BOOK REVIEWS

EDITED BY R. B. KELLOGG

SIAM REV. \begin{matrix}c \\ ◯ \end{matrix} 1998 Society for Industrial and Applied Mathematics Vol. 40, No. 4, pp. 998–1023, December 1998 019

Eigenspaces of Graphs. By Dragos Cvetkovic, Peter Rowlinson, and Slobodan Simic. Cambridge University Press, Cambridge, UK, 1997. $69.95. xiii+258 pp., hardback. ISBN 0-521-57352-1.

This volume continues a series of monographs in algebraic graph theory that specialize to spectral graph theory—the study of interconnections between the properties of a graph and the eigenvalues and eigenvectors of the associated adjacency matrix. The common thread between the two previous works, Spectra of Graphs—Theory and Applications [3] and Recent Results in the Theory of Graph Spectra [2], and the volume being reviewed is the presence of Cvetkovic as one of the authors.

Spectra of Graphs is a self-contained introduction to the subject that includes references to all of the literature available through 1978. Recent Results in the Theory of Graph Spectra was written as an update, intending to survey all of the literature available through 1984. Eigenspaces of Graphs is different in two respects. While the previous volumes focused on eigenvalues, now, as the title indicates, the focus is on the associated eigenspaces. Second, rather than being an exhaustive survey of previous results the authors are more forward-looking, describing their current and ongoing research into graph angles, star partitions, and canonical star bases. The result is a text that is clearly written and instructive (though it is not a textbook) and which should be accessible to experienced researchers and graduate students.

In the early days of spectral graph theory it was hoped that knowledge of a graph’s eigenvalues would determine the

Publishers are invited to send books for review to Book Reviews Editor, SIAM, 3600 University City Science Center, Philadelphia, PA 19104-2688. graph. Since the eigenvalues of a graph are independent of the labeling of the vertices it was hoped that the eigenvalues would allow one to easily determine if two graphs were isomorphic. However, the discovery of pairs of nonisomorphic graphs with identical spectra dashed these hopes. As computing power and techniques increased, many more such pairs were found and it became clear that even within restricted classes of graphs the spectra is far from sufficient for characterizing graphs. For example, Schwenk [4] showed that almost every tree has a cospectral mate.

Despite being insufficent as a characterization tool, eigenvalues are very helpful as a means of investigating a graph’s properties. Translating a question about a graph into a question about its eigenvalues, and then applying techniques from linear algebra, is often a successful approach. Additionally, eigenvalues are useful in a variety of applications, most notably chemistry. It is easy to model a molecule with a graph. (Indeed, some of the early history of graph theory arose in this connection when Cayley related his results on trees to the problem of isomerism [1].) Then, using Hückel theory, the eigenvalues of the chemical graph provide predictions about various properties of the molecule, such as its aromaticity [5]. Applications to chemistry are some of the authors’ main research interests, and the text and bibliography reflect this interest throughout.

In this volume, the authors turn their attention to the eigenspaces of a graph with the hope that this additional information will be enough to characterize graphs. However an eigenspace is a much more complicated object than an eigenvalue so the first problem is to determine a canonical description of an eigenspace. The construction of a star partition of the vertices is designed to overcome this hurdle and it leads to natural bases for the eigenspaces. Another algebraic invariant of a graph is the collection of angles of the eigenspaces and these are discussed thoroughly.

One cannot disassociate eigenvalues from eigenvectors, so the book begins with two chapters about previous results concerning both eigenvalues and eigenvectors and provides an excellent introduction to the current knowledge about these topics. The third chapter deals with the onedimensional eigenspace associated with the largest eigenvalue of a connected graph (this eigenvalue is known as the index). The next three chapters continue to explore geometric properties of eigenspaces that are invariant of the labeling of a graph, such as the angles between eigenspaces.

Chapters 7 and 8 describe the authors’ current research on star partitions and star bases. Here the motivation provided by the graph isomorphism problem intensifies. The final chapter collects some miscellaneous results and is followed by an appendix with a table of graph angles for every connected graph on 5 or fewer vertices. The over 300 references are a comprehensive, but not exhaustive, list of new and old works in spectral graph theory that includes a significant number of references to the literature in mathematical chemistry.

This book is a very well-written exposition of an active area of research in spectral graph theory and is recommended to anyone with an interest in algebraic graph theory. With the authors’ long-standing interest in applications to chemistry it should also be of interest to those working in the intersection of mathematics and chemistry. Given its two introductory chapters, with frequent pointers to the existing literature, this is the kind of text that a graduate student could use to carry them from the basics of spectral graph theory to an understanding of a number of open questions.

REFERENCES

[1] A. CAYLEY, On the mathematical theory of isomers, Philosophical Magazine, (4) 47 (1874), pp. 444–446.

[2] D. CVETKOVIC, M. DOOB, I. GUTMAN,

AND A. TORGASEV, Recent Results in the Theory of Graph Spectra, North– Holland, Amsterdam, 1988.

[3] D. CVETKOVIC, M. DOOB, AND H. SACHS,

Spectra of Graphs—Theory and Applications, Academic Press, New York, 1980. (Third edition, Johann Ambrosius Barth Verlag, 1995.)

[4] A.J. SCHWENK, Almost all trees are cospectral, New Directions in the Theory of Graphs, Proc. 3rd Ann Arbor Conf. on Graph Theory, F. Harary, ed., Academic Press, New York, 1973, pp. 275– 307.

[5] N. TRINAJSTIC, Chemical Graph Theory, Vol. 1, CRC Press, Boca Raton, FL, 1983.

ROBERT A. BEEZER

University of Puget Sound

Lattice-Gas Cellular Automata: Simple Models of Complex Hydrodynamics. By Daniel H. Rothman and Stephane Zaleski. Cambridge University Press, Cambridge, UK, 1997. $69.95. xxi+297 pp., hardback. ISBN 0-521-55201-X.

A lattice gas is, as the name tells, a gas of molecules occupying the sites of a lattice. The name cellular automata is related to the early visions of John von Neumann and Stanislaw Ulam, who envisaged computers in which reality is simulated by exactly solvable models, stated in terms of Boolean logic. Although most of the lattice gases in use retain the feature of a cellular automaton, some do not, and thus the latter term (which has acquired a technical meaning, designing discretely valued dynamical systems evolving in discrete time steps on a regular lattice by nearest neighbor interactions according to simple rules) tends to disappear in the current terminology and one simply talks of lattice gases. Perhaps one should even talk of lattice fluids, since compressibility, the typical feature of gases as opposed to liquids, is the hardest to model with lattice gases (see below).

The starting point is the remark, well known to people who work in statistical mechanics and even more to the historians of continuum mechanics, that the detailed model of the elementary constituents (molecules) has little influence on the results obtained at a macroscopic level. Thus, in 1986, Uriel Frisch, Brosl Haslacher, and Yves Pomeau [1] came out with the idea of simulating fluid motion by the dynamics of particles which, in 2D, may possess just 6 possible velocities, which may be oriented as the sides issuing from a site of a hexagonal lattice. The particles live on these sites and, at a tick of a clock, may hop and scatter, i.e., they move along a side of a hexagon of the lattice and, if two or more particles arrive at the same site, a scattering collision may occur. Since care is exercised to maintain the number of particles and momentum conserved at a scattering, the main ingredients of incompressible fluid dynamics are there, and the Navier–Stokes equations follow, because coarse-graining produces them as asymptotic limits, in much the same way as in the kinetic theory of gases [2]. The most surprising feature is that the equations turn out to be isotropic, in spite of the fact that the basic lattice is highly anisotropic. In fact a predecessor of the 1986 model, proposed by Jean Hardy, Olivier de Pazzis, and Yves Pomeau [3] in 1976 and based on a square lattice, lay dormant for a decade because the corresponding hydrodynamic equations were anisotropic. It is interesting to remark that, at variance with its continuous version, the Boltzmann equation for a gas on a lattice yields a global in time solution with moderate effort [4].

The lattice-gas model went through a stage of wild excitement, because people thought that it would provide a cheap route to the simulation of turbulence. This turned out not to be the case, as any person, simultaneously expert in the various fields involved (kinetic theory, turbulence, and numerical fluid dynamics), could easily predict. On the contrary, none less than the Washington Post stated in a front-page article that the potential of the latticegas method was “one thousand to one million times faster than previous methods” and even reported that the United States Department of Defense had considered whether the method “should be classified to keep it out of Soviet hands” [5].

A more serious debate at a scientific level started at that time. The possibility of simulating incompressible fluid dynamics— with certain, well-understood corrections— emerged. The statement concerning turbulence turned out to be seriously exaggerated, though the possibility of gaining in efficiency with special-purpose machines remains open to this date. In this respect, lattice gases are just another method to solve the incompressible Navier–Stokes equations. Important, unexpected ramifications of the use of lattice gases have, however, emerged in the realm of complex fluids. Specifically, fluid mixtures including interfaces, exhibiting phase transitions, and allowing for multiphase flows can be simulated with lattice gases, thanks to the usually rich phenomenology of the latter. This favorable feature is enhanced when the geometry is also complicated as in the case of multiphase flows through disordered media, such as a porous rock.

The book under review wisely avoids the temptation of rhapsodizing the hopes and, after paying a scarce lip service to the great expectations in a historical introduction (Chapter 1), it concentrates on the solid results, from both the scientific and practical viewpoint.

Chapter 2 describes the two routes— based on molecular and continuum models—to fluid dynamics. Then the plane lattice model and the corresponding inviscid fluid dynamics are described in some detail. Here one meets the first difficulties, related to the lack of Galilean invariance of the lattice gas; the macroscopic equations are not quite the Euler equations. The deviations are, however, inessential in the incompressible case, when the density can be assumed to be constant and the speed small with respect to the speed of sound. Chapter 3 deepens the subject matter of the previous chapter. More refined tools, such as the exclusion principle and the use of rest particles, are introduced. The latter may be used to alleviate the difficulties with Galilean invariance and might be essential if an extension to compressible fluids is sought. In fact, as mentioned in the bibliography, a simple thermodynamic argument shows that thermodynamics must be seriously modified [6, 7] in order to describe compressible fluids with discrete velocity models. Viscosity makes its first appearance in Chapter 4.

The jump to three dimensions is made in Chapter 5. There is no lattice in 3D with the required level of isotropy. A sufficiently large symmetry group is, however, possessed by a 24-cell 4D polytope, whose 3D projection is called the FCHC lattice [8] (FCHC stands for face centered hyper-cubic). This circumstance is exploited thanks to a simple correspondence between the 24 velocities of FHCH and a fundamental pattern of the cubic lattice involving first and second neighbors. A discussion of possible pair and triple collisions concludes the chapter.

Chapter 6 describes what may be considered a useful deviation from the purity of lattice-gas models, the lattice-Boltzmann method. Rather than following the dynamics of the pseudomolecules, one may resort to a statistical description via a Boltzmann collision operator. Here a simple version of this tool (akin to the BGK model in kinetic theory [2]) is described. The model provides smoother approximations to fluid dynamics than the pure lattice method. Both the Euler and the Navier–Stokes asymptotics are briefly described.

Chapter 7 addresses many simple but practically important issues of the lattice-Boltzmann method, such as rescaling, stability, convergence, and boundary conditions. At solid boundaries the “bounceback” boundary condition is suggested. The problem is that this boundary condition does not have first-order slip, but exhibits a second-order slip [9] due to the presence of a Knudsen layer. In order to avoid difficulties, one may think that the physical wall is located between the mathematical boundary and the wall sites; the actual location depends on the slope of the wall with respect to a lattice direction. In addition one may have troubles if heat is exchanged at a wall (this is incompatible with bouncing-back [10]). Although this aspect is not explicitly mentioned in the book, a reference to the relevant case of simulations of Rayleigh–Benard instability is given.

Chapter 8 addresses the case of two fluids, which will become an important theme in the rest of the book. Here the assumption that the fluids are miscible is made and the models are a straightforward extension of those for a single gas.

Chapter 9 introduces immiscible fluids in the plane case and thus enters the area where the lattice gases have achieved a remarkable success in applications. The main difference is that, now, if we have a blue and a red fluid, the collisions are no longer impartial with respect to the color, but tend to move particles of a given color in the direction of the gradient of that color. An immediate and interesting consequence is that a surface tension, agreeing with Laplace’s law, arises with a related jump in the stress tensor at the interfaces.

Chapter 10 describes the lattice-Boltzmann method for immiscible fluids and compares theory with simulation.

Chapter 11 extends the treatment in Chapter 9 to 3D. The difficult task is simplified by clever geometric arguments. Phase separation and drag on a bubble are given as examples of the method.

Chapter 12 introduces interaction at a distance, thus making a study of gas-liquid phase transitions possible. The microdynamics is remarkably simple in both 2D and 3D.

Chapter 13 takes another important step into the applications of the latticegas method by introducing two-phase flows through porous rocks and discussing the validation of the gas-lattice method in this case. As is well known, this is a very important practical issue in oil production, when a fluid such as water is injected into an oil reservoir in order to displace and recover the oil.

Chapter 14 has a distinctly theoretical flavor and addresses the foundations of equilibrium statistical mechanics of a lattice gas. The (pseudo) Liouville equation (a sort of master equation) for a Markov process is introduced, together with a (generalized) H-theorem. The concept of ergodic component is briefly discussed. The ergodic conjecture is introduced and rejected by means of counterexamples. It is then adopted as a working tool to introduce the microcanonical and canonical ensembles.

In order to avoid the difficulties of a theory based on the Liouville equation, Chapter 15 introduces the Boltzmann equation (not to be confused with the lattice-Boltzmann method discussed in Chapter 7). The hydrodynamics are then recovered via the Chapman–Enskog method. Here the spurious invariants (different from those related to classical conservation laws, such as mass and momentum) are discussed. Their presence makes the validity of hydrodynamics doubtful. Thus special initial and boundary conditions must be introduced, but shock waves may still produce problems.

Chapter 16 deals with one of the most fascinating aspects of immiscible lattice-gas mixtures, the spontaneous separation of two initially mixed fluids. After characterizing the phase separation, the authors detail some of the ways in which their models reproduce the phenomenon.

Chapter 17 returns to the interfaces between 2D immiscible lattice gas. After computing the surface tension, the authors present a detailed view of interface fluctuations.

Chapter 18 discusses how the ideas presented in the previous chapters may be used to produce fluids of even greater complexity (“Bubble, bubble, toil, and trouble”). The topics range from pattern formation to simulations of multiphase flows.

The book is completed by 6 technical appendices dealing with tensor symmetry, polytopes and their symmetry group, classical compressible flow modeling, the incompressible limit, the derivation of Gibbs distribution, and the hydrodynamic response to forces at fluid interfaces.

The volume contains a list of interesting problems in each chapter. The references are accompanied by explicative notes.

All in all, this is a good introductory book and an almost compulsory reference for any future paper on the subject.

REFERENCES

[1] U. FRISCH, B. HASLACHER, AND Y.

POMEAU, Lattice-gas automata for the Navier–Stokes equations, Phys. Rev. Lett., 56 (1986), pp. 1505–1508.

[2] C. CERCIGNANI, The Boltzmann Equation and Its Applications, Springer-Verlag, New York, 1988.

[3] J. HARDY, O. de PAZZIS, AND Y. POMEAU,

Molecular dynamics of a classical lattice gas. Transport properties and time correlation functions, Phys. Rev. A 13 (1976), pp. 1949–1961.

[4] C. CERCIGNANI, W. GREENBERG, AND

P.F. ZWEIFEL, Global solution of the Boltzmann equation on a lattice, J. Statist. Phys., 20 (1979), pp. 449– 462.

[5] P.J. HILTS, Discovery in flow dynamics may aid car, plane design, Washington Post, November 19, 1 (1985).

[6] C. CERCIGNANI, On the thermodynamics of a discrete velocity gas, Transport Theory Statist. Phys., 23 (1994), pp. 1–8.

[7] C. CERCIGNANI, Temperature, entropy and kinetic theory, J. Statist. Phys., 87 (1997), pp. 1097–1109.

[8] D. d’HUMIÈRE, P. LALLEMAND, AND

U. FRISCH, Lattice gas models for 3D hydrodynamics, Europhys. Lett., 2 (1986), pp. 291–297.

[9] C. CERCIGNANI, Kinetic theory with “bounce-back” boundary conditions Transport Theory Statist. Phys., 18 (1989), pp. 125–131.

[10] C. CERCIGNANI AND S. CORTESE, Val-

idation of a Monte Carlo simulation of the plane Couette flow of a rarefied gas, J. Statist. Phys., 75 (1994), pp. 817–838.

CARLO CERCIGNANI

Politecnico di Milano

Chaos in Discrete Dynamical Systems: A Visual Introduction in 2 Dimensions. By Ralph H. Abraham, Laura Gardini, and Christian Mira. Springer-Verlag, New York, 1997. $59.95. xxv+246 pp., hardcover. CD-ROM included. ISBN 0-387-94300-5.

With the advent of readily available computer graphics, the study of chaotic dynamical systems has exploded in many directions. For example, interest in complex analytic dynamics was rejuvenated once images of the Mandelbrot and Julia sets appeared. The study of chaotic attractors moved forward after publication of pictures of the Hénon and Lorenz attractors. And research on iterated function systems and their applications began in earnest once images of various “chaos games” became well known.

Given the rapid advances in visualization technology including CD-ROM and the web, it is little wonder that we now have a book that is billed as a “visual” introduction to chaos in discrete dynamical systems. Whatever this book is, it certainly is spectacularly visual. The book really consists of three interdependent components: the book, a CD-ROM, and a website (http://www.vismath.org/chaos). The book itself contains over 100 illustrations. Many of these are linked together in a dozen full color animations included on the CD-ROM. These are a pleasure to look at and think about. Undoubtedly, this type of format (or something resembling it) will become standard in dynamics books in the future. I applaud the authors for their vision and for taking the lead in this movement.

Unfortunately, there are some problems with the authors’ implementation of their vision. For example, when I visited the website, I found only a short list of the four publications of the Visual Math Institute, including synopses and tables of contents, a one paragraph history of the Institute, and a link to only one other related site. Presumably, the site will be enhanced in the future.

The book itself is somewhat narrowly restricted to the study of the chaotic dynamics associated with noninvertible planar mappings. There is no mention of invertible planar maps such as the Hénon or standard map, nor is there any discussion of complex analytic functions. Rather, the entire book focuses on the iteration of just two pairs of quadratic maps of the plane. These maps are given by

u = ax + y,
v = b + x^{2}

and

u = (1 − c)x + 4cy(1 − y),
v = (1 − c)y + 4cx(1 − x),

where a, b, and c are parameters. In each of the four main chapters, the authors investigate the behavior of a specific one parameter family of one of these maps with an eye toward understanding the “exemplary bifurcation sequences” each exhibits.

Bracketing these main chapters are a brief introduction to one-dimensional dynamics and a lengthy series of appendices which mostly deal with the history of the subject.

The reader should be forwarned that this is not a book for mathematicians, pure, applied, amateur, or otherwise. The authors specifically target working scientists and their students who have limited mathematical background. In particular, calculus and linear algebra are not used in the book; these tools are replaced by computer experiments and graphics. There are no theorems, few definitions, and little that is rigorous. Rather, the authors basically give a series of observations of what happens dynamically as the parameters in these maps are varied. Much of the book reads like the following statement which forms the script of one of the animations on the CD: “Stage 5: The boundary contacts a critical curve. … Stage 6: A hole appears. … the holes become more numerous… the holes accumulate on the boundary of the basin. … Stage 7: The basin boundary is weakly fractal.” Basically, the book continues in this fashion for over 100 pages. Spectacular pictures and great animations, but not a lot of traditional mathematics. On the other hand, the pictures and animations are quite interesting, and they did lead me to think quite a bit about what was transpiring on the screen.

Perhaps working scientists will benefit from this glimpse of the complexity inherent in planar dynamical systems by realizing that “simple” dynamical systems like the quadratic maps studied in this book have extremely complicated behavior. Of course, they could quickly come to the same conclusion if they looked at one-dimensional or complex quadratic maps.

Unlike the CD, the book appears to be hastily or carelessly put together. The reader will encounter such terms as Neimark–Hopf bifurcation, Cantor set, period doubling bifurcation, contact bifurcation, and many other terms with little or no definition or explanation whatsoever. As a single example, the term contact bifurcation first appears on page 60. Not knowing what this term means, I looked it up in the index. The reference to contact bifurcation is two pages later. Turning to page 62, I found no reference to this term, only four pictures. Oh well, I read on. On page 64 there is the phrase “a point of contact, h_{0}, between the boundary of d′ and F.” Maybe that was it. But what is the boundary of d′? Well, the index lists 4 references to “boundary.” The first is on page 7 where I read “the boundaries of these basins are particularly important in applications.” That is the only mention of boundary on that page. On page 22 I learned that “boundaries of the basins, also called frontiers or separatrices, are of primary interest in dynamical systems theory.” Not much help there. On to the references on pages 169 and 181, but page 169 contains only a history of functional equations and page 181 describes applications in control theory. Alas, no description of boundary. I gave up and read on.

I encountered numerous subsequent mentions of contact bifurcations with no further illumination until I finally found a section entitled “What is a contact bifurcation?” on page 134, some 75 pages after the term first surfaced. I read “A contact bifurcation involves a contact between the boundary of a chaotic attractor and the boundary of its basin of attraction.” I bought that, but what is a chaotic attractor? At the bottom of the page I read “A chaotic attractor is a chaotic area which is attracting.” Thanks for the help. Turning to the index, the reference to chaotic attractor led me back to page 37 where, again, no such word appears. However, on the previous page I found the statement that “chaotic attractors are more complicated sets [than a finite set] that are attractive.” That, to my mind, means that a lot of what I would call nonchaotic sets are actually chaotic attractors. So maybe the problem is with the word chaotic—what does this mean?

Back to the index—“chaotic dynamic,” see page 226. Unfortunately, nothing there, only a historical treatment of Poincaré’s analytic theorem. No further mention of chaos, so I read on. Finally, on page 160, buried at the end of Appendix 2, I found it, a one line definition of chaos involving sensitive dependence, topological transitivity, and density of periodic points. I got it, but will the intended audience? I doubt it.

Should I look up density in the index? No way.

There are many other examples of such carelessness in the book. Roughly half of the index appears to be wrong. There is a table of notations in Appendix 1, including a number of symbols such as ∀, ∃, and ∂ (boundary), but many of these symbols apparently never appear in the book. There is an extensive bibliography, but when the reader encounters a reference in the text such as M1, GARF, or G1, there is no annotation in the bibliography to indicate the correct reference. Does M1 refer to May or Marotto or Mira? To find out, one has to know that there is a separate section of the bibliography containing frequently used references by code. While the main part of the book deals with only two maps, these formulas are written in the text only once. These formulas must be memorized in order to make sense of the book. Occasionally, the authors do refer back to the formulas, saying on page 137, for example, “see equation 2 in Section 1.5.” Don’t bother going there; the real reference is equation 2 in Section 1.7, not 1.5. Frustrating!

Finally, the book also contains two lengthy appendices on the history of dynamical systems, both written almost twenty years ago and appearing in print elsewhere. Both deal with the early history of dynamics (pre-1980). For a book that touts itself as presenting the frontier of research in noninvertible maps, I cannot help but wonder why these completely unrelated histories are republished here at all.

To summarize, while this project clearly possesses both vision and visualization, both the website and the book have some defects.

ROBERT L. DEVANEY

Boston University

Algebraic 3-D Modeling. By Andreas Hartwig. A. K. Peters, Ltd., Wellesley, MA, 1996. $59.00. x+222 pp., hardcover. ISBN 1-56881-023-7.

“This text deals with difficulties of geometric modeling” is a quote from p. 8 that seems to amply describe the overall aim of this book. Hartwig realizes that precision limitations inherent in floating-point representations can raise serious topological problems. In order to overcome these, he proposes an algebra which is a superset of standard Boolean algebra (as used in CSG modelers). In this algebra, the result of any set operation on geometric objects is again a valid geometric object. Consequently, he produces nonmanifold topologies. The algebra is devised to improve the stability of topological operations, leading to a “correct modeler.”

This modeler is called “GECO,” for “Geometric Construction.” It is a PC package (not tested by this reviewer) and is capable of handling nonmanifold objects. A web site is supplied from which the software can be downloaded.^{1} While the book makes use of this particular sample modeler, the concepts that are presented do apply to a broader class of possible modelers.

The following is an overview of the contents:

Chapter 2 deals with polyhedra. Both the CSG and the Brep forms are combined in order to allow stable handling of nonmanifold objects.

Chapter 3 covers boundary models with the intent of arriving at a stable point/solid classification.

Chapter 4 is the description of the GECO modeler. The objects produced by this modeler are hybrid in the sense that both CSG and Brep forms are used. The modeler is capable of the standard Boolean operations, but can also model convex hulls.

Chapter 5 (Algebraic Model) describes how to weaken standard Boolean algebra, arriving at a “natural modeling algebra.”

Chapter 6 (Computation of Algebraic Manifolds) deals with computational and numerical aspects. Discretization and rounding issues are addressed, and all computations are based on a discretized dot product. The number of significant digits necessary to maintain topological correctness is discussed as well. Numerical operations involving quadrics are used to demonstrate those concepts.

1In this reviewer’s experience, that ftp site does not work. Luckily, the author’s e-mail address is provided.

Chapter 7 deals with correct topological specifications. The aim here is to avoid model instabilities due to instabilities in numerical algorithms.

Chapter 8 (Modeling Principle) summarizes and extends the insights gained in the previous chapters. Separation and connection sets are introduced in order to handle algebraic, topological, and constructive operations.

An appendix gives the details of the GECO modeler.

I find that this book addresses an important issue, namely that of topological correctness in a finite precision environment. As a drawback, I noted that the references are outdated: the latest citation is from 1991. While this book deals with nonmanifold topology, the actual term is never used, presumably a consequence of missing references to Weiler’s work on the subject.

GERALD FARIN

Arizona State University

Conformal Differential Geometry and Its Generalizations. By Vladislav V. Goldberg and Maks A. Akivis. John Wiley & Sons, Inc., New York, 1996. $82.00. vii+383 pp., hardcover. ISBN 0-471-14958-6.

In the monograph under review the authors systematically present the results of their own research as well as that of others in multidimensional conformal differential geometry. Conformal structures are of great significance at present in differential geometry and mathematical physics research. Many published papers are devoted to investigating these structures, and the problems of their geometry are studied from different points of view. But up to now there has not been a unified basis for such investigations. The last monograph devoted to such problems was published by S. Sasaki about 50 years ago.

The monograph discussed here (by M. A. Akivis and V. V. Goldberg) has filled this gap. Not only problems of conformal and pseudoconformal differential geometry as well as questions of Grassmann and almost Grassmann structures are considered from a unified point of view but also a close connection between these problems and the theory of multidimensional webs is demonstrated. In fact, the authors develop a specific apparatus for a systematic investigation of the structures mentioned above. The methods are based on deep mathematical techniques and ideas, such as the method of exterior forms and moving frames of É. Cartan, F. Klein’s idea of the equivalency of conformal and Grassmann geometries, and the connection between geometries of multidimensional webs and almost Grassmann structures established by the authors. Their approach allows the authors both to develop a universal set of methods to investigate the structures considered and to construct numerous realizations of these structures. This has greatly enriched their theory.

It should be mentioned that the authors devote considerable attention to the study of pseudoconformal structures on four-dimensional manifolds. The study of pseudoconformal structures generated by axially symmetric metrics which are widely used in relativity theory (such as Kerr’s, Schwarzschield’s, and Reissner– Nordström’s metrics) is of special interest in modern physics. Though the authors do not show any immediate physical applications of the results obtained, they are undoubtedly extremely valuable for theoretical physicists.

Let us describe the contents of the book chapter by chapter. Chapter 1 contains preliminaries. Some basic facts of geometry of the n-dimensional pseudoconformal space C_{q}^{n} of signature q are recalled. Two approaches for the description of the geometry of the conformal space are considered: the polyspherical coordinates and the Cartan method of moving frames. The authors study the geometry of straight lines in a three-dimensional projective space and the Lie geometry of hyperspheres in the Euclidean space E^{n} by applying the fact that these geometries are equivalent to geometry of the conformal spaces C_{2}^{4} and C_{1}^{n+1}, respectively.

The theories of submanifolds of codimension one and arbitrary codimension of the conformal space C_{q}^{n} are treated in Chapters 2 and 3, respectively. In particular, an invariant normalization of a submanifold is constructed and its geometrical meaning is elucidated. A system of tensors which determines a submanifold up to an equivalency is found, and the rigidity theorem is proved. The authors also study some important special submanifolds: the canal hypersurfaces of different kinds and the submanifolds carrying a net of curvature lines. As examples, they consider submanifolds of C_{2}^{4} which correspond to the line submanifolds of the projective space P^{3}. In Chapter 4 the differential geometry of general (not necessarily flat) conformal structures on a manifold is developed. The structure equations are derived and the tensor of conformal curvature is defined and studied. Here the authors consider also the conformal structure induced on a submanifold of a conformal space and on a hypersurface of a projective space. They describe a wide class of such submanifolds which carry a (locally) flat conformal structure and are connected with the canal hypersurfaces.

Chapter 5 treats a conformal structure of arbitrary signature on a real fourdimensional manifold. Two isotropic fiber bundles of two-dimensional α- and β-planes associated with a conformal structure are studied. Some relations between properties of isotropic bundles and the structure of the conformal curvature tensor are established. In particular, a geometric characterization of self-dual and anti-self-dual conformal structures is given. The close relation between four-dimensional conformal structures and four-dimensional webs is described and used. The conformal structure of some classical Einstein metrics of general relativity is investigated.

The last two chapters are devoted to analytic and differential geometry of Grassmannians G(m, n) of m-planes in the projective space P^{n} and the differential geometry of more general manifolds with almost Grassmann structures modeled on such a Grassmannian.

Some remarkable classes of submanifolds of the Grassmannian G(m, n) are described and studied in Chapter 6. Different normalizations of the Grassmannian are constructed and its generalized stereographic projection onto the Segre-affine space is defined.

In Chapter 7 the authors define and study an almost Grassmann structure on a manifold and derive its structure equations and the structure tensors. They prove that vanishing of the structure tensors is a necessary and sufficient condition for the flatness of an almost Grassmann structure. The notion of semi-integrable almost Grassmann structure is defined and studied. The authors construct some examples of semiintegrable and locally flat almost Grassmann structures using the theory of webs and the relation between almost Grassmann structures and webs.

In conclusion, we remark that the monograph contains a very detailed bibliography with citations of reviews of referenced articles in major mathematical review journals.

The language of the monograph is highly professional, clear, and understandable. The book is read easily and with great interest. It can be recommended for a wide range of researchers of modern differential geometry from beginners to experts in differential geometry and mathematical physics.

A. T. FOMENKO

Russian Academy of Sciences

V. F. KIRICHENKO

Moscow State Pedagogical Institute

D. V. ALEKSEEVSKY

Center “Sophus Lie”

Some Points of Analysis and Their History. By Lars Gårding. AMS, Providence, RI, 1997. $16.00. vii+88 pp., paperback. University Lecture Series. Vol. 11. ISBN 0-8218-0757-9.

This delightful little book (only 88 pages) is a somewhat augmented collection of lectures given during a visit to China in 1994. The 12 lectures survey some of the most important results in mathematical analysis of our century, with emphasis on matters close to the heart of the author. Much of the material is concerned with the development of the modern theory of general linear partial differential equations, in which the author has played a leading role, and also with the tools from mathematical analysis which have had a particularly decisive effect on this development, such as distribution theory, the Riesz–Thorin interpolation theorem, the Tarski–Seidenberg theorem, etc. A pervading feature is the presence of historical comments and very personal recollections from someone who was actually there and very much part of the development. As is natural both because of the culture the author comes from and the importance of the contributions from his part of the world, work of Swedish mathematicians play a major role; the following list of people quoted gives a flavor of this: Beurling, Carleman, Carlsson, Fredholm, Holmgren, Hörmander, Mittag-Leffler, and M. Riesz, and one could add Ahlfors, Lindelöf, and Nevanlinna from neighboring Finland. The selection of topics is well described by reproducing the table of contents below:

1. Picard’s great theorem.
2. On Holmgren’s uniqueness

theorem.

3. The Phragmén–Lindelöf principle.
4. Nevanlinna theory.
5. The Riesz–Thorin interpolation

theorem.

6. The mathematics of Wiener’s

Tauberian theorem.

7. The Tarski–Seidenberg theorem.
8. Intrinsic hyperbolicity.
9. Hypoellipticity.
10. Dirichlet’s problem and

Gårding’s inequality.

11. A sharp form of Gårding’s

inequality.

12. The impact of distributions in

analysis.

Most personal are perhaps the accounts of the areas in which Lars Gårding made his own major contributions, such as the lecture on hyperbolic operators with its elegant study of the corresponding polynomials and that on Dirichlet’s problem, where the historical development culminating in Gårding’s inequality is lucidly expounded. Also fascinating is the final lecture in which the author recounts, along with a description of the theory, his personal awakening to the importance of distributions.

The style is elegant and lively, and the book provides exciting and instructive reading for anyone who is interested in the pure mathematics behind and often exploited in the analysis of many of the applied problems that are relevant to the readers of SIAM Review. The reviewers are nostalgically reminded of privileged times as students of their warmly admired teacher.

BRUCE KELLOGG

University of Maryland

VIDAR THOMÉE

Chalmers University of Technology

Noncoercive Variational Problems and Related Results. By D. Goeleven. Addison–Wesley Longman, Ltd., Essex, UK, 1996. $40.33 iv+168 pp., softcover. Pitman Research Notes in Mathematics. Vol. 357. ISBN 0-582-30402-4.

This book deals with the study of noncoercive variational problems, which can be formulated as the minimization of convex or nonconvex energy functionals. Some additional results on nonvariational problems are also considered.

The main objective of the work is to develop a general mathematical theory applicable to the study of the existence of solution of noncoercive mathematical models arising in nonsmooth mechanics. As a matter of fact, various engineering problems lead, in general, to noncoercive problems. The lack of coercivity is due to the fact that, e.g., because of the boundary conditions, the kernels of the energy bilinear forms of the problem are nonempty linear subspaces. This is the case, e.g., in classical elasticity, if the boundary conditions permit rigid body displacements. The author directs his efforts particularly to the study of problems which can be formulated in terms of variational inequalities and hemivariational inequalities.

The approach presented in this book is relying on the asymptotic behavior of the sets, functions, and operators which are in a considered problem. It is called the “recession approach” and it appears throughout this work as a particularly useful approach to deal with semicoercive unilateral problems. Also, several types of nonlinear perturbations of noncoercive inequality problems are considered. Other variational problems are considered, too. Indeed, besides the study of noncoercive problems one scope of this work is also to generalize some known results and to connect several theories.

In Section 2, certain definitions and general properties of the recession functions, weak-coerciveness conditions, and other preliminaries are given. In Section 3.1, the author considers an abstract minimization problem in a general framework and combines the regularization of Tychonov with a general recession approach. The result is a constructive theory. In Section 3.2, the results of Section 3.1 are modified when one replaces the Tychonov regularization by a general viscosity regularization. Section 3.3 presents an extension of the foregoing recession analysis for the study of nonvariational problems with possibly nonconvex sets of constraints. A general class of hemivariational inequalities is also investigated. Section 4 deals with the applications of the abstract recession method. The author shows that the same methodology can be used in order to study many different problems. In Section 4.1, the author considers an elliptic problem with unilateral boundary conditions. Section 4.2 deals with the study of a model for the description of a masonry-like problem. In Section 4.3, D. Goeleven discusses some basic global minimization theorems, and in Section 4.4, he refines the recession analysis when a separability property holds for the generally nonsmooth energy functional. Section 4.5 deals with the study of noncoercive unilateral problems arising in nonlinear elasticity and involving locking constraints. In Section 4.6, it is shown how weak convergence methods can be used in order to check a compactness condition used in the recession analysis. In Section 4.7, the minimization of noncoercive functionals on manifold-type constraints is studied. Section 4.8 deals with the study of some nonlinear and noncoercive partial differential equations. Finally, in Sections 4.9, 4.10, 4.11, and 4.12, the author applies the recession analysis in order to obtain new results for variational and hemivariational inequalities. In Section 4.13, the book ends with the presentation of a linear ill-posed system.

The book presents in a rigorous and clear manner a general theory for the existence of solutions of noncoercive variational problems and of constrained problems formulated as variational inequalities and hemivariational inequalities. The book illustrates very successfully recent mathematical approaches and results with various examples from mathematics and mechanics.

The book is written with clarity and originality and is addressed to the applied mathematician and to the mathematically oriented engineers.

All the applications are carefully selected by the author. They are problems of general interest which highlight the presented theory in the best possible way. Carefully chosen references are included at the end of the book, helping the reader to broaden his knowledge on the subject of the book. Another strong point of the book is that it unifies ideas for the treatment of various noncoercive problems and contains unpublished original results for variational inequalities and hemivariational inequalities. It is addressed to an audience of students, researchers, and teachers in applied mathematics, mechanics, and engineering sciences who are interested in the formulation, analysis, and computation of equilibrium problems.

In conclusion, the results of the book are original, very interesting, and of high mathematical value. The book is a significant contribution to the very few related published works, it is properly structured, and it is written with accuracy and clarity, with appropriate referencing. It should therefore be considered an essential piece of work to be read by everyone familiar with the field studied therein.

P. D. PANAGIOTOPOULOS

Aristotle University and RWTH

Probabilistic Methods in Combinatorial Analysis. By Vladimir N. Sachkov.

Cambridge University Press, Cambridge, UK, 1997. $64.95. x+246 pp., hardback. ISBN 0-521-455112-X.

Some books delight. This is one such book. I personally have a penchant for small books, ones like the green Methuen series or the old Oliver and Boyd University Mathematical Texts. Probabilistic Methods in Combinatorial Analysis is not quite that small (x+246 pp.), but is of such a size that one does not feel completely cowed upon picking it up the first time.

The book is a mine of information and organized so as to make possible the reading of chapters independently of one another. It deals with the following generic problem. If a_{k}(n) is the number of objects (permutations, matrices, subsets, trees, etc.) having a certain property, and ∑_{k≥0} a_{k}(n) = A(n), then the numbers p_{k} = a_{k}(n)/A(n) constitute a discrete probability distribution. If ξ_{n} is a random variable having P {ξ_{n} = k} = p_{k}, then often one can use methods of probability theory to show

P \{ \frac{ξ_{n} − μ_{n}}{σ_{n}} ≤ x\} → F (x) as n → ∞,

where F is a specific cumulative distribution function and Eξ_{n} = μ_{n}, V arξ_{n} = σ_{n}^{2}. One of the best known and oldest examples of such a result is due to DeMoivre. It is, a_{k}(n) = \binom{n}{k}, the number of ways to choose k items from n. The probabilities, p_{k} = \binom{n}{k}/2^{n}, and

\begin{matrix}P \{ \frac{ξ_{n} − n/2}{\sqrt{n}/2} ≤ x\} \\ → ∫_{−∞}^{x} exp(−t^{2}/2)dt/\sqrt{2π} \end{matrix}

as n → ∞, the simplest and earliest form of the central limit theorem.

Another example is a_{k}(n) equal to the number of permutations of {1, 2, …, n} having exactly k inversions, the number of pairs out of order. Clearly A(n) = \frac{n(n−1)}{2} . Now

\begin{matrix}Eξ_{n} = \frac{n(n − 1)}{4} = μ_{n}, \\ V arξ_{n} = \frac{2n^{3} + 3n^{2} − 5n}{72} = σ_{n}^{2}, \end{matrix}

and again if

P {ξ_{n} = k} = a_{k}(n)/n!,

P \{ \frac{ξ_{n} − μ_{n}}{σ_{n}} ≤ x\}

→ ∫_{−∞}^{x} exp(−t^{2}/2)dt/\sqrt{2π}

as n → ∞.

The first chapter contains the prerequisite probability theory needed for the remainder of the book and discusses the standard analytical tools (probability generating functions, moment generating functions, the Lindeberg–Levy statement of the central limit theorem, etc.). Additionally one finds discussion of binomial moments, factorial moments, central moments, various inversion formulae, and the relations amongst and between. Weak convergence of cumulatives and implied convergence of generating functions and the converse are covered in the needed degree of detail to enable utilization later in the book.

Mixed with the joy engendered by the clarity of exposition and the taste exhibited in the selection of topics I must admit to minor pangs of depression. I thought I knew this area reasonably well, but over and over I found the author producing nuggets of insight of which I was here-to-fore unaware.

Now to a more detailed rendering of the book’s contents. Chapter 2 discusses combinatorial properties of random nonnegative matrices. As an example, let A(p) = [a_{ij}] be an n by n matrix whose elements a_{ij} are independent random variables taking the values either 0 or 1 with probability q or p, respectively. If P_{n}(p) denotes the probability that the permanent of A(p) is positive, then

lim_{n→∞} P_{n}(p) = exp(−2e^{−C})

whenever

p = \frac{log n + C}{n} + o ( \frac{1}{n} ) as n → ∞.

Chapter 3 deals with far reaching generalizations of the standard n distinguishable (indistinguishable) beads are randomly dropped into m distinguishable (indistinquishable) cells. Among the applications are the distribution of and asymptotics for the number of runs of heads in a random sequence of n heads and m tails. The circular version of this problem and others is treated as well. In this material there is one of the very few typographical errors. At the bottom of p. 97 m^{−n} should be n^{−m}, which causes confusion at first. Chapter 3 also contains a brief but insightful formulation and discussion of a matching problem.

Chapter 4 studies random partitions of sets. Bell numbers obviously play a vital role. There is a neat idea inherent in the proof of Theorem 1.1. A polynomial probability generating function is factored into ( \frac{x}{1+α_{1}} + \frac{α_{1}}{1+α_{1}} ) ⋯ ( \frac{x}{1+α_{n}} + \frac{α_{n}}{1+α_{n}}), all α_{i} are positive, and each factor then associated with an independent two point binomial ξ_{j} having

\begin{matrix}P \{ξ_{j} = 0\} = \frac{α_{j}}{1 + α_{j}} , \\ P \{ξ_{j} = 1\} = \frac{1}{1 + α_{j}} . \end{matrix}

An example of a typical result from Chapter 4 is Theorem 2.1. Denote by κ_{m}(l) the number of subsets of size l in a random partition on an m-element set. If r is the unique solution of the equation r e^{r} = m, then for fixed l the distribution of the random variable

\frac{κ_{m}(l) − r/l!}{\sqrt{r^{l}/l!}}

converges to the standard normal distribution as m → ∞.

Chapter 5 deals with the classic random permutations. The author derives the limiting distribution for the number of cycles in a random permutation, the number having even length, the number of cycles whose lengths are multiples of a given integer, and much more.

The final chapter, 6, deals with random graphs which the author approaches from 3 points of view. They can be regarded as the result of randomly deleting edges from a complete graph, or as the result of randomly joining initially isolated vertices with edges, or finally by postulating a uniform distribution on some set of specified graphs. Most results of this chapter deal with some form of labelled graph. Asymptotic formulae are derived for the number of trees of bounded height, the number of connected components of a graph, and much more.

The book finishes with some 125 useful references. It has no sections devoted to problems and a minimal number of typographical errors. On p. 34 Δ^{n}0^{k} are referred to as “Morgan numbers” rather than differences of zero, which I found a bit startling. I could find no reference to Morgan numbers in Comtet, Riordan, Netto, or a number of other texts, so I have no idea who Morgan was (or is).

Page 82 has the, to me, odd notation g ∘ f ∗ h for the mapping x ↦ g(f(h(x))). But these are all very minor considerations in a book so scholarly and lucid.

I found the book informative and a joy to read. I hope others will as well.

ROGER PINKHAM

Stevens Institute of Technology

Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision, and Materials Science. By J. A. Sethian. Cambridge University Press, Cambridge, 1996. $39.95. xviii+218 pp., hardback. ISBN 0-521-57202-9.

Level set methods are numerical techniques for tracking interfaces that arise in a wide collection of phenomena. Physical phenomena involving moving interfaces are ubiquitous, including oil-water boundaries, propagating flames, flow of plastics in molds, the growth of snowflakes, and semiconductor manufacturing. At the same time, nonphysical problems which rely on moving interfaces are equally important, including the geometry of singularities of evolving surfaces, problems in grid generation, image processing, and computer vision, as well as robotic path navigation and construction of first arrival times in seismic analysis. This new book, Vol. 3 in the Cambridge Monograph Series in Applied Mathematics, admirably covers the theory, numerics, and application of level set methods to this wide range of topics. It accomplishes the difficult task of serving both as an introductory resource and as a state-of-the-art reference point. One can read this book both as monograph for understanding the theory of evolving interfaces in a collection of settings and as an explicit manual for writing high-resolution interface methods.

As an example of the difficulty involved in interface modeling, consider the problem of tracking the evolution of a flame burning in a swirling fluid. The burning speed and position of the flame depends on a wide collection of factors, including the underlying fluid mechanics, chemical reactions at the front, heat diffusion away from the interfaces, and vorticity production at the flame front. A closer look at the interface reveals complex behavior: large stretching and contortion, formation of cusps and singularities, as well as changes of topology as the front breaks and merges. In developing computational techniques for tracking the evolving interface in three dimensions, these problems become even more daunting.

Traditional attempts to track this evolving interface are often based on discrete parameterizations of the problem, that is, so-called marker methods, which break the interface up into a large collection of connected points whose position at any time can be used to reconstruct the front. While powerful, these techniques run into considerable difficulty when tracking the formation of cusps and corners, when topological changes occur, and when trying to track moving surfaces in three dimensions and higher.

Level set methods offer a different approach. They come in two varieties, both of which rely on Sethian’s work at Berkeley on the theory of curvature-dependent interface motion. The first type, known as “time-dependent level set methods” and developed by S. Osher and Sethian, exchange this Lagrangian description for an Eulerian, fixed domain perspective. This leads to an initial value partial differential equation, which is solved by using the theory of viscosity solutions to produce upwind methods that satisfy an appropriate entropy condition. The method chooses the correct weak solution when corners and cusps develop in the evolving front. These methods are particularly appropriate for the most complex motions, including fluid interface problems and combustion and flame simulations. The second type, known as “fast marching methods,” developed more recently by Sethian, provide extraordinarily fast ways of computing interface motion problems which can be recast into the Eikonal equation, including problems in robotic navigation, photolithography, and computation of seismic travel times.

This book covers the theory, algorithms, and applications of level set methods. It is aimed at an introductory graduate level and relies on only minimal coding skills and analysis background. Divided into four parts, it follows a progression, from theoretical aspects of the surface evolution through algorithms and details, ending with a series of applications.

Part I of the book analyzes the theory of moving interfaces for both curves and surfaces. Here, the link between hyperbolic conservation laws, corner and cusp formation, and entropy conditions is developed; ending with a nice formulation of both the time-dependent and stationary level set formulations. Each turns out to be appropriate for a particular type of interface motion problem.

Part II develops the technical details required to develop numerical methods to approximate level set equations. It begins with a discussion of the relevant theory of hyperbolic conservation laws necessary for the development of upwind, entropysatisfying methods. This section, while familiar to more advanced readers, provides an introduction to shock capturing methods in computational fluid mechanics, including viscosity methods, Godunov methods, and Riemann solvers. After these preliminaries have been developed, complete algorithms for solving the time-dependent level set equations are presented and then followed with the details of fast adaptive methods, multiple level set functions, and the construction of extension velocities necessary for more sophisticated motions. Without cluttering up the flow of the text, this section provides the necessary details to begin coding a level set method, if the reader so desires.

Part III returns to the theory of viscosity solutions which underlies the level set framework. This is used as a launching point for “fast marching methods,” which are O(N log N) methods for solving the Eikonal equation. Applicable to problems in seismic travel time arrivals, robotic navigation, photolithography, and other wave problems, these methods are presented in complete detail.

Part IV, which comprises over half the book, is devoted to applications. The applications come in three distinct categories: pure geometry problems, interface techniques in which significant interplay occurs between interface evolution and the underlying physics and chemistry of the interface, and “nontraditional” interface problems. Under the pure geometry problems, the focus is on interface techniques whose motion is controlled only by the shape of the interface and includes motion of the curves and surfaces under curvature, sintering, grid generation, image denoising, and shape recovery in computer vision.

Complex interface evolution, in which the motion of the interface depends on the interplay between the underlying physics/chemistry and the interface, requires a careful representation of the appropriate mechanisms at the front. Sethian presents three different applications, each chosen to demonstrate a different technique for linking the material interface with the underlying physics of the problem. First, a problem of dendritic solidification is given. In this approach, a boundary integral formulation of the interface motion which contains such factors as the latent heat release, anisotropy, and curvature-dependence, is reviewed and then used to develop a level set formulation. Next, a combustion/flame stability problem is given in which the flame front produces exothermic effects and fluid vorticity. Here, the interplay between the single-step irreversible flame model and the hydrodynamic motion produces the physical observed change in flame brush and flame wrinkling, as the front becomes turbulent. In this set of examples, the interplay between the interface and the physics comes through elliptic solvers which record delta functions along the front. The third example of a complex interface motion, is an application from two phase flow, namely a falling bubble subject to surface tension. Here, the surface tension—which depends on local curvature at the fluid interface— acts as source term in the Navier–Stokes equations. A projection method is used to compute the new fluid velocity, which then advances the position of the interface. These three separate problems, and the different techniques used to model them, do an excellent job of illustrating possible approaches for modeling fronts whose evolution in time depends on some complex set of physical conditions, which in turn depend on the location of the front. Sethian provides references to a variety of state-of-the-art applications, as well as a sketch of how various practitioners approach these problems.

The final set of examples involve problems which can be recast as front propagation problems, even though one may not typically think of them as such. These applications exploit the fast marching method and include determination of optimal robotic navigation paths around obstacles, shape-from-shading problems in computer vision, construction of geodesics on manifolds, and calculation of first arrival times in seismic analysis.

The final chapter weaves together much of the methodology presented in the previous chapters to develop a framework for simulating topography simulations of etching and deposition in microchip fabrication. This is probably the most elaborate level set application in current practice and forms the core of several industrial semiconductor codes. As a summary piece, it contains aspects of fast adaptive level set methods, methods for modeling nonconvex problems, effects of visibility and masking, integral equations to determine fluxes, and surface diffusion.

If the underlying theory is the heart of this book, the applications chapters provide the legs, perfectly illustrating the wide applicability of the technique. Enough detail is provided for an able graduate student to begin applying these methods to new and exciting applications of their own. Armed with this resource as a guide, the interested reader should be able to tackle a host of additional interface problems.

In summary, this book serves as an inviting and enthusiastic launching point for advanced techniques in tracking moving interfaces. The book has over 200 references, which lead to a large and varied body of level set applications, algorithms, and theory, and provide pointers to a variety of different ways to approach these problems. This book should be on the shelves of all those seriously studying numerical solutions to interface problems and should open up a host of new applications for numerical analysis.

E. G. PUCKETT

University of California, Davis

Variational Methods for Potential Operators Equations. By Jan Chabrowski. Walter de Gruyter, Berlin, 1997. $89.95. ix+290 pp., hardcover. ISBN 3-11-0152699-X.

This well-written book discusses methods of the variational calculus directly related to elliptic partial differential equations. The situation considered throughout the book is that in which the Euler–Lagrange equation is the difference of two potential operators. The first chapter describes minimization of a functional subject to constraints. Both potential operators are assumed homogeneous. An important consideration here is the genus of symmetric sets; an infinite number of critical points is obtained. The next chapter continues this topic, but allows one of the potential operators not to be homogeneous.

Chapter 4 deals with the situation when both potential operators satisfy a covariance condition. The homogeneity assumption is dropped. In Chapter 5 the author investigates constrained minimization in which the constraints depend on a parameter. The dependence of eigenvalues and eigenvectors on the parameter is studied. In Chapter 6 the author considers generalizations of the mountain pass and saddle point theorems. The approach is to consider boundary conditions which keep the paths inside a bounded region. When these boundary conditions are satisfied one need not be concerned about unbounded Palais– Smale sequences. On the other hand, if one drops the boundary conditions but still requires the paths to be bounded, one is led to the Hampwile theorem which finds eigenvalues and the mountain pass, saddle point, and Hampwile alternatives in which it is shown that either one obtains critical points or rich families of eigenvalues and eigenfunctions. Problems in which not all of the requirements of the mountain pass theorem are satisfied are also considered.

In Chapter 7 the author considers several variants of the mountain pass theorem for locally Lipschitz (nondifferentiable) functionals. Generalized derivatives are introduced and the notion of critical point is suitably modified.

The last two chapters deal with weak methods of convergence—when compactness is missing—due either to problems on unbounded regions or problems involving critical Sobolev exponents. Various approaches are described in treating such problems.

Many examples involving elliptic boundary value problems are described in detail throughout the book. The exposition is clear and easily followed. Background material concerning Sobolev spaces is given in an appendix.

MARTIN SCHECHTER

University of California, Irvine

Mathematical Modelling of Immune Response in Infectious Diseases. By Guri I. Marchuk. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997. $174.00. x+347 pp., hardcover. ISBN 0-7923-4528-2.

Audaciously, a Russian school of theoreticians led by the author have attempted medically significant models of the immune system’s fight against disease. The fashion in the West, by contrast, has been developing fragmental submodels to enhance understanding of the enormously complex immune system.

The models in this book take the form of nonlinear delay differential equations. In the model for viral hepatitis B, for example, there are 10 equations and 45 parameters. The book recounts methods for efficient solution of the initial value problem when the equations are stiff. Parameter identification and sensitivity analysis are discussed extensively.

Marchuk’s models fulfill his sensible goal of reproducing a quantitative “generalized picture” of the course of disease. They pioneer in their incorporation of the effects of disease damage. They indicate a direction for studying the influence of drugs. Yet the modeling is indeed at “an initial stage,” as Marchuk himself asserts several times.

I take issue with the formulation in terms of delay equations. Delays there are, many of them. For example, there is a delay between reception by a B cell of a “go” signal by pathogen binding and cross-linking of its receptor molecules and the eventual secretion of pathogen-fighting antibody molecules by the B cell. In my judgement, however, rather than employ fixed delays it is preferable to introduce additional differential equations for new state variables whose lagged following of changes in their surroundings phenomenologically account for the various delays [1]. Mathematically and computationally, it is worth adding differential equations if one can avoid the serious difficulties (with which the author skillfully grapples) caused by the presence of differences. Biologically, appropriately chosen new variables can well approximate the phenomenology that is responsible for the observed delays.

Audaciously, it seems, the publishers charge 50 cents a page for a book full of minor spelling errors and laden with unidiomatic, often murky English. But perhaps a high price is justified, for who will read this book? All but a handful of physicians will find the mathematics opaque and significant medical conclusions few. Applied mathematicians wishing an introduction to immunology should probably look elsewhere among the 358 publications cited, but they will be helped by a 25 page literature survey. Working theoretical immunologists will probably be alternately and inconsistently uncomfortable with oversimplifications and with the proliferation of equations and parameters. Yet these theorists should have the book available, for they will be inspired to and assisted in further work by the author’s first steps along a road that they too will wish eventually to take. One day, large scale mathematical models of the immune system will surely play a role in fighting disease.

REFERENCE

[1] L. A. SEGEL AND A. S. PERELSON (1991), Exploiting the diversity of time scales in the immune system: A B-cell antibody model, J. Statist. Phys., 63, pp. 1113–1131.

LEE A. SEGEL

The Weizmann Institute of Science

Numerical Analysis: An Introduction. By Walter Gautschi. Birkhauser, Boston, MA, 1997. $64.50. xiii+506 pp., hardback. ISBN 0-8176-38895-4.

In the spring of 1996, I volunteered to test this textbook by teaching a numerical analysis course out of it. (My students were, of course, the real guinea pigs, although they had considerably less choice in the matter.) The book was at that time accessible for free downloading via the Internet, although parts of it, such as the preface, were not yet available. Had I been able to read the preface, I would have been reassured by the information that the text has arisen out of 30 years’ worth of courses taught at Purdue University and the summer school in Perugia, Italy by Walter Gautschi.

And it shows. The book reads like an unfolding story, especially in the first four chapters (on machine arithmetic, approximation/interpolation, numerical differentiation/integration, and nonlinear equations). Topics are motivated with the same care and ingenuity that might be given to establishing the drive behind characters in a good novel. Elegant presentations abound (see, for instance, the way both the discrete and continuous least squares problems are treated simultaneously by using appropriate measures, or the way complex analysis is used to bring out the dependence of Lagrange interpolant convergence on the choice of interpolation points). Yet clarity is never sacrificed for elegance. Above all, the pace is always lively and brisk, the writing concise, and the author never lets the exposition bog down. As he states in the preface, his goal is to present “what is essential, interesting and intellectually pleasing, and teachable.”

One of the things this means is that the book does not attempt to cover everything a standard numerical analysis textbook might. For instance, the vast area of numerical linear algebra is wisely omitted, because, as the author points out, this is a separate area by now, presumably with a separate course devoted to it (as in my department). However, the author manages to fit in a surprising amount of information without slowing down the pace, by moving a wealth of historical notes, extensions, and references to the end of each chapter. As a bonus, interesting tidbits on various prominent figures in numerical analysis are sprinkled as footnotes throughout the text (such as the fact that Peano tried to make Latin the world language for scientific publication).

In keeping with the energetic nature of the exposition, a number of the theorems in the first four chapters are stated and proved only informally, as part of the text. For instance, in the whole chapter on approximation and interpolation, there are only three theorems that are formally stated. Clearly, this will not fit everyone’s requirements for a good textbook (students, in particular, will have difficulty culling theorems if using this as a reference). Overall, however, I think the risk pays off, since the first four chapters in particular are engaging as few textbooks are. (However, even I was perturbed to see Weierstrass’s theorem described only in passing as this piece of text (page 56): “One reason why [polynomials are used for approximation] is Weierstrass’s theorem, which states that any continuous function can be approximated on a finite interval as closely as one wishes by a polynomial of sufficiently high degree.”)

For me, the main price that was to be paid for the lively pace came in the last three chapters of this book. These chapters, designed to be a separate course in numerical ODEs, are consistent with the rest of the book in that they are “uncluttered with numerical examples and other illustrative material,” as the author puts it. However, both the material and the style are more formal now. What I found while reading Chapter 5 (which deals with one-point methods for initial value problems) was that I had a craving to see some actual numbers, to see how some of the methods suggested actually work in practice, to see some comparison between a true solution and a computed one. What does a truncation error look like? What does stability mean—in particular, how does instability manifest itself? How does error monitoring and step control look in practice? What does an actual stiff problem look like? These were some of the questions that students complained they had trouble visualizing the answers to, since there were no illustrative examples.

Actually, examples are provided, but they require work—they can be found in the exercises. In fact, both the theoretical problems and the machine assignments are a great resource, since it is difficult to come up with good problems in numerical analysis. (Setting problems for the annual numerical analysis comprehensive exam becomes a lot easier after investing in this book.) However, no answers are provided to any of the problems—the reason being that the solutions (of which the author informed me he has a complete prepared set) form a volume of about the same thickness as the text.

I will point out that the preface describes this book as a graduate text, and this should be taken seriously. As I found out in my cross-listed advanced undergraduate/introductory graduate level course, the book demands a level of mathematical maturity that has usually not been attained by undergraduates students.

But for graduate students, for numerical analysts, or for mathematicians interested in enhancing their knowledge of the subject, this is a stylish, lucid, and engaging book. A lot of thought and effort have gone into the writing, and the book successfully conveys the author’s interest and experience in the subject to the reader.

MANIL SURI

University of Maryland Baltimore County

The Control Handbook. By William S. Levine, Editor. CRC Press, Inc., Boca Raton, FL, 1996. $145. xvi+1548 pp., hardcover. ISBN 0-8493-8570-9.

Control theory is a rich and varied field of science, within which one finds theoretical work of a mathematical nature involving a broad spectrum of techniques, such as linear algebra, function theory, ordinary and partial differential equations, functional analysis, differential geometry, algebraic geometry, stochastic analysis, and combinatorics. This mathematical work marks one end of a continuous spectrum that, at the opposite extreme, contains applications to such things as “regulating the amount of water in a toilet tank, controlling the flow and generation of electrical power over huge geographic regions, regulating the behavior of gasoline engines, controlling the thickness of rolled products as varied as paper and sheet steel, and hundreds of controllers hidden in consumer products of all kinds,” as the editor of The Control Handbook explains in his preface. Practitioners of the art and science of control theory work mostly in engineering departments, although in recent years there have been signs that the mathematical community may be starting to appreciate the beauty of the subject and its potential as an area for active involvement by mathematicians. The size of the field can be measured by the fact that the membership of the Control Systems Society—one of the professional societies within the Institute of Electrical and Electronic Engineers—is about half that of the entire American Mathematical Society. The depth and importance of control theory, both as a scientific field and as a key area of technology, is sometimes underappreciated by the general public, to whom computers with blinking lights often appear to be much more real than abstract scientific theories.

Mathematicians sometimes find it difficult to communicate with control engineers and to understand the problems that their engineering colleagues are trying to solve, because control questions are often design problems, in which one isn’t asking whether some statement is true or false, but how something should be done.

Here is a simple example: suppose you have a system governed by a second-order differential equation ẍ+cẋ+kx = u, where u is an external force or “control” that can be chosen by the controlling agent. Suppose you can measure position but not velocity. (That is, you have direct access to x but not to ẋ.) Suppose your goal is to control the system by choosing u so that all the trajectories of the resulting “controlled dynamics” will go to zero as t → +∞, but you don’t know the values (or even the signs!) of the real numbers c and k. Can the goal be attained? And if so how? An engineer, trying to explain this to a mathematician, might say something like “I want to choose u, depending on x but not on ẋ, c, or k, in such a way that, no matter what the values of c and k are, the solution t → x(t) will go to zero as t → +∞, for all possible initial conditions.” The mathematician, after some thinking, will probably answer as follows: “Theorem: it cannot be done. Proof: If a function x → u(x) has the properties you want, then you should in particular be able to choose c = k = 0. In that case, making the reasonable assumption that u is continuous, and letting U(x) = ∫_{0}^{x} u(s)ds, it is easy to verify that the function V (x, ẋ) = \frac{ẋ^{2}}{2} −U(x) is constant along all trajectories, so if you start with an initial condition x(0) = α, ẋ(0) = β, such that V (α, β) ≠ 0, then the solution cannot go to zero.” This, however, will not satisfy the engineer, who will most certainly reply: “I never said that u had to be a function of x in the usual mathematical sense. It’s true that I want u to ‘depend on x,’ but I see no reason for not allowing u at a time t to depend on past values of x as well as current values. So, if we end up choosing u at time t as a functional of the whole past of x up to time t, rather than as a function of the value of x at time t, that would still be acceptable.” In fact, it turns out that our engineer’s intuition was right and our mathematician’s impossibility proof is wrong: one can construct an “adaptive controller” that does the job, as long as one allows a choice of u that “depends on x only” in the more general sense proposed by the engineer. (Rather than show how this can be done, we prefer to let the mathematical readers figure it out by themselves, so they can see if they find it easy or hard and develop some appreciation for this kind of question. The following hint will help. What has to be done is to start with a formula for u as a function of c, k, x, and ẋ that would result in the desired behavior. For example, we could take u = (c − 1)ẋ + (k − 1)x, which produces the equation ẍ + ẋ + x = 0, all whose solutions obviously go to zero. Then, instead of just taking u to be a function of x and the true values of c, k, and ẋ, which are not known to us, we take u to be a function of x, y, ĉ, and k̂, where y, ĉ, and k̂ are estimated values of ẋ, ĉ, and k̂. This should be supplemented with a set of equations governing the time evolution of y, ĉ, and k̂, chosen so that the estimated values converge to the true values.)

The point of the above example is twofold. First, to show that even simplelooking control problems can be interesting, and second, to show that the mathematical questions that control engineers ask are often not exactly of the “prove or disprove” kind—with a precise mathematical formulation already given to them—that mathematicians normally expect to be asked. From these two facts it follows that there is a need for good books that explain the questions and results of control theory in a way that mathematicians can understand while doing justice to the engineering side and giving many examples of real applications.

The Control Handbook is an ambitious and spectacularly successful effort to respond to this need. It is a most welcome addition to the literature in the field, which should be especially interesting to mathematicians who want to understand control theory in its own terms and to be shown a broad perspective of the mathematical content of the theory and of its numerous applications. In an ideal world, this should mean most, or perhaps even all, mathematicians.

Its monumental size should not deter anyone from taking a look. The book has 1548 very large two-column pages and contains 121 articles. So this is a not a text one would read from cover to cover. But for those who are curious and willing to browse and explore, there is plenty here to satisfy all tastes. Most people who are “specialists in control theory” are really specialists in one rather narrow subfield, and the articles in other subfields will certainly contain new insights for them. As a reviewer, I have to confess that I have not read the whole book, cover to cover, but I have browsed and read several articles, both in areas close to my own interests and on subjects far removed from them. The articles are carefully written and make enjoyable reading, a rather remarkable and unusual fact for collective works of this kind.

The book is divided into three parts devoted, respectively, to the fundamentals of control, advanced methods, and applications.

Part A contains articles on mathematical foundations (ordinary differential and difference equations, Fourier and Laplace transforms, linear algebra, and complex variables). This is followed by articles on models for dynamical systems (inputoutput, state space, graphical models, determining models), articles on continuoustime systems (stability tests such as Routh– Hurwitz and Nyquist, Nyquist and Bode plots, root locus, PID control, pole placement), digital control, analysis and design for nonlinear systems, and software.

Part B deals with more advanced methods of control and contains articles on MIMO (multi-input multi-output) linear systems (both time-varying and timeinvariant), Kalman filter and observers, linear quadratic control, H_{∞} control, adaptive control, nonlinear control, system identification (a detailed article by L. Ljung), stochastic control, and control of distributed parameter systems. To give an idea of the thoroughness of the coverage of the various topics, let me use my own area, nonlinear control, as an example: the book has chapters by V. Jurdjevic on Lie brackets; by H. Khalil on two-time-scale and averaging and on Lyapunov stability; by F. Lamnabhi–Lagarrigue on Volterra and Fliess series; by A. Teel, T. Georgiou, L. Praly, and E. Sontag on input-output stability; by A. Isidori on feedback linearization; by Isidori and C. Byrnes on zero dynamics; by Isidori on nonlinear output regulation; by R. Freeman and P. Kokotovic on Lyapunov design; by R. DeCarlo, S. Zak, and S. Drakunov on sliding modes; by E. Abed, H. Wang, and A. Tesi on control of bifurcations and chaos; by J. Baillieul and B. Lehman on oscillatory inputs; by M. Krstic and Kokotovic on adaptive nonlinear control; by K. Passino on intelligent control; by Passino and S. Yurkovich on fuzzy control; and by J. Farrel on neural control.

Part C, on applications, is the most removed from my own research work, and it is perhaps for that reason that I found it most enjoyable. One would hardly expect toilet tanks to be an exciting reading topic, but the article by B. Coury on “Water level controls for the toilet tank: A historical perspective,” turns out to be precisely that. For example, the author explains that “the Greeks and Romans had developed the types of technologies needed for the operation of modern toilets, but the widespread adoption of that technology did not occur for another 1800 years. One wonders, then, why a control technology so obviously useful took centuries to be adopted.” His discussion of possible answers to this question is entertaining and informative. I know that from now on my courses on introduction to control theory for mathematicians are going to start with the toilet tank example. The process control section has several other examples that will be of particular interest to those looking for “real life” applications: temperature control in large buildings, control of pH, control of pulp and paper making process, and an 11-author 22page article on “Control for advanced semiconductor device manufacturing: A case study.” Then there are articles on automotive control systems, aerospace controls, control of robots and manipulators, power electronic control, control of power plants, and control of power transmission.

Every reviewer feels obliged to make at least one critical comment, so I shall conclude with my own: I was disappointed to see how little represented optimal control theory is in The Control Handbook. For a mathematician, optimal control is a venerable part of control theory, both because of its links with the calculus of variations and because of the tremendous amount of mathematical power needed to analyze optimal control problems. This is so to such an extent that many mathematicians automatically think of “control” as meaning “optimal control,” and are not aware of the fact that optimal control is merely one of many areas within the vast field of control.

In view of the disproportionate importance attached to optimal control by mathematicians, a book like this one should be of great help to us, by giving us a healthier and more realistic perspective of how what matters to us is related to what the rest of the world truly cares for. Yet, I still believe there is more that is worth knowing about optimal control than one would gather from looking at the contents of this magnificent book. So, perhaps, my critical argument should be turned around and be read, constructively, as a call for a sequel— The Control Handbook II?—together with the wish that Part II be planned and edited with the same care, good taste, attention to detail, and breadth of scientific vision as Part I was.

HÉCTOR J. SUSSMANN

Rutgers University

Iterative Methods for Solving Linear Systems. By Anne Greenbaum. SIAM, Philadelphia, PA, 1997. $41.00. xiv+220 pp., softcover. Frontiers in Applied Mathematics. Vol. 17. ISBN 0-89871-396-X.

For a long time the scientific community was well served by Varga’s book on iterative methods, published in 1961, and later by Young’s book. Emphasis in that time was on successive overrelaxation methods. In the 1970s the attention shifted to the initially less popular Krylov subspace methods. These methods, of which conjugate gradients is the archetype, had an unlucky start mainly because their behavior in finite precision arithmetic differed significantly from the expected behavior in exact arithmetic. In combination with preconditioning techniques the Krylov subspace methods became rapidly more popular and at the moment they are the methods of choice for large classes of problems.

In the “old” days, the user of iterative methods for the solution of a particular class of problems, mostly discretized PDEsystems, had to worry most about the choice of a relaxation parameter. Nowadays, the situation is much more complicated since there is a huge variety of Krylov subspace methods and it needs an expert only to explain what all the different acronyms stand for: CG, Bi-CG, QMR, Bi-CGSTAB, GENCG, FOM, CGS, TFQMR, and the list can be made almost arbitrarily long. If these methods all behaved more or less the same and the differences mattered only for fine-tuning, then the user would not have a big problem. On the contrary, however, where many methods fail for one particular problem a few other methods may do an excellent job and vice versa. Therefore, the choice for a method has to be based on considerable understanding of the underlying theory and on experimentally obtained insight.

After this rather long sketch of the situation at the frontiers of iterative methods for linear systems, we can discuss the merits of Greenbaum’s text. Although there are still many open problems regarding those iterative methods, there seems to be a wind silence in their further development, although we have seen quite a few books published recently. In 1994 some of the most representative and successful methods were collected in the so-called Templates, published also by SIAM, and these methods are now part of Matlab as well. The presentation in Templates was kept intentionally as simple as possible, sacrificing depth and essential understanding of the background of these methods. Anne Greenbaum has focused mainly on these methods, as is emphasized in the introduction: This book could serve as a supplement to the Templates documentation, providing a deeper look at the theory behind these algorithms. Of course, this made me very eager to read this text and it has been a breathtaking experience. This is really how a text on a subject with so many experimental side-aspects should look. All the required theory is there, very well represented, and always put clearly in perspective. The why and how of theory and heuristic arguments is nicely explained, which motivated me to read many of the proofs of theorems, although I most often skip proofs at first reading of a text. The theorems and proofs in the presentation of Anne Greenbaum are almost logical parts of the text, rather than slackening technical interruptions.

The author gives much room for detailed discussions on the expected and observed behavior of the different methods and to that end she helps the reader by discussing at some length the experimental results for two well-chosen model problems. She also shares her experience and insight in the selection problem. Much attention is given to the interpretation of residual reduction histories, especially in view of finite precision arithmetic effects. In this area the author has built a reputation. In principle, all Krylov subspace methods are based on orthogonality concepts, and in order to reduce work, clever tricks are employed to avoid the necessity to orthogonalize complete sets of vectors. In Chapter 4, Anne Greenbaum discusses the effects of finite precision arithmetic, which may lead to a severe loss of orthogonality among the basis vectors of the search space. Since most of the methods assume exact orthogonality, it is kind of a surprise that even a severe loss of orthogonality does not necessarily lead to catastrophes. Leaning on Paige’s breakthrough result on the Lanczos algorithm, Greenbaum shows how observations for the related Krylov solvers may be interpreted.

Another aspect that receives much attention is how the different methods are mathematically interrelated. Some elegant recent results that express the residual norm for some methods in terms of the residual norms of other methods are given and these results are nicely interpreted. Guided by an experienced interpretation we learn why some methods lead to almost identical results at some points in their convergence history, almost irrespective of what happened before these points. The presentation of this aspect reaches its climax in Section 5.7: Which Method Should I Use? This section is refreshing in its simplicity in describing what we know and how we can learn from what we have seen so far, without obscuring the fact that there are still so many open problems and that the answer to the question in the section name is largely unknown. I recommend that everyone who is interested in the usage of iterative methods takes knowledge from that section.

The main text on iterative methods is contained in the first half of the book. The second half gives a rather condensed overview on preconditioning techniques including domain decomposition concepts and multigrid. In this part the author clearly restricts herself to some basic concepts and she presents the most relevant theoretical results. This part may serve as an introduction to the field rather than a more or less complete overview. Books published by Axelsson and by Saad go much deeper into the details and possibilities of this minefield.

Of course, there are also points of criticism, almost unavoidable for a text with this modest length. An important aspect of Krylov subspace methods is their superlinear convegence behavior, often observed if the (preconditioned) matrix of the system has relatively isolated eigenvalues. This aspect gets almost no attention. Instead, some attention is given to the, in my view, less relevant situation where the eigenvalues are clustered in separated intervals. Greenbaum discusses the negative effects of large intermediate residual norms on the final solution, but she has not included an elegant solution to this, which was first suggested by Neumaier in 1994 and later worked out in more detail by Sleijpen and myself. Then she shares some of the misunderstanding with respect to a method that should have been more popular: SYMMLQ. SYMMLQ is often regarded as a variant on the Lanczos method in which possible singularity of the reduced system is cured by the LQ algorithm rather than the more standard LU splitting (as in CG and Bi-CG). On page 46 it is stated that SYMMLQ will not be further analyzed because it has no nice mimimization property. But it has: it minimizes the error! A similar lack of understanding is apparant from the Templates text, for which I may blame myself.

A very annoying typo is the reference to Hackbusch [78] on page 94, which should refer to the excellent survey by Gutknecht [74].

In summary, this is a book that anyone working in this field should have. It is also an excellent textbook for classes on this subject and almost ideal in combination with the Templates text and modern Matlab facilities.

HENK VAN DER VORST

University Utrecht

SELECTED COLLECTIONS

Applications of Computational Algebraic Geometry. Edited by David A. Cox and Bernd Sturmfels. AMS, Providence, RI, 1997. $35.00. ix+172 pp., hardcover. Proceedings of Symposia in Applied Mathematics. Vol. 53. ISBN 0-8218-0750-1.

The seven lectures in this volume were given as a short course in San Diego in 1997. They were written to bring the ideas of computational algebraic geometry to a wide audience of mathematicians. The topics include Gröbner bases, methods for solving polynomial systems, and applications to computer aided geometric design, information systems, integer programming, and coding theory.

Finite Element Methods. Superconvergence, Post-processing, and A Posteriori Estimates. Edited by M. Křížek, P. Neittaanmäki, and R. Stenberg. Marcel Dekker, New York, 1998. $165.00. xii+348 pp., softcover. Lecture Notes in Pure and Applied Mathematics. Volume 198. ISBN 0-8247-0148-8.

Superconvergence refers to the fact that a finite element approximation is often more accurate at certain points of an element. This increased accuracy is important in developing local a posteriori error estimates and mesh refinement strategies. The 23 reviewed papers in this volume come from a conference at the University of Jyväskylä, Finland. There is also a bibliographic paper listing and categorizing 600 papers on superconvergence.

Genetic Algorithms in Engineering Systems. Edited by A. M. S. Zalzala and P. J. Flemming. Institute of Electrical Engineers, London, UK, 1997. $79.00. xiii+263 pp., hardcover. Control Engineering series 55. ISBN 0-85296-902-3.

Genetic algorithms are stochastic search and optimization methods based on metaphors of natural biological evolution. This book contains 10 papers on genetic algorithms and applications to engineering problems. The applications include learning behaviors of neural nets, chaotic system identification, job shop scheduling, motion planning for robotic systems, optimal wing shape design, and optimal VLSI layout. In the paper on optimal wing shapes, comparisons are reported with gradient-based search methods, simulated annealing, and a genetic algorithm, the latter proving the best for the problem at hand.

Modern Software Tools for Scientific Computing. Edited by Erlend Arge, Are Magnus Bruaset, and Hans Petter Langtangen. Birkhäuser, Boston, MA, 1997. $69.95. 380 pp., hardcover. ISBN 0-8176-3974-8.

The 17 articles in this interesting volume are divided into three parts. First come 11 articles on computational differential equations. Object-oriented languages and problem solving environment tools are discussed in a variety of situations, including various applications (finance, convection dominated flow, the electrochemical industry) and various development situations (parallelism, the management of numerical software libraries). The second part consists of two articles on computational geometry. Finally comes four articles on software development, which includes articles on debugging and on the conversion of Fortran 77 code to object-oriented code.

Monte Carlo and Quasi-Monte Carlo Methods 1996. Edited by Harald Niederreiter, Peter Hellekalek, Gerhard Larcher, and Peter Zinterhof. Springer-Verlag, New York, 1998. $49.95. xii+448 pp., softcover. Lecture Notes in Statistics 127. ISBN 0-387-98335-X.

The papers in this volume come from a 1996 conference on Monte Carlo methods in scientific computing. The topics range from theoretical issues in Monte Carlo and simulation to applications to quadrature, integral equations, binary search, optimization, mathematical finance, computer graphics, etc.

Network Design: Connectivity and Facilities Location. Edited by Panos M. Pardalos and Dingzhu Du. AMS, Providence, RI, 1998. $79.00. xiii+461 pp., hardcover. Series in Discrete Mathematics and Theoretical Computer Science. Volume 40. ISBN 0-8218-0834-6.

Connectivity and facilities location arise in network design, with applications in data communications, transportation, production planning, and VLSI design. The subject involves combinatorial design and combinatorial optimization problems for which no polynomial time algorithms are known. The 26 papers in this volume are concerned with combinatorial, algorithmic, and applicational aspects of these problems.

Numerical Methods in Finance. Edited by L. C. G. Rogers and D. Talay. Cambridge University Press, Cambridge, UK, 1997. $54.95. x+326 pp., hardcover. ISBN 0-521-57354-8.

This volume contains 16 articles on mathematical finance, written by a mixture of economists, probabilists, and applied mathematicians. Some of the topics are computation of option prices; numerical solution of portfolio management strategies; Monte Carlo simulations; viscosity solutions of partial differential equations related to portfolio management; and backward stochastic differential equations in finance. It is a good volume for someone who wants to become acquainted with the variety of techniques in this important subject.

The Practise of Data Analysis, Essays in Honor of John Tukey. Edited by D. R. Brillinger, L. T. Fernholz, and S. Morgenthaler. Princeton University Press, Princeton, NJ, 1997. $49.50. viii+337 pp., hardcover. ISBN 0-691-05782-6.

This volume contains a collection of essays from a meeting on the 80th birthday of John Tukey. There is also a short bibliography of Tukey’s publications and some interesting biographical contributions. Many of the contributions in this volume have a broad interest for the scientific enterprise.

Topics in Semidefinite and Interior-Point Methods. Edited by Panos M. Pardalos and Henry Wolkowicz. AMS, Providence, RI, 1998. $69.00. xx+250 pp., hardcover. Fields Institute Communications. ISBN 0-8218-0825-7.

This volume is concerned with semidefinite programming, by which is meant problems of the following form: find a semidefinite symmetric matrix X which minimizes the trace of CX, subject to the equality constraints that the trace of A_{j}X is specified, where C and A_{j}, j = 1, …, m are given semidefinite symmetric matrices. Problems of this type arise in control theory, VLSI design, statistics, and other areas. The book contains 16 papers on theory, applications, and algorithms for semidefinite programming. There is also a list of 250 references to applications and theory.

Topological Nonlinear Analysis II. Degree, Singularity, and Variations. Edited by Michele Matzeu and Alfonso Vignoli. Birkhäuser, Boston, MA, 1997. $120.00. vi+601 pp., hardcover. ISBN 0-8176-3886-5.

This volume is a continuation of a similarly titled volume published in 1995 and is intendend to survey the progress in topological methods in nonlinear analysis over the last three years. The nine articles are Classical Solutions for a Perturbed N–Body System, by G. Dell’Antonio; Degree Theory Old and New, by H. Breziz; Global Structure for Nonlinear Operators in Differential and Integral Equations I, Folds, and Global Structure for Nonlinear Operators in Differential and Integral Equations II, Cusps, by P. T. Church and J. G. Timourian; Degree for Gradient Equivariant Maps and Equivariant Conley Index, by K. Geba; Variations and Irregularities, by U. Mosco; Singularity Theory and Bifurcation Phenomena, by B. Ruf; Bifurcation from the Essential Spectrum, by C. A. Stuart; and Rotation of Vector Fields, Definition, Basic Properties, and Calculation, by P. P. Zabrejko.

LATER EDITIONS

Green’s Functions and Boundary Value Problems, Second Edition. By Ivar Stakgold. John Wiley & Sons, New York, 1998. $69.95. xvii+689 pp., hardcover. ISBN 0-471-61022-4.

Problems and Theorems in Analysis I, Classics in Mathematics. Reprint of the 1978 Edition. By George Pólya and Gabor Szegö. Springer-Verlag, Berlin, 1998. $39.95. xix+389 pp., softcover. ISBN 3-540-63640-4.

Problems and Theorems in Analysis II, Classics in Mathematics. Reprint of the 1978 Edition. By George Pólya and Gabor Szegö. Springer-Verlag, Berlin, 1998. $39.95. xi+392 pp., softcover. ISBN 3-540-63686-2.

Theoretical Fluid Dynamics, Second Edition. By Bhimsen K. Shivamoggi. John Wiley & Sons, New York, 1997. $69.95. viii+553 pp., hardcover. ISBN 0-471-05659-6.