The research is Supported by Natural Science Foundation of Fujian Province, China (2007J0183). Abstract Under the condition of arc-wise smooth path of integration, the Plemelj formula of Cauchy-type integral on random process with second order moment is obtained.

Let f ∈ Lloc(R) ∩ S ′, S ′ is the Schwartz class of distributions, and ∫ σ(D) f(x)dx = 0 ∀σ ∈ G, (∗) whereD ⊂ Rn, n ≥ 2, is a bounded domain, the closure D̄ of which is C1−diffeomorphic to a closed ball. Then the complement of D̄ is connected and path connected. Here G denotes the group of all rigid motions in Rn. This group consists of all translations and rotations. It is conjectured that if f ...

We present an operator approach to Rogers-type formulas and Mehler’s formulas for the Al-Salam-Carlitz polynomials Un(x, y, a; q). By using the q-exponential operator, we obtain a Rogers-type formula which leads to a linearization formula. With the aid of a bivariate augmentation operator, we get a simple derivation of Mehler’s formula due to AlSalam and Carlitz. By means of the Cauchy companio...

We consider a generalized Pompeiu equation in the space of Schwartz distributions and as an application we find the locally integrable solutions of the equation. © 2005 Elsevier Ltd. All rights reserved. MSC: 39B22; 47H15; 46F05

We introduce a quantization map and study the quantization of Schubert and Grothendieck polynomials, monomials, elementary and complete polynomials. Our construction is based on the quantum Cauchy identity. As a corollary, we prove the Lascoux–Schützenberger type formula for quantum Schubert polynomials of the flag manifold. Our formula gives a simple method for computation of quantum Schubert ...

Thus for "most" choices of three squares /]1, P2, P3 the validity of (1) for -{P~, P2, P3} implies f ---0. The Pompeiu problem has an extensive history which we will not discuss here. The reader is referred to the paper [13] of L. ZALCMAN for an excellent account of the problem, including the "two-circles theorem" of Delsarte for harmonic functions, which is the motivation for our Theorem 1. We...

In the present paper we study lightlike submanifolds of almost paracontact metric manifolds. We define invariant lightlike submanifolds. We study radical transversal lightlike submanifolds of para-Sasakian manifolds and investigate the geometry of distributions. Also we introduce a general notion of paracontact Cauchy-Riemann (CR) lightlike submanifolds and we derive some necessary and sufficie...

We prove a representation formula for solutions of Schrödinger equations with potentials multiplied by a temporal real-valued white noise in the Stratonovich sense. Using this formula, we obtain a dispersive estimate which allows us to study the Cauchy problem in L or in the energy space of model equations arising in Bose Einstein condensation [1] or in fiber optics [2]. Our results also give a...

This is a Cauchy's formula for fields 0 on Bo satisfying (4.1), the kernels being aap and ZOp. We have seen that Bo always possesses a singularity corresponding to complex-analytic forms of type (p, 0), and formula (4.2) therefore provides a Cauchy representation for such forms in which Z)p = 0. Since by (4.2) a closed Kahler manifold has a singularity Op for O = 1/2 II A if and only if all for...