{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "###### Content under Creative Commons Attribution license CC-BY 4.0, code under BSD 3-Clause License © 2017 L.A. Barba, N.C. Clementi" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Bird's-eye view of mechanical vibrations\n", "\n", "Welcome to **Lesson 4** of the third module in _Engineering Computations_. This course module is dedicated to studying the dynamics of change with computational thinking, Python and Jupyter.\n", "The first three lessons give you a solid footing to tackle problems involving motion, velocity, and acceleration. They are:\n", "\n", "1. [Lesson 1](http://go.gwu.edu/engcomp3lesson1): Catch things in motion\n", "2. [Lesson 2](http://go.gwu.edu/engcomp3lesson2): Step to the future\n", "3. [Lesson 3](http://go.gwu.edu/engcomp3lesson3): Get with the oscillations\n", "\n", "You learned to compute velocity and acceleration from position data, using numerical derivatives, and to capture position data of moving objects from images and video. For physical contexts we used free-fall of a ball, and projectile motion. Then you faced the opposite challenge: computing velocity and position from acceleration data, leading to the idea of stepping forward in time to solve a differential equation. \n", "\n", "Our general approach combines these key ideas: (1) turning a second-order differential equation into a system of first-order equations; (2) writing the system in vector form, and the solution in terms of a state vector; (3) designing a code to obtain the solution using separate functions to compute the derivatives of the state vector, and to step the system in time with a chosen scheme (e.g., Euler, Euler-Cromer, Runge-Kutta). It's a rock-steady approach that will serve you well!\n", "\n", "In this lesson, you'll get a broader view of applying your new-found skills to learn about mechanical vibrations: a classic engineering problem. You'll study general spring-mass systems with damping and a driving force, and appreciate the diversity of behaviors that arise. We'll end the lesson presenting a powerful method to study dynamical systems: visualizing direction fields and trajectories in the phase plane.\n", "\n", "Are you ready? Start by loading the Python libraries that we know and love." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true }, "outputs": [], "source": [ "import numpy\n", "from matplotlib import pyplot\n", "%matplotlib inline\n", "pyplot.rc('font', family='serif', size='14')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## General spring-mass system\n", "\n", "The simplest mechanical oscillating system is a mass $m$ attached to a spring, without friction. We discussed this system in the [previous lesson](http://go.gwu.edu/engcomp3lesson3). In general, though, these systems are subject to friction—represented by a mechanical damper—and a driving force. Also, the spring's restoring force could be a nonlinear function of position, $k(x)$.\n", "\n", " \n", "#### General spring-mass system, with driving and damping." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Newton's law applied to the general (driven, damped, nonlinear) spring-mass system is:\n", "\n", "\\begin{equation}\n", " m \\ddot{x} = F(t) -b(\\dot{x}) - k(x)\n", "\\end{equation}\n", "\n", "where\n", "* $F(t)$ is the driving force\n", "* $b(\\dot{x})$ is the damping force\n", "* $k(x)$ is the restoring force, possibly nonlinear\n", "\n", "Written as a system of two differential equations, we have:\n", "\n", "\\begin{eqnarray}\n", "\\dot{x} &=& v, \\nonumber\\\\\n", "\\dot{v} &=& \\frac{1}{m} \\left( F(t) - k(x) - b(v) \\right).\n", "\\end{eqnarray}\n", "\n", "With the state vector,\n", "\\begin{equation}\n", "\\mathbf{x} = \\begin{bmatrix}\n", "x \\\\ v\n", "\\end{bmatrix},\n", "\\end{equation}\n", "\n", "the differential equation in vector form is:\n", "\n", "\\begin{equation}\n", "\\dot{\\mathbf{x}} = \\begin{bmatrix}\n", "v \\\\ \\frac{1}{m} \\left( F(t) - k(x) - b(v) \\right)\n", "\\end{bmatrix}.\n", "\\end{equation}\n", "\n", "In this more general system, the time variable could appear explicitly on the right-hand side, via the driving function $F(t)$. We'll need to adapt the code for the time-stepping function to take the time as an additional argument. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The `euler_cromer()` function we defined in the previous lesson took three arguments: `state, rhs, dt`—the state vector, the Python function computing the right-hand side of the differential equation, and the time step. Let's re-work that function now to take an additional `time` variable, which also gets used in the `rhs` function." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": true }, "outputs": [], "source": [ "# new version of the function, taking time as explicit argument\n", "def euler_cromer(state, rhs, time, dt):\n", " '''Update a state to the next time increment using Euler-Cromer's method.\n", " \n", " Arguments\n", " ---------\n", " state : state vector of dependent variables\n", " rhs : function that computes the RHS of the DE, taking (state, time)\n", " time : float, time instant\n", " dt : float, time step\n", " \n", " Returns\n", " -------\n", " next_state : state vector updated after one time increment'''\n", " \n", " mid_state = state + rhs(state, time)*dt # Euler step\n", " mid_derivs = rhs(mid_state, time) # update derivatives\n", " \n", " next_state = numpy.array([mid_state[0], state[1] + mid_derivs[1]*dt])\n", " \n", " return next_state" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Case with linear damping\n", "\n", "Let's look at the behavior of a system with linear restoring force, linear damping, but no driving force: $k(x)= kx$, $b(v)=bv$, $F(t)=0$.\n", "The differential system is now:\n", "\n", "\n", "\\begin{equation}\n", "\\dot{\\mathbf{x}} = \\begin{bmatrix}\n", "v \\\\ \\frac{1}{m} \\left( - kx - bv \\right)\n", "\\end{bmatrix}.\n", "\\end{equation}\n", "\n", "Now we need to write a function to compute the right-hand side (derivatives) for this system.\n", "Even though the system does not explicitly use the time variable in the right-hand side, we still include `time` as an argument to the function, so that it is consistent with our new design for the `euler_cromer()` step. We include `time` in the arguments list, but it is not used inside the function code. It's thus a good idea to specify a _default value_ for this argument by writing `time=0` in the arguments list: that will allow us to also call the function leaving the `time` argument blank, if we wanted to (in which case, it will automatically be assigned its default value of 0).\n", "Another option for the default value is `time=None`. It doesn't matter because the variable is not used inside the function!" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def dampedspring(state, time=0):\n", " '''Computes the right-hand side of the spring-mass differential \n", " equation, with linear damping.\n", " \n", " Arguments\n", " --------- \n", " state : state vector of two dependent variables\n", " time : float, time instant\n", " \n", " Returns \n", " -------\n", " derivs: derivatives of the state vector\n", " '''\n", " \n", " derivs = numpy.array([state[1], 1/m*(-k*state[0]-b*state[1])])\n", " return derivs" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's try it!\n", "The following example is from section 4.3.9 of Ref. [1] (an open-access text!). \n", "We set the model parameters, the initial conditions, and the time-stepping conditions.\n", "Then we initialize the numerical solution array `num_sol`, and call the `euler_cromer()` function in the `for` statement.\n", "Notice that we pass the time instant `t[i]` to the function's `time` argument (which will allow us to use the same calling signature when we solve for a system with driving force)." ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": true }, "outputs": [], "source": [ "m = 1.0\n", "k = 1.0\n", "b = 0.3" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": true }, "outputs": [], "source": [ "x0 = 1 # initial position\n", "v0 = 0 # initial velocity" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": true }, "outputs": [], "source": [ "T = 12*numpy.pi\n", "N = 5000\n", "dt = T/N\n", "\n", "t = numpy.linspace(0, T, N)" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": true }, "outputs": [], "source": [ "num_sol = numpy.zeros([N,2]) #initialize solution array\n", "\n", "#Set intial conditions\n", "num_sol[0,0] = x0\n", "num_sol[0,1] = v0" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": true }, "outputs": [], "source": [ "for i in range(N-1):\n", " num_sol[i+1] = euler_cromer(num_sol[i], dampedspring, t[i], dt)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Time to plot the solution—in our plot of position versus time below, notice that we added a line with [`pyplot.figtext()`](https://matplotlib.org/api/_as_gen/matplotlib.pyplot.figtext.html?highlight=matplotlib%20pyplot%20figtext#matplotlib.pyplot.figtext) at the end. This command adds a custom text to the figure: we use it to print the values of the spring-mass model parameters corresponding to the plot. See how we print the parameter values in the text string? We used Python's string formatter, which you learned about in [Module 2 Lesson 1](http://go.gwu.edu/engcomp2lesson1).\n", "If we were to re-run the solution with different model parameters, re-executing the code in this cell would update the plot and the text with the proper values. (We don't want to rely on manually changing the text, as that is error prone!)" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "image/png": 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HjWdk0b4DIC4+bioAf52f1tu5i4jktFQG8loze8bMvmBmx5pZPlCawv1ntZ7O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\nuSnAo8C08lwfxQfDrKTYcPYkxc70x8vXr5P0TWB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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig = pyplot.figure(figsize=(6,4))\n", "\n", "pyplot.plot(t, num_sol[:, 0], linewidth=2, linestyle='-')\n", "pyplot.xlabel('Time [s]')\n", "pyplot.ylabel('Position, $x$ [m]')\n", "pyplot.title('Damped spring-mass system with Euler-Cromer method.\\n')\n", "pyplot.figtext(0.1,-0.1,'$m={:.1f}$, $k={:.1f}$, $b={:.1f}$'.format(m,k,b));" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The result above shows that the oscillations die down over a few periods: the oscillations are _damped_ over time.\n", "And our plot looks pretty close to [Fig. 4.27](https://link.springer.com/chapter/10.1007%2F978-3-319-32428-9_4#Fig27) of Ref. [1], as it should." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Case with sinusoidal driving, and damping\n", "\n", "Suppose now that an external force of the form $F(t) = A \\sin(\\omega t)$ drives the system. This is a typical situation in mechanical systems. Let's find out what a system like that behaves like. The example below comes from section 4.3.10 of Ref. [1].\n", "\n", "We're showy, so we decided to use the Unicode character for the Greek letter $\\omega$ in the code… because we can! \n", "With a handy table of [Unicode for greek letters](https://gist.github.com/beniwohli/765262), you can pick a symbol code, type it into a code cell, and out comes the symbol. Then, it's a copy-and-paste job to reuse the symbol in the code. And using greek letters for some variable names is very chic." ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'ω'" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u'\\u03C9'" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": true }, "outputs": [], "source": [ "A = 0.5 # parameter values from example in 4.3.10 of Ref. [1]\n", "ω = 3" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "More than showy, we're snazzy, and so we build a one-line function using the [`lambda`](https://docs.python.org/3/reference/expressions.html#lambda) keyword.\n", "It's just too cool.\n", "In Python, you can create a small function in one line using the assignment operator `=`, followed by the `lambda` keyword, then a statement of the form `arguments: expression`—in our case, we have the single argument `time`, and the expression is the sinusoidal driving.\n", "The sine mathematical function is avaible to us from the [`math` library](https://docs.python.org/3/library/math.html). Check it out." ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": true }, "outputs": [], "source": [ "from math import sin\n", "F = lambda time: A*sin(ω*time)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is really a function: we can call `F()` at any point in our code, passing a value of time, and it will output the result of $F(t) = A \\sin(\\omega t)$.\n", "\n", "Now, let's write the right-hand side function of derivatives for the driven spring-mass system (with damping). Notice that we use the lambda function `F()` inside this new function, and the `time` variable explicitly as the argument to `F()`. Some powerful Python kung fu!" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def drivenspring(state, time):\n", " '''Computes the right-hand side of the spring-mass differential \n", " equation, with sinusoidal driving and linear damping.\n", " \n", " Arguments\n", " --------- \n", " state : state vector of two dependent variables\n", " time : float, time instant\n", " \n", " Returns \n", " -------\n", " derivs: derivatives of the state vector\n", " '''\n", " \n", " derivs = numpy.array([state[1], 1/m*(F(time)-k*state[0]-b*state[1])])\n", " return derivs" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Here is where the power of our code design becomes clear: solving the differential equation via time-stepping inside a `for` statement looks just like before, with the only difference being that we pass another right-hand-side function of derivatives.\n", "The code cell below solves the driven spring-mass system with the same model parameters we used for the damped system without driving." ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": true }, "outputs": [], "source": [ "for i in range(N-1):\n", " num_sol[i+1] = euler_cromer(num_sol[i], drivenspring, t[i], dt)" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "image/png": 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dkM8Bfg6kA38EaoNwjqiQHQHF1mv3VgEwZWj7YcgzkuL53rnjAbjvjQ1sL60N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\nmVkBOCGbmZkVwP8DfnHjnKgxMKwAAAAASUVORK5CYII=\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig = pyplot.figure(figsize=(6,4))\n", "\n", "pyplot.plot(t, num_sol[:, 0], linewidth=2, linestyle='-')\n", "pyplot.xlabel('Time [s]')\n", "pyplot.ylabel('Position, $x$ [m]')\n", "pyplot.title('Driven spring-mass system with Euler-Cromer method.\\n')\n", "pyplot.figtext(0.1,-0.1,'$m={:.1f}$, $k={:.1f}$, $b={:.1f}$, $A={:.1f}$, $\\omega={:.1f}$'.format(m,k,b,A,ω));" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "And our result looks just like [Fig. 4.28](https://link.springer.com/chapter/10.1007%2F978-3-319-32428-9_4#Fig28) of Ref. [1], as it should. You can see that the system starts out dominated by the spring-mass oscillations, which get damped over time and the effect of the external driving becomes visible, and the sinusoidal driving is all that is left in the end.\n", "\n", "##### Exercise:\n", "\n", "* Experiment with different values of the driving-force amplitude, $A$, and frequency, $\\omega$.\n", "* Swap the sine driving for a cosine, and see what happens." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "An interesting behavior occurs when the damping is low enough and the frequency of the driving force coincides with the natural frequency of the mass-spring system, $\\sqrt{k/m}$: **resonance**.\n", "\n", "Try these parameters:" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": true }, "outputs": [], "source": [ "ω = 1\n", "b = 0.1" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": true }, "outputs": [], "source": [ "for i in range(N-1):\n", " num_sol[i+1] = euler_cromer(num_sol[i], drivenspring, t[i], dt)" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "image/png": 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JMYRE5PVYz8prlVIZPaOUUhuw0nk+JiIn2fuqx4qafgzrGs4bpdSDWKlT12uDRiz+H9Y8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\n3hEOdQ+xIMvI746BUcZiceoriigt8teb4URtbm0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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig = pyplot.figure(figsize=(6,4))\n", "\n", "pyplot.plot(t, num_sol[:, 0], linewidth=2, linestyle='-')\n", "pyplot.xlabel('Time [s]')\n", "pyplot.ylabel('Position, $x$ [m]')\n", "pyplot.title('Driven spring-mass system with Euler-Cromer method.\\n')\n", "pyplot.figtext(0.1,-0.1,'$m={:.1f}$, $k={:.1f}$, $b={:.1f}$, $A={:.1f}$, $\\omega={:.1f}$'.format(m,k,b,A,ω));" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As you can see, the amplitude of the oscillations grow over time! (Compare the vertical axis of this plot with the previous one.) Our result matches with [Fig. 4.29](https://link.springer.com/chapter/10.1007%2F978-3-319-32428-9_4#Fig29) of Ref. [1]." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Solutions on the phase plane\n", "\n", "The spring-mass system, as you see, can behave in various ways. If the spring is linear, and there is no damping or driving (like in the previous lesson), the motion is periodic. If we add damping, the oscillatory motion decays over time. With driving, the motion can be rather more complicated, and sometimes can exhibit resonance.\n", "\n", "Each of these types of motion is represented by corresponding solutions to the differential system, dictated by the model parameters and the initial conditions.\n", "\n", "How could we get a sense for all the types of solutions to a differential system?\n", "A powerful method to do this is to use the _phase plane_.\n", "\n", "A system of two first-order differential equations:\n", "\n", "\\begin{eqnarray}\n", "\\dot{x}(t) &=& f(x, y) \\\\\n", "\\dot{y}(t) &=& g(x, y)\n", "\\end{eqnarray}\n", "\n", "\n", "with state vector\n", "\n", "\\begin{equation}\n", "\\mathbf{x} = \\begin{bmatrix}\n", "x \\\\ y\n", "\\end{bmatrix},\n", "\\end{equation}\n", "\n", "is called a _planar autonomous system_: planar, because the state vector has two components; and autonomous (self-generating), because the time variable does not explicitly appear on the right-hand side\n", "(which wouldn't apply to the driven spring-mass system).\n", "\n", "\n", "For initial conditions $\\mathbf{x}_0=(x_0, y_0)$, the system has a unique solution $\\mathbf{x}(t)=\\left(x(t), y(t)\\right)$. This solution can be represented by a planar curve on the $xy$-plane—the **phase plane**—and is called a _trajectory_ of the system." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "On the phase plane, we can plot a **direction (slope) field** by generating a uniform grid of points $(x_i, y_j)$ in some chosen range $(x_\\text{min}, x_\\text{max})\\times(y_\\text{min}, y_\\text{max})$, and drawing small line segments representing the direction of the vector field $(f(x,y), g(x,y)$ on each point.\n", "\n", "Let's draw a direction field for the damped spring-mass system, and include a solution trajectory. We copied the whole problem set-up below, to get a solution all in one code cell, for easy trial with different parameter choices." ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": true }, "outputs": [], "source": [ "m = 1\n", "k = 1\n", "b = 0.3\n", "\n", "x0 = 3 # initial position\n", "v0 = 3 # initial velocity\n", "\n", "T = 12*numpy.pi\n", "N = 5000\n", "dt = T/N\n", "\n", "t = numpy.linspace(0, T, N)\n", "num_sol = numpy.zeros([N,2]) #initialize solution array\n", "\n", "#Set intial conditions\n", "num_sol[0,0] = x0\n", "num_sol[0,1] = v0\n", "\n", "for i in range(N-1):\n", " num_sol[i+1] = euler_cromer(num_sol[i], dampedspring, t[i], dt)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To choose a range for the plotting area of the direction field, let's look at the maximum values of the solution array." ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "4.0948277569088525" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "numpy.max(num_sol[:,0])" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "3.0" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "numpy.max(num_sol[:,1])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "With that information, we choose the plotting area as $(-4,4)\\times(-4,4)$. Below, we'll create an array named `coords` to hold the positions of mesh lines on each coordinate direction. Here, we pick 11 mesh points in each direction.\n", "\n", "Then, we'll call the very handy [`meshgrid()`](https://docs.scipy.org/doc/numpy-1.13.0/reference/generated/numpy.meshgrid.html) function of NumPy—you should definitely study the documentation and use pen and paper to diligently figure out what it does!\n", "\n", "The outputs of `meshgrid()` are two matrices holding the $x$ and $y$ coordinates, respectively, of points on the grid. Combined, these two matrices give the coordinate pairs of every grid point where we'll compute the direction field." ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": true }, "outputs": [], "source": [ "coords = numpy.linspace(-4,4,11)\n", "X, Y = numpy.meshgrid(coords, coords)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Look at the vector form of the differential system again… with our two matrices of coordinate values for the grid points, we could compute the vector field on all these points in one go using array operations:" ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": true }, "outputs": [], "source": [ "F = Y\n", "G = 1/m * (-k*X -b*Y)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Matplotlib has a type of plot called [`quiver`](https://matplotlib.org/examples/pylab_examples/quiver_demo.html) that draws a vector field on a plane. Let's try it out using the vector field we computed above." ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "image/png": 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v6ygcx5Vjn332Gfz9/XHkyBFYW1uLx6XupQFAz549xXZ//PFHxMXFYceOHao5\nD5JZuXIl6tSpg5iYGPz444+lPl+FLmivX7/GyJEjMWvWLPTs2ZN1HI7jyjlBENC/f3/cvXsXa9as\ngbGxMfz8/BAQECBZhpycHOzcuROxsbEQBEE8/ujRI1y+fFmyHABgbGyMrVu3AgD++OMPnD9/vlTn\nq7AF7c6dO2jbti327duHAQMGoFKlSqwjcRxXQejp6WHWrFl4+PAhJk2ahCVLlkjWtq6uLszNzXHo\n0KEiPTJPT0/Jcqj069cPw4YNAwBMmDABmZmZSEtL+6DiKkjdxXwbW1tbCgoK0ng7O3fuxOTJk5GZ\nmQkA8PX1Re/evTXeLsdxXHFu374NS0tLmJiYSNZmbGwsRo8ejZMnT4rHKleujJiYGBgZGUmWAwDi\n4uLQtGlTvHr1Shx6TExMxB9//AEAEAQhmIhs33miklxok+qh6Ukh6enpNHr06AIXZKtXr065ubka\nbfd9BQQEML3dA8dxFYNCoaANGzYUWIC+fft2Jln+/PNP8dY7Ojo6VKtWLXEnEfBJIQVFRESgffv2\n2LlzZ4HjQ4cOhba2NptQhURERMDJyQlHjhyBlpYW6zgcx5VzMpkMrq6uuH79OmxsbABIP+yoUCjg\n6emJiIgIVKpUCUqlErm5uYiOjkZoaOh7natCFLTMzEz89ttvqFy5cpHnRowYwSBRQfHx8ZgyZQqa\nNWuG8+fPq2W2D8dxXEm1bNkSQUFB+Pbbb3H58mU8ePBAsra1tLRgbW2NrVu3IiMjo8Bz//zzz3ud\nq0IUNAMDA2zYsAGTJk0qcLx+/fpo3749o1RAVlYWVq9ejQYNGmDLli1QKBT44YcfYGZmxiwTx3EV\nU6VKlbB161YcOXIEx44dk7Ttzp074+rVq2jYsGGB47ygvUFERASmTp0KAJg0aRKMjIwwfPjwAtNW\npUJEOHDgAGxsbDB79mykpqYCAExMTDBjxgzJ83Acx6n0798fU6ZMkbzdhg0b4urVq3BwcBCPBQYG\nIjExscTnqBAFLSsrC0OHDkVGRgbat2+PTZs2YcaMGRg+fLjkWSIiImBvb49hw4bh2bNnBZ5zc3ND\n1apVJc9UHLlcjjNnzmDy5Mm4ffs26zgcx0nIwMCASbvVqlXDqVOnMGrUKACAUqnEqVOnSvz+ClHQ\n3NzccPv2bRgbG8PLyws6OjqYM2eOeBFUSo0bN8a2bduKLOSuUaMGpk+fLnme/HJzc+Hn54cJEybA\nwsICjo5Ntc6OAAAgAElEQVSOsLKyQosWLZjm4jiu4tDV1YWnpyeWLVsG4P2GHcvG9D4N+uuvv8RN\nQX///XfUq1cPQN7mmCwIgoCYmJgiK+Lnzp1b7KQVTcvJycHZs2fFDUKTkpLE5wYNGgQ3NzfJM3Ec\nV7EJgoD58+ejQYMG7/UZVK4XVj9//hwtW7ZEUlISxo8fLy7SY+n8+fPo06cPsrKy4OjoCBMTE1y5\ncgWPHj2SvMh6eHjAzc0NycnJRZ6zsbHBtWvXJF9gyXEcl19gYCDs7e1LtLC63PbQ5HI5RowYgaSk\nJNjY2ODXX39lHQlXrlxBv379kJWVBQcHBxw9ehSXLl1Ct27dmPQYx40bh8qVK2PkyJFQKpXicWNj\nYxw5coR5MVMqlUhLS0NKSgqSk5ORkpKCnJwcdO3aFTJZhRgt57gKz87OrsSvLbcFbdmyZbh06RL0\n9PRw4MAB5ns1Xr9+HX369EF6ejrs7Ozw999/o1KlSujevXuBYiKloKAguLu7F2l/9+7daNy4saRZ\n1q1bh6NHj4qFKyUlBampqQX2mmvSpAm8vb15MeM4rngl2U5Eqoe6tr46f/48yWQyAkBbtmxRyzlL\nIyQkhKpWrUoAyNbWlpKTk5nmyc3NpSVLlpC2tjYBIFNTU2rdujUBoAULFjDJlJGRQV988UWRe0Wp\nHiNHjqS0tDRJssjlctq7dy9FRkZK0h7HcW+HEm59xbyI5X+oo6AlJCRQ7dq1CQANGDBA3AuMlTt3\n7pCZmRkBoBYtWtCrV6+Y5nn06BHZ2dmJhaJ3794UHR1NK1eupN69e0u+h6RSqaTr16/T1KlTxd9T\n/oe+vj5t375d8r/j999/TwCoYcOG9O2339LBgweZ/u1ycnJIoVAwa5/jWKqQBU2pVJKTkxMBoLp1\n61JiYmKpzlda9+/fJ3NzcwJANjY2FBcXxzTPzp07ydDQkACQgYEB/e9//xMLxfXr1yX9wM7KyqLl\ny5dTkyZN3tgra9KkCd2+fVuSPP/++y95e3vTrl27aNu2bbRo0aIieQRBoDZt2tDs2bPp1KlTlJGR\nobE80dHRBe6inpGRQQ4ODnT//n2Ntfk22dnZTNp9E9ZfVDlpVciClp2dTWPHjiUtLS26fPlyqc6l\nDvfu3aOaNWtSgwYNKDo6mnUcWrlyJQGgNm3aUHh4ONMsSqWS6tWrRwBIW1ubnJycyNvbmyZMmEAA\n6Ouvv5ZsiJGI6NSpU28srMU99PX1afjw4fTs2TON5Bk4cCAZGhrSwIEDaceOHRQbG0uNGjUifX19\nWrNmjaQ96adPn1Lt2rXp999/LxO9xKVLl9KcOXPKRFGLioqir776ipKSklhHIaVSSTt37qQDBw6w\njkJERJGRkTRv3jy13M2kQhY0lbt376rlPOrw4MEDioqKYh2DiPKuDW3bto1ycnJYRyEiIg8PD/r1\n118pNjZWPDZ8+HDasWOH5B9WwcHB1KpVK7K3t6fu3bvTF198QYIgFChiurq69OWXX9LevXspNTVV\nY1kUCgVZWloW6R3mv8VH+/btJft3/t1334ntdurUie7duyc+d+DAAUmL3JUrV8Qsrq6uTIuaQqGg\n5s2bEwDq378/8wK7bds2AkBmZmaUkJDANEtmZiaZmpoSANq1a1epz1ehCxr38WJ9jVHlxIkTBIB0\ndHSoX79+tHv3bkkn8+Tk5NC5c+do5syZ1KhRo2J7ibq6uvTzzz9r/H5+WVlZ9NNPP4kFVVdXlxYv\nXkxZWVlkZ2dHc+fO1Wj7ha1YsUL8HUyaNIlprzH/BLTVq1czy0GUNyytGvWYOHEi0yxERPPnzycA\nZG1tXeov0bygcVwprFy5kjw9PcvEUJJCoaA+ffq8cfizdevWFBoaqvEc4eHh1KlTpwLXOKtVq0YA\naNu2bRpvP7/169eLOUaPHs30hriqoXwtLS26cOECsxxE//9FDAAFBgYyzfLq1SsyNjZWy01DeUHj\nuHJi0aJFZGJi8taHubk5bdy4UeO9FYVCQX/88Ye4DEX10NLSIl9fX422XdiWLVvE9ocNGyb2AqKi\noiQd/lMqleTs7EwAyMLCgvn18oEDBxIAatmypcZ77++yePFiAkBWVlalmljECxrHcRqhUCho+fLl\nRXqKhoaGdOvWLUmz7NixQ7zWOWDAAMrKyqJJkybRiRMnJM2RlJRE9evXJwDUuXNnpoUkKiqKKleu\nTABo/fr1zHIQ5f1eVF9+StOL5wWN4zi1UyqVtHTpUtLX1y92+LN27dr0/PlzSTPt27ePtLS0CAB9\n8cUXZGFhQTY2NpIXlZs3b5Kenh4BoB9//FHStgtbs2aN+CXjxYsXTLMsXbqUAJClpWWBpSjvgxc0\njuM0JiMjg06ePEnff/892djYFChqLVq0oJSUFEnzHDp0iHR0dArkYLFLkIeHh9j+0aNHJW9fJScn\nh5o1a0YAaMiQIcxyEBGlpqaK11o3b978QefgBY3jOMk8e/aMtm3bRgMGDCBjY2Pq1auXZMtD0tPT\naf/+/UUW6ZuZmTHZZm7MmDEEgKpUqUKPHj0iIqKHDx9KnuPy5cvi78LPz0/y9vNTTZypVavWB21I\nwAsax3FM5OTk0MWLFyksLEyS9kJCQqhXr17FDoHOnj1bkgz5ZWRkUMuWLcWJGenp6dS6dWt6+vSp\n5FnGjh1LAKhBgwaUmZkpefsqaWlpVL16dQJAGzZseO/384LGcVyFcvHiRercuXORtXosCsnDhw/F\nKeu2trYEgNzd3SXPER8fLw73sWg/v19++YUAkLm5OaWnp7/Xe0ta0Ph9ODiOKxc6deqE8+fP4/Tp\n02jXrh2AvDuyz507V/IsDRo0wJYtWwDk3aYJADw9PSW/VZSZmRlWrVoFAFixYgUePXokafv5fffd\ndzA3N0dsbCy2bt2qkTZ4QeM4rtwQBAE9evTA1atXcezYMbRo0QL79+/H1atXJcuQkpKC3r1745tv\nvilwPDIyEufPn5csh8rYsWNhZ2eH7OxsTJkyJW9oDkBGRoakOSpVqiR+uVi1ahVev36t9jbUXtAE\nQegnCMI/giCcFQThqiAIvoIgtFB3OxzHcW8iCAKcnJwQEhKCAwcO4I8//hA/yDWtSpUq8PLywsCB\nA4s8t2PHDkky5CeTybB161ZoaWnh1KlT8PHxwZ07d/DTTz9JnmXSpEmoVasW4uPjsXnzZrWfX1D3\nH1kQhAQA04lo338/rwQwDkAzIop923ttbW1J1T3XJCKCIAgab4fjuLJBLpcDALS1tSVrk4iwY8cO\nTJ8+XewN6evrIyYmBlWrVpUsh8qMGTOwfv161KxZE3Xr1kVGRgZu374teY4tW7ZgypQpqFatGp4+\nfQpjY+N3vkcQhGAisn3X6zQx5HhRVcz+sxaAGYCeGmjrg2zfvp11BI7jJKStrS1pMQPyeonjxo3D\nzZs30apVKwBAVlYWDhw4IGkOAMjNzcW3334LY2NjxMTE4Nq1awgLC0Ns7Fv7GBoxbtw4WFpaIjEx\nERs3blTrudVe0IiocD8787//q6futj7Erl27sHv3btYxOI6rIBo3bozAwEDMnDkTgPTDjkqlEt9/\n/z0aN26M1NTUAs/5+/tLmgUA9PT0sGDBAgDA2rVrkZycrLZzSzEpxA5AFoBjErT1VmfOnMH48eNR\nv3591lE4jqtA9PT08Msvv8DPzw9RUVG4e/euZG3LZDJs3rwZy5YtK/Lc2bNnJcuR3+jRo2FlZYXk\n5GRs2LABAODt7V3q82q0oAl5F6oWAlhARHFveM1EQRCCBEEIio+P11iWsLAwDBo0CHK5HNbW1hpr\nh+M47k169uyJ0NBQxMTESNquIAiYP38+tm/fDi0tLfH4mTNnJJssk5+uri4WLVoEAFi/fj3+/PNP\njB8/vtTLGtQ+KaTAyQVhBYA6RPR1SV6vqUkhL1++RIcOHfDixQsAwN69ezF8+HC1t/OhkpKSYGJi\nwjoGx3EVwIkTJzBkyBBxosqjR4+YfMmXy+Vo0qQJHj9+LB57UxaWk0JUAb4HYANgjKbaKInU1FT0\n7dtXLGYAykwPjYiwZMkShISEsI7CcVwF0bdvX5w7dw5mZmYA8nppUlu4cCEMDQ3x5MmTAsdLO+tS\nIwVNEITxAL4AMJSI5IIg1BcEoYcm2nqb3NxcDBkyBKGhoQWOl4WCprpQu2zZMnTo0IF1HI7jKpB2\n7dohICAA9erVY3IdbcmSJRg9enSR4c4yV9AEQRgGYD6AnwE0FwTBFoAjgI7qbuttiAjfffcd/Pz8\nChw3MjKCqamplFGKyM3NxahRo7Bx40a0a9cOlSpVYpqH47iKp2HDhggMDERKSorkW3IJgoAtW7Zg\n5MiRBY4X7ny8L0300P4EYAXgPIAb/z1+00A7b5WcnIy+ffsWWXNmbW3NdFF1ZmYmBg0ahD179gAA\nOnfuzCwLx3EVm7m5OQ4ePIjMzMx3v1jNZDIZPD09C+yoUuZ6aESkQ0RCMQ93dbf1NiYmJhgwYADu\n378PAGjZsiUcHR2ZDjeq9ng7fvy4eMzBwYFZnsIUCgX++usv1jE4jpOQkZERKleuzKRtbW1teHl5\noU+fPgCAx48fl2qPx3K9OXFqaip++y2vc+jm5ob9+/ejZ082G5bExcWha9euuHjxonhMS0sL9vb2\nTPIUdv/+fXTq1Inpbtwcx1U8urq6OHToELp06QIgb4nVhyrXBc3DwwOpqamwtLTEkCFDUK1aNUyc\nOFHyHFFRUejUqVOR2YytW7eGkZGR5HnyUygUWLNmDVq2bInQ0FCMHTuWaR6O4yoeAwMDHDt2DO3b\nty/VsGO5LWi5ubniCvTvv/8eOjo6TPP8/vvv+OqrrwocYz3ceO/ePdjb2+PHH39EdnY2vv76a74e\njuM4JoyMjODr61tg4ff7KrcFzcfHB8+fP4exsTHGjx/PNEvdunXRvHlzccal6uaDrAqaXC7HihUr\n0KpVK1y/fl08PnXqVCZ53iY3N5d1BI7jJGJiYoJx48Z98PvLZUEjIqxZswYAxB2mWVu4cCESExNR\nv359nD9/Hl27dkXHjpKuZACQNz7doUMHzJs3Dzk5OeLxrl27olmzZpLnKc6jR4+wYsUK9OvXDwkJ\nCazjcBwnodLMQpf2fgoS8ff3x61bt6CtrY3p06ezjoPQ0FBxcsr69ethYGAAHx8fye+JlJGRIS4X\nKGzatGmSZins/v37OHjwIHx8fBAaGgptbW2cP38eNWvWZJqL47iPR7ksaL/88gsAYPjw4ahduzbT\nLESEadOmQalUolevXnBycgIAJou7K1WqhFWrVuGXX35BcHCwePyTTz4Rc0kpPDwcPj4+OHjwYJGZ\nTRs2bMDnn38ueSaO4z5iRFRmHm3atKHSun37NgEgAHT79u1Sn6+09u3bRwBIR0eHIiIiWMeh48eP\nkyAIBIAGDBhAAGjVqlWSZlAqlfTTTz+Jf6fCj2+++YaUSqWkmc6dO0d//fUXXbt2jSIjIyk7O1vS\n9ovz9OlT1hE4rkwAEEQlqCHMi1j+hzoK2qhRowgA9erVq9TnKq20tDSqXbs2ASA3NzfWcej27dtk\naGhIAGjYsGGkVCrp888/p4SEBCZ5duzYUaSYtWrVijIyMiTPEh4eLv5uVI9q1apR06ZNqVu3bjR8\n+HA6fPiwpIW2W7du9NNPP1FOTo5kbb7Lq1evWEfgKqAKWdBiYmJIR0eHANDp06dLdS51WLZsGQEg\nCwsLSklJYZpFLpeTjY0NAaC2bduKRSM6OppJnvDwcNLT0ytSQFj1Ss6fP0916tQptsfYrFkzOn78\nuKTF7MSJE9SpUycCQC1btqTg4GDJ2i7OpUuXxJ71kSNHmGbJzMykpKQkphnyY/EF7G0UCgXrCCJ1\n/TdTIQuaUqmkM2fO0JQpUyQfsipOWloazZs3j/bt28c6ChERXb9+ndq1a8esiOWnVCrJxcWFOnbs\nSG3btiWZTEanTp1ilufSpUtFCpmVlRXt3r2b5HK55HnatGlTIIuWlhbNnz+fsrKyJM+ye/duAkBj\nxoyh7t27k5mZGf3777+S5yAievLkCbVu3Zp69erF5O+SX3Z2Nrm7u5OZmRlFRUUxzUKUN2zetWtX\nWrhwIesolJKSQrNmzSIbGxu1FPwKWdC4dysLhV4lLS2N5HI5ubq60ooVK5hmyczMpM2bNxMAql69\nOm3cuJFJ8SDK+xtNmzat2N5i06ZN6dq1a5LmWbp0aZEcTk5OTP4tBQQEkLa2NgEgd3d3ydvPT6FQ\nUOvWrcXrvqxt375d/PebmZnJNEtubi5Vr16dAJC3t3epz8cLGvfRuHv3bpkotMuWLaMlS5ZQamoq\n0xyRkZFUt27dN06akclk5ObmJulQ15w5c4rk8PDwkKz9/H799VcCQIIgkJ+fn3iMBX9/f/H3wXpY\nODMzk8zMzAgA7dixg2kWIhK/lDk7O5f6XLygcdx7ys3NZR2BiPImXjx+/Pidj+TkZI1nSU9PJxcX\nl2ILq6GhIT1+/FjjGQpTKpU0ZMgQAkCmpqZ08uRJEgSBHjx4IHkWIiInJycCQF26dGH+xWzBggXi\ndVfWWa5evSrO8C7txDNe0DiOU4vs7Gzy8fGhHj16FClqnTp1YnItKzU1lZo0aSL2WAHQvHnzJM9B\nRHTv3j3S0tIiAHT8+HEmGVRevnwpDsmeO3eOaRalUkkNGjQgALR169ZSnaukBa1cbn3FcZz66Orq\nwsXFBadPn8bDhw/h5uYGMzMzAMClS5ewbt06SfMkJSXh4sWLaNu2LQCId1vetWsXFAqFpFkAwMbG\nRryLh5ubG+RyueQZVGrVqoWhQ4cCAH799VdmOYC8LaxGjBgBANi7d680jZak6kn14D00jvs4ZGVl\nkZeXFzk4OJCenh6FhoZK1nZaWhqNGTOm2GFQX19fyXLkFxsbS0ZGRgV6I6yWN1y/fl28xvjkyRMm\nGVQePHgg/m1KsyQHvIfGcZym6OnpYdiwYTh//jxu3bqFkJCQvGsYEjA0NMSOHTvg5eWFKlWqFHhu\nx44dkmQorEaNGpgzZw4AYNGiRTh69ChGjx7NpMfYtm1b2NnZgYiwefNmydvPr2HDhuLdRfbt26fx\n9gSp/hGWhK2tLQUFBbGOwXHcR+LZs2cYMWIEAgICAOQNj0ZHRzPZKzUzMxONGjXCixcvIAgCiAhh\nYWFM7mLh7e2NoUOHwtjYGC9evGB6I+GNGzfC1dUVNjY2uHv37gftpi8IQjAR2b7rdbyHxnHcR8vK\nygoXLlzA4sWLIZPJkJOTI0lPoLBNmzahU6dOSE9PBwCxt3r16lXJswDAgAEDUKdOHaSmpmLXrl1M\nMqgMHToUWlpaCA8Px61btzTaFi9oHMd91LS1teHu7o4LFy6gbt268PT0lDzDd999h4YNGyIpKanA\ncVYFTUdHB1OmTAGQ10NSKpVQKpVF8knB3Nwcjo6OAPDG21epCy9oHMeVCx07dkRoaCgaNmyIkJAQ\nSdvW1tbGnj178M033xQ4zqqgAcCECRNgYGCAhw8fwtfXF+7u7jh16hSTLKrZjl5eXhq9rsgLGsdx\n5UbVqlWxf/9+1KlTR/K2tbS04OnpifHjx4vH7t27h5SUFMmzAHn3XPz6668BAJMnT8bSpUsRFRXF\nJEv//v1RqVIlxMTE4Ny5cxprp9wXNLlcDj7RhOMqDkEQUL16dSZty2QybNu2DZMnTwaQdy3txo0b\nkucICAiAvb292ENUFbLIyEjJswB5M1P79+8PQLNr0sp9Qdu5cycuXrzIOgbHcRWETCbD5s2b8cMP\nPwBgM+xob2+PHj164Pbt2wWOs+qhAcDIkSMBAIcOHUJmZqZG2ijXBS09PR2LFi1CfHw86ygcx1Ug\ngiBg7dq1mD17NgIDA5lk+Omnn8RdQ1RY9dAAwNHREdWrV0daWhqOHz+O169f459//lFrG+W6oK1b\ntw4xMTG8oHEcJzlBELBixQr06tVLskXnhdv39PRE+/btxWMse2ja2tpigd2+fTucnZ1x5swZtbZR\nbgtabGwsVq9eDQBlqqCx3OeN4zhpCYKA6dOnM2vfwMAAR48eRd26dQEAycnJSE1NlTxHWFgY3Nzc\nkJOTAwA4deoUzp07h7S0NLW2U24L2pIlS/D69WsAZaugsd4wlOM46X3I7hjqYmFhgePHj8PQ0BAA\nm15as2bN8OTJE/z+++8FjvOCVgL379/Htm3bxJ/LSkG7du0aNmzYwDoGx3EVTIsWLbB//37IZDIm\n19EEQYCHh4fYU1RRd2+xXBa0uXPnFli8VxYKmlwux6RJkzQ2u4fjOO5t+vbti3Xr1jG7jmZiYgIv\nLy9oaWmJxz6aHpogCLqCIKwQBEEuCIKVptop7MqVKzhy5EiBYykpKeLYLSu//vorQkNDkZ2dzTQH\nx3EV1/Tp09GlSxdm7dvb22PZsmXizx9FQfuvgF0AUAuA1ltfrEZEhAMHDsDNzQ3W1tYAAH19fQBA\nQkKCVDGKiIqKwqJFiwCgTBU0IsLLly9Zx+A4TiKCIMDGxoZphh9//BE9evQA8JEUNACGAL4GIOku\noYIgYOPGjVi5ciXi4uIAAH/99RdcXFyYDjtOmzYNGRkZAIDc3FzxDrusrV+/vsjCS47jOE2SyWT4\n888/UaNGjY+joBHRHSJ6pIlzl0RERATS0tKgra2NTp06wdvbGw0bNmSS5ejRozh27FiBY6yHPwEg\nMDAQs2fPRtWqVVlH4TiugrGwsMCePXvEmejqUi4nhVy/fh1A3sweAwMDCIKASpUqSZ4jLS0N06ZN\nK3Kc9bDjq1evMHToUMjlcl7QOI5jwtHREa6ursjNzVXbOZkXNEEQJgqCECQIQpC6hgWvXbsGAOKt\nv1lZvHgxUlJSxPUfKiwLmlKpxDfffIPnz58DAC9oHMcxs2TJErWu0WNe0IjodyKyJSJbde2Qreqh\n5d/yRWpEhClTpiA5ORnVqlUDkDdtFmBb0NasWVNg/7SyUtAUCgWTOw1zHMeOjo4OtLW11XY+9Z2p\njMjMzBQnOrDsoQmCAGtra9y/fx9RUVEQBAG7d+/GyZMnmU0KuXz5MubPny/+rKenBwMDAyZZ8svJ\nycE333zD7Donx3HlQ7kraCEhIZDL5TAyMkLjxo1Zx8Hp06cBALa2tqhWrRqGDx/OJEd8fDyGDh1a\nYMF5Weidpaenw8XFBSdPnmR2N12O48qHclfQVNfP2rZtW2BFOiuqgubo6Mgsg1KpxMiRIxEdHV3g\nOOuClpSUhH79+iEgIADa2tqwt7dnmofjuI+bphZW6wqCcB6AauPC/YIgHNZEW4Wprp+xnhAC5K05\nU91uvGfPnsxypKSkYPHixTh79qx4TFtbm2lBi4mJgYODAwICAgDk9WArV67MLE9hr1+/ZrIrOcdx\nH05T69ByiKgLEX1GRAIRdSCigZpoq7CyMCFE5dq1a0hLS0PlypVhZ2fHLIeJiQns7e3F4vr5559j\n7969MDU1ZZLnyZMn6NixI8LCwsRjDg4OTLLkl5aWBi8vLwwcOBDDhw9nstSD47gPV66GHOPj4/Hk\nyRMAZaOHphpudHBwgK6uLtMsSqUSu3fvBgCMGjUKQ4YMwWeffSZ5DoVCgd27d8Pa2lr8WwFA586d\nJc+ikpOTg+nTp2Pnzp3Izs5G3bp1ERwcrNbZV+8rJiYGVapUKVNFNS4uDjVq1GAdg+PeiPm0fXWK\njIyEubk56tSpg1q1arGOg1evXkFbW5vp9TOVFy9eQFtbG/r6+hgyZAgAoFGjRpLn0NLSgru7O1xd\nXaGrq4vq1atDJpOhY8eOkmdR0dXVhb29PbKzs6Gnp4fDhw/DzMyMWR4A8Pb2RsuWLXHx4kWmOQAg\nIyMDZ8+ehbOzs1oXwX6o8PBwREREAAD+/fdfplmys7PF6/ZlwcuXL5n/TlRycnJw//59aRslojLz\naNOmDZWWUqmkV69elfo86pKamkopKSmsYxBR3u8mIiKCdQzRo0ePKDIykpydnVlHoYyMDJoxYwZ5\nenqyjkJERD179iQABICmTp1KaWlpzLJMmDBBzPLXX38xy0FEtHz5cgJAgwYNokuXLpGLiwuzLM+f\nPycjIyOSyWSUkJDALIeK6u+0cOFC1lEoIiKCdHR0SE9Pj3Jyckp9PgBBVIIaIuS9tmywtbWloKAg\n1jE4iaWlpcHIyIh1DKSmpsLY2Jh1DMjlchw9erTAekUrKyvJh9GJCD/99BPWrFkjbq795Zdf4ujR\no5LmyC8wMBD29vaQyWQwMTFBnTp1cOvWLSZZiAjW1tZ4+vQp/vzzT4wcOZJJDpW1a9di1qxZ6Nq1\nK/z9/ZlmycnJgaGhIXJzcxEWFoZmzZqV6nyCIAQTke27XleurqFxH6eyUMwAlIliBuTNQHVxcWEd\nA4IgYNGiRTAyMsLcuXORm5uLEydOIDY2Fubm5pJmef36NQYNGoT4+HgIggClUolXr14VWFcpNUEQ\n4OTkhI0bN+LYsWPMC5pq2P7atWvIzc2Fjo4Osyy6urr49NNPcevWLdy6davUBa2kytU1NI7j1Esm\nk2HmzJm4evUqGjVqBLlcjj179kiew9DQEOvXr8fjx4+Rf1QpOTmZ6fIKZ2dnAMDJkyeZ30WjVatW\n0NfXR0ZGBrNea36qSWchISGStckLGsdx79S6dWsEBwdj3Lhx8PT0BItLFU2bNoW3t3eRDROioqIk\nz6LSqVMnGBsbIy0tDRcuXMD9+/fF2c1S09XVFZcrXblyhUmG/Fq1agUAkhZXXtA4jisRQ0NDeHh4\nYPHixXj0iM3tDnv16oWNGzcWOMaqoBERdHV10adPHwB5N8zt3LmzeCcLFlTDjpcvX2aWQUXVQ7t1\n65ZkX4B4QeM47r0MHjyY6UbSkydPxvTp08WfIyMjmeTYs2cPmjdvLm4Q4Ovri7i4OCa9V5XPP/8c\nQF4PjfWEv5YtWwIAEhMT8eLFC0na5AWN47iPztq1a8WeEase2siRI1GrVi3cu3ePSfvFsbOzgyAI\n+OyhXTcAACAASURBVPfffwtsXMBClSpVUK9ePQDSXUfjBY3juI+OtrY29u/fj08//ZRZD00QBPzx\nxx9FZumy7BlVrVpVnFGouo6WmZnJLI/U19F4QeM47qNkbGyMv//+G1lZWcwy1K1bF2vXri1wjPVQ\nX/7raIcPH8aSJUuYZcl/HU0KvKBxHPfRsrKywi+//MI0w/jx49GjRw+mGQDA398f8+bNE3/28vLC\noEGDmK5H4wWtlFhO4eU4Tnr169dn2r5q6NHQ0BAAux5ax44dcfDgQWzduhVA3mJ0IO9uG6yoCtrT\np0+RnJys8fbKXUFzd3dHSkoK6xgcx1UgVlZWWLNmDQB2BU1XV7fI8CfA9ka+derUEW9TFRoaqvH2\nylVBk8vl+Ouvv3Dy5EnWUTiOq2AmTpyIbt26Mb2G1q9fvyJ392BZ0ARBKDDsqFAoNFrYylVBCwwM\nRGJiIo4fP846CsdxFYxMJoOHh4c49MiCIAhYt24dZLL//2hnVdDCw8ORnJwsFrSzZ8+iT58++Pvv\nvzXWZrkqaKpflK+vL+RyOeM0HMdVNPXq1cPQoUOZZmjWrBkmTZok/syqoCUkJMDMzAz79u0DABw/\nfhynT59G7dq1NdZmuSpoqp5ZYmIiAgMDGafJuzszH/7kuIqF5axClSVLlqBKlSoA2BW0Tp06oUeP\nHoiJiSlwnBe0Enj8+DHCw8PFn8vCsGNgYCA8PT1Zx+A4roIxMzPD4sWLAbC9hrZs2bIix3hBK4HC\nBawsFLTjx4+XidtKcBxX8UyZMgVNmjRhep8/W1tb9O/fv8AxXtBKoHABi4iIYLYjuMrx48eRmpqK\nS5cuMc3BcVzFo6uri23bthW53Y7Uli5dCkEQAACVK1fWaIEtFwUtJSUFFy9eLHJck7Np3iX/EGhZ\n6C0CQEZGBusIHMdJqHPnzqwjoFmzZvjqq68A5PXOVMVNE8pFQbty5QpmzJiBdevWAQA6dOgAb29v\nPHv2jFmm/MX0+PHjzPd3A/J2KGe57x3HcRWTu7s7tLS0NDrcCJSTgtanTx+sWrUK1atXB5C3HmTw\n4MFYv349s0z5e2VPnjxBREQEsywqXl5euHDhAusYHMdVMA0bNsTo0aN5QSuJN3VhNdm1fZuUlJQi\nhYP1sKNqCJR1Do7jKqaFCxdqfN/NclHQyho/P78iC7tZFxJV+2Vl+JPjuIrlk08+wbRp0zTaBi9o\nGnD8+HHUqlVL3AKnSZMmCAgIwKtXr5hmAvLuRqC6ZTxr+dcNchxX/pmZmWn0/LygacAXX3yBx48f\no2bNmgCADRs2IDAwELGxsUzyFJ4Fyrq3CACPHj3Chg0bWMfgOK4c4QVNA7766ivo6+sXONauXTs0\nbdqUSZ7CQ6AslzOoHD9+HH///Tcf/uQ4Tm14QasACvfIrl27hri4OEZp8vz999+Ijo7GzZs3mebg\nOK780FhBEwTBWRCEG4IgXBQE4YogCLaaaot7M7lcjn/++afAMSLCiRMnGCUqOARaFoY/AXY3ZeQ4\nTn00UtAEQWgDYB+AUUTUGcAKAH6CIFhooj3uzQIDA1G3bl2MHz8eQN56kO+++w6nTp1ilunkyZPi\nEGhZGP4E8tboKZVK1jE4jisFTfXQ5gLwI6J7AEBEfwOIBTBFQ+1xb9C0aVPcvHkT3bt3B5C38/aW\nLVuwadMmZpny98qCg4MRHR3NLIvK9u3bcf36ddYxOI4rBU0VtB4AggoduwHAsZjXchpkampa7AJz\nTU+ffRO5XA5fX98Cx1j30lRDoGVl+JPjuA+j9oImCEI1AFUAxBR66l8Aml0mzpV5gYGByM3Nhbm5\nOYC8IVDWhUQ1BMq6sKo8ePAABw4cYB2D4z462ho4Z+X//m92oePZACppoD1R06ZN4ebmhk8++UST\nzZTY+PHjkZCQACsrK9ZR0KRJE7i5ucHS0pJpjipVquDp06c4fvw47t27h5kzZyIiIgJExGyrMmtr\na2zYsAG1a9dGTk4OdHV1meRQmT9/Pg4ePIhmzZrh008/ZZolPDwct27dgoWFBezs7IosR5HKo0eP\nULNmTZw9exYRERH45ptvEBcXh+bNmzP7d/P48WMcOnQI1apVE69Rs7Rnzx5ER0ejX79+zJYIqTx7\n9gze3t6oWrUqJk6cKF3DRKTWB4BqAAjA6ELHVwGIK+b1E5E3PBlUt25d4riK7ubNm7R582bWMYiI\nyMvLiwCQtbU1KZVKZjmuXLlC1atXp08++YQAUNu2balz587M8hARnThxggBQ48aNmeZQ6dChAwGg\nvXv3so5Cp06dEv/dqAOAICpB/VH7kCMRJQJIBlB4RqMFgMfFvP53IrIlIlvVbvkcV5G1atUKU6aU\njflTvXr1gpaWFpycnJj1hACgffv2ICJERkYCAG7cuIF+/foxy8OVTZqaFHIGQOF1Z7b/Hec47iNh\n8n/snXdYU9cfhz+HsGU4QNFaUFHcgoKDohWBVqtgW0Vr1da9996KVls3tXXiqD+1rqq1itUq4rYV\ncU/cs0hB2TvJ9/cHzS0BVNTkngDnfZ77lHtN7nlJaD45u0wZtGjRAgEBAVw9FAoF2rVrp3WNV6Cl\npqYWOG8xNTWVg40gN/oKtLkAWjPGagMAY6wtgIoAlumpPIFAoCe6d++O5s2b89bQCjBnZ2fUqlWL\ni8f169fRsmVLnDhxAgCQkZGBOXPmoHv37lx8BP+hj0EhIKJzjLFuADYwxtIBKAC0JqJn+ihPIBDo\nj549e0KhUPDWwMcffwwTExNkZ2dzbQJ1d3fH7du3pUB7+PAhpk6divnz53PxEfyH3pa+IqI9RNSY\niD4kIi8iOquvsgQCgf4whDADABsbG3h7ewMA1yZQIyOjAps7eTfLCsTixAKBoAgREBAAGxsb7k2g\neQPN2dkZNWvW5GQj0CACTSAQFBkCAgLQpk0b7nMF/fz8YGZmJp3zbAL9448/kJ2drXXt4cOHBrOR\nr5yIQBMIBEWGKlWqYMKECbw1UKpUKWl9VIBvc+Pu3btRs2ZN3L9/HwCwePFi1KhRAyYmJtyceFHk\nAy0xMZG3gkAgkJFGjRrxVgDwX4jZ2tqiRYsWXD3u37+PmJgYADkLfjs5OZXIJtAiHWgZGRn45ptv\neGsIBIISiKYfrU2bNlxrQz4+PrC01F5VkPdEeF4U6UD7+++/sWzZMoPYfkQgEJQsKleuDDc3N+4r\nlpibm+Ojj7Q3MimpIy6LdKA9ffoUGRkZmD17Nm8VgUBQAvn888/xySef8NbQCjBbW1tuo0DPnj2L\nGTNmID4+XroWERGBkSNHylJ+kQ80AFizZo3UISoQCARyMWLECJQrV463Btq2bSv9zLMJ1NXVFcHB\nwfjyyy8BAPfu3UPTpk1hZCRP1BSLQMvOzsasWbM42+R0xha0xptAICie2Nra8lYAAFSsWBGNGzcG\nwLe50dTUFK1bt4ZarQYA6fNQrmbZYhFoALBhwwZERUVxtAH27t2L3bt3c3UQCAQlk4CAACgUCu5N\noHkDVc5RoMUm0NRqNWbMmMHRBkhISMDYsWORmZl3b1OBQCDQLwEBAfDy8kLZsmW5erRt21ariVHO\nJtBiE2gAsG3bNly+fJmTDRAfH4979+5hyZIl3BwEAkHJxNXVFYMGDeKtATs7O3h6ekrncjaBFptA\nMzIyQqlSpTBt2jRuPgkJCQCA2bNnS5MceaJpxxYIBMUfxhi++OIL3hoA/gsxIyMjWZtAi2ygERHi\n4uIwduxY6fzx48cYMWIEMjIyuDhpAi05ORlTp07l4pCbpUuX8lYQCAQyYiiTqTWDQORuAi2ygZae\nno6wsDDMmzcPVlZWICLcunULPj4+MDc35+KkCTQAWLt2LS5evMjFQ8O8efMQGRnJ1UEgEJQ86tSp\ng6pVq8o+4rLIBpqlpaU0v8HV1RUAcOHCBa5OuQONiDBy5Eiuw/hVKhV69eolBqkIBAJZYYwhICBA\nBNrb0LBhQwDgXiPKHWgAcOzYMfz666+cbHL+qK5evSpWUhEIBLIzZMgQ2RdILhaB5ubmBoBvoKlU\nKiQlJUn7NNnb22P27NkICwuDSqXi5gUA3333Hc6fP8/VITo6Wkw6FwhKEC4uLrL36RWrQLt8+TK3\n8EhKSoKnpyeOHz8OAIiNjcWgQYOwfPlyblvYa/6YVCoVevbsiaysLC4eAHDp0iUsWLCAW/kCgaD4\nUywCrW7dulAoFEhPT8etW7e4OFhYWCA8PBxNmjTBe++9BwA4ffo0FxcNub8dXblyBXPmzOHm4uTk\nhIkTJ3JtghUIBMWbYhFo5ubmqFOnDgB+zY7m5uYwNzcHYwxeXl4AgJMnT3Jx0ZC3uv/tt99ye30c\nHR1BROjWrRvOnTvHxUEgEBRvikWgAYbRj6ZBs3XDqVOnOJv8h0KhQJ06dTBw4EBkZ2fLXn6pUqVQ\nrlw5pKeno3379vlWeREIBIJ3RQSaHtAE2tmzZ7kOmWeMoV27dgBy+tF2797Ntdbo6OgIIGdj1vbt\n2yM1NZWby6ZNm7iWLxAIdE+xC7QLFy5wH01Xv359WFlZITMzk2vzWv/+/fHbb7/B3d0dQM5uAMbG\nxtz2SnJycpJ+Pn/+PLp3785teS4iQqNGjbiP/hQIBLqj2AVabGwsoqOjuboYGxtLi3PyrBFNmzYN\nCoVCmty4Z88ebi7AfzU0Dbt378bkyZO5uHTs2BHR0dFo1qwZFi5cKNa9FAiKAcUm0MqWLSt9YBpC\ns6NmYIgh9KO1b98eQM5E78TERG4euWtoQM57tn79emzdulV2F0tLS3Tp0gXZ2dkYN24c2rRpw/2L\nkEAgeDeKTaABhtmPdurUKe5NoG5ubqhcuTKUSiUOHDjAzUPzhcPGxgYA0LJlSzx79gxdunTh4tOr\nVy/p50OHDqFFixa4e/eu7B7JyckIDw/XOu7cuSO7R0HwWug7L5mZmaIWLXgtxTLQeO6JpqFp06ZQ\nKBR4/vw57t27x9VFs64aABw9epSbh5OTE1q2bInff/8dAHD48GGuNcZmzZqhVq1aAHK2uTh06BCc\nnZ1l97C2tsaLFy/QpUsX+Pr6wtfXF8+ePZPdIy8ZGRno0qULHj16xFsFR44cwdq1a3H//n3uXxA1\nJCUlIT09nbeGhCENcsrOzubzPhGRwRzu7u70Ljx8+JCuXLlC2dnZ73QfXXH06FF6/vw5bw0iIoqK\niqLIyEhSq9XcHBISEiguLo6IiDZs2EDx8fHcXDTMmzePgoKC6LfffuOtQrGxsfTll19S9erV6eHD\nh7x1qG/fvgSAXFxcSKVScXWZOHEi2djYUOnSpaldu3Zc/441zJgxg2xtbWnBggW8VUilUlH58uXJ\n1dWVbty4wVuHfvrpJypTpgwNGzZMJ/cDEEmFyBDuIZb7eNdAEwjelJiYGEpLS+OtocWFCxd4K1BI\nSAgBIADk4OBAJ06c4Orj6ekp+bz//vs0cuRIOnfunOweL168oJSUFFKpVFS1alUCQCtWrJDdIy/n\nz58nAGRiYkLJycm8dWjQoEEEgAYOHKiT+xU20IpVk6NA8KaUL18eFhYWvDW00DSd8+LMmTP45ptv\nMHDgQBw+fBiPHz+W+oR5kJaWhrNnz0rnjx8/hkql4vI63bp1Cx06dEB4eDju378PMzMzg9gl+tCh\nQwCADz74AFZWVpxtcv6GAKBJkyaylmssa2kCgcDgKVOmDO7fv89tUe28/Pnnn1AqldJ5UFAQpk+f\nzmV35oSEBBw8eFAavezv74/MzEzcvn0bNWrUkN1HgybQPvroI24OGtLT06VxDE2bNpW1bL3U0Bhj\nDoyxvYyxB/q4v0Ag0B8uLi4GE2ZAznQTDT/88ANmzJjBJcwAID4+HsB/AzB27tyJihUr4vbt21x8\ngJwAOXHiBADDCLQLFy5AqVTC2tq66O+Hxhj7GMA+AIbzf4RAICiyHDt2DAqFAhs3bsSwYcO4uuTd\nxBcAJkyYgLZt23KwyeHkyZPIzMxEmTJlpFWBeBIREQEAaNy4sexfjPRRQ1MC8AYQoYd7CwSCEkRG\nRgYuX76MX3/9Fd27d+etky/Qmjdvzm1H+J9//hmpqalSc6Ovr69B1Kx59Z8BeuhDI6JwIP/WJQKB\nQPCmREVF4bfffsOHH37IWwWAdqDZ2dlh69atMDbmMxQhLCwMP/zwA5KSkgDkNDcqlUqkpqbC1taW\nixPwXw2NR6AVqVGOPLY9EQgE/GjQoIHBhBnwX6AxxrBp0yZpM18emJmZISIiAjdv3gQALF68GBUq\nVMDz58+5OcXGxkoLScg9IAQwgEBjjPVnjEUyxiJjY2Nf+diQkBDExcXJZCYQCHhjaC09mkEhkydP\nRuvWrbm6mJmZaZ1HRUVhzJgxqFatGicjSNMr3nvvPVSqVEn28gsVaIyx2Ywxes3h/TYCRBRCRB5E\n5GFvb//KxyYlJaFXr14Gs/SNQCAoWSQkJMDb2xtBQUG8VWBubq51XqdOHYwdO5aLi6b1TNPcyKN2\nBhS+hjYfwPuvOf7Uh2BurK2tERoaiiVLlui7KIFAIMiHubk5Nm/ezK3fLDd5a2grV66EqakpF5cR\nI0bgr7/+4jogBChkoBFREhE9ec2h962Zra2tAQDjx4/nunGmQCAomSxcuBAVK1bkrQFAO9B69+6N\nFi1acHPJzMxEy5YtpTmDsbGxGDhwICIjI2X14N6H9iZoAi07OxtffPGFNLqHByqVyiBWIRcIBPLB\nczWQvGgCrVy5cpg3bx5XF3t7e2RlZUm7DyxatAhXr16VfV6cPiZWN2GMHQXQE4ADY+woY2y6Lu6t\nCTQAuHv3LgYOHMitP02hUKBXr15ITk7mUr5AICjZaAJt4cKFsLOz4+pS0PiHBQsWyD6oR+eBRkQR\nRORNRFWIyPzfn2fp4t65Aw0AtmzZgvXr1+vi1m9FfHw8unbtCpVKxc1BIBCUTMzMzNCyZUv06NGD\nt0q+QOvYsSM8PT1l9yiSTY65GTp0KG7cuMHBJmcH5tDQUIwfP55L+QKBoORiY2ODlStXGsTUhtyB\nZmxsjO+++46LR5EKNBsbG61zR0dH9O/fH9u2bePS9Ojk5AQgZ0JjSEiI7OULBIKSy+effy7tuM6b\n3IE2cOBAbn2NRSrQrK2twRjDyJEjAQDPnj3DtGnTEBQUxOVbiqOjo/Tz4MGDERYWJruDhj/++APP\nnj3jVr5AIJCXvMP2eaIJNGtra0ybNo2bR5EKNCsrK4SEhGDx4sWoWrUqsrKysGPHDm4+mhoakDPq\nMTAwUFqGRm6sra3h7e2Np0+fcilfIBCUXDSBNmHCBJQvX56bR5EKNGNjY/Tt2xeMMXTr1g1AzorT\nvMhdQwOAxMRE+Pv7c1meq2HDhrh37x5atmwpphMIBAJZsbS0RM2aNTFq1CiuHkUq0HKjCbTjx49z\n+wDPXUMDAAsLC1hYWGDUqFGyj3y0sLCAm5sb7t69i5YtW+L+/fuyli8QCEo2y5cvh6WlJVeHIhto\ntWrVkibtbd68mYuDvb09zMzMpDkgKpUKYWFh2LhxI5d9iZo1awYAePDgAT788ENuu+gSkZifJxCU\nMHx8fHgrFN1AAyBt+Ldp0yYuoxyNjIzg5uaG06dPo27dusjKysLKlStl99CgCTQAePLkCVq2bMll\nSgNjDGPHjsWpU6dkL1sgEJRcinSgdenSBUZGRrh27RouX77MxWHnzp2oUaMGRowYAQBYsWIFMjP1\nvqxlgeSdyBgdHQ1vb29cvXpVdhcfHx98+OGHmDlzJpRKpezlCwSCkkeRDjQHBwf4+fkB4Dc4RLPB\nX7du3VC2bFnExMRg+/btXFyqVKmSb4SRkZERBgwYIPuQ/k8//RQ2NjYICgqCt7c3Hj58KGv5AoGg\n5FGkAw34b3DI5s2buS5BZWlpiQEDBgAAvv/+ey5NoIwxrWZHAJg7dy5OnToFBwcHWV3Mzc2l9+bU\nqVNwdXXFtm3bZHUQCAQliyIfaJ9//jksLCzw9OlTHD9+nKvL4MGDoVAocP78eW79R82aNYOHhwcm\nTJgAAJgyZQrS0tK4uPTq1Uv6OTExEV26dOGyoPPt27fxxx9/QK1Wy1quQCCQlyIfaNbW1vj0008B\n8J2TBgCVK1dGYGAgAHDbhNTb2xsbNmzAlClTUL58eTx9+hTBwcFcXBo1aoQGDRpoXVu/fj0aNWok\nbdUuB9WrV8ePP/6I6tWr47vvvkNMTIxsZQsEAvko8oEG/Dfa8ZdffkFGRgZXF83gkF27dnHpN/L0\n9ETt2rVhbW2NWbNyNjmYO3culw9xxphWLQ3ImerQt29fpKamyuqxZs0aJCUlYfLkyahcuTI6d+6M\nw4cPi1qbQFCMKBaB9vHHH8POzg5JSUnYt28fV5dmzZqhSZMmUKvVWLZsGVeXPn36oE6dOkhJScGM\nGTO4OHTr1g0mJibSeWxsLJydneHt7S2rh4ODA1atWgUAUCqV+OWXX+Dn54eaNWti4cKFXFZ3EQgE\nuqVYBJqJiQm++OILADlz0njCGJNqaatXr5a1JpIXY2NjLFiwQHK5fv267A729vZo37492rdvj/79\n+wPI2S7+1q1bsrt07NhRqs1ruHPnDsaNG4dq1arh4MGDencgIiQlJRXq4LV5bV4MxUMgeC1EZDCH\nu7s7vS2nT58mAGRqakrPnz9/6/vogszMTKpYsSIBoBUrVnB1UavV5OvrSwCoXbt2XBxCQ0PpzJkz\nlJ6eTo0aNSIAVK9ePUpJSZHdJT4+nipXrkwApMPe3p7u3r0rm8O+ffvovffe03LIe4wcOZLUarVs\nTgVx7do16tixI8XHx3P1ICK6f/8+bwUBRwBEUiEyhHuI5T7eJdDUajVVq1aNANCqVave+j66Yvbs\n2QSAatWqRSqViqvLhQsXiDFGACgsLEz28nN/MN+7d4/KlClDAKh79+5cPrTDwsLyBYi7uzs9efJE\nNoeEhATq169fgWFWqVIlioqKks0lLw8fPqSePXuSkZERNWrUiGJiYri5EOW8Vi4uLpSUlEQZGRlc\nXQR8KGygFYsmRwDSCvzu7u4oV64cbx30798fZcqUQYsWLbgNm9fg5uaGHj16oHbt2lr9WXKRe6+6\nqlWrYuPGjTAyMoKTkxOX5ixfX1+pWXjq1KmwtrZGQkKCrAur2traIiQkBIcOHcq3yPXff/+Na9eu\nyeaiIS4uDqNHj0aNGjWwfv16qNVqnD9/Hj169JDdRYNarcbXX3+NW7duYd68eXB2dkZUVBQ3n9ws\nXboU06dPN4iBRSqVCoMHD+byd1MQJ06cwIoVK+T/7CtM6sl1vEsNjYhIqVS+0/N1TXp6Om8FicTE\nRMrOzuatIcGzBkJElJaWRq6urpSenk43btyga9eucXNJSkqiwYMHSzW0oUOHUlJSkmzlJycn08yZ\nM8na2rrAGmOHDh1kc8nLrFmz8vl069aNi8uLFy+kFoU//viDjIyMCABt3ryZi09uli9fTgDI1tZW\n1r+dl+Hn50cAaMiQITq5H0pak6NA8KY8ffqUt4IWR44coWrVqtGxY8dkK/Off/6hQYMGkZeXFzk7\nO5OlpWW+ABk/frxsPrnZt2+f1FSuOerWrcvtA3v69Ol06NAhunHjBtna2hIA+vzzz7l3KcTFxVHZ\nsmUJAM2ZM4erC1FOF4fm/dLVF8XCBhojDk0+L8PDw4MiIyN5awgE3EhNTUVMTAyqVavGzSElJQXP\nnj1DdHQ0nj17hmfPnuGrr75C6dKlZXO4e/cuPDw8kJCQkO/f6tati927d6N69eqy+WRlZcHR0RG1\na9fGkydPcOfOHbi5ueHkyZMoVaqUbB4FMXjwYKxYsQLOzs64evUqzM3Nufp0794dP//8M9q1a4fQ\n0FCd3JMxdo6IPF77wMKknlyHqKEJBIKUlBSqX7++Vs1MoVCQn58frVq1issglU2bNmn5VKhQgR49\neiS7R14uXLggNX3u2bOHtw49evSIFAoFAaAjR47o7L4oZA3NWCfxKRAIBDqAiNCvXz9cuXIFCoUC\nvr6+6NSpEz777DNpI10e/PDDD1rnCoUC33zzDbp3744PP/xQVpfk5GRYW1uDiDB8+HCo1Wp88skn\n8Pf3l9WjIL7//nuoVCq4u7ujZcuWspcvAk0gEBgMy5cvR3x8PNauXYtPP/3UIEYsR0REICIiQuta\ndHQ0rKys4OHx+lYwXfPtt9/i008/xf3793HixAmYmJjg+++/1xpNzIOEhASEhIQAAMaNG8fFx2AD\njYi4v0ECgUBevv76awwZMoS3hhY//vij1rmLiwt++uknfPDBB1x8jhw5gj179iAxMREAMGrUKLi4\nuHBxyc3q1auRkpKCKlWqoGPHjlwcDHYeWmxsLH799VfeGgKBQEasra15K2jx7NkzaR8/IyMjjBs3\nDhcvXuQWZikpKYiMjMT169fx9OlTmJubo02bNrh79y7XneGzsrKkHUZGjRoFY2M+dSWDDTR7e3sM\nHToUBw4c4K0iEAhKKCEhIcjOzkbt2rVx+vRpzJ8/HxYWFtx8Tp8+rbWRcUZGBnx8fLBy5UooFApu\nXlu3bsXTp09RunRp9O7dm5uHwQYaYwy1atXC559/jmPHjvHWEQgEJYysrCysWbMGEydOxPnz59G0\naVPeSgVuYvzdd99h/vz5snfRpKenY+PGjSAiLFy4EAAwaNAgWFlZyeqRG4PtQwOABg0aIDw8HP7+\n/jh8+DCaNGnCW0kgEJQQbty4gZ07d6Jx48a8VSRyf7lnjGHlypXSLhZyc//+ffTr1w+PHj3ClStX\nYGpqimHDhnFx0WCwNTQA0m7HKSkpaN26NS5dusTNpaAJngKBoPji6upqUGGWnp4ujbY0MTHB1q1b\nuYUZkDP5PTMzE1OnTgUAeHt748GDB1w/p3UaaIwxM8bYYMbYMcZYOGPsHGNsNWPsrSaQuLq6Sj8n\nJCTgo48+ws2bN3Xm+yZs3LgRv/zyC5eyBQKB4MyZM8jKyoKlpSX27t2Lzp07c/W5e/eu1vnBDdRw\nFQAAIABJREFUgwfh6+uLjIwMTka6r6HVADAXQH8i8gHgBaA6gF1vc7M6derAyOg/xdjYWPj6+uLe\nvXu6cH0jmjdvji+++CLfEF6BQCCQg2PHjqF06dIICwtD69ateevkCzTGGDZv3sy1r1HXgZYOYBUR\nRQEAEWUAWA6gBWPs/Te9mbm5OWrWrKl17e+//4afnx+ePHmiC99C4+rqirJly2L48OGYOHFizsrO\nAoFAIBO3b9/G8ePH4enpyVsFQP5AW7JkCT777DNONjnoNNCI6C4RjctzOf3f/5q9zT1zNzsCwE8/\n/YSrV6+iQoUKb3O7t8bIyAg+Pj4AgHnz5qFHjx7Izs6W1UEgEJRcFixYgPr16/PWkMjdUjZq1Cju\nA0IAeQaFeAK4QER33ubJDRo0QIMGDTB48GAAOW+qmZkZl40q/fz8pJ83btwIf39/JCcny+6RkZGB\nI0eOyF6uQCDgR8WKFXkrSKhUKty/fx8A0LFjR2nYPm/0GmiMsfIA+gIY/IrH9GeMRTLGImNjY/P9\nu6enJ7Zt24bp06fD0tIS169f5zY4w9fXV+v84MGDaNWqFWJiYmT1MDc3R3BwMMaNG4esrCxZyxYI\nBIKnT58iKysLnp6e0g70BkFhluQHMBsF7GSb5/DO8xxTAEcB9C1MGVSI7WPGjx9PAKhmzZpcdl9W\nq9VUpUqVfL97tWrV6Pbt27K67Ny5kwBQo0aN6ObNm7KWLRAISjbh4eFUvXp1io2NlaU8FHL7mMLG\n6nwA77/m+FPzYMaYAsBmAL8T0Zq3StoCGDduHKysrBAVFYUtW7bo6raFhjGWr5Y2fPhwHDp0CJUq\nVZLVxd/fH3Z2djh//jwaNWqEtWvXioEqAoFAFpKSkrB//36uW/oURKECjYiSiOjJa45MAGA566+s\nA3CdiOb/e82PMfbOW/Da2dlh+PDhAICZM2dyWYxT04/WqlUrAMCmTZtgY2MDS0tLWT1MTU3x1Vdf\nAQDS0tLQt29fdO7cGfHx8bJ6ANBaW04gEBR/AgICZN0xvLDoo+FzKYCKAPYwxjwYYx4AOgNw1MXN\nx4wZA2tra9y9excbN27UxS3fCB8fH7Rv3x6///47nJ2d8eLFC0yYMEF2DwDo1auX1vmOHTvg6upa\n4Hpv+uTw4cOYOnWqWE1FICghGEyfWV4K0y5Z2AM5E6kL1cdW0PG6PjQN06dPJwBUpUoVysrKerPG\nWB0QHx9PREQHDhyQfr+TJ0/K7kFE5OHhke+1NjMzo23btsnmoFarqXnz5lS6dGmaPXs2JScny1a2\nQCAo/qCQfWg6DbR3PQobaPHx8VS6dGkCQCEhIW/x8uiOTp06EQCqV68el3BdtmyZVpjVq1ePUlJS\nZPeIiIiQHOzs7GjRokWUlpYmu4dAICh+FDbQDLTe+GpKly6NMWPGAABmz56NzMxMbi7BwcGwsrLC\n1atX8cMPP8he/pdffgkzMzM4OTnB1NQUV69exdy5c2X3aNy4Mbp16wYAiIuLw5gxY+Ds7Izly5eL\nqQUCgUAeCpN6ch2FraERESUmJlLZsmUJAC1btuwN8163BAcHEwAqVaoUPXr0SPbyv/zyS9q6dSut\nWLFCqiXt27dPdo+HDx+Subl5viZQJycnWrdunSxTLZRKJYWFhYlmT4GgGIHiXEMDABsbG4wbl7PK\n1pw5c7iu8Dx06FC4uroiNTUVI0eOlL38yZMnIzAwEAMGDED37t0BAN27d8eDBw9k9XB0dMTo0aPz\nXff19YWzs7MsHckKhQJJSUmws7ODj48P5s+fj0uXLokpDQJBSaAwqSfX8SY1NCKi5ORksrOzIwC0\nZMmSN3qurjl9+jTX2pGGlJQUqlu3LgEgDw8PysjIkLX8pKQkKl++vFYN7dNPP5XdQ1Nr1hwVK1ak\nnj170tatWykuLk5WF4FA8G6gOA8Kyc2CBQsIADk4OFBqauobP1+X9OvXT1o5hOeAiJs3b5KVlRUB\noIEDB8pe/qpVqwgAtWvXjhQKBQGgNm3ayPqaqNVqGjZsWIEjbhlj5OXlRXfu3JHNRyAQvD0lJtBS\nU1OpQoUKBIAWLVr0xs/XJXFxcVKNcerUqVxdtm/fLn2Ab9y4Udays7OzqV69evTw4UPatWsXmZiY\nEADy8fGRdQSmUqmkgICAfIHm7OxMV65ckc1DIBC8GyUm0IiIvv/+ewJA9vb2XIas52bdunUEgExM\nTLivsThixAgCQBYWFrJ/gOf+3fft20dmZmYEgLy8vCgxMVE2j5SUFHJ3d88Xal999ZUs69Ddvn2b\n2rRpQzVq1Hjl4e7uTg8ePNC7j0BQFClRgZaWlkaVKlUiADR37ty3uoeuUKlU1Lx5cwJAvr6+pFar\nublkZmaSp6cnASAXFxdZgyQvYWFhZGlpSQCoSZMm9OLFC9nKjo6OJicnJwJAderUkUKtXLlytGHD\nBr2/RykpKTRixAhijBXYBKpQKOjIkSN6dSgsd+/epblz51JUVBRvFYFAokQFGhHR0qVLCQCVLVuW\n6wc3EdGVK1fI2NiYANDmzZu5ujx+/FhqBg0MDOQasMePH5f69tzc3Oiff/6Rrexr166Rra0tHThw\ngMLCwsjZ2VkKFD8/P1n6006ePEkuLi4Fhpq9vT317NmTduzYQUlJSXp3yc2DBw9o/vz50qoz9evX\n5/p3okGtVnPvFxcYBiUu0DIyMqhy5coEgBYuXPjW99EV48aNIwD0/vvvc1lBJDeHDh2Sagc7duzg\n6vLnn3+Sra0tAaDWrVvLWnZ4eDjduHGDiHJq9RMnTpQGrVhYWNDjx4/17pCWlkZjx44lIyMjrUEq\nucPN1NSUNmzYoFePR48e0aJFi6hp06b5wtXNzY26detGHTp0oLZt29LYsWP16lIQ8fHx1KFDB7pw\n4YJ07cqVKzRp0iSDWYHmzp07dP78ed4aEpcuXeKtIJGYmKjTL2YlLtCIiDZv3kxLly6VfYh4QSQn\nJ1OXLl3o2rVrvFWIiGjOnDk0a9YsUqlUvFXo3LlzVLduXe59jEQ5HwJNmjShHj16yFruX3/9JTV/\nrl+/nnbv3k19+/YlBwcHAkDnzp3TeZkqlYrWr18vNUMX9mjevLnOXV5FREQEVa1alQDQtWvXKDg4\nmBo2bCj5yLlOaUGo1WpavXo1lSpVipydnQ1iEv/atWsJAM2ZM4e3CqnVaurcuTNVr15dZ3/HJTLQ\nBEUHQwhWDUqlksuHUkZGBk2ePJlmzZolXVOpVHT27Fm9NfllZ2fTmTNnaM6cOdSqVSsyNTXNF2Cu\nrq40c+ZMmj9/Pv3www/066+/6sUlL2q1mr7//ntpVKymfzF3TdbPz48OHz4si4+GP//8U/o5JiaG\n2rdvLzl5eHhwWR0oN5s2bZJq+b169eLeXLx69Wrp/dLVeyUCTSAoIsjdZ5ab1NRUOnjwII0fP57c\n3d2JMUa2trayTz5/8eIFffbZZwXWEGvXrk1z586VpUk4L+fOnSN7e3tSq9UUGhoqLRpgZGREU6dO\n5d6d8Msvv0ih37VrV1IqlVx9rl27RhYWFgSAJk+erLP7ikATCARvTFxcHO3YsUPW7ZDOnDlDVapU\nKTDMLC0ttWpIchIbGyuNju3atavkVK1aNTp16hQXp9zs2bNHGnzWsWNHWdZKfRVpaWlUr149AkCe\nnp46DXsRaAKBwKApqIkxd1Nj2bJlqVq1atSiRQvZm/Wys7PJz88vn1fv3r251ahzB8SBAwek5mJ/\nf3/KzMzk4pSbQYMGEQAqXbo03b9/X6f3LmygGUMgEAg4cO/ePVSoUAG//fYbbG1tUbp0aem/lpaW\nYIxxc5s6dSrCwsK0rvn4+OD777+HtbW17D5KpRJDhw7FqlWrcPToUXz22WfIysrCRx99hF9++QWm\npqayO+Vm586dWLFiBQBgzZo1qFKlCh+RwqSeXIeooQkEAt7s2LHjpSM+PT09KSYmRnantWvXEmOM\ntm3bRqVKlSIA1LJlS4OYp/fgwQNpw+UBAwbopQwUsobGch5rGJQpU4aePn0KS0tL3ioCgaAEcv36\ndTRt2hQpKSnSNWdnZ3Tq1AmdOnVCw4YNZa85ZmZmwsXFBY8ePZKuffDBB/jjjz9gZWUlq0telEol\nWrZsidOnT6Nu3bo4e/YsLCwsdF4OY+wcEXm87nEG1eSYkJAAb29v7NmzBw4ODrx1BAJBCSIxMRGf\nf/45UlJS4OLigk6dOiEwMBCurq5cmz/Xrl2rFWbGxsYYM2YMzMzMuPgolUoYG+dER1BQEE6fPg0L\nCwts27ZNL2H2JhhUDc3MzIyysrLg6OiI0NBQ1K9fn7eSQCAoAajVakyZMgUmJiYIDAxE/fr1uYaY\nhrS0NFSvXh3R0dFa1xUKBaZMmYKgoCDZPceNG4cZM2YgIiICfn5+ICKsWrUK/fv311uZha2hGVSg\nlS9fnmJjYwEA1tbW2L59O9q0aSO7R+5vIAKBoPij+Rw0hBDLzaJFizB27FitawEBAZg/fz5q1aol\nu09aWhoqVKiA/v37Y8uWLYiOjkanTp2wbds2vb52hQ00I70ZvAW2trbSz8nJyWjXrh2WL18uu0d8\nfDyGDx+OtLQ02csWCATywxgzuDBLTk7G3LlzpXM3NzccPnwYe/bs4RJmALB3716kpKRg8eLFiI6O\nhpOTE0JCQgzmtTOoQLO2toa5ubl0rlarMWTIEIwaNQoqlUo2D3t7ezx58gQeHh64dOmSbOUKBAKB\nhiVLliAuLg4VK1bEunXrEBkZCR8fH65OP//8s9Y5YwyjR4/GkSNHOBlpY1CBZmRkhFatWmld8/Ly\ngqmpKa5cuSKrS+/evXHjxg00adIES5YsgSE1zQoEguJNfHw8VqxYgaCgINy+fRu9evWCQqHg6vT8\n+XPs379f69qDBw9ga2sLLy8vTlbaGFSgAUDbtm0BQHrzMjMzMWfOHLi5ucnq0aZNGzg4OCArKwsj\nR45Eu3bt8M8//8jqIBAISiYPHjzA2bNnMWPGDJQqVYq3DgBgx44dUCqV0rmtrS127tyJ4OBg7hO7\nNRhkoFlYWCAsLAyWlpaIjIzEkiVLZPcwNjbG119/LZ3v378fDRo0wIEDB2R3uXz5MtRqtezlCgQC\nPjRs2BCVKlXiraFF7ubGRo0a4dy5c+jQoQNHo/wYXKBVq1YNP/30E7y9vfHNN98AAKZNm4a7d+/K\n7tKrVy+t85iYGHzyyScYPXo0MjMzZfOIjo5G06ZNcfLkSdnKFAgEAg2PHj3CiRMnAACDBw/GqVOn\n4OzszNkqPwYXaADwxRdfAABGjBiBxo0bIz09Hf3795e9H6tWrVrw9PTUulazZk0AObUmufj4449h\namqKFi1a4IsvvsCDBw9kK1sgEAi2bNkCKysrbNmyBcuWLdMavGdIGGSgaVAoFFizZg2MjY0RHh6O\nn376SXaH3r17a51nZmZi2rRpaNy4sWwOjDHMmjULALB9+3bUqlULU6ZMQXJysmwOAoGg5HL16lVE\nRkaiS5cuvFVeiUEHGgA0aNAAEydOBACMHj0634x5fdO5c2dYWlpi+PDhqFixIh48eICvv/5a9j4t\nHx8ftGjRAkBOqH777bdwcXHBTz/9JPrXBAKB3lCpVAgJCZFapwwZgw80IGcrh1q1aiExMRFDhw6V\ntWwbGxv069cPs2bNwrZt26BQKBAaGop58+bJ6pG7lqbh2bNn6N27Nxo3biy1b+ublJQUTJ06FRcu\nXJClPIFAwBeFQsF9jcbCovNAY4z1YoyFM8YOMcb+Yoz9yRj7+F3uaWZmhrVr14Ixhl27dmHXrl26\n0i0U8+fPh62tLVq0aCEF2dSpU2WfTOjt7Z1vnl7lypUxdOhQ2b49WVlZwc3NDY0aNULTpk2xbt06\npKamylK2QCAQvJLC7DHzJgeAGwA+zHU+DEAGALvXPfd1+6ENHTqUAJCDgwO9ePHi7TbWeUfUajV1\n6NCBAFD58uXpyZMnspZ//PjxfHs0rVu3TlYHIqLu3btL5dvY2NCQIUPo8uXLsnsIBILiDwq5H5o+\nAq1pnvP6/37wub3uua8LtKSkJHr//fcJAPXp0+cdX6K3JyEhgapXr04AyMvLS2trdDn46KOPqFKl\nStSrVy8CQEZGRrR161ZZHeLj46ly5coFboD4v//9j9LS0mT1EQgExRdugaZ1c6AUgDUAwgEoXvf4\nwuxY/fvvv0sfnmFhYe/wEr0bly5dIgsLCwJAo0ePlrXs06dP07hx40ipVNIXX3xBAMjY2Jj27Nkj\nq0dYWFiBu/ra29vT0qVLSa1W693hwYMHlJ6ervdyBAIBP7gHGoDdAFIB7AVQtjDPKUygERF169aN\nAFC1atW4bkG+fv166UN8x44dspYdHx9PRERZWVkUEBBAAMjMzEz2kB8xYoRWmBkZGVFoaKhs5d+5\nc4eqVq1KHh4eNGTIENqwYQNFRUXJEqYCgUAeuAdajgPMACwFcAtAuZc8pj+ASACRjo6OhfrlYmNj\nyc7OjgDQmDFj3v5V0gH9+vUjAGRtbU23bt3i4pCenk5+fn4EgCwtLenkyZOylZ2Wlka1a9fWCjWF\nQkGLFy+WLVRu3bpFFStW1HIoU6YMtW7dmqZPn0779u2j2NhYWVwEAoHu0WmgAZhdUNNSnsP7Jc81\nBRAHYNbryilsDY2IaPPmzVKNICIi4l1eq3ciPT2dGjVqRACofv363GqMKSkp5OXlJQ3SOHfunGxl\nR0ZGkrGxMS1YsIDat28v/U1069ZNttfjxo0bVKFChQL/Nk1NTWn+/Pmi1iYQFFF0HWg2ACq/5jAD\nwACYFPD8MwB2vq6cNwk0tVpN7dq1k4IkMzPznV6wd+HevXtUunRpAkA9evTg9sGZkJAghWu5cuXo\n6tWrspU9c+ZMunjxIqlUKpo5c6YUJg0bNqQHDx7I4nDt2jWyt7fPF2ijRo0ipVIpi4NAINA9hQ00\nlvNY3cAYqwJgKRH557n+CMBuIhr+qud7eHhQZGRkoct7/Pgx6tati+TkZHzzzTeYOnXqW1jrhtDQ\nUAQEBAAAQkJC0K9fPy4ecXFx8Pb2xrVr1+Dg4IATJ06gevXqei9XqVSCMSZt+7N37150794dSUlJ\nKFeuHLZv3y7L5oRXr16Ft7c3nj9/rnW9bt26mD9/Pj755BO97a6bmZmJBQsW4Pr16699rL+/P7p2\n7aoXD4GguMEYO0dEHq99YGFSr7AHgCrImXPWONe1IQCyAXi87vlvUkPTsGzZMqlZ6fr162/8fF0y\nadIkaXCGnE1+efn777+laQWOjo708OFDLh43b96kWrVqyd6vdvHiRSpbtiwBoJEjR1KpUqWk2pqP\nj49e35v09HSaNGkSKRSKlzbP+/r6ipGZAsEbAB6DQgCYA5gE4CyA4wD+BHAMgF9hnv82gaZSqah5\n8+bSnDCVSvXG99AV2dnZ1KpVKwJAVapU4Tb5m4jo4cOH5OjoSACoRo0aFB0dzcUjMTGRPv30U9n7\n1c6dO0elS5empKQkio6OpgEDBmiFTLdu3fTaFBoZGUn169cvMNBq1qxJEyZMoJMnT3JtCn3x4gX9\n9ttvsg7gEQjeBi6B9q7H2wQaUU5NwMzMjADQ0qVL3+oeuuLZs2fSiDt/f3+uAXvr1i1ycHAgAFSv\nXj2Ki4vj4sGrXy0iIoKys7Ol8+vXr0tTHDQ16XHjxunti0dmZiZNnz6djI2NX1pbs7Ozo6+//pp+\n+eUXSkxM1IuHhri4ONq1axeNGDGC3NzciDFGAOh///ufXssVCN6VEhVoRERz5swhAGRlZcWtiU3D\niRMnpNrAt99+y9XlypUrUvObu7s7JSQkcHPZu3cv2djYSINWDh8+zMXj6NGj1LhxYylUypYtS4sX\nL6aMjAy9lHfx4kVq2LAhAaBWrVrRqlWryN/fn8zNzbXCzcTEhPz8/GjJkiU6WVItJiaGtm/fTkOG\nDKF69eoVGKgKhYIGDx5MkyZNom+//ZZ+/PFHWr9+Pe3cuZP++OMPOn36tGzTUaKjo187Yjk1NdWg\nBvjw/MJakihxgZaVlUWurq5kY2ND+/fvf+v76IqFCxcSAOrQoQP35pzIyEiysbGh0qVL04ULF7i6\n5O5XGzVqFDcPlUpFW7ZsoSpVqkirm+izhpSVlUWzZ88mNzc36Vpqairt3buX+vfvT5UqVdIKmvXr\n179zmdHR0RQcHEweHh6vm3LzyqNp06bv7PIqEhISaMqUKWRpaan1JSc1NZVOnTpFP/zwA/Xo0YPq\n1atHRkZGdPHiRb36vA61Wk0nTpygzz77jAYPHszVRcPly5fJz8+Pnj17xluFiIi2bNlCK1as0Nn9\nSlygEeU0Kcm9WPDLUKvVtHPnTu5hpuHkyZPcw0xDYmIizZgxQ/Y1MAsiIyODFi1aRD/99JMs5d28\nebPAvwm1Wk3nzp2joKAg8vDwoJiYGJ2WGxUVRTNmzJAGC2kOW1tbGj58OPXu3Zs6depEbdq0IS8v\nL2rQoAFVrVqV7O3tqV27djp10ZCWlkYLFy6UWhAA0IIFC7TCq6CAXbt2rV58clPQe5SdnU3btm2j\nJk2aSC7m5ubSqj08UKlUtGjRIjI1NSUA1KVLF24uGlavXi01Z+tq5aISGWgCgeDVqNVqOnPmDA0f\nPpzKly9PAGSv8WRnZ9OaNWsKXNw672FtbU0tW7ak0aNH088//0w3b97UezPfvXv36LvvvpPOk5KS\nKDg4WKrNa5pqu3TpQmfPntWry6t4/Pgx+fj4SE4ffvihbHM+X0ZwcLDk8+WXX+rsS6sINIFA8Eqy\ns7PpwIED9Ntvv8lSnlqtph07dlDNmjVfGmC1a9eWNbzycubMGSpfvjwNHDiQHj9+TOPHjydbW1ut\ngB01ahT34NiyZYu0mIOJiQnNmzePa9+iWq2m2bNnS69T3759deojAk0gEBgMmZmZNGfOHGratCk5\nOTlJo5LzHtWrV+c2cGnXrl3SDhqVKlXSGp1auXJlWrBgAddBVUQ5i5J37dpV8qpbty73rgS1Wk0T\nJkyQnEaOHKnzrhYRaAKBwGBRq9UUHx9P169fp/DwcNq8eTMtXryYxo8fT9u2bZPdZfHixVK/T+6j\nYcOG9PPPP3Pp71Wr1Vr7HIaHh0v7QWqCg/cEfZVKRYMHD5acpk2bppdxAyLQBAKB4DUolUoaNmxY\ngbVFd3d3rttTLVu2jFxcXCgjI4PGjh0rBe57771Hhw4d4ualITs7m3r06CG9XvPmzdNbWSLQBAKB\n4BWkpKRoTbQv6PD39+cSaqdOnSJjY2MyNjamBg0aSD6dO3em58+fy+6Tl8zMTOrYsaPktWzZMr2W\nV9hAM4ZAIBCUMJ49ewZ/f3/cvHkT9evXR7Vq1fIdTk5OsLCwkN0tOjoagYGBUCqVAIDLly/DxsYG\ny5YtQ7du3fS2uParUKlU0sLj6enp6NixI/bv3w8jIyOsW7cOPXr0kN2pIAwq0BISEngrCASCEkBW\nVhb2798POzs7LgHxMrKystCpUydER0drXW/VqhV8fX25uU6ePBkzZ85EdnY2AgICcOzYMZiYmGDz\n5s0IDAzk4lQQRrwFcnP37l30798fqampspb7448/4u7du7KWKRAI+OHo6Ah7e3uDCjMAGDNmDE6d\nOpXv+qFDhzBlyhSkpKTI7nT16lUsXLgQO3bsgJ+fH44dOwZzc3P89ttvBhVmAAyrDw34bzXymzdv\n6qcxtgCOHTtGZmZmNH36dEpLS5OtXIFAINDwv//9T6v/rlSpUtSlSxfauXMnt8EparWaPvroIwIg\nrdxiZWVFR44ckdUDRXFQCP6dkPf333/r51V5CWq1mry8vAgAVa1alfbs2SNr+QKBoGRz/vx5Mjc3\nJ2tra+ratSv9+uuvBvHleu/evfkGyuSeSiAXhQ00ne5Y/a4wxsjS0hIRERGoW7eurGXv378fbdu2\nlc79/f2xZMkSVKtWTVYPgUBQssjIyMC3334LDw8PfPzxxzA3N+etBCCnP69evXq4fft2vn8bMGAA\nfvzxR5iYmMjiUtgdqw0q0KysrCg1NRU1a9bE2bNnYW1tLVvZlDNtABcuXJCumZmZYdKkSRg/frxs\no51evHgBa2tr2f5QBAKBoCCCg4MxevRorWs1a9ZEUFAQOnXqJI16lIPCBppBDQqpVq0a7OzsEBUV\nhb59+0LOsGWMYfLkyVrXMjMzERQUBFdXV0RFRcniYWpqilatWuGHH35AWlqaLGUKBAJBbmJjYzFz\n5kzpvEaNGti0aROuXbuGLl26yBpmb4JBBZqpqSk2b94Mxhi2b9+OH3/8UdbyP//8c9SsWVPrWmBg\nIP7888981/WFlZUVRo8ejREjRsDJyQlz5swR0xkEAoGszJgxA4mJiXB2dsb//vc/XL9+Hd26dTPY\nIJMoTEebXIdmpZCZM2cSADI2NqbTp0/rtHPxdaxfv17aHkLzX7kHiajVavrkk0+0VvgeP3687INl\nBAJByePy5ctUvXp1WrdunUHsWUhURAeFeHh4UGRkJNRqNdq1a4cDBw6gcuXKOH/+POzt7WVxyM7O\nRo0aNTBw4ECcPXsWu3btgrm5OcLCwuDl5SWLA5AzJ69u3brIzMyUrpmZmaFnz54YP368GKwiEAj0\nwrVr1+Di4mJQ/fhFsg9Ng5GRETZt2gRHR0c8efIEXbt2hUqlkqVsExMTTJkyBb169cLPP/8Mb29v\nZGRkwN/fH1evXpXFAQCcnZ0xadIkrWuZmZlYtWoVAgICcO3aNVk8du3ahRMnTkCtVstSnkAg4Evd\nunUNKszeBIOsoWmIiIhA8+bNkZ2djWnTpmHWrFmyeBCRtIJAYmIiWrZsiUuXLqFSpUo4ffo0nJyc\nZPHIyMhAvXr1tFYx+eCDD3DkyBGYmprK4hATEwN3d3eo1Wp07NgRgYGBaN68ueG3pQsEgmJDYWto\n3PvNch8Frba/bNkyqS/p999/f/fG2LcgOjqaqlWrJq1iEhsbK1vZ+/fvzzexsU2bNpTjdCF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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig = pyplot.figure(figsize=(7,7))\n", "\n", "pyplot.quiver(X,Y, F,G);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "OK, that's not bad. The arrows on each grid point represent vectors $(f(x,y), g(x,y))$, computed from the right-hand side of the differential equation. \n", "\n", "_What are the axes on this plot?_ Well, they are the components of the state vector—which for the spring-mass system are _position_ and _velocity_. The vector field looks like a \"flow\" going around the origin, the values of position and velocity oscillating around. If you imagine an initial condition represented by a coordinate pair $(x_0,y_0)$, the solution trajectory would follow along the arrows, spiraling around the origin, while slowly approaching it." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We'd like to visualize a trajectory on the vector-field plot, and also improve it in a few ways. But before that, Python will astonish you with a splendid fact: you can also compute the vector field on the grid points by calling the function `dampedspring()`, passing as argument a list made of the matrices `X` and `Y`.\n", "\n", "_Why does this work?_ Study the function and think!" ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "collapsed": true }, "outputs": [], "source": [ "F, G = dampedspring([X,Y])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The default behavior of `quiver` is to scale the vectors (arrows) with the magnitude, but direction fields are usually drawn using line segments of equal length. Also by default, the vectors are drawn _starting at_ the grid points, while direction fields ususally _center_ the line segments. We can improve our plot using by _scaling_ the vectors by their magnitude, and using the `pivot='mid'` option on the plot. A little transparency is also nice.\n", "\n", "To plot the improved direction field below, we drew ideas from a tutorial available online, see Ref. [2]. To compute the magnitude of the vectors, we use the [`numpy.hypot()`](https://docs.scipy.org/doc/numpy-1.13.0/reference/generated/numpy.hypot.html) function, which returns the triangle hypotenuse of two right-angled sides.\n", "\n", "We should also add axis labels and a title!" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "data": { "image/png": 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FC7jvvvs8osNisZCVlUVKSgoXL160LZFERkaycuVKjxqXqqoqjh8/zsmTJ6mr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\nVw0XESZPnsy4ceOC2vv54MGDrFix4qrhqampZGZm0rdvX79+P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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "M = numpy.hypot(F,G)\n", "M[ M == 0] = 1 # to avoid zero-division\n", "F = F/M\n", "G = G/M\n", "\n", "fig = pyplot.figure(figsize=(7,7))\n", "\n", "pyplot.quiver(X,Y, F,G, pivot='mid', alpha=0.5)\n", "pyplot.plot(num_sol[:,0], num_sol[:,1], color= '#0096d6', linewidth=2)\n", "pyplot.xlabel('Position, $x$, [m]')\n", "pyplot.ylabel('Velocity, $v$, [m/s]')\n", "pyplot.title('Direction field for the damped spring-mass system\\n')\n", "pyplot.figtext(0.1,0,'$m={:.1f}$, $k={:.1f}$, $b={:.1f}$'.format(m,k,b));" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "And just for kicks, let's re-do everything with zero damping:" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "data": { "image/png": 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HNztdRzAajRw/fpzdu3fX8Wo89thjPP300y5NH7SxIklJSRw+fNiqMgvw8ssv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\ndGVlJfPnz+fChatzH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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "m = 1\n", "k = 1\n", "b = 0\n", "\n", "x0 = 3 # initial position\n", "v0 = 3 # initial velocity\n", "\n", "T = 12*numpy.pi\n", "N = 5000\n", "dt = T/N\n", "\n", "t = numpy.linspace(0, T, N)\n", "num_sol = numpy.zeros([N,2]) #initialize solution array\n", "\n", "#Set intial conditions\n", "num_sol[0,0] = x0\n", "num_sol[0,1] = v0\n", "\n", "for i in range(N-1):\n", " num_sol[i+1] = euler_cromer(num_sol[i], dampedspring, t[i], dt)\n", " \n", "F, G = dampedspring([X,Y])\n", "\n", "M = numpy.hypot(F,G)\n", "M[ M == 0] = 1 # to avoid zero-division\n", "F = F/M\n", "G = G/M\n", "\n", "fig = pyplot.figure(figsize=(7,7))\n", "\n", "pyplot.quiver(X,Y, F,G, pivot='mid', alpha=0.5)\n", "pyplot.plot(num_sol[:,0], num_sol[:,1], color= '#0096d6', linewidth=2)\n", "pyplot.xlabel('Position, $x$, [m]')\n", "pyplot.ylabel('Velocity, $v$, [m/s]')\n", "pyplot.title('Direction field for the un-damped spring-mass system\\n')\n", "pyplot.figtext(0.1,0,'$m={:.1f}$, $k={:.1f}$, $b={:.1f}$'.format(m,k,b));\n" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "##### Challenge task\n", "\n", "* Write a function to draw direction fields as above, taking as arguments the right-hand-side derivatives function, and lists containing the plot limits and the number of grid lines in each coordinate direction.\n", "\n", "* Write some code to capture mouse clicks on the direction field, following what you learned in [Lesson 1](http://go.gwu.edu/engcomp3lesson1) of this module.\n", "\n", "* Use the captured mouse clicks as initial conditions and obtain the corresponding trajectories by solving the differential system, then make a new plot showing the trajectories with different colors." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## What we've learned\n", "\n", "* General spring-mass systems have several behaviors: periodic in the undamped case, decaying oscillations when damped, complex oscillations when driven.\n", "* Resonance appears when the driving frequency matches the natural frequency of the system.\n", "* We can add formatted strings in figure titles, labels and added text.\n", "* The `lambda` keyword builds one-line Python functions.\n", "* The `meshgrid()` function of NumPy is handy for building a grid of points on a plane.\n", "* State vectors of a differential system live on the _phase plane_.\n", "* Solutions of the differential system (given initial conditions) are _trajectories_ on the phase plane.\n", "* Trajectories for the undamped spring-mass system are circles; in the damped case, they are spirals toward the origin." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## References\n", "\n", "1. Linge S., Langtangen H.P. (2016) Solving Ordinary Differential Equations. In: Programming for Computations - Python. Texts in Computational Science and Engineering, vol 15. Springer, Cham, https://doi.org/10.1007/978-3-319-32428-9_4, open access and reusable under [CC-BY-NC](http://creativecommons.org/licenses/by-nc/4.0/) license.\n", "V\n", "2. [Plotting direction fields and trajectories in the phase plane](http://scipy-cookbook.readthedocs.io/items/LoktaVolterraTutorial.html?highlight=direction%20fields#Plotting-direction-fields-and-trajectories-in-the-phase-plane), as part of the Lotka-Volterra tutorial by Pauli Virtanen and Bhupendra, in the _SciPy Cookbook_. " ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n" ], "text/plain": [ "" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Execute this cell to load the notebook's style sheet, then ignore it\n", "from IPython.core.display import HTML\n", "css_file = '../style/custom.css'\n", "HTML(open(css_file, \"r\").read())" ] } ], "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": "3.6.2" } }, "nbformat": 4, "nbformat_minor": 2 }