A Criterion for Relative Compactness of a Sequence of Measure-Valued Random Processes

A criterion for relative compactness of a sequence of measure-valued processes is proved and exemplified.     

一个非严格双曲型方程组粘性逼近解的收敛性

本文讨论非严格双曲型方程组的粘性逼近解的收敛性，当ｆ满足一定条件，初值为有界可测函数时，利用补偿紧致的理论及对前进熵波的分析，在没有对粘性解导数一致界估计的情况下，证明了粘性逼近解存在一个子序列在强拓扑意义下收敛．     

Spot Size Technique: A Theoretical Approach To Optimize Dispersion Compensated

Using a spot size based optimization technique, a bend loss versus dispersion diagram has been obtained theoretically for the most widely used step-index disper sion compensated optical fibers (DCF). The Rayleigh scattering loss can be incorpo rated with bend loss to calculate figure of merit (FOM), thus giving an FOM versus dispersion curve on a single diagram. The plot of the FOM-dispersion curve on a single diagram is useful from the designer's point of view, as it provides a quick reference for choosing the best suitable DCF, which should have large negative dispersion as well as maximum possible value of FOM. The effect of fiber manufac turing losses such as absorption loss on the maximum value of FOM has also been estimated.     

Subelliptic Sobolev Spaces for Manifolds with Symmetries

Given a group of symmetries on a (generally non-compact) differentiable manifold, we construct invariant locally subelliptic operators, prove Sobolev inequalities and verify existence of correspondent minimizers. The results apply, in particular, to a class of singular elliptic operators on domains in R^N with singularity on the boundary. Mathematics Subject Classifications (2000) 35H20, 35J20, 40A30, 22E30, 43A85.K. Tintarev: Research supported by a grant from Vetenskaprådet.     

Weighted composition operators between μ -bloch spaces on the unit ball

In this paper, necessary and sufficient conditions are given for the weighted composition operator T _ψ,φ to be bounded or compact from the space β_μ to β_υ (or β_μ,0 to β _υ,0)on the unit ball of C ^rn. At the same time, a series of corollaries are also obtained.     

A Maximum Principle for an Explicit Finite Difference Scheme Approximating the Hodgkin-Huxley Model

We analyze an explicit finite difference scheme for the general form of the Hodgkin-Huxley model, which is a nonlinear partial differential equation coupled to a set of ODEs. The system of equations describes propagation of an electrical signal in excitable cells. We prove that the numerical solution is bounded in the L^∞-norm and L² converges to a unique solution. The L^∞-bound, which is the key point of our analysis, is proved by showing that the discrete solutions are invariant in a physically relevant bounded region. For the convergence proof we use the compactness method. AMS subject classification (2000) 65F20     

Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains

We introduce a notion of an asymptotically compact (AC) random dynamical system (RDS). We prove that for an AC RDS the -limit set of any bounded set is nonempty, compact, strictly invariant and attracts the set . We establish that the D Navier Stokes Equations (NSEs) in a domain satisfying the Poincaré inequality perturbed by an additive irregular noise generate an AC RDS in the energy space . As a consequence we deduce existence of an invariant measure for such NSEs. Our study generalizes on the one hand the earlier results by Flandoli-Crauel (1994) and Schmalfuss (1992) obtained in the case of bounded domains and regular noise, and on the other hand the results by Rosa (1998) for the deterministic NSEs.

     

Weak solution of the merged mathematical equations of the polluted atmosphere

Considered as a geophysical fluid, the polluted atmosphere shares the shallow domain characteristics with other natural large-scale fluids such as seas and oceans. This means that its domain is excessively greater horizontally than in the vertical dimension, leading to the classic hydrostatic approximation of the Navier–Stokes equations. In the past there has been proved a convergence theorem for this model with respect to the ocean, without considering pollution effects. The novelty of this present work is to provide a generalization of their result translated to the atmosphere, extending the fluid velocity equations with an additional convection–diffusion equation representing pollutants in the atmosphere.     

Convergence of the regularized short pulse equation to the short pulse one

We consider the regularized short‐pulse equation, which contains nonlinear dispersive effects. We prove that as the diffusion parameter tends to zero, the solutions of the dispersive equation converge to discontinuous weak solutions of the short‐pulse one. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the setting.     

Essential spectra of self-adjoint relations under relatively compact perturbations

This paper is concerned with the stability of essential spectra of self-adjoint relations under relatively compact perturbation in Hilbert spaces. Relationships between relative boundedness and relative compactness of linear relations are established, and some necessary and sufficient conditions of relative compactness and relative boundedness of linear relations are given. It is shown that the essential spectra of self-adjoint relations are invariant under either relatively compact perturbation or a more general perturbation. The results obtained in the present paper generalize the corresponding results for operators to relations, and some of which weaken certain assumptions of the related existing results.