Classes of Operators Similar to Partial Isometries

The present paper deals with operators similar to partial isometries. We get some (necessary and) sufficient conditions for the similarity to (adjoints of) quasinormal partial isometries, or more general, to power partial isometries. We illustrate our results on the class of n-quasi-isometries, obtaining that a n-quasi-isometry is similar to a power partial isometry if and only if the ranges are closed. In particular if n = 2, these conditions ensure the similarity to quasinormal partial isometries of Duggal and Aluthge transforms of 2-quasi-isometries. The case when a n-quasi-isometry is a partial isometry is also studied, and a structure theorem for n-quasi-isometries which are power partial isometries is given. The second author was partially supported by Romanian 2-CEX Research grant. no. 06-11-34/2006.     

Rank equalities related to outer inverses of matrices and applications

A matrix X is called an outer inverse for a matrix A if XAX=X. In this paper, we present some basic rank equalities for difference and sum of outer inverses of a matrix, and apply them to characterize various equalities related to outer inverses, Moore-Penrose inverses, group inverses, Drazin inverses and weighted Moore-Penrose inverses of matrices.     

Rank equalities related to outer inverses of matrices and applications

A matrix X is called an outer inverse for a matrix A if XAX=X. In this paper, we present some basic rank equalities for difference and sum of outer inverses of a matrix, and apply them to characterize various equalities related to outer inverses, Moore-Penrose inverses, group inverses, Drazin inverses and weighted Moore-Penrose inverses of matrices.     

Generalized inverses and Douglas equations

In this paper we study the general concept of reduced solution for a Douglas type equation and we parametrize these solutions using generalized inverses. On the other hand, we characterize different sorts of generalized inverses by means of solutions of Douglas type equations.

     

Partial isometries and an invariant of <Emphasis Type="Italic">C</Emphasis>*-algebras

In this paper, we will discuss some properties of biprojection-commutative elements which are relevant to the classification of certain infinite C*-algebras, and define an important invariant s(A) of C*-algebra A as well as give some basic properties with regard to s(A). Moreover we prove that the invariant s(A) has continuity.     

Generalized inverses and the total stopping times of collatz sequences

Let f(n) be defined on the set N is even, and f(n)=3n+1 if nie: is odd. A well-known conjecture in number theory asserts that for every n the sequence of iterates eventually reaches the cycle (4,2,1). We recast the conjecture in terms of a denumerable Markov chain with transition matrix P. Assuming that (4,2,1) is the only cycle, but allowing for the possibility of unbounded trajectories, we establish the complete structure of a particular generalized inverse X of I?P and show that the entries of X describe the trajectories and "total stopping times" of integers n. Moreover, the infinite matrix X satisfies properties which, in the case of finite matrices, are the defining properties of the unique group generalized inverse (I?P)^#. The result extends to dynamical systems on ? consisting of points that are fixed, eventually fixed, or have unbounded trajectories. As a consequence, we obtain a generalized inverse that encodes the dynamics of such systems, and for cases in which known general criteria for the existence of (I?P)^# do not apply.     

Moore-Penrose广义逆矩阵与线性方程组的解

线性方程组的逆矩阵求解方法只使用于系数矩阵为可逆方阵,对于一般线性方程组可以应用Moore-Penrose广义逆矩阵来研究并表示其通解,本文主要探讨Moore-Penrose广义逆矩阵及一般线性方程组通解和最小范数解.     

Some Properties of the Optimal Preconditioner and the Generalized Superoptimal Preconditioner

<正>The optimal preconditioner and the superoptimal preconditioner were proposed in 1988 and 1992 respectively.They have been studied widely since then.Recently, Chen and Jin[6]extend the superoptimal preconditioner to a more general case by using the Moore-Penrose inverse.In this paper,we further study some useful properties of the optimal and the generalized superoptimal preconditioners.Several existing results are extended and new properties are developed.     

Generalized partial derivatives and optimality conditions for non-differentiable functions

In [9] a generalization of the concept of a classical gradient was presented, which exists for a broad class of non-differentiable functions. This concept differs from traditional “set-valued”" generalizations by an expansion of a directional derivative into a special orthogonal series. Investigating the coefficients of this series and interpreting them as a kind of partial derivatives formal analogies to properties of classical partial derivatives could be drawn. Moreover the new concept reveals relations between questions appearing within the theory of optimization and results in other branches of mathematics, especially to the theory of harmonic and subharmonic functions, The aim of this paper is to show that with the new tool of generalized partial derivatives necessary and sufficient conditions for optimality can be deduced in the case, that a function under consideration is not differentiable at the point of interest. In such a way criteria of optimality can be formulated using the language of these generalized partial derivatives. Numerical considerations show, that the generalized partial derivatives can (approximately) be calculated without solving integrals by hyperinterpolation on the sphere     

A partial inverse Sturm-Liouville problem: Matching a step function potential to a finite eigenvalue list

We investigate a certain inverse problem involving the Sturm-Liouville equation. In particular, given a finite list of target values, when can a potential function of a given form be found that produces these numbers as eigenvalues?     

求鳞状因子循环矩阵的逆阵及广义逆阵的快速付氏变换法

借助快速付立叶变换(FFT),本文给出一种求n阶鳞状因子循环矩阵的逆阵、自反g-逆、群逆、Moore-Penrose逆的快速算法,该算法的计算复杂性为O(nlog2n),最后给出的两个数值算例表明了该算法的有效性.     

Approximate inverses of multidiagonal matrices and application to the block PCG method

A fast method to compute high-order approximate inverses based on truncated elimination is constructed for multidiagonal matrices of diagonal dominance. Together with the block preconditioned conjugate gradient method, it can be used for the numerical solution of elliptic partial differential equations and related problems. Through numerical experiments it is shown that the method is robust and efficient.     

Generalized filters, the low-pass condition, and connections to multiresolution analyses

We study generalized filters that are associated to multiplicity functions and homomorphisms of the dual of an abelian group. These notions are based on the structure of generalized multiresolution analyses. We investigate when the Ruelle operator corresponding to such a filter is a pure isometry, and then use that characterization to study the problem of when a collection of closed subspaces, which satisfies all the conditions of a GMRA except the trivial intersection condition, must in fact have a trivial intersection. In this context, we obtain a generalization of a theorem of Bownik and Rzeszotnik.     

Inverse sorting problem by minimizing the total weighted number of changes and partial inverse sorting problems

In this paper, we consider two types of inverse sorting problems. The first type is an inverse sorting problem by minimizing the total weighted number of changes with bound constraints. We present an O(n ²) time algorithm to solve the problem. The second type is a partial inverse sorting problem and a variant of the partial inverse sorting problem. We show that both the partial inverse sorting problem and the variant can be solved by a combination of a sorting problem and an inverse sorting problem. Supported by the Hong Kong Universities Grant Council (CERG CITYU 103105) and the National Key Research and Development Program of China (2002CB312004) and the National Natural Science Foundation of China (700221001, 70425004).     

Inequalities related to partial trace and block Hadamard product

In this paper, we present inequalities related to partial trace and block Hadamard product for positive semidefinite matrices. Some interesting results involving block matrices will be also derived.     

A graph-theoretic approach to a partial order of knots and links

We say that a link L₁ is an s-major of a link L₂ if any diagram of L₁ can be transformed into a diagram of L₂ by changing some crossings and smoothing some crossings. This relation is a partial ordering on the set of all prime alternating links. We determine this partial order for all prime alternating knots and links with the crossing number less than or equal to six. The proofs are given by graph-theoretic methods.