Category: Science, Engineering and Technology

Our goal is to demonstrate that, for the Hermite polynomial $latex H_{2n+1}(x)=\sum_{j=0}^{n}f(x_{j})H_{j}(x)+\sum_{j=0}^{n}f'(x_{j})\hat{H}_{j}(x) $ where $latex H_{j}(x)=[1-2(x-x_{j})L'_{j}(x_{j})]L_{j}^{2}(x) $ $latex \hat{H}_{j}(x)=(x-x_{j})L_{j}^{2}(x) $ the error function is given by the equation $latex f(x)-H_{2n+1}(x)=\frac{f^{\left(2n+2\right)}\left(\eta\left(x\right)\right)}{\left(2n+2\right)!}\overset{n}{\underset{j=0}{\prod}}\left(x-x_{j}\right)^{2} $ where $latex f\in C^{2n+2}[a,b] $ Let us begin by considering a point $latex \hat{x}\in[a,b]$ where $latex \hat{x}\neq x_{j}$, i.e., it is not equal to …

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The objective is to show that back substitution is backward stable.  Consider the system $latex Rx=c $ where $latex R$ is an upper triangular $latex m\times m$ matrix and $latex x,c$ are $latex m$ column vectors. For a $latex 3\times3$ matrix system, this would look like: $latex \left[\begin{array}{ccc} r_{1,1} & r_{1,2} & r_{1,3}\\ 0 & …

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The purpose of this piece is to document the numerical integration of a function in a two-dimensional space using the conjugate gradient method. Basic Problem Statement and Closed Form Solution The boundary value problem is as follows: $latex {\frac{\partial^{2}}{\partial{x}^{2}}}u(x,y)+{\frac{\partial^{2}}{\partial{y}^{2}}}u(x,y)=-5\, y\sin(x)\sin(2\, y)+4\,\sin(x)\cos(2\, y) $ where $latex u(x,0)=0 $ $latex u(x,\pi)=0 $ $latex u(0,y)=0 $ $latex u(\pi,y)=0 …

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The object of this is to solve the differential equation $latex {\frac{d^{3}}{d{x}^{3}}}y(x)-\mu\,\left(1-\left(y(x)\right)^{2}\right){\frac{d^{2}}{d{x}^{2}}}y(x)+2\,\mu\, y(x)\left({\frac{d}{dx}}y(x)\right)^{2}+{\frac{d}{dx}}y(x)=0 $ for the following boundary conditions and parameters: $latex \mu=\frac{1}{2}$ $latex y\left(0\right)=0$ $latex {\frac{d}{dx}}y(0)=\frac{1}{2}$ $latex y\left(2\right)=1$ Conventional wisdom would indicate that, because of the high order of the derivatives, this problem cannot be solved using a scalar implementation of simple shooting. However, …

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This from the "godfather of global warming", among other things: Lovelock blasted greens for treating global warming like a religion. “It just so happens that the green religion is now taking over from the Christian religion,” Lovelock observed. “I don’t think people have noticed that, but it’s got all the sort of terms that religions …

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This, from A.W. Grabau's 1913 classic Principles of Stratigraphy: The chief agents in retaining the sun's heat within the earth's atmosphere are the carbon dioxide and the water vapor, which act as thermal blankets. It has been estimated by Arrhenius (2) that if the amount of CO2 in the atmosphere were increased 2.5 to 3 …

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It isn't Stuxnet or even Flame, but... The malware known as 'Flame,' that was described by the analysts who discovered it as a super-cyberweapon, is actually a tool for cyberespionage that has been running inside Iranian data centers and labs for as long as five years without being discovered or causing significant damage. Contrast that …

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