New Integral Representations for a Hypergeometric Function

We obtain new integral representations for a hypergeometric function.     

关于ζ函数的积分表示

由Riemannζ函数的函数方程得到Hurwitzζ函数的Hermite公式,再从Hermite公式得到Γ(s)的Binet′s第二表达式,从而由ζ函数推得Γ(s)的性质.     

Integral Representations for Harmonic Functions of Infinite Order in a Cone

A harmonic function of infinite order defined in an n-dimensional cone and continuous in the closure can be represented in terms of the modified Poisson integral and an infinite sum of harmonic polynomials vanishing on the boundary.     

Integral Representations and Approximations for Multivariate Gamma Distributions

Let R be a p×p-correlation matrix with an “m-factorial” inverse R ⁻¹ = D − BB′ with diagonal D minimizing the rank m of B. A new -variate integral representation is given for p-variate gamma distributions belonging to R, which is based on the above decomposition of R ⁻¹ without the restriction D > 0 required in former formulas. This extends the applicability of formulas with small m. For example, every p-variate gamma cdf can be computed by an at most -variate integral if p = 3 or p = 4. Since computation is only feasible for small m, a given R is approximated by an m-factorial R ₀. The cdf belonging to R is approximated by the cdf associated with R ₀ and some additional correction terms with the deviations between R and R ₀.     

Banach空间非线性二阶脉冲奇异积分-微分方程积分边值问题的正解

The existence of positive solutions to a boundary value problem of second-order impulsive singular integro-differential equation with integral boundary conditions in a Banach space is obtained by means of fixed point theory.Moreover,an application is also given to illustrate the main result.     

Estimates of Integral Kernels for Semigroups Associated with Second-Order Elliptic Operators with Singular Coefficients

In this paper we obtain pointwise two-sided estimates for the integral kernel of the semigroup associated with second-order elliptic differential operators –(a)+b ₁+b ₂+V with real measurable (singular) coefficients, on an open set R ^N . The assumptions we impose on the lower-order terms allow for the case when the semigroup exists on L ^p () for p only from an interval in [1,), neither enjoys a standard Gaussian estimate nor is ultracontractive in the scale L ^p (). We show however that the semigroup is ultracontractive in the scale of weighted spaces L ^p (,²dx) with a suitable weight and derive an upper and lower bound on its integral kernel.     

Integral Representations in Linearly Convex Polyhedra

We introduce the notion of mixed Levian (Levi determinant) of a system of k Hermitian and k linear forms. In terms of these Levians we obtain an integral formula for holomorphic functions in linearly convex polyhedra.Original Russian Text Copyright © 2005 Krivokolesko V. P. and Tsikh A. K.The first author was supported by a grant of the President of the Russian Federation and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-1212.2003.1); the second author was supported by the Ministry for Education of the Russian Federation (St. Petersburg, Grant 02-1-138).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 579–593, May–June, 2005.     

Second-Order Cone and Semidefinite Representations of Material Failure Criteria

A class of important problems in structural mechanics leads to optimization problems with linear objective functions and constraints consisting in (a) linear equalities and (b) inequalities imposed by the material strength, the so-called failure criteria. It is shown that a wide variety of failure criteria can be represented as systems of either second-order cone or semidefinite constraints, giving rise to respective optimization problems. Work partially supported by Air Force grants.     

Integral Representations for Continuous Linear Functionals in Operator-Initiated Topologies

On a given cone (resp. vector space) we consider an initial topology and order induced by a family of linear operators into a second cone which carries a locally convex topology. We prove that monotone linear functionals on which are continuous with respect to this initial topology may be represented as certain integrals of continuous linear functionals on . Based on the Riesz representation theorem from measure theory, we derive an integral version of the Jordan decomposition for linear functionals on ordered vector spaces.     

Integral Representations for Generalized Difference Kernels Having a Finite Number of Negative Squares

An integral representation is derived for matrix-valued generalized difference kernels which have a finite number of negative squares. The representation is used to extend such kernels to the real line with a bound on the number of negative squares. The main results are obtained by means of an operator interpolation theorem. The nondegenerate case is assumed.      

Sturdy Harmonic Functions and their Integral Representations

Sturdy harmonic functions constitute all but the least tractable of the positive harmonic functions in potential-theoretic settings. They are the uniform limits on compact sets of positive, bounded harmonic functions and are also produced by a simple integral representation on the boundary of a natural compactification of the space on which they are defined. The boundary of that compactification is metrizable, and more regular for the Dirichlet problem, in general, than is the Martin boundary if that boundary is even defined in the setting.     

Some Integral Representations and Singular Integral over Plane in Clifford Analysis

In this paper, Cauchy type integral and singular integral over hyper-complex plane \({\prod}\) are considered. By using a special Möbius transform, an equivalent relation between \({\widehat{H}^\mu}\) class functions over \({\prod}\) and \({H^\mu}\) class functions over the unit sphere is shown. For \({\widehat{H}^\mu}\) class functions over \({\prod}\) , we prove the existence of Cauchy type integral and singular integral over \({\prod}\) . Cauchy integral formulas as well as Poisson integral formulas for monogenic functions in upper-half and lower-half space are given respectively. By using Möbius transform again, the relation between the Cauchy type integrals and the singular integrals over \({\prod}\) and unit sphere is built.