{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Pseudo-Riemannian manifolds in SageMath\n", "## The Schwarzschild spacetime example" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This notebook demonstrates some SageMath tools for pseudo-Riemannian geometry, developed through the [SageManifolds project](https://sagemanifolds.obspm.fr/), by these [authors](https://sagemanifolds.obspm.fr/authors.html). " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This notebook requires a version of SageMath at least equal to 9.0:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'SageMath version 9.8.beta4, Release Date: 2022-11-21'" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "version()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First we set up the notebook to display outputs via LaTeX rendering:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "%display latex " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Since some computations are quite heavy, we ask for running them in parallel on 8 \n", "threads:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "Parallelism().set(nproc=8)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We introduce the **Schwarzschild spacetime**, which is the spacetime of a **static black hole** in general relativity, as a **4-dimensional Lorentzian manifold** $M$:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle M\$" ], "text/latex": [ "$\\displaystyle M$" ], "text/plain": [ "4-dimensional Lorentzian manifold M" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M = Manifold(4, 'M', structure='Lorentzian')\n", "M" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "print(M)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$M$ is in the category of smooth manifolds over the real field:" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathbf{Smooth}_{\\Bold{R}}\$" ], "text/latex": [ "$\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathbf{Smooth}_{\\Bold{R}}$" ], "text/plain": [ "Category of smooth manifolds over Real Field with 53 bits of precision" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.category()" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\Bold{R}\$" ], "text/latex": [ "$\\displaystyle \\newcommand{\\Bold}[1]{\\mathbf{#1}}\\Bold{R}$" ], "text/plain": [ "Real Field with 53 bits of precision" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.base_field()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "At the moment, the real field is modeled by 53-bit floating-point approximations, but this plays no role in the manifold implementation:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Real Field with 53 bits of precision\n" ] } ], "source": [ "print(M.base_field())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Coordinate charts" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The function `Manifold` generates a manifold with no-predefined coordinate chart, so that the manifold (user) atlas is empty:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\right]$" ], "text/plain": [ "[]" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.atlas()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We introduce the standard **Schwarzchild-Droste coordinates** $(t,r,\\theta,\\phi)$ on $M$, via the method `chart`:" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [], "source": [ "SD. = M.chart(r\"t r:(0,+oo) th:(0,pi):\\theta ph:(0,2*pi):\\phi:periodic\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that the argument of `chart()` is a raw string (hence the prefix `r` in front of it), which defines the range of each coordinate, if different from $(-\\infty, +\\infty)$, as well as its LaTeX symbol, if different from the Python symbol to denote the coordinate. The Python variables for each coordinate are declared within the `<...>` operator on the left-hand side, `SD` denoting the Python variable chosen for the coordinate chart." ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(M,(t, r, {\\theta}, {\\phi})\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(M,(t, r, {\\theta}, {\\phi})\\right)$" ], "text/plain": [ "Chart (M, (t, r, th, ph))" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SD" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Chart (M, (t, r, th, ph))\n" ] } ], "source": [ "print(SD)" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle t :\\ \\left( -\\infty, +\\infty \\right) ;\\quad r :\\ \\left( 0 , +\\infty \\right) ;\\quad {\\theta} :\\ \\left( 0 , \\pi \\right) ;\\quad {\\phi} :\\ \\left[ 0 , 2 \\, \\pi \\right] \\mbox{(periodic)}\$" ], "text/latex": [ "$\\displaystyle t :\\ \\left( -\\infty, +\\infty \\right) ;\\quad r :\\ \\left( 0 , +\\infty \\right) ;\\quad {\\theta} :\\ \\left( 0 , \\pi \\right) ;\\quad {\\phi} :\\ \\left[ 0 , 2 \\, \\pi \\right] \\mbox{(periodic)}$" ], "text/plain": [ "t: (-oo, +oo); r: (0, +oo); th: (0, pi); ph: [0, 2*pi] (periodic)" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SD.coord_range()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Thanks to the SageMath operator `<...>` used in the chart declaration, the coordinates are immediately available:" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle {\\theta}\$" ], "text/latex": [ "$\\displaystyle {\\theta}$" ], "text/plain": [ "th" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "th" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "They are SageMath's symbolic expressions:" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\text{SR}\$" ], "text/latex": [ "$\\displaystyle \\text{SR}$" ], "text/plain": [ "Symbolic Ring" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "th.parent()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "They are also accessible as items of the chart:" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(t, {\\phi}\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(t, {\\phi}\\right)$" ], "text/plain": [ "(t, ph)" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SD[0], SD[3]" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(t, r, {\\theta}, {\\phi}\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(t, r, {\\theta}, {\\phi}\\right)$" ], "text/plain": [ "(t, r, th, ph)" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SD[:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The manifold (user) atlas is no longer empty: " ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\left(M,(t, r, {\\theta}, {\\phi})\\right)\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\left(M,(t, r, {\\theta}, {\\phi})\\right)\\right]$" ], "text/plain": [ "[Chart (M, (t, r, th, ph))]" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.atlas()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us introduce a second chart on the manifold, that of **Eddington-Finkelstein coordinates** $(T,r,\\theta,\\phi)$:" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(M,(T, r, {\\theta}, {\\phi})\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(M,(T, r, {\\theta}, {\\phi})\\right)$" ], "text/plain": [ "Chart (M, (T, r, th, ph))" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "EF. = M.chart(r\"T r:(0,+oo) th:(0,pi):\\theta ph:(0,2*pi):\\phi:periodic\")\n", "EF" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The transition map from Schwarzschild-Droste coordinates (chart `SD`) to Eddington-Finkelstein ones (chart `EF`) depends on a parameter $m$, the mass of the Schwarzschild black hole:" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [], "source": [ "m = var('m')\n", "assume(m > 0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We provide the explicit coordinate transformation via the method `transition_map`:" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left\\{\\begin{array}{lcl} T & = & 2 \\, m \\log\\left({\\left| \\frac{r}{2 \\, m} - 1 \\right|}\\right) + t \\\\ r & = & r \\\\ {\\theta} & = & {\\theta} \\\\ {\\phi} & = & {\\phi} \\end{array}\\right.\$" ], "text/latex": [ "$\\displaystyle \\left\\{\\begin{array}{lcl} T & = & 2 \\, m \\log\\left({\\left| \\frac{r}{2 \\, m} - 1 \\right|}\\right) + t \\\\ r & = & r \\\\ {\\theta} & = & {\\theta} \\\\ {\\phi} & = & {\\phi} \\end{array}\\right.$" ], "text/plain": [ "T = 2*m*log(abs(1/2*r/m - 1)) + t\n", "r = r\n", "th = th\n", "ph = ph" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SD_to_EF = SD.transition_map(EF, [t +2*m*ln(abs(r/(2*m)-1)), r, th, ph])\n", "SD_to_EF.display()" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left\\{\\begin{array}{lcl} t & = & 2 \\, m \\log\\left(2\\right) + 2 \\, m \\log\\left(m\\right) - 2 \\, m \\log\\left({\\left| -2 \\, m + r \\right|}\\right) + T \\\\ r & = & r \\\\ {\\theta} & = & {\\theta} \\\\ {\\phi} & = & {\\phi} \\end{array}\\right.\$" ], "text/latex": [ "$\\displaystyle \\left\\{\\begin{array}{lcl} t & = & 2 \\, m \\log\\left(2\\right) + 2 \\, m \\log\\left(m\\right) - 2 \\, m \\log\\left({\\left| -2 \\, m + r \\right|}\\right) + T \\\\ r & = & r \\\\ {\\theta} & = & {\\theta} \\\\ {\\phi} & = & {\\phi} \\end{array}\\right.$" ], "text/plain": [ "t = 2*m*log(2) + 2*m*log(m) - 2*m*log(abs(-2*m + r)) + T\n", "r = r\n", "th = th\n", "ph = ph" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SD_to_EF.inverse().display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "There are now two charts in the manifold atlas:" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\left(M,(t, r, {\\theta}, {\\phi})\\right), \\left(M,(T, r, {\\theta}, {\\phi})\\right)\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\left(M,(t, r, {\\theta}, {\\phi})\\right), \\left(M,(T, r, {\\theta}, {\\phi})\\right)\\right]$" ], "text/plain": [ "[Chart (M, (t, r, th, ph)), Chart (M, (T, r, th, ph))]" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.atlas()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "One of them is the so-called *default chart*: it is the chart used by any function that requires a chart as argument and none is provided by the user. At this stage, the default chart is the first chart defined on the manifold, but this can be changed by the manifold method `set_default_chart`. " ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(M,(t, r, {\\theta}, {\\phi})\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(M,(t, r, {\\theta}, {\\phi})\\right)$" ], "text/plain": [ "Chart (M, (t, r, th, ph))" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.default_chart()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "One can plot the `SD` chart in terms of `EF` one:" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "data": { "image/png": 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ojCqN4IIFCygoKIjq1q3r6KEIHEFmJmTsK1WCF/jnn0RHjxJVq+bokRlHv6UkY8gb5BUkn34KI7diBaa+q1ZBpUVla2SKkZpKdP8+7lWEKr99IaCQg7l+HWt9p0+jzG3GDPvV+Crdo4MnMz99iiDGH38gqFGuHKawn3xiPHotcAiqNIKCHEhaGgzUDz9gynT8OHr22hM/P+W0/jIzMWYi1OfmykX0/vtI4m7WLEdIUjkrwggKHM+FC5gq3rpF1LAhpr9ly9r/fZQoa4uIIFq2DMYuLAzbvvkGeXz2nsXk0BQWpRGXJ4HjSE8nmjiRqH59nHh//glvKiZGmffjZW225gkyhrXKbt0QpZ0xA03XV63C8927298AEiHR++efca8E1aoRPX9u/7VXjQZR54MHiXr1gqevInKWyReohxs3UOZ29SrRhAlE48fjRFEz8fGoSlmwAKktb76J3MVevWD0Xr5EmoxSyczOksLCGL4f3nv46FFEtD08IN2vst7jwhMUZC+MEf36K9JBUlKIzpxB3p+Hh6NHZpg7dyDCWqIEpqPVq+PEvnYNeYvc68uTB0nYSvXUVXMKy6NH0FLs1QvfU+XKEG598gTlfgcOYLmjQQNU/agI4QkKso9nz4j69kWp25AhkLrPnVt6Xk0CB4yhr+6cORhvkSKQ4v/iC5zkciidzKw0lqSwxMYSHT4M43bgAFFwMNJ/ataEIWzVCoEtbY/14kUo/bzzjjKitlYijKAge+DrQZmZUEXp0CHrPlzgQCnM6dGRmkq0Zg3W3m7cIKpRA3l9H35oureHSltK2oX0dKQt7duH2/nzWOsLDESz9enTkQjuhInswggKlCUzE1f/qVPhHaxaZdgLULpvr7GytpgYdJ/79Vd4rJ07Y+2vWbOcq9F3/z4qWvbuxdpeYiJRoUKo4BkwAPevQd6jMIIC5Xj2jOjjjzFtmjIFaSPG1I4d0beXl6wtWQKD3acP1v0qVsy+MZiL0iksycm4nzkTaUv37uG9GjfG/6RtW0x3X7OcR2EEBcpw6hSShTMysGbUsqWjR4RytmbNkCuYOzc8zlWrEK0cMQLBDzWsRxqCp7DYk3v3pN7DR45g26lTSPiePRv/N3spOBUoAJEIlXX5U6URFM3XnZwlSxD4qFcPishq6W7Gy9pGj0bQw88POX4DBtgnbUOlLSV1SEsjOnECKtU7dyKgkSsXLg7Tp6MHccWKyiwBBARAPUdtOLrTkzFEtzknIzUVHc+I0LUsNdWy1yvZDe7SJcZatsTx/f0Z++03dGlzJq5fZ6x8edxbwvPnjK1cyVj37oz5+EjfQf/+jG3dKnV/i4hgbOJE3CtBcjJjwcG4VxHCCArsw/PnjDVvzpiHB2PLlll3jCdPGJs2Dff24soVxt57Dyd+yZK4P33afsfXRumWkpZcJIKDGZs9m7FmzRhzdcXr6tZlbPJkxi5eZEyjse341qDSlqeqnA4LnIxbt4g6dUJFxaFD1gsf2FPg4NYtJGFv3IhWlCtWIIG3fn3lErMd2VKSq1Rv2YLmUzdvSk3XFy3C/0dFuXlqQhhBgW0cPYo+tyVKIAASEGD9sewhcBAaCuP355/IOVy2DBJW7u4oa1OyR0d2k5GB9b3Nm2H4wsKQp9e5M1KS2rQhypvX0aNUPcIICqxn7VqovzRtiibgtooG2NK3NyoKJ/5vv8GAzpmDgIenp7QPL2tzdk6dgne3dSuixSVKoO9w1674X+QwFRhbEd+WwDp+/pnoq6+QV7dkiePaWL58CRGDGTMwJfz2W5S3yQkMOGtZW3o6lhlWr0YUe8gQJJP37Qslm7p17ZO7p9IUFsVx9KKkMURgRIVoNIyNHo0F7rFj5RfYrcWShfPMTMZWrUKww8ODsf/9j7Fnz+x3fGu4coWxwoVxbyvp6Yzt38/Y558zVrAgxl2hAr5zQ4ENgVUomvqdkZFBEyZMoICAAMqdOzeVK1eOJk+eTBqNRsm3FShFZiammD/+CE9w2jTHlJSdPIkAR+/euL91C95gkSLZPxZtbNXjYwyfbcgQTHHffhse4IABEB84ehQet5+fMt97SgqSp1NS7H9sNaOkhZ06dSorVKgQ27lzJwsJCWGbNm1i+fLlY3PmzDHr9cITVBHp6Yx9/DHSLVasUOY9btxgLCgI93KEhWEMRIzVqsXYsWOWHV+lKRrs+nV4eGXLYnwlSjA2ciRjZ8/qenzOnsJy+zZjDRrgXkUouiZ46tQp6tKlC3Xs2JGIiMqWLUvr1q2j8+fPK/m2AnuTloYa4G3bsPjepo0y72NI4CAtDZ7nlClY61u2DOthaqthvXEDgZ1t2yC4aownTxBYWr0a5XwFCkCR+uOPEdxQ22ezB0lJUKJJSnL0SHRQ9Jtu0qQJHTx4kO7+JwJ55coVOnHiBHWQk1EiotTUVIqPj9e5CRxMejpkpHbsQGH9X3/hBM4uDh7E9HLCBEwL794l+uwz64yE0mVtpvT4Xr6E4WvXDuk748dDxHTrVgiNLl5M1Lz562kAVYyinuDo0aMpLi6OKlWqRG5ubpSZmUk//PAD9ejRQ3b/6dOn0/fff6/kkASWkJ6O5uf//IMkXKWTbbUFDvz8EH1euxae0V9/QXTVFniPjuyEMaS0LF+OxO34eKiyLFoEz89eDZ8EVqPoJWfDhg20evVqWrt2LV28eJFWrlxJs2fPppUrV8ruP3bsWIqLi3t1C+Pdu4iwKJyRoeRwBdpkZiLwsG0bDNB/SxqKwgUO/v4b1R379sF4HD1quwEkgh7eqVO4V5rISASQKleG0du3D3LzwcFIcO7f33IDmFNTWJRGyQXHkiVLsvnz5+tsmzJlCqtYsaJZr38VGLl1izFPT8aqVWPs8WMlhirQRqNBaoabG2N//y1tT01FcMJSYQRz2bIFC/NEjPXty1hUlH2Pr/TC/9mzOH7z5vjuvLwQyDlwACk9OZ3oaKQ1RUc7eiQ6KOoJvnz5klz11jfc3NwsT5Hx90cp1NWrmCIxZr9BCrIyZgzR778jANG1q7Sdy9/bOzE6M5Pop5+IPvoIjxcuhAdYqJB930cpHj/G7/O99/A4JgYpOxERkOpv3do+63zOnsJSsCBaLKhNgl9JC9unTx9WokSJVykymzdvZoULF2ajRo0y6/U6KTL8Kk7E2JQpSg47ZzNzJr5juTSm+/cZe/993NuLO3eQNuHiIqW/OEMKSGYmY7t3M/bOO0gbypcPMmLnzimXyOzsKSwPHiAVyFRSezajqBGMj49nw4YNY6VLl2ZeXl6sXLlybPz48SzVzOmUrBH84gvcWyvXJDDMunX4bseNk3/e3kZk3jzGcudGJcSJE4wlJeHYSUm2H18Oe4w/OpqxWbMYK1cOx6penbFFixiLj1dej8/Z8gQTExnbu5exMWMYq18fSwREjJ0/b5/j2wnnKZvj/6Dz5xkbOBBf6N69jh7i68ORI4zlysVY796GPRl7nSSPHzP21ls41pAhOFmyA1vK2i5fZqxfP6zz5crFWK9e0A58nZKZbT1+SgpjR4/iQtC0KcoZiRgrVoyxDz/ExVUYQcvQMYLai/Lp6Yx16ACV3GvXHD1M5+f2bcby52esVSvjQQ97nISbN6MWtnhxxvbt030uNBSK1KGh1h/fnmRkYLzNm0uirD/8wNjTp/L7q91I2fv4mZmoY545k7E2beDVEzFWoABjXbsy9uuvqP7hFwqVVuw4j4qMfk/a9esh3tm5M9HZs46vG3VWYmOlZth//62cGkxyMtHIkciPe+89oqVLswY+1NK3NyEBQaF584hCQpDisnEjdBOVEmR1Fh49Qn+WffuQyB4dDYmypk2JJk9GW9UaNZwr4dvRVtgYOp6g3KL8w4eMFS0KCXGl0jZeZ9LTGXv7bVy5g4NN72+t/P3Nm4xVrYqp5KJFyk+3DWGqR8fjx4yNGsWYry9j7u6M9eyJQIe5qNTTMRu5FJaEBMZ27GBs6FDGKlbE53N1xRrf+PFYRjG3V8vdu/AY795VZvxW4jxG0NAP7MQJrD18+aVjBunMDB+OtdUDB5R7jzVrGMuTh7HKlRm7etX4vo6a7t24wVifPjB8vr4whGFhlh//wQMYzgcP7DFax6DRYM30xx/RmIqv65Uti7X4v/5iLCbG0aO0K85vBBljbPFiPKeUusnryPr1+M7mzjX/NbGxjG3bhntTpKTgwkSEIII5wY/sNoKnTiHFha/3/fQTorxqRakUlhcvYNw+/hgXASJcuDp2xLre3bv2SfvJyGAsLg73KkKVRnD+/PmscuXK7I033jDPCDLG2GefYbp1+XL2DtYZuXULeW0ffWTZj9tcIxUejpM1Vy7j0199wsIYGzHCOi/MHPj4f/uNsRYt8HelSrh42mM5RemWkva6SGg0WKKYORPfg7s7jhsQgPuFC5VpR6rS5QJVGkGO2Z4gY4y9fMlYjRrIORP6g4ZJSIBmX+XKlreGNOdH/O+/jPn5QRPvzBnbxmpPNBrkJfKE+9q1Efm1Zzmb2qK32qSkIBo/dKiU45g7N7y9hQsZCwlR9/gVxHmiw/7+UDL295d/Pnduok2b0Ehn4ECojzhC9Vjt/O9/6Mh27px8Hw5bWLECclcNGuB/UayYZa9PTCS6do2oalX7jY0xoj17UNZ29iwEDaZMQTng6/77iIkh2rULIhh79yLqXaoU2m926kTUsiXOG+39cyDOE8fmPWn9/AzvU6ECmv6sX48TUqDLxo2oyZ0/H8bAXmRmEo0ahc5zffui9aalBpBI6tv7n/6kTTCGVI6GDYk6dEAHtv37IXzardvrawAfPkSnvZYtoZvYuzcuet98A6my0FCkIXXooGsAczDOYwRfvCDavh33xvjoI5yMQ4ei2FwAHj2Cl/bBB+gQZw1eXlB/9vKStr18SfT++xBA+OUXCIM6qvMc599/YQS4Ava+fZCvevNNou+/z15RWHtStizRqlW45zAG7/n774lq1kTf59GjYeAWLCAKD4cH/O23RNWrGzf+3t74zry9lf4k6sLR83FjWLQmqE1CAvLBGjRALlxOJyMDuZSlS9s3veHpU8bq1WMsb17Gdu60/Xi2rhldvcpYp05STe/Ona9XWRtHo4Fs1+jRWAMnQvVUjx6Mbdyo3gh3WhrEE9LSHD0SHZxnTdAS8uXDFbNJE0jCjxvn6BE5ll9/JTp+nOjIEfsJcgYHQyb+5UuIntaubZ/jWkNYGDydP/9EP9516+DxZnfVQq1aysm88bXN2bPx3YeFoeLm3XdR2dKqlW6jeWvIzET/j7x5idzc7DJsHTw81FnZ5WgrbAyrPUHO2LFI9jSVpPs6c/cuooDDhtl+rEuXGPP2htpM0aKoIAgJsf24HEsFDuLioFDi5cVYkSKMzZ9v3MtQaXTSIBoNIuxffQUvnke233+fsYMH7T/LUfr7uXePsc6dca8iXm8jmJLC2JtvIh0iJ06LMzMZa9IESwP2UGrh/4O8eRmrW5ex589tP6Y1ZGQg/7BIERj4b781bwroDHp8vGJjzBipBWfRoowNGiQVBTjrdF6lFyHnmQ7LLcqbwtOT6I8/ECGcM4fo668VG54qWbAAAYEjRzDFsZV//8V9lSqItDpiAf3wYaLhw6Ey3rs30qa0hTWMoXSPDltaSj54gGn82rVEN29CfblbN3T6a94c0e2LF+0/ZoETTYdtYfhwlAE5c02npTx+jKqQL76wz/G2bJEqC06dss8x9TEmcPDoEaaBRIw1bIjAgNqw1NN5/pyxBQvwebiH3bMnAjpyFSzO7qmp1BN0nhQZW5gyBVfWIUNyTn+Sr76CxNG0abYf66+/kAbTogUeK5UCI9e3Ny2NaPp0okqV4NWuWgWPtG5dy4+vhh4dqamQLOvSBfJlw4bht7luHdHTp2jG3rGj/HecU1NYlMbRVlgO2dphvih/6ZJ1B928GVehLVvsOFKVsm8fPuuff9p+rL//htLMxx8jEJGd8veHD6O2180NwQFbZwSO8nR4gGPQIMiWEWFNdd48dfXbUDqFJTISIhWRkcoc30pUaQQ5OtPhM2ds+wFrNIy1a4fF5pcv7TtQNZGSwlhgINSQbVX+2LoVU+APP8yewBI3IgcOMPbJJ/i7SRP7qYcrbQT19fgiI9GPJCgI7+vvj9y+mzetO75KVVicHecwgsHBWCuy9Qd85w5SZiZPtt8g1casWfCcDAmHmsv+/VCB6dZNMoBKy9+fP4//cf788Jh+/925BA4Yw3e1cydj776LC4inJy4ie/bYbrycPYUlJgbJ3CrTI3QOIxgdLUkfLV1q20G//hpBktexiXtUFPTgbBWYPXUKi/Tt2+su0Ct5Ej56BJVrIhheJaZMSo4/NBRT9vz5pYqVX3+1b6NxZw9ciMCIDbi7Swv8w4ejENxaJkxAwODbb+0xMnUxeTKRRgPFFGu5fh3F9TVqICCidB0wY2j0/uabEDfYvh3va40AQ3aTmUm0cyf63AQEEP32G2rbV60iunQJgTi1NRoXZME5jCARir+PHSN64w2i9u2hlmENvr5EEydCZebaNXuO0LEEB0MdZNw460uTHj9GKVzp0ji58+Sx7xj1iYhAJLR/f0SfDx4kunBBOYEDXtZWq5Ztx3n2DBHr8uVhAJ88QQOpvXvxfFDQ66tSYymMIQdy2TLVOh7OYwR5R6s9e/B3u3bW658NGIAf8Nix9h2jIxk7Ft5TQoJ1RiQ+HgbJ1RUadPnz232IOmzYgKTrS5dgcP/4A3XIalZ5OXMGCdqlSkmd1c6eJTp/HoZc6YuGs/D4MbzhTz+F4k358jjnQkLwvNrS1Bw9HzeGTnRYe1H+7l3GChVCBNRaGfB167A+ceKEXcfsEC5exGeZNMm6NZe0NKzH+foaD6jYQ/4+Lg4N3okY++ADrGNy1FjWlprK2OrVUMvhEvSzZ8uv9Sk9frWmsERFMbZpExLzAwOlGudq1VCowPvS3LzJWM2a1kfHFcJ5jKD+D+zECUQv+/a1LhUkMxOL1y1b2nXMDuGdd/DjszaNaNAgRDIPHVJmfJzTp2FEvL2Rw6j/f1PTwvzz54xNmYJWAUS4SGzfbjzCq9KWknYnORnZA6NHoy7fxQXf0RtvwBBu2uS4unIrcF4jyBhysoiQFmINW7bg9UeO2GW8DoF/L3/+aZ0RWbAAr1myxPS+CQmMnTxpeW+SzEw09XF3hydmqHxRDUbw9m20lvTywm3AALTkVAOOSmHhog6zZ8PIe3nheyxWDGV+f/yB6L6T4txGkDGobbi6Ig/LUjQaqMyULctYRIR9Bp3ddO6MK3B6OqZuYWHmd047eBA5hUOHmre/NUYqKgqpNkTwHIxN5ZTu22ts/CdOSO03ixWDF2ipN6N0MnN2XiSeP2ds7Vr0Yy5eXGrM1LYtpsxXr1o+A7t4EbO3ixcVGb61OL8RzMhgrEMH5Gfdv2/5m8yaheP+8Yd9Bp2d8OTiVassf+2jR1hXfest86tBLD0JT59mrFQpvI81Fyl7oz9+jYaxHTsYa9wY2ytXZmzZMuvXmdXgyVpLRgZajxLBMeBT3GrVGPvmG0x/bW0lKvIEbaRoUaIRI3CvjZsb0Zo1UNnt1o0oOdmy43JRgOXL7TLMbOXHH9FcqkcPPH7wgKh7d9wbIy0N++XJg8J9dzsrqjGGlJGmTYlKlEAEuG1b069TWuCA9+goWRKSVdWrI8WFMXRku36d6LPPbFdodhaio/E99OqFzIK+fbHd3x8pLeHhRFeuQJ39rbcsk7FzJhxtheWQFVAwxZUrWKvo18+yN+NXJyL71ahmBw8eYBlg4UJpm7lX2qFDUT5oaV9gc46fkoL/ARFjQ4ZY1tQ8O6Kry5ZJfTnatWPs2DH7HV/tnqBGg2ns9Onwfl1dcbyaNRkbNw7fjZrHrxCqNIIcnemwOYvyf/xh+fQwNRUGpWRJrH84C8OGYZqprehizo9s40bss2CB5e9pSv4+IoKx+vVRL7t8ueXHV+okSUtDuSWXqO/QQZkTUemT3JoUltRUqAoNGSIpVefNi9rmpUsZCw+X9lU6hUUYQcuxWF5fo0EOWt68lqcpzJoF78gZAiQxMfiMEybobjf1HYWGIhfwgw9sV5jR5+JFXEj8/S33MDn2PknS0iDCULYs1rh4bfLr7unExjK2Zg3+z97eGFPp0owNHoy1WVvX9qzl5UvkoapMxcl51gTNwcUFkvL+/kQff4y1L1PwdbS330ad7G+/KT9OW1myhCg9HbWp5pKZibUfX1/0BrZnWde2bejs5+eHCop69ex3bGvIzMQ6cVAQ0eefQ4D16lWiGTOUfd+qVVFSV7WqMsePjSXatAn3+oSHo2zy7bdRNtmzJwRqedP1hw+J5s/H2qyj1vZy50aNuNqavjvaChvD6kZLZ88iJ23cONP7ah938GA077E2OpgdpKUxVqKE/NrnkyeMTZuGe32mTMEakC1rYHLy93Pnwst6/33bxVbtsea1fTtjVargOF26MHb5sv2O72j0x3/vHvIvGzTAdnd3eLsLFlhX1aN0CsvDh/jdPnyozPGt5PU0goxJJ/3Jk8b30z7urVvWp5xkF1wh29y2lIzhs7m5oSubLWh/V5mZKKEjgjyZPXX/rOHECSnVpWVLpOfo4+x6fHz8gwah2onn7r33HpLlbdXpU3tgRyGcZzrs7k5UuLD56RxjxmBa1qcPCvPNoVIlFMUvWmT9OJVm6VKi+vWJqlXL+tyLF5CievFC2paWhtSHqlXtp+KRno7p1pw5mGLNmpX9jc45d+4QvfcepuNJSVByOXgQ35E+SvfoiIsj2rED9/bkzh30yfngAzxeuZKocmVMjZ8/J9q8GcIOSnXRe91xtBU2hs3d5m7fRtrMyJGG99G/Om3apN50mYcPMfX8/Xf55+WutN99h2mS9rTQWvjxGzbEtOmvv2w/pjaWCBw8f45UH3d3LPqvXu14b9Sens69e4z98AOSlYnQObBdO/z977+2H18O4QkqQ3h4OPXq1YsKFSpEefLkoRo1atCFCxeUfltQsSIkj+bMwYK9HP7+EGz198fjd95BQvbSpdkzRkv44w/0D/7wQ/P2v3QJn238eCQG20pCAu6vXCHavRvJ6fbEnL696elEc+cSBQbCI5o6FZ5Sz56mvdHMTEiGZWbad9z24skT/Fbr10cS/A8/wOPbvBkBl59/JqpZU3h89kZJCxsTE8PKlCnD+vbty86cOcNCQkLYgQMH2D0z10x0PEFjPWmNkZ4OpYuqVc2XIBo1irGCBdUVIElPR0Bk4EDD+2hfaTMy8LmrVbMsYdkQUVFYh8qXTzm1GVOewt696D7n6orv4elT+x7fVqw5/osXyG9t3Rpefq5cCOisX89YYqIy4zSE0iksjx+j1l9lrS0UNYKjR49mTZo0sfr1NgVGtLl4ESfOjz9mfS42VtI74/AAycaN1g7d/uzYgTGdP294H+3viKvD2KNRemQkIq5FilgWkLEUQ//jhw8Z69oVzzVrZn3bVbUkM6el4f/5wQdYrnFxQTBn6VLVNSHKCShqBCtXrsyGDx/O3n//fVakSBFWo0YNtsSIZFNKSgqLi4t7dQsLC7OPEWQM4o558mTtlGbouPXro7JALXTtyliNGsb3uXED7R2PHkVSdP/+tr/vs2coqC9eHMedOFG5hHL9/0VqKkq8cudGEvbatbYleTtyTUqjwcV42DBcTIgwO5k50/x0FmdPYYmPRy/p+Hhljm8lihpBT09P5unpycaOHcsuXrzIFi1axLy8vNjKlStl9584cSIjoiw3uxjBuDicyN266W43dNyFC5FWYumUSwni4lCKNnu2efv37o2SOm3VZmt4/hwnarFi8I6VNiLafXuPHYNBd3NDYMseJ47S45fT43v2jLFffpECHEWLIrXImkCVGqfzajq+lShqBD08PFjDhg11tg0dOpQ1aNBAdn+DnuCLF1JOmi1f4OrVOMaBA9I2Q/+YqCiU0c2bZ/372Ys//8QYzRGuPH4c+xqKIJtLbCw8zyJFJFHR7PgRx8bCg+VRaHtOv7PrJD97lrFdu3DB9fDArVs3TIFtkcZ3diOlUiOoaHS4ePHiFBQUpLOtcuXK9OjRI9n9PT09ycfHR+dGRESRkZBAeuMN21oxfvwxUePGaNuZkWF830KFUGK0bp3172cv1q/HuEuVMr7fxYtEzZsjovjpp9a/38uXRJ06EYWGIudO73+oGCtWEJUpg+984UKiEyfk8yGtRemyNt4gqnNntC29exc5lBERaCPaqRORh4cy7y2wGkWNYOPGjenOnTs62+7evUtlypSx7EDFixPt2wdj+MknRKmp1g3IxYXol1+gG/fHH9jm5YWTXK6eskcPolOnrG/vaQ9iYvDZP/rI9L5796Lv8IgR1icvp6Wh/eXly+g6p5TB0Ob5c6T9fPopUlg2bSIaNMj+CdgeHqirtachyshAgnqHDjB+RLhgnT2LVKJhw5DkL1AvSrqZZ8+eZe7u7uyHH35gwcHBbM2aNSxPnjxs9erVZr1eJzocEYGmSl5ejHXvbltibM+eWOcy1SsjPh7vN3Om9e9lK0uXIrItVw+sTXKyJINuS+1tz55YfN+/P+vzSsjf//035LkKFmRs6lTnKWsLD2fs+++hnEPEWN26UPVRcvzOnMKSkoJUoAIFoGmoIhSvGNmxYwerUqUK8/T0ZJUqVTIaHdZHNkVm9mwYhWHDrB/Uw4c40b//3vS+772HSLGjaN0aN1PMnIkggi0n4bhxeP2GDda93hJiYxnr1Qvv9+67SCtR+5qURoMIeffuqFTJkwfrl/x4Km0p6RDS0lDZMmUKY61aSc2ZypWzv4ybjThP2Zz2D3jhQvw9Z471Bx85Elprhw7h3lDuGQ+mOKKbVmQkDP7Spcb3e/ECPVa6d7f+JF+8WLrIGCI5mbHgYNv16A4fRu8RX1/d1ptqNYJJSfgfVK2K11esiIDZixfKjNMQak5hyczE9zprFhpr5c2L78rHh7FOnZA/efGick2obMA5jSBjUC5xcYF0kjU8f47qh08+MX5ixMbiqj9/vnXvYwvLluEzmup69v33uNLeu4fPYamk1d69MLZDhhi/SttqpNLSMN1ycWGsefOsOZtK9+21dPxhYRhvwYIYc+fOWCZwlCejpouERoPf26JFkFErWFBStWnThrEZMyCuq93E6+pVVD3ltOmwLRg1gpmZmKrmy2f9lzphAvLvTP3j337bvCmpvenWDWkixoiNhUdl7fLA7dt4fYcOpq/StpyEDx5gWcHdHSeIIzwCc8d//jxjPXpgrD4+SM8yp5Oh0snMjjaCMTEQGBkwQJLqd3PDb/Tbb7FUYKzUNCemyNiVAgVQJM+Lx11dkTZToQJED6KiLD/myJHmSXO98w7R0aOIXGYXaWmICnfoYHy/uXMRLR89mujRI6LBg3FvDrGx+Gz+/ug65uZm+7jl+PtvFP4/e0b0778Yq9x7KS1wUKIE0U8/4V4fjYZo506ili2J6tSBkMPs2USPH0O4oFw508dnDP83xuw/dkeQkYH/13ffETVogCh39+5Ex44hEr5tG7IXTp6EUEmzZs7Zqc/RVtgYZklphYYiobdVK/P752rDO6NpJ1Dr8/Bh9tcSHzpk+qrJvcARI/DYkittRgYaaRcogHU+c7D0Sp6ainJFIqxXmlpDc4SnkJqKfrtvvon3btAAEmHWeKqO9tTsdfwJEzAL8fXF44IFUee8bJlta+PCE7QRQz1pS5cm2rgRntr48ZYf97vv0PNg3z7D+5Qpg94Iu3ZZfnxr2bUL+ZE1axreZ/58eIGjRll+/ClT8JnXr4c3bW8eP0ZP5wULiH79lWjDBvQ3cSTaPTpeviSaNw+fvW9fooAAouPHkRfarZtyXrEtFCsGsWBbCgb0SU/HuTN6tCTRNm0aErxHjiQ6cwYe/IYN6MlsKmHfGXG0FZZDtu+wqavI7Nl4fssWy99wxAhEV41Fxb75hjE/v+xbFA8KYuyzzww/n5yMOtQvvpC2mXul3b0bC/2TJ9tnrPocO4axlSxpmYpNdnk6gwcjN9HNDWk69hLQVamnk4WnT+H9vv8+1jx5TXOfPpDwio5W5n1zooCCrVgkpaXRQGnF19e8RWxOaCgixO7uCOMb4uBBvL89FJpNERKC9zKm3Pz77zBkd+5I28w5CR89grhC+/bKKDEvXIjvsnlzden9xcZCg5AItbwDB9o36Zsx5ZOZrTUiGg1qsKdORXDKxQW3unUZmzSJsXPnHK/K7UBeHyPIGH7oAQGM1atnvpAoP27HjpBpN7SumJKC5Fg5TUJ7ww2JoTU0jQaeYpcuutvDwuDVGpJmyshgrGlT5OhZozBjTP4+LQ0NgIgge2+NUIASRjA2Fie6r6+UCbB7t/2On51Y8v2kpSGdZ+hQxsqUkST6u3VjbPlyec1DpVNYcqKoqq1YJap65gwMyOjR5r0JP+7ataarJdq3R7qM0nTqxFiLFoaf37ULY7W0faatbTcN/Q9iYpBC5O7OmAUVQVlIS4P0lC1KK5z4eHg++fMjh3L4cDQeV3K6qnQys6lzID4ev98ePaSgRqlSUtN1U0rpzh7YsZLXzwgyhjw0FxdMHUyhfdzmzRlr1MjwvrNn44SytWLCGOnpuGJPm2Z4n9atMZXRX59MSECLUbma6JMnbW+7Kfc/ePAAkvcFC5r3fStNcjL0+4oUQc7ekCGo82VM+bI2RxiR588Rte3USfJ0q1eH+O3Fi5atYQsjqD6s7jaXkQGDVrq0ZWkZf/9t/J906RKeV/Jkv3gR73H8uPzzN2/i+XXrsj5n6EeWkICazYYNrUsjMnT8s2exoF6+vH2qPGwROMjIwDSvdGkY+88/z1qRojTZZUT27EH7hFat8FldXLDM8csvtq1z5lAj6DwpMpbg5oZOZC9eQDvQGEWLQnqqaFEkDpcsCS07OapVQ7L2kSN2HrAWx48T5cqFhF05li6FHFTXruYfc8wYSZPR3L7Npti9Gykw5cohrSQw0PZjWtO3lzGkE9WoASmuevWIbtzA91S6tO1jUgsREUgFy5WLqH17SHS5u+O3+uQJEpiHD0eqj8AyHG2FjaHjCVrSk5azbBmuPDt3mv+ayZMRADHkQb77LrxMpejenbHGjeWfS07GtPPrr+Wfl7vSHj6MbfZQyOby94sWYf2vc2fL65SNYamncOkSvCEi/E/OnDG+v7OVtUVGoma9aVN4ex4eKG9cvlyZNBalU1iUXjO1Eucxgtb8wDQaNKwuUcKwUdNfRwsPxxRjwQL5/X/+GeuCSrTj1GigCThmjPzzPHhj6EKg/x0lJiJa3rSp/VIg5s3De/Tta9vUWg5z/8dPniCH0sUFii7bt5u39qX0dMwe0c/oaCjWtG6NIJa7u2T4tDsiCuzG620EGcO6UL58SN+QQ+6477zDWK1a8vufO4f9//3XsnGYw717xj3XFi2Me6FXriAJmPflGD4cqh7mlsWZgouGDhqkTNK4qf9xSgpSlPLlQ67jr79aFklW6ZoUS0pCkvI778Dbc3WFEVy6VNfjc/YUFqXzKK3k9TeCjMF7cXGRN1xyx922zXBidHo6tNKUyBdcsQLjlOs9e+cOxmSmKverXsv2UMXWaBj77ju8P5Hx3se2YKxv7+7djAUGwkv/3/+smw4qbQQtmU5mZCCPr08fGHUi5LfOnWtYRdzZAxcqvQi9noERfb78kqh2baIvvjDdYIkIyi1Fi6Lxjz7u7kT16yMYYG9OnCCqUkVSytFm2TJs79bN9HE0GvToqFzZdGDIFIyhJnvyZKKhQ7HNxcW2YxqiWDHUq2rXxoaGEr33HoIBJUuib8fcuUQFCyozBlsIDoYKTXCw4X1u3ECdbunSRG+/DQWWb77Ba86cIfrf/4j8/LJvzAJlGy1Zy4IFCygoKIjq1q0rbSxbFtHNsmUtP6CbG9GiRWiwNH++6f3d3SHbtWYNCsz1adgQP157SyadOEHUpEnW7RoNOrB9+KF8QyjOjRsQBJgyBSfUb7/Z3lRo0iSi6dMhK9W3r23HMoW2wEF6OtHMmWiCdfYshB4OHoSQhbVUrozfQOXK9huzOcTE4HdXpw4ucr//DsN++jTRnTsQ8VBCxEJgHo52RY2hMx02VApmCV98gYJx7emW/joah+fr7dqV9Tg7duA5e9aePnuGY65Zk/W5f//Fc0ePGj8Gn274+mKaZSuTJ+N4fOqfXdOlFSsgZe/qinVNlRXcG0T7+8nIQD5f9+6ISLu7Y83v77+tD6o5+3RVpdNh5zCCt25h3eSdd2AsrCUqCvp5/fqZ3lejYaxyZaiM6MMNllzCsrVs2YJjyum1DR3KmL+/6Qgv/5HlzWu5eIE+v/yCY02dKm1TWv6eN453cWGsTh37p7JkV1nbgAEoVyOCTuFPP9n+/2DM+VNYlE5RshLnMIJxcfhhEaEjmi3wIImhxkraTJ4M4ysXzQoIgJdiL8aNQ3qMPhkZaA/KhVONsXmzJGBgCytW4DijRtl2HEvYv19qGTpypP3TbxhTzhNJS8N337gxjs+70J0+rbrOaoKsOI8RPH9e8hI2bbL+oGlpyC1r3Ro/0OvXUfZ1/XrWfXlE9u+/sz73wQfIv7MXHTtCoEEfLuF1+rTpY7RogX1PnrR+HNu2SWVn+idwRgZjcXH27Q8SFweDQQTvz5mmY48eoRabG+969SBxZqqftbW8DiksiYnKHdtKnMcI8h9w27ZwqY8csf7APAVmzx7TJ0aNGox9+GHW7TxfzV4GoVQpeeUb3tTGlEdx9Cg+x+jRMCzWcPo08gq7dZP/XPY2IocPQ+Ypb17GfvsNJ6DaBQ4yMxnbtw+VQ66u+A188YV5MwtbccY1u8hILBsNHIjfsZeX6voyO58RPH0apVL581v/ZWo0jDVpAgPHk58N/eOnTMEPXV855sABvM4e/9DoaEnOS5u0NJTJGaog4Wg08ELq1rW+MuTePQSIGjc2rJJjr5MkORnTeyLGmjWzTATXFmwZf1wcllIqVsQxqlSB4dZen1M6mdkZjGB0NJYGhgyB5iXPLa1cGUrWSuaZWokqU2Rk8fYmatMG+WGbNyNnrEMH9D+wFBcXpH1cvkx04IDxfbt1I0pMJNq/X3c77/1x+bLl76/P1au4r1ZNd/uRI0iv+OAD46/ftQtpJF99hXy+J08se//oaOThFSyIDmLG0nBs5cYN5FkuWIDOb4cPm9fJzR5Y06MjOBhiBSVLQmijenX05Lh6FXmn3t7SvunpROHh8mlVrysvX6JXzahRyMUtXBjiHrt3EzVqhC6GERFEN28SjR2L1yiVZ2otjrbCxjAqpfXwIQIGDRtan3LQvr3UP9WYbH9goHxEuXRp+wQP5s6FFpx+MGD4cPTpMDYV1miwltakibRuasmVPC2NsZYt4QWa8shs8RQ0GnhOXl6ImMpV46gleqjRoNtf585Ygy5cmLHx402vxTmDp2br8TMyMHv64Qf8bnLlwmv8/Bjr2ROiJYaiyypNkbGTrlI2kJlJlJRElDcvkp/LlIHX0rw50cCBRMuXW36F+f57SC+NHm04WdXFhejddyHNpdGg3zGnZk2iS5es/kivuHIFScD6Mle7dsHbNfa5du4kOn8eicTWXGGHD4d818GDynlkL14Q9e9P9NdfqGT56Sd0+NOHMWX79iYkEF24AI9F24PjpKdDruqnn/B/ffNNSHL17Kmsd2wuhQoR9euHeyVwcYFUl/7vKCwM3t6+fZg5xcQQ5csHKbVZs4hat0ZSu9o8PHNxtBU2hlm1w6tXY/ucOda9SYcOUEY2FuA4dkw+Qvvtt/BGbaV2baiyaBMcjPfcutXw6zQaCD00a4a/Lb3S/vYb9jdXEt8a+fsLF5BOlD+/fJRdf19HeDoJCVAH4rl9bdowtnev5ektKvV0LOblSwQNhw/HWh4RgkANGuA3f/y4dS0QYmLQu1uuNt6BOL8RZIyxr75CWoc1EWNe/WHMEKSnI8laX5p+0ya81pZE2PR0TBF/+UV3+9y5mGoYS7fYvh3vz5WuLTkJT52CYsngwdaO3DgaDWOLF2OaX6cOOuiZIruN4PPn+J8WKICKjt69s1YOWYLSycxKprDcvYvfXLt2+D0SYSmmXz/8zlVmuOyJKo2gxX2H09OxPlGsmNRPwlz4cStVMn7l/+gjnMza3L6N1+7fb9l7asPl8g8d0t3etq3ppk5Nmuj2RHnwAOsypsr5nj3DD7xhQ/O78jFmvvx9cjJjn36Kz/Xll+av2WaXEdy9m7Fhw5DUnCcPPJ7sluK3Bnt+Pykp8HaHDcOaNxEuBPnyocf29ev2T/Q2phLkQFRpBDkWSWlFRqK0rGlTy6oN+HGJkPZiiJUrsY922V5GBjyduXPNfz991q3DcbVbYCYm4rj63qE2Z87gdZs3W/Z+GRkwroULW16Pbc5JGBaGVB0vL3xnlqB0si73+j08MD3/7jt4g/ZC6WRmW41gZCQCF++9h9xM7u0NGIBlF1626KyBHStxnhQZUxQrBqWRkyehfGIpFStikdcQb7+Ne+1UGTc3okqVkPZhLVevEpUoobvYffgwUWoqgiKG+OknovLl0ReFk5JCdO8e7g0xfTqCIOvXI+3Dnpw6BaWUyEgo4nzyiWWvz50bwQi5oIktPHxINGAAlFtcXSGtFhqKwFjhwvZ7n6dPiWbMwL0aYIzo2jWiH35AWpKfH9Hnn2N848YhvevRI6LFi4m6dCHKk8fRI3YIzmMEq1ZFTmDVqob3adoUeXLTphEdOmTZ8Xv1Itq717BBK14ceXz79uluDwpCDpS1XLmSNT9w1y4YOEPNix4+RKR15EgYYs7Nm3iNofGcOoULxPjxiOjZkzVroKVXoQKi1bVrW36M0FCcpKGh9hlTWBhy+QIDibZuxQUgPp5ozhwiHx/7vIfayMjARXT4cET7q1WDYS5VChkOz54R/fsvjGD16s4b0bUjzmMEPTzQZc2UPt7o0Qjd9+6NJGBTFCiAFIju3WHo5s0zvO9bb8GL0k7hePNNGE5r0zquXMGPUZu9e4natTP8A507lyh/fsv0/eLj8Tnr1YN+nb1gjOjbb3ER6dED30/RotYdKzoa4rHm/N+M8fQpEpwrVMDFYto0opAQoq+/RoqVs2IohSU5GelifftiRtSqFT53+/ZEe/YQRUXh8Sef2NfzfV1w9HzcGDprgpb0pA0LQ8Sve3fLFncnT0btrKFI2D//YE3jzh1pG48QWyPx9eJFVsn88HBsMyQSER+Pxevx47M+Z2zNpVcvxry9bdNA1F/YTk3FcYnQ8N7WhXRb14xevEAflLx5oak4derrVdamTVwc1pPff19a3wsKghrR2bPWlU8qncJiS19pBXEeI2jpD2zDBsMipdokJyMnLzkZJ7eHB050OeLjkYqzeLG07epVvI+hZunG4M3cT53KOm5DfSYWL0bOllxQw9B3xJvK//mn5WM0xIsXUsXA+vX2Oaa1RiQlBUGkQoVwERs9Wr4HidJGSmk9vthYBJs6d0bgjCvvTJ9uWStagQ6vrxFkjLEePeARRkQY3kf/uB99hJQBQ15N/frYh/PyJUqrfv/d/HFxuP6fdsrA0KGMVahg+DV16jDWqZP8c3LfUVQUY0WLMtali+2eGvcUbt1irHp1RFiPHbPtmNpY+j/WaGCAAwJwYejf33iKlEqjk0aJi8PFq1MnpLAQQWnn55/tb2yVTmGxJtk+G3i9jWBUFHIHjRkA/eMeOSKft8f55huk4mgfr2xZbLeUn35Cnpr2sWrWzFo9wuGS/9u2mf8eH39s+kJgLvy7KlkSGnrXrtl+TG0sSTE5eRIVDETwjG7cMP0apY2gvVJ8EhNh3N99V/L4GjVCUYAzp7Co9CKUbYGR6dOnk4uLCw23tfuZJRQqRLRwIRaNN20y7zXNmiFdZskSw89HRGChnRMYSHT/vuXjCwlB4yi+0B0fj0CJXLMlItSx+vsbT53RZts2qHjMm4egj63wz+ziglSkKlVsP6Y2JUoggluihOF9Hj9GgKdRI6QRHTpEtH07ovSO5tYtfCe3bln+2rQ01IF//DECSx99hN/ZtGlIY/n3XzwnsDvZYgTPnTtHS5YsoWr6qSCWUKIEcuOMnSBydO2K2//+hy5mpnBxQZrGli3y+zdujH2OH5e2VaiA/DxLefiQKCBAenzqFEQamjbNum9SEtHq1USffZZVaIFz5w464d25A7GAwYOJOnaE0bCVK1cggkCEbmnWdP0zRUIC5MMSErI+l5JCNHUqLlAHDiCKfO4c0nLMJTAQ6SOGUo+yG8Zg3AYNwsWtc2fkjY4fj4vqmTNIgypVytEjfa1R3AgmJiZSz549aenSpVRArp+uucj1pDWXefOge8b1zEzRuzfyrdavz/pcgQLwOrT7DnMjaGmaTEiIrhE8cQJpQHIn6caN0DXs18/w8ZKS0MYxKQn5krzVo625YBcuwNjw716pNAtDfXt37YKH9f33SHQODsbFQDtH0hy8vZE+Jacgk50EByNNqXx5eP07d+L/euUKWoKOG5d9GosC5Y3g4MGDqWPHjvTWW2+Z3Dc1NZXi4+N1bq/Q7klrKSVKwItYsgTegza1asF41aolbStWDHl6K1fKH69RI0wHORUqwMhaImbKmDQd5hw/jpNCzmitXIkEZ3M8sHv3iH75Bfl7tnpsly6hWiYwEM3oa9a0f0WHIcLC4MV37IiLxbVrqOqxNtE5PBwXwvBw+47THF68QGVGo0ZEb7yBXM+WLeGZhoYS/fhj1qR5fQICcDHUvnDaE19feKO+vsocX60oueC4bt06VqVKFZb8n1x78+bN2bBhwwzuP3HiREZEWW5WB0a0SU9HRNNcCXqeqiLXXnLZMkSEudjrtWvY98QJ88fz9Clew+WlUlNRbyuXnhMejvf74w/jx+TfUY0aEISwRBxBjitXkHZSpw7SM5SGj//MGUQ/8+ZFAGb9evsU8yu9MK8vCsv7kfTogf+tqytUWtatU7aZkVpRolGXHVDMCD569IgVLVqUXdZSEDZlBFNSUlhcXNyrW1hYGIxgRAQ03mz9AfMCcW1jcvs2ooz6eVYvX6JR+8SJWY9z/TqOc/AgHick4PGqVeaP5fRpvIafMKdO4fHZs1n3nTMH+YumDJG2GAQfm7Xcvs1YkSKIVmeXjBIff+XKMPpDh1rfNMrY8ZWOToaGMjZpEpTHuULRjz9arnCkTw5NYVEaxYzgli1bGBExNze3VzciYi4uLszNzY1lmHE1eJUiExoqNbgxJjJqDj16IG2GVxIYOzH69GHsjTfkW096ezM2bZq0rUgRVJyYC1eP4Ybt55+R6Cv3A2zQAI3nTREeDnWYDh3MH4ccoaEQGA0K0lW3UVL+PjkZicZEyNM0p8WopShpBHnv4XbtYMDz5cPnOXnSfpJUzp7CcvcunBm52ZUDUWxNsHXr1nTt2jW6fPnyq1udOnWoZ8+edPnyZXKzZFE7f34s8BMh4hkZaf3AZswgiovDGowpevQgunsXC9bauLlBLeX8eWlbQADRgwfmj+PhQ3yu/Pnx+PJliEPo10aHhCDY8dFHpo+5ejXWnn75xfxx6PPsGdYA3dwgFqGtbqOU/P3p01iTXbWKaMoUqRmTMxAaSjRhAlHp0li/PHZManb1+++I1guRApCQgN+UXPTfgShmBL29valKlSo6t7x581KhQoWoijX5ZSVKQKwgPR2F4dpBE0soXRpdw37+2fQCeatW6MC2YUPW5+SM4MOH5o9DPzIsJ6RAhIXw3LmxYG2M588R/GncGBFsa0hKIurUCReJ/fstT0eylNRUBCoaN0bPiosXYVBMiWRYi716dGRmImLdqRP+h7/+CgO4fj0CZB064PMIQGYmMgz+/NPRI5HFeVRkKldG+sDBgzAg3brBK7GG0aOhJmJKd9DDA02W/v47q/dTuzaSWJ8/x+PSpRHNNBdtI5ieDvkruejghg042UydVJMnY4xHj1o2Dk5GBjzfmzfRLtFQ4yl7ceUKUd26yP2cOhXRdo0GGofXrinznmXKwDsrU8a618fEEM2ejUh5x45IZl6yBPcLFqgn/9DRaDTId5wzB3qXhQrBaVi0CM8r1UjLSrLVCB45coTmzJlj20GqVIE23NGj0Iqz5gv19UUu1vLlOPlXrTKcStKtG/K6rl/X3c5Tani3OW4ENRrzxqCdHnP7NgyhvicYHIzjm5oKh4Yi/cISaS1tGEMy+a5dkFziPZWVQKOB4atXD9PE8+fhDbq7K9+3NzkZU+3kZMted+0aRFlLlkQic+PGyBO9cAGJ9dklz6XWFBbGkJa1eDF6ZBcrht/ymDHIbf3qK6R/HT2K/dW2PODoRUlj6NQO6y/K//knFnGnT7fu4MnJqIHt0cP4fikpWOSeMkV3e2YmgiM//ojHW7caV3/RJiMD0d5ff8XjVavw2hcvdPf76SekViQlGT/egAEIiFgrjz5nDl63dKnx/WytjY2IYOytt/BeX3+dtfeImhb+MzPRyKpVK7zG3x+/AWNNtVRaG2s2lqSwPH3K2Nq1CP6UKYPP7eaGIN748ai9/y817hXPnjE2f751snMK4jxGUO4HNmECInHWRowXLcLrx483/o/p3h1tMfVp0kRSlOHiBmfOmH7fsDDsu3MnHn/zDX5I+rRujQbxxnj4EOoiM2dadxLu3Yv8ta+/Nv811rB7NyLoxYsbbkylBiOYlMTYwoVS86H69RHJNydtRGk9PkemsCQn4//2zTfIt+WpWFWqoFnTjh32TWfKRpzbCGZmMtatGzy169ctf4PUVMb8/EyfGNxT08/zGjwYaSSMoWEPEWN//WX6fXkfY6580qZN1hSY+Hhdb9EQ/ftDKisx0fIUhDt3IIfVvr15V39r9PLS06HvR4T3MXaxcaQRfPoU7TcLFcJF4f33kd6iJrIzhUWjwe/z55/R+ZC34fTzY+yTT3BOWKpMFB2N18lpPToQ5zaCjCFRuWpVaPBZU9UwZozprm1RUTgx9HsTL16MKUByMn405nae453r+DS3WLGsPY23bME+9+8bPk5ICLzA2bNNv6c+8fFI4q1UKes03BCWnoQREej+5+bG2KxZpit1lO7bKzf+Bw/QFtTLC7JmQ4dar76tdDKz0kbw6FEc/733pEb0np7oTjhrFgSEbcl5VOlygfNEhw2RLx8UX6Ki0EPB3MAEh3drM1QnTIToVsOGiJpqU7Uqwv+3bmGx19/fvLrUkBAsHufJg34YT59mjQz/8w8UU4wV0s+ahXSYQYPwODMTqUOZmcbfnzEs6D9+jCCTEgvtx44hwHL/PpRhvv4and6MobTAgXaPjuvX0RclMBBpSOPGIdo/b571tbnh4QgCOKI22RoYQ6Bo1izUMbdqhe0XLyLlZ9cuRMT37cP/r2pV9QU17IGjrbAcss3XTS3K8/4flgZK+NXJ3d24mOfUqQiEaK/HxMfjtby/bpMm6Llhir59sdbEGGpL9WuUNRosxI8YYfgYUVGoMPn++6yfxdSVdu5c431MDGHO8TUaxubNw/fZsqXxQII+SvftZYyx8+chVkoEb2fePNOBJ3NRw5qmKZKTGdu1C94vD2jkzs1Yx47SsoWax68AqjSCHJ3psDmMG4dpqyWS73fvIgLo7W1cHfr8efwDjx7V3V66NH48jDH24Yc48U3RsaO0BjhrFqZh2mtyly/jvQwFEBiDUfby0l1jM+dHdvo0DJQxA2sIU8dPSYGBJ8Lx09Pte3xbOHMG5YREWDpZtsx2gQl91GoEnzxB5L9LF/zWiKCGPngwAlbcsVDr+BXGeYygOYvy6emMNWuG1BftmldzGDUKHcoMrUdlZiINZdw43e3t2kHenTGc+G+8Yfq9GjRg7NNP8Xfv3pJXyJk2DcEe/RQSTkoKFqgHDtTdbupHFheHfhwNGlgXYTTmqT19yljjxlhDsrahkxInyblzuOgQMVauHLwfZ+02Z24Ki0aDWdMPP+C3RQTnoEkTpHTduCG/tqd0CoshsRIH4zxG0NwfWFgYYwULYnHXnEVc/sN6+BAL+PPmGd73ww+zGqyvvsLJxRi8Om9v0+8ZGCilpFSrhjw/bZo2xZTNEH/8ge9C/8dk6juyR9tNOa5ehXEpVsw24QN7GpErV+D5cBWXtWuh0KOkkXJkS8nMTEi5ffUVY+XL43PmzYvsiZUrkb0gkMX5AyP6lCyJ0qgtWyDBboorVxAYiI4mev991IEaCq60bg1R1rg4aVvlygh0pKQQ+fmhOPzlS+PvGR2NmuT0dARVtIMiyckQFOCL1PowhoqLd95B4MRc1qyBwMJvv1m/8C8nf3/gAIRg8+fHd+No4YN799CLo3p1BD/+/BP3PXpYrkRtKeXLo99J+fLKHD84mKhtW0l5Oz0dNd6DBqHOu0kTVD+1aoXAmqVN12Ni8BuJiVFm/GrF0VbYGDZ1m/vsM0wpjaWYMKZ73BMn8Pfu3fL73r+P57dvl7bxKo1r17CGR2Tc08rIQIL24sXwGIgQHOHwNIVLl+Rfz5+X64ZnKJn20SNoI/bsaXhc5qD/P1i+HOuL7drZJ63Flr69kZGMDRqE8fj74/vV/x6Unq4qnczMx//LL5B5y58fjwMCGBs5Er9fWwRLc+ia4OvnCXLmzMHVr29f89NmGjWCB/Hbb/LPBwSg6c2RI9I27o3duSP14DAm9RUXB2+uUCGpe5t+nxEfH6QjyLF0KdI6WrTI+pyHB3qUaKuwMIYGSd7ekhyZrTCGrnCfforbjh32SWuxRuAgMRG9R8qXh4rLtGnwBgcMUE6NxhDXrqFTnL0FIFJT4WFOmIDHI0YQnT1LNGQI0lnu38fsoHFj5b3d1xDnMYLFiqEg29xGS97e6Ilx/Lj5J7+LC6YWO3cih07u+RYtdI1g4cLI1bt9G9NhIuT9GSI6GvfcCLq4QHyBc/w4jLHcjzkmBn1WPv9cPl/r/n1Mk7Xbfy5fTrR3L9ROuHahrfzyC/LqJk1C0byh7neWYonAQWYmljsCA2GQv/wSn/ubbwz3QFG6R4c9SU9Hnl6fPjCsXbpA25IIv4GbN6G9WLPm65m7l404jxE0pyetPs2bQ4R17Fjztf4+/hgnkaH1xGbNIIDK9QxdXNA45949GDY3N+OeoL4RLFkSCbxEOLFPnjTcd3j1ani1htRi4uLglfE1y8eP4TX07Wt+r2JjZGTgfu1aXFgmTrTvCWhu394jR6Di8/nnSPK9fZto5kzTOooFChB172693qLSaDT4bAMH4oLasSM8vhEjcHHYuBH7iU50dsV5jKCxnrTGmD4dBseQ7FbVqlBT5tNPb28soi9fLj+NbtoU2+Vabrq64qptjidYsGDWvsPXr8O4yvUdZgzeXJcueA9zGDwYFTU//2ze/sZIT0f3OiJ8p4MH235MS3n4EMGrli1RbXP6NAyyuR31nj7Fd2Hs/5PdMIYp7ddfY0bQsiWCHQMHImh38yY87qAgLMXMn69cH+K8eYkaNMg+aTC14OhFSWPYFBjRZudOvHbtWvP2P3nScLKyRoN8we++k7ZNnIi8PcaQ8jJ4sOFj87rh5GTkTPXpIz33668QTZCriuFj2rvX8LG1v6Nt28wXdDBFSgpSPzw8UNOsFIb+x8nJqIzx8kLQY/Vq62pY1bTw//Ah8vgqV8ZrihRhbMgQNNyyV08StZGczFhwcFaJLQfjPJ6gLXTsCHHUESN001uI5NfRGjTAFFeuntjFRb7vcGQkvNTChZGaYIiYGHgxXl5Z+w6fOAEFXrk1rWXLsK8Z/ZspORkiqe3aoQbUFlJT8d3t34/F+Xffte14lrJrF9oqTJ1KNHw4AlA9e6pzHax6dfy+5NokEOH3sXw5vL2yZfGZatbEZwwPR3pWgwaGP5uzp7B4eeFc8fJy9Eh0UKURXLBgAQUFBVHdunXtd9A5cxBJnDhRd7v+OhoRfoR9+hBt3ozX6NOwIaZiXKiA54WFhMAIcsl9OaKjMT1/+RLTMj4dZgxBEbmpcGoqJP579TIuQlCiBKKEGzfCKP/6q23GIi0NSsEHDhBt24bjKyl/ry1wEB6O9TvtxuvTp6u7d4ebGyL72kEtvs73ySdY5+vXD//DFSvw/1+zBj1zzIlkP3xI1Lu3Zb1sLOHiRXz3Fy8qc/yQEPyGeVaESlClERw8eDDdvHmTzp07J208c8a2g5YsiTWt+fOxyGyKjz+GodqyJetzDRrAOPIFfG7IQkKQomLME+RGkP+Q+WtDQ9GrQi4osm8fush9+KHxMRcrhn4kv/2GYJAtfUIyMvAd7NmD76BNG+Xl72vWhBd78iSS0E+cIFq3Dl6oJYnhjkI7mTksDNHbChXg+Z06hYj6w4fok9Onj3JqOWolNhZGPzbW0SPRQZVGMAsREZgK5c5tWo7JGCNGILI2fLh8kESbsmVhkNaty/pcnToYBzfMxYphbOZ6ggULZs0RPHEC940aZX3Nhg2YEprq0hcbi6minx+aSVmLRoPcwm3bUHHQvr31x7KEW7fgCQ8ZAgN86xb6q9hr6qt0jw4uO/Xpp8h1nDEDGQrHjiG9Zfx43XQogSpwDiPo749E2NRUdKyytltVrlzoFnbgAMqKTNGjB37U+p5dvnwwStwIurjAaD54IHmChsao7Ql6eOCzEaFpT/nyWdtBvnwJY2RO3+G//kLjoi++sH7dhTFo4q1ciZIzU60+7UFGBqa61atLrRkXLbJfXiNHqbK2kBB4eR074nFcHCL5kZFYA2zaVJ1rmAIichYjSET03nuoJli8OOu6niV07ozpyahROPn4Oppc/mG3bjAKW7dmfa52bZywnIAAyRNMTzfcF5kbwZAQeAt8/ejqVfkF9V27MPU2NRXWaKSm623bGt/XGNOmYf10wQJcBJTm5k2ssU6YAO8vNRUXGCVIT4eXbo/pfGYmDGq7djCqCxdKQauVK5HDaO/prrOnsFjbK1xpHB2eNoZOiszVq4yVKMHY8OFIKVi0yPoDc21AU93VGIM+4NtvZ90+fz5SRrjc1aBBaEBjqn64dGk0duraVTquRoPeFtoCqZxu3eSbPOmzerXU/MbaFBCevjNpkvzz9pS/z8iA6o6nJ9JEzpxRVwqLIZ4+RWpL6dI4Vr16UPVJSlJtbazZ2DuF5cUL1NmPHMlYzZqomff2ztqrx8E4jxHkP7Dz55FP5eoKNWlr6dEDRjU83HiHsPnzUZSv3xzm1ClpPIxB0bpAAcnAGjoR8uZF85qaNdEkiTHo8xFl7ZoXH4/cuJkzjX+W1FRIWbVsaf1JeOAAPme/fsrnqYWGMta8OU6Kr75yDlHPs2chRZYrF/4nn30m/e85Km0pmW0kJUEMZMwYXBxcXSUF708+geBGSIijR5kF5zOCFy7Ai3jnHRgUQ2orpggOhn7giBHGT4zwcDyvLxSalIR/8u+/4/GaNdjv6lXDidYpKXhuxQoogEybhu27dsl7jxs3YrupH87ixTAo27bBuN68afLj63DjBlRm2rY1roBiD/n7desgXluqFGNHjug+pzYjmJaG8TZoIKm1zJ7tuG5pSn8/Dx5Aachcvcn0dDgDU6cy1qIFLhBE0JXs0QONye7dU33yt3MaQcbQYrJWLZxM5jQ8l+Pzz+G9mfph1auH6as+lStL1SG8jSb3EDduzLo/N6jr1kn3jMGL9PHJ+mPp25exN980/hlSUzE1+/BD4/sZIioKorBVqpie5tpyEiYmQk2bCL2a5ToDKt2319zxx8QwNmMGFMqJ0H5h2zbTMlVKt5R09EVCo0E7igULIFjr44P9fXzglMydC0VrlRs9fewk/+EA8ubFwnTduqiKOHyYyNPTsmNMmICkVVN06YKAQWqq7nvUqAExBSIp9SEmBpFAuax+Xjecmop7nh5z5QqEVbUjiBoNutv16WN8bMuXIydNvxOeOaSnIyE5Ph4Rc6Xy1q5dQ2AnNBTj7dNHPlrKBQ4cxcOHCAr9/ju+m549kU6l3wnQ2Ot790bArGBB5caZnbx4gbzGfftwe/gQqkENG6Le+e23kTJmLyUhB+A80eHAQBi6wEBpW4kSSOS9cAG5ZZZSpoyUAmJMvqlzZ6KkJF0JLSKcHNeuIRxRvDi2PXmCk1kuIZQbRl6Fwo3g1atZT7RLl1BRYEz9JTWV6IcfkD4TFITXeHri3hxGjkSVyl9/KScvtWIFUb16SAe6cAGKNobSRZQWODBU1nbpEiLh5ctDmXnECBjsP/4w3wC+Tty4gUTvxo2R7fD++/jtd+6M6qqYGOQ+fvstotVObACJyImiw8bgPTfMifbqs3cvXjt2rOF9NBpMOf/3P93tXJiBKyEXLYoIb/ny8p3r/v4b+0+ejK5fGg0icW5uWNfTZvJkTDOMrdHxtUC+BmjJdIlHkxcuNL0vx5Ljv3yJIAsRlh0MtUq19vi2otFgTbJtW6n72q+/YtpuLY6erlpLVBTERXg3Pj7Ffe89/MZUGMywJ6o0grJ9h00tyg8ciHQL/YidOfTsCSNnzOAMHIhWjdqEhurK7desiaZJdevixNdnyRIYrS+/lNb6eDT51CndfRs0YOz99w2PJzMTne26dZO2mXuSXL8OI/zJJ5at35grfx8airQeLy9EBM1FaSNy9y7SkpYsYaxhQ7xXtWowAJa2B5VD6fHbK4VFo0Fb16lTGWvUSIrivvkmOtJt2aJciwAVokojyLFISislhbE6dRDBk1t0NwaP6K5aZXifrVuxj3bPEo0GV8wZM/C4Uye0d2zTRtc4cXgaTdeu2IcxeLEuLroeyLNn2PbHH4bHw6WyTp6UtplzEsbHM1axImNVq9qv6bg2R45AFqpMGcYuXrTstUoaEY0GkV3u6TRoAE/enov4Km0pyRiDJ75zJy7mPOCTLx+8vd9/V13uXnbiPGuCpvD0hHpKTAzqXs0trbt0CQu7jRqhpM7Q61q2xNrH3r3SNhcXrMVxQYYSJVDn7Osrnx3Pq0X4PRGCIhUq6FYB7N2LcRir2f3pJyxON2xo3ufkfPklxvjXX5D0sgRT8veLF6NqompVlO/VrGnZ8ZWAV/zUqoWFfCIITJw8iTI3e5azVawIoQSlxB4sVWF59gzrml264PfWqRMCYO+/D1GK6GgoJfXrJ5Vv5kBeHyNIhMX9Zctwgi9dat5rGINkVK9eMEj6wQ+Ojw8Mzv79utuDglD6RYQfUng49tXXLSSCgZYzgvqL77t2oSyP9yzR5/x5LEzzk5pTuTLUqStXln+ddtvNN96Q38cYhuTvMzKIhg1DzfLAgTDi5rR41MeeAgeMoT68Th2UXBYogHpeIgRqnLGW1xwVlrt30WqgcWP8fvr3x+9u0iT834KDUV751ltSW4cczutlBIlQ7ztggCTAaS716uEEnzvX8D6tWyNCzXUEiWAEb91CSoufH8QTvL2Ne4LcGDKWtWaYMVytjdX//vQTIplduuhuz50bdbdyoqwhIWgi1bMnbvYiMRHjWLAA9bPz51sfLbSXwMHhw/DsO3WCh33kCNGhQ7iwKInSenxycHn+CRPwW6xYEQavaFF4gZGRyAAYNYqoUiXnNP4K4zxGsFAhuO36Kity/Pwz+jD06mV+sbyLC9HQoTgJDYlWtm6NvCntFJSKFaH0Eh4OI6jRwAgYMoIFC0rG8PlzGERtwYDgYGxv1kx+DBER6DY2bFjWjnShoSjcDw3V3Z6Zie+icGEYK3sREYFxHj8O73XQINuOZ6vAwcWLuHi0agXvdN8+oqNHIWdFpHyPjuxCoyH691+kOAUEwLgvXIgL+datuBBv2YJ0pCJFHD1a1eM8RtCSnrR582Lad+kSkpzNpWdPTGUXLZJ/vl49eFmG+g7z6Stj8tPh6GgcPzXVcN/h48ehVWhorW/5cqx/fvKJ/PGXLZOSsjk//4y1qtWr7aeld/s2xvjsGbQQ27Sx/ZjW9u0NCYECTe3aRI8eQYX77Fkk8mp7PkWKoEGUMxqGzEzJw+zQAVqX69fj7wMHkFu5YgW8ckvXenM4zmMELelJS4RKknHj0MfBWPKw9jpa3ryoZvjjD6mqQ5tcubDWcviwtC0gAInA2kZQo0Fytfa0mQjGiev8yQmrEsGgVKsGY6mPRoMLwYcfmm/Mbt9GUuvIkfKCrZbA5e9v3MBJ6O2NNgOOSiiOjYX2YcWKuDAtXQoD2rWr/LTP2Xp0aDT4PQwdCmX0/v3xu3j3XWx//BgeYOvW2d9o/jXCeYyguT1pteHrJJ99ZniKpb+O9sUXmJJt2ya/f7NmmIpwA+fujjWsO3ekVpj8vbTbgzKGk48vRnNPMH9+XfHQEyfk+4wQ4Yr/8CFOBnPIzJRUjqdMMe81xqhZE8GGQYNgeI4dw8mZ3aSnE82bh6j64sVE332HZYTPPze+Hql0jw57wBgCX199hVLMpk0xtf3oI0S0Y2MR4Gnc2DaVdcErFP0Wp0+fTnXr1iVvb28qWrQovfvuu3THkmCFreTKBa/u2jUEE+TQX0erXBlejqHocrNmmOpevy5tCwxEtzovLxi0ly+xXXtKnJCAdSq+jsfVpbW9wMhI9C821Hx9yRJcCBo0MPXJwdy5UL/+4w/5YIml7NiBtJKmTbHe5oj62L174XkOHw6P7949XOzUIDQaFARjHBRk+Wvv3IFY8BtvYBazejWi2sePY4r/yy9YfhCGz+4o+o0ePXqUBg8eTKdPn6b9+/dTRkYGtWnThpKSkpR8W11q10Yt6Pff67bV5Mito/Xrh6Jx/QADkVQHy3uCEEnN14ngDaak4G/t4Ij+Oh33BPWnwkS4yuvz9Cm80/79DUf4ihUjGjMG92Fh8JCGDJE/nqVs2gSjkycPemdkt9F58ADrXe3a4fNdvIgLlaE0IkdgaUvJJ09g3GrXRuR27lxcZPfvl1pwNmkiDJ/SZGdm9rNnzxgRsaNHj8o+n5KSwuLi4l7dwsLCzK8YMUZiIioY2rbNWiEgd9yEBGgVTp4sf7z69VFqx1mwACrT6ekoQ+rSBcc8cULa59w5bBszBrXCGg3K8EaOlPYZNgwVL3LMmIEyNHNlmrp1Q0P4Fy/M298Ya9eitKpdO2XLwjIyGIuL05WsevkSze09PSGbtnGj9VUeatDjS0qC9mTbtvhOc+VC1cZff6muKXlOIVsvMXH/TQ8LGphGTZ8+nXx9fV/dSmmnMmj3pLWUvHlxVd27FxnypsiXD1n1K1fKV5A0aoR1QU6FClinCgtDGkpiIqY1f/yBqz0RoqhFimCKXLAgFr1DQ7N6gobWA9eswYK4sSloQgICBH//jdsvv9geDd6wAek1n3xCNHmybccyhX7f3r17UX0ybRoCO7duQWrL2lw3pXt0GEpmZgz/288/h+fasyd+IwsXYglk82bkt6qsKXmOIbusrUajYZ07d2ZNmjQxuI9RT9AeBd2dOsGb0K6ZNeQdHDyYtTaXs2EDnnv6FI/v3sXjgwchMFCvXtbj8scff8xYpUqMPXqExzt34vn4eHgGS5Zkfb/r17Hvtm3GPx9/j5IlGWvd2va62L/+gtfauze8s+wQOGjTBt/5Rx9JgqZqrMWVQ//7efwY/UgCAyWlmu++g9qyQDVkmyc4ZMgQunr1Kq2T6+P7H56enuTj46NzIyKsh1WpIt/1zRLmzMGxfvxR2qa9jqZN8+Yog1u7Nutx6tfHPW+5Wbo0vBNz+g7HxcnnCJ4+De9QLiiyYQM8OnO7yD15gqRgW6oD9uxBRPKDD5CbqJ+YrQTx8Qi4tGuHSPiqVbh3hsbr2hw6hABS6dJI0WrQAGlV9+9jbdreLT8FtpEdlnbIkCGsZMmS7IG5vQv+45WKTEwM1lBcXBhbtsy2wYwezVju3PDETDFyJDQC9WWWNBr0URg/XtpWsiQez56NjlqGPMEmTSBFzju7ca90yhQozOh7bxoNPIm+fU2P98ABSb7eFo4fx3f0zju6HriS8vchIfCgieCxR0XZ/z2U9GTv3cP/iKvU1KsHLT5TWpgCh6OoEdRoNGzw4MHM39+f3b171+LX60hpnT6NH1eePAgyWEtcHAxb7954bKyN5Nmz0jRXn06dYJg5TZuiucyKFXhNfDxjYWHoAcIY7sPCEFT59FO0tSxWTHr9+++jWY0+Fy/ieHv2mP5sH3yAfQ8cML2vIS5fRiOkFi2yZ6E+MxOBpbx5EchxJlHStDQsGbz9No7r6wvjJ/d7EagWRafDgwcPptWrV9PatWvJ29ubIiMjKTIykpLNrfrQhmfElyuHUiGekmIpPj6Ykqxejf4gwcGQyQoOzrpvnTqYrm7YIP/c+fNS4KRsWQQ6uHpKfDwSiXlydK5ceBwTI1WLaAdFrlyRb76+fj2O2aqV8c91+zaCIT4+UtK2pTx6BPmuChWQjqO/UG9v+ftHj1BuN3gwkpg3brTPcZUmIgK/obJlEUBLTETJ2pMnWCIx9b8SqAslLSwRyd6Wm6k2LCuqeuAAFJUrVGDs+XPrBpaWhmO0a2faO/j6a3iO+p3GduzQldYfPx5Bl3//xfZdu+DdcRHW+/fxmLfabNZMmrYmJspP9bms/xdfmP5MXbpg4d1a7y02lrGgIBwjMlJ+H3t5UhoNlgN8fLCMsG8ftivdt9eW8Ws0jB09ylj37ujPnDcv/i9Xrth/nIJsRVFPkGG6neXWt29f6w9aoAA6q8XFIWWEJyZbgocHGhTt2WNa9qhrV6S3aKfEEEGkk0iqSy5dWtISJJKES1+8wOMXL/BYOzDCPcHr1+FR6nuC587BW/rwQ+NjPH8entuUKdalWaSloTrhyRN8t/pBInsSG4uAS58+SH6+dg1CB0TqFDhITkaqU82aCJZdvQpvODwcuow5sRHTa4bzpKIHBGC6FBCAKfH27ehe9vnn5qtIa9O1K37YCxca369+feR26dcSFy+Ok1XbCGo0kvCCnJQWEcbq64vid+2Wm66uupJaRKjTLVDAcBkdZ9IkRFCrVMGU2xIVFsagNn3yJD5jpUrmv9ZSjh+Hod+3D0sMf/6pWzettMCBJWVtkZEQnihdGr+xUqWQt3jrFgQN7KXGI3A4zmMEeU/aAgXwuEEDJDOvWYMyLktxdUXy76VL8MwMqXC4uhK98w7Sc7SNrYsLjCg3gjyxmxs/OSktTkYGjqXdcrNixaxe3D//IC3GmCjA2bPY77vvYITDwy3T45s3D2WDS5YYTtS2lcxMpIq0aIF1tKtXkXqjj9ICB+aUtV2/DtGJ0qWRbN6jB9Sad+zA+qUQJX3tcB4jKLco/8EHOPnHj4chsJSOHVG3GRSEygRDdO6M2tXbt3W3V68ueV3cCEZGoiJBW0FGH147re0J6k+FIyPh6XbsaPwzfP89vDdTU2Y59u1DJcZXX5lu8k5knfz9s2cw5N99B8/q0CHHiZoa6tHBGMbVrh1+B/v3Y7nk8WNJrUbw+uLIBUlTmNVtLjMTQQEfH1QcWArvInfsmOF9Xr5E3tzMmbrb//wTr+X1ufnyMTZrFhb7hw9HAOTJEzz35AlaXBIx9uOPqA5JS5M61k2bpnvs5csRLDEWJDhzBsdbuxaPLVn4DwlBXmK7dlmDPvbi+HHG/P2RCnTokOn9s7tvb2YmUlzq1JHab65alaPaTQpeh25zrq5YWypeHOt8lirUlC2L6ea4cYb3yZ0baTS7d+tu594jl9Xi3eYKFsSUdOxYSeXEzw/il0SQPy9ZElPw0FBMofU9wV27oFhjLEjw44+oUZabWhojNRVLC76+qIgxtxrEXPl7xqCI0qIFvKhLl/D9qYX0dKS0BAUhxcXbG0Gyy5fhKQqB0hyFKo3gggULKCgoiOrWrWveC3x8kCP34AEW+S0JlGRmYo3uxAlMSw3Rvj320Z7mVq4MA8KNIO82V6AAprPbt+tGh48dgxRVRITUJoC/p7YRTE/HVLVDB8PjuX8fYptffSUZscBAlGcFBhr/zMOHYxr/99/SGqs5mCN/n5yMqfXw4bgdPIgLlDkoLXCQlob7997Dul+lSihXPHQIU3ax3pczcbQragyLmq8zhqkMkfGm5frw4/r5Mdarl+H9goOx3/btutsrVmTsf//D3716Mda4MabnjRvLl80VK4ZKk65dsf377xkrWFC3XO7IEexrrDJmyBA0OX/50vzPyhhknIjkhRpMYep/EBbGWO3aWDrgU3Q1kJyM/MOiRTH+tm0Zu3rV0aMSqARVeoKymLMo36sXpPSHDMkaxDDFxx+jOiM8XP758uURyNi3T3e7dt/hEiXwel9fw9NyX1/dvsO85aa2F7JrFzwunouoT0wMcte+/FJXMTo8HFNwQ58hJATtA3r2RNqHPTl7FtN3nlPZo4d9j28NaWnI5atQgeh//0P0e/BgCEIYC4QJchTOYwTN7Uk7bx7SG3r0kG+WZIguXWBQfvtN/nkXFzSsPnhQd/ubb6LxEBGmfZGRWGMy1wjKRYb37EGk0pCi8KJFmMZ/+aXu9qdPkS4kV9aWkQHjV7gwciPtOfXbtAmJxGXKwBjWrGndcezVtzcjA4bujTdg9Fq0wIXqr7+grmPu9FyQI3AeI2juonzevFjsv3EDaRmm4OtoNWtinWjxYsNVKG+9hWTZiAhpW6VKqLSIj0fwIyUFeWiJifLH4EawYEH0Irl/X9criY3Fmpuh+tO0NJzIffpYViM8ZQoM1Jo18p3srIExotmzEZjp2hXfoyPl7hmDoatSBTOCevWwXrt6tfPJcQmyDecxgpb0pK1ZEyf9rFmoUjCGtzc8BW9veA1RUfKCCURShFOu7/Ddu5IBcHGBJxgUJCXmennhVqAADCZvtMSYbh7ayZPYZqhKZMcOGN2hQ41/Lm3OnkWy8nffGe5nbA7VqyMJvHp1eKJDhxJ98w3yNFevdqwy8uHDMHrdu2PZ4uJFVBhZ0/RIkKNwHiNoKV9/DRn8vn2Np81or6O98QbqWA01Xy9SBNNf7b7Db7yB+9u3pZpb3nf4+nXpJAwKghEvUQKPDTVfP3ECxrRcOfkxLFkCQ1alitGP/4qUFHi4NWsaTwMyBy5/n5GB5OxFizCeqVMdF1m9cQMJ5a1aYfngyBGkMlk7JRfkOF5fI+jmJskbGTv59dfRBg1C2oShdJnmzYmOHpUe+/hgjUm7+XpGBjwl3nqTEx2dte+whwdSazi8z4icUQkJQTWDob7DhQqhUx5fbyRCaWBwML4LY+V35hAcjFzHFi1QobN5s/k9kO3N06dEAwdCwODOHaxLnj6N/49AYAGvrxEkwjTzhx/QZOnkSfNe06kTjJmhvsNNm8IYREZK2wIDoW/o60vk6SkFZIoVQwIuEaakSUlS601uBMuUkfL8UlKwn6Gp8LJlmLYbSo4uU4bo99+lHMTz55FQPXGi+Z6jMR4/Rk7dtWsQE3jnHduPqY05AgepqUQzZ+I737QJ/aRv3kTSs8jzE1jB620EiZAaUbcuPBaeLGsMDw9MH9eskQ+QcJEBub7DLi6Y8nIPMCkJU2OirB3IChbM2nz9/HmMUc4IZmQgLaZnT8PJxMnJmB4mJ2P//v2xfjdqlOnPbYpnz+B5EWEK3KyZ7cfUx5TAwc6dMObjxmGZIzgYCdncuxYIrMB5jKD2orwluLnBq7tzB5FMc/j0U1R4yDV2KlECpXanTknbAgNxQjKGdUM55WxeOcKNIleXLltW2ufECXh6chp1//yDqf2AAYbHfesWjMStW0iDuXIF0W5by8CePME0k0tcKSW3ZUjg4P59eOidO+OicfUqUqG0p/0CgZU4jxHU70lrCdWqEY0YgYixvkyT3DpaYCCCKitXyh+vUSPd6XWFCjDQMTFS32F9uMRWZiY+h4dHVon948cR9JBbu1uxAoo3NWqY/rzPnxNNmIDEaHNLDw0RGYmgQ2IiptpKot+3NyUFKjlvvimV+e3dKyK+AruiSiMoWzscHIz6TrleIOYwcSIM3fDhutv119E4n3yC6hC5xOOGDSFzxafLPJL74AE8wcRETF214ak0Gg3G8eIFbtwIajSotJCbCsfFoYqkZ0/zPusvv2BK+cMP5u1viGfPEAiJj0dEvHFj5ChmhxTWwYO4eP3wg9R4vWtXse4nsD+Ortszxqva4dhY1MraKrO0fj2OsXevtO3lSzQ316/BjYlhLFcuxn75JetxeBe6U6ekfYnQlH3YMMYqV0YrzQsXpJaa/PGnn0K6iXeRO30az1+9iseHD2d9vxUrIKv1+LHxz8dre4nwGluIjmasalXUVN+5Y9uxzIWPv1Mn3DdrxtiNG9nz3oIciyo9wSw8fUrEm7YbEys1xQcfYEF/+HAEDoh019G0KVAAyjFyzeKrV0cUmDdfL1AAkeGQEHiCUVFQi6lVC/dE0uO4OCkoQiR5gv/+i2lwvXpZ32/9egRkeI6hIRiDp1SlCjxZa0lMhIJNRAQ8Mp4LqaT8PWNSXfbRo/DOjxwRU1+B4jiHESxenGjBAvw9YoR84MEcXFwwVbx9GxFOU/TogZQV/YX6XLmQjHvunLQtIADT4cKFkQbz8CEqUB49wvOPHuFxRISUHpMnj6QXeOkSTnhuNDlRUcgN/Ogj0+O9dQvGZNEi66eNqamQmrp5M+v6m1Ly95GRRN26IWk9KAhro/36iamvIFtwDiNIJGnk3bwJxRfuyVlKrVo4kSdNMtwMidOxI0QV5Prh1q6NdUFOQACMQ+HCWN978AAR2qgoPB8VhcfPn+t2m+Mn+tWr8lHhzZth2Lp1Mz7W5GSkjnTrhrU7a9Bo4EEePw6xitq1rTuOJaxfj8DHv/+i7vfGDcszAAQCG3AeI1iqFBbl//gD9bNDh1rXZY4IZV7x8Ui0NUa+fJgW/vVX1ufq1EHaDTekpUvD2ytYEI8NGVheN6ydHqPRIPppqPl669amxRLmzEEqy61bWaf25jJqFBKQ161DVYiSREej9K5HDwhT3Lhh2tALBArgPEaQ96T9+GNM9xYtMj/vT59SpZBE/dNPSMfIlcvw1KtbNyQxh4bqbq9VC0aYl9dxI8iVmg0ZQbm+w/fvI7Fa3xN88gTrYqamwi9eoDLk/ffhKVuzXPDrr/g+5s7FdFhJ9u+Hcs6BAzDyGzbAgxYIHIDzGEHtRfnPP8fUb9QoTBetYcwY5Bzu2oV1MEMF9x07IqdPv+9w5coIjvCyuNKlUSnCNQANGcG0NBhK7WqRq1dxr+8J/vMPjHOXLsY/yy+/4Liffmp8P0P88w/RsGFIRTGmTmOr/H1qKt6jTRsEb65ds65LnkBgR5zHCOovyk+ZAtmk3r2l3r+WULAgTsiFC3X1AfXx8UGysH71iIcHTmTt5utEuonSI0ZI09iiRbHYT4Qo8MuXui03ixWTVGg4u3YhJ9FYZURsLKbCgwZZ503dvIkpaefOkB4zRsWKqJSxRpsvOBhJ5gsWwGjv2aMrHCEQOAjnMYL6uLpKHcO6dEFir6XwutMaNYyvo73zDoIFvPSNU6OG5AnyBOLnz2HkXF3RJ7lkSWwvWRLTeSJpuqptBPWnwmlpmDYaa7ZEhPdIT7euPjg6Gp+tTBl42YaUrG1l7VosHyQkQOll+HDl3ksgsBDn/iXmyYOOa2lp8ApNqU7r4+uL9bbnz6GQYojOnRGN3rtXd3vVqvCkMjLg6bm5SS03IyPhNXHPMDFREl3gU2UeGOF9RrQ5flzK1zNETAzW8AYPhhcZEIBItnYpniEyMzEVffECkWBvb9OvsVT+PjUVLQB69sSF6sIFofMnUB3ObQSJ4GH99RdqeUePtvz1POiwdq3hfUqVgsHT7ztcrRpO9Hv3YACLF5dabj54gOnf3bvY9+5dBGOIYLzy58ctLg5TfLm+w/7+xtNFfv0VBvibb/C4QAFcDMxpo/nddyiFM9doWkpoKEoAly2DiMOqVeYZWoEgm1GlEZStHTa2KN+kCSKbv/yCFA9L4N3rNm2CQTJE+/YwglwFhkjqDcIDG9rN1w0FRtzcEPXVD4roT4d37YIXaChqnZyMlKHPPpPWHZ8+xfRYrt5Zm507iaZNQ12uoV4mtnD4MFKIoqJwcRowQCQ+C1SLKo3g4MGD6ebNm3ROuyLD1KL80KEoi+vXD56ZpaSmGq8iad8e647aitOFC8MA6bfczJ/fsBH08YGR4h3Prl5FkEVbnurBA1S1GJsKr1qFNT1tQYjwcDRjN9Ryk0gKMHXubB+dQW0YQ5Dm7bexXnr+fPYkXAsENqBKI2gVLi7QDSxWDGtd5rbb5OtoH3yA9TVD64qNGsEL1V8X1O477O8v9R3Wl9bnyLXcDArSFQbdvRuGsXVr+WNoNPD43ntPt0mTKTIykGeZPz9kwuwZnEhLQ+rSiBG47d4t9P4EToHzGEFzFuV9fJB8e+0aOqCZA19HGzMGBkyuOoQIRqpFCyT4aqPdd5hPh318kPxcuLCkDejujrzC/PmzGkG5qXDTpoZbY+7ahWqVr74y7zNyeNvNtWvNWzfUx5D8fVQUvL/Vq2FcZ82yvZ+JQGCEK1euUO/evalJkya0c+dOio2NpeHDh9OgQYPovffeo8s8a8McHC1jY4xXUlpxcZLMkjlSWrNnY9/9+03vGxnJ2E8/4b5VK8bq1ze8788/M+blxVhysrRt/nzGPDwYS09n7I8/8L4jRzIWGJj19R07MvbOO4wVLcrYlCmMZWYylicPxstJT2csXz7Gpk83PI4WLRhr0CDrdmPf0bFjjLm6MjZ5suHjWkNwMGMVKjBWpAhjJ07Y99gCgQE+/fRTlp6ezqZMmcKKFCnCunXrxh4/fswOHTrEPDw82JAhQ8w+lvN4gpYwYgSmkn37Zu3toY/2Otr//gd5LO21SG1at4aQqra0fsWKmEKHhEjd5lxd5dcEedP1mBh4guHhmDZrrwdeuYLUGN7LRJ/r11FKN2JE1ud8fbHWx4M9nPh4rAM2bmxb2019+ftTp5DM7eaG/D9rhRsEAgu4f/8++fv7k7u7O0VERFBMTAyNGzeOSpQoQdHR0ZQvXz7q3Lmz2cd7PY0gT6ROTJTSUsyhUydUfvz2m/zzVarAeGk3X+cGTLvlJmPIv6tQQZoq37iBPLnUVKzN8R4jRFn7Dnt6Iroqx9KlCMa8+27W58qXR85f+fK620eNggH+80/r2hNwtOXvd+xAZLlSJUSADfVJFgjszNOnT6nbf2IbJ06coCZNmlCtWrWIiOj999+nmJgYatOmjdnHez2NIBHyB+fNwzqVft2vIdzckM6xfr28B+nqClFW7ebrJUogYKJtBDMzYezu35cCNKmp8Bj5Wpl283X9Zkv16sEQ6pOcjKhw377yHdbS05H4rR3cOXAAeXqzZum+jy1s346gTIcOqGrhyjkCQTbQqFEjqlmzJkVFRdH169ephY2KR85jBM3pSatP797w7gYNylryZoh+/WBE1qyRf75ZMwQXuHFzcZG6zRUpgsdyrT15fqG+ESxWTBJSZQyVIob6Dv/9N4zz55/LP3/tGrzEa9fwOCEB+7ZqZbxLnaV8/z2+p40bDbfHFAgU5vDhw8QYy0FG0FRPWjlcXDC1TUqSqir00V9H8/PD42XL5Pdv2hQGUHvdkPcddndHRFhOyoq3BeBpKYUKZe07fP8+cggNrQcuXUrUsqUkMGuKceMQuf39d9vTYRiT8ig//RRSZrZMrQUCGzl8+DB5enpSgwYNbDqO8xhBQz1pTVGyJNGMGTAEx49nfV5uHa1fPwgjyKXjVK8OsdV//5W2cSNIBG9QLkeQV6NwIVh9TUEiTIVdXBBs0Of2baJjx8z36C5cgGLLlCm2l8UxBoO6eDG8yh9+EBUgAodz+PBhatCgAXnZOBtRVTJXamoqpaamEmOMEhISKOE/7yk+Pp4oLAxT1AEDLE/C7dEDitT9+8N4aTcjT0+HgfL1lbY3bIhp5ZIlRDNnZj1e7dowSIMG4XHJkqiVffZMqgcmggEqWhTRWe6JeXnhfTIy4PnVqSNFkg8exHRfLrq8cCFy+3gLTDm4WENcHCLeQUFEffqYbiNgDMaIvv0WdcrTp0MQgci2YwoEenh7e5OLBRfWyMhIun37Nn1oDz1KxRJ5rGDixImMiMRN3MQth93i4uIsshWXL19mxYoVYzfs0JLVhTFrG3XYH31P8MmTJ1SvXj26efMmleAqxEePoi7VAHXr1tWtOdbmm2/gTV68KEVyL18mat4863GvXUOAYuNGNH3XYnjlyjQnIgKvDQiAl1qlCqpN/vkHx9+4kWj5ckro3p1K1q5Njy9cIO9Nm7DvxYsYR61aEGtt2RJR3QoViJYto7o//qj7GU6eRO3ynj2yU+VXnzkzEx5pixbQCZw/37rviTN1KtGsWZTy3XdUbPJkCgsLIx9DVSzWHD8b94+Pj6dSpUo57WewZvxKj8nS/Y19Bks9QXuiqumwp6cnef6XGuKrlfDrk5REPn374kG+fIbLyYjIzc3N8I9kxgxEWKdNQx4hP57ccRs1Qknc1q0oq9Pidt685EOEmuHq1ZEr5+GBoEaJEkhLSUwkmjGDXNu3x+dJTKR8M2YQtWuHKTLvQlelCt734EE8btOG3GbP1v0MO3diyv3227IBDp3PPH481ut+/tn674kIKTWzZqF3yRdfEE2eTD4+PmafgCaPn837c5z9M1gy/uwYU3Z8BqVxisAI8/NDDwwieFpGGMzVm+UoUADrdCtX6rbLlMPFBWIDW7dmCXT0HDYMgRTefN3NDUnWISGIDnMDJwdvtPTwIQwal+U/dw4GtGRJ3c+QkQGZrw8/NBjhfbX/7t2I2g4caFJq3+j3tHQpEqwnTLBaacbo8R2wvzWIz2D//VWJzRNqBQkLC2NExMLCwhjTaBj7+mvUxq5da/1B09MZCwpC/a1GY7ze9t49PLdxY9bnPvyQsSZNpMdvvcVY166MrV6N15w4wRgRSzh6lNF/94wI7/3554yNHctY6dLS6zt0wE2f/fvxunPnTH+2pk2x76lTpvc1xF9/ocZ48GB8P0yvhttJcfbP4OzjZ0y9n0HVniCfGnt6esIzmzkT0c4+faTpo6W4u+M4R45AjaV6dXhncgrO5ctj3U5OqLV2baztZWbicdmyaLnJI9f/JWfnypWLJk6cSLl4hYd2y03tCg5DzdfXr8c4TOnyHTokpQDJVZOYw4kTkML/4ANU2/y3RuPp6UkTJ0589f9wRpz9Mzj7+IlU/BkcbYWNIXvlSEtjrG1bxry9Gbt61boDazSMNW/OWNWqjGVkGN/3hx8Yy5uXsZcvdbcfOgSvi0envv8e6jBnzmD7P/8w1rMnYw8e4PkHD/A4Xz7GZs6EWk2fPnguOhqvWbdO9z1SUxkrUICxceNMf57atRmrUsV8pR19bt9mrGBBfC8pKZa/XiBwUlTtCcri4QHPrFw59AR+8sTyY7i4IEhy7Ro8nrZtUfYmR7duqDjZt093O48ka/cdfvZMKoHLnRt1yzxROSCAaPlyBEz0E6W5WrW+N3r4MMrkPvjA+OfZsQNrnEOGmPrk8jx/juiznx8aV6ntSi0QKIjzGUEiNOzZuRNT0XfflS9TM0WDBqgrnjMHBo6XtelTsSKiv/oiDAUKYDqr33eYj+XpU1SRpKTgcUqKVIGSNy8MpnafEU/PrOVwu3bhuHLTZA5jRJMmIc2na1ekxfD2n+aQliYZ+l27rBNbFQicGOc0gkRIGdm+Hd7c558TWZPu+P33WMczRZcu8Lb4+h9Hu+8wN4JcfebmTRg1Lr1/8yYML5EksKDtCVapklWN2VSzJSIY50uX8FmKFEH7zSJFTH8mInxngwcjyr1lC/oPCwQ5DOc1gkQIFqxYAbn42bMtf32tWvCgiLIaOG06d0bay9mzuturVdPtNEeEfsO+vsbLynh5m7YR1J8KBwfDkzTWbEmjgRfYsiU+R0wMpuAxMYZfo838+aipXrQIeZECQQ7EuY0gEdbLxozBzZqIcb9+uDf22gYNsI6nn6NYrRqmtU+fYg2wQAGp77AxIxgbi7VNf3/kAd64kXXK+88/iPIaa4m5fTsM6KRJeMw7yT18aPg1nBMnoE49fDhUYQSCHIoqjaBs32FjTJ1K9NZbaKT++LFlb9auHdb81qwxPKV2c0PwRL/5ulzf4fBwooIF6fZ/XmOTJk2oaNGiNHLkSOl1z55h6unqiqbsqanyzddbtJDvs8yZNQuyW82amf95iWC0P/gA3p+cQIQM06dPJxcXFxqu3eLTCQgPD6devXpRoUKFKE+ePFSjRg26YCpRXkVkZGTQhAkTKCAggHLnzk3lypWjyZMnk0a7/7XKOHbsGHXu3Jn8/f3JxcWFtm7dqvM8Y4wmTZpE/v7+lDt3bmrRogXd4ArsDkCVRlC277Ax3NxgxLy8UFlhqG2mHEWKQHLq+nWoJBuifXsENrQbm5cvj/fU7ztcoAAlRUYSEdHKlStp//79lPnfdJvlzYuINl9D5JFhbU8wMRG1zB07Gh7PqVOoKf76a/M/KxE8zx49MJXesEFXUccA586doyVLllA1YwEaFRIbG0uNGzcmDw8P2r17N928eZN++uknyp8/v6OHZjY//vgjLVq0iObPn0+3bt2imTNn0qxZs+jXX3919NAMkpSURNWrV6f5BmrXZ86cST///DPNnz+fzp07R35+fvT222+/Uo3Kdhydo2MMizPMT55kzN2dsVGjzH+T6GjG/vyTserVUfVhiMhI5OCtWqW7vUYNxvr3x9+ffcZYvXqoHGnbVme3Z8+esWlE7GXx4oy1bs3YBx/giTFjGCtZUveY27bhvYKDDY+nWzfG3ngDHes45nTk++47xtzcGDtyxPA+WiQkJLDAwEC2f/9+1rx5czZs2DCzXqcGRo8ezZpoV/U4IR07dmSfffaZzrauXbuyXr16OWhElkFEbMuWLa8eazQa5ufnx2bMmPFqW0pKCvP19WWLFi1ywAidMU/QGA0bQvBz5kzjXp02Dx8SffIJpocHDkjS9PoUK4ZosH6+oHbzde4JygRG4uLiqBAR1gv1+w7LTYUDAw03Vn/wANHcESN064nz5sX6paEp9PHjWDqYOFEKCJlg8ODB1LFjR3rrrbfM2l9NbN++nerUqUPdu3enokWLUs2aNWnp0qWOHpZFNGnShA4ePEh3794lIvTbPXHiBHUwFjBTMSEhIRQZGanTCMnT05OaN29OJ0+edMiYXi8jSITp4dtvw7A9f27+61q3hhGbO9fwPm+9hQCK9trhm2/CCDKGZOOnT5HHGBUFo3znDjHG6KcBA+hDd3fKzY0gb04k13x93z6sVRpizhy8/pNPdLdXrIhpcsWKWV/z4gWUuS1ou7l+/Xq6ePEiTZ8+3az91caDBw/ot99+o8DAQNq7dy998cUX9L///Y/+/PNPRw/NbEaPHk09evSgSpUqkYeHB9WsWZOGDx9OPXr0cPTQrCLyv2WiYsWK6WwvVqzYq+eym9fPCLq6QiUmPR1qKubmD3p4QDV5zRoYKTlatyaKiEBnOU7Fioj2RkXBCGZk4Fjx8ejFm5REQ4YMoUe3bpFvRgYku7gn+OIFjscDLEQI7ISEGPbU4uKgkv3ll1J1ijkMGoTXrl5tVm+QsLAwGjZsGK1evdpm+XJHodFoqFatWjRt2jSqWbMmDRw4kPr370+/GWqpqkI2bNhAq1evprVr19LFixdp5cqVNHv2bFq5cqWjh2YT+tqBjDGH6Qm+fkaQiKh4cUjjb9mCXrvm0r8/AgZ//CH/fJMmMCDafYe513X7tiTU6uLyKhfwxx9/pO3bt9MS3qTI2xvSXNotN7V79vLeJYY6zq1di+qTgQOzPnfxIt5bvzfKpk0QYli0SArImODChQv07Nkzql27Nrm7u5O7uzsdPXqU5s2bR+7u7q8CPWqmePHiFKTXnbBy5cr0yJwEeZXwzTff0JgxY+ijjz6iqlWrUu/evWnEiBFO6537/XeO6Ht9z549y+IdZhevpxEkQglZ797QITSWNqO9jlakCARUFy+WWmRqky8fUd26un2HK1SA4blzB+uGRESZmcT+0yA8dOgQHTp0iErwZGpel6ttBLUbIR0/jvVAuR8E7/jWsSNyDM3h+XN4je+/jxQiM2ndujVdu3aNLl++/OpWp04d6tmzJ12+fJncnKDTXOPGjemOttdORHfv3qUyTlQZ8/LlS3LV05F0c3NTdYqMMQICAsjPz4/2a63Zp6Wl0dGjR6mRoxL2HRKOMROb9cdiYhjz94dO33/aeCY5fhwR1v375Z8fPZoxPz/d4wUEQOswIQGvHTIE90Ts/JIl7MmTJ+z5nj2MEbH0Xr0kzb/ZsxnLk0f3WNWrM/bpp/Lvfe4cXrtzp/zzctHhDz5grFAhxp4+Ne/zG8HZosNnz55l7u7u7IcffmDBwcFszZo1LE+ePGz16tWOHprZ9OnTh5UoUYLt3LmThYSEsM2bN7PChQuzUZZkQGQzCQkJ7NKlS+zSpUuMiNjPP//MLl26xEJDQxljjM2YMYP5+vqyzZs3s2vXrrEePXqw4sWLs/j4eIeM9/U2goxJ6SZr1pi3v0YD4VOewqLPzp1Z01fatGHsvffwd758EE0lYkOJWIH/GsmU+M8oXmvTBq+/cwfG8s03peO8eMGYiwtjf/wh/94DBiCdxpD8l74R/OsveYkuK3E2I8gYYzt27GBVqlRhnp6erFKlSmzJkiWOHpJFxMfHs2HDhrHSpUszLy8vVq5cOTZ+/HiWmprq6KEZ5PDhw7LNlPr8Jx2n0WjYxIkTmZ+fH/P09GTNmjVj165dc9h4X38jyBgMWpEijEVFZX1Oznv66SfGcuWS3z82FoZq+XJp25dfQpuQMcbKlYNuoLbWIGOMhYZi2/jxuI+KYqxjR8Y6dZL22b0bz929m/V9ExJgYCdONPw5tT9LXBxjxYsz1qWL+V6wQJADeX3XBLWZOxfKLaNHm7d/r15YE1y/Putz+fMjLUY7p4k3X2cMvT14T5Jly6Q0Hd6cPSEBa4j582dVlz5+HE2Y5PID16+H3NVnnxked1AQhBeCgpALGBenoxAtEAiykjOMoJ8fOswtW4Y8OlMULQr1Ft6RTp9GjXSPU6ECdAQjIhBc4eU/P/+MFptEkhGMjUXCtKsrErW1gyInTqAWWM5o/fknUZs2xqO7Xl4Yy+3bMH4TJ5odDRYIciqqNIIWCyiYw8CBkN4aMsS4bBbnk0+Izp/XzQnkNGwI5Ze4ODzmKS6821xSEnIEtalTB/eMITL87Bk8Rm4EU1Mh1SWXGvP4MbzEjz82PuaQEPQI6dcPohBOJnYgEDgCVRpBiwUUzMHNjejXX5FDt2yZ6f07dkT525o1WZ+rXx/G7Px5POZT2pAQeIJRUYaFCeLjUe2hnx5z4QLy/+SM4MaNkNXq0sX4mGNjkUd48SLRwoXWN1wSCHIQqjSCitGwIdb7JkyQvDjtdTRtvLyQa7huXdaqk4oVYSB53+G8eTGFtrTvMJFkBE+dQgUI712izYYNmJ5rNaSXhUv7t25tdm2wQJDTyVlGkIho+nRUc8yYgcd8HU2uNKxHD6zl6VdguLpieqvtqQYEQNigcGF4ZF5eWMPz9sbz3t54nJAgJUoXKCAZtitXUD6nL7H/4AGmyeYkOq9bh/uhQ03vKxAIiCgnGsGSJYm++goiBGFhMEa9eknTU21atoTB2rgx63O1a2MKywkIgHdXuDAeFypEtHev1DwpMBCPk5Pl+w7LqckQwQvMkwdNoYzx9Cm62RFZ1mhJIMjh5DwjSEQ0ahQ8s4kT4bWtWSM1SNLG3Z3ovfeI/v4765S4Th0YUS6yWro0HnN1mKgorP/xIExmJh5z8QTtlptpaUS3bskbwfXr0ePEmMI0EST2PTzw2YoXN/urEAhyOjnTCHp7E337LdRm5DxAbbp1I7p/X5LQ59SqhXveba5UKRhBPr09exZ/c+XoK1fwmMtoaRvB27eheqMvqXX7Nt7X1FQ4JAQNk8aPJ/rxR2EEBQILyJlGkIhowABMjRcvNr5fq1ZEPj5Z+w4HBMCYavcdTk+XPD9DjZYyM7EW+OhR1ubr+kZw5040cGrb1vgYf/gBhvXLL43vJxAIspBzjaCnJ7xBUwrUuXIhMqvXLIZcXRHJ1W++zo2fsW5zjMFgahvBgAAYW2127YIRzp3b8LEePEBS9+jRpqfMAoEgCznXCBIR9emDaWxQkPEpZOfOMHbh4brba9SQpsk8GPH8OSLDxowgL6vjRvDq1axeYHw8EqRNyahPnYpgzBdfGN9PIBDIkrONoIcHvMFbt6DybIj27ZFsvXOn7vaqVdEyMyUF01FPT6nvsLHOWdxA8uiwXGT4wAGoVLdvb/g49++jnG7MGMtUpgUCwStythEkQnmcvz8CCoYoUAD1wnJ9hzUa9BhxcUGPkogIGEQPD5TGcen8qlWJuKx7VBREU3PnJoqMxH76nuCuXUSVK+vWFuszaxa8QDmVaYFAYBaqNIKK1A4bwtMTeYNr1khiB3K0b48mS2lp0rYqVXDPO9T5+0ueYFwcSuh4+ZyHB9YBc+WCoeReIJ9Oa3uCjMEIGpsKP3+O6PaQIcbXDAUCgVFUaQQVqR02Rv/+kM6fM8fwPu3aodJEW0IrXz6iMmVkm6/T48dE77yDKSsR7ufPh4TW8+eSfP6VKwhoaPcZuXwZDdqNGcHffoP3OWiQFR9YIBBwVGkEs518+WBMli41HNCoXh31wXv36m6X6zvs44Pj7Ngh1SjHxWH90Ns7a9/hqlV1+wfv2oX9DDVbSkmBQf30U+k4AoHAKoQR5AwZgpI2Q53mXF0hTHDwoO72N9+ErBYRpsORkTCC/3Wby0L+/EQxMZLxuno1a1Bk9270ODakArNqFdYVR4ww66MJBALDCCPI8fcn+uADyG0Z6uTVujXqhbVL7CpXRs1wcjKmuAkJiNQaMoI+PpInyMvltIMiL1+i2uStt+RfzxjRL79AVktOgVogEFiEMILaDB2K5GP9KDCndWsYyOPHpW0VK8Iw3bun23c4KUn+GD4+MKKFCqFqJCOD6I03pOfPnUMAxdBU+MgRGE4hmCoQ2AVhBLWpXx81wYsWyT9ftiwCIdp9h3nz9Tt3JCPIGIzg7NlYJyTCfcGCiBgzJi+sSgQD6+uLabYcS5fiPZs1s/pjCgQCCWEEteHR1n/+IQoNld+neXN4Y5zChWHQbt+WjGBGBry5wYOlKHCxYtheoAAecyUZV1fdPiAnThA1bozkbH2ioqBo07+/aJ4kENgJxYzgw4cPqV+/fhQQEEC5c+em8uXL08SJEylNO89OjXz0EVJWDEnwN2uGiC6P+hJBK/DePRhDNzf0CyEiWr1aWj989gwR44wMPObq0iVLSrmEmZlIwTE0FV61Cl7kJ5/Y/DEFAgFQzAjevn2bNBoNLV68mG7cuEG//PILLVq0iMaNG6fUW9qHfPnQ0GjFCvmGTE2bwhBp5wtyI+jqijQaLnPfv7805eVJ0TwFR19TkO+TkCBvBBkjWrIEkv9Fitj8MQUCAVDMCLZr146WL19Obdq0oXLlytE777xDX3/9NW3evFmpt7Qf/fqhekQ/HYYIBq9IEfm+w0R4jhtBbbjnyMVZ+Zqgtrr0iRNIi5GrlPn3X0y5+/e36iMJBAJ5snVNMC4ujgpy5WUZUlNTKT4+XufmEOrWRctKub7DLi6oI9Y3gk+fwosrUkQ+PYYLNGRmIoXGyyurJ3jiBN5brt/JsmWoKmnZ0pZPJhAI9Mg2I3j//n369ddf6Qsjkk/Tp08nX1/fV7dSjuqV4eICma2tW+XVYBo1Qqc5Pl0uXx73vNFSUhKUXbThUWKNBlPhpCSsE3IjyBgiw3JT4dRUos2b0QvFVcSyBAJ7YvEZNWnSJHJxcTF6O8/78f5HREQEtWvXjrp3706ff/65wWOPHTuW4uLiXt3CjAkaKE3PnpjWyk3f69eHEePlctyQ8ZabMTEIXtSsKYkblCmDx2lp8i03Q0JQL9y0adb327MHa4kffmjXjygQCIjcTe+iy5AhQ+gjEz0vymqtc0VERFDLli2pYcOGtGTJEqOv8/T0JE9PT0uHpAylSiESvG4dvEJtateGR3b6NOp+ixbFFFe7+XrlyrqtOvnjDz+UN4InTuC+UaOsY1m/Hu+j3xtZIBDYjMVGsHDhwlSYt5U0QXh4OLVs2ZJq165Ny5cvJ1dnm8p9/DFy/Z4/143I5ssHGa2zZ6WcvbJlYQQrVoQRZEw+l0+725yHB8r1iGAgAwOlPEJOUhLR9u1ooiQQCOyOYlYpIiKCWrRoQaVKlaLZs2fT8+fPKTIykiIjI5V6S/vTtSvu5abEdepk7Tv84AGMZXo61vc8PaUeJJcu4fHjx1JkuEwZaY3PUN/hf/5BPbGYCgsEiqCYEdy3bx/du3ePDh06RCVLlqTixYu/ujkNRYogGrtpU9bnatcmun4dslZEMGjafYfj4rD+x1NiGMPjFy+y5ggyJt9nhAhT4bp1peCLQCCwK4oZwb59+xJjTPbmVHTrhjK56Gjd7bVrw+O7fh2PS5eGIAI3gobSe+LishrB8HAEU/Q9wfh4aAua6jssEAisxskW6RzAu+8irUWuyZKrq27z9RcviNz/W2Y1ZARTUrIaQd53WN8I7tmD9Jj337fDBxEIBHIII2gKPz+kxGzZors9Tx4kVOv3HebtNLVri/Xx9MTz2i03fX11hRSI4AVWrZp1u0AgsBvCCJpDly5o0s7X/zg1akieIDdU0dHwEPPmxVS5cmVsr1xZCrDwsjptT7BaNd1oskYDXUNTfYcFAoFNCCNoDp07w8PT1hEkguG6dg2BjeLFYcSePIGEfmIiNAF5snTu3FL6Cy+r0zeC2ly4gIqSjh0V+1gCgUAYQfMICkIe4K5duturVkVZXWgocv6KFkWQo2BBbPv8c0mXMDSUaMoU/B0Tg+l04cLwCu/ezboeuGsXpsgNGyr+8QSCnIwqjWC29h02BxcX9B3Wl93n3huXydJuufn0KUQPeFQ5Opro0CEcKzISXqCLC5o0aTTyRrBtWynQIhAIFEGVRjDb+w6bQ/v26B3MJbOIYPR8fHRbbkZEwAgaig77+MBAliyJx1euwBhqy+k/e4ZeI2I9UCBQHFUaQVXSogW8Mu2+w9x48Zab3BP09TXeaEm77/DVqyiXy5tX2mfPHqwztmunyEcRCAQSwgiai7c3xA3279fdrt183d9fMoKGWm76+mZtvq4fFNm1C1UivD+JQCBQDGEELeGtt1A9wvuEEMEI3rqFdT0/P4gn5MuHJOcxY3QbLVWogOBJdDSCJ4xlrRlmDIrWbdpk60cTCHIqwghaQuvWSHLmCdJEUI1JToYwgp8fjJi7O6bD06frttwsVgz33BN89gxVJtoSWXfuwJCKlpoCQbYgjKAl1K2L1BbtlpvafYe1p6/x8diPK1MnJMBQenlJwqq8CVO5ctLrTpxAsnWDBkp+EoFA8B/CCFqChwfk77WTpsuWxXbt5usZGfAEW7YkCg7GtuBg5ArysjptI6jfZ6RGDQRQBAKB4ggjaCnNmqHzG+8v4u6Otb67dyVPMD096+u4eg7vMcyNYIECCJZwDPUZEQgEiiCMoKU0bYqpLk+QJpJabnp6omROv8aYSIoWcxHVggUhsa/tBUZEQJhVGEGBINsQRtBS6tVDb2DeE4RIar5OpNt8XRuuKsNFErgnqN13+N9/cS+MoECQbQgjaCleXkS1ahGdOiVtq1ABBi0jA/XA3OsrXFia/vK1QB499vGR7ztcvjzEGAQCQbagSiOoutphfeSar2dkQFm6SBEERWrWJNqxAyILfJ8aNTANLlgQeYWPHukaQbEeKBBkO6o0gqqsHdamQQNEennTKP2+w/Hx6B6nneZSpw7yCzMyMBUOD0cAhb82Ph6J03J9hwUCgWKo0giqnvr1cX/mDO5Ll8Zan3bf4WvXIJJw7Rr24Y/v3ZO6zRFJRvD0aXiHwhMUCLIVYQStoVQprNtxI5grFwwc9wSfP4eXx709IulxbKxu83UeGDlzBsbxjTey+9MIBDkaYQStwcXFeN/hxETUDsuh3W3Oz09SnuY1xHIN2wUCgWIII2gtdeoQnT8vJUGXLYtAB1eHMdRoKT5ePj3GUPN1gUCgKMIIWkvt2pDJ59NaW/oOJyZCsFWu+bpAIFAUYQStpVYt3Gt3m4uIgIwWEUrhDh9GIjUR7g8cgHHkgRFuBK9fh0cpPEGBINsRRtBaihdHrTCX1SpVCtHdtDQ8TkmBGrW3Nx57eyNPkDEYyPBw3W5zbm66kloCgSBbEEbQFmrUyNp8na8FPnhANHYsjB0R7seNw98ZGTCG2kawYkVUowgEgmxFGEFbqF5dygMsVQr3T59iShwWRjRjBh7z7UuW4G/ef4QbwatXxVRYIHAQwgjaQtWqqByJi8N0N18+qeWmocAIEdSkXV2lKfTVqyIoIhA4CGEEbYEbLu4NavcdNmYEnz9HcrWHB6LLCQnCExQIHIQqjaDqBRQ4lSohoHH9Oh5zI1iwoGEj6OWFKHKZMnjMdQmFERQIHIIqjaDqBRQ4uXIh9UW/+Xr+/IgS9+snJU8XKgRDV7AgaouLFMH2K1fwnJDPEggcgiqNoFOh3XeYe4I+PqgV/v13yeMrUwYSXLzlpnbfYVEuJxA4DGEEbeXNN4lu3MDfxYtDXsvbG8GPGzcklenkZKTNFCiAShNuBEVkWCBwKMII2kqlSjB88fFInk5JgShCdDRRlSpozE6E+717sYbIPcGXL1Eux4VXBQJBtiOMoK1o9x3mLTeJpFxAfby9dUUUiKA6LRAIHIIwgrbC9f9u35aMoEZj2AjyqhA5YVWBQJDtCCNoK97eRP7+WZuv877E+uTKhXvuCebKhdcLBAKHkC1GMDU1lWrUqEEuLi50mauuvE4EBmJtz8cHvYe5oKqHhxT15cIKbm645+rSZcpIvYgFAkG2ky1n36hRo8j/dfZ2KlQgCg6GwStWTGqvefUqus4RSQILPB9QTlhVIBBkO4obwd27d9O+ffto9uzZSr+V4+DN1xlDjxGeFqNdNRIdjXuNBvf6moICgcAhKGoEnz59Sv3796dVq1ZRnjx5TO6fmppK8fHxOjenoEIFiChER6MSJCEB2z/+WEqRuXgR95GRmDa7uwsjKBCoAMWMIGOM+vbtS1988QXVqVPHrNdMnz6dfH19X91KcXkqtVOuHO4fPIAnmJhItHo11gm5V1i+PO7T0zEVjo2F4RRGUCBwKBYbwUmTJpGLi4vR2/nz5+nXX3+l+Ph4Gjt2rNnHHjt2LMXFxb26hYWFWTo8x6DdfL1IEVSEVK6suw/3hBMTdXMEhREUCByKu6UvGDJkCH300UdG9ylbtixNnTqVTp8+TZ6enjrP1alTh3r27EkrV67M8jpPT88s+zsF+fPjpt132BA8UZo3aBJGUCBwKBYbwcKFC1PhwoVN7jdv3jyaOnXqq8cRERHUtm1b2rBhA9WvX9/St1U/AQEwgrwLnaE8wbg4VJmEhBDlzQujKRAIHIbFRtBcSvOUkP/I918XtvLly1PJkiWVelvHwY1g27aIABcoQLRxo+TpBQTg8cSJUmS4bFmhHiMQOBiRpWsvSpdGXxHed5gxou7dYQyJcN+9O9Rl9PsOCwQCh6GYJ6hP2bJliTGWXW+X/fDm6/nz4/G9e0Q7dhD17IkE6qdPETHmCjIhIURvveXQIQsEAuEJ2o/SpVEpwkvg7t8n+uor3ZabX3+N8rmCBREYEZ6gQOBwhBG0FzynkSdKG0v0dnND/qAwggKBwxFG0F7wQNDz5xBOMGYEeW2xMIICgcMRRtBeFC2KUriICNMtN/lzQjxBIHA4wgjaC1dXKMRERGDNLyWFqHNnIl9fPO/rS1SrFv6OiYGh5M8JBAKHoUoj6DR9h/Xx95earzNGtH27VDNcvjzR55/DW3zyREyFBQKVoEoj6DR9h/XhLTcLFEAqzPPnEEwgwv2jR/ASnz0TfYYFApWgSiPotGj3HX7yBOuE167huWvXiGbMgJCCdt9hgUDgUIQRtCf+/jB+vr6GGy35+gojKBCoCGEE7YmfH3QC8+Y1bATz54cR5OV1AoHAoQgjaE94tzlXV+gGyuHjg+iw8AQFAlUgjKA9KVYM94wZ9gTz5IHMljCCAoEqyDYBhRwB9wQzM9F28/lzSUWmenX8XaIEHgsjKBCoAuEJ2pMiRaAPyPsOE0l9hokgo8WVs4URFAhUgTCC9sTdHUrRKSl4/N576EdMRHThAqbJL17gsQiMCASqQBhBe1OkiNRh7sQJSVWGS2pxL1F4ggKBKhBG0N4UKSIfGeaiCZmZmBKb0YdZIBAojyqNoNPWDhNhOpyURNStm+52b2/cMwYvUPQWEQhUgSqNoNPWDhPBE3zxgmjcON3tXDYrNVVMhQUCFaFKI+jU8L7DpUoRzZ8vKU7zx+npIigiEKgIYQTtTZEiRFFRMIaDB+Mx3z54MNYLhScoEKgGYQTtDU+RCQtDd7mYGGyPicHjp0+FERQIVIQwgvaGG7hr14h690ZXOSLc9+4NL1EYQYFANQgjaG94mZyhHiO8+bpAIFAFwgjaGx70MGQEk5JEYEQgUBHCCNobU54gkfAEBQIVIYygvcmfH/epqUQNGkBglQj31arhb2EEBQLVIIygvXFzg4S+pyfRqVNEFStie8WKRAsW4G9hBAUC1SCMoBIULCilxmgTHY17YQQFAtWgSiPo1LXDRFgXDA5GffDFi9h28SLRu+9KzwsEAlWgSiPo1LXDRDBycXHyz+XNS+Thkb3jEQgEBlGlEXR68uc33m1OIBCoBmEElcDHx3jfYYFAoBqEEVQCHx/DLTeFERQIVIXoNqcEvr7IEwwOJipZEtuCgojq1ZOktQQCgSoQRlAJfHzQW6RCBWmblxfRy5dERYs6blwCgSALYjqsBNwI9uxJFBKCbSEhRPfvE7mKr1wgUBPijFQCX1/0Elm7lig2FttiYtCFjvcdFggEqkBxI/jPP/9Q/fr1KXfu3FS4cGHq2rWr0m/peHx8sm7jvYhFYEQgUBWKrgn+/fff1L9/f5o2bRq1atWKGGN07do1Jd9SHcgZQd50XRhBgUBVKGYEMzIyaNiwYTRr1izq16/fq+0VuaDA64ycoeMVJMIICgSqQrHp8MWLFyk8PJxcXV2pZs2aVLx4cWrfvj3duHHD4GtSU1MpPj5e5+aU+PggANK7N1Hx4tjm5YVa4sBAx45NIBDooJgRfPDgARERTZo0iSZMmEA7d+6kAgUKUPPmzSlGTmGFiKZPn06+vr6vbqWcNaeuZEmijAyiP/+UjGDXrthWv75jxyYQCHSw2AhOmjSJXFxcjN7Onz9PGo2GiIjGjx9P3bp1o9q1a9Py5cvJxcWFNm3aJHvssWPHUlxc3KtbWFiYbZ/OUbi44KaPq6v8doFA4DAsXhMcMmQIffTRR0b3KVu2LCUkJBARUVBQ0Kvtnp6eVK5cOXr06JHs6zw9PclTpJAIBIJsxGIjWLhwYSpcuLDJ/WrXrk2enp50584datKkCRERpaen08OHD6lMmTKWj1QgEAgUQLHosI+PD33xxRc0ceJEKlWqFJUpU4ZmzZpFRETdu3dX6m0FAoHAIhTNE5w1axa5u7tT7969KTk5merXr0+HDh2iAkJZWSAQqAQXxhhz9CAMER8fT76+vhQXF0c+cgnIAoFAYCOidlggEORoVO0JMsYoISGBvL29yUWklggEAgVQtREUCAQCpRHTYYFAkKMRRlAgEORohBEUCAQ5GmEEBQJBjkYYQYFAkKMRRlAgEORohBEUCAQ5GmEEBQJBjub/EAjNNoatgSIAAAAASUVORK5CYII=\n", "text/plain": [ "Graphics object consisting of 30 graphics primitives" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "plot1 = SD.plot(EF, ranges={t:(0, 8), r:(2.1, 10)}, fixed_coords={th:pi/2, ph:0}, \n", " ambient_coords=(r,T), style={t:'--', r:'-'}, parameters={m: 1}) \\\n", " + SD.plot(EF, ranges={t:(0, 8), r:(0.1, 1.9)}, fixed_coords={th:pi/2, ph:0}, \n", " ambient_coords=(r,T), number_values={t: 9, r: 3},\n", " style={t:'--', r:'-'}, parameters={m: 1})\n", "plot1" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Manifold points" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To create a point on $M$, we use SageMath's *parent/element* syntax, i.e. the call operator `M(...)` acting on the parent `M`, with the point's coordinates in some chart as argument: " ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Point p on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "p = M((m, 8*m, pi/2, 0), name='p')\n", "print(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Since the chart has not been specified, the default chart (i.e. `SD`) is meant:" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(m, 8 \\, m, \\frac{1}{2} \\, \\pi, 0\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(m, 8 \\, m, \\frac{1}{2} \\, \\pi, 0\\right)$" ], "text/plain": [ "(m, 8*m, 1/2*pi, 0)" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SD(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Thanks to the transition map declared above, the coordinates of $p$ in the Eddington-Finkelstein chart can computed:" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(m {\\left(2 \\, \\log\\left(3\\right) + 1\\right)}, 8 \\, m, \\frac{1}{2} \\, \\pi, 0\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(m {\\left(2 \\, \\log\\left(3\\right) + 1\\right)}, 8 \\, m, \\frac{1}{2} \\, \\pi, 0\\right)$" ], "text/plain": [ "(m*(2*log(3) + 1), 8*m, 1/2*pi, 0)" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "EF(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Manifold points have a `plot` method:" ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 32 graphics primitives" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "plot1 += p.plot(EF, color='blue', ambient_coords=(r,T), \n", " parameters={m: 1}, label_offset=0.4, fontsize=14)\n", "plot1" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Vector fields" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "When a chart is declared, the manifold is automatically endowed with some vector fields, those of the **coordinate vector frame**:" ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(M, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(M, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right)$" ], "text/plain": [ "Coordinate frame (M, (∂/∂t,∂/∂r,∂/∂th,∂/∂ph))" ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SD.frame()" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(M, \\left(\\frac{\\partial}{\\partial T },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(M, \\left(\\frac{\\partial}{\\partial T },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right)$" ], "text/plain": [ "Coordinate frame (M, (∂/∂T,∂/∂r,∂/∂th,∂/∂ph))" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "EF.frame()" ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\left(M, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(M, \\left(\\frac{\\partial}{\\partial T },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right)\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\left(M, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(M, \\left(\\frac{\\partial}{\\partial T },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right)\\right]$" ], "text/plain": [ "[Coordinate frame (M, (∂/∂t,∂/∂r,∂/∂th,∂/∂ph)),\n", " Coordinate frame (M, (∂/∂T,∂/∂r,∂/∂th,∂/∂ph))]" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.frames()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As for charts, there is a *default frame*, i.e. a vector frame that is used by default in functions having a vector frame in their arguments. The default frame can be changed by the method `set_default_frame`." ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(M, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(M, \\left(\\frac{\\partial}{\\partial t },\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right)$" ], "text/plain": [ "Coordinate frame (M, (∂/∂t,∂/∂r,∂/∂th,∂/∂ph))" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M.default_frame()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The first vector of the Schwarzschild-Droste coordinate frame:" ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\frac{\\partial}{\\partial t }\$" ], "text/latex": [ "$\\displaystyle \\frac{\\partial}{\\partial t }$" ], "text/plain": [ "Vector field ∂/∂t on the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vt = SD.frame()[0]\n", "vt" ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\frac{\\partial}{\\partial t } = \\frac{\\partial}{\\partial t }\$" ], "text/latex": [ "$\\displaystyle \\frac{\\partial}{\\partial t } = \\frac{\\partial}{\\partial t }$" ], "text/plain": [ "∂/∂t = ∂/∂t" ] }, "execution_count": 35, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vt.display()" ] }, { "cell_type": "code", "execution_count": 36, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\frac{\\partial}{\\partial t } = \\frac{\\partial}{\\partial T }\$" ], "text/latex": [ "$\\displaystyle \\frac{\\partial}{\\partial t } = \\frac{\\partial}{\\partial T }$" ], "text/plain": [ "∂/∂t = ∂/∂T" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vt.display(EF.frame())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The second vector of the Schwarzschild-Droste coordinate frame:" ] }, { "cell_type": "code", "execution_count": 37, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\frac{\\partial}{\\partial r }\$" ], "text/latex": [ "$\\displaystyle \\frac{\\partial}{\\partial r }$" ], "text/plain": [ "Vector field ∂/∂r on the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 37, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vr = SD.frame()[1]\n", "vr" ] }, { "cell_type": "code", "execution_count": 38, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\frac{\\partial}{\\partial r } = \\left( -\\frac{2 \\, m}{2 \\, m - r} \\right) \\frac{\\partial}{\\partial T } +\\frac{\\partial}{\\partial r }\$" ], "text/latex": [ "$\\displaystyle \\frac{\\partial}{\\partial r } = \\left( -\\frac{2 \\, m}{2 \\, m - r} \\right) \\frac{\\partial}{\\partial T } +\\frac{\\partial}{\\partial r }$" ], "text/plain": [ "∂/∂r = -2*m/(2*m - r) ∂/∂T + ∂/∂r" ] }, "execution_count": 38, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vr.display(EF.frame())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Creating a vector field from scratch, by providing its components with rest to a given vector frame:" ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle k = \\left( -\\frac{r}{2 \\, m - r} \\right) \\frac{\\partial}{\\partial t } -\\frac{\\partial}{\\partial r }\$" ], "text/latex": [ "$\\displaystyle k = \\left( -\\frac{r}{2 \\, m - r} \\right) \\frac{\\partial}{\\partial t } -\\frac{\\partial}{\\partial r }$" ], "text/plain": [ "k = -r/(2*m - r) ∂/∂t - ∂/∂r" ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" } ], "source": [ "k = M.vector_field(1, -1, 0, 0, frame=EF.frame(), name='k')\n", "k.display()" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle k = \\frac{\\partial}{\\partial T }-\\frac{\\partial}{\\partial r }\$" ], "text/latex": [ "$\\displaystyle k = \\frac{\\partial}{\\partial T }-\\frac{\\partial}{\\partial r }$" ], "text/plain": [ "k = ∂/∂T - ∂/∂r" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "k.display(EF.frame())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Plot with respect to Schwarzschild-Droste coordinates (default chart):" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 81 graphics primitives" ] }, "execution_count": 41, "metadata": {}, "output_type": "execute_result" } ], "source": [ "k.plot(ambient_coords=(r,t), fixed_coords={th: pi/2, ph: 0}, \n", " ranges={t: (0, 6), r: (0.1, 6)}, number_values=9,\n", " parameters={m: 1}, color='green', scale=0.3)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Plot with respect to Eddington-Finkelstein coordinates:" ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [ { "data": { "image/png": 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QtUMIiA3A7VzVBTvm+uZICElAN+duEicjsTzOf4wByQOQfuHfUzrGu4/Hgh4LoKOlI2GyyqnKNHsAcOLWCfjF+AnTHSNdI2zouwGBjQIr5PHZ7FV+T589xYi0EYg5HSPUhrYciiV9lkBfR1/CZCSm2NOxGLZxmLAcS93qdZEekQ6Xmi4SJyOxXMm6Av8Yf5y4fQKAarqzqOciRLo/f6U3yVNRcRE+2vYRvtr/lVDr4dwDccFxMDcwlzBZ5VOlmj1ANd0JigvCwWsHAaimO7O7zMYHXh+88XSHzV7VoFQq8eXuLzFt5zSh5u3ojZSwFE53NEjm9UwExAYIF+yY6ZshPjgePer1kDgZiSWnIAcDkgcg7XyaUIt0i8TCngs53dEgK4+txJiMMXhW/AwA0LhGY6RHpMPZwlniZJVHlWv2ANXVuSPSRiD6VLRQG9JiCH70/bFc052oqChERUWhqKgIFy5cYLNXRcSficeQ1CHCdMepmhMy+mdwuqNBrmZdhX+sP47fOg5AddXeop6LSizlQ/JWVFyEqdumYt7+eUKtu3N3xAXHoZpBNemCkah2XdqFfvH9cP/pfQCApaElksOS0b52e4mTVQ5VstkDVNOdmXtm4tMdnwo1LwcvpISloKZxzXI9Jid7Vc/h64cREBuAmzk3AaimO3HBca+8LRvJS25BLgamDETquVSh9rbr21jYcyFvnq5BVh9fjdHpo4XpTqMajZAekY56FvUkTkZi+evBX/CL8cPZe2cBqNby/dH3RwxrNUziZNKrss3ePxLOJGBI6hA8LVQttlunWh1kRGSgiVWTMj8Wm72q6Vr2NfjH+OPYrWMAVNOdb3p8g/Hu43nhhoYoVhbj420fY86+OUKta92uiA+OR3XD6hImIzHtubwHQXFBwnTHwtACyaHJ6FCng8TJSCxZeVkITQzF5r82C7X32r2HOV3nPLd+nyap8s0eABy5cQQBsQG48fgGAMBUzxRxwXHoVb9XmR6HzV7VlVuQi8Gpg5F8NlmojWkzBt/2+pbTHQ2y5vgajEofJUx3Glo2RHpEOupb1pc4GYnl74d/wy/GD7/f/R2AarqzxHcJhrcaLnEyEkthcSHe/fVdfJf5nVDzb+iPDX03vHYdP7mSRbMHANezr8M/1h9Hbx4FoJruLOi+ABM8JpR6usNmr2orVhbj0+2fYtbeWUKti1MXJIQkcLqjQfZe2YuguCDce3IPAFDdoDqSQpPQyamTxMlILFl5WQhPCscvf/4i1Djd0Tw/HP4B438ejyJlEQCguXVzpEekw9HcUeJk4pNNswe8eLozuvVoLO69uFTTHTZ78rDuxDqMTB8p3EqpgWUDZERkcLqjQS4+vAi/GD+cuXsGAKCjpYPve3+PUW1GSZyMxFJYXIjJv07Gt5nfCjXfBr6I7hsNU31TCZORmLb8tQUhCSHIys8CAFgZW2Fj+Ea0tW8rcTJxyarZA1483ens1BkJIQmwMLR45b5s9uRj35V9CIoLwt0ndwGopjuJoYno7NRZ4mQkluz8bIQnhuPnP38Wau+0fQdfdfuK0x0NsuTIEozbNE6Y7jSzaob0iPTX3qKP5OPcvXPwjfbFXw//AgDoa+tjZcBK9G/WX+Jk4pFds/eP9SfXY0TaCGG6U9+iPjL6Z6CBZYOX7sNmT14uPboEvxg/nL5zGoBquhPVOwqj24yWOBmJpbC4EO9vfh8LDy0Uan3q90F0v2iY6fM1rim2/r0VIQkheJT3CIBqupMalop2Du2kDUaiuf/kPoITgrHz0k6h9mn7T/F5x8+hpdCSLphIZNvsAc/fMLmaQTUkhiSiS90uL9yezZ78ZOdno39Sf/z0x09CbaLHRHzd/WtOdzTIj0d+xLifx6GwuBAA0NSqKdIj0lGnWh1pg5Fozt87D78YP/zx4A8AgJ62Hlb6r8SA5gMkTkZiKSgqQORPkVh+bLlQC3YJxprANTDSNZIwmfrJutkDVDdM9o3xFaY72gptRPWOwluubz23LZs9eSoqLsL7W97HNwe/EWq96vVCbHAspzsaZPvF7QiOD8bDvIcAgJpGNZESlgIvRy+Jk5FYHjx9gOD4YOy4tEOofezzMb7o9IVGTHdItUbvwoMLMXnzZCihan9c7VyxMXwj7EztJE6nPrJv9gDVDZP7J/dHxoUMoTbRYyLmd59f4pY6bPbkbdlvy/D2preF6U6Tmk2QHpEOp+pOEicjsVy4fwF+MX64cP8CANV0Z7nfcgxqMUjiZCSWZ0XPELkpEsuOLhNq/Rr3w5rANTDWM5YwGYnppws/ITwpHDkFOQAAO1M7pIWnoY1dG4mTqYdGNHuAarozZesUfH3ga6HWs15PxPaLFW6YzGZP/nZc3IF+8f2E6U4NoxpICUuBt6O3xMlILA+fPkRwQjC2X9wu1D7y/ggzOs/gdEdDKJVKLDq0CJM3T0axshgA0Nq2NdLC01DLrJbE6Ugsp++chm+0Ly5nXQYAGOoYYl3QOvRz6SdxsoqnMc3eP1YcXYExP40RpjsuNV2QHpGOutXrstnTEH/c/wO+Mb4lpjvL/JZhcIvBEicjsTwreobxP4/Hj7/9KNT6Nu6LtYFrOd3RIJv+2ITwxHA8LngMQP7THXrendw7CIoLwv6r+4XajE4zMNVnqqzuwKRxzR4A7Ly0E/3i++HB0wcAVNOd5NBktKjegs2ehnj49CFCE0Ox9e+tQu1Drw8xs8tMTnc0hFKpxHeZ3+GdX98RpjutbFohLSIN9mb2EqcjsZy+cxp+MX649OgSANV0Z23QWgS7BEsbjESTV5iH0emjse7kOqE2oNkALPdfDgMdAwmTVRyNbPYA4M8Hf8Ivxg/n7p0DoLqlzqJOi/C2z9ts9jTEs6JnmPTLJHx/5HuhFtgoEOuC1mnsLXU00c9//IywxDBhumNrYouN4RvhVstN4mQklru5dxEUF4R9V/cJtS87fYmPfT6W1XSHXk6pVGLO3jmYun2qUGtn3w4pYSmwNrGWMFnF0NhmDwAe5T1CaEIotvy9RVXIAzAHePjoIaqZV5MyGoloceZiTPxlojDdaWnTEmnhaXAwd5A4GYnlzJ0z8Ivxw8VHFwEABjoGWBu4FiFNQiRORmLJL8zH6IzRWHtirVDr36w/VvivkM10h14v+WwyBqUMwpNnTwAAjuaOyIjIQDPrZhInezMafbyqmkE1bBqwCW+7vl2iPiBpgHCFDsnfOPdx2NR/k7AMy/Fbx+G+3B2Z1zMlTkZiaWLVBIdGHhIu1MkrzENoYii+3PUlNPjzsEbR19HH6oDVmN1ltlCLPhWNTms64XbObQmTkZj6Nu6LPcP2oJap6kKdK1lX4LnSs8RqHlWRrJq9wsJCfPLJJ3BycoKhoSHq1q2LL774AsXFxS/dR0dLB1F9orC412LhXK1Nf2yC90pvXMm6IlZ0kliPej1wYMQB1K1eFwBwK+cWOqzugLjTcRInI7HUNK6JrYO2YkiLIULts52fYUDyADx99lTCZCQWhUKBD70/RHJosrDI7sFrB+G+3B0nb5+UOB2JpbVta2SOyoSrnSsAIKcgB/4x/lhwYEGV/fAnq2Zv7ty5WLJkCRYvXoyzZ89i3rx5+Oqrr/Ddd9+9dt9I90gkhiYKX5+4fQLuy9xx6NohdUamSsSlpgsOjTyE9rXbA1BNd8KTwvH5zs+r7AucykZfRx+rAlZhbte5UEB1rlbM6Rh0WtMJt3JuSZyOxBLUOAh7h+0VLtS5knUFnis8kX4+XeJkJBY7UzvsGroLoU1CAQBKKDF582SMSh8l3Ia1KpHVOXu+vr6wtrbGihUrhFq/fv1gZGSEdevWvWJPlX+WXnGa64SLT1Xn7uhr62N14GqENw1XW26qXAqKCjAmYwxWHV8l1MKahGFVwCoY6hpKmIzElHouFQOTByL3WS4AwMHMAekR6Whh00LiZCSWm49vIiA2AIdvHAYAKKDAvG7zMLndZF64oSGKlcWYvnM6vtj9hVDrULsDkkKTYGlkKWGyspHVZM/b2xvbtm3DhQuq9dNOnDiBvXv3onfv3mV6nO2Dt6ND7Q4AgPyifEQkRWDhwYUVHZcqKT1tPazwX4Gvun0lTHfizsSh45qOyCvMkzYciSawUSD2Dt8LBzPVhTpXs6/Ca6UX9lzeI3EyEoutqS12Dd2FsCZhAFTTnfe3vI9Jv0ySNhiJRkuhhemdpiO6bzT0tfUBALsu74LHco8qdS6nrJq9KVOmICIiAo0aNYKuri5atWqFSZMmISIi4oXb5+fnIzs7u8Q/ALAwssDmQZsxotUIYdvPdnwmXK1J8qdQKPCe53tIDU+Fsa5qkd3M65lV/iRdKpuWNi2ROSoT7rXcAQC5z3IxZ98ciVORmAx1DRHTLwbTOkwTat9mflul3ujpzUU0i8CuobtgbaxahuWvh39hzYk1EqcqPVk1e3FxcVi/fj2io6Nx9OhRrFmzBvPnz8eaNS9+QmbPng1zc3Phn4PDv0tt/HNXhYU9FqKWaS2MbD2Si+1qIP+G/tg/Yj/ca7mjuXVzeDp4Sh2JRGZjYoOdQ3ZidOvRsDWxxYBmA6SORCJTKBT4vOPniO0Xi7rV6yKoURCsjK2kjkUi87D3QOaoTHSr2w3O1Z3Rw7mH1JFKTVbn7Dk4OODDDz9EZGSkUJsxYwbWr1+Pc+fOPbd9fn4+8vPzha+zs7Ph4ODARZWJiIhINnSkDlCRnjx5Ai2tktM3bW3tly69oq+vD319fTGiEREREUlCVs2en58fZs6cCUdHRzRp0gTHjh3DggULMHz4cKmjEREREUlCVodxHz9+jE8//RQpKSm4c+cO7OzsEBERgc8++wx6enqv3f+fpVd4GJeIiIjkQlbN3ptis0dERERyw8tLiYiIiGSMzR4RERGRjLHZIyIiIpIxNntEREREMsZmj4iIiEjG2OwRERERyRibPSIiIiIZY7NHREREJGNs9oiIiIhkjM0eERERkYyx2SMiIiKSMTZ7RERERDLGZo+IiIhIxtjsEREREckYmz0iIiIiGWOzR0RERCRjbPYAREVFwcXFBW5ublJHISIiIqpQCqVSqZQ6RGWRnZ0Nc3NzZGVlwczMTOo4RERERG+Mkz0iIiIiGWOzR0RERCRjbPaIiIiIZIzNHhEREZGMsdkjIiIikjE2e0REREQyxmaPiIiISMbY7BERERHJGJs9IiIiIhljs0dEREQkY2z2iIiIiGSMzR4RERGRjLHZIyIiIpIxNntEREREMsZmj4iIiEjG2OwRERERyRibPSIiIiIZY7NHREREJGNs9oiIiIhkjM0egKioKLi4uMDNzU3qKEREREQVSqFUKpXqeOD4M/EYkjoEBjoGCHYJRre63eDj6ANbU1t1/LgKkZ2dDXNzc2RlZcHMzEzqOFVefmE+GixugNs5t+Fh74HgxsHwqe2DZlbNoK2lLXU8EsnbP72N5UeXw9bEFgOaD0CH2h3QzqEdzPT5GtMUR24cQac1nQAAfer3Qe/6veHj6IM61epAoVBInI7EoFQq4bbMDSdvn0RTq6YIbxoOH0cftLFrAz1tPanjyZ7amj3vld7Yd3Xfc3Xn6s7wqe0DH0fVv3oW9SrNi53NXsXafnE7uqzt8lzdTN8MXg5eqt+B2j5ws3ODvo6+BAlJDOZzzJGdn12ipqXQQkublsLfAZ/aPrAytpIoIanbsNRhWH1i9XP1Wqa1SrwfNLFqAi0FDzjJ0ZOCJzCebfxc3VDHEG3t2wp/B9rat4WJnokECeVNbc3eymMrMSJtxGu3szGxgbejt/Bib27dXLKpD5u9ilVQWIBqc6vhaeHTV26nr60P91ruwovd08GTUx8ZCY4PRtLZpNdu18CyQYnmz6maU6X5IEhvJvN6JjyWe7x2u+oG1eHl6CX8HnDqIy+1FtTCjcc3XrmNtkIbrW1bC38HvB29UcOohkgJ5UttzV6xshj1v62Pvx/9Xab9zPTN4OngCR9HH4Q2CUU9i3rqiPdCbPYq3szdM/HJjk/KtI+WQgstrFvAx9EHPer1QK96vfimX4VdzbqKOovqoFhZXKb97EzthDf9Ac0HoJpBNfUEJFF0WdsF2y9uL9M+hjqG8LD3gI+jD4IaBaGVbSs1pSMxxJ6ORURSRJn3a1yjMXwcfdDJqRNCXEJ4GlA5qK3ZA4CozCiM+3lcufc30DHApYmXYG1iXYGpXo7NXsW7/+Q+7L+xR15hXrkfY2GPhZjYdmIFpiKxhSWGIf5MfLn3b2HdAsfHHK+4QCS6n//4Gb2je5d7fwUUyByVCVc71wpMRWJ6VvQMTouccP3x9XI/xgT3CVjUa1EFptIMaj05YkjLITDXNy/3/vmF+cgvyq/ARCQ2SyNLDGw28I0e43HB4wpKQ1KZ6PFmzXpOQQ7U+LmURNCjXg80sGxQ7v2VUOLJsycVmIjEpquti0i3yDd6jOyC7NdvRM9Ra7NnomeCka1Hlnv/mZ1nwtHcsQITkRQmeEwo974tbVrinbbvVGAakkI7+3blnsjoaOlguf9yHsqv4rQUWm/U9I9sNRI+jj4VmIikMLrNaBjoGJRrX3sze8zqPKuCE2kGtV/2NM59XLmurhrUfBA+9P5QDYlIbM2sm6GzU+cy72djYoP0iHQY6z1/BRdVLQqFotxv9D/0+QEd63Ss2EAkicEtBpfraE/HOh0R1SeKDb8MlPdoj5GuEdLC0yr18m2VmdqbvTrV6iCgYUCZ9vF08MQyv2V8YctIWd/oDXQMkBaeBnszezUlIrGFNgmFjYlNmfZ5t+27b3R0gCqX8hztqWdRD4khibwqV0bKc7RnfdB6XqDzBkRZ0Kgsb/R1qtVBSlgK112TmT71+6Bu9bql3n5N4Bq41eIdTeRET1sPY13Hlnr7PvX7YF63eWpMRFIoy9Eec31zpEekw9LIUs2pSExlPdozu8tsBDUOUmMi+ROl2Wtfuz1aWLco1bZ1qtWBka6RmhOR2LS1tDHefXyptjXWNUZt89pqTkRSeKvNW6We0DSzasYFdmWoLEd7bE1sueSOTJV2CKSjpQOXGi5qTiN/ovwlLcv5Ojsv7YTXSi9cybqi5lQktmEth5VqZfTcZ7nosLoDYk/HipCKxGRtYo2IpqVbZ2vOvjnon9wfT5+9elFuqnpK+35w7v45uC9zx4lbJ9SciMRW2qM9hcWFCIwLxPz983lF/hsQ7WNzRLMI1DSq+dLvv9XmLeHE3ZO3T8J9mTsOXjsoVjwSgbmBOYa1HPbS77e2bQ0vBy8AQH5RPiKSIjBtxzS+wGXmVW/0etp6eNv1bSigOl839nQsOq7piJuPb4qUjsTwuqM9YU3ChPN1r2ZfhddKL6SdTxMrHongdUd7apvXhn9DfwCqZXfe3/I+RqaNREFRgVgRZUW0Zs9AxwBvtXnrhd8b1XoUfujzAw6OPAjn6s4AgNu5t9FxdUfEnIoRKyKJYLz7eOGN/L/szeyREZGB7UO2Y3jL4UL9i91fIDwpnNMdGWll2wrta7d/4feW+y1HVJ8opIanwlhXdRV25vVMuC93x/Fbx0VMSer0qqM93Z27Y33f9cgcmQn3Wu4AVNP+wNhAfLXvK374k5HhrYbDVM/0ubqpniky+mcgNSwV0zpME+orj69E93Xdcf/JfTFjyoKoJ8SMdRsLHS2dErWOdTpice/FUCgUaFSjEQ6NPCQss5BflI/+yf0xbce0Mt9qiSqn+pb10adBnxK1/15Sr6eth+X+yzG/23yhKYw/E48OqztwuiMjL3qj/8j7IwxqMQgA4N/QH/uG74ODmQMA4Fr2NXiv9MbGcxtFzUnq86KjPY1qNEJccBx0tHRga2qLnUN2IrxpOADVdOeDrR9gRNoITndkwkzf7LmjPVoKLcQGx6KpVVMoFAp83vFzxPSLEdbm23V5FzyWe+Ds3bNSRK6yRG327EztENYkTPi6nkU9JIUmlThh29LIEr8O/BUjW/17ef4Xu79AeGI4V0+Xif++0SugwIa+G0pcUq9QKDDZczI2hm8UzvE7fOMw3Ja54djNY6LnpYoX0DCgxEU4fRv3xYzOM0ps08KmBTJHZcKjlgcA1XQnKC4Ic/fO5XRHBgx0DDDGdYzwtaWhJTIiMkpckGGoa4jovtH4vMPnQm3V8VXourYr7j25J2JaUpfxHiWP9nzd/Wv0rl/ytnrhTcOxc8hOWBurbp3618O/0G5FO2z+a7OoWasy0S91m9ZhGmxMbOBc3RnpEemwMLR4bhs9bT0s9VuKr7t/LfwSJPyegA6rO+DG4xtiR6YK1sWpC/o17gd9bX182+tbBDYKfOF2fg39sG/4PuEuKtcfX4f3Km+knE0RMS2pg7aWNhb0WAA9bT10rdsVawPXvvDKWxsTG+wculO4qEMJJT7c9iGGbRyG/ELeSrGqm+AxAY1rNIaVsRVSwlLgbOH83DYKhQLTOk5DbL9YYbqz58oeeCz3wO93fxc7MlWwehb18G67d6GjpYMpXlNeenjfw94Dh0cdFs71zMrPQu8NvbE4c7GYcasshVKCj8hKpRLFymJoa2m/dtuMCxmISIpATkEOAKCWaS2kRaShtW3rCs+VnZ0Nc3NzZGVlwczMrMIfn0oqKi4q1e/A7ZzbCIoLwoFrB4Ta7C6zMcVrChferuKKiougpdB67fOoVCoxY/cMfLbzM6Hm7eiN5NBk1DR++YVfVDWU9m9B5vVMBMQG4FbOLQCqw4BxwXHoWa+nuiOSmpX2dyCnIAcDkwdi4/l/T+l42/VtLOq16LnTxOhfkjR7ZXXq9in4xfjhctZlAIChjiHW912Pvo37VujPYbNXeeUV5mFk2khsOLVBqA1uMRhLfZdyAW4NknAmAUNSh+BpoeqCHadqTkiPSEcTqyYSJyOxXMu+Br8YP+GCHS2FFhb2WIhx7uP44U9DFCuLMXXbVMzdN1eoda3bFQkhCVyX8SWqRLMHAHdy7yAwNrDEdGdm55n4yPujCnuBs9mr3JRKJWbtmYVPdnwi1LwcvJASlsLpjgY5cuMI/GP8cTNHdcGOqZ4p4oLj0Kt+L4mTkVhyC3IxMGUgUs+lCrWxrmOxqOci6GrrSheMRLX6+GqMTh+NZ8XPAAANLRsio38G6lnUkzhZ5VNlmj3gxdOdgc0HYpnfMuFcjjfBZq9qSPw9EYNTBgvTnTrV6iA9Ih1NrZpKnIzEci37Gvxj/HHsluqCHS2FFhZ0X4AJHhM43dEQxcpifLztY8zZN0eoda3bFfHB8ahuWF3CZCSmPZf3ICguCPefqpZjsTC0QFJokrCqB6lUqWYPUE13Zu+djY+3fyzUPB08kRKWAitjq3I9ZlRUFKKiolBUVIQLFy6w2asCjtw4goDYAOGCHVM9U8QGxz53FRfJV25BLganDkby2WSh9labt/Bdr+843dEga0+sxaj0UcJyLA0tGyI9Ih31LetLnIzE8vfDv+EX4ydcsKOjpYMlfZZgROsREierPKpcs/eP5LPJGJg8UJju1DavjfSIdDSzblbux+Rkr2q5nn0dAbEB+O3mbwBU05353eZjUttJnO5oiGJlMT7d/ilm7Z0l1Do7dUZCSMILr/Qnedp7ZS+C4oKE5ViqG1RHUmgSOjl1kjgZiSU7PxvhieH4+c+fhdq7bd/FvG7zSnXhh9xV2WYPAI7ePAq/GD9humOiZ4LYfrHPLdpbWmz2qp4nz55gcMpgJJ1NEmqjWo9CVO8oTnc0yPqT60sstlvfoj4y+meggWUDiZORWC4+vAi/GD+cuXsGgGq6833v7zGqzSiJk5FYCosL8d7m97Do0CKh5tvAF9F9o2Gq//ydOjRJlW72AODG4xsIiA3AkRtHAKgW6Z3ffT7eaftOmac7bPaqpmJlMabtmIYZe/5dlLdTnU5IDE3kdEeD7L+6H4Gxgbj75C4A1XQnMTQRnZ06S5yMxJKdn42IpAhs+mOTUHun7Tv4qttXnO5okB+P/IhxP49DYXEhAKCpVVOkR6SjTrU60gaTUJVv9gDVdGdo6lAk/J4g1Ea2GomoPlEl7s7xOmz2qrYNJzdgRNoI5BepFtutZ1EPGREZaFijocTJSCyXH12Gb4wvTt85DUA13VncazHecn3xfblJfoqKi/De5vew8NBCoda7fm/E9IuBmT7/rmuK7Re3o198PzzKewQAqGlUE6nhqfB08JQ2mERk0ewBqunO5zs/x5e7vxRqHet0RGJIIiyNLEv1GGz2qr4DVw8gMC4Qd3LvAACqGVRDQkgCutbtKnEyEsvj/MeISIrAT3/8JNQmekzE/O7zueiqBln621JEborkdEeDXbh/Ab7RvvjjwR8AVHfnWuG/AgObD5Q4mfhk0+z9I/pUNIZvHF5iupMekY5GNRq9dl82e/Jw+dFl+MX44dSdUwAAbYU2FvdeXOI+nCRvRcVFmLJ1Cr4+8LVQ61WvF2L6xcDcwFzCZCSm7Re3Izg+GA/zHgJQTXdSwlLg5eglcTISy4OnDxCSEILtF7cLtaneU/Fl5y9feItGuZJdswcAB68dRGBsIG7n3gYAmOubIyEkAd2cu71yPzZ78vE4/zH6J/dHxoUMoTbBfQK+7vE1pzsaZMXRFRjz0xhhuuNS0wUZERlwqu4kcTISyx/3/4BvjC8u3L8AQDXdWe63HINaDJI4GYnlWdEzjP95PH787Ueh1rdxX6wNXAtjPWMJk4lHls0eAFzJugK/GD+cvH0SgGq6822vb/G229sv3YfNnrwUFRfhw60fYv6B+UKth3MPxAXHcbqjQXZe2ol+8f3w4OkDAEANoxpICUuBt6O3xMlILA+fPkRIQgi2Xdwm1D7y/ggzOs/QqOmOJlMqlfj20Ld4d/O7KFYWAwBa27ZGWngaapnVkjid+sm22QNUN0wekDwAaefThNo4t3H4puc3L5zusNmTp5XHVmJMxhjhljqNazRGRv8M1K1eV+JkJJY/H/wJ32hfnL9/HgCgq6WLZX7LMKTlEImTkVieFT3DhJ8nYMlvS4RaUKMgrAtapzHTHQJ+/uNnhCWG4XHBYwCArYkt0iLS4GrnKnEy9ZJ1sweopjsfbfsIX+3/Sqh1d+6OuOC4526YzGZPvnZd2oW+8X2F6Y6loSWSw5LRvnZ7iZORWB7lPUJoQii2/L1FqE3xmoJZXWZxuqMhlEolFmcuxqRfJwnTnVY2rZAWkQZ7M3uJ05FYztw5A78YP1x8dBEAYKhjiDWBaxDSJETiZOoj+2bvH6uOrcJbGW8J051GNRohIyIDzhbOwjZs9uTtrwd/wTfGF+funQOgmu786PsjhrUaJnEyEkthcSEm/jwR3x/5XqgFNgrEuqB1MNEzkTAZiemXP39BWGIYsvOzAQA2JjZIC0+DWy03iZORWO7m3kXf+L7Ye2WvUPui4xf4pP0nsrwDk8Y0e8DzN0z+3+kOmz35y8rLQmhiKDb/tVmove/5PmZ3mc1FVzVIVGYUJv4yEUXKIgBAS5uWSAtPg4O5g8TJSCy/3/0dfjF++Pvh3wAAAx0DrAlcg9AmoRInI7HkF+bjrYy3sObEGqHWv1l/rPBfAQMdAwmTVTyNavYA1Q2TfaN9cfbeWQCq6c4S3yUY3mo4mz0NUVhciHd+eQeLDy8Wav4N/bGh7wZOdzTI5r82IzQhFFn5WQBU052N4RvhXstd4mQklntP7qFvXF/subJHqE3vOB2ftv9UltMdep5SqcRX+7/Ch1s/hBKqdsijlgdSw1NhY2IjcbqKo3HNHqCa7oQlhuHXv34Vau+1ew9T3afCoroFmz0N8f3h7zHh5wnCdKe5dXOkR6TD0dxR4mQklrN3z8Ivxg9/PfwLgGq6sypgFcKbhkucjMSSX5iPsT+Nxarjq4RaeNNwrPRfCUNdQwmTkZhSz6ViYPJA5D7LBQA4mDkgPSIdLWxaSJysYmhksweopjuTf52MbzO/FWo97Xvil5G/sNnTIFv+2oKQhBBhumNtbI3U8FS0tW8rcTISy/0n99E3vi92X94t1KZ1mIZpHaZxuqMhlEol5u+fjylbp8h6ukOvdvzWcfjH+ONq9lUAgLGuMaL7RcO/ob/Eyd6cxjZ7/1hyZAnGbRqnmu7kAZgDnLp8Ck0dm0odjURy7t45+Eb7CtMdfW19rApYhYhmERInI7EUFBVgbMZYrDy+UqiFNQnDqoBVnO5okLTzaeif1F+20x16vVs5txAQG4DM65kAAAUUmNt1Lt7zfK9Kf/jT+PUGxriOwS8DfymxDEuntZ1w4OoB6UKRqBrVaIRDIw+hY52OAID8onz0T+6PaTumCcszkLzpaethuf9yzO82Hwqo/qDHnYlDh9UdcPPxTYnTkVj8G/pj3/B9cDBTXahzNfsqvFZ6YeO5jRInI7HYmNhg55CdwqkcSijxwdYPMCJtBAqKCiROV36ya/auX7+OgQMHwtLSEkZGRmjZsiV+++23V+7TtW5XHBxxUFhk917uPXRa0wkbTm4QIzJVApZGlvh14K8Y2WqkUPti9xcITwzHk2dPJExGYlEoFJjsORkbwzcKF+ocvnEY7svdcezmMYnTkVha2LRA5qhM4VSO3Ge5CIoLwrx986DhB8I0hqGuIaL7RmN6x+lCbdXxVei6tivuPbknYbLyk9Vh3IcPH6JVq1bo1KkTxo4dCysrK/z111+oU6cOnJ2dX7v/pduX4GTjBHwI4P+vuv7E5xNM7zSdi65qCKVSiYUHF2Ly5snCuTuudq7YGL4RdqZ2EqcjsZy8fRJ+MX64knUFAGCka4QNfTcgsFGgtMFINHmFeRi+cThiTscItaEth2JJnyXQ19GXMBmJKf5MPIakDkFeYR4AoG71ukiPSIdLTReJk5WNrJq9Dz/8EPv27cOePXtev/EL/LP0ypDYIVhz7t91d4JdgrEmcA2MdI0qKipVcj9d+AnhSeHIKcgBANQyrYW0iDS0tm0tcTISy+2c2wiKC8KBa6pTOhRQYHaX2fjA64Mqfe4OlZ5SqcTMPTPx6Y5PhZqPow+SQpNQ07imhMlITIevH0ZAbABu5qhO6TDTN0NccBx61uspcbLSk1Wz5+Ligh49euDatWvYtWsXatWqhbfffhujRo164fb5+fnIz88Xvs7OzoaDgwMePXqEVWdXYfLmycI5W34N/JAWkfbCxyF5On3nNHyjfXE56zIA1XTnbORZLs2iQfIK8zAybSQ2nPr3lI753eZjsudkCVOR2BLOJGBI6hA8LXwKQLVM09HRR7kQuwa5ln0N/jH+OHZLdUqHlkIL+4bvqzIrN8jq2OTff/+NH374AfXr18evv/6KMWPGYMKECVi7du0Lt589ezbMzc2Ffw4OqpNyFQoFJrWdhPSIdJjqmQIA9l7Zy5P1NUxTq6bIHJUJTwdPAMCTZ09w9OZRiVORmAx0DLAuaB1mdJoh1HZf2f2KPUiOQpqEYPew3cKpHKdun8LDvIcSpyIx2ZvZY8+wPejbuC8AoFhZjP1X90ucqvRkNdnT09ODq6sr9u//9wmYMGECDh8+jAMHnr+69mWTvf+us3fu3jks/W0petbrie7O3dX/fwRVOvmF+VicuRh5hXl4z/M9nq+jobb+vRUZFzIw1nUsGtZoKHUcksD17OtYnLkYzayboX+z/lLHIQkUK4ux8thK/P3wb3zg9UGJlTwqM1k1e7Vr10a3bt2wfPlyofbDDz9gxowZuH79+mv35+3SiIiISG5kdRjXy8sL58+fL1G7cOECateuLVEiIiIiImnJqtl75513cPDgQcyaNQt//vknoqOjsXTpUkRGRkodjYiIiEgSsjqMCwAZGRn46KOP8Mcff8DJyQnvvvvuS6/G/V88jEtERERyI7tm702w2SMiIiK5kdVhXCIiIiIqic0eERERkYyx2SMiIiKSMTZ7RERERDLGZo+IiIhIxtjsEREREckYmz0iIiIiGWOzR0RERCRjbPaIiIiIZIzNHhEREZGMsdkjIiIikjE2e0REREQyxmaPiIiISMbY7BERERHJGJs9IiIiIhljswcgKioKLi4ucHNzkzoKERERUYVSKJVKpdQhKovs7GyYm5sjKysLZmZmUschIiIiemOc7BERERHJGJs9IiIiIhljs0dEREQkY2z2iIiIiGSMzR4RERGRjLHZIyIiIpIxNntEREREMsZmj4iIiEjG2OwRERERyRibPSIiIiIZY7NHREREJGNs9oiIiIhkjM0eERERkYyx2SMiIiKSMTZ7RERERDLGZo+IiIhIxtjsEREREckYmz0iIiIiGWOzR0RERCRjbPaIiIiIZEytzd7Pf/6M4zePQ6lUqvPHvLGoqCi4uLjAzc1N6iiy88f9P5B+Ph15hXlSRyGJ5BfmI+5MHK5nX5c6Ckloz+U92HdlH4qVxVJHIYncenwLCWcSkJ2XLXUUjaNQqqkTi/wpEt8f+R4AUM2gGro4dYGPow98avughXULaGtpq+PHvpHs7GyYm5sjKysLZmZmUsep8q5mXUXthbWhhBJaCi20s2+H9rXbw8fRB54OnjA3MJc6IonAJcoFZ++dBQDUqVYHnet0hk9tH/g4+qBu9bpQKBQSJyR1+/7w94jcFAkAMNI1Qqc6nYT3A1c7V+hp60mckNStqLgIRrOMUFBUAAUUaGnTEh3rdISPow+8Hb1R07im1BFlTW3NXs/1PfHrX7++8HumeqbwdPAUXuzutdxhoGOgjhhlwmavYv124ze4LnN94fe0FFpobt1c9Tvw/78HNiY2IickMdT8qibuPbn3wu/ZmtgKjZ+Pow+aWjWtlB8E6c1M+mUSFh1a9MLvGegYwKOWh/B3oJ19O5jqm4qckNQt71keDGcZvvT7jWo0KvF+UNu8Nj8IViC1NXsnb51Eix9blGpbPW09uNm5CU+yl4OXJFMfNnsVr+Hihrhw/0Kptq1nUa/Ei925ujNf7DIwd+9cfLjtw1Jta65vDi9HL+H3wNXOFfo6+mpOSOr28OlDWM+3xrPiZ6/dVluhjZY2LYW/A96O3rAythIhJalbt3XdsPXvraXa1t7MvsT7gUtNF2gpeJlBeamt2QOA3ht64+c/fy7zfgoo0Ny6OYa0GIJJbSeJ9obPZq/iJf6eiJCEkHLta2tii+7O3TGn6xxO/aqwvMI82C+wx/2n98u8r4GOAdxrueMj74/Qs15PNaQjsYzfNB6LDy8u174NLRsi2CUYn3X4jId8q7CjN4+izdI25drXwtACHWp3wJyuc9DAskEFJ5M/tbbJk9pOKtd+Sihx4vYJvLv5XeFcH6qaAhsFwtHcsVz73sy5iTUn1uDr/V9XcCoSk4GOAca4jinXvnmFedh9eTcGpQyq9Bd60auN9xhf7n3P3z+PmXtm4tc/X3xqEFUNrW1bw9vRu1z7Pnj6ACnnUvDRto8qOJVmUGuz161uNzSu0bjc+1sZW6FOtToVF4hEp6Olg0i3yDd6jLb2bSsoDUnlbbe3oaOlU+79PWp58JB+FdfAsgF61+9d7v0NdAzQ3Lp5BSYiKUz0mPhG+3vU8qigJJpFrc2eQqEo9xOrp62HlLAUGOkaVXAqEtvI1iNhqPPyE3NfZbz7ePRz6VfBiUhsdqZ2CHEp3+F8p2pOWBWwqoITkRQmeUwq975rA9eidrXaFReGJPEmR3t8G/hicrvJFZxIM6j9bMdBLQahukH1Mu+30n8lPB081ZCIxGZhaIHBLQaXeb8ezj2woMcCNSQiKZTng5+pninSI9K5LINMdK3bFS41Xcq83xcdv0BIk/J9WKDKpbxHe5pbN0d032herV9Oam/2jHSNMKr1qDLt87HPxxjQfICaEpEUJnhMKNP2jWs0Rlxw3Bsd+qPKxcPeo0yHYLQUWogLjkMTqyZqTEViUigUmOBetr8FEU0j8En7T9SUiKRQ1qM9VsZWSAtP45I8b0CU65gj3SOhrShdN96rXi980ekLNScisbnUdEG3ut1Kta2JngnSI9K56LIMlWW6N6fLHPSq30uNaUgKZTna09SqKVb4r+D5mjJTlqM9Olo6SAlN4SH8NyRKs+do7oigxkGl2vbIjSM4eO2gmhORFEp7dXZOQQ5WH1/N2yrJULBLMOxM7Uq1bdyZONx4fEPNiUhsZTna89eDv166OD9VbaU92lNYXIhlx5YhvzBfzYnkTbQVCl/3if6fyd/dJ3fRaU0nrD+5XoxYJKKe9XqWen2kGXtmICwxDE+ePVFzKhKTrrbua8/X+edvwW83f4P7MnccvXlUjGgkotcd7fln8dynhU8RFBeEOXvncOkdmXGp6YLuzt1Lte3q46vRdV1X3M29q+ZU8iVas+fl4IXWtq1f+D1DHUP8OvBXdHbqDAAoKCrAoJRB+Hjbx5zuyIiWQgvj3V++1tZ77d7DNz2+Ef7QJ/6eiPar2uN69nWxIpIIRrcZ/dLbI9a3qI+dQ3aitrnqkM31x9fhvdIbyWeTxYxIavaqoz0KKBAXHIf+zfoLtY+2fYShG4dyuiMzrxoCBbsEI6ZfjPC3Yu+VvfBY7oEzd86IFU9WRGv2FArFSy+7Xxu0Fl3qdsEvA37B6NajhfqsvbMQkhCC3IJckVKSug1pMQRm+s/fncS/oT/mdJ2DSW0nqU7E1VOdiPvbzd/gvtwdv934TeyopCY1jGpgQLPnL8CqZlANGf0z4F3bG5mjMtHOvh0A1XSnX3w/zNozi9MdGXnZ+8HcrnMR7BKM9UHrMaPTDKG+9sRadFnbhdMdGXnZ0Z42tm2wJnANwpuGY/fQ3bA1sQUAXHx0EZ4rPfHzH2W/M5emE/VGc6FNQmFtbF2i9mWnLxHsEgxAdYhnie8SLOyxUJjuJJ9NRvvVnO7Iham+KUa0GlGi1ty6OTb03SBcUt+nQR/sH7FfWFD7xuMb8Fnlg8TfE8WOS2ryv5/otRXaSAxJFP7wWxlbYfuQ7RjYfKCwzcfbP8bg1MHIK8wTNSuph6eDJ9rYlrx11rCWw/Ce53sAVAOCj9t/jISQBOHKzX1X98F9uTtO3zktel6qeC862mNnaoeN4RuFNXbdarkhc1QmWtm0AgBk52fDN8YXiw4u4oe/MhC12dPX0cfbbm8LXw9oNgAf+3xcYhuFQoGJbSciIyJDmO4cvXkUbsvccOTGETHjkpqMcx8nNPPWxtZIj0iHiZ5JiW2aWjXFoZGH4OXgBUA13QlJCMHM3TP5ApeBZtbNhNM2ACCqdxS61O1SYhsDHQOsDVyLmZ1nCrX1J9ejy9ouuJN7R7SspB4KhaLERVvta7fHEt8lz115G+wSjN3DdgsX9lx6dAmeKzyx6Y9NYsYlNRnSYgjM9VUrLxjqGCItPA21zGqV2MbezB57hu1B38Z9AQDFymJM+nUSxv40Fs+KnomeuSpSKEV+58wrzMOULVNgoGOA6Z2mv/TcHQA4c+cM/GL8cPHRRQCqX4Q1gWvUtrhmdnY2zM3NkZWVBTOz5w81UsWJPhWNn/74CR95f4SmVk1ful1+YT5GZ4zG2hNrhdqAZgOw3H/5K393qPK79OgSpm6bio51OmJ0m9Gv3Dbp9yQMShmEp4VPAQC1zWsjPSIdzaybiRGV1KRYWYwvdn2B2zm3MbPLTFgYWrx02+vZ1+Ef6y9csKOl0MLX3b/GRI+JXJqlitvy1xYsP7YckW6RaF+7/Uu3K1YW49Ptn2LW3llCrbNTZySEJLzyd4ckaPbK6m7uXfSN74u9V/YKtS86foFP2n9S4S9wNnuVk1KpxNx9c0vcALutfVukhqXC2sT6FXuSnBy9eRT+Mf64/lh1SoeJngli+sXAt4GvxMlILE+ePcHglMFIOpsk1Ea1HoWo3lHQ1daVMBmJaf3J9RiRNgIFRQUAVBd2ZfTPKPVqD5qo0jd7gGq681bGW1hzYo1Q69+sP1b4r6jQ6Q6bvcot5WwKBqYMFJZjcTR3RHpEOm+OrkFuPL6BgNgA4ZQOBRSY330+3mn7Dqc7GqJYWYzPdnyGmXv+PbzfqU4nJIYmcrqjQfZf3Y/A2EDcfaK6YKe6QXUkhiaWOD2E/lUlmj1ANd35av9X+HDrh1BCFdmjlgdSw1NhY2JTIT+DzV7ld+zmMfjH+uNa9jUAqulOdN9o+DX0kzgZieXJsycYmjoUCb8nCLWRrUYiqk8U9LT1JExGYtpwcgNGpI1AfpFqOZb6FvWRHpGOhjUaSpyMxHLp0SX4xfgJF+zoaOlgca/FeMv1LYmTVT5Vptn7R+q5VAxIHiBMdxzMHJAekY4WNi3e+LHZ7FUNNx/fREBsAA7fOAxANd2Z120eJrebzOmOhihWFmP6zun4Yve/t1bsULsDkkKTYGlkKWEyEtOBqwcQGBcoXLBTzaAaEkMSn7vYh+Trcf5jRCRF4Kc/fhJqEz0mYn73+by3+n9UuWYPAI7fOg6/GD9humOsa4zoftHwb+hfrseLiopCVFQUioqKcOHCBTZ7VcDTZ08xdONQxJ+JF2rDWw7HD74/cLqjQWJOxWDYxmHCdMe5ujMy+megUY1GEicjsVx+dBl+MX44decUANUyPot7L8YY1zESJyOxFBUX4YMtH2DBwQVCrVe9XojpF8N7rP+/KtnsAarpTmBcIDKvZwJQTXfmdp2L9zzfK/d0h5O9qkWpVGL6rumYvmu6UON0R/McvHYQgbGBuJ17GwBgrm+OhJAEdHPuJnEyEsvj/Mfon9wfGRcyhNoE9wn4usfXnO5okOVHl2PsT2NRWFwIQHVLtoyIDDhVd5I4mfSqbLMHqKY7w9OGI/Z0rFAb1nIYlvguKdd0h81e1RR7OhbDNg4TFtt1ru6M9Ih0NK7ZWOJkJJYrWVfgH+OPE7dPAFBNd77t9W2JdT1J3oqKi/Dh1g8x/8B8odazXk/E9ovldEeD7Ly0E/3i++HB0wcAVHfsSQlLgbejt8TJpFWlmz1ANd35YtcX+HzX50LNx9EHyWHJqGFUo0yPxWav6jp07RACYgNKTHfiQ+JLfaNtqvpyCnIwIHkA0s6nCbVIt0gs7LmQ0x0NsuLoCoz5aYww3WlcozEy+megbvW6Eicjsfz54E/4Rvvi/P3zAABdLV0s81uGIS2HSJxMOlW+2ftH3Ok4DN04VJju1K1eF+kR6XCp6VLqx2CzV7VdzboKvxi/EtOdhT0XYpz7OImTkViKiovw0baP8NX+r4Rad+fuiAuOQzWDatIFI1HturQLfeP7CtMdS0NLpISlwKe2j8TJSCwPnz5EaGIotv69VahN8ZqCWV1mCXdw0iSyafYAIPN6JgJiA3Ar5xYAwEzfDHHBcehZr2ep9mezV/XlFORgYPJAbDy/Uai97fo2FvVaxOmOBll1bBXeyngLz4pVt1JqVKMRMiIy4GzhLHEyEsufD/6EX4wfzt07B0A13VnqtxRDWw6VNhiJ5lnRM0z6ZRK+P/K9UAtsFIh1Qeueu0Wn3Mmq2QOAa9nX4Bfjh+O3jgNQ3VJnYQ/VdOd1F26w2ZOHYmUxpm6birn75gq1rnW7IiEkgdMdDbLn8h4ExQXh/tP7AAALQwskhyajQ50OEicjsTzKe4SwxDBs/muzUHvf833M7jIb2lraEiYjMUVlRmHiLxNRpCwCALSwboH0iHQ4mDtInEw8smv2ACC3IBcDUwYi9VyqUBvrOhaLei565S112OzJy+rjqzE6fbQw3Wlo2RAZ/TNQz6KexMlILH8//Bu+0b44e+8sANV0Z4nvEgxvNVziZCSWwuJCTPplEqIORwk1/4b+2NB3g8ZNdzTZ5r82IyQhBNn52QAAGxMbbAzfCPda7hInE4csmz1ANd35ZPsnmL13tlDrWrcr4oPjUd2w+gv3YbMnP3uv7EVQXBDuPbkHQDXdSQpNQsc6HaUNRqLJystCWGIYfv3rV6H2Xrv3MKfrHE53NMiLpjtpEWlwNHeUOBmJ5ezds/CN8cXfD/8GABjoGGBVwCqENw2XOJn6ybbZ+8faE2sxKn2UcMPkBpYNkBGRgfqW9Z/bls2ePP398G/4xfjh97u/A1DdUueHPj9gZOuREicjsRQWF2Lyr5Pxbea3Qs2vgR829N0AU31TCZORmLb8tQUhCSHIys8CAFgbWyM1PBVt7dtKnIzEcv/JffSN74vdl3cLtc/af4bPO34u6zswyb7ZA56f7lQ3qI6k0CR0cupUYjs2e/KVlZeF8KRw/PLnL0Lt3bbvYl63eZzuaJAlR5Zg3KZxwnSnuXVzpIWnoXa12hInI7Gcu3cOvtG++OvhXwAAfW19rApYhYhmERInI7EUFBVgbMZYrDy+UqiFNgnF6oDVMNQ1lDCZ+mhEswcAFx9ehF+MH87cPQNANd35vvf3GNVmlLANmz15KywuxPub38fCQwuFWp/6fRDdLxpm+ny+NcXWv7ciJCEEj/IeAQCsjK2QGpaKdg7tpA1Gorn/5D76xffDrsu7hNpn7T/DtI7TNHJZDk2kVCqx4MACvL/lfSihaoPc7NywMXwjbE1tJU5X8TSm2QOA7PxsRCRFYNMfm4TaJI9JmN99PrS1tNnsaYgfj/yIcT+PExZdbWrVFOkR6ahTrY60wUg05++dh2+ML/588CcA1XRnhf8KDGg+QOJkJJaCogJE/hSJ5ceWC7UQlxCsDlwNI10jCZORmNLPp6N/cn/kFOQAAGqZ1kJ6RDpa2baSOFnF0qhmD1Atuvre5vdKTHd61++NmH4xQD7Y7GmI7Re3o198P2G6U9OoJlLDU+Hp4CltMBLNg6cPEBwfjB2Xdgi1T3w+wfRO0znd0RBKpRLfHPwG721+T5juuNq5YmP4RtiZ2kmcjsRy8vZJ+MX44UrWFQCAka4R1getR1DjIImTVRyNa/b+sfS3pYjcFFliuhPdJxrNazdns6chLty/AN9oX/zx4A8AgJ62Hlb4r8DA5gMlTkZieVb0DJGbIrHs6DKhFuwSjDWBazjd0SAZFzIQkRRRYrqTFpGG1ratJU5GYrmdcxuBcYE4eO2gUJvdZTameE2RxYUbGtvsAarpTnB8MB7mPQQAWGpZ4v5n99nsaZAHTx8gJCEE2y9uF2pTvafiy85fcrqjIZRKJRYdWoTJmyejWFkMAGhj2wYbwzeillktidORWE7dPgW/GD9czroMQDXdWRe0Dn0b95U4GYklrzAPI9NGYsOpDUJtcIvBWOq7FPo6+hIme3Ma/W7W2akzDo08hAaWDQCoTtoFgNhTsVLGIhFZGFrglwG/4K02bwm1WXtnISQhBLkFuRImI7EoFApMajsJaeFpMNVTLcPy283f4L7cHb/d+E3idCSWZtbNkDkqE+3sVRfqPHn2BP3i+2HWnlnQ4JmIRjHQMcC6oHWY0WmGUFt7Yi26rO2Cu7l3JUz25jR6svePh08fIiQhBNvObgPmAPgQ+KjrR5jReQanOxpCqVTi20Pf4t3N7wrTnda2rZEWnsbpjgY5fec0/GL8cOnRJQCAoY4h1gWtQz+XftIGI9G8aLozqPkgLPNbVuWnO1R6ib8nYnDKYDwtfAoAqFOtDtIj0tHUqqnEycpH1p3M7NmzVZ/aJ0165XbVDavj5wE/l7iF0uy9sxEcH8zpjoZQKBSY2HYiMiIyhOnO0ZtH4bbMDUduHJE4HYmlqVVTHBp5CF4OXgCAp4VPEZwQjJm7Z3K6oyH+me7M7DxTqK07uQ6d13bGndw7EiYjMQW7BGP3sN3ChTqXHl2C5wrPEqt5VCWybfYOHz6MpUuXonnz5qXaXldbFwt6LAAA4WTMlHMp8F7ljWvZ19SWkyqXXvV74cCIA3Cq5gQAuJlzEz6rfJBwJkHiZCQWK2MrbBu8DYOaDxJqn+z4BINSBiGvME/CZCQWhUKBqT5TkRiSCEMd1SK7+6/uh/syd5y6fUridCQWVztXZI7MRBvbNgCAxwWP4Rfjh28OfFPlPvzJstnLycnBgAEDsGzZMlSv/uL74L7IP01eQkiCsMju8VvHOd3RME2smuDQyEPwdvQGoDqsE5oYii93fSlxMhKLvo4+1gSuwazOs4TahlMb0HkNpzuapJ9LP+wdvleY7lzOugzPlVV3ukNlV8usFnYP241+jVWnchQri/Hu5nfxVsZbwmoeVYEsm73IyEj06dMHXbt2feV2+fn5yM7OLvEPALo5d8OBEQdQt3pdAMCtnFsITQgVzuUi+atpXBNbB23FkBZDhNpnOz/Dzks7pQtFolIoFPjI5yMkhSYJy7AcuHYAH2z5QOJkJKbWtq1xeNRhuNq5AgByCnIQnhgurNFJ8meka4T4kHh84vOJUFt2dBnWnlgrYaqykV2zFxsbi6NHj2L27Nmv3Xb27NkwNzcX/jk4OAjfc6npgkMjD8HH0QcAYKJnwos1NIy+juqemXO7zoXi//+fsa6x1LFIZH0b98XeYXtRy1R1oc4/53SS5rAztcOuobsQ4hICQHXazz+Hd0kzaCm08GXnL7E+aD30tVUX6lSlvwWyuhr36tWrcHV1xebNm9GiRQsAQMeOHdGyZUssXLjwue3z8/ORn58vfJ2dnQ0HB4cS6+wVFRch83ommlo1hal+1XliqWJdenQJT589ReOajaWOQhLJLcjF8VvH0da+LbS1tKWOQxJQKpU4dusYHMwcUNO4ptRxSCK3c27jxuMbVeqWarJq9lJTUxEUFARt7X//EBcVFUGhUEBLSwv5+fklvve/eG9cIiIikhsdqQNUpC5duuDUqZJXSg0bNgyNGjXClClTXtnoEREREcmRrJo9U1NTNG1acsFDY2NjWFpaPlcnIiIi0gS84oCIiIhIxmR1zt6b4jl7REREJDec7BERERHJGJs9IiIiIhljs0dEREQkY2z2iIiIiGSMzR4RERGRjLHZIyIiIpIxNntEREREMsZmj4iIiEjG2OwRERERyRibPSIiIiIZY7NHREREJGNs9oiIiIhkjM0eERERkYyx2SMiIiKSMTZ7RERERDLGZg9AVFQUXFxc4ObmJnUUIiIiogqlUCqVSqlDVBbZ2dkwNzdHVlYWzMzMpI5DRERE9MY42SMiIiKSMTZ7RERERDLGZo+IiIhIxtjsEREREckYmz0iIiIiGWOzR0RERCRjbPaIiIiIZIzNHhEREZGMsdkjIiIikjE2e0REREQyxmaPiIiISMbY7BERERHJGJs9IiIiIhljs0dEREQkY2z2iIiIiGSMzR4RERGRjLHZIyIiIpIxNntEREREMsZmj4iIiEjG2OwRERERyZjamr0b2TfQZU0XDEgagF2XdiGvME9dP+qNRUVFwcXFBW5ublJHkZ3Jmyej4+qOWHVsFW7l3JI6Dklgy19b4LXSCx9s+QAnbp1AUXGR1JFIZE8KniAgJgABsQH4+Y+f8Tj/sdSRSAJf7/8a3iu9sfDgQlx6dAlKpVLqSBpDoVTTf+2ua7ti28Vtwtd62npws3ODj6MPfGr7wNPBE9UMqqnjR5dbdnY2zM3NkZWVBTMzM6njVHmHrh1C2xVtS9TqWdRT/Q78/++Bc3VnKBQKiRKSGCznWeLB0wfC1+b65vBy9BJ+D1ztXKGvoy9hQlK3yJ8i8f2R74WvtRRaaGXTSvg74O3oDStjKwkTkro9ffYURrOMStTszexLvB+41HSBloIHHNVBbc3e/764n/vBUKC5dXPhSfZx9IGtqa06opQam72KdSvnFmy/fvVzamNiU+LF3syqGbS1tEVKSGJwiXLB2XtnX/p9Ax0DuNdyF34PPB08YapvKmJCUrcfj/yIMT+NeeU2DS0blng/qFOtDj8IykhxcTEMZhrgWfGzl25T3aA6vB29hd+D1ratoaetJ2JK+VJbs5ednw3r+dZlOnzrXN1ZeKH7OPqgnkU9UV/sbPYqXkhCCBJ/Tyz19ub65vB08BRe7G52bpz6VHE7L+1EpzWdSr29lkILLW1alvgQwKlP1VasLIbDNw648fhGqfepZVqrxPtBE6smnPpUcVO3TcXsvbNLvb2hjiHa2rcV/g60tW8LEz0TNSaUL7U1ewAw6ZdJWHRoUbn3b2ffDjuG7BDtzZ7NXsU7cesEWv7Ystz7VzOohtSwVHSo06HiQpGolEol2ixtg2O3jpX7MSZ5TMI3Pb+pwFQktm8PfYuJv0ws9/6NajTCjiE7YGNiU4GpSEx3c+/C4RsH5Bfll2t/Ax0DrPRfiYhmERWcTP7U+jFpvPt4KFD+ydyBawdw4f6FCkxEYmth0wIdape/UXuU9whp59MqMBGJTaFQYKJH+d/kAWDNiTU8mbuKG9ZyGMz0y/8h+ty9czh07VAFJiKx1TSuif7N+pd7/7zCPMSdiavARJpDrc2es4Uz/Br6lXv/Lk5d0NSqaQUmIim8yRu9iZ4JRrQeUYFpSArhTcPf6FDspLaTeP5WFWeqb4rhLYeXe3+Xmi7o5tytAhORFN7k/UBLoYXx7uMrMI3mUPsJEOV9YhtYNkBCSAL/wMuAf0N/1KlWp8z7KaBAdN9ouNR0qfhQJCp9HX2MafPqE/RfJsQlBJ+0/6SCE5EUxnuU72hPDaMayIjIgJGu0es3pkrtTY72LOyxEF3qdqngRJpB7c1epzqdyjydq25QHRkRGahuWF1NqUhM2lraGOc2rsz7fdXtqzeaDFPlMtZtLHS1dMu0j6udK1YHruaJ+TJRt3rdMr+m9bT1kBKWAqfqTmpKRWIrzxBorOtYjHMv+/sIqaj9L2hZz9fR0dJBUmgS6lvWV2MqEtuI1iNgrGtc6u2HtxyOd9u9q8ZEJDYbExuENQ0r9fa1TGthY/hGTnNkpqxv9Et9l8Lb0VtNaUgKZT3a07VuVyzquYhH+t6AKB+XBzQbAEtDy1JtO7DZQHRyKv0yDVQ1VDOohiEthpRq2xqGNTC/+3y+sGWotG/0Cigwp+sc2JnaqTkRia0sR3t8HH0wuMVgNScisZXlaI+BjgGiekdBV7tsRwWoJFGaPUNdQ4xuM7pU264+sRofbv0QxcpiNacisU3wmFCq7e49vYfOazvjWvY1NScisbnaucLLweu12ymhxOj00Ug4kyBCKhJTWY727LmyB4NSBlXq221S+ZT2aE9eYR66r+uOU7dPiZBKvkQ7EeZtt7ehrXj5nREczR2F/z1331z0i++HnIIcMaKRSBrWaIhe9Xq99PvGusbCi//4reNwW+aGzOuZYsUjkbzujb62eW0AwNPCpwhNDMWXu77ksisy87qjPbYm/955Z8OpDei8pjNu59wWIxqJpJpBNQxtOfSl39dWaKOGYQ0AwOWsy/Bc6YmMCxkipZMf0Zo9ezN7BLsEv/B7TWo2wamxp7C412KhIUw9lwrvld64mnVVrIgkgpe90WsrtJEclozMUZmoW70uANXt1jqs7oC401xXSU6CGgfBwczhhd8b2HwgzkWeK3HI/7Odn2FA8gA8ffZUrIikZq862mNjYoNDIw8hOTRZOF/zwLUDcF/ujpO3T4oZk9TsVcuo/NDnB5wYewKudq4AgJyCHPjH+GPBgQX88FcOol7iNqntpOdqNYxqID0iHWb6Zoh0j8SmAZuEhTdP3D4B9+XunO7ISHfn7mhUo9Fz9YU9F6K7c3e41HTBoZGH4OPoA0A1wg9PCsf0ndP5ApcJHS0dRLpFPlf3dPDEMr9lMNA1wKqAVZjbda6wTEfM6Rh0WtMJt3JuiR2X1ORtt7eho6VTomagY4CN4RvhYO6AoMZB2DtsL2qZ1gIAXMm6Aq+VXkg/ny5FXFKDlx3teaftOxjVZhTsTO2wa+guhLiEAFCd3jF582SMTh+NgqICseNWaaI2e23t28K9lrvw9Ysuqe/u3B0HRxx8broTezpWzKikJgqFAhPcS56797br2yUuqa9hVANbBm3BsJbDhNrnuz5H/+T+nO7IxKg2o2CoYyh87WjuiJSwFBjoGABQ/Z584PUBksP+ne4cun4I7svcceLWCUkyU8V60dGe1QGrS7xHtLJthcOjDsPNzg2AaroTEBuAr/d/zQ9/MvG/R3t61++Nr7p9JXxtpGuE2OBYfNb+M6G2/Nhy9FjfA/ef3BctZ1Un+uJVU72nCv/7ZZfUN67ZGJkjM9G+dnsAqulORFIEpu2Yxhe4DAxuMVj4tN6tbjcs7LnwuW30dfSxwn8F5nWdJ0x3Yk/HouOajrj5+KaIaUkdLAwtMNZ1LADAVM8UGREZL7zDRmCjQOwbvg/2ZvYAgKvZV+G10ou30JOJ9z3fF9Ze/LzD5y9cmsfW1Ba7hu5CaJNQAKrpzntb3sPItJGc7shAd+fuwqHaplZNEdMvBtpaJc/v11JoYXqn6YjuGw19bX0AwM5LO+Gx3APn7p0TPXNVpFBK0D3tubwHBjoGcKvl9srtCooKMDZjLFYeXynUQpuEYnXAahjqGr5iz/LJzs6Gubk5srKyYGZW/ns40utdybqCYzePoXf93q+9pD7tfBr6J/VH7rNcAKqJQHpEOlratBQhKalLYXEhMi5koLVt6xIXaL3IrZxbCIgNEE7p+Gdplvc93+cSPVXcsZvH8Cjv0WuX3FIqlZi+azqm75ou1DrU7oCk0CRYGpVuaS+qnO49uYddl3ahZ72eMNZ79RW6B68dRGBsIG7nqi7YMdc3R0JIAm+l9xqSNHtloVQqseDAAry/5X0ooYrqZueGjeEbYWtq+5q9y4bNXuV14tYJ+MX44Wq26oIdI10jbOi7AYGNAqUNRqJ5+uwphqcNL3FKx7CWw7DEdwn0tPUkTEZiij0di6GpQ5FflA8AcK7ujPSIdDSu2VjiZCSWK1lX4B/jjxO3Vad0aCu08W2vb/G229sSJ6u8Kn2z94/08+non9xfWI6llmktpEeko5Vtqwr7GWz2KrdbObcQFBeEg9cOAlBNd2Z3mY0PvD7gdEdDKJVKfLn7S0zbOU2o+Tj6IDksGTWMakiYjMR06NohBMQGlJjuxIfEo7tzd4mTkVhyCnIwIHlAiVM6It0isbDnwucu/KEq1OwBwMnbJ+EX44crWVcAqKY764PWI6hxUIU8Ppu9yi+vMA/DNw5HzOkYoTakxRD86Psj9HX0JUxGYoo/E48hqUOExXadqjkho38GXGq6SJyMxHI16yr8YvxKTHcW9lzI+6dqkKLiIny07SN8tf/fCzq6O3dHXHAcqhlUky5YJVSlmj0AuJ1zG4FxgcJ0BwBmd5mNKV5T3ni6w2avalAqlZixewY+2/nv1Vnejt5IDk1GTeOaEiYjMWVez0RAbICwHIuZvhniguPQs15PiZORWHIKcjAweSA2nt8o1N52fRuLei3idEeDrDq2Cm9lvIVnxc8AAI1qNEJGRAacLZwlTlZ5VLlmD1BNd0amjcSGUxuE2uAWg7HUd2m5pjtRUVGIiopCUVERLly4wGavikg4k4AhqUPwtFC1HItTNSekR6SjiVUTiZORWK5lX4N/jD+O3ToGQHXV3sIequkOD+1rhmJlMaZum4q5++YKtW51uyE+JJ7THQ2y+/Ju9I3ri/tPVcuxWBhaIDk0GR3qdJA4WeVQJZs9QDXdmbVnFj7Z8YlQ83LwQkpYSrmnO5zsVT1HbhyBf4w/buaolmMx1TNFXHAcetV/+W3ZSF5yC3IxKGUQUs6lCLUxbcbg217f8ubpGmTN8TUYlT5KmO40tGyIjP4ZqGdRT+JkJJa/H/4N32hfnL13FgCgq6WLJb5LMLzVcImTSa/KNnv/SPw9EYNTBgvTnTrV6iA9Ih1NrZqW+bHY7FVNL5ruLOi+ABM8JnC6oyGKlcX4ZPsnmL13tlDr4tQFCSEJqG5YXcJkJKY9l/cgKC6oxHQnKTQJHet0lDYYiSYrLwthiWH49a9fhdp77d7DnK5znlu/T5NU+WYPUE13AmIDcOPxDQCq6U5scCx61+9dpsdhs1d15RbkYnDqYCSfTRZqb7V5C9/1+o7THQ2y7sQ6jEz/d7HdBpYNkBGRgfqW9SVORmL5++Hf8Ivxw+93fweguj3fkj5LMKL1CImTkVgKiwsx+dfJ+DbzW6Hm18APG/pugKm+qYTJpCOLZg8ArmdfR0BsAH67+RsA1XRnfrf5mNR2UqmnO2z2qrZiZTE+3f4pZu2dJdQ6O3VGQkgCLAwtJExGYtp3ZR+C4oJw98ldAEB1g+pICk167aK9JB9ZeVkITwrHL3/+ItTebfsu5nWbp9HTHU2z5MgSjNs0DkXKIgBAM6tmSI9IR+1qtSVOJj7ZNHsA8OTZEwxOGYyks0lCbVTrUYjqHVWq6Q6bPXlYf3I9RqSNEKY79S3qI6N/BhpYNpA4GYnl0qNL8I32xZm7ZwCopjtRvaMwus1oiZORWAqLC/He5vew6NAioebbwBfRfaM1drqjibb+vRUhCSF4lPcIAGBlbIXUsFS0c2gnbTCRyarZA1TTnWk7pmHGnhlCrVOdTkgMTXztdIfNnnzsv7ofgbGBwnSnmkE1JIYkokvdLhInI7Fk52cjIikCm/7YJNQmeUzC/O7zOd3RID8e+RHjfh6HwuJCAKr7r6ZHpKNOtTrSBiPRnL93Hn4xfvjjwR8AAH1t1b3XBzQfIHEy8ciu2fvHhpMbMCJthHBLnXoW9ZARkYGGNRq+dB82e/Jy+dFl+Mb44vSd0wBUi65G9Y7CW65vSZyMxFJUXIT3t7yPbw5+I9R61++NmH4xMNPna1xTbL+4Hf3i+wnTnZpGNZEangpPB09pg5FoHjx9gOD4YOy4tEOofezzMb7o9AW0FFoSJhOHbJs9ADhw9QAC4wJxJ/cOANV0JyEkAV3rdn3h9mz25Odx/mNEJEXgpz9+EmoTPSZifvf5XHRVgyz7bRne3vS2MN1pUrMJ0iPS4VTdSeJkJJYL9y/AN9pXmO7oaethhf8KDGw+UOJkJJZnRc8QuSkSy44uE2rBLsFYE7gGRrpGEiZTP1k3e4BquuMX44dTd04BUE13FvdejDGuY57bls2ePBUVF2HK1in4+sDXQq1XvV6I6RcDcwNzCZORmHZc3IF+8f3wMO8hAKCGUQ2khqXCy9FL4mQklgdPHyAkIQTbL24XalO9p+LLzl9qxHSHVGv0Ljq0CJM3T0axshgA0Ma2DTaGb0Qts1oSp1Mf2Td7gGq60z+5PzIuZAi1Ce4T8HWPr0tMd9jsyduKoysw5qcxwnTHpaYL0iPSUbd6XYmTkVj+uP8HfGN8ceH+BQCq6c5yv+UY1GKQxMlILM+KnmH8z+Px428/CrW+jftibeBaGOsZS5iMxLTpj00ITwzH44LHAAA7UzukhaehjV0biZOph0Y0e4BquvPh1g8x/8B8odbDuQfiguOE6Q6bPfnbeWkn+sX3w4OnDwCopjvJocnwqe0jcTISy8OnDxGSEIJtF7cJtY+8P8KMzjM43dEQSqUS3x76Fu9ufleY7rS2bY208DRZT3eopNN3TsMvxg+XHl0CABjqGGJt0FoEuwRLG0wNNKbZ+8f/Tnca12iMjP4ZqFu9Lps9DfHngz/hG+2L8/fPA1DdUmeZ3zIMaTlE4mQklmdFzzDxl4n44cgPQi2oURDWBa3jdEeD/PzHzwhLDBOmO7YmtkiLSIOrnavEyUgsd3LvoG9cX+y7uk+ozeg0A1N9psrqDkwa1+wBwK5Lu9A3vq8w3bE0tERyWDJaVm/JZk9DPMp7hNCEUGz5e4tQm+I1BbO6zOJ0R0MolUoszlyMSb9OEqY7LW1aIj0iHfZm9hKnI7GcuXMGfjF+uPjoIgDVdGdN4BqENAmROBmJJb8wH6MzRmPtibVCbUCzAVjuvxwGOgYSJqs4GtnsAcBfD/6Cb4wvzt07B0A13VnYcSEi20ey2dMQhcWFmPjzRHx/5HuhFtAwAOv7roeJnomEyUhMv/z5C8ISw5Cdnw0AsDGxQVp4GtxquUmcjMRyN/cu+sb3xd4re4XaFx2/wCftP5HVdIdeTqlUYu6+ufho20dCra19W6SGpcLaxFrCZBVDY5s9QDXdCUsMw+a/NqsKeQDmAA8ePkD1arx5uqaIyozCxF8mCrfUaWHdAukR6XAwd5A4GYnl97u/wy/GD38//BsAYKBjgDWBaxDaJFTiZCSW/MJ8vJXxFtacWCPU+jfrjxX+K2Qz3aHXSzmbgoEpA/Hk2RMAgKO5I9Ij0tHcurnEyd6MRh+vqmZQDT/1/wnj3MaVqA9IHoCcghyJUpHYIt0jsWnAJpjrqy7UOXH7BNyWueHQtUMSJyOxuNR0waGRh+DjqLpQJ68wD2GJYZi+czo0+POwRtHX0ceqgFWY23UuFFBN86JPRaPj6o64lXNL4nQklqDGQdg7bK9wKseVrCvwWumF9PPpEid7M7Jq9mbPng03NzeYmprCysoKgYGBOH/+/Cv30dHSwXe9v0NU7yjhXK2f//gZXiu9cCXrihixqRLo7twdB0YcgHN1ZwDA7dzb6LC6A2JPx0qcjMRSw6gGtgzagqEthwq1z3d9jv7J/fH02VPpgpFoFAoFPvD6AMlhycIiu4euH4L7MnecuHVC4nQklla2rZA5MhNudqpTOXIKchAQG4D5++dX2Q9/smr2du3ahcjISBw8eBBbtmxBYWEhunfvjtzc3Nfu+7bb20gKTRK+Pnn7JNyXuePgtYPqjEyVSOOajXFo5CG0r90eAJBflI+IpAhM2zGtyr7AqWz0dfSx0n8l5nWdJ0x3Yk/HotOaTpzuaJDARoHYN3yfMN25mn0VXiu9kHY+TeJkJBZbU1vsGrpLOJVDCSXe3/I+RqaNREFRgcTpyk7W5+zdvXsXVlZW2LVrF9q3b//a7f9ZesVprhMuPlVdmaWvrRrtRzSLUHdcqiQKigowNmMsVh5fKdRCm4RidcBqGOoaSpiMxLTx3EYMSB6A3GeqD4sOZg5Ii0hDS5uW0gYj0dx8fBOBcYHIvJ4JAFBAgbld5+I9z/d44YaGUCqVmL5rOqbvmi7U2tduj6TQJNQwqiFhsrKR1WTvf2VlZQEALCwsyrTf9sHb0bFORwCq6U7/5P745sA3r96JZENPWw/L/Zdjfrf5wnQn/kw8OqzugLzCPInTkVgCGgVg3/B9cDBTXahzNfsqvFd6Y/fl3RInI7HYmtpi55CdCG8aDkA13flg6weY+MtEiZORWBQKBT7v+Dli+sUIF+rsvrwbHss9cDvntsTpSk+2zZ5SqcS7774Lb29vNG3a9IXb5OfnIzs7u8Q/ALAwssCvA3/FyFYjhW2n7ZwmrMVF8qdQKDDZczI2hm8UlmE5fONwiVvukfy1sGmBzFGZ8KjlAQDIfZaLufvmSpyKxGSoa4jovtH4vMPnQu27zO+q1Bs9vbnwpuHYOWQnrI1Vy7D8/fDvElduV3aybfbGjRuHkydPIiYm5qXbzJ49G+bm5sI/B4d/l9rQ09bDUr+lWNRzEezN7DGq9SgutquB/Br6Yd/wffCo5YEW1i3g6eApdSQSmY2JDXYM2YG32rwFO1M7DGw2UOpIJDKFQoFpHachLjgOztWd0a9xP1gZW0kdi0TmYe+Bw6MOo7tzd9SzqIee9XpKHanUZHnO3vjx45Gamordu3fDycnppdvl5+cjPz9f+Do7OxsODg5cVJmIiIhkQ0fqABVJqVRi/PjxSElJwc6dO1/Z6AGAvr4+9PX1RUpHREREJD5ZNXuRkZGIjo7Gxo0bYWpqilu3VEslmJubw9CQV1ESERGR5pHVYdyXXQq/atUqDB069LX7/7P0Cg/jEhERkVzIarIno76ViIiIqELw8lIiIiIiGWOzR0RERCRjbPaIiIiIZIzNHhEREZGMsdkjIiIikjE2e0REREQyxmaPiIiISMbY7BERERHJGJs9IiIiIhljs0dEREQkY2z2iIiIiGSMzR4RERGRjLHZIyIiIpIxNntEREREMsZmj4iIiEjG2OwRERERyRibPQBRUVFwcXGBm5ub1FGIiIiIKpRCqVQqpQ5RWWRnZ8Pc3BxZWVkwMzOTOg4RERHRG+Nkj4iIiEjG2OwRERERyRibPSIiIiIZY7NHREREJGNs9oiIiIhkjM0eERERkYyx2SMiIiKSMTZ7RERERDLGZo+IiIhIxtjsEREREckYmz0iIiIiGWOzR0RERCRjbPaIiIiIZIzNHhEREZGMsdkjIiIikjE2e0REREQyxmaPiIiISMbY7BERERHJGJs9IiIiIhljswcgKioKLi4ucHNzkzoKERERUYVSKJVKpToeOP5MPAanDIahriH6Ne6HbnW7wae2D+xM7dTx4ypEdnY2zM3NkZWVBTMzM6njVHn5hfmo/1193M69jba12qKfSz/4OPqguXVzaGtpSx2PRDI2YyxWHFsBGxMbDGg2AB3qdICngyfM9Pka0xSHrx9GpzWdAAB96vdB7/q94VPbB07VnKBQKCROR2JQKpVwXeqKU3dOoYlVE4Q3CUf72u3Rxq4N9LT1pI4ne2pr9rxXemPf1X3P1etWrwsfRx/Vv9o+qG9Rv9K82NnsVaztF7ejy9ouz9XN9M3g6eAp/B641XKDgY6BBAlJDOZzzJGdn12ipqXQQgvrFsLfAR9HH1ibWEuUkNRtWOowrD6x+rm6naldifeDplZNoaXgASc5elLwBMazjZ+rG+oYwsPeQ/g9aOfQDiZ6JhIklDe1NXsrj63EiLQRr93O2tga3o7ewou9hXULyaY+bPYqVkFhAarNrYanhU9fuZ2eth7ca7kLL3ZPB0+YG5iLlJLULTg+GElnk167XX2L+iWav7rV61aaD4L0ZjKvZ8Jjucdrt6tmUA1eDl7C74GrnSunPjJiv8Ae1x9ff+U22gpttLJtJbwfeDt6o6ZxTZESypfamr1iZTEafNcAfz38q0z7meqZClOf0CahqG9ZXx3xXojNXsWbuXsmPtnxSZn20VJoobl1c/g4+qCHcw/0rt+bb/pV2NWsq6izqA6KlcVl2s/WxFZo/AY2H4hqBtXUE5BE0XVtV2y7uK1M+xjoGMCjlmrq07dxX7SybaWmdCSG2NOxiEiKKPN+jWo0go+jDzrV6YSQJiHQ0dJRQzp5U1uzBwDfH/4ekZsiy72/vrY+Lk26BBsTmwpM9XJs9ire/Sf3Yf+NPfIK88r9GN/0+AaT2k6quFAkuvDEcMSdiSv3/s2tm+PEmBMVmIjE9sufv6DXhl7l3l8BBQ6NPAS3WryQrqp6VvQMdb+ti2vZ18r9GOPdx+PbXt9WYCrNoNaTIwa3GAxz/fIfjisoKkBBUUEFJiKxWRpZYlDzQW/0GDkFORWUhqQy0WPiG+2fW5ALNX4uJRF0d+6OhpYNy72/EsrXnhJClZuuti4i3co/AAKAxwWPKyiNZlFrs2eiZ4KRrUeWe/+ZnWfC0dyxAhORFCZ4TCj3vq1sWuGdtu9UYBqSQlv7tnCzK99ERkdLB8v9l/NQfhWnpdB6o78FI1uNhI+jTwUmIimMaj0KhjqG5drX3sweszrPquBEmkHtlz2Ncx9XrqurBjUfhA+9P1RDIhJbU6um6OL0/FW5r2NjYoO0iDQY6z1/BRdVLQqFotzTvSV9lqBjnY4VG4gkUd6jPR3rdERUnyg2/DJgaWSJgc0Hlnk/I10jpIWnwdbUVg2p5E/tzV6danUQ0DCgTPt4Onhimd8yvrBlpKxv9AY6BkgLT4O9mb2aEpHYQpqEwNakbH+o3237Lka0fv1V/VQ1lOdoTz2LekgMSeRVuTJSngnv+qD1vEDnDYiyoFFZ3ujrVKuDlLAU6OvoqzERia1Pgz5wru5c6u3XBK7hidgyo6eth7GuY0u9vW8DX8zrNk+NiUgKZTnaY65vjvSIdFgaWao5FYmprEd7ZneZjaDGQWpMJH+iNHvta7dHC+sWpdq2tnntch/Pp8pLS6GF8e7jS7Wtsa4xz9WUqbdc3yr1hKZpTS6wK0dlOdpja2LLJXdkqrRDIB2FDhrXaKzmNPInyl9ShUJR6qUzdl3eBe9V3rj86LJ6Q5HohrUaBlM909dul/ssFx1Xd0TMqRgRUpGYrIyt0L9Z/1JtO2ffHEQkReDpM16BKTelfT84d/8c3Je548QtLrsjN6U92lOoLERQXBC+2vcVr8h/A6J9bA5vGo6aRi9fBXtMmzHCJ7iTt0/Cfbk7Dl47KFI6EoOZvhmGtRz20u+3tm0NbwdvAEB+UT76J/fHtB3TyrwYL1Vur/pEr6eth0i3SCigOl837kwcOqzugJuPb4oVj0Tg4+iDljYtX/r9sCZhcDBzAABczb4Kr5VeSDufJlI6EsPrjvbUNq8tTICVUOKDrR9gRNoILsdWTqI1ewY6BhjjOuaF3xvVehS+7/M9Do44iHoW9QAAd3LvoOPqjog+FS1WRBLBeI/xwhv5f9mb2SMjIgPbhmzDyFb/nsD9xe4vEJEUgSfPnogZk9SopU1LtK/d/oXfW+G/Aot7L0ZqeCqMdVVXYR++cRjuy91x7OYxMWOSGr3q6uwezj2wvu96ZI7KhEct1S3Wcp/lIjA2EPP2zeN0R0ZedrTHVM8UP/X/CSlhKfi8w+dCfdXxVei2rhvuPbknYkp5EPWEmLGuY6GrpVui1qlOJ0T1Vl1S37BGQxwccVBYZiG/KB8Dkgfg0+2fcrojE/Us6qFPgz4lasa6xkiPSIetqS30tPWw1G8pvu7+tdAUxp+JR4fVHXDj8Q0pIpMavOiNfqr3VGFJBv+G/tg/Yr9w7ua17GvwXuWN1HOpYsYkNXrR0Z7GNRojLjgOOlo6sDGxwY4hOxDRVHV7LSWUmLJ1CoanDUd+Yb4UkamCvehoj5ZCC3HBcWhi1QQKhQLTOk5DbL9YGOgYAAB2X94Nj+Ue+P3u71JErrJEbfZsTW0R2iRU+LqeRT0khiZCV/vfBtDSyBK/Dvy1xHRnxp4ZCEsM43RHJv77Rq+AAuv7ri9xSEehUODddu8iLSINJnomAIAjN47AfZk7jt48KnZcUoOAhgGobV5b+Lpv4774svOXJbZpbt0cmSMz0da+LQDgybMnCIoLwpy9czjdkYH/PdpjaWiJ9Ih0mBv8uw6foa4hNvTdgOkdpwu11cdXc7ojI/97tGdB9wXoVb/kbfXCmoZh19Bdwq1T/374N9qtaIdf//xV1KxVmeiXuk3rMA22JrZwru6MjIgMWBhaPLfNP9OdBd0XCFfjJf6eyOmOTHRx6oJgl2AY6Bjgu17fIbBR4Au3823gi/3D9wtNwfXH1+GzygfJZ5NFTEvqoK2ljW96fAN9bX10q9sNawPXvvDKW2sTa+wYsqPERR0fbfsIQzcO5XRHBiZ4TIBLTRdYGVshJSwFzhbPn7CvUCjwWYfPEBccJ0x39lzZA/dl7pzuyEA9i3qY3G4ydLV0McVrykvX4HOv5Y7MkZnCYCA7Pxu9o3tjceZiEdNWXQqlBB+RlUolipXF0NbSfu22P134CRFJEcL98OxM7ZAWnoY2dm0qPFd2djbMzc2RlZUFMzOzCn98KqmouKhUvwN3cu8gMDYQB64dEGqzOs/Ch94fcuHtKq6ouAhaCq3XPo9KpRKz9szCJzs+EWpeDl5ICUtBTeOXX/hFVUNp/xYcvn4YAbEBuJmjumDHTN8MccFx6Fmvp7ojkpqV9ncgtyAXA1MGljil423Xt7Gw58ISRwmpJEmavbI6fec0fKN9cTlLtRyLoY4h1gWtQz+XfhX6c9jsVV55hXkYmTYSG05tEGqDmg/CMr9lXIBbgyT+nojBKYPxtFC1HEudanWQHpGOplZNJU5GYrmWfQ3+Mf44dkt1wY6WQgvf9PgG493H88OfhihWFuPjbR9jzr45Qq1r3a6ID45HdcPqEiarvKpEsweopjtBcUHYf3W/UJvRaQam+kytsBc4m73KTalUYvbe2fh4+8dCzdPBEylhKbAytpIwGYnpyI0jCIgNEE7pMNUzRWxwLHrX7y1xMhJLbkEuBqUMQsq5FKE2ps0YfNvrW053NMia42swKn0UnhU/AwA0tGyI9Ih01LesL3GyyqfKNHuAarozOn001p1cJ9QGNBuA5f7LhXM53gSbvaoh6fckDEoZJEx3apvXRnpEOppZN5M4GYnlevZ1+Mf6CxfsaCm08HX3rzHRYyKnOxqiWFmMT7d/ill7Zwm1Lk5dkBCSwOmOBtl7ZS+C4oKEC3aqG1RHUmgSOjl1kjhZ5VKlmj1ANd2Zs3cOpm6fKtTa2bdDSlgKrE2sy/WYUVFRiIqKQlFRES5cuMBmrwo4evMo/GL8hOmOiZ4JYvvFPresC8nXk2dPMDhlMJLOJgm1Ua1HIap3FKc7GmTdiXUYmT5SWGy3gWUDZERkcLqjQS4+vAi/GD+cuXsGAKCjpYPve3+PUW1GSZys8qhyzd4/ks8mY1DKIGE5FkdzR2REZLzRdIeTvarlxuMb8I/xx283fwOgmu7M7zYfk9pO4nRHQxQri/HZjs8wc89ModapTickhia+8Ep/kqd9V/YhKC4Id5/cBaCa7iSGJqKzU2eJk5FYsvOzEZ4Yjp///FmovdP2HXzV7atSXfghd1W22QNU0x3/GH9cf3wdgGq6E9MvBr4NfMv1eGz2qp4nz55gSOoQJP6eKNRGtR6Fxb0XQ09bT8JkJKb1J9eXuJVSfYv6yOifgQaWDSRORmK59OgSfKN9S0x3onpHYXSb0RInI7EUFhfi/c3vY+GhhUKtT/0+iO4XDTN9zX5Pr9LNHqCa7gTEBuDIjSMAVIv0zu8+H++0fafM0x02e1VTsbIYn+/8HF/u/ndR3o51OiIxJBGWRpYSJiMxHbh6AIFxgbiTewcAUM2gGhJDEtGlbheJk5FYsvOzEZEUgU1/bBJqkzwmYX73+ZzuaJAfj/yIcT+PQ2FxIQCgqVVTpEeko061OtIGk1CVb/YA1XRn2MZhiD8TL9RGtBqB7/t8X6bpDpu9qi36VDSGbxyO/CLVYrv1LOohPSIdjWo0kjgZieXyo8vwi/HDqTunAADaCm0s7r34pfflJvkpKi7C+1vexzcHvxFqver1QmxwrMZPdzTJ9ovbERwfjId5DwEANY1qIiUsBV6OXhInk4Ysmj1ANd2ZvnM6vtj9hVDrULsDkkKTSj3dYbNX9f3vdMdc3xyJoYnoWrerxMlILI/zH6N/cn9kXMgQahPcJ+DrHl9DR0tHwmQkpmW/LcPbm94WpjtNajZBekQ6nKo7SZyMxHLh/gX4xfjhwv0LAFR351rutxyDWgySOJn4ZNPs/SPmVAyGbRwmTHecqzsjo39GqaY7bPbk4fKjy/CP9cfJ2ycBqKY73/X6DmPdxkqcjMRSVFyEKVun4OsDXwu1nvV6IrZfbIl7r5K87bi4A/3i+wnTnRpGNZASlgJvR2+Jk5FYHj59iJCEEGy7uE2ofeT9EWZ0nvHCWzTKleyaPQA4dO0QAmIDcDv3NgDVdCc+JB7dnbu/cj82e/LxOP8xBiQPQPqFdKE2zm0cvun5Dac7GmTF0RUY89MYYbrjUtMF6RHpqFu9rsTJSCx/3P8DvjG+JaY7y/yWYXCLwRInI7E8K3qGCT9PwJLflgi1oEZBWBe0DsZ6xhImE48smz0AuJJ1Bf4x/jhx+wQA1XRnUc9FiHSPfOk+bPbkpai4CB9t+whf7f9KqPVw7oG44DhOdzTIrku70De+Lx48fQAAsDS0REpYCnxq+0icjMTy8OlDhCaGYuvfW4Xah14fYmaXmRo13dFkSqUS32V+h3d+fQfFymIAQCubVkiLSIO9mb3E6dRPts0eAOQU5GBA8gCknU8TapFukVjYc+ELpzts9uRp5bGVGJMxRrilTuMajZEekQ5nC2eJk5FY/nzwJ/xi/HDu3jkAgK6WLpb6LcXQlkOlDUaieVb0DBN/mYgfjvwg1AIbBWJd0DqY6JlImIzE9MufvyAsMQzZ+dkAAFsTW2wM3wi3Wm4SJ1MvWTd7gOrCjY+2foR5++cJtW51uyE+JB7VDKqV2JbNnnztvrwbfeP64v7T+wBU053ksGS0r91e4mQklkd5jxCaEIotf28Rah94foDZXWdzuqNBFmcuxsRfJgrTnZY2LZEWngYHcweJk5FYfr/7O3yjfXHx0UUAgIGOAdYErkFok1CJk6mP7Ju9f6w+vhqj00cL051GNRohPSId9SzqCduw2ZO3vx78Bb8YP5y9dxaAarrzo++PGNZqmMTJSCyFxYWY9MskRB2OEmoBDQOwvu96Tnc0yK9//orQxFBhumNjYoON4RvhXstd4mQklru5d9E3vi/2Xtkr1KZ3nI5P238qyzswaczH2aEth2Lb4G2wNFQtw3Lu3jl4LPfArku7JE5GYnG2cMaBEQfQw7kHAOBZ8TMMTxuO9ze/j6LiIonTkRh0tHSwuPdiLO61GNoK1SK7G89vhPdKb1zJuiJxOhJLj3o9cGDEAeFCnVs5t9BhdQfEnY6TOBmJpaZxTWwdtLXEqRzTdk7DgOQBePrsqXTB/uPEiRMYNGgQvL29kZGRgYcPH2LSpEkYO3YsgoKCcPz48VI/lsY0ewDgU9sHmaMy4VLTBQDw4OkDdFvXDSuOrpA4GYnF3MAcGf0zMN59vFCbf2A++sb3xeP8xxImIzFFukdi04BNMNdXXahz4vYJuC9zx6FrhyRORmJxqemCQyMPCady5BXmITwpHNN3ToeGHPDSePo6+ljpvxLzus6DAqppXszpGHRa0wm3cm5JnA5YtGgRVq1ahZ49e2L48OEYNWoU3n//fYSGhuKnn37CihWl7100qtkDgLrV62L/8P3oWa8nANV0Z2T6SLy3+T1OdzSEjpYOvu31Lb7v/b0w3Uk7nwbvVd64/OiyxOlILN2du+PAiANwrq66UOd27m10WN0BsadjJU5GYqlhVANbBm3BsJb/nsrx+a7PEZEUUWmmO6ReCoUC73u9j5SwFBjrqpZhOXT9ENyXuePErROS5frrr79gZ2cHHR0d3LhxAw8ePMDUqVNRq1Yt3L9/HyYmJvDz8yv142nMOXv/q7C4EJN/nYxvM78Vaj3se+DXkb/ynD0NsuWvLQhJCEFWfhYAwMrYChvDN6KtfVuJk5FY7j+5j37x/bDr8r+ndHzW/jNM6ziNF25oCKVSifn752PK1ilQQvWW6F7LHalhqbA1tZU4HYnl+K3j8I/xx9XsqwAAY11jRPeLhn9Df9Gz7N+/H4aGhmjVqhWaN28OCwsL7Ny5s9yPp7F/yXS0dLCo1yL80OcHYbrz65+/AgCuPOK5O5qim3M3HBx5ULhQ507uHXRc3RHRp6IlTkZisTSyxOZBmzGi1Qih9sXuLxCRFIEnz55ImIzE8s90JzU8VZjuZF7PhPtydxy7eUzidCSWljYtkTkqEx61PAAAuc9yERgbiHn75ol+aN/T0xOtWrXCvXv3cPr0aXTs2PGNHk9jm71/jHEdg18H/lpiGZZOazth/9X90oUiUTWq0QgHRxxExzodAQD5RfkYkDwAn27/VFiegeTtn7sqzO82Xzh3J/5MPDqu7oibj29KnI7E4t/QH/uG74ODmWoZlmvZ1+C9yhup51KlDUaisTGxwY4hOxDRNAIAoIQSU7ZOwfC04cgvzBc9z44dO6BUKtnsVYQudbvg4IiDwpVZ93LvodOaTthwcoPEyUgslkaW+HXgrxjZaqRQm7FnBsISwzjd0RAKhQKTPScjLSJNWIbl8I3DcFvmxumOBmlh0wKZozKFUzmePHuCvnF9MXfvXF64oSEMdQ2xoe8GTO84XaitPr4a3dZ1w70n90TNsmPHDujr66Nt2zc7tUhjz9l7kUu3L8HJxgn4EICBqvaxz8f4otMXPHdHQyiVSiw8uBDvbXlPmOq1sW2DjeEbUcuslsTpSCwnb5+EX4yfsByLka4R1getR1DjIImTkVjyCvMwIm1EiVM6hrQYgh99f4S+jr6EyUhM8WfiMSR1CPIK8wAATtWckNE/Q1jVQ90aN24Ma2vrNzpfD+BkrwQLQwsAKLHuzsw9MxGaEIrcglyJUpGYFAoF3mn3DtLC02CqZwoA+O3mb3Bf7o7fbvwmcToSS3Pr5sgcmYl29u0A/P90J74vZu+ZzemOhjDQMcD6oPX4stOXQm3NiTXosrYL7ubelTAZiSm0SSh2D90NWxPVhToXH11EuxXt8Mufv6j9Z9+6dQvnzp1740O4QCkne0qlEo8fy28Nsvz8fOTn/3sM/vHjx3BxccGVK1ew4cIGTN02VfjD3qNeD8SHxEsVlSTw+93fEZoQiqtZqiuzDHUNkTkyE47VHCVORmLJK8zD+E3jEX/m39f+l52/xASPCRKmIrGlnE3BmIwxwnTHpaYL9g7fC20tbYmTkViuZ19HeGI4Tt4+CUA1GNg8cDPc7Ut/1xVTU9My3Z3jxIkT6NGjB7Zv3w4XlzebJJaq2fvnNmJEREREVHZSLuv2xpM9Nzc3HD58uMKDifFz/neyd/PmTbi7u+P3339HrVqq87Mu3L+A1cdWo0vdLuhSt0uF/vyq/N9Oqp+TnZ0NBwcHXL16Ve0vmn/+78kvzMeyo8tUUx738RV+vo6cnh9AmudI3dzc3DAvfh5++fMXjGw9EvUt66vt58jxNaRuYv2cVu1bIWB6AFysXBDaJFRtP4fPUeX9OcXKYizLXIYPZnyA0ytPw6GmQ6n3LetkryLplGYjhULx0l84bW1tUTpVsX4OoHpC/vlZrmaucHVyVcvPkdt/OzGfIzMzM7X/rP/+3zO161RRfo46ifn8AOI/R+r+OQEtAhDQIkDtP0euryE5/Bz9An0sCFig9p/zDz5HlfPnvOXxFj7Y/QEcajpUmRswvPEFGpGRkRWRo9L8HDHJ7b+d3J4juf13k9vzA8jvv53cniO5/XeT2/MDyO+/nRyfo4rApVf+49q1a8Lo3N7eXuo49AL/nD/KW9pVXnyOKjc+P5Ufn6PKrSo+P1x65T/09fVL/P9U+ejr62PatGl8jioxPkeVG5+fyo/PUeVWFZ8fTvb+oyp260RERESvwskeERERkYyx2SMiIiKSMR7G/Y9/1hOUci0cIiIioorEZo+IiIhIxngYl6qE2bNnw83NDaamprCyskJgYCDOnz8vdSx6idmzZ0OhUGDSpElSR6H/uH79OgYOHAhLS0sYGRmhZcuW+O2336SORQAKCwvxySefwMnJCYaGhqhbty6++OILFBcXSx1NY+3evRt+fn6ws7ODQqFAampqie8rlUp8/vnnsLOzg6GhITp27IgzZ85IE/Y12OxRlbBr1y5ERkbi4MGD2LJlCwoLC9G9e3fk5uZKHY3+x+HDh7F06VI0b95c6ij0Hw8fPoSXlxd0dXXx888/4/fff8fXX3+NatWqSR2NAMydOxdLlizB4sWLcfbsWcybNw9fffUVvvvuO6mjaazc3Fy0aNECixcvfuH3582bhwULFmDx4sU4fPgwbGxs0K1bt5feXlZKPIxLVdLdu3dhZWWFXbt2oX379lLHof+Xk5OD1q1b4/vvv8eMGTPQsmVLLFy4UOpYBODDDz/Evn37sGfPHqmj0Av4+vrC2toaK1asEGr9+vWDkZER1q1bJ2EyAlS3jU1JSUFgYCAA1VTPzs4OkyZNwpQpUwAA+fn5sLa2xty5c/HWW29JmPZ5nOxRlZSVlQUAsLCwkDgJ/VdkZCT69OmDrl27Sh2F/kdaWhpcXV0REhICKysrtGrVCsuWLZM6Fv0/b29vbNu2DRcuXAAAnDhxAnv37kXv3r0lTkYvcvHiRdy6dQvdu3cXavr6+ujQoQP2798vYbIX05E6AFFZKZVKvPvuu/D29kbTpk2ljkP/LzY2FkePHsXhw4eljkIv8Pfff+OHH37Au+++i6lTpyIzMxMTJkyAvr4+Bg8eLHU8jTdlyhRkZWWhUaNG0NbWRlFREWbOnImIiAipo9EL3Lp1CwBgbW1dom5tbY3Lly9LEemV2OxRlTNu3DicPHkSe/fulToK/b+rV69i4sSJ2Lx5MwwMDKSOQy9QXFwMV1dXzJo1CwDQqlUrnDlzBj/88AObvUogLi4O69evR3R0NJo0aYLjx49j0qRJsLOzw5AhQ6SORy/xv8u0KZXKSrl0G5s9qlLGjx+PtLQ07N69G/b29lLHof/322+/4c6dO2jTpo1QKyoqwu7du7F48WLk5+dDW1tbwoRka2sLFxeXErXGjRsjKSlJokT0X++//z4+/PBDhIeHAwCaNWuGy5cvY/bs2Wz2KiEbGxsAqgmfra2tUL9z585z077KgOfsUZWgVCoxbtw4JCcnY/v27XBycpI6Ev1Hly5dcOrUKRw/flz45+rqigEDBuD48eNs9CoBLy+v55YrunDhAmrXri1RIvqvJ0+eQEur5FuytrY2l16ppJycnGBjY4MtW7YItYKCAuzatQuenp4SJnsxTvaoSoiMjER0dDQ2btwIU1NT4XwJc3NzGBoaSpyOTE1Nnzt/0tjYGJaWljyvspJ455134OnpiVmzZiE0NBSZmZlYunQpli5dKnU0AuDn54eZM2fC0dERTZo0wbFjx7BgwQIMHz5c6mgaKycnB3/++afw9cWLF3H8+HFYWFjA0dERkyZNwqxZs1C/fn3Ur18fs2bNgpGREfr37y9h6pdQElUBAF74b9WqVVJHo5fo0KGDcuLEiVLHoP9IT09XNm3aVKmvr69s1KiRcunSpVJHov+XnZ2tnDhxotLR0VFpYGCgrFu3rvLjjz9W5ufnSx1NY+3YseOF7ztDhgxRKpVKZXFxsXLatGlKGxsbpb6+vrJ9+/bKU6dOSRv6JbjOHhEREZGM8Zw9IiIiIhljs0dEREQkY2z2iIiIiGSMzR4RERGRjLHZIyIiIpIxNntEREREMsZmj4iIiEjG2OwRERERyRibPSIiIiIZY7NHREREJGNs9oiIiIhkjM0eERERkYz9H/x1/9UAaN+YAAAAAElFTkSuQmCC\n", "text/plain": [ "Graphics object consisting of 25 graphics primitives" ] }, "execution_count": 42, "metadata": {}, "output_type": "execute_result" } ], "source": [ "k.plot(chart=EF, ambient_coords=(r,T), chart_domain=EF,\n", " fixed_coords={th: pi/2, ph: 0}, ranges={T: (0, 8), r: (0.1, 10)}, \n", " number_values=5, parameters={m: 1}, color='green')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Vector fields as sections of the tangent bundle" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tangent bundle TM over the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "TM = M.tangent_bundle()\n", "print(TM)" ] }, { "cell_type": "code", "execution_count": 44, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Vector field on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "k1 = TM.section({EF.frame(): [1, -1, 0, 0]})\n", "print(k1)" ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 45, "metadata": {}, "output_type": "execute_result" } ], "source": [ "k1 == k" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### The set of all vector fields on $M$ as a $C^\\infty(M)$-module:" ] }, { "cell_type": "code", "execution_count": 46, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Free module X(M) of vector fields on the 4-dimensional Lorentzian manifold M\n" ] }, { "data": { "text/html": [ "\$\\displaystyle \\mathfrak{X}\\left(M\\right)\$" ], "text/latex": [ "$\\displaystyle \\mathfrak{X}\\left(M\\right)$" ], "text/plain": [ "Free module X(M) of vector fields on the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 46, "metadata": {}, "output_type": "execute_result" } ], "source": [ "XM = M.vector_field_module()\n", "print(XM)\n", "XM" ] }, { "cell_type": "code", "execution_count": 47, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle C^{\\infty}\\left(M\\right)\$" ], "text/latex": [ "$\\displaystyle C^{\\infty}\\left(M\\right)$" ], "text/plain": [ "Algebra of differentiable scalar fields on the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 47, "metadata": {}, "output_type": "execute_result" } ], "source": [ "XM.base_ring()" ] }, { "cell_type": "code", "execution_count": 48, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Algebra of differentiable scalar fields on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "print(XM.base_ring())" ] }, { "cell_type": "code", "execution_count": 49, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} & M & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & 2 \\\\ & \\left(T, r, {\\theta}, {\\phi}\\right) & \\longmapsto & 2 \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} & M & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & 2 \\\\ & \\left(T, r, {\\theta}, {\\phi}\\right) & \\longmapsto & 2 \\end{array}$" ], "text/plain": [ "M → ℝ\n", "(t, r, th, ph) ↦ 2\n", "(T, r, th, ph) ↦ 2" ] }, "execution_count": 49, "metadata": {}, "output_type": "execute_result" } ], "source": [ "XM.base_ring().an_element().display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Vectors at a point" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The value of a vector field, and more generally of any tensor field, at a given point is obtained by the method `at`: " ] }, { "cell_type": "code", "execution_count": 50, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tangent vector k at Point p on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "kp = k.at(p)\n", "print(kp)" ] }, { "cell_type": "code", "execution_count": 51, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle k = \\frac{4}{3} \\frac{\\partial}{\\partial t } -\\frac{\\partial}{\\partial r }\$" ], "text/latex": [ "$\\displaystyle k = \\frac{4}{3} \\frac{\\partial}{\\partial t } -\\frac{\\partial}{\\partial r }$" ], "text/plain": [ "k = 4/3 ∂/∂t - ∂/∂r" ] }, "execution_count": 51, "metadata": {}, "output_type": "execute_result" } ], "source": [ "kp.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The parent of `kp` is the tangent space at `p`:" ] }, { "cell_type": "code", "execution_count": 52, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle T_{p}\\,M\$" ], "text/latex": [ "$\\displaystyle T_{p}\\,M$" ], "text/plain": [ "Tangent space at Point p on the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 52, "metadata": {}, "output_type": "execute_result" } ], "source": [ "kp.parent()" ] }, { "cell_type": "code", "execution_count": 53, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tangent space at Point p on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "print(kp.parent())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It is accessible from the manifold via the method `tangent_space`:" ] }, { "cell_type": "code", "execution_count": 54, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 54, "metadata": {}, "output_type": "execute_result" } ], "source": [ "kp.parent() is M.tangent_space(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$T_p M$ is the fiber over $p$ in the tangent bundle $TM$:" ] }, { "cell_type": "code", "execution_count": 55, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 55, "metadata": {}, "output_type": "execute_result" } ], "source": [ "kp.parent() is TM.fiber(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Tangent vectors have a method `plot`:" ] }, { "cell_type": "code", "execution_count": 56, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 34 graphics primitives" ] }, "execution_count": 56, "metadata": {}, "output_type": "execute_result" } ], "source": [ "plot1 += kp.plot(EF, color='green', ambient_coords=(r,T), \n", " parameters={m: 1}, scale=2, label_offset=0.5, \n", " fontsize=16)\n", "plot1" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Vector field defined on an open subset" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us introduce the exterior $E$ of the black hole as an open subset of $M$:" ] }, { "cell_type": "code", "execution_count": 57, "metadata": {}, "outputs": [], "source": [ "E = M.open_subset('E', coord_def = {SD: r>2*m})" ] }, { "cell_type": "code", "execution_count": 58, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle t :\\ \\left( -\\infty, +\\infty \\right) ;\\quad r :\\ \\left( 2 \\, m , +\\infty \\right) ;\\quad {\\theta} :\\ \\left( 0 , \\pi \\right) ;\\quad {\\phi} :\\ \\left[ 0 , 2 \\, \\pi \\right] \\mbox{(periodic)}\$" ], "text/latex": [ "$\\displaystyle t :\\ \\left( -\\infty, +\\infty \\right) ;\\quad r :\\ \\left( 2 \\, m , +\\infty \\right) ;\\quad {\\theta} :\\ \\left( 0 , \\pi \\right) ;\\quad {\\phi} :\\ \\left[ 0 , 2 \\, \\pi \\right] \\mbox{(periodic)}$" ], "text/plain": [ "t: (-oo, +oo); r: (2*m, +oo); th: (0, pi); ph: [0, 2*pi] (periodic)" ] }, "execution_count": 58, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SD.restrict(E).coord_range()" ] }, { "cell_type": "code", "execution_count": 59, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 59, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p in E" ] }, { "cell_type": "code", "execution_count": 60, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle u = \\left( \\frac{1}{\\sqrt{-\\frac{2 \\, m}{r} + 1}} \\right) \\frac{\\partial}{\\partial t }\$" ], "text/latex": [ "$\\displaystyle u = \\left( \\frac{1}{\\sqrt{-\\frac{2 \\, m}{r} + 1}} \\right) \\frac{\\partial}{\\partial t }$" ], "text/plain": [ "u = 1/sqrt(-2*m/r + 1) ∂/∂t" ] }, "execution_count": 60, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u = E.vector_field(name='u')\n", "u[0] = 1/sqrt(1-2*m/r)\n", "u.display()" ] }, { "cell_type": "code", "execution_count": 61, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(E,(t, r, {\\theta}, {\\phi})\\right) \\rightarrow \\left(E,(T, r, {\\theta}, {\\phi})\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(E,(t, r, {\\theta}, {\\phi})\\right) \\rightarrow \\left(E,(T, r, {\\theta}, {\\phi})\\right)$" ], "text/plain": [ "Change of coordinates from Chart (E, (t, r, th, ph)) to Chart (E, (T, r, th, ph))" ] }, "execution_count": 61, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SD_to_EF.restrict(E)" ] }, { "cell_type": "code", "execution_count": 62, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left[\\left(E,(t, r, {\\theta}, {\\phi})\\right), \\left(E,(T, r, {\\theta}, {\\phi})\\right)\\right]\$" ], "text/latex": [ "$\\displaystyle \\left[\\left(E,(t, r, {\\theta}, {\\phi})\\right), \\left(E,(T, r, {\\theta}, {\\phi})\\right)\\right]$" ], "text/plain": [ "[Chart (E, (t, r, th, ph)), Chart (E, (T, r, th, ph))]" ] }, "execution_count": 62, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E.atlas()" ] }, { "cell_type": "code", "execution_count": 63, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle u = \\left( \\frac{1}{\\sqrt{-\\frac{2 \\, m}{r} + 1}} \\right) \\frac{\\partial}{\\partial T }\$" ], "text/latex": [ "$\\displaystyle u = \\left( \\frac{1}{\\sqrt{-\\frac{2 \\, m}{r} + 1}} \\right) \\frac{\\partial}{\\partial T }$" ], "text/plain": [ "u = 1/sqrt(-2*m/r + 1) ∂/∂T" ] }, "execution_count": 63, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u.display(EF.frame().restrict(E))" ] }, { "cell_type": "code", "execution_count": 64, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 25 graphics primitives" ] }, "execution_count": 64, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u.plot(ambient_coords=(r, t), fixed_coords={th: pi/2, ph: 0}, \n", " ranges={t: (0, 6), r: (2.1, 6)}, number_values=5, \n", " parameters={m: 1}, scale=0.3)" ] }, { "cell_type": "code", "execution_count": 65, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle u = \\frac{2}{3} \\, \\sqrt{3} \\frac{\\partial}{\\partial t }\$" ], "text/latex": [ "$\\displaystyle u = \\frac{2}{3} \\, \\sqrt{3} \\frac{\\partial}{\\partial t }$" ], "text/plain": [ "u = 2/3*sqrt(3) ∂/∂t" ] }, "execution_count": 65, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u.at(p).display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Metric tensor\n", "\n", "We define next the **metric tensor** $g$ from its non-vanishing components in the manifold's default frame, namely the coordinate frame associated to Schwarzschild-Droste coordinate:" ] }, { "cell_type": "code", "execution_count": 66, "metadata": {}, "outputs": [], "source": [ "g = M.metric()\n", "g[0, 0] = - (1 - 2*m/r)\n", "g[1, 1] = 1/(1 - 2*m/r)\n", "g[2, 2] = r^2\n", "g[3, 3] = r^2*sin(th)^2" ] }, { "cell_type": "code", "execution_count": 67, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle g = \\left( \\frac{2 \\, m}{r} - 1 \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( -\\frac{1}{\\frac{2 \\, m}{r} - 1} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\$" ], "text/latex": [ "$\\displaystyle g = \\left( \\frac{2 \\, m}{r} - 1 \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( -\\frac{1}{\\frac{2 \\, m}{r} - 1} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$" ], "text/plain": [ "g = (2*m/r - 1) dt⊗dt - 1/(2*m/r - 1) dr⊗dr + r^2 dth⊗dth + r^2*sin(th)^2 dph⊗dph" ] }, "execution_count": 67, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display()" ] }, { "cell_type": "code", "execution_count": 68, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(M, \\left(\\mathrm{d} t,\\mathrm{d} r,\\mathrm{d} {\\theta},\\mathrm{d} {\\phi}\\right)\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(M, \\left(\\mathrm{d} t,\\mathrm{d} r,\\mathrm{d} {\\theta},\\mathrm{d} {\\phi}\\right)\\right)$" ], "text/plain": [ "Coordinate coframe (M, (dt,dr,dth,dph))" ] }, "execution_count": 68, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SD.coframe()" ] }, { "cell_type": "code", "execution_count": 69, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(\\begin{array}{rrrr}\n", "\\frac{2 \\, m}{r} - 1 & 0 & 0 & 0 \\\\\n", "0 & -\\frac{1}{\\frac{2 \\, m}{r} - 1} & 0 & 0 \\\\\n", "0 & 0 & r^{2} & 0 \\\\\n", "0 & 0 & 0 & r^{2} \\sin\\left({\\theta}\\right)^{2}\n", "\\end{array}\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(\\begin{array}{rrrr}\n", "\\frac{2 \\, m}{r} - 1 & 0 & 0 & 0 \\\\\n", "0 & -\\frac{1}{\\frac{2 \\, m}{r} - 1} & 0 & 0 \\\\\n", "0 & 0 & r^{2} & 0 \\\\\n", "0 & 0 & 0 & r^{2} \\sin\\left({\\theta}\\right)^{2}\n", "\\end{array}\\right)$" ], "text/plain": [ "[ 2*m/r - 1 0 0 0]\n", "[ 0 -1/(2*m/r - 1) 0 0]\n", "[ 0 0 r^2 0]\n", "[ 0 0 0 r^2*sin(th)^2]" ] }, "execution_count": 69, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[:]" ] }, { "cell_type": "code", "execution_count": 70, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle -\\frac{1}{\\frac{2 \\, m}{r} - 1}\$" ], "text/latex": [ "$\\displaystyle -\\frac{1}{\\frac{2 \\, m}{r} - 1}$" ], "text/plain": [ "-1/(2*m/r - 1)" ] }, "execution_count": 70, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[1,1]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$g_{rr}$ is diverging at $r=2m$: this is a singularity of the Schwarszchild-Droste coordinates." ] }, { "cell_type": "code", "execution_count": 71, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{lcl} g_{ \\, t \\, t }^{ \\phantom{\\, t}\\phantom{\\, t} } & = & \\frac{2 \\, m}{r} - 1 \\\\ g_{ \\, r \\, r }^{ \\phantom{\\, r}\\phantom{\\, r} } & = & -\\frac{1}{\\frac{2 \\, m}{r} - 1} \\\\ g_{ \\, {\\theta} \\, {\\theta} }^{ \\phantom{\\, {\\theta}}\\phantom{\\, {\\theta}} } & = & r^{2} \\\\ g_{ \\, {\\phi} \\, {\\phi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\phi}} } & = & r^{2} \\sin\\left({\\theta}\\right)^{2} \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{lcl} g_{ \\, t \\, t }^{ \\phantom{\\, t}\\phantom{\\, t} } & = & \\frac{2 \\, m}{r} - 1 \\\\ g_{ \\, r \\, r }^{ \\phantom{\\, r}\\phantom{\\, r} } & = & -\\frac{1}{\\frac{2 \\, m}{r} - 1} \\\\ g_{ \\, {\\theta} \\, {\\theta} }^{ \\phantom{\\, {\\theta}}\\phantom{\\, {\\theta}} } & = & r^{2} \\\\ g_{ \\, {\\phi} \\, {\\phi} }^{ \\phantom{\\, {\\phi}}\\phantom{\\, {\\phi}} } & = & r^{2} \\sin\\left({\\theta}\\right)^{2} \\end{array}$" ], "text/plain": [ "g_t,t = 2*m/r - 1 \n", "g_r,r = -1/(2*m/r - 1) \n", "g_th,th = r^2 \n", "g_ph,ph = r^2*sin(th)^2 " ] }, "execution_count": 71, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display_comp()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The components of $g$ with respect to the Eddington-Finkelstein frame are evaluated via the methods `comp` or `display`:" ] }, { "cell_type": "code", "execution_count": 72, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle g = \\left( \\frac{2 \\, m}{r} - 1 \\right) \\mathrm{d} T\\otimes \\mathrm{d} T + \\frac{2 \\, m}{r} \\mathrm{d} T\\otimes \\mathrm{d} r + \\frac{2 \\, m}{r} \\mathrm{d} r\\otimes \\mathrm{d} T + \\left( \\frac{2 \\, m + r}{r} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\$" ], "text/latex": [ "$\\displaystyle g = \\left( \\frac{2 \\, m}{r} - 1 \\right) \\mathrm{d} T\\otimes \\mathrm{d} T + \\frac{2 \\, m}{r} \\mathrm{d} T\\otimes \\mathrm{d} r + \\frac{2 \\, m}{r} \\mathrm{d} r\\otimes \\mathrm{d} T + \\left( \\frac{2 \\, m + r}{r} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} \\sin\\left({\\theta}\\right)^{2} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}$" ], "text/plain": [ "g = (2*m/r - 1) dT⊗dT + 2*m/r dT⊗dr + 2*m/r dr⊗dT + (2*m + r)/r dr⊗dr + r^2 dth⊗dth + r^2*sin(th)^2 dph⊗dph" ] }, "execution_count": 72, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(EF.frame())" ] }, { "cell_type": "code", "execution_count": 73, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(\\begin{array}{rrrr}\n", "\\frac{2 \\, m}{r} - 1 & \\frac{2 \\, m}{r} & 0 & 0 \\\\\n", "\\frac{2 \\, m}{r} & \\frac{2 \\, m + r}{r} & 0 & 0 \\\\\n", "0 & 0 & r^{2} & 0 \\\\\n", "0 & 0 & 0 & r^{2} \\sin\\left({\\theta}\\right)^{2}\n", "\\end{array}\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(\\begin{array}{rrrr}\n", "\\frac{2 \\, m}{r} - 1 & \\frac{2 \\, m}{r} & 0 & 0 \\\\\n", "\\frac{2 \\, m}{r} & \\frac{2 \\, m + r}{r} & 0 & 0 \\\\\n", "0 & 0 & r^{2} & 0 \\\\\n", "0 & 0 & 0 & r^{2} \\sin\\left({\\theta}\\right)^{2}\n", "\\end{array}\\right)$" ], "text/plain": [ "[ 2*m/r - 1 2*m/r 0 0]\n", "[ 2*m/r (2*m + r)/r 0 0]\n", "[ 0 0 r^2 0]\n", "[ 0 0 0 r^2*sin(th)^2]" ] }, "execution_count": 73, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[EF.frame(),:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that these components are regular at $r=2m$, contrary to the components in Schwarzschild-Droste coordinates." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The metric tensor is a twice-covariant tensor (actually a field of symmetric bilinear forms):" ] }, { "cell_type": "code", "execution_count": 74, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left(0, 2\\right)\$" ], "text/latex": [ "$\\displaystyle \\left(0, 2\\right)$" ], "text/plain": [ "(0, 2)" ] }, "execution_count": 74, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.tensor_type()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It can be thus be applied to a pair of vector fields, for instance $u$ and $k$:" ] }, { "cell_type": "code", "execution_count": 75, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle g\\left(u,k\\right)\$" ], "text/latex": [ "$\\displaystyle g\\left(u,k\\right)$" ], "text/plain": [ "Scalar field g(u,k) on the Open subset E of the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 75, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s = g(u, k)\n", "s" ] }, { "cell_type": "code", "execution_count": 76, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Scalar field g(u,k) on the Open subset E of the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "print(s)" ] }, { "cell_type": "code", "execution_count": 77, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} g\\left(u,k\\right):& E & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & -\\frac{\\sqrt{r}}{\\sqrt{-2 \\, m + r}} \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} g\\left(u,k\\right):& E & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & -\\frac{\\sqrt{r}}{\\sqrt{-2 \\, m + r}} \\end{array}$" ], "text/plain": [ "g(u,k): E → ℝ\n", " (t, r, th, ph) ↦ -sqrt(r)/sqrt(-2*m + r)" ] }, "execution_count": 77, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s.display()" ] }, { "cell_type": "code", "execution_count": 78, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle -\\frac{\\sqrt{r}}{\\sqrt{-2 \\, m + r}}\$" ], "text/latex": [ "$\\displaystyle -\\frac{\\sqrt{r}}{\\sqrt{-2 \\, m + r}}$" ], "text/plain": [ "-sqrt(r)/sqrt(-2*m + r)" ] }, "execution_count": 78, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s.expr()" ] }, { "cell_type": "code", "execution_count": 79, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} g\\left(k,k\\right):& M & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & 0 \\\\ & \\left(T, r, {\\theta}, {\\phi}\\right) & \\longmapsto & 0 \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} g\\left(k,k\\right):& M & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & 0 \\\\ & \\left(T, r, {\\theta}, {\\phi}\\right) & \\longmapsto & 0 \\end{array}$" ], "text/plain": [ "g(k,k): M → ℝ\n", " (t, r, th, ph) ↦ 0\n", " (T, r, th, ph) ↦ 0" ] }, "execution_count": 79, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g(k, k).display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$u$ is a unit timelike vector:" ] }, { "cell_type": "code", "execution_count": 80, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} g\\left(u,u\\right):& E & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & -1 \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} g\\left(u,u\\right):& E & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & -1 \\end{array}$" ], "text/plain": [ "g(u,u): E → ℝ\n", " (t, r, th, ph) ↦ -1" ] }, "execution_count": 80, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g(u, u).display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Scalar products returned by the method `dot` are actually those formed with $g$:" ] }, { "cell_type": "code", "execution_count": 81, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle {u}\\cdot{u}\$" ], "text/latex": [ "$\\displaystyle {u}\\cdot{u}$" ], "text/plain": [ "Scalar field u.u on the Open subset E of the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 81, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u.dot(u)" ] }, { "cell_type": "code", "execution_count": 82, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} {u}\\cdot{u}:& E & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & -1 \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} {u}\\cdot{u}:& E & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & -1 \\end{array}$" ], "text/plain": [ "u.u: E → ℝ\n", " (t, r, th, ph) ↦ -1" ] }, "execution_count": 82, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u.dot(u).display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Levi-Civita connection" ] }, { "cell_type": "code", "execution_count": 83, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Levi-Civita connection nabla_g associated with the Lorentzian metric g on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "nabla = g.connection()\n", "print(nabla)" ] }, { "cell_type": "code", "execution_count": 84, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\begin{array}{lcl} \\Gamma_{ \\phantom{\\, t} \\, t \\, r }^{ \\, t \\phantom{\\, t} \\phantom{\\, r} } & = & -\\frac{m}{2 \\, m r - r^{2}} \\\\ \\Gamma_{ \\phantom{\\, t} \\, r \\, t }^{ \\, t \\phantom{\\, r} \\phantom{\\, t} } & = & -\\frac{m}{2 \\, m r - r^{2}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, t \\, t }^{ \\, r \\phantom{\\, t} \\phantom{\\, t} } & = & -\\frac{2 \\, m^{2} - m r}{r^{3}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, r \\, r }^{ \\, r \\phantom{\\, r} \\phantom{\\, r} } & = & \\frac{m}{2 \\, m r - r^{2}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\theta} \\, {\\theta} }^{ \\, r \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} } & = & 2 \\, m - r \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\phi} \\, {\\phi} }^{ \\, r \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & {\\left(2 \\, m - r\\right)} \\sin\\left({\\theta}\\right)^{2} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, r \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, {\\theta} \\, r }^{ \\, {\\theta} \\phantom{\\, {\\theta}} \\phantom{\\, r} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & -\\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right) \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, r \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\phi} \\, r }^{ \\, {\\phi} \\phantom{\\, {\\phi}} \\phantom{\\, r} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\phi} \\, {\\theta} }^{ \\, {\\phi} \\phantom{\\, {\\phi}} \\phantom{\\, {\\theta}} } & = & \\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)} \\end{array}\\)" ], "text/latex": [ "$\\displaystyle \\begin{array}{lcl} \\Gamma_{ \\phantom{\\, t} \\, t \\, r }^{ \\, t \\phantom{\\, t} \\phantom{\\, r} } & = & -\\frac{m}{2 \\, m r - r^{2}} \\\\ \\Gamma_{ \\phantom{\\, t} \\, r \\, t }^{ \\, t \\phantom{\\, r} \\phantom{\\, t} } & = & -\\frac{m}{2 \\, m r - r^{2}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, t \\, t }^{ \\, r \\phantom{\\, t} \\phantom{\\, t} } & = & -\\frac{2 \\, m^{2} - m r}{r^{3}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, r \\, r }^{ \\, r \\phantom{\\, r} \\phantom{\\, r} } & = & \\frac{m}{2 \\, m r - r^{2}} \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\theta} \\, {\\theta} }^{ \\, r \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} } & = & 2 \\, m - r \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\phi} \\, {\\phi} }^{ \\, r \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & {\\left(2 \\, m - r\\right)} \\sin\\left({\\theta}\\right)^{2} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, r \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, {\\theta} \\, r }^{ \\, {\\theta} \\phantom{\\, {\\theta}} \\phantom{\\, r} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & -\\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right) \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, r \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\phi} \\, r }^{ \\, {\\phi} \\phantom{\\, {\\phi}} \\phantom{\\, r} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\phi} \\, {\\theta} }^{ \\, {\\phi} \\phantom{\\, {\\phi}} \\phantom{\\, {\\theta}} } & = & \\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)} \\end{array}$" ], "text/plain": [ "Gam^t_t,r = -m/(2*m*r - r^2) \n", "Gam^t_r,t = -m/(2*m*r - r^2) \n", "Gam^r_t,t = -(2*m^2 - m*r)/r^3 \n", "Gam^r_r,r = m/(2*m*r - r^2) \n", "Gam^r_th,th = 2*m - r \n", "Gam^r_ph,ph = (2*m - r)*sin(th)^2 \n", "Gam^th_r,th = 1/r \n", "Gam^th_th,r = 1/r \n", "Gam^th_ph,ph = -cos(th)*sin(th) \n", "Gam^ph_r,ph = 1/r \n", "Gam^ph_th,ph = cos(th)/sin(th) \n", "Gam^ph_ph,r = 1/r \n", "Gam^ph_ph,th = cos(th)/sin(th) " ] }, "execution_count": 84, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nabla.display()" ] }, { "cell_type": "code", "execution_count": 85, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\nabla_{g} k\$" ], "text/latex": [ "$\\displaystyle \\nabla_{g} k$" ], "text/plain": [ "Tensor field nabla_g(k) of type (1,1) on the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 85, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Dk = nabla(k)\n", "Dk" ] }, { "cell_type": "code", "execution_count": 86, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field nabla_g(k) of type (1,1) on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "print(Dk)" ] }, { "cell_type": "code", "execution_count": 87, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\nabla_{g} k = \\left( \\frac{m}{2 \\, m r - r^{2}} \\right) \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} t + \\left( -\\frac{m}{4 \\, m^{2} - 4 \\, m r + r^{2}} \\right) \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} r + \\frac{m}{r^{2}} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} t + \\left( -\\frac{m}{2 \\, m r - r^{2}} \\right) \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} r -\\frac{1}{r} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} {\\theta} -\\frac{1}{r} \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} {\\phi}\$" ], "text/latex": [ "$\\displaystyle \\nabla_{g} k = \\left( \\frac{m}{2 \\, m r - r^{2}} \\right) \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} t + \\left( -\\frac{m}{4 \\, m^{2} - 4 \\, m r + r^{2}} \\right) \\frac{\\partial}{\\partial t }\\otimes \\mathrm{d} r + \\frac{m}{r^{2}} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} t + \\left( -\\frac{m}{2 \\, m r - r^{2}} \\right) \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} r -\\frac{1}{r} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} {\\theta} -\\frac{1}{r} \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} {\\phi}$" ], "text/plain": [ "nabla_g(k) = m/(2*m*r - r^2) ∂/∂t⊗dt - m/(4*m^2 - 4*m*r + r^2) ∂/∂t⊗dr + m/r^2 ∂/∂r⊗dt - m/(2*m*r - r^2) ∂/∂r⊗dr - 1/r ∂/∂th⊗dth - 1/r ∂/∂ph⊗dph" ] }, "execution_count": 87, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Dk.display()" ] }, { "cell_type": "code", "execution_count": 88, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{lcl} \\nabla_{g} k_{ \\phantom{\\, t} \\, t }^{ \\, t \\phantom{\\, t} } & = & \\frac{m}{2 \\, m r - r^{2}} \\\\ \\nabla_{g} k_{ \\phantom{\\, t} \\, r }^{ \\, t \\phantom{\\, r} } & = & -\\frac{m}{4 \\, m^{2} - 4 \\, m r + r^{2}} \\\\ \\nabla_{g} k_{ \\phantom{\\, r} \\, t }^{ \\, r \\phantom{\\, t} } & = & \\frac{m}{r^{2}} \\\\ \\nabla_{g} k_{ \\phantom{\\, r} \\, r }^{ \\, r \\phantom{\\, r} } & = & -\\frac{m}{2 \\, m r - r^{2}} \\\\ \\nabla_{g} k_{ \\phantom{\\, {\\theta}} \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, {\\theta}} } & = & -\\frac{1}{r} \\\\ \\nabla_{g} k_{ \\phantom{\\, {\\phi}} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\phi}} } & = & -\\frac{1}{r} \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{lcl} \\nabla_{g} k_{ \\phantom{\\, t} \\, t }^{ \\, t \\phantom{\\, t} } & = & \\frac{m}{2 \\, m r - r^{2}} \\\\ \\nabla_{g} k_{ \\phantom{\\, t} \\, r }^{ \\, t \\phantom{\\, r} } & = & -\\frac{m}{4 \\, m^{2} - 4 \\, m r + r^{2}} \\\\ \\nabla_{g} k_{ \\phantom{\\, r} \\, t }^{ \\, r \\phantom{\\, t} } & = & \\frac{m}{r^{2}} \\\\ \\nabla_{g} k_{ \\phantom{\\, r} \\, r }^{ \\, r \\phantom{\\, r} } & = & -\\frac{m}{2 \\, m r - r^{2}} \\\\ \\nabla_{g} k_{ \\phantom{\\, {\\theta}} \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, {\\theta}} } & = & -\\frac{1}{r} \\\\ \\nabla_{g} k_{ \\phantom{\\, {\\phi}} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\phi}} } & = & -\\frac{1}{r} \\end{array}$" ], "text/plain": [ "nabla_g(k)^t_t = m/(2*m*r - r^2) \n", "nabla_g(k)^t_r = -m/(4*m^2 - 4*m*r + r^2) \n", "nabla_g(k)^r_t = m/r^2 \n", "nabla_g(k)^r_r = -m/(2*m*r - r^2) \n", "nabla_g(k)^th_th = -1/r \n", "nabla_g(k)^ph_ph = -1/r " ] }, "execution_count": 88, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Dk.display_comp()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The acceleration $\\nabla_k k$ of the vector field $k$:" ] }, { "cell_type": "code", "execution_count": 89, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Vector field on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "Ak = Dk.contract(k)\n", "print(Ak)" ] }, { "cell_type": "code", "execution_count": 90, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle 0\$" ], "text/latex": [ "$\\displaystyle 0$" ], "text/plain": [ "0" ] }, "execution_count": 90, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ak.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$k$ is thus geodesic vector. The field lines of $k$ are actually the ingoing radial null geodesics of Schwarzschild spacetime." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Instead of the method contract, one can use index notations to compute $\\nabla_k k$ as $k^b \\nabla_b k^a$:" ] }, { "cell_type": "code", "execution_count": 91, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 91, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ak == Dk['^a_b']*k['^b']" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The acceleration $\\nabla_u u$ of the vector field $u$:" ] }, { "cell_type": "code", "execution_count": 92, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\frac{m}{r^{2}} \\frac{\\partial}{\\partial r }\$" ], "text/latex": [ "$\\displaystyle \\frac{m}{r^{2}} \\frac{\\partial}{\\partial r }$" ], "text/plain": [ "m/r^2 ∂/∂r" ] }, "execution_count": 92, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Au = nabla(u).contract(u)\n", "Au.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$\\nabla g$ is identically zero, since the connection $\\nabla$ is compatible with $g$:" ] }, { "cell_type": "code", "execution_count": 93, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field nabla_g(g) of type (0,3) on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "Dg = nabla(g)\n", "print(Dg)" ] }, { "cell_type": "code", "execution_count": 94, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\nabla_{g} g = 0\$" ], "text/latex": [ "$\\displaystyle \\nabla_{g} g = 0$" ], "text/plain": [ "nabla_g(g) = 0" ] }, "execution_count": 94, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Dg.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Curvature" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The **Riemann curvature tensor** is computed as" ] }, { "cell_type": "code", "execution_count": 95, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field Riem(g) of type (1,3) on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "Riem = g.riemann()\n", "print(Riem)" ] }, { "cell_type": "code", "execution_count": 96, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\begin{array}{lcl} \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, t} \\, r \\, t \\, r }^{ \\, t \\phantom{\\, r} \\phantom{\\, t} \\phantom{\\, r} } & = & -\\frac{2 \\, m}{2 \\, m r^{2} - r^{3}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, t} \\, {\\theta} \\, t \\, {\\theta} }^{ \\, t \\phantom{\\, {\\theta}} \\phantom{\\, t} \\phantom{\\, {\\theta}} } & = & -\\frac{m}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, t} \\, {\\phi} \\, t \\, {\\phi} }^{ \\, t \\phantom{\\, {\\phi}} \\phantom{\\, t} \\phantom{\\, {\\phi}} } & = & -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, r} \\, t \\, t \\, r }^{ \\, r \\phantom{\\, t} \\phantom{\\, t} \\phantom{\\, r} } & = & -\\frac{2 \\, {\\left(2 \\, m^{2} - m r\\right)}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, r} \\, {\\theta} \\, r \\, {\\theta} }^{ \\, r \\phantom{\\, {\\theta}} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & -\\frac{m}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, r} \\, {\\phi} \\, r \\, {\\phi} }^{ \\, r \\phantom{\\, {\\phi}} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, t \\, t \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, t} \\phantom{\\, t} \\phantom{\\, {\\theta}} } & = & \\frac{2 \\, m^{2} - m r}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, r \\, r \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & -\\frac{m}{2 \\, m r^{2} - r^{3}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\theta} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\frac{2 \\, m \\sin\\left({\\theta}\\right)^{2}}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, t \\, t \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, t} \\phantom{\\, t} \\phantom{\\, {\\phi}} } & = & \\frac{2 \\, m^{2} - m r}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, r \\, r \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, r} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & -\\frac{m}{2 \\, m r^{2} - r^{3}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & -\\frac{2 \\, m}{r} \\end{array}\\)" ], "text/latex": [ "$\\displaystyle \\begin{array}{lcl} \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, t} \\, r \\, t \\, r }^{ \\, t \\phantom{\\, r} \\phantom{\\, t} \\phantom{\\, r} } & = & -\\frac{2 \\, m}{2 \\, m r^{2} - r^{3}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, t} \\, {\\theta} \\, t \\, {\\theta} }^{ \\, t \\phantom{\\, {\\theta}} \\phantom{\\, t} \\phantom{\\, {\\theta}} } & = & -\\frac{m}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, t} \\, {\\phi} \\, t \\, {\\phi} }^{ \\, t \\phantom{\\, {\\phi}} \\phantom{\\, t} \\phantom{\\, {\\phi}} } & = & -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, r} \\, t \\, t \\, r }^{ \\, r \\phantom{\\, t} \\phantom{\\, t} \\phantom{\\, r} } & = & -\\frac{2 \\, {\\left(2 \\, m^{2} - m r\\right)}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, r} \\, {\\theta} \\, r \\, {\\theta} }^{ \\, r \\phantom{\\, {\\theta}} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & -\\frac{m}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, r} \\, {\\phi} \\, r \\, {\\phi} }^{ \\, r \\phantom{\\, {\\phi}} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, t \\, t \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, t} \\phantom{\\, t} \\phantom{\\, {\\theta}} } & = & \\frac{2 \\, m^{2} - m r}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, r \\, r \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & -\\frac{m}{2 \\, m r^{2} - r^{3}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\theta} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\frac{2 \\, m \\sin\\left({\\theta}\\right)^{2}}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, t \\, t \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, t} \\phantom{\\, t} \\phantom{\\, {\\phi}} } & = & \\frac{2 \\, m^{2} - m r}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, r \\, r \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, r} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & -\\frac{m}{2 \\, m r^{2} - r^{3}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & -\\frac{2 \\, m}{r} \\end{array}$" ], "text/plain": [ "Riem(g)^t_r,t,r = -2*m/(2*m*r^2 - r^3) \n", "Riem(g)^t_th,t,th = -m/r \n", "Riem(g)^t_ph,t,ph = -m*sin(th)^2/r \n", "Riem(g)^r_t,t,r = -2*(2*m^2 - m*r)/r^4 \n", "Riem(g)^r_th,r,th = -m/r \n", "Riem(g)^r_ph,r,ph = -m*sin(th)^2/r \n", "Riem(g)^th_t,t,th = (2*m^2 - m*r)/r^4 \n", "Riem(g)^th_r,r,th = -m/(2*m*r^2 - r^3) \n", "Riem(g)^th_ph,th,ph = 2*m*sin(th)^2/r \n", "Riem(g)^ph_t,t,ph = (2*m^2 - m*r)/r^4 \n", "Riem(g)^ph_r,r,ph = -m/(2*m*r^2 - r^3) \n", "Riem(g)^ph_th,th,ph = -2*m/r " ] }, "execution_count": 96, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Riem.display_comp(only_nonredundant=True)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The component $\\mathrm{Riem}(g)^t_{\\ \\, rtr} = \\mathrm{Riem}(g)^0_{\\ \\, 101}$ is returned by " ] }, { "cell_type": "code", "execution_count": 97, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle -\\frac{2 \\, m}{2 \\, m r^{2} - r^{3}}\$" ], "text/latex": [ "$\\displaystyle -\\frac{2 \\, m}{2 \\, m r^{2} - r^{3}}$" ], "text/plain": [ "-2*m/(2*m*r^2 - r^3)" ] }, "execution_count": 97, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Riem[0,1,0,1]" ] }, { "cell_type": "code", "execution_count": 98, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\begin{array}{lcl} \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, T} \\, T \\, T \\, r }^{ \\, T \\phantom{\\, T} \\phantom{\\, T} \\phantom{\\, r} } & = & \\frac{4 \\, m^{2}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, T} \\, r \\, T \\, r }^{ \\, T \\phantom{\\, r} \\phantom{\\, T} \\phantom{\\, r} } & = & \\frac{2 \\, {\\left(2 \\, m^{2} + m r\\right)}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, T} \\, {\\theta} \\, T \\, {\\theta} }^{ \\, T \\phantom{\\, {\\theta}} \\phantom{\\, T} \\phantom{\\, {\\theta}} } & = & -\\frac{m}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, T} \\, {\\phi} \\, T \\, {\\phi} }^{ \\, T \\phantom{\\, {\\phi}} \\phantom{\\, T} \\phantom{\\, {\\phi}} } & = & -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, r} \\, T \\, T \\, r }^{ \\, r \\phantom{\\, T} \\phantom{\\, T} \\phantom{\\, r} } & = & -\\frac{2 \\, {\\left(2 \\, m^{2} - m r\\right)}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, r} \\, r \\, T \\, r }^{ \\, r \\phantom{\\, r} \\phantom{\\, T} \\phantom{\\, r} } & = & -\\frac{4 \\, m^{2}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, r} \\, {\\theta} \\, r \\, {\\theta} }^{ \\, r \\phantom{\\, {\\theta}} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & -\\frac{m}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, r} \\, {\\phi} \\, r \\, {\\phi} }^{ \\, r \\phantom{\\, {\\phi}} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, T \\, T \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, T} \\phantom{\\, T} \\phantom{\\, {\\theta}} } & = & \\frac{2 \\, m^{2} - m r}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, T \\, r \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, T} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & \\frac{2 \\, m^{2}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, r \\, T \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, T} \\phantom{\\, {\\theta}} } & = & \\frac{2 \\, m^{2}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, r \\, r \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & \\frac{2 \\, m^{2} + m r}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\theta} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\frac{2 \\, m \\sin\\left({\\theta}\\right)^{2}}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, T \\, T \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, T} \\phantom{\\, T} \\phantom{\\, {\\phi}} } & = & \\frac{2 \\, m^{2} - m r}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, T \\, r \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, T} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & \\frac{2 \\, m^{2}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, r \\, T \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, r} \\phantom{\\, T} \\phantom{\\, {\\phi}} } & = & \\frac{2 \\, m^{2}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, r \\, r \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, r} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & \\frac{2 \\, m^{2} + m r}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & -\\frac{2 \\, m}{r} \\end{array}\\)" ], "text/latex": [ "$\\displaystyle \\begin{array}{lcl} \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, T} \\, T \\, T \\, r }^{ \\, T \\phantom{\\, T} \\phantom{\\, T} \\phantom{\\, r} } & = & \\frac{4 \\, m^{2}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, T} \\, r \\, T \\, r }^{ \\, T \\phantom{\\, r} \\phantom{\\, T} \\phantom{\\, r} } & = & \\frac{2 \\, {\\left(2 \\, m^{2} + m r\\right)}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, T} \\, {\\theta} \\, T \\, {\\theta} }^{ \\, T \\phantom{\\, {\\theta}} \\phantom{\\, T} \\phantom{\\, {\\theta}} } & = & -\\frac{m}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, T} \\, {\\phi} \\, T \\, {\\phi} }^{ \\, T \\phantom{\\, {\\phi}} \\phantom{\\, T} \\phantom{\\, {\\phi}} } & = & -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, r} \\, T \\, T \\, r }^{ \\, r \\phantom{\\, T} \\phantom{\\, T} \\phantom{\\, r} } & = & -\\frac{2 \\, {\\left(2 \\, m^{2} - m r\\right)}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, r} \\, r \\, T \\, r }^{ \\, r \\phantom{\\, r} \\phantom{\\, T} \\phantom{\\, r} } & = & -\\frac{4 \\, m^{2}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, r} \\, {\\theta} \\, r \\, {\\theta} }^{ \\, r \\phantom{\\, {\\theta}} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & -\\frac{m}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, r} \\, {\\phi} \\, r \\, {\\phi} }^{ \\, r \\phantom{\\, {\\phi}} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & -\\frac{m \\sin\\left({\\theta}\\right)^{2}}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, T \\, T \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, T} \\phantom{\\, T} \\phantom{\\, {\\theta}} } & = & \\frac{2 \\, m^{2} - m r}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, T \\, r \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, T} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & \\frac{2 \\, m^{2}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, r \\, T \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, T} \\phantom{\\, {\\theta}} } & = & \\frac{2 \\, m^{2}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, r \\, r \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & \\frac{2 \\, m^{2} + m r}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\theta} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\frac{2 \\, m \\sin\\left({\\theta}\\right)^{2}}{r} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, T \\, T \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, T} \\phantom{\\, T} \\phantom{\\, {\\phi}} } & = & \\frac{2 \\, m^{2} - m r}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, T \\, r \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, T} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & \\frac{2 \\, m^{2}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, r \\, T \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, r} \\phantom{\\, T} \\phantom{\\, {\\phi}} } & = & \\frac{2 \\, m^{2}}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, r \\, r \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, r} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & \\frac{2 \\, m^{2} + m r}{r^{4}} \\\\ \\mathrm{Riem}\\left(g\\right)_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & -\\frac{2 \\, m}{r} \\end{array}$" ], "text/plain": [ "Riem(g)^T_T,T,r = 4*m^2/r^4 \n", "Riem(g)^T_r,T,r = 2*(2*m^2 + m*r)/r^4 \n", "Riem(g)^T_th,T,th = -m/r \n", "Riem(g)^T_ph,T,ph = -m*sin(th)^2/r \n", "Riem(g)^r_T,T,r = -2*(2*m^2 - m*r)/r^4 \n", "Riem(g)^r_r,T,r = -4*m^2/r^4 \n", "Riem(g)^r_th,r,th = -m/r \n", "Riem(g)^r_ph,r,ph = -m*sin(th)^2/r \n", "Riem(g)^th_T,T,th = (2*m^2 - m*r)/r^4 \n", "Riem(g)^th_T,r,th = 2*m^2/r^4 \n", "Riem(g)^th_r,T,th = 2*m^2/r^4 \n", "Riem(g)^th_r,r,th = (2*m^2 + m*r)/r^4 \n", "Riem(g)^th_ph,th,ph = 2*m*sin(th)^2/r \n", "Riem(g)^ph_T,T,ph = (2*m^2 - m*r)/r^4 \n", "Riem(g)^ph_T,r,ph = 2*m^2/r^4 \n", "Riem(g)^ph_r,T,ph = 2*m^2/r^4 \n", "Riem(g)^ph_r,r,ph = (2*m^2 + m*r)/r^4 \n", "Riem(g)^ph_th,th,ph = -2*m/r " ] }, "execution_count": 98, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Riem.display_comp(EF.frame(), only_nonredundant=True)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The **Kretschmann scalar** is the \"square\" of the Riemann tensor defined by \n", "$$K = R_{abcd} \\, R^{abcd}, \\qquad R := \\mathrm{Riem}(g)$$\n", "To compute it, we must first form the tensor fields whose components are $R_{abcd}$ and \n", "$R^{abcd}$. They are obtained by respectively lowering and raising the indices of the components $R^a_{\\ \\, bcd}$ of the Riemann tensor, via the metric $g$. These two operations are performed by the methods `down()` and `up()`. The contraction is performed by summation on repeated indices, using LaTeX notations:" ] }, { "cell_type": "code", "execution_count": 99, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Scalar field on the 4-dimensional Lorentzian manifold M\n" ] }, { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} & M & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & \\frac{48 \\, m^{2}}{r^{6}} \\\\ & \\left(T, r, {\\theta}, {\\phi}\\right) & \\longmapsto & \\frac{48 \\, m^{2}}{r^{6}} \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} & M & \\longrightarrow & \\mathbb{R} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & \\frac{48 \\, m^{2}}{r^{6}} \\\\ & \\left(T, r, {\\theta}, {\\phi}\\right) & \\longmapsto & \\frac{48 \\, m^{2}}{r^{6}} \\end{array}$" ], "text/plain": [ "M → ℝ\n", "(t, r, th, ph) ↦ 48*m^2/r^6\n", "(T, r, th, ph) ↦ 48*m^2/r^6" ] }, "execution_count": 99, "metadata": {}, "output_type": "execute_result" } ], "source": [ "K = Riem.down(g)['_{abcd}'] * Riem.up(g)['^{abcd}']\n", "print(K)\n", "K.display()" ] }, { "cell_type": "code", "execution_count": 100, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\frac{48 \\, m^{2}}{r^{6}}\$" ], "text/latex": [ "$\\displaystyle \\frac{48 \\, m^{2}}{r^{6}}$" ], "text/plain": [ "48*m^2/r^6" ] }, "execution_count": 100, "metadata": {}, "output_type": "execute_result" } ], "source": [ "K.expr()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Since $\\lim_{r\\to 0} K = +\\infty$, we may say that $r=0$ is a **curvature singularity** of Schwarzschild spacetime." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Ricci tensor" ] }, { "cell_type": "code", "execution_count": 101, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Field of symmetric bilinear forms Ric(g) on the 4-dimensional Lorentzian manifold M\n" ] } ], "source": [ "Ric = g.ricci()\n", "print(Ric)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We check that the Schwarzschild metric is a solution of the vacuum **Einstein equation**:" ] }, { "cell_type": "code", "execution_count": 102, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{Ric}\\left(g\\right) = 0\$" ], "text/latex": [ "$\\displaystyle \\mathrm{Ric}\\left(g\\right) = 0$" ], "text/plain": [ "Ric(g) = 0" ] }, "execution_count": 102, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ric.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Geodesics" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First, for graphical purposes, we introduce the **Euclidean space** $\\mathbb{E}^3$ and some map $M\\to \\mathbb{E}^3$:" ] }, { "cell_type": "code", "execution_count": 103, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\begin{array}{llcl} & M & \\longrightarrow & \\mathbb{E}^{3} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & \\left(x, y, z\\right) = \\left(r \\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right), r \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right), r \\cos\\left({\\theta}\\right)\\right) \\\\ & \\left(T, r, {\\theta}, {\\phi}\\right) & \\longmapsto & \\left(x, y, z\\right) = \\left(r \\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right), r \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right), r \\cos\\left({\\theta}\\right)\\right) \\end{array}\$" ], "text/latex": [ "$\\displaystyle \\begin{array}{llcl} & M & \\longrightarrow & \\mathbb{E}^{3} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & \\left(x, y, z\\right) = \\left(r \\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right), r \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right), r \\cos\\left({\\theta}\\right)\\right) \\\\ & \\left(T, r, {\\theta}, {\\phi}\\right) & \\longmapsto & \\left(x, y, z\\right) = \\left(r \\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right), r \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right), r \\cos\\left({\\theta}\\right)\\right) \\end{array}$" ], "text/plain": [ "M → E^3\n", " (t, r, th, ph) ↦ (x, y, z) = (r*cos(ph)*sin(th), r*sin(ph)*sin(th), r*cos(th))\n", " (T, r, th, ph) ↦ (x, y, z) = (r*cos(ph)*sin(th), r*sin(ph)*sin(th), r*cos(th))" ] }, "execution_count": 103, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E3. = EuclideanSpace()\n", "X3 = E3.cartesian_coordinates()\n", "to_E3 = M.diff_map(E3, {(SD, X3): \n", " [r*sin(th)*cos(ph), r*sin(th)*sin(ph), r*cos(th)]})\n", "to_E3.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## A timelike geodesic" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us consider the geodesic starting at point $p$ and having the following tangent vector at $p$ (note that the tangent vector $v_0$ is constructed by means of the call operator `()` acting of the parent, which is the tangent space to $M$ at $p$): " ] }, { "cell_type": "code", "execution_count": 104, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle v_0 = 1.30000000000000 \\frac{\\partial}{\\partial t } + \\frac{0.0640000000000000}{m} \\frac{\\partial}{\\partial {\\phi} }\$" ], "text/latex": [ "$\\displaystyle v_0 = 1.30000000000000 \\frac{\\partial}{\\partial t } + \\frac{0.0640000000000000}{m} \\frac{\\partial}{\\partial {\\phi} }$" ], "text/plain": [ "v_0 = 1.30000000000000 ∂/∂t + 0.0640000000000000/m ∂/∂ph" ] }, "execution_count": 104, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v0 = M.tangent_space(p)((1.3, 0, 0, 0.064/m), name='v_0')\n", "v0.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We declare the geodesic with such initial conditions, denoting by $s$ the affine parameter (proper time), with $(s_{\\rm min}, s_{\\rm max})=(0, 1500\\,m)$:" ] }, { "cell_type": "code", "execution_count": 105, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mbox{Integrated geodesic in the 4-dimensional Lorentzian manifold M}\$" ], "text/latex": [ "$\\displaystyle \\mbox{Integrated geodesic in the 4-dimensional Lorentzian manifold M}$" ], "text/plain": [ "Integrated geodesic in the 4-dimensional Lorentzian manifold M" ] }, "execution_count": 105, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s = var('s')\n", "geod = M.integrated_geodesic(g, (s, 0, 2000), v0)\n", "geod" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that the initial point `p` is not explicitely passed in the argument list of `integrated_geodesic`, because this piece of information is contained in `v0`:" ] }, { "cell_type": "code", "execution_count": 106, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\mathrm{True}\$" ], "text/latex": [ "$\\displaystyle \\mathrm{True}$" ], "text/plain": [ "True" ] }, "execution_count": 106, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p is v0.parent().base_point()" ] }, { "cell_type": "code", "execution_count": 107, "metadata": {}, "outputs": [], "source": [ "sol = geod.solve(parameters_values={m: 1}) # numerical integration\n", "interp = geod.interpolate() # interpolation of the solution for the plot" ] }, { "cell_type": "code", "execution_count": 108, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "execution_count": 108, "metadata": {}, "output_type": "execute_result" } ], "source": [ "geod.plot_integrated(chart=X3, mapping=to_E3, plot_points=1000, \n", " thickness=2, label_axes=False) \\\n", "+ p.plot(chart=X3, mapping=to_E3, size=4, parameters={m: 1}) \\\n", "+ sphere(size=2, color='grey')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A 2D view by suppressing $z$ from the ambient coordinates:" ] }, { "cell_type": "code", "execution_count": 109, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 4 graphics primitives" ] }, "execution_count": 109, "metadata": {}, "output_type": "execute_result" } ], "source": [ "bh_plot = circle((0, 0), 2, edgecolor='black', fill=True, facecolor='grey', alpha=0.5)\n", "geod.plot_integrated(chart=X3, mapping=to_E3, ambient_coords=(x,y), plot_points=1000, \n", " thickness=2) \\\n", "+ p.plot(chart=X3, mapping=to_E3, ambient_coords=(x,y), size=4, parameters={m: 1}) \\\n", "+ bh_plot" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Null geodesics" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us consider a null geodesique $\\mathscr{L}$ in the equatorial plane ($\\theta = \\pi/2$). \n", "The null vector $v$ tangent to $\\mathscr{L}$ and associated to some affine parameter $\\lambda$ is given by the following first integrals of the geodesic equation:\n", "$$\n", " v^t = \\frac{\\mathrm{d}t}{\\mathrm{d}\\lambda} = \\left(1 - \\frac{2m}{r} \\right)^{-1}\n", "$$\n", "$$\n", " v^r = \\frac{\\mathrm{d}r}{\\mathrm{d}\\lambda} = \\pm \\sqrt{1 - \\frac{b^2}{r^2} \\left(1 - \\frac{2m}{r} \\right) }\n", "$$\n", "$$\n", " v^\\theta = \\frac{\\mathrm{d}\\theta}{\\mathrm{d}\\lambda} = 0\n", "$$\n", "$$\n", " v^\\varphi = \\frac{\\mathrm{d}\\varphi}{\\mathrm{d}\\lambda} = \\frac{b}{r^2}\n", "$$\n", "where the constant $b$ is related to the conserved energy $E$ and conserved angular momentum $L$ along the geodesic by \n", "$$\n", " b = \\frac{L}{E} . \n", "$$\n", "For a geodesic arising from infinity, $b$ is the **impact parameter**." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To set up the initial vector for the computation of a null geodesic, let us define a function that takes $b$ and some initial radius $r_0$ as input and returns the initial vector $v_0$. We take advantage that SageMath is built atop Python to construct this function as a pure Python function:" ] }, { "cell_type": "code", "execution_count": 110, "metadata": {}, "outputs": [], "source": [ "def initial_vector(r0, b, phi0=0, inward=True):\n", " r\"\"\"\n", " Evaluate the initial tangent vector along a null geodesic. \n", " \n", " INPUT:\n", " \n", " - r0: radial SD coordinate of the initial point\n", " - b: impact parameter\n", " - phi0: azimuthal SD coordinate of the initial point (default: 0)\n", " - inward: determines whether the geodesic has initially v^r < 0 (default: True)\n", " \n", " \"\"\"\n", " vt0 = 1/(1 - 2*m/r0)\n", " vr0 = sqrt(1 - b^2/r0^2*(1 - 2*m/r0))\n", " if inward:\n", " vr0 = - vr0\n", " vth0 = 0\n", " vph0 = b / r0^2\n", " p0 = M((0, r0, pi/2, phi0), name='p_0') # initial point\n", " return M.tangent_space(p0)((vt0, vr0, vth0, vph0), name='v_0')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us use this function to construct the initial vector for $r_0 = 10m$ and $b=7m$:" ] }, { "cell_type": "code", "execution_count": 111, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle v_0 = \\frac{5}{4} \\frac{\\partial}{\\partial t } -\\frac{2}{5} \\, \\sqrt{\\frac{19}{5}} \\frac{\\partial}{\\partial r } + \\frac{7}{100 \\, m} \\frac{\\partial}{\\partial {\\phi} }\$" ], "text/latex": [ "$\\displaystyle v_0 = \\frac{5}{4} \\frac{\\partial}{\\partial t } -\\frac{2}{5} \\, \\sqrt{\\frac{19}{5}} \\frac{\\partial}{\\partial r } + \\frac{7}{100 \\, m} \\frac{\\partial}{\\partial {\\phi} }$" ], "text/plain": [ "v_0 = 5/4 ∂/∂t - 2/5*sqrt(19/5) ∂/∂r + 7/100/m ∂/∂ph" ] }, "execution_count": 111, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v0 = initial_vector(10*m, 7*m)\n", "v0.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us check that $v_0$ is a null vector:" ] }, { "cell_type": "code", "execution_count": 112, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle 0\$" ], "text/latex": [ "$\\displaystyle 0$" ], "text/plain": [ "0" ] }, "execution_count": 112, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p0 = v0.parent().base_point()\n", "g.at(p0)(v0, v0)" ] }, { "cell_type": "code", "execution_count": 113, "metadata": {}, "outputs": [ { "data": { "image/png": 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ygFXHV7H86HL2pO8xtttr7RkdNJopPabwQPADuNq7qlilaIgVK1bw1ltvER8fj6Oj400Dn/Tw3ZjeoOeLw1/w6o5XKawqRIOGuf3m8vbQt2npIPunOZPAJ4SwCpU1lWw4tYHlR5fzY9qP1BhqgLrbmg32H8wjIY8wsdtEmZusGcnIyKBv375s3bqV0NBQgJsGvj+y5mv4biS/Mp+//vxX4hLjAGjXoh2fjv6U8V3Hq1qXuH0S+IQQFqvWUMu2M9v45tg3rD2xlvLqcuOysLZhzOg1g2k9p9GuZTsVqxS3a926dUyYMAGtVmts0+v1aDQabGxs0Ol09ZZdiwS+G9t2ZhtPbHqCtMI0ACZ0ncC/R/1b3jPNkAQ+IYRFURSF+Kx4lictZ2Xyynq3NvNv5c+MXjOY3ms63Vt3V7FK0RjKyso4f/58vbbo6Gi6du3KCy+8QM+ePW/6HBL4bq6qpoo3d7/JP/f+k1pDLS3sW/De/e/xp75/kqmImhEJfEIIi5BakMryo8v55ug3pBamGts9nTyZ0mMKM0JmENk+Eo1Go2KVwtTklK7pHM05yuMbH2f/hf0ADA0YyqIHF9GxVUeVKxMNIeOthRDNVn5lPiuOrWBZ0jIOZh40tjvZOjG+63hm9JrB8M7DsdPaqVilEJahl3cv9kTvYUH8Al785UW2nd1Gr//04qMRHzEnfI78MWXmpIdPCNGs6Gp1bEzZyNKkpWxO3UytoRYArUbLsM7DmNFrBuO7jpcRtuKGFixYwIIFC9Dr9aSkpEgP3y1KLUgl6oco9mbsBWBU4Ci+HPulXNtnxiTwCSHMnqIo7Luwj6VHlrIyeSXFl4qNy/r49GFmyEym9pyKt6u3ekWKZklO6d4+vUHPJ/s/4ZXtr6DT63BzcON/R/0vM0NmSm+fGZLAJ4QwW2eKzrDsyDKWJS2rd+eLdi3aMTNkJjNDZ8rgC3FHJPDduRN5J5i9bjbxWfEATO4+mYUPLMTdyV3lysSVJPAJIcxK8aVivkv+jmVJy+pNiuxi58JD3R9iVsgsBvsPRmtz4+k2hGgICXyNo9ZQyz9/+yev7XyNWkMtfi39WDZhGff636t2aeL/SOATQqiuRl/DT6d/YumRpaw/tR6dvu5OCDYaG+7vdD8zQ2YyoesEXOxdVK5UWBoJfI0rPjOeGWtnkFqYigYNL939EvMHz5eBU2ZAAp8QQhWKonA4+zDLjizj22PfkleZZ1zWs01PZoXMYnqv6XIRuDApCXyNr7y6nHlb5hGbGAtARLsIlk9cTqBHoMqVWTcJfEKIJpVRksHyo8tZemQpJ/JPGNvbuLRhRq8ZzAqdRah3qFz0LZqEBD7TWX18NY9teIziS8W42rvy6ahPmRU6S97bKpHAJ4QwufLqctYcX8OypGVsP7sdhbqPHUdbR8Z3Hc+skFkM6zwMWxuZGlQ0DZmWpWlklGQw8/uZ7Dq/C4Dpvaaz8IGFMm2SCiTwCSFMwqAY2H1+N3GJcaw+vpqKmgrjsns73sus0Fk81O0h3BzdVKxSWDvp4TM9vUHPP3/7J3/b8Tf0ip6uXl1ZNXkVPdvc/NZ3ovFI4BNCNKozRWdYemQpS44s4VzxOWN7kEcQs0Jn8UjII/i38letPiGuJIGv6fyW/htTVk8hsywTJ1snPhvzGVFhUWqXZTUk8Akh7liZrozVx1cTdySO3ed3G9tbOrRkSo8pRIVFyX1shVmSwNe08irymPn9TH46/RMA0WHRfDr6U5ztnFWuzPJJ4BNC3BaDYmDXuV3EHak7ZVtZUwmABg3DOg8jKjSK8V3H42TnpHKlQlyfBL6mZ1AMvPPrO/x9598xKAZ6tunJ6smr6eLVRe3SLJoEPiHELTldeJolR5aw9MhSzpecN7YHewYTFRrFIyGP4Ofmp2KFQjScBD717Di7g2lrppFTkYOLnQtfjP2C6b2mq12WxZLAJ4S4qTJdGauOryIuMY5f0381trs5uBlP2Q5oP0BO2YpmRwKfui6WX2T6munsOLcDgLl95/LxyI+x19qrXJnlkcAnhLgmg2Jgx9kdxB2JY+2JtfVO2Q7vPJyosCjGdRknp2xFsyTTspgPvUHP67te583db6KgcJffXayavAqfFj5ql2ZRJPAJIepJK0xjSeISliYtJb0k3djexbMLUWF1p2zbt2yvYoVCNB7p4TMfm1I2MWPtDEp0Jfi4+rD64dUM9BuodlkWQwKfEIJSXSnfJX9HXGIcv2X8Zmx3c3BjWs9pRIVFEdEuQk7ZCosjgc+8pBakMmHlBJLzkrGzseNfI//FE32fkM+eRiCBTwgrpTfo2XFuB3GJdadsq2qrALDR2NSdsg2NYlzXcTjaOqpcqRCmI4HP/JRXlxPzQwyrjq8C6qZu+WzMZ/JZdIck8AlhZVIKUoynbC+UXjC2d/PqZjxl69vCV8UKhWiYd955h7Vr13Ly5EmcnJwYOHAg7733Hl26NHx6Dwl85klRFD7Y+wEvbnsRg2Kgr29f1jy8hg5uHdQurdmSwCeEFSjVlbLy2ErijsSxN2Ovsb2VYyvjKdt+vv3ktIloVkaOHMnUqVPp168ftbW1vPLKKxw9epTjx4/j4uLSoOeQwGfefj79M1PXTKWwqhAvZy9WT17Nvf73ql1WsySBTwgLdXli5MWJi1l9fHW9U7YjA0cSFRrF2C5j5TSJsBh5eXm0adOGXbt2MWjQoKuWK4pCbm4u6enplJeXU1tbS0lJCZMnT2bbtm24u7vTpk0bvL29sbW1VeEViGs5V3yOCSsnkHgxEVsbWxaMXsDjfR5Xu6xmRwKfEBbmfPF54hLjWHJkCWeLzxrbu3p1JTosmpkhM2W6A2GR0tLSCAoK4ujRo/Ts2ZPa2lpSU1PJzMwkMzOTc+fOsWPHDo4dO4atrS02NjbY2NigKApjx47FxsYGe3t7HB0dad++PX5+fvj6+hIYGIi7u7vaL8+qVdZUEvNDDCuTVwLw/yL+Hx+N+AhbGwnmDSWBTwgLUFVTxdoTa1mcuJjtZ7ejUPe2bmHfgmk9pxEdHk3/dv3llK2wWIqiMG7cOIqKili/fj2HDh1i//795ObmotVqadmyJR4eHrRq1YpWrVrh5FQ3f6ROp+Pjjz/mxRdfxNbWlsLCQgoKCigoKKC4uJjKykqcnZ0JCQmhf//+BAYGYmNjo/KrtU6KovDWr2/xtx1/A2BYp2GsnLQSdycJ4w0hgU+IZkpRFA5kHmBxwmJWJK+gVFdqXHZfwH1Eh0UzsdtEuSm5sApz587lhx9+4Pnnn+fChQtcunQJPz8/unTpgpub23XX0+l0vPvuu7z44os4ODhctby2tpazZ8+SmppqfM7IyEh69+6Ns7O8t9Sw9sRaZn4/k8qaSoI8gtgwbYPch7cBJPAJ0cxcLL/IsiPLWJy4mBP5J4ztHd06EhUWxezQ2QS4B6hYoRBN6/HHH2fNmjWMHz8eDw8PgoODCQgIaNB1eDcLfFfKy8vj1KlTXLx4EW9vbyZOnEiPHj0a62WIW5B4MZEHv32QjNIM3Bzc+G7ydwzvPFztssyaBD4hmoFqfTWbUjYRmxjLltQt6BU9AE62TjzU/SGiw6IZ7D8YG42cahLWw2AwMG3aNLZt28a4ceMYPHgw7du3v6VLF24l8F1WVVXFgQMHKC4uZsCAAYwdO7bBo4JF48kpz2HidxPZm7EXG40NHw3/iL/0/4tcunIdcrWjEGYsKSeJxQmL+fro1+RX5hvbB7QfQExYDA/3eBg3x+ufrhLCUpWUlPDQQw9x9OhRpk+fzuDBg7G3t6eiogIHBwfs7OxMtm0nJyfuvfdezp07x969e0lNTZXePhV4u3qzfdZ2ntj0BHGJcTz909Mcyz3GgjELsNfaq12e2ZEePiHMTGFVId8e/ZbFiYs5nH3Y2N7WtS2zQmYRFRZFt9bdVKxQCHVlZ2fz1Vdf8eWXX+Lj43PVtXTjxo0jLCysQc91Oz18V7qyt++BBx5g6NCh0sPUxBRF4aN9H/H8L89jUAwM6jiItQ+vxdPZU+3SzIoEPiHuQFFVEU52Tnc8l53eoOfnMz+zOHEx606uo1pfDYCdjR1ju4wlJiyGEYEjZAoCYfXOnz9PbGwsVVVV3HfffTg63t577+DBg8THx6MoCgUFBbcd+KAucBw/fpyTJ08yfPhwHnjgAQl9Kticupmpq6dSVl1GkEcQm2dsJtAjUO2yzIYEPiFuU3JuMnfF3sWErhNYPH7xbT1HakGqcc68zLJMY3uIdwgxYTHMCJmBl7NXY5UsRLN24cIFFi5ciF6vN57CvVN32sN3pdTUVJKSkiT0qehY7jHGfDOG9JJ0PJ08WTd1HXd3uFvtssyCdBcIcRvOF59n6NKhlOhK+PbYt3wy8pMGX0tXpitj1fFVLE5czJ70PcZ2d0d3ZvSaQXR4NOFtw+WXhRBXyMnJYdGiRdTU1HD//feb5Z0wgoKCANi6dSvOzs4MHTpU5YqsT882PTnw6AHGfjuWQ1mHGLp0KIvHLWZ6r+lql6Y683vHCGHm8iryuG/pfcZBFNX6alYmr7zhrX4UReHX9F9ZnLiYVcmrqKipAOpucza883BiwmJ4sMuDONjeWQ+DEJaourqapUuXUl5ezrBhw8wy7F0WFBREdXU1GzduxNvbm549e6pdktVp69qWXVG7eGTtI3x/8ntmrJ3B6cLTvDroVav+Q9p83zVCmKEyXRnDlg3jfPF549QoAF8c/uKagS+jJIMlR5YQlxjH6aLTxvYgjyCiw6KZFTqLdi3bNUntQjRXW7du5fz584wYMaJRTuOaWo8ePcjLy+P7778nICBApmxRgbOdM6sfXs0LP7/AB/s+4O87/05aURpfjv3SakfwSuATooEu1V5i7LdjOZZ7rF7YU1A4nH2Y43nH6d66O5dqL7Hu5DoWJy7m59M/G29z5mrvypQeU4gOi2ag30Cr/ktTiIY6e/Ys27dvp0ePHrRo0ULtchqsf//+bNmyhQ0bNjBlyhR5v6vARmPD+8Pfp7NHZ57a/BRLjywlvSSdNQ+vwcPJQ+3ympwEPiEaQG/QM23NNH5N/xWDYrhquQ02vP3r27R0aMm3x76l+FKxcdm9He8lOiyaSd0n4WIvf+kL0VDV1dWsXr0aR0dHunbtqnY5t8TJyYnw8HD2799Pz5495dSuip7o+wQBrQKYvGoyO8/tJHJRJJunb6azR2e1S2tSMkpXiJtQFIXHNz7Oot8XGXvrbsavpR+zQ2cTFRZldR8qQjSWzZs3s3nzZkaMGNHovXuNOS3LjezcuROtVsuzzz4r995V2dGco4z5ZgwZpRl4OXuxbso67upwl9plNRm5D5MQN/HK9lf46vevGhT2BnUYxNZHtnJ23lneuO8NCXtC3KaKigp+/fVXgoODTXIqNyIigieffJLHHnus0Z/7Sv379yc3N5fDhw/f/MHCpHp59+LAowfo49OH/Mp8hi4dyopjK9Quq8lI4BPiBj7Z/wnv7HmnQY+1wQY3RzeGdR6G1kZr4sqEsGwJCQmUlZURHBysdil3xMnJCW9vb/bt24fBcPXlIKJp+bTwYVfULsZ1GYdOr2Pamml8sPcDrOFkpwQ+Ia5j2ZFlPPPTMw1+vAEDm1M3k1OeY8KqhLB8BoOBffv20bZt29u+k4Y56dq1KxcuXCA1NVXtUgTgYu/CmofXMK//PAD++vNfefrHp9Eb9DdZs3mTwCfEHyiKwtu/vs2sdbNufV0Uvk762gRVCWE90tLSyMjIoEuXLmqX0ii8vLxwdHTkwIEDapci/o/WRssnIz/hw+EfAvC/B/+X0ctHU1FdoXJlpiOjdIX4PxdKL7AkcQn/OfSferc5uxUGxcAXv3/B/0T+j0zDIMRtOnDgAI6OjrRu3VrtUhpNcHAwR44cobCwEA8P65sSxFz9T+T/0K5FOx75/hG2ntlKy3dbMqHrBMZ1GcewzsNo69pWtdqKiop4/fXXqa2tJS0tjYcffpjp06fz17/+FUVRKCoq4pVXXqF79+4Nej4JfMKq6Wp1bEjZwKKERWw9vdU45YqNxgZPJ0+c7ZwpuVRCsa643noaNNja2KKgUGuorbcspSCF+Kx4ItpFNNXLEMJi1NbWcvz4cfz9/dUupVH5+/uTkJBAWloaERHy2WBOpvScQivHVoxcPhKDYmDNiTWsObEGgG5e3RgTNIYRgSO4u8PdONo2zSUG1dXVzJ07lw8//BBfX1/Onz9PQEAAP/zwA5988gmpqamMGTMGd3d3Pv300wY9pwQ+YZWScpKITYjl66SvKagqMLYP6jiImLCYq+bMq9ZXk1OeQ1ZZFlllWWSXZ9d9L8vmQtkFLpRcILs8m6JLRQAczjosgU+I25Cbm0tVVZXJe/eunJalKdja2uLs7ExWVlaTbE/cmhGBI3ik1yN8fbT+JTkn8k+QWpjKB/s+wF5rz6COgxgdOJrhnYfTvXV3k53J+fzzz4mOjsbX1xcAR0dHFEXB39+fgIAATp48SVBQENOmTWvwc0rgE1ajqKqIb499S2xCLIez/ztFgm8LX6JCo4gKiyLIM+ia69pr7fFz88PPze+G26jWV5NfmY+Pq0+j1i6EtcjOzqampgZ3d3eTbiciIoKIiAh0Oh3vvvuuSbd1mbu7OxkZGU2yLXHrXrz7xasCH2A8i1Otr2b72e1sP7sdw1YDbVzaMDpoNCM6j2BYp2F4Ons2Wi3u7u4MHz7c+POhQ4cAGDlyJACjRo1i1KhRt/ScEviERTMoBraf3U5sQixrT6xFp9cBYGdjx7iu44gJi2FY52HY2jTOW8Fea49vC99GeS4hrFFWVhbOzs7Y2lrerydPT0/Onj1LTU0NdnZ2apcj/qBHmx708+3HoaxD15139co7LeVW5PJ10tfEJcbh5exFznM52GgaZyzszJkz6/28Y8cOtFotd999920/p+W9o4QAzhWfIy4xjrjEOM6XnDe292rTiznhc5gRMgMvZy8VKxRCXEt6ejpubm5ql2ESnp6eHD9+nIsXL+Lnd+OzBUIdf+77Z2LWxzT48Zd7//7U50+NFvauZfv27fTp0+eOJiGXaVmExaiqqeKbo99w/9L7CfhXAK/vep3zJedxc3Djz33/TPxj8Rx54gjzBsyTsCeEGVIUhczMTIsdxeru7k5NTQ25ublqlyKu4+EeD+Ni1/B7nmvQEB0WzRtD3jBZTUVFRRw5coTBgwfXa//qq69u6Xkk8IlmTVEUDmUdYu6mufh86MOMtTPYdnYbAEMDhrJ84nKyn83mszGf0de3r0yVIoQZMxgM1NTUYG9vr3YpJqHVatFqtVRXV6tdirgOF3sXHgl5BFvNzU+AajVaRgWO4ouxXzTq75a8vDwiIiJ4/fXXAfjxxx8xGAz1Rnfn5eWxd+/eW3reBp3SVRSFsrKyW3piIUypoLKA75K/Y1nSMpJzk43t7d3a80ivR5jeazodW3UEoKaqhpqqGrVKFUI00KVLl9DpdOj1enQ6nUm2UVtbi17/3zsqXN6Oqbb3R3q9npKSEkpLS5tke+LWTQ+ezsLfFt7wMRo0hLQN4asRX1FZXnnT52zRokWDQ+GuXbuIj49n9OjRVFVVsXLlSnx9fSkvLwfq7jP9l7/8hffee69Bz2esWWnAmPTS0lKLvaZCCCGE+XBxcaF79+64uDT8tFpzkpaWxoULF9QuQzSxkpISWrZs2aDHlpWV8cwzz2Bvb095eTkvvfQSpaWlvPzyy3Ts2JHq6mqef/55QkJCbqmGBgW+a/Xw9evXj/j4+FvaWGlpKX5+fmRkZDT4hd/J9prLeneyX253m81lvdLSUvxC/Hjqy6dYc2oN2WXZxmVhbcN4JOQRJnWfhLvT1VM4NIfXd7vryXvp2mS/XFtz+Yyprq7mjTfe4Pjx4zz33HO3vL0vv/ySxx577IaP+WMPX1lZGV999RVz58695X3TkO390Q8//MDGjRs5duzYLa0HzeeYaQ7viZut99Xhr3h267NXtWvQoFQr7J2zlx7tezR4W7fSw2cqDTqlq9ForvoP12q1t/XBAdCyZctbXvd2t9dc1oPb2y93sk1zXq+iuoLVx1fz5aEvYR58eqxuJnEPdw9mhswkOiya0Lahqtep5nog76Xrkf1ybeb+GWMwGHBwcEBRFBwcHG55exqN5qbrXW+5g4PDLW+zIdv7IxsbG4s/ZprL67vRenMi5/Dyby8bp/KCujswOdo6Uvm/lfR4rcdt71O13Pa0LE8++WRj1mGy7TWX9e5Ec3mNN1tPURT2X9hPbEIsK5JXUF5dd70CBhjWaRiP93ucscFjcbBt2Aesub2+xl7vdjWX1yf7pXHXuxNNVauNjQ2urq507tz5trbXr1+/21rvdt3q9i5duoSiKEyfPv22ttdcjpnm8p640Xpujm5M7TmV5UeXG6desdHYsHLcSsa+Mva2tqe2Bp3SbSyXrwW8lXPZ1sDa98vF8ossO7KM2MRYTuafNLZ3cu/E9G7TeXPSm5RkWOe+uR5rP2auR/bLtTWn/bJkyRJOnz7Nfffd1yTbKy0t5eOPP+aZZ54x+b7JzMzk8OHDvPLKK3h6Nt5dGUyhOR0zprI3Yy93xd4F1J3KXTFpBSP9Rjbb/dKkEy87ODjw2muv3VZXvSWzxv1So69hc+pmYhNj2ZSyCb1Sd02Nk60Tk3tMJiYshns63kNNdQ3aZ7RWtW8awhqPmYaQ/XJtzWm/tG/fnoSEhCbbnlarrffdlAoKCmjRokWzmGewOR0zphLZPpJgj2BSClP4ZOQnPNzjYXQ6XbPdL03awyfEibwTxCbEsjRpKbkV/518dED7AcSExTCl5xRaOjSvv5qEEI3nxIkTLFiwgNGjR+Ps7Gzy7V2+l+6LL75o8l/i27dvJyAggOjoaJNuRzSened2klqQymN9bm1wjjmSW6sJkyvVlfJd8nfEJsSy78I+Y3sblzbMCplFdHg03Vt3V7FCIYS58PHxwd7enoKCApMGvoMHDxIfH09T9nmUlZXRrl27JtueuHOD/Qcz2H+w2mU0Cgl8wiQUReHX9F+JTYhl1fFVVNbUTUyp1WgZEzyGmLAYRgeNxk4rNxAXQvyXm5sbHh4eZGdnm/R+sxEREURERBh7+EytrKyMmpoaCXxCNRL4RKPKLM1kyZElLE5cTFphmrG9i2cX5oTPYWboTNq6tlWxQiGEOdNoNERERLBhwwZ69+6Nra1l/Jo6deoUXl5eBAcHq12KsFKW8U4SqtLV6tiQsoHYhFh+Ov0TBsUAgKu9K1N7TCUmPIYB7QeoPumkEKJ56Nu3Lz/99BPnzp0jMDBQ7XLumF6vJz09nTFjxmBnJ2c1hDok8InblpSTRGxCLF8nfU1BVYGxfVDHQcSExTCp+yRc7C3z9khCCNPx8PAgNDSU5ORkiwh8Z8+exc7Ojr59+6pdirBiNk21oQcffJAOHTrg6OiIj48PM2fOJCsrq95j0tPTGTt2LC4uLnh5efGXv/yF6urqpiqxyZ07d445c+YQEBCAk5MTnTt35rXXXrvqNWs0mqu+Pv/8c1VqLr5UzGfxn9H3i76Efh7Kvw78i4KqAnxb+PLy3S+T8lQKu6J2MTts9h2FvYbuG2s7ZgDeeustBg4ciLOzM61atbrmY8zpmGkqDdkv1ni8XIu/v/9Vx8eLL76odln19O/fn6qqKvLz89Uu5Y6lpqYSEhJi9nPvAcyfP/+qY6NtW+u7DGf37t2MHTsWX19fNBoN69atq7dcURTmz5+Pr68vTk5ODB48mOTkZHWKbaAm6+EbMmQIL7/8Mj4+PmRmZvLcc88xadIk9u7dC9R1eY8ZM4bWrVuzZ88eCgoKmD17Noqi8O9//7upymxSJ0+exGAwsHDhQgIDAzl27BiPPfYYFRUVfPDBB/Ueu3jxYkaOHGn82c3NrcnqNCgGdpzdQWxiLGtPrOVS7SUA7GzseLDLg8SExzC883BsbRrvcGrIvrHGYwbq7jc6efJkIiMjWbRo0XUfp+Yxo4ab7RdrPV6u5x//+Ee9+8C6urqqWM3VgoKCaN++PUlJSQwZMqTZXhKSlZVFVVUV/fv3V7uUBuvRowe//PKL8eemmKPQ3FRUVBAaGkp0dDQPPfTQVcv/+c9/8tFHHxEXF0dwcDBvvvkmw4YN49SpU7Ro0UKFim9OtXn41q9fz/jx49HpdNjZ2bFlyxYeeOABMjIy8PX1BWDFihVERUWRm5vb7Ga0vl3vv/8+//nPfzhz5oyxTaPR8P333zN+/PgmreV88XniEuNYnLiY8yXnje092/RkTvgcZvSaQWuX1k1Wzx/3jbUfM3FxcTz99NMUFxdftUytY8YcXG+/WPvxciV/f3+efvppnn76abVLuaETJ06wcOFCQkJC6NSpU6M+95XTshQUFJhkHr7q6mo2bdpEWFgYUVFRzSK0zp8/n3Xr1pGYmKh2KWbjj5+niqLg6+vL008/zQsvvADUzefo7e3Ne++9x5/+9CcVq72+Jjule6XCwkKWL1/OwIEDjRew7tu3j549exo/iAFGjBiBTqfj8OHDapSpipKSkmvOwv7UU0/h5eVFv379+PzzzzEYDCbZflVNFd8e/ZZhy4YR8K8A5u+az/mS87g5uPHnvn8m/rF4kp5I4ukBTzdp2IOr940cMzfWVMdMcyHHS33vvfcenp6ehIWF8dZbb5nlqe1u3bpx1113kZCQQFVVVaM+d0REBE8++WS9Xs7GdvjwYVq2bMn48eObRdi7LDU1FV9fXwICApg6dWq9DghRd03mxYsXGT58uLHNwcGBe++913jW0hw16aCNF154gU8//ZTKykoGDBjAxo0bjcsuXryIt7d3vce7u7tjb2/PxYsXm7JM1Zw+fZp///vffPjhh/Xa33jjDYYOHYqTkxPbtm3j2WefJT8/n1dffbVRtqsoCr9n/05sQizfHPuG4kvFxmVDA4YSEx7DhK4TcLJzapTt3Y5r7Rs5Zq7P1MdMcyTHy3/NmzeP3r174+7uzsGDB3nppZc4e/YsX331ldqlXWXMmDGkpKSwb9++ZnVqNysri6ysLGbNmoW7u7va5TRY//79Wbp0KcHBweTk5PDmm28ycOBAkpOTm8U1iE3h8ufFHz9PvL29OX/+/LVWMQt31MN3rYs7//h16NAh4+P/+te/kpCQwNatW9FqtcyaNaveLOfXeiMritJs3uCX3ep+gboPh5EjRzJ58mQeffTResteffVVIiMjCQsL49lnn+Uf//gH77///h3XmV+Zz7/2/4uwhWH0/bIvnx36jOJLxXRw68Br977Gmb+c4ZdZvzC91/RGC3uNvW+s+Zi5EVMdM02tsfeLpRwv13Ir++qZZ57h3nvvJSQkhEcffZTPP/+cRYsWUVBQcJOtND1nZ2cmTpxIaWkpp0+fVrucBqmurubAgQP07t272Y3MHTVqFA899BC9evXi/vvvZ9OmTQAsWbJE5crMzx8/N8z9s+SOevieeuoppk6desPH+Pv7G//t5eVlnHiyW7du+Pn5sX//fiIjI2nbti0HDhyot25RURE1NTVXpWhzd6v7JSsriyFDhhAZGckXX3xx0+cfMGAApaWl5OTk3PK+0Rv0bD29ldjEWH44+QM1hhoAHLQOTOw2kZjwGO4LuA8bjWnO9jfmvrHmY+ZW3ckxo6bG3C+WdLxcy53sqwEDBgCQlpZmlr043bp1Y9CgQWzfvh1nZ+d6p+XNTU1NDdu2bcPT07PZncq9FhcXF3r16kVqaqrapZiNy6OWL168iI+Pj7E9NzfXrD9L7ijwXQ5wt+Nyz55OpwMgMjKSt956i+zsbOMO3Lp1Kw4ODvTp0+dOymxyt7JfMjMzGTJkCH369GHx4sXY2Nw8aCUkJODo6HjdqSeuJa0wjcUJi1lyZAmZZZnG9t4+vZkTPodpPafh7mT60w6NuW+s9Zi5HbdzzJiDxtwvlnS8XMud7KuEhASAer+8zM24ceMoLy9n37593HPPPbRp00btkq6i1+vZuXMn9vb2zJkzp1mdyr0enU7HiRMnuOeee9QuxWwEBATQtm1bfv75Z8LDw4G6Xt1du3bx3nvvqVzd9TXJNXwHDx7k4MGD3H333bi7u3PmzBn+/ve/07lzZyIjIwEYPnw43bt3Z+bMmbz//vsUFhby3HPP8dhjj1ns6LmsrCwGDx5Mhw4d+OCDD8jLyzMuu/wXxIYNG7h48SKRkZE4OTmxY8cOXnnlFR5//PGbjiirqK5gzYk1xCbEsuv8LmO7h5MHj/R6hOjwaMLahpnktd2phuwbazxmoG4uucLCQtLT09Hr9cbRdIGBgbi6ut7RMdOc3Wy/WOvx8kf79u1j//79DBkyBDc3N+Lj43nmmWeMc6WaK61Wy7Rp06iurubXX38lMjLSrHr6qqur2blzJxqNhpiYmGZ7z9znnnuOsWPH0qFDB3Jzc3nzzTcpLS1l9uzZapfWpMrLy0lL++/tQc+ePUtiYiIeHh506NCBp59+mrfffpugoCCCgoJ4++23cXZ2Zvr06SpWfWNNMi3L0aNHmTdvHkeOHKGiogIfHx9GjhzJq6++Wu9NkZ6ezty5c9m+fTtOTk5Mnz6dDz74wGJ/ScXFxREdHX3NZZf/W3788Udeeukl0tLSMBgMdOrUiUcffZQnn3zymveYVBSFg5kHiU2I5dtj31JWXQaABg0jAkcQExbDg10exMHWvPdpQ/YNWN8xAxAVFXXN62l27NjB4MGDb/mYsRQ32y9gncfLH/3+++/MnTuXkydPotPp6NixI1OnTuX555/H2dlZ7fJuqqamhm+//ZaDBw8SHh5Op06dbuu0aWNOy1JeXs7u3btxcnJizpw5Zh2cb2bq1Kns3r2b/Px8WrduzYABA3jjjTfo3r272qU1qZ07dzJkyJCr2mfPnk1cXByKovD666+zcOFCioqK6N+/PwsWLKBnz54qVNswqs3DJxpXbkUuXyd9zaKERRzPO25s7+TeiZiwGGaFzsLPzU/FCoUQonHo9Xp++OEHdu/ejZubGwMGDMDJ6fYGlul0Ot59993bCnyKopCSksKxY8do3749s2bNMuvT4sK6We6f+1ag1lDLT2k/sShhERtSNlBrqAXAydaJSd0nERMew6COg0w2AEMIIdSg1WqZOHEiXbt2Ze3atWzevJnw8HACAgKabJBEeXk5e/fupaqqivvvv58RI0bg6OjYJNsW4nZI4GuGUgtSWZxYNwAjq+y/9yPu59uPOeFzmNpzKm6Oln0bLSGE6N69Ox07dmTTpk3s3buXc+fOERoaatKRxtXV1aSkpHDq1CnatWtHTEwMgYGBJtueEI1FAl8zUVFdwerjq1mUsIhf0381tns5ezEzZCbRYdH08u6lYoVCCNF0zp07xxtvvMH27du5ePEiXl5edOvWjYKCAlxdXQkKCiIgIKDR7gNbWFjIiRMnyM7OxsXFhWHDhjF8+HDp1RPNhgQ+M6YoCgcyD7Do90WsSF5BeXU5ADYaG0YGjiQmLIaxXcZir7VXuVIhhGhaJ0+exGAwsHDhQgIDAzl27BiPPvoo7dq1IyIigqSkJJKSkvDz88PHxwdPT89bus5Pr9dTVFREfn4+586do7KyEl9fXyZPnkzv3r1p0aKFCV+dEI1PBm2YoZzyHJYlLSM2IZYT+SeM7Z3dOxMTXjcAo33L9ipWKIQQ5uf999/nP//5D2fOnKGgoIBDhw5x8OBBioqKqK6uxtbWFldXV9zc3HBxcUGr1aLX6+vNClBUVERxcTGVlZVotVqcnJzo2rUr/fv3p0uXLg2aK1UIcyQ9fGai1lDLj2k/sihhERtTNtYbgDG5x2RiwuoGYDT3WduFEMJUSkpK8PDwAMDT05MRI0YwfPhwiouLjfe2/eyzz9iyZQtardZ4yzlFUTh58iROTk60a9eOHj164OPjg6+vL97e3tjZ2an8yoS4c9LDp7KUghRiE2JZcmQJF8v/ewP3/u36ExMew5QeU2QAhhBC3MTp06fp3bs3H3744VX33L6STqfj0qVLGAwGampqKCoqomvXruTn5+Ph4SF/VAuLJT18KiivLmdV8ipiE2PZk77H2N7auTUzQ2YSEx5DjzY9VKxQCCHUMX/+fF5//fUbPiY+Pp6+ffsaf87KymLkyJFMnjz5hmEPwMHBod58e5cHXdjZ2UnYExZNeviaiKIo7Luwj9iEWFYmr6w3AGNU4ChiwmN4IPgBGYAhhLBq+fn55Ofn3/Ax/v7+xqCWlZXFkCFD6N+/P3Fxcbd8jV1paSlubm6UlJRY1S32hPWRHj4TyynPYemRpcQmxnIy/6SxPdAjkJiwGGaHzca3hfncD1IIIdTk5eWFl5dXgx6bmZnJkCFD6NOnD4sXL5YBFULcgAQ+E6jR17AlbQuxCbFsTNmIXtED4GznzOTuk5kTPoe7O9wtpw+EEOI2ZWVlMXjwYDp06MAHH3xAXl6ecVnbtm1VrEwI8ySBrxGdzD/J4oTFLE1aWm8AxoD2A5gTPoeHezxMSwc5ZSCEEHdq69atpKWlkZaWRvv29aepkiuVhLiaXMN3h8qry/ku+TtiE2L5LeM3Y3tr59bMCp1FTHgM3Vt3V7FCIYQQf7RgwQIWLFiAXq8nJSVFruETFk8C321QFIW9GXuNAzAqaiqAugEYo4NGMyd8DmOCxmCnlbmbhBDCnMmgDWEt5JTuLbhYfrFuAEZCLKcKThnbgz2DiQmLYWboTBmAIYQQQgizI4HvJmr0NWxO3cyihEVsTt1cbwDGlB5TiAmP4S6/u2QAhhBCCCHMlgS+6ziZf5LYhFiWHllKTkWOsT2yfaRxAEYLB7l5thBCCCHMnwS+K5Tpyvgu+TsWJSxi34V9xvY2Lm2YHTqb6LBourXupmKFQgghhBC3zuoDn6Io/JbxG7EJsXyX/J1xAIZWo2VM8BhiwmIYHTRaBmAIIYQQotmy2sCXXZZtvANGSkGKsT3YM5g54XOYGTITnxY+KlYohBDCVK6clkUIa2BV07LU6GvYlLqJRQmL2JK6xTgAw8XOxTgAY6DfQBmAIYQQVkKmZRHWwip6+E7knagbgJG0lNyKXGP7QL+BzAmfw+Tuk2UAhhBCCCEslsUGvlJdqXEAxv4L+43t3i7edQMwwqPp6tVVxQqFEEIIIZqGRQU+RVHYk76HRQmLWHV8FZU1lUDdAIwHgh8gJjyGUYGjZACGEEIIIayK6oFv5bGVLEtaxvdTvr/tIJZVlsWSxCXEJsaSVphmbO/i2aVuAEboTNq6tm2skoUQQgghmhVVA9/m1M3MWDsDvaLnx7QfGdtlbIPXrdZXsynl/wZgpG3BoBgAcLV3NQ7AiGwfKQMwhBBCCGH1VAt8+zL2MXHlRAyKAa1Gy5e/f9mgwJecm0xsQizLkpaRV5lnbL+7w93EhMUwucdkXO1dTVm6EEKIZk6mZRHWRpVpWZJzkxkYO5Dy6nJjz5yNxobM/8m85qnXUl0pK46tIDYhlgOZB4ztbV3bMjt0NjHhMQR7BjdZ/UIIISyDTMsirEWT9/Cll6QzdOlQKqorjGHvsq+Tvua5gc8BdQMwdp/fTWxiLKuSV1FVW1VXsI1t3QCMsBhGBY3C1kb1yxCFEEIIIcxak6al/Mp87ltyH/mV+cZJjy8zKAYWHl7I1B5TWZq0lMWJi+sNwOjq1dV4BwxvV++mLFsIIYQQollrslO65dXlDFo8iKScpKvC3pVsNDb1BmBM7TGVOb3n0L9dfxmAIYQQ4io6nY7+/ftz5MgREhISCAsLa/C6ckpXWIsm6eGr1lczbsW4m4Y9qOvpu6fDPcwJn8Ok7pNwsXdpihKFEEI0U88//zy+vr4cOXJE7VKEMFsmD3wGxcDMtTPZeW7nVdfsXYuD1oHNMzbLSFshhBA3tWXLFrZu3cqaNWvYsmWL2uUIYbZsTPnkiqLwly1/YdXxVQ0KewA6vY5VyatMWZYQQggLkJOTw2OPPcayZctwdnZu0Do6nY7S0tJ6X0JYA5MGvjd3v8mC+AUoNPwyQRuNDV/+/qUJqxJCCNHcKYpCVFQUTzzxBH379m3weu+88w5ubm7GLz8/PxNWKYT5MFnge/7n5/n7zr/f8noGxcC+C/tIKUgxQVVCCCHM2fz589FoNDf8OnToEP/+978pLS3lpZdeuqXnf+mllygpKTF+ZWRkmOiVCGFeTDZKt+PHHUkvTa/XZqOxQavRYlAMNxy8odVoeW7gc7x7/7umKE0IIYSZys/PJz8//4aP8ff3Z+rUqWzYsKHe7A16vR6tVsuMGTNYsmRJg7Yno3SFtTDptCzl1eVkl2WTXZ5Ndlk2F8sv1v27PJvM0kwulF4gpyKH4kvFV63r7+bP2afPmqo0IYQQzVh6enq96++ysrIYMWIEq1evpn///rRv375BzyOBT1gLk47SdbV3JcgziCDPoBs+rlpfTU55Dtnl/xcKy7JlcmUhhBDX1aFDh3o/u7rWzezQuXPnBoc9IayJWdyXzF5rj5+bH35ucvGsEEIIIURjM4vAJ4QQQtwJf39/mujGUUI0S012azUhhBDC3CiKQllZGS1atJDbdwqLJoFPCCGEEMLCmXTiZSGEEEIIoT4JfEIIIYQQFk4CnxBCCCGEhZPAJ4QQQghh4STwCSGEEEJYOAl8QgghhBAWTgKfEEIIIYSFk8AnhBBCCGHhJPAJIYQQQli4/w+/2Zkhb9/KzgAAAABJRU5ErkJggg==\n", "text/plain": [ "Graphics object consisting of 6 graphics primitives" ] }, "execution_count": 113, "metadata": {}, "output_type": "execute_result" } ], "source": [ "geod = M.integrated_geodesic(g, (s, 0, 40), v0)\n", "sol = geod.solve(step=0.01, parameters_values={m: 1}) \n", "interp = geod.interpolate() \n", "plot2 = geod.plot_integrated(chart=X3, mapping=to_E3, ambient_coords=(x,y), \n", " plot_points=500, color='green', thickness=1.5, display_tangent=True, \n", " color_tangent='green', plot_points_tangent=4, scale=1) \n", "plot2 += bh_plot\n", "plot2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A null geodesic plunging into the black hole:" ] }, { "cell_type": "code", "execution_count": 114, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle v_0 = \\frac{5}{4} \\frac{\\partial}{\\partial t } -2 \\, \\sqrt{\\frac{1}{5}} \\frac{\\partial}{\\partial r } + \\frac{1}{20 \\, m} \\frac{\\partial}{\\partial {\\phi} }\$" ], "text/latex": [ "$\\displaystyle v_0 = \\frac{5}{4} \\frac{\\partial}{\\partial t } -2 \\, \\sqrt{\\frac{1}{5}} \\frac{\\partial}{\\partial r } + \\frac{1}{20 \\, m} \\frac{\\partial}{\\partial {\\phi} }$" ], "text/plain": [ "v_0 = 5/4 ∂/∂t - 2*sqrt(1/5) ∂/∂r + 1/20/m ∂/∂ph" ] }, "execution_count": 114, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v0 = initial_vector(10*m, 5*m)\n", "v0.display()" ] }, { "cell_type": "code", "execution_count": 115, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 10 graphics primitives" ] }, "execution_count": 115, "metadata": {}, "output_type": "execute_result" } ], "source": [ "geod = M.integrated_geodesic(g, (s, 0, 13), v0)\n", "sol = geod.solve(step=0.01, parameters_values={m: 1}) \n", "interp = geod.interpolate() \n", "plot2 += geod.plot_integrated(chart=X3, mapping=to_E3, ambient_coords=(x,y), \n", " plot_points=500, color='green', thickness=1.5, display_tangent=True, \n", " color_tangent='green', plot_points_tangent=3, scale=0.2)\n", "plot2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### The photon orbit" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The photon orbit corresponds to $r_0=3m$ with the following critical value of the impact parameter: \n", "$$\n", " b_{\\rm c} = 3\\sqrt{3}\\, m\n", "$$" ] }, { "cell_type": "code", "execution_count": 116, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle 5.19615242270663\$" ], "text/latex": [ "$\\displaystyle 5.19615242270663$" ], "text/plain": [ "5.19615242270663" ] }, "execution_count": 116, "metadata": {}, "output_type": "execute_result" } ], "source": [ "bc = 3*sqrt(3)*m\n", "n(bc/m)" ] }, { "cell_type": "code", "execution_count": 117, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle v_0 = 3 \\frac{\\partial}{\\partial t } + \\frac{\\sqrt{3}}{3 \\, m} \\frac{\\partial}{\\partial {\\phi} }\$" ], "text/latex": [ "$\\displaystyle v_0 = 3 \\frac{\\partial}{\\partial t } + \\frac{\\sqrt{3}}{3 \\, m} \\frac{\\partial}{\\partial {\\phi} }$" ], "text/plain": [ "v_0 = 3 ∂/∂t + 1/3*sqrt(3)/m ∂/∂ph" ] }, "execution_count": 117, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v0 = initial_vector(3*m, bc)\n", "v0.display()" ] }, { "cell_type": "code", "execution_count": 118, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 11 graphics primitives" ] }, "execution_count": 118, "metadata": {}, "output_type": "execute_result" } ], "source": [ "geod = M.integrated_geodesic(g, (s, 0, 100), v0)\n", "sol = geod.solve(step=0.01, parameters_values={m: 1}) \n", "interp = geod.interpolate() \n", "plot2 += geod.plot_integrated(chart=X3, mapping=to_E3, ambient_coords=(x,y), \n", " plot_points=500, color='green', thickness=1.5)\n", "plot2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A geodesic with $b$ close to $b_{\\rm c}$ wraps around the circular orbit: " ] }, { "cell_type": "code", "execution_count": 119, "metadata": {}, "outputs": [], "source": [ "v0 = initial_vector(10*m, 5.2025*m)" ] }, { "cell_type": "code", "execution_count": 120, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 16 graphics primitives" ] }, "execution_count": 120, "metadata": {}, "output_type": "execute_result" } ], "source": [ "geod = M.integrated_geodesic(g, (s, 0, 40), v0)\n", "sol = geod.solve(step=0.01, parameters_values={m: 1}) \n", "interp = geod.interpolate() \n", "plot2 += geod.plot_integrated(chart=X3, mapping=to_E3, ambient_coords=(x,y), \n", " plot_points=500, color='orange', thickness=1.5,\n", " display_tangent=True, color_tangent='orange', \n", " plot_points_tangent=4, scale=1)\n", "plot2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Using SymPy as the symbolic backend\n", "\n", "By default, the symbolic backend used in tensor calculus is SageMath's one (Pynac + Maxima), implemented via the symbolic ring `SR`. We can choose to use SymPy instead:" ] }, { "cell_type": "code", "execution_count": 121, "metadata": {}, "outputs": [], "source": [ "M.set_calculus_method('sympy')" ] }, { "cell_type": "code", "execution_count": 122, "metadata": {}, "outputs": [], "source": [ "v = 2*k" ] }, { "cell_type": "code", "execution_count": 123, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\left( \\frac{2 r}{- 2 m + r} \\right) \\frac{\\partial}{\\partial t } -2 \\frac{\\partial}{\\partial r }\$" ], "text/latex": [ "$\\displaystyle \\left( \\frac{2 r}{- 2 m + r} \\right) \\frac{\\partial}{\\partial t } -2 \\frac{\\partial}{\\partial r }$" ], "text/plain": [ "2*r/(-2*m + r) ∂/∂t - 2 ∂/∂r" ] }, "execution_count": 123, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v.display()" ] }, { "cell_type": "code", "execution_count": 124, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\frac{2 r}{- 2 m + r}\$" ], "text/latex": [ "$\\displaystyle \\frac{2 r}{- 2 m + r}$" ], "text/plain": [ "2*r/(-2*m + r)" ] }, "execution_count": 124, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v[0]" ] }, { "cell_type": "code", "execution_count": 125, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\verb|2*r/(-2*m|\\verb| |\\verb|+|\\verb| |\\verb|r)|\$" ], "text/latex": [ "$\\displaystyle \\verb|2*r/(-2*m|\\verb| |\\verb|+|\\verb| |\\verb|r)|$" ], "text/plain": [ "2*r/(-2*m + r)" ] }, "execution_count": 125, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v[0].expr()" ] }, { "cell_type": "code", "execution_count": 126, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\verb|<class|\\verb| |\\verb|'sympy.core.mul.Mul'>|\$" ], "text/latex": [ "$\\displaystyle \\verb||$" ], "text/plain": [ "" ] }, "execution_count": 126, "metadata": {}, "output_type": "execute_result" } ], "source": [ "type(v[0].expr())" ] }, { "cell_type": "code", "execution_count": 127, "metadata": {}, "outputs": [], "source": [ "M.set_calculus_method('SR')" ] }, { "cell_type": "code", "execution_count": 128, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\$\\displaystyle \\verb|<class|\\verb| |\\verb|'sage.symbolic.expression.Expression'>|\$" ], "text/latex": [ "$\\displaystyle \\verb||$" ], "text/plain": [ "" ] }, "execution_count": 128, "metadata": {}, "output_type": "execute_result" } ], "source": [ "type(v[0].expr())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Going further\n", "\n", "\n", "Visit the [SageManifolds examples](http://sagemanifolds.obspm.fr/examples.html) and take a look at the [video tutorials](https://www.youtube.com/playlist?list=PLnrOCYZpQUuJlnQbQ48zgGk-Ks1t145Yw)." ] } ], "metadata": { "kernelspec": { "display_name": "SageMath 9.8.beta4", "language": "sage", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.10" } }, "nbformat": 4, "nbformat_minor": 4 }