不可约矩阵的弱指数

作者与Ｒ．Ａ．Ｂｒｕａｌｄｉ教授提出了不可约矩阵弱指数的概念（见［Ｂｒｕｌｉｕ２］）．本文给出了弱指数（弱本原指数，弱完全不可分解指数，弱Ｈａｌｌ指数）的上确界以及它们的指数集．     

Weak ergodicity and products of random matrices

A representation for a weakly ergodic sequence of (nonstochastic) matrices allows products of nonnegative matrices which eventually become strictly positive to be expressed via products of some associated stochastic matrices and ratios of values of a certain function. This formula used in a random setup leads to a representation for the logarithm of a random matrix product. If the sequence of random matrices is in addition stationary then automatically almost all sequences are weakly ergodic, and the representation is expressed in terms of an one-dimensional stationary process. This permits properties of products of random matrices to be deduced from the latter. Second moment assumptions guarantee that central limit theorems and laws of the iterated logarithm hold for the random matrix products if and only if they hold for the corresponding stationary process. Finally, a central limit theorem for some classes of weakly dependent stationary random matrices is derived doing away with the restriction of boundedness of the ratios of colum entries assumed by previous studies. Extensions beyond stationarity are discussed.     

Weak interlacing properties of totally positive matrices

We prove certain interlacing inequalities for the eigenvalues of totally positive matrices.     

Controllability,inertia, and stability for tridiagonal matrices

Criteria are given for the controllability of certain pairs of tridiagonal matrices. These criteria may be used, with the Chen-Wimmer theorem, to obtain inertia results. Also, a characterization is given of those nonsingular tridiagonal matrices with certain principal minors nonnegative which are positive stable. This extends a previous characterization of the real D-stable tridiagonal matrices.     

Weak amicable T‐matrices and Plotkin arrays

We introduce the class of weak amicable T‐matrices and use it to construct a class of orthogonal designs, for p = 1 and for p a prime power ≡ 3 (mod 4), and all odd q, q ≤ 21. This class includes new Plotkin arrays of order 24, 40, 56 and for the first time, of orders 8q, q ∈ {9,11,13,15,17,19,21}. © 2006 Wiley Periodicals, Inc. J Combin Designs 16: 44–52, 2008     

On the stability of large matrices

The distance rstab(A) of a stable matrix A to the set of unstable matrices and the norm of the exponential of matrices constitute two important topics in stability theory. We treat in this note the case of large matrices. The method proposed partitions the matrix into two blocks: a small block in which the stability is studied and a large block whose field of values is located in the complex plane. Using the information on the blocks and some results on perturbation theory, we give sufficient conditions for the stability of the original matrix, a lower bound of rstab(A) and an upper bound on the norm of the exponential of A. We illustrate these theoretical bounds on a practical test problem.     

Linear stability of some symplectic matrices

This paper is devoted to a study on the linear stability of some basic normal forms of symplectic matrices. Some necessary and sufficient conditions are given.     

Weak asymptotic stability at first approximation for periodic differential inclusions

Weak asymptotic stability of an equilibrium position for a periodic differential inclusion is studied. First the weak asymptotic stability for a discrete-time inclusion generated by the original differential inclusion is investigated with the help of first approximation techniques. Then using the results for discrete-time inclusions the weak asymptotic stability for the differential inclusion is derived from the properties of its first approximation.     

A method of solving the stability problem for complex matrices

The aim of this paper is to determine the location of eigenvalues of a general complex matrix without explicitly computing them. Essentially, the method consists in finding a Hermitian matrixH which has the same inertia as the given matrixM, and determining the inertia ofH. H is found as the solution of the Liapunov equation , where is a matrix in upper Hessenberg form which is similar toM. The method remains applicable even if the solution of the Liapunov equation is affected by rounding errors, as long as a given criterion is satisfied. A programmed version of this method has been run on our computer with satisfactory results.This work was performed under the terms of the agreement on accosiation between the Max-Planck-Institut für Plasmaphysik and EURATOM.     

Weak exponential stability of stochastic differential equations

In this paper we introduce weak exponential stability of stochastic differential equations. In particular, we introduce weak exponential stability in mean, weak exponential asymptotical stability in mean and weak uniform asymptotical stability in mean. We also derive some results related to the above concepts     

Weak stability of almost regular contact foliations

We prove that on a compact manifold, a contact foliation obtained by a smallC ¹ perturbation of an almost regular contact flow has at least two closed characteristics. This solves the Weinstein conjecture for contact forms which areC ¹-close to almost regular contact forms.Supported in part by NSF Grant DMS 90-01861     

Weak stability of factorizations of operator pencils

Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 51, pp. 141–143, 1989.     

Weak stability bounds for approximations of invariant measures with applications to queueing

Methodology and Computing in Applied Probability - This paper investigate the approximation of invariant distributions for countable space Markov chains using truncations of the transition matrix....     

Reduced conservatism in stability analysis of interval matrices

This paper presents new sufficient conditions for the stability of interval matrices. An iterative algorithm, based on the Lyapunov stability approach, is developed. In this way, the stability bounds are increased substantially over the ones recently reported in the literature. Three numerical examples are given to demonstrate the merits of the proposed approach.This research was supported in part by the US-Yugoslav Joint Fund for Scientific and Technological Cooperation in cooperation with DOE under Grant No. PP-727.     

Weak stability of the Ishikawa iteration procedures for -hemicontractions and accretive operators

Let X be an arbitrary Banach space, K be a nonempty closed convex subset of X, and T : K → K be a Lipschitzian and hemicontractive mapping with the property lim inf_t→∞((t)/t) > 0. It is shown that the Ishikawa iteration procedures are weakly T-stable. As consequences, several related results deal with the weak stability of these procedures for the iteration proximation of solutions of nonlinear equations involving accretive operators. Our results improve and extend those corresponding results announced by Osilike.     

Polynomial stability of semigroups generated by operator matrices

In this paper, we study the stability properties of strongly continuous semigroups generated by block operator matrices. We consider triangular and full operator matrices whose diagonal operator blocks generate polynomially stable semigroups. As our main results, we present conditions under which also the semigroup generated by the operator matrix is polynomially stable. The theoretical results are used to derive conditions for the polynomial stability of a system consisting of a two-dimensional and a one-dimensional damped wave equation.     

Weak real integral characterizations for exponential stability of semigroups in reflexive spaces

Let (t _n ) be a sequence of nonnegative real numbers tending to ∞, such that 1≤t _n+1?t _n ≤α for all natural numbers n and some positive α. We prove that a strongly continuous semigroup {T(t)} _t≥0, acting on a Hilbert space H, is uniformly exponentially stable if $$\sum_{n=0}^\infty\varphi\bigl(\bigl|\bigl\langle T(t_n)x, y\bigr\rangle\bigr|\bigr)<\infty, $$ for all unit vectors x, y in H. We obtain the same conclusion under the assumption that the inequality $$\sum_{n=0}^\infty\varphi\bigl(\bigl|\bigl\langle T(t_n)x, x^\ast\bigr\rangle\bigr|\bigr)<\infty, $$ is fulfilled for all unit vectors x∈X and x ^?∈X ^?, X being a reflexive Banach space. These results are stated for functions φ belonging to a special class of functions, such as defined in the second section of this paper. We conclude our paper with a Rolewicz’s type result in the continuous case on Hilbert spaces.     

Equilibration of matrices to optimize backward numerical stability

An equilibration is introduced that minimizes the maximum ratio of the upper bounds for the backward rounding error and the inherent error in the given data. This can be applied to the solution of systems of linear equations and to the linear eigenproblem.     

基于亚正定阵的Volterra系统稳定性的若干新判据

将亚正定阵引入到V o lterra系统,获得了判定该系统在正的平衡态全局稳定、扇形稳定及关联稳定的另一些新方法.从应用情况看,本文方法是有效的而且更为实用.