{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
" \n",
"\n",
"# Lennard-Jones Potential\n",
"\n",
"### Example - Chemistry\n",
"\n",
"By Magnus A. Gjennestad, Vegard Hagen, Aksel Kvaal, Morten Vassvik, Trygve B. Wiig and Peter Berg\n",
" \n",
"Last edited: March 22nd 2018\n",
"___"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Introduction to the Lennard-Jones Potential\n",
"The Lennard-Jones potential (also referred to as the L-J potential or 12-6 potential) is a simple physical model that approximates the interaction between a pair of neutral atoms or molecules.\n",
"\n",
"The most common expressions for the L-J potential are:\n",
"\n",
"\\begin{equation}\n",
"V_\\mathrm{LJ} = 4\\epsilon \\left[\\left(\\frac{\\sigma}{r}\\right)^{12} - \\left(\\frac{\\sigma}{r}\\right)^6\\right]=\\epsilon \\left[\\left(\\frac{r_m}{r}\\right)^{12} - 2\\left(\\frac{r_m}{r}\\right)^6\\right],\n",
"\\label{potential}\n",
"\\end{equation}\n",
"\n",
"where r is the distance between the particles. Here, $\\epsilon$ is the depth (i.e. the minimum) of the potential well, reached at $r = r_m$, and $\\sigma$ is the distance at which the inter-particle potential is zero. It can be shown that $r_m = 2^{1/6}\\sigma$."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"%matplotlib inline\n",
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"from matplotlib import rc"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"newparams = {'figure.figsize': (12, 4), 'axes.grid': False,\n",
" 'lines.markersize': 6, 'lines.linewidth': 3,\n",
" 'font.size': 20, 'mathtext.fontset': 'stix',\n",
" 'font.family': 'STIXGeneral'}\n",
"plt.rcParams.update(newparams)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"After importing the necessary packages, we obtain the following graph for $\\epsilon = \\sigma = 1$:"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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7mF9lQb9uWbtg4jndaAAAQLFKK8Cb2eun2PR/nXMPO+c+6py7QdICSXdKOiHpnZKuzkiVU9e1TIkbZX/hnPuwc+5dkh6StM3MWi9+NPJNajea/yTAAwCAIpXuFfjPmtmbLmx0zo1c8LzXOffvzrk7JS2S9OdzL/GiPiUpLukrKW1fVOLr+ocs/9uYZzevaVZpIPEt++yZQR3qHPK4IgAAgPmXboAflfQeM/uFma1J5wCXuMvwk7OubBpmtkiJLjoPuZQ7Gp1zMUk/k/S7ZtaYrX8f86+yNKDnXdI88ZxuNAAAoBilG+D/0jl3uaS/lvQNM/uImZVMd5BzLjKn6i5uiyST9Owk2/ZJCkq6Pov/PjyQOqnT/XtOeVgJAACAN9IK8M65zyWX/yXpWkl9kraa2c3ZK21aS5PLrkm2dSaXKy/cYGZ3mdkOM9vR2dl54WbkuFsvbVHQb5KkPR0DeopJnQAAQJGZ8Sg0zrmYc+5fJN0h6a7kCDANmS9tWuXJZXiSbWPJZdWFG5xzX3DObXLObWpubr5wM3JcbXlQv33ZuZtZv7ztiHfFAAAAeCAw2wOdcycl/Z6Z3SrpB2b2JefcV6Y7LoPGr7wHJ9k23jY8T7VgHt15fbt+8MRJSdKPnjyp99+2Vs3VpR5XBQAAClUs7hSJxZMPp2gsrnAsrmjMTbRFYnFF4+Pbz+0fjWd+WqS0AryZLXHOHZ9sm3PuQTP7haQPmdlPJb3dOXcgk0VOYbwD9GQ3qjYll/NRB+bZVUvrdeWSOj1xvE/hWFzfevSY3nXraq/LAgAAs+ScUyTmNBaNKRxNhONwNPEYu+D5hdvH1yMpy7FYXJGoUzgWUyTqUtrOhfDxY1ODeTh67nk05ia2ZyGDz0m6V+DfLOluaWL0lzXJxyUp68sk+SXtNrO/dc59NOPVnm+7EkNITpbc1kiKSdqa5RrgkTfe0K53//sTkqSvbT+qt/3GSpUE5jovGQAAxSsaiysUjSsUiSkUiWksuT6xjMQ1Fo0plFyOReMTban7praNTYTwWMp6XGORmMKx5L7J4I30pRvg329mr1QipFemtFtyOSJpl6SnUx5Z5ZzrNLOfSLoltd3MfJKeL+mHzrmebNcBb9y2oVV/Xf2Mzg6OqXNwTPfvOaU7rmzzuiwAADIuHI1rNBLTaDim0UhMI+GoQpGYRsPJ9khMoXCifTSSaBsbb4/EEm3hmMaiidcIjS9TAnkoEstKV49CYSYF/T4FfaZgwKeAz6cS//i6Jbb5fQr4TUGfT8GAKeDzKehPLD+f4XrSDfAlktZLCkl6QucH9aedc4czXFe63ivpMTN7rXPu68m2Nytx8+r7PKoJ86Ak4NNrr1umf/7pc5Kkf9t6hAAPAPCMc06jkZiGxxJBenw5Ej7/+XA4lmgbi2okGcqHx6LJYJ5YDyXXx0N7sQTroN9U4vepJOBTacCvkkBifbxtYpmyHpzYPxGWU9tK/BesB5KhO+XYoH98m51b9ydeK5g8LuAzBfxz+yv/51+fof+kpHQD/MOS3inpyeRESTnBOfecmV0v6SNmtlGJvwi0SbrOOXfQ2+qQbb937VL9n4cOKByL68njfdp1rFdXLa33uiwAQJ6IxOIaHotqMBTV0FjKIxTVcMrzxHoiXA+PRTWcDOTnrYejcnmes82ksoBfZUGfyoN+lQb9Kg34VBr0q+yCZWnAp7JgImgn1pP7JreX+M+1lVzQXhr0pbQn9/H75PPZ9EVCUvoB/h3OuV1ZrWSWnHNPS3qV13Vg/jVVler2Kxbpe4+fkCTds/UIAR4AikQ4GtdAKKKB0YgGQlENhiIaTFmea0+0pYbzgVBUQ2MRhSL50e/a7zNVBP0qL0k+kusVyfWy4Lm2ieclibBdXuJPBulzbee2J8J6aTCxLPH7ZEaIzgdpBfhcDe/AG29onwjw9z11Sh968aVqqSnzuCoAwHSccxoai6pvJKL+0UQQ7x+NaCCUWJ57jIfxxD6DoagGQrkZvksDPlWVBlRR6ldlSUAVJX5VliaXJQGVJ5+XB/2qLPWrvCSgitT1ZACvKPGroiQwEcoZpAEXmvU48EAu2NBWq83t9XrsSK+icaevbz+qP/2tNV6XBQBFwzmn4XBMvcNh9Y1E1DsSVu/IufXxgN43Ek4sRyPqH0ksYx737faZVFUaUHVZUJWlflWVBlRZGlB1WUCVJQFVlQUm2ipLA6pOhvFzbf6JbRVB/5z7SQPpIsAj773xhuV67EivJOmbjxzTH92ySmVBv8dVAUB+GovG1DMcVvdQWD3DiTDeMxxW73BYPSNh9Q5HzmvvG4koHJv/q+EBn6mmPKiaskQAry4LJB/BiWXNJG1VyYBelQzjdBlBPiLAI+/91roWLaot08n+kLqHw/rRkyf1O5uWeF0WAOSEWNypdySsrqExdQ8lll3JZc9QWN3DYXUPj6lnOKyeobAGx6LzVltFiV+15UHVlgdVk1xOPC8LqrY8oNqK8fVgMrAHVVOe6F5C+EaxIsAj7wX8Pr1uS7v+7oF9kqQvbzuiV21czBs7gIIVjzv1jUbUOTimrqHEfBidg2PqTFlPBPVEMM92T5WyoE/1FSWqqyhRfUUwuX5uWVdRorryoOoqkgE9uSwN8NdSYDYI8CgIr9m8RJ988DmFInE9fXJAjx3p1TXLG7wuCwBmJB536hkJ63R/SJ2DYzo7GNKZgdTlmDoHQjo7OJa1scEDPlN9ZYkaK0vUkPKor0guK0vUUFGi+srgRDvdFoH5RYBHQaivLNHLr2rTtx49Lkn68rbDBHgAOWUsGtOZ/jGd6h/V6YGQTveHdHogpDPJ9fGgHollPpjXlgfVVFWipqrS5KNEjVWlaqxKBPXGqlI1JEN7TVmQ8biBHEeAR8F4w/XtEwH+P58+o46+UbXVlXtcFYBiEI3FdXZwTB19ozrZN6pT/SGdGl/2h3Sqf1RdQ+GM/pvVZQE1V5equapUTcllc3XKIxnWGypLGIYQKDAEeBSMtQtrdP3KRm072K1Y3OlrDx/V+29b63VZAArAaDimjr4RHe8dVUdvIqSf7BtNBvbElfRMDYlYUxbQwtoytdSUqbm6VAuqy9RSk1guqClVS3WivbyEbitAsSLAo6DceX27th3sliR969Fjevetq/mQAzCtUCSmjr5RHesZ0YneUZ3oHV+OqqN3JCNXz30mLaguU2tdmRbWJAL6wtpfX+c9C8B0CPAoKLde2qIlDeU63jOq/tGIvvHIUb3lphVelwXAY845dQ2FdaxnRMd7RnQs5XG8Z0SnB0Jyc7yA3lRVqra6Mi2qK1drbblaaxNhvbW2TK215VpQXcpEPwAyggCPguL3md50w3J95Ed7JUmffHC/Xn5VmxqrSj2uDEC2xeNOZwfHdKR7WEe7h3WkeySx7Eosh8OxWb92wGdaVFeuxfXlaqsrV1t9uRbVJdfryrWwtoyRWADMGwI8Cs7vX7tMX334qA53DWswFNU//tez+ptXXO51WQAypG8krIOdwzrcNazDXUM63DWsQ53DOtI9rFBkdjOC+kxqrS3XkoZyLW2o0OL6Ci2uL59YttSUyc/ILAByBAEeBack4NNfvmSd3vjlxyRJ//7Ycf3eNct02eJajysDkK5ILK5jPSM6eHZIBzuHdeDs0ERY7x2JzOo1q0oDWtZYoWWNFVrSUKGlKY/W2nJGagGQNwjwKEi3rF2gW9Y062fPdso56e4fPa3vvm0Ls7MCOWYkHNXBs8Paf3ZQB84O6WBnIrAf7R6e1XjodRVBLWusVHtjxa8tGypLeA8AUBAI8ChYH37JOv3qwP8oEnPaebRXP3jipF52VZvXZQFFaTyoP3dmUM+dHdSBM0N67uygTvSOzvjm0bKgT8ubqrSiqVLLxx/NlVreWKn6ypLsfAEAkEMI8ChYK5qr9KYbluvz/3NIkvQ39z+jF6xrUWUp3/ZAtkRjcR3uGta+04N69vSgnj2TWB7vHZlxUG+tLdOqBVVa2Vyllc2VWtFcpRXNlWqpLmOmUABFjSSDgvaO56/Svbs61Dk4pjMDY/rMzw7oz17E5E5AJnQOjmnvqQE9c2pAz54e1L7Tgzp4dkjhWPo3kvp9pmWNFVq9oEqrF1Rr5YJKrWqu1vLmSlXxyzYATIp3RxS06rKg/vxFa/Xe/3hSkvSlXx7Wqzcv0bLGSo8rA/LH+FX1vacGkoF9UHtPDqhraCzt1xgP6pcsqNbqliqtbqnWJS1VWt5UqdIAwy8CwEwQ4FHwXnFVm762/aiePN6ncCyuj//kGX3x9Zu8LgvISWPRmJ47PaQ9J/u1p6Nfe04OaN+pAY1F07+q3lpbpjULq7VmYbXWLqzWmpYarVxAUAeATCHAo+D5fKaPvHS9XvaZrZKkn+49o/95rlPPu6TZ48oAb42GY9p7aiAR1JNhff+ZQUXj6XVWLw/6tba1Wpe21ujShdVas7BGa1qqVVsRzHLlAFDcCPAoClcuqdOrNi7Wd3eekCR95EdP64H3PE9BpjVHkQhH49p3ekC7T/TrqRP9evJEn/afHVIszbDeWluWCOqt1VrXWqtLW6u1rLGSyY0AwAMEeBSNP3vRGj2w57SGxqI62Dmsr2w7orfctMLrsoCMi8edDnYOadfxPu0+0afdJ/q179Rg2jeXtjdWaH1brTYsqtWGthqtX1SrBoZnBICcQYBH0VhQXaZ33bpKn7hvnyTpk/+9Xy+7qk1NVaUeVwbMTefgmJ443qcnjvfqieN92n28X4Nj0WmPM5NWNFXqsrZabUg+1i2qUU0ZXWAAIJcR4FFU7rx+uf790eM61DWswbGoPvGTZ/TPr77S67KAtEVice09OaDHj/Vq59FEYD/RO5rWsUsbKnT54trko07rF9WomrAOAHmHAI+iUhLw6cO3r9Mb73lMknTvrg5tbK/X71+7zOPKgMl1D43p8WN92nm0V48f69XuE30KRabvCtNUVaIrl9TpyiV1unxxnS5fXKu6CrrBAEAhIMCj6NyyZoFeflWbvr+rQ5L0Vz94Wpe0VGtze4PHlaHYOed0sHNYO470aMfRXu040qMj3SPTHlca8GlDW+1EYL9ySZ0W15fLjBtMAaAQEeBRlP7mFZdp/9lB7ekYUDTu9Pav79SP3nmjWmvLvS4NRSQcjWvPyX7tONKjx44kusT0DIenPW5xfbk2LqvX1UsTj7Wt1YyoBABFhACPolQW9Ovzr9ukl376V+oeDqtrKKw/+NpOfecPtqgsyGQzyI6RcFS7jvXpkcM9evRwt3Yd65t2gqQSv0+XLa7V1UvrJkL7gpqyeaoYAJCLCPAoWm115frM71+t137pEUXjTrtP9OuD339K//Q7V9D1ABkxEIpo55HeicC++0T/tJMk1VUEtWlZgza312tTe702tNUygykA4DwEeBS161Y06i9vX6e//MHTkqR7H+/QhkW1etONyz2uDPmofzSixw73aPuhbm0/3K29Jwc03TxJSxsqtKm9XpvbE6F9RVOVfEyOBAC4CAI8it7rrlumPR39+s6OxCytf33fM1qzsFo3rGryuDLkuoFQRDuO9Ojhg93afqhHT5/snzawX9JSpWuWN+ia5Y26pr1BC2vpDgMAmBkCPIqemeljL9ug584M6YnjfYrFnd7xzcf1w3fcqCUNFV6XhxwyEo7qsSO92nawS9sPduupjosHdp9J6xbV6Jr2xmRob2BGUwDAnBHgAUmlAb8+/7qNesmnf6XOwTH1jkR019d26ntv36KKEn5MitVYNKYnjvVp28FubTvYpSeO9ykSmzqx+0xav6hWW1Y26roVDdrU3sCspgCAjCOZAEktNWX619du1Gu+8LAiMadnTg3ofd/drU+/5ir6JBeJWNxpT0e/th7s0sMHu/XYkZ6LTppkJq1fVKPrljdqy8pGbWpvUG05gR0AkF0EeCDFxmX1+tgdG/T+e5+SJP1k9ylJ0j/9zhUML1mAnHM63DWsrQe69KsDidA+EIpe9Jg1LdW6flWjtqxo1LXLG1VbQWAHAMwvAjxwgddcs1RPnxzQ17YflZQI8Wf6Q/ri6zepnv7Lee/sQEhbD3Zp64FubT3QpVP9oYvu395YoS0rm3T9ykZdt6JRzdWl81QpAACTI8CaDz6nAAAPhElEQVQDk7j7pevlM+krDydC/I6jvXrl57bpnjdu1rLGSo+rw0wMjUX16OFu/XJ/l7Ye6NJzZ4Yuuv+C6lLdsCoR2K9f1aS2OmbnBQDkFgI8MAm/z3T3S9drSUOF/vq+Z+ScdKhrWK/47DZ96Q2bdNXSeq9LxBQisbh2n+ibCOy7jvVddPKk6tKArl3RqBtXNeqGVU1ataCKibwAADmNAA9Mwcz0lptWqK2uXO/59hMai8bVPRzW//ridn3yNVfphesXel0ilOjHfuDskH51IBHYtx/q0dDY1P3Yg37T1UvrdeOqJt2wukmXt9Uq4PfNY8UAAMwNAR6Yxm2XtWpBTZne8pXH1DsSUSgS19u+vlMffvE6Zmz1yOn+kLYmA/uvDnTp7ODYRfe/tLVGN65q1I2rm7W5vZ6hQQEAeY1PMSANG5fV694/vEFvvOdRHekekXPSR3+8Vyd6R/WhF18qP8NMZlX/aETbD3Vr24EubT3YrQNnL96Pva2ufOIK+/UrG9VUxY2nAIDCYc5NM+93Adu0aZPbsWOH12Ugj/QMh/WWrzymx4/1TbRdvbROd790vS5fXOdhZYVlNBzTjqM92nqgWw8f7Jp2xtOasoCuX9mkG1c36cZVTVrWWEE/dgBAzjCznc65TRl7PQI8AR4zE4rE9MfffkL37zk90WYmvXrTEr33hWu42jsL4WhcT57o0/aD3dp6sEuPH+1TODb1BEolfp82tdfrhlWJwL6hrZa/ggAAchYBPoMI8JiteNzpkw/u1+d+fvC8oFldFtAf/+Ylet2WZQpyY+SUwtHESDEPH+zW9sPd2nm0d9oZTy9rq9X1yfHYN7c3qLyEibUAAPmBAJ9BBHjM1eGuYX38x3v14L6z57WvXlClv7p9vW5c3eRRZbklFIlp94l+PXq4W9sP9WjH0Z6LBnZJWrWgKjEW+8ombVnBjKcAgPxFgM8gAjwy5Wf7zuqjP96rw13D57W/cH2LPnDbpWpvKq7Jn3qGw9pxpEc7jvZqx5EePdXRr0js4u81SxsqdN2KBm1JhvaWmrJ5qhYAgOwiwE/BzMqccxefE/0CBHhkUjga1z1bD+tTD+7XcDh23rZrlzfoFVe36bbLWlVTVlhXkqOxuA50Dmn38X7tPNqrx4726FDn8LTHjQf261Y06toVjcx4CgAoWAT4C5jZMkkflNTunHvhTI4lwCMbzg6E9LcP7NO9j3f82rbSgE+/ua5Fr7y6TTetbs67fvLxuNOR7mHtPtGffPTp6ZMDGo3Epj12RXOlNi2rJ7ADAIoOAT7JzHySflfSb0p6s6RfOOdunslrEOCRTTuP9upTD+7XL/d3TjoEYmNliW6/YpHuuHKRNrTV5lyY7x+NaP+ZQe0/O6T9Z4a07/SAnuro12Bo6llOxwX9psvaarW5vUEbl9Vr47J6NTI6DwCgSBHgL2BmQUlhEeCRo84OhPTDJ0/q3sc7tPfUwKT7lAR8WtNSrQ1tNVq3qFbrF9Xo0oU1WR9pJRSJ6XR/SCf7R3Wka0TPnRnUgbNDeu7M4LSzm6ZqrS3TZW21umJJnTa3N+jyxbUqCzJKDAAAEgF+UmbmRIBHHth3ekDff7xD/++JDp0ZuHhA9pm0srlK6xfVqLWuXDVlQdWUB5LLoGrKAsllUCV+n0LRmEKRmEYjMYUicY2GY4m2cEzD4ZjODIR0qn80Edj7Qjo9EFLPcHjGX0NDZYkuX1yryxfX6YrFtbpsca0WVHPDKQAAUyHAT4IAj3wTizs9fLBb39/Voe2HutXRN+p1Sb+mJODTyuYqrV5QpUtaqrRqQeIvBG115cxyCgDADGQ6wAcy9UL5wszuknSXJC1dutTjalCs/D7TjaubJsaJ7x0Oa++pAT19sl97OhLLQ13Dyvbv136faWFNmRbWlmlxfbkuaanWqgVVuqSlWkvqyxXIsX75AACAK/BcgUfOGh6Lat/pAe07Pai+kYgGRiMaCEU0MBpNLiMaCEU1MBpROBpXWYlfZUGfyoN+laU8ypNtzdWlaq0tV2ttmVrrEsumqlL5fVxNBwAgmwryCryZ/VzSb6S5+1udc1/KYjlATqgsDWjjsgZtXNbgdSkAACCH5ESAl/R6SRVp7nsqm4UAAAAAuSwnArxz7pjXNQAAAAD5gDvUAAAAgDyS9wHezKqSq0FPCwEAAADmQV4HeDN7uaR7kk83m9kHzOw6L2sCAAAAsikn+sDPlnPu+5K+73UdAAAAwHzJ6yvwAAAAQLEhwAMAAAB5hAAPAAAA5BECPAAAAJBHCPAAAABAHiHAAwAAAHmEAA8AAADkEQI8AAAAkEcI8AAAAEAeIcADAAAAeYQADwAAAOQRAjwAAACQRwjwAAAAQB4hwAMAAAB5hAAPAAAA5BECPAAAAJBHCPAAAABAHiHAAwAAAHmEAA8AAADkEQI8AAAAkEcI8AAAAEAeIcADAAAAeYQADwAAAOQRc855XYNnzGxQ0rNe14FZaZLU5XURmDXOX37j/OUvzl1+4/zlrzXOuepMvVggUy+Up551zm3yugjMnJnt4NzlL85ffuP85S/OXX7j/OUvM9uRydejCw0AAACQRwjwAAAAQB4p9gD/Ba8LwKxx7vIb5y+/cf7yF+cuv3H+8ldGz11R38QKAAAA5Jtiv4kVOczMypxzIa/rwOxw/vIX5y6/zeb8mZlJerGkF0jqkPS0c+4n2agPF8fPH9JRkF1ozGypmd1jZh8zs0+Z2XfMbFkaxz3fzL5qZp8ws8+a2U/M7Lr5qBnnmFm1mX1A0okZHDOrc47Mm+X5u9PMdptZyMyOmtkXzKwpi2ViErM5dxccHzSzHWZ2Z2YrQzpme/7MbIOkxyW9XdI/Ouf+nvA+/2b53kluyQGz/QybS3YpuCvwyS/8YUkfdM59Odn2NknbzGyTc+7UFMfdLumrSozTeTbZ9jxJ/21mVznn9s/LF1DkzOxmSTdL+jNJ5WkeM6tzjsyb5fl7i6R3S/qepBJJd0h6q6SbzexK59xIVorFeWZz7iZxt6SNmakIMzHb82dmN0n6saR7Jb3J0a/WE7N87yS35IDZfobNNbsUXB94M/uBEh8gS8bfiMzMr8RvtA865157keMWOec2X9DeJeljzrlPZrdypDKzrZKud85ZGvvO6pwje9I9f2ZWLulHkl7snBtLtgUl/UzSDZLe4Jz7arbrxTkz+dm74LgbJP2ppJdLeuP4BxLm1wzfOxdJelLSWUlXOefC2a4PFzeLzz5yi4fm8hk21+xSUF1okm9Gt0t6KPUqgnMupsR/5u+aWeMUh4clrTeztpTXq5RULel09qrGFCLp7DTHc47sSev8Sdoi6UPjb3yS5JyLSPpc8unCTBeGaaV77iaYWbWkD0j6YObLwQzN5Pz9lRIze36c8J4zZnL+yC3em9VnWCayS0EFeCX+I03Ss5Ns2ycpKOn6KY79NyX+bHW/mS1L3tDzD0r8afG7WagVmTGXcw6POececs49MsmmvuRy73zWg1n7R0kflsSNd3nCzEolvVaSk7TUzLab2YiZHTSzP09+BiK3kVs8NofPsDlnl0IL8EuTy65JtnUmlysnO9A5d7+kd0haL2mPpK2StjvnXpn8jQi5adbnHDntMknHJT3gdSG4ODN7haTDzrldXteCGdkkqULSgKRtzrnrJC2S9EtJfyvp4x7WhjSQW3LadJ9hc84uhRbgx2/8mOxPgeN/3qia6mDn3GckfVGJm3u3SPojM1s61f7ICXM658g9ZhaQ9AZJb3PORb2uB1Mzs1ZJr5P0917XghlblFx+yTn3S0lyzvVJukuJAPEnya5RyGHkltyT5mfYnLNLoQX48d9kgpNsG28bnuxAMwuY2b9JGlHiN6PvSLpGibuB2yY7Bjlh1uccOet9kr7hnLvP60IwteSf6z8p6Y+dc3Gv68GMjV+hPZnamOwL/1+SyiRdMd9FIX3klpyVzmfYnLNLoQX48SF3Juv4Pz4e54Epjv2ApOdJeq9zrtM592olhnNqU6JvJ3LTXM45coyZ3SGpyTnHn+9z3x9JelDSaTMrM7MySaXJbcFkW8ENVVxAjieXk41VPR7qK+apFswOuSXHzOAzbM7ZpdAC/HZJcUmrJ9m2RokrDlunOPZVknakXklyzv2DpCeUGAoIuWku5xw5xMxulXSjc+5Pva4FaXmVpH+VNJry2Jfc9oXk87/wpjSkYY8S/d+XT7Jt/Arg8Um2IXeQW3LIDD/D5pxdCirAO+c6Jf1E0i2p7Wbmk/R8ST90zvUk2y7sIzYmacEkL3tA0mDmq8VspZ67mZxz5IbJ+mcmJzH5befc+y5orzezWy7cH9644Nz9oRJ9blMfr0hu+3jy+ZfmtUBc1AXvnaOSviXpNjMruWDXtZKe07lfyJADyC25K53PsExnl4IK8EnvldRgZqkD4L9ZiZsB3idJZvbnko6aWep/9P+WdJOZXTvekJwG9yZJn8p61bhQlTQxIcKEKc7dtOcc8y7t85ecCfIeSX1m9hcpj49KekjSU/NYN9I8d865vc657akPSeMj0RxMtqU9JTwyZibvnR9TYtzx1J/HDZJ+Q9IfMiurJ2Zy/sgtOSCdz7BsZJeC65/onHvOzK6X9BEz26jEOJttkq5zzh1M7nZa0pBSJjpwzn3TzLolfczMjknqVeKmkDudcwxlN0/MbIukF0u6Otn0WTP7sXPuB8nnk527dM455sFMz1/yfN2nxBvWRyd5yW845yYbZgsZNpufPeSOWb53diSP+zsz+7akHkktkl7knPuf+aseszx/5BaPpfsZZmYZzy7GL9gAAABA/ijELjQAAABAwSLAAwAAAHmEAA8AAADkEQI8AAAAkEcI8AAAAEAeIcADAAAAeYQADwAAAOQRAjwAAACQRwjwAAAAQB4hwAMAAAB5hAAPAAAA5BECPABgWmZ2pZn9i5ntMrMPmFmtmd1rZmfMrNXr+gCgmBDgAQDTcs49Ielbkq6U9JikuyV1SKqXVOJdZQBQfAjwAIB0rZY0JmmTpE87594pqcU5d9TbsgCguBDgAQDpukXSAUlh59whSXLO9XpbEgAUH3POeV0DACAPmNlRSeWSVjrnBr2uBwCKFVfgAQDTMrNLJC2V9FXCOwB4iwAPAEjHC5LLr3taBQCAAA8ASMsLJB1OjkYDAPAQAR4AcFFmFlDiBtYfe10LAICbWAEAAIC8whV4AAAAII8Q4AEAAIA8QoAHAAAA8ggBHgAAAMgjBHgAAAAgjxDgAQAAgDxCgAcAAADyCAEeAAAAyCMEeAAAACCPEOABAACAPPL/AR8M/umAFq6GAAAAAElFTkSuQmCC\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"def plotV(r, eps, sig, xlim, ylim):\n",
" \"\"\" Function for plotting the Lennard-Jones potantial with given parameter values. \"\"\"\n",
"\n",
" V_LJ = 4*eps*((sig/r)**12-(sig/r)**6)\n",
"\n",
" plt.plot(r, V_LJ)\n",
" plt.ylim(ylim)\n",
" plt.xlim(xlim)\n",
" plt.ylabel(r\"$V_{LJ}(r)$\")\n",
" plt.xlabel(r\"$r$\")\n",
" \n",
"\n",
"xlim = [0.8, 2]\n",
"ylim = [-1.5, 2]\n",
"r = np.linspace(0.5, 2, 100)\n",
"eps = 1\n",
"sig = 1\n",
"plotV(r, eps, sig, xlim, ylim)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"As we can see, we find a global minimum and expect oscillations about this equilibrium as long as the total energy, $E_\\mathrm{total} = E_\\mathrm{kin} + E_\\mathrm{pot}$ , does not exceed zero."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Parameters for our Particular Example\n",
"In this example, we will consider the Lennard-Jones Potential for two H2O molecules. The parameters are:\n",
"- $\\sigma=0.326 \\times 10^{-9}$ m\n",
"- $\\epsilon=1.08\\times 10^{-21}$ J\n",
"\n",
"Note that the orientation-dependent interaction between the water dipoles is not included here explicitly. We obtain the following plot:"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"r = np.linspace(3e-10, 7e-10, 100)\n",
"eps = 1.08e-21\n",
"sig = 0.32e-9\n",
"ylim = [-1.5e-21, 1.5e-21]\n",
"xlim = [2.5e-10, 7e-10]\n",
"\n",
"plotV(r, eps, sig, xlim, ylim)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### The Problem\n",
"We will now analyze oscillations about the equilibrium in a classical manner.\n",
"- First, we want to examine the frequency of small oscillations near $r = r_m$.\n",
"- To do this, we approximate the force near $r = r_m$.\n",
"- Thereafter, we want to compute the period of oscillations for larger perturbations when the linearization is a poor approximation."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Deriving the Force from the Potential\n",
"Let us calculate the force from the potential by taking the derivative of the potential:\n",
"\n",
"\\begin{equation}\n",
"F = -\\frac{\\mathrm{d}V}{\\mathrm{d}r} = \\frac{12\\epsilon}{r_m}\\left[\\left(\\frac{r_m}{r}\\right)^{13} - \\left(\\frac{r_m}{r}\\right)^7\\right].\n",
"\\label{force}\n",
"\\end{equation}\n",
"\n",
"Note that $F(r_m)=0$, hence $r=r_m$ is an equilibrium point. Now we Taylor expand the force around this point."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We write\n",
"$$F(r_m + \\Delta r) = F(r_m) + F'(r_m)\\Delta r + \\mathcal{O}(\\Delta r^2) = 0 + F'(r_m)\\Delta r + \\mathcal{O}(\\Delta r^2).$$\n",
"\n",
"Now, we need to determine $F'(r)=\\frac{\\mathrm{d}F}{\\mathrm{d}r}$:\n",
"\n",
"\\begin{equation}\n",
"F'(r) = \\frac{12\\epsilon}{r_m^2}\\left[-13\\left(\\frac{r_m}{r}\\right)^{14} + 7\\left(\\frac{r_m}{r}\\right)^8\\right].\n",
"\\label{forcederivative}\n",
"\\end{equation}\n",
"\n",
"Substituting $r=r_m$ yields\n",
"$$F'(r_m) = -\\frac{72\\epsilon}{r_m^2}.$$\n",
"Using this result, we find the force near $r_m$,\n",
"$$F(r_m + \\Delta r) \\approx -\\frac{72\\epsilon}{r_m^2}\\Delta r.$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### The Frequency $f$ of Small Oscillations\n",
"We can now compute the frequency of small oscillations about the equilibrium by writing\n",
"$$m\\frac{\\mathrm{d^2}\\Delta r}{\\mathrm{d}t^2}= F(r_m + \\Delta r) \\approx -\\frac{72\\epsilon}{r_m^2}\\Delta r,$$\n",
"where $m$ is the effective mass of the water-water molecule system. This describes harmonic oscillations with a frequency\n",
"$$\\omega^2 = (2\\pi f)^2 = \\frac{72\\epsilon}{mr_m^2}$$\n",
"We use the values for $\\epsilon$ and $r_m = 2^{1/6}\\sigma$ from above, and $m\\approx 9m_\\mathrm{proton}$ to calculate the frequency:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"f = 1.0079e+12\n"
]
}
],
"source": [
"m = 9*1.67e-27\n",
"r_m = 2**(1/6)*sig\n",
"\n",
"f = np.sqrt(72*eps/(m*r_m**2))/(2*np.pi)\n",
"print(\"f = %.4e\" % (f))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The result is\n",
"$$f = \\frac{\\omega}{2\\pi} = \\frac{1}{2\\pi}\\sqrt{\\frac{72\\epsilon}{mr_m^2}}=1.01 \\times 10^{12}\\,\\mathrm{Hz}.$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Expanding the Problem\n",
"So far, we have examined the frequency near $r = r_m$ where the dynamics can be approximated by that of a harmonic oscillator.\n",
"This was simple and we did not need any computational tools.\n",
"However, what happens if we look at oscillations further away from $r = r_m$?\n",
"If we look at our plot of the L-J potential, we can see that the graph is not symmetric about the equilibrium. During oscillations, we would expect the molecules to spend more time to the right of the equilibrium $(r > r_m)$ then to the left of it. Let us study this phenomenon numerically."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Nonlinear Oscillations\n",
"Now, we will solve the exact dynamic equation\n",
"$$m\\ddot{r}=F(r),$$\n",
"where the dots indicate time derivatives.\n",
"\n",
"We transform this ODE into a set of first-order ODEs and employ Euler’s method to solve it. We do this by introducing new variables $v$ and $w$, with $v = r$ and $w = \\dot{r}$. Substitution into the equation \\eqref{force} gives\n",
"$$\\dot{w} = \\ddot{r} = \\frac{F(r)}{m} = \\frac{12\\epsilon}{mr_m}\\left[\\left(\\frac{r_m}{v}\\right)^{13} - \\left(\\frac{r_m}{v}\\right)^7\\right]$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Euler's Method: Set of Equations\n",
"This gives the following set of equations\n",
"\n",
"\\begin{align}\n",
"\\dot{v} &= w,\\\\\n",
"\\dot{w} &= \\frac{12\\epsilon}{mr_m}\\left[\\left(\\frac{r_m}{v}\\right)^{13} - \\left(\\frac{r_m}{v}\\right)^7\\right].\n",
"\\end{align}\n",
"\n",
"For our initial conditions, we choose $v(0) = r_m$ and $w(0)>0$. Following the notation from the module [Euler's Method](https://nbviewer.jupyter.org/urls/www.numfys.net/media/notebooks/eulers_method.ipynb), we can now write the Euler method as (with $v_1 = v(0), w_1 = w(0)$)\n",
"\n",
"$$\n",
"\\begin{align}\n",
"v_{n+1} &= v_n + h\\cdot w_n,\\\\\n",
"w_{n+1} &= w_n + h\\cdot \\frac{12\\epsilon}{mr_m}\\left[\\left(\\frac{r_m}{v_n}\\right)^{13} - \\left(\\frac{r_m}{v_n}\\right)^7\\right],\n",
"\\end{align}\n",
"$$\n",
"\n",
"where $h$ is the time step.\n",
"\n",
"This can be implemented as follows:"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [],
"source": [
"def euler_oscil(w0, h, n_tot):\n",
" \"\"\" A function which uses euler's method to calculate oscillations and then plots it.\n",
" \n",
" Arguments:\n",
" w initial value of w\n",
" h time step\n",
" n_tot total number of time steps\n",
" \"\"\"\n",
" v = np.zeros(n_tot)\n",
" w = np.zeros(n_tot)\n",
"\n",
" v[0] = r_m\n",
" w[0] = w0\n",
"\n",
" for n in range(n_tot-1):\n",
" v[n+1] = v[n] + h*w[n]\n",
" w[n+1] = w[n] + h*12*eps/(r_m*m)* ( (r_m/v[n])**13 - (r_m/v[n])**7 )\n",
"\n",
" plt.plot(v)\n",
" plt.plot([0, n_tot], [r_m, r_m],'--')\n",
" plt.legend([r\"$v(t)$\", \"$r_m$\"])\n",
" plt.ylabel(r\"Position of Molecule, $v$\")\n",
" plt.xlabel(r\"Number of Time Steps\")\n",
" plt.grid();"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Oscillations near $r=r_m$\n",
"We choose step size $h=10^{-15}\\,\\mathrm{s}$, and set $\\frac{1}{2}m\\omega^2 = 0.001\\epsilon$. This corresponds to very small oscillations because the kinetic energy is much smaller than $\\epsilon$\n",
"and, therefore, we are deep down in the potential well where the oscillations are harmonic. We plot the vector $v$ which represents the position of the molecule vs. time steps."
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
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\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"w0 = np.sqrt(2*0.001*eps/m) # Initial value for w\n",
"h = 1e-15 # Time step\n",
"n_tot = 1100 # Total number of time steps\n",
"\n",
"euler_oscil(w0, h, n_tot)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We find the following features:\n",
"- The oscillations around the equilibrium $r = r_m$ seem harmonic.\n",
"\n",
"- One period seems to be approximately 1000 time steps, each of length $10^{-15}$, i.e. we have $T\\approx 1000\\cdot 10^{-15} = 10^{-12}$, in good agreement with the result from the linear analysis."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Oscillations further up the Potential Well\n",
"Now let’s see what the oscillations look like when we add kinetic energy. We set $\\frac{1}{2}m\\omega^2 = 0.3\\epsilon$ and $n_\\mathrm{tot} = 1270$. We have to increase the number of time steps $n_\\mathrm{tot}$ because we expect the period to be longer."
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"w0 = np.sqrt(2*0.3*eps/m) # Initial value for w\n",
"h = 1e-15 # Time step\n",
"n_tot = 1270 # Total number of time steps\n",
"\n",
"euler_oscil(w0, h, n_tot)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"- We can clearly see how the molecule spends more time on the right side $(r > r_m)$ of the potential well which we expected from the L-J plot.\n",
"- In this case, the period is about $T=1270\\cdot10^{−15}$ s. Since $T$ increases with the amplitude, we no longer have harmonic oscillations.\n",
"- Note that we treat this microscopic system classically and not quantum mechanically. However, it gives us some ideas about how a full quantum mechanical treatment would differ from that of a quantum mechanical harmonic oscillator."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Repeating Oscillations\n",
"So far, we have only looked at oscillations for one period. If we increase the number of oscillations, we would expect neither diminishing nor increasing amplitudes since energy is conserved. Unfortunately, Euler’s method contains errors which can be large enough to clearly violate energy conservation in our solution. Hence, the amplitude may not be constant."
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"w0 = np.sqrt(2*0.3*eps/m) # Initial value for w\n",
"h = 1e-15 # Time step\n",
"n_tot = 10000 # Total number of time steps\n",
"\n",
"euler_oscil(w0, h, n_tot)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"As the number of oscillations increases, the molecules move further away from equilibrium with each oscillation which would correspond to an increase in energy."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Error in the Euler Method\n",
"Remember that this method is an approximation of the solution. Each time we calculate a new step, we make an error. This error results in an increase in the amplitude of each new oscillation. The error can be reduced by using better but more\n",
"complicated approximations, or by reducing the step size $h$.\n",
"\n",
"In the next plot,we have reduced the step size to $h = 10^{−16}$ s. We have to increase the number of calculations to $n_\\mathrm{tot} = 100 000$ so as to obtain the same amount of oscillations. This demands more computational time but reduces the error."
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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9D6yYybREIszRgTAPbDhQ8DbqxN6jg6zZbsWqCgFf+quTl8t62+KpzG+qAqxEtIkWe/6HjW2pCcmM+nLeuazlpMefNrWavzo97T3/uSLsJwL3vbg/lbx+6cJGTp9Wc9LjywJ+PqXkg9z34sTqv36pJIFedeY0ptSc2Auc5DOXpMeD1duPcLh34ohNtf+5cslUmrKw16waP8sTIaDRuOTBE1SnKkYeWH+AaCKZf8XsOk6bWp3V5z50frqi0gMbDkxIJ1a+yVWcXw1cIKX8O+DLwP8TQlyVv2YZskVKyd3PppftPnXx3JPu1hUs8fE5xdty7wv7JlRljZf3d7PtsOWdLAv4+ORFc056fLDEZ5vs3Lt234SK21TFwFsWNTGvseqkx/t8gs8qYuD+9a3EJ8j9JaXkF+v2pf7+cJZbW6vi6Q8b2xgcmRirWaFIjP9TJrv/74LsNst53/IZqdWI1w/1Tpi44L5QhEc2plcKsrXXnIbKVEGAWFzy4ARxMIQiMR5RqktlJs2ejA8q5TvvX986Ifp8KSUPvZyeiIxl86pLFzalnH7dQ5GUQ8eQOzmJcynlC1LKvsTvA1LK7wJDQohP57V1hlOyfl83uzutGK/KoD+rDujac1qYlEisOtg9PKGy+H+hiM13Lm2htuLUuw2+f8UMKhMTnt2dg8fVRC9WhsMxHnxZFU+zs/rc1Wc3p2L1D3QN23bGLGbeaOtL5X2Ulvh43/LsNhdaMbuOeY1WGNrASNSWHFnMrNp2JBWbP72ufNQKGqMxqSLIO5RNdSbKMvrjb3QQilhJ2YumVnNOIoE9G65TxOaDLx+cEGJz9bYjqQpAcxoqOW9O9hvwXXXmNGrKrPysA13DE6IKyRttfew7ZtXDryot4colU7P+rN8nuPac9Crho5vaTnK0IRvymRC6FtgjhPh2Hr/TcAruVwamv1nWkkr4PBllAb9NOEyU0IPeoQh/3pyOp/tIlp6nytISrlma7ngmihh4apu1iRXA7MkVvHl+dlvNlwX8vEsJ57h//cSw1yOvpb2aV5057aThUipCCD6wQhVPE8Oz+YeN6QH83edMH9P23x9QdlX94+vtRGLFv1GYaq93LmsZtULLiXjb4ilUJ8aG1q6h1CSymPmDIhCvPrt5TPYqC/ht4vTRCTBhVv+PV5wxhbLAiVfgR+NqpWrLU1uPTMh8tnySa0LoW4QQXxdCPCSE2CyE6AWGgceBrwohfpLXVhpGpS8U4bHX0w/UdcoAfyrUXfee3tE5Icre/fmNdsKJQfys6bUsbh49sXE0rnuTXQxMhI7n96/ZxYBvLOJJsdeTWzuK3l5xKW2D2zVLm09y9PG865yWVL3gDfu76SjyXVb7QxGe2pbeGVUd2LNhxex6pibih7uHIrxQ5KszXYNhntt1NPW3unKQDaUlfq5YnM5tePT14vZsDoxEWaXeX2eNPS3ur89K35OPbWov6vA8KSWPKffEWO8vsCrdLJxihT0OR2I8sWVi7w0yXnL1nN8L3Ay0AK8CPwA+C3wI+Bvg/ry0znBSntzSYas9feb07MXmrMmVqbrn0bjkj68Xv2fg9xvtu1uOhTNbalmQSHQMReKs3lbcoUC9w/a4wbHaa9HUGk6bYiUThSJx20BZjOzsjtOeSLSrqwhkvcqQpKGqlPMTccFSwp+K/Hl8YksHYaXvSiYRZ4vPJ7hKERDF3n/9aXN7KjfonJmTmFF/4vKJJ+KvFYH62Kb2og5tWbXtSCoE6LQp1SyYkl1io8oF8yan6uof7gvxSmvxhjO+0dbHga5hAKrLSrh44dj6ryTqhMaI8/GRqzh/GJgipbxASvkRKeU3pZQ/lVLeL6X8g5RyTR7baDgB6oA0Vs8T2L0J6pJpMXKkP5Tyrglh70SyQYiJJQYef+NwapVhSUsNc0+RCDoaE8leLx1Orwxcdea0UevAnwrVXo8Vub3+pISX5dJ3AbzjrHTYwZ/fOFzUoS1PKkLnHWPsu5K8eX4j1Yk46oPdw0Ud2rJKKc95VQ5eYICA38fbFqfvsWJ2MKxW/m9vWdREacnYQlqSqPs2PLOjk2gRP5OFJldxfncyIdTgDv2hCM/sSC9zXnVm9skbSd5xVnNqKf2lfV1FvZT+l82HSa5Krphdz9TaU5fUyuQdymTmqW3FHaqhej2uzlEMqPfkqm3FG4MopeS1I+nSYbksCYNV6k0NbSnW+soj0RjPKyEab1PCLcbCshl1qdCWnqFIatfMYiMUidmSqq84PTd7BUt8vGVROul2zfbiFJuxuOSZnen76y2nZ5doPBqXK/ZaXcQVSFYr90K2idmjsWhqdar0cF8omqrJbxg7uVZr2ZzvhhjGxlNbj6Q8m4uba065i9doNFbbl9KL2TOg/t+uGkMWusrCKdW20BZ1clRMjERjtvjWvzojNzGwIMNea3cVZ1zwriMDHAtZM7/q0hJWjKEqhEpDVWlqd0wpKdr766W9XQyFrcnM7MkVOa3KgBXacplNPBVn//XCnmOp8MV5jZXMnDz2kJYkqr3WFGmVrk0He1I5VA1VpZxxitr5J+Oi+Q0E/NaMeWt7X1HWiO8ZCqeq0QgBlyxszPm7hBA2cV/MmqLQ5LNai8FBnsjDsh3AWxQvTLE+SJmep8sX5SY2wb5s9/SO4rSXKp5mTa5g7ii7zWbL5aerYqA47aU+NxcvbMgppCWJOrAVq2dTzdcYj5fO+nxaSBRrbWU15OCycdrr4gWNqa3WX23tpmeo+AoBqB7ulac1jimRPZPK0hLOmzM59XcxPpPP7DyaWlU+e/ok6rOsMnUibKsNRaopnMCIcw8Si0ueK8Cy3XM7jxblzl4v7M6f58kunjqLMqlqVYYYGEsJskxWLpxY9sqn2Hx259GijNlUPdyqJzcXMj2bxRaaJ6W03V+Xj9Ne9ZVBls6YBEBcYgv/KBae3p6/yQzYn8liXJ1ZYwtpyd1rnuSi+ZNTz+T2jn46+0fG/Z0TESPOPcjGgz2pLcKn1JSmqmLkwpyGypRndDgSY92e4gs9UDvU8Q5u58yclEqqau8NsaNjYFzfpyOr8ygGzp1Vl9rA6WD3MHuODo7r+3SjLxSxbUo13sFt0dRqptSUAlbFnI0Hi2v3y/3HBtmbuAfKA/4xbQwzGlWlJalQIICni8x7vv/YEAe7rSoalUE/y2ePz15gnzAXm716hyJsSuwY6xPw5gW5VR1RUZ/pdXu6iqqkopTSVob00nGEtCSpCJawbEZ6g6xi1BROYMS5B1F39LxkQeO4PJtgF2DFuDS8Oo+elBK/j4uVDr/YQlsOdA2ldomrCPo5b+74xECwxMdFSlnBYru/XtrTlSpxt6SlhqbqsScaqwghbAPk00UWF6wO1OfNrR/zRiejodpr7e7i8gS/uDdtrzfNqSdYMv4hWxWs6vcXA+v3dZFcnFvSUpvaqXg8zGusoqEqPWHedrh/3N+pCwe7h1MlYCuDfs5syb4c88k4Xxk3jDjPDSPOPYg6YI8neSPJpYpnoNg28zjQNZSq31pRAM9TsSXtqR3pitn1OZfUUlHvL7VKRzGg2uvCeeP30oH9mS62gW3dnq7U7xfMnXySI7Pngnnp71m3p6uoQqdUe52fJ3udNb2W8kB6Netg91BevlcH1MlGvuwlhChasan+X86dXU/JOPJlVM5XnskXisheTpJ3cS6E+JQQYqMQ4l4hxFk5fkdACLFBCPGxLI6dKYT4HyHEbUKIHwohHhBCZLcvuwfpHYqwMZFZ7ROMebOT0Vg+q94WI3Z0oHhixNTOZ/ns/HueXt7fXVT1lQshBlQRtn5v2tNcDKyziYHxT/zA8pAmee1AT9HkgUgpMzzn+bm/zphWk9qa/nBfiP3HikNsSil5cU/+xWbA72P57HTYwYvKM+91Xtyb/r+MN2RKRbV9MYnzlwpkr3Nm1qXG2j2dg0WXC+IEeRXnQggfcCewGPgE8A4hxNk5fNWtwLlZnG8WsA54Wkp5i5Ty74FVwFohRO4lTDRm/b6uVGb1mS211I0zsxqgPOhPJQlBcXU+amedL/HUPKmcGfXlgBWnv6lI4oIzxVO+7DWnoZLGamtZuH8kytb24tgioXc4whtt1v9FQF5WZQCaqsuY22jlgYSj8VSZM69zoMu+hL6kOfcSdyolmWKzSEI1DnQN05awV1VpCYvzZC+wi81isVd/KMLmRLy5EPl7HiHTXsUTd16I8RGgLOBnWZFqCqfIqziXUsaxdg/9jZQyJqX8NlA6lu8QQlwEnJ7l4T8E4sC9ymt3Y/2/vjeW83qF9fvSD9Ob8jjTVb2bxRTasq4AnifAVl6rWDqeg93DHOpJJ5/lK/5QCGHzyqgDgpdZvzcd3zq7xkdN2fjjW5PY7FUkns3MVax8LaGD3Qu/rljstVe1V11+7TVHDdMoDntt2N+dclydMa0mL/HmSeY1Vtrizrce9r6Dob13mNYua5WpLODjzJZJp/jE2FDH25eVpHlDduQ9rEVK+UEp5fuUv1/K9rNCiGrga8BNWRzbDFwNrJJKkKGUMgasBt4nhMifGtMEVdisyKdnYF7xifMDXelKBxV5FJtAUYpNNTZwxZzCiacXi2Qyo4rN0+rHH5uvok7+isWzWaiJcub3vbjnWFHEnauTMvV+yAdnTZ9EWcB6vlu7hmjvHc7r97tBIe2V6WAohp0v1ZAWNQwlX5wzK72a9UqrEedjZdxXIxHKghCiRgixdJxf9x/ALUA2AUoXYK0mbx/lvW1AALhwnO3RiqFwNLVsB/kV57YYsaODHCmCGDFVNJ87q25cm8NkYvMK7OsqinrULxYg3jz1fcrA9tK+4lgWVlexFtXnd2BTq+S80tpNOOr9+2v9fkU85XEJHWBJc02qZGdbbygVDuJlVEGTz1VSsKooqeXuXi0CsfmqzV51JzkyN5bNTHuWXy0Csale83yGACVRQ2W3tvczHC6O3BmnyHlEEULcKITYD7wEIKXsA1YKIb6b4/ddC+yVUr6a5UdmJv4drfxDspzJvFzaoiuvtfYQTYiahVOq8hJvnqQs4Gfp9PTDVAyegfV7Cyc2p9eV01xrlc0bDMdSscdeRh1wVszO7+A2v6mKyYn7tWcowq5Ob9eHD0VibFFi5+dPyq/nfFptOq8hFInbzuVFjg2MpKomBUt8LGnO3yoWWHHnZyn912se7796hsKpevABv8hrvHmSpYrY9HpeQywueV1xXC2dUQhxnv5Or99fYO2XkkSND88XteUB5jdVAcdfH8OpKcnlQ0KIrwDfSfyZKvQspfyBEOJ3QohPSil/NobvmwZ8BHj3GJpRnvh3tP2Hk+VGqk5wvs8AnwGYMmUKa9asGcNp88PAwMCYz/vbnen/akswlPd2N4j09z/y/CbKjm7L6/c7zXPb0lUbSrr3s2bNwbx+/8yKCG2J/uaBVevpnnV8jGMu19kNBsKSPUcte/kFHNu1kTV7x1c/P5OZlTGOJfYguv+JdVwyPX8xoU6zqztGJGZNlKdWCAgP5v06t5SGOZD4/cFV6+kZ5f7yCq8diaZ+n1EJa597Ju/nmKwMBb9fu4nKrtEWVXPHyWd5U2faXtOrBOuefzbv5wj0ps+x5vV9XFjRkfdzOMWB/jhDCc9sXalg26vryHX0OtF1DsckfgExaa0uP/r4aqqC+e0jnSIal7x+MD0+9re+wZrDW/J+nmmBEXYlfn9ozQaG9ufPoThedB+bcxLnwN8CXwR+Afwg470/AZ8HshLnwtpB53bghkRCabYkPeajjVjJ10bdjlBKeRdwF8Dy5cvlypUrx3Da/LBmzRrGet67dq4DrLjNd160hJVLW/LappHGw/xx78sAHJVVrFzp3aig/lCEtr88DoDfJ/jIX19KRTDX23109gb2su4PVoc2WNrAypXLjjsml+vsBtYWzusBa/OOt77lzXk/xxtyF6/+xRJMwxVTWbnyzLyfwyl2PbsH2ArAhac1U1XVk/frvKdkL+sete6vobJGVq4cb9Sge7z8+HZIDNOXLJnJypWL836OcONhHt1j9V/HZHXe+y8nn+XXntwB7ATgkjNmsHLlkryf44z+ED989SkAWgfgoosvyWvon5Pc/1Ir8DoA582fwsqVpyz2dkJOdp2XbH0utWtg0+xjAAAgAElEQVRv1azFrBznpnZuselgD9HHnwdgZn0F17z1soKcp72ilWd/Y12X3pJ6Vq5cXpDz5ILuY3OuT+JeKeUdUspeIDN4dBKwYAzf9TngKeCwEKJMCFFGusJLIPHaaKqqPfHvaPEKyULUu0Z5z5PE49JWsu/cWYWNqdt0sNfTca6vH+xNVdJYOKU678Ic7DF1r3p8WViNP1SXb/OJai+vL6Or7VfDA/JJMYUd2OxVgCV0sNvr9UO9nt5/wIn7q6m6jJZJ6dCp7R7e+VK119kFur/A3jd6OU5/o2P2sofKFkOitlPkKs7VuzK1riOECALXAWMJkHwP8BNgWPlJrkjdlfj75lE+tw6rjOJoE4HTgBjw/BjaoTV7jg4wMGItQzZUBVOdaj5pqi5jep31vSPRuKfrUb/qgBg4o7mGYMLTtP/YEF2Do0VYeQMnxNNZ02sRid5i++E+hsLRk39AY1R7LStAfCtY5eCSm4PtPTpIz5A37694XNrsdU6BJn+q2ByJxtnW7k2xKaW0iadCxE8nsSU5engC6ET/BcVkr7Sj7+zp+c3/UFnQVE1VYoOwzv6R1D4HhlOTqzj/oxDiV0KI0yG1o+cy4PfAmcD/jeG7rseqvKL+XJt4718Sf/80cZ5kEihSyk7gMcC2HpOoHnM58HspZXHUuAM22h6mSQhRmFg3deD0cvkju3gqTGddWuLndCVRa6NHO2sppSODW3VZgPmNVhpIXFqrG16ks38kVaKztMTHomnVBTlPWcDPGdPS95dXved7jg7QH7ImYpMrgykHQCFYZltt8Gb/tf/YEN1DEQAmVQSYPbmiYOeyrf55tL8fHImyo8OaiPmE5QQoFOpEfOMB73qC1WdjWYFWZsAKKVWTmYuhcIJT5CTOE8meW4HXgI9ilT7cALwVSzB/bQzftUVKuU79AZIVW3YnXjsohPgqsF8I8Y/Kx78M1AshPqy89kmsRFD1OM+zScmsVqsS5JtzbOWivCkGMsVmITufZUUQ2rL36CC9w5YYqKsIMMshMeBVsam2e0lLbUHjdIvBXmo/snRG4RwLye8f7bxewhaiUUBHDBwfyuhFXj/Um9p8aOGUaipL8x/CmGRGfXlqc6Pe4Uhqku4l+kIRdnda6XiWeC7cZAawff9mU7Ela3IeVaSUtwFzgE9g1Sb/ErBcSnlNntqWyWFgIPFvsg07sGqZv1MI8X0hxA+wJgjnSyl3F6gdrrBR6TjPmlG4h2mp4jn36oN0qGeYzn6rYE91aQnzGkct2pMXikE8ZXrNCyqeiiCOeqNDS+hQHPayl7grrL3U+FmveulUexUyHhjg9Gk1qVCzPZ0DnqxFrY5ThfSag7UZkbqa5cV7bIvS5tOmVFMWyG8Z2EyWtHjbXm4xrimmlLINuGeUt94P3DuO792HEsueeO3e0b5TSvkGVtx60RKO2uscn11Az/miqdX4fYJYXLLn6CD9oQjVedyW3Ak2ZUxkfD5nPHWbDlrLnIUUt4Vg86H0vVXIVRmw28urdW/faHNODKhbam/x6MCmtntJge2VFJtSwq7OAUKRWMHFR75R7VWI+uYqFcES5jZUsrtzkLiErYf7CpYTUCjUsbHQXmDrHDWp3ZS3tPVy5ZKpBT9nPnHy/gJrdTGJ2ncaTs4pxbkQohz4ERli+SQEgLcwDnFusLP9cH+qcsqM+nLq87j5UCZlAT8LmqrYlsjc39ren/fd6QqN2gGo4qYQzJpcQXVpCf0jUXqGIrT3hmguQLJuIdnSnrZXoTvrBU3VBPyCSExysHuY3qEItRXemvzZxUBh7TWnoZLygJ/hSIwj/SN09o/QWF166g9qQjwubYnli6cV1l5VpSXMnlzJ3qODxOKS7Yf7C+59zidSStv9dUaB7QWWoE2GObzR5kFxrojNMxwXm96bMKv31+kO3F9zGyopLfExEo3T3hvi2MAIk6u804e5xSnDWqSUw8B5wMey/PkQMCWvrZzgbHQo3jyJ12PEnOyshRC2pFCvddZSSkftFSzxsXBKOoHSaztfHh0YoaPPCpkqC/iY01C4kCmwYkLVhFOv2Wt/1xCDiVCJhqqgIxML9R72mr3aekOp/I+aspKCJs8mUSeYXludGYnG2HUkvdvwoqmFSc5WUe212YOe4K0Oi/MSv892Hq+NkW6Rbcz548AZUkrfqX6ACozXPK+oArmQZY+SqDFiXhTnbzi8bOflwe1g9zB9iUoateWBgpTozET1BnpNPKnXd9HUGvwFDJlKYq924K3nUbWXFXJSeHvZY4K9a68zmp2xl+qM2eIxe+3sGCCayAadWV/hSAjm3MYqygKWdOroG+HowMgpPqEPkVicnR3pyYwTKzOQoSk8do+5Rbbi/D4pZVa74UopQxy/a6hhHGxVNodwIqZOXbbz2oPU2T/CkUQyaHnAz+zJlQU/p5fFQOZExhkx4F172UIOHJj4AZwxTRVPHpvMKCFTTtnLy6XbbOJ8WuH7erDba9vhfqIe2rzJ6RAgSKxmTfXmPba7c4Bw4vq2TCp3LKRQ1S1vHPKOvdwkK3EupXw52y8UQtQDbTm3yGDDiptUvXWFX7Y7Q8ng33XEWxn86pLdomnVDnk2FfHkNU+wC4PbGc0eFpttztvLyysz7tgrfX9ta+8nFvdOLWo3JjN1lUGaa8sAa/OmZPy5F3A63jyJV1eXM1eynGKJh8dIt8iplKIQ4pIT/WCVVLw6v82cuOw7NkgoYs10m6pLHUmkqCy1MviBVAa/V3jDBTEwv6kqtVNoMsnRK6jL2ItbnLHX6UoM9a4jA4xEvTP5Uz39ToRMAZw2NT3J3HtskMER7+ys6nQlDYDG6lKaErHtw5EYe496SGy6MFkGu7D10mqWW/byqkNmq81ehXf0JVkwpSrl8Nt/bJBQxDt9vlvkWud8DbD6BD9fB96bj8YZnE/eSGLLSPeSZ8CFsINgiY8FU9KJgV7qrN9wYRm9uiy90VE0Lm0xkDozFI6yJyH0fALb0nYhKQv4mddoTZalhG0emSwfnzxb+BCzJF4MneodjnCgy9rUJuAXzG8qbLKxirqatdUj/ZeUkq0uec5PU1awd3b0n+RIvXBjfASrD0uGmMYltiRew+jkKs5XA5/G2oBI/bkReBT457y0zuCaOFfPtd1LnY/Ns+mM2LTO5T0x0DUYpr03BFgTjKQAdAIv2mvb4X6Su3XPbayiPOhc/WxbEq1HQltsIWYOJc8msfVfh73Rf21T7LWgqZpgSeF2ns1EDZfc4ZHJ8sHuYfoTq0iTKgJMS4TmOMECZeK0p3MwVepYZ6SUbG1PPwtO6gmAhYoDa4eHNIVb5Pr0f0dK+XMp5b0ZPz8AHgMuymMbJzTbbA+Tc8tQpynl7rwyuGV6NtX/Q6HxYgWSbe32XIaSAm5Dn4lqL3XA0JltLg5sXpwsu+VYAGzlOr0iNtXrusjBvh68KZzUcWnR1GpHN3+rLktXtorGJfuO6R861TkwQtdgGIDKoJ8ZdRWOnt+mKTxyj7lJTqOxlPKJk7y9Dvhibs0xZOLWALdwql2cS6l/UtWOjoGUZ3NOQ6Wjns1FyrXxSpiG2kE6OZGBTPHkjY5abacTidkqNnsd9sb9pT4Hp01xLkQD7PbaecQb95fdXs7eX7MmV6byZtp7Q/SF9M+b2amERix02F7WOdP3tBccWLuU+2v+lOqC7pw9Gqqm2OEBe7lNXl1lQogy4DOAsz1xkdIzFKYtGXbg96WSNJ2gubaM6lJrA9m+UDQVO6ozdvHknqdu55F+4h6oEKF6FJ0e3Lzo2VRF3gIH44EBW07DjiMemSy7KJ7mNlaS1B6tXUOeqDil9l8LHJ7MBPw+5iphbV6Io3bzeQS72PSGvdLPoxv2Os2Dfb6b5FqtJTbaDzAI/B3wUF5bOUFRl/sXTKlyNOxACGHrfLyQhLbTxcGtvjJIQ1UQgFAkzoHuIUfPnwu7jrhnrxn1FZQmYmqPDozQnVhu1RnVs7nAYbHZMqmcysRKUM9QhE7NNz6RUrJLeR7nO3x/lQX8zG5IJ9F6IQFtl008ueEJ9pZ4Uu013wV7eS1Mw+3JzOyGSgJ+a8Z8qGeYfg+szrhJrmrvMPAr4BcZPz8Frk/8GMaJKoidjtkEe0a6F0IP3PQEg31A1X1wk1K6ai+/z16NQvf7q3coktrcqrTEx8x6Z+M1hRC2CYHuoVNtvSEGE97qSRUBGh0oAZvJwibviKdjAyMcS0xQywN+R3bqzcRLYRrxjCpPCx2e/Fnn9E5/D5nOBeftFfD7mNeo9vn628xNchXnn5FSfkRK+fGMn89KKf9bSumdQrwaoy5DOR2DmHnO7R6Ic93l8rKdl5KqOvtH6B22PBfVpSWOVjpIYhvcNPds7lC8TvMaqxytPJLES/eXbRWrqcrRZL0kXgo72GnzAlc5Hg8M3orTP9QzzHCiVnZ9ZdCR/T8ymd/krdrdbq/MgDdzjdwi14TQxzJfE0L4hRDB8TfJkMS2bOfCTFf1nG/v0DusZWAkyqEeq0ZwiU+klrSdxO7Z1Lvj2dFhv7fcEE+q90Z3e7ntdQJveercDAFK4tnJjAb3l+7OmF0Zkxk38FLtbnVlpizgc2VlBjI0hearM26Ta8z5VCHEc0KIf1Ne9gM3CSEuyE/TDLYOqNHdZbudHQNab4Ot2mpOQyUBB+Pzk9gnM/p21JCRfObS4LawyTteFLV9boRMgbcmf27Ht4LHJjMaeDVn1FdQFvBGHogO91fmuXXuwzInM26szIDdXrs79X4m3SZXBfM94AKsBFAApJRhrM2HviOEeFMe2jah6RoMp2qSuhWDWF8ZpDGxDfZINE5rl75JjjqIJ1Vs7u7UezLjdhmyzPPqHkOtgxiwxQR36F2xxe38D4DZk72TgGbvv9y5v7yUB6LD/ZV57p0ae851mPyBtXlbkt0a20sHchXnjcAMKeW/qi9KKePAH4Fvj7dhEx11pju3sdK1me5pHokR02GZs7YiQFNiMhOOxtmv8cYU9mV0dzrr6XXllAesCiTHBsMc07gCiQ5hGlNryqgus8qb9mtc3lRK6Xr+B1i73s5Rwtt0Fk86xAOD3cGgs73cLguYZF5T+v7ao7EnWIfxEWBmfUUqX6etN8RQ2KQnnohcxflmKWXbCd4rA1bk+L2GBDp46sD+IO/p1Fds6uA5zzy3rkvpVqUW9z11vuM8dXraS63UEnShUksSIYQnEqrae0MMJLZVry0PpFbf3EAVurr2X12DYY4OpOOBp9e5Ew8MMM8D/b3bZTpV5jboby/QR08ES3zMUvrPvUf1tZnb5CrOK4UQx2XcJV77CNA7rlYZtJnpqhtT6BwjpkPCXua5dY0LPtI/Ql/IEk/VpSVMrXG+UksS1V67NK0QoUOlliRq7omunrrMfAY3ko2TqP2XrvZS+wk344EB20rDnqN62qtdgzKdSdT7a/+xIW1DGXVY+Uti1xRGnJ+IXMX5z4A/CiHeLISoFEI0CCGuAV4A5gD35q2FExRdxLlal1RXcX5cpZbJzldqSWJbadDUK5BZBchN8WS/v/S3l5teJ8gQm5reX+p1dHOiDJniXE97qdfRjcR/FS/YSx2H5jW6239Vl6VXhsKxOAc13HyuL6Ss/Pl9zHBxZQbsfb6uE2YdyLWU4gbgR8DvgT6gA/gtsARrc6Jv5quBE5XdGorzPZ2DWiahqbaa3VBJsMT5Si1JbMucmoontV3zXBYD8xQxoOsS516N7DW3Uf9l9L2Kx1V9Htxgju151FMIqPfXXJfvr9mTK1O1uw92DzES1a92t81eLpTMzURtg47P5F6lTbMbKhzdaXw0jOc8O3K+SlLKB4G5wIeBrwNfApYmNifS74n2EAMjUdp6Q4DlCZ7loid4Sk1patvw3uFIqlaqTtjFk7uddeYyuo6TGbWznuPy4OYF8aQOuHNcvr/U6+WFyYzb95f6PO7TNOxgj008uWuvMqUyWFxC6zH9PMG2+8vl5xEyKpBo6AnepxQmcHNVOclc4znPinFNoaSUPcD/SSm/gxXK4u6UrEjI9AS7UbM7iRBC+/JHdjHgruepqTo9mekPRenTby5jE8Fue55mTa5QPHXDWu6yt1cje6nVDg71DDMc1tBeGk1masoCNFSlKyi1JcLfdEKn+wsyxaZ+E0Bbf6+B2JyneaiZTpNlOH41Pq7hhFkHclZ9QogbhRD7gZcApJR9wEohxHfz1biJitubD2Xipc7H7cEtczJzeDDuYmtGRyfPU1nAn6pOISXa1dKPxuz1/d32bAZL7DGj+zQr1zkcjqVW/fw+wYw6dyrbqOic1K7b/QUZYRoarmbt06j/Av1jqHUT5/WVQSZVBAAYjsQ43BdyuUV6kusOoV/B2ohoBpCa9kgpfwCcJoT4ZH6aNzGxJbw0uf8wecpzrkFnrXaAuonzcDTOgYQYEEKPZU5baItmg9uhnmEiMauLa6oupaq0xOUW6R13rk4WZtSVu5r/kWSexkmOet5f+torHI1zoDu9+qFD/6WzvcA+mdFh8gfHe88Nx5Nrz/m3wBeBOmBzxnt/Aj4/nkZNdNQBzu0wDdC7YouUUjvPgNpZtw/qtWTX2jVEchWxubacssQmQG5i99Tp1VHv0ezegswENL2eR92eRdC7PKCO9pqr8WT5QHc6b6C5tkyL/mt6XQXBROjpkf4RrXai1XF8BHsfppum0IVcxfleKeUdUspeFM95gknAgvE1a2Kz92h6mXNOg/vLwrZd0DQTT50DI6kNT6rLSphcGXS5RXbPZseQXp5ze2UITTpqjT1Pavy0Lvaao3GFG53yP5LovFGMlsJJ4zBG3UJawArfmjU5PU7rdI91DYZTe1pUBP2pHazdxiSFnppcxXmP8nuqyKgQIghch1Ve0ZADUkrbtu86LNup5bUOdA1plbSXWXnEzZq3SVSvQLtmYS1q8pk2YkART3qLTf3stVtne2kinuZ6ZjKjh72m1pRRnvBI9wxF6NaoQpeO9oLMCY0+YjOzUosO4yPYr90+DSsC6UCuAW5/FEL8CrgNQAgRwKpx/m3gTOCH+WnexONI/whDiQoM1WUl1GvgCU4m7R3oGiYurZ3QTpvq7i5jSXTsrNV2dA5JIrG4qxV3VHS0l867OGrpCVbFZqJcpy6Drk7J2Ulm1FdQ4hNE45L23hBD4SgVQfdju0HP59HnE8xpqGRLu+Vj23N0gHMr611ulYXqydfBcZXE8gR3APaV70IyMjJCV1cX/f39xGIncJiFo9x9zTTA8pxv3brVkbadimbiqXYF/MKVdtXW1o77vH6/n+rqaurr6yktze+qRE49lJTyZ0KIZuA1IAD8v8RbAngU+Fp+mjfx2HdUP08wWMLkQJeViLPv2KAR5yehsrSEqTVlHO4LEZNWiUBd2rZboxrnSabWlFEW8BGKxOlOeOrqNJiUgp73V7Jc52A4Rl8oyrHBcKpcoNvoaK+A38fMyRWpcIM9nYMsaal1uVUWezQMmwKrLSlx3jnIubP0EOf7NAzLA5hVnw5raXWggtLIyAitra3U1dUxe/ZsAoHAqFrhcO8wgcTuoE3VZUytLSt427IhFpfQ1guAQLCopcZxrdPf3091de46RkpJJBKhr6+P1tZWZs6cmVeBPp5NiG4D5gCfAG7B2oRouZTymjy1bUKi24YBSWYrMXX7NSrfpqMYAH29wXbPph6eYMtTp9/OqqFIjEOJuth+n2Bmvfv5H2CV65yjYZx+z1CYrkQIRFnAx9QaPYQA2Oth79dkGT0UidHWa91fPmF5+HVBjaHWqbypjitZADPV8dEBe3V1dVFXV0dDQwPBYPCEwnYkmg6r1KFyUhK/T6R2KpVYq8teQwhBMBikoaGBuro6urq68vr9492EqE1KeY+U8t8SCaKvJN56fx7aNiFRl8R0KXsE2HYp1SlGTEexCRkVIjQRT/2hCJ0JL0rAL2hR6mW7jY4VSHQsC5hEFSa61DrfmxFy4PPpseoHmeJJD3u1dg2R3EC4pa6c0hL3K48kmVWv32RmOByjXamhP12j/kt1pDmxq2p/fz81NTWnPC6siPNSjfovgFIl1FOdRHiRmpoa+vv78/qdpwxrEUKUAz9CSfw8BQHgLVg7hhrGiD2sRR9Pio6e81hc2gaO2RrZy57wooe99ikTv1mTK1M7TeqAeu108dRlJhvrhH0ZXRN7aRpyAHraa0+nnl5gcN4TnA1qPzqzvkKbPB6wQvOCJT7C0TjHBsP0hyJUlwUKdr5YLEYgcPLvl1LaRK9u4jxY4iOZaxz2uDgPBAInjvvPkVOKcynlsBDiPOCMMXyvXsWdPYSuYS2zbOJcj866rWeYcGI5rLG6tKCd4VhRVxp0EZt7NKzUkkRHT90eTZfQQVPxpGmyHtifR13uL/s29Po4FiAjrEUT54KuIYxghebNqCtP5fTsPzZU8LyGU8VoR+OSeGJpxu8TWjljwB5m43VxXoh4+WynUo8DZ0gpfaf6ASowXvOciMdlxgZE+nRA0+sqUuUU23qGGYm6X05Rxw1ikqjxybqIc9VjOFszMaCj2DzQpa406GUvpxPQskG9bvqJcw2fxy77SpZOTKkuS4mn7qEIvcPub6yj2kuX/A8V3RwymV5zXYpLJFE9+V4PaykE2Yrz+6SU27I5UEoZAn6Qe5MmLh39IUIR6yatLQ8wqUKPihVglVNsrrVi/OKJCiRu02pbZdCrs1YHj0Pdw0Q1SHixDW6aiQFVPB3QYGADu4d1pmb3l80TrIm97PeXXvaaXldB0nHY1quHc0HnyZ8vIwFah1Ag/cW5XqvLqjc66NcnnyGJzXOuwfioG1mJcynly6O9LoSoE0K8SwjxKSHElUKIssTxm8baECHEJUKINUKIXiHEYSHEz4QQp6zfJIS4XAjxCyHEvwkh/ksI8ZgQ4vyxnl8H9mmaDJpE7RB1iDtXO0DdPE/lym5s0bikrSfkcovsIk63wU311HUlYjbdRmcx0FRdmvI89Wji2Tygsb2CJT6mJZwLUhfngsb2gow8Iw2SaHW+v8C+mqXD+KgK3mCJXl5zgKDfHtYipYmGVsk5Q0AI8RmgFXgI+G/gMWC/EOKdOXzX5cDXge8BbwV+ilWi8Wen+NzVwMPAl6WUN0kprwe+CzwphFgw1na4jS2kRTNPCtiT9nTwDKiDm05lyJLMMoNb1iRjNpO4fX+Fo3HaE2XuhICWSfpUhoDjPZturzYMjkQ5OmBldwX9PqZoVEYxySyNktqjsXiqTCdYnn3dmKlZHojOKzOgX15DRNMyiklK/L5UHHxcSqJxI85VcrpiQoi/Bn4CHMIKYfki8A/AI8DPhBBvHsN3CSxBfpWU8jEp5YtSypuB1VhVX07Gp4BdUsojyReklM8AIeCqMfyXtMCWUKWh51y3zkd3z5NOg1soEuNwn+W992koNkGvmM22Hms3XEhukqTfsrBOy+gHutPnn15Xrl3yGehlr/bekLURC9YqSHlQ7/vL7bCWWFxySFntmKHjZEazvAZbjXONKtuoFFNSaL7JdQ/jrwFfBf5TSmmzqBDiO8C/AM9l+V0C+JaUMjMIsBc4VXhMGFgshGiRUh5KnL8SqAYOZ3l+bdA1GTSJuszpdnlAKaXWnmDQKwntUM9wqqbytFq9anYn0SmJVveJH2RM/lxemVHFm46rWKDXZNkT95dGK3/tvcMpz2qjppOZ6XXl+ISVk5XMa3Czdn0kprfnHKDU72cYS/qNRONUFnij46GhIcrKyvD5RrfH4OAgFRUVWiTP5nrFuqWU38sU5gBSyj1Ad7ZfJKWMSyltAYBCiJlYu49+7BQf/zlQDvxJCDEr4YX/HvAoVriNp2jtSptBxw5bp8GtazDMYNh6qKtKS6ir0KeMYhKdltG9IAZ08mx6wV4z69OrH257Nr1gL50my56wl0YJoV6wV2mJX5u8hng8veumQGhVE17F7jkvbJL2888/zze+8Y2T1iPv7e3lpptuyvtun7mQ6xXbfor35+bypUIInxDi3cCziXOctEeQUv4J+DywGNgMPA+sk1K+exRPvNZIKTnooRjqg91DrlYgyeysdZjpZmJPoHV5cDum/+BmF0/uTmZ0X5UBvcLMvGAvnRLadc+XAXuFm/a+EKGIe0OqF+4v0MchoyaDBkqEluMj2MV5IcspPvzww3z/+9/nu9/97qibN7344osANDc384//+I986EMfor29vWDtyYZcw1qmCCGmSCk71BeFEEGskJZcFyc+A5wNRID3ARcIIRZLKU+4L6qU8k4hxJnAR4ELAL8QYo2UsvVEn0kks34GYMqUKaxZsybH5ubOwMCA7bwDYUn/SBSAoB9eX79WyweqtlTQOyKJxCS//csaGivcmZGva4umfi+PD7pyDU9Ffzid4LK3s5/Vq1e7dk3Xbh1J/R7rPcyaNe57BjI5PJDunLcf6nL1mm7Ynq6uM9CxnzVrDp30+Mzn2QlUe+1oc9der+5K26uvfQ9r1pyw+3WN4Wj6edx/bJBVq1fjG8PzmM9r/PK2tL2GjuxnzZq2vHxvvqkrFRwLSaSE3/zlaZqr3Onvn90RTv0e6ztS0Ht9PNc5MJLuZ59atwnf4cKs6NbW1p50u/ihSPpe9yPzvrV8vogpz2QoHC1IO9evX8+Xv/xlnn32WYaG0pO8WCxGf38/a9eu5cc//jG//OUvAWu3z1tuuYXrrruO3/3ud/izLEMZCoXyel/mKs7vATYLIe4D9gFlwOnA24FJnDqRc1SklD8By4MO3AFcj1W15fbRjhdClAB3AT3ATOBHWKJ+rRDivGQc+ijnuSvxOZYvXy5XrlyZS3PHxZo1a1DPu+lgD6x6HoA5DdVcdtkljrcpGxZsXcuG/VbU0pT5Z3LJwkZX2rF51U7YtAOAc0+bxcqVp7vSjpMhpeQfn/4joRiMxGDJ8gtprC5wUN0JuK91A2DNpS9dvoSVZze70o6TESyxJdoAACAASURBVIrE+Przf0ZK6ApJLnzzJa7FSv77xmeBPgDe9uZzOWdm3UmPz3yenWAkGuOmpL1GJBe8+WLXYlxve3kNYHkK337xmzijucaVdpyK+heeoGswTDQOi5adT/MYEqPzeY3/v83PYaVVwVsvOpcVs09ZNdgVTtu5jrW7jwEwZd5iVp4+xZV2PNz+KmBNYC5edjorl88o2LnGc523spunD1pbwgTqm1m5cnEeW6acZ+tWqqurT/j+yMAIDFlhNeWlAaqr9VxtKI3GaR+0+tmYFCf9P+VCLBbjc5/7HDfeeCPTp0+3vdfb28tDDz3EF77wBW699VbbuS+88ELq6+v5r//6L2666aaszlVWVsayZcvy1vacxLmU8kkhxD9gieZJylsdwHullM+Op1FSyrgQ4qtY3u2FJzn0a8AlwMJE/Pv7hRAbgH8HbgH+djztcJIDSrz5jHr9KmkkmTm5IiXO1QoNTuOFZWEhBE0VPlr7LQ9na9ega+LcC8vCZQE/U2vKaO8NWUlVPcOuVC3yQrIxWDGuzbXlqWTfg93DzGuscrwd8bjkQLdH+q/6CroGLS/s/mNDYxLn+cQLMdRghWkkxbmboVNesZfaNrdizmf/02OunDcX9n77KoQQSCmJxuPE4jLrSk+vv/46d999Nw8//DD33HMPV1xxBQBr167l6quv5uGHH+bIkSNs27aNa6+99rjP33nnnTzyyCMMDg6ybt06brjhBr7//e+n3r/mmmu44YYb+OIXv0hlpfPjUM5uKSnlL4HpwJXAh7G85bOklI/ko2FSygGsxNKTrSe/B9igJqZKKb8HvAZclI92OIW9FJm+nY9awkqdUDiNVzrrpop0R+PW4Cal9Iy9bHHBLiXt9QxFUiFmFUE/kyv12ak3Ex0q3HT0h1Jl0Oorg1SX6ZecnUSHvIbe4Qg9Q9amUaUlPhqr3JmwZ4NaBMBNZ8wBzWucJ5mu7NVw0EV7eQUhBAF/eoyMjCGP7cwzz+S2227j2LFjrFu3LvX64sWLaWpq4qWXXuKhhx6isbGR5ubjV4q/8IUvsHz5cpqamnjwwQdtwhxg2bJl9PT08Oijj+bwPxs/WYlzIcTU0V6XUg5JKR+XUv5KSrlaShk+2fFjQQgxHagFHlBem5lx2AjQNMrHdwF6BlmdAC94gsHeNjc7Hy8kOAI0KTH5bomnowNhhhKVbarLSpikYWWbJPbayu6IJy8kGyfRoRa1F8ooJlH7ikMueTYPZPT1Pg1rwiexi0137DUwEk2tdgT9PqZU67fBVRLVXoe6h82ul1mQuVPoWKitrWXp0qW8+uqrttcuvfRS3v72t7N+/frjwllUVq1axcqVK0ft45Ofe/bZcQWC5Ey2YS2fwwoTyZbrgW9kc6AQohb4DVZd9P+QUvYnapX/N/AlKeWuxHFfBb4jhPhKwjsO1gZI9ybiy19MHNcAXAx8aQztdR0vLKODvfM54FJnPRKN0Z7YUEfH3RtVVM+5a+LJU2LT/QokXpkog719bu0S6pVVGdBDbHqlrwf97DW9vlzryUx9ZZCKoJ+hcIz+kSh9w1FqHXaG7P32VbzR1kc8MTE4Y1oNJZqWUoQMcZ5DBbj58+fzyiuvpP7u7+8nHo9z5pln0t7efkJxfvjwYbZt28aXvjS6VEzGoO/bt2/MbcoH2YrzLyRE9NEsjm0EPkKW4hxrI6EI8GXgM0KIvwBHgFuklK8oxx0GBlA2F5JS/koIcQy4TQjRihUGMxP4mJTyz1meXwsOeiRm0+Y5d0kMWB4J6/dmTTfUSdJYnm6bDoObEQOnRhWbs4y9Ton9/tK37wJ7yKAO95f+z6P7K6VespcQgul15ezoGACsUKDailpH2xCNy5Qw9wuh5W69Kur4PZawliRz587lkUfS0dR33nkn3/zmNwFOunLxzDPPAHDZZZeN+n5JiSWPw+HwqO8XmmzFeQ2W9xysHT1PhEy8n/VaTmIDoiuzOO5e4N5RXv8L8Jdsz6cjXtiaOMnUmjJKfIJoXHJsMMxQOEpFMNeiP7lh92zqLQYaytOPixncTo1NDPS4H6ahc3wraGIvT91f7scEe2llpqEqSGmJj5FonP5QlN7hCLXlznqCveRcAOuZTIrzg91DLGlxVpxHomqNc5/WK6VgtTHJWMNaAGbMmMHAwAChUIgXXniBJUuW0NLSAlilsgcGBkb93DPPPMP06dNZuHD0miPDw5Yma2hoGHOb8kG2qqoDyyP+G+CNkxwnsGLA3z/Odk0oOvpCqeWc+soglaXOit2x4PcJmieVpwaYg93DLJyS3/JHp8JLnXV9uUAIa8e4w31W4pzTnv79HooJnqGBJ9hL4snYa2xMqy13/Xn00mQm6Qne3WnlfxzsHqK23Fmxabu/NHZcJXF7NUsNDQlqHM6SZDwx5wBVVVaFqueee47du3fz2c9+NvXe8uXLee6550b93DPPPMNll11GOBxmy5YtLF261Pb+kSNHADj77LPH3KZ8kO2Vm4lV1nARsAJYLaX81ig/t0opr8eqUW7IkswEId1RvdVueJ+8NLgFfCKVwBSXcLg3dIpP5B8vTWYaqkpTYqlnKMLASPQUn8g/XhIDmfbqD0Ucb0OrUrVJ9/srWOJjao0+z6PuK3/gfiiQlyZ/oIE4VwSuziGfScYb1pIsc/jEE0/YhDnAtddeS2dn53G7fba3t7Nv3z6WLl3KHXfcwaJFi4773i1btgBWSUU3yOrKSSkjUsqfSynPwtro5xtCiBeFEO8Wo6+Z3JPPRhY7thrBdR7orCe5W07R252185OZQz1qPoPe9vL5BNMn2SseOEksLjnclxZs0zV/Ho+zV4+z9gpFYhwdsHZF9PtESvjqjJvPYzwuaetV7y+9n0dwX2yqfYDuzyO4H6fvNXFe4hOp0JtoXBKLj63CTXV1NZ/+9Kf5zne+c9x7733ve2lpaeGxx+y138PhMHV1dTz00EO8973vpazs+H7riSee4Morr+T0093Z4HDMV05K+Wcp5V8BnwXeCbwhhPg7IUSZcsy+/DWx+DGe87FhT57V315uDm6ZYnNarf7iqcVF8dTRF0oNDg1VQcoC7uy4ORZa6tybzLQrQnNqTZnWVSGSuOkJPjYYTomn2vIAVRqHMCZxU2xKKWnr8Zo4N2EtY0EIMa6KLZdddhl33XXXqLH1paWl3H777Xz/+98nGk2vws6aNYv9+/ezdu1aZs7MrNANR48e5cEHH+SHP/zhmNqST8azCdFrUsqPAG8F5gCbhBC3CCEm5611E4QDHlpGh8zybS54UtTOWuMyikncHNy8KDbdHNxUIeDW7pFjxU2xabeX/hM/cNdz7s37y73nsW84ymBij4aKoN/xZNRcmJHxPDpd6zwSS58v4AFxDtg3Isoh7vxkvPvd7+aqq67i5ptvzur4WCzGpz/9ae6++24WLFiQ17aMhXFfOSnlQeB+rI1/bgX2CyHcm254EHXnNd1jNiGz1rmzg9vgSDS1u17Q76NB4931khixOTbcnMyoE7/mWq/Yyz2xqdpL5/0GVHR5Hls8OZlx1l6HMvov3SuPAEyqCFAZtJwgA8p45QRSSlvcdqBEf3uBPfwml1rnp+J73/se5eXl3H777Sc9LhKJcOONN/KlL32Jq666Ku/tGAvjWlMTQlwKfA24IvFSDHgI+PE42zWhUL3PXkgQyvQMOEl7b/p80yaVab0hRRI3PZveF5tOi6d0mEaLB5bQwW17eX3y567Y9AJuTpa9eH9ZFW4q2N5hbVJ+sHuYusqgI+eOZdY498BkBjLEeZ4950m++c1vsn79esLhMMHg6Nfj4MGD3Hzzza6VT1TJyXMuhLhaCLEWWIUV1hIC7gTmSyk/JqXcmsc2FjWhSIyOfksQ+IQ3OiC1QkTvcIQ+BytEHFLEkxfFptMJe14Xm87by3tiwIjzseFuWIvSf3nEXsla50Cq1rlTtPV6b6UB3LvH7F5z/WucJ1FjznOp2JItK1asOKEwB5gzZ44WwhzGIM6FED4hxIeEEJuA3wHnA73AvwKzpJR/L6VsLVA7i5ZDPendLqfVlnsiRsznE/bOx8G4cy+KgWnKoNLeO1zQzicTL9pLlxhqr4gBdz2byuTPI/dXstY5pGudO4UXn8dkrfMkTt5jXlz5A/cmzGEPxpuDva1OPo86k9XVE0JcD+wEfgEswdqU6KvATCnlLVLKoxnHfyrfDS1W1OoKXvFsgl0QOBl3brOXR8RTaYmfKTVWbLzTtZUPeVBsNlaVpjwpXYNhBh2sde7FsAPVXt0O14b3oth0s9a5dz3B7kyYvbjSAO5NmG2ec783vOZQ+JhzL5Lt1OpHwGxgN1YJxVlSyu9JKW37ogqLmcDH89rKIqbNY5VHkqj12NVqM4XGi2IA3JvMeNFePp+wVf5wMrTFi+Lc5xOulFOUUmbYy0ti0x1PsBefR3DPE6zaa5pn7y/n7BXxWBnFJCU+gS+xnBWLS6JxI9DHcvUGgABwE7BdCLEn8wdoBXZghbwYssC2jO4hz7laTtHJzueQR+3lVmftxWoa4I7nqS8UoT9keZ1LS3xMdiiJKx+o19Ype3UNhhlJLEFXl5VQXaZ/mbskbniCrQ2bwoC1YVNTtZfEpjue4DaP9l9ujY9qSIiXwlqEELb2RqLOlp/UkWyrtTwAXCezKNiZ2IzowXG1agJxyLPLdi55nnqN5ylbMsVmvYfEphv2as+In/ZKMhW4Yy8vxpsncaP/ytywye+BSlNJ3Li/IrE4HcoGalM9sIFaksxyw1JKR/oTW41zD+wOqhLwC5IReZFYnHL035OjkGR79X6fjTAHkFKGgPtyb9LE4lBPemDwlthMewbUCUYhicWlTUB5K0HIec9TptfJq2LTqTANr4YcgDti04shQEncmcx40wsM7tiroy9Ecif3xupSSku8I9Zqy9O1zofCMccq3NjDWrzT34NzFVu8QlbiXEr5q7F8qZTy/tyaM/Gwe5+84xmwxQQ7JAY6+0eIJnrr+sog5UHvdNZuiwHviSfnl4W9Gj8NmZNlY69TYe6vseGGc0FdafBa/yWEsLXZiWcynrEBUYmHwlrA7uk3SaF52CHUkDvxuLRtquOlDqihMl3rvC8Upd+BWudejZ+GDPHkmBhQBzeviQHnPcHensyYyd9YcCNG38v2aqgKpvr7fof6ey+WNVWZplzjdgdWl6Mxe7y5z0MrpWCPkVfDcyYqRpy7yNGBkdRNWFcRoCI4rg1bHcXnEzTXqvW7C9/5tHnY89TsQq1z++BWcZIj9aPFiM0xYasG5FD1JC+HaaiVPzr6R2zCplB4+f4Swvn+3qs1zpOoEwo1V6pQeLXGeRI1DCdiap0bce4mXo7ZBBxftvPy4OZGrXPVQ++1yUxTdVmqTu+xwTBD4cLX7vbyykxTdWnKXt1DEYbDsYKf08vivLTET0OV9TzG4pIj/SMFP6eXE2jB9PdjRZ1QOGEvr9Y4TxIwMec2st2E6HIhxHWFbsxEo5jEeZsDnY+XxRPY2+z04OY1e/l9gmm16v3lxMqMd2NcfT7BlBpnPXVerTSVxObZNGLzlDjd33v5eYRMexW+/1K9zV6qcZ4kM6wlyxokRUu2V/DHwLXJP4QQtYVpzsTCy+IJMsSmA6EHXreXLQbRAfHkfTHgnHiKxuIcVsq2TfNQ2bYkToonq2a35W22anaXFvR8hcBJT7CXN2xK0uxwDLXX+3unJzNqEqXXyiiC5WAo8VntlsgJH3ee7RUMAZ9U/v7+yQ4WQizOuUUTCK8vc7Y47jn3tielxUFPynFi04tioNa5++tI/wixRCWghqpSygLeqQSUxMnn8XBGzW6vVYYAbCszhY6h9vKGTUnUmHOnV0q9OZlx1l4Rj8ecAwRLlLjzCR7aku0V/K2Usm8M33t5Lo2ZaBRXWIvTCaEetFetc1vSd/SPeLZGcBLb/VVg8eT1yhCQKQacs5cXhRM4K5687ogBZ1cavLyBWhJ106SOvlDBk469XOM8iYk7T5NteZCQEOIPQGfi7zcLIX5+gmOrgSuAO8bbuGLHywl7kFHrvMCd9cBINLWRQ9BjW6sncXKZ035veV8MFNxeHp/4ARkx+sZep8LJlYZisJd9slxYe3l5t94kpSV+GqtL6Uw4Sjr6Rwo6MVNjzr3qOVfbPdFrnWclzqWU3xFCfBn4DDALCADzT/aRPLSt6FE7OLV0nFdQO+vDCc9AoZa3bZ662jJ8Htr6OomTYrP4PMHOeTa9Kp5aHBRPxWAvuyfYrDScCrXdh3tDxOOyYP2w11dJkzRPKqczUQmorWe4YOI8Fo8TSyRQ+oTA/8x34OnvZvfhcz4K1/zQ/trv/x5euTe7z1/6T3DZ1+yv/er9sOPPY/sMY691/tprr3HvvfeyZs0a3ve+93H99dfz8Y9/nOeff57XXnuNadOmZfd/0JCslZSU8j+klAullKXAvVJK32g/WJ7znxSsxUVCKCrpGUp4gv0+Giq9l1BVFvDTUGV5sAtdjsxWqcWDExlwNqHK6zWCIcNeBQ5rOdSTrg3uVTHg5P1VDOJpWsbeA4WkGOxVESxhUoUVKx+JyVRCcCHwerx5EqcqAmXWOBd4z3kFY691vnTpUq677jpee+01VqxYwa233kpLSwvd3d2Ew+FCNrXg5OrmXHuiN6SUg8Avc/zeCUNXKP0wTZvkTU8wOOcNbisCsVlXEaAskNhlbyRKXwF32WsrgsnMtIwY/UKW1iqOmGAH7dXr/ZWZhsrSVMm5nqEIgyOFq6Vvt5c37y9wrnZ3MUxmwB5qVkh72UNavKklwF5lJtuwlp07d1JaWsqGDRv4whe+wB133EFHRwezZs0qVDMdIactKaWUd///7Z15nFxVlfi/p7qrt/SardPp7IGwg0hQQZaAqIygogKDDIPoKPpjnBmEUdTBdZgZFRdwHEURhXEbZ4RxxwBCK4vARCIYQhLIQpJOJ+kkva/VVef3x3tV/V51dXctr7rrvXe/n8/7dNV99726fe+795177jnnAohIE7AOmAfsBdpUdVhVn/SshAHl0ND4g+dXYROssj+3twewBp+1RfqdIAzWIsLixmp2dA4A1v9Uv6g4URuCYONaVxWlvqqc3uExRscSHB4YTW0c4zV+D9sGVn3VVZbTNzLGyFiCIwOjzCtSfQXh+YpEhJbGKl4+bK2adPQMcdTCuqL8lt8jTSVZ3FjN5g4rNkRHzzCnFul3gjDew8wpr9zOoBHLZCSD2UjWvOWrE01dcuHKH+d1WT4OoY888ghHHXUUFRUVrFq1CoCmpqa8fr+UyNtAWESuBXYDPwG+CfwKeFlELvGobIHGqTn3q2YTZi5ii9PB0a/CE8ycE1oQhE2YuZdbUJbRZ8IUSFWDIzy5NJvFG78CU18zZKbh8mnwsfLKuapUTFMzv8c4T1IekZTzbzyhqfC2U/Hggw9y8OBB3ve+9xW7eDNKXq0oIhdj2ZW3A7cB/wDcAPwMuEtEzvKshAHl8ND4Q+fnwdo5sXDa7XqNy+zAx5MZt6lG8YSnIERrgZmZ/AUhbFuSmYigdGRglOHYeMzueh/G7E7SMgPC5shYPOUUGBFo9uGGTUlmKpxiECfLRTVrCUCMc7BWl11259Noz7dt28bu3bu5+uqrqasbX/Xq6Ojguuuu46KLLmL37t188IMf5Pjjj2dsbIyvfOUrrFy5kscee6xo/4cX5NuKHwNuAo5X1RtV9WuqeruqXgucDlznWQkDymGn5tzHg0/rDMVWDsIyOqQ77RVnsO4dHmNgNA5AVTRCU41/haeZ0NQFIWxbkpYZWGkIgn1+ktYZ6I9B2LApyUysZMUT6tpALSjjfVHNWgJicw65mbY8+OCDAFx11VWu9JaWFmpra1mzZg2PPfYYN954Iz09PfzoRz/ib/7mb2hqaip5h9F8R4kuVb1VVSfUnKruALoKK1bwOey0OTeDz5SkD9Z+3Fo9yUzUV7pJi6+FzRmI3R0UkwOYGbOpoEyUYWbCKQapvlxmGkUymzrYN5wyZ5g3p8KXu/UmmTenggrbzKR3eIy+IgUBmGBz7mNcsc6nidjy4IMPsnLlSl7xildMOPfoo48SiUS48soreeyxx6ivr+fcc88lFouxbds2zjzzTM/L7iX5tuLWac6vyvO+ocGtOffvgD0Ty3YHeocdW6v7e7BunQEzjaCYtECaZrNIwsDegNjnQ9pKQ5HqKwgxu5PMzGQ5GFpgMJPlXBER187QxRjDVDUwZi2QfazzsbExHnnkES6++OIJ5zo7O9m4cSPXXWcZcaxfv57LLruMZcuW8dBDD3H22WdTVVXaY1de0VqAZhFpVtUDzkQRqQBuAfxrVDcDxBNK13AwbM7nzamgsjzCyFiCvmErPKDXNqhBcW6EieEBi0FQwrbBzEz+giQMLDbCU044BadibdwUpPpaWFdJWUSIJ5RD/aMMx+KeK0vckW1KW4DKhsWN1eyyIwK1dw+xptnbiECxuKL2vo/lkYhvwzInqSjPzua8vLycnp6ejOd+/etfc8EFF7B69WoSiQTr169n48aNADz00ENceOGFPPXUU1xwwQXeFt5D8hXO7wY2icgPgF1AFXAc8BdAI/A6LwoXVDr7RkhOCP2+bCcitDZWs+NQ8cIDBmlZ2Fn+5IpAmceDabDqq/g250YTnBtBmvy1pK3MFGPXyyDs1pukvCzCovqq1Bizv2eYFfPnePobHQEav6D4m4M5BVi/25tDmllLluEU0/n5z3/OpZdeCsCTTz7JkiVLWLJkCQCHDx/mhRde4KKLLiq8sEUk3zjnD4nIDcDtWMJ4kgPAZar6qBeFCypBEp7A+h+cwvmxi+o9vX+QloWromXMm1PB4YFRxhJKZ98Iizy2oQ+Sw15zfRUioAqd/SOMjiVSNpxeEaSVGWd9HewbIRZPeL7MHZSY3QC1leU0VEfpGYqlYukv8DiaStDG+5aGceF8X/eQ58J5kPojFH/C7LI393EYxST5xDpP59577019PvPMM1Nac4D77rsPgL6+vjxLODPk3ZKq+j1gCXAhcBWWtny5qv7Mo7IFlvbADT7FDQ8YpGVhKL6pRnuX/7eiTxIti9BcZz1fqtZqg9cEafJXUR5hoS1cqrojhXhFkPtjMYSnINdXUcavAPVHSI9oVlzh3O/25jDR5ryYOx2XMgW1pKoOquoDqvpDVX1EVWctNo1YXCwit4vIR0SkZNcsgjxYF2PwaQ/QsjAU31QjSJpzKG7s7rF4whUJyOtVjNmgmE57QYrZncRld+5xfVkbNgVL2Cz23gNBez+2NBR3MjMaIGdQgLKIUG6blqkqY1lsRBRESqYlReQcEWkTkR4R2S8id4nI3CyvPRF4Bvh/wBdV9Quq+quiFrgAgmTjCjOreWptrPH8/jNNMYWnWDzBgT7rhSkCzQ3+F56KGbv7YN+IIxJQpa/9P5IUM8JNkGJ2JymmJrh7MMZQzNpzoLaynPqqfN28Sgd3OMUijPc9AX4/FqG+ghTjPIlLez5NOMWgUhIjq4icD/wTcCvwBuDbwHuAu7K49mzgceBPwMWquqeIRfWEoNnULXG+3LqKqzkPwmBdbOEpuQq4oLaSynIjbE6Fa1XGxzvPOinmSkPQ7KchzWGviM/X4sYqX+85kKSYmuCBkTG6B61Y4BVlEebP8b9ywdkf99tOx14SpBjnSbxwCp2MkVic4VicRImby8z6NF6s0eoNwJtUNW4nPyUiZzJN1BcRWQzcB+wF3q8+MU5qD8hW9EmKqXkK0tbqSYpZX4EUNosYfjJIkTSSFHMla1/Axi4orplZ0Ew0oLjPl1MT39JY5fuwgAA1FeU01UTpGowRiyud/SM013s31jiF12gAHELB/X9MFes8Hw70jtA9ZFlgLy0fpammNGWKUmhJAT7jEMyT9ADPTXPtp4D5wC2zae+eK0Fy2AO3ne6B3mHGPJzpBmm3yyRGGMiNYpq1uDSbDQGpr4YiaoIDtMFVkuJOZoJXX+krWV7qxFzOoAHpj1CcZ0xViSc0ZZYnMm6r7XcqyrKLdZ4PLgdaj+qrGHrhogjntplKVqhqQlVdT6uILANWAtdM8RuVWFFiFFgmIk+KyKCIbBeRm6REpbi+4Ri9tia4ojzCvABogquiZcyvtZYfEwoHbIcxL9gXRE3wDAkDQTCZguJuSR904cnU1/S4V7K8ncw4d2kNSn+sry5nToVlLjc4GqdnyLst6YP4fIH3TrRlZWXEYrEJMc5LVOzJGS/CKU5GMaLbxGIxysq8NSHN26xFRE4HXoMV59z5RJQDl2FtSpTrPSPA24AvA08Cg1NkXwvUYGnYn1DVz4tII3Ab8DmgHsuOvaToSBusg9KZWpuqOdRvCeX7uoc8exEFUZOyoLaSaJkQiytdgzGGRuNUV3jTsd31FTwzDa838QhaJA0ors25ewOiYDxfzXWVRMRSLBzqH2FkLO6Zr0bQ/GXA0tC2NFbz0sF+APZ2DdHokWlAEM3MwPuIQHV1dfT29lJZ25BKC0KkliQum3MPHUJV1WUm41Wd9fb2Ulfn7c6veQnnIvJJLJOSySTLfHX81wKnADHgcuAMETlBVTNFi19s//12ctMjVe0WkWuBNwE3iMjnMl1r57kWoLm5mba2tjyLmzvPdY6lPlfr0Iz+djGJjo4LOQ8+8QwDu7xxZ3hy67i10mj3ftrajnhy35miv78/Yxs3VkCnPUb/9IHf0VLrzSCxacd4O3S1b6et7WVP7jubqCrRCMQS0Dcyxq8ffISaqDeT2m17x+f/+7c/T9uhLXndZ7J2ng1c9TU8xv0PPUJ1uTf19eK+8fpqf/F52va/4Ml9Z5vGSuHIsPXa+un639E8Z2J/zKeNt7w8Logd3LmVtp6XCipnqVCVcIz3j/8fhxZ6M94/s2V81bX3wG7a2jo8uW8uFKMvDx0Zf489/fyLHJ3YXdD9RIS6ujrqmbSdkwAAIABJREFUmhagkWqIlCGJeMlvrJMtzvCJo2Pe/V9jCUVt8bRMYGCgP+97qSrxeJz+/n46Ozvp6+tj69atnpQTQPKxlRGRTmA98EWgK+10PfBZVX1b3oWyNOj/DlwHXK+qt2fI83bgXuBGVf1y2rnvA38FnK2qj031W2vXrtUNGzbkW9Sc+f6TL3PzTzdxfflPuL78vuwueuW74C1fdaf9/O/hmXuyu/7cj8J5H3On/fAvYdtvsrv+4ttg7bvdad88Bzqeze76d/4XHPMX7rQvHgP9+7P7+ZFb2KSruPXSk7ls7VIr8dMNU1/k5IYtUN8y/r23A758bPbXf7rH/X3fRvjWuqwuHaloovLju9yJW++HH12R3W+3nALv/707bcN34ZfXZ3f9mgvhyh+70x75N/jd57K7vkSevfO/2JbahXbLkn+j6tCfs7u+wGePa9tg8anuNJ88ewe0kVePfJ0HPnQOa5ptrU4Bz56q8qlP/SOfjXw7u+sD8uwlaWtrY93WT+Y07r363igHei2B89GPnMfS75waimeP2kXwj2mCSg7P3p8TKzhy1UOcu2bBeKKPx71fPLuP6p9cyQVlG6e4yEEW79yR6maOrHwLfYvOIB6tdeedswCiaSuBve2QSHftm4S6RVCWthrSncOEor4VIo7Vp0Tc+v0s0YZl7OseSml5W+vKkGz7TaTM+n0nsSEY6HQl9TOH2sZ5WZcpE2VlZdTV1TF37lwqK7OLLCQif1TVtdPly3e6OwzcrKq7JvnxG/O8L2DZoYvITVja7TWTZEuGTJyf4dw++2/JBcUuRhzwsBAUm3ND7ixurE4J57F4guAseBef9u6hceG8AJLRJkoijIAPiMUTHLT9b0SCscHVTBIos5bG6glazEKpHDpAy+Y7adl858STGZUSl+Q4MUyzTP70a7IvXMaJ4Wuzv/7TPbzvcw+nzMIe++smlqy/PLtrJ5sY3ueeGP5XxaVc8fFpo3XPGvkK5/8BnArsmuR8Y573TaGq/SLSBUw23doE9GI5jqYTtf+WXMzz845dSHW0jCUbImDk9JwIikOVIXec9rpex70NOl4pBIxiITe6B2OoWq+i5rqqQNkEzwQtAfExAuvd5bVwHnRaG6tTwnln3whLPL5/9awHEp+afM1aTgVuxhLSx9JOR4AbVPUtBRVMZAmwHThBVV+y05ap6m5HnjuAK4CFzlCKInI/sAo4drrY5zNt1pKkra2NdevWzfjvFotN7T1c/O+WBdGa5loe+NC5Bd9zLJ5gzc33kzQ/23rLhb7bVGeydv7BUy/zT/+7CYBLT1vCFy87peDfen5fDxd91WqDoxbW8tANhbdBqfCVB7dx+29fBOC6dav5yIU5LM9PwsNbDvCeu62+f/bR8/ne37w673uVWn921tffnreaD7+x8Ppa//x+3v+9PwJw3jEL+O67X1XwPUuF7/1hF5/42fMAXHH6Uj73jpMn5Mm1jf+w/TDvvPNJAF65rJH7rstBc1jiFON/O9g7zKv+9bcANNZE+dMn31DwPfOhGH05nlCOufn+lC31ln++0JPdiNfd+gi7Dlt+IA9+6ByO9mCFrFS4/r828tM/WUYQX7j0ZC5PmrQWwGd/sZnvPL4TgMvWRLn1PTP/jGVr1pLvVP4erKgqDwKPpB0PAxdleyMRaRCR34rIZ0Skzk6bA3wTy948KZjfBLwsIh92XP7PWM6jH3bc70TgXOA6v2xKFARc4ci6hjyJ+3mgbyQlmC+oC8Zul0mKEU7RGYM6aKsMxQgPGMRIQElaPQ7dZt0nmGHuoDgbgwW5vorxfAVxz4EkZRFxmTV5MYYlEuoK1dkSsGesGO9I5yZXc6tKeyUrX8X+vcADWKYl6VJYNfDeHO41iiVg/yNwrYisBw4Cn1DVZxz59gP99l8AVLVdRM4APi8iPwaOAM3Ahaqa5kVnKCZNNVGqohGGYwkGRuP0Do/RUB2d/sIpCOKGJ0mKIWwGWRjwOk6wdZ9w1JcRNqenGIJAEPccSNLcUIkIqMKBvmFi8UTBZjtBDGvqZHFjNXvtd9q+7mFWLaid5oqpOTwwmgoz2FAdpbayxO00cqQofdIxmZlbVdphrPNtzR8CPap6MNNJEXkmU3om7A2ILswi3z1YGvv09JeAd2T7e4biICK0NlazvdNy2tvXPVSwcO7skEsCNlinC5uJhBa8VbV7w5PgOFNBcWJ3B3GDqyTF2IXWKTwtCVx9ufujqha8B4UzJnzQhM3K8jIW1FZysG8EVdjfM8zSuYXFXwhqjPMkXitknPdoCaCzsXsM80Yh09Ht1JyXtnCe11RXVV9U1YMi0iQibxOR94rIhSJSZZ9/2ttiGvxAumlLoQRxA48ktZXlqcnLaDzBoYHCd1VtD4lmc3/vMGMeOIW6V2aC9Xy56qtnOLXFdyEE+fmqrypPaR6HYnG6Bwvf9bI9BJrgJF4Im0F+vsAtQHuhYAjyygx4/3yNjiXo7B+PntQUROEcUhv57AZ+gmUf/issm/BLPCqbwWe4NAM93g4+QRysvTbVCPJgXRUtY36tFXc3ntBUiLpCCEt9jSWUg33ePl9B648i4vnqzL4AKxfAjPe5YiYzubE47fkq1I/tYN8wyVssqK2kvMCV6mKTl3AuIhcDd2CFObwN+AfgBuBnwF0icpZnJTT4Bq/tXIM+WLd6vdIQYBt98PblNhZPsL93XGANYgxqL+trZCyemhBFxNryPmh4WV+qGujJH3hvdtDRE+yVBu8nM8Gur/qqKHX2atZwLEFXgatZHT5zns1Xc/4x4CbgeFW9UVW/pqq3q+q1wOlYO3saQobXmuD2gL/cWj20C3YKT0Hd8KTVw8lfkCMBJXFGvGgvsD/ud7zYFtVXUR7AmN1eKhd6hmIMjlq7MdZUlBXsf1OKFFMZE8Tx3uv3Y0dPsFdmwNsJs8tGv7706yvfEbZLVW9V1QmGn6q6A0y8/TDi1gQPFnQvVQ10aEDw9uV2oGfczCOoG554WV9BX5UBb19sQV9CB28d9tLrq1Dn0lLEy+drOBbn8IC1VUl5RFgQyJUZtzKmUDONoE9mwNtAAM4JkR8CAOT7Bt86zflVed7X4GO8jH3bOzzGgK15qo6W0VgTPM2Tc4AodOAJsvNskmJpUYIYGQK8jdgS9CV08NZMIwz15eVkxnn9ooYqykrcHjgf6qqi1FVZZhojYwmODIxOc8XUOFfD/GCmkQ8tRXrG/NAn8xXOm0WkOT1RRCpE5AtA8Ka9hmlJxr6F8di3+ZIeScNonqbGZQLUVFhIs1LFSxv9vc7nK2AbniTxsr5cS8JBncw0FGllJoAmZuDtxnNB3oDIiVcKrOFYnEN25JGyiATSBwSKNwH0g0Im3zjndwObROQHwC6gCjgO+AugEXidF4Uz+AsvY9+GQdhcUjQzjdIfePLBy5WZIMc4T+LpyoxDuA/angNJircyE8z68nLjuTD0R7CesS37+wCrT560pCGv+3SEwAcE0lazegob89NNzY4cKuh2RScv4VxVHxKRG4DbsYTxJAeAy1T1US8KZ/AfrU3VKcfEfd1DeQvnYXi5za+tJFomxOJK92CMgZEx5uS5y1s4hCfvzDTCYEPtqbDZE3zhaVFDVWrXy4N9I4yMxfN2FN4bAmHTCj9ZzQ4PNp5zx4QPpnIBvBvDwqCMAfcqitdmLUcKulvxyXu6parfA5Zg7e55FZa2fLmq/syjshl8iFdOe+0+W4LKh0hEaHEMPh0FhNcKg7A5d04FVVFryOobGaNnKP/QWkF3NgaYN6eCinKrvnqHx+gb9qa+gvp8RcsiNNeNjzVOJ+tcCUN9gbvvFDR+ufpjMFdKAdd4X4iwGfRIZkm8UjD0DcfoHR4DoKI8wrw5FQWXrdgUtBaiqoOq+oCq/lBVH1HVUQARucqb4hn8hlc2Yq7BOqCaJ3DX194C7ILDsCyc1NQlyff5So9BHbSt6JOISJrwlN+ysKqGSBjwJjpEGFb+wLtwnWHRBHsV6zwsk7/kahZYq1mjY/n5sTnHvlafRE/Kag1dRN4EoKq/tr9fApw8SfZq4HLg+14U0OAvFru2KM5/sHYLA8HVpHgR+zZMwlNr2jL6cS31Od+jZyiWigQU1BjUSRY3VrHzkFVf7d1DrGmuy/kehwdGGbFfivVV5dRVBbm+qnlmdzeQ/+QvfcOmIO45kMQrzWZ7CCbLkL6y7NVkJrj1FS2LsLCukgO9lh/bgd78/NjSA0z4gWwNXH8EKOP25U3Ap6fIX1gAT4NvcTpvejVY+6Uz5YNTy51vfR3qHxee6gIuPHmxEVH6RMYPWpR8cWk281yZCYuWDrxZ+XNu2NRcH8w9B5J4YUOdSGjahjrBfcY8szkPgQ9IksWN1RzotSa77Xn6sfkxGlC2wvkVgNMz5qfARcBHgXRDxhrgnwovmsGPeLEsPByL09k3HiZqkQ9288qXVg/qKyxac/DGpyFUwqYHk78wmAAlcdVXnmYHoXq+PJjMdPaPEItb+rymmig1FfkGkSt9muuriAgkFDoLcDoOg89MksWN1Wy0V7Py9Wvw40pDVr1AVe9P+94lIrer6kuZ8ovInV4UzuA/0gdrVc1ZMxmGrcKTeCFshkl48sIMyB2mM0z1Vfjkzy8vtnzxwoY6rJNlL/pj0J+vaFmE5vqqlA30gZ4Rls3LTROcSKgrrGBLgM2mwJsQun70ASkkWkvGcIki8kpV/V3+RTL4mYbqKDUVliZgcDSeV0SNsJi0gDcbxYRVU9feNZjXPfw4UOeLFy+28AqbHqxkBXzy57Sn3987zFgeG8/5UatZCIUqZA4PjKYcI4PuAwLpfmz5KrD8t2NvXsK5iHxnitMqIv+SZ3kMPic9QkQ+nSlsS3ZJ9vcOE0/k7q4RJuHJCJu5YcyAciPTyl+uhEnYrIqWMb/W2p0yntCUI2wuhGm8h8IngO7IXMENlpCkJaQTZk/tBUSkDFgG/D8v72vwF+nbOudKGDbwSFIVLUvFXLVebrkLnH4cePLFGVrrQN8wsTw0dWEJ0wnuJe98NZther7qq8uZ4+HKX1A3BHPSWqCTY5hWsqBwp9Aw7AHipFC/hnhC2d/rPzOgrIVzETlWRDaLSBx4l4jE0w9gFLgP2FesAhtKHy81A0HXPEHaNut5TGbCpHmqKLdCa4G1k+P+PGJ3t/twiTNfLM2mc/KXu2YzTMJTeiz9fFYb/LiEXgiF1leYJn+QtutlHg6OYXs/FurXcLBvfEV6fm0FVdH8dv2dabJ2i1bVLSJyOlZYxeOAxzJkiwN7gbu8KZ7Bj7g0KfkITyESNsEarJ/b2wNYL6q1OV4fJjMNKCy01nAszqH+8UhAzbagH2RaG6s51D8KWC/2XF7og6NjdA1a2uNombCgNvj1tbixmhcP9gOWMHDC4oasr00kNFQ+M1C48BSmyTIUHus8bON9U02UqmiE4ViC/pExeodj1OdgZ+/XyUxOMYtUdUBELgXeqar3FKlMBp/j0gQXqEkJevQRSA93l9tg3e/Yxr6iLJKy/wwyztBaua7MOPMHPRJQksWN1Tyb5+TPWV8tDdVEIsGNCZ+kkJW/sDnrQeErpU7H7jAImwWbtYTIBwTGV7Ocm8/VL8q+X7kmfz6JcQ552Jyr6qgRzA1T4Vq2y3HwCdOGFEncmpTcIpCkb3sdBuHJacebqxmQc/ITBkEACtNs7g3ZKhYUZkPtNtEIvrMeFFZffcMxeofHAMtkLel/E2QKdTreF7L3IxRmd+5XzXlR1EYiclUx7mvwB4U4hIZpQ4ok7pdbbsJT2LQokCZs5miz6Zz8hMG+FQqd/DkmM6Gsr/wFgTA460Fh9dXR454sh0G5kB5uuHdoLKfrnX0yDCvL4HbizNUUKF2B5ReyknxE5E0Aqvpr+/slwMmTZK8GLge+70UBDf4jGVFDFQ7muAva3hBF0kjS2jiuYct1MhM2+0MozGbTPZnxz0BdCAVN/hzCfCgnf7lqzkO40uBVfYWlPybNNF6y/Rrau4doqMnOTGNwdIwjA5b/SFh8QMDLZ8w/fTJbteSPAAUa7e9NwKenyJ97cFhDYIiWRWiuq0qFL8plFzSXM5WP7MMKoRAbxLBFOoDCNiJqd5m1hMPswKsXWxjCAkJhsfTDtNtlknlzKqgojzA6lqB3eIy+4VjWtvZhHO8Bl3C+r3uI4xfXZ3Wdy2emIRxmjODuSx05jmF7HO+IpT4yNcvWrOUKwGmq8lOskIlrgJVpxwlYwrwhxOTrFLrnyHhHWpZDFA4/M3dOBVVRqyv2ORw8s8FZX2HR1KULT7nYbO7tCp9ZS/outLnVV/iEzeZ6dyz9pINnNoRx5S9947lcJjR7QtgfIT2iWS7vx/G8fhI0CyXf50tV3XU21z/PWFbCuarer6q/dHzvAm5X1ZdU9eW04wXgzmIV2OAPnJ1pTw7azd2HHcJ5ltp2vyMiLHEMtE6BezqceZfPm+NpuUqV+upy6iqtRb+hWDwVJjAbdjvrK0STv6SNa9/IGN2D2U/+XPUVkv5YUR5hUb0lPKka5UI2OMf7vTmM93tC+HyBe5Vgbw6mjH7VAhdKvn4Nh/pHGYrFAairLKeh2j/Rk/JyCBWRk4EeEVlqf4+IyBdF5DkR+TrwtJeFNPgP54vJKXBPh1MYyCV+td9xCoov51lfYREGRMQ1cdt9ZCCr64Zj8ZSpVUTCowkWEXd/zHLyNzQaT21aVBYR3+ys5wX51JeqhrI/gnuszmX8cmo1w1pfuShjwqi8gvx3OnZNZubWIOIfM6B8o7X8EcvpM+lm/EngQ1gbEEWALxZeNIOfcQ4cL+cy+JiXW9bCQO9wLLVBjHPnzDDg1LJlKwy0dw+RtOhoaaimojz4Mc6TuISnLJ+vvWnxp8MQEz6JW7mQ3eTPpaWr8peWrlCWz8t9/ErPGyZNcN6Tma5wKq+qomWp91s8oa4oP1Oxx6Xs85cyJt/R9h5VvVlVO0RkDfBR4BFVfZOqfgA47F0RDX5keR4vt9GxRCrGuUh4bKgh/eWWXX25Bp6mcIQhS5LPZCasEz/Irz+Gur7yEDbT68tPWrpCyWeloWcwlvKvqSyPsCCkyoU9Rwaz9gPZ7bI5D8/7EdzPWLYTGqfJkN8mf/kK5392fP4qVtSXjzjScgvcaQgcTvvnbDV17d1DJOwxalF9FVXR7MIvBoF8NMHOJc6w2JsnWT53/P/N1mwqrPbAQJoZUO7CZpi0dJCfZjPUz1cewrlTCxy2ycy8ND+Qriz8QCznxhA/Y67V+DwUWD6rr3yF81NE5K0icgfwBuA/VPUZABGpB670qoAGf7KwrpJK22yg26EhmYowCwPLHMJm1sJ5iAfq5XmYTYXVXhPy0zqF0Rk0iXOya1Zmpid98pdITK8JDrOgme4H8nIWq1ndgzH6Ryy9Z01FGXNDsJuqk3wUMrtDaNbyReATwDuBbwAfEpGoiHwWeAE42qPyGXxKJCI5O4WG+eW2pKk6Fb6to2coq/Bt4Z7MFCZshu35cgqb2TqghVl4StcEZ2N2EOb+WF8VpcneSGd0LJFyJJ6KMNcX5L7aEGazKchvdTl9dcZP5CWcq+pmVV2rqg2q+reqmlDVmKp+UlVbVTU89giGSVme4zKUK6yWzzpSoVRFy1Lh2xJZhm8Ls7DZ0lBFuW1jf6h/hMHR6S3pwlxfrY3VJF0SOnqHGRmLT3tNmOurqSaaCtc5OBrn8MD04TrDPJmB9Anz9ON92IVzl19DFsJmWJ1Bk+QaZGIsnnDFRF8SEptzAESkSUTeJiLvFZELRSQ8sbYM05KrqUaYzQ6gsJdb2MwOyssiLHE4RE2neQpzmDuwovm02LGVVd0h7DKRXl9hEwZEJGe789AL5zmaAoW5P0Jh9eU350YvcCrssnGi7egZJm6bVy2oq/SdD1vewrmIXAvsBn4CfBP4FfCyiFziUdkMPidXzUCYhQHILULEWDzh2lo9jIO18+U2nfB0eGCUwdHxzSgaa8IT5i5JeoSIqejsH2E4ZplWNVRHQxUWMEku9TUyFqcjhDH0nSybm/1kGcxkxqWMybm+wvd8zZ1TQa29mtU/MsaRaVaz0qOZ+Y18NyG6GLgDaAduA/4BuAH4GXCXiJzlWQkNviUX72pVZZdDWxw2sxbIbeOmjp5hxmytwPzaSqor/KUV8ILlOdSXU3hfEkJ7TchtZcZZn35zpPKKXPwa9naFN4Z+EpfDXjbKhW5/bqvuFema4OkIu/JqghPtNHW20zHGrfBhNLN8R5CPATcBx6vqjar6NVW9XVWvBU4Hrsv1hiJyjoi0iUiPiOwXkbtEZG6O94iKyAYRuSbX3zd4Ty7C0/7e4ZRms6E6GjpPdEjTBE8z8Gzv7E99XrXAfwOPF+TiULXD1FdONps7OsdfbCvn1xatTKVMLuEnnfW1Yn74BCfIbe+BvV1DxOLWbGZhXSU1FeVFLVspstjhB7K/d5jh2NR+IO4+GdIxLAeZwllffhzz8xXOu1T1VlWdEFJCVXcAXbncTETOB/4JuBUrNOO3gfcAd+VYrk8Dp+V4jaFILGmqcTmhTTX4bD/o7khh1Gw6JzO7Dk2t2dzuGHhW+3Dg8QKn8LRrGk3wDkd9rg7pi215Dj4grslfSOvLKQhM+3y56stMZqZ7vnYcMpPlivJIyvxJ1b0jbzqDo2OpXTHLIxJKzTnkFrFl5yF/KxjyFc63TnN+VbY3EksKewPwJlX9lao+pao3A48Ar8vhPq8Fjss2v6H4VJRHUoOIqruzpOMcrFcv8F9H8gLnS2rX4QFi8cnDKTqFgbDWl/P/fulg/xQ5YftBpzAQzvpyPl/T1pfPtU5e4HxOtnf2T+mAtsNMlmmpr6IqaokURwZG6ZrCJthdX+Hsj+A2t3DWSTrOc8vm1RAtC5/ZFKRPAKeeMLuFc//1yXxbuFlEmtMTRaRCRL4A5LIPrwCfUdV0tWoP8FxWNxCpwzK1+XgOv2uYAY5eOD7wvjiFQOD3JSgvqKuK0tJgBTyKxXVKzYAxa7G0KMlwih09w6kNOjLh0pyHVBhYOX9OaiVrT9fglCtZZrIMixuqUrs4dg/G6JvC/8zdH8NZX5GIuCfMnZOP9+7JXzjrC+CobN+PjvErrCsz4P7ft0+h7IvFEy7TKj+amuUrnN8NbBKR20TkehH5qIjcA+wFrgc+le2N7BjprrheIrIMWAlck+VtkpsiDU+X0TCzHLWwLvX5pQN9k+bbbjTBgHuwnkq7aTRPEC2LsMKhEdk+SX2NxRMuLcvKkE5mqqJlrpWsyTR1sXjCZc8Z1smfiLj6476BKVayDhnlAmQ/fhnlgsVqR31NNn6B8ZlJ4nq+DvRNupq1+8hgKoyiNcn2n09DvpsQPYQVneVq4MvAvwJ/DcSBy1T10XzuKyIREXkH8CiW6cy0Lswi8nZgp6puzOc3DcUlH815WJeFIf3llnky0zscS+3AV1EW8d3mCl6SzfO1x+F8tqi+KhWOK4w462syzebuI4OpSEAtPn2xecVRjonvvv7MwnnXwGgqrFtVNMLihvBFHknirK8XD2Q53odYE3xUlisNrpVlH5poeMX82opUGNyB0XjKDj+dnZ3+V8bkPeqq6vdE5F7gLGA+0AE8rqrTb6U2OdcCpwAx4HLgDBE5QVUzSiki0oI1KXhHLj9ix2i/FqC5uZm2trYCipwf/f39s/K7M01Pz/jS+bM7D2T8n0fimgqrFRHYtWkDeyPBcAjNtZ0TXbHU58f/vJ0TI+0T8uzoHq/TBdXKo7//XUFl9DPlQ+PDzcMbNjO/76UJeTYeHDd3aSofLUq/80t/rhger6/fPr2J+q5tE/I8c6D49eUXIv3j9fVy10jGunixy9Efq+D3Ie6Pw53jz87TW3fTVndwQp7BmHKo31IulEfgpeeeYkcJBQCYyb7cMzKu+d3W0cMjjzySMRjCszvHjQt69r5I2+COGSlfKbKgIk63rba998HHOWnBRDH2wZ3j79HKkZ6M7VnqY3bWwrmInATcAqwGNgAfU9UO4AGvCqOqd9i/FQH+HSsk43uA2zOUR+z0D2WKGjPN73wL+BbA2rVrdd26dYUVPA/a2tqYjd+dadaOjPHZP6wH4OAQnHnWORNiAD+7pxsefByAFfPncMH562a4lMUj13au2XmEezb/AYBemcO6dWdPyNO5YQ9Jd4yTVzSzbl14AxT1NLbzi+1/AmC0ai7r1q2dkOeFtu3AFgDWrlnKunUnel4Ov/Tnzto9/Hqn9ezEauZlfHY2t71E0uf/tKOLU19+YWTBfn7y4h8BODRalrGND/zfbnjqzwCcvLKZdeteOZNFLClaD/TxH3/6PQBdYxUZ62vj7i747ROAZW9+/nnnzmQRp2Um+7Kq8qmnHqR7MMZwHI459TUTNrBKJJTOh9envr/jDWcxvzYXt75gsf7Ic7z49B4AqhetYt3ZE+OP/LLzWSwrazj7FWtY99qVE/KU+pidlVmLiCwBHgPeDByPZc7ysIgUZT3dFrZvAsaANZNk+1vgt8B+EakSkSrGHVGjdlp412NLhNrKclrtwWYsoRk9rDd39KY+H9dSP2NlK0WcZi3bO/tJJCba1L3QMb6QdOwiU19JJjMDesHxfB3bUpcxT1jIxibY9XyZ+kp93tef2b7VWV/HNIe7vpbPm0OZverZ3j3EQAYnbWd9rQl5fYmI2xQoQ5/cfWQwtQfI/NqKUAvm4PZj2z6JKdCW/Y4x36fvyGxtzv8OS1C+AXgL8HngKCznz6Kgqv1Y8dInrutbXIq1S+mQ49hin/uW/f3mYpXPkD3OF9yW/RMFKKfwdHzIhfO5cyqYZ2/ANBxLZNzMY3NHT+rz8YvDXV+rF9SSXAXefWSQodGJEUheMJO/FM6+uPPQAKNjExcdTX2Ns3xuDdEy6wHrGlF6hmIT8mw241eKivIIKxzh7jJNAM3z5Wa6CbNLueBTQdNLXBGqczDUAAAamUlEQVRuMvg1jMUTrvRjF/lzApitcH4BcLGq3qaqv1TV5A6hry9WwWxtfQPw3460ZY4s1wFnpB1vt8/dYn//drHKZ8gepwC5qb1nwnn3YO3PjuQlzvr6c1p9qSqb95n6SlIVLUs5SCXULSgBDMfiKe2KiH8Haq+oq4qmtkqPxZVtaRGUhmPxVGQIU19QXhbhaIem7vl9E/uja/wK+WQZ3ALk8/t6J5w3yhg3ztWDzZnqy6HQCvt4D26n9q0ZIrbsOjzIiK10WFRfRZNPdxvPVjivVtU/pKV9C1hUaAFEpEFEfisin7HjlSMic4BvAter6kt22k3AyyLyYQBV3ayqTzoPIBmxZbudtrfQ8hkK5+TWhtTn5/ZOfLlt6XAOPmawPnnJeH2lC+f7eobpHbaWiuurxk2GwszJSxpTn/+8t9t1btuBPpKWQSvnzQl15JEkJ7eO11d6fzT1NRFnf0xXLrR3D9Hn6I+L7X0KwsyJrZOPX4mEulZPzXjvrq/plFdGc25FkGqyI7b0DY9N2A/EadJyjI+VC9kK54fSE2yzk4yhDkXkvTmUYRQrOss/AttE5G6sOOmfUNVvOPLtB/rtvwYfcfLScWFgU3uPy4561+FB+my7xMaaKIvqzcvtJOfLLU14cg7ex7XUZ/TsDxvO+nou7eX2/D6zhJ7OVMLTZlNfEzhxCuXCC2mKBdMf3f0xXdjc2zWU2iysqSZKc3247acBTlhcnzLNe/FgH4Ojbjt9YwbkRkQ4yaGQSR/zg+JjlK1wPlkPmiCci8hcLLv0rFDVIVW9UFXnqGqLql6jqh9R1WfS8t2jqnWq+r0p7rVLVUVV78729w3FZ3FDVcqOum9kjF0Op9A/vtyV+nzq0kbzcgPXwJM+mXnGUV+nOCY9Yca10pAmPDmfr5Mc+cKMa/LX7l5peGb3eH2d0GoEAZha2NzoqC+nEB9mnPW1ZX8vI2PjfiAb9zier8UNZrwH5lSWp5xCE+oWLg/1j7C3ywqjWFEecdlbhxnXavwe9xi2cff49xMX+7dPZiucny4i8fQDODNDWidwUfGKbPAb1kx3vJM4O88fXz6S+nza8qYZLVepkj6ZcW6lvsEhbJr6sjh+cX1qW/rtnf30DY877TmF87WmvgC38LR1fx/DsXHhaYOrvubOaLlKlWMW1VFuP2C7Dg/SMzj+fJn+OJGGmijL7J1oY3Flq8OMxalceKWprxTOPvnsnvEJoLO+Tm5tmBCGOKw45Qmn5jyeUCs0s42fn7FcWlpyOAwGF07B6PGXxq2kNuxyvtyMMADWZOY0V30dBixnPadm2AgDFjUV5anl3oTCUzusCd+h/hF22tuqV5RFjGbTpqEmmtoCPBbX1ATmyMBoaifCaJm4ViTCTFW0jBMcjp5P7rT6YyyecAkCZvI3jnNV7+md4wqYDWaynBFnX3MqFP6420z+MnGKy8+oJxV1atuBPgbsiF3N9ZW+9gHJVjj/JTBXVSPTHcBc4MfFK7LBj5x19ILU58deOoSqsq97KBXXtaIswilLjTCQ5Kyj56c+P2ZPZjbs6mI0bg1CK+fPCX28WydnHTWxvv6w/XAq7cTWeqqiZTNerlIlU309ucNZXw2mvhycsXq8vp6w6+vZPd2pqBBL51az0PjLpDhz9bzU5+Tz1TMUS5lsiMArlhmzvCTO5+vx7YdSpozOiY2ftcBe01xfyZImKxjCUCyeMi9z1tepS5t8bTaVrXD+dVXtnj4b2Pnuyb9IhiByUmsDDdWWh/XBvhFe6Ojj4S3jWzu/etVcExnCgVN4enL7YUbG4jz0woFU2rlrFmS6LLS81lFfv3+xE1V1PV/nmPpy4ayvx160hCdnfZ3tOG9w98dHX5pYX2euMvXlxFlfT+04wuhYgt9v60xFAjqptYH6qugsla70WNNcm1K2dA/GeH5fL0cGRvmTvTITEXjVCrOynEREODtN4QfQttXRJ4+aN+E6P5GVcK6qv8nlpqq6fvpchjBRFhGXgPS/G/fy82f3pb6ff+zC2ShWybJy/pxUPOq+kTEeeP4A658fD1T0uuNMfTk5fcVcqqLWcLajc4BndnfzW8dk5nXHNs9W0UqS16yal7Kj/nN7D9sOuCfL5x9n6svJ2hVN2I8XOzoH2LK/lwc2O54v0x9dLJ1bk7I7H4rF+f22Th501JcZ792ICGc5hMkHNu/nkS0HSYbwPnVZk2/jdReLsx2ryw9vOcjg6Bh/cKz+rVvj72fMeBcYZoy3v7I19fnOR3emlqDKIsJFJ7XMVrFKEhHhbacuSX3/ux9tpKNnGIB5cyp49Up/awW8prqijAtPGN924bI7nkjFg1/SVM2JJvKIi4bqKOuOGX95XX3X0xwZGAWsOMInG/t8F1XRMl65cNzM5x//59nUbo7V0TKXGZrB4k2OMf3OR3fwG4dy4QIz+ZvAGx3j133PtPPj/9uT+m4mfxN57VHzUw6yz+/r5UsPbGM4ZpmZHb2wlmWOnWr9iBHODTPGOUcvSGmDnVx4wiJjr5mBy05bQllkos3cX56+1HjtZ+Dy05emPjuiT3Llq5f52vawWFy2dnzyt793OPX5itOXEcnw3IWds1rHze42tY+Hu7vk1MXGJC8Dl542rox5aueRlNPeia31xjk7A+cft5BGe3Od9u4hnt41rrx6u0NRY7BoqI66FDJ3PbYz9fkdp/m/vswb3jBjlEWET158giutOlrGh994zCyVqLRZOreGa85c4UqbX1vBteesmp0ClThnrJo3wba8ub6Sq89YMTsFKnFef1wzp6RFZJk7p4J3nbl8lkpU2pwwv4xT05wYK8oivO9s0x8zcdTCOt54wkQN+QfOXT0LpSl9KsvLePeZKyekv+WUxSzycdSRYnLNa1dMSKutLOcyI5wbDLnx+uObuf2KV3B8Sz2vWjGX77/3VayYP2e2i1Wy3HThsVx9xnLqq8o5sbWeu9/9KhprjO1hJkSEL19+Cq9ZZTlOLZ9Xw13vOp3aSqPVzEQkInz1naey2g6r2FQT5WtXnmqer0mIiPCly06htdFa/SuPCLdcciKrFpiNYSbj0285wbVa+o5XLjEmjFPw/nNXuSbMrY3VfOxNx85iiUqbVy5r4l1nuJUJn7j4OOYFIJKZeWsZZpy3vqKVt76idfqMBirKI3z2rSfy2beeONtF8QXzayv5r2vPYGg0TlU0YsxZpmH5vDmsv/4c9nYN0VxfRXWFCZ84FasW1PLQDefy3N5uls6tYXHjRDM9wzgtDdWsv/4c2rZ20lRTwWtWzTV9cgqqomX8+P1n8PM/7WMknuDNJ7eYyfI0fOrNJ/CKZY1sau/lnDULAhPJzAjnBoMhcBghM3vKyyJm9SoHqivKePUq45CdLTUV5S7nUMPUVEXLXP4zhqmJRKzgCW87dbZL4i3GrMVgMBgMBoPBYCgRjHBuMBgMBoPBYDCUCEY4NxgMBoPBYDAYSgQjnBsMBoPBYDAYDCWCEc4NBoPBYDAYDIYSwQjnBoPBYDAYDAZDiWCEc4PBYDAYDAaDoUQQVZ3tMswqItIJvDwLPz0fODQLv2uYWUw7hwPTzsHHtHE4MO0cDmarnZer6rQ7JYVeOJ8tRGSDqq6d7XIYiotp53Bg2jn4mDYOB6adw0Gpt7MxazEYDAaDwWAwGEoEI5wbDAaDwWAwGAwlghHOZ49vzXYBDDOCaedwYNo5+Jg2DgemncNBSbezsTk3GAwGg8FgMBhKBKM5NxgMBoPBYDAYSoTy2S6AwWAwGAxeIyICXAS8HmgHnlfVX81uqQwGw3SISJWqDs92OWYTI5zPMCKyDPgMsBdoABYBH1bV2Yi1bgBE5BrgBmANcABYD3xcVQ858mTVbrOVz5AbIhIF/gB8TVXvdqSbdg4AInIi8D1gH/ABVd2Tdt60s48RkfOBa7DqsRFYDvyzqj7pyGPa2GeISB3wQeBGrDjk6edLuk09bXtVNccMHVgDyD7gGkfaB7C0Oi2zXb4wHsB7gT8Dnwb+FXgeUGAbUJNLu81WPnPk1e7/Yrezs25NOwfgAM4GeoDvYvtVpZ037ezjA3gz0AUsdKSdA/QDR5s29ucBrMN6Dw8CmuF8Sbep120/6w0SpgP4GdaMShxpZUAH8P3ZLl/YDqAaeAiodKRFgcewBLerc2m32cpnjpzb/bXAfUwUzk07+/wAFgOdWJPsiknymHb28WHX4/9lSD8E/INpY38fwONkFs5Luk29bvtZb4iwHPZLIwH8Z4ZzPwRGgXmzXc4wHcD5wKszpP8VluD2kWzbbbbyzXYd+u0A6oBfAsfiEM5NOwfjAL5pt+s7Jzlv2tnnB/A/WNrVVkfaHGAE+EvTxv4+gDbShPNSb9NitL2xOZ85zgAE2Jrh3BYsje2ZwC9mslBhRlUfnuRUt/13M9m3W8Us5TPPS258EfgEkO5sZNrZ54hIJXAVlnC+TESeBE7G0lx9C/gCpp2DwHeAS4H7ReTNwG7gVqxJ90+ASzBtHDRKvd963vZGOJ85ltl/D2U412n/XT1DZTFMzUnAHuA3wN/ZadO1m8xSPkOWiMjbgZ2qulFEVqSdzrZ/mnYuXdYCNVj25k+o6udFpBG4DfgcUM94PZt29imqer+IfBD4KrAJy2foDlX9T0g55YFp4yBR6m3qedubOOczR7X9dzTDuRH7b+0MlcUwCSJSDrwLK8LDGNm322zlM2SBiLQAf42lPc2EaWf/s9j++21VfRRAVbuBa7FekDdgCehg2tnXqOp/AHdiKRjPAP7WIZSbvhw8Sr1NPW97I5zPHMkZVTTDuWTawAyVxTA5HwZ+oKq/tr9n226zlc8wDXa869uBD6lqYpJspp39T9z+u8+ZqKqjwANAFZbJC5h29i0iUi4i38GyO18G/DfwKuAJEWnF9OUgUupt6nnbG7OWmaPD/jsvw7lkPM+XZqgshgyIyFuB+ap6oyM513abrXyGyflb4LfAfhGpstMq7b9RO820s/9JxjKfEB+ZcYH9WfuvaWf/8jGs0Ilr7Mn2X4rIBqxVsU8AyY2mTBsHB7+Mz561vRHOZ44nsbx5j85w7hgsrc/jM1oiQwoReR1wVppgDtm3W9ks5TNMz6XAucAdGc59yz6+jmlnv7MJ6AVWZjiX1F4dwbSz37kU2OBcBVPVW0XkSqwwqZ/AtHHQKPX3sPdtP9thc8J0AD/HckhzpkWwYmPeN9vlC+uBtfnBlzKkNwHnZdtus5XPHNO27/HAa9KOt2GZOPyz/X2JaWf/H1gTsG7SYpwD92NFUhDTzv4+gKeBhzOk/w+WI/CstZ1pY0/at43Mcc5Luk29bvtZb4gwHVjbw/cAVznS3me/TFbPdvnCeGDtJrgTS9tys+P4LLARa0kqq3abrXzmyKvdVzBxEyLTzj4/gFYs589/cqSdiGWf/DrTzv4/gCuBGI49Kuxxej9whWljfx/ABntsjqall3Sbet32Yt/AMEOIyAnAZ7DsIwXrZfIJVd0yqwULISJyGtYsfTIv6h+o6lV23qzabbbyGXLDDqW4E3i3qt7tSDft7HNE5Cjg88AYlhlLM3Cbqv7ekce0s48RkTcCN2LFOO/Ccgz9rqr+xpHHtLGPEJEzgIuAj2PV27eBX6rqzxx5SrpNvWx7I5wbDAaDwWAwGAwlggmlaDAYDAaDwWAwlAhGODcYDAaDwWAwGEoEI5wbDAaDwWAwGAwlghHODQaDwWAwGAyGEsEI5waDwWAwGAwGQ4lghHODwWAwGAwGg6FEMMK5wWAwGAwGg8FQIhjh3GAwGGYAESkXkfNE5O9nuywGg8FgKF2McG4wGAKBiFwuIo+LiIrIHhFpzpDnIhG5186zS0RuEpGyGShbFLgJ+BXwnmL/XqGIyFki8iUR+aGIdInIlzPkuc2uxwER2SIim0Rk1E7bKiKbReSw/f1P9jWLRaRDRP5t5v+riYhIi4jcY5d3k4gM2uVVETlptstnMBjCiRHODQZDIFDV/wYuAA4BS4B7baHYmedXqvoOrG2/r1PVz6tqfAbKFlPVfwGeKfZvFYqILAfWA59T1SuBW4BFk2T/BtCoqseq6onAPjv9jap6vKrOA64GkltRC1Dm+D5r2M/Gr4DVwFq7/IuAO+0si9PyzxGRpTNbSoPBEEbKZ7sABoPB4BWqOiQiA0Af8Frga8D7M2TdDWybybLZjM3Cb+bKVUCNqnYCqOqXJskXBz6pqrGpbqaq3xOR19mf24GFXha2AF4DnAr8var2AahqL3CtiDQBrWn57wDuAvbMaCkNBkPoMJpzg8EQRN6CJaBfKyIfyHA+jj8E5dlgeZb57lTVQ1nmLQkzljSSk4Q3ZTj3KRzCuYh8HGvSYjAYDEXHCOcGgyFwqOom4K+ABPBVETlrsrwiskZEfmLbGV/jSPuUnXa3ndYgIu8TkftF5BYReYOIbBCRYRF5RkSOEpGFIvJtETkkIvuS98vwm+eKyNMiMiQiL4rIe9POR0Tko7Z9/BO2TfR7HOePFpFPisjPRaTVtrXvFpG1U/yfi0TkGyLyHyLyHRH5k4j8i4hU2eePt//X8+3vd9vHmyep4y2T/VaGvFvte1aKyJtF5AZHuV4jIl8WkR12Gb8mIp0i0iMin7LzXG7X8YCIrBeReWn/24ki8gO7PnbZ7bmYqXkSGAEuFJEfO++pqpuxTHYQkdcwLsB/1K6T4+1z1SLyOfv3nrHr9GL7nIjI2SJyh33+dBF51NHmf5n2P1xr18NHReSppJ2+wWAIIapqDnOYwxyBOYBdjs8fwbJvPgAsdaS3ASsc3y+w813jSEvaRt9tf28CLrTT/gRcj2WjfBLQC/wB+DKwApgP/BoYAprSfnc/8BPgIuAaLBMbBd7vyHcX8AXH96/YeS4DarCExQPATiyt9DXA81i205nqpMX+nf/nSDsG6AJ+D1Q40u+2Xg351b1dzhUZzq0GPmefb3OkrwF+aad/AzgTqAe+bqfdCbwDqAXeiDXh+pzj+uPtelhqf18M9APPAZFpyns11gqKAoftNi3PkO8aO886R1oEeADLdwEse/r77PudYf8P78SaAOwFvmj/b1dh+UUkgLPsa08GjgBif68G7p3tvmQOc5hjdo5ZL4A5zGEOc3h54BDO7e/32ILVBqDKTmvDLZyvI004t9NTwrn9Xey076Xl+7mdHnWkvc1Oe4Mjrc0WiMsdacfbglonlh/Q6baAV+/Ic4p9r0cdab8DBoA5WdTJnUA3UJaW/gn7vk6h/W6KIJw78rTjEM7ttM/Y1612pL3STvuXtLzPAk84vv8KuDUtz//a174uizKfAjxi509OvE5Oy5NJOL8Ma4IkjrS3pj8fwMvAU2n3e5Odb739/e3299c68rx2urKbwxzmCOZhHEINBkPQuRY4GkubeSfw1/neSFVVRMCyWXdyxD7vdI7ssv+mO0D2qGrK3l1VN4vI74FzsQT1i7G0rV+1fwssLf6zwKCzOECXqg5kUfSLgd06MTLNz4DPYgmL38jiPl6QyYE0Gb3FWb4jk+TvworGg4hUAq8HGpPmRza1WPVVP11hVPVZ4DzbfOdrWML6oyLyOlXdMMWlF9t/v+top2r7d53+DIq1guLkfqzJ2Kn294eBDuAhEfk81srA49OV3WAwBBMjnBsMhkCjqiMi8jbgaeCqWbDlzWacfQlLOK/BMsvoU9VrPCzDfCwH2XR22n/nZThXyiTrdD4QxdJU31HIDVX1FyLyKNYqyNnAV7HMUCZjMbAzn3ayJ3m7sEyiUNVuEVkH/DeWM+rVIvIeVW3L9d4Gg8H/GIdQg8EQeFT1AFYElwHg81ga6lJDge1YQvRCEVmQnsF2MqzL494dwIqk86eD5AZMe/O4ZymQnHCckOmkiDRMdqGIfCs9TVW7gaRz7iuy+O01khZL3753pa3Vn4oa7HCeIlKhqtuAtcAH7HMPiMhp09zDYDAEECOcGwyGoFEuGXb9tM0X/hpr3EsXfEftvzXJBBFpLFoJJ3Iylv1xJ/A4lm37F0QkNUaLZTvxb0A+wvl9WBrmi9PSk5OUn+Rxz0zI9Fm8Q6245JuAd0vajp4icizwwSkuXy4iV2RI77T/bnekJc1tnIL441hOwjen/W4FlvPnpDvPihVHfTXwfTvpnSKyWlXHVPWbwKuxJmvnTVF+g8EQUIxwbjAYAoOIzMGy8V6V6byq/i+WE2Q6W7Dsgv9ORM4UkQuwIqQcAE6xBb2k4AUThdBy+7xTM53M6xxnx4B6EXFOAv4Ky0QiuVnSz4CNWE6IvxORG0Tkg1jOpHtUNbkLZxRI14RPxmeBF4FbkiEDbcH/Jiznyf9x5J1jn89pEmBPHprsr01TZK22DyeZ6jVpupL+P1bgrtPPYJX5DyLyJbHCXd4K/CfWxkGT8SJwt4i8OzkJsv9+HMtB9+OOvMmVhTeKyBLbPv0uO/2TIvJLEfmgHSLyCeAhVXX6B6wRkRbH91uwbNNvd/zfNzjO78d6Hp+YovwGgyGozLZHqjnMYQ5zeHEA78KKXa3AZuDaKfL+kLSIIljRN3ZiheH7H6zwg7/GEv5OwwqR+AX7/ruwwuRFgCuwInIocBtWaMDTsJz+FMvZ7xz7N07A0pY+DfzI/p2vA81pZVmAFWXmiF2ep4B32OfqgI8Bw/b9Pw+clkX9zMcSVjcA37Lr4KPYkWOAZiyTii77vt8FLsmy7i8BvsN4xJNHgb/HHaLxOCwts2I5vN6AJcS/Ddhqp99u5zsOy+Y7aepzBZZQ/kEsoTVuf55n3/tyrNCJI1g7eH4jvU4zlPkfHOXtwAop+X92u706LW8ZVrjHAazJzFw7fZX9vRcrGs7DwHlp1+7CigBzD/ADrPCLXwXqHHmuscvxv3ab3AX81Wz3KXOYwxyzcyRjqhoMBoPBYPAY2/Fzl6qum+WiGAwGn2DMWgwGg8FgMBgMhhLBCOcGg8FgMBSPXHwDDAaDwQjnBoPBYDB4jYgsFpF/w3L2PU1EbhaRU6e7zmAwGIzNucFgMBgMBoPBUCIYzbnBYDAYDAaDwVAiGOHcYDAYDAaDwWAoEYxwbjAYDAaDwWAwlAhGODcYDAaDwWAwGEoEI5wbDAaDwWAwGAwlwv8HIO/a5X7FZQIAAAAASUVORK5CYII=\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"w0 = np.sqrt(2*0.3*eps/m) # Initial value for w\n",
"h = 1e-16 # Time step\n",
"n_tot = 100000 # Total number of time steps\n",
"\n",
"euler_oscil(w0, h, n_tot)"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.5.4"
}
},
"nbformat": 4,
"nbformat_minor": 1
}