由给定的子空间构造新的子空间

本给出并证明了若干个子空间的并以及两个子空间的基构成子空间的充要条件，从而本质地揭示了除子空间的交与和是构造新的予空间的方法外，集合的其它运算不能构造新的子空间，最后分析了子空间直和的两种不同定义的优缺点，指出了张禾瑞教材中子空间直和定义推广时应注意的一个问题。     

有限空间的子空间计数

本文讨论了有限空间Vn 中包含给定l维子空间Vl 的m 维子空间的计数问题,以及给定子空间Vm 与Vn 相交恰为l维子空间的计数问题     

欧氏空间上典型变换的循环子空间

运用实线性空间上线性变换的循环子空间分解的结果,证明了欧氏空间的一些典型线性变换的循环子空间只有1维和2维的.也就是说,在这些典型线性变换下,欧氏空间可分解为两两正交的1维和2维循环子空间的直和,进而得到欧氏空间的这些典型变换的正交相似标准型.     

初始的L-外部空间

研究了初始的L-外部空间.构造了L-外部空间范畴的初始结构,并由此定义了L-外部空间的乘积结构和子空间结构,证明了L-外部空间范畴是L-预外部空间范畴的余反射子范畴.     

有限维线性空间上子空间并的性质的一个注记

对于特征为零的域上的有限维线性空间的子空间的并,我们知道下述性质:有限个互不包含的非平凡子空间的并不是原来的线性空间.一方面,本文通过介绍有限维线性空间中任一子空间与齐次线性方程组解子空间的关系,及商空间的维数公式,给出了上述性质的一个改进证明.另一方面,本文把仿射簇的概念和子空间联系起来,并根据仿射簇的一个简单性质,给出了上述性质的另一个更为简洁的证法.     

子空间Parseval框架的一个基本恒等式

本文研究了子空间框架的一个基本恒等式.利用算子理论的两个基本结果,得到了子空间Parseval框架的基本恒等式,同时给出了恒等式的几种变形,包括一般子空间框架的情况.     

线性空间的亚子空间

本文给出了线性空间中亚子空间的概念，讨论了亚子空间的性质和维数，并将其应用到有解的非齐次线性方程组中去．     

有限空间的子空间计数

本讨论了有限空同V．中包古给定l维子空间V1的m维子空间的计数问题,以及给定子空问Vm与Vn相交恰为l维子空问的计数问题．     

（A,B）不变子空间的分解

本文从(A,B)特征子空间出发建立了(A,B)根子空间的新概念。证明了任意一个(A,B)不变子空间一定能分解成一些根子空间的直和。讨论了这种分解的唯一性问题。而且应用这些结论给出了允许配置的特征结构的充分必要条件。     

置换空间PBBs中的一些逼近性质

H.Mohebi与Sh.Rezapour在研究Banach空间中的逼近问题时,提出了quasi-Chebyshev子空间及ε-weakly Chebyshev子空间等概念,在文中作者就这些逼近性质在置换空间PBBs中进行了讨论.     

On the effect of friend feedbacks

Given ?? an (A,B)‐invariant subspace, we prove that the set of friend feedbacks is a linear variety, which can be considered as the direct sum of the feedbacks of the restriction to ?? and the co‐restriction to ??^⊥. In particular, when the natural controllability hypothesis hold, both pole assignments are simultaneously possible by means of a convenient friend feedback. Copyright © 2010 John Wiley & Sons, Ltd.     

Expansion of solution in terms of generalized eigenfunctions for a hyperbolic system with static boundary condition

This paper studies a linear hyperbolic system with static boundary condition that was first studied in Neves et al. [J. Funct. Anal. 67(1986) 320-344]. It is shown that the spectrum of the system consists of zeros of a sine-type function and the generalized eigenfunctions of the system constitute a Riesz basis with parentheses for the root subspace. The state space thereby decomposes into topological direct sum of root subspace and another invariant subspace in which the associated semigroup is superstable: that is to say, the semigroup is identical to zero after a finite time period.     

Wavelet frames for (not necessarily reducing) affine subspaces II: The structure of affine subspaces

This is a continuation of the investigation into the theory of wavelet frames for general affine subspaces. The main focus of this paper is on the structural properties of affine subspaces. We show that every affine subspace is the orthogonal direct sum of at most three purely non-reducing subspaces, while every reducing subspace (with respect to the dilation and translation operators) is the orthogonal direct sum of two purely non-reducing ones. This result is obtained through considering the basic question as to when the orthogonal complement of an affine subspace in another one is still affine. Motivated by the fundamental question as to whether every affine subspace is singly-generated, and by a recent result that every singly generated purely non-reducing subspace admits a singly generated wavelet frame, we prove that every affine subspace can be decomposed into the direct sum of a singly generated affine subspace and some space of “small size”. As a consequence we establish a connection between the above mentioned two questions.     

Dense subspaces of quasinormable spaces

Dense linear subspaces of quasinormable Fréchet spaces need not be quasinormable, as an example due to J. Bonet and S. Dierolf proved. A characterization of the quasinormability of dense linear subspaces of quasinormable locally convex spaces and several consequences are given. Moreover, an example of a dense linear subspace of a countable direct sum of Banach spaces, which is not quasinormable, is provided. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

闭极大线性子空间正交补的判别及应用

继续前面的工作,证得对于赋范线性空间中固定线性子空间成为迫近子空间的充分必要条件.特别的,对于闭极大线性子空间来说,的迫近性等价于的正交可补性.做为其直接推论,给出Banach空间成为自反空间的5个等价条件,其中4个条件为经典结果,一个条件为已有文献中给出的条件,但给出全新的证明.     

Some topologies on a Šerstnev probabilistic normed space

In this paper we will introduce two other topologies, coarser than the so-called strong topology, on a class of Šerstnev probabilistic normed spaces, and obtain some important properties of these topologies. We will show that under the first topology, denoted by τ₀, our probabilistic normed space is decomposable into the topological direct sum of a normable subspace and the subspace of probably null elements. Under the second topology, which is in fact the inductive limit topology of a family of locally convex topologies, the dual space becomes a locally convex topological vector space.     

Continuity properties of Markov semigroups and their restrictions to invariant <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-spaces

We consider Markov semigroups on the cone of positive finite measures on a complete separable metric space. Such a semigroup extends to a semigroup of linear operators on the vector space of measures that typically fails to be strongly continuous for the total variation norm. First we characterise when the restriction of a Markov semigroup to an invariant L ¹-space is strongly continuous. Aided by this result we provide several characterisations of the subspace of strong continuity for the total variation norm. We prove that this subspace is a projection band in the Banach lattice of finite measures, and consequently obtain a direct sum decomposition.     

A linear system solver based on a modified Krylov subspace method for breakdown recovery

Despite its usefulness in solving eigenvalue problems and linear systems of equations, the nonsymmetric Lanczos method is known to suffer from a potential breakdown problem. Previous and recent approaches for handling the Lanczos exact and near-breakdowns include, for example, the look-ahead schemes by Parlett-Taylor-Liu [23], Freund-Gutknecht-Nachtigal [9], and Brezinski-Redivo Zaglia-Sadok [4]; the combined look-ahead and restart scheme by Joubert [18]; and the low-rank modified Lanczos scheme by Huckle [17]. In this paper, we present yet another scheme based on a modified Krylov subspace approach for the solution of nonsymmetric linear systems. When a breakdown occurs, our approach seeks a modified dual Krylov subspace, which is the sum of the original subspace and a new Krylov subspaceK _m (w _j ,A ^T ) wherew _j is a newstart vector (this approach has been studied by Ye [26] for eigenvalue computations). Based on this strategy, we have developed a practical algorithm for linear systems called the MLAN/QM algorithm, which also incorporates the residual quasi-minimization as proposed in [12]. We present a few convergence bounds for the method as well as numerical results to show its effectiveness.Research supported by Natural Sciences and Engineering Research Council of Canada.     

线性定常系统规范结构分解问题的研究

单输入单输出线性定常系统的状态空间按能控性和能观测性进行结构分解,能全面、完整的反映系统的各个子系统特性,理论上揭示状态空间的本质特征.通过规范性分解,不完全能控和不完全能观测的连续时间线性时不变系统可分解为"能控能观测","能控不能观测","不能控能观测","不能控不能观测"四个子空间,但是其分解后子空间的维度受非奇异变换矩阵P_c的影响.提出了一种约束变量的方法,用来选取P_c中列向量的规则,以控制原系统分解后子空间的表现形式.经实例验算,方法简单有效,具有一定的教学和理论意义.     

Classification of solvable Leibniz algebras with null-filiform nilradical

In this article we classify solvable Leibniz algebras whose nilradical is a null-filiform algebra. We extend the obtained classification to the case when the solvable Leibniz algebra is decomposed as a direct sum of its nilradical, which is a direct sum of null-filiform ideals and a one-dimensional complementary subspace. Moreover, in this case we establish that these ideals are ideals of the algebra as well.