A10_Abhishek_1911007.ipynb

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      "name": "python"
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    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
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      "source": [
        "<a href=\"https://colab.research.google.com/github/physicsme1729/Numerical-methods-in-physics/blob/main/11_swarup_kumar_Giri_phy_P745.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
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    {
      "cell_type": "markdown",
      "source": [
        "# ***Problem-1:***"
      ],
      "metadata": {
        "id": "D5ke9D_5-DVc"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Find the minimum of the function $F(x, y, z) = 2x^2 + 3y^2 + z^2 + xy + xz − 2y$\n",
        "\n",
        "and confirm the result analytically."
      ],
      "metadata": {
        "id": "Kq6Guuoc-DdU"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# ***Solution:-***"
      ],
      "metadata": {
        "id": "uHkqtJ_9_jAV"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**Analytical solution:**\n"
      ],
      "metadata": {
        "id": "mbGd2wN6_oAR"
      }
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    {
      "cell_type": "markdown",
      "source": [
        "\n",
        "\n",
        "To find the minimum of the function $F(x, y, z) = 2x^2 + 3y^2 + z^2 + xy + xz - 2y$, we can use calculus. The minimum occurs at a point where the gradient of the function is zero. The gradient of the function is a vector of its partial derivatives with respect to each variable:\n",
        "\n",
        "$\n",
        "\\nabla F = \\left(\\frac{\\partial F}{\\partial x}, \\frac{\\partial F}{\\partial y}, \\frac{\\partial F}{\\partial z}\\right)\n",
        "$\n",
        "\n",
        "Let's compute the partial derivatives of \\(F\\) with respect to \\(x\\), \\(y\\), and \\(z\\):\n",
        "\n",
        "$\n",
        "\\frac{\\partial F}{\\partial x} = 4x + y + z\n",
        "$\n",
        "\n",
        "$\n",
        "\\frac{\\partial F}{\\partial y} = 6y + x - 2\n",
        "$\n",
        "\n",
        "$\n",
        "\\frac{\\partial F}{\\partial z} = 2z + x\n",
        "$\n",
        "\n",
        "Now, we can set each of these partial derivatives equal to zero to find the critical points:\n",
        "\n",
        "$\n",
        "\\frac{\\partial F}{\\partial x} = 4x + y + z = 0\n",
        "$\n",
        "\n",
        "$\n",
        "\\frac{\\partial F}{\\partial y} = 6y + x - 2 = 0\n",
        "$\n",
        "\n",
        "$\n",
        "\\frac{\\partial F}{\\partial z} = 2z + x = 0\n",
        "$\n",
        "\n",
        "Solving these equations simultaneously will give us the critical point(s) where the gradient is zero.\n",
        "\n",
        "Let's first solve for \\(x\\) in the third equation:\n",
        "\n",
        "$\n",
        "2z + x = 0 \\implies x = -2z\n",
        "$\n",
        "\n",
        "Now, we can substitute this expression for \\(x\\) into the first and second equations:\n",
        "\n",
        "$\n",
        "4x + y + z = 0 \\implies 4(-2z) + y + z = 0\n",
        "$\n",
        "\n",
        "$\n",
        "-8z + y + z = 0\n",
        "$\n",
        "\n",
        "$\n",
        "-7z + y = 0 \\implies y = 7z\n",
        "$\n",
        "\n",
        "$\n",
        "6y + x - 2 = 0 \\implies 6(7z) + (-2z) - 2 = 0\n",
        "$\n",
        "\n",
        "$\n",
        "42z - 2z - 2 = 0\n",
        "$\n",
        "\n",
        "$\n",
        "40z - 2 = 0\n",
        "$\n",
        "\n",
        "$\n",
        "40z = 2\n",
        "$\n",
        "\n",
        "$\n",
        "z = \\frac{2}{40} = \\frac{1}{20}\n",
        "$\n",
        "\n",
        "Now that we have found values for \\(x\\), \\(y\\), and \\(z\\), we can calculate the corresponding minimum value of \\(F\\):\n",
        "\n",
        "$\n",
        "x = -2z = -2\\left(\\frac{1}{20}\\right) = -\\frac{1}{10}\n",
        "$\n",
        "\n",
        "$\n",
        "y = 7z = 7\\left(\\frac{1}{20}\\right) = \\frac{7}{20}\n",
        "$\n",
        "\n",
        "Now, substitute these values into \\(F\\):\n",
        "\n",
        "$\n",
        "F\\left(-\\frac{1}{10}, \\frac{7}{20}, \\frac{1}{20}\\right) = 2\\left(-\\frac{1}{10}\\right)^2 + 3\\left(\\frac{7}{20}\\right)^2 + \\left(\\frac{1}{20}\\right)^2 - \\frac{1}{10}\\left(\\frac{7}{20}\\right) - \\frac{1}{10}\\left(\\frac{1}{20}\\right) - 2\\left(\\frac{7}{20}\\right)\n",
        "$\n",
        "\n",
        "Now, one can  find the minimum value of \\(F\\) from  above $F\\left(-\\frac{1}{10}, \\frac{7}{20}, \\frac{1}{20}\\right)$ .\n",
        "\n",
        "\n"
      ],
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    },
    {
      "cell_type": "markdown",
      "source": [
        "***solution in python using  \"Downhill method\"***"
      ],
      "metadata": {
        "id": "Iaal8KEITqoE"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "import math\n",
        "\n",
        "def downhill(F, xStart, side=0.1, tol=1.0e-6):\n",
        "    n = len(xStart)  # Number of variables\n",
        "    x = np.zeros((n + 1, n))\n",
        "    f = np.zeros(n + 1)\n",
        "    x[0] = xStart\n",
        "    for i in range(1, n + 1):\n",
        "        x[i] = xStart.copy()\n",
        "        x[i, i - 1] = xStart[i - 1] + side\n",
        "\n",
        "    # Compute values of F at the vertices of the simplex\n",
        "    for i in range(n + 1):\n",
        "        f[i] = F(x[i])\n",
        "\n",
        "    # Main loop\n",
        "    for k in range(500):\n",
        "        iLo = np.argmin(f)\n",
        "        iHi = np.argmax(f)\n",
        "\n",
        "        d = (-(n + 1) * x[iHi] + np.sum(x, axis=0)) / n\n",
        "        if math.sqrt(np.dot(d, d) / n) < tol:\n",
        "            return x[iLo]\n",
        "\n",
        "        xNew = x[iHi] + 2.0 * d\n",
        "        fNew = F(xNew)\n",
        "        if fNew <= f[iLo]:\n",
        "            x[iHi] = xNew\n",
        "            f[iHi] = fNew\n",
        "\n",
        "            xNew = x[iHi] + d\n",
        "            fNew = F(xNew)\n",
        "            if fNew <= f[iLo]:\n",
        "                x[iHi] = xNew\n",
        "                f[iHi] = fNew\n",
        "        else:\n",
        "            if fNew <= f[iHi]:\n",
        "                x[iHi] = xNew\n",
        "                f[iHi] = fNew\n",
        "            else:\n",
        "                xNew = x[iHi] + 0.5 * d\n",
        "                fNew = F(xNew)\n",
        "                if fNew <= f[iHi]:\n",
        "                    x[iHi] = xNew\n",
        "                    f[iHi] = fNew\n",
        "                else:\n",
        "                    for i in range(len(x)):\n",
        "                        if i != iLo:\n",
        "                            x[i] = (x[i] - x[iLo]) * 0.5\n",
        "                            f[i] = F(x[i])\n",
        "    print(\"Too many iterations in downhill\")\n",
        "    return x[iLo]\n",
        "\n",
        "def F(x):\n",
        "    return 2.0 * x[0] ** 2 + 3.0 * x[1] ** 2 + x[2] ** 2 + x[0] * x[1] + x[0] * x[2] - 2.0 * x[1]\n",
        "\n",
        "x = [0.0, 0.0, 0.0]\n",
        "result = downhill(F, x)\n",
        "x = result  # Updated x with the optimized values\n",
        "\n",
        "print(\"x, y, z =\", x)\n",
        "print(\"F(x, y, z) =\", F(x))\n",
        "\n"
      ],
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          "base_uri": "https://localhost:8080/"
        },
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      },
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      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "x, y, z = [-0.10000075  0.34999998  0.05000037]\n",
            "F(x, y, z) = -0.34999999999900455\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "# ***Problem 2:-***"
      ],
      "metadata": {
        "id": "-ew_bWKbT3IN"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "\n",
        "***Kiusalaas Chapter 3, Problem Set 3.2, Problem 10***"
      ],
      "metadata": {
        "id": "UiR33ksYT8Q-"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# *Python code-*"
      ],
      "metadata": {
        "id": "8-fiYwGjUUo_"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "import math\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "def gaussPivot(a, b, tol=1.0e-12):\n",
        "    n = len(b)\n",
        "    s = np.zeros(n)\n",
        "    for i in range(n):\n",
        "        s[i] = max(np.abs(a[i, :]))\n",
        "\n",
        "    for k in range(0, n - 1):\n",
        "        p = np.argmax(np.abs(a[k:n, k]) / s[k:n]) + k  # Row interchange, if needed\n",
        "        if abs(a[p, k]) < tol:\n",
        "            raise ValueError('Matrix is singular')\n",
        "\n",
        "        if p != k:\n",
        "            swapRows(b, k, p)\n",
        "            swapRows(s, k, p)\n",
        "            swapRows(a, k, p)\n",
        "\n",
        "        for i in range(k + 1, n):\n",
        "            if a[i, k] != 0.0:\n",
        "                lam = a[i, k] / a[k, k]\n",
        "                a[i, k + 1:n] = a[i, k + 1:n] - lam * a[k, k + 1:n]\n",
        "                b[i] = b[i] - lam * b[k]\n",
        "\n",
        "    if abs(a[n - 1, n - 1]) < tol:\n",
        "        raise ValueError('Matrix is singular')\n",
        "\n",
        "    # Back substitution\n",
        "    b[n - 1] = b[n - 1] / a[n - 1, n - 1]\n",
        "    for k in range(n - 2, -1, -1):\n",
        "        b[k] = (b[k] - np.dot(a[k, k + 1:n], b[k + 1:n])) / a[k, k]\n",
        "\n",
        "    return b\n",
        "\n",
        "def polyFit(xData, yData, m):\n",
        "    a = np.zeros((m + 1, m + 1))\n",
        "    b = np.zeros(m + 1)\n",
        "    s = np.zeros(2 * m + 1)\n",
        "\n",
        "    for i in range(len(xData)):\n",
        "        temp = yData[i]\n",
        "        for j in range(m + 1):\n",
        "            b[j] = b[j] + temp\n",
        "            temp = temp * xData[i]\n",
        "        temp = 1.0\n",
        "        for j in range(2 * m + 1):\n",
        "            s[j] = s[j] + temp\n",
        "            temp = temp * xData[i]\n",
        "\n",
        "    for i in range(m + 1):\n",
        "        for j in range(m + 1):\n",
        "            a[i, j] = s[i + j]\n",
        "\n",
        "    return gaussPivot(a, b)\n",
        "\n",
        "def stdDev(c, xData, yData):\n",
        "    def evalPoly(c, x):\n",
        "        m = len(c) - 1\n",
        "        p = c[m]\n",
        "        for j in range(m):\n",
        "            p = p * x + c[m - j - 1]\n",
        "        return p\n",
        "\n",
        "    n = len(xData) - 1\n",
        "    m = len(c) - 1\n",
        "    sigma = 0.0\n",
        "\n",
        "    for i in range(n + 1):\n",
        "        p = evalPoly(c, xData[i])\n",
        "        sigma = sigma + (yData[i] - p) ** 2\n",
        "\n",
        "    sigma = math.sqrt(sigma / (n - m))\n",
        "    return sigma\n",
        "\n",
        "def plotPoly(xData, yData, coeff, xlab='x', ylab='y'):\n",
        "    m = len(coeff)\n",
        "    x1 = min(xData)\n",
        "    x2 = max(xData)\n",
        "    dx = (x2 - x1) / 20.0\n",
        "    x = np.arange(x1, x2 + dx / 10.0, dx)\n",
        "    y = np.zeros(len(x))\n",
        "\n",
        "    for i in range(m):\n",
        "        y = y + coeff[i] * x ** i\n",
        "\n",
        "    plt.plot(xData, yData, 'o', x, y, '-')\n",
        "    plt.xlabel(xlab)\n",
        "    plt.ylabel(ylab)\n",
        "    plt.grid(True)\n",
        "    plt.show()\n",
        "\n",
        "xData = np.array([1718, 1767, 1774, 1775, 1792, 1816, 1828, 1834, 1878, 1906]) * 1.0\n",
        "yData = np.array([0.5, 0.8, 1.4, 2.7, 4.5, 7.5, 12.0, 17.0, 17.2, 23.0])\n",
        "coeff = polyFit(xData, yData, 1)\n",
        "\n",
        "print(\"Coefficients are:\\n\", coeff)\n",
        "print(\"Std. deviation =\", stdDev(coeff, xData, yData))\n",
        "plotPoly(xData, yData, coeff, 'Year', 'Efficiency (%)')\n",
        "\n"
      ],
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        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 501
        },
        "id": "gzqezeotbWQp",
        "outputId": "56dbd597-bf64-417a-d9b0-5610f4bd6c94"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Coefficients are:\n",
            " [-2.40391746e+02  1.37688935e-01]\n",
            "Std. deviation = 2.8552022884971544\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": 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LrYeBXassKhKVGhERD6MzACWQkw0rX4Dv3gAMqHYl/HmGeYdgqXBUakRExDud3Atzh8CR781xi7uhx4sQWMnSWGIdlRoREfEuhgE//Be+eBwcZyG4CvT5l/l0banQVGpERMR7nDsFX4yE7XPNcd0b4JapEFHT0ljiGVRqRETEO+xfB589AGkHwOYHnZ+Gdo+B3c/qZOIhVGpERMSz5ebA6gmw+hUwnFC1LgyYATVbWp1MPIxKjYiIeK7f98Nn98PBDea46e3Q6xUIDrc2l3gklRoREfFM2z6Fz0dAVjoEhUPvidDkVqtTiQdTqREREc+SdRq+ehKSPjDHNVvBgOnmZSeRS1CpERERz3F4C8y9D1J/BZsdOvwdOjwBfvrrSi5Pf0pERMR6zlzzrsArXwBnDoTXNM/O1GlrdTLxIio1IiJirfQj5nObkleb44R+kPg6hFS1MpV4IZUaERGxzs7PYeHDcO53CKgEN70C190BNj3AU0pOpUZERMpfdgZ8/RRsmWWOY68z7z1TvYGlscS7qdSIiEj5SvnRfBDliV/Mcbu/QqdnwD/Q2lzi9exWB8izevVqEhMTiYuLw2azMX/+/AKv33333dhstgK/evbsaU1YEREpOacT1k2Gd7uYhaZyDNw5H7qNU6ERl/CYMzVnz56ladOm3HvvvfTv37/IfXr27MmsWbPyx0FBQeUVT0REyuLMcZg/DPYsM8cNb4I+b0GlatbmEp/iMaWmV69e9OrV65L7BAUFERMTU06JRETEJX5ZAgsegrO/gX8w9HgBWg7RYmBxOY8pNcXxzTffEB0dTdWqVencuTPPP/881apdvOVnZWWRlZWVP05PTwfA4XDgcDjcnrck8vJ4Wi530Xx9m+br24o935xM7Cuew2/TNACM6ARy+r0DUVdDTo67Y7qMjq/1ipvFZhiG4eYsJWaz2Zg3bx79+vXL3/bhhx8SGhpKfHw8e/fu5amnnqJy5cqsW7cOP7+iHzs/ZswYxo4dW2j7nDlzCA0NdVd8EZEKL+zcYVrsm0xE5kEA9kZ1Z0fc/+G0a+2MlFxGRgYDBw4kLS2N8PCLP8zUa0rNhX799Vfq16/PsmXL6NKlS5H7FHWmplatWpw4ceKSvylWcDgcLF26lG7duhEQEGB1HLfTfH2b5uvbLjlfw8C+dRb2Zc9iy8nECK1ObuKbGA26WRPWBXR8rZeenk716tUvW2q86vLT+erVq0f16tXZs2fPRUtNUFBQkYuJAwICPOZAXciTs7mD5uvbNF/fVmi+Z0+aN9Lb9aU5rt8FW78p+IfVsCagi1X442uh4ubw2lJz6NAhTp48SWxsrNVRRETk12/gswfhzFHwC4SuY6H1ULB7zJ1DpALwmFJz5swZ9uzZkz9OTk4mKSmJyMhIIiMjGTt2LAMGDCAmJoa9e/fyxBNP0KBBA3r06GFhahGRCi4nG1Y+D9/9CzCg+lXmnYFjm1idTCogjyk1mzdvplOnTvnjkSNHAjB48GCmTJnCjz/+yPvvv8+pU6eIi4uje/fuPPfcc7pXjYiIVU7ugQVDISXJHLe4B3q8CIH6IIZYw2NKTceOHbnUmuWvv/66HNOIiMhFGQa1T67Gf8YwcJw1n6bd501olGh1MqngPKbUiIiIFzj3O34L/0qzAwvMcd0boP87EB5nbS4RVGpERKS49q+Fzx7AnnYQJ34YnZ7C74YRYC/6XmEi5U2lRkRELi03B1a/AqsngOHEqBrPmqi7aNv2EfxUaMSD6LN2IiJycb/vg1m9YNXLYDjhujvIGbKCU5XqW51MpBCdqRERkaL9+Al8MRKy0iEoHG6eBNf+GTzomUAi51OpERGRgjLT4cu/w48fmuNaraH/dKhax9pcIpehUiMiIv9zaAvMvde87GSzQ4cnoMPfwU9/XYjn059SEREBZy589zqsfBGcORBRyzw7U6eN1clEik2lRkSkoks7DPMehH1rzPE1/c31MyFVLI0lUlIqNSIiFdmOhbDwEcg8BQGVoPer0PR2sNmsTiZSYio1IiIVUfZZWDwKtr5vjuOamQ+irKaPaov3UqkREaloUn6AT4fAyd2ADdo/Bh2fAv9Aq5OJlIlKjYhIReF0woYpsGwM5GZDWCzcMhXqdbQ6mYhLqNSIiFQEp4/B/KGwd4U5btjbfLJ2pWrW5hJxIZUaERFf98vXMP8hyDgB/iHQ4wVoea8WA4vPUakREfFVjkxY+ixsnGaOazQ2FwNHX21tLhE3KXGpycrKYsOGDezfv5+MjAyioqJo1qwZ8fHx7sgnIiKlcXynuRj4+E/m+E8PQZfREBBsbS4RNyp2qfnuu+944403WLRoEQ6Hg4iICEJCQkhNTSUrK4t69erxwAMPMHToUMLCwtyZWURELsYwYNO7sOQZyMmESlHQbwpc2c3qZCJuZy/OTn369OEvf/kLdevWZcmSJZw+fZqTJ09y6NAhMjIy2L17N8888wzLly/nqquuYunSpe7OLSIiFzp7Ev57O3z5uFloGnSFYWtVaKTCKNaZmt69ezN37lwCAgKKfL1evXrUq1ePwYMHs2PHDlJSUlwaUkRELmPvSpg3FM4cBb9A6DoWWg8Fe7H+7SriE4pVah588MFiv2FCQgIJCQmlDiQiIiWQkw0rnoO1/zLH1RvCn2dAzLXW5hKxQJk+/bR9+3ZWrVpFbm4u7dq1o0WLFq7KJSIil3NiD8wdAilJ5rjlvdD9BQgMtTSWiFVKfV7y7bffpkuXLqxatYqVK1fSuXNnXnjhBVdmExGRohgGbP0PTLvBLDQhVeEvH5hP1lahkQqs2GdqDh48SK1atfLHb731Fj/99BPVq1cHYN26dfTp04enn37a9SlFRMR07ndY9BjsmG+O694A/d+B8DgrU4l4hGKfqenatStvvPEGhmEAUK1aNRYvXkxWVhanT59m2bJlREVFuS2oiEiFt38tTGlvFhq7P3QdA3ctUKER+UOxS82mTZvYtWsXrVu3JikpiXfeeYdJkyYREhJClSpV+Oijj3j//ffdmVVEpGLKdcCK5+G93pB+CCLrwZAl0H4E2P2sTifiMYp9+Sk8PJzJkyezdu1a7r77bjp37syaNWvIzc0lNzeXKlWquDGmiIj3yXUabExO5fjpTKLDgmkVH4mfvYTPW0pNhs/uh0ObzPF1d0CvlyFINzkVuVCJP/3Utm1bNm/ezEsvvUSzZs2YOHEivXv3dkc2ERGvtXh7CmMX7SAlLTN/W2xEMKMTE+jZOLZ4b/LjJ/D5CMg+DUERcPNEuPbPbkos4v2KffkpJyeHyZMn88gjj/Dee+/x1FNPsWjRIl577TVuvfVWjh075s6cIiJeY/H2FIbN3lqg0AAcTctk2OytLN5+mRuUZqbDZw/CZ/eZhabWn2DoGhUakcsodqkZMmQIb731FpUqVWLWrFmMGDGCq666ihUrVtCzZ0/atGnDlClT3JlVRMTj5ToNxi7agVHEa3nbxi7aQa6zqD2AQ5vNj2r/+CHY7NBxFNz9BVSt467IIj6j2KVmwYIFzJ07l/Hjx7N06VK++OKL/NeGDBnC+vXrWbNmjVtCioh4i43JqYXO0JzPAFLSMtmYnFrwBWcurJ4AM7rD7/sgojbc8xV0/Af4lek+qSIVRrF/UmrUqMGSJUuoX78+K1asoFq1agVej46OZs6cOS4PKCLiTY6fvnihueh+aYfgswdg/3fmuPEA6D0RQqq4PqCIDyt2qXnrrbe44447GDlyJLGxsXz88cfuzCUi4pWiw4JLtt+OBbDwUcg8BYGV4aZXoeltYCvhp6REpPilplu3bhw7dowTJ07oJnsiIhfRKj6S2IhgjqZlFrmuxgbERATT6oogs8xs/eP+XnHNYcC7UK1+ecYV8SklevaTzWZToRERuQQ/u43RiQmAWWDOlzd+rb2B3/Qb/yg0Nmg/0ryZngqNSJkUq9T07NmT9evXX3a/06dP8/LLL/P222+XOZiIiLfq2TiWKYOaExNR8FJUbHggX7VKou3K2+DkHgiLg8ELoeto8AuwKK2I7yjW5adbb72VAQMGEBERQWJiIi1btiQuLo7g4GB+//13duzYwbfffsuXX35J7969mTBhgrtzi4h4tJ6NY+mWEJN/R+Er/NJp8f0obD+uNHe4+mbo8yaERlobVMSHFKvUDBkyhEGDBvHJJ5/w0Ucf8c4775CWlgaYl6QSEhLo0aMHmzZtolGjRm4NLCLiLfzsNtrUrwa7FsOChyDjJPiHQM+XoMXdWgws4mLFXigcFBTEoEGDGDRoEABpaWmcO3eOatWqERCg06YiIoU4zsHSZ2HjO+a4xrXw5xkQ1dDaXCI+qtR3dIqIiCAiIsKVWUREfMexHTB3CBzfYY7/NNxcO+MfZG0uER+m21SKiLiSYcCmd+HrpyE3CypFQ78pcGVXq5OJ+DyVGhERVzl7Ar58DH5ZbI6v7A59J0Nl3QpDpDyo1IiIuEBU+nb8pz8OZ4+DXyB0ew5aP6jFwCLlSKVGRKQscrKxLx9D271/3J8r6moYMANiGlubS6QCKtEdhQEGDx7M6tWr3ZFFRMS7nNgNM7rit94sNLnN74H7V6rQiFikxKUmLS2Nrl27cuWVV/Liiy9y+PBhd+QSEfFchgFb/w3TOkDKDxghkWyo9xjOXhMgMNTqdCIVVolLzfz58zl8+DDDhg3jo48+om7duvTq1YtPP/0Uh8PhjowiIp7j3O/wyWBY+Ag4MiD+RnLuX8XRiOZWJxOp8EpcagCioqIYOXIkP/zwAxs2bKBBgwbceeedxMXFMWLECHbv3u3qnCIi1tv3LUxpBzsWgN0fuo2DO+dDWKzVyUSEUpaaPCkpKSxdupSlS5fi5+fHTTfdxLZt20hISGDSpEmuyigiYq1cB6x4Ht67GdIPQ2Q9GLIU2v0V7GX636iIuFCJP/3kcDhYuHAhs2bNYsmSJTRp0oTHHnuMgQMHEh4eDsC8efO49957GTFihMsDi4iUq9Rk+Ox+OLTJHDcbBD1fhqDK1uYSkUJKXGpiY2NxOp3cfvvtbNy4keuuu67QPp06daJKlSouiCciYqEfPoIv/gbZpyEoAhJfh8b9rU4lIhdR4lIzadIkbr31VoKDgy+6T5UqVUhOTi5TMBERy2Smm2Vm28fmuHYb6P8OVKltbS4RuaQSXwzu06cPGRkZhbanpqaSnp5e6iCrV68mMTGRuLg4bDYb8+fPL/C6YRg8++yzxMbGEhISQteuXbUgWURc7+AmmNreLDQ2P+j0NAz+XIVGxAuUuNTcdtttfPjhh4W2f/zxx9x2222lDnL27FmaNm3K22+/XeTrr7zyCv/617+YOnUqGzZsoFKlSvTo0YPMzMxSf08RkXzOXFg1AWb2gFP7zRJzz1dw4xPgp5uvi3iDEv+kbtiwgYkTJxba3rFjR55++ulSB+nVqxe9evUq8jXDMHj99dd55pln6Nu3LwD//ve/qVGjBvPnzy9TmRIR4dRBmPcg7P/OHDf+M9w8EYIjrM0lIiVS4lKTlZVFTk5Ooe0Oh4Nz5865JNSFkpOTOXr0KF27ds3fFhERQevWrVm3bt1FS01WVhZZWVn547zLYw6Hw+NuFJiXx9NyuYvm69u8ab62nQvx+3IEtsw0jMBK5PacgNH4VvNBlMXM703zdQXN17d54nyLm8VmGIZRkjfu1KkTjRs35s033yywffjw4fz444+sWbOmJG9XdCibjXnz5tGvXz8A1q5dS7t27Thy5Aixsf+7ydX//d//YbPZ+Oijj4p8nzFjxjB27NhC2+fMmUNoqG5lLlKR+eVmce3h2dQ5uQqA30PrsbnuMDKCalicTEQulJGRwcCBA0lLS8u/fUxRSnym5vnnn6dr16788MMPdOnSBYDly5ezadMmlixZUvrEbjBq1ChGjhyZP05PT6dWrVp07979kr8pVnA4HCxdupRu3boREBBgdRy303x9m8fPNyUJ//kPYkvdi4ENZ9vHqNzhCTr6lS6rx8/XxTRf3+aJ8y3uB5FKXGratWvHunXrmDBhAh9//DEhISE0adKEGTNmcOWVV5Y4aHHExMQAcOzYsQJnao4dO1bkfXLyBAUFERQUVGh7QECAxxyoC3lyNnfQfH2bx83X6YR1b8HyceB0QFgctv7v4Bd/A34ueHuPm6+bab6+zZPmW9wcpVrSf9111/HBBx+U5ktLJT4+npiYGJYvX55fYtLT09mwYQPDhg0rtxwi4sVOHzUXA//6jTlulAiJ/4LQSEtjiYjrlKrUOJ1O9uzZw/Hjx3E6nQVe69ChQ6mCnDlzhj179uSPk5OTSUpKIjIyktq1a/PYY4/x/PPPc+WVVxIfH88///lP4uLi8tfdiIhc1K6vYMFwyDgJ/iHQazw0H2wuBhYRn1HiUrN+/XoGDhzI/v37uXCNsc1mIzc3t1RBNm/eTKdOnfLHeWthBg8ezHvvvccTTzzB2bNneeCBBzh16hTt27dn8eLFl7yzsYhUcI5zsOQZ2PSuOY65FgbMhKirrM0lIm5R4lIzdOhQWrZsyRdffEFsbCw2F/1Lp2PHjoVK0vlsNhvjxo1j3LhxLvl+IuLjjv0Enw6B33aa4zYPQ5dnwb/wOjsR8Q0lLjW7d+/m008/pUGDBu7IIyJSNoYBG9+BJf+E3CyoFA23TIEGXS//tSLi1Upcalq3bs2ePXtUakTE85z5DRY8BLv/uL3Eld2h72SoHGVtLhEpFyUuNY888gh/+9vfOHr0KNdee22hj1k1adLEZeFERIptzzKYNwzOHge/IOj+HLR6QIuBRSqQEpeaAQMGAHDvvffmb7PZbBiGUaaFwiIipZKTZd53Zt1b5jiqEfx5BtS4xtpcIlLuSlxqkpOT3ZFDRKTkfvsF5t4LR7eZ4+vvg+7PQ0CItblExBIlLjV16tRxRw4RkeIzDNj6Pnz1D8g5ByGR0PdtuPomq5OJiIXspfmi//znP7Rr1464uDj2798PwOuvv86CBQtcGk5EpJCMVPj4Tlj0V7PQ1OsIw9aq0IhIyUvNlClTGDlyJDfddBOnTp3KX0NTpUoVXn/9dVfnExH5n+Q1MKUd7FwE9gDo9hwMmgfhsZf/WhHxeSUuNW+++SbTp0/n6aefxs/vf4+Aa9myJdu2bXNpOBERAHId5mLg9xPh9BGIrA/3LYV2j4K9VCecRcQHlWqhcLNmzQptDwoK4uzZsy4JJSKSL/VXmHsfHN5ijpvdCT3HQ1Bla3OJiMcpcamJj48nKSmp0ILhxYsX06hRI5cFE5EKzjDgx4/gi79B9hkIjoDEN+CaW6xOJiIeqsSlZuTIkQwfPpzMzEwMw2Djxo3897//5aWXXuLdd991R0YRqWgy08wys+0Tc1y7LfR/B6rUsjaXiHi0Epea++67j5CQEJ555hkyMjIYOHAgcXFxvPHGG9x2223uyCgiFcnBjTB3CJw6ADY/6DgKbhgJdr/Lf62IVGglLjUAd9xxB3fccQcZGRmcOXOG6OhoV+cSkYrGmQtrXoNvxoORC1Vqw4AZUKuV1clExEuUqtTkCQ0NJTQ01FVZRKSiOnUQPnsADqw1x9feCr1fM9fRiIgUU7FKTfPmzVm+fDlVq1alWbNm2C7xgLitW7e6LJyIVAA/zTNvpJeZBoFhZplp+herU4mIFypWqenbty9BQUEA9OvXz515RKSiyDoDi5+E72eb4ytawoDpEFnP2lwi4rWKVWpGjx5d5H+LiJTKke/h0yGQuhewwQ1/g47/AL8Aq5OJiBcr8ZqaTZs24XQ6ad26dYHtGzZswM/Pj5YtW7osnIj4GKcT1r0Jy58DpwPCrzA/ql23vdXJRMQHlPj+4sOHD+fgwYOFth8+fJjhw4e7JJSI+KD0FJh9Cyx91iw0jfrA0G/LrdDkOg3W7T3JgqTDrNt7klynUS7fV0TKT4nP1OzYsYPmzZsX2t6sWTN27NjhklAi4mN+/hIWDIdzqRAQaj7moPldcIkPHbjS4u0pjF20g5S0zPxtsRHBjE5MoGdjPQxTxFeU+ExNUFAQx44dK7Q9JSUFf/8yfUJcRHyN45x5Z+APbzcLTUwTeGAVtBhcroVm2OytBQoNwNG0TIbN3sri7SnlkkNE3K/EpaZ79+6MGjWKtLS0/G2nTp3iqaeeolu3bi4NJyLeK/zcAfxndoVNfzw+pc3DcN8yiLqq3DLkOg3GLtpBURea8raNXbRDl6JEfESJT628+uqrdOjQgTp16uQ/rTspKYkaNWrwn//8x+UBRcTLGAb2TdPpsGssNsMBlWtAvynQoEu5R9mYnFroDM35DCAlLZONyam0qV+t/IKJiFuUuNRcccUV/Pjjj3zwwQf88MMPhISEcM8993D77bcTEKCPY4pUaGd+gwUP4bd7CQDOBt2x3zIFKlW3JM7x0xcvNKXZT0Q8W6kWwVSqVIkHHnjA1VlExJvtWQbzhsHZ4xh+QWyL/QuN/u817IGBlkWKDgt26X4i4tmKVWoWLlxIr169CAgIYOHChZfct0+fPi4JJiJeIicLlo+DdW+Z4+gEcvpOJXnzPhqV02Lgi2kVH0lsRDBH0zKLXFdjA2IigmkVH1ne0UTEDYpVavr168fRo0eJjo6+5GMSbDYbubm5rsomIp7ut10wdwgc3WaOWz0A3cZh/q9ln4XBTH52G6MTExg2eys2KFBs8urW6MQE/OzWli8RcY1iffrJ6XQSHR2d/98X+6VCI1JBGAZsngXTbjQLTWg1uP0juGkCBIRYna6Ano1jmTKoOTERBS8xxUQEM2VQc92nRsSHFOtMTWRkJL/88gvVq1fn3nvv5Y033iAsLMzd2UTEE2WkwsJH4OfPzXG9TnDLVAiLsTbXJfRsHEu3hBg2Jqdy/HQm0WHmJSedoRHxLcU6U5OdnU16ejoA77//PpmZ+qSASIWUvBqmtDMLjT0Auj8Pgz7z6EKTx89uo039avS97gra1K+mQiPig4p1pqZNmzb069ePFi1aYBgGjz76KCEhRZ9injlzpksDiogHyHXAyhfh20mAAdUawIAZEHed1clERPIVq9TMnj2bSZMmsXfvXgDS0tJ0tkakoji5F+beB0e2muPmd5nPbgqsZG0uEZELFKvU1KhRg/HjxwMQHx/Pf/7zH6pV0903RXyaYcAP/4Uv/w7ZZyC4CvT5FyT0tTqZiEiRSrxQuFOnTgRaeDMtESkH507BFyNh+1xzXKc99J8GETUtjSUicilaKCwiBR3YAFNvMAuNzQ86/xMGL1ShERGPp4XCImLKzYE1r8Gq8WA4oWpdczFwzZZWJxMRKZYSLxS22WxaKCzia04dgLn3w8H15rjJbeaN9ILDrc0lIlICWigsUtFtnwuLRkBWGgSGwc0Tocn/WZ1KRKTESvyU7uTkZHfkEJHylnUavnoSkj4wxzWvh/7TITLe2lwiIqVUrIXCADfddBNpaWn54/Hjx3Pq1Kn88cmTJ0lISHBpOBFxk8NbYFoHs9DY7NDhCbjnKxUaEfFqxS41X3/9NVlZWfnjF198kdTU1PxxTk4Ou3btcm06EXEtpxO+fR1mdIfUXyG8Jgz+HDo/DX4BVqcTESmTYl9+MgzjkmMR8XDpR2Deg+bzm8C8iV7iGxBS1dpcIiIuUuI1NSLihXZ+DgsfhnO/Q0Ao9HoFmg0Cmx7qKCK+o9ilxmazYbvgf4AXjkXEw2RnwJKnYfMf94+KbWree6b6ldbmEhFxgxJdfrr77rsJCgoCIDMzk6FDh1KpkvlQu/PX24iIBzi6DT4dAif+WOvW9lHz7sD+esyJiPimYpeawYMHFxgPGjSo0D533XVX2ROJSNkYBmyYCkufhdxsqBwDt0yB+p2tTiYi4lbFLjWzZs1yZw4RcYUzv8H8YbBnqTm+qhf0fQsqVbc2l4hIOdBCYRFfsXsZzB8KZ38D/2Do/jxcf58WA4tIhaFSI+LtcrJg2RhYP9kcRyeYi4Fr6GaYIlKxqNSIeLPfdpmLgY9tM8etHoRuYyEgxNpcIiIWUKkR8UaGAVtmweKnIOcchFaDvpOhYU+rk4mIWKbYj0mw2pgxY/LvlZP36+qrr7Y6lkj5y0iFjwbB5yPMQlO/Mwxbq0IjIhWeV52pueaaa1i2bFn+2N/fq+KLlN2vq8xHHZxOAXsAdB0Df3oI7F7z7xMREbfxqlbg7+9PTExMsffPysoqcFPA9PR0ABwOBw6Hw+X5yiIvj6flchfNt4RyHdhXj8e+9l/YMDCqNSCn3zsQ0wRyc81fHkTH17dpvr7NE+db3Cw2w0ueTDlmzBgmTJhAREQEwcHBtGnThpdeeonatWtf8mvGjh1baPucOXMIDQ11Z1wRl6mUdYwW+6ZQNeNXAPZV68j2K+4g1y/I4mQiIuUjIyODgQMHkpaWRnh4+EX385pS89VXX3HmzBkaNmxISkoKY8eO5fDhw2zfvp2wsLAiv6aoMzW1atXixIkTl/xNsYLD4WDp0qV069aNgIAAq+O4neZbDIaBbdtH+H39JLbssxjBVcjtPQnj6kSX58t1Gmze/zvHT2cRHRZEyzpV8bOX/v42Or6+TfP1bZ443/T0dKpXr37ZUuM1l5969eqV/99NmjShdevW1KlTh48//pghQ4YU+TVBQUH5z6o6X0BAgMccqAt5cjZ30Hwv4twp+GIkbJ9rjuu0x9Z/Gv4RNV2eafH2FMYu2kFKWmb+ttiIYEYnJtCzcWyZ3lvH17dpvr7Nk+Zb3Bxeu7qwSpUqXHXVVezZs8fqKCKudWA9TL3BLDQ2P/MhlIMXgpsKzbDZWwsUGoCjaZkMm72VxdtTXP49RUTcxWtLzZkzZ9i7dy+xsWX7l6SIx8jNgZUvwaxekHYAqtaFIUugw+Ng93P9t3MajF20g6KuP+dtG7toB7lOr7hCLSLiPaXm8ccfZ9WqVezbt4+1a9dyyy234Ofnx+233251NJGy+30/vHcTrBoPhhOa3AYProGaLd32LTcmpxY6Q3M+A0hJy2RjcqrbMoiIuJLXrKk5dOgQt99+OydPniQqKor27duzfv16oqKirI4mUjbbPjVvpJeVDoFhcPNEaPJ/bv+2x09fvNCUZj8REat5Tan58MMPrY4g4lpZp+GrJyHpA3Nc83oY8K552akcRIcFu3Q/ERGreU2pEfEph7bA3CHwezLY7HDD43Djk+BXfj+SreIjiY0I5mhaZpHramxATEQwreIjyy2TiEhZeM2aGhGf4MyFNRNhZnez0ITXhMGfQ+eny7XQAPjZbYxOTADMAnO+vPHoxIQy3a9GRKQ8qdSIlJPg7FT85gyA5WPBmQMJfWHYt1C3nWWZejaOZcqg5sREFLzEFBMRzJRBzct8nxoRkfKky08i5cD28xd0+vlp7LlnISAUer0CzQaBzfqzID0bx9ItIYaNyakcP51JdJh5yUlnaETE26jUiLhTdgZ8/RT+W2YB4Ixpiv3PM6F6A4uDFeRnt9GmfjWrY4iIlIlKjYi7pPxoLgY+8QsAu6Nvou7d72IPrmRxMBER36RSI+JqTidsmArLRkNuNlSOISfxLXb8nEFdv0Cr04mI+CyVGhFXOnMc5g+DPcvM8VW9oO9bGIER8POX1mYTEfFxKjUirvLLEljwEJz9DfyDoccL0HKIuRjY4bA6nYiIz1OpESkrRyYsGwMbppjj6GvgzzMgupGlsUREKhqVGpGyOP6zuRj42HZz3OpB6DYOAvRoARGR8qZSI1IahgGbZ8LXT0FOJoRWh36T4aoeVicTEamwVGpESursSVj4COz6whzX7wz9pkJYDWtziYhUcCo1IiXx6zfw2YNw5ijYA6DbWGg9DOx64oiIiNVUakSKIycbVr4A370BGFD9KhjwLsQ2tTqZiIj8QaVG5HJO7jUXAx/53hy3uBt6vAiBujOwiIgnUakRuRjDgKQ58OXfwXEWgqtAnzchoU+J3ibXabAhOZUtJ2xUS06lTYNoPSxSRMQNVGpEinLuFHz+GPw0zxzXvQFumQYRV5TobRZvT2Hsoh2kpGUCfvx792ZiI4IZnZhAz8axrk4tIlKhaXWjyIX2r4Op7c1CY/eHLs/CXQtKVWiGzd76R6H5n6NpmQybvZXF21NcmVpEpMJTqRHJk5sDK1+E926CtINQNR7uXQI3/A3sfiV7K6fB2EU7MIp4LW/b2EU7yHUWtYeIiJSGLj+JAPy+Hz67Hw5uMMdNb4ebJkBQWKnebmNyaqEzNOczgJS0TDYmp9KmfrVSfQ8RESlIpUZk26fw+QjISoegcOg9EZrcWqa3PH764oWmNPuJiMjlqdRIxZV12vxk0w//Nce1WkP/d6Bq3TK/dXRY8Z79VNz9RETk8lRqpGI6tMW898zvyWCzQ4cnoMPfwc81PxKt4iOJjQjmaFpmketqbEBMRDCt4iNd8v1EREQLhaWicebCmtdgZnez0ETUgru/hE6jXFZoAPzsNkYnJgBmgTlf3nh0YoLuVyMi4kIqNVJxpB2Gf/eF5ePAmQPX3AJDv4U6bdzy7Xo2jmXKoObERBS8xBQTEcyUQc11nxoRERfT5SepGHYsNJ+snXkKAirBTa/AdXeAzb1nSno2jqVbQgzr9hxnyZoNdL+hte4oLCLiJio14tuyz8LXT8GW98xxXDMYMAOq1S+3CH52G63jIzm506B1fKQKjYiIm6jUiO9K+QHm3gcnfgFs0O6v0Olp8A+0OpmIiLiBSo34HqcTNkyBZWMgNxsqx0D/aVCvo9XJRETEjVRqxLecPgbzh8He5ea4YW/zydqVdNdeERFfp1IjvuOXr2H+Q5BxAvyDoceL0PJety8GFhERz6BSI97PkQlLn4WN08xxjcbmYuDoqy2Nles02JicyvHTmVQL9UfPrhQRcS+VGvFux3fCp0Pg+E/muPUw6DoGAqx9/MDi7SmMXbSjwEMtqwT6EVD3GDdfV9PCZCIivkulRryTYcCmd2HJM5CTCZWioO9kuKq71clYvD2FYbO3Fno8wqlseOTDH/D399ON90RE3EB3FBbvc/YkfDgQvnzcLDQNusKwtR5RaHKdBmMX7SjyeU95D0gYu2gHuboWJSLicio14l32roQpbWHXl+AXCD1egoGfQOVoq5MBsDE5tcAlpwsZQEpaJhuTU8svlIhIBaHLT+IdcrJhxXOw9l/muPpV5mLg2CbW5rrA8dMXLzSl2U9ERIpPpUY834k9MHcIpCSZ4xb3mB/XDgy1NFZRosOKt0C5uPuJiEjxqdSI5zIM+H42fPUEODIgpCr0eQsa3Wx1sotqFR9JbEQwR9Myi1xXY8N8Sner+MjyjiYi4vO0pkY807nf4ZO7YeHDZqGpe4O5GLicCk2u02Dd3pMsSDrMur0ni72w189uY3RiApC3LPh85nuMTkzQQy1FRNxAZ2rE8+xfC3Pvh/RDYPeHzs9A20fB7lcu376oe8zERgQzOjGhWB/F7tk4limDmhdxnxp4vn9TfZxbRMRNVGrEc+TmwKqXYc2rYDghsh4MeBeuaFFuES52j5mjaZkMm72VKYOaF7vYdEuIKXBH4d92rKfHNTXcE1xERFRqxEP8vs88O3Noozm+7g7o9TIEhZVbhEvdY8bAvJw0dtEOuiXEFOvykZ/dRpv65oM0HQ4HX+50aVwREbmASk0Znf98n+gwcwGo1kuU0I+fwBcjISsdgsLh5klw7Z/LPUZJ7jGTV1ZERMRzqNSUQVnXXlR0/rnn8Fv4EGz72NxQ60/Q/x2oWseSPLrHjIiId9Onn0opb+3Fhf+yz1t7sXh7ikXJvIPt8BY6/vxP7Ns+BpsdOo6Cu7+wrNCA7jEjIuLtVGpK4XJrL0DP97koZy6sfhW/92+iUvZxjIhacM9X0PEf4GfticO8e8xc7OKhDfNMnO4xIyLimVRqSkHP9ymltEPwfh9Y8Rw2I5dDVf5Ezn3fQO0/WZ0MuPQ9ZvLGuseMiIjnUqkpBa29KIUdC2BKO9j/LQRWJifxbbbUHQbBEVYnKyDvHjMxEQUvMcVEBBf749wiImINLRQuBa29KIHss7B4FGx93xzHNYMBMzDCa8OhL63NdhEX3mNGn2oTEfEOXnem5u2336Zu3boEBwfTunVrNm7cWO4ZtPaimFJ+gGk3/lFobNB+BNy7BKrVtzrZZeXdY6bvdVfQpn41FRoRES/gVaXmo48+YuTIkYwePZqtW7fStGlTevTowfHjx8s1h9ZeXIbTCWvfguld4ORuCIuDwQuh6xjwD7Q6nYiI+CivKjUTJ07k/vvv55577iEhIYGpU6cSGhrKzJkzyz2L1l5cxOlj8MEAWPI0OB1w9c0w7DuI72B1MhER8XFes6YmOzubLVu2MGrUqPxtdrudrl27sm7duiK/Jisri6ysrPxxeno6YN6y3uFwlDlTl4bV6XjlDWze/zvHT2cRHRZEyzpV8bPbSvz+efu7IpdVbLuX4Pf5o9gyTmD4h5Db7XmMZneBzQYXzMsX5lsSmq9v03x9m+ZrveJmsRmG4RU3Uzly5AhXXHEFa9eupU2bNvnbn3jiCVatWsWGDRsKfc2YMWMYO3Zsoe1z5swhNDTUrXkrErszm2sOf0i9E8sASAupzea6wzgTfIXFyURExBdkZGQwcOBA0tLSCA8Pv+h+XnOmpjRGjRrFyJEj88fp6enUqlWL7t27X/I3xQoOh4OlS5fSrVs3AgICrI5TfMd34j//AWwnzKc15rYaSminf9LBP+iSX+a18y0lzde3ab6+TfO1Xt6VlsvxmlJTvXp1/Pz8OHbsWIHtx44dIyYmpsivCQoKIiio8F+uAQEBHnOgLuTJ2QowDNj0Lix5BnIyoVIU9JuK35Vd8SvB23jNfF1E8/Vtmq9v03ytU9wcXrNQODAwkBYtWrB8+fL8bU6nk+XLlxe4HCXl4OwJ+O9t8OXjZqFp0A2GrYUru1qdTEREKjCvOVMDMHLkSAYPHkzLli1p1aoVr7/+OmfPnuWee+6xOlrFsXcFzBsKZ46BXyB0ew5aP2guBhYREbGQV5Wav/zlL/z22288++yzHD16lOuuu47FixdTo0YNq6P5vpxsWDEO1r5pjqOuhgHvQsy11uYSERH5g1eVGoCHH36Yhx9+2OoYFcuJ3TB3iHmHYICWQ6D78xCoT5CJiIjn8LpSI+XIMOD7/8BXT4IjA0KqQt+34ereVicTEREpRKVGinbud1j0V/Pp2mDeEfiWaRAeZ20uERGRi1CpkcL2fQefPQDph8DuD53/CW0fBbvXfFhOREQqIJUa+Z9cB6x6Gda8BoYTIuuZi4GvaGF1MhERkctSqRFTajJ8dj8c2mSOrxsEvV6GoMrW5hIRESkmlRqBHz+Gz0dC9mkIioDESdB4gNWpRERESkSlpiLLTDfvCvzjR+a41p9gwHSoUtvaXCIiIqWgUlNRHdxk3nvm1H6w2eHGf8ANfwM//ZEQERHvpL/BKhpnLqyZCN+8BEYuRNQ2z87U/pPVyURERMpEpaYiSTtkflR7/3fmuPEA6D0RQqpYGktERMQVVGoqip/mw6JHITMNAivDTa9C09v0IEoREfEZKjW+Lvus+ZiD7/9jjuOam/eeqVbf2lwiIiIuplLjy44kmYuBT+4BbNB+BHR6CvwCrE4mIiLicio1vsjphHVvwfJx4HRAWBz0fwfib7A6mYiIiNuo1Pia00dh3lD4daU5vvpm6PMmhEZam0tERMTNVGp8ya6vYMFwyDgJ/iHQ8yVocbcWA4uISIWgUuMLHOdgyT9h03RzHHMtDJgBUQ2tzSUiIlKOVGq83bGf4NMh8NtOc9zmYejyLPgHWZtLRESknKnUeCvDgI3TYckzkJsFlaLhlinQoKvVyURERCyhUuONzp4w1878stgcX9kd+k6GylHW5hIREbGQSo232bMc5g+DM8fALwi6PwetHtBiYBERqfBUarxFTpZ535l1b5njqKvNxcAxja3NJSIi4iFUarzBid3w6b1w9EdzfP190P15CAixNpeIiIgHUanxZIYBW/8Ni/8BjgwIiYS+b8PVN1mdTERExOOo1HiqjFRY9FfYudAcx98It0yD8Fhrc4mIiHgolRpPtO9b+OwBSD8M9gDo8k9o8wjY7VYnExER8VgqNR7EZuRg/+ZF+G4SYEBkffjzDIhrZnU0ERERj6dS4yl+30f7X17AL2OvOW52J/QcD0GVrc0lIiLiJVRqPMEPH+H/xUgis89gBEdgS3wDrrnF6lQiIiJeRaXGSplp8MXjsO1jbMCJSg2JuOcjAqrHW51MRETE66jUWOXgRph7H5zaDzY/cjs8wXdpDbkpoqbVyURERLySPk5T3py5sGoCzOxpFpoqteHexTjb/w1sOhwiIiKlpTM15enUQfOj2gfWmuNrb4Xer0FwBDgc1mYTERHxcio15eWneebN9DLTILAy9J4ITf9idSoRERGfoVLjbllnYPGT8P1sc3xFCxjwLkTWszaXiIiIj1Gpcacj35uLgU/uAWxww9+g4z/AL8DqZCIiIj5HpcYdnE5Y9xYsHwdOB4RfAf3fgbrtrU4mIiLis1RqXC09BeYPhV+/MceNEiHxXxAaaWksERERX6dS40o/fwkLhsO5VAgINR9z0PwusNmsTiYiIuLzVGpcwXEOljwDm941xzHXwoCZEHWVtblEREQqEJWasspIhVk3wW87zXGbh6HLs+AfZG0uERGRCkalpqxCqppnZDJOwi1ToUEXqxOJiIhUSCo1ZWWzQeIbkJsDlaOsTiMiIlJhqdS4QkhVqxOIiIhUeHqCooiIiPgElRoRERHxCSo1IiIi4hNUakRERMQnqNSIiIiIT1CpEREREZ+gUiMiIiI+QaVGREREfILXlJq6detis9kK/Bo/frzVsURERMRDeNUdhceNG8f999+fPw4LC7MwjYiIiHgSryo1YWFhxMTEFHv/rKwssrKy8sfp6ekAOBwOHA6Hy/OVRV4eT8vlLpqvb9N8fZvm69s8cb7FzWIzDMNwcxaXqFu3LpmZmTgcDmrXrs3AgQMZMWIE/v4X72Vjxoxh7NixhbbPmTOH0NBQd8YVERERF8nIyGDgwIGkpaURHh5+0f28ptRMnDiR5s2bExkZydq1axk1ahT33HMPEydOvOjXFHWmplatWpw4ceKSvylWcDgcLF26lG7duhEQEGB1HLfTfH2b5uvbNF/f5onzTU9Pp3r16pctNZZefvrHP/7Byy+/fMl9du7cydVXX83IkSPztzVp0oTAwEAefPBBXnrpJYKCgor82qCgoAKv5fW3c+fOecyByuNwOMjIyODcuXPk5ORYHcftNF/fpvn6Ns3Xt3nifM+dOwf87+/xi7H0TM1vv/3GyZMnL7lPvXr1CAwMLLT9p59+onHjxvz88880bNiwWN/v0KFD1KpVq1RZRURExFoHDx6kZs2aF33d0jM1UVFRREVFleprk5KSsNvtREdHF/tr4uLiOHjwIGFhYdhstlJ9X3fJuzR28OBBj7s05g6ar2/TfH2b5uvbPHG+hmFw+vRp4uLiLrmfV3z6ad26dWzYsIFOnToRFhbGunXrGDFiBIMGDaJq1arFfh+73X7JhucJwsPDPeYPUXnQfH2b5uvbNF/f5mnzjYiIuOw+XlFqgoKC+PDDDxkzZgxZWVnEx8czYsSIAutsREREpGLzilLTvHlz1q9fb3UMERER8WBe85gEXxcUFMTo0aMv+kkuX6P5+jbN17dpvr7Nm+frNfepEREREbkUnakRERERn6BSIyIiIj5BpUZERER8gkqNiIiI+ASVGhdavXo1iYmJxMXFYbPZmD9/foHXbTZbkb8mTJgAwL59+xgyZAjx8fGEhIRQv359Ro8eTXZ2dv577Nu3r8j3sOIj72WdL5hPX7/w9fHjxxd4nx9//JEbbriB4OBgatWqxSuvvFIe0yukrPP95ptvLrrPpk2bAO86vmfOnOHhhx+mZs2ahISEkJCQwNSpUwvsk5mZyfDhw6lWrRqVK1dmwIABHDt2rMA+Bw4coHfv3oSGhhIdHc3f//53S543U9b5pqam8sgjj9CwYUNCQkKoXbs2jz76KGlpaQXep6jj++GHH5bHFAtwxfHt2LFjobkMHTq0wD6+cnwv9rNps9n45JNP8vfzluN77Ngx7r77buLi4ggNDaVnz57s3r27wD7e9PObR6XGhc6ePUvTpk15++23i3w9JSWlwK+ZM2dis9kYMGAAAD///DNOp5Np06bx008/MWnSJKZOncpTTz1V6L2WLVtW4L1atGjh1rkVpazzzTNu3LgC+z3yyCP5r6Wnp9O9e3fq1KnDli1bmDBhAmPGjOGdd95x69yKUtb5tm3bttA+9913H/Hx8bRs2bLAe3nD8R05ciSLFy9m9uzZ7Ny5k8cee4yHH36YhQsX5u8zYsQIFi1axCeffMKqVas4cuQI/fv3z389NzeX3r17k52dzdq1a3n//fd57733ePbZZ90+vwuVdb5HjhzhyJEjvPrqq2zfvp333nuPxYsXM2TIkELvNWvWrALHt1+/fu6cWpFccXwB7r///gJzOf8fHb50fGvVqlXo53fs2LFUrlyZXr16FXgvTz++hmHQr18/fv31VxYsWMD3339PnTp16Nq1K2fPns3fz5t+fvMZ4haAMW/evEvu07dvX6Nz586X3OeVV14x4uPj88fJyckGYHz//fcuSOk6pZ1vnTp1jEmTJl30ayZPnmxUrVrVyMrKyt/25JNPGg0bNixL3DJzxfHNzs42oqKijHHjxuVv86bje8011xTIbhiG0bx5c+Ppp582DMMwTp06ZQQEBBiffPJJ/us7d+40AGPdunWGYRjGl19+adjtduPo0aP5+0yZMsUIDw8vcMzLW2nmW5SPP/7YCAwMNBwOxyXf22qlne+NN95o/PWvf73o+/r68b3uuuuMe++997LvbbULM+3atcsAjO3bt+dvy83NNaKioozp06cbhuG9P786U2ORY8eO8cUXXxT5r7jzpaWlERkZWWh7nz59iI6Opn379oX+5eSJLjXf8ePHU61aNZo1a8aECRMKnLpct24dHTp0KPCk9h49erBr1y5+//33csleGsU5vgsXLuTkyZPcc889hV7zhuPbtm1bFi5cyOHDhzEMg5UrV/LLL7/QvXt3ALZs2YLD4aBr1675X3P11VdTu3Zt1q1bB5jH99prr6VGjRr5+/To0YP09HR++umn8p3QZVxuvkVJS0sjPDwcf/+CN28fPnw41atXp1WrVsycORPDA28XVtz5fvDBB1SvXp3GjRszatQoMjIy8l/z5eO7ZcsWkpKSivwZ9/Tjm5WVBUBwcHD+NrvdTlBQEN9++y3gvT+/XvGYBF/0/vvvExYWVuBU3oX27NnDm2++yauvvpq/rXLlyrz22mu0a9cOu93O3Llz6devH/Pnz6dPnz7lEb1ULjbfRx99lObNmxMZGcnatWsZNWoUKSkpTJw4EYCjR48SHx9f4GvyfoCOHj1aogealqfiHN8ZM2bQo0ePAg9Z9abj++abb/LAAw9Qs2ZN/P39sdvtTJ8+nQ4dOgDm8QkMDKRKlSoFvq5GjRocPXo0f5/z/4eY93rea57kcvO90IkTJ3juued44IEHCmwfN24cnTt3JjQ0lCVLlvDQQw9x5swZHn300fKYRrEVZ74DBw6kTp06xMXF8eOPP/Lkk0+ya9cuPvvsM8C3j++MGTNo1KgRbdu2LbDdG45vXjkZNWoU06ZNo1KlSkyaNIlDhw6RkpICeO/Pr0qNRWbOnMkdd9xRoCmf7/Dhw/Ts2ZNbb72V+++/P3979erVCzzI8/rrr+fIkSNMmDDB4/7SO9/F5nv+XJo0aUJgYCAPPvggL730klfeojvP5Y7voUOH+Prrr/n4448LbPem4/vmm2+yfv16Fi5cSJ06dVi9ejXDhw8nLi6uwL/ufEVJ5puenk7v3r1JSEhgzJgxBV775z//mf/fzZo14+zZs0yYMMGj/tKD4s33/MJ27bXXEhsbS5cuXdi7dy/169e3KnqplOT4njt3jjlz5hQ4lnm84fgGBATw2WefMWTIECIjI/Hz86Nr16706tXL484qlZRKjQXWrFnDrl27+Oijj4p8/ciRI3Tq1Im2bdsWa0Fs69atWbp0qatjuszl5nu+1q1bk5OTw759+2jYsCExMTGFVtvnjWNiYtySt6yKM99Zs2ZRrVq1YhUVTzy+586d46mnnmLevHn07t0bMEtpUlISr776Kl27diUmJobs7GxOnTpV4F97x44dyz92MTExbNy4scB7e+LxLc5885w+fZqePXsSFhbGvHnzCAgIuOR7t27dmueee46srCyPKfIlme/5WrduDZhnmevXr++Txxfg008/JSMjg7vuuuuy7+2JxxegRYsWJCUlkZaWRnZ2NlFRUbRu3Tr/Qwve+vOrNTUWmDFjBi1atKBp06aFXjt8+DAdO3akRYsWzJo1C7v98ocoKSmJ2NhYd0R1iUvN90JJSUnY7Xaio6MBaNOmDatXr8bhcOTvs3TpUho2bOixl54uN1/DMJg1axZ33XXXZf/CA888vg6HA4fDUejPp5+fH06nEzD/pxkQEMDy5cvzX9+1axcHDhygTZs2gHl8t23bxvHjx/P3Wbp0KeHh4SQkJJTDTIqnOPOF/31aLzAwkIULF170TN35kpKSqFq1qkf9hVfc+V4oKSkJIP/Pq68d3zwzZsygT58+REVFXfa9PfH4ni8iIoKoqCh2797N5s2b6du3L+DFP7+WLE/2UadPnza+//574/vvvzcAY+LEicb3339v7N+/P3+ftLQ0IzQ01JgyZUqhrz906JDRoEEDo0uXLsahQ4eMlJSU/F953nvvPWPOnDnGzp07jZ07dxovvPCCYbfbjZkzZ5bLHM9X1vmuXbvWmDRpkpGUlGTs3bvXmD17thEVFWXcdddd+fucOnXKqFGjhnHnnXca27dvNz788EMjNDTUmDZtWrnM8XxlnW+eZcuWGYCxc+fOQq950/G98cYbjWuuucZYuXKl8euvvxqzZs0ygoODjcmTJ+e/x9ChQ43atWsbK1asMDZv3my0adPGaNOmTf7rOTk5RuPGjY3u3bsbSUlJxuLFi42oqChj1KhRXjfftLQ0o3Xr1sa1115r7Nmzp8DPb05OjmEYhrFw4UJj+vTpxrZt24zdu3cbkydPNkJDQ41nn33W6+a7Z88eY9y4ccbmzZuN5ORkY8GCBUa9evWMDh065H8PXzq+eXbv3m3YbDbjq6++KvQ9vOn4fvzxx8bKlSuNvXv3GvPnzzfq1Klj9O/fv8B7eNPPbx6VGhdauXKlART6NXjw4Px9pk2bZoSEhBinTp0q9PWzZs0q8uvP757vvfee0ahRIyM0NNQIDw83WrVqVeAjd+WprPPdsmWL0bp1ayMiIsIIDg42GjVqZLz44otGZmZmgf1++OEHo3379kZQUJBxxRVXGOPHj3f31IpU1vnmuf322422bdsW+Zo3Hd+UlBTj7rvvNuLi4ozg4GCjYcOGxmuvvWY4nc789zh37pzx0EMPGVWrVjVCQ0ONW265pUBJNwzD2Ldvn9GrVy8jJCTEqF69uvG3v/2twEegy0tZ53uxrweM5ORkwzAM46uvvjKuu+46o3LlykalSpWMpk2bGlOnTjVyc3O9br4HDhwwOnToYERGRhpBQUFGgwYNjL///e9GWlpage/jK8c3z6hRo4xatWoVecy86fi+8cYbRs2aNY2AgACjdu3axjPPPFPoY9je9PObx2YYXr4qSERERAStqREREREfoVIjIiIiPkGlRkRERHyCSo2IiIj4BJUaERER8QkqNSIiIuITVGpERETEJ6jUiIiIiE9QqRERERGfoFIjIh7FMAy6du1Kjx49Cr02efJkqlSpwqFDhyxIJiKeTqVGRDyKzWZj1qxZbNiwgWnTpuVvT05O5oknnuDNN9+kZs2aLv2e5z8FXkS8l0qNiHicWrVq8cYbb/D444+TnJyMYRgMGTKE7t2706xZM3r16kXlypWpUaMGd955JydOnMj/2sWLF9O+fXuqVKlCtWrVuPnmm9m7d2/+6/v27cNms/HRRx9x4403EhwczAcffGDFNEXExfRASxHxWP369SMtLY3+/fvz3HPP8dNPP3HNNddw3333cdddd3Hu3DmefPJJcnJyWLFiBQBz587FZrPRpEkTzpw5w7PPPsu+fftISkrCbrezb98+4uPjqVu3Lq+99hrNmjUjODiY2NhYi2crImWlUiMiHuv48eNcc801pKamMnfuXLZv386aNWv4+uuv8/c5dOgQtWrVYteuXVx11VWF3uPEiRNERUWxbds2GjdunF9qXn/9df7617+W53RExM10+UlEPFZ0dDQPPvggjRo1ol+/fvzwww+sXLmSypUr5/+6+uqrAfIvMe3evZvbb7+devXqER4eTt26dQE4cOBAgfdu2bJluc5FRNzP3+oAIiKX4u/vj7+/+b+qM2fOkJiYyMsvv1xov7zLR4mJidSpU4fp06cTFxeH0+mkcePGZGdnF9i/UqVK7g8vIuVKpUZEvEbz5s2ZO3cudevWzS865zt58iS7du1i+vTp3HDDDQB8++235R1TRCyiy08i4jWGDx9Oamoqt99+O5s2bWLv3r18/fXX3HPPPeTm5lK1alWqVavGO++8w549e1ixYgUjR460OraIlBOVGhHxGnFxcXz33Xfk5ubSvXt3rr32Wh577DGqVKmC3W7Hbrfz4YcfsmXLFho3bsyIESOYMGGC1bFFpJzo008iIiLiE3SmRkRERHyCSo2IiIj4BJUaERER8QkqNSIiIuITVGpERETEJ6jUiIiIiE9QqRERERGfoFIjIiIiPkGlRkRERHyCSo2IiIj4BJUaERER8Qn/D93EOjbAaN8HAAAAAElFTkSuQmCC\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "The predicted efficiency in year 2000 is -240.39 + $0.13769\\times 2000$ = 35.0%"
      ],
      "metadata": {
        "id": "qDclAxWhbuhP"
      }
    }
  ]
}