On pathwise uniqueness of stochastic evolution equations in Hilbert spaces

The pathwise uniqueness of stochastic evolution equations driven by Q-Wiener processes is mainly investigated in this article. We focus on the case that the modulus of the continuity of the coefficients is not controlled by a linear function. Additionally, we show that the corresponding diffusion process is Feller.     

On embedding well-separable graphs

Call a simple graph H of order nwell-separable, if by deleting a separator set of size o(n) the leftover will have components of size at most o(n). We prove, that bounded degree well-separable spanning subgraphs are easy to embed: for every γ>0 and positive integer Δ there exists an n₀ such that if n>n₀, Δ(H)?Δ for a well-separable graph H of order n and δ(G)?(1-1/2(χ(H)-1)+γ)n for a simple graph G of order n, then H⊂G. We extend our result to graphs with small band-width, too.     

Stability of localized operators

Let ?p, 1?p?∞, be the space of all p-summable sequences and Ca be the convolution operator associated with a summable sequence a. It is known that the ?p-stability of the convolution operator Ca for different 1?p?∞ are equivalent to each other, i.e., if Ca has ?p-stability for some 1?p?∞ then Ca has ?q-stability for all 1?q?∞. In the study of spline approximation, wavelet analysis, time-frequency analysis, and sampling, there are many localized operators of non-convolution type whose stability is one of the basic assumptions. In this paper, we consider the stability of those localized operators including infinite matrices in the Sjöstrand class, synthesis operators with generating functions enveloped by shifts of a function in the Wiener amalgam space, and integral operators with kernels having certain regularity and decay at infinity. We show that the ?p-stability (or Lp-stability) of those three classes of localized operators are equivalent to each other, and we also prove that the left inverse of those localized operators are well localized.     

Existence of T-solution for degenerated problem via Minty’s lemma

We study the existence result of solutions for the nonlinear degenerated elliptic problem of the form, -div（a（x, u,△↓u）） = F in Ω, where Ω is a bounded domain of R^N, N≥2, a ：Ω×R×R^N→R^N is a Carathéodory function satisfying the natural growth condition and the coercivity condition, but they verify only the large monotonicity. The second term F belongs to W^-1,p′（Ω, w^＊）. The existence result is proved by using the L^1-version of Minty＇s lemma.     

Homogenization of the Incompressible Navier-Stokes Fluid With Oscillation Coefficient

We study the homogenization of the incompressible Navier-Stokes equations with periodic oscillating coefficient in a bounded non-homogeneous media. To do that, we introduce a generalized compensate compactness result and a suitable class of test function to this problem. By passing the limit, we obtain the homogenized model of this problem.     

An application of weighted Hardy spaces to the Navier–Stokes equations

In this article, we consider the mapping properties of convolution operators with smooth functions on weighted Hardy spaces H^p(w)$H^p(w)$ with w   belonging to Muckenhoupt class _A∞$A_{\infty }$. As a corollary, one obtains decay estimates of heat semigroup on weighted Hardy spaces. After a weighted version of the div–curl lemma is established, these estimates on weighted Hardy spaces are applied to the investigation of the decay property of global mild solutions to Navier–Stokes equations with the initial data belonging to weighted Hardy spaces.     

Multiple positive solutions for Robin problem involving critical weighted Hardy–Sobolev exponents with boundary singularities

This paper deals with the existence of positive solutions for Robin elliptic problems involving critical weighted Hardy–Sobolev exponents with boundary singularities. Using the Caffarelli–Kohn–Nirenberg inequalities and variational methods, we prove the existence and multiplicity of positive solutions.     

Imposing symmetry in augmented linear systems

This paper discusses the methods of imposing symmetry in the augmented system formulation (ASF) for least‐squares (LS) problems. A particular emphasis is on upper Hessenberg problems, where the challenge lies in leaving all zero‐by‐definition elements of the LS matrix unperturbed. Analytical solutions for optimal perturbation matrices are given, including upper Hessenberg matrices. Finally, the upper Hessenberg LS problems represented by unsymmetric ASF that indicate a normwise backward stability of the problem (which is not the case in general) are identified. It is observed that such problems normally arise from Arnoldi factorization (for example, in the generalized minimal residual (GMRES) algorithm). The problem is illustrated with a number of practical (arising in the GMRES algorithm) and some ‘purpose‐built’ examples. Copyright © 2014 John Wiley & Sons, Ltd.     

与年龄相关的模糊随机种群扩散系统的指数稳定性

讨论了一类与年龄相关的模糊随机种群扩散系统,系统受两种不确定性因素的影响,即随机和模糊.在有界和Lipschitz条件下,利用Ito公式和Gronwall引理,建立了均方意义下与年龄相关的模糊随机种群扩散系统指数稳定性的判定准则并通过数值例子对所给出的结论进行了验证.     

微分方程解的存在区间的确定

从六个方面说明了确定一阶微分方程解的存在区间的方法.