notes/III.5.Interpolation.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# III.5 Interpolation and quadrature\n",
    "\n",
    "_Polynomial interpolation_ is the process of finding a polynomial that equals data at a precise set of points.\n",
    "_Quadrature_ is the act of approximating an integral by a weighted sum:\n",
    "$$\n",
    "\\int_a^b f(x) w(x) {\\rm d}x ≈ \\sum_{j=1}^n w_j f(x_j).\n",
    "$$\n",
    "In these notes we see that the two concepts are intrinsically linked:  interpolation leads naturally\n",
    "to quadrature rules. \n",
    "\n",
    "1. Polynomial Interpolation: we describe how to interpolate a function by a polynomial and a set of points.\n",
    "2. Interpolatory quadrature rule: polynomial interpolation leads naturally to ways to integrate\n",
    "functions, but onely realisable in the simplest cases.\n",
    "\n",
    "\n",
    "## 1. Polynomial Interpolation\n",
    "\n",
    "We already saw a special case of polynomial interpolation, where we saw that the polynomial\n",
    "$$\n",
    "f(z) ≈ ∑_{k=0}^{n-1} f̂_k^n z^k\n",
    "$$\n",
    "equaled $f$ at evenly spaced points on the unit circle: ${\\rm e}^{{\\rm i} 2π j/n}$. \n",
    "But here we consider the following:\n",
    "\n",
    "**Definition 1 (interpolatory polynomial)** Given $n$ distinct points $x_1,…,x_n ∈ ℝ$ \n",
    "and $n$ _samples_ $f_1,…,f_n ∈ ℝ$, a degree $n-1$\n",
    "_interpolatory polynomial_ $p(x)$ satisfies\n",
    "$$\n",
    "p(x_j) = f_j\n",
    "$$\n",
    "\n",
    "The easiest way to solve this problem is to invert the Vandermonde system:\n",
    "\n",
    "**Definition 2 (Vandermonde)** The _Vandermonde matrix_ associated with $n$ distinct points $x_1,…,x_n ∈ ℝ$\n",
    "is the matrix\n",
    "$$\n",
    "V := \\begin{bmatrix} 1 & x_1 & ⋯ & x_1^{n-1} \\\\\n",
    "                    ⋮ & ⋮ & ⋱ & ⋮ \\\\\n",
    "                    1 & x_n & ⋯ & x_n^{n-1}\n",
    "                    \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "**Proposition 1 (interpolatory polynomial uniqueness)** \n",
    "The interpolatory polynomial is unique, and the Vandermonde matrix is invertible.\n",
    "\n",
    "**Proof**\n",
    "Suppose $p$ and $p̃$ are both interpolatory polynomials. Then $p(x) - p̃(x)$ vanishes at $n$ distinct points $x_j$. By the fundamental theorem of\n",
    "algebra it must be zero, i.e., $p = p̃$.\n",
    "\n",
    "For the second part, if $V 𝐜 = 0$ for $𝐜 ∈ ℝ$ then for $q(x) = c_1 + ⋯ + c_n x^{n-1}$ we have\n",
    "$$\n",
    "q(x_j) = 𝐞_j^⊤ V 𝐜 = 0\n",
    "$$\n",
    "hence $q$ vanishes at $n$ distinct points and is therefore 0, i.e., $𝐜 = 0$.\n",
    "\n",
    "∎\n",
    "\n",
    "Thus a quick-and-dirty way to to do interpolation is to invert the Vandermonde matrix\n",
    "(which we saw in the least squares setting with more samples then coefficients):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2023-03-29T16:48:04.452000Z",
     "iopub.status.busy": "2023-03-29T16:48:03.987000Z",
     "iopub.status.idle": "2023-03-29T16:48:17.363000Z",
     "shell.execute_reply": "2023-03-29T16:48:17.345000Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"utf-8\"?>\n",
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"600\" height=\"400\" viewBox=\"0 0 2400 1600\">\n",
       "<defs>\n",
       "  <clipPath id=\"clip640\">\n",
       "    <rect x=\"0\" y=\"0\" width=\"2400\" height=\"1600\"/>\n",
       "  </clipPath>\n",
       "</defs>\n",
       "<path clip-path=\"url(#clip640)\" d=\"M0 1600 L2400 1600 L2400 0 L0 0  Z\" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n",
       "<defs>\n",
       "  <clipPath id=\"clip641\">\n",
       "    <rect x=\"480\" y=\"0\" width=\"1681\" height=\"1600\"/>\n",
       "  </clipPath>\n",
       "</defs>\n",
       "<path clip-path=\"url(#clip640)\" d=\"M193.936 1486.45 L2352.76 1486.45 L2352.76 47.2441 L193.936 47.2441  Z\" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n",
       "<defs>\n",
       "  <clipPath id=\"clip642\">\n",
       "    <rect x=\"193\" y=\"47\" width=\"2160\" height=\"1440\"/>\n",
       "  </clipPath>\n",
       "</defs>\n",
       "<polyline clip-path=\"url(#clip642)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"255.035,1486.45 255.035,47.2441 \"/>\n",
       "<polyline clip-path=\"url(#clip642)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"764.19,1486.45 764.19,47.2441 \"/>\n",
       "<polyline clip-path=\"url(#clip642)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"1273.35,1486.45 1273.35,47.2441 \"/>\n",
       "<polyline clip-path=\"url(#clip642)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"1782.5,1486.45 1782.5,47.2441 \"/>\n",
       "<polyline clip-path=\"url(#clip642)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"2291.66,1486.45 2291.66,47.2441 \"/>\n",
       "<polyline clip-path=\"url(#clip640)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:4; stroke-opacity:1; fill:none\" points=\"193.936,1486.45 2352.76,1486.45 \"/>\n",
       "<polyline clip-path=\"url(#clip640)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:4; stroke-opacity:1; fill:none\" points=\"255.035,1486.45 255.035,1467.55 \"/>\n",
       "<polyline clip-path=\"url(#clip640)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:4; stroke-opacity:1; fill:none\" points=\"764.19,1486.45 764.19,1467.55 \"/>\n",
       "<polyline clip-path=\"url(#clip640)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:4; stroke-opacity:1; fill:none\" points=\"1273.35,1486.45 1273.35,1467.55 \"/>\n",
       "<polyline clip-path=\"url(#clip640)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:4; stroke-opacity:1; fill:none\" points=\"1782.5,1486.45 1782.5,1467.55 \"/>\n",
       "<polyline clip-path=\"url(#clip640)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:4; stroke-opacity:1; fill:none\" points=\"2291.66,1486.45 2291.66,1467.55 \"/>\n",
       "<path clip-path=\"url(#clip640)\" d=\"M217.338 1517.37 Q213.727 1517.37 211.898 1520.93 Q210.093 1524.47 210.093 1531.6 Q210.093 1538.71 211.898 1542.27 Q213.727 1545.82 217.338 1545.82 Q220.972 1545.82 222.778 1542.27 Q224.607 1538.71 224.607 1531.6 Q224.607 1524.47 222.778 1520.93 Q220.972 1517.37 217.338 1517.37 M217.338 1513.66 Q223.148 1513.66 226.204 1518.27 Q229.282 1522.85 229.282 1531.6 Q229.282 1540.33 226.204 1544.94 Q223.148 1549.52 217.338 1549.52 Q211.528 1549.52 208.449 1544.94 Q205.394 1540.33 205.394 1531.6 Q205.394 1522.85 208.449 1518.27 Q211.528 1513.66 217.338 1513.66 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M237.5 1542.97 L242.384 1542.97 L242.384 1548.85 L237.5 1548.85 L237.5 1542.97 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M262.569 1517.37 Q258.958 1517.37 257.13 1520.93 Q255.324 1524.47 255.324 1531.6 Q255.324 1538.71 257.13 1542.27 Q258.958 1545.82 262.569 1545.82 Q266.204 1545.82 268.009 1542.27 Q269.838 1538.71 269.838 1531.6 Q269.838 1524.47 268.009 1520.93 Q266.204 1517.37 262.569 1517.37 M262.569 1513.66 Q268.38 1513.66 271.435 1518.27 Q274.514 1522.85 274.514 1531.6 Q274.514 1540.33 271.435 1544.94 Q268.38 1549.52 262.569 1549.52 Q256.759 1549.52 253.681 1544.94 Q250.625 1540.33 250.625 1531.6 Q250.625 1522.85 253.681 1518.27 Q256.759 1513.66 262.569 1513.66 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M292.731 1517.37 Q289.12 1517.37 287.291 1520.93 Q285.486 1524.47 285.486 1531.6 Q285.486 1538.71 287.291 1542.27 Q289.12 1545.82 292.731 1545.82 Q296.365 1545.82 298.171 1542.27 Q300 1538.71 300 1531.6 Q300 1524.47 298.171 1520.93 Q296.365 1517.37 292.731 1517.37 M292.731 1513.66 Q298.541 1513.66 301.597 1518.27 Q304.676 1522.85 304.676 1531.6 Q304.676 1540.33 301.597 1544.94 Q298.541 1549.52 292.731 1549.52 Q286.921 1549.52 283.842 1544.94 Q280.787 1540.33 280.787 1531.6 Q280.787 1522.85 283.842 1518.27 Q286.921 1513.66 292.731 1513.66 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M726.991 1517.37 Q723.38 1517.37 721.552 1520.93 Q719.746 1524.47 719.746 1531.6 Q719.746 1538.71 721.552 1542.27 Q723.38 1545.82 726.991 1545.82 Q730.626 1545.82 732.431 1542.27 Q734.26 1538.71 734.26 1531.6 Q734.26 1524.47 732.431 1520.93 Q730.626 1517.37 726.991 1517.37 M726.991 1513.66 Q732.802 1513.66 735.857 1518.27 Q738.936 1522.85 738.936 1531.6 Q738.936 1540.33 735.857 1544.94 Q732.802 1549.52 726.991 1549.52 Q721.181 1549.52 718.103 1544.94 Q715.047 1540.33 715.047 1531.6 Q715.047 1522.85 718.103 1518.27 Q721.181 1513.66 726.991 1513.66 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M747.153 1542.97 L752.038 1542.97 L752.038 1548.85 L747.153 1548.85 L747.153 1542.97 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M766.25 1544.91 L782.57 1544.91 L782.57 1548.85 L760.626 1548.85 L760.626 1544.91 Q763.288 1542.16 767.871 1537.53 Q772.477 1532.88 773.658 1531.53 Q775.903 1529.01 776.783 1527.27 Q777.686 1525.51 777.686 1523.82 Q777.686 1521.07 775.741 1519.33 Q773.82 1517.6 770.718 1517.6 Q768.519 1517.6 766.065 1518.36 Q763.635 1519.13 760.857 1520.68 L760.857 1515.95 Q763.681 1514.82 766.135 1514.24 Q768.588 1513.66 770.625 1513.66 Q775.996 1513.66 779.19 1516.35 Q782.385 1519.03 782.385 1523.52 Q782.385 1525.65 781.574 1527.57 Q780.787 1529.47 778.681 1532.07 Q778.102 1532.74 775 1535.95 Q771.899 1539.15 766.25 1544.91 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M792.431 1514.29 L810.787 1514.29 L810.787 1518.22 L796.713 1518.22 L796.713 1526.7 Q797.732 1526.35 798.75 1526.19 Q799.769 1526 800.787 1526 Q806.574 1526 809.954 1529.17 Q813.334 1532.34 813.334 1537.76 Q813.334 1543.34 809.861 1546.44 Q806.389 1549.52 800.07 1549.52 Q797.894 1549.52 795.625 1549.15 Q793.38 1548.78 790.973 1548.04 L790.973 1543.34 Q793.056 1544.47 795.278 1545.03 Q797.5 1545.58 799.977 1545.58 Q803.982 1545.58 806.32 1543.48 Q808.658 1541.37 808.658 1537.76 Q808.658 1534.15 806.32 1532.04 Q803.982 1529.94 799.977 1529.94 Q798.102 1529.94 796.227 1530.35 Q794.375 1530.77 792.431 1531.65 L792.431 1514.29 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M1235.65 1517.37 Q1232.04 1517.37 1230.21 1520.93 Q1228.4 1524.47 1228.4 1531.6 Q1228.4 1538.71 1230.21 1542.27 Q1232.04 1545.82 1235.65 1545.82 Q1239.28 1545.82 1241.09 1542.27 Q1242.92 1538.71 1242.92 1531.6 Q1242.92 1524.47 1241.09 1520.93 Q1239.28 1517.37 1235.65 1517.37 M1235.65 1513.66 Q1241.46 1513.66 1244.52 1518.27 Q1247.59 1522.85 1247.59 1531.6 Q1247.59 1540.33 1244.52 1544.94 Q1241.46 1549.52 1235.65 1549.52 Q1229.84 1549.52 1226.76 1544.94 Q1223.71 1540.33 1223.71 1531.6 Q1223.71 1522.85 1226.76 1518.27 Q1229.84 1513.66 1235.65 1513.66 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M1255.81 1542.97 L1260.7 1542.97 L1260.7 1548.85 L1255.81 1548.85 L1255.81 1542.97 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M1270.93 1514.29 L1289.28 1514.29 L1289.28 1518.22 L1275.21 1518.22 L1275.21 1526.7 Q1276.23 1526.35 1277.25 1526.19 Q1278.26 1526 1279.28 1526 Q1285.07 1526 1288.45 1529.17 Q1291.83 1532.34 1291.83 1537.76 Q1291.83 1543.34 1288.36 1546.44 Q1284.89 1549.52 1278.57 1549.52 Q1276.39 1549.52 1274.12 1549.15 Q1271.88 1548.78 1269.47 1548.04 L1269.47 1543.34 Q1271.55 1544.47 1273.77 1545.03 Q1276 1545.58 1278.47 1545.58 Q1282.48 1545.58 1284.82 1543.48 Q1287.15 1541.37 1287.15 1537.76 Q1287.15 1534.15 1284.82 1532.04 Q1282.48 1529.94 1278.47 1529.94 Q1276.6 1529.94 1274.72 1530.35 Q1272.87 1530.77 1270.93 1531.65 L1270.93 1514.29 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M1311.04 1517.37 Q1307.43 1517.37 1305.6 1520.93 Q1303.8 1524.47 1303.8 1531.6 Q1303.8 1538.71 1305.6 1542.27 Q1307.43 1545.82 1311.04 1545.82 Q1314.68 1545.82 1316.48 1542.27 Q1318.31 1538.71 1318.31 1531.6 Q1318.31 1524.47 1316.48 1520.93 Q1314.68 1517.37 1311.04 1517.37 M1311.04 1513.66 Q1316.85 1513.66 1319.91 1518.27 Q1322.99 1522.85 1322.99 1531.6 Q1322.99 1540.33 1319.91 1544.94 Q1316.85 1549.52 1311.04 1549.52 Q1305.23 1549.52 1302.15 1544.94 Q1299.1 1540.33 1299.1 1531.6 Q1299.1 1522.85 1302.15 1518.27 Q1305.23 1513.66 1311.04 1513.66 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M1745.3 1517.37 Q1741.69 1517.37 1739.86 1520.93 Q1738.06 1524.47 1738.06 1531.6 Q1738.06 1538.71 1739.86 1542.27 Q1741.69 1545.82 1745.3 1545.82 Q1748.94 1545.82 1750.74 1542.27 Q1752.57 1538.71 1752.57 1531.6 Q1752.57 1524.47 1750.74 1520.93 Q1748.94 1517.37 1745.3 1517.37 M1745.3 1513.66 Q1751.11 1513.66 1754.17 1518.27 Q1757.25 1522.85 1757.25 1531.6 Q1757.25 1540.33 1754.17 1544.94 Q1751.11 1549.52 1745.3 1549.52 Q1739.49 1549.52 1736.41 1544.94 Q1733.36 1540.33 1733.36 1531.6 Q1733.36 1522.85 1736.41 1518.27 Q1739.49 1513.66 1745.3 1513.66 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M1765.46 1542.97 L1770.35 1542.97 L1770.35 1548.85 L1765.46 1548.85 L1765.46 1542.97 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M1779.35 1514.29 L1801.58 1514.29 L1801.58 1516.28 L1789.03 1548.85 L1784.15 1548.85 L1795.95 1518.22 L1779.35 1518.22 L1779.35 1514.29 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M1810.74 1514.29 L1829.1 1514.29 L1829.1 1518.22 L1815.02 1518.22 L1815.02 1526.7 Q1816.04 1526.35 1817.06 1526.19 Q1818.08 1526 1819.1 1526 Q1824.89 1526 1828.27 1529.17 Q1831.64 1532.34 1831.64 1537.76 Q1831.64 1543.34 1828.17 1546.44 Q1824.7 1549.52 1818.38 1549.52 Q1816.21 1549.52 1813.94 1549.15 Q1811.69 1548.78 1809.28 1548.04 L1809.28 1543.34 Q1811.37 1544.47 1813.59 1545.03 Q1815.81 1545.58 1818.29 1545.58 Q1822.29 1545.58 1824.63 1543.48 Q1826.97 1541.37 1826.97 1537.76 Q1826.97 1534.15 1824.63 1532.04 Q1822.29 1529.94 1818.29 1529.94 Q1816.41 1529.94 1814.54 1530.35 Q1812.69 1530.77 1810.74 1531.65 L1810.74 1514.29 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M2243.73 1544.91 L2251.37 1544.91 L2251.37 1518.55 L2243.06 1520.21 L2243.06 1515.95 L2251.32 1514.29 L2256 1514.29 L2256 1544.91 L2263.64 1544.91 L2263.64 1548.85 L2243.73 1548.85 L2243.73 1544.91 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M2273.08 1542.97 L2277.97 1542.97 L2277.97 1548.85 L2273.08 1548.85 L2273.08 1542.97 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M2298.15 1517.37 Q2294.54 1517.37 2292.71 1520.93 Q2290.9 1524.47 2290.9 1531.6 Q2290.9 1538.71 2292.71 1542.27 Q2294.54 1545.82 2298.15 1545.82 Q2301.78 1545.82 2303.59 1542.27 Q2305.42 1538.71 2305.42 1531.6 Q2305.42 1524.47 2303.59 1520.93 Q2301.78 1517.37 2298.15 1517.37 M2298.15 1513.66 Q2303.96 1513.66 2307.02 1518.27 Q2310.09 1522.85 2310.09 1531.6 Q2310.09 1540.33 2307.02 1544.94 Q2303.96 1549.52 2298.15 1549.52 Q2292.34 1549.52 2289.26 1544.94 Q2286.21 1540.33 2286.21 1531.6 Q2286.21 1522.85 2289.26 1518.27 Q2292.34 1513.66 2298.15 1513.66 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M2328.31 1517.37 Q2324.7 1517.37 2322.87 1520.93 Q2321.07 1524.47 2321.07 1531.6 Q2321.07 1538.71 2322.87 1542.27 Q2324.7 1545.82 2328.31 1545.82 Q2331.95 1545.82 2333.75 1542.27 Q2335.58 1538.71 2335.58 1531.6 Q2335.58 1524.47 2333.75 1520.93 Q2331.95 1517.37 2328.31 1517.37 M2328.31 1513.66 Q2334.12 1513.66 2337.18 1518.27 Q2340.26 1522.85 2340.26 1531.6 Q2340.26 1540.33 2337.18 1544.94 Q2334.12 1549.52 2328.31 1549.52 Q2322.5 1549.52 2319.42 1544.94 Q2316.37 1540.33 2316.37 1531.6 Q2316.37 1522.85 2319.42 1518.27 Q2322.5 1513.66 2328.31 1513.66 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><polyline clip-path=\"url(#clip642)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"193.936,1445.72 2352.76,1445.72 \"/>\n",
       "<polyline clip-path=\"url(#clip642)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"193.936,1106.28 2352.76,1106.28 \"/>\n",
       "<polyline clip-path=\"url(#clip642)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"193.936,766.846 2352.76,766.846 \"/>\n",
       "<polyline clip-path=\"url(#clip642)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"193.936,427.411 2352.76,427.411 \"/>\n",
       "<polyline clip-path=\"url(#clip642)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"193.936,87.9763 2352.76,87.9763 \"/>\n",
       "<polyline clip-path=\"url(#clip640)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:4; stroke-opacity:1; fill:none\" points=\"193.936,1486.45 193.936,47.2441 \"/>\n",
       "<polyline clip-path=\"url(#clip640)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:4; stroke-opacity:1; fill:none\" points=\"193.936,1445.72 212.834,1445.72 \"/>\n",
       "<polyline clip-path=\"url(#clip640)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:4; stroke-opacity:1; fill:none\" points=\"193.936,1106.28 212.834,1106.28 \"/>\n",
       "<polyline clip-path=\"url(#clip640)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:4; stroke-opacity:1; fill:none\" points=\"193.936,766.846 212.834,766.846 \"/>\n",
       "<polyline clip-path=\"url(#clip640)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:4; stroke-opacity:1; fill:none\" points=\"193.936,427.411 212.834,427.411 \"/>\n",
       "<polyline clip-path=\"url(#clip640)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:4; stroke-opacity:1; fill:none\" points=\"193.936,87.9763 212.834,87.9763 \"/>\n",
       "<path clip-path=\"url(#clip640)\" d=\"M50.9921 1446.17 L80.6679 1446.17 L80.6679 1450.1 L50.9921 1450.1 L50.9921 1446.17 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M91.5706 1459.06 L99.2095 1459.06 L99.2095 1432.7 L90.8993 1434.36 L90.8993 1430.1 L99.1632 1428.44 L103.839 1428.44 L103.839 1459.06 L111.478 1459.06 L111.478 1463 L91.5706 1463 L91.5706 1459.06 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M120.922 1457.12 L125.807 1457.12 L125.807 1463 L120.922 1463 L120.922 1457.12 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M145.992 1431.52 Q142.381 1431.52 140.552 1435.08 Q138.746 1438.62 138.746 1445.75 Q138.746 1452.86 140.552 1456.42 Q142.381 1459.96 145.992 1459.96 Q149.626 1459.96 151.431 1456.42 Q153.26 1452.86 153.26 1445.75 Q153.26 1438.62 151.431 1435.08 Q149.626 1431.52 145.992 1431.52 M145.992 1427.81 Q151.802 1427.81 154.857 1432.42 Q157.936 1437 157.936 1445.75 Q157.936 1454.48 154.857 1459.08 Q151.802 1463.67 145.992 1463.67 Q140.181 1463.67 137.103 1459.08 Q134.047 1454.48 134.047 1445.75 Q134.047 1437 137.103 1432.42 Q140.181 1427.81 145.992 1427.81 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M51.9875 1106.73 L81.6633 1106.73 L81.6633 1110.67 L51.9875 1110.67 L51.9875 1106.73 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M101.756 1092.08 Q98.1447 1092.08 96.316 1095.64 Q94.5104 1099.19 94.5104 1106.32 Q94.5104 1113.42 96.316 1116.99 Q98.1447 1120.53 101.756 1120.53 Q105.39 1120.53 107.196 1116.99 Q109.024 1113.42 109.024 1106.32 Q109.024 1099.19 107.196 1095.64 Q105.39 1092.08 101.756 1092.08 M101.756 1088.38 Q107.566 1088.38 110.621 1092.98 Q113.7 1097.57 113.7 1106.32 Q113.7 1115.04 110.621 1119.65 Q107.566 1124.23 101.756 1124.23 Q95.9456 1124.23 92.8669 1119.65 Q89.8114 1115.04 89.8114 1106.32 Q89.8114 1097.57 92.8669 1092.98 Q95.9456 1088.38 101.756 1088.38 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M121.918 1117.68 L126.802 1117.68 L126.802 1123.56 L121.918 1123.56 L121.918 1117.68 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M137.033 1089 L155.39 1089 L155.39 1092.94 L141.316 1092.94 L141.316 1101.41 Q142.334 1101.06 143.353 1100.9 Q144.371 1100.71 145.39 1100.71 Q151.177 1100.71 154.556 1103.89 Q157.936 1107.06 157.936 1112.47 Q157.936 1118.05 154.464 1121.15 Q150.992 1124.23 144.672 1124.23 Q142.496 1124.23 140.228 1123.86 Q137.982 1123.49 135.575 1122.75 L135.575 1118.05 Q137.658 1119.19 139.881 1119.74 Q142.103 1120.3 144.58 1120.3 Q148.584 1120.3 150.922 1118.19 Q153.26 1116.08 153.26 1112.47 Q153.26 1108.86 150.922 1106.76 Q148.584 1104.65 144.58 1104.65 Q142.705 1104.65 140.83 1105.07 Q138.978 1105.48 137.033 1106.36 L137.033 1089 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M100.76 752.645 Q97.1493 752.645 95.3206 756.21 Q93.515 759.752 93.515 766.881 Q93.515 773.988 95.3206 777.552 Q97.1493 781.094 100.76 781.094 Q104.395 781.094 106.2 777.552 Q108.029 773.988 108.029 766.881 Q108.029 759.752 106.2 756.21 Q104.395 752.645 100.76 752.645 M100.76 748.941 Q106.571 748.941 109.626 753.548 Q112.705 758.131 112.705 766.881 Q112.705 775.608 109.626 780.214 Q106.571 784.798 100.76 784.798 Q94.9502 784.798 91.8715 780.214 Q88.816 775.608 88.816 766.881 Q88.816 758.131 91.8715 753.548 Q94.9502 748.941 100.76 748.941 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M120.922 778.247 L125.807 778.247 L125.807 784.126 L120.922 784.126 L120.922 778.247 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M145.992 752.645 Q142.381 752.645 140.552 756.21 Q138.746 759.752 138.746 766.881 Q138.746 773.988 140.552 777.552 Q142.381 781.094 145.992 781.094 Q149.626 781.094 151.431 777.552 Q153.26 773.988 153.26 766.881 Q153.26 759.752 151.431 756.21 Q149.626 752.645 145.992 752.645 M145.992 748.941 Q151.802 748.941 154.857 753.548 Q157.936 758.131 157.936 766.881 Q157.936 775.608 154.857 780.214 Q151.802 784.798 145.992 784.798 Q140.181 784.798 137.103 780.214 Q134.047 775.608 134.047 766.881 Q134.047 758.131 137.103 753.548 Q140.181 748.941 145.992 748.941 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M101.756 413.21 Q98.1447 413.21 96.316 416.775 Q94.5104 420.316 94.5104 427.446 Q94.5104 434.553 96.316 438.117 Q98.1447 441.659 101.756 441.659 Q105.39 441.659 107.196 438.117 Q109.024 434.553 109.024 427.446 Q109.024 420.316 107.196 416.775 Q105.39 413.21 101.756 413.21 M101.756 409.506 Q107.566 409.506 110.621 414.113 Q113.7 418.696 113.7 427.446 Q113.7 436.173 110.621 440.779 Q107.566 445.363 101.756 445.363 Q95.9456 445.363 92.8669 440.779 Q89.8114 436.173 89.8114 427.446 Q89.8114 418.696 92.8669 414.113 Q95.9456 409.506 101.756 409.506 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M121.918 438.812 L126.802 438.812 L126.802 444.691 L121.918 444.691 L121.918 438.812 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M137.033 410.131 L155.39 410.131 L155.39 414.066 L141.316 414.066 L141.316 422.539 Q142.334 422.191 143.353 422.029 Q144.371 421.844 145.39 421.844 Q151.177 421.844 154.556 425.016 Q157.936 428.187 157.936 433.603 Q157.936 439.182 154.464 442.284 Q150.992 445.363 144.672 445.363 Q142.496 445.363 140.228 444.992 Q137.982 444.622 135.575 443.881 L135.575 439.182 Q137.658 440.316 139.881 440.872 Q142.103 441.427 144.58 441.427 Q148.584 441.427 150.922 439.321 Q153.26 437.215 153.26 433.603 Q153.26 429.992 150.922 427.886 Q148.584 425.779 144.58 425.779 Q142.705 425.779 140.83 426.196 Q138.978 426.613 137.033 427.492 L137.033 410.131 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M91.5706 101.321 L99.2095 101.321 L99.2095 74.9555 L90.8993 76.6222 L90.8993 72.3629 L99.1632 70.6963 L103.839 70.6963 L103.839 101.321 L111.478 101.321 L111.478 105.256 L91.5706 105.256 L91.5706 101.321 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M120.922 99.3767 L125.807 99.3767 L125.807 105.256 L120.922 105.256 L120.922 99.3767 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M145.992 73.775 Q142.381 73.775 140.552 77.3398 Q138.746 80.8814 138.746 88.011 Q138.746 95.1174 140.552 98.6822 Q142.381 102.224 145.992 102.224 Q149.626 102.224 151.431 98.6822 Q153.26 95.1174 153.26 88.011 Q153.26 80.8814 151.431 77.3398 Q149.626 73.775 145.992 73.775 M145.992 70.0713 Q151.802 70.0713 154.857 74.6777 Q157.936 79.261 157.936 88.011 Q157.936 96.7378 154.857 101.344 Q151.802 105.928 145.992 105.928 Q140.181 105.928 137.103 101.344 Q134.047 96.7378 134.047 88.011 Q134.047 79.261 137.103 74.6777 Q140.181 70.0713 145.992 70.0713 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><polyline clip-path=\"url(#clip642)\" style=\"stroke:#009af9; stroke-linecap:round; stroke-linejoin:round; stroke-width:4; stroke-opacity:1; fill:none\" points=\"255.035,87.9763 257.073,88.0103 259.112,88.1123 261.151,88.2824 263.189,88.5204 265.228,88.8264 267.267,89.2003 269.305,89.6422 271.344,90.1518 273.383,90.7293 275.421,91.3746 277.46,92.0875 279.499,92.868 281.537,93.7161 283.576,94.6316 285.615,95.6145 287.653,96.6646 289.692,97.7819 291.731,98.9662 293.769,100.217 295.808,101.535 297.847,102.92 299.885,104.371 301.924,105.889 303.963,107.473 306.001,109.123 308.04,110.839 310.079,112.62 312.117,114.467 314.156,116.38 316.195,118.357 318.233,120.4 320.272,122.507 322.31,124.679 324.349,126.916 326.388,129.216 328.426,131.58 330.465,134.008 332.504,136.5 334.542,139.054 336.581,141.672 338.62,144.352 340.658,147.094 342.697,149.899 344.736,152.765 346.774,155.693 348.813,158.682 350.852,161.732 352.89,164.843 354.929,168.014 356.968,171.245 359.006,174.535 361.045,177.886 363.084,181.295 365.122,184.762 367.161,188.288 369.2,191.872 371.238,195.514 373.277,199.213 375.316,202.969 377.354,206.781 379.393,210.649 381.432,214.574 383.47,218.553 385.509,222.587 387.548,226.676 389.586,230.819 391.625,235.016 393.664,239.266 395.702,243.569 397.741,247.924 399.78,252.332 401.818,256.791 403.857,261.301 405.896,265.861 407.934,270.472 409.973,275.133 412.012,279.843 414.05,284.601 416.089,289.408 418.128,294.263 420.166,299.165 422.205,304.114 424.244,309.11 426.282,314.151 428.321,319.238 430.36,324.369 432.398,329.545 434.437,334.765 436.476,340.028 438.514,345.333 440.553,350.681 442.592,356.071 444.63,361.502 446.669,366.973 448.707,372.484 450.746,378.035 452.785,383.625 454.823,389.254 456.862,394.92 458.901,400.623 460.939,406.363 462.978,412.14 465.017,417.952 467.055,423.798 469.094,429.679 471.133,435.594 473.171,441.542 475.21,447.523 477.249,453.536 479.287,459.58 481.326,465.655 483.365,471.76 485.403,477.894 487.442,484.058 489.481,490.25 491.519,496.47 493.558,502.716 495.597,508.99 497.635,515.289 499.674,521.613 501.713,527.962 503.751,534.334 505.79,540.73 507.829,547.149 509.867,553.59 511.906,560.052 513.945,566.535 515.983,573.037 518.022,579.56 520.061,586.101 522.099,592.66 524.138,599.237 526.177,605.83 528.215,612.44 530.254,619.065 532.293,625.704 534.331,632.358 536.37,639.026 538.409,645.706 540.447,652.398 542.486,659.102 544.525,665.817 546.563,672.542 548.602,679.276 550.641,686.019 552.679,692.77 554.718,699.529 556.757,706.294 558.795,713.065 560.834,719.842 562.873,726.623 564.911,733.409 566.95,740.198 568.989,746.989 571.027,753.782 573.066,760.577 575.104,767.373 577.143,774.168 579.182,780.963 581.22,787.756 583.259,794.547 585.298,801.335 587.336,808.12 589.375,814.901 591.414,821.677 593.452,828.447 595.491,835.212 597.53,841.969 599.568,848.719 601.607,855.461 603.646,862.193 605.684,868.917 607.723,875.63 609.762,882.332 611.8,889.022 613.839,895.701 615.878,902.366 617.916,909.018 619.955,915.655 621.994,922.278 624.032,928.885 626.071,935.476 628.11,942.05 630.148,948.606 632.187,955.145 634.226,961.664 636.264,968.164 638.303,974.643 640.342,981.102 642.38,987.539 644.419,993.955 646.458,1000.35 648.496,1006.72 650.535,1013.06 652.574,1019.38 654.612,1025.68 656.651,1031.95 658.69,1038.19 660.728,1044.4 662.767,1050.59 664.806,1056.75 666.844,1062.88 668.883,1068.98 670.922,1075.05 672.96,1081.09 674.999,1087.1 677.038,1093.07 679.076,1099.02 681.115,1104.93 683.154,1110.8 685.192,1116.64 687.231,1122.45 689.27,1128.22 691.308,1133.96 693.347,1139.65 695.385,1145.31 697.424,1150.94 699.463,1156.52 701.501,1162.06 703.54,1167.57 705.579,1173.04 707.617,1178.46 709.656,1183.84 711.695,1189.18 713.733,1194.48 715.772,1199.74 717.811,1204.95 719.849,1210.12 721.888,1215.25 723.927,1220.33 725.965,1225.36 728.004,1230.35 730.043,1235.29 732.081,1240.18 734.12,1245.03 736.159,1249.83 738.197,1254.58 740.236,1259.29 742.275,1263.94 744.313,1268.54 746.352,1273.09 748.391,1277.6 750.429,1282.05 752.468,1286.45 754.507,1290.79 756.545,1295.09 758.584,1299.33 760.623,1303.52 762.661,1307.65 764.7,1311.73 766.739,1315.76 768.777,1319.73 770.816,1323.65 772.855,1327.51 774.893,1331.31 776.932,1335.06 778.971,1338.75 781.009,1342.38 783.048,1345.95 785.087,1349.47 787.125,1352.93 789.164,1356.33 791.203,1359.67 793.241,1362.95 795.28,1366.17 797.319,1369.34 799.357,1372.44 801.396,1375.48 803.435,1378.46 805.473,1381.38 807.512,1384.23 809.551,1387.03 811.589,1389.76 813.628,1392.43 815.667,1395.04 817.705,1397.58 819.744,1400.06 821.782,1402.48 823.821,1404.84 825.86,1407.13 827.898,1409.35 829.937,1411.52 831.976,1413.61 834.014,1415.65 836.053,1417.61 838.092,1419.52 840.13,1421.35 842.169,1423.12 844.208,1424.83 846.246,1426.47 848.285,1428.04 850.324,1429.55 852.362,1430.99 854.401,1432.37 856.44,1433.67 858.478,1434.91 860.517,1436.09 862.556,1437.2 864.594,1438.23 866.633,1439.21 868.672,1440.11 870.71,1440.95 872.749,1441.72 874.788,1442.42 876.826,1443.06 878.865,1443.62 880.904,1444.12 882.942,1444.55 884.981,1444.92 887.02,1445.21 889.058,1445.44 891.097,1445.6 893.136,1445.69 895.174,1445.72 897.213,1445.67 899.252,1445.56 901.29,1445.38 903.329,1445.13 905.368,1444.81 907.406,1444.43 909.445,1443.98 911.484,1443.46 913.522,1442.87 915.561,1442.21 917.6,1441.49 919.638,1440.7 921.677,1439.84 923.716,1438.91 925.754,1437.92 927.793,1436.86 929.832,1435.73 931.87,1434.54 933.909,1433.28 935.948,1431.95 937.986,1430.55 940.025,1429.09 942.064,1427.56 944.102,1425.97 946.141,1424.31 948.179,1422.58 950.218,1420.79 952.257,1418.93 954.295,1417.01 956.334,1415.02 958.373,1412.97 960.411,1410.85 962.45,1408.67 964.489,1406.43 966.527,1404.11 968.566,1401.74 970.605,1399.3 972.643,1396.8 974.682,1394.24 976.721,1391.61 978.759,1388.92 980.798,1386.17 982.837,1383.35 984.875,1380.48 986.914,1377.54 988.953,1374.54 990.991,1371.48 993.03,1368.36 995.069,1365.18 997.107,1361.94 999.146,1358.64 1001.18,1355.28 1003.22,1351.86 1005.26,1348.39 1007.3,1344.85 1009.34,1341.26 1011.38,1337.61 1013.42,1333.9 1015.46,1330.14 1017.49,1326.32 1019.53,1322.44 1021.57,1318.51 1023.61,1314.52 1025.65,1310.48 1027.69,1306.38 1029.73,1302.23 1031.76,1298.02 1033.8,1293.76 1035.84,1289.45 1037.88,1285.09 1039.92,1280.67 1041.96,1276.21 1044,1271.69 1046.04,1267.12 1048.07,1262.5 1050.11,1257.83 1052.15,1253.12 1054.19,1248.35 1056.23,1243.54 1058.27,1238.67 1060.31,1233.76 1062.34,1228.81 1064.38,1223.81 1066.42,1218.76 1068.46,1213.66 1070.5,1208.52 1072.54,1203.34 1074.58,1198.12 1076.62,1192.85 1078.65,1187.53 1080.69,1182.18 1082.73,1176.78 1084.77,1171.35 1086.81,1165.87 1088.85,1160.35 1090.89,1154.79 1092.92,1149.2 1094.96,1143.56 1097,1137.89 1099.04,1132.18 1101.08,1126.44 1103.12,1120.66 1105.16,1114.84 1107.2,1108.99 1109.23,1103.1 1111.27,1097.18 1113.31,1091.23 1115.35,1085.24 1117.39,1079.22 1119.43,1073.17 1121.47,1067.09 1123.5,1060.98 1125.54,1054.85 1127.58,1048.68 1129.62,1042.48 1131.66,1036.26 1133.7,1030.01 1135.74,1023.73 1137.77,1017.43 1139.81,1011.1 1141.85,1004.75 1143.89,998.369 1145.93,991.97 1147.97,985.548 1150.01,979.103 1152.05,972.638 1154.08,966.152 1156.12,959.646 1158.16,953.121 1160.2,946.577 1162.24,940.015 1164.28,933.436 1166.32,926.84 1168.35,920.228 1170.39,913.601 1172.43,906.959 1174.47,900.303 1176.51,893.633 1178.55,886.951 1180.59,880.257 1182.63,873.551 1184.66,866.835 1186.7,860.109 1188.74,853.373 1190.78,846.629 1192.82,839.876 1194.86,833.117 1196.9,826.351 1198.93,819.578 1200.97,812.801 1203.01,806.019 1205.05,799.233 1207.09,792.443 1209.13,785.652 1211.17,778.858 1213.21,772.063 1215.24,765.268 1217.28,758.472 1219.32,751.678 1221.36,744.885 1223.4,738.094 1225.44,731.306 1227.48,724.522 1229.51,717.742 1231.55,710.967 1233.59,704.197 1235.63,697.434 1237.67,690.678 1239.71,683.929 1241.75,677.189 1243.79,670.457 1245.82,663.736 1247.86,657.024 1249.9,650.324 1251.94,643.635 1253.98,636.959 1256.02,630.296 1258.06,623.646 1260.09,617.011 1262.13,610.39 1264.17,603.786 1266.21,597.197 1268.25,590.626 1270.29,584.073 1272.33,577.537 1274.37,571.021 1276.4,564.524 1278.44,558.048 1280.48,551.592 1282.52,545.158 1284.56,538.746 1286.6,532.358 1288.64,525.992 1290.67,519.651 1292.71,513.334 1294.75,507.043 1296.79,500.778 1298.83,494.54 1300.87,488.329 1302.91,482.146 1304.95,475.991 1306.98,469.865 1309.02,463.77 1311.06,457.704 1313.1,451.67 1315.14,445.667 1317.18,439.696 1319.22,433.758 1321.25,427.854 1323.29,421.983 1325.33,416.147 1327.37,410.346 1329.41,404.581 1331.45,398.853 1333.49,393.161 1335.53,387.506 1337.56,381.89 1339.6,376.312 1341.64,370.773 1343.68,365.274 1345.72,359.815 1347.76,354.397 1349.8,349.02 1351.83,343.685 1353.87,338.393 1355.91,333.143 1357.95,327.937 1359.99,322.775 1362.03,317.657 1364.07,312.584 1366.11,307.557 1368.14,302.576 1370.18,297.642 1372.22,292.754 1374.26,287.914 1376.3,283.122 1378.34,278.378 1380.38,273.684 1382.41,269.039 1384.45,264.443 1386.49,259.898 1388.53,255.404 1390.57,250.961 1392.61,246.57 1394.65,242.23 1396.68,237.944 1398.72,233.71 1400.76,229.53 1402.8,225.404 1404.84,221.332 1406.88,217.314 1408.92,213.352 1410.96,209.445 1412.99,205.594 1415.03,201.799 1417.07,198.061 1419.11,194.38 1421.15,190.756 1423.19,187.19 1425.23,183.682 1427.26,180.232 1429.3,176.841 1431.34,173.51 1433.38,170.237 1435.42,167.025 1437.46,163.873 1439.5,160.781 1441.54,157.749 1443.57,154.779 1445.61,151.87 1447.65,149.023 1449.69,146.238 1451.73,143.515 1453.77,140.854 1455.81,138.256 1457.84,135.721 1459.88,133.249 1461.92,130.841 1463.96,128.496 1466,126.216 1468.04,123.999 1470.08,121.848 1472.12,119.76 1474.15,117.738 1476.19,115.78 1478.23,113.888 1480.27,112.061 1482.31,110.3 1484.35,108.605 1486.39,106.975 1488.42,105.412 1490.46,103.915 1492.5,102.484 1494.54,101.12 1496.58,99.8227 1498.62,98.5922 1500.66,97.4286 1502.7,96.3321 1504.73,95.3028 1506.77,94.3408 1508.81,93.4462 1510.85,92.619 1512.89,91.8594 1514.93,91.1675 1516.97,90.5432 1519,89.9867 1521.04,89.498 1523.08,89.0772 1525.12,88.7243 1527.16,88.4394 1529.2,88.2224 1531.24,88.0734 1533.28,87.9925 1535.31,87.9795 1537.35,88.0346 1539.39,88.1577 1541.43,88.3488 1543.47,88.6079 1545.51,88.935 1547.55,89.3299 1549.58,89.7928 1551.62,90.3235 1553.66,90.922 1555.7,91.5882 1557.74,92.3221 1559.78,93.1236 1561.82,93.9925 1563.86,94.9289 1565.89,95.9326 1567.93,97.0036 1569.97,98.1416 1572.01,99.3467 1574.05,100.619 1576.09,101.957 1578.13,103.363 1580.16,104.834 1582.2,106.373 1584.24,107.977 1586.28,109.647 1588.32,111.383 1590.36,113.185 1592.4,115.053 1594.44,116.985 1596.47,118.983 1598.51,121.046 1600.55,123.173 1602.59,125.365 1604.63,127.621 1606.67,129.942 1608.71,132.326 1610.74,134.773 1612.78,137.284 1614.82,139.858 1616.86,142.495 1618.9,145.195 1620.94,147.956 1622.98,150.78 1625.02,153.665 1627.05,156.612 1629.09,159.62 1631.13,162.689 1633.17,165.819 1635.21,169.008 1637.25,172.258 1639.29,175.567 1641.32,178.935 1643.36,182.363 1645.4,185.848 1647.44,189.393 1649.48,192.994 1651.52,196.654 1653.56,200.37 1655.59,204.144 1657.63,207.973 1659.67,211.859 1661.71,215.8 1663.75,219.797 1665.79,223.848 1667.83,227.954 1669.87,232.114 1671.9,236.327 1673.94,240.594 1675.98,244.913 1678.02,249.284 1680.06,253.708 1682.1,258.182 1684.14,262.708 1686.17,267.284 1688.21,271.911 1690.25,276.587 1692.29,281.312 1694.33,286.085 1696.37,290.907 1698.41,295.777 1700.45,300.694 1702.48,305.657 1704.52,310.667 1706.56,315.722 1708.6,320.823 1710.64,325.968 1712.68,331.157 1714.72,336.39 1716.75,341.667 1718.79,346.985 1720.83,352.346 1722.87,357.749 1724.91,363.192 1726.95,368.676 1728.99,374.2 1731.03,379.763 1733.06,385.365 1735.1,391.005 1737.14,396.683 1739.18,402.398 1741.22,408.149 1743.26,413.936 1745.3,419.759 1747.33,425.617 1749.37,431.508 1751.41,437.433 1753.45,443.392 1755.49,449.382 1757.53,455.405 1759.57,461.459 1761.61,467.543 1763.64,473.657 1765.68,479.801 1767.72,485.973 1769.76,492.174 1771.8,498.402 1773.84,504.657 1775.88,510.938 1777.91,517.245 1779.95,523.577 1781.99,529.933 1784.03,536.313 1786.07,542.716 1788.11,549.142 1790.15,555.589 1792.19,562.058 1794.22,568.547 1796.26,575.056 1798.3,581.584 1800.34,588.131 1802.38,594.695 1804.42,601.277 1806.46,607.876 1808.49,614.49 1810.53,621.12 1812.57,627.764 1814.61,634.422 1816.65,641.094 1818.69,647.778 1820.73,654.474 1822.77,661.181 1824.8,667.899 1826.84,674.627 1828.88,681.364 1830.92,688.109 1832.96,694.863 1835,701.624 1837.04,708.391 1839.07,715.164 1841.11,721.942 1843.15,728.725 1845.19,735.511 1847.23,742.301 1849.27,749.093 1851.31,755.887 1853.35,762.682 1855.38,769.478 1857.42,776.273 1859.46,783.067 1861.5,789.86 1863.54,796.65 1865.58,803.438 1867.62,810.221 1869.65,817.001 1871.69,823.775 1873.73,830.543 1875.77,837.306 1877.81,844.061 1879.85,850.808 1881.89,857.547 1883.92,864.277 1885.96,870.997 1888,877.707 1890.04,884.406 1892.08,891.092 1894.12,897.767 1896.16,904.428 1898.2,911.076 1900.23,917.709 1902.27,924.327 1904.31,930.929 1906.35,937.514 1908.39,944.083 1910.43,950.634 1912.47,957.166 1914.5,963.68 1916.54,970.173 1918.58,976.646 1920.62,983.099 1922.66,989.529 1924.7,995.937 1926.74,1002.32 1928.78,1008.68 1930.81,1015.02 1932.85,1021.33 1934.89,1027.62 1936.93,1033.88 1938.97,1040.12 1941.01,1046.32 1943.05,1052.5 1945.08,1058.65 1947.12,1064.77 1949.16,1070.86 1951.2,1076.93 1953.24,1082.95 1955.28,1088.95 1957.32,1094.92 1959.36,1100.85 1961.39,1106.75 1963.43,1112.62 1965.47,1118.45 1967.51,1124.24 1969.55,1130 1971.59,1135.72 1973.63,1141.41 1975.66,1147.06 1977.7,1152.67 1979.74,1158.24 1981.78,1163.77 1983.82,1169.27 1985.86,1174.72 1987.9,1180.13 1989.94,1185.5 1991.97,1190.83 1994.01,1196.12 1996.05,1201.36 1998.09,1206.56 2000.13,1211.71 2002.17,1216.82 2004.21,1221.89 2006.24,1226.91 2008.28,1231.88 2010.32,1236.81 2012.36,1241.69 2014.4,1246.52 2016.44,1251.31 2018.48,1256.04 2020.52,1260.73 2022.55,1265.37 2024.59,1269.96 2026.63,1274.49 2028.67,1278.98 2030.71,1283.42 2032.75,1287.8 2034.79,1292.13 2036.82,1296.41 2038.86,1300.63 2040.9,1304.81 2042.94,1308.92 2044.98,1312.99 2047.02,1317 2049.06,1320.95 2051.1,1324.85 2053.13,1328.69 2055.17,1332.48 2057.21,1336.21 2059.25,1339.88 2061.29,1343.49 2063.33,1347.05 2065.37,1350.55 2067.4,1353.99 2069.44,1357.37 2071.48,1360.69 2073.52,1363.96 2075.56,1367.16 2077.6,1370.3 2079.64,1373.39 2081.68,1376.41 2083.71,1379.37 2085.75,1382.27 2087.79,1385.11 2089.83,1387.88 2091.87,1390.59 2093.91,1393.25 2095.95,1395.83 2097.98,1398.36 2100.02,1400.82 2102.06,1403.22 2104.1,1405.55 2106.14,1407.82 2108.18,1410.03 2110.22,1412.17 2112.26,1414.25 2114.29,1416.26 2116.33,1418.21 2118.37,1420.09 2120.41,1421.91 2122.45,1423.66 2124.49,1425.34 2126.53,1426.96 2128.56,1428.52 2130.6,1430 2132.64,1431.42 2134.68,1432.78 2136.72,1434.07 2138.76,1435.29 2140.8,1436.44 2142.83,1437.52 2144.87,1438.54 2146.91,1439.49 2148.95,1440.38 2150.99,1441.2 2153.03,1441.94 2155.07,1442.63 2157.11,1443.24 2159.14,1443.79 2161.18,1444.26 2163.22,1444.67 2165.26,1445.02 2167.3,1445.29 2169.34,1445.5 2171.38,1445.64 2173.41,1445.71 2175.45,1445.71 2177.49,1445.64 2179.53,1445.51 2181.57,1445.31 2183.61,1445.04 2185.65,1444.7 2187.69,1444.3 2189.72,1443.82 2191.76,1443.28 2193.8,1442.67 2195.84,1442 2197.88,1441.25 2199.92,1440.44 2201.96,1439.56 2203.99,1438.61 2206.03,1437.6 2208.07,1436.52 2210.11,1435.37 2212.15,1434.15 2214.19,1432.87 2216.23,1431.52 2218.27,1430.11 2220.3,1428.62 2222.34,1427.08 2224.38,1425.46 2226.42,1423.78 2228.46,1422.03 2230.5,1420.22 2232.54,1418.35 2234.57,1416.4 2236.61,1414.39 2238.65,1412.32 2240.69,1410.18 2242.73,1407.98 2244.77,1405.72 2246.81,1403.39 2248.85,1400.99 2250.88,1398.53 2252.92,1396.01 2254.96,1393.43 2257,1390.78 2259.04,1388.07 2261.08,1385.3 2263.12,1382.47 2265.15,1379.57 2267.19,1376.62 2269.23,1373.6 2271.27,1370.52 2273.31,1367.38 2275.35,1364.19 2277.39,1360.93 2279.43,1357.61 2281.46,1354.23 2283.5,1350.79 2285.54,1347.3 2287.58,1343.75 2289.62,1340.14 2291.66,1336.47 \"/>\n",
       "<polyline clip-path=\"url(#clip642)\" style=\"stroke:#e26f46; stroke-linecap:round; stroke-linejoin:round; stroke-width:4; stroke-opacity:1; fill:none\" points=\"255.035,87.9763 257.073,102.781 259.112,117.476 261.151,132.061 263.189,146.536 265.228,160.903 267.267,175.161 269.305,189.31 271.344,203.352 273.383,217.286 275.421,231.114 277.46,244.835 279.499,258.449 281.537,271.958 283.576,285.361 285.615,298.66 287.653,311.854 289.692,324.944 291.731,337.93 293.769,350.812 295.808,363.592 297.847,376.269 299.885,388.844 301.924,401.317 303.963,413.689 306.001,425.959 308.04,438.13 310.079,450.199 312.117,462.17 314.156,474.04 316.195,485.812 318.233,497.485 320.272,509.06 322.31,520.537 324.349,531.917 326.388,543.199 328.426,554.385 330.465,565.474 332.504,576.468 334.542,587.366 336.581,598.169 338.62,608.877 340.658,619.491 342.697,630.011 344.736,640.437 346.774,650.771 348.813,661.011 350.852,671.159 352.89,681.214 354.929,691.179 356.968,701.051 359.006,710.833 361.045,720.525 363.084,730.126 365.122,739.637 367.161,749.059 369.2,758.392 371.238,767.636 373.277,776.792 375.316,785.86 377.354,794.841 379.393,803.734 381.432,812.541 383.47,821.261 385.509,829.895 387.548,838.444 389.586,846.907 391.625,855.285 393.664,863.578 395.702,871.788 397.741,879.913 399.78,887.955 401.818,895.914 403.857,903.791 405.896,911.584 407.934,919.296 409.973,926.926 412.012,934.475 414.05,941.943 416.089,949.331 418.128,956.638 420.166,963.865 422.205,971.013 424.244,978.082 426.282,985.072 428.321,991.983 430.36,998.817 432.398,1005.57 434.437,1012.25 436.476,1018.85 438.514,1025.38 440.553,1031.83 442.592,1038.2 444.63,1044.5 446.669,1050.72 448.707,1056.86 450.746,1062.94 452.785,1068.93 454.823,1074.86 456.862,1080.71 458.901,1086.49 460.939,1092.19 462.978,1097.82 465.017,1103.39 467.055,1108.87 469.094,1114.29 471.133,1119.64 473.171,1124.91 475.21,1130.12 477.249,1135.25 479.287,1140.32 481.326,1145.31 483.365,1150.24 485.403,1155.1 487.442,1159.89 489.481,1164.61 491.519,1169.27 493.558,1173.85 495.597,1178.37 497.635,1182.83 499.674,1187.21 501.713,1191.54 503.751,1195.79 505.79,1199.98 507.829,1204.11 509.867,1208.17 511.906,1212.16 513.945,1216.1 515.983,1219.96 518.022,1223.77 520.061,1227.51 522.099,1231.19 524.138,1234.81 526.177,1238.36 528.215,1241.86 530.254,1245.29 532.293,1248.66 534.331,1251.97 536.37,1255.22 538.409,1258.41 540.447,1261.54 542.486,1264.61 544.525,1267.62 546.563,1270.58 548.602,1273.47 550.641,1276.31 552.679,1279.08 554.718,1281.81 556.757,1284.47 558.795,1287.08 560.834,1289.63 562.873,1292.12 564.911,1294.56 566.95,1296.94 568.989,1299.27 571.027,1301.54 573.066,1303.76 575.104,1305.92 577.143,1308.03 579.182,1310.08 581.22,1312.09 583.259,1314.03 585.298,1315.93 587.336,1317.77 589.375,1319.56 591.414,1321.3 593.452,1322.99 595.491,1324.63 597.53,1326.21 599.568,1327.75 601.607,1329.23 603.646,1330.67 605.684,1332.05 607.723,1333.38 609.762,1334.67 611.8,1335.91 613.839,1337.09 615.878,1338.23 617.916,1339.33 619.955,1340.37 621.994,1341.37 624.032,1342.32 626.071,1343.22 628.11,1344.07 630.148,1344.89 632.187,1345.65 634.226,1346.37 636.264,1347.04 638.303,1347.67 640.342,1348.25 642.38,1348.79 644.419,1349.29 646.458,1349.74 648.496,1350.14 650.535,1350.51 652.574,1350.83 654.612,1351.11 656.651,1351.34 658.69,1351.53 660.728,1351.69 662.767,1351.8 664.806,1351.86 666.844,1351.89 668.883,1351.88 670.922,1351.83 672.96,1351.73 674.999,1351.6 677.038,1351.42 679.076,1351.21 681.115,1350.96 683.154,1350.67 685.192,1350.34 687.231,1349.97 689.27,1349.57 691.308,1349.12 693.347,1348.64 695.385,1348.12 697.424,1347.57 699.463,1346.98 701.501,1346.35 703.54,1345.69 705.579,1344.99 707.617,1344.25 709.656,1343.48 711.695,1342.68 713.733,1341.83 715.772,1340.96 717.811,1340.05 719.849,1339.11 721.888,1338.13 723.927,1337.12 725.965,1336.08 728.004,1335 730.043,1333.89 732.081,1332.75 734.12,1331.58 736.159,1330.38 738.197,1329.14 740.236,1327.87 742.275,1326.57 744.313,1325.24 746.352,1323.88 748.391,1322.49 750.429,1321.07 752.468,1319.62 754.507,1318.14 756.545,1316.63 758.584,1315.1 760.623,1313.53 762.661,1311.93 764.7,1310.31 766.739,1308.66 768.777,1306.98 770.816,1305.27 772.855,1303.54 774.893,1301.78 776.932,1299.99 778.971,1298.18 781.009,1296.33 783.048,1294.47 785.087,1292.58 787.125,1290.66 789.164,1288.71 791.203,1286.75 793.241,1284.75 795.28,1282.74 797.319,1280.69 799.357,1278.63 801.396,1276.54 803.435,1274.42 805.473,1272.29 807.512,1270.13 809.551,1267.94 811.589,1265.74 813.628,1263.51 815.667,1261.26 817.705,1258.99 819.744,1256.69 821.782,1254.38 823.821,1252.04 825.86,1249.68 827.898,1247.31 829.937,1244.91 831.976,1242.49 834.014,1240.05 836.053,1237.59 838.092,1235.11 840.13,1232.61 842.169,1230.09 844.208,1227.56 846.246,1225 848.285,1222.43 850.324,1219.84 852.362,1217.23 854.401,1214.6 856.44,1211.95 858.478,1209.29 860.517,1206.61 862.556,1203.91 864.594,1201.2 866.633,1198.47 868.672,1195.72 870.71,1192.96 872.749,1190.18 874.788,1187.39 876.826,1184.58 878.865,1181.75 880.904,1178.91 882.942,1176.06 884.981,1173.19 887.02,1170.31 889.058,1167.41 891.097,1164.5 893.136,1161.57 895.174,1158.63 897.213,1155.68 899.252,1152.71 901.29,1149.74 903.329,1146.75 905.368,1143.74 907.406,1140.73 909.445,1137.7 911.484,1134.66 913.522,1131.61 915.561,1128.55 917.6,1125.47 919.638,1122.39 921.677,1119.29 923.716,1116.19 925.754,1113.07 927.793,1109.94 929.832,1106.81 931.87,1103.66 933.909,1100.5 935.948,1097.34 937.986,1094.16 940.025,1090.98 942.064,1087.78 944.102,1084.58 946.141,1081.37 948.179,1078.15 950.218,1074.93 952.257,1071.69 954.295,1068.45 956.334,1065.2 958.373,1061.94 960.411,1058.68 962.45,1055.41 964.489,1052.13 966.527,1048.85 968.566,1045.56 970.605,1042.26 972.643,1038.96 974.682,1035.65 976.721,1032.33 978.759,1029.01 980.798,1025.69 982.837,1022.36 984.875,1019.02 986.914,1015.68 988.953,1012.34 990.991,1008.99 993.03,1005.64 995.069,1002.28 997.107,998.92 999.146,995.556 1001.18,992.188 1003.22,988.817 1005.26,985.443 1007.3,982.066 1009.34,978.686 1011.38,975.304 1013.42,971.919 1015.46,968.532 1017.49,965.143 1019.53,961.752 1021.57,958.359 1023.61,954.965 1025.65,951.569 1027.69,948.172 1029.73,944.774 1031.76,941.375 1033.8,937.975 1035.84,934.575 1037.88,931.174 1039.92,927.773 1041.96,924.373 1044,920.972 1046.04,917.571 1048.07,914.171 1050.11,910.771 1052.15,907.373 1054.19,903.975 1056.23,900.578 1058.27,897.182 1060.31,893.788 1062.34,890.395 1064.38,887.004 1066.42,883.615 1068.46,880.228 1070.5,876.843 1072.54,873.46 1074.58,870.08 1076.62,866.703 1078.65,863.328 1080.69,859.956 1082.73,856.587 1084.77,853.222 1086.81,849.859 1088.85,846.501 1090.89,843.146 1092.92,839.795 1094.96,836.447 1097,833.104 1099.04,829.766 1101.08,826.431 1103.12,823.101 1105.16,819.776 1107.2,816.456 1109.23,813.141 1111.27,809.83 1113.31,806.525 1115.35,803.226 1117.39,799.932 1119.43,796.644 1121.47,793.361 1123.5,790.085 1125.54,786.814 1127.58,783.55 1129.62,780.292 1131.66,777.041 1133.7,773.797 1135.74,770.559 1137.77,767.328 1139.81,764.104 1141.85,760.887 1143.89,757.678 1145.93,754.476 1147.97,751.282 1150.01,748.095 1152.05,744.916 1154.08,741.745 1156.12,738.583 1158.16,735.428 1160.2,732.282 1162.24,729.144 1164.28,726.015 1166.32,722.895 1168.35,719.783 1170.39,716.681 1172.43,713.587 1174.47,710.503 1176.51,707.428 1178.55,704.362 1180.59,701.306 1182.63,698.26 1184.66,695.224 1186.7,692.197 1188.74,689.181 1190.78,686.175 1192.82,683.179 1194.86,680.193 1196.9,677.218 1198.93,674.254 1200.97,671.3 1203.01,668.357 1205.05,665.425 1207.09,662.504 1209.13,659.595 1211.17,656.697 1213.21,653.81 1215.24,650.934 1217.28,648.071 1219.32,645.219 1221.36,642.379 1223.4,639.55 1225.44,636.734 1227.48,633.93 1229.51,631.139 1231.55,628.359 1233.59,625.592 1235.63,622.838 1237.67,620.097 1239.71,617.368 1241.75,614.652 1243.79,611.949 1245.82,609.259 1247.86,606.583 1249.9,603.919 1251.94,601.269 1253.98,598.633 1256.02,596.01 1258.06,593.401 1260.09,590.805 1262.13,588.224 1264.17,585.656 1266.21,583.102 1268.25,580.563 1270.29,578.038 1272.33,575.527 1274.37,573.03 1276.4,570.548 1278.44,568.081 1280.48,565.628 1282.52,563.19 1284.56,560.767 1286.6,558.359 1288.64,555.966 1290.67,553.588 1292.71,551.225 1294.75,548.877 1296.79,546.545 1298.83,544.229 1300.87,541.927 1302.91,539.642 1304.95,537.372 1306.98,535.118 1309.02,532.88 1311.06,530.658 1313.1,528.451 1315.14,526.261 1317.18,524.087 1319.22,521.93 1321.25,519.788 1323.29,517.663 1325.33,515.555 1327.37,513.463 1329.41,511.387 1331.45,509.329 1333.49,507.287 1335.53,505.262 1337.56,503.253 1339.6,501.262 1341.64,499.288 1343.68,497.331 1345.72,495.391 1347.76,493.469 1349.8,491.564 1351.83,489.676 1353.87,487.805 1355.91,485.952 1357.95,484.117 1359.99,482.299 1362.03,480.499 1364.07,478.717 1366.11,476.953 1368.14,475.206 1370.18,473.478 1372.22,471.767 1374.26,470.075 1376.3,468.401 1378.34,466.745 1380.38,465.107 1382.41,463.487 1384.45,461.886 1386.49,460.303 1388.53,458.739 1390.57,457.193 1392.61,455.666 1394.65,454.157 1396.68,452.668 1398.72,451.196 1400.76,449.744 1402.8,448.311 1404.84,446.896 1406.88,445.5 1408.92,444.124 1410.96,442.766 1412.99,441.427 1415.03,440.108 1417.07,438.807 1419.11,437.526 1421.15,436.265 1423.19,435.022 1425.23,433.799 1427.26,432.595 1429.3,431.411 1431.34,430.246 1433.38,429.1 1435.42,427.974 1437.46,426.868 1439.5,425.782 1441.54,424.715 1443.57,423.667 1445.61,422.64 1447.65,421.632 1449.69,420.644 1451.73,419.676 1453.77,418.728 1455.81,417.8 1457.84,416.892 1459.88,416.003 1461.92,415.135 1463.96,414.287 1466,413.459 1468.04,412.65 1470.08,411.863 1472.12,411.095 1474.15,410.347 1476.19,409.62 1478.23,408.913 1480.27,408.226 1482.31,407.559 1484.35,406.913 1486.39,406.287 1488.42,405.682 1490.46,405.097 1492.5,404.532 1494.54,403.988 1496.58,403.464 1498.62,402.96 1500.66,402.477 1502.7,402.015 1504.73,401.573 1506.77,401.152 1508.81,400.751 1510.85,400.371 1512.89,400.012 1514.93,399.673 1516.97,399.354 1519,399.057 1521.04,398.78 1523.08,398.523 1525.12,398.288 1527.16,398.073 1529.2,397.878 1531.24,397.705 1533.28,397.552 1535.31,397.42 1537.35,397.309 1539.39,397.218 1541.43,397.148 1543.47,397.099 1545.51,397.07 1547.55,397.063 1549.58,397.076 1551.62,397.11 1553.66,397.165 1555.7,397.24 1557.74,397.336 1559.78,397.453 1561.82,397.591 1563.86,397.75 1565.89,397.929 1567.93,398.129 1569.97,398.35 1572.01,398.592 1574.05,398.854 1576.09,399.137 1578.13,399.441 1580.16,399.766 1582.2,400.111 1584.24,400.477 1586.28,400.864 1588.32,401.272 1590.36,401.7 1592.4,402.149 1594.44,402.619 1596.47,403.11 1598.51,403.621 1600.55,404.152 1602.59,404.705 1604.63,405.278 1606.67,405.871 1608.71,406.486 1610.74,407.121 1612.78,407.776 1614.82,408.452 1616.86,409.148 1618.9,409.866 1620.94,410.603 1622.98,411.361 1625.02,412.14 1627.05,412.939 1629.09,413.758 1631.13,414.598 1633.17,415.458 1635.21,416.338 1637.25,417.239 1639.29,418.16 1641.32,419.102 1643.36,420.063 1645.4,421.045 1647.44,422.047 1649.48,423.07 1651.52,424.112 1653.56,425.175 1655.59,426.257 1657.63,427.36 1659.67,428.483 1661.71,429.626 1663.75,430.788 1665.79,431.971 1667.83,433.173 1669.87,434.396 1671.9,435.638 1673.94,436.9 1675.98,438.182 1678.02,439.484 1680.06,440.805 1682.1,442.146 1684.14,443.506 1686.17,444.886 1688.21,446.286 1690.25,447.705 1692.29,449.143 1694.33,450.601 1696.37,452.079 1698.41,453.575 1700.45,455.091 1702.48,456.627 1704.52,458.181 1706.56,459.754 1708.6,461.347 1710.64,462.959 1712.68,464.589 1714.72,466.239 1716.75,467.908 1718.79,469.595 1720.83,471.301 1722.87,473.026 1724.91,474.77 1726.95,476.532 1728.99,478.313 1731.03,480.113 1733.06,481.931 1735.1,483.767 1737.14,485.622 1739.18,487.495 1741.22,489.387 1743.26,491.297 1745.3,493.224 1747.33,495.17 1749.37,497.134 1751.41,499.116 1753.45,501.116 1755.49,503.134 1757.53,505.17 1759.57,507.223 1761.61,509.294 1763.64,511.383 1765.68,513.489 1767.72,515.613 1769.76,517.754 1771.8,519.912 1773.84,522.088 1775.88,524.281 1777.91,526.491 1779.95,528.718 1781.99,530.962 1784.03,533.223 1786.07,535.502 1788.11,537.796 1790.15,540.108 1792.19,542.436 1794.22,544.781 1796.26,547.142 1798.3,549.52 1800.34,551.914 1802.38,554.325 1804.42,556.751 1806.46,559.194 1808.49,561.653 1810.53,564.128 1812.57,566.618 1814.61,569.125 1816.65,571.647 1818.69,574.185 1820.73,576.739 1822.77,579.308 1824.8,581.892 1826.84,584.492 1828.88,587.107 1830.92,589.737 1832.96,592.382 1835,595.043 1837.04,597.718 1839.07,600.408 1841.11,603.112 1843.15,605.832 1845.19,608.566 1847.23,611.314 1849.27,614.077 1851.31,616.854 1853.35,619.645 1855.38,622.45 1857.42,625.269 1859.46,628.103 1861.5,630.95 1863.54,633.81 1865.58,636.685 1867.62,639.573 1869.65,642.474 1871.69,645.389 1873.73,648.316 1875.77,651.258 1877.81,654.212 1879.85,657.179 1881.89,660.158 1883.92,663.151 1885.96,666.156 1888,669.174 1890.04,672.204 1892.08,675.247 1894.12,678.302 1896.16,681.368 1898.2,684.447 1900.23,687.538 1902.27,690.641 1904.31,693.755 1906.35,696.881 1908.39,700.018 1910.43,703.167 1912.47,706.327 1914.5,709.498 1916.54,712.68 1918.58,715.873 1920.62,719.077 1922.66,722.291 1924.7,725.516 1926.74,728.752 1928.78,731.998 1930.81,735.254 1932.85,738.52 1934.89,741.796 1936.93,745.082 1938.97,748.378 1941.01,751.684 1943.05,754.999 1945.08,758.323 1947.12,761.657 1949.16,765 1951.2,768.352 1953.24,771.712 1955.28,775.082 1957.32,778.46 1959.36,781.847 1961.39,785.242 1963.43,788.646 1965.47,792.057 1967.51,795.477 1969.55,798.905 1971.59,802.34 1973.63,805.783 1975.66,809.234 1977.7,812.691 1979.74,816.157 1981.78,819.629 1983.82,823.108 1985.86,826.594 1987.9,830.087 1989.94,833.586 1991.97,837.092 1994.01,840.604 1996.05,844.123 1998.09,847.647 2000.13,851.177 2002.17,854.714 2004.21,858.255 2006.24,861.802 2008.28,865.355 2010.32,868.913 2012.36,872.475 2014.4,876.043 2016.44,879.615 2018.48,883.193 2020.52,886.774 2022.55,890.36 2024.59,893.95 2026.63,897.544 2028.67,901.142 2030.71,904.744 2032.75,908.35 2034.79,911.959 2036.82,915.571 2038.86,919.186 2040.9,922.804 2042.94,926.426 2044.98,930.05 2047.02,933.676 2049.06,937.305 2051.1,940.936 2053.13,944.569 2055.17,948.204 2057.21,951.841 2059.25,955.48 2061.29,959.12 2063.33,962.761 2065.37,966.404 2067.4,970.047 2069.44,973.691 2071.48,977.336 2073.52,980.982 2075.56,984.628 2077.6,988.274 2079.64,991.92 2081.68,995.565 2083.71,999.211 2085.75,1002.86 2087.79,1006.5 2089.83,1010.14 2091.87,1013.79 2093.91,1017.43 2095.95,1021.07 2097.98,1024.71 2100.02,1028.34 2102.06,1031.98 2104.1,1035.61 2106.14,1039.24 2108.18,1042.87 2110.22,1046.5 2112.26,1050.12 2114.29,1053.74 2116.33,1057.36 2118.37,1060.98 2120.41,1064.59 2122.45,1068.2 2124.49,1071.8 2126.53,1075.4 2128.56,1079 2130.6,1082.59 2132.64,1086.17 2134.68,1089.76 2136.72,1093.33 2138.76,1096.91 2140.8,1100.48 2142.83,1104.04 2144.87,1107.59 2146.91,1111.15 2148.95,1114.69 2150.99,1118.23 2153.03,1121.76 2155.07,1125.29 2157.11,1128.81 2159.14,1132.32 2161.18,1135.83 2163.22,1139.33 2165.26,1142.82 2167.3,1146.3 2169.34,1149.78 2171.38,1153.24 2173.41,1156.7 2175.45,1160.16 2177.49,1163.6 2179.53,1167.03 2181.57,1170.46 2183.61,1173.87 2185.65,1177.28 2187.69,1180.68 2189.72,1184.06 2191.76,1187.44 2193.8,1190.81 2195.84,1194.17 2197.88,1197.51 2199.92,1200.85 2201.96,1204.17 2203.99,1207.49 2206.03,1210.79 2208.07,1214.08 2210.11,1217.36 2212.15,1220.63 2214.19,1223.88 2216.23,1227.12 2218.27,1230.35 2220.3,1233.57 2222.34,1236.77 2224.38,1239.97 2226.42,1243.14 2228.46,1246.31 2230.5,1249.46 2232.54,1252.59 2234.57,1255.72 2236.61,1258.82 2238.65,1261.92 2240.69,1264.99 2242.73,1268.06 2244.77,1271.1 2246.81,1274.14 2248.85,1277.15 2250.88,1280.15 2252.92,1283.14 2254.96,1286.1 2257,1289.06 2259.04,1291.99 2261.08,1294.91 2263.12,1297.81 2265.15,1300.69 2267.19,1303.55 2269.23,1306.4 2271.27,1309.23 2273.31,1312.04 2275.35,1314.83 2277.39,1317.6 2279.43,1320.36 2281.46,1323.09 2283.5,1325.81 2285.54,1328.5 2287.58,1331.18 2289.62,1333.83 2291.66,1336.47 \"/>\n",
       "<circle clip-path=\"url(#clip642)\" cx=\"255.035\" cy=\"87.9763\" r=\"14.4\" fill=\"#3da44d\" fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"3.2\"/>\n",
       "<circle clip-path=\"url(#clip642)\" cx=\"764.19\" cy=\"1310.72\" r=\"14.4\" fill=\"#3da44d\" fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"3.2\"/>\n",
       "<circle clip-path=\"url(#clip642)\" cx=\"1273.35\" cy=\"574.277\" r=\"14.4\" fill=\"#3da44d\" fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"3.2\"/>\n",
       "<circle clip-path=\"url(#clip642)\" cx=\"1782.5\" cy=\"531.526\" r=\"14.4\" fill=\"#3da44d\" fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"3.2\"/>\n",
       "<circle clip-path=\"url(#clip642)\" cx=\"2291.66\" cy=\"1336.47\" r=\"14.4\" fill=\"#3da44d\" fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"3.2\"/>\n",
       "<path clip-path=\"url(#clip640)\" d=\"M1770.82 302.578 L2280.8 302.578 L2280.8 95.2176 L1770.82 95.2176  Z\" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n",
       "<polyline clip-path=\"url(#clip640)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:4; stroke-opacity:1; fill:none\" points=\"1770.82,302.578 2280.8,302.578 2280.8,95.2176 1770.82,95.2176 1770.82,302.578 \"/>\n",
       "<polyline clip-path=\"url(#clip640)\" style=\"stroke:#009af9; stroke-linecap:round; stroke-linejoin:round; stroke-width:4; stroke-opacity:1; fill:none\" points=\"1794.8,147.058 1938.73,147.058 \"/>\n",
       "<path clip-path=\"url(#clip640)\" d=\"M1979.22 128.319 L1979.22 131.861 L1975.14 131.861 Q1972.85 131.861 1971.95 132.787 Q1971.07 133.713 1971.07 136.12 L1971.07 138.412 L1978.08 138.412 L1978.08 141.722 L1971.07 141.722 L1971.07 164.338 L1966.79 164.338 L1966.79 141.722 L1962.71 141.722 L1962.71 138.412 L1966.79 138.412 L1966.79 136.606 Q1966.79 132.278 1968.8 130.31 Q1970.81 128.319 1975.19 128.319 L1979.22 128.319 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M1982.34 154.106 L1982.34 138.412 L1986.6 138.412 L1986.6 153.944 Q1986.6 157.625 1988.04 159.476 Q1989.47 161.305 1992.34 161.305 Q1995.79 161.305 1997.78 159.106 Q1999.8 156.907 1999.8 153.111 L1999.8 138.412 L2004.06 138.412 L2004.06 164.338 L1999.8 164.338 L1999.8 160.356 Q1998.24 162.717 1996.18 163.875 Q1994.15 165.009 1991.44 165.009 Q1986.97 165.009 1984.66 162.231 Q1982.34 159.453 1982.34 154.106 M1993.06 137.787 L1993.06 137.787 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M2034.38 148.689 L2034.38 164.338 L2030.12 164.338 L2030.12 148.828 Q2030.12 145.148 2028.68 143.319 Q2027.25 141.49 2024.38 141.49 Q2020.93 141.49 2018.94 143.69 Q2016.95 145.889 2016.95 149.685 L2016.95 164.338 L2012.67 164.338 L2012.67 138.412 L2016.95 138.412 L2016.95 142.44 Q2018.48 140.102 2020.54 138.944 Q2022.62 137.787 2025.33 137.787 Q2029.8 137.787 2032.09 140.565 Q2034.38 143.319 2034.38 148.689 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M2061.53 139.407 L2061.53 143.389 Q2059.73 142.393 2057.9 141.907 Q2056.09 141.398 2054.24 141.398 Q2050.1 141.398 2047.8 144.037 Q2045.51 146.652 2045.51 151.398 Q2045.51 156.143 2047.8 158.782 Q2050.1 161.398 2054.24 161.398 Q2056.09 161.398 2057.9 160.912 Q2059.73 160.402 2061.53 159.407 L2061.53 163.342 Q2059.75 164.176 2057.83 164.592 Q2055.93 165.009 2053.78 165.009 Q2047.92 165.009 2044.47 161.328 Q2041.02 157.648 2041.02 151.398 Q2041.02 145.055 2044.49 141.421 Q2047.99 137.787 2054.05 137.787 Q2056.02 137.787 2057.9 138.203 Q2059.77 138.597 2061.53 139.407 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M2073.15 131.051 L2073.15 138.412 L2081.93 138.412 L2081.93 141.722 L2073.15 141.722 L2073.15 155.796 Q2073.15 158.967 2074.01 159.87 Q2074.89 160.773 2077.55 160.773 L2081.93 160.773 L2081.93 164.338 L2077.55 164.338 Q2072.62 164.338 2070.74 162.509 Q2068.87 160.657 2068.87 155.796 L2068.87 141.722 L2065.74 141.722 L2065.74 138.412 L2068.87 138.412 L2068.87 131.051 L2073.15 131.051 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M2087.53 138.412 L2091.79 138.412 L2091.79 164.338 L2087.53 164.338 L2087.53 138.412 M2087.53 128.319 L2091.79 128.319 L2091.79 133.713 L2087.53 133.713 L2087.53 128.319 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M2110.74 141.398 Q2107.32 141.398 2105.33 144.083 Q2103.34 146.745 2103.34 151.398 Q2103.34 156.051 2105.3 158.736 Q2107.3 161.398 2110.74 161.398 Q2114.15 161.398 2116.14 158.713 Q2118.13 156.027 2118.13 151.398 Q2118.13 146.791 2116.14 144.106 Q2114.15 141.398 2110.74 141.398 M2110.74 137.787 Q2116.3 137.787 2119.47 141.398 Q2122.64 145.009 2122.64 151.398 Q2122.64 157.764 2119.47 161.398 Q2116.3 165.009 2110.74 165.009 Q2105.17 165.009 2101.99 161.398 Q2098.85 157.764 2098.85 151.398 Q2098.85 145.009 2101.99 141.398 Q2105.17 137.787 2110.74 137.787 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M2151.25 148.689 L2151.25 164.338 L2146.99 164.338 L2146.99 148.828 Q2146.99 145.148 2145.56 143.319 Q2144.12 141.49 2141.25 141.49 Q2137.8 141.49 2135.81 143.69 Q2133.82 145.889 2133.82 149.685 L2133.82 164.338 L2129.54 164.338 L2129.54 138.412 L2133.82 138.412 L2133.82 142.44 Q2135.35 140.102 2137.41 138.944 Q2139.49 137.787 2142.2 137.787 Q2146.67 137.787 2148.96 140.565 Q2151.25 143.319 2151.25 148.689 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><polyline clip-path=\"url(#clip640)\" style=\"stroke:#e26f46; stroke-linecap:round; stroke-linejoin:round; stroke-width:4; stroke-opacity:1; fill:none\" points=\"1794.8,198.898 1938.73,198.898 \"/>\n",
       "<path clip-path=\"url(#clip640)\" d=\"M1962.71 190.252 L1966.97 190.252 L1966.97 216.178 L1962.71 216.178 L1962.71 190.252 M1962.71 180.159 L1966.97 180.159 L1966.97 185.553 L1962.71 185.553 L1962.71 180.159 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M1997.43 200.529 L1997.43 216.178 L1993.18 216.178 L1993.18 200.668 Q1993.18 196.988 1991.74 195.159 Q1990.31 193.33 1987.43 193.33 Q1983.99 193.33 1982 195.53 Q1980 197.729 1980 201.525 L1980 216.178 L1975.72 216.178 L1975.72 190.252 L1980 190.252 L1980 194.28 Q1981.53 191.942 1983.59 190.784 Q1985.68 189.627 1988.38 189.627 Q1992.85 189.627 1995.14 192.405 Q1997.43 195.159 1997.43 200.529 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M2010.14 182.891 L2010.14 190.252 L2018.92 190.252 L2018.92 193.562 L2010.14 193.562 L2010.14 207.636 Q2010.14 210.807 2011 211.71 Q2011.88 212.613 2014.54 212.613 L2018.92 212.613 L2018.92 216.178 L2014.54 216.178 Q2009.61 216.178 2007.74 214.349 Q2005.86 212.497 2005.86 207.636 L2005.86 193.562 L2002.74 193.562 L2002.74 190.252 L2005.86 190.252 L2005.86 182.891 L2010.14 182.891 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M2046.69 202.15 L2046.69 204.233 L2027.11 204.233 Q2027.39 208.631 2029.75 210.946 Q2032.13 213.238 2036.37 213.238 Q2038.82 213.238 2041.12 212.636 Q2043.43 212.034 2045.7 210.83 L2045.7 214.858 Q2043.41 215.83 2041 216.34 Q2038.59 216.849 2036.12 216.849 Q2029.91 216.849 2026.28 213.238 Q2022.67 209.627 2022.67 203.469 Q2022.67 197.104 2026.09 193.377 Q2029.54 189.627 2035.37 189.627 Q2040.61 189.627 2043.64 193.006 Q2046.69 196.363 2046.69 202.15 M2042.43 200.9 Q2042.39 197.405 2040.47 195.321 Q2038.57 193.238 2035.42 193.238 Q2031.86 193.238 2029.7 195.252 Q2027.57 197.266 2027.25 200.923 L2042.43 200.9 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M2068.71 194.233 Q2067.99 193.817 2067.13 193.631 Q2066.3 193.423 2065.28 193.423 Q2061.67 193.423 2059.73 195.784 Q2057.8 198.122 2057.8 202.52 L2057.8 216.178 L2053.52 216.178 L2053.52 190.252 L2057.8 190.252 L2057.8 194.28 Q2059.15 191.918 2061.3 190.784 Q2063.45 189.627 2066.53 189.627 Q2066.97 189.627 2067.5 189.696 Q2068.04 189.743 2068.68 189.858 L2068.71 194.233 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M2077.3 212.289 L2077.3 226.039 L2073.01 226.039 L2073.01 190.252 L2077.3 190.252 L2077.3 194.187 Q2078.64 191.872 2080.68 190.761 Q2082.74 189.627 2085.58 189.627 Q2090.3 189.627 2093.24 193.377 Q2096.21 197.127 2096.21 203.238 Q2096.21 209.349 2093.24 213.099 Q2090.3 216.849 2085.58 216.849 Q2082.74 216.849 2080.68 215.738 Q2078.64 214.603 2077.3 212.289 M2091.79 203.238 Q2091.79 198.539 2089.84 195.877 Q2087.92 193.192 2084.54 193.192 Q2081.16 193.192 2079.22 195.877 Q2077.3 198.539 2077.3 203.238 Q2077.3 207.937 2079.22 210.622 Q2081.16 213.284 2084.54 213.284 Q2087.92 213.284 2089.84 210.622 Q2091.79 207.937 2091.79 203.238 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M2113.31 193.238 Q2109.89 193.238 2107.9 195.923 Q2105.91 198.585 2105.91 203.238 Q2105.91 207.891 2107.87 210.576 Q2109.86 213.238 2113.31 213.238 Q2116.72 213.238 2118.71 210.553 Q2120.7 207.867 2120.7 203.238 Q2120.7 198.631 2118.71 195.946 Q2116.72 193.238 2113.31 193.238 M2113.31 189.627 Q2118.87 189.627 2122.04 193.238 Q2125.21 196.849 2125.21 203.238 Q2125.21 209.604 2122.04 213.238 Q2118.87 216.849 2113.31 216.849 Q2107.74 216.849 2104.56 213.238 Q2101.42 209.604 2101.42 203.238 Q2101.42 196.849 2104.56 193.238 Q2107.74 189.627 2113.31 189.627 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M2132.27 180.159 L2136.53 180.159 L2136.53 216.178 L2132.27 216.178 L2132.27 180.159 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M2157.23 203.145 Q2152.06 203.145 2150.07 204.326 Q2148.08 205.506 2148.08 208.354 Q2148.08 210.622 2149.56 211.965 Q2151.07 213.284 2153.64 213.284 Q2157.18 213.284 2159.31 210.784 Q2161.46 208.261 2161.46 204.094 L2161.46 203.145 L2157.23 203.145 M2165.72 201.386 L2165.72 216.178 L2161.46 216.178 L2161.46 212.242 Q2160 214.603 2157.83 215.738 Q2155.65 216.849 2152.5 216.849 Q2148.52 216.849 2146.16 214.627 Q2143.82 212.381 2143.82 208.631 Q2143.82 204.256 2146.74 202.034 Q2149.68 199.812 2155.49 199.812 L2161.46 199.812 L2161.46 199.395 Q2161.46 196.455 2159.52 194.858 Q2157.6 193.238 2154.1 193.238 Q2151.88 193.238 2149.77 193.77 Q2147.67 194.303 2145.72 195.367 L2145.72 191.432 Q2148.06 190.53 2150.26 190.09 Q2152.46 189.627 2154.54 189.627 Q2160.17 189.627 2162.94 192.543 Q2165.72 195.46 2165.72 201.386 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M2178.71 182.891 L2178.71 190.252 L2187.48 190.252 L2187.48 193.562 L2178.71 193.562 L2178.71 207.636 Q2178.71 210.807 2179.56 211.71 Q2180.44 212.613 2183.11 212.613 L2187.48 212.613 L2187.48 216.178 L2183.11 216.178 Q2178.17 216.178 2176.3 214.349 Q2174.42 212.497 2174.42 207.636 L2174.42 193.562 L2171.3 193.562 L2171.3 190.252 L2174.42 190.252 L2174.42 182.891 L2178.71 182.891 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M2193.08 190.252 L2197.34 190.252 L2197.34 216.178 L2193.08 216.178 L2193.08 190.252 M2193.08 180.159 L2197.34 180.159 L2197.34 185.553 L2193.08 185.553 L2193.08 180.159 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M2216.3 193.238 Q2212.87 193.238 2210.88 195.923 Q2208.89 198.585 2208.89 203.238 Q2208.89 207.891 2210.86 210.576 Q2212.85 213.238 2216.3 213.238 Q2219.7 213.238 2221.69 210.553 Q2223.68 207.867 2223.68 203.238 Q2223.68 198.631 2221.69 195.946 Q2219.7 193.238 2216.3 193.238 M2216.3 189.627 Q2221.85 189.627 2225.03 193.238 Q2228.2 196.849 2228.2 203.238 Q2228.2 209.604 2225.03 213.238 Q2221.85 216.849 2216.3 216.849 Q2210.72 216.849 2207.55 213.238 Q2204.4 209.604 2204.4 203.238 Q2204.4 196.849 2207.55 193.238 Q2210.72 189.627 2216.3 189.627 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M2256.81 200.529 L2256.81 216.178 L2252.55 216.178 L2252.55 200.668 Q2252.55 196.988 2251.11 195.159 Q2249.68 193.33 2246.81 193.33 Q2243.36 193.33 2241.37 195.53 Q2239.38 197.729 2239.38 201.525 L2239.38 216.178 L2235.1 216.178 L2235.1 190.252 L2239.38 190.252 L2239.38 194.28 Q2240.91 191.942 2242.97 190.784 Q2245.05 189.627 2247.76 189.627 Q2252.23 189.627 2254.52 192.405 Q2256.81 195.159 2256.81 200.529 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><circle clip-path=\"url(#clip640)\" cx=\"1866.77\" cy=\"250.738\" r=\"23.04\" fill=\"#3da44d\" fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"5.12\"/>\n",
       "<path clip-path=\"url(#clip640)\" d=\"M1981.14 242.856 L1981.14 246.883 Q1979.33 245.957 1977.39 245.495 Q1975.44 245.032 1973.36 245.032 Q1970.19 245.032 1968.59 246.004 Q1967.02 246.976 1967.02 248.92 Q1967.02 250.402 1968.15 251.258 Q1969.29 252.092 1972.71 252.856 L1974.17 253.18 Q1978.71 254.152 1980.61 255.934 Q1982.53 257.694 1982.53 260.865 Q1982.53 264.476 1979.66 266.582 Q1976.81 268.689 1971.81 268.689 Q1969.73 268.689 1967.46 268.272 Q1965.21 267.879 1962.71 267.068 L1962.71 262.67 Q1965.07 263.897 1967.37 264.522 Q1969.66 265.124 1971.9 265.124 Q1974.91 265.124 1976.53 264.106 Q1978.15 263.064 1978.15 261.189 Q1978.15 259.453 1976.97 258.527 Q1975.81 257.601 1971.86 256.744 L1970.37 256.397 Q1966.42 255.564 1964.66 253.851 Q1962.9 252.115 1962.9 249.106 Q1962.9 245.448 1965.49 243.458 Q1968.08 241.467 1972.85 241.467 Q1975.21 241.467 1977.3 241.814 Q1979.38 242.161 1981.14 242.856 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M2001.09 254.985 Q1995.93 254.985 1993.94 256.166 Q1991.95 257.346 1991.95 260.194 Q1991.95 262.462 1993.43 263.805 Q1994.93 265.124 1997.5 265.124 Q2001.05 265.124 2003.18 262.624 Q2005.33 260.101 2005.33 255.934 L2005.33 254.985 L2001.09 254.985 M2009.59 253.226 L2009.59 268.018 L2005.33 268.018 L2005.33 264.082 Q2003.87 266.443 2001.69 267.578 Q1999.52 268.689 1996.37 268.689 Q1992.39 268.689 1990.03 266.467 Q1987.69 264.221 1987.69 260.471 Q1987.69 256.096 1990.61 253.874 Q1993.55 251.652 1999.36 251.652 L2005.33 251.652 L2005.33 251.235 Q2005.33 248.295 2003.38 246.698 Q2001.46 245.078 1997.97 245.078 Q1995.74 245.078 1993.64 245.61 Q1991.53 246.143 1989.59 247.207 L1989.59 243.272 Q1991.93 242.37 1994.12 241.93 Q1996.32 241.467 1998.41 241.467 Q2004.03 241.467 2006.81 244.383 Q2009.59 247.3 2009.59 253.226 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M2038.55 247.069 Q2040.14 244.198 2042.37 242.833 Q2044.59 241.467 2047.6 241.467 Q2051.65 241.467 2053.85 244.314 Q2056.05 247.138 2056.05 252.369 L2056.05 268.018 L2051.76 268.018 L2051.76 252.508 Q2051.76 248.782 2050.44 246.976 Q2049.12 245.17 2046.42 245.17 Q2043.11 245.17 2041.18 247.37 Q2039.26 249.569 2039.26 253.365 L2039.26 268.018 L2034.98 268.018 L2034.98 252.508 Q2034.98 248.758 2033.66 246.976 Q2032.34 245.17 2029.59 245.17 Q2026.32 245.17 2024.4 247.393 Q2022.48 249.592 2022.48 253.365 L2022.48 268.018 L2018.2 268.018 L2018.2 242.092 L2022.48 242.092 L2022.48 246.12 Q2023.94 243.735 2025.98 242.601 Q2028.01 241.467 2030.81 241.467 Q2033.64 241.467 2035.61 242.902 Q2037.6 244.337 2038.55 247.069 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M2068.66 264.129 L2068.66 277.879 L2064.38 277.879 L2064.38 242.092 L2068.66 242.092 L2068.66 246.027 Q2070 243.712 2072.04 242.601 Q2074.1 241.467 2076.95 241.467 Q2081.67 241.467 2084.61 245.217 Q2087.57 248.967 2087.57 255.078 Q2087.57 261.189 2084.61 264.939 Q2081.67 268.689 2076.95 268.689 Q2074.1 268.689 2072.04 267.578 Q2070 266.443 2068.66 264.129 M2083.15 255.078 Q2083.15 250.379 2081.21 247.717 Q2079.29 245.032 2075.91 245.032 Q2072.53 245.032 2070.58 247.717 Q2068.66 250.379 2068.66 255.078 Q2068.66 259.777 2070.58 262.462 Q2072.53 265.124 2075.91 265.124 Q2079.29 265.124 2081.21 262.462 Q2083.15 259.777 2083.15 255.078 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M2094.63 231.999 L2098.89 231.999 L2098.89 268.018 L2094.63 268.018 L2094.63 231.999 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M2129.98 253.99 L2129.98 256.073 L2110.4 256.073 Q2110.67 260.471 2113.04 262.786 Q2115.42 265.078 2119.66 265.078 Q2122.11 265.078 2124.4 264.476 Q2126.72 263.874 2128.98 262.67 L2128.98 266.698 Q2126.69 267.67 2124.29 268.18 Q2121.88 268.689 2119.4 268.689 Q2113.2 268.689 2109.56 265.078 Q2105.95 261.467 2105.95 255.309 Q2105.95 248.944 2109.38 245.217 Q2112.83 241.467 2118.66 241.467 Q2123.89 241.467 2126.92 244.846 Q2129.98 248.203 2129.98 253.99 M2125.72 252.74 Q2125.67 249.245 2123.75 247.161 Q2121.86 245.078 2118.71 245.078 Q2115.14 245.078 2112.99 247.092 Q2110.86 249.106 2110.54 252.763 L2125.72 252.74 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip640)\" d=\"M2153.5 242.856 L2153.5 246.883 Q2151.69 245.957 2149.75 245.495 Q2147.8 245.032 2145.72 245.032 Q2142.55 245.032 2140.95 246.004 Q2139.38 246.976 2139.38 248.92 Q2139.38 250.402 2140.51 251.258 Q2141.65 252.092 2145.07 252.856 L2146.53 253.18 Q2151.07 254.152 2152.97 255.934 Q2154.89 257.694 2154.89 260.865 Q2154.89 264.476 2152.02 266.582 Q2149.17 268.689 2144.17 268.689 Q2142.09 268.689 2139.82 268.272 Q2137.57 267.879 2135.07 267.068 L2135.07 262.67 Q2137.43 263.897 2139.73 264.522 Q2142.02 265.124 2144.26 265.124 Q2147.27 265.124 2148.89 264.106 Q2150.51 263.064 2150.51 261.189 Q2150.51 259.453 2149.33 258.527 Q2148.17 257.601 2144.22 256.744 L2142.73 256.397 Q2138.78 255.564 2137.02 253.851 Q2135.26 252.115 2135.26 249.106 Q2135.26 245.448 2137.85 243.458 Q2140.44 241.467 2145.21 241.467 Q2147.57 241.467 2149.66 241.814 Q2151.74 242.161 2153.5 242.856 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /></svg>\n"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots, LinearAlgebra\n",
    "f = x -> cos(10x)\n",
    "n = 5\n",
    "\n",
    "x = range(0, 1; length=n)# evenly spaced points (BAD for interpolation)\n",
    "V = x .^ (0:n-1)' # Vandermonde matrix\n",
    "c = V \\ f.(x) # coefficients of interpolatory polynomial\n",
    "p = x -> dot(c, x .^ (0:n-1))\n",
    "\n",
    "g = range(0,1; length=1000) # plotting grid\n",
    "plot(g, f.(g); label=\"function\")\n",
    "plot!(g, p.(g); label=\"interpolation\")\n",
    "scatter!(x, f.(x); label=\"samples\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But it turns out we can also construct the interpolatory polynomial directly.\n",
    "We will use the following which equal $1$ at one grid point\n",
    "and zero at the others:\n",
    "\n",
    "**Definition 3 (Lagrange basis polynomial)** The _Lagrange basis polynomial_ is defined as\n",
    "$$\n",
    "ℓ_k(x) := ∏_{j ≠ k} {x-x_j \\over x_k - x_j} =  {(x-x_1) ⋯(x-x_{k-1})(x-x_{k+1}) ⋯ (x-x_n) \\over (x_k - x_1) ⋯ (x_k - x_{k-1}) (x_k - x_{k+1}) ⋯ (x_k - x_n)}\n",
    "$$\n",
    "\n",
    "Plugging in the grid points verifies the following:\n",
    "\n",
    "**Proposition 2 (delta interpolation)**\n",
    "$$\n",
    "ℓ_k(x_j) = δ_{kj}\n",
    "$$\n",
    "\n",
    "We can use these to construct the interpolatory polynomial:\n",
    "\n",
    "**Theorem 1 (Lagrange interpolation)**\n",
    "The unique  polynomial of degree at most $n-1$ that interpolates $f$ at $n$ distinct\n",
    "points $x_j$ is:\n",
    "$$\n",
    "p(x) = f(x_1) ℓ_1(x) + ⋯ + f(x_n) ℓ_n(x)\n",
    "$$\n",
    "\n",
    "**Proof**\n",
    "Note that\n",
    "$$\n",
    "p(x_j) = ∑_{j=1}^n f(x_j) ℓ_k(x_j) = f(x_j)\n",
    "$$\n",
    "so we just need to show it is unique. Suppose $p̃(x)$ is a  polynomial\n",
    "of degree at most $n-1$\n",
    "that also interpolates $f$. Then $p̃ - p$ vanishes at $n$ distinct points.\n",
    "Thus by the fundamental theorem of algebra it must be zero.\n",
    "\n",
    "\n",
    "∎\n",
    "\n",
    "**Example 1** We can interpolate $\\exp(x)$ at the points $0,1,2$:\n",
    "$$\n",
    "\\begin{align*}\n",
    "p(x) &= ℓ_1(x) + {\\rm e} ℓ_2(x) + {\\rm e}^2 ℓ_3(x) =\n",
    "{(x - 1) (x-2) \\over (-1)(-2)} + {\\rm e} {x (x-2) \\over (-1)} +\n",
    "{\\rm e}^2 {x (x-1) \\over 2} \\\\\n",
    "&= (1/2 - {\\rm e} +{\\rm e}^2/2)x^2 + (-3/2 + 2 {\\rm e}  - {\\rm e}^2 /2) x + 1\n",
    "\\end{align*}\n",
    "$$\n",
    "\n",
    "\n",
    "**Remark** Interpolating at evenly spaced points is a really **bad** idea:\n",
    "interpolation is inheritely ill-conditioned. \n",
    "The labs have explored this issue experimentally.\n",
    "\n",
    "\n",
    "## 2. Interpolatory quadrature rules\n",
    "\n",
    "By integrating an interpolant exactly we get a simple approach to approximating\n",
    "integrals. Using the Lagrange basis we can rewrite this procedure as a simple\n",
    "weighted sum:\n",
    "\n",
    "**Definition 4 (interpolatory quadrature rule)** Given a set of points $𝐱 = [x_1,…,x_n]$\n",
    "the interpolatory quadrature rule is:\n",
    "$$\n",
    "Σ_n^{w,𝐱}[f] := ∑_{j=1}^n w_j f(x_j)\n",
    "$$\n",
    "where\n",
    "$$\n",
    "w_j := ∫_a^b ℓ_j(x) w(x) {\\rm d} x\n",
    "$$\n",
    "\n",
    "\n",
    "**Proposition 3 (interpolatory quadrature is exact for polynomials)** \n",
    "Interpolatory quadrature is exact for all degree $n-1$ polynomials $p$:\n",
    "$$\n",
    "∫_a^b p(x) w(x) {\\rm d}x = Σ_n^{w,𝐱}[p]\n",
    "$$\n",
    "\n",
    "**Proof**\n",
    "The result follows since, by uniqueness of interpolatory polynomial:\n",
    "$$\n",
    "p(x) = ∑_{j=1}^n p(x_j) ℓ_j(x)\n",
    "$$\n",
    "\n",
    "∎\n",
    "\n",
    "**Example 2 (arbitrary points)** Find the interpolatory quadrature rule for $w(x) = 1$ on $[0,1]$ with  points $[x_1,x_2,x_3] = [0,1/4,1]$?\n",
    "We have:\n",
    "$$\n",
    "\\begin{align*}\n",
    "w_1 = \\int_0^1 w(x) ℓ_1(x) {\\rm d}x  = \\int_0^1 {(x-1/4)(x-1) \\over (-1/4)(-1)}{\\rm d}x = -1/6 \\\\\n",
    "w_2 = \\int_0^1 w(x) ℓ_2(x) {\\rm d}x  = \\int_0^1 {x(x-1) \\over (1/4)(-3/4)}{\\rm d}x = 8/9 \\\\\n",
    "w_3 = \\int_0^1 w(x) ℓ_3(x) {\\rm d}x  = \\int_0^1 {x(x-1/4) \\over 3/4}{\\rm d}x = 5/18\n",
    "\\end{align*}\n",
    "$$\n",
    "That is we have\n",
    "$$\n",
    "Σ_n^{w,𝐱}[f]  = -{f(0) \\over 6} + {8f(1/4) \\over 9} + {5 f(1) \\over 18}\n",
    "$$\n",
    "This is indeed exact for polynomials up to degree $2$ (and no more):\n",
    "$$\n",
    "Σ_n^{w,𝐱}[1] = 1, Σ_n^{w,𝐱}[x] = 1/2, Σ_n^{w,𝐱}[x^2] = 1/3, Σ_n^{w,𝐱}[x^3] = 7/24 ≠ 1/4.\n",
    "$$\n",
    "\n",
    "**Example 3 (Chebyshev roots)** Find the interpolatory quadrature rule for $w(x) = 1/\\sqrt{1-x^2}$ on $[-1,1]$ with points equal to the\n",
    "roots of $T_3(x)$. This is a special case of Gaussian quadrature which we will approach in another way below. We use:\n",
    "$$\n",
    "\\int_{-1}^1 w(x) {\\rm d}x = π, \\int_{-1}^1 xw(x) {\\rm d}x = 0, \\int_{-1}^1 x^2 w(x) {\\rm d}x = {π/2}\n",
    "$$\n",
    "Recall from before that $x_1,x_2,x_3 = \\sqrt{3}/2,0,-\\sqrt{3}/2$. Thus we have:\n",
    "$$\n",
    "\\begin{align*}\n",
    "w_1 = \\int_{-1}^1 w(x) ℓ_1(x) {\\rm d}x = \\int_{-1}^1 {x(x+\\sqrt{3}/2) \\over (\\sqrt{3}/2) \\sqrt{3} \\sqrt{1-x^2}}{\\rm d}x = {π \\over 3} \\\\\n",
    "w_2 = \\int_{-1}^1 w(x) ℓ_2(x) {\\rm d}x = \\int_{-1}^1 {(x-\\sqrt{3}/2)(x+\\sqrt{3}/2) \\over (-3/4)\\sqrt{1-x^2}}{\\rm d}x = {π \\over 3} \\\\\n",
    "w_3 = \\int_{-1}^1 w(x) ℓ_3(x) {\\rm d}x = \\int_{-1}^1 {(x-\\sqrt{3}/2) x \\over (-\\sqrt{3})(-\\sqrt{3}/2) \\sqrt{1-x^2}}{\\rm d}x = {π \\over 3}\n",
    "\\end{align*}\n",
    "$$\n",
    "(It's not a coincidence that they are all the same but this will differ for roots of other OPs.) \n",
    "That is we have\n",
    "$$\n",
    "Σ_n^{w,𝐱}[f]  = {π \\over 3}(f(\\sqrt{3}/2) + f(0) + f(-\\sqrt{3}/2)\n",
    "$$\n",
    "This is indeed exact for polynomials up to degree $n-1=2$, but it goes all the way up to $2n-1 = 5$:\n",
    "$$\n",
    "\\begin{align*}\n",
    "Σ_n^{w,𝐱}[1] &= π, Σ_n^{w,𝐱}[x] = 0, Σ_n^{w,𝐱}[x^2] = {π \\over 2}, \\\\\n",
    "Σ_n^{w,𝐱}[x^3] &= 0, Σ_n^{w,𝐱}[x^4] &= {3 π \\over 8}, Σ_n^{w,𝐱}[x^5] = 0 \\\\\n",
    "Σ_n^{w,𝐱}[x^6] &= {9 π \\over 32} ≠ {5 π \\over 16}\n",
    "\\end{align*}\n",
    "$$\n",
    "We shall explain this miracle in the next chapter."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Julia 1.8.3",
   "language": "julia",
   "name": "julia-1.8"
  },
  "language_info": {
   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
   "version": "1.8.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}