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SIAM Review Vol. 56, Issue 3 (September 2014) Book Reviews Introduction, 547 Featured Review: Spectral and Dynamical Stability of Nonlinear Waves (Todd Kapitula and Keith Promislow), Arnd Scheel, 549 Mathematical Foundations of Imaging, Tomography and Wavefield Inversion (Anthony J. Devaney), Brett Borden, 553

A Primer on Mapping Class Groups (Benson Farb and Dan Margalit), Danny Calegari, 554

Fundamentals of Object Tracking (Subhash Challa, Mark R. Morelande, Darko Mušicki, and Robin J. Evans), Chee-Yee Chong, 557

Introduction to Partial Differential Equations (Peter J. Olver), Bernard Deconinck, 559

Classical and Multilinear Harmonic Analysis. Vols. I and II (Camil Muscalu and Wilhelm Schlag), Gerald B. Folland, 561

Mathematics of Two-Dimensional Turbulence (Sergei Kuksin and Armen Shirikyan), Eleftherios Gkioulekas, 561

Mathematical Physics: A Modern Introduction to Its Foundations. Second Edition (Sadri Hassani), Eleftherios Kirkinis, 565

Numerical Methods in Finance with C++ (Maciej J. Capiński and Tomasz Zastawniak), Kjell P. Konis, 567

Analytic Perturbation Theory and Its Applications (Konstantin E. Avrachenkov, Jerzy A. Filar, and Phil G. Howlett), Robert E. O’Malley, Jr., 568

Beautiful Geometry (E. Maor and E. Jost), Robert E. O’Malley, Jr., 569

Five Decades of Tackling Models for Stiff Fluid Dynamics Problems: A Scientific Autobiography (R. Zh. Zeytounian), Robert E. O’Malley, Jr., 569

Introduction to the Network Approximation Method for Materials Modeling (L. Berlyand, A. G. Kolpakov, and A. Novikov), Grigory Panasenko, 569

Number Theory. A Historical Approach (John J. Watkins), John Stillwell, 571

Numerical Methods with Worked Examples: MATLAB Edition. Second Edition (C. Woodford and C. Phillips), Chris Vogl, 572

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Traveling wave solutions to PDEs, especially spectral and dynamic stability of non-linear waves, is the focus of a new Springer book by Kapitula and Promislow. ArndScheel knowingly writes about it in our featured review. Other reviews provide areader’s potpourri of topics ranging from mapping class groups to texts on PDEs, math-ematical physics, numerical methods in finance, and numerical methods in general. Youcan also learn about object tracking, 2-D turbulence, and analytic perturbation theory.Our reviewers again provide their valuable recommendations to help you select whatto read and consult.

Bob O’MalleySection Editor

[email protected]

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SIAM REVIEW c! 2014 Society for Industrial and Applied MathematicsVol. 56, No. 3, pp. 549–574

Book Reviews

Edited by Robert E. O’Malley, Jr.

Featured Review: Spectral and Dynamical Stability of Nonlinear Waves. By ToddKapitula and Keith Promislow. Springer, New York, 2013. $79.95. xiv+361 pp., hardcover.ISBN 978-1-4614-6994-0.

Undergraduate courses on partial di!erential equations (PDEs) rarely venture be-yond the idea of inherently linear eigenfunction expansions or separation of variablestechniques. A glimpse into the world of nonlinear PDEs is sometimes o!ered whenstudying special solutions, either explicit or of a particular form. Standing or travel-ing waves, such as pulses or fronts, can indeed be constructed in many examples usingquite elementary techniques. Beyond direct numerical simulations, looking at simpleone-dimensional setups and focusing on simple solutions of the form u(x! ct) is oftenthe only route to gain insight into the temporal dynamics. The traveling wave ansatzis commonly known as the “ODE reduction,” since it reduces the PDE to an ordinarydi!erential equation (ODE) for which one then looks for bounded solutions, in particu-lar, periodic, homoclinic, or heteroclinic orbits. The method is particularly successfulif one can reduce the ODE to a planar system, where the Poincare–Bendixson theo-rem helps to classify bounded orbits. Famous examples of such special traveling-wavesolutions arise as solitary waves in dispersive equations, such as water wave problemsor equations in nonlinear optics, or as pulses and fronts in reaction-di!usion-typesystems. From the perspective of this book, one might mention solitary waves

u(t, x) =c

2sech2

!"c

2(x ! ct)

"

in the KdV equation

!tu = !!xxxu! 6u!xu

and excitation pulses (u, v)(x ! c"t), c" ="2(12 ! a) + O(") > 0, in the FitzHugh–

Nagumo system

!tu = !xxu+ u(1! u)(u! a)! v,

!tv = "(u! #v).

The first and quite relevant objection to the traveling-wave reduction approach is thatsuch solutions correspond to very particular initial conditions, so they are almost neverobserved in experiments or simulations. The present book can be viewed in many waysas the natural response to this objection. In short, traveling waves are relevant andinformative when open sets of initial conditions stay close to a traveling wave for long

Publishers are invited to send books for review to Book Reviews Editor, SIAM, 3600 Market St.,6th Floor, Philadelphia, PA 19104-2688.

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550 BOOK REVIEWS

time intervals. Such information can be obtained by studying the linearization at atraveling-wave solution, in particular spectral properties. This book by Kapitula andPromislow provides a quite unique entry point into this area, suitable for graduatestudents and young researchers who are interested in entering the field.

The question of the relevance of traveling waves has a counterpart: why shouldone be at all surprised to see traveling waves, or, more generally, coherent behav-ior? In damped-driven systems, this question can be rephrased from a higher vantagepoint as a search for the emergence of self-organized behavior in far-from-equilibriumsystems. Striking examples of such self-organized behavior are, for instance, spiralwaves in chemical reactions. In conservative, Hamiltonian systems, Poincare’s recur-rence theorem excludes asymptotic stability. Linearized or spectral stability still givegood indications of stability when one can establish Nekhoroshev-type estimates, thusguaranteeing that solutions stay close to the equilibrium for exponentially long timespans. Stronger stability results exploit energetic stability or constrained stability.A di!erent route, not taken here, would rely on dispersive estimates to show evenstronger, asymptotic stability [10].

Many of the ideas around spectral, linear, and nonlinear stability became accessi-ble to a larger audience through Dan Henry’s lecture notes [8]. Inspired by dynamicalsystems ideas, spectral properties are viewed as universal criteria, independent ofcoordinate changes, when the goal is to characterize dynamics in the vicinity of equi-libria. Henry’s book is also one of the first references to consider characterizations ofspectral properties of the linearization of traveling waves and, using center manifolds,the nonlinear stability of traveling waves. In fact, both spectral properties and non-linear stability are complicated by the fact that traveling-wave problems are typicallyposed on the whole real line. The loss of compactness is reflected in the presence ofessential spectra that need to be controlled and understood. The translation symme-try on the real line causes the presence of a neutral eigenvalue at the origin, so thatthe linear evolution does not cause exponential decay, but merely exponential decayto a constant.

The authors discuss both of these aspects carefully in Chapters 3 and 4 aftersetting up some basic preliminaries and notation from functional analysis in Chapter2, which also includes a nice review of Sturm–Liouville theory. Chapter 3 contains,in particular, a characterization of essential spectra for operators with asymptoticallyconstant coe"cients (homoclinic and heteroclinic traveling waves) and for operatorswith periodic coe"cients (periodic wave trains). It also explains in an informal waythe e!ect of domain truncation, following [11], a phenomenon that is revisited inChapter 10.

Chapter 5 covers orbital stability in Hamiltonian PDEs, motivated largely by thework of Grillakis, Shatah, and Strauss [6, 7]. In the absence of e!ective dissipationmechanisms, one can strive to show stability using the conservation of the Hamil-tonian. In fact, if an equilibrium is a strict minimum of the Hamiltonian, nearbytrajectories necessarily stay in a neighborhood, which roughly establishes Lyapunovstability of the equilibrium. In the case of traveling waves for Hamiltonian PDEs,equilibria come in families due to translation symmetry. Also, equilibria are typicallynot minimizers of the energy. On the other hand, translation symmetry correspondsto another conserved quantity by Noether’s theorem, the momentum. It is thereforesu"cient to show that the Hamiltonian possesses a strict minimum when restrictedto a level set of the momentum. Such ideas can lead to proofs of stability of travelingwaves in some contexts and, more generally, to counting arguments for unstable eigen-values of the linearization. Both aspects are carefully covered in Chapters 5 and 7,

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BOOK REVIEWS 551

although the most general result, the Hamiltonian Krein index theorem, is stated butnot proved in full generality. Chapter 7 also contains a very interesting collection ofperturbation results, covering, for instance, symmetry-breaking e!ects and dissipativeperturbations. Chapter 6 develops classical perturbation theory for eigenvalues in thepoint spectrum, a necessary prerequisite for various results in subsequent chapters.

Chapters 8 through 10 are concerned with the Evans function, which was intro-duced in the 1970s in a series of articles to study the stability of pulses in nerve axons[2, 3, 4, 5]. It gained traction later when it was used to prove stability of the FitzHugh–Nagumo pulse [9] and the KdV soliton [10]. The paper by Alexander, Gardner, andJones [1] today is viewed as the basic (and first) reference for properties of the Evansfunction. While the Evans function is widely used, in hyperbolic conservation laws,nonlinear optics, water wave models, and in reaction-di!usion systems, there are fewexpository articles, let alone textbooks, from which beginning graduate students canlearn the subject. In this respect, Chapters 8 through 10 are a very valuable asset forthe community. The construction here is slightly more hands-on and explicit than inother review articles. Instead of relying on di!erential forms as in [1], the authors usespecific Jost solutions to build the Wronskian for the Evans function.

Roughly speaking, the Evans function gives a reduction of an eigenvalue problemof the form Lu = $u, with L a kth-order di!erential operator on the real line or abounded interval, to an analytic function E($), so that roots of E coincide with eigen-values of L. The construction writes the eigenvalue problem as a first-order di!erentialequation and uses the linear evolution of this di!erential equation to transport bound-ary conditions to the origin x = 0 as linear subspaces. One then tracks intersectionsof these subspaces by forming a determinant of basis vectors in these subspaces.

While a reduction of an eigenvalue problem to finding the roots of a single analyticfunction is obviously beautiful and striking, it is only the fact that this function (orat least some properties) is computable that turns the Evans function into a powerfultool in the study of stability of nonlinear waves. The authors illustrate this nicely,relying mostly on explicitly solvable scattering problems.

Another strength of the Evans function (and this book) is the possibility of track-ing eigenvalues when they disappear into the essential spectrum. The simplest exam-ple is that of small localized potentials in Schrodinger operators,

d2u

dx2+ "V (x)u = $u, V (x)e!|x| # 0 for |x| # $,

for some % > 0. The Evans function turns out to be an analytic function of # ="$

for all " %= 0 and all &# > !%. For " = 0, the Evans function E(#) = 2# possesses asimple root at the origin. A perturbation calculation gives

E($) = 2# ! "

#V +O("2).

The eigenvalue $ ' ("$V/2)2 turns into a resonance pole when "

$V < 0, with

corresponding eigenfunction u ' e|"x|. The book spends quite some time on carefullyintroducing the Riemann surface branch cuts more generally and provides us withmore explicit, integrable examples. One of the highlights of the book is Chapter 10.4,where those results are applied to a perturbation calculation for eigenvalues locatedat the edges of the essential spectrum in integrable systems.

A somewhat simpler application of the Evans function is that of parity indices.Computing the sign of the Evans function for small $ > 0 and for 0 < $ # $, one

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552 BOOK REVIEWS

can deduce the parity of the number of positive eigenvalues. This has been exploitedin numerous contexts to give instability criteria [12, section 6], but also to showstability [9]. Key to these results is the connection between the Evans function andits derivatives at $ = 0 and the geometry of stable and unstable manifolds in theconstruction of traveling waves. In the simplest case, the derivative of the Evansfunction is given by a Melnikov integral that measures the splitting distance betweenstable and unstable manifolds as the wave speed is varied as a parameter in thetraveling-wave ODE.

The authors also consider the truncation problem, when pulses or fronts are con-sidered on a large but finite interval in a comoving frame with suitable separatedboundary conditions [11]. Spectra converge exponentially as the domain size increases,and the authors give a constructive proof based on exponential convergence of Jostsolutions and Evans functions. An indirect argument is used to show a clustering re-sult for eigenvalues at the absolute spectrum. Although this result is slightly weakerthan the original convergence result in [11], it serves as an excellent stepping stonefor those interested in diving deeper into the theory.

Overall, I found this book to be an extremely valuable asset for students andyoung researchers who want to get into this field. One can find endless lists of topicsthat should have been covered in such a book. Numerous prerequisites from functionalanalysis and classical PDE theory are covered only tangentially. Similarly, results fromand connections with inverse scattering theory are only touched upon. However, itis a strength of the book that it has narrowed the scope. As is, the book does anexcellent job at stimulating readers to get their hands dirty and play around withthe examples o!ered. The book is, surprisingly, largely self-contained, and it provesmost of the key results rigorously, sometimes restricting to the simplest case. Anextensive bibliography and plenty of remarks at chapter endings then serve as a guideto history and current literature. This field has needed such a book as an entry pointfor graduate students, and the authors deserve a huge thanks from the communityfor putting it together.

REFERENCES

[1] J. Alexander, R. Gardner, and C. Jones, A topological invariant arising in the stabilityanalysis of travelling waves, J. Reine Angew. Math., 410 (1990), pp. 167–212.

[2] J. W. Evans, Nerve axon equations. I. Linear approximations, Indiana Univ. Math. J., 21(1971/1972), pp. 877–885.

[3] J. W. Evans, Nerve axon equations. II. Stability at rest, Indiana Univ. Math. J., 22(1972/1973), pp. 75–90.

[4] J. W. Evans, Nerve axon equations. III. Stability of the nerve impulse, Indiana Univ. Math.J., 22 (1972/1973), pp. 577–593.

[5] J. W. Evans, Nerve axon equations. IV. The stable and the unstable impulse, Indiana Univ.Math. J., 24 (1974/1975), pp. 1169–1190.

[6] M. Grillakis, J. Shatah, and W. Strauss, Stability theory of solitary waves in the presenceof symmetry. I, J. Funct. Anal., 74 (1987), pp. 160–197.

[7] M. Grillakis, J. Shatah, and W. Strauss, Stability theory of solitary waves in the presenceof symmetry. II, J. Funct. Anal., 94 (1990), pp. 308–348.

[8] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840,Springer, Berlin, New York, 1981.

[9] C. Jones, Stability of the travelling wave solution of the FitzHugh-Nagumo system, Trans.Amer. Math. Soc., 286 (1984), pp. 431–469.

[10] R. Pego and M. Weinstein, Eigenvalues, and instabilities of solitary waves, Philos. Trans.Roy. Soc. London Ser. A, 340 (1992), pp. 47–94.

[11] B. Sandstede and A. Scheel, Absolute and convective instabilities of waves on unboundedand large bounded domains, Phys. D, 145 (2000), pp. 233–277.

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BOOK REVIEWS 553

[12] K. Zumbrun, Multidimensional stability of planar viscous shock waves, in Advances in theTheory of Shock Waves, Progr. Nonlinear Di!erential Equations Appl. 47, BirkhauserBoston, Boston, MA, 2001, pp. 307–516.

ARND SCHEEL

University of Minnesota

Mathematical Foundations of Imaging,Tomography and Wavefield Inversion. ByAnthony J. Devaney. Cambridge University Press,Cambridge, UK, 2012. $82.00. xviii+518 pp.,hardcover. ISBN 978-0-521-11974-0.

Inverse problems and, in particular, imag-ing problems are a staple of applied mathe-matics, engineering, and physics, and manyscientists earn their bread and butter ad-dressing issues of medical imaging, sonar/radar-based remote sensing, nondestructivetesting and evaluation, etc.. Since these ap-plied problems are all grounded in wavefieldinversion methods, it might be expectedthat a common interdisciplinary text ap-propriate to all students would have ap-peared long ago. But—perhaps because ofthe considerable prerequisite training re-quired to understand the physics of radia-tion and scattering, the issues of ill-posedproblems and regularization methods asso-ciated with the solution of integral equa-tions, and the practical problems of con-structing reliable computer algorithms—ithas been hard to strike that Goldilocks bal-ance between “too much of this” and “toolittle of that.” The more common resultis a book written for a smaller audienceconsisting of mathematicians or physicistsor engineers. Tony Devaney has now pro-duced just such an interdisciplinary bookaddressed to all audiences.

Of course, this task requires both physicsand mathematics preliminaries, and De-vaney starts by introducing the fundamen-tal tools for dealing with initial and bound-ary value problems for the scalar wave andHelmholtz equations. Roughly constitutingthe first one-third of the book, he includesa review of the the Sturm–Liouville prob-lem, multipole expansions, radiation fields,dispersive media, and other preliminaryconcepts in both the time and frequencydomains (this alone could be a valuable,if succinct, reference). Throughout he also

includes an extensive discussion of Greenfunctions for the radiation problem, andthis consolidated exposition is a wellspringfor the uninitiated student. Devaney hasstaged these basic results up-front so thatthey can be referenced when needed laterin the text and, owing to the interdis-ciplinary nature of the book, the layoutmakes good sense. While so much uninter-rupted background material (four chapters)concentrated in one place might disheartena student who is anxious to “get on withimaging,” the formal feel of these chaptersis significantly lessened by the inclusion ofvery informative examples—many of whichanticipate later results. To complementthis necessarily compressed backgroundoverview, each chapter includes thought-fully crafted exercises to amplify the mainpoints of the exposition.

Chapter 5 begins the text’s subject inearnest with the inverse source problem(ISP). After developing the ISP integralequation, Devaney proceeds systematicallyto discuss solutions for a number of impor-tant sources and geometries. Along the wayhe introduces singular value decomposition(SVD) methods and discusses their imple-mentation. Mindful to discuss the conse-quences of nonideal data, this chapter pro-vides a wealth of valuable information, andI was especially glad to see the SVD-basedexposition both here and continued throughthe remainder of the book (something thatis not as common as it should be in suchtexts).

Chapter 6 is preliminary to the inversescattering problem (ISCP) and introducesscattering and di!raction. After developingthe Lippmann–Schwinger integral equationand discussing various general propertiesof the scattered field, Devaney moves on tothe various schemes for approximating so-lutions (Born, Rytov, physical optics, etc.)and the inversion techniques based on theseapproximations in Chapters 7 and 8. Here,

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554 BOOK REVIEWS

as before, his approach is to illustrate ISCPmethods by examining specific problemsinvolving data constraints and scatteringgeometries.

The final three chapters deal with some ofthe odds and ends that have been deferred tothis point so as to more expediently developthe main themes of the book. Chapters 9and 10 address the very important problemof wavefield inversion in inhomogeneousme-dia, while Chapter 11 addresses the vectorHelmholtz equation associated with electro-magnetic fields (a topic too often brushedaside in standard imaging texts). Startingwith an overview of the Maxwell equations,Devaney develops electromagnetic radiationand scattering in an amazingly concise tourde force that describes the similarities aswell as the di!erences between scalar andelectromagnetic waves.

This is the kind of remarkable and com-prehensive book that I would expect to bepenned by one of the foremost experts inthe field and Prof. Devaney is to be con-gratulated. As a reference source this bookis certain to prove its value. As a text,however, the book (in its entirety) might bea bit challenging for a one-semester intro-ductory graduate-level course o!ered to stu-dents who really do have only “rudimentaryfamiliarity with the wave and Helmholtzequations in a homogeneous medium.” Thereason, of course, is the sheer amount of ma-terial covered—the book intends to “presentthe mathematical (and physical) founda-tions of imaging,” and that is a tall order.A preliminary course in Fourier optics couldhelp ease the passage into these more ad-vanced topics. However, the book’s expo-sition allows it to be tailored to a coursewith more modest goals. Other than that,I believe the author has found a marvelousbalance between physics and mathematics,and, even though he laments that this wasaccomplished “at the expense of mathemat-ical rigor,” I think he got it just right.

BRETT BORDEN

Naval Postgraduate School

A Primer on Mapping Class Groups.By Benson Farb and Dan Margalit. Prince-ton University Press, Princeton, NJ, 2012.

$75.00. xiv+472 pp., hardcover. ISBN 978-0-691-14794-9.

The classification of surfaces (or, more for-mally, of closed two-dimensional manifolds)has been known since the 1860s and de-scribes how they are all obtained fromspheres by adding finitely many handlesor crosscaps. Restricting to oriented sur-faces gives an even simpler picture: sur-faces are classified by a single integer—thegenus—which can be any nonnegative in-teger; this integer can be computed easilyfrom any combinatorial description of thesurface (e.g., from the Euler characteris-tic !, which is related to the genus g bythe formula ! = 2 ! 2g). Thus, a closedoriented surface of genus 0 is a sphere, aclosed oriented surface of genus 1 is a torus,and so on. However, as is so often the casein mathematics, the real interest lies not somuch in the objects themselves, but in theirgroups of symmetries.

If S is a closed, oriented surface, thegroup Homeo+(S) of orientation-preservingself-homeomorphisms is extremely compli-cated as an abstract group. However, itmay be topologized in an obvious way (i.e.,with the compact-open topology) by how itacts on S, and then as a space it becomesmuch more manageable (at least up to ho-motopy). If we denote by Homeo0(S) theconnected component of Homeo+(S) con-taining the identity, then the situation is asfollows:

1. The group SO(3,R) of rotations ofEuclidean 3-space acts by isometrieson the round sphere S2 and definesan inclusion SO(3,R) " Homeo0(S

2),which is a homotopy equivalence.

2. The square torus R2/Z2 acts on itselfby translations and defines an inclu-sion R2/Z2 " Homeo0(T ) which is ahomotopy equivalence.

3. For S a surface of genus at least 2,Homeo0(S) is contractible.

One can already see from this a fundamen-tal distinction in the study of surfaces: thecase of genus 0 and 1 is exceptional, whereasall surfaces of genus at least 2 exhibit simi-lar phenomena (for surfaces with puncturesor boundary components there is a similardistinction, but in this case what is im-

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BOOK REVIEWS 555

portant is not the genus, but the sign ofthe Euler characteristic). Because the ac-tion of Homeo+(S) on itself is continuous,the group Homeo0(S) is a normal subgroupof Homeo+(S), and the quotient Mod(S) iscalled themapping class group. Thus—for asurface S of genus at least 2—understandingthe group Homeo+(S) up to homotopy istantamount to understanding the mappingclass group of S.

Surfaces and their symmetries are ubiqui-tous throughout geometry and, even morebroadly, throughout mathematics. In thefirst place, surfaces arise as Riemann sur-faces and therefore can be found whereverone finds the complex numbers (which is tosay, everywhere). Moreover, surfaces fre-quently arise in families, and the globalstudy of these families is governed by map-ping class groups. For example, a meromor-phic function on a complex surface (whichis topologically four-dimensional) has fiberswhich are (possibly singular) Riemann sur-faces, and the monodromy of the nonsin-gular fibers is a subgroup of a mappingclass group. More generally, any symplec-tic 4-manifold admits the structure of aLefschetz pencil, whose nonsingular fibersare surfaces, and these can also be studiedbymapping class groups. Many 3-manifoldshave the structure of a surface bundle over acircle, so that they are described exactly upto homeomorphism by giving a single con-jugacy class in some mapping class group.The dynamics of flows on 3-manifolds is cap-tured by return maps to a two-dimensionalcross-section, and many important featuresof flows are encoded in the relative braidingof orbits around each other; such braiding isparameterized by elements of braid groups,which are mapping class groups for certainpunctured surfaces. So the importance ofthe theory of mapping class groups to manydi!erent mathematical constituencies is be-yond question. What has been sorely miss-ing is a clear, readable introduction to thesubject that lays out the basic facts with aminimum of fuss and with an eye to theirapplications—in short, a primer.

The view of surfaces as geometric objectshas a lot to recommend it, and one of thecanonical routes to the understanding ofmapping class groups is through the studyof geometric structures on surfaces, espe-

cially holomorphic structures, and Rieman-nian metrics. Every surface admits a canon-ical smooth structure, and every smoothmanifold admits a Riemannian metric. Foran oriented surface, any conformal classof Riemannian metric determines a uniquecompatible holomorphic structure (i.e., arealization as a Riemann surface) and, fur-thermore, every conformal class of metriccontains a unique representative of constantcurvature !1, 0, or 1 up to the ambiguity ofscale in the case of curvature 0; this is moreor less a restatement of the Riemann map-ping theorem. The sign of the curvature isthe same as the sign of the Euler charac-teristic !, and the distinction between phe-nomena associated with surfaces of genus atleast 2 and surfaces of genus 0 or 1 can beexplained by geometry. Thus one is quicklyled to study the moduli space of metrics onS of constant curvature !1 (at least whenthe genus of S is bigger than 1). Here oneobtains one of the most fruitful interpreta-tions of mapping class groups: as (orbifold)fundamental groups of moduli spaces.

The geometry and topology of modulispaces and their compactifications is verybeautiful, but also very complicated, so it isnatural to look for simpler spaces that cap-ture some of the information, but are easierto study directly. Associated to a Riemannsurface in a natural way (taking the quo-tient of H1(S;R)! by the period lattice)is a certain kind of high-dimensional toruscalled a Jacobian. Families of Riemann sur-faces give rise to families of Jacobians, andJacobians are examples of what are calledprincipally polarized abelian varieties, whichhave their own moduli space; this is mucheasier to understand than the moduli spaceof a surface. For example, the moduli spaceof principally polarized abelian varieties has(orbifold) fundamental group equal to thesymplectic group Sp(2g;Z), where g is thegenus, so we obtain a natural “forgetful”map Mod(S) " Sp(2g;Z). This map be-tween groups may be seen more easily justby thinking about the action of a mappingclass on the one-dimensional cohomologyH1(S;Z), which is naturally made into alattice in a symplectic vector space by thecup product. Thus we can achieve insightinto the algebraic structure of the mappingclass group by studying both the symplec-

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556 BOOK REVIEWS

tic group Sp(2g;Z) and the kernel of themap Mod(S) " Sp(2g;Z), which is calledthe Torelli group Ig . This latter group isvery mysterious, but some of its structureis revealed by studying its homomorphismsto abelian groups, following the approachof Dennis Johnson.

Finally, a concern somewhat complemen-tary to the study of mapping class groupsthemselves is the problem of finding a rep-resentative self-homeomorphism that is thebest avatar of its mapping class or hasproperties that make it especially usefulfor applications or computation. Here onehas the Nielsen–Thurston classification the-orem, which says that each mapping classhas a representative " which is of one of thefollowing types:

1. " is finite order—i.e., "m = id for somepositive m;

2. " is reducible—i.e., it leaves invarianta finite collection of pairwise disjointessential simple closed curves in S; or

3. " is pseudo-Anosov.

Finite-order automorphisms are very easyto understand. Reducible automorphismsrestrict to automorphisms of simpler sur-faces with boundary (obtained by cut-ting along the invariant system of curves).Therefore, the interesting automorphismsare those which have pseudo-Anosov rep-resentatives. The simplest way to define apseudo-Anosov homeomorphism is to saythat in local charts it looks like an a"nemap of the plane which stretches one axisby some factor # > 1 and the other axis bya complementary factor 1/#, except thatthere are finitely many points where the dy-namics looks more like a “branched cover”of such an a"ne map, of degree n/2 forsome integer n # 3. A more rigorous defi-nition is to say that there is a holomorphicquadratic di!erential $ on the underlyingRiemann surface that defines singular foli-ations tangent to the directions in which asquare root

$$ is real or imaginary, and

that " stretches the leaves of the real folia-tion by # and the leaves of the imaginary fo-liation by 1/# (the singularities correspondto the zeros of $). Pseudo-Anosov repre-sentatives are very nice to work with formany reasons; for instance, they admit niceMarkov partitions, and they have the sim-

plest dynamics (as measured by entropy) ofany representative of their mapping class.

The book of Farb and Margalit does anawesome job of covering all these di!erentperspectives (and much more) and explain-ing the basic phenomena, examples, and ap-plications clearly, unambiguously, and ac-cessibly. The book is divided into threesections; we briefly discuss the contents ofeach section.

The first section is concerned with thealgebraic structure of mapping class groupsand their relation to the combinatorialtopology of two-dimensional surfaces. Themost important results in this sectionare the Dehn–Nielsen–Baer theorem thatMod(S) is isomorphic to the outer auto-morphism group of %1(S) (up to Z/2Zparameterizing orientations) and the factthat Mod(S) is finitely presented. Oneof the highlights here is that the authorsgive several di!erent finite (and infinite!)presentations and discuss the virtues andshortcomings of each one. This section alsodescribes the relationship between Mod(S)and Sp(2g;Z), and it contains a discus-sion of the Torelli group and the Johnsonhomomorphism.

The second section is concerned with thegeometry of moduli space and its universal(orbifold) cover Teichmuller space. Com-pared to the size of this subject, this sec-tion is comparatively brief; nevertheless, itcontains a good introduction to the the-ory of quasi-conformal maps, Teichmuller’stheorem on the existence and uniqueness ofmaps minimizing the dilatation in a givenhomotopy class of homeomorphism betweenRiemann surfaces, andMumford’s compact-ness criterion and the topology of the “end”of moduli space.

The third section is concerned with theNielsen–Thurston classification of mappingclasses, and especially with the structure ofpseudo-Anosov elements. These can be un-derstood in many di!erent ways: with holo-morphic quadratic di!erentials, measuredsingular foliations, transversely measuredgeodesic laminations, weighted train-tracks,and so on. Much of this approach is due toThurston, and the book presents Thurston’sapproach to the subject and sketches hisproofs of key results.

The authors manage throughout to con-vey a coherent, comprehensive, and inte-

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BOOK REVIEWS 557

grated vision of the theory of surfaces andmapping class groups that explains how allthe di!erent ways of thinking about theseobjects fit together, and how they connectwith the rest of mathematics. This bookis thus timely and welcome, and should beof tremendous value to mathematicians ofevery stripe.

DANNY CALEGARI

University of Chicago

Fundamentals of Object Tracking. By Sub-hash Challa, Mark R. Morelande, Darko Musicki,and Robin J. Evans. Cambridge University Press,Cambridge, UK, 2011. $95.00. xii+375 pp.,hardcover. ISBN 978-0-521-87628-5.

This book deals with the important areaof multiple object tracking or multitar-get tracking. Advances in sensor technol-ogy have provided data sources for track-ing moving objects in civilian applicationssuch as air tra"c control or video surveil-lance, and military applications such as in-telligence, surveillance, and reconnaissance(ISR). The objects of interest may move un-derwater, on the surface or land, or in theair or space. Sensors may include radar, in-frared, video, acoustic, seismic, etc. Track-ing problems also exist in the cyber domain,where the objects are network activities in-stead of physical entities and sensors areintrusion detection devices on the commu-nication networks and computers. Some ex-amples are given in Chapter 1 of this book.

Despite the importance of object track-ing, there are not many books that providethe fundamentals of object tracking for stu-dents or engineers who are new to the field.Until the 2011 book by Bar-Shalom, Wil-lett, and Tian [1], the most recent bookwas [2], published in 1999. Readers whowant to learn about object tracking have toread papers scattered in many journals andconference proceedings. Reference [1] is en-cyclopedic and includes many algorithms,but it is huge at over 1200 pages. Thus,there was a need for a more compact bookthat provides the basics of object track-ing without burdening the reader with toomany details. These authors have satisfiedthis need with a well-written book thatstarts with “the generic Bayesian solution”

and “then shows systematically how to for-mulate the major tracking problems . . . andhow to derive the standard tracking solu-tions.” The systematic treatment includesexplicit statements of assumptions such aspoint objects with dynamics described bya Markov process and point measurementsfrom sensors with infinite resolution. Eventhough these assumptions are not valid forproblems such as video surveillance, theyare good approximations for many problemsand necessary for developing implementablealgorithms.

A unique feature of multiobject trackingis that the sensor measurements frequentlydo not come with the identities of the ob-jects that generate them. Thus associationof measurements to objects is needed be-fore object states can be estimated. Dataassociation is the main di!erence betweenobject tracking and state estimation or fil-tering, where algorithms such as Kalmanfiltering assume known measurement ori-gins. Traditional object tracking algorithmsinclude two main components: state esti-mation of object tracks given associatedmeasurements, and association of measure-ments to object tracks. The major part ofthis book discusses algorithms for these twocomponents using a Bayesian approach.

The Bayesian approach is introduced inChapter 1, which formulates object trackingas a general Bayesian estimation problemand presents recursive equations for predic-tion and update of conditional probabilitydensities. These equations are only nota-tional because the object states and mea-surements are random sets and not randomvectors. In fact, the same equations appearlater in Chapter 6: multiple object trackingin clutter, a random-set-based approach.Similar equations for general Bayesian mul-tiple object tracking appear in [3], whichshould have been referenced if only becauseof its title, Bayesian Multiple Target Track-ing.

Chapters 2 and 3 focus on the fil-tering component of object tracking andpresent algorithms for generating state esti-mates and associating measurements. Eventhough the chapter titles include the word“tracking,” the origins of the measurementsare assumed known. Chapter 2 presentstraditional algorithms such as the Kalmanfilter and the extended Kalman filter and

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558 BOOK REVIEWS

more recent nonlinear algorithms such asunscented filter, point mass filter, and par-ticle filter. An example shows the excellentperformance of particle filters, but a com-parison of the computational costs wouldhave been useful. Chapter 3 presents fil-ters for maneuvering objects including thegeneralized pseudo-Bayesian filters, the in-teracting multiple model (IMM) filter, itsvariable structure variant, and the particlefilter for maneuvering objects.

Chapters 4 and 5 address the data associ-ation problem in object tracking. Chapter4 deals with measurement association to anexisting object track, e.g., whether the mea-surement is from the object or clutter. Itpresents standard approaches such as near-est neighbor filter, probabilistic data asso-ciation filter (PDAF), and particle filter inclutter. Chapter 5 is the longest chapter (al-most one quarter of the book) and addressessingle andmultiple object tracking using theobject existence paradigm. This paradigmwas developed by the authors to assess thequality of a track that is updated using thePDAF and extended to handle more com-plex problems. For nondeterministic objectexistence, when the number of objects isnot known, joint integrated probabilisticdata association (JIPDA) deals with single-scan association for multiple objects, andjoint integrated track splitting (JITS) treatsmultiscan association for multiple objects.Single object algorithms just drop “joint”(J) from the names, and multiple modelalgorithms add IMM before the names.

The JITS algorithm for multiobject mul-tiscan tracking is similar to multiple hy-pothesis tracking (MHT). Even thoughJITS does not use the standard MHT ter-minology of tracks and hypotheses, it alsogenerates new branches of a track tree ateach scan to account for di!erent associ-ation alternatives and computes the asso-ciation probabilities recursively. Since thenumber of alternatives (called track com-ponents in the book) grows rapidly, practi-cal implementation also requires standardMHT control techniques such as mergingand pruning. It is understandable that theauthors focus more on their own algorithmsin their book, but more discussion of MHTand reference to classical papers such as[4] would have been useful because MHT

is a more popular approach than JITS formultiobject tracking.

Chapter 6 takes readers back to the re-cursive Bayesian approach introduced inChapter 1 by discussing the random set ap-proach [5] that does not require separationinto filtering and association. The last threechapters of the book deal with Bayesiansmoothing using the augmented state ap-proach, tracking with time-delayed and out-of-sequence measurements, and some prac-tical issues in tracking. Three appendicesprovide the necessary mathematical back-ground.

The authors have organized the contentwith a very readable format. Most chap-ters follow the same structure of problemstatement, derivation, algorithms, perfor-mance bound, and illustrative example. All43 algorithms are summarized in tables thatprovide the necessary equations for imple-mentation. Most chapters end with an ex-ample that shows the implementation of thealgorithms and compares algorithm perfor-mance. More examples would be desirable,especially for the unconventional applica-tions discussed in Chapter 1, but this wouldbe di"cult given the small size of the book,which also forces the authors to skip impor-tant issues such as tracking with multiplesensors, extended objects, unresolved mea-surements, etc.

In summary, this book achieves its objec-tive of providing the fundamentals of objecttracking. By starting with the Bayesian ap-proach, it systematically presents the mostimportant algorithms given some fairly gen-eral assumptions, and it does this in a smallpackage. One might wish that it had in-cluded more discussions of multiobject mul-tiscan algorithms besides those developedby the authors; however, it contains almosteverything most readers need to learn aboutthe fundamentals of object tracking.

REFERENCES

[1] Y. Bar-Shalom, P. K. Willett, and X.Tian, Tracking and Data Fusion: AHandbook of Algorithms, YBS Press,Storrs, CT, 2011.

[2] S. Blackman and R. Popoli, Design andAnalysis of Modern Tracking Systems,Artech House, Norwood, MA, 1999.

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BOOK REVIEWS 559

[3] L. D. Stone, C. A. Barlow, and T.L. Corwin, Bayesian Multiple Tar-get Tracking, Artech House, Norwood,MA, 1999.

[4] D. Reid, An algorithm for tracking mul-tiple targets, IEEE Trans. Automat.Control, 24 (1979), pp. 843–854.

[5] R. P. S. Mahler, Statistical Multisource-Multitarget Information Fusion,Artech House, Norwood, MA, 2007.

CHEE-YEE CHONG

Los Altos, CA

Introduction to Partial Differential Equa-tions. By Peter J. Olver. Springer InternationalPublishing, Switzerland, 2014. $69.99. xxvi+635 pp., hardcover. ISBN 978-3-319-02098-3.

There are plenty of good textbooks for afirst course on partial di!erential equations(PDEs) suitable for one-semester courses orfor full-year courses. The classic and widelyadopted text by Haberman [6] comes tomind, as do texts by (in subjective order ofincreasing sophistication; for the record, Ilike all of these for many, sometimes verydi!erent reasons) Farlow [4], DuChateau[3], Asmar [1], Weinberger [11], and Strauss[9]. Why should one add another to thelist? The answer is obvious, although prob-ably provocative: because this new addi-tion is the best one yet. In the interestof full disclosure, I should mention that Ihave taught from the notes that precededthis book, and I am mentioned in the ac-knowledgments because of this. In fact, itis useful to describe the level of the course Itaught. The course had two sections: Onesection contained undergraduate students,mostly seniors, largely majoring in appliedmathematics, mathematics, or physics. Theother section, somewhat bigger, was aimedat graduate students from a large varietyof science departments (atmospheric sci-ence, geosciences, oceanography, various en-gineering departments, etc.). The courselectures I prepared were identical: my levelof presentation was somewhat more sophis-ticated for the graduate section, which re-ceived a few additional harder homeworkproblems. Such a course is not atypical ofthose o!ered in many departments. Whatdo I want from a textbook when I teach

this course? Here is an incomplete list: (a)I want to be able to cover all of the ma-terial required by the standard first-coursesyllabus: separation of variables, heat equa-tion, Laplace equation, wave equation, andso on; (b) the text should point to the nu-merous connections to the application fieldswhere much of this material originated; (c)the writing should be clear and understand-able; (d) there should be enough materialto entice students to go further, by takinganother more advanced course, for instance;and (e) the book should have a wealth ofhomework problems.

This book easily covers all the materialone might want in a course aimed at first-time students of PDEs. The three maincomponents of a first course (heat equa-tion, wave equation, Laplace equation) arepresent, and their various standard solv-able boundary value problems are discussed.There are several very nice chapters onproblems in multiple dimensions. All classi-cal solution techniques are introduced: sep-aration of variables leading to Fourier se-ries, Fourier transforms, Green’s functions,and d’Alembert’s solution. One classicalmethod that is missing is the Laplace trans-form. I am fine with this; I was never a bigfan, but others may miss it. If so, [6] andothers can be used to supplement this mate-rial. There are many references to the physi-cal origins of the problems treated, althoughthe approach is less grounded in applica-tions from physics or engineering comparedto, say, [6]. I should mention that somedelightful modern applications are includedsuch as the Black–Scholes equation frommathematical finance, image processing anddenoising, and some quantum mechanics.It has always surprised me how underrep-resented the latter is in the standard PDEtexts. There are many smaller sections onrelated and more advanced mathematicaltopics inviting both students and teachersto go beyond what is available here: twochapters on numerical methods (see below)are an obvious example, but there are alsoglimpses of symmetry methods (did you seewho wrote this book?) and nonlinear equa-tions in short sections on the Burgers equa-tion and the Korteweg–de Vries equation.

This book does not follow the approach ofthe typical first course in PDEs from which

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560 BOOK REVIEWS

the average student walks away thinkingthat they know how to solve PDEs: it mighttake be some work, but it is doable. In fact,the typical first course in ODEs often hasthe same e!ect. This is the main reason Ichose to teach from this book (or the notes,before the book was available) rather thanany other. It presents the best first courseon solution methods for PDEs, while ad-mitting that we cannot solve most PDEsanalytically. This admission is made in afew di!erent ways. First, there is quite abit of material on numerical methods forPDEs, in chapters on finite di!erence meth-ods and finite elements. The latter chapteralso discusses weak solutions. This fits inwith the second approach, where a slowtransition to qualitative methods is madethrough function spaces, adjoints, and max-imum principles. This material allows for aneasy transition to a more advanced coursewhich in many other cases can be quite dis-joint from a first course. The book strikes anear-perfect balance between rigor and ac-cessibility, given the audience it is aimed at.

The book places less focus on complexanalysis than a classic like [11] does. Notmany newer books do, and this is not sur-prising. I am unhappy with this trend, as Ihave yet to see a complex analysis techniqueI dislike. I am not alone in this [7, 10]. I be-lieve that a recent method due to Fokas [2, 5]will be part of the standard first-course PDEcurriculum within a decade and it shouldfind itself in textbooks soon. Since it re-quires a bit of complex analysis, perhapsthe trend will reverse.

Let me discuss the chapters on numericalmethods a bit more. Both are well done andfit in well with the other chapters. I believestudents truly benefit from the treatmentgiven here, within the context of PDEs.If you are teaching a one-quarter or one-semester course, it is unlikely your syllabuswill have room for the inclusion of thesechapters. On the other hand, if you havean entire academic year at your disposal,incorporating these chapters into your syl-labus is a great idea.

The book is written very well. It is clearthat the author is serious about conveyingthe message transparently, and it is equallyclear that the text has gone through a num-ber of iterations, resulting in very few if any

errors, typographical or otherwise. I recallteaching from the precursor notes two yearsago and finding very few errors at thatpoint, having covered at least parts of 8 of12 chapters. The text has many homeworkproblems, which range from run-of-the-millboundary value problems, necessary so stu-dents can repeat steps shown in class with-out much change, to harder and usuallymore interesting ones. I found myself goingoutside the homework problems available inthe notes on only a few occasions. The bookis published by Springer, listed at $70 (hard-cover) or $50 (e-book). That price is quitea bit lower than [6] or [9]; it is comparableto [1] (paperback edition), but more expen-sive than [3, 4, 11], all of which are Doverpaperbacks. The hardcover version seemsto be of good quality, although the coverhas started to separate from the book onboth of my copies. Some may be interestedto know that an instructor solution manualcontaining detailed solutions to about halfof the problems will be available in the nearfuture.

In addition to other higher-level texts,Peter Olver has now given us a second un-dergraduate textbook, following [8]. Likethe latter linear algebra textbook, I recom-mend this one highly: It provides the bestfirst-course introduction to a vast and ever-more relevant and active area. Students,and perhaps instructors too, will learn muchfrom it. If they wish to go beyond the ma-terial taught in a first course, this text willprepare them better than any other I know.

REFERENCES

[1] N. H. Asmar, Partial Di!erential Equa-tions with Fourier Series and Bound-ary Value Problems, 2nd ed., PearsonPrentice Hall, Upper Saddle River,NJ, 2004.

[2] B. Deconinck, T. Trogdon, andV. Vasan, The method of Fokas forsolving linear partial di!erential equa-tions, SIAM Rev., 56 (2014), pp. 159–186.

[3] P. DuChateau and D. Zachmann, Ap-plied Partial Di!erential Equations,Dover, New York, 2012.

[4] S. J. Farlow, Partial Di!erential Equa-tions for Scientists and Engineers,Dover, New York, 1993.

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BOOK REVIEWS 561

[5] A. S. Fokas, A Unified Approach toBoundary Value Problems, CBMS-NSF Regional Conf. Ser. in Appl.Math. 78, SIAM, Philadelphia, 2008.

[6] R. Haberman, Elementary Applied Par-tial Di!erential Equations. WithFourier Series and Boundary ValueProblems, 5th ed., Prentice-Hall, En-glewood Cli!s, NJ, 2012.

[7] P. D. Lax and L. Zalcman, ComplexProofs of Real Theorems, Univ. Lec-ture Ser. 58, AMS, Providence, RI,2012.

[8] P. J. Olver and C. Shakiban, Ap-plied Linear Algebra, Pearson Pren-tice Hall, Upper Saddle River, NJ,2006.

[9] W. A. Strauss, Partial Di!erential Equa-tions. An Introduction, 2nd ed., JohnWiley & Sons, Chichester, UK, 2008.

[10] L. N. Trefethen, Review of VisualComplex Functions: An Introductionwith Phase Portraits, SIAM Rev., 55(2013), pp. 791–797.

[11] H. F. Weinberger, A First Course inPartial Di!erential Equations withComplex Variables and TransformMethods, Dover, New York, 1995; cor-rected reprint of the 1965 original.

BERNARD DECONINCK

University of Washington

Classical and Multilinear Harmonic Anal-ysis. Vols. I and II. By Camil Muscalu and Wil-helm Schlag. Cambridge University Press, Cam-bridge, UK, 2013. $135.00. 760 pp., hardcover.ISBN 978-1-107-03262-0.

This two-volume set is a noteworthy ad-dition to the expository literature on har-monic analysis. The authors have assem-bled a large amount of important materialinto a compact and well-organized package,and their exposition is terse but clear andwell motivated.

The first volume o!ers a unified treat-ment of a number of important topics inclassical harmonic analysis, beginning withFourier series and leading up to more re-cent results such as Fourier restriction theo-rems and the T (1) theorem, with emphasison techniques such as Calderon–Zygmundtheory and Littlewood–Paley theory thatcontinue to play an important role in re-search. It provides an excellent resource for

readers who are familiar with the basicsof Fourier analysis and wish to acquire adeeper understanding of the subject. Thesecond volume deals with multilinear har-monic analysis, with emphasis on work ofthe first author and his collaborators. Thecentral theme is the notion of paraprod-uct, and the high points include proofs ofthe by-now classic theorems on Calderoncommutators, Cauchy integrals on Lipschitzcurves, and almost everywhere convergenceof Fourier series.

The reviewer has written a more detailedreview of these books for Mathematical Re-views (review MR3052498 on MathSciNet).

GERALD B. FOLLAND


Mathematics of Two-Dimensional Turbu-lence. By Sergei Kuksin and Armen Shirikyan.Cambridge University Press, Cambridge, UK,2012. $85.00. xvi+320 pp., hardcover. ISBN978-1-107-02282-9.

Two-dimensional turbulence has been avery active and intriguing area of researchover the last five decades, since the publi-cation of Robert Kraichnan’s seminal pa-per [1] postulating the dual cascade the-ory. Some reviews are given in [2, 3, 4].The original motivation for studying two-dimensional turbulence was the belief thatit would prove to be an easier problemthan three-dimensional turbulence and thatmathematical techniques developed for thetwo-dimensional problem would then beused for the three-dimensional problem. Itwas also believed that two-dimensional tur-bulence theory could explain flows in verythin domains, such as the large-scale phe-nomenology of turbulence in planetary at-mospheres.

In general terms, theoretical studies ofturbulence use a wide range of strategies,including phenomenological theories, ana-lytic theories that depend on hypothesesestablished experimentally or via numericalsimulations, and mathematically rigoroustheorems on the Navier–Stokes equations.With a phenomenological approach, onemakes a series of hypotheses based on ex-perimental evidence and physical intuitionfrom which conclusions can be drawn about

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562 BOOK REVIEWS

universal features of turbulence. The Kol-mogorov theory of three-dimensional tur-bulence [5, 6, 7] and Kraichnan’s theory oftwo-dimensional turbulence [1] typify thisapproach. In both cases, minimal contactis made with the Navier–Stokes equations.Nevertheless, a lot of successful numeri-cal and experimental work has been moti-vated by phenomenological theories. Withthe more rigorous strategy of formulatinganalytical theories of turbulence, one usesthe governing equations as a point of de-parture to formulate perturbative closuremodels or nonperturbative strategies. Themathematical foundation for the most ad-vanced of these theories is the Martin–Siggia–Rose formalism [8, 9] (hereafterMSRformalism), with reviews given in [10, 11].These theories cannot be completely rigor-ous on their own since the use of the MSRformalism entails certain assumptions: (a)existence and uniqueness of a deterministicsolution for the velocity field given a choiceof deterministic forcing field; (b) the as-sumption that the system was initialized attime t " !% and has already converged tostatistical steady state. Furthermore, somelack of rigor stems from the dependence onFeynman path integrals. Finally, to con-nect theoretical predictions about ensembleaverages with numerical simulations and ex-periments requires the additional assump-tion of ergodicity. Having made these as-sumptions, the payo! is that it is possibleto make considerable inroads toward clar-ifying, explaining, and predicting the phe-nomenological behavior of turbulence forthe two-dimensional as well as the three-dimensional case. Finally, another strategyis to prove mathematically rigorous the-orems about the Navier–Stokes equationsusing functional analysis and dynamicalsystems theory techniques. This approach,pioneered by distinguished mathematicianslike Leray, Foias, Temam, and many others,has successfully yielded solid results. Theprice is that it is too di"cult to venture asfar as one can go using less rigorous strate-gies that incorporate hypotheses evidencedby experiment or numerical simulations.

Ultimately, all of the above strategieshave strengths and weaknesses that com-plement one another. A curious ironyof two-dimensional turbulence research is

that whereas the phenomenology of two-dimensional turbulence is richer and posesmany more challenges than that of three-dimensional turbulence, two-dimensionalturbulence has turned out to be far moreamenable to the pure mathematician’stoolbox. The current book under reviewby Kuksin and Shirikyan surveys recentdevelopments in the mathematical the-ory of the two-dimensional Navier–Stokesequations that are, with no exaggeration,quite breathtaking. The authors use therandomly forced two-dimensional Navier–Stokes equation with a regular Laplaciandissipation term at small scales as theiransatz. Three types of random forcing areconsidered: (a) kick forcing, consisting of,equispaced in time, delta function spikeswith random amplitudes; (b) white noise,i.e., random Gaussian delta-correlated intime forcing, commonly used in MSR theo-ries; (c) compound Poisson processes, whichare random kick forces where both the am-plitude and the temporal separation be-tween the delta function spikes are ran-domized.

The authors begin in Chapter 1 with avery terse yet comprehensive review of es-sential concepts, needed for the proofs ofthe main results, from the areas of functionspaces, measure theory, and Markov ran-dom dynamical systems. A solid graduateeducation in functional analysis is neces-sary to follow the chapter, but the authorsprovide citations to many other books thatexplain underlying concepts in more detail.Chapter 2 begins with a review of the clas-sical Leray results on the existence, unique-ness, and regularity of solutions for thecase of the two-dimensional Navier–Stokesequations with deterministic forcing. Forthe case of stochastic forcing, a series ofimportant general results are proved thatculminate in proving the existence of atleast one stationary measure. In physicalterms, a stationary measure describes thesteady-state solution to the randomly forcedNavier–Stokes equations when a dynamicalbalance has been established between forc-ing and dissipation and the ensemble av-erages for all observables become constantwith respect to time.

The argument continues in Chapter 3with an array of results establishing the

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uniqueness of the stationary measure aswell as the property of exponential mix-ing, both for periodic flows on an infinitedomain and for flows on a bounded domainfor various random forcing configurations.In physical terms, the property of expo-nential mixing means that regardless of theinitial condition, the randomly forced two-dimensional Navier–Stokes equation willstatistically converge to the steady-statesolution at an exponential rate. This con-vergence has been established for both thevelocity field itself and relevant observables,dependent on the velocity field, such as theenergy spectrum. The authors also estab-lish that if the random force is homoge-neous, then the velocity field at steady statewill also be homogeneous. The chapter con-cludes with a literature review as well as aphysical summary of the main results.

Chapter 4 establishes ergodic theoremsas well as some interesting limiting theo-rems. In particular, the authors establishthat the time average of observables, depen-dent on the velocity field, quickly convergeto the ensemble average as one extends thetime interval over which the time average istaken. The authors also establish a centrallimit theorem that shows that the velocityprobability distribution is close to Gaus-sian, in agreement with experiments andnumerical simulations (see [3] for a review).Furthermore, the authors prove that thestatistical properties of the velocity field atsteady state will vary continuously as onevaries the statistical parameters of randomforcing. Finally, the authors prove that thesteady-state solution of a system forced byrandom kicks will converge to the steadystate of the system forced by white noise ifthe time gap between kicks is shrunk by afactor &, taking the limit & " 0+ , as long asthe amplitude of the kicks is also decreasedby a factor of

$&.

Having established the existence anduniqueness of a stationary measure for thecase of finite viscosity, in Chapter 5, the au-thors investigate the stationary measure un-der the limit ' " 0+ of viscosity approach-ing zero. For technical reasons, instead ofusing a continuous limit it is necessary towork with the discrete limit 'k " 0+ withk & N for some chosen viscosity sequencethat converges to zero. The authors prove

that for every such viscosity sequence, thecorresponding stationary measures have anontrivial limit, as long as forcing is moder-ated by an

$'k factor. It remains an open

question whether all possible sequences suchthat 'k " 0+ with k & N lead to a uniquelimit for the stationary measure. However,it is proved that all stationary measuresobtained from any viscosity sequence limitto zero will satisfy certain universal prop-erties from which a phenomenology of two-dimensional turbulence can be deduced.From these universal properties, if we intro-duce the assumption that the energy spec-trum follows a power law, downscale fromthe forcing range, it is predicted that the en-ergy spectrum will scale as k"a with a # 5,where k is the wavenumber. The authorsalso identify an unproven conjecture thatwould rigorously imply a = 5.

Finally, in Chapter 6 the authors outlinewithout proof a number of incomplete re-sults whose development is the subject ofcurrent active research. A special highlightis a result that shows that the stationarymeasure of the three-dimensional Navier–Stokes equations, defined in a quasi-two-dimensional domain in which the verticaldirection is very thin, and also randomlyforced by random kicks, will converge tothe corresponding stationary measure ofthe two-dimensional Navier–Stokes equa-tions. However, it remains an open questionwhether this result can be extended for thecase of white noise forcing.

In light of the foregoing discussion, thesignificance of these results is clear. In everywell-known theory of two-dimensional andthree-dimensional turbulence, one takes forgranted the existence and uniqueness ofthe statistical steady-state solution, thata forced dissipative system will always con-verge to the steady-state solution, that theensemble average can be exchanged with atime average, and that the discrete kick forc-ing typically used by numerical simulations,where time is discretized, properly approx-imates the case of continuous white noiseforcing. These are all assumptions that un-derlie every theoretical e!ort to understandthe phenomenology of turbulence, but theyare also assumptions that are not easy toprove. It is very reassuring to see that dur-ing the last decade, at least for the case of

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two-dimensional Navier–Stokes turbulence,all of these assumptions have been provedrigorously. This is a major achievement,and the authors are leading experts whohave played a key role in the developmentof many of these results.

It is also worth commenting on the phe-nomenology of the k"5 energy spectrumpredicted in Chapter 5. This is not an en-tirely new result. It was first proposed byTran and Shepherd [12] and Tran and Bow-man [13], who predicted a k"5 spectrumdownscale from the forcing range and a k"3

spectrum upscale from the forcing range.This phenomenology is inconsistent withKraichnan’s theory [1] of a downscale en-strophy cascade with k"3 scaling and anupscale inverse energy cascade with k"5/3

scaling. As was explained by Tran andShepherd [12], the Kraichnan cascades willfail to materialize in the absence of a dis-sipation term at large scales in a boundeddomain flow. On an infinite domain, en-ergy can simply cascade forever to largerand larger scales, and enstrophy can cas-cade to small scales and be dissipated bythe small-scale di!usion term. However, ona finite domain, if there is no mechanism todissipate the upscale energy cascade beforeit hits the largest possible scales, then thecascade configuration will collapse and tran-sition to the conjectured joint k"3 and k"5

configuration. The results by the authorsvindicate the work of Tran et al. [12, 13, 14]by eliminating unproven assumptions thatthey made in order to establish their pre-dictions. As important as this developmentis, the greater challenge of understandingthe robustness of the Kraichnan cascadesremains an open question.

Finally, I should like to make some com-ments about the book itself. It has beenwritten primarily for an audience of puremathematicians who wish to familiarizethemselves with this research area so theycan make further contributions. The writ-ing style is very concise; however, the au-thors provide complete proofs for almostall of their results. An extensive array ofvery general preliminary results needs to beestablished before the main results can beproved. The preliminary results are usefulin and of themselves and can be used forthe future investigation of systems other

than the randomly forced two-dimensionalNavier–Stokes equations. The authors men-tion the complex Ginzburg–Landau equa-tion as a possible area of exploration. Anextensive bibliography of more than 200 ref-erences is given, and I very much appreciatethe reverse citation system in which for eachitem in the bibliography the authors givethe page numbers where the given item iscited in the text. Much heavy notation isused throughout the book; however, theauthors provide a very useful summary ofnotation conventions at the end. Last butnot least, in Chapters 3, 4, and 5 wherethe main results are discussed, the authorsconclude each chapter with a very clear dis-cussion of the physical implications of theirresults. These sections are essential to mak-ing this work accessible to a more appliedaudience. A very detailed literature reviewis also given at the end of every chapterfor those who wish to consult the originalresearch papers.

In summary, this is an excellent bookpresenting and proving a body of resultsthat are of fundamental importance in thedevelopment of theories of two-dimensionalturbulence. For pure mathematicians, thereis much to be learned from the techniquesused to prove the theorems that can beapplied to a wider range of problems. Forapplied mathematicians, it is certainly use-ful to have some understanding of what hasbeen proved rigorously for two-dimensionalNavier–Stokes. The results themselves arevery interesting and their physical implica-tions are clearly explained. While this isnot a book for the faint of heart, I findit an excellent addition to my library andstrongly recommend it to everyone engagedin theoretical research on two-dimensionalturbulence.

REFERENCES

[1] R. H. Kraichnan, Inertial ranges in twodimensional turbulence, Phys. Fluids,10 (1967), pp. 1417–1423.

[2] R. H. Kraichnan and D. Montgomery,Two dimensional turbulence, Rep.Prog. Phys., 43 (1980), pp. 547–619.

[3] P. Tabeling, Two-dimensional turbu-lence: A physicist approach, Phys.Rep., 362 (2002), pp. 1–62.

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[4] E. Gkioulekas and K. K. Tung, Re-cent developments in understand-ing two-dimensional turbulence andthe Nastrom-Gage spectrum, J. LowTemp. Phys., 145 (2006), pp. 25–57.

[5] A. N. Kolmogorov, The local structureof turbulence in incompressible vis-cous fluid for very large Reynoldsnumbers, Dokl. Akad. Nauk. SSSR,30 (1941), pp. 301–305 (in Russian);English translation, Proc. Roy. Soc.London Ser. A, 434 (1991), pp. 9–13.

[6] A. N. Kolmogorov, Dissipation of en-ergy in the locally isotropic turbulence,Dokl. Akad. Nauk. SSSR, 32 (1941),pp. 16–18 (in Russian); English trans-lation, Proc. Roy. Soc. London Ser.A, 434 (1991), pp. 15–17.

[7] G. K. Batchelor, Kolmogorov’s theoryof locally isotropic turbulence, Proc.Camb. Phil. Soc., 43 (1947), pp. 533–559.

[8] P. C. Martin, E. D. Siggia, and H. A.Rose, Statistical dynamics of classi-cal systems, Phys. Rev. A, 8 (1973),pp. 423–437.

[9] R. Phythian, The functional formalismof classical statistical dynamics, J.Phys. A, 10 (1977), pp. 777–789.

[10] V. S. L’vov and I. Procaccia, Ex-act resummations in the theory ofhydrodynamic turbulence: Part 0.Line-resummed diagrammatic pertur-bation approach, in Fluctuating Ge-ometries in Statistical Mechanics andField Theory, Proceedings of theLes Houches 1994 Summer Schoolof Theoretical Physics (Amsterdam),F. David and P. Ginsparg, eds.,North-Holland, Amsterdam, 1996,pp. 1027–1075.

[11] E. Gkioulekas, On the elimination of thesweeping interactions from theories ofhydrodynamic turbulence., Phys. D,226 (2007), pp. 151–172.

[12] C. V. Tran and T. G. Shepherd, Con-straints on the spectral distributionof energy and enstrophy dissipationin forced two dimensional turbulence,Phys. D, 165 (2002), pp. 199–212.

[13] C. V. Tran and J. C. Bowman, On thedual cascade in two dimensional tur-bulence, Phys. D, 176 (2003), pp. 242–255.

[14] C. V. Tran and J. C. Bowman, Robust-ness of the inverse cascade in two-dimensional turbulence, Phys. Rev.E, 69 (2004), 036303.

ELEFTHERIOS GKIOULEKAS

University of Texas–Pan American

Mathematical Physics: A Modern Intro-duction to Its Foundations. Second Edi-tion. By Sadri Hassani. Springer Interna-tional Publishing, Switzerland, 2013. $99.00.xxxii+1205 pp., hardcover. ISBN 978-3-319-01194-3.

Mathematical Physics: A Modern Introduc-tion to Its Foundations is intended for adop-tion in a course or a sequence of “methods ofmathematical physics” at the advanced un-dergraduate or beginning graduate level. Inthis respect it will compete with other stan-dard “methods” textbooks such as those ofBoas [4], Arfken and Weber [2], Mathewsand Walker [11], Byron and Fuller [5], andothers.

The textbook has been reviewed posi-tively by Pure and Applied Geophysics, Zen-tralblatt Math, and the European Mathe-matical Society newsletter, and excerpts ap-pear on its Amazon and Springer web pages.

The title may be somewhat misleading.What are considered to be foundations bymathematicians may be quite di!erent fromthe corresponding perception of physicists.The book does not discuss mathematicalfoundations in the spirit of Elliot Lieb’sAnalysis [10], for example. Instead, the textis written in a formal (but not abstract) waythat emphasizes a general formulation of atopic before it is illustrated with numerousexamples, usually drawn from the under-graduate physics curriculum. In this sense itdi!ers significantly from the aforementioned“methods” texts, which are more informallywritten. Thus, the student of physics willbe exposed to a necessary formalism whoseapplications are met repeatedly in the studyof physics.

Particularly nice is Chapter 4, where theauthor addresses the algebra of endomor-phisms of a vector space. The author definespolynomial operators p(T) such as rotationoperators, the exponential operator exp(T),commutators, projectors, and their calcu-lus, all of which are used in a course ofquantum mechanics, at the level of Saku-rai [13]. Matrix representations of the op-erators are developed in Chapter 5 alongwith diagonalization and change of basis.The Dirac bra-ket notation is used through-out. Orthogonal polynomials are developedas vectors in Hilbert space rather than as

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566 BOOK REVIEWS

series solutions of second-order di!erentialequations with singular points. The authordevelops the main points and theorems ofinfinite-dimensional vector spaces withouthiding their essence behind puristic clut-ter. Chapter 6 discusses spectral decom-positions and projection operators and hasa very nice discussion of simultaneous di-agonalization, which, for example, can beused classically in small oscillations or quan-tum mechanically for simultaneous mea-surements. A very nice discussion of thepolar decomposition theorem follows, fa-miliar to applied mathematicians from ba-sic continuum mechanics, but here the au-thor makes a connection with numbers andquantummechanical operators. Basic infor-mation about Hilbert space and the Diracdelta function appear in Chapter 7 as wellas a short discussion of distributions. PartIV discusses di!erential equations includingexistence theorems, power series solutionswith applications to the quantum harmonicoscillator, and the WKB method. Part Vdiscusses operators in Hilbert space withSturm–Liouville systems as a special case,where the author develops the method ofseparation of variables.

I particularly liked Part VI, which is de-voted to Green’s functions, i.e., a topicthat is scarce in the mathematics curric-ula unless you have the luck to be taughtby an exceptional academic like WernerWeighlhofer, my Green’s function teacher.Again, the author first builds the neces-sary formal background before illustratingthe main points with examples drawn fromclassical electrodynamics and quantum me-chanics. Part VII discusses basic group the-ory with emphasis on topics of interest inhigh energy physics such as group charac-ters and tensor products of representations.Part VIII builds the formalism of tensorcalculus, forms, symplectic geometry, andCli!ord algebras. The forms are somewhatclose in character to that of Flanders [6].

Part IX discusses Lie groups, algebras,and their representations. Two chapters ofthis part deserve special mention. Chap-ter 32 discusses symmetries of di!erentialequations in the spirit of Peter Olver [12].This important topic is written in an ap-proachable manner that makes connectionswith standard quantum mechanical observ-

ables, e.g., angular momentum. Chapter33 discusses calculus of variations in a verywell written manner including the notion offunctional derivatives that other compet-ing texts prefer to avoid. In some sense itsexposition is somewhat more rigorous thanthat of Gelfand and Fomin [7], but also morerestricted in scope. In section 33.1.4 the au-thor discusses the ideas of divergence andnull Lagrangians on which one of the mostimportant recent advances in existence the-orems of nonlinear elasticity by John Balland coworkers [3] is based.

Part X discusses fiber bundles, gauge the-ories, and Riemannian geometry. This lasttopic is very close to the exposition of Stew-art [14]. Not only students of relativity, butalso those of condensed matter field theorywho read, for example, [1] and [9] will ben-efit, obtaining all required geometry foun-dations.

What are missing from this textbook aremethods met in equilibrium and nonequi-librium statistical mechanics written in arigorous but clear manner. I have foundno such exposition in any textbook. Ba-sic probability and stochastic processes atthe level of Papoulis and Ross (models ofprobability) would be advantageous in thetext. Also, perturbation theory is includedin the Green’s function sections but it wouldbe more pedagogically correct if elemen-tary ideas such as its application to singu-larly perturbed nonlinear oscillators couldbe discussed earlier, for example, in the dif-ferential equations part. In addition, someelementary results of perturbation theoryand their associated diagrams might also bebeneficial for the reader. The text would bemore adaptable for self-study if the numberof references was increased and citationsappeared more frequently.

To conclude, this is a textbook that ev-ery library must have (with a large num-ber of copies in reserve), and it will bea valuable aid for committed students, re-searchers, and instructors of science andapplied mathematics. It is sold at almosthalf the price of some popular hardbackdi!erential equations textbooks o!ered bysome mainstream publishers at extortion-ate prices. This is to the merit of Springer,and of the author for choosing a (relatively)low-cost publisher.

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A few words about the author of Math-ematical Physics: A Modern Introductionto Its Foundations. Sadri Hassani receivedhis Ph.D. from Princeton University inhigh energy physics and is currently Pro-fessor of Physics at Illinois State Univer-sity. He has strong and adamant viewsabout the state of education in the UnitedStates. Being a skeptical educator, hisviews can be read in his personal blog atwww.skepticaleducator.org.

Ralph Boas [8], reviewingTheMathemat-ics of Physics and Chemistry by Margenauand Murphy, characterized its content as“pidgin mathematics,” i.e., “a clumsy andinept parody of mathematics” that can “beread by mathematicians who want to ac-quire a smattering of physics to impresstheir friends.” From Ralph’s standpoint,Hassani’s textbook also falls into this cat-egory. However, V. I. Arnold in his ad-dress in Palais de Decouverte in Paris onMarch 7, 1997, stated that “the return ofmathematical teaching at all levels from thescholastic chatter to presenting the impor-tant domain of natural science is an espe-cially hot problem.” Hassani’s expositiondoes exactly that.

REFERENCES

[1] A. Altland and B. D. Simons, Con-densed Matter Field Theory, Cam-bridge University Press, Cambridge,UK, 2010.

[2] G. B. Arfken and H. J. Weber, Mathe-matical Methods for Physicists, Inter-national Student Edition, AcademicPress, 2005.

[3] J. M. Ball, J. Currie, and P. J. Olver,Null Lagrangians, weak continuity,and variational problems of arbitraryorder, J. Funct. Anal., 41 (1981),pp. 135–174.

[4] M. L. Boas, Mathematical Methods inthe Physical Sciences, John Wiley &Sons, 2006.

[5] F. W. Byron Jr. and R. W. Fuller,Mathematics of Classical and Quan-tum Physics, Courier Dover, 2012.

[6] H. Flanders, Di!erential Forms with Ap-plications to the Physical Sciences,Courier Dover, 2012.

[7] I. M. Gelfand and S. V. Fomin, Calculusof Variations, Courier Dover, 2000.

[8] K. Iwasawa, W. Rudin, M. Lees,R. Boas, J. Dekker, Y. N. Dowker,J. Youngs, R. Pierce, J. Green,E. Pitcher, et al., Miscellaneousfront pages, Bull. Amer. Math. Soc.,65 (1959).

[9] A. Kamenev, Field Theory of Non-Equilibrium Systems, CambridgeUniversity Press, Cambridge, UK,2011.

[10] E. H. Lieb and M. Loss, Analysis, Grad.Stud. Math. 14, AMS, Providence,RI, 2001.

[11] J. Mathews and R. L. Walker, Math-ematical Methods of Physics, W.A.Benjamin, New York, 1970.

[12] P. J. Olver, Applications of Lie Groupsto Di!erential Equations, Grad. TextsMath. 107, Springer, New York, 2000.

[13] J. J. Sakurai, Modern Quantum Mechan-ics, Addison-Wesley, Reading, MA,1994.

[14] J. Stewart, Advanced General Relativity,Cambridge University Press, Cam-bridge, UK, 1993.

ELEFTHERIOS KIRKINIS

Northwestern University andQueensland University of Technology

Numerical Methods in Finance with C++.By Maciej J. Capinski and Tomasz Zastawniak.Cambridge University Press, Cambridge, UK,2012. $39.99. x+166 pp., softcover. ISBN 978-0-521-17716-0.

Numerical Methods in Finance with C++ ispart of the series “Mastering Mathemat-ical Finance” and is an introduction toC++ programming targeted primarily atstudents in quantitative and mathematicalfinance masters programs. This book relieson the theory of option pricing to developa sequence of programming tasks designedto introduce the reader to the C++ lan-guage, as well as to nonlinear solvers, MonteCarlo methods, and finite di!erence meth-ods. The tasks in the first two (and, to somedegree, the third) chapters are a particularlywell-thought-out introduction to the C++language and could easily form the core partof a syllabus for a financial programmingcourse. The tasks in the numerical methodsportion of the text serve to introduce thereader to several finance problems that lend

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themselves to numerical solution, and alsoto reinforce earlier C++ concepts.

The authors take the somewhat novel ap-proach of placing a code solution near (or,in one case, at) the beginning of each pro-gramming task, then following it up with aline-by-line description of what that code isdoing. By using this design, they are able tointroduce only the C++ language featuresand tools needed to tackle the task at hand,which prevents the reader from being over-whelmed by a comprehensive list of C++capabilities. On the other hand, it doesmean that some features are left out (e.g.,protected members). However, these omis-sions are not too big of a drawback since,as the authors point out, there are manyexisting C++ manuals and online resourcesthat the interested reader may turn to.

The book is organized as follows. Thefirst three chapters comprise the introduc-tion to the C++ portion of the book.An option pricer based on the binomialmodel for asset prices and the Cox–Ross–Rubinstein procedure is employed to intro-duce C++ language features: classes andsubclasses, inheritance, virtual functions,multiple inheritance, and class templates.Chapter 4 uses nonlinear solvers (bisectionand Newton–Raphson) to solve the impliedvolatility problem. Chapter 5 uses MonteCarlo methods to price path-dependent op-tions. Finally, Chapter 6 uses finite di!er-ence methods to solve parabolic partial dif-ferential equations with the goal of pricingan American option.

This book falls roughly in the middle ofthe “Mastering Mathematical Finance” se-ries and therefore assumes that the reader isalready familiar with the underlying finan-cial topics (in particular, option pricing).Further, while the book does start fromscratch with respect to C++ (the author’sHi there version of the ubiquitous Helloworld program appears on page 3), readerswho are already able to program (in somelanguage) will likely have a much easier timeworking through the programming tasks.

My only criticism of this book is thelack of mention of a C++ compiler or inte-grated development environment (IDE). AnIDE is such an integral part of C++ devel-opment that its use should be encouragedfrom the beginning. In particular, an IDE

makes debugging much easier. Presently,readers self-studying this book must dealwith the significant additional step of figur-ing out how to compile their first program.

This book could be used as the primarytext for a C++ programming course in aquantitative or computational finance pro-gram. The text is well organized and easyto follow, and the exercises are at the appro-priate level. Proficiency with this subjectmatter should be viewed as prerequisite fora career in quantitative finance and/or pro-gramming.

KJELL P. KONIS


Analytic Perturbation Theory and Its Ap-plications. By Konstantin E. Avrachenkov, JerzyA. Filar, and Phil G. Howlett. SIAM, Philadelphia,2014. $89.00. x+369 pp., hardcover. ISBN978-1-611973-13-6.

Konstantin Avrachenkov received a Ph.D.in 1999 from the University of South Aus-tralia, with Jerzy Filar and Phil Howlett asadvisors. Since that time, the authors havefurther developed and applied the subject ofanalytic perturbation theory, together witha variety of coauthors, in both finite- andinfinite-dimensional contexts. The end re-sult is an authoritative SIAM monograph,intended as a textbook, filled with outstand-ing exposition and explicitly detailed exam-ples.

The classical topic, linear and polynomialsystems of algebraic equations dependinganalytically on a small parameter, is animportant and challenging subject. Singu-lar perturbations occur when the solutioninvolves a Laurent expansion. (Thus, thesetting is far removed from the usual di!er-ential equations formulations.) The focushere is on generalized inverses, Puiseux se-ries, Groebner bases, the Newton polygon,mathematical programming, and Markovdecision processes, in lovely series and ma-trix equations. Who would guess that theGoogle PageRank can be e"ciently de-scribed as a singularly perturbed Markovchain?

Such problems becomemore varied as lin-ear operators in Hilbert and Banach spaces,

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where some functional analysis must be in-troduced. The authors clearly explain andmotivate their development, leaving prob-lems at chapter ends as well as bibliographicnotes. If your interests include linear anal-ysis, complex variables, optimization, andtheir applications, you’ll find much of prac-tical value here, eloquently presented.

Readers will find that there’s muchnew to learn since Kato and Vishik andLyusternik appeared in the 1960s!

ROBERT E. O’MALLEY, JR.


Beautiful Geometry. By E. Maor and E. Jost.Princeton University Press, Princeton, NJ, 2014.$27.95. xii+187 pp., hardcover. ISBN 978-0-691-15099-4.

This is a great book to buy your high-school age child (or grandchild) or theirmath teacher. It consists of 51 short chap-ters and an appendix with short proofs.The unusual topics vary from various ge-ometry problems (with many references toEuclid and Coxeter) to numbers, symme-try, and the Reuleaux and Sierpinski trian-gles. Each chapter is well written and illus-trated using detailed figures by the mathhistorian/writer Eli Maor and is enhancedby colorful related plates (not necessarilymathematical) by the Swiss artist EugenJost. Compass constructions inspired, forexample, “Seven Circles a Flower Maketh,”while the number eleven suggests a Celticmotif. No college math is required, butthinking is.



Five Decades of Tackling Models forStiff Fluid Dynamics Problems: A Sci-entific Autobiography. By R. Zh. Zeytou-nian. Springer, New York, 2014. $129.00.xxvi+153 pp., hardcover. ISBN 978-3-642-39540-6.

Radyadour Zeytounian was born in Parisin 1928, became an apprentice tailor, andemigrated to Armenia in 1947, where he fin-

ished secondary school, got a math degreefrom Yerevan State University, and taughthigh school for a year. He moved to Moscowin 1957, where he wrote a Kandidat thesison lee winds downstream of a mountainwith I. A. Kibel of Moscow State’s depart-ment of dynamic meteorology, married aRussian, had a daughter, and worked from1961 to 1965 at the Moscow MeteorologicalComputing Center. Taking advantage of avisit from DeGaulle, the family moved toParis when Zeytounian was 38.

Zeytounian worked in the aerodynamicsdepartment at the French space agency ON-ERA from 1967 to 1972 and as a professorat the University of Lille-I from 1972 to1996. He earned a doctorate in 1969, af-ter his committee member Paul Germainforced him to justify the Boussinesq ap-proximation via asymptotics. He learnedmatched asymptotic expansions from Jean-Pierre Guiraud and they planned bookson rational asymptotic modeling (RAM),which ultimately failed to reach Guiraud’sstandard of perfection. Nonetheless, Zey-tounian wrote many books on related topicsand Guiraud kindly wrote the foreword tothis one. The book outlines his unusual lifeand work, justifying some of it by quoting(even somewhat critical) reviews.

The most outstanding chapter developsthe basic Navier–Stokes–Fourier equationsin a historical context. Overall, readers willrealize the unique value of Zeytounian’swork and perspective and will come to ap-preciate his willingness to tackle very di"-cult problems, aiming to help fluid dynami-cal “numericians.” An example is modelingturbomachinery using the number of bladeson a rotor as a large parameter. Readers willneed to overlook the author’s di"culties inwriting and Springer’s lack of copyediting.



Introduction to the Network Approxima-tion Method for Materials Modeling. By L.Berlyand, A. G. Kolpakov, and A. Novikov. Cam-bridge University Press, Cambridge, UK, 2013.$80.00. xiv+243 pp., hardcover. ISBN 978-1-107-02823-4.D

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The book introduces and analyzes a newe!ective method for the computation ofthemacroscopic properties of heterogeneousmaterials containing dispersed discs. Thismethod is based on network approxima-tion, i.e., the discs are replaced by theircenters forming the so-called Delaunay–Voronoi graph, and the partial di!erentialequation in the continuous, but heteroge-neous, media is replaced by a natural dis-crete problem on the graph. This methodis justified in the case of closely spaced per-fectly conductive discs.

In physics and engineering, network mod-els became classical after the book by Bornand Huang [1] and have been intensivelyused ever since. The best-known exam-ple of the structural approximation is the“sprin” model used to describe a great va-riety of phenomena from molecular dynam-ics to composite materials; see examples inSahimi [2].

Generally, the main tool of the multi-scale modeling of composite materials isthe homogenization method. However, thismethod is very sensitive to the periodicity ofthe inclusions: if the inclusions are not pe-riodically spaced, then the result of the ho-mogenization becomes much less “algorith-mic.” This means that in nonperiodic cases,theoretically one can justify the existence ofan equivalent homogeneous medium, but itis very di"cult to compute its macroscopicproperties.

Another issue complicating the applica-tion of the homogenization method is thepresence of multiple small parameters in theproblem, in particular, contrasting proper-ties of the components of the compositematerial, thin regions spacing the discs,and a strong concentration of local fieldsin these regions. More specifically, the con-trasting properties (such as heat or electri-cal conductivity) of the components mayfall beyond the limits of applicability of thehomogenization approach see [3].

That is why the rigorous analysis of theproposed methods is very important. In thepresent book the reader finds an excellentintroduction to a new method that is appli-cable in exactly this most di"cult case ofa nonperiodic composite with contrastingproperties, presented by the authors of thismethod.

The book’s preface discusses the rela-tion between the discrete and continuousmodels. Chapter 1 reminds the reader ofmathematical notions and facts used laterin the book in the description and justifi-cation of the method of network approxi-mations. This review is brief, but containsall necessary references for the reader whowants to study the question more deeply.Chapter 2 provides an introduction to thetheory giving the motivation and some basicsimple examples explaining the main ideaof the method. Chapter 3 is the centralone. It introduces the rigorous formulationof the problem and contains the completeproof of the main theorem (Theorem 3.14)justifying the network approximation for-mula (3.8.17) in the case of a finite numberof closely spaced perfectly conductive inclu-sions. The main idea of the proof is the con-struction of upper and lower bounds for thee!ective coe"cient in such a way that theleading terms of the asymptotic expansionscoincide. The network approximations givean asymptotic solution to the problem ofcapacity of an arbitrary (random) systemof many closely placed discs. The asymp-totics of capacity may be computed fromthe solution to an algebraic N ' N linearsystem, where N is the number of the disks.

The advantage of the book is that it con-tains not only theoretical results, but thenumerical experiments as well, and Chap-ter 4 is devoted to numerics for percolationusing the method of network approxima-tions. It contains very interesting resultsfor the e!ective conductivity. Chapter 5considers the case of an infinite number ofclosely spaced perfectly conductive inclu-sions; it contains the rigorous formulation ofthe problem and the main theorem, whichshows that the error of the network ap-proximation is determined not by the totalnumber of particles (as is usual in homog-enization theory), but by the perimeter ofthe spaces between the groups of neighbor-ing particles. Chapter 6 generalizes the net-work approximation approach to the case ofnonlinear settings for the field equation (p-Laplacian). The previous chapters treatedconductivity of the material “as a whole,”that is, as a characteristic integral. Chap-ter 7 studies local properties of the networkapproximation approach. It shows, in par-

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ticular, that the potentials of the nodes inthe network approximation correspondingto the centers of closely spaced perfectlyconducting discs approximate the real po-tentials of the discs in the continuous model.Chapter 8 describes complex variable meth-ods in the analysis of two-dimensional prob-lems for closely spaced discs.

The book is well written. It gives an easybut detailed and comprehensive introduc-tion to the subject. The methods are ex-plained in simple examples. The proofs aresupplied by physical and geometrical inter-pretations so that the book is comprehensi-ble to a wide spectrum of readers who maynot be specialists in the topic. In particu-lar, it will be interesting for Ph.D. studentsin mathematics, physics, chemistry, or en-gineering. The price is low at around £50in Cambridge University Press catalogue.Manufactured on demand, it is supplied di-rectly from the printer and a second editionwill be available in paperback.

REFERENCES

[1] M. Born and K. Huang, Dynamical The-ory of Crystal Lattices, Oxford Univer-sity Press, Oxford, 1954.

[2] M. Sahimi, Heterogeneous Materials,Springer, New York, 2003.

[3] G. P. Panasenko, Homogenization of pro-cesses in strongly heterogeneous media,U.S.S.R. Dokl., 298 (1988), pp. 76–79(in Russian); English translation, So-viet Phys. Dokl., 33 (1988), pp. 20–22.

GRIGORY PANASENKO

Universite de Saint-Etienne

Number Theory. A Historical Approach.By John J. Watkins. Princeton University Press,Princeton, NJ, 2013. $75.00. xvi+592 pp., hard-cover. ISBN 978-0-691-15940-9.

This book could well have been titled Num-ber Theory and Its History, if Oystein Orehadn’t already used that title for a bookpublished in 1948. Ore’s book is somethingof a classic, but I believe that the modernreader will find Watkins’ book much moresatisfying. It is far richer, deeper, and morecomprehensive—which shows that numbertheory has become a much more main-

stream subject than it was in 1948. Notonly has it become a standard undergradu-ate course, it is almost required reading fora public concerned about the security andprivacy of the internet.

Watkins’ Number Theory is ideal for avariety of readers: it is well suited to anundergraduate course and also as generalreading for anyone from a bright high schoolstudent to a mathematician with an interestin number theory. In my opinion it main-tains the perfect balance between mathe-matics and history: most of the theoremsare proved in complete and rigorous fash-ion, as one would expect in a mathematicscourse, but the history explains where theideas came from and why they were of inter-est to mathematicians. Also, of course, thehistory of number theory is particularly richin characters and anecdotes (think Hardy,Ramanujan, and the number 1729).

The mathematical content includes theusual core topics: Pythagorean triples, ex-istence and uniqueness of prime factoriza-tion, the Euclidean algorithm, linear andquadratic Diophantine equations (particu-larly the Pell equation), congruences, sumsof squares, and quadratic reciprocity. Thesetopics ensure that the book could be used asan undergraduate number theory textbook,but there are many other topics that en-hance its flavor and usefulness: algorithmsfor factorization and primality testing, cryp-tography (more than just RSA), the Fi-bonacci numbers, continued fractions, andpartitions.

The methods used are mostly elemen-tary (though they grow in complexity asthe book unfolds) with essentially no alge-braic or analytic number theory. Thus, asWatkins says,

there is almost nothing in terms of pre-requisites that readers need to bringalong with them except enthusiasmand curiosity.

I wished at first that Watkins had takenthe opportunity to describe the evolution ofalgebraic and analytic methods that tookplace in response to the evolution of numbertheory, but on reflection I think he is wiseto stick to elementary methods. The bookis already quite hefty, and it would havebecome overweight if algebra and analysis

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were treated as full partners of numbertheory.

The book is written in a pleasant re-laxed style, with frequent historical excur-sions that motivate the material and pro-vide relief from the more technical passages.Topics are developed in roughly chronolog-ical order, starting with Fermat and visit-ing the highlights of his work and that ofhis successors Euler, Lagrange, and Gauss.There are also “flashbacks” to the work ofEuclid, Diophantus, Fibonacci, and otherswho helped pave the way for Fermat andother modern number theorists. For exam-ple, Chapter 1 proves quite ameaty theoremof Fermat—that the area of an integer-sidedright-angled triangle cannot be a square—with the help of a flashback to Euclid’streatment of Pythagorean triples.

Watkins’ narrative style of generally mov-ing forward, but with flashbacks, seems verye!ective. It allows the reader to reach ad-vanced results quite quickly, but also torelax occasionally by going back to a moreelementary level. Also, by allowing ideasto be revisited, it reinforces key ideas byreturning to them again and again. For ex-ample, we see Euclid’s proof that there areinfinitely many primes in Chapter 2, an ex-tension of the idea to primes of the form4n + 1 in the exercises in Chapter 6, anda more sophisticated proof that there areinfinitely many primes (by Euler) in Chap-ter 10. Likewise, the Fermat theorem inChapter 1 is revisited in Chapter 7, whereit is used to prove Fermat’s last theorem forfourth powers.

The historical content is very attractivelypresented and it will undoubtedly be abonus for most readers. As far as I cansee it is pretty accurate. The only slip Inoticed was on page 335: naming NicolasBernoulli, rather than Daniel, as the discov-erer of the so-called “Binet formula” for theFibonacci numbers. However, I must regis-ter a mild protest against the lack of refer-ences. There are no references to primaryhistorical sources (finding such a referencewould have picked up the Bernoulli errorabove) and this is particularly annoying inthe case of quotations. For example, we aretold on page 253 that Gauss was amazedthat Archimedes did not develop a decimal,or similar, notation for numbers, and that

Gauss said, “To what heights would sci-ence now be raised if Archimedes had madethat discovery!” If true, this is a remarkablenugget of information about Gauss—but nosource is given. (The quote also appears inE.T. Bell’s Men of Mathematics, and Bellgives no source either.)

The exercises are a particularly ad-mirable feature. They are not just plen-tiful, and with plentiful hints and solutions,but well thought out and interesting. Veryoften, substantial proofs are organized intoguided sequences of exercises. A notable ex-ample is Zagier’s notorious “One-sentenceproof that every prime p ( 1 (mod 4) is asum of two squares,” which appeared in theAmerican Mathematical Monthly of 1990and has caused many headaches since. Onpage 162Watkins expands Zagier’s sentenceinto an eight-step exercise, spread over twoand a half pages!

The book concludes with a brief introduc-tion to the free Sage software—perhaps notsu"cient to allow readers to start program-ming, but enough to whet the appetite—some well-annotated suggestions for furtherreading, and a useful pronunciation guide tothe names of prominent number theorists.(We mathematicians tend to forget that thepronunciation of Leibniz, Euler, and Jacobiwill not be obvious to our students.)

To sum up: this is a very rich, well-organized, and highly readable book. Itshould be accessible to a wide spectrum ofreaders and is particularly suitable for afirst course in number theory.

JOHN STILLWELL

University of San Francisco

Numerical Methods with Worked Exam-ples: MATLAB Edition. Second Edition.By C. Woodford and C. Phillips. Springer, Dor-drecht, 2012. $59.95. x+256 pp., hardcover.ISBN 978-94-007-1365-9.

One approach in numerical methods text-books, which may be considered the stan-dard, is to present methods first with anal-ysis followed by examples, either workedout by the author(s) or left as exercisesto the reader. While this approach is cer-tainly suitable for the reader whose inter-

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ests are principally in scientific computing,the analysis may prove unnecessary or evendiscouraging to other readers. Thus, Wood-ford and Phillips focus more on the ap-plication and implementation of numericalmethods instead of the analysis. Through-out the book, they present material in whatthey call the “problem-solution-discussion”order. For the reader who is looking to in-corporate numerical methods into their fieldwithout diving too deep into the analysis,the authors’ order of presentation is veryappealing.

The authors begin each chapter by out-lining its purpose and/or aims and finisheach one with a summary of what waslearned, supported by numerous exercisesfor the reader. In this way, the book isnicely compartmentalized so that the readermay pick and choose chapters of interestwithout being confused by previously intro-duced notation. The first chapter providesa well-written introduction to MATLABand includes variable creation, condition-als, flow control, and I/O. This ensures thatall readers can take advantage of the manyprogramming exercises provided through-out the book.

The following two chapters are on lin-ear and nonlinear equations. The authorsmotivate each of these chapters e!ectivelywith examples. Gaussian elimination, piv-oting, Gauss–Seidel iteration, the bisection,secant, and Newton’s methods are all nicelycovered, while the analysis is kept to a levelwhere the reader does not need linear alge-bra experience, aside from familiarity withmatrices and vectors. The conjugate gradi-entmethodmight have been worthmention-ing after the authors introduced symmetric,positive-definite systems; however, maybethe corresponding analysis would have beentoo heavy. Additionally, it is worthy of notethat there appears to be a typo in (2.45),where the authors seek to demonstrate thee!ect of round-o! error by exchanging therows in a linear system. The line shouldread 1.234x1 + 0.996x2 = 1.23 if “System(1)” and “System (2)” are to be equivalent.

The book proceeds with a chapter oninterpolation, motivated by the need for in-formation between measured values in anexperiment. The authors introduce poly-nomial interpolation using either Newton

divided di!erences to construct the poly-nomials or Neville’s method for tabulateddata. They state that higher-order polyno-mials come with higher computation costs,no guarantee of accuracy, and mention thatextrapolation can be a “risky proposition.”However, some discussion about the oscil-latory nature of higher-order polynomialscould have helped further illuminate thebreakdown of high-order polynomial inter-polation. They finish the chapter with somewell worked through examples on splines,linear least squares, and polynomial leastsquares interpolation.

The next two chapters are on numericalintegration and di!erentiation. For integra-tion, the authors stay true to their focus onapplication over analysis by first present-ing the trapezoid/trapezium and Simpson’srules with a graphical interpretation. Onlythen do they introduce some analysis witherror terms to explain why the errors inhigher-order methods decrease faster. Theconcepts of Gaussian and adaptive quadra-tures are also introduced. For di!erentia-tion, the analysis is deeper. Taylor series areused to introduce finite di!erences for firstand second derivatives. The finite di!er-ences themselves, derivation of error terms,and the use of Cauchy’s formula to avoidmachine precision issues (through reformu-lating high-order derivatives into integrals)are all very well written, albeit a slightdeviation from the authors’ focus on appli-cation. There is also a typo worth noting onpage 121, where the remainder R1 shouldbe at most h2/2 times f ##(#) (not h/2 asprinted). Otherwise, the error R1/h wouldappear to be proportional to 1 instead of h.

The next two chapters are on linear pro-gramming and general optimization. Thelinear programming chapter is thoroughlymotivated by word problems, including theubiquitous Traveling Salesman, that arethen translated into the appropriate formfor the reader. The authors cover the con-cept of feasible regions and the details of thesimplex method with graphical descriptionsrather than rigorous proof, keeping the anal-ysis to a minimum. Additionally, they coverinteger programming and decision making,all accompanied by real-world examples, be-fore moving to nonlinear optimization. Thissection is equally well motivated by physical

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problems and their corresponding quanti-ties to be optimized. Grid search methodsare generalized to the higher-dimensionalsteepest descent and quasi-Newton meth-ods. The authors close the chapter withpenalty and Lagrangian approaches to con-strained optimization, while keeping the lin-ear algebra and vector calculus analysis toa minimum.

The authors include ODE numericalmethods in the following chapter. After anice introduction to how physical problemslead to equations to be solved, the authorscover both Euler andRunge–Kutta (second-and fourth-order) methods for initial valueproblems. The authors use computed errortables instead of lengthy truncation erroranalysis to show the order of the methods,while mentioning how the error would bedetermined. They include how to reformu-late higher-order ODEs into first-order sys-tems and how to then apply the methods.The chapter is wrapped up with a discus-sion on boundary value problems and howshooting-type or finite di!erence methodscan be used. Although potentially outsidethe scope of this book, it might have beeninstructive to add some discussion on sta-bility or step-size limitations.

The last two chapters are on numericalmethods for finding the spectrum of a ma-trix and a light introduction to stochasticmethods. In introducing the power and QRmethods for finding eigenvectors and eigen-values, the authors understandably dive a

little deeper into linear algebra than theprevious chapters (including a simple proofof the linear independence of eigenspaces).This comes only after a nice motivation ofwhy the spectrum is important in engineer-ing, mainly dealing with resonance. Thestochastic methods chapter introduces theconcepts of distributions, expected value,variance, covariance, and correlation, alongwith how each one is computed. The au-thors also include introductory randomnumber generation using the modulus oper-ator and close the book with a descriptionof Monte Carlo integration.

The chapters of this book cover almosteverything that one would expect in a nu-merical methods book. The authors holdtrue to their endeavor to teach by exam-ple, including numerous motivating prob-lems and worked solutions. They routinelychoose graphical and intuitive motivationover proof and theory to e!ectively keep thereader focused on implementing the meth-ods. The downside to this approach is thatsome aspects of the methods are lost inthe balance of implementation with analy-sis. Thus, while this approach might leavesome readers looking for more detailed ex-planation or mathematical motivation, itpresents the book nicely to the reader moreinterested in seeing the methods work thanunderstanding why they work.

CHRIS VOGL


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siam review vol. 56, issue 3 (september 2014)people.math.gatech.edu/~dmargalit7/siamreview.pdf ·...

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