Applied Combinatorics on Words by M. LothaireReview by: Milós BónaSIAM Review, Vol. 48, No. 3 (Sep., 2006), pp. 600-601Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/20453845 .

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600 BOOK REVIEWS

problems and results. He concludes with a rueful sketch of the post-Perestroika disper sion of Soviet mathematics.

Faddeev's essay on mathematical physics begins with a very brief sketch of the relations of mathematics and theoretical physics to the 1950s and the author's stu dent days in Leningrad. He came to the view that "the main purpose [of mathe matical physics] is to use mathematical in tuition to derive genuinely new results in basic physics." He contrasts views of Lan dau and Pauli with those of Dirac and Yang, discusses the history of gauge field theory and the use of mathematical intuition by Yang and Mills and by Feynman, recounts Landau's opposition to quantum field the ory, and notes work of Faddeev and Popov. The triumph of the Yang-Mills theory and of the standard model, and continuing joint efforts by physicists and mathematicians, are seen as testimony to the usefulness of mathematical intuition.

In his essay "Computerization... Let's Be Careful," Krasnoshchekov discusses the positive and negative aspects of the avail ability of computer technology, with stress on the latter. Two central concerns are the relation between technology and creative thinking in science, education, and business, and the fact that the availability of more information and more choice can make in telligent decision-making more rather than less difficult. Krasnoshchekov proposes a hierarchical organization of planning and decision to deal with the latter problem.

These accounts give some feeling for what it was like to be a mathemat ics student in the USSR, particularly in

Moscow around the 1950s the many sem inars, the intensity of the experience. They give only fleeting and oblique glimpses of less pleasant aspects of the Soviet sys tem and of some of the mathematicians. Anosov remarks cryptically that "Pontrya gin's shortcomings... were later to manifest themselves more seriously." Vitushkin re

marks that his own treatment "shows that Vinogradov's position towards Jews left much to be desired." Nikol'skii writes in re gard to Kolmogorov becoming Secretary of the Physics and Mathematics Division im

mediately after his election to the Academy,

"These matters were never debated during Soviet times: they were preliminarily dis cussed by the state authorities. It looks very much as if Stalin himself took part in this." Shil'nikov notes that "Because of the sad circumstances of the time, Vitt's name was taken off the list of authors in the first edi tion of [Theory of Oscillations]." Parshin recounts of a paper by Lapin that was writ ten "when he was in detention" and of a three-man committee of the Politburo that included Beria that gave permission to pub lish the paper under a pseudonym.

This is, obviously, a very cursory look at a collection with a great deal of information and many fascinating features, mathemati cal and nonmathematical, that often make one wish for more. (One can find more in [1] and [2].)

REFERENCES

[1] YA. SINAI, ED., Russian Mathematics in the 20th Century, World Scientific, River Edge, NJ, 2003.

[2] S. ZDRAVKOVSKA AND P. DUREN, EDS., The

Golden Years of Moscow Mathematics, AMS, Providence, RI, 1993.

RICHARD BEALS Yale University

Applied Combinatorics on Words. By M. Lothaire. Cambridge University Press, Cam bridge, UK, 2005. $125.00. xvi+6 10 pp., hard cover. ISBN 0-521-84802-4.

This is the third book written in En glish and devoted to the combinatorics of words that has been edited by the group of authors working under the name M. Lothaire, following the footsteps of Marcel Paul Schuitzenberger. The first one, Com binatorics on Words, was first published in 1983, and the second one, Algebraic Combi natorics on Words, appeared in 2002. Un like its predecessors, the present book does not focus sharply on one subject, but has a very broad scope, with applications coming from many disciplines.

The first two chapters prepare the reader for the applications that follow. Chapter 1,

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BOOK REVIEWS 601

written by Berstel and Perrin, describes the core algorithms that are used through out the book and introduces notations that are used throughout the text. Chapter 2, written by Crochemore, is devoted to data structures.

The next two chapters discuss the first area of applications, namely, linguistics. Their subjects are symbolic vs. statistical natural language processing. The former, written by Laporte, describes the passage from letters to words and from words to sen tences and is very easy to read. The latter,

written by Mohri, is more technical, which is to be expected for its subject. It contains an application to speech recognition.

This is followed by a block of applications to biology. Chapter 5 is devoted to applica tions to molecular biology, in particular to inference of network expressions. A network expression is a regular expression without Kleene closure on the alphabet of the input words. The authors, Pisanti and Sagot, de scribe the star model and the clique model, then consider a few algorithms within each model. The chapter is not easy to read for a novice in the subject. Chapter 6, "Statistics on Words with Applications to Biological Sequences," is one of the longest chapters in the book. When studying occurrences of a word in a long sequence, it is natural to ask how many times a word occurs, and where in the sequence the word occurs. For practical reasons, most results in the chap ter are asymptotic formulae, and the proofs have a probabilistic flavor. The authors are Reinert, Schbath, and Waterman.

Then comes a block on applications to algorithms, consisting of Chapters 7 and 8. The former, written by Jacquet and Sz pankowski, is devoted to pattern matching and its variations. Unlike most chapters in the book, this chapter contains exer cises. The latter, written by Kolpakov and Koucherov, analyzes periodic structures of words, starting with the Fine-Wilf theo rem. Counting maximal repetitions is the next topic, with Fibonacci words being a thoroughly studied example. Then various algorithms that find specific types of repe titions are discussed.

The last block, comprising two chapters, is devoted to applications to mathematics.

Both chapters are well written, and the first one is easy to read. Chapter 9 is devoted to encoding walks and maps by words, and then enumerating them. The authors are Poulalhon and Schaeffer. This is a fascinat ing and rapidly evolving topic, which the authors addressed in their previous book, Algebraic Combinatorics on Words. This chapter is meant to be a continuation of Chapter 11 of that book, though it is self contained. A few exciting areas discussed here contain walks on the slit plane and conjugacy classes of trees. This is very en joyable combinatorics. This is the other chapter of the book that has exercises.

Chapter 10 is more difficult to read. Its subject is applications of words to number theory. A central theorem (Christol's theo rem) here is that a formal power series over a finite field is algebraic if and only if the sequence of its coefficients is automatic. So algebraic power series can be characterized in terms of the sequence of their coeffi cients, where the sequence is interpreted as a sequence of letters. The authors, Al louche and Berthe, develop the necessary

machinery, prove Christol's theorem, and show applications to proving that certain power series are not algebraic.

It goes without saying that the authors did a great service to the scientific com munity by writing this book. My few criti cal remarks are exclusively concerned with the reader-friendliness of the book. The goal of the editors was to produce a book that is accessible by a wide scientific au dience. Some authors achieved that goal better than others. A few chapters in the middle of the book become very difficult to read after one page. Some very basic notions such as the Kleene star are used, but not defined, and are not included in the index. In a book intended for such a wide audience, the index should be more inclusive. The typical reader of this book will have a good background in one of the blocks, and much less background in the other blocks; a more reader-friendly style would have benefited every reader. These concerns could easily be addressed in a sec ond edition.

MIKLOS BONA University of Florida

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