The problem of approximating a multivariable transfer function G(s) of McMillan degree n, by Ĝ(s) of McMillan degree k is considered. A complete characterization of all approximations that minimize the Hankel-norm is derived. The solution involves a characterization of all rational functions Ĝ(s) + F(s) that minimize where Ĝ(s) has McMillan degree k, and F(s) is anticavisal. The solution to the latter problem is via results on balanced realizations, all-pass functions and the inertia of matrices, all in terms of the solutions to Lyapunov equations. It is then shown that where σ k+1(G(s)) is the (k+l)st Hankel singular value of G(s) and  for one class of optimal Hankel-norm approximations. The method is not computationally demanding and is applied to a 12-state model.

All optimal Hankel-norm approximations of linear multivariable systems and their L, ∞ -error bounds†

One-dimensional Bose-gas One-dimensional Heisenberg magnet Massive Thirring model Classical r-matrix Fundamentals of inverse scattering method Algebraic Bethe ansatz Quantum field theory integral models on a lattice Theory of scalar products Form factors Mean value of operator Q Assymptotics of correlation functions Temperature correlation functions Appendices References.

Quantum Inverse Scattering Method and Correlation Functions

This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.

A first course in noncommutative rings, by T. Y. Lam. Pp. 385. £37 (pb), £62.50 (hb). 2001. ISBN 0 387 95325 6 (pb), 0 387 95183 0 (hb) (Springer-Verlag).

Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows

The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving them in a number of diverse communities, and to review some problems that remain open. An important contribution of our work is to bring together material from several areas of research and to present results in a unified manner. We begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions. Closely related to the concept of stability are the notions of exponential growth rates and converse Lyapunov theorems, both of which are discussed in detail. In particular, results on common quadratic Lyapunov functions and piecewise linear Lyapunov functions are presented, as they represent constructive methods for proving stability and also represent problems in which significant progress has been made. We also comment on the inherent difficulty in determining stability of switched systems in general, which is exemplified by NP-hardness and undecidability results. We then proceed by considering the stability of switched systems in which there are constraints on the switching rules, through both dwell-time requirements and state-dependent switching laws. Also in this case the theory of Lyapunov functions and the existence of converse theorems are reviewed. We briefly comment on the classical Lur'e problem and on the theory of stability radii, both of which contain many of the features of switched systems and are rich sources of practical results on the topic. Finally we present a list of questions and open problems which provide motivation for continued research in this area.

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Stability Criteria for Switched and Hybrid Systems

Stability.- Exponential and ordinary dichotomies.- Dichotomies and functional analysis.- Roughness.- Dichotomies and reducibility.- Criteria for an exponential dichotomy.- Dichotomies and lyapunov functions.- Equations on ? and almost periodic equations.- Dichotomies and the hull of an equation.

Dichotomies in Stability Theory

We have already determined the integers which can be represented as a sum of two squares. Similarly, one may ask which integers can be represented in the form x 2 + 2y 2 or, more generally, in the form ax 2 + 2bxy + cy 2, where a, b, c are given integers. The arithmetic theory of binary quadratic forms, which had its origins in the work of Fermat, was extensively developed during the 18th century by Euler, Lagrange, Legendre and Gauss. The extension to quadratic forms in more than two variables, which was begun by them and is exemplified by Lagrange’s theorem that every positive integer is a sum of four squares, was continued during the 19th century by Dirichlet, Hermite, H.J.S. Smith, Minkowski and others. In the 20th century Hasse and Siegel made notable contributions. With Hasse’s work especially it became apparent that the theory is more perspicuous if one allows the variables to be rational numbers, rather than integers. This opened the way to the study of quadratic forms over arbitrary fields, with pioneering contributions by Witt (1937) and Pfister (1965–67).

The Arithmetic of Quadratic Forms

Preface.- The Expanding Universe of Numbers.- Divisibility.- More on Divisibility.- Continued Fractions and their Uses.- Hadamard's Determinant Problem.-Hensel's P-Adic Numbers.- Notations.- Axioms.- Index.- The Arithmetic of Quadratic Forms.- The Geometry of Numbers.- The Number of Prime Numbers.- A Character Study.- Uniform Distribution and Ergodic Theory.- Elliptic Functions.- Connections with Number Theory.- Notations.- Axioms.- Index.

Number Theory: An Introduction to Mathematics

It was shown by Hadamard (1893) that, if all elements of an n × n matrix of complex numbers have absolute value at most μ, then the determinant of the matrix has absolute value at most μ n n n/2. For each positive integer n there exist complex n × n matrices for which this upper bound is attained. For example, the upper bound is attained for μ = 1 by the matrix (ω jk )(1 ≤ j, k ≤ n), where ω is a primitive n-th root of unity. This matrix is real for n = 1, 2. However, Hadamard also showed that if the upper bound is attained for a real n × n matrix, where n > 2, then n is divisible by 4.

Hadamard’s Determinant Problem

It was already shown in Euclid’s Elements (Book IX, Proposition 20) that there are infinitely many prime numbers. The proof is a model of simplicity: let \(p_1, \ldots, p_n\) be any finite set of primes and consider the integer \(N = p_1 \ldots p_n + 1\). Then \(N > 1\) and each prime divisor p of N is distinct from \(p_1, \ldots, p_n\), since \(p = p_j\) would imply that p divides \(N - p_1 \cdots p_n = 1\). It is worth noting that the same argument applies if we take \(N = p^{\propto_1}_1 \cdots p^{\propto_n}_n + 1\), with any positive integers \(\propto_1, \ldots, \propto_n\).

W. A. Coppel

Papers

Dichotomies in Stability Theory

The Arithmetic of Quadratic Forms

Number Theory: An Introduction to Mathematics

Hadamard’s Determinant Problem

The Number of Prime Numbers