to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

This article reviews the generalization of field theory to space-time with noncommuting coordinates, starting with the basics and covering most of the active directions of research. Such theories are now known to emerge from limits of M theory and string theory and to describe quantum Hall states. In the last few years they have been studied intensively, and many qualitatively new phenomena have been discovered, on both the classical and the quantum level.

Noncommutative field theory

Scaling level-spacing distribution functions in the “bulk of the spectrum” in random matrix models ofN×N hermitian matrices and then going to the limitN→∞ leads to the Fredholm determinant of thesine kernel sinπ(x−y)/π(x−y). Similarly a scaling limit at the “edge of the spectrum” leads to theAiry kernel [Ai(x)Ai(y)−Ai′(x)Ai(y)]/(x−y). In this paper we derive analogues for this Airy kernel of the following properties of the sine kernel: the completely integrable system of P.D.E.'s found by Jimbo, Miwa, Mori, and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painleve transcendent; the existence of a commuting differential operator; and the fact that this operator can be used in the derivation of asymptotics, for generaln, of the probability that an interval contains preciselyn eigenvalues.

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Level spacing distributions and the Airy kernel

Extended tanh-function method and its applications to nonlinear equations

The (G' G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics

Solitons have been of considerable interest to mathematicians since their discovery by Kruskal and Zabusky. This book brings together several aspects of soliton theory currently only available in research papers. Emphasis is given to the multi-dimensional problems arising and includes inverse scattering in multi-dimensions, integrable nonlinear evolution equations in multi-dimensions and the ∂ method. Thus, this book will be a valuable addition to the growing literature in the area and essential reading for all researchers in the field of soliton theory.

Solitons, Nonlinear Evolution Equations and Inverse Scattering

Solitons, Nonlinear Evolution Equations and Inverse Scattering: References

Some new similarity reductions of the Boussinesq equation, which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering, are presented. These new similarity reductions, including some new reductions to the first, second, and fourth Painleve equations, cannot be obtained using the standard Lie group method for finding group‐invariant solutions of partial differential equations; they are determined using a new and direct method that involves no group theoretical techniques.

New similarity reductions of the Boussinesq equation

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Symmetry reductions and exact solutions of a class of nonlinear heat equations

THE DIRECT METHOD IN SOLITON THEORY (Cambridge Tracts in Mathematics 155) By RYOGO HIROTA (translated and edited by ATSUSHI NAGAI, JON NIMMO and CLAIRE GILSON): 200 pp., £40.00 (US$65.00), ISBN 0-521-83660-3 (Cambridge University Press, 2004)

Peter A. Clarkson

Papers

Solitons, Nonlinear Evolution Equations and Inverse Scattering

Solitons, Nonlinear Evolution Equations and Inverse Scattering: References

New similarity reductions of the Boussinesq equation

Symmetry reductions and exact solutions of a class of nonlinear heat equations

THE DIRECT METHOD IN SOLITON THEORY (Cambridge Tracts in Mathematics 155) By RYOGO HIROTA (translated and edited by ATSUSHI NAGAI, JON NIMMO and CLAIRE GILSON): 200 pp., £40.00 (US$65.00), ISBN 0-521-83660-3 (Cambridge University Press, 2004)