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Applied Combinatorics. by Alan Tucker Review by: Andy Liu SIAM Review, Vol. 30, No. 2 (Jun., 1988), pp. 337-339 Published by: Society for Industrial and Applied Mathematics Stable URL: http://www.jstor.org/stable/2030828 . Accessed: 12/06/2014 12:52 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . Society for Industrial and Applied Mathematics is collaborating with JSTOR to digitize, preserve and extend access to SIAM Review. http://www.jstor.org This content downloaded from 91.229.229.212 on Thu, 12 Jun 2014 12:52:16 PM All use subject to JSTOR Terms and Conditions

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Page 1: Applied Combinatorics.by Alan Tucker

Applied Combinatorics. by Alan TuckerReview by: Andy LiuSIAM Review, Vol. 30, No. 2 (Jun., 1988), pp. 337-339Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2030828 .

Accessed: 12/06/2014 12:52

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Society for Industrial and Applied Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to SIAM Review.

http://www.jstor.org

This content downloaded from 91.229.229.212 on Thu, 12 Jun 2014 12:52:16 PMAll use subject to JSTOR Terms and Conditions

Page 2: Applied Combinatorics.by Alan Tucker

BOOK REVIEWS 337

interest, the theory is still far from being recognized as a general method of analyzing differential equations. It has been my expe- rience to see a puzzled face when I suggest, to a mathematician or physicist, the use of Lie's method to solve differential equations. Indeed, it seems to be little known that Lie's motivation of introducing continuous groups of transformations had to do with differential equations. Compared to modem theories of Lie groups, Lie's original theory is surprisingly practical and provides explicit algorithms for finding the symmetry group and using it for any differential equations. It certainly deserves more attention. This book is an important contribution to the exposi- tion of Lie's theory of differential equations, its applications, and recent developments.

The book consists of seven chapters. The first three deal mainly with subjects related to Lie's work. Chapter 1 introduces basic concepts of Lie groups of continuous transformations. Chapter 2 develops the the- ory of group analysis of differential equations and looks at the problem of using the sym- metry group to integrate ordinary differential equations. The group enables us to reduce the order of the equations and brings us closer to the solution. Chapter 3 concerns a method of constructing solutions for partial differential equations. With the symmetry group we can reduce the number of variables in the equations; this could lead to a family of solutions called group-invariant solutions. Well-known scaling solutions (similarity so- lutions) are a special case. The remaining four chapters are centered, more or less, around the analysis of conservation laws and symmetries, reflecting the author's research interest. Chapters 4 and 5 look at the prob- lem from the Euler-Lagrange viewpoint; Chapters 6 and 7, from the Hamiltonian viewpoint. In Chapter 4 the reader will find an excellent exposition of Noether's theorem on conservation laws. Chapter 5 looks at generalized symmetries (so-called Lie-Back- lund groups) and their relation to conserva- tion laws. Chapter 6 provides some basic working knowledge of finite-dimensional Hamiltonian systems and moves to the dis- cussion of conservation laws and the reduc- tion of order of Hamiltonian systems. Chap- ter 7 generalizes the concepts to infinite- dimensional Hamiltonian systems for the study of evolution equations such as the Korteweg-deVries and the Boussinesq equa-

tions. These last four chapters provide the most detailed discussion on symmetries and conservation laws among the available liter- ature. This part clearly distinguishes this book from others on Lie's theory of differ- ential equations.

This book, part of the Graduate Texts in Mathematics series, is written in consid- eration of readers in applied mathematics. It strikes a remarkable balance between math- ematical sophistication and practical appli- cations. Though the level of mathematics is more demanding than that of other books on the subject (for instance, Ovsiannikov [ 1], Bluman and Cole [2]), definitions, theorems, and new concepts are richly illustrated by explicit examples, many of which are taken from applied problems in such fields as fluid mechanics, classical mechanics, and non- linear wave propagation. These examples should be of great help in enabling readers to grasp key ideas firmly and apply the theory to their own problems. Careful attention is given to actual algorithmic procedures. If the reader wants exercise problems, there are plenty at the end of each chapter. Many of these problems are taken from recent litera- ture and provide a good overview of the activity in the field. A word of caution: many of these exercises require real muscle to solve. Each chapter ends with lengthy notes that include interesting historical accounts.

This book is recommended for those who are interested in Lie groups, symmetries of differential equations, and particularly conservation laws.

REFERENCES

[1] L. V. OVSIANNIKOV, Group Analysis of Dif- ferential Equations, Academic Press, New York, 1982.

[2] G. BLUMAN AND J. COLE, Similarity Methods for D.ifferential Equations, Springer-Verlag, Berlin, New York, 1974.

SUKEYUKI KUMEI Shinshu University

Applied Combinatorics. Second Edition. By Alan Tucker. John Wiley, New York, 1984. xii + 447 pp. $31.95. ISBN 0-471-86371-8.

In the last decade, discrete mathematics has gained a strong foothold in the under- graduate mathematics curriculum in North America, spawning quite a number of text- books in this field (see [1]).

This content downloaded from 91.229.229.212 on Thu, 12 Jun 2014 12:52:16 PMAll use subject to JSTOR Terms and Conditions

Page 3: Applied Combinatorics.by Alan Tucker

338 BOOK REVIEWS

This development is largely attributed to the applications of discrete mathematics in other areas, notably computing science and the social sciences, and the students in discrete mathematics courses tend to come from diverse backgrounds. As a result, many textbooks in this field are application-ori- ented, and attempt to meet the need of the widest possible audience. The one under re- view falls into this broad category.

In this second edition, the contents have been reorganized into three parts. The intro- ductory chapter of the first edition is replaced by five appendices covering set theory and logic, mathematical induction, probability, the pigeonhole principle, and the game mastermind.

Four chapters on graph theory (Chap- ters 7-10 in the first edition) constitute the first part of this edition; they were moved forward following a recommendation by the Mathematical Sciences Panel of the Mathe- matical Association of America. The head- ings are Elements of Graph Theory (Chapter 1), Covering Circuits and Graph Coloring (Chapter 2), Trees and Searching (Chapter 3), and Network Algorithms (Chapter 4).

Four chapters on enumeration tech- niques (Chapters 2-5 in the first edition) constitute the second part of this edition. The headings are General Counting Methods for Arrangements and Selections (Chapter 5), Generating Functions (Chapter 6), Re- currence Relations (Chapter 7), and Inclu- sion-Exclusion (Chapter 8).

The third part of this edition consists of three independent chapters grouped together under additional topics. Chapter 9 (Chapter 6 in the first edition) is on P6lya's enumera- tion formula, following a cursory review of rudimentary abstract algebra. Chapter 10 (new material) is on combinatorial modeling in theoretical computing science, with a sketchy discussion of a number of topics, including NP-completeness. Chapter 11 (Chapter 11 in the first edition) is on games with graphs, analyzed primarily via Grundy functions.

The contents are fairly standard, as most textbooks in this field also emphasize graph theory and enumeration techniques. However, other popular combinatorial top- ics, such as block designs and coding theory, are not even mentioned here.

The second edition contains quite a bit of material related to computing science.

Readers will obtain the maximum benefit from this textbook by a concurrent course on algorithms and their implementation on the computer. Some problems are trivial if one does not worry about the complexity and efficiency of the algorithms which are used to solve them.

The level of this textbook is quite ele- mentary. When one attempts to cope with a wide range of ability and maturity, this is a common end result. While the author envis- ages possible use at the beginning graduate level, a lot of supplementary material would be required to make such a course viable.

The organization of the material in this textbook is very methodical. Each section contains a large number of examples worked out in full detail. These are supplemented with an ample supply of exercises, ranging from simple variations on the examples to theoretical developments beyond the treat- ment in the textbook. There is a short sum- mary for each set of exercises. Rounding off each chapter is a section entitled Summary and References, and additional references are given in a bibliography section at the end of the book. Answers to selected exercises are provided.

The reviewer feels that the strongest point of this textbook is the thorough atten- tion the author pays to heuristics. A case in point is the first section of Chapter 7, which is devoted to setting up recurrence relations. Since their derivation is not a mechanical process, many textbooks in the field ignore it completely, dealing only with the process of solving the recurrence relations once ob- tained. Of course, students can learn to set up recurrence relations by working on the exercises, but they probably would welcome some preliminary discussion and plenty of illustrative examples.

The main weakness of this textbook is its emphasis on heuristics at the expense of theory, which is neither necessary nor desir- able. A case in point is the fourth section in Chapter 7, where the solution of recurrence relations by the method of characteristic equations is presented. Although the process is largely mechanical, it is important for stu- dents to achieve some understanding of its justification. The author does not attempt this.

In his defense, it is the author's stated intention to place problem-solving above theory-building. Under this premise, the

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Page 4: Applied Combinatorics.by Alan Tucker

BOOK REVIEWS 339

book is a very fine piece of work, as reviews of the first edition attest (see [2] and [3]).

However, one cannot help but raise the question of whether taking shortcuts in the- ory is a sound technique in problem-solving. A clearer understanding of the concepts and processes involved also contributes to a bet- ter appreciation of the subject. While appli- cations enhance the liveliness of discrete mathematics, the latter has a strong intrinsic appeal which somehow does not seem to radiate from this textbook. This is probably due to the heavily algorithmic approach adopted by the author.

Despite the above objections, the re- viewer feels that this textbook could serve very well for an introductory course in dis- crete mathematics for computing science students. They will find that it is very read- able and that it contains much detail. The perceived shortcomings can be rectified by supplements from a conscientious instructor, who would appreciate the fine collection of problems.

REFERENCES

[1] E. H. LUCHINS, What is discrete mathemat- ics?-A comparative review, SIAM Rev., 28 (1986), pp. 267-269.

[2] V. STREHL, Review, A. Tucker, Applied Combinatorics, First edition, Math. Rev., (1982), pp. 2329-2330.

[3] D. WELSH, Review, A. Tucker, Applied Com- binatorics, First edition, Bull. London Math. Soc., 13 (1981), p. 383.

ANDY Liu University of Alberta

The Finite Element Method and Its Appli- cations. By Masatake Mori. Macmillan Pub- lishing Company, New York, 1986. xiv + 188 pp. $34.95. ISBN 0-02-948621-1.

This short book is written in a way that can be understood by an advanced under- graduate or first-year graduate student in mathematics, but it is not really a textbook. Rather, it is a monograph at an elementary level. It has only a very short set of references to current literature and no "problems." Per- haps it is fairer to say that the book is all problems-posed by the author and worked out by him on the spot. In that sense it is a do-it-yourself finite element kit for the math- ematically skilled. No book I have seen is

entirely self-contained. Rudin's analysis book [W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, New York, 1964] comes close to being so, and I think I would be only stretching slightly to make a point were I to say that a student could read Rudin and then Mori and go all the way from the axioms of the real numbers to "mixed" finite elements for problems with constraints.

The content and spirit of the first ten chapters of this book are very similar to that of Strang and Fix [G. Strang and G. J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1973], though the reviewed text attempts to be more self-contained. For example, there is enough discussion of variational calculus in Chapters 2 and 14 so that I do not believe any addi- tional reading would be required to under- stand its application to finite elements else- where in the text. One gets the impression that the reader is expected to be able to follow every step of every derivation and take noth- ing on faith. What is lost in such an approach is the ability to review the current state of the art and provide the broader perspective given by Strang and Fix. Chapters 1 1-14 of Mori's book deal with some interesting ap- plications-advection-diffusion problems, the Stefan problem, and minimal surface problems. These topics are not usually treated in texts at this level, but are quite easily understood in the context of what has gone before. These topics, particularly min- imal surfaces, give this book a unique flavor.

I certainly can recommend this book to students of mathematics. Not all of the meth- ods derived are state of the art, but the book teaches by example; it shows how to begin with a common core of concepts (Fourier series, variational calculus, norms, and inner products) to derive a fundamental principle (the maximum principle). The principle is then a fundamental constructive tool; schemes are devised and modified, if neces- sary, to obey a discrete maximum principle a priori so that the new scheme has a prior error estimate, or at least a stability bound, from the beginning. A good example of this is the derivation of the "upwind finite ele- ment" scheme in ??1 1.3-5. The scheme is derived as a modification of the usual Gal- erkin formulation in a way which leads to satisfaction of a maximum principle. In a simple one-dimensional context, the connec- tion is made between the use of an upwind

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