Convergence of non-cyclic infinite products of operators

Using a geometric approach (in particular, angles between subspaces), we establish useful estimates regarding convergence of infinite products involving orthogonal projections and other (possibly nonlinear) nonexpansive operators in Hilbert space.     

关于半准素环

交换环Ｒ称为（受限制的）半准素环，如果对Ｒ的每个（非零）主理想Ａ，都有A^1/2是Ｒ的素理想，本文刻画了受限制的半准素环，给出了有单位元的Ｎｏｅｔｈｅｒ受限制的半准素环的分类以及半准素整环是伪赋值整环的一个条件     

On the Projections of the Mutual Multifractal Rényi Dimensions

In this paper, we compare the mutual multifractal Rényi dimensions to the mutual multifractal Hausdorff and pre-packing dimensions. We also provide a relationship between the mutual multifractal Rényi dimensions of orthogonal projections of a couple of measures $(\mu,\nu)$ in $\mathbb{R}^n$. As an application, we study the mutual multifractal analysis of the projections of measures.     

Zero weights and non-zero slacks: Different solutions to the same problem

This paper re-assesses three independently developed approaches that are aimed at solving the problem of zero-weights or non-zero slacks in Data Envelopment Analysis (DEA). The methods are weights restricted, non-radial and extended facet DEA models. Weights restricted DEA models are dual to envelopment DEA models with restrictions on the dual variables (DEA weights) aimed at avoiding zero values for those weights; non-radial DEA models are envelopment models which avoid non-zero slacks in the input-output constraints. Finally, extended facet DEA models recognize that only projections on facets of full dimension correspond to well defined rates of substitution/transformation between all inputs/outputs which in turn correspond to non-zero weights in the multiplier version of the DEA model. We demonstrate how these methods are equivalent, not only in their aim but also in the solutions they yield. In addition, we show that the aforementioned methods modify the production frontier by extending existing facets or creating unobserved facets. Further we propose a new approach that uses weight restrictions to extend existing facets. This approach has some advantages in computational terms, because extended facet models normally make use of mixed integer programming models, which are computationally demanding.     

A note on linear preservers of certain ranks of tensor products of matrices

In this note, we give a simple proof as well as an extension of a very recent result of B. Zheng, J. Xu and A. Fosner concerning linear maps between vector spaces of complex square matrices that preserve the rank of tensor products of matrices by using a structure theorem of R. Westwick on linear maps between tensor product spaces that preserve non-zero decomposable elements.     

Convergence of minimum norm elements of projections and intersections of nested affine spaces in Hilbert space

We consider a Hilbert space, an orthogonal projection onto a closed subspace and a sequence of downwardly directed affine spaces. We give sufficient conditions for the projection of the intersection of the affine spaces into the closed subspace to be equal to the intersection of their projections. Under a closure assumption, one such (necessary and) sufficient condition is that summation and intersection commute between the orthogonal complement of the closed subspace, and the subspaces corresponding to the affine spaces. Another sufficient condition is that the cosines of the angles between the orthogonal complement of the closed subspace, and the subspaces corresponding to the affine spaces, be bounded away from one. Our results are then applied to a general infinite horizon, positive semi-definite, linear quadratic mathematical programming problem. Specifically, under suitable conditions, we show that optimal solutions exist and, modulo those feasible solutions with zero objective value, they are limits of optimal solutions to finite-dimensional truncations of the original problem.     

因子von Neumann代数上导子的等价刻画

设R是含非平凡幂等元P的素环,C∈R,C=PC.本文证明可加映射△:R→R在C可导,即△(AB)=△(A)B+A△(B),A,B∈R,AB=C当且仅当存在导子δ:R→R,使得△(A)=δ(A)+△(I)A,A∈R.没有I_1型中心直和项的von Neumann代数上的可导映射也有类似结论.利用该结论证明了,若非零算子C∈B(X),使得ran(C)或ker(C)在X中可补,则可加映射△:B(X)→B(X)在C可导当且仅当它是导子.特别地,证明了因子von Neumann代数上的可加映射在任意但固定的非零算子可导当且仅当它是导子.     

LU分解与广义Schmidt正交化方法

利用对称内积的Schmidt正交化方法证明了各阶主子式不为零对称阵的LDLT分解.引入两个向量组关于弱内积广义正交的概念,并构造了将两组含相同个数向量的线性无关组化为广义正交组的广义Schmidt正交化方法.最后应用这一方法证明了各阶主子式不为零矩阵的LDU分解及一些相关的结果.     

Equalities of ideals associated with two projections in rings with involution

In this article we study various right ideals associated with two projections (self-adjoint idempotents) in a ring with involution. Results of O.M. Baksalary, G. Trenkler, R. Piziak, P.L. Odell and R. Hahn about orthogonal projectors (complex matrices which are Hermitian and idempotent) are considered in the setting of rings with involution. New proofs based on algebraic arguments, rather than finite-dimensional and rank theory, are given.     

Surfaces in {\mathbb{R}^4} with constant principal angles with respect to a plane

We study surfaces in ${\mathbb{R}^4}$ whose tangent spaces have constant principal angles with respect to a plane. Using a PDE we prove the existence of surfaces with arbitrary constant principal angles. The existence of such surfaces turns out to be equivalent to the existence of a special local symplectomorphism of ${\mathbb{R}^2}$ . We classify all surfaces with one principal angle equal to 0 and observe that they can be constructed as the union of normal holonomy tubes. We also classify the complete constant angles surfaces in ${\mathbb{R}^4}$ with respect to a plane. They turn out to be extrinsic products. We characterize which surfaces with constant principal angles are compositions in the sense of Dajczer-Do Carmo. Finally, we classify surfaces with constant principal angles contained in a sphere and those with parallel mean curvature vector field.     

局部环上正交群中一类子群的扩群

定出了局部环上正交群中一类子群的扩群,得到了如下结果:设R是局部环,M是R的唯一极大理想,O(2m,R)为R上正交群.对R的任意理想S,G(2m,S)表示子群{A BC D∈O(2m,R)|B∈Sm×m}.如果char(R)≠2,m≥3,G(2m,0)≤X≤G(2m,M),那么存在R的理想S,使得X=G(2m,S).     

Angles between Euclidean subspaces

The angle between two subspaces of dimensions p and q in a Euclidean space is considered by using exterior algebra. Some properties of angles are obtained. The relation between such a higher dimensional angle and the usual principal angles is also given. And finally, an application to Grassmann manifolds is briefly discussed.Supported by the National Natural Science Fund of P.R. China.     

Angles between infinite dimensional subspaces with applications to the Rayleigh-Ritz and alternating projectors methods

We define angles for infinite dimensional subspaces of Hilbert spaces, inspired by the work of E.J. Hannan, 1961/1962. The angles of Dixmier and Friedrichs, and the gaps are characterized. We establish connections between the angles corresponding to orthogonal complements. The sensitivity of angles with respect to subspaces is estimated. We show that the squared cosines of the angles from one subspace to another can be interpreted as Ritz values in the Rayleigh-Ritz method. The Hausdorff distance between the Ritz values, corresponding to different trial subspaces, is shown to be bounded by a constant times the gap between the subspaces. We prove a similar eigenvalue perturbation bound that involves the gap squared. An ultimate acceleration of the classical alternating projectors method is proposed. Its convergence rate is estimated in terms of the angles. We illustrate the acceleration for a domain decomposition method with a small overlap for the 1D diffusion equation.     

正则WB-环

引进了WB-环,研究了正则环为WB-环的等价刻画．如果A是正则环R上的有限生成投射右模而且M_n(R)都是WB-环(n∈N),若B,C是任何右R-模而且A⊕B≌A⊕C,证明了存在正交理想I,J,使得B/BI■~⊕C/CI且C/CJ■~⊕B/BJ．这也给出了QB-环上新的模比较性质．     

Hilbert空间上两个正交射影的乘积

该文研究Hilbert空间H上正则射影对(P,Q)的性质和结构,给出H上有界线性算子A表示为两个正交射影乘积的充分必要条件.     

ON SOME SPECIAL CLASSES OF PRIME RINGS

Abstract

Given a non-zero cardinal α, a ring R is said to be SP(α) if a is the first cardinal for which every non-zero element of R has an insulator of cardinality less than α + 1.

It is shown that the class of SP(α) rings is a special class (in the sense of Andrunakievi?) for each α. A theorem of Groenewald and Heyman (also Desale and Varadarajan) to the effect that the class of all strongly prime rings is a special class is obtained as a corollary. Every SP(α) ring has an SP(α) rational extension ring with identity.     

相配群胚上的诱导联络

本文引入了相配群胚和相配主丛的概念,它们是一对具有相同传递函数的局部平凡李群胚（Г→_→P,α,β）和主丛（B,P,π,G）.我们首先考察了相配群胚Г的内子群胚GГ的局部平凡化,利用这个局部平凡化证明了在B和GГ之间自然存在着一个丛同构.通过这个丛同构以及B的联络日逐步得到了H在GГ和Г上的诱导联络,进而定义了Г上分别以α,β为投影的左联络和右联络,这两个联络都是在Г上整体有定义的,与以往李群胚上的联络只是定义在李代数胚上不同.     

On the power of combinatorial and spectral invariants

We extend the traditional spectral invariants (spectrum and angles) by a stronger polynomial time computable graph invariant based on the angles between projections of standard basis vectors into the eigenspaces (in addition to the usual angles between standard basis vectors and eigenspaces). The exact power of the new invariant is still an open problem. We also define combinatorial invariants based on standard graph isomorphism heuristics and compare their strengths with the spectral invariants. In particular, we show that a simple edge coloring invariant is at least as powerful as all these spectral invariants.     

On the Local Homology Theory for Artinian Modules

Let A be a non-zero Artinian R-module. For an arbitrary ideal I of R, we show that the local homology module Hpx(A) is independent of the choice of x whenever 0 : AI = 0 : A(x1,...,xr). Thus, identifying these modules, we write HpI(A). In this paper we prove that there is a certain duality between HiI(A) and the local cohomology modules and provide some information about the vanishing local homology module HiI(A) which gives an improved form of the main results of [22].