{ "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: $P_i\\ge0$;\n", " - unit normalization: $\\sum_{i=1}^N P_i=1$;\n", " - nesting: if $A\\subset B$ then $P(A)\\le P(B)$;\n", " - additivity (for non-disjoint events): $P(A\\cup B)=P(A)+P(B)-P(A\\cap B)$;\n", " - complementarity $P(not\\, A)=1-P(A)$.\n", "\n", "### What if we are studying more than one random variable?\n", "Bivariate distributions (a special case of multivariate ones) $P(x,y)$ defined the probability of both events happening. It contains all of the information about the variables, including\n", " - The marginal distributions: $P(x)=\\sum_{y\\in Y} P(x,y)$ or $P(y)=\\sum_{x\\in X} P(x,y)$ \n", " - The conditional distributions, which can be defined as $P(y|x)=P(x,y)/P(x)$ or $P(x|y)=P(x,y)/P(y)$, so that the probability of both events is the probability of the first event happening, and then the probability of the second happening given that the first one has happened.\n", "The conditional distributions are related using the Bayes theorem, which says $P(x,y)=P(x|y)P(y)=P(y|x)P(x)$, so that $P(x|y)=\\frac{P(y|x)P(x)}{P(y)}$. \n", "\n", "We can now formalize the intuitive concept of dependence among variables. Two random variables are considered to be statistically independent if and only if $P(x,y)=P(x)P(y)$, or, equivalently, $P(x|y)=P(x)$ or $P(y|x)=P(y)$.\n", "\n", "## Probabilities of continuous variables\n", "**To be written**\n", "\n", "## Random numbers module in Python\n", "**To be written**" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "['Lock',\n", " 'RandomState',\n", " '__RandomState_ctor',\n", " '__all__',\n", " '__builtins__',\n", " '__cached__',\n", " '__doc__',\n", " '__file__',\n", " '__loader__',\n", " '__name__',\n", " '__package__',\n", " '__path__',\n", " '__spec__',\n", " 'absolute_import',\n", " 'beta',\n", " 'binomial',\n", " 'bytes',\n", " 'chisquare',\n", " 'choice',\n", " 'dirichlet',\n", " 'division',\n", " 'exponential',\n", " 'f',\n", " 'gamma',\n", " 'geometric',\n", " 'get_state',\n", " 'gumbel',\n", " 'hypergeometric',\n", " 'info',\n", " 'laplace',\n", " 'logistic',\n", " 'lognormal',\n", " 'logseries',\n", " 'mtrand',\n", " 'multinomial',\n", " 'multivariate_normal',\n", " 'negative_binomial',\n", " 'noncentral_chisquare',\n", " 'noncentral_f',\n", " 'normal',\n", " 'np',\n", " 'operator',\n", " 'pareto',\n", " 'permutation',\n", " 'poisson',\n", " 'power',\n", " 'print_function',\n", " 'rand',\n", " 'randint',\n", " 'randn',\n", " 'random',\n", " 'random_integers',\n", " 'random_sample',\n", " 'ranf',\n", " 'rayleigh',\n", " 'sample',\n", " 'seed',\n", " 'set_state',\n", " 'shuffle',\n", " 'standard_cauchy',\n", " 'standard_exponential',\n", " 'standard_gamma',\n", " 'standard_normal',\n", " 'standard_t',\n", " 'test',\n", " 'triangular',\n", " 'uniform',\n", " 'vonmises',\n", " 'wald',\n", " 'warnings',\n", " 'weibull',\n", " 'zipf']" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "#Initialization\n", "import numpy as np\n", "import numpy.random as nprnd #note the new module we will be using\n", "import matplotlib.pyplot as plt\n", "\n", "%matplotlib inline \n", "dir(nprnd) # see which new functions exist in this new module\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Build-in routines for pseudo-random numbers generation\n", "We will first experiment with the build-in routines for generating uniform standard random numbers (floating point numbers between 0 and 1).\n" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[0.24136295 0.2801732 0.5891325 0.01700196 0.32371912]\n", "[0.47389554 0.47665976 0.88100618 0.21900017 0.29347262]\n", "[0.76832343 0.78360339 0.72824669 0.15279959 0.20291618]\n", "[[0.36050912 0.36699367 0.84493731]\n", " [0.63971783 0.65280219 0.87609947]\n", " [0.05881411 0.73427181 0.57383518]\n", " [0.90222722 0.65104119 0.5585237 ]\n", " [0.17616828 0.29237934 0.73131861]]\n" ] }, { "data": { "image/png": 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\n", 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" ] }, "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", "The linear congruential generator (LCG) illustrated above, with good choices of the parameters, is the simplest good generator in use today. There are many others, somewhat better ones, in use today, which produce sequences that have fewer correlation properties. However, while they may seem a lot more complicated, usually the core remains the same: we use properties of integer arithmetic, such as remainder of an integer division, to redistribute numbers pseudo-randomly in the segment $[0,1)$. The principal problem with the LCG, and the main reason why more complicated versions exist, is not the quality of the sequences that it generates at good choices of parameters. Rather the problem is that there's a strong dependence of the quality of the sequence on the choice of parameters, and there are very few proofs that guarantee the quality for any specific choice of parameters. \n", "\n", ">### Your work\n", "Experiment with the linear congruential random number 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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ltYylxv6pzvscsA+pT/8RUlffQZWYTiI9ZnMBMI3lP9CPBqblrp73RcSNpMR1EinxX8ULZ1T7ky62zwN+DhyVr4+siINIF9XvJO3PC3mhC+01wA1KJfKnk7ouXbG4n7kWkw0JSreLbh0RB/RxuXGkO2mG1bw2YrbK8BmEmZkVOUGYmVmRu5jMzKzIZxBmZlY0qAuPjRw5MsaNG9fpMMzMBpWbbrrpkYho9d2gwZ0gxo0bx4wZMzodhpnZoCKpVuUBdzGZmVmRE4SZmRU5QZiZWZEThJmZFTlBmJlZkROEmZkVOUGYmVmRE4SZmRU5QZiZWdGg/ib1yhh3xK87tu3Zx/9Lx7ZtZlaXzyDMzKzICcLMzIqcIMzMrMgJwszMipwgzMysaJW9i8lsqOrUHXq+O2/o8RmEmZkVOUGYmVlR2xKEpLGSrpB0l6Q7JH0qjz9a0oOSZubX3pVlviBplqR7JL29XbGZmVlr7bwGsQT4TETcLGk4cJOky/K0kyLihOrMkiYA+wHbAaOB30naJiKWtjFGMzPrQdvOICJifkTcnN8vBu4CxvSyyCTg3Ih4NiLuB2YBO7crPjMz692AXIOQNA7YEbghjzpM0q2SzpC0QR43BphTWWwuhYQiaYqkGZJmLFy4sI1Rm5mt2tp+m6uk9YCfAodHxJOSfgB8FYj881vABwEVFo/lRkRMBaYCTJw4cbnpZmYDZagX/WzrGYSkYaTk8OOI+BlARCyIiKUR8TxwGi90I80FxlYW3xSY1874zMysZ+28i0nA6cBdEXFiZfwmldneBdye308H9pO0lqQtgPHAje2Kz8zMetfOLqZdgQOB2yTNzOOOBPaXtAOp+2g28BGAiLhD0vnAnaQ7oD7uO5jMzDqnbQkiIq6hfF3h4l6WOQ44rl0xmZlZff4mtZmZFTlBmJlZkROEmZkVOUGYmVmRE4SZmRU5QZiZWZEThJmZFTlBmJlZkROEmZkVOUGYmVmRE4SZmRU5QZiZWVHbHxhkBp17sMpAPFTFbKjyGYSZmRU5QZiZWZEThJmZFTlBmJlZkROEmZkVOUGYmVmRE4SZmRU5QZiZWZEThJmZFTlBmJlZkROEmZkVOUGYmVmRE4SZmRU5QZiZWZEThJmZFTlBmJlZkROEmZkVtS1BSBor6QpJd0m6Q9Kn8vgNJV0m6d78c4M8XpJOljRL0q2SdmpXbGZm1lo7zyCWAJ+JiJcDuwAflzQBOAK4PCLGA5fnYYB3AOPzawrwgzbGZmZmLbQtQUTE/Ii4Ob9fDNwFjAEmAdPybNOAffP7ScCZkVwPjJC0SbviMzOz3g3INQhJ44AdgRuAjSNiPqQkAmyUZxsDzKksNjePa17XFEkzJM1YuHBhO8M2M1ultT1BSFoP+ClweEQ82dushXGx3IiIqRExMSImjho1qr/CNDOzJm1NEJKGkZLDjyPiZ3n0gkbXUf75cB4/FxhbWXxTYF474zMzs5618y4mAacDd0XEiZVJ04HJ+f1k4JeV8Qflu5l2ARY1uqLMzGzgrdHGde8KHAjcJmlmHnckcDxwvqRDgQeA9+ZpFwN7A7OAZ4BD2hibmZm10LYEERHXUL6uALBHYf4APt6ueMzMrG/8TWozMytygjAzsyInCDMzK3KCMDOzIicIMzMrcoIwM7MiJwgzMytygjAzsyInCDMzK3KCMDOzIicIMzMrcoIwM7MiJwgzMytygjAzsyInCDMzK3KCMDOzIicIMzMrcoIwM7OilglC0oYDEYiZmXWXOmcQN0i6QNLeknp6xrSZmQ0xdRLENsBU4EBglqSvS9qmvWGZmVmntUwQkVwWEfsDHwImAzdKukrS69oeoZmZdcQarWaQ9BLgANIZxALgE8B0YAfgAmCLdgZoZmad0TJBANcBZwH7RsTcyvgZkk5tT1hmZtZpdRLEthERpQkR8c1+jsfMzLpEnYvUl0oa0RiQtIGkS9oYk5mZdYE6CWJURDzRGIiIx4GN2heSmZl1gzoJYqmkzRoDkjYHil1OZmY2dNS5BvFF4BpJV+Xh3YAp7QvJzMy6QcsEERG/lbQTsAsg4NMR8UjbIzMzs46qW6xvLeAxYBEwQdJurRaQdIakhyXdXhl3tKQHJc3Mr70r074gaZakeyS9va8NMTOz/lXni3LfBN4P3AE8n0cHcHWLRX8EnAKc2TT+pIg4oWkbE4D9gO2A0cDvJG0TEUtbxWdmZu1R5xrEvqTvQjzblxVHxNWSxtWcfRJwbt7G/ZJmATuTvqRnZmYdUKeL6T5gWD9u8zBJt+YuqA3yuDHAnMo8c/O45UiaImmGpBkLFy7sx7DMzKyqToJ4Bpgp6YeSTm68VnB7PwC2ItVxmg98K48vlRHv6dvbUyNiYkRMHDVq1AqGYWZmrdTpYpqeXystIhY03ks6DbgoD84FxlZm3RSY1x/bNDOzFVPnNtdpktYBNouIe1ZmY5I2iYj5efBdQOMOp+nATySdSLpIPR64cWW2ZWZmK6fOXUzvBE4A1gS2kLQDcGxE7NNiuXOA3YGRkuYCRwG75+UDmA18BCAi7pB0PnAnsAT4uO9gMjPrrDpdTEeT7ii6EiAiZkpq+QyI/IChZqf3Mv9xwHE14jEzswFQ5yL1kohY1DTOtZjMzIa4OmcQt0v6ALC6pPHAJ4Fr2xuWmZl1Wp0ziE+QvuH8LHAO8CRweDuDMjOzzqtzF9MzpIquX2x/OGZm1i3q3MV0BYVrDhHxlrZEZGZmXaHONYjPVt6vDbybdCuqmZkNYXW6mG5qGvWHysODzMxsiKrTxbRhZXA14NXAS9sWkZmZdYU6XUw3ka5BiNS1dD9waDuDMjOzzqvTxdTyW9NmZjb01Oli+rfepkfEz/ovHDMz6xZ1upgOBV4P/D4Pv5lUl2kRqevJCcLMbAiqkyACmNAo0y1pE+B7EXFIWyMzM7OOqlNqY1zlGQ4AC4Bt2hSPmZl1iTpnEFdKuoRUhymA/YAr2hqVmZl1XJ27mA6T9C5gtzxqakT8vL1hmZlZp9U5gwC4GVgcEb+T9CJJwyNicTsDMzOzzmp5DULSh4ELgR/mUWOAX7QzKDMz67w6F6k/DuxKeg4EEXEvsFE7gzIzs86rkyCejYjnGgOS1sCPHDUzG/LqJIirJB0JrCPprcAFwK/aG5aZmXVanQRxBLAQuA34CHAx8KV2BmVmZp3X611MklYHpkXEAcBpAxOSmZl1g17PICJiKTBK0poDFI+ZmXWJOt+DmE16itx04OnGyIg4sV1BmZlZ5/V4BiHprPz2/cBFed7hlZeZmQ1hvZ1BvFrS5sADwHcHKB4zM+sSvSWIU4HfAlsAMyrjRfoexJZtjMvMzDqsxy6miDg5Il4O/G9EbFl5bRERTg5mZkNcy+9BRMRHByIQMzPrLnW+KLdCJJ0h6WFJt1fGbSjpMkn35p8b5PGSdLKkWZJulbRTu+IyM7N62pYggB8BezWNOwK4PCLGA5fnYYB3AOPzawrwgzbGZWZmNbQtQUTE1cBjTaMnAdPy+2nAvpXxZ0ZyPTAiP/vazMw6pJ1nECUbN55vnX82yoaPAeZU5pubxy1H0hRJMyTNWLhwYVuDNTNblQ10guiJCuOKJcUjYmpETIyIiaNGjWpzWGZmq66BThALGl1H+efDefxcYGxlvk2BeQMcm5mZVQx0gpgOTM7vJwO/rIw/KN/NtAuwqNEVZWZmnVGnWN8KkXQOsDswUtJc4CjgeOB8SYeSSni8N89+MbA3MAt4BjikXXGZmVk9bUsQEbF/D5P2KMwbpGdfm5lZl+iWi9RmZtZlnCDMzKzICcLMzIqcIMzMrMgJwszMipwgzMysyAnCzMyKnCDMzKzICcLMzIqcIMzMrMgJwszMipwgzMysyAnCzMyKnCDMzKzICcLMzIqcIMzMrMgJwszMipwgzMysyAnCzMyKnCDMzKzICcLMzIqcIMzMrMgJwszMipwgzMysyAnCzMyKnCDMzKzICcLMzIqcIMzMrMgJwszMipwgzMysaI1ObFTSbGAxsBRYEhETJW0InAeMA2YD74uIxzsRn5mZdfYM4s0RsUNETMzDRwCXR8R44PI8bGZmHdJNXUyTgGn5/TRg3w7GYma2yutUggjgUkk3SZqSx20cEfMB8s+NSgtKmiJphqQZCxcuHKBwzcxWPR25BgHsGhHzJG0EXCbp7roLRsRUYCrAxIkTo10Bmpmt6jpyBhER8/LPh4GfAzsDCyRtApB/PtyJ2MzMLBnwBCFpXUnDG++BtwG3A9OByXm2ycAvBzo2MzN7QSe6mDYGfi6psf2fRMRvJf0ROF/SocADwHs7EJuZmWUDniAi4j7gVYXxjwJ7DHQ8ZmZW1k23uZqZWRdxgjAzsyInCDMzK3KCMDOzIicIMzMrcoIwM7MiJwgzMytygjAzsyInCDMzK3KCMDOzIicIMzMrcoIwM7MiJwgzMytygjAzsyInCDMzK3KCMDOzIicIMzMrcoIwM7MiJwgzMytygjAzsyInCDMzK3KCMDOzIicIMzMrcoIwM7MiJwgzMytygjAzsyInCDMzK3KCMDOzIicIMzMrcoIwM7OirksQkvaSdI+kWZKO6HQ8Zmarqq5KEJJWB74HvAOYAOwvaUJnozIzWzV1VYIAdgZmRcR9EfEccC4wqcMxmZmtkhQRnY7hnyS9B9grIj6Uhw8EXhsRh1XmmQJMyYPbAves4OZGAo+sRLjdxG3pTkOlLUOlHeC2NGweEaNazbTGCq68XVQYt0wGi4ipwNSV3pA0IyImrux6uoHb0p2GSluGSjvAbemrbutimguMrQxvCszrUCxmZqu0bksQfwTGS9pC0prAfsD0DsdkZrZK6qoupohYIukw4BJgdeCMiLijTZtb6W6qLuK2dKeh0pah0g5wW/qkqy5Sm5lZ9+i2LiYzM+sSThBmZlY05BNEq9IdktaSdF6efoOkcQMfZT012nKwpIWSZubXhzoRZyuSzpD0sKTbe5guSSfndt4qaaeBjrGuGm3ZXdKiyjH5ykDHWIeksZKukHSXpDskfaowz6A4LjXbMliOy9qSbpR0S27LMYV52vcZFhFD9kW60P0XYEtgTeAWYELTPB8DTs3v9wPO63TcK9GWg4FTOh1rjbbsBuwE3N7D9L2B35C+F7MLcEOnY16JtuwOXNTpOGu0YxNgp/x+OPDnwu/XoDguNdsyWI6LgPXy+2HADcAuTfO07TNsqJ9B1CndMQmYlt9fCOwhqfSFvU4bMmVIIuJq4LFeZpkEnBnJ9cAISZsMTHR9U6Mtg0JEzI+Im/P7xcBdwJim2QbFcanZlkEh7+un8uCw/Gq+s6htn2FDPUGMAeZUhuey/C/KP+eJiCXAIuAlAxJd39RpC8C78+n/hZLGFqYPBnXbOli8LncR/EbSdp0OppXcRbEj6b/VqkF3XHppCwyS4yJpdUkzgYeByyKix+PS359hQz1BtCzdUXOeblAnzl8B4yJie+B3vPBfxWAzWI5JHTeT6t68Cvgu8IsOx9MrSesBPwUOj4gnmycXFuna49KiLYPmuETE0ojYgVRZYmdJr2iapW3HZagniDqlO/45j6Q1gPXpzi6Dlm2JiEcj4tk8eBrw6gGKrb8NmZIrEfFko4sgIi4Ghkka2eGwiiQNI32g/jgiflaYZdAcl1ZtGUzHpSEingCuBPZqmtS2z7ChniDqlO6YDkzO798D/D7y1Z4u07ItTf3B+5D6Xgej6cBB+a6ZXYBFETG/00GtCEkvbfQHS9qZ9Df3aGejWl6O8XTgrog4sYfZBsVxqdOWQXRcRkkakd+vA+wJ3N00W9s+w7qq1EZ/ix5Kd0g6FpgREdNJv0hnSZpFyrr7dS7intVsyycl7QMsIbXl4I4F3AtJ55DuIhkpaS5wFOniGxFxKnAx6Y6ZWcAzwCGdibS1Gm15D/BRSUuAvwH7dek/ILsCBwK35f5ugCOBzWDQHZc6bRksx2UTYJrSw9RWA86PiIsG6jPMpTbMzKxoqHcxmZnZCnKCMDOzIicIMzMrcoIwM7MiJwgzMytygjAbAJKulNTWB8yb9TcnCLMul78dazbgnCDMKiSNy88ROC3X379U0jrVMwBJIyXNzu8PlvQLSb+SdL+kwyT9p6Q/Sbp5gK2kAAABWklEQVRe0oaV1R8g6VpJt+dv7yJpXaVnSvwxLzOpst4LJP0KuHSAd4MZ4ARhVjIe+F5EbAc8Aby7xfyvAD5AKsl+HPBMROwIXAccVJlv3Yh4Pal+/xl53BdJpRFeA7wZ+H+S1s3TXgdMjoi39EObzPrMp65my7s/IholGm4CxrWY/4r83IHFkhaRquoC3AZsX5nvHEjPkJD04lxj523APpI+m+dZm1wSglTauRsLR9oqwgnCbHnPVt4vBdYh1bdqnHGv3cv8z1eGn2fZv7HmujZBKtX87oi4pzpB0muBp/scuVk/cheTWT2zeaF8+ntWcB3vB5D0BlIl1EWk4oufqFQW3XEl4zTrN04QZvWcQKr+eS2wos8NeDwvfypwaB73VVL111sl3Z6HzbqCq7mamVmRzyDMzKzICcLMzIqcIMzMrMgJwszMipwgzMysyAnCzMyKnCDMzKzo/wAXm5AESxp3xAAAAABJRU5ErkJggg==\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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\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()\n", "\n", "plt.hist(-5.0*np.log(nprnd.random(n_rands)), 30,log=True)\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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\n", 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PHV/CJ39zDGRwuMJK7lm5lC7sKuLEWPM7uoodxm+Y29kRHMGtI0kUeI6ZiZwx5Jg22KRv/dypM+PBREcaYR/Ze7iBObhcVrl2hRA+m6HYpsbKyqX0nmuWYeDJA6hNTH15sUCYWyw0MYxqbaLpuyRCGFUeP3H+IGcMOdoCndiuWboIO/ZVjNI30KzW4MARHV3tJASsaif1Pm5XYItYHhyueEvFNkLJMazQ3EfcruXO7fu92iihMipXG7qZXYp8Rp6/6dxBHseQo+Xg/OpNM29hVxE/rY7XXUxdMMUu+MQRyOt0QrV1zyE2RgLgmZTp+Ry4OIyFXUUM331V03Hf7/GB7d2na5PGdwDA5l0jTURfbcPgcAUDTxxospsUC4StN6wAgMy+IUdy+MYx5F5JOVoOzm3UhBNjNW+mwKmEfHIFcdHTNmO1yRuKe74NnFfVPdcsM16/eddIZrmPbO/mIrjv+sZB405AbcPWPYeamAIAzJ/TaVXD3bF9f+69NAORq5JysMhq65+lQbVrTifbHtleH08kjlAViIyMaXF3qf6uOxh1jI9aS7b7geuXe0cxm4iy7X02uAzjehuu3PI0a39Q28C15WRsZLe1NfdemnnIGUMOI7J0PeTcRjl1kglSqvW1DXDtkOAI1YQQKBULrKuplH59o5a5fnzg+uVNKjATWmGo9omgdjFZvQ3cGLvOS1RrE9i0cyS3QcwQ5KqkHEZwEvWmnSPOACY9yGnN0kVG9cWtq3saVBfdpSLbHqm24FQOpvbq71PVThxRlSoUXaWiEihbgJ3+7Zt2plMDJTFUp4WqZrNB7VNX0KFNDScxWq3lNSRmCPIdQw4jOII0Wq3VffBNuwiThLxjXwXrVpXxzAvHrNIgZ8CUsO1abAS0bHifLQjNJVFz6hjA35vKRXQlOEm7u1RsmTTtYrLy/ZuuXdakgnKpqHx2IRKqK3Du0dRe5IxhFsO22Fxbfwndj5/baTzzwjGn6qS/t2z0frG9T4Jrr8lrSWJecSqoSyd0LujMY3C4gs88fsDbcC5rL9hUY5KI6iq3UrGATdeaDdVZIJTJSvgw1P7eMptZlmtLHlHdfuSqpFkKV00Dn62/hEpIQoKcTHl1Ri1MwfaskHoF8ttVBnRmfNL5Xg7yeb5MAbDna9JVOWoNXJNqK2vY1GzPbfhgw+4wSV6kApmqUfBtySvStR85Y5ilcC02UwGahV1mG4BKSHwTyHGMqZt5h+1ZXHs5Apo1ofFRvZjAMVHOvVfd/bQyUZ0Pk01TV/qWKy4xHu/Q+IV8Zx5R3X5kokoioo8A+F0ABQBfEkJs0c5vA7Am/rMLwFuEEN3xuQkAMuT1sBDi2izalMMOn8WmqwY2Dh5syPEvsWbpovpvnwRynNqlWpvA3M6OJq8gFeqzTKowH0+fJITGpnZLSqA4Jmprn0mtcuf2/Rh65XjqGhMhSQptzgmu3cx9/cvxo2Nv4LkXp4o1XvnOC3FjXw+biiS0dkWOdEjNGIioAOD3AXwYwKsAvk9EO4UQP5DXCCHuVK7/TQC9yiOqQoiVaduRIyzugNPJL7B4Bj3zwjHncZ8Ecja1y8lqDdvWr6zf3zWngLGzExCIVBDrVpXrxsikemeXa6WOjYMHG/ItyXcNvXIcz7xwLFG2bFu+Jlv7bDUmQovwqPNlQamIU2fH67mVTEkK1eu5bx6t1pzpQQaHK/ibwycbjv3N4ZO4sQ9Gxu4jbOTIFlmokt4P4IdCiJeEEGcBPAbgOsv1twB4NIP35lAQurUfWLsERX3vDuDU2XH2Hl9Ju7+3jOc2fBA/2nJ1g04acKtdZCDZcxs+iG3rV2JSTBleJ4TAjn2VOoFKqg4KtUfoSfjkux7Ze5g10Nu06AUiq/st1741Sxex7wutMaHPl9FqrSHhHtDYn/r1NmzeZS8G5Bo73XYBwFtNqH+j+pyNgwfzWhGeyEKVVAZwRPn7VQBXmC4koksBXA7gaeXwPCIaAjAOYIsQYpC593YAtwNAT09PBs0+v+CT+VPfUczp7EDtbOM9tQm+JkCopG2CTe2iE2fbN6XRO8usq2otaLkT0bF1zyGWEHLHy4YkgUCUNwgCdXfckLTd8nk2hKi0fO0iMkFgiB3F5lUGhKvKQgICJUzPUdWguWeTHVkwBpNwxK2ZmwE8KYRQZ1iPEOIoEb0DwNNEdFAI8WLTA4V4CMBDQJREL22jzycMDlecaSBMC4UDt3CTbOl1ZsRl4CwQNUmBNgKShkmZakHv2FcxqmJC7QeEKXVI36UXNqUO1+sw+Prqu1JTANkxaB2uiHITrtzyNKvSDFWVJanp4MPI8loRPLJQJb0KQHUzeDuAo8y1N0NTIwkhjsb/vwTgWTTaH3I4IAk+B0ksQiQ+jsCEeP6obVPVW2+cHm+qdVwqFvCFm1YYYxO49plUYcUOv5rJIWqoUAPn4u5SXYUhU1xvW78SA2uXsMV5KoqkzKkCXYQ8ROc+OFxBR4DLaBKPK5tK06bKy8oDyff63LPJjCwYw/cBvJuILieiOYiI/079IiJaAmAhgL9Wji0kornx74sAXAngB/q9OXjYCL5KLHwXgIvA2OwHPm2rTQrMn9PpxVxMBISgeEHptM2T1oUQH64NV77zQtYOoBP4gScPYOCJA2x7CkROZmVjUDZVmA6b8d9kc8oCOtO1CRje9bId8L0+92wyI7UqSQgxTkS/AWAPInfVLwshRojoXgBDQgjJJG4B8JhoLADxcwD+iIgmETGpLao302xFiHeRjeCrBNeWWmH+XD5jaRrYMm7uv6e57oAOUwU2AWDHvgr+/MBrTcbS2oTAHdv3Y+ueQ9bvWFAqGqV3Lj4CMHtZcfUcmpjhhF3zOSGEk1mZ1Hjq/ZwqTIdNkKhNCja7LHdcnpsUIqjGNhclnZUH0sDaJdb0KkmfO1uQSRyDEOKbAL6pHbtb+3uT4b7vAEjnfH2eIdQNk9PZl5U00QC/4ELSQIQiiR1AJ7ZjZ8eNHkE29YatzwaHKzh1drzpHl0N5cOcJXGT13L1pV2wFQESiPT1aqpu03W++nLXzpEj/txxWYhHvrf33qesVdxccLk7+8KVXmVhV9GarXe2I498nmEI0X8PDlfwxulmIgcAp840up2G2gdCwKVGCHELlc/R1TAuDxcOXJ9t3XPIKMFfMK+zKdbCx/U3xI3TBNkfthQkKqN7bsMHWY2Zj7owS9XJwq5iA1Pg5mOx4Gf7kQhRV9pgS6/SNaczZwoW5En0ZhhC9N9c1Swg8kvXpeaQPPy+0trgcKWh6LzUqavv9X1eiIHcVIpSh6nPuP49MVarpwiXbqwqOIncp826m6r6DbrU6rMjSOORZVNJ+YJLTOiq4qaj1RlTXaqtPGMrj5wxzDCELHqXhChLJ7p07hJJook37xox6vo37xqpMyJfo6hvOma1DKYtjbOpz2zEQvd1N51fufmpun1iIaPGkyBMeVDJtvqopi7fsNu4+7DZG1w7MfXdMgW6b39z7fA9ftJgz2llag+JgbVLWPXeglIxz9hqQa5KmmEIUb/4qgV8E5wliSbmCGOICsjlcttdKhpVYFLl8OD6lUY1TGW02hThquZ1SgLVaG37xnJ3qUEVEqIecXnmhKgFTWqxHfsqWLN0UZPbsC9CPYdMx22pPbKKSO7vLePW1T1NqrdSsQCiZjfcPGPrFPIdwwxDiPolRC3gY5ycriyWLpdbl4Hc5L0koUuCXL4nF0LKkALpGJDPjsB3J8Yx+69/9zBMWsgOAgoE1Jgs5LadSchOhptTMrVHVlL7ff3LmwIN5U7ChDyuIULOGGYgfBe9ykRMBV10uCZ9Et11N+P6aSvTGdIuXwO5LZmdyhSTLHybq6atPSb4ejsB6T1zAL5vOS/OScGfA+zjEdLuENdWG0K8x1Skydg6G2wTuSrpHICtIIpUUby85WrjtlmFa9KHehEBwKZrlxkjkEMqjNkKw/gmSnPpyyWxCfXKkVHZ5cD7uMJEPt5OWRKerAO4XO3wVZkNrF3CzlXfNqepCZFkrqd957mEnDHMQKiMYOXmpzDw5AGviWiTmn0mfRKX1v7eMrbeuKLhnq03Nqe3sMEZ4czAt2g9MEVsuHchbvttq3uM389lo3W9T4WPDSdrwhOq0uouFdmCTNzxJLDp/31dW9Nk2E3qvj1bqsnlqqQZBt1bw6Sm4ewFWahkfNVYae/R7+cinG3RvCHurZLYJFXTyPObdo40eCVd/d6Lm7KohurV1eNZJZGTCLGpqDs91QUZiFxupSeYijS7G07/73t/WptYknnLPVs6OpwvaqWcMcww+BI70wTl9LY+Kpnphmm34yKIvgSgu1RseIaJICTVVQ8OR+k55JjZImp9bDi+xM6XIPv2UZmp0mZ7fhL35qRV90zP4HbHrcx/5HJ3Pl9cXnPGMMPgu5BNk78dla5aZXhLIv3ZFqmE9GqyIWk1OP0+ADjNufPAb3x8mEdIe336CGiunOYjTYfubnzbbZtjejU9Ha3Of+TyBDxfUnnnNoYZBh9ph5v8rUx7AbTW8MZ9d3dXkTW8m+wFxQ7Cwq5iW/TGoff5jM+apYuceveQCminzjSnOdeR1HZgU6uY5kRaGwtXTU8i6/lugjqGHM4Hl9d8x5AhspCmTRJJsYNwwbxOjI6ZC7OrkJKemtjNN/LZhaz13yqM310gvHF6vB5IpkuYWbl1unYrGwcPNlR7u+WKS3Bf//JEuxybJC4LCKmEj4CmlNohFdBGq7U6szQF5HG2Ax/YdiOmnUAaG8vmXSP4abU5oaKEWiApC9jWshxDzhvufEjlnTOGjJCmOL0Kk5HzgnmdrN7aNIEBZNIWHa0MgDMReVfFM3lfWqZkU99sHDzYkCZjQgg8vPcwHt57GMQEjiwoFRMJCVw0sG5ADq2AVpsU6JrTieG7r3K2Sz2/oFQEEViBxKZWMQkMaWwsrkj6LImx71puh+p2upAzhozASTp3feP5RBLtmfEpXfWJscaEeHLx6kFtcgLP7exoiWSfRc1nG1QiPzhcwR0po1N9ibNtgX/mcb7ADhfzVpuYNBKWoVeO45kXjrHt8WW8tva6InpdOxbOI85EHOX/3Dipc2VwuIJTZ5ozr/raWGwgIFNi7LszzjIQcaYhZwwZgVvU1dpkfaL7Su4uXay6eENqFaSV7NslIfmWK/V5hknqA5oXs6x1oC9wjujZcOqsWYLWi9HLqm5yLvgyXhtBShPR6/KI44gj904C6rYG087C5MEVmv2VANy6uidTYhyyM85ixzoTkQljIKKPAPhdRBXcviSE2KKd/ziArQCkRer3hBBfis99DMDG+Ph9Qog/zaJN7YavpJM2Z1GI776pjWnQLgnJlTtpzdJF1mLz3DOkrlpN1y0ZxgPXLzfqqJOkw/BFbVJg086RettDGK9OkNQIcF3D5cu8fQQHrvypKYupzHsEmOtGm2oi+KoUgWhsTLXCTQhR7dkY9GxIhwFkwBiIqADg9wF8GMCrAL5PRDsNJTq3CyF+Q7v3QgD3AOhDNI/2xfeeSNuudmNg7RJv6TJNziKfxWuqVZCVZJ+VhGRbYLZvfF/PgoaAMm4XFqKrrtYm6ioj/dtuueISaypuHaViAXM7O4yEzAT1uqSMV98dCUwl/TPFJ3DwEW648qdJ1H7cORPTMzFMXw+kUPsfx6DXLF3krD1yviALd9X3A/ihEOIlIcRZAI8BuM7z3rUAviWEOB4zg28B+EgGbWo7+nvL3m5/aXIWue4lAPdcs6zBpa5AVN+pzIScLi63V9s3fufF414uoqG7owkhGtwipbvnMy8cw5XvvBAF8kuH8cD1y7Hp2mVsNTYX+nvDq5dxRmtZUMeXaNmqyAF24YJz31zcXQpO060jrRt2Vm7Fu5831xnfvGvEqx3nErJQJZUBHFH+fhXAFYbr1hHRLwL4OwB3CiGOMPcaR5uIbgdwOwD09PRk0Ozscc81y7z0o678NS7J0fYOgUbppVXFSNJsqV3GPZuemVPq+BpobdJ8tTaBTTtHcGa8UdV0/NRZfOGmFdFzLQXm9QhzteawLfOtnkpB79s1SxdlYrT2wbzilONCqdiBecWCl5u0Sw2W1jbls1Pl5mRWbsXcrihp+dmZjCwYg0mU0tder49HAAAgAElEQVTALgCPCiHOENGnAPwpgA963hsdFOIhAA8BQF9fX2uUvimhEnRbGmxbDiB9cm9bv9LoCfGZxw8Ydd+q5NaquAPT1nzgyQPYtHMEJ6s1JzFzLVSXt4sJvgZawM5YbbmppA1CdSWWMNU1VqOgbRNWN4zrfasbrXXmnoW3mEldA1BDpTxbTIyPGkyuC3UHq96bBjZ1Uau96UxtOdftEFkwhlcBXKL8/XYAR9ULhBCvK3/+NwD/Ubn3A9q9z2bQpmmDKmnoPvASHHH21YVyOwddCmtV3IHRV35C1Imli5j5LNT+3rKRAAPN0jfBnMTMJmVyjJWDzrSadg7ao0KdBFRC6bpPnz9ZeIuFGOu5Epy2/jbN2Sx3sDYhKCtvOp/aI1nFM003srAxfB/Au4nociKaA+BmADvVC4joYuXPawH8bfx7D4CriGghES0EcFV8bMZBr4mwcfAgm6pBXr9jH6/PNxHnEF2oj941rW43pO0uqN/hkwt/cLiCU2eb/d6LHYRb49TYAIxxHC47Sn9vGV+4aYWxDZydaLG2E9PVSbVJ0TBOSfro6GjV+z71urQ6eP15Kk6M1Yz2iyQlOLNIWW2qTWKrFy6/a15xitSR8t6Qb/CpPXK+pOVOvWMQQowT0W8gIugFAF8WQowQ0b0AhoQQOwH8FhFdC2AcwHEAH4/vPU5En0fEXADgXiHE8bRtyhomKcC1vXdJjAsMFc58JPyQbapJUvKpc+BCkiAkYCqHjuwb6QrKZfbUDX1AFAUuJVVTSgLbbswndgHIZieWpI8k8wmpLyGR1lsstL1JSnCm3cEaVZhPHDArpGMsKBWbxlMXJAA/ab6/N0oPr6ZHWf/+S1pm75lOZJJETwjxTSHEzwoh3imEuD8+dnfMFCCEuEsIsUwIsUIIsUYI8YJy75eFEO+K//1JFu3JGj5qAV0qcE0Ek5OLS8IPTWLX31vGulXlhnUj6xy4JCVb1TiX9woHAhoK60wIUSe6+sLk+m9UMfSFpKg29RuAJg8gKX2r6oEz4xO4Y/v+ej/47MRCma/sB5++bUVQIbeLs5VoDSV2aXewXLoPkwABRO0nsqvmQqR5qQWQKsgJIZrWUqt26e1Gnl3VA74LoKJJjDaoBE7CpWJJsk211TlQEVI1TlddLOwqelU3E2hepFz7fRaYLSOrCt9+k31wx/b9OKnokqXWSPbDmqWLmjKWFgtUD7y7fMNuPPrdI/CFqvoxqYW4qnJZglNHbbp2WVAJzlCBIoTJhTKiB65fblxnSZ/rM4/SfuNMQZ4SwwO+22yZAsDlcimfqcPl2ZFkm+qrngqtGmcKQpLt7giMFuaiaTmVjporyoQ3To/Xx4F7vn7cFCRmQrU2gd3Pv9Z0wcSEwPbvH6lLr67vtwVomfrWVYktrSeM7X69up5sv07sXIbXtJHzIeou6Tpsmyfqc33gM4/0b+zuKkIIZJrluB3IGYMHfPO3qHpXOfiqL7uETYKw6YqTuN353OPrQePr9z04XDGmSLC10fQ82bYQd1NgyhAc4s4Z4kVk8lufBDDJqDR0FIgyjdpN6wnjut+3BKePe3QaW4jvOlTXl+se9VoXcw3JZdXfWz6nPZRyVZIHTNtsDrr0MHz3VXhw/cpMVAFJtqk+96SpGmdCf2/ZmymouY909YMpCjgJE+P0/erxdhkHiwX//D6An/oirSeMz/0+Edm2wj1ZQK5DWxR6d6mIecUO3BnbhAA0qT27S82FnHzsd6Hr71z2UMp3DJ7QJZ2QIh2qBOEKFHK1AQjbivvc41siM0RPWmae2V0qYv7czoZAJ1VN4ZKqkjAxTg0jjw8OV7zVX6G5kIim0nPb6kFz8FFfpPWEycqThptHqoo1LWwxPOtWlZvyaA08eQDz53TWAy+59eK725HX+qy/c9lDKWcMCREaNJPFdt8WEW2Da/uetmpcyDOJUI8Kl4SYM46b3tfNVCJTEeJeKsfFxBRkjITuVgs0E6ZiBwGEBg+ZkERvHHzUF2kje33vd6laXFlWffvB9R6OQPsEXnJrzpeI6+/euucQW2cjZFxmWrR0zhgSIlR6sEWWup4xOFxpaVbHtEZBn2cuKBVx6uxUmU6XbG5aqIPDFbxxujnordBBeNNcXiq0LVBONeWT0tlk/8h6cfsIIGkje33u9xFs+nuTZVlVERL9r/ctV6RIhSp0+DhMqK7iplQ3tpgm33GZibaInDGkQIghjVPVnBirsTWNJTbvGmGzOmY1cdIGSLmeeeWWp73VL4BZqjJFHAPAm+Z2Yv89V7HPsqVR5lJqTwrRYNw1EXzOoygpbFJjEgk6ZEfput837xanQkwTq+Cb38vXa0ndKcp3mZiC6gXn47Gmt1fm1srCaN9u5IwhA/jU0bVl11RhmhCc6iRNVkdVArJFIKeFy7XUBE7a5aTOkw6GYyJ8a5YusqYs0YMKWy3N+bp6ckjL2F33+6pa0u5e0ujlfWuiuHaKk0I0rOMrtzwdlPdKba/PuMxEW0TOGFLCh3Bs3XPI20tHPkOtUJY1OGkpa6Jnzthphk9hmTS6dJPzgI8LY7ukuZkoNaoIcdUEmpmwdLpw7WbSjjGXeFHCVRt7Ugj8aMvV9b9teZhsCFm37c7+6oPcXdUCWxSnhMslLcnEktlCpdsch1Ix2fDZXD6zdKdzuZZKp8NybEx/2VGcJm1UqTqetn5VDcZJpDmfeeP7PHk8yTOzBNf3Jldj1bV1YO0S7NhX8U7jYnpPsUA4dWbc69tNRZLUeSbH1iey3lV7nIOa7ddnnAbWLjFG0k9ntHS+Y2Dgq0Lw8XgJhe/uYl7CKmGuLWpWW1jbc5KorVRpNDSvv27At7UrSVCT+p4kqifbe2aCcdKmjlPbdcf2/di8a6TulptkJzS3c6pY0Pw5BZwdn2Q9i3yTI+rv8lF5hQQ9qghJ0jc4XDHaEINUDC1AvmNg4BucYpM8kk4sX/jkgTHBtUXtIMpEIuXeI0tOAgiWgvt7y3WpUleB2e43Lj4Npt1Hu4KabO+ZKYFSepDbMy8cM87vE2O1+niE7LgkA1RVQWNnJ5ocDtSU2b7JEU3f4kpVHiogmVK228ZJtt9kK9TTuLcbOWNgEGJs4xa0TV3BxW761hYGkusgXRk81frHaWDrm9BMsSqSEEqbod4Wke5DQFQkNSTa3jMTjZOu98vxCMk2ahpXWynXtAzTFc0dsr7K3SVWUOP6ySU4Vkar06Y6zFVJDNIY2+TW1VYlzHTUFL0J8MFTaXSQam1fE7IwfNr6xmT89X2nLfVC771PNUUXuxaVamzkvsM3CJHzh+/uKjY4FJjUG9x72mWcDA2ycrmHHh2tYtv6ld5eSiGMbnF3qeUM0xikWSBAoGEXo+7sQsbJp536TqhdqsOcMTAICU7hFlNIhlE1sZopYRng76dua1OIp1BIUBL3Po7YpVnUNoJ0YqzWFPxnkyC5egOhRNIWQQ0AJ5l4Fdk+VW+vR9GmdQH1QRI7BhfpLLFYsdf49KUtpYYps2soIQ4F13b9mPS6MtV5t41TSLbYdnuokQggXuxDiD4C4HcRVXD7khBii3b+PwD4JKIKbscA/LoQ4pX43AQAuUoOCyGudb2vr69PDA0NpW43B18ffxORVdMgcPmUTCC4JVfftmfVJtUWIJ/tk+3UJ50G1w79nb7faHvO5Rt2s8TrQUNqEVcfmhDSrxLdpSLOjE9av0O+F0gewObD5JKOx2UbdrPnTH1rax9HWNetKhtTTiQZp6zhmovdpSI2XWvOjxUipAHZ0Agi2ieE6HNdl3rHQEQFAL8P4MMAXgXwfSLaKYT4gXLZMIA+IcQYEf2fAP4TgPXxuaoQYmXadmQFk48/V2Vs084RqzrEN+AGyE7KcbXJdxegu8txEqVJJVWbFA3S8R3b92PTzpGGBcJFHftUPpPPsPWt+p2cZLawq2hcsK4+dL3PFz6R4GoUbRJiZyyH+eQBbNo50pBCJNRILBlNgVOdlcx962qfgF9MC2AutbluVfYR/Da47ASj1Rq78+J2JK3eCfkgC+Pz+wH8UAjxkhDiLIDHAFynXiCEeEYIMRb/uRfA2zN4b0vga9AaHK6wC1uNevRBVmoBnzZ5Ty5trXP94ht9LReI1Pe7Mp76wGaoV7+TM4Lfc80y/TavPnS9L2tw700aZyMTy6n6a1MNcsCcSE91GuBSSWy6trlvTdi8q5kJS6bgYoY+pTZbDR+BwGYQNxnA08brZIEsGEMZgFrH8NX4GIdPAPgL5e95RDRERHuJqJ+7iYhuj68bOnbMn3iEglMH6Mdtemt1MZlc2ABAVsJUffHTTmifNpkmnYm86u5yWRj01AWSxsbg0ufru50QzyLfcdXh8vTSUSoW2Lnh815fry5fwkUEL2JkSyXhW29EMrTLNuxmBQufdmftxpskiNBXIAhZP6GecK1AFsZnE10xrlgiug1AH4BfUg73CCGOEtE7ADxNRAeFEC82PVCIhwA8BEQ2hvTNNoPbGuvSqW2g1cV0zzXLmgKrigXC+p+/pCk4KK3nga1NY2ejyNHF3aUmnS3HDH3UMUCzcdCnjWk8bWzbd67mga9nke+46tCD72x9QgDWrYqcDFw6ZjWKVjVMm7yfqrUJfObxAw2pJ3wNnKNjNWxbv7IhpcQ8Q2Q91z96KgkOvnp1n3mQpVdS0iBCk2OACQKRHcfXNpQ291VaZLFjeBXAJcrfbwdwVL+IiD4E4HMArhVCnJHHhRBH4/9fAvAsgN4M2pQYnBSqH+cmrq637u8tY+sNKxq4/9YbVhiDg0KlHV3C4dQBQOStIyXLHfsqGFi7pL595SrS6eoYDnLrT4h0y3p4vwoZPLdm6aImiYLgZ2PgFj4BGL77qgYPrBAJULqbmsDZI1RItcDLW67GtrhqnwkCkcrMJBnetrqnfp8pvbNNhYP4uLqDWLN0kddORo71mfHJ+jE1UE2/jrvfBd+gTx+1Sdq2uNrlsx77e8tYt6rsFX8UEqsz3chix/B9AO8mossBVADcDOCj6gVE1AvgjwB8RAjxE+X4QgBjQogzRHQRgCsRGaanDVzaYH3bv2bpImOBdJPe2sT9uSRevqoUvZZ0ZbSKYoFQ7KAGH2uT5KobUn3cIW0JykzeS6Za10BEuAaeOABQc7sEgB37Kui79MLE6SPUNoRIgDb1FDeu8j615gQRGryxOHdOV/bNJF5OOqq1CTzzwrF6igjueVL15pO+olXZU1X4Gq45B4bL3hzOGJLuPnQ7hwvV2gQ27XTXYJlupN4xCCHGAfwGgD0A/hbA40KIESK6l4ik6+lWABcAeIKI9hPRzvj4zwEYIqIDAJ4BsEXzZmo7TAmtAOCN0+N1Ti8ngzoVpHrAV1rtZvTLLmnHGkY/IXDBvM4GCdQWOSrhq9M0JShTI5nl927dcwj3XLMMD65faZSkapOCTU+RNn2ERKgEaNOdc/pdXc8/Wq017MxCjLo6ssxXJXcy3A5m/pxO7wjrUP13yK4WsBuu9Wf9+YHXjNd958XjwVI5Nx4CwMZBPt9ZkrQ3o9Vaooj/diKTADchxDcBfFM7drfy+0PMfd8BsDyLNoTAFZBlkoylMZZLDCbVA/L5NmmVq0Rmy6joW9dgdKyG4bunitb41qb20WnaAn5M3/vA9csxmSBOxvWNPkFToRKgTXfO9YuLKFRrE5hX7ECpWAiSsG0R1Dpk/QCfCmRcv8p6FiHR/klcUV27Wpt7qulZHEJLiQJ2W4HcldzX30yqsnTKmEm7hlkX+eyjYuAKv8hJ4CIuri351j3mSmRScnO12QZ9EWcdNWsiCrb0FiHRnRI+xeNdxCnUuJ3EGO5DFE6M1bCwq1jvH1vAE+D2uFKhBnNxwV5qXioOHUS4fMNuLIjtQ1mlXuFcZRd2FdE1pzNIlRIqmYcSbBkTwVX0e/S7R4yMIcn8NmG6817pmHVJ9HxUDC6jluu8LZePrRYAx5B8F4VpEbfD9c3GKDnVnA1S4kuKweEKTp1p3pHZiFwS33EfIyehMYGfatw1wabSkoZp0zjaxtk1f6TBerRaA0RkT0uiJvJ1lR0dq7HJ67hnhhLOUAO0VA9z4Bi1zf1bdSaQ/cm5KE9nUR4TZt2OwUeV4JKyB9YuwcATBxqk/mLHlBrIJkXY5MA0ybZskmirXd9s0rbNaG1DUgmK211xbqwSITl9JFy5gnwM/zpsKi2TxKp/Q0heKhNqkwJdczob1JEm+Oy8uXkhPdNcqUjUZ3Z3Fb2DKXWG7pMSxMU8Oa+j0Hlj29nNJMw6xuCjMvAabJOvZQxf32YVSZNtZVWnOTRpnApXegtXTWYTXEZKDtwCPzFWcxb0CWWg/b1lPDF0GM+9eLzpXFexA2M18+7AFUGdRToEdTx97RU+7ZNI6sEETKV1BxrHwvZM3+brzgIbBw82eA+aign5fPMtV1zCnguZN0kEkOnArGMMvjp322Bv3XOoyaumNjFlnNYH3zWnXdLswNol+MwTBzCh2SXkLiULppCmQpgrvUUSPeyps+NOO4MJtgXeivTFL79uft/C+XOxEGYjqSuCOq1Eacr3paNULGBescMohUubg41o+XowATCmn6/WJrB5V6Pbpi3Q0petqc4Cg8OVJpdyiRNjNdy5fT/u2L4f5e4SFpSK7K72ttU9zt1aCKY7eM0Hs87GwOliAf9qYr6LQupROTdBiS7G6KxCZwpAdlWefOwuNn2yqz9C00UAU4w2FK6dRpqUCSa47Cuhdgs5P1Vd9NzOsGXK7ZpUbci8Ygeufu/FxnHRg+RMa8E3uKy/t8x6pp0Ya3Tb5LC4u+RdwEp9/9Y9h6wMRd1FjFZrTcSwVCzgwfUrjUwhSfqMcwmzjjEAzYmrADjzzqgTgYuO5RaLizC6trE2QibTJaSZoC7C7srLw323JNIqMwb46nW+7TJhcLiClZuf8rJlZOkB4iKQKlFf2FX0NvyfVtRQegJCF7jvU+nzibEaduyrYN2qslVw4RhpCNNLY1iVz/T10lLfHzrOk4hsdTbD++BwBb33PoU7tu+f8bEIaTArGYMOl8Tsm1GSkwQlYeSknjTBTjKXjpygd27fbw3ICXm/PO7qn4G1S6IqcxqkOgjg00UYbmPbxUlpGwcP4s7t+70N3Fl6gHAEcs3SRU31i0+M1bB514iTgKRNDuf7fTIyOong4vJ2U8dq7Oy4cX5wMD3TtevuoMZa0ECycZ4/t5Mt9WkLLpURzeq15/KOYtbZGExIEpcATAUX+RiQ5Lkk+uMQLycB4JG9h51pJVS49Nqu/unvLRtTYKh2FxVSx2qLzzB5lpjsIEOvHGf1yCYk8VixgTMm2ozgLjuHy92Za6et4A2Ho6Pm2skqOALL6cr1sToxVkOxQCgVO1BlDPIqTIWBOAcHCalpVe1ISZxA1L7X58apM+PO2guSAaSx2c0E5IwBbk8Qn4ySUkKwEZikHgmmCW5b+KGRn652+XjKhBZCl+9zpaGwRXxXaxN49LtHnASQY+A2o7tsn884mQgklwtLtts2Pi5BwERoXAVvTp0ZN+6oFnfztZOBZK6UXGDbpJsnwLSpHhyuYPv3jjSfYKAWN5LtUXNZ2dxe1Uhx30hrFXJXF1LoKa1w0grMWsagJz/TIz4B4NSZSBXiIowhXj0ujwTbJPGt9AQki/y0eUW5djpZRg5LzxKfiG+X7rlYIMyf02l0meVUNpt2jjSU3JQquqFXjnt7p7g8sVzpvV3frRMaLk2LTHC4cfAg61L8zAvHjG01MWgf4sV9m4+d4NYrepqOcZkCbHAlKTT1h5qSJkkOJPW9vufSegS2CrOSMeiDMVqtodhBmD+ngFNnpyaDNPqtW1VuqJ0ANBJGH5/uJO3SJ4npWVyAVZZ6dI4xAajvkrq7ik05cNQ+MhEWFzPxWZxc/QwAmFMgnI2rlQHN/clG5hqYSKiKzkXcbePj6+6stt+l7rO5FHOMn0u34WKUfGDblMpHR4EIt1xxSWb5iFzzv+/SC7H9e0caGY7yM6mDgnyvr5CUFe3IGrPS+Gzc6k6KBk8QCTV1MWdoy6pgSKjRsb+3jFtX9zR5+bQiktLlyXVirAaQ2auD82oy1QrwsW2o195yxSXGlARXvvNCYwZXtT9DmWdIqg5pnC0Zit0A7roTPu7OC0pFb0852xx1GZK53cgjew8bjaomY3axYMizHh+/bXUP3rZgHh7Ze9hoqLWNE8lnKzBl/NWfa9qFqO7ftnornHedfG+Ix1aWxYayxKzcMYRudeXiCdUJ+xIeV+ZU2yS5r385+i69sGU6Sk6FwOmRiab01nKRcQxPrRUQYtsAGiO+Td9v82FX4ytMkjIX+KXe6wPZT6ZvCKltbWpnsYNw6uyU3cDlKeeao7b5zbq/wmzLMu0wORtHsYOaKhneoQSeSUKrp6ABIoaw9YYVTe+yZfyV7fOJvTHNDTUQ1aVe81mTWUW6Z41ZyRi4weDUEq5BShOt6qNHt72fm5xJDVq67eXU2fG65K0uLm5hnRir1YmqvJ77NhfDdak4JEKNvyoxBMxEJSsVXRYSoamdY2fHjcyLM7SnmaOuqGSuzeqYXL5ht/E6Lm0I0Ji+feuNKxpybslsAYCZANsy/vb3lr0YJfds7htt388h6+zHWWFWMgbOy2dCiCZvH59BUidRZbSKAlGDysImYbj06Lb3m3S/A08cwOf++8EGW4k8rrbF53kmCS80nXa1NpGY4abJK8O1j9BYNtK0gAeHK+jS7E3yXhlQmLYdoQzGl9BytZflvapbsW9EtS1ZoO93JE1PrXoYcXEFpl1B0h2Ba27o70+7U3fN8enyWMqEMRDRRwD8LoACgC8JIbZo5+cC+CqAVQBeB7BeCPFyfO4uAJ8AMAHgt4QQe7Jokw69g9etKte9MVRmoLv5uTIlqs9cs3RR07ZYdX80TWIbU3C9n7OV1M42P7M2KbBp54h1Uvl6YhwdreLW1T3e8QMTQgQXq5Hwlbx0cMz/1tU9Dc9zjaEKNYWCr+dIUonQRRCSMhxTRLXrO/p7y8Z4kWKBcOrMuDOvEsD3w9zODmdgIkfkbTa5LHYENmTpTeQbD9JOjyUSCSpsNTyAqADg7wB8GMCriGpA36KW6CSiTwN4rxDiU0R0M4B/LYRYT0TvAfAogPcDWAzgLwH8rBDCSp36+vrE0NCQdxu5VLe2Wrh6HWOfZ3KxBdJYFaK+sr3ft5qbCS8bpEmJyzfs9iL0C7uKOF2b9Pp2IPrGL9y0ou2Sj4u4hoyhCa454tsO0/UuFZrPNTq4an5JvqO7q4g3To83eaG56jfo/WArjuNqHzdfCcC29StZwSBtQrzB4YoxMaCtrfr9PvMh7XiZQET7hBB9ruuy2DG8H8APhRAvxS9+DMB1ANTazdcB2BT/fhLA7xERxccfE0KcAfAjIvph/Ly/zqBdddgki6Q6YM5TI/RZodJ0SDW3UPhs90vFAoRoDuCxEdOJOC6h3e53rneGjKEJldGqVwbY0G/3cWFMIvGmtXeo33HllqebbBwuN0tTP7i8vGxrgcuIuri7ZNzlCAA79lWCsgLocFXYc/VlyC5gOj2WsmAMZQBqWOKrAK7grhFCjBPRSQBvjo/v1e41jhgR3Q7gdgDo6WkOgrHB1sFJt+Qhg2PzbZbqIl2dsXXPIdy5fb8xyC0pU+CqR0lw3i8XzOvE6Fit3habYdcEV56bVsElmWWxwEK39hsHD+LR7x7BhBCs774vQQhlOFl6wPi00Ucyto2BqkrVn3XZm0usl5NkJM+8cCy4UJILSdOHAPxOg2vTdHosZcEYTA7U+nhw1/jcGx0U4iEADwGRKimkgbYOTqoDthk3OeO1SdKXJSjl1tAlUdgWkq04TLFAdS8ODr5SKKfGIkJTQZVWeFj4EJw0FcZ0SB98U/Qtt6h91CYTQtT/Vl1uucI6aQlClh4wWWUD4J6jqksGhysYePJAg3ccN24XzJtKYd8KiTtp+pDB4QoGnjCrn7jnTqfHUhYBbq8CUMsbvR3AUe4aIuoEsADAcc97U8MWcOIK7gl95q1MXV75Hl1q11Mqu4LcbIE3gkloXe4uYesNK7ykJD2QzXQPl41Tn/MhaaZ94UoBLuETLOhaYHIMt96wAuvfz1fw0hc110ZOl/7I3sPO7L2qoTdpts6kc90EVxCXb7CmTzDY5l0jxmBFE06MTSWy49ZKCIPVg+S6mV23XjlOx6adI9a0HqY2qeMl3yH7sNXZWrPYMXwfwLuJ6HIAFQA3A/iods1OAB9DZDu4AcDTQghBRDsBfJ2IvojI+PxuAN/LoE0NcEnCSfTfSXS8MuDJpptN6mZn0vsDzZJXFgZgeY+rlrNPAaJQbN414pVCwEda7O/l61Hr/WYrFK8vao4ochDMeRmTIA29XGqPEGRl63HN/xB1mHwO5+rtW+tZwpZdNUTiNu16ih2EQgc1FM6SgXa2frWtE1fKfvlN7fROSs0YYpvBbwDYg8hd9ctCiBEiuhfAkBBiJ4A/BvC12Lh8HBHzQHzd44gM1eMA/r3LIykpWmH8TPJMbsFI/3iu6LnLzY7T+8v3Ze36JpmcbcJnbSQbHK54RyTbVB16emod+kK16ZVNizqr75YxCUkMve2Abf6H6MddxC8U1doEPvP4gTpTndvZgZPVWrAwxLmDNyGdY6dz1zYd+ZQyiWMQQnwTwDe1Y3crv08DuJG5934A92fRjnMBNr22PK5LJD6BN5ze35aQLnRy6TsOl34+ayOZzYOlg6jBQ4iTFmUBHXnctKb1wC8boV+3qtzkKJA0mEuHzWnB1a7pgB41ryfNUw3DOmzzs9tSj5mDVMedGKuhVCxg2/qVmQlxOmSOJdvzFzICX5p2tHL8Z2USvekEp59XMTEpsLDLXmLQ57khxXZcMOnNbTW5WmEkc7n9qrYGTp/+zAvHnF5dut3HZtfZsa/ilRzQv37Z1PUyERx373Tn01Ghz7bOMjwAACAASURBVI/Raq05k6qlE2w76WWL39R0vANuLzsJWyJKG1z1w1W41pHN8cPVtixsJaHIGUMGcJXxU89v3XPIWWcXiPTzNgOwjv7eMtatKtfLhxaIsG7V1M4i7eQK8fnvLhUbMqtmVeLQ1VadAJgM6b6MUH0Wx3RNdh0uG++tq3ucAoEKgSl1HdfPY2fTGaN1pBkrHzdqWdHP9F4uOywAPPfi8Ya/CcBHV/dg+O6r8OD6lV79GipdDw5XcOrsuPf1vuldTHC1LSRba1aYlbmS0sI30ZwkjLruVBZht6WUSDKRd+yr1LfQE0I0BPOkNcTZ2sOlEEli17AZyF01DlztBMJy9shnJbHrmNR90iXV9/1c5KuEnqxQbWsoNg4ebJiPoc/0na/6dfp7fSAwlZ1WHxtfV1+fzKgmTygC0KnVHfHNoVVOEZcwt7OjPu9lAsFW2pdyxhCIkERzANiAFldJytBtIqejlfmR0uaGsRFUtVKYT5s4faxvoSIbcU2SCZeD+qwkdh0d8hm+0eshtZtD7EWm/FAm4iyNuLLtNvgyXLVvBocrwUxBQvcuk+2zVapT3+sqPGRLNS7rjsi178tMTf3sEs5Mc8VUNyZrzGpVUpKts2/ksZwkXECLrcxhkm0iN5HVAuU+MQocXLYRky+/zchu6ncf33f5DSYVgm8mXN03HOCLrtiQZos/TyngUyp2sPpyLgrUhMpo1TmXTbYiG3HWbTccfGxnJm+vpA49HPO1VapT32tSi6qFh2wChtxJmMaFs2fIHb36vQQ0qHtN8I0HyRqzljH4Bkrp8N0yS39sDtxiNwXK+DAw20TOYhJJglpwVAqT7bW5Gsqtt97vIQbypMFaqrRc7i7hCzetwMtbrsa29SuDn5WkDbJvGj1Uoqh0bk7IHZl8h83oKvv0zu37cRlTuSw0P5QPITL1xW1MsKdEUq8aG/N1uYPb5plaoc/F6EartSA1MNfvu59/zbq2pytf0qxVJSV13/RNNGdjCsUOwjgTBXnLFZc0MQUfPf3A2iW4wxHLkBYmf3MgzO/fpBrxTZVsak/IridJTW0XQu9LkipaV9P5qKI49UbSueCjJgrtixB7T7m75KUCdbmD3/WNg2zyPaDZrsRlUbXBNF9DClup75+ufEmzdseQlBObJIliBzW5l9q8jooFYqUNfSvsu5Xs7y2zkqQpjD+pB4qPlGzrQ5uU1WrvC64vP/P4Aa/vz8LDyjbvfL9fV4e54JNSxaWu4naKLtj6zEf9JN/tqwJ1PbNamwAR/726XekLN60wjgm31vRCUKbn2mBK29JujyRgFjOGpO6bJsK49cYVGL77qoaJO7B2CTv5XOUMVYQwsHuuWdZUGB0ATip2BiC5Gk3CZavg+rBAxC6oDiIMvXK8Ibgs61xLXF9OCIE7t+/HxkFe/ZW2zyS4vpFqDOnK7FJNyTEIsT9cueVpY4yFmuOLQ6jUDLj7TF9LSd6tMx4ATqY5OlbDrat7vOxKnCB0zzXLjLEqeiEoCV8mCDQb1rPKbxWCWatKSuO+6bNl7u8ts6odGwhoiOANTS3wO994vsnNblJEOYZUr56sQ+z1Ii5FzaUPiBb4G6fHUSxQUxvVTKMSWXtfuDyrHtl7mM3Vn1Wf2byipCtzyMIPUceortLPvHDMqJqxFYcJhW9diaTv5lSDD1y/HM9t+CD7vMXdJdzXv7who61NRWVb7yFefrrLqRBmr0Z9bbcinY8Ls5YxhLhvJk0+x/ktLzRUv5KQkqN8/pqli5zudyq43ciJsRqu3PI0jsbSmwk+BMbUF0Cj3eHEWA3FAhntCbVJge5SEf90etwphWadD4brSwm971VkZQR0udyGfnOI+618/jMvHGMrgGWZ6jm0z0Lf7WI8ruelJbi+93Mup+tWlZvKyLYrrbYLs5YxAH4DazNYAnbGwk1MGR7vYyz2cb/zhYvw67sVHVxfqJKQhC1N8smAvDcmIpKUUfv0GUe0sjQCynnHlab0qQKmfv+6VeV68R8f2J6fJN6FG48kzgQh73YxnrSxO1mBY2AyQn6622fCrGYMPrAFjp0ZnzQyDHWbLJ9hGnifAKlQqStJwjEJgUaVE9C46E1RpdXahLe0KuFKDme6Vm1P0iyxPtI9R7SM1e2UGgn62PowryTMhoukv+WKS5qkTy44zseO5kucbOORZPcR8m6fDLqy/5Mk0csKtjU8HWoiH8xqxpCm9KAt4ll9Bjfwg8OVevU2FfrCCZn8A2uXYNO1yxLZNiRkoRNThG6oAXJhVxGna5NNhGHN0kXY/fxrzvtN2TjT6Ppd+nifvPiqHYWrkQD45c9PQjhDpM81SxcZmQWnhkwC23hIdVWrJGLfDLqV0SoGnjiAzbtGGkrUhrYj6U416W4zq/opSTBrGUPa0oMcfKRSzg/dlAMlZPJLw1taSCLrG+XNMQCpMnMRKw7FAnkzap9+N/Ull+fJBN1QytVIkL9N53ShAQgjnEmkTzWyWQANObTSwked00piNq84pcbsLhWx6dplbB0Fn9xSHDHm6MXQK8dZQ74EN+9s+ZWyrp8SilnLGHwlT85gOX9OAafONhM3H50zR3BNFc844tHKUHm5qH2ILccA9Ap5Elduedpb9TRWm2yyeaTR9fsQYl8pLQmD4iK4swgK6yAyqrSeeeEYG1CYBYGZrgAsk3B1ZjxyvPAR5Ex9YCPG3HrzSTyoq45VFR93Tys8B0OQijEQ0YUAtgO4DMDLAG4SQpzQrlkJ4A8A/AyACQD3CyG2x+e+AuCXAJyML/+4ECK5HiQAvgubM1gWCx0oFdEkJft4FLjC9qWqQgjUK0+tWbqoLpnYksgdHa2mKgoCTC1q125JT98hJStZuGbrnkMN7U5SwEZfNGm8ZlxE30dKk8/glGo2+4kPsXS1cWDtEgw8ecDo7mtqs6soVFpk4cWURGViI5wFJsOqDr0PbM+0JtUzXG8S8Pp7y0YXWtM905UKQyLtjmEDgG8LIbYQ0Yb4789q14wB+FUhxN8T0WIA+4hojxBiND4/IIR4MmU7guEr6bhsDHISlmPiLYmiTRrlpqzcXgJoIOyV0WrDrsWWdVO+VycehQ5CB5jShApUvb7LFXJSCCdh9W03B5PfOxCut/Yh+i4pzZWKQiWISYilt/rA0YFqmzkimTSSWUdaz5+kKhMb4fSdXx1KF7iSPnKu59z1l2/YjQWlIohQt2usWbrIKtCpmK6dmERaxnAdgA/Ev/8UwLPQGIMQ4u+U30eJ6CcAFgEYxTTCV9JxSbkTQtR1/qru3FWXQUcowZRZN00pfOWC2rxrpM5g3jS3E/9qxcV16Z3LW3/BvM4mIszli9EnqY9NwtRuF/RFE+o14xsz4JLSbN8n9du6OiArKVglwC7mDkztPpNk9w1FGjuCDzNO4grrQ8RlN7qSPhaIrPYpE2QVOwldSOLaLZFlPEkSpE2J8VYhxGsAEP//FtvFRPR+AHMAvKgcvp+InieibUQ013Lv7UQ0RERDx46F+/Dr6O/1CzX3CWWv1qL6Cjadv42olLtLidIP61k39farkcOj1Rp27KtgYO0S/GjL1ZhkCMOopoLq7+XzxeiT1Hebq7f7wfUr8eD6lV6ZW0OgpmTgoLbZlSbF9n1Svy3R3xue4txHfRCiApI7NBOSRDK3ArZvtqXUsOUQCkk/AbgFmol4Z5y2Kh+HkFQcM8YriYj+EsDbDKc+F/IiIroYwNcAfEwIIVfRXQD+ARGzeAjRbuNe0/1CiIfia9DX15eJuOMj6cjzLhdQTgJzGXIJsIbv22AqjiPhksRCU23IZybxyw9pd5ZSks8ORv1el5Rm+74sDIOuMZH1n7PaWcpnTmeAle2b07rCqrYBU591xzWdXQKNGvip901oVT4TVII/3eMh4WQMQogPceeI6MdEdLEQ4rWY8P+Eue5nAOwGsFEIsVd5tnRmP0NEfwLgt4Na30LoA+Qy6HK6XJchV54PTW0gF/fGwYP1qNcCEW654hLc17/cKX1yW2POx92Hifp8Q0isQNqF4Vrweltc73d9X1LDoKrushFxm33KFtgo0GgLc7lgAu1xiQTszNhWOlW2kWunem5wuIKBJw40qOCKHYRN10bedC6BRqDZCUJ9D9As0Pii3F1qaOd0j4dEWhvDTgAfA7Al/v/P9AuIaA6A/w7gq0KIJ7RzkqkQgH4A/ytlezKBaYCKHWRM/iax+h0L8TeHT7LSpk/eFqAxgIrzSpIEa+iV4w16SzURnYsR9feWMfTK8SYf90f2HsbDew97+fXrMBFWU7t9YwUksg4sAvi4BRexAew2l9C26nNNlfD1NtoYz/57rrLuOqUtTI+8b5VLZEg/mGIR+nvLXpkBfJCW4QN8vwwOVxLVbAD86pi000VVRVrGsAXA40T0CQCHAdwIAETUB+BTQohPArgJwC8CeDMRfTy+T7qlPkJEixCthf0APpWyPZmAC5CxJX97+fWqNe+JjzQcasST9Xh1PPrdI/jCTSucahmTj7vLv9q14NMYIk1IKkUNDldw4tQZ47nbVvfUa/uGwiYhjo6dbZBMfdpqmmtcDW2O0ZU9d52hxnYXQoPBgOa5xMUiAPYEkrZ3m45z81Je7yPt630v229jCgu7ig1eSc+8cAyV0Wq9wqO0Qfb38kWU2uWiqiIVYxBCvA7gXxqODwH4ZPz7YQAPM/ebFc3TDG4gbMnf9MhT1Z9fn6BZweZx4sOIXBNOJyTTsdVNIkVNtdOcaTZJAkIV8r2bdo40qG9MAY/V2gQ27xphxyGEGPjuOm0SrG5sTxNvERoMpo+Z6zpunHY//5rRA3DoleNWz0DXN7hAhHqc0eLuEk6dGbfea2Lutn6bbhdVFbM28tmGJK5w6uC1i4C6fNRdjMjHWKwSkjRb3aTqIF/C6Ur2p0INJExqx5CqDp+EhbbyjVk7AfT3llndvP7cNC6RSYLB9OOu67jzJluf9Aw0JXnk5qfvTkFCiKm171o3XD/a+m26XVRVzNoKbjaEusKF6AqzxC1XXBJ0XIfPhEuT6VUiTfUzlwup6fkufa8MJExTiQ1IvsVX54LPfFLh4wbL9ZledjKNS6Qter/D0+3YNbahkrLPLsnneFoQIrvJndv3N5UzdeW6mk4XVRU5YzDANkA+g9euiXhf/3LctrqnvkMoEFn156YyiFypTcCc6dUE1wJOwyiTMmIOJnfPEKat9iFHAH2getZkTQxMfcaVnZSMZtv6lQBgJGYm2MbcRKBNzM41ttx56WaqIzQOphUqmmIHobNAODFWMwoerrYkiX9pBXJVEgNfVzgT2qUrHByu4JkXjmFSc0PkrjWpt0xVpIAoSeD9/7qRQNm2ujZVURpGKb2nVJfcdavKXoxYh82lM0lWXBMBLBYI8+d01r3JTp0ZZ8s3tqpmAKdyAmBUoSVRffp48hSIMCkEq65zqcb08zLFxImxmtGtN7QiWqiLuAkLu4romtNZb//Y2XE2625/r7uq3ExBzhhagKwG30ZsXYtZv9dkKKvWojz+61aVG9xWgal0ASpsBMfWlrRGzh37KnUiPCFEU9po7vkmwmSrA+wCtzOxEUCTgdOWNh3Ixg6lCy+6L7+sUcB9l8t2pM4FTt8+KQR+tOXqoHZy533den3rOOvfIF3EuZK7JsjMwurzL9+w23itukNU3zmdQWw2kMgwZ0q70NfXJ4aGhqa7GVakjWDkCIpUM9gKp4dIQgS7GyQXoazC1pbnNnzQ+S1png24+0odC9Pi920LV4qTACsBNM0FjqD69nkoVm5+yrhz6S4VcbJaYwPnXnYQdsBvjNKiHe8A7HEJ+u7AtKbb1c6kIKJ9Qog+13X5jqFFSOua6pLibOqZEJ374u5SapuIK9+NbI8p+jbNsyVsUpjONE6M1VAsUJ0gZhEw59ptmOaCK6o3a3AqtNFqzZo5dOPgQWfMByeInDozbq0hHoJ22e24OBVCNHe65nRaVX4+2oKZkvbChpwxtBlZFIIZHK6wLpk2Qq9DTti0EaYcwSRqzDFlir5N+my9bRwjNgYrTgjMn9uJ/fdc5dUGiSz1w6FMppXEZGDtEjYX2KPfPeJkDLIdajZfIGI6WanH2unjr6vJfArrmO5NogKeKci9ktqIELfNBYznRanYgYEnzFtdSaR8FotaZCfUZVIHl83SpKoNddsNbZvueeWb/94HLg8i/d02z56Q70rj7ivBeZ8t7CpaCZJvqof+3jK65jTLmVm5abv6K6TvfSC9g0yZj13fZPMsapcre1rkO4Y2IsTIx3lCjjHRvISpLI16DiQTZNpttWLc3M6OYPUK0Cwl+QSY+cLHK0nCJI3ZChr5wCSpm3TFSSRBLkeQjixy6NxzzbKm4k3FAtXLsmZR0KeV6h6ToViISCW3eddIg90oqRSuj3VIYR1fzKS0FzbkO4YApJFKXBWiLtuwGys3P1V/pl4XwQWBKW+kHfsqztTMC0rFBin0xFgNZ8YnsW39ykT+06qUxNV6kJBpjH3AeSWZ7ufyDumkLaQUqK+kHiIJyueqahe9noMKWzCZL/p7y9h6w4qG3c7WG1bUxzltsCTQ+ngBNebidG0So7HR/MRYrcmTKFQKN411SGEdX7QzpiIN8h2DJ0IkQpPksWOfmxCOVmt1F0Lf2gY6fAzPBODs+ERTLqGsMjn6pDH2fU+ItMwRUOnSGKqf5959R1zPOmmsRugOgLXhYKpOgA9sDhHSjmBK4e6LpHEuLvi4XpsQIoWHOGykiTvI4xjOM/guZhMDcal1VNQmhTVvSgeZk7VJHbLPYhDgVVK2HES+C9rHXdY3oCxkK5/W7VaHjbmlidUIVSfI2gSmTLhy/mVhnL6vf3lw1ln9vetWlZvSrAPNcS4DTx7App0jTtWlaT35IkQKD2EiaSLTz5U4hpwxeMJ3MXPqjNB32YLJbLripDsNiSySAep2Add7THDV4TXdn7U0xundJZJGtIZ62PT3llmvIemlpo/Tndv3Y+iV40ZCH5qymoPpvTv2VZoI55VbnjZ6hkkX2spoFXfEtgI9YCw00Z1EyLjbvPx0dJemjPV6fIxaP8UVWDfTGIGOnDF4glvMHUS4fMPu+mTIwoik5k3hJpCeJuDOWL0h1VZZLKakRk/dLmB6z5qli9j0DLZoWglTpbmspTEfQpEkojUJA+NiDRZ3m0tgyqJLaoQ4YCbmd27fjyeGDjcUmkpaS8I0P3zXxImxRvdW245Rh56KxBSBbotz8fW+krZ4U3yMxEx1QQ1Bzhg8walH5ISSk2EBk49H944pdhAmAUxoRrNiBzklHS5NgJTY9O386NhZVv1ki+RM6kFhk/LKis3FlU/fBi5Xf5bSmC3wS0KV8n3fnYSB2ZgJFyxnsuVwTOS5F4833e8SAnznR8guVjUa23aMPlHIEqG1I2yQTiGu+7Ky100XUjEGIroQwHYAlwF4GcBNQogThusmAMhRPiyEuDY+fjmAxwBcCOBvAPwbIcTZNG1qFXxcMqu1CcwrdqBULDQtYE73qgYFcS6LnLTDSWzPvHCsIV2ENGirUNVP8tlyQcr3Jwkqskl5Mu2zKeVAtTZh9QLRoRKfLNKPmO532UrSqKlCGZhNtWhTg/jWQOBgu953foQmq3NF75tyFNmQpHYEB/ltPvfNNBfUEKTdMWwA8G0hxBYi2hD//VnDdVUhxErD8f8IYJsQ4jEi+kMAnwDwBynb1DKoi5lLljU6VsO29SuDE3lxsEk7PhLb1j2HjEnB5sfBSKZnD71yvF6C0Facnmsrh3nFjqBtuw1ygaaJJB0crjRF65ru19V2slRju42GOjPxUYOYaiBkZcD1VYnp/egafVf0fqjx1+bu67IjqVB3891dRWPBIBUzzQU1BGkZw3UAPhD//lMAz8LMGJpARATggwA+qty/CTOYMaiwSUsuadCVNdUnK+rWPYe8JDZuUZys1lhJSvWi4rJYconhbFLhmfFJYzR0EkgbQxo7CCfFqve3wlCYVXoLV39zNRBMHk4muNSaISoxtR9tEemuNC3leH2FwObuGySkKPaFN06PWy+diS6oIUjLGN4qhHgNAIQQrxHRW5jr5hHREIBxAFuEEIMA3gxgVAghe/hVAOyIE9HtAG4HgJ6enpTNTo+kHjA2CRdoluA5HB2tYtv6lc422JiHze9f/9uWzVR+g0tVkIQpcJHLf37gNdzXv7wldhCf+5MiTTyMLkDY5geXqNA3Mh4ALpjX6STCHOO0tZ1TLS3sKjaoiLLyMDO9j5tXHRR5o5l22bUJUVe3ms7LQMqZ6oIaAidjIKK/BPA2w6nPBbynRwhxlIjeAeBpIjoI4KeG69i5KoR4CMBDQJR2O+DdLUFSDxhXhGxIVlSfNtgYmI/3j4SunjJ9g2tbHrJtd10/Wq1h5ean2AmT1A4isaBUTF0XWr7Ldwfo8h7SBQgOrriN+/qXN9Qt4PowNPrep+0qI7HNW/UaqfJR10la+ww3/kIAW29aYXUPtsFVf+JcgZMxCCE+xJ0joh8T0cXxbuFiAD9hnnE0/v8lInoWQC+AHQC6iagz3jW8HcDRBN8wbUiiZsgiV4oqOaltkEToju37G1JcmwzfnFTmk1uIa6vMnmpibrYKW9zxB65fbmVeXCrpNHYQIFKhnDo73uBnr97jKwyEBGf5xMNUaxPYvGsEP62OW12BbVK1qWpc2uy6OnzUez5rxzRHk7qC6u+zFWySjh2m8wtKRcyf29m2TK/ThbS5knYC+Fj8+2MA/ky/gIgWEtHc+PdFAK4E8AMRVQh6BsANtvvPN9hypXDnFnYVnTWB1VwvQKMb7Y59FQysXdKU7bG/tzlb6K2re5xZP7l2Siak5yYiAOtWlXFf//Km90mmJXcc8jlq5tcQuGomu1RI3aUiLpjX2RBACESEbdPOEa/cSTKn1h3b9wftAFVwzPfEWM2667J9O5f7ac3SRamy6+rIMlFcq7KRDqxdgmKhcaYWC1M2lYG1S1DsaE4geOrseOb9NROR1sawBcDjRPQJAIcB3AgARNQH4FNCiE8C+DkAf0REk4gY0RYhxA/i+z8L4DEiug/AMIA/TtmeGQ+bamfoleNGl82r33uxM1WBjeDZjLEmyc1VHtH0DcUC4dSZcWP7BabiDvQdjl5DWQa/bd1zCHdu34/F3SXMn1MwxmHoIMCZ+sJGnB6MC7CwHmeGHYretzajtg2nzoxj4+DB+s7ONxJXhSsTqs29We7OsggMtNm1Qg3vLc1GajKmxejvLTd5rAGRnSHr/pqJSMUYhBCvA/iXhuNDAD4Z//4OACNVE0K8BOD9adpwrsGmX+WkIC6YC5hSDbh05raF5JtamvsGWS6TU+1w7+cIlcpcKqNVFDsIxQI1SfE6fLbytnxKrvgNDi77iw6Z10ovaqN+dxKX3gkhrGoWG5HN0vuKE35cda5N87BVBXpMbtwyT5nsB87GknV/zUTkabenAf29jYU8ALsLH7egdfWRDbbKYLp64c7t+7FxsFkPr6Ydl4n+tq1fiZ9W3QXUQ5LJ6ahNCsyf01lXQS3sKjZt83238qaCL8UOwtjZ8Xo6dZOqwAYpCdvGUG3nPdcsMxa1MaFAVFe7dTPFm1TY1CztSvlsUlE+cP3yusrQ1N52qbkkfHYi50qK7FYgZwxtAlfLwYe4cxPRN5zftpBseXbUNvbe+xTu2L6/YeHesX0/7ti+3yndyohn3+8y4WS1Vmemw3dfha03rnDaXUzQiVZ3qQhQJL3L75JpReQ1NhWNKgm7mILaTl+mOClEXYDYdO0yL4Zly9I6nbpxGzF2qbmSjLUNPkR/uvtrOpHnSmoD0uZqMSWMA+wSt+qVZAtKs8UyqDlrkiTlU5/lm0yOg76Q02zl9WArXQWmpxXhbA4A6rpmV6CZTsy4nFo69FxMgLtSni1Lq3p/q3Tj3HznvtkWU+Ortgm1XfjEIbWrv2YicsbQBqTN1cLZGELqD4QuVsCds8YXZQ9CZZO2Wyml2QiSJDbcfkjaJbgkdvIaU6bPU2ftkbMAn17CZuh29VU7dOPcfDflEQOAsbPjbIoJn11lkrQovkT/fLclcMhVSW2Ajfj4TPzKaNVYTjRkq8stVqLm0pcSrpw1PnDFFNQN2KVik/sgEKl6slAdcOjuMuvt1dKnJqjfZXPfNZVJ3brnkNOQDky5nnJqSFUtBqAhCCyk7GzW4ObM6FgND1y/vMlWcmKshjdOjzeNv69AkNSlVbpDy3m+dc8hbBw8mLh87/mEnDG0ATZ9pom4m2Dym+eMfCEeKaNjNdy6uoetixxqaCsWCN2lorM9urFxtFoDRGRYlvc+uH4l9t9zlTPnUZo63FzOm9rEpDV1uPpdLgatt9HX40kSKlvshCRupWKhKQX8dBE123zv7y1j/txmRUVtUmBcYZYLu/wFgqQurVydZ58a3+c7clVSG2DTZ8qJ/zvfeJ4tt6miWpvAZx4/UPfxd7mWStjc/vQUCa7oaA56rhsbNu8aaXpmbVLgp1W3ikUiTWZVgM88C5jLpwLmWAmbWsLURi66XAdXFlaPnQhNJJhVEj8OLv29T46u0x5rQSLEpVX9dp9YEbnegHO36E4S5IyhDfDRZwpWodMMXTJU38HBtVg5XSrXdtf32DA4XGFTFvt8my12I6RAShI1mc2wqxr4ZUW9sbPNuZFCIhS4a9W2h0jMaZmpD1zz3SdOJGQcfRNamgIqfeCKDzkfQSKD3PjtRl9fnxgaGpruZmQGTr3gm3TOt9h9qyVF33eHRvWqBtyNgwedmUEJfsnMQtQ6gNm7SEXSqOckUMfcNn++cNMKrxxBvnMoC/j2k+84yme65nboeOtoZx+1CkS0TwjR57ou3zHMACRJSudzv47p8rBIKqlJqAWEfNJFd3f5ZUYNrSz2wPVRAD/37BAPrpBMs65iSdx3mCTdlqaYcKDB2aCriLmdHThZrQW73ZrgM7fT1NprSgAAE+hJREFUfuO5XJEtFDljmAGwuZ3K9NhJ/NZN2Dh4EI9+9wgmhECBCLdccYkzD1NaZOHyWq1N4NHvHnEyhWKB8Mbp8bqqyqYqkX9v2jnijCmQnjQ2NYwv4eAyyhY7Ihcx1WPJVhZWZVDrVpXr46pCV8mkTTGRdNepCwcnxmooFQvYtj4q7JhV7QUbQlOdmO6fLcgZwwyAyzidxm9dxcbBg035eOTfIcwhhDj41D7whUvCLndH9Q5MAWuuJIKunFO1iUm2VrV8Nkd4uuNUzTph3/38a/WxlPW+AbftxmQn2LGv4lX3OWmBKe69Nt27S30o+06qZ1qt5uS+PW32gPMRuY1hhsCX2KaxE7zzrm8aiUeBCC8+8Cve7TQtLtXnXq2RfOrsOOuzXyDCpBD1AjYuqd2mfrltdQ/u61+OyzfsNu4qfPXV3P02yGe7+kbC9zoOoTYpXTeuz6E1Sxex9Tp83ivfoVeY81XTvdyG4jYq49ezAnACgTo/z5eI59zGcI7BV/+fxk7AEdUQnb8rmEglBjZCr6tIuuPEeJz7qLyeszHI6PC0qpIk6gb5bN9o2qR1qiVCbFKh0dOV0SoGnjiAzbtGMDpWa/gGm6pM3z34qg8pbkMria7JxkWIUs1wLtkhjPp8RM4YzkEklfg4idKVx18FRxwqgekzdB37ibFaPTjuZLWG7q4ihIiS56nfZKr3oLYrjaqEu98GU36dpEZQXxuFr03KR9I1jVltUhhtNC6mqcbY+IoaMieXvqNKo1bS7+dchh/Zexh9l144q3MicUjFGIjoQgDbAVwG4GUANwkhTmjXrAGwTTm0FMDNQohBIvoKgF8CcDI+93EhBJ94JodRwtPrF3B631uuuMRIWG+54hLv93PEgWAvXami3F0ypmCuTQjMn9uJ/fdcZb3XtiNIu8hN93NqrgJRsFQ5OFxJ7UTga5PygQ8zkgRfSto2op+kjkQlzkvFBQT6xhAMDleaiuvY5qTKlGZrTiQOaXcMGwB8WwixhYg2xH9/Vr1ACPEMgJVAnZH8EMBTyiUDQognU7Zj1sBHKufUEtLAnMYraWDtEqNEKODngikJGFds3cVcfLNiplnk+v1pbQL6c0x9JFUb6rUcczMxL73qnS8z9FWdyTa3yiJpyzbso2ZLGkMym1xQQ5CWMVwH4APx7z8F8Cw0xqDhBgB/IYQYS/neWQvficxdd1//8lTuqf29ZZaom3TcxQ7CBfM6m/TVJg8fwK3Wmo5tf1bvtDF1AWDHvgr6Lr0QgN0tVv5vsxOkiYifDriyDbvmfVKX6NnkghqCtIzhrUKI1wBACPEaEb3Fcf3NAL6oHbufiO4G8G0AG4QQZ1K26byGr4TXygnPqXNCdNw+hnBOap6ObX/ad/q47apGfJPUfEecZkPv0zTGbJ3puTzJWgk5zkmcB1yMo6vYgWpt0hoomGMKTsZARH8J4G2GU58LeRERXYyo9vMe5fBdAP4BwBwADyHabdzL3H87gNsBoKenJ+TV5xV8JLxWT3hTG6SNwUS4TLAxF6A9OX3aBfktPpBJ9mzns45mNqnOQtOXLOwqYnSsZq1dIdvPQSCqzWDyTjt15n+3d+4xelTXAf8dL2uwFSV+xG1tB2ITUZJUETZdURRLeVAaU9pgJwHiRLROS1SRVpHSKjRGIMVESmKKIvqKQmiUtxQIjnAXJZEDsVEkGlIQNgEDBhta4rWLt5glimzMYp/8MXfM/Wbnztx5ffutfX7Sar/vzuvsmdl77px77jmvFkYvFQ2Y5gwP8QW3at0mmOMoNQyqenFom4g8LyKL3dvCYuBgwamuBO5S1RMzQ+nbBnBURL4BfLpAjttIjAcjIyMzb/FFS4T8yzFRSTHERIT4MmSzhcZ24GVzBWWj4CI5pzMnVB5V3BwCwaI1KdmMn01DdLOULarMktaxDrkY4bXSrmXnSqPT5g7P6sk2PHFksvC5Cg2Y0oWD2fkZo5hGC9xE5GbgBW/yeYGq/mNg3weA69xkdNqWGhUhiVx6WVU3lF33ZFzgNgjUmWQNLXpKV/v6C4rmzRlGhBPzDUUGrWih2i0fXhGUE/I7nyopwdum6qK5eXOGOfpquB5EStHf3GYcfmx4dNGzkEaa5S00y6NosV7IXTloA4JBJHaBW1PDsBD4PnAW8BxwhaoeEpER4BpV/bjbbxlwP3Cmqh73jt8GLCL5f9/pjvlN2XXNMHRDncybdVYKpxR1XjEGJ09OCLsrpmvRUtWsnqnxKyt5Cq/dm0HoFOsMLOo8P3mL+E7lxWhV6IthmC7MMHRDnXQSK278SVRR+xAho7NlxxjX3vnIFF/z8JAEJ0ZT33zREz1/7jBzZ5/W1w60aihldoRddGyV1NRVaJIsr8pxVdN7xKb9MPKxlBhGZer4qissms4lNEG6duXSKYuVIFkEF+oclkRMcL54eDIq82oedTvLvHmhZQvncP/eQ7n7Tx47PuXYUHhvF9FnTSb+q0Zvheaazj/rDfzX3kNToohCBtLWI7SL1Xw2TlBWuziPiYJJ0hhmiQRr6obOna6X8EnljK2hnRJTNB7y6wNXqQe8duVS7t9wEc9u+jOuXX0uDz/3UnDfbFnRtSuX8qUrz8v9uw6/8mqpDFXrYpflw2qTtSun1i3/0B8u5eHnXuoxCkKSRmVpwBDaeoR2sTcG4wR1FnIVpciIcVIWlU1skhMopsZCytjEEVZt2lY4Gd408Z1PTJTSqk3bclc7Z/+uFw8XR+vUGf33u5hP9i1j1aZtubmNtj853jgXlhGHGQajh7ZcAWnm1GxU0q9fniSbQNUfjWajX7LFbGJyAqXbfNdPmZEqyznVZmcZc4x/faBnXUGWIgNVJ+y37fDXqhTp2hLe9QczDEYjqv6jLt/ww9z2tCPMFp/Jq15WJSFe2vmFfPRFxBTh8TvLbC0KPzTXlzt29fqRyWNsHN3VE7oaU4zHJ3SdNHFd3ttEXnW5uqPyOvMyZbq2hHfdY4bBaEyVf9TQP/2QSO7IdvuT442iTYoS18UQm84728n67p7s20eV/ESx7rDQaL4ouif0NrH9yXG++MF3NB6V153ENnfR9GOGwWhE1RFh1fKKTf3aTetNx6bzLruO//ZRJbV3DMNDEuw0i3JSlblssmky/BrTMYai7ryMuYumHzMMRm3qjAhD//ShxVxN/dpNDEuVdN4x1/H3iU3tfcbwrML0GEBwlv+GLeH8TEWLAbM6j73P2UFCyI0VoytzF00vZhiM2jQZEebmu8ksaBueFR4JF1EnCdy8OcP8+XmLe0qNqsLfBzKaZomZNygyciGDCeX5hSaP6xSd37Dl0WC1O9/gxbhsbrx7V+l9zjMeocg0Cy0dfMwwGLWpEqkT5XLKBtx432NdVnn1fbOEakRs2THG9ifHUejJFBrzJlQ2byDAsoVzCt0xRaPksugqX+dbdowFjQIwJX1EkV637BgLvrH418wbJChTw5ZtrmBmYIbBqE1sWGOMK+LmrbunpLqYPKYnwlhjXVYbR6eObiGZbD2uGjQqWRmzHXDZm1B2xH+Gy/+fotCz0rnuauJQColU5zEpvrPGCBK9jU0c4VN37OTGu3edSDhYtKjNv8+hQYKSuK1srmBmYYbBqE1s9EiMy6no7SPWZbVlx1hwAve4amFOoZhJ6jLfeDY8tozQ+o2izrNOunKfbIW8vJxULx6e5NrNifxFf7N/n4sWI1oOo5mHpcQwapOXziAvy2WMyynkd17iRptlxwPRo9uYc9U5B1QPj03fHGJTbZTpvOzv+Mgfndnz/eatu6ckKoTX3tZCf/O8OcM997lOOhVjcLE3BqMRMdEjMS6nopFwbMRS7Oi2ioxZWcqoGh4bWr9R5rYKza8UTbZfdeFZU+p9F+ls/8SRYO2LjZf9wRSZwEJMTxbMMBidE+NyKutYYlxWoc59/tzh0g4qVK409ZHHdnJVwmPbXL9R9KZSVK+gyCAumTenUodvIaYnD2YYjM6J7VxCHUvs8SED9Nn3945um8hYRlFHu+otC/ifF460vn6jKOXHkEhhEZtrV58brHuRGl7r8E89mlZwuwLYCLwNuEBVc6vniMglwL8AQ8DXVHWTa18O3A4sAB4G/kJVXym7rhXqMUJMZyWzLTvGCrO6+iN3v8RlXkhnbEWyNgr5ZOWezjKoRrf0q1DPY8AHga8WCDIEfBn4E2Af8KCIjKrq48BNwC2qeruI3ApcDXyloUzGKcx0jW5jqrT5UUjZ0Ng6bison9OIefOwNwIjSyPDoKpPAEhxGa8LgD2q+ozb93ZgjYg8AVwEfNTt9y2Stw8zDMaMI3bSORR+mxqFqqGdRXMRFhVk1KUf4apLgV953/e5toXAhKq+mmnPRUT+RkQeEpGHxsfHOxPWMOoQO1lcJfw29nx5lM0tGEYRpYZBRO4VkcdyftZEXiPvdUIL2nNR1dtUdURVRxYtWhR5acPoDzEum3QEX7Rmoyqh9QNfuvI8MwpGbUpdSap6ccNr7AP8VTVvAvYD/w/ME5HT3FtD2m4YM468iKhQTiaIC7+NwdYPGF3Qj3DVB4FzXATSGLAO+KiqqohsBy4niUxaD/xnH+QxjNapGu8fu2/stc0QGG3SNFz1A8C/AYuACWCnqq4WkSUkYamXuv0uBf6ZJFz166r6edd+Nq+Fq+4ArlLVo2XXtXBVwzCM6sSGqzYyDNOFGQbDMIzqxBoGS6JnGIZh9GCGwTAMw+jBDINhGIbRgxkGwzAMo4cZOfksIuPA/9Y8/I0kaygGjUGVCwZXNpOrOoMqm8lVnTqyvVlVS1cIz0jD0AQReShmVr7fDKpcMLiymVzVGVTZTK7qdCmbuZIMwzCMHswwGIZhGD2ciobhtukWIMCgygWDK5vJVZ1Blc3kqk5nsp1ycwyGYRhGMafiG4NhGIZRgBkGwzAMo4eT0jCIyBUisktEjotIMJxLRC4Rkd0iskdENnjty0XkFyLytIjcISKzW5JrgYjc4857j4jMz9nnvSKy0/t5WUTWum3fFJFnvW0r+iWX2++Yd+1Rr70TfcXKJiIrROTn7p7/UkQ+7G1rVWehZ8bbfrrTwR6nk2Xetutc+24RWd1Ejhpy/YOIPO7081MRebO3Lfe+9lG2j4nIuCfDx71t6929f1pE1vdZrls8mZ4SkQlvW2c6E5Gvi8hBEXkssF1E5F+d3L8UkfO9be3oS1VPuh/gbcC5wH3ASGCfIWAvcDYwG3gEeLvb9n1gnft8K/CJluT6J2CD+7wBuKlk/wXAIWCu+/5N4PIO9BUlF/CbQHsn+oqVDfh94Bz3eQlwAJjXts6Knhlvn78FbnWf1wF3uM9vd/ufDix35xnqo1zv9Z6jT6RyFd3XPsr2MeDfc45dADzjfs93n+f3S67M/p8kKRnQD529CzgfeCyw/VLgxyRVMC8EftG2vk7KNwZVfUJVd5fsdgGwR1WfUdVXSOpCrBERAS4CNrv9vgWsbUm0Ne58see9HPixqh5u6fohqsp1go71FSWbqj6lqk+7z/uBgyQ1Qtom95kpkHcz8MdOR2uA21X1qKo+C+xx5+uLXKq63XuOHiCpmNgPYnQWYjVwj6oeUtUXgXuAS6ZJro8A32vp2oWo6s9IBoQh1gDf1oQHSCphLqZFfZ2UhiGSpcCvvO/7XNtCYEKTcqN+exv8rqoeAHC/f6dk/3VMfRg/714fbxGR0/ss1xki8pCIPJC6t+hWX1VkA0BELiAZAe71mtvSWeiZyd3H6eQlEh3FHNulXD5Xk4w4U/Lua1vEyvYhd482i0haCnggdObcbsuBbV5zlzorIyR7a/rqR2nPThCRe4Hfy9l0varGlAiVnDYtaG8sV+w53HkWA+8AtnrN1wH/R9Lx3QZ8BvhcH+U6S1X3S1J5b5uIPAr8Ome/SjHQLevsO8B6VT3ummvrLO8SOW3Zv7WT56qE6HOLyFXACPBur3nKfVXVvXnHdyTb3cD3VPWoiFxD8sZ1UeSxXcqVsg7YrKrHvLYudVZG58/YjDUMqnpxw1PsA870vr8J2E+SlGqeiJzmRnxpe2O5ROR5EVmsqgdcJ3aw4FRXAnep6qR37gPu41ER+Qbw6X7K5dw0qOozInIfsBL4AQ301ZZsIvJ64IfADe71Oj13bZ3lEHpm8vbZJyKnAW8gcQvEHNulXIjIxSTG9t3qldAN3Ne2OrlS2VT1Be/rfwA3ece+J3Psff2Sy2Md8Hd+Q8c6KyMke2v6OpVdSQ8C50gSUTOb5OaPajKLs53Evw+wHoh5A4lh1J0v5rxTfJquY0z9+muB3KiFLuQSkfmpG0ZE3gisAh7vWF+xss0G7iLxu96Z2damznKfmQJ5Lwe2OR2NAuskiVpaDpwD/HcDWSrJJSIrga8Cl6nqQa899762JFesbIu9r5cBT7jPW4H3ORnnA++j9w26U7mcbOeSTOT+3GvrWmdljAJ/6aKTLgRecgOg9vTV1cz6dP4AHyCxnkeB54Gtrn0J8CNvv0uBp0gs/fVe+9kk/7R7gDuB01uSayHwU+Bp93uBax8BvubttwwYA2Zljt8GPErSuX0XeF2/5ALe6a79iPt9ddf6qiDbVcAksNP7WdGFzvKeGRLX1GXu8xlOB3ucTs72jr3eHbcb+NOWn/kyue51/wupfkbL7msfZfsisMvJsB14q3fsXztd7gH+qp9yue8bgU2Z4zrVGcmA8IB7pveRzAldA1zjtgvwZSf3o3iRl23py1JiGIZhGD2cyq4kwzAMIwczDIZhGEYPZhgMwzCMHswwGIZhGD2YYTAMwzB6MMNgGIZh9GCGwTAMw+jht1v6YAavBibtAAAAAElFTkSuQmCC\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Generated 1021 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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yZUvuu+++apdx5pln0rNnTyZPnkxxcTFNmzatMH9xcTFTp07l7bffZurUqaxenfptp9nZ2bzxxhuMGjWq9BLPuHHjOOKII1i0aBHjx49n+PDhpfkXLlzIs88+y+OPR2+cXbx4MY8//jjz58/nuuuuo1mzZrz55pv06dOHxx57DIAhQ4awYMEC3nrrLbp06cLDD9ft42C1kBZJpbBV2T9JW8eOHTn66KMBOP/883nlleq8vrxmTjjhBFq1akVWVhZdu3blww9T9zk3ZMgQAI488khWrlwJwCuvvMIFF1wAwPHHH8+GDRvYtCm6bDhw4MAygem4446jRYsWtG3bllatWnH66acD0K1bt9LyFi9eTN++fenWrRuTJ09myZIl1GUKDiKyWyQ/Wlky3LhxY3bu3AlQ7efy582bR/fu3enevTvTp0+Pjd9nn31KPzdq1Ijt27fH8iTmS8yTqoFwSZ333Xffcuez1157lQ7vtddepeVdeOGF3Hvvvbz99tuMGzeuzrdUV3AQkd1i1apVvPbaawBMmTKFY445Boiu+S9cuBCAadOmVVpOixYt2Lx5MwD5+fkUFxdTXFyc8uZ1Oo499lgmT46e0p8zZw7Z2dm0bNmyxuVt3ryZdu3asW3bttJy67I98za7iFQq+cmrdK289dQKx3fp0oVJkybxox/9iM6dOzNq1CggurZ/8cUXM378ePLz8yudz4UXXsill15K06ZNee211yq971BThYWFXHTRReTl5dGsWTMmTZqUVnk33ngj+fn5HHTQQXTr1q00wNVV9fYd0upbSWpTVR9Rrc+Psi5btowuXbqUDu/u4CC7X/I2h1rsW0lERBoeBQcREYnRPQeRXaA+X34SAZ05iMgeoKCggPp0DzKxVXhdpeAgIhm1Y8eOTFehXimvrUZt02UlkQYqk08XNW/enKuvvpoXX3yRX//617z00kv8+c9/ZuvWrXzve9/jt7/9LWZGQUEB+fn5zJ49m08//ZSHH36Yvn37snXrVi666CKWLl1Kly5dyvSxNGXKFMaPH4+7c+qpp3LbbbeVzvPyyy9n1qxZtGnThvHjx3PNNdewatUqJkyYEGsnMWfOHAoLC8nOzmbx4sUceeSR/OEPf8DMyMnJoaioiOzsbIqKivjpT39amv+DDz7go48+4p///Cd33nknr7/+Oi+88ALt27fnz3/+c+lrO3/1q18xe/ZsAB5//HG+853vsG7dOi699FJWrVoFwIQJEzj66KMpLCxk7dq1rFy5kuzsbK677jouuugivv76a3bu3Mm0adPo3LlzrW4jnTmIyG73+eefk5uby7x58zjmmGMYPXo0CxYsYPHixWzdupXnnnuuNO/27duZP38+EyZM4Je//CUA999/P82aNWPRokVcd911pY3o1q5dy9ixY3nppZcoLi5mwYIFPPPMM6XzLCgoYOHChbRo0YJf/OIXzJw5k6effprrr78+ZT3ffPNNJkyYwNKlS1mxYgWvvvpqpcv2/vvvM2PGDJ599lnOP/98jjvuON5++22aNm3KjBnf3Itq2bIl8+fPZ/To0Vx55ZUAjBkzhquuuooFCxYwbdo0LrnkktL8if05PfDAA4wZM4bi4mKKioro0KFDrB7pUnAQkd2uUaNGDB06tHR49uzZ5Ofn061bN1566aUy/Q6l6vfo5Zdf5vzzzwcgLy+PvLw8ABYsWEBBQQFt27alcePGDBs2jJdffhmAvffemwEDBgBRn0f9+vWjSZMmZfo/Sta7d286dOjAXnvtRffu3cvNl+jkk08uLXfHjh1l5pk4/bnnnlv6v6Tl+KxZsxg9ejTdu3dn4MCBfPbZZ6WN5RL7c+rTpw/jx4/ntttu48MPP9wlDQF1WUlEdrusrCwaNWoERP0pXXbZZRQVFdGxY0cKCwvL9DuUqt8jSP0azIoa9TZp0qR0mvL6P0pWXt9MFfUHlVhu8jzLq3/J5507d5bb6juxP6fzzjuP/Px8ZsyYwUknncRDDz3E8ccfX+6y14TOHEQko0p+XLOzs9myZUuVXtaT2O/R4sWLS9+1kJ+fz9y5c1m/fj07duxgypQp9OvXr9brXN3+oFKZOnVq6f8+ffoA0L9/f+69997SPMXFxSmnXbFiBQcffDBXXHEFAwcOLF3+2qTgICIZ1bp1a374wx/SrVs3Bg8eTK9evSqdZtSoUWzZsoW8vDxuv/12evfuDUC7du245ZZbOO644zj88MPp0aMHgwYNqvU6jxs3jjFjxtC3b9/SM6Dq+uqrr8jPz+fuu+/mrrvuAuCee+6hqKiIvLw8unbtygMPPJBy2qlTp5Kbm0v37t155513yrxroraobyURar9vpbrYCC5VPzuyZ1PfSiIiUqsUHEREJEbBQUREYhQcREQkRsFBRERi1AhOGpbCVknD6jpbJBWdOYiIVNPKlSvJzc3NdDV2KQUHEanzarObancv7fpCyqfgICK7xZ133klubi65ublMmDABiB+B33HHHRQWFgLRC3x+/vOf069fP+6+++4yZRUWFvKDH/yAgoICDj74YO65555K59OlSxcuu+wyevTowerVq2nevDljx47lyCOP5MQTT2T+/Pml5U2fPr10ur59+9KjRw969OjBP/7xj125iuqUSu85mNkjwGnAx+6eG9K+BUwFcoCVwPfd/ROLeo+6GzgF+AK40N3fCNOMAH4Rir3J3SeF9COBiUBT4HlgjNfXZttSpyS3UgZYmZWBiggLFy7k0UcfZd68ebg7+fn59OvXjzZt2lQ43aeffsrcuXNTjnvnnXeYPXs2mzdv5pBDDmHUqFEsWrSo3Pm8++67PProo9x3333AN11433bbbZxxxhmlXXgvXbqUESNGMHDgQPbff39mzpxJVlYW7733Hueee269euNcOqpy5jARGJCUdi3wN3fvDPwtDAOcDHQOfyOB+6E0mIwD8oHewDgzK9kr7g95S6ZLnpeI1HOvvPIKZ5xxBvvuuy/NmzdnyJAh/P3vf690urPPPrvccaeeeir77LMP2dnZ7L///vznP/+pcD4HHXQQRx11VOn0VenCe9u2baX9Pp111lksXbo0jbVQv1QaHNz9ZWBjUvIgYFL4PAkYnJD+mEdeB1qbWTvgJGCmu29090+AmcCAMK6lu78WzhYeSyhLRPYQ5V0MSOz6GuLdXyd2U50sVXfaFV10SC6rKl1433XXXRxwwAG89dZbFBUV8fXXX5db/p6mpvccDnD3jwDC//1DentgdUK+NSGtovQ1KdJTMrORZlZkZkXr1q2rYdVFZHc79thjeeaZZ/jiiy/4/PPPefrpp+nbty8HHHAAH3/8MRs2bOCrr74q8wa42pxPTW3atIl27dqx11578fvf/75Bve+6tts5xN++AV6D9JTc/UHgQYh6Za1JBUUkSG7zkXZ55bcZ6dGjBxdeeGFp19qXXHIJRxxxBADXX389+fn5dOrUiUMPPTStKpQ3n6q8wS2Vyy67jKFDh/LHP/6R4447rsIzmT1NTYPDf8ysnbt/FC4NfRzS1wAdE/J1ANaG9IKk9DkhvUOK/CKyh7n66qu5+uqrY+lXXHEFV1xxRSx9zpw55ZZV8kRTicWLF1c4n5ycnDJ5ALZs2VJueSXjOnfuXOZFOrfccku55e1panpZaTowInweATybkD7cIkcBm8JlpxeB/mbWJtyI7g+8GMZtNrOjwpNOwxPKEhGRDKnKo6xTiI76s81sDdFTR7cCT5rZxcAq4KyQ/Xmix1iXEz3KehGAu280sxuBBSHfDe5ecpN7FN88yvpC+BMRkQyqNDi4+7nljDohRV4HLi+nnEeAR1KkFwF7djt0EdmlCgoKuOOOO+jZs9IXnNUJOTk5FBUVkZ2dnemqlEsd74nUJaluEu/hnQPu2LGjxu9hboi2b99O48a7/qdb3WeIyG7XvHnz0qeUXnvtNW644QZ69epFbm4uI0eOLG2vUFBQwNixY+nduzff/e53Sxu0bd26lXPOOYe8vDzOPvtstm7dWlr2lClT6NatG7m5uYwdO7bMPCvrLiPRnDlzKCgo4Mwzz+TQQw9l2LBhpfXKyclh/fr1ABQVFVFQUABEN7ZHjBhB//79ycnJ4U9/+hPXXHMN3bp1Y8CAAWzbtq20/F/96lf07t2b3r17s3z5cgDWrVvH0KFD6dWrF7169eLVV18tLXfkyJH079+f4cOHs2TJEnr37k337t3Jy8vjvffeq61NU0pnDiINVQbPSD7//HNyc3O54YYbAOjatSvXX389ABdccAHPPfccp59+OhAdKc+fP5/nn3+eX/7yl8yaNYv777+fZs2asWjRIhYtWkSPHj0AWLt2LWPHjmXhwoW0adOG/v3788wzzzB48OAqdZeR7M0332TJkiUceOCBHH300bz66qscc8wxFS7b+++/z+zZs1m6dCl9+vRh2rRp3H777ZxxxhnMmDGDwYOjdr4tW7Zk/vz5PPbYY1x55ZU899xzjBkzhquuuopjjjmGVatWcdJJJ7Fs2TIg6oLklVdeoWnTpvz4xz9mzJgxDBs2jK+//nqXtL9QcBDJoOT+nxpK30+NGjVi6NChpcOzZ8/m9ttv54svvmDjxo0cdthhpcFhyJAhABx55JGl7RVefvnl0sdf8/LyyMvLA2DBggUUFBTQtm1bAIYNG8bLL7/M4MGDY91l7LPPPrHuMpL17t2bDh2ip+27d+/OypUrKw0OJ598cmm5O3bsKDPPxPmce+65pf+vuuoqAGbNmlWmi47PPvuMzZs3AzBw4ECaNm0KQJ8+fbj55ptZs2YNQ4YMoXPnzhXWqSYUHERkt8vKyiq9z/Dll19y2WWXUVRURMeOHSksLCzTjUZJtxYlXWSUKOn6IlFF3WdUpbuMZKm66ICy3X4kd/mRWG7yPMurf8nnnTt38tprr5UGgUSJDfDOO+888vPzmTFjBieddBIPPfQQxx9/fLnLXhO65yAiGVXy45qdnc2WLVt46qmnKp3m2GOPZfLkyUDUAK6koVp+fj5z585l/fr17NixgylTptCvX79ar3NOTg4LFy4EYNq0aTUqY+rUqaX/+/TpA0D//v259957S/MUFxennHbFihUcfPDBXHHFFQwcOLBMQ73aouAgIhnVunXr0p5PBw8eTK9evSqdZtSoUWzZsoW8vDxuv/320u4y2rVrxy233MJxxx3H4YcfTo8ePRg0aFCt13ncuHGMGTOGvn371vhJq6+++or8/Hzuvvtu7rrrLgDuueceioqKyMvLo2vXrjzwwAMpp506dSq5ubl0796dd955h+HDh9d4Wcpj9fXVCT179vSG0q+61Ezq9zmcVzYh3JSNX/uvI/kS8qZr2bJldOnSpVbKkvoh1TY3s4XuXmmDEJ05iIhIjIKDiIjEKDiIiEiMgoNIA1Jf7zFK9aW7rRUcRBqIrKwsNmzYoADRALg7GzZsICur5q0q1QhOpIHo0KEDa9asQa/YbRiysrJKW3fXhIKDSAPRpEkTOnXqlOlqSD2hy0oiIhKj4CAiIjEKDiIiEqPgICIiMQoOIiISo6eVpN6JdVZ366kZqonInktnDiIiEqPgICIiMQoOIiISo+AgIiIxuiEt9V9hq6Th2nlzmkhDltaZg5ldZWZLzGyxmU0xsywz62Rm88zsPTObamZ7h7z7hOHlYXxOQjk/C+nvmtlJ6S2SiIikq8bBwczaA1cAPd09F2gEnAPcBtzl7p2BT4CLwyQXA5+4+3eAu0I+zKxrmO4wYABwn5nV7I3dIiJSK9K959AYaGpmjYFmwEfA8cBTYfwkYHD4PCgME8afYGYW0p9w96/c/QNgOdA7zXqJiEgaahwc3P1fwB3AKqKgsAlYCHzq7ttDtjVA+/C5PbA6TLs95N8vMT3FNGWY2UgzKzKzIvVJLyKy66RzWakN0VF/J+BAYF/g5BRZS147ZeWMKy89nuj+oLv3dPeebdu2rX6lRUSkStK5rHQi8IG7r3P3bcCfgO8BrcNlJoAOwNrweQ3QESCMbwVsTExPMY2IiGRAOsFhFXCUmTUL9w5OAJYCs4EzQ54RwLPh8/QwTBj/kkcvs50OnBOeZuoEdAbmp1EvERFJU43bObj7PDN7CngD2A68CTwIzACeMLO89l1wAAALkElEQVSbQtrDYZKHgd+b2XKiM4ZzQjlLzOxJosCyHbjc3XfUtF4iIpK+tBrBufs4YFxS8gpSPG3k7l8CZ5VTzs3AzenURUREao+6zxARkRgFBxERiVFwEBGRGHW8J1JP6A14sjspOIjUV+qNVnYhXVYSEZEYBQcREYlRcBARkRgFBxERiVFwEBGRGAUHERGJUXAQEZEYBQcREYlRcBARkRgFBxERiVFwEBGRGAUHERGJUXAQEZEYBQcREYlRcBARkRgFBxERiVFwEBGRGAUHERGJUXAQEZEYvUNa6oyca2eUGV5566kZqomI6MxBRERi0goOZtbazJ4ys3fMbJmZ9TGzb5nZTDN7L/xvE/Kamd1jZsvNbJGZ9UgoZ0TI/56ZjUh3oWQPUdiq7J+I7DbpnjncDfzF3Q8FDgeWAdcCf3P3zsDfwjDAyUDn8DcSuB/AzL4FjAPygd7AuJKAIiIimVHj4GBmLYFjgYcB3P1rd/8UGARMCtkmAYPD50HAYx55HWhtZu2Ak4CZ7r7R3T8BZgIDalovERFJXzpnDgcD64BHzexNM3vIzPYFDnD3jwDC//1D/vbA6oTp14S08tJjzGykmRWZWdG6devSqLqIiFQkneDQGOgB3O/uRwCf880lpFQsRZpXkB5PdH/Q3Xu6e8+2bdtWt74iIlJF6QSHNcAad58Xhp8iChb/CZeLCP8/TsjfMWH6DsDaCtJFRCRDahwc3P3fwGozOyQknQAsBaYDJU8cjQCeDZ+nA8PDU0tHAZvCZacXgf5m1ibciO4f0kREJEPSbQT3Y2Cyme0NrAAuIgo4T5rZxcAq4KyQ93ngFGA58EXIi7tvNLMbgQUh3w3uvjHNeomISBrSCg7uXgz0TDHqhBR5Hbi8nHIeAR5Jpy4iIlJ71EJaRERiFBxERCRGwUFERGLUK6vIHka920ptUHAQ2dMld1pYuCkz9ZB6RZeVREQkRsFBRERiFBxERCRGwUFERGIUHEREJEbBQUREYhQcREQkRsFBRERiFBxERCRGwUFERGIUHEREJEbBQUREYhQcREQkRsFBRERiFBxERCRGwUFERGIUHEREJEZvgpNdLvbayqzzymbQm8lE6hydOYiISIyCg4iIxCg4iIhITNrBwcwamdmbZvZcGO5kZvPM7D0zm2pme4f0fcLw8jA+J6GMn4X0d83spHTrJCIi6amNM4cxwLKE4duAu9y9M/AJcHFIvxj4xN2/A9wV8mFmXYFzgMOAAcB9ZtaoFuolIiI1lFZwMLMOwKnAQ2HYgOOBp0KWScDg8HlQGCaMPyHkHwQ84e5fufsHwHKgdzr1EhGR9KR75jABuAbYGYb3Az519+1heA3QPnxuD6wGCOM3hfyl6SmmKcPMRppZkZkVrVu3Ls2qi4hIeWocHMzsNOBjd1+YmJwiq1cyrqJpyia6P+juPd29Z9u2batVXxERqbp0GsEdDQw0s1OALKAl0ZlEazNrHM4OOgBrQ/41QEdgjZk1BloBGxPSSyROIyIiGVDjMwd3/5m7d3D3HKIbyi+5+zBgNnBmyDYCeDZ8nh6GCeNfcncP6eeEp5k6AZ2B+TWtl4iIpG9XdJ8xFnjCzG4C3gQeDukPA783s+VEZwznALj7EjN7ElgKbAcud/cdu6BeIiJSRbUSHNx9DjAnfF5BiqeN3P1L4Kxypr8ZuLk26iIiIulTC2kREYlRcBARkRh12S3SQKkrdamIzhxERCRGwUFERGIUHEREJEbBQUREYhQcREQkRsFBRERiFBxERCRGwUFERGIUHEREJEbBQUREYhQcREQkRsFBRERiFBxERCRGvbJK7SlslTSsXj1F6isFB6mxeJfPGaqIiNQ6XVYSEZEYBQcREYlRcBARkRgFBxERiVFwEBGRGAUHERGJUXAQEZEYBQcREYmpcSM4M+sIPAb8F7ATeNDd7zazbwFTgRxgJfB9d//EzAy4GzgF+AK40N3fCGWNAH4Rir7J3SfVtF4iUrtijR1vPTVDNZHdKZ0W0tuBn7j7G2bWAlhoZjOBC4G/ufutZnYtcC0wFjgZ6Bz+8oH7gfwQTMYBPQEP5Ux390/SqJuI7CrqJqVBqPFlJXf/qOTI3903A8uA9sAgoOTIfxIwOHweBDzmkdeB1mbWDjgJmOnuG0NAmAkMqGm9REQkfbVyz8HMcoAjgHnAAe7+EUQBBNg/ZGsPrE6YbE1IKy891XxGmlmRmRWtW7euNqouIiIppB0czKw5MA240t0/qyhrijSvID2e6P6gu/d0955t27atfmVFRKRK0goOZtaEKDBMdvc/heT/hMtFhP8fh/Q1QMeEyTsAaytIFxGRDKlxcAhPHz0MLHP3OxNGTQdGhM8jgGcT0odb5ChgU7js9CLQ38zamFkboH9IExGRDEnnaaWjgQuAt82sOKT9HLgVeNLMLgZWAWeFcc8TPca6nOhR1osA3H2jmd0ILAj5bnD3jWnUS0RE0lTj4ODur5D6fgHACSnyO3B5OWU9AjxS07qIiEjtUgtpERGJ0WtCpQy1hhURUHCQyiS3hgW1iBVpAHRZSUREYhQcREQkRsFBRERiFBxERCRGwUFERGL0tJKI1IrYY9BZ58Uz6Um3ekNnDiIiEqPgICIiMQoOIiISo+AgIiIxCg4iIhKj4CAiIjF6lLWBqPQxQz1iKCIJdOYgIiIxCg4iIhKj4CAiIjG65yAiu53ugdV9OnMQEZEYBQcREYnRZaV6TqfnIrIrKDiISJ0VO/i59dQM1aThUXAQkfqjsFXSsM6MdxUFhzpKR0wikkl1JjiY2QDgbqAR8JC735rhKtUtOmISkd2oTgQHM2sE/Ab4H2ANsMDMprv70szWTETqAz2YUfvqRHAAegPL3X0FgJk9AQwC6kVwqM67c7UTi2ROVb9/yfkqyrunMnfPdB0wszOBAe5+SRi+AMh399FJ+UYCI8PgIcC7u6hK2cD6XVR2faV1UpbWR1laH3F1dZ0c5O5tK8tUV84cLEVaLGq5+4PAg7u8MmZF7t5zV8+nPtE6KUvroyytj7j6vk7qSgvpNUDHhOEOwNoM1UVEpMGrK8FhAdDZzDqZ2d7AOcD0DNdJRKTBqhOXldx9u5mNBl4kepT1EXdfksEq7fJLV/WQ1klZWh9laX3E1et1UiduSIuISN1SVy4riYhIHaLgICIiMQoOlTCzn5qZm1l2puuSSWb2KzN7x8wWmdnTZtY603XKBDMbYGbvmtlyM7s20/XJNDPraGazzWyZmS0xszGZrlNdYGaNzOxNM3su03WpKQWHCphZR6IuPVZlui51wEwg193zgH8CP8twfXa7hG5eTga6AueaWdfM1irjtgM/cfcuwFHA5VonAIwBlmW6EulQcKjYXcA1pGiQ19C4+1/dfXsYfJ2oLUpDU9rNi7t/DZR089JguftH7v5G+LyZ6AexfWZrlVlm1gE4FXgo03VJh4JDOcxsIPAvd38r03Wpg34AvJDpSmRAe2B1wvAaGvgPYSIzywGOAOZltiYZN4HooHJnpiuSjjrRziFTzGwW8F8pRl0H/Bzov3trlFkVrQ93fzbkuY7oUsLk3Vm3OqJK3bw0RGbWHJgGXOnun2W6PpliZqcBH7v7QjMryHR90tGgg4O7n5gq3cy6AZ2At8wMoksob5hZb3f/926s4m5V3vooYWYjgNOAE7xhNpBRNy8pmFkTosAw2d3/lOn6ZNjRwEAzOwXIAlqa2R/c/fwM16va1AiuCsxsJdDT3etiD4u7RXgZ051AP3dfl+n6ZIKZNSa6GX8C8C+ibl/Oy3Br/oyy6OhpErDR3a/MdH3qknDm8FN3Py3TdakJ3XOQqroXaAHMNLNiM3sg0xXa3cIN+ZJuXpYBTzbkwBAcDVwAHB/2i+Jw1Cz1nM4cREQkRmcOIiISo+AgIiIxCg4iIhKj4CAiIjEKDiIiEqPgICIiMQoOIiIS8/8BAigXVBcOs3EAAAAASUVORK5CYII=\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 0.33435.\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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\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 $(3/2)^{1/2}$ 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; we will return to this multiplier later." ] }, { "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 = (3/2)**(1/2)*(2*nprnd.random(N) - 1) # displacements along x\n", " dy = (3/2)**(1/2)*(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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BnpZStkcVnHrWK0e/lPJRKeUAKeWA1q2TjpWJiYmJqfF89hnMDCwflR1yKUQW4qwM155EddWpqOI/SCmnAvUILr1ZbcnPz6dfv3706tWL7bbbjrvuuouKClXlc8OGDRx33HH06dOH3r17s/vuu7Nu3brA/SZOnEi/fv3o168fjRo1onv37vTr148TTvAqCJcanTp1YtmyZfz222/07t07fIeYmJgqo7wcvvjCXh4yBHLxmObSO+troJsQojOqvOVI4FjXNn8Aw4CnhRA9UELEW19Vzalfvz7fffcdAEuWLOHYY49l9erV3Hjjjdx77720adOGGTNmADBnzpxKL6eg/fbdVxVnHDp0KGPGjGHAgAE5+GUxMTFudt8dpkxRHX1elobq110Ht92mBMlOUcuRZYGczUSklGWoet8TUdXMxkkpZwohbtJlM1GlG08XQnyPKmd6Um1It7zFFlvw6KOP8sADDyClZNGiRbRrZ5uDunfvTnFxceh+UTjnnHN48803ATj00EM55ZRTAHjiiSe45pprADjkkEPo378/vXr14tFHg8ugzJ8/n+23356vv/460vljYuoiU6yaoW+9lb1z3Habel+0KHvniEJO40SsmI93XOuuMz7PAnbL5DkvvBCsgX3G6NcP7rknuX222WYbKioqWLJkCaeccgr77LMPr776KsOGDePEE0+kW7duofu1adMm9DyDBw9m8uTJHHTQQfz5558ssu64zz//nJEjRwLw5JNP0qJFCzZu3MjAgQM5/PDDadmyZcKx5syZw8iRI3nqqafo169fcj84JqYOUhJYWDYzaJ+W4mK44ILsn89NHLGeQ/Rsol+/fsyfP59LL72UFStWMHDgQGbPnh26XxQGDRrE5MmTmTVrFj179qRNmzYsWrSIqVOnsuuuuwJw3333sd1227HzzjuzYMECfv7554TjLF26lIMPPpjnnnsuFiAxMQGMGWN/zpYqy8QyrVJeDvn52T+fm1qXxTeMZGcM2WL+/Pnk5+ezxRZbANCoUSMOO+wwDjvsMPLy8njnnXfo0aNH6H5htGvXjpUrV/Lee+8xePBgVqxYwbhx42jUqBGNGzfmk08+4cMPP2Tq1Kk0aNCAoUOHesZ7NG3alA4dOjBlyhR69eqV3o+PianFXHqp/bkqOvXfflPv5eVQkIMePZ6J5IClS5dy1llncd555yGEYMqUKaxcuRKAkpISZs2aRceOHUP3i8ouu+zCPffcw+DBgxk0aBBjxoxh0KBBAKxevZrmzZvToEEDfvrpJ74w3T0MioqKGD9+PM888wwvvPBCCr86JqbuURVC5O671WxEyngmUqvZuHEj/fr1o7S0lIKCAo4//nguvvhiAH755RfOPvtspJRUVFSw//77c/jhh4fuF5VBgwbx/vvv07VrVzp27MiKFSsqhcjw4cN5+OGH6du3L927d2fnnXf2PU7Dhg1566232HvvvWnYsCEHH+xOMBATE2OSTXXWttvC3LkwfLiahUBuhEitq7E+YMAA6S5KNXv2bE/VUExmia9zTF3HrSA47TT4v/+DZs0yf64ddoBvv1WfN26E+vWVx9aVV6Z2PCHEdCll0nECsTorJiYmJks8/ji0bZvZYy5bBn//7Zx1aM/82CYSE1ONeOop+O9/c92KmJrC5s3e6zduzOx5WreGrbZyrtOuvbFNJCbGBylh3Tpo3LjqzmnFZVLLNL4xWWLYMO/1TZtm53zaDmKSCyESz0RiagS33AJNmqipfExMdURHqbtZvTpz5zAHNNoeYpKLZNqxEImp9lx5pcoTBLB4cW7bEhOTS6y8rL7owMOqJBYiMdWef/87fJt0qKiAdu3g4Yeze54Yg2+/rdV6wv32y85xw+wrZWXZOW8QsRCpIv7++29GjhxJly5d6NmzJ/vttx9z586loqKC888/n969e9OnTx8GDhzIr7/+CqhU7H369KFPnz707NmTa665hs2bNzNjxozKNPAtWrSgc+fO9OvXj7322itj7R06dCjaVbpRo0YZO266ZKPf2bwZ/voLzj4788eO8eCdd5R/6hNP5LolWSMkj2nKrF8f/H0uhEhsWK8CpJQceuihnHjiibz00ksAfPfddyxevJjp06fz119/8cMPP5CXl8fChQtp2LBh5b6TJk2iVatWrFu3jjPOOIMzzjiDsWPHVqaHP+mkkzjggAM44ogjcvLbqppM6pc1pops1ars+PTHGOy/v3q3Sh/URlq3hgkT4MADM3vc998P/v7TT+GKKzJ7zjDimUgVMGnSJAoLCznrrLMq1/Xr149BgwaxaNEittpqK/Ks0Nb27dvTvHnzhGM0atSIhx9+mPHjx7NixYpI5x09ejT33XcfABdddBF77rknAB999BGjRo0C4Oyzz2bAgAH06tWL66+/PvB4y5YtY5ddduHtt9+OdP5ssPvu2T3m+PHq3c9dMyYNvviiVquwTAoL4YADMn9cnSfLj4ULM3/OMOreTCQHueB//PFH+vfv7/ndUUcdxe67787kyZMZNmwYo0aNYvvtt/fctkmTJnTu3Jmff/6ZnSJUoRk8eDB33nkn559/PtOmTWPz5s2Ulpby+eefV6Y9ufXWW2nRogXl5eUMGzaMH374gb59+yYca/HixRx00EHccsst7L333qHnziZffQU77pi54/35p/1Zl3G56KLMHT8GeP11OPxw6NPHXpeLWq5VRLa8pLp3D/7+/POzc94g4plIjmnfvj1z5szh9ttvJy8vj2HDhvHRRx/5bp9Mmpr+/fszffp01q5dS3FxMbvssgvTpk1j8uTJlUJk3Lhx7LDDDmy//fbMnDmTWbNmJRyntLSUYcOGMXr06JwLEFBV3MK8VKLiznp/++3q3eMyxKTDnDnq3VRhzZ+fm7ZUIYZmOiOcfLL/dz17wqGHZvZ8Uah7M5Ec5ILv1asXr776qu/3xcXFjBgxghEjRtCmTRvGjx/PMI/IpbVr1/Lbb7+x7bbbRjpvYWEhnTp14qmnnmLXXXelb9++TJo0iV9++YUePXrw66+/MmbMGL7++muaN2/OSSed5JkGvqCggP79+zNx4kSGDBkS/YdnkWXLIBP2fvel1H3c2rXpHzvGwMv3NBdBDVWAmQr+vPNUlt2qIFcTu3gmUgXsueeebN68mccee6xy3ddff82nn37KN998w19//QVARUUFP/zwg2ca+HXr1nHOOedwyCGHeNpM/Bg8eDBjxoypTAP/8MMP069fP4QQrFmzhoYNG9K0aVMWL17Mu+++63kMIQRPPvkkP/30E//Otr9tRE49NTvH1RG/33xjrzvzzOycq04xb17iusLCqm9HltlzTxg92l4uLlbVDbNlCurZUxnTc+mjEAuRKkAIwRtvvMEHH3xAly5d6NWrFzfccANt27ZlyZIlHHjggfTu3Zu+fftSUFDAeeedV7nvHnvsQe/evdlxxx3ZeuuteeSRR5I6tzbe77LLLrRp04Z69epVqrK22247tt9+e3r16sUpp5zCbrv5VyLOz8/npZdeYtKkSTz00EOpXYgM8vHH2Tmul4zMlrtmneLppxPXHXNMlTcjmzRoAG7Tp7axZatMbkUFDB4MvXtn5/hRqHvqrBzRtm1bxo0bl7C+W7duDB8+3HOf38JcMYCnvR5Og2HDhlFaWlq5PHfu3Ej7f/LJJ5Wf11kGiKKiIiZOnBjapupOSYl6uG+4IfG7bOU5qvO0aAFur8JaNhMpLU3MoltUpN43b7YFSiapDs5u8UwkptqT6YdPR/3edVfid15J7WIywD//mbguFzk6skRpqXq99ZZz/ZtvqnfTAzCTxEIkJiYCmRYiuu/yegDXr1feqDEZxktgGDPkKAihbADVEcusmWCbOPJI9Z6XB/fem/RPtnn+efjyy4TzDUi6hFTmqTPqLCllUnXJY5IjmxUyvQL/Cgth9mzlXRWW/rq83FYzSOmdGmL+fNhmG6W6//HHtJsc48ZripdCjzp7dgbakgV0Gdxzz3Wu/+AD9b7HHrBokboMSVa3VljBwSA54QRVT+Srr3JrC9HUiZlIvXr1WL58eVY7urqMlJLly5dTr169jB+7osJbiHTrpkalUdTqX3/tXPbquzp3VuVFa3H8W27544/EdSkPy53cc48dhpIr9E9xB8FqobdokXqPmGzCl3r17IJUAweqe7aSIUPg+OPTO0EK1ImZSPv27Vm4cCFLly7NdVNqLfXq1aN9+/YZP66pWmrVyq4nooMBo4wLdtnFuWzORFq3ttOeNGyY+Sp0MRbPPZe4Lgkh4vc/b96ssgvcdhssWZJi2zKA9r5yD2p23NHp3ZzuOLasLKAE7mefqfdnn03vJElSJ4RIYWEhnTt3znUzYlLAHLl5FaRq2FCVBh09OrrtxOy78vOVcIJa5yxU/UlCiPg5PGihn+vgUP1TtDeW5oor4IUXwvd/5x2Vjemqq4K3CxQiOaJOqLNiai7ajOVnzlq/Hu67L7nBl37g166Fv/+21WWxEKkizj4bttwyKSHitek334COu81FWVgTv5mIe2Djl8p9//3h6qvV5z/+UPe7OyWPJhYiBkKI4UKIOUKIeUIIzwTGQoijhBCzhBAzhRARZHpMbUJPIF97TdlB/IjqLbp+faJhXXu6BI1mV62KdnyAn37yL5Uag8q1X1iYlBBZudL+rP9rM7AvrM5GttFCxC003ELl3nvDj6UTVhhJvx34CpEuXeDYY8NPkGFyJkSEEPnAg8AIoCdwjBCip2ubbsCVwG5Syl7AhVXe0DrMmjVqpJ5LFixQ7+3apZZqya2DLi9P7LsGD1bvZkflJplOqkeP7KSsr7G4/wQpA4WIlHD//c46L4ccYn/OkD0+o2gh4lZnuZfDOOgg+/PHH0uuFzdy/tbjHdv4CpEc2XxzORPZEZgnpZwvpSwBXgIOdm1zOvCglHIlgJQyh6azukevXrYnSK445RT1XlycWmya27NLB4WZRFFjVTcVQo3CPcWrqAgUInPmqJTmI0fa60wPu5okRJJVkU6YYH/enm+5kRu4b4EzNa/nMcvK1KgvigEmw+RSiLQDFhjLC611JtsC2wohpgghvhBCeOYHEUKcIYSYJoSYFntgZYZ16+wCNy+8kHt1QVGR4SqfBGvWOJfffju6ELFSjAHwxhvJnzvGwpxGgJoOSgkbNnhurtWNft5WXkb27bZLo30ZQA9W3EIkHVtNId7S0nNAk63kXBHIpRDxUk64HeAKgG7AUOAY4HEhRELxUinlo1LKAVLKAa1bt854Q+sijRvbn487TnlA5ZLiYrjuOtXvXHll4vd+rpNt2jiXTzwxUYj4zTLMfJRxydw0mDRJvevUJ4ceCnPnqhwhQjikdd++8Pjj6rPff+olRL7/PoPtTQFd7NN9L5nPUbKIhO7Q+xxAboqrW+RSiCwEOhjL7YG/PLb5r5SyVEr5KzAHJVRiqhivWLGqJC9P9Tf168Pllyd+H6bqMm0U7v39ZiJmfEk8NskAAwcqyeDOFv3554BySJgxwzY++wmRHPaXvjz8sHp3F0tzz0y22CL6MfPwvqljIWLzNdBNCNFZCFEEjATedG0zHtgDQAjRCqXeqv3l0Koh1SlXXoMGievCSqyYtba++sr5nVcUcUGB8kLVxEGIGcC8iPvtl/D1bbc5l3/6yfswNTlJ5ooVakDmVV7FjfdMRMZCRCOlLAPOAyYCs4FxUsqZQoibhBDaR2EisFwIMQuYBFwqpVyemxbXbXLx4Jrn7NTJ/uz1EGkV1SuveAdHT5vmfx49ijSpX985EvYo+JgUV10F11yT3jFqPKb09hiV+MX6uFP2e92LHTokrqtKTjtNve+8c/B2ZWXKhTfIXR2gNUs4m/8krC+kNFiI5FV9l55TnxMp5TvAO6511xmfJXCx9YqpIrxcXY3yIlXGL7+o9759neu9XH1LS+G99+Coo9Ty0KEQNQuLlxvzlClOu28qM5FNm1SuI7Brt99yS/LHqTWccIL9+b33Iu92443OZd1ftm9vO3/k0K4MKBd08B7gfPghXHaZs1pmGH/SjkISZxdFlFBQ4OE3rC+KUT21qogj1mMSSOZmzzSrV6vOYepUeyZwhWcYqpPZs+H66+3lzZudM4lzzvHf12sy4MF3AAAgAElEQVRk26ePeuk67qloC3QHV+fRSQF1QE6a6P/LdJDIpQt2aakt6LwGOMOGwfTpKn4oKl4CBKAZq5wC84UXlNeHHuXk4ELEQiQmgVwaLqdOVQV8rr/ebofXczFrltOtc8wYp7ZEp93WaGHgxRlnqHftYaNp0MDWy59ySvL5maIIvzpDWO46n+APL1tc165qsFFSolKvH3NM+tlx0yFqqeYjjnAKmVSSMY7hX07V7AUXqIuhQxtiIRJTHchlMJf5kOl2eD0XPXokpng3Oflk5+8IcgzQFQ73209lXT36aPs703PrwAP9j+HF77+rd3dnsWGDisiuTs4KWcWrbqyLWd94G5381IhTpqjDFhfDiy+q7dwOE1VFVGFQVOTc9ttv1Xsy98EWLHEGpusHJuhhyTKxEIlJwM+I7l6/erW6h++6K3NlOvUzYRaP8nPBDYoGnjfPmacoSGfesKH9+csv4aWX7GXzmfz0U/vzwIFw+un+xwQ7AM0dU3f11Soiu84EMJaWhoZuD9nZW4j4xCPyxBOJh81VRcpkhIjJc88pe18yAYkFlHH33R5f6KjaWIjEVAeW+/i/uVOIaDfFSy5RTiGZrgiY7uDKDEr0K6saJgj8+r5p0+ygOD+0R5d7ZqfrbR9xRHKJHWssZWWJF9IVMFGP5GYir7+eKESS8aAbO9auSZMuUWfuH37oXL77bmc5XTOhpMm13FT5uYgSHLXftATTWQHmV30ERCxE6ghSqpoFUVx1Tz3Ve737gXb3C5mIan/+efX+7bfhMxGAHXaIdtw993QuH3CAej/88OD90hnY6Wh508b09tvKDVlz4ompH7/G4KXOevJJx6KfEDFnIief7PxOp+DSRM14JCWcdJLKDZcJoubHCouqnz7de/17DOe/qKiHemxy2vfc06CoRXUySCxE6ggTJqiaBclkinaPtN35s7QLbibRbrlbbBFsWNf4PXgmLVqol8mrr8LLL8M++wTvm2wCPW2kB1tAmSNVLbw0ZqbaWouXOmv//R2eEVGESNOmid+bh42Se3D58syEUqxfrxwnysuj99sPPJD6+f600gqWUBSs/spBUZxYiNQRdM2MceOi63C7d3fmntJ1DjT33ZeZtpno0eHs2arSG6Sv5n38cWjZEu68015XXKxiSsLSyyebQM9009cG06CQiC+/TO74NYoNG5Su84MPvIOPjKF5a7ynEaYQ8fqv1q9XaqGO/MY/Cb8hH3nEuXzccaG7eNKoEdxxhxoUuNW8fgwcGPz9YYd5r2/DYm5DlTx8FlcNda9aB1VMLETqIOXl8P774fmwyspUWqNhw7y/d3ewqdT78DonKGP3xVaIabpCRAeCudUhUUjnN+nnecyY1I9Ro7n/ftv1bc6cwE2f5iTP9aYQ8ZpB3HorbLONUvncxwWhOq2JE53L6WZOf+89vA3dHrhT85ip7oP4hS6sR3l//AOfXDCaTHm4JEEsROoIZme4zz6w776w/fbB+xQWKp/8Pn28vzcL6LjPkSq64zVVZ+k+F1rdkKmZftQgQv1bcl26tcq58UYVoh8WKGPk9t/aURXCxrTDedW2ueoqpeZqymq1IiDeZP16+Oyz4CalwgcfRNvOrY4zvQDB27vs1/wu/MQ/KEG5dp3LQ84N3A9HDtKexEKkjmDeazozt1+Altbl77qreq9f33s7d2Zbt/dJKngFOqYb/Ki9WdIVIloY+QWXudupl3NdHbLKueEG1buHsffenqtHjYKLLlKfzZmIlxNE9+4weLvVSF1Zwu3LvXIllJZy1VXBAafVAT3YMFPjdCr5GRCU4nPzutWEsYtvTHWgUSNl/9Azi969vbfLhvrVzIvkXueHV2p4k0zNRLTu+3//s9fpXF3m9xp9feI6aT4cfzxD+ASA77ETpD37rJ2m5tVX1fuZZzr/P60ays8H0bwZbVmkVpSUqChUbZBq0QKOPZaxYwObkTTZKIJ11FFqsHf11fY6kaceQl8h4iYH7n6xEKkjJKNqKitzDmiOPNL+bM5ooswQ1q5NLnmhPqapMvJK/W5yxRXBz47WcGRipl9R4TTOmu66biEyf371rH1RbRCCzxjCCprzGSqvls6Gq2cNWsWzapUzWE/fUwk1Ny65RKUduOUW++K/+mrgf++V9dmPKVOUXcV01w27P6OSkMXYoQ70eYBNa/2ee8YuvjHZw8uu0KqV97Zuj0zzszn7cHeQRxyReKwmTZTKIQpPPaUiud3suGPwfs2awdNPw5tvqmzD7kjwLl2inT8K777r/5072O2ZZ6LNfoSAn39Or101mRas5J8o/1dtJtG1XPQ9+vLL3oGFReWuEcpbb6n3xx5z5MXJlKlg991huKtIt1a5JWNv8SinYtvOdDSq6U5oYj7MZuRh1CReGSYWInUErxGxI/LVta2fatW0W7qP6WcAX+BtM03glFOibefHgQfCkCEqeNeM/0i3A5kwwf7sjvMwSafmiNtrqC7glTPKHKTUr2+7mDdrlpg2BKDN3MneB69f39EJu50bHuM0JmOXu1y9Omqr/TGq/Pqiizruu6+97pmtr+avTrvaK3TwkF8o/Guv2Z9zMPNwEwuROsCsWXZ5axO/xG9eAcZaR+0lRBYsUA97UPoHr1lKNvEL4ttpp+Rdbg84AM46K3y7dIRIukWvqg1+Iwkzv4eFu0ZNMZscQqSgwPYMfvJJ50ykAev5kV4MeNDHb/vnnx2drU6GqYXSaTzB7kyp/H72bO/DZJpJk5SXmO77L+Rujv/jNrb6baq9kR7dWRb2adNcsUZmWUQvyVrFxEKkDjDZZ7D2l7uiPUq1MmFCohpG1xgxbRX6gW/YUKkf3PWldU44UM9zVUZn+6WY+OILpTZPlq5dw7fRNhEvgR3GpZcmv0+1xC/wYtttE1a584a1ZLmj5kZhoT1QadTIObBZTyN6MYt6q5JzfVP3YKKgixow6MeAAdG2KyxUNhTd99/tVW9PN8aKvO3f3zlzcVANUkHHQqQOEDVOwXyQ3DMRPVIz8w1pgVJQoDptt0rWncjRLWSi4CcAq5oglZgWlno2sf/+2W9PtcUvD42HcWjJEudyxy1LaN/C9uk178F69TIThwSqsJObVO5Nk2TrmaxYAXm43Bt32AHmzrUfRD9VlXkhqsEUNhYidQA/IVJcDP362SNC8350F2D6j1HuWbvia7WQn/3E7ZWVSifgMYDNCUEDPv279PXzszXVCfzyuHj8+W7Hjv/9vY2a1lqVwMxBTTrXdHM7Z0Gsg3gzYRtXUuGkSTZ57mOPQTGu6c+338LNN9s/3E9VZaoMU6nbnGFiIVIH8BMimzerGYT2ODL10e607ma0rTueyxQipl0k2UqAXrgTJ0Yl0378QSPVqZY6WwsRrwGk6dFz//22E1GtwwyiCcG3xoul0zIHNc7+NGIKA0sylDdyhorvvdXMhE2rOlvIsGGwAx51qMvL7QvjNxP55htliJwwoVokX4uFSB0gLIj1o4/Ue1BlODNq3T34MYWU+Z07mDaVBzXVANyWLVPbzw+dAsqLa69Vz/S116plr1Fzs2b253PPVSqvXXdN3K7Gc/bZkTcNq8NhzkRMbdg57tQffnToAIAodUqr4tVLEjYNKlqWDc44uZTP8XDnevFFf3WWTtL1yiuw9daJvuzZyOkSgViI1AHMyn1ePPGEeh8xwn8bs2N0C5G8PDsg0fzObRMJ6zS22Sb4+2TQ2pNMqcOCDOtffaWeaW0O8BIipjDUbdPR2JpakRZeB3hMmqQCZQKCfJIJxNRCZPlyGNU04jTOmsrUn++sPjVkwzsJm0YxrLvv3xdfjNYMLyoW/On/pa6x4BYi7joO7vrQUXyMs0AsROoAUSq4hRkGzfv5T4/7XwsgUwXhPmZQ6gnwTrCXKo88orKk/vBDZo7nDljcay//bb20EF5Bh26Vdy5r22cMrRMdMkTlE/niC9/8OMmkzdFCuEULyBMhU9ptt1Wpm32MzmaUeyOUztUxE1mxwvPPcGfrjZqF14uuHQKk1oUXqnf3jeS+iTJdSjRFYiFSB4iSCy8s2Mr0TtIJHE20usucifzjH85t/v3v4HO4OxW/xI9R6NJFjRQzFYv11FPOZXcGY5MGDRIFopdazq1yy5T3UU6pqFA3i/4xQvi6tiUzE6nMpTZvHjutConMLCpSN5OPEGmFPUW+/zLlYlgZQb5+vfpjjjkmYb8vvrA/u4uZJZuXrfGHb4Rv5B5lpPNAZJFYiNQhtO0jXUyfeJ3jSKtwTCGSrD1DC5Hnn1fPsl+t91zQsKEzkeKZZ6pktV40agTXX+9cV1Cg0rKMH+9/jmrg8p8+5eWRUwTo/3vNC8HqqZ12Mjpptw7Qi8JCKC9n82olRNbin7730Bk3OVfovCtmVLiFaYK42BXekXS6fx26DomFRjTuEVC9es4qcdWEWIjUIXRxJi/cs3f3Q2IaxWfOtAeauiaJHiSZOaCSNVaWl9slfBs0qH4DL9MltbAwUVBo6tdXqVdMCgtVWpaDD/Y/fg6K0mWeiorIPaqeieQ1DPbfdcSTRB3yl5dTvkEJkXwjHuNQnEU76s1zRdJPnUoUtDpTD3TSSq3jVfURvKfRZm78akIsROoAF1yg3rt3989P9fbbzuXffnMum0LEFDA6GFB3+EcfbX/npfby488/lediTSngFKR6ystLLEAURXVTK4TIHXdEDv/WvzevZbAft2Og7r5Io0fD998z+TqjMtS338KKFdSTalpcgH3xt2WuY3edar2SCMF7HTrY96mWaUmrIqPcEF5CJBN+8xkmp0JECDFcCDFHCDFPCOFbBk0IcYQQQgohIiYXqKMIoYa7LkpLbf27nxBxBxqfe65z2a+6ocZr1nDHHcH7mGgV9LffRt+nOuMeMEdJN14rhEgSVM5EOm0duJ3DQcMoHrNg2EkqX0zfvqzsn+jpkCeVfrCIUiSCPiR6WYhCl841QiZGs5aWFh5Jz0RSFSJe/PFHYvh/FZIzISKEyAceBEYAPYFjhBA9PbZrDJwP5D6qpjqjk7J5RLGVlto2ut12847XcBeZ6+n6JwoLExPmmaSretIzmqgZf3OJu6KjF+4ZVZR8WnVNiFSWD67vMiC7Os/zzvPev8MNp1Z+9rO/mfaQhzmLDTiLfwjD//2PP0IabHH//fbnhg3hsstSCNEwhYhfRG1U1V379tFuyiyRy5nIjsA8KeV8KWUJ8BLgpTG+GRgN5D5JTHUmIHK1pCTR0cMMfgM7waLGaxAUFCzojo1YsQI6W9kmaoXB2GLDhtQEnV+sTmcjI0etKWAVsYCL/r35xZYEqF9fzSys4X2PHvAso7jkXz66ot3tVO6mEKnItzvftTSu/NyapczGyPAIVBxhl6a87bZIzXYUoRJCzbj79Yu2byXmn+2lw3W7NgaRY7e+UCEihGgjhHhCCPGutdxTCHFq2H4RaAeYj+NCa5157u2BDlLKQPcNIcQZQohpQohpS+NapAmUlCQOanR0tea++5zLXkLECgB28H//p97dM5HDD4dff1Wfa4XrqkX9+tG1DGaUu5+6Y/582xHIrwZRjUMHy4VQXm55ANezRjjXXadGI5s3g5R06ACjeD7SsQoK4Die4/sbx/N2uZ3y1iwr2415doyI5WJYeG1IbeVsoYXIl1+qPPdudtjBe78991TvmYzMTZMoM5GngYlAW2t5LnBhBs7t1bVUjnWFEHnA3UBo4m4p5aNSygFSygGtczitq45s2qQSg7qFyEUXwckn+6sBvDpKrwHm1pY62y1EglRfXlQ3T6xMcNFF9ueg2Zj+D8KCMWsblcXP8vPVNPeKK5QQkRJKS52Dj9JSfy8m1HFe4DgW7HAwFUa31hGnjqpSiNx+OwAiyJBeXk5FhT0D798/gxma9c3RoAH07p34vZ+HyYcfqguXbMbHLBJFiLSSUo4DdfWllGXgzmGcEgsBc2zbHjArXDQGegOfCCF+A3YG3oyN6z74DHUHD1ZG8/XrneuFUAMgv1iMqF5SWq9tCqNUdPs6vsSsWV4T+OADuy54EK+/7v9dTfFICyVJfVx5ucdv16OXTZuct/SmTepi+6Dvv82b4S38y0++ixVVaIxaGlsaL+FK7CgHDCA/364/46UWThldJaugwE4hfN119vcJBdcthKh2N0wUIbJeCNESa5YghNgZyEAxSb4GugkhOgshioCRYOdollKullK2klJ2klJ2Ar4ADpJSTsvAuWsfDRp4rtbpdfyMhu4Zxz33VGbijoQWGObDlewsxMQdX1Hd2WsvldbbrLPiRZA9qZr1CakTMTBo3Tp1X3qWYdbTj9WrOfFEY/1rrwWmUjaFyEtEyEdiGKm01+yBX17jbMp33wEq3cnEid5q4bQRQj27UsKNN2b44FVDFCFyMapz7yKEmAI8g/KWSgtrRnMeSlU2GxgnpZwphLhJCBGQVCLGE1MftGkTq1cru0QYbiHSvLmKJ/Fj2DDv9aZh3UtgXXpptJoQ+auTrO5TTfjxRyV83XnFli5V1zQoW0CAlqZmoeNDfMvwKUaOVHkZV670EKCPPKLex4xxxBzx7LOBktgUIqYdxBeH/lTSmDXs952/ZX34cFWqN+PVaNMtqVgNiCJEZgJDgF2BM4FeQBJjVX+klO9IKbeVUnaRUt5qrbtOSplQNUZKOTSehQRgKt3PPpshQ4JVKCZmJoWwkdZJJ9mfzz0XDjsscRsvV/uyMqWRCBJQ+/IeolXLxBKJNYTu3aFtW+e6Vq2Up5o7+NDErWqsseiZSEBYvpR2YOuqVR55DrVe0x1UN3BgYKF7LUSef94pRA5ggvcOhjT411Yv0JhoQXyhcUyLF0fL19Oxo3rfOjhGpiYQRYhMlVKWSSlnSil/lFKWAtFyA8RUHaY++umnfWuMe2GOBr3qrvtt+8AD3lo001340EPVuy69PXcusGiR56hyV6yCRp9/Ht7omOpHWEU+nGaNN97wyOIxerR6d09577gj0Oai78uPPoIK7Jv0bQ5IsHW42/h/i0bRkGiSfGZiPSsnW26pRg5z5wZvt/feasTh9rXXBKTQd+BOD58DfIWIEGJLIUR/oL4QYnshxA7WayjgrYCPyR1pBBmYgkGXvPUjaJKgR5imTVA/C2qkKPmUwerhcddGBcr1w1/Hou7C6r3UGMIq8hHet1Z6KgXpPvv2TUhc5udl6Ouo4dKjNWFNSMOSJEwI6GzHfnhknvAk0yU8UyBoJrIvMAblNXUXcKf1uhiIkFw8pkoxhEjF6Wcmtav5ALpzaLnxcmnXeFUT1OqvwkI10xiMFZpu5LKo6tKk1Y1u3XLdggyhVVEBAiD0v9b6VC2Q9LH22MPe5p57ElIo+wkRt3qxkkbOzL7NMQxTejaUDmHpU8KESFjAps4AadZdzhG+v0JKOVZKuQdwkpRyD+N1kJQyorY9psq48srKj2Me95ki+2AOyirrNvgQFO/gHoB27WoXuisq8h/tJeMNVhvp31+9e4UL1CgWLVLvAenKQ7MXaDVTaakSSjqOw0yg5RG96idEhPApuOY6RkthCBEjEj4p3F4Vr77qLzU/+cTbA0ULsLCI9Q8+UN5qffsm3cxME2oTkVK+JoTYXwhxmRDiOv2qisbFRGT+fDt3FpAvE6uy3Xuv/+6mEAnzoAq6Z92qcFNdW1wMA3GV87RIxn5TGxFCOTTVeLWWzhYRIERCzV3mTGTKFHu9eZN4SAw/IbJ4sZUF2G17MIXIkUfSKn+l93fJ4DbwHHmkv3eLO0225l//goUL7RoLQVSTGyZK2pOHgaOBf6KizI8EOma5XTHJMMHpgVKE01//7rvh/ACnbDNSOsw767331LtXniG3u+Y7RinroiK4Ce8CHFoLckrzgIpNtZzi4hru7fnWW0rNBIEjkdCaUuZMpHFj7206dUpY5SVEPmWw7Sjy3/86vzTtbs2a0SrPmK507co9qPoJgiQSv3kJnyOOiL6/PkZQ4Z9qSJTac7tKKfsKIX6QUt4ohLgTiNVZ1YkLnVlolBCR6MwyHs+cg512sj+HVSPcaiv/GbpbxWt6eh18MOBKN8+cOdC9e2VOro4rVXCXozxiHaGoCL77Dv7zHzj77Fy3JgVMQ3A6wRTmTMRvROOhc3Xftw1YTymFrNWlD8yb8/rrlYBq21bdpC++yA0ldiCjbNmKpaj0SQWUUYr9e/zipADPuuyACogpL1dTIp3ipRYRxcVXP9EbhBBtgVKgc8D2MTnmTB5FksdQVFWooGp6bpItaWvinomY7r9XeFWLsfS+CSPwZEsi1gJ0v3vOObltR0ZIR4jofUtKwl0FDdz33kYasM22hfakyKyQpo3y2hfdFQlfWmrHmhTiFAyV9vw1a5xlPPWOXrRooVK1X3WV0q899VTo76lJRBEibwkhmgH/B3wD/IZK2x5TzZmEyviZjIo3nYy7bq9dw0wTKJx+cNcKCrPu10JqTRp48BUikSLz9ezjsstUNGJEvAb3Dndi7c1k0qaNMxe/xRdf2ELErIoIxn3ctClsu61zx7A/8fXXlafJqZlIgl59iGJYv1lKuUpK+RrKFvIPKeW1YfvF1CwyYaNr0sS5bGaWEGv9/fATOoCPP1Yj0b//VlLt+WjpwGsy48blugVp4JYOPnEip58e4VimCstrROPjIRKUEQDw96N2q8wGDGDIEDsZYzHOaXJCmhZz1uw3E9GYo6paRKSiVEKIXYUQx6IM7AcLIU7IbrNikiIkV1EUlixRM/RMcoCZTDUgK6N2nCnZzYoFePddZRiYNUstm0XdY6of7jBun2mnrpsSiCk4TM8MjY9uNrQ8rSsupBL3CMYKLLkEVdzlWF5wfJ0gRLT9rqQkXIi40cV4ajhRvLOeRQUd7g4MtF5xOvbqhBmI5WKffaIdokEDf2eYVHE8I9f5e4VXZsuYYuitn3zSfkBzWD+6qtA1hpKukJdrPv00saJZBJ2o7ruTLv+TqtHOL4rebddo0YI+faAFylurF04BmXD6jRvh1lvVD0lWiCRTvbAaE+UfGQD0lLKWuRTUJrQu9uOP7cpnFhMn5qA9Fh1NR/Bly3y383X7rAXJ6aKig/BqXFr4oUNT3vX996Fnzwgb7rgjfPWV+hxBiHTrpmTDAHOoqwXbUUd57lNJ8+bk5YG0PBtP53HanH04B/9HRYZ7zkSusVLIu569UAJdvWoOUdRZPwJbZrshMRH5+2+YPdu5Tt/EQ4Ykbv/jj9lvkwf77Ycabo4bp6YaVi/5MokP8f/+53OQWmVtDkYLkSRsybnjjz+UQfr999M6zN57RwiJOPBA53Q6gueX1n7qTACVbN4ML74YuO+UacV8/z3MxTaaH/SfEZWffdVZqVBLynlGqmwIzBJCTBRCvKlf2W5YjA9bb+0cvpnJrrwUw336qHQMVdQ7nWL55d9xByql6tFHq5QsVkqMWfTkOJ6LdrA6JER0UatffoGp1T1H9nPPqYjrJGxxZoB2nz5JnOuXX5yzj+bNfTd99lllTtPaXUc9ElACKMR4cuNktfPheBtwEiZCf/8deLy6QBR11g3ZbkRMErj1rgc4S4FuOGwUDV53ddLaZXbtWn8DY4Z49FFVTrRnT2CWFQV8992V39/PP1lJC55nlFpRUYHvWMYUIt98AzvskJU2VwdefNF2MPjqK9hll6o79/z5amIR2b07yAPjAO/StGYoRoJLdxDauQLgmGMCNx01yv6cqvL9U9Rsfi22gVD2H1AZKJswEaolKql0iOLi+6nXqyoaF5M8m7sEKJkDyotmivz8YD13Ca6n0BKKW7f3SC9huk/WcuO66aKarH02Hb77TiWMDcqtloDbl1tz882J6UVQPhWhKeCD0G5dIaqoTFBmjatXY/8hYj9bnVVURJ2aIUchqJ7I59b7WiHEGuO1VgiRYWfQmCiUl4TX2SjL96/lkFYkYSp4VLjSNUNuRtlxyterLK2LFno8mGbJvxqWTygdqjKHlg5duOwyNXqfNCnCKN6vHsCJJyaoi37+WcmWKKWafalCn54Kq0sswXiOjNQLhYXYmYWj8OCDGWpZ9SUoFfzu1ntjKWUT49VYSukzFInJJhu+nGEv+DxYq3cK8OkNzcOdYTxynehI4CUFyh+/bK0yTLojgwHYf3/78403Zr591ZQXXgjfJlPo26i0FB57TDkY+RZy0vzyi/d6D3tFRjLYZNMAnSAQPQZaX9vZp4uKCJbyBxwA06bBRRepaZ7frK0WETQTaRH0qspGxihKpWHC8plSr+vUm6b4GNGrumKgR2Gdcktd0LKd6hhWzl3Kr3RiN6y03351tCNFqtUOTDNAVaJVTr//HrLhJZd4rw9LAR0Vt5BKw404lJNPDt/GSOdevz7BQmS33ZRb2F13qaqDM2YkbvPBB6o2cC0hyCYyHZhmvS8F5gI/W5/d+VhjqoCCU463F3wU5yUlsAafHBBVrcvVZQ0NdAc5+3clRFoduDOd+J0PsGZQkQIHaidm/fGqwpzQ6lsqNBSjgU91bA8hcvXVKTRqm22c9g8dqJpOdtAgPDr6H3/EM7apoIBgIeJ+xs70qDK6115wyCHJtbEa4/uvSCk7Q2U9kTellO9YyyMAj2xmMemwZIkyrurA2rFjVfSyWUK5yS/f2QvnnONMeNWjB2B3BBX5BeSVu27oqpyJbNyYWGb0hBN0M+mBinUp2Owq5FNQAN27qzTxdYyAWk5Vgu4bwwqT+drWPNxn3Xb2SPmzAAYPVu+NG9sNylbpR4/jKpdrq97z0Ufz3slGQa0gIbJ2rXO5lsSCBBElTmSgFiAAUsp3AY+otph0aNMGRoxQI8PJk9Ug3kyBkVBKeexYeOghe3m6mhxqHfRXz85NzBZalUJk330TR3LGCLYtiUZ3QAmR0F6sdpJR9fmECaqzD0mda85EKtPPhMXzpeE+lhAA6IfZiIICFXOUi6na1ltDvXrsu69yEAACsy/wyCPO5YdzS+8AACAASURBVFyPDKqAKEJkmRDiGiFEJyFERyHE1cDybDesLlBWpioO6sDfSZPg5ZftQZhJaPoSa8Sjn+/yrTurkGAT09sp20yeHPj1aC7z/iIvT3UYJu3bp173ugbRsaPKLp6Rweutt6r3EAOLqX3RA5BQIZKG+1jkyYRWjWkpt+eeibUGqoKFCxPvx3/9y39797Vxz85uuikz7apGRBEixwCtgTesV2trXUyaTJgA998PI0fa69LV4jg6Arcr4rvvpnfwdDFSupvBXA6khJYtlbuopkGDqg2eyCFz5ypNYNrFHb/8Ur2HCF+zz9MGdb9chZVEdHH1SsK7226Rdq16d3Q/KiqUIDEJmjKGGepPOy39NlUzAoWIECIfuFJKeYGUcnsp5Q5SygullCuC9ouJxmGHJa5z2+UKC5Nzk9d9bWEhiULjtdeUp0tVVw7Uhh3DBXQzPj3VBstGYhZ+r1+/zggRTVXFipi3whTLQS40rXrExpke2knTuLHKW2Km9akCOncOTIqt0AMcr0JXus68F1Kq+tK1jMDbRUpZDkTVYsZkALcQKStLzqnKMROxaiNU8s03KnW3O4Fjtrn9dvVupb4uKQkQIl6xLHVQiJid+8aNKiPtF19k/jzz5yeuC73fPGYibxzljCZPOz5QCHjpJW/dbjb47DN4+mnmz1fJsBPQ92VpqR3e787GOGtWeqWBayhR1FnfWkkXjxdCHKZfmTi5EGK4EGKOEGKeECIhMk0IcbEQYpYQ4gchxEdCiI5ex6npmLbPFR5zvMWLox/LMRPxK1dY1UGHXbuqdyvLcGEhbMLHeO5l/C8urtNCZMYM5Tfxz39m7vj33afGEl51kQKFSEWFQy35LKMQSA4bN5I//7Q3q3F/16BBThWqG/2D7r/fDj40hcjee1d6SNY1ogiRFihD+p7AgdbLO8taEliqsgeBEUBP4BghhDtI4FtggJSyL/Aq4PIZrRo2bFADI48A7JTx0wg8+mjiugFJlADT93pREXYNaLezflXHi3TrptK4Oi6gj87bS4g0bVr1KrgccdVV6t38uVq9FFn2e5UEMCgpgQsuUMUjvQi8PaY7Q8TGYBuZTcekGidEwtB/iJnKx7SNXFt3K4ZHScB4ssfrlAyce0dgnpRyvpSyBHgJcNS+lFJOklLqQIIvgPYZOG/S6Mqud9yRuWOuXh1928WLYRCf+W9gxWMsX25HHRcWotKaTpumXMBMli5NrrGZoGPHCMp2bCGyxRb2unbtMl+7t5qi06SbgwxtY/7224gH+SzgXsEWEp/6pFENFCKGnkog+QE7kMl0Yqo1QkR7vWghYno83HqrGqCVlKiZTB0lNARUCNEeuB/YDZDA58AFUsqFgTuG0w5YYCwvBHYK2P5UwNO9SAhxBnAGwNZZqIanU3RnEi/PlSA+CwrNsepOd+liC6eiIlTv079/ol/7/vtXXVI7n1J9l18OeAllPdw2dcv162fAXalmoH+2KUQefli9Z+ovCwsXChQiAV5TZkB5rREi3bur9223VaM08z5s0QJuuSX8GO+8ExxbUsOJos56CngTaIvq+CdY69LF6270fEyEEKNQZXo9K9tLKR+VUg6QUg5onXTR5nCykW2hMnApAt0IyaNtPfXm7MaRgSJMfdW3L9x5Z/QGRUEXxHBH8Fr8+9+uFV27Ko8cnTvLTJlSv76KcakDFZp1jIgpRB5/PM2Duq5bWkLEsIe4+ewz+OkndTqvY7gnxDWCp6yuThsrTSEStWMYMQKOPz58uxpKFCHSWkr5lJSyzHo9jYoVSZeFQAdjuT0khjELIfYCrgYOklJWYZJsG526wc9OnQpeXjF+1CdkFO4xOnQ4iYQJ1hkzggOoUqFxY9h55+iRcz//rFRWLa1UEzprb/PmqpIeZMc9qZqhg/WTyTbuwEtPOnw4jB9fuRg2pvCdRZSXhxYe6dFD1SDzOkbCwKEm4BYUG4w0PZlKOFnDiRqxPkoIkW+9RpGZiPWvgW5CiM5CiCJgJGrGU4kQYnvgEZQAyVlVottuU++58t6rCPubPFJwO+5vH5VSAskYasIoLw+1gfxGgLNdXp5SAfz+ux0FpwPoajE60M8vmXEoOlLd5P334dBDKxfDZiIXXwzPPBPx2B589lmiEJGyhqaRMr1a3njDOZDJVkLIGkYUIXIKcBTwt/U6wlqXFlLKMuA8YCIwGxgnpZwphLhJCHGQtdn/AY2AV4QQ3+W6tnumhEiy6n1TiJTe4PEgb7llQqR7oNwww4bNYalZCDtdPvpIGfUD8KwhYtKypZrRaKqgMmOu0TOQn34K3m7XXX2C4jZs8FjpJEoKNU9v1+uvD98RZdaqNTYRM3hwzhxnhc0ojiJ1ASllrXr1799fZho1jpKyffvMHO+nn+xj+r0KC+3P/2CWvWA2yFj34osJq7x/hH4tXqzWr1sXsmOKRDjer3SUEuT9nBt8rGbN1LGy8N9WN6ZMSbx07r9nxgyfyztxYvBNZfHHH+H3X9g9dDm3S5Cyc+fE/fbfX8offsjObZUTIl+gmg0wTabQ54aKUiFEeyHEG0KIJUKIxUKI1yyPrTpHplSgHmWoHZx5ptOHX/rFVBh8/32SjdABKWGNySL6d92JT5Ejjc4f/s03WW5R7vHSkKi05Da+lQ9DSxIqUkrm7JoFTkJNg777LjF+xT0TOe+8FM5XXTnuuFy3oNqRS++sOstdd9mf//e/xO/vuUe99P26I1+FHtOdsDcBM5wYbJ1aSLbdbHIkrzCOI/mDELdsbW2uLkn5ski3bv7rtL3E/CsrKpSavqKCUH3r8uXqErqLRH4VfnupEHcDrWJt0iTxb1m+3BYiL70UnE6qxlEHBjLJkkvvrGrP1KnOB6SpT8HAZDFtItoT1qSwUJ1X9wlPcGriRi5CjZbuPFo6LXwO7QzTGcDRjKPTNhEN/zLExfeDD5wV8Wogbh+J0aNtx6qSEnUJzAnHtdeqe+j440mcKp/iNF3++KN6d6c6KS6GgQO92/Pzz9aHBx5wrM/DOf3QAbmg7Dna1Na8eXS/jhpBVeedqwHk0jur2vPkk87l0BTZEQkLvtYPne4zC8MM0ETMZGIaBXWBEu0+q0nZt9RA6zd02pUQQjOa6IpcJ5wQvN0++8Cxx0Y6Z03h8svtz1IqVZQ5CNEety+8QKIubL/9PI/pTlhQVORtI/7gA/UXPv88sGiR47tO/MYZZ9jLpopszRpXDrfayDnn5LoF1YZkvbMWkSHvrJqAewSV6cKAWii5JwmahIG3+dS6iCREzHgRv8j+TOSo0j1IUEI7oFMn9e4u15CA1ud4pletvXhp7957z7nsqDPmlgRmxOKee1Yez23DaNzY+1zjx8PevE+76xIf968ZyA8/2MvuZ0OXLa8VQsQtjPv3hwcfzE1bqiFRcmf9IaU8SErZWkq5hZTyECnl71XRuFzjrnRZXq762HTVolp4XHSRev/9d2deN02CEAmo7BY5p+I226h3P/fEkhLVo6RTSVA3JqQHCckTaKMvxIIFwdtpPv206jMVVxGBabHMmciPPzo7vwAB3K6dfYlNYdKuHbzPvgydn2gC/ZVtHCYY9+WukdHpfhx8sHO5juRxi0oU76yxQohmxnJzIcSTQfvUVsrKVDru/v3t+LdUOMiKgtFB2QUFjvLjlSQIkYCOsVJ3jR0c6YlWsO+5p3O9TjSng6t0haJU0DORkGAsrYoJrS2ebNbhoUPhmmuS26eGEFjt1jSM9eqVUKs+SFPZr596N3cJk8Pnnmt/7tDBe5swM1aNwK0iNR+2mEjqrL5SylV6QUq5Etg+e02qvsyYYXvGphPcrQ2j5kjOa9CuH8D1LSyParevp8X48fYD/dBDTj16AlqH5NZfDBum3tORjpq//1bvITORJk3gyiu9PdQcmBI2aq90++0RXNZqHsuDrJFu74riYjjGqmTdti377uu/6z33qHGDaW/xy27egT8AZ7BjT3cRB4taMSFs1CjXLajWRBEieUKISp8RIUQLImT/remsWhX8faa9Tb36W23CWNJ/hPrg4aM+hV3NjBb07BkSSKtnB+6SiREinSOji/PMnBm4mRBq1uQjG23MNMq6IFAUPvww+rY1hMCZiP7jdREwISh75gXWDxgSmoq6Xj0VBR+GbNmShVbKO7ejideMslYIkZhAogiDO4H/CSFeRWXZPQqIlkSnBhOWASQdg+FhhyV6Cnppfm69FbbtJul8zmNqhUtyFVCaEIgY2i69QWmpPWPIFpECEJKkjvdKgep4rUacNAlQqceOPho+muZdOGTECFVyxmT33eHzz32Of8YZzDz8erBmNO57bc6cxBLitcKwHhNIFMP6M8DhwGJgKXCYlPLZbDcsl0jpLLDjRTppc8rKEuPCvGY29evDWSd4zBAefZQNx51OOQVU4HQh84o7caBdzkpLnZ5Y2XjaqzK3UCZck6sxobM1sIVImzaAmlkEObS9/Xaiun9uUNWBvfbirW9sV0L3LbPlljg8tsCZpi2mdhJJLSWlnAUETaRrFW+/HZ4ZPZ0Ks2Vl3n32iy96ZG330qudfjobDj0dPEo7hKrZhFAnd6uzsiFEOgZk6c00tUiIfPkl7OQqz3boocHawfpssI0Y1rQ2zP7rda988YXtwJdA375sabgUewUR6sqMQeeo8aTjuVgLidNQehBFy5NKltJNm5QBc8MGb/XVyJG2fbsSn4poaT2cBQXqB5g/IhthxW7dRibwM6xnOognh+y4o3O5Wzc7fZgft3GVveC6OZbi7xrupnNn5/JaDKNy9+6Vhf4mTKilAiIKsRBxEAsRD6JoYXQKiWR44gkVG/LJJ0n0eT5CRNduSonCQiVAdMDL2LHePzqVH2mSySpEOoufX9qAZAzuNQDT0aqkJHyieAJeBUAUT3IKm0gt3UI9rBnedqqWuo52z1VtnWpBVc6wawBR4kTOM72z6gJeI6wxY5zLboNkFBYvtj9H7vMCajO7gyEjo4WITiPSurX31Ojll+3PUsL06eHHXrtWGWb22iuzpSC1f6qfYf2xx7zXe00Z58yp9gZ6czbctGm4EJlHV9/vNlNMPTbjU306kMUo+4q+4U87TS1mw2eixnDyybluQbUiykxkS+BrIcQ4IcRwIWr/JNbrFyaomVIgcuDVpk1KoS2ldyi7xeDBKTZE20TMZS91lin1Xn5ZBSKagsXNjBnKz3Pq1MwPVU3XZC/81rstxTNmwD/+4Sx2VQ1p0gROtfJu3n13uBDphr8BZLM1CynEFqgrV0Zrx2qa8jb7UX6hStfvVWq8TqB/OGQuiV4tIYp31jVAN+AJ4CTgZyHEbUKILlluW87wEiKhUdURiGxHOfZYlfnu5ptVrdJ0j+dm+XJnFr6iIm8paSZs1FbaoMIlpuX3nXdSbJwPWsiFCZEePZzp7d2SWw/xN2xQKXKrMffdB6++qpILmDK5RQtlkzBpjn9gkxYixWzmACYg16wNDBt54w37cxElrKIZpWXqodBxn3XOdbd5c3j66dDYp7pIJJuIVfVKl8ctA5oDrwohqvdTmCQNGsCll3oLkUwMXCPnNtRPsVmOtEWLhM3cGplBgyIev7TU2VPk5yuVlq7ZpjFzYDz9tHo3k/q5yaaiXM9E/IxJul3Nmzvb4d7e/E0h5XtzTYMGcPjh6rPZaZ9zDhxwQPTjlKCuRy9mMoGDElLEM3mymj2SeK5iNlNCUWX5ZZ0N4cwz/c93xRXR21ajOPFE/9D8OkwUm8j5QojpwGhgCtBHSnk20B8VP1Jr2LhR2T68bMyZmImY/W/SalWP5IumO+WPPzprOiTFl186l3VQgjnVmT8/cZ2bbCZKClNn6YtbXOz8A91CxOwhQ3J7ZYWKipSyA5jaxmQd6bTn1kk8rVb8+qtzg8GDHeHqxXn2f1xECSUUVaZW0/I5aFB1++3JtS+mZhNlJtIKFWC4r5TyFSllKYCUsgJIYjxUczAGZZVkYpBtGkv79vXZyM/k5KrnAM6+smHDJGL73Olz3TmXtEHcq8MOCpAxp0bz5kVsTETC1Fl6Gnb55XbaD6/tzQ58r70y176o3Hijur5JJl8zb4vnXfFBo+8whLfOjWbQCBXcUenBZTpIuG1Gixez135F/ITy5W1YWMJmiisj5fXlDJO/zZrBhRcGbxNTO4hiE7nOL/W7lLJWlvkyTQEPPKAcjlJyJ3jkEUf+lNdft79KembjV3rOIikh504ZfOSRzmXt9rXPPvDHH85OJ2gmYgqRLhk2mYXNRLS6b7fdVA+msxW7i26ZNU7S9R6rqFA3hk7HHIVnrWQPAV53YWw5bzJMm0bXrkp717eH8Z+4Z5XAR6iMzd+wQ+LBdOAHKH2rpfPvzlzyKaNx6UrasLiybknELP+sXKkcAmJqP3GciAdmX7jPPikm8Vy3Ds46KyHlekd+Y1emeB8zSB3kLrPoIilDp3vm4VaVtW9vt6djRzs9PAQbdrLpNhtmE3EPkbVh//77nduZFbDcv+XLL9X+pidOEFqFdsMN0baH8BlVBCYzGAYO5Kef1ICnXy9DiHgIxltQafEX0j74wEuWOEZLc1GVKY9mXKUQ0WOIqsxoE1O9iW8FD8y+UBfVc+Nh53aiOztX7u5f6MIUdvfup4KCR0KkRFIzkbBplT7X448nfhfFOyAbXuBaOBx9tKucn4VbiPjqCw3cs6qdd1b/W9Tsv6m4x2khkmyE/UMPsZJmmLEe+fnq57ZpYbTDcBzQNjPtnTUSwz3bKxB0/XrHPbgNtu1EX96yMnXO2u/oHxOVWIh4oDUhfuyyS2KOoAR0J2E9bVow5aM+aDu1gzQ8n5Kaibz2mv3Zq4KVPpeXO29QLnLds3gZldJFd76bNsFddyV+r3s5vZ1fKWFzCO0nBIJmCY8+att7UiklrN2cAnO6e3DuuTRjNSfjqjJYUeHUGxlWd33PlXmlyLvyysR1q1eHFKNR5WbSyRsXU/uIhUgKTJ2qKrAGOiO5hIg7P2BlJbg1a+DPPx3behIiJVL227/sssR1QQIrSNDpTjmtnCw+mJbc665L/L68XAkIfQ39LohZt9VPCPjZcxYvVr6tenpq7r9hQ3gSSDNw1G2HisiTnOpccd99cNNNAFxY/xHHPantbp5CxAsfY/8lqHQNpaXwwgtJNTemDhALEQsvgRDUX4JyCV61yjmwr8Q1XHN7VR61nTUi7dXLtkEE5e0OkRJJeauaBhkv5XaQD+k++/h/p4VINiLRwn6g1rN4Yf65ZhCkeyaijcx+qqZx45zLv/xif27YMDF7oRt3MZB0a3WvXu0wpK/eWFjZya9apQY7XbuGCBGzEtXq1Z73Q2nrdowaZfsExMSYxELEwm0TzstL7AsXLXKWcn3kETjpJDjiCGd/AiQIEXel1harLH2WaegNMp57FWE3SEpHvfXWKe5I8PRL5+nIhhAJC44IEiKmUDBdWrUQmT5d/S59LfxmKOYsBhJd7MLSP7s76KZNg7cPo1kzx41XSiEffAATJ9qJDpo0gXoNPa6LthmZ7t6PP+7pHLF26Uaeey69ktAxtZecChErF9ccIcQ8IURCnKsQolgI8bL1/ZdCiE7Zakt5OUgEY1FJCevVS+xft9zSWfRp6VKl1gIPW6zuoKx6IO4wj8KD91OFS0xS8G7q1y/pXdKziroV4jNnKvuDWZs9G0Ik7JhlZf6Cxk8olJTYOcFef92+LmFTUFBR3l5CIyj9fTas0UbVpxKKGDsWhg+HpyzTiZTw36c9EmX16qV+pylgJ070PMW/LHVWQAaemDpMzoSIECIfeBAYAfQEjhFCuHMKnAqslFJ2Be4G7shWe/SzdAJqzh6UY625ldP49tvtmlFnneXaKIr10Z27IqEiVTgffxwtua6DKP6ZfumuH37Y/vzRR9C7t8qgawa5ZUOImHaawz0SJXjNRLTE9zOgl5badYpnzEhOiAwerHprN0GzkVRdoM1cYG6M+6yUxOuenw9bFS1PWM+LLyoXwwgeZiKF7L8xdYdczkR2BOZJKedLKUuAl4CDXdscDIy1Pr8KDMtWFuFlc50+t2aKJTeRYtRS8VBauzbpXZo3hx08YsgCiSJEwtIWl5b6R3xnW4h4Cdvy8kQhomu/es1EiorUb9DpbP/zn+SESBAeAX9AeLT/smUq8abbcSAoXbPx27xsH/Pm4X+DbNgQabDzBoeGbhNTd8mlEGkHLDCWF1rrPLeRUpYBq4EE1x8hxBlCiGlCiGlLzey0SdCsYJ1jOUiIWM4wCTjSY3sYTVuzJGGdg7B6pplCd5ZenlmaMEHw6KOp75sKpoDw8oLymolowWOOtvv3h/33V20sKbE948w0BekKEb9Mr16jfm1juuMOJRx//lllb47Ke+9VfvQSIqtW4Zl3rRJ3MKYHN3Nt9PbE1DlyKUS8ZhTueXOUbZBSPiqlHCClHNA6BZUQQOOeHSo/78HHNC7y70h23tl7/YMPGgtmOgmLpqRgmTzwQJV7JZPoa3T88f7bGJ2TJ0FtyoYQMaPskxUi5kxEb6cLc5kqJq3TTFeIeCVYHD1aqZC82g1ONWEy/PFH5ccKv8c5aEQUgc2kt39M7SYHaUwrWQh0MJbbA+4KTHqbhUKIAqApEDEnRZIIgezRAzF7Nh8zjDvX3QVc5LmpnzaovZlVwkNN0CKo6bpUrZuxY20jTKZ47jllUP7/9s49WorizuPfH1wvIKgXiCgIPgBRFBWUgBpfhwjGFxhjFEIirA84Rlg16iriro9zuIfER1jO+hYXo2w0YqKAGnAR18REFI8PUHmuGjAoCsY1YnhI7R9Vdae6p7pnpufRc2e+n3Pu6e7qmp6qrrn96/rV72Gj9fpwF8rDfPQRsHJldnn//voBWg6No3vTfUJkx45sIWKFmU+IWHWW29Zc6qzhw4Hnn8/d1rCfyWuvRTvx7dih2+kIAwBBa7E8aYts0+RBg8xOz55BS8B8aWzErn8wzAmJJs2fxmsADhaRg0SkEcBoAPNCdeYBsBHzzgPwgsltUhZkn31a9odsfTGyXpQaObBW4qk0H2dHf/kxx/jLC437nQ/dugGTJyd/2F9wgb988+Y8XPlLgE+IbNuWbQ0Rnol8/rleSLczke3bgybLVuhEpe0LxxyLIrxo9l5MnNIdO/xCK9+wKP36tey2QfbC/RtvmJ2k2fh+8AOIBC3M6S9CXFITImaNYxKAhQDeA/AbpdQ7InKriIw01WYB6CoiawH8DEB50904b7Infh6WZxmi/r8Dzx6PEOmGAtZr+vbVSanSSuMa/l43Gm74rdmyaVNlcnT4HvJxQuSII/QKc5cuWn1lnYDCybmsEInK9zEv+jcRIDz2ccJ6y5ZglklLPnG5evUKhL33zURaSKpiNCbErqw16dYJAZCyn4hS6lmlVD+lVB+l1DRT9m9KqXlm/x9KqR8qpfoqpYYopXwRp0pH+B/t44+9tvNRrgCLFzsHiXPXGk45RUeHTSvS3eTJmX2lgLFjgQu1Dw3Wr/d/Bsg/Am4SXnlFe88VOhMBgo48mzfrc+Hr5HI2zMUVV+htWIiEdUG9HC3uc8/5HThztWHYMG2G7fThlO9kv7gMGRJqg888Og5ju153OdVJ3lDT6RIWIt27e30Bwmv3l1+utwGDJfdBkiR3RNpR7nxK8HwEWjmfNkOHauHqEyILFwJvvx0sc8fTnT4uWqTPPfFEsL6950lfAMaMCV7HEr6Xrsnu9u3+78vVhoYG/T3OvfjnK7PHp2WiZdsQZUzxwAP+8pA6tZwZkEnrhELEJZ/YSx7cdBIt6+PugyTJg7XQUOGl5u9/zy57+OHsMiAYi6vcM6f27aMDHYbL3SeezxDA5fvfz4zZzxP6tNrfjzUbtoTvybhxmZAn1nM8jE+IzJ0b/K6dOzNxeDp0QNuTTwxUP/RQoEcPc2CFSGOjNnEOc8klmf2YjGnnnRd5itQpFCIuUYuPOYSAu946eza0Smf8+JayF+Z/hX2Rnd7Wi/X8jrIjrhRxvgUubdtq81XLq6+Wpz2WDh2ihUhYz+gKkWeeCVq5hX05dt+9+NmfFSLhMPQ+wfrYY3rbqZO/Py0r4g6uKsoKEcvWrVlT5MDP1hUijz0Wv3YVE6o3zrWI1CcUIi5R9vQvvZRVdPXV/o/t3ImsUOjDLu+PjeiRKZg2LboNBxygF4EvuyyPBpcRn0nqjTdml3XtWtlZU9RM5MADs6Ncuuqs99/PL4hjFLaP114bXcd9MD/+OLBihd6/7bZgPaWCmRp9M5H9wn63hqee0nlewkLEQ8BK2xUinToBd9wR/cGArXoQqrNIGAoRlygLlhtu0G+Tzhvl7bdnTrvPptmz8/ieG26IPrdsmfYzSDt1nO9N1Xd//vSneBPWUhMlRHzOhmHhFhd6fc6cTLIpIHuNwC50+2LejBsHXHpp8PtHj86YO4eDm+3alak7cqRWpYWJWlgfNUpH4G1o8Bo4LF8OLFni+ZyN9mtVs3FxvD74IPJUJYzvSOuCQsQl6jXLp1qIIO/lj0mT/OXl8AtJyqJFQam4777Zdfr08ZeXizghEr53YaFnH8w9eiCLsNAOq6SsmnHRouzPzp6trSryFfydO2fa9vnn/qCNX3+tw9ZHhcJpaPDGZh8wIBgLswVb18Z0i1vni0rKBQoRkg2FiEuBtvT33actNAvC6sGivssJ7Z06w4frt2yLu/jqUkl1Vrt2Wv0Tfgj6AjBGCWRfet1cPqw2tpabUCaMzxhh/vzssqFDcz+NTz5Zh87p10/77ITvfYxeqalJb71xF+26n9tf1wcI0JIIAA4LB9VOHoiY1C4UIi4FCpEJE7ItgMXjNRzAGu67ebFd/rWKg91Fxb6wpq2V4O679fbZZ4HevTMpI33qrKj2XnBBJoFWKfF56wechxwK+a3ttlu20UeMB3pTk14bD/hGWqFjow1YaXDNNdoHKMxbbwF//GNWcdpG9jeFRAAAD7hJREFUg6T6oBBxKcGqYTvkCN6XK3RGta9c+nJoHHKIfuhUAutzc9ZZWoBY+2qfOitqcRoATjih9G3zCYbwOowVXoXohb75JrtvrhBxreMMY8aEum8X0q3Rh3V4dFP6jh2beYk58siANdu6dcCUKZn08oRYKERciliP6NZNbzsgx6KINeU655zscxddlCA5SIWJ8iGwb+HlTn83bFjw2IYM8c1E4gJXhvOl50s4t0f4O8OzkTffzOzfdFMm9HqxQsQ1CcwnptekSVqFZV9Szj9f62LdbGqPPhqZ56B3b6C5OX17D1J9UIi45OMnsGCBt3jOHL39CXJEp7Nvq0OHZsr69tU68Fmzqv+/NEopLqIfUnGmo6UgnLDJOuXt3FmYiiipV7oTqwoLFmRbpoXXtNxF6ptvzjzwC2nrrl3xM5EkIXZF9KyS4XlJkfAX5GL8N+bgR9F1zvZH4rVJ/v4dV7aUfQSPFdDy5Xrrvj2uWQO8+GIhLU0Pd0E2jYBK4bWAI47Q7fCFgo/DzvgmTdKhVHy4CwA20+PMmcC552rfjzPPDAoVIHvR3nqZh2NW+dq6fHkm02K4HeGHvTsTqfYXD1LTUIh4OKPP6mzHtQSsRd/sQquOqfa1jyhsXvh33y062VEiwvftqKMybtSvvJL/dWwwycmTo4XhV19l9puadIiSjh2BJ5/UC9I+olRLDz4YPPbNRAYMyJhWuWzbFj8ToRAhKUIh4qHzN59pz+AiuQU34exwihRrxG/XFjwZEKua8eO1z0H//ul8v8/3w4Yw8QmDcLRMOzM4/HA9q+rXL9rS6Q9/0NuXX9aCIyrtbT6EhUvcrMnJa9NCnBApRyZJQvKEQsTHZ58Fs/AkYD7OwhIMwzqEHLfsP/yoUcCIEdpUtbURE6Cv7IQfmBMmZFy0fbOA1aszIdqBoDWS75ruwrJ1SnzoocLaaP0sor7Dd+ziM/+OW1hvrbNaUhNQiLjYt0PrNBaXItbDgqczOvRfQatLeu8XCl9hHx6dOunw5b17J2pq3RK3EOwTIk1NwXLfA9e95tSpmX07VqWIIlDIArZvlhL+vDsTsSpGQlKAQsTloov01sa73n//6Lrr1wNPPx0oOnN15g1yLvQ1ln0UCglC/XX5yCeUv0+I2PPNzfphfe+9+jhpcqq4301UO1yh8OWX2fWj1FkjRsSbMhNSZihEXJqbtcXNPffE11NKe56HfT1eftk50MLiE+xb/vDo9UTcrCAqdIn7wPapkazZ8re/rbd2LcR6xxcq+B95JLcKbI89gjOIgQMz+75+xKmzCEkRChGXrl217b+bS2PixOx627f7g+ZtjMgZwoXP0pFE/XfaaZl930zE+nJYb247A7He8YWm/O3SJRhzLIp58zKxtdyZiO+lw40wDGSESjUF7CR1CYVILqxqw8WNImvfYq++Gli61H+NmKiopEBEgOOP958bPNhf7joA+oTIjBn6YT5okD625rtWEPjUS7nIZw1EJCM83BcNXz6PWbOCx3a2QvUoSRkKkUKw6o0XXsiUffKJ3voiw1r22KN8bapHorIuxiWMsvjMeXffPahasgvxEycCv/+9NoAAdLj3JLihRcJYz3lXiBx3XO5rWiFCj3OSMvwFFoJ9uJx7bqbsxz/O77NXXAHcdVfp21SPRC2g56M2dPPBR2Gv8+mnwOmnZ8p98c7y4aijos8deaTeuqHeTz1V+6/EwZjspEpgiplC8C1m2jwTDv+Dk7LrzZhRhgbVKT5nvFw0N+u4W/m8uUcJo6QZmWLCtqNnT/9CeijFckuET4v1Y+rSJVmbCCkRnInkgxUePj8ET37sE7bMb9GA2JhapIQ0NwPTp0dGnPUyZUr+QRcrKUSiCM96wmFWTj1Vv5jMnJmsTYSUCAqRfPj6a/226JuJeHwJ2nbeE8OHA7fcoq09SYlpagKuuw648cbyXD/K4impEHnmmcI/c801epZr13kOPTR4XkSrSPfaK1mbCCkRVGcVgm8mEhYi5sEmkh21nJSYclkmRQmRJDMKIFnMrTZtdNyvW2/VaWrplU6qFM5ECiFKiLjTjST6epKcdev0Ni6LYaH4ZhzTpxduCWUjNtsow0lo314HvaQpL6lSUhEiItJFRJ4XkTVmmxW3QUQGisifReQdEXlbRC5Io60BfOqso48OmvySymIf7KV0uivVtaxjZDiSMCE1RFozkesBLFZKHQxgsTkOsxXAhUqpwwF8D8AMEfEkW0iZIUOA2bMzx1GhN0h56NhRb+PMaAvFJ0SSCJZf/lL7FtG6gtQwaa2JjAJwitl/GMCLAK5zKyilVjv7fxWRTQD2BvC3yjTRg09VtWxZ5dtBMuy9N/DSS8HYU8XiExhJwq136gRcdlnx7SGkiklLiOyjlNoIAEqpjSLSLa6yiAwB0AhgXcT5CQAmAMD++URQTYrP9NPa8lrSzLVRr5x4Ymmv5xMiF19c2u8gpEYomxARkf8GsK/n1FRPWdx1ugN4BMA4pZTXTVcpdT+A+wFg8ODB5dMn5YqcOnFi/h7spHrxCRGrNiOEBCibEFFKRSqCReQTEeluZiHdAWS7fet6ewJ4BsCNSqkCEmiXiQ4d9BtpOBieZepURlWtBcJjSPNaQiJJa2F9HgAbK3scgKfDFUSkEcDvAPxKKfVEBdsWz+TJ2WU22muvXpVtCykPYVPeIlMlE1LLpCVEpgMYLiJrAAw3xxCRwSLyoKlzPoCTAIwXkTfNXwlXTxPim2m0aeMP301qgzFj0m4BIVVLKgvrSqnNAL7rKV8G4BKz/yiARyvctNz4HNF27GDiqVrm8MPTbgEhVQs91gvFFSJTpmjhsXNn8rhKpLq5806gb9+0W0FI1cInX6G46qzGRj0L2b6dM5Fa5aqr0m4BIVUNhUihuDMO64D2+uvJUqiS6qV9+2DyMUKIFwqRQnFnInb28eGH6bSFlI8NG5jWmJA8oBApFHcmwsCLtUs4syAhxAsX1gvFDcm9Zk167SCEkCqAQqRQOjtR6xmxlxBS51CIFIq7JjJ0aHrtIISQKoBCpFBcddbNN2f2O3WqeFMIISRtKESKwZ2VUIgQQuoQCpGkHHJIMFAfvZoJIXUITXyTsGWLjuzqZrubNi299hBCSEpQiCTBtdAihJA6huqsYhk5Um937ky3HYQQkgKciRTLAw/oUOEnn5x2SwghpOJQiBRLt25Ac3ParSCEkFSgOosQQkhiKEQIIYQkhkKEEEJIYihECCGEJIZChBBCSGIoRAghhCSGQoQQQkhiKEQIIYQkRlSNZecTkU8BfFjEJb4F4LMSNae1UG99rrf+AuxzvVBMnw9QSu1d6IdqTogUi4gsU0oNTrsdlaTe+lxv/QXY53ohjT5TnUUIISQxFCKEEEISQyGSzf1pNyAF6q3P9dZfgH2uFyreZ66JEEIISQxnIoQQQhJDIUIIISQxFCIGEfmeiKwSkbUicn3a7SkGEeklIktE5D0ReUdErjDlXUTkeRFZY7adTbmIyEzT97dF5GjnWuNM/TUiMi6tPuWDiLQVkTdEZIE5PkhElpq2Py4ijaa8nTlea84f6FxjiilfJSKnpdOT/BCRJhGZKyIrzVgfVwdjfJX5Ta8QkV+LSPtaG2cReUhENonICqesZOMqIseIyHLzmZkiIkU1WClV938A2gJYB6A3gEYAbwE4LO12FdGf7gCONvt7AFgN4DAAvwBwvSm/HsDPzf4ZAJ4DIACOBbDUlHcB8L9m29nsd067fzH9/hmA/wKwwBz/BsBos38vgMvM/k8B3Gv2RwN43OwfZsa+HYCDzG+ibdr9iunvwwAuMfuNAJpqeYwB7AfgfQAdnPEdX2vjDOAkAEcDWOGUlWxcAbwK4DjzmecAnF5Ue9O+YdXwZ27oQud4CoApaberhP17GsBwAKsAdDdl3QGsMvv3ARjj1F9lzo8BcJ9THqhXTX8AegJYDGAYgAXmH+QzAA3hMQawEMBxZr/B1JPwuLv1qu0PwJ7mgSqh8loe4/0ArDcPxgYzzqfV4jgDODAkREoyrubcSqc8UC/JH9VZGvvjtGwwZa0eM4UfBGApgH2UUhsBwGy7mWpR/W9N92UGgH8BsMscdwXwN6XUTnPstr2lX+b8F6Z+a+pvbwCfAvhPo8J7UEQ6oobHWCn1EYDbAfwFwEbocXsdtT3OllKN635mP1yeGAoRjU8n2Optn0WkE4AnAVyplPq/uKqeMhVTXlWIyFkANimlXneLPVVVjnOtor+GBmiVxz1KqUEAvoJWc0TR6vts1gFGQaugegDoCOB0T9VaGudcFNrHkvedQkSzAUAv57gngL+m1JaSICK7QQuQOUqp35riT0SkuznfHcAmUx7V/9ZyX74DYKSIfADgMWiV1gwATSLSYOq4bW/plzm/F4AtaD39BXRbNyillprjudBCpVbHGABOBfC+UupTpdQOAL8FcDxqe5wtpRrXDWY/XJ4YChHNawAONlYejdCLcPNSblNijLXFLADvKaXudE7NA2CtNMZBr5XY8guNpcexAL4wU+aFAEaISGfzFjjClFUVSqkpSqmeSqkDocfuBaXUWABLAJxnqoX7a+/Deaa+MuWjjVXPQQAOhl6ErDqUUh8DWC8ih5ii7wJ4FzU6xoa/ADhWRHY3v3Hb55odZ4eSjKs596WIHGvu4YXOtZKR9gJStfxBWzmshrbUmJp2e4rsywnQU9S3Abxp/s6A1gcvBrDGbLuY+gLgLtP35QAGO9e6CMBa8/dPafctj76fgox1Vm/oh8NaAE8AaGfK25vjteZ8b+fzU819WIUirVYq0NeBAJaZcX4K2gqnpscYwC0AVgJYAeARaAurmhpnAL+GXvPZAT1zuLiU4wpgsLl/6wD8B0LGGYX+MewJIYSQxFCdRQghJDEUIoQQQhJDIUIIISQxFCKEEEISQyFCCCEkMRQihBSJiab7U7PfQ0Tmpt0mQioFTXwJKRITn2yBUmpAyk0hpOI05K5CCMnBdAB9RORNaGew/kqpASIyHsA50KkGBgC4Azpk+08AbANwhlJqi4j0gXYY2xvAVgCXKqVWVr4bhBQO1VmEFM/1ANYppQYCuDZ0bgCAHwEYAmAagK1KB0z8M3TICQC4H8BkpdQxAK4BcHdFWk1ICeBMhJDyskQp9SV0vKIvAMw35csBHGkiLR8P4AknwVy7yjeTkGRQiBBSXrY5+7uc413Q/39toPNhDKx0wwgpBVRnEVI8X0KnIS4YpfO8vC8iPwRacmYfVcrGEVJOKEQIKRKl1GYAL4vICgC3JbjEWAAXi8hbAN6BTrxESKuAJr6EEEISw5kIIYSQxFCIEEIISQyFCCGEkMRQiBBCCEkMhQghhJDEUIgQQghJDIUIIYSQxPw/zwEoD/JYX8YAAAAASUVORK5CYII=\n", "text/plain": [ "
" ] }, "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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\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", ">### Verification\n", "In fact, the linear scaling of the walk variance (MSD) can be used for verification of computational models of random walks. Let's denote by $\\sigma^2_1$ the variance of a single step of the walk. Then the variance after $T$ steps is $\\sigma^2_T= T\\sigma^2_1$. If the computational model gives a different result, it is wrong.\n", "\n", ">### Your turn\n", "Derive the factor of $(3/2)^{1/2}$ used in the generation of the CSDT random walk above.\n", "\n", "Processes that have the linear scaling of the MSD with time are called *diffusive* processes. They include different random walks, Brownian motion, diffusion per se, and many others. One way to think about diffusion is a limit of a CTCS random walk with finite variances of durations and lengths of jumps, when the mean duration and length go to zero, and the number of steps goes to infinite in such a proportion that the MSD stays linear with time, $${\\rm MSD}=2dDT,$$ where $d$ is the dimensionality of the process, $T$ is its overall duration, and $D$ is what's called the *diffusion coefficient*, defined by the expression above.\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", "## Central Limit Theorem\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).\n", "\n", "Now we can explain why the error of MC methods scales as $\\sim 1/\\sqrt{N}$. If we are calculating a quantity numerically that is **TO BE CONTINUED**" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Projects\n", "\n", "There are three projects available for you to choose from in this module. We will first introduce them, and then we will introduce a few more theoretical ideas and modeling topics that are needed to properly address them\n", "\n", ">### Project 1: \n", "**Diffusion-limited aggregation (physics)** Read about diffusion limited aggregation (DLA) __[here](http://paulbourke.net/fractals/dla/)__. Implement DLA as follows. Start with a large square lattice (at least $100\\times 100$, but generally larger; make this a changeable parameter in the model). Make the central point of the lattice the beginning of the DLA cluster (a stationary dot). Release a walking dot on a discretized circle slightly larger than the size of the stationary cluster. Let it perform a random walk, and at each step check if the walking dot comes in contact with a cluster. If it does, the walking dot becomes stationary itself and joins the cluster. If the dot tries to exit the circle, from which it was released, do not let it. Once the dot becomes stationary, repeat the cycle by releasing the next walking dot. Keep repeating this until the cluster consists of a larger number of stationary dots (e.g., 1000 -- again, make this a changeable parameter. At every attachment event, record the state of the lattice (which points have a stationary dot, and which don't), and combine these frames into a movie, as shown in Chapter 8.2 of the book. In your report, instead of a movie, show a few time-lapsed images of the cluster. Finally, measure the maximum horizontal extent of the cluster at every frame and plot it as a function of time. How does the size grow with time? Is this a power law? Which one? The DLA cluster models the deposition of molecules on surfaces -- think of snowflakes, but also growth of bacteria in a bacterial film, where the movable dot is now a food particle, and the stationary dots are bacteria that divide when the food gets eaten.\n", "\n", ">### Project 2:\n", "**Annihilation in different dimensions (chemistry)** A chemical annihilation reaction, $A+A\\to0$, is one of the simplest reactions one can think of. But it also illustrates how properties of random walks depend strongly on the dimension of the system. First, write down the chemical kinetics equation for the system and solve it analytically. At $t\\to\\infty$, how does the concentration of A decrease? Now let's simulate this problem. Initiate a lattice of a linear size $L$ in 1, 2, and 3 dimensions (it will be an array of size $L$, $L\\times L$, and $L\\times L\\times L$). Play with different values of $L$, as long as it is much larger than 1. Release $N$ random walkers onto this lattice at random points (again, explore different ''even'' values of $N$). As the walkers do their 1, 2, and 3 dimensional walks, you should check if any pair of them ends on top of each other. When this happens, the two walkers annihilate each other and disappear. As time increases, measure the number of remaining walkers (you will need to average them over many repeats when this number is of order 1 or less). Does the number of walkers in different dimensions decrease with time as a power law? What is the power? Does the power depend on the dimension? Does any dimension match what we would expect from the analytical, deterministic solution? This model is used in various material science contexts, but also in population biology scenarios.\n", "\n", ">### Project 3:\n", "*Simulating a bistable gene (biochemistry)** Most cells in our bodies have the same DNA, and yet they have very different phenotypic properties. This is because different genes are active or inactive (express corresponding proteins or not) in different cells. Often this is achieved in cells by positive autoregulation: an expressed protein activates expression of additional copies of itself, so that if a large number of proteins is expressed, then they produce even more of themselves, resulting in a fully active gene. On the contrary, if originally only a small number of proteins is expressed, then their effect is insufficient to produce many more, and the gene is inactive. We model such dynamics as a combined effect of two processes. First, existing proteins $p$ degrade with the propensity $rp$, where $r$ is the rate. Second, existing proteins activate production of new proteins with the propensity $A_0 + \\frac{A_1p^2}{1+(p/p_0)^2}$, where $A_0$ is the basal expression rate, $A_1$ is the expression amplitude, and $p_0$ is the half-maximal expression protein concentration. Simulate this system using the Gillespie algorithm, discussed in class. Explore how the dynamics of expression depends on time and on the initial conditions. Find parameter values where, depending on the initial condition, there are two stable states: inactive, with $p\\approx 10$ and active with $p\\approx 25$. Does the system stay in either of the stable states forever?\n", "\n", "How do we simulate such spatially extended probabilistic systems? A key question is: what should be the dynamical degrees of freedom that we should use to represent the system? This is not a simple question, and it requires a serious discussion.\n", "\n", "### Cellular Automata vs Agent-based simulations\n", "The same systems (usually stochastic, but often also deterministic) can sometimes be simulated in multiple different ways. Consider, for example, Project 2 for this module: the annihilation reaction, where two diffusing individuals of the same species annihilate each other when they meet. We can model this system by choosing random positions of $N$ individuals on the $L\\times L$ lattice. We would then run a loop over time steps, and in each of the time steps we would loop over the individuals. For each individual, we would first generate a step in a random direction, and then we would check if the individual is at the same point on the lattice as some other. If the latter is true, both individuals would then be removed from the future consideration. \n", "\n", "This is called an *agent-based* simulation. To specify such a simulation, one needs to specify the number of agents, their state (the coordinates on the lattice in our case), and the rules, by which the agents evolve with time. The agents are the primary units, the degrees of freedom of the system. The behavioral rules can include the rules for motion, the rules for creation, destruction, for interaction with other agents, and so on. The state of each agent may be much more complex than just the position, and it can include such attributes as reactivity, age, or whether the agent is mobile or not (which is relevant to Project 1, growth of the DLA cluster). The rules and states do not necessarily have to put the agents on the lattice; instead, they can move through the real space. The rules can be deterministic or probabilistic, or a mixture of both. All of these combinations are possible. Such agent-based simulations are used routinely in many different fields, and are a staple of modern science. We can track motion and interactions of individual molecules, including their creation and destruction in chemical reactions. We can track position of individual animals or humans in behavioral or sociological experiments. \n", "\n", "An alternative way of simulating the same annihilation system is to take not the individual, but rather the space itself (always represented as a lattice on a computer) as the main variable. The lattice consists of many cells, and each cell has a certain number of individuals in it. The simulations proceed by, again, looping over time. However, now in each time step, we loop over the lattice sites, the cells, and not the individuals themselves. Each cell has rules for its change based on its own state, as well as the state of the surrounding cells, with which it interacts. In our annihilation problem, the rules could be the following. First if there are more than two individuals in a cells, then the number of them is decreased by two (two are annihilated). And then a random number of individuals from each cell (distributed according to some probability law) is removed from a cell and added to one of the neighboring cells at random. It's pretty clear that one can match the rules, by which we model the cells, to the rules, by which agents are modeled, so that the results of simulations are the same.\n", "\n", "This type of simulations is called *cellular automaton* simulation (that is, there is an automaton that obeys a certain rules in every cell). To specify such simulation, we need to specific the number and the geometric arrangement of the cells, as well as the rules by which the cells evolve in time. As in agent-based approaches, the rules can be deterministic or probabilistic, and the state of the lattice can be either very simple (e.g., occupied or not) or very complex (how many individuals and of which type are in a cell). Cells may be real physical cells, or they can be discretized bins of the continuous space, for example exchanging molecules. These type of approaches are also used routinely in applications. For example, we can predict forest fires by viewing a forest as consisting of lattice sites that have states such as *on fire*, *combustible*, or *barren*, and with rules that define how fires emerge and how they spread to new combustible cells. Spread of epidemics can also be models by cellular automata. And even diffusion of molecules through space can be viewed as cells with rules for exchanging molecules.\n", "\n", "Cellular automata and agent-based simulations are not exclusive. Some problems may require the use of both, as we will illustrate below using the example of the DLA cluster growth. \n", "\n", "### Which method to use?\n", "If both methods can be used often to achieve the same results for the same systems, which one ''should'' be used? Both methods can be made equally accurate, and both are similarly easy (or hard) to code. So often the choice is left to the scientist, and either choice is reasonable. However, sometimes there is a big distinction: the execution time. And here common sense rules. In both methods, the main loop is over time. But then the first loops over the agents, and the second over the cells. It is clear that the number of operations per each time step that would need to be performed in each cycle of an agent-based simulation would scale at least as $N$, the number of agents, and may require even more time if many pairs of agents interact, e.g., $N^2$. Let's generalize this and denote the scaling of the time in the agent-based formulation as $N^a$. In contrast, for cellular automata, this internal cycle would scale with the number of cells, $\\propto L^d$ in a $d$-dimensional system. If the agents and the cells have similarly complex rules, than the best type of simulations is determined by whether $N^a$ is larger than $L^d$ or not -- the smaller of the two should be chosen. For example, if we have only a handful of individuals diffusing on a big lattice and they only rarely interact with each other, then agent-based approaches are the way to go. If, on the other hand, each cell, on average, has $\\gg1$ individuals, then we can move many of them from one cell to another in just a few computer commands, rather than a few commands per individual -- and cellular automata simulations will be faster in this case.\n", "\n", "In my experience reviewing papers for scientific journals, the choice of the representation between cellular automata and agents is probably one of the most common and the most costly mistakes an computational scientist can (and do) make. Choose wrong, and your code will spend idle cycles moving millions of individuals one at a time, or analyzing cells where nothing at all happens." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Coding considerations for the DLA Project\n", "The most important thing we should do is to decide how to represent the data. Let's consider the costs of different choices.\n", "\n", "*Cellular automaton*. There's only one walker at any given time. Thus nothing will be happening over most of the lattice, and running a cellular automaton would be unnecessarily costly. \n", "\n", "*Agent-based simulation*. From the first glance, this seems like a way to go. Since only one walker is moving, let's represent it as an agent. However, at every step of the walker, we need to check if it is touching the DLA cluster. Thus if the cluster is represented by agents that are inactive, and the walker is an active agent, then every step of the walker will require checking if it is touching any of the previous agents. This will require $O(N^2)$ steps, where the first power in $N^2$ comes from having to launch $N$ agents, and the second from having to check every other agent every time. This will be costly too.\n", "\n", "*Mixed simulation*. The best solution here is to represent the walker as an agent, but the cluster as a cellualr automaton, where, say, a 0 on the lattice at a certain point means that it does not belong to the cluster, while 1 sayd that it does. Then to decide whether an agent is touching the cluster, one only needs to check the state of the cell under it, which does not scale with $N$. Thus the time complexity of the problem decreases to $O(N)$ from $O(N^2)$ in this representation.\n", "\n", "The code below implements the simplest DLA cluster using this representation. You should feel free to use this in your solution for Project 1. However, this is not the whole solution -- a lot of modifications to the code are still needed to answer the questions posed by the Project." ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Attached walker No. 0\n", "Attached walker No. 100\n", "Attached walker No. 200\n", "Attached walker No. 300\n", "Attached walker No. 400\n", "Attached walker No. 500\n", "Attached walker No. 600\n", "Attached walker No. 700\n", "Attached walker No. 800\n", "Attached walker No. 900\n" ] }, { "data": { "image/png": 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# initialization block\n", "N_walkers = 1000 # how many walkers in a cluster\n", "L = 100 # lattice size\n", "lattice = np.zeros((L,L)) # the lattice itself\n", "lattice[int(L/2), int(L/2)] = 1 # seeding the DLA cluster\n", "\n", "# The following function starts the walker somewhere on the lattice.\n", "# For now, I coded the function to start a walker on one of the edges of the lattice.\n", "# However, as the accompanying text describes, starting the walker on the edge is not really ok.\n", "# Indeed, this will break the circular symmetry of the cluster. Further, it will take the walker a \n", "# a very long time to find the claster if it starts far away. You will need to re-write this function\n", "# to start the walker at a circle that is slightly larger than the current cluster size. This is why\n", "# we supply the lattice as the argument to this function.\n", "def start_walker(lattice):\n", " L = lattice.shape[0] # getting the lattice size\n", " side = np.floor(4*nprnd.random()) # which side the walker will be releazed on\n", " start_coord = 1+int(np.floor((L-2)*nprnd.random())) # which site on the chosen side the walker \n", " # will be released on. Notice that we don't let the walkers to be released at edges, but only\n", " # with coordinates [1,L-2]. This makes it easier later to check if the walker is approaching an edge,\n", " # which we block.\n", " if (side==0): # left side\n", " x = 1\n", " y = start_coord\n", " elif (side==1): # top\n", " x = start_coord\n", " y = L-2\n", " elif (side==2): # right\n", " x = L-2\n", " y = start_coord \n", " else: # bottom \n", " x = start_coord\n", " y = 1\n", " \n", " return (x,y) # return the walker initial position\n", "\n", "\n", "# The following function runs a walker till it ends up joining the cluster.\n", "# At this point, the function checks if the walker is trying to leave the finite range\n", "# of the lattice. Instead, to speed things up, you should recode this so that\n", "# the walker is not allowed to leave the circle that is slightly larger than the \n", "# current cluster size.\n", "def run_walker(lattice):\n", " L = lattice.shape[0] # getting the lattice size\n", " x, y = start_walker(lattice) # we initiate the walker position\n", " ii=0\n", " # keep running the walker till it hits the cluster and gets stuck\n", " while (True):\n", " # Keep generating plausible steps until an acceptable one (not hitting a wall)\n", " # is generated.\n", " while(True): \n", " # this code is copied from the DSDT_2d_walk() function above\n", " directions = int(np.floor(nprnd.random(1)*4)) # (0,1,2,3) directions of motion on each time step\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", " # we now verify that the walker doesn't approach the edge. If it doesn't, we break the loop.\n", " # If it does, we continue pulling random numbers and trying different steps.\n", " if((int(x+dx)>0)&(int(x+dx)0)&(int(y+dy)0): # the walker is near the cluster\n", " lattice[x,y] = 1 # add the walker to the cluster\n", " break\n", " \n", " return (lattice)\n", "\n", "# release N_walkers walkers and build the DLA cluster on the lattice. \n", "for i in np.arange(N_walkers):\n", " lattice = run_walker(lattice)\n", " if (np.round(i/100)==i/100): # signal attachment of every 100th walker\n", " print('Attached walker No. '+str(i)) # we do this to monitor progress\n", "\n", "# now we show the final cluster lattice\n", "plt.imshow(lattice, cmap='Greys', interpolation='nearest')\n", "plt.title('DLA cluster')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ ">### Your turn:\n", "Use the code above and make the necessary improvements to it, and then use it to answer questions posed in Project 1." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Coding examples for Project 2, more to come" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Coding examples for project 3, more to come" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# import additional modules, which\n", "from scipy.integrate import odeint\n", "from mpl_toolkits.mplot3d import Axes3D\n", "from matplotlib import cm\n", "\n", "N_events = 100\n", "N_trials = 10\n", "a = 10.0\n", "r = 1.0\n", "\n", "N_part = np.zeros((N_events, N_trials))\n", "t_events = np.zeros((N_events, N_trials))\n", "\n", "for j in range(N_trials):\n", " for i in range(1, N_events):\n", " prop_birth = a\n", " prop_death = N_part[i-1, j]*r\n", " prop_tot = prop_birth + prop_death\n", " prob_birth = prop_birth / prop_tot\n", " which_event = (nprnd.random(1) < prob_birth)\n", " if which_event:\n", " N_part[i, j] = N_part[i-1, j] + 1\n", " else:\n", " N_part[i, j] = N_part[i-1, j] - 1\n", "\n", " t_events[i, j] = t_events[i-1, j] + nprnd.exponential(1)*(1/prop_tot)\n", "\n", "plt.plot(t_events, N_part, '-')\n", "plt.xlabel('time')\n", "plt.ylabel('number of particles')\n", "plt.title('Individual trajectories in the birth-death process')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "> Your turn and Verification\n", "Calculate the mean of the trajectories at large $t$. Does the mean agree with the theoretical prediction? Remebering our lecture about Random Walks, what should the variance of $n$ be as a function of the mean? Do the simulations agree?\n", "\n", "## Dynamics of the the probability distribution" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "def dpdt(p, t, a, r):\n", " dpdt = 0*p\n", " maxn = p.size\n", " n = np.arange(maxn)\n", " dpdt[1:maxn] = a*p[0:maxn-1]\n", " dpdt[0:maxn-1] = dpdt[0:maxn-1] + r*n[1:maxn]*p[1:maxn]\n", " dpdt = dpdt - a*p\n", " dpdt = dpdt - r*n*p\n", " return dpdt\n", "\n", "maxn = 20\n", "p = np.zeros(maxn)\n", "p[0] = 1\n", " \n", "t = np.arange(10)\n", "n = np.arange(maxn)\n", "pp = odeint(dpdt, p, t, args=(a, r))\n", "T, N = np.meshgrid(t, n)\n", "ax = Axes3D(plt.figure())\n", "ax.plot_surface(T, N, np.transpose(pp),cmap=cm.coolwarm)\n", "plt.title('Evolution of the probability for the birth-death process')\n", "plt.xlabel('time')\n", "plt.ylabel('particle number')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ ">### Your turn and verification\n", "Compare the mean and the variance of the solved probablity distribution with the results of individual simulations.\n", "\n", ">### You turn\n", "Figure out how to change the plot above to make it into a 2-d color intensity plot." ] }, { "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 }