{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Numerical Differentiation\n", "\n", "## Overview, Objectives, and Key Terms\n", " \n", "From [Lecture 1](L01_GettingStarted.ipynb) through the first half of the course, we used Python to assemble models, solve linear systems, check residuals, and interpret response quantities. We now shift toward numerical calculus: computing derivatives from formulas, black-box functions, or sampled data.\n", "\n", "In this lesson, we use Taylor series to derive **finite-difference** approximations for derivatives, then compare three practical derivative estimates: forward/backward differences, central differences, and the **complex-step** method. We finish by moving from a one-dimensional derivative to the **gradient** of a scalar field, which is the derivative object we need for many balance-law and transport applications.\n", "\n", "We will use the symbol $\\Delta$ in formulas and `Delta` in code for the finite-difference step.\n", "\n", "### Objectives\n", "\n", "By the end of this lesson, you should be able to\n", "\n", "- Derive first- and second-derivative finite-difference approximations using Taylor series.\n", "- Explain the difference between truncation error and floating-point roundoff error.\n", "- Compare forward, backward, central, and complex-step derivative estimates for a callable function.\n", "- Estimate a gradient from sampled two-dimensional data using `np.gradient`.\n", "- Use plots as evidence for accuracy, order of accuracy, and derivative direction.\n", "\n", "### Key Terms\n", "\n", "- finite difference\n", "- stencil\n", "- step size, $\\Delta$\n", "- forward difference\n", "- backward difference\n", "- central difference\n", "- truncation error\n", "- roundoff error\n", "- cancellation\n", "- absolute error\n", "- order of accuracy\n", "- first-order approximation\n", "- second-order approximation\n", "- complex-step derivative approximation\n", "- scalar field\n", "- partial derivative\n", "- gradient\n", "- sampled data\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## If, in the beginning, $\\Delta$ didn't go to 0...\n", "\n", "Recall that the derivative of $f(x)$ is\n", "\n", "$$\n", " \\frac{df}{dx} = \\lim_{\\Delta \\to 0} \\frac{f(x+\\Delta)-f(x)}{\\Delta} \\, .\n", "$$\n", "\n", "Here, $h$ (the ubiquitous \"step\" used in calculus books around the world) is swapped for $\\Delta$ (a better symbol that represents \"small change\"). What if that limit is not taken and, instead, a \"small\" value of $\\Delta$ is used? The result is our first **finite-difference** approximation, the **forward difference**:\n", "\n", "$$\n", " f'(x) \\approx \\frac{f(x+\\Delta)-f(x)}{\\Delta} \\, .\n", "$$\n", "\n", "This is called the *forward* difference approximation because $f'(x)$ depends on information (1) at point $x$ and (2) to the right (or forward) of $x$ at $x+\\Delta$. Intuition suggests that this approximation improves with smaller $\\Delta$. \n", "\n", "An equally valid approximation is the **backward difference**:\n", "$$\n", " f'(x) \\approx \\frac{f(x)-f(x-\\Delta)}{\\Delta} \\, .\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**Exercise**: Use the forward-difference approximation to approximate the derivative of $e^x$ at $x = 1$ for $\\Delta = 1$, $0.1$, $0.01$, and $0.001$. Plot the **absolute error** as a function of $\\Delta$ using a log-log scale.\n", "\n", "*Solution*:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "execution": { "iopub.execute_input": "2026-07-05T18:02:05.576729Z", "iopub.status.busy": "2026-07-05T18:02:05.576647Z", "iopub.status.idle": "2026-07-05T18:02:06.109268Z", "shell.execute_reply": "2026-07-05T18:02:06.108462Z" } }, "outputs": [ { "data": { "image/png": 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8SeSYKJWOxnXr1k2NE1noGWeK+EPJjebll182ylGUwAULFqjjck2iHMmjyOno0aNGPzDTXFumSMTc5cuXVYTe77//riIGf/rpJzz55JNOJy2VuSSfk8jdMJfEt0rmkpyPEBJcfPPNN8olQJQjidaV+5X4TAqiDMkfVvntkEdvKEcCFSQfRUwb8oMkPiCSSkAe5Uctu5cYxUFXrkXyMsmKgvixyI+4KDuOEK1ffkhNlSGZ7NJmulojib+ef/55FW4vryFJGkU5cIQoNbLCIb43ErLev39/FRrviPfff1998bp06aKUk2bNmtn0o3EGUQJldaxp06bqWmTFTEyHoiiJo7e9cQafMhknPjiyyuJonCXi6C4hsOKoLit+otgYfJTEF01MeKJkSy4tMW+KYiJ+TrbCXMX5XJa65fMSp+maNWsqhcrg5O0sYl7t2bOnSgEhn7MocfL56ZlLhJDAYdq0aeoPofxBqlevnvp9k3uTz8EYf9cwhO5bhglKCKGEMUsYsoRPCxKabAhvd2XfXbhyXr1j7PWzdcxauz/LSu84T8nKsi1QZSXfq/3796sUFhIiLN85U2Tfsl1Pm+m+tf7uwpVz6xnjqI9euVhrp6z0yzDQZOWon945ZPjudu7c2Riy/9RTT2nLli3LdlnZ+v22hCtIhBBCCPEoN2/eVD6oUkbLsKIsyXWz6t/oSYJaQRJziPiwiIOr5PchhBBCiHuRSDVxuRC/IwmSEd9U8Zn0Zo5APfiu6uZhJM+O+GuI7VN8MMQnQvx7JCcOIYQQQrKOhO+Ln6n40MrvqyQPthZp64sE7QqSROWII6tEiknyPcm9IM7HhBBCCMk6EiUswSeiHEm0skTQSiSzv+C3CpKUR5DoKom0kWU6a2HhYt+UAqkSAi1RSvLhGJDCnaZh9IbkjIQQQgjJGhJ9LSWoJIxf0phIjjpxZ/En/FZBktpOkjNBEhFaQ3LtSAI6yYuze/duVRFcEk5JjhlXszATQgghxDby2/r222+r/HLiyiKpPSTnnK2M/XZOBG/jtz5IouzIZgspUyEJASWvjCH7tJjQpNSC5PaR1SPTFSNJlmdac8wSybUjmwFDPTDJBiybAcNzmSSSFVo2gzJmaHN23124cl69Y+z1s3XMWrs/y0rvOE/JyrLNdGwgycrwXuTmK5h+/0z3rX0v7bXZ6+NOrL2OO8Y46qNXLtbaKSv9Mgw0WTnqZ3pMNkkwK0WpBcl1JM8lUs30t9LRY+qlY4j8/nkUiGzlUVk5IkRi/eHnyMpPTEyM+jAEKUYqifGWLl2K7t27G/tJkjspbSAZhOXmKsnyJHmewUlbMnlaFgw1IEuFUivM2jKiZUFYmQzFixdXhUGlvAIhxH3I9/vkyZOqPItBSSKEeN+q895776mqBpJIVqou2FvEsEXJK7+izsl5iEy7ifioMlhf7W35kXd7ygFJxhsfH28zUa5fryDZIy4uTmX/tcxCLPtyUzUoMR988IEKPZR/pFJM1ZZyJLzyyisq6s10BUkUIBlvKmC5YYuvk/g9SXkFeRSkcKtpJXZn992FK+fVO8ZeP1vHrLX7s6z0jvOUrCzbAlFWiYmJ6nslK76S6Vu+g6a5VOQ7KNGppu162kz3Bcv+7sLatbhjjKM+euVirZ2y0i/DQJOVo35y7Ouvv1Y1RKWEkPzuzZ8/X1UpcEpWabcQvuZVRB5bltG3eD1sL9gLre+91+2yMliAgnIFyeCALfWsxDnMgJShkNpUEnaYlRTpsokCJgVOuYJESPbCFSRCfIcjR44onyMp+i1lg6TkUcWKFZ06R4GEw2hwbCZyJV+AhhAcKt4FB4t3hRbimTUcvStIAVFqRN5GTEyMcT8pKUkLCwvTVqxYYdZv2LBhWosWLdyaqjwuLs6YFl22hIQEVWrkr7/+Us+lREJqaqoqlyCPruy7a3PlvPbGXLhwQStSpIh25MgRu/0sj/Xo0UObNGmS1THukJU879+/v1agQAH1Ge3cuTNbZKV3nDOystfuqM2X51VWZCXfKynlc/nyZVWKQPYtv4OW7XraTPet9XfX5sq59Yxx1EevXBzJJlhl5co8CgRZJdvpJ225cuVS99nq1aur3wJnZbVu0Yda+pgCmjY6r5Y2uYa28cuJHpeV/G4HbakR8fuRsP41a9aYtcu+FBIljpGlT1mVs1e1XVLFd+rUCeXLlzdrl5U7qb5sy/4s/zCkoKreZU5nkWrxX3zxhcraKo74UmSVEEKI+5g5c6ZKrmyIKP/555+V24mzxOcsj7SKbZBe80EkPv4zLueu6jMfk9+a2G7cuKEqywtSDVii1sSOKZk6y5Ytq8L8pWr4p59+qsxss2bNwmeffYZ9+/ahXLlyLr9uMJjYRK4Gp3RRYuSLIEqP+IFIbin5MkgVeHFyF9tzo0aNzMYPGzZMLWFKzZ0dO3ZY/dK0atVKfT4SaajXrKJXlvJZf/zxx9i7d6/Od5y11yPZC01shHiP9PR05WNkyD3Ypk0bDBo0SL+fkKah5NXfcCFvbaSGZfx2hqanID00AtlFwJvY1q1bZ6wIbLr169fP2GfatGlauXLltMjISK1+/frahg0b3Pb6gWpiO3XqlFa2bFntzz//1LZu3ao9++yzypQh7/euu+5Sy6jSb+nSpVrhwoUznVv65c6dW9u2bZvWvn17bcyYMVZfd/To0VqzZs1smojEFDp48GDtueee0woWLKg1b95cHU9JSVHnrFChghYVFaXVrl1bmzdvnvEcffv2NZsP8vkbzvvuu++ajVuyZInxdVu2bKleb/jw4VqhQoXU68l1yOs5Gjd06FDthRdeUCa9YsWKaaNGjTJ7T3KOCRMmaJUqVVJzsUyZMtprr71m9nmbvkaNGjXMXoMmNprYaGKjic0XTGxxcXFa9+7djffXsWPHajdu3NBvjrx0Rjv5YbsMc9rSp7xmjtRrYvPbKDZZgXC0+DV48GC1+QzJCRnhinLdKTeB5NDb+6mJAEyieUyPWxISCkTkND+vNSJzOX2Jzz//vIoOkjIsQ4cOVRGBEydOVA7uonFLJN+lS5dUpJ6YMS2RFSVZPZNVJek/evRolazTkoYNGyoTneSWshXFJP9SBgwYoExmhlU6Mc+tWLFCJQitUqWKyqgufWSVSuaE5LuSlPayWvjbb78pU59hnJgMZfVPsrnKuL59+6rkZS1btjS+3sCBA9V7M+TckXHyT8nROJGbZIqV7cknn1TX0q5dO3X81VdfVcWQJWqyWbNmKohAQmENmF5b5cqVlSnY8jUIIcSbxMfHq5JcUqZLVtdlpV7u8bpzCp3YiqiVg1D62mloIWHQClX2iWSQAWli8xZZMbHln2LbtJdSvjUSumUk1xLyfVINIam3rPZNLXU3bjy4xLifd2Y9hN66nKnf1eHHnXpvZ8+eVf46X331lfpxX7lyJZ599lmlyMgXQd7366+/rvx6JMeFmDPFlGVK+/btlanz5ZdfVmbQqlWrYuHChUphsKzRI9nNRVEQk6gl4tsk5j1RSAyIrVsUCLkuU7OewaQnSoggyUBl+/PPP3WPc/X1ZJzMhe+//954XJac5b2JmVJC1EWxkvwgovRYovc9kdvQxEZI9nL69GmMGzcO58+fR+7cudX9Xa9vZ4iWimpnY3DH+VUIgYYbkUWxq/wgXMlVCd4iqPMgeRLJEiqb/Jjmy5dPKRKmAhZtWvI6SJ4WmUjyKDqo/FA6QlZSJGWBI501LDzM7DVtlUix/OAN12F4HUu+++47tXIiqzsyVnyEHnvsMeMYUQjlmCh/8j6lzfAacu5du3apfxfiIC3tsnXp0kX5g4mCZPq6hrTz4stk2m64Rln5EYVBjhle/+DBgyoHjjgGWv5g1q1b13gtInNJVGbYtzdO/Neknyi18nqW70fPuNq1a5uNk3xbhi+epJSQVbIHHnjArI+e92R4DVufnaM2wfS4o8/fVVw9r55x1vqIvHLmzKkCLkShbdu2LSIibvsvyNyUVTjTdj1tpvuCZX93Ye1a3DHGUR+9crHWTlnpl2GgyWrz5s1qVfzy5csqIEeCX6pVq+bwHNK+edUitLk8H2HnM1bMU2s9ig0hrdC6QxevykpvgBAVpCwiH5y1D09u5vIjLZuxRMLLp4z7165fR948eYz7CTcSkPe/MbIfP2CX8Xjmc4cixLR9uHVnZMuxhuswXJu1FSRBCgAbjhvGyI+UrHLIKpIcEwVHihCa9pMcU6JAmX55RMGSWjxS3kV+7A39ZaxQuHBhs+sxvUZRMA0/iqY/oKJwyaqMQVayUiVJPg3nMPS1fI+iAJoWKBZy5Mhh7CevZ3odhmtxNE5WCU3HyevLo7RJ0jTDtVh7j5bXZng/ci2iBFgbo7fNgOG4o8/fVVw9r55x1vrIo+wbnEJtfQettetps3zu7ptzVs6tZ4yjPnrlYq2dstIvw0CQ1eLFi9GvXz/1h01cGn755ZdM90J750gLi0Lo9bNAVH6g81RoVR5A6urVXpeV3vNRQcoiTtVii4hGumGFKCLNfD88zbxmVkS08bhVTGtXhed03EdH7auiRYuqRynoK348pmOk8K/4zzzyyCNqrKzYiOnMcB5536K4jBo1yuzcov3Lv3/xTRo5cqTxmJi/SpcurRQbW7XEDJthXxQvUUykzESHDh0yrYpYnsOwbxh37NgxZfrKLKbb40yfi3nQlXGmx0WOoujIPyFDXUBr78nwGpYrJnrrrrEW221Yi4212Ezngbvq1gVTLTa5p7z//vvKpUIQS0Dv3r1VIkiHMkm6DuTIo/aTw/MgqfvnCCtQFshb0udk5Qj6IDlJIIf5yxLq3XffrZQa8S2Sf+ryYy1fkosXLyrHZ8OKiKRLELPZ4cOHkT9/frUCIqtF4mxsULQMyOrR8ePHzfx7xHlezGiWPkwGxLenVq1aaqwpkrF17ty56lGuVa5PnLFlxeXRRx+16oOkZ5yrr2dtnNxIxPwqKREEcXKXdBPSRxzgxfFdTG9iwtT7nsht6INEiPuQ3zIpEWLIhC0rReL7uOa/PIKiHMkqkiHgxR6Fr+9D/eOzsL/kwzhV0HdzDgZ8mL+3CdQw/927d2tt2rTR6tSpo8I5ZZs7d67VMXfffbc2ffp09fyBBx6wmnbBdPv999+NmZDz5s2rbd682W6Yv2Q+t7xmQ9h91apVtYiICJXJW673l19+MZ5j8uTJxvB+0/NOmTLFbFy7du1UughDuL68nuV1yOs5O+7+++9X6QZMw/zfeustdU1yDkmj8MYbb5h93qavIekTTF+DYf4M82eYP8P8PRHmv2jRIpXWxPQ+nSNHDvUYGhqq7kuOzi1t36xYqiWvGqXC91UI/8xWWoKV8H9fyToe8GH+/uiDZMsXxNG+u9BzXjGdSUZU0zHi0GZtjISnv/DCCyrMXhz3pJ+pn5HlOQzHZLVEVlIkgafluQ3XKI7utmQhr/fiiy/e9ueyeF0JuZfNkueee05t1li/fr1VWck5nR0npkfT65FHWYUzLFdbk6nhNay9H9ProQ8SfZDog+Rdfy1b7d72q3H23JIuRVa7LYOCJKhEeOmllzLd96yeO+4Qmh8ai4hbJzL273oSoe3GIyIkwmdlRR8kX/RBMvUxcmLfXbhyXntjxA9IzIziEyT+RLb6WZ5DlmqnTp2qy6/Gn2Sld5y9PraO0QfptjwMn7uUw3HVTyRQfEWc6aNXLtbaKSv9MvQHWYlZTVKJ2IuY/vLLL1UeO7lfWz23piF051yEr30T+VMToeUsiLRO4ojd0W9k5Qj6IDlJIPsgEeIP0AeJkKwhZZjEAuCIt956S/lYWiPfzaNodXC0en4hT03sKvcMkiLy+8VHwzxIPpgHyVY+Gkf77sKV8+odY6+frWN6cvv4k6z0jvOUrCzbhECUFfMgMQ+SN3JGBVIeJL15gMqVK6eyZ9s6d9q6K0iLKohtl8uibbv2fiMr5kHKJgLNB8nVMfb62Tqmx6/Gn2Sld5ynZGXZZiDQZMU8SMyDpAf6INlG7/e0TJkyxt+30PRk5NjwFsIaPwMUrJDRod1YpIu5ykpuI3ufA32QCCGEEOJTiHP2oEGD7PaRPySlS5e+nf/t/D60PDgaYYmngTM7gSd/tF4nNMBgFJsHnbTFTymQnbSzw/HYn2SldxydtLMmK/le0Uk7M3TS9qys/MXx2Na5xXdPCmd/9NFHal8S4YofraWjtmH1edKkSUhPSwV+nY7wX8Yhb1oytFxFkNb0eWj/BUj4u6wcQSdtDzhpywQrUaKEctS2VameEOIa4pN07tw5VRrHXhQOISSDCxcuKIVHfrOE7t27qxD/7du3q6SQly5dMopKyj899dRTaH1XNdQ7/hmKXv9LtZ/LWxe7y/ZHcoSdxIp+gl4nbSpILmJw0pasyJZO2uJYJoVGpaaW1CyTUhPygUgWaoOzrFRx17vvLlw5r94x9vrZOmat3VnZ+JKs9I7zlKws24RAkpUg3yPJ6i7fuYIFC7rsSGutzRedaT3teKzXGZmy0i9DX5OVVDmQYrOGTNlz5sxRFQAMY3744Qf1J1++V/LHXkpKhcUdQPjC7gi5dRlaeE6ktB6N7y+WQNt27TJdtz86tMvvtyiCjhQkmtiyiC3HNFk9Eq1cFCi5yUvVelGUDD9Uzuy7C1fOq3eMvX62jllr92dZ6R3nKVlZtgmBKCu5ycv3y5AHicVqzWGxWs/Kylcdjy3PLd8PSU4rpY4EKSQuNTHLly9v1l/yHLVp08b8eorfCeQrpbaQnnMQkr+iXUdsf5MVE0V6GYOZTeqSyQ1+w4YNaNGihVFLlrpkevfdhSvn1TvGXj9bx6y1OysbX5KV3nGekpVlmxBospJNT00oQoKZ06dPq8LimzdvVvuSFFKKz9rNy3d+P1D4DiAsAgjPAfT6GogulPHcA35A/gBXkDyM3MylWrto85ITyXCDd2bfndfi7Hn1jrHXz9Yxa+3OysaXZKV3nKdkZdkmBLKsCCGZEbOUFJc1uH98/vnn6Nmzp21RaekI3foRsOEdoPlIoPWrGe15Swa9eKkgEUIIIX6OBA599dVXyowmpmmpq7l06VJUrlzZ9qBrp9H0n4kIu/F3xv7Fg6qESDCE8OuBCpKHwvyzEtroTE0dZ6/V2fPqHWOvnyfDQH1JVnrHeUpWetsoK/h0OLarYxz10TtfrLVTVvpl6A1ZnT9/Ho899phy5RCefvppfPDBB2rl1dZrhfy9EuHfjUCRpHhoEdFIa/cOtDq9ZdnZZ+7vnv4OOoJRbB4I8yeEEEKyq67a5MmTVZSaKESSBLJly5Y2+4en3UKtU1+i7OVNav9KdEXsLDcQCVHFg+YDu6kzzF+W4ogLxMfHSwIWLS4uTktOTjZuCQkJWmxsrHq01ebsvrs2V86rd4y9fraOBZqs9I7zlKws2ygr/bJyJDtvz62szitn5pCz37tgkZUr88iTskpMTNTGjRunhYaGqt+iO++8U/vkk08cn/vsPi39rWJa+uh8WvIPb2grY5b55P09wYPzSn63RWbyO24PmtiyiLtDG7MjFNSV8+odk51hoL4qK73jPCUrQ5ut45SV47nmaVm5eu6szitbx/XON8pKvww9KStxwO7Tp4/KYSQ8/vjjmDJlCtavX2/93KZ+RcWrA10+znDCLtUImoPwfW/d3/W8pqvoPZ/7KlYSQgghxKNs3bpVJSIW5Ujyg0mU2ty5c227elw5Dsx7ADi+9XZb7QeB8vfwk3IAFSRCCCHEx5HINCkXIv5Fp06dUrXUfvvtNzzxxBO2B/35NfBpM+D4FuC7FzJWkohuaGIjhBBCfBhxwBYz2jfffKP2H330UcycOdN2rc/EeOC7kcDepRn7ZRoDPWYxfN9JqCARQgghPooUlH3ooYdw7NgxlQn7o48+wjPPPGOzPE/IyV+BlYOB+BNASBjQ6mWg2QggjD/3zkKJEUIIIT5oUpOUMiNGjFB5eypWrKgSP9avX9/mmHw3jyJsweMqOzYKlAd6zAbKNMzW6w4kqCBlESaKvC0H00c9x5gokokibc0ZT88ra21MFGlbhpSV++5Zjua8IPl5Bg4ciOXLl6v97t27Y9asWciXL5/dRKDxOcsjrXI7hOYsoBI/Ikceu3XUfD0RsMBEkX4EE0USQgjxFP/++68qLHv27FmEh4cr36MHHnjAuklN01D6yjacy1sHqeG5VFNIeiq0UK592IOJIj0ME0V6P5GYq4nkPJWgzh1J6pgo0juyCtbkh0wU6TuJIpOSkrQZM2ZoOXLkUEkMy5Ytq23evNn2vIg/r6V99Zimjc6rpS3ppyXcuBFQiYATmCjS/wmW5Ie+mEjMV2WldxwTRfqOrJgokoki3T23nJlXN27cUCVCvvzyS7XfqVMnfPHFFyhYsKDV68LRjcCKAcD1M0BoOEJL1DaeL1Du776QKJLrcIQQQoiHkfqdmzdvxsaNG5ErVy60bt0aYWFh2LdvH/73v//hwIEDan/ChAkYOXIkQkOtpClMTQZ+eQvY+rEsMgEFKwE9ZwOl6tv1NSKuQQWJEEII8SDbtm3DkCFDcPr0abUvxWVLly6Nbt26Yc6cObh16xZKliyJJUuWoFmzZtZPcuUYsKQPcO7PjP36/YAOE4DIDN8j4n6oIBFCCCEeIiYmBhMnTszULtmwP/nkE/W8Xbt2yrxWpEgR2yfKkRdIuAjkLJhRS+3OTvzMPAwVJEIIIcRDZjXJY2QPCd1ftWqVdb8YyYgtipFEsEUXBB5emFFkNm8Jfl7ZAGuxEUIIIR5g06ZNRrOavZxHW7ZsyXzg8M/AJw2BPYtvt5VuQOUoG6GCRAghhHgAyWXkdL+UROD7l4GFPYEb54Edc1hk1kvQxEYIIYR4gBIlSjjX7/w+YHl/4ML+jP1GzwBtx7HIrJeggkQIIYS4mfT0dFVo1h6SHVui2ZpL5NqvnwJr3gTSkoBcRYCu04Eq7fi5eJGgNrFJfZsCBQqoHBSEEEKIO7h69Sq6dOmCl156ydhmWSrEsD9lyhSEXdgL/PByhnJ0R3tg0DYqRz5AUK8gDRs2DE8++aTKWEoIIYRklZ9++gnPPfeccr7OmTMnPvjgAxw7dgwLFy40c9iWlSNRjnr06JHR0OJFIHdRoGF/mtR8hKBWkCST6fr16719GYQQQvycpKQkvPzyyyoJpFCzZk2V+PGOO+7A6tWrMXbsWJVJ+/vvv8cD7e5Fq7QNCG1c5/YJ7n3NexdP/MvEJunYO3furLKLylJkbGxspj7Tp09HhQoVEBUVhQYNGqiQSkIIISQ7kZWhFi1aGJWj+++/H1u3bkX16tWNfaSMSMuWLdH5rrJofWg0Qn+fCcQMYISaD+OzK0gJCQmoU6cOnnjiCfTs2TPTcdHMhw8frpSke+65BzNnzkTHjh2xf/9+lC1bVvURpUm0emtLoKJ4OYOcx/Rc165dU48pKSlqM2B4bq/N2Ud34cp59Y6x18/WsUCTld5xnpKV3jbKyr5crMnOH76HjvronS/W2ikr63LRNA1z585VtdMSExNRqFAhzJgxA5GRkUohMv19SElOQvj2mWhxaDxCtDRouYshrfkoaKmpcAeBeH/39HfQESGafMI+jqwgSbp2qVtjoHHjxqhfv76ajAbuvPNO1UeK/elFTGyS7n3ZsmV2+40ZM0YtkVqyaNEiREdH6349Qggh/o/8if/000+NlotatWqpP+2iJFkSlXwZ9Y/PQpEbGeH7Z/M1wB9ln0RyeJ5sv24C3Lx5E7169VJ+Ynnz5rUtEs0PkMuMiYkx7iclJWlhYWHaihUrzPoNGzZMa9GihVPnXrdundazZ0+H/RITE7X4+HjjdvLkSXVdcXFxWnJysnFLSEjQYmNj1aOtNmf33bW5cl69Y+z1s3Us0GSld5ynZGXZRlnpl5Uj2Xl7bmV1Xjkzh5z93gWLrEzb5XejQoUK6jdAfot69+6tXbt2zerYm0e3a0njSmja6Lxa+tvFtN2zn9MSbtzwG1kle+n+7sl5Jb/b8tnJb7k9fNbEZo+4uDhV46ZYsWJm7bJ/7tw53edp3749du3apf4JSESBrFI1bNjQat8cOXKojRBCSHAivztLly7F4sWL1XPxgf3888/VSoSY1KyhFaqMm5FFEJa3EpI7T8fxnf+iikXIP/FN/NLEdubMGZQqVUo5wTVp0sTYb/z48ViwYAEOHDjgsWuZNm2a2uTLcejQIZrYCCEkCLh06ZIKy9+7d6/ab968OQYOHIhcuXJl6pv35glcz1kSWkjGGkSOlKtIDssNLdQv1ySC1sTml59W4cKFlbZuuVp04cKFTKtK7mbIkCFqEydtqcIsqQJMBZyamop169ap9vDwcKttzu67C1fOq3eMvX62jgWarPSO85SsLNsEykqfrCznkqXs3ImnvoeO+uidQ9bag11WEqYvSR9FSZKo6Q8//BB9+vRRf97N+oaGIHTbR4jc/QFSGg9GWstX/V5W3rq/e1JWhiCrgFxBMjhpS5SaRLEZkJDKrl27OuWk7SxcQSKEkOAgOTkZ8+bNUwqSULFiRRWxJhYMS3Imx6H+sZkonHBQ7Z8s0AS7yg1k0kcfxO+dtK9fv67t3r1bbXKZkydPVs+PHz+uji9evFiLiIjQ5syZo+3fv18bPny4litXLu3YsWPZcn3i3EUnbe868bnqmOgph0d3OIjSSds7sgpWx2M6aduW1ccff6zVqFFD3edle/7557XLly9bleH2z0dp6e+UznDEHl9S2/H5i0ZHbH+fV8l00vY9duzYYVxmE0aMGKEe+/XrpzT6hx9+WC13jhs3DmfPnlVZS0XLL1euXLZeZ0REhNr0tFu2Obvv6Wt2xxh7/YJFVnrHeUpWhjZbxykrx3PN07Jy9dxZnVe2juudb8EgKzGqfPbZZ2qlSFaQihYtqspRdejQwZg/xzguMR5hq0firuNLMwaXboTULtNxatt+1I6MDChZZff93dF5s4Le8/msD1KrVq3URLXH4MGD1eZNmCjythxMH/UcY6JIJoq0NWc8Pa+stTFRpG0ZBousLl++rByvDZUb7rvvPpUIUnxbzZI+GsZdPYPwg6ugIQQp94xESIsXkJImv1v7A0ZWAhNFEl3QB4kQQgKPffv2qVIhYpkQp+DHHnsMXbp0QWioRUUu+eNuEqZf8spvuBVREFdy35H9F02C0wfJ16EPkvdt1K7a3T1lz3eHrwh9kLwjq0DwFaEPkmuyunnzpvb6669roaGhyteocuXK2ubNm62e4+apfVrcew20W3+vceme5Y/zKjmIfZB8tlgtIYQQ4klOnDihzGhvv/020tPTVej+b7/9hnr16pl31DSE/LkYUfPuQ6GEw4hc+waLzAYBfhHm70vQxEYIIf6PJBqW+7lUUsiZM6fyPWrZsmWmfhGpCahzci5KXf1d7cflropd5QbgVmRhL1w1cQc0sdHEFvBLsK6+J08tV+sdRxOb78kqO5f3aWLzrqyuXLmiPfHEE8bw/UaNGmkHDhyweg4xpaVPqpYRvj+2oJa0doIWG7PCJ81GgXZ/T/ABE5vPRrH5C8ESuu6LYaC+Kiu94xjm7zuyCqRwbIb525bDnj178Mgjj6hyVJKA+MUXX1TmNWsyzX/zX+RYPBYhokcVrISQnp8hpGhtYPVqnwxdD7T7uwGG+fsxDPO/LQfTRz3HGObPMH9bc8bT88pam7+HYzvqo1cu1tr9TVZSK3Pt2rW4ePEiSpQogXvuuQczZ87EqFGjVG6j4sWLqxQxkuvI2vll/2rOCkircj9Cowsgre3bQGRut9yznH0/zrxvZ8/NMH/70AfJSeiDRAghvsu2bdswe/ZsFa5vugphUAbuuusuDBs2LHN4t6ahzOXNOJevPlLCMwrQhmipxoKzJHCgDxJ9kALeRu3qe/KUPV/vOPog+Z6sstP/gT5InpPVkiVLtJCQEKN/keUmvkdJSUmZz3v1rJa28BHla5S2uI8qE+JvfjWBdn9PoA+S/xMsfjW+aKP2VVnpHUcfJN+RVSD5igSrD5KY1cRkZiswW3yOfv75Z4SFhannxvOe2ATEDAJunAPCIhFappHxtfzJrybQ7u++4IPEPEiEEEL8nk2bNuHUqVM2j4vidPLkSdVPCE1PQejPbwALumcoR4WrAv3XAk2HmmXKJsELjauEEEL8njNnzujqJ8XNceUYWhwcg7DEkxmNDfsDbd8CIqM9e5HEr6CClEUYxXZbDqaPeo4xio1RbLbmjKfnVSBGGwVzFNuVK1dUlJoeihQpgpSwaESm3YAWXQhpnT6Cdkd7wwvafV1GsaXo/oz8QVaOYBSbkzCKjRBCfKvI7Icffoi4uDi7/fJHAeG5C2HmzFnKDynfzaNIjCiApIj82XatxDdgFBuj2AI+ysHV9+SpiBC94xjF5nuyys4IGkaxuUdW0jZq1Chj1JoUmX3nnXesRrE9UCVcO/9Cbu23mcN0fQb+GJkVaPf3BEax+T/BEpnli1EOviorveMYxeY7sgqkaKNgiGI7cuQIevXqhd9/z6iP9sQTT+Cjjz5C7ty5UalSJZUEUvIgRYUD77eNwtBGkapf0bC/gfDwLMnKVruvysqdYyKCLIqNPkiEEEL8AolEmz9/PoYOHYobN24gf/78mDVrFh588EFjn+7duyM8PBwlQy/jzv3vI1/yuYwDdw8B2rzJCDWiGypIhBBCfJ6rV69i4MCBWLJkidpv0aIFFixYgLJly5p31NJRJe4n1Di/HCFpyUDuYkC3GUDlNt65cOK3UEHKIoxiuy0H00c9xxjFxig2W3PG0/PKHyKznB0TyFFs69evR//+/XHixAnlYD169GhVaFaeW75u6qndqHFmsSoym16lI9IemAJEFzJGqNl734EQmcVabI5hFJuHYBQbIYRkD6mpqWrFaPny5UhPT1dFZkeMGIEqVarYHVflbCySIvLieKHWNKmRTDCKjVFsAR/l4Op78lREiN5xjGLzPVllZwQNo9j0yenvv//WGjZsaIxE69u3r3bp0qXMfW9c1lK/Ga4ln/vbLfPKk/csf5xXyYxiI64SLJFZvhjl4Kuy0juOUWy+I6tAijby9yg2ccQW36IhQ4YoR+zo6GjliN27d+/MnU/vBJY/DVw+grBzfwL9f3bqehjF5txnF8EoNkIIIcQ7jtiDBg3C4sWL1X6zZs3Qt29fPPTQQ+Yd09OALVOAde8A6alA3lLAfWNoTiNuhcVqCSGEeJ3Nmzejbt26SjkS5+u33noLa9asQdGiRc07Xj0JfNEZWDsuQzmq3g0YtAWo0Nxbl04CFEaxEUII8aoj9rhx4zB+/HjliF2xYkUsXLgQd999d+Zoo3N7gXkPAInxQGRuoON7QN1eXDkiHoEKEiGEEK9w9OhR5Vu0bds2tS/mtI8//hh58+a1PqBINaBgRSAkDOgxCyhUKXsvmAQVVJAIIYRkO19++aUqCXL9+nXky5cPn376KR555JFM/aSoLCThozgRh0UAvb4GchbIeE6IB6EPEiGEkGwjPj5erRr16dNHKUfiiL1nz57MylFaKkI3TkTLg2MQuvG92+25i1I5ItkCV5CyCDNp35aD6aOeY8ykzUzatuaMp+dVIGY89odM2mJK69evH44dO6YcsV9//XWMGjVK1U4zG3/lGMJWDkLY6e1qV4s/jZTkZF2+Rp6Sla12ZtJO8VtZOSJE8im6/dUDGGbSJoQQ50hLS8PXX3+NpUuXKkfsYsWK4fnnn0e1atXMO2oaylzeglqn5iMiPREpoTmxp0w/nC7YlCInboOZtJlJO+Azrbr6njyVldbXskNTVsyk7coccnd26EOHDmlNmjQxZsTu3bu3FhcXl7lv/AUtbUlfTRudV21ps9tpN88c8Mg9i5m0ff/+nuDBrOMy/9TCZHy8XU2BJrYsEizZoX0x06qvykrvOGbS9h1ZMZO2ZzJpL1q0SCV+vHbtmopME0fsRx991PqHfO0acHhNRoRa61cQ2mwEwtPSARzwqazjttq9kXXcmWt2x5gIZtImhBBC9JnOJMHjxo0bkStXLrRu3Vr5FolCJKVCJFJNuOeee9Tz8uXLm59APDwMfkUSst99RkZW7NJ3/fcCoiAR4h0YxUYIIcRpxOG6cuXKaNu2LSZPnqweRQGaMGGCyogtClFoaCjGjBmD9evXZ1aOLh0BZt8HHN10u61619vKESFeRreJrUKFCgjREUFgyfDhwzFs2DCnxxFCCPFNYmJiMHHixEztp06dwquvvqqei0IkGbGbNm2aedVo9wLg+1FAyk3gh5eBgZuZDZv4r4I0b948l14g078GQgghfm1WGzFihN0+0dHR2LlzJwoWLGh+4OZl4JtngQOrMvbLNwe6z6RyRPxbQWrZsqVnr4QQQojPs2nTJpw+fdphGPWff/6JVq1a3W48sg6IHQRcPwuERgBt3gCaPAuE0tOD+CbhWUm0dO7cOfVFKFKkSOZ/CoQQQgKOs2fPOt/v9C5gQbeM54XuAHrOBkrW9dAVEuIFBenGjRvKpvzVV1/h999/R1JSEiTPpPgmlS5dGu3atcMzzzyDhg0buunyCCGE+BIlSpRwvl/JekCN7hk11NqNByKjPXeBhLgJ3WubH374ofIn+uyzz3DvvfdixYoV+OOPP3Do0CEVzTB69GikpqaqSIYOHTrg8OHD8GVOnjypln+rV6+O2rVrqwyvhBBCbCP3+HXr1tkVkfxhLlOmNFrkPprhc5TRCPSYDXT6kMoRCbwVpK1bt6ovRq1ataweb9SoEZ588kmVCGzOnDnYsGED7rjjDvgqUvtnypQpKhz1woULqF+/Pu6//36Vy4MQQog558+fV3+Of/31VzNlyLRalewXiQ7Bb89VQOiq4cCRX4CH5mcoSGHMS0z8C90zVu8KS44cOTB48GD4OrL8a1gCLlq0qPKhunz5MhUkQgixQFwrJGXLrVu3VEbsTz75BPv27VO5jkwdth+7uxhm3R+OqOt7gLAcQLl7KEvit7gUPnD9+nV4GsnM2rlzZ5QsWVL9K4mNjc3UZ/r06So/U1RUFBo0aKCiK1xhx44dqoBimTJl3HDlhBASGFy9ehW9e/fGE088oZQjyYi9Z88ePPLII2jSpAn++ecfrFmzBi+NGIbjnz6C+e1uIir1GlC0OvDMOuDugQzhJ8GlIDVv3lxFsHmShIQE1KlTR/1TscaSJUvUP5rXXnsNu3fvVtfUsWNHnDhxwthHlKaaNWtm2s6cOWPsc+nSJfTt2xezZs3y6PshhBB/Yv/+/SrgRuqpSfmQXr164eeffzbLbSftLWuVwWtF1qDs2dUZjY0HAU+vA4rV8N7FE+IGXDIK33XXXWjcuDF+/PFHVKtWzdguioooLKtX//dFyQKi7MhmC0lt/9RTT6F///5qX/yJ5HpmzJihUt0LkqjMHhKF1717d7zyyiuZs71a6SubAak1ZEh3IJsBw3N7bc4+ugtXzqt3jL1+to4Fmqz0jvOUrPS2UVb25WJNdv7wPXTUx5n5Mm7cOLz//vtqZb1ixYr4/PPP1WqS7Jve89RjRB6Ep92ClqsI0jp/Aq1SG8OJdL+/rLzv7JKVrXZX7lH+NK+8eX+39ZpZRe85QzRTDzsnGDt2LD7++GNl+hIfntdffx3Lly9Hly5dVBp6dyImNjlnt24ZeTSSk5NVplbxixIFx8Bzzz2nIuvEQdwR8rblH1HVqlVVrSBHSB95z5bIvyu5FkII8Xckd5FELEt0siBO2U8//TRy5sxp1i8i9TpSwnIbzWd5bx5HYkQBJEfk9cp1E+IMkr9Rfv/j4+OVT50tXA4rkLD+yMhIFdYvqefbt2+P7du3q2gwTxMXF6des1ixYmbtsq/X9LdlyxZlppMQf4N/04IFC2xG6ckqk2l6fVlBEp8lqV5tKmBDGKy0S6SctTZn992FK+fVO8ZeP1vHAk1Wesd5SlaWbQJlpU9WlnPJUnbuxFPfQ0d97MlAUp4sXrwYL774osp3ly9fPqUYvfHGG5lkk+PoGkR8/zr2FO2BSv8bEzSystXuzD3KH2Xlrfu7J2VlsAB5ZAVJ/mWIGWv27Nm48847ceDAAeXDI858nsByBUl8iEqVKqVSD4ijoIHx48crJUeux1NMmzZNbaKgyb8sriARQvwZUYgk4EXup0KNGjWUf6dUSDAlLC0JNU8vQvlLGXmQLuWqgs13vEYnbBKwK0hianKaqKgorW7dutqqVavU/g8//KDlzZtXe/fddzVPIJcZExNj3E9KStLCwsK0FStWmPUbNmyY1qJFCy07iI+PV9cVFxenJScnG7eEhAQtNjZWPdpqc3bfXZsr59U7xl4/W8cCTVZ6x3lKVpZtlJV+WTmSnbfnVlbnla3jcu8uVKiQupeFh4drb7/9tnbr1q1MfW8e2apdf6eKpo3Oq7bk717WvlmxNKhk5co8CoR5leyl+7snZSW/2zLn5XfcHi5Fsc2dO1c5ZD/wwANqX8xrshQ2derUbMmBJKY9iVCT8FJTZN+RszUhhAQ74scpfqMSCCORvJUrV1ZpUl566SUVmWYkPQ2hWz9C1JedkDvpHNJzF0dqrxVIafUG0qXgLCEBjMtO2tY4duyYykYt4aHuWPaVHBtCvXr1VNSa2CIloWPZsmWV/1CfPn1U5m4xs4mJT8qgSPKycuXKwVPQxEYI8WcksaM4Yhvur+JHKlUQLB2xhXw3j6HlwdEIgYYz+RvijzKPIyU8jxeumhAfNrEdP35cV7/Lly+rx1OnTmlZYd26dWoJzHLr16+fsc+0adO0cuXKaZGRkVr9+vW1DRs2aNkFTWzeX4J1dVnZU8vVesfRxOZ7ssrO5X1vmY1u3LihDRkyRIuOjlb30gIFCmgLFy50KMPUX97VErfN1mJjYoJGVv5oNqKJLdl7JjZJGCZRDb///rvNPqKNLVu2TCVjlGK2WUGiKmRxy3KbN2+esY+Y82TVSvITSc6jFi1aZOk1CSEkEBEzmgTRyAq4/HuW1Xi5ZxoCX4wkXUfo6pEIuZSxuiSkNxuBtNqP0hmbBB26TWxSp+ydd95RCcMiIiJUskgpAyJlPq5cuaLMamLeknaDbTsQoYmNEOJPSGkQ8Q+Ve7iES4ui1LVrV4SGmv8/LpDwD+of+xS5ky/gSnRFbKwymkoRCUg8FsUmEQ7Lly/Xhg8frnXr1k1r37691rt3b23SpEna3r17tWCBJrbsMYXQxOa62YjmyOA2sYlJbcSIEVpISIgyJ1SpUkX74IMPMslgZcxyLWnNW1r6mAIqQi39g+rarQNrA9psRBObc/JNCNIoNqezL8mKUY8ePdRGoFbTZNPTbtnm7L67cOW8esfY6xcsstI7zlOyMrTZOk5ZOZ5rnpaVq+fWO6+OHDmi/iFLtLEwYMAATJw4EevXrzc/x9UTuOfwO4hMOJyxX/N/CHngA4SF5wIOrNb1vcvK+/EFWdnr4+57ljPXFuz3d0/KSu/5nFKQJJmY+PmIjxHJgLXYbsvB9NEU1mLLHlnpbWMtNgRkzSw5Jh4TEtk7atQo3Lp1C4UKFcLMmTNVCahM5zi3F+ELuqBQ8nVokbmR1vF9aDUftPp6gSgre338ub4Ya7F5qRabpJ2XjNaSYXXo0KGqDpqlHTvQoQ8SIcQXkfIJn3zyiTGQpk6dOqo+paRGsUaIlopmh8ZDCwnFrnIDcTOHeeZsQgIVt/sgvfTSS8r/SDhz5oz2xRdfaGPGjNGCFfoged9G7aqPgqd8H3wtdJ2yCh4fpO+++04rVqyY8quQtCfvv/++lpiYmOkc6xd+oCVcu3K77eJx5YPkaL4Fkqz09PFHvxpPySqZPkiOkcRiI0eOVD5IL7/8sjK35cqVC8FOsPjV+KKN2ldlpXccfZB8R1b+4Csi9R+lXtrFixdRokQJNG/eXBX3lELacn8WpIC2pFiRaGLzwSkI3TIJLQ5NRvrWywhr/3ZGe74S0ELCdMvQX2Sldwx9kJyTbwR9kKwjxWF37dqFDh064Msvv8T7779PBYk+SEbog+RYDp6Wld42+iDB73xFJL/cs88+q/IZGShWrBhy5MiBEydOGB2x7733XlSvXt383JePIGzlIISd2aV2tRtxSElOViH8euebP8mKPkjulZU37++2XtPnfJDE9+b5559X/0x+++03TJo0Cffccw9q1aplNUV9oEIfJEJIdrJt2zYVgWaL6OhodW+WZL5maBrKXt6IWqe+RHh6EpLDorGnzBM4U6Cx5y+akGDLg/TXX39p48ePV3k1KlWqpIWFhakK0NWqVdMefvhhbcKECdrq1au1YIA+SN63Ubtqd/eUPd8d/g/0QfKOrHzVV0T8PkuVKmW17JJhK1mypOpndt7481raV71VXiPZ0uZ00G6ePaRLLs5+73xFVu76DmZ1bgWarJLpg6SPGjVqqE2yaf/666/KxPbnn3/ijz/+UNvKlSsxfvx4XL9+HcFCsPjV+KKN2ldlpXccfZB8R1a+6lezZcsWVVzWHmfOnFH3Y1nRN543MQH4dx0QGg7c+zpCmw5DeFo6gP265WKt3Zdl5coY+iA5J98I+iA5xlAFWmjcuLHaDDiRNYAQQogdzp49q7+fJgrQfxSsAHSfCeQrBZSsl9GmFCRCiF6czqTtiJCQEAQTTBR5Ww6mj9nlxEfHYzppu8uh3Rcdj/WuxpfLlYTQOfehcJ6OSElpm9FYub3hBW2+Lp20mShSz1xMoZM20QOdtAkhniY9PR3fffcdvvjiCxXKb4+RLQvg3dZhCNeSER9VBuurvc0is4R4o1gtoZO2rzjxueqY6CmHR3c4iNJJ2zuy8iVn2uPHj2vt2rUzOmHXr1/fWHDWdCuSK1Rb+Ui00RE7dW4n7Yelc51yPKaTNp206aRtmyyb2CQ3ktRmi4yMRDASLI7HvujE56uy0juOTtq+IytfcTz+5ptv8NRTTyEuLk6lT5k8ebLKb7R06VIMHjzYmAepXaUwLOiRG0WjNSAsErhvDNIb9Efi9z+45HhMJ239stIrQ1+aV+4aE0EnbeeQ3Bt///03qlSp4uRIQgghhiV/qVQghWaFunXrYtGiRbjzzjvVvtS9DA8PV+aAtJM70P74hIyFpCLVgJ6zgeK1jL5GhBD3kOUVJEatEUJI1lbhxR/i4MGDav/FF1/EW2+9pbJkmxIWFoaWLVsiIrwNsPwwEF0QaDsOiAieRL2E+HUUGyGEEH2O2FKy6bXXXlNRQiVLlsT8+fPRpk0by44I3TkXkanRGfsSKdxjFhAaRjET4kGoIGURhvnfloPpo55jDPNnLTZbc8bT88rbYf7iY9S+fXts2LBBtXXr1g0zZsxAoUKFzF/z+jmErRqGsH9/QZ18DZCS3P32MYu8Ro7SBzDMP2uy0itDf0gf4eyYFIb5u0ZoaCgOHDgQND5IDPMnhGSFrVu3Yvr06bhx4waioqLQv39/tWpkmUOuePwu1D0xBzlSryMtJAJ/lXoUxwq3YQg/If4S5i/hpwcPHtSCDdZiY5g/w/wDJyVCdoT5X758WXv88ceNYfoNGjTQ9u3bl7lvwlUtdeUwY/h++rSm2s3juzxSX4xh/gzzZ5i/bWhiyyLBErrui2GgviorveMY5u87svJ0OPbvv/+O3r17qzJNslLUs2dPlQQyOvo/vyIDl44AXz0CxB3K2G8yFCFt3kS4FgrsPpFlWdk6zjB/52TJMH//lpXe81FBIoQQD5GWloZ3330Xo0ePVs/LlCmDefPmqRIiVm/SuQoDKYlA7uJA9xlApXsz2hnCT0i2k2UFSb74hQsXds/VEEJIgHD8+HH06dMHmzZtUvsPP/ywynOUK1curF69+nbHhDggulCGb1FUPuDRr4C8JTPC+AkhXiPUHQpSwYL8IhNCiIGvvvoKderUUcpRnjx5VPi+tOXPn99cSPtXAh83AHbOu91WvCaVI0ICQUEihBCSgUTFyKqRIUKmSZMm+OOPP1SbaZRaWFoiwlY9B3zdF0i8CuxdJhEzFCMhPgR9kAghxA1s2bIFjz32GI4dO6bSn7z55psqCaSUCDEl5PQutDr4BkKTzsse0HwE0OoVhu8T4mNQQcoiTBR5Ww6mj3qOMVEkE0XamjOenlfuTOiXmpqK8ePHY8KECSo7doUKFZQjtqweSSkm4znS0xC6dSrCNk5Ebi0N6XlKIr3rDGjl7gEk52O6a3LQ24eJIrMmK70yZKJI/5GVI0Ik1t/trx7AMFEkIcTAuXPn8OGHHxrrqLVu3RpPP/105vB9APluHkPLg6MRAg2n8zfCnjJPICU8F4VJSKAlipQkYydOnNAOHDigXbp0SQs2mCiSiSJ9PfmhvUSI7kp+GKyJIpOSkrQ5c+ZouXPnVkkf8+XLpy1YsMDhe05dP0lL/G2uFhsT45S8siorJorMuqxcmUfOzqvs/h7qHZPgoe+ht2QVFxenvrfyO24Pp0xskhp/4cKFKhpDEp8lJSUZj5UuXRrt2rXDM888g4YNGyJYCJbkh0wU6X4ZM1Gk78jKmSR1V65cwaBBg7BkyRK136JFCyxYsABly5Y1v4DEeODHV4Gmw4AiVTPaWo5Euizvn13tse8hE0V6Vla+mvww0O7vvpAoUncUmywjly9fHp999hnuvfderFixQkVnyNLytm3bVLi/2OLbtm2LDh064PDhw1m5fkII8TmkuKyE74tyJM7X77zzDn755ZfMytHxbcCMZsDuL4GYAYxQI8QPCXemwOK6detQq1Ytq8cbNWqEJ598UiVCmzNnjrqR3HHHHe68VkII8TiS8Xrz5s3YuHGjSuoofkXSNmbMGJUVW9w25d4mq+mZVsvTUoAN7wGbJgFaOpC/HNBhIiPUCAlkBWnp0qXG55ImX5KfWSNHjhwYPHiwe66OEEKyEVkNHzJkCE6fPq32J0+ejOLFiytF6ciRI6qtf//+akU9d+7c5oMv/wssfxo4vSNjv86jQMf3gCg7TqCEkMAK82/evDl++OEHdeMghJBAICYmBhMnTrQaqSaIQiQFZnv06JF58Nk/gbkdgeQbQI58QKfJQK3/ZcdlE0J8KZP2XXfdhcaNG+PAgQNm7bt378b999/vrmsjhJBsQUxoI0aMsNtHwoG7du1q/WDR6kCxGoDkNBq0hcoRIcGqIM2ePVv5GzVr1kzZ6g8dOoSHHnpIKU5iYiOEEH9CaqYZzGq2OHPmjLHwrOLEb0Dqf5G8YeHAo4uBft8C+ct4+GoJIT6dSVui1iIjI1XUmvz7at++PbZv34769eu79woJIcTDnD17Vn+/1GRg3Xhgy1SgyRCg/fiMg9Es2k0Ign0FSW4Sw4YNw1tvvYXq1aurnAKPPPKIXylH4mguESh169ZVkXmSvoAQEpyY5nSzR8V8acCc+4AtUyTPHJB0nSH8hAQoLq0gVaxYEdWqVVORbQ888AB+/PFHZWI7deoURo0aBX9ASgFIKgJ5lLTjNWvWVM6XhQoV8valEUKyCQnZl9QkL774ot1+ISEhePHeomi0+0Ug5SaQswDQ+SOgehd+VoQEKC4pSHPnzlUrRgbEvCY5kjp16oTjx49j+vTp8HXCwsKM9ZISExOVmZBl6QgJHi5cuKCKzO7YkRGWLwkg9+zZo5Qh03tB4ehQzOqcA92r3QKkxmWFlkD3T4G8Jb149YQQnzSxmSpHBsS8Jskk169f747rUknaOnfujJIlS6obVmxsbKY+oohJ5eyoqCg0aNDA3IFSB1evXlU3RSmT8tJLL6Fw4cJuuXZCiG/z/fffq3uWKEcSWDJ16lT89ttvagVc7jmm3FGuBDpVzwuERgDt3gb6xFI5IiQI0K0gnThxwmEfKUWyZcsW9dxRRIgjEhISlPLyySefWD0uqf6HDx+O1157TaUXkNxMHTt2NLtOUZrEdGa5STSKkD9/fvWP8ejRo1i0aBHOnz+fpWsmhPg2t27dUv6Tko5EVpCkRIj8sZO20NBQNGnSBP/88w/W/PSDCvtfs2YNNu09joiH5wJP/wI0fRYIdel/JSEkUE1s4tDcpUsXPP3006qsiDXi4+OxbNky9W9swIABePbZZ12+MFF2ZLOFZLh96qmnVFZbYcqUKcoXasaMGZgwYYJq27lzp67XKlasGGrXrq1WrR588EGbTpymjpzXrl1TjykpKWozYHhur83ZR3fhynn1jrHXz9axQJOV3nGekpXetmCV1bFjx3D33Xfj77//Vm2S8b9Vq1bKn9L0e6xd+Bv3Hh6L6Lr3o27TpkhPT0d6xfsMJ3LqPTr7nrIyxlEfvfPF1e+du+eVvevzhqxstQe6rLx5f7f1mllF7zlDNJ2ON5cvX1aFGT///HMVtSY5j2QpWsxbUt16//792Ldvn2p//fXX7So3ziImNsly261bN7WfnJys/IfESbx79+7Gfs8995wqoCvO146Q1aKcOXOq5G+i7Mg/x6+++kopStaQOkxjx47N1C4rTwZfJkKI7yEKzqpVqzB//nxVUFtWjmXFKFPUraahQtzPqHF6McK0FMRHlcH6am+zjhohAYYEZvXq1Ust6ogOYBPNSW7duqUtX75cGz58uNatWzetffv2Wu/evbVJkyZpe/fu1TyBXGZMTIxx//Tp06pty5YtZv3Gjx+vValSRdc5d+zYodWpU0erXbu2VqtWLW369Ol2+ycmJmrx8fHG7eTJk+oa4uLitOTkZOOWkJCgxcbGqkdbbc7uu2tz5bx6x9jrZ+tYoMlK7zhPycqyjbJK0I4fP661bdtWfVdlu//++9X9I5P8Lp/SUr/ormmj86otdV5X7ful89w+rzz5PXTUR+8ccvZ756nvoK/JSq8MA01WyV66v3tSVvK7LfcD+S23h9NRbLJiJOHwVusRZTOysmSK6FKWbbYQ/yRZbdKLOHIySzgh/oOsGokZ7dKlS2q1uF+/fpg0aZJKcGtK6JGfEf798wi5GYe0kAiktRmD1HpPIOnnn7127YQQ76PbxOaImTNnKr8jT+AJE5urTJs2TW2SFkBKrNDERohvIWk7JBWJ+CQKEukqDtdlymQuAZL/5r9oeXCMei4mtZ3lB+F6ztLZfs2EEN8zsblcasSSbdu24a+//lIO2hINcvDgQbz99ttYsGAB3I38A5QVIIkwMVWQZN9mMUk3MWTIELWJ31K+fPnQunVrMwGLj4PkhJL28PBwq23O7rsLV86rd4y9fraOBZqs9I7zlKws24Rgk5VEtD7xxBM4fPiw+mMlf5refPNNlffMuqyA5Dt7IiRXIYQ0fRHXN22zKjt34qnvoaM+eueQtXZ7+8EiK1vtgS4rb93fPSkrQ5BVtq0gGSLLfvrpJ+UEKTcoyU5rLWeSHm7cuKHCbYV69eqpc4ugChYsqEJzJcy/T58+KguuOFjPmjVLlQsRR/Fy5crBU3AFiRDfQ1Z0V65ciYULF6rnkhFflKNMQRdaOirErcXp/I2RHPHfHxstHQhh6D4hwcJNTzlp22Lnzp1ajx49tLp162oVKlTQjh07lqXzrVu3zuhYabr169fP2GfatGlauXLltMjISK1+/frahg0btOxCnLvopO1dJz5XHRM95fCodxydtN0rq4MHD2otW7Y03iMkeGT+/PmZ5tUPX3+upX7+gHLCTvvyIS3hxo2Acqalk7ZnZeWOe5Y/zqvkIHbSdtvfJnGGlLxEssS9ePFiZeoyJI10BclPIotbltu8efPMXlNym0h+Isl51KJFCze9G0KIPyD3GMltJH6HuXLlUivJX375ZaZ/hWGHVqP1gdcQdnwTtPCcSK/c1mvXTAjxD9xqYjNFMmlL0kXJUhtI0MRGiG9kxJ49ezbWrl2r9u+44w48//zzmcqEhKUloubphSh/KSNw42rO8thZfiBuRLGOGiHBys3sNrGlpKRoCxYs0CZPnqz98MMPWnp6usqZFKjQxOb9JVhXl5U9tVytdxxNbFmT1ebNm7VKlSqpJfKQkBDtpZdesjqHbp78U0ufWleZ1NJH59MOTXtUS7h2Rddc80dTCE1snpWVO+5Z/jivkoPYxOY21/BHH31UOUZK2n4pNyJFHyUMX/7ZBTKSVVw2Pe2Wbc7ue/qa3THGXr9gkZXecZ6SlaEt0GQl0bFSRkiy2osjtgRrSGoRCQaxNj48fymEpKcBeUoircs07N9/HeWjcumaa1l9T978Hjrqo3cOWWunrPTLMNBkld33d0fnzQp6z+c2BUnC+v/880/j/q5du1TdtvXr1yOQYS2223IwfdRzjLXYWIvN1pyxnCMS0Sr3E4Nf40MPPYQPP/wQ27dvNx9/4zxSIgtkjA2NAh5aCOQuhpTw3MD+NU7VhXJ0bYFQX4y12FiLTc9cTGEttqzRtGlTlfOoUqVKxjYJsTVVmgIB+iARkr2IA7YkohW/AcmILatGLVu2zJQ1v+SVX1H35DzsL/EgjhVpw4+JEOIbPkhSh61mzZraiy++qH3++efq8YEHHtACFfoged9G7ard3VP2fL3j6IOkT1YXL17UHnroIWP4fpMmTVRIfyY5Xj6npS17+nYdtTkdtdiYGLvzypv+D/RB8k9ZueOe5Y/zKpk+SFmvli1h9jt27EBsbCz+/vtvtZIkvgKBTrD41fiijdpXZaV3HH2QbMth8+bNeOyxx3D8+HHle/T666/jjTfeyJRRt0DCYeT84g2EXD2ekeyxxYtIbzIc+HGNU/5ageQrQh8kz8rKV/1qAu3+HjA+SHIDk9pHUgzy4YcfRjBBH6TbcjB91HOMPkjB7YMkNdP27t2LK1euoHTp0mjWrJn6syUliiZOnKiely9fHgMHDsSzzz6r8qAZXyM9FdrGSWh2aDJCkA4tX1mkdZ0BrUzjLMuFPkiOZROMsrLVHuiyEuiDlEUkak2KQQ4dOhSBDH2QCHFP7UbJY3Tp0iVjm5QoEh+js2fPqn0pLSSO2VKY2pK8N4+j5cHRCEU6ThZoij/L9EVqWOZ+hBDidR+kjh07amXKlFGlPx599FHtnXfe0b799lstUKEPkvdt1K7a3T1lz3eH/0NWZOXI38FXZLVkyRKVv8haKSHZoqOjtS+//NLhuZPWfaBt//wlt8jKkewC1a9G73yjrNxzz/LHeZVMH6Sss3r1amOV3L/++kttP//8Mzp16oRAJlj8anzRRu2rstI7Lhh9kCR/0ciRI5W5zBb58uVTRa7DwsJunzvtJiJ+fB245zmg6J2qLeWeZ3E6fjXquEFWgeQrQh8kz8rKV/1qAu3+HjA+SKbIcpWE/MtGCCGmbNq0CadOnbIrFDGxST+pxygUun4A4Z+9Alw7DVz4G3hmPWAR4k8IIe7GbQqSOFtK4rarV6+iVq1a6N+/v/JJCnTopH1bDqaPeo7RSTv4nLRPnjyp65zSLyXxJrB+Au7552OEQINWoALS2k+Elppq93XppM1EkY7mI520nfvupjBRZNaoWrUqXnvtNdx5550qi/ann36KSZMmoU2bwErYRidtQlznl19+wUcffeT4ezbueTya4xcUuHlU7R8v2AJ/le6N1LCcFD8hxL+ctBs2bGi2f+HCBa127dpaoEInbe878bnqmOgph0d3ONMGqpN2UlKSNm3aNC0qKsqmc7ah+Gz7OiW09LeLZRSZnVBW+23uqx6XlSPZeXtu0Unb+7JyZR4FwrxKppN21qlYsSImT56M559/XpUAKFiwIKKiohDoBIvjsS868fmqrPSOCxYn7YsXLyqT+zfffGMsQSQmecHUWdtQOuSZNz5CSPxsIDQcqZ2n4ezmP1DPw7IKJGdaOml7Vla+6ngcaPd3X3DSDnXXCyYlJSnzk1TY7tChA2rWrKnMa6dPn3bXSxBC/IyffvpJKUSiHEVGRuKDDz7A7t27sWzZMpQsWdLYr1nZMFQqV0q19+j5P+CRRUDfb4C8t/sQQohfOmnHxMSox4SEBFWg1rBJuO6ZM2dw5MgRd70UIcTHkSzZr7zyCqZMmaL2xTdx0aJFqFu3rtrv0aMH7r//fnz4/rvoknsPasT/gvTGvRDasUfGCaILZjympXntPRBCghu3RrHJzVDKBhii2KTqNiEkuJAcaOIAaTCjSXb99957T2XJNiXs8j8YHLkc+eJPqP3Q9FSxuTGEnxASWArS//73P7Moti5dugRkFJslDPO/LQfTRz3HGOYfWGH+ycnJysz+8ssvK5N70aJFMWvWLLVSZPa6mobQnXMRvvZN5EtNhJazINI6TYVWpSPwXwi/vffj6BjD/Bnm72ieM8xfnxwMMMw/izRq1Ai///67mWPmfffdhz179iCQYJg/IZmRleOPP/5Y/TkS6tevj2HDhqn6aqZEplxDvROfofi1jPvC+Ty1sLvc00iKMO9HCCEBE+b/8MMPax988IGWnp6u9lNTU7VGjRppgQrD/L0fBupqaKunQmaDNcx/2bJlWr58+VSYfo4cObQpU6aosH6r/S8e0dLfKa2ljyuiJW2cqsXGrPAJWTmSnbfnFsP8vS8rV+ZRIMyrZIb5uy+KTbJp16hRA8ePH0f37t1VFFupUqUQqARL6LovhoH6qqyCJcxf/oW98MILmDFjhtoX38OvvvpKff/NSEsFwv6z5heuCPScA+QrhZCCVaSIo0/IKpDCsRnm71lZ+WroeqDd330hzJ9RbIQQp5FQfVmiPnDggNoXn8OFCxcid+7c5h3P7QVWPAO0HQfc0TajrUq7jEc3+0IRQog7cZuClJqaqgpMSnLI6tWro0mTJu46NSHER0hPT1cJYV999VXluFmiRAnMmTNHOWjnyJHDtCPw2wzg5zFAWjLw81ig8n2MUCOEBGcUW6FChRAbG6ucnuRGKkvuq1atctdLEEK8iJjL+/Xrh7Vr16r9rl27Yvbs2ciXLx9Wr159u+O1s0DsIODfdRn7Ep3W9RMqR4SQ4FSQjh49qpSjnTt34o8//sDUqVNVZAshxP9Zvnw5nnnmGVy+fBnR0dEq55nkOpPyIGYhwn+vAr55Frh1GQjPCbQfD9z1JJUjQkjwKkiGJHBSTkCW25977jm0bt3aXacnhHiBGzduYPjw4cqMJjRo0ED5GlWtWjVT35Azu4ElvTN2itfKcMYukrkfIYQElYIk2XLl36WUEBgyZAiaNm2KY8eOIdBhosjbcjB91HOMiSJ9O1Hkjh070LdvX/zzzz9qpUgi1kaPHq3+BFk7f3KRmgit/Si06EJIb/kKEJ7DriO2nuthokj3yMrWcb3zzdGjo2tzFV+Sla32QJeVwESRbmTevHnYt28f+vTpowpVBhJMFEkCnbS0NKxYsQKLFy9Wz8W3UFaRxKfQDC0dFS/+hFMFmiI54r9kaywVQggJxkSR06ZN0/bu3eu57It+BBNFej+RmKvJ0TyVdC0QEkUePnxYa9asmUr6KFvPnj218+fPZ+4bd1RL+7yjpo3Oq6V+0V2LjYnxa1k5kp235xYTRXpfVq7Mo0CYV8lMFKmPs2fP4pFHHkGRIkWUSU0SQYaGhiKYCZbkh76YSMxXZeWviSIlyeOgQYPUvyrJZ/TJJ58oE5uY18zYFwN8OxxIvApE5IJWvStwyr9lFUgJ/Zgo0rOy8tXkh4F2f/eFRJG6tZtRo0apYrRSqXvRokVISEjAW2+9lZVrJIT4AKIQiTncsOR89913q0hUCek3U46SrgOxg4Glj2coRyXrAwM3QavTi1FqhJCAQ7eCJCVErl27pp5Lte6ePXsqh01CiO8jvkQbNmzAxo0b1aPsC1u2bEHdunXx5ZdfqtVg+U5LwtdKlSqZnyDuMPBpM+CPhRKvBjR/AXjqJ6CQRT9CCAk2BUnqqRkqdcvNVFaQCCG+T0xMDMqXL4+2bduqLNjyKPsPPfQQWrRooaJNZV+UpzFjxiA83Epwa57iGYpRvjLA498Bbd4AwjxjIiCEEL9SkCTEV+otSfi+ILlQfv/9d9y6dcuT10cIyQLbtm1TfoOnTp0ya5f9pUuXqoz3Yl4Tk9o999xjPlgyYktUmpAjD9BrCTBwM1Deoh8hhASzgiS5jaRAZadOnSTyTYW7i7IkIXJ33nmnugm/++67+P777z17xYQQXYgZTUqByPfVFhLCP3fuXFUuxIw9S4BPGgK/f3a7TZI+5sxP6RNCggKnQtBq1KihilRWrFgRv/76K65fv47NmzerHCkFChTAypUr1bI9IcT7yHfz0qVLdvvIcfE5MnLrKrC8PxDzDJB8HTiw6vYqEiGEBBEuZdKWzLoGGjdurDYD9v6tEkKyD0nL4VS/41uBFQOA+BNASBjQ6mWg2QhGqBFCghK3lRoxkClnCiHEKxQuXFhXv5LFigBr3wI2T1bZsVGgPNBjNlCmocevkRBC/F5BqlChgkvKj5jfhg0bBl9OOS4+VA8++CAmTZrk7cshxC0cPHhQ5S2zh3yfS5cujWZViwBzpmQoR3V7Ax0nZjhlE0JIEBPuTH01V5DwYV9m/PjxZiZCQvwZMXH/9NNP6vsqyn9UVBQSExOVMmRq/jb82ZkyZQrCStUB2r0N5C4G1OzhxasnhBA/VJBatmyJQOPw4cM4cOAAOnfurDKEE+LPxMXF4amnnsI333yj9lu3bo3evXsjT548GDlypDHUv0AU8Pn/CiJ3hzdwX4//FKK7B3nz0gkhxOfw2UJqkrROFJeSJUuqf7uxsbGZ+kyfPl2Z/uRfcoMGDcyjcXTmdpowYYIbr5oQ77BmzRrUrl1bKUeS6HHixIkq5Yb4IUnNREkGKX2mv/gIzrxRHt0qpeC+G8sZoUYIIdnlpO0uJFN3nTp18MQTT6iyJpYsWbJE+TeJkiQJ7mbOnImOHTti//79KFu2rOojSlNSUlKmsWKC2L59O6pUqaK2rVu3OrweOY/puQxlV1JSUtRmwPDcXpuzj+7ClfPqHWOvn61jgSYrvePcKSvZZHXo448/Vm1Vq1bFgAED1GYoJyJ9IkI1tEz+BW2iv0dIsgatYCWkdvwASE116v25+p5dHeepeWWtzV4ff/geOuqjVy7W2ikr/TIMNFl58/5u6zWzit5zhmh+EJcvK0hSLqFbt27GNvEbql+/PmbMmGFsE2dr6aNnVeiVV15RJVPCwsJw48YN4w/Nm2++abW/lGAYO3ZspnYp3BsdHe3yeyPEVU6cOKFKh8jqkNChQwf1hyJHjhxm/XInnkGDY58i/62MfkcLtca+Ur2QFmbejxBCgoGbN28ai3NLsmubaH6AXGZMTIxxPykpSQsLC9NWrFhh1m/YsGFaixYtnD7/3LlztZEjR9rtk5iYqMXHxxu3kydPquuKi4vTkpOTjVtCQoIWGxurHm21Obvvrs2V8+odY6+frWOBJiu947IqK5n/U6dO1aKiotQcLFy4sPouWPaTbd2iKVr6W0U1bXReLf3dctqv894IKlm5Otc8Na9clVdWZeWMXBzJJlhl5co8CgRZJXvp/u5JWcnvttw75bfcHj5rYnPkjCrmg2LFipm1y/65c+c88pryr9zynzkh2c358+dV2R9DSZ969eph2bJlKFOmjNX+8TnLIr14XYRERCGpw4c49+te1MrmayaEEH/EL01sZ86cQalSpZTvUJMmTcxC9hcsWKAi0zyF1KCTTRS0Q4cO0cRGso0dO3YoXyNZFo6IiEC/fv1w//33IzTUPNai8PX9uJyrMtJDI9V+eGoCUsNyAiE+G5NBCCHZBk1sHkaW5mhi8+4SrKvLyp5artY7zllZyVwbMGCAmm+y1axZU9u1a1fmvjevacnfjFDmtOSVzwelrPTOK28u7/uS2YgmNprYaGKzjV/+pYyMjFQRahK2bIrsN23a1GvXRYi72bNnD+6++24VpSmIeU1WTmvWrGne8cJ+hH/eFhE7Z99u8/3FYUII8Vl81sQmkWWGorjiZyHROpL4rmDBgiqMX8L8+/Tpg08//VSZ2WbNmoXPPvsM+/btQ7ly5Tx2XTSxkewgPT0d3377rTIZp6amokCBAqpkj3wXzNA0VLy4BtXPLEGYloLE8LzYXfZpXMhXhx8UIYQEoolt3bp1RpOC6davXz9jn2nTpmnlypXTIiMjtfr162sbNmzItuujiS17TCHBaGL7/PPPtXvvvdc45zt16qQdO3Yss6wuHNPOTWqmTGqypS3oqSVcOG7TTBSIsqKJjSa27DJHuuOe5Y+m22RGsfkerVq1MqsdZY3BgwerjZBAQVaNnnvuOVy/fh05c+ZUBZT79++vVpEsCUlLRoGb/0ALj0J6m3FIb/CEWxI/EkII8WETm69CExvxBFJQds6cOUa/uooVK2LEiBEoXbq0Wb8QLQ1aSJhxv8i1P5EYUQjXc5biB0MIIcFgYvN1aGLz/hJsoJiNtm3bpt1xxx3KnBYSEqJ1795du3LlSqYxkvQx7aMGWsq+VbrkF4iycte8CkRTCKPYPCsrV+ZRIMyrZJrYCCHZjeTS+vDDD1UZGzGhSW4vCTaQsjcSqXn7X0w6wn+fgRaHxiNUS0PIhglIrdyWHxghhHgQmtichCY24g4uXryIKVOmqKhLQSIxxZ8uT548Zv2iki+j/vFZKHJjv9o/k68B9pR9Esnh5v0IIYTogyY2mtgCfgnW1ffkqeVqvePmz5+v5cqVS5nU5PGzzz5T9dUsx6f8uVxLn1A2o47a28W0XbOHawk3buiWXyDIiiY298hK7/ctWMxGNLH5/v09gbXY/B8p+SCbnnbLNmf3PX3N7hhjr1+wyMrWOIlMe/bZZ/HFF1+o/YYNG6pSNZUrV840PvLiXwhf/kTGTom6SO36KU78dgg1IyMdysrQ5s+yyq55Za0tO2Tl6rmzKitbx/XKkLLSL8NAk1V2398dnTcr6D2fXxarJcSX/Yr27t2La9euqQKyzZs3R1hYGLZt24bHHnsM//77r6qd1rNnT8ybNw/R0dFWz6OVrAfU7wtEFwZavSLu2wAOZfv7IYSQYIUKUhYRh1rZTPdNH621OfvoLlw5r94x9vrZOhZospKCyhKaf/r0aWObOF5L+ZsVK1Yo5UmyvM+ePRsJCQnm509PQ+hv05FSrcft9g4fSKVmlS5Sj6z0tvmCrPSO89S8stZmr48/fA8d9dErF2vtlJV+GQaarLx5f7f1mllF7znppO0kdNIm1pAVookTJ9oVTosWLTBgwADkypXLrD1nchwaHPsUhRIO4XyeWvi10gsZihEhhBC3QydtOmkHvBNfVhyE3ekceuvWLa1UqVJWS+MYtoIFC6p+ludO2f2Vlv5OqQxH7PEltcTf52mxMTEuycqRk6MvyMqdzrRZmVfedBD1JcdjOmkzD5Kv3t8T6KTt/wSL47EvOvH5iqy2bNliZlazxuXLl/Hrr7+qEjpCeNpNRK0ehtC/lmZ0KN0IIT1mITRPaeDMarc4HvvDvNI7jk7a7pGVreN00obfOx4H2v3dAJ20CfFjzp4961y/uMNodeB1hCbHASGhQMtRQPMXgLBwMY579mIJIYTogk7aWYRO2rflYPoYTE7aBQoU0HXOIkWKZMyXnEVUPbX0fGWR3u1TaKUbAekakJ7idsdja2100kZAOtPSSduzsvIXx2M6aTuGTtoegk7axJQzZ85g8uTJ+Oeff+wKpnb5Qhg96VOEhWcsHedOPIvEiPxIDctJgRJCSDZCJ206aQe8E19WHISz+r4l8/XMmTONGbGjo6ONxWZNnbNl//G6EVry2CJa6uaPPSorR06OdNKmkzadtF13aPd1x+NAu78n0Enb/wlUx2NXxwSDk/alS5fwzDPPqNxGgjhez58/H9u3b8ewYcOMDtv5o4AFDxVEpwopQHoS8O9ahDUdouua6aStTx7uklUgOdPSSduzsvJVx+NAu78boJM2IX7C2rVr0bdvX2VaCw8Px/jx4zFy5EiVLVsyZ99///2YNGkSGhVNQotLCxGVdAkIDQdavwrcM5z5jQghxE+gkzYhOkhKSsIbb7yhlB9N01ClShVVR61BgwZm/cKQhocL7kflM6sRIla2gpWAnp8Bpcz7EUII8W2oIGURRrHdloPpYyBFsf39999q1WjPnj2q7emnn8Z7772nMmJbvm7qub9R6cIPSjlKr/sY0tq+DUTmNgvfz87yGdbaGMWGgIw2YhSbZ2Vlq51RbCl+KytHsNSIkzCKLXiQlaIffvgBc+fORXJyMvLkyYOhQ4eicePGdsdVuLhGRaidzd8w266VEEKIPhjFxii2gI9y8GRk1tGjR7W77rrLGI3Wtm1b7fjx45n7Xj2rpS3uoyWf3KX7ehjFpn9usdSIe2Sl9/vma9FGvlSWxR33LH+UVTKj2IirBEpklrvGBEIUm6waPf744zh//jwiIyNVEVqJTgsNDTXveOQXIGYQcOMcQq8cBQZsdOp6GMXmO7IKpGgjRrF5Vla+GpkVaPd3A4xiI8QHSExMxKhRo/DRRx+p/bJly6pQfktHbKQkAmvHAb9Oy9gvXBXoOo0RaoQQEkDQSZsQAHv37kWvXr3w119/KXkMGTIELVq0QO3atc3lc+FvYHl/4HxGPzR8Gmg7DoiMphwJISSAsLAZEBJcpKenY+rUqWjYsKFSjooWLYrvvvsOH374IXLkyGHe+cwfwKxWGcpRdGHg0SXAA5OoHBFCSADCFSQStJw9exZPPPEEfvzxR7XfqVMnzJkzRylJVsNAi9cGSjcEwqOAbtOB3EWz/6IJIYRkC1SQSFDyzTff4KmnnkJcXByioqJUwdmBAwciJCTErF/Iv+uBCvdkrBKJk/Yji4AceehvRAghAQ5NbCTo8l8MGjQIXbt2VcpRnTp1sHPnTtVmphyl3EKtk/MR/tX/gDVv3G6PykvliBBCggCuIGURZtK+LQfTR1/MpL1792706dMHhw4dUvvPP/88xo0bp3yNzMaf24uw2GdQ8dJhtZsWEo705GRdilFWs/gyk7b3ZMVM2rZlSFm5Po/8fV558/5u6zWzCjNpewhm0vZPR+yVK1di4cKFSE1NRYECBTB8+HC1emSGlo5KF35E9bNfI1RLQ2J4Puwq9wwu5q3lrUsnhBDiZphJm5m0Az7Tqp7r+/fff7XWrVsbM2J37dpVO3v2bOa+l05oaXM7adrovGpLWfCgtnrZAr/ODp1dWcddPa8vycqR7LwtL2bS9r6sXJlHgTCvkplJm7iKv2SH1nPNgZZJe9myZXjmmWdw5coVREdHqwSQTz75ZCZHbEV4GHDuTyA8J9BhArTavZH8/fd+nR3aH+aV3nHMpO0eWdk6rne+BVp2aGbSdk6+EcykTYh/c/36dTz33HOqyKxw1113KfNalSpVzDumJgPhkRnP85YEHpwH5C0FFKkiRmovXDkhhBBfgVFsxG9JS0vDhg0bsHHjRvUo+7/99hvq1aunlCNZKXr11VexdevWzMrR6Z3AjCbAgdW32yq1zlCOCCGEBD2MYiN+SUxMDEaOHIlTp06pfcljlDdvXty4cUM5ZZcpUwYLFixAy5YtzQempwFbpgDr3gHSU4H1E4CqHRm6TwghxAwqSMTv2LZtG9577z1omvhd3+batWvq8Z577sG3336rotXMuHoSiBkAHN+SsV+9G9B5CpUjQgghmaCCRPwKMaPNnj07k3JkyokTJ9Rqkhl/LQe+fR5IigcicwMd3wPq9qJyRAghxCr0QSJ+xebNm3Hp0iW7fU6ePIlNmzbdbji9C1j2ZIZyVOouYMBGoF5vKkeEEEJswhUk4ncFZp3uV6o+cNeTQHRhoOVLQJhnQpEJIYQEDkGtIIWHh6NmzZrGUHAx3RDfRdLD//jjjw77hYUAjZO3ANdaAXlLZDQ+MJkrRoQQQnQT1ApS/vz58ccff3j7MogOjhw5gscff1yF8dujYoFQLHk4HyoeXQDEngL6xGQoRjrqqBFCCCEG6INEfBpxxl67di0aNmyolKN8+fKhS5cuKseRZUbsvnUisHtALtxVLA3IkReoSz8jQgghAaYgSfK/zp07o2TJkuqHMDY2NlOf6dOno0KFCoiKikKDBg3MHXN1IGHhMq5Zs2Yq0SDxLS5fvoxevXrh448/VvmNWrRogZ07d6pyIYsXL0apUqVUv/xRwFc9c+KLbjmRN0cIULYJMHAzUPtBb78FQgghforPKkgJCQmq2vonn3xi9fiSJUtURfbXXnsNu3fvRvPmzdGxY0cV4m1AlB/xMbLczpw5o44fO3ZM/eB++umn6Nu3rzGPDvE+69evV5//8uXLERYWhrfffhu//PILypYtq453795dfX5bVs7Fvy+WwCM1I6CFhAH3vg48/h1QoJy33wIhhBA/xmd9kETZkc0Wkjn5qaeeQv/+/dX+lClTlAPvjBkzMGHCBNUmyo89ZHVKEKWpevXqOHTokHLWtkZSUpLaDBiUKXEcls2A4bm9Nmcf3YUr59U7xl4/W8estYtiPH/+fJUpW8xrlStXxsCBAzFo0CCVIdt0jBROrNuiE9L2vYv0nEWQ3m0mtFINgLT0jC0LuPoZ6BnnLlm52uYL88oXZWWvjz98Dx310SsXa+2UlX4ZBpqsPPk99LasHBGi2cu45yOIiU1+MLt166b2k5OTVXX2pUuXqpUEA1KgVJyu9ZjLDBXec+TIocpVSPZlWYkqWLCg1f5jxozB2LFjM7UvWrRInYdkndOnT+ODDz7Av//+q/bbtm2rzGk5c+Y065czOQ63IgoCIRkLoLmSziMxPB/SwqL4MRBCCLHLzZs3lftGfHx85qTC/rCCZI+4uDiVUblYsWJm7bJ/7tw5Xef4+++/MWDAAISGhioFbOrUqTaVI+GVV17BiBEjzFaQpN5X69atzQScmpqKdevWqXZJI2Ctzdl9d+HKefWOsdfP1jFDe6tWrdSq0ahRo3Dr1i3kyZNHrQSK8ms2NiwMIX8sRMTa15DafBTSGw/2KVnpHZcVWdmbV5Ztgq/OK1+TlaV8BE/ISu/7dmWMoz565WKtnbLSL8NAk5Unv4fekpVedxq/XEESHyJx0JUq7U2aNDH2Gz9+vCpQeuDAAY9dy7Rp09QmCpqY5LiClDVEgxd5/v7772pf/I5kJdBSWY1IvY66Jz5HyfgMs+n5vLXxa8WRDN8nhBDikRUk8fPweeQyY2JijPtJSUlaWFiYtmLFCrN+w4YN01q0aJEt1xQfH6+uKy4uTktOTjZuCQkJWmxsrHq01ebsvrs2V86rd4y9fraOrVy5UitQoICSY2RkpPbee+9p169fzySLLfPf1tLer6Jpo/Nq6WMLaXtnPqMl3Ljuc7LSO84VWemZV5ZtvjyvfE1WjmTnbXllVVbOyMWRbIJVVq7Mo0CQVbIHv4fekpX8bsvvjvyO28Nno9jsERkZqSLU1qxZY9Yu+02bNvXadRF9JCYm4oUXXkDXrl2VL1i1atVUjTWJShSTp5HUJESsG4umR95D6I1z0ApVRuJjq3Ck2P1G/yNCCCHEE/isiU3y3vzzzz/qeb169VTUmtgixfQiod4S5t+nTx8Voi9mtlmzZuGzzz7Dvn37UK6c50K8aWLLGsePH1efpTwK999/P/r166ec5S3Jc+skWh18E6FaGo4Wvhf7Sj2KtNDM/QghhJCgMbGtW7dOLYFZbv369TP2mTZtmlauXDllnqlfv762YcOGbLs+mticW4IVE+mkSZO0HDlyqM+xSJEi2rJlyxwuwSZtnq79+sWbfmGOdMfyPk1s3pFVsJqNaGKjiY0mNtv4bBSbRDY5WtwaPHiw2ohvI5GFb731Fnbt2qX2O3TooFb7ZDXQzEyacBGR341AvtDbjvep9frhXNwa1PLGhRNCCAlafNbE5qvQxOYc27dvV9nQZSlTEjtKwVkxq1nWUSsavwf1TnyGqNRruJqzLDZUfYsRaoQQQtyO35vYfB2a2Owv31+9elUbOHCg0TRavnx5bfv27ZnGfLPiay155XAVoSZb2seNtLVffazbpEYTG6PYaGKjiS07zJGumGoDwXSbHMRRbD5rYvMXZFVENj3tlm3O7nv6mt0xRvqIo7xo55KMU5C8RpKpXHIcmZ3j/F9oeXA0IhJPZ+w3HoS0Vq/h+k+/+LWs9I6z1ycr88rQZus4ZeX4e+lpWbl67qzOK1vH9c43ykq/DANNVp68Z2W3rPSejwpSFmEttttykFpp77//virLIvslSpTA7NmzlT+Z+BqZ1b85uwfhX3RE3rRkaLmKIK3zJ9AqtfHrunV6x7EWm2/Kyt9rZrEWm2dl5S/1xViLLchqsfkS9EGyzqVLl1S5lj///FPtN2rUCEOHDrVt39XS0eTI+0gLicQfZZ9CcoQdOzAhhBDiJuiDRB+kbLNRL1myRCtYsKCy6UZHR2szZsxQ2c4tz3Fr7zda8o0rt9uvXNBiY2JctlHTB4mZtBnmzzB/V+5Z9EGiDxLog+R5gsEHSerOSd27ixcvKrNZ8+bNERYWppJ5SvbrOXPmqH6VKlVCbGwsatasaX6C5ATUOTEXUbvXAQ2eADpPyWjPlV9FqgWSrJwZRx8k35FVIPmK0AfJs7LyVb+arJybPkjWoQ9SFgl0H6Rly5bh2WefVSY0A1IoeMiQIfj8889VtnMJ2R8xYgTuvvtuVKxY0cLX6A+ExQxA+StH1G5aRC6kJycrxSjQZKV3HH2QfFNW/u4rQh8kz8rKVjt9kFL8VlaOoA+SkwSTD9K2bdswceJEu30KFSqkVpFq1bJI5ailo/KF1bjzzHKEIg23IgpgV7lnEJenhmcvmhBCCLEDfZDog5QlH6Rbt25ppUqVslruxbDlzJlTO3PmTOZzxx3X0j7vaMxtlLLwEe27ZQvcnieDPkj0QaIPEn2Q6IPEPEjJzIPkmwSqD9KWLVtw+vR/+YlscOvWLRw8eFD5JZmdOzUCuPA3EJEL6DgRWs2HkfL99wErK1fH0QfJd2QVSL4i9EHyrKxstdMHKcJvZKX3fPRBIlY5e/asc/1Sk2435i0BPDgPyFcaKFRJDL6UMiGEEL+CClIWCVQn7SJFiug6p/RLPbYNYbEDUKJAV6SktM04UKap4UU96vDoC7JydhydtH1TVnTSti1Dysr1eeTv88qT30Nvy8oRdNJ2kmBw0pb3tnjxYixdutRuv2JFCuGXsQ+g2vlvEIp0XM1ZHhuqjmWRWUIIIT4LnbTppO2S8+KBAwe0u+++265zdkhIiFY+f6h24d36tx2xF/fTVi1bmK3FDOmkTSdtOmnTSZtO2nTSTqaTtm8SKI7HUnFmwYIFKr+RJIDMly8fZsyYoRJCDh482CwP0tAWRfHBvUDErX+AyDxAp8nQ7uyO1NWrPeZM60uyctc4Omn7jqzopM1ite6eW4E2r4TsvL87Om9WoJM20c2VK1cwaNAgLFmyRO1LpmxRlsqVK6dsteHh4aqmmmTSrpLrBhrsegFIFz+ju4EeM4EC5emITQghJKCgk3aQs2HDBvTp0wcnT55Uq0Vjx47Fyy+/rJ4bkOctW7a8rXWHHwJyFQGajQDCOIUIIYQEHvx1C1KSk5MxZswYvPvuu8q8JnXUxOG8UaNG5h3TUlDlXCxwrR5QqGxGW8f36IhNCCEkoKGCFIRh/vv378dTTz2FnTt3qv3HH38ckydPRu7cuc1f8/IRhMYOxJ1ndyPtm4tI6b3CpmLkrTBQhvm7Hs7uLgIlJYK/h2M76qNXLtbaKSv9Mgw0WQkM8ycBH+YvK0U///wzZs+ejaSkJKUQiQN206ZNLTui7OWNqHXqS4SnJyE5LBp7yjyBMwUae+vSCSGEELfAMH+G+ZuFPp49e1br0qWLMVS/devW2r///ps5PDL+vJb2VW9j+H7q7PbaD1/PcVi/zV4KAYb5Z4+sLNuYEkG/rBzJzp2bK+fWM8ZRH71zyNn0GsEiK1fmUSDIKtlL93dPyiouLk79DsbHx9vVFGhiyyL+ELq+Zs0a9OvXT5UFkYi0t99+Gy+++CJCQ0PNO0r9tAU9gOtngNBw4N7Xkd5wEBJ/+NEnw0AZ5m9ffpQVw/z1fg/1fjcDLXTdUR9/Cl3Pyrl98f5ugGH+xCOIGe3VV19V/kVC1apVMWDAAAwdOjSzciTkLwfkyANEVgZ6zgZK1mP4PiGEkKDEyq8kCQTEEbtx48ZG5UjyHP3222+oWLGiecfLR4H0tIznkdFAryXAgI0ZyhEhhBASpFBBCjDEEXv69Olo0KAB9uzZg8KFC+Obb75RbWbO5JoGbJ8DTG8CbP34dnvBCkBkLq9cOyGEEOIr0AcpgLhw4QKefPJJfPfdd2q/ffv2mDdvHooXL27eMSEOWP08cOj7jP0T2wDtOeY2IoQQQv6DK0gBwvfff49atWop5ShHjhyYOnUqVq9enUk5KnrtT4R/1iJDOQqLBNpPAB75isoRIYQQYgJXkPw8UeStW7eUI7bkZhJq1KiB+fPnK2VJcjXJpkhNBH4egyZHZqtdrUg1pHadCRSrAUgfQz8r70/PtTBRpG8mP7TWxkSRCMiEfkwU6VlZ2WpnosgUv5WVI0Ik1t/trx7A+FKiyKNHj+LDDz/EiRMn1H6nTp3Qt29fREZGZuqb59YptDz4JsK0VPxbpC32lXwY6aGZ+xFCCCGBDBNFBnCiyMTERO3999/XIiMj1TUUK1ZM+/bbbx0mx0rcOlPb+sXYgEkk5mpyNE8lXdM7jokifU9WwZr8kIkimSjSV+/vCUwU6f9kd/LDM2fOqNppkvxR6Ny5M+bMmYMiRYqYX8D188C3zwGtRhlD9lPuegIXLqwOmERiTBRpX36UFRNF6v0eMlEk/D75YVbO7Yv3d19IFEknbT8iNjYWtWvXVspRzpw5MWPGDKxcuTKzcnRgNTCjSYYj9spnM0L6CSGEEKIbOmn7GOLbtGHDBmzcuBG5cuVC69atkZiYiBEjRmDWrFmqT926dZXv05133mk+OPkm8NNrwI7PM/aL1crIiB0S4oV3QgghhPgvVJB8iJiYGIwcORKnTp1S+5IFu2jRoggLC1N11ASpofbWW2+pUH4zzu4BlvcH4g5l7DcZCrR5Ewi36EcIIYQQh1BB8hG2bduG9957T2XCtkz+KBQoUABLly5FmzZtMg8+sxuY3RZITwFyFwe6zwAq3Ztdl04IIYQEHFSQfMSsNnv27EzKkSmSSqBVq1bWDxavA1RoDkREA10+BqILeu5iCSGEkCCATto+wObNm3Hp0iW7fU6fPo1Nmzbdbjj4A5B0I+N5aCjw8JcZG5UjQgghJMtQQfIBDP5FuvqJUrRyCPDVw8CPr9w+KAVm6YxNCCGEuAWa2HyAEiVK6OpXJdd1YGZz4PK/kgQdyFUkI4SfihEhhBDiVoJ6BUlKdUgYffXq1VXtsoSEBK9cR7NmzVCoUCGE2FB0wkJDMLFTEdTf/XKGcpS3NPD4qowoNSpHhBBCiNsJagVJMlKPGzcO+/fvV7mHMoXOZxMSxt+/f3/13FJJKp03FL/0zYmXGiQhREsDavQABm0GyjfzyrUSQgghwUDQKkj79u1T6cabN2+u9gsWLIjwcO9ZHJs0aYLFixejVKlSZu1Fi5dA44r5gcjcQLdPgf99DuQs4LXrJIQQQoIBn1WQJJO01BkrWbKkWlWRMhuWTJ8+HRUqVEBUVBQaNGhgHuXlgMOHDyN37tzo0qUL6tevj3feeQfepnv37jh27BjW/vidypwtJUV+338cOfosAQZuAuo+SpMaIYQQEsxO2uIPVKdOHTzxxBPo2bNnpuNLlizB8OHDlZJ0zz33YObMmejYsaMyl5UtW1b1EaUpKSkp09iffvoJKSkpSqH6448/VLbqDh06oGHDhmjbtq3V65HzmJ7r2rVr6lHOI5sBw3N7bfYeI8LD0erIO8hdqyVqNm2K9PR0pJdqbDgRsoK1a3PXGHv9bB3LqqxcfU96cPW8esZ5SlZ62ygr+3KxJjt34qnvoaM+eueLtXbKSr8MA01W3ry/23rNrKL3nCGaveyEPoKsIEkZjm7duhnbGjdurFZ+pGCrAalNJn0mTJigK3P12LFj8cMPP6j9999/31jKwxpjxoxR/S2RmmiSxNGdlLzyO+44vwobqo7lihEhhBDiRm7evIlevXohPj4eefPmtd1R8wPkMmNiYoz7SUlJWlhYmLZixQqzfsOGDdNatGih65wpKSla3bp1tcuXL2tpaWlap06dtG+//dZm/8TERC0+Pt64nTx5Ul1XXFyclpycbNwSEhK02NhY9WirTc/+Nyu+NjuHOzZr1+auMfb62TrmLlk5+548JSu94zwlK8s2ykq/rBzJzttzK6vzypk55Oz3Llhk5co8CgRZJXvp/u5JWcnvtvx+y2+5PXzWxGaPuLg4VZ6jWLFiZu2yf+7cOV3nEIds8Ttq0aKFKvHRrl07dOrUyWZ/iXDLzii39NDIbHstQgghhASAie3MmTMq2mvr1q0q+svA+PHjsWDBAhw4cMBj1zJt2jS1iYJ26NAhj5jYCCGEEOJdE5tfriAVLlxY5Q6yXC2SyveWq0ruZsiQIWoTJ+18+fKpRJOmAk5NTcW6detUuyFtgGWbs/vuwpXz6h1jr5+tY4EmK73jPCUryzaBstInK8u5ZCk7d+Kp76GjPnrnkLV2ykq/DANNVt66v3tSVoYgq4BcQTI4aUuUmkSxGZCM2F27dtXlpO0qXEEihBBC/Be/d9K+fv26tnv3brXJZU6ePFk9P378uDq+ePFiLSIiQpszZ462f/9+bfjw4VquXLm0Y8eOZcv1iXOXJ520fcXx2Jed+HxNVnrH0Unb92QVCM60dNL2rKzccc/yx3mVTCdt32PHjh3GZTZBEicK/fr1w7x58/Dwww/j0qVLqlSIVLmvWbMmVq9ejXLlymXrdUo2btn0tFu2Obvv6Wt2xxh7/YJFVnrHeUpWhjZbxykrx3PN07Jy9dxZnVe2juudb5SVfhkGmqyy+/7u6LxZQe/5fNYHqVWrViq6zB6DBw9Wmzdxd6JIW+dw17UGWiIxX5KV3nFMFOmbsvL3hH5MFOlZWflL8sNAur8LTBTpR9AHiRBCCPFf/N4HydehD5L3bdSu2t09Zc/3Nb8ayoo+SK7MoWDwq6EPEn2QoCNRpM8WqyWEEEII8RZ+EebvS9DERgghhPgvNLHRxJZty9U0sdHE5u/myGA1G9HExjB/X72/J7AWm/8TLKHrvhgG6quy0juOYf6+I6tACsdmmL9nZeWroeuBdn83wDB/P4Zh/rflYPqo5xjD/D0fum6tjSkREJDh2Azz96ys/CV0nWH+jtErI/ogueiDJDVjDh8+jNmzZ7NYLSGEEOJHPkj9+/fH1atXVU1VW1BBcpFTp06hTJkyrg4nhBBCiBc5efIkSpcubfM4FSQXSU9Px5kzZ5AnTx5VTNeUhg0bYvv27Xbb7O1LpWFRvuTDs5vEygWsXZu7xtjrZ+tYoMlK7zhPycqyjbJyTn6GfU/OK1vX4o4xjvronUPW2ikr/TIMNFl54/7uSVlJ8P7169dRsmRJhIaG+l+pEV9HhGpL8wwLC8v0gVq2OdoXZN/dE8Pa67hrjL1+to4Fmqz0jvOUrCzbKCvn5Ge574l5Zeta3DHGUR+9c8haO2WlX4aBJitv3t89JSt7pjUDTBTpAYYMGeKwzdG+p3DldfSOsdfP1rFAk5XecZ6SlWUbZeWc/HxZXlmdV7aO651vlJV+GQaarALx/q4Hmth8EFlaFO3WYZ0YQllxXvE76GV4v6KsAnVecQXJB8mRIwdGjx6tHgllxXnF76Avw/sVZRWo84orSIQQQgghFnAFiRBCCCHEAipIhBBCCCEWUEEihBBCCLGAChIhhBBCiAVUkAghhBBCLKCC5OccPXoUrVu3RvXq1VGrVi0kJCR4+5J8lvDwcNStW1dtUqiQOC7oWK5cObzwwgsUlQ2kXIGURpA5Jd+/zz77jLKyg5SNaNWqlbpf1a5dG0uXLqW87NC9e3cUKFAA//vf/ygnC1atWoWqVavijjvuUEXjPQHD/P2cli1b4u2330bz5s1x+fJllVBLFAGSmcKFCyMuLo6i0clrr72Gw4cPo2zZspg0aRLlZoW0tDQkJSUhOjpaKZQ1a9ZUdaQKFSpEeVnh7NmzOH/+vFIoL1y4gPr16+PgwYPIlSsX5WWFdevW4caNG/jiiy+wbNkyyug/UlNTlZIt8pHfPJlHv/32GwoWLAh3whUkP2bfvn2IiIhQypEgk4PKEXEHohgdOHAA999/PwVqB6kbJcqRkJiYqBQmKYRJrFOiRAmlHAlFixZV9yz5Y0esI9YBKYhOzPn9999Ro0YNlCpVSslH7lM//vgj3A0VJA+yceNGdO7cWVUMDgkJQWxsbKY+06dPR4UKFRAVFYUGDRpg06ZNTv2I5c6dG126dFEa9DvvvAN/xdOyMqSul3HNmjXDhg0b4K9kh6zErDZhwgT4O9khq6tXr6JOnTqqePVLL72kVir9leyQl4EdO3YgPT1dVWz3R7JTVoHGxizK7syZM0o5MiDfvdOnT7v9OqkgeRDxB5Ib5yeffGL1+JIlSzB8+HBlyti9e7daCerYsSNOnDhh7CMTQ5btLTeZICkpKWrSTJs2Ddu2bcOaNWvU5o94WlbCsWPHsHPnTnz66afo27evUpj8EU/LauXKlahSpYra/J3smFf58+fHnj17lD/gokWLlAnJX8kOeQmXLl1S38FZs2bBX8kuWQUiCVmUnbVVWlG03I5GsgURdUxMjFlbo0aNtIEDB5q1VatWTXv55Zd1nXPr1q1a+/btjfvvvfee2vwdT8jKkg4dOmjbt2/X/B1PyEr6lS5dWitXrpxWqFAhLW/evNrYsWM1fyc75pWc6+uvv9YCAU/JKzExUWvevLk2f/58LVDw5Nxat26d1rNnTy1QgQuy27Jli9atWzfjsWHDhmkLFy50+7VxBclLJCcnq9WMdu3ambXL/tatW3WdQ6Jn5N/qlStX1FK1LFveeeedCDTcISuRkTjTCqdOncL+/ftRsWJFBBrukJWY1iTaSFbcxDn76aefxptvvolAwx2yku+fYSVSHuU7KJE1gYg75CW/h48//jjuvfde9OnTB4GKO2QVrCTrkF2jRo3w119/KbOaRJKuXr0a7du3d/u1MNzJS0g0lTh0FitWzKxd9s+dO6frHOKQLX5HLVq0UDcemUCdOnVCoOEOWf39998YMGAAQkND1VLs1KlT3R7xECiyChbcIStRtp966in1/ZNt6NChKnw9EHGHvLZs2aLMJyIjg9/JggULVIqEQMJd30P50d+1a5cySYmfTUxMjPpjHMjE6ZCd/PZ98MEHyoldFgfE988TkaNUkLyMpd1UbrLO2FLFLitbMJAVWTVt2hR79+5FsJDVeWVA/u0HOlmRlfiQ/PHHHwgmsiIvCZCQH7RgIavfQ09EZgWK7Lp06aI2T0ITm5eQSBcJEbb8NyG5QSw152CHsqKsOK+8D7+HlFWwzTMqSF4iMjJS/fu0jDqTfVntIJQV5xW/g74E71mUVbDNM5rYPIhkQP3nn3+M+xIGLMvx4vsi2YlHjBihHBXvuusuNGnSRIW8ShjjwIEDEWxQVpQV55X34feQsuI8M8HtcXHELDxTRGy59evXz9hn2rRpKpw6MjJSq1+/vrZhw4aglCBlRVlxXnkffg8pK86z27AWGyGEEEKIBfRBIoQQQgixgAoSIYQQQogFVJAIIYQQQiyggkQIIYQQYgEVJEIIIYQQC6ggEUIIIYRYQAWJEEIIIcQCKkiEEEIIIRZQQSKEEEIIsYAKEiGEmLB161aEhISgQ4cOlAshQQxLjRBCiAn9+/fHzZs3sXz5chw+fFgVliaEBB9cQSKEkP9ISEjAkiVLMHz4cNx7772YN28eZUNIkEIFiRBC/kOUo+LFi6NRo0bo3bs35s6dC03TKB9CghAqSIQQ8h9z5sxRipHQrVs3XLhwAWvXrqV8CAlC6INECCEADh48iGrVqqnHKlWqKJn06tVLPS5atIgyIiTI4AoSIYT8t3rUsGFDo3IkyGrSihUrcOXKFcqIkCCDChIhJOhJTU3F/PnzjStGBtq3b488efJg4cKFQS8jQoKNcG9fACGEeJtVq1bh/PnzqFmzJv766y+zY82bN1erS0OHDvXa9RFCsh/6IBFCgp7OnTsrJckeO3fuRP369YNeVoQEC1SQCCGEEEIsoA8SIYQQQogFVJAIIYQQQiyggkQIIYQQYgEVJEIIIYQQC6ggEUIIIYRYQAWJEEIIIcQCKkiEEEIIIRZQQSKEEEIIsYAKEiGEEEKIBVSQCCGEEEIsoIJECCGEEGIBFSRCCCGEEJjzf5Dhz0SdZCsZAAAAAElFTkSuQmCC", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "\n", "x = 1.0\n", "dfdx_exact = np.exp(x)\n", "Deltas = np.logspace(0, -6, 13)\n", "dfdx_approx = np.zeros(len(Deltas))\n", "\n", "for i, Delta in enumerate(Deltas):\n", " dfdx_approx[i] = (np.exp(x + Delta) - np.exp(x)) / Delta\n", "\n", "plt.loglog(Deltas, abs(dfdx_approx - dfdx_exact), 'k-o', label='forward difference error')\n", "plt.loglog(Deltas, Deltas, 'C1--', label=r'$\\mathcal{O}(\\Delta)$ reference')\n", "plt.xlabel(r'$\\Delta$')\n", "plt.ylabel(r\"$|f'_{approx}(x)-f'(x)|$\")\n", "plt.grid(True, which='both')\n", "plt.legend();\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**Exercise**: Repeat the previous exercise but use the *backward* difference. Do you observe any difference?\n", "\n", "
\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Back to Order\n", "\n", "The solved exercise above suggests that the forward-difference approximation yields a better approximation as $\\Delta$ grows smaller, but *how much better*? To answer this, we consider the **Taylor series** for $f(x+\\Delta)$ about the point $x$:\n", "\n", "$$\n", " f(x+\\Delta) = f(x) + f'(x)\\Delta + \\frac{1}{2}f''(x)\\Delta^2 + \\mathcal{O}(\\Delta^3) \\, .\n", "$$\n", "\n", "By isolating $f'(x)$ on one side, we have\n", "\n", "$$\n", "\\begin{split}\n", " f'(x) &= \\frac{f(x+\\Delta) - f(x)}{\\Delta} - \\frac{1}{2}f''(x)\\Delta + \\mathcal{O}(\\Delta^2) \\\\\n", " &= \\frac{f(x+\\Delta) - f(x)}{\\Delta} + \\mathcal{O}(\\Delta) \\\\\n", "\\end{split}\n", "$$\n", "\n", "In other words, the forward difference leaves out terms proportional to $\\Delta$ and higher powers of $\\Delta$. Moreover, as $\\Delta$ shrinks, $\\Delta \\gg \\Delta^p$ for $p > 1$. The terms left out represent the **truncation error**, and because this error goes as $\\Delta^1$ for small $\\Delta$, the *order of the error* is 1, i.e., the forward-difference approximation is a **first-order approximation**.\n", "\n", "
\n", "\n", "**Exercise**: Show that the backward-difference approximation is also first order.\n", "\n", "
\n", "\n", "Can we do better than first order? It turns out that we can. Consider the following Taylor series:\n", "\n", "$$\n", " f(x-\\Delta) = f(x) - f'(x)\\Delta + \\frac{1}{2}f''(x)\\Delta^2 + \\mathcal{O}(\\Delta^3) \\, .\n", "$$\n", "\n", "Subtraction of this series from the one above for $f(x+\\Delta)$ leads to\n", "\n", "$$\n", " f(x+\\Delta)-f(x-\\Delta) = 2 f'(x)\\Delta + \\mathcal{O}(\\Delta^3) \\, ,\n", "$$\n", "\n", "or\n", "\n", "$$\n", " f'(x) = \\frac{f(x+\\Delta)-f(x-\\Delta)}{2\\Delta} + \\mathcal{O}(\\Delta^2) \\, .\n", "$$\n", "\n", "This latter approximation for $f'(x)$ is a **central difference** and is a **second-order approximation** because $\\Delta$ is raised to the second power. \n", "\n", "> **Note**: Forward and backward differences yield first-order approximations to $f'(x)$, while the central difference yields a second-order approximation for $f'(x)$.\n", "\n", "\n", "
\n", "\n", "**Exercise**: Consider the Taylor series for $f(x-\\Delta)$, $f(x+\\Delta)$, $f(x-2\\Delta)$, and $f(x+2\\Delta)$. Try to develop a fourth-order approximation for $f'(x)$ that involves these four values of $f$.\n", "\n", "
\n", "\n", "**Exercise**: Consider a function $f(x)$ for which you are given values only at $x = i\\Delta$ for $i = 0, 1, 2, \\ldots$ for small $\\Delta$. Suppose that some application requires that you evaluate $f(x)$ at some middle point $(i+1/2)\\Delta$. Of course, the natural thing to do is *average* the two adjacent values at $i\\Delta$ and $(i+1)\\Delta$. What order is this *averaging* approximation? \n", "\n", "
\n", "\n", "**Exercise**: Apply the *central* difference to the function $f(x) = \\sin(2x - 0.17) + 0.3 \\cos(3.4x + 0.1)$ at $x = 0.5$. Plot the error for $\\Delta = 10^{-1}, 10^{-2}, \\ldots, 10^{-7}$.\n", "\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## When Smaller Is Not Always Better\n", "\n", "The Taylor-series argument says smaller $\\Delta$ should reduce truncation error. Floating-point arithmetic adds another effect: **roundoff error**. If $f(x+\\Delta)$ and $f(x)$ are nearly equal, the subtraction can cancel many shared digits before division by $\\Delta$ amplifies the remaining noise.\n", "\n", "For a smooth real function that can also accept complex inputs, the **complex-step derivative approximation** avoids that subtractive cancellation:\n", "\n", "$$\n", " f'(x) \\approx \\frac{\\operatorname{Im}\\left[f(x+i\\Delta)\\right]}{\\Delta}.\n", "$$\n", "\n", "Why does this work? Taylor expand into the complex plane:\n", "\n", "$$\n", " f(x+i\\Delta) = f(x) + i\\Delta f'(x) - \\frac{\\Delta^2}{2} f''(x) + \\mathcal{O}(\\Delta^3).\n", "$$\n", "\n", "The imaginary part is $\\Delta f'(x) + \\mathcal{O}(\\Delta^3)$, so dividing by $\\Delta$ gives a second-order derivative estimate without subtracting two nearly equal real numbers.\n", "\n", "Use this method for analytic callable functions that support complex inputs. Do not use it for already-sampled data, functions with non-analytic operations such as `abs`, or code paths that reject complex numbers.\n" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "execution": { "iopub.execute_input": "2026-07-05T18:02:06.125193Z", "iopub.status.busy": "2026-07-05T18:02:06.125020Z", "iopub.status.idle": "2026-07-05T18:02:06.230512Z", "shell.execute_reply": "2026-07-05T18:02:06.229417Z" } }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "def forward_difference(f, x, Delta):\n", " return (f(x + Delta) - f(x)) / Delta\n", "\n", "\n", "def central_difference(f, x, Delta):\n", " return (f(x + Delta) - f(x - Delta)) / (2 * Delta)\n", "\n", "\n", "def complex_step_difference(f, x, Delta):\n", " return np.imag(f(x + 1j * Delta)) / Delta\n", "\n", "\n", "def f(x):\n", " return np.exp(x)\n", "\n", "x = 1.0\n", "dfdx_exact = np.exp(x)\n", "Deltas = np.logspace(0, -16, 17)\n", "\n", "error_forward = np.array([abs(forward_difference(f, x, Delta) - dfdx_exact) for Delta in Deltas])\n", "error_central = np.array([abs(central_difference(f, x, Delta) - dfdx_exact) for Delta in Deltas])\n", "error_complex = np.array([abs(complex_step_difference(f, x, Delta) - dfdx_exact) for Delta in Deltas])\n", "\n", "plt.loglog(Deltas, error_forward, 'o-', label='forward')\n", "plt.loglog(Deltas, error_central, 's-', label='central')\n", "plt.loglog(Deltas, error_complex, '^-', label='complex step')\n", "plt.gca().invert_xaxis()\n", "plt.xlabel(r'$\\Delta$')\n", "plt.ylabel('absolute error')\n", "plt.grid(True, which='both')\n", "plt.legend()\n", "plt.title('Truncation error improves, then roundoff takes over');\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Approximating the Second Derivative\n", "\n", "So far, the finite differences developed represent approximations to the *first* derivative, $f'(x)$. Approximations for the *second* derivative can be derived in a similar fashion. Consider again the Taylor series for $f(x+\\Delta)$ and $f(x-\\Delta)$. Whereas these series were *subtracted* to yield the central-difference approximation for $f'(x)$, they can instead be *added* to yield the following:\n", "\n", "$$\n", "f(x+\\Delta)+f(x-\\Delta) = 2f(x) + f''(x)\\Delta^2 + \\mathcal{O}(\\Delta^4) \\, ,\n", "$$\n", "\n", "where both the $f'$ and $f'''$ terms cancel. By isolating $f''(x)$, we have\n", "\n", "$$\n", "f''(x) = \\frac{f(x+\\Delta)-2f(x)+f(x-\\Delta)}{\\Delta^2} + \\mathcal{O}(\\Delta^2) \\, ,\n", "$$\n", "\n", "which is the central-difference approximation for the second derivative. Like the central-difference approximation for the first derivative, the truncation error is second order." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "How can we establish order of accuracy? Earlier computational experiments used plots to reveal a trend; the same habit works here. Although order of accuracy is not the same thing as algorithmic complexity, the technique is similar: we test the approximation for different values of $\\Delta$ and see whether the error goes as $\\Delta$ or $\\Delta^2$ or something else entirely. By \"see\", I mean literally: use a graphic!\n", "\n", "
\n", "\n", "**Exercise**: Confirm graphically that the central-difference approximation for the second derivative is, in fact, second order. Use $f(x) = e^x$ for demonstration.\n", "\n", "*Solution*: I set us up for success with the last solved exercise, which is adapted here." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "execution": { "iopub.execute_input": "2026-07-05T18:02:06.231487Z", "iopub.status.busy": "2026-07-05T18:02:06.231401Z", "iopub.status.idle": "2026-07-05T18:02:06.403996Z", "shell.execute_reply": "2026-07-05T18:02:06.403300Z" } }, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "<>:10: SyntaxWarning: invalid escape sequence '\\m'\n", "<>:11: SyntaxWarning: invalid escape sequence '\\m'\n", "<>:12: SyntaxWarning: invalid escape sequence '\\D'\n", "<>:15: SyntaxWarning: invalid escape sequence '\\D'\n", "<>:10: SyntaxWarning: invalid escape sequence '\\m'\n", "<>:11: SyntaxWarning: invalid escape sequence '\\m'\n", "<>:12: SyntaxWarning: invalid escape sequence '\\D'\n", "<>:15: SyntaxWarning: invalid escape sequence '\\D'\n", "/tmp/ipykernel_1101175/3659422081.py:10: SyntaxWarning: invalid escape sequence '\\m'\n", " plt.loglog(Delta, Delta, 'r--', label='$\\mathcal{O}(\\Delta)$')\n", "/tmp/ipykernel_1101175/3659422081.py:11: SyntaxWarning: invalid escape sequence '\\m'\n", " plt.loglog(Delta, Delta**2, 'g--', label='$\\mathcal{O}(\\Delta^2)$')\n", "/tmp/ipykernel_1101175/3659422081.py:12: SyntaxWarning: invalid escape sequence '\\D'\n", " plt.xlabel('$\\Delta$', fontsize=16)\n", "/tmp/ipykernel_1101175/3659422081.py:15: SyntaxWarning: invalid escape sequence '\\D'\n", " plt.title('The error follows $\\Delta^2$, so confirmed!')\n" ] }, { "data": { "image/png": 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", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "x = 1.0\n", "fpp_ref = np.exp(x) # fpp for \"f prime prime\"\n", "Delta = np.logspace(0, -3, 4)\n", "d2fdx2_approx = np.zeros(len(Delta))\n", "for i in range(len(Delta)):\n", " d2fdx2_approx[i] = (np.exp(x+Delta[i])-2*np.exp(x)+np.exp(x-Delta[i]))/Delta[i]**2\n", "plt.loglog(Delta, abs(d2fdx2_approx-fpp_ref), 'k-o', label='$|f''_{approx}(x)-f''(x)|$')\n", "plt.loglog(Delta, Delta, 'r--', label='$\\mathcal{O}(\\Delta)$')\n", "plt.loglog(Delta, Delta**2, 'g--', label='$\\mathcal{O}(\\Delta^2)$')\n", "plt.xlabel('$\\Delta$', fontsize=16)\n", "plt.legend()\n", "plt.grid(True)\n", "plt.title('The error follows $\\Delta^2$, so confirmed!')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## From Derivatives to Gradients\n", "\n", "A one-dimensional derivative measures how a scalar function changes as one coordinate changes. For a scalar field $T(x,y)$, the derivative information becomes a **gradient**:\n", "\n", "$$\n", " \\nabla T = \\left[\\frac{\\partial T}{\\partial x}, \\frac{\\partial T}{\\partial y}\\right].\n", "$$\n", "\n", "The gradient points in the direction of steepest increase. Its components are partial derivatives: change $x$ while holding $y$ fixed, and change $y$ while holding $x$ fixed.\n", "\n", "For sampled data on a grid, NumPy's `np.gradient` applies finite-difference ideas along each array axis. The array convention matters: for a mesh built by `X, Y = np.meshgrid(x, y)`, the field `T` has rows in the $y$ direction and columns in the $x$ direction, so\n", "\n", "```python\n", "dTdy, dTdx = np.gradient(T, y, x, edge_order=2)\n", "```\n", "\n", "returns the $y$ derivative first and the $x$ derivative second.\n" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "execution": { "iopub.execute_input": "2026-07-05T18:02:06.405761Z", "iopub.status.busy": "2026-07-05T18:02:06.405669Z", "iopub.status.idle": "2026-07-05T18:02:06.493106Z", "shell.execute_reply": "2026-07-05T18:02:06.492385Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "sample point: 1.0 0.75\n", "np.gradient dT/dx: -1.3699581032055743 exact: -1.3714832918927073\n", "np.gradient dT/dy: 0.15000000000000036 exact: 0.14999999999999994\n" ] }, { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "def temperature_field(x, y):\n", " return np.sin(np.pi * x) * np.exp(-y) + 0.15 * x * y\n", "\n", "\n", "def exact_dTdx(x, y):\n", " return np.pi * np.cos(np.pi * x) * np.exp(-y) + 0.15 * y\n", "\n", "\n", "def exact_dTdy(x, y):\n", " return -np.sin(np.pi * x) * np.exp(-y) + 0.15 * x\n", "\n", "x = np.linspace(0.0, 2.0, 81)\n", "y = np.linspace(0.0, 1.5, 61)\n", "X, Y = np.meshgrid(x, y)\n", "T = temperature_field(X, Y)\n", "\n", "dTdy, dTdx = np.gradient(T, y, x, edge_order=2)\n", "\n", "# Compare at an interior grid point.\n", "j, i = 30, 40\n", "print('sample point:', X[j, i], Y[j, i])\n", "print('np.gradient dT/dx:', dTdx[j, i], 'exact:', exact_dTdx(X[j, i], Y[j, i]))\n", "print('np.gradient dT/dy:', dTdy[j, i], 'exact:', exact_dTdy(X[j, i], Y[j, i]))\n", "\n", "skip = (slice(None, None, 6), slice(None, None, 6))\n", "plt.figure(figsize=(7, 4.8))\n", "contours = plt.contourf(X, Y, T, levels=18, cmap='viridis')\n", "plt.colorbar(contours, label='T(x, y)')\n", "plt.quiver(X[skip], Y[skip], dTdx[skip], dTdy[skip], color='white', scale=35, width=0.004)\n", "plt.xlabel('x')\n", "plt.ylabel('y')\n", "plt.title('Gradient vectors point toward increasing scalar field');\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "## Optional Extension: Higher-Order Stencils\n", "\n", "The finite-difference approximations explored were identified pretty easily by manipulating the Taylor series of a function. Approximations that are higher order or that can be used for higher derivatives are somewhat more challenging to develop.\n", "\n", "### Simple Rules\n", "\n", "Generally, approximations that use values of $f$ at more values of $x$ (e.g., $x-\\Delta$ and $x+\\Delta$) are better than those that use fewer. \n", "\n", "*A more specific rule-of-thumb*: an $n$th-order approximation for the $m$th derivative requires that $f$ be evaluated at $n+m-1$ evenly-spaced $x$ values if those values are symmetric about $x$ (e.g., $x-\\Delta$ and $x+\\Delta$ as used for the central difference) or $n+m$ points, if the points are not symmetric about $x$ (e.g., $x-\\Delta$ and $x$ as used for the backward difference).\n", "\n", "
\n", "\n", "**Exercise**: Does the backward-difference approximation for $df/dx$ follow this rule? \n", "\n", "*Solution*: The backward-difference approximation for the first derivative ($m=1$) is first order ($n=1$). The points are not symmetric about $x$. Hence, the rule requires $m+n = 2$ points, consistent with the approximation. \n", "\n", "
\n", "\n", "**Exercise**: Does the forward-difference approximation for $df/dx$ follow this rule? \n", "\n", "
\n", "\n", "**Exercise**: Does the central-difference approximation for $df/dx$ follow this rule? \n", "\n", "
\n", "\n", "**Exercise**: Does the central-difference approximation for $d^2f/dx^2$ follow this rule? \n", "\n", "
\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Further Reading\n", "\n", "None for now." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.13.13" } }, "nbformat": 4, "nbformat_minor": 2 }