{ "cells": [ { "cell_type": "markdown", "metadata": { "id": "8iQQ9gr5jK8g" }, "source": [ "# Problem 9\n", "The integral $$I(\\alpha) = \\int_0^2 (2 + \\sin(10\\alpha)) \\; x^\\alpha \\sin(\\frac{\\alpha}{2-x}) dx $$ depends on the parameter $\\alpha$. What is the value of $\\alpha$ in $[0,5]$ in which $I(\\alpha)$ achieves its maximum?" ] }, { "cell_type": "markdown", "metadata": { "id": "tghHic8Dol6x" }, "source": [ "# Builtin Routines\n", "\n", "As a baseline, we use builtins from `scipy` to attempt to maximize the above integral" ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 452 }, "id": "WdKR3QK9iYil", "outputId": "08367203-98b3-4dfc-c6d4-c4c2d10a4dcc" }, "outputs": [ { "data": { "image/png": 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\n", 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\n", 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V6HGzvBI6S3OD/ZviL2LRlhSDfYfOX2/y95nu74bNiZcBAOWVtQPN0/d0x1sP+ojblXoBkz/Yh/JKPbxdtPj5z4w6X3tUn87Ye+Jqrf1X8m4PMj6ZdQOLt6bi2q1g9G4U4Oemw7Du9gju64ThPRzEtqnp+XDRaeBgY9H4jlKd3ok6hs9iz+Pl4F54Obi31OUQtTgOFCZZulFa+/JTe1zcsrCkHNPW/AG/f/6KJyPiUXErbBSXVdQKNDWpzZR4eHDtVeo7WVUFo1nD3HEu/H5YmN3+X/z/HvbFnr+NAmD81l13e0uDbZVSgZ8X3IudL98HS/O6x8t8/vQwvDOlv9Fj2YWl4uM/L+eJgabakcv5+GTfOTz6yQFE/XkFoZ8dgufrP+OBD/dj4qp9WP/7GTyyNhYpl/Nxo7SCs0I302ex5wEA/911yuhxnrwhU8czNSRLxe308lOlXsAPhy8j8tAl2FiY4eC5XNy8NRZm36kc7DqejRG9HNG/xmUcAPB10+Hb5wIRczwb3i62cLS2gM7KHN8fviy2OfbOBFipzXAqqxAeDlZQKBQY27eLeCeYUqlAd0frOmuzt65995RKqQCggFk9Y2HMVAp4OFhh1aMDsTAy2eDYut/OiI/P3rql/N5ejlj3hD8Wb03FlqR0AFUh68VNSQbPvVpYive2pwEAHly9H2qVEsE+XfDR4/511mJKpP9trE2msx5QB8JQQ7JUamS8iFShJreoDDpLc1To9fgw5jRW11hk804xx7MMLtkM726PiDlDoVYpoTZTYrKvi0H7D2cNwktfJ+Gdqf1gpa7637mXk614fOm0AVAqFJg51N3o9wvs4YDQEd2x50Q2JvV3MdoGqA43dRy79ef91IGueDfqGHJu3D4bszstW3xcPU9ONwcrWFuYYeXMgQjoYY8KvYB3o46hpLzqZ/bwYDeDsFatrFKP7SmZ+DzuPLQac1wrKkNZhR5P3eMJTT1nkgRB4PgRog6CoYZkSW/kT862mnxPEAQIQtWZkYj95/Bu1LG7PueV4N5YueskzuYU4WT2DQCArcYMS//SHzYWdf9v+qBfV9zXu3Ot8TjV7KzVWP3YYIN943ycEH1rYcr+rloE+zgh2Mep3vrqO1NTM/BsejYA41f+Xu9r+bp1Eh/PHOoBAPgu8bK4Vtc7U/th+SO++GD3KaOXSd768ajBduyZHGx8apjRO6tybpRi6uo/8NBgV/x1fJ966yIi08dQQ7JkLL9U6mufvWlp6Xk38cT/4nEupwi2GjMUGrmDCKgKIz/VuN062KcLVu46icQLVQOBVUoF9vxtFBwbMFC2rkBTl09ChuDtH1NxNqcIz4/yatBzVMq6h9/VDBO9nWzRxdbCYCxNTc+M6G50LFDNn5f1rRD3cnBvvBzcG8euFCCvuAxZhSV45ZsjtZ6771QOeizaji9Ch+HeXp0Njq3bewbpeTfx4e7TDDVEHQBDDclOdkEJ0mtcwqlm5IafZsktKsO5nBv46UgGzl8rwoR+zgj74fbg3pqBxquLDU7fOgMDAF41bmV+aYwXujtaQ6VUiJfI7uvl2KBA01T/nGp8YG9dzFR3v/xUbe0Tg7Hg62TYW6uRkp4v7l/+iC+mDzF+Gay+c2g+XbVVbQQBnazU8HPrhJXRJ/HFgQsG7Z6MOIjTSyfBTKXE9pQMXMm7ies1FuNsrsvXi3H0StV8PLycRdQ+MdSQrAiCgGHvxRg91lJnagRBwIGzuXhm4yEU1Zjkz9gtzdW2PB+EAUt+Fbd7dL49YHdif2dYqc0w1a8rfrg1cPbRYR4tUmtLqXdMzR3H/LvZ44/XxyD/Zjn8/lnV52Hd7esMNAAaNEJVoVBgdJ8uAAArC+NjaPz++SuS3hqPFzYdbvFBr89sTEBaZiGWPTSg3f18iKgKb+kmWbnzluGamnNLd6VewFfxF/DD4ctYtiMNsz45YBBo7ubOcTEONrfvNKq+fLQwuBes1Cp0c7DCGO8uTa61Ndx5NqYmZR3HdJbmOPPe/fj3I74If2hAva8//9ZlsAf9ujaoHmu18b/HisoqsTUp3Wig+fnPDNz7793YcyK79sEGqJ5t+fUfUpB/lzNAGfk3ceIuszMTUcvjmRqSlZv1BI2mzHEiCALiz+Xi+8TL4sR1NW178R4McNXh2c8TsOt43R+Wd16usKrxoVwdaro5WCPmryOhMVPBXNW+/t5ozJmaO4/NqO8MzS0T+ztj919Hwt3eqkH1WKkNz9TMGuaOrw9eAgC8/sOfRp9TvaTEUxsOIeqlEejvqmvQ9wJQK8Qs3X4M/37Er872geG7AQCxr49B106WdbYjopbVvt45qcP7Kv4C/vLRH7Wm42+o0oq6Q01jztTo9QKWbDuK7mHb8ej6A0YDzZehw+Hr1unWfDD13z0EAFNunYUIHdEdTtrb42VqnsVx0VnCzsh8MVKr/+6nlvkePTrbNDjM3dnOtZMllj/iCwBoyI/5gQ/3452fjjXoNv9KvYC9Jw0D67cJtX8fjDmeUVDnMY7KIWp5PFND7cobW1IBAKt2nWz0YFYAuFlW97iZ+j7AMvJvorRcD09HaxSWlGP972fF2VeNUZspEdjz9rT+D/p1xZcHLuDoldofYtXjZxY/0BeTfV0wrq8TlEoFPn7SH7YWZiYx6FRVz0Dhui4/tSWFQoEx3l1gZ2Xe4MHBn/5xDhn5N7FkSj841Vjh/HR2IdQqFTwcqs4afRZ7vtZt+X1qzAV0J6EJg3k4lw5Ry2CooXbJ2DIH1U5n30DsmRzMGuZR6y/2knrO1NQVavR6AX9ZE4vMgpJaxyb7umBiP2fk3CiFp4M1/rf/LApuVuCx4R4Gl11sLMzw1TPDMfCdaADA/0KGwFZjhp1HszA7qBsAoIutBhP6OYvPqfm4vat3RuF6bvduLXd+/isUgIONBb6bH4SE81Wrj//je+PLTGx5Pghzv0jE1cJS7EjNxIVrxdi+8F4s35mGrUlXkJ53E3ZW5vjj9TGwUpvhv9EnxecOdO+E5Et59dZWXnn796yh+UYvAPXkRiJqIIYaapfS84rx2uYjmD+qZ62VnIPf/w1A1UKQz9zbw+BYSXk9oaaOT5g9J7KNBpqena2x5o6J60bXM4D3zsHAw3s4GCzSaMrqn6emDQupQ1/nqtu+e3a2Qc/ONtidlmW0nbu9JQZ52MHb2RZXb82lcyyjACXllViz5/aSDteLy+Hz1k542FuhsMaSG48N80DypTyjq8BXq3kJ9OeUjDonNqz521h1doephqi5GGqoXTpwNhcHkIvNiZdxftlkFJVWoLCkAs6625cJDp3PrRVq6hsoXFnjL+gvD1zA+9Encb24rM6/pusbCGqMWY2zRvXN62KKGjqjcFvxqDGg+P8eHoBRfQwn3bM0N/7WVr3wZicrw3FLR6/kG2uOi7nF4uNdr44UzxAVltR9iausxhIdW5LS8c+p/aDV1D9BIpdcImoZDDXU7t0orcB9/96D3KIyxIWNEfcbCyMlRtZ8qrYi+iTW7D2Nsgq90cGkwX27iHcwjezdGf7d7Bpd61P3eOLYlQLc4+XY6Oe2Z8aWIKhW3+3erWVk785YPLkv+rpojf5b33l3VLXqVcw73TEL88Nr44y27+FojbM5RXC0UcPd3lK8C+pGaUWd42DuXHesuLTy7qGGqYaoRTDUULt3+MJ18W6oA2evifuNBZP6Lj9VHa/6wFEoABetBjorNcZ4d8ZzI3tCqzGH5+s/i8eb4u0H+zXtie1cfWdq6gs8rUWhUNQ6S1dTXaGm+kxfdbi5k5PWAi+O6YU3t1YNWP/1lfuQf7McAgALMxVsNFW/dHoBuFleCUEALM1VBv8GZXeEmpt3+Z0EAIHnaohaBEMNtXupNS4NKAzGHdT+ILhbqKm27YURGOBW9zwl/MvZUENW6W5PLI2EGjc7S7joquaMqWu9rA1zhsHT0QqRBy9iSDc7mKmUcKixXIWluQpKRVWo8XlrJ4CqO98+nDVIbHPnmZqiesbfVOPvG1HLYKihdq/mxGfRx28PAG3MmRpvZ1t8+cxwpGUUYkQveV0aagvt7UzN3RgLLZ4Ot5emGOJpLz7e9/fRcLBRI7eoDG52VWN1fl5wr9HXVSgUmOxruBjpT0eu4INHB4qXou48U1N8xzivs1dvYM+Jqygtb/0FVok6GoYaaveu3ri94vPPf2aIj2vOB5KZXwKtpRl2pxmf1fe7+UGwsTDDiF4NWyTS26XueUg6oqbOKCwVW405Vj82CFFHMvDL0UwAhoO3B7p3wvJHfHG9uEycxdiqjqUX7vThrEGwsVCJMxgDwCf7zuJ6cTnm3dez1gSQRWWGZ2rGvv9brTMzPFND1DIYaqjdu3bD+OzC1Wdqzl69gYn/3YcyI8twu3ayxE8vjah1u3Vdvp8fhO0pGVgwpleT65Wj+uaiaY+XnwDgAd+umNjPGV5v7ABQ+2xTvQts3sW/pg3A0/d0x7OfJ+D8tWK8tz0NALB27xm8NMbLoG3xHXMuGQswHFND1DIYaqjdqGttpmtFpUb36wUBRaUVWLPnjEGgmdjPGYsf6AvXW2vuNGamVv9udk2660nu6rtFvT3MU1OXmmeRWvKMkkqpQC8nW7wyrjcWRiYbHPs87oLB9p1naozhmRqilsFQQ+1Gud74GIOcQuNnavadysGgd6PFMQwPDXLF65O80aXGlPfUMuoLeu31TA1gGGhbY+bjqQNdkVdcjiU/HYWNhRkKSyqQf9NwDpvihgwUbvHKiDomhhpqN2pOL19TXWdqgKpBmT0crfHiGC/8ZZAr189pJY42FvgkZAhSLufhg92nDY61xzE1xrTWhIizgzwxbaArbDVmePx/8YirMe0AACz56RjMzZTo42QLX7dORl+jKetFEVFtDDUkqfAdx/Hr0Sx8/vSwOse91BV2JvRzwl8GuWFCPyeGmTYwzscJ43ycaoUaU/m3b80zSrpb896snDkQAeExtY5XL9SqrmMV8rNXi+Dn3qnV6msoRisyde34ajh1BB//dhbncorw0tdJKDcy0LcuTwR44OMnh2Bif2eT+VAlabXFGSVnnQZ7/jYKfV20WDmz9jIbxgazA8DzXx1u7dKIOgSeqaE2JwgCXv4mGQ7Wt2+vPnolH+V1DBS+07fPBWJYd/u7NySqoa3W4+ruaI0dC6vmuXGy1eCr+It4+0Ef2FmrMf/Lw9h1vPZim+l5N9uktrvhnwdk6hhqqM2dzLqBH5OvGOwrrxQMJtmri7ezLQMNNYkUY3+CvBwRVGNtqrVPDMbTnx3CvlM5bV4LUUfAy0/U5uq6xfXtbakG22qVEr41ljL478yB2PRsQKvWRvLVGnc/NZa5SonPnx6GFdP9MMijk9TlEMmO9P+XU4dRfYdHSZnxpQwOnb9usP30iO74+El/2FqYYbq/G6YNcoW9tbrV6yR5qm+ph7akUCjwsL8btjx/DxZP7ivun7p6f4PXLiMi43j5idrEgbPXsDAyCZMHdMVQz4ZNbvfC6J6w1Zgj8c1xMG+j8RAkX6p2+Dv0zL09cCrrBr5JuIQjl/Px8W9n4WZniQn9nRs8CzYR3cYzNdTqbpZV4tH1B5BVUIoNseeQd/PuY2cSFgfDVlN1m6zaTMk7nKjZvDrbSF2CUfNG9RQfr9x1En/dfAT9396Jd6OOIbuwRMLKiEwPQw21upi023d7CAJwMbfYaLuhnnb4fn4g4sLGwNGmYQtPEt3NN3MD8Oq43nhosJvUpRjV3dEaX4YOr7U/Yv85zPz4AK4WlrbZ5Hycp4ZMHc9vUqvbf8edHmv3njHYvn+AMx4f3g1DPe2hNmPOppY1vIcDhvdwkLqMeo3o5YhNzwzH2ZwiXMotRvTxLJy9WoRzOUUYunQXXhzthb9N6CN1mUTtHj9BqNXdbQ6Op+7pjnu8HBloTMT7M2pPKkfNF+TliCcCuiHs/r7Y/ddR+GZuAKzVKgDA6j2n8czGBKRlFrRqDbzIS6aOnyLU6jLz6x8XMNST886YkocGu+GlMV5SlyF7w3s4IGXJBDw61B0AsOt4Fp7/kjMPE9WHoYYabefRTJy9eqNBbdMyC3Aqu6rtaxP6QK1S4qHBruLx+3p3bpUaqXXxL/q2oVQqsPQvAzDOxwlA1VnP7xMv48fkdC6CSWQEx9RQo/x+8iqe+yIRAHB+2WSjbQRBwNrfzuDfv5ww2D87yBMvjK76C/+poO744sB5/G08xwmYIpdOllKX0GGolAos/Ut/RB/LQmmFHn/dfAQA8P3hdHz0+OAWvfWbMYlMXaPO1Kxduxa+vr7QarXQarUIDAzEjh076n3O5s2b4e3tDY1GgwEDBmD79u0Gx5csWQJvb29YW1vDzs4OwcHBiI+PN2gzZcoUeHh4QKPRwMXFBU8++SSuXDGcZp/aRsKF63dtk3jheq1A42ZnafDmO8BNh38/4ocuWk2L10it7xF/N8wJ8sQnIUOkLqVDsFbXDi6/n7yKpzYcRGUD10wj6ggaFWrc3NywbNkyJCYmIiEhAWPGjMHUqVNx9OhRo+1jY2Mxa9YshIaGIikpCdOmTcO0adOQmnp7OvzevXtj9erVSElJwf79++Hp6Ynx48fj6tWrYpvRo0fj22+/xYkTJ/D999/jzJkzeOSRR5rYZWoOfT1voHq9gPiz17A7LVvc19nWAuN8nLDuCf+2KI/aiLlKiSVT+omXRah1WZqrjO4/dP46QjceEmciLq/U460fU/FLakaTvg8vK5KpUwjNvDBrb2+P5cuXIzQ0tNaxmTNnoqioCFFRUeK+gIAADBw4EOvWrTP6egUFBdDpdNi1axfGjh1rtM22bdswbdo0lJaWwtzcvEF1Vr9ufn4+tFptg55DtS3bkYZ1v1Xdkn3n5aeV0SexKuaUuH1vL0d8YWT+DSJqPO83d6CkXA8AePqe7hjk0Qkvf5OMSr2ANx/wQeiI7og8eBGv/5ACwPjlYc/XfxYfGzs+dOkuXC0srfM4kRQa8/nd5IuxlZWV2Lx5M4qKihAYGGi0TVxcHF599VWDfRMmTMDWrVuNti8rK8P69euh0+ng52f8ttHc3Fx89dVXCAoKqjfQlJaWorS0VNwuKGjdWyE7ikq93uj+ikq9QaDp66LFsod926osItmzVpuhpLwMAGCjMcODfl2RVVCCf/18HO9GHcP/9p1FRo07DfV6AcpmrHclCAJn8iaT0+i7n1JSUmBjYwMLCwvMmzcPW7ZsgY+Pj9G2mZmZcHIyPD3t5OSEzMxMg31RUVGwsbGBRqPBypUrER0dDUdHR4M2//jHP2BtbQ0HBwdcvHgRP/74Y711hoeHQ6fTiV/u7u6N7SoZUWHk8pNeLyD2zDVxe8HYXvj5pRFw5WBSohZjqb59Ccr21vi0x4d3E8eqZdwxdUJOUSmagzdXkSlqdKjp06cPkpOTER8fj/nz52P27Nk4duxYs4oYPXo0kpOTERsbi4kTJ2LGjBnIzs42aPPaa68hKSkJv/76K1QqFUJCQuq9pTEsLAz5+fni16VLl5pVI1WpOaameoDi4h9TEfLpQQDAeB8nvDqud7P+QiSi2qxqhBobTVWQsVSr0NvJ+JpWl6/XP+nl3TDTkClqdKhRq9Xw8vKCv78/wsPD4efnh1WrVhlt6+zsjKysLIN9WVlZcHZ2NthnbW0NLy8vBAQEICIiAmZmZoiIiDBo4+joiN69e2PcuHGIjIzE9u3bceDAgTrrtLCwEO/Sqv6i5qt5pqasQo/vEy9jU/xFcd/iycbP2hFRy6l5J+E7U/vD1sIMiyf3xaZnbo9he+ijWEz87+8oLLn7ArLVav4pwnlwyBQ1e/I9vV5vMHalpsDAQMTExBjsi46OrnMMTkNes/o4gHrbUOvQ13ijS7iQiyXbbt/5lrg4GB4OVlKURSR7nW1vL/JaM9T0d9Uh5Z8T8My9PRDk5Yjv599+f03LLMT8Lw/jywMXsDAy6a7fo2aM4Z3iZIoaNVA4LCwMkyZNgoeHBwoLC7Fp0ybs3bsXO3fuBACEhITA1dUV4eHhAICFCxdi5MiRWLFiBSZPnozIyEgkJCRg/fr1AICioiIsXboUU6ZMgYuLC3JycrBmzRqkp6dj+vTpAID4+HgcOnQII0aMgJ2dHc6cOYM333wTPXv2vGs4opZXVnH7ne7JiKpLTuYqBQ6EjYUDV9YmajVzgrrjj9PX4GZnicEednW28+9mD425UrxTav/pHOw/nVNn+7oIvABFJqhRoSY7OxshISHIyMiATqeDr68vdu7ciXHjxgEALl68CKXy9smfoKAgbNq0CYsXL8aiRYvQq1cvbN26Ff379wcAqFQqpKWlYePGjcjJyYGDgwOGDh2Kffv2oV+/fgAAKysr/PDDD3j77bdRVFQEFxcXTJw4EYsXL4aFBT9E29rN8opa+2JeHcVAQ9TKxvk4Nfg267HeTvg5pWlz1VTj1ScyRc2ep8ZUcJ6a5rlRWoEFXycZTKwHAAvGeOFVLnVA1K7k3yzH+7+ewOMB3RB9LAvLd56o1eZu89Qcf2eiwR1XRFJpk3lqqOO4XlSGv3z0B85fK651bGJ/FwkqIqL66CzN8c+pVWfEezhaw1ylwHvb0wzanM8pgqejdZ2vwctPZIq4SjfV69ejmZjxcZzRQAMAXl2M305KRO2DmUqJuff1rLWkxRtbU+p9HgcKkyliqKE67TmRjblfJOJU9g1Yq1WYOaT2BIZqM/4KEZmCPk62BtuXcm+itKKyzvYdZGQCyQw/kcioSr2A5bdW2nazs0Tk3EAM8TS846KTVcPW3SIi6c0c6g5fN524fTG3GAPe/hVPRsTjSl7tifoYacgUMdSQgaLSCizZdhTD34vBsYwC2FqY4acXR2CAmw62GsMhWK9P9JaoSiJqLHd7K7w/w3BNvbJKPfadykHQst34dP85g2OC8WXeiNo1DhQmUaVewN82H8GO1Kq1ucxVCvzfI76ws1YDgMHidhvmDMVo7y6S1ElETWOprvst/18/HzMYR8OBwmSKeKaGAFTd4fTYJwfEQAMAkXMDcP+A23c3acxv397JQENkelw7WWLNY4Nx/4DbS9WEBHbDfb071xoYzIHCZIoYagixp3Mw6N1oxJ/LBQCEjuiOuLAx8O9mb9AuqKcDAnrYY06QpwRVElFLmOzrgmfu7SFu93C0xurHBsHd3tKg3ZFLeW1cGVHzMdR0cJV6AW/VWL/JSq3CX8f3hovOslZbc5USkXMDsWRKv7YskYhamFun2/9/22jModWY4/WJfQ3aPPXZIcz7IhH5xQ1fEJNIagw1HdyqXSdxOvsGAGD5I76IemkErOq57k5Eps+xxrImJeVVt3XfP8AZG+YMNbj1+5ejmVj+a1qt5xO1V/z06qByi8ow9/MEJFy4DgB4d1p/TDcyDw0RyY9SeXvQv0/XqmnnFQoFRnt3wWjvLpjxcRwO3rocfTyjUJIaiZqCZ2o6oITzuRjyr2gx0CwY44UnA7pJXBURtaWYv47E/0KGGF3xe+rAruLjxAvXEfbDnyir4D3e1P4x1HQwhy9exyPr4sQ7G2YOceeClEQdUM/ONgi+Y+mEajOHuOOF0T3Fy1RfH7yE4Pd/MzpJH1F7wlW6O5CoP6/gxU1J4vb/QobU+aZGRCQIAr6Kv4jFW1MBAI42aqx7wh9DPO3v8kyiltOYz2+eqZGx1PR87E7LgiBUTapXM9AAQEBPB4kqIyJToFAo8ERANyx/xBc2FmbIuVGGR9bFYdWuU1KXRmQUBwrL2AMf7gcAdHe0xrmcInH/C6N7YlJ/F9hY8MdPRHc3fYg7HvDtivuW78HVwlL8cjQTC4N7SV0WUS08UyNT126Uio9rBhoAeGlML/R31d35FCKiOlmqVfjsqaEAgOMZBZi0ah+mr4tFTo33GiKpMdTI1J/p+XUeq7ncARFRQ3W2vT2/zfGMAhw6fx0zP45DVkGJhFUR3cZQI0NHr+Rj/peJUpdBRDLjYG1Ra9+Zq0WY8XEcZx6mdoGhRmZOZRXi0fUHUFKuRycrczx1j6fB8Xen9ZemMCIyeSqlQhyLp1AA0a/cB9dOlrhwrRiRhy5KXB0RQ42s/Hk5D+NW/o7Ckgr06mKDmFdHol/X22NnHvTrykn2iKhZ/ja+N8b7OGH/P8agl5MtQkd0BwC8H30S6ZzHhiTGUCMjU1b/IT5+bLgHHGwsYK2+PX6m5mMioqaYc093rA8ZAtdbi2I+NtwD3s62KK3Q455lu/H2j6kSV0gdGUONTHxzx6lf61uLUlrVuG3bmrdwE1EL05ir8HJwb3F7Y9wFXMotlrAi6sgYamTiH9+nGGxrbp2V4ZkaImptwX27GGw/szEBer3hZPVZBSWo1HeICexJQgw1JqyiUl/nInOWt27btlLfPjtjxTM1RNQKzFSGHyUnsgrRY9F2XC2smsMm8cJ1DH8vBgu+TjL2dKIWw1Bjoir1Aib893dMWvU7KiprBxsrtcrgvwDgotO0WX1ERJ/sO4vSiko8vDYWAPBzSobEFZHc8U93E3X5ejHOXK2aKTgjv/bEV9UT7DnrNHC0UUOtUmJCP+c2rZGIOqZxPk6IPpaF9b+fxdmrN6QuhzoQnqkxQV/FX8DI5XvF7ezC2tOUV19+0pirsPe10fj11ZGcSZiIWk0nK3MAgGsnS6x/0h/3eFUtmLvreLaUZVEHw1BjYn4/eRVvbDG8ZTLbyBTlNS872ViYcfFKImpVXz0zHGO9uyBizhAoFAq8Oq630XY3yyrbuDLqSBhqTMj+UzkI+fRgrf1HrxTU2mfJO52IqA3166pDxJyh8HbWAgD8u9lj4djaK3nnFpe1dWnUgTDUmIhfUjPxRES80WOr95yutY+XmohIagvG9sL2Bffi+/lB4r61e2u/XxG1FIYaE3A+pwjzjCxQ2d9VW+dzLBlqiEhiKqUCPl218O9mh5eDq87abE26gox8LqdArYOhph3743QOfjh8GXM21L7kBAA6S3Oj+xdP7gu1GX+0RNR+LBjTC/1dtbhRWoE3t3IpBWod/ORrp9IyC/BERDxe/fYIzl8zPuX4uVu3dAOAmVIhPn4ykItWElH7olQq8N+Zg6BUVN0R9dgnB7A1KV3qskhmGGraIb1ewNs/HoVwa0ZxPzcdwiZ512p3pcb8NG52luJjBRS12hIRSc2riw0Ge9gBAGLPXMPL3yTjehEHDlPLYahph75JuIT4c7kAgK+fDcCPL47A2Bprq4zwcsTyR3zxoF9XABD/W03JTENE7dTc+3oYbH954IJElZAcKQRB6BArjBUUFECn0yE/Px9abd0DbKV2KbcYE//7O4rKKvHG/X3xbI03gGU70uBgrRb35ReXY9fxLEzs74zJH+wTL1Odfe9+KJlsiKidSryQi/e2pyHxwnVx37fPBWJYd3sJq6L2qjGf3zxT044Ul1Xgxa+TUFRWiaGednh6RHeD469P8jYIOTorczzs7wbrOybWUzDPEFE75t/NHt/MDUCvLjbivue/qn2HJ1FjMdS0E4Ig4JmNCThyKQ/WahVWTB8IVSPOtihqJBkFUw0RtXNmKiW+CB0OW03VH2U5N8qw4Oskowv0EjUUQ007kJF/E499Eo/YM9egUAAfPjYIHg5WUpdFRNSqnHUaHH5znLi97cgVrN17BseuFKCc4YaagKGmHfjH9ymIO3sNALD8ET+M8XZq9Gt0kKFRRCQz5iol/j6xj7i9Ivok7v9gH1bv5szD1HgMNRJLungdv5+8CgB4f4YfHvF3k7giIqK29fwoL6S9OxHPj+op7lsVcwpFpRUSVkWmiKFGQh//dgZ/+SgWADDGuwseGsxAQ0Qdk8Zchb9P9MaGOUPFff3e3omCknIJqyJTw1AjkVNZhQjfkSZuL3mwn4TVEBG1D6O9u2DlTD9x++PfzkhYDZkahhqJfFjjevEnIUOaPTCYdzwRkVw84Ht7QtE1e87g64MXJayGTAlDTRvLLijBpFX7sO3IFQDAzwtGYJxP4wcGExHJlblKicWT+4rbYT+k4Iu487hZVilhVWQKGGra0LIdaRj2XgyOZxQAAGYOcUe/rjqJqyIian8CejgYbL/541GMW/kb8oq5VhTVjaGmDVy+XoyXvk7CuhrXhp21Grw7rX+LfQ/e0k1EctLfVYfv5gVi399HY2I/ZwDA5es38eaPR/l+R3VqVKhZu3YtfH19odVqodVqERgYiB07dtT7nM2bN8Pb2xsajQYDBgzA9u3bDY4vWbIE3t7esLa2hp2dHYKDgxEfHy8eP3/+PEJDQ9G9e3dYWlqiZ8+eePvtt1FWZhpp/czVGxjxf3vw063LTQE97HH0nxOw/x+joTZjpiQiqssQT3u421th3ZP+2PJ8EFRKBX46cgWLtqSgUs9gQ7U16lPVzc0Ny5YtQ2JiIhISEjBmzBhMnToVR48eNdo+NjYWs2bNQmhoKJKSkjBt2jRMmzYNqampYpvevXtj9erVSElJwf79++Hp6Ynx48fj6tWquVvS0tKg1+vx8ccf4+jRo1i5ciXWrVuHRYsWNaPbbeOjvacxdsVv4nZXnQabngmAtYUZzFQtG2g4UJiI5GyQhx3+PqFqkr6vD17CjtQMg+N5xWW4kndTitKoHWn2Kt329vZYvnw5QkNDax2bOXMmioqKEBUVJe4LCAjAwIEDsW7dOqOvV70a565duzB27FijbZYvX461a9fi7NmzDa6zrVfpvlFagf5v7xS3R/bujP9M90NnW4tW+X6j/7MX53KKAADnl01ule9BRCSl6jXyYtKy0bOzNX5ecC805ioAQI+wn6EXgKQ3x8HOWi1xpdSS2mSV7srKSkRGRqKoqAiBgYFG28TFxSE4ONhg34QJExAXF2e0fVlZGdavXw+dTgc/Pz+jbQAgPz8f9vb1L1FfWlqKgoICg6+2dORSnsH2p3OGtlqgISLqCBQKBcIfGoDOthY4c7UIr333JwRBQHmlHtVXo9IyC6UtkiTV6FCTkpICGxsbWFhYYN68ediyZQt8fHyMts3MzISTk+Htyk5OTsjMzDTYFxUVBRsbG2g0GqxcuRLR0dFwdHQ0+pqnT5/Ghx9+iOeee67eOsPDw6HT6cQvd3f3RvSy+RIvXDfYbsyK20REZFwXrQbvTKmarPSnI1fw0d4zyL95e9bhszk3sP73Mygp5+3fHVGjQ02fPn2QnJyM+Ph4zJ8/H7Nnz8axY8eaVcTo0aORnJyM2NhYTJw4ETNmzEB2dnatdunp6Zg4cSKmT5+OZ599tt7XDAsLQ35+vvh16dKlZtXYGKUVlYj680qbfT8ioo5k0gAXzAnyBAD859cTiD1zTTz2xpZUvLc9DR/uPiVRdSSlRocatVoNLy8v+Pv7Izw8HH5+fli1apXRts7OzsjKyjLYl5WVBWdnZ4N91tbW8PLyQkBAACIiImBmZoaIiAiDNleuXMHo0aMRFBSE9evX37VOCwsL8S6t6q+28o/v/sTJrBtt9v2IiDqatx7wwZBudhAEYMHXSbWO1ww61HE0+xYcvV6P0tJSo8cCAwMRExNjsC86OrrOMTh1vWZ6ejpGjRoFf39/bNiwAUpl+7wVWq8X8L99Z7E1mWdpiIhak1KpwMdP+sPOylzqUqgdMWtM47CwMEyaNAkeHh4oLCzEpk2bsHfvXuzcWXWXT0hICFxdXREeHg4AWLhwIUaOHIkVK1Zg8uTJiIyMREJCgnimpaioCEuXLsWUKVPg4uKCnJwcrFmzBunp6Zg+fTqA24GmW7du+M9//iPe6g2g1hkfqb23/Tj+t/8cAKCvi1acOZiIiFqeg40F1j7hj0fXH6h1jPPzdUyNCjXZ2dkICQlBRkYGdDodfH19sXPnTowbNw4AcPHiRYOzKEFBQdi0aRMWL16MRYsWoVevXti6dSv696+aSVelUiEtLQ0bN25ETk4OHBwcMHToUOzbtw/9+lUNBIuOjsbp06dx+vRpuLm5GdTTnmaV3HUsSww0ALBgjBfmf3VYwoqIiOQvoIcDvgwdjici4g32t59PB2pLzZ6nxlS05jw1J7MKMW3NHyguq8ScIE88fU93eDhYwfP1n8U2rT13DOepIaKOLOrPK3hx0+2xNX5uOvz44ggJK6KW0pjP70adqaHa9HoBy3eeQHFZJfy72SHsfm9YmKmkLouIqEMZ3acLlAqI89VcKzKNpXSoZbXPEbcm5H/7zyL6WNUdXm894MNAQ0QkAWsLM/z6ykgE9+0CoGrxy2s3jN/EQvLFUNNMjw/vBv9udvjvzIHwc+8kdTlERB2WVxcb/G/2UHh1sQEAHDyXK3FF1NYYaprJ2sIMm58LxLRBrrWOBfV0AAD0cLRu67KIiDqs0X06AwDe2JqKC9eKJK6G2hJDTQtQ1rEEwqpHB2HB2F744pnhbVwREVHH9WSAJ+yt1cgtKsPDa+NQUFJ+9yeRLDDUtKLOthZ4dVxvuHaylLoUIqIOw8PBClEvjYDO0hw5N0rxS0rm3Z9EssBQQ0REstO1kyXm3tcDAPCvn4/hZBZX7+4IGGpkgmuAExEZCh3RHX1dtCgoqcD4lb/jFION7DHUEBGRLGnMVXh/hp+4/dLXSSitqJSwImptDDUy0SGmhSYiaqS+LlpsunWzRlpmIT6MOS1xRdSaGGqIiEjWgrwcse6JwQCA9fvOYt+pq3d5BpkqhhoiIpK9Cf2cMaSbHcoq9Hgy4iBe//5PlFfqpS6LWhhDDRERyZ5CoUDE7KHofmsy1MhDlzBnw0GOsZEZhhqZ4N1PRET101mZI/qV+7D8EV8AwB+nr6HP4l+wMfa8tIVRi2GoISKiDsNMpcT0Ie4Im+Qt7vshKV3CiqglMdQQEVGH8+y9PfDahD4AgJTLeci/yaUU5IChRiZ4SzcRUcMplQo8P6onXHQa6AXg/35Jk7okagEMNURE1CEpFAr838NV42s2xV/EpviLEldEzcVQQ0REHdZ9vTvjyYBuAIAl244iM79E4oqoORhqZIJ3PxERNc07U/vBv5sdyir12JxwSepyqBkYaoiIqENTKBSYOcQdALAi+iR6LtqOA2evSVwVNQVDDRERdXiTBjiLjyv1AmZ9ckDCaqipGGqIiKjDs9WYY8NTQ8VtQQBS0/MlrIiagqGGiIgIwOg+XfDN3ABxO+yHFFRwfSiTwlBDRER0y/AeDjj4xlhoNWZISc/Hmj1npC6JGoGhhoiIqIYuthq8+YAPAGDlrpNc9NKEMNTIhEsnjdQlEBHJxkOD3aBWVX1ETlq1D4LAedtNAUONTCx/xA/BfZ2w6ZnhUpdCRGTyVEoFJvSvuiPq7NUifH+Yi16aAoXQQeJnQUEBdDod8vPzodVqpS6HiIjaufJKPf7+3Z/YcmsV759eHIEBbjqJq+p4GvP5zTM1RERERpirlFgwtpe4PWXNflTqO8R5AJPFUENERFQHTwcr2GrMAFTNXfP7qasSV0T1YaghIiKqg0KhwLYXR8DRRg0AeGrDIVzKLZa4KqoLQw0REVE9ujta44NZg8Tt5TtP8G6odoqhhoiI6C6Cejrih+eDAADbjlzBvlM5EldExjDUEBERNcBgDzvMGla1mvffv/sTV/JuSlwR3YmhhoiIqIHm3tcTluYqZBaUIGjZbvx05IrUJVENDDVEREQN1N3RGitn+onbr3yTzIHD7QhDDRERUSNM7O+C7QvuBQBU6AW88k0yBw63Eww1REREjeTTVYs9fxsFK7UKCReu4+1tR6UuicBQQ0RE1CTdHa2x6P6+AIDP4y7g/lX7UFxWIXFVHRtDDRERURM9EdANL43xAgAcyyjAnjTOOCwlhhoiIqJm+Ov4Phjh5QgA+DGZq3lLiaGGiIiomd560AdKBfDrsSwkXsiVupwOi6GGiIiomXo72eIRfzcAwIKvk5FdUCJxRR0TQw0REVELeON+H/RwtEZ63k08+3kCb/OWAEMNERFRC9BZmePTOUOhMVfiyOV87EjNlLqkDoehhoiIqIV4OlrjqXu6AwCW/nwcFZV6iSvqWBhqiIiIWtDCsb1gZ2WO9Lyb+GjvGanL6VAaFWrWrl0LX19faLVaaLVaBAYGYseOHfU+Z/PmzfD29oZGo8GAAQOwfft2g+NLliyBt7c3rK2tYWdnh+DgYMTHxxu0Wbp0KYKCgmBlZYVOnTo1pmQiIqI2pTFXiZPyffzbGVwvKpO4oo6jUaHGzc0Ny5YtQ2JiIhISEjBmzBhMnToVR48anx46NjYWs2bNQmhoKJKSkjBt2jRMmzYNqampYpvevXtj9erVSElJwf79++Hp6Ynx48fj6tXbExiVlZVh+vTpmD9/fhO7SURE1HYe8XdDXxctisoq8dp3f3Km4TaiEJo5PNve3h7Lly9HaGhorWMzZ85EUVERoqKixH0BAQEYOHAg1q1bZ/T1CgoKoNPpsGvXLowdO9bg2GeffYaXX34ZeXl5ja6z+nXz8/Oh1Wob/XwiIqLG+P3kVTz12SFU6gU8NNgV788YKHVJJqkxn99NHlNTWVmJyMhIFBUVITAw0GibuLg4BAcHG+ybMGEC4uLijLYvKyvD+vXrodPp4OfnZ7RNQ5WWlqKgoMDgi4iIqK3c17szlj/iCwD44XA6fj3Ku6FaW6NDTUpKCmxsbGBhYYF58+Zhy5Yt8PHxMdo2MzMTTk5OBvucnJyQmWn4g42KioKNjQ00Gg1WrlyJ6OhoODo6NrY0A+Hh4dDpdOKXu7t7s16PiIiosR4a7IapA7sCAD7+/SwKSsolrkjeGh1q+vTpg+TkZMTHx2P+/PmYPXs2jh071qwiRo8ejeTkZMTGxmLixImYMWMGsrOzm/WaYWFhyM/PF78uXbrUrNcjIiJqioVje0GtUiLxwnXM+fQgb/NuRY0ONWq1Gl5eXvD390d4eDj8/PywatUqo22dnZ2RlZVlsC8rKwvOzs4G+6ytreHl5YWAgABERETAzMwMERERjS3NgIWFhXiXVvUXERFRW+vR2QZfzx0OWwszHL6Yh13Hs+7+JGqSZs9To9frUVpaavRYYGAgYmJiDPZFR0fXOQanIa9JRERkavy72WPWcA8AwDs/HUNJeaXEFcmTWWMah4WFYdKkSfDw8EBhYSE2bdqEvXv3YufOnQCAkJAQuLq6Ijw8HACwcOFCjBw5EitWrMDkyZMRGRmJhIQErF+/HgBQVFSEpUuXYsqUKXBxcUFOTg7WrFmD9PR0TJ8+Xfy+Fy9eRG5uLi5evIjKykokJycDALy8vGBjY9MS/w5EREStau59PfB53HlcyS9B8qU8BPRwkLok2WlUqMnOzkZISAgyMjKg0+ng6+uLnTt3Yty4cQCqwodSefvkT1BQEDZt2oTFixdj0aJF6NWrF7Zu3Yr+/fsDAFQqFdLS0rBx40bk5OTAwcEBQ4cOxb59+9CvXz/xdd566y1s3LhR3B40aBAAYM+ePRg1alSTO09ERNRWHG0sMMa7C7anZCLxwnWGmlbQ7HlqTAXnqSEiIql9ceAC3tyaCm9nW+xYeC8UCoXUJbV7bTJPDRERETXOg74uUKuUSMssxF+/PSJ1ObLDUENERNRGOlmpMWlA1R3APySlY8+J5k1fQoYYaoiIiNrQO1P6o7ujNQDgsz/OS1uMzDDUEBERtSGdlTk+enwwAOC3k1eRX8xZhlsKQw0REVEb8+pyezqSse/vxS+pGSir4EzDzcVQQ0RE1MbMVUr0d626kyfnRhnmfXkYr33HgcPNxVBDREQkgY1PDcOw7vbi9o/JV+C1aDtS0/MlrMq0MdQQERFJwMHGAm894GOwr0Iv4LXv/pSoItPHUENERCSRfl21GO/jhFF9OmN2YDcAwPGMApy5ekPiykxTo5ZJICIiopajUCiwPmQIAEAQBGxPzcTVwlK8tvkIfnj+HomrMz08U0NERNQOKBQKvBLcGwCQdCkPF64VSVyR6WGoISIiaiceG+6Bkb07QxCAjbEXpC7H5DDUEBERtSMPDXYFACRcyJW4EtPDUENERNSODHK3AwD8eTkfB85ek7ga08JQQ0RE1I6421tihJcjAODR9QdwIrNQ4opMB0MNERFRO6JQKPD86J7i9tOfHUJhCdeHagiGGiIionZmeHcHTOrvDABIz7uJDVzNu0EYaoiIiNoZlVKBtU/444NZgwAAEfvPcTXvBmCoISIiaqcmD3BBz87WyL9Zjve2H5e6nHaPoYaIiKidUikVWPawLwBgc+IlHDzH27zrw1BDRETUjg31tMdDg12hF4APd5+Supx2jaGGiIionVs4thcUCmDfqRy8H31S6nLaLYYaIiKidq6bgzWeu6/qNu8PYk7hg5hT+CU1Q+Kq2h+GGiIiIhPw4hgv8fH70Scx78vDOHIpT7qC2iGGGiIiIhNgY2GGVY8ONNi3JSldmmLaKYYaIiIiEzGhnzPu7eUobn8Wex4XrxVLWFH7wlBDRERkIjTmKnwROhyJi4PFffct34MreTclrKr9YKghIiIyMQ42Fgbbr/+QAkEQJKqm/WCoISIiMkGfPTVUfPz7yavYdypHwmraB4YaIiIiEzSqTxecXzYZDw12BQCEfHoQN8sqJa5KWgw1REREJmyAq058vDstW8JKpMdQQ0REZMKm+HUVH0ceutihx9Yw1BAREZkwBxsLbHvxHgBVyyg89km8xBVJh6GGiIjIxPm6dcKoPp0BAHFnr+HajVKJK5IGQw0REZEMvDahj/g4MHw3MvNLJKxGGgw1REREMtCvqw4jvKpmGy6r1OOTfWclrqjtMdQQERHJRGBPB/Fx/LlrHW6mYYYaIiIimXjuvh74YNYgAEBqegFGLt+DrIKOcxmKoYaIiEgmzFRKPDDABRZmVR/v5ZUCjlzKk7aoNsRQQ0REJCNKpQIvjvYSt3/6MwMVlXoJK2o7DDVEREQy89LYXnhuZA8AwE9HrmDUf/aipFz+Sygw1BAREcnQIHc78fHl6zex7rczElbTNhhqiIiIZGi8jxNcO1mK2wfOXpOwmrbBUENERCRDSqUCu14dKW53ttVIWE3bYKghIiKSKUu1Cssf8QUA5BTKf+kEhhoiIiIZc7OzAlC1JtTp7BsSV9O6GGqIiIhkrK+LLWw1ZgCABV8noVIvSFxR62GoISIikrFOVmrsWHgvLM1VOJZRgCOX86QuqdU0KtSsXbsWvr6+0Gq10Gq1CAwMxI4dO+p9zubNm+Ht7Q2NRoMBAwZg+/btBseXLFkCb29vWFtbw87ODsHBwYiPjzdok5ubi8cffxxarRadOnVCaGgobtyQ9yk0IiKiluJmZwWvLjYAgF+PZklcTetpVKhxc3PDsmXLkJiYiISEBIwZMwZTp07F0aNHjbaPjY3FrFmzEBoaiqSkJEybNg3Tpk1Damqq2KZ3795YvXo1UlJSsH//fnh6emL8+PG4evWq2Obxxx/H0aNHER0djaioKPz++++YO3duE7tMRETU8VipVQCAdb+dQezpHImraR0KQRCadXHN3t4ey5cvR2hoaK1jM2fORFFREaKiosR9AQEBGDhwINatW2f09QoKCqDT6bBr1y6MHTsWx48fh4+PDw4dOoQhQ4YAAH755Rfcf//9uHz5Mrp27dqgOqtfNz8/H1qttgk9JSIiMl1PbTiIPSeqThh0sjJH4uJxUCkVEld1d435/G7ymJrKykpERkaiqKgIgYGBRtvExcUhODjYYN+ECRMQFxdntH1ZWRnWr18PnU4HPz8/8TU6deokBhoACA4OhlKprHWZqqbS0lIUFBQYfBERERGQV1yOBV8nSV1Gi2t0qElJSYGNjQ0sLCwwb948bNmyBT4+PkbbZmZmwsnJyWCfk5MTMjMzDfZFRUXBxsYGGo0GK1euRHR0NBwdHcXX6NKli0F7MzMz2Nvb13qdmsLDw6HT6cQvd3f3xnaViIhINirvuC7zc0oGLuUWS1NMK2l0qOnTpw+Sk5MRHx+P+fPnY/bs2Th27Fizihg9ejSSk5MRGxuLiRMnYsaMGcjOzm7Wa4aFhSE/P1/8unTpUrNej4iIyJTpjdzK/Xnc+bYvpBU1OtSo1Wp4eXnB398f4eHh8PPzw6pVq4y2dXZ2RlaW4SjrrKwsODs7G+yztraGl5cXAgICEBERATMzM0RERIivcWfAqaioQG5ubq3XqcnCwkK8S6v6i4iIqKMyNj/N94fTUVohn9W7mz1PjV6vR2mp8amXAwMDERMTY7AvOjq6zjE4xl4zMDAQeXl5SExMFI/v3r0ber0ew4cPb2b1REREHYOxUJNbVIboY/K5xdusMY3DwsIwadIkeHh4oLCwEJs2bcLevXuxc+dOAEBISAhcXV0RHh4OAFi4cCFGjhyJFStWYPLkyYiMjERCQgLWr18PACgqKsLSpUsxZcoUuLi4ICcnB2vWrEF6ejqmT58OAOjbty8mTpyIZ599FuvWrUN5eTlefPFFPProow2+84mIiKijq7zjZufujtY4l1OEFzclobRcj4f93SSqrOU06kxNdnY2QkJC0KdPH4wdOxaHDh3Czp07MW7cOADAxYsXkZGRIbYPCgrCpk2bsH79evj5+eG7777D1q1b0b9/fwCASqVCWloaHn74YfTu3RsPPvggrl27hn379qFfv37i63z11Vfw9vbG2LFjcf/992PEiBFiMCIiIqK7e/qe7obbI25vb06Ux7jTZs9TYyo4Tw0REXVkgiBg08GLeGNL1QS4nz89DHtOZGPDH+ehUAAn/zUJ5qr2t3pSm8xTQ0RERKZDoVCgf1eduG1hpsQwT3sAgCAAr3yTLFFlLYehhoiIqIOwML/9sa8xV0Fza+kEAIj6MwNbki5LUVaLYaghIiLqICzMbocYjbkKluYqg+OvfHMEpjwqhaGGiIiog7AwUxo8vjPUAMC1orK2LKlFMdQQERF1EOqaocZcKa7cXdNfvz2C1PT8tiyrxTDUEBERdRBmNVblVioU0Bg5U/PbyatYGX2yLctqMQw1REREHYStxhxajRlsLMzgYK2GZY0zNT4ut2+XTuGZGiIiImrPVEoFDr4RjITFwTBTGY6pmT+qp/jYpZOlFOU1G0MNERFRB6IxV4mXnWpefnLSarDx6WEAgLIKvSS1NRdDDRERUQelqjHGxtPBCupbMwqXVVSiUi8gv7jcpAJOoxa0JCIiInmJ+etI3CipQBetBpeu3wQAnLlahJ6LtgMA7vFywFfPBEhZYoMx1BAREXVgPTvbiI9rzmNT7Y/T11BaUWkwcV97xctPREREBMBwHpua5nx6qI0raRqGGiIiIgIAcUzNneLOXmvjSpqGoYaIiIgA1H2mBgD0+va/JhRDDREREQGoP9T0WLQdv6RmtGE1jcdQQ0RERABqh5oejtYG2/O+PNyW5TQaQw0REREBqD2m5oXRXngluLdE1TQeQw0REREBqB1qzM2UeHGMl7iad10DiduL9l0dERERtRmlUgFz1e1ZhtUqJVRKBf74xxgAQFmlHoUl5VKVd1cMNURERCSqeTZGbVYVcOys1ehsawEAOJ19Q5K6GoKhhoiIiEQ1Bwub1wg4fZxsAQCJF663eU0NxVBDREREIq2lufi45lmbfq5aAMC/fj6O6GNZbV5XQzDUEBERkchJqxEfm9c4a/PwYDfx8bOfJ2BT/MU2rashGGqIiIhI5Fwj1NQ8U9PbyRaLJ/cVtxdtScGprMI2re1uGGqIiIhI5KyrEWrumIzvmXt7YPIAF3F77W9n2qyuhmCoISIiIpFLjVBjbmRemhUz/MTHPxxOxxP/i2+TuhqCoYaIiIhEPTvbiI+NrQWlMVfh99dGi9v7T+cgM7+kTWq7G4YaIiIiEvXscjvUKBXG23g4WOG310aJ2wHhMbhZVtnKld0dQw0RERGJXLQadLa1QBdbCzjaWNTZrpuDNZy0t49/Hne+DaqrH0MNERERiZRKBfb/YzT2/G2U0TE1Nf3fw77i4//uOoWE87mtXV69GGqIiIjIgIWZCtYWZndtN6pPF5xaOgn39e6Mm+WVWBiZjLIKfRtUaBxDDRERETWZuUqJj5/wx6T+zvj4SX+jg4vbyt1jGBEREVE9LNUqrH3CX+oyeKaGiIiI5IGhhoiIiGSBoYaIiIhkgaGGiIiIZIGhhoiIiGSBoYaIiIhkgaGGiIiIZIGhhoiIiGSBoYaIiIhkgaGGiIiIZIGhhoiIiGSBoYaIiIhkgaGGiIiIZKHDrNItCAIAoKCgQOJKiIiIqKGqP7erP8fr02FCTWFhIQDA3d1d4kqIiIiosQoLC6HT6eptoxAaEn1kQK/X48qVK7C1tYVCoWjR1y4oKIC7uzsuXboErVbboq/dHrG/8sb+yhv7K39y67MgCCgsLETXrl2hVNY/aqbDnKlRKpVwc3Nr1e+h1Wpl8QvUUOyvvLG/8sb+yp+c+ny3MzTVOFCYiIiIZIGhhoiIiGSBoaYFWFhY4O2334aFhYXUpbQJ9lfe2F95Y3/lryP2uVqHGShMRERE8sYzNURERCQLDDVEREQkCww1REREJAsMNURERCQLDDUA1qxZA09PT2g0GgwfPhwHDx6ss+2oUaOgUChqfU2ePFlsc+PGDbz44otwc3ODpaUlfHx8sG7dOoPXKSkpwQsvvAAHBwfY2Njg4YcfRlZWVqv1saa27m9ubi5eeukl9OnTB5aWlvDw8MCCBQuQn5/fqv2sJsXPt5ogCJg0aRIUCgW2bt3a0l0zSqr+xsXFYcyYMbC2toZWq8V9992Hmzdvtkofa5Kiv5mZmXjyySfh7OwMa2trDB48GN9//32r9bGmlu5vVlYW5syZg65du8LKygoTJ07EqVOnDF5HyvcroO37LLf3rIb8jKtJ8Z7VooQOLjIyUlCr1cKnn34qHD16VHj22WeFTp06CVlZWUbbX7t2TcjIyBC/UlNTBZVKJWzYsEFs8+yzzwo9e/YU9uzZI5w7d074+OOPBZVKJfz4449im3nz5gnu7u5CTEyMkJCQIAQEBAhBQUGt3V1J+puSkiI89NBDwrZt24TTp08LMTExQq9evYSHH35Ylv2t6f333xcmTZokABC2bNnSSr28Tar+xsbGClqtVggPDxdSU1OFtLQ04ZtvvhFKSkpk2d9x48YJQ4cOFeLj44UzZ84I7777rqBUKoXDhw+bVH/1er0QEBAg3HvvvcLBgweFtLQ0Ye7cuYKHh4dw48YN8XWker8SBGn6LKf3rIb+jKu19XtWS+vwoWbYsGHCCy+8IG5XVlYKXbt2FcLDwxv0/JUrVwq2trYGvxz9+vUT3nnnHYN2gwcPFt544w1BEAQhLy9PMDc3FzZv3iweP378uABAiIuLa0537kqK/hrz7bffCmq1WigvL29kDxpHyv4mJSUJrq6uQkZGRpu9QUjV3+HDhwuLFy9uZvWNJ1V/ra2thc8//9ygjb29vfDJJ580pRsN1tL9PXHihABASE1NNXjNzp07i32R8v1KEKTpszGm+p7VmP5K8Z7V0jr05aeysjIkJiYiODhY3KdUKhEcHIy4uLgGvUZERAQeffRRWFtbi/uCgoKwbds2pKenQxAE7NmzBydPnsT48eMBAImJiSgvLzf4vt7e3vDw8Gjw920KqfprTH5+PrRaLczMWm/5MSn7W1xcjMceewxr1qyBs7Nzy3WqHlL1Nzs7G/Hx8ejSpQuCgoLg5OSEkSNHYv/+/S3bwTtI+fMNCgrCN998g9zcXOj1ekRGRqKkpASjRo1qsf7dqTX6W1paCgDQaDQGr2lhYSH+/KR6vwKk67Mxpvqe1dD+SvGe1SokjVQSS09PFwAIsbGxBvtfe+01YdiwYXd9fnx8vABAiI+PN9hfUlIihISECAAEMzMzQa1WCxs3bhSPf/XVV4Jara71ekOHDhX+/ve/N7E3dydVf+909epVwcPDQ1i0aFHTOtJAUvZ37ty5QmhoqLiNNvirR6r+xsXFCQAEe3t74dNPPxUOHz4svPzyy4JarRZOnjzZMp0zQsqf7/Xr14Xx48eLbbRarbBz587md6oerdHfsrIywcPDQ5g+fbqQm5srlJaWCsuWLRMACOPHjxcEQbr3K0GQrs93MuX3rIb2V4r3rNbQYVbpbg0REREYMGAAhg0bZrD/ww8/xIEDB7Bt2zZ069YNv//+O1544QV07drVIIGbmpbob0FBASZPngwfHx8sWbKkDatvvKb2d9u2bdi9ezeSkpIkqrxpmtpfvV4PAHjuuefw1FNPAQAGDRqEmJgYfPrppwgPD2/zvjREc36f33zzTeTl5WHXrl1wdHTE1q1bMWPGDOzbtw8DBgyQojt3Zay/5ubm+OGHHxAaGgp7e3uoVCoEBwdj0qRJEGQw2XxL9NnU37Ma0l9Tfc8ySuJQJanS0lJBpVLVSqMhISHClClT6n3ujRs3BK1WK/z3v/812F9cXCyYm5sLUVFRBvtDQ0OFCRMmCIIgCDExMQIA4fr16wZtPDw8hPfff79pnWkAqfpbraCgQAgMDBTGjh0r3Lx5s+kdaSCp+rtw4UJBoVAIKpVK/AIgKJVKYeTIkc3uV12k6u/Zs2cFAMIXX3xh0GbGjBnCY4891sTe3J1U/T19+nStMQqCIAhjx44VnnvuuSb25u5ao7815eXlCdnZ2YIgVI3reP755wVBkO79ShCk63M1Obxn1VRXf6V6z2oNHXpMjVqthr+/P2JiYsR9er0eMTExCAwMrPe5mzdvRmlpKZ544gmD/eXl5SgvL4dSafhPq1KpxL9o/f39YW5ubvB9T5w4gYsXL971+zaHVP0Fqv7aGT9+PNRqNbZt22Zwfbe1SNXf119/HX/++SeSk5PFLwBYuXIlNmzY0AI9M06q/np6eqJr1644ceKEQZuTJ0+iW7duzelSvaTqb3FxMQDc9Xe+pbVGf2vS6XTo3LkzTp06hYSEBEydOhWAdO9XgHR9BuTznlVTXf2V6j2rVUidqqQWGRkpWFhYCJ999plw7NgxYe7cuUKnTp2EzMxMQRAE4cknnxRef/31Ws8bMWKEMHPmTKOvOXLkSKFfv37Cnj17hLNnzwobNmwQNBqN8NFHH4lt5s2bJ3h4eAi7d+8WEhIShMDAQCEwMLB1OlmDFP3Nz88Xhg8fLgwYMEA4ffq0we2HFRUVrddZQbqf753Qhrd0S9HflStXClqtVti8ebNw6tQpYfHixYJGoxFOnz7dOh29RYr+lpWVCV5eXsK9994rxMfHC6dPnxb+85//CAqFQvj5559br7NC6/T322+/Ffbs2SOcOXNG2Lp1q9CtWzfhoYceMmgj1fuVIEjTZ7m9ZzXkZ3yntnrPamkdPtQIgiB8+OGHgoeHh6BWq4Vhw4YJBw4cEI+NHDlSmD17tkH7tLQ0AYDw66+/Gn29jIwMYc6cOULXrl0FjUYj9OnTR1ixYoWg1+vFNjdv3hSef/55wc7OTrCyshL+8pe/CBkZGa3Svzu1dX/37NkjADD6de7cudbqpkiKn++d2vINQqr+hoeHC25uboKVlZUQGBgo7Nu3r8X7ZowU/T158qTw0EMPCV26dBGsrKwEX1/fWrd4t5aW7u+qVasENzc3wdzcXPDw8BAWL14slJaWGrSR8v1KENq+z3J7z2rIz/hOphpqFIIgg9FgRERE1OF16DE1REREJB8MNURERCQLDDVEREQkCww1REREJAsMNURERCQLDDVEREQkCww1REREJAsMNURERCQLDDVEREQkCww1REREJAsMNURERCQLDDVEREQkC/8Pq8o6NJp1ST0AAAAASUVORK5CYII=\n", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import warnings\n", "\n", "from scipy.integrate import quad\n", "from math import sin\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "%matplotlib inline\n", "\n", "def I(a, eps, limit):\n", " out, _ = quad(lambda x: (2+sin(10*a))*x**a*sin(a/(2-x)), 0, 2, epsabs=eps, epsrel=eps, limit=limit)\n", " return out\n", "\n", "# To suppress IntegrationWarning warning about maximum subdivisions reached\n", "with warnings.catch_warnings():\n", " for x, eps, limit in zip([np.linspace(0,5,1000), np.linspace(0.77,0.81,1000), np.linspace(0.78,0.795,1000)], \n", " [1e-5, 1e-5, 1e-12],\n", " [50, 100, 1000]):\n", " plt.plot(x, np.vectorize(lambda x: I(x, eps=eps, limit=limit))(x))\n", " plt.show()\n", " plt.clf()" ] }, { "cell_type": "markdown", "metadata": { "id": "Q9sgz1bgvLCX" }, "source": [ "From initial findings, we see that we attain a maximum of $I(\\alpha) \\approx 3.033 $ where $\\alpha \\approx 0.786 $ " ] }, { "cell_type": "markdown", "metadata": { "id": "fo_Vfbb86omN" }, "source": [ "We now attempt to use out-of-the-box maximization routines that the `scipy` module provides on the integral." ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "FtAudwDH6phm", "outputId": "150c8199-2abc-42e4-9f58-e2d5571cdb35" }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "CPU times: user 2min 11s, sys: 24.9 ms, total: 2min 11s\n", "Wall time: 2min 11s\n" ] }, { "data": { "text/plain": [ " fun: -3.033733163355978\n", " message: '\\nOptimization terminated successfully;\\nThe returned value satisfies the termination criteria\\n(using xtol = 1.48e-08 )'\n", " nfev: 28\n", " nit: 24\n", " success: True\n", " x: 0.7859422948318388" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from scipy.optimize import minimize_scalar\n", "\n", "def I(a):\n", " out, _ = quad(lambda x: (2+sin(10*a))*x**a*sin(a/(2-x)), 0, 2, epsabs=1e-15, epsrel=1e-15, limit=100000)\n", " return -out\n", "\n", "%time minimize_scalar(I)" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(3.033733163355978, 2.564722480374352e-06)" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a = 0.7859422948318388\n", "quad(lambda x: (2+sin(10*a))*x**a*sin(a/(2-x)), 0, 2, epsabs=1e-15, epsrel=1e-15, limit=100000)" ] }, { "cell_type": "markdown", "metadata": { "id": "n2A8IMBO7_Rn" }, "source": [ "Note that when calculating the integral with the above parameters, we have an error of $\\approx 10^{-6}$ - it stands to reason that only 6 of the digits we have for our maximum can be accurate - it turns out that only 4 of them are." ] }, { "cell_type": "markdown", "metadata": { "id": "WpVKTx_1wjWP" }, "source": [ "# Attempt 2: Manipulating the Integral\n", "We have that $$ I(\\alpha) = \\underbrace{(2+\\sin(10\\alpha))}_{:=f(\\alpha)} \\underbrace{\\int_0^2 x^a \\sin(\\frac{\\alpha}{2-x}) dx}_{:=I_1(\\alpha)}$$\n", "\n", "Note that by substituting $u = \\frac{1}{2-x}, du = \\frac{1}{u^2}$, we can rewrite $$I_1(\\alpha) = \\int_{1/2}^\\infty \\frac{(2-1/u)^\\alpha}{u^2} \\sin(\\alpha u) du$$.\n", "\n", "We use adaptive integration - i.e. we integrate between the zeros of the integrand until we achieve the desired error. This is done with the builtin [`quadosc`](https://mpmath.org/doc/current/calculus/integration.html#oscillatory-quadrature-quadosc) function in the `mpmath` module" ] }, { "cell_type": "code", "execution_count": 122, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 1000 }, "id": "_SC3JDK2x9yf", "outputId": "99163fad-d08a-4b8e-d9ad-a73e7d6c65b1", "scrolled": false }, "outputs": [ { "data": { "text/plain": [ "mpf('0.78593367378800233421807')" ] }, "execution_count": 122, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import mpmath\n", "from mpmath import quadosc, findroot, diff\n", "\n", "mpmath.mp.dps = 20\n", "\n", "def integrand_f(a):\n", " return lambda u: (2-1/u)**a / (u*u) * mpmath.sin(a*u)\n", "\n", "def I(a):\n", " return (2 + mpmath.sin(10*a)) * quadosc(integrand_f(a), \n", " [1/2, mpmath.inf], \n", " zeros = lambda n: 0.5 if n == 0 else n*mpmath.pi/a)\n", "findroot(lambda a: diff(I, a), 0.7)" ] }, { "cell_type": "markdown", "metadata": { "id": "MiDFWRfFURtG" }, "source": [ "With this, we are able to get 8 accurate digits." ] }, { "cell_type": "markdown", "metadata": { "id": "5bh19EgDGrS5" }, "source": [ "# Analytically Solving the Integral\n", "\n", "Instead of solving the integral numerically, we attempt to solve it analytically:\n", "\n", "We have that \n", "\n", "$$ \\begin{align} \n", "I(\\alpha) &= \\left(2+\\sin{(10 \\alpha)} \\right) \\int_0^2 x^\\alpha \\sin{\\left(\\frac{\\alpha}{2-x}\\right)} dx\\\\\n", "&= \\underbrace{\\left(2+\\sin{(10 \\alpha)} \\right)}_{f(\\alpha)} \\underbrace{\\int_0^2 (2-x)^\\alpha \\sin(\\alpha/x) dx}_{I_1(\\alpha)} & (x \\mapsto 2-x)\n", "\\end{align}$$ \n", "\n", "Using the `sympy` module, we try to integrate directly by representing $I_1$ with the [Meijer G-function](https://en.wikipedia.org/wiki/Meijer_G-function):\n" ] }, { "cell_type": "code", "execution_count": 138, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "yp6BxdeUYi9j", "outputId": "16e987ad-05df-4e37-e4c1-63e5edbe6ffc" }, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\frac{\\sqrt{\\pi} \\alpha {G_{2, 4}^{3, 0}\\left(\\begin{matrix} & \\frac{\\alpha}{2} + \\frac{1}{2}, \\frac{\\alpha}{2} + 1 \\\\0, 0, \\frac{1}{2} & - \\frac{1}{2} \\end{matrix} \\middle| {\\frac{\\operatorname{polar\\_lift}^{2}{\\left(\\alpha \\right)}}{16}} \\right)} \\Gamma\\left(\\alpha + 1\\right)}{4}$" ], "text/plain": [ "sqrt(pi)*alpha*meijerg(((), (alpha/2 + 1/2, alpha/2 + 1)), ((0, 0, 1/2), (-1/2,)), polar_lift(alpha)**2/16)*gamma(alpha + 1)/4" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle \\left|{\\arg{\\left(\\alpha \\right)}}\\right| = 0 \\wedge \\operatorname{re}{\\left(\\alpha\\right)} > -1$" ], "text/plain": [ "(re(alpha) > -1) & Eq(Abs(arg(alpha)), 0)" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "from sympy.integrals import meijerint\n", "from sympy.abc import x, alpha\n", "import sympy as sp\n", "\n", "from IPython.display import display\n", "\n", "_ = [display(i) for i in meijerint.meijerint_definite((2-x)**alpha*sp.sin(alpha/x), x, 0,2)]" ] }, { "cell_type": "markdown", "metadata": { "id": "thn0K4EAgtUI" }, "source": [ "We therefore have a closed-form solution for $I$:\n", "\n", "$$I(\\alpha) = \\frac{\\sqrt{\\pi} \\alpha}{4} \\left(\\sin{\\left (10\\alpha \\right )} + 2\\right) {G_{2, 4}^{3, 0}\\left(\\begin{matrix} & \\frac{\\alpha}{2} + \\frac{1}{2}, \\frac{\\alpha}{2} + 1 \\\\0, 0, \\frac{1}{2} & - \\frac{1}{2} \\end{matrix} \\middle| {\\frac{\\alpha^2}{16}} \\right)} \\Gamma{\\left(\\alpha + 1 \\right)} $$" ] }, { "cell_type": "code", "execution_count": 144, "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 279 }, "id": "Xg4E68tv-mz8", "outputId": "82eb1137-ee84-4808-e2eb-6cb93e4b1c7a" }, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "mpmath.mp.dps = 100; mpmath.mp.pretty = True\n", "\n", "def I(a):\n", " return 4*mpmath.sqrt(mpmath.pi)*mpmath.gamma(a)*(2+mpmath.sin(10*a))*mpmath.meijerg([[],[(a+3)/2, (a+4)/2]],[[1,1,1.5],[0.5]],a*a/16)\n", "\n", "mpmath.plot(I_1,[0,5])" ] }, { "cell_type": "markdown", "metadata": { "id": "NEYxbiqyRg3i" }, "source": [ "To find the maximum of this integral, we simply find the root of $I'(\\alpha) = 0$ near $\\alpha = 0.7$ - with this, we are able to get the answer to 100 digits." ] }, { "cell_type": "code", "execution_count": 155, "metadata": { "colab": { "base_uri": "https://localhost:8080/" }, "id": "9uOmZ3RXOR4F", "outputId": "6205c2f3-3252-4428-b969-1149454e9609" }, "outputs": [ { "data": { "text/plain": [ "0.7859336743503714545652439863275455829623954590618668175812318070989103971494123651167706337659944953" ] }, "execution_count": 155, "metadata": {}, "output_type": "execute_result" } ], "source": [ "I_prime = lambda x: diff(I,x)\n", "findroot(I_prime, 0.7)" ] } ], "metadata": { "colab": { "collapsed_sections": [], "name": "Problem 9", "provenance": [] }, "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.10.6" } }, "nbformat": 4, "nbformat_minor": 1 }