we show that  higher derivations on a hilbert$c^{*}-$module associated with the cauchy functional equation satisfying generalized hyers--ulam stability.

Here we will consider examples of conformally flat manifolds that are conformally equivalent to open subsets of the sphere S. For such manifolds we shall introduce a Cauchy kernel and Cauchy integral formula for sections taking values in a spinor bundle and annihilated by a Dirac operator, or generalized Cauchy-Riemann operator. Basic properties of this kernel are examined, in particular we exa...

In this work we prove a new Lp holomorphic extension result for functions defined on product Lipschitz surfaces with small Lipschitz constants in two complex variables. We define biparameter and partial Cauchy integral operators that play the role of boundary values for holomorphic functions on product Lipschitz domain. In the spirit of the application of David-Journé-Semmes [DJS85] and Christ’...

This paper consider the Cauchy problem for Schr$ {\rm \ddot{o}} $dinger equation coupled with stochastic Benjamin-Ono equation. A priori estimates integral and nonlinear terms corresponding to coupling system are achieved by using Fourier transform restriction method introduced Bourgain. It is shown that locally well-posed as initial data in appropriate Sobolev spaces.

A new class of solutions to Laplacian growth (LG) with zero surface tension is presented and shown to contain all other known solutions as special or limiting cases. These solutions, which are time-dependent conformal maps with branch cuts inside the unit circle, are governed by a nonlinear integral equation and describe oil fjords with non-parallel walls in viscous fingering experiments in Hel...

In our lectures on integral solutions to differential equations using Laplace kernels ,we encountered integrals of the type-∫ = + C n t xt t f x y 1) exp() () (where t=γ+iτ and C is a closed contour within the complex plane. To evaluate this type of integrals under conditions where the curve partially lies at infinity, one makes use of complex variable methods and in particular the Cauchy Integ...