{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "[Table of Contents](http://nbviewer.ipython.org/github/rlabbe/Kalman-and-Bayesian-Filters-in-Python/blob/master/table_of_contents.ipynb)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# One Dimensional Kalman Filters" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "#format the book\n", "%matplotlib inline\n", "from __future__ import division, print_function\n", "from book_format import load_style\n", "load_style()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now that we understand the discrete Bayes filter and Gaussians we are prepared to implement a Kalman filter. We will do this exactly as we did the discrete Bayes filter - rather than starting with equations we will develop the code step by step based on reasoning about the problem. \n", "\n", "\"One dimensional\" means that the filter only tracks one variable, such as position on the x-axis. In subsequent chapters we will learn a more general form of the filter that can track many variables simultaneously, such as position, velocity, and accelerations in x, y, and z. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Problem Description\n", "\n", "As in the **Discrete Bayes Filter** chapter we will be tracking a moving object in a long hallway at work. Assume that in our latest hackathon someone created an RFID tracker that provides a reasonably accurate position of the dog. The sensor returns the distance of the dog from the left end of the hallway in meters. So, 23.4 would mean the dog is 23.4 meters from the left end of the hallway.\n", "\n", "The sensor is not perfect. A reading of 23.4 could correspond to the dog being at 23.7, or 23.0. However, it is very unlikely to correspond to a position of 47.6. Testing during the hackathon confirmed this result - the sensor is 'reasonably' accurate, and while it had errors, the errors are small. Furthermore, the errors seemed to be evenly distributed on both sides of the true position; a position of 23 m would equally likely be measured as 22.9 or 23.1. Perhaps we can model this with a Gaussian.\n", "\n", "We predict that the dog is moving. This prediction is not perfect. Sometimes our prediction will overshoot, sometimes it will undershoot. We are more likely to undershoot or overshoot by a little than a lot. Perhaps we can also model this with a Gaussian." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Beliefs as Gaussians\n", "\n", "We can express our belief in the dog's position with a Gaussian. Say we believe that our dog is at 10 meters, and the variance in that belief is 1 m$^2$, or $\\mathcal{N}(10,\\, 1)$. A plot of the pdf follows:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "scrolled": true }, "outputs": [ { "data": { "image/png": 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VdC8Yx/U0L1xP6brVYeotwahSDwAymaxJj91IEATMmTMHe/fuxebNm9G1a9dG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\ng1c2nwAAPBfmg/HBnURORCQ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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import filterpy.stats as stats\n", "stats.plot_gaussian_pdf(mean=10, variance=1, \n", " xlim=(4, 16), ylim=(0, .5));" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This plot depicts our uncertainty about the dog's position. It represents a fairly inexact belief. While we believe that it is most likely that the dog is at 10 m, any position from 9 m to 11 m or so are quite likely as well. Assume the dog is standing still, and we query the sensor again. This time it returns 10.2 m. Can we use this additional information to improve our estimate?\n", "\n", "Intuition suggests we can. Consider: if we read the sensor 500 times and each time it returned a value between 8 and 12, all centered around 10, we should be very confident that the dog is near 10. Of course, a different interpretation is possible. Perhaps our dog was randomly wandering back and forth in a way that exactly emulated random draws from a normal distribution. But that seems extremely unlikely - I've never seen a dog do that. Let's look at 500 draws from $\\mathcal N(10, 1)$:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Mean of readings is 9.865\n" ] }, { "data": { "image/png": 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BNNGMXQERThXHnDqXzZaycIuClNdmBbrsBBpzzjGvYfxRGzoqSZfo7wALDh6V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\n247lNjXSz/pXXtKqUce8whBzu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+dO/MZv/MaluuzzsiLEfPfabtz/4VvwX752FH/09Wey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\nS0ycJB1hzcqSD0bHPLzOh/Y4+WkDQ7TNImXJT2Pu94yOQWMeI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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "from numpy.random import randn\n", "import matplotlib.pyplot as plt\n", "\n", "xs = range(500)\n", "ys = randn(500)*1 + 10.\n", "plt.plot(xs, ys)\n", "print('Mean of readings is {:.3f}'.format(np.mean(ys)))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Eyeballing this confirms our intuition - no dog moves like this. However, noisy sensor data certainly looks this way. The computed mean of the readings is almost exactly 10. Assuming the dog is standing still, we say the dog is at position 10 with a variance of 1." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Tracking with Gaussian Probabilities\n", "\n", "The discrete Bayes filter used a histogram of probabilities to track the dog. Each bin in the histogram represents a position, and the value is the probability of the dog being in that position.\n", "\n", "Tracking was performed with a cycle of predictions and updates. We used the equations \n", "\n", "$$\\begin{aligned} \n", "\\bar {\\mathbf x} &= \\mathbf x \\ast f_{\\mathbf x}(\\bullet)\\, \\, &\\text{Predict} \\\\\n", "\\mathbf x &= \\mathcal L \\cdot \\bar{\\mathbf x}\\, \\, &\\text{Update}\n", "\\end{aligned}$$\n", "\n", "to compute the new probability distributions. Recall that $\\bar{\\mathbf x}$ is the *prior*, $\\mathcal L$ is the *likelihood* of a measurement given the prior $\\bar{\\mathbf x}$, $f_{\\mathbf x}(\\bullet)$ is the *process model*, and $\\ast$ denotes *convolution*. $\\mathbf x$ is bold to denote that it is a histogram of numbers, or a vector.\n", "\n", "This method works, but led to histograms that implied the dog could be in multiple places at once. Also, the computations are very slow for large problems.\n", "\n", "Can we replace $\\mathbf x$, the histogram, with a Gaussian $\\mathcal N(x, \\sigma^2)$? Absolutely! We've learned how to express belief as a Gaussian. A Gaussian, which is a single number pair $\\mathcal N(\\mu, \\sigma^2),$ can replace an entire histogram of probabilities:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "scrolled": true }, "outputs": [ { "data": { "image/png": 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+fHk+++wzmjRp4lDftm0bAwcO5MKFCyZ1JiJFSWFdRK5OGvARxiqll2oK9ESr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\nadasGdOmTSMrK8uEDkWKjp/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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import kf_book.kf_internal as kf_internal\n", "kf_internal.gaussian_vs_histogram()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "I hope you see the power of this. We can replace hundreds to thousands of numbers with a single pair of numbers: $x = \\mathcal N(\\mu, \\sigma^2)$. \n", "\n", "The tails of the Gaussian extend to infinity on both sides, so it incorporates arbitrarily many bars in the histogram. If this represents our belief in the position of the dog in the hallway, this one Gaussian covers the entire hallway (and the entire universe on that axis). We think that it is likely the dog is at 10, but he could be at 8, 14, or, with infinitesimally small probability, at 10$^{80}$. \n", "\n", "In this chapter we replace histograms with Gaussians:\n", "\n", "$$\\begin{array}{l|l|c}\n", "\\text{discrete Bayes} & \\text{Gaussian} & \\text{Step}\\\\\n", "\\hline\n", "\\bar {\\mathbf x} = \\mathbf x \\ast f(\\mathbf x) & \n", "\\bar {x}_\\mathcal{N} = x_\\mathcal{N} \\, \\oplus \\, f_{x_\\mathcal{N}}(\\bullet) &\n", "\\text{Predict} \\\\\n", "\\mathbf x = \\|\\mathcal L \\bar{\\mathbf x}\\| & x_\\mathcal{N} = L \\, \\otimes \\, \\bar{x}_\\mathcal{N} & \\text{Update} \n", "\\end{array}$$\n", "\n", "where $\\oplus$ and $\\otimes$ is meant to express some unknown operator on Gaussians. I won't do it in the rest of the book, but the subscript indicates that $x_\\mathcal{N}$ is a Gaussian. \n", "\n", "The discrete Bayes filter used convolution for the prediction. We showed that it used the *total probabability theorem*, computed as a sum, so maybe we can add the Gaussians. It used multiplications to incorporate the measurement into the prior, so maybe we can multiply the Gaussians. Could it be this easy:\n", "\n", "$$\\begin{aligned} \n", "\\bar x &\\stackrel{?}{=} x + f_x(\\bullet) \\\\\n", "x &\\stackrel{?}{=} \\mathcal L \\cdot \\bar x\n", "\\end{aligned}$$\n", "\n", "This will only work if the sum and product of two Gaussians is another Gaussian. Otherwise after the first epoch $x$ would not be Gaussian, and this scheme falls apart." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Predictions with Gaussians\n", "\n", "We use Newton's equation of motion to compute current position based on the current velocity and previous position:\n", "\n", "$$ \\begin{aligned}\\bar{x}_k &= x_{k-1} + v_k \\Delta t \\\\\n", " &= x_{k-1} + f_x\\end{aligned}$$\n", "\n", "I've dropped the notation $f_x(\\bullet)$ in favor of $f_x$ to keep the equations uncluttered. \n", "\n", "If the dog is at 10 m, his velocity is 15 m/s, and the epoch is 2 seconds long, we have\n", "\n", "$$ \\begin{aligned} f_x &= v\\Delta t = 15\\cdot 2\\\\\n", "\\bar{x}_k &= 10 + (15\\cdot 2) = 40 \\end{aligned}$$\n", "\n", "We are uncertain about his current position and velocity, so this will not do. We need to express the uncertainty with a Gaussian.\n", "\n", "Position is easy. We define $x$ as a Gaussian. If we think the dog is at 10 m, and the standard deviation of our uncertainty is 0.2 m, we get $x=\\mathcal N(10, 0.2^2)$.\n", "\n", "What about our uncertainty in his movement? We define $f_x$ as a Gaussian. If the dog's velocity is 15 m/s, the epoch is 1 second, and the standard deviation of our uncertainty is 0.7 m/s, we get $f_x = \\mathcal N (15, 0.7^2)$.\n", "\n", "The equation for the prior is \n", "\n", "$$\\bar x = x + f_x$$\n", "\n", "What is the sum of two Gaussians? In the last chapter I proved that:\n", "\n", "$$\\begin{gathered}\n", "\\mu = \\mu_1 + \\mu_2 \\\\\n", "\\sigma^2 = \\sigma^2_1 + \\sigma^2_2\n", "\\end{gathered}$$\n", "\n", "This is fantastic news; the sum of two Gaussians is another Gaussian! \n", "\n", "The math works, but does this make intuitive sense? Think of the physical representation of this abstract equation. We have \n", "\n", "$$\\begin{gathered}\n", "x=\\mathcal N(10, 0.2^2)\\\\\n", "f_x = \\mathcal N (15, 0.7^2)\n", "\\end{gathered}$$\n", "\n", "If we add these we get:\n", "\n", "$$\\begin{aligned}\\bar x &= \\mu_x + \\mu_{f_x} = 10 + 15 &&= 25 \\\\\n", "\\bar\\sigma^2 &= \\sigma_x^2 + \\sigma_{f_x}^2 = 0.2^2 + 0.7^2 &&= 0.53\\end{aligned}$$\n", "\n", "It makes sense that the predicted position is the previous position plus the movement. What about the variance? It is harder to form an intuition about this. However, recall that with the `predict()` function for the discrete Bayes filter we always lost information. We don't really know where the dog is moving, so the confidence should get smaller (variance gets larger). $\\mu_{f_x}^2$ is the amount of uncertainty added to the system due to the imperfect prediction about the movement, and so we would add that to the existing uncertainty. \n", "\n", "Here is our implementation of the predict function, where `pos` and `movement` are Gaussian tuples in the form ($\\mu$, $\\sigma^2$):" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def predict(pos, movement):\n", " return (pos[0] + movement[0], pos[1] + movement[1])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's test it. What is the prior if the intitial position is the Gaussian $\\mathcal N(10, 0.2^2)$ and the movement is the Gaussian $\\mathcal N (15, 0.7^2)$?" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(25, 0.530)" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "%precision 3\n", "predict((10, .2**2), (15, .7**2))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The prior states that the dog is at 25 m with a variance of 0.53 m$^2$, which is what we computed by hand." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Updates with Gaussians\n", "\n", "The discrete Bayes filter encodes our belief about the position of our dog in a histogram of probabilities. The distribution is discrete and multimodal. It can express strong belief that the dog is in two positions at once, and the positions are discrete.\n", "\n", "We are proposing that we replace the histogram with a Gaussian. The discrete Bayes filter used this code to compute the posterior:\n", "\n", "\n", "```python\n", "def update(likelihood, prior):\n", " posterior = likelihood * prior\n", " return normalize(posterior)\n", "```\n", "\n", "which is an implementation of the equation:\n", "\n", "$$x = \\| \\mathcal L\\bar x \\|$$\n", "\n", "We've just shown that we can represent the prior with a Gaussian. What about the likelihood? The likelihood is the probability of the measurement given the current state. We've learned how to represent measurements as a Gaussians. For example, maybe our sensor states that the dog is at 23 m, with a standard deviation of 0.4 meters. Our measurement, expressed as a likelihood, is $z = \\mathcal N (23, 0.16)$.\n", "\n", "Both the likelihood and prior are modeled with Gaussians. Can we multiply Gaussians? Is the product of two Gaussians another Gaussian?\n", "\n", "Yes to the former, and almost to the latter! In the last chapter I proved that the product of two Gaussians is proportional to another Gausian. \n", "\n", "$$\\begin{aligned}\n", "\\mu &= \\frac{\\sigma_1^2 \\mu_2 + \\sigma_2^2 \\mu_1} {\\sigma_1^2 + \\sigma_2^2}, \\\\\n", "\\sigma^2 &= \\frac{\\sigma_1^2\\sigma_2^2}{\\sigma_1^2+\\sigma_2^2}\n", "\\end{aligned}$$\n", "\n", "We can immediately infer several things. If we normalize the result, the product is another Gaussian. If one Gaussian is the likelihood, and the second is the prior, then the mean is a scaled sum of the prior and the measurement. The variance is a combination of the variances of the prior and measurement. Finally, the variances are completely unaffected by the values of the mean!\n", "\n", "We put this in Bayesian terms like so:\n", "\n", "$$\\begin{aligned}\n", "\\mathcal N(\\mu, \\sigma^2) &= \\| prior \\cdot likelihood \\|\\\\\n", "&=\\mathcal{N}(\\bar\\mu, \\bar\\sigma^2)\\cdot \\mathcal{N}(\\mu_z, \\sigma_z^2) \\\\\n", "&= \\mathcal N(\\frac{\\bar\\sigma^2 \\mu_z + \\sigma_z^2 \\bar\\mu}{\\bar\\sigma^2 + \\sigma_z^2},\\frac{\\bar\\sigma^2\\sigma_z^2}{\\bar\\sigma^2 + \\sigma_z^2})\n", "\\end{aligned}$$\n", "\n", "If we implemented that in a function `gaussian_multiply()` we could implement our filter's update step as" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def gaussian_multiply(g1, g2):\n", " mu1, var1 = g1\n", " mu2, var2 = g2\n", " mean = (var1*mu2 + var2*mu1) / (var1 + var2)\n", " variance = (var1 * var2) / (var1 + var2)\n", " return (mean, variance)\n", "\n", "\n", "def update(prior, likelihood):\n", " posterior = gaussian_multiply(likelihood, prior)\n", " return posterior" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Perhaps this would be clearer if we used more specific names:\n", "\n", "```python\n", "def update_dog(dog_pos, dog_var, measurement, measurement_var):\n", " estimated_pos = gaussian_multiply(\n", " (dog_pos, dog_var), \n", " (measurement, measurement_var))\n", " return estimated_pos \n", "```\n", "\n", "That is less abstract, which perhaps helps with comprehension, but it is poor coding practice. We are writing a Kalman filter that works for any problem, not just tracking dogs in a hallway, so we won't use variable names with 'dog' in them. Still, the `update_dog()` function should make what we are doing very clear. \n", "\n", "We have the majority of our filter implemented, but I fear this step is still a bit confusing. I've asserted that we can multiply Gaussians and that it correctly performs the update step, but why is this true? Let's take a detour and spend some time multiplying Gaussians." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Understanding Gaussian Multiplication\n", "\n", "Let's plot the pdf of $\\mathcal{N}(10,\\, 1) \\times \\mathcal{N}(10,\\, 1)$. Can you determine its shape without looking at the result? What should the new mean be? Will the curve be wider, narrower, or the same as $\\mathcal{N}(10,\\, 1)$?" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "scrolled": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "(10.0, 0.5)\n" ] }, { "data": { "image/png": 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zoP6vb/SzX7+p/6uO2WymTRvHNrxzKFlPSkqiQ4cORY4HBAQUnC+JzWbj9ttv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\nmKyv4btC+MVKfUFINHQcA85i6PaQfuGoHDSFqN/MNmjRo/J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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "z = (10., 1.) # Gaussian N(10, 1)\n", "\n", "product = gaussian_multiply(z, z)\n", "\n", "xs = np.arange(5, 15, 0.1)\n", "ys = [stats.gaussian(x, z[0], z[1]) for x in xs]\n", "plt.plot(xs, ys, label='$\\mathcal{N}(10,1)$')\n", "\n", "ys = [stats.gaussian(x, product[0], product[1]) for x in xs]\n", "plt.plot(xs, ys, label='$\\mathcal{N}(10,1) \\\\times \\mathcal{N}(10,1)$', ls='--')\n", "plt.legend()\n", "print(product)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The result of the multiplication is taller and narrow than the original Gaussian but the mean is unchanged. Does this match your intuition?\n", "\n", "Think of the Gaussians as two measurements. If I measure twice and get 10 meters each time, I should conclude that the length is close to 10 meters. Thus the mean should be 10. It would make no sense to conclude the length is actually 11, or 9.5. Also, I am more confident with two measurements than with one, so the variance of the result should be smaller. \n", "\n", "\"Measure twice, cut once\" is a well known saying. Gaussian multiplication is a mathematical model of this physical fact. \n", "\n", "I'm unlikely to get the same measurement twice in a row. Now let's plot the pdf of $\\mathcal{N}(10.2,\\, 1) \\times \\mathcal{N}(9.7,\\, 1)$. What do you think the result will be? Think about it, and then look at the graph." ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "image/png": 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IYSEBAQG88MILJdb/8uXL+fLLL0us/9Jm1gqmkZGRNGzYELU6+2HNmjXL2t+h\nQ4c8jz9z5gwff/wxoaGhODs7mx3s6dOnMRgMZh8nik6r1Wb9e+LECStHI0qbnP+SV4lKnIgr4s/W\naMQmMwHb1BhsU29hm3oTm8xEtPaeaB2rkvn3l0HjVKTuCzr/RqORhHQDt1P13EnRcStFT0qmAU9H\nG7wdbajspKaykw2OmqLXMujt3hudqw6vBC/5P1iKKvprX6PRkJqaau0wrMZoNGb9WxI/B6PRiE6n\nK7Gf8dKlSzlz5gyvvPJKifRfWElJSTlePyqViho1apjVj1nJelxcHLVq1cqx3cPDI2t/XgwGAyNH\njiQ4OJjevYs2P1On0xVqqo0oGfd/eYuKSc6/5RiNRhRFMfs4xaCj0p0jeFz/FcfEC9im3cZGX3Cl\nFJ3GmUyHqiR5NCWu+hOkutYHM8fXarVoDUaO3czg96vpRCXoiE3Vk1mIX8nOtgrejjY0qmxL5xr2\n1HRTF/r5t3FuY3pgAK1B/g9aQ0V87avV6qyEtaJ78OeQlpaGg4NDifRdEqx9Do1GY47Xj42N+Z+k\nmpWsA/n+gs1v33//+1/++usvNm3aZO6QWdRqNSqVVJssTQ/+J9NopFpFRSPnv2R8c+UbdEYdA6oM\nwM/BL//GRiMO9/7E/eoO3K79iibzXo4mWntPMh2qkulUFb1tJTRpsWhSb2Kbegu1NhG1Nhm19gKO\niReoErWedOca3K3Rg7vVA9E6VctzaK1Wi9Fo5Hy8lt+vZfL7lTSSMrN/uqkA7g4qvJ3UeDvZ4GSr\nIj5Vz+0UPbdTdKRojSRnGknO1HHpno6f/0qlhquaLgGOPB7giJej2X+GRCmo6K99RVHyzGnSdGl5\nHqdSVNjZ2BWqraIo2NvYF6ltcXzyySdMnz6dffv2MXPmTMLDw1EUhV69ejFz5kwqV66M0WikUaNG\nNGrUiOeee44ZM2Zw7tw5Xn/9daZNm0Z6ejrTp09n7dq13LhxAy8vL5588kkmTZqEm5tb1lharZbJ\nkyezfPlykpKSaNGiBTNmzMj2vB6MKSUlJVusS5cu5dVXX+XMmTP4+/tnbV+1ahXffvstp0+fBqBW\nrVqMHj2a559/nqCgIH77zVRB6sFZHP/uuzQoipLj9VOUPNas35Kenp65Xj2Pj48H/rnC/m9Xr15l\n4sSJzJgxA1tbW+7dM/2x0el0GAwG7t27h52dXYHv1ho3bizJeik7ceIEWq0WjUZD8+bNrR2OKGVy\n/i0vKTOJIyePkGnIZNxj46jvUT/3hukJcGgBnFwFsef/2e5UGZoMhLqB4B4ArtXRqO3QALlOdslI\ngnvREHcBzmyEP7dgn3yVamci9HZcAAAgAElEQVT+R7Uz/wP/jtAiBJoNAZt//iTcS81k5voIwi+n\ncDP5n8vn3i529G/hw+P1vPHzcKCaqwO26rx/Lyela7l+L42Lt1PYeiqGnWdvcTVBx48nEll6MpFH\na3kyuLUf/Zr7oFLlnhwlZCSw4cIGriVd48P2H+b9wxUWU9Ff+9HR0Tg6Oua6r92Sdnke95jvY3zz\nxDdZ33f5qUueSXjrKq1ZFLQo6/uglUHczbiba9vGno1Z2df8Mtm5uZ88PvPMMwwePJjXX3+d06dP\nM2HCBM6fP09ERETWm7Xjx49z/vx5PvroI2rWrImTkxMODg4EBwfz66+/8p///IfHHnuMkydPMmnS\nJA4fPsyBAwewszO9YXnhhRf48ccfee+99wgMDCQyMpJnnnmGpKQk1Gp11s/4fkz//pnf78fBwSFr\n38SJE5k2bRrBwcGMHTsWV1dXIiMjiYmJwdHRkW+//ZZXXnmFixcvsn79+qy+8jqfJcnFxQU/v+wX\nZAwGA0lJSWb1Y1ay3rRpU1asWIFOp8s2b/3UqVMAedZMv3TpEmlpaYwZM4YxY8bk2O/u7s6YMWPK\n1c0AQgiRm/DocDINmdRyrUU993o5GxgMcHIl7JwIKXdM29T20KAPNBsKtbuCOTXZ7VygSiPTV6N+\nkJ4IZzebxrj8G1zZZ/o6OB96f4berz2rDkfz2fY/uZtq+oNtZ6PQu5kPAx7xpWMdL2zySKpz42Kv\noUFVDQ2qVqJPs2okpGnZdiqG0GPXOXQ5nv0X49h/MY5F+y4zpX8TWvi55egjWZvM50c+R0Hh5WYv\n4+3oXfjnL4TIVXBwMLNmzQKgR48eVKlShZCQEFavXs2AAQMAuHPnDmfOnKFevX9+V23fvp3t27cz\na9Ysxo4dC0BgYCB+fn4MGTKEH3/8kZdffpk///yTJUuW8M4772SNExgYmDVOUVy+fJnp06cTEhLC\nsmXLsrYHBgZmPW7UqBFubm7Y2dnRvn37Io1T1piVrA8YMIAFCxawbt06hgwZkrV9yZIl+Pj40K5d\n7u82W7RoQXh4zjv53377bRISEli0aBHVq1c3M3QhhHj4bL28FYCgmkE5P2a/cRy2joVrh0zfe9aB\nTu9Aw35gX8kyAdhXgkdCTF8J101J+76v4NYpWBTEXtuufJE4kLu44++qpl89Rzr6O9O+dQuLDO/q\noGFo2xoMbVuD6PhUQo9e53+/XeLEtQSemrePwa2rMy6oAV7O/0wl8HX2pXnl5py4c4IdUTsY3mi4\nRWIRoiginsm78t2/KzvtHrw7z7YqJfsnUmFPhxW6rSX8O2EePHgwzz//POHh4VnJepMmTbIl6gC7\ndu0CyFHNZdCgQYwcOZJff/2Vl19+OSvvy2ucoti5cyd6vZ7XX3+9SMc/rMxK1nv16kVgYCCjR48m\nMTGROnXqsGLFCsLCwli2bFnWpPkXX3yRJUuWcPHiRfz9/XFzc6NLly45+nNzc0On0+W6Twghypu7\n6Xc5eOMg8E8NcQBS42HXNDiyCDCCxgm6vA/tRoPatuQCcvWFx/6P2HrDuLByHG3jf6ZrZji77Q5w\nut5o1A36YDAqaIpRzSU/fh6OjHmiLsPa+TFz2znWHb3G6iPX2BZ5k/8LrMfw9v6obUxj96rZixN3\nTrAtapsk68KqHDWFn05RUm0toWrVqtm+V6vVOaY7/7sNmIqJqNVqKleunG27oihUrVo16/j7/+Y1\nTlHcuWP6tLGiXeA1+zdwaGgozz77LBMnTiQoKIiIiAhWrFiR7Z2TXq9Hr9db/S5cIYQoS3Ze2Yne\nqKehR0MCXANMG4+vgLkt4cgPgBGaDoI3j0DHMSWbqGOqVLA84ipd559iaMww+mdO44pDI5yUdNr+\n9QWNwkfidPdMicYA4O1iz+zBzVk3+lEa+1QiKV3H5M1n6Dv3dyKvJwDQM6AnKkXFyTsnuZZ0rcRj\nEqK8u3nzZrbvdTodcXFx2RLp3G6y9fT0RKfTZSXO9xmNRm7evImXl1dWu/zGeZC9venm2YyMjGzb\nY2Njs31//w3CtWsV63eA2cm6s7Mzc+bMISYmhoyMDE6cOMHQoUOztVm8eDFGo5GAgIB8+9q9ezeR\nkZHmhiCEEA+lbZe3AdC7Zm/Qa2HL/8GGVyHtLng3hhe2wtP/g0o+JR5Lhk7P++tO8sH6UySl62ji\nW4kpo4fjP3Yf9J8Hjl7YJ1+l/oF38Yj6ucTjAWjl78GmNzrxyYAmuDlq+PNmEk/P38+GY9fxcvCi\nTVVTGcewqLynCwghCuenn37K9v3q1asLNduhe/fuANnmjAOsW7eOlJSUrP33+8lrnAfdzxdPnjyZ\nbfvmzZuzfd+jRw9sbGyYP39+vjHa2dmRlpZ3dZ2HjdTMEkKIUmA0Guni14VUXSo9vVvDj/1NN3YC\ndPkPPPZetmosJelWYjqjlv7B8eh7qBQY27MBr3Su9c+No48MhwZ9uffjcNxifsPv2CxQx0PPT0v8\nar+NSiGknT+9m1Tj3dXHCT93h7dXHef0jQR61A4iIiaCsMthvNT0pRKNQ4jyLjQ0FLVaTWBgYFY1\nmObNmzN48OAcyfSDAgMD6dmzJ++//z6JiYl07NgxqxrMI488wrPPPgtAw4YNGT58OF9++SUajYYn\nnniCyMhIPv/8cypVyn4PTu/evfHw8ODFF19k6tSpqNVqFi9eTHR0dLZ2AQEBfPDBB0ybNo20tDSG\nDRuGq6srZ86cITY2lilTpgCmgiihoaHMnz+fVq1aoVKpaN26tYV/gqVHknUhhCgFiqLwfOPned69\nGSwbAonXwNYFgr+HBkVbKK4o/rhyl1eX/cGdpAxcHTTMHfYInetVztnQwY0r7aaRcnYJvucWweH/\nwa0zMPhHcM6lvYW5O9nyv+fb8N+d55gXfpEFv13m0ZvuONo74efiR7ouHXu1ZepOC1ERhYaGMnny\nZObPn4+iKDz55JN8+eWX2Nra5pusK4rChg0bmDx5MosWLeKTTz7By8uLZ599lunTp2eVWwRYuHAh\nVapUYfHixXz11Ve0aNGCdevW5ZiRUalSJcLCwnj77bcZPnw4bm5uvPTSS/Tq1YuXXsr+xnzq1KnU\nrVuXuXPnEhISglqtpm7durz11ltZbcaMGcPp06f54IMPSEhIwGg0PtRTsxVjGY0+tzqULi4uUme9\nlFX0WrsVnZx/Czu5Gja9Cbp0U6WXoSugci7lG0vIykNXmbAxEq3eSP0qLnz/XCv8PXOtzg78c/49\nYw9R8+h0yEyCStVh6DLweaTU4t5yMob31pwgTavHz0PDgufa06CqharjiFxV9Nd+dHR0jvrY5cXk\nyZOZMmUKd+7cyZpf/m+pqalZKy1boz55eZHb/6Oi5LeS+QohRAmLSbrOpg3Pkbz+FVOiXrcHvPRr\nqSXqOr2BCRsiGR96Cq3eSFDjqoS+1iHfRP1BidU6wsu7TG8wEq/BD0Fwck0JR/2PPs2qEfpaB/w8\nHIiO1xL8zX7CImNKbXwhhLAmSdaFEKIkGfRs2fIqHyYc4z1vL9Pc9GErwSHn4j8lQac38Paq4yw9\neAVFgfd61GP+8JY42Zk5C7JyPVPCXren6Q1H6Et/l5osHQ2rVWLT653oWMeT1Ew9r6/ezpJDx0tt\nfCGEsBZJ1oUQoqQY9LDxDbYnngcgsP4g6D4B/rVwSknR6Q2MWXWcn0/GoLFRmB/Sije61c21HFuh\n2Lua3mi0e9X0/c9v/11ysnS4O9myZERbmjXbi2Pt2Uz/bSHrj1WsEm5CFNfkyZMxGo15ToERZY8k\n60IIURIMetj4OlGnV/OnnS02KHTv8H6pDa/VGxiz8jhbHkjUg5rkXODEbCoVBM2A9q+Zvv/5HTi8\nsPj9FpLaRsXo9l1Njyud5N3Vx1n3hyTsQojyS6rBCCGEpRn0sOE1OLmSHW6uALT36YCbfelMfTEl\n6sfYeupmVqL+RKMqlhtAUaDndFBUcOBr2PIuYIQ2pVNO8XG/ztjb2JNuG49id5331ioYgYGtKtaq\nhkKIikGurAshhCU9kKij2BDmY7qJtGdAz1IZXqs38NYKU6Jua6Pi2+EWTtTvUxTo8TE8+obp+y3/\nB4cWWH6cXDhqHOlcvTMAzetfwWiEsWtPsOZIdAFHClE4KpUKrVZr7TDEQ0yv11usL0nWhRDCUgx6\n2DA6K1G/1GcGf6XdQq1S061GtxIf/n6ivi3SlKh/92wrujcsgUT9vvsJe4c3Td9vfa/UEvagmkEA\npNr+QUg7P4xGGLfuJKslYRcW4OHhQUxMTFYJQyHModfruX79OpUrW2ZNCpkGI4QQlmA0mm64PLkK\nVGoY+APHNaYrK49WexRXO9cSHd5gMPLemhPZEvWuDbxLdEzAlLAHTgMU2P+VKWHXOMIjISU6bCff\nTjioHbiRfIMhvUGl+LP04BXeX3cSR1sb+jbzKdHxRfnm4OCAr68vcXFxxMXFWTucUpeUlJRVZ93F\nxcXa4TyUvL29sbe3zMJtkqwLIYQl/P5fOPqjaR73wB+gUX+CgY4+HUnWJpf48P/deZ6Nx2+gViml\nl6jfpygQOBUwwv65sPktcK0OtR4vsSEd1A50qd6FbVHb+OXKL0zt/w5GjCw7eJV3V5+gmqs9rfw9\nSmx8Uf7Z2Njg7V2Kr6My5MFFscrr4lAPE5kGI4QQxXVqLfw61fQ4aCY06p+1q4pTFWq71S7R4Vcf\njubr8AsAfBrctHQT9fsUBZ6YCk2eBoMOVj0Lt/8s0SGfbfQsX3b5ktdavIaiKEzp14QnGlYhU2fg\n5R//ICo2pUTHF0KI0iDJuhBCFMeVA6Z56mAqZ9juFQC0htK5Oe33v2L5YP0pAN7sVodBra14FUyl\ngv7fgF97yEiAnwZB0q0SG65p5aZ09++Ovdr0UbONSuGrYS1o6utKfEomIxYf5m5KZomNL4QQpUGS\ndSGEKKq4i7ByGOgzoUFf082WgNFoZPDmwYzaOYroxJK74fHczSRGL/sDncFI/xY+vBtYr8TGKjSN\nPQxdDh61IeEqrBgKmamlNryjrZqFz7fG182By7EpjFr6Bxk6y1VlEEKI0ibJuhBCFEVKHPw0ENLu\ngk9LCF6QtTLp+bvnuXDvAkduHsHd3r1Ehr+dmM7IxYdJytDRtqYHswY2K/rKpJbm5Akha8DBA24c\nhdCXTZVySkByZjLzjs/jxe0vYjAaAPCuZM+iEW1wsVNzKCqecWtPSkUPIcRDS5J1IYQwlzYdVj4D\n8ZfAtQY8swpsHbN2b4/aDkBH34442zpbfPjUTB0vLjnC9Xtp1PJy4vtnW2GntrH4OMXiWRuGrQAb\nO/jzZ9gxoUSGsbWxZdmZZRy6eYjjt49nba9XxYX5w1uhVilsPH6D/+48XyLjCyFESZNkXQghzGE0\nwsbXIPog2LmariA7ez+w28iOKzsACAoIsvjwBoORt1Yc59T1BDycbFk0og1ujrYWH8ciarSHAfNN\njw/OK5Ea7LY2tlk17O+/SbqvU10vpg9oCsDcXRdk0SQhxENJknUhhDDHgXkQuc5US33IUvBukG33\nubvnuJJ4BTsbOx73s3zpwq/DL/DL2VvYqlUseK41/p5OFh/Dopo8Dd0nmR6HjYerBy0+xP3VYXde\n2Yn+X9NtBrfx442udQD4cEMkp64lWHx8IYQoSZKsCyFEYUXtg50TTY+DZuRaRzzschgAj/k+hpPG\nson0nvN3+OIX03SO6QOa0sq/ZObDW1ynd6DJQFNJxzUvQPJti3b/aLVHcbF14U7aHY7dPpZj/7uB\n9XiioTeZOgOjf/qDe6lSIUYI8fCQZF0IIQoj6SasHQFGPTQdDG1eytHEaDRmTcW4f7XXUq7dTWXM\nymMYjfBMuxoMbFXdov2XKEWBJ+dA5QaQFANrR4JeZ7HuNTYautfoDkBYVFiO/SqVwuzBLajh4ci1\nu2m8veo4BoPccCqEeDhIsi6EEAXRa/++InwLvBvBk1+aEtB/NzPqGdFkBB19O9K5emeLDZ+u1fPa\nT0e5l6qlWXVXJvZtZLG+S42dMwxeCrbOEPUb/DrFot3nNxUGwNVBw/zhLbFTq9h97g5f7frLouML\nIURJkWRdCCEK8stkuHoA7Cr9nXDmPr1FrVIzuP5gvn3iWxw1jrm2KYopm89w8loCbo4avglpib2m\njFV+KazK9aD/PNPj/V/BmU0W67pdtXZUd67OY76PkaxNzrVNYx9XPvn7htM5v/7F7nOWnY4jhBAl\nQZJ1IYTIz+n1cOBr0+OnvgGvOqU6/Joj0aw4dBVFga+GPkJ1d8u9CbCKxk/Bo2+YHm94DWItc4Vb\no9KwNXgrH3f6GFc71zzbDWxVnWfa1cBohLdXHSc6vvQWbBJCiKKQZF0IIfJy5xxs/Dux7DgGGj6Z\nZ9Nz8edYfnY5sWmxFhv+9I0EPtoQCcA7T9Sjc73KFuvbqp6YAv4dITMJVj0LmSkW6bawi0JNerIR\nzaq7ci9Vy2s/HSVdKyucCiHKLknWhRAiNxnJfyeSyRDwGHSbmG/zDRc28OmhT5l9ZLZFhk9I1TJ6\n2VEydAa61q+cVX6wXLBRw8AfwLkK3DkLm94y1a+3AKPRyOnY01y8dzHPNnZqG74JaYmbo4ZT1xOY\nsvm0RcYWQoiSYHaynpyczNtvv42Pjw/29va0aNGClStXFnjcL7/8QmBgID4+PtjZ2eHt7U23bt3Y\nunVrkQIXQogStXUsxJ4Dl2qmxNJGnWdTvUGfVQXGEgshGY1Gxoee5Gp8KtXdHfhiSAtUqsJdNX5o\nuFSFQYtBsYHItXD0R4t0+/Xxrxm6ZSg/RP6Qb7vq7o58NfQRFAVWHIpm04kbFhlfCCEszexkPTg4\nmCVLljBp0iS2bdtGmzZtGDZsGMuXL8/3uLi4OBo3bswXX3zBjh07+O6779BoNPTp04dly5YV+QkI\nIYTFnVoLJ5aDooKnF2ZboTQ3R28f5U7aHVxsXejg06HYw688HM22yJuoVcrfV4DL6AqlxeXfAbr/\n/YlF2Hi4c77YXXb06QjArqu7yNBn5Nu2c73KvHl/waTQUzJ/XQhRJpmVrG/dupWdO3fyzTffMGrU\nKLp27cqCBQsIDAxk7Nix6PV5z/sbMmQIX375JUOGDOHxxx9nwIAB/Pzzz/j6+vL9998X+4kIIYRF\n3I2Cn98xPe48FgI6FnjItsvbAAj0D0RjoynW8BduJ2dNyxjbsz7NqrsVq78yr8NbUPNx0KbCupGg\nyz/BLkgL7xZUcaxCsjaZfdf3Fdj+re51aeXvTlKGjjErj6HTG4o1vhBCWJpZyfr69etxdnZm0KBB\n2baPGDGCGzduEBERYdbgGo0GNzc31Oq8P14WQohSo9fBupchIxH82kHncQUeojVo+eXKL0DxF0LK\n0Ol5a8Ux0rUGOtXx4uXHahWrv4eCSgUDvgMHD7h5Cn4pXv11laLKOg/3V5PNj9pGxZdDWuBip+bo\n1Xt89avUXxdClC1mZcmRkZE0bNgwR3LdrFmzrP0dOuT/EbDBYMBgMHD79m2+++47zp8/z8yZMws1\n/unTpzEY5KpHadJqtVn/njhxwsrRiNJW0c5/1TP/o8q1Q+g1zpxr+C7ayIJvPDyZeJK7GXeppK6E\n/W17Ttwp+s9p4dF7nIlJppKdihebaDh16mSR+7KE0jz/lZqPpebB/8DBeVxSBZBUpV2R+6qtqw2Y\npsJEHI3A3sa+wGNeaVWJ2fvj+Tr8AlWVBBp72xV5/PKgor32RXZy/kuOSqWiRo0aZh1jVrIeFxdH\nrVo5r/R4eHhk7S9I79692b7ddCNWpUqVWLVqFX369CnU+DqdLt+pNqJk3X/xioqpvJ9/57gTeJ9b\nCkBU07dJtfWCQjzn6NRobBQbWrm0wqAzYKBoFxSO3cxg0znTYj6vtaqEi9qAVlt2Lk6U9PmP82qL\nY8BTVInagN+RTzj9+AJ0dh5F6stP7UdlTWXuaO9w5O4R2rkWnPg/6qOhq7894VfSmb0/js8DPXGx\nlYJpUP5f+yJ/cv4ty8bG/EXtzJ5/kl8d28LUuJ07dy737t0jJiaGZcuWMWTIEJYsWcKwYcMKPFat\nVqNSyS/P0vTgi1SjKd5cXPHwqSjn3yYzkVrHP0XBSHyN3iT796Cwz7ZP1T487vU4WqO2yD+je2l6\nvj6caOqvrhOP+jsXqR9LK+3zf6vZ61SKP4FD4mVqnfycy4/ONN3kWwTt3duz+fZmIlMi6eTVqVDH\njGrjwbn429xI0vH90WTe7+RR6Nrt5U1Fee2L3Mn5LzlFyWPNStY9PT1zvXoeHx8P/HOFPT9169bN\netyvXz969erF66+/zpAhQwp8Ao0bN5ZkvZSdOHECrdaUhDRv3tza4YhSViHOv9EIq5+FtDvgWQeP\nkAV42JVesmwwGBmx+DAJGQYaVHVh9nMdsdeYf+WlJFjl/FdfAd93odKtCJqnHYBHXytSN561POmX\n3I/WVVqjVhX+T913VRMInr+PA9fSOJvpwbC25n1cXV5UiNe+yJOc/5JjMBhISkoy6xizMt+mTZty\n9uxZdDpdtu2nTp0CoEmTJmYNDtC2bVvu3r3LnTt3zD5WCCGK7Y/FcHYzqDTw9P/AjEQ9ISOh2MP/\nsO8ye87fwU6t4qthj5SZRN1qvBtCz09Mj3+ZBDFFm7df3aU67au1NytRB2ha3ZWxPesDMGXzaS7c\nNu+PqhBCWJpZyfqAAQNITk5m3bp12bYvWbIEHx8f2rUz74Ygo9HInj17cHNzw9PT06xjhRCi2OIu\nwvYPTI+7TwSfRwp9aJoujR5rexCyNYS76XeLNPy5m0nMCjsHwEd9G1GvikuR+il3Wr8I9fuAPhNC\nXwZterG6M5q5OupLnWrxWF0v0rUG3ll1Aq2UcxRCWJFZyXqvXr0IDAxk9OjRLFiwgPDwcF555RXC\nwsKYNWtW1qT5F198EbVazZUrV7KO7d+/PxMnTiQ0NJQ9e/awYsUKgoKC2LNnD5988omUbxRClC69\nDta/aqrvXbMzPPqGWYfvvbaXVF0qcWlxuNmZXws9U2fgnVXHydQb6N7Am+HtKuZ0i1wpCvSbC07e\ncOdP2DWtSN1oDVpmHppJr9Be3Eu/V+jjVCqFzwc1x81Rw6nrCczddaFI4wshhCWYPQE8NDSUZ599\nlokTJxIUFERERAQrVqwgJCQkq41er0ev12e7mtGxY0fCwsJ46aWX6N69O2+++SaKovDzzz/z2mtF\nm5MohBBFtu9LuHYI7CpB/29M9b7NcL+Gd1BAUJFuQpzz63nOxCTi7qjh06ebVtgbGfPk5GlK2AEO\nzIOo383uQqPScOTWEa4nX+fXq7+adWyVSvZ8/JRpaue88Ascjy58si+EEJZkdrLu7OzMnDlziImJ\nISMjgxMnTjB06NBsbRYvXozRaCQgICBr27hx4zh06BDx8fHodDpiY2MJCwsrdNlGIYSwmJgTsPtT\n0+Nes8DNz6zDkzOT+e36b6bDa/Yye/g/rsQzf/dFAKYPaIq3S8F1wCuk+kHQ8jnACOtHQ3qi2V0E\nBQQBsC1qm9nH9m3mQ7/mPugNRt5ddZy0TCkdLIQofVJaRQhRsWjTIXQUGHTQ8EloPrTgY/4lPDqc\nDH0GAZUCqOdez6xjUzJ0vLv6BAYjBD/iS6+m1cwev0LpOR3c/CHhKoT9x/zD/17N9PDNw8SmxZp9\n/LT+TahSyY5LsSnM2HbW7OOFEKK4JFkXQlQs4R/DnbOm+dB9vzTNjzZTWJRpCkyvmr3Mnr4yfetZ\nrsSlUs3Vnkn9Gps9doVj5wIDvgUUOL4M/txi1uHVXarTzKsZBqOBHVE7zB7e1VHDZwNNpeuWHLjC\nb39J5TIhROmSZF0IUXFE/Q77vzY97vcVOHmZ3cW99Hvsv7Ef+GeKRWGFn7vNTxFXAfh8UHNcHWSx\nkULx7wAd3zI93vQWJJuXMN+/ur7tsvlTYQA616vMc4/6AzB2zUkSUmVFRyFE6ZFkXQhRMaQnmuY9\nY4RHnoX65s81B3DQOPBpp08JaRhCLbdahT7ubkom76811Qx/oUMAHeuY/0ahQuv6IXg3htRY2DzG\ntJhVIQXVDEKlqDh+5zjRSdFFGn58rwbU9HLiZmI6EzdFFqkPIYQoCknWhRAVw/b/mOY9u/lD0KdF\n7sbOxo6gmkGMbzu+0McYjUY+2hjJ7aQMald2YnyvBkUev8JS20Hwd6bFq85tgeM/FfpQb0dvetXs\nxfCGw9GoivZphqOtmv8Obo6NSmHj8Rv8fPJGkfoRQghzSbIuhCj/zoXBsWWAYpr/bFe6iw9tPhnD\nlpMx2KgUvhjSQlYpLaqqTaHbh6bH28bDvauFPnTGYzN4v+37VHWqWuThH6nhzutdagPw0YZIbicV\nb7EmIYQoDEnWhRDlW2o8bP57vnOHN0zzn4tozfk1LDi5gJspNwt9zO2kdCZuNE2beKNrHZpVN38B\nJfGADm+BXzvITIKNb4ChdFcXfbN7XRr7VOJeqpYPQiPNXh1VCCHMJcm6EKJ82zoWkm+BV33o+lGR\nuzEajSyOXMxXx77i8M3DhT7mg9BT3EvV0tinEm90q1Pk8cXfVDbw1HxQO8DlPXBkYaEPNRgNRMRE\nsOWSeRVlHqSxUTF7cHM0Ngq/nL1F6NHrRe5LCCEKQ5J1IUT5dWYjRK4FxQYGzAdN0RcfOhV7iqtJ\nV3FQO9C9RvdCHbPu6HV+OXsbjY3yd4Inv3ItwrM2PDHZ9HjnRIi/VKjDDt44yEs7XmLW4VloDUWv\n6NKgaiXefsJUX3/y5tPEJKQVuS8hhCiI/OUQQpRPyXfg53dMjzu9Db6titXdz5d+BqBbjW44ahwL\nbB+TkMaUzacBePuJejSoWqlY44t/afsKBDwG2lTY8HqhpsO0rdYWT3tP4tPjOXDjQLGGH9W5Fs39\n3EhK1zF+3SmZDiOEKDGSrAshyh+jEba8C6lxpnJ/j79frO60Bi1hl00LIT1Z68lCDG/k/XWnSErX\n0dzPjVGdC1/iURSSSg84eycAACAASURBVAX9vwZbZ7i6HyK+LfAQtUpNr5qmkp2bL24u1vBqGxWz\nBzXDVq1iz/k7rDpctJKQQghREEnWhRDlT+Q6OLsJVGpT9Re1XbG62399P3cz7uJp70m7au0KbL/y\ncDR7z9/BTq1i9qDmqGX6S8lwD4Ae00yPf50CsX8VeEjf2n0BCI8OJzkzuVjD1/F2YWyP+gB8vOUs\n1+6mFqs/IYTIjfwFEUKUL0k3Ycv/mR53HgfVmhW7y/tTYHrX6o1apc63bXR8Kh//fAaAsT3rU8fb\nudjji3y0GgG1uoIuHTaMBoM+3+aNPBpRy7UWGfoMfrn6S7GHH9mpJq393UnO0DFu7UkMBpkOI4Sw\nLEnWhRDlh9FoWt0y/R5UawGPvWuRbp00TjioHehbq2++7QwGI+PWniQlU09rf3dGdKxpkfFFPhTF\nNB3GrhJcOwz7vyqguZJ1Hn+++HOxh7dRKXw+qDkOGhv2X4xjWcSVYvcphBAPkmRdCFF+HF8O58PA\nxtZU3s+maKtV/tvkDpPZPXg3DT0a5ttu6cErHLgUh4PGhs8HmVa7FKXAtfo/q9KGT4dbZ/Jt3rtW\nbwBupd4iXVf8hY0CvP5ZlfbTrX8SFZtS7D6FEOI+SdaFEOVDwnUIG2963OU/UKWRRbt31DiiKHkn\n31GxKczY9icA43s1IMDLyaLjiwK0CIF6QaDPNE2H0eddmtHX2ZfQfqFsemoT9uqil/N80LPt/Xm0\nlidpWj1j156Q6TBCCIuRZF0I8fAzGmHTG5CRCL6tTatcWkB8ejzn4s8V2E5vMDJ27QnStHoereXJ\ns+39LTK+MIOiwJNzwN4NYo7D71/k27yue91833yZS6VSmDWwGU62NhyOussP+y5brG8hRMUmyboQ\n4uH3x2K4uAvU9qbqLzb53wRaWP/P3n2HR1GtDxz/ztb0nhASQi+BAAGkC6gg1UJXECyIBcu9ol79\nqXjBq95rL6ioKApRBKSqKKBUBektQIAACTWEkN6zdX5/TBKIECCYXUrez/Pss5uZM3PO7mRm3z1z\nyg+HfmDY4mH8+89/XzDd9D8Ps+VINt4mPW8Pa41Omr9cGb7hMOBd7fXvb0HqrotuYnVYybPmVUv2\nUUFeTLhNu6Pzzq+JJKX/vdFmhBACJFgXQlzrso/Cby9rr3tNhJAm1bJbVVXLx+KODY2tNN2h0wW8\n/atW+/7y7S2ICrr4hEnChVoNg+Z3gNOuNYexWytNOu/APG6eezNf7vqy2rIf2TGKHk1DsdidPDs3\nHrvj4pM1CSHEhUiwLoS4djmd8OMTYC2Aul2h02PVtusD2Qc4lHMIo85In/p9zpvG7nDy7Lx4rHYn\nPZqGMqJDVLXlLy6TosBtH4BXMKTt0WrYKxHkEUS+NZ8lyUtwXGTIx0vPXuGtoa3w9TCw83gOX6xN\nrpb9CiFqLgnWhRDXri1fwpG1YPSCQVO0WS2ryQ+HfgDg5qib8TP5nTfNF2uTiT+eg6+HgbeGtqrW\nNtDib/AJhdve116v+wBStp03WffI7vib/TldfJoNqRuqLfva/p5MvF1rDvPh8oMknsqvtn0LIWoe\nCdaFENemzCRYPkl73ftVCGpYbbu2OCwsTtaawAxuPPi8aRJP5fPhcm3GzEl3xFDb37Pa8hfVIGYQ\ntBwKqgMWPQa2c4doNOlN5WOuLziwoFqzH3ZDHXpFh2F1OHlm7k5s0hxGCHGZJFgXQlx7nI7S9sjF\n0KAHtB9brbtfcXQFuZZcwr3D6RrR9Zz1ttIAzOpw0is6jKHtIqs1f1FNBrwL3mGQkQir/3veJEOa\nDAFgzfE1ZBRnVFvWiqLwxpBW+HsaSTiZx5TVh6pt30KImkWCdSHEtWfDFDi+CUy+MLB6m78ArDy2\nEtBq1fU6/Tnrp6w+RMLJPPw9jbwxRJq/XLW8grThHAHWfwzHNp2TpGlgU1qHtMau2vkp6adqzT7M\nz4NXB8YA8MmqQ+xJya3W/QshagYJ1oUQ15bT+2DV69rrvv+FgLrVnsXbPd7m454fM7TJ0HPW7T6R\nyyertFrSVwfGEOZXPZPqCBeJHgCxIwEVfhgH1nNnFx3aVDvOiw4uQlWrdzKjO2Mj6N8yHLtT5dm5\n8ZTYqqcjqxCi5pBgXQhx7bBbYeEj4LBA497Q7j6XZGPQGbg56mZqedeqsLzE5uDpuTuxO1X6twzn\nztgIl+Qvqlm/N8EvErKSYfnEc1fX78dDrR7ik16fVPtdEkVReH1QS0J8TCSm5fPB8gPVun8hxPWv\nysF6QUEB48ePJyIiAg8PD9q0acOcOXMuut3ChQsZOXIkjRs3xtPTk/r16zNq1CgOHjx4WQUXQtRA\nf7wDp3aBZyAM/EQbpq8aOZwO7E57pevf/TWRQ6cLCPEx89/B0vzlmuEZoP2/AGyZBodWVFjtZfTi\nqXZPUc/PNTPPBvuY+d/gVoA2gtDmw1kuyUcIcX2qcrA+ZMgQ4uLimDRpEkuXLqVDhw6MHDmSWbNm\nXXC7t956i6KiIiZMmMCyZct4/fXX2bFjB+3atSMhIeGy34AQooY4sRXWvqe9vu19bbbKarbmxBr6\nLujLNwnfnLNuQ1ImX5VOIf/2sFYEeZuqPX/hQo16QsdHtNc/PgnF2W7Nvk9MOMNvqIOqwrPzdlJg\nqfxHoRBCnK1Kc3IvWbKE5cuXM2vWLEaOHAnALbfcwtGjR3nuuee4++670evP7YwFsHjxYsLCwios\n69mzJ/Xr1+eDDz5g2rRpl/kWhBDXPWsRLHpUG4av5TBoOcQl2Sw4sIDTRafJKKk4Kkh+iY1/zYtH\nVbUZKntG16pkD+Kqdut/IGkVZB6CJc/B0IrfO9vTtvPt3m/pEtGFu5rdVe3ZT7yjBeuTMjmeVcx/\nf9nLG0NaV3seQojrT5Vq1hctWoSPjw/Dhw+vsHzMmDGcPHmSTZvO7Wlf5q+BOkBERAR16tTh+PHj\nVSmGEKKmWfGKFmD51oYB77gki1OFp/jz5J8ADGlc8cfAaz/vJSWnmKggTybc1sIl+Qs3MHnB4Kmg\n6GD3PNizsMLqfVn7WHFsBd8nfl/tHU0BfD2MvDs8FoDZm4+zan9atechhLj+VClY37NnD82bN8dg\nqFgh37p16/L1VZGcnMzRo0eJiYmp0nZCiBokaTVsnqq9HviJNhyfCyw6tAin6qR9rfbU969fvnz5\n3jTmbj2BosB7w9vgY67SDUlxtanTHro/q73+5RnIP1W+6vaGt2PSmTiQfYCETNc0z+zSKJix3RoA\n8Pz83WQVWl2SjxDi+lGlb53MzEwaNjx3lsCgoKDy9ZfKbrczduxYfHx8ePrppy9pm4SEBJxOmQXO\nnWw2W/lzfHz8FS6NcLcrffx11nyarXoYE5DRYBAphaHggnI4VSff7/0egPbm9uXvNbfEwb+WaLWf\ng5r5YM47Tnx8zbkTeKWPv6sogf1oHPAjXjkHyJt5P4e7vFXeWbm9f3vWZ6/ni41fMDaqeifbKtMv\nQuU3PwPH8yw8OWMdz98YdNV1Vr5ej724NHL8XUen01G3btWGHK5yFdGFLiiXerFRVZWxY8eydu1a\nFixYQFRU1CVtZ7fbcThkjNorpezkFTXTlTj+9Xd+iKk4nRKvSI41ewini8qwu2A3mbZMvHRetPVu\ni81mQ1VVPtmUS67FSV0/A8Obe9Xoc+B6e+/Jsf9Hi7Xj8EvbiH/Sj2TUuw2A7v7dWZ+9ng3ZG7gr\n9C489NU/jr4CPNnBj5dWZbH+eDGrk/PpXtez2vOpLtfbsRdVI8e/elXWt/NCqhSsBwcHn7f2PCtL\nG4aqrIb9QlRV5aGHHmLmzJnExcUxcODAS87fYDCgq+aZCsWFnX2SGo3GK1gScSVcyePvn7KGkJTl\nqOg43n4Cek8/qn6JuzTrctYBcGPQjXibvQFYlVzI5pMWDDp4umsQ3h41b/SX6/n8dwQ1IbXFI0Tu\nmULU3s8oDm+P1acOLf1bUstUizRrGtsKt3Fz8M0uyT86zMjdLe3M2p3HtB35tAr3ItT76mlidT0f\ne3Fxcvxd53Li2CpdGVq1asXs2bOx2+0V2q3v3r0bgJYtW15w+7JAffr06Xz11VeMHj26SoWNiYmR\nYN3N4uPjsdlsGI1GYmNjr3RxhJtdseOfewKWasM0Kt2fpknPUS7N7pnIZ1hwcAEjokfQNLApRzIK\nmbZgLQBP927G4JsbuzT/q9V1f/63agUFu9AfWUvzhHdh7G+gNzJSP5IPt3/I5pLNPBX7lMuyj2np\nZG/OBnYez+GL3VZmP9wOve7qaA5z3R97cUFy/F3H6XSSn59fpW2qFPkOHjyYgoICFixYUGF5XFwc\nERERdOrUqdJtVVXl4YcfZvr06UydOpUxY8ZUqaBCiBrC6YCFj0JJLkS0g5tfdHmWMSExTOwykaaB\nTbE5nDz1/U4KrQ46NQhi3E2NXJ6/uEJ0Ohj8OXj4w8ntsOYNAAY2HkjzoOYMaDAAp+q6flIGvY7J\nI9rgbdKz+XAWn/+e5LK8hBDXrirVrPfv35/evXvz2GOPkZeXR+PGjZk9ezbLli1j5syZ5e1wxo4d\nS1xcHElJSdSrp80I989//pOvvvqKBx98kFatWrFx48by/ZrNZtq2bVuNb0sIcc36czIcXQdGb20c\nbL17b8F+tPIg8cdz8PMw8MHdba6amk7hIv514I7JMO8BWPs+NOpFSP0bmXvHXLdkXy/Ym1fujOG5\n+bv4YPkBbmwcQpuoALfkLYS4NlS5TcnChQu59957mThxIv369WPTpk3Mnj2bUaPO3KZ2OBw4HI4K\n49QuXrwYgK+//pouXbpUeAwePLga3ooQ4pqXsg1W/1d7PeBtCHZtrfbaE2uZsG4CiVmJAGw+nMWU\n1YcAeGNIayICrt5Of6IaxQyGNqMBFRY+4vbZTYfdUIfbWtfG7lQZP2cHhTK7qRDiLFUO1n18fJg8\neTKpqalYLBbi4+MZMWJEhTQzZsxAVVXq169fvuzIkSOoqnrex5EjR/7u+xBCXOssBbDgYXDaocUg\naOPaduoAcQlx/JT0E78k/0JusY2nv9+JU4XhpcGTqEH6vwVBDSHvBPz8NKgqJfYSFhxYwE9JP7k0\na0VR+N+gVkT4e3Aks4hXfnLNGO9CiGuT9NYUQlwdlv0fZCWBXyTc8WH5uNeukpiVyKZTm9ArekY0\nG8GERbtJySmmXrAXk+6UidpqHLOP1uxKZ4CERbBzFr8e+ZVXNrzCR9s/wuZ07fB1/l5GPri7DYoC\n87ad4JddqS7NTwhx7ZBgXQhx5SX8ADtmAgoM+QI8A12e5bd7vwXg1nq3suGAk593pWLQKUwe0VZm\nKa2pIm8406F56fP082tKkEcQaUVprDy60uXZd2oYzOM3a02/Xly4i5M5xS7PUwhx9ZNgXQhxZeWe\ngMWlw+N1exrqd3N5lhnFGSw5vASA3pHDmfjjHgCe7t1UOvfVdN2ehnrdwFqA+ccnuLvJMODMjztX\nG39rU2KjAsgrsfP09ztxONWLbySEuK5JsC6EuHIc9tJhGnO0YRpveckt2c5NnIvNaaNVSGs+/dVO\nodVBRxmmUQDo9DBkqjacY8o27kpPwagzsitjF/Hprp923ajXMflubTjHTYez+LS0w7MQouaSYF0I\nceWseUMbptHk47ZhGi0OC98nfg+Ad3FP4o/n4O9plGEaxRn+deCOjwAI2fAZA4LbAO6rXa8f4s2r\nA7VJBj9YcYANSefOHC6EqDkkWBdCXBmHVsBabZZS7vzI5cM0lnE4HYxuPpq63s1ZvjUUgPeGxxIp\nwzSKs8UMgg4PA3Dvvt8BWHF0BakF7un4OfSGOgy7oQ5OFf45Zwfp+Ra35CuEuPpIsC6EcL+8k9p4\n1qjQfiy0HOq2rL2MXvSNHMWxPWMBPY/0aMitLWq5LX9xDen7X6gdS7P8DDo5TXSp3Zliu/s6fb42\nsCVNa/mQnm+R9utC1GASrAsh3Mthh/ljoSgTwltD3/+5NXuL3cGTs7eTX2KnXd0AnuvbzK35i2uI\nwQzDZ4DZj0+PHuIzwmgY0NBt2Xua9Ey5px2eRj3rDmXwySppvy5ETSTBuhDCvVb/F46tB5OvFggZ\nPdyW9eTtk3nyhzh2ncgmwMvIJ/e0w6iXy6C4gKCGMPATTAB/TobEZW7NvkktX14fpLVf/3DlAdYf\nynBr/kKIK0++pYQQ7nNwOax7X3s98GO3tVMHOJJ7hGm7p7Gh4EMUYxbv3xVLhLRTF5eixUDoNA6A\nUz89xrTN7+JUnW7LfugNdbirfR1UFf45Zyen80vclrcQ4sqTYF0I4R65J0rbqaN13IsZ7NbsP98x\nAwBHQTSPdO1Iz2hppy6qoPer2CLacnewJ5P3xfHH0VVuzf4/d7akWS1fMgosPDVb2q8LUZNIsC6E\ncD2HTWunXpwFtWO1jntulFaQxZIjiwGob+zHv/pIO3VRRQYzxuEzGFhsB+Cb9a+7NXtPk54po9rh\nZdKzITmTySsOuDV/IcSVI8G6EML1lr0IxzeC2U9rp24wuy1rVVV5+Mf3UBUrWCOYdtcIaacuLk9g\nfe65cSIGVWWLLZOt699za/aNw3z43+BWAHy06hDL96a5NX8hxJUh31hCCNfa/g1s+VJ7PeQLrcOe\nG322dhfJVq1T4LjWjxEZ6OXW/MX1JbzNaIb4aP/Dn+7+AlJdP6vp2Qa1jeT+LvUAePr7nRw6ne/W\n/IUQ7ifBuhDCdY5vgV+e1V7fMgGa9Xdr9uuTMvho65coOithpkY83nGgW/MX16eH+36GEYUtHiY2\nzx8Fhe4doeXl21vQsUEQBRY7D3+zjdxim1vzF0K4lwTrQgjXyEuF70eDwwrRt0P3f7k1++NZRTzx\n3XZshY3wVRowsdvTKIri1jKI61O4byTDGmsdpKeYLKjz7tf6ZbiJUa/j01HtiPD34HBGIePn7JAO\np0JcxyRYF0JUP7sF5t4LBacgtDkM/hx07rvcFFsdPPLtNrKLbDQPaM/KEQvpUaeH2/IX17+H2j6B\nv9GHVjYV+5F18Nu/3Zp/iI+Zqfe2x2zQsToxnfeXJ7o1fyGE+0iwLoSoXqqqNX05sQU8/GHEd2D2\ndWP2Ks8v2MW+1DyCvU1Mvbc9niaD1KqLahXmFcaKu1bzr14fYATY9BnsnOXWMrSq489bQ1sDMGV1\nEr/sSnVr/kII95BgXQhRvbZ+BTu+BUUHw75268RHAF/8kczi+JN4hC5nQPcEArzdN3mNqFk8DB7Q\n/A646f+0BYvHw4ltbi3DoLaRPNy9AQD/mhfPvtQ8t+YvhHA9CdaFENXnyDpYWhq49JoEjW91a/Zr\nEk/z1rL9KIZcPEL/4IejX7ErfZdbyyBqnp3N+/FGozaoDovWTyP/lFvz/79+0XRvEkKxzcEj324l\nq9Dq1vyFEK4lwboQonqkJ8Kce8Bph5ZD4can3Jp9wslcnvhuO04VWjTfgkO10S6sHZ1rd3ZrOUTN\nkmfN45GV45jlzOL3Wo0g/yR8NxwsBW4rg0Gv4+ORbakb5MXxrGIeittCic3htvyFEK4lwboQ4u/L\nT4OZw6AkF+p0gIFTwI1txFNyihkzfQuFVgc3NISTzjUAPNn2SWmrLlzKz+THyOiRAEwJr4PqFQKn\ndsG8B8Bhd1s5ArxMfP1Ae/w8DGw/lsP4OTtlhBghrhMSrAsh/h5LAcwaDrnHtAmPRs4Bo6fbss8t\ntjFm+mZO51toWsuH6OabsTvtdArvRIfwDm4rh6i5Hoh5AC+DF/vzDrPq1v8DgyccWg6/PK11uHaT\nxmG+fHlfe0x6HcsSTvG/JfvclrcQwnUkWBdCXD6HHeY/qM3i6BUMo+aDd4jbsrfanTw2cxsH0goI\n8zXzxl2RLDn8EwCPt3ncbeUQNVugRyCjmo8CYMqJZTiHTtM6WG//Bta+59aydGoYzLt3xQLw1brD\nfL3usFvzF0JUPwnWhRCXR1Vh6XNw8FcweMDI79068ouqqrywYBfrkzLxNun5+oEOLDw8Hbtqp2tE\nV9rVaue2sghxf8z9+Bh9OJh9kJ9NTuj/trZi1WsQ/71by3JnbAQv9I8G4LVf9rJsj3s7vAohqpcE\n60KIy/Pnh7D1a0CBodMgyr1NTt5ffoCFO1LQ6xSmjGpHy0h/Hm/zOD2jevLPdv90a1mE8Df7M7bV\nWADe2/oeBW1GQtd/aCt/fAIO/+HW8jzaoyGjO9dFVeGpOTvYdjTbrfkLIaqPBOtCiKrbPR9WvKK9\n7vemNta0G83ZfIyPVx0C4L+DWnJzszAAIn0imdxzMjHBMW4tjxAA97e4nxtq3cCz7Z/F2+gNt74K\nLQaB0wZzRkPaXreVRVEUXrkjhl7RYVjsTh6K28LhjEK35S+EqD5VDtYLCgoYP348EREReHh40KZN\nG+bMmXPR7U6cOMH48eO56aabCAgIQFEUZsyYcTllFkJcSYlLYdGj2uvOT0DncW7N/uddJ3lp0W4A\n/tGzMSM61iWzONOtZRDifIx6I9P7TufORndqoxDpdDB4KtTtApZc+HYwZCa5rTwGvY6P72lL6zr+\nZBfZGD1tEyk5xW7LXwhRPaocrA8ZMoS4uDgmTZrE0qVL6dChAyNHjmTWrAtPs3zo0CG+++47TCYT\nAwYMuOwCCyGuoEMrYe59pWOpD4M+r7s1+98STjF+zk6cKtzdPopnejfldNFpblt0GxPWTaDIVuTW\n8gjxV2cPFVpkK0I1mGHELAiLgYJTEHcnZB91W3m8TAa+ur8DDUO8Sckp5p4vN5KWV+K2/IUQf1+V\ngvUlS5awfPlyPv30Ux599FFuueUWvvzyS3r37s1zzz2Hw1H5JAw9evQgPT2d5cuX88wzz/ztggsh\n3OzIOpgzChxWrdnL4KlazaGb/H4gnSdn7cDuVBnUJoL/DWmFoii8u/VdCm2FHMk7ok3/LsRV4Lcj\nv3H7ottZnLwYvILgvh8gpCnknYBv7oS8k24rS6ivme8e7kRUkCdHM4u458uNZBRY3Ja/EOLvqdI3\n7aJFi/Dx8WH48OEVlo8ZM4aTJ0+yadOmyjNy45e6EKJ6eWXuge/uAnsxNOkLQ78GvcFt+a9PyuCR\nb7ZidTgZ0Cqcd4fHotcpbE7dzNLDS9EpOiZ0moBOkeuMuDqcKDhBenE67219jzxrHviEwX0/QmB9\nyD6i1bAXnHZbeWr7ezLroc7U9vcgKb2Q0dM2kVNkdVv+QojLV6Vv2z179tC8eXMMhoqbtW7dunx9\n165dq690f5GQkIDT6XTZ/sW5bDZb+XN8fPwVLo1wN5vNhlfOARpsfA7sheSHtudw83+hJrhvspV9\n6RZeWZOBxa7SIdKDB5vrSdizG7tqZ2LiRAB6BffCdsJG/An5H61Ocv5fvlhnLBHmCE6WnOQ/y//D\nfXXuA8DY8W0a//EPTJkHKf6iD0ndJuMwB7itXBO7+/PSCiv7T+Uz9JM1vHZLKN6mc3/kyrGv2eT4\nu45Op6Nu3bpV2qZKwXpmZiYNGzY8Z3lQUFD5eley2+0XbGojXKvs5BU1h2deMk03PY/BXkh+UCsO\n3vAKTqdOG93CDQ5l2Xj1j2xK7Cqtw0w83dEPnHZsTliWsYyUkhR89b4MDBko/58uJp9v1Y0KH8U7\nR99hecZybvS7kbqedbEZg0ns/A7N1j+NZ95hGvz5LAc6v4vD6OOWMoV6wMQegUxak0VSlo3/rEnn\n5e4BeBoqvyslx75mk+NfvfR6fZW3qfJ97LM7z1RlXXUwGAzSnMbNzj5JjUbjFSyJcDeP3CQabnwO\ngy2fgoDmHOn6DnqjF1W/zFyeQ1lWXl+XTZFdpWWYiZdvCsFcGlBkWbP4Mf1HAEZGjCTAw301kzWJ\nnP9/T2xgLJ1zO7MxZyMz02by78b/RqfocAbUJ7nbBzRa+0+8cw/SdPOLHO76Dg6Tr1vK1TDYyH96\nhvLyynQSM228+WcuL98UgpfxzPerHPuaTY6/61xOHFulYD04OPi8tedZWVnAmRp2V4mJiZFg3c3i\n4+Ox2WwYjUZiY2OvdHGEuxzfDEvHgzWXQr/GHL7xXVq17+K27DckZTJp4VYKrCpt6wbw7dhO+JjP\nXK62ntqKd7I30b7RPHHLE9JW3UXk/P/7Xm/8Onf8cAcHCw9yxPsIg5sMLl0TC00aQdzteGfvpeWW\n5+HeheAb7pZyxQING+UwatomEtKtvLGxkBljOhLkbQLk2Nd0cvxdx+l0kp+fX6VtqvQN16pVK/bt\n24fdbq+wfPdubczjli1bVilzIcRV6NBK+GYglORSGBTDgc7v4nRTjR/A8r1p3D99MwUWO10aBp8T\nqAO0D2/P4sGLeaP7GxKoi6taLe9aPB77OACHcw9XXBneEh74BXxqwekE+LovZB0+z15cIzYqgFkP\ndyLQy8iuE7kM/3w9J2UcdiGuOlX6lhs8eDAFBQUsWLCgwvK4uDgiIiLo1KlTtRZOCOFmexbCrLvB\nVgSNepF84/tuuzUPsGDbCcbN3IbV7qR3i1pMH9OhQqCuqmr5a1+TL3V867itbEJcrtEtRvNlny95\npv15hi2uFQMP/npmlJiv+0FagtvK1rpOAPPGdS0fJWb45xtISi9wW/5CiIurUrDev39/evfuzWOP\nPcaXX37J6tWreeSRR1i2bBlvv/12eaP5sWPHYjAYOHq04sQP8+fPZ/78+axatQqArVu3li8TQlxh\nW6fD/Ae1zqMxQ2DkHJwGT7dl/9W6wzw7Lx6HU2XYDXX4bFQ7PIxnWshbHVYe+u0hfkn+xW1lEqI6\nGHQGOtfuXP732T86AQhqoAXsYS20iZOm99eaorlJ4zAf5j/WtXzipLs+30BSlgzrKMTVosodTBcu\nXMiECROYOHEiWVlZREdHM3v2bEaMGFGexuFw4HA4zrkg/XV89ilTpjBlyhTgPBcvIYR7qCqs+wBW\n/kf7+4YxcNt7oHNPV1JVVXl/+QE+XnUIgLHdGjBhQHN0uood1j/c/iGbT20mMTuRbpHd8Df7u6V8\nQlSn00WneWndb4OfAAAAIABJREFUSzwQ8wDdIrudWeEbDmOWaPMZnNisNUW7eyY07uWWckUGeDJ3\nXBcemL6ZPSl5TFiZzgs3BtAmQjoXCnGlVbmxp4+PD5MnTyY1NRWLxUJ8fHyFQB1gxowZqKpK/fr1\nKyxXVbXShxDiCnDYYdkLZwL17v+C2z9wW6ButTt5adHu8kD9ub7NePm2cwP1P078wbd7vwXgta6v\nSaAurlkz985kU+omJqybQEZxRsWVnoHaTKeNemlN0WbdDTtnu61sIT5mZj/cmU4Ngii2q/x3bTZ/\nHityW/5CiPOTnllC1FRFWfDdMNj0ufZ3n9eh17/BxUOwlskssDB62iZmbz6OosBrg1ryxC2NzxkC\n9nTRaV5e9zIA90Tfwy11b3FL+YRwhcfbPE6TwCZklWTx4toXcap/mejP5A0j52hN0Zw2+GEc/DoB\nnO6ZY8TXw0jcgx3pGOmB1Qlv/5nFe78l4nRKpZoQV4oE60LUROmJMK0XJK8Goxfc9Q10/Yfbst97\nMo87P/mTzUey8DUb+Or+9tzbud456RxOBy+tfYlsSzbNApudv4OeENcQD4MH7/Z4Fw+9BxtTNzJ9\nz/RzExlMMPQr6PGc9veGT2DWXVCc454yGvW80C2YO5p4AfDxqkM8OnMbBRb7RbYUQriCBOtC1DQH\nfoUve0FWMvjXhbG/QYuBbst+6e5Uhn62npScYuoHe7Hoia70jK513rRf7/maTac24Wnw5J2b3sGs\nN7utnEK4SsOAhrzQ8QUAPtnxCbvSd52bSKeDni/DsOlg8IRDK7Qf2BkH3VJGvU7h/lhfnuociEmv\nY/neNIZ+up5jmdIsRgh3k2BdiJqirCPprLvBmg/1boRHVkN4K7dk73SqfLD8AI99t51im4PuTUL4\n8YluNA6rfGjIQlshAC91eokG/g3cUk4h3GFIkyH0q98Pu2rn+T+eP7f9epmWQ2Dsr+BXBzIPaT+0\nDy53Wzl7NvBmzqOdCfU1k5iWz51T1rE+qZKyCiFcQoJ1IWoCSz4seAhWvAKo0P5BuPcH8A5xS/a5\nRTYe+24bk1dqtYJjuzVg+gMd8Pe68EgT428Yz7w75jGwkftq/oVwB0VRmNhlIlG+UZj1ZuzOCzQx\nqR2r/bCO6gyWXK1JzNr3wemsfJtq1K5uIIuf7EbrOv7kFNm496vNfLXusAwOIYSbSLAuxPXuxFb4\nvBvsmQ86gzYs4+0faO1i3WBTcib9J//BrwlpmPQ63h7Wmn/f3gKD/vyXn5SClPIadYDooOhzOp0K\ncT3wNfkytfdU4vrFEe4dfuHEPmFw/2Joey+oTm0Ep28HQd5Jt5Q13N+DuY92YWCbCBxOldd+3suY\nGVtIz7e4JX8hajIJ1oW4Xjkd8Ps78FUfbWZE/yi4/2fo8JBbsrc5nLzz635GfLmRk7kl1Av2Yu64\nLtzVPqrSbdKL0hn761jG/jqWrJIst5RTiCspyjeKAI+A8r+3p23HUdnILwYT3Pkx3PGR1jH88O/w\nWVfYt9gtZfUw6vnw7ja8ckcLTAYdaxLT6T/5D1bvP+2W/IWoqSRYF+J6lHMMZtwGq18H1QEth8K4\ndVCvi1uyP5JRyLDPNzBldRKqCsNvqMMv/+xOm6iASrfJs+YxbsU4UgpSyLPmnTuknRDXubmJc3lg\n2QP8d9N/K29ioihww/3w6B9a85jibPh+NPz0T7AWnn+baqQoCg/c2IDFT3ajWS1fMgqsjJmxhVd+\nSqDE5p7hJYWoaSRYF+J6s3s+fNYNjm0Aky8MnqoNA+dZeaBcXVRVZd7W49z20Vrij+fg52Fgyj3t\neGd4LD7myidMLrGX8I+V/+BA9gFCPEOY2nsqIZ7uaU8vxNUiwKydo/MOzGPKzikXThzSBMaugBuf\nAhTYHgdTb4KTO11fUKBZuC8/PnkjD3StD8CM9UcY+MmfJJ7Kd0v+QtQkEqwLcb3ITdFq2BaM1Tqh\n1ekA49ZC7Ai3THR0PKuIsXFbeW7+LgqtDjo2CGLp+B7c1rr2BbezO7XRMLaf3o6P0YfPb/2cKN/K\nm8oIcb3qU78PL3fWJgCbumsqs/bNuvAGBhP0fhXu+xF8a0PmQW14xxWvgNX1Qyx6GPW8cmcM08d0\nIMTHRGJaPnd8vI73lx+QWnYhqpEE60Jc6xx2WP8JfNJBa7uq6KHH8zBmKQS5frhDq93JlNWH6P3B\n76zafxqjXuG5vs2Y/XBnIgM8L7ytw8pL615i9fHVmHQmPu75Mc2Cmrm8zEJcre5qdhePt3kcgDc3\nv8ncxLkX36jhTfDYemh+Jzjt2hCtUzpB4lIXl1ZzS7Mwlo3vQa/oMKwOJx+tPEi/D//gjwPpbslf\niOudBOtCXMuOb4YvboLfJoCtEKI6aW1Ze04A/YWHRawOG5IyGfDRWt75NZESm5PODYNY+lR3nril\nMXrdxWvz86x5bE/bjl7R8/ZNb9M+vL3LyyzE1W5c63GMjB6JisprG1/j/W3vX3yYRK8guPtbGDFb\n60yeewxmj4A5oyDnuMvLHOJjZtr97ZlyTztq+Zk5klnEfV9v5olZ20nLK3F5/kJczypvRCqEuHoV\nZWm3urfHaX97Bmq3w9uM1mY+dLGMAgv/W7KPhdtTAAj2NvHy7c0Z1CaySsMshniGMKXXFDKLM+ka\n2dVVxRXimqIoCi92fJFAj0A+3fkp3gbvSz+vogdoNe2/vwUbpsD+nyFpFdz8AnR+3KU/4hVF4bbW\ntenRNIT3lx8gbv0RftmVyu+J6Tzbpyn3dq5X6ZCtQojKSbAuxLXEUgCbPoc/P9LapYMWoPd+FbyD\nXZ59fomNaWsPM21tMoVWB4oC93Ssy/N9oy86wVGZ7WnbySjOoE/9PgDS7EWI81AUhcdiH6Nz7c60\nCW1TtY1N3to1ofUI+OVZOLYelk+ErdPhlpe00aF0etcUHPD1MDLpjhiGtqvDyz/sYefxHP6zeC/f\nbjzKs72b0b9lOLpLuPMmhNBIsC7EtcBWAtumw9r3oLC0HWhYDNz2LtRzfY20xe5k6u9JfPZ7EjlF\nNgBaRfrz6sAY2tYNvOT9LElewst/ah3oInwiaBnS0iXlFeJ60TasbfnrIlsRE9ZN4Mm2T9IooNHF\nN67VAsYsgZ2zYMUkyD4MCx/W2rT3fBmaDXBp5/OWkf4sfKwrs7cc491fE0lOL+SJWduJifDjX32b\ncXPTUJnwTIhLIMG6EFczhx3iZ8GatyDvhLYssAHcMqG0dsy1t5RtDpVfk4qYv7+Q7GJtpsRGod48\n26cZ/WIuvXbM4XTwxa4v+DT+UwB61e11acGGEKLch9s/ZMWxFWxK3cTbN71Nt8huF99IUaDtKGgx\nEDZPhT8nw+m9MOceiLwBek2Ehje7rMw6ncKoTvW4MzaCr9YdZtrawySczGPM9C10qB/Ic32j6dgg\nyGX5C3E9UNSL9lq5MpxOJ/n5Fcdr9fX1ReeG9rjijPj4eGw2G0ajkdjY2CtdnJrDWgjxs7U2p1nJ\n2jLfCLjpeWg72uWdRwssduZtPc5nqxI5XagNwRYZ4Mn4W5swuG1kldqdJuck8+/1/2ZX+i4AHoh5\ngKdveBqdIufy1U7O/6tLdkk241ePZ/vp7QAMazqMZ254Bl+T76XvpDgb1n8MGz8DW+nwjvVuhC5P\nQNN+5c1jXHXsswqtfP57EnHrj2CxaxOfdWscwtjuDbipSag0j7lKyLnvOpcT30qwLi5ITlg3yz0B\nm7+EbTOgJEdb5hUM3Z+F9mPB6OHS7I9nFRG3/gjfbzlOvsUOQIBZx10t/Xh2cGfMhqq1c41LiGPy\n9snYnDa8jd680PEFBjUe5IqiCxeQ8//qY3VYeWfLO8xJnANAmFcYk7pMokedHlXbUcFprVnd1q/B\nYdWWBTaATuOg7Sji9ye79Nifyi3h41UH+X7LcexOLQxpFOrNmBsbMKRdJF4mufF/Jcm57zoSrItq\nJyesmxzfAhs/hb0/glo6mUhgA+j8GLS5B8xVqDmrIlVV2Xo0m6/WHua3vaco/d6kYag3feoZ6V7H\nhI+n6bKO/4w9M3hv23t0j+zOxC4TCfcOr+bSC1eS8//qtfXUViatn8Sx/GMAvNzpZe6OvrvqO8pN\ngc1fVKwgMPtxOqofp+oORPWPcumxP18Fgb+nkZEd63Jfl3pEXGSuBuEacu67jgTrotrJCetCeamw\nex7s+h7S9pxZXr+7NsRa074uHbHhZE4xP+xMYdH2FA6eLihf3r1JCA92025J7969q0rH3+KwkF6U\nTh3fOoDWVv2PE39wc9TN0pHsGiTn/9Wt2F7MlB1T+Dn5ZxYOXEiQx99o+13W9G7j59pMqICKQkHo\nDfh2exiibwezTzWV/FxlTe9mrD/C0UyteY5ep9C9SQiD20bSp0U4nibXXQ9FRXLuu44E66LayQlb\nzayFsO9n2DUHkteAqrXZRG+CVsO1W9C1W7ss+wKLnaW7U1m0I4UNyZmUnf1mg44h7SIZc2MDmtY6\nU4t/qce/2F7MvMR5zEiYgbfRm3l3zMPD4NomO8L15Py/NhRYC/AxaYG0qqq8uvFVekT2uLwfyU4n\nJK0kf/mb+J7eema50Qua3wGt79Y6pLqoIsHhVFm1/zRfrUtmY3JW+XIfs4F+LcMZ0jaSzg2DpW27\ni8m57zqXE99KozAhXK04B5JWalN/71+izTRaJqoTxI6AFoO0GQhdIKfIyprEdJbvS2PlvjRKbM7y\ndR0bBDG0XST9W9XGz6PqnVaLbEXMSZxDXEIcWSXaF6tep+dI3hGig6Kr7T0IISpXFqgDbEzdyPwD\n85l/YD7RQdE80voRetXtdekdunU6aNKb5KIwlJyjhJ1aTXjaGq2j+67vtYdPuDa6TLN+WudUg7na\n3otep9C7RS16t6jF4YxCFm0/waKdKRzPKmb+thPM33aC2v4e9G9Zm1ubh9GhQRBGmWhJXOekZl1c\nkPy6vkyZSVpwfmAZHNsATvuZdYH1IXYktL4LghpWe9aqqpKUXsDKfadZue80W49mlbdDB2gY4s2Q\ndpEMbBNJVJDXBfdV2fEvsBYwa/8svt37LTkWrZ1rpE8kD7d6mDsb3YnRxaPVCPeQ8//ak1WSxTcJ\n3zB7/2yK7FpzksYBjXmk9SP0rtcbg+7S6ugqHPvWreHEVq2ZTMJCbUSZMiYfaNRTG0mmSR/wCa32\n91TWr2bh9hP8vCuV/JIz11Nfs4EezUK5tXkYNzcNI9DbVO3510Ry7ruONIMR1U5O2EuUmwJH18PR\nP+HwH5CVVHF9SDOtDXr07RDVsdonIjmRXcTmw1lsPpzFhuTM8jafZZrV8qVX8zD6xIQTW8f/km+N\nV3b8E7MSGbZ4GAD1/OrxcKuHGdBwAEadBOnXEzn/r105JTnM3DeTWftmkW/TvkuDPYKZc/ucS+ro\nXemxt1vh0ApIXAIHf4OCtLO2UiCyHdTvptW4R3UCz4BqfV8lNgdrEtNZsS+N1ftPk1loLV+nU6BN\nVACdGgbTsUEQ7esF4nsZdwyFnPuuJM1ghHAHpwMyDsKJLWcC9JyjFdPojFD/Rq22qWnfaq1Btzuc\nHEovYMexnPIAPSWnuEIak15H50bB9IoOo2d02EVr0CvjVJ0cKDzA5vzN1CqpxYudXgSgWVAz+tbv\nS8+onvSt3xe9CzvCCiGqLsAjgCfbPsl9Mfcxa98sZu2bhZfRi1petcrT/H78dxr6NyTKL+rSd2ww\nQfQA7eF0QuoOOPCrdifx1C5I2aY9/pwMKBDeEup21WZajmwH/lF/q7LCw6inX8tw+rUMx+lU2Xki\nh1X7TrNiXxr7T+Wz/VgO24/l8NmaJHQKtIjwo2P9YDo2CCQ2KoBwPw/p7C6uOVKzLi6oxv+6tlsh\nfR+kxp95nNoD9orBMYoOasdqtUn1umojunj4/e3sS2wODqTlk3Ayjz0puew5mcf+1LzyyUTK6HUK\nrSL96dQgiA71g+jcKBgf8+X9Fk8rTGPzqc1sObWFP47+QaYtEwBPgydr7lqDl/H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NOu32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\n/BSHdSIiIiIiP/UPGKpfVEWEna0AAAAASUVORK5CYII=\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "def plot_products(m1, v1, m2, v2): \n", " plt.figure()\n", " product = gaussian_multiply((m1, v1), (m2, v2))\n", "\n", " xs = np.arange(5, 15, 0.1)\n", " ys = [stats.gaussian(x, m1, v1) for x in xs]\n", " plt.plot(xs, ys, label='$\\mathcal{N}$'+'$({},{})$'.format(m1, v1))\n", "\n", " ys = [stats.gaussian(x, m2, v2) for x in xs]\n", " plt.plot(xs, ys, label='$\\mathcal{N}$'+'$({},{})$'.format(m2, v2))\n", "\n", " ys = [stats.gaussian(x, *product) for x in xs]\n", " plt.plot(xs, ys, label='product', ls='--')\n", " plt.legend();\n", " \n", "z1 = (10.2, 1)\n", "z2 = (9.7, 1)\n", " \n", "plot_products(z1[0], z1[1], z2[0], z2[1])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "If you ask two people to measure the distance of a table from a wall, and one gets 10.2 meters, and the other got 9.7 meters, your best guess must be the average, 9.95 meters if you trust the skills of both equally.\n", "\n", "Recall the g-h filter. We agreed that if I weighed myself on two scales, and the first read 160 lbs while the second read 170 lbs, and both were equally accurate, the best estimate was 165 lbs. Furthermore I should be a bit more confident about 165 lbs vs 160 lbs or 170 lbs because I now have two readings, both near this estimate, increasing my confidence that neither is wildly wrong. \n", "\n", "This becomes counter-intuitive in more complicated situations, so let's consider it further. Perhaps a more reasonable assumption would be that one person made a mistake, and the true distance is either 10.2 or 9.7, but certainly not 9.95. Surely that is possible. But we know we have noisy measurements, so we have no reason to think one of the measurements has no noise, or that one person made a gross error that allows us to discard their measurement. Given all available information, the best estimate must be 9.95.\n", "\n", "In the update step of the Kalman filter we are not combining two measurements, but one measurement and the prior, our estimate before incorporating the measurement. We went through this logic for the g-h filter. It doesn't matter if we are incorporating information from two measurements, or a measurement and a prediction, the math is the same. \n", "\n", "Let's look at that. I'll create a fairly inaccurate prior of $\\mathcal N(8.5, 1.5)$ and a more accurate measurement of $\\mathcal N(10.2, 0.5).$ By \"accurate\" I mean the sensor variance is smaller than the prior's variance, not that I somehow know that the dog is closer to 10.2 than 8.5. Next I'll plot the reverse relationship: an accurate prior of $\\mathcal N(8.5, 0.5)$ and a inaccurate measurement of $\\mathcal N(10.2, 1.5)$." ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "image/png": 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F0fHRaA1aGnk1Itgz2HaJnP0Jrh0CjQt0f80sTfau25uTySdRKszQD6RxNi6W\ntH6ScXaYNmPB2bvi7QpRyaWnpxMXF0evXr3o3LkzW7eWfHP80ksvsWHDhmKzwBQKCgqia9eurF+/\nnueff77E/okTJxIXF1f08w8//MAPP/wAwOXLlwkODi7at3btWt58801mzpxJamoqISEhREZGMmLE\niKJjDAYDOp0OvV5f9NjGjRsB2LJlC1u2bCmRg+Gee1WysrIA45uQspQ3FzBO8bhq1So+++wzsrKy\n8PHxoUuXLnz77be0a9eu2LHR0dGkp6czevToMvMxJ4XBYJ937+j1ejIzM4s95u7uXmICf2FZsbGx\nFBQUoNFoaNWqla3TEVYm19/8Xt7+MtHXonmx1YtMDJ9omyR0WvhfB0g5D91eLXVOc1Ovf74uH5VC\nhUpZsRvXiuh1sKgz3DpjXDip72zztCseqLK/9uPj4wkKCrJ1GhYTFRXF9evXeeqpp0oM0Sh0+/Zt\nXFxccHQsuQrxmjVrGD58OHFxcdSuXbvE/pycHAwGAwqFwmqrdJamcNx9bGwsLVq0sGkuAGPHjuXS\npUsPNXXj/f4flqe+lcpXCCGsRG/QczPnJgB96tlwCEzMt8ZC3cUXOk02W7MOKgfzFepgXCypzx9j\nWQ98BunXzde2EJVYz549GTNmTKmFOoC3t/d9C3WAoUOH0q5dO95//31LpWg2UVFRjBgxwi4K9YsX\nL7Jq1Srmz59v1bhSrAshhJUoFUq+H/Q9GwdvpJFXI9skkZ8N0fOM292mgZOH2UPoDXqupF8xT2NN\n+kPdTsaVVaPtv7AQojJQKBQsXryYwMDAEkNC7M2HH37IihUrbJ0GAFevXmXhwoUlVjW1NCnWhRDC\nyoI9g01aMMSs9i+CrBvgVQ8injN782l30uj7Q1+GbBhCZn7mg094EIXi7vCXY8sh6WzF2xRC0Lx5\nc9544w0ZXmyCnj173necv6XJFRJCCCso0BWQq8198IGWlJ0Cez8xbveaAer7f0ReEV5OXrhoXNDq\ntey9bqbluIMegZCBYNDDdhm3LoSoXqRYF0IIK4i+Fk33Vd2Zf9C6Yx2L2f0R5GVAzZbQfJjFwvSs\na5zSbEf8DvM12meWcT74c5sgbp/52hVCCDsnxboQQljB1rit5GpzUSttNGPu7StwcLFxu+9ssOBH\n372CegGw59oeCvQF5mnUr7Fx+kaAbW8b54kXQohqQIp1IYSwsDxdHruu7QJsOAvMjndBXwANekDD\nXhYN1cKvBT5OPmQWZHL4xmHzNdz9dVA7Q/wBOPuz+doVQgg7JsW6EEJY2IHEA2QXZBPgEkALPxtM\nP3bzFJz43rjdx/JjvlVKFT3EPhPLAAAgAElEQVSCegAQFR9lvoY9akHHScbt7XPAzmexEEIIc5Bi\nXQghLGxn/E4Aegb1NM/qnqba+3/G72FPQmC4VUL2DDKOW4+Kj8Ksa+91ngyOnpB8Dn4vufKhEEJU\nNVKsCyGEBRkMBnZeMxbr3ep0s34C6dfh5GrjduepVgvboVYHng17lve6vIcBMxbrTp53p5z8bYH5\n2hVCCDslxboQQljQudvnuJlzE2e1M+1rtbd+AgcWgV4LwV2hdhurhXVSO/FKu1doV7Od+T9NaP8C\nKDVw9Te4ZsYx8UIIYYekWBdCCAvydfJlSpspjA4djaPK/POal+lOOhxeatzu9LJ1Y1uSRy1o+Yxx\nu3DeeCGEqKKkWBdCCAvyd/Hnby3+xpQ2U6wf/MgyyM8E/xBo1Nf68THeXPvu/ndJyEowb8OFbz7O\nbITUS+ZtWwhRpuDgYP76179arP0VK1bw8ccfW6z9ykaKdSGEqIq0+bB/kXG740sWnVe9LJ/FfsbK\ncyvNOysMQEDoH29ADLDvU/O2LYSwKSnWi5NiXQghLOTQjUP8dOkn0vPSrR/81FrITAC3GneHjNhA\n0awwV81crINxZhiAmOWQnWL+9oWoYnJzc22dgigHKdaFEMJClp9ZzvTd01lxZoV1AxsMd6drbD8B\n1FYeK3+PnnWNxfrhm4fN/6YluCvUagXaXDj0pXnbFlVWTkFOqV95uryHPvaO9k65j62IWbNmoVAo\niImJYejQoXh4eODp6cmYMWO4detW0XHBwcEMHDiQtWvX0rp1a5ycnJg927jOwp07d5g+fTr169fH\nwcGB2rVrM2nSJNLS0orFKigoYNq0adSsWRMXFxe6dOnCwYMHS83pz5YuXYpCoeDKlSvFHl+xYgUd\nO3bEzc0NNzc3wsPDWbJkCQA9evTg559/Ji4uDoVCUfRVndlo3WshhKja8nR5/JbwGwDdg7pbN/jF\nHZB0CjSuEDHOurH/JMg9iEZejbiQdoHd13czsMFA8zWuUECnybBmPBz8wtjTrnE2X/uiSmq/ovRZ\nmbrW7sr/+vyv6Oce3/cgV3v/3uiIGhF83f/rop/7r+nP7bzb9z22mW8zVg5cWc6M72/IkCE888wz\nvPDCC5w6dYoZM2Zw+vRpDhw4gEajAeDo0aOcOXOGt956i/r16+Pq6orBYGDw4MFs376d6dOn07Vr\nV44fP87bb7/Nvn372LdvX1GMSZMmsWLFCl555RX69u3LyZMnGTp0KJmZmeXOe+bMmcydO5ehQ4fy\nr3/9C09PT06ePElcXBwA//vf/3j++ee5ePEi69atq9g/UhUhxboQQljA4RuHydXmEuAcQKhPqHWD\n//ZHr3qbv4Czt3Vj30fPoJ5cSLvAjqs7zFusA4QNhm2zIf0qxEba/M2JENYydOhQPvjgAwD69etH\njRo1GD16NN9//z2jR48GICkpidOnT9OkSZOi83755Rd++eUXPvjgA1599VUA+vbtS1BQEMOHD+eb\nb75h9OjRnDt3juXLl/OPf/yjKE7fvn2L4pTH5cuXee+99xg9ejTfffdd0eN9+969AT4sLAwvLy8c\nHR3p0KFDueJUNVKsCyGEBUTHRwPQLaibdT/CTTwOl6JBoYIOE60Xtww9g3qy+MRi9iXso0BXgEal\nMV/jKjV0fBG2vA6/LYQ2f7XZzbSicjgw6kCp+1RKVbGfo5+JLvXYP68fsGVY6SvqWmLl4j8XzM88\n8wzPPvssUVFRRftatmxZrFAH2LFjB0CJ2Vyefvppxo0bx/bt2xk9ejS7du0qM055bN26FZ1Ox6RJ\nk8p1fnUlxboQQpiZwWBg1zXjH7rudaw8BKZwVc9mg8G7nnVjl6KZXzN8nHxwVDlyPes6wZ7B5g3Q\neixEvw+pF+HcJgg1c++9qFJcNC42P9YcatasWexntVqNr68vKSl3b7auVatWifNSUlJQq9X4+/sX\ne1yhUFCzZs2i81NTU8uMUx6FY+rr1KlTrvOrK+l+EEIIMzufdp6E7AQcVY7WXbU0/RqcXGPctqNF\nkJQKJasHreaXYb+Yv1AHcHSDiPHG7cI3K0JUcTdu3Cj2s1arJSUlpVghfb9P9Xx9fdFqtcVuRgVj\nJ8ONGzfw8/MDwMfHp8w493JycgIgL6/4DbrJycnFfi58g3Dt2rWyn5woRop1IYQwsxO3TgDQvlZ7\nnNVWvOFx/yIw6IyzpAS2tl7ch+Dv4m/Z4UDtJ4DKAeL3Q/why8URwk4sX7682M/ff/89Wq2WHj16\nlHle7969AYqNGQdYs2YN2dnZRfu7du1aZpx7BQcHA3D8+PFij2/cuLHYz/369UOlUrFo0aIyc3R0\ndJRpJu8hw2CEEMLMhjUZRtc6XcnKz7Je0IJciPnjj2/Hl6wX10RavRatXouT2sm8DbvXhBZPw7Hl\ncGgxBLUzb/tC2Jm1a9eiVqvp27dv0WwwrVq14plnyl5XoW/fvjz66KO89tprZGRk0Llz56LZYFq3\nbs3YsWPR6XSEhIQwYsQIPv74YzQaDX369OHkyZN89NFHeHh4FGtzwIAB+Pj4MH78eObMmYNarWbp\n0qXEx8cXOy44OJg33niDuXPnkpuby8iRI/H09OT06dMkJycXTS3ZokUL1q5dy6JFi2jbti1KpZKI\niAjz/gNWItKzLoQQFhDgEkADrwbWC3hyLdxJA8+60Ljvg4+3gS9PfEn3Vd1Zc36NZQK0+2MozKl1\nskiSqPLWrl3L2bNnGTp0KDNnzmTQoEH8+uuvODg4lHmeQqHgxx9/5J///Cdff/01AwYM4KOPPmLs\n2LHs2LEDR8e76zIsWrSIf/7znyxdupQnnniC77//njVr1uDtXXyWKQ8PD7Zs2YK7uztjxozhhRde\noHnz5rz55psl4s+ZM4dvvvmGuLg4Ro8ezeDBg/n666+pX79+0TFTpkzhqaee4o033qBDhw60a1e9\n33xLz7oQQpiRwWCwzQIeh40LihDxHPxpRgt7oVFqyMjPYNe1XYwOLd/Ub2Wq3dY4/CchBmK+hS5T\nzR9DCDtRt25dNmzYUOr+Py9EdC8nJyfmzZvHvHnzyozh4ODARx99xEcfffTAttu1a8fevXtLPD5+\n/PgSj40dO5axY8eWGtfb25sffvihzNyqE+lZF0IIM3p99+tM2DqB2Fux1guaEAPXj4BSY5wZxU51\nq9MNgEM3DpFTkGOZIIU3mh75GvR6y8QQQggrkmJdCCHMpEBXQHR8NL8l/IZKYcXe7UN/9Ko3Gwxu\n/mUfa0PBHsEEuQdRoC9gX+K+B59QHs2HgZMn3L5iXMlVCCEqOSnWhRDCTA7dPESONgc/Zz/CfMOs\nEzQ3DU6sNm5HlPy42Z4oFIqi3vXd13ZbJoiDC4T/McTm0JeWiSGEDc2aNQuDwVA0xaKo+qRYF0II\nMylcCKlbnW4WWbHwvmJXgjYXAsKgrv0vzd2t9t1i3WAwWCZIxDjj9/O/QFp82ccKIYSdk2JdCCHM\n4N5VSwt7j60Q9O6Npe3Ggy1ubDVRRM0InNXOJOUmcTb1rGWC+DWG+t3AoIcjSy0TQwghrMTkYj0r\nK4upU6cSGBiIk5MT4eHhrFy50uTAb731FgqFgubNm5t8rhBC2JsrGVeIz4xHo9TQsVZHKwXdDcm/\ng4MbtBxunZgV5KByYGTISF5o9QJejl6WC1Q4JOjoMtDmWy6OsEtKpZKCggJbpyGqMZ1OZ7a2TJ66\ncejQoRw6dIh58+bRpEkTVqxYwciRI9Hr9YwaNeqh2jh27BgfffQRNWrUMDlhIYSwR4W96hE1InDR\nuFgnaOGNpS2fAUd368Q0g3+0/Yflg4Q8Dm41IesGnN1ovPFUVBs+Pj4kJibi5+eHs7OzbaZTFdWW\nTqfj+vXrBAQEmKU9k4r1TZs2sXXr1qICHaBnz57ExcXx6quvMnz4cFSqsmdA0Gq1PPfcc0yYMIHY\n2FiSk5PLn70QQtiJOu516BTYiV5BvawTMPMGnP3JuG3nN5bahEoDbZ+FnfPh0FdSrFczzs7O1K5d\nm5SUFFJSZIEsU2VmZhatGeHuXnk6AuxJQEAATk7mWanZpGJ93bp1uLm58fTTTxd7/LnnnmPUqFEc\nOHCATp06ldnGvHnzSE1N5d1332XgwIGmZyyEEHaod93e9K7b23oBj34Dei0EdYCalW84YXZBNvsT\n9uPr7Et4QLhlgrR5FnZ9BHF7IOksBIRYJo6wSyqVymw9m9VNbGwsBQUFaDQagoKCbJ1OtWdSsX7y\n5ElCQ0NRq4uf1rJly6L9ZRXrp0+f5p133mHt2rW4ubmZnOypU6fQyyIXVlU45q+goIDYWCsu8iLs\nglx/O6XXErr/CxyAuBp9SbPQtbHk9f8h8QfW31zPI16PMDl4slnbvle9mp3wStxN8pb5XG8lK5o+\nLHntV29y/S1HqVRSt25dk84xqVhPSUmhQYMGJR738fEp2l8avV7PuHHjGDp0KAMGDDApyUJardas\nA/aFaeRmnepNrn/pzmWfI8AhAG+Nt1Xied3Yi0PuLQocvEj274TBCtfG3Ne/hUsL1rOeExknyM3P\nRa0w+Raqh5JUdyBeibvxuvoLV5uMQ692tkicqkxe+9WbXH/zetBw8fsx+bdjWTdplLXvP//5D+fP\nn2fDhg2mhiyiVqtRKmW2SWu690Wq0WhsmImwBbn+D6Y36Fl0bRHp2nRmNp5JE9cmFo9Z4+pGAG7X\nG4DaydVicSx5/Zt4NMFd5U6mLpPLeZcJc7fMIlK5tdqT51obx+zr+N/YSWr9QRaJU9XIa796k+tv\nOeWpY00q1n19fe/be56amgrc7WH/s6tXrzJz5kzmzZuHg4MDaWlpgLGnXK/Xk5aWhqOjI87OZfd4\nNGvWTIp1K7t33FqrVq1snY6wMrn+D3Yq5RTpsem4qF0Y3H4wGpWF/7ClXoakQ4CCgAGvEeAdbLFQ\nlr7+PTJ7sPHSRhKcEhjZaqTZ2y+S8yL8+iZBN38laPBblotThchrv3qT6285er2ezMxMk84xqfJt\n0aIFZ86cQavVFnv8xIkTAKXOmX7p0iVyc3OZMmUK3t7eRV979+7lzJkzeHt7M336dJMSF0IIe1A4\nZWPHwI6WL9QBYr4zfm/YCyxYqFtD4eJRhf+GFtNqJKgcIDEWEo5ZNpYQQpiZScX6kCFDyMrKYs2a\nNcUeX7ZsGYGBgbRv3/6+54WHhxMVFVXiq1WrVgQHBxMVFcVLL71U/mchhBA2svvabsBKq5bqtHBs\nuXG77bOWj2dhnWp3QqVQcSn9EvGZ8ZYL5OoLoX8Mfzn6jeXiCCGEBZg0DOaxxx6jb9++TJw4kYyM\nDBo1akRkZCRbtmzhu+++Kxo0P378eJYtW8bFixepV68eXl5e9OjRo0R7Xl5eaLXa++4TQgh7l5Kb\nwsnkkwB0rd3V8gEvbIXMRHDxgyaPWT6ehXk4eNA6oDWHbx7myM0jBLlbcIq4Nn+Bk2vgxA/Q7x1w\nsNLCVUIIUUEm32C6du1a3nzzTWbOnElqaiohISFERkYyYsSIomN0Oh06nQ6DwWDWZIUQwp7sub4H\nAwZCfULxd/G3fMAjy4zfw0eC2sHy8azg1Xav4q5xJ8jDwnM5B3czDhu6fQVO/wjhD7fithBC2JrJ\nd2u6ubnxySefkJiYSF5eHrGxscUKdYClS5diMBgIDg4us63o6GhOnjxpagpCCGEXdl+34hCYjAQ4\n/4txu03lHwJTKMw3zPKFOoBSCa3HGrdlKIwQohKRqVWEEKKcZnSYwfyu83m8weOWD3ZsORj0ULcT\n+DW2fLyqKHw0KFRwdR/cOmfrbIQQ4qFIsS6EEOXk6ejJgAYDqO9Z37KB9Ho4+q1xuwrcWPpnx5KO\nMXnHZOYdnGfZQB61oMmjxm3pXRdCVBJSrAshhL27vBPS4sDRE0KfsHU2ZpejzSEqPopfr/yK3qC3\nbLDCIUSxkaDNs2wsIYQwAynWhRCiHKbvns6XJ74kPS/d8sEKe4FbPlMlZzGJqBGBi9qFW7m3OJN6\nxrLBGvUB91qQkwLnNlk2lhBCmIEU60IIYaJrmdf46dJPLIxZiEKhsGyw7BQ4+5Nxu81fLBvLRhxU\nDnQM7AhYYYEklRpajzFuF86uI4QQdkyKdSGEMFHhLDDhAeF4OHhYNtjxlaDLh8DWUKulZWPZUPc6\n3QHYFW/hYh3uFuuXooxTOQohhB2TYl0IIUxU2Ptr8SkbDYa7Q2CqaK96oa51jItKnUw5SXJusmWD\neQdDg57G7Zjllo0lhBAVJMW6EEKYIKcgh0M3DgHQrbaFi/X4g3DrLGhcoPlTlo1lY37OfjTzbQbA\n7mu7LR+w8M1PzHeg01o+nhBClJMU60IIYYL9ifvJ0+VR2602Db0aWjZYYa96s6HgZOHhNnagZ1BP\n2gS0wcPRCs815HFw8YXMBLi43fLxhBCinKRYF0IIE+y8thMwjrG26M2ldzLg1FrjdhUfAlPo+ZbP\ns+yxZfSu29vywdSO0GqkcVtuNBVC2DEp1oUQwgQqhQpntTM9gnpYNtDJ1VCQA/4hEPSIZWPZCYvP\nrPNnhW+Cft8CmTesG1sIIR6SFOtCCGGCmR1nsnvEbtrVbGfZQIVDYFqPBWsXsTaWdieNUymnLB/I\nvykEdQCDDo6tsHw8IYQoBynWhRDCRI4qR9RKteUCJB6HhBhQOdwdqlFNHEw8SPfvu/PqzlcxGAyW\nD1jYu370G9BbePVUIYQoBynWhRDiISVkJVgnUGGveshAcPW1Tkw70cyvGUqFkvjMeK5kXLFCwMHg\n6AG3L0PcHsvHE0IIE0mxLoQQDyEhK4FH1zzKkPVDKNAVWC5QQS4c/964XU1uLL2Xq8aVdjWMQ4ws\nvpopgIMrtPhjWszCN0lCCGFHpFgXQoiHEB0fDYCHgwcalcZygU5vgLx08KoL9btbLo4d6x70x2qm\n1ijW4e6botMbICfVOjGFEOIhWXDQpRBCPFieVkdi2h0y72jJvFNAxh/fs/K0ZOdpcVArcXfS4O6k\nxs1RjbuTBg8nNf7ujni5OFgtz8IpG3sG9bRsoKIbS/8CyurZn9KtdjfmMY+jN4+SmZ+Ju4O7ZQPW\nCoeaLeDGCTjxA7SfYNl4QghhAinWhRBWodMbiEvJ5vebmZy7kWX8fjOTy8nZ6PTlu5HQ392RpjXc\naVLDnaY13WhSw53GNdxxczTvr7bsguyiVUsLe30tIvmCcdy0QgnhoywXx84FeQRR37M+l9Mv81vC\nbzwa/KhlAyoU0OZZ2PSKcc71R56vdjPwCCHslxTrQgiLSUjLZefvt9h57hZ7LySTmXf/Zd2dNSo8\nnY2958YvDW5Oatwc1OTr9EU97ll3tGTmFZB5R0taTgG3MvO4lZnHngvJRW0pFNCyjhc9mvjTvak/\nrep4oVJWrPD6LeE3CvQF1POoR33P+hVqq0wxf/SqN+oLnrUtF6cS6F6nO5fTLxMdH235Yh2M49Z/\nfQuSTsH1o1CnreVjCiHEQ5BiXQhhNlqdnv2XUtn5exLR525xPimr2H4njZImhT3hNdxpUtOdJjXc\nqOnhZPKCOFl5Ws7fzCzqqT+flMm5G5kkZeYRG59GbHwan2w/j5eLhq6N/enexJ9eIQH4uJo+dKZw\nvHr3OhbsVdcV3J3ru+2zlotTSQxqOIhgj2DLfpJxL2dvCHsSjq+Co8ukWBdC2A0p1oUQFXY5OZvv\nD8ez5sg1kjLzih5XKqB1XW+6NzEWy81re1a4l7uQm6Oa1nW9aV3Xu9jjN9LvsOv3W+z8/Ra7z98i\nLaeAjbEJbIxNQKNS0Ce0Bs+0C6JbY/+HykWn17H72m4Ay65a+vsWyL4FbjWgcT/Lxakkmng3oYl3\nE+sGbfMXY7F+cg08+h44ulk3vhBC3IcU60KIcsnN17HpRCKrDsdz8PLdGTR8XB3oHRJA96b+dGnk\nZ9WbQAFqejrxTLsgnmkXhFan51h8Gjt/v8X2M0mcTsxg88kbbD55g1qeTjzVtg7PRAQR5ONSansG\nDMzuNJu9CXtpHdDacokX3lgaPgosOduMKF29zuDTAFIvwal10GasrTMSQggp1oUQpklIy+WLXZdY\nc+Ra0Rh0pQK6NfFneEQQvUNr4KC2j1lM1ColEcE+RAT78K9+TTmTmMGqQ/H8eOw6iel3WLDjAgt2\nXKBLIz9e6N6Qzo18SwzHUSvV9Kzbk551LTgLTPo1uLDNuN1aCsRCOQU5rLuwjpikGD7s9qHJQ6VM\nplAYe9e3zTK+eZJiXQhhB6RYF0I8lKspOSzaeYHVR65RoDPO3hLk48wzbYN4KqIOtTydbZzhg4XW\n8mDWE814/bEQtp6+yapD8ey5kFz0FR7kxcu9GtErJMDyheG9jq0Agx6Cu4JvQ+vFtXNKhZJPjn5C\nrjaXcc3HEeYbZvmgrUbBjnfg2kFIOgMBoZaPKYQQZZBiXQhRpusZWtb9nsnuq9FFUyx2aODDxB6N\n6NrID6WZxqBbk5NGxaBWgQxqFUh8ag5L9lwm8uBVjsWnMX7ZYUJrefByr0Y0r6vlx4vr6FW3F839\nmlsmGb0ejn5r3K6GK5aWxUntROfAzmy7uo0dV3dYp1h3rwFN+sPZn4y96/3ft3xMIYQog318Vi2E\nsDtXkrP5YG8KU39NIfpKDjq9ge5N/PnhhY6sfL4j3Zv4V8pC/c+CfFyY9UQz9rzWiwndG+DqoOJM\nYgYvLj/K8OVfsvjEYv575L+WS+BSFKRfBScvCH3CcnEqqV51ewEQFR9lvaBt/piNJzYStHllHyuE\nEBYmPetCiGIy7hTw6Y4LfLX3ctFwl/a1nXhzSFta1vGycXaW4+/uyPTHQnmhW0O+/u0KX++9TBrH\nUANx8cGcvJ5O89qe5g9ceGNpy+GgcTJ/+5VctzrdUClU/H77d+Iz4wlyD7J80Ea9wT0QMhPgzEbj\nHOxCCGEj0rMuhACMK4xGHrxKr4+i+XzXJQp0BsJrOvJRHx/e6OZXpQv1e3m7OvDPvk3Y8s92aFzj\nALgYV49BC/fw2urj3Mo0Y09rVpJxuAXIEJhSeDp6ElEjAoCoq1bqXVeqoPUY4/aRpdaJKYQQpZBi\nXQjBvospDFywh+lrT5CclU8DP1e++msEs3r4EexVPacRjE3ejwEddd3qMyisBQYDrDocT8+PolkU\nfZE8ra7iQWK+A70W6rSDmhYaE18FFM7EsyN+h/WCtvkLKJRwZTckn7deXCGE+BMp1oWoxlKy8ng5\nMoaRi/dzJjEDDyc1MwaGsWVqN3qF1LDujCh2ZttV41SK/er35v9GtmbNxI60rONJVp6W+VvO8uh/\nd7HvYkr5A+j1cORr43bEODNkXHX1DOqJRqnBTeOGTm+GN0kPwyvo7uJU0rsuhLAhGbMuRDVkMBjY\nEJvArA2nuJ1TgFIBo9rX5Z99m+Ljat1FjOxRrjaXPdf3ANCnXh8A2tbz4ccXO7Mu5jrzt5zlSkoO\nIxfvZ+QjdZk+IAQPJxM/gbi4A9KugpMnNBti7qdQpQS6BbJnxB5cNKUvXmUREeOMK8seWw69Zsg9\nBUIIm5CedSGqmcT0XMYvO8yUlce4nVNASE13fpzUmXcGt5BC/Q9X0q/gpHKitlttwnzuTheoVCoY\n1rYO2/7VnVHt6wIQefAq/f6zix1nb5oWpLBXvdUo0Nj/HPW2ZvVCHaBRH/AMgtzbcHq99eMLIQRS\nrAtRbej1BlYcKCwsk9CoFPyzbxM2vNSl2tw8+rBCfUPZ8cwOFvdbfN+hQB5OGt4b0oLIv3egnq8L\nNzLuMG7pYaasjCE1O//BAdKvw7nNxu2I58ycfdWWmJXIHe0d6wRTqu5O43j4K+vEFEKIP5FiXYhq\nICEtlzFLDvDGuhNk5mkJD/Li58ldmdy7MQ5q+TVwP2ql+oHTBHZs6MuWKd14vlsDlApYfyyBPv/Z\nyS+nbpTdeMy3YNBBvS7g39SMWVdtU6Om0m9NP/Ze32u9oG3GgkIF8fvh5mnrxRVCiD/IX2khqrhN\nJxJ57JPd/HYxBSeNkrceD2XNxE40qeFu69TsUlZ+FnqD/qGPd3ZQ8caAUNa+2JmmNdxJzc5nwrdH\neGPdCXLz73MzpE57d2516VU3SaBbIGDlWWHca0LIAON24dAlIYSwIinWhaiicvK1vLb6OC8uP0p6\nbgEt63iyeUo3/ta1AaoqsPKopbx/8H36ru7LtrhtJp0XHuTFxpe7MKFbAwBWHLjKwAW7OZWQXvzA\n879CxnVw8YXQQeZKu1roFWRczTQ6PhqtXmu9wIWz9cSuhPxs68UVQgikWBeiSjpxLZ2B/7eHVYfj\nUSjgxR4NWTOxE/X9XG2dml0r0BUQFR9FUk4S3k7eJp/voFYyfUAo341vT4C7IxdvZTPk09/4cvcl\n9HrjarBFY59bjwG1oxmzr/rCA8LxdvQmIz+DozePWi9w/R7gXR/yMuDkGuvFFUIIpFgXokrR6w18\nvvMiQxft5VJyNjU9nFjxtw5M6x+CRiUv9wc5cOMAmfmZ+Dr5Eu4fXu52ujT2Y8vUbvQJrUG+Ts87\nP5/h2a8Pkhz/O1z4o8e+8MZF8dDUSjXdg7oDEBVvpdVMAZRKaPtX4/ZhGQojhLAu+estRBVxOzuf\nccsO8f7msxToDPRvVpMtU7vSsaGvrVOrNAqHvvSp1weVUlWhtnxcHVj8l7a8M7g5Tholu88ns+Gr\n9wEDNOgJvg3NkHH1UzgUZvvV7RgMBusFbj0GlBpIOAoJMdaLK4So9qRYF6IKiI1PY+CCPUSfu4Wj\nWsn7Q1uwaEwbvFxk3vSHpdVr2X51O3B3IaSKUigUjOlQj40vdaFZDWcG6Y3tb3EecHdYjDBJx8CO\nuKhdSMxO5HjycesFdvWDsCeN29K7LoSwIinWhajEDAYD3+6P4+nP9nE9LZdgXxfWvdiZkY/Uve/8\n4KJ0R24eIS0vDS9HLyJqRJi17cY13FnbKw1/RTpJBi9eOlKD5789THpOgVnjVAdOaidebv0yH3b/\nkCbeTawbvHD2nhOr4aDyMvoAACAASURBVE6GdWMLIaotKdaFqKRy8rX8Y9UxZvx4knydnn5hNdjw\nchfCAj1snVqltDVuKwC96vZCrVSbvX3HY0sBSGr0DEqVA9vOJDFw4W5OXk8v+0RRwpiwMfQP7o+z\n2sorv9brDH5NoCAbTnxv3dhCiGpLinUhKqGLt7IY/OlefjyWgEqp4I0BIXw+ti0eThpbp1ZpPdHw\nCUaHjmZgg4Hmbzz5AlzeCQolzQdNZs3ETtTxdiY+NZehi35j5cGr5o8pzE+huDuN46GvwJpj5oUQ\n1ZYU60JUMtvP3GTwwr38fjMLf3dHVvytPc93ayjDXiqopX9LXn/kddrVbGf+xg9+bvzeuB94BdGi\njic/vdyFXiEB5Gv1vL72BG+uO0G+9uEXY6rubmTf4IvjX7D4+GLrBm41AjQukHQKruyxbmwhRLUk\nxboQlYTBYGDhjvP87ZvDZOZpeSTYh58nd6F9A5ntxa7dSYdjK4zb7V8oetjLxYEv/xLBK/2aoFDA\n8gNXGf3lfm5l5tko0crlcvplFsQsYNnpZRTorTj239kbWg43bh/4zHpxhRDVlhTrQlQC2XlaJq04\nyke//o7BAGM61OW7v7UnwN3J1qlVenqDnv8e+S8HEg+g0+vMHyDmO8jPAv9QaNCj2C6lUsFLvRqz\n5NkI3B3VHLpymycW7uHENRnH/iDtarbDx8mH9Lx09ifst27wwjddZ3+G21esG1sIUe1IsS6EnYtP\nzWHYot/YdOIGGpWC94e24J3BLXBQy8vXHE4mn+Srk18xecdktAYzL2Gv18GBP4bAtJ9gHPN8H71C\narBuUmca+LmSmH6Hpz77jfXHrps3lypGrVTTr14/ALZc2WLd4AEh0LAXYICDVh6GI4SoduSvvRB2\n7LcLyTyxcA9nb2Ti5+ZI5N87MPKRurZOq0opXAipe53uOKoczdv471sgLa740IlSNApwY92kzvRs\n6k+eVs+Ulcd4b9MZdDIfe6kGNBgAGBdIytNZefhQ+4nG70e/hbws68YWQlQrUqwLYae+3R/3/+zd\nd3wU1drA8d9sS9v0SggQEjqBECD0LggoSBMBwVcBEbCXK3LVq6Bey71eFRUFRAUugnRRBFSESw0o\nLZDQAyEQQnrfbJ/3j4UAQoDAJpNyvnz2s5PZM2eezbCZZ8+cOYdHvvmDXIOFVnW9+fHprrQP91M6\nrBrFLttLW2WdNRHSNXZ/6Xhu+yjo3G9Z3NtNy/xHY3myl2N203nbTjNx4Z8UGMV47DcSHRhNiEcI\nxZZidpyv5Js9G/UF/0Zgyof4pZW7b0EQahWRrAtCFWO12XlzbQL/+CEBm11maJtQVkzpTKhPJY8p\nXQvsS99HWnEaeq2eHmE9nFv5xQRI3g6SGjpMuu3N1CqJaQOa8dmYGFy1Kv53PJMRX+wiJdvg3Phq\nAJWkYkD4AAA2JG+o5J2roMNkx/KeOWAXI/kIglAxRLIuCFVIfomF8Qv+ZGHcWQBe7t+Uj0e1wVWr\nVjiymunHpB8B6B/eH1eNk2/WvTxSSPPB4B1W7s0HR4eyfHJngr1cOJlRxNAvdvLHmRznxlgDDGg4\nAHeNO3qtvvJ33mYMuHhB9ilI+r3y9y8IQq0gknVBqCLOZhcz/IudbD+ZhZtWzZxxbXmqdyMxfnoF\nKbGW8GvyrwAMjhzs3MqLs+HwCsdyp6l3XE3rMB/WPtWNVnW9ySk2M3b+blbsPeekIGuGFn4t2Dpq\nKzO6zKj8nbt4QswjjuXLXZ4EQRCcTCTrglAF7D6dzZDZO0nKLCbEy5UVUzozIKqO0mHVaOcLz+Pr\n6ktdfV1igmKcW/m+b8FqhDptoF7Hu6oqxNuV5ZM7c1+rECw2mZdXHuK9DeLG08skSXL+VZHy6DAJ\nkBwt65knlItDEIQaSyTrgqCw5XvP8cjXe8gzWGgd5s3ap7sSVddb6bBqvMa+jdkwfAMLBixAJTnx\nT6HNAn9+7VjuNLXM4RrLw02n5vMxbXmmTyMA5m49zZTF+yg2OXmoyWpMlmWO5xynyFzJI7P4NYSm\njlFpxCRJgiBUBJGsC4JC7HaZDzYeY9rKQ1hsMve3rsOyJzoT7CUmOqoskiQR4hHi3EqPrIXCC+AR\nBC2HOa1alUripXubMmt0G3QaFb8dSeehuXFczDc6bR/V2Qv/e4EHf3qQ387+Vvk773RpkqT4pVCS\nW/n7FwShRit3sl5UVMTzzz9PaGgorq6utGnThu+///6W261evZoxY8bQqFEj3NzcCA8PZ+zYsZw8\nefKOAheE6sxosfH00v18+b8kAJ7t04jPRsfgphM3klaGcwXnMNvMFVP55dbV2ImgcfK47cCQNnVZ\nOqkT/h46Ei8UMHT2ThIviBlPW/i3AGDDmUoeFQYgvDsER4HF4Bh3XRAEwYnKnawPHz6chQsX8uab\nb7JhwwZiY2MZM2YMS5Ysuel2H3zwAQaDgddee42NGzfyzjvvcODAAdq2bUtiYuIdvwFBqG4yCo2M\nmre7dEbS/4yM5sV7m6JSiRtJK4Msyzy75Vn6rOjDvvR9zq38/D44/yeoddB+gnPrvkq7Br6sebIr\njYL0XCwwMnJOHL8fTa+w/VUHA8MHArDn4h6yS7Ird+eS5JihFhwzmtpE9yRBEJynXMn6+vXr+e23\n3/jiiy+YPHkyvXv35quvvqJfv368/PLL2Gy2Mrf96aefWLt2LePHj6dnz56MGzeOTZs2YTKZ+Pjj\nj+/6jQhCdXD8YiHDZu8i/lwePu5aFk/syIh25R/WT7hzx3OPcyrvFAaLgUY+jZxbedznjueoEaAP\ncm7df1Hf351VU7vQrVEABrONSYv28s2OM8hy7bzxtJ5XPaL8o7DLdtafWV/5AbQaCW5+kJ8CR3+s\n/P0LglBjlStZX7NmDXq9npEjR16zfvz48Vy4cIE9e/aUuW1Q0PUnrtDQUMLCwjh3TgxFJtR8W09k\nMuLLXaTmldAwwIM1T3alY4S/0mHVOpfHVu9VrxfeLk68kTfrFCSucSx3ftp59d6Et5uWb8fHMqZD\nPewyvLXuCG/+mIjVVjsn6BnaaCgAq06sqvwvLVo36PCEY3nHR1BLvzQJguB8mvIUTkhIoHnz5mg0\n127WunXr0te7dOly2/WdPn2as2fPMnTo0Nsqn5iYiF3MElepLBZL6XN8fLzC0VRfG04WMW9fHnYZ\nooJ0TO/mTUHqKeJTlY7s5mra8bfJNtaeWAtAK1Urp76nsP3v449MfkgXktNtkF55v69RETIuZm8W\nHsxnUdxZEpPT+VtXP9y1dzeGQHU7/vVt9XFRuZCUn8TKuJU08WhSqftX67vRXO2G+uJhTv86l8KQ\nzpW6f2eqbsdecC5x/CuOSqWifv365dqmXMl6dnY2ERER16338/Mrff12Wa1WJk6ciF6v54UXXrjt\nbW7W1UaoWJc/vMLts8kyiw8V8dNJx1TxPRu4MqWdF1qVDYulev1frgnHP74wngJrAXq1nuZuzZ32\nnnQl6fim/ALAhcgxivyuBjVyJdANZv2Rz740I6/8lsGrXX0IcHfOTcvV4fhr0RLrFcuOvB3syt5F\nQ13DSt2/RXIjs8FgQk4vJ+jYInL82jll6E6lVYdjL1QccfydS60u/9/kciXrwE1nU7zdmRZlWWbi\nxIls376dVatWUa9evdvaTqPRoFKJ0SYr09UfUq1Wq2Ak1Y/RauejuBz2pDqG1hvb2ouRLTyr1Yyk\nNe347y7YDUAX3y646dycVm+dI6tQyTYKA9tiDopGqd9Ut3AtwZ4u/HNbFin5Vv6+JYfXewTQyE93\nR/VVx+M/KHgQnfw60dqztXPHz79N2U1HE5T8A/rcRHwKjlAc0KbSY3CG6njsBecRx7/i3EkeW65k\n3d/f/4at5zk5OcCVFvabkWWZxx9/nMWLF7Nw4UKGDBly2/tv2bKlSNYrWXx8PBaLBa1WS3R0tNLh\nVBvpBUYeX7iXw6lGdBoVH46M5oHoUKXDKreadPwNFgMHDh8AYEKnCbT0b+mciosy4KefAfAc+CbR\nEcr+nqKBzm1LmLjgT45dLOT1zdl8MroN/VuWfzz56nj8o6kCcWY+Anu/plHqGrjnUaWjuSPV8dgL\nziOOf8Wx2+0UFhaWa5tyZb6tWrXi6NGjWK3XDkt1+PBhAKKiom66/eVE/dtvv2X+/PmMGzeuXMEK\nQnVwNM0x9vXh1Hz8PHQsndSxWibqNY271p1lg5bxYrsXaeHXwnkVx80GqxHqtoeGPZ1X712o6+PG\niimd6dkkkBKLjSmL9/HVttO1bqQYq92qzHvu+hxIaji9BVKdPDyoIAi1TrmS9WHDhlFUVMSqVauu\nWb9w4UJCQ0Pp2LFjmdvKssykSZP49ttvmTt3LuPHj7+ziAWhCttyPIMHv9xFWr6RiEAP1jzZhXYN\nbn3FSagckT6RjI8a77yuSCW58OfXjuXuL1Wp/smerlq+frQ94zrVR5bhn+uP8toPCVhqyUgxc+Ln\n0G9lP45kH6n8nfs2gNYPOZa3f1T5+xcEoUYpVzeYgQMH0q9fP6ZOnUpBQQGNGjVi6dKlbNy4kcWL\nF5d2mp84cSILFy4kKSmJBg0aAPDss8/y9ddfM2HCBFq1asXu3btL63VxcSEmJsaJb0sQKt+CnWd4\na90R7DJ0jvBnzrh2eLuLvn5VgSzLFXOvwB9fgbkQglpCkwHOr/8uadQq3h4SRbi/B/9cf5Qle1I4\nl2Pg84fb4u1Ws/9vnsk/Q1ZJFitOrKBlgJO6PJVHtxch/ns4tg7Sj0CwE6/mCIJQq5S7A/jq1at5\n5JFHeOONNxgwYAB79uxh6dKljB07trSMzWbDZrNdc/nxp59+AuCbb76hc+fO1zyGDRvmhLciCMqw\n2uy8sTaBGT85EvVR7euxcEIHkahXIR/v+5iX/vcSx3OOO69SUxHs/sKx3P1FqKL300iSxOPdI5j3\nSHvcdWq2n8xixJe7SMk2KB1ahXqwyYMAbDizAYNFgfca2ARaPOBY3iEm/hME4c6V++yi1+uZNWsW\naWlpmEwm4uPjGT169DVlFixYgCzLhIeHl65LTk5GluUbPpKTk+/2fQiCIgqMFiYu3MuiuLNIEvx9\nYDPeH9EKnaZqJm61kcFiYMWJFfx69lcyDBnOq3jfAkc3GL8IaFn1Gxz6tQhm+eTOhHi5ciqjiKFf\n7GRvco7SYVWY9sHtaeDVAIPVwIYzG5QJovtLjueElZBzWpkYBEGo9kRGIQh36FyOgQe/3MXWE5m4\nalV8ObYdk3tGVquhGWuDH5N+pMhSRAOvBnSt29U5lVqMsOszx3K3F0DlnLHMK1pUXW/WPt2VVnW9\nySk28/BXe1hz4LzSYVUISZIY0XgEAKtOrrpF6QpSJxoa9QPZDjs+USYGQRCqPZGsC8Id2Hc2l6Gz\nd3IivYggTxdWTO7CgKjyD40nVCy7bGfJsSUAjGk2xnnjbscvgaKL4FUXWo++dfkqJNjLlWWTO9G/\nZTBmm50XlsXzn1+PY7fXvJFiHoh8AI1Kw+Gsw87tAlUel1vXDy6BggvKxCAIQrUmknVBKKfV+88z\nZt5usovNtKjj5WipDPNWOizhBnZf2M2Z/DO4a9wZEnn7czrclNV8pZW0y7OgubMJh5TkrtPw5dh2\nTOkZCcBnm0/x9NL9GMzWW2xZvfi7+dO7Xm9Awdb1Bp2hQVewW2DnLGViEAShWhPJuiDcJptd5v0N\nx3hxeTxmm51+LYJZMaUzdbydNxOm4FzfHfsOgKGNhqLX6Z1T6b5vIe8seARB2/9zTp0KUKkkpg9s\nxr8ebI1WLbH+8EVGzonjQl6J0qE51ZhmYxjVdFRplxhF9HjZ8fzn15CbrFwcgiBUSyJZF4TbUGSy\nMvm/e5mzNQmAp3pHMndcOzxcyjX6qVCJzhacZdv5bYAjYXMKYwFs/cCx3PvvoHN3Tr0Keqh9PZZM\n6oSfh47ECwU88PlODqTkKh2W08SGxPJ6p9dp6tdUuSAie0NEb0fr+uZ3lItDEIRqSSTrgnAL53IM\njPhiF5uOZqDTqJg1ug0v92+GSiVuJK3K/F39eSX2FR5q8hDh3uHOqXTnLDBkg39jiKm+rep/FRvu\nx9qnutIsxJOsIhOj5u2usTeeKqbfTMfz4RVw4YCysQiCUK2IZF0QbmLP6WyGzN7J8fRCAj1dWD65\nM0Pa1FU6LOE26HV6xrUYxz86/8M5FRZcgLjZjuW+M0Bds66q1PNzZ+XULvRtHozZ6rjxdNHBfOxy\nzbjx9FjOMaZtncafF/9UJoA60dB6lGP5tzeghvxeBUGoeCJZF4QyfLfnLOO+3kNOsZmoul78+HRX\n2tTzUTosQSn/ew+sJVCvIzS7X+loKoTeRcO8R9oxtZfjxtNVRwv5YFcexWa7wpHdvZUnVrIheQPz\nD89XLojer4FaB2e2wanflYtDEIRqRSTrgvAXJquNv68+xGtrErDYZO5vXYcVk7uIG0mrCbts5+Wt\nL/Nj0o9YbBbnVJpxDA4sdiz3extq8Fj6KpXEKwOa8fGoaLQq2Jdm5uVfMziVUaR0aHflsZaPoZbU\n7Lqwi8TsRGWC8G0AHZ5wLG96E+w2ZeIQBKFaEcm6IFwlvcDI6Hm7WfrHOSQJXhnQjM/HxOCmqx6T\n3giO4Ro3Jm/k3T3vYrabnVPpphmOiW2aDYL6HZ1TZxU3LCaM9/oG4e+mIrXQytDZO/ntSLrSYd2x\nMM8wBjQcAMA3h79RLpDuL4GrN6QnwKFlysUhCEK1IZJ1Qbhk39lcBn22gwMpeXi5avj2sVim9hIz\nklY3l4drHNZoGB5aj7uvMHknnNgAktrRV70Waeyv41/3+NMyUEeRycqkRXv5ZNOJajuB0oSoCQD8\ndvY3kvOTlQnC3e/KREmb/+mYDVcQBOEmRLIuCMDSP1IYPS+OzEITTYL1/Ph0N3o1DVI6LKGcLg/X\nKCE5Z7hGWYbfLt2g2u5RCGh893VWM96uKt7qE8ijnRsA8Mmmk0xevI9Co5O6GFWiJr5N6BnWExmZ\nBYkLlAukw2TwCoOC8/DHXOXiEAShWhDJulCrGS2O/ul/X30Yi01mYFQIq5/sSniAE1pkhUr33yP/\nBaB7WHfqe9W/+wqP/ACp+0DrAT2n33191ZRGJTFzSBT/frA1Oo2K346kM3T2Tk6kFyodWrk93upx\nANYmrSW9WKFuPVpX6PO6Y3n7f8CQo0wcgiBUCyJZF2qtlGwDI77cVdo//W/3NuGLsW3Ri4mOqqXz\nhedLp5R/rOVjd1+h1Qy/v+VY7vIMeAbffZ3V3Mj29VgxuTMhXq4kZRYz5POdrD2YqnRY5dImqA0D\nwgfwXMxzeOo8lQuk9UMQHAXGfEfCLgiCUAaRrAu10m9H0rn/s+0kXijAz0PHogkdeLpPY9E/vRr7\nMv5LrHYrnet0JjYk9u4r/GMu5JwGjyDo8vTd11dDRNfzYd2z3ejWKIASi43nvj/I6z8cxmStPiOb\n/Lvnv3ks6jHctQrOQKtSX5ko6Y95kHVSuVgEQajSRLIu1CpWm533Nhxl0qK9FBqtxNT3Yd0z3eje\nOFDp0IS7NLTRUNoEtuGZmGfuvrLcZNjyrmP5nn+Ai4ItsFVQgN6FhRM68GyfRgAs3p3CyDlxnMsx\nKBxZNRN5DzS+F2xm+Ok5sFf/8ewFQXA+kawLtUZGgZGH5+9h7tbTAIzvGs6yJzoT6iPGT68JYkNi\nWTRwEa0CW91dRbIM614AiwHCu0PMI84JsIZRqyRevLcp346Pxcddy6Hz+Qz6bAebj1WP4R1tdhsb\nkzcy6ddJGCwKfcmQJLj/P457Is7uhAOLlIlDEIQqTSTrQq2w9UQm9326nT/O5OChUzP74ba8Obgl\nOo34CFR38lXTtjulG9Oh5ZC0GdQuMHhWjZ4AyRl6Nw1i3TPdiA7zJr/EwoQFe3lv/VHM1qrdSiwj\nM2vfLHan7WbNqTXKBeJT/8rNpr++AYUXlYtFEIQqSWQqQo1mttp5d/1RHv3mD7KKzDQN9uTHZ7px\nf+s6SocmOMm0bdP4dP+nFJgL7r6y4mz45e+O5Z7TwD/y7uusBcJ83Vk+pTP/d2l4x7nbTvPgnF0k\nZxUrHFnZNCoN46PGA7AgcQFmm5Mm0LoTHSdDaFsw5cOGacrFIQhClSSSdaHGSs4q5sE5u5i3zdHt\nZVyn+qx9uiuRgXqFIxOc5UDGATYmb+SbhG/IM+bdfYW/vAqGbAhqCV2fu/v6ahEXjZq3hkQxZ1w7\nvN0c3WLu/3Q7PxyouqPFDGk0hEC3QC4WX+S7o98pF4hKDQ986ph468haOLZeuVgEQahyRLIu1Ehr\nDpzn/k+3c+h8Pt5uWuY+0o53hrbCVatWOjTBSWRZ5tP9nwKOm0vvelz1U7/Doe8ByZE4qbV3H2Qt\nNCAqhA3PdadDuB/FZhvPLzvIi8sPUmSyKh3adVzULjzb9lkA5h6aS1ZJlnLBhLSCro5Y+PklMDrh\nSpEgCDWCSNaFGqXQaOHFZQd5YVk8xWYbHRr6seG57vRvGaJ0aIKTxaXFsTd9L1qVlinRU+6uMnMx\nrHvesdxxMoS1v/sAa7FQHzeWPtGJF/o2QSXB6v2pDPp0O/HnnHD1w8keiHyAVgGtKLYUM2v/LGWD\n6fkK+EVA4QX4faaysQiCUGWIZF2oMXadymLAJ9tZfSAVlQQv9mvC0kmdxGgvNdDVreqjmo4ixOMu\nv4xteRfyUhxTwF++2U+4K2qVxHN9G7NscmdCvV1JzjYw/MtdfPTbCSy2qnPzqUpS8UqHVwD44dQP\nnMk/o1wwWjcY9Ilj+c+vIWWPcrEIglBliGRdqPZKzDZm/JjIw/P3kJpXQn0/d5ZP7syz9zRGrRIj\nedREm89tJjE7ETeNGxNbTby7yi4cgN1fOJYHfSTGVHey2HA/NjzXg0Gt62Czy3z6+0mGzt7J8YuF\nSodWKjowmsmtJ/Nl3y9p6N1Q2WAiekKbcYAMPz0LVpOy8QiCoDiRrAvV2v6UXO7/dDsLdiUDMLZj\nfTY815324X7KBiZUqAUJCwAY13wcAW4Bd16R2QA/PAmyHVoOhyb9nROgcA1vdy2fP9yWz8bE4OOu\nJfFCAYM/28GcrUnY7PKtK6gET8c8Tbe63ZQOw+Het8EjEDKPwZZ/Kh2NIAgKE8m6UC2ZrDb+/csx\nHvxyF6ezignxcmXhhA78c1grPFw0SocnVLAPe37IoIhBPNry0TuvRJbh5xch4wh4BMHAD5wXoHBD\ng6ND+fWFHtzTLAizzc77G44xam5clRviMaskS7mJkgDc/eD+jxzLO2eJ0WEEoZYTybpQ7ew7m8vg\nz3Ywe0sSdhmGx9Tll+d70LNJoNKhCZUk2COY97q/h7eL951Xsn8RxC8FSQUPfgP6IOcFKJQpyNOV\n+Y+2518jWqN30bD3bC4DZm1j3rYkrFWgL/vaU2sZvGYw8w7NUzaQFg9Apycdyz9MgdxkRcMRBEE5\nIlkXqo0Co4V//JDAg3N2cSK9CH8PHXPGteWjUW3wdhfD7NV0NruNvRf3OqeytEOw/mXHcp/XoWF3\n59Qr3BZJkngoth4bn+9Ol0h/jBY7764/xpDZOzl0XtkRY7x0XhRZilh0ZBHnCs4pGgt9Z0JYLBjz\nYfn/gcWobDyCIChCJOtCtbAx4SL9PtrKf3efRZbhofZh/P5STwZEiZlIa4vFRxcz/pfxvLvn3bur\nqCTPkfjYTNC4P3R9wTkBCuUW5uvOd4935F8PtsbbzdGXfejsnby97gjFCo3L3qteL7qEdsFit/Dv\nvf9WJIZSGh2MXABufpAWf2V2XUEQahWRrAtVWlp+CU8s2suUxftILzAR7u/Okkkd+deD0fi465QO\nT6gkZwvO8tmBzwBo6tv0ziuSZVj7FOSeAe/6MGwOqMSfQSVJksRD7evx+0s9GdImFLsMX+84w70f\nb2PLsQxF4pkWOw21pGbLuS3sSN1R6TFcwzsMRnwFSLD3Gzi0XNl4BEGodOIsJVRJJquNOVuT6Puf\nrfx6JB2NSuLp3o3Y+HwPukTexegfQrVjl+28uetNTDYTHet0ZHjj4XdeWdxsOLYO1Dp4aIHjRj6h\nSgjQuzBrdAwLxscS5utGal4J4xf8yeT/7uVcTuXe7BnpE8mYZmMA+MfOfyg7sylAo77Qc5pj+afn\nIOOYsvEIglCpRLIuVCmyLPP70XT6f7yN9zcco9hsI6a+Dz8/252/9W+Kq1atdIhCJVtxfAX70vfh\npnFjRucZSNIdjp1/Ng5+e8Ox3P9dqNvOeUEKTtOraRC/vtCDSd0bolZJ/JKYzj0fbeXDX45jMFde\n15hn2z5LI59GZJVk8dqO17DLCt/82vMViOgFFgMsfwRMRcrGIwhCpRHJulBlJGUW8di3fzJx4V6S\nsw0Eerrw4choVk3pQtMQMVFNbXSh6AIf7XMMYfdc2+cI8wy7s4ryU2HleJBtEPUgxD7uxCgFZ3PX\naXjt/hasf7Y7XRv5Y7ba+XzLKfp8uJW1B1OR5Yofm91N48aHPT/EXeNOVEBUpezzplRqGPE1eIZC\n1glY+yTYbcrGJAhCpRDJuqC4/BIL//z5CP0/3sbWE5lo1RKTe0aw5W+9eLBdGCoxC2mtJMsyb8W9\nhcFqICYoprRbQrkVZcKiIVCYBgFNYfAsuNPWeaFSNQ3xZPHEjswZ144wXzcuFhh57vuDjJwTx+Hz\n+RW+/0ifSNYPX88zMc+gVlWBq3oeATDyW1Bp4chaxzwBSn+JEAShwonZYwTFGC02Fu5K5ov/JZFf\nYgHgnmZBvD6oBQ0DPBSOTlCaJEmMbDqS5IJkZnaZiUq6g7aFkjxYPAyyT4JXGIxbBS565wcrVBhJ\nkhgQFUKvpoHM336a2VuS2Hs2l8Gf72BQ6zq8dG/TCv174e/mX7pstpkx2Ux46hS80le/k+OG05UT\nYN8CcPGCfm+JL6CCUIOJZF2odFabnRX7zvPJphOkF5gAaByk59X7m9O7qZiYRrjinvr30DOsJxrV\nHfypMhfDkofg+VW0tQAAIABJREFU4mHH1O3/txZ86jk/SKFSuGrVPN2nMSPahfGvjcf54WAq6w6l\nsSHhIqNi6/HcPY0J9nKtsP2fKzjHS1tfIsg9iM/6fHbn9044Q8thjj7rPz4Nuz4FVy/o8bJy8QiC\nUKFEsi5UGrtdZkPCRf7z63FOX5pevK6PGy/0a8KwmLqoRXcXATiecxytWkuEdwTAnSXqVhN8PxbO\n7QFXb3jkBwho5ORIBSXU8Xbj41FtmNQ9gg9/Pc7mYxks2ZPCqn3nGd+1IVN7RlbIJGnF1mKS8pI4\nmnOUxUcX80iLR5y+j3Jp+wiYCh1jr29+B3Se0GmKsjEJglAhRJ91ocLZ7DLrD6cx6LMdPLVkP6ez\nivH30PHGoBZs/ltPHmwXJhJ1AYBMQyZP/f4U49aPIyEr4c4qsVkdXQRObwGtB4xdBSFRzg1UUFyL\nUC++eSyW5ZM7076BLyarnTlbk+j2r83859fj5BSbnbq/Zn7NeDnW0Xr90b6P7vz/pzN1fhJ6XZoo\naeMrcOA7ZeMRBKFCiGRdqDAWm52V+85z78dbefK7/RxJK8BDp+aFvk3YOq03E7o1xEVTBW7aEqqE\nEmsJz2x+hnRDOgFuAdT3ql/+Sux2R9eAy2Opj1kC9WKdH6xQZXRo6MeKKZ355rH2NAvxpNBo5bPN\np+j2wWbeWXeE9AKj0/Y1quko+jXoh9Vu5fktz3O+8LzT6r5jPV+BTk85ln982nHjqSAINYroBiM4\nndFiY8W+88zdmsT53BIAvFw1PNa1IeO7hOPrIWYeFa5ll+28uv1VErMT8XHxYXaf2XjpvMpXidUE\nPz4Lh74HSe2Ypj2iVwVEK1Q1kiTRp1kwvZoE8UviRT7fcorECwXM33GGRXFnGdk+jCk9I6nn537X\n+5nRZQan8k5xJv8Mj//6ON/2/5Y6+jpOeid3FBT0/yeYCuDAf2HlRBhSAtGjlYtJEASnEsm64DSZ\nhSaW/pHCf3efJbPQceNogF7HxG4RjOtUH09X5/cjFWqGT/d/yqaUTWhVWmb1nkU9r3LeCFqcdamP\n+m5Hoj5sLjS7v2KCFaoslUpiYKs6DIgK4X8nMpm9+RR7z+by3Z4Uvv/zHPe1qsP4ruHE1PO54xtE\nvXRezL93PhN+mcDZgrPMjJvJnH5znPxOykmSHEOSWkogYSWsmewYi73366ASF9AFoboTybpw1xJS\n8/lm5xnWxadhtjlm+Qv1dmVyz0hGxdYTs44KN7X8+HK+TvgagJldZtI2uG35Ksg46hj1JS8FXLzh\noQUQ2cf5gQrVhiRJ9G4aRK8mgew5k8PsLafYfjKLn+Iv8FP8BaLDvHmsazj3twpFpyl/MhvkHsT8\ne+czM24mMzrPcP4buBMqNQz/Cnzqw46PYPt/IOuk44ur7u6uKAiCoCyRrAt3xGKz80viRRbsTGbv\n2dzS9W3q+TC+azgDo+rc0UlQqF1kWWZH6g4Anmj9BIMjB5evgpObYMVjYC4E34bw8HIIbOL8QIVq\nSZIkOkX40ynCn4TUfBbsSubHgxeIP5/PC8vieXf9McZ2rM/DHesT5Fm+YR9DPEL4su+X16yz2q13\nNnqRs6hU0PdNCGgCPz4DR390fIkdsxS8QpWLSxCEuyKSdaFcTqYXsmLfeVbvP09WkWO0Ba1a4v5W\ndXi0Szgx9X0VjlCoTiRJ4oMeH/DDqR8Y3bQcfWxlGfbMdQxbJ9uhQVcYtRjc/SouWKFai6rrzYcj\no5k+sBnfX+qul15g4pNNJ/l88yl6Nwviofb16NU0EK26/A0N60+v59vEb5nXbx6+rgr/HWwzBnzD\nYdlYSDsIX/WBMd9DaBtl4xIE4Y5Islw15yq22+0UFhZes87T0xOV6H9XqeLj48k3mNhzwcKudDiQ\nklf6WoDehbEd6zO2Y32CKnAyEkE58fHxWCwWtFot0dHRTqmz2FLMD6d+4OFmD99Zv2FTIfzyKuxf\n5Pg5Zhzc/zFoxI3LzlYRx7+qsNjsbEy4yIJdyey76upggN6FEW3rMrJ9GI2Cbm+m0hJrCQ/88AAX\niy/S1Lcps++ZTbBHcEWFfvtyzsCSUZB1HDRuMOgjiB5zW7Od1uRjL9yaOP4V507yW9GyLtyQyWpj\nx8ksFu3KIe68AbPNsV6tkuhzly1QQu11eRz1ozlHKTIXMTl6cvkqOLUJfnoe8s8BEvSbCV2eFVOt\nC+WmVasYHB3K4OjQv1wxNDF322nmbjtNTH0fhkSHcl+rOjdtkHDTuDGv3zzGbxzP8dzjjPxpJO92\nf5dudbtV4ju6Ab+G8Phvjq5iSZvhh6mQsBoGfSxm8xWEakS0rAulzFY7O09lse5QGr8euUih0Vr6\nWpiXhke7NWZoTF0CPV0UjFKoTM5sXUnKS2LqpqmkFafh5+rH7HtmExVwm5MVGXLgl9cgfonjZ58G\n8MCnYmjGClbbWtcsNjtbjmWwfO95thzPwGZ3nB4lCTqE+zGodR0GRNUp829gSkEKL219iWM5xwCY\n1GoST7Z5Utl+7OCYKGzXp/C/98FmAp3e8UW33YQyR4upbcdeuJY4/hXnTvJbkazXcsUmKztPZbHp\naDq/JKaTX2IpfS3Yy4UOIVo619XRItiNNm1Ef8faxhl/sO2ynZUnVvLxvo8pshQR7hXOF32/oJ7n\nbbbsHVkLP/8NijMACTpOgXv+ATqPO4pHuH21+YSdUWjkp/g0fj50gf1Xdf9TSdCxoT8DW4XQu2nQ\ndWO3m2wm/v3nv1l2fBkAsSGxfNXvK9SqKjAqVuYJx42n53Y7fm7QFR74DPwjrytam4+9II5/RRLd\nYITbkpJtYPOxdH4/lsGe0zmlwy0CBHq6cF9UCIOiQ2lX35fDhw9hsVjueExioXZLzk/mjV1vcCDj\nAABtg9oyq/csfFx9br1xdhJsmuEY0QIgoCkM+Rzqdai4gAXhkiBPVyZ2a8jEbg1JzSthw+E01h1K\n4+C5POJOZxN3OhtIpHGQnj7Ng+jTNIh2DXxxUbvweqfXaR/cnhlxM4gNia0aiTo4RkoavwH+nO/4\nbJ3dCV92gR4vO74Eu+iVjlAQhBsQyXotkF9i4Y8zOcQlZbP1RAZJmcXXvF7fz50+zYIYEBVCbLgf\napVIzAXnsMt2ErIScNO48Vzb5xjddPStE5esk7DtQzi83DHSi0oD3V5wJBQa0QVLqHx1fdx4vHsE\nj3eP4FyOgQ0JaWw6msG+s7mczCjiZEYRc7eexstVQ/cmgXSNDKBzZHdWDV5FiEdIaT3Hco4R4BZA\ngFuAcm9GpYKOT0CT/vDTc3B6C2x+G+JmQ5enIXYSuJZz9mBBECqUSNZroGKTlT+THcl53OlsElLz\nsV/V2UmtkogN96VPsyD6NAsmMtBDtJwLTnOh6AKheseYzhE+Ebzb7V2iA6NvPSV7xlFHkp6wCrj0\nH7Zxf0eXl5BWFRu0INymen7uPNEjkid6RJJvsLD1ZCZbjmWw5XgGeQYLPx9K4+dDaQCEeLnSJTKT\nTpH+dGjow993/J3UolTGNR/H+KjxeOpub7SZCuHbAB5ZA4dXOPqy5yTB72/Bzk+h81Oo3LuAJEb5\nEoSqQPRZr+ZkWeZMVjEHUvLYn5LLgZQ8jqcXlt4YdVnDAA86RfjTtZE/3RsH4u2mva36Rb+12u12\nj78sy+xN38uiI4vYfn47i+9bfHs3j8oynPsDdn/h6Jt+OUlvej/0fBlCY5zzRoQ7Ij7/t89mlzl4\nLpdtJ7KIO53NwZS8a7oYSupCvMIXY9edBUCv8ebxVo8zruUYXNQKXzGyWSFxNWz7N2SdcKzSeJDe\ncBh5EUNo0fleZeMTKp347Fcc0We9hrPbZc7lGjhyoYAjaQUcTs3n4Lk88gyW68qG+brRJdKfzpH+\ndI4IIMRbtJAIzmexW/gl+RcWJS7iaM7R0vW703bfPFnPToJDy+HQMsg9c2V98wcc3V3qtK7AqAXB\n+dQqiXYN/GjXwI8XgBKzjX1nc4k7nUVcUjaHzkvkJ01Boz+CLugXisjgkwP/4bN939DSfRj96g8k\nJiyEpiGeuGoruY+7WgOtH4KoEXDkB9j6b9SZRwk9uZg6J7+D492g9ShoMUR0kREEBYhkvYrKLjKR\nlFnMqYwijl90JOdH0wopMlmvK+uiUdGqrjcx9X2Iqe9LTH0f6ni7KRC1UFuYbCYWH1nMkmNLyDBk\nAOCidmFI5BDGthhLhHfE9RsVZzta7w4tg/N/Xlmv9XAkAV2egeAWlfQOBKFiuenUdGscQLfGjv7p\nBrOVw+fz2Z/SnP1ne7I3ZxNmzw2gzeWQ8RvifnHFbqyPSoLIQD0tQr1oXseLxkF6IgP11PNzr/j7\niVRqR8LeYhjJGz/DL2kVXtnxkLzd8Vj/N2h6nyNxj+wjJiIThEoiknUFFZusnMs1cDbbQEq2gaTM\nIpIyiziVUUTuDVrLAXQaFU2DPWlRx4sWoV7E1PehWYgXOo3oHiRUrKySrNIb49SSmmXHl5FhyCDA\nLYAxzcYwssnIa6dZNxsgJQ7ObIXT/4O0Q5R2c5FUENEbokdDs/vFMIxCjeeu09Axwp+OEf5AJLLc\ngTPZTzL3wHfEZ+0loF4MRy4UklNs5qy8ipQLVn462ha7KQSQ0KlVNAzwoFGQnshADxoGelDfz516\nfu4E6l2ce9+RSkV+3V5kBXXF3ZJDC+thiF/mmAk1cbXjofWABp2hYU+I6AnBrcocs10QhLtT7mS9\nqKiI119/neXLl5OTk0OzZs2YPn06o0ePvuW2GRkZTJs2jXXr1mEwGIiOjuadd97hnnvuuaPgqzJZ\nlskpNpOWb+RivpG0/BLS8o2k5pWQkmPgXI6BrCLzTeuo6+NGoyA9jYP0tKzrRYs63kQEeohZQ4VK\nYbAYOFJ4hP15+zlYdJCChAK2j96OTq1Do9LwXNvnsNqtDGw4EJ1KC4UXIeVXuHAAzmyD83+A7S//\nx0NaO1rlWj0IniE33rEg1AKSJBER4MsH/Z4uXSfLMufy8hnx89sYbcXo/LejsntjKQ7HUtSQk7kR\nHE8PBK5NzF21Kur7uVPfz50wX3fqeLtSx8fN8eztSrCX6x2fNyzuwRB9L3R7EdIOOpL2hFWOeQ9O\nbXI8ANz8oGF3CO/u+JwHtwAXBW+gFYQapNzJ+vDhw/nzzz95//33adKkCUuWLGHMmDHY7XYefvjh\nMrczmUzcc8895OXlMWvWLIKCgpg9ezYDBgxg06ZN9OzZ867eSGWw2WXyDGZyDRbyDGayi81kFZnI\nKjSTWWQks9BEVpGZzEITFwuMmK32W9bp464tbR2JDPAg8tIlz8hAPW66KjI2r1Br7Ejdwc+nfyYx\nO5Hk/GRkrtyorFFpOJFznCj3UMhL5v7CQrh4GHZ+DekJYMi+vkKvMEerW8Oe0LAHeN1iRBhBqMUk\nSaKOtwcf9HiXH5N+ZNv5bVjIR+0Zj9ozHoCG7rG0UD3P2WwDZ3OKSTekYjT7ciK9iBPpRWXUCwF6\nF4I8XQj0dCFAf+2zv4cOX3cdvh5afN11N+4zL0mOG75DY6D/u5CRCKe3Oq6cnd0FJTmOm8SPrL2y\njW9DCIm6lLxHgV8E+NQTV9IEoZzKNRrM+vXruf/++0sT9MvuvfdeEhMTSUlJQa2+cYL5xRdf8NRT\nT7Fr1y46d+4MgNVqJTo6Gr1ez549e64pX1VGgzGYrdw3azu5Bss1s3vergC9C3W8XQnxdiX0UmtH\ng0vJeT0/99selUUp4o7w6k+WZYosReQZ87houEhqUarjUeh4frXjqzT1awpWMwvj5/Bhwlel2wZI\nrrSxu9PDDP2sJvR558B844QASQX+jR3DLDboAhG9HCdnMSxotSU+/8oyWo0czjrM3ot72Zu+l/jM\neB5t+SjPxDwDOLqm9V7eG61KR6BrGJ7qumhsgVjNXhQb3Mkv8CczR4/FVr5B39y0ajy0oNdJ6HVq\n6gT44OWqxctNg5erFk9XDZ6uGtx1GvQuGvRaGf+8w/imx+GeeRBtZgKqootl78A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sFscnk5p9DV9XBybXcJnGPysvZqP9o/FpUruxla39XYnt1hyAWRtP2nSzpEHN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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "prior, z = (8.5, 1.5), (10.2, 0.5)\n", "plot_products(prior[0], prior[1], z[0], z[1])\n", "prior, z = (8.5, 0.5), (10.2, 1.5)\n", "plot_products(prior[0], prior[1], z[0], z[1])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The result is a Gaussian that is taller than either input. This makes sense - we have incorporated information, so our variance should have been reduced. And notice how the result is far closer to the the input with the smaller variance. We have more confidence in that value, so it makes sense to weight it more heavily.\n", "\n", "It *seems* to work, but is it really correct? There is more to say about this, but I want to get a working filter going so you can it experience it in concrete terms. After that we will revisit Gaussian multiplication and determine why it is correct." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Interactive Example\n", "\n", "This interactive code provides sliders to alter the mean and variance of two Gaussians that are being multiplied together. As you move the sliders the plot is redrawn. Place your cursor inside the code cell and press CTRL+Enter to execute it." ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "scrolled": false }, "outputs": [ { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "9059fd1da4fb4d0fa87bd0db474ac9bc", "version_major": 2, "version_minor": 0 }, "text/plain": [ "A Jupyter Widget" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "from ipywidgets import interact\n", "\n", "interact(plot_products,\n", " m1=(5, 10., .5), m2=(10, 15, .5), \n", " v1=(.1, 2, .1), v2=(.1, 2, .1));" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## First Kalman Filter\n", "\n", "Let's get back to concrete terms and implement a Kalman filter. We've implemented the `update()` and `predict()` functions. We just need to write some boilerplate code to simulate a dog and create the measurements. I've put a `DogSimulation` class in `kf_internal` to avoid getting distracted with that task. The filtering is performed by 3 lines of code inside the `for` loop." ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "scrolled": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "PREDICT\t\t\tUPDATE\n", " x var\t\t z\t x var\n", " 1.000 401.000\t1.354\t 1.352 1.990\n", " 2.352 2.990\t1.882\t 2.070 1.198\n", " 3.070 2.198\t4.341\t 3.736 1.047\n", " 4.736 2.047\t7.156\t 5.960 1.012\n", " 6.960 2.012\t6.939\t 6.949 1.003\n", " 7.949 2.003\t6.844\t 7.396 1.001\n", " 8.396 2.001\t9.847\t 9.122 1.000\n", " 10.122 2.000\t12.553\t 11.338 1.000\n", " 12.338 2.000\t16.273\t 14.305 1.000\n", " 15.305 2.000\t14.800\t 15.053 1.000\n", "\n", "final estimate: 15.053\n", "actual final position: 14.838\n" ] } ], "source": [ "import kf_book.kf_internal as kf_internal\n", "from kf_book.kf_internal import DogSimulation\n", "\n", "np.random.seed(13)\n", "\n", "process_var = 1. # variance in the dog's movement\n", "sensor_var = 2. # variance in the sensor\n", "\n", "x = (0., 400.) # dog's position, N(0, 20**2)\n", "velocity = 1\n", "dt = 1. # time step in seconds\n", "process_model = (velocity*dt, process_var) \n", " \n", "# simulate dog and get measurements\n", "dog = DogSimulation(\n", " x0=x[0], \n", " velocity=process_model[0], \n", " measurement_var=sensor_var, \n", " process_var=process_model[1])\n", "\n", "# create list of measurements\n", "zs = [dog.move_and_sense() for _ in range(10)]\n", "\n", "print('PREDICT\\t\\t\\tUPDATE')\n", "print(' x var\\t\\t z\\t x var')\n", "\n", "# run the filter\n", "xs, predictions = [], []\n", "for z in zs:\n", " # perform Kalman filter on measurement z\n", " prior = predict(x, process_model)\n", " likelihood = (z, sensor_var)\n", " x = update(prior, likelihood)\n", "\n", " # save results\n", " predictions.append(prior[0])\n", " xs.append(x[0])\n", " kf_internal.print_gh(prior, x, z)\n", "\n", "print()\n", "print('final estimate: {:10.3f}'.format(x[0]))\n", "print('actual final position: {:10.3f}'.format(dog.x))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Here is an animation of the filter. Predictions are plotted with a red triangle. After the prediction, the filter receives the next measurement, plotted as a black circle. The filter then forms an estimate part way between the two. " ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support.' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", " 'Firefox 4 and 5 are also supported but you ' +\n", " 'have to enable WebSockets in about:config.');\n", " };\n", "}\n", "\n", "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", " this.id = figure_id;\n", "\n", " this.ws = websocket;\n", "\n", " this.supports_binary = (this.ws.binaryType != undefined);\n", "\n", " if (!this.supports_binary) {\n", " var warnings = document.getElementById(\"mpl-warnings\");\n", " if (warnings) {\n", " warnings.style.display = 'block';\n", " warnings.textContent = (\n", " \"This browser does not support binary websocket messages. \" +\n", " \"Performance may be slow.\");\n", " }\n", " }\n", "\n", " this.imageObj = new Image();\n", "\n", " this.context = undefined;\n", " this.message = undefined;\n", " this.canvas = undefined;\n", " this.rubberband_canvas = undefined;\n", " this.rubberband_context = undefined;\n", " this.format_dropdown = undefined;\n", "\n", " this.image_mode = 'full';\n", "\n", " this.root = $('
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');\n", " var titletext = $(\n", " '
');\n", " titlebar.append(titletext)\n", " this.root.append(titlebar);\n", " this.header = titletext[0];\n", "}\n", "\n", "\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "\n", "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "mpl.figure.prototype._init_canvas = function() {\n", " var fig = this;\n", "\n", " var canvas_div = $('
');\n", "\n", " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", "\n", " function canvas_keyboard_event(event) {\n", " return fig.key_event(event, event['data']);\n", " }\n", "\n", " canvas_div.keydown('key_press', canvas_keyboard_event);\n", " canvas_div.keyup('key_release', canvas_keyboard_event);\n", " this.canvas_div = canvas_div\n", " this._canvas_extra_style(canvas_div)\n", " this.root.append(canvas_div);\n", "\n", " var canvas = $('');\n", " canvas.addClass('mpl-canvas');\n", " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", "\n", " this.canvas = canvas[0];\n", " this.context = canvas[0].getContext(\"2d\");\n", "\n", " var backingStore = this.context.backingStorePixelRatio ||\n", "\tthis.context.webkitBackingStorePixelRatio ||\n", "\tthis.context.mozBackingStorePixelRatio ||\n", "\tthis.context.msBackingStorePixelRatio ||\n", "\tthis.context.oBackingStorePixelRatio ||\n", "\tthis.context.backingStorePixelRatio || 1;\n", "\n", " mpl.ratio = (window.devicePixelRatio || 1) / backingStore;\n", "\n", " var rubberband = $('');\n", " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", "\n", " var pass_mouse_events = true;\n", "\n", " canvas_div.resizable({\n", " start: function(event, ui) {\n", " pass_mouse_events = false;\n", " },\n", " resize: function(event, ui) {\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " stop: function(event, ui) {\n", " pass_mouse_events = true;\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " });\n", "\n", " function mouse_event_fn(event) {\n", " if (pass_mouse_events)\n", " return fig.mouse_event(event, event['data']);\n", " }\n", "\n", " rubberband.mousedown('button_press', mouse_event_fn);\n", " rubberband.mouseup('button_release', mouse_event_fn);\n", " // Throttle sequential mouse events to 1 every 20ms.\n", " rubberband.mousemove('motion_notify', mouse_event_fn);\n", "\n", " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", "\n", " canvas_div.on(\"wheel\", function (event) {\n", " event = event.originalEvent;\n", " event['data'] = 'scroll'\n", " if (event.deltaY < 0) {\n", " event.step = 1;\n", " } else {\n", " event.step = -1;\n", " }\n", " mouse_event_fn(event);\n", " });\n", "\n", " canvas_div.append(canvas);\n", " canvas_div.append(rubberband);\n", "\n", " this.rubberband = rubberband;\n", " this.rubberband_canvas = rubberband[0];\n", " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", " this.rubberband_context.strokeStyle = \"#000000\";\n", "\n", " this._resize_canvas = function(width, height) {\n", " // Keep the size of the canvas, canvas container, and rubber band\n", " // canvas in synch.\n", " canvas_div.css('width', width)\n", " canvas_div.css('height', height)\n", "\n", " canvas.attr('width', width * mpl.ratio);\n", " canvas.attr('height', height * mpl.ratio);\n", " canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;');\n", "\n", " rubberband.attr('width', width);\n", " rubberband.attr('height', height);\n", " }\n", "\n", " // Set the figure to an initial 600x600px, this will subsequently be updated\n", " // upon first draw.\n", " this._resize_canvas(600, 600);\n", "\n", " // Disable right mouse context menu.\n", " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", " return false;\n", " });\n", "\n", " function set_focus () {\n", " canvas.focus();\n", " canvas_div.focus();\n", " }\n", "\n", " window.setTimeout(set_focus, 100);\n", "}\n", "\n", "mpl.figure.prototype._init_toolbar = function() {\n", " var fig = this;\n", "\n", " var nav_element = $('
')\n", " nav_element.attr('style', 'width: 100%');\n", " this.root.append(nav_element);\n", "\n", " // Define a callback function for later on.\n", " function toolbar_event(event) {\n", " return fig.toolbar_button_onclick(event['data']);\n", " }\n", " function toolbar_mouse_event(event) {\n", " return fig.toolbar_button_onmouseover(event['data']);\n", " }\n", "\n", " for(var toolbar_ind in mpl.toolbar_items) {\n", " var name = mpl.toolbar_items[toolbar_ind][0];\n", " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", " var image = mpl.toolbar_items[toolbar_ind][2];\n", " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", "\n", " if (!name) {\n", " // put a spacer in here.\n", " continue;\n", " }\n", " var button = $('