BOOK REVIEWS 337

interest, the theory is still far from beingrecognized as a general method of analyzingdifferential equations. It has been my expe-rience to see a puzzled face when suggest,to a mathematician or physicist, the use ofLie’s method to solve differential equations.Indeed, it seems to be little known that Lie’smotivation of introducing continuousgroups of transformations had to do withdifferential equations. Compared to moderntheories of Lie groups, Lie’s original theoryis surprisingly practical and provides explicitalgorithms for finding the symmetry groupand using it for any differential equations. Itcertainly deserves more attention. This bookis 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 subjectsrelated to Lie’s work. Chapter introducesbasic concepts of Lie groups of continuoustransformations. Chapter 2 develops the the-ory ofgroup analysis ofdifferential equationsand looks at the problem of using the sym-metry group to integrate ordinary differentialequations. The group enables us to reducethe order of the equations and brings uscloser to the solution. Chapter 3 concerns amethod of constructing solutions for partialdifferential equations. With the symmetrygroup we can reduce the number ofvariablesin the equations; this could lead to a familyof solutions called group-invariant solutions.Well-known scaling solutions (similarity so-lutions) are a special case. The remainingfour chapters are centered, more or less,around the analysis of conservation laws andsymmetries, reflecting the author’s researchinterest. Chapters 4 and 5 look at the prob-lem from the Euler-Lagrange viewpoint;Chapters 6 and 7, from the Hamiltonianviewpoint. In Chapter 4 the reader will findan excellent exposition ofNoether’s theoremon conservation laws. Chapter 5 looks atgeneralized symmetries (so-called Lie-B/ck-lund groups) and their relation to conserva-tion laws. Chapter 6 provides some basicworking knowledge of finite-dimensionalHamiltonian 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 thestudy of evolution equations such as theKorteweg-deVries and the Boussinesq equa-

tions. These last four chapters provide themost detailed discussion on symmetries andconservation laws among the available liter-ature. This part clearly distinguishes thisbook from others on Lie’s theory of differ-ential equations.

This book, part of the Graduate Textsin Mathematics series, is written in consid-eration of readers in applied mathematics. Itstrikes a remarkable balance between math-ematical sophistication and practical appli-cations. Though the level of mathematics ismore demanding than that of other bookson the subject (for instance, Ovsiannikov ],Bluman and Cole [2]), definitions, theorems,and new concepts are richly illustrated byexplicit examples, many of which are takenfrom applied problems in such fields as fluidmechanics, classical mechanics, and non-linear wave propagation. These examplesshould be of great help in enabling readersto grasp key ideas firmly and apply the theoryto their own problems. Careful attention isgiven to actual algorithmic procedures. Ifthereader wants exercise problems, there areplenty at the end of each chapter. Many ofthese problems are taken from recent litera-ture and provide a good overview of theactivity in the field. A word ofcaution: manyofthese exercises require real muscle to solve.Each chapter ends with lengthy notes thatinclude interesting historical accounts.

This book is recommended for thosewho are interested in Lie groups, symmetriesof differential equations, and particularlyconservation laws.

REFERENCES

[1] L. V. OVSIANNIKOV, Group Analysis ofDif-ferential Equations, Academic Press, NewYork, 1982.

[2] G. BLUMAN AND J. COLE, SimilarityMethodsfor Differential Equations, Springer-Verlag,Berlin, New York, 1974.

SUKEYUKI KUMEIShinshu University

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

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

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338 BOOK REVIEWS

This development is largely attributedto the applications of discrete mathematicsin other areas, notably computing scienceand the social sciences, and the students indiscrete mathematics courses tend to comefrom diverse backgrounds. As a result, manytextbooks in this field are application-ori-ented, and attempt to meet the need of thewidest possible audience. The one under re-view falls into this broad category.

In this second edition, the contents havebeen reorganized into three parts. The intro-ductory chapter ofthe first edition is replacedby five appendices covering set theory andlogic, mathematical induction, probability,the pigeonhole principle, and the gamemastermind.

Four chapters on graph theory (Chap-ters 7-10 in the first edition) constitute thefirst part of this edition; they were movedforward following a recommendation by theMathematical Sciences Panel of the Mathe-matical Association of America. The head-ings are Elements ofGraph Theory (Chapter1), Covering Circuits and Graph Coloring(Chapter 2), Trees and Searching (Chapter3), 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 Methodsfor Arrangements and Selections (Chapter5), Generating Functions (Chapter 6), Re-currence Relations (Chapter 7), and Inclu-sion-Exclusion (Chapter 8).

The third part ofthis edition consists ofthree independent chapters grouped togetherunder additional topics. Chapter 9 (Chapter6 in the first edition) is on Prlya’s enumera-tion formula, following a cursory review ofrudimentary abstract algebra. Chapter 10(new material) is on combinatorial modelingin theoretical computing science, with asketchy discussion of a number of topics,including NP-completeness. Chapter 11(Chapter 11 in the first edition) is on gameswith graphs, analyzed primarily via Grundyfunctions.

The contents are fairly standard, asmost textbooks in this field also emphasizegraph 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 bitof material related to computing science.

Readers will obtain the maximum benefitfrom this textbook by a concurrent courseon algorithms and their implementation onthe computer. Some problems are trivial ifone does not worry about the complexityand efficiency of the algorithms which areused to solve them.

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

The organization of the material in thistextbook is very methodical. Each sectioncontains a large number ofexamples workedout in full detail. These are supplementedwith an ample supply of exercises, rangingfrom simple variations on the examples totheoretical developments beyond the treat-ment in the textbook. There is a short sum-mary for each set of exercises. Rounding offeach chapter is a section entitled Summaryand References, and additional referencesare given in a bibliography section at the endof the book. Answers to selected exercisesare provided.

The reviewer feels that the strongestpoint of this textbook is the thorough atten-tion the author pays to heuristics. A case inpoint is the first section of Chapter 7, whichis devoted to setting up recurrence relations.Since their derivation is not a mechanicalprocess, many textbooks in the field ignoreit completely, dealing only with the processof solving the recurrence relations once ob-tained. Of course, students can learn to setup recurrence relations by working on theexercises, but they probably would welcomesome preliminary discussion and plenty ofillustrative examples.

The main weakness of this textbook isits emphasis on heuristics at the expense oftheory, which is neither necessary nor desir-able. A case in point is the fourth section inChapter 7, where the solution of recurrencerelations by the method of characteristicequations is presented. Although the processis largely mechanical, it is important for stu-dents to achieve some understanding of itsjustification. The author does not attemptthis.

In his defense, it is the author’s statedintention to place problem-solving abovetheory-building. Under this premise, the

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BOOK REVIEWS 339

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

However, one cannot help but raise thequestion of whether taking shortcuts in the-ory is a sound technique in problem-solving.A clearer understanding of the concepts andprocesses involved also contributes to a bet-ter appreciation of the subject. While appli-cations enhance the liveliness of discretemathematics, the latter has a strong intrinsicappeal which somehow does not seem toradiate from this textbook. This is probablydue to the heavily algorithmic approachadopted by the author.

Despite the above objections, the re-viewer feels that this textbook could servevery well for an introductory course in dis-crete mathematics for computing sciencestudents. They will find that it is very read-able and that it contains much detail. Theperceived shortcomings can be rectified bysupplements from a conscientious instructor,who would appreciate the fine collection ofproblems.

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, AppliedCombinatorics, 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 LIUUniversity ofAlberta

The Finite Element Method and Its Appli-cations. By MasatakeMori. 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 thatcan be understood by an advanced under-graduate or first-year graduate student inmathematics, but it is not really a textbook.Rather, it is a monograph at an elementarylevel. It has only a very short set of referencesto current literature and no "problems." Per-haps it is fairer to say that the book is a//problems--posed by the author and workedout by him on the spot. In that sense it is ado-it-yourself finite element kit for the math-ematically skilled. No book have seen is

entirely self-contained. Rudin’s analysisbook [W. Rudin, Principles ofMathematicalAnalysis, McGraw-Hill, New York, 1964]comes close to being so, and I think I wouldbe only stretching slightly to make a pointwere I to say that a student could read Rudinand then Mori and go all the way from theaxioms ofthe real numbers to "mixed" finiteelements for problems with constraints.

The content and spirit of the first tenchapters of this book are very similar to thatof 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 moreself-contained. For example, there is enoughdiscussion ofvariational calculus in Chapters2 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 impressionthat the reader is expected to be able to followevery step of every derivation and take noth-ing on faith. What is lost in such an approachis the ability to review the current state ofthe art and provide the broader perspectivegiven by Strang and Fix. Chapters 11-14 ofMori’s book deal with some interesting ap-plications--advection-diffusio problems,the Stefan problem, and minimal surfaceproblems. These topics are not usuallytreated in texts at this level, but are quiteeasily understood in the context of what hasgone before. These topics, particularly min-imal surfaces, give this book a unique flavor.

certainly can recommend this book tostudents ofmathematics. Not all ofthe meth-ods derived are state of the art, but the bookteaches by example; it shows how to beginwith a common core of concepts (Fourierseries, variational calculus, norms, and innerproducts) to derive a fundamental principle(the maximum principle). The principle isthen a fundamental constructive tool;schemes are devised and modified, if neces-sary, to obey a discrete maximum principlea priori so that the new scheme has a priorerror estimate, or at least a stability bound,from the beginning. A good example of thisis the derivation of the "upwind finite ele-ment" scheme in 11.3-5. The scheme isderived as a modification of the usual Gal-erkin formulation in a way which leads tosatisfaction of a maximum principle. In asimple one-dimensional context, the connec-tion is made between the use of an upwind

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