SIAM REVIEW Vol. 49,No. 3,pp. 511–542ramanujan.math.trinity.edu/polofsson/research/SIAM.pdf ·...

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SIAM REVIEW c© 2007 Society for Industrial and Applied MathematicsVol. 49, No. 3, pp. 511–542

Book Reviews

Edited by Robert E. O’Malley, Jr.

Featured Review: Classical Mechanics. By R. Douglas Gregory. Cambridge UniversityPress, Cambridge, UK, 2006. $120.00. xii+596 pp., hardcover. ISBN 978-0-521-53409-3.

Writing a book on classical mechanics is a very daring enterprise. Hundreds of peoplehave done it before you, so you must be ready to face critical remarks from differentperspectives. Be sure that roughly half of your readers will say that there is not enoughphysics in the book, and the other half will complain about a lack of mathematicalrigor. I belong to the second half, but let’s start with the good news: this is one ofthe better books, in many respects; it deserves a place in every library which coversthe subject and on the list of recommended reading accompanying an undergraduatemechanics course.

In his preface, the author says that “a previous course in mechanics is helpful butnot essential.” That struck me as interesting, because I personally have found myselffor years in a similar position: teaching theoretical mechanics both to students whohave had such a preliminary course and others who haven’t. The preliminary courseis offered to physics students and is typically taken from volume 1 of some series ongeneral physics. In my opinion, there are many excellent books one can take as aguideline for such a purpose. The situation is different, however, when it comes tofinding a good book for a first course on theoretical mechanics, which is supposedto start from scratch, but should cover the subject in more generality, should payparticular attention to the fundamentals of the theory, and bring it to a point fromwhich other courses on mathematical physics can take off. I must admit I have neverfound one that gives me complete satisfaction, and there must be other people, likeDouglas Gregory perhaps, who have had the same feeling and come to the conclusion,“Why don’t I write my own?”

If you do teach an undergraduate course on theoretical mechanics to mathemat-ics and physics students, which nowadays is becoming a privilege, you better takethe issues of basic principles and subsequent consistent mathematical developmentsseriously, because there will be no opportunity to come back to it in greater depth ina later course. Far from wanting to insinuate that Gregory would not take this issueseriously, my feeling is that the book has a number of shortcomings in that respect. Iwas put somewhat on alert (a warning of bigger disasters to come later) with Defini-tion 1.1 on the first page in the book, which says, “If a quantity Q has a magnitudeand a direction associated with it, then Q is said to be a vector quantity.” If thatis supposed to be a mathematical definition, I can only raise the question, “Whaton earth is a quantity?” And it is a pity because, if you turn the page, you find anice figure showing different representations of vectors forming a parallelepiped, whichhas all the ingredients for a proper definition of the concept of a vector. Putting thataside, the first two chapters give a good buildup of the prerequisites concerning vector

Publishers are invited to send books for review to Book Reviews Editor, SIAM, 3600 UniversityCity Science Center, Philadelphia, PA 19104-2688.

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calculus and kinematics, except for the following remark. When we get to the discus-sion of reference frames in relative motion, only translational motion is considered,which after all is not very common in practice. We have to wait until Chapter 17,long after we have been told about Lagrangian and Hamiltonian systems, before theimportant concept of the angular velocity vector is introduced.

Then comes Chapter 3, which is of course about Newton’s laws, and which remainsthe most delicate part of such a book. I believe that more than 300 years after themaster, nobody can give a completely indisputable account of the basic principles ofNewtonian mechanics, in the sense of setting up a mathematical model, and Newton’slaws should somehow be the axioms from which everything else in the model can bededuced. Many books can be caught on real inconsistencies and circular argumentsin this respect. The present author is aware of this danger and generally does agood job. But having discussed the axiom about existence of inertial frames and theintroduction of the concept of mass, Gregory chooses the easy way out by consideringforce as being defined as mass times acceleration. I have a problem with such aninterpretation, because it is somehow contrary to the whole purpose of setting up amathematical model. It condemns us to being useless spectators in the whole theory:all we can do is observe motions, measure mass and acceleration at each instance oftime, and then conclude that the product of the two is what we have defined to bethe force at that time! The whole purpose of setting up a model is of course to beable to make predictions, rather than simply observing. In addition, our intuitionis that force is an independent concept. Having made this point, I feel obliged tooffer an alternative, knowing that everything I say now can be used against me. Ipostulate the principle in question roughly as follows: Every nonisolated particle withmass m moves, with respect to an inertial frame, in such a way that ∀t, we havema(t) = F(t), where F(t) represents a physically interpretable force. A force beingphysically interpretable then essentially means that, from experimental observationswhere one measures ma(t) for different values of time (in reproducible experiments),one has to discover a “force law,” i.e., a vector function F(x,v, t) such that at all timesduring the observed motion, the measured F(t) is equal to F(x(t),v(t), t). This is thestep which makes the model useful: from then on, we become intelligent creatureswho can predict phenomena by solving differential equations. Moreover, the fact ofdiscovering a force law serves at the same time as justification for the conclusion thatthe reference frame which was used in the process is a sufficiently good approximationof an inertial frame (at least for the kind of interaction under consideration). Thereis an interesting consequence of such an approach! (I am beginning to realize thatwriting a review of a book on classical mechanics is an equally daring enterprise,because I am about to overthrow a sacred principle!) At some point, say, based onthe experimental observations which led to Kepler’s laws, our fantastic gravitationalforce law, with magnitude mMG/r2, is discovered. Later on, for certain phenomenabelieved to be attributed to gravitation, more precise measurements might indicatethat the results are not quite in agreement with the theoretical predictions. Thenthere is a variety of possible explanations: (i) maybe the assumption that we hada good approximation of an inertial observer should be improved (which is what wedo when discussing the effect of the rotation of the earth on falling objects); (ii) thegravitational constant G in the model is perhaps not quite a constant and we shouldthink of something better; (iii) even more dramatic changes to Newton’s gravitationalmodel should be considered; or there is another mysterious force in action, etc. Butthere is absolutely no logic in arguing from the beginning that the mass factors mM


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in Newton’s gravitational law should relate to something different from the conceptof mass introduced in the first few axioms, then setting up all sorts of complicatedexperiments to test this (in an earthly environment even!), only to conclude that,miraculously(?), inertial and gravitational mass are the same. In other words, theprinciple of equivalence is a complete artifact (in classical mechanics).

In Chapter 4, we find a good collection of examples of solving relatively simpleproblems. The exercises look very good and challenging. Chapter 5 offers the clas-sical treatment of simple, damped, and driven oscillations, but includes a few moreadvanced topics as well, such as the use of Fourier series and coupled oscillationsand normal modes (in two dimensions). I miss a couple of features in the qualitativediscussion of one-dimensional conservative motion in Chapter 6, for example, the factthat an unstable equilibrium can be approached only in an infinite time. Also, forthree-dimensional motion, there is no mention of the test conditions for a force lawto be conservative. Chapter 7 is devoted to central forces and thus is important. It isrich in content, including, for example, a nice discussion of Hohmann transfer orbitsand Rutherford scattering. I also appreciate that elements of perturbation analysisare used throughout the book, when it is suitable. Concerning the discussion of theparadigm central force problem, which is of course Kepler’s problem, there is a nicederivation of Kepler’s equation for determining the position of the planet on its orbit,for example. Yet, it is a pity that there is no mention of the Laplace–Runge–Lenzvector constant (which is understandable though, in the author’s conception, becauseeven the angular momentum vector is only introduced much later). Chapter 8 is aninteresting digression on nonlinear effects. It may look a bit strange that there is somuch focus on phase space analysis in two dimensions (with the Poincaré–Bendixsontheorem, for example). But the justification is that it makes the link with the linearoscillations of Chapter 5 and with conservative one-dimensional motion (though thislink should have been made more explicit by also discussing the phase portrait, asderived from the study of the graph of the potential energy function).

Part 2 of the book carries the title “Multi-particle Systems and ConservationPrinciples.” It is a nice idea to organize this material in three chapters, in whichconsecutively the emphasis is on the impact of conservation of energy, conservation oflinear momentum, and conservation of angular momentum. It is unfortunate, however,that the author calls these conservation theorems or the equations from which theyderive “principles.” The term principle should be reserved for the fundamental laws,or axioms, or assumptions of the theory (whatever you want to call them). Forexample, the author talks about d’Alembert’s principle later on, which is fine becausethis concerns an additional assumption which is thrown in at some stage and whichis valid for all systems. . . for which it is not violated. This brings us back to theissue of fundamentals, and I must say that there is again an alarm signal at the verybeginning of Part 2. In the description of “Key Features” of Chapter 9, we read,“In particular, multi-particle mechanics is needed to solve problems involving therotations of rigid bodies.” In the author’s defense, he defines rigid bodies at firstas rigid arrays of particles, which is fine. But that is not how rigid bodies presentthemselves in applications, and to tacitly make the transition from a finite array ofparticles to a continuous mass distribution means sweeping a couple of conceptualdifficulties under the carpet. Having proved that gravity acts on a system of particlesas though the total mass is concentrated in the center of mass, and then taking thisfor granted when also applied to something like a uniform plank (p. 233), is stillacceptable, because a standard limit process can make the transition from a finite


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sum to an integral rigorous in this case. But things are different regarding specificforces (for example, springs attached to specific points of the body), which are notforce fields defined by a certain force density. Then, to prove for a system of particlesthe well-known theorems about the motion of the center of mass and the rate ofchange of angular momentum, and to proclaim these to be the (proved) rigid bodyequations (Chapter 11), is rather misleading, in my opinion. It is not so easy toget around this difficulty: my feeling is that we don’t want to start talking in such acourse about distributions (force fields which have a kind of δ-function density, roughlyspeaking). My preferred way out, therefore, is to introduce the above-mentionedequations, called Euler’s laws, as the axioms for rigid body motion, on the same levelas Newton’s laws for particles, and of course duly motivated by the fact that theybecome theorems in the case of a rigid array of particles. I further regret that we arestill not taught about the nature of general rigid body motion in this part. True, thereis a paragraph which carries the title “Rigid body in general motion” in Chapter 9, butit unfortunately adds to the haziness about the subject. Indeed, the author discussesthe splitting of the total kinetic energy into the kinetic energy of the center of massand a remainder term, which he then claims to have the “nice physical interpretation”as having the form 12Iω

2. But a general motion of a rigid body will of course notbe a (translational) motion of its center of mass G plus a rotation about a fixed axisthrough G superimposed on this. By the way, there is a reference in this chapter to anappendix where moments of inertia are computed. They are defined as a finite sumagain, but then computed via integrals (as something which is self-evident) for thetypical examples of bodies with symmetry and a constant mass density. In betweenall these remarks about basic principles, I should not forget to say that this part ofthe book again contains a large number of interesting topics, among which I mentionthe two-body problem, collision and scattering problems, the use of the center of massframe in general, the spherical pendulum, and rigid body static problems.

Part 3 also has three chapters and is called “Analytical Mechanics.” The firstchapter introduces generalized coordinates, d’Alembert’s principle and Lagrange’sequations (for systems of particles) as derived from it, and finally Noether’s theorem,or at least a reduced version of it. The latter is a bit strange, since Noether’s theoremcan be dealt with so much better in the context of the calculus of variations, whichis the subject of the next chapter. The treatment of the calculus of variations andHamilton’s principle is quite acceptable, though I don’t see the need for limiting theanalysis first to one-degree-of-freedom systems and then repeating the arguments formany degrees of freedom. As before, there are interesting and challenging exercises,almost always originating from realistic sounding problems. Concerning the transi-tion to Hamilton’s equations in Chapter 14, I again don’t see a compelling need fordiscussing the Legendre transform first for the two-variable case. I further abstainfrom comments about the author’s usage of “imagined substitutions,” something Ihave never seen before. But it seems to me that a simplified account of the inverseand implicit function theorem would be welcome here. A point of merit in this chap-ter is the discussion of Liouville’s theorem and recurrence. Incidentally, I spotted anobvious factor 2 mistake in Exercise 14.9, which proposes a variational principle forHamilton’s equations, but there is a slightly more annoying factor 2 problem in thenext chapter on the theory of small oscillations.

Chapter 15 is actually the first one of Part 4 on “Further Topics.” It has all theusual contents about small oscillations and normal modes, treated within a Lagrangiansetup. The author strangely omits the factor 12 in the quadratic term of the Taylor


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expansion of the potential energy function. I have a sneaking suspicion that thisis the reason why he chose earlier to represent the quadratic term of the kineticenergy in generalized coordinates without a factor 12 as well, which is permitted butrather unusual. This has the effect of making things fall in the right place in thischapter. Another remark is that the proof about a sufficient condition for stability ofan equilibrium point, in my opinion, requires this equilibrium to be a strict minimumof the potential energy, and then the strange extra assumption that it should be aminimum of the approximate potential goes away as well. Chapters 16 and 17 areactually very good: at last we are told about general rigid body kinematics and generalfeatures of relative motion and particle dynamics with respect to noninertial frames.In my opinion, however, these are not issues one can classify under “further topics”;they really should come at the very beginning of the book. The author’s opinionbecomes clear when he says, “We now prove the fundamental theorems of rigid bodykinematics. The details of the proofs are mainly of interest to mathematics students.”With all due respect, I totally disagree with this point of view: if we are talking about“fundamental theorems” (and indeed we are!), then they must be of interest to physicsstudents as well!

Chapter 18: the dreaded disaster strikes! This chapter is about tensor algebraand the inertia tensor. The author warns that “Any account of tensor algebra ismathematics, not mechanics, and some readers may find this indigestible.” He is rightbut for the wrong reasons: it is the pseudomathematics which makes it intolerable.I know of course that in numerous physics books, tensors are defined essentially asthings, with little things attached to them (called indices), which respond in a certainway when we change coordinates. Well, we can have a little fun with this, but it isusually not all that dramatic if one quickly passes to the calculations with tensorsand doesn’t start asking silly questions like, “What on earth is a quantity?” But herethe text is a bit harder to swallow, because the definitions start with mathematicalnonsense of the following kind: “Let φ be a real number defined in each coordinatesystem . . . ” (for a scalar), or “Let {v1, v2, v3} be a set of three real numbers definedin each coordinate system . . . ” (for a vector). Anyhow, the point I would really liketo make (again) in this context is the following: physics students should not be scaredaway from a little bit of elementary mathematics. Aren’t they supposed to get a basiccourse on linear algebra, for example? Well then, what’s wrong about talking about areal vector space V (the term is never used in this book) and its dual V ∗: the space oflinear maps from V to R? Then a tensor becomes a multilinear map from a number ofcopies of V and V ∗ to R, etc. What physics students are supposed to understand lateron, when they study more advanced theories such as quantum mechanics, relativitytheory, string theory, etc., often involves a level of abstract thinking which is muchmore abstract than any abstract mathematical subject one can think of! Coming backto the issue under consideration here, I agree to some extent with the author that onecan deal with rigid body dynamics without knowledge of tensors. This is all the moretrue, for example, if all one says about the inertia tensor (as many books do) is thatit is a certain matrix or, a little bit better, a linear map which transforms the angularvelocity vector into the angular momentum vector. Then why use the term tensor?The main reason for talking about an inertia tensor (rather than an inertia matrix) isin fact its appearance as a bilinear form in the kinetic energy: it is the metric tensorof rigid body motion.

Let us pass to the final Chapter 19, which treats problems in rigid body dynamics.As I said before, what I called Euler’s laws are taken for granted as starting equations


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here, even though they were proved to hold only for systems of particles. Not sur-prisingly (in fact this was already the case in Chapter 12), Lagrange’s equations arealso taken for granted as a model for rigid body motion, even though their validitywas proved to follow from Newton’s principles and d’Alembert’s principle only forsystems of particles. Putting these remarks aside, the selection of topics and theirtreatment in this chapter is extremely well done, apart from one slip of the tongue:as an illustration of the “deficiency of Euler’s (dynamical) equations,” the authorclaims that the right-hand sides of these equations, in the case of the spinning top,are unknown! This is not true, as it is easy to express the vertical unit vector (neededfor the computation of the moment of the gravitational force) in terms of the bodyframe, using Euler’s angles. It is to be regretted also that there is no mention at all ofEuler’s kinematical equations, especially in the context of the last question the authoraddresses: if the ωi(t) are known, what is the corresponding motion of the body?

In summary, my list of comments may create the wrong impression that Gregory’sbook is very bad, but that is not the case: I did mean what I said at the beginning.The point is that a number of my critical remarks are not specific to this book,but apply to many books on classical mechanics. As such, they may well be rathercontroversial, in which case I would hope they stimulate further critical examinationof basic principles. If you are not so sensitive about fundamentals of the theory, andabout a minimal care for a good mathematical education of physics students, then youwill not be faced with stumbling blocks and can find lots of interesting topics and ideasin this book. Besides, the style of writing is very good, so it certainly makes pleasantreading. As for myself, I have still not found the book I want to adopt for my courses.

WILLY SARLETGhent University

Embedded Robotics: Mobile Robot De-sign and Applications with EmbeddedSystems. Second Edition. By ThomasBräunl. Springer-Verlag, Berlin, 2006. $59.95.xiv+458 pp., softcover. ISBN 978-3-54034-318-9.

Robotics is continuously evolving, movingfrom the industrial to the everyday-life en-vironment. Robots are progressively invad-ing human spaces; as an example, todaya large number of robotic vacuum clean-ers are in use in private habitations. Therobotics community is spending significanteffort toward making robotic devices coexistwith humans. Following this trend, in re-cent years several universities have startedoffering classes that give the background forautonomous, embedded robotics.

This book provides an interesting over-view on embedded robotics. The author

imparts his experience acquired duringseveral years of robot laboratory coursesand from the results of an internationalproject, named EyeBot, that involved insti-tutions from Australia, Canada, Germany,New Zealand, and the U.S.

One of the achievements of the projectwas the development of an embedded con-troller, named EyeCon, developed with acustomized operating system, named Ro-BIOS. Starting from this controller, duringthe research and the teaching activities, anumber of customized as well as commer-cial sensors and actuators were used on thedesign and implementation of several au-tonomous robots. The overall result is amodular robotic system used for mobile aswell as walking or underwater robots.

The book presents most of the practicalaspects related to the design and control ofan autonomous robot. In the first part of


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the book the reader gains a good overviewof the considerations to be made when se-lecting the sensors and actuators, while inthe second part several robotic mechanicsand applications are given. The last partrefers to higher level aspects of autonomousrobotics such as localization and naviga-tion, and exploration with advanced con-cepts such as neural networks or geneticprogramming. It is appreciated that theauthor gives the book a practical slant byreporting technicalities useful for practicalimplementations, thus avoiding the use ofheavy formalisms. Moreover, the correctemphasis is placed on the importance ofhardware in the loop simulations as part ofthe robotic design.

In the reviewer’s opinion, this book issuitable as a textbook for a laboratory classon robotics. A student will learn most ofthe capabilities required to select the com-ponents and build, program, and control anautonomous robotic system. Since it is ob-viously not easy to produce a self-containedbook for such a wide topic, and in order toreally appreciate the book’s contents, thereader should have a basic background incontrol/robotics.

The book might also be of interest toPh.D. students when first reading aboutautonomous robotics; however, in the re-viewer’s opinion the book would not be ofspecial interest to researchers with a ma-ture background in this topic. In fact, inseveral chapters the bibliography does notseem to address the relevant citations; asan example, Chapter 12 contains six ci-tations and four of them are theses; thePeters citation in Chapter 7 is in German.Furthermore, there is a certain nonhomo-geneity in the depth of the various chapters’bibliographies, with some containing semi-nal books and journals and others reportingonly conference papers.

Another comment concerns the relativeweights among the topics: it is the re-viewer’s opinion that maybe classical con-trol approaches are a little under-evaluatedcompared to genetic or neural network ap-proaches.

In conclusion, the book is very well or-ganized and it is written in a pleasantlyconcise style. Undergraduate and graduatestudents and researchers interested in em-

bedded robotics will find it useful and richin valuable material.

GIANLUCA ANTONELLIUniversità degli Studi di Cassino

A Java Library of Graph Algorithms andOptimization. By Hang T. Lau. Chapman &Hall/CRC, Boca Raton, FL, 2007. $99.95. x+386 pp., hardcover. ISBN 978-1-58488-718-8.

As the title suggests, this text describesa collection of routines in the Java pro-gramming language solving basic problemsin graph theory, such as locating the con-nected components of a graph, and in graphoptimization, such as finding a maximumnetwork flow. Each of the 55 routines is pre-ceded by an introduction about a page inlength. This is split between a descriptionof the problem to be solved, with some ref-erences to the literature, and a descriptionof the parameters passed to the routine.This is followed by the verbatim listing ofthe code. The one-page conclusion of eachroutine’s description is a small example,typically a graph on 8 or 10 vertices, andthe program listing for a small driver pro-gram and its output.

So roughly 250 pages are given over toprogram listings. The use of the word “li-brary” in the title would suggest some sortof uniformity to the routines and an interde-pendence, utilizing common data structuresand exploiting the object-oriented featuresof Java for efficiency in use and presenta-tion. This is not the case. There is not evena graph object. Typical data structures arearrays of integers, so graphs are imple-mented as paired one-dimensional arraysof endpoints of edges (nodei[], nodej[])or as a two-dimensional array that is theadjacency matrix. Worse, the code takes noadvantage of Java’s strengths and is writtenin a style reminiscent of C, Pascal, or FOR-TRAN. Arguments to routines are oftenlong lists of arrays, there are no objects insight anywhere, and the output requires theuser to interpret the contents of the arrayscontaining the results. Besides questionsof style, the particular choice of algorithmsalso raises questions. The description ofdepth-first search ignores the possibility of


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a recursive approach. The long section ongraph isomorphism makes no mention ofMcKay’s algorithm (implemented as thenauty package and freely available for non-military use), and instead uses an algorithmfrom a FORTRAN90 library.

Of the roughly 100 pages that are notcode, there is little explanation of thechoice of algorithms or data structures forthe problem at hand. A similar effort tothis book is Skiena’s Implementing Dis-crete Mathematics (Addison–Wesley, 1990),which describes the Combinatorica packagefor Mathematica. The reader who expectsthe work under review to match Skiena’scareful attempts to teach and explain will bevery disappointed. The book’s introductionsays, “The library of programs is intendedto be used for educational and experimentalpurposes.” While the use of such a librarycould be beneficial in an educational setting(or for rapid prototyping), the book itselfmakes very little effort to teach or inform.

Included inside the back cover is a copy-righted compact disc. There is no referenceto this disc in the book, no description of itscontents, no guidance on allowed uses. Theclosest thing to a mention of how the discmay be used is the standard boilerplate onthe copyright page which says, “No part ofthis book may be utilized. . . in any informa-tion storage or retrieval system. . .withoutwritten permission from the publishers.”Hazarding a copyright violation, an exami-nation of the contents of the disc reveals itcontains the class files of the routines. Theseare the intermediate, machine-readable filescreated by a Java compiler, and not theoriginal source files as printed. So mod-ifications to these files (perhaps for “ex-perimental purposes”) are not made easily.For a project of this nature, it would bemore useful to release the source code inelectronic form with a license explainingclearly what the purchaser is allowed to dowith the code and with any programs thatmight incorporate it. A search uncoveredno web locations that might also be host-ing the source code. Using Skiena’s workas an example again, consider that his Al-gorithm Design Manual (TELOS, 1998) isavailable online in a hypertext-linked edi-tion and is supported by the comprehensiveStonybrook Algorithm Repository website.

There is little to recommend in thisproject. The text and the code are notinstructive. Practical use of the class fileson the compact disc for experimental pur-poses is limited at best, and prohibited asdescribed in the front matter. Whether ornot the algorithms are carefully chosen andimplemented, whether or not the style ofcoding is appropriate or instructive, deliv-ery as a printed book with an electronic sup-plement containing only machine-readableimplementations seems ill-advised.

ROBERT A. BEEZERUniversity of Puget Sound

Handbook of Mathematics for Engineersand Scientists. By Andrei D. Polyanin and Alexan-der V. Manzhirov. Chapman and Hall/CRC, BocaRaton, FL, 2007. $99.95. xxxii+1509 pp., hard-cover. ISBN 978-1-58488-502-3.

How many hours are there in a day for An-drei Polyanin? Given his production overthe last decade, simply measured in termsof reference books [3, 4, 5, 6, 7], the an-swer has to exceed 24. And now arrives themost massive of them all, the 1500-pagehandbook that is the subject of this re-view. The book is coauthored with Alexan-der Manzhirov, who was also a collaboratoron [5]. That previous book has a far morelimited scope: it focuses on a single topic,broad as it may be, and presents a seem-ingly exhaustive list of solutions to differentintegral equations. In addition, the user (Ihad written “reader,” but corrected my-self) finds sections on the methods leadingto those formulae. The other reference texts[3, 4, 6, 7] are in the same vein. The currentbook is different. It is by far the most classi-cal, and there are plenty of books availablefor direct comparison (see [1, 2, 8], for in-stance). Of these, CRC’s own [8] is perhapsthe best known. The most accurate com-parison is probably with [1]. Both booksnot only list lots of facts, but also discussmethods, give theorems, and even includethe occasional proof. For reference bookssuch as this, one’s favorite will often be theone first used, as this is the book that allowsthe user to find what is needed the fastest.Therefore I cannot recommend that we all


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toss our favorites, but I can safely recom-mend this volume to anyone who wants areference.

What is to be found here? All the clas-sical culprits are present (algebra, elemen-tary functions, geometry, calculus, complexanalysis, integral transforms, ordinary andpartial differential equations, special func-tions, probability and statistics), as aresome not-so-standard ones (integral equa-tions, difference equations, calculus of vari-ations). The book is split into two parts:methods (about 1100 pages) and tables(about 400 pages). Both parts are wellstructured, and well written. I have somequibbles with notation (often, the authorsuse y′x to denote the derivative of y(x);similarly y′′xx denotes the second deriva-tive), but these are minor. Also, occasion-ally theorems are attributed to names thatmay not be canonical for the Western lit-erature. For instance, Gauss’s theorem isreferred to as the Ostrogradsky–Gauss the-orem. That may be neither here nor there,but it does affect how quickly one retrievesinformation using the index. An easy solu-tion would be to have multiple index entriesfor these cases. On the plus side, the cov-erage of the topics included is excellent,often beyond what one would expect froman applied mathematics reference (Stieltjesintegrals, inverse scattering, or game the-ory, anyone?). Due to the well-thought-outlayout of the different chapters, the extrainformation provided does not seem to in-terfere with the effectiveness of getting tothe desired information.

One may argue that some parts of thematerial included in the tables section ap-pear somewhat arbitrary. After a first lookI felt this second part of the book was aprolonged advertisement for the other ref-erence books mentioned above [3, 4, 5, 6, 7],but that is unfair. What is somewhat arbi-trary is where one draws the line on whatto include in the table, and what to omit.I am not clear on how the authors reachedtheir decision, and it is likely that the pub-lisher had a say. Most important is thatthe elementary results one expects to findare present. Maybe there is a question ofrelevance for the tables section: Maple andMathematica and other computer algebrasystems are now so widespread that they

are the first reference for elementary (andnot so elementary) results for whole newgenerations of students. I think this criti-cism can be applied to several chapters ofthe tables section included here (integralsand ordinary differential equations), but itdoes not (yet) apply to most of the others:Maple andMathematica still resort to tablelookup themselves for integral transforms,and they offer limited assistance with suchthings as integral equations and symbolicsolutions of partial differential equations.

For many of the results in the meth-ods section, one may be tempted to goonline to free sources such as wikipedia(http://www.wikipedia.org) or MathWorld(http://mathworld.wolfram.com), and onemay wonder whether $99.95 is a wise in-vestment for even such a fine reference textas this. Admittedly, this is something I amdoing myself more and more. Maybe it issomething Dr. Polyanin is doing as wellthese days: he is the main editor of Eq-World (http://eqworld.ipmnet.ru), anotherfree resource. The contents of this text donot appear there, but there is serious over-lap with [3, 4, 5, 6, 7]. Maybe there areplans to include the material of this text.This should be welcomed by the scientificcommunity. If so, Dr. Polyanin may wantto consult Eric Weisstein, who has someexperience with this publisher. Until then,this is a fine reference text, offered at a veryreasonable price in a good quality hardback.

REFERENCES

[1] I. N. Bronshtein and K. A. Semendya-yev, Handbook of Mathematics, 4thed., Springer, New York, 2004.

[2] G. A. Korn and T. M. Korn, Mathemat-ical Handbook for Scientists and En-gineers: Definitions, Theorems, andFormulas for Reference and Review,2nd ed., Dover, New York, 2000.

[3] A. D. Polyanin, V. F. Zaitsev, and A.Moussiaux, Handbook of First-OrderPartial Differential Equations, Chap-man & Hall/CRC, Boca Raton, FL,2001.

[4] A. D. Polyanin, Handbook of Linear Par-tial Differential Equations for En-gineers and Scientists, Chapman &Hall/CRC, Boca Raton, FL, 2001.

[5] A. D. Polyanin and A. V. Manzhirov,Handbook of Integral Equations, Chap-


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man & Hall/CRC, Boca Raton, FL,1998.

[6] A. D. Polyanin and V. F. Zaitsev, Hand-book of Exact Solutions for OrdinaryDifferential Equations, 2nd ed., Chap-man & Hall/CRC, Boca Raton, FL,2002.

[7] A. D. Polyanin and V. F. Zaitsev, Hand-book of Nonlinear Partial Differen-tial Equations, Chapman & Hall/CRC,Boca Raton, FL, 2003.

[8] D. Zwillinger, CRC Standard Mathemat-ical Tables and Formulae, 31st ed.,Chapman & Hall/CRC, Boca Raton,FL, 2002.

BERNARD DECONINCKUniversity of Washington

Applied Fuzzy Arithmetic: An Introduc-tion with Engineering Applications. ByMichael Hanss. Springer-Verlag, Berlin, 2005,$69.95. xiv+256 pp., hardcover. ISBN 978-3-54024-201-7.

Introduction. This book deals with theapplication of fuzzy set theory to mechan-ical engineering. More precisely, it focuseson an extension of interval methods whereintervals are changed into fuzzy intervals.The thrust of this monograph is the pre-sentation of a practical but general methodfor computing functions of fuzzy intervals.The main originality lies in the attempt togo beyond rational functions expressed bymeans of arithmetic operations, toward asystematic handling of general nonmono-tonic functions. This is not the first bookaddressing the topic of fuzzy arithmetic,although there are not many of them. Thefirst systematic treatment of this problemcan be found in the 1980 book of Duboisand Prade [5], with developments in an-other book [8] (and with more details in[7]). The first dedicated monograph is thatof Kaufmann and Gupta [13] from 1985. Amore recent one is that of Mares [16]. Anextensive account of more recent theoreticaland practical developments along with anextensive bibliography is the survey paperby Dubois et al. [9]. The originality of Ap-plied Fuzzy Arithmetic lies in its stress oncomputational methods which can be usedbeyond the four operations of arithmetic,

and in the inclusion of reports on real ap-plications. In this sense, this is an originaland welcome addition to the literature.

Analysis of Contents. The content of thebook can be described as follows. It is di-vided into two parts, one on the basics offuzzy interval analysis and computationalmethods, and one devoted to applicationexamples.

The first chapter recalls the necessarybackground on fuzzy set theory, especiallythe notion of a cut set of a fuzzy set, set-theoretic operations, basic facts on fuzzyrelations, and the formal underpinnings ofany computation with fuzzy intervals: theextension principle of Zadeh. This principleenables the image of fuzzy sets by a nu-merical function (whose arguments becomefuzzy sets) to be defined. It is the fuzzycounterpart to the principle that prescribeshow to carry a probability measure fromone set to another via a function. In fact,the calculus of fuzzy intervals is a max-min–based counterpart to the calculus ofrandom variables. On the other hand, theextension principle comes down to apply-ing standard interval analysis to all cut setsof fuzzy intervals appearing as argumentsof the function. This approach is useful inrepresenting incomplete knowledge in scien-tific calculations and evaluating its impacton the knowledge of the results.

The second chapter deals with elemen-tary fuzzy arithmetic. It surveys variousshapes of fuzzy intervals and the general so-called LR-fuzzy numbers. These are speci-fied by the choice of shape functions L andR, and differ from each other by a transla-tion and a homothety. The shape of suchfuzzy intervals is then invariant under addi-tion. Closed forms for subtraction, division,and multiplication of LR-fuzzy numbers canbe obtained [8]; however, the shape is nolonger invariant for the two last operations,and L-R approximations of products andquotients are recalled in this chapter. Theymay lead to significant loss of information.In contrast, another approach is recalled,based on the use of cut sets. A fuzzy inter-val with a continuous membership functioncan be discretized by considering a finiteset of membership grades (including 0 and1). In fact the author considers so-called


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discretized fuzzy numbers characterized bya sequence of pairs each made of a realvalue and a membership grade. The ex-tension principle is then applied directly tosuch discrete representations. However, thistechnique has many drawbacks as it is notalways obvious how to preserve this repre-sentation through computations, nor howto retrieve the right membership grades.Hanss then considers so-called decomposedfuzzy numbers, understood as a finite nestedfamily of cut sets. Fuzzy arithmetic appliedto these decomposed fuzzy numbers comesdown to interval arithmetic applied to thecorresponding cut sets. Both discretizedand decomposed fuzzy numbers yield thesame results with arithmetic operations intheory, but using cuts of fuzzy sets is by farthe most convenient and general represen-tation. In fact, the difficulty with domaindiscretization of membership functions hasbeen known since at least 1978 (see Duboisand Prade [4]).

The second chapter is devoted to exhibit-ing the limitations of standard fuzzy arith-metic as it is still used by a majority ofauthors. Such limitations were repeatedlypointed out in the past by several authorslike Dubois and Prade [7], Dong and Wong[2], and Klir [14], among others. Namely,given a rational function involving fuzzyarguments, the fuzzy set image of the func-tion cannot generally be obtained by sub-stituting parameters by their fuzzy valuesand performing the four arithmetic opera-tions in the order prescribed by the formof the rational function. In general, whatis obtained is more imprecise than (i.e., acovering approximation of) the actual out-put fuzzy interval. Worse, different formsof the same rational function, which wouldyield the same result with precise numbers,yield different results with fuzzy sets whenusing fuzzy arithmetic, and sometimes noneof them yields the exact result. Very simpleexamples of this phenomenon can be laidbare. For instance, substituting fuzzy inter-vals A and B for x and y in the expressionx+y−y is not equivalent to substituting Afor x in the simplified expression without y.This is because A − A is computed as thefuzzy range of y − z, where y and z are in-dependent variables known to be restrictedby the fuzzy set A.

One way of making sure of obtaining themost precise output is to express the func-tion so that each fuzzy argument appearsonly once. But this is not always possi-ble. When the same argument appears inseveral places, and independent variablessubstituted for them would not display thesame monotonies (like y−y becoming y−z),then fuzzy arithmetic cannot be applied.This phenomenon appears when comput-ing a weighted average either as a rationalfraction with the same weights appearingin the numerator and the denominator (arecent paper along this line being [11]), oras a convex sum, where the normalizationconstraint expresses a linear dependence be-tween weights [6]. The book under reviewexhibits these limitations on a mechanicalproblem (computing the displacement of across section in a one-dimensional staticproblem involving a massless rod under atensile load), as well as other simpler exam-ples of polynomials and rational fractions.

Chapters 3 and 4 contain the meat ofthe book. They propose a general compu-tational scheme, termed the transformationmethod, for computing functions of fuzzyvariables without resorting to standardfuzzy arithmetic, which can deal with non-monotonic functions. This approach con-siders a selection of m equidistant cuts ofthe fuzzy inputs and relies on computingprecise values of the function using a suit-ably chosen sampling of input values in eachconsidered cut. When the function is knownnot to contain extrema inside the supportsof the fuzzy inputs, the function needs tobe evaluated only on vertices of the hy-percubes formed by each tuple of cuts ofthe fuzzy inputs. These vertices are tuplesof endpoints of intervals generated by thecuts. This is the restricted transformationmethod.

When the monotony of the function isunknown, inner points of the hypercubesmust be explored. The originality of thetransformation method lies in the order inwhich the sample of evaluation points is gen-erated. The technique starts with the coresof the fuzzy intervals that are singletons,and hence form a crisp point that will bethe core of the result. Then it proceeds topdown through decreasing values of member-ship thresholds, thus considering wider and


522 BOOK REVIEWS

wider cuts. Due to the nestedness of cuts,points computed at a certain membershiplevel can be reused for exploring the nextmembership level down, thus avoiding re-dundancy that would appear if cuts wereindependently explored. New inner pointsin a cut are obtained as the arithmetic aver-age of adjacent values in intervals formingthe cuts of fuzzy input values at the pre-vious level. In this way, each fuzzy inputis systematically sampled in a regular way.At each membership level the greatest andleast output values form the approximationof the corresponding cut of the fuzzy out-put; this can be done recursively, using theendpoints of the output cut of the previoushigher level and the new evaluated samplepoints. However, all intermediate resultsare retained from one level to the next,to avoid recomputing them and for possi-ble subsequent use like sensitivity analyses.This is the backbone of the general trans-formation method. Note that while, dueto its sampling scheme, the transformationmethod avoids overestimating the outputimprecision, it is not immune to underesti-mating it for functions having a capriciousbehavior, since only part of the Cartesianproduct of the supports of fuzzy inputs isscanned by the method.

In Chapter 5, some refinements of thegeneral scheme are suggested. First, whenthe monotonicity properties of the functionunder study are partially known, say, thefunction is monotonic with respect to somevariables and nonmonotonic with respectto others, the restricted method can be hy-bridized with the general one. The latter isonly applied to the nonmonotonic variables,for the sake of better efficiency. This is theso-called extended transformation method.Next, it is pointed out that the results ofcomputing the output of the function on thegenerated sample of points enable a sensi-tivity analysis of the function to be carriedout. More precisely, gain factors for eachinput variable can be computed from theresults and express the relative sensitivityof the output to the variations of each input.This is also useful in providing informationabout local monotony of the function. Morerefinements in the computational schemesare proposed, exploiting the structure ofthe arrays of input sample values, when

fuzzy inputs have symmetric membershipfunctions and via the use of sparse gridinterpolation techniques.

Interestingly, the author points out, viaan example, that the general transformationmethod can also be applied to solve dif-ferential equations with fuzzy coefficients.What the method computes is the fuzzyrange of solution functions for all parame-ter values restricted by the fuzzy intervals.It can be done by fuzzifying an analyticalsolution, but also, more importantly, by us-ing any numerical solution technique suchas a Runge–Kutta method.

Chapter 6 presents additions to fuzzyarithmetic as a miscellany of issues. Firstit discusses the role of fuzzy arithmeticfor the nonprobabilistic handling of un-certainty, and points out the connectionwith possibility theory [8]. It also stressesthe fact that the transformation methodis less computationally demanding than afull-fledged Monte Carlo sampling method:fewer points are evaluated, in a nonran-dom way. Besides, the author argues thatthe type of uncertainty encountered in me-chanical engineering is not really the onecaptured by single probability distributionsand is more akin to standard interval anal-ysis. However, the book does not accountfor the existing bridges between possibilitytheory and probability theory, in terms ofimprecise probabilities (see [3] for a bibli-ography), nor for techniques that combinefuzzy arithmetic and probabilistic methods,on functions having both random and fuzzyarguments [12], [17].

One section is devoted to a preliminarydiscussion of the potentially important in-verse problem with fuzzy intervals: given aset of fuzzy outputs to a set of functions,find the fuzzy range of arguments yieldingsuch fuzzy output intervals. However, theproposal in the book remains ad hoc and un-connected to the existing theoretical workson this topic like Biacino and Lettieri [1]or Friedman, Ming, and Kandel [10]. Theimpact of the transformation method forcomputing a defuzzified value representinga fuzzy interval and the degrees of impreci-sion and of asymmetry (called eccentricity)of a fuzzy interval are also discussed.

The second part of the book proposesseveral applications of fuzzy interval anal-


BOOK REVIEWS 523

ysis and the transformation method tomechanical, geotechnical, biomedical, andcontrol engineering problems. I shall notcomment on this part, leaving it to spe-cialized engineers to assess its relevanceand contribution to the specific case stud-ies. Nevertheless, the presence of detailedaccounts of such applications is a veryimportant feature of this book, likely toattract the interest of mechanical or civilengineers and to bring these methods totheir research areas.

Final Comments. From the point of viewof its contribution to fuzzy set theory, thestress put on the limitation of standardfuzzy arithmetic is a very good asset of thebook, likely to draw the attention of fuzzyset researchers to the importance of bridgingthe gaps between fuzzy interval computa-tion and interval analysis methods (on thistopic see [15]). One shortcoming of the bookis the chosen style of presentation, whichis perhaps both too application-dependentand tends to overuse discretized encodingsof fuzzy intervals, close to implementationnotations. For instance, the mechanical en-gineering example in section 3.2, despiteits usefulness as an application example, isperhaps not of interest to the general readerand is not really needed to show the limi-tations of standard fuzzy arithmetic. Sim-pler, more universal examples can do thejob just as well. Section 2 of Chapter 4 pro-vides the algorithm of the transformationmethod in full generality right away, us-ing complicated notation, based on arrayswhose entries are symbols with sophisti-cated upper and lower indices (e.g., pp. 101,117), which will hardly be palatable forsome readers in a first pass. The transfor-mation method could have been introducedin abstracto, in a more concise and ele-gant way, at a more basic, application-free,implementation-free level. Despite the con-cern for practical implementation of fuzzyinterval analysis, there is also a relative lackof elementary examples that readers couldrecompute step by step. The descriptionof sensitivity coefficients and their compu-tation schemes is also hard to grasp at aglance, and needs careful, if not tedious,scrutiny. The scope of the book will makeit attractive for mechanical engineers, but

may possibly deter others. Despite these de-batable presentation choices, this book is anoticeable step toward a mature processingof fuzzy intervals, namely, a better under-standing of how to correctly handle themand what to do with them in practice.

REFERENCES

[1] L. Biacino and A. Lettieri, Equationswith fuzzy numbers, Inform. Sci., 47(1989), pp. 63–76.

[2] W. Dong and F. Wong, Fuzzy weightedaverages and the implementation ofthe extension principle, Fuzzy Setsand Systems, 21 (1987), pp. 183–201.

[3] D. Dubois, Possibility theory and statisti-cal reasoning, Comput. Statist. DataAnal., 51 (2006), pp. 47–69.

[4] D. Dubois and H. Prade, Comment on“Tolerance analysis using fuzzy sets”and “A procedure for multiple aspectdecision-making,” Internat. J. Sys-tems Sci., 9 (1978), pp. 357–360.

[5] D. Dubois and H. Prade, Fuzzy Setsand Systems: Theory and Applica-tions, Math. Sci. Engrg. 144, Aca-demic Press, New York 1980.

[6] D. Dubois and H. Prade, Additionsof interactive fuzzy numbers, IEEETrans. Automat. Control, 26 (1981),pp. 926–936.

[7] D. Dubois and H. Prade, Fuzzy num-bers: An overview, in The Analysisof Fuzzy Information, Vol. 1: Mathe-matics and Logic, J. C. Bezdek, ed.,CRC Press, Boca Raton, FL, 1987,pp. 3–39.

[8] D. Dubois and H. Prade, PossibilityTheory, Plenum Press, New York,1988.

[9] D. Dubois, E. Kerre, R. Mesiar, andH. Prade, Fuzzy interval analy-sis, in Fundamentals of Fuzzy Sets,D. Dubois and H. Prade, eds., TheHandbooks of Fuzzy Sets Series,Kluwer Academic, Boston, MA, 2000,pp. 483–581.

[10] M. Friedman, M. Ming, and A. Kandel,Fuzzy linear systems, Fuzzy Sets andSystems, 96 (1998), pp. 201–209.

[11] S.-M. Guu, Fuzzy weighted averages re-visited, Fuzzy Sets and Systems, 126(2002), pp. 411–414.

[12] D. Guyonnet, B. Bourgine, D. Dubois,H. Fargier, B. Come, and J.-P.Chilès, Hybrid approach for address-ing uncertainty in risk assessments,J. Environ. Engrg., 129 (2003), pp.68–78.


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[13] A. Kaufmann and M. M. Gupta, In-troduction to Fuzzy Arithmetic: The-ory and Applications, Van NostrandReinhold, New York, 1985.

[14] G. J. Klir, Fuzzy arithmetic with requisiteconstraints, Fuzzy Sets and Systems,91 (1997), pp. 165–175.

[15] W. A. Lodwick and K. D. Jamison,eds., Special issue: Interfaces betweenfuzzy set theory and interval analysis,Fuzzy Sets and Systems, 135 (2003),April.

[16] M. Mares, Computation Over FuzzyQuantities, CRC Press, Boca Raton,FL, 1994.

[17] B. Möller and M. Beer, Fuzzy Ran-domness: Uncertainty in Civil Engi-neering and Computational Mechan-ics, Springer, Berlin, Heidelberg, NewYork, 2004.

DIDIER DUBOISUniversité de Toulouse

Optimal Stopping and Free-BoundaryProblems. By Goran Peskir and Albert Shiryaev.Birkhäuser, Basel, 2006. $59.95. xxii+500 pp.,hardcover. ISBN 978-3-7643-2419-3.

Optimal stopping problems arise in manyfields. One of the most basic is the so-calledsecretary problem in which one wishes tohire the best secretary from a set of can-didates whom one interviews sequentially.However, the hiring rules are such that onehas to decide whether to reject or hire eachcandidate immediately after their interview.That is, one cannot interview all candidatesand then choose the best, one has to makean irreversible choice and select a success-ful candidate at some stage in the interviewprocess. Here an optimal stopping rule pro-vides a number N∗, which states one shouldinterview N* candidates and then choosethe best candidate after those N∗.

In continuous time there are impor-tant examples of optimal stopping problemsfrom financial engineering, the most basicof which is the optimal exercise time foran American put. An American put is acontract which allows one to sell some as-set S, whose price at time t is St, for afixed price $K at any time between nowand a future time T , the expiration time.The question is: When is the best time to

exercise this option, that is, to sell for $K?Does one sell S when its price is $5 be-low $K? The contract would then providea profit of $5 because one can sell for $Ksomething whose price is $(K−5). Or doesone hold the contract in the hope that theprice will drop even more below K? If thedynamics of S are described by the standardgeometric Brownian motion model popularin financial engineering, the solution of thisproblem is reduced to solving a free bound-ary problem for a parabolic partial differ-ential equation. That is, there is a criticalprice curve S∗(t). If at time t the price S(t)is less than S∗(t), one should sell, makinga profit of K − S∗(t). If S(t) is greater orequal to S∗(t), one should continue to holdthe contract.

Free boundary problems have a long his-tory and are related to determining theboundary between ice and water when icemelts. This is the so-called Stefan problem.

The book under review comprises an ex-tensive study of optimal stopping and re-lated free boundary problems in the frame-work of the modern theory of stochasticprocesses. While the emphasis is on con-tinuous time, the discrete time case is alsodiscussed. A classic contribution to discretetime is the 1971 book by Chow, Robbins,and Siegmund [2].

Turning to the book under review, af-ter a general description of the problem inChapter 1, Chapter 2 provides a review ofsome of the theory of stochastic processes.Stochastic integrals are briefly defined andstatements of the Itô differentiation rule andsome of its extensions, such as Tanaka’s for-mula for local time, are given. However, thetreatment is rather brief and few proofs areprovided.

Chapter 3 presents various optimal stop-ping and free boundary problems. Methodsfor their solution are given in Chapter 4.These include the relation of optimal stop-ping to free boundary problems and meth-ods which involve a time change, spacechange, or measure change. Optimal stop-ping problems which arise in the theory ofstochastic processes are discussed in Chap-ter 5 and applications in mathematicalstatistics, such as detecting properties of adiffusion or Poisson process, are presentedin Chapter 6. The final two chapters give


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applications to finance, including the Amer-ican put option described above.

The book contains an excellent compi-lation of problems and descriptions of thefield. Its style implies that it will be usedmore as a reference than as a text. Thismeans it will not usually be read sequen-tially. Unfortunately, the presentation isoften not user friendly for this purpose. InChapter 2 the concept of uniform integra-bility is repeatedly used. However, “uni-form integrability” does not appear in theindex. The notation D appears to repre-sent both cadlag processes and also “theDirichlet class” on page 556. + and − signsoccur in, for example, equations (3.3.30)and (3.5.5). It is unclear what they meanand they do not appear in the list of sym-bols. Section 6 on page 124 is headed “MLSformulation of optimal stopping problems.”However, the reference to “MLS” in the in-dex points just to this section. Only twothirds down the page do we learn that Mstands for Mayer, L for Lagrange, and S forsupremum.

Unfortunately for beginners in the fieldthis book would be difficult to read. Itwould be of interest to see more connectionsdrawn between the contents of the book andthose in Bensoussan and Lions [1]. Also thenice little book by Wong [4] extends op-timal stopping concepts to the case wherethe process can be stopped only on subsetsof the time interval. The neglected workof Davis and Karatzas [3] reduces optimalstopping to the deterministic case by intro-ducing a Lagrange multiplier which enforcesthe nonanticipative nature of the stoppingtime.

However, for those with some backgroundin optimal stopping problems the book isa valuable addition to the literature andit should be on the shelf of probabilists,statisticians, and financial engineers.

REFERENCES

[1] A. Bensoussan and J. L. Lions, Ap-plications of Variational Inequalitiesin Stochastic Control, North–Holland,Amsterdam, New York, 1982.

[2] Y. S. Chow, H. Robbins, and D. Sig-mund, Great Expectations: The Theoryof Optimal Stopping, Houghton Mifflin,Boston, MA, 1971.

[3] M. H. A. Davis and I. Karatzas, ADeterministic Approach to OptimalStopping, Wiley Ser. Probab. Math.Statist., Wiley, Chichester, UK, 1994.

[4] D. Wong, Generalized Optimal StoppingProblems and Financial Markets, CRCPress, Boca Raton, FL, 1996.

ROBERT J. ELLIOTTUniversity of Calgary

Surface Evolution Equations. By YoshikazuGiga. Birkhäuser, Basel, 2006. $119.00. xii+264 pp., hardcover. ISBN 978-3-7643-2430-8.

The level set method has gained incredi-ble popularity since Stanley Osher (UCLA)and James Sethian (UC Berkeley) pub-lished the seminal paper on the topic (al-though there were some previous less-knownworks). Their original paper, and most oftheir subsequent papers and books on thetopic, focused on the computational aspectsof level set and related methods, typicallyonly presenting enough theory to justifythe computational approach. That is, thetheory was driven by the computation asopposed to vice versa, and as such manyelegant mathematical results were omittedsimply for the sake of exposition. This newbook by Yoshikazu Giga takes a step backand derives a theory of level set methodsfrom the ground up in a more formal math-ematical framework than what exists in theworks of Osher, Sethian, and their manycollaborators.

The book is obviously intended as amathematical text and does not discuss nu-merical discretizations or practical imple-mentation issues. Instead, it first presentsthe basic theory behind surfaces that un-dergo motion in the direction of the localunit normal including the effects of cur-vature. After discussing things from thisgeometric standpoint, the text moves tothe theory of viscosity solutions, which isof course a requirement for the level setmethod and any numerical discretizationthereof, but is often taken for granted sinceresearchers in this area typically have train-ing in methods for hyperbolic conservationlaws. The author devotes the entire nextchapter to the comparison principle, andhere one gets a notion of how much he


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enjoys both the subject matter and therelevant mathematics for its own beauty.Again, this book is not intended for usersor implementation, but instead to explorethe mathematical foundations behind themethod—and in that sense I found the bookquite enjoyable to read. After setting thestage in the first three chapters, the levelset method is discussed in the second-to-lastchapter of the book.

Each of the five chapters concludes witha notes and comments section, which ischatty in nature and briefly sketches outa bit of the published literature. This is agood format that gives the reader a place tostart looking for further and related results.There is a community of researchers some-where in between practioners of pure math-ematics and those actively doing numericalcomputations, and this book will serve thatcommunity well. It can also be appreci-ated by the more computational at heart aslong as they have some formal training andongoing interest in pure mathematics.

RONALD FEDKIWStanford University

Cooperative Stochastic DifferentialGames. By David W. K. Yeung and Leon A.Petrosyan. Springer-Verlag, New York, 2006.$79.95. xii+242 pp., hardcover. ISBN 987-0387-27260-5.

This interesting book aims at the synthesisof two major branches of game theory: thecooperative and differential games. The for-mer were introduced by von Neumann andMorgenstern in their famous 1944 treatiseThe Theory of Games and Economic Be-haviour, while the latter were pioneered byIsaacs from the late 1940s until the pub-lication of his seminal 1965 book Differ-ential Games. Separately, these theoriesflourished over subsequent decades, withcooperative games being studied predomi-nantly by economists and differential gamesby engineers. The latter trend was not sur-prising as the classical cooperative gamesaddressed the issue of division of benefits(or costs) of cooperative ventures, whereasdifferential games focused on controls influ-encing the dynamics of situations of con-

flict, such as, for instance, those in pursuit-evasion problems. However, it was clear tomany researchers that it would be desir-able to develop a unified theory that wouldcombine the cooperative and strategic anddynamic elements of these two importantbranches of game theory. After all, coopera-tive ventures are usually conducted in a dy-namic way, over time. Conversely, in manycontexts, players contributing to controls ofcomplex dynamics will ultimately need toshare the benefits or costs of such “coop-eration.” Despite this, it was also recog-nized that a variety of both conceptual andtechnical challenges needed to be overcometo achieve a synthesis of these topics. Ofcourse, these challenges generated new ideasand approaches, many due to the authorsof the present book and their associates.

Consequently, Professors Petrosyan andYeung are making a significant and timelycontribution by summarizing both theabove-mentioned challenges and an emerg-ing new theory of cooperative (stochastic)dynamic games that addresses these chal-lenges. A key issue that needed resolutioncan be summarized as follows: whateversolution concept of cooperative game the-ory players may wish to adopt in a dynamicsetting ought to be time consistent (or dy-namically stable). Roughly speaking, therationale used for a division of benefits (orcosts) of cooperation at the beginning ofthe game should still apply at any instantduring the course of play until the endof the game. A significant conceptual ad-vance that is used repeatedly in the bookis based on the recognition that Isaac’s“tenet of transition” or, equivalently, Bell-man’s “optimality principle” of dynamicprogramming contains the essence of theabove time consistency requirement. Thisinsight also provides the basis for develop-ing mathematical tools for the derivation ofdynamically stable solutions to cooperativestochastic differential games.

The book provides a state-of-the-art sum-mary of what has been achieved so farin exploiting the above advance for vari-ous classes of dynamic cooperative gamesincluding those involving stochasticity. Inthe latter case, the ordinary differentialequations describing the dynamics of theprocess are replaced by stochastic differen-


BOOK REVIEWS 527

tial equations. Models involving both finiteand infinite time horizons are discussed anda number of interesting examples inspiredby applications from economics, resourcemanagement, environmental pollution con-trol, and technological change are treatedin some detail. Importantly, the authors’analyses often result in computable solu-tions. In particular, in many cases, ex-act formulae for “equilibrating” transitorycompensation are obtained for specific co-operative game solution concepts. In thecase of two-person cooperative stochasticdifferential games the notion of time con-sistency leads to a more stringent notion ofsubgame consistency. The authors derivetransitory compensation and payoff distri-bution procedures under uncertainty andillustrate this technique. Challenging ex-tensions to cooperative stochastic differen-tial games with nontransferable payoffs arealso discussed at some length.

The presentation is partially self-con-tained in the sense that the authors pro-vide an introduction to basic techniquesinvolving dynamic programming, optimalcontrol and stochastic control, and generaldifferential games and their solution con-cepts and methods. The style is rigorousand derivations of important formulae arepresented in a detailed way. The text alsoincludes quite a few problems that may as-sist students and other researchers enteringthis topic. However, this book is not fora novice or the faint-hearted; a significantamount of mathematical sophistication andstamina is required.

Because of the complex nature of thesubject studied, the mathematical notationis also complex. Similarly, because of thenovelty of the topic, a substantial amount ofnew terminology is introduced. Arguably,the latter reflects an orientation towarddynamic rather than classical cooperativegame theory. For instance, a solution con-cept of cooperative game theory such asthe Shapley value is, at times, referred toas an “optimality principle” that in the au-thors’ terminology essentially correspondsto an agreed-upon rationale for the divi-sion of benefits of cooperation. It can beexpected that the terminology of this fasci-nating new subject will evolve as its popu-larity increases.

Overall, this is an important new bookon an important—still emerging—topic. Iam happy to recommend it to all seriousgame theory students and researchers as itis likely to stimulate a lot of further devel-opments.

JERZY A. FILARUniversity of South Australia

Mathematical Methods in Robust Con-trol of Linear Stochastic Systems. By VasileDragan, Toader Morozan, and Adrian-Mihail Sto-ica. Springer-Verlag, New York, 2006. $79.95.xii+312 pp., hardcover. ISBN 978-0-387-30523-8.

This book considers linear time varyingstochastic systems, subjected to whitenoise disturbances and system parame-ter Markovian jumping, in the context ofoptimal control (using H2 and H∞ for-mulations), robust stabilization, and dis-turbance attenuation. The book has twomain objectives, both successfully met: (a)to develop mathematical theory for theabove classes of linear time varying sys-tems, and (b) to develop design techniques,including numerical algorithms for solvingcorresponding problems (in general, givenin terms of coupled matrix algebraic Riccatiequations). The material presented in thebook is organized in seven chapters.

Chapter 1 presents a review of relevantresults from probability theory and stochas-tic differential equations.

Chapter 2 deals with exponential sta-bility in the mean square sense of lineartime varying stochastic systems subjectedto additive white noise and Markovian pa-rameter jumping. An equivalence of thisconcept and the exponential stability of acorresponding class of deterministic sys-tems (Lyapunov differential equations) isestablished.

Concepts of stochastic stabilizability, de-tectability, observability, and controllabilityfor the considered class of linear stochasticsystems are presented in Chapter 3.

In Chapter 4, Riccati equations of sto-chastic control are studied, including theirminimal, maximal, and stabilizing solu-tions, as well as iterative numerical al-


528 BOOK REVIEWS

gorithms for finding such solutions. Bothalgebraic and differential coupled Riccatiequations are considered.

The linear-quadratic optimal control forthis class of systems is presented in Chapter5. H2-optimal controllers and the trackingproblem are studied in detail. The stateand control multiplicative white noise prob-lem formulations are also included in thischapter. In addition, the formulations whenthe performance criterion weighting matri-ces are not positive (semi)definite are con-sidered.

Robustness properties of stable linearstochastic systems are studied in Chapter 6via stochastic versions of the bounded reallemma, the small gain theorem, and sta-bility radius theory (robust stability withrespect to linear and nonlinear structuraluncertainties).

The central part of Chapter 7 is the dis-turbance attenuation problem. Necessaryand sufficient conditions are derived for theexistence of stabilizing γ-attenuating con-trollers (using H∞ problem formulation)in terms of solutions of game-type coupledRiccati equations and inequalities.

The book is very well written and orga-nized. It is mostly based on the authors’recent research work, which has been al-ready published in very respected journals.In addition, some new research results ap-pear for the first time in this book.

The book is a valuable reference for allresearchers and graduate students in ap-plied mathematics and control engineeringinterested in linear stochastic time varyingcontrol systems with Markovian parameterjumping and white noise disturbances.

ZORAN GAJICRutgers University

Automata Theory with Modern Applica-tions. By James A. Anderson. Cambridge Uni-versity Press, Cambridge, UK, 2006. $45.00.viii+255 pp., softcover. ISBN 978-0-521-61324-8.

Anderson’s book gives an excellent, clearlywritten account of automata theory. Thematerial in Chapters 1–5 deals in a fairlystandard way with finite state automata,

pushdown automata, and Turing machines,and their connections with regular, context-free, and decidable languages. His ap-proach, however, is outstanding for the clar-ity of the exposition, the excellent diagrams,the well-chosen worked examples, and themany exercises.

Authors (and indeed publishers) of text-books have a difficult choice to make re-garding solutions of exercises: if they areall solved, the teacher making use of thebook has to set new exercises; if noneare solved, the self-studying student canbe discouraged. Some authors compromiseby providing solutions to a selection of theexercises, but it is then often hard to guessthe rationale behind the selection process.Interestingly, Anderson offers teachers a setof solutions on a website—an excellent idea,though one suspects that enterprising stu-dents will quickly jump the “teachers only”barrier.

Chapter 6, titled “A Visual Approach toFormal Languages,” contains less familiarmaterial. It begins with a brief and rela-tively well-known account of combinatoricson words, such as one might find in [5],but moves on to more recent material dueto Tom Head [1]. (The book is subtitled,With Contributions by Tom Head.) Givena finite alphabet Σ, a language in its mostgeneral form is simply a subset of Σ+, theset of all nonnull words in the alphabet Σ.The details are too complicated to be repro-duced here, but the visual approach of thechapter title associates a language L with asketch consisting of a set of 1 by 1 black andwhite squares in the upper half plane. Forexample, if Σ = {a, b} and L = {w ∈ Σ+ :a and b occur equally often in w}, then thesketch of L consists of a completely blackleft quadrant and a completely white rightquadrant. This chapter is undoubtedly thehardest to read.

The final chapter, “From Biopolymersto Formal Language Theory,” is the mostintriguing of all. From the epoch-makingdiscovery of the DNA double helix, it didnot take mathematicians long to realize that(if one ignores the geometry) a single strandof DNA is a word in the alphabet

{A = adenine, C = cytosine,G = guanine, T = thymine .


BOOK REVIEWS 529

The Crick–Watson pairing, in which A andT pair only with each other and C andG pair only with each other, means thatif a single strand is known, its companionstrand is also known; thus a strand

TTTTGGAAACCTTT

has a companion strand

AAAACCTTAGGAAA .

The resulting double strand is de-noted by ttttggaaaccttt (or of course byaaaaccttaggaaa). Motivated by molecu-lar biology, Anderson exemplifies a “splic-ing” procedure involving the strand u =ttttggaaccttt and another strand v =tttggaacctttt. Specifically, u and v are“cut,” so that u = u1u2 and v = v1v2,where u1 = ttttgga and v1 = tttgga. Twonew strands,

x = u1v2 = ttttggaacctttt

andy = v1u2 = tttggaaacttt,

are formed. The book ends with a study ofsplicing rules, schemes, systems, and lan-guages. Among the references are [2], [3],and [4].

In a way Anderson has written two books.Chapters 1–5 constitute a beautifully writ-ten textbook on automata, languages, andmachines, while Chapters 6 and 7 couldbe expanded into a valuable research-levelmonograph. It seems highly likely that ad-vances inmolecular biology will benefit froma sound mathematical underlay.

REFERENCES

[1] T. Head, Visualizing languages using prim-itive powers, inWords, Semigroups andTransducers, M. Itô, Gh. Paun, andS. Yu, eds., World Scientific, Singa-pore, 2001, pp. 169–180.

[2] T. Head, Cylindrical and eventual lan-guages, in Mathematical Linguisticsand Related Topics, Gh. Paun, ed., Ed-itura Academiei Romane, Bucharest,Romania, 1995, pp. 179–183.

[3] T. Head, Formal language theory andDNA: An analysis of the generativecapacity of specific recombinant behav-iors, Bull. Math. Biology, 49 (1987),pp. 737–759.

[4] T. Head and D. Pixton, Splicing and reg-ularity, in Formal Languages and Ap-plications, II, Z. Esik, C. Martin-Vide,and V. Mitrana, eds., Springer-Verlag,Berlin, 2006.

[5] M. Lothaire, Combinatorics on Words,Encyclopedia Math. Appl. 17, Addi-son–Wesley, Reading, MA, 1983.

JOHN M. HOWIEUniversity of St. Andrews

Models of Mechanics. By Anders Klarbring.Springer-Verlag, New York, 2006. $77.00. xii+206 pp., hardcover. ISBN 978-1-4020-4834-0.

Mathematical modeling is usually describedas an art rather than a science. Modelsof Mechanics, by Anders Klarbring, showsthat there is indeed a science of mathemat-ical modeling. While the book is obviouslyfocused on developing models specificallyfor mechanics, it has a broader value as ageneral exposition of mathematical model-ing. For this reason, this small book belongson the bookshelves of mathematical model-ers who do not work in mechanics, as wellas those that do.

Part I is titled “General Background,”but an equally valid title would be “A Gen-eral Theory of Mathematical Modeling.”This theory is constructed around two ba-sic principles. First, the author discussesthe relationship between the real world ofobservation and experiment and the idealor conceptual world of theory. This shouldbe the starting point of any presentationof mathematical modeling, but it is gen-erally lacking in other texts and nowherepresented as well as it is here. The secondkey principle is that models are constructedfrom assumptions having two different lev-els of validity: universal laws that apply inall settings, and particular laws that supplydetails that vary according to the setting.The author classifies the variety of particu-lar laws as kinematic constraints, force laws,and constitutive laws, with a full discussionpostponed until Chapter 8.

The main portion of the book is dividedinto two roughly equal parts: a part on ge-ometry and universal laws and a part on


530 BOOK REVIEWS

particular laws and complete models. Klar-bring divides model construction into twostages. In the first stage, one writes downthe correct “open scheme,” which consistsof the three conservation laws (mass, lin-ear momentum, and angular momentum) inthe appropriate geometric setting. In prac-tice, one need only choose the right openscheme with which to begin a model. Tothat end, the five chapters of Part II presentthe conservation laws in every possible ge-ometry, including systems of discrete parti-cles, one-dimensional continua, and three-dimensional continua, with fluids and solidstreated separately.

The second stage of modeling by Klar-bring’s method is to complete the model byadding particular laws to the appropriateopen scheme. The variety of final modelsarises from the variety of physical settings.Part III begins with a chapter that dis-cusses particular laws in general; like thechapters of Part I, this chapter is of valueto modelers who work outside mechanics.The last four chapters present the particularlaws needed to complete small displacementmodels, pipe flow models, and models inthree-dimensional fluid and solid mechan-ics. These are given as examples that areuseful in themselves and also serve to illus-trate the process of completing models.

The author describes his book as beingat an intermediate level, and this warningshould be taken seriously. This is not a bookfor readers who have little prior experiencewith mathematical modeling in mechanics.Its unified approach gives it levels of gener-ality and abstraction that would be incom-prehensible to the beginner, while at thesame time providing an attractive synthesisand broadening for readers already famil-iar with models of particle mechanics, pipeflow, fluid mechanics, and linear elasticity.The author’s explanation of the distinctionbetween spatial and material time deriva-tives is clear, but his notation for the mate-rial derivative is somewhat confusing. Thisis one unfortunate drawback of Klarbring’swork, but it is a relatively minor point incomparison with the value of the theoreticalframework and generally clear writing.

GLENN LEDDERUniversity of Nebraska–Lincoln

Probability: Modeling and Applicationsto Random Processes. By Gregory K. Miller.Wiley-Interscience, Hoboken, NJ, 2006. $110.xxiv+459 pp., hardcover. ISBN 978-0-471-45892-0.

This is an introductory probability text thatis somewhat different from most introduc-tory probability texts. On the one hand,it covers the standard introductory topics(sample spaces, laws of probability, con-ditioning, discrete and continuous randomvariables, expected value, basic distribu-tions, joint distributions) for an audienceof students knowing multivariable calculusand a bit of matrix algebra. It is suitablefor third-year undergraduates majoring inmathematical sciences, computer science,or engineering. On the other hand, it in-troduces various topics in modeling andstochastic processes as soon as the requiredprobability concepts are introduced. Thismakes for an engaging text that clearlyshows students why probability is impor-tant and ubiquitous.

For example, discrete-time Markovchains are first used to illustrate the calcu-lation of probability by conditioning. Thedefinition of a Markov chain is delayed untilthe next chapter, in which random variablesare introduced. The Poisson process is in-troduced and explored immediately afterthe Poisson distribution, and revisited oncethe exponential distribution has been dis-cussed. In between these two sections onthe Poisson process, the student learns abit about the M/M/1 queue and its equi-librium distribution. Brownian motion isdiscussed on the heels of the normal distri-bution, and is revisited in a later chapter.The author also devotes significant spaceto modeling issues: given a sample of datapoints from an unknown distribution, whatdistribution fits the model well? Simpleparameter estimation is discussed for themain distributions, and the χ2 goodness-of-fit test is described. Also, the reader isgiven tastes of a few more specialized top-ics, including Fisher’s exact test, the sec-retary problem from optimal stopping the-ory, time series models, and self-avoidingrandom walks.


BOOK REVIEWS 531

This makes for a very full course, buta satisfying one for students who will takeonly one undergraduate course in probabil-ity (e.g., engineering or computer sciencemajors). The author has based it on thecourse of 15–16 weeks that he teaches athis university. However, my university hascourses of 12–13 weeks, so I would not beable to use this book without omitting a lotof material.

Each chapter contains a large number ofexercises of various kinds. Some are theoret-ical, some treat applications, and a few in-volve data analysis. Most should be withinreach of the average student. Some exercisesare refreshingly original (e.g., the geometricdistribution is illustrated by a high schoolbasketball coach who ends each practice bychoosing one boy to shoot free throws andmaking the team run sprints until the boysucceeds), but not too many of the exer-cises will really challenge the outstandingstudents. Occasionally, the applications areless than satisfying, such as the followingexample, in which the application is irrel-evant to the question: “Brownian motionhas been used in optimal switching prob-lems which arise in manufacturing whenone phase of a production process mustbe switched on and off due to economicfactors. Suppose in a context such as thisa Brownian motion process {X(t)} withσ2 = 10 is implemented. What is the den-sity function associated with X(t)? Whatis the probability that X(24) < 25?” Thetext often says that a particular proof orexample will be dealt with in the exer-cises, but does not give the number of theexercise. Since the longer chapters have50 to 100 exercises, this is not conve-nient for the reader. Also, the author choseto number examples and sections but nottheorems; thus, for example, an exerciserefers to “the last theorem of this chap-ter,” which appears 16 pages earlier. Bet-ter cross-referencing would have been worththe trouble.

The author makes an effort to write witha conversational style. This shows up insome places more than others. It usuallydoes not draw attention to itself, and the neteffect is positive. An exception is the preface“To the Student,” which would probably be

fine if spoken by the author to his class onthe first day of the term, but on the printedpage it tends to come out as condescending.

Regretfully, the book contains a largenumber of errors. Some are editorial errors,such as misspellings of some mathemati-cians’ names (e.g., James Stirling of the fa-mous formula is written “Sterling”), wrongwords (“State and prove and extension. . . ”instead of “an extension”), and missingor superfluous words (“. . . show that thatif . . . ”). Such errors may not hinder thereader’s understanding, but they are irri-tating and distracting. I am surprised thatthey were not caught before publication.

There are also more serious errors. Refer-ences to equations occasionally contain thewrong equation number. There are someimproper uses of mathematical terms (e.g.,we read “the limit of this sequence ap-proaches one” instead of “the limit equalsone” or “the sequence approaches one”).The author uses the fact that the ex-pected value of a sum is the sum of theexpected values, but never states this factexplicitly (e.g., as a theorem). An exer-cise in Chapter 6 requires the fact thatE(XY ) = E(X)E(Y ) for independent ran-dom variables X and Y , but this fact is notmentioned in the text until Chapter 8. Thedefinition for “countably infinite” is wrong,since it does not exclude finite sets. An ex-ercise asks for an argument that any statein a Markov chain with Pii > 0 is recurrent(which is of course false). The author statesthat a particular 4-state Markov chain satis-fies the conditions for limiting probabilitiesto exist, but does not notice that the chainis periodic.

In summary, I feel that this would be auseful book for some audiences, such as anintensive third-year probability course formath or engineering students. The model-ing and introductory stochastic processesare effectively integrated into the text. Un-fortunately, the editing leaves much to bedesired. In particular, there are many er-rors, more than should appear in a text-book. I hope that there will be a secondedition or a reprinted version with correc-tions.

NEAL MADRASYork University


532 BOOK REVIEWS

Understanding and Implementing the Fi-nite Element Method. By Mark S. Gocken-bach. SIAM, Philadelphia, 2006. $87.00. xvi+363 pp., softcover. ISBN 978-0-898716-14-6.

This book will make an excellent text for anundergraduate applied mathematics classon the numerical solution of partial differen-tial equations by the finite element method.It contains a balanced

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