Average quadrature formulas of Gauss type

In the average quadrature formulas the values of a given functionat given points are replaced by its averages over some distinctintervals. If all the intervals are of the same length, thequadrature formulas of interpolatory type and in particular,of Newton-Cotes type were constructed in Omladi (1978). Here,we construct average quadrature formulas of Gauss type for intervalsof the same length. The middle points of the intervals are zerosof polynomials, orthogonal in a technical sense.     

Gauss legendre quadrature formulas over a tetrahedron

In this article we consider the Gauss Legendre Quadrature method for numerical integration over the standard tetrahedron: {(x, y, z)|0 ≤ x, y, z ≤ 1, x + y + z ≤ 1} in the Cartesian three‐dimensional (x, y, z) space. The mathematical transformation from the (x, y, z) space to (ξ, η, ζ) space is described to map the standard tetrahedron in (x, y, z) space to a standard 2‐cube: {(ξ, η, ζ)| ? 1 ≤ ζ, η, ζ ≤ 1} in the (ξ, η, ζ) space. This overcomes the difficulties associated with the derivation of new weight coefficients and sampling points. The effectiveness of the formulas is demonstrated by applying them to the integration of three nonpolynomial, three polynomial functions and to the evaluation of integrals for element stiffness matrices in linear three‐dimensional elasticity. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

On Gauss quadrature formulas with equal weights

Summary It is well known that the Chebyshev weight function (1–x ²)^–1/2 is the only weight function (up to a linear transformation) for which then point Gauss quadrature formula has equal weights for alln. In this paper we describe all weight functions for which thenm point Gauss quadrature formula has equal weights for alln, wherem is fixed.     

一种Gauss型求积公式的收敛性

构造一种有理插值型求积公式(RIQFs),并证明其收敛性.该方法是Gauss求积公式在有理函数空间(Γ)2n中的推广.     

Explicit formulas for Hecke Gauss sums in quadratic number fields

We derive an explicit formula for Hecke Gauss sums of quadratic number fields. As an immediate consequence we obtain a quadratic reciprocity law in quadratic number fields which generalizes the classical one given by Hecke. The proofs use, apart from the well-known formulas for ordinary Gauss sums, only elementary algebraic manipulations.     

Upgraded Gauss and Lobatto formulas with error estimation

Recent generalizations of the Euler-Maclaurin formula are used to produce upgraded Gauss and Lobatto formulas for numerical integration; comparison of companion formulas provides error estimates.     

Summation formulas involving generalized harmonic numbers

By means of the derivative operator and three hypergeometric series identities, several interesting summation formulas involving generalized harmonic numbers are established.     

Interior and boundary monotonicity formulas for stationary harmonic maps

We derive monotonicity formulas for stationary harmonic maps. The proofs are given in coordinates.     

Certain summation formulas involving harmonic numbers and generalized harmonic numbers

Harmonic numbers and generalized harmonic numbers have been studied since the distant past and involved in a wide range of diverse fields such as analysis of algorithms in computer science, various branches of number theory, elementary particle physics and theoretical physics. Here we aim at presenting further interesting identities about certain finite or infinite series involving harmonic numbers and generalized harmonic numbers by applying an algorithmic method to a known summation formula for the hypergeometric function ₅F₄(1).     

On exact quadrature formulas for harmonic functions on polyhedra

A classical quadrature result for analytic functions of a complex variable due to Motzkin and Schoenberg is extended to higher dimensions. A general scheme for integrating on polyhedra solutions of partial differential equations is discussed.

     

Rigidity and mean curvature flow via harmonic Gauss maps

We investigate properties of harmonic Gauss maps and their applications to Lawson-Osserman’s problem, to the rigidity of space-like submanifolds in a pseudo-Euclidean space and to the mean curvature flow.     

Koppelman formulas on flag manifolds and harmonic forms

We construct Koppelman formulas on manifolds of flags in ${\mathbb{C}^N}$ for forms with values in any holomorphic line bundle as well as in the tautological vector bundles and their duals. As an application we obtain new explicit proofs of some vanishing theorems of the Bott–Borel–Weil type by solving the corresponding ${\bar{\partial}}$ -equation. We also construct reproducing kernels for harmonic (p, q)-forms in the case of Grassmannians.     

Four families of summation formulas involving generalized harmonic numbers

In terms of Abel’s transformation on difference operators, we establish four families of summation formulas involving generalized harmonic numbers. They include several known and numerous new harmonic number identities as special cases.     

Convergence of Gauss type product formulas for the evaluation of two-dimensional Cauchy principal value integrals

The authors consider product rules of Gauss type for the numerical approximation of certain two-dimensional Cauchy principal value integrals with respect to generalized smooth Jacobi weight functions.Convergence results for these rules are given, and some asymptotic estimates of the remainder are established.Work sponsored by Italian Research Council and by the Ministero della Pubblica Istruzione of Italy.     

Surfaces in a pseudo-sphere with harmonic or 1-type pseudo-spherical Gauss map

We give a complete classification of Riemannian and Lorentzian surfaces of arbitrary codimension in a pseudo-sphere whose pseudo-spherical Gauss maps are of 1-type or, in particular, harmonic. In some cases a concrete global classification is obtained, while in other cases the solutions are described by an explicit system of partial differential equations.     

An existence theorem for harmonic maps

The aim of this article is to prove the existence of a harmonic map in a given homotopy class from a non-compact manifold into another one, while this class fulfills a geometrical simplicity condition.