Maple: A Comprehensive Introduction. by Roy Nicolaides; Noel WalkingtonReview by: Patrick FitzpatrickSIAM Review, Vol. 39, No. 4 (Dec., 1997), pp. 792-793Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2132705 .

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792 BOOK REVIEWS

forms and Their Applications, CRC Press, Boca Raton, FL, 1995. Sneddon, I. N., The Use of Integral Transforms, McGraw-Hill, New York, 1972.

LOKENATH DEBNATH University of Central Florida

Maple: A Comprehensive Introduc- tion. By Roy Nicolaides and Noel Walk- ington. Cambridge University Press, Cam- bridge, UK, 1996. $39.95. xix+465 pp., hardcover. ISBN 0-521-56230-9.

Maple is often referred to as a "computer algebra" package, but it does a lot more than that appellation indicates. This book provides an excellent overview of its capa- bilities. Here the reader will discover how to do basic and advanced calculus and lin- ear algebra, to use numerical methods and computational algebra, and to plot graphs in two and three dimensions. At the less elementary levels there are introductions to several of Maple's packages, such as those on graphs and combinatorics, finance and statistics, and number theory. As well as being an interactive computing tool, Maple is also a high-level programming language which permits the user to build complex libraries of procedures for a vast array of different applications, and the book intro- duces the necessary techniques.

The authors recommend Maple as an ex- tremely valuable aid for routine calculations and for constructing examples in their re- search. Computer packages such as this are also becoming essential tools for students of mathematics, science, and engineering, and the authors believe that Maple is a good choice because of its simple structure. (I agree with both these opinions.) In this book they aim to give a broad overview of the whole package as well as explain how to use Maple as a programming language. This goal is more ambitious than that of the majority of texts on Maple which in- troduce the student to the package using applications in one particular area, such as calculus or linear algebra.

The first chapter introduces the basic ideas of entering commands and interpret- ing responses, using assignments, typing

and type conversion, and beginning graph- ics. The next three chapters follow the sequence precalculus, calculus, and linear algebra, and the fifth chapter deals with numerical techniques. Chapters 6, 7, and 9 introduce more advanced graphics, compu- tational algebra, and some of Maple's pack- ages, while Chapters 8 and 10-12 explain how to use Maple as a programming tool.

For the independent researcher the book provides an excellent resource, with good cross-referencing between sections, a com- prehensive index, and summary reference sections at the end of each chapter. The explanations are very clear and are well illustrated with examples. Several pitfalls and idiosyncracies of Maple are highlighted, and I especially liked the detailed accounts of the problems of parameter passing, evaluation (particularly in relation to ar- ray structures), and the introduction to user-defined libraries and packages. For the mature reader the book is both an excellent beginner's guide to Maple and a valuable continuing reference at the inter- mediate and advanced levels.

In providing an undergraduate textbook on Maple the authors are a little less suc- cessful, at least in the early stages. The difficulty is that they do not feel free to begin with some motivating problems and solutions and to present some attractive pictures that would draw the new reader in and display what Maple can achieve. Instead, they want to develop the early material logically from scratch and, having covered the basics, then allow the reader to choose and browse through further material from the later chapters. I think it would have been preferable if the book had started out with a little more motivation, perhaps in a preliminary chapter describing some "test problems" that were later solved with Maple, when sufficient experience had been gained. From the same viewpoint, several topics from Chapter 1 could well have been left until much later (for example, the inter- face, map, and assume commands), when they were required in practice.

Another aspect of the presentation that might pose initial problems for students is that the authors do not discuss windows interfaces to Maple at all. Our experience with undergraduates is that they usually begin by using a Windows (or X-Windows)

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BOOK REVIEWS 793

Worksheet environment with Maple in in- teractive mode, and they invariably want to "get their hands dirty" right away. An ini- tial chapter giving a quick tour through some of Maple's impressive capabilities, based on a worksheet sample, would have been useful in addressing this, as well as providing initial motivation.

However, once they get going, I have no doubt that students of mathematics, science, and engineering will find the book easy to read and to use, either as a class textbook or for self-study, and that it will be useful as a reference after they have mas- tered the essentials. Exercises are included in the first four and the last three chapters, and some of these are sufficiently demand- ing to motivate further investigation.

Finally, I wonder if the authors would consider experimenting with the typeface used in presenting Maple's responses to commands. They have adopted the usual practice of displaying the standard output derived from a basic DOS or UNIX in- terface. This is fine for the experienced user who may well find these environments more efficient than using worksheets (for ex- ample, in terms of memory requirements), especially when it comes to building signif- icant applications. But many (if not most) beginning students will only see Maple's colorful "prettyprinted" output and won- der where those primitive responses in the book came from!

In summary, I think that the authors have done an excellent job of producing a comprehensive introduction to Maple. It will immediately suit the established re- searcher and, with some initial extra moti- vation and assistance from the instructor, would be a very good textbook for under- graduates.

PATRICK FITZPATRICK University College Cork

Probability Theory and Combinato- rial Optimization. By J. Michael Steele. SIAM, Philadelphia, PA, 1997. $26.50. viii+159 pp., softcover. ISBN 0-89871- 380-3.

A typical problem discussed by the book is as follows: let the edge weights X = (Xe) of a complete graph Kn be a collection of random variables. Consider the random variable Z = Z(X) equal to the length of the minimum spanning tree of Kn with the given edge weights. What can we say about Z? What is its expectation? Is it concentrated around its mean? Is there a central limit theorem? The author has de- voted great efforts to this type of question, and so the book is a concise but very in- formative description of the current state of knowledge of the area. It is well worth acquiring.

As the author states, there are three themes: the first is the use of martingale tail inequalities. These have had a major impact on the area, much as they have had in the land of the probabilistic method. A second and related theme is that of Tala- grand's isoperimetric theory of concentra- tion inequalities, which promises to be just as useful and significantly more powerful. Finally, there is the use of subadditivity in sequences and stochastic processes.

There are six chapters in all. Chapter 1 is introductory and introduces some of the problems to be discussed, most notably the longest increasing sequence (LIS) problem and the maximum common subsequence problem. Subadditivity is introduced, and Azuma's inequality is proved along with an application to the LIS problem.

Chapter 2 introduces the reader to Eu- clidean problems, a favorite with the au- thor. We see how Azuma's inequality has made life simpler in relation to the fa- mous Beardwood-Halton-Hammersley the- orem on the length of the shortest tour through n randomly chosen points in Rd, the travelling salesman problem (TSP). These are some interesting results related to space-filling curves. Chapter 3 continues this discussion and generalizes it to subad- ditive Euclidean functionals. Thus results on the TSP are extended to minimum span- ning trees (MST), Steiner trees, minimum matchings, etc. There are a wealth of recent results showing that the area is still very active, containing interesting and nontrivial problems.

Chapter 4 deals with random linear programs-in particular, the (so-called)

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