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...

Abstract In this article, different types of Gaussian quadrature methods have been presented to find the numerical integration a neutrosophic valued function. A new definition distance between two number has defined and it proved that set all form complete metric space. Also, continuity on closed-bounded interval in sense $$(\alpha ,\beta ,\gamma )$$ <mml:math xmlns:mml="http://www.w3.org/1998/...

In order to approximate the Riemann–Stieltjes integral ∫ b a f (t) dg (t) by 2–point Gaussian quadrature rule, we introduce the quadrature rule ∫ 1 −1 f (t) dg (t) ≈ Af ( − √ 3 3 ) + Bf (√ 3 3 ) , for suitable choice of A and B. Error estimates for this approximation under various assumptions for the functions involved are provided as well.

This proceeding is intended to be a first introduction to spectral methods. It is written around some simple problems that are solved explicitly and in details and that aim at demonstrating the power of those methods. The mathematical foundation of the spectral approximation is first introduced, based on the Gauss quadratures. The two usual basis of Legendre and Chebyshev polynomials are then p...

We introduce a new class of polynomials {Pn}, that we call polar Legendre polynomials, they appear as solutions of an inverse Gauss problem of equilibrium position of a field of forces with n + 1 unit masses. We study algebraic, differential and asymptotic properties of this class of polynomials, that are simultaneously orthogonal with respect to a differential operator and a discrete-continuou...

The p-adic integrals evaluated here were explicitly introduced only in the later 20th century, starting with Tate and Iwasawa in 1950, by MacDonald in the 1950s, in the 1960’s by Gelfand and Piatetski-Shapiro, Jacquet, Shalika, and then by Jacquet-Langlands in 1970, although shadows of them appeared long ago in the work of Lagrange, Legendre, Gauss, Galois, and Dirichlet. From our vantage, they...

In preparation for computational work with molecular orbitals written as linear combinaions of atomic orbitals, for first and second row atoms, we have developed a general procedure to parametrize linear combinations of Gaussian orbitals. We used a Gaussian transform, and a Gauss-Legendre quadrature, to express hydrogenic orbitals (with the correct long-range dependence for collisions) as contr...

In this paper, we present a computational method for solving boundary integral equations with logarithmic singular kernels which occur as reformulations of a boundary value problem for Laplace’s equation. The method is based on the use of the Galerkin method with CAS wavelets constructed on the unit interval as basis. This approach utilizes the nonuniform Gauss-Legendre quadrature rule for appr...

Filon-Simpson quadrature rules are derived for integrals of the type ∫ b a dx f(x) sin(xy)/(xy) and ∫ b a dx f(x) 4 sin(xy/2)/(xy) which are needed in applications of the worldline variational approach to Quantum Field Theory. These new integration rules reduce to the standard Simpson rule for y = 0 and are exact for y → ∞ when a = 0 and f(0) 6= 0. The subleading term in the asymptotic expansio...