Core inverse of matrices

This article introduces the notion of the Core inverse as an alternative to the group inverse. Several of its properties are derived with a perspective towards possible applications. Furthermore, a matrix partial ordering based on the Core inverse is introduced and extensively investigated.     

UMD空间及其应用

UMD空间是被广泛研究的一类新型的Banach空间，它具有一系列良好的几何性质与分析性质并且与向量值调和分析、随机分析有着广泛深刻的联系。本扼要介绍这类空间的有关问题，主要是以下几个方面：（1）引言（定义与产生背景）；（2）UMD空间的几何特性与分析特征；（3）此类空间的例；（4）在向量值调和分析理论中的应用；（5）关于鞅不等式的最优系数问题。     

The lower bound for the variance of unbiased estimators for one-directional family of distributions

Summary In this paper we introduce the concept of one-directionality which includes both cases of location (and scale) parameter and selection parameter and also other cases, and establish some theorems for shapr lower bounds and for the existence of zero variance unbiased estimator for this class of non-regular distributions.     

Banach空间值鞅的两种新型BMO空间和Sharp算子

借助于 ｐ 条件均方算子和 ｐ 均方算子, 首先引入了 Ｂａｎａｃｈ 空间值鞅的两种新型 Ｂ Ｍ Ｏ 空间 Ｂ Ｍ Ｏσ（ｐ ）ｒ （ Ｘ）和 Ｂ Ｍ Ｏ Ｓ（ｐ ）ｒ （ Ｘ）以及两种新型 ｓｈａｒｐ 算子 ｆσ（ｐ）ｒ 和 ｆ Ｓ（ｐ）ｒ ． 并且讨论了 Ｂ Ｍ Ｏσ（ｐ ）ｐ （ Ｘ）与 Ｂ Ｍ Ｏ＋ｐ （ Ｘ）, Ｂ Ｍ Ｏ Ｓ（ｐ）ｐ （ Ｘ）与 Ｂ Ｍ Ｏｐ （ Ｘ）的相互嵌入关系, 以及ｆσ（ｐ ）ｐ 与 ｆ＃ｐ , ｆ Ｓ（ｐ）ｐ 与ｆ＃ｐ 之间的凸Φ不等式, 所得结果给出了 Ｂａｎａｃｈ 的凸性和光滑性的一种新刻划     

Sharp bounds on the ranks of negativity of certain sums

If M is a complex vector space and 〈·, ·〉 a Hermitian sesquilinear form on M with a finite rank of negativity k (i.e., k is the maximal dimension of any linear subspace E of M satisfying 〈x, x〉 < 0 for each nonzero x in E), if n is a positive integer, and if a ₁, …, a _n are endomorphisms of M, then it is easy to see that the Hermitian sesquilinear form $ (x,y) \mapsto \sum\limits_{v = 1}^n {\left\langle {a_v x,a_v y} \right\rangle } $ on M has rank of negativity at most nk. It is also fairly easy to see that the bound nk cannot be improved in general. Less trivial is the fact that it cannot be improved by making the following assumption (a) the space M is the *-algebra A:= (C[[w ₁, w ₂]] of polynomials in two self-adjoint non-commuting indeterminates; there is a (necessarily Hermitian) linear form φ on A such that 〈x, y〉 = φ(y* x) (x, y ∈ A); and a _v is just left multiplication by some element of A (which we may denote by ‘a _v ’ at no great risk of confusion). Now suppose that, with M, 〈·, ·〉, k, n, and a ₁ , …, a _n as initially, the following two conditions are satisfied:

1. each a _v has a formal adjoint a* _v , being an endomorphism of M such that $ \left\langle {a_v x,y} \right\rangle = \left\langle {x,a_v^* y} \right\rangle (x,y \in M); $
2. the mappings a ₁, …, a _n , a*₁, …, a* _n commute pairwise.

Then the bound nk can be replaced by k (regardless of how large n may be). This result cannot be improved in general since it may happen that each a _v is a scalar multiple of the identical mapping of M into itself (not all a _v equal to 0), in which case the form (1) is a positive multiple of 〈·, ·〉 itself. There are ties with the subjects of ‘positive semidefinite submodules’ (‘positive semidefinite left ideals’) and ‘definitisation’.     

A sharp interface method for high-speed multi-material flows: strong shocks and arbitrary materialpairs

A general framework is developed for solving high-speed and high-intensity multi-material interaction problems on adaptively refined Cartesian meshes. The framework is applicable for interfaces separating materials with very different properties and in the presence of strong shocks. A sharp interface treatment is maintained through a modified Ghost Fluid Method. The embedded boundaries are tracked and represented with level sets. A tree-based Local Mesh Refinement scheme is employed to efficiently resolve the desired physics. Results are shown for situations that cover varied combination of materials (fluids, rigid solids and deformable solids) with careful benchmarking to establish the validity and the versatility of the approach.     

保持特征的三角形有限元网格简化方法

基于二次误差的网格简化算法,提出了一种三角形有限元网格模型简化算法。在网格质量评估中考虑折叠代价,避免过于狭长的三角形出现,保证新生成单元的质量。同时,通过改进特征检测机制,加入用户控制,对模型上的重要细节特征进行精细定位,使算法适用范围更广泛,简化结果更能满足网格简化过程中细节特征的保持要求。     

Nonlinear Schrodinger equation with combined power-type nonlinearities and harmonic potential

This paper discusses a class of nonlinear Schrodinger equations with combined power-type nonlinearities and harmonic potential.By constructing a variational problem the potential well method is applied.The structure of the potential well and the properties of the depth function are given.The invariance of some sets for the problem is shown.It is proven that,if the initial data are in the potential well or out of it,the solutions will lie in the potential well or lie out of it,respectively.By the convexity method,the sharp condition of the global well-posedness is given.