{ "cells": [ { "cell_type": "markdown", "id": "ac717206", "metadata": {}, "source": [ "---\n", "format:\n", " html:\n", " other-links:\n", " - text: This notebook\n", " href: L18-19 Gaussian Quadrature.ipynb\n", "---\n", "\n", "\n", "# Gauss Quadrature\n", "\n", "
Note. These notes are mainly a record of what we discussed and are not a substitute for attending the lectures and reading books! If anything is unclear/wrong, let me know and I will update the notes. \n", "
" ] }, { "cell_type": "markdown", "id": "7d08ed8c", "metadata": {}, "source": [ "Chapter 1: How computers add\n", "\n", "* Week 1: L1: Introduction\n", "* Week 2: L2 Floating point numbers, L3 Stability & Conditioning\n", "\n", "Chapter 2: Solving nonlinear equations in 1d\n", "\n", "* Week 3: L4 Solving nonlinear equations in 1d, L5 Exercises\n", "* Week 4: L6-7 Solving nonlinear equations in 1d II & III\n", "\n", "Chapter 3: Polynomial interpolation\n", "\n", "* Week 5: L8 Polynomial Interpolation, L9 How to choose $X$? & Barycentric formula\n", "* Week 6: L10 Hermite Interpolation, L11 Exercises\n", "\n", "Chapter 4: Numerical Integration\n", "\n", "* Week 7: L12 Newton-Cotes quadrature\n", "\n", "---\n", "\n", "* Week 8: L14 Revision, L15 Midterm Exam\n", "* Week 9: L16-17 Group Project: Integrating Differential Equations\n", "\n", "---\n", "\n", "* Week 10: L18 Gaussian quadrature\n", "\n", "Next - Chapter 5: Linear systems of equations \n", "\n", "\n", "\n", "---\n", "\n", "* December, 15 at 10:30: Final Exam (Keller Hall 3-125)" ] }, { "cell_type": "code", "execution_count": 2, "id": "193b2d8d", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "✓ file included! \n", "\n", "using: Plots, LaTeXStrings, Polynomials, PrettyTables \n", "\n", "Functions included: \n", " simple_iteration, \n", " Newton, \n", " orderOfConvergence, \n", " ChebyshevNodes \n", "\n", "Use @doc <> for help\n" ] } ], "source": [ "include(\"preamble.jl\");" ] }, { "cell_type": "markdown", "id": "c50cbd71", "metadata": {}, "source": [ "## Previously...\n", "\n", "Recall that we are considering *quadrature rules* of the following form:\n", "\n", "\\begin{align}\n", " \\int_{a}^b f &\\approx \\sum_j w_j f(x_j). \n", "\\end{align}\n", "\n", "where $\\{w_j\\}$ are the *quadrature weights*, $\\{x_j\\}$ are the *quadrature nodes*, and $(\\{w_j\\}, \\{x_j\\})$ or $\\sum_{j} w_j f(x_j)$ is a *quadrature rule*.\n", "\n", "Suppose $p$ is the polynomial of degree $\\leq n$ interpolating $f$ on $X = \\{x_0,\\dots,x_n \\}$: that is $p(x) = \\sum_{j=0}^n \\ell_j(x) f(x_j)$. We simply approximate the integral of $f$ by instead integrating $p$:\n", "\n", "\\begin{align}\n", " \\int_{a}^b f &\\approx \\int_a^b p = \\sum_{j=0}^n \\left[\\int_a^b\\ell_j(x)\\mathrm{d}x \\right] f(x_j) \n", " %\n", " \\tag{$\\star$}\n", "\\end{align}\n", "\n", "which gives a quadrature rule with weights $w_j := \\int_a^b\\ell_j(x)\\mathrm{d}x$ and nodes $x_j$.\n", "\n", "### Newton-Cotes \n", "\n", "When the interpolation nodes $\\{x_0,\\dots,x_n\\}$ are equispaced, this is a *Newton-Cotes* quadrature rule.\n", "\n", "We saw the following examples:\n", "\n", "* Rectangular rule:\n", "\n", "\\begin{align}\n", " \\int_a^b f \\approx (b-a) f(a)\n", "\\end{align}\n", "\n", "* Midpoint rule (from exam):\n", "\n", "\\begin{align}\n", " \\int_a^b f \\approx (b-a) f\\big( \\tfrac{a+b}2 \\big)\n", "\\end{align}\n", "\n", "* Trapezoid rule: \n", "\n", "\\begin{align}\n", " \\int_a^b f \\approx \\frac{b-a}{2} \\left( f(a) + f(b) \\right)\n", "\\end{align}\n", "\n", "* Simpson's rule:\n", "\n", "\\begin{align}\n", " \\int_a^b f \\approx \\frac{b-a}{6} \\left( f(a) + 4 f\\big( \\tfrac{a+b}2 \\big) + f(b) \\right)\n", "\\end{align}\n", "\n", "
Definition. \n", "\n", "We say the quadrature rule $(\\{w_j\\}, \\{x_j\\})$ is *exact for all polynomials of degree $N$* if \n", "\n", "\\begin{align}\n", " \\int_a^b p(x) \\mathrm{d}x = \\sum_{j=0}^n w_j p(x_j)\n", "\\end{align}\n", "\n", "for all $p \\in \\mathcal P_N$ (all polynomials of degree $\\leq N$).\n", "\n", "
\n", "\n", "Let us consider the quadrature rule $(\\star)$. For all polynomials $P$ of degree $\\leq n$, we have \n", "\n", "\\begin{align}\n", " \\sum_{j=0}^n w_j P(x_j) &= \\sum_{j=0}^n \\left[ \\int_a^b \\ell_j(x) \\mathrm{d}x\\right] P(x_j) \\\\\n", " %\n", " &= \\int_a^b \\sum_{j=0}^n \\ell_j(x) P(x_j) \\mathrm{d}x \\\\\n", " %\n", " &= \\int_a^b P\n", "\\end{align}\n", "\n", "In the final line, we have used the fact that $P(x) = \\sum_{j=0}^n \\ell_j(x) P(x_j)$ beacuse $P$ has degree $\\leq n$. That is, the Newton-Cotes quadrature rules given by $(\\star)$ are exact for all polynomials of degree $n$.\n", "\n", "Therefore: \n", "\n", "* Rectangular rule ($n=0$) is exact for all polynomials of degree $0$\n", "* Trapezoid rule ($n=1$) is exact for all polynomials of degree $1$\n", "\n", "You have also seen that: \n", "\n", "* Midpoint rule ($n=0$) is exact for all polynomials of degree $1$\n", "* Simpson rule ($n=2$) is exact for all polynomials of degree $3$\n", "\n", "This is a general fact: Suppose $n$ is even and $X = \\{x_0, \\dots, x_n\\}$ is a set of interpolation nodes symmetric about $\\frac{a+b}{2}$ (that is, $x_j + x_{n-j} = a+b$ ). Then, the quadrature rule $(\\star)$ is exact for all polynomials of degree $n+1$.\n", "\n", "Suppose we have the quadrature rule $(\\star)$ with equispaced points $X = \\{x_0,\\dots,x_n\\}$ and it is exact for all polynomials of degree $N$ ($N = n$ for odd $n$ and $N = n+1$ for even $n$). Let $P$ the polynomial interpolant of $f$ on $\\tilde{X}$ such that $P$ has degree $N$ and $\\tilde{X}$ contains $X$ (i.e. $P(x_j) = f(x_j)$ for all $j=0,\\dots,n$). Then, \n", "\n", "\\begin{align*}\n", " \\left| \\int_a^b f - \\sum_{j=0}^n w_j f(x_j) \\right| \n", " %\n", " &= \\left| \\int_a^b (f - P ) \\right| \\\\\n", " %\n", " &\\leq \\frac{\\|f^{(N+1)}\\|_{L^\\infty([a,b])}}{(N+1)!} \\int_a^b \\left| \\ell_{\\tilde{X}}(x) \\right| \\mathrm{d}x \n", "\\end{align*}\n", "\n", "*Composite Rules.* Fix $K$ and define $h := \\frac{b-a}{K}$. Then, split $[a,b]$ into the $K$ intervals $[a + kh, a + (k+1)h]$ for $k = 0,\\dots,K-1$: \n", "\n", "\\begin{align}\n", " \\int_a^b f = \\sum_{ k=0 }^{K-1} \\int_{a + kh}^{a + (k+1)h} f\n", "\\end{align}\n", "\n", "and apply the quadrature rule on each subinterval $[a + kh, a + (k+1)h]$. Using the error estimate on each sub-interval and summing over $k$, we saw that errors in the composite rules are as follows: \n", "\n", "* Rectangular: $O(h)$,\n", "* Midpoint, Trapezoid: $O(h^2)$,\n", "* Simpson: $O(h^4)$" ] }, { "cell_type": "markdown", "id": "833dfc81", "metadata": {}, "source": [ "## L18-19: Gaussian Quadrature\n", "\n", "
Remark. \n", "\n", "By considering the change of variables $y = \\frac{2}{b-a} \\left( x - \\frac{a + b}{2} \\right)$ and defining $\\tilde{f}(y) = \\frac{b-a}{2} f\\big( \\tfrac{b-a}2 y + \\tfrac{a+b}2 \\big)$, we have\n", "\n", "\\begin{align}\n", " \\int_a^b f(x) \\mathrm{d}x = \\int_{-1}^1 \\tilde{f}(y) \\mathrm{d}y. \n", "\\end{align} \n", "\n", "Moreover, if $p$ is the polynomial interpolation of $f$ on $X \\subset [a,b]$ then $\\tilde{p}$ is the polynomial interpolation of $\\tilde{f}$ on $\\tilde{X} \\subset [-1,1]$ where $\\tilde{X} = \\{ \\frac{2}{b-a} \\left( x - \\frac{a + b}{2} \\right) : x \\in X \\}$. As a result, we may without loss of generality consider the interval $[-1,1]$.\n", "\n", "
\n", "\n", "Before: Fix $X$ find $\\{w_j\\}$ such that the quadrature rule\n", "\n", "\\begin{align}\n", " \\int_a^b f \\approx \\sum_{j} w_j f(x_j)\n", "\\end{align}\n", "\n", "is exact on $\\mathcal P_n$ for maximal $n$.\n", "\n", "**Idea:** Choose $X$ **and** $\\{w_j\\}$.\n", "\n", "
Example. \n", "\n", "Choose $w_0, w_1$ and $x_0, x_1$ such that \n", "\n", "\\begin{align}\n", " \\int_{-1}^1 f \\approx w_0 f(x_0) + w_1 f(x_1)\n", "\\end{align}\n", "\n", "is exact for all polynomials of degree $3 = 2n+1$.\n", "
\n", "\n", "
Answer. \n", "\n", "Using the functions, $f = 1, x, x^2, x^3$, we want to solve\n", "\n", "\\begin{align}\n", " 2 &= w_0 + w_1 \\\\\n", " 0 &= w_0 x_0 + w_1 x_1 \\\\\n", " \\frac{2}{3} &= w_0 x_0^2 + w_1 x_1^2 \\\\\n", " 0 &= w_0 x_0^3 + w_1 x_1^3\n", "\\end{align}\n", "\n", "We have \n", "\n", "\\begin{align}\n", " 0 &= ( w_0 x_0 + w_1 x_1 )( x_0 + x_1 ) \\\\\n", " &= (w_0 x_0^2 + w_1 x_1^2) + ( w_0 + w_1 ) x_0 x_1 \\\\\n", " &= \\frac23 + 2 x_0 x_1\n", "\\end{align}\n", "\n", "and so $x_0 x_1 = -\\frac13$. Similarly, we have \n", "\n", "\\begin{align}\n", " \\frac23 (x_0 + x_1) &= ( w_0 x_0^2 + w_1 x_1^2 )( x_0 + x_1 ) \\\\\n", " &= (w_0 x_0^3 + w_1 x_1^3) + ( w_0 x_0 + w_1 x_1 ) x_0 x_1 \\\\\n", " &= 0 + 0 x_0 x_1 = 0\n", "\\end{align}\n", "\n", "and so $x_0 = - x_1$ and $x_0 = - \\frac{1}{\\sqrt{3}}$, $x_1 = \\frac{1}{\\sqrt{3}}$.\n", "\n", "Finally, we have $0 = w_0 x_0 + w_1 x_1 = (w_0 - w_1) x_0$ and so $w_0 = w_1 = 1$.\n", "\n", "
" ] }, { "cell_type": "markdown", "id": "05251005", "metadata": {}, "source": [ "The above example gives the quadrature rule\n", "\n", "\\begin{align}\n", " \\int_{-1}^1 f \\approx f\\big( -\\tfrac{\\sqrt3}{3} \\big) + f\\big( \\tfrac{\\sqrt3}{3} \\big)\n", "\\end{align}\n", "\n", "which is exact for all polynomials of degree $\\leq 3$.\n", "\n", "Q: How to generalise this??\n", "\n", "It turns out that one can choose the interpolation nodes and weights so that the quadrature rule \n", "\n", "\\begin{align}\n", " \\int_{-1}^1 f \\approx \\sum_{j=0}^n w_j f(x_j)\n", "\\end{align}\n", "\n", "is exact for all polynomials of degree $2n+1$. The solution in the general case requires a set of *orthogonal polynomials*: " ] }, { "cell_type": "markdown", "id": "287a43c7", "metadata": {}, "source": [ "## Legendre Polynomials\n", "\n", "
Definition. \n", "\n", "Let the *Legendre polynomial $P_n$* be the *monic* polynomial of degree $n$ such that \n", "\n", "\\begin{align}\n", " (P_n, q)_{L^2} := \\int_{-1}^1 P_n(x) q(x) \\mathrm{d}x = 0\n", "\\end{align}\n", "\n", "for all $q \\in P_{n-1}$. \n", "\n", "
\n", "\n", "
Remark. \n", "\n", "We say that $f$ is orthogonal to $g$ if $(f,g)_{L^2} = 0$.\n", "\n", "
\n", "\n", "
Remark (An equivalent definition). \n", "\n", "$P_n$ is monic and $(P_n, P_m) = 0$ for all $n \\not= m$.\n", "\n", "
\n", "\n", "\n", "It may not be immediately clear that such polynomials exist, but we may compute the first few explicitly:\n", "\n", "
Example. \n", "\n", "When $n=0$, there is only one monic polynomial: $P_0(x) = 1$,\n", "\n", "When $n=1$, we require $P_1(x) = x + a$ to satisfy\n", "\n", "\\begin{align}\n", " 0 = \\int_{-1}^1 P_1(x) 1 \\mathrm{d}x = 2 a\n", "\\end{align} \n", "\n", "and so $P_1(x) = x$.\n", "\n", "When $n=2$, we have $P_2(x) = x^2 + a x + b$ and so \n", "\n", "\\begin{align}\n", " 0 &= \\int_{-1}^1 P_2(x) 1 \\mathrm{d}x = 2b + \\frac23 \\\\\n", " 0 &= \\int_{-1}^1 P_2(x) x \\mathrm{d}x = \\frac{2a}3 \n", "\\end{align}\n", "\n", "therefore $P_2(x) = x^2 - \\frac13$.\n", "\n", "
" ] }, { "cell_type": "markdown", "id": "3a2514b3", "metadata": {}, "source": [ "
Theorem. \n", "\n", "The Legendre polynomials exist for all $n$.\n", "\n", "
\n", "\n", "(we will come back to the proof)" ] }, { "cell_type": "markdown", "id": "83d11e77", "metadata": {}, "source": [ "
Remark. \n", "\n", "The roots of $P_2$ are $\\{ -\\tfrac{\\sqrt3}{3}, \\tfrac{\\sqrt3}{3} \\}$ (which are the nodes used in the quadrature rule we had above)\n", "\n", "
\n", "\n", "Recall $P_n$ are the Legendre polynomials and define the set of *Legendre points* $X$ by \n", "\n", "\\begin{align}\n", " X := \\{ \\text{zeros of } P_{n+1} \\}.\n", "\\end{align}\n", "\n", "
Claim. \n", "\n", "* $X$ is a set of $n+1$ distinct points in $[-1,1]$\n", "\n", "
\n", "\n", "
Proof. \n", "\n", "Notice that $P_{n+1}(x) = \\prod_{j=0}^n(x-x_j)$ (i.e. the monic polynomial with roots $X$). \n", "\n", "Suppose that $x_0 = x_1$ and define $q(x) := \\prod_{j=2}^n (x - x_j)$. Then, since $P_{n+1}$ is orthogonal to $q$ (as $q$ is of degree $n-1$), we have \n", "\n", "\\begin{align}\n", " 0 &= \\int_{-1}^1 P_{n+1}(x) q(x) \\mathrm{d}x \\\\\n", " %\n", " &= \\int_{-1}^1 \\left(\\prod_{j=0}^n (x - x_j) \\right) \\left( \\prod_{j=2}^n (x - x_j) \\right) \\mathrm{d}x \\\\\n", " %\n", " &= \\int_{-1}^1 (x - x_0)^2 \\prod_{j=2}^n (x - x_j)^2 \\mathrm{d}x >0,\n", "\\end{align}\n", "\n", "a contradiction (the integral of a non-negative polynomial on a positive interval is positive). By relabeling the indices in $X = \\{x_0,\\dots,x_n\\}$, we see that $X$ is a set of pairwise distinct nodes.\n", "\n", "Suppose $x_k \\not\\in [-1,1]$. Then, on defining $q(x) := \\prod_{j=0 \\,:\\, j \\not= k }^n (x - x_j)$, a polynomial of degree $n$, we have \n", "\n", "\\begin{align}\n", " 0 &= \\int_{-1}^1 P_{n+1}(x) q(x) \\mathrm{d}x \\\\\n", " %\n", " &= \\int_{-1}^1 \\left(\\prod_{j=0}^n (x - x_j) \\right) \\left( \\prod_{j\\not=k} (x - x_j) \\right) \\mathrm{d}x \\\\\n", " %\n", " &= \\int_{-1}^1 (x - x_k) \\prod_{j\\not=k} (x - x_j)^2 \\mathrm{d}x.\n", "\\end{align}\n", "\n", "Now, since $x_k \\not\\in [-1,1]$, we have $(x - x_k)$ is either strictly positive or negative for $x \\in [-1,1]$. Therefore, $\\int_{-1}^1 (x - x_k) \\prod_{j\\not=k} (x - x_j)^2\\mathrm{d}x$ is either strictly positive or negative, a contradiction. As a result, each $x_k$ is in the interval $[-1,1]$.\n", "\n", "
\n", "\n", "
Definition (Gaussian Quadrature) \n", "\n", "Let $p$ be the polynomial interpolation of $f$ on the $n+1$ Legendre points. Then, we may define the quadrature rule\n", "\n", "\\begin{align}\n", " \\int_{-1}^1 f \\approx \\int_{-1}^{1} p = \\sum_{j=0}^n w_j f(x_j)\n", "\\end{align}\n", "\n", "
\n", "\n", "
Theorem. \n", "\n", "Gaussian quadrature is exact for polynomials of degree $2n+1$.\n", "\n", "
\n", "\n", "
Proof. \n", "\n", "First, note that we have already seen that Gaussian quadrature is exact for all polynomials of degree $n$.\n", "\n", "Let $X = \\{x_0,\\dots,x_n\\}$ be the zeros of $P_{n+1}$, $w_j$ the corresponding quadrature nodes, and let $P$ be a polynomial of degree $2n+1$. On dividing $P$ by $P_{n+1}$, there exists $q_n, r_n \\in \\mathcal P_n$ such that \n", "\n", "\\begin{align}\n", " P(x) = P_{n+1}(x) q_n(x) + r_n(x).\n", "\\end{align}\n", "\n", "Notice that $P(x_j) = P_{n+1}(x_j) q_n(x_j) + r_n(x_j) = r_n(x_j)$ so that $r_n$ is a degree $n$ polynomial interpolant of $P$ on $X$. Using this, combined with the fact that $P_{n+1}$ is orthogonal to $q_n$, we have \n", "\n", "\\begin{align}\n", " \\int_{-1}^1 P &= \\int_{-1}^{1} \\left[ P_{n+1}(x) q_n(x) + r_n(x) \\right] \\mathrm{d}x\n", " \\\\\n", " &= 0 + \\int_{-1}^{1} r_n(x) \\mathrm{d}x \\\\\n", " %\n", " &= \\sum_{j=0}^n w_j r_n(x_j) \\\\\n", " %\n", " &= \\sum_{j=0}^n w_j P(x_j) \n", " % \n", "\\end{align}\n", "\n", "Therefore, Gaussian quadrature is exact for all polynomials of degree $\\leq 2n+1$.\n", "\n", "
" ] }, { "cell_type": "markdown", "id": "ed1bc741", "metadata": {}, "source": [ "
Claim. \n", "\n", "* $\\sum_{j=0}^n w_j = 2$\n", "* $w_j \\geq 0$, \n", "\n", "
\n", "\n", "
Proof. \n", "\n", "Notice that $\\sum_{k=0}^n \\ell_k(x)$ is a polynomial of degree $n$ which is equal to $1$ on the set of $n+1$ points $X$ and so $\\sum_{k=0}^n \\ell_k(x)= 1$ for all $x$. Therefore, because the quadrature rule is exact for polynomials of degree $2n+1$, we have\n", "\n", "\\begin{align}\n", " 2 &= \\int_{-1}^1 \\mathrm{d}x \\\\\n", " &= \\int_{-1}^1 \\sum_{k=0}^n \\ell_k(x) \\mathrm{d}x \\\\\n", " &= \\sum_{j=0}^n w_j \\left[ \\sum_{k=0}^n \\ell_j(x_k) \\right] \\\\\n", " &= \\sum_{j=0}^n w_j.\n", "\\end{align}\n", "\n", "Moreover, $\\ell_k(x)^2$ is a polynomial of degree $2n$ and the quadrature rule is exact for all polynomials of degree $2n +1$, we have \n", "\n", "\\begin{align}\n", " 0 &\\leq \\int_{-1}^1 \\ell_k(x)^2 \\mathrm{d}x \\\\\n", " &= \\sum_j w_j \\left[ \\ell_k(x_j)^2 \\right] \\\\\n", " &= w_j \n", "\\end{align} \n", "\n", "Therefore, the weights are positive and sum to $2 = \\int_{-1}^1 1 \\mathrm{d}x$.\n", "\n", "
" ] }, { "cell_type": "markdown", "id": "6c21f622", "metadata": {}, "source": [ "
Theorem. \n", "\n", "The Legendre polynomials $P_n$ exist.\n", "\n", "
\n", "\n", "
Proof. (Presentation) \n", "\n", "We will use the notation\n", "\n", "\\begin{align}\n", " \\|f\\|_{L^2} := \\left( \\int_{-1}^1 |f(x)|^2 \\mathrm{d}x \\right)^{1/2}.\n", "\\end{align}\n", "\n", "Let $p_0 = 1$ and $p_1 = x$. \n", "\n", "If $p_0, p_1, \\dots, p_n$ have been defined, we define \n", "\n", "\\begin{align}\n", " a_n &:= \\frac{ \\int_{-1}^1 x p_{n}(x)^2 \\mathrm{d}x }{\\|p_{n}\\|_{L^2}^2} \\qquad \\text{and}\\\\\n", " b_{n} &:= \\frac{ \\int_{-1}^1 x p_{n}(x) p_{n-1}(x) \\mathrm{d}x }{\\|p_{n-1}\\|_{L^2}^2}\n", "\\end{align}\n", "\n", "and thus $p_{n+1}(x) := (x - a_n) p_n(x) - b_n p_{n-1}(x)$ for all $n \\geq 1$.\n", "\n", "We claim that $p_n(x)$ is the $n^\\text{th}$ Legendre polynomial. We prove this by induction: the statement is true for $n=0$ and $n=1$. Suppose the statement is true for $n$. \n", "\n", "Then, notice that $p_{n+1}(x)$ is a monic polynomial of degree $n+1$.\n", "\n", "Moreover, we have: for all $j < n-1$,\n", "\n", "\\begin{align}\n", " \\int_{-1}^1 p_{n+1}(x) p_{j}(x) \\mathrm{d}x\n", " %\n", " &= \\int_{-1}^1 \\left[ (x - a_n) p_{n}(x) - b_n p_{n-1} \\right] p_j(x) \n", " \\mathrm{d}x \\\\\n", " %\n", " &= \\int_{-1}^1 p_{n}(x) \\left[ (x - a_n) p_j(x) \\right] - b_n \\int_{-1}^1 p_{n-1} p_j \\\\\n", " %\n", " &= 0.\n", "\\end{align}\n", "\n", "Here, we have used the fact that $(x-a_n) p_j$ is a polynomial of degree $j+1 < n$, $p_j$ is a polynomial of degree $j < n-1$, and $p_n$ is the $n^\\text{th}$ Legendre polynomial.\n", "\n", "Moreover, we have \n", "\n", "\\begin{align}\n", " \\int_{-1}^1 p_{n+1}(x) p_{n-1}(x) \\mathrm{d}x\n", " %\n", " &= \\int_{-1}^1 \\left[ (x - a_n) p_{n}(x) - b_n p_{n-1} \\right] p_{n-1}(x) \n", " \\mathrm{d}x \\\\\n", " %\n", " &= \\int_{-1}^1 x p_{n}(x) p_{n-1}(x) - b_n \\int_{-1}^1 | p_{n-1}(x) |^2 \\mathrm{d}x \\\\\n", " %\n", " &= 0\n", "\\end{align}\n", "\n", "and\n", "\n", "\\begin{align}\n", " \\int_{-1}^1 p_{n+1}(x) p_{n}(x) \\mathrm{d}x\n", " %\n", " &= \\int_{-1}^1 \\left[ (x - a_n) p_{n}(x) - b_n p_{n-1} \\right] p_{n}(x) \n", " \\mathrm{d}x \\\\\n", " %\n", " &= \\int_{-1}^1 (x - a_n) p_{n}(x) p_{n}(x) \\\\\n", " %\n", " &= 0\n", "\\end{align}\n", "\n", "As a result, we have $p_{n+1}$ is monic and orthogonal to all polynomials of degree $\\leq n$.\n", "\n", "
" ] }, { "cell_type": "markdown", "id": "872038e6", "metadata": {}, "source": [ "We may use the recursion in the proof to compute the Legendre polynomials: here we plot $P_n$ for $n \\leq 8$ and the set of Legendre points " ] }, { "cell_type": "code", "execution_count": 3, "id": "b14d3ac3", "metadata": {}, "outputs": [ { "data": { "image/png": "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", "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# | echo: false\n", "\n", "function Legendre( n ) \n", " x = Polynomial([0,1])\n", " m = n-1\n", " if (n==0)\n", " return Polynomial([1])\n", " elseif (n==1)\n", " return x\n", " else\n", " return ( (2m+1) * x * Legendre(m) - m * Legendre(m-1) )/(m+1)\n", " end\n", "end\n", "\n", "plot( Legendre(0), -1,1, label=L\"P_0\" )\n", "plot!(Legendre(1), -1,1, label=L\"P_1\" )\n", "plot!(Legendre(2), -1,1, label=L\"P_2\" )\n", "plot!(Legendre(3), -1,1, label=L\"P_3\" )\n", "plot!(Legendre(4), -1,1, label=L\"P_4\" )\n", "plot!(Legendre(5), -1,1, label=L\"P_5\" )\n", "plot!(Legendre(6), -1,1, label=L\"P_6\" )\n", "plot!(Legendre(7), -1,1, label=L\"P_7\" )\n", "plot!(Legendre(8), -1,1, label=L\"P_8\" )\n", "\n", "p = Legendre(8)\n", "r = roots(p)\n", "scatter!( r, p.(r), markersize=5, primary=false )" ] }, { "cell_type": "markdown", "id": "c8854823", "metadata": {}, "source": [ "Next we approximate the integral \n", "\n", "\\begin{align}\n", " \\int_{-1}^{+1} \\frac{1}{1 + e^{10(x - 0.5)}} \\mathrm{d}x\n", "\\end{align}\n", "\n", "by using *(i)* the composite trapezoid rule, *(ii)* composite simpson rule (from last time) and *(iii)* using Newton quadrature. " ] }, { "cell_type": "code", "execution_count": 6, "id": "cac67b0a", "metadata": {}, "outputs": [ { "data": { "image/png": "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", "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# | echo: false \n", "\n", "a, b = -1, 1\n", "f = x -> 1/( 1 + exp( 10*(x-.5) ))\n", "df_max = 5/2\n", "d2f_max = 50/(3*sqrt(3))\n", "d3f_max = 125/3\n", "\n", "exact = (5 + log( 1 - exp(5) + exp(10) ) ) / 10\n", "\n", "plot(f, a, b, label=L\"f(x) = (1 + e^{10x})^{-1}\")\n" ] }, { "cell_type": "code", "execution_count": null, "id": "1b916ede", "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAlgAAAGQCAIAAAD9V4nPAAAABmJLR0QA/wD/AP+gvaeTAAAgAElEQVR4nOydZ1wUVxeH7/ZKZ+lL711FiiKKsaNRRIO9oRJb1EQTNYktxhJNotGYaBJLxBJFsSMqNqoUQZQqTZfeFrb3eT8M77oiKtKWcp8fH2bvzNw5W5gz995z/geDIAiAQCAQCKS/glW3ARAIBAKBqBPoCCEQCATSr4GOEAKBQCD9GugIIRAIBNKvgY4QAoFAIP0a6AghEAgE0q+BjhACgUAg/RroCCEQCATSr4GOEAKBQCD9GugIIRAIBNKv6UGOkMfjbd68ue3HS6XSrjMGAoH0FhAEkcvl6rYC0ovpQY6wrq7u1KlTbT9eJBJ1nTEQCKS3IJfL4WMxpCP0IEcIgUAgEEj3Ax0hBAKBQPo10BFCIBAIpF+DV7cB7UEoFN6/fz/23j0nR8fx48ebmpqq2yIIBAKB9FZ634hwx67dukzDGWvCIjKurf/9B1sPx4FDhwqFQnXbBYFAIJBeSS8bEe779dedB3e7bRpMMaShLQqp4sXJPDefwYVZz9VrGwQCgUB6I71sRLhl1zbXdYOUXhAAgCVgHcKcq3nVV65cUaNhEAgEAuml9CZH+OTJE5wmUdULNoMBJqPNDx45og6jIBAIBNK76U2OMC8vj2zwlhcEAABAMSCX11Z1sz0QCAQC6QP0pjVCKysrKad1NRkpV2Koq9fN9kAgkC5CIBDcvHkTQZC2HCyXyxUKBYFA6GqrIOrF1dXVycmpK3ruTY7Q19dXXMWTCaR4astffHV85YbVX6jFKggE0unExcWtWrVq2LBh6jYE0lNgsViWlpZnz57tis57kyPEYDCrP//i6O/HndcMxBJeT+pWJ1QQG3Fz585Vo20QCKQTUSgUAwYMOH/+vLoNgfQU/vvvv6ioqC7qvDc5QgDArh0/NjY1nvzmlIG/GY1JkzRK6lKrdYDm85QMdZsGgUAgkF5JbwqWQfnj4O8lWS/WBoQ7V1t+Zjbx9olrxdkFurq66rYLAoFAIL2SXjYiRDE0NFy9evWiRYs0NDTUbQsEAoFAeje9b0QIgUAgEEgnAh0hBAKBQPo1vXJqFAKBQN6PRCKJiYlJfJwqFEtszM2CgiZYW1ur2yhIDwU6wn4EgiClpaX5+flGRkZOTk4kEkndFkEgXUJsbOzMhUslpp4cq2EIgULKL/1uz9ig0SP+/v0AlUrtSM9SqTQtLa2qqsrc3NzDwwOPV9sttKSk5OLFi+vWrfuosxoaGlgsVotGPB7v4uLSeaa1lWfPnsXHxy9btqxF+59//jl8+PAuyp1vFegI+wuxsbHhaz7H6hLwhiSEK+cVN4bNC9u88XssFk6PQ/oU9+7fnzo/nBN2HjDd0RYxAOKg76OiNr76NOTR7Rvt/s0nJCTMmjVLS0vL0dGxpKSksbHxxo0b9vb2nWf7R9DU1JSWloZuu7i43Lt3z9DQ8INnxcXF/fDDDwCAxsbGsrIyV1dXAICOjs6dO3e61NpWqaure/r06dvtZ86cMTU17U5H2FtvgvX19cnJyUVFRQqFQt229ALu3b+3cG0YY5m18VJbxmSmwRxLq289zjy+8MW61eo2DQLpTBQKxdzFyzgLTyu9YDM4gmjavqwm/Okz7ZQmqaysnDhx4po1a7Kyss6fP5+amnry5Em5XI7ulUqlhYWFEomkxVlcLjc/Px89TCwWFxQUvH1McXFxU1NTi8aXL1+y2WzVFqFQmJeXpxzPeXh4nDp1Ct0uKSmRyWSqB4tEoqKiohaNAIDJkyenpaWlpaXt2bPH3Nwc3Ua9IJfLzc3NRQ+rr68vKCgQi8UtThcIBIWFhcp3jSKTyYqLi9tSFLa8vLy2tlb5cvjw4QcPHlS+5HK5hYWFbdTV61x6nyNMS0sb4uk2fbjv+a1r14VOcre1PLBvr1o+u17Eyq9WmSy1J2q9ngvF4DDGn1nfeHizsLBQjYZBIJ1LcnKyUMMEWAxsdS83cO2hf061r+cTJ05YWlquWbNG2TJkyBB01HLq1CkTE5P58+ebmpoePXoU3Tt06NAlS5b4+flNnjzZ29s7MTHR19d35syZNjY2paWlAICEhAQbG5uJEyfOmzfPzs5u69at6Im5ublOTk4hISGenp6hoaGoN4qMjLS1tV25cuXUqVNnzJgBAIiPjx84cCAA4PPPPxeLxRMmTPDy8oqLi0MQZNOmTXZ2dgsXLrSysoqJifngW/P19V28ePHAgQOnT5+uUCimTZsWGBi4ZMkSc3Nzpa+dPXv2jBkzvL29586da2VlpRzJnTlzxtLScsGCBba2tvv37wcApKWlef0fJpPp5eUFAGCxWIMHDx4/fvyQIUPGjRuHOv7Lly+PGTMG7Wf//v3W1tYLFy4cOXIkj8dr33fUfpAeQ0lJiaWl5fuPSU1JGWDNjF8wjLV6LPpXtHL0Am+HjV+u6R4jeyOlpaW2QxwCIia9/ee0dMC+X/ep20AIpCU3b94cP358O07866+/yBO+An9JWv/7vVHb0Kx9Jn322WdLlix5u/3Vq1d0Oj0jIwNBkNzcXBqNVlBQgCDIkCFDpkyZIpPJ5HK5t7e3p6cnh8NBEGTJkiXffPMNgiDx8fEAgKioKARBqqqq9PT0UlJSEATx9/fftm0bgiBCodDX13ffvn0Igvj4+MTExKBX5PP5CII8evTI1dUVbaFQKGVlZej2uXPnBg8ejB6Tnp5uYmIiFArfNjsyMtLOzg7d9vHxGTNmjEQiQV/W1NSgGy9evNDR0UFPnzVrloODA5fLRRBk//793t7eCILk5+cbGxu/fPkSQZD6+nozM7Ps7GzlJSorKy0tLc+cOYMgyLRp05YvX44giEwmCwoKWrduHYIgFy9eHDFiBNqPpqZmSUkJgiAPHz4EAFy9erWFwefOnQsNDf3w99QuetmI8Kvl4b8Ns7TQer3cTcRht/tZxt28Akc276KyspKg23pcDF6XVMp62c32QCBdh0QiUeCI79yNI8hl0vb1LJVKKRTK2+3x8fGenp6enp4AAEdHR39///v376O7ZsyYgcPhsFjsgAEDxowZgwqAeHl5FRUVoQfo6elNmTIFAGBoaDhx4sS7d+8KhcKEhITPP/8cAEAmkxcsWIDOWzo6Om7ZsuXYsWMVFRXvj/e5evWqo6PjjRs3Lly4gM6Ovnjx4oPvbt68ecryHUKhcPv27QsWLNi0aZNYLC4pKUHbQ0ND6XQ6ACAsLCw1NZXD4dy6dcvCwuLx48cXLlyIjY21sLBITExUdhIcHPz555/PnDkTAHD37t3w8HAAAA6HW7JkSYslyQcPHowYMcLS0hIAEBAQ0P3Lrr3JEVZXV+OEHEvtlj8CDABTLTSuRl1Si1U9H11dXRm39X9+GVdibGDczfZAIF2HpaUltf7d9/3qF8ZMy/b1bG1tnZ+f/3Y7h8NRlbjS1NTkcDjoNo3WXD+VQCAot/F4vHLpDvUrKBoaGlwuFx1yKTvU1NTkcrkAgKNHjy5ZsuT69et2dnbr169/j51sNpvH4xX/ny+//FJbW/uD705HRwfdaGpqGjp0KJVKXbx48TfffEOj0QQCQQtrqVQqDofjcrlsNhtdIESZNGmSm5sbAEChUMyePdvZ2fmbb75BX/L5fE1NzRZvSgmXy23xUXzQ4M6lN0WNlpeXm9LJre4y1yDHF8MRYevY2dlJq0Wtlq+SZPAmLB6vFqsgkK5gxIgRioVLAacaaLYSQklJPjFr2uT29Txz5kxfX190AQxt4XA4QqHQ0dFx+/btEomESCTK5fL09PT58+e3sc/y8vKqqiojIyMAQHp6+rJlyxgMhp6eXmpqakBAAAAgJSXF0dERAEAkEhctWrRo0aKioiJ7e/udO3eq9kMikaTS5oddNze3qqoq1AO1g+zsbE1NTTQro6ysrL6+XrkrPT0d3cjMzKTRaMbGxm5ubhcuXFi3bh0Oh1PtZP369Ww2+9y5c+hLLBZrZ2eXmpqKjvlSUlJaRIQ6Ojr+888/CIJgMBgOh9PqA0eX0pscoba2dqOkZRAUSoNQossw6GZ7egsYDObHLTu+/W2z2RIHLPH177XuXrmLqRM6pQOB9A2oVOrmTd9sOzafuzQSkOiquzCZ17Tzbn51Pr19PQ8aNGjnzp2BgYHh4eHOzs7FxcURERHnzp0LCAhwcHCYMWPGjBkzLl26xGAwxo9v68MlnU6fP3/+kiVLHj58WFdXN336dAwGs2XLlrCwsM2bN7NYrFOnTsXFxQEAFi5c6OfnZ2Jicvv27UGDBrWoQjxo0KBNmzZ5eHiEhISsX7/ey8vriy++GDFiRGNj4/Xr1y9d+ojZMjs7u4qKin379pmbm//xxx+qg7OEhIQtW7Y4Ozvv2LFjw4YNWCx2ypQphw8fDg4Onj17tlQqjYuLW7ly5cuXLw8dOrRjx44TJ04AAPT19adOnbp58+a1a9dyOBwej7d3797Lly+rXnTcuHGbN28OCwsLCgo6ceJEB3M920FvcoRWVlYsnoQrlmmQWpp9u0q4Ziwc2byTWTNmcfm8HXt+1HDRQRhYwEVE+VwfF+8T/x5Tt2kQSCfz1epVtXX1h3f78Ed8obALACQaqCmkp5/RqUiLjbmuOgX3saxfv3706NFRUVGpqalMJjM6Ohod2URHRx87diw+Pn7o0KHHjx9H8xSXLVumHPd8+umnypu7l5eXcq7SyMjoxx9/jIiIYDAYCQkJZDIZALBq1Sp7e/s7d+5oaGgkJyfb2toCAKZNmxYXF5eZmWlnZ7dt2zYAgLW1tTKbPjIy8tq1a5WVlQAAfX39rKyskydP3rt3T0NDA12Zext3d/cNGzag2ytWrEDHnQAABoNx7969U6dOVVdXHzp0KD4+3tTUFN21bt06Eon06NGj7777LjQ0FACAxWJjYmLOnj2bnJyMwWACAwNtbW0VCsX27dsVCgWa/oEKd4SGhpqaml67do1AINy7d8/d3R0A4ObmhmbT43C4e/fuHT58+MGDB998883Lly+7M4kQAIBBekziQWlpaWBgoHJhtlXORJw6+9O2A8OtibjXq5tXCmuiRVrX797veht7NxwO5/Hjxzm5OWamZoMHDzY3N1e3RRBI60RHRx88ePDmzZvt7uHZs2eHjh6LS04VCYUWFhazgoPmzZ3To9SUEhISFi9erEzd6+HMnj3bz89v5cqV6jIALcyrnG7tXHrTiBAAMGvO3CZ2w9Rffx5nrmVNx9eL5XE1IoqZ9dmoC+o2rRegqak5evTo0aNHq9sQCKTLcXNzO3LwV3Vb8T7weLwyfqTnQ6PR0AFrn6SXOUIAwLJVqz+bNefhw4fPnqSbW9vs9fFRi0oeBALpyQiFwitXrqQmJQh4PAtb+4mTJqFyYj0HHx+fx48fq9uKtqIUCuiT9D5HCADQ09ObOnXq6NGjYWFeCATyNtevXVu+JMzXWMtbn0Qj4EsKkqf/vt/TZ8iR4yc7OAgTCoVxcXG1tbXm5uY+Pj5E4rtzFruYoqKis2fPfvfdd+04VyaTxcXFVVZW6urqDh48WE9PDwBw6NAhX19fZUxsv6JXOkIIBAJ5F7eio9csXfTvWEdbHZqyccVA819Scz8dPyb2UUKLWP+2ExsbO3v2bEtLS1R0++XLl8p4me6Hz+crE/NNTEzS09ONjduUE9zQ0BAQEKChoeHu7l5RUREfH49GtXR/iErPATpCCATSd5DL5SvDl/zxib2qFwQA4DCY9d6Wn8cWnPr35IKFi9rRc1lZWXBw8N69e5VxmE+ePFEWsuDxeKhIpHKaisvlkslkDofDYrGcnZ2JRKJAIHjx4oWNjQ0auSqTyQQCgYaGRk5Ojp6eHppNiCKTyYqKiuh0ujJiEwDAZrNfvnxJoVDs7OywWKyzs/OBAwfQC7HZbDabTSaT6XQ6mlnR2NjIYrFsbW3fVsP5559/0NBQDAaDno62b9myBQ0mEolECoUCj8fn5OSYm5vr6uoiCJKTk6Orq6v0tY2NjRoaGpWVlQKBwM7ODu0KANDU1FRaWkokEu3t7ZUPHPX19RUVFfb29spgJTabraWlVVFRweVyVY9UF9ARQnoHaMKyuq2A9HQSExPNaHgn/dYXTcKcDA79/Vf7HOGJEyccHBxUsxFQzWsAwF9//fXdd9+5u7tnZWVt2rRp9erVAIBx48bZ2Ng8f/5coVBIpdKDBw+uW7eOSqUWFRXFxcXZ2to+fvx4zpw5qJzY8+fPZ8+e/dNPPwEAsrKygoODTU1Nq6qqHB0dz507R6VSIyIiNmzYMGjQIDabbWRkdP78+aSkpOXLlz979mzbtm0SieTzzz8nkUg//PCDj4/P2rVrL1++7OjomJ2d/fvvv3/66aeqb4TH4yEIolAoUPej9NzTpk2bP3/+zJkzDxw4EBsb29jYSKVSMzMz//vvv59//lkikWRlZe3fv3/evHkAADs7u3HjxhUVFYlEIhKJdPPmTR0dnaioqFWrVg0cOBB9CIiOjgYArF69+uLFiw4ODjk5OceOHUOTLI2NjefPn5+ZmcnlcrW0tB4+fKjm/+4u0jBtB20R3VYFVbCF9G3y8vJGTZlh4uxl4Opn6uw1btpstFALpG/TbtHto0ePLve1V4ryt/h7sWKUib5e+0yaPn16eHj42+0lJSU0Gi0nJwdBkKKiIuX2kCFDPvvsM4VCoVAo/Pz8BgwYIBAIEARZvnw5KjmNim7fvHkTQZC6ujoDA4PExEQEQXx8fPbs2YMgiEQiGTZs2K5duxAE8fb2vnPnDnpFsViMvFt0+9SpU0OGDBGJRAiCPHv2zMjICL2uqsFMJtPCwmLBggX//vsv6hcRBBk7diwqkL17925jY+O6ujoEQXbt2kWhUDIzMxEEiYmJUep06+vro5+GQqEICQlBZcRHjRp18eJFVSOvXr1qbm7OZrMRBLl27ZqBgQGqBk4ikXbs2IEgiEwmc3d3v3z58gc/fyi6DemnJCQm+X86867zyoq1iTWrH5avTbxlHTZkwrT0J0/UbRqkhyKTyfD/n6Z7GxwGI1fI37X3gz23OmpJSEgYNGgQurpmbW0dEBDw6NEjdFdISAgGg8FgMB4eHmPHjkVnKT09PdEyTAAAfX19dISkp6cXFBR0//59gUCQkpKyaNEiAACBQJg7dy4q4e3h4bF+/fpDhw4VFRW9f/B048YNExOTkydPHj16NDExUSaTFRQUqB5gaWmZl5f3888/a2lpbd68ecCAAS0KHwIARo4ciUbQeHp6Wlpaenh4oDYoLQcAzJ07FwCAwWCURnp6em7evPnXX3/Nz89HjXzw4EFISAgqIDBx4kQMBvP8+XP0dLSYFA6HGzBgwPvTx7sB6AghPRS5XD5r6aq6xZeAlffrVjv/mgX/TV+4DOl2IQg+n9/NV4S0A2tr6yJe60KMAIBCNt+SyWxfz7a2tq0mv/P5fFVJMBqNpiynp2zH4/HKtTocDqcU3W5xIp/PRwdMynZlb4cPH96wYUNycvLAgQNXrFjxHjs5HI6qbsCPP/5oYNBSfpJKpYaEhOzfvz8nJ0cqlZ4927JYsdIAHA6n+i5UK/0q26lUKvrfsWfPnu3bt2dlZfn5+aEzqG9/OMr/I2U7gUBQCqWqi17sCOvLEwUclrqtgHQViYmJPLPBQMe05Q6GFVffMSMjo3vMKC8vD5k9zcrVxtXfw9LFym/k0ITEhO65NKQdDB8+PLOqsYonanXvuYK6qTNmta/nOXPmPHjwICHh9bff0NBQUVHh7OyckZGBls+VyWSpqaltz2yuqKgoLy9Htx8/fuzi4sJgMBgMhjK/MDk5Ge0Nj8eHhoZGRERkZWX9+eefLcrcq4pue3p6KhSKpSq0iCZVLSVPoVC0tLQ+6nNQkpqaim6kpKSgRmKx2KlTpx4/fryoqCgiIqKhocHZ2Tk5ORk9rLy8vLy8XKnl1qPoxcEyvIa85w++0jHytHafZ2T1CQaj5rgjSOdS8OJFE6P1GwrPwKWgoEAZqtB1FBYWfjJptPYUY/MJzZYIq/gzl83+6dvdMz6b0dVXh7QDMpn8/bbtK/f+eHysUwtR4msvqhPrJQe/+KJ9Pbu7u+/fv3/s2LHz5s1zcnIqLS09f/78pUuX/P39Bw4cGBwcHBoaeunSJQsLi7Fjx7axT01NzXnz5s2bN+/Ro0d8Pn/atGkAgB07dixcuHDjxo0sFuvs2bNohb/Q0FBfX18zM7Pbt28PGTKkxeyoj4/PmjVr/Pz8QkJC1q1b5+PjExYWNnLkyPr6+ps3b966dUv14K1bt2ZmZgYGBmpqasbGxtbU1EydOrUdH8jhw4fFYrFIJNq/fz96ifnz57u5uZmbmyckJLi5ueno6CxatOjAgQPh4eE+Pj6HDx8ODw9vY45HN4PbunWrum1oprGx8eTJk2vWrGnj8RRtFxffFTg8ufjpyecJuyTCeg0dWwIJptj3EXJzcq7lseXWvm/vIr24/9lAs254tJw4bRJpso6mg66yhUAnag7Si/zxzMI5C/uw4pTaKSwsTElJmT17djvO9Ro8uJ4vWn08SiaTE7EYrlSWXtn0c0Z5bL3i+u27DAaj3VZ5e3uHhIRUVlZWVFQwmcw9e/agI6HQ0FCFQpGTkzN06NB9+/bh8XgAgL6+vqenJzre0tLScnFxQX0AlUq1s7OzsbFhsVixsbHHjh27f/++hYXFwYMH0bSKQYMGDRw4MDMzU1NT89ChQ1ZWVgAAPT29oqKiwsJCNze3PXv2EIlEIpFoYWGByuUEBwejU6/29vZGRkZhYWGNjY05OTkAgDlz5lhYWKi+i8GDB2tra1dUVNTW1np5ef3xxx/6+voAAG1tbQ8PD11dXQ0NDQcHB/QsMplsbW2NroBisVgmkzlo0CAAwN69e8+ePVtYWMjlcnfv3j1gwAAAAIPBQI20tbX95ZdfKBQKkUhcsGBBZWXly5cvZ86cuXbtWtQGY2NjLy8vNNNDW1vbxcVFNXukVbKzs/Py8tBnhU6nl4luq8LlcpWBv0212cVZ/5YVXDUwH+bos1ZLv5+mhfYlCgoK/OeurV1y+e1djN/HPrn2r5mZWZcaUFlZ6T95uNnqVtxtTfSrb8etmzNnTpca0J/puOj2ixcvjv11JDUxgc/nW1haTZ4eOm369BbVi9RL7xLdbgGDwYiPj3dwcOi2K0LR7Q+jxXAZ8Mket2Hfv8q7JORWQEfYB7C3t3fSIzY8uyF3C1JtJ6RfGGRt1NVeEABQVFREMm6ZjIyCNyI9y3ve1QZAOoKdnd2un/ap24r3QSaTme2N3FE7FhYWPeqpooP0EUeIgifSrd3nqbYUpP+hb+qja9Tli0mQruDy6WMjgqaWFsdxXD4F+lagrljz6UU7Qf756xe74epkMhmRtD5fIpfINfTaX9YO0g1wOJzIyMi4pDgun2dnbTd1cvDgwYPVbdQbDBo06Pbt2+q2op2kpaWp24TOpE85wrfB4Smp0avwRKq1+3ymYzCeQPvwOZAeg46OzpO4u5GXoq7FXip8Uupgaz156SdTJh9Q6lp1Ka6urrziRoYCwWBb5qXJC0TDQoZ1gw2Q9nH+/PllXyzX8tQn2dBwmvinr/KPzz7hbuf2X8Q5HR2djvTM5XIfPHhQWVlpZGQ0ePDgrgv9CA8P37Vrl66u7ocP7RYuXLhAIpFaiNT0Gfq4I7TxWGDjsaCuPKUo8+/n8T+a2k208Vig9Y5YREgPBIfDhU6fFjq9S1bI3w+ZTJ42OeTWrfsGE96oYMwtYFMbiQEBAd1vEqQtXLlyZcXXq2w3eJIZrzPYwBjLkuvFI8d+kpqYggaztIMbN27MmzfP3d3d2dn51q1b4eHh69ev//LLLzvH7jepqalRKBRd0XP7SE5OptPpbXSE2dnZoaGhytz5nk8vziOsS+VWxNVzSgUKyQd+Lvqm3j5BR0fPvU/RMEm4Mu/Bf5821fXKBWpIN7N3514nuU3Z0YKGpzXCan5TfkN1VKn4GvvGpeuYd8uXQNSITCYLX/m55QqXN7wgAAADTCZZsyncf479076eS0pKPvvss99+++3+/fu///77pUuXcnNznZ2d0b1SqfTp06ePHz/mcDjKU16+fKlM76uvr1cKuLDZ7ISEhLS0NGVKn0AgePz4cXJycmNjI9py+PBh5eD11atXCQkJqqouTU1NNTU1IpEoMTHx1atX77K5qKgoNTVVLpezWCw007GqqkqZ78/j8SorK1vYr9TgVpqdmJio+qb4fH5lZaVUKn38+DGaBNnQ0JCYmJiTkyOXN6v2sFis+vr64uLi4uJiuVxeW1vb1NSE7hIKhehZCIIUFxcjCJKXl/f06VPlO01JSVFa2G304hEhSYcgLBLXpDcKq8RkfSKdSaEzKRpmFJopGYNr5SZFphs5+ax19P6iuvQ+nCPtXchksurqaiMjo25Wqcfj8ZfOXYyLi7t0PSrnUQ7T1PqTaSOnhUzrS2ECfYyEhASSEYVq0voKrs5Io7///Sd8aXire9/PiRMnPDw8VDM6tLW1x40bBwBQKBR2dnYuLi5YLDYtLe3ff/8dPXo0AMDLyyspKcnW1hYA8OOPP2pqam7duvX69evh4eGoHCiHw3n48GFOTs7YsWMHDBiAx+OfPHlSUlKCwWDs7Oxyc3OZTOamTZtiY2PNzc0zMjKGDRt2/PhxAMDx48cvX74sFAoZDEZSUtLevXtRVTZVwsPD79y54+7uzuFwioqKLly44O3tvWTJktDQUDTg+caNG8ePH79161YL+yMiIj755BMAQGRk5LJly/z9/UtLS/X09IYOHYqetXv3bgKBgCZBamlpbdy40cHBoaKiQiaT3bt3T1NTc/fu3Ww2GxUov3Dhwtdff+3u7o7mTiQmJn799dfp6ekCgcDGxmbWrFllZWVubm779u2bO3dufn6+tbX1kydPjh07NmrUqHZ8Te2jFztCDVuyyQAGAACRI/xKEY8l5LGEVUkNcqHC6zv7d52FweCMrF5/vrR6aoMAACAASURBVDKp4Mnd9Wb2k4ytRmGwvfjT6KskJSd98fWaWnYtUYskaRIb6Rke/uX3bkilV2XYsGHDhsEVwd5Bfn4+wfSd+Z10c83sF8nt6zknJweV3AQAiESiiooKAACRSDQzM8Nisbm5uWgmX0xMzIYNG1BH2CrHjx/ftWsXqkCGDqH++++/GTNm7N27F21pMdnw/fff79y5EwAglUrd3d2Tk5N9fX0BALm5uWj9ppiYmJUrV7ZwhPfu3YuOjs7OztbQ0IiJiUEd9rvAYDBK+6Ojozds2JCamioQCJYtWxYVFeXv789msx0dHVFHCADIzs5++vQpmsgrFAqV86Xz588/evTounXrfvvtt6CgoDt37nzwU/Xz81u5ciUAANUWf/r0KQaDSU5OnjFjhuoIuKvpC7d+DA5DN6PQzSjAr+UuYY0489ciqjFZg0lBh4xUAxJQ+ZnhCRRj69Ev0v/MvLfRym2OpetsCv0DeZ2QbuNG9M3Pv15mvNDWwtAQbRFU8CbPDT556PjIwJHqtQ3SM1EoFOCt4CYlGCwGUbQzcxot0Ydu5+fnh4WFoSUAs7OzAQBJSUmnT5+urKzk8/l5eXnv6cfb23vz5s0lJSXjx49HA1m9vLwWLlwol8uDgoICAgJaTHtwOJydO3dmZ2ejdQdzc3NRRxgQEIDqYg8YMODly5ctrhIXFxcUFIRmWo8dO/b9QTcYDCYxMfHMmTOq9ufm5pLJZH9/fwCAjo6Oqiv19PRUyllgsdjffvstOTm5traWxWJ97PpraGgouhETE2NkZIQWogIAVFZWVlZWdpsMTV9whO+BYkDy3urIKxPyyoTsPC7rTo2EI0O9poG3Ns2YDACG6TCF6TCFyy56mX0uNmKUFsPFym2Oqd0EqNmmXiQSyYovVzDXOBHor9WkqCZ0sxWOS1YuLcjKV3sxT0gPxMbGRl4leddefhnX3Mr8XXvfj52dnTJnwMPDIy0t7eLFi5s3bwYAJCcnL1iw4M8//7S3t6+trUWdBwAAg3mtWKKUq/7mm2+8vb0vX748depUV1fXmzdvTpo0KTY2NjIyctWqVQCAhIQE1dDWSZMmjR07dvv27VQqNTw8XCRqllFV1cVWLs4pEQqFqspHraogKc9KTExctGjRn3/+aWdnV11dPWLEiPf3oKmpqdzeuHHjy5cvN2zYoKen99dff7FYH9B/bmGqsisej6ejo6N84wcPHny7pHDX0ccdIQAAR8Jq2dC0bJoXBWVCOa9MyGMJZYLX3wc7jyeXMOxcvnb0WcvKjypIO5SdsMs/+AxN21I9RkMAiI+Pp9hrqnpBFKIWiWhOSU1NRZ+LIRBVAgICeKWN4nohSa+V2yj7UdWKGUvb1/OcOXN++eWX+/fvBwYGttj1+PHjwMDACRMmAACeqNQI09fXLy8vt7OzAwCkp6cr50sDAwMDAwP37Nmjo6Pz6tUrtM6Rh4fHDz/84O7uHh8fP2nSJPRIiUSSlpYWGxuroaEhFovbLkPj6up66NAhdLu4uLiqqgrdZjAYSpnv9PR0VfvRglBKZ48u+1VUVJiYmCAIkpSUFBwc/PaFEhMTt27dig5tc3NzUZU4CoWCxua856ItGDRoEA6HW7q0nd9OB+n7jrAFeApO246ubffGWrpcLK9NbyqOqkTkCJ3pb8McjdF5iVW0X5MQ0nGKS4oV+q3PcSEMXElJSV9yhFevXf1x386auhoEIBo0jbC5YauWr4RD3nZAIpF+2vXTxl3fWq9xJ2i88RRV86gM+0qxYvn7ahi9B1dX18OHD0+aNCkkJMTFxUUgEFy+fNnT0xMAMHTo0K1bt/7yyy8SieTKlSvKU0JCQlavXh0WFpaUlNTQ0IA2LliwwNLS0tLS8smTJ1ZWVmZmZlu2bOHz+S4uLiwWq66uDhXzRCESiT4+PkuXLh05cmRkZGTb5W0/++yzn376ae7cuUOHDj19+rRypBUSErJw4UI8Hl9WVnb//n1U4XPo0KHbt2//9ddfxWLx5cvNooYMBmPZsmWTJk1aunRpbGysQCBo9ULDhg3bvHlzeXl5YmJiXl6el5cXAMDCwoJEIoWFhTEYjO+++y44OHjChAn6+vpNTU3v0hDYsmWLv79/WFiYj49PTU1NcnLy9evX2/hmO06/c4Stou+hpe+hBQCQcGQ8lpDLEnKzDCpvsHBE7MANdnxucWr0SguXUHOnEAIRinp3EzQqTcZrvUqZpElIo/WduN/13359Mf6KwTQLc10DAIBcKPsz5tjVm1dvX4tpd8Zbf2bh/IVNTZztW7frBhjR7bWxZJywis9PadBRaN29+6AjUumLFy8eO3bstWvXysvLTU1N//77b/S+7+XlFR0dHR0dbWBgcPXqVaUv2bZtm6OjY0FBwfLly/F4PFoyYu3atQ8fPkTls3fs2IHH4xcuXBgTE1NUVGRgYJCWlmZiYgIAOHLkCLoEGB0dfeLEiYqKil27drHZbHTZ7JNPPlFG7tBotD///LOFqUQiMS4uLiIioq6u7vjx40FBzTqFQUFB586de/jwoZ+f34oVK9Cavd7e3miRCgMDg2vXril9+d69ey9dupSVlbV06VIajYYGS3t5ean+9+3atevs2bMFBQVTpkz58ssvq6ur0atnZGTExcXV1tbicDg/P78bN27cvn3b2dl52bJl6KCTRCIdOXJE+Qs3MTHJysqKjIwsLi5mMBh79uxp99fUDvqI6HYXIeXLCTQcAKC2LLHwycmqovtapE9MLWYZ2g+gm1HwFPjA3oU8ePBg6qKpbj/4v70rc8PDh9ceog/jvZ2kpKSZa+ZarHICb8YKVl8pXTJk/ldrvlKXYeql46LbLBbr2InjcclxfD7f1sZ2+uRpEydO7B5Noh6Ig4PDqVOnvL29P3xoTwWKbqsN1AsCABhmQxhmQzjlFYVpZ16UrC0spJM5QZr4sZpMHTqToueiSdZvuZQF6SDl5eXGCkz9A5beiDeEiWtjSs3xxNLS0r7hCH878pvOeCPwVnq+/lizv3//p986wo7DZDK3fL9Z3Va0CYFAoFrGvStgMpmqZeshLeinz0ftQ9PUZODkdUErUgZ9+h3F/VmV9iwNO4y4QcopfT17LqgWc0sFCmkP0kbqpYhFos+sjHQeVZYdyqhLreSVNtWlVrIOpBun1X1qYaiMnevtZOfm0C1aKRGOI+MFotZXZSBq58mTJ9evX0cTJ5TI5fLz589/bFeRkZGpqalLlixBF+GSk5O7In/u7t27yqlUyNtAR/jRYDBYI6tRQyb/O37xYxMfM+tgYwMvbQCaZ5h5ZcKiqMrk7/My9hW++K+8KrGBVyZE5D1l/rkXYWdvXyoCZ8d77jUzGZPeZBpZMu4JZ7818+QY90K+wt7+nZoJvQsCgaCQveuxCaq49TgyMjI2bdpEo9GCgoJKSkqUlWYBAEeOHPlYMRS5XB4bGzt8+HBDQ8OUlBQAgK+v7/nz55W5FpDuAU6Nth9Vnbaqktj0O+ssXWdauc3xHGSDyBF+hQiNu6lMbBDWSWjGZOfFFsq5VsgHGTJkyBcN4gquyNtUx9v0dVrVyyZBsVCBVsTuA/h5+z3ITdEbZNiiXdwg0tfVV4tJkHeRnJy8e/fuixcvogG9EydOPHPmzK1bt8aNG1deXs7n8z+2WAQOh/vjjz8QBHnx4sX69evRxk8//fTYsWPqSiTon0BH2DkYWY0a/tml0udn750Zp6XvbOU2x9R2PJ1JQVVqFFKFoFpMoDZ7QYVEkXviFYVBalXsBoKCw+H+OHFq6ZzQrV4mXsbaaGNyOXt7euXJi1f6jOb1V6u+vBI0QttZD0dR+WdEQM350l/W/6Q+uyAtEYlEs2bNio6OVk1rsbKyio+PHzdu3IULF9A8PBQ2my2RSAwNDWtqaiQSCVpHOi8v79mzZ8pjMBhMcHAwDoe7cOHC9u3btbSaZ8gdHR1/+eUX6Ai7k+5whDKZbNGiRVlZWVgsds6cOV1UtUTt0LWtXf2/dfZbX1EUU/IsIuvhZnOnaTaeiyh0YywBSzd7nd6LJWKZYwy4pQJ2LvfV7RopT0Y3bVaA07alETTg00kzvn5+56Pvbl7/5babOXhEIcfiXNzcr8Ses7KyUrdpnYaVldWBH39d/e1azVEGGrbaWBKOV9rEuVszO2jmlMlT1G0d5DVnzpxhMpkODg6qjWVlZWgBitTU1C+++AJtVCgUN27cuH79+qxZs2xtbY8fPz59+nRvb29HR0elMpmS2NhYExMTgUCQmZmpjP8SCoUymQwmz3Qb3fRBL1iwYOTIkXw+39fXNyAgAE2+6ZNgcUQz+0lm9pMaa54VZ51Murpg5KyYtw/TtKRqWjbHicmEclQxvP5pk7BabD7OoLmdL5dLFCSdfl3owN7e/tyV6wCAPnxfmBo81WuQ1+9HD6fGpPL53KEeA5ccWdzNwuI9HxlfXvWYrVyMV4VmQtFxbJbIEFSLG7I5bx8DANB20KD/X4+78QWfx2o9FsnAS4eo2cov7f79+y201+Vy+f3791G1a6FQqMzNKCsrGz58+OHDh/38/BgMBp1OV+qqtEAulz969AgVYfn2229fm6qtXV1dbWpq2upZkE6nO+4seDx+5MiRAAAajWZqasrn87vhompH28Bt4Kh9qi0SUSMGgyWQNFsciafgtO3p2vYtC8c05HJLr1chCkBnUpSi4a3+i/YH+qoXRDE3N9+zY7e6rejRIAgiE8pb84NANUgbkSMyQevBR6oxa4hM0fphGPAuVW65XG5gYKDacuXKFUtLy+HDh6PmKdvNzc0bGxvJZDKDwQAA3L9/f/ny5a32icPhtm3b1mo7jJfpTjr55pKTk6MsNYni7Oys1E69du0aj8frtxVtKotish5tMbGZYO0xT8fwwzlwBl7aBl7akiYplyXksYSVCfU8lhCDx2qYU2ymGhO1+vVIse9RVVWVlZUlFApdXV1tbGzUbU6Pg0DHWwa1DCl6G5oJmWbyYe0YHScNHaePk+MYOnSoqtRndXX1zz//fOrUKfRliyy9R48eodLbFRUVZDJZJpOVlJS0fUq/oaEBFT+DdA+d7AjPnz+PlulS8v333zOZTADAw4cPt27dGh0d3W/FHSxcQk1sx5cVXE2LWY3Dk63c5jIdgz9YIpioRdDTIui5No8jRQ0SQaUYpxS1QUDhxQqyLlqXmIwnw6jU3kd9ff2MsOVZr+rEFt4yAoVWcVJPUnfx5BEnJyd1mwZ5TXh4+Lx58+7evTty5MjExMTz58+fPn3a0tIS3evo6Kjq6uLi4lAN7qqqKk9Pz9jYWNWKvh8Eg8HA/PfupK2O8MmTJ0lJSSwWKzg42MfHR9n+9OnTAwcOsNns4ODgefPmbd26tdXTk5KS1q5de/369RZzC/0NAknTym2OpeusmldxxVknn8fvZDpOtfFcqKHT1hEAWZdI1lVRscEAHQc6p5j/KobLLxcStQiv51FNyVhiP33m6EVIpVL/cZNfDFknn9isBskHoKYiZ+SUmamx19FoQ0hPAI/HnzlzJiMj4+bNmy4uLr/99pvq3tDQ0JiYmM8//xx9+eWXX6JDuoEDB2poaJibm7c9zrmsrAw+A3UzbXWEP/zwA5VKjY+Pt7S0VDrCysrKESNGbNq0ydHRcdWqVWh06NvncrnccePG+fv7o7PhCxYs8PN7q4RufwKDwRpaDDe0GC7kVZU8iyjK+Mdz5M5296bnpqnnpgkAQBSIsEaCJi/WZjSR9YkOs5tvo4gCQRQAi+8jKQd9ib9P/MuyGCV3C3qj1cS5esKOtd/9cOHEETXZBWmdAQMGtJrD6uTkFBUVJZFIUGVt1YqyaBmmtnPmzJl3rSlCuoi2OsKoqCgAQItCXMeOHQsICEDzQAUCwY4dO1p1hBQK5d69e8qXysmEFigUCoFA8N9//ylb3N3d3yMgIpfL3y5H2bsgUhgO3muBSrHKptrneKImTaudtUNJDDyJoaE3sHnxQ9lt/VNO4flKsj5Bg0mhmZHpTArViITBQb+ofiIuXecPbyVfEHEek/Dz9739F95uFIreJ1K4YsWKiIiIVu+BbScpKSkgIEC18i1ECYIg7fiPwGKxHxyOd2iNMC0tTVmL2d/f//nz50Kh8O2ywng8XrXC1rsQi8V8Pl9VrA+DwVhYWLzreIlEolr7sW9QW55ekPqLlr6rufMshkUgBtM5a350R5L7JgtRlYRfIW4q4VfGN0jYMrIBgWpKMgzQJmr15YDMHk5NdTXQNm5lBwYjA7i+9wtvI1KptOcUxmkjWlpaCxYsaN+5yjunj49Pv42i+CAKhaId/xFEIvGDMecdugNWV1crJYXQ0lmVlZXW1tbt641CoTAYjIsXL7bxeLlc3tWS7d2Pw6BFdp5z0JT87IQtFk7TrD0WUDU6J52IrknT//8AWyFR8MpF/HIhhUIhUZsDUCse1RNoODqTQmFAsZtugsEwKGyqAnpvPfAhCB6j6Hu/8DZCIpF6snhQWlpaVVWVhYWFm5ubslEul0dGRoaGhn5sb1KpdNmyZSdOnAAApKSkGBoa9iXJiE4Eh8N10X9Ehx49qFSqsggAmjVBp7dMhoN8LGhK/rCQ//yDz0glvNiI0cnXF7Orszr5KkSsphXV2F+PpE1QbazP5mb/9TLp25xnh0tKr1XVZTaJ6iWde2mIKjMmj6Okt1JiDZN332dQH9FT7Uukp6dv3LhRS0tr4sSJLBZr9erVyl1//vnnmDFj2tFncnIyurIIAPD19Y2MjJRKWy9JDekiOjQiZDKZyoohpaWlFApFXx/KBHcaWvpOA0buchv2HSsvitvwQsfQvauvaOSrY+SrAwCQCeTKoJuSq1VkfaLbcviI2iWEhy08+NfwIqYX4vTJ69aaQsb1Db/GRKnPrmaePXsWczfmeV62raVNYEDg0KFD1W2ROklMTPzpp5+UotsTJkyIiIiIjo4eP358eXm5QCDQ0dH5YCctePXqlbW1terwFxXdDg8P70zTIe+lQ45w+vTpK1as+P777zU1NY8dOzZt2jQ4u93p4Ak0K7c5qi0lz09r6TnqGn942bX9F6XitB3o2g7N43tVSY7GfF7BmTL6/5Vu6EwKEYqjdgASiRR/68rUeUtexB8UmPvI8GTN6mda7MLI8yfeFVbWPcjl8oXhi+KfJ5IHapKMqSmVz/7dcsYEb3Q98mr/nPgRCoVz5sy5detWC9HthISE8ePHX7hwAU0cRGloaJBIJEZGRtXV1RKJBM2lzs3Nff78uWqfU6ZMSU5O9vLy4nA41dXVhoaGAAAHB4d9+/ZBR9idtPUWFhYWFhUVxeVyU1JSNm3adOrUqaCgIDQpwtXV1djYuL6+/s6dO11qKwQFg8GmxnyBx1Os3OebOwbjiV1+V1KNL9V2oHt+acN9JeSxhJXx9VyWEEfAoh7R0Lt1kUbI+zE0NEyIuVpSUpKZmSkWi52dx7m5ual9hezrb79O4WYyVykT2nSAD6h/XD19zmfRl2+q0zI1cfbsWXNz8xZx7CwWy8XFBQCQkpKinCZVKBQ3b95ERbft7e2PHTsWEhLi4+Pj5OTUIkFQoVCgwss8Hk+1XSQS9WFx3R5IWz/o3377bd++18qZ6CMhFos9depUcXFxU1OTm5sb/Nq6B0uXmZYuM9nVWYUZf2Un7DS1m2jtsUCb4dJtBhC1CHpuBDR5EQAgapCgouESjlTpCNm5XLTmBo4MJwnahJWVVc8JkeDz+eevRFp/23I2XtfH8EV6fk5ODlpyoTsRC+pePDnaqui2jtEAU9vm0VhTbTYr/3KrPZjYjtc1apYyryq5W1f+uNXD0Ioxb7e/S3R78eLFAACxWKx8dlGKbg8ZMkRfX59KpbbQ21KCxWKtra2fPn3q7e1dVVWFjggBADo6OlB0uztpq+ui0Wg0WutiYO0OE4V0BB1D98HjDor41aXPzyZdmU/RMHEfvlX5f96doGI3+h5aqo2CKnH9cw6/QkTUIigVw2mmZBwUu+kNpKWl0e21QWujUrwjNS4urvsdIQaLJ5K1Ws2pwBNeRxJicUQCSevtYwAAWNxr0TIcnvKuw96VsySXy1ERbSWXL1+2trYOCAgAbyY+KkW30ZiJBw8erFix4l3vCwDg4eHh4eHxhqlYLBTd7k7gGK53Q6YZOvqscfBeVVUSi8X2IBlu00B900B9RIEIq8XcMiGPJazNaBRUijUsKK7Lesq4B/IueDwehtz63CyGgmVzGrvZHgAAkaxt7/U+d4KioWvnoPthJRcGcyiD+XGBP/7+/tnZ2cqXVVVVv/zyS0REBPqSTH5D6Vspul1eXk6hUKRS6ceKbqtq00C6GugI+wIYDM7Y+o247YzYrw0sRphYj8Fg1fkVY7AYqjGZakw2HKwDAEDkiJT/WhiCVyYsPF9BNyPTzal0JoUGxW56DBYWFrKad6TN1MrsR36cZljfYOnSpfPnz79z5w4quh0ZGXnmzBml4oejo2NxcbFyekwpul1dXe3u7n7v3r2PEt3GYrHKhApINwAdYd/EwHx4UcbfT+9/a+k6y8ptTqtrHt0PBodRjaahm1JsppnwWEJuqaAyrl5UL6Eak9BJVF1nTQINVtJQG66urhg2Iq4XkvTeEIqSi+W8p42j/hqlLsPUCB6PP336dGZm5q1bt1xcXA4cOKC6NzQ09Pbt20rR7a+++gpd8Bs4cKCWlhaTyWx79BOLxXq7kD2kS4GOsG9iahdkahfEYxeXZp+NjRitxXCxcptjajehszTbOgcM0DCnaJg332rlEgW/XMRjCRsL+AQqXtelWTFVUC3GYAFFH4rddCvHDv8zI2yG0UIbqmnzFyFhiyqOF/743Q/9WQnT09PT07OVYqKOjo5RUVFisRgtn6RaTfBjq0uePXv2/WuKkE4HOsK+DF3H2tX/WyffL1l5UQVpvz9P2OkbdFTbwO3DZ6oDHBGraUXVtGopoVSfxalOYcv4cjqTTDej0M0pdCbljVpUkC5g6JAh185eXbo6/GVDKVmfKmaLaATqX7vaKZ7SH1i+fPnp06c7KLqdmJg4fPhwDY2PKxoM6SDQEfZ9cHiKpessS9dZTXU5FHrvC8hmjmYwRzNei908aSq5UiWXKjQtqc6LLPrSMLG2tlYgELxHaL6b8fT0THn4WCqVvnr1ytTUtEU8CKQFHRHdVuLr6wtlSbof6Aj7EVr6r0PexcKGpKsLmA6TzZ2mE0i9YKarhdiNlCsT1UuUXlDcKC2OqqSZkNG6xIReJXYjk8l++vGHc6dOGlKJZDyutJE/JGD4nv0HtbW11W0aAAAQCISPndwDACgUinPn/7t883J+Qb6pqemogFHhi5e+KwWrz9BxHwa9oFroTfcLSAcpKyuLjb2XnvPC2sxoqK+3q/+moqcncpL2mtpOtHKf1w1app0IQQOv6u2ImniDwdq8V8KKuHoeS4gjNovdaJhTNC2p2J6dvDhz6mQrzqurExzw2GbHfr0oZ8ywIbGJj3vpFJlAIBg9cXSdBofurasxwrSOLfo749Rhn8O3r8bAtGNIDwQ6wv7Ct9t3Hf3vatOAUKm+DzajWifyVwcS9+aFCM8R0tLsc49vLCGSta3c5jIdp6qmJ/cWMFiMnqumnuv/xW7qm8VuWHdr9dw0TYbpoe0yoRyDxeBIPcgvXr1yhVJd/IXfG9OhE20YPFn1nu1bd+z9WV2GdYQlK5ZwHCVGQ5vT5ihGNIoRjWfXNGn65GepT+GgB9LTgI6wX3Dg8JHfHxQ0rb4PsDgAgAKA+iHzUjKiJkyfk3D7msPglfZey6tL7xdnnXwev/OT2bepmmbqNrlDkPWIZD2ivmdL6ZCKR/XlD+pUxW7opmT1jhfPHv87zE737fYQW8a0a1d6oyNsamqKS02w2thygoFuqcU3anj06NGIESPUYRcE8k6gI+z7KBSK3Qf+aFr9EPWCSmQDgvOzryYlJfn5+WEwWCOrT4ysPhELG0iUjy4l01swH2vAHM0QVIvR8WLtk0ZBlZisR9SwoFhPMVaLR3z1stTKv5UsTwIOi8h6ZVG658+f023eoV5mRUxOSYaOENLTgI6w75Ofn68wdgbEViY86x2DomMf+vn5KVtIlNejE3Z1VuKVuRYuM6zd5lA1md1ha9eDwWJoxmSaMdnQu1nshl8pElSLMfjm9Tm5WPEyuppqRNZgUqhdL3ZDo9G5YhmJ2ko2iKJ3RsRKpVLwjg8Ni8NKZG2t85yfn79hw4bOswvSu8nLy+u6uGXoCPs+HA5HTn1H/CFVu5ad9a4TdQzdh38WVfr87L2zE7T0na3c5pjajlevZlung8Fh6GYUutlr/RQsHkM1IHFLBBWP6sUNEqpxcyQqnUmhGBAx2E52TsNHjYlNvhLqaNSiPb+eZ27ZK0VZ7e3tRSx+q7sUFRL3kW2KyfLz81u+fHkbhacVCoVCoYDVb/o2fn5+qL55V4BpVc1dLZSWlgYGBpaUlLTxeC6X20tj6rqZ8vLygcFhNUuvvb2LEHd0nzf2ixXL3t+DQi6pKIopeRbBaSiwcJpm47GQomHSNcb2LJRiN1yWkMcS6rlqWk5sLpSDyBEMFtPxMVtDQ8PwwQOOBFiaa712xnypfMGdggMRF7y9vTt6AXUwbHQAzw/RdHhjjl3SJK74rSAvI4dCobzrxPYhk8lkMhlMc4S0G/gM1fcxNTXVx4pqagqBge0bOxRyrfSI4B3nP9gDFkc0s59kZj+psTa7+OmJ++cmjlv0GIvrQcUuuoh3id0AAMof1JXF1tGZZPr/427aJ3ajq6t7KvLygtBpgUaUAbokEg6XyxZeKm3avHtvL/WCAIAzx04PHxcoDRTpehuhc8tN+Q31F16d/ONEp3tBCKTjwBFhvyD9yZNxs5bUzT4BjP8v5iviap1ftXqcx7ZNX3ekZ7lMpFBICcT++EXI+HIuS8grE6KhNwqpgs6kaJhTTQP1vwu9PQAAIABJREFUPzZDQywWX79+PfNxklAocBvkPenTT3V1Wwkl7UU0NDR8t/372AexIqkYj8V7unns2rqzRXn3zgKOCCEdBDrC/sLTp0/nLV9bxZMiDBsstwbPrd66fvXihfM62G1lUUxqzBcmtuOt3eeppSxwz0HClfFYQn6FyHioLp6CAwAABFTE11P0iXQmhUCHsy9dBXSEkA4CHWH/gs/nFxUVGRkZGRgYdFafUjGnrOBqYcZfCIJYu8+1dJ2FJ/RxJa22goBXt2s4JQIeS4gjY18nLzIpzZ6yyxCJRMdOHo+5f7u0tMTO1m7SmImzZ83uq+Ek0BFCOgh0hJDOAUEUNa/iirNO1pUlmTlMsfFYoKnnoG6jehCo2A2XJeS9EvLKhYwB2rbTuyrgqLKycuT4T4Azke6pQ9Kniqr5vPQGagXh3s3YHqJf2rlARwjpINARQjoZIa+q9Pnppro834l/qduWngoCFHIE+//MxapkdsXDumZxVCaFZkrGEjqU2u8T4CsNxGvYv7HK2JheY1aqf+PS9Y703DOBjhDSQfrmVAlEjVDoRk6+X6m2cNlFWAyOpm2pJot6Hhig9IIAAENvbQ1zChpxU5PeKKgSo8uKGuYUxiBt3Efq3WRmZrJxHGN72xbt2oMMch7llJeXm5r2vlJcEEiXAh0hpMtprH769OEWbYartcd8Y6tRfSwlv+NgsBiaCZlmQjb0+b/YTYUI9Yva9nScXnNWBjuXS9QiUI1I70/qT09Px1q3nshBsaFnZGRARwiBtADekiBdDtNxqqndpIqiW0WZxzNiN1g4T7d2n99bdL2lUukffx+LvHH3ZWmprp7uJ8P8vlm9gsFgdN0VMTgMOk3aor0hl9v0gi9ulFKNycq4G6oBqUVSv0QqAe8YQyI4jFTaK/VLIZAuBTpCSHeAxRHQlPymutzirH9jT4/WN/OzH7RMz2Swuk17HxwOx3/sp8UmAfxhu8AUi1e8+ufZMaeHjbn13wkPD49uNsZmqgkAQC5W8MqEvDIhO4/LulMj4coYA7Vtp72Ou3FydALX5a32IC8XOTo6troLAunPwGAZiBqQSfmsvCiphGs/6APqbupl6pyw6/RAqVfoG621xeYRs15kJBGJ7ZGS6VxkQrlMKFeK2rBzuWUP6k7fOVU3mFNrVNNIbFQeyX/FkV5ufPo4sx1XEYlEPTkUBQbLQDoIHBFC1ACeQLNym6PaUlF0i0jW1TftQaJiTU1N8RnZ0rV/ttzBsG60CYy5fXvSxInqsOsN8BScakqili0NADCZPDHl7hPb2ik4PKGcVlZBLc+ryn58K+/GpVb0Zt/Dq1evVnz9fcazHCmOhJOJrC2Yh3Zt8fT07OT3AIGoG+gIIT0CuUz05O5XGAze2mOeudO0nqDZlpeXp7BoXSuHw/RNfpLVExxhC7AErI6Thp/TQONA3Q1bNhbmFJvTrWx1HCcyJ/7473eWDs21tATVYgIdT6C9L6k/Ozv7k6mzayb9hIwKRFsqy56NmhV+6tcfxo8d0+XvBALpRqAjhPQImA5TmA5T2NVZhRl/ZcfvMrYZazdwqbaBmxpNUigU4F3VJTBYhULRveZ8HJaWludOnn3X3vJ7dfXPOXgK7rVouFlLsZvPFi2vnv0vMHF63WTmVr/0ctjqCUUZw6B2NqQvAR0hpAehY+g+eNxBEb+mNPts0rVFZKqBq/9GBtNfLcY4ODhgWa2vqGmUp3mP6kGzuB+L3UxTO8RUWCdBkzRYt2t5ZUKiJl7HScN6ijEA4MWLF3UEvTe8IApdn+84NjY2dmLPGw1DIO0GOkJIj4NMM3D0Xu0weFVVSSzAdEhjpSPo6uoOtLe4++yG3C3ojR3sco286HFjv1eTXZ0EBlAYRAqDyBioBQAACBDUiKW85kK4hYWFJmZDwyqqCsikPAq5kEwS//+L4DBcsvNf9Bw/mJ6enpaeJhAKvAZ6+fn59VU9VUiXAn80kB4KBoM1th6t2vI8YZc2w9XEdhwW202lEE8fPeg3euKrumKxzxxA1QEyMSb3LuPWtsiTf/S1uUEMoBqSgCEJfUUikWpEdc+oFAehaFwTx1IsKSMS8sjkfAo5H8HRekZ8ZnFxcfDMqSJNGdacCHDg6N3jkhLB6X8ifH181W0apJcB0ycgvYbK4tsv0o9w2UWWrjOsXOd0T0q+UCjc/ctvl27eZjdyKGSSv8/g7Ru/ZDKZ3XBpNcJms+2Hjq37MgF9SUAQG5HYUSRyEIpcKgscxup5T2mOHZUJ5TgS9v1iN10Bj8fz8PXUnmFKt9JSNorrheV/FDy4fs/Gxqab7YH0aqAjhPQyeOzi0uyzpc/PajFcrNzmmNqOh5ptb6NQKG7FxFyPjSsseeVsbz15TGBgYOBH9TB7yYpLUmeR/1LVRmzevQGpP6c9vK1sKThdVv+cQzMm098tdtMV/Ljnx9MFlxgjW8rFNeXW25WaXog43+UWQPoQ0BH2O3p4cnQbkcuErPwrJVn/igR1XmP3M8yGqNuiHkRTU9PISdOKqHZNzpOALhPUleg8jfSk8W5Gnm77Vy8SicZNnflUrNPoGQoM7UADSyP3pmlFwsMbl1oUs2wWu2EJ0TpTUp6MbkbRsqGZj+20mpdv4zPCFxeqRdAktdyBgFc7n5fkFHfdpSF9D+gI+wvV1dXfrfsyPSWZhEHEcsTQxPT7nXuGDB2qbrs6SmPNcxJVj0I3VrchPYjAiSHxljNlnpNVG8nxf03GZp079pY+wLtBEOTWrZiomHvZ+YXWluYTRgyZHjL1g9EoMqGcxxKK2VJUQxwAIGmSViWxaaZkOpNC0u6c9V0HD0ej9XYA08rYs3Tns5fZbb2NQCAABsv0E16+fPnpqMA1LrpbxtuhLSWNgvWLZq/a8uOMWbPVa1sH0TZwVW4jClncpZkmNmMtnD8jkDTVaJUayc/Pz66TyKZMbtEu8l9yb//wuro6fX39NnaFwWCsrCwH2FvSxI0Wdma21lZticnEU3Da9nTVFiwRi8GC6hR20cUKBAFKxXA6k0LUaNnhs2fPHqekllfXebo4DB8+/F2VhA2NDEV1QjKD2qIdkSN4zPuEAiCQt4Ejwn7BpFGBYXpCL+M37il8qTz0VsG91AxdXd13ndjrqK9ILco8Xv3ynonNBGuPeTqG/U4P7MTJk0tus2WBK97epXlz67kl/uPHj29LPwqFYs3y8KxHsRPN6FZalHKe6F61CGNoEREZRaPR2m2epEnKZTXPo/JYQvOxhsb+zT8/Npv96cwFeTxCo/UIGVWfXpdLybryw4a14Yvmv93Pkb+O/Hr3sOGnFi3aGx5X+WO8/jhw+D3vC4tVW04OpGcCR4R9n/r6enb5Ky9XuxbtNAJuiqXWlcuXFy5apBbDugI9k8F6JoOlYs7LnP8e3/icSNaycpvLdAzGE9p/7+5d8PkCOaHlOAlFhqcIhcI29rPt242Y7KSIMfbKlqkO4OKLmrDZM85d/jjNUlWIWgQ9LYKea8vxOoIgx9ZEfeG0NVdLN49CzieT8ylTaj/5auMfMw31dad8OqnF8WELw44cP9qQXKXra6Rs5BSw+bH1Pzzc/vZ1MzMz13//dVFJkRwo8BjcsCEBP/2wu8ViJ6TfAp+M+j7FxcV2Oq3fGe21SC+eZ3WzPd0AgaRpO2DJ2IWJLkM2VJXERv/t1VjzTN1GdRN2tjZa9bmt7qLV5rUxr0AgEFz+7+yXA1vGZIbYGTQU5eXl5XXUyre4e/fuzsrHX9nZx2vQ9WWyRbX1lwqKT7+qWjlu/80jSeCteSs8Hv/g1n2nRquSXVk1Z0prL7x89XO2VjL+4a0Hb8/9XrtxfdKcyewhEuZGF8uNbmbfOKfTsn2G+5aWlnb6G4H0RuCIsO9DIpFE8taFMf/H3n0GNHn0AQC/7IRAQtgrAcLeGxXFibgHIu49697baquvWq1ba6u2WvfeW8QBKiooMmWvsEcY2fv9EBogBEQJ0/t9knvG3aOYf+6eu//xxRKchuoY2QUgEEhDi36GFv0E3FIMXvWrpq6nT58+hCVrK9llQLN+PCjL1mSmubq6Nucm0dHR3YzJKFVTUfob4SPCw9W+r+HNR2FMp1FMLJaBxYaSSQAApExmLhTZ8/j2BIOS4hIDIwMAgEQoLf1URTTGE03xJBLp6rkrLBYrISGBw+F4enqqHORns9mLVy2mrXREE/+bp4MAOl6GaDJ2ypypr59FqPdBoM4IBsKuz97ePqGUJZHJGn6uvSkVjPPv0y6taks4jdoN5XmsgpdXR1HtRlu6TCGSld8wdQE4HO7Ukd+nLB9dPuFkbbLQ7Gj964sunzuOUBXbGmKxWCSM6jO10Mjqyoqv3kEsFp8+e/7K/dDMzAxjY5OB/t1XLJpPJpMbO7+EWQmounVLpAhEFg6bhcN+LP/wU/VQeSCUSWWcPF7RWya3RKChj5XPuHE0dcXqIglE1bl+Hjx4oOGuXRsF/0OypuTd/1JUVGRkZKTyQujHAQNh14fFYoPGTTj27tES93ojXbEl1YlcREBAQHs1rF0QtEz8g69mJ1x8cXk4Wc+xSy7JHxw48Ol53QVrNuYUFAKiDoJbYUu3OH77YvO7cTQaLZstUnkohyf1pX9lfJXFYvkPHpVh0IPtvQkMpmdVFkYlPf67R7+wW5fs7OxUXmJFNUGUZctU7nvFZChiFRqPshprAgCQimWcAj6bwWPl8AoiygVMEW2QgWk/FRNi477Eo0xUb6GMMyWmpaXBQAh1qf//UGO2/G/nzEkZS17Fj7UgWekQSznC8EL2izLRjYdPUKgfbq65pralc69Njj3WFGQ8yYo/H/tqi7nDWLrbDA0t5VdinZenp+f7sPsAgIqKCgqF8q2XOzs7Z3EkRWy+kWa9Bfh8seRxbtWmgQMbu1Bu6vylSS6zRN7ja37WMRP1mpNP9xs6blrqp7cqf+Umjx11au6Gcq8xygfyE80oRBJJeXINEo3QohG0aAQAgFgsFnCFeFxtU3OfllSmsDWpBC0qQRetLxM1MjdeLMNiVcdI6IcCJ8v8ENBo9LmrN5YfPBln7rc7D/2IYOM6e/WbT7Hm5l1wbLCZkCisme0I/+ArvUafFwmqn18IFHBL27tR6vcdURAAgEAgDv51cv6LzKxKrqKwjCuc/zx99c9bmxjhBABUVFS8jU+pjYIKJo4Vpj4vX75UeZWbm9swT2vNOxuAVFxbWpSqf2n2v0d+/2qDUVgkClf7aWbaV898iCGWjClPZPnw+x5E/TE7dc7gvMFOlc4I2X9DvjLAyqh0dnZWfUfoRwLXEUKQMplULBZxf9gl+QrR0dGrFswTVFfSyIRCNl+Iwv68c/fwESObvur169ejfrvGHL1XxbGoq9vphZvXrVZ5oVQq/fW3vX/9ex6YOImJ+ujiL3oo4cUTh93c3JquUSwWi8XixrLHSaVSv57+VgNdrMi2BnyD+9R7PBQPAMAMKwzWHLVh2zp1JbuBOi84NApByphFMW9uTzGmB9Ldpusae7d3c9qNt7f3q6hPfD4/KyvLzMysmd87xWKxDNnIeDsCJRZLGrsQiUT+unHtlnWrMjIySkpKHB0d1ZLqAYlEXr1yaeCIwE/O7zQcyVg2nl/MYb+rsMFahPScEnsgAwCg2WSyG6jLg//kEKRM18RnyOyovNS7n0JXy2QSc8fxli5TsfimxgO7MDwe7+DQYKv6xtnb22PyVCxpBwCQCmO8hvVq+nIUCmVra2tra9v0aXIymezjx49R0Z94Ar63h7ufn5/KJHA0Gi32/eez58+GhT9n5DM87RzGLA0aOnSofA6toFIkT3NT+LqcxeChMEgTf12V826grgoOjUJQE2QluRGZcWdLGa/NbEfR3aaT9Rzbu0mdgP+Q0W+cF8rs6m/8VFlg+s/o9JhIdW1+kpmZOXzCjFIty0qzblIkRrs4lpgTeeWfP3p0b9HGvHymEMgAXrdmEk1JdCUziSWfd6NpRkDh4byKLggGQgj6Oj67KCvxUnHW874T7oI22G2vkyssLPQLHJnvM1fkMwFgCEAqRnx5ofdw0+1/j/n16KGWKlgsllO3PozgP0HdFRfMPP1TwZEPrqlxY14RR1KZwmLl8tgMHqeAjyVjaoIilUCy0IC/C10DDIQQ9D34nGKJiEfUtmjvhnRQ1dXVv/62797T52weD4/BdPf23Ll5jYWFhbruv/23vf/7ghf6z1cqR6S8GlFw8c6FU+qqqC6ZVMYrFrDyeGwGj5PPt51kpug48kqFOAoGiYaBsVOCgRCCvkdR1rPopyvJeg5012nGVoFIJJx52KY8+gz6PPoU0GqQNVsmMznQIz/pY5u2RgbijmSyC/gaBjjFpBuiEQ6BgnGxc4CTZSDoexhZBgyd87Eg43Fm3NnPLzaZO4bQXadpkKjt3a7Oraio6MDRg6/fv2GWM+1sbScFTwwZG6IyLVxVZaVyMlU5BELSyHhlaWnpp0+f8vPzbWxsvLy8NBrPssvn8/Pz82k0GgbTvO83COC6lC4Vyzj5fDaDx8riFoSXC5hCDWO8vifZxF/363eA2hXsEUJQS1WXp2bGnWUk39Az7W7tOVffzK+9W9QpvY18O2HWJFKAvpaDDkYTwy1gV0eUWiDNHty83zAg9Qgc9W7APqDbICOEVGx2yJ+RGF23TCwWL1y14XbYG6FtXxbRWLsyA53y4n8bVs2dOU3p6qehz5Zu3FYpkAKKKWDmGlM0Tx3a4+Hh8R2PIxFKOfl8mVRGtqrZAqwqnVOeUK1FI2hSCQQ9HHy/2HHAQAhB6iEWcRkptziVOc69NrZ3WzofFovl7ONivNgWR6k3p7TkEWOEeeCeHbuVzv/r71OrH2Zxhm5RKkdHX5uB/XTycL3l/FPmLb5Voc8N3AAUnUsBR/vc9MNLJ02dVJsB59zFK8t/P8GcfBpoG9cUFSbrXZx18+RB/149lSqqrKxkMBiWlpaamprNfEZBpag0porN4LFzeWKuRJOKV4yj4nVgprf21KaBMDw83MbGxtjYWOVRGAihLqY07y0CgdQzbdFs/h/E36f+2ff8iMEQmlK5TCrL/l9cVlKmUoZSsVjs6R+Q7Dpb5DNBUYhIf2N6d+XniFBd3drRyLS0tJ4TFpYueKhcJa/a5I8ARtJH+Yb1bDbb2qtX8dIXAFc/sFXkW5wblxn7XjFCe/vuvZU/7+CiNaUUM2R5NgUj/fvgbz39vm0YQMyRsBg8dh6PzeCx83j202nyvKkAAKlIisTARRptqu3eEV6/fn3evHkHDhyYPn16m1UKQe1ILGDFv96BQCDprtNoDmNhzrYmvP7wBm+tomuFQCIIRsS8vDylvLhoNPr1k7szF62M2H9ERvOQovHovFhrA9KVp3frRkEAwMMnoUyXsSqqJJBExk5JSUnydKPPnj3jOI1QjoIAAIopR9c2ISHBxcUFAHDy33PrjlyomHlL8YayuDxn1LxpFw9uDwwY0PznRRNRFHtNir2KR447kiWsFsl7ivJ1GhhNOJmjdbXR3295efnJkyfHj2+QhxeCui5jq0HGVoMqiuOy4s89PuVrQOtt7TFX18SnvdvVEUkkEtDYXokIhESiIjEbiUS6ce5vNpudkJDA5XI9PH5RmWGcUVQq0VKdrVSsZVhaWpNpPS0rh61jrfI0nq5Vdna2i4sLi8Xa/Nv+imWvAKbO3oe65uWzrs5ZPjo77oO8cwkAiImJeRj6PD41y86SOrBPz169vpJPpy73lVaKZDf5r8rZDB4Kj9SkEnQctQx9vyeFOvRVbRQIV6xYsXPnzjNnzrRNdRDUcVAMXSmGvzv5rc1OuPTh0SIcQdexxyojyx9rG8iv8nL1/Pg5Hlg3+KCXAW4R28zMrLELNTU1vb29m0i6bWFiiGbki1UdwlQVKDYj1NYkogTVKhOhYoQs+YvAsLAwrtOIelFQjmTIM3T6/Pmzp6enWCweP3P+q7TScrfxwLA7yMs7+vMJB+yexzcuNv9tIk4bg9PG6LrUDCHwy4VsBk8qrn2NxS0WVCSxNGGyGzVRZyCMj4//5Zdf6pbY29vv2LHj/v37xsbGXl5eMBBCPyychr6d71Jbn8XF2S/EIk57N6fDmTxh0r4/9lO6G6I16k0QLX9bGNBnQEt2DRw+dPCvx6eV+c9TPsAux5amKDYr9vfvpfP3qtI+PymfJpOiUl56e/8KAMjIzmVr01XWwtOh5+bmenp6Llq98bGQzp3zV80BS1+m15ioqEvB0+Y+uXlJXiYSiSIiImITkvA4rKe7m6+vr8olIgp4Xaxi5b4cEoMQVInKE6rlyW60FEnDTfFILIyL3+wbAqFYLGaxWEqDD2KxODk5WV9f39DQ0MbGZv/+/XWPyr+jHTp0iMfjDRw4MCUl5fnz58bGxoGBgWppPQR1LggE0siy3puk9JiTeA0DE+uhSNQPvSRfX1//4K79K7as0g2mkqy1AQIh4YnLwwvxqYjDoYdacmcLC4ux/Xucv72OPXInUGyLwa2gnJ16+LdfFRHI3t7ew0TrZdQloc/EupcTn+waP3KwfF4ehayF4VeIVNWCFVSRyeSqqqqbT55zV75VOir0mRhz4nJqaqqtrW14xOtJ85dy6L2rDN2QYj7l2l96VasfXDlDp6sOsSrhdbD00cYAAJlUxi0WyMdRSz9VcosE9tOpFAc4i/DbNGvWaGJi4pw5cz5//iwWi0Wi2l+D5OTkoUOHksnk/Pz8mTNn7t6tPMVZjsViicViAMC6des8PDxmz56t8vsdnDUK/YCKc16mRP3BYqaaO02gu0z5wZfkJyQk/Lxja0JivEgippC1x44KXr18NQ6Ha/qqpvcjBABIpdKNv+48deWmzLIbV8uEyMzA5MUe2rV1bNDouqdxOJwhwZMS+VpM+2GAYoYozdSJuxrgTL1w8g/5nNWMjIwe435SMQFVKtHb1yPrY8S7d+9CjjypHLGzYRvQL48dH0Dq6efnP3py6eybtSs0AAC5MdQbC5Lev5KPnUZHR99/EvY5Od3a3GxQH7+AgICm+4t1ySQyBBKhWKGYeDJHxBIr5t1oGOMQSLh6UYVmBcKCgoK4uDgsFjto0KC6gXDUqFGOjo67du0qLCx0cXEJDQ1teuXpH3/8YWtrO3DgQJVHYSCEOqbIyMjHd2+nJiaYmpv36NN/1OjRKvf6aQlWRUZW3NncL9cpRh501+lGlv0RiEa29IMa+GoglGOz2YmJiQUFBdbW1g4ODo39I759+zb01evU7Dx3e6vBAf3lk0UVRk+a+QTrw+81t7ZIJtN8sHWeE3bfjl9v37495UIsZ5CqhaSR5/e5ssPeRD2kzwbWymstsK/+3OIqXrdi6bgZ88IzmOXuE4GBFWDmaac8tBJkP7tzVVtbuxl/E8qkYhknn8dm8FgMHpvBEzBFGsZ4TSpB25qo6wrnMNf6hnWE8fHxnp6eikAoHyZNS0uztLQEAMycOdPAwKCxTmFzJCUlubu7m5qaKkqWLFkyd+7cxs5ns9nNf/kMQd9BIpEsnjurKjU+2IJkqa1RzBGEF3GiqsH5G7cVkyzUWZ2YX5z5kPHlkoBX6uC3RZ/WX+1VdEnNDIRqwefzJ89d/KmIX+EwTKpDQ5dlkWOvDvV1OvL7DiQSGRcXN3rTsbIp/za8UOv+lr8neSz9eWfxmigV960q9Lg7z9vN6UKhNnfQ+rpH0LF3fZJPP7l5SSqVXrxy7eqD0KysLApFp3c3r2U/zdbX129+46VCGa9QwC0QSnhSo/41kVXMlbIyeBomOJwOuksmu8Hj8V9Nlff9gTA5OdnZ2VkoFMpnDG/fvj0xMfHy5cvf3dzs7Gx/f/9Xr14pSgwMDJoIdbBHCLW2n9etEbx/vNTdtG7h+4KKw9nil++jmz9g9a0qSxNRKJxWI7P5ISVtGQjlYmNjw1+/TcrMdbOjB/TvZ21d8y8lk8ksXX1zpl0DlHq/M4BXbXi0f8ant3TvPiWrIlXcUSq2/KMPWyApXfUWIJRnu+idnvDw4Pol67cmaTiwus0CBlaAU47+Eqb3av/dC3/7eHvLT1M5jeOr+OXC7PvFbAZPzJNomv036YaK/6GS3Xz/CA+Hw8HhcIp1MxoaGiwWq6WtQaO/6Y0xBLUePp9/59qVeyOUN2fvZkKh5uRGRET07t27larW1neq++P7hz/pGHmaO47D4r9nfAxSO0dHRxQKZWWZa21tLR8Sk0MgEOf+PDhmbkjZxH+AyX+/Ocw8nQszD+3YSiQSMUAKpGKAbPDBW5ajpaFRRXVrGAUBAEybwGXrt3ymDhf0+m/uq5aB2HdikVXPoGkhaR/f3H/0ePOuA1U8IcBrITnlvbr5HNu7Q09PVVLyBvC6WPvpVACAiCOpmXTzsTLzNk8mllqFmOr9GCOo3x8IDQwMuFwun8+XfxErLy9vjcEiCGoviYmJbgYklKpuX089bGREeOsFQiU2nvMyPp/68m6fidVguus0HWOvtqn3hyUUCouKikxNTZXyuskd+uPwviP7iOYkhDZKWibmF3B/3747ZGyI/Kh/r56hl07OXr42r7QCoWMGKgt1NLBHD/7Sv19fAMCgAX3PxNyWeClnutF8/2//Ht6nslVPC5Ki8XFp2YIJDd4T6dIqHYZPmbPgeRa7ctJloFUzTHoz9t67PoM+hD0wMjLKzMyMiorKzM13cbD18/PT0dFp7Kkx9ZPdCKvFKFxtVM68VchnChXrNLpYspvvfxgTExNjY+PIyMh+/foBACIjI0NCQtTXMAhqZzwej9DIPqsENKqI3dLxj+bTMfLUGewpElTnpd79GLpSJpPRXadaOE9CY4ht1oYfxPMXL1ZuXFnJqcJq4wVMnqmhyV8H/nR1dVWcsHPPzn+enDVf76JIByrmitbsWS+SiCeNr1l34e7u/vHlY7FYnJ+fb2xsXHeS/O5fNj7tHZhPMpLZ1Oaawb4/Z8OMnrN2c8isAAAgAElEQVTz0IX5W1W2SiPnHYrmqjLzDods/jD8IX/t69qVIQBI3EbkobGzl6zG4XCvU/JYVv35WiZa7z/gV29Zu3D26qWLmvNXgSXViw7UgfrVmVxWnWQ38qBIstQg0Tv972GzAiGPxzt37lx+fr5UKj1x4gSRSJw8eTIKhVq0aNGKFSsOHjz4/v37hISE27dvt3ZzIajN0On09EqeykPp1SJ7R+c2bg8GR7J0mWLpMrk4Jzwz7syXd/u7DTtuQGujXmnbkEqlT58+jY58U1yQ7+jhPXjIkLoDj63t8tXLa3duMJllba5bs4iFk8caNmHElX8u+/XoAQBgMpl//XvcYr1r3R130RoYs3l2G7ZuGBccUncmKhqNVsqPCgDQ09OLfHp3wpxFaY+3SkxdkBIhyI0J8PM98ei2pqYmmVtQWpoJ9Ou/HuKzCGkvEFbdVDc6PVLQf3ndKCgncxr07OIS5KCV/Pk1S/tZALAC1u24MJukqWlJNf3foeOZ2dkyGTA1Nlo5f/r4EFXpWOvAaKJ1XUmKuab8MqE8aXh+eDnJkiifZSOTyFgMHtEYX7cr2Sk0a7IMi8VavXq14kdtbW357FCJRHL48OGnT5/q6emtX7/eycmp8Xt8HVw+AXU0AT27r7ZAOuvXe03CFUlCHqe8jI791lkJ6sXnlKCxxK7UKczPzx83YqgTXtxNF6ergU1lcm9kVwVNm7n+51+avlAtk2U4HI6DlyNtjRMKX697ICjnVZxipHxOBgBcvXp16+2dBsOUt8gAAJRcyP5368lu3WrCVUJCwrPnYXFf4mzptv379PP19VU6v7Ky8suXLzgczsHBgUCoydkW/fHj0MlzS8f9BWj/rUMrz9W5OHvf6rlrdx0sbbBOHwCA+n2AZMIBQHVVPpCfiLy8XLoqVLlcwNbc4Y0xd60Yth0Y2QEAAJNBfroj0Ax15fRxBoPx4cOHbEa+s4Ndt27dvvU3XFAhSjnH4BTycdoYxeJFoim+42+m0aweoZaW1vHjxxuWo1CoFStWrFixQt2tgqAO4djps2OHBG7yEPc0q3mzwqjmrXmTs2H7rvaNggAAPNFA8WexiBt2PsDEeoilyxRN7bbrQqmRRCIZO2zwWjuij3HNhCBvY+3xjsZrbl08Y0qbPmtWazfg2bNnRFdtpSgIAMDpEoAOKiEhwdnZmZGfJyM38pmujSosLAQASCSSGfNmvk54i/ck4fQ1InM+/732tBXJ4tblmxoaGrWna2v36NFD6R7eXl6vbl2YvWxd5vUCQDYGrFJ9ksbhQ7/269vn7pPndz/dkHgG17uAXYZhFUiEqjL2fQmT9lDedhgAABixHDJVNuty7awcHWrVhL+eXlrQI2BYJpPPsR3A1TQlRUTglm1Yv3jeyiUL5GcJBIKvZjbAUTCuS+kyqYxbVJPspuRjJbdIQNDDUgfq67mRm768HXWpF54QpF62trYPXkasX75k1/3PJCyaI5Lo6BvsOnnWv62myTQTGqPRK+hiVsKFl1dGkfUcLF2mmFgPRiI7U862+/fuuROliigoh0IgtnWnTdy9ow0CYVpmOtBTncQAaYjNyspydnY21DcAn6Qqz0GwpfLtn1atXx3FiaMtdaw91gMUvCmcNHPy7Su3vtoMBweHt0/vymSywsJCAwMDxVjrP0f29hg4PKcyj99jJiCQgFQM0t7o3V8/Y2rIn6nPOFbKMRVUlwBDGxUVhP8jG7u74dzUKg4niuovnVgz8lcNAAjcsP387OLiwlfvY3LzCyUoHEYq9PPxPLJ7m6GhYROPgEAiiCZ4ognesBsFACCTyDgFfIxWbazJf1XGLxXKu4waRh0i2Q0MhBDUFCqVeuHGbQBAVVUViURqvbWDLUTUtnDutcmxx9qCjMdZ8edjX/5s7hhCd52uQWp034YOJTz0SV8jFcO8mli0Pg5VWFjY2Ibe6kLWJMn4jQQ5vlS+oLlfv36b9vwMBgGlhedSkbQ6ucLHx6e6uvrG/Zv0TcoDlTo9jeMOJ6SnpytWHDYNgUCYmJjULaFQKJ9fh/22//C1i2Mrq1k4DMbbw23H3Us0Gu2aV0+OaxAwrfPSWiYjZL0V6JiqeJ6iVEBtsClVUQrgVkkHra5XiMZVOgzbe2avdP4lYGQrL7sR/+BNn8HXTx8LffX67cd4sUTi5+kye+rEhm9Da58FhdCk1tuvQ9eJxPzCqsrg5L8sE1SKNIzx8nk3WuYEgv5XOp2tBAZCCGoWMrnjDuwoIFEYM9sRZrYjqsuSM+POhl0Y2CfkFknPvr3b9XVsVpUWVvXHkRYWzWKxWjsQ9urV67dTv4OGe+vKZKzkSi8vLwCAqanpsP5Dn995bTjKXBELZRJp4YWM5QuX4vH4N2/eaDpoq5zeiXEgRryOaGYgVAmPx/+yce0vG9cqlYfeuhQ4ZlKp7RCOpT/Q0kcUJOl8+DtogNftV1fK+vyk3BipBMikANTv+yaFAc8g5fq4leDpAem6lwBfOxVD6jKsIC+p95hpsmEbJW5rAQL5Iufjn0NCdq5dMnrY4E+fPuUyGNZWVj4+Pk3kQsHrYU38azZPlgik7DweO49XkczKf1nmuoSumGgjrBYrzVxtPTAQQlAXRNKzd++/07Xvr3UHSCViPgrddulXvomFjV1mZJKroYrl24xqXt3Mi63EwcHBxsCq4H0xpVu9cb+Sh4xRQ0aSSDUNO7Lv8NLVy+7veaDhpC0jAyQTsBMq5k+ft3r5agAAi8VC4FSPGSAIyKqqqtZouY2NTXL06+s3br5497QgqbSbm2PIsqOOjo7kzb+cvLq4Ong/QP/XzaosxAqrhZnvgU39jYI55UC3QZcu7iHwGVc3CgIAQGEy+HxXvPkDwNXEOamhTZnbiCWbu6377bDEcSBb04x89z4mZcWm5QuXLpjH4/Gys7NpNBqRqHpWFwqHJFsRyVbKR0UcSdzhTDFPoph0o0kl4CitNdr/DSnWWhucNQpBrYdVkfHy8jAjy4F012m6Jj7t3RxlaWlpM0YEXhpki6zfg4kurPy3UvP2k2dNXKuuFGssFmt48IhCWQnGiYjTIfBLOIJPLB9rr/Onzill6C4qKvrw4UNWdpadrZ2vr69ilXpMTMyEtVONZqhIj1VyO2f31O0jR45sYSObTyaT7Tl4ZP+ffwNTFxHJGFuWTmAVbFg8d+PR8+ULHwJ0nQxqr04CsQAMWFzv+rvbgKUPcBlSr/D8IuAdAuz71iu8ugZgCWDUr7W9TxFf86/RmvwyGZ4k07UAzFwSSvzHb78GDvy2/ahFbLF80o08abhMCjSpBKPuFMWWxeoCe4QQ9EPQolgNnvUhL/VuTNg6qVRk7jje0mUqFt9RxnttbGwCx05c/fD6tu40zf/GSD8UVP7yqehO6Iu2aYOWltarpy9fvXr1IvxFek6Go43j0DlD3N3dG55pZGSkMqS5u7tLigTCCj6WUi8qS/hidlxl//5tmkUdgUCsW7F01ZKF6enphYWF1tbWVCoVAMATy7b/ObR84CZg4Q1QGJD7WTv+hqCayeu/qN44KhoLRHzlm+Z+BpMO1yspzwU5MWBN/W8qaa/ZHC577g1FR7OksnDc0vFWBr+XlDOFEhkBg+rTs/vebZubThqO0URTHLQU2ysKq0QsBg+rpf6wBXuEEPSjkZUy3mbGnS3JDTezHUF3nU7Wb9EKYDU6e/r04d934WUSHQI2n8WzsLE78NdJGk3Fur262j7pdhNevHwxbdEM41nWhP/m/ggr+fn/pO1au2PyhEnt2zaFuLi4PUdPfoqLl0ikLk6OK+ZOvfUo9MSHQtaY/bU9xY83ELH3ZbNO17vyf93B5nf1Sl6fBkIe6L+wtkQqATt6gOUPFCnfAAAg7TW4vApM3A+sewIAgEyK+nxH5+EWX1/ftMxssVjsYG+3Yu7UAW37XUEBBkII+kHxOSXZiZfy0+73DrmJwXag/0osFqu8vJxGoyly+jetQwVCAMCHqA8/LV/ArK7AG2gImHwNFOHAzn2BgYHt3a6myGSyXfsOHTp+CtA8BCQTXFkajpklk0ryxxyTWdYZSD8aBMbvB/p11qo+/h3o0YF3nQWOWR/A63/B1GO1JWIB2OUPlt4B5DoznsL/Bu8ugNHbgKU3QKJBfgIl9DcvHRnF0DgjK8vC3HzEgF6TJ4z/6g5KagEDIQRBNUSCagGvvNMtye9ogVBOKBRmZWVRqdS66+g7OKFQmJqaWlBQYGNjY2FhwWAw+o0YW2w9mGM/CJBNQHEK8ckugTZNPONU7TVvzwJOBRi4rLYk+joozwGDVtWWxD8CqREgeGdtSV48uLQcrHhU2wHlVYNjIQhje5nfVKBrDpgMzYR7BumPgkYMS0jL1tEm9+vhOWn8uMYm3bQQDIQQBNUoL4h6d38uSdeW7jrN2GpQZ1mS3zEDYdcgEAguXrn6OPxdXn6hi4NtyPDAg8dPv+LoswZtAgQSAACUZiL+HCf7+UPtIv2ExyDtLQjaVnuX0EOAZAi6TagtubAE+IQA2zqJKU5OBR6j6/UsH+wCCU/AgMXA1BnwqggZEeSYi5ODhmcVMbU0ib193MaNDVbX3uwwEEIQVEsqEcmX5FeXp5g7htBdp2mQqO3dqK+AgbAtyWSyw3+eOHziX45IChAIDTTCzNjgs9SEFby/Zp9FbgXYPxRsels79eb5MYDXBH51Ur7t7gtWPamzrqMA/DMTrHpSe8KHKyDuIZh9uja+vjkDXhwDfeYBmgcQcvHZ7yixV2+e+at7t0bSkX8LOGsUgrqs4uLi7Oxsa2trefav5lAsyWcx0zPjzoZdGKRr6mvtPquLbXMBfTcEArFs4fxlC+dLJBKZTIZGo2Uy2fbd+47u6yGz9OVpmWpUZHIETMmjHYKhm2uuMfcA4f/UC4QyWb30PDkxwNa/XjXPjoDl92qjYMor8OEKWB+uiJ18m16F3hNHTxuV8CasmVsQN6GjJwWHIOg73Lxx3dPeempg7+Mr54zr293X2f5ZaIONCJqkpWPt1nfb0LmfTKwG53653krthDovFAolX16JQCC2rF+d8zny0f/mnZ/s9vzgmvKspFHkIt3jI5Fv/gWpEYiSdFRaOMh8X3uxiQPI+Vj7o1hQb12jiAdQGKBRJ6/9471gytHaHqQcxbTMf/n+P1RsCPGtYI8Qgrqaf/85eXH/rnP9LMm4mpd8JRzBksVz+HuPDB/xbQu6UWi8hdMEC6fatztVZUkiQbWeaXd1thjq/AgEgre3t7e3t/zHK6ePJyQkhL0M/5z8wNnVwinkz4VrVhQ5jeE5DgUUM2DbC3F9g2xNWM1OioY24FOdjORCHsDWS08KqkuUt2kEAAAgcQh4+mD+zoYHvhEMhBDUpbDZ7P07tt8caodH1+aTNCDijveznrxy2ZChw1Ao1XssNJOAWx77cjMAgO46jeYQgsGpOccH1GU4Ozs7O9emAk/s0+fkv2cfvdzPYDBsbWyQ3WyfnxxTMXQ7MHMGZi6gJB0UpdRskahBAdXFQCarecsok4HGFtLgNblcbsubCgMhBHUpz58/D6CS60ZBOW08xk1PIzo6ulvLJhcY0PwHTntVURyXFX/u8SlfA1pvS5cp8A0i9FUEAmHpgvlLF8xXlIRHRGw/sC/1ZrpUJtMzJWf9HVIdtE/mNBAgEMDWH0RdAb4TAAAAgQBSKRAL6w2fyhWm2FhZtbxtMBBCUJfCyM01a2QrGyoBxWAwWhgI5SiGrhTD35381mUnXvr0bA0WT7HzWWJqM6zld4Z+HL39/UP9a+fIMBiM5Zu2vT+wRSgFGJm0MvGhUCYR+04ECCRwGwZenwJ9f1K6AyX88KKfZ7a8JTAQQlCXok2hJItUH6oUybS1tVUf+y44DT07nyW23ouKc17yWAVqvDP0A6JSqTfOnlT8WF1dvXzD1icHjogxRCDiV5YWSjFYsd+MmqmkAo7Ww18CLImBAwe2vGq4jhCCuhQGgzE+oPfVIXZK5RKZbOS9pNexSW3wvyY3+SYCgTS1HoZEtcWSfLiOsGvjcDh4PJ7P56/Y+Mu9J89kmnpAxMdJBcvnz1y+6Ce17JUNe4QQ1KVQqVRXP/+T8bFzXWrzOsoA2BOdN3r8pLb57kjQNEp+fzD25RYLpwmWLlOI5K9kzYagJsjTqhGJxBOHfgcAVFRU4PF4AoHwteu+AewRQlBXIxKJfpo5LSsmKsCYYELEMjjCx3lsv8Bhuw8ebmYaa7VgV2Rmxp/NTbpGMXSnu04zogcgEC2asNqYTtcjLCkpuXX71rtP7zUIhB5ePYKCglophSbUTDAQQlDXlJqa+joiIic91drByb93bwsLi3ZphkQsyEu9mxl3hscudPHfTLULUnsVnSsQnrtwbuP/NhO76xDMiVKRVJDF5cdUXzp9sUd3uDSz3cBACEFQW6gsTZSIebrG3mq/czMDYUlJyadPnwoKCqytrb29vdtlU4j379+HzJ9gvswRia3tHAuY/PyjKR8jogwMDNq+SRCAgRCCoHYR+3KLBsnM3HEcFt/SiaxfDYRisXjt8qVvQx/1MNIyQMtyBOBNQdWqTVtmzJ7Twqq/Vb8h/dl9ZRpmyh9cZZEFI7QDfvvfb23cHkgOTpaBIKgd0ByC02NOfnm3z8RqMN11mo6xV+vVtWTeHHLmx2tD7BTzC3lio6UHd+EJhAmTJrdevQ1l5mRZmLk0LCe76IVdeN6WLYHqgkm3IQhqBxRDN5/BR4fMjtI18fkYuurpmd4pUUeF/Cq1V5SWlpb8LmKxu2ndWfYENOqAv+WuLZulUqnaa2yCDKgegUPj0RyOGlKFQd8HBkIIgtoNBkeydJkycNoL977bK4pinpzunp92X71VPH38aJiZiu1bNbFoWwohKSlJvdU1DYvCSEUqQi+3kG1padGWLYHqgkOjEAS1CqFQmJCQkJaWZmpq6urqSiI1kZ4bYWDex8C8D59TjECo+dt5SWGBBV71B50+Hl1aWqre6po2LijkdsRj/f5mSuVVL0q2LFrZli2B6oKBEIIg9bty6eL/Nm/wMNCyICCeC8GHwqpREyZv2b6j6YWMeKJh3R9fXBqqZ9rd0nWKpraKLXiayciMWhSlOulcEU9kZGT03Xf+DpvWbbrf90E5tlC3p5F8awWpSFr6gOFIsh058tt2yILUCM4ahSBIza5fvXL81w1H+tA1sTVftSUy2Z5oBtF7wJ5DR5p/H05VTmbcuZykK2R9J7rrNBN6IAKp4rt707NGs7OzJw3qd6XOTBm5Cr5o6vPsmJR0teToaj42m71yw6onoU8wZJxULAUC2Zxps9etWtvC7bGgloCBEIIgdZJKpR52VpcCrEi4ekFLBsD4RymXnr40Nzf/thtKRAUZj7Piz1eXp5g7htBdp2uQ6g0tfnX5xLrlS6veh27wpaH+i3lVAtHCFxmr9xwaMXLUNzVGjUpLS7FYLJlMbv4lJSUlL1++jImPMTcz796tu7u7e+s174cCAyEEQeoUFxf3vzkT9vupyC96NiFfL2TxvPnzGx5qjurylMy4s4zkm92H/61P7ako/2oglEqlO3/dev38GQ9DsiEOkcOVJDG5v+7ZNzpozPe1pFWlpaWFPQ+L/RJnbWHdp1dvxZ7vew/uPXTiiKYnBWWIk1QLxal8qobprcs3vimUQirBQAhBkDo9e/bs7raVaz1NGh56klFS6Dls86/bW3J/qUSIQKIVc2qkEqFUhmxOZhkOh5OYmJifn29jY2Nvb49Gd7gZElKpdOHyRY9fPyV4kbAGBFGFQJjIsSJb3Lx04+r1a9tP7TKbbYNA1b5krYgq1k7Chz991Y5t7ho63K8CBEGdmr6+fjFfrPJQCV+sb2zawvsjUbXblAv5FU9O+xmY96M6TDSx9G/iKgAAkUj09fVtYe2t6udff37BeG2+0gko3lr2BAVvi8ZPm5D4Jcl0pV3dKAgAoPgYFiSkv379ulevXm3f2q4EriOEIEidXFxcvjC5bKGKWPiAwQocNEiNdWHxlEEz31EM3eJfbQg92y8j9rRIUK3G+7clHo939vI5wyALUH/ujo6fUVxWApKCQhFU9FuwLpoPQx+1URO7LhgIIQhSJyQSuXHbjuXhWVyRRFEolcl+j2b4DBhkaWmp3uqweDLdbXafCc+8Bx2qKk169I/P+wfzygui1FtLG/j48aOWHQWBUjGFFUnDSbGq32FhNDFlzLJWblrXB4dGIQhSswmTJouEouBtW3yMyJYEUCwCkQXVg4PGbvttT+tVSjF0pRj+7uS3NivhYvTT5f5jLmuQqK1XndqxWCyAV90zweriqlNVRzthKd/axro12/VDgIEQgiD1mzpjxtjx42NjY1NTU3uamW338KBQKG1QL05D3953mb3vMkWJVCLksYuIZBWzWDsUGo0mLhWoPITkIYhIAq+IQzCqv3+vDHDfVwSv6YhzXzsXGAghCGoVBAKhe/fu3dt7v1l2ZVbEjfFaFCu66zQT66FIFKZ929MYJycnwJQImHycTr3pr1KRlPOp4tzxM1MXTDeeba2IhVKhpPByZsiQsVZWVu3R3i4FLp+AIKhza8Y6QlFB+uPMuLMsZqqF00RLlylKS/I7iPCIiMk/TTGebU0wrIl2IpYw/1Tapvnr5s2eFxMTM3PBrGoRC2dMlLJEvCLOqiUrly5a2saZcbokGAghCOqsxGJxWFjYp/fv+Hyeu0+3wMBAIpHYxPmsioysuLO5X67rGHtZuc00tOjXZk1tpg9RH+Yv/amSV4XTJwiZAowE/duvu4JGjVacUFVVlZKSYmho+K0JeqAmwEAIQVCnFB8fP2NcsK8u1o2MxqKQSVWiJ3nV+44dDxw0uOkLJWJ+XurdgvRH3Yad6JgjpTweLzMz08zMDGaNaRswEEIQ1Pkwmcx+vl5HeprRKbVdwAq+aOaz9DN3Hjo5OX3T3ThVuVxWvr5ZdwDgMOOPCK4jhCCo8zl26MAMG3LdKAgAoOAxW71N/rdp/bfeTcAri32x6emZ3ukxJzvvknzou8FACEFQ5/PiyeOB5roNyz2MyF8SE771bjpGngFTn3sN3F9RHPv4lO/HpyuYRTHqaCbUOcDlExAEdT7VLJYWTkflIST4ztc9uiY+uiY+Ah4zJ/Hyh0cLMVgtG6+faPZwlV7XBwMhBEGdj6mpaW4Vz1JbQ6lcKJEi0ViVlzQTjqBj673Qxuunktxw2C/8QcChUQiCOp+QaTMuppU3LL+VVjJ4xMiW3x+BQBqa93XotkJRUpzzMjf5plQibPnNWwmfzy8sLGzvVnRKMBBCENT5TJw0OQuj829iobTOvPfQ7PJLDP7azVtao0YMjpSbdPXh317xEf/jVGa3RhXf7dzF8w6ejg7dnXqP7W/uYDFm4piioqL2blRnApdPQBDUKfH5/C3r1jy8e9taRxODRKQz2S5ePgf+PKGrq2ISjbqwK7Oy4s7lJF3VNnChu003tgxAINv5BdOmrZsuh183mkhHE2vWRFbGlbHuFUeEhpuatnT3xx8EDIQQBHViUqk0PT2dx+M5OTm12abzUokwL/VeZtwZbjXDoftqS5fJbVNvQykpKYEThliscgL1E61VJpSZp+vfvXanvRrWucDJMhAEdWJIJJJOp4vFYrVEwcrKymOHDkY8f1ZSUkylUgcMGT53wcKGWUyRKCzNIZjmEFxV9oXPKWl5vd/t7KVzWr11QYN0o9rOerF34wQCAQ6Ha5eGdS7wHSEEQRAAAKSnp/f19cJG3t5lh70zyGoDVVp879++3bzLyhrd+Zas52Bo3kfxY0rUkdSPfwr5FW3SXgAASE5LxhupTq+K0yPAN4XNBAMhBEEQkEqlU4JH/+5rNMHeSE8DCwAw1cL/5GqykK4xd+qkZt7EyKJ/dVny41Pdox4vLi+Ias321tAkakr4YpWHJDxx0ynIIQUYCCEIgkBkZKQNQeagpzztoL+5TkVuRkFBQXNuQtZ38h50aPCs99oGrh+frnh2rn9m7BmxkN0K7a0xuN8gXhKrYbmYKwIcqZ6eXutV3ZW0USDMyckZO3asl5fX+PHj26ZGCIKg5vscE+NBQqk85K5HjI+Pb/6tsHhtG895gTMi3PpuK817++gfn6yEC2pqprLg4GBZqoCdXVWvVCYrupS1ac3GVqq062mLyTIikWj48OEnT57s3r07HLOGIKgDkkoljW1wiwRAKpV++y0R+tRe+tRefE6JVCJoWesahcViH99+NCRoKIdegbUlojUxvHw2923FtDFTZs2Y1UqVdj1t0SN88uSJm5ubVCq9c+cOXPAAQVAH5OziGlctUXkojsn71n2d6sITDTRIVMWPb+9Mjwv/hV2R+d03VGJpaRkfFbdr6q/9JN3sMsxm204KuxG6fet2dd3/R6DOHqFAIFBaBYjD4SwtLTMzM8PDw6lUKgaD2bp1a2RkJIFAUGO9EARBLeTv77+6SpxdybWon7/0fUEFWseQRqOpqyL3fjuy4s+/uhZE0rWnu04ztgpEIlu6OTAGgwkKCgoKClJLC39A6gyEhYWF27fX+xpiZma2e/duPB5vZ2e3a9cuAEBSUtLz58+HDRumxnohCIJaCI1Gn758bUrQiMWOuv3NdfFoFEckuZdRejGHe+/ZCzVWpEEyc+q53qH7qoL0hxmfT31+udnSaaKlyxSClokaa4G+SbMCoVAojIyM/PjxY0lJybZt27DY2uTu165du3v3LplMXrp0qa2t7YULKt4Ju7i4XL9+Xf5nRGPD8BAEQe3K1dX16Zv3v/9v2+nX4QIeV4tE7hc45OXVDSQS6VtvFfbsWeiDe5mpKTQ63X9A4IiRI5HIeu+hkCiMmd0oM7tRLGZaZty5Z+cDPAb8ZmarhnTh0HdoVoq1+Pj4KVOm2NraXr9+ncViaWpqysvPnj27cePGvXv3fvny5dixY8nJyY1l+Rs+fLiHhwcOh7t169abN+0B7J8AACAASURBVG8aZmoAMMUaBEHfRSwWi8VilZ8qbY/P508YPYJQzhhJ1aKRCfksflgBJ1lCuPXoqY5OzQaKnz9/vnDqZFzMJwJBw83bZ/ZPC01NDBAIJBLVog2koO/2DblG8/LyqFRq3UDo6uq6fv36SZMmAQCGDh3av3//1atXq7xWKBQ+f/6cy+UOHjxYQ0N5CzG5zMzMHj16HDlyRFHi4+Njbm7eWHtgIIQgCHSwQLhg5nR6UcJkB8O6hc9zmJerCPefvQAA7Nr2y9Mr5+bZ6zobkPhi6afiqpPJzM279o4ODpafLJOKH5/upmfS3dJ1qp5p93Z4hq5FqS+u0ve/I+RyufHx8X361KQX6tOnz7t37xo7GYvFDh48uOkbCgQCDodz+fJlRQmfz9fX12/sfB6Ph0KpXvcDQdCPQx4Iv2uFg5oxmcxPb8N/HuagVN7fXOfKi8zo6GhGTs7bGxfOBdoi/3tJZKqF70fTnbhupa2Dg4WFhbywZ/DDgrTbMWHrAQBUh4kmNqPRWHV+6U9KSvrj72Ox8bFisdjRwfGnGfN9fX3VeP8OBY/HfzUP7fcHQvmKQMVYqJ6eXgvXCBIIBH19/Zs3bzbzfJlMpuibQhD0w+o4PcK3b9/6GZNVHuqlj4uPi7t06sQOXyqy/lQJTSx6mavh6b+O7Tt6TFGm3W2BY7cFFcVxWfHnIq70M6D1tvaYq2vi0/JG/nH82J4/f9ceaqw12xiBQqRl58/eOG/8wJDftu9q+c07qe9fRyhfAiEQ1CwU5fP5jY15QhAE/Qh4PB6ukc9UPArB5XLKS0tMtVQE7O4m2tHvVYyoUQxdPQN+D5z+WtvANfrpcmbhxxa2MDY2dvex381XOGk76aEIaCQWRbLVoS12vPri+oMHD1p4887r+wOhgYEBFovNzc2V/5ibm2tmZqamVkEQBHU+VlZWaSyRykPpXJmNrR1oZEoGBoUUi1VfCADAaejZ+SweNOONjrHXf2UybjXjO1r424HdOqNMEej6n/wIoD/W/H/7dn7HDbuG7w+EKBRq9OjRZ8+eBQBwOJwbN24E//eyF4Ig6Afk7OxcKEJmVXKVypk8YUQBq2/fvjiCBluoYrOIpFKWnb3ym8Um8NjFL66MDL82hpFyWyppNII2FBMbQ7KmNCzH62sUlfy4+S+bFQjFYrGVlVXPnj0BAC4uLs7OzvLybdu2Xbp0acCAAR4eHu7u7kOGDGnFlkIQBHV4f545vzgiO7qwUlGSUs6eFZa+78/jeDx+8qw5x+OV440MgKMJJbMWLpH/+DwsbP2KZSFDB61c9NON69fFYhWBk6BpNHROtH235Yzkmw9OuMa+/JlTldOc5smkMtDYWu5mryDoepq1fEImk9Vd3odAICwtLeV/5vF40dHRZDLZ1dW1hU2B6wghCPoOHWeyjFxGRsaG5UvSkr8QMUiuSGpKo23be9DDwwMAIBaLxwwdbMUvmu9spIlFAwCKOYId0Xn2/Yfv3HdAKBRODBqFLsocZU6yIGsUsHkvCjif+Zjbj0ObmD/PrsjMjD+bm3SNYuhOd5tuTA9som2jxo/Oc6nQtFSe0SOsEggulH96E62Ov4DO5xvWEbY2GAghCPoOHS0QKrDZ7IYz2yUSyV9/HL14+m8ui4VAInX09Jas3TBqdBAAYOn8uQaZUTOcjOqeH8FgnipBh0a8bbouiViQl3onN/mm7+CjOI1GtyF8Ff5qxtrZtMUOoP7M1YLz6ZsnrZs+dfq3PWFXAQMhBEGdW4cNhN+kurq6r6fLneEODUcul0ZkbzpxwcvLS8VljRPwmGWFCXdCE56/fpGfn29vax88fMywYcPW/bz+2sub+kE0ghERACBg8ssf5HkZuF06c/GHTYHZFvsRQhAEQU2LiYnxNSKpDEQ9dTHvIiPlgZDL5aanpxsYGBgZGak6t1Zm2uewa9MwRGDgosvsYfC5JPX1kXWH/jp8/8a9Af79dx/ak52bBAAwMjTcNu/nieMnqv+ROg8YCCEIgtofj8fDo1R3yAhoJJfDTkhIWDp3pqCSaUEmlPFERVzRktVr5sxfoPISiUQyduZS/BgrGyOkC6+0l+BLpiElfqJhwquyZWuW/3X4z0GDBrXm03QyMBBCEAS1P2tr67Qq1RvZp7GlFnjClFFD9/vRbHVt5IVckWTLiQMFeXlbtu9oeMmLFy8kxghNC3IhAIUYLbxM7MgrC6zO6u+JeBZ2j8vdB/Of1AUDIQRBUPuztrauQuHTmGwbnXrza6oEomf51boXzx3oaW6jQ1SUa2BQe3pZTrh2ceqsOTQa7cmTJ1FvX5cVFzl6eA8dNuzNuzco69qXpnwE+pOGUYyGEVVYRbVkJCUlenurIVtbl/H9C+ohCIIgNfrrzIVlbxjv8isUJWlMzszQtA3bdgB2Zd0oKIdEIMZakP499U9PT7e7O9bQ4h4PqE5i3forJKB3+KuXCJTyx7sMgFws+XGGllBYswafWRTDSL4pEdfriVZXV7fCw3VosEcIQRDUITg7O98Ne7VxxdIdD+KJaCRXLDWh0o5duoFCo6mqMpQCAEw1sUdP/nWon42HYc3SQG9j7XEOxiF34wQUXeCp4hJ+Psfa2lr+ZxQKm5N0LfblFnPHcUyxw7rt+4tKi5E4pJQvdXN2PfL7YRqN1jrP2rHAQAhBENRRWFhYXLx1FwDA4XCIxJouYEZGRpVQovL8V9nlgTSKIgrKoZGIPwbY9bsTbRBARRMxdQ9Vp1RQdc0MDAzkP5L1nXqNucRjF967vIFb+tfEkfg4kkkmXlsKkDlJpf6BvZ/eeWJnZ6f+5+xg4NAoBEFQh6OIggAAOp2eXc3niFTEwsii6kEWOg3LTbTwlkRS4v/esXP+G+eUgbL3BXmnU879fVbpZKFEY/W+l6cN3BJJxu684hnl8R7cYrKjrv50i6lzp6ntkTow2COEIAjq0BAIxIr1Gzcf2/N7LzoaWbvE4ml2OR+Blqdqa0jA4822MYw6n5JVzZMgAVYGuhlq83W0Ht+/N2/horpnPnjwgOihDfCYFKCbgtfVFfP0xFwAAJFGyhcW5Ofnm5qatuoDtjsYCCEIgjq6mXPmlpeWBP9zfKQ5ma6FK+OJ3pYL2Vr6o8dPykx+4aCnnGMrqYxlSECv8LFQKueJJWP27Zk9/yeZTJaeni4UCu3t7RNSElFGWMU55WhCOZog/zPWmJD4/iSHZmDuNAFHUNH17BpgIIQgCOoEVm/YNGn6zLBnz1LiY42o5mu6d/f19U1KSloQfHuIlYHSrveXEvJCHIwb3oSARtlSNKZPHBcX/cFOVwuLQn4pZQEtkqQXtuHJAACZUKal58ZiRjw53cPIMoDuOk3PtFurPF67goEQgiCoczAxMZk6rd5LO0dHx94jg9c+u/NLN5pijPRtXsWrAnY/CxWpt8VSWWx2/nxd7K6RTorIeexjzqmoCoMeDcY/ZTJWeqW7z2ACIci1zy95qXc/P98glQjNnSZYukzB4rXV/XztBibdhiCoc+saSbdb4vTfJ4/u3UNASCkEbH41z8bJ2cuvlyj04nRnE6UzryblpzE5m3rZ1i2UAdDzaqTOLGeyg27d8pKHuYNN+x3Yc6DuuaWMt5lxZ0tyX9n7Lrfx+qm1HqltwUAIQVDnBgOhHIvFYjKZVCoViUTm5eWN6dfr+lA7pSHTSbei9wY4mzRYlZjH4gfciqL4GOl4G+J0CLwidlForrOhQ+iDp2i0ioFDPqdEwCsn6zm04vO0Ibh8AoIgqCvQ0tIyNzdHIpEAADMzs9FTpq95ncUT1y66yGPx0it4DaMgAOB+amFPHa21KE3XZ0Va/37p9aZ8h5kRJzMj8q3qfRDxRIO6UTDm+frYl1tYzHR1P1Mbge8IIQiCuqCNW389aWgUtGeXpTbRSAOdWcXnILAmpqZsoVhpxUUJR/AoveTmOF8UAhEEaqfYeBiRFy2cf+Xew6ioqHxGrp2Do5+fn7a2ileD9j5LM+PPhl8fq6VjTXedZmI1BInCNDytw4JDoxAEdW5waLQJMpksJyenoKDAzs5OV1d364b1Op8fB9vV28vwcmIeRySZ7W6udK1AIh12LdpAm9zHRMsAi8jmy57lVi5YufqnxUtV1iWVigrSH2XGnWUx0yycJlq6TNYgUVvrwdQKBkIIgjo3GAibr7y8vK+v1x+9qHRKbeaanW9SfYwpA+n6SicvexLvZkiaVSdACiXS5eGZ41ZunjJ9RhO1sJjpmfFnc5OuO/qtsXKbqdYnaBVwaBSCIOhHoaure/H2vRnjg720sW5kFBqJTKwWP8iptqEob23xpYzFFopn1e8mYlHIPT0tRm/ZHB/zKS7mo0QicfXwmj7vJxcXl7qnaelYu/XZ5txzo0ymOkVqRwMDIQRB0A/ExcXlfVzSy5cvP3+MFouEgzy8phka/jx7UohjvdNeM5hDbQwbXv4wvRjJYzlkv51oR0IiQCzj/YKQh+PmLlq6arXSmSh0vT76s3MDtHSs6K7T9Kk9AUCAjgQGQgiCoB8LGo0OCAgICAhQlJAs7C6n5E2wq418FTxhwx0QE0qrrybl3x/fDfvfZof9LPT8aboz//7Dw8fXzd09Ly+PTqerHKbuM/527pcbn1/8LJOJ6a7TzB3HY3CkVni47wEDIQRB0I/u9KUrsydPfPY8qbcB3kgDk8US3cuuNNYiKJ/2OXdDT1ts/S1/0UjEAGONKWNGmhvqGRNxudU8HImy788TXl5edU/DYLWs3GZYuc0oL4jKjDvz5d0+E6shlq7TdIw8Wv3xvgYGQgiCoB+dhobGpVt3kpKS3kW+zcvOcnFxu2JhsWzq+Cku9QYxU8rZnsbKyyeuJuW/zC69F+ypQ6hJWJpZwZk3PujImUt+PXs2rEvXxEfXxEfAY+YkXYl6tMjF/2cT6yGt9WDNAwMhBEEQBAAAjo6Ojo61rwoHjByzJfzB1m7mir2fxFKp0su9aoH439jcW+O64ep0E+kU4rE+9OmTJw4bNiw9NdnSyrr3wEGjRgehUCjFOTiCjq3XAluvBXXvJuCV4wj10ry1DRgIIQiCIBW279l7YI/+8L/+cNInGeBR2WwhW4rMqOBYUYhfylhfytkSqayUIxhqbYirP1jKFUk2PE90I+K7F38cT9PIK48L3f/20J7fbj8OpVAojVUnFnHDLgQSSTS62zRT62FIlOoNMVoDXEcIQVDnBtcRtiqRSJSamlpYWGhjY5MQH39k7eI8LktgSEDZaCOQCPaXchmDfS7Q1UZHU3HJyqfxPWm6wfb1Un4/yym/ztLcse8gg8GwsbGxsrJqmMVUJhUXZoZmxJ2pKk00dxxPd51KJCsv828NMBBCENS5wUDYZiorK+mOdMtFLpoWZEUhN59VeOjT/WFeehpYAEApV7DwYdy1sT5K1576nHPyc25PG5oZEZPDEadW8n7ZvTdoTLDKitiVmVlx53OSrmobuNDdpptYDWrVFRdwaBSCIAhqlt37dhuPsqobBQEAGqZalLG2+yKzdvnZAQA+F1X3MFMe/zwalZlbzQuf2hPz3yBqlUC0eNNqmVQ2ZuxYiURS9/UhAEBTm+7Se4uj37r8tPsZn09ratNJurag1cDdJyAIgqBmuff4PsXHoGE5xcvwTWGF/M8CiQSPrhfVyrjCp5klv/V3xNR5lUjGYYaaEpfOneFpY9Hdwbq7i+PBvXskknqZaFBoHM0h2D/4iiIKSsT8svx3AKh5IBP2CCEIgqBm4fH5ujhUw3IEClkulNxILjTVwqUyuQklVXWPhueWDbcxUtoZceurZJ5Y8nCcr3zRBVckOXHv3KjHj28/Vr0DopyQx4wP3+7ca6M+VcXCjO8Ge4QQBEFQs2gSNcVcUcNyqVCio6svC5z61sDTZuryUrRWThVXcbSEIzDWrPcG9zWjvJQj2DPASbH0UAODWuhqLGak9PZy83V2CB4y8NC+vRwOR6kigpZJv4kP1BsFAewRQhAEQc0UEjT24tsb+gFmSuXMDyXBo4OXr6xJN9qtW7eZ44J+8TL2MaEAACh4bClXUPf860kFi3zo9e7AE866FzPAQn+kna4ZiVDA4j99dLb3PyfvPA0zM1OuTu3grFEIgjq3dpk1mpeXl5uba21tbWCg4p1ZV8Xlcj16eOGHUrSd9RSF1SlM1s3i6NdRdffszcjI2LRyWcqXJDxSxhJIEHz2o/G+iqOjrry/EeKrWKcPAJh9L2aaK7WPee1tAQC3U4v+SCr38HDX1qZ49+ozfsIEIlE5/alawB4hBEHQN7h04fxvv2wx08SaaGCyWYIKMWLXwSMDAwPbu11tQUNDIyL01eTZU9KfJBEsNAESIcjlmGmb3Hv8Qmnneisrq8t37gMA+Hw+Ho9fOn/uwZjIZR5m8tCHRiGEEikaWfO6MY/FE0qkSlHw8IeMNwzmcleaHamSxSv7cD7Gb/eOM1dvuru7q/25YCCEIAhqrhPHjt7588CVgXRNbM2HZylXsHjJXOG+o8OGj2jftrUNAwOD0HtPS0pKEhISpFKps7OzkZFRE+fLe+oHjv21dtmSkIcPe5mQTLAAicK8zWMGWNZsBZxUyvIxqRdH76QUZlRwL43xVkyx8TAidzOqGjUoYNjwYVpaZM/ufqNGj1bXMAAcGoUgqHNrs6HRqqqqPp6uN4faKW2/UMEXTXmeFZOcjkT+oNMPs7OzN2//OepjlEAk1MATBg0I3Lpxq1IfEQBQVFQUFRWVm5OjqaV14NfNlwfbyb9PPEwvzqzgLK7z1nDklXfnRnuRcRhFye2UwhOfskMcTZ31tfhi6adyXmgh99Sla57197j4PrBHCEEQ1CzPnj0LpJKUoiAAgILHOFMInz598vb2bpeGta/Id5HjZkzQCTI1XGWDQCKkIunTD6/v9fJ5+fiF0jwXIyOjESNq+s3aJNLE1ctm2Op4GpJIOPTHokrFaXyxBI1E1o2CUQWVlxPzbob4KlYo+tNAsBVv+rgxr6JidHR0WvgIP+j3FwiCoG+Vl5triled6IuqgcrLy2vj9nQEQqFw8qwppovstJ31EEgEAACJQer1NNYaYzRx5qQmLhwVFHQ/PJLrF3S0jHiFTWKI0PEl1fJDfLEUj64Xm/6IztzZz1Fpnb4ZiTDDhnL86JGWPwUMhBAEQc2iTaFUCqUqD1WIZA1HAn8EYWFhWDsiTkd5XJpkRymoKsrPz2/iWlPT/7d3p2FR1YsfwM9sDAPDMszIqsjmwrAYi2AimCBiwlXvlVRM6VKBy99bPZZFPnmvlnWzp1Kv2SKVmYWaYrdcsIuGorG4RYCAg+wM+xLLALOe/4vpEhcnRZiZM8N8P6/m/M7MOV99wfc52++4vLr9H8fPZH77w8Wz2VdSrzd/f7dVrlTZmLPa+2XDL9q1SqQePA33iy6Ywsu5mDX+fwWKEABgVB5bsCCrceQj3gRBKFRkrvjXkJCQe1dNeL+UFNFcWBpXsV0tysrKRrkdT0/Pi3nXajzDEi41LDsnkjLM/n2nSb2KJAjaH0y4zTVjSvr6Hj71SLhGCAAwKlOnTvWZE55W/Euyn9PQoIok37xetybpGQsLCwqzUYXFZJHKP7jjUkWMmEr7/gQCwTv7fjvP2dXVtSh8LklvXe41iU6jkSShUJHDnztUq+ySuHt43LOlh4YjQgCA0fog7TOxs29iVsWhEvF/qlrTisQrM+84hMembv8H1dGoERwYRNbINK6S3O3x9/cf22Z5PN7F3IKqqbOXZlasPC+SEIyjtzWcZf1C1JHwdPLYdjEcHp8AAOOm/5llKioq8nJza+6Kpvv4zZs3Tw9zgBkslUrlN9vffIWAO9V6+HhnQYtXq8u3x06Nfxf9/f0ymSwmIizR1Xyp1yT1UaFUqdr3s7jLeeaRbzLGvwsUIQAYN7yYl1qVlZWLlsaw51hb+dmx7cwHWvp7r3VYilnZ53+0trZ+8O9Hp6en59UtL+T8eMHJylyqUPUoyHVPJ7+w9WWtPLuJIgQA44YipFxvb+8HHx+4mHOxUdzo4enx5yXLn1r31H3epjQezc3NHA7HxsbmwV8dNRQhABg3FCGME26WAQAAk4YiBAAAk4YiBAAAk4YiBACA+6mtrc3Pz+/s7KQ6iK5gZhkAANDswMcf7t7zDsfRgmHNkjYPcEj2pwfS5oTOoTqXlqEIAQBAg+2v//1ozjdTU33oZr/NlDbY1v/E06uOpR0NmzuX2mzahVOjAAAwUkNDw5cnj7j8ddpQCxIEYT7JwjllesrfUigMpgt6KsK+vr7Tp0+fO3duYGBAP3sEAIAx++777yxm8+5974O5gCMzV9TW1lKSSkf0UYTd3d3BwcGFhYW5ubmhoaFSqVQPOwUAgDGrrKtiCsw0rmIJ2BPsLcT6uEZYXFzs6em5fft2giB+/PHHyspKoVCoh/0CAMDYCHgCZa1c4yqVRDnB3kKsjyIMCgqSyWQHDx6USqV2dnbTp0/Xw04BAGDMFkfHfP78YSJs5LhKpuyv7/X29qYilK5oswhbWlq2bt06fITP5+/Zs0ehUHC5XJFIJJfLuVyuUqnU0WSsAACgFYGBgZ52bo0/NdmF/f4WYlJJNn5VmbolVSvvfDAcWiik+vp6DocjEAh4PN4rr7wyfBWLxSII4uTJk05OTu+++y5BEGvWrDl//vyyZcvGv18AANCdU+kZK9etKv/wDmumBdOGpWyV9d3q2vBUyqb1G8e2QbFYXFBQUF1b4z1jZmhoKJ/P127gMRtVEba3tycmJt64caOtrU0sFjs7O6vHOzo64uLiGhsbJRJJfHz8Rx995OPjc+/PBQJBRUWFQqFQqVSVlZUCgUCb/wIAANABS0vLs6fO3L59Ozcvt6ahdtYc//nvzXdwcBjDphQKxYbnNl7IvcjxsSat6fR8Ve+LXZue2fjqS6lajz0GoypCJpO5du3anTt3hoSEDB9/8803XVxccnNzu7u7AwICzp07Fxsbe+/P4+Lifvrpp4iICJIkly5dGhZ2z1lnAAAwSD4+PhqPcB5K8v+l5PfemrrVd2hEEDv508NfcNicF/72/Dg3Pn4P8T7C7u5uW1vb4UeEjo6O6enpkZGRBEG89tpr1dXVX3/99ZijlJaWBgYGDr+hNCkpKSkp6Y++39fXx+Vyx7w7AJgY8D5CA1dXV/f4k3FuL/mOGFfJVdVvFRXlF5qZaX5OQyvMzc0feFfK2K8RDgwMtLS0TJs2Tb04bdq07OzsMW+NIAgLCws+n5+WljY04ubmdp+qI0kSRQgAKEIDl5eXx3nE+t5xOotuNc1WJBLNmUPx5KVjL8Le3l6CIDgcjnrRwsKip6dnnGnMzMyCgoLGuREAADAczW0tdK7mrqFx6YbwUoux3wLL5/MZDEZXV5d6sbOzc2wXUQEAYAJznTxF1aXQuErVpXByctK4Sp/GXoQMBsPX17egoEC9WFBQ8Mgjj2gpFQAATBAxi2Ikt7pI1cj7URR9cml9/6xZsyhJNdxoT42eOnVKfS70zJkzPB4vPj6eRqNt3rx5x44d7u7udXV1GRkZ169f12VUAAAwPvb29snrnv3y8NdOa73orN+OvuR9MvFB0Xu73hGJRIODg97e3mw2m6qEo71rdO3atTKZbGjx2LFjdDqdJMkDBw6cOHGCy+Vu2bIlKipqPFFqamoWLFhQXV09yu/39vZaWVmNZ48AMAHgZhmjsO+Dfe/uf99yihWdx1K2yWTNAzNmzCgpv82dYk1j0SV1PbOE/p8d+NTe3l7/2R7i8QldQxECwBigCI2FXC6vqKior6/38PBYl/xU73SZIMKZ+O+Lnn4tah/I7Lh+5ZqNjY2eg02o+eIAAMBgsVgsoVAYExNzNffqr/w+wfzfW5AgCFt/gdk8651v7dR/MBQhAADo1eFjX9qEazgFahfqcDrzjP7zoAgBAECvmhqbzAWce8dpTLpMqfkliDqFIgQAAL3iWnHlEs2FRydpGsd1CkUIAAB6tST68e5b7feO94g6/f0oeKwQRQgAAHq15bktkpzO/sa+4YPyHmn7ifrdO/+p/zx4UzwAAOgVj8c7l3HmL2tW9E5m0l3NaCy6SizrL+k5/NGhmTNn6j8PihAAAPRNKBTevlly6dKlm4W3BqWDQYsCo6KiqHoYFEUIAABaJhaLX/l7av61fJlSzqIzAx8J3P362x4eHsO/w2AwoqKixjklmVagCAEAQJtKS0sf/8sSm6WOLqne6hFRacP8JQsyjpwImR0y+u3k5+cfOf7VLyW/2NjYRMwJfzbpWT6fr4vAmGINAIwbplgzNH6z/c2f4Fu4/M/f58H2gY5Pa+4UljMYjNFsZOPzm85fy7J6TGA5xVrZL+8TdUtyOtI/Tw+bO1frgXHXKAAAaE1JSYnMSjmiBQmCMBdwmFM5ubm5o9nIx2kfXxBdct3szfOdZGbD5jhxJ813cd48fWXiqoyMjJycnKFX4WoFTo0CAIDW3Llzh+5kpnEV6UgvKysLDw9/4Ebe/df7ji94DR8ZaJZUfF5EMyde/nI7m87qr+oN8gn8/KPPtDJDN4oQAAC0hsViESrNq2hKwoytuSOHk0gkSqaSYf57PUm7Bsv235ixPsDS1XpoUFRQH/l41LUrBaM813ofODUKAABaExgYOCjq0bhKWTkYHBT8wC3IZDI683+6qe5bkdsq7+EtSBCEXaiDxFl25Ksj40mrhiIEAACtmTx5srfrzK6brSPGu0s7JzH4vr6+D9yCra2trEdGDLuRs7fqVzs/DW+rsJ4jSD91bJyBCRQhAABo19Ev0ln5iuaMmr7qbkW/XFLX03K6TnG++9/Hvx3Nz2k02tIlf2q/0vjbMknQaDRC01zcbD6nublp/IFRhAAAoE02NjYFOfmvrdjqJXIaPNzmVjrppUWbb+XdtLfXcFSn0e433mYWKtsuNpAKFUEjVErNVx1l3VKBQDD+wLhZBgAAfBBazgAAB89JREFUtIzJZK57ct26J9eN7edcLrfgcv7Ot17PeP/UoHyQIaN13+m0mWE34ms919vXxj477rA4IgQAAMNjbm7+z9ffEv1SXldac/3ytY4T9dKOgeFf6C7rIG7LUp5JHv++cEQIAAAGbfr06ScPf/PkM+uYLmzChUmTkYpaqT2Tn5l5kc1mj3/7KEIAADB0oSGhdwrLbt68WVxczOVyZ82apcUXNqEIAQDACDAYjJCQkJCQh5i2e5RwjRAAAEwaihAAAEwaihAAAEwaihAAAEwaihAAAEwaihAAAEwaihAAAEwaihAAAEwaihAAAEwaihAAAEwaihAAAEwaihAAAEwaihAAAEwaihAAAEwaihAAAEwa3kcIAADUk8lknx767PQPp2traidPmbw4cvGG5PUWFhZ62DWOCAEAgGLt7e1Bc4P/denjjjCp4AX3X+cr0m5+GfBooFgs1sPecUQIAAAUW5W4mh7FdfAXqBc5DpacGMtej67lq/58/eo1Xe8dR4QAAECl2tra6vZa2/+24BCrabxuVl9hYaGuA6AIAQCASoWFhWxPS42r6G5mPxf+rOsAKEIAAKCSQqH4wy6i0xRyha4DoAgBAIBK3t7eigap5nWNCqFQqOsAKEIAAKCSUCi0lHL6arpHjA80S8hG+aOPPqrrAChCAACg2Mmvvun4qr6joJlUqgiCIJVkV2Frc1rliSPf0Ok67ykjLsLvvvtOIpFQnQIAKHb37t1r13R+hz3olJeX1/Wcggj67NY9lbVvlbTuuRvUK8y7+JO/v78e9m7EzxHu3r3b399fP/9NAGCwsrKySktLIyIiqA4C4zJp0qQP9x6gZNdGfEQIAAAwfihCAAAwaShCAAAwaTSSJKnO8Jvy8vKAgAAfH59Rfr+0tNTd3Z3D4eg0FQAYuLa2toGBAVdXV6qDgCFKSEh48cUX7/8dAypCgiB++OEHgWDkdHN/pKWlxd7enkaj6TQSABi4gYEBmUxmY2NDdRAwRC4uLo6Ojvf/jmEVIQAAgJ7hGiEAAJg0FCEAAJg0FCEAAJi0iVOELS0tKSkpK1asuHDhAtVZAIAyxcXFO3fuXL9+fWdnJ9VZwDhMnCJcvXr1kiVL9u/f/9xzz9XW1lIdBwCo0dra6u7unpWV1dfXR3UWMA4TpAgbGxubm5uXL1/u7Oy8evXqjIwMqhMBADWioqISExO5XC7VQcBoTJAibGhomDJlivqzq6trfX09tXkAAMBYTJAiNDMzUygU6s9yuZzNZlObBwAAjIURvIapqqrqxo0b3d3dycnJw8dzcnLUM9EkJSW5ublVVVWRJEmj0crLy4VCIVVpAUBHFApFSUlJUVERn8+PjY0dGlcqlUePHi0pKfH19U1ISGAwGBSGBGNk6EeEOTk5wcHB77333oYNG4aPHzt27IknnrC1tS0oKAgLC+NwOAsXLty6devRo0czMzPj4+OpCgwAOrJr165ly5a9/fbbe/fuHT6+cePGvXv3Ojo67tu3b9OmTSKR6ODBg52dnenp6biHHEbD0KdYUygUTCaztLTUz89PqVQOjfv5+W3bti0hIYEkyYCAgJdffnnlypXp6ekdHR3x8fFD1wsBYMJQ/zXYv3//999/n5WVpR4Ui8Wenp5VVVXOzs7Nzc1ubm5ZWVllZWXqtR4eHgsXLqQuMhgHQz81ymRqSNjW1lZSUhITE0MQBI1GW7Ro0aVLl9asWZOYmKj3gACgJxr/Gly9elUoFDo7OxME4ejo6OPjIxaLU1JS9J4OjJihnxrVqKmpicVi8Xg89aKDg0NjYyO1kQCAEk1NTfb29kOLDg4OTU1NFOYBY2SURchkMlUq1dBJXYVCwWKxqI0EAJRgMBgqlWpoUalUajxwBLgPoyxCJycnpVLZ2tqqXmxsbHRycqI2EgBQwtnZWSwWDy2KxWL1aVKA0TPKIuTxeGFhYerpY6RS6dmzZ+Pi4qgOBQAUiIyMrKmpKS0tJQiivLy8uro6MjKS6lBgZAz9HEJPT8+KFSskEglJktHR0XZ2dsePHycI4o033oiPjy8sLCwuLnZ3d1+8eDHVSQFAty5fvrxr1676+vq2trbo6OioqKjU1FQej7dt27bFixfHxcWdPXt227ZtQ3cPAIySoT8+IZfLL1++PLTIZrPDw8PVn+vq6rKzsx0cHBYuXIirAgATXktLS3Fx8dCio6Ojr6+v+vONGzeKi4v9/PyCg4MpSgdGzNCLEAAAQKeM8hohAACAtqAIAQDApKEIAQDApKEIAQDApKEIAQDApKEIAQDApKEIAQDApKEIAQDApKEIAQDApKEIAQDApGGKTgCjcejQIV9fX3d39yNHjjQ1NQmFwlWrVnE4HKpzARg3zDUKYDSsra1jY2OvXLni5OTEZrPz8vIiIiKys7OpzgVg3FCEAEbD2tq6v78/MzMzOjqaIIhPPvlkw4YN2dnZjz32GNXRAIwYrhECGJPY2Fh1CxIEkZiYSKPRioqKqI0EYOxQhADGZMaMGUOfORyOpaVle3s7hXkAJgAUIYAxYbFYwxdpNBpVSQAmDBQhAACYNBQhAACYNBQhAACYNMaOHTuozgAAozI4ODhv3jwvL68RI56enhSmAjB2eI4QAABMGk6NAgCASUMRAgCASUMRAgCASUMRAgCASUMRAgCASft/cprXeXjz4MsAAAAASUVORK5CYII=", "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# | echo: false \n", "\n", "N = 30\n", "T = [] # Composite trapezoid rule \n", "S = [] # Simpson\n", "G = [] # Gaussian rule\n", "\n", "for n ∈ 1:N\n", " \n", " h = (b-a)/n\n", " x = [a + j*(b-a)/n for j in 0:n]\n", " push!( T, sum( [ (1/2)*( x[j+1] - x[j] ) * ( f(x[j]) + f(x[j+1]) ) for j in 1:n] ) )\n", " \n", " h = (b-a)/(2*n)\n", " y = [a + j*(b-a)/(2*n) for j in 0:(2*n)]\n", " \n", " p = 0\n", " for j in 0:2*n\n", " if j==0\n", " p = p + f( y[j+1] )\n", " elseif j==2*n \n", " p = p + f( y[j+1] )\n", " elseif mod(j,2)==0\n", " p = p + 2*f( y[j+1] )\n", " else\n", " p = p + 4*f( y[j+1] )\n", " end\n", " end\n", " push!( S, h*p/3 )\n", "\n", " r = roots( Legendre(n) )\n", " p = fit( r, f.(r) )\n", " I = integrate( p )\n", " push!( G, I(1) - I(-1) )\n", "\n", "end\n", "\n", "scatter( abs.(T .- exact ), xlabel=L\"n\", xaxis=:log, yaxis=:log, markersize=5, label=\"Composite Trapezoid\" )\n", "scatter!( abs.(S .- exact), markersize=5, label=\"Composite Simpson\")\n", "\n", "scatter!( abs.(G .- exact ), markersize=5, label=\"Gaussian quadrature\" )\n", "\n", "#plot!( (1/12)*d2f_max*(b-a)^3*(1:N).^(-2), linestyle=:dash, label=\"error bound (Trapezoid)\" )\n", "plot!( (1/25)*(1:N).^(-2), linestyle=:dash, label=L\"O(n^{-2})\" ) \n", "#plot!( (1/(27))*d3f_max*(b-a)^4*(1:N).^(-3), linestyle=:dash, color=:red, label=\"error bound (Simpson)\" )\n", "plot!( (1/15)*(1:N).^(-4), linestyle=:dash, label=L\"O(n^{-4})\" )\n", "# plot!( (1/2880)*d4f_max*(b-a)^5*(1:N).^(-4), linestyle=:dash, label=\"n^{-4}\" )\n" ] }, { "cell_type": "code", "execution_count": 5, "id": "330b01c4", "metadata": {}, "outputs": [ { "data": { "image/png": "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", "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# | echo: false \n", "\n", "scatter( abs.(G .- exact ), yaxis=:log, markersize=5, label=\"Gaussian quadrature\" )\n", "plot!( exp.(-(1:N)* 3/4 ), linestyle=:dash, label=L\"\\exp( - μ N )\" )" ] }, { "cell_type": "markdown", "id": "a1694f81", "metadata": {}, "source": [ "## Error estimate for $n=1$ \n", "\n", "We have $X = \\{x_0, x_1\\} = \\{ -\\frac{\\sqrt3}{3}, \\frac{\\sqrt3}3 \\}$. Quadrature in these points is exact for all polynomials of degree $2(1) + 1 = 3$. We therefore let $p$ be the polynomial of degree $3$ such that $p(x_j) = f(x_j)$ and $p'(x_j) = f'(x_j)$ for $j=0,1$ and notice that the error is \n", "\n", "\\begin{align}\n", " &\\left| \\int_{-1}^1 f - \\left( f(x_0) + f(x_1) \\right) \\right| \\\\\n", " %\n", " &= \\left| \\int_{-1}^1 f - \\left( p(x_0) + p(x_1) \\right) \\right| \\\\\n", " %\n", " &= \\left| \\int_{-1}^1 (f - p) \\right|.\n", "\\end{align}\n", "\n", "Since $p$ is the Hermite polynomial interpolant on $\\{\\{ x_0, x_0, x_1, x_1 \\}\\}$, there exists $\\xi_x$ such that $f(x) - p(x) = \\frac{f^{(4)}(\\xi_x)}{4!} (x-x_0)^2 (x-x_1)^2$ [this is the generalisation of the Taylor remainder theorem we saw when we were considering Hermite interpolation]. Therefore, using the mean value theorem, there exists $\\xi \\in [-1,1]$ such that \n", "\n", "\\begin{align}\n", " &\\left| \\int_{-1}^1 (f - p) \\right| \\\\ \n", " &= \\left| \\frac{f^{(4)}(\\xi)}{4!} \\right| \\int_{-1}^1 (x-x_0)^2 (x-x_1)^2 \\mathrm{d}x \\\\\n", " &= \\frac{8}{45} \\frac{1}{4!} \\left| f^{(4)}(\\xi) \\right| \\\\\n", " &= \\frac{1}{135} \\left| f^{(4)}(\\xi) \\right|.\n", "\\end{align}\n", "\n", "\n", "
Example. \n", "\n", "Approximate $\\log 2 = \\int_1^2 \\frac{1}{x} \\mathrm{d}x$ using Gauss quadrature and evaluate an upper bound for the error.\n", "\n", "
\n", "\n", "
Answer. \n", "\n", "First notice that \n", "\n", "\\begin{align}\n", " \\log 2 &= \\int_1^2 \\frac1x \\mathrm{d}x \\\\\n", " %\n", " &= \\frac{1}{2} \\int_{-1}^1 \\frac1{ \\frac{1}2 x + \\frac32 } \\mathrm{d}x \\\\\n", " %\n", " &= \\int_{-1}^1 \\frac1{ 3 + x } \\mathrm{d}x \\\\\n", " %\n", " &\\approx \\frac1{ 3 - \\frac{\\sqrt{3}}3 } + \\frac1{ 3 + \\frac{\\sqrt{3}}3 } \\\\\n", " %\n", " &= \\frac9{13}\n", "\\end{align}\n", "\n", "The error here is $\\log 2 - \\tfrac{9}{13} \\approx 8.39 \\times 10^{-4}$. \n", "\n", "The error is $\\frac{1}{135} f^{(4)}(\\xi)$ where $f(x) = \\frac{1}{3+x}$ and $\\xi$ is some point in $[-1,1]$. Taking derivatives of $f$, we find that $f^{(4)}(x) = \\frac{24}{(3+x)^5}$ and so we actually have the error between $\\log 2$ and the approximation using Gauss quadrature with $n=1$ is \n", "\n", "\\begin{align}\n", " \\log 2 - \\left( \\frac1{ 3 - \\frac{\\sqrt{3}}3 } + \\frac1{ 3 + \\frac{\\sqrt{3}}3 } \\right)\n", "\\end{align}\n", "\n", "and belongs to $[\\frac{24}{135} \\min_{x\\in [-1,1]} (3+x)^{-5}, \\frac{24}{135} \\max_{x\\in [-1,1]} (3+x)^{-5}] = [ \\frac1{5760}, \\frac{1}{180} ] \\supset [ 1.74 \\times 10^{-4}, 5.55 \\times 10^{-3} ]$. The actual error belongs to this interval!\n", "\n", "
\n", "\n", "
Exercise. \n", "\n", "Compare this to rectangular and its composite rule with 2 sub-intervals and the trapezoid and Simpson rules. \n", "\n", "
\n", "\n", "\n", "## Error estimates \n", "\n", "If $w_j \\geq 0$ and the quadrature rule is exact for all polynomials of degree $N$ (this is true for Composite Trapezoid rule with $N = 1$ and Gaussian quadrature with $N = 2n+1$), then\n", "\n", "\\begin{align}\n", " &\\left| \\int_{-1}^1 f - \\sum_{j=0}^n w_j f(x_j) \\right| \\\\\n", " %\n", " &= \\min_{q \\in \\mathcal P_{N}}\\left| \\int_{-1}^1 (f - q) - \\sum_{j=0}^n w_j \\left( f(x_j) - q(x_j)\\right) \\right| \\\\\n", " %\n", " &\\leq 4 \\min_{q \\in \\mathcal P_{N}} \\left\\| f - q \\right\\|_{L^\\infty([-1,1])}\n", "\\end{align}\n", "\n", "*c.f.* Approximation theory [we will come back this this].\n", "\n", "## How to compute the roots of $P_{n+1}$?\n", "\n", "Recall, $P_0(x) = 1, P_1(x) = x$, and \n", "\n", "\\begin{align}\n", " xP_n(x) = a_n P_n(x) + P_{n+1}(x) + b_n P_{n-1}(x)\n", "\\end{align}\n", "\n", "It turns out that that $X = \\{ \\text{zeros of } P_{n+1} \\}$ is the set of eigenvalues of a tridiagonal matrix $T$ (see: Assignment 7). Next chapter, we will see methods for finding the eigenvalues of this matrix!" ] }, { "cell_type": "markdown", "id": "c7852cc6", "metadata": {}, "source": [ "## Exercises\n", "\n", "
Exercise 1. \n", "\n", "Show that Gauss quadrature with nodes $X = \\{x_0, \\dots, x_n\\}$ is **never** exact for all polynomials of degree $N > 2n+1$.\n", "\n", "Hint: Apply the quadrature rule to $\\ell_X(x)^2$.\n", "\n", "
\n", "\n", "
Exercise 2. \n", "\n", "1. Write down the Gaussian quadrature rule with $n=2$.\n", "2. What is the absolute error in approximating $\\log 2$ using the $n=2$ scheme and following the same arguments as in class? \n", "3. How would you prove an error bound for the $n=2$ rule in general?\n", " \n", "
\n", "\n", "
Exercise 3. \n", "\n", "Try to understand the proof that roots of Legendre polynomials are contained in $[-1,1]$.\n", " \n", "
\n", "\n", "## Next...\n", "\n", "* Assignment 7: I'll set A7 on Gaussian quadrature for you to do in the week beginning November 10\n", "* Next week we start Chapter 5: Solving Linear Equations - it would be very useful if you read through Section 6.1 of @Burden before the start of class on Monday. \n", "* Presentations: I will make a list of ideas for topics that you can present on. You'll then be able to sign up to give a presentation. If you have a topic in mind, let me know and I can advise if your topic is suitable." ] } ], "metadata": { "kernelspec": { "display_name": "Julia 1.11.6", "language": "julia", "name": "julia-1.11" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.11.6" } }, "nbformat": 4, "nbformat_minor": 5 }