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02-1 Boundary value problem (linear ODE case).ipynb
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02-1 Boundary value problems (linear ODE case).ipynb
{
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{
"metadata": {},
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"source": "# 境界値問題 (線形常微分方程式の場合)\n\n* Author: 黒木玄\n* Date: 2019-04-15\n* Repository: https://github.com/genkuroki/DifferentialEquations\n$\n\\newcommand\\Z{{\\mathbb Z}}\n\\newcommand\\R{{\\mathbb R}}\n\\newcommand\\C{{\\mathbb C}}\n\\newcommand\\eps{\\varepsilon}\n\\newcommand\\QED{\\text{□}}\n$\n\nこのファイルは [nbviewer](https://nbviewer.jupyter.org/github/genkuroki/DifferentialEquations/blob/master/02-1+Boundary+value+problems+(linear+ODE+case).ipynb) でも閲覧できる.\n\n[Julia言語](https://julialang.org/) と [Jupyter環境](https://jupyter.org/) の簡単な解説については次を参照せよ:\n\n* [JuliaとJupyterのすすめ](https://nbviewer.jupyter.org/github/genkuroki/msfd28/blob/master/msfd28genkuroki.ipynb?flush_cached=true)\n\n[Julia言語](https://julialang.org/) 環境の整備の仕方については次を参照せよ:\n\n* [Julia v1.1.0 の Windows 8.1 へのインストール](https://nbviewer.jupyter.org/github/genkuroki/msfd28/blob/master/install.ipynb)"
},
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"cell_type": "markdown",
"source": "<h1>目次<span class=\"tocSkip\"></span></h1>\n<div class=\"toc\"><ul class=\"toc-item\"><li><span><a href=\"#微分方程式の分類\" data-toc-modified-id=\"微分方程式の分類-1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span>微分方程式の分類</a></span></li><li><span><a href=\"#初期条件や境界条件\" data-toc-modified-id=\"初期条件や境界条件-2\"><span class=\"toc-item-num\">2&nbsp;&nbsp;</span>初期条件や境界条件</a></span><ul class=\"toc-item\"><li><span><a href=\"#調和振動子\" data-toc-modified-id=\"調和振動子-2.1\"><span class=\"toc-item-num\">2.1&nbsp;&nbsp;</span>調和振動子</a></span></li><li><span><a href=\"#調和振動子の初期値問題\" data-toc-modified-id=\"調和振動子の初期値問題-2.2\"><span class=\"toc-item-num\">2.2&nbsp;&nbsp;</span>調和振動子の初期値問題</a></span></li><li><span><a href=\"#調和振動子のDirichlet型境界値問題\" data-toc-modified-id=\"調和振動子のDirichlet型境界値問題-2.3\"><span class=\"toc-item-num\">2.3&nbsp;&nbsp;</span>調和振動子のDirichlet型境界値問題</a></span></li><li><span><a href=\"#調和振動子のNeumann型境界値問題\" data-toc-modified-id=\"調和振動子のNeumann型境界値問題-2.4\"><span class=\"toc-item-num\">2.4&nbsp;&nbsp;</span>調和振動子のNeumann型境界値問題</a></span></li></ul></li><li><span><a href=\"#BVProblemの使い方\" data-toc-modified-id=\"BVProblemの使い方-3\"><span class=\"toc-item-num\">3&nbsp;&nbsp;</span>BVProblemの使い方</a></span><ul class=\"toc-item\"><li><span><a href=\"#境界条件1\" data-toc-modified-id=\"境界条件1-3.1\"><span class=\"toc-item-num\">3.1&nbsp;&nbsp;</span>境界条件1</a></span></li><li><span><a href=\"#境界条件1-(別の方法)\" data-toc-modified-id=\"境界条件1-(別の方法)-3.2\"><span class=\"toc-item-num\">3.2&nbsp;&nbsp;</span>境界条件1 (別の方法)</a></span></li><li><span><a href=\"#境界条件2\" data-toc-modified-id=\"境界条件2-3.3\"><span class=\"toc-item-num\">3.3&nbsp;&nbsp;</span>境界条件2</a></span></li></ul></li><li><span><a href=\"#調和振動子の数値解\" data-toc-modified-id=\"調和振動子の数値解-4\"><span class=\"toc-item-num\">4&nbsp;&nbsp;</span>調和振動子の数値解</a></span><ul class=\"toc-item\"><li><span><a href=\"#初期値問題\" data-toc-modified-id=\"初期値問題-4.1\"><span class=\"toc-item-num\">4.1&nbsp;&nbsp;</span>初期値問題</a></span></li><li><span><a href=\"#斉次Dirichlet境界条件\" data-toc-modified-id=\"斉次Dirichlet境界条件-4.2\"><span class=\"toc-item-num\">4.2&nbsp;&nbsp;</span>斉次Dirichlet境界条件</a></span></li><li><span><a href=\"#斉次Neumann境界条件\" data-toc-modified-id=\"斉次Neumann境界条件-4.3\"><span class=\"toc-item-num\">4.3&nbsp;&nbsp;</span>斉次Neumann境界条件</a></span></li><li><span><a href=\"#非斉次Dirichlet境界条件\" data-toc-modified-id=\"非斉次Dirichlet境界条件-4.4\"><span class=\"toc-item-num\">4.4&nbsp;&nbsp;</span>非斉次Dirichlet境界条件</a></span></li><li><span><a href=\"#非斉次Neumann境界条件\" data-toc-modified-id=\"非斉次Neumann境界条件-4.5\"><span class=\"toc-item-num\">4.5&nbsp;&nbsp;</span>非斉次Neumann境界条件</a></span></li></ul></li></ul></div>"
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"cell_type": "code",
"source": "using DifferentialEquations\nusing Plots\ndefault(:bglegend, plot_color(default(:bg), 0.7))\ndefault(:fglegend, plot_color(ifelse(isdark(plot_color(default(:bg))), :white, :black), 0.2))\n\ngr()\nENV[\"PLOTS_TEST\"] = \"true\";",
"execution_count": 1,
"outputs": []
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{
"metadata": {},
"cell_type": "markdown",
"source": "## 微分方程式の分類\n\n偏微分が出て来ない微分方程式を **常微分方程式 (ordinary differential equation)** と呼ぶ.\n\n**例.** 調和振動子: $\\omega > 0$ に対する微分方程式\n\n$$\n\\ddot u = -\\omega^2 u\n$$\n\nは常微分方程式である. $\\QED$\n\n**例.** 単振子: $\\omega^2$ に対する微分方程式\n\n$$\n\\ddot u = -\\omega^2\\sin u\n$$\n\nも常微分方程式である. $\\QED$\n\n偏微分が出て来る微分方程式を **偏微分方程式 (partial differential equation)** と呼ぶ.\n\n**例.** 熱方程式: 偏微分方程式\n\n$$\nu_t = u_{xx}\n$$\n\nを空間次元1次元の **熱方程式 (heat equation)** と呼ぶ. $\\QED$\n\n**例.** 熱方程式: $c>0$ に対して, 偏微分方程式\n\n$$\n\\frac{1}{c^2}u_{tt} = u_{xx}\n$$\n\nを空間次元1次元の **波動方程式 (heat equation)** と呼ぶ. $\\QED$\n\n**例.** Korteweg--de Vries 方程式: 偏微分方程式\n\n$$\nu_t + u_{xxx} + 6uu_x = 0 \n$$\n\nを **KdV方程式** と呼ぶ. $\\QED$\n\nさらに, 微分方程式には線形と非線形の区別もあって, 調和振動子と熱方程式や波動方程式は線形であり, 単振子やKdV方程式は非線形の微分方程式である.\n\n一般に unknown function $u$ に関する線形作用素 $P$ によって $Pu=0$ と書ける微分方程式は **線形微分方程式 (linear differential equation)** もしくは **斉次線形微分方程式 (せいじ, homogeneous linear differential equation)** と呼ぶ. さらに与えられた函数 $f$ のデータが追加されて, $Pu=f$ と書ける微分方程式を **非斉次線形微分方程式 (ひせいじ, inhomogeneous differential equation)** と呼ぶ\n\n例えば, $P=(d/dt)^2+\\omega^2$ のとき, $Pu = \\ddot u + \\omega^2 u$ なので, $Pu=0$ は調和振動子の微分方程式になる. 調和振動子の微分方程式は斉次線形常微分方程式である.\n\n例えば, $P=(\\frac{1}{c}\\partial/\\partial t)^2 - (\\partial/\\partial x)^2$ のとき $Pu = c^{-2} u_{tt} - u_{xx}$ なので $Pu=0$ は波動方程式になる. 波動方程式は斉次線形偏微分方程式である.\n\n非斉次な線形微分方程式の典型例は外部から強制力が与えられている場合に得られる微分方程式である. 調和振動子にが与えられているとき, 微分方程式は\n\n$$\n\\ddot u = -\\omega^2 u + f(t)\n$$\n\nの形になる. $f(t)$ が外力を表す. この微分方程式は $P=(d/dt)^2+\\omega^2$ とおくとき, $Pu=f$ と書ける. 外力が与えられた調和振動子は非斉次な線形常微分方程式である.\n\n微分方程式が時間発展を意味するとき, 微分方程式中に時間変数 $t$ が明示的に含まれているとき, その微分方程式は **非自励的 (non-autonomous)** と呼ばれる. 微分方程式中に時間変数 $t$ が明示的に含まれないとき, その微分方程式は **自励的 (autonomous)** であるという. 自励的な微分方程式で記述される系は **自励系 (autonomous system)** と呼ばれ, 非自励的な微分方程式で記述される系は **非自励系 (non-autonomous system)** と呼ばれる.\n\n**例.** Painlevé I and II: 非自励的な常微分方程式\n\n$$\n\\frac{du^2}{dt^2} = 6y^2 + t\n$$\n\nは **Painlevé I 方程式 (the first Painlevé equation)** と呼ばれ, \n非自励的な常微分方程式\n\n$$\n\\frac{du^2}{dt^2} = 2y^3 + ty + \\alpha\n$$\n\nは **Painlevé II 方程式 (the second Painlevé equation)** と呼ばる. 他にも Painlevé III～VI 方程式があり, どれも2階の非自励的非線形常微分方程式である. $\\QED$\n\n微分方程式の階数は微分方程式が含む最高次の導函数の階数のことである.\n\n**例.** 楕円函数の満たす微分方程式: \n\n$$\n\\left(\\frac{dy}{dt}\\right)^2 = 4y^3 + 2ay + b\n$$\n\nの形の自励的な非線形常微分方程式の解は **楕円函数** と呼ばれるクラスの函数になることが知られている. この微分方程式の解として楕円函数を定義することもできる. この微分方程式の両辺を微分すると次の微分方程式が得られる:\n\n$$\n2\\frac{dy}{dt}\\frac{dy^2}{dt^2} = 12 y^2\\frac{dy}{dt} + 2a\\frac{dy}{dt}.\n$$\n\n両辺を $2dy/dt$ で割ると\n\n$$\n\\frac{dy^2}{dt^2} = 6y^2 + a.\n$$\n\n上の Painlevé I 方程式はこの楕円函数が満たす微分方程式における定数 $a$ を時間変数 $t$ に置き換えた形をしている. $\\QED$\n\n**おまけ:** 次のノートブックでは熱方程式, KdV方程式, Schrödinger方程式, Smith方程式を数値的に解いている:\n\n* [FFTを用いた偏微分方程式の数値解法(in-place版)](https://nbviewer.jupyter.org/github/genkuroki/DifferentialEquations/blob/master/A01+Solving+heat%2C+KdV%2C+Schroedinger%2C+and+Smith+eqations+by+in-place+FFT.ipynb)\n\n非常に沢山の微分方程式があって, 数値的に解くための様々な方法が存在する. それらをマスターするには自分で試行錯誤してみるのが一番早い. 他人の話を聴くのではなく, 自分でやってみることが重要である."
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"source": "## 初期条件や境界条件\n\n微分方程式を解くときには, 与えられた微分方程式を満たすすべての解を求めるのではなく, 目的に応じて適切な条件を満たす解を求めることが多い.\n\n「目的に応じた適切な条件」として現われる条件の典型例が **初期条件** や **境界条件** である.\n\nそれらについて例を用いて説明したい."
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"cell_type": "markdown",
"source": "### 調和振動子\n\n$\\omega>0$ であると仮定する. 次の微分方程式を調和振動子の微分方程式と呼ぶ:\n\n$$\n\\ddot x = -\\omega^2 x.\n$$\n\nこの微分方程式の任意の解は定数 $A$, $B$ によって次のように表わされる:\n\n$$\nx(t) = A\\cos\\omega t + B\\sin\\omega t.\n$$\n\nこの式の右辺が調和振動子の微分方程式の解になっていることは時間変数 $t$ で2回微分してみればすぐにわかる. 任意の解がこのように表わされることは2階の線形常微分方程式の解空間(解全体のなすベクトル空間)が2次元になるという事実からわかる."
},
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"metadata": {},
"cell_type": "markdown",
"source": "### 調和振動子の初期値問題\n\n任意に与えられた定数 $a,b$ に対して, 調和振動子 $\\ddot x = -\\omega^2 x$ の解で初期条件\n\n$$\nx(0) = a, \\quad \\dot x(0) = b\\omega\n$$\n\nを満たすものが唯一つ存在することを示そう. この初期条件に一般解\n\n$$\nx(t) = A\\cos\\omega t + B\\sin\\omega t\n$$\n\nを代入することによって, \n\n$$\nA = a, B = b\n$$\n\nを得る. これで\n\n$$\nx(t) = a\\cos\\omega t + b\\sin\\omega t\n$$\n\nが上の初期条件を満たす唯一つの解になる."
},
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"source": "### 調和振動子のDirichlet型境界値問題\n\n$T>0$ と与えられた定数 $a,b$ に対して, 調和振動子 $\\ddot x=-\\omega^2 x$ の解で境界条件\n\n$$\nx(0) = a, \\quad x(T) = b\n$$\n\nを満たすものをすべて求めてみよう. これに一般解 $x(t) = A\\cos\\omega t + B\\sin\\omega t$ を代入すると\n\n$$\nA = a, \\quad A\\cos\\omega T + B\\sin\\omega T = b.\n$$\n\n$A$ は $A=a$ によって一意的に決まるが, $B$ については $\\sin\\omega T$ が $0$ であるか否かによって場合分けが必要になる.\n\n$\\sin\\omega T\\ne 0$ の場合. そのとき, 任意の $a,b$ の組み合わせに対して, $A,B$ が次のように一意的に決まる:\n\n$$\nA = a, \\quad B = \\frac{b - a\\cos\\omega T}{\\sin\\omega T}.\n$$\n\nすなわち,\n\n$$\nx(t) = a\\cos\\omega t + \\frac{b - a\\cos\\omega T}{\\sin\\omega T}\\sin\\omega t\n$$\n\nが上の境界条件を満たす調和振動子の微分方程式の唯一つの解になる.\n\n$\\sin\\omega T = 0$ の場合. そのとき $\\omega T = m\\pi$, $m\\in \\Z_{>0}$ と書け, $\\cos\\omega T = (-1)^m$ となる. 上の境界条件を満たす解が存在するための必要十分条件は\n\n$$\nb=(-1)^m a\n$$\n\nとなる. そのとき, \n\n$$\nx(t)=a\\cos\\omega t + B\\sin\\omega t = x(0)\\cos\\omega t + B\\sin\\omega t\n$$\n\nは上の境界条件を満たす調和振動子の微分方程式の解になる($B$ はなんでもよい)."
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"cell_type": "markdown",
"source": "### 調和振動子のNeumann型境界値問題\n\n$T>0$ と与えられた定数 $a,b$ に対して, 調和振動子 $\\ddot x=-\\omega^2 x$ の解で境界条件\n\n$$\n\\dot x(0) = a\\omega, \\quad \\dot x(T) = b\\omega\n$$\n\nを満たすものをすべて求めてみよう. これに一般解 $x(t) = A\\cos\\omega t + B\\sin\\omega t$ から得られる\n\n$$\n\\dot x(t) = \\omega(-A\\sin\\omega t + B\\cos\\omega t)\n$$\n\nを代入すると\n\n$$\nB = a, \\quad -A\\sin\\omega T + B\\cos\\omega T = b.\n$$\n\n$B$ は $B=a$ によって一意的に決まるが, $A$ については $\\sin\\omega T$ が $0$ であるか否かによって場合分けが必要になる.\n\n$\\sin\\omega T\\ne 0$ の場合. そのとき, 任意の $a,b$ の組み合わせに対して, $A,B$ が次のように一意的に決まる:\n\n$$\nB = a, \\quad A = \\frac{a\\cos\\omega T - b}{\\sin\\omega T}.\n$$\n\nすなわち,\n\n$$\nx(t) = \\frac{a\\cos\\omega T - b}{\\sin\\omega T}\\cos\\omega t + a\\sin\\omega t\n$$\n\nが上の境界条件を満たす調和振動子の微分方程式の唯一つの解になる.\n\n$\\sin\\omega T = 0$ の場合. そのとき $\\omega T = m\\pi$, $m\\in \\Z_{>0}$ と書け, $\\cos\\omega T = (-1)^m$ となる. 上の境界条件を満たす解が存在するための必要十分条件は\n\n$$\nb=(-1)^m a\n$$\n\nとなる. そのとき, \n\n$$\nx(t)=A\\cos\\omega t + a\\sin\\omega t = A\\cos\\omega t + \\dot{x}(0)\\sin\\omega t\n$$\n\nは上の境界条件を満たす調和振動子の微分方程式の解になる($A$ はなんでもよい)."
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"source": "## BVProblemの使い方\n\nhttps://docs.juliadiffeq.org/latest/tutorials/bvp_example.html\n\n単振子:\n$$\n\\ddot\\theta = -\\frac{g}{\\ell}\\sin\\theta.\n$$"
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"source": "### θ = u[1]\n### dθ/dt = u[2]\n\nconst g = 9.81\nconst ℓ = 1.0\ntspan = (0.0,pi/2)\nfunction simplependulum!(du,u,p,t)\n θ = u[1]\n dθ = u[2]\n du[1] = dθ\n du[2] = -(g/ℓ)*sin(θ)\nend",
"execution_count": 2,
"outputs": [
{
"data": {
"text/plain": "simplependulum! (generic function with 1 method)"
},
"execution_count": 2,
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"output_type": "execute_result"
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"source": "### 境界条件1\n\n$[0,T]$ 上で\n\n$$\n\\theta(T/2) = -\\frac{\\pi}{2}, \\quad\n\\theta(0) = \\frac{\\pi}{2}.\n$$"
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"source": "function bc1!(residual, u, p, t)\n residual[1] = u[end÷2][1] + pi/2 # the solution at the middle of the time span should be -pi/2\n residual[2] = u[end][1] - pi/2 # the solution at the end of the time span should be pi/2\nend\nbvp1 = BVProblem(simplependulum!, bc1!, [pi/2,pi/2], tspan)\nsol1 = solve(bvp1, GeneralMIRK4(), dt=0.05)\nplot(sol1, label=[\"\\$\\\\theta\\$\", \"\\$d\\\\theta/dt\\$\"], lw=1, legend=:topleft, size=(400, 200))",
"execution_count": 3,
"outputs": [
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"execution_count": 3,
"metadata": {},
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}
]
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### 境界条件1 (別の方法)\n\n$[0,T]$ 上で\n\n$$\n\\theta(T/2) = -\\frac{\\pi}{2}, \\quad\n\\theta(0) = \\frac{\\pi}{2}.\n$$"
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "u₀_2 = [-1.6, -1.7] # the initial guess\nfunction bc3!(residual, sol, p, t)\n residual[1] = sol(pi/4)[1] + pi/2 # use the interpolation here, since indexing will be wrong for adaptive methods\n residual[2] = sol(pi/2)[1] - pi/2\nend\nbvp3 = BVProblem(simplependulum!, bc3!, u₀_2, tspan)\nsol3 = solve(bvp3, Shooting(Vern7()))\nplot(sol3, label=[\"\\$\\\\theta\\$\", \"\\$d\\\\theta/dt\\$\"], lw=1, legend=:topleft, size=(400, 200))",
"execution_count": 4,
"outputs": [
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1180.69,353.68 1182.06,353.224 1183.43,352.767 1184.81,352.311 1186.18,351.855 1187.55,351.399 1188.93,350.943 1190.3,350.487 1191.67,350.031 \n 1193.05,349.575 1194.42,349.12 1195.79,348.664 1197.17,348.209 1198.54,347.754 1199.91,347.299 1201.29,346.844 1202.66,346.389 1204.03,345.935 1205.41,345.48 \n 1206.78,345.026 1208.15,344.573 1209.53,344.119 1210.9,343.666 1212.27,343.213 1213.65,342.76 1215.02,342.307 1216.39,341.855 1217.77,341.403 1219.14,340.952 \n 1220.51,340.5 1221.89,340.049 1223.26,339.599 1224.63,339.148 1226.01,338.699 1227.38,338.249 1228.75,337.8 1230.13,337.351 1231.5,336.903 1232.87,336.455 \n 1234.25,336.008 1235.62,335.561 1236.99,335.114 1238.37,334.668 1239.74,334.222 1241.11,333.777 1242.49,333.333 1243.86,332.888 1245.23,332.445 1246.61,332.002 \n 1247.98,331.559 1249.35,331.117 1250.73,330.675 1252.1,330.235 1253.47,329.794 1254.84,329.354 1256.22,328.915 1257.59,328.477 1258.96,328.039 1260.34,327.601 \n 1261.71,327.165 1263.08,326.728 1264.46,326.293 1265.83,325.858 1267.2,325.424 1268.58,324.991 1269.95,324.558 1271.32,324.126 1272.7,323.694 1274.07,323.264 \n 1275.44,322.834 1276.82,322.405 1278.19,321.976 1279.56,321.548 1280.94,321.121 1282.31,320.695 1283.68,320.27 1285.06,319.845 1286.43,319.421 1287.8,318.998 \n 1289.18,318.576 1290.55,318.155 1291.92,317.734 1293.3,317.314 1294.67,316.895 1296.04,316.477 1297.42,316.06 1298.79,315.644 1300.16,315.228 1301.54,314.814 \n 1302.91,314.4 1304.28,313.987 1305.66,313.575 1307.03,313.164 1308.4,312.754 1309.78,312.345 1311.15,311.937 1312.52,311.529 1313.9,311.123 1315.27,310.718 \n 1316.64,310.313 1318.02,309.91 1319.39,309.507 1320.76,309.106 1322.14,308.705 1323.51,308.306 1324.88,307.907 1326.26,307.51 1327.63,307.113 1329,306.718 \n 1330.37,306.323 1331.75,305.93 1333.12,305.538 1334.49,305.146 1335.87,304.756 1337.24,304.367 1338.61,303.979 1339.99,303.592 1341.36,303.206 1342.73,302.821 \n 1344.11,302.438 1345.48,302.055 1346.85,301.673 1348.23,301.293 1349.6,300.914 1350.97,300.535 1352.35,300.158 1353.72,299.782 1355.09,299.408 1356.47,299.034 \n 1357.84,298.661 1359.21,298.29 1360.59,297.92 1361.96,297.551 1363.33,297.183 1364.71,296.816 1366.08,296.451 1367.45,296.086 1368.83,295.723 1370.2,295.361 \n 1371.57,295.001 1372.95,294.641 1374.32,294.283 1375.69,293.926 1377.07,293.57 1378.44,293.215 1379.81,292.862 1381.19,292.509 1382.56,292.158 1383.93,291.808 \n 1385.31,291.46 1386.68,291.113 1388.05,290.767 1389.43,290.422 1390.8,290.078 1392.17,289.736 1393.55,289.395 1394.92,289.055 1396.29,288.717 1397.67,288.38 \n 1399.04,288.044 1400.41,287.709 1401.79,287.376 1403.16,287.044 1404.53,286.713 1405.9,286.383 1407.28,286.055 1408.65,285.728 1410.02,285.403 1411.4,285.078 \n 1412.77,284.755 1414.14,284.434 1415.52,284.113 1416.89,283.794 1418.26,283.476 1419.64,283.16 1421.01,282.845 1422.38,282.531 1423.76,282.219 1425.13,281.908 \n 1426.5,281.598 1427.88,281.29 1429.25,280.982 1430.62,280.677 1432,280.372 1433.37,280.069 1434.74,279.768 1436.12,279.467 1437.49,279.168 1438.86,278.871 \n 1440.24,278.574 1441.61,278.279 1442.98,277.986 1444.36,277.694 1445.73,277.403 1447.1,277.113 1448.48,276.825 1449.85,276.538 1451.22,276.253 1452.6,275.969 \n 1453.97,275.686 1455.34,275.405 1456.72,275.125 1458.09,274.847 1459.46,274.569 1460.84,274.294 1462.21,274.019 1463.58,273.746 1464.96,273.475 1466.33,273.204 \n 1467.7,272.936 1469.08,272.668 1470.45,272.402 1471.82,272.137 1473.2,271.874 1474.57,271.612 1475.94,271.352 1477.32,271.092 1478.69,270.835 1480.06,270.578 \n 1481.43,270.323 1482.81,270.07 1484.18,269.818 1485.55,269.567 1486.93,269.318 1488.3,269.07 1489.67,268.823 1491.05,268.578 1492.42,268.334 1493.79,268.092 \n 1495.17,267.851 1496.54,267.611 1497.91,267.373 1499.29,267.136 1500.66,266.901 1502.03,266.667 1503.41,266.434 1504.78,266.203 1506.15,265.973 1507.53,265.745 \n 1508.9,265.518 1510.27,265.292 1511.65,265.068 1513.02,264.845 1514.39,264.624 1515.77,264.404 1517.14,264.186 1518.51,263.968 1519.89,263.753 1521.26,263.538 \n \n \"/>\n<polyline clip-path=\"url(#clip9502)\" style=\"stroke:#e26f46; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n 149.361,631.839 150.734,631.427 152.107,631.009 153.48,630.586 154.854,630.156 156.227,629.721 157.6,629.28 158.974,628.833 160.347,628.38 161.72,627.922 \n 163.093,627.458 164.467,626.988 165.84,626.513 167.213,626.032 168.586,625.545 169.96,625.054 171.333,624.556 172.706,624.053 174.08,623.545 175.453,623.032 \n 176.826,622.513 178.199,621.988 179.573,621.459 180.946,620.924 182.319,620.385 183.692,619.84 185.066,619.29 186.439,618.735 187.812,618.175 189.186,617.61 \n 190.559,617.04 191.932,616.465 193.305,615.885 194.679,615.301 196.052,614.712 197.425,614.118 198.798,613.519 200.172,612.916 201.545,612.308 202.918,611.696 \n 204.292,611.079 205.665,610.457 207.038,609.831 208.411,609.201 209.785,608.566 211.158,607.927 212.531,607.284 213.904,606.636 215.278,605.985 216.651,605.329 \n 218.024,604.669 219.398,604.005 220.771,603.337 222.144,602.665 223.517,601.989 224.891,601.309 226.264,600.625 227.637,599.937 229.01,599.246 230.384,598.551 \n 231.757,597.852 233.13,597.149 234.504,596.443 235.877,595.733 237.25,595.02 238.623,594.304 239.997,593.583 241.37,592.86 242.743,592.133 244.116,591.402 \n 245.49,590.669 246.863,589.932 248.236,589.192 249.61,588.449 250.983,587.702 252.356,586.953 253.729,586.2 255.103,585.445 256.476,584.686 257.849,583.925 \n 259.222,583.161 260.596,582.393 261.969,581.623 263.342,580.851 264.716,580.075 266.089,579.297 267.462,578.516 268.835,577.733 270.209,576.947 271.582,576.158 \n 272.955,575.367 274.328,574.573 275.702,573.777 277.075,572.979 278.448,572.178 279.822,571.375 281.195,570.57 282.568,569.762 283.941,568.952 285.315,568.14 \n 286.688,567.326 288.061,566.51 289.434,565.691 290.808,564.871 292.181,564.049 293.554,563.224 294.927,562.398 296.301,561.57 297.674,560.74 299.047,559.908 \n 300.421,559.074 301.794,558.239 303.167,557.401 304.54,556.563 305.914,555.722 307.287,554.88 308.66,554.036 310.033,553.191 311.407,552.344 312.78,551.495 \n 314.153,550.645 315.527,549.794 316.9,548.941 318.273,548.087 319.646,547.232 321.02,546.375 322.393,545.517 323.766,544.657 325.139,543.797 326.513,542.935 \n 327.886,542.072 329.259,541.208 330.633,540.342 332.006,539.476 333.379,538.608 334.752,537.74 336.126,536.87 337.499,535.999 338.872,535.128 340.245,534.255 \n 341.619,533.382 342.992,532.508 344.365,531.633 345.739,530.757 347.112,529.88 348.485,529.002 349.858,528.124 351.232,527.244 352.605,526.365 353.978,525.484 \n 355.351,524.603 356.725,523.721 358.098,522.838 359.471,521.955 360.845,521.071 362.218,520.187 363.591,519.302 364.964,518.417 366.338,517.531 367.711,516.644 \n 369.084,515.757 370.457,514.87 371.831,513.982 373.204,513.094 374.577,512.206 375.951,511.317 377.324,510.427 378.697,509.538 380.07,508.648 381.444,507.757 \n 382.817,506.867 384.19,505.976 385.563,505.085 386.937,504.194 388.31,503.302 389.683,502.411 391.057,501.519 392.43,500.627 393.803,499.734 395.176,498.842 \n 396.55,497.95 397.923,497.057 399.296,496.164 400.669,495.271 402.043,494.379 403.416,493.486 404.789,492.593 406.163,491.7 407.536,490.807 408.909,489.914 \n 410.282,489.021 411.656,488.128 413.029,487.235 414.402,486.342 415.775,485.449 417.149,484.556 418.522,483.663 419.895,482.771 421.269,481.878 422.642,480.986 \n 424.015,480.093 425.388,479.201 426.762,478.309 428.135,477.417 429.508,476.525 430.881,475.634 432.255,474.742 433.628,473.851 435.001,472.96 436.375,472.069 \n 437.748,471.178 439.121,470.288 440.494,469.397 441.868,468.507 443.241,467.618 444.614,466.728 445.987,465.839 447.361,464.95 448.734,464.061 450.107,463.172 \n 451.481,462.284 452.854,461.396 454.227,460.508 455.6,459.621 456.974,458.733 458.347,457.846 459.72,456.96 461.093,456.073 462.467,455.187 463.84,454.302 \n 465.213,453.416 466.587,452.531 467.96,451.647 469.333,450.762 470.706,449.878 472.08,448.994 473.453,448.111 474.826,447.228 476.199,446.345 477.573,445.463 \n 478.946,444.581 480.319,443.699 481.693,442.817 483.066,441.936 484.439,441.056 485.812,440.175 487.186,439.295 488.559,438.416 489.932,437.536 491.305,436.657 \n 492.679,435.779 494.052,434.901 495.425,434.023 496.799,433.145 498.172,432.268 499.545,431.391 500.918,430.515 502.292,429.639 503.665,428.763 505.038,427.888 \n 506.411,427.013 507.785,426.138 509.158,425.264 510.531,424.39 511.905,423.516 513.278,422.643 514.651,421.77 516.024,420.897 517.398,420.025 518.771,419.153 \n 520.144,418.282 521.517,417.41 522.891,416.54 524.264,415.669 525.637,414.799 527.011,413.929 528.384,413.059 529.757,412.19 531.13,411.321 532.504,410.453 \n 533.877,409.584 535.25,408.716 536.623,407.849 537.997,406.981 539.37,406.114 540.743,405.248 542.117,404.381 543.49,403.515 544.863,402.649 546.236,401.784 \n 547.61,400.918 548.983,400.053 550.356,399.189 551.729,398.324 553.103,397.46 554.476,396.596 555.849,395.732 557.223,394.869 558.596,394.006 559.969,393.143 \n 561.342,392.28 562.716,391.418 564.089,390.556 565.462,389.694 566.835,388.832 568.209,387.97 569.582,387.109 570.955,386.248 572.329,385.387 573.702,384.526 \n 575.075,383.666 576.448,382.805 577.822,381.945 579.195,381.085 580.568,380.226 581.941,379.366 583.315,378.507 584.688,377.647 586.061,376.788 587.435,375.929 \n 588.808,375.07 590.181,374.212 591.554,373.353 592.928,372.495 594.301,371.636 595.674,370.778 597.047,369.92 598.421,369.062 599.794,368.204 601.167,367.346 \n 602.541,366.488 603.914,365.631 605.287,364.773 606.66,363.916 608.034,363.058 609.407,362.201 610.78,361.344 612.153,360.486 613.527,359.629 614.9,358.772 \n 616.273,357.915 617.647,357.057 619.02,356.2 620.393,355.343 621.766,354.486 623.14,353.629 624.513,352.772 625.886,351.915 627.259,351.058 628.633,350.2 \n 630.006,349.343 631.379,348.486 632.753,347.629 634.126,346.771 635.499,345.914 636.872,345.057 638.246,344.199 639.619,343.341 640.992,342.484 642.365,341.626 \n 643.739,340.768 645.112,339.91 646.485,339.052 647.859,338.194 649.232,337.336 650.605,336.478 651.978,335.619 653.352,334.761 654.725,333.902 656.098,333.043 \n 657.471,332.184 658.845,331.325 660.218,330.466 661.591,329.606 662.965,328.747 664.338,327.887 665.711,327.027 667.084,326.167 668.458,325.307 669.831,324.446 \n 671.204,323.585 672.577,322.724 673.951,321.863 675.324,321.002 676.697,320.14 678.071,319.279 679.444,318.417 680.817,317.555 682.19,316.692 683.564,315.829 \n 684.937,314.967 686.31,314.103 687.683,313.24 689.057,312.376 690.43,311.512 691.803,310.648 693.177,309.784 694.55,308.919 695.923,308.054 697.296,307.189 \n 698.67,306.323 700.043,305.457 701.416,304.591 702.789,303.725 704.163,302.858 705.536,301.991 706.909,301.124 708.283,300.256 709.656,299.388 711.029,298.52 \n 712.402,297.651 713.776,296.782 715.149,295.913 716.522,295.044 717.895,294.174 719.269,293.303 720.642,292.433 722.015,291.562 723.389,290.691 724.762,289.819 \n 726.135,288.947 727.508,288.075 728.882,287.203 730.255,286.33 731.628,285.456 733.001,284.583 734.375,283.709 735.748,282.835 737.121,281.96 738.495,281.085 \n 739.868,280.21 741.241,279.334 742.614,278.458 743.988,277.581 745.361,276.704 746.734,275.827 748.107,274.95 749.481,274.072 750.854,273.194 752.227,272.315 \n 753.601,271.436 754.974,270.557 756.347,269.677 757.72,268.797 759.094,267.917 760.467,267.036 761.84,266.155 763.213,265.274 764.587,264.392 765.96,263.51 \n 767.333,262.628 768.707,261.745 770.08,260.862 771.453,259.978 772.826,259.095 774.2,258.21 775.573,257.326 776.946,256.441 778.319,255.556 779.693,254.671 \n 781.066,253.785 782.439,252.899 783.813,252.013 785.186,251.126 786.559,250.239 787.932,249.352 789.306,248.465 790.679,247.577 792.052,246.689 793.425,245.801 \n 794.799,244.912 796.172,244.023 797.545,243.134 798.919,242.245 800.292,241.355 801.665,240.465 803.038,239.575 804.412,238.685 805.785,237.795 807.158,236.904 \n 808.531,236.013 809.905,235.122 811.278,234.231 812.651,233.339 814.025,232.448 815.398,231.556 816.771,230.664 818.144,229.772 819.518,228.88 820.891,227.987 \n 822.264,227.095 823.637,226.202 825.011,225.31 826.384,224.417 827.757,223.524 829.131,222.631 830.504,221.738 831.877,220.845 833.25,219.952 834.624,219.059 \n 835.997,218.166 837.37,217.273 838.743,216.38 840.117,215.487 841.49,214.594 842.863,213.701 844.236,212.809 845.61,211.916 846.983,211.023 848.356,210.131 \n 849.73,209.239 851.103,208.346 852.476,207.454 853.849,206.562 855.223,205.671 856.596,204.779 857.969,203.888 859.342,202.997 860.716,202.106 862.089,201.215 \n 863.462,200.325 864.836,199.435 866.209,198.546 867.582,197.656 868.955,196.767 870.329,195.879 871.702,194.991 873.075,194.103 874.448,193.215 875.822,192.329 \n 877.195,191.442 878.568,190.556 879.942,189.671 881.315,188.786 882.688,187.901 884.061,187.018 885.435,186.135 886.808,185.252 888.181,184.37 889.554,183.489 \n 890.928,182.608 892.301,181.728 893.674,180.849 895.048,179.971 896.421,179.093 897.794,178.216 899.167,177.34 900.541,176.465 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},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
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]
},
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"metadata": {},
"cell_type": "markdown",
"source": "### 境界条件2\n\n$[0,T]$ 上で\n\n$$\n\\theta(0) = -\\frac{\\pi}{2}, \\quad\n\\theta(T) = \\frac{\\pi}{2}.\n$$"
},
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"metadata": {
"trusted": true
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"cell_type": "code",
"source": "function bc2!(residual, u, p, t) # u[1] is the beginning of the time span, and u[end] is the ending\n residual[1] = u[1][1] + pi/2 # the solution at the beginning of the time span should be -pi/2\n residual[2] = u[end][1] - pi/2 # the solution at the end of the time span should be pi/2\nend\nbvp2 = TwoPointBVProblem(simplependulum!, bc2!, [pi/2,pi/2], tspan)\nsol2 = solve(bvp2, MIRK4(), dt=0.05) # we need to use the MIRK4 solver for TwoPointBVProblem\nplot(sol2, label=[\"\\$\\\\theta\\$\", \"\\$d\\\\theta/dt\\$\"], lw=1, legend=:topright, size=(400, 200))",
"execution_count": 5,
"outputs": [
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"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {},
"cell_type": "markdown",
"source": "## 調和振動子の数値解\n\n調和振動子\n\n$$\n\\ddot x = -\\omega^2 x.\n$$\n\nを数値的に解いてみよう."
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "function harmonicoscillator!(du, u, p, t)\n ω = p[1]\n du[1] = u[2]\n du[2] = - ω^2*u[1]\nend\n\nfunction plot_harmonicoscilator(; ω=5.0, a=0.0, b=1.0, T=2π)\n prob = ODEProblem(harmonicoscillator!, [a,b], (0,T), [ω])\n sol = solve(prob, dt=0.005)\n L = round(Int, 8π/T)\n kpiL(k) = isone(L) ? \"\\$$(k)\\\\pi\\$\" : \"\\$\\\\frac{$(k)\\\\pi}{$L}\\$\"\n xticks_num = 0:π/L:T\n xticks_str = [\n \"\\$0\\$\"\n \"\\$$(kpiL(\"\"))\\$\"\n [\"\\$$(kpiL(k))\\$\" for k in 2:length(xticks_num)-1];\n ]\n P1 = plot(title=\"\\$x(t):\\\\; \\\\omega = $ω\\$\", titlefontsize=12, legend=false)\n plot!(xticks = (xticks_num, xticks_str))\n plot!(sol, vars=(0,1), label=\"\\$x\\$\", xlabel=\" \", size=(400, 165), lw=1, color=:blue)\n P2 = plot(title=\"\\$dx(t)/dt:\\\\; \\\\omega = $ω\\$\", titlefontsize=12, legend=false)\n plot!(xticks = (xticks_num, xticks_str))\n plot!(sol, vars=(0,2), label=\"\\$dx/dt\\$\", xlabel=\" \", size=(400, 160), lw=1, color=:red)\n plot(P1, P2, size=(700, 160))\nend",
"execution_count": 6,
"outputs": [
{
"data": {
"text/plain": "plot_harmonicoscilator (generic function with 1 method)"
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### 初期値問題\n\n$$\nx(0) = a, \\quad \\dot x(0) = b.\n$$"
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "plot_harmonicoscilator(ω=1.1, a=1.5, b=2.3, T=10π)",
"execution_count": 7,
"outputs": [
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1591.59,362.945 1592.76,368.705 1593.94,374.396 1595.11,380.012 \n 1596.28,385.547 1597.46,390.993 1598.63,396.345 1599.8,401.595 1600.98,406.738 1602.15,411.767 1603.32,416.676 1604.5,421.46 1605.67,426.112 1606.84,430.628 \n 1608.01,435.001 1609.19,439.227 1610.36,443.301 1611.53,447.217 1612.71,450.971 1613.88,454.559 1615.05,457.976 1616.23,461.217 1617.4,464.28 1618.57,467.16 \n 1619.74,469.854 1620.92,472.359 1622.09,474.671 1623.26,476.789 1624.44,478.709 1625.61,480.429 1626.78,481.948 1627.96,483.263 1629.13,484.373 1630.3,485.277 \n 1631.48,485.973 1632.65,486.461 1633.82,486.739 1634.99,486.809 1636.17,486.668 1637.34,486.318 1638.51,485.759 1639.69,484.992 1640.86,484.017 1642.03,482.837 \n 1643.21,481.451 1644.38,479.862 1645.55,478.072 1646.73,476.083 1647.9,473.898 1649.07,471.519 1650.24,468.949 1651.42,466.191 1652.59,463.248 1653.76,460.124 \n 1654.94,456.822 1656.11,453.347 1657.28,449.702 1658.46,445.892 1659.63,441.921 1660.8,437.795 1661.98,433.517 1663.15,429.094 1664.32,424.53 1665.49,419.831 \n 1666.67,415.003 1667.84,410.052 1669.01,404.983 1670.19,399.803 1671.36,394.517 1672.53,389.132 1673.71,383.656 1674.88,378.093 1676.05,372.451 1677.23,366.736 \n 1678.4,360.955 1679.57,355.115 1680.74,349.224 1681.92,343.287 1683.09,337.312 1684.26,331.306 1685.44,325.276 1686.61,319.23 1687.78,313.175 1688.96,307.119 \n 1690.13,301.068 1691.3,295.031 1692.48,289.013 1693.65,283.023 1694.82,277.066 1695.99,271.152 1697.17,265.286 1698.34,259.476 1699.51,253.728 1700.69,248.051 \n 1701.86,242.449 1703.03,236.931 1704.21,231.501 1705.38,226.168 1706.55,220.937 1707.73,215.815 1708.9,210.807 1710.07,205.92 1711.24,201.159 1712.42,196.531 \n 1713.59,192.041 1714.76,187.693 1715.94,183.494 1717.11,179.447 1718.28,175.559 1719.46,171.834 1720.63,168.277 1721.8,164.891 1722.98,161.682 1724.15,158.652 \n 1725.32,155.806 1726.49,153.146 1727.67,150.677 1728.84,148.4 1730.01,146.319 1731.19,144.436 1732.36,142.753 1733.53,141.272 1734.71,139.995 1735.88,138.923 \n 1737.05,138.058 1738.23,137.401 1739.4,136.951 1740.57,136.71 1741.74,136.678 1742.92,136.856 1744.09,137.242 1745.26,137.838 1746.44,138.641 1747.61,139.652 \n 1748.78,140.869 1749.96,142.291 1751.13,143.916 1752.3,145.743 1753.48,147.768 1754.65,149.99 1755.82,152.406 1756.99,155.013 1758.17,157.807 1759.34,160.786 \n 1760.51,163.946 1761.69,167.282 1762.86,170.79 1764.03,174.466 1765.21,178.306 1766.38,182.306 1767.55,186.46 1768.73,190.764 1769.9,195.212 1771.07,199.8 \n 1772.24,204.522 1773.42,209.371 1774.59,214.344 1775.76,219.433 1776.94,224.633 1778.11,229.937 1779.28,235.34 1780.46,240.835 1781.63,246.414 1782.8,252.073 \n 1783.98,257.803 1785.15,263.598 1786.32,269.45 1787.49,275.352 1788.67,281.298 1789.84,287.279 1791.01,293.29 1792.19,299.323 1793.36,305.371 1794.53,311.426 \n 1795.71,317.482 1796.88,323.53 1798.05,329.564 1799.23,335.576 1800.4,341.56 1801.57,347.507 1802.74,353.413 1803.92,359.268 1805.09,365.067 1806.26,370.802 \n 1807.44,376.467 1808.61,382.055 1809.78,387.56 1810.96,392.974 1812.13,398.291 1813.3,403.504 1814.47,408.608 1815.65,413.595 1816.82,418.459 1817.99,423.196 \n 1819.17,427.8 1820.34,432.266 1821.51,436.588 1822.69,440.76 1823.86,444.777 1825.03,448.635 1826.21,452.329 1827.38,455.855 1828.55,459.207 1829.72,462.383 \n 1830.9,465.379 1832.07,468.19 1833.24,470.815 1834.42,473.248 1835.59,475.489 1836.76,477.534 1837.94,479.38 1839.11,481.026 1840.28,482.47 1841.46,483.709 \n 1842.63,484.743 1843.8,485.571 1844.97,486.192 1846.15,486.604 1847.32,486.808 1848.49,486.802 1849.67,486.587 1850.84,486.163 1852.01,485.53 1853.19,484.688 \n 1854.36,483.639 1855.53,482.384 1856.71,480.924 1857.88,479.261 1859.05,477.397 1860.22,475.334 1861.4,473.075 1862.57,470.622 1863.74,467.979 1864.92,465.148 \n 1866.09,462.134 1867.26,458.94 1868.44,455.57 1869.61,452.029 1870.78,448.322 1871.96,444.451 1873.13,440.422 1874.3,436.239 1875.47,431.908 1876.65,427.433 \n 1877.82,422.82 1878.99,418.074 1880.17,413.201 1881.34,408.206 1882.51,403.096 1883.69,397.876 1884.86,392.553 1886.03,387.133 1887.21,381.622 1888.38,376.027 \n 1889.55,370.354 1890.72,364.611 1891.9,358.804 1893.07,352.941 1894.24,347.028 1895.42,341.074 1896.59,335.085 1897.76,329.069 1898.94,323.032 1900.11,316.982 \n 1901.28,310.925 1902.46,304.869 1903.63,298.822 1904.8,292.79 1905.97,286.781 1907.15,280.803 1908.32,274.862 1909.49,268.965 1910.67,263.12 1911.84,257.332 \n 1913.01,251.61 1914.19,245.959 1915.36,240.387 1916.53,234.899 1917.71,229.503 1918.88,224.205 1920.05,219.011 1921.22,213.928 1922.4,208.962 1923.57,204.119 \n 1924.74,199.405 1925.92,194.826 1927.09,190.387 1928.26,186.093 1929.44,181.949 1930.61,177.96 1931.78,174.131 1932.96,170.466 1934.13,166.971 1935.3,163.649 \n 1936.47,160.505 1937.65,157.542 1938.82,154.764 1939.99,152.174 1941.17,149.776 1942.34,147.571 1943.51,145.564 1944.69,143.755 1945.86,142.147 1947.03,140.742 \n 1948.21,139.542 1949.38,138.548 1950.55,137.76 1951.72,137.181 1952.9,136.811 1954.07,136.649 1955.24,136.696 1956.42,136.951 1957.59,137.415 1958.76,138.087 \n 1959.94,138.966 1961.11,140.053 1962.28,141.344 1963.46,142.84 1964.63,144.538 1965.8,146.436 1966.97,148.533 1968.15,150.826 1969.32,153.312 1970.49,155.988 \n 1971.67,158.852 1972.84,161.899 1974.01,165.126 1975.19,168.529 1976.36,172.104 1977.53,175.846 1978.71,179.751 1979.88,183.814 1981.05,188.029 1982.22,192.391 \n 1983.4,196.894 1984.57,201.534 1985.74,206.306 1986.92,211.204 1988.09,216.222 1989.26,221.354 1990.44,226.594 1991.61,231.935 1992.78,237.372 1993.96,242.897 \n 1995.13,248.506 1996.3,254.19 1997.47,259.943 1998.65,265.759 1999.82,271.631 2000.99,277.551 2002.17,283.512 2003.34,289.508 2004.51,295.531 2005.69,301.574 \n 2006.86,307.629 2008.03,313.689 2009.2,319.746 2010.38,325.792 2011.55,331.821 2012.72,337.826 2013.9,343.801 2015.07,349.737 2016.24,355.627 2017.42,361.465 \n 2018.59,367.244 2019.76,372.955 2020.94,378.593 2022.11,384.15 2023.28,389.621 2024.45,394.998 2025.63,400.275 2026.8,405.447 2027.97,410.506 2029.15,415.448 \n 2030.32,420.266 2031.49,424.954 2032.67,429.507 2033.84,433.92 2035.01,438.187 2036.19,442.302 2037.36,446.262 2038.53,450.06 2039.7,453.692 2040.88,457.154 \n 2042.05,460.443 2043.22,463.555 2044.4,466.486 2045.57,469.231 2046.74,471.788 2047.92,474.154 2049.09,476.325 2050.26,478.299 2051.44,480.073 2052.61,481.645 \n 2053.78,483.013 2054.95,484.176 2056.13,485.132 2057.3,485.88 2058.47,486.42 2059.65,486.75 2060.82,486.87 2061.99,486.781 2063.17,486.482 2064.34,485.975 \n 2065.51,485.259 2066.69,484.336 2067.86,483.207 2069.03,481.873 2070.2,480.337 2071.38,478.599 2072.55,476.663 2073.72,474.529 2074.9,472.201 2076.07,469.681 \n 2077.24,466.973 2078.42,464.078 2079.59,461.001 2080.76,457.745 2081.94,454.313 2083.11,450.711 2084.28,446.942 2085.45,443.01 2086.63,438.92 2087.8,434.677 \n 2088.97,430.287 2090.15,425.754 2091.32,421.085 2092.49,416.284 2093.67,411.358 2094.84,406.314 2096.01,401.158 2097.19,395.896 2098.36,390.533 2099.53,385.077 \n 2100.7,379.533 2101.88,373.908 2103.05,368.209 2104.22,362.442 2105.4,356.616 2106.57,350.736 2107.74,344.809 2108.92,338.843 2110.09,332.845 2111.26,326.821 \n 2112.44,320.778 2113.61,314.724 2114.78,308.667 2115.95,302.612 2117.13,296.567 2118.3,290.54 2119.47,284.538 2120.65,278.567 2121.82,272.637 2122.99,266.753 \n 2124.17,260.923 2125.34,255.155 2126.51,249.456 2127.69,243.832 2128.86,238.288 2130.03,232.832 2131.2,227.47 2132.38,222.209 2133.55,217.055 2134.72,212.015 \n 2135.9,207.095 2137.07,202.3 2138.24,197.636 2139.42,193.109 2140.59,188.725 2141.76,184.488 2142.94,180.403 2144.11,176.475 2145.28,172.709 2146.45,169.109 \n 2147.63,165.679 2148.8,162.424 2149.97,159.348 2151.15,156.454 2152.32,153.745 2153.49,151.226 2154.67,148.899 2155.84,146.765 2157.01,144.828 2158.19,143.09 \n 2159.36,141.554 2160.53,140.22 2161.7,139.092 2162.88,138.171 2164.05,137.458 2165.22,136.954 2166.4,136.659 2167.57,136.575 2168.74,136.701 2169.92,137.036 \n 2171.09,137.582 2172.26,138.337 2173.44,139.299 2174.61,140.469 2175.78,141.844 2176.95,143.422 2178.13,145.202 2179.3,147.181 2180.47,149.357 2181.65,151.727 \n 2182.82,154.287 2183.99,157.035 2185.17,159.967 2186.34,163.08 2187.51,166.371 2188.69,169.835 2189.86,173.469 2191.03,177.268 2192.2,181.228 2193.38,185.344 \n 2194.55,189.612 2195.72,194.026 2196.9,198.581 2198.07,203.272 2199.24,208.093 2200.42,213.039 2201.59,218.104 2202.76,223.281 2203.93,228.564 2205.11,233.948 \n 2206.28,239.424 2207.45,244.988 2208.63,250.63 2209.8,256.345 2210.97,262.126 2212.15,267.965 2213.32,273.857 2214.49,279.795 2215.67,285.77 2216.84,291.776 \n 2218.01,297.806 2219.18,303.852 2220.36,309.907 2221.53,315.964 2222.7,322.016 2223.88,328.056 2225.05,334.076 2226.22,340.069 2227.4,346.029 2228.57,351.949 \n 2229.74,357.821 2230.92,363.639 2232.09,369.395 2233.26,375.082 2234.43,380.695 2235.61,386.225 2236.78,391.667 2237.95,397.012 2239.13,402.255 2240.3,407.389 \n 2241.47,412.407 2242.65,417.305 2243.82,422.078 2244.99,426.718 2246.17,431.221 2247.34,435.582 2248.51,439.793 2249.68,443.852 2250.86,447.752 2252.03,451.488 \n 2253.2,455.058 2254.38,458.455 2255.55,461.676 2256.72,464.718 2257.9,467.577 2259.07,470.249 2260.24,472.732 2261.42,475.022 2262.59,477.116 2263.76,479.013 \n 2264.93,480.71 2266.11,482.205 2267.28,483.496 2268.45,484.582 2269.63,485.462 2270.8,486.135 2271.97,486.6 2273.15,486.856 2274.32,486.904 2275.49,486.742 \n 2276.67,486.37 2277.84,485.79 2279.01,485.001 2280.18,484.004 2281.36,482.8 2282.53,481.391 2283.7,479.778 2284.88,477.964 2286.05,475.95 2287.22,473.738 \n 2288.4,471.332 2289.57,468.734 2290.74,465.948 2291.92,462.977 2293.09,459.825 2294.26,456.495 2295.43,452.993 2296.61,449.323 2297.78,445.489 2298.95,441.496 \n 2300.13,437.349 2301.3,433.052 2302.47,428.61 2303.65,424.028 2304.82,419.312 2305.99,414.468 2307.17,409.501 2308.34,404.417 2309.51,399.222 2310.68,393.923 \n 2311.86,388.524 2313.03,383.034 2314.2,377.457 2315.38,371.802 2316.55,366.074 2317.72,360.28 2318.9,354.428 2320.07,348.524 2321.24,342.575 2322.42,336.59 \n 2323.59,330.575 2324.76,324.538 2325.93,318.487 2327.11,312.429 2328.28,306.369 2329.45,300.316 2330.63,294.276 2331.8,288.257 2332.97,282.266 2334.15,276.311 \n 2335.32,270.399 2336.49,264.537 2337.67,258.731 2338.84,252.99 2340.01,247.318 2341.18,241.724 2342.36,236.213 2343.53,230.792 2344.7,225.468 2345.88,220.247 \n 2347.05,215.135 2348.22,210.137 2349.4,205.261 2350.57,200.512 2351.74,195.896 2352.92,191.419 2354.09,187.086 2355.26,182.902 2356.43,178.873 2357.61,175.004 \n 2358.78,171.296 2359.95,167.757 2361.13,164.389 2362.3,161.198 2363.47,158.186 2364.65,155.359 2365.82,152.719 2366.99,150.27 2368.17,148.015 2369.34,145.956 \n 2370.51,144.095 2371.68,142.436 2372.86,140.98 2374.03,139.728 2375.2,138.682 2376.38,137.844 2377.55,137.214 2378.72,136.793 2379.9,136.581 2381.07,136.578 \n 2382.24,136.785 2383.42,137.2 2384.59,137.824 2385.76,138.655 2386.93,139.693 2388.11,140.936 2389.28,142.382 2390.45,144.031 2391.63,145.88 2392.8,147.927 \n 2393.97,150.171 2395.15,152.608 2396.32,155.236 2397.49,158.052 2398.66,161.052 2399.84,164.233 2401.01,167.592 2402.18,171.124 2403.36,174.824 2404.53,178.69 \n 2405.7,182.715 2406.88,186.895 2408.05,191.226 2409.22,195.7 2410.4,200.313 2411.57,205.059 2412.74,209.932 2413.91,214.925 2415.09,220.033 2416.26,225.25 \n 2417.43,230.571 2418.61,235.989 2419.78,241.497 2420.95,247.088 2422.13,252.757 2423.3,258.496 2424.47,264.299 2425.65,270.158 2426.82,276.068 2427.99,282.02 \n 2429.16,288.008 2430.34,294.025 2431.51,300.064 2432.68,306.118 2433.86,312.178 2435.03,318.239 2436.2,324.293 2437.38,330.333 2438.55,336.35 2439.72,342.337 \n 2440.9,348.288 2442.07,354.194 2443.24,360.048 2444.41,365.844 2445.59,371.576 2446.76,377.237 2447.93,382.82 2449.11,388.317 2450.28,393.723 2451.45,399.03 \n 2452.63,404.232 2453.8,409.323 2454.97,414.297 2456.15,419.148 2457.32,423.871 2458.49,428.459 2459.66,432.907 2460.84,437.211 2462.01,441.365 2463.18,445.363 \n 2464.36,449.203 2465.53,452.877 2466.7,456.384 2467.88,459.717 2469.05,462.873 2470.22,465.849 2471.4,468.639 2472.57,471.242 2473.74,473.656 2474.91,475.877 \n 2476.09,477.901 2477.26,479.727 2478.43,481.353 2479.61,482.775 2480.78,483.992 2481.95,485.003 2483.13,485.806 2484.3,486.401 2485.47,486.785 2486.65,486.96 \n 2487.82,486.924 2488.99,486.678 2490.16,486.223 2491.34,485.558 2492.51,484.684 2493.68,483.604 2494.86,482.317 2496.03,480.827 2497.2,479.134 2498.38,477.241 \n 2499.55,475.151 2500.72,472.866 2501.9,470.389 2503.07,467.723 2504.24,464.871 2505.41,461.836 2506.59,458.622 2507.76,455.232 2508.93,451.67 2510.11,447.942 \n 2511.28,444.05 2512.45,439.999 2513.63,435.795 2514.8,431.442 2515.97,426.945 2517.15,422.31 2518.32,417.542 2519.49,412.646 2520.66,407.63 2521.84,402.498 \n 2523.01,397.257 2524.18,391.914 2525.36,386.474 2526.53,380.946 2527.7,375.336 2528.88,369.651 2530.05,363.897 2531.22,358.08 2532.4,352.208 2533.57,346.288 \n 2534.74,340.327 2535.91,334.331 2537.09,328.31 2538.26,322.268 2539.43,316.215 2540.61,310.156 2541.78,304.099 2542.95,298.05 2544.13,292.018 2545.3,286.009 \n 2546.47,280.031 2547.65,274.089 2548.82,268.192 2549.99,262.347 2551.16,256.56 2552.34,250.838 2553.51,245.189 2554.68,239.619 2555.86,234.137 2557.03,228.747 \n 2558.2,223.458 2559.38,218.276 2560.55,213.205 2561.72,208.251 2562.9,203.421 2564.07,198.72 2565.24,194.155 2566.41,189.731 2567.59,185.453 2568.76,181.326 \n 2569.93,177.356 2571.11,173.547 2572.28,169.904 2573.45,166.431 2574.63,163.131 2575.8,160.01 2576.97,157.07 2578.15,154.315 2579.32,151.748 2580.49,149.373 \n 2581.66,147.192 2582.84,145.208 2584.01,143.422 2585.18,141.838 2586.36,140.457 2587.53,139.28 2588.7,138.308 2589.88,137.543 2591.05,136.986 2592.22,136.638 \n 2593.39,136.499 2594.57,136.57 2595.74,136.851 2596.91,137.341 2598.09,138.041 2599.26,138.949 2600.43,140.064 2601.61,141.386 2602.78,142.912 2603.95,144.642 \n 2605.13,146.571 2606.3,148.699 2607.47,151.023 2608.64,153.54 2609.82,156.246 2610.99,159.139 2612.16,162.213 2613.34,165.467 2614.51,168.894 2615.68,172.491 \n 2616.86,176.254 2618.03,180.178 2619.2,184.259 2620.38,188.493 2621.55,192.874 2622.72,197.397 2623.89,202.056 2625.07,206.846 2626.24,211.762 2627.41,216.798 \n 2628.59,221.948 2629.76,227.205 2630.93,232.564 2632.11,238.018 2633.28,243.561 2634.45,249.187 2635.63,254.887 2636.8,260.657 2637.97,266.487 2639.14,272.373 \n 2640.32,278.305 2641.49,284.277 2642.66,290.281 2643.84,296.309 2645.01,302.356 2646.18,308.415 2647.36,314.477 2648.53,320.537 2649.7,326.585 2650.88,332.616 \n 2652.05,338.621 2653.22,344.593 2654.39,350.526 2655.57,356.413 2656.74,362.245 2657.91,368.018 2659.09,373.723 2660.26,379.354 2661.43,384.904 2662.61,390.368 \n 2663.78,395.737 2664.95,401.007 2666.13,406.171 2667.3,411.222 2668.47,416.154 2669.64,420.962 2670.82,425.638 2671.99,430.179 2673.16,434.576 2674.34,438.826 \n 2675.51,442.924 2676.68,446.866 2677.86,450.647 2679.03,454.262 2680.2,457.706 2681.38,460.976 2682.55,464.066 2683.72,466.974 2684.89,469.696 2686.07,472.229 \n 2687.24,474.569 2688.41,476.714 2689.59,478.661 2690.76,480.408 2691.93,481.952 2693.11,483.294 2694.28,484.429 2695.45,485.358 2696.63,486.079 2697.8,486.592 \n 2698.97,486.896 2700.14,486.991 2701.32,486.876 2702.49,486.552 2703.66,486.019 2704.84,485.278 2706.01,484.329 2707.18,483.173 2708.36,481.813 2709.53,480.249 \n 2710.7,478.483 2711.88,476.518 2713.05,474.355 2714.22,471.998 2715.39,469.448 2716.57,466.71 2717.74,463.786 2718.91,460.681 2720.09,457.397 2721.26,453.94 \n \n \"/>\n</svg>\n"
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
]
},
{
"metadata": {},
"cell_type": "markdown",
"source": "### 斉次Dirichlet境界条件\n\n$T = 2π$ とし,\n\n$$\nx(0) = x(T) = 0\n$$\n\nと区間 $[0,T]$ の両端で $0$ になる微分方程式\n\n$$\n\\ddot x = -\\omega^2 x\n$$\n\nの解を探してみる. $\\omega = n/2$, $n=1,2,3,\\ldots$ のとき, \n\n$$\nx(t) = \\sin\\frac{nt}{2}\n$$\n\nは上の条件を満たす解になっている."
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "for k in 1:10\n plot_harmonicoscilator(; ω=0.5k, a=0.0, b=1.0, T=2π) |> display\nend",
"execution_count": 8,
"outputs": [
{
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1830.07,176.745 1831.22,177.095 1832.37,177.446 1833.51,177.799 1834.66,178.152 1835.81,178.507 1836.96,178.863 1838.1,179.221 \n 1839.25,179.579 1840.4,179.94 1841.54,180.301 1842.69,180.663 1843.84,181.027 1844.98,181.392 1846.13,181.759 1847.28,182.127 1848.43,182.495 1849.57,182.866 \n 1850.72,183.237 1851.87,183.61 1853.01,183.984 1854.16,184.359 1855.31,184.735 1856.45,185.113 1857.6,185.492 1858.75,185.872 1859.89,186.253 1861.04,186.635 \n 1862.19,187.019 1863.34,187.404 1864.48,187.79 1865.63,188.177 1866.78,188.566 1867.92,188.955 1869.07,189.346 1870.22,189.738 1871.36,190.132 1872.51,190.526 \n 1873.66,190.922 1874.81,191.318 1875.95,191.716 1877.1,192.115 1878.25,192.516 1879.39,192.917 1880.54,193.319 1881.69,193.723 1882.83,194.128 1883.98,194.534 \n 1885.13,194.941 1886.27,195.349 1887.42,195.759 1888.57,196.169 1889.72,196.581 1890.86,196.993 1892.01,197.407 1893.16,197.822 1894.3,198.238 1895.45,198.655 \n 1896.6,199.073 1897.74,199.493 1898.89,199.913 1900.04,200.334 1901.19,200.757 1902.33,201.18 1903.48,201.605 1904.63,202.031 1905.77,202.458 1906.92,202.885 \n 1908.07,203.314 1909.21,203.744 1910.36,204.175 1911.51,204.607 1912.65,205.04 1913.8,205.474 1914.95,205.909 1916.1,206.345 1917.24,206.782 1918.39,207.221 \n 1919.54,207.66 1920.68,208.1 1921.83,208.541 1922.98,208.983 1924.12,209.426 1925.27,209.87 1926.42,210.315 1927.57,210.761 1928.71,211.209 1929.86,211.657 \n 1931.01,212.105 1932.15,212.555 1933.3,213.006 1934.45,213.458 1935.59,213.911 1936.74,214.365 1937.89,214.819 1939.03,215.275 1940.18,215.731 1941.33,216.189 \n 1942.48,216.647 1943.62,217.106 1944.77,217.566 1945.92,218.028 1947.06,218.49 1948.21,218.952 1949.36,219.416 1950.5,219.881 1951.65,220.346 1952.8,220.813 \n 1953.95,221.28 1955.09,221.748 1956.24,222.217 1957.39,222.687 1958.53,223.158 1959.68,223.629 1960.83,224.102 1961.97,224.575 1963.12,225.049 1964.27,225.524 \n 1965.41,226 1966.56,226.476 1967.71,226.954 1968.86,227.432 1970,227.911 1971.15,228.391 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2043.41,260.023 2044.55,260.544 \n 2045.7,261.065 2046.85,261.587 2048,262.11 2049.14,262.632 2050.29,263.156 2051.44,263.68 2052.58,264.204 2053.73,264.728 2054.88,265.253 2056.02,265.779 \n 2057.17,266.305 2058.32,266.831 2059.47,267.358 2060.61,267.885 2061.76,268.413 2062.91,268.941 2064.05,269.469 2065.2,269.998 2066.35,270.527 2067.49,271.057 \n 2068.64,271.587 2069.79,272.117 2070.93,272.648 2072.08,273.179 2073.23,273.71 2074.38,274.242 2075.52,274.774 2076.67,275.307 2077.82,275.839 2078.96,276.372 \n 2080.11,276.906 2081.26,277.44 2082.4,277.974 2083.55,278.508 2084.7,279.043 2085.85,279.578 2086.99,280.113 2088.14,280.648 2089.29,281.184 2090.43,281.72 \n 2091.58,282.257 2092.73,282.793 2093.87,283.33 2095.02,283.867 2096.17,284.405 2097.31,284.942 2098.46,285.48 2099.61,286.018 2100.76,286.557 2101.9,287.095 \n 2103.05,287.634 2104.2,288.173 2105.34,288.712 2106.49,289.251 2107.64,289.791 2108.78,290.331 2109.93,290.871 2111.08,291.411 2112.23,291.951 2113.37,292.492 \n 2114.52,293.032 2115.67,293.573 2116.81,294.114 2117.96,294.655 2119.11,295.196 2120.25,295.738 2121.4,296.279 2122.55,296.821 2123.69,297.362 2124.84,297.904 \n 2125.99,298.446 2127.14,298.988 2128.28,299.53 2129.43,300.073 2130.58,300.615 2131.72,301.158 2132.87,301.7 2134.02,302.243 2135.16,302.785 2136.31,303.328 \n 2137.46,303.871 2138.61,304.414 2139.75,304.956 2140.9,305.499 2142.05,306.042 2143.19,306.585 2144.34,307.128 2145.49,307.671 2146.63,308.214 2147.78,308.757 \n 2148.93,309.3 2150.07,309.843 2151.22,310.386 2152.37,310.929 2153.52,311.472 2154.66,312.015 2155.81,312.558 2156.96,313.101 2158.1,313.644 2159.25,314.187 \n 2160.4,314.73 2161.54,315.272 2162.69,315.815 2163.84,316.357 2164.99,316.9 2166.13,317.442 2167.28,317.985 2168.43,318.527 2169.57,319.069 2170.72,319.611 \n 2171.87,320.153 2173.01,320.695 2174.16,321.237 2175.31,321.778 2176.45,322.32 2177.6,322.861 2178.75,323.403 2179.9,323.944 2181.04,324.485 2182.19,325.025 \n 2183.34,325.566 2184.48,326.106 2185.63,326.647 2186.78,327.187 2187.92,327.727 2189.07,328.267 2190.22,328.806 2191.37,329.345 2192.51,329.885 2193.66,330.424 \n 2194.81,330.962 2195.95,331.501 2197.1,332.039 2198.25,332.577 2199.39,333.115 2200.54,333.653 2201.69,334.19 2202.83,334.727 2203.98,335.264 2205.13,335.801 \n 2206.28,336.337 2207.42,336.873 2208.57,337.409 2209.72,337.945 2210.86,338.48 2212.01,339.015 2213.16,339.549 2214.3,340.084 2215.45,340.618 2216.6,341.152 \n 2217.75,341.685 2218.89,342.218 2220.04,342.751 2221.19,343.283 2222.33,343.815 2223.48,344.347 2224.63,344.878 2225.77,345.41 2226.92,345.94 2228.07,346.471 \n 2229.21,347 2230.36,347.53 2231.51,348.059 2232.66,348.588 2233.8,349.116 2234.95,349.644 2236.1,350.172 2237.24,350.699 2238.39,351.226 2239.54,351.752 \n 2240.68,352.278 2241.83,352.804 2242.98,353.329 2244.13,353.854 2245.27,354.378 2246.42,354.902 2247.57,355.425 2248.71,355.948 2249.86,356.47 2251.01,356.992 \n 2252.15,357.514 2253.3,358.035 2254.45,358.555 2255.6,359.075 2256.74,359.595 2257.89,360.114 2259.04,360.632 2260.18,361.15 2261.33,361.668 2262.48,362.184 \n 2263.62,362.701 2264.77,363.217 2265.92,363.732 2267.06,364.247 2268.21,364.761 2269.36,365.275 2270.51,365.788 2271.65,366.301 2272.8,366.813 2273.95,367.324 \n 2275.09,367.835 2276.24,368.345 2277.39,368.855 2278.53,369.364 2279.68,369.873 2280.83,370.381 2281.98,370.888 2283.12,371.395 2284.27,371.901 2285.42,372.406 \n 2286.56,372.911 2287.71,373.415 2288.86,373.919 2290,374.422 2291.15,374.924 2292.3,375.426 2293.44,375.927 2294.59,376.427 2295.74,376.927 2296.89,377.426 \n 2298.03,377.924 2299.18,378.422 2300.33,378.918 2301.47,379.415 2302.62,379.91 2303.77,380.405 2304.91,380.899 2306.06,381.393 2307.21,381.885 2308.36,382.377 \n 2309.5,382.869 2310.65,383.359 2311.8,383.849 2312.94,384.338 2314.09,384.826 2315.24,385.314 2316.38,385.801 2317.53,386.287 2318.68,386.772 2319.82,387.256 \n 2320.97,387.74 2322.12,388.223 2323.27,388.705 2324.41,389.187 2325.56,389.667 2326.71,390.147 2327.85,390.626 2329,391.104 2330.15,391.581 2331.29,392.058 \n 2332.44,392.534 2333.59,393.009 2334.74,393.483 2335.88,393.956 2337.03,394.428 2338.18,394.9 2339.32,395.371 2340.47,395.84 2341.62,396.309 2342.76,396.778 \n 2343.91,397.245 2345.06,397.711 2346.2,398.177 2347.35,398.641 2348.5,399.105 2349.65,399.568 2350.79,400.03 2351.94,400.491 2353.09,400.951 2354.23,401.41 \n 2355.38,401.869 2356.53,402.326 2357.67,402.783 2358.82,403.238 2359.97,403.693 2361.12,404.147 2362.26,404.599 2363.41,405.051 2364.56,405.502 2365.7,405.952 \n 2366.85,406.401 2368,406.849 2369.14,407.296 2370.29,407.742 2371.44,408.187 2372.58,408.631 2373.73,409.074 2374.88,409.516 2376.03,409.957 2377.17,410.398 \n 2378.32,410.837 2379.47,411.275 2380.61,411.712 2381.76,412.148 2382.91,412.583 2384.05,413.017 2385.2,413.45 2386.35,413.882 2387.5,414.313 2388.64,414.743 \n 2389.79,415.172 2390.94,415.6 2392.08,416.027 2393.23,416.452 2394.38,416.877 2395.52,417.301 2396.67,417.723 2397.82,418.144 2398.96,418.565 2400.11,418.984 \n 2401.26,419.402 2402.41,419.819 2403.55,420.235 2404.7,420.65 2405.85,421.064 2406.99,421.477 2408.14,421.888 2409.29,422.299 2410.43,422.708 2411.58,423.116 \n 2412.73,423.523 2413.88,423.929 2415.02,424.334 2416.17,424.738 2417.32,425.14 2418.46,425.542 2419.61,425.942 2420.76,426.341 2421.9,426.739 2423.05,427.136 \n 2424.2,427.532 2425.34,427.926 2426.49,428.319 2427.64,428.711 2428.79,429.102 2429.93,429.492 2431.08,429.881 2432.23,430.268 2433.37,430.654 2434.52,431.039 \n 2435.67,431.423 2436.81,431.805 2437.96,432.187 2439.11,432.567 2440.26,432.946 2441.4,433.323 2442.55,433.7 2443.7,434.075 2444.84,434.449 2445.99,434.822 \n 2447.14,435.193 2448.28,435.563 2449.43,435.932 2450.58,436.3 2451.72,436.666 2452.87,437.032 2454.02,437.395 2455.17,437.758 2456.31,438.119 2457.46,438.48 \n 2458.61,438.838 2459.75,439.196 2460.9,439.552 2462.05,439.907 2463.19,440.261 2464.34,440.613 2465.49,440.964 2466.64,441.314 2467.78,441.662 2468.93,442.01 \n 2470.08,442.355 2471.22,442.7 2472.37,443.043 2473.52,443.385 2474.66,443.725 2475.81,444.065 2476.96,444.402 2478.1,444.739 2479.25,445.074 2480.4,445.408 \n 2481.55,445.74 2482.69,446.071 2483.84,446.401 2484.99,446.73 2486.13,447.057 2487.28,447.382 2488.43,447.707 2489.57,448.03 2490.72,448.351 2491.87,448.671 \n 2493.02,448.99 2494.16,449.307 2495.31,449.623 2496.46,449.938 2497.6,450.251 2498.75,450.563 2499.9,450.874 2501.04,451.183 2502.19,451.49 2503.34,451.796 \n 2504.48,452.101 2505.63,452.405 2506.78,452.706 2507.93,453.007 2509.07,453.306 2510.22,453.604 2511.37,453.9 2512.51,454.195 2513.66,454.488 2514.81,454.78 \n 2515.95,455.071 2517.1,455.36 2518.25,455.647 2519.4,455.934 2520.54,456.218 2521.69,456.501 2522.84,456.783 2523.98,457.064 2525.13,457.342 2526.28,457.62 \n 2527.42,457.896 2528.57,458.17 2529.72,458.443 2530.86,458.715 2532.01,458.985 2533.16,459.253 2534.31,459.52 2535.45,459.786 2536.6,460.05 2537.75,460.312 \n 2538.89,460.574 2540.04,460.833 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862.44,405.722 863.573,406.619 864.706,407.514 865.839,408.404 866.972,409.29 868.104,410.172 \n 869.237,411.05 870.37,411.925 871.503,412.795 872.636,413.661 873.769,414.523 874.902,415.381 876.035,416.235 877.168,417.084 878.3,417.929 879.433,418.77 \n 880.566,419.606 881.699,420.438 882.832,421.266 883.965,422.089 885.098,422.908 886.231,423.722 887.364,424.532 888.496,425.337 889.629,426.138 890.762,426.934 \n 891.895,427.725 893.028,428.512 894.161,429.293 895.294,430.071 896.427,430.843 897.56,431.61 898.692,432.373 899.825,433.131 900.958,433.883 902.091,434.631 \n 903.224,435.374 904.357,436.112 905.49,436.845 906.623,437.572 907.756,438.295 908.888,439.013 910.021,439.725 911.154,440.432 912.287,441.134 913.42,441.831 \n 914.553,442.523 915.686,443.209 916.819,443.89 917.951,444.565 919.084,445.236 920.217,445.9 921.35,446.56 922.483,447.214 923.616,447.862 924.749,448.505 \n 925.882,449.143 927.015,449.775 928.147,450.401 929.28,451.022 930.413,451.637 931.546,452.246 932.679,452.85 933.812,453.448 934.945,454.041 936.078,454.627 \n 937.211,455.208 938.343,455.783 939.476,456.353 940.609,456.916 941.742,457.474 942.875,458.026 944.008,458.572 945.141,459.112 946.274,459.646 947.407,460.174 \n 948.539,460.696 949.672,461.212 950.805,461.722 951.938,462.226 953.071,462.724 954.204,463.216 955.337,463.702 956.47,464.181 957.603,464.655 958.735,465.122 \n 959.868,465.583 961.001,466.039 962.134,466.487 963.267,466.93 964.4,467.366 965.533,467.796 966.666,468.22 967.799,468.638 968.931,469.049 970.064,469.454 \n 971.197,469.852 972.33,470.245 973.463,470.63 974.596,471.01 975.729,471.383 976.862,471.749 977.995,472.11 979.127,472.463 980.26,472.811 981.393,473.151 \n 982.526,473.486 983.659,473.813 984.792,474.135 985.925,474.45 987.058,474.758 988.191,475.06 989.323,475.355 990.456,475.643 991.589,475.925 992.722,476.201 \n 993.855,476.47 994.988,476.732 996.121,476.988 997.254,477.237 998.387,477.479 999.519,477.715 1000.65,477.944 1001.79,478.166 1002.92,478.382 1004.05,478.591 \n 1005.18,478.793 1006.32,478.988 1007.45,479.177 1008.58,479.359 1009.72,479.535 1010.85,479.703 1011.98,479.865 1013.11,480.02 1014.25,480.169 1015.38,480.31 \n 1016.51,480.445 1017.65,480.573 1018.78,480.694 1019.91,480.809 1021.04,480.916 1022.18,481.017 1023.31,481.111 1024.44,481.198 1025.58,481.279 1026.71,481.352 \n 1027.84,481.419 1028.97,481.479 1030.11,481.532 1031.24,481.578 1032.37,481.617 1033.51,481.65 1034.64,481.676 1035.77,481.695 1036.9,481.707 1038.04,481.712 \n 1039.17,481.71 1040.3,481.702 1041.44,481.686 1042.57,481.664 1043.7,481.635 1044.84,481.599 1045.97,481.557 1047.1,481.507 1048.23,481.451 1049.37,481.388 \n 1050.5,481.318 1051.63,481.241 1052.77,481.157 1053.9,481.067 1055.03,480.969 1056.16,480.865 1057.3,480.754 1058.43,480.637 1059.56,480.512 1060.7,480.381 \n 1061.83,480.243 1062.96,480.098 1064.09,479.946 1065.23,479.788 1066.36,479.622 1067.49,479.45 1068.63,479.272 1069.76,479.086 1070.89,478.894 1072.02,478.695 \n 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1839.87,277.975 1841.01,279.044 1842.14,280.114 1843.27,281.186 1844.4,282.258 1845.54,283.332 1846.67,284.406 1847.8,285.482 1848.94,286.558 \n 1850.07,287.635 1851.2,288.714 1852.33,289.793 1853.47,290.872 1854.6,291.953 1855.73,293.034 1856.87,294.116 1858,295.198 1859.13,296.281 1860.26,297.364 \n 1861.4,298.448 1862.53,299.532 1863.66,300.617 1864.8,301.702 1865.93,302.787 1867.06,303.872 1868.19,304.958 1869.33,306.044 1870.46,307.13 1871.59,308.216 \n 1872.73,309.302 1873.86,310.388 1874.99,311.474 1876.12,312.56 1877.26,313.646 1878.39,314.732 1879.52,315.817 1880.66,316.902 1881.79,317.987 1882.92,319.072 \n 1884.06,320.156 1885.19,321.239 1886.32,322.322 1887.45,323.405 1888.59,324.487 1889.72,325.568 1890.85,326.649 1891.99,327.729 1893.12,328.808 1894.25,329.887 \n 1895.38,330.965 1896.52,332.041 1897.65,333.117 1898.78,334.192 1899.92,335.266 1901.05,336.339 1902.18,337.411 1903.31,338.482 1904.45,339.551 1905.58,340.62 \n 1906.71,341.687 1907.85,342.753 1908.98,343.817 1910.11,344.88 1911.24,345.942 1912.38,347.002 1913.51,348.061 1914.64,349.118 1915.78,350.174 1916.91,351.228 \n 1918.04,352.28 1919.17,353.33 1920.31,354.379 1921.44,355.426 1922.57,356.471 1923.71,357.515 1924.84,358.556 1925.97,359.596 1927.1,360.633 1928.24,361.669 \n 1929.37,362.702 1930.5,363.733 1931.64,364.762 1932.77,365.789 1933.9,366.814 1935.04,367.836 1936.17,368.856 1937.3,369.874 1938.43,370.889 1939.57,371.902 \n 1940.7,372.912 1941.83,373.92 1942.97,374.925 1944.1,375.928 1945.23,376.928 1946.36,377.925 1947.5,378.919 1948.63,379.911 1949.76,380.9 1950.9,381.886 \n 1952.03,382.87 1953.16,383.85 1954.29,384.827 1955.43,385.802 1956.56,386.773 1957.69,387.741 1958.83,388.706 1959.96,389.669 1961.09,390.627 1962.22,391.583 \n 1963.36,392.535 1964.49,393.484 1965.62,394.43 1966.76,395.372 1967.89,396.311 1969.02,397.247 1970.15,398.179 1971.29,399.107 1972.42,400.032 1973.55,400.953 \n 1974.69,401.871 1975.82,402.785 1976.95,403.695 1978.08,404.602 1979.22,405.504 1980.35,406.403 1981.48,407.298 1982.62,408.189 1983.75,409.076 1984.88,409.96 \n 1986.02,410.839 1987.15,411.714 1988.28,412.585 1989.41,413.452 1990.55,414.315 1991.68,415.174 1992.81,416.029 1993.95,416.879 1995.08,417.725 1996.21,418.567 \n 1997.34,419.404 1998.48,420.237 1999.61,421.066 2000.74,421.89 2001.88,422.71 2003.01,423.525 2004.14,424.336 2005.27,425.142 2006.41,425.944 2007.54,426.74 \n 2008.67,427.533 2009.81,428.32 2010.94,429.103 2012.07,429.881 2013.2,430.655 2014.34,431.423 2015.47,432.187 2016.6,432.946 2017.74,433.7 2018.87,434.449 \n 2020,435.193 2021.13,435.932 2022.27,436.666 2023.4,437.395 2024.53,438.119 2025.67,438.838 2026.8,439.551 2027.93,440.26 2029.06,440.963 2030.2,441.661 \n 2031.33,442.354 2032.46,443.042 2033.6,443.724 2034.73,444.401 2035.86,445.073 2037,445.739 2038.13,446.4 2039.26,447.055 2040.39,447.705 2041.53,448.35 \n 2042.66,448.988 2043.79,449.622 2044.93,450.25 2046.06,450.872 2047.19,451.489 2048.32,452.1 2049.46,452.705 2050.59,453.305 2051.72,453.899 2052.86,454.487 \n 2053.99,455.07 2055.12,455.647 2056.25,456.218 2057.39,456.783 2058.52,457.342 2059.65,457.896 2060.79,458.443 2061.92,458.985 2063.05,459.521 2064.18,460.051 \n 2065.32,460.574 2066.45,461.092 2067.58,461.604 2068.72,462.11 2069.85,462.609 2070.98,463.103 2072.11,463.591 2073.25,464.072 2074.38,464.547 2075.51,465.016 \n 2076.65,465.479 2077.78,465.936 2078.91,466.387 2080.04,466.831 2081.18,467.269 2082.31,467.701 2083.44,468.126 2084.58,468.545 2085.71,468.958 2086.84,469.365 \n 2087.98,469.765 2089.11,470.159 2090.24,470.546 2091.37,470.927 2092.51,471.302 2093.64,471.67 2094.77,472.032 2095.91,472.388 2097.04,472.736 2098.17,473.079 \n 2099.3,473.415 2100.44,473.744 2101.57,474.067 2102.7,474.383 2103.84,474.693 2104.97,474.996 2106.1,475.293 2107.23,475.583 2108.37,475.867 2109.5,476.144 \n 2110.63,476.414 2111.77,476.677 2112.9,476.934 2114.03,477.185 2115.16,477.429 2116.3,477.666 2117.43,477.896 2118.56,478.12 2119.7,478.337 2120.83,478.547 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1712.6,205.053 1713.72,206.358 \n 1714.83,207.673 1715.95,208.996 1717.07,210.328 1718.19,211.669 1719.31,213.018 1720.43,214.377 1721.55,215.743 1722.67,217.118 1723.79,218.501 1724.91,219.892 \n 1726.03,221.291 1727.15,222.698 1728.27,224.113 1729.39,225.535 1730.51,226.965 1731.63,228.401 1732.75,229.845 1733.87,231.297 1734.99,232.755 1736.11,234.219 \n 1737.22,235.691 1738.34,237.169 1739.46,238.653 1740.58,240.144 1741.7,241.641 1742.82,243.143 1743.94,244.652 1745.06,246.166 1746.18,247.686 1747.3,249.212 \n 1748.42,250.742 1749.54,252.278 1750.66,253.819 1751.78,255.365 1752.9,256.916 1754.02,258.471 1755.14,260.031 1756.26,261.595 1757.38,263.164 1758.49,264.736 \n 1759.61,266.312 1760.73,267.893 1761.85,269.476 1762.97,271.064 1764.09,272.654 1765.21,274.248 1766.33,275.846 1767.45,277.446 1768.57,279.048 1769.69,280.654 \n 1770.81,282.262 1771.93,283.872 1773.05,285.485 1774.17,287.1 1775.29,288.717 1776.41,290.335 1777.53,291.955 1778.65,293.577 1779.77,295.2 1780.88,296.825 \n 1782,298.45 1783.12,300.076 1784.24,301.704 1785.36,303.331 1786.48,304.96 1787.6,306.589 1788.72,308.217 1789.84,309.846 1790.96,311.475 1792.08,313.104 \n 1793.2,314.732 1794.32,316.36 1795.44,317.987 1796.56,319.614 1797.68,321.239 1798.8,322.864 1799.92,324.487 1801.04,326.109 1802.16,327.729 1803.27,329.347 \n 1804.39,330.964 1805.51,332.579 1806.63,334.192 1807.75,335.802 1808.87,337.41 1809.99,339.016 1811.11,340.619 1812.23,342.219 1813.35,343.816 1814.47,345.41 \n 1815.59,347 1816.71,348.588 1817.83,350.172 1818.95,351.752 1820.07,353.328 1821.19,354.901 1822.31,356.469 1823.43,358.033 1824.55,359.593 1825.66,361.148 \n 1826.78,362.699 1827.9,364.245 1829.02,365.786 1830.14,367.321 1831.26,368.852 1832.38,370.378 1833.5,371.898 1834.62,373.412 1835.74,374.92 1836.86,376.423 \n 1837.98,377.92 1839.1,379.411 1840.22,380.895 1841.34,382.373 1842.46,383.845 1843.58,385.309 1844.7,386.767 1845.82,388.218 1846.93,389.662 1848.05,391.099 \n 1849.17,392.529 1850.29,393.951 1851.41,395.366 1852.53,396.773 1853.65,398.172 1854.77,399.563 1855.89,400.946 1857.01,402.321 1858.13,403.688 1859.25,405.046 \n 1860.37,406.396 1861.49,407.737 1862.61,409.069 1863.73,410.392 1864.85,411.707 1865.97,413.012 1867.09,414.308 1868.21,415.594 1869.32,416.871 1870.44,418.139 \n 1871.56,419.396 1872.68,420.644 1873.8,421.882 1874.92,423.11 1876.04,424.328 1877.16,425.535 1878.28,426.732 1879.4,427.919 1880.52,429.095 1881.64,430.26 \n 1882.76,431.414 1883.88,432.558 1885,433.691 1886.12,434.812 1887.24,435.923 1888.36,437.022 1889.48,438.109 1890.6,439.185 1891.71,440.25 1892.83,441.303 \n 1893.95,442.344 1895.07,443.373 1896.19,444.391 1897.31,445.396 1898.43,446.389 1899.55,447.37 1900.67,448.339 1901.79,449.295 1902.91,450.239 1904.03,451.17 \n 1905.15,452.088 1906.27,452.994 1907.39,453.887 1908.51,454.767 1909.63,455.634 1910.75,456.488 1911.87,457.329 1912.99,458.157 1914.1,458.972 1915.22,459.773 \n 1916.34,460.561 1917.46,461.335 1918.58,462.096 1919.7,462.843 1920.82,463.577 1921.94,464.297 1923.06,465.003 1924.18,465.695 1925.3,466.373 1926.42,467.037 \n 1927.54,467.687 1928.66,468.323 1929.78,468.945 1930.9,469.552 1932.02,470.146 1933.14,470.724 1934.26,471.289 1935.38,471.839 1936.49,472.374 1937.61,472.895 \n 1938.73,473.402 1939.85,473.894 1940.97,474.371 1942.09,474.833 1943.21,475.28 1944.33,475.713 1945.45,476.131 1946.57,476.534 1947.69,476.922 1948.81,477.295 \n 1949.93,477.653 1951.05,477.996 1952.17,478.325 1953.29,478.638 1954.41,478.935 1955.53,479.218 1956.65,479.486 1957.76,479.738 1958.88,479.975 1960,480.197 \n 1961.12,480.404 1962.24,480.596 1963.36,480.772 1964.48,480.933 1965.6,481.078 1966.72,481.209 1967.84,481.324 1968.96,481.423 1970.08,481.508 1971.2,481.577 \n 1972.32,481.63 1973.44,481.669 1974.56,481.692 1975.68,481.699 1976.8,481.691 1977.92,481.668 1979.04,481.63 1980.15,481.576 1981.27,481.507 1982.39,481.422 \n 1983.51,481.322 1984.63,481.207 1985.75,481.077 1986.87,480.931 1987.99,480.77 1989.11,480.594 1990.23,480.402 1991.35,480.195 1992.47,479.973 1993.59,479.736 \n 1994.71,479.483 1995.83,479.216 1996.95,478.933 1998.07,478.635 1999.19,478.322 2000.31,477.994 2001.43,477.651 2002.54,477.293 2003.66,476.92 2004.78,476.532 \n 2005.9,476.129 2007.02,475.712 2008.14,475.279 2009.26,474.832 2010.38,474.369 2011.5,473.893 2012.62,473.401 2013.74,472.895 2014.86,472.374 2015.98,471.839 \n 2017.1,471.289 2018.22,470.724 2019.34,470.146 2020.46,469.553 2021.58,468.945 2022.7,468.324 2023.82,467.688 2024.93,467.038 2026.05,466.374 2027.17,465.696 \n 2028.29,465.004 2029.41,464.298 2030.53,463.578 2031.65,462.845 2032.77,462.097 2033.89,461.337 2035.01,460.562 2036.13,459.774 2037.25,458.973 2038.37,458.158 \n 2039.49,457.33 2040.61,456.489 2041.73,455.635 2042.85,454.767 2043.97,453.887 2045.09,452.994 2046.2,452.088 2047.32,451.169 2048.44,450.237 2049.56,449.294 \n 2050.68,448.337 2051.8,447.368 2052.92,446.387 2054.04,445.394 2055.16,444.388 2056.28,443.371 2057.4,442.342 2058.52,441.3 2059.64,440.247 2060.76,439.183 \n 2061.88,438.107 2063,437.019 2064.12,435.92 2065.24,434.809 2066.36,433.688 2067.48,432.555 2068.59,431.411 2069.71,430.257 2070.83,429.092 2071.95,427.916 \n 2073.07,426.729 2074.19,425.532 2075.31,424.325 2076.43,423.107 2077.55,421.879 2078.67,420.642 2079.79,419.394 2080.91,418.136 2082.03,416.869 2083.15,415.592 \n 2084.27,414.306 2085.39,413.01 2086.51,411.705 2087.63,410.391 2088.75,409.068 2089.87,407.736 2090.98,406.395 2092.1,405.046 2093.22,403.688 2094.34,402.321 \n 2095.46,400.946 2096.58,399.563 2097.7,398.172 2098.82,396.773 2099.94,395.367 2101.06,393.952 2102.18,392.53 2103.3,391.1 2104.42,389.663 2105.54,388.219 \n 2106.66,386.768 2107.78,385.31 2108.9,383.845 2110.02,382.374 2111.14,380.896 2112.26,379.411 2113.37,377.92 2114.49,376.423 2115.61,374.92 2116.73,373.411 \n 2117.85,371.897 2118.97,370.376 2120.09,368.851 2121.21,367.32 2122.33,365.783 2123.45,364.242 2124.57,362.696 2125.69,361.145 2126.81,359.589 2127.93,358.029 \n 2129.05,356.465 2130.17,354.896 2131.29,353.323 2132.41,351.747 2133.53,350.166 2134.64,348.582 2135.76,346.995 2136.88,345.404 2138,343.81 2139.12,342.213 \n 2140.24,340.612 2141.36,339.009 2142.48,337.404 2143.6,335.796 2144.72,334.185 2145.84,332.572 2146.96,330.957 2148.08,329.34 2149.2,327.722 2150.32,326.101 \n 2151.44,324.479 2152.56,322.856 2153.68,321.232 2154.8,319.606 2155.92,317.98 2157.03,316.352 2158.15,314.725 2159.27,313.096 2160.39,311.468 2161.51,309.839 \n 2162.63,308.21 2163.75,306.581 2164.87,304.952 2165.99,303.324 2167.11,301.696 2168.23,300.069 2169.35,298.443 2170.47,296.818 2171.59,295.194 2172.71,293.571 \n 2173.83,291.949 2174.95,290.329 2176.07,288.711 2177.19,287.094 2178.31,285.48 2179.42,283.867 2180.54,282.257 2181.66,280.649 2182.78,279.043 2183.9,277.441 \n 2185.02,275.841 2186.14,274.244 2187.26,272.65 2188.38,271.059 2189.5,269.472 2190.62,267.888 2191.74,266.308 2192.86,264.731 2193.98,263.159 2195.1,261.59 \n 2196.22,260.026 2197.34,258.466 2198.46,256.911 2199.58,255.36 2200.7,253.814 2201.81,252.273 2202.93,250.737 2204.05,249.206 2205.17,247.68 2206.29,246.16 \n 2207.41,244.645 2208.53,243.136 2209.65,241.633 2210.77,240.136 2211.89,238.645 2213.01,237.161 2214.13,235.682 2215.25,234.211 2216.37,232.746 2217.49,231.288 \n 2218.61,229.836 2219.73,228.392 2220.85,226.955 2221.97,225.526 2223.08,224.104 2224.2,222.689 2225.32,221.282 2226.44,219.883 2227.56,218.492 2228.68,217.108 \n 2229.8,215.733 2230.92,214.366 2232.04,213.008 2233.16,211.658 2234.28,210.317 2235.4,208.985 2236.52,207.661 2237.64,206.347 2238.76,205.041 2239.88,203.745 \n 2241,202.459 2242.12,201.181 2243.24,199.914 2244.36,198.656 2245.47,197.408 2246.59,196.17 2247.71,194.942 2248.83,193.724 2249.95,192.517 2251.07,191.319 \n 2252.19,190.133 2253.31,188.957 2254.43,187.791 2255.55,186.637 2256.67,185.493 2257.79,184.36 2258.91,183.239 2260.03,182.129 2261.15,181.03 2262.27,179.942 \n 2263.39,178.866 2264.51,177.801 2265.63,176.749 2266.75,175.708 2267.86,174.678 2268.98,173.661 2270.1,172.656 2271.22,171.663 2272.34,170.682 2273.46,169.714 \n 2274.58,168.758 2275.7,167.814 2276.82,166.883 2277.94,165.965 2279.06,165.059 2280.18,164.167 2281.3,163.287 2282.42,162.42 2283.54,161.566 2284.66,160.725 \n 2285.78,159.898 2286.9,159.083 2288.02,158.282 2289.14,157.495 2290.25,156.721 2291.37,155.96 2292.49,155.213 2293.61,154.48 2294.73,153.76 2295.85,153.055 \n 2296.97,152.363 2298.09,151.685 2299.21,151.021 2300.33,150.371 2301.45,149.736 2302.57,149.114 2303.69,148.507 2304.81,147.914 2305.93,147.335 2307.05,146.771 \n 2308.17,146.221 2309.29,145.685 2310.41,145.164 2311.53,144.657 2312.64,144.165 2313.76,143.688 2314.88,143.225 2316,142.777 2317.12,142.344 2318.24,141.926 \n 2319.36,141.522 2320.48,141.134 2321.6,140.76 2322.72,140.402 2323.84,140.058 2324.96,139.729 2326.08,139.416 2327.2,139.117 2328.32,138.834 2329.44,138.566 \n 2330.56,138.313 2331.68,138.075 2332.8,137.852 2333.91,137.645 2335.03,137.453 2336.15,137.276 2337.27,137.115 2338.39,136.969 2339.51,136.838 2340.63,136.722 \n 2341.75,136.622 2342.87,136.538 2343.99,136.468 2345.11,136.415 2346.23,136.376 2347.35,136.353 2348.47,136.345 2349.59,136.353 2350.71,136.376 2351.83,136.415 \n 2352.95,136.468 2354.07,136.538 2355.19,136.622 2356.3,136.722 2357.42,136.838 2358.54,136.969 2359.66,137.115 2360.78,137.276 2361.9,137.453 2363.02,137.645 \n 2364.14,137.853 2365.26,138.075 2366.38,138.313 2367.5,138.566 2368.62,138.834 2369.74,139.118 2370.86,139.417 2371.98,139.73 2373.1,140.059 2374.22,140.403 \n 2375.34,140.762 2376.46,141.136 2377.58,141.524 2378.69,141.928 2379.81,142.347 2380.93,142.78 2382.05,143.229 2383.17,143.692 2384.29,144.169 2385.41,144.662 \n 2386.53,145.169 2387.65,145.69 2388.77,146.226 2389.89,146.777 2391.01,147.342 2392.13,147.921 2393.25,148.514 2394.37,149.122 2395.49,149.744 2396.61,150.38 \n 2397.73,151.03 2398.85,151.695 2399.97,152.373 2401.08,153.065 2402.2,153.771 2403.32,154.49 2404.44,155.224 2405.56,155.971 2406.68,156.732 2407.8,157.506 \n 2408.92,158.294 2410.04,159.095 2411.16,159.91 2412.28,160.737 2413.4,161.578 2414.52,162.432 2415.64,163.299 2416.76,164.179 2417.88,165.072 2419,165.978 \n 2420.12,166.897 2421.24,167.828 2422.35,168.771 2423.47,169.727 2424.59,170.696 2425.71,171.677 2426.83,172.67 2427.95,173.675 2429.07,174.692 2430.19,175.722 \n 2431.31,176.763 2432.43,177.816 2433.55,178.88 2434.67,179.956 2435.79,181.044 2436.91,182.143 2438.03,183.254 2439.15,184.375 2440.27,185.508 2441.39,186.652 \n 2442.51,187.806 2443.63,188.972 2444.74,190.148 2445.86,191.335 2446.98,192.532 2448.1,193.74 2449.22,194.958 2450.34,196.186 2451.46,197.424 2452.58,198.672 \n 2453.7,199.93 2454.82,201.198 2455.94,202.476 2457.06,203.762 2458.18,205.059 2459.3,206.364 2460.42,207.679 2461.54,209.003 2462.66,210.336 2463.78,211.677 \n 2464.9,213.027 2466.02,214.386 2467.13,215.753 2468.25,217.129 2469.37,218.512 2470.49,219.904 2471.61,221.304 2472.73,222.711 2473.85,224.126 2474.97,225.549 \n 2476.09,226.979 2477.21,228.416 2478.33,229.861 2479.45,231.312 2480.57,232.77 2481.69,234.235 2482.81,235.707 2483.93,237.185 2485.05,238.67 2486.17,240.161 \n 2487.29,241.658 2488.41,243.16 2489.52,244.669 2490.64,246.184 2491.76,247.704 2492.88,249.23 2494,250.761 2495.12,252.297 2496.24,253.838 2497.36,255.384 \n 2498.48,256.935 2499.6,258.49 2500.72,260.05 2501.84,261.614 2502.96,263.183 2504.08,264.755 2505.2,266.332 2506.32,267.912 2507.44,269.496 2508.56,271.083 \n 2509.68,272.674 2510.79,274.267 2511.91,275.864 2513.03,277.464 2514.15,279.067 2515.27,280.672 2516.39,282.28 2517.51,283.89 2518.63,285.503 2519.75,287.117 \n 2520.87,288.734 2521.99,290.352 2523.11,291.972 2524.23,293.594 2525.35,295.216 2526.47,296.841 2527.59,298.466 2528.71,300.092 2529.83,301.719 2530.95,303.346 \n 2532.07,304.975 2533.18,306.603 2534.3,308.232 2535.42,309.861 2536.54,311.49 2537.66,313.118 2538.78,314.747 2539.9,316.375 2541.02,318.002 2542.14,319.628 \n 2543.26,321.254 2544.38,322.878 2545.5,324.502 2546.62,326.124 2547.74,327.744 2548.86,329.363 2549.98,330.98 2551.1,332.595 2552.22,334.209 2553.34,335.819 \n 2554.46,337.428 2555.57,339.034 2556.69,340.638 2557.81,342.238 2558.93,343.836 2560.05,345.431 2561.17,347.022 2562.29,348.61 2563.41,350.195 2564.53,351.776 \n 2565.65,353.353 2566.77,354.926 2567.89,356.496 2569.01,358.06 2570.13,359.621 2571.25,361.177 2572.37,362.728 2573.49,364.275 2574.61,365.816 2575.73,367.353 \n 2576.85,368.884 2577.96,370.409 2579.08,371.93 2580.2,373.444 2581.32,374.953 2582.44,376.455 2583.56,377.952 2584.68,379.443 2585.8,380.928 2586.92,382.406 \n 2588.04,383.878 2589.16,385.343 2590.28,386.801 2591.4,388.252 2592.52,389.696 2593.64,391.134 2594.76,392.563 2595.88,393.986 2597,395.401 2598.12,396.808 \n 2599.23,398.207 2600.35,399.598 2601.47,400.982 2602.59,402.357 2603.71,403.724 2604.83,405.082 2605.95,406.432 2607.07,407.773 2608.19,409.105 2609.31,410.428 \n 2610.43,411.743 2611.55,413.048 2612.67,414.343 2613.79,415.63 2614.91,416.907 2616.03,418.174 2617.15,419.432 2618.27,420.679 2619.39,421.917 2620.51,423.145 \n 2621.62,424.363 2622.74,425.57 2623.86,426.767 2624.98,427.953 2626.1,429.129 2627.22,430.294 2628.34,431.448 2629.46,432.592 2630.58,433.724 2631.7,434.845 \n 2632.82,435.956 2633.94,437.054 2635.06,438.142 2636.18,439.218 2637.3,440.282 2638.42,441.335 2639.54,442.376 2640.66,443.405 2641.78,444.422 2642.9,445.427 \n 2644.01,446.42 2645.13,447.4 2646.25,448.369 2647.37,449.325 2648.49,450.268 2649.61,451.199 2650.73,452.118 2651.85,453.023 2652.97,453.916 2654.09,454.796 \n 2655.21,455.663 2656.33,456.517 2657.45,457.357 2658.57,458.185 2659.69,458.999 2660.81,459.8 2661.93,460.588 2663.05,461.362 2664.17,462.122 2665.29,462.869 \n 2666.4,463.602 2667.52,464.322 2668.64,465.027 2669.76,465.719 2670.88,466.397 2672,467.061 2673.12,467.71 2674.24,468.346 2675.36,468.967 2676.48,469.574 \n 2677.6,470.167 2678.72,470.746 2679.84,471.31 2680.96,471.86 2682.08,472.395 2683.2,472.916 2684.32,473.422 2685.44,473.913 2686.56,474.39 2687.68,474.852 \n 2688.79,475.299 2689.91,475.732 2691.03,476.149 2692.15,476.552 2693.27,476.94 2694.39,477.313 2695.51,477.671 2696.63,478.013 2697.75,478.341 2698.87,478.654 \n 2699.99,478.952 2701.11,479.234 2702.23,479.502 2703.35,479.754 2704.47,479.991 2705.59,480.213 2706.71,480.419 2707.83,480.611 2708.95,480.787 2710.06,480.947 \n 2711.18,481.093 2712.3,481.223 2713.42,481.338 2714.54,481.437 2715.66,481.521 2716.78,481.59 2717.9,481.644 2719.02,481.682 2720.14,481.705 2721.26,481.712 \n \n \"/>\n</svg>\n"
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1841.01,471.266 1842.14,471.996 1843.27,472.701 1844.4,473.379 1845.54,474.031 1846.67,474.658 1847.8,475.258 1848.94,475.832 \n 1850.07,476.379 1851.2,476.9 1852.33,477.394 1853.47,477.862 1854.6,478.302 1855.73,478.717 1856.87,479.104 1858,479.464 1859.13,479.797 1860.26,480.103 \n 1861.4,480.382 1862.53,480.634 1863.66,480.859 1864.8,481.057 1865.93,481.227 1867.06,481.37 1868.19,481.486 1869.33,481.575 1870.46,481.636 1871.59,481.67 \n 1872.73,481.676 1873.86,481.656 1874.99,481.608 1876.12,481.532 1877.26,481.43 1878.39,481.3 1879.52,481.143 1880.66,480.958 1881.79,480.747 1882.92,480.508 \n 1884.06,480.243 1885.19,479.95 1886.32,479.63 1887.45,479.283 1888.59,478.91 1889.72,478.509 1890.85,478.082 1891.99,477.628 1893.12,477.147 1894.25,476.639 \n 1895.38,476.106 1896.52,475.545 1897.65,474.959 1898.78,474.346 1899.92,473.707 1901.05,473.042 1902.18,472.351 1903.31,471.634 1904.45,470.891 1905.58,470.123 \n 1906.71,469.329 1907.85,468.51 1908.98,467.666 1910.11,466.796 1911.24,465.902 1912.38,464.982 1913.51,464.038 1914.64,463.07 1915.78,462.077 1916.91,461.06 \n 1918.04,460.018 1919.17,458.953 1920.31,457.864 1921.44,456.751 1922.57,455.615 1923.71,454.456 1924.84,453.274 1925.97,452.069 1927.1,450.841 1928.24,449.591 \n 1929.37,448.319 1930.5,447.024 1931.64,445.708 1932.77,444.37 1933.9,443.011 1935.04,441.63 1936.17,440.229 1937.3,438.807 1938.43,437.364 1939.57,435.901 \n 1940.7,434.418 1941.83,432.916 1942.97,431.393 1944.1,429.852 1945.23,428.291 1946.36,426.711 1947.5,425.113 1948.63,423.496 1949.76,421.861 1950.9,420.209 \n 1952.03,418.539 1953.16,416.851 1954.29,415.147 1955.43,413.426 1956.56,411.688 1957.69,409.934 1958.83,408.164 1959.96,406.378 1961.09,404.577 1962.22,402.761 \n 1963.36,400.93 1964.49,399.085 1965.62,397.225 1966.76,395.351 1967.89,393.464 1969.02,391.563 1970.15,389.649 1971.29,387.723 1972.42,385.784 1973.55,383.832 \n 1974.69,381.869 1975.82,379.895 1976.95,377.909 1978.08,375.912 1979.22,373.904 1980.35,371.886 1981.48,369.858 1982.62,367.821 1983.75,365.774 1984.88,363.718 \n 1986.02,361.654 1987.15,359.581 1988.28,357.5 1989.41,355.411 1990.55,353.315 1991.68,351.212 1992.81,349.103 1993.95,346.987 1995.08,344.865 1996.21,342.737 \n 1997.34,340.604 1998.48,338.466 1999.61,336.324 2000.74,334.177 2001.88,332.026 2003.01,329.872 2004.14,327.715 2005.27,325.554 2006.41,323.391 2007.54,321.225 \n 2008.67,319.058 2009.81,316.889 2010.94,314.718 2012.07,312.547 2013.2,310.376 2014.34,308.204 2015.47,306.032 2016.6,303.861 2017.74,301.69 2018.87,299.521 \n 2020,297.353 2021.13,295.188 2022.27,293.024 2023.4,290.863 2024.53,288.705 2025.67,286.55 2026.8,284.399 2027.93,282.251 2029.06,280.108 2030.2,277.97 \n 2031.33,275.836 2032.46,273.708 2033.6,271.585 2034.73,269.468 2035.86,267.357 2037,265.253 2038.13,263.156 2039.26,261.066 2040.39,258.984 2041.53,256.909 \n 2042.66,254.843 2043.79,252.785 2044.93,250.737 2046.06,248.697 2047.19,246.667 2048.32,244.647 2049.46,242.636 2050.59,240.637 2051.72,238.648 2052.86,236.67 \n 2053.99,234.704 2055.12,232.749 2056.25,230.807 2057.39,228.876 2058.52,226.959 2059.65,225.054 2060.79,223.163 2061.92,221.285 2063.05,219.422 2064.18,217.572 \n 2065.32,215.737 2066.45,213.917 2067.58,212.112 2068.72,210.323 2069.85,208.548 2070.98,206.79 2072.11,205.048 2073.25,203.322 2074.38,201.613 2075.51,199.92 \n 2076.65,198.245 2077.78,196.588 2078.91,194.948 2080.04,193.327 2081.18,191.724 2082.31,190.139 2083.44,188.573 2084.58,187.027 2085.71,185.499 2086.84,183.991 \n 2087.98,182.503 2089.11,181.036 2090.24,179.588 2091.37,178.161 2092.51,176.754 2093.64,175.369 2094.77,174.005 2095.91,172.662 2097.04,171.341 2098.17,170.041 \n 2099.3,168.764 2100.44,167.509 2101.57,166.276 2102.7,165.066 2103.84,163.878 2104.97,162.714 2106.1,161.572 2107.23,160.455 2108.37,159.36 2109.5,158.289 \n 2110.63,157.243 2111.77,156.22 2112.9,155.221 2114.03,154.247 2115.16,153.297 2116.3,152.371 2117.43,151.471 2118.56,150.595 2119.7,149.745 2120.83,148.919 \n 2121.96,148.119 2123.09,147.345 2124.23,146.596 2125.36,145.872 2126.49,145.175 2127.63,144.503 2128.76,143.858 2129.89,143.238 2131.02,142.645 2132.16,142.078 \n 2133.29,141.537 2134.42,141.023 2135.56,140.535 2136.69,140.074 2137.82,139.639 2138.96,139.231 2140.09,138.85 2141.22,138.496 2142.35,138.169 2143.49,137.869 \n 2144.62,137.596 2145.75,137.35 2146.89,137.131 2148.02,136.939 2149.15,136.775 2150.28,136.638 2151.42,136.528 2152.55,136.445 2153.68,136.39 2154.82,136.363 \n 2155.95,136.362 2157.08,136.389 2158.21,136.443 2159.35,136.525 2160.48,136.634 2161.61,136.77 2162.75,136.934 2163.88,137.125 2165.01,137.343 2166.14,137.589 \n 2167.28,137.861 2168.41,138.161 2169.54,138.488 2170.68,138.842 2171.81,139.223 2172.94,139.631 2174.07,140.066 2175.21,140.527 2176.34,141.015 2177.47,141.53 \n 2178.61,142.072 2179.74,142.639 2180.87,143.234 2182,143.854 2183.14,144.501 2184.27,145.174 2185.4,145.872 2186.54,146.597 2187.67,147.347 2188.8,148.123 \n 2189.94,148.924 2191.07,149.751 2192.2,150.602 2193.33,151.479 2194.47,152.381 2195.6,153.307 2196.73,154.258 2197.87,155.233 2199,156.233 2200.13,157.256 \n 2201.26,158.304 2202.4,159.375 2203.53,160.47 2204.66,161.588 2205.8,162.73 2206.93,163.895 2208.06,165.083 2209.19,166.293 2210.33,167.526 2211.46,168.782 \n 2212.59,170.059 2213.73,171.359 2214.86,172.68 2215.99,174.023 2217.12,175.387 2218.26,176.773 2219.39,178.179 2220.52,179.606 2221.66,181.053 2222.79,182.521 \n 2223.92,184.009 2225.05,185.516 2226.19,187.043 2227.32,188.59 2228.45,190.155 2229.59,191.739 2230.72,193.342 2231.85,194.963 2232.98,196.603 2234.12,198.26 \n 2235.25,199.934 2236.38,201.626 2237.52,203.335 2238.65,205.061 2239.78,206.803 2240.92,208.562 2242.05,210.336 2243.18,212.127 2244.31,213.932 2245.45,215.753 \n 2246.58,217.588 2247.71,219.438 2248.85,221.302 2249.98,223.18 2251.11,225.072 2252.24,226.977 2253.38,228.895 2254.51,230.826 2255.64,232.769 2256.78,234.724 \n 2257.91,236.691 2259.04,238.669 2260.17,240.659 2261.31,242.659 2262.44,244.67 2263.57,246.69 2264.71,248.721 2265.84,250.761 2266.97,252.81 2268.1,254.868 \n 2269.24,256.934 2270.37,259.009 2271.5,261.092 2272.64,263.182 2273.77,265.28 2274.9,267.385 2276.03,269.496 2277.17,271.613 2278.3,273.736 2279.43,275.865 \n 2280.57,277.999 2281.7,280.138 2282.83,282.282 2283.96,284.429 2285.1,286.581 2286.23,288.736 2287.36,290.894 2288.5,293.055 2289.63,295.218 2290.76,297.384 \n 2291.9,299.551 2293.03,301.72 2294.16,303.89 2295.29,306.061 2296.43,308.232 2297.56,310.404 2298.69,312.575 2299.83,314.746 2300.96,316.915 2302.09,319.084 \n 2303.22,321.251 2304.36,323.416 2305.49,325.579 2306.62,327.739 2307.76,329.896 2308.89,332.05 2310.02,334.201 2311.15,336.348 2312.29,338.49 2313.42,340.628 \n 2314.55,342.76 2315.69,344.888 2316.82,347.01 2317.95,349.126 2319.08,351.235 2320.22,353.338 2321.35,355.435 2322.48,357.523 2323.62,359.605 2324.75,361.678 \n 2325.88,363.743 2327.01,365.799 2328.15,367.847 2329.28,369.885 2330.41,371.913 2331.55,373.932 2332.68,375.94 2333.81,377.938 2334.94,379.924 2336.08,381.9 \n 2337.21,383.864 2338.34,385.815 2339.48,387.755 2340.61,389.682 2341.74,391.596 2342.88,393.497 2344.01,395.384 2345.14,397.259 2346.27,399.119 2347.41,400.965 \n 2348.54,402.796 2349.67,404.613 2350.81,406.415 2351.94,408.201 2353.07,409.971 2354.2,411.726 2355.34,413.464 2356.47,415.186 2357.6,416.891 2358.74,418.579 \n 2359.87,420.249 2361,421.902 2362.13,423.537 2363.27,425.153 2364.4,426.752 2365.53,428.331 2366.67,429.892 2367.8,431.434 2368.93,432.956 2370.06,434.458 \n 2371.2,435.941 2372.33,437.404 2373.46,438.846 2374.6,440.267 2375.73,441.668 2376.86,443.048 2377.99,444.407 2379.13,445.744 2380.26,447.059 2381.39,448.353 \n 2382.53,449.625 2383.66,450.874 2384.79,452.101 2385.92,453.305 2387.06,454.487 2388.19,455.645 2389.32,456.781 2390.46,457.892 2391.59,458.981 2392.72,460.045 \n 2393.86,461.086 2394.99,462.103 2396.12,463.096 2397.25,464.064 2398.39,465.007 2399.52,465.926 2400.65,466.821 2401.79,467.69 2402.92,468.534 2404.05,469.353 \n 2405.18,470.146 2406.32,470.914 2407.45,471.657 2408.58,472.374 2409.72,473.065 2410.85,473.73 2411.98,474.369 2413.11,474.982 2414.25,475.57 2415.38,476.13 \n 2416.51,476.665 2417.65,477.173 2418.78,477.654 2419.91,478.109 2421.04,478.537 2422.18,478.939 2423.31,479.313 2424.44,479.661 2425.58,479.981 2426.71,480.275 \n 2427.84,480.541 2428.97,480.78 2430.11,480.992 2431.24,481.177 2432.37,481.335 2433.51,481.465 2434.64,481.568 2435.77,481.643 2436.9,481.691 2438.04,481.712 \n 2439.17,481.705 2440.3,481.671 2441.44,481.61 2442.57,481.521 2443.7,481.405 2444.84,481.261 2445.97,481.09 2447.1,480.892 2448.23,480.667 2449.37,480.414 \n 2450.5,480.134 2451.63,479.827 2452.77,479.493 2453.9,479.132 2455.03,478.744 2456.16,478.328 2457.3,477.886 2458.43,477.418 2459.56,476.922 2460.7,476.4 \n 2461.83,475.851 2462.96,475.276 2464.09,474.675 2465.23,474.047 2466.36,473.393 2467.49,472.713 2468.63,472.008 2469.76,471.276 2470.89,470.519 2472.02,469.736 \n 2473.16,468.928 2474.29,468.094 2475.42,467.236 2476.56,466.352 2477.69,465.444 2478.82,464.51 2479.95,463.553 2481.09,462.571 2482.22,461.565 2483.35,460.535 \n 2484.49,459.481 2485.62,458.403 2486.75,457.302 2487.88,456.177 2489.02,455.029 2490.15,453.858 2491.28,452.665 2492.42,451.448 2493.55,450.209 2494.68,448.948 \n 2495.82,447.664 2496.95,446.359 2498.08,445.032 2499.21,443.683 2500.35,442.314 2501.48,440.923 2502.61,439.511 2503.75,438.079 2504.88,436.626 2506.01,435.153 \n 2507.14,433.66 2508.28,432.147 2509.41,430.615 2510.54,429.064 2511.68,427.493 2512.81,425.904 2513.94,424.297 2515.07,422.671 2516.21,421.027 2517.34,419.365 \n 2518.47,417.686 2519.61,415.989 2520.74,414.276 2521.87,412.546 2523,410.799 2524.14,409.036 2525.27,407.258 2526.4,405.464 2527.54,403.654 2528.67,401.829 \n 2529.8,399.99 2530.93,398.136 2532.07,396.268 2533.2,394.386 2534.33,392.491 2535.47,390.582 2536.6,388.661 2537.73,386.726 2538.86,384.78 2540,382.821 \n 2541.13,380.851 2542.26,378.869 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1770.77,448.946 1771.9,450.518 1773.03,452.056 1774.16,453.558 1775.3,455.025 1776.43,456.455 1777.56,457.849 1778.7,459.206 1779.83,460.526 1780.96,461.809 \n 1782.1,463.054 1783.23,464.261 1784.36,465.429 1785.49,466.559 1786.63,467.65 1787.76,468.702 1788.89,469.714 1790.03,470.687 1791.16,471.619 1792.29,472.512 \n 1793.42,473.364 1794.56,474.175 1795.69,474.946 1796.82,475.676 1797.96,476.364 1799.09,477.011 1800.22,477.616 1801.35,478.18 1802.49,478.702 1803.62,479.182 \n 1804.75,479.62 1805.89,480.015 1807.02,480.368 1808.15,480.679 1809.28,480.948 1810.42,481.173 1811.55,481.357 1812.68,481.497 1813.82,481.595 1814.95,481.65 \n 1816.08,481.663 1817.21,481.633 1818.35,481.56 1819.48,481.444 1820.61,481.286 1821.75,481.085 1822.88,480.842 1824.01,480.556 1825.14,480.228 1826.28,479.858 \n 1827.41,479.445 1828.54,478.991 1829.68,478.494 1830.81,477.956 1831.94,477.376 1833.08,476.754 1834.21,476.091 1835.34,475.386 1836.47,474.64 1837.61,473.854 \n 1838.74,473.027 1839.87,472.159 1841.01,471.251 1842.14,470.302 1843.27,469.314 1844.4,468.287 1845.54,467.219 1846.67,466.113 1847.8,464.968 1848.94,463.784 \n 1850.07,462.562 1851.2,461.302 1852.33,460.005 1853.47,458.67 1854.6,457.298 1855.73,455.889 1856.87,454.444 1858,452.963 1859.13,451.446 1860.26,449.894 \n 1861.4,448.307 1862.53,446.686 1863.66,445.03 1864.8,443.341 1865.93,441.619 1867.06,439.864 1868.19,438.077 1869.33,436.257 1870.46,434.407 1871.59,432.525 \n 1872.73,430.613 1873.86,428.671 1874.99,426.7 1876.12,424.699 1877.26,422.67 1878.39,420.612 1879.52,418.527 1880.66,416.415 1881.79,414.277 1882.92,412.112 \n 1884.06,409.922 1885.19,407.708 1886.32,405.468 1887.45,403.205 1888.59,400.919 1889.72,398.61 1890.85,396.279 1891.99,393.927 1893.12,391.553 1894.25,389.159 \n 1895.38,386.745 1896.52,384.312 1897.65,381.86 1898.78,379.39 1899.92,376.903 1901.05,374.399 1902.18,371.878 1903.31,369.342 1904.45,366.792 1905.58,364.226 \n 1906.71,361.647 1907.85,359.055 1908.98,356.451 1910.11,353.834 1911.24,351.207 1912.38,348.569 1913.51,345.921 1914.64,343.265 1915.78,340.599 1916.91,337.926 \n 1918.04,335.246 1919.17,332.559 1920.31,329.867 1921.44,327.169 1922.57,324.467 1923.71,321.762 1924.84,319.053 1925.97,316.342 1927.1,313.629 1928.24,310.914 \n 1929.37,308.2 1930.5,305.485 1931.64,302.771 1932.77,300.059 1933.9,297.349 1935.04,294.642 1936.17,291.939 1937.3,289.24 1938.43,286.546 1939.57,283.857 \n 1940.7,281.175 1941.83,278.499 1942.97,275.832 1944.1,273.172 1945.23,270.521 1946.36,267.88 1947.5,265.249 1948.63,262.629 1949.76,260.021 1950.9,257.424 \n 1952.03,254.84 1953.16,252.27 1954.29,249.713 1955.43,247.172 1956.56,244.645 1957.69,242.134 1958.83,239.64 1959.96,237.163 1961.09,234.704 1962.22,232.263 \n 1963.36,229.841 1964.49,227.438 1965.62,225.055 1966.76,222.694 1967.89,220.353 1969.02,218.035 1970.15,215.739 1971.29,213.466 1972.42,211.217 1973.55,208.992 \n 1974.69,206.792 1975.82,204.617 1976.95,202.468 1978.08,200.345 1979.22,198.249 1980.35,196.18 1981.48,194.139 1982.62,192.126 1983.75,190.143 1984.88,188.188 \n 1986.02,186.264 1987.15,184.37 1988.28,182.507 1989.41,180.675 1990.55,178.875 1991.68,177.107 1992.81,175.372 1993.95,173.67 1995.08,172.001 1996.21,170.366 \n 1997.34,168.766 1998.48,167.2 1999.61,165.67 2000.74,164.175 2001.88,162.716 2003.01,161.293 2004.14,159.906 2005.27,158.557 2006.41,157.245 2007.54,155.97 \n 2008.67,154.733 2009.81,153.534 2010.94,152.374 2012.07,151.253 2013.2,150.17 2014.34,149.127 2015.47,148.123 2016.6,147.159 2017.74,146.235 2018.87,145.351 \n 2020,144.508 2021.13,143.705 2022.27,142.943 2023.4,142.222 2024.53,141.542 2025.67,140.904 2026.8,140.307 2027.93,139.752 2029.06,139.239 2030.2,138.767 \n 2031.33,138.337 2032.46,137.95 2033.6,137.604 2034.73,137.301 2035.86,137.041 2037,136.823 2038.13,136.647 2039.26,136.514 2040.39,136.423 2041.53,136.376 \n 2042.66,136.371 2043.79,136.409 2044.93,136.489 2046.06,136.612 2047.19,136.778 2048.32,136.986 2049.46,137.238 2050.59,137.531 2051.72,137.867 2052.86,138.246 \n 2053.99,138.667 2055.12,139.13 2056.25,139.635 2057.39,140.182 2058.52,140.771 2059.65,141.402 2060.79,142.074 2061.92,142.788 2063.05,143.542 2064.18,144.338 \n 2065.32,145.175 2066.45,146.052 2067.58,146.969 2068.72,147.927 2069.85,148.925 2070.98,149.962 2072.11,151.038 2073.25,152.154 2074.38,153.308 2075.51,154.501 \n 2076.65,155.732 2077.78,157 2078.91,158.307 2080.04,159.65 2081.18,161.03 2082.31,162.447 2083.44,163.899 2084.58,165.388 2085.71,166.911 2086.84,168.47 \n 2087.98,170.064 2089.11,171.692 2090.24,173.354 2091.37,175.049 2092.51,176.778 2093.64,178.539 2094.77,180.332 2095.91,182.157 2097.04,184.014 2098.17,185.901 \n 2099.3,187.819 2100.44,189.766 2101.57,191.743 2102.7,193.75 2103.84,195.784 2104.97,197.847 2106.1,199.937 2107.23,202.054 2108.37,204.198 2109.5,206.368 \n 2110.63,208.563 2111.77,210.783 2112.9,213.027 2114.03,215.296 2115.16,217.587 2116.3,219.901 2117.43,222.238 2118.56,224.595 2119.7,226.974 2120.83,229.373 \n 2121.96,231.792 2123.09,234.23 2124.23,236.687 2125.36,239.162 2126.49,241.654 2127.63,244.162 2128.76,246.687 2129.89,249.227 2131.02,251.781 2132.16,254.35 \n 2133.29,256.932 2134.42,259.527 2135.56,262.134 2136.69,264.753 2137.82,267.382 2138.96,270.022 2140.09,272.671 2141.22,275.33 2142.35,277.997 2143.49,280.671 \n 2144.62,283.352 2145.75,286.04 2146.89,288.734 2148.02,291.432 2149.15,294.135 2150.28,296.841 2151.42,299.55 2152.55,302.262 2153.68,304.975 2154.82,307.689 \n 2155.95,310.403 2157.08,313.117 2158.21,315.83 2159.35,318.541 2160.48,321.25 2161.61,323.956 2162.75,326.658 2163.88,329.355 2165.01,332.048 2166.14,334.735 \n 2167.28,337.416 2168.41,340.09 2169.54,342.756 2170.68,345.414 2171.81,348.063 2172.94,350.702 2174.07,353.331 2175.21,355.95 2176.34,358.556 2177.47,361.151 \n 2178.61,363.733 2179.74,366.302 2180.87,368.856 2182,371.396 2183.14,373.92 2184.27,376.428 2185.4,378.92 2186.54,381.395 2187.67,383.851 2188.8,386.289 \n 2189.94,388.708 2191.07,391.108 2192.2,393.486 2193.33,395.844 2194.47,398.18 2195.6,400.494 2196.73,402.785 2197.87,405.053 2199,407.298 2200.13,409.518 \n 2201.26,411.713 2202.4,413.884 2203.53,416.028 2204.66,418.146 2205.8,420.237 2206.93,422.3 2208.06,424.335 2209.19,426.342 2210.33,428.32 2211.46,430.268 \n 2212.59,432.186 2213.73,434.074 2214.86,435.93 2215.99,437.756 2217.12,439.549 2218.26,441.31 2219.39,443.038 2220.52,444.733 2221.66,446.395 2222.79,448.022 \n 2223.92,449.615 2225.05,451.173 2226.19,452.696 2227.32,454.184 2228.45,455.635 2229.59,457.051 2230.72,458.429 2231.85,459.771 2232.98,461.076 2234.12,462.343 \n 2235.25,463.572 2236.38,464.762 2237.52,465.915 2238.65,467.028 2239.78,468.103 2240.92,469.138 2242.05,470.134 2243.18,471.09 2244.31,472.005 2245.45,472.881 \n 2246.58,473.716 2247.71,474.51 2248.85,475.264 2249.98,475.976 2251.11,476.647 2252.24,477.277 2253.38,477.866 2254.51,478.413 2255.64,478.918 2256.78,479.382 \n 2257.91,479.803 2259.04,480.183 2260.17,480.52 2261.31,480.814 2262.44,481.067 2263.57,481.276 2264.71,481.444 2265.84,481.568 2266.97,481.65 2268.1,481.689 \n 2269.24,481.686 2270.37,481.639 2271.5,481.55 2272.64,481.419 2273.77,481.244 2274.9,481.027 2276.03,480.767 2277.17,480.465 2278.3,480.12 2279.43,479.733 \n 2280.57,479.303 2281.7,478.831 2282.83,478.317 2283.96,477.761 2285.1,477.163 2286.23,476.524 2287.36,475.843 2288.5,475.12 2289.63,474.357 2290.76,473.552 \n 2291.9,472.707 2293.03,471.821 2294.16,470.894 2295.29,469.928 2296.43,468.922 2297.56,467.876 2298.69,466.791 2299.83,465.667 2300.96,464.504 2302.09,463.302 \n 2303.22,462.063 2304.36,460.786 2305.49,459.471 2306.62,458.119 2307.76,456.731 2308.89,455.306 2310.02,453.845 2311.15,452.349 2312.29,450.818 2313.42,449.251 \n 2314.55,447.65 2315.69,446.015 2316.82,444.346 2317.95,442.643 2319.08,440.907 2320.22,439.139 2321.35,437.339 2322.48,435.507 2323.62,433.644 2324.75,431.75 \n 2325.88,429.826 2327.01,427.872 2328.15,425.889 2329.28,423.877 2330.41,421.836 2331.55,419.768 2332.68,417.672 2333.81,415.549 2334.94,413.4 2336.08,411.225 \n 2337.21,409.025 2338.34,406.8 2339.48,404.55 2340.61,402.277 2341.74,399.981 2342.88,397.662 2344.01,395.321 2345.14,392.959 2346.27,390.576 2347.41,388.173 \n 2348.54,385.75 2349.67,383.308 2350.81,380.847 2351.94,378.369 2353.07,375.873 2354.2,373.361 2355.34,370.832 2356.47,368.289 2357.6,365.731 2358.74,363.158 \n 2359.87,360.573 2361,357.974 2362.13,355.364 2363.27,352.742 2364.4,350.11 2365.53,347.467 2366.67,344.815 2367.8,342.155 2368.93,339.487 2370.06,336.81 \n 2371.2,334.127 2372.33,331.438 2373.46,328.743 2374.6,326.043 2375.73,323.339 2376.86,320.631 2377.99,317.921 2379.13,315.209 2380.26,312.495 2381.39,309.78 \n 2382.53,307.065 2383.66,304.351 2384.79,301.637 2385.92,298.926 2387.06,296.218 2388.19,293.512 2389.32,290.811 2390.46,288.114 2391.59,285.422 2392.72,282.736 \n 2393.86,280.057 2394.99,277.384 2396.12,274.72 2397.25,272.064 2398.39,269.417 2399.52,266.78 2400.65,264.153 2401.79,261.538 2402.92,258.934 2404.05,256.342 \n 2405.18,253.763 2406.32,251.198 2407.45,248.647 2408.58,246.111 2409.72,243.59 2410.85,241.085 2411.98,238.597 2413.11,236.127 2414.25,233.674 2415.38,231.24 \n 2416.51,228.824 2417.65,226.429 2418.78,224.054 2419.91,221.7 2421.04,219.368 2422.18,217.057 2423.31,214.77 2424.44,212.506 2425.58,210.266 2426.71,208.05 \n 2427.84,205.86 2428.97,203.695 2430.11,201.556 2431.24,199.445 2432.37,197.36 2433.51,195.302 2434.64,193.273 2435.77,191.272 2436.9,189.3 2438.04,187.357 \n 2439.17,185.445 2440.3,183.563 2441.44,181.712 2442.57,179.893 2443.7,178.106 2444.84,176.351 2445.97,174.629 2447.1,172.94 2448.23,171.285 2449.37,169.664 \n 2450.5,168.077 2451.63,166.526 2452.77,165.01 2453.9,163.529 2455.03,162.085 2456.16,160.677 2457.3,159.305 2458.43,157.971 2459.56,156.674 2460.7,155.415 \n 2461.83,154.194 2462.96,153.011 2464.09,151.867 2465.23,150.762 2466.36,149.695 2467.49,148.669 2468.63,147.682 2469.76,146.734 2470.89,145.827 2472.02,144.961 \n 2473.16,144.135 2474.29,143.349 2475.42,142.605 2476.56,141.901 2477.69,141.239 2478.82,140.619 2479.95,140.04 2481.09,139.503 2482.22,139.007 2483.35,138.554 \n 2484.49,138.142 2485.62,137.773 2486.75,137.447 2487.88,137.162 2489.02,136.92 2490.15,136.72 2491.28,136.563 2492.42,136.448 2493.55,136.375 2494.68,136.345 \n 2495.82,136.358 2496.95,136.413 2498.08,136.511 2499.21,136.651 2500.35,136.834 2501.48,137.06 2502.61,137.328 2503.75,137.638 2504.88,137.991 2506.01,138.386 \n 2507.14,138.824 2508.28,139.303 2509.41,139.825 2510.54,140.389 2511.68,140.994 2512.81,141.641 2513.94,142.329 2515.07,143.059 2516.21,143.83 2517.34,144.642 \n 2518.47,145.495 2519.61,146.388 2520.74,147.322 2521.87,148.295 2523,149.309 2524.14,150.362 2525.27,151.455 2526.4,152.587 2527.54,153.757 2528.67,154.967 \n 2529.8,156.214 2530.93,157.499 2532.07,158.822 2533.2,160.182 2534.33,161.579 2535.47,163.013 2536.6,164.482 2537.73,165.988 2538.86,167.529 2540,169.105 \n 2541.13,170.715 2542.26,172.359 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1700.53,356.459 1701.66,359.582 \n 1702.79,362.687 1703.93,365.773 1705.06,368.839 1706.19,371.884 1707.32,374.906 1708.46,377.904 1709.59,380.878 1710.72,383.827 1711.86,386.749 1712.99,389.643 \n 1714.12,392.509 1715.25,395.345 1716.39,398.15 1717.52,400.923 1718.65,403.664 1719.79,406.371 1720.92,409.043 1722.05,411.68 1723.18,414.281 1724.32,416.844 \n 1725.45,419.368 1726.58,421.854 1727.72,424.299 1728.85,426.703 1729.98,429.065 1731.12,431.384 1732.25,433.66 1733.38,435.892 1734.51,438.078 1735.65,440.218 \n 1736.78,442.312 1737.91,444.358 1739.05,446.355 1740.18,448.304 1741.31,450.204 1742.44,452.053 1743.58,453.851 1744.71,455.598 1745.84,457.292 1746.98,458.934 \n 1748.11,460.522 1749.24,462.056 1750.37,463.536 1751.51,464.961 1752.64,466.331 1753.77,467.644 1754.91,468.901 1756.04,470.101 1757.17,471.244 1758.3,472.33 \n 1759.44,473.357 1760.57,474.326 1761.7,475.235 1762.84,476.086 1763.97,476.878 1765.1,477.609 1766.23,478.28 1767.37,478.892 1768.5,479.442 1769.63,479.932 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1839.87,398.148 1841.01,395.343 1842.14,392.507 1843.27,389.642 1844.4,386.748 1845.54,383.827 1846.67,380.878 1847.8,377.904 1848.94,374.906 \n 1850.07,371.884 1851.2,368.84 1852.33,365.774 1853.47,362.688 1854.6,359.583 1855.73,356.46 1856.87,353.319 1858,350.163 1859.13,346.992 1860.26,343.808 \n 1861.4,340.611 1862.53,337.402 1863.66,334.184 1864.8,330.957 1865.93,327.721 1867.06,324.48 1868.19,321.233 1869.33,317.981 1870.46,314.727 1871.59,311.471 \n 1872.73,308.214 1873.86,304.957 1874.99,301.701 1876.12,298.448 1877.26,295.199 1878.39,291.955 1879.52,288.716 1880.66,285.486 1881.79,282.263 1882.92,279.05 \n 1884.06,275.848 1885.19,272.658 1886.32,269.481 1887.45,266.318 1888.59,263.17 1889.72,260.039 1890.85,256.925 1891.99,253.829 1893.12,250.754 1894.25,247.699 \n 1895.38,244.665 1896.52,241.655 1897.65,238.669 1898.78,235.707 1899.92,232.772 1901.05,229.864 1902.18,226.983 1903.31,224.132 1904.45,221.312 1905.58,218.522 \n 1906.71,215.764 1907.85,213.04 1908.98,210.35 1910.11,207.695 1911.24,205.076 1912.38,202.494 1913.51,199.951 1914.64,197.446 1915.78,194.98 1916.91,192.555 \n 1918.04,190.172 1919.17,187.83 1920.31,185.532 1921.44,183.278 1922.57,181.068 1923.71,178.905 1924.84,176.787 1925.97,174.717 1927.1,172.694 1928.24,170.721 \n 1929.37,168.796 1930.5,166.922 1931.64,165.098 1932.77,163.325 1933.9,161.605 1935.04,159.937 1936.17,158.322 1937.3,156.761 1938.43,155.254 1939.57,153.802 \n 1940.7,152.405 1941.83,151.064 1942.97,149.779 1944.1,148.551 1945.23,147.38 1946.36,146.267 1947.5,145.211 1948.63,144.214 1949.76,143.276 1950.9,142.396 \n 1952.03,141.576 1953.16,140.815 1954.29,140.114 1955.43,139.473 1956.56,138.893 1957.69,138.373 1958.83,137.913 1959.96,137.514 1961.09,137.177 1962.22,136.9 \n 1963.36,136.684 1964.49,136.53 1965.62,136.437 1966.76,136.405 1967.89,136.435 1969.02,136.526 1970.15,136.679 1971.29,136.893 1972.42,137.169 1973.55,137.505 \n 1974.69,137.903 1975.82,138.362 1976.95,138.882 1978.08,139.463 1979.22,140.104 1980.35,140.805 1981.48,141.566 1982.62,142.387 1983.75,143.268 1984.88,144.208 \n 1986.02,145.206 1987.15,146.263 1988.28,147.378 1989.41,148.551 1990.55,149.781 1991.68,151.068 1992.81,152.411 1993.95,153.81 1995.08,155.264 1996.21,156.773 \n 1997.34,158.336 1998.48,159.953 1999.61,161.622 2000.74,163.344 2001.88,165.118 2003.01,166.942 2004.14,168.817 2005.27,170.742 2006.41,172.716 2007.54,174.738 \n 2008.67,176.808 2009.81,178.925 2010.94,181.088 2012.07,183.297 2013.2,185.551 2014.34,187.848 2015.47,190.189 2016.6,192.572 2017.74,194.996 2018.87,197.461 \n 2020,199.965 2021.13,202.509 2022.27,205.09 2023.4,207.709 2024.53,210.363 2025.67,213.053 2026.8,215.777 2027.93,218.534 2029.06,221.324 2030.2,224.145 \n 2031.33,226.996 2032.46,229.877 2033.6,232.786 2034.73,235.722 2035.86,238.685 2037,241.673 2038.13,244.684 2039.26,247.719 2040.39,250.776 2041.53,253.853 \n 2042.66,256.95 2043.79,260.066 2044.93,263.198 2046.06,266.347 2047.19,269.511 2048.32,272.689 2049.46,275.879 2050.59,279.082 2051.72,282.295 2052.86,285.518 \n 2053.99,288.749 2055.12,291.988 2056.25,295.232 2057.39,298.482 2058.52,301.735 2059.65,304.99 2060.79,308.247 2061.92,311.504 2063.05,314.76 2064.18,318.014 \n 2065.32,321.265 2066.45,324.511 2067.58,327.752 2068.72,330.986 2069.85,334.212 2070.98,337.429 2072.11,340.636 2073.25,343.832 2074.38,347.015 2075.51,350.186 \n 2076.65,353.341 2077.78,356.481 2078.91,359.604 2080.04,362.709 2081.18,365.796 2082.31,368.862 2083.44,371.907 2084.58,374.929 2085.71,377.929 2086.84,380.904 \n 2087.98,383.854 2089.11,386.776 2090.24,389.672 2091.37,392.538 2092.51,395.375 2093.64,398.181 2094.77,400.956 2095.91,403.697 2097.04,406.404 2098.17,409.077 \n 2099.3,411.713 2100.44,414.314 2101.57,416.877 2102.7,419.402 2103.84,421.888 2104.97,424.333 2106.1,426.738 2107.23,429.1 2108.37,431.42 2109.5,433.696 \n 2110.63,435.928 2111.77,438.115 2112.9,440.255 2114.03,442.349 2115.16,444.395 2116.3,446.393 2117.43,448.342 2118.56,450.241 2119.7,452.09 2120.83,453.887 \n 2121.96,455.633 2123.09,457.327 2124.23,458.968 2125.36,460.556 2126.49,462.09 2127.63,463.569 2128.76,464.993 2129.89,466.362 2131.02,467.674 2132.16,468.93 \n 2133.29,470.129 2134.42,471.271 2135.56,472.356 2136.69,473.382 2137.82,474.349 2138.96,475.258 2140.09,476.108 2141.22,476.898 2142.35,477.628 2143.49,478.299 \n 2144.62,478.909 2145.75,479.46 2146.89,479.949 2148.02,480.379 2149.15,480.747 2150.28,481.055 2151.42,481.301 2152.55,481.486 2153.68,481.61 2154.82,481.672 \n 2155.95,481.673 2157.08,481.613 2158.21,481.491 2159.35,481.308 2160.48,481.063 2161.61,480.757 2162.75,480.389 2163.88,479.961 2165.01,479.471 2166.14,478.921 \n 2167.28,478.31 2168.41,477.638 2169.54,476.907 2170.68,476.115 2171.81,475.264 2172.94,474.353 2174.07,473.383 2175.21,472.355 2176.34,471.268 2177.47,470.123 \n 2178.61,468.921 2179.74,467.662 2180.87,466.346 2182,464.974 2183.14,463.546 2184.27,462.064 2185.4,460.526 2186.54,458.935 2187.67,457.291 2188.8,455.594 \n 2189.94,453.844 2191.07,452.044 2192.2,450.193 2193.33,448.291 2194.47,446.341 2195.6,444.342 2196.73,442.295 2197.87,440.201 2199,438.06 2200.13,435.873 \n 2201.26,433.641 2202.4,431.365 2203.53,429.045 2204.66,426.683 2205.8,424.278 2206.93,421.833 2208.06,419.348 2209.19,416.824 2210.33,414.261 2211.46,411.66 \n 2212.59,409.024 2213.73,406.351 2214.86,403.644 2215.99,400.903 2217.12,398.129 2218.26,395.324 2219.39,392.488 2220.52,389.621 2221.66,386.726 2222.79,383.803 \n 2223.92,380.854 2225.05,377.879 2226.19,374.878 2227.32,371.855 2228.45,368.809 2229.59,365.741 2230.72,362.653 2231.85,359.546 2232.98,356.421 2234.12,353.279 \n 2235.25,350.121 2236.38,346.949 2237.52,343.763 2238.65,340.565 2239.78,337.356 2240.92,334.138 2242.05,330.91 2243.18,327.675 2244.31,324.433 2245.45,321.186 \n 2246.58,317.934 2247.71,314.679 2248.85,311.422 2249.98,308.164 2251.11,304.906 2252.24,301.65 2253.38,298.397 2254.51,295.148 2255.64,291.903 2256.78,288.665 \n 2257.91,285.434 2259.04,282.212 2260.17,278.999 2261.31,275.798 2262.44,272.608 2263.57,269.431 2264.71,266.268 2265.84,263.12 2266.97,259.989 2268.1,256.875 \n 2269.24,253.78 2270.37,250.704 2271.5,247.649 2272.64,244.615 2273.77,241.605 2274.9,238.618 2276.03,235.657 2277.17,232.721 2278.3,229.812 2279.43,226.931 \n 2280.57,224.08 2281.7,221.258 2282.83,218.468 2283.96,215.71 2285.1,212.985 2286.23,210.294 2287.36,207.638 2288.5,205.019 2289.63,202.436 2290.76,199.892 \n 2291.9,197.387 2293.03,194.922 2294.16,192.497 2295.29,190.114 2296.43,187.774 2297.56,185.476 2298.69,183.222 2299.83,181.012 2300.96,178.848 2302.09,176.731 \n 2303.22,174.66 2304.36,172.638 2305.49,170.664 2306.62,168.739 2307.76,166.864 2308.89,165.04 2310.02,163.268 2311.15,161.547 2312.29,159.879 2313.42,158.265 \n 2314.55,156.704 2315.69,155.197 2316.82,153.745 2317.95,152.349 2319.08,151.008 2320.22,149.724 2321.35,148.497 2322.48,147.327 2323.62,146.214 2324.75,145.16 \n 2325.88,144.164 2327.01,143.226 2328.15,142.348 2329.28,141.529 2330.41,140.77 2331.55,140.07 2332.68,139.431 2333.81,138.852 2334.94,138.334 2336.08,137.876 \n 2337.21,137.479 2338.34,137.143 2339.48,136.869 2340.61,136.655 2341.74,136.503 2342.88,136.412 2344.01,136.382 2345.14,136.414 2346.27,136.506 2347.41,136.66 \n 2348.54,136.875 2349.67,137.151 2350.81,137.488 2351.94,137.887 2353.07,138.346 2354.2,138.865 2355.34,139.446 2356.47,140.086 2357.6,140.787 2358.74,141.548 \n 2359.87,142.369 2361,143.249 2362.13,144.188 2363.27,145.186 2364.4,146.242 2365.53,147.357 2366.67,148.529 2367.8,149.759 2368.93,151.045 2370.06,152.388 \n 2371.2,153.787 2372.33,155.242 2373.46,156.751 2374.6,158.315 2375.73,159.933 2376.86,161.604 2377.99,163.328 2379.13,165.104 2380.26,166.931 2381.39,168.809 \n 2382.53,170.737 2383.66,172.715 2384.79,174.741 2385.92,176.814 2387.06,178.935 2388.19,181.102 2389.32,183.314 2390.46,185.571 2391.59,187.872 2392.72,190.215 \n 2393.86,192.6 2394.99,195.026 2396.12,197.493 2397.25,200 2398.39,202.545 2399.52,205.128 2400.65,207.748 2401.79,210.404 2402.92,213.095 2404.05,215.82 \n 2405.18,218.578 2406.32,221.368 2407.45,224.189 2408.58,227.041 2409.72,229.921 2410.85,232.83 2411.98,235.766 2413.11,238.728 2414.25,241.715 2415.38,244.726 \n 2416.51,247.76 2417.65,250.816 2418.78,253.893 2419.91,256.989 2421.04,260.105 2422.18,263.237 2423.31,266.386 2424.44,269.551 2425.58,272.73 2426.71,275.922 \n 2427.84,279.126 2428.97,282.34 2430.11,285.565 2431.24,288.798 2432.37,292.038 2433.51,295.284 2434.64,298.536 2435.77,301.791 2436.9,305.049 2438.04,308.308 \n 2439.17,311.567 2440.3,314.825 2441.44,318.081 2442.57,321.333 2443.7,324.581 2444.84,327.822 2445.97,331.058 2447.1,334.286 2448.23,337.505 2449.37,340.713 \n 2450.5,343.911 2451.63,347.096 2452.77,350.268 2453.9,353.425 2455.03,356.566 2456.16,359.69 2457.3,362.795 2458.43,365.882 2459.56,368.948 2460.7,371.993 \n 2461.83,375.015 2462.96,378.014 2464.09,380.988 2465.23,383.936 2466.36,386.858 2467.49,389.752 2468.63,392.617 2469.76,395.452 2470.89,398.257 2472.02,401.03 \n 2473.16,403.77 2474.29,406.476 2475.42,409.148 2476.56,411.784 2477.69,414.384 2478.82,416.947 2479.95,419.471 2481.09,421.956 2482.22,424.4 2483.35,426.804 \n 2484.49,429.166 2485.62,431.485 2486.75,433.761 2487.88,435.993 2489.02,438.179 2490.15,440.319 2491.28,442.412 2492.42,444.458 2493.55,446.456 2494.68,448.404 \n 2495.82,450.303 2496.95,452.151 2498.08,453.949 2499.21,455.695 2500.35,457.39 2501.48,459.032 2502.61,460.62 2503.75,462.155 2504.88,463.635 2506.01,465.06 \n 2507.14,466.43 2508.28,467.744 2509.41,469.001 2510.54,470.201 2511.68,471.343 2512.81,472.428 2513.94,473.454 2515.07,474.421 2516.21,475.33 2517.34,476.179 \n 2518.47,476.969 2519.61,477.699 2520.74,478.368 2521.87,478.977 2523,479.526 2524.14,480.013 2525.27,480.44 2526.4,480.805 2527.54,481.109 2528.67,481.352 \n 2529.8,481.534 2530.93,481.653 2532.07,481.712 2533.2,481.709 2534.33,481.644 2535.47,481.518 2536.6,481.331 2537.73,481.082 2538.86,480.772 2540,480.401 \n 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1700.53,404.548 1701.66,407.689 \n 1702.79,410.783 1703.93,413.826 1705.06,416.82 1706.19,419.761 1707.32,422.648 1708.46,425.48 1709.59,428.256 1710.72,430.975 1711.86,433.634 1712.99,436.232 \n 1714.12,438.769 1715.25,441.243 1716.39,443.653 1717.52,445.997 1718.65,448.275 1719.79,450.486 1720.92,452.628 1722.05,454.701 1723.18,456.703 1724.32,458.633 \n 1725.45,460.491 1726.58,462.275 1727.72,463.985 1728.85,465.621 1729.98,467.18 1731.12,468.663 1732.25,470.068 1733.38,471.395 1734.51,472.644 1735.65,473.814 \n 1736.78,474.904 1737.91,475.914 1739.05,476.842 1740.18,477.69 1741.31,478.456 1742.44,479.139 1743.58,479.74 1744.71,480.259 1745.84,480.694 1746.98,481.046 \n 1748.11,481.315 1749.24,481.5 1750.37,481.601 1751.51,481.619 1752.64,481.553 1753.77,481.403 1754.91,481.17 1756.04,480.854 1757.17,480.454 1758.3,479.971 \n 1759.44,479.405 1760.57,478.756 1761.7,478.026 1762.84,477.213 1763.97,476.32 1765.1,475.345 1766.23,474.29 1767.37,473.154 1768.5,471.94 1769.63,470.646 \n 1770.77,469.274 1771.9,467.825 1773.03,466.298 1774.16,464.695 1775.3,463.017 1776.43,461.265 1777.56,459.438 1778.7,457.539 1779.83,455.567 1780.96,453.525 \n 1782.1,451.412 1783.23,449.231 1784.36,446.981 1785.49,444.664 1786.63,442.282 1787.76,439.835 1788.89,437.325 1790.03,434.752 1791.16,432.118 1792.29,429.425 \n 1793.42,426.674 1794.56,423.865 1795.69,421.001 1796.82,418.083 1797.96,415.112 1799.09,412.09 1800.22,409.018 1801.35,405.897 1802.49,402.73 1803.62,399.517 \n 1804.75,396.261 1805.89,392.962 1807.02,389.622 1808.15,386.244 1809.28,382.828 1810.42,379.377 1811.55,375.891 1812.68,372.373 1813.82,368.824 1814.95,365.246 \n 1816.08,361.641 1817.21,358.01 1818.35,354.356 1819.48,350.679 1820.61,346.982 1821.75,343.266 1822.88,339.534 1824.01,335.786 1825.14,332.026 1826.28,328.254 \n 1827.41,324.474 1828.54,320.685 1829.68,316.891 1830.81,313.094 1831.94,309.295 1833.08,305.496 1834.21,301.698 1835.34,297.904 1836.47,294.115 1837.61,290.334 \n 1838.74,286.561 1839.87,282.8 1841.01,279.051 1842.14,275.317 1843.27,271.599 1844.4,267.9 1845.54,264.22 1846.67,260.563 1847.8,256.928 1848.94,253.32 \n 1850.07,249.738 1851.2,246.185 1852.33,242.662 1853.47,239.172 1854.6,235.715 1855.73,232.294 1856.87,228.91 1858,225.564 1859.13,222.259 1860.26,218.996 \n 1861.4,215.777 1862.53,212.602 1863.66,209.474 1864.8,206.395 1865.93,203.365 1867.06,200.387 1868.19,197.461 1869.33,194.589 1870.46,191.773 1871.59,189.013 \n 1872.73,186.312 1873.86,183.669 1874.99,181.088 1876.12,178.568 1877.26,176.111 1878.39,173.719 1879.52,171.393 1880.66,169.134 1881.79,166.942 1882.92,164.819 \n 1884.06,162.767 1885.19,160.785 1886.32,158.875 1887.45,157.038 1888.59,155.275 1889.72,153.587 1890.85,151.974 1891.99,150.437 1893.12,148.977 1894.25,147.594 \n 1895.38,146.29 1896.52,145.064 1897.65,143.918 1898.78,142.852 1899.92,141.867 1901.05,140.962 1902.18,140.139 1903.31,139.397 1904.45,138.738 1905.58,138.161 \n 1906.71,137.667 1907.85,137.256 1908.98,136.927 1910.11,136.682 1911.24,136.52 1912.38,136.441 1913.51,136.447 1914.64,136.535 1915.78,136.707 1916.91,136.963 \n 1918.04,137.302 1919.17,137.725 1920.31,138.23 1921.44,138.818 1922.57,139.489 1923.71,140.243 1924.84,141.078 1925.97,141.995 1927.1,142.993 1928.24,144.071 \n 1929.37,145.23 1930.5,146.468 1931.64,147.785 1932.77,149.181 1933.9,150.654 1935.04,152.204 1936.17,153.83 1937.3,155.532 1938.43,157.308 1939.57,159.157 \n 1940.7,161.079 1941.83,163.073 1942.97,165.137 1944.1,167.271 1945.23,169.473 1946.36,171.742 1947.5,174.078 1948.63,176.479 1949.76,178.944 1950.9,181.472 \n 1952.03,184.062 1953.16,186.712 1954.29,189.422 1955.43,192.189 1956.56,195.013 1957.69,197.892 1958.83,200.825 1959.96,203.81 1961.09,206.846 1962.22,209.932 \n 1963.36,213.066 1964.49,216.247 1965.62,219.472 1966.76,222.741 1967.89,226.052 1969.02,229.404 1970.15,232.794 1971.29,236.221 1972.42,239.684 1973.55,243.181 \n 1974.69,246.709 1975.82,250.268 1976.95,253.856 1978.08,257.47 1979.22,261.11 1980.35,264.772 1981.48,268.456 1982.62,272.16 1983.75,275.881 1984.88,279.617 \n 1986.02,283.368 1987.15,287.132 1988.28,290.906 1989.41,294.689 1990.55,298.479 1991.68,302.274 1992.81,306.072 1993.95,309.871 1995.08,313.67 1996.21,317.467 \n 1997.34,321.259 1998.48,325.046 1999.61,328.824 2000.74,332.593 2001.88,336.35 2003.01,340.094 2004.14,343.823 2005.27,347.535 2006.41,351.229 2007.54,354.902 \n 2008.67,358.553 2009.81,362.18 2010.94,365.782 2012.07,369.356 2013.2,372.901 2014.34,376.415 2015.47,379.897 2016.6,383.345 2017.74,386.757 2018.87,390.131 \n 2020,393.466 2021.13,396.76 2022.27,400.012 2023.4,403.22 2024.53,406.381 2025.67,409.496 2026.8,412.561 2027.93,415.576 2029.06,418.539 2030.2,421.45 \n 2031.33,424.307 2032.46,427.107 2033.6,429.851 2034.73,432.536 2035.86,435.161 2037,437.725 2038.13,440.227 2039.26,442.665 2040.39,445.038 2041.53,447.344 \n 2042.66,449.584 2043.79,451.756 2044.93,453.858 2046.06,455.89 2047.19,457.851 2048.32,459.739 2049.46,461.555 2050.59,463.296 2051.72,464.962 2052.86,466.553 \n 2053.99,468.068 2055.12,469.506 2056.25,470.865 2057.39,472.147 2058.52,473.349 2059.65,474.472 2060.79,475.515 2061.92,476.477 2063.05,477.358 2064.18,478.157 \n 2065.32,478.875 2066.45,479.51 2067.58,480.063 2068.72,480.533 2069.85,480.921 2070.98,481.226 2072.11,481.447 2073.25,481.586 2074.38,481.64 2075.51,481.612 \n 2076.65,481.499 2077.78,481.303 2078.91,481.024 2080.04,480.661 2081.18,480.215 2082.31,479.686 2083.44,479.074 2084.58,478.379 2085.71,477.602 2086.84,476.743 \n 2087.98,475.803 2089.11,474.782 2090.24,473.68 2091.37,472.498 2092.51,471.236 2093.64,469.896 2094.77,468.477 2095.91,466.981 2097.04,465.409 2098.17,463.76 \n 2099.3,462.036 2100.44,460.238 2101.57,458.366 2102.7,456.422 2103.84,454.407 2104.97,452.321 2106.1,450.166 2107.23,447.942 2108.37,445.652 2109.5,443.296 \n 2110.63,440.876 2111.77,438.392 2112.9,435.845 2114.03,433.237 2115.16,430.569 2116.3,427.842 2117.43,425.058 2118.56,422.217 2119.7,419.322 2120.83,416.373 \n 2121.96,413.373 2123.09,410.322 2124.23,407.222 2125.36,404.074 2126.49,400.88 2127.63,397.642 2128.76,394.361 2129.89,391.038 2131.02,387.675 2132.16,384.274 \n 2133.29,380.837 2134.42,377.364 2135.56,373.858 2136.69,370.321 2137.82,366.753 2138.96,363.158 2140.09,359.536 2141.22,355.889 2142.35,352.219 2143.49,348.529 \n 2144.62,344.819 2145.75,341.092 2146.89,337.349 2148.02,333.593 2149.15,329.825 2150.28,326.047 2151.42,322.262 2152.55,318.47 2153.68,314.674 2154.82,310.875 \n 2155.95,307.074 2157.08,303.275 2158.21,299.478 2159.35,295.686 2160.48,291.901 2161.61,288.124 2162.75,284.357 2163.88,280.603 2165.01,276.862 2166.14,273.137 \n 2167.28,269.43 2168.41,265.742 2169.54,262.075 2170.68,258.431 2171.81,254.811 2172.94,251.218 2174.07,247.652 2175.21,244.116 2176.34,240.612 2177.47,237.14 \n 2178.61,233.704 2179.74,230.303 2180.87,226.941 2182,223.618 2183.14,220.337 2184.27,217.098 2185.4,213.903 2186.54,210.755 2187.67,207.654 2188.8,204.602 \n 2189.94,201.601 2191.07,198.651 2192.2,195.756 2193.33,192.915 2194.47,190.131 2195.6,187.405 2196.73,184.738 2197.87,182.131 2199,179.585 2200.13,177.102 \n 2201.26,174.682 2202.4,172.327 2203.53,170.039 2204.66,167.818 2205.8,165.665 2206.93,163.582 2208.06,161.57 2209.19,159.629 2210.33,157.76 2211.46,155.965 \n 2212.59,154.245 2213.73,152.6 2214.86,151.03 2215.99,149.537 2217.12,148.122 2218.26,146.785 2219.39,145.526 2220.52,144.347 2221.66,143.248 2222.79,142.229 \n 2223.92,141.292 2225.05,140.435 2226.19,139.66 2227.32,138.967 2228.45,138.357 2229.59,137.829 2230.72,137.384 2231.85,137.023 2232.98,136.744 2234.12,136.549 \n 2235.25,136.438 2236.38,136.41 2237.52,136.465 2238.65,136.604 2239.78,136.826 2240.92,137.13 2242.05,137.518 2243.18,137.989 2244.31,138.543 2245.45,139.179 \n 2246.58,139.897 2247.71,140.697 2248.85,141.579 2249.98,142.541 2251.11,143.585 2252.24,144.709 2253.38,145.912 2254.51,147.195 2255.64,148.556 2256.78,149.996 \n 2257.91,151.512 2259.04,153.105 2260.17,154.774 2261.31,156.518 2262.44,158.337 2263.57,160.228 2264.71,162.192 2265.84,164.228 2266.97,166.334 2268.1,168.509 \n 2269.24,170.753 2270.37,173.064 2271.5,175.441 2272.64,177.882 2273.77,180.388 2274.9,182.956 2276.03,185.584 2277.17,188.273 2278.3,191.019 2279.43,193.823 \n 2280.57,196.682 2281.7,199.594 2282.83,202.559 2283.96,205.576 2285.1,208.643 2286.23,211.759 2287.36,214.921 2288.5,218.129 2289.63,221.381 2290.76,224.675 \n 2291.9,228.01 2293.03,231.384 2294.16,234.796 2295.29,238.244 2296.43,241.725 2297.56,245.24 2298.69,248.785 2299.83,252.36 2300.96,255.962 2302.09,259.59 \n 2303.22,263.242 2304.36,266.916 2305.49,270.611 2306.62,274.325 2307.76,278.056 2308.89,281.802 2310.02,285.562 2311.15,289.333 2312.29,293.114 2313.42,296.903 \n 2314.55,300.698 2315.69,304.498 2316.82,308.299 2317.95,312.101 2319.08,315.902 2320.22,319.699 2321.35,323.49 2322.48,327.274 2323.62,331.049 2324.75,334.813 \n 2325.88,338.564 2327.01,342.302 2328.15,346.023 2329.28,349.727 2330.41,353.411 2331.55,357.073 2332.68,360.712 2333.81,364.326 2334.94,367.914 2336.08,371.472 \n 2337.21,375 2338.34,378.496 2339.48,381.958 2340.61,385.385 2341.74,388.774 2342.88,392.125 2344.01,395.435 2345.14,398.704 2346.27,401.929 2347.41,405.109 \n 2348.54,408.242 2349.67,411.327 2350.81,414.363 2351.94,417.348 2353.07,420.28 2354.2,423.159 2355.34,425.983 2356.47,428.75 2357.6,431.459 2358.74,434.109 \n 2359.87,436.698 2361,439.226 2362.13,441.69 2363.27,444.091 2364.4,446.425 2365.53,448.694 2366.67,450.894 2367.8,453.025 2368.93,455.087 2370.06,457.078 \n 2371.2,458.998 2372.33,460.845 2373.46,462.619 2374.6,464.319 2375.73,465.944 2376.86,467.492 2377.99,468.964 2379.13,470.359 2380.26,471.675 2381.39,472.912 \n 2382.53,474.07 2383.66,475.148 2384.79,476.145 2385.92,477.06 2387.06,477.894 2388.19,478.647 2389.32,479.316 2390.46,479.903 2391.59,480.407 2392.72,480.828 \n 2393.86,481.165 2394.99,481.419 2396.12,481.589 2397.25,481.675 2398.39,481.678 2399.52,481.596 2400.65,481.431 2401.79,481.183 2402.92,480.851 2404.05,480.435 \n 2405.18,479.937 2406.32,479.356 2407.45,478.692 2408.58,477.947 2409.72,477.12 2410.85,476.211 2411.98,475.222 2413.11,474.153 2414.25,473.004 2415.38,471.776 \n 2416.51,470.469 2417.65,469.084 2418.78,467.621 2419.91,466.082 2421.04,464.467 2422.18,462.777 2423.31,461.012 2424.44,459.174 2425.58,457.263 2426.71,455.28 \n 2427.84,453.226 2428.97,451.102 2430.11,448.909 2431.24,446.648 2432.37,444.32 2433.51,441.927 2434.64,439.469 2435.77,436.947 2436.9,434.363 2438.04,431.718 \n 2439.17,429.014 2440.3,426.251 2441.44,423.431 2442.57,420.555 2443.7,417.625 2444.84,414.642 2445.97,411.608 2447.1,408.524 2448.23,405.391 2449.37,402.212 \n 2450.5,398.989 2451.63,395.721 2452.77,392.412 2453.9,389.063 2455.03,385.677 2456.16,382.253 2457.3,378.793 2458.43,375.3 2459.56,371.775 2460.7,368.219 \n 2461.83,364.635 2462.96,361.024 2464.09,357.387 2465.23,353.728 2466.36,350.047 2467.49,346.346 2468.63,342.628 2469.76,338.893 2470.89,335.144 2472.02,331.382 \n 2473.16,327.609 2474.29,323.827 2475.42,320.038 2476.56,316.244 2477.69,312.446 2478.82,308.646 2479.95,304.846 2481.09,301.048 2482.22,297.254 2483.35,293.465 \n 2484.49,289.683 2485.62,285.91 2486.75,282.149 2487.88,278.4 2489.02,274.666 2490.15,270.948 2491.28,267.248 2492.42,263.569 2493.55,259.912 2494.68,256.278 \n 2495.82,252.67 2496.95,249.09 2498.08,245.54 2499.21,242.02 2500.35,238.534 2501.48,235.081 2502.61,231.664 2503.75,228.285 2504.88,224.944 2506.01,221.644 \n 2507.14,218.386 2508.28,215.173 2509.41,212.005 2510.54,208.884 2511.68,205.811 2512.81,202.789 2513.94,199.819 2515.07,196.901 2516.21,194.038 2517.34,191.231 \n 2518.47,188.481 2519.61,185.789 2520.74,183.158 2521.87,180.587 2523,178.079 2524.14,175.634 2525.27,173.254 2526.4,170.939 2527.54,168.692 2528.67,166.512 \n 2529.8,164.402 2530.93,162.361 2532.07,160.391 2533.2,158.494 2534.33,156.669 2535.47,154.918 2536.6,153.242 2537.73,151.641 2538.86,150.116 2540,148.669 \n 2541.13,147.299 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1630.29,202.531 1631.42,205.981 1632.55,209.495 1633.69,213.073 \n 1634.82,216.712 1635.95,220.409 1637.09,224.162 1638.22,227.969 1639.35,231.827 1640.48,235.734 1641.62,239.687 1642.75,243.684 1643.88,247.723 1645.02,251.8 \n 1646.15,255.914 1647.28,260.061 1648.41,264.24 1649.55,268.446 1650.68,272.679 1651.81,276.934 1652.95,281.21 1654.08,285.503 1655.21,289.811 1656.34,294.132 \n 1657.48,298.461 1658.61,302.798 1659.74,307.138 1660.88,311.479 1662.01,315.819 1663.14,320.155 1664.27,324.484 1665.41,328.803 1666.54,333.109 1667.67,337.401 \n 1668.81,341.674 1669.94,345.927 1671.07,350.156 1672.2,354.359 1673.34,358.534 1674.47,362.677 1675.6,366.787 1676.74,370.859 1677.87,374.893 1679,378.885 \n 1680.14,382.832 1681.27,386.733 1682.4,390.585 1683.53,394.385 1684.67,398.131 1685.8,401.821 1686.93,405.452 1688.07,409.022 1689.2,412.529 1690.33,415.97 \n 1691.46,419.344 1692.6,422.648 1693.73,425.88 1694.86,429.039 1696,432.121 1697.13,435.126 1698.26,438.05 1699.39,440.893 1700.53,443.652 1701.66,446.326 \n 1702.79,448.913 1703.93,451.412 1705.06,453.82 1706.19,456.137 1707.32,458.36 1708.46,460.489 1709.59,462.523 1710.72,464.459 1711.86,466.296 1712.99,468.035 \n 1714.12,469.672 1715.25,471.208 1716.39,472.642 1717.52,473.972 1718.65,475.198 1719.79,476.318 1720.92,477.333 1722.05,478.242 1723.18,479.043 1724.32,479.737 \n 1725.45,480.323 1726.58,480.8 1727.72,481.168 1728.85,481.428 1729.98,481.578 1731.12,481.619 1732.25,481.55 1733.38,481.373 1734.51,481.086 1735.65,480.69 \n 1736.78,480.186 1737.91,479.573 1739.05,478.852 1740.18,478.024 1741.31,477.088 1742.44,476.047 1743.58,474.9 1744.71,473.648 1745.84,472.292 1746.98,470.833 \n 1748.11,469.272 1749.24,467.609 1750.37,465.845 1751.51,463.983 1752.64,462.023 1753.77,459.965 1754.91,457.813 1756.04,455.566 1757.17,453.226 1758.3,450.795 \n 1759.44,448.274 1760.57,445.665 1761.7,442.969 1762.84,440.188 1763.97,437.325 1765.1,434.38 1766.23,431.355 1767.37,428.254 1768.5,425.077 1769.63,421.826 \n 1770.77,418.504 1771.9,415.113 1773.03,411.655 1774.16,408.132 1775.3,404.547 1776.43,400.901 1777.56,397.197 1778.7,393.437 1779.83,389.624 1780.96,385.76 \n 1782.1,381.848 1783.23,377.889 1784.36,373.887 1785.49,369.844 1786.63,365.763 1787.76,361.645 1788.89,357.494 1790.03,353.312 1791.16,349.102 1792.29,344.866 \n 1793.42,340.608 1794.56,336.329 1795.69,332.033 1796.82,327.722 1797.96,323.399 1799.09,319.068 1800.22,314.729 1801.35,310.388 1802.49,306.046 1803.62,301.706 \n 1804.75,297.371 1805.89,293.042 1807.02,288.724 1808.15,284.419 1809.28,280.13 1810.42,275.858 1811.55,271.608 1812.68,267.382 1813.82,263.182 1814.95,259.011 \n 1816.08,254.873 1817.21,250.768 1818.35,246.7 1819.48,242.672 1820.61,238.686 1821.75,234.744 1822.88,230.85 1824.01,227.004 1825.14,223.211 1826.28,219.472 \n 1827.41,215.789 1828.54,212.165 1829.68,208.603 1830.81,205.104 1831.94,201.67 1833.08,198.305 1834.21,195.01 1835.34,191.787 1836.47,188.638 1837.61,185.565 \n 1838.74,182.571 1839.87,179.655 1841.01,176.822 1842.14,174.073 1843.27,171.409 1844.4,168.832 1845.54,166.344 1846.67,163.946 1847.8,161.641 1848.94,159.429 \n 1850.07,157.311 1851.2,155.29 1852.33,153.367 1853.47,151.542 1854.6,149.816 1855.73,148.192 1856.87,146.67 1858,145.25 1859.13,143.934 1860.26,142.722 \n 1861.4,141.616 1862.53,140.615 1863.66,139.721 1864.8,138.934 1865.93,138.255 1867.06,137.684 1868.19,137.221 1869.33,136.866 1870.46,136.62 1871.59,136.483 \n 1872.73,136.455 1873.86,136.536 1874.99,136.726 1876.12,137.025 1877.26,137.433 1878.39,137.95 1879.52,138.575 1880.66,139.308 1881.79,140.148 1882.92,141.096 \n 1884.06,142.15 1885.19,143.31 1886.32,144.575 1887.45,145.944 1888.59,147.417 1889.72,148.992 1890.85,150.669 1891.99,152.447 1893.12,154.323 1894.25,156.298 \n 1895.38,158.37 1896.52,160.537 1897.65,162.798 1898.78,165.151 1899.92,167.595 1901.05,170.129 1902.18,172.75 1903.31,175.457 1904.45,178.248 1905.58,181.122 \n 1906.71,184.077 1907.85,187.11 1908.98,190.221 1910.11,193.407 1911.24,196.666 1912.38,199.996 1913.51,203.395 1914.64,206.86 1915.78,210.391 1916.91,213.983 \n 1918.04,217.637 1919.17,221.348 1920.31,225.114 1921.44,228.934 1922.57,232.805 1923.71,236.725 1924.84,240.69 1925.97,244.699 1927.1,248.749 1928.24,252.837 \n 1929.37,256.961 1930.5,261.118 1931.64,265.305 1932.77,269.52 1933.9,273.76 1935.04,278.022 1936.17,282.304 1937.3,286.601 1938.43,290.913 1939.57,295.237 \n 1940.7,299.57 1941.83,303.908 1942.97,308.25 1944.1,312.592 1945.23,316.932 1946.36,321.266 1947.5,325.593 1948.63,329.909 1949.76,334.212 1950.9,338.498 \n 1952.03,342.766 1953.16,347.013 1954.29,351.235 1955.43,355.431 1956.56,359.597 1957.69,363.732 1958.83,367.832 1959.96,371.895 1961.09,375.919 1962.22,379.9 \n 1963.36,383.837 1964.49,387.726 1965.62,391.566 1966.76,395.354 1967.89,399.088 1969.02,402.764 1970.15,406.382 1971.29,409.937 1972.42,413.428 1973.55,416.853 \n 1974.69,420.209 1975.82,423.496 1976.95,426.71 1978.08,429.85 1979.22,432.913 1980.35,435.899 1981.48,438.803 1982.62,441.626 1983.75,444.365 1984.88,447.018 \n 1986.02,449.583 1987.15,452.059 1988.28,454.445 1989.41,456.738 1990.55,458.938 1991.68,
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