By Tolimieri R., An M., Lu C.
This graduate-level textual content offers a language for realizing, unifying, and enforcing a large choice of algorithms for electronic sign processing - specifically, to supply ideas and techniques which may simplify or perhaps automate the duty of writing code for the most recent parallel and vector machines. It hence bridges the distance among electronic sign processing algorithms and their implementation on various computing structures. The mathematical idea of tensor product is a habitual topic during the e-book, when you consider that those formulations spotlight the knowledge move, that is specially very important on supercomputers. as a result of their value in lots of functions, a lot of the dialogue centres on algorithms regarding the finite Fourier rework and to multiplicative FFT algorithms.
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Additional resources for Algorithms for discrete Fourier transform and convolution
Let x be the distance from the center of mass of the region to the y-axis, and let A be the area of the region (that is, its mass). If the region is rotated about the y-axis, then the theorem of Pappus says that the volume of the resulting solid of revolution is V = 2π xA . The center of mass (or centroid) is x = ,y = Summary In this lesson, we study moments and centers of mass. In particular, we develop formulas for finding the balancing point of a planar area, or lamina. First, we study one- and two-dimensional examples and then develop the formulas for arbitrary planar regions.
Example 1: A Sine Substitution Evaluate the integral ∫x dx 2 9 - x2 . Solution We begin by making the substitution x = 3sin θ . Then, we have x 2 = 9sin 2 θ and 47 dx = 3cos θ dθ , 9 - x 2 = 9 - 9sin 2 θ = 3 1 - sin 2 θ = 3 cos 2 θ = 3cos θ . x to the variable θ , as follows. Next, go from the variable ∫x dx 2 9− x 2 =∫ 3cosθ dθ ( 9sin θ ) ( 3cosθ ) 2 = 1 1 1 1 dθ = ∫ csc2 θ dθ = − cot θ + C . 2 ∫ 9 sin θ 9 9 1 1 9 - x2 Finally, return to the original variable x : - cot θ = + C . That is, 9 9 x dx 9 − x2 = − + C.
This problem would be more difficult to solve using the disk or shell method. Study Tips • For a region of uniform density, the center of mass is often called the centroid of the region. • The moment of a point is large if either the distance from the other point P is large or if the mass m is large. • The center of mass for the system in Example 1 is not at the origin. The system would be in equilibrium if the fulcrum were located at x = 1. • The same principles apply to the study of centers of mass of planar regions of nonuniform density.