On irreducible factorizations of rational matrices and their applications

This paper is an extension of our studies of the computational aspects of spectral problems for rational matrices pursued in previous papers. Methods of solution of spectral problems for both one-parameter and two-parameter matrices are considered. Ways of constructing irreducible factorizations (including minimal factorizations with respect to the degree and size of multipliers) are suggested. These methods allow us to reduce the spectral problems for rational matrices to the same problems for polynomial matrices. A relation is established between the irreducible factorization of a one-parameter rational matrix and its irreducible realization used in system theory. These results are extended to the case of two-parameter rational matrices. Bibliography: 15 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 219, 1994, pp. 117–156. This work was carried out during our visit to Sweden under the financial support of the Chalmer University of Technology in Góterborg and the Institute of Information Processing of the University of Umeă. Translated by V. N. Kublanovskaya.     

Pair correlation of roots of rational functions with rational generating functions and quadratic denominators

For any rational functions with complex coefficients A(z),B(z), and C(z), where A(z), C(z) are not identically zero, we consider the sequence of rational functions H _m (z) with generating function ∑H _m (z)t ^m =1/(A(z)t ²+B(z)t+C(z)). We provide an explicit formula for the limiting pair correlation function of the roots of $\prod_{m=0}^{n}H_{m}(z)$ , as n→∞, counting multiplicities, on certain closed subarcs J of a curve $\mathcal{C}$ where the roots lie. We give an example where the limiting pair correlation function does not exist if J contains the endpoints of $\mathcal{C}$ .     

Zero-flee regions for a rational function with applications

In this paper, we study a rational function which plays an important role in several problems of interest (eigenvalue problems, linear control theory, ... ). Our main interest is to determine zero-free regions. We also derive upper and lower bounds for this function.     

Zero-flee regions for a rational function with applications

In this paper, we study a rational function which plays an important role in several problems of interest (eigenvalue problems, linear control theory, ... ). Our main interest is to determine zero-free regions. We also derive upper and lower bounds for this function. Communicated by T.L. Freeman     

A rational Arnoldi process with applications

The rational Arnoldi process is a popular method for the computation of a few eigenvalues of a large non‐Hermitian matrix and for the approximation of matrix functions. The method is particularly attractive when the rational functions that determine the process have only few distinct poles , because then few factorizations of matrices of the form A ? z_jI have to be computed. We discuss recursion relations for orthogonal bases of rational Krylov subspaces determined by rational functions with few distinct poles. These recursion formulas yield a new implementation of the rational Arnoldi process. Applications of the rational Arnoldi process to the approximation of matrix functions as well as to the computation of eigenvalues and pseudospectra of A are described. The new implementation is compared to several available implementations.     

Constraint satisfaction problems: Algorithms and applications

A constraint satisfaction problem (CSP) requires a value, selected from a given finite domain, to be assigned to each variable in the problem, so that all constraints relating the variables are satisfied. Many combinatorial problems in operational research, such as scheduling and timetabling, can be formulated as CSPs. Researchers in artificial intelligence (AI) usually adopt a constraint satisfaction approach as their preferred method when tackling such problems. However, constraint satisfaction approaches are not widely known amongst operational researchers. The aim of this paper is to introduce constraint satisfaction to the operational researcher. We start by defining CSPs, and describing the basic techniques for solving them. We then show how various combinatorial optimization problems are solved using a constraint satisfaction approach. Based on computational experience in the literature, constraint satisfaction approaches are compared with well-known operational research (OR) techniques such as integer programming, branch and bound, and simulated annealing.     

Computing generalized inverses of a rational matrix and applications

In this paper we investigate symbolic implementation of two modifications of the Leverrier-Faddeev algorithm, which are applicable in computation of the Moore-Penrose and the Drazin inverse of rational matrices. We introduce an algorithm for computation of the Drazin inverse of rational matrices. This algorithm represents an extension of the papers [11] and [14]. and a continuation of the papers [15, 16]. The symbolic implementation of these algorithms in the package mathEmatica is developed. A few matrix equations are solved by means of the Drazin inverse and the Moore-Penrose inverse of rational matrices.     

On algebraic automorphisms and their rational invariants

Let X be an affine irreducible variety over an algebraically closed field k of characteristic zero. Given an automorphism Φ, we denote by k(X)^Φ its field of invariants, i.e., the set of rational functions f on X such that f o Φ = f. Let n(Φ) be the transcendence degree of k(X)^Φ over k. In this paper we study the class of automorphisms Φ of X for which n(Φ) = dim X - 1. More precisely, we show that under some conditions on X, every such automorphism is of the form Φ = ϕ_g, where ϕ is an algebraic action of a linear algebraic group G of dimension 1 on X, and where g belongs to G. As an application, we determine the conjugacy classes of automorphisms of the plane for which n(Φ) = 1.     

Maximal inequalities for demimartingales and their applications

In this paper, we establish some maximal inequalities for demimartingales which generalize and improve the results of Christofides. The maximal inequalities for demimartingales are used as key inequalities to establish other results including Doob’s type maximal inequality for demimartingales, strong laws of large numbers and growth rate for demimartingales and associated random variables. At last, we give an equivalent condition of uniform integrability for demisubmartingales.     

Lattices, Diophantine equations and applications to rational surfaces

In this article,we study certain quadratic Diophantine equations in Picard lattices of blow-ups of the complex projective plane,and describe their relations with root systems and Weyl group orbits of quasiminuscule fundamental weights.We apply these to study the geometry of certain rational surfaces.     

Joint bounds for the Perron roots of nonnegative matrices with applications

Given a finite set {A_x}_{x ∈ X} of nonnegative matrices, we derive joint upper and lower bounds for the row sums of the matrices D⁻¹ A^(x) D, x ∈ X, where D is a specially chosen nonsingular diagonal matrix. These bounds, depending only on the sparsity patterns of the matrices A^(x) and their row sums, are used to obtain joint two-sided bounds for the Perron roots of given nonnegative matrices, joint upper bounds for the spectral radii of given complex matrices, bounds for the joint and lower spectral radii of a matrix set, and conditions sufficient for all convex combinations of given matrices to be Schur stable. Bibliography: 20 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 30–56.     

Means and their applications

General considerations about means and expectations are preceded by a short historical overview. The existing approaches to the definition of means are classified into three groups: approximational, functional, and axiomatic. In some particular cases all of them are equivalent. The problem of meaningfulness of means is discussed for ordinal data and for some important cases of metric data. A survey of the main areas of applications: decision theory, group decision, insurance, economical equity and inequality is also provided. This revised version was published online in June 2006 with corrections to the Cover Date.     

Universal elements for non-linear operators and their applications

We prove that under certain topological conditions on the set of universal elements of a continuous map T acting on a topological space X, that the direct sum T⊕Mg is universal, where Mg is multiplication by a generating element of a compact topological group. We use this result to characterize R₊-supercyclic operators and to show that whenever T is a supercyclic operator and z₁,…,zn are pairwise different non-zero complex numbers, then the operator z₁T⊕?⊕znT is cyclic. The latter answers affirmatively a question of Bayart and Matheron.