Robust D-stability

Motivated by the paper by Johnson, the 13 sufficient conditions are extended to robust D-stability of n × n matrices including variant types of matrices.     

Stability and D-stability

In this paper, we prove a new sufficient condition for D-stability of a matrix. We then use it to show that (i) a nonderogatory stable matrix A has a Schwarz canonical form which is D-stable, and (ii) a nonderogatory stable matrix A has a Routh-canonical form which is totally stable.     

Characterization of acyclic d-stable matrices

Necessary and sufficient conditions for D-stability of acyclic matrices are given. Special cases of this characterization are recent results on tridiagonal matrices.     

A new efficient approach to the characterization of D‐stable matrices

The concept of D‐stability is relevant for stable square matrices of any order, especially when they appear in ordinary differential systems modeling physical problems. Indeed, D‐stability was treated from different points of view in the last 50 years, but the problem of characterization of a general D‐stable matrix was solved for low‐order matrices only (ie, up to order 4). Here, a new approach is proposed within the context of numerical linear algebra. Starting from a known necessary and sufficient condition, other simpler equivalent necessary and sufficient conditions for D‐stability are proved. Such conditions turn out to be computationally more appealing for symbolic software, as discussed in the reported examples. Therefore, a new symbolic method is proposed to characterize matrices of order greater than 4, and then it is used in some numerical examples, given in details.     

An extension of the Kreiss stability theorem to families of matrices of unbounded order

The Kreiss matrix theorem asserts three necessary and sufficient conditions for a family of matrices of fixed finite order to be L₂-stable: a resolvent condition (R), a triangularization condition (S) and a Hermitian norm condition (H). We extend the Kreiss theorem to families of matrices of finite but unbounded order with the restriction that the degrees of the minimal polynomials of all matrices in the family are less than a fixed constant. For such matrix families, we show that (R) and (H) remain necessary and sufficient for L₂-stability, while (S) must be replaced by a somewhat stronger “block triangularization” condition (S′). This extended Kreiss theorem permits a corresponding extension of the Buchanan stability theorem.     

Stability of acyclic matrices

Necessary and sufficient conditions for stability of acyclic matrices whose principal minors are nonnegative are given. Also, a characterization is given of additive D-stable acyclic matrices.     

The Problem of Topologies of Grothendieck and the Class of Fréchet T-Spaces

We examine the interpolation with periodic polynomial splines of degree d and defect r (d ≦ r) on equidistant partitions of the real axis and generalize known results for r = 0. We prove necessary and sufficient conditions for the existence and a certain L²-stability of the interpolants as well as their approximation properties in the scale of the periodic SOBOLEV spaces.     

Holomorphic Triples on Elliptic Curves

Here we study the α-stability for holomorphic triples over curves of genus g = 1. We provide necessary and sufficient conditions for the moduli space of α-stable triples to be non-empy and, in these cases, we show that it is smooth and irreducible. Received: January 10, 2007.     

OnA-stable implicit Runge-Kutta methods

An elementary proof is given of theA-stability of implicit Runge-Kutta methods for which the corresponding rational function is on the diagonal or one of the first two subdiagonals of the Padé table for the exponential function. The result is extended to give necessary and sufficient conditions for theA-stability ofn-stage methods of order greater than or equal to 2n–2.     

Critical Cases of the π-Stability of a Nonautonomous Quasilinear Equation of the nth Order

We establish sufficient conditions for the n-stability of the trivial solution of a quasilinear equation of the nth order.     

Positive invertibility of matrices and stability of Itô delay differential equations

We study the global exponential p-stability (1 ≤ p < ∞) of systems of Itô nonlinear delay differential equations of a special form using the theory of positively invertible matrices. To this end, we apply a method developed by N.V. Azbelev and his students for the stability analysis of deterministic functional-differential equations. We obtain sufficient conditions for the global exponential 2p-stability (1 ≤ p < ∞) of systems of Itô nonlinear delay differential equations in terms of the positive invertibility of a matrix constructed from the original system. We verify these conditions for specific equations.     

w-Stability and Clifford w-Regularity of Polynomial Rings

We investigate the transfer of w-stability and Clifford w-regularity from a domain D to the polynomial ring D[X]. We show that these two properties pass from D to D[X] when D is either integrally closed or it is Mori and w-divisorial.     

关于Drazin逆逆序律的一个注记

本文利用矩阵间秩的关系给出Ｄｒａｚｉｎ逆逆序律成立的一个充分必要条件     

An algebraic characterization ofB-convergent Runge-Kutta methods

Summary In the analysis of discretization methods for stiff intial value problems, stability questions have received most part of the attention in the past.B-stability and the equivalent criterion algebraic stability are well known concepts for Runge-Kutta methods applied to dissipative problems. However, for the derivation ofB-convergence results — error bounds which are not affected by stiffness — it is not sufficient in many cases to requireB-stability alone. In this paper, necessary and sufficient conditions forB-convergence are determined.This paper was written while J. Schneid was visiting the Centre for Mathematics and Computer Science with an Erwin-Schrödinger stipend from the Fonds zur Förderung der wissenschaftlichen Forschung     

FACTORIZATION IN MONOID DOMAINS

In this paper, we determine necessary and sufficient conditions for the group ring D[G] to be a BFD (resp., an FFD, an SFFD). Also we give necessary and sufficient conditions for the monoid domain D[S] to be a BFD (resp., an FFD, an SFFD). In addition, we characterize when themonoid domain D[S] is a UFD in terms of 2-factoriality.     

Hadamard products of matrices

The entry-wise product of arbitrary n × ncomplex matrices is studied. The principal tools used include the Kionecker product, field of values and diagonal multiplications. Inclusion theorems for the field of values and spectrum are developed in the general case and refined in special cases. These are employed to obtain inequalities involving the real parts of the characteristic roots and the numerical radius, and previously known results are found to be special cases of several of the theorems. In addition, the case of positive stable matrices is considered and a new class of nonnegative stable matrices is introduced, studied and related to D-stability.     

A Class of Multiderivative Hybrid One-step Methods

This paper presents a class of hybrid one-step methods that are obtained by using Cramer's rule and rational approximations to function exp(q). The algorithms fall into the catalogue of implicit formula, which involves sth order derivative and s 1 free parameters. The order of the algorithms satisfies s 1≤p≤2s 2. The stability of the methods is also studied, necessary and sufficient conditions for A-stability and L-stability are given. In addition, some examples are also given to demonstrate the method presented.     

A 0-Stability and stiff stability of Brown's multistep multiderivative methods

Summary Brown [1] introducedk-step methods usingl derivatives. Necessary and sufficient conditions forA ₀-stability and stiff stability of these methods are given. These conditions are used to investigate for whichk andl the methods areA ₀-stable. It is seen that for allk andl withk1.5 (l+1) the methods areA ₀-stable and stiffly stable. This result is conservative and can be improved forl sufficiently large. For smallk andl A ₀-stability has been determined numerically by implementing the necessary and sufficient condition.     

Representations of bounded harmonic functions

Summary An open subsetD ofR ^d ,d≧2, is called Poissonian iff every bounded harmonic function on the set is a Poisson integral of a bounded function on its boundary. We show that the intersection of two Poissonian open sets is itself Poissonian and give a sufficient condition for the union of two Poissonian open sets to be Poissonian. Some necessary and sufficient conditions for an open set to be Poissonian are also given. In particular, we give a necessary and sufficient condition for a GreenianD to be Poissonian in terms of its Martin boundary. Supported by NSF DMS86-01800.     

The semigroup generated by a unitary orbit of a singular matrix

We investigate the structure of the multiplicative semigroup generated by the set of matrices that are unitarily equivalent to a given singular matrix A. In particular, we give necessary and sufficient conditions, in terms of the singular values of A, for such a semigroup to consist of all matrices of rank not exceeding the rank of A.