Multigrid algorithms are developed to solve the discrete systems approximating the solutions of operator equations involving pseudo-diierential operators of order minus one. Classical multigrid theory deals with the case of diierential operators of positive order. The pseudo-diierential operator gives rise to a coercive form on H ?1=2 ((). EEective multigrid algorithms are developed for this pr...

Studying classical wave propagation in periodic high contrast photonic and acoustic media naturally leads to the following spectral problem: − u = λεu, where ε(x) (the dielectric constant) is a periodic function that assumes a large value ε near a periodic graph in R2 and is equal to 1 otherwise. High contrast regimes lead to appearence of pseudo-differential operators of the Dirichlet-to-Neuma...

The article proposes a method for analyzing the Cauchy problem wide class of evolutionary linear partial differential equations with variable coefficients. By applying (inverse) Fourier transform, original equation is reduced to an integro-differential equation, which can be considered as ordinary in corresponding Banach space. selection this space carried out such way that principle contractio...

This paper gives a generalization of the Pseudo-Laplace transform. In the special cases of semirings, the pseudo-exchange formula is proved. Also, for these semirings the forms of the Pseudo-Laplace transform and inverse operator are given. The results can be applied in dynamical programming for finding the maximum and minimum of the utility functions. AMS Mathematics Subject Classification (19...

In this paper we extend the definition of generalized Sobolev space and subsequent theoretical results established recently for positive definite kernels and differential operators in the article [21]. In the present paper the semi-inner product of the generalized Sobolev space is set up by a vector distributional operator P consisting of finitely or countably many distributional operators Pn, ...

But what is it exactly? Let us motivate the problem in a mathematical way: consider the Gel’fand-Naimark duality theorem, a mathematical theorem which establishes a one-to-one correspondence between topological (Hausdorff) spaces X and commutative C∗-algebras A=C(X), where A is the algebra of continuous complex-valued functions f : X → C with the pointwise multiplication (f.g)(x) = f(x).g(x). G...

We consider the homogenization of parabolic equations with large spatiallydependent potentials modeled as Gaussian random fields. We derive the homogenized equations in the limit of vanishing correlation length of the random potential. We characterize the leading effect in the random fluctuations and show that their spatial moments converge in law to Gaussian random variables. Both results hold...

A detailed analysis is presented of all pseudo-differential operators of orders up to 2 encountered in classical potential theory in two dimensions. Each of the operators under investigation turns out to be a sum of one or more of standard operators (second derivative, derivative of the Hilbert transform, etc.), and an integral operator with smooth kernel. This classification leads to an extrem...

We use the calculus of adiabatic pseudo-differential operators to study the adiabatic limit behavior of the eta and zeta functions of a differential operator δ constructed from an elliptic family of operators with base S. We show that the regularized values η(δt, 0) and tζ(δt, 0) have smooth limits as t → 0, and we identify the limits with the holonomy of the determinant bundle, respectively wi...