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{ | |
"cells": [ | |
{ | |
"cell_type": "markdown", | |
"id": "e87159c9-8dbf-4956-bd63-daea7654f678", | |
"metadata": {}, | |
"source": [ | |
"[2024-08-16 Fiddler](https://thefiddler.substack.com/p/how-high-can-you-jump)\n", | |
"====================\n", | |
"Let $a+b = 1$.\n", | |
"\n", | |
"The height is" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 1, | |
"id": "c874f9dc-bc5e-4af1-a3b7-9a0a612ce8a2", | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"data": { | |
"text/html": [ | |
"<html>\\(\\displaystyle \\theta \\ {\\mapsto}\\ \\cos\\left(\\theta\\right)\\)</html>" | |
], | |
"text/latex": [ | |
"$\\displaystyle \\theta \\ {\\mapsto}\\ \\cos\\left(\\theta\\right)$" | |
], | |
"text/plain": [ | |
"theta |--> cos(theta)" | |
] | |
}, | |
"execution_count": 1, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"%display latex\n", | |
"y(theta) = cos(theta)\n", | |
"y" | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"id": "94e0d3f7-a38e-4605-8e17-42e83fe04cda", | |
"metadata": {}, | |
"source": [ | |
"Then, the vertical center of mass is" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 2, | |
"id": "b22acbba-d3de-4f41-b1f5-f10d02f8df56", | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"data": { | |
"text/html": [ | |
"<html>\\(\\displaystyle \\left(\\frac{2}{\\pi}, 0.636619772367581\\right)\\)</html>" | |
], | |
"text/latex": [ | |
"$\\displaystyle \\left(\\frac{2}{\\pi}, 0.636619772367581\\right)$" | |
], | |
"text/plain": [ | |
"(2/pi, 0.636619772367581)" | |
] | |
}, | |
"execution_count": 2, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"a = integral(y,theta,-pi/2,pi/2)/pi\n", | |
"(a,numerical_approx(a))" | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"id": "5bccd820-6994-47fc-8ca1-d7e8c187473e", | |
"metadata": {}, | |
"source": [ | |
"And the ratio is" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 3, | |
"id": "08e873a4-b034-425a-9e51-e4f80b629bfa", | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"data": { | |
"text/html": [ | |
"<html>\\(\\displaystyle \\left(\\frac{2}{\\pi - 2}, 1.75193839388411\\right)\\)</html>" | |
], | |
"text/latex": [ | |
"$\\displaystyle \\left(\\frac{2}{\\pi - 2}, 1.75193839388411\\right)$" | |
], | |
"text/plain": [ | |
"(2/(pi - 2), 1.75193839388411)" | |
] | |
}, | |
"execution_count": 3, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"b = 1-a\n", | |
"((a/b).simplify_full(),numerical_approx(a/b))" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 4, | |
"id": "69814159-39f5-474b-8907-6aff7d26b1a0", | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"data": { | |
"image/png": 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", | |
"text/plain": [ | |
"Graphics object consisting of 5 graphics primitives" | |
] | |
}, | |
"execution_count": 4, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"plot(sqrt(1-x^2),x,-1,1,axes=false) + circle((0,a),0.02,fill=true,color='black') \\\n", | |
" + plot([0,a,1],x,-1.25,1.25,axes=false,linestyle='dashed',color='black')" | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"id": "492277e3-e73f-40a3-9be4-4cd56207fc9a", | |
"metadata": {}, | |
"source": [ | |
"Extra credit\n", | |
"------------\n", | |
"Let $a + b = 1 - \\cos\\phi$." | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 5, | |
"id": "a2afcaa7-43c2-4e22-bdd8-c9163f78952b", | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"data": { | |
"text/html": [ | |
"<html>\\(\\displaystyle -\\frac{\\sin\\left(\\phi\\right)}{\\phi} + 1\\)</html>" | |
], | |
"text/latex": [ | |
"$\\displaystyle -\\frac{\\sin\\left(\\phi\\right)}{\\phi} + 1$" | |
], | |
"text/plain": [ | |
"-sin(phi)/phi + 1" | |
] | |
}, | |
"execution_count": 5, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"phi = var('phi')\n", | |
"b = 1 - integral(y, theta, -phi, phi)/(2*phi)\n", | |
"b" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 6, | |
"id": "dae56957-92f1-45d9-827c-415bbc6bf2b3", | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"data": { | |
"text/html": [ | |
"<html>\\(\\displaystyle -\\frac{\\phi \\cos\\left(\\phi\\right) - \\sin\\left(\\phi\\right)}{\\phi - \\sin\\left(\\phi\\right)}\\)</html>" | |
], | |
"text/latex": [ | |
"$\\displaystyle -\\frac{\\phi \\cos\\left(\\phi\\right) - \\sin\\left(\\phi\\right)}{\\phi - \\sin\\left(\\phi\\right)}$" | |
], | |
"text/plain": [ | |
"-(phi*cos(phi) - sin(phi))/(phi - sin(phi))" | |
] | |
}, | |
"execution_count": 6, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"a = 1 - b - cos(phi)\n", | |
"(a/b).simplify_full()" | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"id": "171d4c59-a718-463f-b1f1-c83df6574b84", | |
"metadata": {}, | |
"source": [ | |
"It looks like the ratio approaches 2 for small $\\phi$" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 7, | |
"id": "7671bfef-57c6-481d-8422-a80f2d1885ef", | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"data": { | |
"image/png": 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", | |
"text/plain": [ | |
"Graphics object consisting of 1 graphics primitive" | |
] | |
}, | |
"execution_count": 7, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"plot(a/b, phi, 0, pi/2)" | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"id": "af10e236-83ad-4e3f-a0a6-8e5050be54f2", | |
"metadata": {}, | |
"source": [ | |
"But, zooming in on small $\\phi$, there's wild oscillation. That might be numerical noise." | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 8, | |
"id": "4aba9122-4b13-4a44-b10f-a89fbc3407ac", | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"data": { | |
"image/png": 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", | |
"text/plain": [ | |
"Graphics object consisting of 1 graphics primitive" | |
] | |
}, | |
"execution_count": 8, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"plot(a/b, phi, 0, 1e-6)" | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"id": "3097c62d-97ff-472f-a812-32730e7a57b1", | |
"metadata": {}, | |
"source": [ | |
"Taking the power series expansion around $\\phi=0$," | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 9, | |
"id": "051c2019-e9e7-43eb-b4ee-19180b7d7ca7", | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"data": { | |
"text/html": [ | |
"<html>\\(\\displaystyle -\\frac{1}{10} \\, \\phi + \\frac{2}{\\phi} + \\mathcal{O}\\left(\\phi^{3}\\right)\\)</html>" | |
], | |
"text/latex": [ | |
"$\\displaystyle -\\frac{1}{10} \\, \\phi + \\frac{2}{\\phi} + \\mathcal{O}\\left(\\phi^{3}\\right)$" | |
], | |
"text/plain": [ | |
"-1/10*phi + 2/phi + Order(phi^3)" | |
] | |
}, | |
"execution_count": 9, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"(a/b).series(phi,3).simplify()" | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"id": "4af868f2-24cc-498a-ab13-10b6e38638fc", | |
"metadata": {}, | |
"source": [ | |
"The $2/\\phi$ term seems like some weird artifact. Let's try something" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 10, | |
"id": "b067b894-b6a8-4f91-90a9-4ac759552e41", | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"data": { | |
"text/html": [ | |
"<html>\\(\\displaystyle 2 + {(-\\frac{1}{10})} \\phi^{2} + \\mathcal{O}\\left(\\phi^{3}\\right)\\)</html>" | |
], | |
"text/latex": [ | |
"$\\displaystyle 2 + {(-\\frac{1}{10})} \\phi^{2} + \\mathcal{O}\\left(\\phi^{3}\\right)$" | |
], | |
"text/plain": [ | |
"2 + (-1/10)*phi^2 + Order(phi^3)" | |
] | |
}, | |
"execution_count": 10, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"(a/b).simplify_full().series(phi,3)" | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"id": "f42237da-c418-4992-9a76-5914335cf28e", | |
"metadata": {}, | |
"source": [ | |
"That convinces me that the ratio approaches 2 rather than $2/\\phi$, and there's some bug\n", | |
"in the power series algorithm. That also cleans up some of the numerical noise." | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 11, | |
"id": "2d43aed3-b774-4c74-b016-ea829222a918", | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"data": { | |
"image/png": 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", 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"Graphics object consisting of 1 graphics primitive" | |
] | |
}, | |
"execution_count": 11, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"plot((a/b).simplify_full(), phi, 0, 1e-6)" | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"id": "1bd74699-2028-436b-890d-fdc2d4a9e555", | |
"metadata": {}, | |
"source": [ | |
"Look at some plots for a few different values of $\\phi$" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 12, | |
"id": "6cfceb56-1f50-4af9-86aa-60670c42a7a9", | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"data": { | |
"image/png": 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15bqW16+rTkf0dLiOHRmVzEy5sPCSJYCjIzBxIjBsGGBvrzoZERm7nBxgzRpg7lx5rRk7Vl5jnJxUJyN6fCzsyCgUFson6qlTgYwM4MMPgfHjecElotJ37x6wcKHcrqxCBbmo+YABfBtAxoGvYsngHTwIBAbKV68dOgBxccCsWSzqiEg/ypeX15jYWDnDfsgQOQaPS6SQMWBhRwYrOhro3FmuRVe+vFzKZMsWwMNDdTIiMgeensDWrXLXCltbuURK9+5y1xoiQ8XCjgxOejowejQQEABcvAh8/bVcLZ77PRKRCi1ayOJu61a5LqafHzBunBwWQmRoOMaODEZREfDll8BHH8n1paZOBd5/X24PRERkCB48ABYvlpO4HB3lOLx+/bi8EhkO9tiRQQgLA1q2BAYOlGNaYmNlgceijogMiYODfOiMi5OLG7/zDtC2LRAVpToZkcTCjpTKygI++EAOTM7MlIOT//c/wN1ddTIior/m7g5s2wb8/LPc7aZxY/kwmpWlOhmZO76KJWV27waGDwfu3AE+/hgYMwawtladikxJQUEBfv75Z1y9ehUAUKtWLXTs2BFWVlZqg5FJyc2Vr2RnzwZcXIBly+TCx3w9SyqwsKMyd+MGMGqU3MrnX/8CVq8GatdWnYpMSXZ2NhYvXoxVq1YhKSkJmv+/wwohUK1aNQwfPhwffPAB7LmyNZWi+Hhg5Ehgzx7gtdfkYsecxU9ljYUdlZnCQlnETZ4MlCsnn2p79uRTLZWujIwMvPLKKzh79iyKiooe+XssLCzQvHlz/PTTT6hQoUIZJyRTJgSwcycwYoRc6PiTT+QanBYc+ERlhD9qVCaiooAXX5SzXPv1k2vU9erFoo5KlxAC3bt3x7lz5/6yqAOAoqIinDt3Dt27dwefbak0aTRAt27ymte7t9zysH17QKtVnYzMBQs70quCAmDePKBJE/n0evy47LWrWFF1MjJFv/76Kw4dOoTCwsJ//L2FhYU4dOgQjhw5UgbJyNxUqAB89hnwyy9AYiLQsKEch1dQoDoZmToWdqQ3ERFyYc8pU+TEiLAw2WtHpC+rVq16ookRVlZWWLVqlR4Tkblr3x64cEH23I0fL6+BERGqU5Ep4xg7KnX5+cCCBcDMmYCXF/DFF9w1gvRPCAFbW1vk5+c/0Z+zsbFBTk5O8QQLIn05dQoYNAi4dEk+8E6aBHCCNpU2vRV2t27dwq1bt0r8mrOzM2rXro2cnBxcvHhR5880adIEABAbG4v79++XaKtVqxYqVaqE27dvIzExsUSbk5MTvLy8UFhYiPPnz+scNyAgANbW1rh8+TIy/rQHjLu7O1xdXZGWlob4+PgSbfb29vDz8wMAhIWF6YzF8fPzg729Pa5du4a7d++WaHN1dYW7uzsyMzOh/dPgCmtrawQEBAAAIiIidG5EXl5ecHJywo0bN5CcnFyirXLlyqhZsyays7MRHR1dok2j0SAwMBAAEB0djezs7BLttWvXhrOzM5KTk3Hjxo0SbRUqVEDdunWRn5+PiEc8TjZq1AiWlpbQarXIzMws0ebh4YEqVaogNTUVhw9fxdSpcoHhAQOA0aPLISDABwAQGhqqc1x/f3/Y2dkhPj4eaWlpJdrc3Nzg5uaGe/fu4dKlSyXabG1tUb9+fQDAhQsXUPCn9xve3t5wdHTE9evXkZKSUqLNxcUFnp6eePDgAWJiYkq0WVhYoHHjxgCAixcvIicnp0R7nTp1ULFiRSQlJeHmzZsl2ipWrIg6deogLy8PkZGROt9r48aNYWFhgbi4OGT9abErT09PuLi44M6dO0hISCjR5ujoCG9vbxQVFSE8PFznuA0aNICNjQ2uXLmC9PT0Em3Vq1dHtWrVkJ6ejitXrpRos7Ozg7+/PwAgPDxcZ0yar68vHBwckJCQgDt37pRoq1q1KmrUqIGsrCzExcWVaLOyskLDhg0BAFFRUcjNzS3RXq9ePZQvX77UrxHVqlWD+1MugHjv3j04OTnxGlEG14iHS888VK5cOfj4mM81wt29Dj7+OA8LFkSifn1g1iy5Jy3Aa8RD+rpGGFod8fDnt9QJPZk+fboAUOLTt29fIYQQWq1Wp+2PUVq0aKHTtnnzZiGEECtXrtRpCwoKEkIIkZGR8cjjpqSkCCGE6NKli07b4sWLhRBC7NixQ6ctMDCwOJONjY1Oe2RkpBBCiMGDB+u0TZgwQQghxOHDh3Xa3N3di4/r7u6u03748GEhhBATJkzQaRs8eLAQQojIyEidNhsbm+LjBgYG6rTv2LFDCCHE4sWLddq6dOkihBAiJSXlkX+HGRkZQgghgoKCdNpWrlwpioqEeOedzTptLVq0KM70qONqtVohhBB9+/bVaZs+fboQQoj9+/frtNWtW7f4uC4uLjrtJ0+eFEIIMWbMGJ224OBgIYQQISEhOm1OTk7Fx/X399dp37lzpxBCiLlz5+q09ejRQwghRGJi4iO/15ycHCGEEG3atNFpW7dunRBCiHXr1um0tWnTRgghRE5OziOPm5iYKIQQokePHjptc+fOFUIIsXPnTp02f3//4u/VyclJpz0kJEQIIURwcLBO25gxY4QQQpw8eVKnzcXFpfi4devW1Wnfv3+/EKL0rxFffvmlsLCweOSf+6dPcnKyEILXCH1eI4QQYvNmXiOE4DVC1TXC0OqIhz+/pY09dnwaB/BsT+M2Nh4YP74K9u1LRa9eVzFqFPBweTBzexpnj53ap/HXX38dJ0+e/NsZsX9kYWGBhg0b4syZM7xGgD12D5XVNeLBA+DTT+Wanq1aAdu3N4abG68R7LF7NhxjR8/k+++B996T40Q+/xzo1El1IjJnO3bswFtvvfXEf6Znz556SkT0z3bvBoYMAYQA1q8HunZVnYiMGQs7eipZWcDo0cCGDXLNps8+A6pUUZ2KzF1eXh58fX2RkJDwj0ueWFlZwcPDA7GxsbDmXnakWEqKfEjetQsYPBhYuhRwdFSdiowRlzuhJ3b+PNCsGfDVV/Lp8rvvWNSRYbCxscGBAwdQuXLlv132xMrKCpUqVcKBAwdY1JFBqFoV+OEHeU396it5jX3EG0Gif8TCjh6bEMB//ws8/zxgZweEhsonS64SQYakXr16OHfuHP71r39Bo9HA0tISlpaWAOT4KI1Gg3/96184d+4c6tWrpzgt0e80GnlNDQ2V19jnnwfWrpXXXqLHxVex9FgyMuRrgq+/BoKDgcWL5YWHyJAlJCRg06ZNCAkJwZ49e/Dee+9hwoQJ8Hy4vgSRgcrJAT74QO7U06uXHO7CbY3pcbCwo3909izw1lvA3btyTF2PHqoTERGZh6+/lhMrXFyA7dvlK1qiv8NXsfSXhABWrABeekleVMLCWNQREZWlnj3ltbdSJbkd2YoVfDVLf4+FHT3SgwfAO+8Ao0bJV6/HjwN16qhORfR0Ll68iPr16z9y3SsiQ1enDnDihLwWjxoF9O8vr9FEj8LCjnTEx8teum+/BbZuldPubWxUpyJ6eg8XM/3zYrJExsLGRl6L//c/uX7oCy8Aly+rTkWGiIUdlfDTT0DTpsC9e3LD6rffVp2IiIge6tNHXpvv35fj7fbuVZ2IDA0LOwIgx2zMmwe8+irQogVw7hzw/zu+EBGRAQkIkNfoli2Bzp2BGTOAx9xFj8wACzvCvXvAm28CkyYBkyfL7W2cnVWnIiKiv1KxIrBzJ/Dxx7Kw69oV+NM2sGSmuNyJmYuNlVuC3bgBbN4MvP666kREpS89PR1Hjx5F69atUbFiRdVxiErVvn1y2Iyrq3ww9/JSnYhUYmFnxg4ckAtfVq8uB+P6+KhORERET0Orla9lb9+WE9/atVOdiFThq1gztXIl0KmTXBfp1CkWdWTakpKSMG/ePCQlJamOQqQXXl7yWt60KRAUJHeqIPPEws7MFBQAw4cDI0fKz65dQPnyqlMR6dfNmzcxadIk3Lx5U3UUIr1xdpazZIcOBf79b2DMGKCwUHUqKmtWqgNQ2UlPl69eDx8G/vtf+Q+fiIhMh7U1sGoV4O8PvP++HEf91Vd8gDcn7LEzE9euydeuZ8/KtepY1BERma7hw2Xv3cmTcjHja9dUJ6KywsLODJw/L/9hZ2fLMRjt26tORERE+hYUJK/52dlyfdLwcNWJqCywsDNxBw8CrVrJma+//cZJEmSeKlasiB49enCpEzI7vr7y2u/uDrRuDfz8s+pEpG9c7sSEbdkCDBwIdOgAfP014OioOhEREamQlSXHWP/8M/D550D//qoTkb6wx84ECQHMny//4fbvL2e+sqgjc5aXl4fr168jLy9PdRQiJRwd5U4V77wjP/PmyXsFmR4WdiamsBAYMQKYOBGYNg3YsEHOkiIyZ5GRkfDw8EBkZKTqKETKWFsD69fLbcgmTZITLLgciunhcicmJDdXbiuzc6dcnPK991QnIiIiQ6LRANOnAzVqyNURUlKArVsBW1vVyai0sMfORNy/LzeB3rNHbg/Goo6IiP7K4MHyXvHjj0CXLnIMHpkGFnYmICMDeOUVuV7Rvn3yHykREdHf6dIF2L9fLonSsSOQmqo6EZUGFnZG7vZtuS5dVJRc2oQbPxMR0eNq2xY4dAjQaoE2bYBbt1QnomfF5U6M2M2bcimTu3flFPaGDVUnIjJMRUVFyM/Ph7W1NSws+DxL9GfR0XJBYzs7Weh5eKhORE+LhZ2Rio+XRV1+vuyp8/ZWnYiIiIxZfLx8A6TRyOKuVi3Viehp8NHVCMXEAC1bAhYWwLFjLOqI/klcXBzatm2LuLg41VGIDFbt2sCRI/Le0ro1cOmS6kT0NFjYGZnoaDkmwtlZFnU1a6pORGT4srKycOTIEWRx6h/R3/L0BI4eBRwc5Ji7mBjViehJsbAzItHRcnJElSrA4cNAtWqqExERkampXl323Dk7y44EruttXFjYGYmYmN+LukOH5FciIiJ9cHX9vQOhXTvg4kXViehxsbAzAjEx8qmpShXgl19Y1BERkf497EioXh14+WWAQ1SNAws7A/ewp87FRRZ1VauqTkRkfDw9PbFu3Tp4enqqjkJkVCpVkisvVKokZ8xevqw6Ef0TLndiwC5dkjOTKlWST00s6oiISIWkJDmZIidHTq7gxD3DxR47A5WYKLu+y5dnTx3Rs7pz5w7Wr1+PO3fuqI5CZJSqVZMdDFZWsufu+nXVieivsLAzQMnJcvFhCwvZBe7qqjoRkXFLSEjAe++9h4SEBNVRiIyWu7ss7goKZMcDtx8zTCzsDExqqtyMOTNTFnU1aqhOREREJNWsKWfL3r//+5aWZFhY2BmQe/eAf/1LPgUdPAjUras6ERERUUl16sghQikpwGuvySKPDAcLOwORkwO8/joQGwv89BPg7686ERER0aP5+AD79wNRUcCbbwJ5eaoT0UMs7AxAYSHQty9w6hSwZw/QpInqRESmxdHREW3atIGjo6PqKEQmo2lTYOdO+Wp2wACgqEh1IgK43IlyQgDDhwNr1wLffw907ao6ERER0eP79lugVy8gOBhYvhzQaFQnMm/ssVNs9mxgzRpZ2LGoI9KPoqIi5ObmoohdCkSl7s035X1s5Upg1izVaYiFnULr1gHTpsl/CEOGqE5DZLrCw8NhZ2eH8PBw1VGITNLQobKjYvp0WeSROlaqA5irnTuB//xHvoadPFl1GiIiomczaZKcKTtihFyqq0sX1YnME3vsFDh7FujTB3jjDWDZMo5HICIi46fRAEuWyBUeevcGzp1Tncg8sbArY9euyaeYRo2AzZsBS0vViYiIiEqHpSWwZQsQEAB07gxcvao6kflhYVeGMjLkD7qDg3wVa2+vOhEREVHpcnAAdu0CypUDOnUC0tJUJzIvXO6kjOTny6Lu9Gngt98APz/ViYjMR15eHlJSUlC1alXY2NiojkNkFuLigBdeABo2BA4cAKytVScyD+yxKwNCACNHys2Tv/2WRR1RWbOxsUGNGjVY1BGVIW9v4IcfgBMn5D2Q3Uhlg4VdGViyRK5Tt3Yt8PLLqtMQmZ8rV66gZ8+euHLliuooRGalVavf12pdvVp1GvPAwk7PfvwRGDcOmDABGDRIdRoi85Seno5vvvkG6enpqqMQmZ3Bg4HRo4H33wd++UV1GtPHwk6PoqOBt9+WU7/nzFGdhoiISI2FC4EOHYCePQGtVnUa08bCTk/S0mRB5+kJfPklYMG/aSIiMlNWVsBXXwFVqsjtMzMyVCcyXSw39KCwUC5AfPeuXNbEyUl1IiIiIrUqVgR27waSkuQ9srBQdSLTxMJODyZMAA4eBHbsAOrWVZ2GiKpXr465c+eievXqqqMQmTVvb2D7duCnn+S9kkof17ErZdu3y61UPv1UDhYlIiKikj79FBg7Fvj6a6BHD9VpTAsLu1IUHQ00by7H1m3Zwj1giQxFeno6jh49itatW6NixYqq4xCZPSFkJ8jevXJPWR8f1YlMBwu7UpKVBTz3nCzmzpyRW6kQkWEIDQ1F06ZNERISgiZNmqiOQ0QAMjPlfdPCQu7K5OioOpFp4Bi7UiAE8N57QGKi3FmCRR0REdHfc3KS98xr14ChQ7kzRWlhYVcKVq2S07g3bAB8fVWnISIiMg7+/sD69cC2bdyZorRYqQ5g7E6dkgNA338f6NVLdRoiIiLj0rs38NtvwJgxQNOmQIsWqhMZN/bYPYPbt+Uq2s2aAZ98ojoNEf0VOzs7+Pv7w87OTnUUInqEhQvlvbRnT3lvpafHyRNPqagI6NQJCAkBwsKAGjVUJyIiIjJe168DTZrIz9693LHpafGv7Sl9+qlcYHHrVhZ1REREz6pGDWDzZnlvXbZMdRrjxcLuKYSEABMnAh9+CAQFqU5DRP8kPDwc5cuXR3h4uOooRPQ3XnlFjlv/6CP5NoyeHAu7J5SVJfe4CwgA5sxRnYaIHkdRUREyMzNRVFSkOgoR/YO5c4EGDeS99v591WmMDwu7J/T++8DNm3Jqto2N6jRERESmxdZW3mMTE+U9l54MC7snsGMH8PnnwIoVciNjIiIiKn0+PsDy5XJ92K+/Vp3GuLCwe0wPV8bu1QsYMEB1GiIiItM2aJBc/mToUCAhQXUa48HlTh5DQQHQtq3sFj5/HuAe4kTG5cGDB4iJiYGvry8cHBxUxyGix5SWBjRuDHh6AocPA1bcVuEfscfuMXzyiVwVe+tWFnVExsjBwQFNmjRhUUdkZJyd5b335Elg0SLVaYwDC7t/cOEC8PHHwPjxQMuWqtMQ0dNISEjA8OHDkcD3OURGp2VLYNw4YNo0ICJCdRrDx1exfyMvD3j+efkq9tw5OVOHiIxPaGgomjZtipCQEDRp0kR1HCJ6Qrm5ch9ZGxu5RztXpfhr7LH7G3PnApGRwKZNLOqIiIhUsbWV9+KICK4h+09Y2P2F0FD5wzN5sty3joiIiNRp2lTek+fMkTtA0aOxsHuE3FzgnXfkyteTJqlOQ0RERIAs7Bo2lPfonBzVaQwTC7tH+PhjIC4O+PJLvscnMgVVq1bFmDFjULVqVdVRiOgZWFvLe/OlS/JeTbo4eeJPzp2TEyZmzWJvHRERkSGaO1fOkj17FggMVJ3GsLCw+4P8fKB5c8DCAjhzhgshEpmKrKwsREREICAgAI6OjqrjENEzys+XY+6srYHTp3m//iO+iv2DJUvkjJt16/hDQmRK4uLi8OKLLyIuLk51FCIqBdbW8l4dFgYsW6Y6jWFhYff/Hr6vHzNGPgUQERGR4Xr+eWDkSPlKNj5edRrDwcIOgBDAf/4DVKsGzJihOg0RERE9jtmzgcqV5T2cA8skFnaQix7+8gvw3/8C5cqpTkNERESPw8kJWLMGOHBA7ilLLOxw5w7wwQdA377AK6+oTkNE+mBlZQUXFxdYcfAskcl57TXgrbfkUKo7d1SnUc/sZ8W+9x7wzTdAbCzAJa6IiIiMT3Iy4OcHdOsGfP656jRqmXWP3ZkzwIYN8h09izoiIiLj5Ooqtxr74gvg1CnVadQy2x67wkKgRQugoEAuSmxpqToREelLVFQUXn/9dezcuRP169dXHYeI9KCwEHjuOfnfZ86Y733dbHvsNmyQBd2qVeb7P5/IXOTm5uLy5cvIzc1VHYWI9MTSEli5EggNBdavV51GHbMs7O7eBSZOBN59F3jxRdVpiIiIqDS88AIwYIDcEvTuXdVp1DDLwm7yZNllu2CB6iRERERUmubPl/f4yZNVJ1HD7Aq7c+eAtWsLMHOmHGxJREREpsPVFZg5U97rQ0JUpyl7elvU6datW7h161aJX3N2dkbt2rWRk5ODixcv6vyZJk2aAABiY2Nx//79Em21atVCpUqVcPv2bSQmJpZoc3JygpeXFwoLC3H+/Hmd4wYEBMDa2hqXL19GcHAirK374/LlHjh9ujdq1aoFV1dXpKWlIf5Pe5LY29vDz88PABAWFoY/zzPx8/ODvb09rl27hrt/6vN1dXWFu7s7MjMzodVqS7RZW1sjICAAABAREYH8/PwS7V5eXnBycsKNGzeQnJxcoq1y5cqoWbMmsrOzER0dXaJNo9EgMDAQABAdHY3s7OwS7bVr14azszOSk5Nx48aNEm0VKlRA3bp1kZ+fj4iICJ2/w0aNGsHS0hJarRaZmZkl2jw8PFClShWkpqbi6tWrJdrKlSsHHx8fAEBoaKjOcf39/WFnZ4f4+HikpaWVaHNzc4Obmxvu3buHS5culWiztbUtHgR/4cIFFBQUlGj39vaGo6Mjrl+/jpSUlBJtLi4u8PT0xIMHDxATE1OizcLCAo0bNwYAXLx4ETk5OSXa69Spg4oVKyIpKQk3b94s0VaxYkXUqVMHeXl5iIyM1PleGzduDAsLC8TFxSErK6tEm6enJ1xcXHDnzh0kJCSUaHN0dIS3tzeKiooQHh6uc9wGDRrAxsYGV65cQXp6eom26tWro1q1akhPT8eVK1dKtNnZ2cHf3x8AEB4ejqKiohLtvr6+cHBwQEJCAu78aXGoqlWrokaNGsjKytLZf9XKygoNGzYEICct/HlcW7169VC+fPkyvUZoNBrs378ftWvXfuTP4R+vERkZGSXa3N3deY0ArxEP8RrxO0O9RuTl5SEhYQosLTfh1Vefx75906HRaEqtjiita8TDn99SJ/Rk+vTpAkCJT9++fYUQQmi1Wp22P0Zp0aKFTtvmzZuFEEKsXLlSpy0oKEgIIURGRsYjj5uSkiKEEKJZs2Y6bYMHDxZFRUVix44dOm2BgYHFmWxsbHTaIyMjhRBCDB48WKdtwoQJQgghDh8+rNPm7u5efFx3d3ed9sOHDwshhJgwYcIj8wohRGRkpE6bjY1N8XEDAwN12nfs2CGEEGLx4sU6bV26dBFCCJGSkvLIv8OMjAwhhBBBQUE6bStXrhRCCLF582adthYtWhRnetRxtVqtEEKIvn376rRNnz5dCCHE/v37ddrq1q1bfFwXFxed9pMnTwohhBgzZoxOW3BwsBBCiJCQEJ02Jyen4uP6+/vrtO/cuVMIIcTcuXN12nr06CGEECIxMfGR32tOTo4QQog2bdrotK1bt04IIcS6det02tq0aSOEECInJ+eRx01MTBRCCNGjRw+dtrlz5wohhNi5c6dOm7+/f/H36uTkpNMeEhIihBAiODhYp23MmDFCCCFOnjyp0+bi4lJ83Lp16+q079+/XwhhmNeILl266LQtXrxYCCF4jeA1QgC8RvzxY4jXiF27dgkvL69HthnaNeLhz29p09tyJ4bWY1dYWIh69eqhXr16WLJkCS5duoTFixfjzJkzGDt2LKZMmcKncT6N82n8Dwz1aRx48mtEdnY2fv75ZwwZMkTn5wFgj91DvEZIvEZIxnaN+Oqrr7Bw4UJ06NABo0aNgru7O95//33Ex8fjm2++gbe3t1n02JnNOnYLFizAlClTEBkZWXwREUJg165dqF69Opo3bw6tVgsXFxc4OzsrTktEpSk0NBRNmzZFSEhIcXFIRMYvLS0NFy5cQJs2bXD37l2cOHECXbp0gUajASAfYAICAjB//nx8+OGHitOWDbOYPJGSkoI5c+YgODi4uKgD5NPr66+/jubNmwMAhg0bBi8vL6xevVrn6Y6IiIgMQ0FBAdasWQMvLy/0798f+fn5qFy5Mrp27Vpc1AGy13zo0KFYu3YtCgsLFSYuO2ZR2H3++eewtLTEtGnT/vb3bdmyBV27dsWIESMQGBiIQ4cOlVFCIiIiehyHDh1CkyZNEBwcjK5du+L06dOwtrb+y98/Z84cnDt3DpZmshuBWbyKFUJAq9XC29v7sX7/uXPn8P777yMmJgbXrl2Do6OjnhMSkT7xVSyRaRBC4KWXXgIALFu2rPiN2+NITk5GXl4ePDw89BXPIJh8j11cXBw0Gs1jF3UA0KxZMxw/fhxnzpyBo6MjUlJSMHXqVJ0BwURkHJydndG3b1+OnyUyQpmZmZg0aRKOHj0KjUaD3bt348SJE09U1Akh0KFDB7z//vt6TGoYTLqwCwkJgY+PD3766acn/rMajQZ169YFAJw+fRqLFi2Ct7c3Nm3apDMziIgMW+3atbFlyxbUrl1bdRQiekxFRUXYtGkTfHx88OmnnyI2NhaAnPn9x3F0j0Oj0WDcuHH4/vvvcfbsWX3ENRgm/Sr2lVdeQUJCAiIiImBl9WxrMSckJOCjjz7CV199hebNm+Ozzz4rnvJORIYtJycH169fR40aNWBnZ6c6DhH9g8uXL6NPnz44e/YsevXqhU8++QQ1a9Z8pmMWFhYiICAAHh4eT9XhYyxMtsfu0KFDOHDgAObMmfPMRR0g1xDatm0bjh49CiFE8Vo07L0jMnwXL16El5fXI9fGIyLD8XBdQBcXF1SqVAlHjx7F9u3bn7moAwBLS0vMnj0bBw4cwK+//vrMxzNUJtljJ4RAixYtIITA6dOnn7jL9nGOr9FoUFBQgFatWqFTp0748MMPYW9vX6rnIaLSwckTRIYtOzsbixcvxsqVKxEaGorq1avr5TxCCIwZMwZDhgxBgwYN9HIO1Uyyxy4vLw9t27bFggULSr2oA1B8zPz8fLRu3RqzZs2Cr68vduzYobPyPBERET2aEALffPMN/Pz8MHPmTPTr1w/lypXT2/k0Gg2WLl1qskUdYKKFna2tLRYsWIB27drp9Tz29vZYsGABoqKi0LhxY7z11lvo3bu3Xs9JRERkKiZOnIiePXsiICAAkZGRWLRoESpUqKD38x4/fhx9+/Y1yeFUJlfY7dmzB7Nnzy7T/1leXl7YuXMnDhw4gL59+wKQ6+U8ak9KIiIic3b79m1cuHABAPDOO+9g37592L179xMtS/asNBoN/ve//+Gbb74ps3OWFZMaY1dUVITAwEBUqVIFBw8eVJpl6NCh2L59O6ZNm4aRI0fCxsZGaR4iIiKV8vLysGrVKsyYMQN+fn747bfflOZ55ZVXcPPmTZw/fx4WFqbTz2U63wmAXbt24cKFC5g+fbrqKJg3bx7eeecdfPTRR2jQoAF+/PFHjr8jIiKztG/fPjRs2BAffvgh+vTpg127dqmOhGnTpiEyMhI7d+5UHaVUmUyPnRACTZs2RYUKFXD48GHVcYpFRkZi9OjROHLkCC5dulQqU7aJ6MnExsZiwIAB2LhxI3x8fFTHITIrubm58PLyQr169bB06VI0bNhQdaRi7du3h62tLfbt26c6Sql59gXeDMSRI0cQFhZmUEUdADRo0AA///wzoqKiULNmTeTl5WHu3LkYNWoUKlWqpDoekVm4f/8+Tp06hfv376uOQmQW0tPTMWfOHAQHB6N27do4deoU3Nzc9LJSxbPYtGkTXFxcVMcoVSbzKrZNmzY4ffo02rZtqzqKDo1GUzy1+vz581iyZAm8vLywevVqFBQUKE5HRERUOgoLC/HZZ5/By8sLa9asQXh4OACgevXqBlfUAYCHhwfs7e1x69YtkxkuZRKFXWZmJjQaDZ577jnVUf5R8+bNERcXh27dumHEiBEIDAzEsWPHVMciIiJ6JuHh4WjatCn+/e9/o1OnToiLi8Mbb7yhOtY/io6ORs2aNfHzzz+rjlIqTKKw69y5M4YPH646xmOrVq0aNmzYgLNnz6JChQpITk4GAJN5WiAiIvPx8M2To6Mjypcvj1OnTmHTpk162z2itPn6+iIwMBCzZ89WHaVUGP3kiVOnTuGFF17A999/j27duqmO88Qe/vVrNBr06dMHtWrVwqRJk+Dk5KQ4GZHpSE1Nxd69e9GpUyeObSUqJffv38f8+fPx7bffIjQ0FHZ2dqojPbWdO3eiW7du+O2339CiRQvVcZ6J0ffYLVy4EN7e3ujatavqKE9Fo9FAo9GgqKgIvr6+WLZsGby9vbFx40aTXBGbSIVKlSqhX79+LOqISoEQAlu3boWPjw8WLlyI7t27G/39qkuXLsXfj7Ez6sIuLi4O33//PT788EOjX1zQwsIC06dPR0xMDNq2bYuBAweiTZs2Rv+PhcgQ3L59G6tWrcLt27dVRyEyev3790e/fv3QokULREdHY/bs2XBwcFAd65lYWFhg/PjxsLGxQWFhoeo4z8SoX8Xu2rULEydOREhIiFF3AT/K8ePHcfHiRQwdOhQPHjzA3bt34eHhoToWkVEKDQ1F06ZNERISgiZNmqiOQ2R0bt68ifz8fNSsWROHDx+GRqMxyFUoyMh77Lp27YqIiAiTK+oAoGXLlhg6dCgAYMWKFfDx8cHMmTPx4MEDxcmIiMhc5OTkYN68efD29saECRMAAO3atTPZoi4/Px9bt2416t59oy3sfvnlF9y+fdvoX8E+jmHDhmHkyJGYPXs2/Pz8sGPHDs6gJSIivRFC4Pvvv4e/vz+mTZuGoUOHYs2aNapj6d29e/cwdOhQrFy5UnWUp2aUVVF2djbeeustzJs3T3WUMlG+fHksWLAAUVFRaNy4Md566y2cOHFCdSwiIjJRKSkp6Nu3L3x9fREREYElS5agYsWKqmPpXeXKlTF48GCsWrXKaN+QGWVht23bNqSmpiI4OFh1lDLl5eWFnTt34uzZs2jZsiWEEFiyZEnxOnhE9GhOTk4ICgriMkJEf+POnTsYP348srKy4OrqisjISOzduxe+vr6qo5Wp999/H6mpqdi2bZvqKE/F6CZPCCHQpEkTuLu748cff1QdR6mEhAQ0btwYBQUFmDZtGkaNGgUbGxvVsYiIyIjk5+djzZo1mD59OoQQ+PHHH9GyZUvVsZTq3Lkzbty4gdDQUIPcCu3vGF2P3fHjxxEeHo5Ro0apjqKcp6cntFot3n33XUyYMAH169fH3r17VcciMjiFhYW4d++e0S9jQFTajh8/jkaNGmH06NHo1asX4uLizL6oA4BJkyZh5MiRRrnkmNEVdhUqVMDw4cPRoUMH1VEMQuXKlbFixQqcP38etWvXLt5wmYh+d/78eVSoUAHnz59XHYXIIDwsWHJzc1G1alWEhoZi7dq1qFq1quJkhuHFF1/EoEGDYGlpqTrKEzO6V7H014QQKCoqgqWlJaZMmYLMzExMnz6dq+2T2eM6dkRSRkYGZs+ejfPnz+Onn36CRqOBEMLoXjeWhVu3bmHixImYM2cO3N3dVcd5bEbVY7dhwwbs2rVLdQyDpdFoip8uXFxc8MUXX8DLywurV68u3qSZiIjMT2FhITZs2ABvb2+sXr0arVq1Kh6awKLu0cqVK4dvv/0Wa9euVR3liRhNYZeXl4eJEyfil19+UR3FKIwePRpxcXHo1q0bRowYgcDAQGRmZqqORUREZUwIgY4dO2LIkCEICgpCbGwspk6dCisrK9XRDFr58uXxzjvvYO3atcjNzVUd57EZTWG3c+dO3L59G//+979VRzEa1apVw4YNG3D27Fn07NkTTk5OEEIgISFBdTQiItKzhIQEpKenQ6PRYOjQoTh58iQ2b96MGjVqqI5mNEaMGIGUlBR89913qqM8NqMZY9exY0dkZ2fj+PHjqqMYtW+++QZ9+/bFmDFjMHnyZK7rRWYhPz8f6enpqFixIqytrVXHIdKrBw8e4JNPPsGCBQswduxYzJkzR3Uko9a6dWvY2Njg4MGDqqM8FqMo7K5cuYK6deti06ZNeOedd1THMWoPHjzAwoULsWDBAlSoUAHz5s3DO++8YxZbsxERmTIhBLZv345x48YhJSUFY8eOxaRJk/gA/4zCwsLg7OyMWrVqqY7yWIzibu7q6op169ahR48eqqMYPQcHB0yfPh0xMTFo27YtBg4ciC1btqiORaRXly9fRteuXXH58mXVUYj05vz58+jTpw+aN2+OixcvYt68eSzqSkFgYKDRFHWAkfTYkf6cOnUKTZs2hbW1Nb788ku0a9cOHh4eqmMRlSoud0KmKikpCevWrcPkyZNhYWGByMhINGjQQHUsk7N3717MmjULx44dM/hJJwbfY7dnzx4MGjTIqGakGJMWLVrA2toa9+/fx6RJk+Dj44OZM2ca7ebHRETmIDc3F5988gm8vb2xdOlSaLVaAGBRpydubm44deoU9u3bpzrKPzL4wm7VqlWIjIyEra2t6igmrVy5crh48SJGjhyJ2bNnw9fXFzt27FAdi4iI/mT//v2oX78+Jk2ahIEDB0Kr1cLHx0d1LJMWGBiIpk2bYt26daqj/CODLuxu3bqFn376CYMHD1YdxSyUL18eCxYswMWLF9GkSRPuO0tEZEAejpx6OKHwwoULWLZsGXcXKiNDhgzBnj17cPPmTdVR/pZBj7FbvHgxJk2ahKSkJDg7O6uOY3by8/NhbW2NL774AidPnsTs2bPh6uqqOhbRE0tOTsbWrVvRt29f/gyT0UlNTcX06dNRWFiI1atXo6ioCBqNhjtGlLGMjAy4urpiyZIlCA4OVh3nLxl0j92WLVvQpUsXFnWKPFzvy8LCAt9++y28vLywaNEi5OXlKU5G9GRcXV0xduxYFnVkVAoKCrBq1Sp4eXlh06ZNqFevHgB5TWZRV/YqVKiACxcuYNiwYaqj/C2D7rG7cuUK8vLy4OvrqzqK2bt79y4+/vhjrFmzBrVr18bBgwdRs2ZN1bGIHktaWhoOHjyIDh068EGRjEJubi6ee+45REREYNCgQZgzZw4fTAxIYWFh8d7shsage+zq1KnDos5AVK5cGStWrMD58+fxyiuvFG9Jk5SUpDgZ0T+Lj49Hr169EB8frzoK0d+Kj49HQUEBbG1tMWDAAJw9exbr169nUWdAunTpgsmTJ6uO8ZcMsrArLCxEy5YtsWfPHtVR6E/q16+PlStXwtLSEqGhofD09MSoUaOQmpqqOhoRkdHKzMzEhAkT4Ovri02bNgEAxowZg6ZNmypORn/m6emJrVu3oqioSHWURzLIwu7QoUM4ceIEqlSpojoK/Y369etjzpw52LhxI7y8vLB69WoUFBSojkVEZDSKioqwceNGeHt7Y/ny5Zg0aRL69OmjOhb9jX79+uH69es4evSo6iiPZJCF3datW+Ht7Y3mzZurjkJ/w9bWFuPGjYNWq8Ubb7yBESNGYO7cuapjEREZjV27dmHgwIFo27YtYmJiMH36dDg4OKiORX+jRYsWqFOnjsFux2lwhV1ubi5++OEHvPXWW5z1YyRcXV2xfv16nDt3rngK+N69e7kvJxkMe3t7BAYGwt7eXnUUIly/fh1r1qwBAHTt2hVnz57Ftm3b4OnpqTgZPQ6NRoN+/frhwIEDBvk61uBmxe7fvx+vvvoqLly4gICAANVx6CkUFRWhSZMmiI6OxpgxYzB58mRuRE1EZi87OxuLFi3C/Pnz4eTkhIsXL3JxYSN179492NraGuSuWAbXYxcUFIQzZ85wvzsjZmFhgZMnT2LSpElYvnw5vLy88MUXXxjkkw0RUVn49ttv4evri1mzZmH48OGIi4tjUWfEypcvD1tbW+Tn56uOosPgCjsLCws0b96cr2GNnIODA6ZPn47Y2Fi0b98e69ev5/9TUiYsLAy2trYICwtTHYXM1PHjx9GoUSNERUXhk08+Qfny5VVHome0detWeHh4GNyi/QZV2O3btw8vvfQSMjIyVEehUuLh4YH//e9/+OWXX6DRaHD06FG8/fbbSExMVB2NzIgQAnl5eTCwkSdkwlJSUjB06FAsWbIEALBw4ULs2rULXl5eipNRaWnYsCGSk5Pxyy+/qI5SgkEVdtu3b8fdu3f5JGOC7OzsAMi99g4dOgQfHx/MmDEDDx48UJyMiKj05OXlYfHixfDy8sLXX39dfD+zsrJSnIxKW4MGDeDt7Y2vv/5adZQSDKawy8vLw86dO9GzZ0++sjNhXbp0QVxcHEaNGoW5c+fC19cXoaGhqmMRET2z5ORkNGjQAOPHj0f//v1x6dIlDBkyRHUs0hONRoMePXrghx9+MKixdgZT2B06dAjp6eno2bOn6iikZ+XLl8f8+fMRFRWFdu3aoW7dugDkfrRERMbm4dCSqlWrolu3bggPD8fKlStRuXJlxclI33r27Il79+4hPDxcdZRiBrPcyYgRI7Bnzx5cuXKFPXZmKCkpCT4+PujZsyc3u6ZSl52djStXrqBOnTpcy45KTVpaGmbMmIFVq1bhwIEDaNeunepIVMaEEEhNTTWoIt5gCrsHDx4gPj4e9evXVx2FFCgoKMDatWsxbdo05OfnY+rUqRg1apRBrhFEROatoKAA69evx5QpU5Cbm4spU6Zg9OjRvF6ZscLCQlhYWBhEx5TBvIp1cHBgUWfGrKysMHz4cGi1WgwYMAATJ07EsGHDVMciE3Ht2jUMGTIE165dUx2FTMDKlSsxbNiw4jHDH330EYs6M3bx4kVUq1YN58+fVx0FgIH02C1atAiRkZHYuHGj6ihkIKKiomBtbQ1vb2+cPn0ajo6OLPzpqYWGhqJp06YICQlBkyZNVMchI3TlyhWEhISgZ8+eyMrKQnR0NPczJwBy8qeLiwvGjx+PKVOmqI5jGD1227dvR05OjuoYZEDq168Pb29vAMDs2bPRqFEjjBo1CqmpqYqTEZE5ycrKwuTJk+Hv74/JkycjPz8fjo6OLOqomI2NDV555RX8+OOPqqMAMIDC7saNGzh37hy6dOmiOgoZqG+++Qbz5s3Dxo0b4eXlhVWrVqGgoEB1LCIyYUIIfPnll/D29saSJUswfvx4hIWFwdraWnU0MkCdO3fGmTNnkJycrDqK+sJuz549sLS0xKuvvqo6ChkoW1tbjBs3DlqtFm+88QY++eQTg9vChYhMi0ajwfbt29GqVSvExMRg5syZKFeunOpYZKAe1jC//vqr2iAwgDF23bt3x+3bt3Hs2DGVMciIZGVlwdHREVevXsW4ceMwb9481KtXT3UsMmA3btzAypUrMWLECLi7u6uOQwbqxo0bmDhxInr27IkuXbogNzeXkyLosSUmJsLDw0N1DPU9dqtWrcLq1atVxyAj4ujoCAC4fv06Tp8+jfr16+Ojjz7CvXv3FCcjQ+Xu7o558+axqKNHys7Oxpw5c+Dt7Y39+/cXj/lmUUdPwhCKOsAACjs3NzcEBASojkFGqGXLloiJicGkSZOwYsUKeHt74+DBg6pjkQHKzMzEr7/+iszMTNVRyMBER0fD398fH3/8MYYNGwatVssdkOipXL9+HX5+fjh+/LjSHEoLu5UrV+KDDz5QGYGMnIODA6ZPn47Y2Fh07NgRNWvWBAD23lEJWq0W7dq1g1arVR2FDMTt27cBALVr10aHDh0QGRmJRYsWoUKFCoqTkbFyc3NDcnIyfv75Z6U5lBZ227Zt44KhVCo8PDywefNmeHl5IScnB40bN8bbb79dvIcjEREgC7phw4ahZs2auHz5Muzs7LBu3Tr4+PiojkZGztLSEi+//LL5Fnbp6ek4ffo0goKCVEUgE2VjY4OpU6fi0KFD8PHxwYwZM/DgwQPVsYhIofz8fCxbtgze3t7Ytm0b5s6dC09PT9WxyMR07NgRZ86cQUZGhrIMygq7w4cPo7CwkIUdlToLCwsMHDgQcXFxGDVqFObOnYvOnTurjkVECo0fPx5jx47FW2+9Ba1Wi9GjR3NNOip1HTp0QGFhodJlT5QtdzJs2DAcOnQIsbGxKk5PZuTy5cu4ffs2WrRogUuXLiEjIwNNmzZVHYvKUEREBF599VXs27ePk7XMSGxsLG7duoW2bdsiISEBaWlpaNSokepYZOJCQkLQqFEjWFlZKTm/ssLu4sWLSEpKQvv27VWcnszUiBEjsHr1agwaNAhz5syBq6ur6khEVMrS09Mxa9YsLF++HC+88AKOHj2qOhJRmVH2Ktbf359FHZW5pUuXYsWKFfj+++/h5eWFhQsXIjc3V3UsIioFhYWFWLduHby9vbF27VrMmDEDBw4cUB2LzExkZCTatm2LpKQkJedXUtj9+uuvmDRpEvf7pDJnZWWF4cOHQ6vVYsCAAZg1axZSUlJUxyI9i4iIQI0aNRAREaE6CulRUVERPv30U/zrX/9CbGwsJk2aBDs7O9WxyMxUqlQJR44cUbajlpLC7rvvvsOOHTuUvX8mqlSpEpYvX45r167Bw8MDWVlZePvttxEVFaU6GulBfn4+bty4gfz8fNVRqJRdvXoVvXv3RkREBKytrXH27Fl8+eWX3GWElKlevTrq1q1rXoXd0aNH0aZNGxWnJirB2dkZAHDt2jWcPXsWjRo1wsiRI5Gamqo4GRH9nfv372Pq1Knw8/PD0aNHi197lStXTnEyIqBVq1bKxnaWeWGXlpaGCxcuoHXr1mV9aqK/VL9+fURGRmL+/PnYtGkTvLy8sGPHDtWxiOgRTp48CR8fHyxcuBAffPAB4uLi0LFjR9WxiIq1bt0aFy5cQHp6epmfu8wLu+PHj0MIwcKODI6trS0+/PBDaLVavPHGG8W9eVzcmMgwPFz0tXbt2mjTpg2io6Mxe/ZsODo6Kk5GVFKnTp2wa9cuJWM8y3y5k+joaHz33XeYNGkSNBpNWZ6a6IkJIdCuXTtUrFgRixYtQr169VRHoqeQmZmJkJAQNG3aFE5OTqrj0BO6desWJk2ahP379yMmJob7uRL9jTLvsfPz88PkyZNZ1JHRCA4ORmhoKPz9/fHRRx/h3r17qiPRE3JyckLbtm1Z1BmZnJwczJ8/H97e3ti9ezemT5/OMXRkNH766SdMmzatzM9bpoVdQUEBli9fjhs3bpTlaYmemkajQa9evRATE4MpU6ZgxYoVaNasGQoLC1VHoydw48YNTJw4kdceI/P2229j6tSpGDJkCLRaLf7zn/9wNQUyGhcvXsTChQvLfDZ+mb6KDQ8PR2BgII4dO4aWLVuW1WmJSk1iYiIuXLiA1157DWlpaYiKiuLPshEIDQ1F06ZNERISgiZNmqiOQ38jMjISANCgQQOEh4fD1tYWfn5+ilMRPbkTJ06gZcuWCAsLQ+PGjcvsvGXaY3f69GlYWlrywkpGy8PDA6+99hoAYMOGDWjVqhX69OmDhIQExcmIjNvdu3cxYsQINGrUCPPnzwcANG7cmEUdGa3AwEBYWlrizJkzZXreMi/sAgIC4ODgUJanJdKLsWPH4osvvsDhw4fh6+uLGTNmcAYt0RPKz8/HihUr4OXlhc2bN+OTTz7B559/rjoW0TNzcHBAQECAaRd2Z86cwfPPP1+WpyTSGwsLCwwYMABxcXEYNWoU5s2bh/Pnz6uORWRU7t27hxkzZqBHjx7QarX44IMPYGNjozoWUakYO3YsOnXqVKbnLLPCTgiBF198Ea+++mpZnZKoTJQvXx7z589HQkICXnjhBRQVFSE4OBghISGqo9H/q1y5MgYPHozKlSurjkIAtFot3n77bdy5cweVK1fGpUuX8Nlnn6Fq1aqqoxGVqv79+6N79+5les4yX8eOyNTdunULQUFBiIqKwoABAzB37lxUq1ZNdSwi5e7du4fZs2dj6dKlcHNzw7fffotmzZqpjkWkN/n5+fjhhx8QGBhYZuugllmP3eXLl3Ht2rWyOh2RMm5ubggLC8PKlSuxc+dOeHt7Y82aNapjmbXs7GxERUUhOztbdRSztWfPHnh5eWHlypWYOnUqYmJiWNSRyXs4ZOeHH34ou3OW1Yk+/vhjvPXWW2V1OiKlrKysEBwcDK1Wi4EDB8LS0hIAkJubC3aSl73o6Gg0aNAA0dHRqqOYnfv37wMAXF1d0aFDB8TGxmLq1Kmwt7dXnIxI/ywtLdGwYcMyHX9dZoXd+fPn0ahRo7I6HZFBqFSpEpYtW4ahQ4cCAEaOHIlXXnkFUVFRipMR6VdCQgJ69+6NVq1aobCwEM2aNcPWrVvh4eGhOhpRmWrUqJHpFXa5ubmIjo5mYUdmr2vXroiPj0ejRo0wcuRIpKamqo5EVKoePHiAjz/+GL6+vjhy5AhGjRrFLSTJrDVs2BDR0dHIzc0tk/OVSWEXHR2NgoICFnZk9jp37ozIyEjMnz8fmzZtgo+PD9LT01XHIioVQgi0adMG8+bNw/vvv4+4uDgMGDAAFhZlvi05kcF47rnnEBQUVGbX+jLZdC85ORlVqlRBw4YNy+J0RAbN1tYWH374Ifr3748DBw6gYsWKKCgowMmTJ9G6dWvV8UySRqOBjY0Ne470JCQkBNWrV4ebmxvmzZuH2rVro27duqpjERmEZs2aYc+ePWV2Pi53QmQAvvnmG/Ts2RNdu3bF4sWLy2xaPNGzSEpKwuTJk/HFF19gwoQJmDt3rupIRAYpKysLqamp8PT01Pu5yqR/vKCgoCxOQ2S03nzzTWzfvh3h4eHw9/fHRx99hHv37qmORfRIubm5WLhwIby9vfHDDz9g5cqVmDlzpupYRAarf//+GDRoUJmcq0wKu1q1amHFihVlcSoio6TRaNCrVy9ER0djypQpWLFiBXbt2qU6lsmIjo5GkyZNuNxJKbl8+TKmTp2KAQMGQKvVIjg4GFZWZTKyh8goNWrUCBEREWVyLr0Xdmlpabhx4waqVKmi71MRGT0HBwdMmzYNly5dwttvvw0AmDlzJo4fP644mXHLzs5GWFgYFyh+BlFRURg6dCjy8/Ph7++PhIQELF++HJUqVVIdjcjg+fr6IiUlBWlpaXo/l94Lu9jYWADymyKix1O9enVYWFggOzsbP/74I1q1aoU+ffogISFBdTQyM6mpqRg1ahQaNWqEQ4cOFf8Mcl9Xosf3sAZ6WBPpk94Lu5iYGACAt7e3vk9FZHLs7e1x6tQpfPHFF/j111/h6+uL2bNnq45FZmLr1q3w8vLCxo0bMW/ePERFRXG2K9FT8PLygrW1NW7cuKH3c+l9UERMTAxq1qwJBwcHfZ+KyCQ93GvwzTffxNy5c5GZmQkAKCwshIWFBZfwoFKXl5cHGxsbWFlZoVu3bpgzZw6qVaumOhaR0SpXrhwePHhQJmNR9b7cSWZmJpKSkuDl5aXP0xCZnUWLFuGHH37AsmXL0LRpU9VxDFpaWhoOHjyIDh06wNnZWXUcg3X58mV8+OGHsLa2xo4dO1THIaKnoPdXsU5OTizqiPSgadOmyMjIQPPmzTF48GAkJSWpjmSwnJ2d0bNnTxZ1fyEzMxMTJ06Ev78/zp07h+7du6uORGRyli1bhjZt2uj9PHot7PLz89G1a1ecPHlSn6chMkvt2rVDWFgYVq5ciR9++AHe3t7QarWqYxmk5ORkLFmyBMnJyaqjGJzc3FwEBARg6dKlmDhxImJjY9G7d2/VsYhMjpWVFX777Te9r+2r18Lu2rVr2L17Nx48eKDP0xCZLSsrKwQHB0Or1WLWrFnFO1acOnUK3FTmdzdu3MAHH3xQJgOXjcWpU6eQk5MDW1tbzJs3D7Gxsfj44485HppIT3x8fJCfn4+rV6/q9Tx6Lewehq9du7Y+T0Nk9ipVqoT3338fGo0GISEheOGFFxAUFITIyEjV0cjAXL9+HX379sULL7yATZs2AQD69OlTJlsdEZmzOnXqAIBxF3bx8fGwsLCAh4eHPk9DRH/QpEkT7N69G1evXkXjxo0xcuRIpKamqo5FimVnZ2PWrFnw8fHBwYMHsWHDBgwZMkR1LCKzUaNGDVhYWBh3YXf16lW4u7vDxsZGn6choj/QaDTo3LkzoqKiMH/+fGzatAnLly9XHYsUO3ToEGbNmoXhw4dDq9Vi0KBBsLS0VB2LyGzY2Njg559/RpcuXfR6Hr0ud3LixAlcunQJ7777rr5OQUT/IDk5GeXKlYOjoyPWr1+PmjVromPHjqpjlanLly9jzJgx+PTTT81qgd2wsDB88803mDNnDoQQuH79Ot+gEJk4va9jR0SGQQiBV199FT/99BO6du2KxYsXF0+2INOSkpKCKVOmYP369fDz88OxY8e4pyuRAdi9ezeioqIwYcIEvZ1Dr69iN27cyOUXiAyERqPBvn37sH37doSHh8Pf3x/jx4/X+9R7Q5Cfn4/bt28jPz9fdRS9W716Nby8vPD1119j2bJlCA8PZ1FHZCDOnj2LFStW6PUceivssrOzMXDgQJw4cUJfpyCiJ6TRaNCrVy/ExMRg6tSpiI+Ph5WVFYQQKCoqUh1PbyIiIlC1alVERESojqI3hYWFAOQuG/369YNWq8XIkSNhbW2tOBkRPVS7dm3cvHkTubm5ejuH3gq769evAwBq1qypr1MQ0VOyt7fH1KlTi7eN2r17N5o3b47jx48rTkZPKjo6Gq+++iqmT58OAJg8eTJWrVoFFxcXxcmI6M8e1kQJCQl6O4feCrubN28CAKpXr66vUxDRM9JoNAAANzc3WFpaolWrVujdu7deLzpUOtLS0jBmzBg0bNgQcXFxeO6551RHIqJ/4O7uDgC4deuW3s6ht8LuYWg3Nzd9nYKISknz5s1x6tQpbNy4EUeOHIGvry+OHj2qOhb9heTkZHh7e2PdunWYNWsWoqKi0LVrV9WxiOgfuLu7Y+TIkahSpYrezqG3ws7R0RHt27eHk5OTvk5BRKXIwsIC7777LuLi4jBlypTiHqDz589zezIDce7cOQgh4OrqimnTpiEuLg4TJkyAnZ2d6mhE9BgcHR2xfPly+Pn56e0cXO6EiP7SrVu3UKtWLTRv3hxLly5Fs2bNVEd6KoWFhbh//z7KlStnlIvyxsfHY9y4cfj222/x008/ISgoSHUkInpKUVFRsLS0hK+vr16Or7ceu4yMDLNYRoHIlLm5uWHv3r3IyMjAc889h0GDBiEpKUl1rCdmaWmJ8uXLG11Rl5WVhSlTpsDPzw+//fYbtmzZYnaLSxOZmv/85z+YM2eO3o6vt8Kue/fu6Nevn74OT0Rl5OWXX0ZYWBhWrVqFXbt2Ydy4caojPTGtVotXXnnF6NbV/OKLL7Bo0SKMGzcOsbGx6Nu3b/GEFyIyTm5ubnqdPGGlrwPfvHkTDRs21NfhiagMWVlZYdiwYejdu3fx+kt79uxBQUEBunbtavDFRmZmJg4cOIDMzEzVUf7R6dOncfbsWYwYMQL//ve/0aVLF9SqVUt1LCIqJdWqVcPFixf1dny99dilpKTA1dVVX4cnIgWcnZ1RrVo1AMB3332Hbt26ISgoCJGRkYqTGb+bN2/i3XffRYsWLbBp0ybk5+fDxsaGRR2RidF3j51eCrvCwkKkpaWhcuXK+jg8ERmA9evXY/fu3bh27RoaN26MESNGICsrS3UsoyOEwLx58+Dt7Y29e/fis88+w6lTp7hjBJGJqlGjBipUqKC33X70UthlZGRACMHCjsiEaTQadO7cGZGRkViwYAHOnTsHW1tbAODyKI9BCAEhBDQaDWJjY/Hvf/8bWq0W7733ntFN8iCix9e/f39cuXIFFhb6eWmql+VOhBDIyMiAnZ0d11ciMhMPi5TIyEj07dsXixYtMpgZnLdv38aOHTvQq1cvvS4M+rguXLiA0aNH491338W7775b/HdHRPSs9FIuajQaVKxYkUUdkRl5WJhYWlqiYsWKCAoKQteuXQ1iJmqVKlUwfPhw5UXd7du3MWzYMAQGBuLmzZuoUaMGALCoIzIjMTExqF69OkJCQvRyfL0Udr/99hs6deqEtLQ0fRyeiAyYn58ffv31V+zYsQPnz59H/fr1sWPHDqWZUlNTsWXLFqSmpirLEBERAW9vb2zbtg2LFy9GREQEXn75ZWV5iEgNBwcH3Lp1C3fu3NHL8fVS2F2+fBn79u1jjx2RmdJoNOjZsydiYmIwbdo0tGzZEgAQFxeHwsLCMs9z9epV9O/fH1evXi3zc0dFRQGQBe/o0aOh1WoxevRoTo4gMlOVKlUCAL09aOqlsEtNTYWdnR3s7e31cXgiMhL29vaYMmUKqlevjpycHLRu3RrNmzfHsWPHVEfTu7i4OHTu3BkBAQGIiYmBlZUVpk+frvx1MBGpVa5cOVhZWRlXYZeRkYEKFSro49BEZKTs7Ozw3XffwdLSEq1bt0bv3r2RkJCgOlapS09PxwcffID69esjKioKX3/9NXx8fFTHIiIDodFoUKlSJb0NV9NLYZeVlQVHR0d9HJqIjNiLL76I06dPY+PGjThy5Aj69OmjOlKpmzFjBtauXYsZM2YgOjoab775JidHEFEJu3fvxrvvvquXY+tluZOTJ08iPj4effv2Le1DE5GJyMzMRFJSEry8vBAaGorY2Fj07t1bL0VQbGwsBgwYgI0bN+ql9+zYsWNISUnBm2++iTt37iA3Nxfu7u6lfh4ion+il8KOiOhJTJs2DbNmzcJLL72EpUuXolmzZqojPZZr165h/Pjx2LFjB1599VXs3btXdSQiMgL//e9/4ejoiH79+pX6sfXyKvbQoUM4cuSIPg5NRCZo5syZOHjwIDIyMvDcc89h0KBBelsKoDTk5eVh2rRp8PX1xbFjx7Bp0yb8+OOPqmMRkZH47rvvsGvXLr0cWy+F3cKFC7Fs2TJ9HJqITNTLL7+MsLAwrFq1qsSs2dJ4qRAaGgqNRoPQ0NBnOs7DLFZWVjhy5AjGjh2LuLg4vPPOO3rbHoiITI+DgwPu37+vl2Pr5Up0//59lCtXTh+HJiITZmVlhWHDhiEmJgYuLi64c+cOnnvuOezcuVP5/rPnzp1Dq1atcPLkSVhYWODw4cOYM2cOJ4oR0RMrV64cHjx4oJdjc1YsERkcS0tLAMCDBw9QuXJldOvWDR07dkRkZGSZZ0lKSsKgQYPQvHlz3Lt3r7hnjj10RPS0ypUrZ1w9drm5ubC1tdXHoYnIjHh6emLfvn348ccfkZCQgEaNGmH58uVldv5ffvkFXl5e2LVrF9asWYPQ0FC0aNGizM5PRKapffv2eOONN/RybCt9HLRhw4aoW7euPg5NRGZGo9HgtddeQ8eOHbFixQq88MILAIDExERUq1at1LfmEkIgPj4ederUQZMmTRAcHIwJEybA2dm5VM9DROard+/eejs2lzshIqMjhEDz5s2RnZ2NpUuXomPHjn/7+3NycnD9+nXUqFHjb/ewjoyMxOjRo3HmzBlcvXq1eE9HIqLSdOvWLdy5cwcBAQGlfmy9vIotLCxUPtCZiEyXRqPBunXr4OLigqCgIHTt2hVarfYvf7+dnR3q1av3l0Xd3bt3MWLECDRq1AgJCQnYtm0be+iISG/++9//4rXXXtPLsfVS2Hl6emLmzJn6ODQREQAgMDAQv/76K3bs2IHz58+jffv2yM/Pf+TvjY+PR79+/RAfH//I9sGDB2Pz5s345JNPEBkZiddee43bgBGR3lhaWqKgoEAvx9ZLYVdQUAArK70M3yMiKqbRaNCzZ0/ExMRg165dsLa2xs2bN7F+/XoUFhYW/760tDRs3bq1xKbbP//8M06cOAEAWLx4MeLi4vDBBx/AxsamzL8PIjIvVlZWJa5RpYmFHREZPXt7ewQGBgIA9uzZg/feew/NmzcvsdDxQ5cuXcLrr7+OoKAgbNy4EQBQt25duLq6lmVkIjJj7LEjInpM7733Hk6ePAkrKyu0bt0avXv3RnJyMgBg2bJl8Pf3R3h4OLZv347PPvtMcVoiMkf29vZ628iBhR0RmZwXXngBp06dwsaNG3HixAlkZmYCkPtYT506FTExMejVqxfH0RGREqNGjUJCQoJejq2X5U7i4+Ph7OyMihUrlvahiYieSH5+Pu7cuYO1a9di0KBB8PT0VB2JiEhvuI4dERERkYngZodEREREJoKFHREREZGJYGFHREREZCJY2BERERGZCBZ2RERERCaChR0RERGRiWBhR0RERGQiWNgRERERmQgWdkREREQmgoUdERERkYlgYUdERERkIljYEREREZkIFnZEREREJoKFHREREZGJ+D9ifyPrHABP6gAAAABJRU5ErkJggg==", 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"text/plain": [ | |
"Graphics object consisting of 9 graphics primitives" | |
] | |
}, | |
"execution_count": 12, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"def make_plot(phi0,xmax=1,ymin=0,aspect_ratio=1):\n", | |
" return plot(sqrt(1-x^2),x,-sin(phi0),sin(phi0),axes=false,aspect_ratio=aspect_ratio,\n", | |
" xmin=-xmax,xmax=xmax,ymin=ymin,ymax=1) \\\n", | |
" + ellipse((0,1-b(phi=phi0)),0.025*sin(phi0),0.025*sin(phi0)/aspect_ratio,fill=true,color='black') \\\n", | |
" + plot([cos(phi0),1-b(phi=phi0),1],x,-xmax,xmax,linestyle='dashed',color='black') \\\n", | |
" + plot(sqrt(1-x^2),x,-1,-sin(phi0),linestyle='dashed',color='black') \\\n", | |
" + plot(sqrt(1-x^2),x,sin(phi0),1,linestyle='dashed',color='black') \\\n", | |
" + plot(abs(x*cot(phi0)),x,-sin(phi0),sin(phi0),linestyle='dashed',color='black') \\\n", | |
" + line([(0,0),(0,1-b(phi=phi0))],linestyle='dashed',color='black')\n", | |
"make_plot(pi/3)" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 13, | |
"id": "a5081ec3-230e-47ab-8b69-3e5957a657de", | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"data": { | |
"image/png": 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jLDIyUqE2ZcoUxhhjJSUlnf4Mq6qqGGOMBQYGKtQ2b97MGGMsOjpaoebn59eRqbPj5uTkMMYYCw4OVqiFhYUxxhiLi4tTqDk6OnYc18rKSqGemJjIGGMsJCREobZkyRLGGGNJSUkKNRMTk47jurm5KdRjY2MZY4xFREQo1CSSmSwoiLEdOwo7/V4bGhoYY4wFBAQo1KKiohhjjEVFRSnUAgICGGOMNTQ0dHrcwsJCxhhjM2fOVKhFREQwxhiLjY1VqLm5uXV8ryYmJgr1pKQkxhhjS5YsUaiFhIQwxhhLTExUqFlZWXUc19HRUaEeFxfHGOu+MeKTTz7p9HPv9qIxQtwYYWHhx7S1GdPRUf8xYubMmYwxxgoLaYxoJ2KMUJY+ov33t6vRjB2djQOgGbt293I2Xl1tjk2bbiIm5jpu3gTs7IDp04G5c80xYgSdjSvD2fiQIUNw8+ZNhc+/U9++ffHDDz8AoDGinagxwsJiMPbuBTZvTsbly3z5lKAg/uDFuHGqN0bQjJ3yjxHK0Ed014wd3WNHyD1oa+N7sn76KfDdd4CRETBnDjB/PjByJN03p2zWr1+PVatWKbwRtZNKpYiIiMDKlSt7OBm5G8aAs2eBbdv4/XjNzXx9vBde4Lut0CLIhPw1auwIuYvKSmDXLmDrVuDSJb5EydKl/EEIWjhYeVVXV2PkyJHIyclBa2urXE1LSwtOTk44e/YsTE1NBSUkf6WyEti9mzd5GRmAgwOwaBGwcCFtpUfI3VBjR0gnUlOBLVv4UiVNTcATT/CGzt+fZudURVlZGRYvXoyDBw92zNxJpVLMmDEDW7duhZWVleCE5F4wxpcNap/Fk0r5E7WvvAJ4eIhOR4jyocaOkN+0tfGdIDZuBE6eBGQyfglo4UJ+zw9RTUVFRYiJicGKFSvw448/4pFHHhEdiTyg0lJg+3Z+0nX9Ot9yb/lyvtMFXaYlhKN/CkTj1dfz2QBXV36zdksL8J//APn5wFtvUVOn6mQyGSZMmACA39BNVFfv3sCqVfzfZkwMXy5lyhRg8GDgk0/4ji6EaDpq7IjGKi0FwsMBW1vgxRf5/XOJifw1cyagoyM6ISGkM7q6fNHvs2f5Zdphw4CQEP5vOTwc+MMDmoRoFGrsiMbJzQUWL+ZvAu+/z59uzcnh+1uOGiU6HSHkfvj58V1ecnP5vXfvvQcMHMjvwbt6VXQ6QnoeNXZEY2Rm8oHf2Rk4dAhYvRooKOCXcBwdRacjhPwdAwcCmzbxf9NvvMEffHJ0BJ5+mm/zR4imoIcniNpLTgbeeQf45hvAxgZYuZKvP2dgIDoZIaS71NYCO3bwh6EKC4EZM4C33wa8vEQnI6R70YwdUVvnzgGTJwNDhwJpacDnn/MNyJcupaaOEHVnZMQvx+bmAl98wZcw8vbmSxd1srEAIWqDGjuidtLS+NOtI0YAeXl8kdPsbGDBAn7TNdE8Fy9exKhRo3Dx4kXRUUgP09EBnn+ejwF3NngzZvD/JkTdUGNH1EZmJjB7Nh+0L1wAvvySr1gfHAxoa4tOR0Sqra3FmTNnFPaOJJrjzgZv505+Aujjwxs8ugePqBNq7IjKu3yZ3yDt4cGXP/j8cyAri/+ZlpbodIQQZaKjAzz3nHyD5+UFPPMMcOWK6HSE/H3U2BGVVVICvPQSX1j4+HFg82a+n+uCBbQGHSHk7u5s8DZvBv77X77QcUgIrYNHVBs1dkTl1NYC69bxpQx27wYiIvis3ZIlgJ6e6HSEEFWio8PHjsuX+U4zO3YADg58jKEr90QV0XInRGW0tAC7dvElC8rK+GzdqlVAr16ikxFld+vWLRw+fBiPPfYYLC0tRcchSqy0lJ8sbtkCWFgAYWHAP/9J9+kS1UGNHVF6jAGHDwMrVvAHJObO5evS2duLTkYIUVdXrvCmLjoacHMDPvwQmDRJdCpC/hpdiiVK7eJF4LHHgMcfB/r04WvTxcRQU0fuT2lpKT799FOUlpaKjkJUhJ0d8O9/A+fP85m7wEC+jNLly6KTEXJ31NgRpVRdDbz+On/S9eJF4OBB/oDEsGGikxFVVFhYiJdeegmFhYWioxAV4+sL/PQTsH8/X/fOzY1fPaiuFp2MkM5RY0eUSlsbv4/O2RnYuhUID+eXX6dPByQSweEIIRpJIuFrZGZn8wcsPv0UcHLiSyu1topOR4g8auyI0vjlF2DUKL6I6EMP8UF01SpAX190MkII4VsRvvUWv4owaRJ/qGLECD52EaIsqLEjwpWX87XnRo4EGhuBhARg717AxkZ0MkIIUTRgAF9qKTGRP9zl5we8+CJQUSE6GSHU2BGBGOPbfrm4AF9/zS9vJCUB48aJTkbUjYmJCQIDA2FiYiI6ClEjo0bx2bqPPgL27OFjWXQ0H9sIEYWWOyFCXLrEz3CPHwfmzOFLCfTtKzoVIYQ8mOvXgVdf5Q9ZjB/P18FzdRWdimgimrEjPaqxEVi7FvD0BPLzgbg4ftmVmjrSnVpbW1FdXY1WutOddJP+/YF9+/jWZNeu8f1nw8OBpibRyYimocaO9JjTpwFvb+D//g947TUgIwN4+GHRqYgmSEtLg5mZGdLS0kRHIWpu0iQgPR1YuZIvpD50KD1cQXoWNXak29XV8Y21x44FzMyAlBS+ZY+hoehkhBDS9fT1+QlsUhLfv3rUKH4yW1cnOhnRBNTYkW518iS/JPHZZ8D77/NZOw8P0akIIaT7DRkCnDkDrF/P77nz9ATi40WnIuqOGjvSLWprgWXLgIAAwNoaSEvjZ6xaWqKTEUJIz9HWBt54A/j1V75Myj/+ASxeDNy+LToZUVfU2JEul5DAz1SjooAPPuD/29lZdCpCCBHHyYnP1m3ZwpdE8fICTp0SnYqoI2rsSJdpaOCP+48fD8hk/Ax1+XKapSPieXp6oqSkBJ6enqKjEA0mlfJlntLS+EoA48bxfWcbGkQnI+qE1rEjXSI9HQgO5lvtvPsub+ikdNpACCGdam0FIiP5FmVOTnwWz8dHdCqiDuitl/wtbW18ceFhw/hq6+fP81k7auqIMsnNzcXUqVORm5srOgohAPiVjBUrgHPn+H+PGMFXC6ClFsnfRW+/5IEVFfF16F59FVi6lA9QdKWLKKOqqip89913qKqqEh2FEDlDhvCx8/XXgdWrgQkTgMJC0amIKuu2S7E3btzAjRs35P7MwsIC9vb2aGhoQGZmpsLn+Pr6AgAuXryI2tpauZqdnR0sLS1RWlqKwj/81puYmMDJyQmtra2dLkDq6ekJHR0d5ObmKgzsMpkM1tbWqKioQH5+vlzNwMAArr/tCZOSkoI//qhcXV1hYGCAq1evory8XK5mbW0NmUyGmpoa5OTkyNV0dHQ67vVJT09Hc3OzXN3JyQkmJiYoKipCcXGxXK1Xr14YOHAg6uvrkZWVJVeTSCTw+W0uPysrC/X19XJ1e3t7WFhYoLi4GEVFRXI1MzMzODo6orm5Genp6fgjLy8vaGlpIScnBzU1NTh6FFi3jq/RtGWLDWbN6o1bt27hypUrcp9nZGSEwYMHAwCSk5MVjuvm5gZ9fX3k5+ej4g87aPfr1w/9+vVDdXU1Ll++LFfT09ODu7s7AODXX39FS0uLXN3Z2RnGxsa4du0aSkpK5GpWVlawtbVFXV0dsrOz5WpSqRTe3t4AgMzMTDT84eYXBwcHmJub4+bNm7h+/bpczdzcHA4ODmhqakJGRobC9+rt7Q2pVIpLly7h9h8eibO1tYWVlRXKyspQUFAgVzM2NoazszPa2tqQmpqqcFwPDw/o6uoiLy8PlZWVcrX+/fujb9++qKysRF5enlxNX18fbm5uAIDU1FS0tbXJ1V1cXGBoaIiCggKUlZXJ1fr06YMBAwbg9u3buHTpklxNW1sbQ4YMAQBcuHABjY2NcvVBgwbB1NS0R8eIgoICBAUF4ZdffoFWJzd90hjBdeUYcScbGxv07k1jxF+NEZWV3nj2WSmqqi7hzTcr8fDD2h01GiM4deoj2n9/uxzrJmFhYQyA3Cs4OJgxxlhOTo5C7c4ofn5+CrXo6GjGGGObN29WqAUGBjLGGKuqqur0uCUlJYwxxqZMmaJQi4yMZIwxduDAAYWaj49PRyZdXV2FekZGBmOMsQULFijUQkNDGWOMxcfHK9RkMlnHcWUymUI9Pj6eMcZYaGioQm3BggWMMcYyMjIUarq6uh3H9fHxUagfOHCAMcZYZGSkQm3KlCmMMcZKSko6/RlWVVUxxhibMGGCQm3z5s2MMcaio6MVan5+fh2ZOjtuTk4OY4yx4OBghVpYWBhjjLG4uDiFmqOjY8dxraysFOqJiYmMMcZCQkIUakuWLGGMMZaUlKRQMzEx6Tium5ubQj02NpYxxlhERIRCbebMmYwxxgoLCzv9XhsaGhhjjAUEBCjUoqKiGGOMRUVFKdQCAgIYY4w1NDR0etzCwkLGGGMzZ85UqEVERDDGGIuNjVWoubm5dXyvJiYmCvWkpCTGGGNLlixRqIWEhDDGGEtMTFSoWVlZdRzX0dFRoR4XF8cY69kxov3vJyQkdHpcGiO6bowIDAxUqNEYce9jRHk5YwYGtgo1GiO6d4wQ0Ue0//52NZqxo7NxAPd2Nn7p0iVMmzYN5eXlmDBhFr76aju2bNmCJ554gs7GQTN27ZTxbJxm7Diasfudso4R8fHxmDRpEoYNG4+0tNOwt7fH+vXr4e/vT2ME1KuP6K4ZO3oqltyTmJgYLFq0CAMHDsSBAwfg4uKCwMBAZGVl4ddff4WVlZXoiIT8qeLiYuzZswfBwcGwtrYWHYeQTpWWlsLLywtubm7473//i/T0dMyePRvXr19HVFQU5syZIzoiUQH08AT5Sxs3bkRwcDBmzJiBX375Be7u7tDS0kJ0dDSamprw/PPPK8xUEKJMrK2t8eqrr1JTR5Ra+4zyl19+CalUCi8vL5w/fx5TpkzpdHaKkM7QjB35U4wxSCQS5OXlIT4+HvPnz4dEIpH7O99//z22b9+OvXv3wsjISFBSQu6uoqICR48excSJE2FhYSE6DiF/qqGhAfr6+nJ/xhgDYwxSqRT//ve/MWrUKDjTdj7kT1BjRzoVGxuLiIgI/O9//4Opqeld/257A9j+kRBlk5ycjKFDhyIpKanjHhxClEVqaiq+/PJLvPvuu9DT0/vTv9fU1ARfX18UFhYiOjoaU6dO7cGURFXQpVgip6WlBStXrsT06dMhk8nu6RKrRCJBQkICxo8fr3CzKiGEkD9XW1uLOXPmID4+/i//rq6uLn7++WdMnDgR06ZNw9tvv63wQAMh1NiRDrdu3cJjjz2GyMhIbNy4EV9//TXMzMzu6XOtra1x/vx5LF++vHtDEkKIGlm2bBkKCwuxb9++u87WtTMxMcFXX32Fd999F+vWrcO6det6ICVRJdp//VeIpjh37hxSU1Nx5MgRTJgw4b4+18XFBR9//DEWLlyISZMmYfbs2d2UkhBC1MP+/fuxY8cO7Nixo2PJl3shkUgQGhqK4cOHdyxf09TUBF1d3e6KSlQIzdgRnD59GowxPPzww8jNzb3vpq7d/PnzMXv2bCxatEhhrSpCRDIwMICPjw8MDAxERyGkw5kzZ/Dkk0/i+eeff6DPnzBhAiwtLXHt2jW4uLhg//79XZyQqCJq7DRYW1sbVq1aBX9/fxw8eBAAn+Z/UBKJBNu2bcPIkSMVFiglRCRXV1ckJyd3LBRKiDL48MMPER0d/bcfOrO0tMTo0aMxZ84cuu+O0FOxmqq2thbBwcH49ttvsX79erzxxhtd/kQrPSVLCCGKIiIiYGNjg6effrrLjskYw3vvvYc333wTTzzxBP7973/D0NCwy45PVAfN2GmgW7duYezYsTh27Bi+++47rFixossbsNLSUowaNQonTpzo0uMS8iBSUlKgp6eHlJQU0VGIhouPj8fq1asVtrT6uyQSCVauXImvv/4aJ06cUNjGi2gOauw0kJmZGfz8/HDq1ClMnjy5W76GpaUlDAwMEBwcrLCHICE9jTGGpqYm2iGFCFVWVoZ58+Zh/PjxWLlyZbd8jaCgIOTn58Pb2xtNTU24cOFCt3wdoryosdMg3377LY4fPw4tLS1s2bIFXl5e3fa1tLS0sHv3bjQ0NGDBggX0hkoI0WiMMcyfPx+NjY2Ijo6GlpZWt30tY2NjAHw7yOHDh+Prr7/utq9FlA81dhqAMYYPPvgA06dPx549e3rs68pkMuzcuRPffvsttmzZ0mNflxBClE1VVRVKSkqwa9cuyGSyHvmaISEhmDJlCmbOnIn333+fTrA1BK1jp+ZaWlrw8ssv47PPPkNoaCjeeeedHv36U6dORWhoKO3PSQjRaObm5khMTIRU2nPzKQYGBti3bx+cnJywYsUKFBYW4qOPPurRDKTn0VOxam7JkiXYvn07tm/fjvnz54uOg9bW1m69BEFIZ+rr65GXlwcHBwday470qNraWjz11FNYu3Ztt97+8le2bduGzMxMfPTRR7RagZqjxk7NZWZm4urVq3j00UdFR8HSpUvR0tKCbdu2iY5CCCE9YtGiRdi9ezeSkpKUZh3FI0eOYPjw4bC0tBQdhXQDmo9VQ4WFhXj22WdRW1sLNzc3pWjqAMDb2xvbt2/HV199JToK0TBXr17FwoULcfXqVdFRiAb5z3/+g6ioKHz88cdK09TV19dj/vz5GDNmDP17UFM0Y6dmLly4gEceeQRaWlqIj4+Hvb296EgdGGOYPXs2jh49itTUVAwcOFB0JKIhkpOTMXToUCQlJcHX11d0HKIBrl69Ci8vLwQGBmL//v1KdfkzJycHjzzyCOrr6xEXF4chQ4aIjkS6EM3YqZHTp0/D398flpaWSExMVKqmDuALaG7fvh2mpqYIDg5GS0uL6EiEENItrl27hsGDB2P79u1K1dQBgJOTExITE9GvXz8EBATg3LlzoiORLkSNnZooKCjAxIkT4e3tjZ9++gn9+/cXHalTFhYWiImJwYgRI2g/Q0KI2hozZgzOnDkDc3Nz0VE6ZW1tjePHj2Py5MlK+35BHgw1dmrC1tYW0dHR+PHHH2FmZiY6zl2NGTMGH3zwAXR1ddHU1CQ6DiGEdJmEhARMnToV1dXVSjdT90dmZmbYvXs3ZDIZSktLERcXJzoS6QLU2Km46OjojsV/Z86cCX19fcGJ7t3Bgwfh5uaG8vJy0VGImrO2tkZoaCisra1FRyFqrLy8HMHBwaipqYGRkZHoOPfl448/xuOPP47o6GjRUcjfRI2dCvvss8/wzDPPIDU1VSVXFB8+fDgqKiqwcOFClcxPVIdMJsO7777bYyv+E83DGMOCBQtQX1+P3bt3q9x6neHh4XjuuefwzDPP0E5BKo4aOxUVGRmJF198Ea+88go+++wzpZ/y78yAAQPwxRdf4NChQ/jss89ExyFqrKamBidOnEBNTY3oKERNbd26FbGxsdi5c6dKnkBoaWkhKioKISEhWLp0KTZs2CA6EnlA1NipoB07duD111/Hv/71L5XfHmbatGlYsmQJQkJCkJ6eLjoOUVM5OTl46KGHkJOTIzoKUVONjY1Yvnw5pk6dKjrKA5NIJIiMjER4eDh69+4tOg55QLRXrAqaMWMGGGNYuHCh6ChdYuPGjWhra0OvXr1ERyGEkPvCGINEIkFISIjoKF1CIpEgLCys43+fPHkSY8eOFZiI3C/VnerRQJs2bcKVK1dgYWGhNk0dwDeq3rp1K/r374+6ujrRcQgh5J4tXboU69atEx2jW5w6dQrjxo3D2rVrRUch94EaOxWxdu1aLF++HIcPHxYdpdtcv34dzs7O+Oabb0RHIYSQv/T1119j69at6NOnj+go3cLf3x/r1q1DWFgYwsPDRcch94guxaqA9n9Y77zzDpYsWSI6Trfp168fRo4ciQULFmDYsGGwtbUVHYmoCR0dHchkMujo6IiOQtREQUEBFi5ciCeeeAL//Oc/RcfpNqtWrYJUKsW//vUvMMawZs0a0ZHIX6C9YpXchg0bEBoaiv/7v//D6tWrRcfpdrdu3YK3tzfs7OwQHx+vcksGEELUX0tLCx566CFcvXoVaWlpsLCwEB2p223YsAFxcXE4cuQIdHV1Rcchd0GXYpWcu7s71q1bpxFNHQBYWlpiz549OH36tNret0IIUW2NjY2wsbFBTEyMRjR1ALBy5Ur873//g66uLqqqqkTHIXdBM3ZK6vTp0xg9erRKrk/XFTZv3gwfHx+MGTNGdBSiBtLT0/Hoo4/ixx9/hKenp+g4RIW1tbWp9BJTf1dRURGGDh2KdevWqdVDfOpEc387ldju3bsxduxYHDhwQHQUYV566SWMGTMGLS0tuH37tug4RMU1NzejqKgIzc3NoqMQFdZ+q8jx48dFRxGmf//+CAoKwqJFi7B3717RcUgn6OEJJRMbG4vnnnsOzz33HGbNmiU6jnAzZ86ElpYWvvrqK42dvSSEiNe+dui1a9fg5OQkOo4wEokEn376Kerq6vD000/D0NAQ06ZNEx2L3IFm7JTIsWPHMHv2bAQFBSEqKkqjp/vbPfPMM/jmm2+wfft20VEIIRps27ZtOHjwIHbs2AEbGxvRcYSSSqXYsWMHgoKCMH/+fFRXV4uORO5A99gpkeeffx43b95EbGwsPXV0hxdffBG7du3C+fPn4e7uLjoOUUHJyckYOnQokpKS4OvrKzoOUTEZGRkYPnw4nn/+eWzZskV0HKXR1NSE7OxsDBkyRHQUcgdq7JRA+824LS0taG5uhoGBgehISqW+vh7Dhw+HRCLB+fPnoaenJzoSUTE1NTVISkrC0KFDYWJiIjoOUTHZ2dkIDw/Hzp07aXzuRFNTE1599VW88sorcHZ2Fh1H41FjJ9j169fxyCOPYMuWLfD39xcdR2llZGTg9OnTWLRoEd1rRwjpMS0tLdDWptvR76aiogJjxoxBfX09EhMT0a9fP9GRNBrdxCVQVVUVHn30UVRUVMDOzk50HKXm4eGBF154ARKJBKWlpaLjEBVTVFSEN998E0VFRaKjEBVy6NAh+Pj4oLy8XHQUpWZhYYEjR46gubkZjz76KK1zJxg1doI0NjYiKCgIBQUFiIuLw4ABA0RHUglffPEFXFxccO3aNdFRiAopLi7G+vXrUVxcLDoKURGFhYWYP38+nJycYGlpKTqO0rOxscGRI0dw9epVTJ8+HY2NjaIjaSxq7ARZvnw5Tp8+jdjYWHog4D5Mnz4dhoaGmDdvHlpbW0XHIYSoodbWVsybNw9GRkb4/PPP6faPe+Tu7o7vvvsOvr6+tC+zQNTYCfLKK6/gP//5D8aNGyc6ikqxtLTE7t27cfLkSURERIiOQwhRQxERETh16hT27NlDs3X3yd/fH5GRkZBKpcjNzRUdRyNRY9fDDh8+jNraWri6umLq1Kmi46ikgIAArF69GuHh4Thz5ozoOIQQNePh4YENGzbQifffkJ2dDVdXV2zbtk10FI1Dj/r0oLi4OEyZMgXvvfceXnvtNdFxVNpbb70FY2NjWj+J3JNevXphwYIF6NWrl+goRIk1NDRAT08PQUFBoqOoPBcXF7zwwgtYunQp7O3tERgYKDqSxqDlTnrIhQsXMHr0aIwdOxaxsbHQ0tISHUltFBYWYsCAAXQfDCHkgTHGMGvWLFhaWtJON12kpaUF06ZNw6lTp5CYmEj3k/cQuhTbA0pLS/H4449j4MCB2Lt3LzV1XaigoAAuLi74/PPPRUchSqy+vh4XLlxAfX296ChESUVFReHrr7/Gww8/LDqK2tDW1sa+ffswcOBALFmyRHQcjUGNXQ/4/vvvUV9fj++++45Wve9itra2mDdvHpYtW4bMzEzRcYiSysrKgoeHB7KyskRHIUooMzMTy5cvxwsvvIAnnnhCdBy1YmJigh9++AH79+8XHUVj0KXYHlJRUQELCwvRMdRSXV0dhg8fDi0tLfzyyy/Q19cXHYkoGdorlvyZhoYGjBgxAq2trTh37hwMDQ1FR1JbN2/exK5du7By5Uq6daYb0YxdN3r//fexadMmAKCmrhsZGhpi3759uHTpElavXi06DiFEhWhrayMoKAj79u2jpq6bnTp1Cm+++SY++ugj0VHUGjV23SQuLg4rV66k7a96iKenJ/bu3YuXXnpJdBRCiIq4ffs2tLW1sWbNGnh6eoqOo/ZmzpyJFStW4PXXX8fRo0dFx1Fb1Nh1g7y8PDz11FN49NFHsWbNGtFxNEZQUBDs7OxQXV1NW0cRORKJBLq6unT5h3QoKirCoEGD8M0334iOolEiIiIwadIkPPnkk8jLyxMdRy3RPXZdrLa2FqNHj0ZdXR3OnTsHc3Nz0ZE0TkBAAKRSKY4ePUpPIBNCFLS2tmLChAm4fPky0tLSaH3DHlZRUQF/f3+Eh4dj1qxZouOoHZqx62KNjY2wt7fHwYMHqakTZM2aNUhISMD69etFRyGEKKF3330XP/30E/bs2UNNnQAWFhZITU3taOpofqlrUWPXhRoaGmBpaYlDhw7Bw8NDdByNNX78eKxatQphYWFITEwUHYcogaysLPj6+tJyJwSJiYkIDw/H6tWrERAQIDqOxtLR0QFjDK+99ho2bNggOo5aocauixw7dgyDBg2iTY+VRFhYGEaMGIF58+ahqalJdBwiWH19PVJSUmiBYgInJyesWLECb7/9tugoGk8ikcDAwACrVq3CiRMnRMdRG3SPXRcoKiqCt7c3fH19cfjwYbqvS0lcuXIFly9fxsSJE0VHIYLROnaEMYaKigpYWlqKjkLu0NraikmTJiEzMxOpqano27ev6Egqj2bs/qaWlhbMnTsXenp62L17NzV1SsTOzg4TJ05EW1sb0tPTRcchhAi0Y8cOuLi4oKSkRHQUcgctLS3ExMRAKpVi7ty5aGlpER1J5VFj9zeFh4cjMTER+/btQ+/evUXHIZ345JNP4Ofnh+zsbNFRCCECZGVlYdmyZZg+fTr69OkjOg75g759+2Lfvn1wdHSkxq4L0KXYv+ncuXNISUnBokWLREchf6Kurg7Dhg2Drq4uzpw5Q1uOaaCKigocPXoUEydOpF1gNExDQwP8/PzQ1NSE8+fP0+4SKqCxsRF6enqiY6gsauweUHl5OUxNTaGjoyM6CrkHaWlpGDlyJBYvXkzb2RCiQUJCQrB161acPXsWXl5eouOQvxAfH49nn30Wp06dgq2treg4KokuxT6AlpYWTJ8+HU899ZToKOQeeXl5YePGjdi0aRMtgaKBiouL8cEHH9COJBro6aefxo4dO6ipUxHt/z/NmzcPra2tgtOoJpqxewCrV6/G+vXrkZCQgDFjxoiOQ+4RYwwHDx7E9OnTIZXSOY0moadiNU95eTlMTEygq6srOgq5TydPnsT48eMRFhZGy9I8AHp3u08//fQTIiIisGbNGmrqVIxEIsGMGTMglUpx9uxZOhskRE21trZi1qxZmD17tugo5AGMHTsWq1evxpo1a3D69GnRcVQONXb34fbt23j66afh7++P0NBQ0XHIA8rPz8eYMWPw3nvviY5CCOkGGzZswIkTJ7Bs2TLRUcgDeuutt7B48WJa1+4BUGN3H4yMjLB27VpER0fTenUqzN7eHitXrsRbb72FM2fOiI5DCOlCP//8M95++23861//wkMPPSQ6DnlA2tra+PTTT+Ho6Ii6ujraT/Y+0D1296ioqAgymUx0DNJFmpubMW7cOBQXFyMlJQVmZmaiI5FulJubi5CQEHz44YdwdHQUHYd0k7q6Ori7u6Nfv35ISEigVQvUQFVVFYYNG4Y333wT8+fPFx1HJdCM3T3Iz8+Hm5sbvvjiC9FRSBfR0dFBTEwMysvLER4eLjoO6WaOjo749ttvqalTc4aGhli3bh1iYmKoqVMTZmZmGDt2LJYtW4YrV66IjqMSaMbuL7S0tGD8+PG4du0a0tLSaGZHzSQmJsLDwwOmpqaio5Bu1NzcjMrKSpibm9Mbvpq6cuUK7OzsRMcg3aC6uhpDhgyBnZ0djh8/Tqsa/AX66fyFyMhI/Pzzz9i9ezc1dWpo9OjRMDU1RX5+PnJyckTHId0kPT0dffr0oT2D1VR2djbc3d2xa9cu0VFINzA1NcXOnTuRkJCATZs2iY6j9LRFB1Bm2dnZePvtt/Hqq6/C399fdBzSTRhjmDt3LhobG3HmzBnayoYQFdLQ0IA5c+bAxsYGs2bNEh2HdJOHHnoI69atg6urq+goSo9m7O5i4MCBCA8Px9q1a0VHId1IIpFg69atyMzMpGVsCFExoaGhyMrKwr59+2BkZCQ6DulGq1atwiOPPALGGK1DehfU2P2J6upqGBgY4M0334SBgYHoOKSb+fj44L333sNHH32EH374QXQcQsg9OH78ODZt2oT3338f3t7eouOQHtDW1obp06dj3bp1oqMoLWrsOpGRkQFbW1ucPHlSdBTSg1555RVMnjwZL730Epqbm0XHIYT8hTFjxmD79u14+eWXRUchPUQqlWLIkCF45513kJGRITqOUqKnYv+gubkZo0aNQn19PZKSkqCvry86EulBpaWlKC8vh4uLi+gopAu1traitrYWRkZGtLi4Gmhra0NOTg4GDx4sOgoRoLGxEd7e3jA1NUViYiL9m/4DmrH7g40bNyIlJQW7du2ipk4D9e7dGy4uLqirq8ORI0dExyFdREtLC6ampvQGoCbaL73euHFDdBQigJ6eHnbs2IFz587h448/Fh1H6VBjd4fLly9jzZo1eO211zB8+HDRcYhAW7ZsweOPP45ffvlFdBTSBXJycvDwww/TkjZq4OzZs1i9ejVCQkLQr18/0XGIIKNHj8Ybb7wBiUQiOorSoUuxdygoKMDatWuxadMmerpKwzU3N8Pf3x9lZWVISUmhBYxVXHJyMoYOHYqkpCT4+vqKjkMeUFVVFXx8fNCnTx+cPHmSFpsmpBM0Y/cbxhhsbW3x+eefU1NHoKOjg71796K0tBQvvvgibUBNiBIIDQ1FeXk59u7dS00dAcDfu0NDQ7Fz507RUZQGNXYAysrKMHz4cJw/f150FKJEHBwcsG3bNsTExODcuXOi4xCi8d566y188803sLe3Fx2FKAmJRILi4mKEhISguLhYdBylQI0dgNdeew15eXmwtbUVHYUomblz5yI5ORkjRowQHYUQjXXlyhWUl5ejf//+mDBhgug4RMls3LgR2traeOONN0RHUQoa39gdPXoUX375Jd5//3306dNHdByihHx8fMAYw969e9HY2Cg6DnkANjY22Lx5M2xsbERHIfepsbERM2bMwOzZs0VHIUqqV69e2LBhA6Kjo5GQkCA6jnAa/fBEQ0MDPD09IZPJEB8fT0/XkD91+fJluLm54eWXX0ZkZKToOIRojNdeew2bN2/GmTNn4OPjIzoOUVJtbW3w9/fH+PHjERERITqOUBo9Y1dQUAAdHR1s3bqVmjpyV4MGDcKGDRvwwQcfIC4uTnQccp9u3bqF3bt349atW6KjkPvw448/4oMPPsCGDRuoqSN3JZVKcfToUY1v6gANn7EDeJcvlWp0f0vuUVtbGx5//HEkJSUhLS0Nffv2FR2J3CNa7kT1NDQ0wMHBAT4+Pvj+++/p5JvcE8YYYmJiMHbsWI29b15jO5oNGzbg4sWL1NSReyaVSrFr1y5IJBJ89NFHouMQotb09fXx1VdfYefOndTUkXtWW1uLFStWYNmyZaKjCKORXc2xY8cQGhpKS1iQ+9a+MOo777wjOgohauv06dNoaWnB6NGj6aE2cl+MjY3x4Ycf4tChQxp724zGNXbNzc145ZVX4O/vj+DgYNFxiApycnKClpYWTp8+jaSkJNFxCFEr586dw/jx47F9+3bRUYiKmjVrFgICAhASEoLm5mbRcXqcxjV2n376KbKzs/HJJ5/Q9D55YIwxvPHGG3jyySdRU1MjOg75C0ZGRvDz86NdZZRcdXU15syZA19fX/zzn/8UHYeoqPbbZS5evIjo6GjRcXqcRj080djYCDs7OwQFBWHLli2i4xAVl5ubCx8fH0ybNk0jBw9CuhJjDPPmzcN3332HlJQUODo6io5EVNyxY8cwbtw4jdt+TqMaOwC4du0aDA0NYWlpKToKUQN79uzBvHnz8OWXX+Lpp58WHYcQlXXs2DFMnDgRe/bswVNPPSU6DlEjxcXFsLa2Fh2jx2hMY3f9+nWYmprC2NhYdBSiZp599ln88ssvSE9Ph7a2tug4pBO03InyY4whPj4e//jHP0RHIWrk6NGjmDx5MpKSkuDh4SE6To/QmMaOEEIIIUTdadzDE4QQQggh6ooaO0IIIYQQNUGNHSGEEEKImqDGjhBCCCFETVBjRwghhBCiJqixI4QQQghRE9TYEUIIIYSoCWrsCCGEEELUBDV2hBBCCCFq4v8BxcF2AvgYXToAAAAASUVORK5CYII=", | |
"text/plain": [ | |
"Graphics object consisting of 9 graphics primitives" | |
] | |
}, | |
"execution_count": 13, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"make_plot(pi/5,1,0.5)" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 14, | |
"id": "f05d9176-3a1c-4895-8234-24f94b413faf", | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"data": { | |
"image/png": 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", | |
"text/plain": [ | |
"Graphics object consisting of 9 graphics primitives" | |
] | |
}, | |
"execution_count": 14, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"make_plot(pi/10,0.8,0.8)" | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"id": "ed883d8a-6375-46f4-83b2-8a2370a8c8e6", | |
"metadata": {}, | |
"source": [ | |
"For even smaller $\\phi$, I have to stretch the plot vertically to be able to see the\n", | |
"different heights." | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 15, | |
"id": "cf5e8667-f91b-4e83-8286-97a5e859fe47", | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"data": { | |
"image/png": 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", | |
"text/plain": [ | |
"Graphics object consisting of 9 graphics primitives" | |
] | |
}, | |
"execution_count": 15, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"make_plot(pi/50,0.2,0.995,50)" | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"id": "9c494307-20c6-44fa-b5f4-2e5029e1e4ea", | |
"metadata": {}, | |
"source": [ | |
"" | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"id": "c86e06fa-d3d5-49f4-89f8-ddc6b8474ef9", | |
"metadata": {}, | |
"source": [ | |
"And $2-\\phi^2/10$ is very good approximation for the ratio all the way up to $\\phi=\\pi/2$." | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 16, | |
"id": "a5c3d9c4-ad6b-4dce-80c7-5aeaf3336c65", | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"data": { | |
"image/png": 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", | |
"text/plain": [ | |
"Graphics object consisting of 2 graphics primitives" | |
] | |
}, | |
"execution_count": 16, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"plot([a/b,2-phi^2/10],phi,0,pi/2,legend_label=['$a/b$','$2-\\phi^2/10$'])" | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"id": "c36ca969-5e0a-46f8-af4d-42f2f377f3ff", | |
"metadata": {}, | |
"source": [ | |
"Making the rounds\n", | |
"-----------------\n", | |
"### Some thoughts\n", | |
"It's always possible to construct a number in all positions of _non_-primitive triples.\n", | |
"If $(a,b,c)$ is a triple, then $(abc,b^2c,bc^2)$, $(a^2c,abc,ac^2)$, and $(a^2b,ab^2,abc)$\n", | |
"are triples with $abc$ in the three spots.\n", | |
"\n", | |
"In a primitive triple, at least two of the numbers must be odd, otherwise 2 would be a\n", | |
"common factor. One of the numbers must be even, since the sum of two odd numbers must\n", | |
"be even and the sum of an odd and an even must be odd." | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 17, | |
"id": "b8139229-803b-41d6-8177-97156e0d95ed", | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"data": { | |
"text/html": [ | |
"<html>\\(\\displaystyle 4 \\, i^{2} + 4 \\, j^{2} + 4 \\, i + 4 \\, j + 2\\)</html>" | |
], | |
"text/latex": [ | |
"$\\displaystyle 4 \\, i^{2} + 4 \\, j^{2} + 4 \\, i + 4 \\, j + 2$" | |
], | |
"text/plain": [ | |
"4*i^2 + 4*j^2 + 4*i + 4*j + 2" | |
] | |
}, | |
"execution_count": 17, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"i, j = var('i,j')\n", | |
"((2*i+1)^2 + (2*j+1)^2).expand()" | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"id": "2fa8e110-931c-45e6-a6fa-fa8b03f0cecb", | |
"metadata": {}, | |
"source": [ | |
"The sum of two odd squares is never a multiple of 4, so $c$ must be odd, so any number in all three spots\n", | |
"must be odd." | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 18, | |
"id": "12c0691e-0b15-4628-995c-8d32fd4baaa9", | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"data": { | |
"text/html": [ | |
"<html>\\(\\displaystyle 4 \\, i^{2} + 4 \\, j^{2} + 4 \\, i + 1\\)</html>" | |
], | |
"text/latex": [ | |
"$\\displaystyle 4 \\, i^{2} + 4 \\, j^{2} + 4 \\, i + 1$" | |
], | |
"text/plain": [ | |
"4*i^2 + 4*j^2 + 4*i + 1" | |
] | |
}, | |
"execution_count": 18, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"((2*i+1)^2 + 4*j^2).expand()" | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"id": "4dc441c7-1f56-42c9-868d-76d52f8198bb", | |
"metadata": {}, | |
"source": [ | |
"The sum of an odd square and an even square is one more than a multiple of four, so\n", | |
"the square of any number in all three spots must also be one more than a multiple of four." | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"id": "51c9d948-3b6f-486b-9baa-686e4dac8b3f", | |
"metadata": {}, | |
"source": [ | |
"For a number to be in two spots, $x^2 + y^2 + z^2 = c^2$, where $x^2 + y^2$ is the square of\n", | |
"the number, and $x$ and $c$ are odd, and $y$ and $z$ are even. For the number to be in all\n", | |
"three, need to be able to find $z$ and $c$ such that $x^2+y^2 > z^2$ and another $z$ and $c$\n", | |
"such that $x^2+y^2 < z^2$. I think it will come down to proving that at least one of the $c$\n", | |
"or $z$ must have a common factor with $x^2+y^2$. Or maybe a $c$, $z$ pair must share a common\n", | |
"factor. I don't know to prove that, but if I could that would mean there is no number that is\n", | |
"in all three spots." | |
] | |
} | |
], | |
"metadata": { | |
"kernelspec": { | |
"display_name": "SageMath 10.3", | |
"language": "sage", | |
"name": "sagemath-10.3" | |
}, | |
"language_info": { | |
"codemirror_mode": { | |
"name": "ipython", | |
"version": 3 | |
}, | |
"file_extension": ".py", | |
"mimetype": "text/x-python", | |
"name": "python", | |
"nbconvert_exporter": "python", | |
"pygments_lexer": "ipython3", | |
"version": "3.11.8" | |
} | |
}, | |
"nbformat": 4, | |
"nbformat_minor": 5 | |
} |
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