Let A be a n

Here we introduce more general definitions of weak block diagonally dominant ma-trix and weak block H-matrix which permit block triangular factorizations and extend the theory to the block diagonally dominant matrices and the block H-matrices. Fur-thermore,by the theory of weak block H-matrix, we prove that any partitioned block form of a pointwise H-matrix has a block triangular factorization. 1998 Elsevier Science Inc. All rights reserved.

An upper bound and a lower bound for ®0 are given suchthat ®I+B2M1for®>®0and®I+B62M1for®•®0, where B is a nonnegative matrix and satises that for any positive constant, I+B is a power invariant zero pattern matrix.

Let X and Y be two real Hilbert spaces with the dimension of X greater than 1. Several cases about the Aleksandrov Rassias problem for T: XY preserving two or three distances are presented and geometric interpretations of these cases are also given.

Let X and Y be normed real vector spaces. A mapping T: X Y is called preserving the distance r if for all x; y of X with kx ykX = r then kT(x) T(y)k = r. In this paper, we provide an overall account of the developmentof the Aleksandrov problem, especially the Aleksandrov Rassias problem for mappings which preserves distances with a noninteger ratioin in Hilbert spaces.

Let A be an n × n matrix, q(A) = min{|λ|: λ ∈ σ(A)} and σ(A) denote the spectrum of A. From Fiedler and Markham [Linear Algebra Appl. 101 (1988) 1], Song [Linear Algebra Appl. 305 (2000) 99] and Yong [Linear Algebra Appl. 320 (2000) 167], for the Hadamard products of n×n M-matrices and their inverses, the infimum of q(A◦ A−1) is 2/n. In this paper the following results are presented: if q(Ak◦ A−1 k ) tends to the infimum 2/n for n×n (n>2) M-matrices Ak, k=1, 2,..., then the spectral radius ρ(Jk) of the Jacobi iterative matrix of Ak tends to 1. That is, if q(A◦ A−1) is close to 2/n, then ρ(J) is close to 1; and another lower bound is given for A being an n×n M-matrix, q(A◦ A−1) max1−ρ(J)2, +ρ(J)1 n+21+(n−1)ρ(J) 1n+2 where ρ(J) is the spectral radius of the Jacobi iterative matrix of A. Furthermore, if A is an H-matrix, then q(A◦A−1) (1−ρ(Jm(A))2)/(1+ρ(Jm(A))2), where ρ(Jm(A)) is the spectral radius of the Jacobi iterative matrix of the comparison matrix m(A).

Let X be a uniformly smooth infinite dimensional Banach space, and (Ω,Σ,μ) be a σ –finite measure space. Suppose that T :X→L∞(Ω,Σ,μ) satisfies 1−ε)‖≤‖Tx‖≤‖x‖Vx ∈ X, or some positive numberε<1/2 with δX∗ (2−2ε)>13/14. Then T is close to an isometry U :X→L∞(Ω,Σ,μ) such that ‖T‖≤16(1-δX∗ (2 −2ε)1/2∈ here δX∗ (t ) is the modulus of convexity of the conjugate space X∗.

The A. D. Aleksandrov problem and the A. D. Aleksandrov-Th. M. Rassias problem are two very essential open problems for mappings preserving distances. In this paper, we provide an account of some of the development of these problems.

A method for integral transformations of highly oscillatory functions, Bessel functions, is presented.It is based on the Filon-type method and the decay of the error can be increased as increases.The effectiveness and accuracy of the quadrature is tested for both large arguments and higher orders of Bessel functions in the case where the orders are nonnegative integers.

We give new error bounds for the linear complementarity problem where the involved matrix is a P-matrix. Computation of rigorous error bounds can be turned into a P-matrix linear interval system. Moreover, for the involved matrix being an H-matrix with positive diagonals, an error bound can be found by solving a linear system of equations, which is sharper than the Mathias-Pang error bound. Preliminary numerical results show that the proposed error bound is efficient for verifying accuracy of approximate solutions.