{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Foldy-Lax\n", "The Foldy-Lax formulation provides a way of calculating the wavefield around isotropic scatterers. In the case of $N$ point scatterers only an $N \\times N$ matrix equation needs to be solved. The wavefield, $\\phi$, must satisy the Helmholtz equation in free space:\n", "\n", "$$\\nabla^2 \\phi + k^2 \\phi = 0,$$\n", "\n", "where $k$ is the wavenumber. If the incident waves satisfy the Helmholtz equation, so must the scattered waves. \n", "Therefore when a plane wave $e^{i\\vec{k}.\\vec{r}}$ is incident upon a point scatterer at $\\vec{r_1}$, the scattered wavefield, $\\phi_s$, must be a solution to\n", "\n", "$$\\nabla^2 \\phi_s + k^2 \\phi_s = \\delta(\\vec{r} - \\vec{r_1}) \\sigma_1 \\phi_1$$\n", "\n", "since a point scatterer of strength $\\sigma_1$ in a wavefield of local amplitude $\\phi_1$ acts like a source $\\delta(\\vec{r} - \\vec{r_1}) \\sigma_1 \\phi_1$. We therefore have\n", "\n", "$$\\phi_s(\\vec{r}) = \\sigma_1 \\phi_1 G(\\vec{r}, \\vec{r}_1),$$ \n", "\n", "for a single point scatterer, where $G$ is the Green's function of the Helmholtz equation. In 3D, \n", "\n", "$$G(\\vec{r}, \\vec{r_i}) = \\frac{1}{4\\pi} \\frac{e^{ik|\\vec{r}-\\vec{r_i}|}}{|\\vec{r}-\\vec{r_i}|}.$$ \n", "\n", "For multiple scatterers we can simply add the scattered waves, this is because the Helmholtz equation is linear. The overall wavefield at $\\vec{r}$ is\n", "\n", "$$\\phi(\\vec{r}) = \\phi_{inc}(\\vec{r}) + \\sum_j G(\\vec{r}, \\vec{r}_j) \\sigma_j \\phi_j,$$\n", "\n", "where the sum is over point scatterers. Now to get the matrix equation out from that, set $\\vec{r} = \\vec{r_i}$ and exclude the self-interaction at scatterer $i$:\n", "\n", "$$\\phi_i - \\sum_{j \\neq i} G_{ij} \\sigma_j \\phi_j = \\phi_{inc}(\\vec{r_i}),$$\n", "or in matrix form\n", "\n", "$$\\sum_j M_{ij} \\phi_i = \\phi_{inc}(\\vec{r_i}),$$\n", "\n", "$$M_{ij} = \\delta_{ij} + (\\delta_{ij} - 1) \\sigma_j G_{ij}.$$\n", "\n", "This matrix equation can be solved for the wave amplitudes at each scatterer. The amplitude at $\\vec{r}$ is then given by the sum of the incident, $\\phi_{inc}$, and scattered, $G(\\vec{r}, \\vec{r}_j) \\sigma_j \\phi_j$, waves." ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [], "source": [ "#NAME: Foldy-Lax Formulation\n", "#DESCRIPTION: Calculating the wavefield around isotropic scatterers using the Foldy-Lax formulation.\n", "\n", "%matplotlib inline\n", "import numpy as np\n", "from numpy import linalg as la\n", "import matplotlib.pyplot as plt\n", "\n", "#set the incident wavevector\n", "k = [3.0, 0.0, 0.0]\n", "#and wavenumber\n", "K = la.norm(k)\n", "\n", "#define a point scatterer\n", "class Scatterer:\n", " def __init__(self, position, strength):\n", " self.r = np.array(position)\n", " self.s = strength\n", " self.phi = 0.0 #wave amplitude at scatterer\n", " \n", "#Green's function for 3D Helmholtz\n", "def green(r1, r2):\n", " return np.exp(1.0j*K*la.norm(r1-r2))/(4.0*np.pi*la.norm(r1-r2))\n", "\n", "def incident_wave(r, form):\n", " #r is a position vector\n", " incident_amplitude = 0.0j\n", " if form == 'plane':\n", " incident_amplitude = np.exp(1.0j*np.dot(k,r))\n", " elif form == 'spherical':\n", " incident_amplitude = np.exp(1.0j*K*la.norm(r))/la.norm(r)\n", " return incident_amplitude\n", "\n", "#scattered amplitude from a single scatterer\n", "def scattered_wave(scatterer, r):\n", " return (scatterer.s)*(scatterer.phi)*green(r,scatterer.r)\n", "\n", "def Foldy_Lax(scatterers, incident_form, x, y):\n", " #x and y are 1D arrays of points at which amplitude is calculated\n", " #incident_form = 'spherical' or 'plane', form of incident wave\n", " #scatterers is a list of instances of Scatterer\n", " \n", " N = len(scatterers)\n", " #calculate the amplitude of the incident plane wave at each scatterer\n", " incident_amplitudes = [incident_wave(scatterers[i].r, incident_form) for i in range(N)]\n", " \n", " #calculate the elements of the matrix M (see above)\n", " M = np.zeros((N,N), dtype = np.complex128)\n", " for i in range(N):\n", " for j in range(N):\n", " if i == j:\n", " M[i][j] = 1.0\n", " else:\n", " M[i][j] = -(scatterers[j].s)*green(scatterers[i].r, scatterers[j].r)\n", " \n", " #solve for the amplitudes at each scatterer\n", " scatterer_amplitudes = la.solve(M, incident_amplitudes)\n", " for i in range(N):\n", " scatterers[i].phi = scatterer_amplitudes[i]\n", " \n", " #calculate the total wave amplitude at all values of x and y\n", " amplitudes = np.zeros((y.shape[0], x.shape[0]), dtype = np.complex128)\n", " for i in range(y.shape[0]):\n", " for j in range(x.shape[0]):\n", " r = [x[j],y[i],0.0]\n", " amplitudes[i][j] = incident_wave(r, incident_form)\n", " for scatterer in scatterers:\n", " amplitudes[i][j] += scattered_wave(scatterer, r)\n", " \n", " return amplitudes" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Scales\n", "The idea of separation of scales is endemic in physics. In continuum mechanics we ignore the underlying atomic structure of matter; in atomic physics we often treat the nucleus as a point. \n", "In the context of interference and scattering, the wavelength sets the length scale of the problem. We therefore anticipate that structures on much smaller scales than this will be unimportant.\n", "\n", "In the units of the plot axes, the wavelength is ~2." ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "collapsed": true }, "outputs": [], "source": [ "x = np.linspace(-8,8,80)\n", "y = np.linspace(-8,8,80)\n", "\n", "#single point scatterer\n", "position = np.array([0.0,0.0,0.0])\n", "strength = 1.0\n", "p1 = [Scatterer(position, strength)]\n", "\n", "#calculate the amplitudes\n", "amplitudes1 = Foldy_Lax(p1, 'plane', x, y)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Single Scatterer\n", "With only one point scatterer, the only lengthscale present in the problem is the wavelength. As a consequence, all of the features which appear in the interference pattern are on this scale.\n", "\n" ] }, { "cell_type": "code", "execution_count": 29, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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jDQgykRJZY5Eoayxa7i2Othf1b8hZnskaA/zMsUy9sWwkgVtXLOizt3+AnyGZ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\nADBrDIYAAMCs/R/fqYnr+JT60wAAAABJRU5ErkJggg==\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig1 = plt.figure(figsize = (10,8))\n", "plt.pcolor(x,y,np.log10(np.absolute(amplitudes1)**2), cmap = 'inferno')\n", "plt.colorbar()\n", "plt.title(\"Logarithmic intensity for a plane\\nwave incident on a single point\")\n", "plt.xlabel('x')\n", "plt.ylabel('y')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Interference and Symmetry\n", "In the following demonstrations, a single point source emits spherical monochromatic waves. The source is surrounded by perfect scatterers which reflect some of the incident radiation, creating interference fringes which respect the symmetry of the system. As the separation between the scatterers decreases below the order of a wavelength, circular symmetry begins to emerge in the wavefield." ] }, { "cell_type": "code", "execution_count": 30, "metadata": { "collapsed": true }, "outputs": [], "source": [ "#create a list of 5 point scatterers in a ring\n", "N2 = 5\n", "p2 = []\n", "for i in range(N2):\n", " position = np.array([5.0*np.sin(2*np.pi*i/N2), 5.0*np.cos(2*np.pi*i/N2), 0.0])\n", " strength = 1.0\n", " p2.append(Scatterer(position, strength))\n", "\n", "#calculate the amplitudes\n", "amplitudes2 = Foldy_Lax(p2, 'spherical', x, y)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Ring of 5 Scatterers\n", "The separation of the scatterers is much larger than a wavelength. The wave can therefore resolve different scattering centres and the interference fringes are not circular. This can be seen from the shapes of the minima around each of the scatterers." ] }, { "cell_type": "code", "execution_count": 35, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 35, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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0mawpJ22jOmJZmXpfvSBDitUtA+ICzlEtrCgrL4teq2gGWNCenQd1ZF4xmSyr\nzPiaK4GXp5geERERkcUwR3d6REREpDEduNOjSY+IiIh0YtKjx1siIiLSCXNzp2dt8FsmKDgdQNxg\ncHLcBwnKDvrIBtL2MvsjsTxTWiJbhmLjSzpUH0emRAPDSjdksIBZViDBgnFv6cVtD1mqfv7fOIhT\nXAfD9Qf/Rti5QY9hQ8eLBSazoOcoSJq1Zfsj2kY+jujnYyN6zkz2kr12RNefIQlY5uVBgoXJgOVo\nfKx0Bu0j+L0QlbcovhEvDvuepTIUSlkXERERWQxzc6dHREREmmOK6RERERGpl5m938y2mdk/T2nz\n12Z2kZmda2YPrGO9mvSIiIhIkb3VxH+xUwA8jn3TzJ4A4F7ufh8ALwbwv+rYRD3eEhERkQ1NWXf3\ns83sqClNTgBwatn222Z2iJkd7u7b1rPeuZn0rI2+z5SWyGZpZfpgciUkqmekZMY8K7JZWtG2pLPW\ngvZLntt3UeZPJrOM4Rk3cRpHJruJ9c0yZiJ32FK5KW68OV4elSvIlmKI9n/mWAHNHS9WUoMdq4Fl\nsmJIFlPQdy+RJTQMSlO0gV6rSNZUtI1hRhdQS6kINr4w2yubHDjrZShmyxEAto59fUW5rBuTHhER\nEWlQTXd6/vGHq/jaD+Oabm3TpEdERERqc/z9l3H8/W8tPv1nH9u5L91cAeBuY18fWS5bl9l9JiIi\nIiIbZ+TN/McZ+AO+TwN4PgCY2XEAblhvPA+gOz0iIiKywczsNADHAzjMzC4DcBKATQDc3d/j7v9g\nZk80sx8DuAXAiXWsV5MeERER2ejsredWaPOSutc7N5OetRlOqdpbyYynuH5X9fpYTDYDKRp3psZN\n0X6yjx7ZH71EjSy2LeH6Ella2fXxWkzVM39YVlc/HAfJEkqkYLDMnyjjCcjWfor7Xg0yd24aroRt\nL/hF9cvCTh9UXh8bW5+dd4naZ9GxmtY+wsY3CsbBjtWArG4prL3Fak0lsvXI8jjjiX2+WQ2woF4V\nOVZOxhxlPPXIPqJPQYL2PGuteuZbKkuLjCNbAyxeYaJt0/RGZhEREZHFMDd3ekRERKRB04OOF4Lu\n9IiIiEgn6E6PiIiIAIk3t8+ruZn0VC5DkQj+5aUsgj4SpSKy42BYcHK4PhZoHZWyoEHIieDkRBBy\nJmCZLc8ELLN1ZgKWAWApWN6n58H63yXPA2knl7OyEryUxeTyXYiDkHeR4OSMaH9s9uWgZS44uWfr\nDyRn2L7O6lawAAAgAElEQVTLlPBgQalRgDMLIAYrnZHYxKhn9nmLApaLPqJrWDzm7DUlwgKcowcS\nLAicicbtHpflSJW4yJbDUMmJ1s3NpEdEREQa1IGYHk16REREpBOTHgUyi4iISCfoTo+IiIjoTo+I\niIjIopibOz0TZSgSpSVo5pWxbJL1l7KoI/Mqt77q48uUm2DL2Zjj0gHJ7K1E+QGeGTbZPltCwhJ9\n9BNpGSwDiQqas1IWTJS9xWSSVjN/NUX7c2rf2f0UyO6nsI/EeeCJc4xlIKWyuhK7yElWWKY8Bcv0\nYuUpovE5K7dCxhe175Mxe1D+pFA9A4xlqEV901IiWI2HkckAa0EHMtZ1p0dERES6YW7u9IiIiEiD\nOhDTo0mPiIiI5J5tzyk93hIREZFO0J0eERER6cSdnrmd9GTqadFMoxrqZrEsrbB+VzLzKu43Wb8r\nMY7Mflry+NTpB6cUa5vNyMr0EWWAsewh2keQPcRGxmtCTbJkVlLYR49l/pA+ZiBDhIULsOusB5lX\n2WtytD8y6wPiY+se79Ao0wsgteOM1U+Lj2L06ezRcSQyvchxcYuylViWaLwtmf3P6tqxbK9IJgOM\nfSZoWEtUP43tu0Q2WyKxUmowt5MeERERqVEHJmCK6REREZFO0J0eERERgY9m4Dl4wzTpEREREQUy\nbxQroo+/C+Byd39q3KZasG8YQJwoCVH0EZRBSAQsA0A/KHHRZBAy76N6YPcSOR36qL4tUdBynwZI\nJ4I+iUwJiSUWIEqO7VLQB2vLRpwJWs70sUS67ZP1RUszwdcMu0ZG5R9IzC2GJIB4EJV/IG3ZOKJ9\nx/oAu05E7+anbclAAjwIOVEyhB7voA8WSE4CqqPPfVZUuiF7DYuCkEek3EQmGHpI9kcmwJmVvaC/\nc8I+OjDTmCEzMekB8HIA/wrg4LYHIiIi0kkdeLzVeiCzmR0J4IkA3tf2WERERGRxzcKdnrcBeBWA\nQ9oeiIiISFcpkLlhZvYkANvc/VwzOx789VlYGVy359/93n5Y6h3Q/ABFREQ2wHC0HcPRjnYHoUlP\n4x4G4Klm9kQA+wE4yMxOdffnr224aen2Gz44ERGRjdDv7Y9+b/89X68Or29xNIur1UmPu78OwOsA\nwMweAeAPoglPJJORlSk3AcSZWizjiZd0CDLAkn2wccd9VC8hQfcH6yPIpmKZXlE21ZKTDDe63ev/\nayPqI5Olxdqz7Kg6guNYplcdfWdKaqT6Zd8I1seytDJ9s2wlmpEVSWS40ZFEGV1AKquLZWlVL0rD\nyy4Mg4wgJ1la7LPswTaOyDWT/SKJymGwkhUsEyra/+yayTKhoqXpTNjgBGHlJlh5iui4ZLOLG8VS\nLBfIDO1tERERkea0/XhrD3f/GoCvtT0OERGRLupCILPu9IiIiEgnzMydHhEREWnRaPHvg2jSIyIi\nIkpZl1vRDCuWAZaom5XJpmJtm8r0KtoHGWAkaySqs8WytFgtrKhuFqsTxYTZWyQ/Z7nHanJVz3hi\n9XrqEO8P1raG9SU6SSVNkeWpemGsgFTy/IhEtZWAeHxDkuViozh7KDp/MzW2gLieGd0f0c+Tz33U\nLxBnFbFrBKseFbXmfbDlk72zz2FmHCybjZ0HPLtM5okmPSIiIgJXyrqIiIjIYtCdHhEREelEIPPi\nb6GIiIgI5uhOT9VXhmde6Z19Dfl6+8iWf4heD8/aZtcZtiXByXEpC1YOoHpbVhYiap89UpkSEstk\nedS+Tx55ZwOtwz7I8qjrbOB0pnmmbxb0GfFkmHXUN4s5YAGsufXFGzMMFtPzkQTFR2UyhqyUBRGV\n4GDB0GHQM/t80/Ink+2d9MEP7WT7KDAZiJMgih4mA4gHlddWCMs/TGldFS+HEfcR/X5yn50A6S68\nnHBuJj0iIiLSoA5MevR4S0RERDpBd3pEREREKesiIiIii0J3ekRERKQTKeua9ASyGVJxH5MnT778\nQ6IMBcuqqDi2aVIZYGHmFcsOIVlTNZShiDKvWJbWMklXitqzYWTKQrC2dWRvsb3UZJmMqvjr/au3\nZ2UvWB5U2AdtG++kpeAHVmknZHko9zm0RM2PzOeQ9xFlXpGSLeT642EJCVJugpWFCMdWvS0AWPid\nXNZUfP2uI29QNpImPSIiIqKUdREREekGBTKLiIiILAjd6REREZFOBDIv/haKiIjITDGzx5vZBWZ2\noZm9Jvj+I8zsBjP7p/K/N9SxXt3pmQE0IyuqeUWyJFhGVibzqilGs7diUaYWz2yq3jfLYGI1ufpB\nJ7T2Vrw4lXmV6TsaG5CrsTUrWE7SMEiMYbkyUX0sIJkBRnZe1PeINB5a3HkvWkzOO2cDDNqztuwz\nt17560n12lvhPgLCk9pZBpiyqdZlowKZrShC9i4AjwZwJYBzzOwMd79gTdOvu/tT61x3+78RRURE\npEuOBXCRu1/q7qsATgdwQtCu9lmYJj0iIiICd2vkv8ARALaOfX15uWyth5rZuWb2OTP7lTq2UY+3\nREREpLZA5m9s3YmzL9+53m6+B+Du7r7dzJ4A4FMAjl5vp5r0iIiISG0efrctePjdtuz5+s3funFt\nkysA3H3s6yPLZXu4+81j//68mf2Nmd3e3a9bz9jmdtKTKRVhtv6yEr0a+qgDC1heShzKTMmKJmUC\nnFnAMu07aE8DlhOBxTTYOFEWYomtjxyWaJ2ZwOlp7TcSCzFlcbv94CPHApZ7pPNBtD52rKpXeaBB\nz+wcG0SLWRAyC84P2o8aDF2PrhPZQOaotATrYxAeLWCjSz2wa/3Qq4+D/s4J+qij7FFdNvCNzOcA\nuLeZHQXgKgDPBvCc8QZmdri7byv/fSwAW++EB5jjSY+IiIjMH3cfmtlLAJyJ4m+y97v7+Wb24uLb\n/h4AzzCz3wWwCmAHgN+qY92a9IiIiMiGlqFw9y8AuO+aZe8e+/fJAE6ue72zcNdbREREpHG60yMi\nIiKdKEOhSY+IiIhsZCBza+Zm0lO8tXr29EjkfVhCgjxNZNH7mewJlpHVC7I7WAaYN5QlEZWVmLY8\nyl5Z/8vv+as9WeZVrpRF9T5YltZyom+apUVSkGbhUsaSo0YkjiBKbqLnAftGopQFE44jka0HxMNL\nJIsBAEaJz0X0uc+uMOyDoGVwgmM7IgPpkWu8B9e2EcmO6pGjG5WtcHLdHaqUxUKbm0mPiIiINGcj\nA5nbMpu3T0RERERqpjs9IiIi0omYHt3pERERkU5YuDs9mdeks8C5WREFB9JAZhJ02A+C9XgZijiw\nb2BD0n596CiSJSeq9rGULUMRDDBTsoItX2aBzL04wDNVDiMRrVrDbqZYaYkICxuNSk6wMhRG/kJd\nrdjvVMHxGpFBs3NsGCzOhstGn07P7GiCBSFHljxXMiEKvgbi6wm7to1m+G9z9juElayItnGWAqej\noPFFs3CTHhEREdkHerwlIiIishh0p0dERESUsi4iIiKyKHSnR0RERDqRsj43k561pRoyZSky2Qms\nfbaPCM+8SmSckej6KEsLAJZ98hAvJbclepX8ig2q/zzLLGPlKaJlyc9imPFEs6bI8kQfLJsq6mOJ\nZGllsrr6pNwEO7IWtG/yNm+Uj8JunbOyBMOg/Sq5KBsrbRCce6sk4YkcFvSijSE7j2WG9YPl6TIU\nwQ/Qz1ANjyk2BdeOZXKdYQZRFhMZ2ohlogXt2TUzKjcBsHwxtjpWLigqZRH3zEv9TLaf1RJLi2pu\nJj0iIiLSHKWsi4iISCd04fHW4k/rRERERKA7PSIiIgKlrDfOzI40s6+Y2b+Y2Xlm9rI2xyMiIiKL\nq+07PQMAr3D3c83sQADfM7Mz3f2CtQ3XRriz6H2zoNYUiY5fmxF2a/vqGQo80r96HywjK9pGlgnF\nauJE2Rabyfb1SCbIjuA5L8u0sGB8/WT2Vqb2FsvqimogRZlU05YvBYdlKdEWiDOvNtHsLbI8yLxi\n2Vt90kc0vCiji7VlWNWg6C9GWmOLxBFE2VvZc2YlWMzqdA3IAMPLB2kb1dgqlk9+g2VvRVlaAM8Q\nDNsGn7noswnwa8em4NfDfr34Vwa7HuzyyWwlVi9sSGr8RddBFnDL9n90PTZ+RsbjCK6bzmpskXF4\nWAuRDKMFutPTMHf/mbufW/77ZgDnAziizTGJiIjIYmr7Ts8eZnYPAA8E8O12RyIiItI9XcjemolJ\nT/lo6+MAXl7e8Zmwc/XqPf9e6h2ATf2DNmh0IiIizVod3ozB6JZWx6D39GwAM1tCMeH5oLufwdpt\nWb7Txg1KRERkAy33D8Ry/8A9X+8aXD2lteyr1ic9AP4WwL+6+zsyPxQFLAOshET1tmx5jwU9J0pL\nZMcR98HGTIIUgyjM/fvxOA4idRCuC6JBV4bVy1BEQcXAtKDUyl3T4OswCJmVkCDLo6DlTMAyEAct\nb6aBzHFQZFS2IgpuBqYFOE/2vUTaZm5wsxjMQRAQORyx4NN4jatBHz1y+z1TyoVj9REmF7E/iJfI\nDomWj5IlNWLscx8sZ2U2SB8H9pcnlt1+U3ztuGk1PneHQV2O6LgCuevgKJlA0gsOIitZEQUbF+uM\n1pctcVS9LEcbuvB4q+2U9YcB+I8AHmVm3zezfzKzx7c5JhEREVlMrd7pcfdvAskKdiIiIlI7payL\niIiILIhZiOkRERGRlnXhTo8mPSIiItKJQOa5mfSsjeCnkf5BthJvyyL9o1eWZ7MFogyw3NPEqDwF\ny7Rgy6PX0W8h6UovvM+V4fKTz7/z5ML4Te1YCrYxm6XFMrIyfYSZV6Qt6yNazrKjWDZVXIYiznSh\nZSiC9ryPeHnfguwt0pa9Qj/CSiYMgkytYS8+71ZJVlcvWE7Lz5CSAlGWD6mCAHa99+B8ZAUMaJmS\nYDkrmVBH/lz0mYs+mwCwSj7Mm4MTgV0j3vuju4bLtwcbmb2GhWV62LWDZmRFGWAsm5Z8LqJ+g88V\nwD8X0e8n1laaMTeTHhEREWlOFx5vKZBZREREOkF3ekRERKQTZSgWfwtFREREoDs9IiIiAmDUgZie\nuZn0sFpbE+0SL3jO1d6qXheGjSPMQpjad5D5kLz9OAyyO24ZxNkaf3P+XcLl1w9WJpaNSNbIcnCc\nlkk6EKvJlcEyjaIENVZji+3RaHg0A4zV5AqyunidrjgTZHN/8nht6sXHcFM/7mNTf7JWGsv0MpKJ\nFmGBj1FG1sowvtz0h9WzK/kZQ+p6BalaQ3Y+ks2OsqzIIaTnQT9oz/pg2WURlukYfeaWR/G1cZfH\ndfSuH6xOLGPXiJVRfD5G1x+GXh+DbaRZWqSPUXAuRfW4AF6TK1d7K94f8e8nlgu48bqQsq7HWyIi\nItIJc3OnR0RERJqjlHURERGRBaE7PSIiItKJOz1zO+mJXucNkCDkRLkJJlNuYtry9RrR156TV8z7\nZEDdzeS958Nh3PcuTAY69sn2LQfHhZWhyGBdZEpIsFGwvntBQC9rmylDwQKIo4BltnxLEJgMAPst\nTwafAnEgc7QMAPpkfJEhKSERBS2vDOPt27G6HC43UuokwsJlh8FFPApuBoBVcnDj8yBuy2LA45Im\ncVtaDiMR4Bx95qLPJgD0ybVju08mMAxXWZkTFkg+2Z4lQbBrWx2i6zEPWGa/FybHx36HZMpTsN9l\nbejCpGd29raIiIhIg+b2To+IiIjUhz0xWCSLv4UiIiIi0J0eERERgV5OKCIiIrIw5uZOD3/dd/2a\nyrzKbkOU5WAk82HV4kycYfSMlmSBsOyJ6PXwyyRrYTnIGmnybwe2R6PlrGQF6yPOuIl3HlselZbI\nlJsAgP2XJjOy9guWAcD+m3aFy7dsmszE2UwyvfpL1dOmhgNS2iDIyNq5sils2yMnZJQ1xbCaQdHy\nVfLXLD+2QQYYGUfmHGvyihYNI/psAvyzPAyylXbY5HkE5Mrj0AxUltWVKGXBr7HrzwyLfi9E+2ie\ndSF7a24mPSIiItKcLkx69HhLRERENpSZPd7MLjCzC83sNaTNX5vZRWZ2rpk9sI716k6PiIiI0MfE\ndTOzHoB3AXg0gCsBnGNmZ7j7BWNtngDgXu5+HzP7dwD+F4Dj1rtu3ekRERGRjXQsgIvc/VJ3XwVw\nOoAT1rQ5AcCpAODu3wZwiJkdvt4V606PiIiIbGRMzxEAto59fTmKidC0NleUy7atZ8WdmPRsZObX\nNE4j/UkNmCjLgSQyOMmRGiYKGPVJFkdUZ2szqWeWqbOVyXtYZjW2yKGNlrOzgNbeisYR1NKatjzK\nCErX3lqazMxjWVoHbNkZL99/+8SyLfvFbZc2xVldkcFKXDdr544tE8v62/ev3C8AjIJzOsxGBDAg\nNcBWg+X0GJLiVoMg24ueM6z2VjC8PmnL+lhN1N4Kx0AGzT7LUd2+EfnUrlr1c4apoyYXv8ZuLP47\nJ1FQbo5997rr8N3rr2t7GKFOTHpERERkurru9Dz40MPw4EMP2/P1uy/+6domVwC4+9jXR5bL1ra5\n217apM3GLRARERFp1citkf8C5wC4t5kdZWabADwbwKfXtPk0gOcDgJkdB+AGd1/Xoy1Ad3pERERk\nA7n70MxeAuBMFDdf3u/u55vZi4tv+3vc/R/M7Ilm9mMAtwA4sY51a9IjIiIiG/pyQnf/AoD7rln2\n7jVfv6Tu9c7NpGcyQC1+MhcFsvHgtjiALwrWY0G+TBjwR84n80TwHemDhkgHgZ9sW5aclJYI2i8l\nApaHNPi6+mv/2doyy2nwaaK0BC9ZwQKcJ4/MEglkXurFQY5bgpITUVkJIA5YBoADD7lpYtn+t5tc\nBgDLB90SLo+s3nRAuHzphoMq9zEkQciD0eR5tzIk526PnLvBvl4h68scW1ZuwshJFp0e2XM6MiTB\n16NE0DP7LC8H1wNn6yPn/zAI3M0EJgPxtZQFVPMSF+sPcM70wX7nzEqgdZfNzaRHREREmqMyFCIi\nIiILQnd6REREZMPKULRJkx4RERHR4y0RERGRRTG3d3pGJOOpH7xSfRS8Th0AikKvQR/BXNDJ68NH\nZN6YWcoi+jNx/j26xsmZeyZLCwA2BfuJZalEWJYWu5XaDxazbJklmpEVLIub8iya4BssS8vI8igz\nrE8yTFh5iiira/Ny/Np/VloiytTa/x5XhW3799gcLo8sX/KL+BuXTC5iJSt27YrXt3Nl0+T6yD5i\n+zTa/+xY8WNbPZuQnWPR+Zg5dxmWpcU+cxH2WY4+906uHSOS1eXBcWHXNZqRlcrIjUXXb7Y+dq2P\njNjvBfI7J247OxldutMjIiIisiDm9k6PiIiI1KcLgcy60yMiIiKdoDs9IiIi0omYHk16REREpBOP\nt+Zm0uNro+HJsclE3mfqo7D4ektkEdC2ZFv6Qd0sJqqxBcR1tqLsNADos2w2sjwSbiFJJImytID4\nmWsdz2F57a3qy1mdrkxmGK/bRLKKgoyl/lJ8ni9tirO6onpaNEvrN98ZL4/G9vcvjdd37eT6WD0u\nti3RdvMsuXh84f6Pm/Jjm8gmpEmNiVpYdHzVuwizurI5QtHnvs/q5ZHrzyi4/rBMr0xNrmyNrVT9\nrsTvhWyNrej3k+pxbay5mfSIiIhIczxV7nY+KZBZREREOqH1SY+ZPd7MLjCzC83sNW2PR0REpIvc\nrZH/Zkmrkx4rXon8LgCPA3AMgOeY2f3aHJOIiIgsprZjeo4FcJG7XwoAZnY6gBMAXLC24drgsh4J\nDIzKQrCgwxELFI7KDyB+hT4vTzGpR4OsEwHLLAiZlJCISk6wchNLNTzP9SBIcUSiO+NRxBJVL9J4\n4Ggi+pSIemClA2Tf8XIM6xedB1F5l7rUca5H15/os5nFrhFRwDJb54gEjGcDi8P10etxFIQct2WB\nxUNMJgmwchN0HEHJiYkknRZ1IXtrr79tzeylZnZoQ+s/AsDWsa8vL5eJiIjIBtLjrcLhAM4xs4+V\n8TeztQUiIiIiFez18Za7v8HM/gjAYwGcCOBdZvYxAO9395+sc/1XALj72NdHlssm7Fy9es+/l3oH\nYFM/fueHiIjIvFkd3ozBaPL9VhupC4+3KsX0uLub2c8A/AzAAMChAD5uZme5+6vXsf5zANzbzI4C\ncBWAZwN4TtRwy/Kd1rEaERGR2bXcPxDL/QP3fL1rcPWU1rKv9jrpMbOXA3g+gGsAvA/Aq9x9tcy8\nugjAPk963H1oZi8BcCaKR23vd/fz97U/ERER2TezFn/ThCp3em4P4Om7M6x2c/eRmT15vQNw9y8A\nuO/e262JeielEcISEiRpwdirzIP2RtZnJCwqXrr+ZDlWbiLK0gLikhM9EpZlNWSkRHs0k6UF1JS9\nEhzDGpJX9mEckxvD3no6HMXHNlo+HMR7dbASZxmu3nTAxLLlS34RtmWlJcKxXbKLrO/2lcfGtiXc\nbrKP2D7d6Nv17ByrI2NvvZ+LOoodsGsEu6ZE5SnYtWrIsltrOIRRNhXPFmOZV9VLSERZWtPay8ap\nEtNz0pTv6a6MiIjIAhh1oAxF2+/pERERkRnQhcdbrZehEBEREdkIutMjIiIinUhZ150eERER6YS5\nvdPDIu/Dmlxk8srqo8Rts9H4k/NJNmaW3RTV2Wqy5o/TSkVBBhJJU+kHWRxsZt0nm8JqpUXY3g9r\nXiX7yAT1ZfoejuJ+ByQzaWU4+THdtRpnQu3csSVcvnRD8DLPS8KmWL62+gvSoiwtANgerI+NjW1L\ntN1sH7F9Gu3/bP5MdB5kz6XM+chEnwv2GRpG2Yuk32EirZFfI9aPXdui6yC7cufqdOWu6dHybP2u\nqM5WZsxNU0yPiIiIyIKY2zs9IiIiUp/ZuefUHE16RERERI+3RERERBbF3NzpmQgYozG31edx7FXh\nfcsWTqgmCshrw4gELo5YeYqgPXvtfIQFJvNyGNWx1/tnylBk+hiQv4SWyPKobMIqOb9WR/H5GAX0\n7lzZFLbtb98/XB5hZSHCoOdkH1HQ8i1kbGxbou1eHbF9lyhlQY4VO7bRecDOmcw5li1NEY2OfYZ6\nNtl5FNw8TXSdoAHcbdR4CbBrbPWUlRz2OyRjosRSi5SyLiIiIrIg5uZOj4iIiDSHFe5dJLrTIyIi\nIp2gOz0iIiLSiZgeTXpEREQkHVw/jxZu0hO90rtPCz3Mhp4395RxFKS59UjqW9S2aB+1jf8iiPa0\n0dfLNydKJhmQDzTNvAras8wfVgZhGGTXsD5WhvEe2TWY/Jj2Lc54YqIspl27Nodt+0vVc12Gg/iz\nFZWWYFlaO1bj5dF2s31Ej0uwnB6rTB80AyxcTLO61ot9huLPHPvcx6KSE+wawcpTsPbrxa6Zwxm/\nUTFLJSe6auEmPSIiIpKnQGYRERGRBaE7PSIiItKJQGbd6REREZFO0J0eERERaSzgfpYs3KQnU9+q\nR+p0WdBHtGz68mYyxlg2xMBIxk3mJE7UM4vqcRXfaOb2KFsduxsb5UiwzWM1iTLZW6tkeT9Y3gsy\nqQBe821nomjSkGS1DILaVLR+V696hkmUFUbXN4jrdEVZWgCwM1F7a4WMIzou7Fjx7K1qywB+jmXO\nx43+xcPqZg2CEQ5JnagByUoaBsvZtaqpTC8gvh4bGXPmWs9+h7D9FP1+ivZRW1hW7iLR4y0RERHp\nhIW70yMiIiJ5rkBmERERkcWgOz0iIiIyMynrZnYogI8COArAJQCe5e43Bu0uAXAjirC5VXc/dm99\nz+2kx0gAWdh2g4ONgThgrcn1OQmGG0WBxcl4QQs+CG7VX2nPXlHPXtnfD4LpWKgfGUYYjsfqyrDA\n0ShYdZWUMOiRAMCVKPgxXl0qhJBdnFgg88pw8tzrk51XRyBzFBQcBTcDwK5gbEA85p2k7SoZR1S2\nYkCOITu20XnAzhl2jkXLadt4cfgpGpCBxCUkWL/ksxz0nQlYBoCRTS5n16o6sGtsL1hn9WIr+fWx\nIGkP1pr5Xda0GUre+kMAX3L3t5rZawC8tly21gjA8e5+fdWOZ2dvi4iIiAAnAPhA+e8PAHgaaWdI\nzmPm9k6PiIiI1GdWHm8BuJO7bwMAd/+Zmd2JtHMAZ5nZEMB73P29e+tYkx4RERGpzQW3XIkfbb9q\nahszOwvA4eOLUExi3hA0Z0/eHubuV5nZHVFMfs5397OnrVeTHhEREakt2uroA+6Kow+4656vP33N\n9yfauPtj2M+b2TYzO9zdt5nZnQFcHbVz96vK//+5mX0SwLEApk56FNMjIiIis+TTAF5Y/vsFAM5Y\n28DM9jezA8t/HwDgsQB+uLeO5+ZOT9XMJyOv8s/oBX30aFbAxs4bo2wIAADJ2rHoriDpo8cygoL5\n/4hs9zDK+CD9sqfHq9Hwkru5FyWtJbNlhkE2zyq9y8oGWH1jMn9lsZIJA5LFtNQLzmmSvRWeM4ST\noxjFBrCxscyraPnORJYWAKwEx3AXXV+4ODwPaCZUojxFpvwJEI+PtY0yI6PPJjDl/A++QzO9yDUl\nKi3Byk3Qa1tD2LXbybXeg/GxchNM+Psp2UeTZujlhG8B8DEzexGASwE8CwDM7C4A3uvuT0bxaOyT\nZuYo5jIfdvcz99bx3Ex6REREpDmzEsjs7tcB+A/B8qsAPLn898UAHpjtW4+3REREpBN0p0dERERm\n6eWEjdGdHhEREemEub3Tkwkg7pHXfPPyFFEJiVwpizoCnEdBIGG23/iV7+sP9qbBiEGwJAugpFsS\nxvWRgFnyCDrqm73tvU/+vAn7YM+8WU2BoBe670gPUXDhKhnHMgnSjUpLsP3PApwjLAYg2hZWsoJt\nSxT4vIsc8BUanBzsOxawTEt7VFsG8NIqw2CdfBzx8tXgHGOfrWh59NkE+PlYhzpKTkTXwazoujlK\nlpDI/F5gv3OiwOeNToaZZlZiepo0O3tbREREpEFze6dHRERE6jM7yfPN0aRHREREZuk9PY3R4y0R\nERHpBN3pERERET3emkdxhP36s5VmBc1kYFlMvrHbHr2m3lmGCekj2kIn2VFLxm7HTi5nTVkWTZiA\nwRvST88AACAASURBVKqAkM7j/cFKN1Qv7bGZZQ+RbewH2U39RJZWVpQJlcmOAuJyEVE2FgAMyPLV\nKPOKtM1kZGUzr3aF2Vtx4wH5vETjYJ+taDkrIbHReMmKxfmVy37nRJlhTq+E0oSFm/SIiIhInmJ6\nRERERBZEa3d6zOytAJ4CYBeAnwA40d1/0dZ4REREuoy+Y3WBtHmn50wAx7j7AwFcBOC1LY5FRESk\n07yh/2ZJa5Med/+S+553cn8LwJFtjUVEREQW36wEMr8IwOkbvdKeNZfZFGUi9ElEP8tm6IVT5I2f\np/ZYalhFLCeD9doLMk8GtDX5OyJYaY9mepFxVF44RZQpFB9YjGhWUfVMqGWSkRVlas1K9harvRVl\narHsKLo80QermzVI1M1aIcujTK1MlhYQZ2Sxz1YdeVDR576NPKOofhe7ZjJNZYax3yFRja150IXa\nW41OeszsLACHjy9C8Vvq9e7+mbLN6wGsuvtp0/ratfrzPf/u9/ZHf+mQ+gcsIiLSgtXhzRiMbml7\nGAuv0UmPuz9m2vfN7IUAngjgUXvra/PyHWsalYiIyGxZ7h+I5f6Be77eNbh6w8cwn/enctrM3no8\ngFcB+HV339XWOERERKQb2ozpeSeATQDOsiLW4lvu/nstjkdERKSzuvBywtYmPe5+n7bWvdvI47C8\nflh/IMZeIR7dJuwl2gIII32NBMj1SITtKAj0nZWiHOwV+qMg4DgKbgaAEQlwjvb0LhbBSvqIVrmJ\n9EAqSITHlgUsk8UYBQHH7OI0ZGUoesE+JUHgmXhvcljC4zKkpRtIaYmohESylEW0nLWlwcnByRSV\ntwD4ORYtZu9EoZ+LRNtZEV1/eFtSniIIWmZt+fW4mfIP7HfIvOrC4y29kVlEREQ6YVZS1kVERKRF\nM37jsBa60yMiIiKdoDs9IiIiQmMkF4nu9IiIiEgnLNydnuiV5SxK38icL+pjiFWyxuVwadTzgPTA\nDkKcARbH1w9I75uCfKNMRgVrn8kAG5EHxdmyEOvFtppl7YQHkSRrsEyc5aAPlqXFskU92E80S4ss\njypfrLe8yDRh1hrNsKqePZctIRG1Z8c7KjcBxJlarI9ZCYlgn7lUH6nMq+pt2bUquu4WfVe/pvO+\nJ9uzDDB2rWfjq7q+bB9t6EKV9YWb9IiIiEieAplFREREFoTu9IiIiIgCmUVEREQWhe70iIiISCdi\neuZ20sMi7/tBXtGI1KuCkYysKNOF3PXjkf6T42AZTyyrK8zUIuNY8vhQDmyy9x6rdWQkNSnaHyzV\nKKhbRm8nkg9YuCU1ZHple4hqRdH6WKSPaBOXWZZWon5XlI0FAMPEfmryNm8mR4XX75pEa2zVkHlF\nM7ISvwiafDgQ1dkakMENgzOPZVgNyfVxGBwBdo1gWUnR9SfTthjH5DU2W3trGFxlR8kMq6jO1ohc\n/+nvnKjtjGd0LZq5nfSIiIhIfbow/dKkR0RERDrxnh4FMouIiEgn6E6PiIiIzMwbxZs0N5OeiQA1\ncnTCQDYWfMqOcBDg7CQwrW9xGYrMs1GjQXmJG3EsgjLYxkEQbMza0r5pOYDJbVki2zEggx4G7ZeT\nNyXjwPN4fZmeWVArC4KNAm9ZCYklFtgdtO+TQVviqtXkBS4T0MvGEQUns3ITmRIS7HOfCVhm5wwP\naJ/sfEhWuEqXB4HF5LoUrW9ArzPx+qKg5SEr/0ACnKMA4kzAMlvOA5lZKYvJ8bG2Q2fJKUEgM93/\nZHkQDM1+t0gz5mbSIyIiIs0ZsazcBaKYHhEREekE3ekRERERvZxQREREuqEL0UV6vCUiIiKdMDd3\neiYi3FlZiOD2HCv/kMnqomUoyO1AC+aTbvE8OmpbiDLDSOkMJpF5NbL4dFgK7nkOSbZGVAZkSOor\n9MgBWA2C6QYeH8UhyUQbRUc9OcVfSpR0YLeFo/ODlbJgmUlR9hbL9GLn6Sxj2VRRRhbbR+xz2NTt\nelb+gS3fFWzkyij+DK2QbJ7VMAMpXl9UcmJErj8sIyvqg2dexcuj9pksrWIck+Pm5X+ql5CgGVZs\nfwTHJZOlBQBDj8pykPI/LejC4y3d6REREZFOmJs7PSIiItIcxfSIiIiILAjd6REREZFOFBzVpEdE\nRERUe2uWTETJJ+pE8Qyr+AlmL8q8IjlgtI8gqyhbvyvOUIjb8m0J6sXQ2ltxBka0T6N9BACjIPy/\nl8iCKlY32feQZbSQzLCVoHDT0OPTfUuP7I9gcXZbov3Bzl2WlxdlkWXqdAFANOwmM72ivxhZZkim\nnhbLjsqo4xiukD+Jd47i83THaPKzFWVjAcCQfJaj+lYseygSZWMBPKsrzpqKrxGsj7BeFa2PxbLL\nglqIM1I3i9YAo32r9lbbFNMjIiIiGHkz/2WZ2TPM7IdmNjSzB01p93gzu8DMLjSz11TpW5MeERER\nmSXnAfgNAF9jDcysB+BdAB4H4BgAzzGz++2t47l5vCUiIiLNmZWXE7r7jwDAbOqz6GMBXOTul5Zt\nTwdwAoALpvWtOz0iIiIyb44AsHXs68vLZVPNzZ2eiSAyNv+LZqqZtojLBPQSgdNArpTFiJVpCNqn\nX50ezGtZ0DPbFssE2oVlL3JlKCIsUJIdwygYdJW89n/V4+Dw/YPSF5tI0HMmKJg942YlBaJAfF6y\nIh7IUhiUHfeR+UuInRnRNkaByQAPTo62m+0jI+dStI1Dsj52XFaC4OTt5FzaQQJpWXByOA52ridk\nylBkAovr6INdq/i1LRgHKfPQZAmJcFsSAcusfSYYvWl1jeTKlctx5crlU9uY2VkADh9fhOKK/np3\n/0xNQ5kwN5MeERERaU5d7+m58/KRuPPykXu+/t7270y0cffHrHM1VwC4+9jXR5bLptLjLREREZlV\n7D76OQDubWZHmdkmAM8G8Om9daZJj4iIiMAb+i/LzJ5mZlsBHAfgs2b2+XL5XczsswDgxXPIlwA4\nE8C/ADjd3c/fW996vCUiIiIzw90/BeBTwfKrADx57OsvALhvpm9NekRERES1t2ZJ5Vd1h9lDibak\nfZTRBWSzuuKniTx6Py59sV4ss6Cpp50sS6tPtm8pyJpaJm2Xrfo+Ypk1230lXD4aTWZ17ReMDchl\ndbELCz/DJ3+AnXZs+Sg4gfvsnK4hEy3KvGJZU2w5y1CL9MiWR0szWVoAsCPI5tlJziWWRXagbY5X\nGlgl2UNR2YqoNEUhKpnQHH5NaWp9ua2J2meytIr2iUy0RIkLNo42zMp7epqkmB4RERHphLm50yMi\nIiLNmZ03BjVHd3pERESkE3SnR0RERDoRyKw7PSIiItIJc3OnZyLqPTMjTdbeymSAsayuKDMpqv8C\nAP1EBtKs6wV1tozMraO2QJyptcXiU/V2y/Hyux0weWAu/EW8n68Z7giXR1ldI1Kna0Q+SstBLSxW\nONgTqROryfMxWk4zEqcWNl7TR6JuVjZ7K2NExjwM/nRdJevbMRqEy3diMlOLZSQe1t8SLj/64Mlz\nb+st8ThuWCX7P8oqZXXEbHJ5j7ZdnEgOdo0N2yaytIA4UyuTpVW0r95HGzpwo2d+Jj0iIiLSHD3e\n2gBm9gdmNjKz27c9FhEREVlcrd7pMbMjATwGwKVtjkNERKTr9HLC5r0NwKtaHoOIiIh0QGt3eszs\nqQC2uvt5LLBz3ESwVzY4ORxE9T4sGWwcBayZtT3H3DdxUN5sbAs73FEgLSsVsTyMj+0OmwxkXvE4\n2JW91WsUnDdL5Hyv8jnYKxasGiyL9hEA9IIgWCZTUoMFatcRxumkhMQgWOcu8tr/FcTHNgr03eyb\nwrbsHIv29az/UZ0t9TDL6ij1EJehqB6wnO2jDbMzkuY0Oukxs7MAHD6+CMVn/Q0AXofi0db496jB\n8KY9/+7ZJvT7+9U3UBERkRaNRrsw8l1tD2PhNTrpcffHRMvN7P4A7gHgB1b8eXskgO+Z2bHufnX0\nM0v9gxobp4iISJt6vc3o4dbCtMPRTVNaN4O9AmGRtPJ4y91/CODOu782s4sBPMjdr29jPCIiIl23\n+FOeWQnMKPZ1DQENIiIiIrGZeDmhu9+z7TGIiIh0WRdeTjgTk555EL2CHOAlFmZF9Kr1Hhkzey17\n1J61jTJdeh7fxGOlAwZBhsMqWd9Ng/i4/OSmyb5Z+YGMFSPZWywTKrqBScpvsDMpc4bRHJXEtmey\nyDKlM+rAsktYLEJ03rAsLXZsWcmJyC3D9Z+P7FwfBMtZCYkoI4i1ZZ/l9bbdl/azgF3rZTFo0iMi\nIiLwDkT1zPZtChEREZGa6E6PiIiIKKZHREREumH+IrDy9HhLREREOmFu7vRMZCOw23BRokXylt08\n1shiWRJ9TNZ+yrQFWEZW9XEMsnXSEseQZe3sCjKQWNsoKwYARsFKWZ2cgcUZH7uizDWLP3ZLLMut\nhldYpQIUG7rFbQ1ux4As3xXUSmPHih3bUfC3ITtndoziDLDM+bhKsodWg+yyIWkbbSP73NPtJtle\nYdsGM8BmWabGFhDva9a2DRudjdmG+fvtLiIiIrIP5uZOj4iIiDRndu45NUd3ekRERKQTdKdHRERE\nOhHTo0nPDBsFQXIsyLpPbtpFr1TntzBXSR9B37YcdxEEjkaBoEUfZBjR5460ZR9SI0HBkWEqCJNd\nFOKAxqiExyoJfoTFgeTR3mO3aGmZhmDcLJA2E/TMgpN7QeButsxGtC1s/7N9Gh1bFvzL+o6OSuac\nARCe0zQomwRaR+NmbQdBqY1swPIwuB7Q8jM0SLp6SYcROy7s8yK10+MtERERkQWhOz0iIiJC7/4u\nEt3pERERkU7QnR4RERFRlXURERGRRTE3d3omXvdNknOshnlclOXA+qWvEE+UsmAZDtE6WQYGf4X+\npGzmTyzO9BoFGUhRBhMAXloiyCZZIs+aRxYv7yVKHrCsnShbhr6a3+NtjLJreiyzjOwPlpmXEWUb\nse2uJXsr2MamtmPa8rgcAzuX2LENlpFDmCvHUP28A4BBUIaCrW+YaMuuP1H7OvrIXsMy68vIln/I\njG8W+t0XszOS5szNpEdERESaw1/JsTj0eEtEREQ6QXd6RERERCnrIiIiIotCd3pERESkEynrCzfp\nCSPya8j0YpH+RuolhX3Q2HhWNyvInmDnZGIbPawmBBjNqpjsY0T66IXZGvH2sWyZJQ9Oy2S2TI9k\nU2VE46PZOYk6Yiuk7aZouxEHF7LstExGVjqLKcD2swfjY+vLbAu7KK8E2UpALgMvdWzJ5zCTU5Qd\nxzCopxVldAGs5t7662bxtqzvyT5oW1JjK5fVtf78o2xWV2YcdfQt67Nwkx4RERHJ60L2liY9IiIi\n0olJjwKZRUREpBN0p0dEREQUyLzo6glOZq9fn7yJ1mNlF8gNt171Sgo0wDnqg4fSsUDCyf3RVNBz\n2XnUcYjvu2D/s4DqVOmAXDBoNA62LSzAOQr0zQZqx0HZ67/AjYIyD0BuzCz4NzPmOko3ZLAg5Mw5\nlg0KjrZlyMrBhAkF1UtFsHFEgcnTxMkYuYDlMKCaXLsnyhXtbp/YHxkKTJ4/nZ70iIiISEExPSIi\nIiILQnd6REREJPWOrnmlOz0iIiKCEbyR/7LM7Blm9kMzG5rZg6a0u8TMfmBm3zez71TpW3d6RERE\nZJacB+A3ALx7L+1GAI539+urdjxHk57b3najxWATmT8WZdaAZAAkS1lEUf0j2gd7LfvkskxGFwAM\ng3GwMfdI1towGB/d7kSmF7BMlgcZKTSzie3/KIssd2OzjlfaR/uOlt8gH8co66lnyRMhIfOXGSsh\nEeXFDEmmV2YcbN8NghINQHPHkJ3/qUyodBmKyc9Fpm2mVMS09mEftZSQIJlXwTWMb8v6yz/wEhLV\nz1+2vqjvTL9Nq+PzUgd3/xEAmO31QmdIPrHS4y0RERGZRw7gLDM7x8z+c5UfmKM7PSIiItKUulLW\nbx5uw83DbVPbmNlZAA4fX4RiEvN6d/9MxVU9zN2vMrM7opj8nO/uZ0/7AU16REREpDYH9g/Hgf1b\n5zPbBudNtHH3x6x3Pe5+Vfn/PzezTwI4FoAmPSIiIjLdjKash3E9ZrY/gJ6732xmBwB4LIA37q2z\nTk96aNBWcIePlaagAXKJgGr6OvSgjygwuWhavQwCWx8tyxGV1CDhZVEPLHCMvUI/DnDOtAX5mNRQ\nfqCGPnjwNSuLUr2kBlvelDoukZnSHjSANRkUnBHv0+bOpUxpCdY2U0KC7tMgwDZTKoKhJSQyQc8s\ncJr2HWxLsoRE1H5WAn/rUsfnpQ5m9jQA7wRwBwCfNbNz3f0JZnYXAO919yejeDT2STNzFHOZD7v7\nmXvru9OTHhEREZkt7v4pAJ8Kll8F4Mnlvy8G8MBs35r0iIiIyMzc6WmSUtZFRESkE3SnR0RERBYu\nRimiOz0iIiLSCXNzp2dt5Dx7OXVYnoK9yDpRnoK+KjzTN2k7ZOMIZt00O4f2Xf0V+iwjJcwAS2R6\nOcms6ZPMqzgjJVGyAoAH47CgRMY0qYyU1Ov2SQkDkiEYHfOozEYxjup/xzSZ6ZXbH9UzsniZB1LC\nIDEOtj+GQd91nEuZrLViHNVLS4QlK5KlIsIxZzOeasjICrP4EllarP1Gl5vgfc/O3ZUZTVmv1dxM\nekRERKQ5CmRumJm91MzON7PzzOzNbY5FREREFltrd3rM7HgATwHwb9x9YGZ3aGssIiIiXZd5pD+v\n2rzT87sA3uzuAwBw92taHIuIiIgsuDYnPUcD+HUz+5aZfdXMHtLiWERERDpt1ND/Zkmjj7emlI5/\nQ7nuQ939ODP7NQAfA3BP1tdwtH2s3yX0sImsM1qazbQIDhLJsKIStbcyGWBDcgLR2ltR5g+5hcnq\ni42CcbDaW1V/vlgfqx81KVeni44k0Zb1kM1eibJ22Pk4CJdGxzDKqAN4Rlw4NpIBVodMFkim1lQd\n+59lXrG+Z6X2VqaeVh11s6JMrToynmgGXiIzLFs/MBp3vn5XPZXm2PrcV1E++GjNrE1QmtDopGda\n6Xgz+38BfKJsd46ZjczsMHe/Nmrf621paJQiIiLtMluG2a1/tAxHu1oczeJqM2X9UwAeBeBrZnY0\ngGU24REREZFmdSGQuc1JzykA/tbMzgOwC8DzWxyLiIiILLjWJj3uvgrgeW2tX0RERG6lmJ4F0WQJ\niVSAc7KPKHCOBf+yPqLAZxpAnAi0ZsHJmQDnoZPg5KAPFnLLgz6bCdLNB9JG7eNgRVYeJNoWHgwd\ni4J3WbBxpjxFrtxE/DkckP2RKd2QCTJlQf/MRp9LtCxHsD/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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig2 = plt.figure(figsize = (10,8))\n", "plt.pcolor(x,y,np.log10(np.absolute(amplitudes2)**2), cmap = 'inferno')\n", "plt.colorbar()\n", "plt.title(\"Logarithmic intensity around a point\\nsource in a ring of 5 scatterers\")\n", "plt.xlabel('x')\n", "plt.ylabel('y')" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": true }, "outputs": [], "source": [ "#create a list of 20 point scatterers in a ring\n", "N3 = 20\n", "p3 = []\n", "for i in range(N3):\n", " position = np.array([5.0*np.sin(2*np.pi*i/N3), 5.0*np.cos(2*np.pi*i/N3), 0.0])\n", " strength = 1.0\n", " p3.append(Scatterer(position, strength))\n", "\n", "#calculate the amplitudes\n", "amplitudes3 = Foldy_Lax(p3, 'spherical', x, y)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Ring of 20 Scatterers\n", "The scatterers are now separated by less than a wavelength, so cannot be resolved. The wave 'sees' a continuum of scatterers around the source; the interference pattern has near perfect circular symmetry." ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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B3IS4gvuo8UYGt3U7gCPMbFSx0X3tfjq03Ye6+ylkPB0Uj8cVY44PZrYNwNnI7pZ8YOjb\nPwLwywNf3z9fVuBZ4c6/cPfjkcUFnYIsJmjUdv8xgHsie4R2GIB9rxzYt+/HCVreDuB1Q/trq7t/\ncHB4Q+032r/DRVYvdvenu/uRAP4awEfMbPMYYxMRQpMemRtmdi8z+00zWwGwjqwSez//9j8CeI2Z\n3d7Mbo+sKvt78+/9AFmF6ftZFox8Kkb84nJ3B/BuAP87Dy5tWBa83EZWnfsUM3tUvnyTZe+zSbmr\ndJvNIssPRjZxuM6yIO3/kS9LHe+odcDdrwbwWQBvzwN4W2b2kKDptwDcnAffbjKzppkdb2YPDPrs\nA/gQgNeZ2dZ8AvMS3Ho8RjKzowF8EcBb3P2dQZMzALzUzO6Ut30pBrK7hvo62czuk99N24Vsn/aC\n7W4PbPfByM6tm8zsCGR3FgftADCcnn710LJ3AvgvlleXN7ODzOxxZnYQ2eyx9+/Atv1efq4DwE5k\n53SftReRjWnSI/NkFcDrAVyD7C7OkcgeAQDAXyILBP03ZJOcbwN4HQC4+4XIAp2/COAnAG6TyUX8\nCYDzAJyLLKbk9cjiYi4H8EQAr8rHcWneln1WNrorwL7/+fzPT5DFf+xG8fHGhuMN1hGt7xnI4lsu\nQPYL/cWFQWYTmccju6tyCbI6Z+/EbR8zDXpRPuafAvgagPe5ezgxCfw+gLsC+LM8G+pmM9v/GMrd\n3wHgk8i29wcAPkEmR0AWu/MRZJOCHyGLx3lf/r3B7b4at273m5DdGbwWWbDwZ4b6fDOAJ+eZXW/K\nl70WwBn5o6knuft3APwBsiyv65Edx2cN9DF81yZ1/wLAYwD8KN83bwTwFFfNOZGJzLz2lpm9BNlF\nsI/sIvccd1+f6aBERERk6cz0Tk/+yOCFAE5w9/shC4586izHJCIiIsupNesBIHtnxUFm1kd2y1m1\nekRERKR0M73T4+5XAvhbZKmvVwC40d2/MMsxiYiIyHKa6Z0eMzsMWdDoNmSBiB8xs6cPp7Ca2WwD\nj0RERKbM3adWO+8udznSL7302qq6v9Td71JV5ylm/XjrPyB7d8X1AGBZbZxfBzD83g4MD9VYlm74\nfrG0G1oWtU+uUVlsH/YLoN/fg0aj+PqN+J12cR+sPiFbZ9wH6TvaloT90entxErriPHHMeH6sj7G\nr9eY0ndjaGx7OjuwuX3U2D8/ehzV1ZhMOQ8G7V6/EltWDjRb/7a8wmxr994B/dw4x6+fMO4sSWvM\ntkgbc9T3Rvt0vXv9/s9e6j5KWV/adrM+2PiCceTrY9fO8daZeD4ewP4fa50kmcjRSex7Mpdeei26\n/r6NGx6Alv2nbRu3mo5Zp6xfBuCk/L0Vhqz8wPkzHpOIiIgsoZne6XH3b5nZRwB8D9lLxb4H4LRZ\njklERKSO+v0Du2O6SGb9eAvu/lpkL/6qLbOZH4ZKNce4/byoWg32At7l0G6GL4leGst+/Jb5swcs\n/7Vz2ty7sx5C5Wb9eEsA3FpNYDkt84W33dw66yFUatknPct+/Jb5swcs/7VTyqdpsoiIiBxwQsAi\nWZhJTyFbK8zSAlJuXtGMloSsqZS+eZZQSoZVWpZWSmZSSh8pmUbljI1s9wSZVxuus4RtrEojOZuw\nKCXDLVWcmZQ25n5CRlBqdl+EZeI0g/1EfzmQcUQZYOycoZlQwSUvqY8ykp/Zy0NS+k7sI0pu4seb\n7Lugj+T3oET7n3TCs7qizODxxyyTW5hJj4iIiFSnr5geERERkeWgOz0iIiJSi+wtTXpEREREk565\nUghcripgOe47NVC1jBISYdvEoOBSSjoE45v34OSqgpDLCCDm45g8sHjaAdVMyjhoAHHCMUwt6RAj\nQchBUDA7R8sIhu6zCjvB+CoLeiZ9MKUEVJcQJE0qOsR5L4mBwmFzdqySApzn4zNbF4sz6REREZHK\neH/57/RoiikiIiK1oDs9IiIiAtQgpkd3ekRERKQWdKdHRERElL21iMIsgsTyD2VkPIUZYIklJMK2\nCVlarH214wgyvRL3XZSRlZKNlY2j2Edq5lW4LTO4OdpI3PbIPGR18Vfzj799fZLxVOX2WVAmgGeL\njZ8BxjJ/okwvIM72qirTK/uBoGli1tq4/aaOg0oqZZHQLxkHHVpCVlfSvqtavzO1VZnZuwA8HsAO\nd79f8P2HAfg4gJ/mi/7Z3f9y0vUu3aRHRERE5t7pAN4C4IwRbb7m7k8oc6Wa9IiIiMhUH2+5+zlm\ntm2DZmWUyb2N2d/zFhERESn6NTP7vpl92sx+qYwOdadHREREgJJeTvj1cy7DOedsn7Sb7wA41t13\nm9ljAXwMwL0m7dScvbd7jpiZm60MLU0I6FUg8xTHoUDmsi1/IPP4WCBzlaJxp5a9iAKZU/dHGMic\n0AcLZGbbQgOfwz5I38GY+ThYH9WMA8n7P+UYkuUJfbjvhbuX/niHMTPfecPLKun70MP/V7gt+eOt\nT0aBzEHbSwD8irtfP8lYFuhOz20v2mn1tCafKKQ+CSyjXlUpk5BSxpFQxyrog9XH4n2MP+aUiQyt\n9UX7rqZ+Vxkac16nK76Qp425H/wyblZY+ywlu6zvaVlMjfBXV0KmFxBGNzQS/s3KMr1YClL0OaST\nvRLqZvFxBJlodFIx/jhS/72fUr8rJauL1+magemXoTCQs8TMjnL3HfnfT0R2k2aiCQ+wUJMeERER\nqcwUA5nN7AMATgZwOzO7DMCpAFYAuLufBuBJZvaHADoA9gB4Shnr1aRHREREpsrdn77B998G4G1l\nr1eTHhEREYGpyrqIiIjIcliYOz2FoMuEjKzUwN2UuWBKJlQZmVfVjoP0kRCcnJJllRJIywKWU4KT\nWWBy0jgSg3GrChZODQ5PkRK4nlqOIW0c1WxjajmMlIBq1ncU+JwW9Az0JywLQctbkGMYZoaVUUKi\njHykCscRZ3oBcUZuwvrY4jLKb5RFd3pERERElsPC3OkRERGRCtXgTo8mPSIiIgKbYsr6rOjxloiI\niNSC7vSIiIgI0J9+mZdp06RnTKn1qsKaV6nlH0ooIRG2TcjSAtJqYc1LzasoU4uWm0gsTxH3Mfn+\nT1Fl9laaKrPIJq/VFWcmkfIPZH3NoD3N0iLjiDKyomwsYNRnPKUGWLGPlPIWQFziIrWURVh2gWyf\n0+UJ50FKVldiBhjP6pJFokmPiIiI6OWEIiIiIstCd3pEREREMT0iIiJSEzV4vLU4k55CMOz4JR1S\ny02UEUCcVP5hzktITBqcnFIqgkktIREFJ/O2k29LSh+pQchlBD4vomireeBurJ90bOOA2ZRgaB6k\nW+w7pYQEQMpQJKyPlbfggd1RMkbKPor78MSA6lJKWaT0kRDg7CwYPaHvWVSbqLPFmfSIiIhIZawG\nj7cUyCwiIiK1oDs9IiIiokBmERERqQc93hIRERFZEgt0p+e287O0bKoqs6YSylCUUspiuiUkgDhT\nKyUji2VeMWEfiRlg4XYnZkFFfVSZecX6bnhKBtLkr/LvB+UOmOmPLbWERHH/s0wjtv+jDKSU9bF1\npu6P6DPO911xHH1yrFgfKaUzpl3KIrWExMTrY+1pyQrSdcq2zILu9IiIiIgshwW60yMiIiJVUUyP\niIiIyJLQnR4RERGpRUyPJj0iIiJSi8dbM5/0mNmhAP4ewH0A9AE8192/eeA9VpU1lVh7q5T6XUEf\nCVlaqeNg9bTCcdA+UmpeJWSA0VpH1dXCKiPzKmxLsmjYfmpV9DFlGUjGaiMFaMZZRU/Oux4XRGyw\nelVhJtrkGWANWmuK7bvJ61hF5wet0xX0wWpv0YysQHIfQfsoowtIzOoqIwOMqTCLLM7qGv/zJpOb\n+aQHwJsBfMbdn2xmLQBbZj0gERGR2tGdnmqZ2SEAHuLuzwYAd+8CuGmWYxIREZHlNOs7PXcFcK2Z\nnQ7glwF8G8CL3X3PbIclIiJSL9Zf/kdts05ZbwE4AcDb3P0EALsB/GnUsN/fu/+Pk+f6IiIii8i9\ng35/z/4/Uo1Z3+m5HMB2d/92/vVHALwiaths3DbUhwX0liGpDEUppSzGLy2RErDMxpESsAwATWuP\n1W82vpRA5pQSBmkBy00Ux5waXJtSYoFJCcpuefxxbARRkWyfMlHAa1ReoSxVjZntu67F/xCKgrJp\n2YWE8bFyGCz0PQ5w7pC2MfYpiqQEPbPrUtS+RwJ3kwKcE0pWACTAucISElTQR0oJFWbf/jdbBbC6\nf3m3tzZx38kU01Mtd99hZtvN7F7u/hMAjwDw77Mck4iISC1p0jMVLwLwfjNrA/gpgOfMeDwiIiKy\nhGY+6XH3HwB40KzHISIiUmcp7+haVLMOZBYRERGZipnf6REREZE5oJie+VHMMKiu/EOKtFIW42dp\nAXE2SWpJjShTKyVLi/U97RISqX1EmVBlZJGlCreFZIVF5Q6yPoppJumZZcW+4xIN5Sgj861vUQYY\nSduhmT9R1trk291k66OZYcFylsVES1wU9ylfX/TzbH2xqD27RqRkdaWUrADirK6kkhWsb3oqjf+L\nn/4OSRhHXJpCqrIwkx4RERGpUA1eTqhJj4iIiNRi0qNAZhEREakF3ekRERERWA0CmXWnR0RERGph\nge703HZ+Vk7Nq4TMK7a+hDpWKVlabJ0pWVp0HAn1erLxjd9HGTWvUvrgmVDFUzu1jyhrKlWczRb3\nW+04in20nWTxJRyvHskJ6gfpK05TWmKNYHxRvwDQCDK9AKAfpMaUUS+JjYNlxIVZVjTjjH0+x6/f\nFf17Nv55oEf6SMkAS6mrllSnK+u82EdKnS4gqfYWywgdu98RfZdSA6xKiukRERERWQ4LdKdHRERE\nKlODOz2a9IiIiEgtJj16vCUiIiK1sLB3etLKP6SVoUgJIGaivlMCltk6UwKWWR/sVfIppSWiYOOs\nj4rKUCQELANxGYqUMg9ManmFKCi4lXguRUGzLICYibZxs8X77pBWfGwjN3XjINg93i0s6yUGMkf7\nrs2OFem6G+yn1H0XBSezUNcocDpbZ5AKTINdi/uOt4+PVUopBdZHFOBMz1yyLT1ngdZFSQHOCSUr\ngDjA2T3eR/RanxJwXEI5jJlQyrqIiIjIcljYOz0iIiJSHlNMj4iIiEi5zOxdZrbDzP5tRJu/M7ML\nzez7Znb/MtarSY+IiIhk2VtV/ImdDuDR7Jtm9lgAd3f3ewJ4PoD/U8Ym6vGWiIiITDVl3d3PMbNt\nI5o8EcAZedtvmtmhZnaUu++YZL0LM+lhGU7j/ezkWVMp5SaAOAMpZX1AnKmVkqUFxJlaKVlaQGJZ\niPD19+PvIyDOkOIlMtiYi+tkWVotUo4hEpVzAIA22caDGsWP2KZmPObdvfiCs6u/XljWtTjLgpVH\nWAmy2di2HLVp/DSVm3fFbaMMqXWLs5LYcWkG58GWRpxptIXs073BPr2lH4+jQzKeGj7+/mDHJcr3\nCktTAOjTz0WwkGU8RT9PRsYV93VKyQoA4fhSMrqAOKsrpWQFEO87Zzl4FZaQcF/+mJkSHQ1g+8DX\nV+TL6jHpERERkQqVdKfnKz/s4Ks/TJvcTosmPSIiIlKak+/Txsn3ufVO4V98aO+BdHMFgDsPfH1M\nvmwiCmQWERGR7LlhFX84A39I+AkAzwQAMzsJwI2TxvMAutMjIiIiU2ZmHwBwMoDbmdllAE4FsALA\n3f00d/+MmT3OzC4CcAuA55SxXk16REREZNrZW08fo80Lyl7vwkx6hrN3eDbV+E/s0up3kYynEup3\npdTTYnWzeHbT+ONIqaeVVHuL1KtKychiWVos8yrKCGJtWRaTB+kaNAOMHMND2sWP2LPvGd+hffsF\nR4bL+8Ht4bCWE0ZlDxXtIllJF+0auwvs8mJmGRBnanUsDmxkx6UfXJ5WGvF+ft5x14TL/+HCowrL\n1tbjC3uPZNZENcPYOcO2JTouNGuQZmQVx2dkzI2gbZ9eZ9gvuuh4jV+nKxtHuMKQk3H0gqcjSXW6\nyDrpdpdQN4vVPguveYkZYJXSG5lFRERElsPC3OkRERGRCo0OOl4KutMjIiIitaA7PSIiIgLU4I3R\nCzPpGQ4YTgpCTixhEQUQs4BlFmAbjS8lYBmIg5ZTApZZ+5QgZNa+FZQ1YOtLKVkBxIHPLIC47fG2\nNMNxxH3/WUcHAAAgAElEQVSw0g3R8j4JUGQBjTd2iuNgAcs39+Jg0KikAzsP2iRoPNr2KFAb4MHJ\nEdZHtD52rJhou9k+Yvt0PSg5seZpZSgiTVY6g10PgqBxdv4zUXmKvrFSFpMFPWc/EC1kb9kdP8CZ\nbTX7dRtdB6PgZiAtwJnt/35CGQoWfJ30O2f5nyjNlYWZ9IiIiEiFahDTo0mPiIiI1GLSo0BmERER\nqQXd6RERERHd6RERERFZFgt7p6eM8g+sjwjN0iJR+lGmFi1lkZIBlthHlHmVkqUFxJlaTXLqRH3z\n8g/jZxqxPlbIOFaD4xKVEwCAPR5npERlE1imV5ccl17wWnej6SGxdtD3JpIJxcphREvNqnv/fZTV\nxbJzurT8Q3E5y7xa68bLo3FEJTKy9cXZW9H52PTVsO2qxedjlO215mR9pDxIVMqi7/H5GGUVsRIl\nUaZX3kkRPWXGz+pKKlmB+LxJKTcExFldTj6zDXKzI8yiZL9DUkpZzJEaZKzrTo+IiIjUw8Le6RER\nEZES1SCmR5MeERER4c+gl4geb4mIiEgt6E6PiIiI1OJOz8JMelKj9W/zs4k1r1Lqd6WMi2YrsQyw\nqAZYYuZVlE2VWgsrpZ5WlGXVpFkS42dvsSytQ1sr4fJjNhf7vnxP/Ine02WZJ+ONDeDZZVEtpjY5\n3m22T4MsqybJvGLLWU2iaWLhAj2SgRQt75K2HXK17gQZUuxYOaljleKgZnyexudjPI6dcXJZmBFE\n68ZF5wHZ/11Wryrhc+9J1+fx63Rl6wzWx2qOkW30IDOPXrvJ/rDoHEvM0vI6pEfNuYWZ9IiIiEiF\nlj+OWTE9IiIiUg+60yMiIiLwxBenLiJNekRERESBzNNiWeTwtwFc7u5PiNs0hr4uIbCYvrI/oTxF\nQpA0C1hOK0ORNuZoOQvg5oHF45eWiIKWUwKWs/EVlzdYgG64FGgF32ABs1GpAgBoB6UeopIQALCZ\nlB9oB+dHqzF+qQggDk6eh8DkVE0y5jY5D6KgVHYM++S4dPvF5R0STLqHnNOdIAiWnTNsfNH5yI43\nO9ctKE9Br1TRZ46dMySOox8ECzv5LPdpMsb4WDB0ShkK9rmIAohZ2x45P6LrILsx4qTESDjuBfws\nL7K5mPQAeDGAfwdwyKwHIiIiUks1eLw180BmMzsGwOMA/P2sxyIiIiLLax7u9LwRwMsAHDrrgYiI\niNSVApkrZma/BWCHu3/fzE7GiKebne6N+//eaGxCq3lQ9QMUERGZgl5/D3r9PbMdhCY9lXswgCeY\n2eMAbAZwsJmd4e7PHG7Ybh029cGJiIhMQ7OxGc3G5v1fd3s3jmgtB2qmkx53fxWAVwGAmT0MwB9H\nE54UYQkJktmR1G9i+FNK+6jcRLZ88m0Js8gSssX4OEjbILuDZWkxUWbMmsfv5r+hG6ee7LypuM6o\nJAEQl4oAgC1BRtYqydZbIRlZy5J5NQvRfmKZTUxU8mOFZCC1+/HytSCbh52PO3tr4fLzbiqOo09L\nasTnKcsYi4SfOZp5FfcbZislZo+SYhHhUppVGixz0rZH+y72EpWmYG1HtY/7INfp6LDMU2mKIENw\n2cw8kFlERERkGmb9eGs/d/8qgK/OehwiIiJ1VIdAZt3pERERkVqYmzs9IiIiMkMkrm2ZaNIjIiIi\nSlmfJ8O1tqqsscXqeoVtE7KpUmpsMWzMTRTrRLH2KTW2WHuWkRUtj2ppATwbZd2KmTFOsjLWyJij\n2mCr5HTfROpmrQYZWSxLaxEzsqoc8vh5RtVKyQBrks99M/hFEC0DgL0kq2u3rxeWdY1laY2fgbTi\n8bkb1q8LW474LEefe3ItZTW54q7jaxXQCZf2gqyp9Gtp1Ae7dpP9H5wfDVq3LF4e1eRKzQyWySzM\npEdERESq40pZFxEREVkOutMjIiIitQhkXv4tFBEREcEC3+lhwclJgcUkcDEKLGMBfLQcQxT0llBu\nIut7/BISKUHIKeUmsuXF57xRoDAArASnFAscZa/yjwI5WdDnKhnHpiBYcnODBDLPeQmJaJUtsk9Z\nlYZo3GxTUraxTwI5o8W0LVneDb5RZYA02+4oeD06NwCgQSJYLVi+mwXnk3O9FWw8SxJYDYLzWdkL\ntlN7wTh4uQl2/QkWsiBf2nd0HWR9xNcDt+K+Zucju9aH7dlnhZSWCH8/zVEYTR1eTriwkx4REREp\nUQ0mPXq8JSIiIrWgOz0iIiKilHURERGRZaE7PSIiIlKLlPWFmfSM+6ruKKuLZRwwUfR+SlZY1n78\nrCnWdzRu2pa8Br4ZZVMllG4AgLYXM6GiLC0AOLS1EqwvdkM3Tp+ISkuwLK2t2BQu39Isjm+VpOew\nTJyqsPW1yY5qBs2jZaOWh+UY4qaliHJXWLZMjy4vDpq17cTJMuixjKUEKaUsWOZP04qfIZbpxbKp\n+sE32uR6cGizuD6yi7AzTqLk2V4Rsin9aK0ka7NPsmkbQR9sWyyhhEdUmiJrS8YRbGOPZGnRLLeg\nj6g0hVRnYSY9IiIiUh2lrIuIiEgtKJBZREREZEnoTo+IiIjUIpB5+bdQRERE5oqZPcbMLjCzn5jZ\nK4LvP8zMbjSz7+Z/XlPGehf2Tk9VNbb4+savsQXEdbZS62Y1g/pRLY8PWZSlxdo3abYYyW6KsqnI\ndh+zudi2RXbzzpvi9UVZZFEtLSDO0gLiTK0qs7RSMrLaZBhsP7Wi7C3SNmULq6wjxjK1IqxpL0iM\nIQl/dJ92ghiFKjO92HmwGh6v+Nzt91bD5XvRKSxjWWTHHlRcYZds995d8We5E2QVRXWwAMBI9mgv\nypCi9arI8rB9cV9kXcTjiGpyRfW4gFH15IrbQn+HJNTkSs0MrtK0ApktS7N+K4BHALgSwLlm9nF3\nv2Co6dfc/Qllrlt3ekRERGSaTgRwobtf6u4dAGcCeGLQrvRZmCY9IiIiAner5E/gaADbB76+PF82\n7NfM7Ptm9mkz+6UytnFhH2+JiIhIieYrkPk7AI51991m9lgAHwNwr0k71aRHRERESvP17XtxzuV7\nRzW5AsCxA18fky/bz913Dfz9s2b2djM7wt2vn2RsCzPpicpLRFJKTrBXxkeBZSxgmZeWmLyURVyG\nIjUYOgqoHj+AmLXvkajDy/cUA/VYgGgUKAkAq8FpubkxfsAyUF3Qcpv0u0IObUogMwtOjkpLpP57\nbMqVNsIxp8YJN4N92iR9RMHerD0r1bHei7/RqSjAOQ5uBnokWcGDCFv2Gbp4V3E5+0ywz3L0uWfX\niK6xUgrF9mFpCgD9hGsbWxu7xkblKXhySty3B0HIrG1KeYoeLaoxfWUFMv/G0ZvxG0dv3v/167+x\nc7jJuQDuYWbbAFwF4KkAnjbYwMyOcvcd+d9PBGCTTniABZr0iIiIyOJz956ZvQDAWcj+Dfcudz/f\nzJ6ffdtPA/AkM/tDZKl6ewA8pYx1a9IjIiIiUy1D4e6fA3Dc0LJ3DPz9bQDeVvZ65ypqSURERKQq\nutMjIiIi85a9VQlNekRERGRqb2SepYWd9IybzQWklZtg7XnWFMt4SsgiK6Ftg7wGPiUDw0hWVz/I\n7tjj8Wvg93SLy51kh0TlLQBgkxVPy00Nkp1WYVpSlKm1iWRppWRvsewhdhakbGJKHw2bPCuJ6Qex\nAWxtLHcl2qVs+6JMLyAte4t+CoOsrqoyugB+rveDrK7d5HN4fW9PYRn7fKdgfaRkddFrFbmmR5la\n7DrIMsOi9lFpiqyPWPQ7ICpNwdqy9im/y2RyCzvpERERkfJMM5B5VjTFFBERkVrQnR4RERGpRUyP\n7vSIiIhILSzMnZ5xSzgYKReR0icrORH3MXkQMh0HCfiL+558hs4CjqNA5o7FAZSRtrfD5VuCgGUA\nWA0COacdsAzEQcsskLlFDlX0mvoyApmbJAiZvRY/LmUR95Gyq1k8bxQM2mNtyfJeQjB0ytnB4kaT\n/gU45ZIVQPy56PXjz9AuXyss69h60jiiz20Z15nUPqLrYJ9coxskDLkXBRDTIx4HJ0e/F6LSFJl4\nefj7ifYxfZ7wO2dRLcykR0RERCqkx1siIiIiy0F3ekREREQp6yIiIiLLQnd6REREpBYp6ws76Ukq\n3UDSNdIyr8bP6Mr6LrZPGXNZoswH9ip5llXRD7IZoowu1keb7LtVclxWgiwVlpWUIiVLiy1nWVot\nMr6w/EPcFK1GvE+jvlnbNlkeZXuxcaSUp4jKTQAse4tkPJELbTfopEuG1mUX64T9z76xibWPlJDV\nxc716HPRI/2uBWUhOogzLtlnOcJKx9DrQXD96QWlKcrCy1NEH3JWsiK+IPSC9rQ8ESupEWRqzeL3\nQp0t7KRHREREyqOUdREREamFOjzeWv5pnYiIiAh0p0dERESglPXKmdkxZvYlM/uRmZ1nZi+a5XhE\nRERkec36Tk8XwEvd/ftmthXAd8zsLHe/4EA7TMnIKqMPFnmflF025bkny6ZqsaJEQWJGl/URZI1s\nJjW2omwUoJxMrah+0QrJ0mLLo0ytlCytbBzFZSzDaoUsXw2Wtxtx5gnL6krJ3ipDSvYWy7zq9Isj\nXCNt18n+jzLDWO4QvRgGO2qFNKWVmIKsLpZ5RYcRbCP7DG324tZ0SA28Lsmmiq4Tq+Sz3CX1o6KM\npzKwa2ZUY4u1d9oHqZsVtk/LRIv68MQ+qqQ7PRVz96vd/fv533cBOB/A0bMck4iIiCynWd/p2c/M\n7gLg/gC+OduRiIiI1E8dsrfmYtKTP9r6CIAX53d8CtY61+z/e7OxBc3WoVManYiISLU6vV3o9m+Z\n6Rj0np4pMLMWsgnPe93946zdavvI6Q1KRERkitrNrWg3t+7/eq378xmOZnnNfNID4N0A/t3d3zyq\nkbEg2ykpJ0A6tZRFUI6BjIONL3pt/EGN+LAf0o6X39gJXiXfj4P9ovW1ybGLgo3L0g5WGS0btTwK\nHE0JWM76LgarbmrGAaybSHDySrO4vE1KRUQBywDQDPqeehmKIDAZAHpkp0ZB2c1e2mv/Q+QWPgsn\njXpm50yP7Lpoea+E+FX2GYo+c+0gyQAAGuQYbm0Uw7UPI9eImzrdcHmnX9zIbmLyRz9czoKN2TV2\n/IBqfq2fn4DjqtTh8dasU9YfDOD3ADzczL5nZt81s8fMckwiIiKynGZ6p8fd/wVIvP0hIiIipVPK\nuoiIiMiSmIeYHhEREZmxOtzp0aRHREREahHIrEnPhFi2QFIZCvJuhKgP3pZkwAR9bGrGfTz7njvC\n5W+/oPi6ACMfjrYV90dryuUmsnGMtyzrY/zl7Kiy0hJRphbL0toUZGkBwErQPsrGAkZkdUV9kLZl\nZG9FJSd6pN8O6aMZZHulP48f/yc8KBUBAOSjFWLnWNR1h5y7KeUp2Gco+syx7K0+WV9U4oJdI077\ncfxKkVZwDDtkh9JrW5CJxt4n07f4cxFdS/slZHrJ4tGkR0RERGrxeEuBzCIiIlILutMjIiIitShD\nsfxbKCIiIgLd6RERERHwxIRlsrCTHguyhGjbOXnpc2rdrJYXD0+TZYsl3Jbc3YuzE6IsLQC4udcZ\nu+92WC+sOqwGUitYTpLWeA2qsF+W6TJ+Pa2ULC0AWG0Wa/60WJ2uRlwfqB20j5YBPDMswuppdaKs\nHdK22Y/P6W5CFlmKPsv8IavrBpmKbGTsHGsFP9BmdbpKKPEU1gsjZ/oa6SP63LNrxDqpxRdh16om\nu6YH+yklSwsAelOum5VSAyzld1nV6pCyrsdbIiIiUgsLe6dHREREyqOUdREREZEloTs9IiIiUos7\nPbWe9DQqDCCLAtlYAF+LHIaotAQrN8GW94MowF399bgtieTsBcF3bRKo1wperc9KRaRgPbASEq1g\nOWvLhhctj/oFgFUW4BwELacELGfti8s3teK2m1tx0PlqsLzFgp7JOCKdHglCDoKT17rtsO0esnxv\nd/LPZ7SnoxIZQBywDMQlJOj5SJYnnY+kj5Sw7ugzF302AaBB9sduL14n1rrduI+EWh2p1zAPlrNr\npnv82eqH1/p4W1Kw3yE9Mo55V4dJjx5viYiISC3U+k6PiIiIZNjrHJbJ8m+hiIiICHSnR0RERKCX\nE4qIiIgsjYW508NeL15oZ5PP4xpR5hVZ/7jjOhBReYqUchNAnHnVtTg7h72qPRrHJo8zbqKskUYJ\n/3hgmScsAyYqB5B6pJpBGQRWhoKVdGgHfbAyD6y0RJSptaUdZ+BtacdFBbasFpevtuNMrybJDIv0\nSIbVWqd4fuxeWw3bGi03sVJY0u+SzCuSddIOlnfp/md9F5exDDB2jkXnI6lGQs/1jo+fvxVtCsui\nbJJR77HiOdYBOWdYNqePn4HHrm398PyYvKwEu3Y7XV7cluj6OnKdwe+necr0qkP21sJMekRERKQ6\ndZj06PGWiIiITJWZPcbMLjCzn5jZK0ibvzOzC83s+2Z2/zLWqzs9IiIigv6U7vSYWQPAWwE8AsCV\nAM41s4+7+wUDbR4L4O7ufk8z+1UA/wfASZOuW3d6REREZJpOBHChu1/q7h0AZwJ44lCbJwI4AwDc\n/ZsADjWzoyZdse70iIiIyDRjeo4GsH3g68uRTYRGtbkiX7ZjkhXXYtITZR/NwryMI6rHBfCsrnaQ\nVdEiWXJl1NmKsG5T6xdF2FGJMmDaJHuLZXVFGWBRRhcQ19gC4npaLEvr4M17wuUHbdldWLZp896w\nbWslztCJdNdJ3aw9mwrLWNYaE12AaX2sfnwUe0kZePHy9WCd7JxhOUXRqFNrwSUV3wrXF3fMPssR\ndo2wOXmT77xcY9k4vISsM5lMLSY9IiIiMlpZd3q+fcP1+PYN149qcgWAYwe+PiZfNtzmzhu0SaZJ\nj4iIiJQWyHzCYbfDCYfdbv/Xp11y8XCTcwHcw8y2AbgKwFMBPG2ozScA/FcAHzSzkwDc6O4TPdoC\nNOkRERGRKXL3npm9AMBZyJ4Wv8vdzzez52ff9tPc/TNm9jgzuwjALQCeU8a6NekRERGRqb6c0N0/\nB+C4oWXvGPr6BWWvd2EnPWUErE076M0TX1ket49f624kdLcRLF/xtMMe9ZES/FsG1m/K+liAaEqQ\ndBSYPHJ5ELzLAnpZKYvVKJA5KCsBxAHLALD10JuLfRxWXAYAra1xH5Huri1xHzcePHYfLAi52y+e\n63u78bm7TvZdM7iIR8uAUce2uIyeS/HiMAi5jHM6BV0fax9sTZuUn2HXlKjEBUukYJHaKdfN1Gvs\npPjvkPEDlucl+LouFnbSIyIiIuVRGQoRERGRJaE7PSIiIjK1MhSzpEmPiIiI6PGWiIiIyLJYujs9\nRrKb0vqY7lywTzIO+lHmg5HMH4+3e7MVDzHL9NpFZvkeZFVYReUmgDgDhq2tjCyyBsnaaQTbzTNd\nxl/OsoRY9lYrKE+x2o5LRbDSElGm1pZtV4dtm3ddCZdHepfcNHZbVrJibW01XN5aK24320dsn0b7\nP/0YBucBOb965DMUtWfrY6dutHzCyhR5v6Q8RXBNYW23WnzORNeOPd4N23bItS3K9mLXzCrFvxfS\nykrEv5+mvy2M7vSIiIiILImlu9MjIiIi6eoQyKw7PSIiIlILutMjIiIitYjp0aRHREREavF4S5Oe\nMZWRFcYyr8wnj96PatwAwCGtYsbMUZviE/uiXXHfu3z9gMdVlqrqEY0SJaixTC8mas/6oDW5msUM\nkWYrzhpprcRZXVE9LZql9f+9JV4eaP7TC+P1XVdcH6vHxbYl3G6yj2gGXsLxYm0rTFQk45ju+pgo\nU4tlad1ja5yZt2NvcZ+udya/3tGMV3KNTcGv9fOTZSUHTpMeERERgfOSuUtDgcwiIiJSCzOf9JjZ\nY8zsAjP7iZm9YtbjERERqSN3q+TPPJnppMfMGgDeCuDRAI4H8DQzu/csxyQiIiLLadYxPScCuNDd\nLwUAMzsTwBMBXDDTUYmIiNRMHbK3NrzTY2YvNLPDK1r/0QC2D3x9eb5MREREpkiPtzJHATjXzD6U\nx9/M1xaIiIiIjGHDx1vu/hoz++8AHgXgOQDeamYfAvAud794wvVfAeDYga+PyZcV7Ons2P/3VuMg\nrLaquvkkIiIyXZ3ezej0bp7pGOrweGusmB53dzO7GsDVALoADgfwETM7291fPsH6zwVwDzPbBuAq\nAE8F8LSo4eb2UROsRkREZH61mwej3bz1BZ57u1fPcDTLa8NJj5m9GMAzAVwL4O8BvMzdO3nm1YUA\nDnjS4+49M3sBgLOQPWp7l7uff6D9iYiIyIGZt/ibKoxzp+cIAP9xX4bVPu7eN7PHTzoAd/8cgOMm\n7adqjvhV+SlZ/w2P2zZKeHNAj7wi/aZusSzBzbviE5uVm3CklV6oQn8GQ/Bgnam3f6P2rI9ePz4P\nOr3ia/F73fhV+d31uBxAd9eWYh+X3BS2ZaUlIr1L4nOmu+uIscfGtiXcbrKP2D5NOV6sbXQeVGkW\n53ok+tyzawQrYRP1wa5VKdg108k1NqU8Bb/WyzIYJ6bn1BHf010ZERGRJdCvQRmKWb+nR0REROZA\nHR5vzbwMhYiIiMg06E6PiIiI1CJlXXd6REREpBaW7k5PFHlviXM7D7MLqpsfskyERhBUxjLA+iTD\nao93C8tY9sS6FduycVSZ0RX1zNbGcjKiDJgm+UcMzfxJWV/C8h5ZX4dkJnX7xSymtU6cCbV3z6Zw\neevGg8PlYdvrdo/dNsrSAoDdwfrY2Ni2RNvN9hHbp+Ucw/Ez8JjofGTrY+d6VZ849lnuWvFayq4z\nTlLcmsG1jfVBs1uDIgDsmlll3lX8eyG1j/nODFNMj4iIiMiSWLo7PSIiIpJu8ntZ80+THhEREdHj\nLREREZFlsbB3enhQ2fjzON5H/Fr8SaUGVKe0Z8GIvWA5C1juWLFkBQC0vRhoWkYAcQr2av6UV/az\ncgKsi17wDRYwS5cHgbc9i9fIgnTXusX9v3ttNWzbbIx/g5qVhUgJemZ9REHLt+wulsIA+LZE200D\nmROWJx/D4HDRcynhfCzjnE5B18faB58Mdo1gVnzyXzHxdTAOCE69xk6qnODm+XmopJR1ERERkSWx\nsHd6REREpDxeg9pbutMjIiIitaA7PSIiIlKLmB5NekRERKSyIPp5UotJz7SztNLHMV1RWQkAaPn4\n+6Pr8bb0gvSVdgnPiVlWTJRZA6S9sj8lE63Tj7elS5b3glfod8i/pppB2QUA2BNkMRnJAGOiLKY1\nlgHWGv9V+b1uPOaotATL0trdiZdH271O9hHbp1FGFjtW7NimlJBgoqNFz92KfvFEn02Af5ajjy27\nRrBryrTNyzV2XsYhRbWY9IiIiMhoCmQWERERWRK60yMiIiK1CGTWnR4RERGpBd3pERERkcqC6OfJ\nwkx6+kPR8E2SedUPMhGalpal1Q/qurCaLkai9MvIC4syAPpBNhAANEhWRTMYd9PjbemT06EXjCNa\nBsQZIiwNspFwJ7VLPo285lVxWZMcFHas4syfuC2rCdVqFMfdJG27JCNrb5ghtRK2ZVWSu0HWU2st\nztJqN8fP3ur04r0XrS+qpQXEWVpAvN3dxNpbUVYXO1Y0Ay/hlj/L2YnOR5a9xc71FNFnjmVvsc9y\nlKnVRHysUrK32Pr6Fi+vKhNq+HfKxsvH/1zQdbJMuTnRVyCziIiIyHJYmDs9IiIiUh12p3iZ6E6P\niIiI1ILu9IiIiEgtUtZrPenpexyY1rTJb4B5EPTWJ/12vRsubwQ34owUWOiT5VEJiC2NOBhxpRGP\n7+Zep7BsjYw5CsJkAZQNEpQdYaGdPBi0uKzJAqoT1hn1CwBrJAi22Sv2Xsbt1X43rRzG3m7xo95u\nkCB8sjxCA4iD5SyAmJWWiIKW10jg9DrrO9j/7FixYxudvqllUaK+yyihwkSfORYgza4dW6wYLH9w\nM752rPfjc2Z3v3jt6CRew6LlXcTXHx4MPXkQcrg+8jtkUc1L8paZHQ7ggwC2AfgZgN91951Bu58B\n2Iksh6Dj7idu1Lceb4mIiMg8+VMAX3D34wB8CcArSbs+gJPd/QHjTHiAmt/pERERkcwcPd56IoCH\n5X9/D4CvIJsIDTMk3rzRnR4RERGZJ7/g7jsAwN2vBvALpJ0DONvMzjWzPxinY93pERERkdJeA/nj\nW67Ej3dfNbKNmZ0N4KjBRcgmMa8JmrNwowe7+1VmdiSyyc/57n7OqPVq0iMiIiKlOe6gO+G4g+60\n/+tPXvu9Qht3fyT7eTPbYWZHufsOM7sDgJ9H7dz9qvz/15jZRwGcCGA5Jz1OouYtyJBikfustERV\n2OvNG2R514IMBVrSgTyLDdpvacbb/bzjrgmXv/2CIwvL1rpx9kQnKp1RSlGOWIf806Qd7I4W2UWs\nPEXUnGVHrZO+GyVkAka6tPwGyWIKMrKapOxFgyyPsBiAqHRDSqkI1p5lae0NsrQAYG/YR1rmW3SK\npZSbyPouLmPnbhmirqPP5ihRptYf3Tu+Rpz24+I1AgB2B6tkGVY9cp2OroOsNAW7xk5bSrYY+102\nC3P0csJPAHg2gDcAeBaAjw83MLMtABruvsvMDgLwKACv3ajjhZ30iIiISHnmKJD5DQA+ZGbPBXAp\ngN8FADO7I4B3uvvjkT0a+6iZObK5zPvd/ayNOtakR0REROaGu18P4D8Ey68C8Pj875cAuH9q35r0\niIiIyNy8nLBKSlkXERGRWtCdngmxgLV+MJ9skoBeFthnHgQFk7Z98or5bhDYt5dEW/7DhUeFy9f7\nUSBhvL5OEJTXJWUG2hYvbyQ8VmYlLqLgWFaGgi0PkbF1SBBsLO3fGtHRarPgXxKE3Azas1GUEcgc\njZkFMkdBz0B8DKOyEkAcsAwAe3vFPtixosHJCSUkOgnL2bmbos/KXgRlIaLPJsA/y1FpCXaN2Nsj\nZWnCxAZSboJd26I+SFsm6qOq0hSLbI5ieiqjOz0iIiJSC7rTIyIiInOS8F8tTXpERERknt7TUxk9\n3nLLxl0AACAASURBVBIREZFa0J0eERER0eOteeLDmUwVvd6frp+eDuOPg2cLxH1E62SvWWfj6wXL\nbwmysQBgbT3uY82L7dejEhkAWl7MyOoEWWgAsOLxdtOSGgmiV/w3WRkKsjw6xdjRpnkgCVldfbI/\nouymblBWAgBajcmzt8oQZm+RW+es/EMnyMhaY2VAaB9BOQyS8cSSqaIMKVZCInX5pHj2YpC9Rcs8\nxMt39deLfZBrRJd8xqPrT2oJCX7tjdpOnpGVsj5ZPAsz6REREZHqKKZHREREZEnM7E6Pmf01gFMA\nrAG4GMBz3P2mWY1HRESkztjLLpfJLO/0nAXgeHe/P4ALAbxyhmMRERGpNa/ozzyZ2aTH3b/gt0Yn\nfwPAMbMai4iIiCy/eQlkfi6AM8voKI68T5vbxX2Qulkk0t+C5Q0yDtYHaz8plsXRIxkYUfse6cOD\nmjh7gowuAGiTeknNoCZXSj0uIM5qWQ/qMAEjalAlNGYfpGgvORkHr6NUbN8iO6TNsreCelpTr73F\namyRzKtou7sJ+4iNg2Vp8b6Ly9ZJkhBbXlWdrag+FgDsCTIuO9aJ+2V19KKzl2W4Tfnf8+yamdI+\ntY8ysrrmPTOsDrW3Kp30mNnZAAYr1Bmyj82r3f2TeZtXA+i4+wdG9bXWuWb/35uNLWi2Di1/wCIi\nIjPQ6e1Ct3/LrIex9Cqd9Lj7I0d938yeDeBxAB6+UV+r7SNLGpWIiMh8aTe3ot3cuv/rte7Ppz6G\n+b4PVY5ZZm89BsDLADzU3ddmNQ4RERGph1nG9LwFwAqAsy17A+833P2PZjgeERGR2qrDywlnNulx\n93tO8vMsCK0ZBBz3SYCuBUG3+3opri+OUGzSEhLF9n3SNhpzWfrBNjbIid1LCEZsYPwPBwucXiPH\npRkEpa40WMmKsYeBDgsmJYHFkU3sGyQqOFxMVseCcaPhsaBbVo4hKrXRIMc7pQoILd0QbCQr/8AC\nuKPAZ74+1sf464sClgFgb3D6RsuAEedYAja+KGiZfYbYZy6S8lmOykqMEl1/qsRLWYy/P9i1Pu6X\nlNQgxyVe3/w8VJqfkVRHb2QWERGRWpiXlHURERGZoRJuVM493ekRERGRWtCdHhEREQlj8ZaN7vSI\niIhILSzMnZ7h6HtjWVNejLw3Gz/DCgD6XmzfpH2wePdie15ugowjWGcz8TXwKXlhRmb5zWB501fH\n7tfJ2NaCV+UDcfZWk6QUNVJSjYgysrpWyPJ2wj8r2OP0aBSspANbXbSbUkt7pIheZ5+aeRUpI3ur\nQxqzEhJRplYZWVoMK1mxFmZvxZ+hKCNrU8JnNhX7jEdSS1ZEGWD02p1wNqWWhOgHv1tSssKA+PfT\nPKlDlfWFmfSIiIhIdRTILCIiIrIkdKdHREREFMgsIiIisix0p0dERERqEdOzsJMeZ7VNgoynlAwr\n1p7WWCE9R9llrMYWryM2fltW46YfnMVdizMIWh6PL6ovtmrxqXNQs7icZaPs7K2Fy/cGGSkNUlOq\nQbLqWLZXijBDh2R0pWQPtcnQmuS+a3RU2C1alhuSkgFWldQLarRPWf2uHjkAnRKyt6rK1GKfi71B\nlhYQfy6YI5qbC8vYZ+KWXtxvlBnGam+xa0qUqUWvVQnZVKn1qqL2rMYWu9anZHsltU2o0yWTW9hJ\nj4iIiJSnDtMvTXpERESkFu/pUSCziIiI1ILu9IiIiEjiu7IX08JMeoaDvczGDwqmAcQkgCx6PX9U\nmoK1BVgwNAszjfuOtqVBA6rj5b1wnfH+YMGIFgS8RqUpAOCYzcVtaZH7iefdFI9jt68Xx0ACmZvW\nDpevBuusLLgZQJ8EOEeBt6y6RYtccVpBexb0TLcw6LvaMhTjt2VNo+DkLmncLSGQmQUWlyHqe43s\npD19EliM4vItFhdAufvW4meL7aOLdrHg/PGTIFhpiej6w65VPFkkJYB4/OBkuj5SKiLqm/0OYVID\nsKV8CzPpERERkepENfOWjWJ6REREpBZ0p0dERET0ckIRERGphzpEHOnxloiIiNTCwt7pYVH6UfkH\nJ9H4PNVl/FIWLEvFgvII0diy5Wx+3Ykax9htySgTjayPlaFoBONeI/v08j3FPtjMOiqRAcQZIrvJ\nmFl5iujUjjK6gHKyuljmTy/YTR2yvjY5hs0oe4ucMlFbIM7UqvJfPNHw2GeFlpaIMt9I21lkZKWs\nL8rU2k2ytHZbXJ4lLOlA1nfZLVHZhRj7LEclJ+Js0FFZXVEf8XZ3LV7eC66DtBwPXR5lXqWVoUhp\nS3/nhH2M37ZqdXi8pTs9IiIiUgsLe6dHREREyqOYHhEREZEloTs9IiIiUouCo5r0iIiIiGpvzZPh\nKHmWCTVcowsAWIJPSk0uVqeIjiOh1guL3o9aN1jbIFss67yYEdFnTzUTkpga5HXlO4MEjAbJVuok\n1MlZI9kh7FPa760WlvU8Pt03NeL9EWV1lVGvKiXTC4gPS4vsU5aIFo2bbUrKNrJ/GUaLaVtWTyv4\nxiwuytG42THc248/41E9LZaltWbF2nNAnF3JPkM7e0HGExnzOsmm6lixD5al1SV9RJ/lvqVeB8ev\nY8ivsZPX70qps0XriyXW6pLyLcykR0RERKpTh8dbCmQWERGRuWFmTzKzH5pZz8xOGNHuMWZ2gZn9\nxMxeMU7fmvSIiIgI3Kv5cwDOA/A7AL7KGlj2BuC3Ang0gOMBPM3M7r1Rx3q8JSIiInPD3X8MAGYj\nX5d/IoAL3f3SvO2ZAJ4I4IJRfS/spIcGhIUBmyzoefzIUV5uYvzyFFFpimx18fJoKQtCpqUsovID\nLPiXvsK9uE4W0BhFmhoJenYSlhrtjxYdMwkoDV5d7+Qg9kmA82oQ4LxCgp7LCHBmolF32D+fEv5V\nVeGQFzILhH3G14Pg5DUSsLw3SBwAgLUg0Jedu6wcTFhih/SxFoyDteUlJIKyF4mBwlHQcmoJiSiw\nOKXcBBCXnOClhcbfxpRyE1kfwTjmKLh5fkYylqMBbB/4+nJkE6GRFnbSIyIiIuUpK5D5yvXLcVXn\n8pFtzOxsAEcNLkL276VXu/snyxlJkSY9IiIiUpo7rRyDO60cs//r7+7+VqGNuz9ywtVcAeDYga+P\nyZeNpEBmERERgVf0Z0LsSfy5AO5hZtvMbAXAUwF8YqPONOkRERGRuWFmv21m2wGcBOBTZvbZfPkd\nzexTAOBZQNULAJwF4EcAznT38zfqW4+3REREZG5eTujuHwPwsWD5VQAeP/D15wAcl9L3wkx6ClHy\nCWUhWHQ8K08RZTexzCv2yvI40yL1FenFPljWAoJspUw7GlzI6H4KMjBI9lCU8ZF6O3ElyKYyMug2\nKSUSlb7okEyL3R7vu16/OA5WfoBldVVVyqIMc3J9q1RKCYkoSwsA1oLPRZQdNcoWWwnGFo+Dl2cZ\n/4hFbVm2GF2eknnFsliD9j1yreLXtqIqy03Q9sH1o4xyEyljrtoBvlNnoejxloiIiNTCwtzpERER\nkerMzz2n6uhOj4iIiNSC7vSIiIjI3AQyV0l3ekRERKQWlu5OTxg1n5DpRftNyLDi7eNMI1YvJhof\nH/H42WV8S8bPwOC1doKd7ay2WHxgokytVYtP1UObQXYagGMPKq7z4l3xfr6+tydcvsvXCsvWSF2k\nzaR+VzvI+mux+l3h0vnOAKtSSuYVO6e7QUZWh2TW7CEZWVE2FTt3j2huDpfffWvxvLnslngcO3tx\ndlNKPa2wbhbJsKKf5ZTPfULdLIa1jfpm10ze93Szqdi2zFOdrUgNbvQs36RHRERE0unx1hSY2R+b\nWd/Mjpj1WERERGR5zfROj5kdA+CRAC6d5ThERETqTi8nrN4bAbxsxmMQERGRGpjZnR4zewKA7e5+\nnkXBr0PGDgALumIByynlKZosCJn0EQWa9llAL9n8KKCO7QVLCL5jM90+KbURlafoGgkkjP6lwA4v\nDXAOxpYcwFpcFgUEA7zERcfWi8vIK/Q7HgdUt4PA52gZALTJkWkF42bbwpbPQ+AzixdgwcnR8i4t\n3RCfCVHpEVbmoWOslEvRJl8Nl7P9H52P7Nxl53pSaYkgaLlHtpt9lsMAYhoMnRKEnBoMnRBQTUrN\nhAkdiUHFYWB3Yh/htpAxz8J8h1mXo9JJj5mdDeCowUXIfi2+BsCrkD3aGvwe1ent3P/3hq2i1dxS\n3kBFRERmqNffi35/76yHsfQqnfS4+yOj5WZ2HwB3AfADy27zHAPgO2Z2orv/PPqZdvPQysYpIiIy\nS83GJjQbm/Z/3V3fOaJ1NdhdxmUyk8db7v5DAHfY97WZXQLgBHe/YRbjERERqbvln/LMPpB5H8cG\nj7dEREREJjEXLyd097vNegwiIiJ1VoeXE87FpGccw1HvSSUkWIQ9zSqKFpEIe3p/qjg+nnFAug6y\nqdhW90hWERBnFYXrI+OLylOwTItutD/I9vUt/kb4in/Sx864cgD27ipmSPVKuHnLsmVYBkzDi9vC\nnpsXi17wPpo002v8jDiWtVaGONMo1iWfz1543qUdw2gcPFsp4XwkbunFJ+RFu4p9rJGsnXXEfUTj\nZmOOrjUpWVqsj9TMq2g5u1bxEhdR5hXL0pq83ATLpop+jyxquYk6W5hJj4iIiFSH1XJbJvMS0yMi\nIiJSKd3pEREREcX0iIiISD3UIRJJj7dERESkFhb2Tk9SdHxClhYAmBUzf1h2QoNlJgXzydT6Rym1\nt9KyulhGF8kAS8nICkbC6vU0yKgbQU2uHss8YbWYggyMlCwcAGiTelphW1KbbWtjpbBspRFv9829\neP/v9mINsD1BXbBRom1vkRpgKVldLPAxJdOIrS4a3xYr7k8AOLgZH6v1fvHc29WP9x2ryZVizePM\nq2jbo+w0gNcAi/pgn62Umlcsi6lrxW1hmVes76h9SpYWa08zr1gNsCjzimRpsW0J10d+D6Vkhs1T\nppfX4I3MutMjIiIitbCwd3pERESkPPNzz6k6utMjIiIitaA7PSIiIlKLmJ6FmfQUgr0Sg5MjUZmH\nrIsgwI2WUiB9R4GEpA8W4NwL2jctDthMCXBOL1kRtCdjjoKTPQhMBuKgZwBohOU30m5K9oPA4tTA\n3Sj4l5V/WLX4o3RYu7j82ffcEbZ9+wVHhsvXusWA0g45hqzUQBSUzbZ7KwkWjuwKgqyBOOiWBeiy\n49IMzkcWsPxH974mXP4PFx5VHMc6+bSQzycLOE5pGx2XHgm6TSkXkVRCIiHoORtf8XiltKXjSAhY\nBoCeR8HQ4wcs83GwYOO0IOm4j5RSFvPzUGl+RlIdPd4SERGRWliYOz0iIiJSHfYKkGWiOz0iIiJS\nC7rTIyIiIqqyLiIiIrIsFuZOTyHCnU1Ig4QUI3M7GjUf9Z2YLRZG+pM+UrK6ooyurGsyfw36SCtZ\nAcRZXSxbIypDEWfnsIysKNuLZXqxfWpBHywrhmUPRVgphS7J+LipU8y8Ou3HcZZWVDIBYFlkJBON\nZMqtePGjzrK07rF1/PIbF+2Kl6ekvrLyINFyto/YPt3bK+5/dqxomYxASvkN1jdr20VcymLSjCya\n8UTLQhSXp2RpZX0EWWtBNtaoPqJxp2RpAWnlH9h+KiPzKiypMUdlKOZnJNVZmEmPiIiIVCdl4r+o\n9HhLREREakF3ekREREQp6yIiIiLLQnd6REREpBYp6wsz6RmOvmd1s+KfTYxJj5JJErLFgDjbqJHa\nR9Ce1eliwkwJWjeL9BFmbIyf4cP6pUclyjhjtc9oLaAoAyPOeGJZNJEGyY5iNZc6/WLfrf7kN1hT\nMs6AuGYYu8Dt2Dt5FlO0viiDLNXufpz5s5ucTN0wAyntesAypMK2ZH9E52NK3Sw2Dt7H+OtLqaeV\nkqUFsLpZaVlTUaZWSpZWNr7xs9l4VleUeZXWh8zewkx6REREpDp1yN7SpEdERERqMelRILOIiIjU\ngu70iIiIiAKZ51mVJSRCrNwBLaUQvTI+7iMlwDmlZAXDXgOfEuCcVrKCYcF+QR/0GMav7A/3HVkf\nC06OyiD0EoKeAaAb7L0OK7uQMI5U0W3rPWTfrXfGD8JkQcFl3CaP+u6Qfnk5hmL71NIB4/Y7ehxR\nQHW8/9P6YJ/D8X5+VB9h2YuEgGXWR0rAMu0jIWA5az/+MS+nlMX44yjjfJTxLeykR0RERMqjmB4R\nERGRJaE7PSIiIpL0TqpFpUmPiIiI6PGWiIiIyDSZ2ZPM7Idm1jOzE0a0+5mZ/cDMvmdm3xqn7wW6\n03Pb2260GGwJJSRSJrvO5o0J40jK6kooWQGkZXXxV8wH/ZI+4lfXx62NlIVAQh99I30E+yMqDQIA\njYSSJqkaYTmSeH1Nsi1R9hbrgyvukE6Ft7LTx1cUl10Yv8wD76O67U4pLdG1OHuLZjeF2VSTl6FI\nKS3BS0gkjCM54ylhW2jfCZlXCaUlUrK0WHu2vlmYo0yy8wD8DoB3bNCuD+Bkd79h3I4XaNIjIiIi\ny87dfwwAZrbRP9sNiU+sNOkRERGRRYzpcQBnm1kPwGnu/s6NfkCTHhERESnNrt4O7OrtGNnGzM4G\ncNTgImSTmFe7+yfHXNWD3f0qMzsS2eTnfHc/Z9QPaNIjIiIipaWsb2kdiS2tI/d/vaN7XqGNuz9y\n0vW4+1X5/68xs48COBHAckx6hgPD2JO+MMA5MWDZEgJbqyyHEQU4p5SsAOIAZxbc3CN9N61YFoJ9\nNKI9l9IWiINS+yToucF6j/YdWSMLIywlGDcKZGbnF9n/HmxM39JuQ8clBaq7ld0IPqCsZAuTMmYW\nFBwFvJZShiLxl0NKCYmU4OSU9aUELLP2vNzE+KUlUgKWs/FFwb9p+6iq8g/pZSii8c1N8HClQf4T\nCH9jmdkWAA1332VmBwF4FIDXbtSZUtZFRERkbpjZb5vZdgAnAfiUmX02X35HM/tU3uwoAOeY2fcA\nfAPAJ939rI36Xpg7PSIiIlKdebnT4+4fA/CxYPlVAB6f//0SAPdP7Vt3ekRERKQWdKdHRERE5unl\nhJXRnR4RERGphYW501OYgZaQeTX2ukasj6qoHEZSyQrSd2rJiiiri2XieJDVQtuSjKyoPTuq/N8l\nxSwTmqWVWuIioY9ohE6ywlhGUCv8mE7+6nr2/D7luT7b7nj55GPugpVuINuSkGWVst0ppSJS15fS\ndxklJPqs7EK4vvGztGgfCVlaWfvxx1FG+YfUjKxJsfXNgqqsi4iISC3MSyBzlWb6eMvMXmhm55vZ\neWb2+lmORURERJbbzO70mNnJAE4BcF9375rZ7Wc1FhERkbpLeQHmoprlnZ4/BPB6d+8CgLtfO8Ox\niIiIyJKb5aTnXgAeambfMLMvm9kDZzgWERGRWutX9N88qfTx1ogqqq/J1324u59kZg8C8CEAd2N9\n9ft7B/ptwWyFtSwsCetx7RtNpIS6WROvj7VPqNMFkKyuhDpdAM/qGrcPmhVGPgxR9hbP9GKZFsU+\njPTBPpRRXS9evyu+LRytk2VIsL5Tsjt4plxCVlFCBgfLRIv2RpVjKyPziq4zrAGWdiGP1smztybP\nyAp/PiFLK1sejLmEWlMpWVpsHGXUvErvY/I6YqPyTd07yB98zMy8TVCqUOmkZ1QVVTP7LwD+OW93\nrpn1zex27n5d1L7R2FTRKEVERGbLrA0bKO7c66/NcDTLa5Yp6x8D8HAAXzWzewFoswmPiIiIVKsO\ngcyznPScDuDdZnYegDUAz5zhWERERGTJzWzS4/7/2ru3UNuqOo7jv9/JRNPULtDFgzciqpN5wcyU\nSryQGOlLD1YUJnT1hkVUGgQ9lRXdpIdIhUSTOkY3uqhoL5GlpeYtFCKvZFhkWGnnnP3vYa5wn73H\nWHuNveZcY805vx/Y6N7MM9Z/rbXX3GOO+f+Pf+yQ9J5ajw8AAJ5FTs9S2f3NyOYPp1omZBJpixKc\n20iGzukwoTqV4FzSskIqS04uSTrMJbam0mBT7S2mjZFKfM4lPZckJ69kkqFL2lCUJkOXtMNYtEVv\nW99GEnLpsSWPWZKcnEpMzh0r5RORZx2j+LVrIXE32TqjIGE5d3xpHGVjzN/KIpewXBIHutGjSQ8A\nAOjKGCZgTHoAAEB29XFIqvbeAgAAWBRWegAAwChub7HSAwAARqE/Kz1rs96z1UOz/rCwqqvDFhJZ\nLVSRJbdfL2zpkKo2yo+x/tiSSq9mjETrhuxrl6nWSFQVlVR65Y+fve1FTq4CLG/91vT5arHhaKN8\nto3Kq/S4s7eKyB1fUo2VG6Po+S1JC4mcNqqmysaY//UoqdLKj708qyulv5N9NPwzJwAAgPq00gMA\nADozhpweJj0AAGAUvbe4vQUAAEaBlR4AAJBNdh8SJj0zyvZY6bBvVvrQ2Suemp8nFvMW3L9rV3Hv\nrUS/KqePzT3vkn5h+Y/5+td6S+bxdmVGKakAK1FSLVYq95qm4+huObyr5oeleQslu9SW9M3KVgnl\nerAV/EEqeY7L3jdr1nGnj91NHOgfJj0AAGAUiczk9AAAgFFgpQcAAIzi1h6THgAA0Fke3TLpzaRn\n7b1GFyTSZvOEi9pTZFoYlCQ4F7aQKG5bscA48om7iSTkbPuHTFK2Z29DUdLiIpf8mEuo3pKIoyxh\nufkXXci1zihRkkheqo3cgJIE4jaUxJzbrr80OTk99vztGJLjFrZMSMfQRvuHsjFmHbetOPLmb2WR\n/GyNoGJqmfRm0gMAALpTMvHsKxKZAQDAKLDSAwAAaEMBAAAwFKz0AACAUeT09GjSs/ubkS1AKmj/\nUFLVla7oyosW2lCkKgCcacdQVOm14DhSbSymSba4yIxR0uKipNJLSp8AFl2lJZVVkeWk417e6qic\nXNVUl5KVP4WvXaoiq/T1SFUblZQYt9FCIjt2QUuHNlpItBFHaZViSSuL7Ngt/B50aZli6Qq3twAA\nwCj0aKUHAAB0ZQw7MrPSAwAARoGVHgAAMIpEZlZ6AADAKAxupSdZaVTYayp5eGEfrFS1V7Kia0oc\nSR327yrphVUSR+4+sTP9o5LVXpkrkFxfr1QcJZVek0jWP16uaq0FudcjF3d6jMVXl82rjYqRLjdV\ny/XCSsdRUk2VjrnLiqyyMebvV1VS8VTexyo1Ri6+2ftmZcdOxjF7lVZ+jOWx7PG1YXCTHgAAUI7b\nWwAAAAPBSg8AABjF7S1WegAAwCj0Z6VnbRawczPS9fO43Oy1JMG5MI+5qJVFNsE5pTRhOZlYXJrQ\nO3scRa0bsmOsT0YsSnqWkomEuaTnkgTu8mToErNfZeUSqksSWHOvaRvaSCwuSSBuQzsJs2nLkJzc\nRtuLdpKNS9tQlLzWsydJl69qJI5vJWF5eVZX2JwQAABgIJj0AAAANatOXXyVsX2p7fts32H7Otv7\nZY47zfYfbd9v+xOzjM2kBwAAKGKlk69NuF7Stog4UtIDkj619gDbWyRdJumtkrZJeqftV200MJMe\nAACwNCLixnh2tnSLpK2Jw46V9EBEPBgROyRdK+nMjcbuTyIzAADozJKWrJ+jZkKz1oGSHl71/SNq\nJkJT9WbSE2vKk/KVV6k3LVfpUlDVVdIqQqWtLDJxJCp0iltZdFkZNue4RVVkBZVezeGJyquC17kZ\nfP4KsBLJdh8ZJa0ppFx12fJWRxWP3WHVSRuVV8ljC39n5q1AqtFCInlscZXWgltIZAfpqgJs3hPs\n8onYqYidU4+xfYOkl6z+kZq/NpdExI8nx1wiaUdEXNNWbL2Z9AAAgC61c2Fib5G957Ojrvx33TER\ncer0MXy2pNMlnZQ55FFJB636fuvkZ1OR0wMAAJaG7dMkfVzSGRHxTOawWyW9wvbBbmZYZ0n60UZj\nM+kBAADNLbwuvsp9XdK+km6w/Xvb35Ak2y+z/RNJiuZ+6HlqKr3ukXRtRNy30cCOHtxPtB1r78Q5\nl+SR3Pa4bG6XzikpnR+uP7501950rkl6jFw+SMlj5nJbUmNk82CS45bt/Dvv4zVjzP6YJWPncnra\nUPo6FY29BNc35PSsOZacnjVjkNOz+9g7FBGF2aSbZzvsvToZO+LphT6XaeqfCQEAABagt4nMa6u5\n/i9deVV6VTDjuNMsSf+u1NVkdkVh0f27clp4vNQVWG5Fp40KsJyiFaqCq9fSFac2qssWrWSFpcQm\nN0vbfYwWVmnyY3ezekPfrLWD1FgtWn+Szf0tq2MpS9ZbVW2lx/YRtn9t+3bbv7V9TK1YatuotK/v\ndq08XTuEzuzc9a/aIXRqx66naofQqaG/f0P+7EnDP3eifTVvb10q6TMRcZSkz0j6QsVYqhr6B3dl\nwCfeXSv/rh1Cp3auDH1SMOz3b8ifPWn4586Fi+jma4nUnPSsSNp/8v8HaIb6egAAgM2qmdNzkaRf\n2P6SmiyO4yvGAgDAqC1XflE3Oi1Zn7bNtKRTJN0cET+w/Q5JH8zt0NiUrAMAMB4LLln/s6SDOxr+\nwYg4pKOxi1Tbp8f2PyLigFXfPxkR+0/7NwAAAJtVM6fnUdtvkSTbJ0u6v2IsAABg4Grm9Lxf0tfc\nbBrztKQPVIwFAAAMXC/aUAAAAMyrN20ohr6Zoe3zbd9n+y7bn6sdTxdsf8z2iu0X1o6lTbYvnbx3\nd9i+zvZ+tWNqg+3TbP/R9v22P1E7njbZ3mr7Jtv3TD5zF9SOqW22t0yaNW7YebpvbO9v+3uTz909\ntt9QO6Y22b7I9t22/2D76kkXcbSgN5MeDXgzQ9snSnq7pMMj4nBJX6wbUftsb5V0qqQHa8fSgesl\nbYuIIyU9IOlTleOZm5v+GZdJequkbZLeaftVdaNq1U5JH42IbZLeKOncgT0/SbpQ0r21g+jIVyX9\nNCJeLekISRt21+4L2y+XdL6koyPidWrSUM6qG9Vw9GnSM+TNDD8s6XMx2V40Ip6oHE8Xvizp47WD\n6EJE3BjPNgi6RdLWmvG05FhJD0TEgxGxQ9K1ks6sHFNrIuIvEXHH5P+fUvNH88C6UbVncpFx8F44\nKgAAA6RJREFUuqRv1Y6lbZOV1DdFxJWSFBE7I+KflcNq23Mk7WN7D0nPk/RY5XgGo0+TnoskfdH2\nQ2pWfXp/Nb3KKyW92fYttm8e4K27MyQ9HBF31Y5lAc6R9LPaQbTgQEkPr/r+EQ1oUrCa7UMkHSnp\nN3UjadX/LzKGmLR5qKQnbF85uX33Tdt71w6qLRHxmKQvSXpIzcX9PyLixrpRDcdSdVmfYTPDC1dt\nZniFmtslvTDluX1azfvwgog4zvbrJX1X0mGLj3LzNnh+F2v392phG261ZdrvZkT8eHLMJZJ2RMQ1\nFULEJtjeV9J2NeeWQXRXtf02SY9HxB2TW+e9+7xtYA9JR0s6NyJus/0VSZ9Uk/bQe7YPULOqerCk\nJyVtt/0uzivtWKpJT25HZkmyfVVEXDg5brvtyxcX2fw2eG4fkvT9yXG3TpJ9XxQRf1tYgHOaspv2\nayUdIulO21Zz6+d3to+NiL8uMMS5THv/JMn22WpuJ5y0kIC696ikg1Z9v1XDuqWsya2D7ZKuiogf\n1o6nRSdIOsP26ZL2lvR829+OiPdWjqstj6hZOb5t8v12SUNKtD9F0p8i4u+SZPv7ato0MelpQZ9u\nbw15M8MfaPLH0vYrJT23TxOeaSLi7oh4aUQcFhGHqjlhHdWnCc9GbJ+m5lbCGRHxTO14WnKrpFfY\nPnhSOXKWpKFVAV0h6d6I+GrtQNoUERdHxEERcZia9+2mAU14FBGPS3p4cq6UpJM1rITthyQdZ3uv\nyYXiyRpQonZtS7XSs4Ehb2Z4paQrbN8l6RlJgzlBJYSGt9z+dUl7SrqhOUfploj4SN2Q5hMRu2yf\np6YybYukyyNiMCde2ydIereku2zfrub38uKI+HndyDCjCyRdbfu5kv4k6X2V42lNRPzW9nZJt0va\nMfnvN+tGNRxsTggAAEahT7e3AAAANo1JDwAAGAUmPQAAYBSY9AAAgFFg0gMAAEaBSQ8AABgFJj0A\nAGAUmPQAAIBRYNIDYB3bx9i+0/aetvexfbft19SOCwDmwY7MAJJsf1ZNw8q91TR4/HzlkABgLkx6\nACRN+hrdKuk/ko4PThYAeo7bWwByXixpX0nPl7RX5VgAYG6s9ABIsv1DSd+RdKikl0fE+ZVDAoC5\n7FE7AADLx/Z7JP03Iq61vUXSr2yfGBG/rBwaAGwaKz0AAGAUyOkBAACjwKQHAACMApMeAAAwCkx6\nAADAKDDpAQAAo8CkBwAAjAKTHgAAMAr/A0HzWI0gcWicAAAAAElFTkSuQmCC\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig3 = plt.figure(figsize = (10,8))\n", "plt.pcolor(x,y,np.log10(np.absolute(amplitudes3)**2), cmap = 'inferno')\n", "plt.colorbar()\n", "plt.title(\"Logarithmic intensity around a point\\nsource in a circle of 20 scatterers\")\n", "plt.xlabel('x')\n", "plt.ylabel('y')" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": true }, "outputs": [], "source": [ "#create a list of 20 point scatterers in an ellipse\n", "N4 = 20\n", "p4 = []\n", "for i in range(N4):\n", " position = np.array([7.0*np.sin(2*np.pi*i/N4), 4.0*np.cos(2*np.pi*i/N4), 0.0])\n", " strength = 1.0\n", " p4.append(Scatterer(position, strength))\n", "\n", "#calculate the amplitudes\n", "amplitudes4 = Foldy_Lax(p4, 'spherical', x, y)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Ellipse of 20 Scatterers\n", "Both regimes are visible in this interference pattern. The scatterers are spaced by less than a wavelength at large $x$, so the wave sees them as continuous. This creates the large curved minima on the left and right of the plot. On the other hand, the scatterers near $x=0$ can be resolved by this wavelength, so the patterns around them are characteristic of point scatterers." ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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MyLO0FojZCIwxxhhjjJkwU9/pMcYYY8wM4Jie+aPEQqLEnqJUpUXLKFTtsGuW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\nbPJabwqb2AAAAABJRU5ErkJggg==\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig4 = plt.figure(figsize = (10,8))\n", "plt.pcolor(x,y,np.log10(np.absolute(amplitudes4)**2), cmap = 'inferno')\n", "plt.colorbar()\n", "plt.title(\"Logarithmic intensity around a point\\nsource in an ellipse of 20 scatterers\")\n", "plt.xlabel('x')\n", "plt.ylabel('y')" ] } ], "metadata": { "anaconda-cloud": {}, "kernelspec": { "display_name": "Python 3", "language": "python", "name": "Python 3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.5.2" } }, "nbformat": 4, "nbformat_minor": 0 }