh32d9jSIv3DP0hlZRaObrvNmlEAAABg5BGKcN06O63l9M6d0qVLdm3CBAtDd91lh6/i2saPl/7qr+zvLfH32doqbd0q/du/2fWamtz9+/T5fCorK+u7DQAAMFJ8Xhb1to1EIgqFQgqHwyotLXVdTtYLh6Vvv5X27pW6u+3apEnJmY38fLf1ZbpLl2zW6Ntv+8+83XefHQb721FeAAAAGEAq2YBQhJR4nu17+fZbO2soHrfrU6daGLrlFvbAjLRYzILnjh1SJGLXCgttFu6ee+zAWAAAAPRHKCIUjbjeXungQQtDLS3J6zfdJN1/P93SxkJPj7U2/+YbW1Yn2d/5vHnSvfdKs2bxbwAAAJBAKCIUjZiODum772wZV0eHXSsokO64w2YppkxxW18u8jzp558toB45krw+ZYr9m9x2W3Z2+IvFYnrvvfckSc8++6z8dO4AAABDSCUb5OiWbQylp0eqr7czho4etTfhku1nWbTIOsmNG+e0xJzm80lz5tj49Vdp1y77tzpzRvroI+njj6X5860pw8yZ2TN75HmeWn+bIsuiz3IAAEAaIBRBkgWflhZ7c/3jj8kucpI0Y4aFofnzaZ6QbiZNkh57TPrDH+xsqL17pXPn7N9x/37rAlhTI915pzRxottaAQAA0hWhKId5nnT6tM0KHTyY3Kci2dk5iTfTN97orEQMU1GR7e267z5rhLF/v3TggNTeLn3xhY2ZMy3YzptHQAIAALgcoSjH9PRIjY0WhOrrk93MJDtc9ZZbLAzNmsVhoZnI55MqK208+qh0+LAFpJ9/lk6csLFtm+0/qqqyUVGRPUvsAAAAfo+0CEUff/yx1q1bp7179yoajaqqqkrPPPOMnn/+eeXxzvy6eJ4tpzpxwt4YHz0qRaPJx/1+6xxXVSVVV3P2TTbx+6Xbb7cRidhsYH29dPKk7T86c0b68kvbKzZvnnUSnDHDZgkBAAByifNQVFdXp9WrV+vxxx/Xp59+qmAwqPfff18vvPCCPvnkE23dulUFBc7LzBi9vbYk7sQJe/N78qTU2dn/OYk3wVVV0uzZ1k0O2a201JbW3Xef/T4cPWoB6cgR6yq4d68NyZbWzZiRHJMmMZMEAACym9O3w9u3b9fq1as1f/58bd26ta/Fbl1dndra2rRhwwbV1dXp1VdfdVlm2urqks6etU/8z561ceqUHfZ5uYICafp021Mybx7LpXJdSYm1VL/jjuRyyqNHLUifPi21tdn4/vvk88vLbcnd5Mn2tazMZqLGks/n04QJE/puAwAAjBSn5xQtX75cn3/+ud555x0999xz/R6rr69XdXW1SktLdfr0aRUXF1/z9bLtnCLPsy5w7e1SOGxf29utDfPZs/33A12uuLj/J/0VFXSNw/BEo9aoITHT2NxswelKPp90ww0Wkm68UQqFrNPdhAl2OxvPSQIAAJklIw5vbW1tVXl5ueLxuBoaGjR37tyrnlNZWanm5mZt2bJFTz755DVfM1NCUTxuS5YuXJAuXrTlTBcvJkdnpwWe9vb++38GMmGCvTFNfIKf+BSfD9IxEhLLMRN7kBIzkhcvDv3nSkrsdzMYtDOtSkrs6+W3i4ulQMACFKEdAACMtIw4vHXPnj2Kx+Py+/2aM2fOgM+prq5Wc3Ozdu/ePaxQlE6iUZvRaW9PLkdK3A6H7c3mcI0bl/wUfsIE2/ORWMoUCIxK+YAkCyvTptm43IULyYCU+N0Oh+12NGrB/sq9bEPx+5MBKRCwJZ+DBfu8PAtWxcX29fIRCPCBAAAA6WDSJPv/6kzhLBQdO3ZMklRWVjZoh7mpU6f2e+6VotGoopdNpUQGW0/mwIED0j//8+CP5+dL48f3/+T88k/Tx4+38BMKjf3eDeBaxo+3MXv21Y8llny2t1t4unwm9PLbly4ll+bFYlfvhbtSb29M+/f/vSSppmaF8vP5HwYAAOnqqaekARaCpS1noSgRYIbaK1RSUiJJCofDAz6+du1arVmzZuSLGwETJ9rSoYkTk7M7l98OBjkHCNmpqMgaM5SXX/u5vb02s9TdbV8TY6B9TN3dns6fb1E8Lj36qKdYLDkj1dVlXy9dGvmfBwAApC7TPtTP6GbMtbW1WrVqVd/9SCSiyspKhxUlzZ4tvfii6yqA9Jafn1z6di3d3bZfTpIWLaKZAwAAGDnOQlFis1NXV9egz+n8bVNCKBQa8PFAIKAAm2oAAAAAXAdnC7gSzRVaW1sVj8cHfM6pU6f6PRcAAAAARpqzULRw4ULl5eUpFosN2kjh8OHDkqRFixaNZWkAAAAAcoizUDR58mQtW7ZMkvTZZ59d9Xh9fb2am5sVDAb16KOPjnV5AAAAAHKE0/5nr732miTp7bffVs8V7abWrVsnSXrppZf6utAByG0lJSX89wAAAIw4n+d5nssC3nzzTb366qt6/PHHtWbNGo0fP17//b//d9XV1emxxx7T//pf/0sFBcPrB5HKqbUAAAAAslcq2cD5STmvvPKK/vznP6urq0t/+MMfdNddd+lf//Vf9d/+23/TRx99NOxABAAAAAC/h/OZopHETBEAAAAAKbVswDQMgIwQi8W0efNmSdJTTz0lf6YdlQ0AANIWoQhARvA8T42NjX23AQAARorzPUUAAAAA4BKhCAAAAEBOIxQBAAAAyGmEIgAAAAA5jVAEAAAAIKfRfQ5AxqANNwAAGA0c3goAAAAg66SSDVg+BwAAACCnEYoAAAAA5DTnoai3t1d1dXUqKirSQw895LocAGmqp6dHmzdv1ubNm9XT0+O6HAAAkEWcNlo4dOiQVqxYoV27drksA0AGiMfjOnLkSN9tAACAkeJspmjjxo1asGCBpk2bprfeestVGQAAAABynLNQtHPnTn344YfaunWrpkyZ4qoMAAAAADnO2fK5d999V/n5+a6+PQAAAABIcjhTRCACAAAAkA6cd58DAAAAAJecdp+7XtFoVNFotO9+JBJxWA0AAACATJTRoWjt2rVas2aN6zIAjIHCwkK9/vrrrssAAABZKOVQ9PLLL+ujjz5K+Rtt3LhRS5YsSfnPDaW2tlarVq3qux+JRFRZWTmi3wMAAABAdks5FLW0tKi+vj7lb3ThwoWU/8y1BAIBBQKBvvue50liGR0AAACQ6xKZIJERhpJyKNq0aZM2bdqUelVjoKOjQ5KYLQIAAAAgyTJCKBQa8jkZvafoShUVFWpqalIwGJTP53NdDgAAAABHPM9TR0eHKioqrvncrApFeXl5mj59uusyAAAAAKSBa80QJXBOEQAAAICc5mymqLu7W+fPn5ckhcPhvmunT5+WJBUXFw872QEAAADA7+XzhtOOYRR88cUXevjhhwd9/Omnn9YHH3wwdgUBAAAAyEnOQhEAAAAApAP2FAEAAADIaWkTinbs2KGqqir5fD41Nja6LgcAAABAjnAeirq6uvTiiy9q6dKlamhocF0OAAAAgBzjNBQdO3ZMNTU12rJli7Zt2+ayFAAAAAA5ymkoOnjwoJYvX64DBw7okUcecVkKAAAAgBzltPtcb2+v8vPzk8X4fJKk48ePa9asWSm/XjweV0tLi4LBYN9rAQAAAMg9nuepo6NDFRUVyssbei7I2eGtkvoFopHQ0tKiysrKEX1NAAAAAJmrqalJ06dPH/I5TkPRSAsGg5LsBy8tLXVcDQAAAABXIpGIKisr+zLCUDI6FEWjUUWj0b77HR0dkqTS0lJCEXAN8bjU1iadO2dfh1pIGwpJkyZJN9wgjfAELwAAwKgazraajA5Fa9eu1Zo1a1yXAaS99nbpxAnp11+T4/x5qbc3tdfJy5MmTrSAdOON9nXGDLvNNj53uru79eabb0qSXnnlFRUWFjquCACAzJLRoai2tlarVq3qu5+YIgNyXW+v1NQkHTkiNTRIra0DP6+gwILNxIl2e7DXam+3INXdbTNL5871f87EidK8edLcudKsWYO/FgAAQDrK6LcugUBAgUDAdRlAWujqsgDU0CAdOyZdupR8LC9PmjZNKi9PzvBMmmTL4oY7w+N5UkeHhaNz5+zrmTMWvtrapJ07bfj90uzZFpCqq6Xx40fn5wUAABgpGR2KgFzneRZK9u6VDh6UenqSj5WUWDCZO1eaM0cqLr6+7+XzSaWlNmbPTl7v7pZ+/jk5K9XRIdXX2/jzn6WqKmnhQquBJXYAACAdEYqADNTVJX3/vYWhy5fGTZ5sszPz5kkVFTZDNNoKC+17VldbSDtzxsJRfb30yy/STz/ZmDDBwlFNjTSMJjBIgc/nU0VFRd9tAACQGqeHt17peg9vjUQiCoVCCofDdJ9DVmpulnbv7j8r5PdLt90m3X23BaF0ek989qwFt++/Ty7ny8uz2aNFi6SbbkqvegEAQPZIJRswUwSkOc+Tjh6VvvrKOsgllJfbzMvtt0tFRe7qG8rkydKf/iT98Y/SoUMWkE6eTM4eTZ0qPfCANH/+2MxqAQAADMR5KGptbVXvFX2BW1tbVfTbu7zy8nIXZQHOxePSgQPS11/bkjTJzgi6/XabZUm3WaGh+P3SnXfaOHtW2rNH+u476dQp6Z/+yc4/uv9+e9zvd10tAADINc6Xz82aNUsnLv/4+wqplMfyOWSDWEzat0/65htrhS3Zvp2775buvdcaHWSDzk5p1y4bnZ12bdw4+xkXLUrf2a90FIvFtH79eknSypUr5SdZAgCQWcvnGhsbXZcApIVo1PYLffONdPGiXUuEhLvvvv7ucemmpER66CGbIfruO2nHDikclj77zJYK3nOP/ewlJa4rTX+e56n9twSdRttEAQDIGM5DEZDrLl2y2ZJvvrGucpIdhnr//dapLds/9C8stAB0993WQOKrr2yJ3ZdfSt9+a7NG999vAREAAGA0EIoAR7q67LDTb79Ndma78UZp6VLbN5RrjQfy86U77rCfvb5e+stfbM/R119baLz7bgtHtPMGAAAjjVAEjLHOTgtCO3fakjlJKiuzMHTrrbkXhq7k89mZR1VVdiDsX/5i5x19840tL1y40MJRKOS6UgAAkC0IRcAYiURs38zevdZMQZKmTLEwNH9+5nSSGys+nx1CO3eudOyYhaOmJguTe/ZYp7oHHrDZNQAAgOtBKAJG2fnztgRs/34p0X2+okJ68EGbESEMDc3nk26+WZozRzp+XNq+3b7u22cNGm691f4up0xxXSkAAMhUhCJglJw9a2/gDxywA1glaeZMmxmaPZswlCqfz/7eZs+2GaPt26WGBvv7PXDAlts9+KA0fbrrSseez+dTWVlZ320AAJAa5+cUjSTOKYJrnicdPWp7ho4dS16fO9fesM+Y4a62bHT6tHWrO3gwGTwrK62V9y23sD8LAIBclko2IBQBI6C7W/r+ewtD587ZNZ/P3pg/+KA0darb+rLduXMWjn74IblEMRSSFi+WFizIvjOeAADAtRGKCEUYI+3t1i56375kW+1AwN6IL15s5w1h7Fy4YE0Ydu9OHoDr91tThnvusS5/AAAgNxCKCEUYRbGYnaOzf78tkUv8L+jGG+2N9513WjCCOz09ts/o229tiV3CzJl2IO78+dn1bxSLxfTee+9Jkp599ln5s/3EXwAAhiGVbECjBWAYPM/Oytm/395sJ2aFJNv4f++9tm+IPe7poaDAws+dd0onTlg4qq+32ydOSH/+swWjmhpp1qzM/3fzPE+tra19twEAQGoIRcAQzp2TfvrJwtCvvyavh0LJN9033OCqOlyLz2ehZ9YsOyfqhx+S/5bff28j8W85f740eXLmByQAAJA6QhFwmXjcZoTq62389uG7JNubkk2zC7mmtFRassQOfL181i8ctoNh//IXacIEa+1dXW2dAvPzXVcNAADGQlqEoo8//ljr1q3T3r17FY1GVVVVpWeeeUbPP/+88uipi1F26ZLU2GghqKEhuUFfspbOs2ZJt9+efftQcpXPZ2cZTZ8u/fVf27/7jz/a/rD2dmnnThtFRbYksqpKuukmadw415UDAIDR4jwU1dXVafXq1Xr88cf16aefKhgM6v3339cLL7ygTz75RFu3blVBgfMykUU6OmxfycmTNs6cSTZLkPq/Gb75ZruP7OT3S7fdZqO7W/r55/7h+McfbUjSpEnWqGHGDBsTJjBbCABAtnCaNrZv367Vq1dr/vz52rp1a1/HpLq6OrW1tWnDhg2qq6vTq6++6rJMZLCLF6WzZ220tFgIamu7+nk33GBBiGVTuauw0P79q6v7L6NsaLDfn19/tbF3rz2/tNR+VyoqpClTbD/S+PEEJQAAMpHTltzLly/X559/rnfeeUfPPfdcv8fq6+tVXV2t0tJSnT59WsXDOH2Rlty5KR6382na260xwtmzNvtz9qxdv5LPJ5WX2xvamTOlykopGBzzspFBOjulpiYL1SdOWMCOx69+XnFxMiBNnmyzSxMmWIAazZXAsVhM69evlyStXLmSltwAAChDzilqbW1VeXm54vG4GhoaNHfu3KueU1lZqebmZm3ZskVPPvnkNV+TUJRdPE+KRu0N6cWLNjo7rYtYOGwhqL3dbg/0BjVh4kR7ozpligWgykr2BuH6xGI2k3TypJ2DdPasBfLB/mual2fBe8KE5AgGbZ/SuHFSSYl9LSpipgkAgJGSEecU7dmzR/F4XH6/X3PmzBnwOdXV1Wpubtbu3buHFYqQWXp7k8Gmra3/144OC0G9vcN7rbw8a608cWLyU/opU6SyMlsWBYwkvz/Z6juhp8e6FV4+U3n+vIX23l77Gg7bTNNg8vIsIJWUWHBPjMLC5G2/f/Dg5PPZbFXiNRK3i4sJWwAADMVZKDp27JgkqaysbNAOc1OnTu333CtFo1FFo9G++5FIZISr/P1++kn65392XUV66+oa/JP1yxUW9v80ffz45KftoVDyU3caFcKlggJp6lQbl/O85PLOxMzm5cE/MRMajSaXgg607PN6+HzMQgEAxtZ//I92wH2mcBaKEgFmqL1CJSUlkqRwODzg42vXrtWaNWtGvrgR0Ntrb3YwNL/fQs3EiTYSt0tLk0GI7RHIZD6fhfZg0JZuDqanJxmQurosJHV329fLRyw20J+N6fPP/17xuHTvvSsUi/nV2WmvF41aMOvqGr2fEQCAKw13tU+6yOhe17W1tVq1alXf/Ugkosqh3nWMoZtvlp5/3nUV6S0x88On14DNNJWW2khVd7enAwdaJEn/+T97/ZaM9vZaIBruzCwAACMhFHJdQWqchaLEZqeuIT6+7PxtqiU0yN9qIBBQIE13zBcVcb4NAPfy823J6fjxrisBACB9OduFkWiu0NraqvggrcNOnTrV77kAAAAAMNKchaKFCxcqLy9PsVhs0EYKhw8fliQtWrRoLEsDAAAAkEOchaLJkydr2bJlkqTPPvvsqsfr6+vV3NysYDCoRx99dKzLAwAAAJAjnDYxfu211yRJb7/9tnp6evo9tm7dOknSSy+91NeFDgAAAABGmtNQtGzZMr3xxhs6ePCgnnjiCe3bt08NDQ1avXq1NmzYoMcee0y1tbUuSwSAjFBSUsIHSAAA/E4+z3PfpHXbtm1at26d9uzZo1gspnnz5mnFihVauXKl8vPzh/06kUhEoVBI4XC4r7sdAAAAgNyTSjZIi1A0UghFAAAAAKTUsoHT5XMAAAAA4Jqzw1sBACMjFotp8+bNkqSnnnpKfr/fcUUAAGQWQhEAZDjP89TY2Nh3GwAApIblcwAAAAByGqEIAAAAQE4jFAEAAADIaYQiAAAAADmNUAQAAAAgp9F9DgCyAG24AQD4/XxeFvVvTeXUWgAAAADZK5VswPI5AAAAADmNUAQAAAAgpzkPRb29vaqrq1NRUZEeeugh1+UAQMbp6enR5s2btXnzZvX09LguBwCAjOO00cKhQ4e0YsUK7dq1y2UZAJDR4vG4jhw50ncbAACkxtlM0caNG7VgwQJNmzZNb731lqsyAAAAAOQ4Z6Fo586d+vDDD7V161ZNmTLFVRkAAAAAcpyz5XPvvvuu8vPzXX17AAAAAJDkcKaIQAQAAAAgHTjvPgcAAAAALjntPne9otGootFo3/1IJOKwGgAAAACZKKND0dq1a7VmzRrXZQCAU4WFhXr99dddlwEAQMZKORS9/PLL+uijj1L+Rhs3btSSJUtS/nNDqa2t1apVq/ruRyIRVVZWjuj3AAAAAJDdUg5FLS0tqq+vT/kbXbhwIeU/cy2BQECBQKDvvud5klhGBwAAAOS6RCZIZIShpByKNm3apE2bNqVe1Rjo6OiQJGaLAAAAAEiyjBAKhYZ8TkbvKbpSRUWFmpqaFAwG5fP5XJcDAAAAwBHP89TR0aGKioprPjerQlFeXp6mT5/uugwAAAAAaeBaM0QJnFMEAAAAIKc5mynq7u7W+fPnJUnhcLjv2unTpyVJxcXFw052AAAAAPB7+bzhtGMYBV988YUefvjhQR9/+umn9cEHH4xdQQAAAABykrNQBAAAAADpgD1FAAAAAHJa2oSiHTt2qKqqSj6fT42Nja7LAQAAAJAjnIeirq4uvfjii1q6dKkaGhpclwMAAAAgxzgNRceOHVNNTY22bNmibdu2uSwFAAAAQI5yGooOHjyo5cuX68CBA3rkkUdclgIAAAAgRzntPtfb26v8/PxkMT6fJOn48eOaNWtWyq8Xj8fV0tKiYDDY91oAAAAAco/neero6FBFRYXy8oaeC3J2eKukfoFoJLS0tKiysnJEXxMAAABA5mpqatL06dOHfI7TUDTSgsGgJPvBS0tLHVcDINN5ntTRIfX2Dv6cceOkwsKxqwkAAAxPJBJRZWVlX0YYSkaHomg0qmg02ne/o6NDklRaWkooAjBs3d3SuXPSr78mvyZux2LX/vOhkDRpUnLceKN9DQYlVvICAODWcLbVZHQoWrt2rdasWeO6DAAZxvOkM2ekhgbpyBGpudmuDSQvTyoY5L+UnmehKRy2cexY/8eDQWnuXBuzZ0uBwMj+HBie7u5uvfnmm5KkV155RYVM7QEArpDRoai2tlarVq3qu5+YIgOAK3V3Sz//bCHoyBEpEun/eEnJwLM9EydaMBpMZ2f/maXE7bY2W3q3b5+N/Hxp5kxp3jwLSTfeOLo/LwAAGL6MDkWBQEABPnoFMAjPkxobpb17pcOHpZ6e5GN+v83eJGZyQqHf9z1KSqQZM2xcrqdHOnEiORt1/ryFsp9/lj7+WJo8WVqwQLrzTqm4+Hf/iAAAYARkdCgCgIFcvCjt329h6Pz55PWJEy0AzZsnzZo1+LK4kVBQIM2ZY+PRR20W6cgRC0knTkhnz1o4+v/+P2n+fGnhQgtW7EEaeT6fTxUVFX23AQC4ktNziq50vecURSIRhUIhhcNhGi0AOcbzpOPHk7NCiY5xgYB0xx02K1Nenh6h49Il6ccfrdbTp5PXy8osHN1xh81AAQCA3y+VbMBMEYCM1ttrAePrr6XW1uT1adOku++Wbr01/VpmFxVJixZZfS0tFo4OHLD6P/5Y+uwz6a67pPvus9ktAAAwughFADJSd7c1MNixI9k0ITErtHChzQqlO5/Pwtu0adJf/3Vy9ujUKWnXLmnPHgt1DzyQGT8PAACZynkoam1tVe8VJyO2traqqKhIklTOOwEAl7l40QLDrl1SV5ddGz/eZlUWLrRZmEwUCNjM0cKF1hzi66+lo0ctKP34o3TzzdKSJdbBLh2WAGaSWCym9evXS5JWrlwpv9/vuCIAQLpxHooWLVqkEydO9Lu2ePHivttptOUJgEORiAWFffuSB6reeKN0//3WwW00myaMJZ9PuukmG6dO2c988KAFpKNHpenTpaVLrWEE4Wh4PM9Te3t7320AAK7k/G1EY2Oj6xIApLFwWPrqKwtDiUnligqbNamuHvoMoUw3dar0H/+j9Ic/SN98I333nR00+3/9X/bYsmVSVRXhCACA6+U8FAHAQNraLAzt358MQzNnWhC46abcCgI33CD9u39nP/s330i7d9ss0v/9f0tTptj1W27Jrb8TAABGEqEIQFo5d87C0PffS/G4XZs925aM/Y5O/Vll/HjpkUes8cI339i+qjNnpH/8R2vnvXSpNWbI5tkzAABGA6EIQFo4c0bavt32zyS2fdx8s73RnzHDbW3ppqREWr7c9lN9+620c6e1896yRfq3f7OlhXfeKeXnu64UAIDMQCgC4FRzs4Wh+vrktXnzLAxNn+6urkxQXCw9/LB13tu50wLS+fPSRx9JX3xhM0oLFkg0WwMAYGiEIgBjzvOk48ctDB0/btd8Pmn+fOnBBzmTJ1VFRbav6L777Gyjb76xbn3btklffinde68dFpup7cqvl8/nU1lZWd9tAACu5POyqD9pJBJRKBRSOBxWaWmp63IAXKG3Vzp0yGY0fvnFruXl2VKvJUusxTauX0+PNaj46ivpt07UKiqyM5AWL5ZCIZfVAQAwNlLJBoQiAKOus1Pau9e6pkUidq2gwJZ2PfAAb9JHSzxuB79+9ZXtOZIshFZX2+xRZSUd6wAA2YtQRCgC0sLZs7bX5fvvbfZCsg5qixZJd98tjRvntr5c4XlSQ4P9W/z8c/J6RYWFo1tvpSkDACD7EIoIRYAzsZh0+LAdNHr5G/CpU5NvwAvYzejMmTMWjn74IRlUg0HprrtsGWM2LmGMxWJ67733JEnPPvus/HSeAICckEo24K0JgOvmedZFbv9+6cABKRq16z6fHSrKUq30MWWK9Dd/Yy29E0saOzqsIcOXX1r785oaa3qRLY0ZPM9T62/rB7Poc0AAwAgiFAH43cJhm3HYv98OXU2YMMHeWNfU2G2kn3HjrO35Aw/YzN7+/dLRo9LJkza2bbNAW1Njh+ZyICwAIJsRigAMm+fZPqH6ehuJDnKSVFhosws1NdLMmcwKZYr8fFvSeOutNmOUCLmtrXb7hx8sQM2bJ1VVSbNn2781AADZJC1C0ccff6x169Zp7969ikajqqqq0jPPPKPnn39eeXw8CTjV22szB4cPWxBKtHhOmDnT9qPMn8+b5UwXDNrM0f33Sy0tyeWQFy/aHrHvvrP9YLNnW0CaN8/+DAAAmc55KKqrq9Pq1av1+OOP69NPP1UwGNT777+vF154QZ988om2bt2qAnZlA2Omp8feEJ88KZ04YV8Te4Qke1M8Z07yTfH48e5qxejw+aRp02w8+qj9DtTXWzBub7dOdg0N9tzyctuHNHOmfSUkAQAykdO0sX37dq1evVrz58/X1q1b+zoC1dXVqa2tTRs2bFBdXZ1effVVl2UCWcvzpAsXpNOnk3tJfvkl2ZUsgeVTuSs/X7rpJht//de2rC4xa/jLL/a7c/q0tGuXPX/ixGRAmjZNmjSJdt8AgPTntCX38uXL9fnnn+udd97Rc8891++x+vp6VVdXq7S0VKdPn1ZxcfE1X4+W3MDgolHbD5QYZ87Y187Oq587blzyje2MGTYbwEpWXOnCheRs4smTFo6u/H+U/Hxr8z1lijR5so0pU+zA3rHadxaLxbR+/XpJ0sqVK2nJDQA5IiPOKWptbVV5ebni8bgaGho0d+7cq55TWVmp5uZmbdmyRU8++eQ1X5NQhFzleVJXl71JbW+3EQ4nb7e3276Qgfh89qZ1+vRkELrhBholIHXRqNTUlAxJp071X3p5Ob/fgtGECTYuv11aasGcldMAgOuREecU7dmzR/F4XH6/X3PmzBnwOdXV1Wpubtbu3buHFYqAbJMIO21tFmza2qRIxGZ3Ll600dlpIx6/9usFg8lP6hNfJ02yN6jA9QoEpJtvtiHZ728kkpyVTHz99Vc75PfXX20M9XrjxkklJfY1cTsQsFFYmLyduD9UmC8qkoqLWc4HALias1B07NgxSVJZWdmgHeamTp3a77lXikajil72MWQkEhnhKn+/X36R9uxxXQUy1eVBaLBP2gdSVJT8tH2gT+CHsQoVGDE+n/3+hUK2Jy2htzc5k3nljGZ7u7UGj8ftdz8alc6fH9m6ioosXJWU2P8mSkpYHgoAI+3ee+3D10zhLBQlAsxQe4VKSkokSeFweMDH165dqzVr1ox8cSOgrc3a1wIjIRi0UDNxor3BvPxT88TXkhKWGyEz5OfbEs0bbhj4cc+TLl26ekY08bW7OxmYLh/d3QO/Xm9vTLt3/716eqSamhW6dMmvS5dGPmwBAJLmzycUjZna2lqtWrWq734kElFlZaXDipKmTJGWL3ddBTJVIGABKDHDw/I25BKfz2Zwiottv9v16u729OabLfI86f/4Pzz19NhsbGLpaVfX8JafAgCGbyT++z2WnIWixGanrq6uQZ/T+VtbrFAoNODjgUBAgUBg5IsbAWVlNgAA6cHns5lVWsoDAK7kbBV1orlCa2ur4oN8RHfq1Kl+zwUAAACAkeYsFC1cuFB5eXmKxWKDNlI4fPiwJGnRokVjWRoAAACAHOIsFE2ePFnLli2TJH322WdXPV5fX6/m5mYFg0E9+uijY10eAAAAgBzhtAnpa6+9Jkl6++231dPT0++xdevWSZJeeumlvi50AAAAADDSnIaiZcuW6Y033tDBgwf1xBNPaN++fWpoaNDq1au1YcMGPfbYY6qtrXVZIgAgC5SUlPABGwBgUD7P8zzXRWzbtk3r1q3Tnj17FIvFNG/ePK1YsUIrV65UfgpHj0ciEYVCIYXD4b7udgAAAAByTyrZIC1C0UghFAEAAACQUssGTpfPAQAAAIBrzg5vBQBgLMRiMW3evFmS9NRTT8nv9zuuCACQbghFAICs5nmeGhsb+24DAHAlls8BAAAAyGmEIgAAAAA5jVAEAAAAIKcRigAAAADkNEIRAAAAgJxG9zkAQNajDTcAYCg+L4v6k6Zyai0AAACA7JVKNmD5HAAAAICcRigCAAAAkNOch6Le3l7V1dWpqKhIDz30kOtyAABZpqenR5s3b9bmzZvV09PjuhwAQBpy2mjh0KFDWrFihXbt2uWyDABAFovH4zpy5EjfbQAAruRspmjjxo1asGCBpk2bprfeestVGQAAAABynLNQtHPnTn344YfaunWrpkyZ4qoMAAAAADnO2fK5d999V/n5+a6+PQAAAABIcjhTRCACAAAAkA6cd58DAAAAAJecdp+7XtFoVNFotO9+JBJxWA0AAACATJTRoWjt2rVas2aN6zIAAGmssLBQr7/+uusyAABpLOVQ9PLLL+ujjz5K+Rtt3LhRS5YsSfnPDaW2tlarVq3qux+JRFRZWTmi3wMAAABAdks5FLW0tKi+vj7lb3ThwoWU/8y1BAIBBQKBvvue50liGR0AAACQ6xKZIJERhpJyKNq0aZM2bdqUelVjoKOjQ5KYLQIAAAAgyTJCKBQa8jkZvafoShUVFWpqalIwGJTP53NdDgAAAABHPM9TR0eHKioqrvncrApFeXl5mj59uusyAAAAAKSBa80QJXBOEQAAAICc5mymqLu7W+fPn5ckhcPhvmunT5+WJBUXFw872QEAAADA7+XzhtOOYRR88cUXevjhhwd9/Omnn9YHH3wwdgUBAAAAyEnOQhEAAAAApAP2FAEAAADIaWkTinbs2KGqqir5fD41Nja6LgcAAABAjnAeirq6uvTiiy9q6dKlamhocF0OAAAAgBzjNBQdO3ZMNTU12rJli7Zt2+ayFAAAAAA5ymkoOnjwoJYvX64DBw7okUcecVkKAAAAgBzltPtcb2+v8vPzk8X4fJKk48ePa9asWSm/XjweV0tLi4LBYN9rAQAAAMg9nuepo6NDFRUVyssbei7I2eGtkvoFopHQ0tKiysrKEX1NAAAAAJmrqalJ06dPH/I5TkPRSAsGg5LsBy8tLXVcDQAAAABXIpGIKisr+zLCUDI6FEWjUUWj0b77HR0dkqTS0lJCEQCMoJ4e6fx56ddfbZw7Z19jscH/THGxNGlS/xEKSddYwQAAwIgazraajA5Fa9eu1Zo1a1yXAQBZp71damiQjh2TWlultjbp9+xAPXGi//2CAumGG6TycmnuXOnmmy08wY3u7m69+eabkqRXXnlFhYWFjisCADcyOhTV1tZq1apVffcTU2QAgNT09kpNTdKRIxaGWluvfk5RkXTjjclZnxtvHDzQeJ504UJyRikxu9TTI509a+OHHySfT6qstIA0d640ZYpdAwBgLGV0KAoEAgoEAq7LAICMFI/bTND330tHj0qXLiUfy8tLhpXp0y0EjRt3fYElHpfCYQtIJ05Y+Dp7Vjp50sZnn0mlpVJVlbRggTR16vX/jAAADEdGhyIAQOoiEem776R9+yykJJSU2HK2efOkOXNGfllbXp40caKNuXOlP/7RlukdOWLj+HGrbfduGxUV0sKF0m23SXz+NTp8Pp8qKir6bgNArnJ6TtGVrvecokgkolAopHA4TKMFALhMYlZo716boYnH7XpxsXTnndKtt0rTprltghCLSY2Ntqzu0CFb0idJhYXS7bdbQPrt/TsAANeUSjZgpggAslg0akFo587+s0IzZ1rImD/fmh+kA78/ubfoT3+yZX1799pyu717bUydKt1/v4U4utgBAEYKM0UAkIUuXLAgtHt3cq9QcbFUU2P7dcrKnJY3bJ5n+4/27u0/ezRxonTffdJdd1mYAgDgShk1U9Ta2qrexP/LXXatqKhIklReXu6iLADISOfPSzt2SPv3W6c3yZokPPCALUFLl1mh4fL5pFmzbPzpT9KePRb22tqkP/9Z+uIL6d57pUWLaO39e8RiMa1fv16StHLlSvlJmABylPP/e1y0aJFOXHGQxeLFi/tup9FEFgCkrdOnpe3bbTYl8Z/N6dOlJUusm1s27KEvKZGWLrUZou++s/DX3i59/rn01Ve2HPD++6VhHFyO33iep/b29r7bAJCrnIeixsZG1yUAQMZqaZH+8hepvj55be5cC0MzZmRHGLqS3y8tXizdfbd08KD09dcWCr/5xpYL3nWX/fyhkOtKAQCZwnkoAgCkrrnZwtCRI3bf57PmAw8+aAeg5oK8PFsSeNtt1lnvyy/tvKPdu63deE2N/X1MmOC6UgBAuiMUAUAGOXnSwtCxY3bf55PuuMPe/E+a5LY2V3w+O19pzhxryvCXv9iZR3v32jK7O++0v58bbnBdKQAgXRGKACDNeZ7088+2Zyix4jgvjzf7V7q8KcOJEzZzdOyYBaPvv7cZpSVLpMmTXVcKAEg3hCIASFOeZ3uFtm+XfvnFruXn27KwJUusLTUGNnOm9F/+S/9lhj/8YOOWWyxMchAsACCBUAQAaSYelw4csI5qZ8/aNb/fzhe6/34aCKRi+nTpqaesIcVXX0k//ZQcc+ZYOJo5MzsbUgyHz+dT2W+HVvly9S8BAJRmh7deLw5vBZDJurttJuPrr+0cHkkKBKzT2r33SuPGua0vG7S2Wjj68UcLn5JUWWnnOM2bZ8sSAQDZIZVsQCgCAMfCYeuYtnev1NVl10pK7DyeRYuk386yxghqa7Nzjr77LnnI7cSJ0j33WEvvQMBtfQCA60coIhQBSHOeZ/tdvv3WlnIlZi1uuCH5xryw0G2NuaCjQ9q5s38gDQTs73/xYppYAEAmIxQRigCkqVhMOnRI2rUr2TxBkm66yZbIzZ3LEi4XEksXd+60JXaS7TOaN89m62bPzs5/l1gspvfee0+S9Oyzz8rv9zuuCABGTirZgEYLADDKPE9qapL275cOHpSiUbteUGCHj957b+4cuJquCgulu++WFi609ufffmsd6+rrbZSWWgv0O+/MrvOgPM9T628pMIs+IwWAlBGKAGCUhMN2Ps7+/dL588nrEyfa8qyFC2mekG58PutKN2eO9OuvNqP3449SJGKt0bdvt8YMNTXSrbey3wsAsgWhCABGUFubzSwcPmwHiCY+fC8stDfRNTXSjBm52wI6k0yaJD32mPRXfyU1NFi4PXrUZv2amqRt22x5XVWVLXssKXFdMQDg90qLUPTxxx9r3bp12rt3r6LRqKqqqvTMM8/o+eefV142LuIGkDU8z87ASSyzOnOm/+M33WRB6JZbaJyQqQoKpPnzbVy4YHuP9u+3M6QOHbKRl2dht6rKBg0aACCzOA9FdXV1Wr16tR5//HF9+umnCgaDev/99/XCCy/ok08+0datW1VQ4LxMAJBkISgctlmgEyds30lHR/Lxy98c33KLNGGCs1IxCsaPtwN077tPOn3aOgcmwnBjo41//VeprMxmj2bMsMEsEgCkN6dpY/v27Vq9erXmz5+vrVu39nW9qaurU1tbmzZs2KC6ujq9+uqrLssEkMM8z2YETp60EHTypO0vuVxhob0BTiyjKi52UyvGjs8nTZ1q4w9/kNrbk7OFjY3Wwa611c5CkiwkzZghzZxpX0MhllACQDpx2pJ7+fLl+vzzz/XOO+/oueee6/dYfX29qqurVVpaqtOnT6t4GO8yaMkN4Hp0dVkASowzZ+zrpUv9n5eXJ1VU2Jvb2bOlWbNsiRUg2e/L0aPS8eMWohMtvi83frx1HJwyRZo82UZZmTTWHbFjsZjWr18vSVq5ciUtuQFklYw4p6i1tVXl5eWKx+NqaGjQ3Llzr3pOZWWlmpubtWXLFj355JPXfE1CEYCheJ508aJ9qh8O29f2dmuOcPbs1TNACYWF1nEssRRq2jT2B2H4OjstHCVGS0vysN7L+Xy2F6mszDoUhkK2/DIx6HQHAKnJiHOK9uzZo3g8Lr/frzlz5gz4nOrqajU3N2v37t3DCkUAclNvr4WccNhCT2enfb389oUL9nhPz9CvFQr1//R+yhR7k0rPF/xeJSVSdbUNyQ7wPXMmOROZmJXs7JTOnbMxkEDAfj/HjUuOkpL+t4uK7HmBgAX3/Pyx+zkBIJM5C0XHjh2TJJWVlQ3aYW7q1Kn9nptJIhHp1CnXVQDZJxq1mZ3EDE9bm/3vbbhz3j6fFAwmP31PfBpfVmYhiE/jMdr8fmn6dBsJiVnMM2fsfKTLZzITYT8atQCVioKCZEgaag9TIGChKjGKi5O3WRoK4PeYNs2WCmcKZ/+pi/y2TmWovUIlv7XrCYfDAz4ejUYVTRwNf9lrpoOTJ6V/+ifXVQC5o6DAAs748QN/gj5unIWf0lI+PUf68fnsd3f8eDs49krd3cnZ0MTs55Uzop2dFpyiUZuNkmxmtKfHnjOQ3t6Y9u//e0lSTc0K5eezpwjAyHjqKWs+lCky+vOftWvXas2aNa7LGFBxcf9PAQGMDL8/OcszcaKNCRPszSTdvJCtCgttNrOsbHjPj8eTASkatVA10Gxqd7enjo4W9fZKf/M3nmIxC1ddXcmg1ds7sj8LgNyQaSsvnIWixGanrq6uQZ/T2dkpSQqFQgM+Xltbq1WrVvXdj0QiqqysHMEqf785cwb+tA8AgNGWl2cfzl2rcWt3d/IsrTvvpIEIgNzlLBQlmiu0trYqHo8PuK/o1G+bcgZrxBAIBBQIBEavSAAAAABZz1k/pYULFyovL0+xWGzQRgqHDx+WJC1atGgsSwMAAACQQ5yFosmTJ2vZsmWSpM8+++yqx+vr69Xc3KxgMKhHH310rMsDAAAAkCOcnrzx2muvSZLefvtt9VxxeMi6deskSS+99FJfFzoAAAAAGGlOQ9GyZcv0xhtv6ODBg3riiSe0b98+NTQ0aPXq1dqwYYMee+wx1dbWuiwRAICsVlJSwoePAHKez/OGe+Th6Nm2bZvWrVunPXv2KBaLad68eVqxYoVWrlyp/BQOFIlEIgqFQgqHw33d7QAAAADknlSyQVqEopFCKAIAAAAgpZYNnC6fAwAAAADXnJ1TBAAA3IrFYtq8ebMk6amnnpLf73dcEQC4QSgCACBHeZ6nxsbGvtsAkKtYPgcAAAAgpxGKAAAAAOQ0QhEAAACAnEYoAgAAAJDTCEUAAAAAchrd5wAAyGG04QYAyedlUQ/OVE6tBQAAAJC9UskGLJ8DAAAAkNMIRQAAAABymvNQ1Nvbq7q6OhUVFemhhx5yXQ4AADmjp6dHmzdv1ubNm9XT0+O6HABwxmmjhUOHDmnFihXatWuXyzIAAMhJ8XhcR44c6bsNALnK2UzRxo0btWDBAk2bNk1vvfWWqzIAAAAA5DhnoWjnzp368MMPtXXrVk2ZMsVVGQAAAABynLPlc++++67y8/NdfXsAAAAAkORwpohABAAAACAdOO8+BwAAAAAuOe0+d72i0aii0Wjf/Ugk4rAaAAAAAJkoo0PR2rVrtWbNGtdlAACQkQoLC/X666+7LgMAnEs5FL388sv66KOPUv5GGzdu1JIlS1L+c0Opra3VqlWr+u5HIhFVVlaO6PcAAAAAkN1SDkUtLS2qr69P+RtduHAh5T9zLYFAQIFAoO++53mSWEYHAAAA5LpEJkhkhKGkHIo2bdqkTZs2pV7VGOjo6JAkZosAAAAASLKMEAqFhnxORu8pulJFRYWampoUDAbl8/lclwMAAADAEc/z1NHRoYqKims+N6tCUV5enqZPn+66DAAAAABp4FozRAmcUwQAAAAgpzmbKeru7tb58+clSeFwuO/a6dOnJUnFxcXDTnYAAAAA8Hv5vOG0YxgFX3zxhR5++OFBH3/66af1wQcfjF1BAAAAAHKSs1AEAAAAAOmAPUUAAAAAclrahKIdO3aoqqpKPp9PjY2NrssBAAAAkCOch6Kuri69+OKLWrp0qRoaGlyXAwAAACDHOA1Fx44dU01NjbZs2aJt27a5LAUAAABAjnIaig4ePKjly5frwIEDeuSRR1yWAgAAACBHOe0+19vbq/z8/GQxPp8k6fjx45o1a1bKrxePx9XS0qJgMNj3WgAAAAByj+d56ujoUEVFhfLyhp4LcnZ4q6R+gWgktLS0qLKyckRfEwAAAEDmampq0vTp04d8jtNQNNKCwaAk+8FLS0sdVwMAAADAlUgkosrKyr6MMJSMDkXRaFTRaLTvfkdHhySptLSUUAQAGFO9vVJ7u/TrrzYuXRr8uYWF0qRJNiZOlAoy+v+NASC9DWdbTUb/Z3jt2rVas2aN6zIAADmms1M6dkw6cyYZgtraLBilyuezYJQISZMnS3PmSMP4YBOjqLu7W2+++aYk6ZVXXlFhYaHjigCMpowORbW1tVq1alXf/cQUGQAAI8nzLAA1NEhHjkjNzXbtSn6/BZsbb5TGjbPAM5DOTuncOQtT0ah0/ryNy4/rmzpVmjtXmjdPqqiQrrFHGABwHTI6FAUCAQUCAddlAACyUG+vzQbV11sQikT6Pz5lijRjhlRWZiFo0iSptHTwIDQQz5MuXLBwlAhJTU1SS4t06pSNL7+USkqSAamqiuV2ADDS+M8qAACXaW+X9u2TvvtO+m2rqiSbBZo928LJ3LlSKHT938vns2VywaB0003J6xcvWhA7ckQ6etRmlr7/3kZxsXTnndLChRbIAADXj1AEAMh5vb22dG3vXpsdSiyNGzdOuvVWm6GZNWvsZmjGjZNqamz09trsUUODdPCgFA5L335rY8YM6e67pVtusdCGkePz+VRRUdF3G0B2c3p465Wu9/DWSCSiUCikcDhM9zkAwDV1dEi7dtms0IULyeuzZ9tMTHW1NMJH6l2XeNxC2969FpLicbuemD1avFi64Qa3NQJAukglGzBTBADIOefOSV9/bcvREh3jxo2T7rpLWrAgfYNFXl5y+V4kIu3fbwEpMXu0c6fNbD3wgDVqAAAMj/NQ1Nraqt4repi2traqqKhIklReXu6iLABAFvrlF+mrr6TDh5NL5GbMkO65J/1mha6ltFRaulRaskT6+WcLRUePSgcO2Jgzx8LRTTel1vwBAHKR8+Vzs2bN0okTJwZ9PJXyWD4HALiS59mSs6+/lo4fT16vqrLQMGOGu9pG2unT9nMePJhcWldRYcGpupq23qmIxWJav369JGnlypXys2kLyDgZtXyusbHRdQkAgCzkebbv5i9/sRbXkoWC22+3MDR5stv6RkN5ufS3fyv94Q/SN99YF72WFukf/9Fahi9dKt12G+FoODzPU3t7e99tANnNeSgCAGAkeZ4tj/vLX2zmRLLObAsXSvfdNzKttNPdxInSY49Jy5ZZI4mdO+0MpK1bpS++sHB0++2ZtVwQAEYToQgAkBXicenQITvs9OxZu1ZYaB3Z7rvPGinkmnHjpIcflu6/38LRN99I589L/8//Y6HxwQetax3hCECuIxQBADJaPG6NBbZvl1pb7VogIN17rzVQKClxW186CAQsAN1zj7R7t7Rjh9TWJn30kYWjJUvsTCS2zQDIVYQiAEBG6umxltRff21v8CWpqMhmhe65x26jv8JC20+1eLG0Z4/93YXD0r/8i4Wj++6zw2ADAdeVAsDYIhQBADJKd7e9of/mGzt8VbJlYvfea2/2eUN/bX5/MgB9910yHH36qbUsv+ce+7tklg1AriAUAQAyQleX7Yv59lu7LdlZPQ88YAeusvQrdX6/hZ+FC6Uff7QliOfOWTOGHTssNN13nxQMuq507Pl8PpWVlfXdBpDdnJ9TNJI4pwgAsk9rq3VP+/57KRazazfeaPtg7riDJgEjKR6XfvrJwlGic19+vnTrrTYTV1Hhtj4ASEUq2YBQBABIO54nHT1qYejo0eT18nILQ/Pnc9bOaEr8/W/fLp08mbw+Y4aFIw6CBZAJMurwVgAAErq7bUYoca6OJPl89ib8nnukmTPtPkaXzyfNnWujpcWWLB48aAHp5Ek762nxYlu2WFzsuloAuH7MFAEAnPI8e6O9f7+98e7utuuBgL3pXrzYDiOFWx0d1uBizx7p4kW7VlBggbWmRpo9O7tmj2KxmN577z1J0rPPPis/m9aAjMNMEQAg7bW326zQ/v3JltqS7RdavNjeaNNJLn0Eg3YQ7IMPWlOGnTtt39GBAzaCQTsI9s47pd/6E2Q0z/PU+tvBV1n0+TGAQRCKAABj5uJFqaFB+uEH6fjx5PVAwDbz19RIlZUskUtnBQXSXXfZv9Xp0xZqf/zRZpK++srG9OnWBKOqypbaAUC6IxQBAEaN59neoPp6G83Ndk2y4HPTTfbm+pZbaKmdaXw+aepUG3/1VxZ29++Xjhyxf+fmZunPf7bHq6pslJcTeAGkp7QIRR9//LHWrVunvXv3KhqNqqqqSs8884yef/555WXTAmUAyAGxmL0hPnLEgtC5c/0fr6iwfSh33CFNmOCkRIyw/HwLtrfcIl24YDNHP/0kNTVJp07Z+OILO1eqqkqaN8862bE8EkC6cB6K6urqtHr1aj3++OP69NNPFQwG9f777+uFF17QJ598oq1bt6qgwHmZAIBBdHYmu5KdPGndyuLx5OP5+TYjVF1tb4bpg5Pdxo+3A1/vu8+WSybC8dGjUiQi7d5tw+eTpkyxjoIzZtjX8eNdVw8gVzlNG9u3b9fq1as1f/58bd26ta+zS11dndra2rRhwwbV1dXp1VdfdVkmAOA3nZ3SmTPS2bP2tanJDle9UmmpBaGqKmnOHGYEctW4cbY8sqbGZhCPH7eA9PPP1lzj9GkbO3fa82+4wfaUTZliY/JkC0osuQMw2pyGotdff12S9MILL1zV6vLFF1/Uhg0b9F//63/VqlWrVMxBCAAwJnp77RP99nZ749ramgxCFy4M/GfKyuzT/sQn/qEQb2TRn99vM4Xz5tn9jg6bWTxxwr6eOSOdP2/jciUlFo4SY+JEW3YZClnTh9Hi8/k04bf1nT5+mYGs5+ycotbWVpWXlysej6uhoUFz58696jmVlZVqbm7Wli1b9OSTT17zNTmnCAAG19trMz0XLya/JkY4bCGovd3erA71/wwTJ9qb0ylTbH9QZaXNCADX49Ilm3n85ZfkTOT580P/LgaDFo4mTLAxbpyNkpL+t1mFD+SmjDinaM+ePYrH4/L7/ZozZ86Az6murlZzc7N27949rFAEALksFksGm7Y2G4nb7e32pnO4CgqSn8ZPmpRcyjR5slRYOCrlI8cVFUlz59pIiMWse2FiprK1NRngu7stwHd0WGOPoQQC9vqFhXb7yjFUT6fCQqm42MLV5aO42Ga/mEQCsoOzUHTs2DFJUllZ2aAd5qZOndrvuVeKRqOKRqN99yORyAhX+fsdOCD90z+5rgIA+vP5rv4Ufdy4/p+2h0J2jTd7cM3vT7b9vpznSV1dyQ8BwmEbl89+JmZD43EpGrUBYOw89VT/DznSnbNQlAgwQ+0VKikpkSSFw+EBH1+7dq3WrFkz8sUBQIYKBJJ7LiZOTN6eMME2rBcXE3aQ+RLhvqTElnAOxvNshrSz0752dycD0uVjoCV6PT0x/eu//r16eqQHHlih7m6/OjvVN3p7R+/nAzD2MnqVbW1trVatWtV3PxKJqLKy0mFFSdXV0ksvua4CQC7Jz7dQROgBjM9nHwT8nl5N3d2edu5skST97d96/ZaNep4FrJ6eESoUyEJFRa4rSI2zUJTY7NTV1TXoczo7OyVJoVBowMcDgYACadrntaCAjZ0AAGQjny+5HwlAdhhia+HoSjRXaG1tVfzyU/4uc+rUqX7PBQAAAICR5iwULVy4UHl5eYrFYoM2Ujh8+LAkadGiRWNZGgAAAIAc4iwUTZ48WcuWLZMkffbZZ1c9Xl9fr+bmZgWDQT366KNjXR4AAACAHOEsFEnSa6+9Jkl6++231XPFbsV169ZJkl566aW+LnQAAAAAMNKchqJly5bpjTfe0MGDB/XEE09o3759amho0OrVq7VhwwY99thjqq2tdVkiAADIUSUlJXwwC+QIn+cN1J1/bG3btk3r1q3Tnj17FIvFNG/ePK1YsUIrV65Ufn7+sF8nEokoFAopHA73dbcDAAAAkHtSyQZpEYpGCqEIAAAAgJRaNnC6fA4AAAAAXON4UQAAgCvEYjFt3rxZkvTUU0/J7/c7rgjAaCIUAQAAXMHzPDU2NvbdBpDdWD4HAAAAIKcRigAAAADkNEIRAAAAgJxGKAIAAACQ0whFAAAAAHIa3ecAAAAGQBtuIHf4vCzqM5nKqbUAAAAAslcq2YDlcwAAAABymvNQ1Nvbq7q6OhUVFemhhx5yXQ4AAACAHON0T9GhQ4e0YsUK7dq1y2UZAAAA/fT09Ogf/uEfJEl/93d/p4ICtmED2czZTNHGjRu1YMECTZs2TW+99ZarMgAAAK4Sj8d15MgRHTlyRPF43HU5AEaZs1C0c+dOffjhh9q6daumTJniqgwAAAAAOc7ZXPC7776r/Px8V98eAAAAACQ5nCkiEAEAAABIB867zwEAAACAS4QiAAAAADkto/tLRqNRRaPRvvuRSMRhNQAAAAAyUcqh6OWXX9ZHH32U8jfauHGjlixZkvKfG8ratWu1Zs2aEX1NAACAwsJCvf76667LADBGUg5FLS0tqq+vT/kbXbhwIeU/cy21tbVatWpV3/1IJKLKysoR/z4AAAAAslfKoWjTpk3atGnTaNSSskAgoEAg0Hff8zxJLKMDAAAAcl0iEyQywlAyek/RlTo6OiSJ2SIAAAAAkiwjhEKhIZ+TVaGooqJCTU1NCgaD8vl8rssBAAAA4Ijneero6FBFRcU1n5tVoSgvL0/Tp093XQYAAACANHCtGaIEZ6Gou7tb58+flySFw+G+a6dPn5YkFRcXD/uHAAAAAIDfy+cNZ+fRKPjiiy/08MMPD/r4008/rQ8++GDsCgIAAACQk5yFIgAAAABIB3muCwAAAAAAl9ImFO3YsUNVVVXy+XxqbGx0XQ4AAACAHOE8FHV1denFF1/U0qVL1dDQ4LocAAAAADnGaSg6duyYampqtGXLFm3bts1lKQAAAABylNNQdPDgQS1fvlwHDhzQI4884rIUAAAAADnKafe53t5e5efnJ4vx+SRJx48f16xZsxxVBQAAACCXODu8VVK/QDQS4vG4WlpaFAwG+wIWAAAAgNzjeZ46OjpUUVGhvLyhF8g5DUUjraWlRZWVla7LAAAAAJAmmpqaNH369CGfk1WhKBgMSrIfvLS01HE1AADgcvG4FA5LnZ2DP6egQJo4USosHLu6AGSnSCSiysrKvowwlIwORdFoVNFotO9+R0eHJKm0tJRQBACAI7GY1Noq/fqrdO6cfU3c7ukZ3muUlkqTJvUfZWXSMN7bAEA/w9lWk9GhaO3atVqzZo3rMgAAyHnt7VJDg3TkiHT8+ODhJz/fgs1g71GiUZtJikRs/Pxz/8cnTZLmzZPmzpVmzLDXw9jq7u7Wm2++KUl65ZVXVMi0HrJARoei2tparVq1qu9+YooMAACMrt5eqanJQlBDg80MXW7cuKtnem68UZowQbrGfmd1dfWfXUrcPn8+eXvHDikQkObMsZB0883S+PGj9uMCyHIZHYoCgYACgYDrMgAAyBnnzkl790r79/ffG5SXJ1VW2gzOvHm21O33NoItLrbXuvJzzkuXpGPHLIQdPSpdvCgdOmTD57OAtHChfX9mkACkIqNDEQAAGH09PdJPP1kYamxMXi8psRmaefMskBQXj24dRUXSrbfa8Dzpl1+SM1WnTllQOnrUZozuuktasMCaNmBk+Xw+VVRU9N0GsoHTw1uvdL2Ht0YiEYVCIYXDYRotAABwnQaaFfL5bDZo4UL7eq2lcGPl/Hlp3z7pu+9sBikhMXtUVcXsEZBrUskGzBQBAIB+mpqkr76S6uuT10pLbeblrrukUMhdbYO54Qbpj3+UHn7Y6t6715baJUYoJN1/v9VPXwAAV3IeilpbW9Xb23vVtaKiIklSeXm5i7IAAMgpnmdL0b7+Wjpxwq4lZoXuvtuWyaXLrNBQ8vOl+fNttLXZ7NG+fXY+0rZt0l/+Ii1ebKOkxHW1ANKF8+Vzs2bN0onEf30HkEp5LJ8DACA1vb3SgQMWhs6etWv5+dKdd9rMyqRJbusbCbGY9P339jO2tdk1v99mvu67zzriYfhisZjWr18vSVq5cqX8fr/jioCBZdTyucbLd2wCAIAx0dNj+2++/trOGJJsWdndd0v33mvL5bKF328/14IF1qnu66+tMcPOndLu3dIdd0gPPmgtw3Ftnuep/bdfmjTamg5cF+ehCAAAjJ1YzJaTff21HY4q2ZlC995rwWG0O8i5lJcn3Xabda/7+WfbN3X8uDWS+P576fbbLRyVlbmuFMBYIxQBAJADurut+cDXX0sXLti10lLpgQdsBiWXVkAlzjSaM8faev/lL9bW+4cfpB9/tP1IS5dKU6a4rhTAWCEUAQCQxaJRac8eaceOZKvqUMhmRGpqpIIcfycwbZr0n/+zLaf7y1+kw4elgwdt3HKLhaOpU11XCWC05fh/CgEAyE6dndKuXbZvpqvLrk2caGHozjs5s+dKU6dK//v/Lp05I335pe09+uknG3Pn2t/bjBmuqwQwWghFAABkkY4O6ZtvbHaou9uu3Xijvam//XbC0LVMmSL9p/8ktbZaODpwwFqVHzkizZxpM0ezZ9sSPADZg1AEAEAWaG+3/ULffWed5SSpvNzC0C23ZMYZQ+mkrEz627+1w2C//tqaMZw4If2P/yFVVNjfa3V1boYjn8+nst+6Ufhy8S8AWcn5OUUjiXOKAAC5prnZlsgdPCjF43atstJmNG6+OTfftI+GSMT2Ze3dax38JDvD6d57raV3YaHb+gBcLZVsQCgCACDD9PbaXpdvv7VQlDBnjs1gzJxJGBotFy9aCN21S7p0ya4VF1sHv8WLrYkFgPRAKCIUAQCyUGennTG0a1fyjKH8fNsrdM89dEkbS9GoLanbuVM6f96u5eXZUsV775WmTyeYAq4RighFAIAs4Xm2l2X/flsil1i6NW6ctGiRHbg6frzTEnNaPG5NGL791g6CTaiokO66yw6LzbYDcWOxmN577z1J0rPPPit/Lh1yhYySSjag0QIAAGmorU36/nsLQ+3tyevl5TYTcdttnDGUDvLypKoqG2fOWDj68UeppcXGxx9bQ4aaGlvemA0NLzzPU2tra99tIBvwn1MAANJENGp7hfbvlxobk9cDAQtBNTUsy0pnU6ZI/9v/Jv3xjxaMvvvOglLiMNjx4+2MqDvvtO52/DsC6YNQBACAQ+Gw1NAgHT5sQai31677fHYeTk2NzTSwQilzjBtns3n33iudPm0h94cfpAsXrL33119LN9xgs0vV1dYtMBtmkIBMlhah6OOPP9a6deu0d+9eRaNRVVVV6ZlnntHzzz+vPP4rAQDIIp5nb5Tr622cOtX/8UmTbCbhjjvoZJYNysulRx+VHnnE9h7t329fz5+3Q3a/+cb2HM2bZyFpzhybGQQwtpyHorq6Oq1evVqPP/64Pv30UwWDQb3//vt64YUX9Mknn2jr1q0qYNE0ACBDxeMWgk6etIYJJ09aW+cEn89mChL7UiZNclcrRk9+vs0KVVfbMsljxywUNzRIXV22f+z77+15U6dKM2ZYa/XKSqmkxHX1QPZzmja2b9+u1atXa/78+dq6dWtf95K6ujq1tbVpw4YNqqur06uvvuqyTAAAhsXzLPCcOWPnB508KTU1Sd3d/Z/n99vBqlVV0ty5ttwKuSMQkObPtxGP2+9Ifb0toTx/3n53mpvtsFjJ9h/NnGlBqbxcuvFGC08ARo7TltzLly/X559/rnfeeUfPPfdcv8fq6+tVXV2t0tJSnT59WsXD6GdJS24AwFi5dElqbZXOnrVx5ox97ey8+rlFRfaJf+KNbUUFneNwNc+zToMnTyZnFn/99ern5edbMJoyRZo8Ofm1tHRs9ibFYjGtX79ekrRy5UpaciNtZcQ5Ra2trSovL1c8HldDQ4Pmzp171XMqKyvV3NysLVu26Mknn7zmaxKKAAAjoafHZnwuXrRDUtvbrx6XLg38Z30+20SfWAI1Y4a9YWWLLH6PixeTIam52YJ3NDrwc/PybB/ahAn9Ryhkne/GjbOATtc75IqMOKdoz549isfj8vv9mjNnzoDPqa6uVnNzs3bv3j2sUAQAwFASy9va2mwkAs6FC3a9s9O+XrncbTDBoAWeyz+tLyujUxxGzrhx0i232JDsdzgc7j87efaszSj19iZ/tweTl2d7lMaNS34tKrIlfQONoZbpFRTYa5SUSIWFhC1kNmeh6NixY5KksrKyQTvMTZ06td9zrxSNRhW97OOSSCQywlX+fidOJNcCAwDc6u21N5Lt7VIsNrw/k59vb/ZKS6/+1D1xu7BwtCoGBubzJX//5s1LXo/HpY4O+x1P/K4nRjhswT8atedduGBjJOXnWxc9QhISli2zpcKZwlkoSgSYofYKlfzWbiUcDg/4+Nq1a7VmzZqRL24EdHTYpkkAQHrx+SzoTJxobywnTkwuLbr80/NAgDd1yByJpXNDtXHv6UnOhia+XrxoYWmwEY9f/Tq9vTF9++3fq7dXuu22FYrH/ertHZ2whcx1992uK0hNRm/zrK2t1apVq/ruRyIRVVZWOqwoqaJC+pu/cV0FAEDqH4RCITp3ITcVFNj/Dq5323V3t6e2thZJ0urVnnw+ayve2Zkcw12Ciuw1ebLrClLjLBQlNjt1dXUN+pzO31r4hAb52CMQCCiQpiec3XCDDQAAgGzm99ugxxUymbNeOInmCq2trYoPNDcr6dRvx3wP1ogBAAAAAK6Xs1C0cOFC5eXlKRaLDdpI4fDhw5KkRYsWjWVpAAAAAHKIs1A0efJkLVu2TJL02WefXfV4fX29mpubFQwG9eijj451eQAAAAByhNOj5F577TVJ0ttvv62enp5+j61bt06S9NJLL/V1oQMAAACAkeY0FC1btkxvvPGGDh48qCeeeEL79u1TQ0ODVq9erQ0bNuixxx5TbW2tyxIBAABwhZKSEj60RlbxeZ7nuS5i27ZtWrdunfbs2aNYLKZ58+ZpxYoVWrlypfJT6JsaiUQUCoUUDof7utsBAAAAyD2pZIO0CEUjhVAEAAAAQEotGzhdPgcAAAAArjk7vBUAAACZJxaLafPmzZKkp556Sn6/33FFwPUjFAEAAGDYPM9TY2Nj320gG7B8DgAAAEBOIxQBAAAAyGmEIgAAAAA5jVAEAAAAIKcRigAAAADkNLrPAQAAICW04Ua28XlZ1EsxlVNrAQAAAGSvVLIBy+cAAAAA5DRCEQAAAICc5jwU9fb2qq6uTkVFRXrooYdclwMAAIAh9PT0aPPmzdq8ebN6enpclwOMCKeNFg4dOqQVK1Zo165dLssAAADAMMXjcR05cqTvNpANnM0Ubdy4UQsWLNC0adP01ltvuSoDAAAAQI5zFop27typDz/8UFu3btWUKVNclQEAAAAgxzlbPvfuu+8qPz/f1bcHAAAAAEkOZ4oIRAAAAADSgfPucwAAAADgktPuc9crGo0qGo323Y9EIg6rAQAAAJCJMjoUrV27VmvWrHFdBgAAQM4oLCzU66+/7roMYESlHIpefvllffTRRyl/o40bN2rJkiUp/7mh1NbWatWqVX33I5GIKisrR/R7AAAAAMhuKYeilpYW1dfXp/yNLly4kPKfuZZAIKBAINB33/M8SSyjAwAAAHJdIhMkMsJQUg5FmzZt0qZNm1Kvagx0dHRIErNFAAAAACRZRgiFQkM+J6P3FF2poqJCTU1NCgaD8vl8rssBAAAA4Ijneero6FBFRcU1n5tVoSgvL0/Tp093XQYAAACANHCtGaIEzikCAAAAkNOczRR1d3fr/PnzkqRwONx37fTp05Kk4uLiYSc7AAAAAPi9fN5w2jGMgi+++EIPP/zwoI8//fTT+uCDD8auIAAAAAA5yVkoAgAAAIB0wJ4iAAAAADktbULRjh07VFVVJZ/Pp8bGRtflAAAAAMgRzkNRV1eXXnzxRS1dulQNDQ2uywEAAACQY5yGomPHjqmmpkZbtmzRtm3bXJYCAAAAIEc5DUUHDx7U8uXLdeDAAT3yyCMuSwEAAACQo5x2n+vt7VV+fn6yGJ9PknT8+HHNmjUr5deLx+NqaWlRMBjsey0AAAAAucfzPHV0dKiiokJ5eUPPBTk7vFVSv0A0ElpaWlRZWTmirwkAAAAgczU1NWn69OlDPsdpKBppwWBQkv3gpaWljqsBAAAA4EokElFlZWVfRhhKRoeiaDSqaDTad7+jo0OSVFpaSigCAABpKxaTzp+Xfv01Oc6dkyIRabCNDXl50sSJ0qRJyXHjjXbtGiuDgJw2nG01GR2K1q5dqzVr1rguAwAAYEidndLRo9KRI1Jzs9TePnj4GUpHh3TyZP9r+fnSDTdIs2ZJ8+bZV79/BIpGyrq7u/Xmm29Kkl555RUVFhY6rgjDldGhqLa2VqtWreq7n5giAwAAcMnzpDNnLAQ1NFgQujIEFRVJZWU225OY+ZkwYfBZn54em006d67/7FIsJrW22ti92wLRTTdJc+daSAqFRv3HBTJeRoeiQCCgQCDgugwAAABJ0tmz0t690k8/2VK4y02ZYkFlzhxp8mSppERKtVluRUX/+54nhcPJAHbkiN1vaLDxL/9i3+u226S77pKGsbUCyEkZHYoAAABci8WkgwctDDU1Ja/7/dLs2RaE5s4dnRkbn89mlyZMkKqqLCSdPZucoWpqsvuffy598YXNHC1caMGMfUgjz+fzqeK35MrxMJnF6TlFV7rec4oikYhCoZDC4TCNFgAAwKg6c8aC0A8/SJcu2bW8PKm6WqqpsUBU4Pjj564uqb5e2rev/16kCROkBQuYPUJ2SyUbMFMEAAAwTJ5nIWPHjv4hY+LEZMgYP95dfVcqLraAVlNjM0b79knff2+NHhKzR9XV0pIlVy/NA3IJoQgAAOAaenulH3+Uvv7aGhpIyVmhhQttVijdV0tNniw9+qi0fLntedqzx4LdoUM2Zs+WHnggM34WYKQ5D0Wtra3q7e296lpRUZEkqby83EVZAAAA6u62JXLffJNsnBAISIsWSffck5lLz/x+6Y47bJw5Y7NeP/4o/fyzjalTbebollvYd5SqWCym9evXS5JWrlwpP73RM4bzULRo0SKdOHGi37XFixf33U6jLU8AACBHdHZKO3dKu3bZvhzJlsXdd5/NDP322W3GmzJFeuIJ6eGHLfjt2yedOiX9z/9pZx898IB0553u90ZlCs/z1N7e3ncbmcP5r3hjY6PrEgAAACRJFy9aONi1y2aJJDtH6P77szscTJgg/elP0rJl9rPv3CmdPy/98z9Lf/mLzRwtWJC9Pz/ArzYAAMh5Fy7YMrLdu63FtiSVl0tLl9q+oVxZRlZSIj30kIXAfftsD1UkIv35z9L27TZztHChLcEDsgmhCAAA5KxIxN74790r9fTYtYoKmzGZNy93Gw4UFkr33ivdfbf03XcWiCIR6eOPk+Ho7rvteUA2IBQBAICc09ZmYei776yznCRNn25h6OabczcMXamgwJpK3HWXtfLevt3aeX/yifTVVxacFi/Onj1WyF2EIgAAkDNaW+2N/YEDUjxu12bOtDB0002EocEUFNiyuZoaO6x2+3bbc/T55xYuFy2ygJROZzQBqSAUAQCArNfSYm/kf/opee3mm6UHH7RQhOHJz7dZozvvlA4etL/Ts2dt1ujbby043X+/FAq5rtQNn8+nsrKyvtvIHD4vi/oFRiIRhUIhhcNhlZaWui4HAAA45Hl27s6OHdKxY3bN57Pzd5Yssb1DuD6eJzU0SF9+Kf3yi13Ly7PQdO+91vIbcCWVbMBMEQAAyCqxmO1/2bnTlstJ9kb99tstDP32QT5GgM8nVVVZU4rjx23m6Phx26v13Xe2JPHee3O7aQUyA6EIAABkhUjEztjZuzd54GphoZ2vc8890sSJbuvLZj6fNHu2jeZmO+vp0CELSMeP20Gw99xje5ICAdfVAldj+RwAAMhYnic1NloQOnQo2Txh4sTkm3A6o7kRDidD6qVLdi0QsD1JCxZIkye7rW80xGIxvffee5KkZ599Vn4OdHKK5XMAACCrnT9vS+T277c33wmzZiWXa+XKgavpKhSSHnnEOvslljP++qs1ZPj2W9vTVVMj3XabHRqbDTzPU+tvazazaN4hJxCKAABARohGbTZo/37pxInk9aIi6dZbrS10ebmz8jCIwkL7t7n7bunoUWnfPqm+3joCtrRI//qvti+ppsY6AhJm4QKhCAAApK3OTunIEXsTfeSINVGQbA/LnDn2RrqqSmKVUvrz+aS5c21cvGhnRe3fL506ZWH30CFp3Diputr+TW+6iX9XjJ20CEUff/yx1q1bp7179yoajaqqqkrPPPOMnn/+eeXxcQEAADnl/Hnp8GELQidP2r6hhEmTLAjdcYfE9uHMNW6c7fm65x7p9GlbXvfDDxaW9u614fdb8E10txs3znXVyGbOQ1FdXZ1Wr16txx9/XJ9++qmCwaDef/99vfDCC/rkk0+0detWFRQ4LxMAAIySCxcs/Jw4YecKJdpoJ5SX2xvjqipp6lRaO2eb8nIbf/yjNc2or7cRDls4PnzY/s2nT7fZoxkzpMpKuthhZDlNG9u3b9fq1as1f/58bd26ta9DR11dndra2rRhwwbV1dXp1VdfdVkmAAAYIZ5nM0EnTyaD0Pnz/Z+Tl2cNExJBaMIEF5VirOXn28zQnDnSn/4knTmTnDE8dUpqarIhWUgqL7eANHOmfR0/3m39yGxOW3IvX75cn3/+ud555x0999xz/R6rr69XdXW1SktLdfr0aRUXF1/z9WjJDQBA+rh0yd7Ynj1rI3E70Z45weeTpkxJvsGdM4c22ugvHLYmDYkw3dZ29XOCQWvzPWWKfZ082Q7qHct9SbFYTOvXr5ckrVy5kpbcjqWSDZyFotbWVpWXlysej6uhoUFz58696jmVlZVqbm7Wli1b9OSTT17zNQlFAACMnXhc6uiwN6zt7f3Hr7/aYaoDyc+Xpk1LhqDKSkIQUhOJ9J9tPHu2/96zBJ/PDo698UabcQyF7GtilJSwHDObZcQ5RXv27FE8Hpff79ecOXMGfE51dbWam5u1e/fuYYUiAABw/WKxZLi5cME2v3d22tfLb3d0JA9LHUwodPWn95MmSWwXxvUoLbXzjW67ze5Ho1fPSJ49a7+r587ZGIjfbzNMJSXWyGHcuP63i4utpXgg0H/k54/dz4qx4ew/SceOHZMklZWVDdphburUqf2em0nOn7fNggAApLPEbE9bm432drs/XPn59gb18k/fQyH7dH7yZGaAMDYCAZtxrKxMXvM8C/Vnz149kxkO2+95LGbv2a7c13YtBQVDB/v8fAtUJSXJkbhfVJQbs1Nz5th/CzKFs1AU+W1Ofai9QiW/HW8cvvyo6stEo1FFo9GrXjMdtLRIH33kugoAAH6fQMACTmnp4J+ijx9vn7JzegbSkc9nv5/B4MCP9/ZaOLpwof9M6OWzoZcu2SxUYiTOyerpsXH1a8a0f//fS5JqalYoPz939xQ99RShaMysXbtWa9ascV3GgIJB65gDAEC6Gz9emjjRxoQJ9rW4ODc+zUbuys+3Gc0bbhj+n4nHpe5uC0gDhaLubk//5//Zonhc+i//xVNPjwWszk6pq8u+XtloJFtl2rlSzkJRYrNTV1fXoM/p7OyUJIUGiZm1tf9/e/cfG3V9x3H81euPa6E/pK1tKZVWwRZoBLHW6fyBA7cgE0UXY9xMprgmENwfgIJEExRh/oiMaDRsrDaD0MUFkM3IKkoFCUj44X7wQygtLc7KdbLw4/jR0V/v/dHcyXFXaK9Hv5R7PpJL2u/n82nfl776vXvffb/fm6dZs2b5v/d6vbru/PdNHZSf33kDAADA1cHl6jz8ravDQltaOt9NlTovK5+Q0GeloZcca4p8F1c4evSoOjo6Qp5X5PF4AuZeyO12y80ndwEAAADoBceOAi4pKZHL5VJra2uXF1I4cOCAJKm0tLQvSwMAAAAQRRxrirKysjRu3DhJUnV1ddB4TU2NGhsblZKSookTJ/Z1eQAAAACihKPXi5k/f74k6Z133lHbBWerLV68WJL03HPP+a9CBwAAAACR5mhTNG7cOC1atEj79u3Tww8/rL///e86ePCgnn/+ef3hD3/QpEmTNG/ePCdLBAAAALptwIABvKDfD8WYmTldRFVVlRYvXqxdu3aptbVVhYWFeuqppzRjxgzF9uAjg71er9LS0nTy5En/1e0AAAAARJ+e9AZXRFMUKTRFAAAAAKSe9QZ8BjUAAACAqObY5xQBAAAAV5PW1lZVVlZKkn7xi18oPj7e4YrQXTRFAAAAQASYmQ4fPuz/Gv0Hh88BAAAAiGo0RQAAAACiGk0RAAAAgKhGUwQAAAAgqtEUAQAAAIhqXH0OAAAAiBAuw90/xdhVdL3AnnxqLQAAAICrV096Aw6fAwAAABDVaIoAAAAARDXHm6L29na9/vrrSkxM1L333ut0OQAAAEBY2traVFlZqcrKSrW1tTldDnrA0QstfPXVV3rqqae0Y8cOJ8sAAAAAeq2jo0O1tbX+r9F/OPZOUXl5uW655RYNGTJEr732mlNlAAAAAIhyjjVF27dv1/Lly/XBBx8oOzvbqTIAAAAARDnHDp/73e9+p9jYWKd+PQAAAABIcvCdIhoiAAAAAFcCx68+BwAAAABOcvTqc7117tw5nTt3zv+91+t1sBoAAAAA/VG/bopeffVVvfzyy06XAQAAACghIUEvvfSS02UgDD1uiubMmaMPP/ywx7+ovLxcd911V4/XXcy8efM0a9Ys//der1fXXXddRH8HAAAAgKtbj5uiI0eOqKampse/6PTp0z1ecylut1tut9v/vZlJ4jA6AAAAINr5egJfj3AxPW6KVq5cqZUrV/a8qj5w6tQpSeLdIgAAAACSOnuEtLS0i87p1+cUXSg3N1fffPONUlJSFBMT43Q5/sP5vvnmG6WmpjpdDvoJcoNwkBuEi+wgHOQG4ejr3JiZTp06pdzc3EvOvaqaIpfLpby8PKfLCJKamsoOAz1GbhAOcoNwkR2Eg9wgHH2Zm0u9Q+TD5xQBAAAAiGqOvVPU0tKiY8eOSZJOnjzp39bU1CRJSkpK6nZnBwAAAADhcqwp+uKLL/SjH/0oYNu2bds0ePBgSdIvf/lL/fGPf3Sgsshxu92aP39+wBXygEshNwgHuUG4yA7CQW4Qjis5NzHWnWvUAQAAAMBVinOKAAAAAEQ1miIAAAAAUY2mCAAAAEBUoykCgH5m5syZiomJUUFBgdOlAACi0BdffKGioiLFxMTo8OHDTpcTETRFl8HHH3+sH//4x0pPT9fAgQN1yy236J133lFHR4fTpcEh//73v/XKK6/ozjvv1KBBgxQfH6/s7GxNmjRJq1evvuja7du3a8qUKbr22muVlJSk4uJiLVy4UOfOneuj6nEl2b59u95+++1uzyU70e3MmTN69dVXVVpaqmuuuUYDBgzQDTfcoIceeki///3vQ64hN9Ht8OHDmjlzpkaPHq1BgwYpMTFRw4cP17Rp09TQ0NDlupqaGj3xxBMaPHiwEhMTNWzYMD333HP+j13B1aG5uVmzZ8/WPffco4MHD3Z7Xbj7FY/Ho2nTpmno0KFyu90aOnSopk2bpiNHjvT2rgQzRNRrr71mkuyBBx6wXbt2WU1Njc2ZM8ck2eTJk621tdXpEtHHNm3aZHFxceZ2u+3FF1+0nTt3Wm1tra1atcpuvPFGk2SPP/64tbe3B62trKy02NhYu/32223z5s1WV1dnb775psXFxVlpaamdOnXKgXsEp5w7d86Ki4tt6NChJsny8/O7nEt2cPDgQcvPz7dbb73V1q5da/X19bZ//35bvHixud1uGzJkSNAachPdqqurLSkpyeLj4+03v/mN7d692/bv32/vvvuupaamWnJysm3bti1o3caNGy0pKcmKiors448/tvr6equoqLDk5GS74YYb7MiRIw7cG0RaXV2dFRYWWn5+vn3yyScmySRZQ0PDRdeFu1/Zt2+fZWRkWE5Ojq1evdrq6+tt9erVlp2dbRkZGbZv376I3j+aogjavHmzSbJRo0ZZS0tLwFhZWZlJsoULFzpUHZyydu1ak2Rvv/120Ni3335rKSkpJsmWLVsWMFZXV2dut9uysrLs+PHjAWOLFi0ySfarX/3qcpaOK8z8+fPtmmuusT/96U8XbYrIDk6fPm0FBQVWWlpqzc3NQeOLFi2ysWPHBmwjNxg9erRJshdeeCFobNmyZSbJRo8eHbD9+PHjlpmZaYmJiXbo0KGAscrKSpNk991332WtG33jr3/9q02fPt3fxHSnKQp3v9La2mojRowwSbZ169aAsS1btpgkGzlyZETfbKApiqDx48ebJFu6dGnQ2IEDB0ySpaam2tmzZx2oDk5Zu3atxcbG2smTJ0OO//znPw/5oDF16lSTZHPnzg1ac+LECXO73eZyuezrr7++LHXjyrJ3715LSEiw8vJy27hx40WbIrKDF154wSTZ+vXru72G3CApKckk2UcffRQ0tmfPHv+T4P/+97/+7QsWLDBJ9thjjwWtaW9vt9zcXJNkW7Zsuay14/Jra2sL+L47TVG4+5UVK1aYJPvBD34Q8ufefvvtJslWrlzZ8zvSBc4pipCjR49q06ZNkqQJEyYEjRcVFSkvL09er1dVVVV9XB2c9NOf/lTHjh1TampqyPG8vDxJ0rFjx/zb2tvb9cEHH0gKnae0tDSVlpaqo6Pjkuckof/r6OjQ008/rbvuuktPP/30ReeSHbS3t6uiokJut1vjx4/v9hpygzFjxkiSvvrqq6Ax37aYmBglJib6t69atUpS6Ny4XC5/Bv/85z9HvF70rdjY2B7N781+5WK5kqT77rtPUmRzRVMUIbt27VJHR4fi4+M1bNiwkHNGjBghSdq5c2dflgaHxcfHd9kQSZ0nEUrSTTfd5N928OBBnThxQtL3ubkQeYoeb731lnbv3q1ly5Zdci7Zwd69e+XxeFRQUCCv16u5c+dq1KhRSk9P17Bhw/Tkk09q3759AWvIDSRp6dKlGjJkiBYuXKhVq1apublZLS0t+vTTTzVr1ixJ0rhx4zRw4EBJnRfy8GWJ3OBCvdmvbN++Pax1vREXsZ8U5Q4dOiRJuvbaa+Vyhe41Bw8eHDAXaGtr06effipJmjFjhn/7+RnJyckJuZY8RYeGhga9+OKLWrBgQZcvuJyP7GD37t2SOq8SVVJSoptuuklLly5VZmamNm/erHnz5un999/XqlWrNHnyZEnkBp1uvvlm1dTUaMmSJZo6darOnDmjmJgY/9Vzx4wZo4qKCv/8hoYG/xi5wYXC3a+cPn1a3333XbfWNTU16ezZsxowYECv66UpihCv1ytJSkpK6nKO7w/G5SnhU1FRoaamJk2fPl2lpaX+7b48xcbGKj4+PuRa8hQdysrKNHLkSM2cObNb88kOjh49KqnzowDGjh2rv/zlL/4X64qLi1VQUKBJkybpiSee0KFDh5SZmUluIEk6fvy4nnzySX300UeaPXu2Hn30Ubndbm3evFlNTU2aPXu2Bg0a5J/vy43U9fMfchO9wt2v9CRXvrWRaIo4fA5wSF1dnZ599lndcccd+u1vf+t0ObgCvffee/r888/13nvv9fhYbkSvs2fP+r9+5plngo5euP/++1VYWCiv16vly5f3dXm4QpmZJk6cqA8//FDPP/+83njjDZWWlmr06NF65plnVFJSopEjR2rdunVOlwpcFjRFEeI7Z6S5ubnLOb4HqrS0tD6pCVcuj8ejn/zkJxo+fLiqqqoCTlqVvs9Te3u7WltbQ/4M8nR183g8evbZZzVnzhz/yc/dQXZw/iuro0aNCjmnpKREkrRjxw5J5AbSZ5995s/D7Nmzg8YffvhhpaamasqUKfrHP/4hSQHny3b1/IfcRK9w9ys9ydWFa3uDpihCfMf6Hz161H987YV8J9R357wAXL2OHDmiCRMmKCsrS9XV1SH/mc/PSFNTU8ifQ56ubp988olOnDihJUuWKDk5OeB2//33S+o8POr87RLZwffH2ktSenp6yDkpKSmSOg+XksgNpC+//FKSlJ2d3WVuRowYoba2NpWXl0uSrr/+esXExEgiNwgW7n4lOTlZWVlZ3VqXk5MTkUPnJJqiiCkpKZHL5VJra2uXJxMeOHBAkgLOHUF0+frrr3XPPfcoKytLGzZsCDg2+3yFhYX+ZsmXmwuRp6vbI488otraWu3evVv//Oc/A26+JyS5ubkB2yWyA2ns2LH+r//zn/+EnOPb7tsHkRv4+JqcUHyHYjY2NkqSBg4cqOLiYknkBsF6s1+57bbbwlrXGzRFEZKVlaVx48ZJkqqrq4PGa2pq1NjYqJSUFE2cOLGvy8MVoLa2VnfffbeGDRumqqoq/yv7UufVos7PRWxsrB555BFJofN08uRJ7dy5Uy6XSz/72c8uf/HocykpKRo+fHjI25AhQyRJcXFxAdslsgNp5MiRKiwslCT961//CjnH95kzP/zhDyWRG8ifmaampi4vilBTUyMp8Ipgjz76qKTQueno6NBnn30WMA/Rozf7lYvlSpI2bNgQMC8iIvYxsLBNmzaZJCsuLrbW1taAsbKyMpNkCxYscKg6OGnv3r2Wk5NjDz30kP3vf/8LGt+4caNd+O9YW1trCQkJlpWVZSdOnAgYW7RokUmyqVOnXta6cWXy5SU/Pz/kONnB8uXLTZKNGjUq6PFo3bp1JskyMjLs+PHj/u3kJrqdOXPGsrOzu3yuUlVVZZJMkq1fv96//dixY5aRkWGJiYlWX18fsKaystIk2fjx4y97/eh7vjw0NDR0OSfc/UpLS4sVFhaaJNu6dWvA2NatW02SFRUVBe3feoOmKMJ8f+AHHnjAvvzyS6upqbG5c+eaJJs0aVJE/3joH/bs2WOZmZkmycaMGWMlJSVBN98//oVWrFhhLpfL7rjjDtuyZYsdOnTI3nzzTYuLi7OSkhLzer0O3CM45bvvvjOPx2Nr1qwxSZaXl2cej8c8Ho+dOnUqYC7Zge/FuMmTJ9uOHTvs8OHDtmLFCsvMzLTU1FTbuHFj0BpyE902bNhgAwcOtLi4OHvllVds//79Vl9fb+Xl5ZaRkWGS7Ne//nXQuurqaktMTLSioiJbv3691dfXW0VFhSUnJ1tBQYF9++23DtwbXA6+xyGPx+Nvinbs2OHfFkq4+5U9e/ZYenq65eTk2Jo1a6yhocHWrFljOTk5lp6ebnv27InofaMpugz+9re/2YQJEywtLc0GDBhgN998s7311lvW1tbmdGlwwJIlS/w7jkvdQtm2bZs9+OCDlpGRYW6320aMGGEvv/yyNTc39/E9gdPy8/O7zM78+fOD5pMdvP/++3bvvfdaWlqaJSQk2PXXX2/Tp08PekX/fOQmutXV1dmMGTOssLDQEhMTLT4+3nJzc23KlCm2bt26Ltft37/fHn/8ccvOzvZnbdasWQHvRqL/u9jj0MUOQAt3v9LY2GhlZWWWl5dnCQkJlpeXZ2VlZdbY2Bjpu2YxZmaROxgPAAAAAPoXLrQAAAAAIKrRFAEAAACIajRFAAAAAKIaTREAAACAqEZTBAAAACCq0RQBAAAAiGo0RQAAAACiGk0RAAAAgKhGUwQAAAAgqtEUAQAAAIhqNEUAAAAAohpNEQAAAICoRlMEAAAAIKrRFAEAAACIav8Hjr4dJ28cG2YAAAAASUVORK5CYII=",
      "text/plain": [
       "<Figure size 1000x1000 with 10 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, axs = plt.subplots(10, 1, figsize=(10, 10))\n",
    "for k in range(10): \n",
    "    axs[k].plot(dydu[10*k].detach(), c='blue', alpha=0.5)\n",
    "    axs[k].vlines(10*k, -1, 1, color='black', alpha=0.5, linestyle='--')\n",
    "    if k < 9: axs[k].set_xticks([])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is no information flowing \"forward\" in the future i.e., from index $k$ to $n > k$, since the gradients are zero. Notice how \n",
    "the \"memory\" is also limited: after $M - 1$ steps, the gradients are zero. Is this a coincidence?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f7603a5e9e0>]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 400x200 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(h)\n",
    "plt.plot(dydu[30].detach())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As discussed in the notes, the gradient is exactly `h`. After all, the convolution is a linear operation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Recall the original \"centered\" convolution. In that case, the operation is not causal, hence care must be taken if the causality property is desired:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "y_centered = centered_toeplitz(h, L, M) @ u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "dydu = torch.zeros(L, L)\n",
    "for n in range(L - M + 1):\n",
    "    dydu[n] = grad(y_centered[n], u, retain_graph=True)[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1000x1000 with 7 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, axs = plt.subplots(7, 1, figsize=(10, 10))\n",
    "for k in range(7): \n",
    "    axs[k].plot(dydu[10*k].detach(), c='blue', alpha=0.5)\n",
    "    axs[k].vlines(10*k, -1, 1, color='black', alpha=0.5, linestyle='--')\n",
    "    if k < 9: axs[k].set_xticks([])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This result should be intuitive by this point (if it is not, no worries, step through the first notebook!): centered convolutions start with the kernel overlapping `u`, with the first output `y[0]` computed as a linear combination of `u[0]`, ...,  `u[M-1]`. \n",
    "\n",
    "Causality is easily reintroduced by appropriate padding: we have previously shown how to pad `u` to recover full convolutions from centered convolutions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The Frequency Domain"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us look at a different basic convolution kernel."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "h = torch.exp(- torch.arange(L))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f7603fa1a50>]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 400x200 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(h)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "H = torch.fft.fft(h)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f7612395630>]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 400x200 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(H.real, H.imag)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is a peculiar symmetry between the real and imaginary components of the frequency response, which we can enforce by taking a reciprocal discrete Hilbert transform: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "def hilbert(u):\n",
    "    L = u.shape[0]\n",
    "    U = torch.fft.fft(u)\n",
    "    L_mid = L - L//2 - 1\n",
    "    U[L//2+1:] = torch.zeros(L_mid)\n",
    "    U[1:L//2] = 2 * U[1:L//2]\n",
    "    v = torch.fft.ifft(U)\n",
    "    return 1j* (v - u)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f7603ca0b20>]"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 400x200 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(H.real, hilbert(H.real).real)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We have recovered the imaginary part of the Fourier transform of `h` directly from its real part via a Hilbert transform. This is directly an application of the Kramers-Kronig relations discussed in the text. Note that this\n",
    "is valid only in the case of causal convolutions!\n",
    "\n",
    "A formal treatment requires results from complex analysis. Our showcase here eschews a formal exposition in favor of intuition and visualization."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  }
 ],
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   "language": "python",
   "name": "python3"
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    "name": "ipython",
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   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
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