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).

To say that this is the best book on the quantum theory of fields is no praise, since to my knowledge it is the only book on this subject But it is a very good and most useful book The original was written in German and appeared in 1942 This is a translation with some minor changes A few remarks have been added, concerning meson theory and nuclear forces, also footnotes referring to modern work in this field, and finally an appendix on the symmetrization of the energy momentum tensor according to Belinfante Quantum Theory of Fields Prof Gregor Wentzel Translated from the German by Charlotte Houtermans and J M Jauch Pp ix + 224, (New York and London: Interscience Publishers, Inc, 1949) 36s

Quantum Theory of Fields

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

Vorlesungen uber theoretische Physik Von Prof. Arnold Sommerfeld. Band 1: Mechanik. Vierte, neubearbeitete Auflage. Pp. xii + 276. 18 D. marks. Band 2: Mechanik der deformierbaren Medien. Pp. xv + 376 + 4 plates. 18 D. marks. Band 3: Elektrodynamik. Pp. xvi + 368. 18 D. marks. Band 6: Partielle Differentialgleichungen der Physik. Pp. xiii + 332. 18 D. marks. (Wiesbaden: Dieterich'sche Verlagsbuchhandlung, 1947–1949.)

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Lectures on Theoretical Physics

Hydrodynamic and hydromagnetic stability

Symmetry reduction is studied for the relativistically invariant scalar partial differential equation H(⧠u,(∇u)2,u)=0 in (n+1)‐dimensional Minkowski space M(n,1). The introduction of k symmetry variables ξ1, ... ,ξk as invariants of a subgroup G of the Poincare group P(n,1), having generic orbits of codimension k≤n in M(n,1), reduces the equation to a PDE in k variables. All codimension‐1 symmetry variables in M(n,1) (n arbitrary), reducing the equation studied to an ODE are found, as well as all codimension‐2 and ‐3 variables for the low‐dimensional cases n=2,3. The type of equation studied includes many cases of physical interest, in particular nonlinear Klein–Gordon equations (such as the sine–Gordon equation) and Hamilton–Jacobi equations.

Symmetry reduction for nonlinear relativistically invariant equations

Various forms of the nonlinear Klein–Gordon equation are seen to have exact, solitonlike solutions when separation of variables is postulated. The family for which these exact solutions are found includes the sine‐Gordon equation as a special case. An interesting conclusion obtained is that both the soliton–antisoliton and breather solutions, hitherto known for the sine‐Gordon equations result from a broader class of Klein–Gordon equations. The method can be extended to other equations.

A family of nonlinear Klein-Gordon equations and their solutions

The often used φ6 model of classical critical phenomena is studied in (3+1)‐dimensional Minkowski and Euclidean spaces. The Euler–Lagrange equations describing the kinetics of the scalar order parameter are in this case nonlinear Klein–Gordon equations. The method of symmetry reduction is systematically applied to derive all the solutions invariant under subgroups with generic orbits of codimension 1. Whenever the obtained ordinary differential equations have the Painleve property, they can be transformed to one of two standard forms. These are then solved in terms of elliptic functions or elementary ones. This results in a large number of new exact solutions. Particularly interesting solutions are found in the immediate vicinity of the tricritical point. Our treatment of the φ6 theory is complete only for the four‐dimensional spaces M(3,1) and E(4), but many of the results are given for the more general cases of M(n,1) and E(n+1).

Exact solutions of the multidimensional classical φ6‐field equations obtained by symmetry reduction

The integrability of a system which describes constant mean curvature surfaces by means of the adapted Weierstrass–Enneper inducing formula is studied This is carried out by using a specific transformation which reduces the initial system to the completely integrable two-dimensional Euclidean nonlinear sigma model Through the use of the apparatus of differential forms and Cartan theory of systems in involution, it is demonstrated that the general analytic solutions of both systems possess the same degree of freedom Furthermore, a new linear spectral problem equivalent to the initial Weierstrass–Enneper system is derived via the method of differential constraints A new procedure for constructing solutions to this system is proposed and illustrated by several elementary examples, including a multi-soliton solution

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The Weierstrass-Enneper system for constant mean curvature surfaces and the completely integrable sigma model

We study some geometrical aspects of two-dimensional orientable surfaces arising from the study of sigma models. To this aim we employ an identification of with the Lie algebra by means of which we construct a generalized Weierstrass formula for immersion of such surfaces. The structural elements of the surface like its moving frame, the Gauss–Weingarten and the Gauss–Codazzi–Ricci equations are expressed in terms of the solution of the model defining it. Further, the first and second fundamental forms, the Gaussian curvature, the mean curvature vector, the Willmore functional and the topological charge of surfaces are expressed in terms of this solution. We present detailed implementation of these results for surfaces immersed in and Lie algebras.

A. M. Grundland

Papers

Symmetry reduction for nonlinear relativistically invariant equations

A family of nonlinear Klein-Gordon equations and their solutions

Exact solutions of the multidimensional classical φ6‐field equations obtained by symmetry reduction

The Weierstrass-Enneper system for constant mean curvature surfaces and the completely integrable sigma model

Surfaces immersed in Lie algebras obtained from the sigma models