Characterizations of Positive Linear Volterra Integro-differential Systems

We first give a criterion for positivity of the solution semigroup of linear Volterra integro-differential systems. Then, we offer some explicit conditions under which the solution of a positive linear Volterra system is exponentially stable or (robustly) lies in L²[0,+∞). The first and last author are supported by the Japan Society for Promotion of Science (JSPS) ID No. P 05049.     

Categorical Abstract Algebraic Logic: Partially Ordered Algebraic Systems

An extension of parts of the theory of partially ordered varieties and quasivarieties, as presented by Pałasińska and Pigozzi in the framework of abstract algebraic logic, is developed in the more abstract framework of categorical abstract algebraic logic. Algebraic systems, as introduced in previous work by the author, play in this more abstract framework the role that universal algebras play in the more traditional treatment. The aim here is to build the generalized framework and to formulate and prove abstract versions of the ordered homomorphism theorems in this framework. To Don Pigozzi and Kate Pałasińska.     

Computing Numerically with Functions Instead of Numbers

Symbolic computation with functions of a real variable suffers from combinatorial explosion of memory and computation time. The alternative chebfun system for such computations is described, based on Chebyshev expansions and barycentric interpolation. For Richard Brent on his 60th birthday     

Some Remarks on the Definition of Boltzmann, Shannon and Kolmogorov Entropy

We give an alternative definition of Shannon and Kolmogorov-Sinai entropies based on the Boltzmann formula S = k log W. We prove the equivalence of those new definitions with the traditional ones using some tools from Information Theory such as information function and empirical entropy. Lecture held in the Seminario Matematico e Fisico on November 29, 2004 Received: April 2005     

Stability of differential equations with piecewise constant arguments of generalized type

In this paper we continue to consider differential equations with piecewise constant argument of generalized type (EPCAG) [M.U. Akhmet, Integral manifolds of differential equations with piecewise constant argument of generalized type, Nonlinear Anal. TMA 66 (2007) 367–383]. A deviating function of a new form is introduced. The linear and quasilinear systems are under discussion. The structure of the sets of solutions is specified. Necessary and sufficient conditions for stability of the zero solution are obtained. Our approach can be fruitfully applied to the investigation of stability, oscillations, controllability and many other problems of EPCAG. Some of the results were announced at The International Conference on Hybrid Systems and Applications, University of Louisiana, Lafayette, 2006.     

Frequency Domain Methods and Decoupling of Linear Infinite Dimensional Differential Algebraic Systems

We discuss the analysis of linear constant coefficient differential algebraic equations on infinite dimensional Hilbert spaces. We give solution concepts and discuss solvability criteria which are mainly based on Laplace transform. Furthermore, we investigate the decoupling of these systems motivated by the Kronecker normal form for the finite dimensional case. Applications are given by the analysis of mixed systems of ordinary differential, partial differential and differential algebraic equations.     

Topologically Algebraic Algebras

We show that a locally convex algebra is topologically algebraic if, and only if, it is algebraic.Thanks are offered to Professor M. Oudadess for many remarks which allowed an improvement of the first version of this paper. We are also indebted to Professor A. Mallios for valuable suggestions.     

An interpolation problem in the class of stieltjes functions and its connection with other problems

This paper contains the proofs and some development of results that were published without proofs in [KN2]. It is completed with comments added by the second author explaining how these results became the basis of statements and the solutions of problems in the theory of entire functions, in the moment problem, in direct and inverse problems of the spectral theory of nonhomogeneous strings and in other problems.     

Existence and Stability of Periodic Solutions of a Duffing Equation by Using a New Maximum Principle

The purpose of this paper is to obtain new criteria for existence and asymptotic stability of periodic solutions of a Duffing equation x + cx + g(t, x) = 0, taking advantage of a new maximum principle with L^p-conditions combined with known relations between upper and lower solutions, topological degree and stability.Partially supported by D.G.I. BFM2002-01308, Ministerio Ciencia y Tecnología, Spain.     

Absolute exponential stability of nonlinear systems of differential equations with lag

Constructive sufficient conditions of absolute exponential stability for a class of nonlinear systems of differential equations with lag are found.     

Vector potential theory on nonsmooth domains in R<Superscript>3</Superscript> and applications to electromagnetic scattering

We study boundary value problems for the time-harmonic form of the Maxwell equations, as well as for other related systems of equations, on arbitrary Lipschitz domains in the three-dimensional Euclidean space. The main goal is to develop the corresponding theory for L^p-integrable bounday data for optimal values of p's. We also discuss a number of relevant applications in electromagnetic scattering.     

Structured Low Rank Approximation of a Bezout Matrix

The task of determining the approximate greatest common divisor (GCD) of more than two univariate polynomials with inexact coefficients can be formulated as computing for a given Bezout matrix a new Bezout matrix of lower rank whose entries are near the corresponding entries of that input matrix. We present an algorithm based on a version of structured nonlinear total least squares (SNTLS) method for computing approximate GCD and demonstrate the practical performance of our algorithm on a diverse set of univariate polynomials. The work is partially supported by a National Key Basic Research Project of China 2004CB318000 and Chinese National Science Foundation under Grant 10401035.