Methods of Applied Mathematics with a MATLAB Overview by J. H. DavisReview by: J. David LoganSIAM Review, Vol. 46, No. 2 (Jun., 2004), pp. 367-368Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/20453521 .

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BOOK REVIEWS 367

love. Rather, it is an imagined or wished-for higher order category theory categories of categories of categories, I think). He guesses, perhaps daringly, that this hoped for "higher-dimensional algebra" will be the unifying theme of 21st-century mathe

matics. The second paragraph of the introduction

to this climactic chapter starts out, "Con sider the following spaces: the Eilenberg

MacLane space, K(Z, 2), whose only non

vanishing homotopy group is 7r2(X) = Z; the space of unit vectors modulo phase in a

Hilbert space of countable dimension; CPO, the direct limit of finite dimension com plex projective spaces, CPn; the classify ing space for principal U(t) bundles; the complex vector space of non-zero rational complex functions in one variable modulo constants; and the space of configurations of integer-labelled points on the surface of a sphere, whose labels sum to zero. Now, in a certain strong sense, namely up to homotopy, these are just different descrip tions of the same space." Or, as a pass ing remark, "The fact that mathematicians can see analogies between the mighty Lang lands programme and topological quantum field theory (Kapranov 1995) should en courage us." No doubt the innocent reader imagines that if he only looks up Kapranov he will understand the "mighty" Langlands programme and topological quantum field

theory! Corfield himself apparently under stands all these subjects well enough to justify dropping them on the reader's head.

Unfortunately he does not seem to realize that such remarks are more likely to intimi date or stupefy than to enlighten. Certainly one must be tremendously impressed by his

mathematical erudition. Because of economic necessity, he has

also had to delve into automated theo rem proving and Bayesian statistics. His

chapters on automatic theorem proving are admirable reportage, on an intriguing,

though perhaps Quixotic, branch of com puter science. His Bayesianism serves to

put forward the suggestion that Bayesian reasoning lies behind or explains the heuris tics of mathematical research. A striking thought, if hard to verify.

Combining the Lakatos critique, the re port on computerized theorem proving, the

Bayesianism, and above all the tremen

dous enthusiasm for up-to-date theoretical physics and category-theoretic algebra into one smoothly coherent book, would have been too much to ask for. There is nothing wrong with a collection of loosely connected chapters. I hope his book is widely read. I wish it had been more coherent and more considerate of the actual potential reader.

REUBEN HERSH University of New Mexico

Methods of Applied Mathematics with a MATLAB Overview. By]. H. Davis. Springer Verlag, New York, 2003. $79.95. xiv+721 pp., hardcover. ISBN 0-8176-433 1-1.

When you see a textbook with "methods of applied mathematics" in the title, you can never be too sure what it means! Texts on this topic can range from mathematical

methods in engineering to abstract meth ods using functional analysis. The present text presents methods of applied mathe

matics that arise out of Fourier analysis. It treats topics that are central in mathemat ical physics and contemporary engineering, and the presentation is a mixture of analyti cal results and applications, both written in a style that will be comfortable for applied scientists and mathematicians.

The level of the text is intermediate, for upper level undergraduates and beginning graduate students. The author states that part of the material served as a one-semester course for engineering students, while the whole of the material can serve as a one year course for students in mathematics, science, and engineering. This reviewer be lieves that an instructor would have to push hard to cover, in detail, all 700 pages in one year.

The book examines classical Fourier se ries, boundary value problems, eigenfunc tion expansions in different geometries, and a substantial number of standard topics as

sociated with Laplace, Fourier, and discrete transforms. Inversion of transforms involves complex analysis, and there is a long chap ter on functions of a complex variable. The final chapter includes material on wavelets and waveform analysis. Thus, the book is

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368 BOOK REVIEWS

more about methods for the solution of boundary value problems than for the anal ysis of signals. There are three brief appen dices, one on linear algebra, one a MAT LAB primer, and one containing tables of transforms. The linear algebra appendix is a quick overview; the MATLAB appendix is brief and really could not replace a man

ual or serve as a tutorial for programming. Several MATLAB programs are written out or outlined in the text. Students will find

many exercises; some are mechanical, some require computation, and some extend the theory and applications not covered in the text. Portions of the book are consider ably more advanced than the typical text books used for the standard undergraduate courses in "elementary partial differential equations" or "Fourier series and bound ary value problems." It may have trouble finding a home in the traditional undergrad uate and graduate offerings at typical U.S. universities.

Overall, this textbook has an attractive format with lots of figures, programs, and formulas, and it presents, in a very tradi tional way, a body of material that is fun damental in applied mathematics, science, and engineering. It would make an excel lent textbook for courses focused around

Fourier analysis and applications to differ ential equations.

J. DAVID LOGAN University of Nebraska-Lincoln

Wavelets Through a Looking Glass: The World of the Spectrum. By Ola Bratteli and

Palle E. TJorgensen. Birkhauser Boston, Boston, MA, 2002. $59.95. xxii+398 pp., hardcover. ISBN 0-8176-4280-3.

Mere words cannot adequately describe all the great features of the new book by Ola

Bratteli and Palle Jorgensen, which has something for everyone of all mathemat ical persuasions. Whatever your feelings about this book, you will be left breath less by its scope. Subband filters, qubits from physics, loop groups, homotopy the ory for wavelets and related index theorems, Cuntz C*-algebras, transfer operators and a Perron-Frobenius theory for their eigen

values, cycles from dynamics, isospectral approximation all these are discussed in addition to the standard theory of wavelets and multiresolution analysis which can be found in the books by Y. Meyer [Me], I.

Daubechies [D], or E. Hernandez and G. Weiss [HW], to name just a few. And I have yet to mention the many informative graphs of wavelets and eigenvalues of trans fer operators, or the tables of definitions

which serve as "dictionaries" between ter minologies for classical theory and quantum theory for wavelet resolution algorithms. In addition to all of the preceding, there are

many lovely pictures and diagrams. Figure 1.11, which relates Edvard Munch's paint ing "The Scream" to qubits, is a particular favorite of mine. And all of this in the first chapter alone!

I am being a bit facetious here, as part of the purpose of the first chapter is to serve as an overview of the entire book. But it is clear that the authors view wavelets almost as living organisms, and one of their aims is to "show that wavelets have a life of their own outside the Fourier Kingdom: that is the world of cascades, algorithms, and the spectrum." Their love of the subject is ev ident from their writing, and is contagious to the reader.

This book has quite a different perspec tive from the other monographs on wavelets

mentioned above, mainly because it em phasizes the Fourier domain as the proper "window" or "looking glass" from which one can most easily study wavelet theory; thus the book's subtitle. Some wavelet theorists

would say that to use the Fourier trans form, or more generally duality for abelian groups, in the study of wavelets, is a circu lar approach, because one of the points of

wavelet theory is to offer an alternative to

ordinary harmonic analysis. On the other hand, at certain stages in the development of wavelet theory, for example, in the con struction of multiresolution analyses of Mal lat [Ma] and Meyer [Me], Fourier analysis has proved critical. This is the key theme of the book under review. The second chapter is where the book really runs away from all competitors in this regard. Not only is the key theme repeatedly emphasized, but new research is introduced, in the form of a one-to-one correspondence between cer

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