176 Book Revie~:~

computing and investigating the asymptotics of eigenvalues of wide classes of differential and pseudo-differential operators. The main tool in constructing this method is (a version of) the Weyl-H~rmander calculus of pseudodifferential operators, which is described in Part 1. Part 2 contains the general theorems obtained by the method, together with proofs. The results of Part 2 are then applied to specific (classes of) operators: elliptic, hypoelliptic, and Douglis-Nirenberg elliptic, SchrOdinger, Dirac, and linearized Navier-Stokes operators, as well as to differential operators with rapidly increasing coefficients or with operator coefficients. In many cases remainder estimates are given. Examples are taken from mathematical physics and mechanics, and it is shown that this general method has a wider range of applications than other methods.

(WFA)

F.R. Harvey, Spinors and Calibrations. Academic Press, Boston, 1990. 323 pp., US$39.95, |SBN 0-12-329650-1.

The author uses many facets of differential geometry as its motivation to provide a collection of examples. They range from (simple) Lie groups to spin groups and normed algebras for general signature, including the exceptional groups G and F and the special Lagrangian and associative calibrations. The book is divided into two parts. Readers who are primarily interested in the spin groups are encouraged to start with Part II M Spinets.

Part I begins with an introduction to certain specific matrix groups wherein the entries are real, complex or quaternionic. Some of these groups are defined by requiring that the matrix fix one of several types of (generalized) inner products. This leads to a discussion of eight types of inner products.

The book will be a welcome supplement to standard texts on Lie theory because it provides an unusually thorough treatment of nonstandard examples. Further it provides concrete material to these who study and contribute to the interesting developments in spin geometry and Twister theory with applications to general relativity.

(WFA)

V.G. Makhankov, Soliton Phenomenology, Mathematics and its Applications (Soviet Series). Kluwer, Dordrecht, 1990. 452 pp., Dfl.260, US$149, UK£91, ISBN 90-277-2830-5.

This volume presents a survey of soliton phenomcnology in physics and mathematics. The first part deals with quantum systems and classical behaviour, and surveys various physical models of importance and some physically interesting nonlinear equations.

Part 2 is concerned with exact results in D = 1 space and concentrates on aspects of the nonlinear SchriSdinger equation as well as the Laudau-Lifshitz equation.

In Part 3, noncompact symmetries and the Bose gas are considered in detail. The last two parts deal with, respectively, the phenomenology of D = 1 so!itons and many-dimensional solitons. The book contains many qualitative discussions and presents the material in a pedagogical way.

Audience: mathematicians and mathematical physicists interested in nonlinear phenomena and their applications.

(WFA)