{ "cells": [ { "cell_type": "markdown", "id": "72d0db40-aec3-4db1-a698-ad38a39c62bf", "metadata": {}, "source": [ "# Coupled Oscillators" ] }, { "cell_type": "markdown", "id": "04bc4b5b-a44c-4756-8e5a-dc0aa22389f5", "metadata": {}, "source": [ "Consider three masses connected by two springs, moving freely along a horizontal line. The masses are $m_1$, $m_2$, and $m_3$, and the spring constants are $k_1$ and $k_2$. Let $x_1$, $x_2$, and $x_3$ be the displacements of the masses from an equilibrium position, such that the springs are relaxed when $x_1 = x_2 = x_3 = 0$. We would like to solve the equations of motion for the displacements $x_i$, $i = 1, 2, 3$. \n", "![oscillators](source/oscillators.png)" ] }, { "cell_type": "markdown", "id": "6fdc0963-5017-4721-b111-23c8c6231f41", "metadata": {}, "source": [ "The Lagrangian of the system is given by:\n", "\\begin{equation}\n", "L = \\frac12 m_1 \\dot{x}_1^2 + \\frac12 m_2 \\dot{x}_2^2 + \\frac12 m_3 \\dot{x}_3^2 - \\frac12 k_1 (x_2 - x_1)^2 - \\frac12 k_2 (x_3 - x_2)^2\n", "\\end{equation}\n", "The Euler-Lagrange equations for $x_i$ are:\n", "\\begin{align}\n", "m_1 \\ddot{x}_1 &= - k_1 (x_1 - x_2) \\\\\n", "m_2 \\ddot{x}_2 &= - k_1 (x_2 - x_1) - k_2 (x_2 - x_3) \\\\\n", "m_3 \\ddot{x}_3 &= - k_2 (x_3 - x_2) \\\\\n", "\\end{align}\n", "These can be written in a matrix form as:\n", "\\begin{equation}\n", "\\left( \\begin{array}{c} \\ddot{x}_1 \\\\ \\ddot{x}_2 \\\\ \\ddot{x}_3 \\end{array} \\right)\n", "= - \\left( \\begin{array}{ccc} k_1 / m_1 & - k_1 / m_1 & 0 \\\\\n", "- k_1 / m_2 & (k_1 + k_2) / m_2 & - k_2 / m_2 \\\\\n", "0 & - k_2 / m_3 & k_2 / m_3\n", "\\end{array} \\right) \\cdot \\left( \\begin{array}{c} x_1 \\\\ x_2 \\\\ x_3 \\end{array} \\right)\n", "\\end{equation}\n", "or simply\n", "\\begin{equation}\n", "\\ddot{\\mathbf{X}} = - \\mathbf{A} \\cdot \\mathbf{X}\n", "\\end{equation}\n", "where the vector $\\mathbf{X} \\equiv (x_1, x_2, x_3)^\\top$ is what we want to solve for as a function of time. For simplicity, we assume that the initial velocities are zero, $\\dot{x}_i(0) = 0$, so we only need to specify the initial displacements $x_i(0)$." ] }, { "cell_type": "markdown", "id": "b756d0cc-3bb2-4cac-a1e5-109c5ea8c402", "metadata": {}, "source": [ "## Brute-force numerical solution" ] }, { "cell_type": "markdown", "id": "482262ef-97bd-4554-97e6-ad8db1d9522e", "metadata": {}, "source": [ "We can of course solve the coupled ODEs directly, using the `scipy.integrate.odeint` function as we did before. Here is the code for doing that. Note that we have to define velocities $v_i \\equiv \\dot{x}_i$ as auxiliary variables in order to turn the equations to first-order." ] }, { "cell_type": "code", "execution_count": 1, "id": "9427c1c0-78d2-47fd-a103-c60486e1dab5", "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "np.set_printoptions(suppress=True)\n", "import scipy.integrate as intgr\n", "import matplotlib.pyplot as plt" ] }, { "cell_type": "code", "execution_count": 2, "id": "47960261-877e-42da-a20b-0dc5dcf05f03", "metadata": {}, "outputs": [], "source": [ "def equations(X, t, m1, m2, m3, k1, k2):\n", " x1, x2, x3, v1, v2, v3 = X # unpack variables\n", " dx1 = v1\n", " dx2 = v2\n", " dx3 = v3\n", " dv1 = -k1/m1 * x1 + k1/m1 * x2\n", " dv2 = k1/m2 * x1 - (k1+k2)/m2 * x2 + k2/m2 * x3\n", " dv3 = k2/m3 * x2 - k2/m3 * x3\n", " dXdt = [dx1, dx2, dx3, dv1, dv2, dv3] # pack derivatives\n", " return dXdt" ] }, { "cell_type": "code", "execution_count": 3, "id": "286cbe9e-da21-498d-83bf-c4e2b2d83418", "metadata": {}, "outputs": [], "source": [ "# choose parameters\n", "m1, m2, m3 = 1, 2, 3\n", "k1, k2 = 2, 1\n", "\n", "# specify initial values\n", "x1i, x2i, x3i = 0, 0.4, -0.4\n", "v1i, v2i, v3i = 0, 0, 0\n", "init = [x1i, x2i, x3i, v1i, v2i, v3i]\n", "\n", "T = 20. # total time to solve for\n", "time = np.arange(0, T, 0.1) # time points to evaluate solution at\n", "\n", "sol = intgr.odeint(equations, init, time, args=(m1, m2, m3, k1, k2)) # solve equations\n", "X = sol[:,0:3] # vector X consists of the first three components of the solution" ] }, { "cell_type": "code", "execution_count": 4, "id": "84e029c4-5497-4e56-9910-bfdab639d0c8", "metadata": {}, "outputs": [ { "data": { "image/png": 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L54OFpovq+iZu/2IHo/+9mgve3cTk/67j2+25NhZWYQ5NOj07Dpcx0sJaTS1JCPXBw9WJPbmVNpBM0R578yoI93MnzNd6R3szA6P9SSusUiHcJqAUiaV0GwzTnoTUZZDyk9mHV9c3cdF7m/hlbz53TE7g3SuGEhfizX3/t4tlu1U/C0eRkl9FTYOO4R30HjEFF2cn+kT6qRDgTmB3XgUDogJsOme/bv7o9JIDhVU2nfdURCkSaxh1K4T0gjXPga7JrEOf+3k/+/MrWXTlcO6f2ZtZ/SNZfNNohsQG8PC3e1Q/Cwex7ZBmqhxhZlmUthgQ5c++IxXo9crhbi9qGprIKqmmXzfb+EeaaZ5v3xG1ouwIpUiswdkFpjwOJWmw6yuTD1u1v5CvtuRw84SeTOv7d0FAV2cn3rhkCAh4Ztk+e0h8etFQA5vehk/Ogf/NgTXPd+jT2pZdRlSAJ5H+luWPtKZ/lD/VDToyVX8Lu3Gg8BhSQp9IX5vOGxPoha+7C/uOqBVlRyhFYi19zoZuQ+CPV7UQ4Q5o1Ol57uf99Ar34d7pvU7aHxPkxXVj41mVUqSW1NZQkQcfzYTfHoZjRdBwDH5/Gd4dB4c2GT1ESsnW7FKzizS2R3M4qjJv2Y+0Am3FYGrPGFNxchL06eanViQmoBSJtQgBY+6A0oNwcHWHw7/dnkv20RoemJnUZoXSq8+Iw9PVmXfXH7S1tKcHteXw8ZlQmgWXLoHbN8NN6+CGVeDiDp/Pg/xdJx2WU1pLUVW9TfwjzSSG+eDu4nQ8qkhhe1ILqvB0dSY2yMvmc/fr5kdqfhU6ZZpsF6VIbEGfc8AnAv56t91h9U063lidzuCYAKa10ywpyNuNS0bGsDT5CIWVdbaW9tRGSvjpLqjIhSu+hd6z/t4XNRSu/QU8A+DLi6Gq4IRDm5sZ2co/AprDvW83P6VI7EhqfhW9Inwtaq3bEf26aRnuWSUqw709lCKxBS5uWqJixiooyWhz2PI9+RypqOPuaYkI0f6X/vJR3WnSS37enW9raU9t9n4L+3+AKY9B7KiT9/tGwGVfQ20ZLH/ghF3bDpXi6+FCrzDb2tr7RvqRml+pMtztgJSS1IJKksJt+5k1oxzupqEUia0YehUIpzad7lJKPt6YTc9Qb5NKkyeE+dA30o+lu1QosMk0NcDqZyBiAIy9u+1xEf1h4gJIWQqpPx/fvDW7jOHdA21+Z5sU4UtlXRMFanVpc4qr6imraaR3hH0USUKYD24uTkqRdIBSJLbCNwJ6ToHdS4w63XccLmd3bgXXnBHX4WqkmbMHdSM5p5ycUhUKbBI7PoHyQzD1KXDq4Kt9xp0Q3h9+eRCa6imtbiCj6JhN/SPN9DY4gVMLVPCErWl+T5NsHLHVjKuzE73DfVXkVgcoRWJLBl0KFTlwaONJuz7ffAhfdxfOHxpt8nRzBmqt6tWqxASaGmDDS9B9rFbKpiOcXWHm89rnte3j41V6R9hDkRjMLmlKkdic5vfU1hFbLelniNxSpsm26RKKRAgxSwiRJoTIEEI8ZGT/vUKI/UKI3UKI1UKI7i326YQQyYbH0s6VvBVJZ4G730nmrWP1Tfy6t4CzB3fD293F5OligrwYFBPAqpRCW0t66pGyFI4VwLh7tUg6U+gxCeInwO8vsetgLm7OTgyMtk2tppb4e7kS4efBgS6qSKobq9ldvJtVh1axPmc9B8sPopf/jD4cKQWVhPm6E+RtXV209ujXzY/ymkaOVCjTZFuYflWzE0IIZ2AhMB3IBbYKIZZKKfe3GLYTGC6lrBFC3Ar8B7jYsK9WSjm4M2VuE1dPLa8kZRnMeU1zwgO/7i2gtlHHvKFRZk85sVcob61Jp6KmEX8vVxsLfAqx5X0I6qGZF81hyhPw4TTC0r5gQPQFeLg620W83hG+Xcq0JaVkQ+4Gvkr7ir/y/6JJf2JlBn93f87ucTaXJV1GjF+Mg6TsmLSCKrv5R5rpZ8gF2pdXQZSFjc46m9qmWjYd2URaWRolNSU4CSeifaNJDEhkVOQonJ1s+z13uCIBRgIZUspMACHEYmAucFyRSCnXthi/GbiiUyU0h75zIfkLyFoPidMB+G5HLt2DvSxqATohMYQ3Vqez8WAJswdE2lraU4P83ZCzGWb+q2PfSGtiRqCLm8CMrO/I73ONXcQDTZFsyjxKk06Pi7NjDQFppWk8velp9pTsIcI7giv6XMHQsKFE+kTSqGsksyKTjXkbWZy2mMVpi7m679XcOvhW3J3dHSp3a5p0etKLjnHNGXF2PU+fCD+chBa5NaNfhF3PZS2F1YUs2r2InzJ/orapFoEgwD2AJtlEVUMVni6ebL5ss83P2xUUSRSQ0+L/XMBI3OZxrgd+afG/hxBiG9AEvCCl/MHYQUKIm4CbAGJjY62Rt316TNLMW/t+gMTp5JXXsinzKHdP7WWyk70lg2MC8PVwYcOBYqVI2mLnZ+DiAYMvs+jwjITr6J19DXPYCAy0rWwGeof70tCkJ/toDQlhPnY5R0dIKflk3ye8vuN1/Nz9eOaMZ5jTcw6uTieudAeEDmBuwlyKa4p5bcdrfLj3Q/488ievTHqFaF/TfXz2JvtoNQ1N+uM+KHvh6eZMj1Af4w53vV5LRN73A9RXQEB3GHkjBMbZVaaTxJB6vkr9ile3v4pO6jin5znMjp/NwNCBeLpoq6iyujKOHDuCk7D9jUxXUCQmI4S4AhgOTGyxubuUMk8I0QNYI4TYI6U8KSVcSrkIWAQwfPhw+3nNXNyh95laVWDda/ywMw8p4bwh5pu1QEtoG9szhN/TS5BSWqSMLKFJ38TOop1sLdhKamkqecfyqG2qxVk4E+wZTK/AXgwLH8b4qPF4udo+o9hkdI2w9zvoNQs8LUskXNXQH70+lqSs/4G8xXQfixk0m1/SCqocokgadY08/ufj/Jz5M9O7T+eJ0U8Q4BHQ7jGhXqE8P+55psVO49E/HuXy5Zfz3vT3SApK6hyhOyAlXzMV2tu0BZqfZEtWqzptVQXw3U2a9cHdX4vcPLACtiyCyY/AuHvsLhdATWMND254kHW565gQPYFHRj1ClM/J15tAj0ACPWyXbNuSrqBI8oCWRthow7YTEEJMAx4FJkop65u3SynzDH8zhRDrgCGAY2uL9J0Lu5cgszbw3Q5nRsYFERts+cV2fK8Qft1XQGZJNT1D7XsRyqnM4cvUL1metZzSulKchBNxfnHE+Mbg4+ZDo66R4tpifsj4ga9Sv8LTxZNzep7Ddf2vo5tPN7vKZpTM9VBTAgMvsniKbYfKqPE6lwdK3tAi7uLG2VBAjYQwH5yEVhfqrIGdu7KsbarlnnX3sDFvI/OHzOfGATeadUMyOXYyX5z1BTeuuJHrfr2O92e+T7/gfnaU2DTSCqpwdhKdopj7dfPjx+QjlFY3aI790iytDE9tOcx5FQZfoflEK4/Arw/Bqqe0Kgvj77WrXGV1Zdyy6hZSS1N5aORDXJZ0WafdbLakKyiSrUCiECIeTYFcApxgoxBCDAHeA2ZJKYtabA8EaqSU9UKIEGAsmiPesfScAi6elOxYysHiGdw4vodV043uobV83ZpVajdFUlRTxFs73+KHjB9wdnJmSswUzow/k1GRo/B1O/mOT6fXsaNoB0sPLuXb9G/5Pv17bhx4I9f1vw43Z/tF0JzEnq/Bwx8Spll0uF4v2XaojJj+Z0PGp7D1A7soEg9XZ+JCvDvd4d6ga+CuNXexOX8zT415inm95lk0T7x/PJ+e+SnX/nott626jc/P/NzhTvjUgiriQ7ztFiDRkr6RmsM9Jb+SseFN8Nm50FQHN6zUEmCb8esGF3wM398Mq5+GkEQtAMcOVDZUcvPKm8msyOSNyW8wMWZixwfZCYeH/0opm4A7gN+AFOBrKeU+IcQzQohzDMP+C/gA/9cqzLcPsE0IsQtYi+Yj2Y+jcfWEHhNxzliBm4tgtpV3oD1CvAnydmPbIdu3bK1rquPdXe8y5/s5/JT5E1f2vZLf5v3Gy5NeZlr3aUaVCICzkzMjIkbw7Nhn+eX8X5gcO5mFyQu56peryDt20oLSPjTWaZnpfedqJkULSCusoqquiUHx3WDIFVqTslY1uGxFUoRvp1Z01ul1PPT7Q2zK38QzY5+xWIk0082nG+9Ofxe91HPLqluobHBstndqQSVJnWDWgr9L1KfklsL/XatVlL782xOVSDNOznDuOxA5CJbda5N23K1p0DUwf/V80svTeW3yaw5VItAFFAmAlHK5lLKXlLKnlPJ5w7YnpJRLDc+nSSnDpZSDDY9zDNv/lFIOkFIOMvz90JGvoyVNPWcQ1HCEq3rW4edhXdiuEILh3QPZlm3bL+SBsgNc+vOlLExeyLiocSydu5QHRjxAmFfbBSWNEeEdwUsTX+K1Sa9xqPIQFy+7mOSiZJvKapSs9Vp5+L5zLZ6iuVDjyPggGH4d6Jtgx2e2kvAEeoX7cqi0hpoG85qgWcpbyW+x8tBK7h9+P+cmnGuTOeP943ljyhscOXaEhzY85LB8k6q6RnLLajtNkQT7uBPh50Hsnjfh8J+aOSt6WNsHOLvC3IVQWworHrOpLFJKnt70NDuKdvDv8f9mXJTtV9Dm0iUUyanIRiftS3ZpgG0WSMPjAsk+WkNxVX3HgztASskXKV9w6bJLKasr451p7/DKpFesNlVM7T6VJXOW4Ofmx40rbuTPvD+tlrVdUpdpEXJxEyyeYktWKRF+HkQHekJwT80suf1jsztemkJShC9SQnqh/SvJ/pr1Kx/s+YB5ifO4qu9VNp17SNgQHhr5EL/n/c5Hez+y6dxIqa0yv7gQ3psAn54L2z7SmpS1oHllZ8+M9tbMCc5j6tHPYfDlMOiSjg+IGACjb4XkL6H4gM3k+Cr1K5YeXMptg25jVtysjg/oBJQisRNfpepII44eZX/YZL7mGlDbD1m3KqlrquOBDQ/wwpYXGNNtDN+e861N72hi/WL59MxP6e7XnbvW3sW2gm02m/sE9DpI+0XL1XGxzCfT3MhqRHzQ3w7KETdAZR4c+NWGwmo019xKs7N5K7U0lcc3Ps6QsCE8OupRuzhfL+p9EdO7T2dh8kLSStNsM2l9leZ7WHwZFKWCT7hWwmbZPfDeeMjdfnxos6+pMyK2AGhq4OaK1yiSAdRNe97048berZm6N9jGdZtyNIWXtr3EhOgJ3DLoFpvMaQuUIrED5TUNrE4tpChyEiLnL5vYSPt388fdxYmt2Zb7SY7WHuX6FdezInsF9wy7hzenvEmwZ7DVsrUmxDOERTMWEekTyR1r7iC9LN3m5yB3K1QXa2VpLCSntJbCynpGtuw/kjgT/KI0p7uNiQ3ywsPVya41t8rryrlzzZ34u/vzyqRXcHW2TzUEIQSPj34cfzd/HvnjERp0DdZNWFeprT6yfofZL8GdO+Hy/4M7tsGV32v+sI/P1MJr0XqQ+Li7aCvJzuDPNwitOcijjddxoNwMxewdouWV7PkGSqz7HdQ11bFgwwICPQJ5buxzDonOagulSOzAT7vzadRJokadB1Kv9SmxEjcXJwZFB7DjsGWKJLMik8uXX86B0gO8MukVrut/nV2/iEEeQSyavggvFy/mr5nP0dqjtj1B2nJwcoWE6RZPsaW5kVV8i0KNzi4w7FrIXAtHbRtF7uwkSAzztZsikVLy1KanKK4t5rXJrxHiGWKX8zQT6BHI02c8zYGyA7yd/LZ1ky1/AI7sgIs+1S68zoaAUiE0c+Mtv0NYkrZayVh1vDRKp1xMK3Lh95ep7jmbNfqh7De3pPyY+ZrPZMv7Vonx5s43ya7M5vlxz9stH8RSlCKxA9/tyCUpwpf4gePBO9RmZpKB0f7sP1JJo848B2dGWQbX/nottU21fDzrY6Z1tyxU1lwivCN4Y8oblNSWsGDDAnR6ne0mT18F3ceAh+U28q1Zpfh7up7cyGrIFSCctbL0NsaeNbe+Tf+W1YdXc9eQu+gf0t8u52jNxJiJzEucx8f7PrY8wGLPN7B7MUx8EPrMMT7GKwiuWgphScj/u5b6gv2dZ9Za+QRIPZ5n/Rsfdxf255upSHxCtYCQXV9BQ7VFIiQXJfPZ/s+4uPfFjI4cbdEc9kQpEhuTWXyMnYfLOX9oFMLJWTOVZKzSMrCtZEC0P/VNerOctQfKDnD9iutxFs58POvjTrvANNM/pD+Pj36cLQVbeHd3+62ITaYiD4r2WZw70szW7FLjjaz8IrXqBDu/0MrT25DEMB9KjtVTXmPbeTMrMnlxy4uMjhzNVf1s61zviOZIv+c2P3dS8ccOqa/SesJEj4Dx97c/1jMALl2M3tmd1/UvMjCkE1Yjh/7Uum6OvQunoDj6RPqavyIBGH491Fdqc5lJk76JZzc/S7h3OPcOs2+Co6UoRWJjvt+Zh5OAuYMNJQp6zYS6CjhsfaG0QdEBAOzOLTdp/IGyA9zw2w24CBc+mvkRPfytS4y0lLkJc5nbcy7v7XqPnUU7rZ/w4GrtrxVmreKqejJLqk80a7Vk+LVaxnzqTxafwxiJ4VpCaUaR7SK3GnWNPLThITxcPHh+3PN2qaXUHt6u3iwYsYC0sjSWpC0x7+A/39Le5zNf/Nuc1R7+0ewa8wYxooip2a9YJrCp6HWwfAH4RR/vuNk30o+U/Er0ejOrLMWOhtA+sO1js8VYkraEA2UHeHDEg44tR9QOSpHYEJ1e8s32XMYlhhLu56Ft7DlZs+VnrLR6/u7BXvh6uLA7r+NubXnH8rh55c24Orvy0ayPiPOPs/r81vDIqEfo5tONJzY+QV2TlX0d0leCbzcI62PxFM35I202suoxRSvAZ8EPvz0SDWa0dBsqkvf3vE9KaQpPn/G02TlAtmJa7DTGRI7hrZ1vUVJbYtpBx4rgzzeh77kQ1U5ORis263rxlu48Qg9+a9Edvsls/xgK98DM58BNu4D36+ZPdYOOw+Z2LRVCM5ke2QElGSYfVlJbwls732Jst7FMjTWhYZuDUIrEhqw/UER+RR2XjmiRj+Huq9ny061XJEIIBkb7d7giKa8r55aVt1Cvq+e9ae/R3a97u+M7Ay9XL54c8yTZldm8vcsKx6yuCTLXQeI0q4orbskqxcPViQFRbTSycnKCYVdD9u9m/fA7IirAE09XZ5vlkqSXpfP+nveZHT+bKbFm9mKxIUIIHh71MHW6Ol7d/qppB21ZBI01MOVxs86VVlDFN16Xauawn+6B8pyODzKX6qOw+lmIG68pOgN9u2k+OYt6uPc/HxCw9xuTD3l1+6vU6+p5eNTDXSpKqzVKkdiQr7bkEOLjzrS+4SfuSJwBRftt8oUfGB1AWkEVdY3GHdd1TXXMXzOfI8eO8MbkN0gITLD6nLZiTLcxzEucxyf7PmFvyV7LJslP1mzNPSZbJcvW7FIGxwTg5tLOT2DwFeDkot2Z2ggnQ5HB9CLrHe46vY4n/3wSX1dfHhz5oA2ks454/3iu6XcNSw8u7diE2VirJRr2ng0h5n1H0wqqSOwWCOe/r0VFfneTZoayJauf1qomzP7vCTcsCWE+uDgJ9udb0MPdr5tWx23PN1riZQfsLNrJ0oNLuabfNV3iZrA9lCKxEYWVdaxJLeLC4dG4tm5c1GzLt4F5a2CUP406aTSEVErJo388yq7iXfx7/L8ZHjHc6vPZmvuG30eIZwiPb3ycY/V15JTWkF9Ra/oEWRu0v3HjLZahqq6RlPxKRsZ3kEPjG67lqSR/qV34bERimI9NfCRfpHzBnpI9PDjyQYI8bN9r3hJuHHAjYV5h/Hfrf9vvcb7n/6DmqJb5bQYNTXoyio5pEVtB8XDWS1rJko2vWyl5C/K2w45PYdQtJ5lPPVydSQjzsczhDjDgAjiaDvm72h0mpeTV7a8S5hnGjQNvtOxcnYhSJDbi003Z6KXk0hFGmmaF9gb/WC1k1Ur6dfu7CmlrPtr7ESsOacmGM+JmWH0ue+Dr5ssTo58gozyD8e8+y/j/rGXMv9dwzcdbTMuvyNoAYX21kEoL2Zpdil7CyLb8Iy0ZcaNWL2n31xafrzUJ4T7kV9RRVWd5JF9OVQ5vJb/FhOgJzI6fbTPZrMXL1Ys7Bt/BnpI9rDi0wvggKeGvRRDe3+xKy5klx2jSy79rbA28WDM9rf0XHEm2SnZAa1T18/3gE6aFIxuhb6Sf+SHAzfQ5RwstT1na7rDf835nZ9FObh508/HGVF0ZpUhsQE1DE59vPszMvhHG+44IoZXyyFwHTdbVyooO9MTbzfmkXISNeRt5fcfrzIqbxTX9rrHqHPYm83AsTcd6o/NfwePnxHDf9F7sPFzOvHf+ZGd7CZdNDZDzl1WrEYCNGUdxc3FieJwJSV1x4yBiIGxaaJI5whSaHe6WrkqklDyz6RmchBOPj368y9nOz+l5DgkBCby+43UajYW9F+zWnNjDrjHbz5Wa36rGlhBaAUXvEM3EZe3KceenmkN8+rNt5ij17eZHYWU9Jccs+C17BUHcWK2eWBvopZ63dr5FtE805yWcZ/45HIBSJDbg/7blUlHbyI0T4tselDgDGqu1uHQrcHIS9I7wPWFFklOZw4INC0gITODpM57ucheWlmzJKuX5n1MY5nM1zk5N5PAd86cm8uvd4wnyduOqj7a0XWo9b7vmnI23VpGUMCIu0LQ+FkLAmDugJA0yVlt13mYSDY2YLI3c+iHjBzbnb+aeofcQ4d31eog7Ozlz77B7yanK4esDRlZyyV+Bsxv0N7+sfWpBFa7Ogh6h3n9v9AqCc9/WPqOVT1oueEUurHhcu1Fpp1Fa30hNwRizCphE77OgOLXNII5Vh1aRUprCbYNvs1uJG1ujFImV1DXqWLQhkyGxAQzr3o6pJH689uOxQfRWUqQfqQVVSClp0DVw3/r7AHh98utdNs4coL5Jxz1LkokJ8uLti2dxSdIlfJf+HamlqUT6e/LVTaNxd3Hizq92Ut9kxHma/TsgoPtYi2UorqontaCKsQlmlA/pdx74RMDmhRaftyUxQV64uThZtCIprinmv9v+y9CwoVzY+0KbyGMPxkWNY1TEKN7b9R5VDS1uDJoatGZkvc/UFICZpBZU0jPU52Q/ZM8pMOpW2PIe7G/fbGQUvR5+vENz2p/zZrsrpebILYv9JEkGU2TayasSnV7HW8lv0dO/Z5cyWXaEUiRW8tmmQ+SV1/LAjN7tD3Tz1swkNnC494nwpaK2kYLKOl7f8ToppSk8O/ZZYnwd27GuI7786zB55bU8O7c/fh6u3DLoFvzd/Xlxy4tIKYkK8OQ/FwwktaCKl34zUlE2a4NWmtuCC1Azfx7UchzGmaNIXNxg1E1wcA0UWt8WwNlJ0DPUh3RjK6+GGi034qe74dsbYd0LULjv+O5//fUv6pvqefqMpzs98dAchBDcM/weyurL+Hhvi6i3jFWak33QZW0f3A5pBVVt9yCZ/rSWj/LDreaXbf/jFa2+2oxnNSd+OwR4uREV4GlZCDBAQKxmLjVi3lqWuYysiizuGHIHzk727/xoK7ruN/EfQHlNA2+uSWdS71DOMOXClDgDSg5o/Z6tIMmwtP4uZTWf7v+Ui3tf7NAcAlOoaWhi4dqDjO4RxNgELVrK392fOwbfwbbCbazNWQvAlKRwLh0Zy4d/ZJFa0OKH2lgHOVsg3vLeI6CZtfw8XI4HLZjMsGvBxdNmq5LEMJ8TTVtSatFhr/WHb66Dfd9BzmZNkbxzBvx8HysP/sSqw6u4dfCtDk8wNYV+wf2YHT+bT/d/SkG1oevk/h/AIwASzE+uq6hpJL+i7ng5/pNwcYeLPgMXD/hiHlTmmzZx+kpY8xz0v0BrbmYCfaxxuAP0mqVVsG5RGbxR18g7u96hb3DfLp18aAylSCxESsmj3++lukHHQ2cmmXZQoiGSyspqwL0jfBHOx/gk/UUSAhK4f3gHNYq6AF9vzaHkWD33z+h9gg9nXq95dPfrzps73zxe1PHBWb3x83TlqaX7/g4hzd0CunqrHO1SStalFTMuMQTn1vW1OsIrCAZfpkVvVR6xWIZmEsN8yC2r1bol1h+DJVdod9LBCXD1T7AgG+7eA/enw+jbqdj+If/6/TH6BPbi6n5XW33+zmL+kPnopZ6FyQu1QJO0XyBpjlYN10yaL9zNbW+N4h8Fl3+tJRR+fn7HbZOz/4AlV0J4PzjnDZOd/327+ZFZfIzaBgvzVxKmajkwWeuPb/o2/VvyjuUxf8j8Lu3nNEaXUCRCiFlCiDQhRIYQ4iEj+92FEEsM+/8SQsS12PewYXuaEGJmZ8n85ZbD/Lwnn/tn9Da9S1twTwiMt9pP4uvuQkDs99Tpq3lxwot4uHhYNZ+9kVKyeGsOA6P9jzfoasbFyYU7Bt9BRnkGv2T/Amimg/tn9GZzZim/7jVcCLI2aGGT3c+wWI59RyopqqpnSlJ4x4ONMfYubeWw4b8Wy9BMc82t7EOH4ONZWln8Gc/Btb9qqy4nw0/TJxRm/YuXh51LGTqeLqvB1TbBY51CtG80lyZdyo8ZP5K+9ystmbTfuRbN1ezcbvZRtEnUMLj0Syg7BO9PhbwdxsftWqJ1YgyI1XqeuHkbH2eEvpF+6KUVTcqihoO7//EAjtqmWhbtXsTQsKGM7Wa5D9BRmKxIhBDPCCH+JYS4SAjRy1YCCCGcgYXAmUBf4FIhRN9Ww64HyqSUCcCrwIuGY/sClwD9gFnA24b57IaUkg9+z+TxH/YyPjGEmyeYWQgxcYZ2UbQiTPGnzJ9o8tiHT/XZ9Aq02UdhN3bnVpBaUMXFI4z7cGbEzSApKImFOxfSqNfCRS8dGUtimA//XZFGk06vNTzqNtiqsvGrU4oQAib3tjAHJbC7VjZlx6dWmycTwnwJpoKopRdr0TuXfQ1nzP9bgbRg05FNfF+ynWvCxtAn60/Y8JJV5+5sbhxwIz6uPryx72Pt4hk/0aJ5UvIrCfFxI8zXhBunHpPgul+0u/73p8APt2l5XIX7IOUn+PwC+P4miBykrQB9zKtR1s9ah7uzC/SYoPndpGRx6mKKa4u5c+id/7jVCJihSKSUTwCvAxXAeUII67q0/M1IIENKmSmlbAAWA3NbjZkLNDeH+AaYKrR3ey6wWEpZL6XMAjIM89mcJp2eh7/bzRkvrOG5n1OY2S+CRVcOP7kEeUckzoCmWsjeaJEcxTXFvLDlBcJckyjMHdFmqZSuxOKtOXi4OnH2oG5G9zsJJ+YPmU/usVy+T/8e0BzSD8zsTWZxNd//dUAL/bUyf2RNaiFDYgII9nG3fJIJD2hFOFc9ZZUs3T2q+dL9X3hVH4bLlmh5Rkaoaazh6U1P092vO7fMeBMGXAS/v2Sb5LtOIsAjgOv6Xc26plJ29BxrcWvklIJK+kSacSMROQhu2wRjbtfKknwxT/M3LblCy2WZ9hRcvUyrYGAm0YGe+Hq4sPeIBaVSmuk5FSrzOHZkJx/u/ZCx3cYyLNz04pVdiQ4ViRDidcNFGylloZTyNynli1JKW+XtRwEti1DlGrYZHSOlbEJTZsEmHguAEOImIcQ2IcS24uJis4V0cdbCNQfHBPDyhYNYeNlQPN0sWPzEjdWctultZP22g5SSZzY/Q4OugSsTFqDTC5uWI7cHDU16lu06wuz+kfh5tG0XHx81niFhQ3hv13vHqwNP7xvO0NgAfl/zM+gbrXK0F1XVsSu3gql9LDRrNeMbAePv1ZzGluaVHCvG9fNz6S4KeS3seejR9h36W8lvkXcsj6fGPKWZMGf/B7xC4Ke7tJDVfwiX+/QitKmJV50r2y+d0gaNOj0HCo6Zp0hA62Ey83l4MBuu/AEu/B9csxzu2Qfj7jGtdL0RhBD07+bPXhMqcbeJIeDgs51vUlFfwfyh8y2fy8GYsiKpApYKIbwBhBAzhRCW3U47ECnlIinlcCnl8NBQy0wbX988hneuGMa8YdHmr0SacfXUckrSV5idKb08aznrctYxf8h8xsdrNYDs1W3PVvx5sISq+ibOGhjZ7jghBHcOuZOi2qLjPS2EEDw4K4k+dTvRCRetp4OF/GbwtUyzVpGA5isJToCf79PCdc3hWBF8MgdKs3g36gV+rmq7YOGu4l18vv9zLu598d910zwDtTvp/GRNmf1D8Dy4mlsrqkk+dph1OevMPj6zuJoGnf54MqDZuHlpLR36nafdzNkg0W9gjD8p+ZXGc55MISCW8pAEPjm6g2mx0+gX3M9qmRxFh4pESvkY8BWwzqBA7gVOcohbQR7Q0ngebdhmdIwQwgXwB46aeKzNsJntstdMKMuCYiO5Em1QXlfOi1teZGDoQK7ocwVxwd64uziRak0IYifw275CvN2cTUoAHB4xnNGRo/lo70fUNGoX6FE9gpnhlc4u2ZOKJstMIgBLdx2hV7iPbdqzurjDnNegLBt+vtf0G4KqQvjfHCg/DJf/HzJuPIdLa4yaJxt0DTz151OEeYVx99C7T9w58CKt3tiaZ23SebNTSPuV80KGEucXx+s7Xje77XLK8Ygty31ktmZQdECbBVRN5aOwKGqkjtv732BDyTofU0xbU4EbgWogBLhTSvm7DWXYCiQKIeKFEG5ozvPWqalLgeaYxwuANVJbHy8FLjFEdcUDicAWG8pmH3q3ndnaFq/ueJWqhiqeHPMkzk7OODeXSinouopEp5es3F/IpN5hppUjAW4ffDuldaV/d9qrq6RHYzp/NPXl3Q0HLZIjr7yWrdllnNOGj8Yi4sfDpIe0PtxbTHAXlh3SViIVuXD5NxA/nsRwH/RSu9tuzaLdi8goz+CJMU/g4+Zz4k4nZ5j6BJRmalV0uzpHD8LRdFx6z2b+kPkcrDjI0oPmZZ+n5Ffi5ux0YmkUB9Pcy2ZXrmXmraKaIr6qz+Ws6hoSKgttKVqnY4pp61HgcSnlJLSL+BIhhM2y3ww+jzuA34AU4Gsp5T5DlNg5hmEfAsFCiAxarIiklPuAr4H9wK/A7VLKru999usG3Ya2W7itJTsKd/Bd+ndc2e/KE6K0+kT4kZJfZZHNuTNIzimj5Fg9M/qZbk4aHDaYsd3G8tHej6hurIbDmxBSh1P8BD7emEVhpfndFZft0vI+2nL2W8yEB7TEsl8e0KrZtsXBtfD+ZG1FcsU3mmmFlt0ST7yjTS1N5cM9H3J2j7OZEN2GX6jXLG1V8udbNismaTcO/Kb97TWD6d2nMyBkAAuTF5rVKXN/fiWJ4UZKoziQ6EBPgrzd2GNi6+vWLNq9iCap57bKGpvVcXMUppi2pkgp/zA834MWpvucLYWQUi6XUvaSUvaUUj5v2PaElHKp4XmdlPJCKWWClHKklDKzxbHPG47rLaX8xZZy2ZWk2VokUgfZt426Rp7d/CyR3pHcMvCWE6eI9KW0uoFiS6qQdgKrU4pwcRJMTjIvtPK2wbdRXl/OV6lfaaHSzu6cM+dcdHrJ66vTzZpLSsl3O/IYFBNA92Ab3806OcNFn2pF+H55QIsGKkrVLuxSwpGdWpmTz84Fr2C4cc0JeTBxIV44O4kTuiU26ht5fOPj+Lv7t9+sSggtGqlon1baoytzcA0EJ0JgHEII7h56N4U1hSxOXWzyFCn5lZb7R+zE3x1LzV+R5Fbl8m36t5yXeD4x3UZo79E/GLPVu5QyH/hn5e93RZLmaH87MG99uv9TMsozeGTUIycVZGxOhGwurd3V+COjhCGxAe1GaxljYOhAxkeN53/7/sexrPUQM5LY8CAuGxnLkq05ZBabHqm2KfMoaYVVXD7KSJ8YW+DiDhd9orWLTV8Fb4+C/8TDv2Ng0STY/6O2crl5w0mdAN1dnOke7HXCiuSjPR+RWprK46M1ZdIuAy4En3CtxH1XpakeDm3UHN0GRkaOZGzUWN7f8z6VDR2bZouq6ig51tCl/CPNDIzy50BhlVahwAze2fUOTjhx08CbtIKTRfu1Fes/FIvWiVJK27WLO10JTdIif/b/2OaQguoC3tv9HlNipjApZtJJ+5sdx22WXXcgZdUN7MmrYFyCZRFytw++nYr6Cr6oO3w87PeOKYm4uzjx8krTC/J9vDGbIG832/pHWuPsChPuhzt3aE74PmfDkCvg7Nfh/jSY8pgWrWeEljW3MsoyeHf3u8yKm8XU7ibcq7m4azXAMlZrPpiuSM5fWun/nidaw+8eejeVDZUnFnRsgxTDjVJXVCSDYgLQS9hjxqokszyTZZnLuCTpEq0NQHOCZrbprueahiZ+2ZPPj8l55JaZGTloB7qOwfF0QwgtFDH7DzhmPK/lle2voJd6FoxcYHR/kLcbIT5uXVKRbDxYgpQwLtGMKrst6BfSj0kBffjEz5fKKC1JK9TXnRvGxfPz7nx2m2CXPny0hlUphVw2MtZkZ79V+HWD4ddqZcjPfEFr3OTZfvOsxDBfDh2tobqhnsc3Po6vqy8Pj3rY9HMOuVz7m/yF5XLbk4Nrtb73rTohJgUlMTt+Np/v/5zC6vbvxJuzx7uaaQtgSKz2+W5vryFbKxYmL8TD2YPrB1yvbYgcBB7+WuM7E/hmey7Dn1vFrV/s4K7FyYx7cS3//S0Vvd5xvjKlSBxJ33O1Eg5G2m7uKNzBL1m/cE2/a4jyMZpjCUCvcF/SCrteUuIf6SX4ergwKNrMKrstuI1Aqpyd+KIy5fi2Gyf0IMTHjQe/3UNDU/sJeS/+loq7ixNXjulusQz2JjHcB51e8ua2D9l7dC8Pj3rYvP7rAbFaYtvOz7VeGl2Ng2sgegS4nxx2PX/IfHRSxxs732h3ipT8Srr5e+Dv1fWaPAV5u9Ej1Jsdh8pNGr//6H5WHFrBFX2v+PtzdnLWqjZkbejw+NdXpXP//+1iYLQ/X944imXzx3HR8GgWrj3I3UuSHRZ4oxSJIwnvpzkh931/wmadXscLW14g3Cuc6/q3X9a6V7gvGYVdL3Lr9/QSxvQIxsWKKJs+ebuZijefpn5BRb1mOvD1cOVf5w0gJb+SN9pxvG86eJSfd+dz68QEwv26blHLhDAfnNyPsDjjfaZ3n86suFnmTzL0KqjM63oO25pSyN91klmrmWjfaK7qexVLDy5lb8neNqdJya/suFCjAxkWG8iOw2Um/QZf2/4aAe4BJ7fDjp8A5Ye03KQ2+GnXEV5ddYDzh0bx2fWjOKNnCP2j/Hlx3kDund6LpbuO8OWWw9a9GAtRisSRNJu3Dm08oTT59xnfk1Kawn3D7+uw42GvcF+qG3TklZvhttI1aoUCy7LtktCWW1ZDXnktZ/QMtnyS6hIo2s+tUdM41niMz/Z/dnzXjH4RXDgsmoXrMli2++SS7hU1jTz6wx6iAjy5ydyimp1MdJArHlFLcBM+lvdf7zVLM43s/db2AoLlJpPMdYCEHpPbHHLDgBsI9gg+3tysNXWNOjJLqrukf6SZod0DKa1uIPto+76Kzfmb2ZS/iRsG3ICvW6sVWrOfpI1VSWbxMR76djdDYwN44fyBJ4RBCyG4Y3IC4xNDeHbZfrJKTs5LsjdKkTiaQZdo5q3dWgJeZUMlb+x4g6FhQ026O+1lKEdukp+kNEsLR/1PD3hrGLw+CF5Ogl8fgdpya17FCWzJ0pr1jIy3QpEY+jT0TjqP6d2n83nK55TXlR/f/czc/gzvHsg9S5L5YWfe8YtQVV0j13+yldzSWl66cJBl9dA6kff2vIWzeyE9uZ5Aj/b9KW3i4q45+FOWaQ3AbEBZdQN3L97JsGdXkvjYL9z82TaT/FIncHCNVu2325A2h/i4+XDn0DtJLk7m1+xfT9qfWlCFTi+7pH+kmWHdtc9tx6G2/SRSSl7b/hoR3hFcknTJyQNCe2sReEYUiZSSx3/ci7OTYOHlQ3FzOfmy7eQkePnCQTgJwcsrTK+YYSuUInE0wT0hZrTWHU9K3kl+h/L6ch4a+ZBJd6eJ4c2RW+34SaSEP9+EhSO1JMh+58K572hO4bhx8Ne7WlXUQ5ts8pK2Zpfi6+FiXTmSzHXaXXa3wdw66FZqGmv4ZP8nx3d7ujnzwdUj6B/lz91Lkrng3U3c+7XmeNx+uIxXLx7MGGtWRJ3A5vzNfLb/M8KYzNGS9tu7dkj/edBQZXXTNNDa2c58bQM/78lnSlIYV47uzpasUi54dxOrU0wMUZVS+wx7TOiwMOLcnnNJCkrile2vUNt04sq6OdlvYEyA+S+kk0gI9cHXw4Vt7SiSFYdWsO/oPm4ffDvuzkaqTwuhmbeyNpyUYPrznnw2ZhzlgZm9ifQ3Hv0HEObnwTVnxLFsd/6J3UU7AaVIugKDL4OSAxxM+4mvUr9iXq959AnuY9Kh/p6uRPh5cKCtej+6Rvj+FljxmFbCfv52TYEMvkyzrV/0CdywUmtP+vn5cOhPq1/OlqxSRsQFmd+FsBkp4eA6zQHp5ExiYCIz42byRcoXlNX9/WP193Tlm1vO4Mmz+1Jd38Tv6SWMiAti6e3jOiwS6Wgq6it47I/HiPOLY0rY9WSVVNOos6Kab9wErSqwleatitpGbvpsGxL44fax/PfCQTx1Tj/W3DeJ3uG+3PzZ9uN979vl6EGoyGnXrNWMs5MzC0YsoKC6gPd3n1huZlduBcHebnTz77p+LicnwYi4IP7KOmp0f6O+kTd3vklCQAJn9zi77YniJ8CxwhNq8NU36fjXzyn06+bHZaM6Dhq5aUIPfN1deH2VeYm71qIUSVeg33lIF09e3P4SXi5ezB9iXjnpxHAfDhQZUSS6Rq3/9+7FMPlRuPhz8DNygY0aBtf9Cv7RWse4wv0WvhAoOVbPweJqRsSZEXnUmrIsqDisNScycOugW6lrquPjfSfmHTg7Ca4dG8+vd09g66PT+ODq4QywIlKss/jXX//iaO1R/j3+3/QJD6ZRJznUgY29XZxdtJXmgV+hwXIb+YPf7OZIeS3vXjH0hL72gd5ufHHjKGKDvbh3yS7Kaxran6jZ8d+Go701IyJGcHaPs/l438dklh8vXMHu3HIGRvt3+WZPY3oEk1lcbbSEz9dpX3Oo8hB3Db0LZ6d2TK3H/SR/t9/9ZnsuRyrqePjMPibdmAV4uXHlmO78tq+AI+b4Ta1EKZKugIcfa/tMYVNTGbf1u8a88E+gd7gvGUXH0LV0ikoJP96hhRbP/DdMXNB+P2qfMLhqKbh6wf9drfURt4Bt2c3+ESsUSabhh9RCkfQI6MHsHrNZnLqYo7XG7/z+KfyQ8QPLs5Zz86Cb6R/S/3jNrQxjNwPm0H+elvyXZlmloD/SS/h1XwH3TO/FsO4nf35+Hq68fvEQSo7V8/iP+9qfLHMtBMZBkOkmu/uG34eXixfPbH4GKSXV9U1kFB1jQHSAeS/EATSbUTcdPPG7WVpXysLkhYyOHM3E6A46QwZ2h4Dux/0kjTo9b689yJDYAMYmmG6mvXRkLBKtoVxnoRRJF6BeV89/dUX0bGjg4grzbZu9wn2pa9STU9rijnb9i3+vRMbcZtpEfpEw7wMoSYdf2qnz1A5bs8twd3E6XhnVIjLXgV+UlvnfglsG3kK9rt6kbOiuSkZZBs9vfp4RESO4cYDWG65nmFYDLN3afKCY0eDbDfZ+Z/ahOr3k+eUpRAd6cv24ti/+A6L9uX1yAj/tOsJWw03DyZM1aq2RTTBrtSTYM5h7h93L9sLt/HjwR/YdqUQvsSoXqbPoE+mHn4fLSYpk4c6F1DTWmOzzpMdELcNdr+P7HXnklddy55REs1ZkMUFeTEgMZcnWw1qb6k5AKZIuwKf7PiW3tpAFbrG4bvvI7JDcxNaRW7uWwLp/w6DLtDpP5tBjIoy7G5I/NylBqjU7D5cxMNrfaGSJSej12tK+x6STVlBx/nHM6TGHxWmLKa4xv8ulo6lprOH+9ffj5erFi+NfPG7m8HJzITrQ83ipFItxctLCyTNWmh2Ft2z3EVLyK3lwVhLuLu1Hut08sQfhfu48/3OK8dyJ3G2a499Es1ZLzks8jyFhQ3h528v8dUjLifgnmCqdnQSjewTzZ+bf/qO00jS+Sf+GS5IuoWdAT9Mmip8IdRU05SWzcF0GA6L8mdTb/DJDl42KpbCynnVpnfM7UYrEwRRUF/D+nveZGjuVM85YoCWW7fys4wNb8HfkVpXWC/7H2zVH9dmvt2/OaosJCzSzxE93mxVOWt+kY29e5fGyERZRsBtqy/62F7fi5oE306Rv4qO9H1l+Dgfxr7/+RWZFJi+Mf4FQrxMvDi1rbllF/3mga4C05SYfIqXk/d8z6RnqzVkDOg5S8HJz4b7pvUnOKedXQ+fJE8hcC8LJotbITsKJx0c/zrHGY3x/+A0i/N0J8+26jvaWjOkZTE5pLTmlNUgpeWHLC/i5+XHroFtNn8TwnqVu+olDR2u4Y0qCRf6hKUlhBHq5snTXyXlW9kApEgfz6vZX0el13D/8fkiYBjGjYP1/odF0R5mPuwtRAZ6UH94Hiy/TlMDFn4GLhR0F3bzgrJeh9CBsec/kw/YfqaRBp2dobIBl54W/HY1t9DGP9Yvl7J5n83Xa1xTVFFl+nk7mx4wf+fHgj9w86GbGdBtz0v7EcF8OFrfyc1lC1FDwi4aUn0w+ZNuhMvbmVXLt2HiTW0jPGxZNfIg3C9dlnLwqObhG67fjGWCG4H+TGJjI7YNvp1huJTo6peMDugjjE7Wbg3VpRfyc9TPbCrcxf8j8jqs4t8QnDBnah9q0tSRF+DLdwtbQrs5OzOofyaqUQmob7F86RykSB7KjcAfLs5ZzTf9riPaN1lYPUx6HqiOwpZ1GSUYYEdrE9YcXaJVor/imw2KBHZIwDRKmw4aXtVIXJrDzcDmAdSuSzHUQ2gd8I9occtPAm9BLPR/s+cDy83QiB8sP8vxfml+kdU+ZZhLCfGhoauXnsgQhoM8c7WJuYvTWR39k4e/pyvlD267p1hpnJ8EtE3uwN6+SDektwoFry7U+OxaYtVoyM/oSmmq6k8Xn5B9rv2dPV6FnqDdxwV78kpLBi1teZFDoIOYlzjN7niy/4fRv2s9dE7ubrNiNcfagSGoadKxJtf8Nl1IkDqJlPa3r+1//94748VrJi7X/1mLxTaGhhgWlTxKgK6Pp4q+0FYktmP6MZuve8JJJw3ccLiMqwNPy2laNdVpSZItoLWPE+MYwN2Eu3xz4hpzKzotMsYSqhiruWXcPni6eJ/hFWpMYpvm5bGLeSpoDTXUmJScWV9WzYn8hF4+Iwcut/cTB1pw3JJpIfw8Wrs34e2PWBq1SQ0/zHO2t2Xm4krojFyGE5LGNj6GXneM0tgYhBFP7hJNc8zHVjdU8c8Yz7Yf7GkGvl3yaH4unaGBmgHXf7VHxwYT6uhstI2RrlCJxEN9lfNd2Pa05r4Kzmxa+q+ugYU5jLXx9FZHVKdzZeAfZHqYlMppEeF8YdCls+xCqjNjCW7HzcDmDrTFr5WyGpto2zVotuW3wbbg4ufDSNtOUnCPQSz2P/P4IOZU5vDzx5ZP8Ii1JOK5IbNASIHYMeAaZZN76MTkPnV5y4bBos0/j5uLE9ePi2ZJVyt48Qz+OzLXg5qNV/LWCHYfK8CCMBcMXsKVgCx/u+dCq+TqL4LADOPvuZmrE5fQIML/O24r9hXxXGofECads84NdWuLsJDizfwRrUovsbt5yqCIRQgQJIVYKIdINf0+yiQghBgshNgkh9gkhdgshLm6x739CiCwhRLLhMbhTX4CFVNRX8OaON9uup+XXDc58EQ7/Ccvuarsnd205fHkRZKziyPh/s1I/3Pa9Scbfpzlv/3yz3WHFVfXkldcyxJpSFgdWgLO7SU7aMK8wbhp4E2ty1rDpiG1Ku9iat5PfZl3uOhaMXMDwiOHtjvX1cCXS34MMW7QEcHbRWjkfWAFN7ScOfrM9l0HR/scDNszlwuExeLk589HGLG3DwbVaoIezdSXftx0qZXBMABf2nseZcWfy5s43+SPvD6vmtDcV9RX8X9brUN8NXdkks4+XUvLmmnSCgkO1+mQWRE22ZnrfcOqb9GzMMKEagRU4ekXyELBaSpkIrDb835oa4CopZT9gFvCaECKgxf4HpJSDDY9kewtsC97a+RYVDRU8POrhtiMyBl8KEx/U+kx8fwvUterAlr0R3huvlTQ5712Cxt+EEHbolhjcEwZcBNs+arMBF3C8oN8gaxRJ+m9a7S8303qrX9n3SqJ8ovjP1v/QpDev1am9WXVoFe/tfo/zEs7jkt5GivQZIcFWkVsASWdDfQW0c1e770gFqQVVzLNgNdKMv6crFwyLZtmufI7mpGlVCaw0a1XXN5GSX8XwuECEEDx1xlMkBiayYMOCLmvKlFIzwZXXlzPK71ZW7iuhrtG8VcCa1CL2Hank9skJiB4TIHerxYnBzYyKD8bH3YXVqfZt4+toRTIXaK7E9wlwbusBUsoDUsp0w/MjQBFgWf/WLsDu4t0sSVvCxb0vJikoqf3Bkx7WlMmer+HN4VpY76+PwAfT4X+zQQLX/gqDLsHTzZnuQV7WJ7UZY8L9mglt01ttDtmVW4GTgH6W9o04ehCOZkCvmSYf4u7szgPDHyCjPIOv07627Lx2YG/JXh7+/WEGhgzk0dGPmhy+mRimVSiwSae7HpM0E1M75q2lyUdwdRacPdC6NsTXnBFHg05P8jpDXx0rHe3bDpWh00uGG8rseLl68drk1xAI7lp3FzWNjm8t25r/7fsf63LWcd/w+7huxDiq6pv4bV/H5uBmpJS8sSaD6EBPzh0SpYW/65vg8Gar5HJzcWJCrxBWpxTZtYOioxVJuJSyOSSjAGg31k0IMRJwA1p6oZ83mLxeFUIYKat5/NibhBDbhBDbiosdk8zWqG/k6U1PE+oVyp1D7uz4ACFg8iNwwyqIMVTu3f6x5keY9SLcvhli/rZFJ4b7kmaPtrshiVp+wpb324zg2pNbTmKYr9kO2+Okr9D+Js4w67ApsVMYFTmKhckLKam17/LdFPKO5XHH6jsI9gzm9SmvG6/02gaJ4T7UNprZW6YtXD0gcTqkLjfaOVFKyc978hmbEEKgt4Vh4gZ6hPowuXcoInMN0khFAnPZmFGCm7MTI+L+tnTH+Mbwnwn/4WD5Qe5bfx+NduijYynbC7fz+o7Xmd59Opf3uZzR8cFEBXjyzfZck+dYub+QXTnl3DE5Qes1EjNK85NmrbNavqlJ4RRV1bP3iOl95c3F7opECLFKCLHXyGNuy3FSC0ZvU2UKISKBz4BrpTwewvEwkASMAIKANut6SCkXSSmHSymHh4Y6ZkHz+f7POVB2gEdGPoKPm4/pB0YNg0u+gAVZ8MgRuOUPGH3LSSag3uG+ZJdUU99kB8fahPuhsRo2LTxpl5SS3bkVDLQmA/nArxDSy6zaTKBFyjwy8hHqmup4dtOzDu0UWdlQye2rbqdB18DCqQsJ8TSvX31z5FaGzcxbc6C6SDORtGJ3bgW5ZbXMNiEB0RSuGx3FCP1usgPPsCwJtgV/pJcwtHvASTclY6PG8vjox/kj7w8e+v0hdF2gtXBJbQkL1i8g2jeaZ854BiEETk6CecOi+SOjxKTCiU06PS/+mkqPUG8uaDYzunlpysQGfpLJSWEIAatT7BcGbHdFIqWcJqXsb+TxI1BoUBDNisLoKxVC+AE/A49KKTe3mDtfatQDHwMj7f16LCW3Kpe3k99mUswkpsRauPQXot0faWK4D016aZ8OaWF9oO9cLb+lVfmNvPJajlY3WK5Iakq12kxJZ1l0eI+AHtwx5A7W5Kzh56yfLZPBSuqa6rhn7T0cqjrEq5NfNb0kRguaI7ds5udKnKHd1Roxby3fk4+rs2Bm37bzdcxhnMdBfEUtX5T2skqZlxyrZ39+5fHkvtZc0OsC7h9+PysOrThe3NFR1DTWMH/1fCobKnl54ssn3BxeOCwaZyFYtCGznRk0lmzL4WBxNQtmJp3Ymjp+AuTvhmrripQGebsxMDqADen2s8Q42rS1FLja8Pxq4MfWA4QQbsD3wKdSym9a7WtWQgLNv9J242cHopd6ntr0FEIIHh1lus3cXHqZ0uTKGsbfB/WVsPXERMA9udqSeaClVVrTloPUQZ9zLBbtqr5XMSh0EP/+69+dnvHeqGvkvvX3sbVgK8+OfZZRkaMsmifAy41wP3fS2uotYy4efpqvJOWnEyL/pJQs36uZtfy9rIuuakZkrEYvXFhcHN9ug6eO+NNQ9HBsQturuav7Xc1NA2/iu/TveGrTUzTqO9/M1ahr5IEND7C/dD//nfhfegf1PmF/TJAX84ZG8+WWwxRUtF1mqLCyjhd+SWVkfBAz+7Wy7PecCkgtpNpKJvYKZVdOORU19nmvHK1IXgCmCyHSgWmG/xFCDBdCNF+tLgImANcYCfP9QgixB9gDhADPdar0JvL5/s/5K/8vFoxYQIS3be4AjdEj1BtnJ0G6PfwkAJGDtLvczW+fkDW9K7cCV2dBUqSFHRH3LwX/2HZbsnaEs5Mzz419jnpdPc9s6rw7VZ1ex8N/PMyG3A08Nvox5vSYY9V8fSP92HfEht3tkuZA+SEo/PseKyW/ipzSWmb1s+F3MWMVMmY0zp7+fNwcCmwBf6QX4+fh0mH16DsG33FcmcxfPZ/qxs7rU96ob+T+9fcf/8wnxUwyLuOUBPR6yRtrjDeZklLy2A97aWjS8+K8gSffYHYbrOUD2aDr5cReIegl/GGnMGCHKhIp5VEp5VQpZaLBBFZq2L5NSnmD4fnnUkrXFiG+x8N8pZRTpJQDDKayK6SUdroVt5wDZQd4fcfrTIqZZFG5BHNwd3EmLtjLdne0xhh/H9Qche1/t73dk1dOUoRfh1VjjVJXqd1x9Tnbatt6nH8cdw29i/W56/ki5Qur5jIFnV7HE38+wW/Zv3H/8Pu5qPdFVs/Zr5s/GcXHzA4dbZPes7UCii3MW6tSChECplpYx+kkKvOhcC/OvaZxycgYfttXaFHAgE4vWZNaxPheoR02cRJCMH/IfJ4a8xSb8zdz9S9XU1BtepSUpdQ01nDnmjtZk7OGh0c+zIW9LmxzbEyQF1eM7s6Xfx1m5f6Tw2/fWJ3Byv2F3DejF/EhRkLenZy1CLiMVVpVbCsYFB2Ar4cLGw7Yx7zl6BXJKU29rp6Hfn8IXzdfnj7j6U7p8tY7wtd2uQjGiB0N3cdpCYpN9ej1VjraU5dpCY9953Y81gSu6HMFk2Mm8/K2l9leuN0mcxqjQdfAAxseYOnBpdw++Hau7nd1xweZQN9ufuj00nZ+Ep9QLdM9ZdnxTatSChkcE0Cor+kRZe3SfMecMJ2rxsQB8OmmbLOn2ZpdSsmxBmb3Nz0AYF6veSycupCcqhzmLZ3HL1mWNfUyhfxj+Vz727X8eeRPnhzzJJf1uazDYx46M4kBUf7c+3UymzM1s12jTs9ba9J5ddUBzh8axY3j28mAT5gG1cVaVWwrcHF2YnxiCBvSi+2yWleKxI68tv010svSeWbsM2Z3PbSUxDBfso9W2+6O1hgT7tMKS+76iuyj1VTVNVmuSJK/hKAeWnizDRBC8Py454nyjeLutXeTXZFtk3lbUtNYw+2rb2floZUsGLGAWwYZL8RoCc15OPttbd4q2gdHD1JQUcfu3Aqm97XRagS0/ie+kRDej6gAT2b2C2fxlhxqGsxLEv1lTz7uLk5m998YGzWWr8/+mjj/OBZsWMAD6x+gvK7crDk6Yu3htVy07CIOVx7mjclvcEGvC0w6zsPVmbcvH0qAlyuXLNrM3IUbmfzSOl5acYCzBkQaN2m1JGGq9jdjpdWv4dzBUZwzqBv1TbavW6YUiZ34OfNnPk/5nMv7XM6EaPP7MlhKr3BfpLRhCKkxekzW/Bl/vMreHM3mapGjvSxb6wY36DKrzVot8XXz5e2pb+MknLhl1S0UVtsuqzf/WD7X/XYdWwu28vy457my75U2mxsgJtALH3cXG/tJDNFwqctYlaK9F5aWJz8JXRMcXKdd8Ayf4XVj46mobeS7HXkmT6PXS37dV8Ck3qF4u5ufi9TdrzufzPqEO4fcyapDq5j9/Ww+2fcJtU3W5eTkH8tnwfoF3Ln2TsK9wlk8ZzETYzquBdeSmCAvVtw9kTunJODp6kRimA+LrhzGwsuHajkj7eETBpGD4cBvlr8IAzP6RfDw7D54uFpggu4ApUjswL6SfTz151MMDRvKfcPv69Rz946wYfG/thACJj4EZdmI5K/wMPw4zGbXEkDAINNKiJhDrF8sb015i/L6cq759RqOHLO+AuofeX9w4bILOVR5iNcmv8Y5PS2PMmsLJydB30g/9ufbUJEEdtcCJVI0RdI92Ot4qLHV5G7VSrEkTD++aVj3QAZE+fO/P7NNzqbeml1KYWU9Z5ph1mqNi5MLNw68ka/P/pqBoQN5adtLzPxmJm/ufJOcKvNKq2RVZPH85ueZ8/0cVh9ezW2DbuOrs76iu193i2TzdHPm3hm9WXzTGD6+diQzzAl0SDpL6zpZZd8yJ9agFImNyanM4bbVtxHsGczLk17G1ck24ZWm0j3YG1dnQVqBneMOes2E6BGMyf2AwZGeJ8a/m4KuSesEGT8BAmLsIuKA0AG8P/19KhoquHz55SQXJVs0T4OugTd2vMFtq247flfaVqSOLejbzY+U/Errm1y1JOlsyN1CekYG0/uE285fl7EShPMJpf+FENw4oQcZRcdYtse0XiKf/3UYXw8XZrQOgbWAxMBE3p32Lp/M+oSBoQN5f/f7zP5uNhf+dCGvbH+FVYdWkVGWQUV9BfW6eqoaqsiuyGZ9znre2PEGF/10Eef8cA7fpn/LWT3OYtl5y7h18K24WlmI0mJ6nwlILWm3i2JhPQuFMXKqcrhhxQ3opZ53pr1jdmazLXB1dqJnqI/9QoCbEQLdpEcJ+fxcrnVbDUwy7/j9P0BFDsz+rx2E+5sBoQM0k8eaO7n2t2u5ZeAtXNf/OpMuClJK/jzyJ//Z+h8yKzI5N+FcHhn1CJ4unnaVuV83P2oadGSVHCMhzMKQ6tb0mQNrn2MyW5jW1/R6Zh2SutxQtj7ghM1zBkTy9toMXlmRxpn9I9o14RRV1vHLnnyuPiPO8hI7RhgaPpSh4UMpqC5gedZy1ues57P9n7Vb4NNZONMvpB/3DbuPOT3nOOQ3fBLh/bXw+LTlMMw2QR22RikSG5Fels6tq26lTlfHoumLiPc3r9SHLUkM92XnYcuTwkwl3XsohbqBTC74GKrvAm8Tf3RSalFfwYmQaMOLWhskBiayeM5int38LG8lv8VPmT9xXf/rmBU36+ReMGhZ6mtz1rI4dTE7inYQ5RPFwqkLO83XNdhQQTk5p8J2iiQ0iQK37pzLZgZ3t7J7ZjMlGVCcotV9a4WTk+D+Gb254dNtLNmawxWj2zYJffHXYXRScmU7Y6whwjuC6/pfx3X9r6NeV096WTqHKw9TWldKbVMtbs5uBHoEEu0TTVJQktHvhEMRQmsLsP1/Wv6WidWxOxOlSGzAiuwVPL7xcbxcvfhwxocnZbl2Nr3CfPhp1xGq65ssclyayu7cShY1XcFKl4dh7fNaQy5TyFoP+cnaeKfOsa76u/vz0sSXOLvH2SxMXsiTfz7J85ufZ0jYEOL84/B08aS6sZrsymx2Fe2iQd9AlE8UD498mAt6XYCbs3WFDc2hR6gPPu4u7Mop/7v2kpU06SXfNo7hVpbgdOwI+Ntg3lRDbkobpW2m9gljdI8gXvw1lWl9wonwP7lzZnFVPR9tzGJK7zDijOVS2Bh3Z3f6h/Snf0h/u5/LpvQ5G/56F9J+gQGmRYx1JspHYgU5lTncu+5e7lt/Hz0DerL4rMUOVyIAvSK0u1i7Rm4Bu/PKKXSLgxE3wLaPtd4oHaHXwW+PgX+M1n2xk5kYM5Elc5bw0cyPuLD3hdQ01bA8azmLUxez4tAK6prquDjpYhZNX8Ty85dzWZ/LOlWJgNbZbmC0P8k55Tabc9uhMr6uG4kTEvZ+Z5tJU5Zp0Xtt+LiEELxw/kAadXoe/m63Ucf7v5anUNeo4+HZNuzseSoSO0YLsd77raMlMYpakZhBVkUWJbUlZFdmsyF3AxtyN+AiXLhzyJ1c0/+aTnest0Vzza20wirrGk11wO7cCvpH+SOmPqE1pfr+Frh1I7i3Y47Z8SkU7oELPgZX+/oa2kIIwYiIEYyIsK4drD0ZFBPA+xsyqWvU2SRcc8W+QvKdu6GLHILz3m9grAltDNqjIg/ytsGUx9sdFhfizUOzknjqp/08sXQvz87tf9zR/92OXL7fmcf8KQm2iyI7VXFyhn7nG4qmloGnjcyTNkIpEjO4+perKavXfA8R3hFc3fdqruh7BWFeYQ6W7ERig7xwd3Gyq8O9vklHSn4l142LB3cfOO89+GiWpkwu+lT74remJANWPgmxZ0C/8+wm26nAoOgAmvSS/fmVDI217qIhpWRlSgHjEkJwTroYfn0ICvdDeF/LJ91rqJ9qwud49Rlx5FfW8d76TA6X1jJvaBR7civ44I8sRvcI4vbJ1vUvOW0YMA82L9RWgkNtm79kLUqRmMGzY5/Fw8WDSO9IYnxjOqXkiSU4OwkSwnxIs1cVYCCtoIpGnWRgVIC2IXY0zPq3dpH6+T446+UTlUltGSy+VNt23rs2TUA8FRkSGwDArpxyqxVJaoFWpPG2SQnQ7yJY8bjWwnnWvyyfdPfXED1Ca8XcAUIIHpqVRIi3O++uP3i83tPcwd14cd5AuyTInZJ0G6pVgdj1lVIk/2TMzWh1JL3CfY/X9rEHu46Xjm9RGmX0rVCVDxtfh+I0mP40hCbB4U2acqnKhyt/0BLkFO0S7udBpL8H2w+Vce1Y6yIAV+xrLtIYBt4eWgTQ7sUw7SlwscD/U7BXqyZ8pumh2825JVeO6U5qQRU9Qr3x8+gapuB/DELA0Kth1ZNQlAphHbTqbk1RCpTnaJ0zbXwjp5ztpyi9wn3Jr6ijss4+/Qd255QT6OVKdGArP8e0pzUzV8Fu+HA6vBADX14ETi5w3W8QP94u8pyKjIgLYktWqdVF9lbsL2BobCBhvoaoqSFXaRWc05ZbNuGer7UkxP7nm32oh6szg2MClBKxlCFXgJOr1nLbXNY+D9/eAPW2N3krRXKK0ivcUCrFTn6S5JxyBscEnGzeE4aSJ3cbHOpTHoNLl2hO+OjhdpHlVGV0j2CKquqt6niZV17LviOVzGhZpLHnZC1qrlWDMpNorNPMYr1mmZ43pLAd3iFapezkr07oCdQhhfu1VgKjbtYantkYpUhOUezZLbGyrpGM4mMMjmnHdu8VpN2xTngAes/qkklUXZ1RPbSK0X9llVo8x8p9Wo+OE6r9OjlrF5Ts3+FIsnkT7v9BW82MvNFimRRWMvJGrb7ZNjNWJb+/DG4+mvnZDihFcooSFeCJl5uzXZpc7c6pQMq/HcIK+9AjxJsQH3f+ssLXtWJ/IQlhPvQIbRVeO/Qq7cKy+W3zJtyySKtI0KK2lqKTiR0N8RNh42umrUoK9sC+72DE9doNnh1QiuQUxclJkBjmY5cqwMk5Wgi0PXNUFJqDelSPIP6y0E9SVt3AX1mlxnuPePhrymTvt1BqYmvcQ5sgbzuMvElF3TmayY9oDa+2vN/+OCnh5/u1vJNx99hNHIcqEiFEkBBipRAi3fDXqK1ECKFr0a99aYvt8UKIv4QQGUKIJUKIzk1B7uL0Cve1SxXg5JxyeoZ64++pHKb2ZnSPYPIr6sg+WmP2sb/sLUCnl5w1oI3S7GfcqTlu1zzX8WRSwupnwCdcc/gqHEvsaEicAev/A6WZbY9L/gJyNsP0Z+yaxOjoFclDwGopZSKw2vC/MWpb9Gtv2QTiReBVKWUCUAZcb19x/1n0ifSj5Fg9RZV1NptTSsnOw+Xt+0cUNmNSL61b4JrUIrOP/TE5j56h3se7Lp6EXySMuV1LLszb0f5kGavg8J8wcQG4dbGihqcrc17VoiG/v0Vry9CaI8naaqT7WK15nB1xtCKZC3xieP4JcK6pBwotXGgK8I0lx58ODDDkeOw9UmGzOXPLajla3cBg5R/pFGKCvOgV7sPqFPOaGh0pr2VLdilzB0e1nzg79i7wCoGf7oKmeuNj6o9piaaBcVrosKJr4B+ttWHI+Qu+uVaLqGsmfzcsvgy8guHC/9m9OKqjFUm4lLK5800B0FZXGw8hxDYhxGYhxLmGbcFAuZSyWRXnAlFtnUgIcZNhjm3FxcW2kL3L0yfSDyFgT67tuu3tNBQSHKL8I53G1D7hbMkqNSsnaNnuI0gJ5wzq1v5ADz+Y+5aW97PyCeNjlj8ARw/C3IWWJTAq7Megi2HmvyBlKbw3Adb+G5beCR9M1QqkXvqV1q7XzthdkQghVgkh9hp5zG05TmrexLY8it2llMOBy4DXhBAd12VohZRykZRyuJRyeGhoqPkv5B+Ij7sL8SHeNl2R7DxchoerE0kRNuqToeiQaX3CaNJL1qeZdgMkpWTJ1hyGxAaYVpq995kw6latTPmqp0Gv17brdbDiMdj1pRbGHTfOilehsBtjboeLP9eKpa5/AfZ9D33OgVv/hMiBnSKC3UukSCmntbVPCFEohIiUUuYLISIBo4ZgKWWe4W+mEGIdMAT4FggQQrgYViXRQJ7NX8A/nAFR/myxIg+hNck55QyMCjC/ta7CYgbHBBLk7cZv+wo4u6MVBrA5s5SDxdW8dOEg008y41loqoU/XoH0FRAzCg5vhqJ9WpuAiQ9a8QoUdqfP2dqjthzc/Tqtz08zjr4aLAWae0deDfzYeoAQIlAI4W54HgKMBfYbVjBrgQvaO/50p383f/Ir6ig51ob92wzqm3TsO1Kp/COdjLOT4OyBkazYX0h5TUOH47/46xB+Hi7MGdhGtJbRk7jCnNfgnLe0/JLkL7Q73HPf0QpwOquyfP8IPAM6XYmA4xXJC8B0IUQ6MM3wP0KI4UKI5voNfYBtQohdaIrjBSnlfsO+B4F7hRAZaD6TDztV+n8A/aMMDvc8681bKflVNDTpj7eCVXQeF4+IpaFJz/c72190F1bW8du+Ai4YFmN+VV0htKqy1/8GjxVqfwfbN9pHcWrg0NsMKeVRYKqR7duAGwzP/wQGtHF8JjDSnjL+0+kXpYV+7smtYFJv65xuyYY+8CqjvfPp282PQdH+LN6SwzVnxLUZifX22gykhGvOiOtcARWnNY5ekSjsjJ+HKz1DvY9HW1nDzpxywv3cifR3TGfD052LR8SSVljFHxklRvfnltXw5ZbDXDg8hthgleuh6DyUIjkNGBobyM7DZVaVI5dSsiWrlOFx9qnVo+iY84dGERPkyXPLUmjS6U/YJ6XkxV/TEAjmT1EdBxWdi1IkpwFDuwdSVtNoVTnynNJa8ivqGB2vFImj8HB15tHZfUkrrOKzzYdO2Pd/23L5adcRbp+cQLcAtWJUdC5KkZwGNLdq3XG43OI5NmdpFWhH9Qi2hUgKC5nZL5yJvUJ5dtl+vvzrMA1Ner7emsPjP+5lbEIwd6jViMIBKEVyGpAY5oOvuws7DM5yS/grs5QgbzcSw3w6HqywG0II3rliKOMTQ3nk+z30euwXFny7m0HRAbx+yRCcnVRVXkXno4LDTwOcnASDYwPYccgKRZJ1lBFxge3XbVJ0Cl5uLrx/1XCW78kns6SauGAvzh0chZNSIgoHoRTJacLQ2EDeXJNORW2j2eXf88pryS2r5bqx8XaSTmEubi5OnDukzdJyCkWnokxbpwljegajl1jUbe/3A1qNpzMSlH9EoVCcjFIkpwlDYgPwdHXmz4PmK5L1B4qJ8POgd7gq1KhQKE5GKZLTBHcXZ0bEB7WZzNYWjTo9f6SXMKl3qPKPKBQKoyhFchoxtmcwGUXHKDSjY+LOw+VU1TcxsdfpUXpfoVCYj1IkpxFjE0IA+POg6auS9QeKcHYSnGE4VqFQKFqjFMlpRN9IP4K93ViTanqHyFX7ixgWG2h2pJdCoTh9UIrkNMLJSTCjXwRrUgqpa9R1OD6toIq0wirOMqevhUKhOO1QiuQ0Y/aACKobdKw/0PGqZNnuIzgJmD1AKRKFQtE2SpGcZozuEUyAlyu/7Mlvd5yUkp92HeGMniGE+rp3knQKheKfiFIkpxmuzk7M7BvBqpQiahvaNm8l55STfbSGswep1YhCoWgfpUhOQ+YNi+ZYfRPf7Mhtc8wHv2fh6+GizFoKhaJDHKpIhBBBQoiVQoh0w99AI2MmCyGSWzzqhBDnGvb9TwiR1WLf4M5+Df9ERsQFMigmgA9+z0SnP7nZVXZJNb/szeeK0d3x9VDRWgqFon0cvSJ5CFgtpUwEVhv+PwEp5Vop5WAp5WBgClADrGgx5IHm/VLK5E6Q+R+PEIKbJ/Tg0NEaVuwrOGn/exsO4uLkxLVj4zpfOIVC8Y/D0YpkLvCJ4fknwLkdjL8A+EVKWWNPoU4HZvaLoGeoN0//tJ+jx+qPb19/oJivtuRw2ahYwnw9HCihQqH4p+BoRRIupWwOHyoAwjsYfwnwVattzwshdgshXhVCqPAiE3F2Erx+yRBKaxq4c/FOCivr2H6olHuXJNM73JeHzkxytIgKheIfgpDyZBu5TU8gxCogwsiuR4FPpJQBLcaWSSlP8pMY9kUCu4FuUsrGFtsKADdgEXBQSvlMG8ffBNwEEBsbO+zQoUPGhp12fL01hwXf7sZJgF5CiI8bi28aTUKYqvSrUCj+RgixXUo53Ng+uze2klJOa2ufEKJQCBEppcw3KIWidqa6CPi+WYkY5m5ezdQLIT4G7m9HjkVoyobhw4fbV3v+g7hoRAwj4oP4dnsuQd5uXDIyBi831e9MoVCYjqOvGEuBq4EXDH9/bGfspcDDLTe0UEICzb+y105yntLEh3hz/8zejhZDoVD8Q3G0j+QFYLoQIh2YZvgfIcRwIcQHzYOEEHFADLC+1fFfCCH2AHuAEOC5zhBaoVAoFH/j0BWJlPIoMNXI9m3ADS3+zwZOalAtpZxiT/kUCoVC0TGOXpEoFAqF4h+OUiQKhUKhsAqlSBQKhUJhFUqRKBQKhcIqlCJRKBQKhVUoRaJQKBQKq1CKRKFQKBRWoRSJQqFQKKxCKRKFQqFQWIVSJAqFQqGwCqVIFAqFQmEVSpEoFAqFwiqUIlEoFAqFVShFolAoFAqrUIpEoVAoFFahFIlCoVAorEIpEoVCoVBYhVIkCoVCobAKpUgUCoVCYRUOVSRCiAuFEPuEEHohxPB2xs0SQqQJITKEEA+12B4vhPjLsH2JEMKtcyRXKBQKRTOOXpHsBc4HNrQ1QAjhDCwEzgT6ApcKIfoadr8IvCqlTADKgOvtK65CoVAoWuNQRSKlTJFSpnUwbCSQIaXMlFI2AIuBuUIIAUwBvjGM+wQ4127CKhQKhcIoLo4WwASigJwW/+cCo4BgoFxK2dRie1RbkwghbgJuMvx7TAjRkQJrixCgxMJj7YmSy3y6qmxKLvNQcpmPJbJ1b2uH3RWJEGIVEGFk16NSyh/tff5mpJSLgEXWziOE2CalbNOf4yiUXObTVWVTcpmHkst8bC2b3RWJlHKalVPkATEt/o82bDsKBAghXAyrkubtCoVCoehEHO1sN4WtQKIhQssNuARYKqWUwFrgAsO4q4FOW+EoFAqFQsPR4b/nCSFygTHAz0KI3wzbuwkhlgMYVht3AL8BKcDXUsp9hikeBO4VQmSg+Uw+7ASxrTaP2Qkll/l0VdmUXOah5DIfm8omtBt7hUKhUCgs459g2lIoFApFF0YpEoVCoVBYhVIkbdBWWZYW+90NZVkyDGVa4jpBphghxFohxH5DaZm7jIyZJISoEEIkGx5P2Fsuw3mzhRB7DOfcZmS/EEK8YXi/dgshhnaCTL1bvA/JQohKIcTdrcZ02vslhPhICFEkhNjbYluQEGKlECLd8DewjWOvNoxJF0Jc3Qly/VcIkWr4rL4XQgS0cWy7n7sd5HpKCJHX4vOa3cax7f5+7SDXkhYyZQshkts41p7vl9HrQ6d8x6SU6tHqATgDB4EegBuwC+jbasxtwLuG55cASzpBrkhgqOG5L3DAiFyTgGUOeM+ygZB29s8GfgEEMBr4ywGfaQHQ3VHvFzABGArsbbHtP8BDhucPAS8aOS4IyDT8DTQ8D7SzXDMAF8PzF43JZcrnbge5ngLuN+Gzbvf3a2u5Wu1/GXjCAe+X0etDZ3zH1IrEOEbLsrQaMxetLAtoZVqmCiGEPYWSUuZLKXcYnlehRbG1mc3fxZgLfCo1NqPlAEV24vmnAgellIc68ZwnIKXcAJS22tzye9RWmZ+ZwEopZamUsgxYCcyyp1xSyhXy76oRm9HytDqVNt4vUzDl92sXuQzXgIuAr2x1PlNp5/pg9++YUiTGMVaWpfUF+/gYww+uAi0EuVMwmNKGAH8Z2T1GCLFLCPGLEKJfJ4kkgRVCiO1CK0fTGlPeU3tyCW3/uB3xfjUTLqXMNzwvAMKNjHH0e3cd2mrSGB197vbgDoPJ7aM2zDSOfL/GA4VSyvQ29nfK+9Xq+mD375hSJP9AhBA+wLfA3VLKyla7d6CZbwYBbwI/dJJY46SUQ9GqNN8uhJjQSeftEKElsp4D/J+R3Y56v05CajaGLhWPL4R4FGgCvmhjSGd/7u8APYHBQD6aGakrcSntr0bs/n61d32w13dMKRLjtFWWxegYIYQL4I9WtsWuCCFc0b4kX0gpv2u9X0pZKaU8Zni+HHAVQoTYWy4pZZ7hbxHwPZp5oSWmvKf24kxgh5SysPUOR71fLShsNvEZ/hYZGeOQ904IcQ0wB7jccAE6CRM+d5sipSyUUuqklHrg/TbO56j3ywWtLcaStsbY+/1q4/pg9++YUiTGMVqWpdWYpWhlWUAr07KmrR+brTDYXz8EUqSUr7QxJqLZVyOEGIn2GdtVwQkhvIUQvs3P0Ry1e1sNWwpcJTRGAxUtltv2ps27REe8X61o+T1qq8zPb8AMIUSgwZQzw7DNbgghZgELgHOklDVtjDHlc7e1XC39aue1cT5Tfr/2YBqQKqXMNbbT3u9XO9cH+3/H7BE9cCo80KKMDqBFfzxq2PYM2g8LwAPNVJIBbAF6dIJM49CWpbuBZMNjNnALcIthzB3APrRIlc3AGZ0gVw/D+XYZzt38frWUS6A1KDsI7AGGd9Ln6I2mGPxbbHPI+4WmzPKBRjQb9PVofrXVQDqwCggyjB0OfNDi2OsM37UM4NpOkCsDzWbe/D1rjlDsBixv73O3s1yfGb4/u9EukJGt5TL8f9Lv155yGbb/r/l71WJsZ75fbV0f7P4dUyVSFAqFQmEVyrSlUCgUCqtQikShUCgUVqEUiUKhUCisQikShUKhUFiFUiQKhUKhsAqlSBQKhUJhFUqRKBQKhcIqlCJRKLoIQohoIcTFjpZDoTAXpUgUiq7DVLQ+FwrFPwqV2a5QdAGEEOPQaiCVA1XA+VLKTIcKpVCYiFIkCkUXQQjxK1r3P7sWPlQobI0ybSkUXYfeQKqjhVAozEUpEoWiC2DogVIh/25vq1D8Y1CKRKHoGsQBRxwthEJhCUqRKBRdg1QgRAixVwhxhqOFUSjMQTnbFQqFQmEVakWiUCgUCqtQikShUCgUVqEUiUKhUCisQikShUKhUFiFUiQKhUKhsAqlSBQKhUJhFUqRKBQKhcIq/h/MIMYqSPbBBwAAAABJRU5ErkJggg==\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.figure()\n", "for i in range(3):\n", " plt.plot(time, X[:,i], label=f'$x_{i+1}$')\n", "plt.ylim(-1, 1)\n", "plt.xlabel(r'$t$')\n", "plt.ylabel(r'$x_i$')\n", "plt.legend(ncol=3)\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "5e5cac20-0383-409b-a1e9-29c504c828d3", "metadata": {}, "source": [ "The dynamics look a bit \"chaotic\". To gain more insight into the dynamics, we will decompose them into **normal modes** using matrix diagonalization." ] }, { "cell_type": "markdown", "id": "1cd79d93-4e1b-47df-a65e-8d9da08d02d9", "metadata": {}, "source": [ "## Normal modes" ] }, { "cell_type": "markdown", "id": "785cb66e-2a61-4c13-9e63-89c2f87a64f2", "metadata": {}, "source": [ "Suppose $\\mathbf{A}$ can be diagonalized as:\n", "\\begin{equation}\n", "\\mathbf{A} = \\mathbf{R} \\cdot \\mathbf{D} \\cdot \\mathbf{R}^{-1}\n", "\\end{equation}\n", "where $\\mathbf{D}$ is a diagonal matrix formed by the eigenvalues of $\\mathbf{A}$, and $\\mathbf{R}$ is a matrix whose columns are the corresponding eigenvectors of $\\mathbf{A}$, i.e.,\n", "\\begin{equation}\n", "\\mathbf{D} = \\left( \\begin{array}{ccc} \\lambda_1 & & \\\\ & \\lambda_2 & \\\\ & & \\lambda_3 \\end{array} \\right) \\,, \\qquad\n", "\\mathbf{R} = \\left( \\begin{array}{ccc} & & \\\\ \\vec{\\xi^{(1)}} & \\vec{\\xi^{(2)}} & \\vec{\\xi^{(3)}} \\\\ & & \\end{array} \\right)\n", "\\end{equation}\n", "We can use $\\mathbf{R}$ to transform $\\mathbf{X}$ to a new set of variables $\\mathbf{X}' \\equiv \\mathbf{R}^{-1} \\cdot \\mathbf{X}$. With this transformation, the equation for $\\mathbf{X}'$ simplifies to:\n", "\\begin{equation}\n", "\\ddot{\\mathbf{X}'} = - \\mathbf{D} \\cdot \\mathbf{X}'\n", "\\end{equation}\n", "Thus, each component satisfies an equation $\\ddot{x}'_i = - \\lambda_i x'_i$, the solution of which is just:\n", "\\begin{equation}\n", "x'_i(t) = x'_i(0) \\cos(\\omega_i t)\n", "\\end{equation}\n", "with $\\mathbf{X}'(0) = \\mathbf{R}^{-1} \\cdot \\mathbf{X}(0)$ and $\\omega_i = \\sqrt{\\lambda_i}$ (we need $\\lambda_i \\geq 0$, which is true for $\\mathbf{A}$, as shown below). Therefore, the solution for the original variables can be written as a sum over **normal modes**:\n", "\\begin{equation}\n", "\\mathbf{X}(t) = \\mathbf{R} \\cdot \\mathbf{X}'(t) = \\sum_i \\vec{\\xi^{(i)}} x'_i(0) \\cos(\\omega_i t)\n", "\\end{equation}\n", "Each mode has its own frequency of oscillation $\\omega_i$, and the normalized eigenvector $\\vec{\\xi^{(i)}}$ represents the \"direction\" of each mode, whereas $x'_i(0)$ represents the \"amplitude\" of each mode." ] }, { "cell_type": "markdown", "id": "522da301-7f0c-4ce2-bcaf-cccf7225c9ac", "metadata": {}, "source": [ "To construct the solution using the normal modes as described above, let us first find the eigenvalues and eigenvectors of the matrix $\\mathbf{A}$. This can be done using the `numpy.linalg.eig` function. Note that $\\mathbf{A}$ will be symmetric only if all masses are equal, in which case we can use `numpy.linalg.eigh` instead." ] }, { "cell_type": "code", "execution_count": 5, "id": "ec679472-cf4d-45ca-ba87-bc79f18efe3c", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "eigenvalues = [3.21034789 0.62298544 0. ]\n", "eigenvectors = \n", "[[ 0.85399876 0.6897631 0.57735027]\n", " [-0.5168178 0.47490692 0.57735027]\n", " [ 0.05987895 -0.54652565 0.57735027]]\n" ] } ], "source": [ "A = np.array([[k1/m1, -k1/m1, 0],\n", " [-k1/m2, (k1+k2)/m2, -k2/m2],\n", " [0, -k2/m3, k2/m3]])\n", "\n", "D, R = np.linalg.eig(A) # calculate eigenvalues and eigenvectors\n", "print(f'eigenvalues = {D}') # each entry `D[i]` is an eigenvalue\n", "print(f'eigenvectors = \\n{R}') # each column `R[:,i]` is the corresponding eigenvector\n", "# D = np.around(D, 8) # remove truncation error if necessary" ] }, { "cell_type": "markdown", "id": "a1ada195-bd62-4254-848e-fe979e1fdb2d", "metadata": {}, "source": [ "Note that all eigenvalues are non-negative (barring truncation error, which can be removed by uncommenting the last line of code). From these we can obtain the frequencies of the normal modes and their amplitudes." ] }, { "cell_type": "code", "execution_count": 6, "id": "1ea43c70-84da-43a2-b50a-12081eb3a98e", "metadata": {}, "outputs": [], "source": [ "freq = np.sqrt(D) # each entry is a normal frequency\n", "amp = np.dot(np.linalg.inv(R), [x1i, x2i, x3i]) # each entry is the corresponding amplitude" ] }, { "cell_type": "markdown", "id": "5e58d372-331e-429e-bd00-743b43c1a6d7", "metadata": {}, "source": [ "Then we can sum over the normal modes to obtain the solution." ] }, { "cell_type": "code", "execution_count": 7, "id": "953d907a-6801-4a19-ba54-d9459e6d2cf0", "metadata": {}, "outputs": [], "source": [ "modes = np.zeros((3, len(time))) # to construct normal modes\n", "for i in range(3):\n", " modes[i,:] = amp[i] * np.cos(freq[i] * time) # each normal mode as a function of time\n", "X_sum = np.dot(R, modes)" ] }, { "cell_type": "markdown", "id": "aa6121db-e263-4837-85f8-772927f5e3dd", "metadata": {}, "source": [ "Let us compare the solutions we constructed using normal modes to those found above by brute-force." ] }, { "cell_type": "code", "execution_count": 8, "id": "0f46e9f8-990f-4924-8910-38b15af01255", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.figure()\n", "for i in range(3):\n", " plt.plot(time, X[:,i], label=f'$x_{i+1}$') # solutions found by brute-force\n", " plt.plot(time, X_sum[i,:], 'k:') # solutions found using normal modes\n", "plt.ylim(-1, 1)\n", "plt.xlabel(r'$t$')\n", "plt.ylabel(r'$x_i$')\n", "plt.legend(ncol=3)\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "e270bedc-b4ee-4223-8232-e8c585dd1b3e", "metadata": {}, "source": [ "As expected, the solutions lie perfectly on top of each other." ] }, { "cell_type": "markdown", "id": "00a9b6d8-4b63-401c-b7fc-f5e815a2db87", "metadata": {}, "source": [ "## Decomposing the motion" ] }, { "cell_type": "markdown", "id": "c944b8cc-5d9e-4fc7-b502-726753f79fc3", "metadata": {}, "source": [ "To better visualize the motion, we can decompose it into superposition of the normal modes. We will plot these modes in separate figures below." ] }, { "cell_type": "code", "execution_count": 9, "id": "d46f20d9-4b87-4fd0-b37e-4e6131edea45", "metadata": {}, "outputs": [ { "data": { "image/png": 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vvvsuX331FUOHDiU2NpZbb73V1iF1qvnz5zNjxgyqqqoYNGhQsxn25q5PpdRqrXXCkYrTlqQfPn5IX2w/jqe+2Jx+GKQvlr74+CD9sP04nvph6Jx7YklgiGOWPXfWQshNs/TDxwvpi4U9k75Y+uLjgfTDwp5Z2g9LDQwhhBBCCCGEEELYPUlgCCGEEEIIIYQQwu5JAkMIIYQQQgghhBB2TxIY4ph2PNZ4EfZPrktxvJFrXtgjuS7F8USud2GP2nNd2kUCQyl1mlJqq1Jqh1LqkWaOv6WUSm742qaUKmxyrK7JsZlHNHBh11xdXcnLy5MOW9gVrTV5eXm4urraOpSDSD8sOov0xcIeSV8sjifSDwt71N5+2LGT4jGbUsoBmAKcAmQAq5RSM7XWmxrP0Vrf1+T8u4C4Jk1UaK1jj1C44igSHh5ORkYGOTk5tg5FiIO4uroSHh5u6zD2k35YdCbpi4W9kr5YHC+kHxb2qj39sM0TGMAwYIfWeheAUup74FxgUwvnXwY8fYRiE0cxJycnoqOjbR2GEEcD6YdFp5G+WAizSV8sOoX0w+JYYg9LSMKA9CZ/zmh47TBKqSggGpjX5GVXpVSiUmq5Uuq8TotSCCGOXdIPCyGE7UlfLIQQbbCHGRiWmAT8pLWua/JalNY6UykVA8xTSq3XWu889I1KqZuBmwEiIyOPTLRCCHHskX5YCCFsT/piIcRxyR5mYGQCEU3+HN7wWnMmAdOavqC1zmz47y5gPgevBWx63sda6wStdUJQUFBHYxZCiGOJ9MNCCGF70hcLIUQb7CGBsQroqZSKVko5Y3TIh1VOVkr1AfyAZU1e81NKuTT8fyAwmpbXCQohhGie9MNCCGF70hcLIUQbbL6ERGtdq5S6E5gFOACfa603KqWeAxK11o0d9yTge33w/j99gY+UUvUYyZiXm1ZqFkII0Tbph4UQwvakLxZCiLap43E/4ISEBJ2YmGjrMIQQ4iBKqdVa6wRbx3EkSD8shLBX0hcLIYRttdYP28MSEiGEEEIIIYQQQohWSQJDCCGEEEIIIYQQdk8SGEIIIYQQQgghhLB7ksAQQgghhBBCCCGE3ZMEhhBCCCGEEEIIIeyeJDCEEEIIIYQQQghh9ySBIYQQQgghhBBCCLsnCQwhhBBCCCGEEELYPUlgCCGEEEIIIYQQwu5JAkMIIYQQQgghhBB2TxIYQgghhBBCCCGEsHuSwBBCCCGEEEIIIYTdkwSGEEIIIYQQQggh7J4kMIQQQgghhBBCCGH3JIEhhBBCCCGEEEIIuycJDCGEEEIIIYQQQtg9SWAIIYQQQgghhBDC7kkCQwghhBBCCCGEEHZPEhhCCCGEEEIIIYSwe3aRwFBKnaaU2qqU2qGUeqSZ49cqpXKUUskNXzc2OXaNUmp7w9c1RzZyIYQ4dkhfLIQQtiX9sBBCtM7R1gEopRyAKcApQAawSik1U2u96ZBTf9Ba33nIe/2Bp4EEQAOrG95bcARCF0KIY4b0xUIIYVvSDwshRNvsYQbGMGCH1nqX1roa+B4418z3TgTmaK3zGzroOcBpnRSnEEIcy6QvFkII25J+WAgh2mAPCYwwIL3JnzMaXjvUhUqpdUqpn5RSERa+F6XUzUqpRKVUYk5OjjXiFkKIY0mn98XSDwshRKvknlgIIdpgDwkMc/wOdNNaD8LIKH9laQNa64+11gla64SgoCCrByiEEMeBDvXF0g8LIUSHyT2xEOK4Zg8JjEwgosmfwxte209rnae1rmr446fAEHPfK4QQwizSFwshhG1JPyyEEG2whwTGKqCnUipaKeUMTAJmNj1BKdWlyR/PATY3/P8s4FSllJ9Syg84teE1IYQQlpG+WAghbEv6YSGEaIPNdyHRWtcqpe7E6GQdgM+11huVUs8BiVrrmcDdSqlzgFogH7i24b35SqnnMTp8gOe01vlH/JsQQoijnPTFQghhW9IPCyFE25TW2tYxHHEJCQk6MTHR1mEIIcRBlFKrtdYJto7jSJB+WAhhr6QvFkII22qtH7aHJSRCCCGEEEIIIYQQrZIEhhBCCCGEEEIIIeyeJDCEEEIIIYQQQghh9ySBIYQQQgghhBBCCLtn811IhBBCiLaUVtWyanc+QV4udA/yxM3ZwdYhiRZkFVWQnl+B1proQA+CvFxQStk6LNGMsqpadueWUVxZg5+7Mz2DPXF0kGdbQggh7JckMIQQQti9HdmlXPelsTugi6OJcb2CuHV8DEOi/G0cmQBjIDx1eSrTVqaRkld+0LEewZ5MGhrBlSOicHWSxJM9WL4rj48X7mLR9hxq6g7sRufh7MBpA7pwy/gYeoV42TBCIYQQonmSwBBCCGH3eoV4MuP2UewtqmTl7nxmrt3DhR8s44K4MJ4+uz8+7k62DvG49d+WbB77ZT1ZRZUMj/bn6pHd6BniidawbV8Jf2/Yywt/bubLpSm8dtFgRnYPsHXIx62Csmoe+2U9f2/YS7CXC9eO6saQKD+83ZzIKali2c48Zq7dwy9JGdw4Nob7T+klSSchhBB2RWmt2z7rGCN7Xgsh7FFre14fazraD5dX1/Lh/J1Mmb+TcD83vrxuGNGBHlaMULRFa827/+7grbnb6BPqxYvnD2hxRsySHbk8+esGUvLKeOT0Ptw8rvsRjlZs3VvCtV+sJLe0intP7sUNY6KbTU4Ullfzyj9bmbYyjfhIXz66KoEgLxcbRGw70hcLIYRttdYPy0JHIYQQRx13Z0fuP7U3028ZQUllLRe8v4Ste0tsHdZxQ2vNc39s4q2527ggLoxf7xjd6nKe0T0CmXnXGE4bEMr//bWFV//ZwvH4AMVWktMLufjDpdRrzYzbRnPHCT1anFnh6+7MSxcM5P0r4tmUVcylHy0ju7jyCEcshBBCNE8SGEIIIY5aQ6L8mXHbKJwdTVzx6QpScstsHdJx4Y3Z2/hiSQrXj47mjUsGm7XMwNPFkfcui+eyYZG8P38nHy7YdQQiFdv3GTMvfN2d+fm2UQwM9zHrfWcM7MLUG4azt7iSyz5ZTlF5TSdHKoQQQrRNEhhCCCGOat0CPfj2xhHU1ddz49eJlFTKQKszTU9MZ/J/O7hsWARPntXXoh1GHEyKF88bwNmDu/LKP1v4Y92eToxU5JdVc+0Xq3ByMDH1huGE+7lb9P6Ebv58fu1Q0vLLuf271dTU1XdSpEIIIYR5JIEhhBDiqNcj2JMpV8SzO7eM+6evleUJnWRDZhFP/LqBsT0Def7cAe3aHtVkUrx+8SCGRPnx0E/r2JEtS386Q1295u5pSeSUVPHJ1QlEBliWvGg0IiaAly4YxJIdebw+a6uVoxRCCCEsIwkMG6qoruP+6cmMfOlfJr61kDVpBbYOySLfLE/lhNfnM/z/5vLZ4t1H1YBhXUYhF36wlPjn53DP90kUllfbOiSzFVfWcM/3SQx7cS5nvbeIpKPoutFa8/HCnZz85gKGvjiXr5elUFd/9Fw3y3flcdnHy+n/1D88OmMdOSVVtg5JNDGqeyCPnt6HOZv2MXV5qq3DOeaUVdVy53dr8Hd35p1JcTg6tP8WwsXRgSmXx+Pu7MAd3yZRVVtnxUgFwMcLd7F4Ry7Pntuf2AjfDrV10ZBwrhwRyUcLd/Hf1mzrBCiEEEK0gyQwbKSiuo7LPlnOL0mZDO3mT3lNLZM+Ws6CbTm2Ds0s7/27nSd/3UCAhzMxgZ48/8cmnpm50dZhmSU5vZArPlnBnsIKxvYM5M91WZz//lLKqmptHVqbyqpque6LVfy5LotR3QPILanmog+XkZiSb+vQzPL+/J38319b8PdwJibQg6d+28jrs4+OJ3rrM4q4/stVpOWXc2LfEH5ancHVn6+kskYGXvbk+tHRjO8VxAt/bma3HdfDKK8pJ3FvIr9s/4Xvt3zP7JTZpBan2nUi+KW/N5OaX847k2Lx93DucHuhPq68dtFgtu4r4c0526wQYefQWrOrcBezUmbxw5Yf+GX7L6zZt4bKWvstbLlxTxFvztnKGQNDmTQ0wiptPnFmP/qEevHQT+vsuh5GcXUxSzKXMGP7DH7b8ZutwxFCCGFljrYO4Hj1wfwdJKcXMuXyeM4c1IXC8mou+WgZj/68jjn3j8fDxX7/atZnFPHm3G2cF9uV1y8ejINJ8czMjXy1LJWJA0IZ1T3Q1iG2qLq2nvt+SMbH3Ynpt4ykq68blyREcOVnK3j1ny08e+4AW4fYqjdmbyMprYApl8dz+sAuFFXUcNZ7i7h/+lr+vmesXV83y3fl8dqsrZwX25U3L4lFKXh0xno+XLCTsT0D7fq6qaiu4+ZvEvFzd+aX20cR7O3KvC1duf7LRP7vr808Z+fXzfHEZFK8dtEgTnpjAU/9toGvrx/WrmUOnWVD7ga+3Pgl/6X9R3X94TO/IrwiuKjXRVza+1I8nOxnW9ilO3OZujyNG8ZEMzwmwGrtntAnmMuGRfDxwl2cPagrA8LMKzB5JBRVFTF963R+3v4zmaWZhx13c3TjhIgTuG7AdfTx72ODCJtXW1fPIz+vx9fdmRfPG2i169/VyYHXLx7MuVOW8Nwfm3jjksFWadcatNYs2bOEbzd/y9I9S6nXRq2OMM8wzu1xro2jE0IIYU0yA8MGUvPK+HDhLs4Z3JUzB3UBDmxbtqeoknf/3W7jCFumtebZ3zcS4OHMc+cNwNHBhFKKR07vS1SAO4/NWG/XRb6mLk9ld24Zz587gK6+boCxvd81I7vx1bJU1mUU2jbAVqTllfPN8hQuSYjg9IHGdePj5sQbF8eSXlDOlP922DjClmmteeWfLYR6u/LyhYMwmRRKKZ46ux9R/u489dtG6u14Kck3y1PIKqrkzUsGE+ztCsCJfUK4dlQ3vl6WKmv47Uywtyv/m9ibRdtz+WNdlq3DASC/Mp+HFjzEZX9exvKs5VzY60KmnDSFvy/4m/8u+Y/pZ03n8eGPE+wezFur3+KsX87ir11/2TpsAGrq6nlm5kYi/N14cGJvq7f/yOl9CfBw5qnfNthFP6C15pftv3DWL2fxbtK7RHhF8NTIp/jp7J/475L/+OuCv3jvxPc4O+ZsFmYs5OLfL+bJJU9SWFlo69AB+HJpCuszi3jm7P74WWGmTFMDwny4dXwMP6/JYOVu+5j5l16czk2zb+K2ubexrWAb1w+4nk9O/YRZF85ixjkzbB2eEEIIK5MEhg18uGAnJgWPn9n3oNeHRPlzQVwYXy9LpajCPqdnzt+WQ2JqAf87tTferk77X3dzduDxM/qSklfO7I37bBhhyyqq63h33nbG9gxkQu+gg479b2JvvFwc+WTRbhtF17a35m7D0WTivlN6HfT6sGh/JvYLZeryVLtdBjNn0z6S0gq59+SeB2236O7syL0n92JHdqndrqsuqazhg/k7Gdcr6LAnz3ed2ANXJ5NsB2mHrhwRxcAwH577YxPFNt6VJHFvIuf/dj5z0+Zy2+DbmHXhLB4b/hjjwscR7hVOoFsgfQP6MqnPJL487Uu+PeNbunp05eFFD/PookdtvlRh6vJUtu0r5Ykz+5m1XaqlfNycePi0PqxJK2RG0uEzHY6k0upSHljwAE8tfYoYnximnzWdT079hIt7XUxv/94EugUS4RXBhIgJPDnySWZdNIvrB1zPHzv/4KLfL2Jtzlqbxp9bWsXbc7dzQu8gzhgY2imfcecJPeni48qzv2+0eQ2jv3f/zYW/X8jGvI08OuxR/rngH+6Jv4cRXUbQ1bMr7k7tK1wqhBDCfkkC4wgrrqzh16Q9nDO4KyENT3Kbun5MNBU1dfy0OsMG0bXt2+VpBHo6c0F8+GHHTu4bQlSAO18utc8kwJ/rsygsr+GOE3ocNqXW08WRS4dG8Nf6LPYUVtgowpbll1Xzx7o9TBoW0ex1c9O4aIora+32uvlqWQphvm5cNOTw6+bMQV3o6uPKRwvtMwnwS1ImBeU13H9I4gggwNOFSUMj+TUp0y6vm2NKfR3UmZ+gczApXjx/ALmlVbxhw50TZu6cyU2zb8Lb2ZsfzvqB22Nvb3NpyKCgQXx1+lfcHns7f+76k+tnXU9+pW2edueVVvHWnG2M7RnIqf1COu1zLowPJz7Sl5f/3myzhFN2eTbX/nMt89Lm8cCQB/jitC/oG9C31fd4O3tz35D7+PbMb3E0OXLdP9cxN3XuEYr4cG/P3UZFTR1PnNWv05ZOuTk78Mjpfdi4p5ifVqd3yme0RWvNh2s/5KGFD9HHvw+/nPsLl/e9HCcHp7bfLIQQ4qhmFwkMpdRpSqmtSqkdSqlHmjl+v1Jqk1JqnVLqX6VUVJNjdUqp5IavmUc2csvNWJ1BRU0dV43o1uzxAWE+DIny45tlKXYxlbapPYUVzNuyj0sSInB2PPzSMZkUV4/sxqqUAjZkFtkgwtZ9vzKNmEAPhkf7N3v8mlHd0Frb5e4FM9ZkUFOnmTQ0stnjQ6L8iYv05aulKXZXBDA9v5wlO/K4JCGi2V0LnBxMXD8mmpW789m2z/6WYvy0OoN+XbxbrOJ/w5hoaus1P9tp8shcdt8P79sAzwfA/4XDh2Pgj/th13yob3nJ2qBwXy4fFsm3K9JIsUFBzx+3/cjjix9nSOgQvjvzO3r69TT7vY4mR24bfBtvnfAW2wu2c8OsG8iryOvEaJv3+uxtlFXX8VRbA+KyXFjxEXx7MbzZD54LML7e7A/TLoOVn0B5y0kYk0nx3LkDyCur5u05R34Z5b6yfVz7z7Wkl6Tz/knvc+2AazEp82+R+gX044ezfqBfQD8eWPAAv+/8vROjbd6O7BKmrUzniuGRdA/ybPnE+jrYMRdm3g3vj4T/C4Nn/eClSPjkRJj9BGSshlZ+l5wzuCtDovx4bdZWSo5wwklrzVtr3mJK8hTO6X4On536GaEenTPbxBbsvi8WQggbs3kCQynlAEwBTgf6AZcppfodcloSkKC1HgT8BLza5FiF1jq24eucIxJ0B3y/Kp1B4T4MDG+5UNlVI6JIyStnlZ3tLPFjYgYauGxY84NoMLZac3YwMWONbacBH2r7vhISUwuYNCyixZvwCH93xvUKYubaPXaVBNBa88OqdGIjfOkd6tXieZckRLArt4yNe4qPYHRt+3lNBkrBhUPCWjzn3NgwTApmJu85gpG1beveEtZlFDU7c6RRhL87w6L9+TU5066uG0scFf2weyBMeAzirgSPIFj7PXx9Lrw/HNb/1OJg656TeuLooHhr7pHd5WJe2jxeWP4C48LHMeWkKXg5t/xvtzUnRZ7ElJOmkFGSwc1zbqak+sgl+bbtK+H7VWlcPTKKniEtxF+cBb/fA2/2hb8fgvxd0G0sjLrL+IoaBdmb4a//wRt94K+HoLT53bYGhPlwaUIEU5enknkEZzQVVRVx4+wbya/M5+NTP2ZU2Kh2tePj4sMnp37C0JChPLnkSf5N+9fKkbbupb+24O7kwD0ntZAoq6+H5O/gvSEw9ULY+At4h0H81TDmfhh0CTi6wfIP4dMT4dOTjSRhM5RSPHVWP3JLq5l8hOsvfbzuY77Y8AWX9r6U50c/f0zNujgq+mIhhLAxmycwgGHADq31Lq11NfA9cFDJaK31f1rr8oY/LgdaHk3Ysd25ZWzZW8K5sS0P5ABO7heCi6OJvzfsPUKRmeev9VkM7eZPhH/La0p93JwY1yuQvzdk2dUMkt/XZWFScH5c65fOWYO6klFQwdoM+5lBsimrmO3ZpVyc0Hrsp/UPxdGk+H2d/SQBtNbMWJPJ6O6BhPu1fN0Eebkwqnsgv6+zr+TRjDUZOJoU58Z2bfW8c2O7sjOnjE1Z9pU8soD998M+YTDhYTj9ZbjqF3hoJ1zwKTg4w883wFdnQ2HaYW8L9nblutHRzFy7h81H6O9nXc46Hl74MP0D+vP6+NdxcXDpUHvDugzjnRPeYVfhLu6bfx819Ufmifc7c7fj7uTA3Sc2MyDWGhK/gMkJkPQtxF0Ft6+Au1bDBR/Byc8YXxd+Avckwy2LjAHyqk+N9yR/12zS6e6GwffkeUdmFkZNfQ0PzH+AzNJMppw0hcFBHdtZw83RjXdPfJf+gf15dNGjbM0/MsuXlu7M5d8t2dxxYg8CPJu53nJ3wBenwa+3gas3XPwlPLgDrvwJTnsJTnoSznwdrvvT+Ld1xutQmm0kCX+8rtnZM4MjfDk/LoyvlqaQXXJk6rTM3DmTycmTOaf7OTw2/DGLZskcJey/LxZCCBuzh54/DGi6iDKj4bWW3AD83eTPrkqpRKXUcqXUeS29SSl1c8N5iTk5zT/96Wz/NCQkThvQ+lRHTxdHxvUK4p8Ne+0mCbA7t4yt+0o4rX/b0zTPGNiFrKJKktILOz8wM83euJeEbv4EebU+kDilXwhODoo/7SgJMHvjPpSCiW387P08nBnTM5A/1mbZTRJg674S0vLLOatht53WnDO4K6l55ayzk+SR1pp/Nu5ldI/A5gcETZwxoAuOJmV3M0gscPT1w05uMOhiuGUhnPU2ZK2FD8bA1r8PO/XWcd3xdHHkjdmdP5jMKMngrnl3EegWyHsnvoebo5tV2h0VNoqnRz3NiqwVvLfmPau02Zote4v5c30W142OPnwni6pS+PEa+ONeCBsCd66Es96E4Fa2Eu0yCM6dDLcvh+C+xkD619ug5uCZFl193bhsWATTEzNIzev8ZT+vrXqNFXtX8PTIpxkSMsQqbbo7ufPOCe/g5ezF3fPupqCywCrttkRrzVtzttHFx5VrR3U7/IQNM+Dj8ZC7Dc77AG5eAP3PB8cW+jVXHxh2E9y5Ck54HDb/Dh+MhszVh51690k9qa6t5+MjUMh4S/4Wnlv2HMNCh/HMqGeOxeQFHI19sRBCHGFHVe+vlLoSSABea/JylNY6AbgceFsp1b2592qtP9ZaJ2itE4KCgpo7pdP9syGLweE+hPm2fUN7xsBQ9hZXkmwn23rO2mgkX07t33YRt5P7heDsYOKv9faxfWFqnjHzxZwCdD5uTozrGcRf6/faTRJgzqZ9DIn0I7CNQTQYM0gyCyvsJgkwp2FHmhP7Brd57sQBxgySfzbax8yjnTmlpOaVc4oZ142fhzOjegQyZ7N97sBjTXbXD5scIOE6uHUR+Ecb9RZWfHTQKT7uTtw6vjtzN2ezOrXzlubV1NXw4IIHqamv4YOTPyDALaDtN1ngvB7ncWnvS/li4xfMS5tn1bYP9fac7Xi5OHLj2OiDD5TsM57kb/4dTnkOrv4N/GPMbzioF1z7J0x41FgG9NU5hz3dv+OEHjiaFO908pbiP237iWlbpnFNv2s4t8e5bb/BAoFugbx7wrvkVuTyvwX/69RZM8t35bMqpYBbx3c/eJcYrWHh6/DTdRDSH25dArGXg7nFPZ1cYfxDcONcMDnCF2fAloO39o0O9OC8uDCmrkjt1FkYRVVF3Pffffi4+PDquFdxMh07y0bay+76YiGEOELsIYGRCUQ0+XN4w2sHUUqdDDwOnKO1rmp8XWud2fDfXcB8IK4zg22vrCJjWcLENmZfNDqprzETwF62JP1nw14Ghvm0ugygkberE6N6BPCvnQzm5mwy4mhrBkOjU/qFkFlYwfbs0s4Myyzp+eVsyio2axANcGKfYJSC+Vvt44nK3M37iI3wJdjr8J1TDuXj5sSQKD87it3Y1vUkM5IvACf0DmJXTtkReWrcCY7+ftivG1z3N/Q506jFsOz9gw5fN7obAR7OvDev89brT06ezIa8DTwz8hm6+XTrlM94aOhD9A/ozxOLnyC9pHN2gNi4p4h/Nu7lujHR+Lo3mX1RnAWfT4S8nXD5dBh9j/mD4aZMDjDhEbjkK2PmzJdnHpTECPZ25eqRUfyalMmO7M6p+bGrcBcvr3yZUV1Hcd+Q+zrlM/oH9ueZUc+wcu9KPkj+oFM+A+C9edsJ8nLh0qERBx+Y/zLMex4GXgzX/G4sw2qPrrFw838Q3A+mXw2b/zjo8F0n9qSmTnfaLIx6Xc8Ti59gb/le3hj/htUTg3bm6O+LhRCik9lDAmMV0FMpFa2UcgYmAQdVTlZKxQEfYXTU2U1e91NKuTT8fyAwGth0xCK3wKJtuQCc1Me8gai3qxPxkX4s3Gb7wVxBWTVrMwrNHsgBjO8VREpeOWl55W2f3Mnmbt5Hn1CvVmt3NDWul/E0YoEdDKQbk0DmJjD8PZwZFO7Lgm3ZbZ/cyfYVV7I2o8js2AEm9A5mc1Yx+4qPzHrq1szdtI8BYd508TFvCcAJvY1/H/aSgLHQsdEPO7sba/v7ngOzHj1oJoa7syPXj4lm/tYcNu6x/gylZXuW8cWGL7io10Wc2u1Uq7ffyNnBmdfHvw4KHlrwELX15m8ta6535m7Hy9WRG8Y0mX1Rng/fnAdlOcasi56ndPyD+p0LV0w3EiJTL4DKAzVKbh3fHRdHB96fv7Pjn3OImvoaHl38KG6Obrw45kUcTA5tv6mdzu5+Nuf1OI/PNnzGupx1Vm8/MSWfpTvzuGVczMGzLxa/BQtehtgr4fyPW14uYi6PQLj6V+gy2Fg+tOm3/YeiAz04L7bzZmH8uPVH5mfM538J/yM2ONbq7duZY6MvFkKITmTzBIbWuha4E5gFbAama603KqWeU0o1VlB+DfAEfjxka6i+QKJSai3wH/Cy1tr6nXXJPqit7lATC7fnEOzlQq+QVrY2O8S4XkFsyiomp6Sq7ZNbojXUduD9wJKduWgNY3uaP82wMQmwcHsHB3Md/LmXV9eyJrVwfzzm6OrrRq8QTxZ0NHlUV9PqNnTmWLwjl0h/d2Ja2xLvEON7BZGcXkhheQd+dlobW+11QGPybUJv83/2jed2OHnUytaa5iiqqGFNWgEnmplwBOgW6EF0oAf/bbV98shSR0U/bC4HJ7joc+hzljETY803+w9dOSIKTxdHPrTyk+KiqiIeX/w40T7RPDT0Iau23Zxwr3CeGvkUG/I28PWmr63a9o7sUmZv2sd1o7rh49YwTb+qFL69CPJ3w2XTIGKY9T4wZoIxE2PvevjuUqg2kt4BnsaMgpnJe9hj5R1JPlr7EZvyNvH0yKcJdAu0atvNeWjoQ4S4h/D44sepqLXu9/LuvB0EeDhz+fAmu4Ot+BjmPgMDLoJz3gWTlW71XH2MArpd4+Gn62H3ov2H7jqxB9W19Xy2eLd1PqtBZmkmb6x+g5FdRnJ5n8ut2rY9Oqb6YiGE6CQ2T2AAaK3/0lr30lp311q/2PDaU1rrmQ3/f7LWOuTQraG01ku11gO11oMb/vtZpwT45/3wcqSxv32u5dOP6+s1S3bkMqZnYItbeDZnbE/jxmrJjlyLP5O6WmP66NuD4IUQowBXkycmlli8PRcvV0cGt7L166FiAj0I83Vr/wySrf8YW7i9EASv94Y5T0ON5U92VqUUUF1Xz5gelt2kjusZxMrd+ZRXt+PpZkGKUbX9pXD4vzDjpjzf8pu62rp6lu/KZ7SFsY/vFUS9hkXb23Hd1FbBvBeN7Q5fCIYpI5otimiOpTvz8Pdwpm+ot9nv6RPqRai3a/uTAJt/h48nwHN+8Gp3499AjeUDhuW78qjXWHzdTOgdxLKdeVTWdCz5Ywt23w9bwsEJLvoCYk6AP+6D1GWAsUzpiuGR/Lluj1WX+ry1+i3yK/N5eezLViva2ZaJURONLVaTprC7yHqDxk8X7cLF0cQ1jcUg6+vgx2thT5KRGIoeZ7XP2q/36XDBx5C2DH67Y3/i94Yx0WjgcysOitfnrOfT9Z9yTvdzODnqZKu12xovZy+eH/08KcUpvLvmXau1uyGziIXbcrhxbAzuzo7Gizvmwj8PGwm88z8ylutYk6s3XPGjUfdk+lXGtrkYCdzTB3bhuxVplFZZZ1aQ1pqnlzyNSZl4dtSzFt0/Hc2Oqb5YCCE6gV0kMOzekOuMfdLTVxqVvHdaVjxt455iCsprGGfBDAaAAV198HN3sjwJUFUKU8+H+S9BUG8Y07C+d/rVMPsJi2YFaK1ZtD2XkTEBODqYf7kopRjXK5ClO/OoqbPwafjit2DapVBRaMQeMRSWvA2fnWy8ZklT23NwdjAxtJu/Re8b3zuI6rp6VuyysOBfRiJ8cpJxExl7OcRdASlL4INRkLrUoqbWZhRRWlVr8SA6NsIXb1dHFluawKgsgqkXwsJXjWnCo+4GNEybBPNesKgprY2k3cjuAZhM5t90KqUY2zOQZbvyLNuBR2tY9Ab8cCVUl8GY+42nxPNfgi/PMv5NWGDZzjzcnByIjfC16H1jegRSVVvPmrTO3XVAmMHRGS7+AnwjjeuiYYvVG8ZE42gy8fFC68zCWLV3FT9v/5mr+19N34C+VmnTHEopHh/+OK6Orjy99GnqdcdmHQFkl1QyY00mFw0JP7DzzvyXYMccOOM16HtWhz+jRQMuhJOfho0zYKmxy0qEvztnDerCtJVpFJV3vAhmbX0tzy57lgC3AB4Z9kiH27PE8C7DubzP5UzdPJXk7GSrtPnJol14ujhyxYiG2Rd5O42ZEcH9jYSQg6NVPucwbr5w2ffG/393qfG7A7h5bAwllbV8v/Lw7Yzb48dtP7Ji7woeSHiALp5t72QlhBDi+CAJDHP0PBnOeBVuXQy+UTD9Gsg1vzr6oh1GAsLSJ+kmk2J0j8CGJRxmDua0ht9uh5TFcO77xh7vJz8NN/0HCdcbN4YrPzE7hpS8cjILK/bPBrHE6B6BlFbVsmlPcdsnN1r/kzH1tf8Fxq4CJz8Dl06FSdMge4uRhKkz/0Z28Y48hkT54eZs2VOohCh/nBwUy3flmf+m4j3w3SXg4mn8vM96y7jpv2M5eIcZuyPkbDO7uSU7clEKRna3rGCZg0kxLDqAFbstiF1rmHGz8QT0/I+NdeknPw23LIK4K2HhaxZdNztzSskuqbI4+QIwIiaAwvIatu6zoHhf0jfw73NGsbrblhqxXzYNLvnaeHL8wxUWLYlZsiOXodH+ODta1kUOjfbHpIxdAYQdcPMzBlp11TDtcqguJ9jblQuHhPPj6owOr9evqqviuWXPEeYZxm2Db7NS0OYLcg/i4WEPk5SdxPSt0zvc3ldLU6ipr+fGsQ27imz5y/i3H3slJNzQ4fbbNPpeoy7G3Kdh538A3DwuhrLqOqauSO1w89O2TGNrwVYeGfYIXs5eHW7PUvfE30OwezAvrnixw7VL9hRW8Oe6LC4dGoG3q5NRP2TaZaAcYNK34OxhpahbENAdLvnGmIHx802gNYMjfBkW7c8XS1Isf3BxiNyKXN5a/RbDuwznop4XWSloIYQQxwJJYFjCNwIu/wEcnI0nembWZ1ixK59eIZ4EeVleRGtYtD/7iqvIKDBzGvzKT4ylIic/Yzz9b+ToDGe8Dr3PgH8eMSq/m6FxAD+qHQPRxlkPq1LMHMzl7YSZd0HECOPpkVOTqdh9zjDW8u5eAIveNKu5/LJqNmcVM6YdyRc3ZwcGh/uyfLeZsdfXGU++airh8h8hsMeBYz7hRiLJ5AgzbjJ7IL1kRy79unjj7+Hc9smHGBHjT0peOVlFZl43yz+Abf/AxP+DwZceeN3RGc56B3pOhFmPmZ2AWbLDuG5Gd7f8Zz88xrhuzE4e5WyDvx82praf/5GxfKBRv3ONRNKu+cb3aIbskkq2Z5cyysLEERjFdweE+bB8pwXJI9G5gnrBhZ/BvvXGDDTgprHRVNfW8+3yjj0p/nT9p6QUp/DkiCeP2NKRQ50dczbDuwznvaT3KKhs/8yfsqpapi5PY2K/UKIDPYz++JdboEssnPl6+3YbsZRSRuI9sLex9WdRJv27+jC2ZyBfLEnp0NKsfWX7mJw0mTFhYzg58sgsHTmUu5M7Dw19iC35W/hh6w8dauurZSnUa821o7oZCeg/7oO8HUYRW78oq8TbpuixMPEl2D4LVnwIGLMwMgsrOryN+tur36ayrpInhj9x3CwdEUIIYR5JYFjKNwLOnQI5W2BF2wOi2rp6VqcWMCzasiUMjRKijPclppoxkC7NNrZM635Sw/T/Q5gc4Lz3jaeSfz5gVqHDlbvzCfR0JibQ8qc5Id6uRPq7m5/A+PthY5B/8ZcHD0IbxV4O/c83lgrktV2ZvvFz2/uzHxETwIbMIvPW8yZ/a8xeOPN1Y8B0KL9uxmyMrOSDdkZoSVVtHUnphYyIad92cY3vM2sJTFGmMXuh1+kw7ObDjzs4wrmTwcndSDCZcd2s2J1HmK8bkQHm7fzSVLifOxH+buYlMBpv3B1djJkjza33jr/aSNzNe96s62ZZQ/KhPQkMgJExASSnF1JRffTVwThm9ToVRt4JiZ/Blj+JCfLkhN5BfLsilara9v09ZZZm8vn6zzm92+mMDhtt5YDNp5TikaGPUFZTxuSkye1u58fEdIoqarh5fIwxy+3nG0GZ4NJvDk4mdzYXT2MGQW01/Hob1Ndzy7ju5JZW8ce69g+KX0t8jdr6Wh4b9phNB8SnRp3KyC4jmZw0mdyKdtQpwkg2fbcijdMHdDF211o3HTb8BBMehZjxVo64DcNuMvrXOU9B1lpO7BNMTJAHnyzaZf7M0UMkZyfz287fuLrf1Z22HbEQQoijlyQw2qP3acYv7PmvQMneVk/dnFVCaVUtw6LbNxjqHeqFl4sjq1LMeLI291mjYOHpr7b8tMzND059ATJWwdppbTa5cnc+Q7v5t/uGL6GbH4kpBW3fyGybZayzHv8weLey1vW0l43B6qzH2vzsVbvzcXY0MciC4qNNDY/xp65eszq1jZ99ZTH8+zxEDIfBl7V8Xv/zjZkM//2fsSVhK9ZnFFFdW29x7Y5Gfbt44+XqaN4yknnPg66H019p+brxDIaJL0L6ctj0a6vNaa1ZubuAod38LA+8wYjoAFbszm+7DsbWvyF1MZzweMvXjVJw5pvG1Op5z7f52YkpBXg4O9Cvi/nFR5saERNAdV09SVIHw76c9LRR2+W3O6B4D9ePiSa3tJo/1rZvUPxm4puYlIn7E+63cqCW6+HXg8v6XMaP235kc95mi9+vteab5anERvgSH+lnJIn3rIGz3zZqiBxpAd3htJeMGXcrPmB0jwC6B3nw1dKUdg2KV2atZFbKLG4cdCMR3hGdELD5lFI8NvwxquqqeDPRvNmEh5qemE5JZS03jI02ikb/9T+IHAljbXAtKmU81HEPhJ+ux1RTxo1jYtiQWWzefcsh6urreGnlSwS7BXPLoFs6IWAhhBBHO0lgtNfEF6G2Epa0XlG8cQA5rJ0DUQeTIj7Kj8S2ZjHk7oC138HwWw5evtCcwZOMacELXzV2K2lBZmEFmYUV7Z7BAMYykryyanbntlLxX2sjAeAf0/wMgKa8QmH0PcZyh8zVrZ66KiWf2HBfXBzbV4V9SJQfjiYz6mAs/wDKso0b7tYSPUrBKc9Cden+InUtabzxa28SwMGkGNbNv+1aDPs2GomsEbe2Pe148GUQ1Mco6tfKMpjUvHJyS6sY2oHrZnhDHYxt2a3UwaivM9bKB/SEIde23qB3FxhxG2z8BbLWtXrqqpR84qP8LCpa29TQaH/iIn2ptaQIqeh8js7GUpLaKvjlVsZ0D6BnsCdfLN1t8aB49b7VzE6dzfUDrifUI7STArbMbbG34efqx0srX7L4+1m2K4+dOWVcOSIKMlbDgldh0KVG0tVW4q+G3mfC3GdQ2Zu4ZlQ31mcWkZxeaFEz9bqe1xNfp4tHF67rf13nxGqhbj7duLrf1fy+63c25m606L319Zovl6YQF+lLfJgXzGgY5HfGjiPmcveHCz8xZrjNeYrz48LwdnXk62UpFjf1y45f2JS3iQcSHsDdyfIZfEIIIY59ksBoL/8YGHQJJH4OpS3vErIqJZ9If3dCfVzb/VFDu/mxbV8pheWt1NxY/JZRm2P0PW03qJQx06EgBda3XPhtVUP9h/bOAjDeawzAE1t7ErN9trE+fez/jEFGW4bdbMwkmf9Ki6eUVdWyYU8xQ6PbPwvA3dmRAWE+rG4t9uoyY+1vr9MhbEjbjQb3hQEXGMtIylpOjKxKyad7kMeBnQDaIaGbP7tzy8gva+26eRucPQ/sVNMak4MxRTl3m1FstQWNS3c6ct0MiTL+3takFrZ80pY/jVhOeLT5JUeHGnUXuPrAgpavm6IKo3ho49Kt9vB0ceSX20czrpdluw6JIyCwp5F83r0AlfQ1147uxobMYhLbmmXVRL2u55WVrxDiHsK1A67tvFgt5O3szd1xd5OUncTs1NkWvffb5Wn4uDlxVl8f+OVm8OpizOSzJaWMukeuvjDjZi4YHIKniyNfLU2xqJnfd/7O5vzN3BN/D66O7f89bG03DrwRf1d/Xk983aKE06IduaTmlRu1L1Z8YMyKO+P1I1f3oiXdxsDIOyDxM9wyl3BxQgT/bNhLdrH5hXLLa8qZnDSZuOA4To8+vRODFUIIcTSTBEZHjH3AmIWx/P1mD2utSUwp6NBADoyBKNDyUoaiDFj3PcRfY0z1N0fv0yF0oJH4aOHmaWVKPl4ujvRt51R6gO5Bnvi5O7GytRkki94AnwgjIWQOV28YcYdROCy7+enSa9IKqKvX7V660yg+0o91mYUtV1Rf/RVU5Fs2dXfcQ1BTZiS/mlFfr0lMye/QzBeA+EhfgJaXMhSkwIafjdkLbmYmevqeY8zCWDa5xesmMaUAX3cnegR5Whxzo24B7vh7OLe8HanWxta6ft2g33nmNermayS/tvzZYi2MNWkFaE2HEl/CzsVfC93GwuwnuaC7wsfNiS+W7Db77Y0D4vuG3Gezwp0tOa/HefTw7cF7Se9RU2/ebk3ZxZXM2riXi4eE47rkNaMQ5HnvG/9ebM0jEM5+B/ZtwDNxMhcNCefP9VnklFSZ9faK2greTXqXAQED7G5A7Onsye2DbydxXyLz0+eb/b6py1MJ8HDmtK7lMO9FYzmrub87O9sJjxsPd2bexdVDgqit13xnwZaqX238irzKPB5IeEAKdwohhGiRJDA6IrAn9DkTVn9p7D5xiPT8CvLKqomP8u3QxwwO98XRpFp+SrjqM6OGwag7zW9UKaOoXe422PVfs6esSS0gNtIXB1P7bySUUiR08295CUzmakhfYcRizlP0RgnXg6NriwUx16QWotSBQXx7xUf5UllTz5asZpYy1NcZT8CiRkPEMPMbDe5jFFpd9UmzO9lszy6luLK2Q7MAAAY1XDctJgGWf2gU6Rt5h/mNmkzGUoy96yB1abOnrErNJyHKD1MHr5u4CN+WY09bZlw7o+6ybNr00BuNQrEtXDerdufjaFLERvhaHrQ4OphMxpP9uhrcZj/IpIRwZm3cx96itp8U19TV8H7y+/QL6McZ0WccgWAt42By4J74e0gtTuWX7b+Y9Z4fVqVTW6+5rnsxLJ1sLN040oUgW9PnDCNJueBVru9TQ02d5nszB8Vfb/ya7PJsHhz6ICZlf7c7F/S6gGifaN5c/aZZCaesogr+3byPi4eE4/LXfcbvzDPfODI7xJjD2R3OmQwFKUQlv8H4XkF8tyLNrC1Vcyty+WLjF5wSdQqDgwYfgWCFEEIcrezvN/rRZtjNxhP4jTMOO5SUbgy+OjoYcnN2YECYT/NJgJpKWPOV8RTG0mJr/c8HjyBY8fFhh8qqatm2r4Q4KwzkhnbzIyWvnOySZgYIqz4DJw+IbaX4ZXM8AmDgxbD2e6g4fJCblF5Az2BPvFwtSIo0Iy6yYSlDcwPpHXOhMK3tuh3NGXEblO5rtiBm44yJ+KiOzQJwc3agbxfv5pdhVJcbNVP6nQPeXS1reNCl4Obf7MyjgrJqduWUdTh2ML7/XTllzS+dWvWZsRxk8OWWNeoVCgMvgqSpRvHVQ6xOLaBfV2/cnR3bGbU4KvjHwElPwrZ/uNF/DXX1mh9Wpbf5thnbZ7CnbA93xd1lt0+Ix4ePJy44jg/WfkB5TXmr59bVa6atTGNcdz/CFj4E7gFwynNHKFILnP4qOLkTufgRxvXwZ+qK1DYHxQWVBXyx8QtOijyJ+JD4IxSoZZxMTtw/5H5SilOYse3we4hDTVuZjgZu8lwMKYvg1Oct7787W7fRMPQmWPEhd/YqJLukitkb97X5tveT36emroZ74+/t/BiFEEIc1SSB0VHR44w961cengRISivEzcmB3iFeHf6Yod38WJteRGXNIcUTN/0K5XnGk2VLObrAkOuMgpgFqQcdWpdRRL0+MIDviP1LYA6tJVGebyxhGHypMRi11PBboLYCkg/eTUVrTXJ6IXERHY+9q48rId4uzScwVn0KnqHGLBxLdT8J/Lsbs3cOkZxeiK+7E93asQXpoeIjfVmbUUjtoTf7G36GyiJIuMHyRp3cIP4qYweQkoNvTJMzCgGs8rOP278EpvDgA6U5sOk3I3nh3I6f0dAbjSU8hyQda+vqWZ9ZZOzCII59w2+FsCEELX6WU7u78f2qtMP/nTRRWVvJx+s+Jj44ntFdbbdtaluUUtw35D5yK3L5dvO3rZ47b0s2e4oqeSzgP8haa2z1bO5ysiPJK8SoXZK2jEeDV7CvuO1B8Rcbv6C8ppy74u46QkG2z/jw8cQHx/PRuo+oqK1o8byaunq+X5nGmTGOBCx93lgGFX/NEYzUAic9BV6hJGx4nig/Z75qo5jn7qLdzNg+g0t6X0Kktw12vRFCCHFUkQRGRyllLGfYk2Ts6NBEUnohA8N92r2bQVMJ3fyprqtnQ2bRwQeSphoD4ZgJ7Ws4/mrjv+t+OOjlxkrvg60wA2NAVx9cnUyHb6m2brpRQ6Q9g2gwanh0jYPk7w56OSWvnMLyGmI7uHwEGpcy+B0+iC5Mh+1zYMg1li19aWQyQdwVkLoE8ncddCgprZDYCF+rPOGNj/KjvLqOrfsOWQKz+gsI6gtRo9rXcNxVoOsOv27SCjEp2r11bVODw43lS4clj5KnQn0NJLRzR4GwIUYdj6SDB3fb9pVSXl0ny0eOFyYHY/p9WQ6Pu80gq6iSeVuyWzz9h60/kF2RzZ1xd9rt7ItGccFxTIiYwOcbPqewsrDF86YuT2WgVym9N082ChH3O/fIBWmp2Cug21j6bHqLAb7VrQ6KcytymbZ5GmfGnEl33+5HLsZ2UEpxd/zd5FTk8MOWH1o879/N+8guqeJR5+lG8egz37SfpSOHcvWGif+H2ruOF8NXsnJ3Plv2Hj7jrdH7ye/j7ODMzYPaMZtRCCHEcUcSGNYw8CJjXX2TgXRVbR2b9xTvf4rcUQkNU/IPSgIUphvTSAdPav+NjG8ERI+F5G8PKsqYlFawv5BiRzk7mhgc7kti6iFLYNZ9D6GDIHRA+xuPvcLYwaTJ1piNSzCs9bOPj/IlLd/YGnS/9dMBDbEWLmFoatAkowZFkxkkpVW1bMsusdogOn7/EpjCAy/mbDPqR8Rf1f7rJrAnhA87/LpJL6RXiBceLh1fguHh4kifUK+DExhaG//OIkZAUO/2NayUcd1krDR+Fg0ak3aSwDiOdI2DoTcQufNbxnpm8u2K5msrlNeU89n6zxjRZQRDQ4ce4SDb5564eyivLeeT9Z80ezwtr5yF23N43edHlK6H01+x3wExGLGd8RqqupTX/X5l5e58Nmc1Pyj+dP2n1NTXcNvg245wkO0zJGQIo7uO5tMNn1JaXdrsOd+uSOMkr3TCdv8EI26HoF5HOEoL9T8fYiYwOvUDwhyL+WZZarOnbSvYxj8p/3Bl3ysJcOtY0W0hhBDHB0lgWINHIPQ6zZhRUFcLwMY9xVTX1VtlKj1AgKcL0YEeBw/m1v9o/LejFchjrzB2pEhbBhhLMJLSC606kIuP8mPTnuIDS2BythmzVgZP6ljDAy40to9tkjxKTi/Ew9mBnsEdX7oDTZIAjUVUtYa1P0DkSGMXjPbyCYOYE2DtNKg3pq6vSy9Ea+sNosP93Aj0dCapaQHYdd8biZMBF3Ws8bgrIGcLZK4BjOtmbXqh1RJHYPzsk9MKqatvSJJkJRuFZzt63Qy6FJSDMZujQXJ6AX7uTkRZYemOOIqc+ATKzZ9X3L5m0fZ9pOUdXjdi6uapFFQV2P1yhKZ6+PXgnO7nMG3LNPaU7jns+LcrUxlp2kzv3Dkw+l7bb8NpjuC+MOI2+mT9ylDHnUxdfvigeG/ZXqZvnc55Pc47qpYj3BV3F0VVRXyz6ZvDju3OLWPx9mz+z+UrY9ni+IdsEKGFlIIz3kDVVfFuwAx+ScqkuPLwQqXvJ7+Pp5Mn1/S30+UwQggh7I4kMKxl8GVQlg075wEH1u1bczAXF+FLUlqhsWe81sb0/Y4OogH6ng3OnsbTdGBPUSU5JVVWqX/RKC7Cl9p6fWAJzLofGgbRF3asYXd/o4Dp+un7d/RISitkUHjHdk9pakCYD44mRVLDE3qy1kLuVmMQ3FGxl0NROqQsBNj/GdZKYCiliIv0O5D4qq+HdT8aiROvkI413v98cHTbnwTYnVtGUUWNlRNfvpRV17GtcQnM2h+MhFX/8zrWsFcI9JpoFIFtSDpac+mOOIq4+cGpz9O1ZD2XOCxk2qqDZ2EUVRXx5YYvmRA+gUFBg2wUZPvcEWvsMPTRuoN33amqrWPGqlRe85hqFH8ec68Nomun8Q+DVxfe8vyamUnplFbVHnT443VGPapbBt1ii+jarX9gf06OPJmvNn112LKfaSvTmOS4gJDSTXDqC+BineR8pwvsAaPvYUjRbAbVrueXNZkHHd6Ut4l/0/7l6n5X4+PS8WWHQgghjg+SwLCWnqcaOzM0JAGS0wsbCkC6Wu0j4iJ9yS2tIqOgwhhE52yxzv7vzh7GgHDjb1BdRnJnJF8akiFJaYUNg+jpDYPo0I43HnuFUch0+2wqa+rYnFVslfoXjVydHOjf1fvADIx1VhpEA/Q5C1x89tdjSEorJCbQA1/3ji/daRQfaewCk1daZcyyKUrr+AwGMAqv9j0b1v8MNRVNknbWS3zFN90Fpq4WNvxkJB6sUWgw9gpjJ5id/1JSWcOOnFJirTRjShxlBl8GkSN5wvl7Zq3cRFXtgWLJX2/6mpKaEu6Is2C7YTsR6hHKJb0v4bcdv5FafGC2wt/r93Jm1Z+EVe+Gif9nFOY9Wrh4wcQXCa/czrl1s/kl6cCgOL0knV+2/8JFvS6ii2cXGwbZPnfE3kF5TTmfb/x8/2uVNXXMWrWJx51/gMhRxpLVo8nYB8A3ilfdvuL7ZTuMBzANpiRPwdvZmyv7XWnDAIUQQhxtJIFhLY7OxraeW/+CigKS0gqsOpCDJkmA9MImg+jzrdP44MuhugQ2/0FSWgHOjib6hHpbp20gyMuFCH83Y2vZxkG0NWYwAHQ/ETxDIPk7NmQWUVuvrbL9a1NxkX6syyiitqYa1ltxEO3kCgMvhM2/oysKSU4vtGryBYydSKChxsO6741ta9uzc0pz4q6AqiLY8ifJ6YV4ujjSPcjTOm0Dkf7uBHg4szq1AHb9B2U5Ru0Qa+g10dhGOOkb1mUUGUt3rPyzF0cJpeDMN/DQZdxQPZVZDTtc5FfmM3XTVCZ2m0gf/z42DrJ9bhx4I04mJ95PPrDt8cyla/mf00/omBOMJOrRpv8F6OhxPOI8nT+Wrts/KP5w7Yc4mBy4aeBNNg6wfXr49eDMmDOZtnkaOeU5APy9IYvra6bhoUuNXWKOthliTm5w+qtE1qUzJv/n/XW8krOTWZixkOsGXIeX81Eyo0QIIYRdkASGNcVeDnXVlCT+QEZBRZtT6StqK8iryCOrNIvM0kzKa8oPejpxqD6hXrg6mUhOzbHuIBoOLEVJ/pbk9EIGhvng7Njy5aG1prCykPSSdPaW7SW/Mp+a+sPXtza1fzePdT8Yg+i+VrpxdnA0kiHbZ7Flp7GjR2sDUa01JdUl5JTnkF6cTk55DjV1bcQe6UtFTR0Zq/82lgpZK/kCxkyA2goKEn8kt7SqzeRLTV0Ne8v2klWaRXZ5NkVVRa1eN4PCfXE0Kdam7DVm2fQ925h1Yw3dxoFPBKz9nqT0AgaF+7S6dKemvoaiqiKySrNIL0mnqKqIet3y1pWNS2D2XzeuvtDzFOvE7uBkJB23zWLTTuPpdGy4b4un1+t6ymrKyC7PJq04jdyK3DavG3EUCekPw27mMsd5LF00F4DP139OZV0ltw++3cbBtV+gWyCX972cv3f/zfaC7WzZW8zErI9wV9Wo0189+gbEAEqhzngdd6q4MP8TVqcWsKtoF3/s+oNJvScR5B5k6wjb7fbBt1NbX7u/+OriRf9xpeO/xvbPHSl4bUu9T6Oux0TudZrBzMWrAWP2hb+rP5f36UAhbCGEEMeljm8VYAVKqdOAdwAH4FOt9cuHHHcBvgaGAHnApVrrlIZjjwI3AHXA3VrrWUcw9IN1GQzB/ahL/h548KAlGFV1VSTuTWRR5iI2520mrcQYAB3KxcGFrp5d6ePXhz4BfRgSMoT+Af1xNDni6GBiULgveuf8hkG0lZ5Eg7Gt56BJ6AWvsK/2Uk4dEbf/UEVtBYl7E1mXu47NeZvZVrCNnIocausPXnvsoBwI9Qgl0iuS/oH9iQuOIzY4Fm9nYyZHXKQvs9amUL/xF0zWHESDMQV86bs4bfmNMN8TCPYylu7U1dexrWAbifsSWZ+znt3Fu0ktTqWituKwJrycveju053e/r3pF9CP4V2GE+YZBhxYyqDXft8wiD7VerGHDQH/7tSvnQ7ce9DMnb1le1m5dyUbczeyMW8j6SXp5FfmH9aEu6M7XT27EuMTw+CgwcQGx9IvoB+OJkfcnB3o28Ub0/bZxmyJwVZMvphMMPAi9JJ3ya66kPHjD1w3ZTVlJGUnsXrfarbmb2V30W4ySzPRHJxsUShCPELo6duTXn69iA+JZ0jIEDycjOsjPsqXZZtT0JV/oGIvA0cX68U/8GJY/j7O2/8gJmg8Pu7Glrh19XVsKdhC4t5ENuZuZFfRLlKKU6iqqzqsCW9nb3r49qCnX08GBA5gRJcRhHpYYWmUDRwzfXE7qRMepWLND1yU/R4rUkbz/dbvOSvmLGJ8Y2wdWodcP+B6pm+dzpTkKfTeM5LbHOdTmXAHrva+k0VrgnpTN+xWLlkxmbf++5s9Mcm4OLhw/cDrbR1Zh0R4R3Buj3P5adtPjA28kEm571Lt4oPbCY/bOrQOcTjjZVzeG8bQ7W/x725Plmct538J/8PdSYomH+p474eFEKItZicwlFLPNZyfDCRrrbe1/g6z23UApgCnABnAKqXUTK31pian3QAUaK17KKUmAa8Alyql+gGTgP5AV2CuUqqX1roOW1AKBl2C79xniDZlE+xXyU/bfmJBxgJWZK2gorYCVwdX+gX0Y0zYGCK9IvFy9sLFwRiQFVYVkleRR1pJGsk5yfyd8jcAXk5eDO8ynHHh4+gb3oW4FbPQ7r4oaz2JbjToEtSClzlNLyYipA/fbv6WRZmLSNybSFVdFSZlIto7mrjgOLp4dCHALQBPJ0/qdB3VddXkVeaRXpJOSlEKX274kk/1p5iUibjgOE6IOIGugQmcaErCVFVsndodTYX0g5CB9M/+h0E9zuDf1H/5N+1fFmQsoLja2Gqvi0cXuvt2JyEkgVCPUNwc3XBxcKGytpLCqkJyKnLYXrCdP3f9yQ9bfwAg0iuS0WGjmRA+gQiPKsL2/gvxl1t3EK0UDLqUgPkvEemYS7lpO2+vXszCzIVsL9gOgJujG339+3Ji5IkEuwcT6BaIo3KkVtdSUVNBVlkWGSUZbMjdwOzU2YAxsB4bPpYJERMYEOHPoKR/0F6hqOjx1osdYNClqMVvcbpaRlRwf6Zumsq/af+SlJ1Ena7DUTkS4xvDgMABnBlzJr4uvrg7uWNSJkqqSyisKiSjJIPthdtZtmcZn234DEflyKCgQYwLH0doQBynmlaiaiusO/MFoGscOqAnA3L/YUDfU/kn5R/mpc5j8Z7FlFQbhUO7enSlu293RnQZQaBbIO5O7rg6ulJWU0ZJdQn7yvaxo3AHf+3666Dr5qGhDzE+wro/687qhxvaPnb64vZy9aH+xKdJmHUP189/mDpTHbcOvtXWUXWYj4sPV/e7mvfXvs8JGfMpdvTH+6RHbB1Whzmd8DDFq6fRK/M1PjfVcdPAm/B39bd1WB12y6BbmLlzJp8ufIyvTNsoO/EtcPO1dVgd4x9DcfztnLP6Hc5e8gJBbkFc2tvK/fkRJPfEQghhO2YnMLTWTymlQoBY4HylVA+ttTUWmg4DdmitdwEopb4HzgWadtbnAs80/P9PwGRlbBVwLvC91roK2K2U2tHQ3jIrxGWxuvo6NoQNYKGvDybPjzjz1zcBY+B8TvdzGBc+jmGhw3B1NK+wZ35lPiv3rmTZnmUszlzM3LS5KBTbwqrY5TGQM0pS6enb0yq7Jmit2aZqmRnak6WmxaRtWgxAN+9uXNzrYsaGjyU2KNbspyUVtRVsyN3Asj3LmJ8xn9cTXwcgJNqB1yq7MNErgAG6HpOyziqmgsoCZocNZKmezfLaO1g8vwYfFx8mRExgVNdRDAkZYvZTca01Owt3smLvCpbuWcov239h2pZpOEc48ki5FyeERDG2shBfV1+rxF5dV83y4G78G+hHpcc73DC7BkflSFxIHA8MeYBRYaPo7tMdB5ODWe1ll2eTlJ3EooxFLMxYyJ+7/sQBB3JDy9jtH8+ZFTlWmyGgtWankxN/BMeQ5DCfGev+A6CHbw+uG3AdQ0OHWnTdVNZWkpyTzIqsFSzJXMLba94GICjGxPNV4ZzmZCKuvhZHk3Umj+VW5vF32ECWOS5iaeXt/LegFn9Xf06KPIkRXUaQEJJAiId5u7VordleuJ0VWStYkbWiU6rqd2I/DMdQX9wRnsOvZsXi90liC2d3O5cIrwhbh2QVV/W7iq/Wfsbf/oUM7v0k3q7Wq3FkM67elI1/mtnrn8MNn2NmK84unl04p9tZ/LpzBis8ejF8+LW2Dskq/Cc+wt9bfiC1Lo1H+z9s9r2QPZJ7YiGEsJ02RwFKqXeAe7VhHzCr4ctawoD0Jn/OAIa3dI7WulYpVQQENLy+/JD3hjX3IUqpm4GbASIjrbc3fE55Div3rmTpnqUsylhEQVUByteHAdWlXB5/H+PCx9HDt0e7kgz+rv6c1u00Tut2GlprtuRvYdayt1m+5z8+q0vls5kXEuYZxrjwcUwIn0BCaALODubvXlFRW0FSdhJLM5fyb9q/ZJRmgCvEV1VySe+bOKHfJCK92/ezcnN0Y2joUIaGDuXu+LvJKMlg/q6/+HfZ60z1cuHrf64iyC2I8RHjOSHiBBJCEiyaSto4WFyYsZAF6QtYl7uOel1PiIszZzhHcM6ExxgSMqRdA12lFD38etDDrwdX9L2CytpKVmSt4I+/H2KlixtzNn+OacuXxAbFMiFiAhMiJhDtE23RZ+wr28eqfatYkL6ARZmLKKspw83DkxFVdZxx2muM7jq63YXNgt2DmdhtIhO7TaSuvo51uev4a/EbLK9O5O3KTbz90yn0D+jPCREnMD5iPD19e5qdHAEjybBy70oWZixkceZiMkszwQMGVVZxX5+bOKnvJUR5R7UrdldHV0Z0GcGILiO4J/4e9pbtZeGO35m39BV+9nBn+uwb8Hb2ZkzYGE6IOIHRYZb9nOrq6/bP9JiXNo+1OWvRaMKcnDjPtTtnjX+EuOA4i34ejZRS9PLrRS+/XlzV7yqL399G253dD8MR6Is7qx+2KpOJqb0GY8pfxmm7c2CcrQOyDo+6Gq7KL+RDfzfy+g2im60DspL8PkOYt8udGwqK8aqtBytOjrOlk3ft5Q+tmdprMMNNx0a5Mu3kxoddowitTGfwzj3GPIGjjNwTCyGE7ZkzuisBZiqlJmmty5RSE4GntNajOzk2q9Jafwx8DJCQkNByxcNmZJZmklOeQ1lNGcXVxaQWp7KzcCdb8reQUpwCGFN0R3cdTU+vYWyYsYC3Hb4C/1jw62mV+JVS9A3oS9+8LDKyFM90f4vThhWyIH3B/hkC7o7uDA4aTHff7nT37U6Aa8D+JSpltWWUVZexp2wPu4p2sb1gOxvzNlLb8DR7eJfhXD/wer6aWc3npXfi0L0c2pm8aE64VzhXVimu3LuPs+qf4carurMwcz5/7fqLn7b9hINyoKdfTwYGDiTCK4Kunl3xd/XHQTlgUiaKq4vZW7aXPaV72JS3iY15G/cvDekf0J9bBt3C7rQozkt8gZGB6ZhCh1mtOJ2royvjfXoxLmsb79ScQ8AFN1FsWsv89Pm8ufpN3lz9JuGe4fQN6EtPv55EeEXg4+yDt4s39bqe8ppyiqqKSC1OZXfRbjbmbSStJA04kKSKcR/B1p/+4EXHb8A1HKxUld3B5GDUIsnfy45MJ97s+ypD+mXyX/p/TE6ezOTkyXg4eTAocBC9/XsT7hlOV8+uuDm64WhypE7XkVuRS055DruKdrExbyPbCrZRW1+Lm6Mbw7sM54aBN/DFjCK+qfwfpl610M7kRXNCPUK5pKySS/blcIZ6lfuu6c6CjPkszFjIX7v/wlE50tu/N739e9PLrxeBboH4ufjh5uhGZV0lFbUV7CvfR2pRKjuLdrI2ey0lNcbSkL7+fbkt9ja27+rGFclPEB+SiQpNsFrsVnbc98NHSkpRCouKVjGyzJ9huT9BzgMQ1NvWYXVY9sxnuLYoj6mBfXkveTJfTPzCKrP3bO29tZPxMHlwXWEGe359kvArJts6pA7T2ZsZsfsnhgUNYKHDKlKKUujm083WYXXYosxF7KrN4upCL7plvgMnXgeeR13BVemLhRDCxtpMYGitn1BKXQ7MV0pVA6WANRfPZgJN5+iGN7zW3DkZSilHwAejcJE57+2w/1vxfyzMWHjQa2GeYfT07cmFPS9kaJeh9PHrg4PJgWkr05hXU0e98/eY1k03CjRaS/Ee2L2Qdf5XsznDxKdXXsRFvS7aP0NgQcYCNuVt4qdtP1FZV9liMz4uPnT36c5V/a5iWOgw4oLj8HDyoKCsmgdz55DedRTd1v0IJz5lFGm0lnXTKfbuyYbsnnR3G8t5E86huq6aVXtXsSZ7Dety1vFPyj/7aw80pzHRcUrUKQwOGsyYsDH7K85PWrmM1d6nMDr/HchcA+FW/Nmv/xGl65lZP5bzCkK4+6Q7uCP2DrJKs/bXONmav5W5qXMPK1LZSKHo6tmVnn49uaT3JQwNHUpvv944mBz4amkKf9cW8ILTd6j1061bbT5vJypjFcl+N7E9050PLr2RGwfeSE55DsuzlrM2Zy1J2Ul8u/nbVneS8XLyol9gP67qdxXDQ4eTEJqAi4ML+4orebDwX/Z2GUrX9dNhwiPW3dlg7Q/k+Q5k095wot2Hc8qYk/fPLFmQvoANeRv4L+0/Zmyf0WITLg4uRHpHcmq3UxkSMoShoUP3L6O5YMkSkrxPZUj2+7B3g11W+j8C/TAcBX3xkfD+2vdxdnBmSN9nKVt4OaZfH8Drxt+Pzt06Gu3bSODmb/hRn8zNsefw5ppXWZa1jFFdR9k6sg5Jyk5iSeYS7om7j1m7ZnHR9m9h7y0QOtDWobWf1pT88gBauzCs3/Mk7nmQD9Z+wCvjXrF1ZB2itWZy0mTCPcPx8boD57VXU/b3U3hc/IGtQ7OI3BMLIYTtmbOE5CTgJqAM6AJcr7XeasUYVgE9lVLRGB3tJODQfbVmAtdgrOO7CJintdZKqZnAd0qpNzEKFvUEVloxNgBuGngTk3pPwsvZC08nT7p6dm1xuUNyWiEmd19U79OMrU5PfcHYrtEa1v8IaMr6XEDmvDL2FVcS4u1qzBCIGL+/YGC9rierLIvCqkJKqkuoqq3Cw8kDT2dPAt0CCXANaPbJW3JGIQDV/S6GxfdA2lLoNsY6sefthIyV6DFPwFxFUnoBA8N9cHZwZnTYaEaHHXh4UVxdTFZpFgVVBdTrerTWeDl7EeIeQoBbQLPLQurqNesyihgUewZs/NDYctOaCYyGZJSppCfJ6YX7X+7i2YVJfSYxqY+xI0x5TTn7yvdRXF1MUVURjsoRdyd3PJ08CfcKb3HNb1JaAY5eQdDtZOO6OekZ6yWP1n4PykRprwvYsaCAoooafNycCHIP4uzuZ3N297MB47rJKc8hqyyLqroq6urrQBnbMAa7BePj4tPsdZOUZvw8qvtdBEsegszVEG6lmQx718O+9dSNexH2wpq0QnoEe+2fWRIXbOx6orUmvzKf/Mp8CqsK9xfMdXF0IcgtiFCP0GbrrVTX1rNhTzFjE86CdR+DtZNHVnIE+mE4CvrizratYBv/7P6H6wdczyX9x/Dugkt4IvNL2PKHsf3w0Uhrav74H6Xajd2D7uWBfkOZtvUbJidNZmSXkUf1LIzJSZMJcA3gin6X8WlafwpXLcZj5gO43DTr6E04bfoV76wlvKCv595RoynYcBlfbPiCmwbeRA+/HraOrt3mps1lc/5mXhj9AvH+J/L5mtO5ZeN3MPJG6/6u7mRyTyyEELZnzgjpceBJrfUEjI7yB6XUidYKQGtdC9yJsYZwMzBda71RKfWcUuqchtM+AwIaChLdT0O2W2u9EZiOUdzoH+COzqi2HBscaxSxDI6lh1+PVms1JKUXEBfhixp0KZTnwq751glCa1j7A4QPpXufwcZnNQwcD2VSJsI8w+gf0J8RXUYwPmI8CaEJ9PHvQ6BbYIs3rElphZgUhI24AJw9jSSAtaybDii8h15GsJcLa1ILWjzV29mb3v69GdFlBKO6jmJ02GgGBQ0ixCOkxZoW2/aVUF5dR7/oCOh9Omz4Gepank1gkX0bYd96GHQpcRG+JKcXonXzsyzcndyJ9olmcNBgxoWPY1TYqP3XTWsFy5LTC4mL9EUNugSKMyF1iXVir6+Hdd9DzAR69TS2TFzbJAHTlEmZCPEIITY4luFdhjMqbBSjuo6il18vfF19W7xuktMLcXJQhI64BBxcGv6urWTt92ByInDYZXi5OrZ4zSulCHALoKdfT4aGDjUK5nYZxuCgwXT17NpisdjNWcVU19bTKyYGup9kJI/q660Xv/V0aj8MR0df3NmmJE3Bw8mD6wZch7erE+WDrmGrjqT+n8eg5vCtl48K63/CKX0pr9VeyoVjBuHs4Mytg29lfe565qfPt3V07bY8azkr967kpkE34eboxvmjB/Ja7SRc9qww/h0fjapKqf/nMTbpblTGXoOniyPX9b8Odyd33l/7vq2ja7e6+jqmJE0h2ieas2LOIsLfnbUxN5ODH/V/PmCvfW5Ljvt7YiGEsLU2Exha6xO11osb/n89cDrwgjWD0Fr/pbXupbXurrV+seG1p7TWMxv+v1JrfbHWuofWelhjdeaGYy82vK+31vpva8ZlqaKKGrZnlxIX6Qc9TgE3P+slAbKSIXsjDJ5E/67eODuYSEpvOQnQHklpBfQK8cLD09t42rjxN6hpeSmK2errYe00iB6L8gknLtKXpBYG0e3VOLCNjfA1tmi1ZvJo7TRQDjDgQmIjfckvqyY933qDmfyyalLyyomN8IPeZ1g3eZS2DArTYPBlDAr3QamWE1/tlZxeQL8u3rh6+Vk3eVRXayRDek3E5BlAbIQvSWnWveYbZ9PERfoa1401k0dWdCT64Ya2j4m+uD025m5kXvo8ru5/9f4dZC4b0Z2nqq/BVJQGS96xcYTtUFmEnv04W0w92Bl+Ab1CjNo6Z3c/m0ivSCYnT6ZeH1WDR8CYcfXumnfp4tGFi3tdDEC4nzu5PS9mEzHoOU9CVctLEe3WwtcwlezhiepruXxEDAC+rr5c2fdK5qTOYUv+FhsH2D5/7f6LnUU7uSP2jv0Fki8a1Yfnqy/HlJUESd/YOELzyT2xEELYnsVbNGitsxqm0IlDGE/mYUiUHzg6Q//zIXmacSPl0sGijGu+AUdXGHARLo4O9A/zJim10Cpxg7EEIzmtkHNiuxovDLrEGLhvnwX9zu1Y4ykLoTAVTnwSgLhIP2Zt3EdeaRUBntYpGb86tYAAD2eiAtzBt0nyqOcpHWu4ttr4O+x9OngEEhthLAdKSi8gMsD8XVNak9yQiIqN8AVnd+hzFmyaCWe8Dk4d3GZu7TQjIdLnTLycnegV7GXVxFdtXT3rMoq4eEi48cKgS2DTr0byqKM/+53zoCwbBl8GGNfN5HnbKa2qxdPFOtuoJqUVEOzlQhcfV3BvkjyKHmuV9juL9MPW917ye/i6+HJV3wM7yAwM96EybAT/FY5lwuK3UIMvAz/rFantdPNfhtJsHqy6ixtHHtgpycnkxO2xt/PIokeYnTKb06JPs2GQlvsv/T/W567nuVHPHbTz1uUjY3h86zX8wtOw8DU45TkbRmmh3O3oZVOY5XgiBA+jX9cD29xe3f9qvtvyHVOSpvDeSe/ZMEjL1dTX8H7y+/Tx78MpUQd+J4zvFcyTXiexWS+g79xnjIcm7v62C7Sdjse++N5PT2FPfY6twxBCHEW6moJ4+8Y5VmuvXYvstdZH6VzazrU6tQCTgsERvsYLgy6F2grY8mfHGq6pMKbE9jsX3Iy24yL8WJdZSE2ddZ6ebc8uoaSq1ki+AESPB88QY9lKR635Blx9968hj2v4+SRbcRbGmrQC4qP8jGUOjcmjzX90/Cnctr+N2Rzx1wDQO8QLNycHq85iSEwpwNGkjAQGGEmAqiLjszuipgI2/mpcN84egDHTICmt5SUwltqy11i6E9943TTOPEr+ruONr50Gbv7Q81TAiL1ew7qGWi3WsCatYemOUkbyqO85xs+susxqn9FZpB+2ntX7VrMkcwnXD7geT2fPg45dMSKKR0svoR4Fsx6zUYTtsHcDrPiIBV5nsce9D6cNCD3o8GndTqOHbw+mJE+htr7WRkFarq6+jveS3qObd7f99Xsaje8ZRK7fIP5zOxWWvQ+5220UpYW0hr8epM7BlcdLL+bKEQcnybydvbm2/7XMz5jP+pz1NgqyfX7d8SsZpRncFXfXQUv5HEyKy0dEcV/JlejKQvjv/2wXZAdJXyyEEEeWdR5jCsB4mts71PvA0+GI4eAbBUlTYfCk9je8aaYxoI078GQwLtKXz5fsZuveEgaE+XQwciP5AhxIYJgcYODFsOJDKM0Gz+D2NVyeD5t/hyHX7J9NMDDcBweTIimtkJP6hnQ49rzSKnbnlnHp0CbFtwdfBomfG8sZhlzb/sbXfA1eXaGH8YDF0cHEwDAfqyZfVqcW0L+rN27OxtRaYiaAd5iR+Ol/fvsb3vInVJccdO3FRfry/ap0dueWERPk2cqbzXPYdePobCTuEj83/u7b+0StotCIf8g1RptAbLgvYCyBGdU9sIORQ3ZJJWn55VzVdLAQdyWs/Q42/Qaxh9ZNs0B1uZEQEXZPa81bq98i2C14fzHeps4e1JUX/gjhL98rOXvLJ8bMoO5WLT9ifVrDnw9Q5+rDfblnM2l8BC6ODged4mBy4M7YO7l3/r38sesPzutxnm1itdA/Kf+wo3AHr41/7bCaSCaT4vJhUTz4z/ms8F6Gw98PwZUz7L+g56bfYNd//BR4J/V1gZwxsMthp1zR9wq+2fQNU5Kn8OEpH9ogSMtV1VXx0dqPGBw0mLFhh89quyQhgrfnRrEq6EKGJX4G8VdDl0E2iFRYwppPUYUQoj2suEfm8a2uXpOUVsiQKN8DLyoF8VdByiJjF472SvoG/LpB1IGdOuIijc+xVk2A1akFBHo6E+nfZNAVfw3U13bsafr6H6Gu6qDki7uzI31CrbeUYU3DbIj9g2iA8KEQ1NdIQLRXYTrs+NcY1JoO3PzHRvqyaU8xVbUdr41VU1fP2ozCAzMYwPisuCuNgVJhWvsbT/oGfCIg6sBOMnGRxudYawZJYmoBod6uhPm6HXgx7iqoq+5YHY8NPxvXTZPki5+HMzGBHlaLfXVKQ/KlW5OffdQo8O9uJI/aq2QvvN7r6C0keJyZlz6PtTlruT32dtwc3Q477ubswIVDwnkkayx1vtHw5/+sUxuoM639HtKXM6fr7RTiyeXDIps97cTIE+kX0I8P135IjbWKHneimvoapiRPobdfb06NOrXZcy5JCKfYwY/ZwdcbfeiWP45wlBaqKoFZj1MT2I8nM4dz6dBIXJ0cDjvNw8mD6wdcz5I9S1i9b7UNArXcj1t/ZF/5Pu6Ku6vZItBBXi5M7B/K/dlnot384a//Gck3IYQQohWSwLCSbftKKG26BKNR7JVGAcg1X7Wv4fxdRgIk7sqDttUM83UjyMvFaoO5pLRC4iP9Dr7JCOoFkaOM2NtTJVxrI4HQJfawpypxkb6sTS+irr7jNyurUwtwclAMbDoTRSnj6X3mamMqdXskfwtoiLvioJfjI32prqtnQ2Zx+4NusGlPMZU19SREHTJTIe5K479JU9vXcN5Oow5F/DUHXTc9gjzxcnFkjbUSXyn5DOl2yHUTOgC6xsPqr9p3M6q1MYMjdKDRThOxkb4kpxdYZQlMYmoBLo4mBnQ95LqJu9LYQjhnW/saXv2VMfOla1yHYxSdq7a+lnfWvEO0TzTn9mi51s8Vw6Moq3Pkz8gHIX+nUV/BXpXnw5wnqQ8byhO7B3FSnxAi/JufDaSU4q64u8gszWTG9hlHOFDL/brjV9JL0rk7/u4WdxYK8HThjIGhPJo+gvqgvvD3w1DZ8b6608x7AYozmR56H3U4cMXw5pNNAJP6TCLYLZg3V79ptWWAnaW8ppxP1n/C8NDhDO8yvMXzrhgeRUalM4k97ob0FUbyTQghhGiFJDCspHFAGB95SALDu4tRADLp2/Y9tVvzDSgTxB48iFZKEW+l3Twal2AclnwBY/lF/i7YPd/yhvckwb4NxiyUQ8RF+FFaVcuO7FLL2z3EmtQC+nf1Ofyp1aBLjcKnqz6xvNH6OiN5EDPBmP3SRONsida2gjXXYUswGvlGQo+TYfWXRiFRSyV+DiZHY0puEyaTYnCEr1USX3sKK9hTVElCc9dNwnWQs9lIvlkqfYVx3Qy98bCp33GRfuSWVpNR0PElx4mpBQwO98XZ8ZBuMO5KcHBu33VTVwOrvzCWGAR073CMonP9tuM3dhft5p74e1rcohmgR7AnI2L8eXV7F/Tgy2DJ27Bv05EL1BKzHoOKAuZ1f4Tc8lquHdWt1dNHdx1NfHA8H6/7mMpa+51ZUlFbwYdrPyQ2KLbZ5QhNXTkiisIqzdwej0PxHvjXTot5ZiTCio+oHXIDb2z246S+LSebANwc3bg99nbW5azj37R/j2Cglvti4xfkV+ZzT/w9rZ43IsafHsGe/N+eeGPm5KzHoFQKRAohhGiZJDCspNklGI2G3WQUglw/3bJGq8uNAWyv08C762GH4yL92J1bRkFZOwa4TTS7BKNR//PAI9goiGaplR+DkwcMuOiwQ41JgI4ugamuNZZgNBu7u3/DbirfQ1meZQ1vnw1F6YclAACCvVyJ9HcnMTW/nVEfsDq1gDBfN0J9mtltZMStULoPNv5iWaNVpUbypc9Z4HV4jZG4SF+27C2mvLpjhftaTL6AUT/FzR+Wt2Ot9spPwMXbaOMQjQVgO5q4q6iuY2Nm0cHLRxp5BsOAC42lU5VFljW89S8oyYKhN3UoPtH5KmoreD/5fQYHDebEiLZrWlw5IoqMggqWdL8XXH3g97uNRKc92TEX1k5Dj76Xdze50iPYk9E9Alp9i1KKO+PuJLsimx+2Wmn75k7wzaZvyC7P5t4h9za7HKGpIVF+9An14p2tvuhhN8OqTyFt+RGK1Ey11TDzLvDqwl/BN5FfVs01I7u1+bZze5xLjE8M76x5h5p6+1z2k12ezVcbv+K0bqcxMGhgq+cqpbhieCRJGcVsH/5/UF0Kfz90hCIVQghxNJIEhpWsSS04fAlGo+jxEDIQlk2xbEr92u+gIh9G3dXsYWvt5tG4BKPZYqCOLjDsZtgxB3K2mt9oUYZR/yL+qv07pzTVLcAdX3enDs8E2LiniKra+uYH0QAjbofaSlj9uWUNL37bqB/R95xmDydE+bE6tWO7eWitSUzNbzn27idBYC9YbuF1s+ZrqCyEkXc2e/jAbh4WDs4PsTq1ADcnB/p28T78oJObMQtj61/GDB5zFaQYCZv4q/fvnNJU71AvXJ1MHU58rc0opLZeNz97BGD4LcaNtCW1MLSGJe8YhXt7TexQfKLzfbv5W7Irsrl/yP1tDogBTu0XSqCnC1+sKYGJL0HGKljx0RGI1ExVJfD7vRDYi+SYm1iXUcQ1I6PM+t6Ghg5lZJeRfLb+M0qqO7hzUyfIrcjls/WfcVLkSQwJGdLm+UoprhgRxcY9xazrfbfRl8+8y75qlyx5B7I3wZlv8FliHjFBHm0mmwAcTY7cG38vKcUp/LLdwuT2ETIleQo19TXcHX+3WedfEB+Oq5OJz7e7wriHYOOMju/eJoQQ4pglCQwryCutIiWvvOWBqFIw6k7I2QLbZpnXaF2tkfAIGwKRI5s9pXE3j47WM2hxCUajhOuNpRhL3jG/0eUfGAO6Ebc3e1gpRVyEb4cLebY6CwAguK+RCFjxkTEzwRxpyyF9uZEAcHBq9pT4KD9yS6tIyy9vT9gAZBZWsK+4qvXrZsTtkLXWKCZqjtpqWDbZKNwZMbTZU+IirFPIMzE1n9gIX5wcWuhGht5k/PwWvWl+o0veNZZMjbyj2cNODiYGhfnunzXUXm1eN13joNtYWPqusR2tOXYvMGqujLn3oKKvwv4UVhby2frPmBAxgfiQ+LbfADg7mrh8eCTztmazu+uZ0HMi/PusZYndzjT3WSNxfM5kvlyRhZeLIxfEh5v99nuH3EthVSEfr/u4E4NsnynJU6iuq+a+IfeZ/Z7z48LwcHbg69V5cNZbkLsN/nuxE6O0QNZaWPAK9D+fZI9RrE0v5OoR5iWbAOO6DY5nSvIUSqs7vgzTmrYVbOPXHb9yWZ/LiPCKaPsNgI+bE+cM7sqvSXsoTrjDeODzx/1QYZ1aTUIIIY4tksCwgsbBVHxLgyEwpqT7dTMKdplTEDP5W+PJ9Zj7W9wCbv9uHh0YzLW6BKORR4BRj2DtNPNu1ov3wKrPGr7nqBZPi4v0Y3t2KcWV7Z8GuybNWIIR4t3MEoxG4x+GshxjSUtbtDb2o3fzb7Z2R6OEhqUHq1Laf4PV5iAajNonPpHw3wvmzcJIngrFmcYgugV+Hs5EB3qwugNLYMqqatmcVdJ67N5dYMh1xlIMc2ZhFGUYS19iL2t2yVSj+Cg/NmYWdWgJTGJKPj2CPfF1d275pPEPG0t4Vn/ZdoNaw4LXwDMUBndg+1VxRExOnkxFbQX3xLW+Pv9QV46IxNGk+GpZKpzzHji5wy+3GLVPbGnbbKNmy4jbyfYbzF/rs7goIRwPl5brehyqX0A/zutxHlM3TyW1OLUTg7XM9oLtzNg+g0l9JhHl3fLvk0N5ujhyfnwYf6zbQ2HYOKOe09L3YHc76vJYU00FzLgZ3APgzDf5elkKHg073ZhLKcWDQx8kvzKfj9bZzywgrTVvJL6Bh5MHtwy6xaL3XjkiioqaOn5Zmw3nTjaW3f5xn+xKIoQQ4jCSwLCCZnfBOJSDE5z4JOxbDxva2F6xuhzmv2QUtOpzZqunxkX6kpxeSG1dO3YJATZlFbe+BKPRmPuMm/V5z7fd6H8vGtuvnvh4q6cNifJD6wMDeUtprVmdWrA/mdCiyOHQ81RjBkl5G4P27XOMJ+njH252CUOjXsFe+Lg5sXK3hbU1mliTWoC7swN9Qr1aPsnRGcY/ZBRE3fx76w1WFsO8F43tdnuc3Oqpw6P9Wbk7v927wKxNL6SuXjdfQ6KpMfcZ1/68F9pudO4zxn/H/q/V04bH+FNbr9t93dQ3vLfF5SONoscaszAWvt72k8DNv0PqYhj3P3BqJZkmbG5z3mZ+3PYjk/pMoodfD4veG+zlytmDu/JjYjrFTv5w9tvGv82Fr3dOsOYozYbfboeQAXDSU3y3Io2aOs3VZtRTONTd8Xfj4uDC66ts+P00obXm1VWv4uHkwa2Db7X4/VeOiKKqtp6fVmfAxP8D/xj45VaoKLR+sOb69zljNuZ5U8ir9+CPdVlcEB+Ol2vzs/1aMiBwAOf3OJ+pm6ayq8iCZXqdaF7aPJbuWcrtg2/Hx6WV+6FmDAr3ZXC4D18tTaE+dDCc8JixnLAj27gLIYQ4JkkCwwpW7M5jYFgrSzAa9b8AQgfB7CdaH0gveNkoBHjysy3Ovmg0PDqA0qpaNuxp3zZxq3YbcbSZwPAIhNH3GAO1rX+3fF76KuOGY9jNh+3ecaj4SD+cHBTLd7UvCZBRYCzBOGznl+ac9LSxRnxWK0mV6nKjArp/d2PZTCtMJsXwaH+W72r/LIbE1ALiIn1xbGkJRqPBlxmDk78far2o5IJXjKdWE19s+7qJ8ae4spYte9t33SSmtrDrzqG8uxjXzYafjQKDLUldZtRMGXVXq7N2wKg/4mBSrGjnz35HTinFlc1sedycif9n1KGZ+2zL51SXw+zHIbi/MeNE2C2tNf+34v/wdfHl9tjml7e15frR0ZRV1zF9VTr0O9fY7Wjha5C61MrRmkFr+O0Oo2+78FOqlTPfrkhjQu8gogNbTsC2JNAtkJsH3cz8jPksyVzSCQFbZlbKLJZnLefO2DstHhAD9An1JiHKj6nLU6l3dIcLPjF+t/75gG2e7O/4F5a/b/x+7HEy361Io7q2nqtHmj+zpKl74u/B1dGVV1a+YvNtVStqK3hl1Sv09OvJpD6T2tXGDWNj2JVbxr9bsmH0vUYC+a8HjW3BhRBCiAaSwOig0qpa1mUUMap7YNsnm0wNUyPzWr6BSlli1AEYci10G91mkyNijKJfy3a2LwmwdGcu3YM8Wl+C0Wj0vcYg7fd7m9/Vo6oEZtwI3uEw4eE2m3NzdiA2wpfl7Yy98Xtu/Bm0KnSAMRtg7XctJ2BmPQZ52+HM142ZD20YERNAWn45mYWWb+lZVF7DpqxiEqL82z7ZwdGYrl66D/5+pPnrZsdco/ZFwvVG/YY2DI82fmbtTcCsSsmnd4gxC6VNY+6HgB7w+33NXzfl+TDjJmOpzJi217h7uToxoKs3K9o5+2VVivE9J3Qz42ffZRAMv83YGnX7nMOPa238Wy5Mg9NfMf6uhN36Y9cfJOckc2/8vXg7N1N81gwDwnwYFu3Pl0tTjBlMZ7xmJGt/vBZK9lk13jYtm2zsmHTK8xDcl7/WZ5FTUsU1bWyd2por+15JpFckL698maq6KuvFaqHS6lJeXfUqff37cmnvS9vdztWjupGSV87czfsgfAhMeNSYBZn4mRWjNUNhGvx8IwT1hZOfpbKmjq+WpTChdxA9Q1qZhdeKALcAbo+9naV7lvJf+n9WDtgyn6z7hKyyLB4f/nirWxK35owBoYT5uvHJol1GHaHzPzJm8P18g30VYBVCCGFTksDooFUN0/BHdTdjEA3QZTBMeMSosj3nqYMHo1lr4YcrjZvhU80rNhbk5ULPYE+WtWMWQ01dPSt355uXfAFjUH/+B8YOF1PPP3gablUpfH+5cZN2wcfGNoNmGBkTwPrMIkraUQdjyc5cAj1d6BXiad4bxj9kzID56XpIX3lIY+8Yg9TR90D3trdUhAOJkxXt+Nkv25WH1jCmp5k/+7B4GPegkYCZ/9LB103mavj5JgjuZ8wYMENXXzci/d3bFXtlTZ1x3ZhRMR8wllSc/xGUZcO0S41E1/7GimD61UZy5pIvwcW8v8vhMQGsTS+issbybSyX7sgjxNuFbgHNbHncnBMfN66bH6+FzDUHXtcaFr1h/J2Mf9hYciLsVlFVEW8kvsGgwEGc2+PcDrV1/ehuZBRUMHvjXqOvu/QbYwnXT9cZBZiPhJ3zjN8hfc+GYTehtebDBTvpGezJ+J5B7W7W2cGZx0c8TkpxCh+ttV19hSnJU8ityOXJEU/i0IGiuGcMCCXcz42PFjYssxh7v7Gk8O+Hj9zWqjWV8MNVxtLKSd+Cszsz1mSSW1rNzeNiOtT0pD6T6O7TnZdWvmSzgp6pxal8ufFLzoo5y6xdYlri6GDiutHdWLk7n3UZheATBue9byzTknoYQgghGkgCo4OW7szF2cHUegHPQ439HyTcYOxw8M35xpKLf5+HL8406i5cNcPsgRzAyO4BJKbkU2NhHYx1GUWUVdeZn3wBIwFzyTewbxN8OMbYbWTlJ/DJicbskfM+gKjmd01pzoiYAOr1gafi5tJas3RnHqO6B5hduR1HF7jyZ/AKhS/PNJaTrP0evr/CGAj0v8CoU2KmPqHGDIT2zH5ZujMXd2cHBof7mv+mCY8aRT0XvALfXgzJ04y6EV+eBS5ecOlUY/tSM42I8WdFO+pgrEkroKq2ntHmJr4AwhPgwk+NZMv7o4xdYVZ8bFw3acvg3IYddyyIvbqu3uI6GPX1miU7cxnTI8j868bZAy7/AVx94fOJxrWSPA2mXWbUhOl/gZHAEHbt1VWvUlhVyBMjnsCkOvar75R+oXQLcOeDBTuNqfsh/eGcdyF1iXF9dLb8XfDjdRDUB877EJRi/rYctuwt4eZxMZhMZl7bLRjVdRTndD+HLzZ8wdb8I7/Lyua8zXy35Tsu7nUxA4MGdqgtRwcTN42NYXVqAYkp+caT/Qs+Ad9II3lanGWlqFugNfz1AGQlG4ncgO7U12s+XbSLgWE+jDRnBmErnExOPDPqGfaV7ePtNW9bJWRL1Ot6nl76NC4OLtw/5P4Ot3fp0Ai8XBz5ZNFu44U+Zxq/+9Z+Z+zMJoQQ4rgnCYwOWrozj/go37brXzSlFJzxOkx8yZh18ettsOh16H4CXPeXUWjMAiNjAiivrmNteqFF71u2MxcwcwlGU71OhWt+Bzc/+OcR+Ot/xk3h5T/AYMvWvsZH+eHsaGLJDsuSADtzSskpqbIs+QLgGQzXzzJ2SFk22dhBIGUxjH/EGGC3sG1qc0wmxciYAJbsyLV4/fGSHbkMi/bH2dGCf4JKGUtJTnvZeHL4663GcqPuJ8KNcyGgu0UxjO4RSFFFDeszW6mr0ULsDibF8BgzlmA01fdsuO5vY0bG3w/B3w+CyQmu+gUGXWJRU8OjA3ByUCzclmPR+zbuKaawvIYxPS28bry7wi0LoPcZxmydX281Ei8nPgkXfS7bptq5BekLmLlzJjcOvJG+AX073J6DSXHL+O6syyg60HcNugSG3QLLp3TuQKuiwEi6KgWTvtuf7P5w/k66+LhybmyYVT7mwYQH8Xbx5tllz1JXb/lMp/aqqqviscWP4e/qz93xd1ulzYsTwvFzd+LDBQ2zMNx84dJvjZmD0yYZs2c6y6LXjd2Vxv4P+pwBwJzN+9iVW8bN42LMT6S2IjY4liv7XckPW39g1d5VHW7PEt9t/o7V+1bz8LCHCXJv/8yfRl6uTkwaFsFf67PIKGjYpnzcQ0a9mTlPNr+UTwghxHFFEhgdkFNSxaasYsueRDcymWDk7XD/JrhrDTyw1ZiG7BtpcVOjugdiUrDAwsHcou259OvijZ9H2/UeDhM1Em5eAPdugDtXw21LoecpFjfj6uTA8Gh/5m/Ntuh9i7cbyZfRPdrxs/cMhvM/hId2wx0r4f7NcMKj7RqEju8dxJ6iSrZnmz91d29RJTtzytp53TjAiNvgoZ0N180WY0qyZ7DFTY3tGYRSsGCrZdfN4h15xEb4Wlw1H4DIEQd+5netgduXQfQ4i5vxcHEkIcrf4mt+8Y4OXDcegXDJV/BIOty6BB7cYew6YoUBiOg8RVVFPLfsOXr69bR4a8fWXBAfRoi3C1P+23HgxdNeMgZasx6DtT9Y7bP2qyyGqRdC3g64+EvwjwYgKa2AFbvzuWFMtGVJ0Vb4uvry8NCHWZ+7nq82fWWVNs0xOWkyOwp38Nyo59pVuLM57s6OXDOqG3M372P7voYlbCH94OIvYN8GI4lRXW6VzzrIyk+MHZgGTYITDhSQ/njhLsL93Dh9QKjVPuquuLuI8Irg6aVPU1FreV2m9kgpSuGdNe8wPnw853bv2LKspq4bHY1JwUeNCSeTyZjdGdLfWIpj661whRBC2JQkMDpg/tZstIYT+lg+eNzPyc14cu7V/hsZH3cnhkT5MW+L+UmAovIaElMLOKFPB56YmEzgGwGBPTo0iDuhdzA7c8pIyzP/BnLe1hyiAz2I8DezjkFz3P0hqDc4t7+NCb2Nn58lCZgF24xzza5/0RxHF+O6aUfiopG/hzODwn2Zv8382PPLqlmfUchoS2e+NKWUMaMhoHuHrpvxvYPYsreEfcXmF3dbvCOH3iFeBHt1YKtTV2+jKKwFs3WEbWiteWnlS+RV5vH86OdxsuLfmYujAzeNjWHZrjzWpDUsZWpcnhA9ztjadMtfVvs8qkqNpWNZa+HiryBmwv5DHy7YiY+bE5cNszwB3prTo0/nlKhTeG/Ne6zNWWvVtpuzau8qvtr4FZf0uoSx4datKXP1yG64OpkOzMIA6DXRqNmUuhSmXwW1Vixaum66sYNG7zOM4t0m43Zr6c5cVqcWcNPYmLZ3oLKAm6Mbz456lvSSdF5Z+YrV2m1JbX0tTyx5AmcHZ54a+ZRVZpI06urrxkVDwvlhVTpZRQ3JGGcPuPIXY5eq7y4xlqwKIYQ4Ltk0gaGU8ldKzVFKbW/472GFJJRSsUqpZUqpjUqpdUqpS5sc+1IptVspldzwFXsk45+3JZsQbxf6d21fNXtrOqFPMBv3FJs9mJu/LZu6es1JfUM6ObK2NSaAzB1Il1bVsnxnHif37UDiyEq6+LjRO8SL+RbMYpizaR9hvm70CW1f5XlrmtAriLXphRSUVZt1/rwt2dRrOLmf7a+b8b2M5JG5szCKymtYsSufCR1J2h2jjva+uCXTt07nz11/csvgW+gf0N/q7V82LBJfdyfe/6/JNo+OLsbyhC6DjaLMa7/v+AeV5xvJi4yVxlK3hqUIAFv2FjN70z6uHhmFh4t1d8FRSvHMqGcI8Qjh4YUPU1zdeUstCisLeXzx40R4RfBAwgNWb9/fw5nLh0Xxa3Imu3PLDhwYcKFRv2THXGMmRtMiw+216jOYcTN0GwMXfbE/2am15u052wnxduHSoREd/5xDDA0dyo0Db+Tn7T8zc+dMq7ff1Ltr3mVtzloeG/4Ywe7W/118+4Qe1Gt9YBYGgGeQsXzVJ9z497BrvtU/15aO1X5YCCGszdYzMB4B/tVa9wT+bfjzocqBq7XW/YHTgLeVUr5Njj+otY5t+Eru7IAbVdfWs2h7Lif2CbHqk4f2OrEhCfCfmbMw/t2cTYCHs2VFJDtJdKAH3QLczZ5BsmhbDtV19ZxsB8kXMGZhrErJN2snlfLqWhZtz+WUfvZx3YzvHUS9Nj8JMGfTXkK9XRkYZp2p3R3RJ9SLUG9X5m4yb+vK/7ZmU1uvmdjfetO2jyFHbV/cknU563h51cuMDRtr1aUjTXm4OHJtw9KEDU1rybh6w9W/QdQoo87O3GehvXUksjfDZ6dCZqKRvOh//kGH35i9DU9nR24YE92B76Rl3s7e/H97dx4eVXn+f/x9hxB2whIMIeyIbMpmBAVEQLSKCriguFSstLQqVUuttdKvWqs/t7oUqVVccd8VLAoCooCsAdkFEtaw77uQkDy/P2aiIcwkIZnlJPm8rmuuzMw5c87NmXNuZu55lid6PsH2w9t5aNZDpzzeT1Fk5WRxz3f3sOunXTx2/mNUrViClnUFuK1XC+IqxPDvKatPXND5Zt9Awmu/g9cugd1rAm+gMNlZvqmuJ4zwte644UPfmD9+36fvZt76PdzR+/RTGzfrFNzR8Q5SElP45+x/krY3LSz7mLh+Iq8vf53rWl3HZc0vC8s+GtWpytWdG/LuvI0n/jBT/TRfEaNWI193qvmvlKXZScpcHhYRCYdoFzAGALmda8cCA/Ov4Jxb7ZxL89/fAuwAov4T6tx1uzl07PjPhYNoa5VYg+RaVfi6CF/msrJz+HbVDnq3Po0KJRytPlT6tE5kVvpu9v9UeBFg8o/bia/i6zbjBRe3SyQr2zG5CMd+Ztoujh3P4SIPtGAA6NiwFvVrVuZ/Swofif9oVjbTV3un+GJmXHpWfb5dvZNDxwqfunLS8m2cVqMSHT1QtPOgUpuLA9lzdA8jvh1BYtVEHjv/sRLPOlKQW3s0o1bVijw5Kd9sHZVqwE2fQuchMPMZeP1S2JUeeCOB5GT7ZuoZ09s3dfXN43ytBfJYlLGPySu287uezalVtRhjGRVRh3oduLPznUzeMJlXlr4S8u0/Nf8p5m6by4PnPUj7eu1Dvv1c9WpU4uZuTRi3eMsvY2Hk6nQT3PQxHNgML10AC944tS/GuYWmuf+Fc2/3zQqVp3uic45np6wmKb5yWFpf5IqNieXJnk9SrWI1Rnw7IuStZtL2pvHA9w/QsV5H/npOeGdfuqP36WTn5GuFAb4ut0O/9nWjmvBnX/efg9vCGkuElKk8LCISLtEuYCQ653K/OW0DCvxWZ2ZdgDgg788jj/qb0T1rZpUKeO0wM0s1s9SdO09t4L9APv9hCzUqxXJ+ScYxCCEz4/L2SUxfvZM9hXQHmL56JweOHvfUL9H9OzYgMzuHScsK/hByNCubKSu2c2Hr00Laf7gkOjWqTXKtKnyxeEuh605cto2alWPp0uwUZ/AIk5gY4zL/eVNY8Wj66p38lJXtmeILwOXtk8g8nsPUHwsuHh3Nyua71Tu5qG1iiaeYLKMikotDnYcDOZJ1hOFTh7Pv2D6e6fVMyAaCDKZm5Yrc3qsF01fvZJZ/ZqefxcbBFf/2jYuxYyW8cC58eS/s3RB8g9nHYcV4GHOBb6aeJt3gDzN9f/N5+utV1KkWx61han2R1y3tbqFfs36M+mEUX64N3dgeH676kPdWvseQtkMYcHroBoIM5vc9W1C1YgWezd8KA3wzOv1+OjToCF/c5ZvmedXEglvP7EqHL+6G/3aHfRt845Nc8thJY+RMT/ONfXFH79OpFBveWYvqVa3Hvy74F5sObeKub+4iM7toXQQLs+PIDv74zR+pVrEaT/d6OqRjygTSuG5VruyUzDtzN7BlX76BSSvHww0fQd9/wOpJ8HwKTPt/cHhX4I2VDqX2M7GISCSF/RugmU0xs2UBbid8UnG+dqlBf+4wsyTgLeA3zrkc/9N/A1oD5wB1gKA/BzjnxjjnUpxzKfXqlaxY/VNmNhOXbaXfWUlhawZaHAM7JXM8xzFhScFfpD9ZuIk61eJ+HoDSCzo0jKdp3aqMW7y5wPUmr9jOgaPHuapzwwhFVriYGOPyDknMSNtV4FgSB49m8eWyrVzeoQEVPVJ8AX8RIDuHr5cXXDz6eMEmEqrHcV5JBvAMsU6NapMUX3gLkknLt3EkM5vLzkqKUGTe44VcHMo8HMhPx3/i7ml3s3z3cp7q+RRt67YN+T4Cufm8piTFV+aJiatO7mJh5ptidfh83zTTqa/Cvzv4fq2f8hAsfNM3Y8nsF+DT38MzrX2/KGce9k3Re9MnAQd5nrN2NzPSdnF7rxZUD/HYF4GYGQ93f5iUxBRGzhzJtI3TSrzNCWsn8MicR+iR3IM/nf2nEERZuDrV4hjaoxlfLt3G8i0BppCu1RhuHu+b9eLwLnjvOni2HXx+B8z5r29wzgVjYfKDvgLH6LN906SeM9Q3w1K7gSdtMifH8a9Jq0iuVYVrU8LX+iKvlPopPNL9EVK3p/Lnb/9c4iLGnqN7+P3k37P36F5G9R4VlnEvArm7b0sAnpy48uSFMTHQ4264fQ40vwC+ewKebgVvDoRpj/nel7QpEYmzqLyQh/3bD2suFhEJp7B/i3LO9XXOnRngNg7Y7k/Cuck44CAIZlYTmACMdM7NybPtrc7nGPA60CXc/x6Ar1ds43BmNld2To7E7oqsTVJNWiXW4LMfghcB9h/JYsqKHfT32JdoM6N/x2RmrdnNtv3BByL9eMEmGsRXppuHvkQD9O/QgOM5ji8KKB79b8lWjmblROwDbFF1bFSLRnWq8MnCTUHX2XHwKN+s3MHVnRt66ryJifG1PPp21Y4CB7B9d+5GmtStyrnNvXXeRFJZzMV5Hcg8wB1T72DO1jk83O1hejfuHbF9V65YgT/1PYPFGfuYtDxIa6Aaib7ZKO5aAr3+5pvxYtbzMP6P8NkwmPQ3WDMVmnT3DQJ6x3xfl5EA3bVychyPfbWSxJqVuOncJmH+1/2iUoVKjL5wNG3qtmHEtyOYsHZCsbf10eqPuH/m/aTUT+GZXs9QoRjTWBfX0PObU7tqRR7+YkXgMT1iYqDjDfDHBXDtm9AwBVZNgIn3wae/gy/uhNmjAYM+/wd/Wg79nvJNtRzARwsyWLp5P3/5VauQTXNbFJc1v4yRXUfy7aZvuWvaXRzJKt5UsdsOb+OWibeQcTCDUX1GcVa9s0IcaXANa1flt+c34/NFW1iUsS/wSnVb+KYTv30unDccDm71FTPG3QETw9vN5VSV9TwsIhIJ0f4mMh4Y4r8/BBiXfwUziwM+A950zn2cb1luojd8fQWXhTNY8PVjfXvOBpJrVaFLU290A8jr6rOTWbhxH0s3BfhlCV/ri8zsHK72UAuGXFd3TsaAN2atD7h8876fmJG2k6vPbui5bgBtk2rSvmE8r3+/nuyckz8QO+f4YH4GLU+rToeG0R8AMy8z46auTZizdk/gXySBTxdu5niO49ow9t0urpvObcLxHN91GcianYeYu24P153TyHPnjYeUulyc19r9a7lxwo38sOMHHj//8Yh0Rcjvqs7JtKhXjce/+pGjWQV0OYhPhl5/hd9/B/dvhbuXwvBUuHcd3JMG146FNpdDheCtKj5MzWBxxj7+eknriLcCrFaxGmMuGkPnxM7cN+M+Ri0cRfYpDFCalZ3FU/Of4uHZD3Neg/MY3Wc0VWKrhDHik8VXqchfftWauev28EVBrbdi46DtAN94Fveug7+sheELfO/ZyO3wu6nQ8x5fcSqI/T9l8eTEVaQ0qc2Ajg3C8K8p2ODWg3nwvAeZvWU2N391MxsPbDyl1y/YvoDr/ncdO4/s5KWLXqJrUtcwRRrcbb1OJ6F6JR75X5CCU67TWsNF/4A75sL9W3zFwutDMAtQ5JTqPCwiEinRLmA8DlxkZmlAX/9jzCzFzHJHCrsW6AncEmBqqHfMbCmwFEgAHgl3wHPW7mH++r0M69nck1+Gru/SmJqVY/nPtJMHizualc2L362hS7M6nJkc/alf82tStxqXnpXEO3M2cCDAjB6jv0mjQowxuEvjKERXMDNjWM/mrNt1OOBgnjPTd7EoYx83ndvEEwNg5je4S2OqxVXglRnrTlp2JPM4r3+/jq7N6tCiXvUoRFewJnWr0bdNIu/M3Rjwi+MrM9YSG2Ncc7b3inYeUupyMcCx7GO8vux1rv3iWvYf28/LF71Mv+b9Cn9hGMRWiOEf/c9k/e4jvBAg/wZ+UZyvy0JCS6haJ2Bri/z2Hs7kiYkr6dK0Dld2ik4rwOpx1Xmh7wtc3fJqXl76Mjd/dTMr9wRo4p/Pkp1LuPHLG3lzxZsMbjWY5/s8H7YZRwpz3TmNOCs5nkcnrCjSIMCYQbW6kHC67z0roMCU16ipaew5kslD/dtFLfdfc8Y1jL5wNFsPb2XQF4N458d3yMopeMyjg5kH+df8f3HrpFupEVeDt/u9zdmJZ0co4hNVrxTLPRefQeqGvUxYWviA04BvANXaTXzXVulRKvOwiEikhb/jbAGcc7uBCwM8nwr81n//beDtIK/vE9YAT94fz01ZzWk1wjOHeyjUqFyRW7o1ZdQ36Szfsp92DX75tf/duRvZcfAY/x7cyZNfogFuu6AFE5Zs5ZUZ6xhx0Rk/P79+12E+TN3ETV0bk1wrsr/WFdUl7erTqE4V/j01jT6tT/u5qXBOjuPJib7+z4O7ePO8ia9SkWvPacRbszfwu/Ob07bBLwWuV2esY/uBY/znhs5RjLBgv+3RjMkrtvPid2u4u+8v583KbQf4YH4Gt3Rrxmk1KhewhfKtNOXi7Jxs0vel823Gt3yc9jHbDm+jV6NePHjegyRUie6gyj1aJnBlp2T++90arujQgJaJNUK+jycnreTA0eM8PDB6X4jB153koW4P0aV+F56Y/wSDvhhE70a9uaLFFZyTeA61KtfCOcfeY3uZt20en6d/zvebv6delXo81+s5Lmxy0ukWURVijH8MaMdVL8zi+W/S+NulbUK+j/QdBxk7az2Dz2nMmVGeerpHcg8+6f8JD856kMfnPc5bK95i0BmD6NWoF01rNqVCTAUyszNZtWcVkzdO5pPVn3Ag8wCDzhjEiLNHUD0uusXrQSmNGDt7A49O+JELzqhHjcrhHUA0GkpTHhYRiaaoFjBKmzHT1zJ33R7+OaCdpwbvzO833Zvx7ryN3PneD4wb3oPqlWL5cesBnv56Fd1a1PXUIIz5nZkcT/8ODfjPtHQuOCOBs5vU4WhWNn/+aDFxFWK4o/fp0Q4xqNgKMYzs15Y/vL2A56as5t5LWgPw0vS1LN28n6cHdQj76PMl8cc+Lfli8VZGfLiIccO7Uym2Asu37OfF79bwq3aJpHiwy1Surs3rMqBjA0Z/k87FbevTtkFNfsrMZuRny6hRuSJ3Xujd80aKZsXuFdw25Tb2H9tPtvO1tOlavysPd/N1RfCKv1/WhmmrdnD/Z0v5YNh5IW2pt3DjXt6fn8HQ7s1oXd8brej6Ne9H9+TuvPPjO7y/8n2mZfgG96wa62tZceS4b9yFhCoJDO84nJva3kS1itWiFm9enRvXZtDZDXlt5joGdEg+oXBbUtk5jns/XkI1f+sBL6hfrT4v9n2RGZtn8Nqy13hu4XM8t/A5Yi2WKrFVOHz8MDkuh1iLpWfDngzrMIx2ddtFO2zAV3B67KqzuOqF73li4koeGRi5cThERMRbrMD+hGVUSkqKS01NLfL6a3ce4q05Gxg7az2XnpnE6Bu824Ih16w1u7jplbm0axDPxW0TeXvuBgxj3PDuJNb09i/RB45mcfmomRw4msWQ85oye+1u5q/fw+jrO3NZe+/PInHfJ0t4f34G16U0Ii42hrfmbODy9kmMGtzJk92O8pr643aGjk3lrOR4LjmzPq/MWEuVihX46LZunm35kmvv4UwuenY6mcezGdqjOd+t3sGijH08N7gT/TtEvu95cZjZAudcSrTjiIRTzcPbDm/jpSUvUbtSbZrFNyMlMYWk6t7MBx+mZnDvx0v458Az+XWIBtk8knmcy0bN5FhWNl+PuCAiM4+cquM5x/lhxw+s2L2CbYd9sxrVr1af9vXa0z6hfUQH6iyqPYczufjZ6SRUj/u5cBsKL323hse+Wslz13VkYJS6+hRm86HNzNs6j4yDGRzOOkz1uOq0rN2SrvW7Urty7WiHF9A//7eCV2eu481bu9DzjPDNnqFcLCISXQXlYRUwiuCGl+cwf/0ermjfgEeuPJOqcd774BjI+MVbeOKrlWze9xPtG8bz+FXtQ/oLUzil7zjE/32+jNlrd5NQPY4RF7Xihq7eG/sikKNZ2fxr0iremLWeGDP6nVWfJ6/pENHR50viy6VbeWDcMnYdyqRVYg1e+vXZNE3wxi+mhdmw+zB3f7CIHzbuo0blWB4ZeCYDOnrzy0Mg+tBcNjjnuPm1ecxbt4fxw3vQqn7Ju5Lc98kSPkjN4N3fnuvpVnSlUW7h9tbuzXjgipJPvbsoYx+DXpzFha0T+e9NnT3/g0dpcjQrm/6jZ7LncBYT7z6fhOqVwrIf5WIRkehSASOfU03W6TsOEl8ljno1wvMfZThlZeew53Cm51tdBOKcY/uBY5xWo5LnWy4EsvdwJlXiKni6u1EwR7OyOZaVQ3zV0tfP2DnH3iNZ1K5asdR9cdCH5rJj58FjXPrvGdSsEstnt3cnvkrxr6X3523kvk+XcnuvFj93TZPQemj8ct6YtZ7RN3Ti8vbFb7G153AmVzw/E4AJd/agVtW4UIUofiu3HWDA6O/p1LgWbw3tGpapvZWLRUSiq6A8XDp+Eo6y00+rUSqLFwAVK8SUyuIF+Gb2qB9fuVQWLwBqV4srlcULgMoVK5TK4gX4zps61eJKXfFCypZ6NSox+oZOZOw5wvB3F5KVnVOs7cxas4sHxi3n/JYJ/PniViGOUnLd368NnRvX4p6PFrNw495ibeNoVja/ezOVnYeO8Z8bO6t4ESat69fk8avPYs7aPTwwblnBU6uKiEiZowKGiIhIGJzbvC6PDjyLGWm7uPO9H065iPHDxr38bmwqTepW5fnrO1GhlBZzS4O42BjG3JxCYs3KDH1jPsu37D+l1x/Nyub2dxayYMNenr22Ix0b1QpPoALAlZ0acnuvFrw3L4PHv1qpIoaISDmiAoaIiEiYXHtOI/5+WRu+WraNYW+mcvBoVpFeN23lDm54eS51qsfx1tCu+jU/AhKqV+LNW7tQpWIFBo+Zw+w1u4v0ur2HMxk6dj7frNzBo1eeWSoGmy4L/vKrVvz63Ca8NH0tf/t0abFbOYmISOmiAoaIiEgY/fb85jx65ZlMT9vFgNHfM2/dnqDrHsk8zmNf/citY+fTLKEan9zWjfrxpbMbYGnUpG41PrqtG6fVqMSNr8zhmcmrOZqVHXT9mWm7uGL0TOav28vTgzpwY9fQzDojhTMz/tG/HcN7n8778zO45sXZpO84GO2wREQkzErHdBoiIiKl2I1dm9CiXnXu+Wgx1740m55n1OPqzsl0aFiLqnEV2LTvJ75dtZN3525k16FjXN+lEf93edtSM+tVWZJcqwrjhvfg758tZdTUND5OzWBwl8ac3zKB5FpVOJyZzaKMvXy8YBPfp++mSd2qfPiH89RtJApiYox7ftWKNkk1Gfn5Ui5+djqXt2/A5e2TaJNUk3o1KpXasahERCQwzUIiIuIRGvm+7DuSeZxXZ6xj7OwN7Dp07IRlZtC9RQJ3921JStM6UYpQ8pq1ZhfPT01n9tqTu5M0iK/Mzd2acku3pvqS7AG7Dh1jzPS1vDd3IwePHQd8xajv7+tzyttSLhYRia6C8rB+2hEREYmQqnGx/PHCltze+3SWb9nPqm0HOZqVTWLNynRsXIvTaqi7iJd0a5FAtxYJbN3/E0s27WfHwWNUjo2hTVJN2iTV1MCqHpJQvRL392vDPRe3YuHGvWzcfQT09oiIlDkqYIiIiERYhRijfcNatG9YK9qhSBEkxVchKb5KtMOQIoiLjeHc5nU5t3ndaIciIiJhoEE8RURERERERMTzVMAQEREREREREc9TAUNEREREREREPE8FDBERERERERHxPBUwRERERERERMTzVMAQEREREREREc+LagHDzOqY2WQzS/P/rR1kvWwzW+S/jc/zfDMzm2tm6Wb2gZnFRS56EZGyQblYRCS6lIdFRIom2i0w7gOmOudaAlP9jwP5yTnX0X/rn+f5J4BnnXOnA3uBoeENV0SkTFIuFhGJLuVhEZEiiHYBYwAw1n9/LDCwqC80MwP6AB8X5/UiIvIz5WIRkehSHhYRKYJoFzASnXNb/fe3AYlB1qtsZqlmNsfMBvqfqwvsc84d9z/eBCQH25GZDfNvI3Xnzp2hiF1EpKyISC5WHhYRCUqfiUVEiiA23DswsylA/QCLRuZ94JxzZuaCbKaJc26zmTUHvjGzpcD+U4nDOTcGGAOQkpISbD8iImWSF3Kx8rCIlGdeyMP+7SsXi0ipFfYChnOub7BlZrbdzJKcc1vNLAnYEWQbm/1/15rZt0An4BOglpnF+ivODYHNIf8HiIiUAcrFIiLRpTwsIlJy0e5CMh4Y4r8/BBiXfwUzq21mlfz3E4DuwArnnAOmAdcU9HoRESmUcrGISHQpD4uIFEG0CxiPAxeZWRrQ1/8YM0sxs1f867QBUs1sMb7k/LhzboV/2V+BEWaWjq//36sRjV5EpGxQLhYRiS7lYRGRIjBf0bZ8SUlJcampqdEOQ0TkBGa2wDmXEu04IkF5WES8SrlYRCS6CsrD0W6BISIiIiIiIiJSKBUwRERERERERMTzVMAQEREREREREc9TAUNEREREREREPE8FDBERERERERHxPBUwRERERERERMTzVMAQEREREREREc9TAUNEREREREREPE8FDBERERERERHxPBUwRERERERERMTzVMAQEREREREREc9TAUNEREREREREPE8FDBERERERERHxPBUwRERERERERMTzVMAQEREREREREc9TAUNEREREREREPE8FDBERERERERHxPBUwRERERERERMTzolrAMLM6ZjbZzNL8f2sHWKe3mS3KcztqZgP9y94ws3V5lnWM9L9BRKS0Uy4WEYku5WERkaKJdguM+4CpzrmWwFT/4xM456Y55zo65zoCfYAjwNd5VvlL7nLn3KIIxCwiUtYoF4uIRJfysIhIEUS7gDEAGOu/PxYYWMj61wBfOeeOhDMoEZFyRrlYRCS6lIdFRIog2gWMROfcVv/9bUBiIesPBt7L99yjZrbEzJ41s0ohj1BEpOxTLhYRiS7lYRGRIogN9w7MbApQP8CikXkfOOecmbkCtpMEnAVMyvP03/Al+ThgDPBX4OEgrx8GDANo3LjxKfwLRERKPy/kYuVhESnPvJCH/a9XLhaRUivsBQznXN9gy8xsu5klOee2+pPxjgI2dS3wmXMuK8+2cyvVx8zsdeCeAuIYgy+hk5KSEvQ/BRGRssgLuVh5WETKMy/kYf+6ysUiUmpFuwvJeGCI//4QYFwB615PvqZy/gSPmRm+voLLQh+iiEiZp1wsIhJdysMiIkUQ7QLG48BFZpYG9PU/xsxSzOyV3JXMrCnQCPgu3+vfMbOlwFIgAXgkEkGLiJQxysUiItGlPCwiUgRh70JSEOfcbuDCAM+nAr/N83g9kBxgvT7hjE9EpDxQLhYRiS7lYRGRool2CwwRERERERERkUKpgCEiIiIiIiIinqcChoiIiIiIiIh4ngoYIiIiIiIiIuJ5KmCIiIiIiIiIiOepgCEiIiIiIiIinqcChoiIiIiIiIh4ngoYIiIiIiIiIuJ5KmCIiIiIiIiIiOepgCEiIiIiIiIinqcChoiIiIiIiIh4ngoYIiIiIiIiIuJ5KmCIiIiIiIiIiOepgCEiIiIiIiIinqcChoiIiIiIiIh4ngoYIiIiIiIiIuJ5KmCIiIiIiIiIiOepgCEiIiIiIiIinqcChoiIiIiIiIh4XlQLGGY2yMyWm1mOmaUUsN4lZrbKzNLN7L48zzczs7n+5z8ws7jIRC4iUnYoF4uIRJfysIhI0US7BcYy4CpgerAVzKwC8B/gUqAtcL2ZtfUvfgJ41jl3OrAXGBrecEVEyiTlYhGR6FIeFhEpgqgWMJxzPzrnVhWyWhcg3Tm31jmXCbwPDDAzA/oAH/vXGwsMDFuwIiJllHKxiEh0KQ+LiBRNbLQDKIJkICPP401AV6AusM85dzzP88nBNmJmw4Bh/oeHzKyw/yTySwB2neJrwkFxeCsGUBxeiwFKbxxNwhVICJQ4F4cgD4M33lsvxACKw2sxgOLIzwtxFCcGr+ZifSY+keLwVgygOLwWA5TeOILm4bAXMMxsClA/wKKRzrlx4d5/LufcGGBMcV9vZqnOuaB9EiNFcXgrBsXhvRgUR2BeyMUlzcPgjWPqhRgUh/diUBzejMMLMeTyQh4GfSYui3F4IQbF4b0YymocYS9gOOf6lnATm4FGeR439D+3G6hlZrH+inPu8yIiko9ysYhIdCkPi4iUXLQH8SyK+UBL/+jKccBgYLxzzgHTgGv86w0BIla9FhEpZ5SLRUSiS3lYRMq9aE+jeqWZbQLOAyaY2ST/8w3M7EsAfyV5ODAJ+BH40Dm33L+JvwIjzCwdX/+/V8MYbomaPYeQ4viFF2IAxZGXF2IAxXFKlItPmRdiAMWRlxdiAMWRnxfi8EIMhVIeLhbF8QsvxACKIy8vxABlMA7zFW1FRERERERERLyrNHQhEREREREREZFyTgUMEREREREREfE8FTDyMbNLzGyVmaWb2X0Bllcysw/8y+eaWdMwxNDIzKaZ2QozW25mdwVYp5eZ7TezRf7bA2GIY72ZLfVvPzXAcjOzUf5jscTMOochhlZ5/o2LzOyAmd2db52wHAsze83MdpjZsjzP1TGzyWaW5v9bO8hrh/jXSTOzIWGI4ykzW+k/7p+ZWa0gry3wPSxhDA+Z2eY8x71fkNcWeE2FII4P8sSw3swWBXltSI6Ff1sBr9FonB9lkfLwSftRLo5yLvZCHi4gjnKZi5WHw0+5+IR9KA/rM3FBMZTLPOzfVuRzsXNON/8NqACsAZoDccBioG2+dW4HXvTfHwx8EIY4koDO/vs1gNUB4ugF/C/Mx2M9kFDA8n7AV4AB5wJzI/D+bAOaROJYAD2BzsCyPM89Cdznv38f8ESA19UB1vr/1vbfrx3iOC4GYv33nwgUR1HewxLG8BBwTxHeswKvqZLGkW/508AD4TwW/m0FvEajcX6UtZvycMBYlIujnIu9kIcLiKNc5mLl4fDelItPikN5WJ+JC4qhXOZh/7YinovVAuNEXYB059xa51wm8D4wIN86A4Cx/vsfAxeamYUyCOfcVufcQv/9g/hGmk4O5T5CZADwpvOZg28O8qQw7u9CYI1zbkMY9/Ez59x0YE++p/O+/2OBgQFe+itgsnNuj3NuLzAZuCSUcTjnvna+0cgB5uCb8z1sghyLoijKNRWSOPzX4bXAe8Xd/inEEewajfj5UQYpD5865eIwX2teyMPB4iiiMpeLlYfDTrn41CgP6zNxYcpcHvbHEfFcrALGiZKBjDyPN3Fykvx5Hf/Fsh/fdFVh4W+O1wmYG2DxeWa22My+MrN2Ydi9A742swVmNizA8qIcr1AaTPALMdzHIleic26r//42IDHAOpE+Lrfiq/oHUth7WFLD/U32XgvSNCySx+J8YLtzLi3I8rAci3zXqBfPj9JGefhkysUn89q1Fs08DOU8FysPh4Vy8YmUh0/mxWtNn4l9yvRnYhUwPMzMqgOfAHc75w7kW7wQX7OxDsDzwOdhCKGHc64zcClwh5n1DMM+isTM4oD+wEcBFkfiWJzE+do+uUjsKxgzGwkcB94Jsko438P/Ai2AjsBWfE3Voul6Cq40h/xYFHSNeuH8kJLzQB4G5eICRftai3IehnKei5WHywcP5GLl4QJ44VrTZ+ITlOnPxCpgnGgz0CjP44b+5wKuY2axQDywO9SBmFlFfCfBO865T/Mvd84dcM4d8t//EqhoZgmhjME5t9n/dwfwGb6mT3kV5XiFyqXAQufc9gBxhv1Y5LE9t0mg/++OAOtE5LiY2S3A5cCN/sRwkiK8h8XmnNvunMt2zuUALwfZdqSORSxwFfBBsHVCfSyCXKOeOT9KMeXhk/ejXHwyT1xr0c7D/u2W21ysPBxWysUn7kN5+GSeudainYvLcx727zOiuVgFjBPNB1qaWTN/dXMwMD7fOuOBIf771wDfBLtQisvfb+lV4Efn3DNB1qmf28/QzLrgey9D9p+GmVUzsxq59/ENkLMs32rjgZvN51xgf56mQqEWtJIY7mORT973fwgwLsA6k4CLzay2vwnZxf7nQsbMLgHuBfo7544EWaco72FJYsjbt/PKINsuyjUVCn2Blc65TYEWhvpYFHCNeuL8KOWUh0/ch3JxYFG/1ryQh/3bLZe5WHk47JSLf9m+8nBgnrjWvJCLy2se9m8j8rnYhWD00bJ0wzeK8Gp8o8SO9D/3ML6LAqAyviZb6cA8oHkYYuiBr5nNEmCR/9YP+APwB/86w4Hl+EawnQN0C3EMzf3bXuzfT+6xyBuDAf/xH6ulQEqY3pNq+JJvfJ7nwn4s8P3nsBXIwtcnayi+vp1TgTRgClDHv24K8Eqe197qP0fSgd+EIY50fH3Gcs+P3FHAGwBfFvQehjCGt/zv+xJ8SSopfwzBrqlQxuF//o3c8yHPumE5Fv7tBbtGI35+lMVboHOGcpiH/ftQLvZALg4SQ0TzcAFxlMtcXMA1qjwcolugc4ZymIuDnbcoD+szcTnPw/7tRTwXm/+FIiIiIiIiIiKepS4kIiIiIiIiIuJ5KmCIiIiIiIiIiOepgCEiIiIiIiIinqcChoiIiIiIiIh4ngoYIiIiIiIiIuJ5KmCIiIiIiIiIiOepgCEiIiIiIiIinqcChkgxmVlDM7su2nGIiJRnysUiItGlPCyRpAKGSPFdCHSOdhAiIuWccrGISHQpD0vEmHMu2jGIlDpm1gMYB+wDDgJXOefWRjUoEZFyRrlYRCS6lIcl0lTAECkmM5sI3OOcWxbtWEREyivlYhGR6FIelkhSFxKR4msFrIx2ECIi5ZxysYhIdCkPS8SogCFSDGaWAOx3zh2PdiwiIuWVcrGISHQpD0ukqYAhUjxNgS3RDkJEpJxrinKxiEg0NUV5WCJIBQyR4lkJJJjZMjPrFu1gRETKKeViEZHoUh6WiNIgniIiIiIiIiLieWqBISIiIiIiIiKepwKGiIiIiIiIiHieChgiIiIiIiIi4nkqYIiIiIiIiIiI56mAISIiIiIiIiKepwKGiIiIiIiIiHieChgiIiIiIiIi4nn/H5+hs2ij8ER/AAAAAElFTkSuQmCC\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "fig, ax = plt.subplots(1,3, figsize=(18,4))\n", "for n in range(3): # plot each mode\n", " for i in range(3): # for each mode, plot all variables x_i\n", " x_i = R[i,n] * modes[n,:]\n", " ax[n].plot(time, x_i, label=f'$x_{i+1}$')\n", " ax[n].set_ylim(-1, 1)\n", " ax[n].set_xlabel(r'$t$')\n", " ax[n].set_ylabel(r'$x_i$')\n", " ax[n].legend(ncol=3)\n", " ax[n].set_title(f'mode {n+1}: $\\omega_{n+1} = {freq[n]:.3f}$')\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "b8f82235-d55d-40cd-9713-fd6abaf1e6dd", "metadata": {}, "source": [ "It can be seen that the three modes oscillate at different frequencies --- actually, the third mode has a zero frequency, meaning that it does not oscillate at all! That is because this mode represents the motion of the center of mass, which is not moving since there is no external force and the initial velocity is zero. For each of the other modes, the three variables oscillate at the same frequency but with different amplitudes, and they can be in or out of phase with one another." ] }, { "cell_type": "markdown", "id": "90be7807-ab28-4518-b88c-9ee9a6ca7699", "metadata": {}, "source": [ "We can make an animation to visualize these modes and their superposition as below." ] }, { "cell_type": "code", "execution_count": 10, "id": "f4ee18f3-4459-4972-ba8b-50ffe04e7033", "metadata": {}, "outputs": [], "source": [ "import matplotlib.animation as anim\n", "\n", "plt.rcParams[\"animation.html\"] = \"jshtml\"\n", "fig, ax = plt.subplots(figsize=(4,4))\n", "ax.set_xlim(-1, 3)\n", "ax.set_ylim(-0.5, 3.5)\n", "ax.axis('off')\n", "ax.text(-1, 0.2, 'all modes')\n", "ax.text(-1, 1.2, 'mode 1')\n", "ax.text(-1, 2.2, 'mode 2')\n", "ax.text(-1, 3.2, 'mode 3')\n", "p1, = ax.plot([], [], 'o-') # plot masses in mode 1\n", "p2, = ax.plot([], [], 'o-') # plot masses in mode 2\n", "p3, = ax.plot([], [], 'o-') # plot masses in mode 3\n", "p0, = ax.plot([], [], 'o-') # plot masses in all modes\n", "offset = np.arange(3) # add natural length of springs\n", "\n", "def animate(t):\n", " p1.set_data(R[:,0]*modes[0,t] + offset, [1, 1, 1])\n", " p2.set_data(R[:,1]*modes[1,t] + offset, [2, 2, 2])\n", " p3.set_data(R[:,2]*modes[2,t] + offset, [3, 3, 3])\n", " p0.set_data(X_sum[:,t] + offset, [0, 0, 0])\n", "\n", "mov = anim.FuncAnimation(fig, animate, frames=len(time), interval=50)\n", "plt.close()" ] }, { "cell_type": "code", "execution_count": 11, "id": "948ebec2-3b68-4533-b760-dfc030fad0a3", "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n", "
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\n", "\n", "\n", "\n" ], "text/plain": [ "" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "mov" ] }, { "cell_type": "markdown", "id": "6eb9b5ad-8b8b-4f40-b691-dadb7bd2f9c2", "metadata": {}, "source": [ "Finally, we can plot the configuration of the system in a 3-D $(x_1, x_2, x_3)$ space to visualize the superposition of the modes." ] }, { "cell_type": "code", "execution_count": 12, "id": "8e8a30b1-73df-455e-855a-a7d091c4e5cf", "metadata": {}, "outputs": [], "source": [ "fig3D = plt.figure(figsize=(8,8))\n", "ax = fig3D.add_subplot(projection='3d')\n", "ax.set_xlim(-1, 1)\n", "ax.set_ylim(-1, 1)\n", "ax.set_zlim(-1, 1)\n", "ax.set_xlabel(f'$x_1$')\n", "ax.set_ylabel(f'$x_2$')\n", "ax.set_zlabel(f'$x_3$')\n", "ax.plot3D([R[0,0], -R[0,0]], [R[1,0], -R[1,0]], [R[2,0], -R[2,0]], 'blue') # plot eigenvector 1\n", "ax.plot3D([R[0,1], -R[0,1]], [R[1,1], -R[1,1]], [R[2,1], -R[2,1]], 'orange') # plot eigenvector 2\n", "ax.plot3D([R[0,2], -R[0,2]], [R[1,2], -R[1,2]], [R[2,2], -R[2,2]], 'green') # plot eigenvector 3\n", "p1, = ax.plot3D([], [], [], 'o') # plot displacements in mode 1\n", "p2, = ax.plot3D([], [], [], 'o') # plot displacements in mode 2\n", "p3, = ax.plot3D([], [], [], 'o') # plot displacements in mode 3\n", "p0, = ax.plot3D([], [], [], '-') # plot displacements in all modes\n", "\n", "def animate3D(t):\n", " p1.set_data([R[0,0]*modes[0,t]], [R[1,0]*modes[0,t]])\n", " p1.set_3d_properties([R[2,0]*modes[0,t]])\n", " p2.set_data([R[0,1]*modes[1,t]], [R[1,1]*modes[1,t]])\n", " p2.set_3d_properties([R[2,1]*modes[1,t]])\n", " p3.set_data([R[0,2]*modes[2,t]], [R[1,2]*modes[2,t]])\n", " p3.set_3d_properties([R[2,2]*modes[2,t]])\n", " p0.set_data(X_sum[0,:t+1], X_sum[1,:t+1])\n", " p0.set_3d_properties(X_sum[2,:t+1])\n", " ax.view_init(azim=30+t)\n", " \n", "mov3D = anim.FuncAnimation(fig3D, animate3D, frames=len(time), interval=50)\n", "plt.close()" ] }, { "cell_type": "code", "execution_count": 13, "id": "0059f103-fc68-48fe-a21a-0f3cda7fabe3", "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n", "
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\n", "\n", "\n", "\n" ], "text/plain": [ "" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "mov3D" ] }, { "cell_type": "markdown", "id": "ad0decdc-5a14-46a3-a18b-14884eb4057c", "metadata": {}, "source": [ "In this plot, the lines represent the directions of the eigenvectors, and the dots represent the instantaneous values of the normal modes, which oscillate along the corresponding directions with different frequencies. We have also plotted the trajectory of the overall configuration of the system, which is the vector sum of the normal modes at every instant. This plot shows that the eigenvectors provide a new set of basis vectors in the configuration space, and the normal modes are the components of the system configuration along these bases. The matrix diagonalization that we did can be thought of as a change of basis in the configuration space, such that the oscillations are decoupled. Note that, if the matrix is symmetric, then the eigenvectors are orthogonal to each other; otherwise, the new basis vectors are non-orthogonal, and the normal modes as the components along these directions are not simple projections of the total displacement (but are projections along the dual bases which are the left eigenvectors of the matrix)." ] }, { "cell_type": "code", "execution_count": null, "id": "35ade3b2-a184-449f-a040-e5351533439d", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.10" } }, "nbformat": 4, "nbformat_minor": 5 }