Reproduce-PRL-99-226801-2007-Hwang.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Reproduce: Density Dependent Exchange Contribution to $\\partial\\mu/\\partial n$ and compressibility in Graphene"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "E. H. Hwang, Ben Yu-Kuang Hu, and S. Das Sarma, <i>Density Dependent Exchange Contribution to $\\partial\\mu/\\partial n$ and Compressibility in Graphene</i>, Phys. Rev. Lett. <strong>99</strong>, 226801 (2007)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Reproduced by <strong>Eunjong Kim</strong> (<a href=\"mailto:ekim7206@gmail.com\">ekim7206@gmail.com</a>)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from numpy import *\n",
    "from scipy.integrate import dblquad\n",
    "import matplotlib.pyplot as plt\n",
    "from matplotlib import rcParams"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "rcParams.update({'font.size': 18, 'text.usetex': True})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Similarly as an electron gas system, the exchange self-energy in graphene is given by"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\hbar \\Sigma_{s}^{(1\\mathrm{E})} (\\mathbf{k}) = -\\sum_{s'} \\int \\frac{d^{2}k'}{(2\\pi)^2} u(\\mathbf{k}-\\mathbf{k}') n_{\\mathbf{k'},s'} F_{s,s'} (\\mathbf{k},\\mathbf{k'})\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where $u(\\mathbf{q}) = 2\\pi e^2 / \\epsilon_0 q$ is the 2D Coulomb potential ($\\epsilon_0$ is the background dielectric constant due to substrates) and $F_{s,s'}(\\mathbf{k},\\mathbf{k'}) = \\left| \\big\\langle \\psi_{\\mathbf{k},s}\\big\\vert \\psi_{\\mathbf{k'},s'}\\big\\rangle \\right|^2$ is the wavefunction overlap factor given by"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "F_{s,s'} (\\mathbf{k},\\mathbf{k}') = \\frac{1}{2} \\left[ 1+ss'\\cos{(\\phi_{\\mathbf{k}}-\\phi_{\\mathbf{k}'})}\\right]\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "in graphene. Here, $s=\\pm1$ is the band index. At zero temperature ($T=0$) and the positive chemical potential ($\\mu=\\epsilon_\\mathrm{F}>0$), "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$n_{\\mathbf{k}, s} = \\Theta(\\varepsilon_F - \\varepsilon_{\\mathbf{k}, s}) = \n",
    "\\begin{cases}\n",
    "\\Theta(k_\\mathrm{F}-k) &\\mathrm{for}\\ s = +1\\\\\n",
    "1 &\\mathrm{for}\\ s = -1\n",
    "\\end{cases}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We separate the exchange self-energy into contributions from the intrinsic electrons, $\\Sigma_{x}^\\mathrm{int}$, and the extrinsic carriers, $\\Sigma_{x}^\\mathrm{ext}$. That is, $\\Sigma_{x,s}(k) = \\Sigma_{x,s}^\\mathrm{int}(k)+ \\Sigma_{x,s}^\\mathrm{ext}(k)$, where"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\hbar\\Sigma_{x,s}^\\mathrm{int}(\\mathbf{k}) = - \\int \\frac{d^2 k'}{(2\\pi)^2} u(\\mathbf{k}-\\mathbf{k}') F_{s,-} (\\mathbf{k},\\mathbf{k}');$$\n",
    "$$\\hbar\\Sigma_{x,s}^\\mathrm{ext}(\\mathbf{k}) = - \\sum_{s'}\\int \\frac{d^2 k'}{(2\\pi)^2} \\delta n_{\\mathbf{k}',s'}u(\\mathbf{k}-\\mathbf{k}') F_{s,s'} (\\mathbf{k},\\mathbf{k}'),$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where $\\delta n_{\\mathbf{k}',s'} \\equiv n_{\\mathbf{k}',s'} - \\frac{1}{2}(1-s')$ is the difference in the electron occupation from the intrinsic $T=0$ case. At zero temperature ($T=0$) and the positive chemical potential ($\\mu=\\epsilon_\\mathrm{F}>0$),"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\delta n_{\\mathbf{k}',+} = n_{\\mathbf{k}',+} = \\Theta(k_\\mathrm{F} - k)\\\\\n",
    "\\delta n_{\\mathbf{k}',-} = n_{\\mathbf{k}',-} - 1 = 0\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and the exchange self-energy contributions from the extrinsic carriers, only $s'=+1$ in the summation remains, and\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\hbar\\Sigma_{x,s}^\\mathrm{ext}(\\mathbf{k}) = - \\int \\frac{d^2 k'}{(2\\pi)^2} \\delta n_{\\mathbf{k}',+}u(\\mathbf{k}-\\mathbf{k}') F_{s,+} (\\mathbf{k},\\mathbf{k}') = -\\int \\frac{d^2 k'}{(2\\pi)^2} \\Theta(k_\\mathrm{F}-k) \\frac{2\\pi e^2}{|\\mathbf{k}-\\mathbf{k}'|} \\frac{1}{2} \\left[1+s\\cos{(\\phi_\\mathbf{k}-\\phi_{\\mathbf{k}'})}\\right].\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For numerical evaluation of the integral, we assume that the wavevector $\\mathbf{k}$ is directed along the $x$-direction, $\\mathbf{k} = k\\hat{\\mathbf{x}}$ ($\\phi_\\mathbf{k}=0$),"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\cos(\\phi_\\mathbf{k}-\\phi_\\mathbf{k'}) = \\cos{\\phi_\\mathbf{k'}},$$\n",
    "$$|\\mathbf{k}-\\mathbf{k'}| = \\sqrt{k^2 + (k')^2 -2kk'\\cos{\\phi_\\mathbf{k'}}}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\hbar\\Sigma_{x,s}^\\mathrm{ext}(k) = -\\frac{2\\pi e^2}{\\epsilon_0} \\frac{1}{(2\\pi)^2} \\frac{1}{2}\\int_{0}^{k_\\mathrm{F}} k'dk' \\int_{0}^{2\\pi} d\\phi' \\frac{1+s\\cos{\\phi'}}{\\sqrt{k^2 + (k')^2 -2kk'\\cos{\\phi'}}}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We define unitless wavevectors $\\tilde{k}\\equiv k/k_\\mathrm{F}$, $\\tilde{k}'\\equiv k'/k_\\mathrm{F}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\frac{\\hbar\\Sigma_{x,s}^\\mathrm{ext}(k)}{e^2k_\\mathrm{F}/\\epsilon_0} = -\\frac{1}{4\\pi} \\int_{0}^{1} d\\tilde{k}' \\int_{0}^{2\\pi}\n",
    " d\\phi' \\frac{\\tilde{k}'(1+s\\cos{\\phi'})}{\\sqrt{\\tilde{k}^2 + \\tilde{k}'^2 -2\\tilde{k}\\tilde{k}'\\cos{\\phi'}}}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We make use of <code>scipy.integrate.dblquad</code> to evaluate the double integral:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def integrand(k,s,k_,phi): # define the integrand\n",
    "    return k_*(1 + s*cos(phi)) / sqrt(k**2 + k_**2 - 2*k*k_*cos(phi))\n",
    "def i(k,s): # double integration of the integrand\n",
    "    return dblquad(lambda k_, phi: integrand(k,s,k_,phi), 0, 2*pi, lambda k_: 0, lambda k_: 1)[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "k = linspace(1e-5, 2, 50)  # range of k/kF values: 0 to 2\n",
    "\n",
    "SE_ext_p, SE_ext_m = zeros(len(k)), zeros(len(k))\n",
    "for idx, k_ in enumerate(k):\n",
    "    SE_ext_p[idx] = i(k_,+1)/(-4*pi)\n",
    "    SE_ext_m[idx] = i(k_,-1)/(-4*pi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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G12U1V0KrK1dzedHlzY7vObwnoDI7qc9JXHbSZfTt1pejux7NMd2O4ehuRzOs97CY3E+4\niDITBEFwAIfrD/Ptnm8bFVXXDl356Qk/bVbv1U2vcsnTlzQ7Pj5zfEBlNqTnEMYOHku/7v3o260v\nfbv1pV/3fpzY58SAcpw16CyKBhW1/YaijNjMQiA2M0EQ4sGeQ3uoPVTLsT2ObXbunS3vcPFTF7Nj\n/44mx38w6Aesu3Zds/pl35Zx6dOX0q97P/of1Z/+3c3r1GNOJffk3JjdgwexmQmCIKQAX9d+zV/e\n+Atbdm/h691fs2X3FnYf2s3IfiMpm1bWrH6PTj3YsX8H7VQ7+nXvx8AeAxlw1ACGHzM8YPtZ/bPY\nfMvmWN9GwiHKTIgLyWSTSARkPKNLW8bzcP1hNn67kc21m/mq9qvGv+1UO56Z9Eyz+vW6nkc2PtLk\nWJf2XejUvlPA9k/odQLf/PYb+nbrS7u0dq2SMRUQZSYIghCEBt3Ad3u/o8pdxfZ927n0xEub1XEf\ndHPmI2c2O96tQze01ijVdMVt4FEDWfSzRQxKH8SxPY7l2B7H0rNzz2b1PHRo14EBRw2Izg0lMWIz\nC4HYzAQh9ThUd4ifL/05le5KKmsqOVR/CID2ae05+IeDzWZHWmvOeuQs+h/Vn+PTj+e49OMa/44e\nMDqokkpWxGYmCIIQI77b+x2bqjexqXoT5dXllNeUU1FTwRuT32i2vNepfSfe3vI27oNuAPp07cNg\n12AyXBnsO7KPHp16NKmvlOKdKe/E7V6EwMjMLAQyM4seYuOJLjKeTdFas23fNvp07UP7tOa/0fve\n1Zft+7Y3O/7ZjZ8xrM+wZuNZUlVC7y69GewazFGdjoql6EmHzMwEQRDC5PnPnmfjdxv5YtcXja89\nh/fw4fUfcuoxpzar//2B32f7vu0M6TmEob2GMqTnEDJ7ZnJc+nEB2x87eGyM70CINjIzC4HMzAQh\n/mit2bpnK5/u/JThfYdzTLdjmtU59/FzKakqaXLM1dlF0cQixmWMi5OkQiBkZiYIQsqy/OPlvLLp\nFT7d+Smf7PiE3Yd2A7B0wlImnjKxWf3LT72cMweeybA+w/he7+/xvd7fo3eX3innbJFobNhgX9+i\nzIS4IDae6OKk8fTMtD7a/hGDXYMZ1qd5DL+SqhIee/+xxnKvLr04+eiT6dK+S8A2p46aGlUZnTSe\ndlNfD+0sh85ly6ChAX7xC1P+97/tk0uUmSAIUWfd5nUs/WgpH27/kP9u+y81B2sA+NOP/sRfzv1L\ns/qTTp3EqcecyklHn8TJR5/M0V2PlllWArBtG+zaBSefbMoPPghffgl//7sp794Na9d6ldm4cfDX\nv9ojq9jMQiA2M0EIjNaaLbu3cKj+EEN7DW12/qH1D3Hjyzc2lnt16cWpx5zKladdSf6o/HiKKkTA\np5/Cxo3wy1+a8tKl5vWMFchk5Uq4+2547TVT3roVtmyB73/f24ZdNjNRZiEQZSYIhh37drCyfCVl\n35bx/rb3ef+796k+UM2EkyewPG95s/qf7fyMf3/+b4b3Hc7wvsPp372/zLQShIYGSEsz7//7X1i8\nGB54wJTffht+/WsoLTXlTZtg4UIoKDDlujpQyrvMGAhRZgmIKLPoITaJ6BKr8QwUfgmMTevcx89t\ncqxXl15cPOxiHrvksWb1nUayPp/19bB5M2RkmPLnn8OkSfD++6a8dSucfjps326U1L59sGQJ3Hxz\n6/sUb0ZBEOKK1prymnJKt5ay/pv1rN+6nsP1hwNGsxjZbySXDLuErP5ZZPXPYkS/EQw8aqDMthKM\nujooLoaf/MSU3W6jrGpqzGwqMxPKy43S6tYN+vdv6rTRrVvbFJmdyMwsBDIzE5IV90E3mfdmNjpm\neEhTaeyevZtuHbvZJJkQCVrDww9Dfj506GCWEHv1Mk4aRx9t6mRnw+OPw7FWqrT9+6Fr19jJJDMz\nQRCiRl1DHR9u+5C3t7xNflY+Hdp1aHLe1dlF1w5d6diuI2MGjmHMAPMaPWC0KLIE55574IoroG9f\nszT48MMwZox5paXBtGlQXe1VZqtXN70+lorMTmRmFgKZmUWPZLVJ2EWg8VxTuYaSqhLe/PpN3v3m\nXfYe3gtA2dQyRvYf2ayN6gPVIVOPpBKJ/Hzee69xeT/tNFO+6CK46irIyzPlxx+HrCzvebuRmZkg\nCG3ijrV3sLrS+zM8s2cmZw86u9mszEOvLr3iJZoQAY88AoMGwXnnmXJ5ORw65FVWt9wCvXt76199\ndfxlTEQcOzNTSk0FdlnFTK31wjCu8WxwGWX9naW1rg1RX2Zmgu1oramoqWDt5rX856v/MOmUSZw/\n9Pxm9R4pe4RPdnzCD477AWcPOpt+3fvZIK0QKc8/bxwyPHu77rnHuMQ/9JApf/QRHD5sZl9OQGZm\nEWApsgatdZFVHqmUWqS1nh7imnytdaFVLLQU2wag+Y5PQUgA1m1ex+INi1lTtYYtu7c0Hu/WoVtA\nZXZd1nXxFE9oJW+/DevXw29+Y8r19fCPf3iV2cSJ8O233vqnNk8CIAQgzW4BWslUrfUST0FrvRHI\nUUqlB6oc6Lil2HoppbJjJ6bgoaSkxG4RHEeVu4p//vefbNm9hd5devPzE3/OPefdw/TR02U8o0ws\nx/Orr8xsy0OXLt5ZFxhvwxkzvOVjjzXOHEJkOG5mppRyAZkBTlUAOUBRgHNDgAKl1FKt9W6/azKi\nL6UghMZ90E1JVQnFFcUoFPdfcH+zOjmZOdz7k3s5d/C5nHLMKaQp72/Pkk9K4iitEAl79xqnjBut\naF7du8Nf/mLKnTqZfV9z5xq3eqUgPd0oNKFtOM5mppTKApZprYf6HV8GrA9mO1NKjdBav+93rBqY\noLV+Pcg1YjMToob7oJuFby6kuLKY0q2lNOgGAI7qeBS7Zu4K6qghJDZawwsvwM9+ZjYm19UZt/kP\nPvDu7XriCfj5z82m5GRHbGbhE8oFq3ewEwEU2QSgPJgi8zB58mQGDx4MgMvlYsSIEY0uvJ6lCSlL\nOZzye2++x91P3c2hQYdon9aeU/adwqj+o8jPzSdNpdkun5TDL+/YARs2lNC5synfdhts3lzCqaea\n8l13wbp1JfTrZ8pXXplY8kez7HlfVVWFnThxZpYDLAoyMyvXWs8Jow0XUAyM81t29K8nM7MoUZLA\n+3iixYEjB3jjqzd46YuX+NOP/8TR3Y5uVqegtIBjexzLjwf/mO4du7e6r1QYz3gSznh6lgUBLr7Y\npD3xOG0sWQJDhsC55wa/PlVIyZmZ5VGYF2b1PB83+kCzMxdeV/2WmIdZXgyqyAQhHDbXbualL17i\n5U0vs7piNQfqDgBw9qCzufy0y5vVnzZ6WrxFFKLAvHlGkc2aZcoTJ0Jlpff8lCn2yCV4ceLMzAVU\na63T/I6XAjNbWjZUSs0Almutq8LoS2ZmQkim/HsKj2x8pLGc1T+LC4ZewJXDrwyYUVlwBuvXwzvv\nmHQoYEJC3X67SUQphEZSwESAUmoTMMp3w7NSapP/0mOA6/KBVb6KTCmVrbVeHaS+KDOBfYf3sX3f\ndjJ6Nnd8ffnLl3l046NccMIF/GToTxhw1AAbJBTayr598NZbMH68KX/5JfzwhybxZPv2xqnj8OHk\njWsYTexSZk7dZzYfaLSNWR6Oq3zKmUqp5b77yyxbW6lHkSmlXNYxIQ74GoudwPZ921lStoQLn7yQ\n3gt6M/2lwPvxLzjhAlZMXMG1I6+NqyJz2ngmIg0N3vevv17ChAkmZQrACScYO5jnt2z79qLIEh0n\nejOitS5USuX7bHjO0lpf71MlExgH9ARqlVKZwErAP6iqtuoIAgBb92zl8qLLWbd5XaPrvEJxsO4g\nDbqhyV4vwblobTYmL10KQ4fCUUfBzJmwYwe4XKbOhRfaK6MQGY5cZowXssyYetQ11NH3rr7sObSH\n8UPG8/MTf86F37tQ4hwmAUVFRnGdfrop33ILDBjQNPqG0HbEZpaAiDJLTjZVb2L5x8u5Lus6jul2\nTLPz72x5h5P6nER654DR0QQHceCACR8FsGABfPopPPaYKe/fb85JBpzoIjYzIamx28ZTUVPB/HXz\nGbV4FCfcfwK/f/33PPPpMwHrnnnsmQmvyOweTyfwwgswYYK3fM018JOfeMtdu3oVmYyn8xFlJiQ9\nd669kyH3DWH26tmUfVvGUR2P4srhVzK873C7RROiyL598OCD3nJODnz2GdRaPs9HHw2TJtkjmxB7\nZJkxBLLMmBy8UfUGP3vyZ1w87OLGXGCd23e2WywhSngic9TXG5vY00/DGWeYc/X1Jl6iED/EZpaA\niDJzBkfqj/Ba+Wts2LqBP4/9c7PzDbqBg3UH6dpBfKuTjWnTTABfz/Lhyy/D4MFw8sm2ipXSiM1M\nSGpiYZN4/7v3ueXVWzj2f4/loqcu4i9v/KVJEksPaSot6RRZqtp4Dh0C33i2Z50FBQXe8gUXtE6R\npep4JhOO3GcmCOf98zxWVTTuk+fEPidy9elXJ53SEpryyitw553w7rtmafGXv4TcXLulEhIBWWYM\ngSwzJi63vHoL//jvP7j81Mu5+vSrGT1gtP+GeCEJqKszrvTXXQdpaSZqx8SJJlPzMc13VQgJgNjM\nEhBRZvZSWVPJzv07GTOweQ75mgM1dO3QlU7tO9kgmRAvPJE6/vhHuPRSu6URwkFsZkJSE65N4kj9\nEZZ/vJzx/xxP5n2Z3PjyjQHr9ezSM6UVWTLbeIqK4KWXzHul4J57jFt9LEnm8YwL33xj0mlfe61t\nIojNTEgIDhw5wNx1c1lStoRv934LQOf2nRnWZxgH6w6KK30K0b07/OY3cP75JsDvj35kt0RCM3bt\ngpISkxvn9dfh88/tlkiWGUMhy4zxo76hnhPuP4FKdyUnH30y00dN58rhV9Kzi8SBTnZqaoyL/ZNP\nGuWltfmeHDtWQk0lDIcOwZtvwqpVsHIlbNzoTSkA5hfIj34E48ahbr1VbGaJhiiz+PLC5y+Q3jmd\nHx73Q3HmSCG0huxsuOoqmDzZbmkEwPxTPv7YKK5Vq+CNN0ygSw+dOsHZZ8O4ceafN3o0dOgAiANI\nQiLKLHqUlJQwaPgg7n/vfk7scyLTRwfODyaER0lJCWPHjrVbjFbz8MPQowdccYUpf/019OnjDQoc\nb5w+nlGhttYsG77yCrz6qslM6svw4SZ76XnnwTnnBE3wZpcyE5uZEFO01qypWsMfV/+Rd954B41m\nsGswU0dNldxgKcyoUXDZZSYQcKdOMGiQ3RKlIFrDhx+asCmvvGJSbdfVec/37WsMl+PHm0CX/RI7\nDZLMzEIgM7O2sXP/TsY9Po4Pt38IQMd2HbnitCu4+YybOb3f6TZLJ8ST/fshP99kb/bMvjZvhuOO\ns1eulOPgQVizxqQUePFFMyX20K6dCany05+a1+mnm819ESIzMyHp6N2lN53bd6Zf937cMPoGpo2e\nFjB/mJD8dO1qfAj+/neYM8ccE0UWJ777ziiuF1809q/9+73n+vY1McB++lMzA/Ok2XYgMjMLgczM\n2k6Vu4r+3fvz9rq3xSYRRZxg49mwAcrLTcQOgG3bzJJiIn5fOmE8I+LLL+G55+DZZ+Gdd5p6Ho4c\nCRddBBdeaNZ7WzH7CoXMzARHUllTyfw359O/e/+AEesHuwbHXyghIejUCW64Ac4808zC+va1W6Ik\nRmsoK/MqsI8/9p7r1MnYvC66CH72Mzj2WPvkjCEyMwuBzMyCs6l6E3euvZN/fPAP6nU9PTv3ZOvv\ntsrm5hTnrbfgtNPgqKNM+Y03jDLrlLrBWmKH1vDee7B8OaxYAV995T2Xnm5mXpdeavLjdO8eN7HE\nNT8BEWXWnLqGOq59/lr+9eG/aNANtFPtuGL4Fcw5Zw4n9jnRbvEEm7nuOujY0bjeCzFAa5MywKPA\nNm/2nuvf3yivSy81O847drRFRFlmFBxB+7T21BysIU2lcc2Ia5hzzhyG9BrS4nVJZ5OwmUQaz4MH\nobM1Ib/7bqPIPNmfnUIijWcztDYRN558EpYta+qBOHCgyYGTl2c2MUfZ/uUkRJkJEXPPefdw/0/v\nF3uYwL59Zlnx5ZfhxBONc4fHW1FoI198AU89ZV6+sQ+PPdZs0MvLM2u4KazAfJFlxhCk8jLj1j1b\nWbd5HROhDxQoAAAgAElEQVRPmWi3KEKCU1hosj/fcYfdkiQB337rVWClpd7jRx8NkybB5ZcnvAIT\nm1kCkorKbNf+Xcx/cz73v3c/9Q31fPHrL2QGJjTh66/hX/+C2bNN2fMRcdKyYkJx8CA8/zw8/ji8\n9prJQArGi+ayy0w67XHjTBRmByD5zARb2X9kP3/7z9/IvC+ThW8t5GDdQS458ZKotS/5oqKLnePp\ncpnsz8uXm7JSzldkcR9PrY3r57RpJkzUL35hQkqlpcEllxjnjm3b4P/+z8RCdIgisxMZIQGAGStn\n8FDpQwCcP+R87hh3B6MGjLJZKiFR0NrEoXW5zIThpZdinzAzKfnuO6OgHn3UbGz2MGoUXH21WUbs\n08c28ZyMLDOGIJWWGb9yf8UVz1zB38b9jbGDx9otjpBgvPAC3HYbrF3r3UMmhEl9vVk+LCw0A1lf\nb44PGABXXmly35xyir0yRhGxmSUgqaTMBCEUWsPvfmec6M4+225pHMJXX5kZ2KOPetOptGtnInHk\n55uI9O3a2StjDBCbmRBz9hzaw4yVM/ho+0dx71tsZtElHuNZVWUmFGBsYvfck7yKLGrj2dBgBu3i\niyEjA/76V6PIhgyBuXPN+2efNcF9k1CR2YnYzFIArTUrPlnBLa/dwjd7vuHjHR/z8hUv2y2WkODs\n3GlWwV591Zh0hBDU1Bhb2EMPwaZN5ljHjmZDc34+/PjHCe1OnwzIMmMIkmGZ8YtdX3DTyzexqmIV\nAKMHjOahCx5izMAxNksmOIG1a02C4fR0uyVJUN5/Hx54wETnOHDAHBs0CKZPhylT4JjUS3kkNrME\nxOnKbN/hfQz630HUHKyhZ+eezM2ey5SsKbRLk+UNITCvvGJSXt19t/Pd7WNGQ4MJeXLPPSbRpYfx\n4+HGG01k+hR2pRebmRB1unXsxpxz5nDtiGv5/KbPmTZ6mm2KTGxm0SVW43nWWWb700svxaT5hCWs\n8dy/3wSePOkk48SxZo2JRv+b35hwUytXmj1iKazI7ERGPcm59exbUfITW2iBhgZj0nG5oKREUrY0\n4dtvzVLiokVQXW2OHXcc3HyzSRMga7AJgSwzhsBJy4xvbn6TswedLYpLiJgvvzQBKF5+WRJoNqGi\nAhYsMOFODh82x844A377WxNmSmZgAZFlRqFV7Ni3g0krJnHOY+fw9EdP2y2O4EBOOMGsjj36qN2S\nJAgffghXXGEGpqAAjhwxyuutt+Cdd2DiRFFkCUhE/xGlVDYwHnABvazD1UA5sEpr/X50xROCobVm\n6cdL+fUrv2bn/p1069CNA3UH7BYrKAmdL8qBRGM89+zxRvP405+8AYNTkZKSEsZ27gx33mmidIBR\nWFddBbNmmfw2QkLTojJTSqUD84FMYBWwHqgAaq0qLiADGK+U+j1Guc3UWu+OicQCuw/tZuoLU1n6\n8VIAxmWMY8lFS8jomWGzZIJTOHgQTj8d7r/fON9BCnsvvv22CW9SVmbKnTsbt/pbb4Xjj7dXNiFs\nQtrMlFK5GCW2WGtdG7Ri02tcQB5QrbUuioqUNpGoNrP9R/YzevFoNtdu5p7z7yE/K19sZULEvPsu\nLF1qPMxTkvXr4c9/NvsRwExTb7rJOHaI8bDVJNw+M0uRlWmtK1vVsFFq2U5WaImqzAA+3v4xndp3\nYmivoXaLIjiII0egQwe7pbCZjRuNEvMsJ3bvbhTYb38LvXqFvlZokYRzANFaF7VWkVnXu52syBKd\nU445xVGKTPaZRZfWjufs2TB1Khw6FF15HMEnnxhHjqwso8i6djX2sMpKSnJyRJE5nFZ5Myql0pVS\n+UqpeUqpRdbfy5RSPaItYKrz3GfPsffwXrvFEJKE2283XuaeIO4pwdatRoOfdpoJ8tu5s5mFVVbC\nvHmSPyxJiHifmbX8OAnjCOL2OeUChgArtdbPRE1CG7FzmfFw/WF++9pveXD9g1wz4hoevUT8poXW\no3UKOnjs2QMLF5rYXPv3myj1+fnGdbN/f7ulS1rsWmZs1WYJrfXEYOcs932hDXy39zvyluexbvM6\nOrbryJgBY9Bai5OH0Cq2bTPB2x97zGydSnqOHDGJMG+/HXbsMMcuu8y43Q8bZqtoQuyIxaZpVwza\nTBne2fIOoxaPYt3mdQw8aiD/mfwfrh9zveMVmdjMoksk43nMMfCrX8H8+bGTJ2F46SU49VQT8HfH\nDpOAbd06KCoKqcjk+XQ+rZmZ1SqllmH2k7mBXUBvvBup50ZPvNSjYEMBW/ds5YfH/ZDlecvp211c\nhIW2oRRMm2bMRknLpk3w//6fN0Ly975n7GGXXpqC66upSatjMyqlcjB70NIxSq1Ca706irLZjh02\ns/1H9nPfu/fxu7N+R4d2qe5DLbSFv//d+DZceaXdksSQvXvN8uHddxvPlh49zPLiTTfJHgSbSLh9\nZgErK5WdKApLKTUVMysEyNRaL4zw+uVa67wW6iTsPjNBaIk33zTf6U88AaecYrc0UUZrs+P71lvh\nm2/MscmTYe5c6NfPVtFSnYTbZxaE8Z43llcjPuX8qEgUBpYia7D2whUBxUqpRRFcnwXktlhRiBpi\nk4gu4YznD34AGzYkoSL7/HM491y4/HKjyEaPNiGpHnus1YpMnk/n0xYHkDF+5Yq2CBIhU7XWSzwF\nrfVGIMeKIxkOtu+OLPu2jAv+dQF7Du2xWxQhyfj8c++m6LRkyotx+DD87W8wfDi88YZZQy0sNHG5\nzjzTbukEm4n0UV+mlHo4yLm4eDFaYbIyA5yqAHLCuD5Xa10cdcEiYGX5Sn78fz/mlU2vsODNBXaK\nEjckYn50CTWe999v9gdXVcVNnNjz7rswahTcdptRatdea7T2lClR0djyfDqfiLwZtdZlSqk0pVQp\nsEspNRjj1TjGesUjfFWm1ac/bgIruUaUUiOBDbEQKlye+O8TXPP8NdQ11HHFaVdw249vs1McIQl5\n4AFYvRqOPdZuSaLAnj3wxz8aDa01DBkCixfDuHF2SyYkGBG75mutpymlZgJz8C4trgi1kTrKhFoi\n7N3CtZmRxoucPHkygwcPBsDlcjFixIjGX3GedfZwylprpj8wncUbFkMGzDx7Jue3P5+31r7Vqvac\nVva1SSSCPE4vtzSe2dmJJW+ryvPnwz33MHb7dmjXjpKJE+HqqxlrKTJ5PhOj7HlfZfdSgNa61S/A\n1ZbrW9lnDrApwPFlwNwQ1+X6lRvC6EtHi4aGBj35ucla3a70ve/cG7V2ncKaNWvsFiGpCDSe11yj\n9UMPaV1XF395osrevVpPm6a1mYtpPWqU1hs3xrRLeT6jh/W9GVe9oLVu/T6zaGB5QIZ0j/chT2td\na+1vW6a1bjJDU0qtxMSFvCtAPxkYxbvR51iD1jrkYnu0XfOP1B9h3eZ1nJtxbtTaFAQPH30Ec+bA\nkiUOTsf17rtmY9ymTdCxI/zP/5igwO1bFXlPsIGE22dmubqXYhRHwmSNthxAqv0VkWXHm6m1fj3A\nNfk0d1CZD8wC3FrrwiB9RVWZCYIQhCNHjKfiHXdAfb3xYHniCeO5KDiKhFNmAEqpCYDHFva0TpBo\n+EqpTcAo7ZP9Wim1SWsddoIvO2ZmqUxJSUnjWrvQdnzHc+VKOOsskyjZkXz+uZmNlZaa0FO/+51R\nbJ06xU0EeT6jR0JumtZar9BaT9TGuaOPUmqZlb/Mblei+RgHFKBxE/Qqn3KmUmp5BPvOos4/P/gn\nNQdq7OpeSCGefRZOOglqnPa4aQ0FBTBypFFkxx0Hr79u0rbEUZEJyUFr8pm5gKkYR4wyzIzt/RjI\n1pIc+Xi9KbO0Tzgry662FDN7q/K7LhuYhokAUgQU6CAhulo7M7vv3fu4+dWbGdV/FG9f97bEWBRi\nzpdfOiy9y969Jvrxk0+a8q9+Zdzv0237/SlEiYRcZmzxYqUyMYphJLAS46JfFR3R7Kc1yqygtIDp\nL0037y8sYOqoZA5VLgit4JNPYMIE+PRT6NbN7Bv75S/tlkqIEgm5zNgSWusKrfUsrfV5wOvAbKXU\nSqXUFKVUj+iI6Bz++cE/GxXZfT+5TxSZD757UoS2U1JSwooVxtlvyxa7pYmAJ5+EMWOMIjv5ZFi/\nPiEUmTyfzidqkdu01mVa6+mWYqsCFkQS/NfpfPDdB0x5YQoAC8cv5Ndn/NpmiYRk58QT4bvv4J13\n7JYkDA4ehOuvhyuugP37zd/33jPGPkGIArbuM0t0IllmPFJ/hN+t/B0A9/30vliKJQjOoqrKLCtu\n2GD2jt13n8kUKkkzkxJH2sySndbYzBp0A2kqmUKVC4mI1g7RBW+9ZbI979gBGRmwfLkJGCwkLY6x\nmSmlMpRS2ZZXoOCHKLLAiE0iejQ0QGZmCTfcYFbsEpZ//cvkHduxA847z8zMElSRyfPpfFrzzbsK\nqAjmzi4IQmxJS4N584zNrEsXu6UJQEODSdVy5ZUmXcuNN8JLL0HPnnZLJiQxrdlnNkNrvdDakJyD\niWJfDRT7RuRIBkItM3647UP6H9WfPl37xFkqQUhg9u+HyZPNcmJaGtx7L9x0k91SCXHErmXG1kTv\ndANYiqtIKbVM+6R/UUqlJ5tS82fb3m389F8/pUO7Drx+1etk9MywWyQhRdi9Gw4cSNBAwt9+C5dc\nYtzte/SApUvhJz+xWyohRYiGgWeVXzncKPiO5HD9YfKW5/HNnm8YeNRABvYYaLdIjkBsEtHh3Xdh\n2DC47roSu0VpykcfwRlnGEU2eLBx/HCQIpPn0/m0ZmY2Synlq7Ay/cqjgCVtEytx+esbf2Xt5rUM\nOGoAKyauoGO7jnaLJKQQ48fD11/Diy/aLYkP771nFFdNDZx9Njz3HBx9tN1SCSlGa2xmy4GngWBr\nohN1/LJOxxR/m9mW3Vs44f4TOFh3kLXXrOWc486xUTpBSABKSuCii0ysxYsugmXLoHNnu6USbMRJ\nNrM7fZNc+qOUKm+DPAnNvz//NwfrDpJ3cp4oMiHuPPccDB0Kp55qtyQWL75oNkMfOmRCUv3f/0EH\nCaot2ENENjMrPNU8v2PZViZnAEIpOqdzw5gbePu6t5mXM6/lykITxCbRdr74wqzmffJJAozn00/D\nz39uFNn06fDPfzpakdk+nkKbidQBpAxY4XvA2m/mUkrlRk2qBObMY88ks2em3WIIKcjMmfDVVyY+\nr614otzX1RmhHnrIuOELgo1EZDNTSuVqrYt8yulABlAJTPXNKZYMSKZpQfDjrrtgxgzz/s47Yc6c\n0PWFlMMR4ay01kVWUkwPlZjUL07LcSsIjmHrVjMRWm13zJ277/YqsgceEEUmJBRBlZlSqkAptV4p\nNVcpdZknP5nWulAplauUugzjudhLa52WbLMyD0fqj9gtQlIgNonW07278Xj3VWZxH8/CQrj1VvP+\n0UdNiKokQp5P5xPUm1FrPU0ptQGoAC4HliilqjGbpMswy4pj4iOmfZz1yFmMGTCGuTlzcXV22S2O\nkIL06GFzRKilS2HaNPP+/vvhmmtsFEYQAhPSZqaUyvYNKKyUysLEYxwPZGMU3SrrVay13h1bceOL\nUkpzOwzqMYgvfv0FndvL/hkhvrz9Npx5po3pXl5+2YSoqquDv/0N/vAHmwQRnIIj85kppTyKLQfo\nobU+IVqCJQIeZfbYJY8xecRku8URUoyaGvjBD+C008zkKO785z9w/vkmS/Stt8KCBQ5JoibYiSMc\nQPzRWhdrrWdprUclmyLzcMrRp/Cr4b+yWwzHIzaJyOnZEz74wOtz4UvMx3PDBrjwQqPI8vOTXpHJ\n8+l8ZHNIC9yZfSft0trZLYaQonToAKNHx7nTTz81u7P37IFJk+Dhh5NakQnJQbNlRqXUSMyy4Qat\n9evWMU/usnKt9ftxl9ImlFK6oaEBJR9kIY7s2GHMVHfcYRI1x5WtW+H734dvvoELLoBnn4WOEkxb\nCJ+EsJlZe8imAaXAaGCX1vp8n/P1WuuUmabIpmnBDrQ2OuT11812rrhx8CCMHWvyzJxzDrz2GnTt\nGkcBhGQgUWxmQ7TWo7XW07XWo4HrlVKLPHvMCB4pXxBCIjaJ8FEKLrsstCKL+nhqbWIsvvsuHH88\nPPNMSikyeT6dj78yaxLxXmtdobWeDkyzlhpTjro6k0BXEGJNfT0UFUFDgw2d33svPP64UWCSj0xw\nIP7LjLmAG7PUmK+1rvU5lwOs1FqnjNOIUkqffLImJ8d81gUhlmzbZmxlgwbB8uVx7Li42LjgNzSY\nfGR5SZ0sXogxCWEzswTJALJ8Awr7nMvSWpfFSzi7UUrpF1/UXHCBOHMJ8aGhwWSSPv74OHVYXg5j\nxphNbX/4g9kYLQhtIFFsZmitKwMpMutcmW/uMjCejsm8BPmzn4kiiwZikwhOfT1s327ep6WFp8ii\nMp579pipYE2NyRL917+2vU2HIs+n84k0OecMIMtK0gmAtRQ50cdJJOmor4fHHjP5CAUh2qxbByNH\nwtq1cey0oQGuugo+/hhOOgmeeEJykgmOJtJ8ZvlW1HwXkO2X2yxfa10YCyHtwuOa/8ILJgDC3Xeb\nLTiCEG1ee81skB43Lk4d3n47/OUv4HLBe+/BCUkZwEewAbuWGYNGzQ+F1tqtlOoVbWESlQsvNC9Z\nbhSiyeHD3v3I558fum5UWbvWLCmmpZnlBlFkQhIQ8bqCUipevx0TBqW8iuzgQdidVLkB4oPYJJpz\n6aVGp7TGFb/V47l3L0yebPaVzZkTZy2auMjz6XwizTRdCCxUSr2GsZ0NhsYQWEOiLl2CUVwMw4YZ\n72VBaCtLlhhnwn374tjpzJlQUQGnnw5/+lMcOxaE2NKqFDBKqfnAVMDjxVgG5GmtK6Mom+34h7P6\n9FOorjZpOQTBcaxcaWZiHTpAaSkMH263REISkjD7zCK62DiCoLV2R02iBKKl2Iz19dAuZSJVCtHg\nww9NarB//AP69o1jx263SYy2ZQvceadZYhSEGJAw+8w8+O8nC4TW2h1KkYXThhNpaDDOYDfcYLck\nzkFsEoaTTzYesQ8/3LZ2Ih7Pm282iuyMMwInSEtx5Pl0PqG8GV1KqdxgG6hbwgqNVdE6sRKb774z\nuQuXLLFbEsEpNDQY58F27eB//sf4X8SN554zU8HOnU38xfatcmIWhIQm5DKjtYyYj4nXuFRrHdKP\nz4oEMhHjDFLgdBtauClgtBa3fSE4DQ0mL9ltt0FOTpw737EDTj3VhBj5+9/NDE0QYkhC28ysIMMT\nAM/esgpgl/W+N5AJuDBKr0BrvTr6osafcJTZ9u0mGe999xmThCAEYu1as0d55co4BtrQ2gQNLioy\necpWr5YoH0LMSWhl1uQCpTIxysvjyegGKpw+CwtEOMrsb38zm19vv12+J0JRUlLC2LFj7RYjrhw4\nYFb2PLP2aM7gwxrPp56CX/4Sunc3nieDB0en8yQkFZ/PWOGYCCBa6wqS1BbWGv7wB1liFAJzyy3Q\nqZNZ3fPdeB8X9u0zAgDcc48oMiHpaZNrfrITrs3Mw6pV8K9/wf/9X+xkEpyD2w1Tp8KDD9qQ63Lu\nXPj97016l3fflV9cQtxIuGVGKzJ+KbCsJcePZCUSZbZ/v/F6fvBB+NGPYiyYkLDs2WOU2KBBNgpR\nUwOZmUaQVats8DoRUpmE22emtZ6OsYctUUotU0pdFj+xnEfXrvD++15F1tAQ5zBFCU6q7ON5/nkT\nZKOmJrb9hBzPBQuMIhs3ThRZmKTK85nMhLSZaa1XACsAlFJTlVLLgGrMbO31OMjnKDzRQLSGG280\nEdHvvddemYT4cuWVZm+y2w09e9ogwLffeh+6O++0QQBBsIfWeDO6MHEZczAxGZ/WWr8fA9lsJ1Kb\nmYedO+H66+GRR6BH0qYsFTwUF5uYnRMn2i0JcNNNZq370kvh2WftlkZIQRLOZhbWxcZNfxowElgJ\nrNBaV0VHNPtprTLzZ8sWqKsTh7Jk5b//Nat5JSUmXJVtVFaatA51dcYV/5RTbBRGSFUSzmYWDlrr\nCq31LK31ecDrwGyl1Eql1BSlVEznJNayZ671CjvYnFJqhlIq37puQixlBBOAIScHXnkl1j0lNslm\nk2hoMPsLwQSfX7sWTjwxfv0HHM8//xmOHIFf/UoUWYQk2/OZikQtSJvWugyYDo0RQxZYGnp6tPrw\noJSaCjR44kYqpUYqpRa11Jdl85vpmT0qpRqUUq5Yemt27GhSSF17bax6EOygsBD+8x944gnj9T5s\nmM0CffSREaZDB7ODXxBSDEfuM1NKlWqtR/sd26S1HhrimqlAD631XT7HBodaFo3WMqMvjzwChw5J\nxH2nc+AAnHcePPoonHCC3dJgbGTPP288jx54wG5phBTGkTYzO7AcUKq11ml+x0uBWcHiQiqlqoGs\nSGx60VZmO3eamK9r1sBJJ0WtWSFOLFwI2dmQlWXKCRNg+p134KyzzP6Q8nLo189uiYQUxhE2M6VU\ntlJqnlJqkbX3bJn1foZSakSshPQjE7P/zZ9qICvQBZYCdJm3Kte6jxlWlP+40acPfPyxV5Ht3Qtv\nvx1PCewjGWwSgwbBFVcY/wqwV5E1Gc8//MH8vflmUWStJBmez1SnRZuZ9YU/H6NEVgHrMbEZa60q\nLiADGK+U+j1GqcyMoR2ql9WHP25MBP9AeBRguo+drRRYDYwOcg0AkydPZrDlhuhyuRgxYkRjQFLP\nB6A1Za3hkktK6NwZXnqp7e1JOTblDz+Em24ai1LQr18Js2dD+/aJIx8bNjD29dfB5aLkrLPAJ2Bu\nQsgn5aQve95XVVVhJy3lM8vFKILFWuvaoBWbXuMC8jBLga1K7NlC+znAIn/7mOXcUa61bpYP3rpm\nJdDE2SOMpcmo28w8NDSYALTTp5vVITChkI46KibdCa2grs6ENpw+HaZNs1uaIFxwgXGVvfNOmNPs\n0ReEuJNwUfMtRVYWqULSWruBQqVUi5mqlVL5GMUXDnk+CrVXgPMuvDnW/KmwZPOfLXqWJuOefy0t\nDX77W2/5gw/g5z+Hzz83DmmCPdTVQVUVDB1qEjI/9ZRxu09IduwwCdLatYMpU+yWRhBsJagya+us\nylJqIdvQWhcChRE2XYpRXP70wkQkCdRPhQpu4IhxFL3wePZZY/rwKLL6em94rGSgxCH5ot57Dy6/\nHD75BLp1M3vH4rl/LFxKSkoY++mn5kH5yU9sCMufXDjl+RSC47h0kpaSrAjgvOFqIV5kmVIqw+9Y\nJkY52s7tt3v3omkNubnw2mu2ipQy7Nrl3QB99tkmc3iFEzL2PfWU+Xv55fbKIQgJQESu+VZamAyt\n9fk+x7KJc6Zpa3lyiNZ6tlXOAvK11tdb5UyM08oUz9KkJWeeZ2O1dc1c33sJ0E/MbGah+Ogj4zW3\nfr3ZdK21eaU57qeHM/jFL2DUKJgRdhyZBODrr+G440wq623bJAiokDA4wjUfs4y3wveA5Tzhsmxs\nccFaniy3XOxzgWyPIrPIBMYBPX2uWQ2sslzyZwATQykyOzn1VK8iA5OSakLMA2+lDgcOmK1ZHv72\nNzPejtpyuXSp+fuzn4kiEwQin5k1ceiwlvoygEpgqtZ6YfRFtA+7Zmb+TJhgXr/4hSnv2GH2rCXE\nht0wSSSbxKZNcOaZ5q8rkPXVAZQMG8bYL76AFSvMmrTQJhLp+XQ6jpiZaa2LrCU+D5WYAMMJ4USR\nrDz5pDe9iNbG3r9ypb0yOY1HHjERWMB4Kv7udyb1lyP58kv44guzj+OCC+yWRhASgqAzM6VUAcZt\nvRizUbrY49puLe1pYLfWujhOssadRJmZ+VJVZeI6vviisaEdOQKPPQb5+c6aqcWDI0e83qE33WS+\n++fOtVemqPDXv5oI+VddBY8/brc0gtCEhJuZaa2nWecrgMuBKqXUJqXUwxg3+DnJrMgSlcGD4eWX\nvc4gL7wA//ynV5E1NNgmWkKxZAn85jfe8m9+A+ecY588UUNr8WIUhAC0tMw4U2tdqLXO01r3AiZi\nlhYnAqM8yk0pdVms85cJgendu2nGj8WLYfZs28QJim/om1hQU2OWEj1ccAG8+qo3juL3vmd8JRzP\nBx/AZ59R0qOHiXosRIVYP59C7AmpzPzDPGmty7TWC7TW462o9dOB3cAfgA2xE1MIxo9/3PQ77amn\nTDJQDwUFsCEJ/zNam3xinlXgLl3g1lvhm29MecAAY1ZqH7WMfQmCZ1Y2dqyEihEEHxyXAiaeJKLN\nrCX27zdbj9LSzBf98ceb0H2exMPPPgvnnutML76dO43S6tbN3NsJJ8CyZd6ULI8/br7jjz/eVjFj\nR0MDZGTA5s1Gk//wh3ZLJAjNSDibWYBoGRETjTaEyOja1WtPq6+HBQvg5JNNed8+uPpq72xGa3jo\nIe9SXKLxzTdmG4KHa6+Fl14y75UyTh0eD0Uw95a0igxMvqDNm+HYY+EHP7BbGkFIKEItM7ZpI7R1\nrQN//ycP7dubvWke5xC32wRW72ltJa+sNBuGPTEg9+6FO+7wXu+JPBINwrFJvPcevPWWt7xggfHU\n9HDeeU3d6f/f/zPHUgbPEuMvfkHJf/5jryxJhtjMnE8ob8aNwGorYkZ+OA4eSql0q+48TMT9jdEU\nVmgbAwc2zRLS0AC33eZVdu++a5YkPWzY0HQlq7LS+30KUFtrEht7OHjQOGJ42L/fxD308OmnsNrH\nCvv000ZheXjvPeOZ6eG880ybHm66yeSfTEnq6mD5cvNevBgFoRlh2cysfGAT8KZeqcCbbqU3JnyU\nC5MAsyBYfjCn4USbWVv47DOzH/eii0z56afhmWeMXQrg+eeNx+C//23KL7wAixZ5l/5WrDDKrqjI\nW37mGbPpG8y5557zRmJ67jnT3gsvmPJ//2sU6DXXxP5eHcfKlXD++cYt87PPZFOhkLAkXD4zX6z9\nZMXQGMQ3E/BErS8jzoGGhdjgn+5k0iS45BJvefBg+OUvveX27WH4cG+5SxdI98ll0K1b0+DIp53W\ndIECeB8AACAASURBVOY2blzT64cPb1oWfPDdWyaKTBCaId6MIUi1mVkskdh3beDgQejbF3bvNrOy\nYcNkPKOMjGf0SDhvRkEQEoSXXzaKbORIGDbMbmkEISFp08xMKTUSs8RYGz2REgeZmQm2ozV8//tQ\nWgr/+7/GhVMQEhinzswmAaM9BaVUruwtE4Qo8u9/G0XWty9MnWq3NIKQsLRVma0CGp2zrVxnWW1s\nU0hCZB9PK2hogD/9ybyfM8fsiLeQ8YwuMp7Op62R68YDs5RSvTDu+usxrvpFIa8SBKFliorMfoWB\nA2HaNLulEYSEpq02s8bM05bLfi5ms7TsMxOEtlBfb/YyfPopPPwwTJ9ut0SCEBZOtZlVWE4gaK0r\ngBWYPWiCILSFp582imzwYBOUUhCEkLRJmWmtN/qFrJJYjEJAxCYRAXV13iR1t90GHTs2qyLjGV1k\nPJ1P2DYzpVS6vwu+/zFLsUk8RkFoC//4B2zaBEOHwlVX2S1Nq1ASpSQpSWSzS0ibmVKq1Hq7lAC2\nMKVUNsYJZAJQo7UeEytB7UBsZkLcOXzYxF/86isTdfnKK+2WqFVYdhO7xRCiSLj/U7tsZi0pswag\nZ0ubopVSWUCplX06aRBlJsSdRYvg+uvhpJPgww+9+Xkchiiz5CPRlVlLymdFONE9tNZlmIDDghAQ\nsUmEwcGDJsEcwF/+ElKRyXgKQlNaUmYVEbQVSV1BEPxZvNik1x4+HHJbnRdXEFKSlpYZZ2itF4bV\nkFKLtNZJtRlGlhmFuLF/P2RmwrZtJtGbb+4dByLLjMmH05cZm2Blkl6mlJobK4EEISX561+NIhs9\nGi6+2G5pBMFxhK3MlFK5QCVmU/QypdQMpVSPmEkmJBVi4wlBWRncdZdJunn//WEl35TxFCKhsLDQ\nbhFiTovKTCnVQym1DFgOzNVaj7Y2Sy8EJimlRsRcSkFIVo4cgeuuM+Grbr4ZzjzTbomEJMPtdlNe\nXt5yRYfTkjIbD1RhZmND/O1nWutCoLc1axOEoEgW3yDcdRe8/74JW+XxZAwDGU8hHNxuN/n5+XaL\nERfC2Wc2qyUnECvI8DKt9ehQ9ZyGOIAIMeWzz2DECDh0CFatgpwcuyWKGuIAEpiePXtSW1vL1CC5\n6SoqKigtLcXtduNyuaiurm48vnjxYsaMGcP69euZM2cO6enprFixglmzZlFTU0NVVRU7d+5k6NCh\n5OXlMW/ePMrKyigoKABg/PjxZGVlkZ2d3SrZE90BBK110BfgCnU+2V9meIRosGbNGrtFSCzq67U+\n5xytQetrron48kQfT/nsBGbFihVaKaWLiopC1lu8eLFOS0trLA8ZMkTX1tZqrbUuKyvTeXl5jefc\nbrceMmSIdrvdury8XBcWFjZpa8GCBXrWrFltlj3c/6lVL+7f1yGXGbXW7paUoVIqW7JLC0KELFoE\n69ZBv35w9912SyPEidzcXHJycpgyZQq1tcHjUeTn55Obm0ttbS2LFy8mMzOTHj2Mv93IkSMpLi5u\nrJuens78+fOZMGECq1evZsqUKTG/j0QkHAeQdL/yCKXUYE9Zm3iNWVacRkEIiNh4fNi8GWbNMu8f\nfBB69oy4CRlP51JQUIDb7WaW5xkIwqRJk6ioqKCiwsSjWL16deNr0qRJTerm5uailMLtbnH+kbSE\n45o/0fPG8mpcgclj1qCUWqqUGqdNgk7JYyYILaG1SbS5d6+J8nHZZXZLJMSZjIwM5s+fz+LFi1m9\nOnge45ycHKqrqxkzZgwul4vs7OzG18MPP9yk7saNG5k1axYFBQVUVlYGbbOoqChq95FoRBoYeL3W\neqg2AYWHAhuAQstRZFTUpROSBtkXZfHkk/DKK+BywQMPtLoZGU9nM2PGDDIzM5k2bVrQOunp6WRn\nZ5Obm0tFRUWTZUlfJeh2uyktLSU7O5uCggLy8vKatJOZmZkSM7ZIlVlj/EWtdYXWeoHWeojWOk0n\nWSgrQYg6W7eavWQA99xj7GVCyrJ8+XIqKiqYPXt2WHXnzp1LUVERRUVF9OrVC4AFCxaQmZnZuBTZ\ns2dPysrKOP/88xsVXm5uLtXV1RQWFuJyJW/+5KCu+UqplUBP4ADwd6AYGAOUa62r4iWgnYhrvhA1\n6upg3DhYuxbOP9/MzpI4gaW45ofH9OnTKSwspL6+3m5RWiTRXfNDzcxmW+c3A9cDbqAAKFNK3err\nBCIIQgv88Y9GkQ0YYDJJJ7EiE8InLy+vcR+Y0DaCKjNtcpTlA29orcdbdrKJwDzgfIwTyCal1MNK\nqcskTqMQipS28bz4Isyfb/KTLV0KxxzT5iZTejyTBLfbzapVq1LWlT7atLTPrEybkFW+5QU+ym06\nsBv4A8YZRBAEX6qq4KqrzPu5c+Gcc2wVR0gc5s2bx7x584Ker6ysDOntKDQlZDirVEdsZkKbOHQI\nfvhDWL8eLrrI5ClLi9Tnypm0xWamlNnBYFc5HhQWFjJ+/HgGDx4ctE5RURGZmZmMHDmyzf0VFRWR\n28aEr062mQmC0BZmzDCKbPBgePzxlFFkQmhWr15N7969QyoyMJurhwwZEpU+169fH5V2Epn2dgsg\npAYlJSWpFbVi+XKTm6xjR1i2rFVRPkKRzOPp/+M/3uVY4na7KSsrY8aMGSHrzJw5k+Li4sYQVrW1\ntSxbtgy3201GRgY1NTW43W4yMzPbPONKFkSZCUK0+eILk6MMzH6yMWPslUdIGCZOnEh1dTWrVq1q\ndq66uhq3291kz5iH4uLixlQuaWlpjc4jghdRZkJcSNZZRDN274YJE2DPHpg4EW64ISbdpMx4Jhkr\nV65s1XVZWVkAlJWVkZWVRY8ePWRG5ocoM0GIFkeOGAX24Ydw4omwZInsJxOiQkaGSUxSXFzM+PHj\nW6w/fXrTgEzFxcXNQlrl5eW1OrdZIuJYb0al1FRgl1XM1C0kELWu8aRcdQG9gbla66B5GMSbMXok\ns40HMIaXadOgsBCOPhreeQcyYxd7O9HHUyKAxIbx48czZ84cxo0bR21tLenp6S1fBMyePTvkNoBw\nEG/GGGApsgatdZEVsb9YKbWohWtmAEu11oVa64Va69lAYahrBCFsFiwwiqxzZ3jhhZgqMiG1KC4u\nZuHChbjdbjZs2ECm9WwtW7assU5ZWRkbN260S8SEwKnLjFO11qM9Ba31RqVUSznnxwSYvVUopdJD\nzc6E6JDIs4g2s2wZzJ5tlhSfeALOOCPmXSb1eApNUNZStSeX2fLly3G5XE2WG5ctW0ZZWVmrbXLJ\ngOOUmVLKBWQFOOVWSmVbyUIDkamUGqm19v354hJFJrSJN9/0RvhYuNDkKBOEKOLJYeYh0CbqefPm\npXy0ECcuM2Zigh77U01gJechH9hgLTdiZcYOuTQpRI+kjCW4aRNccomJ9HHDDfDb38at66QcT6FN\nhMpZNiYFtoc4UZn1wiguf9wYp46AWDOyIcAcpVS1dez9mEgoJD87d8JPfwq7dsEFF8C994rnomAb\nGzdubHTfD0QquPE7bpmxBYK62iilMoFcYDDwe2CVUmqabyDlQEyePLkx7IzL5WLEiBGN9grPr2Mp\nt1weO3ZsQsnTpvKoUXDRRZRs2gRDhzJ26VJo317G068sxI9oxG8MB18vWt/VgZKSEqqqquIiQzBs\ndc23XOXzWqxoyNNa11qOHsu01r382loJrNRa3xWkr0W+2bCVUiOB1Va7ARebxTVfaMa+fWZGtnYt\nHHccvP22yVEmNEFc85OPRHfNt3VmZs2KInWPL8XsE/OnF1AW6ALLPtbEzcfygMwDxmOUmhBDEn1f\nVFgcOAAXX2wU2cCBsGaNbYosKcZTEKKI42xmWms3lku93ymX1vr1EJcG+qVQiXfjtSAE59AhuOwy\neP116NvX/JW9ZIKQMDhOmVnMB+Z4CkqpLGCVTzlTKbXco/CsZcRJAdrJBSRneRxw9Czi8GETpurV\nV6FPH1i9Gr73PVtFcvR4CkIMcHI4q3ygArPk2CSclWVXWwqM0lpXWcfSMQpwF8bz0QUs95wP0ofY\nzFKdujq4/HJYscKkcVmzBk4/3W6pEh6xmSUfiW4zc6wyiweizKKHI2089fVmQ/STT0KPHmZpcdQo\nu6UCEn88RZklH4muzJy6zCgIseXIEbj6aqPIuneH115LGEUmCEJzZGYWApmZpSgHDkBeHrz0klFk\nL78MP/yh3VI5CpmZJR+JPjNLtk3TgtA2amvhoouM+33v3vDKK5IpWhAcgCwzCnHBEbEEt22DsWO9\n+8jWrk1YReaI8RQShsLC5M92JTMzQQD46isYPx6+/BJOOAFWrYLjj7dbKkFoM263m/LycrvFiDky\nMxPiQiJ73vHJJ/CDHxhFNmIErFuX8IosocdTSBjcbjf5+fl2ixEXxAEkBOIAkgK8/baxke3aZZw8\nXngBwkxFLwRHHEAC07NnT2pra5k6dWrA8xUVFZSWluJ2u3G5XFRXVzceX7x4MWPGjGH9+vXMmTOH\n9PR0VqxYwaxZs6ipqaGqqoqdO3cydOhQ8vLymDdvHmVlZRQUmLgQ48ePJysrq0lutEhIdAcQtNby\nCvIywyNEgzVr1tgtQnOeeELrjh21Bq0vvFDr/fvtlihsEnI8fZDPTmBWrFihlVK6qKgoZL3Fixfr\ntLS0xvKQIUN0bW2t1lrrsrIynZeX13jO7XbrIUOGaLfbrcvLy3VhYWGTthYsWKBnzZrVZtnD/Z9a\n9eL+fS3LjELq0dAAf/wjXHmlCVV1ww3w7LPQpYvdkglJTm5uLjk5OUyZMoXa2uBJ7vPz88nNzaW2\ntpbFixeTmZlJjx49AJPupbi4uLFueno68+fPZ8KECaxevZopU6bE/D4SEVFmQlxIGBvP/v0waRLc\ncQe0awcPPAAPPgjtneULlTDjKURMQUEBbrebWbNmhaw3adIkKioqqKioAGD16tWNr0mTmoaazc3N\nRSkVMtt0siPKTEgdvvkGfvQjE2exRw+zGfrGG+2WSkgxMjIymD9/PosXL2b16uDZp3JycqiurmbM\nmDG4XC6ys7MbXw8//HCTuhs3bmTWrFkUFBRQWVkZtM2ioqKo3Uei8f/bu/fYOK56D+Df45KoL9lr\nB1AoBLLjgIoIcP1IePxDiO0USsVrbaf/gFCauL4IARU4dq6AqysQcewqIJDa2hshVIRE7WxAgIqI\nd4sr+CNt1ptEBKjaZB2JBoVQb2YT+iCt/bt/nFnHdnZn1+vZmZ2d70daubNzZnx6fLK/PY85h8GM\nXOH5c1EzM8D27fqnYQAnTgC7dnmbpzXwvDxpTQYGBmAYBh588MGCaRoaGtDR0YFIJIJ0Or2sW3Jp\nEDRNE8lkEh0dHRgbG0NPz/L9jg3DCESLjcGMat/jj+uZiv/4h26ZPfMM8N73ep0rCrjJyUmk02kM\nDQ2VlPbgwYOIxWKIxWJoamoCAIyMjMAwjMWuyMbGRqRSKdxzzz2LAS8SiSCTySAajSIUyrevcW3g\n1HwbnJrvc6++Cnz1q8CRI/p4zx7g0UeB9eu9zVcAcGp+afr7+xGNRjE/P+91Voqq9qn5/hr1JirV\nCy/oxYLPnAFuvVVP9NizB1DuP/5CVEhPTw/a29u9zkZNYDcjucLVMZ5YTG/XcuYMsGWLHh974IGa\nCmQcM/M/0zQxNTUV2Kn0TmMwo9px/Trw0ENAdzdw7RoQiQDJJHeGpqo0PDyM4eHhgudnZ2dtZzvS\nchwzs8ExMx85f14/BH3ihH5m7OGH9XhZDbXG/GQtY2bq//L/zeR/89+v0ukrIRqNoqurC5s3by6Y\nJhaLwTAMtLS0uJYvO9U+ZsaWGfmbCDA+rltfJ04AmzbprVu+9jUGMqpKiUQCGzZssA1kgH64urm5\n2Z1M1QC2zGywZeac6elp51etuHQJ2LtX7wgNAPffr1fzsKYt17KKlKeDOJsxP9M0EY1GMTAwYJtm\n//79OHLkCBYWFgAA2WwWExMTME0T4XAYV65cgWmaMAwDkUjElbxXe8uMsxnJn44dA/r69Gr3oZCe\ncn///V7nishWb28vMpkMpqambjqXyWRgmuayZ8Zy4vH44lYudXV1i5NH6Aa2zGywZVaFslk9Fvb4\n4/q4qwv4yU+Ad7zD23zRMmyZOWt2dhbhcBipVAp9fX1IJpOu54EtMyKn/PrXei3FF1/Uz46NjuoV\n7+s49Eu1LRwOA9AttK6urqLpo9EoZmZmbNMMDg4u3rcWsGVmgy0z56xpjOfiRd0aO3ZMH2/bpltm\nd9/tWP78hmNmwdTV1YUDBw5g586dyGazaHBxI9lqb5nxKy1Vr4UF4JFH9DqKx44Bd94J/PCHenfo\nAAcyCpZ4PI7R0VGYpomZmRkYhgEAmJiYWEyTSqVw6tQpr7JYFdgys8GWmYf+/Gc9wePECX386U/r\nJak2bfI2X1QStsyck0gkkEqlYBgGDMNAPB5HKBRa9pza0NAQUqkUjh8/XrF8VHvLjMHMBoOZB65e\n1RtnHj4MvPEGcNddwI9/DHzuc3xuzEcYzNyXSCTQ0dFRsftXezBjNyO5ouhagvPzenX7d78bGBnR\nx1/+MvDXvwKf/zwD2Qpcm5FWCsKeZXY4m5G89/TTwNe/Dpw+rY8/+lE9NrZtm7f5IvKJU6dOobW1\n1etseIrdjDbYzVhh6TQwMHBjluKmTbpVtns3W2I+x27G2lPt3YxsmZH7Ll8Ghof10lPXrwO33w4M\nDQHf/CZw221e546IfIhjZuSK6elp4MoV4FvfAgwD+MEPdCD7wheA558Hvv1tBrJV4JgZ0XJsmVHl\n/fvfwM9+pmck5gapP/Up4LvfBapkewsi8jeOmdngmNkavfIKMDYGHDwI/Otf+r2Pfxz43vf0JA+q\nWRwzqz0cM6PgyWT0eNiPfgS89JJ+78Mf1s+P7dzpbd6IqCZxzIyc8+KLwDe+AbzzncB3vqMDWXs7\n8JvfYPr732cgcxDHzIiWYzCjtXvuOeCBB/TEjsOHgZdfBnbtAhIJ4Nlngfvu41R7ogqJx+PYsmUL\nhoaGvM6KpzhmZoNjZjbm54Hf/U6vl/j73+v36uqA7m5gcBAI+AOcQccxM3eNjo5ibm4Ow8PDFfsd\nHDOj2jI3pzfDfOQR4MIF/d6ttwJf/KJ+AHrLFk+zR0TBxG5GKk4ESCaBPXv0js779+tAFg7rDTIv\nXtSzFm0CGcd4nMXyJFqOLTMq7NIl4Oc/B376U+Ds2Rvvf+ITwFe+on/ecotn2SPym8bGRmSzWfT1\n9eU9n06nkUwmYZomQqEQMpkM4vE4+vv70dnZiba2NjQ1NWFqauqmnaJN00QsFkMoFEIqlUJ3d3dN\n7SRdlIjwVeCliydgXntN5OhRkfvuE7nlFhHdLhN585tFHnpI5Pnnvc4h+UAg/+2U4OjRo6KUklgs\nZptufHxc6urqFo9HRkakq6tr8dg0TWlsbJRsNisiIocOHZKenp5l92hubhbTNB3Le6l/Uyud65/X\n7GYkPZnjD3/QW67cdZeexPHb3+oZiJ/5DPDLX+quxMOH9RYtRFSWSCSCzs5O7N27F9lstmC6ffv2\nIRKJLEuzdFX8hoYGtLe344knngCgJ13kdqDOMQxj2W7UtY7BLKjeeENPne/v1wFs507g0Uf1A88f\n+IBeO/HiReBXvwI++1lg/fo1/TqO8TiL5elfY2NjME0Tg4ODtul2796NdDpd8LxhGEXPnz9/vux8\n+g2DWZC8+qqeTt/XB7ztbUBnp564cfky0NysV64/dQo4c0bvL/bWt3qdY6KaEw6HcejQIYyPjyOR\nSBRM19nZiUwmU/D8+fPn0dzcXPD83NwctgRodjGfM7NRE8+ZpdM6gD35JPDUU8Brr9049573AD09\nulvxgx/kg83kGD5nVlwu0Jw7d65o2pGRESSTycVuQ9M00d7evnjtyMgIZmZmFrsdc/cv5d6l4nNm\n5K6rV4E//Ul3IT75pF6dY6m2NuDee3UAe//7GcCIPDI5OYm2tjYMDQ0VfdhZWf9OE4kETNPEyZMn\nEY/Hl50/cOAAYrEYANx0PgjYMrPhi5aZaQJ//CPw9NP6lUoBCws3zjc06KWl7r1XT6XfuNGTbE5P\nT2PHjh2e/O5aVO3lyZZZafr7+xGNRjE/P2+bbnR0FJlMBgcPHnQpZzdjy6yClFKtAIZEpLfE9H0A\n5qxDQ0RGK5a5SpifB/72N+DkSb3m4TPPAKdP68nzOW96k16h/mMfAz75Sf3f69Z5l2ciKqinpwft\n7e1F0+Wmn1NhvgxmSqkWALutQ8Mu7ZJr+gAsiEgsdw+l1GMi0l+hbK7N9et6B+azZ/XqG88+q1td\nL7+8PN26dcCHPqSD144dwEc+AtxxhydZtlPNrQg/Ynn6n2mamJqaKtrFmEgkMDExgWw2i127dmEn\nd5/Iy9fdjFZQi4pI0a82SqnkynRKqXMiUnC6jyvdjK+8AszO6hbXX/6iX2fPAi+8oKfPr7R5M7Bt\n243X9u3A7bdXNo9Eq8RuxuKKjZXNzs4inU6jo6PDxVwVxm7GyiqpwJRSIQD5lnE3lVIdIlJ4fuxa\nLCzo57YuXdKvv/9dzy5Mp3UAS6eBf/6zUKb1dPn3vU+vQJ8LXm95S0WyWmnVPsbjNyxPf4tGo+jv\nt+8USqVSNz0ITYX5PZiVygBg5nk/Ax3kCgezkyeB11/XraTXX9ev//wHuHZNzxzMZvXP3Gtp8Lp8\nOX/raql164B3vUtPk9+6VQevrVuBu+9mi4uCpxpm11a4RZlIJLBhwwZs3rzZNt3Y2BiOHj1a0bzU\nkqAEsybowLWSCWCD7ZXbt6/tNzc26hmEGzfqlTYMQ682n/v59rcHYrFetiKcxfL0J9M0kUqlMDAw\nYJtm//79iMfjqK+vdzF3/haUYGbH9mvYl5qasPm224C6OoTWr8d/NTZix8aNQH09pq9dA+64Azu2\nbgUaGjB98SJw553Y0dUFbNyI6eeeA9avX/zgyS1BtOw4nbY/z2Me+/S4LDU+ztbb24tMJoOpqamb\nzmUyGZimubhEVWNj4+K5bDaLiYkJmKaJcDiMK1euwDRNGIaBSCTiWv6Xdm8vXVJtenoaF3L7G3rE\n0wkgSql9AHpKTN4jIstW5rSm5o8XmwCilOoEMCEiTSvePw7guIg8XOC66n/OzCc4xuOsai9PTgBx\nViwWWwxadXV1izMhAbgWzDgBxIaIRAFEXfhVSQChPO83AUi58PuJiMqWWzE/lUqhtbUV9fX1rrbI\n/CAQCw2LiAkgrZRqWHEqJCJPeZGnoKnmVoQfsTyDJbfJZjweR1dXl8e5qU5+D2ZN+d5UShlKqckV\nwesQgANL0rQCuLnjmoioSk1NTS0GM7v90ILIl8FMKRVWSg0DGAbQopR6zBp/yzEA7ASwOIJqdWme\nV0p1KKUiADpE5L9dzXiAcf8tZ7E8gyMej2N0dBSmaWJmZmbx2bMgbbxZCl/OZhSRWQBDNufjyDPl\n3gpoRES+sXTF/EQigcnJSYRCIXY3ruDr5awqjbMZicrD2Yy1p9pnM/qym5GIiGgpBjNyBcd4nMXy\nJFqOwYyIiHyPY2Y2OGZGVB6OmdUejpkRERFVGIMZuYJjPM5ieRItx2BGRES+xzEzGxwzIyoPx8xq\nD8fMiIiIKozBjFzBMR5nsTyJlmMwIyIi3+OYmQ2OmRGVJ7c4LtWWah4z8+Wq+URU3fglkNzGbkZy\nBcd4nMXydBbL0/8YzIiIyPc4ZmaDY2ZERKvD58yIiIjKxGBGruCYhLNYns5iefofgxm54vTp015n\noaawPJ3F8vQ/BjNyhWmaXmehprA8ncXy9D8GMyIi8j0GM3LFhQsXvM5CTWF5Oovl6X+cmm9DKcXC\nISJaJS+m5jOYERGR77GbkYiIfI/BjIiIfI/BjDyhlAorpTq9zgfRUqyX/hXYLWCUUn0A5qxDQ0RG\nK3FNUJRRNm0AxpVSIQAmgCSAQRE5VcFs+oJSygAwLCK9JaZnvbShlGoFMFRiebJeFqGU2mf9Z5v1\nc1BEskWuqXgdDWQwswp2QURi1nGLUuoxEel38pqgKLNsRESalFL1InLVnZxWN6VUC4Dd1qFR4jWs\nlwWUU55gvbSllNonIlHrMGoFthkAW2yucaWOBnI2o1IqKSLtK947B6Ct0DeMcq4JikJlIyJ2FTyS\nq9y0nPUhHF1ZpgXSsl4WscryZL0sQCnVAKB3STDLvZ8B0CMiiQLXrfrzoRyBGzOzug/yfUtLA8jb\nV17ONUFhlU1rnlOmUqrD7fzUiJKe0WG9LJnrzzzVqGYAY0qp+hXvpwGE813g5udDELsZDQCZPO+b\nKNwVUc41QWFAl8NKGehKnPfbGgCsqMytAMbZmlgV1ssKYL3MT0RSSqnWPN2vBnRAy6fsz4fVCmIw\na7I5t8HBa4KiCYU/UO3KJgUAIjILAEqpNIBJALuczmANY710HuulDRFZtr2AUqobwHkRearAJeV+\nPqxa4LoZiyhnADF4g46lK1g2IjKb+8DIHQMwrPENWjvWyzKwXpbO6kIcAlBud6GjdTSowSzfN9oQ\nbkwddeqaoHCqbEzcmO5LpWG9rDzWy/yGAXSXMOvTlToaxGCWhC7IlZpgdTE4dE1QrLpslFKGUmoh\nz6kM8ndJUH6slw5ivSydUmoA+lnIC0WSulZHAxfMRMQEkLammS4VKtTvW841QVFm2cwBeDDP++3g\nh3DJWC8dx3pZAuvZssmlgazQzEQ362jggpnlEIADuQNrhYCpJceGUmpyxR/A9pqAW1V55psZZj1Y\n+UQJ3/SCIO/EDtbLspVUnqyXxVlLfSVz5aGUCi1d/svLOhrIh6aBxW8XuemkrUuXV7H+OE9AP3h6\noZRrgm5J2YSwYrkam/IcgB6PCEGvvPCwq5muMkqpMHTLoBNAC4AogJncQ6qsl6uzhvJkvczDWmbt\nXJ5TAqBRRK6WUEdv+nxwLH9BDWZERFQ7gtrNSERENYTBjIiIfI/BjIiIfI/BjIiIfI/BjIiI5trZ\nuAAAAkBJREFUfI/BjIiIfI/BjMhDSqmxMq55zKl7EdUKBjMiDyilOpRSM8i/caHddd3Q692t+V5E\ntSSI+5kReU5EElYLa7WbaPYC2OvQvYhqBltmRN7ZBeDkai8qsOVGWfciqhVczorII0qpDIDNJewH\nlUvfB+AlETm21nsR1Rp2MxJ5wFo5PJMLPkqpCPQ28q0i0l/gsk4R6XXoXkQ1hd2MRN7ohLUNhlJq\nn4jEoFcfD+dLbG1RX2iDyFXdi6gWsZuRyANKqSkAE9AtqlgJ6fdD7yN104aGdvdSSrUA+B/oLVAO\nWW+HoLdG6RKR2TX9jxBVCXYzEnmjAzqoHCwxfaeIjKz2XiJySin1CwANuX28AEApdRR69iODGdUE\ndjMSuczawDAtItsAXFVKZfJsK780vQEgtYZ7qRXXNLBFRrWGwYzIfV2wxrhEJG69Fwb0A9B50j8I\n4BcO3QvQY2wQkcRqM05UrRjMiNy3OGHDEgIwa41vpfOkbxGR02u8l6GU2qeUmgDQVH7WiaoTgxmR\n++oBxJccDwLoA2Cs7P6zpt0fd+BeaRGJWlP7C82KJPItzmYkqmLWMlXDInJhDffoBtAnIrus4wYR\nyTqURaKqwJYZUXUz1hLI8mEgo1rEYEZUpayZihNrvEcL9OLEbUqpvXazJon8jN2MRFXK6mLcz/UW\niYpjy4yoijGQEZWGLTMiIvI9tsyIiMj3GMyIiMj3GMyIiMj3GMyIiMj3GMyIiMj3GMyIiMj3GMyI\niMj3GMyIiMj3/h9lsBldFs4UpwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x111647c50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(1,1, figsize=(6,6))\n",
    "\n",
    "ax.plot(k, SE_ext_p, label=r'$\\Sigma_{x, +}^\\mathrm{ext}$', lw=2, ls=':')\n",
    "ax.plot(k, SE_ext_m, label=r'$\\Sigma_{x, -}^\\mathrm{ext}$', lw=2, ls='--')\n",
    "ax.plot(k, SE_ext_p + SE_ext_m, label=r'$\\Sigma_{x}^\\mathrm{pb}$', lw=2, ls='-')\n",
    "\n",
    "ax.set_xlim([0, 2]); ax.set_xlabel(r'$k/k_\\mathrm{F}$')\n",
    "ax.set_ylim([-1.1, 0]); ax.set_ylabel(r'$\\hbar\\Sigma_{x}(k)/(e^2 k_\\mathrm{F}/\\epsilon_0)$')\n",
    "ax.legend(loc=0)\n",
    "fig.text(0.08, 0.95, r'Exchange self-energies for graphene with $\\mu>0$')\n",
    "ax.grid(True);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and the Figure 1 of [PRL <b>99</b>, 226801 (2007)] is reproduced."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Version Information"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "application/json": {
       "Software versions": [
        {
         "module": "Python",
         "version": "3.4.3 64bit [GCC 4.2.1 Compatible Apple LLVM 6.0 (clang-600.0.56)]"
        },
        {
         "module": "IPython",
         "version": "3.2.1"
        },
        {
         "module": "OS",
         "version": "Darwin 14.5.0 x86_64 i386 64bit"
        },
        {
         "module": "numpy",
         "version": "1.9.2"
        },
        {
         "module": "scipy",
         "version": "0.17.0.dev0+18ebef6"
        },
        {
         "module": "matplotlib",
         "version": "1.4.3"
        }
       ]
      },
      "text/html": [
       "<table><tr><th>Software</th><th>Version</th></tr><tr><td>Python</td><td>3.4.3 64bit [GCC 4.2.1 Compatible Apple LLVM 6.0 (clang-600.0.56)]</td></tr><tr><td>IPython</td><td>3.2.1</td></tr><tr><td>OS</td><td>Darwin 14.5.0 x86_64 i386 64bit</td></tr><tr><td>numpy</td><td>1.9.2</td></tr><tr><td>scipy</td><td>0.17.0.dev0+18ebef6</td></tr><tr><td>matplotlib</td><td>1.4.3</td></tr><tr><td colspan='2'>Wed Nov 11 16:24:11 2015 KST</td></tr></table>"
      ],
      "text/latex": [
       "\\begin{tabular}{|l|l|}\\hline\n",
       "{\\bf Software} & {\\bf Version} \\\\ \\hline\\hline\n",
       "Python & 3.4.3 64bit [GCC 4.2.1 Compatible Apple LLVM 6.0 (clang-600.0.56)] \\\\ \\hline\n",
       "IPython & 3.2.1 \\\\ \\hline\n",
       "OS & Darwin 14.5.0 x86\\_64 i386 64bit \\\\ \\hline\n",
       "numpy & 1.9.2 \\\\ \\hline\n",
       "scipy & 0.17.0.dev0+18ebef6 \\\\ \\hline\n",
       "matplotlib & 1.4.3 \\\\ \\hline\n",
       "\\hline \\multicolumn{2}{|l|}{Wed Nov 11 16:24:11 2015 KST} \\\\ \\hline\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "Software versions\n",
       "Python 3.4.3 64bit [GCC 4.2.1 Compatible Apple LLVM 6.0 (clang-600.0.56)]\n",
       "IPython 3.2.1\n",
       "OS Darwin 14.5.0 x86_64 i386 64bit\n",
       "numpy 1.9.2\n",
       "scipy 0.17.0.dev0+18ebef6\n",
       "matplotlib 1.4.3\n",
       "Wed Nov 11 16:24:11 2015 KST"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%reload_ext version_information\n",
    "\n",
    "%version_information numpy, scipy, matplotlib"
   ]
  }
 ],
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