{ "cells": [ { "cell_type": "code", "execution_count": 6, "metadata": { "scrolled": false }, "outputs": [], "source": [ "using Plots, ApproxFun, SingularIntegralEquations, DifferentialEquations, ComplexPhasePortrait\n", "gr();" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# M3M6: Methods of Mathematical Physics 2018\n", "\n", "$$\n", "\\def\\dashint{{\\int\\!\\!\\!\\!\\!\\!-\\,}}\n", "\\def\\infdashint{\\dashint_{\\!\\!\\!-\\infty}^{\\,\\infty}}\n", "\\def\\D{\\,{\\rm d}}\n", "\\def\\E{{\\rm e}}\n", "\\def\\dx{\\D x}\n", "\\def\\dt{\\D t}\n", "\\def\\dz{\\D z}\n", "\\def\\C{{\\mathbb C}}\n", "\\def\\R{{\\mathbb R}}\n", "\\def\\CC{{\\cal C}}\n", "\\def\\HH{{\\cal H}}\n", "\\def\\I{{\\rm i}}\n", "\\def\\qqqquad{\\qquad\\qquad}\n", "\\def\\qqfor{\\qquad\\hbox{for}\\qquad}\n", "\\def\\qqwhere{\\qquad\\hbox{where}\\qquad}\n", "\\def\\Res_#1{\\underset{#1}{\\rm Res}}\\,\n", "\\def\\sech{{\\rm sech}\\,}\n", "\\def\\vc#1{{\\mathbf #1}}\n", "\\def\\pr(#1){\\left({#1}\\right)}\n", "\\def\\br[#1]{\\left[{#1}\\right]}\n", "\\def\\set#1{\\left\\{{#1}\\right\\}}\n", "\\def\\ip<#1>{\\left\\langle{#1}\\right\\rangle}\n", "\\def\\iip<#1>{\\left\\langle\\!\\langle{#1}\\right\\rangle\\!\\rangle}\n", "$$\n", "\n", "Dr Sheehan Olver\n", "<br>\n", "s.olver@imperial.ac.uk\n", "\n", "<br>\n", "Website: https://github.com/dlfivefifty/M3M6LectureNotes\n", "\n", "# Solution Sheet 2\n", "\n", "\n", "## Problem 1.1\n", "\n", "\n", "### 1.\n", "\n", "Take as an initial guess\n", "$$\n", " \\phi_1(z) = {\\sqrt{z-1} \\sqrt{z+1} \\over 2\\I(1 + z^2)}\n", "$$\n", "This satisfies for $-1 < x < 1$\n", "$$\n", "\\phi_1^+(x) -\\phi_1^-(x) = { \\sqrt{1-x^2} \\over 2(1 + x^2)} - {- \\sqrt{1-x^2} \\over 2(1 + x^2)} = {\\sqrt{1 - x^2} \\over 1+x^2}\n", "$$\n", "\n", "Further, as $z \\rightarrow \\infty$, \n", "$$\n", "\\phi_1(z) \\sim {z \\over \\I(1+ z^2)} \\rightarrow 0\n", "$$\n", "\n", "The catch is that it has poles at $\\pm \\I$:\n", "\\begin{align*}\n", " \\phi_1(z) = -{\\sqrt{\\I -1} \\sqrt{\\I+1} \\over 4} { 1 \\over z - \\I} + O(1) \\\\\n", " \\phi_1(z) = {\\sqrt{-\\I -1} \\sqrt{-\\I+1} \\over 4} { 1 \\over z + \\I} + O(1) \n", "\\end{align*}\n", "Thus it follows that\n", "$$\n", "\\phi(z) = \\phi_1(z) + {\\sqrt{\\I -1} \\sqrt{\\I+1} \\over 4} { 1 \\over z - \\I} - {\\sqrt{-\\I -1} \\sqrt{-\\I+1} \\over 4} { 1 \\over z + \\I}\n", "$$\n", "is \n", " 1. Analytic at $\\pm \\I$ and off $[-1,1]$ (Analyticity)\n", " 2. $\\phi(\\infty) = 0$ (Decay)\n", " 3. Has weaker than pole singularities (Regularity)\n", " 4. Satisfies\n", "$$\n", "\\phi_+(x) - \\phi_-(x) = {\\sqrt{1-x^2} \\over 1+x^2}\n", "$$\n", " \n", "By Plemelj II, this must be the Cauchy transform.\n", "\n", "_Demonstration_ We will see experimentally that it correct. 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We can check decay as well:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.0005177682933717976 + 0.000517765612545622im" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "φ(200.0+200.0im)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Finally, we compare it numerically it to `cauchy(f, z)` which is implemented in SingularIntegralEquations.jl:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.05303535516221752 + 0.05036581190871381im" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "φ(2.0+2.0im)" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.0530353551622175 + 0.050365811908713795im" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x = Fun()\n", "cauchy(sqrt(1-x^2)/(1+x^2), 2.0+2.0im)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 2.\n", "\n", "Recall that\n", "$$\n", "\\psi(z) = {\\log(z-1) - \\log(z+1) \\over 2 \\pi \\I}\n", "$$\n", "satisfies\n", "$$\n", "\\psi_+(x) - \\psi_-(x) = 1\n", "$$\n", "Therefore, consider\n", "$$\n", "\\phi_1(z) = {\\psi(z) \\over 2 + z}\n", "$$\n", "This has the right jump, but has an extra pole at $z = -2$: for $x < -1$ we have\n", "$$\n", "\\phi_1(x) = {\\log_+(x-1) - \\log_+(x+1) \\over 2 \\pi \\I} {1 \\over 2 + x} = \n", " {\\log(1-x) - \\log(-1-x) \\over 2 \\pi \\I} {1 \\over 2 + x}\n", "$$\n", "hence we arrive at the solution\n", "$$\n", "\\phi_1(z) - {\\log 3 \\over 2 \\pi \\I (2+z)}\n", "$$\n", "We can verify that $\\phi_1(\\infty) = 0$." ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "image/svg+xml": [ "<?xml version=\"1.0\" encoding=\"utf-8\"?>\n", "<svg xmlns=\"http://www.w3.org/2000/svg\" 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"phaseplot(-3..3, -3..3, φ)" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.0416136101650096 + 0.04191488519537722im" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "φ(2.0+2.0im)" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.041613610165009585 + 0.04191488519537721im" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "cauchy(1/(2+x), 2.0+2.0im)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 4. \n", "\n", "We first calculate the Cauchy transform of $f(x) = x/\\sqrt{1-x^2}$:\n", "$$\n", "\\phi(z) = {\\I z \\over 2 \\sqrt{z -1} \\sqrt{z+1}} - {\\I \\over 2}\n", "$$\n", "This vanishes at $\\infty$ and has the correct jump. We then have\n", "$$\n", "-\\I{\\cal H}f(x) = \\phi^+(x) + \\phi^-(x) = -\\I\n", "$$\n", "This implies that \n", "$$\n", "\\dashint_{-1}^1 {t \\over (t-x) \\sqrt{1-t^2}} \\dt = \\pi \\HH f(x) = \\pi\n", "$$" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "3.141592653589793" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f = x/sqrt(1-x^2)\n", "π*hilbert(f, 0.1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Problem 1.2\n", "\n", "### 1.2.1\n", "\n", "From Problem 1.1 part 4, we have a solution: \n", "$$\n", "\\phi(z) = -{ z \\over 2 \\sqrt{z -1} \\sqrt{z+1}} + {1 \\over 2}\n", "$$\n", "All other solutions are then of the form:\n", "$$\n", "\\phi(z) + {C \\over \\sqrt{z-1} \\sqrt{z+1}}\n", "$$" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "#11 (generic function with 1 method)" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "C = randn()\n", "φ = z -> -z/(2*sqrt(z-1)*sqrt(z+1))+1/2 + C/(sqrt(z-1)*sqrt(z+1))" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "1.0 + 0.0im" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "φ(0.1+0.0im)+φ(0.1-0.0im)" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "2.1249130872220166e-8" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "φ(1E8)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 1.2.2\n", "\n", "$\\psi(z) = \\phi(z) - 1$ satifies \n", "$$\n", "\\psi_+(x) + \\psi_-(x) = -2, \\psi(\\infty) = 0\n", "$$\n", "hence we know that $\\psi(z) = { z \\over \\sqrt{z -1} \\sqrt{z+1}} - 1 + {C \\over \\sqrt{z-1} \\sqrt{z+1}}$, giving \n", "$$\n", "\\psi(z) = { z \\over \\sqrt{z -1} \\sqrt{z+1}} + {C \\over \\sqrt{z-1} \\sqrt{z+1}} + 1\n", "$$" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "#13 (generic function with 1 method)" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "C = randn()\n", "φ = z -> z/(sqrt(z-1)*sqrt(z+1)) + C/(sqrt(z-1)*sqrt(z+1)) " ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.0 + 0.0im" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "φ(0.1+0.0im)+φ(0.1-0.0im)" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.9999999983329383" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "φ(1E9)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 1.2.3 \n", "\n", "For $f(x) = \\sqrt{1-\\diamond^2}$, we use the formula\n", "$$\n", "\\phi(z) = {\\I \\over \\sqrt{z - 1} \\sqrt{z+1}} \\CC[\\sqrt{1-\\diamond^2} f](z) + {C \\over \\sqrt{z-1}\\sqrt{z+1}} = {\\I \\over \\sqrt{z - 1} \\sqrt{z+1}} \\CC[1-\\diamond^2](z) + {C \\over \\sqrt{z-1}\\sqrt{z+1}}\n", "$$\n", "We already know $\\CC1(z)$, and we can deduce $\\CC[\\diamond^2]$ as follows: try\n", "$$\n", "\\phi_1(z) = z^2 \\CC1(z) = z^2 {\\log(z-1) - \\log(z+1) \\over 2 \\pi \\I}\n", "$$\n", "this has the right jump, but blows up at $\\infty$ like:\n", "$$\n", "x^2 (\\log(x-1) - \\log(x+1)) = x^2 (\\log(1-1/x) - \\log(1+1/x) )\n", "= -2 x + O(x^{-1})\n", "$$\n", "using\n", "$$\n", "\\log z = (z-1) - {1\\over 2}(z-1)^2 + O(z-1)^3\n", "$$\n", "Thus we have\n", "$$\n", "\\CC[\\diamond^2](z) = {z^2 (\\log(z-1) - \\log(z+1)) + 2 z \\over 2 \\pi \\I}\n", "$$\n", "and\n", "$$\n", "\\phi(z) = {\\I \\over \\sqrt{z - 1} \\sqrt{z+1}} {(1-z^2)(\\log(z-1) - \\log(z+1)) - 2 z \\over 2 \\pi \\I} + {C \\over \\sqrt{z-1}\\sqrt{z+1}}\n", "$$\n", "\n", "_Demonstration_ Here we see that the Cauchy transform of $x^2$ has the correct formula:" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.028322293739596095 + 0.024377589786690298im" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "z = 2.0+2.0im\n", "cauchy(x^2, z)" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.028322293739596032 + 0.02437758978669024im" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(z^2*(log(z-1)-log(z+1))+2z)/(2π*im)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We now see that $\\phi$ has the right jumps:" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "#15 (generic function with 1 method)" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "C = randn()\n", "φ = z -> im/(sqrt(z-1)*sqrt(z+1)) * ((1-z^2)*(log(z-1)-log(z+1))-2z)/(2π*im) + C/(sqrt(z-1)sqrt(z+1))" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "1.1102230246251565e-16 + 0.0im" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "φ(0.1+0.0im) + φ(0.1-0.0im) - sqrt(1-0.1^2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Finally, it vanishes at infinity:" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "-1.0949779687205364e-6 - 0.0im" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "φ(1E5)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 1.2.4\n", "\n", "Let $f(x) = {1 \\over 1+x^2}$. From Problem 1.1 part 1 we know \n", "$$\n", "\\CC[\\sqrt{1-\\diamond^2}] f(z) = {\\sqrt{z-1} \\sqrt{z+1} \\over 2\\I(1 + z^2)} + {\\sqrt{\\I -1} \\sqrt{\\I+1} \\over 4} { 1 \\over z - \\I} - {\\sqrt{-\\I -1} \\sqrt{-\\I+1} \\over 4} { 1 \\over z + \\I}\n", "$$\n", "hence from the solution formula we have\n", "$$\n", "\\phi(z) = {1 \\over 2(1 + z^2)} + {\\sqrt{\\I -1} \\sqrt{\\I+1} \\I \\over 4\\sqrt{z-1} \\sqrt{z+1}} { 1 \\over z - \\I} - {\\sqrt{-\\I -1} \\sqrt{-\\I+1} \\I \\over 4\\sqrt{z-1} \\sqrt{z+1}} { 1 \\over z + \\I} + {C \\over \\sqrt{z-1} \\sqrt{z+1}}\n", "$$\n", "\n", "But we want something stronger: that $\\phi(z) = O(z^{-2})$. To accomplish this, we need to choose $C$. Fortunately, I made the problem easy as every term apart from the last one is already $O(z^{-2})$, so choose $C = 0$:" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "-0.2071067812011923 + 0.0im" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" } ], "source": [ "φ = z -> 1/(2*(1+z^2)) + \n", " sqrt(im-1)sqrt(im+1)*im/(4sqrt(z-1)sqrt(z+1))*1/(z-im) - \n", " sqrt(-im-1)sqrt(-im+1)*im/(4sqrt(z-1)sqrt(z+1))*1/(z+im)\n", "\n", "φ(1E5)*1E5^2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We see also that it has the right jump:" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.9900990099009901 + 0.0im" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "φ(0.1+0.0im) + φ(0.1-0.0im)" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.9900990099009901" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "1/(1+0.1^2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Problem 1.3\n", "\n", "1. From the Hilbert formula, we know that the general solution of $\\HH u = f$ is \n", "$$\n", " u(x) = {-1 \\over \\sqrt{1 - x^2}}\\HH \\left[{ f(\\diamond) \\sqrt{1-\\diamond^2} }\\right](x) - {C \\over \\sqrt{1-x^2}}\n", "$$ \n", "Plugging in $f(x) = x/\\sqrt{1-x^2}$ means we need to calculate\n", "$$\n", "\\HH \\left[{\\diamond}\\right](x)\n", "$$\n", "We do so by first finding the Cauchy transform. Consider\n", "$$\n", "\\phi_1(z) = z \\CC 1(z) = z {\\log(z-1) - \\log(z+1) \\over 2 \\pi \\I}\n", "$$\n", "This has the right jump:\n", "$$\n", "\\phi_1^+(x) - \\phi_1^-(x) = x\n", "$$\n", "but doesn't decay at $\\infty$:\n", "\\begin{align*}\n", "x {\\log(x-1) - log(x+1) \\over 2 \\pi \\I} = x {\\log x + \\log(1-1/x) - \\log x -\\log(1+1/x) \\over 2 \\pi \\I} \\\\\n", "= -x { 2 \\over x 2 \\pi \\I} = -{1 \\over \\I \\pi}\n", "\\end{align*}\n", "But this means that\n", "$$\n", "\\phi(z) = \\phi_1(z) + {1 \\over \\I \\pi} = z {\\log(z-1) - \\log(z+1) \\over 2 \\pi \\I} + {1 \\over \\I \\pi}\n", "$$\n", "Decays and has the right jump, hence is $\\CC[\\diamond](z)$. " ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(0.013174970881571598 - 0.00098617598822645im, 0.013174970881571569 - 0.0009861759882264232im)" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "t = Fun()\n", "z = 2.0+2.0im\n", "cauchy(t, z), z*(log(z-1)-log(z+1))/(2π*im) + 1/(im*π)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Therefore, we have\n", "$$\n", "\\HH[\\diamond](x) = \\I (\\CC^+ + \\CC^-) \\diamond(x) = x {\\log(1-x) - \\log(1+x) \\over \\pi} +{2 \\over \\pi}\n", "$$" ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(0.6302322257442835, 0.6302322257442834)" ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x = 0.1\n", "hilbert(t,x), x*(log(1-x)-log(1+x))/(π) + 2/π" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Therefore, we get\n", "$$\n", "u(x) = - {x(\\log(1-x) - \\log(1+x))+2 \\over \\pi\\sqrt{1-x^2}} - {C \\over \\sqrt{1-x^2}}\n", "$$\n", "This can be verified in Mathematica via\n", "```mathematica\n", "NIntegrate[-((\n", " x (Log[1 - x] - Log[1 + x]) + \n", " 2)/(π Sqrt[1 - x^2] (x - 0.1))), {x, -1, 0.1, 1}, \n", " PrincipalValue -> True]/π\n", "```" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "2. Following the procedure of multiplying $\\CC[\\sqrt{1-\\diamond^2}](z)$ by $1/(2+z)$ and subtracting off the pole at $z=-2$, we first find:\n", "$$\n", "\\CC\\left[{\\sqrt{1-\\diamond^2} \\over 2+\\diamond}\\right](z) = {\\sqrt{z-1}\\sqrt{z+1} - z \\over 2\\I(2+z)} -{\\sqrt{-2-1}_+ \\sqrt{-2+1}_+ +2 \\over 2\\I(2+z)} = {\\sqrt{z-1}\\sqrt{z+1} - z \\over 2\\I(2+z)} +{ \\sqrt{3} -2 \\over 2\\I(2+z)} \n", "$$" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(0.03230545315801244 + 0.032449695183223826im, 0.032305453158012455 + 0.032449695183223784im)" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "t = Fun()\n", "z = 2.0+2.0im\n", "cauchy(sqrt(1-t^2)/(2+t), z), (sqrt(z-1)sqrt(z+1)-z)/(2im*(2+z)) + (sqrt(3)-2)/(2im*(z+2))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Therefore, calculating $\\I(\\CC^+ + \\CC^-)$ we find that\n", "$$\n", "\\HH\\left[{\\sqrt{1-\\diamond^2} \\over 2+\\diamond}\\right](z) = -{x\\over 2+x} +{ \\sqrt{3} -2 \\over 2+x} \n", "$$" ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(-0.17521390115767732, -0.17521390115767752)" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x = 0.1\n", "hilbert(sqrt(1-t^2)/(2+t), x), -x/(2+x)+(sqrt(3)-2)/(2+x)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Thus the general solution is\n", "$$\n", "u(x) = {1 \\over \\sqrt{1-x^2}} \\left( {x\\over 2+x} -{ \\sqrt{3} -2 \\over 2+x} + C \\right)\n", "$$\n", "We need to choose $C$ so this is bounded at the right-endpoint: In other words, \n", "$$\n", "u(x) = {1 \\over \\sqrt{1-x^2}} \\left( {x\\over 2+x} -{ \\sqrt{3} -2 \\over 2+x} +{ \\sqrt{3} -3 \\over 3} \\right)\n", "$$" ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [ { "data": { "image/svg+xml": [ "<?xml version=\"1.0\" encoding=\"utf-8\"?>\n", "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"600\" height=\"400\" viewBox=\"0 0 2400 1600\">\n", "<defs>\n", " <clipPath id=\"clip0800\">\n", " <rect x=\"0\" y=\"0\" width=\"2000\" height=\"2000\"/>\n", " </clipPath>\n", "</defs>\n", "<defs>\n", " <clipPath id=\"clip0801\">\n", " <rect x=\"0\" y=\"0\" width=\"2400\" height=\"1600\"/>\n", " </clipPath>\n", "</defs>\n", "<polygon clip-path=\"url(#clip0801)\" points=\"\n", "0,1600 2400,1600 2400,0 0,0 \n", " \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n", "<defs>\n", " <clipPath id=\"clip0802\">\n", " <rect x=\"480\" y=\"0\" width=\"1681\" height=\"1600\"/>\n", " </clipPath>\n", "</defs>\n", "<polygon clip-path=\"url(#clip0801)\" points=\"\n", "189.504,1503.47 2321.26,1503.47 2321.26,47.2441 189.504,47.2441 \n", " \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n", "<defs>\n", " <clipPath id=\"clip0803\">\n", " <rect x=\"189\" y=\"47\" width=\"2133\" height=\"1457\"/>\n", " </clipPath>\n", "</defs>\n", "<polyline clip-path=\"url(#clip0803)\" style=\"stroke:#000000; 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font-size:48px; text-anchor:start;\" transform=\"rotate(0, 2040.73, 390.184)\" x=\"2040.73\" y=\"390.184\">contour</text>\n", "</g>\n", "</svg>\n" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a = 1.0+2.0im\n", "b = -2.0+2.0im\n", "\n", "@show log(a*b)\n", "@show log(a) + log(b) - 2π*im\n", "scatter([real(1/b)], [imag(1/b)]; label=\"1/b\")\n", "scatter!([real(a)], [imag(a)]; label=\"a\")\n", "scatter!([0.0], [0.0]; label=\"0\")\n", "plot!(Segment(1, 1/b) ∪ Segment(1/b, a) ∪ Segment(a, 1); label=\"contour\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "If the contour passes through the origin, there are three possibility: \n", "1. $[a,1]$ contains zero, hence $a < 0$ \n", "2. $[1,1/b]$ contains zero, hence $b < 0$\n", "3. $[1/b, a]$ contains zero, which can only be true if $a b < 0$ by considering the equation of the line segment.\n", "\n", "1\\. In the case where $a < 0$ and $b < 0$ (and hence $a b > 0$), perturbing $a$ above and $b$ below or vice versa avoids $\\gamma$ winding around zero, so we have\n", "$$\n", "\\log(a b) = \\log_+ a + \\log_- b = \\log_- a + \\log_+ b = \\log_+ a + \\log_+ b - 2 \\pi \\I = \\log_- a + \\log_- b + 2 \\pi \\I\n", "$$" ] }, { "cell_type": "code", "execution_count": 37, "metadata": { "scrolled": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "log(a * b) = 1.791759469228055\n", "log(a + 0.0im) + log(b - 0.0im) = 1.791759469228055 + 0.0im\n", "log(a - 0.0im) + log(b + 0.0im) = 1.791759469228055 + 0.0im\n", "log(a - 0.0im) + log(b - 0.0im) + (2π) * im = 1.791759469228055 + 0.0im\n", "(log(a + 0.0im) + log(b + 0.0im)) - (2π) * im = 1.791759469228055 + 0.0im\n" ] } ], "source": [ "a = -2.0\n", "b = -3.0\n", "\n", "@show log(a*b)\n", "@show log(a+0.0im) + log(b-0.0im)\n", "@show log(a-0.0im) + log(b+0.0im)\n", "@show log(a-0.0im) + log(b-0.0im) + 2π*im\n", "@show log(a+0.0im) + log(b+0.0im) - 2π*im;" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In the case where $a < 0$ and $b > 0$, then $ a b < 0$, but we can perturb $a$ above/below to get\n", "$$\n", "\\log_\\pm(a b) = \\log_\\pm a + \\log b\n", "$$\n", "(and by symmetry, the equivalent holds for $b < 0$ and $a > 0$.)" ] }, { "cell_type": "code", "execution_count": 38, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "log(a * b + 0.0im) = 1.791759469228055 + 3.141592653589793im\n", "log(a + 0.0im) + log(b) = 1.791759469228055 + 3.141592653589793im\n", "log(a * b - 0.0im) = 1.791759469228055 - 3.141592653589793im\n", "log(a - 0.0im) + log(b) = 1.791759469228055 - 3.141592653589793im\n" ] } ], "source": [ "a = -2.0\n", "b = 3.0\n", "\n", "@show log(a*b +0.0im)\n", "@show log(a+0.0im) + log(b);\n", "\n", "@show log(a*b -0.0im)\n", "@show log(a-0.0im) + log(b);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In the case where $a < 0$, if $\\Im b > 0$ we can perturb $a$ below so that $\\gamma$ does not contain zero, giving us\n", "$$\n", "\\log(ab) = \\log_- a + \\log b\n", "$$\n", "similarly, if $\\Im b < 0$ we can perturb $a$ above." ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "log(a * b) = 1.8444397270569681 - 2.819842099193151im\n", "log(a - 0.0im) + log(b) = 1.8444397270569683 - 2.819842099193151im\n", "log(a * b) = 1.8444397270569681 - 2.819842099193151im\n", "log(a + 0.0im) + log(b) = 1.8444397270569683 + 3.4633432079864352im\n" ] } ], "source": [ "a = -2.0\n", "b = 3.0 + im\n", "\n", "@show log(a*b)\n", "@show log(a-0.0im) + log(b);\n", "\n", "b = 3.0 + im;\n", "@show log(a*b)\n", "@show log(a+0.0im) + log(b);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "2\\. In this case, swap the role of $a$ and $b$ and use the answers for $a < 0$. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "3\\. Finally, we have the case $a b < 0$ and neither $a$ nor $b$ is real. Note that\n", "$$\n", "ab = (a_x + \\I a_y) (b_x + \\I b_y) = a_x b_x - a_y b_y + \\I(a_x b_y + a_yb_x)\n", "$$\n", "It follows if $b_x > 0$ we have\n", "$$\n", "(ab)_+ = a_+ b\n", "$$\n", "and if $b_x < 0$ we have\n", "$$\n", "(ab)_+ = a_- b\n", "$$" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "log(a * b + eps() * im) = 0.6931471805599453 + 1.5707963267948966im\n", "log((a + eps() * im) * b) = 0.6931471805599453 + 1.5707963267948968im\n" ] }, { "data": { "text/plain": [ "0.6931471805599453 + 1.5707963267948968im" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a = 1.0 + 1.0im\n", "b = 1.0 + 1.0im\n", "@show log(a*b + eps()im) \n", "@show log((a+eps()im)*b)" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "log(a * b + eps() * im) = 0.6931471805599453 + 3.141592653589793im\n", "log((a - eps() * im) * b) = 0.6931471805599452 + 3.141592653589793im\n" ] }, { "data": { "text/plain": [ "0.6931471805599452 + 3.141592653589793im" ] }, "execution_count": 41, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a = 1.0 + 1.0im\n", "b = -1.0 + 1.0im\n", "@show log(a*b + eps()im) \n", "@show log((a-eps()im)*b)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can use this perturbation to reduce to the previous cases. For example, if $a = 1 + \\I$ and $b = -1 + \\I$, pertubing $ab$ above causes $a$ to be perturbed above, which causes the contour to surround the origin clockwise, hence we have\n", "$$\n", "\\log_+(ab) = \\log(a)_+b = \\log a b - 2 \\pi \\I\n", "$$" ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "log(a * b - eps() * im) = 0.6931471805599453 - 3.141592653589793im\n", "(log(a) + log(b)) - (2π) * im = 0.6931471805599453 - 3.141592653589793im\n" ] } ], "source": [ "a = 1.0 + 1.0im\n", "b = -1.0 + 1.0im\n", "@show log(a*b - eps()im) \n", "@show log(a)+log(b)-2π*im;" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "## Problem 2.2\n", "\n", "Use the contour $\\gamma(t) = 1 + t(1-z)$ to reduce it to a normal integral:\n", "$$\\overline{\\log z } = \\overline{\\int_1^z {1 \\over \\zeta} \\D \\zeta} = \\overline{\\int_0^1 {(z-1) \\over 1+(z-1) t} \\dt} = \\int_0^1 {(\\bar z-1) \\over 1+(\\bar z-1) t} \\dt = \\int_1^{\\bar z} {\\D \\zeta \\over \\zeta} = \\log \\bar z.$$\n", "We then have, since the contour from $1$ to $1/(\\bar z)$ to $z$ never surrounds the origin since both $\\Im z$ and $\\Im 1/(\\bar z)$ have the same sign, we have \n", "$$\n", "2 \\Re \\log z = \\log z + \\overline{\\log z} = \\log z + \\log \\bar z = \\log z \\bar z = \\log |z|^2 = 2 \\log |z|\n", "$$\n", "On the other hand, we have, where the contour of integration is chosen to be to the right of zero and then we do the change of variables $\\zeta = |z| \\E^{\\I \\theta}$\n", "$$\n", "2 \\Im \\log z = \\log z - \\log \\bar z = \\int_{\\bar z}^z {\\D \\zeta \\over \\zeta} = \\I \\int_{-\\arg z}^{\\arg z} \\D \\theta = 2 \\I \\arg z\n", "$$\n", "\n", "\n", "## Problem 2.3\n", "\n", "We first show that it is analytic on $(-\\infty,0)$. To do this, we need to show that the limit from above equals the limit from below: for $x < 0$ we have\n", "$$\\log_1^+ x -\\log_1^- x = \\log_+x - \\log_- x +2 \\pi \\I = 0$$\n", "Then for $x > 0$ and using $\\log_1^\\pm(x) = \\lim_{\\epsilon\\rightarrow 0} \\log(x\\pm \\I \\epsilon)$ we find\n", "$$\n", "\\log_1^+(x) - \\log_1^-x = \\log x- \\log x- 2\\pi \\I = -2 \\pi \\I\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "_Demonstration_ Here we see that the following is the analytic continuation:" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "#19 (generic function with 1 method)" ] }, "execution_count": 43, "metadata": {}, "output_type": "execute_result" } ], "source": [ "log1 = z -> begin\n", " if imag(z) > 0 \n", " log(z)\n", " elseif imag(z) == 0 && real(z) < 0\n", " log(z + 0.0im)\n", " elseif imag(z) < 0\n", " log(z) + 2π*im\n", " else\n", " error(\"log1 not defined on real axis\")\n", " end\n", 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"cell_type": "markdown", "metadata": {}, "source": [ "## Problem 3.1\n", "\n", "1. Because it's absolutely integrable, we can exchange derivatives and integrals to determine\n", "$$\n", "{\\D \\CC f \\over \\D z} = {1 \\over 2 \\pi \\I} \\oint {f(\\zeta)\\over (\\zeta-z)^2} \\D \\zeta\n", "$$\n", "2. There are two different possible approaches:\n", " - the subtract and add back in technique: since $f$ is analytic for $z$ near $\\zeta$, we can write\n", "$$\n", "\\CC f(z) = {1 \\over 2 \\pi \\I} \\oint {f(\\zeta)-f(z) \\over (\\zeta-z)^2} \\D \\zeta + f(z) \\CC 1(z)\n", "$$\n", "Therefore\n", "$$\n", "\\CC^+ f(\\zeta) - \\CC^- f(\\zeta) = f(\\zeta)( \\CC^+ 1(\\zeta) - \\CC^- 1(\\zeta))\n", "$$\n", "But we know (using Cauchy's integral formula / Residue calculus)\n", "$$\n", "\\CC 1(z) = \\begin{cases} 1 & |z| < 1 \\\\\n", " 0 & |z| > 0\n", " \\end{cases}\n", "$$\n", "hence $(\\CC^+-\\CC^-)1(\\zeta) = 1$\n", " - Since $f$ is analytic, we have for any radius $R > 1$ but inside the annulus\n", "$$\n", "\\CC^+ f(\\zeta) = {1 \\over 2 \\pi \\I} \\oint_{|z| = R} {f(\\zeta) \\over \\zeta -z} \\D\\zeta\n", "$$\n", "Similarly, for $\\CC^-f(\\zeta)$ with any radius $r < 1$ but inside the annulus. Therfore,\n", "$$\n", "\\CC^+ f(\\zeta) - \\CC^- f(\\zeta) = {1 \\over 2 \\pi \\I} \\left[\\oint_{|z| = R} - \\oint_{|z| = r} \\right] {f(\\zeta) \\over \\zeta -z} \\D\\zeta\n", "$$\n", "Deforming the contour and using Cauchy-integral formula gives the result. \n", "3. This follow since ${1 \\over \\zeta - z} \\rightarrow 0$ uniformly.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Problem 3.2\n", "\n", "Suppose we have another solution $\\phi$ and consider $\\psi(z) = \\phi(z) - \\CC f(z)$. Then on the circle we have\n", "$$\n", "\\psi_+(\\zeta) - \\psi_-(\\zeta) = \\phi_+(\\zeta) - \\CC_+f(\\zeta) - \\phi_-(\\zeta) + \\CC_+f(\\zeta) = f(\\zeta)-f(\\zeta) = 0\n", "$$\n", "Thus $\\psi$ is entire, and since it decays at infinity, it must be zero by Liouville's theorem.\n", "\n", "## Problem 3.3\n", "\n", "When $k \\geq 0$, we have from 3.1 and 3.2\n", "$$\n", "\\CC[\\diamond^k](z) =\\begin{cases}\n", " z^k & |z| < 0 \\\\\n", " 0 & |z| > 0\n", " \\end{cases}\n", "$$\n", "when $k < 0$ since\n", "$$\\CC[\\diamond^k]^+(\\zeta) - \\CC[\\diamond^k]^-(\\zeta) = \\zeta^k - 0 = \\zeta^k$$. we similarly have \n", "$$\n", "\\CC[\\diamond^k](z) =\\begin{cases}\n", "0 & |z| < 0 \\\\\n", " -z^k & |z| > 0\n", " \\end{cases}\n", "$$\n", "\n", "Therefore,\n", "$$\n", "\\Im \\CC^-[\\diamond^k](\\zeta) =\\begin{cases}\n", "0 & k \\geq 0 \\\\\n", " -{\\zeta^k - \\zeta^{-k} \\over 2 \\I} & k < 0\n", " \\end{cases}\n", "$$\n", "and\n", "$$\n", "\\Re \\CC^-[\\diamond^k](\\zeta) =\\begin{cases}\n", "0 & k \\geq 0 \\\\\n", " -{\\zeta^k + \\zeta^{-k} \\over 2} & k < 0\n", " \\end{cases}\n", "$$\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Problem 3.4\n", "\n", "Express the solution outside the circle as\n", "$$\n", "v(x,y) = \\Im ( \\E^{-\\I \\theta} z + \\CC f(z))\n", "$$\n", "for a to-be-determined $f$. On the circle, this reduces to\n", "$$\n", "\\Im \\CC^- f(\\zeta) = -\\cos \\theta {\\zeta - \\zeta^{-1} \\over 2 \\I} + \\sin \\theta {\\zeta + \\zeta^{-1} \\over 2}\n", "$$\n", "Unlike the real case, we can include imaginary coefficients, thus the solution is\n", "$$\n", "f(\\zeta) = (\\cos \\theta + \\I \\sin \\theta) \\zeta^{-1}\n", "$$\n", "and thus the full solution is\n", "$$\n", "v(x,y) = \\Im ( \\E^{-\\I \\theta} z - \\E^{\\I \\theta} z^{-1}))\n", "$$" ] }, { "cell_type": "code", "execution_count": 54, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "#21 (generic function with 1 method)" ] }, "execution_count": 54, "metadata": {}, "output_type": "execute_result" } ], "source": [ "θ = 0.1\n", "v = (x,y) -> x^2 + y^2 < 1 ? 0 : imag(exp(-im*θ) * (x+im*y) + exp(im*θ) * (x+im*y)^(-1))" ] }, { "cell_type": "code", "execution_count": 58, "metadata": {}, "outputs": [ { "data": { "image/svg+xml": [ "<?xml version=\"1.0\" encoding=\"utf-8\"?>\n", "<svg 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1572.77,1328.02 1570.95,1328.21 1569.12,1328.39 1567.3,1328.58 1565.48,1328.76 1563.66,1328.95 1563.13,1329 1561.83,1329.13 \n", " 1560.01,1329.32 1558.19,1329.5 1556.37,1329.69 1554.54,1329.87 1552.72,1330.06 1550.9,1330.24 1549.6,1330.38 1549.08,1330.43 1547.25,1330.61 1545.43,1330.8 \n", " 1543.61,1330.98 1541.79,1331.17 1539.96,1331.35 1538.14,1331.53 1536.32,1331.72 1536.01,1331.75 1534.5,1331.9 1532.67,1332.09 1530.85,1332.27 1529.03,1332.45 \n", " 1527.21,1332.64 1525.38,1332.82 1523.56,1333 1522.37,1333.12 1521.74,1333.19 1519.92,1333.37 1518.09,1333.55 1516.27,1333.74 1514.45,1333.92 1512.63,1334.1 \n", " 1510.8,1334.28 1508.98,1334.47 1508.67,1334.5 1507.16,1334.65 1505.34,1334.83 1503.51,1335.01 1501.69,1335.19 1499.87,1335.38 1498.04,1335.56 1496.22,1335.74 \n", " 1494.89,1335.87 1494.4,1335.92 1492.58,1336.1 1490.75,1336.28 1488.93,1336.46 1487.11,1336.64 1485.29,1336.82 1483.46,1337 1481.64,1337.18 1481.04,1337.24 \n", " 1479.82,1337.36 1478,1337.54 1476.17,1337.72 1474.35,1337.9 1472.53,1338.08 1470.71,1338.26 1468.88,1338.44 1467.09,1338.62 1467.06,1338.62 1465.24,1338.8 \n", " 1463.42,1338.98 1461.59,1339.16 1459.77,1339.34 1457.95,1339.51 1456.13,1339.69 1454.3,1339.87 1453.04,1339.99 1452.48,1340.05 1450.66,1340.22 1448.84,1340.4 \n", " 1447.01,1340.58 1445.19,1340.76 1443.37,1340.93 1441.55,1341.11 1439.72,1341.28 1438.88,1341.37 1437.9,1341.46 1436.08,1341.64 1434.26,1341.81 1432.43,1341.99 \n", " 1430.61,1342.16 1428.79,1342.34 1426.97,1342.51 1425.14,1342.69 1424.6,1342.74 1423.32,1342.86 1421.5,1343.04 1419.68,1343.21 1417.85,1343.38 1416.03,1343.56 \n", " 1414.21,1343.73 1412.39,1343.9 1410.56,1344.08 1410.19,1344.11 1408.74,1344.25 1406.92,1344.42 1405.1,1344.6 1403.27,1344.77 1401.45,1344.94 1399.63,1345.11 \n", " 1397.8,1345.28 1395.98,1345.45 1395.62,1345.49 1394.16,1345.62 1392.34,1345.8 1390.51,1345.97 1388.69,1346.14 1386.87,1346.31 1385.05,1346.48 1383.22,1346.65 \n", " 1381.4,1346.81 1380.9,1346.86 1379.58,1346.98 1377.76,1347.15 1375.93,1347.32 1374.11,1347.49 1372.29,1347.66 1370.47,1347.83 1368.64,1347.99 1366.82,1348.16 \n", " 1366,1348.24 1365,1348.33 1363.18,1348.49 1361.35,1348.66 1359.53,1348.83 1357.71,1348.99 1355.89,1349.16 1354.06,1349.32 1352.24,1349.49 1350.91,1349.61 \n", " 1350.42,1349.65 1348.6,1349.82 1346.77,1349.98 1344.95,1350.15 1343.13,1350.31 1341.31,1350.47 1339.48,1350.64 1337.66,1350.8 1335.84,1350.96 1335.61,1350.98 \n", " 1334.02,1351.12 1332.19,1351.29 1330.37,1351.45 1328.55,1351.61 1326.73,1351.77 1324.9,1351.93 1323.08,1352.09 1321.26,1352.25 1320.09,1352.36 1319.44,1352.41 \n", " 1317.61,1352.57 1315.79,1352.73 1313.97,1352.89 1312.15,1353.05 1310.32,1353.21 1308.5,1353.37 1306.68,1353.53 1304.86,1353.68 1304.32,1353.73 1303.03,1353.84 \n", " 1301.21,1354 1299.39,1354.16 1297.56,1354.31 1295.74,1354.47 1293.92,1354.62 1292.1,1354.78 1290.27,1354.93 1288.45,1355.09 1288.28,1355.1 1286.63,1355.24 \n", " 1284.81,1355.4 1282.98,1355.55 1281.16,1355.71 1279.34,1355.86 1277.52,1356.01 1275.69,1356.17 1273.87,1356.32 1272.05,1356.47 1271.96,1356.48 1270.23,1356.62 \n", " 1268.4,1356.77 1266.58,1356.92 1264.76,1357.08 1262.94,1357.23 1261.11,1357.38 1259.29,1357.53 1257.47,1357.68 1255.65,1357.83 1255.32,1357.85 1253.82,1357.97 \n", " 1252,1358.12 1250.18,1358.27 1248.36,1358.42 1246.53,1358.57 1244.71,1358.71 1242.89,1358.86 1241.07,1359.01 1239.24,1359.15 1238.35,1359.23 1237.42,1359.3 \n", " 1235.6,1359.45 1233.78,1359.59 1231.95,1359.74 1230.13,1359.88 1228.31,1360.03 1226.49,1360.17 1224.66,1360.31 1222.84,1360.46 1221.02,1360.6 1221.02,1360.6 \n", " 1219.2,1360.74 1217.37,1360.88 1215.55,1361.03 1213.73,1361.17 1211.91,1361.31 1210.08,1361.45 1208.26,1361.59 1206.44,1361.73 1204.62,1361.87 1203.29,1361.97 \n", " 1202.79,1362.01 1200.97,1362.15 1199.15,1362.29 1197.32,1362.43 1195.5,1362.57 1193.68,1362.7 1191.86,1362.84 1190.03,1362.98 1188.21,1363.12 1186.39,1363.25 \n", " 1185.14,1363.35 1184.57,1363.39 1182.74,1363.53 1180.92,1363.66 1179.1,1363.8 1177.28,1363.93 1175.45,1364.07 1173.63,1364.2 1171.81,1364.34 1169.99,1364.47 \n", " 1168.16,1364.6 1166.54,1364.72 1166.34,1364.74 1164.52,1364.87 1162.7,1365 1160.87,1365.13 1159.05,1365.26 1157.23,1365.4 1155.41,1365.53 1153.58,1365.66 \n", " 1151.76,1365.79 1149.94,1365.92 1148.12,1366.05 1147.46,1366.09 1146.29,1366.18 1144.47,1366.31 1142.65,1366.44 1140.83,1366.56 1139,1366.69 1137.18,1366.82 \n", " 1135.36,1366.95 1133.54,1367.07 1131.71,1367.2 1129.89,1367.33 1128.07,1367.45 1127.86,1367.47 1126.25,1367.58 1124.42,1367.71 1122.6,1367.83 1120.78,1367.96 \n", " 1118.96,1368.08 1117.13,1368.2 1115.31,1368.33 1113.49,1368.45 1111.67,1368.57 1109.84,1368.7 1108.02,1368.82 1107.7,1368.84 1106.2,1368.94 1104.37,1369.07 \n", " 1102.55,1369.19 1100.73,1369.31 1098.91,1369.43 1097.08,1369.55 1095.26,1369.67 1093.44,1369.79 1091.62,1369.91 1089.79,1370.03 1087.97,1370.15 1086.96,1370.22 \n", " 1086.15,1370.27 1084.33,1370.39 1082.5,1370.51 1080.68,1370.62 1078.86,1370.74 1077.04,1370.86 1075.21,1370.98 1073.39,1371.09 1071.57,1371.21 1069.75,1371.33 \n", " 1067.92,1371.44 1066.1,1371.56 1065.61,1371.59 1064.28,1371.67 1062.46,1371.79 1060.63,1371.9 1058.81,1372.02 1056.99,1372.13 1055.17,1372.25 1053.34,1372.36 \n", " 1051.52,1372.47 1049.7,1372.59 1047.88,1372.7 1046.05,1372.81 1044.23,1372.93 1043.61,1372.96 1042.41,1373.04 1040.59,1373.15 1038.76,1373.26 1036.94,1373.37 \n", " 1035.12,1373.48 1033.3,1373.59 1031.47,1373.7 1029.65,1373.81 1027.83,1373.92 1026.01,1374.03 1024.18,1374.14 1022.36,1374.25 1020.94,1374.34 1020.54,1374.36 \n", " 1018.72,1374.47 1016.89,1374.58 1015.07,1374.69 1013.25,1374.79 1011.43,1374.9 1009.6,1375.01 1007.78,1375.12 1005.96,1375.22 1004.13,1375.33 1002.31,1375.44 \n", " 1000.49,1375.54 998.667,1375.65 997.595,1375.71 996.845,1375.75 995.022,1375.86 993.2,1375.97 991.377,1376.07 989.555,1376.18 987.732,1376.28 985.909,1376.38 \n", " 984.087,1376.49 982.264,1376.59 980.442,1376.7 978.619,1376.8 976.797,1376.9 974.974,1377.01 973.56,1377.08 973.152,1377.11 971.329,1377.21 969.507,1377.31 \n", " 967.684,1377.41 965.861,1377.52 964.039,1377.62 962.216,1377.72 960.394,1377.82 958.571,1377.92 956.749,1378.02 954.926,1378.12 953.104,1378.22 951.281,1378.32 \n", " 949.459,1378.42 948.843,1378.46 947.636,1378.52 945.813,1378.62 943.991,1378.72 942.168,1378.82 940.346,1378.92 938.523,1379.02 936.701,1379.12 934.878,1379.22 \n", " 933.056,1379.32 931.233,1379.42 929.41,1379.51 927.588,1379.61 925.765,1379.71 923.943,1379.81 923.464,1379.83 922.12,1379.9 920.298,1380 918.475,1380.1 \n", " 916.653,1380.2 914.83,1380.29 913.008,1380.39 911.185,1380.49 909.362,1380.58 907.54,1380.68 905.717,1380.77 903.895,1380.87 902.072,1380.96 900.25,1381.06 \n", " 898.427,1381.16 897.461,1381.21 896.605,1381.25 894.782,1381.35 892.96,1381.44 891.137,1381.54 889.314,1381.63 887.492,1381.72 885.669,1381.82 883.847,1381.91 \n", " 882.024,1382.01 880.202,1382.1 878.379,1382.2 876.557,1382.29 874.734,1382.38 872.912,1382.48 871.089,1382.57 870.887,1382.58 869.266,1382.66 867.444,1382.76 \n", " 865.621,1382.85 863.799,1382.94 861.976,1383.03 860.154,1383.13 858.331,1383.22 856.509,1383.31 854.686,1383.4 852.864,1383.5 851.041,1383.59 849.218,1383.68 \n", " 847.396,1383.77 845.573,1383.87 843.812,1383.95 843.751,1383.96 841.928,1384.05 840.106,1384.14 838.283,1384.23 836.461,1384.32 834.638,1384.41 832.816,1384.51 \n", " 830.993,1384.6 829.17,1384.69 827.348,1384.78 825.525,1384.87 823.703,1384.96 821.88,1385.05 820.058,1385.14 818.235,1385.23 816.413,1385.32 816.32,1385.33 \n", " 814.59,1385.41 812.768,1385.5 810.945,1385.59 809.122,1385.68 807.3,1385.77 805.477,1385.86 803.655,1385.95 801.832,1386.05 800.01,1386.13 798.187,1386.22 \n", " 796.365,1386.31 794.542,1386.4 792.72,1386.49 790.897,1386.58 789.074,1386.67 788.507,1386.7 787.252,1386.76 785.429,1386.85 783.607,1386.94 781.784,1387.03 \n", " 779.962,1387.12 778.139,1387.21 776.317,1387.3 774.494,1387.39 772.671,1387.48 770.849,1387.57 769.026,1387.66 767.204,1387.75 765.381,1387.84 763.559,1387.92 \n", " 761.736,1388.01 760.471,1388.08 759.914,1388.1 758.091,1388.19 756.269,1388.28 754.446,1388.37 752.623,1388.46 750.801,1388.55 748.978,1388.64 747.156,1388.73 \n", " 745.333,1388.81 743.511,1388.9 741.688,1388.99 739.866,1389.08 738.043,1389.17 736.221,1389.26 734.398,1389.35 732.575,1389.44 732.314,1389.45 730.753,1389.53 \n", " 728.93,1389.61 727.108,1389.7 725.285,1389.79 723.463,1389.88 721.64,1389.97 719.818,1390.06 717.995,1390.15 716.173,1390.24 714.35,1390.32 712.527,1390.41 \n", " 710.705,1390.5 708.882,1390.59 707.06,1390.68 705.237,1390.77 704.13,1390.82 703.415,1390.86 701.592,1390.95 699.77,1391.04 697.947,1391.12 696.125,1391.21 \n", " 694.302,1391.3 692.479,1391.39 690.657,1391.48 688.834,1391.57 687.012,1391.66 685.189,1391.75 683.367,1391.84 681.544,1391.93 679.722,1392.01 677.899,1392.1 \n", " 676.077,1392.19 676.006,1392.2 674.254,1392.28 672.431,1392.37 670.609,1392.46 668.786,1392.55 666.964,1392.64 665.141,1392.73 663.319,1392.82 661.496,1392.91 \n", " 659.674,1393 657.851,1393.09 656.029,1393.18 654.206,1393.27 652.383,1393.36 650.561,1393.45 648.738,1393.54 648.017,1393.57 646.916,1393.62 645.093,1393.71 \n", " 643.271,1393.8 641.448,1393.89 639.626,1393.98 637.803,1394.07 635.981,1394.16 634.158,1394.25 632.335,1394.34 630.513,1394.43 628.69,1394.52 626.868,1394.61 \n", " 625.045,1394.71 623.223,1394.8 621.4,1394.89 620.225,1394.94 619.578,1394.98 617.755,1395.07 615.933,1395.16 614.11,1395.25 612.287,1395.34 610.465,1395.43 \n", " 608.642,1395.52 606.82,1395.61 604.997,1395.7 603.175,1395.79 601.352,1395.88 599.53,1395.98 597.707,1396.07 595.884,1396.16 594.062,1396.25 592.679,1396.32 \n", " 592.239,1396.34 590.417,1396.43 588.594,1396.52 586.772,1396.61 584.949,1396.71 583.127,1396.8 581.304,1396.89 579.482,1396.98 577.659,1397.07 575.836,1397.17 \n", " 574.014,1397.26 572.191,1397.35 570.369,1397.44 568.546,1397.53 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1585.53,1297.44 1583.7,1297.63 1581.88,1297.82 1580.06,1298.01 1578.24,1298.21 \n", " 1576.41,1298.4 1574.59,1298.59 1572.8,1298.78 1572.77,1298.78 1570.95,1298.97 1569.12,1299.16 1567.3,1299.35 1565.48,1299.55 1563.66,1299.74 1561.83,1299.93 \n", " 1560.01,1300.12 1559.7,1300.15 1558.19,1300.31 1556.37,1300.5 1554.54,1300.69 1552.72,1300.88 1550.9,1301.07 1549.08,1301.26 1547.25,1301.45 1546.57,1301.53 \n", " 1545.43,1301.64 1543.61,1301.83 1541.79,1302.02 1539.96,1302.21 1538.14,1302.4 1536.32,1302.59 1534.5,1302.78 1533.39,1302.9 1532.67,1302.97 1530.85,1303.16 \n", " 1529.03,1303.35 1527.21,1303.54 1525.38,1303.73 1523.56,1303.92 1521.74,1304.11 1520.17,1304.27 1519.92,1304.3 1518.09,1304.49 1516.27,1304.68 1514.45,1304.87 \n", " 1512.63,1305.05 1510.8,1305.24 1508.98,1305.43 1507.16,1305.62 1506.88,1305.65 1505.34,1305.81 1503.51,1305.99 1501.69,1306.18 1499.87,1306.37 1498.04,1306.56 \n", " 1496.22,1306.74 1494.4,1306.93 1493.53,1307.02 1492.58,1307.12 1490.75,1307.31 1488.93,1307.49 1487.11,1307.68 1485.29,1307.87 1483.46,1308.05 1481.64,1308.24 \n", " 1480.11,1308.39 1479.82,1308.42 1478,1308.61 1476.17,1308.8 1474.35,1308.98 1472.53,1309.17 1470.71,1309.35 1468.88,1309.54 1467.06,1309.72 1466.6,1309.77 \n", " 1465.24,1309.91 1463.42,1310.09 1461.59,1310.27 1459.77,1310.46 1457.95,1310.64 1456.13,1310.83 1454.3,1311.01 1452.99,1311.14 1452.48,1311.19 1450.66,1311.38 \n", " 1448.84,1311.56 1447.01,1311.74 1445.19,1311.93 1443.37,1312.11 1441.55,1312.29 1439.72,1312.47 1439.28,1312.52 1437.9,1312.65 1436.08,1312.83 1434.26,1313.02 \n", " 1432.43,1313.2 1430.61,1313.38 1428.79,1313.56 1426.97,1313.74 1425.44,1313.89 1425.14,1313.92 1423.32,1314.1 1421.5,1314.28 1419.68,1314.46 1417.85,1314.64 \n", " 1416.03,1314.82 1414.21,1315 1412.39,1315.18 1411.48,1315.26 1410.56,1315.35 1408.74,1315.53 1406.92,1315.71 1405.1,1315.89 1403.27,1316.06 1401.45,1316.24 \n", " 1399.63,1316.42 1397.8,1316.6 1397.37,1316.64 1395.98,1316.77 1394.16,1316.95 1392.34,1317.12 1390.51,1317.3 1388.69,1317.48 1386.87,1317.65 1385.05,1317.83 \n", " 1383.22,1318 1383.1,1318.01 1381.4,1318.17 1379.58,1318.35 1377.76,1318.52 1375.93,1318.7 1374.11,1318.87 1372.29,1319.04 1370.47,1319.21 1368.66,1319.39 \n", " 1368.64,1319.39 1366.82,1319.56 1365,1319.73 1363.18,1319.9 1361.35,1320.07 1359.53,1320.24 1357.71,1320.41 1355.89,1320.58 1354.06,1320.76 1354.02,1320.76 \n", " 1352.24,1320.92 1350.42,1321.09 1348.6,1321.26 1346.77,1321.43 1344.95,1321.6 1343.13,1321.77 1341.31,1321.94 1339.48,1322.1 1339.17,1322.13 1337.66,1322.27 \n", " 1335.84,1322.44 1334.02,1322.61 1332.19,1322.77 1330.37,1322.94 1328.55,1323.1 1326.73,1323.27 1324.9,1323.43 1324.1,1323.51 1323.08,1323.6 1321.26,1323.76 \n", " 1319.44,1323.93 1317.61,1324.09 1315.79,1324.25 1313.97,1324.42 1312.15,1324.58 1310.32,1324.74 1308.77,1324.88 1308.5,1324.9 1306.68,1325.07 1304.86,1325.23 \n", " 1303.03,1325.39 1301.21,1325.55 1299.39,1325.71 1297.56,1325.87 1295.74,1326.03 1293.92,1326.19 1293.17,1326.25 1292.1,1326.35 1290.27,1326.51 1288.45,1326.66 \n", " 1286.63,1326.82 1284.81,1326.98 1282.98,1327.14 1281.16,1327.29 1279.34,1327.45 1277.52,1327.61 1277.26,1327.63 1275.69,1327.76 1273.87,1327.92 1272.05,1328.07 \n", " 1270.23,1328.23 1268.4,1328.38 1266.58,1328.54 1264.76,1328.69 1262.94,1328.84 1261.11,1329 1261.04,1329 1259.29,1329.15 1257.47,1329.3 1255.65,1329.45 \n", " 1253.82,1329.6 1252,1329.75 1250.18,1329.9 1248.36,1330.06 1246.53,1330.2 1244.71,1330.35 1244.45,1330.38 1242.89,1330.5 1241.07,1330.65 1239.24,1330.8 \n", " 1237.42,1330.95 1235.6,1331.1 1233.78,1331.24 1231.95,1331.39 1230.13,1331.54 1228.31,1331.68 1227.48,1331.75 1226.49,1331.83 1224.66,1331.97 1222.84,1332.12 \n", " 1221.02,1332.26 1219.2,1332.41 1217.37,1332.55 1215.55,1332.7 1213.73,1332.84 1211.91,1332.98 1210.09,1333.12 1210.08,1333.12 1208.26,1333.27 1206.44,1333.41 \n", " 1204.62,1333.55 1202.79,1333.69 1200.97,1333.83 1199.15,1333.97 1197.32,1334.11 1195.5,1334.25 1193.68,1334.39 1192.23,1334.5 1191.86,1334.53 1190.03,1334.66 \n", " 1188.21,1334.8 1186.39,1334.94 1184.57,1335.08 1182.74,1335.21 1180.92,1335.35 1179.1,1335.48 1177.28,1335.62 1175.45,1335.75 1173.88,1335.87 1173.63,1335.89 \n", " 1171.81,1336.02 1169.99,1336.16 1168.16,1336.29 1166.34,1336.42 1164.52,1336.56 1162.7,1336.69 1160.87,1336.82 1159.05,1336.95 1157.23,1337.08 1155.41,1337.21 \n", " 1154.98,1337.24 1153.58,1337.34 1151.76,1337.47 1149.94,1337.6 1148.12,1337.73 1146.29,1337.86 1144.47,1337.99 1142.65,1338.12 1140.83,1338.25 1139,1338.37 \n", " 1137.18,1338.5 1135.5,1338.62 1135.36,1338.63 1133.54,1338.75 1131.71,1338.88 1129.89,1339.01 1128.07,1339.13 1126.25,1339.26 1124.42,1339.38 1122.6,1339.5 \n", " 1120.78,1339.63 1118.96,1339.75 1117.13,1339.87 1115.39,1339.99 1115.31,1340 1113.49,1340.12 1111.67,1340.24 1109.84,1340.36 1108.02,1340.48 1106.2,1340.61 \n", " 1104.37,1340.73 1102.55,1340.85 1100.73,1340.97 1098.91,1341.08 1097.08,1341.2 1095.26,1341.32 1094.59,1341.37 1093.44,1341.44 1091.62,1341.56 1089.79,1341.68 \n", " 1087.97,1341.79 1086.15,1341.91 1084.33,1342.03 1082.5,1342.14 1080.68,1342.26 1078.86,1342.37 1077.04,1342.49 1075.21,1342.6 1073.39,1342.72 1073.06,1342.74 \n", " 1071.57,1342.83 1069.75,1342.95 1067.92,1343.06 1066.1,1343.17 1064.28,1343.29 1062.46,1343.4 1060.63,1343.51 1058.81,1343.62 1056.99,1343.73 1055.17,1343.85 \n", " 1053.34,1343.96 1051.52,1344.07 1050.76,1344.11 1049.7,1344.18 1047.88,1344.29 1046.05,1344.4 1044.23,1344.51 1042.41,1344.62 1040.59,1344.72 1038.76,1344.83 \n", " 1036.94,1344.94 1035.12,1345.05 1033.3,1345.16 1031.47,1345.26 1029.65,1345.37 1027.83,1345.48 1027.64,1345.49 1026.01,1345.58 1024.18,1345.69 1022.36,1345.79 \n", " 1020.54,1345.9 1018.72,1346 1016.89,1346.11 1015.07,1346.21 1013.25,1346.32 1011.43,1346.42 1009.6,1346.53 1007.78,1346.63 1005.96,1346.73 1004.13,1346.83 \n", " 1003.67,1346.86 1002.31,1346.94 1000.49,1347.04 998.667,1347.14 996.845,1347.24 995.022,1347.34 993.2,1347.45 991.377,1347.55 989.555,1347.65 987.732,1347.75 \n", " 985.909,1347.85 984.087,1347.95 982.264,1348.05 980.442,1348.15 978.823,1348.24 978.619,1348.25 976.797,1348.34 974.974,1348.44 973.152,1348.54 971.329,1348.64 \n", " 969.507,1348.74 967.684,1348.84 965.861,1348.93 964.039,1349.03 962.216,1349.13 960.394,1349.22 958.571,1349.32 956.749,1349.42 954.926,1349.51 953.104,1349.61 \n", " 953.103,1349.61 951.281,1349.7 949.459,1349.8 947.636,1349.9 945.813,1349.99 943.991,1350.08 942.168,1350.18 940.346,1350.27 938.523,1350.37 936.701,1350.46 \n", " 934.878,1350.56 933.056,1350.65 931.233,1350.74 929.41,1350.84 927.588,1350.93 926.525,1350.98 925.765,1351.02 923.943,1351.11 922.12,1351.21 920.298,1351.3 \n", " 918.475,1351.39 916.653,1351.48 914.83,1351.57 913.008,1351.67 911.185,1351.76 909.362,1351.85 907.54,1351.94 905.717,1352.03 903.895,1352.12 902.072,1352.21 \n", " 900.25,1352.3 899.136,1352.36 898.427,1352.39 896.605,1352.48 894.782,1352.57 892.96,1352.66 891.137,1352.75 889.314,1352.84 887.492,1352.93 885.669,1353.02 \n", " 883.847,1353.11 882.024,1353.2 880.202,1353.29 878.379,1353.37 876.557,1353.46 874.734,1353.55 872.912,1353.64 871.089,1353.73 871.012,1353.73 869.266,1353.81 \n", " 867.444,1353.9 865.621,1353.99 863.799,1354.08 861.976,1354.16 860.154,1354.25 858.331,1354.34 856.509,1354.43 854.686,1354.51 852.864,1354.6 851.041,1354.69 \n", " 849.218,1354.77 847.396,1354.86 845.573,1354.95 843.751,1355.03 842.254,1355.1 841.928,1355.12 840.106,1355.21 838.283,1355.29 836.461,1355.38 834.638,1355.46 \n", " 832.816,1355.55 830.993,1355.64 829.17,1355.72 827.348,1355.81 825.525,1355.89 823.703,1355.98 821.88,1356.06 820.058,1356.15 818.235,1356.23 816.413,1356.32 \n", " 814.59,1356.4 812.991,1356.48 812.768,1356.49 810.945,1356.57 809.122,1356.66 807.3,1356.74 805.477,1356.83 803.655,1356.91 801.832,1357 800.01,1357.08 \n", " 798.187,1357.17 796.365,1357.25 794.542,1357.34 792.72,1357.42 790.897,1357.5 789.074,1357.59 787.252,1357.67 785.429,1357.76 783.607,1357.84 783.37,1357.85 \n", " 781.784,1357.92 779.962,1358.01 778.139,1358.09 776.317,1358.18 774.494,1358.26 772.671,1358.35 770.849,1358.43 769.026,1358.51 767.204,1358.6 765.381,1358.68 \n", " 763.559,1358.76 761.736,1358.85 759.914,1358.93 758.091,1359.02 756.269,1359.1 754.446,1359.18 753.541,1359.23 752.623,1359.27 750.801,1359.35 748.978,1359.44 \n", " 747.156,1359.52 745.333,1359.6 743.511,1359.69 741.688,1359.77 739.866,1359.85 738.043,1359.94 736.221,1360.02 734.398,1360.11 732.575,1360.19 730.753,1360.27 \n", " 728.93,1360.36 727.108,1360.44 725.285,1360.52 723.652,1360.6 723.463,1360.61 721.64,1360.69 719.818,1360.78 717.995,1360.86 716.173,1360.94 714.35,1361.03 \n", " 712.527,1361.11 710.705,1361.2 708.882,1361.28 707.06,1361.36 705.237,1361.45 703.415,1361.53 701.592,1361.62 699.77,1361.7 697.947,1361.78 696.125,1361.87 \n", " 694.302,1361.95 693.839,1361.97 692.479,1362.04 690.657,1362.12 688.834,1362.2 687.012,1362.29 685.189,1362.37 683.367,1362.46 681.544,1362.54 679.722,1362.63 \n", " 677.899,1362.71 676.077,1362.8 674.254,1362.88 672.431,1362.96 670.609,1363.05 668.786,1363.13 666.964,1363.22 665.141,1363.3 664.215,1363.35 663.319,1363.39 \n", " 661.496,1363.47 659.674,1363.56 657.851,1363.64 656.029,1363.73 654.206,1363.81 652.383,1363.9 650.561,1363.98 648.738,1364.07 646.916,1364.15 645.093,1364.24 \n", " 643.271,1364.33 641.448,1364.41 639.626,1364.5 637.803,1364.58 635.981,1364.67 634.873,1364.72 634.158,1364.75 632.335,1364.84 630.513,1364.93 628.69,1365.01 \n", " 626.868,1365.1 625.045,1365.18 623.223,1365.27 621.4,1365.36 619.578,1365.44 617.755,1365.53 615.933,1365.62 614.11,1365.7 612.287,1365.79 610.465,1365.88 \n", " 608.642,1365.96 606.82,1366.05 605.881,1366.09 604.997,1366.14 603.175,1366.22 601.352,1366.31 599.53,1366.4 597.707,1366.49 595.884,1366.57 594.062,1366.66 \n", " 592.239,1366.75 590.417,1366.83 588.594,1366.92 586.772,1367.01 584.949,1367.1 583.127,1367.19 581.304,1367.27 579.482,1367.36 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1789.65,1246.16 1787.83,1246.35 1786.01,1246.54 \n", " 1785.72,1246.57 1784.18,1246.74 1782.36,1246.93 1780.54,1247.12 1778.72,1247.31 1776.89,1247.51 1775.07,1247.7 1773.25,1247.89 1772.72,1247.95 1771.43,1248.08 \n", " 1769.6,1248.28 1767.78,1248.47 1765.96,1248.66 1764.14,1248.86 1762.31,1249.05 1760.49,1249.24 1759.76,1249.32 1758.67,1249.44 1756.85,1249.63 1755.02,1249.83 \n", " 1753.2,1250.02 1751.38,1250.21 1749.56,1250.41 1747.73,1250.6 1746.86,1250.7 1745.91,1250.8 1744.09,1250.99 1742.27,1251.18 1740.44,1251.38 1738.62,1251.57 \n", " 1736.8,1251.77 1734.98,1251.96 1734,1252.07 1733.15,1252.16 1731.33,1252.35 1729.51,1252.55 1727.69,1252.74 1725.86,1252.94 1724.04,1253.14 1722.22,1253.33 \n", " 1721.17,1253.44 1720.4,1253.53 1718.57,1253.72 1716.75,1253.92 1714.93,1254.11 1713.11,1254.31 1711.28,1254.51 1709.46,1254.7 1708.39,1254.82 1707.64,1254.9 \n", " 1705.82,1255.09 1703.99,1255.29 1702.17,1255.49 1700.35,1255.68 1698.53,1255.88 1696.7,1256.08 1695.64,1256.19 1694.88,1256.27 1693.06,1256.47 1691.23,1256.67 \n", " 1689.41,1256.86 1687.59,1257.06 1685.77,1257.26 1683.94,1257.45 1682.92,1257.56 1682.12,1257.65 1680.3,1257.85 1678.48,1258.05 1676.65,1258.24 1674.83,1258.44 \n", " 1673.01,1258.64 1671.19,1258.83 1670.23,1258.94 1669.36,1259.03 1667.54,1259.23 1665.72,1259.43 1663.9,1259.62 1662.07,1259.82 1660.25,1260.02 1658.43,1260.22 \n", " 1657.56,1260.31 1656.61,1260.42 1654.78,1260.61 1652.96,1260.81 1651.14,1261.01 1649.32,1261.21 1647.49,1261.4 1645.67,1261.6 1644.91,1261.69 1643.85,1261.8 \n", " 1642.03,1262 1640.2,1262.2 1638.38,1262.4 1636.56,1262.59 1634.74,1262.79 1632.91,1262.99 1632.27,1263.06 1631.09,1263.19 1629.27,1263.39 1627.45,1263.58 \n", " 1625.62,1263.78 1623.8,1263.98 1621.98,1264.18 1620.16,1264.38 1619.65,1264.43 1618.33,1264.58 1616.51,1264.77 1614.69,1264.97 1612.87,1265.17 1611.04,1265.37 \n", " 1609.22,1265.57 1607.4,1265.77 1607.02,1265.81 1605.58,1265.96 1603.75,1266.16 1601.93,1266.36 1600.11,1266.56 1598.29,1266.76 1596.46,1266.96 1594.64,1267.15 \n", " 1594.39,1267.18 1592.82,1267.35 1590.99,1267.55 1589.17,1267.75 1587.35,1267.95 1585.53,1268.14 1583.7,1268.34 1581.88,1268.54 1581.75,1268.55 1580.06,1268.74 \n", " 1578.24,1268.94 1576.41,1269.13 1574.59,1269.33 1572.77,1269.53 1570.95,1269.73 1569.12,1269.93 1569.1,1269.93 1567.3,1270.12 1565.48,1270.32 1563.66,1270.52 \n", " 1561.83,1270.72 1560.01,1270.91 1558.19,1271.11 1556.43,1271.3 1556.37,1271.31 1554.54,1271.51 1552.72,1271.7 1550.9,1271.9 1549.08,1272.1 1547.25,1272.3 \n", " 1545.43,1272.49 1543.74,1272.68 1543.61,1272.69 1541.79,1272.89 1539.96,1273.08 1538.14,1273.28 1536.32,1273.48 1534.5,1273.67 1532.67,1273.87 1531,1274.05 \n", " 1530.85,1274.07 1529.03,1274.26 1527.21,1274.46 1525.38,1274.65 1523.56,1274.85 1521.74,1275.05 1519.92,1275.24 1518.23,1275.42 1518.09,1275.44 1516.27,1275.63 \n", " 1514.45,1275.83 1512.63,1276.02 1510.8,1276.22 1508.98,1276.41 1507.16,1276.61 1505.4,1276.8 1505.34,1276.8 1503.51,1277 1501.69,1277.19 1499.87,1277.39 \n", " 1498.04,1277.58 1496.22,1277.78 1494.4,1277.97 1492.58,1278.16 1492.51,1278.17 1490.75,1278.36 1488.93,1278.55 1487.11,1278.75 1485.29,1278.94 1483.46,1279.13 \n", " 1481.64,1279.32 1479.82,1279.52 1479.56,1279.54 1478,1279.71 1476.17,1279.9 1474.35,1280.09 1472.53,1280.29 1470.71,1280.48 1468.88,1280.67 1467.06,1280.86 \n", " 1466.53,1280.92 1465.24,1281.05 1463.42,1281.25 1461.59,1281.44 1459.77,1281.63 1457.95,1281.82 1456.13,1282.01 1454.3,1282.2 1453.41,1282.29 1452.48,1282.39 \n", " 1450.66,1282.58 1448.84,1282.77 1447.01,1282.96 1445.19,1283.15 1443.37,1283.34 1441.55,1283.53 1440.18,1283.67 1439.72,1283.71 1437.9,1283.9 1436.08,1284.09 \n", " 1434.26,1284.28 1432.43,1284.47 1430.61,1284.65 1428.79,1284.84 1426.97,1285.03 1426.85,1285.04 1425.14,1285.21 1423.32,1285.4 1421.5,1285.59 1419.68,1285.77 \n", " 1417.85,1285.96 1416.03,1286.15 1414.21,1286.33 1413.39,1286.41 1412.39,1286.52 1410.56,1286.7 1408.74,1286.89 1406.92,1287.07 1405.1,1287.25 1403.27,1287.44 \n", " 1401.45,1287.62 1399.79,1287.79 1399.63,1287.8 1397.8,1287.99 1395.98,1288.17 1394.16,1288.35 1392.34,1288.53 1390.51,1288.72 1388.69,1288.9 1386.87,1289.08 \n", " 1386.04,1289.16 1385.05,1289.26 1383.22,1289.44 1381.4,1289.62 1379.58,1289.8 1377.76,1289.98 1375.93,1290.16 1374.11,1290.34 1372.29,1290.52 1372.11,1290.54 \n", " 1370.47,1290.7 1368.64,1290.87 1366.82,1291.05 1365,1291.23 1363.18,1291.41 1361.35,1291.58 1359.53,1291.76 1358,1291.91 1357.71,1291.94 1355.89,1292.11 \n", " 1354.06,1292.29 1352.24,1292.46 1350.42,1292.64 1348.6,1292.81 1346.77,1292.99 1344.95,1293.16 1343.67,1293.28 1343.13,1293.33 1341.31,1293.51 1339.48,1293.68 \n", " 1337.66,1293.85 1335.84,1294.03 1334.02,1294.2 1332.19,1294.37 1330.37,1294.54 1329.12,1294.66 1328.55,1294.71 1326.73,1294.88 1324.9,1295.05 1323.08,1295.22 \n", " 1321.26,1295.39 1319.44,1295.56 1317.61,1295.73 1315.79,1295.9 1314.32,1296.03 1313.97,1296.06 1312.15,1296.23 1310.32,1296.4 1308.5,1296.56 1306.68,1296.73 \n", " 1304.86,1296.9 1303.03,1297.06 1301.21,1297.23 1299.39,1297.39 1299.24,1297.4 1297.56,1297.56 1295.74,1297.72 1293.92,1297.88 1292.1,1298.05 1290.27,1298.21 \n", " 1288.45,1298.37 1286.63,1298.53 1284.81,1298.69 1283.86,1298.78 1282.98,1298.86 1281.16,1299.02 1279.34,1299.18 1277.52,1299.34 1275.69,1299.5 1273.87,1299.65 \n", " 1272.05,1299.81 1270.23,1299.97 1268.4,1300.13 1268.14,1300.15 1266.58,1300.29 1264.76,1300.44 1262.94,1300.6 1261.11,1300.76 1259.29,1300.91 1257.47,1301.07 \n", " 1255.65,1301.22 1253.82,1301.38 1252.05,1301.53 1252,1301.53 1250.18,1301.68 1248.36,1301.84 1246.53,1301.99 1244.71,1302.14 1242.89,1302.29 1241.07,1302.44 \n", " 1239.24,1302.6 1237.42,1302.75 1235.6,1302.9 1235.56,1302.9 1233.78,1303.05 1231.95,1303.19 1230.13,1303.34 1228.31,1303.49 1226.49,1303.64 1224.66,1303.79 \n", " 1222.84,1303.93 1221.02,1304.08 1219.2,1304.23 1218.62,1304.27 1217.37,1304.37 1215.55,1304.52 1213.73,1304.66 1211.91,1304.81 1210.08,1304.95 1208.26,1305.09 \n", " 1206.44,1305.24 1204.62,1305.38 1202.79,1305.52 1201.19,1305.65 1200.97,1305.66 1199.15,1305.81 1197.32,1305.95 1195.5,1306.09 1193.68,1306.23 1191.86,1306.37 \n", " 1190.03,1306.51 1188.21,1306.64 1186.39,1306.78 1184.57,1306.92 1183.22,1307.02 1182.74,1307.06 1180.92,1307.19 1179.1,1307.33 1177.28,1307.47 1175.45,1307.6 \n", " 1173.63,1307.74 1171.81,1307.87 1169.99,1308.01 1168.16,1308.14 1166.34,1308.27 1164.66,1308.39 1164.52,1308.4 1162.7,1308.54 1160.87,1308.67 1159.05,1308.8 \n", " 1157.23,1308.93 1155.41,1309.06 1153.58,1309.19 1151.76,1309.32 1149.94,1309.45 1148.12,1309.58 1146.29,1309.71 1145.44,1309.77 1144.47,1309.84 1142.65,1309.96 \n", " 1140.83,1310.09 1139,1310.22 1137.18,1310.34 1135.36,1310.47 1133.54,1310.6 1131.71,1310.72 1129.89,1310.84 1128.07,1310.97 1126.25,1311.09 1125.51,1311.14 \n", " 1124.42,1311.22 1122.6,1311.34 1120.78,1311.46 1118.96,1311.58 1117.13,1311.7 1115.31,1311.83 1113.49,1311.95 1111.67,1312.07 1109.84,1312.19 1108.02,1312.31 \n", " 1106.2,1312.42 1104.79,1312.52 1104.37,1312.54 1102.55,1312.66 1100.73,1312.78 1098.91,1312.9 1097.08,1313.01 1095.26,1313.13 1093.44,1313.25 1091.62,1313.36 \n", " 1089.79,1313.48 1087.97,1313.59 1086.15,1313.71 1084.33,1313.82 1083.21,1313.89 1082.5,1313.93 1080.68,1314.05 1078.86,1314.16 1077.04,1314.27 1075.21,1314.38 \n", " 1073.39,1314.5 1071.57,1314.61 1069.75,1314.72 1067.92,1314.83 1066.1,1314.94 1064.28,1315.05 1062.46,1315.16 1060.7,1315.26 1060.63,1315.27 1058.81,1315.38 \n", " 1056.99,1315.48 1055.17,1315.59 1053.34,1315.7 1051.52,1315.81 1049.7,1315.91 1047.88,1316.02 1046.05,1316.13 1044.23,1316.23 1042.41,1316.34 1040.59,1316.44 \n", " 1038.76,1316.55 1037.18,1316.64 1036.94,1316.65 1035.12,1316.76 1033.3,1316.86 1031.47,1316.96 1029.65,1317.06 1027.83,1317.17 1026.01,1317.27 1024.18,1317.37 \n", " 1022.36,1317.47 1020.54,1317.57 1018.72,1317.67 1016.89,1317.78 1015.07,1317.88 1013.25,1317.98 1012.58,1318.01 1011.43,1318.07 1009.6,1318.17 1007.78,1318.27 \n", " 1005.96,1318.37 1004.13,1318.47 1002.31,1318.57 1000.49,1318.67 998.667,1318.76 996.845,1318.86 995.022,1318.96 993.2,1319.05 991.377,1319.15 989.555,1319.24 \n", " 987.732,1319.34 986.867,1319.39 985.909,1319.44 984.087,1319.53 982.264,1319.62 980.442,1319.72 978.619,1319.81 976.797,1319.91 974.974,1320 973.152,1320.09 \n", " 971.329,1320.19 969.507,1320.28 967.684,1320.37 965.861,1320.46 964.039,1320.56 962.216,1320.65 960.394,1320.74 959.993,1320.76 958.571,1320.83 956.749,1320.92 \n", " 954.926,1321.01 953.104,1321.1 951.281,1321.19 949.459,1321.28 947.636,1321.37 945.813,1321.46 943.991,1321.55 942.168,1321.64 940.346,1321.73 938.523,1321.82 \n", " 936.701,1321.9 934.878,1321.99 933.056,1322.08 931.968,1322.13 931.233,1322.17 929.41,1322.26 927.588,1322.34 925.765,1322.43 923.943,1322.52 922.12,1322.6 \n", " 920.298,1322.69 918.475,1322.78 916.653,1322.86 914.83,1322.95 913.008,1323.03 911.185,1323.12 909.362,1323.2 907.54,1323.29 905.717,1323.37 903.895,1323.46 \n", " 902.843,1323.51 902.072,1323.54 900.25,1323.63 898.427,1323.71 896.605,1323.79 894.782,1323.88 892.96,1323.96 891.137,1324.05 889.314,1324.13 887.492,1324.21 \n", " 885.669,1324.3 883.847,1324.38 882.024,1324.46 880.202,1324.54 878.379,1324.63 876.557,1324.71 874.734,1324.79 872.912,1324.87 872.721,1324.88 871.089,1324.95 \n", " 869.266,1325.04 867.444,1325.12 865.621,1325.2 863.799,1325.28 861.976,1325.36 860.154,1325.44 858.331,1325.52 856.509,1325.6 854.686,1325.68 852.864,1325.77 \n", " 851.041,1325.85 849.218,1325.93 847.396,1326.01 845.573,1326.09 843.751,1326.17 841.928,1326.25 841.762,1326.25 840.106,1326.33 838.283,1326.41 836.461,1326.49 \n", " 834.638,1326.57 832.816,1326.65 830.993,1326.73 829.17,1326.8 827.348,1326.88 825.525,1326.96 823.703,1327.04 821.88,1327.12 820.058,1327.2 818.235,1327.28 \n", " 816.413,1327.36 814.59,1327.44 812.768,1327.52 810.945,1327.59 810.172,1327.63 809.122,1327.67 807.3,1327.75 805.477,1327.83 803.655,1327.91 801.832,1327.99 \n", " 800.01,1328.07 798.187,1328.14 796.365,1328.22 794.542,1328.3 792.72,1328.38 790.897,1328.46 789.074,1328.54 787.252,1328.61 785.429,1328.69 783.607,1328.77 \n", " 781.784,1328.85 779.962,1328.93 778.19,1329 778.139,1329 776.317,1329.08 774.494,1329.16 772.671,1329.24 770.849,1329.32 769.026,1329.39 767.204,1329.47 \n", " 765.381,1329.55 763.559,1329.63 761.736,1329.71 759.914,1329.78 758.091,1329.86 756.269,1329.94 754.446,1330.02 752.623,1330.09 750.801,1330.17 748.978,1330.25 \n", " 747.156,1330.33 746.057,1330.38 745.333,1330.41 743.511,1330.48 741.688,1330.56 739.866,1330.64 738.043,1330.72 736.221,1330.8 734.398,1330.87 732.575,1330.95 \n", " 730.753,1331.03 728.93,1331.11 727.108,1331.19 725.285,1331.26 723.463,1331.34 721.64,1331.42 719.818,1331.5 717.995,1331.58 716.173,1331.66 714.35,1331.73 \n", " 714,1331.75 712.527,1331.81 710.705,1331.89 708.882,1331.97 707.06,1332.05 705.237,1332.13 703.415,1332.21 701.592,1332.28 699.77,1332.36 697.947,1332.44 \n", " 696.125,1332.52 694.302,1332.6 692.479,1332.68 690.657,1332.76 688.834,1332.84 687.012,1332.91 685.189,1332.99 683.367,1333.07 682.21,1333.12 681.544,1333.15 \n", " 679.722,1333.23 677.899,1333.31 676.077,1333.39 674.254,1333.47 672.431,1333.55 670.609,1333.63 668.786,1333.71 666.964,1333.79 665.141,1333.87 663.319,1333.95 \n", " 661.496,1334.03 659.674,1334.11 657.851,1334.19 656.029,1334.27 654.206,1334.35 652.383,1334.43 650.834,1334.5 650.561,1334.51 648.738,1334.59 646.916,1334.67 \n", " 645.093,1334.75 643.271,1334.83 641.448,1334.91 639.626,1334.99 637.803,1335.07 635.981,1335.16 634.158,1335.24 632.335,1335.32 630.513,1335.4 628.69,1335.48 \n", " 626.868,1335.56 625.045,1335.64 623.223,1335.72 621.4,1335.81 619.974,1335.87 619.578,1335.89 617.755,1335.97 615.933,1336.05 614.11,1336.13 612.287,1336.22 \n", " 610.465,1336.3 608.642,1336.38 606.82,1336.46 604.997,1336.55 603.175,1336.63 601.352,1336.71 599.53,1336.8 597.707,1336.88 595.884,1336.96 594.062,1337.04 \n", " 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1612.87,1235.74 \n", " 1611.04,1235.95 1609.22,1236.15 1607.4,1236.36 1605.58,1236.56 1603.75,1236.77 1602.07,1236.96 1601.93,1236.97 1600.11,1237.18 1598.29,1237.38 1596.46,1237.59 \n", " 1594.64,1237.79 1592.82,1238 1590.99,1238.2 1589.87,1238.33 1589.17,1238.41 1587.35,1238.62 1585.53,1238.82 1583.7,1239.03 1581.88,1239.23 1580.06,1239.44 \n", " 1578.24,1239.64 1577.68,1239.7 1576.41,1239.85 1574.59,1240.05 1572.77,1240.26 1570.95,1240.46 1569.12,1240.67 1567.3,1240.87 1565.48,1241.08 1565.47,1241.08 \n", " 1563.66,1241.28 1561.83,1241.49 1560.01,1241.69 1558.19,1241.9 1556.37,1242.1 1554.54,1242.31 1553.26,1242.45 1552.72,1242.51 1550.9,1242.72 1549.08,1242.92 \n", " 1547.25,1243.13 1545.43,1243.33 1543.61,1243.54 1541.79,1243.74 1541.02,1243.83 1539.96,1243.94 1538.14,1244.15 1536.32,1244.35 1534.5,1244.56 1532.67,1244.76 \n", " 1530.85,1244.97 1529.03,1245.17 1528.76,1245.2 1527.21,1245.37 1525.38,1245.58 1523.56,1245.78 1521.74,1245.99 1519.92,1246.19 1518.09,1246.39 1516.47,1246.57 \n", " 1516.27,1246.6 1514.45,1246.8 1512.63,1247 1510.8,1247.21 1508.98,1247.41 1507.16,1247.61 1505.34,1247.81 1504.13,1247.95 1503.51,1248.02 1501.69,1248.22 \n", " 1499.87,1248.42 1498.04,1248.62 1496.22,1248.83 1494.4,1249.03 1492.58,1249.23 1491.75,1249.32 1490.75,1249.43 1488.93,1249.63 1487.11,1249.83 1485.29,1250.04 \n", " 1483.46,1250.24 1481.64,1250.44 1479.82,1250.64 1479.31,1250.7 1478,1250.84 1476.17,1251.04 1474.35,1251.24 1472.53,1251.44 1470.71,1251.64 1468.88,1251.84 \n", " 1467.06,1252.04 1466.79,1252.07 1465.24,1252.24 1463.42,1252.44 1461.59,1252.64 1459.77,1252.84 1457.95,1253.03 1456.13,1253.23 1454.3,1253.43 1454.2,1253.44 \n", " 1452.48,1253.63 1450.66,1253.83 1448.84,1254.02 1447.01,1254.22 1445.19,1254.42 1443.37,1254.62 1441.55,1254.81 1441.52,1254.82 1439.72,1255.01 1437.9,1255.21 \n", " 1436.08,1255.4 1434.26,1255.6 1432.43,1255.79 1430.61,1255.99 1428.79,1256.18 1428.73,1256.19 1426.97,1256.38 1425.14,1256.57 1423.32,1256.77 1421.5,1256.96 \n", " 1419.68,1257.16 1417.85,1257.35 1416.03,1257.54 1415.83,1257.56 1414.21,1257.74 1412.39,1257.93 1410.56,1258.12 1408.74,1258.31 1406.92,1258.5 1405.1,1258.7 \n", " 1403.27,1258.89 1402.79,1258.94 1401.45,1259.08 1399.63,1259.27 1397.8,1259.46 1395.98,1259.65 1394.16,1259.84 1392.34,1260.03 1390.51,1260.22 1389.61,1260.31 \n", " 1388.69,1260.41 1386.87,1260.6 1385.05,1260.78 1383.22,1260.97 1381.4,1261.16 1379.58,1261.35 1377.76,1261.53 1376.27,1261.69 1375.93,1261.72 1374.11,1261.91 \n", " 1372.29,1262.09 1370.47,1262.28 1368.64,1262.46 1366.82,1262.65 1365,1262.83 1363.18,1263.02 1362.75,1263.06 1361.35,1263.2 1359.53,1263.38 1357.71,1263.57 \n", " 1355.89,1263.75 1354.06,1263.93 1352.24,1264.11 1350.42,1264.29 1349.03,1264.43 1348.6,1264.48 1346.77,1264.66 1344.95,1264.84 1343.13,1265.02 1341.31,1265.2 \n", " 1339.48,1265.38 1337.66,1265.55 1335.84,1265.73 1335.08,1265.81 1334.02,1265.91 1332.19,1266.09 1330.37,1266.27 1328.55,1266.44 1326.73,1266.62 1324.9,1266.8 \n", " 1323.08,1266.97 1321.26,1267.15 1320.9,1267.18 1319.44,1267.32 1317.61,1267.5 1315.79,1267.67 1313.97,1267.84 1312.15,1268.02 1310.32,1268.19 1308.5,1268.36 \n", " 1306.68,1268.53 1306.43,1268.55 1304.86,1268.7 1303.03,1268.87 1301.21,1269.04 1299.39,1269.21 1297.56,1269.38 1295.74,1269.55 1293.92,1269.72 1292.1,1269.89 \n", " 1291.67,1269.93 1290.27,1270.06 1288.45,1270.22 1286.63,1270.39 1284.81,1270.56 1282.98,1270.72 1281.16,1270.89 1279.34,1271.05 1277.52,1271.22 1276.57,1271.3 \n", " 1275.69,1271.38 1273.87,1271.54 1272.05,1271.71 1270.23,1271.87 1268.4,1272.03 1266.58,1272.19 1264.76,1272.35 1262.94,1272.52 1261.11,1272.68 1261.11,1272.68 \n", " 1259.29,1272.83 1257.47,1272.99 1255.65,1273.15 1253.82,1273.31 1252,1273.47 1250.18,1273.63 1248.36,1273.78 1246.53,1273.94 1245.23,1274.05 1244.71,1274.09 \n", " 1242.89,1274.25 1241.07,1274.4 1239.24,1274.56 1237.42,1274.71 1235.6,1274.86 1233.78,1275.02 1231.95,1275.17 1230.13,1275.32 1228.89,1275.42 1228.31,1275.47 \n", " 1226.49,1275.62 1224.66,1275.77 1222.84,1275.92 1221.02,1276.07 1219.2,1276.22 1217.37,1276.37 1215.55,1276.52 1213.73,1276.66 1212.05,1276.8 1211.91,1276.81 \n", " 1210.08,1276.95 1208.26,1277.1 1206.44,1277.25 1204.62,1277.39 1202.79,1277.53 1200.97,1277.68 1199.15,1277.82 1197.32,1277.96 1195.5,1278.1 1194.64,1278.17 \n", " 1193.68,1278.25 1191.86,1278.39 1190.03,1278.53 1188.21,1278.67 1186.39,1278.81 1184.57,1278.94 1182.74,1279.08 1180.92,1279.22 1179.1,1279.36 1177.28,1279.49 \n", " 1176.59,1279.54 1175.45,1279.63 1173.63,1279.77 1171.81,1279.9 1169.99,1280.04 1168.16,1280.17 1166.34,1280.3 1164.52,1280.44 1162.7,1280.57 1160.87,1280.7 \n", " 1159.05,1280.83 1157.84,1280.92 1157.23,1280.96 1155.41,1281.09 1153.58,1281.22 1151.76,1281.35 1149.94,1281.48 1148.12,1281.61 1146.29,1281.74 1144.47,1281.87 \n", " 1142.65,1281.99 1140.83,1282.12 1139,1282.24 1138.3,1282.29 1137.18,1282.37 1135.36,1282.49 1133.54,1282.62 1131.71,1282.74 1129.89,1282.87 1128.07,1282.99 \n", " 1126.25,1283.11 1124.42,1283.23 1122.6,1283.35 1120.78,1283.48 1118.96,1283.6 1117.88,1283.67 1117.13,1283.72 1115.31,1283.83 1113.49,1283.95 1111.67,1284.07 \n", " 1109.84,1284.19 1108.02,1284.31 1106.2,1284.42 1104.37,1284.54 1102.55,1284.66 1100.73,1284.77 1098.91,1284.89 1097.08,1285 1096.46,1285.04 1095.26,1285.12 \n", " 1093.44,1285.23 1091.62,1285.34 1089.79,1285.45 1087.97,1285.57 1086.15,1285.68 1084.33,1285.79 1082.5,1285.9 1080.68,1286.01 1078.86,1286.12 1077.04,1286.23 \n", " 1075.21,1286.34 1073.94,1286.41 1073.39,1286.45 1071.57,1286.55 1069.75,1286.66 1067.92,1286.77 1066.1,1286.88 1064.28,1286.98 1062.46,1287.09 1060.63,1287.19 \n", " 1058.81,1287.3 1056.99,1287.4 1055.17,1287.51 1053.34,1287.61 1051.52,1287.71 1050.19,1287.79 1049.7,1287.82 1047.88,1287.92 1046.05,1288.02 1044.23,1288.12 \n", " 1042.41,1288.22 1040.59,1288.32 1038.76,1288.42 1036.94,1288.52 1035.12,1288.62 1033.3,1288.72 1031.47,1288.82 1029.65,1288.92 1027.83,1289.02 1026.01,1289.11 \n", " 1025.08,1289.16 1024.18,1289.21 1022.36,1289.31 1020.54,1289.4 1018.72,1289.5 1016.89,1289.59 1015.07,1289.69 1013.25,1289.78 1011.43,1289.88 1009.6,1289.97 \n", " 1007.78,1290.07 1005.96,1290.16 1004.13,1290.25 1002.31,1290.34 1000.49,1290.44 998.667,1290.53 998.512,1290.54 996.845,1290.62 995.022,1290.71 993.2,1290.8 \n", " 991.377,1290.89 989.555,1290.98 987.732,1291.07 985.909,1291.16 984.087,1291.25 982.264,1291.34 980.442,1291.43 978.619,1291.51 976.797,1291.6 974.974,1291.69 \n", " 973.152,1291.78 971.329,1291.86 970.377,1291.91 969.507,1291.95 967.684,1292.04 965.861,1292.12 964.039,1292.21 962.216,1292.29 960.394,1292.38 958.571,1292.46 \n", " 956.749,1292.55 954.926,1292.63 953.104,1292.72 951.281,1292.8 949.459,1292.88 947.636,1292.97 945.813,1293.05 943.991,1293.13 942.168,1293.21 940.636,1293.28 \n", " 940.346,1293.3 938.523,1293.38 936.701,1293.46 934.878,1293.54 933.056,1293.62 931.233,1293.7 929.41,1293.78 927.588,1293.86 925.765,1293.94 923.943,1294.02 \n", " 922.12,1294.1 920.298,1294.18 918.475,1294.26 916.653,1294.34 914.83,1294.42 913.008,1294.5 911.185,1294.58 909.362,1294.65 909.318,1294.66 907.54,1294.73 \n", " 905.717,1294.81 903.895,1294.89 902.072,1294.97 900.25,1295.04 898.427,1295.12 896.605,1295.2 894.782,1295.27 892.96,1295.35 891.137,1295.43 889.314,1295.5 \n", " 887.492,1295.58 885.669,1295.65 883.847,1295.73 882.024,1295.81 880.202,1295.88 878.379,1295.96 876.562,1296.03 876.557,1296.03 874.734,1296.11 872.912,1296.18 \n", " 871.089,1296.26 869.266,1296.33 867.444,1296.4 865.621,1296.48 863.799,1296.55 861.976,1296.63 860.154,1296.7 858.331,1296.77 856.509,1296.85 854.686,1296.92 \n", " 852.864,1296.99 851.041,1297.07 849.218,1297.14 847.396,1297.21 845.573,1297.29 843.751,1297.36 842.621,1297.4 841.928,1297.43 840.106,1297.5 838.283,1297.58 \n", " 836.461,1297.65 834.638,1297.72 832.816,1297.79 830.993,1297.87 829.17,1297.94 827.348,1298.01 825.525,1298.08 823.703,1298.15 821.88,1298.23 820.058,1298.3 \n", " 818.235,1298.37 816.413,1298.44 814.59,1298.51 812.768,1298.59 810.945,1298.66 809.122,1298.73 807.856,1298.78 807.3,1298.8 805.477,1298.87 803.655,1298.94 \n", " 801.832,1299.01 800.01,1299.09 798.187,1299.16 796.365,1299.23 794.542,1299.3 792.72,1299.37 790.897,1299.44 789.074,1299.51 787.252,1299.58 785.429,1299.65 \n", " 783.607,1299.73 781.784,1299.8 779.962,1299.87 778.139,1299.94 776.317,1300.01 774.494,1300.08 772.686,1300.15 772.671,1300.15 770.849,1300.22 769.026,1300.29 \n", " 767.204,1300.37 765.381,1300.44 763.559,1300.51 761.736,1300.58 759.914,1300.65 758.091,1300.72 756.269,1300.79 754.446,1300.86 752.623,1300.93 750.801,1301.01 \n", " 748.978,1301.08 747.156,1301.15 745.333,1301.22 743.511,1301.29 741.688,1301.36 739.866,1301.43 738.043,1301.51 737.526,1301.53 736.221,1301.58 734.398,1301.65 \n", " 732.575,1301.72 730.753,1301.79 728.93,1301.86 727.108,1301.94 725.285,1302.01 723.463,1302.08 721.64,1302.15 719.818,1302.22 717.995,1302.29 716.173,1302.37 \n", " 714.35,1302.44 712.527,1302.51 710.705,1302.58 708.882,1302.66 707.06,1302.73 705.237,1302.8 703.415,1302.87 702.731,1302.9 701.592,1302.94 699.77,1303.02 \n", " 697.947,1303.09 696.125,1303.16 694.302,1303.24 692.479,1303.31 690.657,1303.38 688.834,1303.45 687.012,1303.53 685.189,1303.6 683.367,1303.67 681.544,1303.75 \n", " 679.722,1303.82 677.899,1303.9 676.077,1303.97 674.254,1304.04 672.431,1304.12 670.609,1304.19 668.786,1304.26 668.565,1304.27 666.964,1304.34 665.141,1304.41 \n", " 663.319,1304.49 661.496,1304.56 659.674,1304.64 657.851,1304.71 656.029,1304.79 654.206,1304.86 652.383,1304.94 650.561,1305.01 648.738,1305.09 646.916,1305.16 \n", " 645.093,1305.24 643.271,1305.31 641.448,1305.39 639.626,1305.46 637.803,1305.54 635.981,1305.61 635.195,1305.65 634.158,1305.69 632.335,1305.77 630.513,1305.84 \n", " 628.69,1305.92 626.868,1306 625.045,1306.07 623.223,1306.15 621.4,1306.23 619.578,1306.3 617.755,1306.38 615.933,1306.46 614.11,1306.53 612.287,1306.61 \n", " 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1621.98,1205.19 1620.56,1205.36 1620.16,1205.41 1618.33,1205.62 1616.51,1205.83 1614.69,1206.04 1612.87,1206.26 \n", " 1611.04,1206.47 1609.22,1206.68 1608.78,1206.73 1607.4,1206.89 1605.58,1207.11 1603.75,1207.32 1601.93,1207.53 1600.11,1207.75 1598.29,1207.96 1597.01,1208.11 \n", " 1596.46,1208.17 1594.64,1208.38 1592.82,1208.6 1590.99,1208.81 1589.17,1209.02 1587.35,1209.24 1585.53,1209.45 1585.26,1209.48 1583.7,1209.66 1581.88,1209.88 \n", " 1580.06,1210.09 1578.24,1210.3 1576.41,1210.52 1574.59,1210.73 1573.53,1210.85 1572.77,1210.94 1570.95,1211.16 1569.12,1211.37 1567.3,1211.58 1565.48,1211.8 \n", " 1563.66,1212.01 1561.83,1212.22 1561.8,1212.23 1560.01,1212.44 1558.19,1212.65 1556.37,1212.86 1554.54,1213.08 1552.72,1213.29 1550.9,1213.5 1550.06,1213.6 \n", " 1549.08,1213.72 1547.25,1213.93 1545.43,1214.14 1543.61,1214.36 1541.79,1214.57 1539.96,1214.78 1538.32,1214.98 1538.14,1215 1536.32,1215.21 1534.5,1215.42 \n", " 1532.67,1215.64 1530.85,1215.85 1529.03,1216.06 1527.21,1216.28 1526.56,1216.35 1525.38,1216.49 1523.56,1216.7 1521.74,1216.91 1519.92,1217.13 1518.09,1217.34 \n", " 1516.27,1217.55 1514.79,1217.72 1514.45,1217.76 1512.63,1217.98 1510.8,1218.19 1508.98,1218.4 1507.16,1218.61 1505.34,1218.82 1503.51,1219.04 1502.98,1219.1 \n", " 1501.69,1219.25 1499.87,1219.46 1498.04,1219.67 1496.22,1219.88 1494.4,1220.09 1492.58,1220.3 1491.13,1220.47 1490.75,1220.51 1488.93,1220.73 1487.11,1220.94 \n", " 1485.29,1221.15 1483.46,1221.36 1481.64,1221.57 1479.82,1221.78 1479.23,1221.85 1478,1221.99 1476.17,1222.2 1474.35,1222.41 1472.53,1222.62 1470.71,1222.83 \n", " 1468.88,1223.03 1467.27,1223.22 1467.06,1223.24 1465.24,1223.45 1463.42,1223.66 1461.59,1223.87 1459.77,1224.08 1457.95,1224.29 1456.13,1224.49 1455.25,1224.59 \n", " 1454.3,1224.7 1452.48,1224.91 1450.66,1225.12 1448.84,1225.32 1447.01,1225.53 1445.19,1225.74 1443.37,1225.94 1443.15,1225.97 1441.55,1226.15 1439.72,1226.35 \n", " 1437.9,1226.56 1436.08,1226.76 1434.26,1226.97 1432.43,1227.17 1430.96,1227.34 1430.61,1227.38 1428.79,1227.58 1426.97,1227.79 1425.14,1227.99 1423.32,1228.2 \n", " 1421.5,1228.4 1419.68,1228.6 1418.66,1228.71 1417.85,1228.8 1416.03,1229.01 1414.21,1229.21 1412.39,1229.41 1410.56,1229.61 1408.74,1229.81 1406.92,1230.01 \n", " 1406.25,1230.09 1405.1,1230.21 1403.27,1230.42 1401.45,1230.62 1399.63,1230.81 1397.8,1231.01 1395.98,1231.21 1394.16,1231.41 1393.7,1231.46 1392.34,1231.61 \n", " 1390.51,1231.81 1388.69,1232.01 1386.87,1232.2 1385.05,1232.4 1383.22,1232.6 1381.4,1232.79 1381.01,1232.84 1379.58,1232.99 1377.76,1233.18 1375.93,1233.38 \n", " 1374.11,1233.57 1372.29,1233.77 1370.47,1233.96 1368.64,1234.16 1368.15,1234.21 1366.82,1234.35 1365,1234.54 1363.18,1234.74 1361.35,1234.93 1359.53,1235.12 \n", " 1357.71,1235.31 1355.89,1235.5 1355.1,1235.58 1354.06,1235.69 1352.24,1235.88 1350.42,1236.07 1348.6,1236.26 1346.77,1236.45 1344.95,1236.64 1343.13,1236.82 \n", " 1341.84,1236.96 1341.31,1237.01 1339.48,1237.2 1337.66,1237.39 1335.84,1237.57 1334.02,1237.76 1332.19,1237.94 1330.37,1238.13 1328.55,1238.31 1328.36,1238.33 \n", " 1326.73,1238.5 1324.9,1238.68 1323.08,1238.86 1321.26,1239.04 1319.44,1239.23 1317.61,1239.41 1315.79,1239.59 1314.61,1239.7 1313.97,1239.77 1312.15,1239.95 \n", " 1310.32,1240.13 1308.5,1240.31 1306.68,1240.48 1304.86,1240.66 1303.03,1240.84 1301.21,1241.02 1300.58,1241.08 1299.39,1241.19 1297.56,1241.37 1295.74,1241.54 \n", " 1293.92,1241.72 1292.1,1241.89 1290.27,1242.07 1288.45,1242.24 1286.63,1242.41 1286.22,1242.45 1284.81,1242.59 1282.98,1242.76 1281.16,1242.93 1279.34,1243.1 \n", " 1277.52,1243.27 1275.69,1243.44 1273.87,1243.61 1272.05,1243.78 1271.5,1243.83 1270.23,1243.94 1268.4,1244.11 1266.58,1244.28 1264.76,1244.44 1262.94,1244.61 \n", " 1261.11,1244.77 1259.29,1244.94 1257.47,1245.1 1256.39,1245.2 1255.65,1245.27 1253.82,1245.43 1252,1245.59 1250.18,1245.75 1248.36,1245.91 1246.53,1246.07 \n", " 1244.71,1246.23 1242.89,1246.39 1241.07,1246.55 1240.82,1246.57 1239.24,1246.71 1237.42,1246.87 1235.6,1247.02 1233.78,1247.18 1231.95,1247.34 1230.13,1247.49 \n", " 1228.31,1247.65 1226.49,1247.8 1224.74,1247.95 1224.66,1247.95 1222.84,1248.11 1221.02,1248.26 1219.2,1248.41 1217.37,1248.56 1215.55,1248.71 1213.73,1248.86 \n", " 1211.91,1249.01 1210.08,1249.16 1208.26,1249.31 1208.09,1249.32 1206.44,1249.46 1204.62,1249.6 1202.79,1249.75 1200.97,1249.89 1199.15,1250.04 1197.32,1250.18 \n", " 1195.5,1250.33 1193.68,1250.47 1191.86,1250.61 1190.79,1250.7 1190.03,1250.75 1188.21,1250.9 1186.39,1251.04 1184.57,1251.18 1182.74,1251.32 1180.92,1251.45 \n", " 1179.1,1251.59 1177.28,1251.73 1175.45,1251.87 1173.63,1252 1172.76,1252.07 1171.81,1252.14 1169.99,1252.28 1168.16,1252.41 1166.34,1252.54 1164.52,1252.68 \n", " 1162.7,1252.81 1160.87,1252.94 1159.05,1253.07 1157.23,1253.2 1155.41,1253.34 1153.9,1253.44 1153.58,1253.47 1151.76,1253.59 1149.94,1253.72 1148.12,1253.85 \n", " 1146.29,1253.98 1144.47,1254.1 1142.65,1254.23 1140.83,1254.36 1139,1254.48 1137.18,1254.61 1135.36,1254.73 1134.07,1254.82 1133.54,1254.85 1131.71,1254.98 \n", " 1129.89,1255.1 1128.07,1255.22 1126.25,1255.34 1124.42,1255.46 1122.6,1255.58 1120.78,1255.7 1118.96,1255.82 1117.13,1255.94 1115.31,1256.05 1113.49,1256.17 \n", " 1113.15,1256.19 1111.67,1256.29 1109.84,1256.4 1108.02,1256.52 1106.2,1256.63 1104.37,1256.74 1102.55,1256.86 1100.73,1256.97 1098.91,1257.08 1097.08,1257.19 \n", " 1095.26,1257.31 1093.44,1257.42 1091.62,1257.53 1090.97,1257.56 1089.79,1257.63 1087.97,1257.74 1086.15,1257.85 1084.33,1257.96 1082.5,1258.07 1080.68,1258.17 \n", " 1078.86,1258.28 1077.04,1258.39 1075.21,1258.49 1073.39,1258.59 1071.57,1258.7 1069.75,1258.8 1067.92,1258.91 1067.34,1258.94 1066.1,1259.01 1064.28,1259.11 \n", " 1062.46,1259.21 1060.63,1259.31 1058.81,1259.41 1056.99,1259.51 1055.17,1259.61 1053.34,1259.71 1051.52,1259.81 1049.7,1259.91 1047.88,1260 1046.05,1260.1 \n", " 1044.23,1260.2 1042.41,1260.29 1042.04,1260.31 1040.59,1260.39 1038.76,1260.48 1036.94,1260.58 1035.12,1260.67 1033.3,1260.76 1031.47,1260.86 1029.65,1260.95 \n", " 1027.83,1261.04 1026.01,1261.13 1024.18,1261.22 1022.36,1261.32 1020.54,1261.41 1018.72,1261.5 1016.89,1261.59 1015.07,1261.67 1014.84,1261.69 1013.25,1261.76 \n", " 1011.43,1261.85 1009.6,1261.94 1007.78,1262.03 1005.96,1262.11 1004.13,1262.2 1002.31,1262.29 1000.49,1262.37 998.667,1262.46 996.845,1262.54 995.022,1262.62 \n", " 993.2,1262.71 991.377,1262.79 989.555,1262.88 987.732,1262.96 985.909,1263.04 985.503,1263.06 984.087,1263.12 982.264,1263.2 980.442,1263.29 978.619,1263.37 \n", " 976.797,1263.45 974.974,1263.53 973.152,1263.61 971.329,1263.69 969.507,1263.77 967.684,1263.85 965.861,1263.92 964.039,1264 962.216,1264.08 960.394,1264.16 \n", " 958.571,1264.23 956.749,1264.31 954.926,1264.39 953.85,1264.43 953.104,1264.46 951.281,1264.54 949.459,1264.62 947.636,1264.69 945.813,1264.77 943.991,1264.84 \n", " 942.168,1264.92 940.346,1264.99 938.523,1265.06 936.701,1265.14 934.878,1265.21 933.056,1265.28 931.233,1265.36 929.41,1265.43 927.588,1265.5 925.765,1265.57 \n", " 923.943,1265.64 922.12,1265.72 920.298,1265.79 919.802,1265.81 918.475,1265.86 916.653,1265.93 914.83,1266 913.008,1266.07 911.185,1266.14 909.362,1266.21 \n", " 907.54,1266.28 905.717,1266.35 903.895,1266.42 902.072,1266.49 900.25,1266.56 898.427,1266.63 896.605,1266.69 894.782,1266.76 892.96,1266.83 891.137,1266.9 \n", " 889.314,1266.97 887.492,1267.03 885.669,1267.1 883.847,1267.17 883.49,1267.18 882.024,1267.23 880.202,1267.3 878.379,1267.37 876.557,1267.43 874.734,1267.5 \n", " 872.912,1267.57 871.089,1267.63 869.266,1267.7 867.444,1267.77 865.621,1267.83 863.799,1267.9 861.976,1267.96 860.154,1268.03 858.331,1268.09 856.509,1268.16 \n", " 854.686,1268.22 852.864,1268.29 851.041,1268.35 849.218,1268.42 847.396,1268.48 845.573,1268.55 845.327,1268.55 843.751,1268.61 841.928,1268.67 840.106,1268.74 \n", " 838.283,1268.8 836.461,1268.87 834.638,1268.93 832.816,1268.99 830.993,1269.06 829.17,1269.12 827.348,1269.19 825.525,1269.25 823.703,1269.31 821.88,1269.38 \n", " 820.058,1269.44 818.235,1269.5 816.413,1269.57 814.59,1269.63 812.768,1269.69 810.945,1269.76 809.122,1269.82 807.3,1269.88 805.999,1269.93 805.477,1269.95 \n", " 803.655,1270.01 801.832,1270.07 800.01,1270.14 798.187,1270.2 796.365,1270.26 794.542,1270.33 792.72,1270.39 790.897,1270.45 789.074,1270.51 787.252,1270.58 \n", " 785.429,1270.64 783.607,1270.7 781.784,1270.77 779.962,1270.83 778.139,1270.89 776.317,1270.96 774.494,1271.02 772.671,1271.08 770.849,1271.15 769.026,1271.21 \n", " 767.204,1271.27 766.335,1271.3 765.381,1271.34 763.559,1271.4 761.736,1271.46 759.914,1271.53 758.091,1271.59 756.269,1271.65 754.446,1271.72 752.623,1271.78 \n", " 750.801,1271.84 748.978,1271.91 747.156,1271.97 745.333,1272.03 743.511,1272.1 741.688,1272.16 739.866,1272.23 738.043,1272.29 736.221,1272.35 734.398,1272.42 \n", " 732.575,1272.48 730.753,1272.55 728.93,1272.61 727.108,1272.68 727.102,1272.68 725.285,1272.74 723.463,1272.8 721.64,1272.87 719.818,1272.93 717.995,1273 \n", " 716.173,1273.06 714.35,1273.13 712.527,1273.19 710.705,1273.26 708.882,1273.33 707.06,1273.39 705.237,1273.46 703.415,1273.52 701.592,1273.59 699.77,1273.65 \n", " 697.947,1273.72 696.125,1273.79 694.302,1273.85 692.479,1273.92 690.657,1273.98 688.862,1274.05 688.834,1274.05 687.012,1274.12 685.189,1274.18 683.367,1274.25 \n", " 681.544,1274.32 679.722,1274.38 677.899,1274.45 676.077,1274.52 674.254,1274.59 672.431,1274.65 670.609,1274.72 668.786,1274.79 666.964,1274.86 665.141,1274.93 \n", " 663.319,1274.99 661.496,1275.06 659.674,1275.13 657.851,1275.2 656.029,1275.27 654.206,1275.34 652.383,1275.41 651.934,1275.42 650.561,1275.48 648.738,1275.54 \n", " 646.916,1275.61 645.093,1275.68 643.271,1275.75 641.448,1275.82 639.626,1275.89 637.803,1275.96 635.981,1276.03 634.158,1276.1 632.335,1276.18 630.513,1276.25 \n", " 628.69,1276.32 626.868,1276.39 625.045,1276.46 623.223,1276.53 621.4,1276.6 619.578,1276.67 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1824.28,1152.24 1822.46,1152.44 1820.64,1152.64 1818.81,1152.84 \n", " 1816.99,1153.04 1815.95,1153.16 1815.17,1153.24 1813.35,1153.44 1811.52,1153.64 1809.7,1153.84 1807.88,1154.04 1806.06,1154.24 1804.23,1154.45 1803.48,1154.53 \n", " 1802.41,1154.65 1800.59,1154.85 1798.77,1155.05 1796.94,1155.25 1795.12,1155.45 1793.3,1155.66 1791.47,1155.86 1791.09,1155.9 1789.65,1156.06 1787.83,1156.27 \n", " 1786.01,1156.47 1784.18,1156.67 1782.36,1156.88 1780.54,1157.08 1778.8,1157.28 1778.72,1157.29 1776.89,1157.49 1775.07,1157.69 1773.25,1157.9 1771.43,1158.1 \n", " 1769.6,1158.31 1767.78,1158.52 1766.59,1158.65 1765.96,1158.72 1764.14,1158.93 1762.31,1159.13 1760.49,1159.34 1758.67,1159.55 1756.85,1159.75 1755.02,1159.96 \n", " 1754.46,1160.02 1753.2,1160.17 1751.38,1160.37 1749.56,1160.58 1747.73,1160.79 1745.91,1161 1744.09,1161.21 1742.41,1161.4 1742.27,1161.41 1740.44,1161.62 \n", " 1738.62,1161.83 1736.8,1162.04 1734.98,1162.25 1733.15,1162.46 1731.33,1162.67 1730.45,1162.77 1729.51,1162.88 1727.69,1163.09 1725.86,1163.3 1724.04,1163.51 \n", " 1722.22,1163.72 1720.4,1163.93 1718.57,1164.14 1718.55,1164.15 1716.75,1164.35 1714.93,1164.57 1713.11,1164.78 1711.28,1164.99 1709.46,1165.2 1707.64,1165.41 \n", " 1706.73,1165.52 1705.82,1165.63 1703.99,1165.84 1702.17,1166.05 1700.35,1166.26 1698.53,1166.48 1696.7,1166.69 1694.98,1166.89 1694.88,1166.9 1693.06,1167.12 \n", " 1691.23,1167.33 1689.41,1167.55 1687.59,1167.76 1685.77,1167.98 1683.94,1168.19 1683.29,1168.27 1682.12,1168.4 1680.3,1168.62 1678.48,1168.83 1676.65,1169.05 \n", " 1674.83,1169.27 1673.01,1169.48 1671.67,1169.64 1671.19,1169.7 1669.36,1169.91 1667.54,1170.13 1665.72,1170.35 1663.9,1170.56 1662.07,1170.78 1660.25,1171 \n", " 1660.1,1171.01 1658.43,1171.21 1656.61,1171.43 1654.78,1171.65 1652.96,1171.87 1651.14,1172.08 1649.32,1172.3 1648.59,1172.39 1647.49,1172.52 1645.67,1172.74 \n", " 1643.85,1172.96 1642.03,1173.17 1640.2,1173.39 1638.38,1173.61 1637.13,1173.76 1636.56,1173.83 1634.74,1174.05 1632.91,1174.27 1631.09,1174.49 1629.27,1174.71 \n", " 1627.45,1174.93 1625.72,1175.14 1625.62,1175.15 1623.8,1175.37 1621.98,1175.59 1620.16,1175.81 1618.33,1176.03 1616.51,1176.25 1614.69,1176.47 1614.35,1176.51 \n", " 1612.87,1176.69 1611.04,1176.91 1609.22,1177.13 1607.4,1177.35 1605.58,1177.57 1603.75,1177.79 1603.02,1177.88 1601.93,1178.02 1600.11,1178.24 1598.29,1178.46 \n", " 1596.46,1178.68 1594.64,1178.9 1592.82,1179.12 1591.71,1179.26 1590.99,1179.34 1589.17,1179.57 1587.35,1179.79 1585.53,1180.01 1583.7,1180.23 1581.88,1180.46 \n", " 1580.44,1180.63 1580.06,1180.68 1578.24,1180.9 1576.41,1181.12 1574.59,1181.34 1572.77,1181.57 1570.95,1181.79 1569.19,1182 1569.12,1182.01 1567.3,1182.24 \n", " 1565.48,1182.46 1563.66,1182.68 1561.83,1182.9 1560.01,1183.13 1558.19,1183.35 1557.95,1183.38 1556.37,1183.57 1554.54,1183.8 1552.72,1184.02 1550.9,1184.24 \n", " 1549.08,1184.47 1547.25,1184.69 1546.73,1184.75 1545.43,1184.91 1543.61,1185.13 1541.79,1185.36 1539.96,1185.58 1538.14,1185.8 1536.32,1186.03 1535.5,1186.13 \n", " 1534.5,1186.25 1532.67,1186.47 1530.85,1186.7 1529.03,1186.92 1527.21,1187.14 1525.38,1187.36 1524.28,1187.5 1523.56,1187.59 1521.74,1187.81 1519.92,1188.03 \n", " 1518.09,1188.26 1516.27,1188.48 1514.45,1188.7 1513.04,1188.87 1512.63,1188.92 1510.8,1189.15 1508.98,1189.37 1507.16,1189.59 1505.34,1189.81 1503.51,1190.04 \n", " 1501.78,1190.25 1501.69,1190.26 1499.87,1190.48 1498.04,1190.7 1496.22,1190.93 1494.4,1191.15 1492.58,1191.37 1490.75,1191.59 1490.5,1191.62 1488.93,1191.81 \n", " 1487.11,1192.03 1485.29,1192.26 1483.46,1192.48 1481.64,1192.7 1479.82,1192.92 1479.19,1193 1478,1193.14 1476.17,1193.36 1474.35,1193.58 1472.53,1193.8 \n", " 1470.71,1194.02 1468.88,1194.24 1467.82,1194.37 1467.06,1194.46 1465.24,1194.68 1463.42,1194.9 1461.59,1195.12 1459.77,1195.34 1457.95,1195.56 1456.41,1195.74 \n", " 1456.13,1195.78 1454.3,1196 1452.48,1196.21 1450.66,1196.43 1448.84,1196.65 1447.01,1196.87 1445.19,1197.09 1444.93,1197.12 1443.37,1197.3 1441.55,1197.52 \n", " 1439.72,1197.74 1437.9,1197.95 1436.08,1198.17 1434.26,1198.39 1433.37,1198.49 1432.43,1198.6 1430.61,1198.82 1428.79,1199.03 1426.97,1199.25 1425.14,1199.46 \n", " 1423.32,1199.68 1421.73,1199.86 1421.5,1199.89 1419.68,1200.11 1417.85,1200.32 1416.03,1200.53 1414.21,1200.75 1412.39,1200.96 1410.56,1201.17 1409.98,1201.24 \n", " 1408.74,1201.38 1406.92,1201.59 1405.1,1201.81 1403.27,1202.02 1401.45,1202.23 1399.63,1202.44 1398.12,1202.61 1397.8,1202.65 1395.98,1202.86 1394.16,1203.07 \n", " 1392.34,1203.28 1390.51,1203.49 1388.69,1203.69 1386.87,1203.9 1386.13,1203.99 1385.05,1204.11 1383.22,1204.32 1381.4,1204.52 1379.58,1204.73 1377.76,1204.94 \n", " 1375.93,1205.14 1374.11,1205.35 1373.99,1205.36 1372.29,1205.55 1370.47,1205.76 1368.64,1205.96 1366.82,1206.16 1365,1206.37 1363.18,1206.57 1361.69,1206.73 \n", " 1361.35,1206.77 1359.53,1206.97 1357.71,1207.17 1355.89,1207.37 1354.06,1207.57 1352.24,1207.77 1350.42,1207.97 1349.19,1208.11 1348.6,1208.17 1346.77,1208.37 \n", " 1344.95,1208.57 1343.13,1208.77 1341.31,1208.96 1339.48,1209.16 1337.66,1209.36 1336.49,1209.48 1335.84,1209.55 1334.02,1209.75 1332.19,1209.94 1330.37,1210.13 \n", " 1328.55,1210.33 1326.73,1210.52 1324.9,1210.71 1323.55,1210.85 1323.08,1210.9 1321.26,1211.1 1319.44,1211.29 1317.61,1211.48 1315.79,1211.67 1313.97,1211.85 \n", " 1312.15,1212.04 1310.34,1212.23 1310.32,1212.23 1308.5,1212.42 1306.68,1212.6 1304.86,1212.79 1303.03,1212.98 1301.21,1213.16 1299.39,1213.35 1297.56,1213.53 \n", " 1296.84,1213.6 1295.74,1213.71 1293.92,1213.9 1292.1,1214.08 1290.27,1214.26 1288.45,1214.44 1286.63,1214.62 1284.81,1214.8 1283,1214.98 1282.98,1214.98 \n", " 1281.16,1215.16 1279.34,1215.33 1277.52,1215.51 1275.69,1215.69 1273.87,1215.86 1272.05,1216.04 1270.23,1216.21 1268.79,1216.35 1268.4,1216.39 1266.58,1216.56 \n", " 1264.76,1216.73 1262.94,1216.9 1261.11,1217.08 1259.29,1217.25 1257.47,1217.42 1255.65,1217.59 1254.15,1217.72 1253.82,1217.75 1252,1217.92 1250.18,1218.09 \n", " 1248.36,1218.26 1246.53,1218.42 1244.71,1218.59 1242.89,1218.75 1241.07,1218.92 1239.24,1219.08 1239.03,1219.1 1237.42,1219.24 1235.6,1219.4 1233.78,1219.56 \n", " 1231.95,1219.72 1230.13,1219.88 1228.31,1220.04 1226.49,1220.2 1224.66,1220.36 1223.36,1220.47 1222.84,1220.52 1221.02,1220.67 1219.2,1220.83 1217.37,1220.98 \n", " 1215.55,1221.14 1213.73,1221.29 1211.91,1221.44 1210.08,1221.59 1208.26,1221.75 1207.06,1221.85 1206.44,1221.9 1204.62,1222.05 1202.79,1222.2 1200.97,1222.34 \n", " 1199.15,1222.49 1197.32,1222.64 1195.5,1222.78 1193.68,1222.93 1191.86,1223.08 1190.04,1223.22 1190.03,1223.22 1188.21,1223.36 1186.39,1223.51 1184.57,1223.65 \n", " 1182.74,1223.79 1180.92,1223.93 1179.1,1224.07 1177.28,1224.21 1175.45,1224.35 1173.63,1224.48 1172.18,1224.59 1171.81,1224.62 1169.99,1224.76 1168.16,1224.89 \n", " 1166.34,1225.03 1164.52,1225.16 1162.7,1225.29 1160.87,1225.43 1159.05,1225.56 1157.23,1225.69 1155.41,1225.82 1153.58,1225.95 1153.35,1225.97 1151.76,1226.08 \n", " 1149.94,1226.21 1148.12,1226.34 1146.29,1226.46 1144.47,1226.59 1142.65,1226.71 1140.83,1226.84 1139,1226.96 1137.18,1227.09 1135.36,1227.21 1133.54,1227.33 \n", " 1133.38,1227.34 1131.71,1227.45 1129.89,1227.57 1128.07,1227.69 1126.25,1227.81 1124.42,1227.93 1122.6,1228.05 1120.78,1228.16 1118.96,1228.28 1117.13,1228.4 \n", " 1115.31,1228.51 1113.49,1228.63 1112.06,1228.71 1111.67,1228.74 1109.84,1228.85 1108.02,1228.96 1106.2,1229.08 1104.37,1229.19 1102.55,1229.3 1100.73,1229.41 \n", " 1098.91,1229.52 1097.08,1229.62 1095.26,1229.73 1093.44,1229.84 1091.62,1229.94 1089.79,1230.05 1089.13,1230.09 1087.97,1230.15 1086.15,1230.26 1084.33,1230.36 \n", " 1082.5,1230.47 1080.68,1230.57 1078.86,1230.67 1077.04,1230.77 1075.21,1230.87 1073.39,1230.97 1071.57,1231.07 1069.75,1231.17 1067.92,1231.27 1066.1,1231.36 \n", " 1064.28,1231.46 1064.27,1231.46 1062.46,1231.56 1060.63,1231.65 1058.81,1231.75 1056.99,1231.84 1055.17,1231.94 1053.34,1232.03 1051.52,1232.12 1049.7,1232.21 \n", " 1047.88,1232.31 1046.05,1232.4 1044.23,1232.49 1042.41,1232.58 1040.59,1232.67 1038.76,1232.75 1037.07,1232.84 1036.94,1232.84 1035.12,1232.93 1033.3,1233.02 \n", " 1031.47,1233.1 1029.65,1233.19 1027.83,1233.27 1026.01,1233.36 1024.18,1233.44 1022.36,1233.53 1020.54,1233.61 1018.72,1233.69 1016.89,1233.77 1015.07,1233.86 \n", " 1013.25,1233.94 1011.43,1234.02 1009.6,1234.1 1007.78,1234.18 1007.04,1234.21 1005.96,1234.26 1004.13,1234.34 1002.31,1234.41 1000.49,1234.49 998.667,1234.57 \n", " 996.845,1234.65 995.022,1234.72 993.2,1234.8 991.377,1234.87 989.555,1234.95 987.732,1235.02 985.909,1235.1 984.087,1235.17 982.264,1235.24 980.442,1235.32 \n", " 978.619,1235.39 976.797,1235.46 974.974,1235.53 973.671,1235.58 973.152,1235.6 971.329,1235.67 969.507,1235.74 967.684,1235.81 965.861,1235.88 964.039,1235.95 \n", " 962.216,1236.02 960.394,1236.09 958.571,1236.16 956.749,1236.23 954.926,1236.29 953.104,1236.36 951.281,1236.43 949.459,1236.49 947.636,1236.56 945.813,1236.63 \n", " 943.991,1236.69 942.168,1236.76 940.346,1236.82 938.523,1236.89 936.701,1236.95 936.488,1236.96 934.878,1237.01 933.056,1237.08 931.233,1237.14 929.41,1237.2 \n", " 927.588,1237.27 925.765,1237.33 923.943,1237.39 922.12,1237.45 920.298,1237.51 918.475,1237.57 916.653,1237.64 914.83,1237.7 913.008,1237.76 911.185,1237.82 \n", " 909.362,1237.88 907.54,1237.94 905.717,1238 903.895,1238.06 902.072,1238.11 900.25,1238.17 898.427,1238.23 896.605,1238.29 895.357,1238.33 894.782,1238.35 \n", " 892.96,1238.41 891.137,1238.47 889.314,1238.52 887.492,1238.58 885.669,1238.64 883.847,1238.69 882.024,1238.75 880.202,1238.81 878.379,1238.87 876.557,1238.92 \n", " 874.734,1238.98 872.912,1239.03 871.089,1239.09 869.266,1239.15 867.444,1239.2 865.621,1239.26 863.799,1239.31 861.976,1239.37 860.154,1239.42 858.331,1239.48 \n", " 856.509,1239.53 854.686,1239.59 852.864,1239.64 851.041,1239.7 850.857,1239.7 849.218,1239.75 847.396,1239.81 845.573,1239.86 843.751,1239.92 841.928,1239.97 \n", " 840.106,1240.03 838.283,1240.08 836.461,1240.13 834.638,1240.19 832.816,1240.24 830.993,1240.3 829.17,1240.35 827.348,1240.4 825.525,1240.46 823.703,1240.51 \n", " 821.88,1240.57 820.058,1240.62 818.235,1240.67 816.413,1240.73 814.59,1240.78 812.768,1240.83 810.945,1240.89 809.122,1240.94 807.3,1240.99 805.477,1241.05 \n", " 804.46,1241.08 803.655,1241.1 801.832,1241.16 800.01,1241.21 798.187,1241.26 796.365,1241.32 794.542,1241.37 792.72,1241.42 790.897,1241.48 789.074,1241.53 \n", " 787.252,1241.59 785.429,1241.64 783.607,1241.69 781.784,1241.75 779.962,1241.8 778.139,1241.85 776.317,1241.91 774.494,1241.96 772.671,1242.02 770.849,1242.07 \n", " 769.026,1242.13 767.204,1242.18 765.381,1242.23 763.559,1242.29 761.736,1242.34 759.914,1242.4 758.091,1242.45 758.088,1242.45 756.269,1242.51 754.446,1242.56 \n", " 752.623,1242.62 750.801,1242.67 748.978,1242.73 747.156,1242.78 745.333,1242.84 743.511,1242.89 741.688,1242.95 739.866,1243 738.043,1243.06 736.221,1243.11 \n", " 734.398,1243.17 732.575,1243.23 730.753,1243.28 728.93,1243.34 727.108,1243.4 725.285,1243.45 723.463,1243.51 721.64,1243.57 719.818,1243.62 717.995,1243.68 \n", " 716.173,1243.74 714.35,1243.79 713.3,1243.83 712.527,1243.85 710.705,1243.91 708.882,1243.97 707.06,1244.02 705.237,1244.08 703.415,1244.14 701.592,1244.2 \n", " 699.77,1244.26 697.947,1244.31 696.125,1244.37 694.302,1244.43 692.479,1244.49 690.657,1244.55 688.834,1244.61 687.012,1244.67 685.189,1244.73 683.367,1244.79 \n", " 681.544,1244.85 679.722,1244.91 677.899,1244.97 676.077,1245.03 674.254,1245.09 672.431,1245.15 670.89,1245.2 670.609,1245.21 668.786,1245.27 666.964,1245.33 \n", " 665.141,1245.39 663.319,1245.45 661.496,1245.52 659.674,1245.58 657.851,1245.64 656.029,1245.7 654.206,1245.76 652.383,1245.83 650.561,1245.89 648.738,1245.95 \n", " 646.916,1246.02 645.093,1246.08 643.271,1246.14 641.448,1246.21 639.626,1246.27 637.803,1246.33 635.981,1246.4 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1833.39,1120.93 1831.57,1121.13 1829.75,1121.33 1827.93,1121.53 1827.66,1121.56 1826.1,1121.73 1824.28,1121.93 1822.46,1122.13 1820.64,1122.34 \n", " 1818.81,1122.54 1816.99,1122.74 1815.28,1122.93 1815.17,1122.94 1813.35,1123.15 1811.52,1123.35 1809.7,1123.55 1807.88,1123.76 1806.06,1123.96 1804.23,1124.17 \n", " 1802.99,1124.31 1802.41,1124.37 1800.59,1124.58 1798.77,1124.78 1796.94,1124.99 1795.12,1125.19 1793.3,1125.4 1791.47,1125.6 1790.81,1125.68 1789.65,1125.81 \n", " 1787.83,1126.02 1786.01,1126.22 1784.18,1126.43 1782.36,1126.64 1780.54,1126.85 1778.73,1127.05 1778.72,1127.05 1776.89,1127.26 1775.07,1127.47 1773.25,1127.68 \n", " 1771.43,1127.89 1769.6,1128.1 1767.78,1128.31 1766.74,1128.43 1765.96,1128.52 1764.14,1128.73 1762.31,1128.94 1760.49,1129.15 1758.67,1129.36 1756.85,1129.57 \n", " 1755.02,1129.78 1754.85,1129.8 1753.2,1129.99 1751.38,1130.2 1749.56,1130.42 1747.73,1130.63 1745.91,1130.84 1744.09,1131.05 1743.05,1131.17 1742.27,1131.27 \n", " 1740.44,1131.48 1738.62,1131.69 1736.8,1131.91 1734.98,1132.12 1733.15,1132.33 1731.34,1132.55 1731.33,1132.55 1729.51,1132.76 1727.69,1132.98 1725.86,1133.19 \n", " 1724.04,1133.41 1722.22,1133.63 1720.4,1133.84 1719.72,1133.92 1718.57,1134.06 1716.75,1134.27 1714.93,1134.49 1713.11,1134.71 1711.28,1134.93 1709.46,1135.14 \n", " 1708.19,1135.3 1707.64,1135.36 1705.82,1135.58 1703.99,1135.8 1702.17,1136.02 1700.35,1136.24 1698.53,1136.45 1696.74,1136.67 1696.7,1136.67 1694.88,1136.89 \n", " 1693.06,1137.11 1691.23,1137.33 1689.41,1137.55 1687.59,1137.77 1685.77,1138 1685.37,1138.04 1683.94,1138.22 1682.12,1138.44 1680.3,1138.66 1678.48,1138.88 \n", " 1676.65,1139.1 1674.83,1139.32 1674.08,1139.42 1673.01,1139.55 1671.19,1139.77 1669.36,1139.99 1667.54,1140.22 1665.72,1140.44 1663.9,1140.66 1662.86,1140.79 \n", " 1662.07,1140.89 1660.25,1141.11 1658.43,1141.34 1656.61,1141.56 1654.78,1141.78 1652.96,1142.01 1651.71,1142.16 1651.14,1142.23 1649.32,1142.46 1647.49,1142.69 \n", " 1645.67,1142.91 1643.85,1143.14 1642.03,1143.36 1640.62,1143.54 1640.2,1143.59 1638.38,1143.82 1636.56,1144.04 1634.74,1144.27 1632.91,1144.5 1631.09,1144.73 \n", " 1629.6,1144.91 1629.27,1144.95 1627.45,1145.18 1625.62,1145.41 1623.8,1145.64 1621.98,1145.87 1620.16,1146.1 1618.64,1146.29 1618.33,1146.32 1616.51,1146.55 \n", " 1614.69,1146.78 1612.87,1147.01 1611.04,1147.24 1609.22,1147.47 1607.73,1147.66 1607.4,1147.7 1605.58,1147.93 1603.75,1148.16 1601.93,1148.39 1600.11,1148.62 \n", " 1598.29,1148.85 1596.86,1149.03 1596.46,1149.08 1594.64,1149.32 1592.82,1149.55 1590.99,1149.78 1589.17,1150.01 1587.35,1150.24 1586.04,1150.41 1585.53,1150.47 \n", " 1583.7,1150.71 1581.88,1150.94 1580.06,1151.17 1578.24,1151.4 1576.41,1151.63 1575.26,1151.78 1574.59,1151.87 1572.77,1152.1 1570.95,1152.33 1569.12,1152.57 \n", " 1567.3,1152.8 1565.48,1153.03 1564.52,1153.16 1563.66,1153.27 1561.83,1153.5 1560.01,1153.73 1558.19,1153.97 1556.37,1154.2 1554.54,1154.43 1553.8,1154.53 \n", " 1552.72,1154.67 1550.9,1154.9 1549.08,1155.13 1547.25,1155.37 1545.43,1155.6 1543.61,1155.84 1543.1,1155.9 1541.79,1156.07 1539.96,1156.31 1538.14,1156.54 \n", " 1536.32,1156.77 1534.5,1157.01 1532.67,1157.24 1532.42,1157.28 1530.85,1157.48 1529.03,1157.71 1527.21,1157.95 1525.38,1158.18 1523.56,1158.42 1521.75,1158.65 \n", " 1521.74,1158.65 1519.92,1158.89 1518.09,1159.12 1516.27,1159.36 1514.45,1159.59 1512.63,1159.82 1511.08,1160.02 1510.8,1160.06 1508.98,1160.29 1507.16,1160.53 \n", " 1505.34,1160.76 1503.51,1161 1501.69,1161.23 1500.4,1161.4 1499.87,1161.47 1498.04,1161.7 1496.22,1161.94 1494.4,1162.17 1492.58,1162.4 1490.75,1162.64 \n", " 1489.71,1162.77 1488.93,1162.87 1487.11,1163.11 1485.29,1163.34 1483.46,1163.57 1481.64,1163.81 1479.82,1164.04 1479,1164.15 1478,1164.27 1476.17,1164.51 \n", " 1474.35,1164.74 1472.53,1164.97 1470.71,1165.21 1468.88,1165.44 1468.26,1165.52 1467.06,1165.67 1465.24,1165.91 1463.42,1166.14 1461.59,1166.37 1459.77,1166.6 \n", " 1457.95,1166.83 1457.49,1166.89 1456.13,1167.07 1454.3,1167.3 1452.48,1167.53 1450.66,1167.76 1448.84,1167.99 1447.01,1168.22 1446.66,1168.27 1445.19,1168.45 \n", " 1443.37,1168.68 1441.55,1168.91 1439.72,1169.14 1437.9,1169.37 1436.08,1169.6 1435.78,1169.64 1434.26,1169.83 1432.43,1170.06 1430.61,1170.29 1428.79,1170.52 \n", " 1426.97,1170.75 1425.14,1170.97 1424.82,1171.01 1423.32,1171.2 1421.5,1171.43 1419.68,1171.66 1417.85,1171.88 1416.03,1172.11 1414.21,1172.34 1413.79,1172.39 \n", " 1412.39,1172.56 1410.56,1172.79 1408.74,1173.01 1406.92,1173.24 1405.1,1173.46 1403.27,1173.69 1402.66,1173.76 1401.45,1173.91 1399.63,1174.13 1397.8,1174.36 \n", " 1395.98,1174.58 1394.16,1174.8 1392.34,1175.02 1391.41,1175.14 1390.51,1175.25 1388.69,1175.47 1386.87,1175.69 1385.05,1175.91 1383.22,1176.13 1381.4,1176.35 \n", " 1380.05,1176.51 1379.58,1176.57 1377.76,1176.78 1375.93,1177 1374.11,1177.22 1372.29,1177.44 1370.47,1177.65 1368.64,1177.87 1368.54,1177.88 1366.82,1178.09 \n", " 1365,1178.3 1363.18,1178.52 1361.35,1178.73 1359.53,1178.95 1357.71,1179.16 1356.87,1179.26 1355.89,1179.37 1354.06,1179.58 1352.24,1179.8 1350.42,1180.01 \n", " 1348.6,1180.22 1346.77,1180.43 1345.02,1180.63 1344.95,1180.64 1343.13,1180.85 1341.31,1181.06 1339.48,1181.26 1337.66,1181.47 1335.84,1181.68 1334.02,1181.89 \n", " 1332.96,1182 1332.19,1182.09 1330.37,1182.3 1328.55,1182.5 1326.73,1182.7 1324.9,1182.91 1323.08,1183.11 1321.26,1183.31 1320.67,1183.38 1319.44,1183.51 \n", " 1317.61,1183.72 1315.79,1183.92 1313.97,1184.12 1312.15,1184.31 1310.32,1184.51 1308.5,1184.71 1308.11,1184.75 1306.68,1184.91 1304.86,1185.1 1303.03,1185.3 \n", " 1301.21,1185.49 1299.39,1185.69 1297.56,1185.88 1295.74,1186.08 1295.26,1186.13 1293.92,1186.27 1292.1,1186.46 1290.27,1186.65 1288.45,1186.84 1286.63,1187.03 \n", " 1284.81,1187.22 1282.98,1187.41 1282.07,1187.5 1281.16,1187.59 1279.34,1187.78 1277.52,1187.97 1275.69,1188.15 1273.87,1188.34 1272.05,1188.52 1270.23,1188.7 \n", " 1268.51,1188.87 1268.4,1188.88 1266.58,1189.07 1264.76,1189.25 1262.94,1189.43 1261.11,1189.61 1259.29,1189.78 1257.47,1189.96 1255.65,1190.14 1254.51,1190.25 \n", " 1253.82,1190.31 1252,1190.49 1250.18,1190.66 1248.36,1190.84 1246.53,1191.01 1244.71,1191.18 1242.89,1191.35 1241.07,1191.52 1240.01,1191.62 1239.24,1191.69 \n", " 1237.42,1191.86 1235.6,1192.03 1233.78,1192.2 1231.95,1192.36 1230.13,1192.53 1228.31,1192.69 1226.49,1192.86 1224.93,1193 1224.66,1193.02 1222.84,1193.18 \n", " 1221.02,1193.34 1219.2,1193.5 1217.37,1193.66 1215.55,1193.82 1213.73,1193.98 1211.91,1194.14 1210.08,1194.29 1209.2,1194.37 1208.26,1194.45 1206.44,1194.6 \n", " 1204.62,1194.76 1202.79,1194.91 1200.97,1195.06 1199.15,1195.21 1197.32,1195.36 1195.5,1195.51 1193.68,1195.66 1192.69,1195.74 1191.86,1195.81 1190.03,1195.96 \n", " 1188.21,1196.1 1186.39,1196.25 1184.57,1196.39 1182.74,1196.54 1180.92,1196.68 1179.1,1196.82 1177.28,1196.96 1175.45,1197.1 1175.27,1197.12 1173.63,1197.24 \n", " 1171.81,1197.38 1169.99,1197.52 1168.16,1197.65 1166.34,1197.79 1164.52,1197.93 1162.7,1198.06 1160.87,1198.19 1159.05,1198.33 1157.23,1198.46 1156.76,1198.49 \n", " 1155.41,1198.59 1153.58,1198.72 1151.76,1198.85 1149.94,1198.97 1148.12,1199.1 1146.29,1199.23 1144.47,1199.35 1142.65,1199.48 1140.83,1199.6 1139,1199.73 \n", " 1137.18,1199.85 1136.93,1199.86 1135.36,1199.97 1133.54,1200.09 1131.71,1200.21 1129.89,1200.33 1128.07,1200.45 1126.25,1200.56 1124.42,1200.68 1122.6,1200.8 \n", " 1120.78,1200.91 1118.96,1201.02 1117.13,1201.14 1115.5,1201.24 1115.31,1201.25 1113.49,1201.36 1111.67,1201.47 1109.84,1201.58 1108.02,1201.69 1106.2,1201.8 \n", " 1104.37,1201.91 1102.55,1202.01 1100.73,1202.12 1098.91,1202.22 1097.08,1202.33 1095.26,1202.43 1093.44,1202.53 1092.04,1202.61 1091.62,1202.64 1089.79,1202.74 \n", " 1087.97,1202.84 1086.15,1202.94 1084.33,1203.04 1082.5,1203.13 1080.68,1203.23 1078.86,1203.33 1077.04,1203.42 1075.21,1203.52 1073.39,1203.61 1071.57,1203.71 \n", " 1069.75,1203.8 1067.92,1203.89 1066.1,1203.98 1066.03,1203.99 1064.28,1204.07 1062.46,1204.16 1060.63,1204.25 1058.81,1204.34 1056.99,1204.43 1055.17,1204.51 \n", " 1053.34,1204.6 1051.52,1204.69 1049.7,1204.77 1047.88,1204.86 1046.05,1204.94 1044.23,1205.02 1042.41,1205.1 1040.59,1205.19 1038.76,1205.27 1036.94,1205.35 \n", " 1036.66,1205.36 1035.12,1205.43 1033.3,1205.51 1031.47,1205.58 1029.65,1205.66 1027.83,1205.74 1026.01,1205.82 1024.18,1205.89 1022.36,1205.97 1020.54,1206.04 \n", " 1018.72,1206.12 1016.89,1206.19 1015.07,1206.26 1013.25,1206.33 1011.43,1206.41 1009.6,1206.48 1007.78,1206.55 1005.96,1206.62 1004.13,1206.69 1002.87,1206.73 \n", " 1002.31,1206.75 1000.49,1206.82 998.667,1206.89 996.845,1206.96 995.022,1207.02 993.2,1207.09 991.377,1207.16 989.555,1207.22 987.732,1207.29 985.909,1207.35 \n", " 984.087,1207.41 982.264,1207.48 980.442,1207.54 978.619,1207.6 976.797,1207.66 974.974,1207.72 973.152,1207.78 971.329,1207.84 969.507,1207.9 967.684,1207.96 \n", " 965.861,1208.02 964.039,1208.08 963.205,1208.11 962.216,1208.14 960.394,1208.2 958.571,1208.25 956.749,1208.31 954.926,1208.37 953.104,1208.42 951.281,1208.48 \n", " 949.459,1208.53 947.636,1208.59 945.813,1208.64 943.991,1208.7 942.168,1208.75 940.346,1208.8 938.523,1208.86 936.701,1208.91 934.878,1208.96 933.056,1209.01 \n", " 931.233,1209.07 929.41,1209.12 927.588,1209.17 925.765,1209.22 923.943,1209.27 922.12,1209.32 920.298,1209.37 918.475,1209.42 916.653,1209.47 916.151,1209.48 \n", " 914.83,1209.52 913.008,1209.57 911.185,1209.61 909.362,1209.66 907.54,1209.71 905.717,1209.76 903.895,1209.81 902.072,1209.85 900.25,1209.9 898.427,1209.95 \n", " 896.605,1209.99 894.782,1210.04 892.96,1210.09 891.137,1210.13 889.314,1210.18 887.492,1210.22 885.669,1210.27 883.847,1210.31 882.024,1210.36 880.202,1210.4 \n", " 878.379,1210.45 876.557,1210.49 874.734,1210.54 872.912,1210.58 871.089,1210.63 869.266,1210.67 867.444,1210.71 865.621,1210.76 863.799,1210.8 861.976,1210.84 \n", " 861.547,1210.85 860.154,1210.89 858.331,1210.93 856.509,1210.97 854.686,1211.02 852.864,1211.06 851.041,1211.1 849.218,1211.15 847.396,1211.19 845.573,1211.23 \n", " 843.751,1211.28 841.928,1211.32 840.106,1211.36 838.283,1211.4 836.461,1211.45 834.638,1211.49 832.816,1211.53 830.993,1211.57 829.17,1211.62 827.348,1211.66 \n", " 825.525,1211.7 823.703,1211.74 821.88,1211.78 820.058,1211.83 818.235,1211.87 816.413,1211.91 814.59,1211.95 812.768,1212 810.945,1212.04 809.122,1212.08 \n", " 807.3,1212.12 805.477,1212.17 803.655,1212.21 802.822,1212.23 801.832,1212.25 800.01,1212.29 798.187,1212.34 796.365,1212.38 794.542,1212.42 792.72,1212.47 \n", " 790.897,1212.51 789.074,1212.55 787.252,1212.59 785.429,1212.64 783.607,1212.68 781.784,1212.72 779.962,1212.77 778.139,1212.81 776.317,1212.85 774.494,1212.9 \n", " 772.671,1212.94 770.849,1212.99 769.026,1213.03 767.204,1213.07 765.381,1213.12 763.559,1213.16 761.736,1213.21 759.914,1213.25 758.091,1213.3 756.269,1213.34 \n", " 754.446,1213.39 752.623,1213.43 750.801,1213.48 748.978,1213.52 747.156,1213.57 745.7,1213.6 745.333,1213.61 743.511,1213.66 741.688,1213.7 739.866,1213.75 \n", " 738.043,1213.8 736.221,1213.84 734.398,1213.89 732.575,1213.93 730.753,1213.98 728.93,1214.03 727.108,1214.08 725.285,1214.12 723.463,1214.17 721.64,1214.22 \n", " 719.818,1214.27 717.995,1214.31 716.173,1214.36 714.35,1214.41 712.527,1214.46 710.705,1214.51 708.882,1214.56 707.06,1214.61 705.237,1214.66 703.415,1214.7 \n", " 701.592,1214.75 699.77,1214.8 697.947,1214.85 696.125,1214.91 694.302,1214.96 693.559,1214.98 692.479,1215.01 690.657,1215.06 688.834,1215.11 687.012,1215.16 \n", " 685.189,1215.21 683.367,1215.26 681.544,1215.31 679.722,1215.37 677.899,1215.42 676.077,1215.47 674.254,1215.53 672.431,1215.58 670.609,1215.63 668.786,1215.68 \n", " 666.964,1215.74 665.141,1215.79 663.319,1215.85 661.496,1215.9 659.674,1215.96 657.851,1216.01 656.029,1216.07 654.206,1216.12 652.383,1216.18 650.561,1216.23 \n", " 648.738,1216.29 646.916,1216.34 646.746,1216.35 645.093,1216.4 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1581.88,1061.37 1580.06,1061.64 1578.24,1061.91 1576.41,1062.18 1574.59,1062.44 1574.32,1062.48 1572.77,1062.71 1570.95,1062.98 \n", " 1569.12,1063.25 1567.3,1063.52 1565.48,1063.79 1565.04,1063.86 1563.66,1064.06 1561.83,1064.33 1560.01,1064.61 1558.19,1064.88 1556.37,1065.15 1555.82,1065.23 \n", " 1554.54,1065.42 1552.72,1065.7 1550.9,1065.97 1549.08,1066.24 1547.25,1066.52 1546.68,1066.61 1545.43,1066.79 1543.61,1067.07 1541.79,1067.34 1539.96,1067.62 \n", " 1538.14,1067.9 1537.6,1067.98 1536.32,1068.17 1534.5,1068.45 1532.67,1068.73 1530.85,1069.01 1529.03,1069.29 1528.58,1069.35 1527.21,1069.56 1525.38,1069.84 \n", " 1523.56,1070.12 1521.74,1070.4 1519.92,1070.68 1519.62,1070.73 1518.09,1070.96 1516.27,1071.24 1514.45,1071.52 1512.63,1071.8 1510.8,1072.09 1510.71,1072.1 \n", " 1508.98,1072.37 1507.16,1072.65 1505.34,1072.93 1503.51,1073.21 1501.84,1073.47 1501.69,1073.5 1499.87,1073.78 1498.04,1074.06 1496.22,1074.35 1494.4,1074.63 \n", " 1493,1074.85 1492.58,1074.91 1490.75,1075.2 1488.93,1075.48 1487.11,1075.77 1485.29,1076.05 1484.2,1076.22 1483.46,1076.34 1481.64,1076.62 1479.82,1076.91 \n", " 1478,1077.19 1476.17,1077.48 1475.42,1077.6 1474.35,1077.76 1472.53,1078.05 1470.71,1078.34 1468.88,1078.62 1467.06,1078.91 1466.67,1078.97 1465.24,1079.19 \n", " 1463.42,1079.48 1461.59,1079.77 1459.77,1080.05 1457.95,1080.34 1457.92,1080.34 1456.13,1080.63 1454.3,1080.91 1452.48,1081.2 1450.66,1081.48 1449.18,1081.72 \n", " 1448.84,1081.77 1447.01,1082.06 1445.19,1082.34 1443.37,1082.63 1441.55,1082.92 1440.43,1083.09 1439.72,1083.2 1437.9,1083.49 1436.08,1083.77 1434.26,1084.06 \n", " 1432.43,1084.35 1431.67,1084.46 1430.61,1084.63 1428.79,1084.92 1426.97,1085.2 1425.14,1085.49 1423.32,1085.77 1422.9,1085.84 1421.5,1086.06 1419.68,1086.34 \n", " 1417.85,1086.63 1416.03,1086.91 1414.21,1087.19 1414.09,1087.21 1412.39,1087.48 1410.56,1087.76 1408.74,1088.04 1406.92,1088.33 1405.25,1088.59 1405.1,1088.61 \n", " 1403.27,1088.89 1401.45,1089.17 1399.63,1089.46 1397.8,1089.74 1396.36,1089.96 1395.98,1090.02 1394.16,1090.3 1392.34,1090.58 1390.51,1090.86 1388.69,1091.14 \n", " 1387.41,1091.33 1386.87,1091.42 1385.05,1091.69 1383.22,1091.97 1381.4,1092.25 1379.58,1092.53 1378.39,1092.71 1377.76,1092.8 1375.93,1093.08 1374.11,1093.36 \n", " 1372.29,1093.63 1370.47,1093.9 1369.29,1094.08 1368.64,1094.18 1366.82,1094.45 1365,1094.72 1363.18,1095 1361.35,1095.27 1360.09,1095.46 1359.53,1095.54 \n", " 1357.71,1095.81 1355.89,1096.08 1354.06,1096.35 1352.24,1096.62 1350.78,1096.83 1350.42,1096.88 1348.6,1097.15 1346.77,1097.42 1344.95,1097.68 1343.13,1097.94 \n", " 1341.34,1098.2 1341.31,1098.21 1339.48,1098.47 1337.66,1098.73 1335.84,1098.99 1334.02,1099.26 1332.19,1099.51 1331.76,1099.58 1330.37,1099.77 1328.55,1100.03 \n", " 1326.73,1100.29 1324.9,1100.54 1323.08,1100.8 1322,1100.95 1321.26,1101.05 1319.44,1101.31 1317.61,1101.56 1315.79,1101.81 1313.97,1102.06 1312.15,1102.31 \n", " 1312.06,1102.32 1310.32,1102.56 1308.5,1102.81 1306.68,1103.06 1304.86,1103.3 1303.03,1103.55 1301.89,1103.7 1301.21,1103.79 1299.39,1104.03 1297.56,1104.27 \n", " 1295.74,1104.51 1293.92,1104.75 1292.1,1104.99 1291.47,1105.07 1290.27,1105.23 1288.45,1105.46 1286.63,1105.7 1284.81,1105.93 1282.98,1106.16 1281.16,1106.4 \n", " 1280.77,1106.45 1279.34,1106.63 1277.52,1106.86 1275.69,1107.08 1273.87,1107.31 1272.05,1107.53 1270.23,1107.76 1269.73,1107.82 1268.4,1107.98 1266.58,1108.2 \n", " 1264.76,1108.42 1262.94,1108.64 1261.11,1108.86 1259.29,1109.08 1258.31,1109.19 1257.47,1109.29 1255.65,1109.51 1253.82,1109.72 1252,1109.93 1250.18,1110.14 \n", " 1248.36,1110.35 1246.53,1110.56 1246.44,1110.57 1244.71,1110.76 1242.89,1110.97 1241.07,1111.17 1239.24,1111.37 1237.42,1111.57 1235.6,1111.77 1234.04,1111.94 \n", " 1233.78,1111.97 1231.95,1112.17 1230.13,1112.36 1228.31,1112.55 1226.49,1112.75 1224.66,1112.94 1222.84,1113.13 1221.02,1113.31 1221.01,1113.31 1219.2,1113.5 \n", " 1217.37,1113.69 1215.55,1113.87 1213.73,1114.05 1211.91,1114.23 1210.08,1114.41 1208.26,1114.59 1207.22,1114.69 1206.44,1114.76 1204.62,1114.94 1202.79,1115.11 \n", " 1200.97,1115.28 1199.15,1115.45 1197.32,1115.62 1195.5,1115.79 1193.68,1115.96 1192.49,1116.06 1191.86,1116.12 1190.03,1116.28 1188.21,1116.44 1186.39,1116.6 \n", " 1184.57,1116.76 1182.74,1116.92 1180.92,1117.07 1179.1,1117.23 1177.28,1117.38 1176.59,1117.44 1175.45,1117.53 1173.63,1117.68 1171.81,1117.83 1169.99,1117.97 \n", " 1168.16,1118.12 1166.34,1118.26 1164.52,1118.4 1162.7,1118.54 1160.87,1118.68 1159.15,1118.81 1159.05,1118.82 1157.23,1118.95 1155.41,1119.09 1153.58,1119.22 \n", " 1151.76,1119.35 1149.94,1119.48 1148.12,1119.61 1146.29,1119.73 1144.47,1119.86 1142.65,1119.98 1140.83,1120.1 1139.62,1120.18 1139,1120.22 1137.18,1120.34 \n", " 1135.36,1120.46 1133.54,1120.58 1131.71,1120.69 1129.89,1120.8 1128.07,1120.92 1126.25,1121.02 1124.42,1121.13 1122.6,1121.24 1120.78,1121.35 1118.96,1121.45 \n", " 1117.13,1121.55 1117.03,1121.56 1115.31,1121.65 1113.49,1121.75 1111.67,1121.85 1109.84,1121.95 1108.02,1122.04 1106.2,1122.14 1104.37,1122.23 1102.55,1122.32 \n", " 1100.73,1122.41 1098.91,1122.5 1097.08,1122.59 1095.26,1122.67 1093.44,1122.75 1091.62,1122.84 1089.79,1122.92 1089.49,1122.93 1087.97,1123 1086.15,1123.08 \n", " 1084.33,1123.15 1082.5,1123.23 1080.68,1123.3 1078.86,1123.38 1077.04,1123.45 1075.21,1123.52 1073.39,1123.59 1071.57,1123.66 1069.75,1123.72 1067.92,1123.79 \n", " 1066.1,1123.85 1064.28,1123.92 1062.46,1123.98 1060.63,1124.04 1058.81,1124.1 1056.99,1124.15 1055.17,1124.21 1053.34,1124.27 1052,1124.31 1051.52,1124.32 \n", " 1049.7,1124.37 1047.88,1124.42 1046.05,1124.47 1044.23,1124.52 1042.41,1124.57 1040.59,1124.62 1038.76,1124.67 1036.94,1124.71 1035.12,1124.75 1033.3,1124.8 \n", " 1031.47,1124.84 1029.65,1124.88 1027.83,1124.92 1026.01,1124.96 1024.18,1124.99 1022.36,1125.03 1020.54,1125.07 1018.72,1125.1 1016.89,1125.13 1015.07,1125.17 \n", " 1013.25,1125.2 1011.43,1125.23 1009.6,1125.26 1007.78,1125.28 1005.96,1125.31 1004.13,1125.34 1002.31,1125.36 1000.49,1125.39 998.667,1125.41 996.845,1125.44 \n", " 995.022,1125.46 993.2,1125.48 991.377,1125.5 989.555,1125.52 987.732,1125.54 985.909,1125.55 984.087,1125.57 982.264,1125.59 980.442,1125.6 978.619,1125.62 \n", " 976.797,1125.63 974.974,1125.65 973.152,1125.66 971.329,1125.67 970.045,1125.68 969.507,1125.68 967.684,1125.69 965.861,1125.7 964.039,1125.71 962.216,1125.72 \n", " 960.394,1125.73 958.571,1125.74 956.749,1125.74 954.926,1125.75 953.104,1125.76 951.281,1125.76 949.459,1125.76 947.636,1125.77 945.813,1125.77 943.991,1125.78 \n", " 942.168,1125.78 940.346,1125.78 938.523,1125.78 936.701,1125.78 934.878,1125.78 933.056,1125.78 931.233,1125.78 929.41,1125.78 927.588,1125.78 925.765,1125.78 \n", " 923.943,1125.78 922.12,1125.78 920.298,1125.77 918.475,1125.77 916.653,1125.77 914.83,1125.76 913.008,1125.76 911.185,1125.76 909.362,1125.75 907.54,1125.75 \n", " 905.717,1125.74 903.895,1125.74 902.072,1125.73 900.25,1125.73 898.427,1125.72 896.605,1125.71 894.782,1125.71 892.96,1125.7 891.137,1125.7 889.314,1125.69 \n", " 887.492,1125.68 886.544,1125.68 885.669,1125.68 883.847,1125.67 882.024,1125.66 880.202,1125.65 878.379,1125.65 876.557,1125.64 874.734,1125.63 872.912,1125.62 \n", " 871.089,1125.62 869.266,1125.61 867.444,1125.6 865.621,1125.59 863.799,1125.59 861.976,1125.58 860.154,1125.57 858.331,1125.56 856.509,1125.56 854.686,1125.55 \n", " 852.864,1125.54 851.041,1125.53 849.218,1125.53 847.396,1125.52 845.573,1125.51 843.751,1125.51 841.928,1125.5 840.106,1125.49 838.283,1125.49 836.461,1125.48 \n", " 834.638,1125.47 832.816,1125.47 830.993,1125.46 829.17,1125.45 827.348,1125.45 825.525,1125.44 823.703,1125.44 821.88,1125.43 820.058,1125.43 818.235,1125.42 \n", " 816.413,1125.42 814.59,1125.41 812.768,1125.41 810.945,1125.41 809.122,1125.4 807.3,1125.4 805.477,1125.4 803.655,1125.39 801.832,1125.39 800.01,1125.39 \n", " 798.187,1125.39 796.365,1125.39 794.542,1125.38 792.72,1125.38 790.897,1125.38 789.074,1125.38 787.252,1125.38 785.429,1125.38 783.607,1125.38 781.784,1125.38 \n", " 779.962,1125.38 778.139,1125.39 776.317,1125.39 774.494,1125.39 772.671,1125.39 770.849,1125.4 769.026,1125.4 767.204,1125.4 765.381,1125.41 763.559,1125.41 \n", " 761.736,1125.42 759.914,1125.42 758.091,1125.43 756.269,1125.43 754.446,1125.44 752.623,1125.45 750.801,1125.45 748.978,1125.46 747.156,1125.47 745.333,1125.48 \n", " 743.511,1125.49 741.688,1125.5 739.866,1125.51 738.043,1125.52 736.221,1125.53 734.398,1125.54 732.575,1125.55 730.753,1125.56 728.93,1125.57 727.108,1125.59 \n", " 725.285,1125.6 723.463,1125.61 721.64,1125.63 719.818,1125.64 717.995,1125.66 716.173,1125.67 715.488,1125.68 714.35,1125.69 712.527,1125.71 710.705,1125.72 \n", " 708.882,1125.74 707.06,1125.76 705.237,1125.78 703.415,1125.79 701.592,1125.81 699.77,1125.83 697.947,1125.85 696.125,1125.87 694.302,1125.89 692.479,1125.92 \n", " 690.657,1125.94 688.834,1125.96 687.012,1125.98 685.189,1126.01 683.367,1126.03 681.544,1126.06 679.722,1126.08 677.899,1126.11 676.077,1126.13 674.254,1126.16 \n", " 672.431,1126.18 670.609,1126.21 668.786,1126.24 666.964,1126.27 665.141,1126.3 663.319,1126.33 661.496,1126.35 659.674,1126.39 657.851,1126.42 656.029,1126.45 \n", " 654.206,1126.48 652.383,1126.51 650.561,1126.54 648.738,1126.58 646.916,1126.61 645.093,1126.64 643.271,1126.68 641.448,1126.71 639.626,1126.75 637.803,1126.78 \n", " 635.981,1126.82 634.158,1126.86 632.335,1126.89 630.513,1126.93 628.69,1126.97 626.868,1127.01 625.045,1127.05 624.743,1127.05 623.223,1127.09 621.4,1127.13 \n", " 619.578,1127.17 617.755,1127.21 615.933,1127.25 614.11,1127.29 612.287,1127.33 610.465,1127.38 608.642,1127.42 606.82,1127.46 604.997,1127.51 603.175,1127.55 \n", " 601.352,1127.6 599.53,1127.64 597.707,1127.69 595.884,1127.73 594.062,1127.78 592.239,1127.83 590.417,1127.87 588.594,1127.92 586.772,1127.97 584.949,1128.02 \n", " 583.127,1128.07 581.304,1128.12 579.482,1128.17 577.659,1128.22 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570.369,988.865 572.191,988.817 574.014,988.77 575.836,988.723 577.659,988.677 579.482,988.632 \n", " 581.304,988.587 583.127,988.543 584.949,988.501 586.772,988.458 588.594,988.417 590.417,988.377 592.239,988.337 594.027,988.299 594.062,988.298 595.884,988.26 \n", " 597.707,988.223 599.53,988.186 601.352,988.151 603.175,988.116 604.997,988.082 606.82,988.049 608.642,988.017 610.465,987.986 612.287,987.956 614.11,987.927 \n", " 615.933,987.898 617.755,987.871 619.578,987.844 621.4,987.819 623.223,987.794 625.045,987.771 626.868,987.748 628.69,987.726 630.513,987.706 632.335,987.686 \n", " 634.158,987.667 635.981,987.65 637.803,987.633 639.626,987.617 641.448,987.603 643.271,987.589 645.093,987.577 646.916,987.566 648.738,987.555 650.561,987.546 \n", " 652.383,987.538 654.206,987.531 656.029,987.526 657.851,987.521 659.674,987.518 661.496,987.515 663.319,987.514 665.141,987.514 666.964,987.515 668.786,987.518 \n", " 670.609,987.521 672.431,987.526 674.254,987.532 676.077,987.54 677.899,987.548 679.722,987.558 681.544,987.569 683.367,987.582 685.189,987.595 687.012,987.61 \n", " 688.834,987.627 690.657,987.644 692.479,987.663 694.302,987.684 696.125,987.706 697.947,987.729 699.77,987.753 701.592,987.779 703.415,987.806 705.237,987.835 \n", " 707.06,987.865 708.882,987.896 710.705,987.929 712.527,987.964 714.35,988 716.173,988.037 717.995,988.076 719.818,988.116 721.64,988.158 723.463,988.202 \n", " 725.285,988.246 727.108,988.293 727.334,988.299 728.93,988.341 730.753,988.39 732.575,988.441 734.398,988.494 736.221,988.548 738.043,988.604 739.866,988.661 \n", " 741.688,988.72 743.511,988.781 745.333,988.843 747.156,988.907 748.978,988.972 750.801,989.039 752.623,989.108 754.446,989.178 756.269,989.251 758.091,989.324 \n", " 759.914,989.4 761.736,989.477 763.559,989.556 765.381,989.636 766.188,989.673 767.204,989.718 769.026,989.802 770.849,989.888 772.671,989.975 774.494,990.064 \n", " 776.317,990.155 778.139,990.247 779.962,990.341 781.784,990.437 783.607,990.535 785.429,990.634 787.252,990.735 789.074,990.838 790.897,990.942 792.681,991.046 \n", " 792.72,991.049 794.542,991.156 796.365,991.266 798.187,991.377 800.01,991.49 801.832,991.605 803.655,991.722 805.477,991.84 807.3,991.96 809.122,992.082 \n", " 810.945,992.205 812.768,992.33 814.066,992.42 814.59,992.457 816.413,992.585 818.235,992.715 820.058,992.846 821.88,992.979 823.703,993.114 825.525,993.251 \n", " 827.348,993.389 829.17,993.529 830.993,993.67 832.576,993.794 832.816,993.813 834.638,993.957 836.461,994.103 838.283,994.25 840.106,994.399 841.928,994.549 \n", " 843.751,994.701 845.573,994.854 847.396,995.009 849.218,995.165 849.248,995.168 851.041,995.323 852.864,995.481 854.686,995.641 856.509,995.803 858.331,995.965 \n", " 860.154,996.129 861.976,996.295 863.799,996.461 864.675,996.542 865.621,996.628 867.444,996.797 869.266,996.967 871.089,997.138 872.912,997.309 874.734,997.482 \n", " 876.557,997.656 878.379,997.831 879.251,997.915 880.202,998.007 882.024,998.184 883.847,998.361 885.669,998.539 887.492,998.718 889.314,998.898 891.137,999.079 \n", " 892.96,999.26 893.254,999.289 894.782,999.442 896.605,999.624 898.427,999.807 900.25,999.99 902.072,1000.17 903.895,1000.36 905.717,1000.54 906.903,1000.66 \n", " 907.54,1000.73 909.362,1000.91 911.185,1001.1 913.008,1001.28 914.83,1001.47 916.653,1001.66 918.475,1001.84 920.298,1002.03 920.394,1002.04 922.12,1002.21 \n", " 923.943,1002.4 925.765,1002.58 927.588,1002.77 929.41,1002.95 931.233,1003.14 933.056,1003.32 933.911,1003.41 934.878,1003.51 936.701,1003.69 938.523,1003.88 \n", " 940.346,1004.06 942.168,1004.24 943.991,1004.42 945.813,1004.6 947.636,1004.78 947.659,1004.78 949.459,1004.96 951.281,1005.14 953.104,1005.32 954.926,1005.49 \n", " 956.749,1005.67 958.571,1005.84 960.394,1006.02 961.882,1006.16 962.216,1006.19 964.039,1006.36 965.861,1006.53 967.684,1006.7 969.507,1006.87 971.329,1007.03 \n", " 973.152,1007.2 974.974,1007.36 976.797,1007.52 976.919,1007.53 978.619,1007.68 980.442,1007.84 982.264,1008 984.087,1008.15 985.909,1008.3 987.732,1008.45 \n", " 989.555,1008.6 991.377,1008.75 993.2,1008.9 993.314,1008.91 995.022,1009.04 996.845,1009.18 998.667,1009.32 1000.49,1009.46 1002.31,1009.59 1004.13,1009.73 \n", " 1005.96,1009.86 1007.78,1009.98 1009.6,1010.11 1011.43,1010.23 1012.12,1010.28 1013.25,1010.35 1015.07,1010.47 1016.89,1010.59 1018.72,1010.7 1020.54,1010.81 \n", " 1022.36,1010.92 1024.18,1011.03 1026.01,1011.13 1027.83,1011.23 1029.65,1011.33 1031.47,1011.42 1033.3,1011.51 1035.12,1011.6 1036.26,1011.65 1036.94,1011.69 \n", " 1038.76,1011.77 1040.59,1011.85 1042.41,1011.92 1044.23,1012 1046.05,1012.07 1047.88,1012.14 1049.7,1012.2 1051.52,1012.26 1053.34,1012.32 1055.17,1012.37 \n", " 1056.99,1012.42 1058.81,1012.47 1060.63,1012.52 1062.46,1012.56 1064.28,1012.59 1066.1,1012.63 1067.92,1012.66 1069.75,1012.69 1071.57,1012.71 1073.39,1012.73 \n", " 1075.21,1012.75 1077.04,1012.76 1078.86,1012.77 1080.68,1012.78 1082.5,1012.78 1084.33,1012.78 1086.15,1012.77 1087.97,1012.76 1089.79,1012.75 1091.62,1012.73 \n", " 1093.44,1012.71 1095.26,1012.69 1097.08,1012.66 1098.91,1012.63 1100.73,1012.6 1102.55,1012.56 1104.37,1012.52 1106.2,1012.47 1108.02,1012.42 1109.84,1012.36 \n", " 1111.67,1012.3 1113.49,1012.24 1115.31,1012.18 1117.13,1012.1 1118.96,1012.03 1120.78,1011.95 1122.6,1011.87 1124.42,1011.78 1126.25,1011.69 1127,1011.65 \n", " 1128.07,1011.6 1129.89,1011.5 1131.71,1011.4 1133.54,1011.29 1135.36,1011.18 1137.18,1011.07 1139,1010.95 1140.83,1010.82 1142.65,1010.7 1144.47,1010.57 \n", " 1146.29,1010.43 1148.12,1010.29 1148.27,1010.28 1149.94,1010.15 1151.76,1010 1153.58,1009.85 1155.41,1009.69 1157.23,1009.53 1159.05,1009.37 1160.87,1009.2 \n", " 1162.7,1009.03 1163.98,1008.91 1164.52,1008.85 1166.34,1008.67 1168.16,1008.49 1169.99,1008.3 1171.81,1008.11 1173.63,1007.91 1175.45,1007.71 1177.04,1007.53 \n", " 1177.28,1007.51 1179.1,1007.3 1180.92,1007.08 1182.74,1006.87 1184.57,1006.65 1186.39,1006.42 1188.21,1006.19 1188.46,1006.16 1190.03,1005.96 1191.86,1005.72 \n", " 1193.68,1005.48 1195.5,1005.23 1197.32,1004.98 1198.76,1004.78 1199.15,1004.73 1200.97,1004.47 1202.79,1004.21 1204.62,1003.95 1206.44,1003.68 1208.21,1003.41 \n", " 1208.26,1003.4 1210.08,1003.13 1211.91,1002.85 1213.73,1002.56 1215.55,1002.27 1217.01,1002.04 1217.37,1001.98 1219.2,1001.68 1221.02,1001.38 1222.84,1001.08 \n", " 1224.66,1000.77 1225.29,1000.66 1226.49,1000.46 1228.31,1000.14 1230.13,999.825 1231.95,999.503 1233.15,999.289 1233.78,999.176 1235.6,998.847 1237.42,998.513 \n", " 1239.24,998.176 1240.64,997.915 1241.07,997.836 1242.89,997.492 1244.71,997.144 1246.53,996.793 1247.83,996.542 1248.36,996.439 1250.18,996.081 1252,995.72 \n", " 1253.82,995.355 1254.75,995.168 1255.65,994.988 1257.47,994.617 1259.29,994.243 1261.11,993.865 1261.46,993.794 1262.94,993.485 1264.76,993.101 1266.58,992.714 \n", " 1267.96,992.42 1268.4,992.324 1270.23,991.932 1272.05,991.536 1273.87,991.137 1274.29,991.046 1275.69,990.736 1277.52,990.332 1279.34,989.925 1280.46,989.673 \n", " 1281.16,989.515 1282.98,989.102 1284.81,988.687 1286.5,988.299 1286.63,988.27 1288.45,987.849 1290.27,987.427 1292.1,987.001 1292.42,986.925 1293.92,986.574 \n", " 1295.74,986.144 1297.56,985.711 1298.24,985.551 1299.39,985.277 1301.21,984.84 1303.03,984.401 1303.96,984.177 1304.86,983.96 1306.68,983.517 1308.5,983.073 \n", " 1309.6,982.804 1310.32,982.626 1312.15,982.177 1313.97,981.726 1315.16,981.43 1315.79,981.274 1317.61,980.82 1319.44,980.364 1320.67,980.056 1321.26,979.907 \n", " 1323.08,979.448 1324.9,978.988 1326.11,978.682 1326.73,978.527 1328.55,978.064 1330.37,977.599 1331.51,977.308 1332.19,977.134 1334.02,976.667 1335.84,976.199 \n", " 1336.87,975.935 1337.66,975.731 1339.48,975.261 1341.31,974.79 1342.19,974.561 1343.13,974.318 1344.95,973.846 1346.77,973.373 1347.49,973.187 1348.6,972.899 \n", " 1350.42,972.424 1352.24,971.949 1352.76,971.813 1354.06,971.474 1355.89,970.998 1357.71,970.522 1358.02,970.439 1359.53,970.045 1361.35,969.568 1363.18,969.091 \n", " 1363.27,969.066 1365,968.613 1366.82,968.136 1368.52,967.692 1368.64,967.659 1370.47,967.181 1372.29,966.704 1373.76,966.318 1374.11,966.227 1375.93,965.749 \n", " 1377.76,965.273 1379.01,964.944 1379.58,964.796 1381.4,964.32 1383.22,963.844 1384.28,963.57 1385.05,963.369 1386.87,962.894 1388.69,962.42 1389.55,962.197 \n", " 1390.51,961.947 1392.34,961.474 1394.16,961.002 1394.85,960.823 1395.98,960.53 1397.8,960.059 1399.63,959.59 1400.18,959.449 1401.45,959.121 1403.27,958.653 \n", " 1405.1,958.186 1405.53,958.075 1406.92,957.72 1408.74,957.255 1410.56,956.792 1410.92,956.701 1412.39,956.329 1414.21,955.867 1416.03,955.407 1416.35,955.328 \n", " 1417.85,954.948 1419.68,954.491 1421.5,954.035 1421.82,953.954 1423.32,953.58 1425.14,953.126 1426.97,952.674 1427.35,952.58 1428.79,952.223 1430.61,951.774 \n", " 1432.43,951.327 1432.93,951.206 1434.26,950.88 1436.08,950.436 1437.9,949.993 1438.57,949.832 1439.72,949.552 1441.55,949.112 1443.37,948.674 1444.27,948.459 \n", " 1445.19,948.238 1447.01,947.803 1448.84,947.37 1450.04,947.085 1450.66,946.939 1452.48,946.509 1454.3,946.082 1455.89,945.711 1456.13,945.656 1457.95,945.232 \n", " 1459.77,944.809 1461.59,944.389 1461.82,944.337 1463.42,943.97 1465.24,943.554 1467.06,943.139 1467.84,942.963 1468.88,942.726 1470.71,942.315 1472.53,941.906 \n", " 1473.94,941.59 1474.35,941.498 1476.17,941.093 1478,940.689 1479.82,940.288 1480.15,940.216 1481.64,939.888 1483.46,939.49 1485.29,939.094 1486.46,938.842 \n", " 1487.11,938.7 1488.93,938.308 1490.75,937.918 1492.58,937.53 1492.87,937.468 1494.4,937.143 1496.22,936.759 1498.04,936.377 1499.4,936.094 1499.87,935.996 \n", " 1501.69,935.617 1503.51,935.241 1505.34,934.866 1506.05,934.72 1507.16,934.493 1508.98,934.122 1510.8,933.753 1512.63,933.386 1512.82,933.347 1514.45,933.02 \n", " 1516.27,932.656 1518.09,932.295 1519.72,931.973 1519.92,931.935 1521.74,931.577 1523.56,931.221 1525.38,930.867 1526.77,930.599 1527.21,930.514 1529.03,930.163 \n", " 1530.85,929.814 1532.67,929.468 1533.95,929.225 1534.5,929.122 1536.32,928.779 1538.14,928.437 1539.96,928.097 1541.29,927.851 1541.79,927.759 1543.61,927.422 \n", " 1545.43,927.087 1547.25,926.754 1548.78,926.478 1549.08,926.423 1550.9,926.093 1552.72,925.765 1554.54,925.439 1556.37,925.114 1556.43,925.104 1558.19,924.791 \n", " 1560.01,924.47 1561.83,924.15 1563.66,923.832 1564.24,923.73 1565.48,923.515 1567.3,923.2 1569.12,922.887 1570.95,922.575 1572.23,922.356 1572.77,922.265 \n", " 1574.59,921.956 1576.41,921.648 1578.24,921.343 1580.06,921.038 1580.4,920.982 1581.88,920.736 1583.7,920.434 1585.53,920.134 1587.35,919.836 1588.75,919.609 \n", " 1589.17,919.539 1590.99,919.243 1592.82,918.949 1594.64,918.656 1596.46,918.365 1597.28,918.235 1598.29,918.075 1600.11,917.786 1601.93,917.499 1603.75,917.213 \n", " 1605.58,916.929 1606.01,916.861 1607.4,916.645 1609.22,916.363 1611.04,916.082 1612.87,915.803 1614.69,915.525 1614.94,915.487 1616.51,915.248 1618.33,914.972 \n", " 1620.16,914.698 1621.98,914.424 1623.8,914.152 1624.06,914.113 1625.62,913.881 1627.45,913.612 1629.27,913.343 1631.09,913.076 1632.91,912.81 1633.4,912.74 \n", " 1634.74,912.545 1636.56,912.281 1638.38,912.018 1640.2,911.756 1642.03,911.495 1642.94,911.366 1643.85,911.236 1645.67,910.977 1647.49,910.72 1649.32,910.464 \n", " 1651.14,910.208 1652.69,909.992 1652.96,909.954 1654.78,909.701 1656.61,909.448 1658.43,909.197 1660.25,908.947 1662.07,908.697 1662.66,908.618 1663.9,908.449 \n", " 1665.72,908.202 1667.54,907.955 1669.36,907.71 1671.19,907.465 1672.84,907.244 1673.01,907.222 1674.83,906.979 1676.65,906.737 1678.48,906.496 1680.3,906.256 \n", " 1682.12,906.017 1683.24,905.871 1683.94,905.779 1685.77,905.541 1687.59,905.304 1689.41,905.069 1691.23,904.834 1693.06,904.6 1693.86,904.497 1694.88,904.366 \n", " 1696.7,904.134 1698.53,903.902 1700.35,903.671 1702.17,903.441 1703.99,903.212 1704.7,903.123 1705.82,902.983 1707.64,902.756 1709.46,902.529 1711.28,902.302 \n", " 1713.11,902.077 1714.93,901.852 1715.76,901.749 1716.75,901.628 1718.57,901.404 1720.4,901.182 1722.22,900.96 1724.04,900.738 1725.86,900.518 1727.04,900.375 \n", " 1727.69,900.298 1729.51,900.079 1731.33,899.86 1733.15,899.642 1734.98,899.425 1736.8,899.208 1738.54,899.002 1738.62,898.992 1740.44,898.777 1742.27,898.562 \n", " 1744.09,898.348 1745.91,898.135 1747.73,897.922 1749.56,897.71 1750.26,897.628 1751.38,897.498 1753.2,897.287 1755.02,897.076 1756.85,896.866 1758.67,896.657 \n", " 1760.49,896.448 1762.19,896.254 1762.31,896.24 1764.14,896.032 1765.96,895.825 1767.78,895.619 1769.6,895.413 1771.43,895.207 1773.25,895.002 1774.34,894.88 \n", " 1775.07,894.798 1776.89,894.594 1778.72,894.391 1780.54,894.188 1782.36,893.985 1784.18,893.783 1786.01,893.582 1786.69,893.506 1787.83,893.381 1789.65,893.181 \n", " 1791.47,892.981 1793.3,892.781 1795.12,892.582 1796.94,892.384 1798.77,892.186 1799.26,892.133 1800.59,891.988 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2006.54,513.86 2004.71,513.971 2002.89,514.082 2001.07,514.192 1999.25,514.303 1998.68,514.337 1997.42,514.413 1995.6,514.524 \n", " 1993.78,514.634 1991.96,514.743 1990.13,514.853 1988.31,514.963 1986.49,515.072 1984.66,515.181 1982.84,515.291 1981.02,515.399 1979.2,515.508 1977.37,515.617 \n", " 1975.8,515.711 1975.55,515.725 1973.73,515.834 1971.91,515.942 1970.08,516.05 1968.26,516.157 1966.44,516.265 1964.62,516.372 1962.79,516.479 1960.97,516.587 \n", " 1959.15,516.693 1957.33,516.8 1955.5,516.907 1953.68,517.013 1952.45,517.085 1951.86,517.119 1950.04,517.225 1948.21,517.331 1946.39,517.436 1944.57,517.542 \n", " 1942.75,517.647 1940.92,517.752 1939.1,517.857 1937.28,517.961 1935.46,518.066 1933.63,518.17 1931.81,518.274 1929.99,518.378 1928.57,518.458 1928.17,518.481 \n", " 1926.34,518.585 1924.52,518.688 1922.7,518.791 1920.88,518.894 1919.05,518.996 1917.23,519.099 1915.41,519.201 1913.59,519.303 1911.76,519.404 1909.94,519.506 \n", " 1908.12,519.607 1906.3,519.708 1904.47,519.809 1904.05,519.832 1902.65,519.909 1900.83,520.01 1899.01,520.11 1897.18,520.21 1895.36,520.309 1893.54,520.409 \n", " 1891.72,520.508 1889.89,520.607 1888.07,520.706 1886.25,520.804 1884.42,520.902 1882.6,521 1880.78,521.098 1878.96,521.196 1878.76,521.206 1877.13,521.293 \n", " 1875.31,521.39 1873.49,521.486 1871.67,521.583 1869.84,521.679 1868.02,521.775 1866.2,521.871 1864.38,521.966 1862.55,522.061 1860.73,522.156 1858.91,522.251 \n", " 1857.09,522.345 1855.26,522.439 1853.44,522.533 1852.52,522.58 1851.62,522.626 1849.8,522.719 1847.97,522.812 1846.15,522.905 1844.33,522.997 1842.51,523.089 \n", " 1840.68,523.181 1838.86,523.272 1837.04,523.363 1835.22,523.454 1833.39,523.545 1831.57,523.635 1829.75,523.725 1827.93,523.814 1826.1,523.903 1825.08,523.954 \n", " 1824.28,523.992 1822.46,524.081 1820.64,524.169 1818.81,524.257 1816.99,524.345 1815.17,524.432 1813.35,524.519 1811.52,524.606 1809.7,524.692 1807.88,524.778 \n", " 1806.06,524.863 1804.23,524.949 1802.41,525.034 1800.59,525.118 1798.77,525.202 1796.94,525.286 1796.04,525.327 1795.12,525.37 1793.3,525.453 1791.47,525.535 \n", " 1789.65,525.618 1787.83,525.7 1786.01,525.781 1784.18,525.862 1782.36,525.943 1780.54,526.024 1778.72,526.104 1776.89,526.183 1775.07,526.263 1773.25,526.341 \n", " 1771.43,526.42 1769.6,526.498 1767.78,526.575 1765.96,526.653 1764.81,526.701 1764.14,526.729 1762.31,526.806 1760.49,526.882 1758.67,526.957 1756.85,527.032 \n", " 1755.02,527.107 1753.2,527.181 1751.38,527.255 1749.56,527.328 1747.73,527.401 1745.91,527.473 1744.09,527.545 1742.27,527.617 1740.44,527.688 1738.62,527.758 \n", " 1736.8,527.828 1734.98,527.898 1733.15,527.967 1731.33,528.035 1730.27,528.075 1729.51,528.103 1727.69,528.171 1725.86,528.238 1724.04,528.305 1722.22,528.371 \n", " 1720.4,528.436 1718.57,528.501 1716.75,528.566 1714.93,528.63 1713.11,528.693 1711.28,528.756 1709.46,528.819 1707.64,528.881 1705.82,528.942 1703.99,529.003 \n", " 1702.17,529.063 1700.35,529.122 1698.53,529.181 1696.7,529.24 1694.88,529.298 1693.06,529.355 1691.23,529.412 1690.04,529.449 1689.41,529.468 1687.59,529.524 \n", " 1685.77,529.579 1683.94,529.633 1682.12,529.687 1680.3,529.74 1678.48,529.792 1676.65,529.844 1674.83,529.895 1673.01,529.946 1671.19,529.996 1669.36,530.045 \n", " 1667.54,530.094 1665.72,530.142 1663.9,530.189 1662.07,530.236 1660.25,530.282 1658.43,530.327 1656.61,530.372 1654.78,530.416 1652.96,530.459 1651.14,530.501 \n", " 1649.32,530.543 1647.49,530.584 1645.67,530.624 1643.85,530.664 1642.03,530.703 1640.2,530.741 1638.38,530.779 1636.56,530.815 1636.18,530.823 1634.74,530.851 \n", " 1632.91,530.886 1631.09,530.921 1629.27,530.954 1627.45,530.987 1625.62,531.019 1623.8,531.05 1621.98,531.081 1620.16,531.11 1618.33,531.139 1616.51,531.167 \n", " 1614.69,531.194 1612.87,531.221 1611.04,531.246 1609.22,531.271 1607.4,531.295 1605.58,531.318 1603.75,531.34 1601.93,531.361 1600.11,531.381 1598.29,531.401 \n", " 1596.46,531.419 1594.64,531.437 1592.82,531.454 1590.99,531.47 1589.17,531.485 1587.35,531.499 1585.53,531.512 1583.7,531.524 1581.88,531.535 1580.06,531.545 \n", " 1578.24,531.555 1576.41,531.563 1574.59,531.57 1572.77,531.577 1570.95,531.582 1569.12,531.587 1567.3,531.59 1565.48,531.592 1563.66,531.594 1561.83,531.594 \n", " 1560.01,531.594 1558.19,531.592 1556.37,531.589 1554.54,531.586 1552.72,531.581 1550.9,531.575 1549.08,531.568 1547.25,531.56 1545.43,531.551 1543.61,531.541 \n", " 1541.79,531.53 1539.96,531.518 1538.14,531.504 1536.32,531.49 1534.5,531.474 1532.67,531.458 1530.85,531.44 1529.03,531.421 1527.21,531.401 1525.38,531.38 \n", " 1523.56,531.357 1521.74,531.334 1519.92,531.309 1518.09,531.283 1516.27,531.256 1514.45,531.228 1512.63,531.199 1510.8,531.168 1508.98,531.137 1507.16,531.104 \n", " 1505.34,531.07 1503.51,531.035 1501.69,530.998 1499.87,530.961 1498.04,530.922 1496.22,530.882 1494.4,530.841 1493.63,530.823 1492.58,530.798 1490.75,530.754 \n", " 1488.93,530.71 1487.11,530.663 1485.29,530.616 1483.46,530.568 1481.64,530.518 1479.82,530.467 1478,530.415 1476.17,530.361 1474.35,530.306 1472.53,530.25 \n", " 1470.71,530.193 1468.88,530.135 1467.06,530.075 1465.24,530.014 1463.42,529.952 1461.59,529.889 1459.77,529.824 1457.95,529.758 1456.13,529.691 1454.3,529.623 \n", " 1452.48,529.553 1450.66,529.482 1449.81,529.449 1448.84,529.41 1447.01,529.337 1445.19,529.263 1443.37,529.187 1441.55,529.11 1439.72,529.032 1437.9,528.953 \n", " 1436.08,528.872 1434.26,528.791 1432.43,528.708 1430.61,528.624 1428.79,528.539 1426.97,528.452 1425.14,528.365 1423.32,528.276 1421.5,528.186 1419.68,528.095 \n", " 1419.28,528.075 1417.85,528.003 1416.03,527.91 1414.21,527.816 1412.39,527.72 1410.56,527.624 1408.74,527.526 1406.92,527.428 1405.1,527.328 1403.27,527.227 \n", " 1401.45,527.126 1399.63,527.023 1397.8,526.919 1395.98,526.815 1394.16,526.709 1394.03,526.701 1392.34,526.602 1390.51,526.495 1388.69,526.386 1386.87,526.277 \n", " 1385.05,526.167 1383.22,526.056 1381.4,525.944 1379.58,525.831 1377.76,525.717 1375.93,525.603 1374.11,525.487 1372.29,525.371 1371.6,525.327 1370.47,525.255 \n", " 1368.64,525.137 1366.82,525.019 1365,524.9 1363.18,524.781 1361.35,524.661 1359.53,524.54 1357.71,524.419 1355.89,524.297 1354.06,524.175 1352.24,524.052 \n", " 1350.78,523.954 1350.42,523.929 1348.6,523.805 1346.77,523.681 1344.95,523.556 1343.13,523.432 1341.31,523.306 1339.48,523.181 1337.66,523.055 1335.84,522.929 \n", " 1334.02,522.802 1332.19,522.676 1330.81,522.58 1330.37,522.549 1328.55,522.422 1326.73,522.295 1324.9,522.168 1323.08,522.041 1321.26,521.914 1319.44,521.787 \n", " 1317.61,521.66 1315.79,521.533 1313.97,521.406 1312.15,521.279 1311.09,521.206 1310.32,521.153 1308.5,521.026 1306.68,520.9 1304.86,520.775 1303.03,520.649 \n", " 1301.21,520.524 1299.39,520.399 1297.56,520.275 1295.74,520.151 1293.92,520.028 1292.1,519.905 1291.02,519.832 1290.27,519.782 1288.45,519.661 1286.63,519.54 \n", " 1284.81,519.419 1282.98,519.299 1281.16,519.18 1279.34,519.062 1277.52,518.945 1275.69,518.828 1273.87,518.713 1272.05,518.598 1270.23,518.484 1269.81,518.458 \n", " 1268.4,518.371 1266.58,518.259 1264.76,518.148 1262.94,518.039 1261.11,517.93 1259.29,517.823 1257.47,517.716 1255.65,517.611 1253.82,517.508 1252,517.405 \n", " 1250.18,517.304 1248.36,517.204 1246.53,517.106 1246.13,517.085 1244.71,517.009 1242.89,516.914 1241.07,516.82 1239.24,516.727 1237.42,516.636 1235.6,516.547 \n", " 1233.78,516.459 1231.95,516.374 1230.13,516.289 1228.31,516.207 1226.49,516.126 1224.66,516.047 1222.84,515.97 1221.02,515.895 1219.2,515.822 1217.37,515.751 \n", " 1216.33,515.711 1215.55,515.681 1213.73,515.614 1211.91,515.548 1210.08,515.485 1208.26,515.424 1206.44,515.364 1204.62,515.307 1202.79,515.252 1200.97,515.2 \n", " 1199.15,515.149 1197.32,515.101 1195.5,515.055 1193.68,515.012 1191.86,514.97 1190.03,514.931 1188.21,514.895 1186.39,514.861 1184.57,514.829 1182.74,514.8 \n", " 1180.92,514.773 1179.1,514.749 1177.28,514.727 1175.45,514.708 1173.63,514.692 1171.81,514.678 1169.99,514.666 1168.16,514.658 1166.34,514.652 1164.52,514.648 \n", " 1162.7,514.648 1160.87,514.65 1159.05,514.655 1157.23,514.662 1155.41,514.673 1153.58,514.686 1151.76,514.702 1149.94,514.721 1148.12,514.743 1146.29,514.767 \n", " 1144.47,514.795 1142.65,514.825 1140.83,514.859 1139,514.895 1137.18,514.935 1135.36,514.977 1133.54,515.022 1131.71,515.071 1129.89,515.122 1128.07,515.176 \n", " 1126.25,515.234 1124.42,515.294 1122.6,515.358 1120.78,515.425 1118.96,515.494 1117.13,515.567 1115.31,515.643 1113.76,515.711 1113.49,515.723 1111.67,515.805 \n", " 1109.84,515.89 1108.02,515.979 1106.2,516.07 1104.37,516.165 1102.55,516.263 1100.73,516.365 1098.91,516.469 1097.08,516.577 1095.26,516.688 1093.44,516.802 \n", " 1091.62,516.92 1089.79,517.04 1089.14,517.085 1087.97,517.164 1086.15,517.291 1084.33,517.421 1082.5,517.555 1080.68,517.692 1078.86,517.832 1077.04,517.975 \n", " 1075.21,518.122 1073.39,518.271 1071.57,518.425 1071.18,518.458 1069.75,518.581 1067.92,518.74 1066.1,518.903 1064.28,519.069 1062.46,519.238 1060.63,519.411 \n", " 1058.81,519.587 1056.99,519.766 1056.33,519.832 1055.17,519.949 1053.34,520.134 1051.52,520.323 1049.7,520.515 1047.88,520.71 1046.05,520.908 1044.23,521.11 \n", " 1043.38,521.206 1042.41,521.315 1040.59,521.523 1038.76,521.734 1036.94,521.949 1035.12,522.166 1033.3,522.387 1031.73,522.58 1031.47,522.611 1029.65,522.838 \n", " 1027.83,523.068 1026.01,523.302 1024.18,523.538 1022.36,523.778 1021.04,523.954 1020.54,524.021 1018.72,524.266 1016.89,524.515 1015.07,524.767 1013.25,525.022 \n", " 1011.43,525.28 1011.09,525.327 1009.6,525.541 1007.78,525.804 1005.96,526.071 1004.13,526.341 1002.31,526.614 1001.74,526.701 1000.49,526.89 998.667,527.168 \n", " 996.845,527.45 995.022,527.734 993.2,528.022 992.864,528.075 991.377,528.312 989.555,528.604 987.732,528.9 985.909,529.198 984.395,529.449 984.087,529.5 \n", " 982.264,529.803 980.442,530.11 978.619,530.419 976.797,530.731 976.267,530.823 974.974,531.046 973.152,531.363 971.329,531.682 969.507,532.005 968.43,532.196 \n", " 967.684,532.329 965.861,532.657 964.039,532.986 962.216,533.318 960.844,533.57 960.394,533.653 958.571,533.99 956.749,534.329 954.926,534.671 953.477,534.944 \n", " 953.104,535.015 951.281,535.361 949.459,535.709 947.636,536.059 946.301,536.318 945.813,536.412 943.991,536.767 942.168,537.124 940.346,537.483 939.292,537.692 \n", " 938.523,537.844 936.701,538.207 934.878,538.572 933.056,538.939 932.429,539.065 931.233,539.308 929.41,539.678 927.588,540.051 925.765,540.425 925.696,540.439 \n", " 923.943,540.801 922.12,541.178 920.298,541.558 919.076,541.813 918.475,541.939 916.653,542.321 914.83,542.705 913.008,543.091 912.556,543.187 911.185,543.478 \n", " 909.362,543.867 907.54,544.257 906.123,544.561 905.717,544.648 903.895,545.04 902.072,545.434 900.25,545.829 899.766,545.934 898.427,546.226 896.605,546.623 \n", " 894.782,547.022 893.475,547.308 892.96,547.421 891.137,547.822 889.314,548.224 887.492,548.626 887.24,548.682 885.669,549.03 883.847,549.434 882.024,549.84 \n", " 881.052,550.056 880.202,550.246 878.379,550.652 876.557,551.06 874.905,551.43 874.734,551.468 872.912,551.877 871.089,552.286 869.266,552.696 868.788,552.803 \n", " 867.444,553.106 865.621,553.517 863.799,553.928 862.696,554.177 861.976,554.34 860.154,554.752 858.331,555.164 856.623,555.551 856.509,555.577 854.686,555.99 \n", " 852.864,556.403 851.041,556.816 850.56,556.925 849.218,557.229 847.396,557.643 845.573,558.056 844.502,558.299 843.751,558.469 841.928,558.883 840.106,559.296 \n", " 838.443,559.672 838.283,559.709 836.461,560.122 834.638,560.535 832.816,560.947 832.376,561.046 830.993,561.359 829.17,561.772 827.348,562.183 826.297,562.42 \n", " 825.525,562.594 823.703,563.005 821.88,563.416 820.2,563.794 820.058,563.826 818.235,564.236 816.413,564.645 814.59,565.053 814.078,565.168 812.768,565.461 \n", " 810.945,565.869 809.122,566.275 807.927,566.541 807.3,566.681 805.477,567.087 803.655,567.492 801.832,567.895 801.742,567.915 800.01,568.299 798.187,568.701 \n", " 796.365,569.103 795.516,569.289 794.542,569.503 792.72,569.903 790.897,570.302 789.246,570.663 789.074,570.7 787.252,571.098 785.429,571.494 783.607,571.889 \n", " 782.923,572.037 781.784,572.283 779.962,572.677 778.139,573.069 776.546,573.41 776.317,573.46 774.494,573.85 772.671,574.239 770.849,574.627 770.107,574.784 \n", " 769.026,575.014 767.204,575.4 765.381,575.784 763.603,576.158 763.559,576.167 761.736,576.55 759.914,576.931 758.091,577.311 757.026,577.532 756.269,577.689 \n", " 754.446,578.067 752.623,578.443 750.801,578.818 750.372,578.906 748.978,579.192 747.156,579.564 745.333,579.935 743.637,580.279 743.511,580.305 741.688,580.674 \n", " 739.866,581.041 738.043,581.407 736.813,581.653 736.221,581.772 734.398,582.135 732.575,582.498 730.753,582.858 729.896,583.027 728.93,583.218 727.108,583.576 \n", " 725.285,583.933 723.463,584.288 722.882,584.401 721.64,584.642 719.818,584.995 717.995,585.347 716.173,585.696 715.764,585.775 714.35,586.045 712.527,586.393 \n", " 710.705,586.739 708.882,587.083 708.536,587.148 707.06,587.427 705.237,587.769 703.415,588.109 701.592,588.449 701.194,588.522 699.77,588.787 697.947,589.123 \n", " 696.125,589.458 694.302,589.792 693.733,589.896 692.479,590.125 690.657,590.456 688.834,590.786 687.012,591.115 686.146,591.27 685.189,591.442 683.367,591.768 \n", " 681.544,592.092 679.722,592.416 678.429,592.644 677.899,592.737 676.077,593.058 674.254,593.377 672.431,593.695 670.609,594.012 670.577,594.017 668.786,594.328 \n", " 666.964,594.642 665.141,594.955 663.319,595.266 662.583,595.391 661.496,595.576 659.674,595.886 657.851,596.193 656.029,596.5 654.444,596.765 654.206,596.805 \n", " 652.383,597.109 650.561,597.412 648.738,597.714 646.916,598.014 646.154,598.139 645.093,598.313 643.271,598.611 641.448,598.908 639.626,599.203 637.803,599.498 \n", " 637.709,599.513 635.981,599.791 634.158,600.083 632.335,600.374 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402.695,398.874 404.517,398.676 406.34,398.477 408.162,398.277 409.985,398.078 411.807,397.878 \n", " 413.63,397.678 414.675,397.564 415.452,397.478 417.275,397.278 419.097,397.078 420.92,396.877 422.743,396.676 424.565,396.475 426.388,396.274 427.145,396.19 \n", " 428.21,396.072 430.033,395.87 431.855,395.668 433.678,395.466 435.5,395.264 437.323,395.061 439.145,394.859 439.526,394.816 440.968,394.656 442.791,394.452 \n", " 444.613,394.249 446.436,394.045 448.258,393.841 450.081,393.637 451.821,393.442 451.903,393.433 453.726,393.229 455.548,393.024 457.371,392.819 459.194,392.614 \n", " 461.016,392.409 462.839,392.203 464.031,392.069 464.661,391.997 466.484,391.791 468.306,391.585 470.129,391.379 471.951,391.172 473.774,390.966 475.596,390.759 \n", " 476.158,390.695 477.419,390.551 479.242,390.344 481.064,390.136 482.887,389.929 484.709,389.721 486.532,389.512 488.206,389.321 488.354,389.304 490.177,389.095 \n", " 491.999,388.886 493.822,388.677 495.644,388.468 497.467,388.259 499.29,388.049 500.175,387.947 501.112,387.839 502.935,387.629 504.757,387.419 506.58,387.208 \n", " 508.402,386.998 510.225,386.787 512.047,386.576 512.069,386.573 513.87,386.365 515.692,386.153 517.515,385.942 519.338,385.73 521.16,385.518 522.983,385.305 \n", " 523.891,385.2 524.805,385.093 526.628,384.88 528.45,384.668 530.273,384.455 532.095,384.241 533.918,384.028 535.644,383.826 535.74,383.814 537.563,383.601 \n", " 539.386,383.387 541.208,383.173 543.031,382.958 544.853,382.744 546.676,382.529 547.33,382.452 548.498,382.314 550.321,382.099 552.143,381.884 553.966,381.669 \n", " 555.788,381.453 557.611,381.237 558.955,381.078 559.434,381.021 561.256,380.805 563.079,380.589 564.901,380.373 566.724,380.156 568.546,379.939 570.369,379.722 \n", " 570.52,379.704 572.191,379.505 574.014,379.288 575.836,379.071 577.659,378.853 579.482,378.635 581.304,378.417 582.03,378.33 583.127,378.199 584.949,377.981 \n", " 586.772,377.763 588.594,377.544 590.417,377.326 592.239,377.107 593.489,376.957 594.062,376.888 595.884,376.669 597.707,376.45 599.53,376.23 601.352,376.011 \n", " 603.175,375.791 604.902,375.583 604.997,375.571 606.82,375.352 608.642,375.132 610.465,374.911 612.287,374.691 614.11,374.471 615.933,374.25 616.272,374.209 \n", " 617.755,374.03 619.578,373.809 621.4,373.588 623.223,373.367 625.045,373.146 626.868,372.925 627.605,372.835 628.69,372.704 630.513,372.482 632.335,372.261 \n", " 634.158,372.039 635.981,371.818 637.803,371.596 638.906,371.461 639.626,371.374 641.448,371.152 643.271,370.93 645.093,370.708 646.916,370.486 648.738,370.263 \n", " 650.179,370.088 650.561,370.041 652.383,369.819 654.206,369.596 656.029,369.374 657.851,369.151 659.674,368.929 661.431,368.714 661.496,368.706 663.319,368.483 \n", " 665.141,368.26 666.964,368.038 668.786,367.815 670.609,367.592 672.431,367.369 672.667,367.34 674.254,367.146 676.077,366.923 677.899,366.7 679.722,366.477 \n", " 681.544,366.254 683.367,366.031 683.893,365.966 685.189,365.808 687.012,365.585 688.834,365.361 690.657,365.138 692.479,364.915 694.302,364.692 695.116,364.592 \n", " 696.125,364.469 697.947,364.246 699.77,364.023 701.592,363.8 703.415,363.577 705.237,363.354 706.343,363.219 707.06,363.131 708.882,362.908 710.705,362.685 \n", " 712.527,362.462 714.35,362.24 716.173,362.017 717.58,361.845 717.995,361.794 719.818,361.572 721.64,361.349 723.463,361.127 725.285,360.904 727.108,360.682 \n", " 728.836,360.471 728.93,360.46 730.753,360.237 732.575,360.015 734.398,359.793 736.221,359.571 738.043,359.35 739.866,359.128 740.118,359.097 741.688,358.906 \n", " 743.511,358.685 745.333,358.464 747.156,358.242 748.978,358.021 750.801,357.8 751.435,357.723 752.623,357.579 754.446,357.359 756.269,357.138 758.091,356.918 \n", " 759.914,356.698 761.736,356.478 762.796,356.35 763.559,356.258 765.381,356.038 767.204,355.818 769.026,355.599 770.849,355.38 772.671,355.161 774.212,354.976 \n", " 774.494,354.942 776.317,354.723 778.139,354.505 779.962,354.287 781.784,354.069 783.607,353.851 785.429,353.633 785.692,353.602 787.252,353.416 789.074,353.199 \n", " 790.897,352.982 792.72,352.765 794.542,352.549 796.365,352.333 797.248,352.228 798.187,352.117 800.01,351.901 801.832,351.686 803.655,351.471 805.477,351.256 \n", " 807.3,351.042 808.892,350.854 809.122,350.827 810.945,350.613 812.768,350.4 814.59,350.186 816.413,349.973 818.235,349.761 820.058,349.548 820.637,349.481 \n", " 821.88,349.336 823.703,349.124 825.525,348.913 827.348,348.702 829.17,348.491 830.993,348.28 832.497,348.107 832.816,348.07 834.638,347.86 836.461,347.651 \n", " 838.283,347.442 840.106,347.233 841.928,347.025 843.751,346.817 844.489,346.733 845.573,346.61 847.396,346.402 849.218,346.196 851.041,345.989 852.864,345.783 \n", " 854.686,345.578 856.509,345.373 856.628,345.359 858.331,345.168 860.154,344.964 861.976,344.76 863.799,344.556 865.621,344.353 867.444,344.151 868.934,343.985 \n", " 869.266,343.949 871.089,343.747 872.912,343.546 874.734,343.345 876.557,343.145 878.379,342.945 880.202,342.745 881.428,342.612 882.024,342.547 883.847,342.348 \n", " 885.669,342.15 887.492,341.953 889.314,341.756 891.137,341.559 892.96,341.363 894.131,341.238 894.782,341.168 896.605,340.973 898.427,340.779 900.25,340.585 \n", " 902.072,340.392 903.895,340.199 905.717,340.006 907.071,339.864 907.54,339.815 909.362,339.624 911.185,339.433 913.008,339.243 914.83,339.053 916.653,338.864 \n", " 918.475,338.676 920.276,338.49 920.298,338.488 922.12,338.301 923.943,338.114 925.765,337.928 927.588,337.742 929.41,337.557 931.233,337.373 933.056,337.189 \n", " 933.779,337.116 934.878,337.006 936.701,336.823 938.523,336.641 940.346,336.46 942.168,336.279 943.991,336.099 945.813,335.92 947.616,335.743 947.636,335.741 \n", " 949.459,335.562 951.281,335.385 953.104,335.208 954.926,335.031 956.749,334.855 958.571,334.68 960.394,334.506 961.83,334.369 962.216,334.332 964.039,334.159 \n", " 965.861,333.986 967.684,333.814 969.507,333.643 971.329,333.472 973.152,333.303 974.974,333.133 976.47,332.995 976.797,332.965 978.619,332.797 980.442,332.63 \n", " 982.264,332.463 984.087,332.297 985.909,332.132 987.732,331.967 989.555,331.804 991.377,331.64 991.592,331.621 993.2,331.478 995.022,331.316 996.845,331.155 \n", " 998.667,330.995 1000.49,330.835 1002.31,330.676 1004.13,330.518 1005.96,330.36 1007.26,330.247 1007.78,330.203 1009.6,330.047 1011.43,329.891 1013.25,329.736 \n", " 1015.07,329.582 1016.89,329.429 1018.72,329.276 1020.54,329.124 1022.36,328.973 1023.56,328.874 1024.18,328.822 1026.01,328.673 1027.83,328.523 1029.65,328.375 \n", " 1031.47,328.227 1033.3,328.08 1035.12,327.934 1036.94,327.789 1038.76,327.644 1040.58,327.5 1040.59,327.5 1042.41,327.356 1044.23,327.214 1046.05,327.072 \n", " 1047.88,326.93 1049.7,326.79 1051.52,326.65 1053.34,326.511 1055.17,326.373 1056.99,326.235 1058.44,326.126 1058.81,326.098 1060.63,325.962 1062.46,325.826 \n", " 1064.28,325.692 1066.1,325.558 1067.92,325.424 1069.75,325.292 1071.57,325.16 1073.39,325.029 1075.21,324.898 1077.04,324.769 1077.27,324.752 1078.86,324.64 \n", " 1080.68,324.511 1082.5,324.384 1084.33,324.257 1086.15,324.131 1087.97,324.005 1089.79,323.88 1091.62,323.756 1093.44,323.633 1095.26,323.51 1097.08,323.389 \n", " 1097.24,323.378 1098.91,323.267 1100.73,323.147 1102.55,323.027 1104.37,322.908 1106.2,322.789 1108.02,322.672 1109.84,322.555 1111.67,322.438 1113.49,322.323 \n", " 1115.31,322.208 1117.13,322.093 1118.56,322.005 1118.96,321.98 1120.78,321.867 1122.6,321.755 1124.42,321.643 1126.25,321.532 1128.07,321.422 1129.89,321.312 \n", " 1131.71,321.203 1133.54,321.095 1135.36,320.988 1137.18,320.881 1139,320.774 1140.83,320.669 1141.49,320.631 1142.65,320.564 1144.47,320.46 1146.29,320.356 \n", " 1148.12,320.253 1149.94,320.151 1151.76,320.049 1153.58,319.948 1155.41,319.847 1157.23,319.747 1159.05,319.648 1160.87,319.55 1162.7,319.452 1164.52,319.354 \n", " 1166.34,319.257 1166.35,319.257 1168.16,319.161 1169.99,319.066 1171.81,318.971 1173.63,318.876 1175.45,318.783 1177.28,318.69 1179.1,318.597 1180.92,318.505 \n", " 1182.74,318.414 1184.57,318.323 1186.39,318.232 1188.21,318.143 1190.03,318.054 1191.86,317.965 1193.55,317.883 1193.68,317.877 1195.5,317.79 1197.32,317.703 \n", " 1199.15,317.616 1200.97,317.531 1202.79,317.445 1204.62,317.36 1206.44,317.276 1208.26,317.193 1210.08,317.109 1211.91,317.027 1213.73,316.945 1215.55,316.863 \n", " 1217.37,316.782 1219.2,316.701 1221.02,316.621 1222.84,316.541 1223.58,316.509 1224.66,316.462 1226.49,316.384 1228.31,316.305 1230.13,316.228 1231.95,316.15 \n", " 1233.78,316.074 1235.6,315.997 1237.42,315.921 1239.24,315.846 1241.07,315.771 1242.89,315.696 1244.71,315.622 1246.53,315.549 1248.36,315.475 1250.18,315.402 \n", " 1252,315.33 1253.82,315.258 1255.65,315.186 1256.95,315.136 1257.47,315.115 1259.29,315.044 1261.11,314.974 1262.94,314.904 1264.76,314.834 1266.58,314.765 \n", " 1268.4,314.696 1270.23,314.628 1272.05,314.56 1273.87,314.492 1275.69,314.425 1277.52,314.358 1279.34,314.291 1281.16,314.224 1282.98,314.158 1284.81,314.093 \n", " 1286.63,314.027 1288.45,313.962 1290.27,313.898 1292.1,313.833 1293.92,313.769 1294.13,313.762 1295.74,313.705 1297.56,313.642 1299.39,313.579 1301.21,313.516 \n", " 1303.03,313.453 1304.86,313.391 1306.68,313.329 1308.5,313.267 1310.32,313.205 1312.15,313.144 1313.97,313.083 1315.79,313.022 1317.61,312.962 1319.44,312.902 \n", " 1321.26,312.842 1323.08,312.782 1324.9,312.722 1326.73,312.663 1328.55,312.604 1330.37,312.545 1332.19,312.486 1334.02,312.428 1335.26,312.388 1335.84,312.37 \n", " 1337.66,312.311 1339.48,312.254 1341.31,312.196 1343.13,312.138 1344.95,312.081 1346.77,312.024 1348.6,311.967 1350.42,311.91 1352.24,311.853 1354.06,311.797 \n", " 1355.89,311.74 1357.71,311.684 1359.53,311.628 1361.35,311.572 1363.18,311.516 1365,311.461 1366.82,311.405 1368.64,311.35 1370.47,311.295 1372.29,311.239 \n", " 1374.11,311.184 1375.93,311.129 1377.76,311.074 1379.58,311.02 1379.76,311.014 1381.4,310.965 1383.22,310.91 1385.05,310.856 1386.87,310.801 1388.69,310.747 \n", " 1390.51,310.693 1392.34,310.639 1394.16,310.584 1395.98,310.53 1397.8,310.476 1399.63,310.422 1401.45,310.368 1403.27,310.315 1405.1,310.261 1406.92,310.207 \n", " 1408.74,310.153 1410.56,310.099 1412.39,310.046 1414.21,309.992 1416.03,309.938 1417.85,309.885 1419.68,309.831 1421.5,309.778 1423.32,309.724 1425.14,309.67 \n", " 1426.16,309.64 1426.97,309.617 1428.79,309.563 1430.61,309.509 1432.43,309.456 1434.26,309.402 1436.08,309.348 1437.9,309.295 1439.72,309.241 1441.55,309.187 \n", " 1443.37,309.134 1445.19,309.08 1447.01,309.026 1448.84,308.972 1450.66,308.918 1452.48,308.864 1454.3,308.81 1456.13,308.756 1457.95,308.702 1459.77,308.648 \n", " 1461.59,308.594 1463.42,308.539 1465.24,308.485 1467.06,308.43 1468.88,308.376 1470.71,308.321 1472.53,308.267 1472.53,308.267 1474.35,308.212 1476.17,308.157 \n", " 1478,308.102 1479.82,308.047 1481.64,307.992 1483.46,307.937 1485.29,307.882 1487.11,307.826 1488.93,307.771 1490.75,307.715 1492.58,307.66 1494.4,307.604 \n", " 1496.22,307.548 1498.04,307.492 1499.87,307.436 1501.69,307.38 1503.51,307.323 1505.34,307.267 1507.16,307.21 1508.98,307.154 1510.8,307.097 1512.63,307.04 \n", " 1514.45,306.983 1516.27,306.926 1517.32,306.893 1518.09,306.868 1519.92,306.811 1521.74,306.753 1523.56,306.696 1525.38,306.638 1527.21,306.58 1529.03,306.522 \n", " 1530.85,306.463 1532.67,306.405 1534.5,306.346 1536.32,306.288 1538.14,306.229 1539.96,306.17 1541.79,306.11 1543.61,306.051 1545.43,305.992 1547.25,305.932 \n", " 1549.08,305.872 1550.9,305.812 1552.72,305.752 1554.54,305.692 1556.37,305.631 1558.19,305.57 1559.73,305.519 1560.01,305.51 1561.83,305.449 1563.66,305.387 \n", " 1565.48,305.326 1567.3,305.265 1569.12,305.203 1570.95,305.141 1572.77,305.079 1574.59,305.017 1576.41,304.954 1578.24,304.892 1580.06,304.829 1581.88,304.766 \n", " 1583.7,304.703 1585.53,304.64 1587.35,304.576 1589.17,304.513 1590.99,304.449 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1984.66,240.844 1982.84,240.948 1982.81,240.95 1981.02,241.052 1979.2,241.156 1977.37,241.259 \n", " 1975.55,241.362 1973.73,241.466 1971.91,241.569 1970.08,241.672 1968.26,241.775 1966.44,241.877 1964.62,241.98 1962.79,242.082 1960.97,242.185 1959.15,242.287 \n", " 1958.49,242.324 1957.33,242.389 1955.5,242.491 1953.68,242.593 1951.86,242.695 1950.04,242.797 1948.21,242.898 1946.39,242.999 1944.57,243.101 1942.75,243.202 \n", " 1940.92,243.303 1939.1,243.404 1937.28,243.505 1935.46,243.605 1933.78,243.698 1933.63,243.706 1931.81,243.806 1929.99,243.906 1928.17,244.006 1926.34,244.107 \n", " 1924.52,244.206 1922.7,244.306 1920.88,244.406 1919.05,244.505 1917.23,244.605 1915.41,244.704 1913.59,244.803 1911.76,244.902 1909.94,245.001 1908.63,245.072 \n", " 1908.12,245.099 1906.3,245.198 1904.47,245.296 1902.65,245.395 1900.83,245.493 1899.01,245.591 1897.18,245.689 1895.36,245.787 1893.54,245.884 1891.72,245.982 \n", " 1889.89,246.079 1888.07,246.176 1886.25,246.273 1884.42,246.37 1883.01,246.445 1882.6,246.467 1880.78,246.564 1878.96,246.66 1877.13,246.757 1875.31,246.853 \n", " 1873.49,246.949 1871.67,247.045 1869.84,247.141 1868.02,247.236 1866.2,247.332 1864.38,247.427 1862.55,247.523 1860.73,247.618 1858.91,247.713 1857.09,247.808 \n", " 1856.86,247.819 1855.26,247.902 1853.44,247.997 1851.62,248.091 1849.8,248.186 1847.97,248.28 1846.15,248.374 1844.33,248.468 1842.51,248.561 1840.68,248.655 \n", " 1838.86,248.748 1837.04,248.841 1835.22,248.935 1833.39,249.028 1831.57,249.12 1830.14,249.193 1829.75,249.213 1827.93,249.306 1826.1,249.398 1824.28,249.49 \n", " 1822.46,249.582 1820.64,249.674 1818.81,249.766 1816.99,249.858 1815.17,249.949 1813.35,250.04 1811.52,250.132 1809.7,250.223 1807.88,250.314 1806.06,250.404 \n", " 1804.23,250.495 1802.78,250.567 1802.41,250.585 1800.59,250.676 1798.77,250.766 1796.94,250.856 1795.12,250.946 1793.3,251.035 1791.47,251.125 1789.65,251.214 \n", " 1787.83,251.303 1786.01,251.392 1784.18,251.481 1782.36,251.57 1780.54,251.659 1778.72,251.747 1776.89,251.835 1775.07,251.924 1774.72,251.941 1773.25,252.012 \n", " 1771.43,252.099 1769.6,252.187 1767.78,252.275 1765.96,252.362 1764.14,252.449 1762.31,252.536 1760.49,252.623 1758.67,252.71 1756.85,252.797 1755.02,252.883 \n", " 1753.2,252.969 1751.38,253.055 1749.56,253.141 1747.73,253.227 1745.91,253.313 1745.88,253.314 1744.09,253.398 1742.27,253.484 1740.44,253.569 1738.62,253.654 \n", " 1736.8,253.739 1734.98,253.824 1733.15,253.908 1731.33,253.993 1729.51,254.077 1727.69,254.161 1725.86,254.245 1724.04,254.329 1722.22,254.412 1720.4,254.496 \n", " 1718.57,254.579 1716.75,254.662 1716.18,254.688 1714.93,254.745 1713.11,254.828 1711.28,254.911 1709.46,254.993 1707.64,255.076 1705.82,255.158 1703.99,255.24 \n", " 1702.17,255.322 1700.35,255.404 1698.53,255.485 1696.7,255.567 1694.88,255.648 1693.06,255.729 1691.23,255.81 1689.41,255.891 1687.59,255.972 1685.77,256.052 \n", " 1685.54,256.062 1683.94,256.133 1682.12,256.213 1680.3,256.293 1678.48,256.373 1676.65,256.452 1674.83,256.532 1673.01,256.612 1671.19,256.691 1669.36,256.77 \n", " 1667.54,256.849 1665.72,256.928 1663.9,257.006 1662.07,257.085 1660.25,257.163 1658.43,257.242 1656.61,257.32 1654.78,257.398 1653.89,257.436 1652.96,257.475 \n", " 1651.14,257.553 1649.32,257.631 1647.49,257.708 1645.67,257.785 1643.85,257.862 1642.03,257.939 1640.2,258.016 1638.38,258.092 1636.56,258.169 1634.74,258.245 \n", " 1632.91,258.321 1631.09,258.397 1629.27,258.473 1627.45,258.549 1625.62,258.625 1623.8,258.7 1621.98,258.775 1621.15,258.81 1620.16,258.851 1618.33,258.926 \n", " 1616.51,259 1614.69,259.075 1612.87,259.15 1611.04,259.224 1609.22,259.299 1607.4,259.373 1605.58,259.447 1603.75,259.521 1601.93,259.595 1600.11,259.669 \n", " 1598.29,259.742 1596.46,259.815 1594.64,259.889 1592.82,259.962 1590.99,260.035 1589.17,260.108 1587.35,260.181 1587.28,260.183 1585.53,260.253 1583.7,260.326 \n", " 1581.88,260.398 1580.06,260.471 1578.24,260.543 1576.41,260.615 1574.59,260.687 1572.77,260.759 1570.95,260.83 1569.12,260.902 1567.3,260.973 1565.48,261.045 \n", " 1563.66,261.116 1561.83,261.187 1560.01,261.258 1558.19,261.329 1556.37,261.4 1554.54,261.47 1552.72,261.541 1552.3,261.557 1550.9,261.611 1549.08,261.682 \n", " 1547.25,261.752 1545.43,261.822 1543.61,261.892 1541.79,261.962 1539.96,262.032 1538.14,262.102 1536.32,262.172 1534.5,262.241 1532.67,262.311 1530.85,262.38 \n", " 1529.03,262.45 1527.21,262.519 1525.38,262.588 1523.56,262.657 1521.74,262.726 1519.92,262.795 1518.09,262.864 1516.31,262.931 1516.27,262.932 1514.45,263.001 \n", " 1512.63,263.07 1510.8,263.138 1508.98,263.207 1507.16,263.275 1505.34,263.344 1503.51,263.412 1501.69,263.48 1499.87,263.548 1498.04,263.616 1496.22,263.684 \n", " 1494.4,263.752 1492.58,263.82 1490.75,263.888 1488.93,263.956 1487.11,264.024 1485.29,264.091 1483.46,264.159 1481.64,264.227 1479.82,264.294 1479.53,264.305 \n", " 1478,264.362 1476.17,264.429 1474.35,264.497 1472.53,264.564 1470.71,264.632 1468.88,264.699 1467.06,264.766 1465.24,264.834 1463.42,264.901 1461.59,264.968 \n", " 1459.77,265.036 1457.95,265.103 1456.13,265.17 1454.3,265.237 1452.48,265.305 1450.66,265.372 1448.84,265.439 1447.01,265.506 1445.19,265.573 1443.37,265.641 \n", " 1442.34,265.679 1441.55,265.708 1439.72,265.775 1437.9,265.842 1436.08,265.91 1434.26,265.977 1432.43,266.044 1430.61,266.112 1428.79,266.179 1426.97,266.246 \n", " 1425.14,266.314 1423.32,266.381 1421.5,266.449 1419.68,266.516 1417.85,266.584 1416.03,266.651 1414.21,266.719 1412.39,266.787 1410.56,266.855 1408.74,266.922 \n", " 1406.92,266.99 1405.25,267.052 1405.1,267.058 1403.27,267.126 1401.45,267.194 1399.63,267.262 1397.8,267.331 1395.98,267.399 1394.16,267.467 1392.34,267.536 \n", " 1390.51,267.604 1388.69,267.673 1386.87,267.741 1385.05,267.81 1383.22,267.879 1381.4,267.948 1379.58,268.017 1377.76,268.086 1375.93,268.155 1374.11,268.225 \n", " 1372.29,268.294 1370.47,268.364 1368.84,268.426 1368.64,268.434 1366.82,268.504 1365,268.574 1363.18,268.644 1361.35,268.714 1359.53,268.784 1357.71,268.855 \n", " 1355.89,268.926 1354.06,268.996 1352.24,269.067 1350.42,269.138 1348.6,269.21 1346.77,269.281 1344.95,269.353 1343.13,269.424 1341.31,269.496 1339.48,269.569 \n", " 1337.66,269.641 1335.84,269.713 1334.02,269.786 1333.66,269.8 1332.19,269.859 1330.37,269.932 1328.55,270.005 1326.73,270.078 1324.9,270.152 1323.08,270.226 \n", " 1321.26,270.3 1319.44,270.374 1317.61,270.448 1315.79,270.523 1313.97,270.598 1312.15,270.673 1310.32,270.748 1308.5,270.824 1306.68,270.9 1304.86,270.976 \n", " 1303.03,271.052 1301.21,271.128 1300.13,271.174 1299.39,271.205 1297.56,271.282 1295.74,271.359 1293.92,271.437 1292.1,271.514 1290.27,271.592 1288.45,271.671 \n", " 1286.63,271.749 1284.81,271.828 1282.98,271.907 1281.16,271.987 1279.34,272.066 1277.52,272.146 1275.69,272.227 1273.87,272.307 1272.05,272.388 1270.23,272.469 \n", " 1268.47,272.548 1268.4,272.551 1266.58,272.632 1264.76,272.714 1262.94,272.797 1261.11,272.879 1259.29,272.962 1257.47,273.046 1255.65,273.129 1253.82,273.213 \n", " 1252,273.298 1250.18,273.383 1248.36,273.468 1246.53,273.553 1244.71,273.639 1242.89,273.725 1241.07,273.811 1239.24,273.898 1238.75,273.922 1237.42,273.985 \n", " 1235.6,274.073 1233.78,274.161 1231.95,274.249 1230.13,274.337 1228.31,274.426 1226.49,274.516 1224.66,274.606 1222.84,274.696 1221.02,274.786 1219.2,274.877 \n", " 1217.37,274.969 1215.55,275.06 1213.73,275.153 1211.91,275.245 1210.92,275.295 1210.08,275.338 1208.26,275.431 1206.44,275.525 1204.62,275.619 1202.79,275.714 \n", " 1200.97,275.809 1199.15,275.905 1197.32,276 1195.5,276.097 1193.68,276.194 1191.86,276.291 1190.03,276.388 1188.21,276.487 1186.39,276.585 1184.84,276.669 \n", " 1184.57,276.684 1182.74,276.784 1180.92,276.883 1179.1,276.984 1177.28,277.085 1175.45,277.186 1173.63,277.288 1171.81,277.39 1169.99,277.493 1168.16,277.596 \n", " 1166.34,277.699 1164.52,277.804 1162.7,277.908 1160.87,278.013 1160.36,278.043 1159.05,278.119 1157.23,278.225 1155.41,278.331 1153.58,278.438 1151.76,278.546 \n", " 1149.94,278.654 1148.12,278.762 1146.29,278.871 1144.47,278.981 1142.65,279.091 1140.83,279.202 1139,279.313 1137.3,279.417 1137.18,279.424 1135.36,279.536 \n", " 1133.54,279.649 1131.71,279.762 1129.89,279.875 1128.07,279.99 1126.25,280.104 1124.42,280.219 1122.6,280.335 1120.78,280.451 1118.96,280.568 1117.13,280.686 \n", " 1115.51,280.791 1115.31,280.803 1113.49,280.922 1111.67,281.041 1109.84,281.16 1108.02,281.28 1106.2,281.4 1104.37,281.521 1102.55,281.643 1100.73,281.765 \n", " 1098.91,281.888 1097.08,282.011 1095.26,282.135 1094.83,282.164 1093.44,282.259 1091.62,282.384 1089.79,282.509 1087.97,282.635 1086.15,282.762 1084.33,282.889 \n", " 1082.5,283.017 1080.68,283.145 1078.86,283.274 1077.04,283.403 1075.21,283.533 1075.14,283.538 1073.39,283.663 1071.57,283.794 1069.75,283.926 1067.92,284.058 \n", " 1066.1,284.19 1064.28,284.323 1062.46,284.457 1060.63,284.591 1058.81,284.726 1056.99,284.862 1056.32,284.912 1055.17,284.998 1053.34,285.134 1051.52,285.271 \n", " 1049.7,285.409 1047.88,285.547 1046.05,285.686 1044.23,285.826 1042.41,285.965 1040.59,286.106 1038.76,286.247 1038.27,286.286 1036.94,286.389 1035.12,286.531 \n", " 1033.3,286.674 1031.47,286.817 1029.65,286.961 1027.83,287.105 1026.01,287.25 1024.18,287.396 1022.36,287.542 1020.9,287.66 1020.54,287.688 1018.72,287.836 \n", " 1016.89,287.983 1015.07,288.132 1013.25,288.281 1011.43,288.43 1009.6,288.58 1007.78,288.73 1005.96,288.882 1004.13,289.033 1004.13,289.033 1002.31,289.185 \n", " 1000.49,289.338 998.667,289.491 996.845,289.645 995.022,289.8 993.2,289.955 991.377,290.11 989.555,290.266 987.913,290.407 987.732,290.423 985.909,290.58 \n", " 984.087,290.737 982.264,290.896 980.442,291.054 978.619,291.213 976.797,291.373 974.974,291.534 973.152,291.694 972.175,291.781 971.329,291.856 969.507,292.018 \n", " 967.684,292.18 965.861,292.343 964.039,292.506 962.216,292.67 960.394,292.835 958.571,293 956.868,293.155 956.749,293.166 954.926,293.332 953.104,293.498 \n", " 951.281,293.665 949.459,293.833 947.636,294.001 945.813,294.169 943.991,294.338 942.168,294.508 941.948,294.529 940.346,294.678 938.523,294.849 936.701,295.02 \n", " 934.878,295.191 933.056,295.363 931.233,295.536 929.41,295.709 927.588,295.882 927.376,295.902 925.765,296.056 923.943,296.23 922.12,296.405 920.298,296.581 \n", " 918.475,296.756 916.653,296.933 914.83,297.11 913.116,297.276 913.008,297.287 911.185,297.464 909.362,297.642 907.54,297.821 905.717,298 903.895,298.179 \n", " 902.072,298.359 900.25,298.54 899.138,298.65 898.427,298.72 896.605,298.902 894.782,299.083 892.96,299.265 891.137,299.448 889.314,299.631 887.492,299.814 \n", " 885.669,299.998 885.412,300.024 883.847,300.182 882.024,300.366 880.202,300.551 878.379,300.737 876.557,300.922 874.734,301.109 872.912,301.295 871.914,301.398 \n", " 871.089,301.482 869.266,301.669 867.444,301.857 865.621,302.045 863.799,302.234 861.976,302.423 860.154,302.612 858.62,302.771 858.331,302.801 856.509,302.991 \n", " 854.686,303.182 852.864,303.372 851.041,303.563 849.218,303.755 847.396,303.946 845.573,304.138 845.509,304.145 843.751,304.331 841.928,304.524 840.106,304.717 \n", " 838.283,304.91 836.461,305.104 834.638,305.298 832.816,305.492 832.564,305.519 830.993,305.687 829.17,305.882 827.348,306.077 825.525,306.273 823.703,306.469 \n", " 821.88,306.665 820.058,306.861 819.767,306.893 818.235,307.058 816.413,307.255 814.59,307.453 812.768,307.65 810.945,307.848 809.122,308.046 807.3,308.245 \n", " 807.101,308.267 805.477,308.444 803.655,308.643 801.832,308.842 800.01,309.041 798.187,309.241 796.365,309.441 794.552,309.64 794.542,309.641 792.72,309.842 \n", " 790.897,310.043 789.074,310.244 787.252,310.445 785.429,310.646 783.607,310.848 782.107,311.014 781.784,311.05 779.962,311.252 778.139,311.454 776.317,311.657 \n", " 774.494,311.86 772.671,312.063 770.849,312.266 769.753,312.388 769.026,312.469 767.204,312.673 765.381,312.876 763.559,313.08 761.736,313.284 759.914,313.489 \n", " 758.091,313.693 757.478,313.762 756.269,313.898 754.446,314.102 752.623,314.307 750.801,314.512 748.978,314.718 747.156,314.923 745.333,315.129 745.272,315.136 \n", " 743.511,315.334 741.688,315.54 739.866,315.746 738.043,315.952 736.221,316.159 734.398,316.365 733.125,316.509 732.575,316.572 730.753,316.778 728.93,316.985 \n", " 727.108,317.192 725.285,317.399 723.463,317.606 721.64,317.814 721.028,317.883 719.818,318.021 717.995,318.228 716.173,318.436 714.35,318.644 712.527,318.851 \n", " 710.705,319.059 708.971,319.257 708.882,319.267 707.06,319.475 705.237,319.683 703.415,319.891 701.592,320.1 699.77,320.308 697.947,320.516 696.946,320.631 \n", " 696.125,320.725 694.302,320.933 692.479,321.142 690.657,321.35 688.834,321.559 687.012,321.768 685.189,321.977 684.945,322.005 683.367,322.185 681.544,322.394 \n", " 679.722,322.603 677.899,322.812 676.077,323.021 674.254,323.23 672.961,323.378 672.431,323.439 670.609,323.648 668.786,323.857 666.964,324.066 665.141,324.276 \n", " 663.319,324.485 661.496,324.694 660.988,324.752 659.674,324.903 657.851,325.112 656.029,325.321 654.206,325.531 652.383,325.74 650.561,325.949 649.018,326.126 \n", " 648.738,326.158 646.916,326.367 645.093,326.576 643.271,326.786 641.448,326.995 639.626,327.204 637.803,327.413 637.045,327.5 635.981,327.622 634.158,327.831 \n", " 632.335,328.04 630.513,328.249 628.69,328.458 626.868,328.667 625.064,328.874 625.045,328.876 623.223,329.085 621.4,329.293 619.578,329.502 617.755,329.711 \n", " 615.933,329.92 614.11,330.128 613.068,330.247 612.287,330.337 610.465,330.545 608.642,330.754 606.82,330.962 604.997,331.171 603.175,331.379 601.352,331.587 \n", " 601.054,331.621 599.53,331.795 597.707,332.003 595.884,332.211 594.062,332.419 592.239,332.627 590.417,332.835 589.014,332.995 588.594,333.043 586.772,333.251 \n", " 584.949,333.458 583.127,333.666 581.304,333.873 579.482,334.08 577.659,334.288 576.946,334.369 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Then consider\n", "$$\n", "\\kappa(z) = (z-1)^{-\\alpha} (z+1)^{\\alpha-1}\n", "$$\n", "which has weaker than pole singularities and satisfies $\\kappa(z) \\sim z^{-1}$.\n", "For $-1 < x < 1$ it has the right jump\n", "$$\n", "\\kappa_+(x) = (x-1)_+^\\alpha (x+1)^{1 - \\alpha} = \\E^{\\I \\pi \\alpha} (1-x)^\\alpha (x+1)^{1 - \\alpha} = \\E^{2\\I \\pi \\alpha} (x-1)_-^\\alpha (x+1)^{1 - \\alpha} = \\E^{2 \\I \\pi \\alpha} \\kappa_-(x)= \\E^{\\I \\theta} \\kappa_-(x)\n", "$$\n", "and for $x < -1$ it has the jump\n", "$$\n", "\\kappa_+(x) = (x-1)_+^\\alpha (x+1)_+^{1 - \\alpha} = \\E^{\\I \\pi \\alpha}\\E^{\\I \\pi (1-\\alpha)} (1-x)^\\alpha (-1-x)^{1 - \\alpha} = \\kappa_-(x)\n", "$$\n", "hence $\\kappa$ is analytic. \n", "\n", "We need to show this times a constant spans the entire space. Suppose we have another solution $\\tilde \\kappa$ and consider $r(z) = {\\tilde \\kappa(z) \\over \\kappa(z)}$. Note by construction that $\\kappa$ has no zeros. Then\n", "$$\n", "r_+(x) = {\\tilde\\kappa_+(x) \\over \\kappa_+(x)} = {\\tilde\\kappa_-(x) \\over \\kappa_-(x)} = r_-(x)\n", "$$\n", "hence $r$ is analytic on $(-1,1)$. It has weaker than pole singularities because $\\kappa(z)^{-1}$ is actually bounded at $\\pm 1$. Therefore $r$ is bounded and entire, and thus must be a constant $r(z) \\equiv r$, and thence $\\tilde \\kappa(z) = r \\kappa(z)$." ] }, { "cell_type": "code", "execution_count": 59, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.0 + 1.1102230246251565e-16im" ] }, "execution_count": 59, "metadata": {}, "output_type": "execute_result" } ], "source": [ "θ =2.3\n", "α = -θ/(2π)\n", "κ = z -> (z-1)^(-α)*(z+1)^(α-1)\n", "κ(0.1+0.0im) - exp(im*θ)*κ(0.1-0.0im)" ] }, { "cell_type": "code", "execution_count": 60, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.00982876598532333" ] }, "execution_count": 60, "metadata": {}, "output_type": "execute_result" } ], "source": [ "κ(100.0)" ] }, { "cell_type": "code", "execution_count": 62, "metadata": {}, "outputs": [ { "data": { "image/svg+xml": [ "<?xml version=\"1.0\" encoding=\"utf-8\"?>\n", "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"600\" height=\"400\" viewBox=\"0 0 2400 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"\n", "We want to mimic the solution of $\\phi_+(x) + \\phi_-(x)$. So take\n", "$$\n", "\\phi(z) = \\kappa(z) \\CC\\br[{f \\over \\kappa_+}](z) =\\E^{-\\I \\theta/2} (z-1)^{-\\alpha}(z+1)^{\\alpha-1} \\CC[f (1-x)^{\\alpha}(1+x)^{1-\\alpha}](z)\n", "$$\n", "This has the jump\n", "\\begin{align*}\n", "\\phi_+(x) - \\E^{\\I \\theta}\\phi_-(x) = \\kappa_+(z) \\CC_+\\br[{f \\over \\kappa_+}](x) - \\E^{\\I \\theta}\\kappa_-(z)\\CC_-\\br[{f \\over \\kappa_+}](x) = \\kappa_+(x) \\pr({\\CC_+\\br[{f \\over \\kappa_+}](x) - \\CC_-\\br[{f \\over \\kappa_+}](x) }) = f(x)\n", "\\end{align*}\n", "\n", "Thus the general solution is $\\phi(z) + C \\kappa(z)$." ] }, { "cell_type": "code", "execution_count": 63, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "8.881784197001252e-16 - 2.220446049250313e-16im" ] }, "execution_count": 63, "metadata": {}, "output_type": "execute_result" } ], "source": [ "θ =2.3\n", "α = -θ/(2π)\n", "κ = z -> (z-1)^(-α)*(z+1)^(α-1)\n", "\n", "\n", "x = Fun()\n", "κ₊ = exp(im*θ/2)*(1-x)^(-α)*(x+1)^(α-1)\n", "\n", "f = Fun(exp)\n", "\n", "z = 2+im\n", "φ = z -> κ(z)*cauchy(f/κ₊, z)\n", "\n", "φ(0.1+0.0im)-exp(im*θ)*φ(0.1-0.0im) - f(0.1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Problem 4.3\n", "\n", "Note for $x < 0$\n", "$$\n", "x_+^{\\I \\beta} = \\E^{\\I \\beta \\log_+ x} = \\E^{\\I \\beta \\log_- x - 2\\pi y} = \\E^{-2 \\pi \\beta} x_-^{\\I \\beta}\n", "$$" ] }, { "cell_type": "code", "execution_count": 64, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "-3.3881317890172014e-21 + 1.0842021724855044e-19im" ] }, "execution_count": 64, "metadata": {}, "output_type": "execute_result" } ], "source": [ "β = 2.3;\n", "x = -2.0\n", "(x+0.0im)^(im*β) - exp(-2π*β)*(x-0.0im)^(im*β)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We actually have bounded (oscillatory) growth near zero since \n", "$$\n", "|\\E^{\\I \\beta \\log z}| = |\\E^{\\I \\beta \\log |z|} \\E^{-\\beta \\arg z}| = \\E^{-\\beta \\arg z}\n", "$$" ] }, { "cell_type": "code", "execution_count": 67, 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"metadata": {}, "output_type": "execute_result" } ], "source": [ "xx = yy = -1:0.01:1\n", "contourf(xx, yy, abs.((xx' .+ im.*yy).^(im*β)))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Thus if we write $c = r \\E^{\\I \\theta}$ for $0 < \\theta < 2 \\pi$ and define $\\alpha = -{\\theta \\over 2 \\pi} + \\I{\\log r \\over 2 \\pi}$ we can write the solution to 4.1 as\n", "$$\n", "\\kappa(z) = (z-1)^{-\\alpha} (z+1)^{\\alpha-1}\n", "$$\n", "\n", "The same arguments as before then proceed. and the solution to 4.3 is \n", "$$\n", "\\phi(z) = \\kappa(z) \\CC\\br[{f \\over \\kappa_+}](z)+ C \\kappa(z)\n", "$$" ] }, { "cell_type": "code", "execution_count": 68, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.0 + 1.1102230246251565e-16im" ] }, "execution_count": 68, "metadata": {}, "output_type": "execute_result" } ], "source": [ "θ =2.3\n", "α = -θ/(2π)\n", "κ = z -> (z-1)^(-α)*(z+1)^(α-1)\n", "κ(0.1+0.0im) - exp(im*θ)*κ(0.1-0.0im)" ] }, { "cell_type": "code", "execution_count": 69, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "2.220446049250313e-16 + 2.220446049250313e-16im" ] }, "execution_count": 69, "metadata": {}, "output_type": "execute_result" } ], "source": [ "r = 2.4\n", "θ = 2.1\n", "c = r*exp(im*θ)\n", "\n", "α = -θ/(2π) + im*log(r)/(2π)\n", "\n", "κ = z -> (z-1)^(-α)*(z+1)^(α-1)\n", "\n", "\n", "κ(0.1+0.0im)-c*κ(0.1-0.0im)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Problem 5.1\n", "\n", "1. It is a product of functions analytic off $(-\\infty,1]$ hence is analytic off $(-\\infty,1]$, and we just have to check that it has no jump on $(-\\infty,-1)$ and $(-a,a)$. This follows via, for $x < -1$:\n", "$$\n", "\\kappa_+(x) = {1 \\over \\I^4 \\sqrt{1-x}\\sqrt{-1-x}\\sqrt{a-x}\\sqrt{-a-x}} = {1 \\over (-\\I)^4 \\sqrt{1-x}\\sqrt{-1-x}\\sqrt{a-x}\\sqrt{-a-x}} = \\kappa_-(x)\n", "$$\n", "and for $-a < x < a$ we have\n", "$$\n", "\\kappa_+(x) = {1 \\over \\I^2 \\sqrt{1-x}\\sqrt{x+1}\\sqrt{a-x}\\sqrt{x+a}} = {1 \\over (-\\I)^2 \\sqrt{1-x}\\sqrt{1+x}\\sqrt{a-x}\\sqrt{-a-x}} = \\kappa_-(x)\n", "$$\n", "2. This follows via the usual arguments: for $a < x < 1$ we have:\n", "$$\n", "\\kappa_+(x) = {1 \\over \\I \\sqrt{1-x}\\sqrt{x+1}\\sqrt{x-a}\\sqrt{x+a}} = -\\kappa_-(x)\n", "$$\n", "and for $-1 < x < -a$ we have\n", "$$\n", "\\kappa_+(x) = {1 \\over \\I^3 \\sqrt{1-x}\\sqrt{x+1}\\sqrt{x-a}\\sqrt{x+a}} = -\\kappa_-(x)\n", "$$\n", "3. This has at most square singularitie4s which are weaker than poles\n", "4. $\\kappa(z) = {1 \\over z^2 \\sqrt{1-1/z}\\sqrt{1-a/z} \\sqrt{1+a/z} \\sqrt{1+1/z}} \\sim {1 \\over z^2} \\rightarrow 0$. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Problem 5.2\n", "\n", "Ah, this is a trick question! Note that $z \\kappa(z) \\sim z^{-1} = O(z)$ and satisfies all the other properties. Thus consider any other solution $\\tilde \\kappa(z)$ and write\n", "$$\n", "r(z) = {\\tilde \\kappa(z) \\over \\kappa(z) }\n", "$$\n", "This has trivial jumps and hence is entire: for example, on $(a,1)$ we have\n", "$$\n", "r_+(x) = {\\tilde \\kappa_+(x) \\over \\kappa_+(x) } = {-\\tilde \\kappa_-(x) \\over -\\kappa_-(x) } = r_-(x)\n", "$$\n", "But since $\\kappa \\sim O(z^{-2})$ we only know that $\\kappa$ has at most $O(z)$ growth, hence it can be any first degree polynomial. Therefore, the space of all solutions is in fact two-dimensional: $\\psi(z) = (A + Bz) \\kappa(z)$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Problem 5.3\n", "\n", "Here we mimick the usual solution techniques and propose:\n", "$$\n", "\\phi(z) = \\kappa(z) \\CC\\br[{f \\over \\kappa_+}](z) + (A+Bz) \\kappa(z)\n", "$$\n", "A quick check confirms it has the right jumps:\n", "$$\n", "\\phi_+(x) = \\kappa_+(x) \\CC_+\\br[{f \\over \\kappa_+}](x) + (A+Bx) \\kappa_+(x) =\\kappa_+(x) \\left({f(x) \\over \\kappa_+(x)} + \\CC_-\\br[{f \\over \\kappa_+}](x)\\right) - (A+Bx) \\kappa_-(x) = -\\phi_-(x) + f(x)\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "## Problem 6.1\n", "\n", "Let's first do a plot and histogram. Here we see the right scaling is $N^{1/4}$:" ] }, { "cell_type": "code", "execution_count": 71, "metadata": {}, "outputs": [], "source": [ "V = x -> x^4\n", "Vp = x -> 4x^3\n", "\n", "N = 100\n", "λ⁰ = randn(N) # initial location\n", "prob = ODEProblem((λ,t,_) -> Float64[sum(1 ./(λ[k] .- λ[[1:k-1;k+1:end]])) - Vp(λ[k]) for k=1:N], λ⁰, (0.0, 10.0))\n", "λ = solve(prob; reltol=1E-6);" ] }, { "cell_type": "code", "execution_count": 73, "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "┌ Info: Saved animation to \n", "│ fn = /Users/sheehanolver/Documents/Teaching/Imperial/M3M6 Methods of Mathematical Physics/2018/tmp.gif\n", "└ @ Plots /Users/sheehanolver/.julia/packages/Plots/rmogG/src/animation.jl:90\n" ] }, { "data": { "text/html": [ "<img src=\"tmp.gif?0.021308146046194576>\" />" ], "text/plain": [ "Plots.AnimatedGif(\"/Users/sheehanolver/Documents/Teaching/Imperial/M3M6 Methods of Mathematical Physics/2018/tmp.gif\")" ] }, "execution_count": 73, "metadata": {}, "output_type": "execute_result" } ], "source": [ "@gif for t=0.0:0.05:7.0\n", " scatter(λ(t)/N^(1/4) ,zeros(N); label=\"charges\", xlims=(-2,2), title=\"t = $t\")\n", "end" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The limiting distribution has the following form:" ] }, { "cell_type": "code", "execution_count": 74, "metadata": {}, "outputs": [ { "data": { "image/svg+xml": [ "<?xml version=\"1.0\" encoding=\"utf-8\"?>\n", "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"600\" height=\"400\" viewBox=\"0 0 2400 1600\">\n", "<defs>\n", " <clipPath id=\"clip0600\">\n", " <rect x=\"0\" y=\"0\" width=\"2000\" height=\"2000\"/>\n", " </clipPath>\n", "</defs>\n", "<defs>\n", " <clipPath id=\"clip0601\">\n", " <rect x=\"0\" y=\"0\" width=\"2400\" height=\"1600\"/>\n", " </clipPath>\n", "</defs>\n", "<polygon clip-path=\"url(#clip0601)\" points=\"\n", "0,1600 2400,1600 2400,0 0,0 \n", " \" fill=\"#ffffff\" 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\\lambda_k \\over \\D t} = \\sum_{j=1 \\atop j \\neq k}^N {1 \\over \\lambda_k -\\lambda_j} - 4\\lambda_k^3\n", "$$\n", "Rescale via $\\mu_k = {\\lambda_k \\over N^{1/4}}$ gives\n", "$$\n", "0 = N^{1/4} {\\D \\mu_k \\over \\D t} = N^{-1/4} \\sum_{j=1 \\atop j \\neq k}^N {1 \\over \\mu_k -\\mu_j} - 4 N^{3/4} \\mu_k^3\n", "$$\n", "or in other words\n", "$$\n", "0 = N^{-1/2} {\\D \\mu_k \\over \\D t} = {1 \\over N} \\sum_{j=1 \\atop j \\neq k}^N {1 \\over \\mu_k -\\mu_j} - 4 \\mu_k^3\n", "$$\n", "We can now formally let $N\\rightarrow \\infty$ to get our equation\n", "$$\n", "\\dashint_{-b}^b {w(t) \\over x-t} \\dt = 4 x^3\n", "$$\n", "where I've used symmetry to assume that the interval is symmetric. \n", "We want to find $w$ and $b$ so that this equations holds true and $w$ is a bounded probability density:\n", "1. $w(x) >0$ for $-b < x < b$\n", "2. $\\int w(x) \\dx = 1$\n", "3. $w$ is bounded\n", "\n", "\n", "Our equation is equivalent to\n", "$$\n", "\\HH_{[-b,b]} w(x) = -{4 x^3 \\over \\pi}\n", "$$\n", "recall the inversion formula\n", "$$\n", " u(x) = {-1 \\over \\sqrt{b^2 - x^2}}\\HH \\left[{ f(\\diamond) \\sqrt{b^2-\\diamond^2} }\\right](x) - {C \\over \\sqrt{b^2-x^2}}\n", "$$ \n", "In our case $f(x) = -{4 x^3 \\over \\pi}$ and we use\n", "$$\n", "\\sqrt{z-b}\\sqrt{z+b} = z\\sqrt{1-b/z}\\sqrt{1+b/z} = z - {b^2 \\over 2 z} -{b^4 \\over 8z^3} + O(z^{-4})\n", "$$\n", "to determine\n", "$$\n", "2 \\I \\CC \\left[{ \\diamond^3 \\sqrt{b^2-\\diamond^2} }\\right](x) = z^3(\\sqrt{z-b}\\sqrt{z+b} -z + {b^2 \\over 2 z} + {b^4 \\over 8z^3}) = z^3\\sqrt{z-b}\\sqrt{z+b} -z^4 + {b^2z^2 \\over 2} + {b^4 \\over 8}\n", "$$" ] }, { "cell_type": "code", "execution_count": 75, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "71.6687198577313 + 40.090510492433154im" ] }, "execution_count": 75, "metadata": {}, "output_type": "execute_result" } ], "source": [ "b = 5\n", "x = Fun(-b .. b)\n", "(2im)cauchy(x^3*sqrt(b^2-x^2),z)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Therefore,\n", "\\begin{align*}\n", "u(x) = {4\\I \\over \\pi \\sqrt{b^2 - x^2}}(\\CC^+ + \\CC^-) \\left[{ \\diamond^3 \\sqrt{b^2-\\diamond^2} }\\right](x) - {C \\over \\sqrt{b^2-x^2}} \\\\\n", "{4 \\over \\pi \\sqrt{b^2 - x^2}}(-x^4+{b^2 x^2 \\over 2} + {b^4 \\over 8})- {C \\over \\sqrt{b^2-x^2}} \n", "\\end{align*}\n", "We choose $C$ so this is bounded, in particular, we get get the solution\n", "$$\n", "u(x) = {4 \\over \\pi \\sqrt{b^2 - x^2}}(-x^4+{b^2 x^2 \\over 2} + {b^4 \\over 2})\n", "$$" ] }, { "cell_type": "code", "execution_count": 76, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "-0.0012732395447351292" ] }, "execution_count": 76, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u = 4/(π*sqrt(b^2-x^2))*(-x^4 + b^2*x^2/2 + b^4/2)\n", "hilbert(u, 0.1)" ] }, { "cell_type": "code", "execution_count": 77, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "-0.001273239544735163" ] }, "execution_count": 77, "metadata": {}, "output_type": "execute_result" } ], "source": [ "-4*0.1^3/π" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "At least it looks right, we just need to get the right b:" ] }, { "cell_type": "code", "execution_count": 78, "metadata": {}, "outputs": [ { "data": { "image/svg+xml": [ "<?xml version=\"1.0\" encoding=\"utf-8\"?>\n", "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"600\" height=\"400\" viewBox=\"0 0 2400 1600\">\n", "<defs>\n", " <clipPath id=\"clip0800\">\n", " <rect x=\"0\" y=\"0\" width=\"2000\" height=\"2000\"/>\n", " </clipPath>\n", "</defs>\n", "<defs>\n", " <clipPath id=\"clip0801\">\n", " <rect x=\"0\" y=\"0\" width=\"2400\" height=\"1600\"/>\n", " </clipPath>\n", "</defs>\n", "<polygon clip-path=\"url(#clip0801)\" points=\"\n", "0,1600 2400,1600 2400,0 0,0 \n", " \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n", "<defs>\n", " <clipPath id=\"clip0802\">\n", " <rect x=\"480\" y=\"0\" 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362.873,278.448 340.127,353.126 319.501,439.941 301.044,538.326 284.798,647.498 270.803,766.474 \n", " 259.089,894.083 249.686,1028.99 242.613,1169.73 237.889,1314.7 235.525,1462.26 \n", " \"/>\n", "<polygon clip-path=\"url(#clip0801)\" points=\"\n", "1900.86,251.724 2249.26,251.724 2249.26,130.764 1900.86,130.764 \n", " \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n", "<polyline clip-path=\"url(#clip0801)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n", " 1900.86,251.724 2249.26,251.724 2249.26,130.764 1900.86,130.764 1900.86,251.724 \n", " \"/>\n", "<polyline clip-path=\"url(#clip0801)\" style=\"stroke:#009af9; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n", " 1924.86,191.244 2068.86,191.244 \n", " \"/>\n", "<g clip-path=\"url(#clip0801)\">\n", "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 2092.86, 208.744)\" x=\"2092.86\" y=\"208.744\">b = 5</text>\n", "</g>\n", "</svg>\n" ] }, "execution_count": 78, "metadata": {}, "output_type": "execute_result" } ], "source": [ "plot(u; label = \"b = $b\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We want to choose $b$ now so that this integrates to $1$. For example, this choice of $b$ is horrible:" ] }, { "cell_type": "code", "execution_count": 79, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "937.5" ] }, "execution_count": 79, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sum(u)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "There's a nice trick: If $-2\\pi \\I \\CC u(z) \\sim {1 \\over z}$ then $\\int_{-b}^b u(x) \\dx = 1$. We know since\n", "$$\n", "{1 \\over \\sqrt{1+z}} = 1 - {z \\over 2} + {3z^2 \\over 8} - {5z^3 \\over 16} + O(z^4)\n", "$$\n", "\\begin{align*}\n", " {-z^4 + {b^2z^2 \\over 2} + {b^4 \\over 2} \\over \\sqrt{z-b}\\sqrt{z+b}} &= {-z^3 + {b^2z \\over 2} + {b^4 \\over 2 z} \\over \\sqrt{1-b/z}\\sqrt{1+b/z}} \\\\\n", " &= (-z^3 + {b^2z \\over 2} )(1 + {b \\over 2 z} + {3 b^2 \\over 8 z^2} + {5 b^3 \\over 16z^3} )(1 - {b \\over 2 z} + {3 b^2 \\over 8 z^2} - {5 b^3 \\over 16z^3}) + O(z^{-1}) \\\\\n", " &=-z^3 + \\pr({b^2 \\over 2} +{b^2 \\over 4} + {b^2 \\over 2} - {3 b^2 \\over 8} -{3 b^2 \\over 8} ) z +O(z^{-1})\\\\\n", " &= -z^3 + O(z^{-1})\n", " \\end{align*}\n", "Thus we know\n", "$$\n", "\\CC u(z) = {2\\I \\over \\pi} z^3 +{2 \\I \\over \\pi} {-z^4 + {b^2z^2 \\over 2} + {b^4 \\over 2} \\over \\sqrt{z-b}\\sqrt{z+b}}\n", "$$" ] }, { "cell_type": "code", "execution_count": 80, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "40.55106915762885 + 9.075717966750503im" ] }, "execution_count": 80, "metadata": {}, "output_type": "execute_result" } ], "source": [ "cauchy(u, z)" ] }, { "cell_type": "code", "execution_count": 81, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "1.4908359390683472 - 0.0im" ] }, "execution_count": 81, "metadata": {}, "output_type": "execute_result" } ], "source": [ "z =100.0im;\n", "2im/π*z^3 + 2im/π*(-z^4 + b^2*z^2/2 +b^4/2)/(sqrt(z-b)sqrt(z+b))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "taking this one term further we find \n", "$$\n", " {-z^4 + {b^2z^2 \\over 2} + {b^4 \\over 2} \\over \\sqrt{z-b}\\sqrt{z+b}} = -z^3 +{3 b^4 \\over 8 z} + O(z^{-2})\n", " $$\n", "Hence we want to choose $b$ so that\n", "$$\n", "-2\\I\\pi \\CC u(z) = {3 b^4 \\over 2 z} \\sim {1 \\over z}\n", "$$\n", "in other words, $b = ({2 \\over 3})^{1/4}$" ] }, { "cell_type": "code", "execution_count": 82, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.9999999999999997" ] }, "execution_count": 82, "metadata": {}, "output_type": "execute_result" } ], "source": [ "b = (2/3)^(1/4)\n", "x = Fun(-b .. b)\n", "u = 4/(π*sqrt(b^2-x^2))*(-x^4 + b^2*x^2/2 + b^4/2)\n", "sum(u)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "And it worked!" ] }, { "cell_type": "code", "execution_count": 83, "metadata": {}, "outputs": [ { "data": { "image/svg+xml": [ "<?xml version=\"1.0\" encoding=\"utf-8\"?>\n", "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"600\" height=\"400\" viewBox=\"0 0 2400 1600\">\n", "<defs>\n", " <clipPath id=\"clip1000\">\n", " <rect x=\"0\" y=\"0\" width=\"2000\" height=\"2000\"/>\n", " </clipPath>\n", "</defs>\n", "<defs>\n", " <clipPath id=\"clip1001\">\n", " <rect x=\"0\" y=\"0\" width=\"2400\" height=\"1600\"/>\n", " </clipPath>\n", "</defs>\n", "<polygon clip-path=\"url(#clip1001)\" points=\"\n", "0,1600 2400,1600 2400,0 0,0 \n", " \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n", "<defs>\n", " <clipPath id=\"clip1002\">\n", " <rect x=\"480\" y=\"0\" width=\"1681\" height=\"1600\"/>\n", " </clipPath>\n", "</defs>\n", "<polygon clip-path=\"url(#clip1001)\" points=\"\n", "161.394,1503.47 2321.26,1503.47 2321.26,47.2441 161.394,47.2441 \n", " \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n", "<defs>\n", " <clipPath id=\"clip1003\">\n", " <rect x=\"161\" y=\"47\" width=\"2161\" height=\"1457\"/>\n", " </clipPath>\n", "</defs>\n", "<polyline clip-path=\"url(#clip1003)\" style=\"stroke:#000000; 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