We give new proofs of two basic results in number theory: The law of quadratic reciprocity and the sign of the Gauss sum. We show that these results are encoded in the relation between the discrete Fourier transform and the action of the Weyl element in the Weil representation modulo p, q and pq. 0. Introduction Two basic results due to Gauss are the quadratic reciprocity law and the sign of th...

Approximations of matrix-valued functions of the form WT f(A)W , where A ∈ Rm×m is symmetric, W ∈ Rm×k , with m large and k m, has orthonormal columns, and f is a function, can be computed by applying a few steps of the symmetric block Lanczos method to A with initial block-vector W ∈ Rm×k . Golub and Meurant have shown that the approximants obtained in this manner may be considered block Gauss...

The Gauss–Newton algorithm is an iterative method regularly used for solving nonlinear least squares problems. It is particularly well suited to the treatment of very large scale variational data assimilation problems that arise in atmosphere and ocean forecasting. The procedure consists of a sequence of linear least squares approximations to the nonlinear problem, each of which is solved by an...

This paper presents a Gaussian quadrature method for the evaluation of the triple integral ( , , ) T I f x y z d xd yd z = ∫∫∫ , where ) , , ( z y x f is an analytic function in , , x y z and T refers to the standard tetrahedral region:{( , , ) 0 , , 1, 1} x y z x y z x y z ≤ ≤ + + ≤ in three space( , , ). x y z Mathematical transformation from ( , , ) x y z space to ( , , ) u v w space maps th...

Here is a proof for the demeaning method used in lfe, and a description of the methods used for solving the residual equations. As well as a toy-example. This is a preliminary version of [3]. This method, formulated as the Gauss-Seidel algorithm, was noted in [4] without the author noticing it in the first version of this paper.

O experimento do canhão de Gauss é notável por sua transversalidade dentro conceitos relevantes para a Física. Ele simples, rápido, fácil execução e com um rico conteúdo mecânica magnetismo ser observado. Possui uma construção acessível, feita algumas esferas aço ímãs posicionados linearmente sobre trilho. Considerando essas vantagens, este trabalho apresenta estudo teórico experimental aplicaç...

We discuss notions of Gauss curvature and mean curvature for polyhedral surfaces. The discretizations are guided by the principle of preserving integral relations for curvatures, like the Gauss–Bonnet theorem and the mean-curvature force balance equation.

The normalization of energy divergent Weber waves and finite energy Weber-Gauss beams is reported. The well-known Bessel and Mathieu waves are used to derive the integral relations between circular, elliptic, and parabolic waves and to present the Bessel and Mathieu wave decomposition of the Weber waves. The efficiency to approximate a Weber-Gauss beam as a finite superposition of Bessel-Gauss ...