{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Random numbers\n", "Module 4 of Phys212/Biol212 class\n", "### by Ilya Nemenman, 2016-2019\n", "\n", "## General notes\n", "In this notebook, we explore stochastic models, including generation of (pseudo)-random numbers and a variety of models and calculations that employ them. In general, all methods involving generation of random numbers are called *Monte Carlo* methods, referring to the Monte Carlo casino in Monaco, arguably the most famos casino in the world. The history of this term is rather curious. As most modern computational methods we use, claculations with random numbers trace their history to the Manhattan project: making the nuclear bomb in Los Alamos, New Mexico, during the second world war. People like von Neumann, Ulam, Metropolis, Bethe, Feynman -- whose names are familiar to everyone who knows even a bit of modern physics or advanced math -- worked there at the time and are responsible for the bulk of early ideas in computational physics. However, this was a secret laboratory, and projects, whether they involved production of fissile materials or development of algorithms, required codenames. The story goes that the name *Monte Carlo* was suggested by Metropolis as a code name for various stochastic models that were developed at the time by von Neumann and Ulam. And the name stuck.\n", "\n", "A good introduction to probability theory, one of my favorites, but more on the mathematical side, is in the book \n", "__[Introduction to Probability](http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/pdf.html)__ by CM Grinstead and JL Snell." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Why do we need random numbers?\n", "Hopefully an answer to this question will become clearer as we progress through this notebook and see a bunch of their applications. However, even now it is useful to break down various problems where random numbers are used into a few geenral classes.\n", "\n", "First, some processes are **fundamentally random**. Here I am largely talking about quantum mechanics, where all experimental evidence suggests that the randomness is intrinsic and unavoidable. Second, sometimes we simply do not know enough information about the underlying system, and we choose to model this **absence of knowledge** or as randomness, as well. Third, there are processes that lie somewhere invetween these two extremes. For example, motion of molecules of air or interactions of molecules in chemical processes can be described by deterministic Newton's equations. However, these equations give rise to **chaotic** motion (as we discussed in one of the projects in Module 2), which is also not predictable and can be modeled stochastically. Finally, there are some deterministic calculations that are **easier done** using random numbers than using deterministic approaches (e.g., calculating area of a complex object). We will discuss examples of these in subsequent lectures.\n", "\n", ">### Your turn\n", "Think of various physical, chemical, biological processes that involve randomness (e.g., mutations and recombinations in biology, turbulence in physics, and so on) and decide to which of the classes of uses of random numbers deined above they belong." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Introducing concepts of randomness\n", "Our introduction here is deliberately a bit sloppy, with no hard mathematical definitions and proofs. Those interested can find them in the textbook I linked.\n", "\n", "To define the necessary probabilistic concepts, we need \n", " - To define a space of outcomes that a random variable can take (e.g., head or tails, six sides of a dice, etc.). If a random variable is denoted by a small latin letter, then its space of outcomes is typically denoted by the capital version of the same letter, e.g., $x$ and $X$.\n", " - Then we define a probability of a certain outcome $x$ as a limit of frequencies after many random draws, or events. That is, if after $N$ draws, the outcome happened $n_x$ times, then it's frequency is $f_x=n_x/N$, and the probability is $P(x)=\\lim_{N\\to\\infty}f_x=\\lim_{N\\to\\infty}\\frac{n_x}{N}$.\n", " \n", "Probabilities satisfy the following properties, which follow from their definition of limits of frequencies:\n", " - nonnegativity: $P_i\\ge0" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# Examples of random number generation using built-in routines. The r1 variable should be pointing at an \n", "# array of 5 random numbers. We do the same calls multiple times to illustrate that the numbers are, indeed,\n", "# seeminly random (or rather pseudo-random)\n", "r1 = nprnd.random(5)\n", "print(r1)\n", "r1 = nprnd.random(5)\n", "print(r1)\n", "r1 = nprnd.random(5)\n", "print(r1)\n", "\n", "# The following command generates a 5x3 matrix of pseudo-random numbers.\n", "r2 = nprnd.random((5, 3))\n", "print(r2)\n", "\n", "# Finally, the following command generates n_rand pseudo-random numbers and histograms them into 50 bins,\n", "# illustrating that the numbers we generate are, indeed, uniformely distributed.\n", "n_rands = int(1e4) # how many random numbers to generate\n", "x = nprnd.random(n_rands)\n", "plt.hist(x, bins=50)\n", "plt.xlabel('number')\n", "plt.ylabel('frequency')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Most pseudo-random generators fail not because they do not distribute the numbers uniformely along the interval $[0,1)$, but rather because they produce correlations among the generated numbers. For example, making a scatter plot of two consecutive generated numbers against each other, $x_{i+1}$ vs $x_i$, may show some structure, indicative of dependence. We can plot the pairs of consecutive random numbers from the built-in generator against each other to see that no structure is visible. In reality, this generator passed a lot more stringent tests, but this is a good beginning." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.plot(x[0:n_rands-1],x[1:n_rands],'o')\n", "plt.xlabel('Random number i')\n", "plt.ylabel('Random number i+1')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Linear congruential random number generator\n", "Most pseudo-random generators in use today have underneath them some version of a linear congruential generator illustrated above. They may have bells ans whistles on top, but the core remains the same. In **your work** part, experiment with the generator by using different multipliers, moduli, and increments. What happens if the multiplier has common divisers witht he modulus? Do the numbers still look reasonably random?" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "# simple linear congruential random number generator\n", "def lcrng(seed=10, multiplier=7**5, modulus=2**31-1,increment=0,n_rands=1):\n", " # seed,multiplier,modulus, and incrment are the corresponding parameters of the generation\n", " # the default values of parameters are one particularly good generator, similar in quality \n", " # to the built-in one.\n", " # n_rands -- number of random numbers to be generated\n", " rand_flt = np.zeros(n_rands) # allocate memory for our pseudo-random numbers\n", " rand_flt[0] = seed # seed\n", " \n", " for i in np.arange(1, n_rands): # generate the pseudo-random integres\n", " rand_flt[i] = np.mod(rand_flt[i-1] * multiplier + increment, modulus)\n", " \n", " rand_flt = rand_flt / modulus # transform the pseudo-random integers into pseudo-random standard uniform numbers\n", " return rand_flt" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Having defined the generator, we can now check that its default version produces correct spread of numbers over the range $[0,1)$ and shows no correlation between the consecutive samples." ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "n_rands = 1000 # the number of random numbers to generate\n", "nbins = 50 # number of bins to histogram the random numbers\n", "\n", "x = lcrng(n_rands=n_rands)\n", "plt.hist(x, nbins) # show the resulting histogram of random numbers\n", "plt.xlabel('number')\n", "plt.ylabel('frequency')\n", "plt.show()\n", "\n", "plt.plot(x[0:n_rands-1],x[1:n_rands],'o')\n", "plt.xlabel('Random number i')\n", "plt.ylabel('Random number i+1')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we can explore how choosing parameters of the linear congruential generator affects its performance. Make sure to make your own choices, but here's one that is not too good and has a period much shorter than the modulus. Notice the patterns of correlation among consecutive random numbers." ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "n_rands = 1000 # number of random numbers to be generated\n", "nbins = 50 # number of bins to histogram the random numbers\n", "\n", "x = lcrng(seed = 10, multiplier = 77, modulus = 8100, increment = 2, n_rands=n_rands)\n", "plt.hist(x, nbins) # show the resulting histogram of random numbers\n", "plt.xlabel('number')\n", "plt.ylabel('frequency')\n", "plt.title('Histogram of linear congruential pseudo-random numbers')\n", "plt.show()\n", "\n", "plt.plot(x[0:n_rands-1],x[1:n_rands],'o')\n", "plt.xlabel('Random number i')\n", "plt.ylabel('Random number i+1')\n", "plt.title('Correlations between consecutive pseudo-random numbers')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Examples of using the standard uniform pseudo-random number to generate other types of pseudo-random numbers\n", "Here we will study how to generate different types of random pseudo-numbers using just a standard, uniformly distributed pseudo-random number. From this point on, I will be omitting the prefix *psuedo* everywhere, calling the numbers we generate just *random*. However, you should keep in mind that the only such numbers we can generate on a deterministic digital computer are pseudo-random, and this is always implied.\n", "\n", ">### Your turn \n", "Make sure to run each cell below multiple times to convince yourself that the generated numbers indeed change." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Uniform random numbers\n", "The standard linear congruential generator allows us to generate a uniform pseudo-random floating point number between 0 and 1, called the ''standard'' uniform random number. Variables $x$ taken from this generator are denoted as $x\\sim{\\cal U}(1)$. Then how do we generate a number between, say, $a$ and $b$, to be denoted as $y\\sim {\\cal U}(a,b)$? For this, we only need to scale and shift the standard number. That is, if we say that $y=a + (b-a)x$, where $x\\sim {\\cal U}(1)$, then we get $y\\sim{\\cal U}(a,b)$. The probability density for this number is $P(y) = 1/|b-a|$ when $y\\in [a,b)$ and zero otherwise. The code in the next cell illustrates this." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# generating numbers U(a,b)\n", "n_rands = 1000 # number of random numbers to generate\n", "nbins = 30 # number of bins in the histogram to show\n", "a = 1 # left edge of the interval\n", "b = 4 # right edge of the interval\n", "randAB = a+(b-a)*nprnd.random(n_rands)\n", "\n", "plt.hist(randAB, nbins)\n", "plt.title('Histogram of non-standard uniform random numbers.')\n", "plt.xlabel('number')\n", "plt.ylabel('frequency')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Multinomial random number\n", "A dice is an example of a multinomial random number with six equally probable outcomes. In general, a multinomial random variable is a random variable that can have a finite, discrete set of values, with a given probability of each of these values. A binomial random number is a special case of a multinominal random number with only two possible outcomes, such as a coin flip.\n", "\n", "We generate multinominal random numbers by sub-dividing the interval between zero and one into subintervals. The number of the sub-intervals is equal to the number of possible outcomes of the multinomial variable, and the length of each sub-interval is equal to the probability of each outcome. We then generate a uniform floating point number between zero and one and check which of the sub-intervals it fell into. The index of the subinterval is then the generated multinomial number. For example, the cell below produces a multinomial random number with 4 possible outcomes, each with the same probability." ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# generating integer randoms 1...4\n", "n_rands = 1000 # number of random numbers to generate\n", "rand4 = np.floor(nprnd.random(n_rands)*4) # we first multiply a standard uniform number by 4, so that it spans \n", " # the range [0,4). Flooring the result produces a number [0,1,2,3] with equal probabilities\n", "plt.hist(rand4)\n", "plt.title('Histogram of multinomial random numbers\\n with 4 equiprobable outcomes')\n", "plt.xlabel('number')\n", "plt.ylabel('frequency')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As another example, the cell below simulates a 6-sides dice, but the probabilities of coming up with each of the sides are $(0.2, 0.3, 0.1, 0.15, 0.1, 0.15)$ instead of the uniform $1/6$. Make sure you carefully study the code -- especially how we verify, which of the subintervals the generated numbers fall into." ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "image/png": 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UbM8HRueG93bn1Tf5NeOZWLnxwY6Oz1bAc7HyNYTG/d8d2wAnN+yL0azc7Pichuk723ew6vds/W5evyhtX0fxNNNk+J8in443Lk/SgZLuUXpD32JSyafalHlXsfSEbYAfVfbns6TSwsh8InQWqUp4gaSJkjbu4fXXzgmiPreRqlN2JdXv30Yqdnb2A7SmTy0+1I1lNDZZDSs32bxSs8qk6wLtSuuYQyryl9YzWiu/r3d1m4aeAzwZKze/PDQiDoJOm01fSaRGEKcCnwYeycn0LlKLpH+IiKdXI7auzAc2aYhn644mbsIc4LSGffGaXBJsFw3Td7jvmtDM96q0fdVm0KvLaKbJ8JGS1DB+ntLrXa8knQBtEel9K9excsvFXcXSjK62eQ7p2kd1n24QEXcBRMSZEfFmYBfSiUt3WtPtE5wg6nMbqR790fwDNJlUDfNkRCzqYJ4FrNy0d3ddCOwhaf98l8hnSP+IMwrTXkc6iz5S0mBJHyK9TevXefw04HBJQySNAz5YmbfUDPlPgc9JenO+Z3/7XNS+l5RsPp+XtQ+pGuuS1di++4DnJX1B0ga5CeY3SHoLdNpsesltwEm8Wpqb3NBfstrHJ1e3TQG+LmldSW9n9avzIFUPfULSW/P+3lCpCeyhHUzf6b5rQrPb3r59/0S6iH55aaJorsnwzYFP5e/NYaSS8HWkksd6pO/hckkHAo3vmW46lk4sANoaTm6qzgG+qPzecKX3dB+Wu9+Sj037TSrtNxmsVZwg6nMX6eJge2nhUdKXpLPqix8BH8wPQ53Z3RVGxOOkl8KcQ7rgfAjpjqGXCtM+Q/qnOZn0/oHPA++pnD1/hVQieI50V9bPK/Ou0gx5RFyeh/2cdPH1l6QLii8B7yO9B/pp0h1Lx0Rqcru727eC9KM6lnQh+2lSYmp/r0Ox2fQOFncbqY749g76SyYAk/I2r0599pGki8rPkt6vcP5qLAOAiJhCug5xFukYzSS/T7qD6bvad135DukdGIsldfR+6D/nWOaR3qX9iS6Oc1dNht9Lak/sadJ364MR8UxELCU1yX1ZXt+RwDVrGEtJe0J5RtIDjSMj4ipSSfUSpTf2PUL6ngNsTEriz5Gqt54hlXjWKm6LyczWWC4ZXhgRxZfp2NrJJQgzMytygjAzsyJXMZmZWZFLEGZmVtTnGvHqjuHDh0dbW1urwzAzW6tMnTr16YgY0dV0a3WCaGtrY8qUKa0Ow8xsrSKpqaf4XcVkZmZFThBmZlbkBGFmZkVOEGZmVuQEYWZmRU4QZmZW5ARhZmZFThBmZlbkBGFmZkVr9ZPU1j1tp1zbsnXPPv3glq3bzFaPSxBmZlbkBGFmZkW1JQhJ60u6T9KDkqZL+noevq2keyU9IelSSevm4evl/pl5fFtdsZmZWdfqLEG8COwbEW8ivSj9AEl7kF7y/YOIGEN6offxefrjgeciYnvgB3k6MzNrkdoSRCTLcu+Q/AlgX+CKPHwScGjuPiT3k8fvJ0l1xWdmZp2r9RqEpEGSpgELgZuAPwCLI2J5nmQuMDJ3jwTmAOTxS4DNCsscL2mKpCmLFi2qM3wzswGt1gQRESsiYiwwCtgd2Kk0Wf5bKi2s8sLsiJgYEeMiYtyIEV2+EMnMzFZTr9zFFBGLgcnAHsAwSe3PX4wC5uXuucBogDz+tcCzvRGfmZmtqs67mEZIGpa7NwD2B2YAtwIfzJMdC1ydu6/J/eTxt0TEKiUIMzPrHXU+Sb0lMEnSIFIiuiwifi3pUeASSd8Cfgecm6c/F7hA0kxSyeHwGmMzM7Mu1JYgIuIhYNfC8Fmk6xGNw/8GHFZXPGZm1j1+ktrMzIqcIMzMrMgJwszMipwgzMysyAnCzMyKnCDMzKzICcLMzIqcIMzMrMgJwszMipwgzMysyAnCzMyKnCDMzKzICcLMzIqcIMzMrMgJwszMipwgzMysyAnCzMyKnCDMzKzICcLMzIqcIMzMrMgJwszMipwgzMysyAnCzMyKnCDMzKzICcLMzIpqSxCSRku6VdIMSdMlfToPnyDpT5Km5c9BlXm+KGmmpMclvbuu2MzMrGuDa1z2cuDkiHhA0lBgqqSb8rgfRMT3qhNL2hk4HNgF2Ar4raQdImJFjTGamVkHaitBRMT8iHggdy8FZgAjO5nlEOCSiHgxIp4EZgK71xWfmZl1rleuQUhqA3YF7s2DTpL0kKTzJG2Sh40E5lRmm0shoUgaL2mKpCmLFi2qMWozs4Gt9gQhaSPgSuAzEfE8cDbwemAsMB/4fvukhdljlQEREyNiXESMGzFiRE1Rm5lZrQlC0hBScrgoIn4BEBELImJFRLwC/IRXq5HmAqMrs48C5tUZn5mZdazOu5gEnAvMiIgzKsO3rEz2fuCR3H0NcLik9SRtC4wB7qsrPjMz61yddzHtBRwNPCxpWh52KnCEpLGk6qPZwMcBImK6pMuAR0l3QJ3oO5jMzFqntgQREXdQvq5wXSfznAacVldMZmbWPD9JbWZmRU4QZmZW5ARhZmZFThBmZlbkBGFmZkVOEGZmVuQEYWZmRU4QZmZW5ARhZmZFThBmZlbkBGFmZkVOEGZmVuQEYWZmRU4QZmZW5ARhZmZFThBmZlbkBGFmZkVOEGZmVuQEYWZmRU4QZmZWNLjVAbRK2ynXtmzds08/uGXrNjNrlksQZmZW5ARhZmZFThBmZlZUW4KQNFrSrZJmSJou6dN5+KaSbpL0RP67SR4uSWdKminpIUm71RWbmZl1rc4SxHLg5IjYCdgDOFHSzsApwM0RMQa4OfcDHAiMyZ/xwNk1xmZmZl2oLUFExPyIeCB3LwVmACOBQ4BJebJJwKG5+xDg/EjuAYZJ2rKu+MzMrHO9cg1CUhuwK3AvsEVEzIeURIDN82QjgTmV2ebmYY3LGi9piqQpixYtqjNsM7MBrfYEIWkj4ErgMxHxfGeTFobFKgMiJkbEuIgYN2LEiJ4K08zMGnSZICRturoLlzSElBwuiohf5MEL2quO8t+FefhcYHRl9lHAvNVdt5mZrZlmShD3Srpc0kGSSmf5RXnac4EZEXFGZdQ1wLG5+1jg6srwY/LdTHsAS9qroszMrPc1kyB2ACYCRwMzJX1b0g5NzLdXnmdfSdPy5yDgdOCdkp4A3pn7Aa4DZgEzgZ8AJ3RvU8zMrCd12RZTRARwE3CTpHcAFwInSHoQOCUi7u5gvjsoX1cA2K+D9ZzYbOBmZlavLhOEpM2Ao0ilgQXAJ0nVQWOBy4Ft6wzQzMxao5nWXO8GLgAOjYi5leFTJJ1TT1hmZtZqzSSIHXP1zyoi4rs9HI+ZmfURzVykvlHSsPYeSZtIuqHGmMzMrA9oJkGMiIjF7T0R8RyvPv1sZmb9VDMJYoWkrdt7JG1D4QlnMzPrX5q5BvEl4A5Jt+X+vUmtrZqZWT/WzHMQv8nvZtiD9FzDZyPi6dojMzOzlmqmBAGwHvBsnn5nSUTE7fWFZWZmrdbMg3LfBT4ETAdeyYMDcIIwM+vHmilBHEp6FuLFuoMxM7O+o5m7mGYBQ+oOxMzM+pZmShAvANMk3Qz8vRQREZ+qLSozM2u5ZhLENfljZmYDSDO3uU6StAGwdUQ83gsxmZlZH9DMK0ffC0wDfpP7x0pyicLMrJ9r5iL1BGB3YDFAREzD74AwM+v3mkkQyyNiScMwt8VkZtbPNXOR+hFJRwKDJI0BPgXcVW9YZmbWas2UID4J7EK6xfVi4HngM3UGZWZmrdfMXUwvkFp0/VL94ZiZWV/RTFtMt1K45hAR+9YSkZmZ9QnNXIP4XKV7feADwPJ6wjEzs76imSqmqQ2D7qy8PMjMzPqpZh6U27TyGS7p3cDrmpjvPEkLJT1SGTZB0p8kTcufgyrjvihppqTH8zrMzKyFmqlimkq6BiFS1dKTwPFNzPcz4Czg/IbhP4iI71UHSNoZOJx0t9RWwG8l7RARK5pYj5mZ1aCZKqbVemo6Im6X1Nbk5IcAl+R3TjwpaSbp6e27V2fdZgNZ2ynXtmS9s08/uCXrtfo0cxfT/+lsfET8opvrPEnSMcAU4OSIeA4YCdxTmWZuHmZmZi3STBXT8cCewC25/x3AZGAJqeqpOwnibOCbeb5vAt8HPkqqvmpUbM5D0nhgPMDWW2/djVWbmVl3NJMgAtg5IuYDSNoS+HFEfKS7K4uIBe3dkn4C/Dr3zgVGVyYdBczrYBkTgYkA48aNc5tQZmY1aaapjbb25JAtAHZYnZXl5NLu/UD7HU7XAIdLWk/StsAY4L7VWYeZmfWMZkoQkyXdQGqHKUh3G93a1UySLgb2AYZLmgt8DdhH0ti8nNnAxwEiYrqky4BHSXdKneg7mMzMWquZu5hOkvR+YO88aGJEXNXEfEcUBp/byfSnAad1tVwzs76iVXeMQe/cNdZMCQLgAWBpRPxW0mskDY2IpXUGZmZmrdXMk9T/ClwB/HceNBL4ZZ1BmZlZ6zVzkfpEYC/SeyCIiCeAzesMyszMWq+ZBPFiRLzU3iNpMH7lqJlZv9dMgrhN0qnABpLeCVwO/KresMzMrNWaSRCnAIuAh0m3pV4HfLnOoMzMrPU6vYtJ0iBgUkQcBfykd0IyM7O+oNMSRH5YbYSkdXspHjMz6yOaeQ5iNuktctcAf2kfGBFn1BWUmZm1XoclCEkX5M4PkRrVWwcYWvmYmVk/1lkJ4s2StgGeAv6rl+IxM7M+orMEcQ7wG2Bb0st92on0HMR2NcZlZmYt1mEVU0ScGRE7Af8TEdtVPttGhJODmVk/1+VzEBHxb70RiJmZ9S3NPChnZmYDkBOEmZkVOUGYmVmRE4SZmRU5QZiZWZEThJmZFTlBmJlZkROEmZkVOUGYmVmRE4SZmRU5QZiZWVFtCULSeZIWSnqkMmxTSTdJeiL/3SQPl6QzJc2U9JCk3eqKy8zMmlNnCeJnwAENw04Bbo6IMcDNuR/gQGBM/owHzq4xLjMza0JtCSIibgeebRh8CDApd08CDq0MPz+Se4BhkrasKzYzM+tab1+D2CIi5gPkv5vn4SOBOZXp5uZhq5A0XtIUSVMWLVpUa7BmZgNZX7lIrcKwKE0YERMjYlxEjBsxYkTNYZmZDVy9nSAWtFcd5b8L8/C5wOjKdKOAeb0cm5mZVfR2grgGODZ3HwtcXRl+TL6baQ9gSXtVlJmZtcbguhYs6WJgH2C4pLnA14DTgcskHQ88BRyWJ78OOAiYCbwAfKSuuMzMrDm1JYiIOKKDUfsVpg3gxLpisYGr7ZRrW7bu2acf3LJ1m/WE2hKEmVlvaeWJQH/WV+5iMjOzPsYJwszMipwgzMysyAnCzMyKnCDMzKzICcLMzIqcIMzMrMgJwszMipwgzMysyAnCzMyKnCDMzKzICcLMzIqcIMzMrMgJwszMipwgzMysyAnCzMyKnCDMzKzICcLMzIqcIMzMrMgJwszMipwgzMysyAnCzMyKnCDMzKxocCtWKmk2sBRYASyPiHGSNgUuBdqA2cC/RMRzrYjPzMxaW4J4R0SMjYhxuf8U4OaIGAPcnPvNzKxF+lIV0yHApNw9CTi0hbGYmQ14rUoQAdwoaaqk8XnYFhExHyD/3bw0o6TxkqZImrJo0aJeCtfMbOBpyTUIYK+ImCdpc+AmSY81O2NETAQmAowbNy7qCtDMbKBrSQkiIublvwuBq4DdgQWStgTIfxe2IjYzM0t6PUFI2lDS0PZu4F3AI8A1wLF5smOBq3s7NjMze1Urqpi2AK6S1L7+n0fEbyTdD1wm6XjgKeCwFsRmZmZZryeIiJgFvKkw/Blgv96Ox8zMyvrSba5mZtaHOEGYmVmRE4SZmRU5QZiZWZEThJmZFTlBmJlZkROEmZkVOUGYmVmRE4SZmRU5QZiZWZEThJmZFTlBmJlZkROEmZkVOUGYmVmRE4SZmRU5QZiZWZEThJmZFTlBmJlZkROEmZkVOUGYmVmRE4SZmRU5QZiZWZEThJmZFTlBmJlZkROEmZkV9bkEIekASY9LminplFbHY2Y2UPWpBCFpEPBj4EBgZ+AISTu3Niozs4GpTyUIYHdgZkTMioiXgEuAQ1ock5nZgKSIaHUMfyfpg8ABEfGx3H808NaIOKkyzXhgfO7dEXh8NVc3HHjV9KxXAAAEt0lEQVR6DcJdG3mbBwZv88CwJtu8TUSM6Gqiwau58LqoMGylDBYRE4GJa7wiaUpEjFvT5axNvM0Dg7d5YOiNbe5rVUxzgdGV/lHAvBbFYmY2oPW1BHE/MEbStpLWBQ4HrmlxTGZmA1KfqmKKiOWSTgJuAAYB50XE9JpWt8bVVGshb/PA4G0eGGrf5j51kdrMzPqOvlbFZGZmfYQThJmZFQ24BCHpPEkLJT3S6lh6i6TRkm6VNEPSdEmfbnVMdZO0vqT7JD2Yt/nrrY6pN0gaJOl3kn7d6lh6i6TZkh6WNE3SlFbHUzdJwyRdIemx/D/9ttrWNdCuQUjaG1gGnB8Rb2h1PL1B0pbAlhHxgKShwFTg0Ih4tMWh1UaSgA0jYpmkIcAdwKcj4p4Wh1YrSf8OjAM2joj3tDqe3iBpNjAuIgbEg3KSJgH/GxE/zXd7viYiFtexrgFXgoiI24FnWx1Hb4qI+RHxQO5eCswARrY2qnpFsiz3Dsmffn02JGkUcDDw01bHYvWQtDGwN3AuQES8VFdygAGYIAY6SW3ArsC9rY2kfrm6ZRqwELgpIvr7Nv8Q+DzwSqsD6WUB3Chpam6Kpz/bDlgE/E+uSvyppA3rWpkTxAAiaSPgSuAzEfF8q+OpW0SsiIixpCfyd5fUb6sUJb0HWBgRU1sdSwvsFRG7kVqBPjFXI/dXg4HdgLMjYlfgL0Btr0Vwghggcj38lcBFEfGLVsfTm3IRfDJwQItDqdNewPtyffwlwL6SLmxtSL0jIublvwuBq0itQvdXc4G5ldLwFaSEUQsniAEgX7A9F5gREWe0Op7eIGmEpGG5ewNgf+Cx1kZVn4j4YkSMiog2UhM1t0TEUS0Oq3aSNsw3XpCrWt4F9Ns7FCPiz8AcSTvmQfsBtd1s0qea2ugNki4G9gGGS5oLfC0izm1tVLXbCzgaeDjXyQOcGhHXtTCmum0JTMovoVoHuCwiBsytnwPIFsBV6RyIwcDPI+I3rQ2pdp8ELsp3MM0CPlLXigbcba5mZtYcVzGZmVmRE4SZmRU5QZiZWZEThJmZFTlBmJlZkROEWS+QNFlSrS+YN+tpThBmfZykAfe8kvUNThBmFZLachv7P8nvkbhR0gbVEoCk4blJCyQdJ+mXkn4l6UlJJ0n699yQ2j2SNq0s/ihJd0l6RNLuef4N8ztK7s/zHFJZ7uWSfgXc2Mu7wQxwgjArGQP8OCJ2ARYDH+hi+jcAR5LaADoNeCE3pHY3cExlug0jYk/gBOC8POxLpGYx3gK8A/i/ldY53wYcGxH79sA2mXWbi65mq3oyItqbJJkKtHUx/a35PRtLJS0BfpWHPwy8sTLdxZDeSSJp49xW1LtIjex9Lk+zPrB17r4pIgbUu0usb3GCMFvVi5XuFcAGwHJeLXGv38n0r1T6X2Hl/7HGdm0CEPCBiHi8OkLSW0lNOZu1jKuYzJozG3hz7v7gai7jQwCS3g4siYglwA3AJ3OLu0jadQ3jNOsxThBmzfke8G+S7gKGr+YynsvznwMcn4d9k/Q61IckPZL7zfoEt+ZqZmZFLkGYmVmRE4SZmRU5QZiZWZEThJmZFTlBmJlZkROEmZkVOUGYmVnR/wfwcjXPW3fJKgAAAABJRU5ErkJggg==\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# generating integer 1...6 with nonuniform probabilities \n", "n_rands = 1000 # number of random numbers to generate\n", "p1 = .2 # the following lines specify probabilities for each of the 6 outcomes\n", "p2 = .3\n", "p3 = .1\n", "p4 = .15\n", "p5 = .1\n", "p6 = 1 - p1 - p2 - p3 - p4 - p5 # remember that probabilities must sum up to 1, so the probability of the \n", " # last outcome is defined by the probabilities of the five other ones.\n", "\n", "rand6 = nprnd.random(n_rands) # generating n_rands standard uniform numbers \n", "\n", "# Now we could have iterated over all generated numbers and checked which of the subintervals of [0,1) they \n", "# fall into. However, recall that vectorized calculations are always much faster, so we vectorize these checks.\n", "rand6 = (rand6 < p1)*1 + ((rand6 >= p1) & (rand6 < p1+p2))*2 + ((rand6 >= p1+p2) & (rand6 < p1+p2+p3))*3 \\\n", " + ((rand6 >= p1+p2+p3) & (rand6 < p1+p2+p3+p4))*4 \\\n", " + ((rand6 >= p1+p2+p3+p4) & (rand6 < p1+p2+p3+p4+p5))*5 \\\n", " + (rand6>1-p6)*6\n", "\n", "plt.hist(rand6)\n", "plt.xlabel('number')\n", "plt.ylabel('frequency')\n", "plt.title('Histogram of multinomial random numbers\\n with 6 outcomes with different probabilities')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ ">### Your turn\n", "Look at the histogram above. The bins are not positioned uniformly, and are not centered about the integers $1,\\dots,6$. Figure out how to make the figure look nicer." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exponential random numbers \n", "Real-valued variables that have an exponential distribution are also very common. For example, the time between two subsequent nuclear decays, or the time between two nearby spikes in the simplest models of neuronal dynamics is exponentially distributed. Why is that? Nuclear decay has a finite probability $r \\Delta t$ of happening in any interval of time $\\Delta t$. For the next event to happen at time $T$, it must not happen in all of the $T/\\Delta t$ previous windows of the width $\\Delta t$, and then it should happen in the window $[T,T+\\Delta t)$. Combining these, we get the probability of the next event in the interval $[T,T+\\Delta t)$ being equal to $P_{\\rm next event}(T:T+dt)=(1-r \\Delta t)^{T/\\Delta t}r \\Delta t\\to e^{-r T}r \\Delta t$, so that the probability *density* is $P_{\\rm next event}(T)=r e^{-\\rho T}$ (here we used the, so called, *first remarkable limit*; make sure you can do this calculation). More generally, a random variable $x$ is called exponentially distributed with the *mean* $x_0$ (or the *rate* $r=1/x_0$) if the probability density function for this variable obeys $P(x)=\\frac{1}{x_0} e^{-x/x_0}$. We denote such variables as $x\\sim {\\cal E}(x_0)$. The distribution with $x_0=1/r=1$ is called the *standard* exponential distribution, denoted as $x\\sim{\\cal E}(1)$. \n", "\n", "***\n", "Numerical generation of exponential random numbers illustrates a few important properties of probability distributions. First of all, if we have an access to a standard exponential random number, then by just multiplying it by $x_0$ we will get the exponentially distributed number with a mean of $x_0$. That is, ${\\cal E}(x_0)=x_0{\\cal E}(1)$. This means that we only need to figure out how to generate standard exponential numbers, and the rest can be obtained by a simple multiplication. \n", "\n", "**Figure is needed here**\n", "In its turn, for generation of standard exponential numbers, let's go back to our definition of the probability densitiy of a continuous random variable as a limit of an ever-finer discretization of probabilities of observing the variable in segments of the real axis. Suppose we have a real valued variable $x$, which is distributed according to $P(x)$. This means that the probability of having a sample land in the interval $[x,x+\\Delta x)$ tends to $P(x)\\Delta x$. Suppose now we have defined a new variable $y=y(x)$. We can invert the functional dependence and write $x=x(y)$ (note that we are now assuming that such inverse function exists!) Of course, $y$ is also a random variable. How is it distributed? We note that the probability of a sample to land in our interval should not depend on whether we marked its end points as $[x,x+\\Delta x)$, or as $[y(x),y(x+\\Delta x))\\approx [y(x),y(x) + dy/dx \\Delta x)$. In other words, $P(x)|\\Delta x|=P(y)|\\Delta y|$, where we added the absolute value signs to ensure that, even if an interval is negative (is read from right to left), the probability is still positive. This gives, in the limit of small $\\Delta x$, $P(y) = P(x(y))\\left|\\frac{dx}{dy}\\right|$.\n", "\n", "How does this help with generating exponential numbers? Suppose that $x$ is a standard uniform number, $x\\sim {\\cal U}(1)$, so that $P(x)=1$. Let's choose $y=-\\log x$, resulting in $x= e^{-y}$. Then $P(y)=P(x(y))\\left|\\frac{d e^{-y}}{dy}\\right|=e^{-y}$. In other words, the variable $y$ is a random standard exponential variable, $y\\sim {\\cal E}(1)! So generating samples of such random variables is easy -- just generate standard uniform random numbers and take the negative log of them. \n", "\n", "Below we implement such generation of standard exponential random numbers and compare with the built-in function for their generation, `nprnd.exponential()`. For this, first, we generate exponential random numbers using the built-in function. Second, we compare the built-in functio to our own generator. We also do the comparison on the semi-log plot to illustrate that the numbers in both cases are, indeed, exponentially distributed. The built-in and log-uniform are indistinguishable. Maybe this should not be surprising since, under the hood, the built-in function generates the random numbers in exactly the same way as our own! " ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "n_rands=int(1e5)\n", "# First, we generate exponential random numbers using the built-in function.\n", "plt.hist(nprnd.exponential(size=n_rands), 30)\n", "plt.xlabel('number')\n", "plt.ylabel('frequency')\n", "plt.title('Histogram of built-in standard exponential numbers')\n", "plt.show()\n", "\n", "# Second, we compare the built-in function to our own generator. We also do the comparison on the semi-log plot\n", "# to illustrate that the bumbers are, indeed, exponentially distributed. \n", "plt.hist([nprnd.exponential(size=n_rands), -np.log(nprnd.random(n_rands))], 30,log=True)\n", "plt.xlabel('number')\n", "plt.ylabel('frequency')\n", "plt.title('Comparing histograms of built-in\\n and our own standard exponential numbers')\n", "plt.legend(('built-in generator', '-log U(1)'))\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## A uniform random number in a circle\n", "Let's suppose we need a uniform random number in a circle around $(0,0)$ with a radius of 1. This is not a trivial problem -- and, in fact, some of consulting and data science companies use this during interviews. We will explore multiple different ways of generating such numbers. First read the methods and ask yourself: Which of these will work? Which of these will be the fastest? Then we will look at implementations and see if your guesses were correct.\n", "\n", "**First plausible method**: We will generate a random $x$ coordinate and a random $y$ coordinate. We will combine the two to fine the angle defined by them. We will then generate a uniform radius between 0 and 1, and we will put the point at this radius and at the chosen angle. \n", "\n", "**Second plausible method**: We generate the random radius and the angle directly, and generate $x$ and $y$ pairs directly from the angle and the radius.\n", "\n", "**Third plausible method**: This is similar to the second, except that the random variable we generate is $r^2$, not $r$, and we then calculate the distance to the point by taking a square root of the pseudo-random generated number.\n", "\n", "**Fourth plausible method**: We generate a random $x$ coordinate between $(-1,1)$, and the same range for the $y$ coordinate. We then reject all of the points that fell outside the unit circle. In this case, we do not know how many pairs of points to generate precisely, because we do not know how many will be rejected. So we make an estimate: the area of a square is 4; the area of the unit circle is $\\pi$. So if i want to generate $\\mbox{n_nands}$ points, I should generate instead about $4 \\mbox{n_rands}/pi$. I then check how many actually got generated, and if not enough were, then I request the missing number to be generated additionally. This is *not* the fastest way to implement the rejection method (we will discuss this a bit later), but probably the cutest: the function we write for generating numbers on a circle calls itself *recursively* if it needs to generate the missing numbers.\n", "\n", "Once you have thought about these method, run the cell below. " ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n"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SFCQ1HbLG0ZvyEpIaRskYertLIER+hDuvWdL2kUQu12qFRK66Zwf3qrtLRWy47ryaKSyvGlMB+SNoDAFamAhhYt4pj1VQMISIZiGeprwEXntIa1/PYqrySaxV9wD8RxfZQqWJyZhuO0SnBWRD0BgCtNBl//KNZgC9Q1J3rSzjSJBVI7FtmpN30pvqHkAkkfvUflzgoomFzOvpj6AxBGihC6l0cUhmJZ4q/wWPLBqJre9goK8XQ3sP4YGn9mOSMRSIcO357hqKTjsxRStdvH57SxIIbTWxdohOC8iGwBgCtNARqZUbR4znE+CFgPHjUJlZsmgktsRscLiMTbvKNe1okjFs2lVG/xnzrZ9vcLiMVQ/vRnUqukZ5rNKQGGYTrdSuJppmZV4H5IfAGGYpXGLxVZKizhYO+E8gS8YhC+fMqpHYEjMfUUlrN4/WmEKC6hTD2s2jtecT3w0A3PzQ7gZzXaU6iZsfaq9s42ZkXgfki8AYZiF8SZ46806ehECmxSxbtAAbtu3Byo0j6O4qgjHg9Uq1wQyjYoa2xMyHmWSsoujDUKlK382qR3YDTO3D8REO7BN515gKyB/ELGrKtBv6+/vZ0NBQq4cxbaEqO51Gwr918Fncv3OftFF3bxMIwuBwGWs3jyqJLRAR+GvP78WmXWVtVI2NFpV27vhr6764XoMWpkOrS3wEtD+IaBdjrN90nK8ObpcD+BMABQBfZ4ytF/bfBWBZ/LMLwFsZY93xvkkAz8b79jHGrvIxpgA1fDoHdzx3UEno8raB6zKEeVSqkzVnsbidNwGZnKuDw2WMH59o2G7SjmzHOW9OIbeQ3tmIdqj0O12RmTEQUQHAVwG8H8ArAJ4mos2MsR8mxzDGVnLH/y6APu4SFcbY0qzjCLCHT+egiRhVqpNYt2U0lw/SJSpKZYaxJaYq4m5Tatp2nMVCB07rnpNaY0hCiAMxnD6O+naFD43hAgAvMMZeAgAiehDA1QB+qDj+BkQ9oQM8w5YopHEOqq5tckADwOHxqpdaSSJcJGRV8p3IDGUJe73dJYwfn5AS93lzO6XRQ/xc2RL61ytVrL1qsZV2IaJULGDZogVKYgi42/x9MZlWMKt2qPQ7neGDMfQC2M/9fgXAhbIDiegMAGcB2M5tPomIhgBMAFjPGBtUnHsTgJsAYOHChR6G3f5w+aBcJCRX56Du2jb5Bcm9fH+QtkRX5WModhDGj0/grNVbaw5s/piEkejuIQtnFedKbAikex7x3ajKW5eKHZg/b27d+1MRw3VbRnG0OuUkPfuSuFsluYdcimzI7HwmousALGeMfTL+/Z8AXMAY+13JsZ8H8A5+HxGdxhg7QES/gIhhvJcx9qLunrPB+awKyVRlm/p0KLtem5eydfCV05BAZd6ZN6eAYqFDG5V0cqmII8cn6hrN2BJwEbyTXTVXpmvblpfQHXvW6q1O49etDV/rKc912Y73bXc00/n8CoDTud/vAHBAcez1AH6H38AYOxD//xIRPYHI/6BlDLMBrqpwnhKS6dq803bpuseUEUJ8qYnkvCxw1Xz4cV68fnvDONOKSPwzqeZK7Ba3bNEC7HjuoHHcLs/oYrbSjVW3T7Vdpd2qjk/rR7FFyKXIBh+M4WkAZxPRWQDKiIj/R8SDiOgcAD0A/pHb1gNgnDF2jIhOAXAxgD/yMKZpD9cPM89sU5dr29jIfdp60xbMS8Mwu0tFzJvbKZ2L5JlUc5VFUrV9xmWLFjSEDus0FZ2z2uWd68xFqutQfF5e5iSRoZ5cKoIIWLlxBBu27Zm1TnlbZC6ixxibAPAZANsA/AjAQ4yxUSK6nYj40NMbADzI6m1XvwRgiIh2A9iByMegclrPKuiK18mQZ3E3l2sP9PXi2vN7paW3ebTa1quaR9WoS8UC1l61GE+uvlR5zIGxSlOK7MmQlOoQmYBOCxobP45VD++WFg50eQ6ddqt6boYokzvPctwDfb14cvWluGvFUhybmMLh8aq2QGLACXjJY2CMfQfAd4Rttwm/10rO+z6AJT7GMNPgqgrnmW3qcm2xlpAK3V3FzOPi7+n63Cqn+UnFDpxULODweLUuKom/pk6a9lVkzxUuobsJjhxXd4hLtBubeU1rxpRlbNu8S9f3HSKU3BFKYrQp0hD6tGYV2/HYXNuWQPlKuE8b9ZLsE/sqV6pTAAh3r1iqPF/HtH0U2UuDPBoF2b5zHaM0ldrmCbTNu0zzvkOEkjsCY2hj5Eno84Ltx/a6poSFC7JIgwN9vdiwbU8dY7A5X8e0L16/PZN0aisNi8d1dxUbniMtXP1SyxYtwH0790m33y/ZLiJZM7pwW134rml+Q7VXdwTGEOAVtpExvj7KrNJg2vNVTFsXhZPkS+iIvY00LDuu2KH36QB24bhp/CE7njuo3G6zHpK1oJq7w+PVGtNLk8EeIpTcETq4BRjh0r9X1X2Mh8+P0tVJL0Ll6+iao38G1/EAMDo+bTufyY4Ty3iL6O0u4aMXLWx4N8UCobtUdOqPLa4HFeFXOeN58Gshrw5/A31R97ke7l3P7QykT4egMcwCZClJ4GrTVZXEtonZT4Os0qDK13Hk+GSqcEqbTHCV6cNWe3G1jfOhsv1nzM8UoOCS2S3L5FaVRAfk4bY2sH3fR6tTtb/HKtVQO0mDwBhmOLKWJEhjw2+2b2RuZ0dtjD1dRay5Ul/UjofO15EmakUkhCoixxP3weEy1m0Z1eYbiL9lUnp3qYhjE1NaJpn13cjWA0OjmYq/r809ZeG2hKj0xzhH0BMUiDDFmDVzC5FJbgiMYYYj6wfRzhEdsnIRRyVERAedDdzGLyCDmGGtc3wODpex6pHddaU5eMikYZWWtPaqxQDybZBjm9ntel8Vw5nTWQADWZeGcR13O6zjdkRgDC1EM6pOZv0gVITz5FKxZU3pE/iQAlctPwcrN44opfWsZTxMpq4N2/YomUKBqEYAxbVy7fm9SvNcnu8hj8xuQL0eX69UcdeKpdLvxOX7CZFJbgiMoUVoVtXJrB+EzO5b7CAcOT5RqzXUqlr3JqZnQziShDRZuCWPtGYHUz6KjkFPMaaM79+0q+wsNfuAbD34CCYwJQ3Kktx4TavWAhXyNRgik9wQXPMtgm0ESlZkKdEwOFzGxn/a3yBNz+nsaJBy8xi7CbqIpISYyso9iLhjoD5iRQVZiW2baK2kNMPL66/Ak6svrSNcOgad7GvWWjFB5Qewyew2zZXrOl23ZbRhDVYnGdZtGZUen0Qm9XaXnCKwZiuCxtAiNMvmmaVUxtrNo9IwSFkpBaA59lpeC5CFlCbExNXMNGaRHMYTcV8anyo5rNhBNaLYLvZxlR9AlceQwGauXNepKplPl+Q3HRNGW4XAGFqEZto8034QqvLZKuRtrxUJjMigeOl15cYR6TXEaCBTQ5wEovTqynhkneG6454QjffqwJ3XvNuqNlMapPVtpWVQtnMVCHf7IDCGFsHF5unTSe3rWqViQVkrKC+HuqkOEy+9moipyGRkTCEJweyJY+/5ks0u/g2xKVByLxXjnT9vbt2c+bSPZ9F00jKoPDSe7lJROn/dJX/FGWczgo+hRbC1ebrYyk1wvZbK7t7TVZSOHYC3scpgQ0jKYxVcvH47li1aoLVZq5hMgaj2THetWIq7VyzF0eoUxir1JZtVGdMy/8ZYpaqMPLJ5Tp/28Sz+irT+qqzZ6TKsvWpxQxmQYgfVQnYDsiFoDCnhQzK2UZ3ThmTKxud6rTVXLm6IsS8WqJZAJp6TtYCcbuwDfermMSKSqJ0kpDMx3/AEUMVkphjDy+uvMD7T3M4OpdaUpgQ2D16r8a19ZZHe0/qr8ogIch1LM0LDZxICY0iBZjY4T/Mhq8anIlaqa7l+fD5MBrq5tSk3kaBSncSO5w42nJNc72SFKaKDqK4URpr4epV/wwbFAuHIsQmcuXprXTaxrzWW1V+Rxg+QJQDCx1ia+b3OFHhhDER0OYA/AVAA8HXG2Hph/8cBbEDU+hMAvsIY+3q872MAbo2338EY+6aPMeUJn+n1JkkmzYesGl9B4WA1FSCzfSYfTlLd3IrNY5K6Oypb/YGxivJ6JxUbJX4gsv+v3DiCob2HcMfAEuUzdcQd6mRJXS69l4sdhDed1Imx8Sq6u4p44+iJ/BDxTVWqk1i7eTQTMWtVPL/LOvIt3YdyGO7I7GMgogKArwL4AIBzAdxAROdKDt3IGFsa/0uYwnwAawBcCOACAGviPtBtDV/ONBubfxq7rmock4zl2nZy1fJzpHZfl+ub5pbPCRi+7TKMrLkMvQrGc3KpqCTQY+NV3HnNEmkLUgbgvp37lC0ugRPdx2T+E9k5xQ6q+WySe/Z2l7DhuvMwfNtleHn9Feia02mskjpWqeJMiyq3PPgcgg3b9uDa83vbNp5f9k2s3DiCWwefTX3Ndgn3nU7woTFcAOAFxthLAEBEDwK4GoBN7+blAB5njB2Kz30cwOUAHvAwrtzgK3zQRpJJo4bryhYkNnAXaUwnwYnRNw2VisxtAqzGrptbmRScZGfr7gOo6/sDUR7HyJrLAET9iW0bxKQ1nbgQKltzSDtlTdtAlStx/859qbvg5RkaLgtD7s25onAz4IMx9ALYz/1+BZEGIOJaIvp1AD8GsJIxtl9xrnT2iOgmADcBwMKFCz0MOz18qeO2koyohicSoGrR6cbnaiPW2WcB1O2TmXSqk8y5dpHr3MoI8fjxCWWyU6lYwLJFC+qeQ4bkeWzzIsQxuRICFxMUYGcOmW5mFF2RvrRjzst8pgp5Lo9V6pIWp6NPwwdjkMmEohi2BcADjLFjRPQpAN8EcKnludFGxu4BcA8A9Pf3e+oYnA4qiRCAU2G5NJJMHlmkOpjCG20cwS6SsMvYRU0mkdJMxPXOa5Y4Rw75kjpV2tfgcBnjGg1HBdPcTjczio45ph1zXhFMLmuonZmxDD4YwysATud+vwPAAf4Axthr3M//CeAPuXMvEc59wsOYcodMineNfEgjyTQ7i9QHYXElnrqx86q7GLVjKoQHROY028ghPo8jq9QZrY9nUOHKgidrZGjvIWzaVU4V4mqaWx1D0zGpVoV2rlp+Dj6reDdZTD95RDC5Mqp2ZcYy+EhwexrA2UR0FhHNAXA9gM38AUR0KvfzKgA/iv/eBuAyIuqJnc6XxdumHdIkDqVJXGq2BKhLTrL5UH06t3nHJGDuX6wbi83Yr3j3iWWbJclscLiMVQ/vrmMKCSrVSTzw1P5UTMFmblXBC4kpTQx8uHXwWS9Jii7tYEXI2le7BjGkhct37MqoplOJ78waA2Nsgog+g4igFwDcyxgbJaLbAQwxxjYD+G9EdBWACQCHAHw8PvcQEX0REXMBgNsTR3Q7QyZR+W4qr0Kz68qbJGWZ0zcJv2x2SQwTeEJukxMhFodLq4Vt2LZHG22kc4DLQID13KrMKCoC+MBT+6VO9s9y5UBsghXS5g1s2LYHsql600mdTdFaXL5jl7ya6Vbi20seA2PsOwC+I2y7jfv7FgC3KM69F8C9PsaRJ3QmjKREgszZ6ZtgpzFpZDEN2Nhnm2V2yKIVJSakBPxz+bZpA/VzbiL7qvwSGXq6ihi+7TKnsYjvUPfMunHYEvgs2fq68OJmwEXwEtfQTIpKIuYorbQD+vv72dDQUNPuJ2shKULVbzepIeSTeLoQetnY07RGdL1vHlC1yTTB9Lyq66btSmazXhIQgI9etLDBx1AsECanWJ30XCwQNnzovFTvTRyP2KM5gS2T6tX4JFTviIC6ciOmMcrux4dGE8G7Zprme2n1d+ECItrFGOs3HhcYgxk2BIkAaYkEoNHckpYwp4Fq7D1dRXTN6bRezD4ZTFrYENzemDDx0luzn82Fgd140ULcMbBESlwAPwKFajwicygVC7j2/F5rR7jqeBXT0TFa05zNKRCOawoR+lyLrRK8moHAGDzirNVbjeYA1aK3kUazSBymc23GDpgX89J1j0nzFLL2+nXF4HBZmmwGnGDOaT5In1KfzZx3EPDlD9uNNevYdOPp7S4po5JsmJtKw5AxHd36sl2nOjR7LQL+tc28YcsYQhE9C5gSj3Q2fpu6/WkddTbn2iZNmZrM6OoRNRPJ+FZuHGkgJFmSoHyF9wLaVj/hAAAgAElEQVR2c86YXbKTjwJwukx4GfFK5sJGQ1OZnRjkTMd1jC5oRTjodMsTsUXoxyCBGGonq+2fRNSZwhZNtejT1sdPJGfTuapaPzKoFrNuLK0IwRvo61VKlzzDTRsumRU20Se285alfwI/HlONLNl88SG6rkiYjqzPte0YXdGKtZhHr4l2QNAYBKhqyyS1/V3VeVMUUZay2ippjT9XFlV05NiEVANQLWbdWJLGOM12uPUoosAYgL7bH8MbRydqIaKuWpiPPhvrtoxqS3LYhi76kEhNkWUmrSTRHlY9vNtY5A9IF5qZjEU3b77v6QOtqlabNwJjEKCS0HY8dzCVzdD0Ufoqq607VzST3Dr4rDRL+PCRYzhr9VbrMSZodi2YweEy3jiqLh8hIyy24ZKuZhsVI5E1OUqQ9KW2ga+8FZ2pzLaYo4lou+RX6MYoFmY09R4vEDnNqU/4LD/TTgiMQUAeNkPdR5lG4tCNxUZaERO3EozHmblioTybGj4utWCySuWmhDEVTO/QNf5+cLhcR/zLYxWsemQ3gOidr908KiVqW595FXcMLLEas2x9EIBlixZYnW8D2zWvyyVwza/QrQGZIHP/zn1K8+EkY9i0q+xUfdVmDbqUDLERGqdTWGvwMQhots0wTakF1VgKRFZhcjZMrlKdxO8/+gxWbhyxVu1truujh3VaJm16h65Cwbotow0aQXWSYd2WUQBRlzcZDo9XrZ93oK8X157fW1dtMilD7dqXQQXbNa+bvzeOTjj1h3BZA3cMLMFdK5ZqfR0ufheb+6uOSVsyxGfv9mYgMAYBaRrjuEDl5MvqqCsVC/jjD9slP9kyufHqlDIJKu11fThT0zDpYgdh/PiE1hntKhSoGGayvZsrwifC5Xl3PHdQGoEF6AmMrQPeds3rHMTVKWb9TGnriq1afo62vYfO3MnPhU3Qhq5kSJr162PdNxPBlCQgT5uhr96zNs5EWTlq/nfaap7AiU5waRxuPkx1NjVqigXCvDmdeL1SxcmlIo5w/RmSrmBJ+07dddMKBSY/iMvzmo6Vmbtc1prtmk9+q6qf2j5T2jWwYdseba6DSmARTVE2QRu6Loimc132t2tYa2AMEsh8Aj7sgz6bpvBjTMa2cuNIjQjydm+xaYgYZZX0Gra126ftBAf4caYm91EluhWovnTExeu3N9j6ZV3BXIWCboVjtLtUNPpBbJ93cLisTiXmIBIY17Vmm8cx0NerTH6zfaa0a8BERCcZawieGBwua/0Tqvurxpimb7rueu0a1hoYgwV8Sfo+pQZVUT9TBAcgj7ISGZ8qpJWAVJ3gEviSypN725Qj0HUFu/mhE87iBOPHJ2p24LWbRxv2J/iN806VRnf9xnmn4n5Nbwjb503WnU1xApHAZFlrJiEo6zs0na+6v1XiIOq/z3VbRq0z//nxq8YoKwFiW/58OoW1BsZgAV+Svs8uYPwiS1NKwKZ9qCwa5qMXLcxkVlNJ5YBb9zvdtcTzdARlkrEaERnae6iB0I9VqvhcbDoRr6uK7trx3EGtxGlbR8e2zHjCrIETRFW1JkxrrRkdAnXni/kS5bEKVj0cMW+XMteV6qQxvLZAhCnGpOPXjbH/jPnOzz7dwlpDrSQL6Oq43O1Qm8dXwa20VUZ52NRyaVZ4Xd6FyAaHy9ISGjx6uooYG68qjykVOzB/3ty6yp4qopPUbMr6TC71g36y/gpjCQub+7e69o+qJld3qYiRNZfVacou5cpFpK2r5fpNtFuIaqiV5BE6idPFpORLasjqsLJVYcWEo5UOzVpc4EsjU32EA329Um2Ahykkt1Kdqq0Bk7kukcpPKnbUnqu7VMTaqxZ70TBFJGGcOg3Dpsos4G6C8k34VHObbJeZMNMISmk0X1Peiux4Ufv67MYRrNsyijVXuq2FZsMLYyCiywH8CaIObl9njK0X9n8OwCcRdXA7COC3GGN7432TAJJsqn2Msat8jMkndCqsKwFLa5vnYSIYYhe1LE1DfPlXdEhbFkSMvOJtv8lHu3bzKF6vRPPQVeyoJfHlhaRtplg+4sgxc5KgCBvTiU15FQKspX0Xc6fN2miGxKyy38/t7FBqH7YJhjxUeSsrH5KbGlWM+vB4tamVAtIgM2MgogKArwJ4P4BXADxNRJsZYz/kDhsG0M8YGyei3wbwRwBWxPsqjLGlWceRJ3yF6fmCKhs2qWiZdztN2/IStgTB1fciI0iyyJPqJKsRhqymNxskc79282hDRFJ1iuGWR5/JHJasY/I+fFguTlLT2kgjVKhqYPVockJ0fivZs6y9arHyWjqotErG5JYDHV1IG43YLPjQGC4A8AJj7CUAIKIHAVwNoMYYGGM7uON3ArjRw32bCh9hej7HAtibpLJIbVmK/NkShFXLz2moK1QsqJu/ywhSO3jKymNR20yVOaRSncKtg88apdW078tX5MvczhMmsJ6uotLsYVobaYQKWY2pYoGw5ko9Mddp4s2w8cuey6TZt2sOA+CHMfQC2M/9fgXAhZrjPwHgu9zvk4hoCJGZaT1jbFB2EhHdBOAmAFi4cGGmAaeF7YfXDPXZ1iSV1RSURgpdt2VUm+Upk+wmBRVd/M2jFR+UmNCngkkzEXMnRIjJWC7vK6sPS+a8PqoxvZnWRhqhwnf0jg/TbQJV3koC8blMpsB2zWEA/JTEkKUbSr9qIroRQD+ADdzmhbGX/CMA7iaid8rOZYzdwxjrZ4z1L1jgr4CYCwb6zHWNbh18Fis3jrRNTRQTkTbBtUTI4HBZqXIncyHOze8/+gxE8jMF1HIIRGT5oLpLRalZolQsKM0VyXtO05dARNJMSAZVMpbpfQ0Ol7F03WM4c/VWfHbjCA4fOYburiIOxBqM7dpzLdtgWhtp644N9LmViGkW1l61GMUOdVEOsQRKQi+6S/L11q45DIAfxvAKgNO53+8AcEA8iIjeB+ALAK5ijB1LtjPGDsT/vwTgCQB9HsaUG3SLNu2HbQvX5jM6Im0rdYvMsLtUxEnFDqzcOCIdg+45C0RSwqNyCKukMxVBMiGxLw/fdhnujouy8Qx+zZWLlYQuee8+mIOuIZKpAZGIJO6fn6vx6hQOx6G3NoLJ4HAZfbc/ptR2VPc2CUp51x1rNgb6erHhuvOUAoSskOBAXy9G1sjXW7swPBl8mJKeBnA2EZ0FoAzgekTSfw1E1AfgLwBczhj7Gbe9B8A4Y+wYEZ0C4GJEjulpiTQftgo2UTcmE4Ovzmt82KrJLKV7zrQx57LxAI3mBl2fYtEpn9YmLfOHuCJNQyTxHJe+zC6lw13GC+jn0cUs1G7x/iokzyvLt0gKCcrGLZunweFyXRJempDmvJCZMTDGJojoMwC2IQpXvZcxNkpEtwMYYoxtRmQ6ehOAhykqdJWEpf4SgL8goilE2st6IZppWsHlw9bBNurG5MjTjSeN1GbjTFTZnbtLRcyb2yndRwRp2QeZZCYSkLtWLK2NTSwPAvhNlANOELsv/M2zOHI8XRFCVS8F1dzxmc2A3Bdggk5L0TGFrBK+jY2/GSHRvqEqqW4rAMoY8lilWsvybvVze8ljYIx9B8B3hG23cX+/T3He9wG4BxS3KWw/bBNcom6ShSiTuHREOo3UZuNMVDnokxBBVf2ZjU/vN0aiqBKGOoCaj4IhfeiuC4ES6+Ml97Sod6cspbFs0YIGAYAvQ+KiJYhIo6UAqDHVPCV6n8UlfcO1bhM/z7o5UzFkndbRTITMZ49Q5Re4Zlm6mJ1O6y4pCZqq4JcsjtuGKNp8DDbmg7T1Z1QJQ6KHImEKriUcbAmUinHbMAVA/n4Hh8vYtKssZQp3DCxJpSUk0En9upDKnq5IgHCJlErDQNqhJLVs3ACU34RNIUDd96R7tnYIY531jMGnJOQr1E6neYhmklXLz1FGHu147iDuvGaJ1XhsiKJtuK7J7qyywSbjSKJp+O2AW5Jamo/LlkDpqrXaoFTsaCgYqGI2iXZhW1AvQVL7yWTX12Vkv3F0QtlWM2sPCB6tLkmtGjdf0iRB8tyJ0KHTCHTfk44ht0MY66xmDHnYNn3ETZ/5Fvmi+dV3zsdPXqvUOaRVfYWBiIDZjseGKJoYXxYma/MuXIqmJZqUy3hsCZRtDSMVxqtTGI/PT55TRfTLYxWnYnqAWVsSnZ4qVKeYtr5U1h4QCVpdklo1bt07AfTfuul7UgUxFDvUiZ3NxKxmDO1o2xwcLuP7Lx6S7vvJa5XaB29jWnCRPGyJoupjyJKYBdi9C1umUCwQDh05VlfCxGY8NgRqcLiMw0eOyU5PDZMm4MIUTAQ1i0lKhK8eED40bVshQHZcHqYb0/eUjG3GRiVNZ7SDbVOELuSVX2g2pgUXyWPZogVS6VAVQcPDlL/hS2Pp1UjqSdpR0o2uIsmNsBmPrhyE2Cug3WDjbHc1SakgC6jIYhLKomnbav6q47oV9ZmywEbI8JmV7RuzmjGoFjID8M5bvoNJxmrmC9/F6VTQMSW+p62Jeakij1TQNZ1JoJLKfORv2BAVlfrdAeDLcW39vtsf0xJuXaKYqRzEhm36dp2tgktIrg+hRxVQ0SqTUJaggUp1EnM7O6Q9zI9WJ6XrWtVbmofv0h7NxqxmDLpaJonZIvm/WbHVpm5jNselqSBpkthVoaKmeH5bc5athAWo1W9dprdpPDriMrT3EB54ar9zgl4iVGRpKCNCDEAgANeeH82LTQc83bpJyrXr5lAnILWKGGYNGni9UsVdK5Y2jFvVw+OGC0+XXKUR7awRmDCrGYOM0OjQDP/DquXnKLuN9QrSs4yp6aph6mCS2FUmCB1TkCVmqYiGLVExZSzrUOwgjB+fkDaMVxHL8lhF64DVQRQufEC8EgPwd7tftc6KV60bkcHadJ/TNUZqJrIGDZzWXZKOO/mdCAUFItxw4empejlMN8xqxgCcKKdta2PkpY4sDi9dOOfQ3kMNNnuV9OxLOjNJ7K4mCNHckKaXsCxkVQfdGLuKHahOsdp7TjSe33/0mUzlLdoBsqi0SnUSazePKomdad3wvpYOqq/3JWMeyXyu3Tza4EDNu9yFrQkrjanrjoEls4IRiJj1jAFwTygDsju8xON43DGwxCrhy6d0ZiIYLiGaMnODylTDE6+8SoTrynHk3dGtlRirVDE4XHZaNzJtIXGr8O9DpUGOVeq7kzWj3IWLtmlzXABAzKOa2yz09/ezoaEhb9ez7RnLq9O2TdNb3VzdFwaHy0oTFw/Vc+li8e+OHcdZ5koVm5+8M5uxT1d0UGOJjgS6uZNJ8mlLbqjua/NOp0sBvZkAItoVtznQImgM0DuhVVFJWR1erQqJdTF/8clzPV1F/Oo75+P7Lx5SEti0pRcSv41urvhxn1wqgihqtZi8H1k5Ct5u7ovgtSN0gVJlYe5M5R58hLICJ9Z3mqAGk0YxGxlJs5951jAGH45PHj4cXs2Gi/lLjNc/PF7FP/3kMD560cJaz+GEQJtKLwAR8zX1zFaagrqKdePmbeqJY1dGG+fN7ayNR5VNPtPRQXIGwPsQElSqk94iqJL1nSaowVQmXHyelRtH8NmNI00LKfcNE9FvRfVZH4162h7JxKq6qqXhxrZNSNqpWYnqI7z5od11zX9U8frVSYYHntqPVcvPwcvrr8DImsuw5srFOK27ZOwWNtDXq2xwkhAJ1VwxZs4OlqE8VsHF67fjzNVb8aQim3ymY0oyd5XqpLKMyiRjVk2PdODXt2n922jUfIOqmx/araw83OpuiWlgok2Ae2c9H5gVGoNpYtNw4+no8FJ9hGKuho4ITzJWmx/Abe7WXLlYGhWybNGCWgz+yXGHOF4LWanQNEwguBXeA+pt3641imYCknBnmdnOBrKGSIB7UIMqyMOkzeQRUt7qkuOtMEfPCsagmsAkxE6E7eKyjQpql0QXm8giG3MCz1RdzAAyIiF2phurVOvKTSfHpzEDpSHq/H3yKJWQF4od5JSV3dNVxBvHJhpCdd84GlVbFR3WNjWWVI5u3fo3hZCmKeHhk2DmYcbhGY1NxYBWmKO9mJKI6HIi2kNELxDRasn+uUS0Md7/FBGdye27Jd6+h4iW+xiPiDQTmAc3du3Z7BsytV6GScaMC+PAWCWVJDPQV98ze8dzB6Wmgft37qvNj+24fYAvd9COAXuEqGw3cGKsPV1ROK7LNa5496mYN6fxnKRRjIiBPnVjeyDyZaQxjybX5c2Mczs7MLT3kHW0oAifBNO3GUc0HakgloJptjk6M2MgogKArwL4AIBzAdxAROcKh30CwGHG2LsA3AXgD+Nzz0XUI3oxgMsB/Fl8Pa9IQ1h8c2MbW2LeSD7CpCm5ruZLoUA1AiTDad0l5Ry5zJ2ut8FnN47g4vXbMbT3EE7SjIWHuYqNHpOM1Zi2qn1jK8EATEwx3L1iKV6884O48aKFGBuvKn0Gqmts2lVWnpNEMokY6OtVMqCsJaT4ulRjlSru27lPyxSSpSu+b98E07cZx0YDkiWz8t9tb3fJa7taGXyYki4A8AJj7CUAIKIHAVwNgO/dfDWAtfHfjwD4CkXNn68G8CBj7BiAl4nohfh6/+hhXDXwJgzbfAXT4nK1O7ZLiW9erdeZB6qTDG9980lGVT9r0TSTecu1JIUPIT/PqpsmJK1Oxc57PKqTDOu2jAKAtKqtDUzESWUu0RFE3VpWfS+Dw2Xc/NBu50ioTiJs+PB5dXWsCkS49ny/ZlvfZhzd/FF83byTWW3ggzH0AtjP/X4FwIWqYxhjE0T0OoC3xNt3CudKn56IbgJwEwAsXLjQeZDJxJrUU5uQtzR2x3bLZ0ggC1tMkDT6AdxbddpCVxsqD3QVO6yynXWNWvICn3eRZL6r1urh8Spufmh3bvOmElp0jNy2cm3yvQztPYRNu8qpwmOrUwxrN4/i2MRUXU2qTbvK6D9jvjcimqaMhk5oVM1fuyW8+mAMMu1dfNOqY2zOjTYydg+Ae4Ao89llgDx0nZM2XHee1YJKI/03y4HkksBm2+hHV8MoqySjqg2VF9qxBIZMGEn+VuV+AH6L88kgI/Q6Ru5auTZNxVoeqhpR67aMtqxdr0lobHW3Olv4YAyvAODr0L4DwAHFMa8QUSeAkwEcsjzXK5IXmqVzUhrpf9miBcbCeDzShMi5aDImW6dLY/Os4+drQ7nWY9IRzumAnq5ibZ5lfaBbCRmhty3yyMMUJu0bh8erdcUSXaKIslaMNQmN7RS+rkPmWkkxof8xgPcCKAN4GsBHGGOj3DG/A2AJY+xTRHQ9gGsYYx8mosUAvo3Ir3AagO8BOJsxptXhfddK0sGlnoxKHZRJ52JIpul4m2YsLrWGdDH6ogTrWsMo7fhV54pIyl/0dpdw6Mgxabe2nq4iuuZ01mLxXZyzzUaxgwBCnRYrNo5p+pgKhA0fUmvQLoxftX5ss6znzSng+MRUXThuqVjAScUOp/wKm3pbaddtgjNXb1Xu+8n6K6yukSeaVisp9hl8BsA2AAUA9zLGRonodgBDjLHNAP4SwF/HzuVDiCKREB/3ECJH9QSA3zExhWZCJSnLnIM6iUkmRTCou6aldVTr8jVEuNg6ddcVexvoxn/zQ7uxcuOIlpCoch12PHcQ5bFKXU2k8lgFxQJJY/gZQ+0eF6/f3taMQZZ/UKlOSus/5YGO+EZ17FVxY5Eh3BUXQNQdd3KpiGKBGhifycmeoLtrTk0g09V70kGst9XdVQRjUZMeXuDLGiCiYnamrm/tVv8pVFeN4aoZyBaq6kWqpHMC8LJEirA9Xhzz+PEJqQRFQMMH7CId2cST8+faZAy7SmK6cSQx8OKzN4uwzlSIQoJLAx/xuKQ7nFhXyybZS/WdJPfiv4EjxyakQkB3qYhjE1NKJqLT0HT3F5FGY3CZr6wI1VUdoNIMTNE6ti/K1fFsUyZALDFdHqtEJgkJGBpDCV1snbrqswl4yco2w3rtZjcnoUpzST4ekTHMJqYwb04B48flPYrTQpzvLL2Vq1MMXXM6MXzbZXXb+e+o7/bHpIKNLkBD/A5VzItIH6Kry/h3CRDp1WjiKqjmK62fxAdmRRE9E1QLXqX+uSyUweEyjhybaNiuMz3pMh2ThS/7gHQlEWREVcxCVi06McHGdA/bhMKxStUp4U/HSFsd9ttqHDk+ic7YrOYL4nzbBl24hrMC0XeSlOPgUSyQc8QOnwzZXSrizmuWYMzCFyErIJjU27KtVJAmS9lm7eZdNE9EYAzQR01kSUVPiLio2vZ0FbVmFF2mY5raMYAdM1OV7BDVdVVZBD681TbDmodp8cs+ujSF8mYqqpMMbzqpUyudqmCTQWyT6T44XFYKD7o1qKroO29OZ8N3olunotB0bGLKeO8EyXeWzJ/oz7KpVJAmS9lW0Gym8BNMSdA7Ym19CSofhYyId0kWuwiVqcq0OGS2VNtMbl0SEr9d5vCVpfHbZFi7PJ+Ywc5/uDYmFJWanxZdxQ70zJubC2Oa29lRI2ouGBuvYvi2y3Dr4LPWGePFDsKKC06v9dlQrXObGPwN2/Yo38WRYxPSYAVA/d7FsiS60GmdqctkDk2eQ5cIm6W4ps65bGOqBZrbwyUwBuhzDGx8CWl8FGmhs9+XigWsvWoxAPc4aZckpOokQ1exA2/9dydZ3UPmz1A5yk2L3zaDXUTyPtOUX1BhvDqF8ZykuFKxgOv63+GcBNZBhLNWb0WHpZYGwJjYKUbzzO3sqIvmsSHwwImENFlzHdW67iCq61utI/46U5e4BmVRSTbPkebbNeUB2SR5NjsJbtYzhsHhMjbtKte9EAKcaq7ofBRZnVkiVNKFmKTn6qRyTUIar07hDzJESlzx7lOdQn5FuHygfF7GdEmIe71SxR0DS2p5LrZaF18ewga93SWn8i+Hx6soFQvKMFWbwAOg3kSz6uHdWHHB6dLQVb7/x0Cfvv2rKl/l5Nj0mWfAiA42Tvsdzx20zitqBmY9YzDlGNjEF5t8FD7T37NmTqqeR/Uh6JKQbOO7ZRLTpl1lXHt+r9F8oRq3TdJaEkqZjLVdmIIpdBKICJD4zOJ8JfkdB8Yq6EjRktNmLbrG9tuaRXhUpxj+bveruPOaJVKNzibi7bTuEsbGj0uv76A81eCzdIWN9qE6htDYG6MZmPWMQZe8ZVsKwoePQoSpR3Ua6UH3PKoP4drze5W2alupXUVcdjx30GrRy8Yt83PIYr8B+ySoZqAQRw2ZypEsW7RAykxVjsyzNPHzMtiWgXE1qcgEF1VuAY+xShUDfb3Kbn3J/VRm32WLFijXqU1Eks1zpJXabbSPVjTj0WHWMwbVCyFE9ZRkBG3dltG6BaKTLlREXEf4ZYRw1SO7sXbzqNImagOd9JcQaNmYtj7zqpU/QPVMWe210jjvSVZX9kI1Jxev3942TCEp7aAjkgUi3HnNEmODGHGebU04CUyO7eRdqnSQrLkFMly8fruy1Hl3VxFL1z3WMHeJ2VdVRQA44Xdx/W7SCmAibLSPdiuuN+sZg6paJIO6z+3h8WqdQ8xVujBpIipCyDvv0iS8mAi06kNQ9WrmF63umVREq7tLHvYqQkXwkggcGRLCljZiyLaOTwJVljWfdW7jMJ9iTCs5i4ENujItxY7oGWTpLTpzkImQuxIsMZpMhWRfB9U3/ikWCG8cnZCGsyZmX52QIfY058fUDNjQB58aig+EkhjQp7GrkKV+uopAdJeiFo0uFUZdxuBaDI+Hydeiu/aZbynhyRcPNezrAPBlhROTv6/KN+BSxM8FLnV8knGIvasTFDuAiSl7p2zyTK6F52RmS1XkVwK+1AP/fnX+ip6uItZcaV+JWMTgcFla9l5Ed6lY045NpqikwU1e340L2q3mkYhQEsMBaeLbs4ScKks7VNxaNLqOIYu6alKrdb4a1dxOwezATjqVyZBkgosf4trNjSZAGRKfxOHxao3g8hEgSSlwHbEsENV9/KKdOyn86to5UPWuVM9VHmss02LyO/AlVvh76TQlvgUnr5XJ5k8GW+1h3txOjKy5zPo5XJzeeSWKpSlP364IjAHpIimyOIVc7cG+xpCnupr2mUwfqamssswXY5JGgYigm+L3bZL0+HDKrc+8aryvCiJBVb0rHUFduu6xOh+UKeclYUIu2fS8n0PGTERiqOtvoDOt8evC5jl85sqkhY/qrO2CwBhg1y2LR1ankCsj6ukqNthYXUtz5K3epmGuQLaPVOWLsUFiy7dFcqwqnDJLKKyq6qZKS1N1UBN9UCpzmGgOcpWgky5+qnetYh6ybmY23eBUa0t8Dhund5Zv1/Qd+UyKazVCraQYA329yhozPV1Fp9onNvcS66n0KByxvd0lDN92GTZcd16qMSQfh0uxOh/PZINiB9XMQaraNyp0l4qZM8hV91VhoK8XU559cq51jQb6eq3KfyThwOI6u3vFUgzfdllDuLULbIoWqpgHzzQG+nrxq++c33CurLyKzXOIGOjrxbXn99ZqdRWInBJXedh8Rza1pKYLgsbAQWXXzeJsU8FVukkbOudDvbXVOPgxmiJwSsUO3HnNuwGopUpdQb3fOO/UWvMeV6jyBGzswb7MgMk40kivtj4xXXl4scyFrPaVriGVyUegYx7lsQpuHXwW/WfMxz/ve71un6rqgGgq4pmLCklVAz4bfNOuMvrPmO811Du5VruFnGZBJo2BiOYT0eNE9Hz8f4/kmKVE9I9ENEpEzxDRCm7fN4joZSIaif8tzTKerJBJJlm1g1bfO6t6m1bjUJXe7i4VcfeKpfjRFz+gDM011b4BohBF2T2KktLTxQ5CT1exbl53PHdQet+bH9qtfTbbkuImEE70pOi7/TFrrcVlDCpJVXynh8erAEXvhp+j/jPmY27nCRLBVwXWjSEhhjpJ+b6d+3DLo89oqw7oxmyzDk0aiwtsvqNW0g/fyKoxrAbwPcbYeiJaHf/+vHDMOID/zBh7nohOA7CLiKBN6mAAACAASURBVLYxxsbi/asYY49kHIc3+EpqaZd7Z82oTKtx2Dq6dR+cTjqXFUbjs51N91XlCYi1eXTPlUVzSGRzPgrNVmuxHcP48Ym6fJsEKt8MHwkk02D5iCRxDKqoJJ3fSdarG5CviTTr0KfN3/Y7aiX98ImsjOFqAJfEf38TwBMQGANj7Mfc3weI6GcAFgAYwzRAu8clm5BVvc3ycdl8JLoPTueclBVGE8MndbZ4HdMxEZzknlnzJVT35jPrZesPOMH4ZIEJCQ6PV6WMxuad2hBi0/t1DepIIBNa0qxDn2UmZN9R0gvknbd8xypUdzohK2N4G2PsVQBgjL1KRG/VHUxEFwCYA+BFbvOXiOg2AN8DsJoxdkxx7k0AbgKAhQsXZhy2HdLEJbcbI8kaourycbk+++BwGePH1d3tBvqicsSyGjhHBGlYFYuveme62jqAPeMD6ufWh/8hyawHJOG4D+8G6ET01eHxKooFQreiqGBiHlu5ccQYxsq/U51/4OL1263f8UBfr7LUeQcBczvtikymIfImoejWwWdrZc0JQFfcHtWUmSz2Aml1ZnUeMGY+E9HfA3i7ZNcXAHyTMdbNHXuYMdbgZ4j3nYpIo/gYY2wnt+1fEDGLewC8yBi73TRo35nPKrhmCt86+Ky0wJeNndEXQ/HNmGyfyaZRvOjwlEm6suJuql7A/HswObvFd+Z6vC1c+0To7g/Yd6frjZ29poglnVOZf1eq5xBLf9isb1XToBsvWliXRGhqhGVaXzKovgdTIyPdtdOunXYQGr1lPjPG3qe5yU+J6NRYWzgVwM8Ux/07AFsB3JowhfjaSUbQMSL6KwC/ZxpPM+Givg4Ol6WNNmzs8VkyJvnFdnKpiCPHJ2rSZFYJxqVXhcn0IKvrL8O8udGS5KVS1bE2ZYvF/TY1lLJEkqTN5xDhage3ZSB8GKuOSKlMJ2nWd9JTIpHOC0S44cLTa9tt1mZazVdl7nrgqf3a83TPZXo3sncx3bKis5qSNgP4GID18f9/Kx5ARHMA/A2AbzHGHhb2JUyFAAwA+EHG8XiFSn1liAiXmHGpktZMCymtg1dcbCpTQtrMS1OvCh4mJmqbXSsrEqcqUGdTtpjfb+MTSKqbZtHWKtXJOmdsGg3iNEeNwQWy8hkiXMxkNkyMbzqUFmkduzJJ3aZAokoANPW+oPg4fqzTLSs6K2NYD+AhIvoEgH0ArgMAIuoH8CnG2CcBfBjArwN4CxF9PD7v44yxEQD3E9ECRHM5AuBTGcfjFTrpT+T4JieYDmkdvLbENm0imMu4TDZglzHImJHMhGEqWywea5ovW7NfAp1pLGnSpIv5VxXEo/h5gMaonmIH1fkY0qBg2blGJMQqE4ppfbfShKKS1IkAE2+QlZW/5dFnjUyFobEGWBam2gpkymNgjL3GGHsvY+zs+P9D8fahmCmAMXYfY6zIGFvK/RuJ913KGFvCGPv3jLEbGWNvZH8kf+DjkmXgY6JVHwf/kauQNmPSdlGlzbx0GZcsrp0n3lmzPxlQFx9+7flRDkQS/w+g7l0lxI+PJdfNl2vMuSwXQPSXJOtDNTc3XHh6w3YC8NGLFtaI8p3XLKnLip83txMrfvl0pwxzEZOMOeVNJDC9YxGDw2UsXfcYPrtxJPfMexVUknqpU0/6ZN+tS00pfq0NDpeV76pds6JD5rMByQd61uqtUnNGeayCs1ZvlWaP8h+5DmlDSm2iYHzby1XXM9mAs9reeYeeSgq885olWoexrtOeq6PZRVvTzY2N85XPHxirVOs6udk6iUXwRBrwb+fXme18mVBMmsjgcFkTkjyFGy9aKPULqr5bF+meJ/gqM7ON0NgqBMZgCR0RTiTGJGzQtcuaq2NtcLiMtZtHpT4FWXvLtB+g67h0NmDZtWxaPgKNzCitvdZnyQJXbU01Nya7uepZ124elYZOAmamIF4rDzu3iXG6hr2KMDlzk/0qnNZdqvk9VLki4vh03R51Zk7VWmFoT8czEBr11MGl3aYKeTYBScax6uHd0oQm2z6+PseSxXZsM6eJ9MY7LlXaG994xueYZefYZD4nPgsgfR6J6llFJMQpjbPbZt4A9fuSrTvTuNOEvfIwhZLrQkpN91KFxqrCfJPWommaWOVJK2QIjXocIZNAVm4cwdDeQ7hjYEmDxJs2AikrMd2wbY+UKQCR/bmVTj3XvtTinMqiPWRRUFkyWvms5Q3b9mDlxpGaH8DGJFIeq+CzG0dQKnagWCClE5gINaaQJUzRNmkuYQomoqi6hw1UWsBYpTHDWjfutGGvPEyBEbrv0MSAVFqaTZivDNOxuF5gDDFUoZn379xXq8bIq/1pIjR85CvoPvhmRjio6u249qXm51TVqUt8rqwfmst7UBHDSnUKxQ7CvDkFHDneuJ+xqBSELPLIlgiqMsNVSObJxZ+TzJuNwKJbXzaVRoGoEJ9NXooJJuFA508yzbuO6aQJmU2bg9FKhH4MMXR2QFk1RtcIDSB9tUc+AkYH1wgH134EPGw+YtdKlrZRUHy0WJoqli7vQfec1SmG7q45yl4agLpNpo1mecujzxo72PHg/Rm2vTF4rcYUOeQSJSd7R0kPBVWUn03Ya7Jex49PNFbRLRCOHJtQ7rcVHlTj6CBK9a0A0Xw8ufpSvLz+Cjy5+tK2ZgpA0Bhq0Km+su1ppIA88xWKBcKyRQusHXpZMzFtTRwuJg3XKKi0H5eP/AzTtUyQxcjza2n8+IRTBJdK8r9rxVKlpplIzxev327lzDdpIraVRtNofLLMeT7YI8kjSTTWLMEgqueciTWRVAiMIYaukqcskxFwJ05pbeMm4tPTVcQV7z61zjFmWrxZMzFtzRW2yVT8OG0iRMQxuvhuTO9BLDOi8yWkyVAWiaCMSZtQLBDmzemsI3qA3J+ha7gD6Avm8Ujmc92W0QZNxsWUl0aoUpkuiSLH+cXrtzeMSSwlbgsb31c7Zy37QGAMMQb61JU8ZZmMaeA7X0EsIudC6LPWqrd1xtuUHhCvK8aimzQbV+1n1fJzsOqR3Q3EvjxWQd/tj9VlMI9VqkpfAv/ubEptTDEmJYIuiVMAlOWdVWvA5DTVhWGKApHovE9rM3cVqlTMMqlE67P3gjg+W9/XTEJgDBzuGFiirLjoYxGkdULZMBTXgn+qei/dXUWtdK6K+VaVVrbpaawjMjaaTRrtZ1IR2SWz6VenGN7aNQdf+o/naN+dKq/AFB5pu7bSXkeWZJf4Uwb6epXask4gymLKc0WSOawSMTZs2+O194LsGnldu10RGIMAVRy4r0WQV1SD7eLV1XspFghvHJ2oEUde8gbQkFSXhKiCyTUDG23IJO3bMDwX3xAQzaOCLyhhikjh97lK06p3110qYt7czszXkRUQFOdZ1UynFVKxOH9Hjk1ocyIOjFVw14qluYWETsdw06wIjEFAuy4CE0OxHbfKbFGgyGYtZiInWbbHJqak56ls7zZVSgeHy1JNg5f2VcSug6hm5lAVo1P5N9IQOxfBwJX5q96da7Kibg2YtKq8BSJbpPG3JJqNKdEsLaZjuGlWBMYgoF0WgavUKRv3skULaklcyTVURHGSMWV5CpuyFSKmGDMyBV2lSlNMPt+bWXUN1XbbiKoEeQsGiX+L71cg63lhcx1AvnZVPa5185z3c6syytPU0yqPVepqSPlGM01n7YBQEsMTfJYWTtutyuYaJxU7nGLj08KU7u/SBUulWSTHAXLJUtdJS+Z8lqEZZUZ8vG8TbMoyNLM8tuqZszY4cjW/2aIduq/5QCiJkTPy7Jzmo6mH6hoE5vQBJkX5VMykWCBMTjJMCeeYJE2dOUeUVE0Sr8y+DACHjhxD3+2PNRQU1IVdipCVGRGLGPZ0FbHmyvTMoxlNXFQ9rpctWlD720Yq9kUgVc+sMgv2dBXRNecEwVcJFWOVqnP2vQk2BftmAtPgERhDCuTdOc1H6J3q2PG43LCu3y2PYoGw5srFyvIGV7z7VGx8ej+meOlbkbrAf0CqqKjENwHU5y50K0opnMaVOBAJfaU6hUpcslr8mEUiqCr6Js6jrIjh4fEqbn74RJ2o7q4iGIN1YpXvUEsZZF33gKjF5f0791mN02d7Sp1JUxRcSsVCA+O1rQeV5TvUlaFJfG9AtlpY7YpMJTGIaD4RPU5Ez8f/9yiOmySikfjfZm77WUT0VHz+xrgNaNvDV+c0VUmKtI17bI/d8dxBqzBSIGIkAJTlDXY8d7DBJFOdZFi3ZbTu2W4dfLau7IIqiumPP3wegMYSDW8cnUCxoC5xMNDXi645ejlHV6LDds5VRQwnpyIfTVKCPfnb1JgmCR12GVMa6AixbQOdtCVdZFA9W9KEKQkcUPlbZCVpVEjDYG3K0IxVqli7edTbnLQTstZKWg3ge4yxswF8L/4tQ4Xr3nYVt/0PAdwVn38YwCcyjqcp8NE5TewAxn+YaeowidAde2Cs4vRhJRKXrNaLai4Oj1frnu3+nfuU0VBivSNplusUw7w5ndr6SDbvRXWMbD4I9aYW23uIUBEKnQPet+PXhsmYCJpPzUa1xpctWoBNu8q1OZlkDBuf3o+l6x6rE6AG+uzrQaVhsLbCnyowY7onv2U1JV0N4JL4728CeALA521OJCICcCmAj3DnrwXw5xnHlDt8dE7TNWCZN7ezzt6qynTVYaCvV9nMhze/8H6SNIvcpSy0DFOMNfQCUN3v9UpVW97AZiwqIpFEBvEdvRiATbvKteq6ALTzpIPsmXTE5+jEJD5rKAnuAtsSJmnedRrCq4qgcq3am1xH18lO9R3qfANZCft0T37LqjG8jTH2KgDE/79VcdxJRDRERDuJaCDe9hYAY4yxpK7wKwCUq5+IboqvMXTwoNxe2izIpJ1iB6Gnq2hd7VO18MYq1doC5xvKpyEMa69arNU8eC1gZM1lyiqhukXuonnIIKtYmdaUZhqLiVnveO6gtE9AYksGgOrkFNJANnYd8UmUCBsTj02VXFHCVuV4uL7rLJqNTAtNW7VXpfGpWuvqNHbAnrD3dBW9zkm7wKgxENHfA3i7ZNcXHO6zkDF2gIh+AcB2InoWwL9KjlPGDzLG7gFwDxCFqzrc2zt85DrYStpZnGeu45Q5mU2L3LZmEiBv0CKrWLlq+TkNDl5ZpJNM4uNrAp1cKoII1m1Odcz6zNVb0V0qSnsvmKCaQx9rwMUhLGZnyzSII8cmGuoj8fPc3VXE3M4O54qltrCdkwNjlYb375LgZooEs9GwEqd4cr2ZFJWUKY+BiPYAuIQx9ioRnQrgCcaYllUS0TcA/B2ATQAOAng7Y2yCiH4FwFrG2HLTfdsxj8EVtq1CAfvWiz6QNfROp9J/9KKFtQ9XFZWUmM3EPINigbDhQ+dpCVvW2H/Xzmc8+HBK26gklzUAyIvnqcZcIMIff/g87VwMDpeVlVKTeWxGjoU4Jps56S4VG7LxXcZl0x5W/BaWLVqQS2Z1M2Gbx5CVMWwA8BpjbD0RrQYwnzH234VjegCMM8aOEdEpAP4RwNWMsR8S0cMANjHGHiSirwF4hjH2Z6b7zgTGAMhr8MtCMlvRGzYNVITGtW+zbTVZ3310B4fLyppBJty9Yqn3kEgZROJn6q1sStBL2ztZVy2Wf640RFTUUPhqt4A+UdP2/bdTH+ZmwpYxZPUxrAfwfiJ6HsD7498gon4i+np8zC8BGCKi3QB2AFjPGPthvO/zAD5HRC8g8jn8ZcbxTCuINtY1Vzb6BAiReSBN16hmQtVxrLtUxF0rltYxBUBtw03svTLwpp48Yv8H+nq13dhU6C4VU0uOyRq4e8VSK1+NaF832cKTfsyqtWOaxzRhrib7vQn8dzF822XYcN15DdFoY4rERNv379tfMtOQKSqJMfYagPdKtg8B+GT89/cBLBGPife9BOCCLGOYSeDt9WL55nZPnFFF2MgyhwG3vsQJeCKoig46ueRO2HmokvlUSIrdZYVLNjZP/JYtWlAXSSVDpTqJdVtGpe8hbe9k8fo3P7S7VpNL1n3OxVcm0zZEKV6lZZlCxE0+CcDcFGo2IPR8bjJMESSJtNTbXZJGyLRr4oyrBM9HydhAlOZUjeEcGsZZjUu8nC76LEsPbQAY2ntIKQnz4DvNbdpV1jKFBElDGxHLFi2Q5gGMH59Q5tTIwGsQKsamk+aTuTtz9Vas3Dhi1DZcJX6ZFrNpV/R8icYO2PW+ng0IjKGJcFGxm1EmwSfShJgmTNBEywloyH5VEVAbwmpCMq6frL8Cd61YWmfG2HDdeRi+7bKGRL+s5pPB4bJR8gfqiZ9rJVJRqNAxlsPj1ZqGmpQoyQrVWhCzjG0EIjH81hQibpO17TOze7oj1EpqIlyKpU23rlFZyjabzBUMjbV+bObHR3Ez23LLaQvhuTqf+TpSrhFUolBhYizJ+J9cfanTGGXQrQUbBicTiFxKYdsIWtNNGMsTQWNoIlwWXp7OsawmDxlcJTgeNuYKcY5M85NVgndFGqJiU49HxNrNo1j1yG7tOSoNTBQqXEqIqN5Rh+Jm3aWi9VqwGUdWgchGo/VRo2ymIGgMTYSLFuAjiU4GnxUyZWPOmoinInjiHJnmpxmlrMXxuWp4aZrSmMpxlIoFXHt+LzbtKhu1N5cSIqr5BqB01tuuV9M4fAhENhptu3ZvbAVCo54motnJQjK0e/x2ljniTUe6VZ0m58BklkozblMOgiv4BDgbM5opmcxl3k2JcqbzxXEkEXlp6oTp7mMzJzMti5lHaNTThshLC3BBu9tR086RSxaxrYbE2/9NocNpxq2TlBOCuHLjiBXzSBh7YiZMxnCXhgnKCim6lBDhr7Nh254GxmCrodnMXbP8RWm13pmGoDHMMrS7xpAWrs5Y0/PKGvK4XsME1T348h+2z3XjRQvRf8b8lmmkNiUm0qIdNO2ZgmZlPgdMM6iqUE6H7GodXDUe0/FrN49qmUKae4oY6OvFm05qVNqrk6wWImmbR/B3u1/V+lXyCDjgkafjNm0Yad7PPJMRTEkzCDbqdjtlV/u056rMMqoewqYMWZueCz6Inqm0QzIfphpOY5UqXleMOXmvebafzNNxmyXia6a13GwWgsYwQ+ASnmmTXZ23tOU7nFQVvnrDhac7Z8iueni38X5iaGzaubKRtAf6etFtUepDda0CkVbi9vGus4Qrm5BGGwnJatkQNIYZgjThmSqJqxkSpu9wUp0Ds/+M+UrNRNRaDh85ZjQhidE/WeZKVutIxrhMpT56uopKqV3lkE96Gvh613k5btNoI+0eZNHuCIxhhiDNh6Azv+SdA5BXdVTZ+MQGNRu27cHKjSM4uVTEkeMTtb4PJiev2BMCyMbgZCUpZOU/AH2pj2KBsObKxUrmqCs454tB5xnm6TPiazYmq6VBYAwzBGk+hDQSpi+04sMVpWPX3s0iUwCyMTgZUZaV/wD0TJwfl4wJir4k4ITEvVLhu3B5182w57tqIyFZLRuCj2GGIE0JDZVdWFXx1CfRVvXNHj8+YWXrTmMXT5NpnGDenIKybLUMNnOVtUQKALxZEtUEyAvTJdYo3v7vI5qoHe35efo8ZgOCxjBDkDYxTCWJ5S1tyZKrjnAd7HRSZ1oJNYvGM358Emet3towr3kUD9SVSBEzjJNGPPwxgFobEXMvXMcvMxm1qz0/JKulRyaNgYjmE9HjRPR8/H+P5JhlRDTC/TtKRAPxvm8Q0cvcvqVZxjPbkUQbiSWhE9hK2c2StvjxzpvbWdfjGVBLnWkl1CwaDwOk0VO+iwfqiPJAXy+65jTKcrJntyXWLuNXRZJ1K7reBXv+9EVWjWE1gO9xPZ9XI2rXWQNjbAeApUDESAC8AOAx7pBVjLFHMo4jwACdlA3INY12LdWRVkKVScfFDjJGIYkQnbM+igfaanm2z+6qjdiMX8WQ53Z2NPimxHDemVx/aCYiK2O4GsAl8d/fBPAEBMYg4EMAvssYG8943wBHqD7qtZtHcWxiquWJQCpCJpNG0zquXaN2dPBlJnFlKrbPnofzVfXMr1equGvFUinxD4lm0xNZGcPbGGOvAgBj7FUieqvh+OsBfFnY9iUiug3A9wCsZowdk51IRDcBuAkAFi5cmG3UsxCqj1oWmZNneWoVVi0/B6se2d1gTnrjaNRikh9LFqJn61NJonjSZE7nCdtnz6Ngo44pqea12eXPA/zAyBiI6O8BvF2y6wsuNyKiUwEsAbCN23wLgH8BMAfAPYi0jdtl5zPG7omPQX9///Sr/Ndi2NTe59Fsx+FAXy/Wbh5tYFTVKdZARHwTPd31VAXc8gx71JleXJ7dtzmwnRLNgnkqXxgZA2Psfap9RPRTIjo11hZOBfAzzaU+DOBvGGO1Lz/RNgAcI6K/AvB7luMOcITso9ahFRKxqtZP1raONhAJbuLMzUPy1kFmelm5cQRDew/hjoEltTHlTQR1hLfViWbBPJU/spqSNgP4GID18f9/qzn2BkQaQg0cUyEAAwB+kHE8AQL4D/zkUtGaMSxbtCDnkTWildmqJmLTLEIse34G4P6d+9B/xvymFTf0NRe+fR2Dw2Xc/NDuBvNeME/5RVbGsB7AQ0T0CQD7AFwHAETUD+BTjLFPxr/PBHA6gP8tnH8/ES1AZNIdAfCpjOMJ4CDL9BUzYFXYtKvcNEKUoJXZqiZbeJ6mC5smQyweY57vQ8ec+JBYl3nwqXEl8yTz+QCtz5uYScjEGBhjrwF4r2T7EIBPcr9/AqBhJTDGpm9nmGkAVZKTDXNohQTWbLMND50tPG/ThW1Gdp6Ez4Y52RRXVDHQZsxTyJvwh5D5PIOhIiRJBqysho7N+Wlh2y8iT/u96v46M5ZLZE0azcJ2nvMkfDbMiQCjVpUnA9XNU6iD5BeBMcxgqIgdXxZBZbNNzvcFFdEY2nsIO5476FVDkBFnAFqipTNj2RaaS0sYbSLG8iZ8NszJJEDkHZqqKyQY6iD5RSiiN4NhU3JhoK8Xf/zh85wL8LlCRTTu37nPW7MeQF22Yd2WUW0ZDV1pCNtCc7alOsTSJMsWLZAWFOzpKjatAFwWISA5N++aSar1/Mcfbqx6G5ANQWOYwbC12TfDtq8za/HIKmGqiLNNKXGVGcvWKW5DGGVaxaZdZVx7fq+15pSHI9w1nDkBPw95R5W10gc12xAYwwyHrc0+rW3flki5JNhlkTBdz7UhWrYEyUQYdaGWO547WFf1VIW87PjiM3YoMr57uoromtMpnYdmRJWFiqnNATFF6Fc7o7+/nw0NDbV6GLMeqqxgmdlDdqzO8d2bUhq8eP12ec2lUrGuJpRurGmhmw+gsewGDwLw8vorjPdQPZ9YTjsrXN6teF6Q6NsXRLSLMdZvOi5oDAGp4eJslEndyxYtwKZdZSmxTCsJq3oor71qccP9fRMtnWZx8frtXkItm9X7IK3ZJkj0MwOBMQSkhiuRkhGN/jPmG5OqbAmNTQ/lvImWijD6CrXM244fJP4AIEQlBWSAj7aQA31Rsx5S7HeRhF16KKdFmpaigHpOXEMt07RwtYUqoitLlFjA9ERgDAGp4ZNI+WAyeZtZshBOVe2pGy483Uki14XVZkU79m4OaA2CKSkgNXyGD/qIaMnbzJIlgUultaTRZvKy47dr7+aA5iMwhoBM8EWkBvp6MbT3EB54aj8mGUOBqM43YIO8wyWzEM7pQHRbWd02oL0QTEkBbYHEcZzEzk8yhk27yk727TzNLEA2c5esRantuTzS+jhskKf/ImB6IWgMAW0BX3V28gyXTKuRDA6X8cbRiYbtxQI5Ed1bB5+tC8X1XaQuZBYHJAiMIaAtMB1MLa6EU9ffAADmzel0CsUV8zMA/+XRQx5CABAYQ0CboNXd22yJvS3htOlvoGplKsOGbXuaVh49ICCTj4GIriOiUSKairu2qY67nIj2ENELRLSa234WET1FRM8T0UYimpNlPAHTF62yb+cVu2/T38BHKK7rdQICbJDV+fwDANcA+AfVAURUAPBVAB8AcC6AG4jo3Hj3HwK4izF2NoDDAD6RcTwB0xR5O45VyCt23yTFpwnFlYGA4BwO8I6srT1/BABEqrxVAMAFAF5gjL0UH/sggKuJ6EcALgXwkfi4bwJYC+DPs4wpYPqiFfbtvHwbumqyaQoEyhzfBOCjFy0MPoEA72iGj6EXwH7u9ysALgTwFgBjjLEJbrtyhRPRTQBuAoCFCxfmM9KAWYe8fBuqCKa0WlCIGApoJoyMgYj+HsDbJbu+wBj7W4t7yNQJptkuBWPsHgD3AFHZbYv7BgQYkVdSXB6EPEQMBTQLRsbAGHtfxnu8AuB07vc7ABwA8HMA3UTUGWsNyfaAgKYhT0k8EPKA6YpmmJKeBnA2EZ0FoAzgegAfYYwxItoB4EMAHgTwMQA2GkhAgFcEAh4QUI+s4ar/kYheAfArALYS0bZ4+2lE9B0AiLWBzwDYBuBHAB5ijI3Gl/g8gM8R0QuIfA5/mWU8AQEBAQHZEVp7BgQEBMwS2Lb2DEX0AgICAgLqEBhDQEBAQEAdAmMICAgICKjDtPQxENFBAHsdTjkFUXhsu6FdxwWEsaVFu46tXccFhLGlRZqxncEYk/eZ5TAtGYMriGjIxuHSbLTruIAwtrRo17G167iAMLa0yHNswZQUEBAQEFCHwBgCAgICAuowWxjDPa0egALtOi4gjC0t2nVs7TouIIwtLXIb26zwMQQEBAQE2GO2aAwBAQEBAZYIjCEgICAgoA4zhjG0a/9pIppPRI/H132ciHokxywjohHu31EiGoj3fYOIXub2LfUxLtuxxcdNcvffzG3PrWe35bwtJaJ/jN/7M0S0gtvndd5U64bbPzeegxfiOTmT23dLvH0PES3PMo6UY/scEf0wnqPvEdEZ3D7pu23i2D5ORAe5MXyS2/ex+P0/T0Qfa/K47uLG9GMiGuP25T1n9xLRz4joB4r9RER/Go/9pGYjWQAABPtJREFUGSJ6D7fPz5wxxmbEPwC/BOAcAE8A6FccUwDwIoBfADAHwG4A58b7HgJwffz31wD8tqdx/RGA1fHfqwH8oeH4+QAOAeiKf38DwIdymjOrsQF4Q7E9lzmzHRuAXwRwdvz3aQBeBdDte95064Y75tMAvhb/fT2AjfHf58bHzwVwVnydgsd5shnbMm49/XYyNt27beLYPg7gK5Jz5wN4Kf6/J/67p1njEo7/XQD3NmPO4uv/OoD3APiBYv8HAXwXUbOziwA85XvOZozGwBj7EWPM1MG91n+aMXYcUR+Iq4mIEPWffiQ+7psABjwN7er4erbX/RCA7zLGxj3dXwfXsdWQ85xZjY0x9mPG2PPx3wcA/Az/t73zCZGjiOLw9ySYHIISFWUVxCwEvQgGgogeEjTEP4eNosgIQdG9LHjNRZKDCKKecsohURASYYWsBEYMhOgmeMkSclAWBXWjF3VNQCXehiV5Huq11Mz27FQ7VW1i3gdDT7/pqv7tr2u6uqtr9sHIX3X+C2rbzRp654AnzKPdwMeq2lPVn4Alq681bap6OmpPC4SkWG2Q4tswngROqeofqvoncAp46j/S9RIwm2nfI1HVLwkXh8PYDRzRwAIh4dkEGT3733QMidTln76HhvmnG3KXqi4D2PLOEdt3WN0I37ZbxgMisj6TribaNojIeRFZqIa4KOtZE20AiMjDhKu/C1E4l2/D2k3tNubJZYJHKWXHoWn904SrzYq6Y9u2tuftOM2JSJXtsaRvyXXbsNtmYD4Kl/QshWH6s3nWRga3bMg1kn+6ia7UOqyeCeBBQlKjijeA3wgnvcOE5EZvtaztXlX9VUQmgXkRWQT+qtmu0dznzL4dBV5R1asWHsu3wV3UxAb/1iJtK4Hk+kVkD7AN2B6FVx1bVb1QV76Qtk+BWVXticgM4a7r8cSyJXVVdIA5Vb0SxUp6lkLxtnZddQx6jeafXkuXiFwUkQlVXbYT2KU1qnoROK6qK1Hdy/a2JyIfAntTdeXSZsM0qOqPInIG2Ap8wpg5u3NoE5FbgM+A/XZbXdU9lm8DDGs3ddv8LCLrgFsJwwEpZcchqX4R2UnocLeraq+KDzm2uU5yI7Wp6u/R6vvAe1HZHQNlz7SlK6IDvB4HCnuWwjD92Ty70YaS/sk/LWEGTQfoanhyU+Wfhrz5p7tWX0q9q8Yy7aRYjek/C9TOVCilTUQ2VcMwInIH8BjwbWHPUrXdDBwnjLceG/gsp2+17WYNvS8A8+ZRF+hImLW0GdgCnBtDS2NtIrIVOARMqeqlKF57bFvWNhGtThHS/0K4a95lGjcBu+i/ky6qy7TdT3iIezaKlfYshS7wss1OegS4bBdC+Twr+XS9zRfwHKHH7AEXgZMWvxs4EW33DPA9oYffF8UnCV/YJeAYsD6TrtuBL4AfbHmbxbcBH0Tb3Qf8Atw0UH4eWCSc2D4CNmb0bKQ24FHb/9e2nC7tWQNte4AV4Kvo9VAJ3+raDWFoasrebzAPlsyTyajsPiv3HfB0gbY/Stvn9p2oPOqOOrYtansH+MY0nAYeiMq+Zn4uAa+2qcvW3wTeHSjXhmezhBl2K4Rz2jQwA8zY5wIcNO2LRLMwc3nm/xLDcRzH6eNGG0pyHMdxRuAdg+M4jtOHdwyO4zhOH94xOI7jOH14x+A4juP04R2D4ziO04d3DI7jOE4ffwPscO2JOYgK4gAAAABJRU5ErkJggg==\n", 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Generated 996 random numbers.\n", "Generated 5 random numbers.\n" ] }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "n_rands = int(1e3)\n", "x = 2*nprnd.random(n_rands) - 1 # generate random x between -1 and 1\n", "y = 2*nprnd.random(n_rands) - 1 # generate random y between -1 and 1\n", "distance = np.sqrt(x**2+y**2) # calculate the distance the points are away from (0,0)\n", "r = nprnd.random(n_rands) # generating random distance from the center\n", "x = x/distance*r # These two lines keep the angle defined by (x,y) and move the \n", "y = y/distance*r # point to the radius defined by the variable r.\n", "plt.plot(x, y, 'o')\n", "plt.title('Putative random numbers on a circle: Method 1')\n", "plt.show()\n", "\n", "\n", "r = nprnd.random(n_rands) # generate random distance from the center between 0 and 1\n", "phi = 2*np.pi*nprnd.random(n_rands) # generate random angle between 0 and 2*pi\n", "x = r*np.cos(phi) # calculate x\n", "y = r*np.sin(phi) # calculate y\n", "plt.plot(x, y, 'o')\n", "plt.title('Putative random numbers on a circle: Method 2')\n", "plt.show()\n", "\n", "r2 = nprnd.random(n_rands) # generate random distance to the center, squared\n", "phi = 2*np.pi*nprnd.random(n_rands) # generate random angle\n", "r = np.sqrt(r2) # generate r from r^2\n", "x = r*np.cos(phi) # x is the cosine projection of (r,phi)\n", "y = r*np.sin(phi) # y is the sine projection of (r,phi)\n", "plt.plot(x, y, 'o')\n", "plt.title('Putative random numbers on a circle: Method 3')\n", "plt.show()\n", "\n", "# Because generating numbers using Method 4 is a bit complicated, we first \n", "# define a function for this, and will then call the function later for the\n", "# actual generation.\n", "def rnd_circle_rejection(n_rands):\n", " N = int(np.ceil(4/np.pi*n_rands)) # need to generate more points because of possible rejections.\n", " # Note that np.ceil makes its argument a double precision number, which just happens to be\n", " # an integer. We need to actually make it an integer type, hence int()\n", " x = 2*nprnd.random(N) - 1 # generate random x\n", " y = 2*nprnd.random(N) - 1 # generate random y\n", " ind = (x**2+y**2 < 1) # which points fall in the circle?\n", " x = x[ind] # choose x of points inside the circle\n", " y = y[ind] # choose y of points inside the circle\n", " print('Generated ', np.size(x), ' random numbers.',sep='') # we do not need this, and you should feel \n", " # free to remove this line.\n", " # I put it here so that you can see how additional data points get generated if we rejected\n", " # too many numbers initially\n", " if (np.size(x)n_rands): # if too many points were generated originally,\n", " x = x[0:n_rands] # then truncate the arrays to the needed size\n", " y = y[0:n_rands]\n", " \n", " return x, y # return the x and y coordinated\n", "\n", "x,y = rnd_circle_rejection(n_rands) # now let's actually generate the numbers \n", "plt.plot(x, y, 'o')\n", "plt.title('Putative random numbers on a circle: Method 4')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Which one of the generators is correct? \n", "\n", "The first method generates a cross of higher density at $\\pm 45^{\\rm o}$ angle. This is easy to understand: the rays that extend in these directions are longer than in all others, and, in particular, $\\sqrt{2}$ longer than in the horizontal and the vertical directions. Thus we generate more points along these diagonal rays, which, in turn, transforms into a higher density once we pull these numbers first onto the rim of the circle, and then spread them over the whole area of the circle. \n", "\n", "Hopefully you also see that the first two of the tried methods generate too many points at the center of the circle compared to the edges. Those of you who know multivariate calculus should be able to extend the calculation we did for generation of exponential random numbers to the multivariate case and relate the higher density of points near the center of the circle to the Jacobian of the transformation we make when we generate such two-dimensional random numbers from uniform standard numbers by transforming from the $(x,y)$ coordinates to $(r,\\phi)$ ones. One can also reason this without the more advanced math. By generating a uniform random $r$, we are putting the same number of points at $0\\le r<0.5$ as at $0.5\\le r<1$. But the area inside the circle of the radius $0.5$ is $\\pi/4$, while the area of the annulus of the width of $0.5$ and the external radius of $1$ is $3\\pi/4$. In other words, the density of points generated using a uniform random $r$ is 4 times as high in the inner circle as it is in the annulus, and this is what you see in the figures. \n", "\n", "> ### Your work\n", "> Time how long it takes each of the generators to generate $10^6$ points in a circle. Which one is faster? (Focus only on *Methods 3, 4*, because the first two are incorrect. In general, performing complex arithmetic operations (trigonometric functions and roots) takes time, slowing down *Method 3*. On the other hand, memory management in *Method 4*, and specifically recursive calls to the function, as well as creating arrays and then stacking them to produce the final output, is also costly. Which of the methods won in your case? Now write an even faster method, which simply generates a bit too many points to start with, so that it almost never has to do the recursive calls. Suprisingly, even though this is wasteful (we generate and then throw away potentially many numbers), as long as it is not too wasteful, such method will be the fastest. *Note: it only takes a few small changes to the function `rnd_circle_rejection()` to implement this.*" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Generating a standard normal number \n", "A Gaussian (or normal) random variable is a variable with the probability density function $P(x) = \\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp \\left[-\\frac{(x-\\mu)^2}{2\\sigma^2}\\right]$. $\\mu" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" }, { "data": { "image/png": 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\n", 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# generating Gaussian random numbers using built-in function\n", "n_rands = int(1e5) # how many numbers to generate?\n", "nbins = 30 # how many bins to histrogram into?\n", "plt.hist(nprnd.randn(n_rands), nbins) # make sure to explore help(nprnd.randn)\n", "plt.title('Built-in normal random numbers generation')\n", "plt.show()\n", "\n", "# and comparing built-in to our own generator\n", "phi = 2 * np.pi * nprnd.random(n_rands) # generating a random angle\n", "r = np.sqrt(2 * nprnd.exponential(size=n_rands)) # generating a random exponentially distributed r^2, \n", " # multiplying by 2, and taking a square root to get the correctly distributed r\n", "rand_norm = r * np.cos(phi) # the x projection of this (x,y) pair is standard-normal distributed, and we \n", " # discard the y projection for now\n", "plt.hist([nprnd.randn(n_rands), rand_norm], nbins)\n", "plt.title('Comparing built-in in our own normal random numbers')\n", "plt.legend(('built-in normal\\n random numbers', 'our normal\\n random numbers'))\n", "plt.show()\n", "\n", "# Now we show that generating just a single x instead of the (x,y) pair doesn't give us the right distribution\n", "x = np.sqrt(2 * nprnd.exponential(size=n_rands)) # we do the same as above, but generate x as a square root of x^2,\n", " # rather than r as a square root of r^2\n", "si = np.round(nprnd.rand(n_rands))*2 - 1 # this generates a random sign for x -- some will be +1 and some will be -1\n", "x = x*si # this is finally our putative standard normal\n", "plt.hist(x, nbins)\n", "plt.title('Incorrectly generated normal random numbers')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Using random numbers for Monte-Carlo (MC) calculations\n", "We use random numbers to perform computations that are fundamentally random, and we will focus on those in the upcoming weeks. However, as we discussed a few lectures ago, there are computations that are fundamentally deterministic, but may be easier to perform with random numbers. One of these is calculation of areas under the curve (definite integrals), known as Monte Carlo integration.\n", "\n", "Suppose we want to calculate the areas under a curve $y=y(x)$, with the limits $x\\in [a,b]$. We can, of course, do this analytically for some functions $y$ or numerically by subdividing the range of $x$ into small segments $\\Delta x$ and adding up the areas of all small rectangles, $y(x)\\times \\Delta x$. An alternative (and dare I say, more fun?) way is as follows. Suppose we know that $y(x)$ is positive and smaller than $f_{\\rm max}$. Then the area under the curve can be written down as $(b-a)f_{\\rm max}\\alpha$, where $\\alpha$ is the fraction of the rectange $b-a$ by $f_{\\rm max}$ that is actually under the curve. How do we estimate this fraction? Suppose I generate random points (dart throws), uniformly distributed over the rectangle $b-a$ by $f_{\\rm max}$. The fraction of those that end up under the curve (and it's easy to check which ones do!) is an estimate of $\\alpha$. The next cell performs such calculation for an arbitrary $y=y(x)$. Note that our implementation is incomplete: we don't check if the function is ever below 0 or is ever above $f_{\\rm max}$, which a better implementation should do. We will address this later.\n", "\n", "The implementation below first defines the function that will be integrated. You can plug different expression there, to get integrals of different functions. Then we define the MC integrator itself. Finally we integrate the function using the integrator." ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Estimate of the area is 0.33291.\n" ] } ], "source": [ "# a function to integrate using the Monte Carlo approach\n", "def to_integr(x):\n", " return x**2\n", " #return np.exp(np.sin(x))\n", "\n", "# Monte Carlo integrator\n", "def mc_integr(func, a, b, fmax, N):\n", " # func -- function to integrate\n", " # a,b -- the x range of the integration\n", " # fmax -- the maximum value of the integrand\n", " # N -- number of random samples used for the integration\n", " # return -- a real number, estimate of the area\n", " N = int(N) # just making sure that the number of dart throws is integer\n", " x = a+(b-a)*nprnd.random(N) # generating random x coordinated\n", " y = fmax*nprnd.random(N) # generating random y coordinate\n", " counts_under_curve = np.sum(y <= func(x)) # how many dots under the curve?\n", " frac_under_curve = counts_under_curve/N # what is the fraction of dots under the curve?\n", " area = (b-a)*fmax*frac_under_curve # estimate of the area\n", " return area\n", "\n", "# calculating area under the function to_integr using MC integrator\n", "area = mc_integr(to_integr, 0, 1, 3, 100000)\n", "print('Estimate of the area is ', area, '.', sep='')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Above we integrated the function $x^2$ between 0 and 1. The integral should be $1/3$ -- and hopefully after running the cell above, you got a value very close to $1/3$. \n", "\n", ">### Your work\n", "Harden the function `mc_integr()` to be able to handle various user errors. First, expand the function to allow areas under the curve to be negative (this will require you to also have an `fmin` argument, in addition to `fmax`). Second, check if any values of `to_integr(x)` generated in the process of integration fall outside of the range $[f_{\\rm min}, f_{\\rm max}]$. If they do, then expand the range appropriately. Return not just the estimate of the area, but the estimates of $f_{\\rm min}$ and $f_{\\rm max}$ as well.\n", "\n", "## What is the error of MC methods?\n", "The result of the area calculation above came out to be slightly different from $1/3$. Herewithin lies the problem of MC approaches: they are probabilistic, and every time we perform them, the results are different. But how different? We can verify, first of all, that our probabilistic estimate of the area converges to to true one as the number of random samples increases, $N\\to\\infty$ (at least when the true value of the area is known, e.g., when we integrate a function $x^2$). In fact, one can can convince onself that this should be the case rather easily if the numbers are truly uniformley distributed in the rectangle around the curve. \n", "\n", "Can we additionally understand how quickly the convergence happens? For this, we can explore the distribution of the estimates of the area when we perform the calculation with a varying number $N$ of dart throws. That is, we can generate many different estimates with different realizations of $N$ dart throws and see how different they are from each other. The cell below does precisely this. Hopefully you will see that the larger $N$ is, the more concentrated the estimates are. We quantify this decrease of the spread by calculating the standard deviation $\\sigma$ of the estimates calculated at the same value of $N$. We then compare the standard deviations to the $\\sigma = 1/\\sqrt{N}$ line." ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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zBfHMnz+fxo0bc9ddd1G+fHkGDRrERx99RJ06dahSpQpbt24NaHsSy6xhw4bUqFGDGjVqnE9OgwYN4vvvv6datWq8+uqrzJ8/n9atWwPw119/0a5dO6pWrUrdunVZs2YNAMOGDeOBBx6gSZMmlC1b9nyiOX78OK1ateKGG26gcuXKTJqUtKPHEFDVbPPC6TJyXLly5TRYU3+K0/oj5mrpgTO08X9m6bpPnlJ9rqjqiKtUV76veu5c0Ms0JpzWrVsX8LT1R8zVUgNnXPSqP2Jumtf/22+/qYjokiVLVFV137592rBhQz127Jiqqo4cOVKfffZZPXDggJYvX17Pud+hgwcPqqrqfffdp59//vn55eXPn//8cq+//npVVX3vvfe0b9++56dp3bq1Llq0SFVVjx49qmfOnLkgphUrVmjlypX12LFjevToUa1UqZL+9NNPqqpaqlQp3bdv30Xb0a9fP504caKqqp46dUpPnDhxQTzz5s3TQoUK6e7du/XkyZNavHhxfeaZZ1RV9bXXXtNHHnkk4O05fvy4xsfHq6rqpk2btGbNmufX0apVq/Pz+r7v16+fDhs2TFVV586dqzfccIOqqg4dOlTr1aunJ0+e1H379mmRIkX09OnTOnnyZO3Ro8f5ZR06dOiibU5u3wFWaIC/tdmqOq6qTgem16pVq2ew87arXiJJ9dsWsO9emP4ITPsnrPkMWr8GRcuFLmBjMoi/+3XpvY9XqlQp6tatC8DSpUtZt24dN910E+CcMdSrV4+CBQuSN29eevToQatWrc4fSafFTTfdxGOPPUa3bt3o0KEDJUuWvGD8okWLaN++Pfnz5wegQ4cOfP/991SvXt3vMuvVq8fzzz9PXFwcHTp04Nprr71omtq1a3PllVcCcM0113DrrbcCUKVKFebNmxdw/GfOnKFfv36sXr2ayMhINm3alOo8ixYt4osvvgDg5ptv5sCBAxw+fBiAVq1akSdPHvLkycNll13Gnj17qFKlCk888QQDBw6kdevWNGzYMOD4AmWXqlJSrDx0/wravA5/roEx9WHBKDh72uvIjAmKv/t16b2Pl/gDDc7Vi1tuuYXVq1ezevVq1q1bxzvvvEOuXLlYtmwZHTt2JDY2lpYtWwKQK1eu8/dFVJXTp1P/Xg0aNIi3336b+Ph46taty4YNGy4Yr2lotLVr165MmzaN6OhoWrRowXfffXfRNHny5Dn/f0RExPn3ERERnD17NuDtefXVV7n88sv5+eefWbFiRUDbnNw2JdaM8o0rMjKSs2fPUr58eVauXEmVKlUYPHgwzz33XKrrCJYljtREREDN+6Dvcrjudpg3HN5qBDuXeR2ZMQEb0KIC0VGRFwyLjopkQIsKIVtH3bp1+eGHH9iyZQsAJ06cYNOmTRw7dozDhw9z++2389prr7F69WoASpcuzcqVKwH48ssvOXPmzEXLjImJueA+xtatW6lSpQoDBw6kVq1aFyWORo0aERsby4kTJzh+/DhTp05N9Yh727ZtlC1blv79+9O2bdvz9xCCFcj2HD58mCuvvJKIiAg+/PBDEhISkt3OpNuUeEN//vz5FC1alIIFC/qNY/fu3eTLl4977rmHJ554gp9++ilN25OSbHWpKqxiLoc7J0DVzvDV4/DOrVD7QWg2FPL6/xCNyQwSL8OGs3WEYsWKMWHCBLp06cKpU6cAGD58ODExMdxxxx2cPHkSVT1/c7pnz57ccccd1KlTh2bNml1w9pKoadOmjBw5kmrVqjF48GAWLVrEvHnziIyMpFKlStx2220XTF+jRg26d+9OnTp1AOjRo0eKl6kAJk2axMSJE4mKiuKKK67gmWeeSdP2B7I9ffr0oWPHjnz++ec0bdr0/DRVq1YlV65c3HDDDXTv3v2CmIcNG8b9999P1apVyZcvH++//36Kcaxdu5YBAwYQERFBVFQUY8aMSdP2pMT640iLU8fgu+Hw41iIuRJuHwUV037d1pi0WL9+PRUrVvQ6DJMFJbfvWH8c4ZanANw2EnrMhXxFYFI3+LQbHNntdWTGGBN2ljjSo2RNeGg+NB8GW76F0TfC8rchDA9CGWNMZmGJI70io6DB/0GfJVCihnP/472WsHe915EZY0xYWOIIlSJl4R+x0P4t2L8ZxjaE756HM9ZXgjEme7HEEUoicENn6LccKneEhS/C2AawfZHXkRljTMhY4giH/EWhw1twzxRIOA0TWjlPn8cf9DoyY4xJN0sc4VSuGfRZCjc9Aqs+gjfqwC9fQDasAm1ypgceeIDLLrss2WbDlyxZQs+eQbf+c5HY2FjWrVt3wbBHH32UhQsXAtCkSRNq1fq7FumKFSto0qQJ4DzT0L1792SX69uQ4LRp0xg5cmS6Y80pLHGEW+58cMtzTu2rQiVg8gPw8d1waKfXkRmTbt27d2fWrFnJjps1a9b55kXS6uzZsxcljr/++oulS5fSqFGj88P27t3LzJkzL5q/SpUqxMXF8fvvv6e4nrZt2zJo0KB0xZqTWOLIKFdWdZ77aDHCuecx+kZYOgbOJXgdmTFp1qhRI4oUKZLsuLlz59K8eXO/zZY///zzVKhQgebNm9OlSxdeeuklwDmDePLJJ2ncuDEvvPAC06ZNY8CAAVSrVo2tW7cyefLkixLSgAEDGD58eLJxtGnThk8//TTF7ZgwYQL9+vUDnGTYv39/6tevT9myZS9oKn3UqFHUrl2bqlWrMnTo0MAKKRuyJkcyUkQk1OvjPGU+4zGYNQjWTIK2/4MrqiQ7S+yqXWFtJsJkEzMHwZ9rQ7vMK6o4D7qmwf79+4mKiqJQoUIMGTKERx55hG7dunH69GkSEhJYuXIln376KatWreLs2bPUqFGDmjVrnp//0KFDLFiwAIDNmzfTunVrOnXqBMBzzz13/v9E9erVY+rUqcybN4+YmJgLxtWqVYuRI0fyr3/9K+D4//jjDxYtWsSGDRto27YtnTp1Ys6cOWzevJlly5ahqrRt25aFCxdecOaTU9gZhxcKXw3dPodO78LhOHirMXwzFE6fuGCycHS+Y0xGmDNnzvmmx+vVq8d//vMfXnjhBXbs2EF0dDTff/897du3J1++fBQsWJC2bdteMP/dd9/td9l//PEHxYoVu2j4kCFDkj3ruOyyy9i9O7hWHdq1a0dERASVKlViz54957dpzpw5VK9enRo1arBhw4bzZ085TbY64xCRNkCbcuWyQJ8ZIk6V3bJN4duh8MNrsC4WWr8K19wMOA3SxZ+58FJW/JkERs3eaGcd5kJpPDMIl5kzZ/LYY48BTrPlN954I1999RUtWrTg7bffBv5uGjw5yTUQmCg6OpqTJy9+Purmm2/m6aefZunSpRcMP3nyJNHRwTUf79tceWJ7fqrK4MGD6dWrV1DLyo6y1RmHqk5X1YcKFSrkdSiBy1fEuVR13wyIyAUftocpveD4gbB1vmNMOKkqa9asoVq1akDyzZY3atSIqVOnEh8fz9GjR5k+fbrf5SVtcrxixYrnm25P6qmnnuLFF1+8YNimTZuSrfUVrBYtWvDuu++e7w53165d7N2bM7uWzlaJI0sr0xB6/wCNBsAvk+GNWtwf8yNwcdXd9Ha+Y0yodOnShXr16rFx40ZKlizJO++8w8qVK6levfr5M4pJkyZRuXJlqlWrxoYNG7j33nupUaMGd999N9WqVaNjx44p9pnRuXNnRo0aRfXq1dm6dSutWrVi/vz5yU57++23X3QZa968ebRq1Srd23rrrbfStWtX6tWrR5UqVejUqZPfPjSyO2tWPTPas87psjZuGT9oVQadvp+dejngdL4zokMVu1RlMm2z6sOHD6dcuXJ07tw54HmGDRtGgQIFeOKJJwKavkGDBsyYMYPChQunON2pU6do3LgxixYtIleubHVlPl2sWfXs6PJK8MBsaPUydaK28k2egfSOnM7VhaIsaZhMb8iQIUEljbR4+eWXU302A+D3339n5MiRljRCLKAzDhG5BLgWyJs4TFUXhjGudMnyZxy+juyGrwfAhhlweRVo+18oUTP1+Uy2l1nPOEzmF/YzDhHpASwEZgPPun+HBR2pSZuCxaHzR3D3RDixH95uDrMGO70QGmOMBwK5VPUIUBvYoapNgerAvrBGZS5WsQ30/RFqPeA8cf5mXdg0x+uojMey4z1KE16h2GcCSRwnVfUkgIjkUdUNQIV0r9kEL28haPWyc/8jdwH4+E74/H44usfryIwH8ubNy4EDByx5mICpKgcOHCBv3rypT5yCQO4YxYlIYSAW+EZEDgLWubaXrr4Rei2EH/7r9PmxdS7cOhyq/8N5sNDkCCVLliQuLo59++wCgAlc3rx5KVmyZLqWEVR1XBFpDBQCZqnq6XStOYyy1c3x1OzfDNMfhR2LoFQDaPMaFL3W66iMMVlMyKvjikgDEblfVRcASwCrD5pZFL0W7pvuPH2+Zy2MqQ8LXoSzmTavG2OyuEBqVQ0FBgKD3UFRwMRwBmWCFBEBNe6FvsvhutYw73l4qyH8/qPXkRljsqFAzjjaA22B4wCquhuISXEO442Yy+HO96DrZ3D6OLzbwmm+/eRhryMzxmQjgSSO0+rcCFEAEfHfbKXJHMq3cLqsrfswrHzP6TRqvf9G5IwxJhiBJI7PROQtoLCI9AS+BcaHNyyTbnkKQMsR0ONbyFcUJt0Dn3ZznkQ3xph0CLTJkVuAWwEBZqvqN+EOLD1yVK2qQCScgaVvwrwRTtPtzYdCrQedeyPGGENwtapSTBwiEomTKJqHKriMYInDj7+2Ofc8ts2DknWgzX+dBhWNMTleyKrjqmoCcEJEslDPSMavImXhH1Oh/VtwYAu81Qi+Gw5nLu5NzRhj/AnkyfGTwFoR+Qa3ZhWAqvYPW1QmfETghs5Q7haY8xQsHAW/THHOPsr470zHGGMSBXKR+yvgaZwWclf6vExWlv9SaD8W/hELmgDvt4Yv+8GJv7yOzBiTyVkPgAZOn4AFL8Di/zl9oLccCZU7WrtXxuQgoe6P41oRmSwi60RkW+Ir/WGaTCN3PrjlWei1AApdBV88CB/dCQd3eB2ZMSYTCuRS1XvAGOAs0BT4APgwnEEZj1xRxXnuo+VI2LHY6fNjyWhIOOt1ZMaYTCSQxBGtqnNxLmvtUNVhwM3hDct4JiLSeeK8749QuiHMfhLebgZ//Ox1ZMaYTCKgjpxEJALYLCL9RKQ9cFmY4zJeK3wVdJ0End5znjYf1xTmPO3cDzHG5GiBJI5HgXxAf6AmcA9wXziD8iUiZUXkHRGZnFHrNC4RqNwB+i2D6vfA4tedy1db5nodmTHGQ6kmDlVdrqrHVDVOVe9X1Y6qujSQhYvIuyKyV0R+STK8pYhsFJEtIjIolfVvU9UHA1mfCZPoS6Dt69D9K4iMgokdYMpDcHy/15EZYzwQ7saKJgAtfQe4zZiMBm4DKgFdRKSSiFQRkRlJXnZJLDMp3QB6/wCN/uU8NPhGLVj9MWTDKt3GGP/CmjhUdSGQ9ImyOsAW90ziNPApcIeqrlXV1kleewNdl4g8JCIrRGSF9cEcRlF54eanoPf3ULQ8xD4MH9wBB7Z6HZkxJoN40TxqCWCnz/s4UuiKVkQuFZGxQHURGexvOlUdp6q1VLVWsWLFQhetSVbsroI02PsvnjrzAMd+W07C6Hrw/StOS7zGmGwtkAcAy4vI3MT7FCJSVUSGpGOdyT2O7Pdah6oeUNXeqnqNqo5Ix3pNiMSu2sXgKWuJO3yKjxKa0+zki8xNuAHmPgvjmkCctUhjTHYWyBnHeJz+xs8AqOoaoHM61hkHXOXzviRgvQtlIaNmbyT+TML593sowkOnHmVQ1CCnrau3m8HMgXDqqIdRGmPCJZDEkU9VlyUZlp5HiZcD14pIGRHJjZOEpqVjeSaD7T4Un+zwSUerOg8O1u4BP74Fo+vCxlkZHJ0xJtwCSRz7ReQa/u5zvBPwRyALF5FPgCVABRGJE5EHVfUs0A+YDawHPlPVX9MU/cXrayMi4w4fPhyKxRk/iheO9j88b0Fo9RI8OAfyxMAnd8Nn98HRPzM4SmNMuKTaOq6IlAXGAfWBg8BvwD2quj3s0aWRtY4bXon3OHwvV0VHRTKiQxXaVfep53CJFRNKAAAdvklEQVT2NCz+LywYBbnywq3PQfV7rctaYzKhkHUdm2Sh+YEIVc30F64tcYRf7KpdjJq9kd2H4ileOJoBLSpcmDR87d8CMx6F7d/D1fWdTqOKlc/YgI0xKQpp4hCRPEBHoDQ+PQaq6nPpiDGsLHFkQqqwaiLMGQJnTkDDx6HB/0GuPF5HZowhxP1xAF8Cd+DcED/u8zImcCJQ4x/QbzlUbAvzR8DYhrBjideRGWOCFMgZxy+qWjmD4kkXEWkDtClXrlzPzZs3ex2OScnmb2DGY3D4d6h5PzQfBtGFvY7KmBwr1Gcci0WkSjpjyhCqOl1VHypUqJDXoZjUXHsL9FkC9frBT+/D6Bth3ZfW7pUxWYDfxCEia0VkDdAA+MltzXaNz3Bj0idPAWjxPPT8DgpcBp/dC592hcNxXkdmjElBrhTGtc6wKEzOVrw69JwHS9+Eef9xzj6aDYXaDzo9EhpjMhW/ZxxuN7E7gOGJ//sOy7gQTY4QmQtu6g99l8JVdWDmAHjnVtgTkmdDjTEhFMg9jut937j9adQMTzjpY0+OZwOXlIZ7pkCH8XDwN3irEXz7LJxJvpkTY0zGS+kex2AROQpUFZEj7usosBenim6mYzfHswkRqHoX9FsBVe+GRa/AmPqwbYHXkRljSPlS1QhVjQFGqWpB9xWjqpeqqt9+MYwJmXxFoN2bcK9b2+qDthDbx2mB1xjjmUD6HLckYbxVtolTdbfB/8HPn8IbtWHN51Z11xiPWGtzJmuIinYeEuy1EC4pBVN6wMSOcHC7x4EZk/OkdI+jTEYGYkxArqgMD34Dt70IO3+EN+vB4v9BQnq6iDHGBCOlM47JACIyN4NiMSYwEZFwYy+n06gyjZ2GE8c3hd2rvI7MmBwhpQcAI0RkKFBeRB5LOlJVXwlfWGnj01aV16GYjFCoJHT5xGmqZOa/YPzNULcPNH0Scuf3Ojpjsq2Uzjg6AydxkktMMq9Mx6rj5kAicH075+yjxr2w5A2ny9rN33gdmTHZViCt496mqjMzKJ6QsP44crAdi2H6I7B/E1TuBC1HQoFiXkdlTKYXjtZxXxGRFe7rZRGxQ3qTOZWqD70XQeNBziWsN2o5HUhZ1V1jQiaQxPEucBS4y30dAd4LZ1DGpFXsql3c9NIPlJldla65XmZ/vrLwZV94vw0c2Op1eMZkC4EkjmtUdaiqbnNfzwJlwx2YMcGKXbWLwVPWsutQPAosPlKURvsGsPqGofDHz07V3YUvwdnTXodqTJYWSOKIF5EGiW9E5CbAWpwzmc6o2RuJP5NwwbATZ5S+G6tB32VQoSV8928Y1xh2LvcoSmOyvkASR29gtIhsF5HtwBtAr7BGlUbWOm7OtvtQ8sczuw/FQ8Er4a4PoPMnEH8I3rkFvh4AJ49kcJTGZH2BtFX1s6reAFQFqqpqdVXNlD0AWnXcnK144ejUh193u1N1t85DsGy802nUhq8yKEJjsoeA26pS1SOqaodnJtMa0KIC0VEX9hgYHRXJgBYVLpwwb0G4/UXo8S1EX+J0VzvpH3DkjwyM1pisyxo5NNlGu+olGNGhCiUKRyNAicLRjOhQhXbVSyQ/Q8la0GsBNHsGNs2G0XVg+Ttw7lyGxm1MVpPqA4BZkT0AaIJ2YCvMeBR+WwhX1YU2/4XLrvM6KmMyTKgfAERE6otIVxG5N/GVvhCNyWQuvQbunQZ3vAn7N8LYBjDvP3D2lNeRGZPppJo4RORD4CWgAVDbfQWUlYzJUkSgejfouxyubw8LXoAxN8H2H7yOzJhMJaXWcRPVAippdrymZUxyChSDjuOd/s6/+j+YcDvUuA9ueQ6iC3sdnTGeC+RS1S/AFeEOxJhM59rm0Gcp1OsHqz50bp7/OtXavTI5XiCJoyiwTkRmi8i0xFe4AzMmU8idH1o8Dz3nQcwV8Hl3+KQzHI7zOjJjPBNIs+qNkxuuqgvCElE6+HTk1HPz5s1eh2Oym4Sz8ONYmPc8SATc/DTU6en0SGhMFhdMraqAquOKyOU4N8UBlqnq3nTEF3ZWHdeE1cHt8NXjsOVbKFET2rzu9IVuTBYW0uq4InIXsAy4E6dZ9R9FpFP6QjQmC7ukNHSbDB3fgYM7nEYTvx0GZ6ztT5MzBFKr6imgduJZhogUA74FJoczMGMyNRGo0gmuuRm+eRoWvQq/xkLrV+Gapl5HZ0xYBXJzPCLJpakDAc5nTPaXrwjcMRrum+4kkw/bwdSH4fgBryMzJmwCSQCz3BpV3UWkO/AV8HV4wzImiynTCB5eDA0fh7Wfweja8PMkq7prsqVAmlUfAIzDaVb9BmCcqg4Md2DGZDlR0U6Dib0WwiVlYOpDMLED/PWb15EZE1LWyKEx4XAuAVa8C98+C+fOQtPBULcvRAZyW9GYjBeSWlUissj9e1REjvi8joqI9cthTEoiIp1nPPr+6Nws/+YZGN8Edv3kdWTGpJvfxKGqDdy/Mapa0OcVo6oFMy5EY7KwQiWg88dw14dwbB+83QxmPQmnjnkdmTFpFmjruKkOM8b4IQKV2jpnHzW7w9LR8GY92PyN15EZkyaB1Kq63veNiOQCaoYnHGOysejCznMe989ybqR/1AkmPwDHMnVDDMZcJKV7HINF5ChQ1ff+BrAH+DLDIgyCiLQRkXGHDx/2OhRj/CtVD3p/D02ehPXT4Y3a8NOHVnXXZBmBNHI4QlUHZ1A8IWG1qkyWsW8TTH8Efl8MpRtC69egaDmvozI5UEjbqlLVwSJyiYjUEZFGia/0h2mMoVh56P6V08f5H2tgTH1YMArOnvY6MmP8CuTmeA9gITAbeNb9Oyy8YRmTg0REODfN+y2D626HecPhrUawc5nXkRmTrEBujj+C06T6DlVtClQH9oU1KmNyopgr4M4J0OVTOHUU3rkVvnoCTtpjUyZzCSRxnFTVkwAikkdVNwAVwhuWMTlYhdug71K4sTcsfxtG3wjrZ3gdlTHnBZI44kSkMBALfCMiXwK7wxuWMTlcnhi4bST0mOu0wDupG3zaDY7YV894L6i2qtxuZAsBs1Q10969s1pVJrOJXbWLUbM3svtQPMULRzOgRQXaVS8R2MwJZ2DJGzB/JETmhuZDoeYDzr0RY0IkVG1VFXT/Fkl8AWuBRUCBkERqTA4Qu2oXg6esZdeheBTYdSiewVPWErtqV2ALiIyCBv/nNNtevLrTbe17LWHv+rDGbYw/KR2yfOz+XQmsSOavMSYAo2ZvJP5MwgXD4s8kMGr2xuAWdOk1cO+X0G4s7N8MYxvCd8/DmZMhjNaY1Plt41lVW7t/y2RcOMZkP7sPJd8Xub/hKRKBal3g2ltg9lOw8EX4dSq0eQ1KN0hnpMYEJpDnOL4UkS4iki8jAjImuyleODqo4QHJXxQ6vAX3TIGE0zChFUz7J8QfTPsyjQlQIHfXXgEaAutF5HMR6SQiecMclzHZxoAWFYiOirxgWHRUJANahKBWe7lm0Gcp1O8Pqz6CN+rAL19Yu1cmrAJpcmSBqvYByuJ0IXsXYM15GhOgdtVLMKJDFUoUjkaAEoWjGdGhSuC1qlKTOx/c+m94aB4ULO60uPvx3XBoZ2iWb0wSAVXHFZFooA1wN1ADmKGq/wxzbGlm1XFNjpVwFpaNg++GO++bPQ11HnJ6JDQmBSFt5FBEJgHrgZuB0cA1mTlpGJOjReaCen2cJ89L1YdZg5xeB/9Y43VkJhvxW6vKx3tAV1VNSHVKY0zmUPhq6PY5/DoFZg6EcU2gfj9oPMi5tOVK14OJJscK5Ob4QmCwiIwDEJFrRaR1eMNKG+vIyRgfIlC5I/RdBtW6wg//hTH1YOt3QAgeTDQ5ViCJ4z3gNFDffR8HDA9bROmgqtNV9aFChQp5HYoxmUe+InDHG3DfDIjIBR+2hym9GDdreWgeTDQ5TiCJ4xpVfRE4A6Cq8YCENSpjTOiVaQi9f4BGA+CXyUw82Zf2Ed8DF1aQSdODiSZHCSRxnHZrVSmAiFwDnAprVMaY8IjKCzcPgV7fsyuiBK/mHsOHUSO4WvacnyRdDyaaHCGQxDEUmAVcJSIfAXOBf4U1KmNMeF1eia1tvuDZcw9SLWIrs3MPpFfkdGKiNDQPJppsLdVaVar6jYj8BNTFuUT1iKruD3tkxpiwalfjKpDHuWdWPfqcGMvgqE/oU/AnCl02BrCaVcY/vw8AikiNlGZU1Z/CElEI2AOAxqTB+unw9QA4tsfpfbDpU5DHelDIKYJ5ADClM46X3b95gVrAzzhnHFWBHwFritOY7KRiGyjTCOY+B0vHOImk1ctQvoXXkZlMxu89DlVtqqpNgR1ADVWtpao1gerAlowK0BiTgfIWcpLFA7MhdwH4+C74vDsc3ZPqrCbnCOTm+HWqujbxjar+AlQLX0jGGM9dfSP0WghNh8CGr2B0bVj5Ppw753VkJhMIJHGsF5G3RaSJiDQWkfE4bVcZY7KzXLmh8QCny9rLq8D0/vB+a6f3QZOjBZI47gd+BR4BHgXWucOMMTlB0WvhvunQ9n+w5xcYUx8WvAhnT3sdmfFIQM2qZzVWq8qYMDm6x2lx99cpUOw6aPO6c1nLZHkhbVbdGGPOi7kc7nwPun4Gp4/Du7fCjMfgpDUsmpNY4jDGBK98C6fL2rp9YeV7Tpe166Z5HZXJIJY4jDFpk6cAtPwP9PgW8heDz/4Bn3aDI7u9jsyEmd8HAEVkOkmbzfShqm3DEpExJmspUdPp73zpmzBvhHP20Xwo1HoQIuzYNDtK6VN9Cefp8d+AeGC8+zoG/BL+0IwxWUZkFNz0CPRZDCVrwddPwLstYM86ryMzYZBqrSoRWaiqjVIblplYrSpjPKQKaz6D2YOdm+Y3Per0ARKV1+vITApCXauqmIiU9Vl4GaBYWoMzxmRzInDD3dB3OVS5E75/yXn247fvvY7MhEggieP/gPkiMl9E5gPzcB4GNMYY//JfCu3Hwj9iQROcp86/7Asn/vI6MpNOKfbHISIRwBHgWuA6d/AGVbUeAI0xgbmmKTy8BBa8AIv/B5tmQ8uRULmjc3ZispwUzzhU9RzwsqqeUtWf3ZclDWNMcHLng1uehV4LoNBV8MWD8NGdcHCH15GZNAjkUtUcEekoYocGxph0uqKK89xHy5GwYzG8WReWjIaEs15HZoIQSOJ4DPgcOCUiR0TkqIgcCXNcxpjsKiIS6j4MfX+E0g1h9pPwdjP442evIzMBSjVxqGqMqkaoam5VLei+L5gRwQGISDsRGS8iX4rIrRm1XmNMmBW+CrpOgk7vOU+bj2sKc4bA6RNeR2ZSEdBjnSJyiYjUEZFGia8A53tXRPaKyC9JhrcUkY0iskVEBqW0DFWNVdWeQHfg7kDWa4zJIkSgcgfotwyq3+PcPH+zLmyZ63VkJgWpJg4R6QEsBGYDz7p/hwW4/AlAyyTLiwRGA7cBlYAuIlJJRKqIyIwkr8t8Zh3izmeMyW6iL4G2r0P3ryEyN0zsAFMeguP7vY7MJCOQM45HgNrADrcP8urAvkAWrqoLgaSVtusAW1R1m6qeBj4F7lDVtaraOslrrzheAGaq6k8Bb5kxJuspfRP0XgSNB8IvU+CNWrD6Y+dpdJNpBJI4TqrqSQARyaOqG4AK6VhnCWCnz/s4d5g//wSaA51EpLe/iUTkIRFZISIr9u0LKK8ZYzKjqLzQ9EkngRStALEPwwdt4cBWryMzrkASR5yIFAZigW9E5EsgPe0mJ1etN6VWeF9X1Zqq2ltVx6Yw3ThVraWqtYoVsxZRjMnyLrsO7p8JrV6B3audZku+fwUSzngdWY4XSK2q9qp6SFWHAU8D7wDt0rHOOOAqn/clSV8iMsZkVxERUPtB6LsMrr0F5j4L45pA3EqvI8vR/CYOESmS9AWsBRYBBdKxzuXAtSJSRkRyA50B6zrMGONfwSvh7olw90dOW1dvN4OZA+HUUa8jy5FSaqtqJc4lJAGuBg66/xcGfgfKpLZwEfkEaAIUFZE4YKiqviMi/XBqZ0UC76rqr+nZCJ/1tQHalCtXLhSLM8Z4JHbVLkbN3sjuQ/EULxzNgBYVaFe9BFRsDWUawXf/hh/fgvXTodXLUOE2r0POUQLpj2MsME1Vv3bf3wY0V9XHMyC+NLH+OIzJumJX7WLwlLXEn0k4Pyw6KpIRHao4ySPRzuUwvT/sXQeV2sFtL0DMFR5EnD2Euj+O2olJA0BVZwKN0xqcMcakZNTsjRckDYD4MwmMmr3xwgmvqg0PLYCbn4aNM50ua1e8B+fOZWC0OVMgiWO/iAwRkdIiUkpEngIOhDswY0zOtPtQfODDc+WGRk/Aw4vhyqow41GY0Ar2bQpzlDlbIImjC06Pf1NxquRe5g4zxpiQK144OqjhABQtB/dNhztGO5euxt4E80fCWesFIhwCqY77l6o+oqrV3dcjqpopu/ASkTYiMu7w4cNeh2KMSaMBLSoQHRV5wbDoqEgGtEjluWMRp72rfiugYluYPwLGNoQdS8IYbc4UyM3x8sATQGl8amGp6s1hjSwd7Oa4MVmb31pVwdj8Dcx4DA7/DjXvh+bDILpwOMLNFoK5OR5I4vgZGItTPff8HStVzbRP4FjiMMYAcOqYc+ax9E3IXwxuexEq3WFd1iYj1LWqzqrqGFVdpqorE1/pjNEYY8IvTwFo8Tz0/A4KXA6f3wefdoXDcV5HlqUFkjimi0gfEbkyyVPkxhiTNRSvDj3nwS3/hq3zYPSN8OM4OJeQ+rzmIoFcqvotmcGqqmXDE1L62aUqY4xfB7c79z62zoUStZx+QC6/3uuoPBfSS1WqWiaZV6ZMGlaryhiTqktKwz1fQIfxcPA3eKsRfPssnEn++RFzsVTPOABEpDJOb315E4ep6gdhjCtd7IzDGBOQE385/Zyv/giKlIXWr0HZnNkwRkjPOERkKPA/99UUeBFom64IjTEmM8hXBNq9Cfd+6fQy+EFbiO3jJBTjVyA3xzsBzYA/VfV+4AYgT1ijMsaYjFS2CfRZAg0egzWT4I3asOZz67LWj0ASR7yqngPOikhBYC+QKe9xGGNMmkVFQ/OhTsOJl5SCKT1gYkfnZrq5QCCJY4Xbdex4nIcAfwKWhTUqY4zxyhWV4cFvnIcFd/4Ib9aDxf+DhLNeR5ZpBHRz/PzEIqWBgqq6JlwBpYdPR049N2/e7HU4xpis7nAcfPUEbJoJV1R1qu4Wr+51VGER6pvjcxP/V9XtqrrGd1hmoqrTVfWhQoUKeR2KMSY7KFQSunwCd74Px/bA+Jth9lNw+rjXkXkqpT7H87pPiBcVkUt8nhovDRTPqACNMcZTInB9O+i7DGrcB0vegNF1YfO3XkfmmZTOOHrh3NO4zv2b+PoSGB3+0IwxJhOJLkxsyQH0yf08Ww6ehY86svPtrnBsn9eRZbhAmhz5p6r+L4PiCQl7ANAYE2q+faHn5gwPR06jb65YyJ2f3LePgGrdsnSru6FuHfdPEYlxFzxERKaISI10RWiMMVmMb1/op4nivwkdue30SNafLQFf9oX328CBrR5HmTECSRxPq+pREWkAtADeB8aENyxjjMlckuvzfKuWoH38U05TJX+scaruLnwJzp72IMKME0jiSGx3uBUwRlW/BHKHLyRjjMl8/PV5fmXh/FDrfui3DCq0hO/+DeMaw87lGRxhxgkkcewSkbeAu4CvRSRPgPNlOGsd1xgTLqn2hR5zBdz1AXT+BE4ehnduga8HwMkjHkQbXoHcHM8HtATWqupmEbkSqKKqczIiwLSwm+PGmHAIuC/0U0dh7r9h2TiIuRJavQTXtcr4gIMQ0j7HsyJLHMaYTCFuBUzrD3t/hYptnWZMCl7pdVTJCnWtKmOMMWlRshb0WgDNnoFNs2F0HVj+Dpw753Vk6WKJwxhjwikyCho+7jTbXrwafPUYvHcb7N3gdWRpZonDGGMywqXXwL3ToN0Y2L8RxjaAef+Bs6e8jixoljiMMSajiEC1rtB3OVzfHha8AGNugu0/eB1ZUCxxGGNMRitQDDqOh25fQMIpmHC7cxM9/qDXkQXEEocxxnjl2ubQZynU/yes+hDeqAO/Ts30XdZmq8RhDwAaY7Kc3Pnh1uHQc55TVffz7vBJZ6cTqUwqWyUO68jJGJNlFa8GPb6DW5+H3xbC6Bth6Vg4l5D6vBksWyUOY4zJ0iJzQf1+zuWrq+vCrIHwdnP4c63XkV3AEocxxmQ2l5SCbpOh4ztw6Hd4qzF8OwzOXNxCrxcscRhjTGYkAlU6Qb/lUK0LLHrVabZ96zyvI7PEYYwxmVq+InDHaLhvupNMPmwHUx+G4wc8C8kShzHGZAVlGsHDi6HhE7D2MxhdG36e5EnVXUscxhiTVURFQ7OnoddCuKQMTH0IJnaAv37L0DAscRhjTFZz+fXw4By4/SWnp8E368EP/4WEsxmyekscxhiTFUVEQp2e0PdHuKYpfPMMjG8CR/8M/6rDvgZjjDHhU6gEdP4Y7voQCpeC/MXCvkpLHMYYk8XFrt7NTdMKUGZ1V256cQGxq3aFdX25wrr0DCYibYA25cqV8zoUY4zJELGrdjF4ylrizzhNk+w6FM/gKc6T5sn2hx4C2eqMw9qqMsbkNKNmbzyfNBLFn0lg1OyNYVtntkocxhiT0+w+lHwzJP6Gh4IlDmOMycKKF44OangoWOIwxpgsbECLCkRHRV4wLDoqkgEtKoRtndnq5rgxxuQ0iTfAR83eyO5D8RQvHM2AFhXCdmMcLHEYY0yW1656ibAmiqTsUpUxxpigWOIwxhgTFEscxhhjgmKJwxhjTFAscRhjjAmKqAe9R4WbiOwDdrhvCwGHk0ziOyzp+KLA/jCFllwsoZonpen8jUutbPwN831v5WXlZeUV3HSZtbxKqWpgTeuqarZ+AeNSGpZ0PLAiI2MJ1TwpTedvXGplk0IZ+ZaflZeVl5VXDigv31dOuFQ1PZVhyY0Pl7SsK9B5UprO37jUysbfsIwqMyuv4Fh5BcfKK42y5aWq9BCRFapay+s4sgorr+BYeQXHyis4GVVeOeGMI1jjvA4gi7HyCo6VV3CsvIKTIeVlZxzGGGOCYmccxhhjgmKJwxhjTFAscRhjjAmKJY5UiEhZEXlHRCZ7HUtWICLtRGS8iHwpIrd6HU9mJyIVRWSsiEwWkYe9jicrEJH8IrJSRFp7HUtmJyJNROR7dx9rEqrl5sjEISLvisheEfklyfCWIrJRRLaIyCAAVd2mqg96E2nmEGR5xapqT6A7cLcH4XouyPJar6q9gbuAHFntNJjycg0EPsvYKDOPIMtLgWNAXiAuZEFkxFOGme0FNAJqAL/4DIsEtgJlgdzAz0Aln/GTvY47i5XXy0ANr2PPCuUFtAUWA129jj2zlxfQHOiMc2DS2uvYs0B5RbjjLwc+ClUMOfKMQ1UXAn8lGVwH2KLOGcZp4FPgjgwPLhMKprzE8QIwU1V/yuhYM4Ng9y9Vnaaq9YFuGRtp5hBkeTUF6gJdgZ4ikuN+w4IpL1U9544/COQJVQzWdezfSgA7fd7HATeKyKXA80B1ERmsqiM8iS7zSba8gH/iHBUWEpFyqjrWi+AyIX/7VxOgA86X+msP4sqski0vVe0HICLdgf0+P4w5nb/9qwPQAigMvBGqlVni+JskM0xV9QDQO6ODyQL8ldfrwOsZHUwW4K+85gPzMzaULCHZ8jr/j+qEjAslS/C3f00BpoR6ZTnuNC8FccBVPu9LArs9iiUrsPIKjpVXcKy8gpOh5WWJ42/LgWtFpIyI5Ma5ATfN45gyMyuv4Fh5BcfKKzgZWl45MnGIyCfAEqCCiMSJyIOqehboB8wG1gOfqeqvXsaZWVh5BcfKKzhWXsHJDOVljRwaY4wJSo484zDGGJN2ljiMMcYExRKHMcaYoFjiMMYYExRLHMYYY4JiicMYY0xQLHEYE2YioiLyss/7J0RkmIchGZMuljiMCb9TQAcRKep1IMaEgiUOY8LvLDAO+D+vAzEmFCxxGJMxRgPdRKSQ14EYk16WOIzJAKp6BPgA6O91LMaklyUOYzLOa8CDQH6vAzEmPSxxGJNBVPUv4DOc5GFMlmWJw5iM9TJgtatMlmbNqhtjjAmKnXEYY4wJiiUOY4wxQbHEYYwxJiiWOIwxxgTFEocxxpigWOIwxhgTFEscxhhjgmKJwxhjTFD+H9zsVnBZbXjiAAAAAElFTkSuQmCC\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# investigating the dependence of the accuracy of the MC integrator\n", "# on the number of samples\n", "n_trials = 30 # number of area estimates for each N\n", "N = np.logspace(1, 5, 9) # different values of N to be used\n", "areaN = np.zeros((N.size, n_trials)) # array to store all of the estimated area values\n", "\n", "# for each N and each trial within a fixed value of N, calculate and store the area\n", "for i in np.arange(N.size):\n", " for j in np.arange(n_trials):\n", " areaN[i, j] = mc_integr(to_integr, 0, 1, 3, N[i])\n", "\n", "\n", "# plotting areas as a function of N\n", "plt.semilogx(N, areaN, 'o')\n", "plt.xlabel('N')\n", "plt.ylabel('area')\n", "plt.title('Estimated of the area vs. N')\n", "plt.show()\n", "\n", "# plotting the standard deviation of the area estimates as a function of N\n", "# and comparing to 1/sqrt(N)\n", "plt.loglog(N, np.std(areaN, 1), 'o')\n", "plt.loglog(N, 1/np.sqrt(N))\n", "plt.xlabel('N')\n", "plt.ylabel('standard deviation of the area')\n", "plt.title('Standard deviation of MC estimates of the area vs. N')\n", "plt.legend(('results of simulations', '1/sqrt(N) line'))\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Notice how the convergence of the estimators to their true value (the decrease of the standard deviation) goes as $\\sigma = A/\\sqrt{N}$, where $A$ is some constant. The general claim, which is known as the *Central Limit Theorem* is that the $1/\\sqrt{N}$ law holds true for essentially any function being integrated, and only the constant $A$ changes. Moreover, the same scaling is true for almost all simple MC methods, beyond just area estimation: you need to increase the number of smaples by a factor of 100 to get a 10-fold increase in precision. \n", "\n", ">### Your work\n", "Verify that MC integration of different function still converges as $\\sigma=A/\\sqrt{N}$. For this, integrate different functions you can think about. Think about how $A$ depends on the shape of the function being integrated. Suppose you overestimated $f_{\\rm max}$. How would this change $A$? \n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Random walks and the Central Limit Theorem\n", "Every one of you who has taken any science lab can probably recall some mention of the Gaussian (also known as normal) distribution: in many experiments, the stochastic fluctuations around the mean are suppsed to be distributed according to the familiar bell-shaped curve, defined by two parameters: the mean $\\mu$ and the standard deviation $\\sigma$, namely: $P(x) = \\frac{1}{\\sqrt{2\\pi}}\\exp\\left(-\\frac{(x-\\mu)^2}{2\\sigma^2}\\right)$, shown in the next cell.\n", "\n", ">### Your Turn\n", "Vary the mean and the standard deviation, and explore how the distribution changes. Show many different normal distribution on the same axes." ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "image/png": 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jYJfQxgsLC2MiRJs/wJpddVw5aahnc1f01JWThnC4uZ3N+w97XYrpIxYWxkSIjXsPc6zVzxUTI39U18sm5OBLEF4vtX6LeGFhYUyEWL2rlsQEYVZ+ttelhDWwXxLT8zJ5fVed16WYPmJhYUyEWLOrlgtGD2JAiqvTzPSauQXZvFt5hCPNbV6XYvqAq2EhIgtEpFREykTkvm62f05E3hORzSLyhogUhmz7pnNcqYhc62adxnit7lgrWw80cllB5J9VdJpbkIMqvFFmZxfxwLWwcObQfgS4DigEbgsNA8eTqjpFVacDPwJ+4hxbSHDO7vOABcAvO+fkNiYWrXU+cC+Lolnopo0cSEZqImt2WljEAzfPLGYCZapaoaptwBJgUegOqtoYspgGdF6HtwhYoqqtqrobKHOez5iY9PrOWgb1T+K8EQO9LqXHEn0JzM7PZvWuWhuFNg64GRa5wP6Q5Upn3QeIyBdEpJzgmcWXzvDYu0WkWESKa2vtqgwTnVSVNbvqmJ2fjS9CJjrqqbkFORw82kJ57TGvSzEuczMsuvupP+nPD1V9RFXHA98Avn2Gxz6mqkWqWpSTEz2n78aEKq1uorapNaqaoDrNdfpYVltTVMxzMywqgbyQ5ZFA1Wn2XwLcdJbHGhO1Vu8MnhXPjaLO7U55g/szLjuN1TZOVMxzMyw2AAUiMlZEkgl2WC8N3UFECkIWbwB2OY+XAotFJEVExgIFwNsu1mqMZ9bsqmPC0AEMH9jP61LOymUTclhfUU+rv8PrUoyLXAsLVfUD9wDLge3A06paIiIPichCZ7d7RKRERDYD9wJ3OseWAE8D24C/AV9QVftJNDHnRFsHb+1uYG5B9DVBdZpbkE1Le4DiPTb0Ryxz9e4fVV0GLOuy7oGQx18+zbHfA77nXnXGeG/Dngba/AHmRGETVKdLxmWR5JP3O+lNbLI7uI3x0NryOpJ8wsVjB3tdyllLS0lkRt4g1pVbJ3css7AwxkNry+qYMWoQ/ZOjY4iPU5mVn8V7B47a0B8xzMLCGI8cPt5GSVUjs8dHf9PN7PxsVGF9Rb3XpRiXWFgY45E3K+pRhTkFWV6Xcs6m52WSluyzcaJimIWFMR5ZW1ZHWrKPqSMzvS7lnCX5Epg5djDryuzMIlZZWBjjkXXl9Vw8LoskX2z8Gs7Oz6ai7jhVR054XYpxQWz8lBoTZQ4cOcHuuuMxdalp53tZa01RMcnCwhgPdH6gzs6P/v6KThOHppOVlsy6cmuKikUWFsZ4YF1ZHdkDkpk4NN3rUnpNQoJw6fgs1pbV2ZDlMcjCwpg+pqqsLa9n1vhsRKJrSPJw5uRnU9PUSlmNDVkeaywsjOlju2qOUdvUGlNNUJ2s3yJ2WVgY08fWOR+ks2LgZryu8gb3J29wP+u3iEEWFsb0sXXl9eQN7kfe4P5el+KKWeOyWV9RT0fA+i1iiYWFMX2oI6Csr6hn1rjYO6voNCs/i8YWP9uqGr0uxfQiCwtj+tC2qkYaW/zMisH+ik6Xjgu+NxuFNrZYWBjThzo/QDs/UGPRkIxU8ocMsH6LGONqWIjIAhEpFZEyEbmvm+33isg2EXlXRF4VkdEh2zpEZLPztbTrscZEozcr6skfMoAhGalel+KqWeOz2LCngfaOgNelmF7iWliIiA94BLgOKARuE5HCLru9AxSp6lTgz8CPQradUNXpztdCjIly7R0B3t7dENNnFZ0uHZdFc1sH71Ye8boU00vcPLOYCZSpaoWqtgFLgEWhO6jqSlVtdhbXAyNdrMcYT71beYTmtg5mjY/9sLiks9/CRqGNGW6GRS6wP2S50ll3KncBL4Usp4pIsYisF5GbujtARO529imura0994qNcVHnB+clcXBmMSgtmcLhGdZvEUPcDIvuxjHo9sJrEfkEUAQ8HLJ6lKoWAbcDPxOR8Sc9mepjqlqkqkU5OTm9UbMxrllXXk/h8AwGpSV7XUqfmDU+i437DtPS3uF1KaYXuBkWlUBeyPJIoKrrTiIyH7gfWKiqrZ3rVbXK+bcCWAXMcLFWY1zV0t7Bxn2H46IJqtOs/Cza/AE27T3sdSmmF7gZFhuAAhEZKyLJwGLgA1c1icgM4FGCQVETsn6QiKQ4j7OB2cA2F2s1xlWb9h2mzR+I6fsrurpozGB8CWJNUTEi0a0nVlW/iNwDLAd8wOOqWiIiDwHFqrqUYLPTAOAZZ/TNfc6VT5OBR0UkQDDQ/l1VLSxM1HqzvB5fgnDRmMFel9Jn0lOTmDpyIG9WWFjEAtfCAkBVlwHLuqx7IOTx/FMctw6Y4mZtxvSldeX1TB05kPTUJK9L6VOzxmfx6OsVHGv1MyDF1Y8b4zK7g9sYlx1r9bNl/5G4uL+iq0vHZeMPKBv2NHhdijlHFhbGuGzDngb8AY3JIcnDuXD0IJJ9Cbxp/RZRz8LCGJe9WV5Psi+BC0cP8rqUPtcv2ceMUZk2qGAMsLAwxmXryuuYMSqTfsk+r0vxxKzx2ZRUNXKkuc3rUsw5sLAwxkVHm9spqWqMyyaoTrPys1CFt3Zbv0U0s7AwxkXrd9ejSlzdX9HVtJGZ9EvyWb9FlLOwMMZFb5bX0y/Jx7SRmV6X4pnkxASKxgyyfosoZ2FhjIvWlddRNGYQyYnx/as2a3w2O6uPUdvUGn5nE5Hi+yfYGBfVNrWys/pYXPdXdOocE8vu5o5eFhbGuKTzgzGeBg88lfNGZJCemsi6MmuKilYWFsa4ZF1ZHempiZw3IsPrUjyX6Evg4rFZNqhgFLOwMMYla8vruGRcFok++zUDmJ2fxb6GZvY3NIff2UQc+yk2xgX7G5rZ33CC2dYE9b7Z+cG+G7sqKjr1KCxEZKSIfF1EXhCRDSKyWkR+KSI3iIgFjjFdrHXa5js/IA0UDBlATnoKa21e7qgUdsxgEfkfgnNn/xX4IVADpAITgAXA/SJyn6qudrNQY6LJ2vJ6hqSnkD9kgNelRAwRYfb4LN4oq0NVceawMVGiJwPM/1hVt3azfivwnDML3qjeLcuY6KWqvFlex5z8bPtA7GJWfjbPb66itLqJScOs4z+ahG1C6gwKERnSdZuITFTVNlUt6+5YEVkgIqUiUiYi93Wz/V4R2SYi74rIqyIyOmTbnSKyy/m688zeljHeKa1uou5YG7OsCeoknc1y1hQVfc6kv2GNiHysc0FEvgb85VQ7i4gPeAS4DigEbhORwi67vQMUqepU4M/Aj5xjBwMPAhcDM4EHRST+xnc2Uanzg9D6K06Wm9mPMVn97X6LKHQmYTEPuENEnhGR1QT7LGaeZv+ZQJmqVqhqG7AEWBS6g6quVNXO6+jWAyOdx9cCK1S1QVUPAysI9o8YE/HWldUxNjuN3Mx+XpcSkWblZ/PW7gb8HQGvSzFnoMdhoaoHgb8BlwJjgP9V1WOnOSQX2B+yXOmsO5W7gJfO5FgRuVtEikWkuLa2Nux7MMZt7R0B1lfU213bpzF7fHZwqtnKo16XYs5Aj8NCRFYQbBY6H7ge+KmI/MfpDulmnZ7iuT8BFAEPn8mxqvqYqhapalFOTs7pyjemT7xbeYTjbR3WBHUal47PQuTvlxeb6HAmzVCPqOonVfWI0+k9CzjdnwaVQF7I8kigqutOIjIfuB9YqKqtZ3KsMZFmbVk9InDpODuzOJXBackUDs+wsIgyYcNCnGv/VPX50PWq6lfV74bu08UGoEBExjqX1y4GlnZ57hnAowSDoiZk03LgGhEZ5HRsX+OsMyaivbGrjvNHDGRQWrLXpUS0OQXZbNp3mOOtfq9LMT3UkzOLlSLyRRH5wL0UIpIsIleKyO+Bky5tVVU/cA/BD/ntwNOqWiIiD4nIQme3h4EBwDMisllEljrHNgDfJRg4G4CHnHXGRKxjrX427TvMnAJrggpnbn4O7R3KW7vtEtpo0ZOb8hYAnwaeEpGxwBGCd3D7gJeBn6rq5u4OVNVlwLIu6x4IeTz/VC+qqo8Dj/egPmMiwvryevwBZa6FRVhFYwaRkpjAml11XDlpqNflmB4IGxaq2gL8EviliCQB2cAJVT3idnHGRJM1u2rpl+TjwtF2S1A4qUk+Zo4dzJpd1m8RLXrSZ5EqIl8RkV8A/wDUWlAYc7I1u+q4eNxgUhJ9XpcSFS4ryKGs5hgHj57wuhTTAz3ps/g9wcta3yN4yeyPXa3ImChUebiZirrjzC2wS7h7qrNvx84uokNPwqJQVT+hqo8CHwXmulyTMVHnDecD7zLrr+ixScPSyR6Q8v73zkS2noRFe+cD5wonY0wXa3bVMTTDhiQ/EyLC3IJs3iirIxDo9n5dE0F6EhbTRKTR+WoCpnY+FpFGtws0JtJ1BJS15XXMLcixIcnP0NyCbBqOt7HtoH2URLqeXA1lvXXGnMbWA0c50txul8yehTn5f++3OD93oMfVmNOxKVGNOUdv2BSqZ21IRiqThqWzZpcNBBrpLCyMOUerSms4b0QG2QNSvC4lKl02IYfiPTb0R6SzsDDmHBxtbmfTviNcMfGkiSRND82bkENbR4B15Tb0RySzsDDmHKwpq6UjoFwxye6vOFtFYwaTluxjZWlN+J2NZywsjDkHq0prGdgviel5NsTH2UpOTGB2fjavl9aiapfQRioLC2POUiCgvL6zlrkF2fgS7JLZczFv4hAOHDlBWc3pJt80XrKwMOYsbTvYSG1Tq/VX9IJ5E4PNeNYUFbksLIw5S6ucD7bLJlh/xbkakdmPiUPTWVVql9BGKlfDQkQWiEipiJSJyH3dbL9MRDaJiF9EPtplW4czIdL7kyIZE0lWltYydeRActLtktneMG9SDhv2NHDMLqGNSK6FhYj4gEeA64BC4DYRKeyy2z7gU8CT3TzFCVWd7nwt7Ga7MZ450tzGO/sOM8/OKnrNvAlDaO9Qm5s7Qrl5ZjETKFPVClVtA5YAi0J3UNU9qvouEHCxDmN63ZpddQQULrf+il5TNGYQA1ISrSkqQrkZFrnA/pDlSmddT6WKSLGIrBeRm3q3NGPOzcrSGjL7JzE9L9PrUmJGki+BOfnZrCqtsUtoI5CbYdHdtYRn8hMwSlWLgNuBn4nI+JNeQORuJ1CKa2vtrxHTN/wdAVbuqOGKiUPsktleduXkIRw82kJJlY1CG2ncDItKIC9keSRQ1dODVbXK+bcCWAXM6Gafx1S1SFWLcnKs7dj0jY17D3O4uZ2rC4d6XUrMuWrSEERgxbZqr0sxXbgZFhuAAhEZKyLJwGKgR1c1icggEUlxHmcDs4FtrlVqzBl4ZXs1yb4Eu2TWBVkDUrhw1CBe2W5hEWlcCwtnVr17gOXAduBpVS0RkYdEZCGAiFwkIpXALcCjIlLiHD4ZKBaRLcBK4N9V1cLCeE5VWbGtmkvHZzEgJex0MOYsXF04lJKqRg4cOeF1KSaEqz/tqroMWNZl3QMhjzcQbJ7qetw6YIqbtRlzNspqjrGnvpm75o7zupSYNb9wKD94aQevbKvmzlljvC7HOOwObmPOwAqneeTqydZf4ZbxOQMYl5NmTVERxsLCmDOwYls1U0cOZNjAVK9LiWlXFw5lfUU9jS3tXpdiHBYWxvRQTVMLm/cfYb6dVbju6slDae9Qu0EvglhYGNNDr22vQRW7ZLYPzBg1iKy0ZF6xS2gjhoWFMT308rZqcjP7MWlYutelxDxfgnDlpCGsLK2hzW+jAUUCCwtjeuDoiXbe2FXHdecPQ8Tu2u4L100ZRlOL3wYWjBAWFsb0wCvbqmnrCHD91OFelxI35uTnkJ6ayF/fPeh1KQYLC2N6ZNl7BxkxMJUZNnBgn0lOTODqwqGs2HbImqIigIWFMWEcPdHOml11XD9luDVB9bEbpw6n0ZqiIoKFhTFhWBOUd6wpKnJYWBgTxrL3DpKb2c+aoDyQnJjANYXDrCkqAlhYGHManU1QdhWUd26YOsyaoiKAhYUxp9HZBHWDNUF5xpqiIoOFhTGn0dkEZdOneqezKerlbYdo9Xd4XU7csrAw5hTqj7Xy+s5abpxqV0F5beH0ETS1+Fm5w8aK8oqFhTGn8OKWKvwB5eYLTppyxfSx2eOzGJKewnObKr0uJW65GhYiskBESkWkTETu62b7ZSKySUT8IvLRLtvuFJFdztedbtZpTHeIZz1WAAARtklEQVSe3XSA83MzmGhjQXku0ZfATTNyWVlaQ8PxNq/LiUuuhYWI+IBHgOuAQuA2ESnssts+4FPAk12OHQw8CFwMzAQeFJFBbtVqTFc7q5t478BRbp5hZxWR4uYLcmnvUF7cUuV1KXHJzTOLmUCZqlaoahuwBFgUuoOq7lHVd4GuF1BfC6xQ1QZVPQysABa4WKsxH/DcpgP4EoSF00d4XYpxTBqWQeHwDGuK8oibYZEL7A9ZrnTW9dqxInK3iBSLSHFtrXV8md7REVCef+cA8ybkkD0gxetyTIibL8hlS+VRymqavC4l7rgZFt1dPqK9eayqPqaqRapalJOTc0bFGXMqb5bXc6ixxTq2I9DC6SPwJQjPbTrgdSlxx82wqATyQpZHAj1tbDyXY405J89tqiQ9NZGrJg/xuhTTxZD0VC4ryOYv7xwgEOjp356mN7gZFhuAAhEZKyLJwGJgaQ+PXQ5cIyKDnI7ta5x1xrjq6Il2lm09yI1TR5Ca5PO6HNONj1w4koNHW1i9y5qe+5JrYaGqfuAegh/y24GnVbVERB4SkYUAInKRiFQCtwCPikiJc2wD8F2CgbMBeMhZZ4yrnt1YSUt7gI9fPMrrUswpXFM4jOwByTyxfp/XpcSVRDefXFWXAcu6rHsg5PEGgk1M3R37OPC4m/UZE0pVeeKtvUzPy+T83IFel2NOITkxgY8V5fGr18s5cOQEuZn9vC4pLtgd3MY43iyvp6L2OHdcMtrrUkwYt188CgWeesvOLvqKhYUxjife2ktm/yQbYTYKjBzUnysnDmHJhv02z0UfsbAwBqhpbOHlkmpuuXCkdWxHiU9cMpq6Y628vO2Q16XEBQsLY4AlG/bjDyi3X2xNUNHisgk5jBzUjyfW7/W6lLhgYWHiXntHgKfe3sfcgmzGZqd5XY7pIV+CcPvFo1hf0cDOaruj220WFibuLd1cxcGjLfzD7DFel2LO0OKLRtEvycejr1d4XUrMs7AwcS0QUB5dXc7EoelcMdHu2I42g9OSufWiPF7YfICqIye8LiemWViYuLaytIad1cf47OXjbDa8KHXXnLEo8Ns3dntdSkyzsDBx7Vevl5Ob2Y8PTbOhyKNV3uD+fGjqcJ56ex9Hmm1iJLdYWJi4tXFvAxv2HOauOWNJ8tmvQjT77OXjaW7r4A9v2pVRbrHfEBO3/ntVBZn9k1g8My/8ziaiTR6ewRUTc/jduj20tHd4XU5MsrAwcWnrgaO8sr2aOy8dQ/9kV4dIM33kn+blU3+8ze67cImFhYlLP1peSmb/JO6aO9brUkwvmTl2MHMLsnlkZRmNLe1elxNzLCxM3FlXXsfqnbV8YV4+GalJXpdjetE3FkzicHM7v15t9130NgsLE1dUlR/+rZThA1O541Ib2iPWnJ87kBunDuc3a3ZT09TidTkxxcLCxJW/bT3Elv1H+Or8CTZgYIz62jUTae8I8F+vlnldSkxxNSxEZIGIlIpImYjc1832FBH5k7P9LREZ46wfIyInRGSz8/UrN+s08aG9I8DDL5eSP2QAN1+Q63U5xiVjs9O49aI8nnp7H3vqjntdTsxwLSxExAc8AlwHFAK3iUhhl93uAg6raj7wU+CHIdvKVXW68/U5t+o08eN/1u6movY431gwiUS7ryKmffmqAlISE/jOiyWoqtflxAQ3f2NmAmWqWqGqbcASYFGXfRYBv3ce/xm4SmzMBeOCA0dO8NMVu5g/eQjzJ9sYULFuSEYq914zkVWltby01ea76A1uhkUusD9kudJZ1+0+quoHjgJZzraxIvKOiLwuInO7ewERuVtEikWkuLa2tnerNzHlwRdKAPjOwvNsDKg4ceeloykcnsG/vlhCk11Ke87cDIvufiO7ng+eap+DwChVnQHcCzwpIhkn7aj6mKoWqWpRTk7OORdsYtPykkO8sr2ar8wvYOSg/l6XY/pIoi+B7988hZqmVn788k6vy4l6boZFJRA6jsJIoOpU+4hIIjAQaFDVVlWtB1DVjUA5MMHFWk2Mampp5ztLS5g0LJ1Pz7Eb8OLN9LxM7rhkNL9/cw9b9h/xupyo5mZYbAAKRGSsiCQDi4GlXfZZCtzpPP4o8JqqqojkOB3kiMg4oACwu2zMGVFV/uX5rVQ3tvC9D0+xwQLj1NevncjQ9FS+8qfNHGv1e11O1HLtt8fpg7gHWA5sB55W1RIReUhEFjq7/RbIEpEygs1NnZfXXga8KyJbCHZ8f05VG9yq1cSm5zYd4PnNVXz5qglcOHqQ1+UYj2SkJvGzxdPZW3/8/b4rc+YkVi4rKyoq0uLiYq/LMBGiovYYN/7XG5yfO5Cn/vESfAnWqR3vfrJiJ//56i5+dut0bpph99l0EpGNqloUbj87Lzcxp9XfwRefeofkxAR+vni6BYUB4EtX5nPRmEHc/5f37Ga9s2BhYWKKqvKNP79LSVUjP/rIVIYP7Od1SSZCJPoS+NniGST6EvjM/xZztNkupz0TFhYmpvxkxU6e31zF/7t2ItecN8zrckyEyc3sx6N3XMje+uN89oliWv02UVJPWViYmPH0hv3812tl3FqUx+fnjfe6HBOhLhmXxcMfncb6igbue/Y9Gw6kh2yKMBMTVmyr5lt/eY+5Bdn824fPt7u0zWndNCOX/Q3N/HjFTnLSU/jmdZPsZyYMCwsT9V7cUsVX/7SZ80Zk8MjHL7D7KUyP3HNlPjVNrTy2uoI2f4AHbiwkwS6GOCULCxPVninezzeefZei0YP57aeKSLeZ70wPiQgPLTqPlMQEfvPGbk60dfD9m6fY1XOnYGFholIgoPz36+U8vLyUuQXZPHZHEf2SbTIjc2ZEhPtvmEz/lET+89VdHDnRxn/cMs3+6OiGna+bqNPY0s5nn9jIw8tLWThtBL/+pAWFOXsiwr1XT+CBGwt5ZXsNNz2ylrKaJq/LijgWFiaqbD1wlEW/WMvKHTU8cGMhP1883aZHNb3i03PG8sRdF3P0RDuLfrGWFzYfsCulQlhYmKhwoq2DH7y0nUWPrOVYq58n//ESPj1nrF3BYnrVpeOzePGLc5g4LJ0vL9nMZ35fTNWRE16XFRFsbCgT0VSVV7bX8N2/bmNfQzO3FuXxresnM7C/tSkb9/g7Avxu3R5+/PJOEgS+Mn8Cd1w6OibPYns6NpSFhYlIqsqqnbX8dMVO3q08yrjsNL734SlcOj4r/MHG9JL9Dc38ywtbWVVay9CMFO65Ip+PXZRHSmLshIaFhYlKx1r9vLD5AH9cv49tBxsZOagfX7qygA9fkGv3TxjPrCuv46crdrJhz2GGpKew+KI8bp05itzM6B97zMLCRI2W9g7W7KpjeckhXnrvIMfbOpg8PINPXjqaj1wwkuRECwnjPVXljbI6Hn9jN6t21iLA3IIcrp8yjPmTh5I1IMXrEs+KhYWJWP6OANsPNvHW7nrWVzSwrryO5rYOMlITufa8Ydx+8Sim52Va57WJWPsbmvnThv08v/kAlYdPkCBw4ehBXDIui4vHZnHB6Ez6J0fHbWwRERYisgD4OeADfqOq/95lewrwv8CFQD1wq6rucbZ9E7gL6AC+pKrLT/daFhaRpyOgVB05we664+ypP86OQ01sq2pkx6FGWtoDAIzJ6s+cgmyuPW8Yl4zLsqYmE1VUlW0HG1leUs2q0hq2HjhKQMGXIIzPSaNweAaThmcwLjuNMdlpjBrcP+I6yT0PC2cO7Z3A1UAlwTm5b1PVbSH7fB6YqqqfE5HFwIdV9VYRKQSeAmYCI4BXgAmqesrxhC0selcgoLQHArR3KK3tHbR1BGhpD3CirYMT7X6a2zpoavFzrMVPY0s7h5vbaDjeTsPxVqobW6lubKGmqZWOwN9/vjJSEykckUHh8IFMyxvIJeOyGJqR6uG7NKZ3NbW0s3HvYTbtPcy2g41sq2qk6mjLB/YZnJbMsIxUhg1MZXBaMllpyQxKSyYjNYkBqYmkpyaSlpxI/2QfqUk++iX7SPYlkJKUQLIvgSRfQq8OSdLTsHDzPGkmUKaqFU5BS4BFwLaQfRYB33Ee/xn4hQTbHhYBS1S1FdjtzNE9E3izt4s80tzGLb/q9ac9yblE8qkCXUMehO6hqijQeZiiBAJ/3xZQCIT82xFQAgGlQxV/ILgc+iHfE74EYVD/ZAanJTE0I5X8IdkMy0glb3A/xmSlMTorjaEZKda0ZGJaemoS8yYOYd7EIe+vO9rczp764Nn13vpmDjW2UH20hUONLWw/2Ej98Tba/IEzep0EgcSEYGgkJgjT8jJ54jMX9/bb+QA3wyIX2B+yXAl0fTfv76OqfhE5CmQ569d3OfakSXNF5G7gboBRo0adVZEJCULB0AFndeyZEs7hg/IUh3auFvngs4sEt3V+OAeXBRHwSfBfEcGXEFxOcH7oOv9Ncv6CSfIJKYk+UhITSE5MoH+yj37JifRL8pHu/BWUnppEekqijdhpTDcG9k9iWv9MpuVldrtdVd8/U29qaaexxc+Jtg6a2/ycaO+gtT1Aq7+DVn+Ato4A7X6lvSOAP6AEVPF3KCMy3T9DdzMsuvvk6Prn6qn26cmxqOpjwGMQbIY60wIBMlKT+OXHLzybQ40x5pyJCGkpiaSlJDJsYOQ2y7rZm1gJ5IUsjwSqTrWPiCQCA4GGHh5rjDGmj7gZFhuAAhEZKyLJwGJgaZd9lgJ3Oo8/CrymwQb6pcBiEUkRkbFAAfC2i7UaY4w5DdeaoZw+iHuA5QQvnX1cVUtE5CGgWFWXAr8F/uB0YDcQDBSc/Z4m2BnuB75wuiuhjDHGuMtuyjPGmDjW00tn7Q4oY4wxYVlYGGOMCcvCwhhjTFgWFsYYY8KKmQ5uEakF9npdx1nIBuq8LsID8fi+4/E9Q3y+72h6z6NVNSfcTjETFtFKRIp7ciVCrInH9x2P7xni833H4nu2ZihjjDFhWVgYY4wJy8LCe495XYBH4vF9x+N7hvh83zH3nq3PwhhjTFh2ZmGMMSYsCwtjjDFhWVhEEBH5uoioiGR7XUtfEJGHRWSHiLwrIn8Rke6nEosBIrJAREpFpExE7vO6HreJSJ6IrBSR7SJSIiJf9rqmviQiPhF5R0T+6nUtvcXCIkKISB5wNbDP61r60ArgfFWdCuwEvulxPa4QER/wCHAdUAjcJiKF3lblOj/wNVWdDFwCfCEO3nOoLwPbvS6iN1lYRI6fAv9MN9PHxipVfVlV/c7ieoIzIsaimUCZqlaoahuwBFjkcU2uUtWDqrrJedxE8IMz19uq+oaIjARuAH7jdS29ycIiAojIQuCAqm7xuhYPfRp4yesiXJIL7A9ZriROPjgBRGQMMAN4y9tK+szPCP7hF/C6kN7k2kx55oNE5BVgWDeb7ge+BVzTtxX1jdO9b1V9wdnnfoLNFn/sy9r6kHSzLi7OIEVkAPAs8BVVbfS6HreJyI1AjapuFJF5XtfTmyws+oiqzu9uvYhMAcYCW0QEgk0xm0Rkpqoe6sMSXXGq991JRO4EbgSu0ti96acSyAtZHglUeVRLnxGRJIJB8UdVfc7revrIbGChiFwPpAIZIvKEqn7C47rOmd2UF2FEZA9QpKrRMmLlWRORBcBPgMtVtdbretwiIokEO/CvAg4AG4DbVbXE08JcJMG/fH4PNKjqV7yuxwvOmcXXVfVGr2vpDdZnYbz0CyAdWCEim0XkV14X5AanE/8eYDnBjt6nYzkoHLOBO4Arnf/bzc5f2yZK2ZmFMcaYsOzMwhhjTFgWFsYYY8KysDDGGBOWhYUxxpiwLCyMMcaEZWFhjDEmLAsLY4wxYVlYGOMSEbnImasjVUTSnHkdzve6LmPOht2UZ4yLROTfCI4R1A+oVNUfeFySMWfFwsIYF4lIMsGxoFqAWara4XFJxpwVa4Yyxl2DgQEEx8BK9bgWY86anVkY4yIRWUpwZryxwHBVvcfjkow5KzafhTEuEZFPAn5VfdKZh3udiFypqq95XZsxZ8rOLIwxxoRlfRbGGGPCsrAwxhgTloWFMcaYsCwsjDHGhGVhYYwxJiwLC2OMMWFZWBhjjAnr/wMCjMgmExXNoAAAAABJRU5ErkJggg==\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "sigma = 1.0 # standard deviation\n", "mu = 1.0 # mean\n", "x = np.arange(-5,5,0.1)\n", "y = 1/np.sqrt(2*np.pi*sigma)*np.exp(-(x-mu)**2/(2*sigma**2))\n", "plt.plot(x,y)\n", "plt.xlabel('x')\n", "plt.ylabel('P(x)')\n", "plt.title('Normal distribution')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Why were your instructors so confident that the distribution of errors is Gaussian? To understand this, we will start from afar, from random walks.\n", "\n", "Random walks is probably my favorite subject in the modern physics curriculum. Many of the famous names in the late nineteenth and early twentieth century physics and math have contributed to the field, including Einstein, Polya, Wiener, and many others. A single lecture cannot do justice to the subject, and I hope you will pick some of the books to read instead. The \n", "__[Introduction to Probability](http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/pdf.html)__ by Grinstead and Snell is a good starting point. Another great book is __[Random Walks in Biology](https://www.amazon.com/Random-Walks-Biology-Howard-1993-09-07/dp/B01MQZYDZ4)__ by Howard Berg. \n", "\n", "In general, **random walk** is *a stochastic process that consists of a succession of random steps*. The usual and overused analogy is that of a drunkard, who tries to walk, falls, stands up, and, having forgotten which way he came from, chooses a random direction to go to, before falling again and repeating the cycle. There are many different types of random walks. *Discrete time* (DT) walks mean that every step takes a discrete unit of time. In contrast, in *continuous time* (CT) walks, the duration of each step is a real-values variable, sampled from some distribution. Similarly, one can define *discrete space* (DS) or *continuous space* (CS) random walks by whether the step length is fixed or sampled at random from some probability distribution. As we will see later in this lecture details of these probability distributions matter very little (as long as they have finite variances of their own), and we can learn a lot about many different random walks by studying the simplest version of a random walk, the DSDT walk with the duration of a step $\\Delta t=1$ and the step size $a=1$. Such walks can exist in any number of dimensions (with 1-d being, of course, the simplest). However, for illustration purposes, below we will mostly focus on 2-d walks on a square lattice, where the walker at any point in time can move in one of four directions: up, down, left, or right. Let's generate a few such walks of different lengths." ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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\n", 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# This function will generate a 2d DSDT random walk\n", "def DSDT_2d_walk(N):\n", " # Input:\n", " # N - duration of the walk\n", " # Return:\n", " # x,y -- arrays of length N, coordinates of the walker on every step\n", " \n", " directions = np.floor(nprnd.random(N)*4) # (0,1,2,3) directions of motion on each time step\n", " # Do you understand why multiplication by 1.0 is needed in the next two lines?\n", " # Check what happens if this multiplication is removed.\n", " dx = (directions == 0) - 1.0*(directions == 1) # directions == 0/1 corresponds to right/left motion\n", " dy = (directions == 2) - 1.0*(directions == 3) # directions == 2/3 corresponds to top/down motion\n", "\n", " x = np.cumsum(np.hstack((0,dx))) # x and y coordinates of the walker are a cumulative sum of steps\n", " y = np.cumsum(np.hstack((0,dy))) # Notice that the initial position of the walker is always 0. \n", " # Also notice that the walk of N steps has N+1 positions in it -- there is the initial and the final\n", " # position, surrounding N steps.\n", " \n", " return x, y\n", "\n", "\n", "N = np.array((100, 1000, 10000)) # we will explore random walks of these different lengths\n", "for i in np.arange(N.size):\n", " x, y = DSDT_2d_walk (N[i])\n", " plt.plot(x, y, 'b-o')\n", " plt.title('Random Walk of length ' + str(N[i]))\n", " plt.xlabel('x coordinate')\n", " plt.ylabel('y coordinate')\n", " plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Notice how a moving particle comes back again and again, and how random walk eventually covers almost the entire 2-d space, with very few holes in the middle. This is a crucial property of random walks, which we will refer to again in one of the projects for this Module.\n", "\n", ">### Verification\n", "To verify that the function generates random walks correctly, generate very short walks, e.g., `N=4...10` and see if the walker makes any unallowed steps visually. \n", "\n", ">### Your turn\n", "Run the cells multiple times and convince yourself that the walks are not biased. Sometimes they move more to one side or the other, but other times it's the opposite.\n", "\n", "We can now see that the details of how the individual steps are generated in a walk matter much less than one might naively think. For this, we define a CSDT walk, where in each individual step, the walked moves in both $x$ and $y$ directions by steps of sizes taken from ${\\cal U}(0,2^{1/4})$. We then compare the DS and the CS walks to each other. \n", "\n", "Note the weird $2^{1/4}$ multiplier. We do this to ensure that the average steps we take in each of the walks are equal. Indeed, for CSDT walk, the steps, on average, are smaller than 1, but we make steps in both the $x$ and the $y$ direction always. The algrebra is a bit more complex than I want to go over here, but I hope the logic is clear." ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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\n", 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# This function will generate a 2d CSDT random walk with steps taken from the uniform distribution\n", "def CSDT_2d_walk(N):\n", " # Input:\n", " # N - duration of the walk\n", " # Return:\n", " # x,y -- arrays of length N, coordinates of the walker on every step\n", " \n", " dx = 2**(1/4)*(2*nprnd.random(N) - 1) # displacements along x\n", " dy = 2**(1/4)*(2*nprnd.random(N) - 1) # displacements along y\n", " \n", " x = np.cumsum(np.hstack((0,dx))) # x and y coordinates of the walker are a cumulative sum of steps\n", " y = np.cumsum(np.hstack((0,dy))) # Notice that the initial position of the walker is always 0. \n", " # Also notice that the walk of N steps has N+1 positions in it -- there is the initial and the final\n", " # position, surrounding N steps.\n", "\n", " return x, y# comparing different ways of generating random walks\n", "\n", "\n", "N = np.array((100, 1000, 10000)) # we will explore random walks of these different lengths\n", "for i in np.arange(N.size):\n", " xD, yD = DSDT_2d_walk (N[i]) # discrete walk of this length\n", " xC, yC = CSDT_2d_walk (N[i]) # continuous walk of this length\n", " plt.plot(xD, yD, 'b-')\n", " plt.plot(xC, yC, 'r-')\n", " plt.title('Different types of Random Walks of length ' + str(N[i]))\n", " plt.xlabel('x coordinate')\n", " plt.ylabel('y coordinate')\n", " plt.legend(('DSDT walk', 'CSDT walk'))\n", " plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Notice that we can easily see the difference between the two walks when there are only a few steps, and the walks do not move far away from the origin. However, at $N=10000$, when individual steps are invisible, it would be hard to distinguish one walk from the other. \n", "\n", "To formalize this observation, we will look at the means and the variances of displacements in walkers of different types. For this, we will generate many different walks of the same type and the same length and average." ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "N = 10000 # length of a walk\n", "N_walks = 5000 # number of walks for averaging\n", "xD = np.zeros((N_walks,N+1)) # the x coordinates of the N_walks walks of length N, DSDT walk\n", "yD = np.zeros((N_walks,N+1)) # the y coordinates of the N_walks walks of length N, DSDT walk\n", "xC = np.zeros((N_walks,N+1)) # the x coordinates of the N_walks walks of length N, CSDT walk\n", "yC = np.zeros((N_walks,N+1)) # the y coordinates of the N_walks walks of length N, CSDT walk\n", "\n", "for i in np.arange(N_walks):\n", " xD[i,:], yD[i,:] = DSDT_2d_walk(N) # generate a DSDT walk\n", " xC[i,:], yC[i,:] = CSDT_2d_walk(N) # generate a DSDT walk\n", "\n", "# mean x and y positions of both types of walks\n", "mean_xD = np.mean(xD, 0)\n", "mean_yD = np.mean(yD, 0)\n", "mean_xC = np.mean(xC, 0)\n", "mean_yC = np.mean(yC, 0)\n", "\n", "plt.plot(mean_xD, 'b-')\n", "plt.plot(mean_xC, 'r-')\n", "plt.title('Mean x displacements for different walks of length ' + str(N))\n", "plt.xlabel('time')\n", "plt.ylabel('x coordinate')\n", "plt.legend(('DSDT walk', 'CSDT walk'))\n", "plt.show()\n", "\n", "plt.plot(mean_yD, 'b-')\n", "plt.plot(mean_yC, 'r-')\n", "plt.title('Mean y displacements for different walks of length ' + str(N))\n", "plt.xlabel('time')\n", "plt.ylabel('y coordinate')\n", "plt.legend(('DSDT walk', 'CSDT walk'))\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that, in the example above, we make $N=10^4$ steps, and the mean $x$ and $y$ displacements for 5000 random walks are on the order of about only 1 even at the very end of the time interval. This is relatively very small, and probably consistent with the displacement actually being zero, on average.\n", "\n", ">### Your turn\n", "Run the cells above with larger $N$ and convince yourself that the mean $x$ and $y$ displacements decrease as $N\\to\\infty$.\n", "\n", "So, both CS and DS walks result in the zero mean displacement. Let's now look at what's known as the *mean squared displacement* (which, for the unbiased walks like we are dealing with is equivalent to the walk variance), ${\\rm MSD}= \\left$, where $\\left<\\dots\\right>$ denotes averaging over realizations. " ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "image/png": 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D9MASq7R6UCmlVKKhQUsppVSioUFLKaVUovHI3dNyJyQkhMDAQK5fj6qTbfWgUqdOTd68eUmRIoWns6KUSuQ0aAGBgYFkyJCBAgUKcHeHw+phGWM4d+4cgYGBFCxY0NPZUUolcnFWPSgik0TktIjscEnLKiKLRGS/8z+Lky4iMkpEDojINhF52mWdts7y+0WkrUt6ORHZ7qwzSh4i2ly/fp1s2bJpwIoDIkK2bNm0FKuUihVxeU9rCnZIAFe9gCXGmKLAEu4Mr1AP2zVJUaAD8B3YIIftvfxZ7BAgfcIDnbNMB5f1Ir7XfdGAFXf02CqlYkucBS1jh38/HyG5CXZIc5z/TV3SfzDWOiCziDwO1AEWGWPOG2MuAIuAus68jMaYtU7/Wz+4bEsppZSLD/7vBl99BSERx2ZOhOK79WAuZ2C+8AH6cjrpebh7XJxAJy2q9EA36W6JSAcR8RMRvzNnzjz0TsSFZMmS4ePjg7e3N2XKlGHYsGGEhdlRvq9du8arr75KqVKlKFmyJM8//zxXrlyJcr0FCxbg4+ODj48P6dOnp1ixYvj4+NCmjbsBWh9MgQIFOHv2LAEBAZQsWTL6FZRS8coY6NTlBsNveNNrxBaSJ4FWDAllF9zVH5kHSHfLGDMeGA/g6+ubIHsITpMmDVu2bAHg9OnTtG7dmkuXLtG3b19GjhxJrly52L59OwB79+693RIvqvXq1LGDOlerVo0hQ4bg6+vrgT1TSnnC7oOX6dMrIzNnpoJS/Zj1a0qSQk19fJe0TjlVezj/TzvpgUA+l+XyYrvgjyo9r5v0JCFnzpyMHz+eb775BmMMJ06cIE+eOwXJYsWKkSpVqmjXi4nOnTvz559/AvDiiy/y5ptvAjBx4kR697bjWjZt2pRy5crh7e3N+PHjo9zewYMHKVu2LBs2bIjR+yulYt9n49ZRYkJ+ft+8nPfeg6A1rWlepYSnsxUr4ruk9SfQFhjk/J/jkt5FRKZjG11cMsacEJEFwJcujS9qAx8bY86LSJCIVADWY4drHh0bGezWDZyCS6zx8YERI+5vnUKFChEWFsbp06d58803qV27NrNmzaJmzZq0bduWokWLRrterly5on2fKlWqsHLlSho3bsyxY8c4ceIEAKtWraJVq1YATJo0iaxZsxIcHEz58uVp3rw52bJlu2dbe/fupVWrVkyePBkfH5/722Gl1EO7csXw2WfCiG9LQ4OmLJiZjxpJ7KsYl03epwFrgWIiEigib2GDVS0R2Q/UcqYB5gEHgQPABKAzgDHmPNAf2OD89XPSADoB3zvr+AP/xNW+eEp4acnHx4eDBw/Ss2dPzp8/T/ny5dm9e3e068VE5cqVWblyJbt27aJEiRLkypWLEydOsHbtWp577jkARo0aRZkyZahQoQJHjx5l//7992znzJkzNGnShJ9++kkDllIe8NHM8WTtUY0Ro0Ip8kRaDg2fQg2fwp7OVqyLs5KWMeaVSGbVdLOsAd6NZDuTgElu0v2AWL/7f78lorhy8OBBkiVLRs6ctq1K+vTpadasGc2aNcPLy4t58+ZRvHjxaNeLTp48ebhw4QLz58+nSpUqnD9/nhkzZpA+fXoyZMjA8uXLWbx4MWvXriVt2rRUq1bN7TNXmTJlIl++fKxevRpvb++H23mlVIxdvw79+8PgOZlJXT4z/Ydd4b23M5M2radzFjcSSkMM5eLMmTN07NiRLl26ICKsXr2aEiVKkCVLFm7evMmuXbuoVq1atOvFVMWKFRkxYgRLly7l3LlzvPTSS7z00ksAXLp0iSxZspA2bVr27NnDunXr3G4jZcqU/PHHH9SpU4f06dPTunXrB9p3pVTMhNwKoXLvAayfWwy2t+ap4i3496sW5MyZBFpbREGDVgIRHByMj48PISEhJE+enNdff50PPvgAAH9/fzp16oQxhrCwMBo0aEDz5s2jXS+mKleuzMKFCylSpAhPPPEE58+fp3LlygDUrVuXsWPHUrp0aYoVK0aFChUi3U66dOn4+++/qVWrFunSpaNJkyYPeDSUUlG5cgUaNvJifaHFkOci0z5pzcsvS5JoHRgduZ/7H0mBr6+viTgI5O7du91WtanYo8dYqYd39eZV3p7yNdPe6wE3M1C9djAzfk5D9uxx+74istEYkyCemdGhSZRSKhHw94esxbcz7Vh/KPoPAwfCkvlxH7ASGg1aSimVgB08fpH63eZSpAjcPFiBuvv3c3JJSz75hJhXB4aFQUBAXGYz3mjQUkqpBGr2bCj8Ti/+SdeSCtXPcfgw/PNLIWLwCOYdFy9C8eJQpQqEhsZZXuOLBi2llEpg9gSeJE+xkzRvDizryxcFVrBmSTby57+PjRgDv/wCxYrBvn1wnw20EiptPaiUUglESAi81eEGP2b1hTIVea/OTHr3zkXOnPdTtAJWr4Z+/WDhQjvdr5/t7icJ0KCllFIJwJhJF+jVLQtBQanAexhTB5emTf0H2NDMmfDqq5ApEwwcCL16gVfSqVRLOnuSyJ08eZJWrVpRuHBhSpQoQf369dm3bx9hYWF07dqVkiVLUqpUKcqXL8+hQ4cAOzRIqVKlKFWqFCVKlKB3797cuHGD7du33x6WJGvWrBQsWBAfHx9eeOGFWMtvtWrVCH90IH369LG2XaUeNRcuQJ4Kq3n3QH6Csi/lo4/g1raWtKn/1P1t6OhRKFUKWraEbNlgxw745JMkFbBAS1oJgjGGF198kbZt2zJ9+nQAtmzZwqlTp9i4cSPHjx9n27ZteHl5ERgYSLp06W6vu2zZMrJnz86VK1fo0KEDHTp0YOrUqbeHK2nXrh0NGza83cOFUirhCL0VRq1aXhzfVpaMRVvx1++FqVLmPjdiDHz7Lbz3np1u2BCmTYMk+mMyaYXgRGrZsmWkSJGCjh073k7z8fGhcuXKnDhxgscffxwv59dS3rx5yZIlyz3bSJ8+PWPHjuWPP/7g/PmIA0a7N3jwYEaNGgVA9+7dqVGjBgBLlizhtddeA6BTp074+vri7e1Nnz59otze2bNnqVixInPnzo3R+yv1KOu/4FvSv1eFjZtD6dopLZd+nECVMk/c30Y2bYIUKWzAKlkSdu+Gv/5KsgELtKR1Lw+MTbJjxw7KlSvndl7Lli15/vnnWblyJTVr1uS1116jbNmybpfNmDEjBQsWZP/+/Tz77LPRZqtKlSoMHTqUrl274ufnx40bNwgJCWHVqlW3u3EaOHAgWbNm5datW9SsWZNt27ZRunTpe7Z16tQpGjduzIABA6hVq1a0763Uo+rUKfjmGxgwOyd45+bTL67Rv3fG+9vImjX2XtXKlXZ61Ch4990kVxXoTtLfw0Qub9687N27l//97394eXlRs2ZNlixZEuny99MtV7ly5di4cSNBQUGkSpWKihUr4ufnx8qVK28HrRkzZvD0009TtmxZdu7cya5du+7ZTkhICDVr1mTw4MEasJSKxM1bN2k96WOKNPuJAQOAXS0Y98IMBnyWMeYPCd+4AZ07Q6VKsG4ddOkCJ0/aktYjELBAS1r38sDYJN7e3syaNSvS+alSpaJevXrUq1ePXLly8ccff1Cz5j0jvBAUFERAQABPPvlkjN43RYoUFChQgMmTJ/Pcc89RunRpli1bhr+/P8WLF+fQoUMMGTKEDRs2kCVLFtq1a+d2WJLkyZNTrlw5FixYQNWqVWO+40o9IoyBdzslZ9qtlaTOHsLy5a9x31+VzZvh6aft69q1Ydw4KFAgtrOa4D0aoTmBq1GjBjdu3GDChAm30zZs2MC///7Lpk2bOH78OABhYWFs27aNJ564t977ypUrdO7cmaZNm7q95xWZKlWqMGTIEKpUqULlypUZO3YsPj4+iAiXL18mXbp0ZMqUiVOnTvHPP+7H2RQRJk2axJ49exg0aJDbZZR6FF2+cZke8z6hftMgvp/gxdM7FnN4wpD7C1hnz0LXrjZgicCkSbBgwSMZsECDVoIgIvz+++8sWrSIwoUL4+3tzRdffEHu3Lk5ffo0jRo1omTJkpQuXZrkyZPTpUuX2+tWr16dkiVL8swzz5A/f37GjRt3X+8d3tijYsWK5MqVi9SpU9+uGixTpgxly5bF29ubN998k0qVKkW6nWTJkjF9+nSWLVvGmDFjHuxAKJXETJ23i2Hrvmb+/kV07gwb1qYmhuOzwrlzULQo5MgBo0dDiRKwaxe88Uac5jmh06FJ0GEz4oMeY/WoOHvtLJ+OXc3WGU1Yvx7SPn6EJbPzE8VQdHf77jvo2ROuXr2TtmQJVK9+Hz3kxi4dmkQppZKgwYMhx8ufMv7cq6zfeoH27WG/330ErN9+sw0trl6Fl1+GrVvtDbEaNTwWsBIabYihlFIP6djlY/Tr58X4oY9Duv5UStGFuSezkClTDDewdy885fSA4etr71llzRpn+U3MtKSllFIPITjkOk8OKc/4I+9TsybcOJ+TVbNLxTxgDR9+J2CVKgWrVmnAioKWtJRS6gGcvXaWdJKdMmVScy35aFJeKMOf/pAyZQw3YIwNWD162B4stm6FQoXiNM9JgZa0lFLqPq04vIL8w54grfdS9u+HTtWaczmgCGnTxnADZ8/aJus9ekCTJrbXXA1YMaJBSymlYuhW2C0AUpzxJXhdWzhbjMGDYcwYSJUqhhvx84MKFeDIEahfH2bNguRa6RVTGrQSCBHh9ddfvz0dGhpKjhw5aNiwIWD79mvYsCFlypS5PXQJQEBAAGnSpKFs2bIUL16cZ555hqlTp8b4fS9evHjXc1XHjx/XHuGVcmPEuhFUGP88VWuE8lz5tDB3DCvm5qFnzxhuwBgYOdIGrBs37ECNc+dqwLpPerQSiHTp0rFjxw6Cg4NJkyYNixYtIk+ePLfnf/7559SqVYv3338fgG3btt2eV7hwYTZv3gzAwYMHadasGWFhYbwRg4cQw4NW586dAcidO3eUXUop9ag6sS8PfosKw+pgatTIwNSpkDdvDFc+fBjq1LGtBOvXh59/hsyZ4zS/SZWWtBKQevXq3R7WY9q0abzyyiu35504cYK8Lt8Qdz2tAxQqVIhhw4bdHnIkOr169cLf3x8fHx969uxJQEAAJUuWBGDKlCk0bdqURo0aUbBgQb755huGDRtG2bJlqVChwu0hUPz9/albty7lypWjcuXK7Nmz54H2X6mE5EboDbrP784PW3+gWzcY3LYFKf76iWULMrBkyX0ErDVroEwZG7BGjbJDh2jAenDGmEfqr1y5ciaiXbt23TVddXJVM3nzZGOMMTdDb5qqk6uaH7f+aIwx5urNq6bq5Kpm+vbpxhhjLgZfNFUnVzW/7frNGGPMmatnTNXJVc2fe/40xhhzIujEPe/nTrp06czWrVtN8+bNTXBwsClTpoxZtmyZadCggTHGmPnz55tMmTKZatWqmQEDBphjx44ZY4w5dOiQ8fb2vmtbFy5cMKlTpzbGGLN06VJTpkyZe/4qVqzodn3X6cmTJ5vChQuby5cvm9OnT5uMGTOa7777zhhjTLdu3czw4cONMcbUqFHD7Nu3zxhjzLp160z16tWjPcZKJXQ3Q26ZwgMqm9QNexkw5umnjTkRs6+zFRRkzJNPGmMrBo35++84y2tcA/xMArh+G2O0ejAhKV26NAEBAUybNu32PatwderU4eDBg8yfP59//vmHsmXLsmPHDrfbMS5dc1WvXv32KMYPonr16mTIkIEMGTKQKVMmGjVqBECpUqXYtm0bV65cYc2aNbRo0eL2Ojdu3Hjg91PKky4EX2DAioGk9+vDmBEZOHthMdxKSd++8NFH99HYYvZs2zIwIABy5rT3r4oUicusPzI0aLmxvN3y269TJEtx13TaFGnvms6UOtNd09nTZr9r+rH0j93Xezdu3JgPP/yQ5cuXc+7cubvmZc2aldatW9O6dWsaNmzIihUr3A4euXnz5tv9/C1btozu3bvfs0zatGlZs2ZNtPlJ5fIt9fLyuj3t5eVFaGgoYWFhZM6c+aECo1IJxYpdexm2ejTMqELl4o0pVSolgwdDunQxWPn6dVv999FHdvqxx2DePKhbV7tgikUatBKYN998k0yZMlGqVCmWL19+O33p0qVUqFCBtGnTEhQUhL+/P/nz579n/YAolZifAAAgAElEQVSAAD788EPee+89IPqSVoYMGQgKCnrg/IaPljxz5kxatGiBMYZt27ZRpkyZB96mUvHp1JVTrDqyGnY3o2PHCnDjEB1eyc1338VwXMU1a2DoUDsoozOMEAMHQvfukCZNnOb9UeSRhhgi0l1EdorIDhGZJiKpRaSgiKwXkf0i8quIpHSWTeVMH3DmF3DZzsdO+l4RqeOJfYltefPmvd1C0NXGjRvx9fWldOnSVKxYkfbt21O+fHnANoQIb/LesmVL3nvvvRi1HATIli0blSpVomTJkvSMcdvdu/38889MnDiRMmXK4O3tzZw5cx5oO0p5QofpvWn5Szteeu0iWbPCmgW5GTcuhgHr5ZftKMKzZ8Pp0zZYBQXBJ59owIor8X0TDcgDHALSONMzgHbO/1ZO2ligk/O6MzDWed0K+NV5XQLYCqQCCgL+QLLo3j8mDTFU7NNjrBKSgAsB5tjlY2bKFGNIe9qQfbcZM8aYmzdjuIFbt4xp2tQ2sHjsMWOWLzfm2rU4zbMnkYAaYniqyXtyII2IJAfSAieAGkD4A0JTgabO6ybONM78miIiTvp0Y8wNY8wh4ADwTDzlXymVSAWHBOM77ll8enWnXTvIliYH+9c+RadOkCJFDDawe7d91uqPP6BlSzh4EKpW1ZJVPIn3oGWMOQYMAY5gg9UlYCNw0RgT6iwWiC2R4fw/6qwb6iyfzTXdzTp3EZEOIuInIn5nzpyJ3R1SSiUKp66cAuDAnjScnTKWMz8PpkMHCAyMYcO+sDA7arC3t+2J/euvYfp0DVbxLN6DlohkwZaSCgK5gXRAPTeLhrfbdtfsxkSRfm+iMeONMb7GGN8cOXK4zZcxj9YIzvFJj63ytGWHllFgRAFKNFhC6dLAnqZMHPYE48ZB6tTRrGyM7W4pWTKYMgXat7dN2T/8UFsFeoAnWg++ABwyxpwBEJHZwHNAZhFJ7pSm8gJOMxwCgXxAoFOdmAk475IeznWd+5I6dWrOnTtHtmzZED0JY5UxhnPnzpE62iuDUrEv5FYIKZKlIJ9XBVLu6MDuf0uSIQMcOGAfn4pWaKgd4yq8l5e334Zx4zRYeZAngtYRoIKIpAWCgZqAH7AMeAmYDrQFwpug/elMr3XmLzXGGBH5E/hFRIZhS2xFgf8eJEN58+YlMDAQrTqMG6lTp76rCyql4sPg1YP5bddvdE69mu7vpyHk+kjGjbBxJ0Yx5+hRaNTIBqxeveDzz7UqMAGI96BljFkvIrOATUAosBkYD8wFpovIACdtorPKROBHETmALWG1crazU0RmALuc7bxrjLn1IHlKkSIFBQsWfIi9UkolNHnTFmbHspK0m3WDir7JmToVihaNwYohITBiBPTrZ0taX30F//d/cZ5fFTPyqN1v8PX1NX5+fp7OhlIqll0PvU6PBT3wzV2etPvaMWAA7NgBbdrA5MkxeO4qLAwGDLANLK5csb2yf/cd6A9aRGSjMcbX0/kA7RFDKZVEpEyWkv8CdjL75+ycnGZHsP/tN2jWLAYr+/vDSy/Bli2QIQMMGwbduum9qwRIhyZRSiVaZ6+d5f1/3udi8GX6fuGFX9fFnJzWl8GD4dKlGASsgACoUcO2ed+yBbp2tWndu2vASqC0pKWUSrT8z/szbuM4/GbUZs2UBtSpk5x+/eCZ6LoZ2LLFDsQ4cqS9h6U9sScaGrSUUonKscvHWHN0DS28W7Dxz2e5Megwa67m4uuv7WggURaQQkNtL+wjR8KtW5ArF3TuDJ99piWrREKDllIqUemzvA+zds3Cf1FtPu6eiZIlczFkiG03EaXly6F6dfu6cmXbm0Xu3HGdXRXL9J6WUirB8z/vz7HLxwAY9MIguqT04+PumWjSBDZtiiZgHTsG1arZgOXlBe3awb//asBKpDRoKaUStGsh16gwsQI9FvbgyhXo8mZ2BvYoQuPG8Ouv0XRyu3QplC1rg1TXrrYp++TJWhWYiGnQUkolSOElq7Qp0jKx0SQK7h9Khgw2UOXNa5uzuwysfbfly20VYM2akD077Npl72NpjxaJngYtpVSCs/jgYgqOLMjig4tZuBBeLd+IQZ/kIVs2W3g6cgSSu7sjf+MGtG1rqwJXrYIuXeC//6B48XjfBxU3NGgppRKMm7duAvB8/ufp+uz7LP7Jhzp1bK1e5852cODq1SOp3Tt0CKpUgR9+gFdfhcuXYfRo+5SxSjK09aBSKkEYuGIgv+/5nVXt1vLt6NQM/fBrAJo3h/HjIWvWSFa8etX2gjttmh1nZOZM27uFSpI0aCmlEoTiOYqz48gxsucK4erFFIjA2LHR9Mq+ZYsdmHHLFmjVCj79FEqWjNd8q/il1YNKKY8IDgmm/Z/tmbx5MgCPX2zGb2+O4erFtEyaBDdvQocOkQSsmzdtO/eyZe0Nrt9/tyUtDVhJnpa0lFIekSp5Kg5eOEiBzAUZNsz2ZlGwoG1oUaBAFCueP2/7afL3B19f+Ptv27OFeiRoSUspFW9OXTlF57mduXzjMl7ixbhKi1je71N69IA8eWDZsmgC1o4dtnTl7w8dO9qWgRqwHikatJRS8SbgYgBTtkxh9oY1NGsGTxVLxpIl0KcP7NwJTzwRyYrXr8PAgVCqlO0/cN06O9aVPiT8yNHqQaVUnDp88TBrjq7hlVKvkPbCsxT+8whv9M5OsmR2gMa334bnnotiAxs2QKdOsHEj1KplW2cUKhRv+VcJiwYtpVSc6r+iPzO2z+bd2g24cDIjkJ2cOeGvv6IZQmTHDrtAcLDtyeL77+Gtt+Ir2yqB0upBpVSs23N2D4GXAwkNhZzbBhE0ZBMXTmakSxfYvx9OnYoiYC1fDmXK2KrA4GD48EMIDNSApQAtaSmlYtnVm1epNKkST4TW5sx30wgMzE7TptkZOzaaNhNBQXZcq5Ej7XStWjBhQhQ3utSjSIOWUipWHLl0hPyZ8rNvZzoyLP6RzWvKUfZJGD7c9moRZZuJZcugdWs4eRLeeQdGjLC9WygVgQYtpdRDW3BgAQ2nNaTRpX/4fegLQH3eest2v+QV1U2IsDDbV2DHjvYhrTlzornRpR51GrSUUg/seuh1zpxIzYZZVUm3pSe/zytHiRK2gV/lylGsuH8/vPwybN5spytVsgErW7Z4ybdKvLQhhlLqgXyy8AvyfVGJok+F8NnHqSkc8CVzfs3Czp1RBKwjR6BbN3jySRuwihWz961WrNCApWJES1pKqft2+DBM6O/D2XSXaFArlL6fpaBcuShWuHkT+vaFL7+00y1bQuPGdggRpe6DBi2lVIxcC7lGx787kuJYNX7//E0uXGjKhAlNad8+ipVCQuzzVb172z4D06e3JawiReIt3ypp0aCllIqRg/tSM3vRCa5uPcfTBWHhQttfbaSCguCFF2z/gOnS3RlnJMqWGUpFTYOWUipSxy4fo8/yPpQ6PpRunTKBLGD0KC86dYJkyaJYMSDAtgZMlgzGjLHVgXrPSsUCDVpKKbdCQ+Gzr48xxUzHTHuZxx6rxfr1XuTPH82KixbBa6/Z1/PmQe3acZ5X9ejQcrpS6i7+5/2Z6PcjmTLB5AHPkOvnowx8qxaHDxN1wNqxA6pVs0Eqc2bYulUDlop1HglaIpJZRGaJyB4R2S0iFUUkq4gsEpH9zv8szrIiIqNE5ICIbBORp12209ZZfr+ItPXEviiV1Hz410A6/v4B10KD+PZbOHEoC598AilTRrLCuXPw8cd2nKt//7U9W2zdCqVLx2u+1aPBUyWtkcB8Y8xTQBlgN9ALWGKMKQoscaYB6gFFnb8OwHcAIpIV6AM8CzwD9AkPdEqp+7Pj9A6OXjrK3Lmw5KOvSf/TFiZ8m4HOnaNYKSgIOnSA7Nlh0CBo1QpOnICff9YumFScifd7WiKSEagCtAMwxtwEbopIE6Cas9hUYDnwEdAE+MEYY4B1TintcWfZRcaY8852FwF1gWnxtS9KJQVXbl6h8qQqPH6lHrsH/kzJktn4++8o+qkNDISaNWHfPjtduzZ88AHUqRNveVaPLk80xCgEnAEmi0gZYCPwPpDLGHMCwBhzQkRyOsvnAY66rB/opEWWfg8R6YAtpZE/2rvISj0aDl44iNelQrz6anqS3ZrG7u3leOMN+OYbSJvWzQrGwLff2gAVEmJHbvz662hGcFQqdnmiejA58DTwnTGmLHCVO1WB7rjrG9pEkX5vojHjjTG+xhjfHDly3G9+lUpyZu+YR5GRRSlYczFr1sBTyeswZ1p2Jk1yE7CMgb17oUEDeO89qFjRNrpYvVoDlop3nghagUCgMWa9Mz0LG8ROOdV+OP9Puyyfz2X9vMDxKNKVUpG4FnKNf/+F5mVrYFb05vUaz7B3L6xaZXtVusfq1VCyJDz1FCxZYocMWboUvL3jPe9KgQeCljHmJHBURIo5STWBXcCfQHgLwLbAHOf1n0AbpxVhBeCSU424AKgtIlmcBhi1nTSllBu9Fn1Ckf9VolqNEAhNzczOfflhQkaefNLNwmfP2laAzz8Pu3bZoUMOHID334/mqWKl4panHi5+D/hZRFICB4E3sAF0hoi8BRwBWjjLzgPqAweAa86yGGPOi0h/YIOzXL/wRhlKqTuMMfz3n/DbqGc5cT2MRk3CGPctPP64m4XPnYMBA2yJKlky6NnT9huYMWO851spd8Q2ynt0+Pr6Gj8/P09nQ6k4d/XmVdr98QZpT9Tmlw/bExpq+6598003owgbAxMn2uHuT560/QNu3w4lSngk7yphEZGNxpioepqMN9qNk1JJUFgYTPsxDXPXXyZ4+xXyPmZ7VCpVKsKCxsDixbYV4KJFNkjNnm0bWyiVAGnQUioJOXLpCJ1nfsbpH0axYWUmUqT8hynjhTZt3JSuFi+GTp3svaocOWDYMDtA4z0LKpVwaN+DSiURZ85A996nmOs/h31BG5k0CW5cF9q2jRCHQkPtvapatWzAGjjQPjDcvbsGLJXgaUlLqURu79m9/LpmLYNatSM4uDwlyx1h6cKMuH0kcft2aN/ejnH17rswdCikShXveVbqQWnQUioRu3ULWo/5ik1B88iVvQW/jEpHkyYZ7y0wnTplW2H07WtbBU6ZAm21j2mV+GjQUioR2nxiM1vWZuOzrvk5dn4INWp9yZTV6ciXL8KCxsCkSdC5M9y8CZUq2UEZtQd2lUhFG7ScIT/eB8IfBt4NjDLG/BCXGVNKuXf8XBAVx1XnxvaGpDzzEz9OzMqrr0a4HXXpEkydCqNGgb+/HTZk2jQoVizS7SqVGEQZtESkDdAN+ADYhO3v72ngaxFBA5dS8WffuX2EnHySFi0ycOPGLJpXLMf4E5A1q8tCN27AuHEweDAcO2aDVe/edrwrt73gKpW4RFfS6gy8aIwJcElbKiLNgemABi2l4sFfe/+iyfQm8ONCsl58gSUzXqBGjQgLrVkDr7wCR45A+fLw0092JGGlkpDomrxnjBCwAHDStF8XpeLYlZtX2L8fRr5fC7O0H7VLVGDzZu4OWGFh9uHgKlXskCGzZtnWgRqwVBIUXdAKfsB5SqmH1G3ehxTs/zxPeYewdmVqBtXvzd+z09/d2GL9eihTBv7v/6BJE9i5E5o391ielYpr0VUPFheRbW7SBTuYo1IqFhljCAuDuXOFif2qcCVjWl5rbfjfAMib12XB0FBbFThrlr1XNXKkHetKHw5WSVy0QStecqGUIuhGEM1+ep1DCxrgP/NtUqZszIDPG/PppxEW3LIFOnSADRvsA8L/+x9kyOCRPCsV36IMWsaYw67TIpINqAIcMcZsjMuMKfUoMQY+65WOxadDSH0yhNGj4e233XRW8f33NlBlzQq//GJLW0o9QqJr8v430MsYs8MZTXgT4AcUFpHxxpgR8ZFJpZKqg+cP0XzMp5z4fgynDmem2FN/s3CBkD9/hAWvXrXDhgwfbhtczJgBuXJ5JM9KeVJ0DTEKGmN2OK/fABYZYxoBzwJvxmnOlEriVq2CFm3PseXKfLIU38qECbB9m5uAtXMnPP20DVivvmqHENGApR5R0QWtEJfXNbGjCGOMCQLC4ipTSiVlGw7voOK7E6lcGY77+TI07xF2zatK+/aQIoXLgjduwOefg4+P7eFi6VL77FXKlB7Lu1KeFl1DjKMi8h4QiO0JYz6AiKQBUkS1olLqXn//DY2+HwZF5vPRZ634vFc60qZNf++CmzfbUtXu3fDSS7Z1YO7c8Z9hpRKY6EpabwHeQDvgZWPMRSe9AjA5DvOlVJKy9sh/vPvJERo1AhYO5ddq2xjUL929PStduWKft3r6aTh3DubMsfevNGApBUTfevA00NFN+jJgWVxlSqmkZPHKy9SZX4uw3U15662pjBiRhfRuClfMm2dHEj5yxD6UtX69BiulIoiu9eCfUc03xjSO3ewolXTsOLWL7/9XgpEjM5LOew6jPn6aN1q7ef537Vr49FNYtgxKlLANLV54wSN5Viqhi+6eVkXgKDANWI/tCUMpFY1Bv8/h421N4a+FvP56LYYPr0a2bBEWunLF9hk4eLB9IGvgQPjwQ21ooVQUogtajwG1gFeA1sBcYJoxZmdcZ0ypxOjgscu0eTkjq9fVJdlzg/myy/P07BahdHXtGnzwgW0JePUqtGxpG1o89pjH8q1UYhHdPa1b2BaD80UkFTZ4LReRfsaY0fGRQaUSizoj3mfxgWWErfPjlZap+PrrnuTJE2GhAwegdWvw87OtAjt31t7YlboPMRm5OBXQABuwCgCjgNlxmy2lEgdjDOvXw2uvCf7JavHEszmZs0UoU/KeBe0zV199BWnSwOzZ0LSpR/KsVGIWXUOMqUBJ4B+gr0vvGEo98s4GXebZIS9zaG4zUh17m4/eb0ifPg1Jkybigmfhrbfgzz+hRQsYMUJbBSr1gKIrab0OXAWeBLrKnYp5AYwxRgeCVI+csDAYOxa+HpKBgGdTUqECzJkHOXNGWPDKFfj2Wxg0CC5ftg0uPvgAkiXzSL6VSgqiu6cV3cPHSj1SPh6yn7H7P+HiDxMoWSQzv7/8B02bumlUu2mTbWDh7w9Vq9rg5e0d/xlWKonRoKRUDBgDX34Jg4Zf4mLm5Xw8fAfbtuE+YK1caXtiv34dFi+2z19pwFIqVkTbEEOpR93GY5vpMew//h32DilT+nK012FyZonQ/9KtW7bOcOJE229gvnz2oeF7mg8qpR6GlrSUisKJE9D8q2/41wzgrY7XCA7m3oB17Zrtib1LF1skGzwYduzQgKVUHPBY0BKRZCKy2RloEhEpKCLrRWS/iPwqIimd9FTO9AFnfgGXbXzspO8VkTqe2ROVFC3cs4p3Pgogd244PGEo/fNsY8KYtHi5fmOuX7cBqnx5G6Q++siWsnr2hIzaRkmpuODJktb7wG6X6a+A4caYosAFbA/zOP8vGGOKAMOd5RCREkArbC/0dYExIqLNstRDmzHnEnV/bMD4vX0pVw42rs5M7x5Z7u7VYv58KFTIBqqbN+2YI4MGeSzPSj0qPBK0RCQv9oHl751pAWoAs5xFpgLhT142caZx5td0lm8CTDfG3DDGHAIOAM/Ezx6opMgvcCvvvAMvN81EwTVzmffeN/j52VFCbjt/Hho3hnr1IDTUtgrctw8aNPBYvpV6lHiqpDUC+D/ujH6cDbhojAl1pgOB8BsCebCd9uLMv+QsfzvdzTp3EZEOIuInIn5nzpyJzf1QScRPG3+j/EQfxi9aQuvWsH3u89Srme7OAufOwYAB9j7VX3/BJ5/A0aO2G6Z7um1XSsWVeG89KCINgdPGmI0iUi082c2iJpp5Ua1zd6Ix44HxAL6+vm6XUY8eYwwXr1/k9JEsDGjbEMk4nA9bPc/gL+9ayA5z37KlLWVVrgxDh9r7WEqpeOeJJu+VgMYiUh9IDWTElrwyi0hypzSVFzjuLB8I5AMCRSQ5kAk475IeznUdpaLV8a93+XPLSk7220j2LKlYPKobNWq4LHD5MtSvD6tXQ+rU9v9zz3ksv0opD1QPGmM+NsbkNcYUwDakWGqMeRU7EvJLzmJtgTnO6z+daZz5S40xxklv5bQuLAgUBf6Lp91QiZQxBmMM+/bB5E8acHJBW4o96cXGjdwJWGFh8PPPULy4DVT9+9vqQQ1YSnlcQnq4+CNguogMADYDE530icCPInIAW8JqBWCM2SkiM4BdQCjwrjOUilJuXbp+iWYzmvHUrZb88sE7cLUBc2Y1oLHr+NuHD8O778LcufDUUzBjBlSq5LE8K6Xu5tGgZYxZDix3Xh/ETes/Y8x1oEUk6w8EBsZdDlVSsmtzRrauz8TSNanwzmNbqRco4MwMCrJN1ocOtQ0rBg+GHj24+8EspZSn6TdSJWm7z+ymwU9NeLvrBZ57TvCaOZvRb7Vj0yYnYBljRxAuVsx2LvjSS7YJe8+eGrCUSoASUvWgUrHq5k34dnww8y//R9jve3j//YoMHAjpwluy79kD771nO7V95hk7MGOFCh7Ns1Iqahq0VJKzPnA9kxdsZMmgzhw48DQFix5iys+pqVLFWSA4GEaPtr1ZZMliRxP+8EMtWSmVCGjQUklKcDC0HT2OvTeWUSz1G/z9dxrq10995/nfuXPhnXfg2DE7evCmTZArl0fzrJSKOf1pqZKEBfuW0m/0IQoXhr2jhtHu+la2b0pDgwZOhxWXLkH79tCwIWTLBkuW2MClAUupREVLWipRMwbGTrnIuwdexOx4icpFJvJjn8zUqOHSu9LZs1C3LmzZYhtY9OtnHxZWSiU6GrRUorX28EbGfl6OH37IzFMvzOfL931o2iBCV4DLl8Mbb8DJkzBnjnZsq1Qip9WDKlGauG4Gz03x5YeVS+nTB3YuqMiLDdPcCVgBAbaLi+rVIVkyO+S9BiylEj0taalEwxjDueBznPDPztAOTUie/hu+71eZtq+5LBQaah8QHjTIdsc0dKhteJEuXaTbVUolHhq0VKLRdvbbzN+2ngtfbSRzhlTMmfou9es7M8PCbG/sn38Oa9dCnTq2WXvRoh7Ns1IqdmnQUglamLFDrm3Z7MW8Ic04Z0rzYoNkjBsLOXI4C61cCS++aDu1zZQJfvkFXnnFc5lWSsUZDVoqwbp4/SL1fmhE8H+vsm1SR/Lkqc+q6S791968aQdjHD4c8uaF3r1tsNJm7EolWRq0VIJ0/TpMn5KJTatyc2tnRj791Pa4lDOns8CKFfa5q/37oW1bGDnSlrKUUkmaBi2VoKzx387LEz4i+OefOReYhYoVf2X091CunLPAxo0wYgRMmwaFCsGff0KjRh7Ns1Iq/miTd5UgGGNjUZWqoQTe3E4h3wMsXmzHYCxXDrhyBbp0AV9f2yt7hw7g56cBS6lHjJa0lMfN9ltF/wmb2TL+PZ55pix96/lTt1ZKO9MY+Gc+dO9uhwzp2NEOIZIli2czrZTyCA1aymPCwmDSJOg0fzK38qzgiwFv89nHqfHycgLW8ePQtSv89hvky2f7C6xe3bOZVkp5lAYt5RH9ps1n4ldPcmRrISrVGM7YL70o+aRLf4Bz50LjxpA8OfTpY/sM1AeElXrkadBS8W78j+fps7slXgVaMbnbeNq0yXhnKKvLl6F/fxgyxE5v3w5PPumxvCqlEhZtiKHihTGGxXvW8dpr8E6brBRdu5jdw0bTrp0z9qIx8OOPtgeLIUPgtdds4wsNWEopFxq0VLzoM3M6tX6tyLR1S+nbF3YseIYnC6WCkBCYMQMKF4Y2bezAjOvX2wCm1YFKqQi0elDFGWMMp6+e5vefcvHl+83JWXMcs3+sQqWKzgILF9rObAMC7DNXkybZwJUsmSezrZRKwDRoqTjz6qx2LNjmx/n/baZe7ZRMn96BjBmxfQR+/DFMmAAlSsDvv9thQ1Kk8HSWlVIJnAYtFatuhd3i1i1h9Cgv/pjSmuBUz9Gta3KGfA3JTCh8PwV69YKLF+GDD+wowloNqJSKIQ1aKtacDz5PlfH1uLi8Hcf+6ES9enX49FOo9GwofPuNHdvqyBGoXBnGjIGSJT2dZaVUIqNBS8UKY+CXiVnYtaowGU7mYMYMaNEC2LoVnn0TNm2CKlVg1Cj7/NXtIYaVUirmNGiph7Ll5Ba6ze1J8j9+ZcnfWalb9xem/QOZLx+Bd7+CiRMhc2bbQvCllzRYKaUeigYt9cDCwmD+fFi16wCy7RCjR2fl3WYnkP/rY1sChoVBu3bw1VcuIzYqpdSD06Cl7tuyQ8tYvGMrywd1Y80aH0r77Gfqb+CzeiQU/9wOhtW2LfToYVsHKqVULNGgpe7L9evQ65ef2XByNRn3dmLC2OS8kXUOyd7qB9u2QdWqMG4cFCvm6awqpZKgeO8RQ0TyicgyEdktIjtF5H0nPauILBKR/c7/LE66iMgoETkgIttE5GmXbbV1lt8vIm3je18eJX/smcOwyQcpUgT+6z+Mekc2sf/nnbSfWJFkLV+yTdhnzYJlyzRgKaXijCe6cQoFehhjigMVgHdFpATQC1hijCkKLHGmAeoBRZ2/DsB3YIMc0Ad4FngG6BMe6FTsOnbhHC2nvU6P34aQJw8snwl/5/+IHPXL2ybsU6eCvz80b64NLZRScSreqweNMSeAE87rIBHZDeQBmgDVnMWmAsuBj5z0H4wxBlgnIplF5HFn2UXGmPMAIrIIqAtMi7edScKMMfwbsJIjK6vw9dfZCDm3jE/fLkW/Qj/i1eljO9ZV584wYIBtHaiUUvHAox3mikgBoCywHsjlBLTwwJbTWSwPcNRltUAnLbJ0d+/TQUT8RMTvzJkzsbkLSdaXf/9M9R+q0rbPMq5dg4VfPsaAwx3watfGDsi4ahV8840GLKVUvPJYQwwRSQ/8BnQzxlyWyKuV3M0wUaTfm2jMeGA8gK+vr9tlFISZMHYGnGLC8McZPaYl6SrcYlDXCnT0+pbkXT+2rTA++cSOd+WlAwQopeKfR+PH/DAAAA2PSURBVIKWiKTABqyfjTGzneRTIvK4MeaEU/132kkPBPK5rJ4XOO6kV4uQvjwu852U3boFzwx6nS2nthA2ZjOd3k7B/57JQKb+3nDokO16acoU2xu7Ukp5iCdaDwowEdhtjBnmMutPILwFYFtgjkt6G6cVYQXgklN9uACoLSJZnAYYtZ00dR9Cw0L56+8wnnoKNk1uS4mLPTk0bgVjNj5LpjebQ/r0MG8erFihAUsp5XGeKGlVAl4HtovIFiftE2AQMENE3gKOAC2cefOA+sAB4BrwBoAx5ryI9Ac2OMv1C2+UoWJmf+A5yn9Ti0tL21NUOrPk7axUXzwEab/I3rcaO9aOb5UmjaezqpRSgGdaD67C/f0ogJpuljfAu5FsaxIwKfZy92i4eRO++w4Gf52VoAolaVMrExMPvEryj36BrFltb+ydO0Pq1J7OqlJK3UV7xHjETFnoR6c/PuT6lNlU+v/27j08iuqM4/j3TSBRLEqCl0buUmpVVECKwUsttmirVlqLFawCKl5KrUpFgYqPxT4iVrRcShVKtaLgBS9tiFq0KI9aKyJVNCJguGlEAeWiSDFA3v4xJ+2aghHYZHY2v8/zzLMzZ2Z337MnyZtzZnZOl0LmnHAOHW69BFavhhEj4JpriGZqFBHJPLoErIH49NNosuAL+jdi217vccedb/J8+/50GHwGNG8OL78cXRWohCUiGUw9rQbg5hl/47Z7y/ho5hAuvLATY48dQ9PB58CaNVHv6vrrIS8v7jBFRGqlpJXFNm2CIUNg0qpHyDvkJZ6acS49Zw6HPlPhyCOhtBS6dKn9hUREMoSGB7OQuzN06gy+cdxSJk+GqzrcwodtLqLnpUfCtGlR7+qVV5SwRCRx1NPKMuXlcNWvPuLx9gMpOKofrw3oxVETL4Vly6BHDxg3LupliYgkkHpaWaKy0rn0ltl07Ah/L9mf4dum8cEn73DU1T2jWy7NnAmzZythiUiiqaeVBZYsgVOHTmVFpwGceObfKD1wDvuOHRNdXDFqFAwerO9ciUhWUNJKsE83b2f4qA+YPKYFezc5h9+0eZ5fze5Lzrr1MGBAlLCKiuIOU0QkbZS0EmrRIii+rS8b88sYc+z1XLFmFI3HlUXT3d90Exx/fNwhioiknc5pJUzl9koemlFFt27QfF4vJi03fvncuTS27XDfffDMM0pYIpK11NNKkIUrPqT7pB5smj2QcQWVDCq/kZxlFt1IcOBAaKTmFJHspp5WArjDlClQfHRzmi5uwX0+hsvfuZack3vAggVw2WVKWCLSIChpZbgn33iJ/a89nhuuWMDM/J9Q8dgs+q7Pj+5mUVIC7drFHaKISL1R0spQ7tEpqit7f0zev99iZmEx39r0eHRT27IyOP30uEMUEal3GlPKQJPmlDBh2kJOmZLH6zkjaLz0M3LPOx9GjoQ2beIOT0QkNkpaGWTbNrj9dpjw4iT2PegZRuduofGpp2MTxmuqexERlLQygrszZtZ0Hht1GBc8fweLGj9BTlEb8h4eC716ge1somcRkYZFSStmW7bAiFFrGb/1Evrtk8NFthkbPBS7cSTk58cdnohIRlHSikmVVzHmsacpHXIEVy6/kgsO2Ez7g7uQM/dO+OY34w5PRCQjKWnFYN06+NENd/Hc/hczq2ovvpNv5F49OpqxMTc37vBERDKWklY92rp9GxOnruLxq9dy68bfs/Ib0OOI75I7ZwK0bRt3eCIiGU9Jq56sWgVdb/4h2/1FVm7cQG7hV+n2m0fgrLPiDk1EJDGUtOrYv7du4dEHc3npZ9OYcMAL+N4bybv4MnJG3wzNmsUdnohIoihp1aGFK9dw4h+Opd8cmLBpBZu/1p0md02Ezp3jDk1EJJF0G6c6UFXl3Dt+PS98fSRnvbWCkzetp+pPd9Nk/gtKWCIie0A9rTSb/vxzDJ0xgKenfEyHyvX0bno5hS/cCAUFcYcmIpJ46mmlSWUlTPn5q2w/+xcUbl3O6sNaYfPnUzhtghKWiEiaqKeVBqPvv5tX/zyJ6U/NY1N+c8449G4KJvaDHP1PICKSTkpae+DjDVWU/PgeFjYZxPLWWyj/wSAOnXqTrgoUEakjSlq7wd0ZesuvOe53JZy35jU6tyqm5bUT2O/ErnGHJiKS1RI/fmVm3zOzxWZWbmbD6vr93itbzwNHXMjkjTfy985LWHrDPRyx8kUlLBGRepDonpaZ5QITgZ5ABTDPzErcfWG636vys63cesUQLv3jdH7i62jc8jx+cN848g8sTPdbiYjITiS9p9UNKHf3Ze5eCTwA9Er3m2xYsYHR3z6UEQeP54nDini/9F/0fupeJSwRkXqW6J4W0AJ4N2W7Aji25kFmdglwCUDr1q13+U32a70fJ60uZvT6Mznv9dvJyU16rhcRSaakJ60dTenr/1fgPhmYDNC1a9f/21/rm+QYJy2bzkm7Hp+IiKRR0rsMFUCrlO2WwKqYYhERkTqW9KQ1D+hgZu3MLA/oA5TEHJOIiNSRRA8Puvs2M7scmAXkAne5+5sxhyUiInUk0UkLwN2fAJ6IOw4REal7SR8eFBGRBkRJS0REEkNJS0REEkNJS0REEsPcd/m7tolmZmuBlbv59P2BD9MYThKoztmvodUXVOdd1cbdD0hnMLurwSWtPWFmr7h7g7qdu+qc/RpafUF1TjIND4qISGIoaYmISGIoae2ayXEHEAPVOfs1tPqC6pxYOqclIiKJoZ6WiIgkhpKWiIgkhpLWl2Bm3zOzxWZWbmbD4o5nT5hZKzN71szeMrM3zezKUF5oZk+b2dvhsSCUm5mND3V/3cy6pLxW/3D822bWP646fRlmlmtmr5pZadhuZ2ZzQ+wPhqltMLP8sF0e9rdNeY3hoXyxmZ0aT02+PDNrZmYPm9mi0N7ds7mdzWxw+JkuM7P7zWyvbGxnM7vLzNaYWVlKWdra1cyOMbM3wnPGm9mOJtuNj7tr+YKFaMqTpcAhQB6wADg87rj2oD5FQJew3hRYAhwO/BYYFsqHAbeE9dOAJ4lmiS4G5obyQmBZeCwI6wVx1+8L6v1LYDpQGrYfAvqE9TuBn4X1QcCdYb0P8GBYPzy0fT7QLvxM5MZdr1rqfA8wMKznAc2ytZ2BFsByYO+U9h2Qje0MfAvoApSllKWtXYGXge7hOU8C34+7zp+rf9wBZPoSGm9WyvZwYHjccaWxfn8FegKLgaJQVgQsDuuTgL4pxy8O+/sCk1LKP3dcJi1EM1rPBk4GSsMv44dAo5ptTDQ3W/ew3igcZzXbPfW4TFyAfcMfcatRnpXtHJLWu+GPcKPQzqdmazsDbWskrbS0a9i3KKX8c8dlwqLhwdpV/zJUqwhliReGRDoDc4GD3P19gPB4YDhsZ/VP0ucyFrgWqArbzYEN7r4tbKfG/t96hf0bw/FJqi9EIwNrgbvDsOgUM9uHLG1nd38PGAO8A7xP1G7zyf52rpaudm0R1muWZwwlrdrtaDw38d8TMLOvAI8AV7n7x1906A7K/AvKM4qZnQGscff5qcU7ONRr2ZeI+qZoRDSEdIe7dwY+JRo22plE1zucw+lFNKR3MLAP8P0dHJpt7VybXa1nxtdfSat2FUCrlO2WwKqYYkkLM2tMlLCmufujoXi1mRWF/UXAmlC+s/on5XM5HjjTzFYADxANEY4FmplZ9czdqbH/t15h/37AOpJT32oVQIW7zw3bDxMlsWxt5+8Cy919rbtvBR4FjiP727lautq1IqzXLM8YSlq1mwd0CFch5RGdtC2JOabdFq4E+hPwlrvfnrKrBKi+gqg/0bmu6vJ+4SqkYmBjGH6YBZxiZgXhv9xTQllGcffh7t7S3dsStd0z7v5T4FmgdzisZn2rP4fe4XgP5X3CVWftgA5EJ6wzkrt/ALxrZoeGou8AC8nSdiYaFiw2sybhZ7y6vlndzinS0q5h3ydmVhw+x34pr5UZ4j6ploSF6AqcJURXEl0Xdzx7WJcTiLr7rwOvheU0ovH82cDb4bEwHG/AxFD3N4CuKa91IVAelgvirtuXqPu3+d/Vg4cQ/TEqB2YA+aF8r7BdHvYfkvL868LnsJgMu6JqJ/XtBLwS2vovRFeJZW07AyOBRUAZcC/RFYBZ187A/UTn7bYS9YwuSme7Al3DZ7gU+D01LuaJe9FtnEREJDE0PCgiIomhpCUiIomhpCUiIomhpCUiIomhpCUiIomhpCVSh8Kd1geF9YPN7OG4YxJJMl3yLlKHwv0dS929Y8yhiGSFRrUfIiJ7YDTQ3sxeI/ri52Hu3tHMBgA/JJr6piNwG9H0IecDnwGnufs6M2tP9OXQA4DNwMXuvqj+qyGSGTQ8KFK3hgFL3b0TcE2NfR2Bc4FuwE3AZo9ubvtPotvnAEwGfuHuxwBDgD/US9QiGUo9LZH4POvunxDd620jMDOUvwEcFe7EfxwwI2Xy2Pz6D1MkcyhpicTns5T1qpTtKqLfzRyi+aA61XdgIplKw4MidesToOnuPNGjec6Wm9nZEN2h38yOTmdwIkmjpCVSh9z9I+AfZlYG3LobL/FT4CIzWwC8STTRoUiDpUveRUQkMdTTEhGRxFDSEhGRxFDSEhGRxFDSEhGRxFDSEhGRxFDSEhGRxFDSEhGRxPgP03HyFaCuw7QAAAAASUVORK5CYII=\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "mean_R2D = np.mean(xD**2 + yD**2, 0) # MSD for DS walk\n", "mean_R2C = np.mean(xC**2 + yC**2, 0) # MSD for CS walk \n", "\n", "plt.plot(mean_R2D, 'b-')\n", "plt.plot(mean_R2C, 'r-')\n", "plt.plot(np.arange(N+1), 'g:')\n", "plt.title('Mean squared displacements for different walks of length ' + str(N))\n", "plt.xlabel('time')\n", "plt.ylabel('MSD')\n", "plt.legend(('DSDT walk', 'CSDT walk', 'MSD=time'))\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Both of the walks have the same MSDs! And both of the MSDs grow linearly with time. In fact, this is the crucial property of random walks: mean squared displacement grows linearly with time. Irrespective of the type of the walk and the distributions of individual steps (provided those have finite variances), we will always get this scaling.\n", "\n", ">### Your turn\n", "Generate either a CSDT random walk where the distribution of step sizes is exponential, or the DSCT walk where distribution of time steps is exponential and show that the same linear scaling of the MSD is observed. \n", "\n", "Now we are well positioned to answer the original question we started this lecture from: why are distributions of experimental errors Gaussian? Well, errors usually are contributed to by many independent processes that describe irreproducibility of steps taken by the experimentalist and irreproducibility of the conditions that the system experimented on is in. None of the processes dominates the others, and so each produces small contributions, which collectively result in a big effect. Random walks model this: there are many individual steps, each of which is small, but collectively they result in a big displacement. So what is the distribution of displacements in a random walk? To study this, we plot the distribution of $x$ positionsof our CS and DS walkers." ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.hist([xD[:,-1],xC[:,-1]],nbins)\n", "plt.xlabel('x position')\n", "plt.ylabel('frequency')\n", "plt.title('Distribution of x displacements')\n", "plt.legend(('DSDT walk', 'CSDT walk'))\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Shockingly, both of the distributions are indistinguishable Gaussians! This is a demonstration of what is known as a *Central Limit Theorem*, arguably one of the most important theorems in probability theory: a sum of many independent random variables is itself a random variable that tends to be normally distributed with the variance equal to the sum of variances of contributing variables (provided these variances are all finite). It is precisely because of the Central LImit Theorem why we also call Gaussian distributions normal.\n", "\n", ">### Your turn\n", "Show that the CLT property holds for the walks you defined in the previous Your Turn exercise.\n", "\n", ">### Your turn\n", "Verify that the variance of the $x$ displacement of the walk is, indeed, the sum of variances of the individual displacements (for this, you will need to calculate the latter variance analytically)." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.1" } }, "nbformat": 4, "nbformat_minor": 2 }