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Table of Contents

Looking Back . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1General Mathematics. History. Foundations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

MSC sections 00 General • 01 History and Biography • 03 MathematicalLogic and Foundations • 06 Order, Lattices, Ordered Algebraic Structures •08 General Algebraic Systems • 18 Category Theory, Homological Algebra

Number Theory. Algebra. Algebraic Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21MSC sections 11 Number Theory • 12 Field Theory and Polynomials • 13 Com-mutative Algebra • 14 Algebraic Geometry • 15 Linear and Multilinear Algebra;Matrix Theory • 16 Associative Rings and Algebras • 17 Nonassociative Ringsand Algebras • 19 K-Theory • 20 Group Theory and Generalizations

Real and Complex Analysis. Functional Analysis and Operator Theory. . . . . . . . . . . . . . . . 51MSC sections 26 Real Functions • 28 Measure and Integration • 30 Functions ofa Complex Variable • 31 Potential Theory • 32 Several Complex Variables andAnalytic Spaces • 33 Special Functions • 40 Sequences, Series, Summability •41 Approximations and Expansions • 42 Harmonic Analysis on EuclideanSpaces • 43 Abstract Harmonic Analysis • 44 Integral Transforms, OperationalCalculus • 46 Functional Analysis • 47 Operator Theory • 49 Calculus of Varia-tions and Optimal Control; Optimization

Differential, Difference and Integral Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76MSC sections 34 Ordinary Differential Equations • 35 Partial Differential Equa-tions • 37 Dynamical Systems and Ergodic Theory • 39 Difference and FunctionalEquations • 45 Integral Equations

Discrete Mathematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87MSC sections 05 Combinatorics • 52 Convex and Discrete Geometry

Topology and Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90MSC sections 22 Topological Groups, Lie Algebras • 51 Geometry •53 Differential Geometry • 54 General Toplology • 55 Algebraic Topology •57 Manifolds and Cell Complexes • 58 Global Analysis, Analysis on Manifolds

Probability Theory. Statistics. Applications to Economics and Biology. . . . . . . . . . . . . . . . . 109MSC sections 60 Probability Theory and Stochastic Processes • 62 Statistics • 91Game Theory, Economics, Social and Behavioral Sciences • 92 Biology and otherNatural Sciences

Numerical Analysis. Modelling. Computer Science. Algorithms. . . . . . . . . . . . . . . . . . . . . . 117MSC sections 65 Numerical Analysis • 68 Computer Science • 90 Opera-tions Research, Mathematical Programming • 93 Sytem Theory, Control •94 Information and Communication, Circuits

Mathematical Physics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132MSC sections 70 Mechanics of Particles and Systems • 74 Mechanics of De-formable Solids • 76 Fluid Mechanics • 78 Optics, Electromagnetic Theory •80 Classical Thermodynamics, Heat Transfer • 81 Quantum Theory • 82 Statis-tical Mechanics, Structure of Matter • 83 Relativity and Gravitational Theory •85 Astronomy and Astrophysics • 86 Geophysics

Editorial Statement

Starting in January 2010, Zentralblatt will offer its new print service Excerpts from Zentralblatt MATH.There will be one issue per month of roughly 150 pages.As opposed to the previous print edition the Excerpts will not reproduce all reviews from thedatabase, but limit itself to a collection representing all areas of mathematics. Full coverage will ofcourse still be provided by the ZMATH database, as before. Key features of the new print editionwill be:

• A selection of informative reviews in all fields of mathematics and its applications, taken fromZMATH database.

• Reviews chosen with a view towards potential interest to a wide audience.• Extended coverage of monographs; indeed almost all available book reviews will appear in print

in the new periodical.• A section entitled “Looking Back” allowing a fresh look at classical mathematical works; this

section will contain especially commissioned reviews by renowned experts.• New reader-friendly and physically browsable format.

For the production of the reference databases Z-MATH and MathSciNet, it has been agreed onproviding uniform references to reviews in Jahrbuch über die Fortschritte der Mathematik (JFM), Zbland MR.References to reviews in JFM will be given the format JFM vv.pppp.ii, where vv=volume,pppp=paging, and ii=placement on the page. (e.g., JFM 24.0048.02). n the Z-MATH database,these reviews are available athttp://www.emis.de/zmath-item/?vv.pppp.ii

References to reviews in Zbl will be given the current Zbl format, Zbl vvvv.rrrrr, referring to thedatabase item athttp://www.emis.de/zmath-item/?vvvv.rrrrr.References to reviews in the MathSciNet database will be given by the 7-digit MR number,MSCrrrrrrr, referring to the database item athttp://www.ams.org/mathscinet-getitem?mr=rrrrrrr.

Looking Back 1

Looking Back

Abel, N.H.The proof of the impossibility of solving alge-braic equations of degrees higher than four inradicals. (Beweis der Unmöglichkeit algebrai-sche Gleichungen von höheren Graden als demvierten allgemein aufzulösen.) (German)J. Reine Angew. Math. 1, 65-84 (1826).The question whether polynomial equations ofdegree larger than 4 can be solved by radicals wasone of the main questions of algebra at the begin-ning of the 19-th century. Still in 1771 Lagrangein a fundamental paper, presented to the BerlinAcademy, with the title Réflections sur la résolutionalgébrique des équations had written, “il serait àpropos d’en faire l’application aux équations descinquième degré et de degrés supérieurs, dontla résolution est à présent inconnue; mais cetteapplication demande un trop grand nombre derecherches et de combinaisons, dont le succès estencore d’ailleurs fort douteux”.Paolo Ruffini published several papers in whichhe claims that the solution of equations of degreelarger than 4 cannot be solved by radicals. But hisproof is difficult to understand and incomplete.His first published paper is from 1802.Abel was the first to give a complete proof, whichhe published in 1824 at his own expense in a pa-per entitled Mémoire sur les équations algébriques.In the paper at hand he gives a new, more elabo-rated version of his first proof. It consists of foursections.In the first section, Ueber die allgemeine Form alge-braischer Functionen, he defines some fundamen-tal notions like rational functions. In modern lan-guage he takes his base field as the rational func-tion field in several variables over a constant fieldwhich is not specified, but later on he uses thatthe constant field contains the roots of unity.In the second section, Eigenschaften der algebrais-chen Functionen, welche einer gegebenen Gleichung

genugthun, he proves that if the equation is solv-able by radicals, then all irrational quantities oc-curring in the expression of the roots are rationalfunctions of these roots. This theorem was as-sumed by Ruffini without proof. Its proof is theessential new step in Abel’s paper.The third section is entitled Ueber die Zahl der ver-schiedenen Werthe, welche eine Function mehrererGrößen haben kann, wenn man die Größen, vonwelchen sie abhängt, unter einander vertauscht. Inthis section Abel studies the possible values ofrational functions if the variables are permutedand gives the proof of a theorem of Cauchy, pub-lished in 1815, that the number of values a ratio-nal function of five variables attains cannot be 3or 4. This is the only quotation of the whole arti-cle. Ruffini is not quoted at all, but he appears inAbel’s paper of 1824.The fourth section, Beweis der Unmöglichkeit derallgemeinen Auflösung der Gleichungen vom fünftenGrade, then contains the rest of the proof that theequation of fifth degree with coefficients whichare general, i.e., are variables, cannot be solvedby radicals. In this section Abel follows the ideasof Lagrange, Ruffini and Cauchy.Nowadays Abel’s theorem about equations offifth degree is considered as a special case of Ga-lois theory: The general equation of n-th degreehas a Galois group which is the general permu-tation group of n letters. For n > 4 this group isnot solvable and therefore the general equationcannot be solved by radicals. Galois developedhis theory some years after Abel’s paper at handwas published. Galois quotes Abel in his paperof 1831 presented to the Académie des Sciencesde Paris.For more details see e.g. B.L. van der Waerden, AHistory of Algebra, Springer-Verlag Berlin (1985;Zbl 0569.01001). Helmut Koch (Berlin)

2 General Mathematics. History. Foundations.

General Mathematics. History. Foundations.

1167.00001Shahriari, ShahriarApproximately calculus.Providence, RI: American Mathematical Soci-ety (AMS) (ISBN 0-8218-3750-8/hbk). xvii, 292 p.$ 49.00 (2006).“Approximately Calculus” is a somewhat un-usual text for a second semester calculus course.One unusual aspect is the connection with num-ber theory: there are chapters on patterns andinduction, divisibility, primes (with applicationsto cryptography), distribution of primes, and theprime number theorem.The actual calculus course is modeled after thehistorical development of the subject, and is notaxiomatic. Consequently, it does not begin witha discussion of real numbers and the mean valuetheorem, but rather starts with the definition ofthe derivative and then gives the idea of Tay-lor expansion (without the remainder term) andintegrals (without going through the technicali-ties of refining partitions etc.) The main idea inthe development is approximation: the tangentapproximates the function locally, the Taylor se-ries approximates it globally, the Riemann sumsapproximate integrals (and the function x/ ln xapproximates the number of primes less than x).The feared ε’s and δ’s occur in Chapter 17 at theend of the book.There are a huge number of well-chosen exer-cises, in which the reader is rarely asked to provea given result but rather is encouraged to dis-cover a theorem for himself and to apply the ba-sic ideas for solving a problem. Often readers areasked to find approximations to numbers or (val-ues of) functions, or to relate different approachesto one and the same problem.My overall impression is very favorable: the bookis written around the basic and fundamental ideaof approximation, the exposition is excellent, andthere are many exercises that can’t be found insimilar books. Franz Lemmermeyer (Jagstzell)

1167.00003Yang, Xin-SheApplied engineering mathematics.Cambridge: Cambridge International SciencePublishing (ISBN 978-1-904602-57-6/pbk).x, 319 p. (2007).

It is well known fact that modern engineer-ing design and process modelling require bothmathematical analysis and computer simula-tions. There is a vast amount of literatures existson the said subject but so far there is no unifiedbook in the area so far available.This book is anattempt to fill up this gap. This book endeavorsto strike a balance between mathematical andnumerical coverage of mathematical and numer-ical techniques. The book emphasizes the ap-plication of important numerical methods withdozens of worked examples.The applied topicsincludes elasticity, harmonic motion, chaos, kine-matics,pattern formation and hypothetical test-ing. This book can be used as a text book in en-gineering mathematics, mathematical modelingas well as scientific computing in the universitylevel. Prabhat Kumar Mahanti (Ranchi)

1167.00005Andreescu, Titu; Dospinescu, GabrielProblems from the book.Allen, TX: XYZ Press(ISBN 978-0-9799269-0-7/hbk). xvi, 554 p. (2008).Quoting from the preface of the marvelous [M.Aigner and G. M. Ziegler, Proofs from THE BOOK(Springer-Verlag, Berlin) (1998; Zbl 905.00001)],Paul Erdös liked to talk about The Book, in which Godmaintains the perfect proofs of mathematical theorems.If one can imagine an edition of THE BOOK thatcontains also the beautiful problems, then thebook under review deserves, in this reviewer’sopinion, its title. It is worth noting that the con-nection between the book’s title and THE BOOKis not mentioned anywhere; even the cover de-sign groups the title as [PROBLEMS] [FROMTHE] [BOOK] instead of the more informative[Problems from] [THE BOOK].The book contains an excellent collection of prob-lems spread over 23 chapters each dealing witha single topic. These topics belong mainly to therealms of Number Theory, Algebra, and Com-binatorics. Each chapter starts with a brief in-troduction into the theory behind the chapter’sproblems, then a set of well chosen problemswith illuminating and instructive solutions, andfinally, under the heading Problems for Training,a list of practice problems. In all, the book has

General Mathematics. History. Foundations. 3

about 300 solved problems and about 400 prac-tice problems.Unlike a standard problem book, this one doesnot provide solutions for the practice problems.The authors feel that having a solution handywould tempt the reader to take a shortcut. Thismay be wise, but many problem solvers wouldalso love to compare their own solution to a prob-lem with the one that the proposers have in mind.However, the book already has more than 550pages, and including solutions would probablydouble this number.The book contains no Geometry problems. Theauthors suggest that there are already quite afew excellent Geometry problem books around.This may be true, but this reviewer, and proba-bly many other lovers of Geometry and problemsolving, would welcome an appearance of a Ge-ometry Problems from THE BOOK.References for sources of the problems are quiteinadequate. For example, some problems areonly referenced as [AMM], with no hint regard-ing which volume, year, or problem number ofThe American Mathematical Monthly is intended.This reviewer had to assume that this is inten-tional, conforming to the authors’ policy of keep-ing the reader away from the solutions.The book has its good share of misprints.Thus you see [magnificient] for [magnificent],[Matematicorum] for [Mathematicorum], [peri-odica] for [periodic], [Rusia] for [Russia], [Klark]for [Clark], [≤≤] for [≤], and so on. Althoughthese are in general harmless, they may give abrowser the wrong impression that the book hasbeen carelessly written and the wrong idea aboutthe book’s credibility. The authors did get thebook reviewed before getting it published – theythank, among others, Marian Tetiva whose closereading of the manuscript uncovered many errors thatthey would not have liked in the final version. Hername is misspelled as Marin.The book is a must for problem solvinglovers and mathematical competitions teams andcoaches, and it would undoubtedly be of greatinterest to all those who appreciate beauty in el-ementary mathematics. Mowaffaq Hajja (Irbid)

1167.00008Pulte, HelmutAxiomatics and empiricism. A historical-philosophical study of mathematical naturalphilosophy from Newton to Neumann. (Ax-iomatik und Empirie. Eine wissenschaftsthe-oriegeschichtliche Untersuchung zur Mathe-matischen Naturphilosophie von Newton bisNeumann.) (German)Edition Universität. Darmstadt: Wis-senschaftliche Buchgesellschaft (WBG) (ISBN3-534-15894-6/pbk). 502 p. EUR 52.00; SFR 86.50(2005).This is a book on the philosophy of science ofrational, or: analytical mechanics. It studies thedevelopment of the philosophical understand-ing of this field of science, beginning with IsaacNewton and going up to the end of the 19th cen-tury. The Neumann of the subtitle is the Germanmathematician Carl Neumann (1832-1925).The author explains in detail how the philo-sophical understanding of the basic principleschanged from seeing them as axioms in the tra-ditional sense of absolute and obvious truth’sto their understanding as well motivated andempirically based principles which constitute avery fruitful, deductively structured mathemat-ical model to describe the movement of isolatedparticles. Siegfried J. Gottwald (Leipzig)

1166.00001Bryant, John; Sangwin, ChrisHow round is your circle? Where engineeringand mathematics meet.Princeton, NJ: Princeton University Press(ISBN 978-0-691-13118-4/hbk). xix, 306 p. £ 17.95;$ 29.95 (2008).As its subtitle suggests, this book provides il-lustrations where elementary and fundamen-tal mathematical concepts provide a basis forphysics and engineering principles. The math-ematics consists primarily of simple geometricshapes and algebraic analysis encountered in sec-ondary school. Correspondingly the engineer-ing principles are those discussed in beginningphysics and engineering studies.But it is precisely these elementarymathematical-physical associations that give riseto the principles of modeling and approxima-tions forming the bases for modern technologicaladvances.The authors present these concepts in a series ofindependent fascinating illustrations. A reader


can open the book at random and immediately beengrossed in a stimulating discussion of a classi-cal mathematical-physical connection, with pro-found current and future applications.The book itself is divided into 13 chapters span-ning approximately 300 pages. The first threechapters discuss aspects of lines, linkages, andmechanisms. Chapter four discusses scaling andaccuracy. Chapter five extends these ideas to an-gular measurements and divisions.In Chapter six the authors discuss jigsaws, dis-sections, and packing. The subsequent threechapters consider approximations and limitingprocesses. Chapter ten then provides answers tothe title question.The book is likely to be a best-seller of interestto mathematicians, engineers, and physicists. Itwill also be of interest and accessible to the publicat large. It is an “easy-read" and yet a profoundlyinsightful treatise of modelings, approximations,analyses, and inventions.

Ronald L. Huston (Cincinnati)

1166.00002Glaeser, Georg; Polthier, KonradPictures of mathematics.(Bilder der Mathematik.) (German)Heidelberg: Spektrum Akademischer Verlag(ISBN 978-3-8274-2017-6/hbk). xi, 323 p.EUR 34.95; SFR 54.50 (2009).Over the last few years, there have been vari-ous publications in which the visualization ofmathematics played a major role, and “Picturesof Mathematics” is a particularly successful at-tempt at displaying the beauty of mathematicsthrough pictures. Of course we find here classicalexamples such as Escher’s tesselation of the hy-perbolic plane, fractal sets, polyhedra, and soapbubbles; but – and this is one of the strong pointsof the book – the authors also tried their handsat giving portraits of more advanced mathemat-ical objects. There are pictures of the Weierstraß℘-function, Riemann’s zeta function, the addi-tion formula for the sine function, knots, alge-braic surfaces, the Taylor expansion of the sine(to name a few), along with sketches of the math-ematical background.It is a fantastic book, parts of which are accessi-ble to highschool students, without being trivialor boring for professional mathematicians, and itshould be translated into other languages as soonas possible. Franz Lemmermeyer (Jagstzell)

1166.00004Behrends, EhrhardFive-minutes mathematics. Translated from theGerman by David Kramer.Providence, RI: American Mathematical Soci-ety (AMS) (ISBN 978-0-8218-4348-2/pbk). xxvii,380 p. $ 35.00 (2008).For two years beginning in May 2003, ProfessorEhrhard Behrends wrote a weekly column onmathematics, entitled ‘Five Minute Mathemat-ics’, for the German newspaper Die Welt. Thesecolumns were revised, expanded, and collectedinto a book. The American Mathematical Societyhave now published an English language editionof this collection.The author notes that when writing his columnshis aim was to emphasise three aspects of math-ematics: mathematics is useful; mathematics isfascinating; without mathematics one cannot un-derstand the world. This collection certainlyachieves this aim. Each chapter is self-containedand all are interesting and informative. There areexcellent chapters on probability, number theory,infinity, visualising higher dimensions, quantumcomputing, open problems in mathematics; fi-nancial mathematics, the mathematics of CATscans, the history of mathematics, to name buta few. The exposition is very clear and only as-sumes a minimal amount of prior knowledge.One of the main achievements of this book, inthe reviewer’s opinion, is the fact that sophisti-cated mathematical ideas are presented in an ac-cessible manner. The applications of mathemat-ics are given much consideration, but the readerwill also gain an appreciation for the beauty andpower of the subject and the enjoyment that canbe had from engaging with it.This book is aimed at the general reader andwould be an excellent resource for mathematicsteachers or anyone engaged in communicatingmathematics to the general public.

Julann O’Shea (Kildare)

1166.00005Gray, JeremyPlato’s ghost. The modernist transformation ofmathematics.Princeton, NJ: Princeton University Press(ISBN 978-0-691-13610-3/hbk). viii, 515 p.$ 45.00; £ 26.95 (2008).According to the author, the book contains thefollowing main novelties:


1. It gives a new picture of the shift into modernmathematics, one that sees it as a character-istic modernist development.

2. It uncovers a rich interconnected web ofmathematical ideas joining the foundationsof mathematics with questions at the fron-tiers of research.

3. It does this across every branch of mathemat-ics.

4. It shows that these questions, patently inter-nal as they are, forged links in the minds ofcontempory mathematicians between math-ematics, logic, philosophy and language.

Accordingly, it brings to light aspects of the lifeof mathematics not previously discussed notably,interactions which ideas of international and ar-tificial languages, and with issues in psychology.It integrates a number of issues too often treatedseparately: philosophy of mathematics in thehands of Husserl and Frege, Russell and Peirce;interactions between mathematics and physics,theories of measurement. In particular

a. it brings in the important figure of WilhelmWundt, who is almost always ommitted;

b. it always reminds the historian of mathemat-ics of a number of forgotten figures: JosiahRoyce, W. B. Kempe (for once, not for his fal-lacions proof of the four-color theorem);

c. it brings in a number of philosophers hith-erto marginalized in the history of mathemat-ics (Herbart, Fries, Erdmann, and Lotze) oreven completely forgotten, such as W. Tobias,G. F. Lipps, and S. Santerre;

d. it documents how the very names of Leibnizand Kant were profitably taken as markingmajor, reval, and evolving positions in logicand the philosphy of mathematics;

e. it takes the popularization of mathematicsaround 1900 seriously, as indicative of the fun-damental changes mathematics was undergo-ing, and being seen to undergo;

f. it looks as the impoartance of the history ofmathematics, which had a resurgence in theperiod, and considers the ways in which itwas written;

g. it remises the issue of modernism in theology,which coexisted with the situation in mathe-matics but incurred outsight opposition andrepression.

It is not only a collossal book, but also a very goodone. Let us mention here only some titles of sec-

tions in order to give the reader an impression ofthe contents of the book. We found:Mehrten’s moderne Sprache Mathematik, mod-ernism and mathematics, science-mathematics-philosphy, the modernization of mathemat-ics, geometry (Pappos-Desargues-Lobachevskii-Bolyai-Riemann-Beltrami), analysis (Cauchy-Weierstrass-Green), algebra (Kummer), phi-losophy (Kant), mathematicians (Grassmann-Riemann-Kronecker-Helmholz-Boole-Peirce-Ladd). So far we mentioned only things from thebook to and including the Introduction, Chap-ters 1 and 2, a total of 112 pages; but the bookhas 516 pages!It is impossible from the reviewer to mention thecontents, let alone the details, of the book. Thereader is referred to the keywords belonging tothis review.The author made a valuable piece of work inbringing together all the intended material in onebook, namely this one.

R. W. van der Waall (Huizen)

1166.00008Adams, Dennis M.; Hamm, MaryBringing science and mathematics to life for alllearners.Hackensack, NJ: World Scientific (ISBN 978-981-279-163-4/hbk; 978-981-279-164-1/pbk). vii,265 p. $ 65.00, £ 35.00/hbk; $ 42.00, £ 23.00/pbk(2008).This book focuses on strategies designed to in-clude and interest all pupils in a mixed-abilityscience or mathematics classroom. It advocatesthat teachers should have high expectations ofeach student and that different learning stylesshould be accomodated.In Chapter 2, the authors consider collaborativeinquiry and give good examples of group tasks.They stress the importance of involving every-one in the planning and execution of tasks. InChapter 3, they explore collaborative learning.Here students work in groups and the group hasthe responsibility to ensure that everyone partici-pates and understands. Again examples of grouptasks are given. In Chapter 4 differentiated in-struction according to student learning styles isadvocated. Detailed lesson plans for classes seek-ing to explore multiple intelligences are given.The use of portfolios as an assessment tool issuggested. In Chapter 5, the authors give detailsof many activities which straddle subject bound-aries and help students think like scientists about


the big ideas in science. They also give strategiesfor teachers to reach all students in a collabo-rative learning environment. In Chapter 6, thenature of mathematics and problem solving areexplored. Once again, many good ideas on howto make mathematical concepts concrete and rel-evant to pupils are given. The use of projects isconsidered in Chapter 7, and Chapter 8 discussesthe effects of various technologies on the learningenvironment.This book focuses on inquiry and problem solv-ing in collaborative groups. There are manygood, practical ideas that should prove usefulfor a teacher of science or mathematics at pri-mary level. Julann O’Shea (Kildare)

1166.01001Curbera, Guillermo P.Mathematicians of the world, unite!The international congress of mathematicians.A human endeavor.Wellesley, MA: A K Peters (ISBN 978-1-56881-330-1/hbk). xvii, 326 p. $ 59.00 (2009).In 2006 the author organized an exhibit on thehistory of the International congresses of mathe-maticians (ICM), which was successfully shownat the 25th congress in Madrid. The book cantherefore partly be read as a detailed catalogueto this exhibit. The title is alluding to an addresswritten by Felix Klein on the occasion of the un-official congress in Chicago 1893 preceding thefirst official one in Zurich 1897. Curbero claimshere (p.7) an echo of Marx’ Communist Mani-festo, which seems a bit far fetched given thedominant political opinions among the mathe-maticians of the time.The author describes in the preface that he hadto find new material particularly for the firstcongresses before World War II, because thearchives of the International Mathematical Unionin Helsinki did not contain anything about thatperiod. Curbero succeeded, for example in con-tacting the then over one hundred years oldSwiss mathematician J. J. Burkhardt (1903-2006)who had helped to organize the Zurich congressof 1932. Curbero found also, with the help ofmany colleagues abroad, new and interestingmaterial on the Oslo congress of 1936 and onthe origins of the Fields medals in mathematics,which were awarded for the first time in Oslo.The book proceeds chronologically, describingeach congress on the basis of the published Pro-ceedings, usually citing the titles of the plenary

lectures, commenting on the number and the na-tional distribution of participants, and on thefinancial conditions. This is accompanied by awealth of partly new pictorial material of goodquality, which is sometimes, due to lack of betterchoices, slightly anachronistic, particularly withrespect to the age of the mathematicians (for ex-ample showing in the context of the congress of1904 (Heidelberg) the mathematician P. Painlevéwhen greeting 1927 as French prime minister inParis the American pilot Ch. Lindbergh). Thebook has several ‘interludes’: on images of theICM, on awards, on social life at the ICM, and onthe International Mathematical Union. The po-litical strains particularly in the interwar years,when the Germans and their allies during theFirst War were temporarily ousted from the con-gresses, are mentioned too. The book is con-cluded by a bibliography of the various confer-ence proceedings (which in the text are usuallynot directly quoted), secondary literature, and anextensive index.

Reinhard Siegmund-Schultze (Kristiansand)

1166.01002Sivin, NathanGranting the seasons. The Chinese astronomi-cal reform of 1280. With a study of its many di-mensions and a translation of its records. Withthe research collaboration of the late KiyosiYabuuti and Shigeru Nakayama.Sources and Studies in the History of Math-ematics and Physical Sciences. New York,NY: Springer (ISBN 978-0-387-78955-2/hbk). vi,664 p. EUR 46.95/net; SFR 78.00; $ 69.95; 37.99(2009).The present book is a brilliant and wide-rangingstudy of the famous Shoushi li (the Shoushi li isthe famous Chinese astronomical canon whichwas promulgated in 1280, at the beginning ofthe Yuan (Mongol) dynasty; it is generally con-sidered as the summit of Chinese instrumen-tal and mathematical astronomy and, as such,it wholly depends on numerical techniques for-eign to the Greek geometrical tradition). Far fromconsidering his subject under the narrow angleof the technical history of astronomy, on the con-trary, the author views alike technical and non-technical dimensions of the question as elementsof interdependent historical processes (“culturalmanifolds"). While doing so and when manage-able, he never forgets the comparative dimen-sion of the history of astronomy, given what is


presently known of ancient civilisations in thisrespect. Consequently, he provides a sophisti-cated and critical analysis of the complex cul-tural, political, bureaucratic, personal and tech-nical background of the underlying astronomi-cal reform. Although the author always showsa degree of sinological expertise reflecting thestate of the art at its best in such domains (notonly Western but also Chinese and Japanese), hehas succeeded to make his book highly readable,particularly by translating the totality of Chi-nese terms into English, even the names of reign-periods and technical terms but always withoutleaving aside the needs of specialists since he alsoprovides detailed index, showing the correspon-dence between English and Chinese terms. In ad-dition, well beyond the Shoushi li itself, the bookalso contain a general survey of Chinese astron-omy which will certainly become a valuable sub-stitute of most aspects of Joseph Needham’s his-tory of Chinese astronomy for many readers (cf.Science and Civilisation in China, Cambridge,vol. 3, 1959). As regards the translation itself,the author has kept the usual mathematical arse-nal (algebraic and the like) minimal in order toavoid the pitfalls of anachronism often inherentin such a metamorphosis of ancient formulations.To sum up, the present book will constitute at thesame time a necessary basis of further investiga-tions into Chinese astronomy and, more broadly,a landmark with respect to the history of worldastronomy. J.-C. Martzloff (Paris)

1166.01003Anderson, Marlow (ed.); Katz, Victor (ed.);Wilson, Robin (ed.)Who gave you the epsilon? and other tales ofmathematical history.Spectrum. Washington, DC: The MathematicalAssociation of America (MAA) (ISBN 978-0-88385-569-0/hbk). x, 429 p. $ 65.50 (2009).Lately the MAA has engaged in the custom ofreprinting papers on special topics. This book is acontinuation of the bestselling “Sherlock Holmesin Babylon" (2004; Zbl 1035.00005), and therehave been similar collections, like “The Genius ofEuler" (2007; Zbl 1120.01009), “Euler and Mod-ern Science" (translated in 2007; Zbl 1165.01010),and “Musing of the Masters" (2004).The book under review consists of a collection of41 articles on the history of mathematics in the19th and 20th centuries. The articles are takenfrom MAA journals printed between 1900 and

2007. They are written by well-known mathe-maticians (G. B. Halsted, G. H. Hardy, B. L. Vander Waerden, H. Weyl, and others); the length ofthe papers runs between one and some 20 pages.The collection is divided into analysis (10); ge-ometry, topology and foundations (11); algebraand number theory (16), and surveys (4).There is no doubt that after decades the historyof science has made some progress, and in thisbook for each group such gaps are bridged by Af-terwords (2-3 pages) which give a short outlookwhereas Forewords (2-3 pages) for each groupembed and introduce the topics dealt with. Al-though the papers are well chosen and interest-ing to read today, of course the supplied frame,Fore- and Afterwords, is simply too short to beelucidating in a satisfying way.Most of the articles were reviewed individuallyearlier. That’s why we will use three papers byway of example in order to demonstrate the col-lection’s advantage and disadvantage. The bookopens with Grabiner’s article that gave the col-lection its title. This paper was published in 1983,i.e., in a period when non-standard analysis hadalready supplied other views on Cauchy, butsuch controversial views (in English J. Cleaveor G. Fisher, let alone D. Spalt in German) areomitted by the author and the editors (in theirAfterword). Nor is the recently published trans-lation of Cauchy’s “Cours d’Analyse" (1821), thefirst time in English, mentioned. Grabiner states,“Cauchy did not distinguish between point-wiseand uniform convergence" (p. 10) as well as “Butit was Cauchy who gave rigorous definitions. . . and the modern rigorous approach to calcu-lus" (p. 12). If these statements (on continuity)were true, how could Cauchy be rigorous on thebasis of such vague and obviously confused no-tions? Furthermore, we find two comments onthe 2nd ICM in Paris in 1900, one by Halsted(2 pp.) and the other one by Griffith (12 pp.),written a century apart – an interesting historicalcombination.Although the intention of the editors is to presentabove all American papers in the Afterwords,they leave the non-American reader anyhowirritated because this concentration mostly onthe U.S. book market overlooks some essentialworks, like J.-P. Pier’s “Developing of Mathemat-ics" (2000), one that surely should be referenced.

Rüdiger Thiele (Halle)


1167.01005Auffray, Jean-PaulEinstein and Poincaré. On the traces of relativ-ity. (Einstein et Poincaré. Sur les traces de larelativité.) 2nd ed. (French)Essais et Documents. Paris: Éditions Le Pom-mier (ISBN 978-2-7465-0233-8). 284 p. EUR 29.00(2005).This is a second, enlarged edition of an earlierbook devoted to origins and development ofideas which led to the theory of relativity. Em-phasis is put on contributions made by Poincaréand Einstein but the book reminds necessar-ily also of those made by Lorentz, Minkowski,Hilbert and a few others. The author’s sympathylies with Poincaré but he does his best to cast adry light. Narration runs chronologically and thestory, concentrated upon main ideas and fruitfulcontroversies, is told in a vivid and clear man-ner, apparently with a view to a wider audience.Everyone interested in the genesis of modernphysics, including historians and philosophers,will enjoy reading the book.

Roman Duda (Wrocław)

1167.01008Bradley, Robert E. (ed.);Sandifer, C. Edward (ed.)Leonhard Euler. Life, work and legacy.Studies in the History and Philosophy of Math-ematics 5. Amsterdam: Elsevier (ISBN 978-0-444-52728-8/hbk). viii, 534 p. EUR 144.00;$ 175.00; £ 110.00 (2007).The articles in this book discuss “Euler, his workand times”, and aim to “give a more or lesscomplete picture”. Anyone who is aware of thebreadth of Euler’s work knows that this is an im-possible task for a book with little more than 500pages.R. S. Calinger’s outstanding article titled “Leon-hard Euler: Life and Thought” gives a readableand thorough account of Euler’s life and his mainresults. P.Hoffmann explains Euler’s relation toRussia and the Russians, and R. S. Calinger andE. N. Polyakhova look at “Princess Dashkova,Euler, and the Russian Academy of Sciences”.W. Breidert gives an account of “Leonhard Eulerand Philosophy”, and F. Fasanelli presents theknown “Images of Euler”.The next articles are dedicated to Euler’s work.C. Wilson (“Euler and applications of analyti-cal mathematics to astronomy”) and K. Plofker(“Euler and the Indian Astronomy”) discuss

some of Euler’s work in astronomy, T. Koetsier(“Euler and Kinematics”, with connections toquaternions and the four-squares theorem) andS. G. Langton (“Euler on rigid bodies”) take careof his work in physics. V. Katz studies “Euler’sAnalysis Textbooks”, and R. Thiele “Euler andthe calculus of variations”. R.E. Bradley’s articleis about “Euler, d’Alembert and the logarithmfunction” and presents controversy between Eu-ler and d’Alembert on the nature of logarithmsof negative numbers. The article by E. Sandiferon “Some facets of Euler’s work on series” ex-plains the development in Euler’s notation andis extremely useful for those who want to studyEuler’s original articles. H. S. White deals with el-ementary aspects of “The geometry of LeonhardEuler”.O. Neumann’s article “cyclotomy: from Eulerthrough Vandermonde to Gauss” already be-longs to the “legacy” part of this book: only a fewpages are about Euler, the rest is a detailed expo-sition of the work of Vandermonde and Gauss onthe solution of the cyclotomic equation by rad-icals. J. Suzuki talks about “Euler and numbertheory: a study in mathematical invention”; themain topics are Euler’s four proofs of Fermat’s lit-tle theorem and a few results on quadratic formsand diophantine equations.The following articles are devoted to special top-ics: “Euler and Lotteries” by D. R. Bellhouse,“Euler’s science of combinaorics” and “The truthabout Königsberg” (Euler did not use graphs inhis proof) by B. Hopkins and R. Wilson, and “Thepolyhedral formula” F−E+V = 2 by D. Richeson.The rest of the book deals with Euler’s legacy: thearticle by I. Grattan-Guinness is “On the recog-nition of Euler among the French, 1790–1830”,S. Caparrini discusses “Euler’s influence on thebirth of vector mechanics”, and K. Reich stud-ies “Euler’s contribution to differential geometryand its reception”. Finally, D. Suisky tries to use“Euler’s mechanics as a foundation of quantummechanics”.The book has a well-prepared index, and per-haps the only flaw (apart from the high price) isthat the editors did not demand a common for-mat for the references. Yet it is a valuable book,with a lot of information that cannot easily befound elsewhere, and a welcome addition to thelist of books about Euler that have appeared tocommemorate the tercentenary of Euler’s birthin 1707. Franz Lemmermeyer (Jagstzell)


1167.01011Santaló, Luis AntonioSelected works of Luis Antonio Santaló. Editedby Antonio M. Naveira and Agustí Reventósin collaboration with Graciela S. Birman andXimo Gual. Preface by Simon K. Donaldson.Berlin: Springer (ISBN 978-3-540-89580-0/hbk; 978-3-540-89581-7/e-book). ix, 854 p.EUR 149.00/net; SFR 249.00; $ 229.00; £ 136.50(2009).This book, as the title suggests, is a selectionof papers of Santaló. He was a pioneer in theIntegral Geometry. S. K. Donaldson write in thePreface of this book: “. . . the theory of IntegralGeometry, to which Luis Santaló contributed somuch".Luis Antonio Santaló (1911, Girona, Spain - 2001,Buenos Aires, Argentina), under the influenceof Julio Rey Pastor (1888-1962) and Esteban Ter-radas (1883-1950) he moved to Hamburg to workwith Wilhelm Blaschke (1885-1962). In 1936 San-taló achieved the doctoral degree with a memo-ria on Integral Geometry. After the Spanish CivilWar (1936-1939) he moved to Argentina. First ob-tained a position at the city of Rosario and, later,at Buenos Aires. He is the main figure of the set ofSpanish mathematicians which after of SpanishCivil War went America.The editors of this volume have classified the pa-pers included in the selection in five parts.The Part I is dedicated to Differential Geometry(11 papers, 1945-74), with comments by E. Teufel,which say: “in many aspects his works served asan inspiration and a starting point for mathemat-ical research to the present day".The Part II is devoted to Integral Geometry (26papers, 1936-80), with comments by R. Laugevin.He recall that “Santaló is known for a topic (In-tegral Geometry) and a book (Integral Geometryand Geometric Probability)", and also that “manyof Santaló’s papers deal with hyperbolic geomet-ric".In the Part III, dedicated to Convex Geometry (9papers, 1940-88), R. Schneider, in his comments,affirms that this topic is “a smaller part of San-taló’s mathematical work" and he to point theBlaschke-Santaló inequality as a notable result.The Part IV is dedicated to Affine Geometry(5 papers, 1946-62) with comments by K. Le-ichtweiss, which say “Santaló was an especiallyversatile geometer".Finally, in the Part V, devoted to Statistic andStereology (9 papers, 1940-84), L. M. Cruz-Orive

write in his comments: “Integral Geometry re-ceived a fundamental push in the 1930’s from theHamburg School of W. J. E. Blaschke, whose dis-ciples exerted a great influence. A distinguishedmember of that School was Luis A. Santaló,whose career extended for over sixty years. His1976 book is the undisputed reference on IntegralGeometry and Geometric Probability".This book contains comments on some Santaló’spapers by several mathematicians and somebook reviews by Santaló.Santaló showed a great interest in the teachingof Mathematics and worked very actively withthe societies of teachers. Some papers of San-taló are devoted to Mathematics Education. Hemaintained a near relation with the Spanish ge-ometers, especially with Enrique Vidal-Abascal(1908-94). The book ends with a part dedicated tothe correspondence between Santaló and Vidal-Abascal. Santaló received numerous awards inArgentina, Spain and other countries.The editors have done a wonderful job of se-lection of papers and colleagues who have con-tributed with their comments and opinions.

Antonio Martinón (La Laguna)

1167.01012Wußing, Hans6000 years of mathematics. A cultural-historicaljourney. 1. From the beginnings to Leib-niz and Newton. With the collaboration ofHeinz-Wilhelm Alten and Heiko Wesemüller-Kock. (6000 Jahre Mathematik. Eine kul-turgeschichtliche Zeitreise. 1. Von den Anfän-gen bis Leibniz und Newton.) (German)Berlin: Springer (ISBN 978-3-540-77189-0). xiv,529 p. EUR 29.95; SFR 49.00 (2008).In 1 Mathematik am Anfang und Ethno-mathematik geht es um den Bereich bis8000 v. d. Z., denn der “grundsätzlicheWandel gesellschaftlichen Lebens in den. . . Frühkulturen . . . zog die Notwendigkeit desZählens und der Bildung von Zahlwörtern nachsich. Deshalb kann dieser Zeitraum im weitestenSinn als Anfang der Mathematik angesehen wer-den.” (S. 6) Die Ursprünge eigentlichen mathe-matischen Denkens freilich liegen erst 6000 Jahrezurück. – Im Nachgang zu seiner Geschichte derNaturwissenschaften, Leipzig 1983, stellt HansWußing mit dem nun vorliegenden Teil 1 dieanhängige Zeitspanne einem breiten Leserkreisteils in beinahe erzählerischem Stil, teils in knap-per Formulierung vor. Er spricht hierbei den


Wissensstand während einzelner dieser Epocheneher pauschal an, gibt ihn für andere jedochauch anhand von Rechenexempeln detailgetreuwieder. Meistens leitet er die Abschnitte mitübersichtlichen Zeittafeln hinsichtlich der allge-meinen und der Kulturgeschichte des gerade be-trachteten Bereichs ein und fasst wesentliche In-halte jeweils am Schluss zusammen. Schritt fürSchritt führt er trotz des Mutes zur Lücke seineLeserschaft in sein Metier ein, indem er sichzwar schon hierbei auf zum Teil nur stichwortar-tige Erläuterungen beschränkt, andererseits je-doch in einem weiten Bogen bereits auf die hier-für relevanten Länder unserer Erde eingeht; eruntermauert seine Aussagen laufend mit Hin-weisen auf Sekundärliteratur, bringt sie durchBilder nahe, markiert deutlich auf Landkartendie betreffenden Kulturkreise und aktualisiertdies sehr reizvoll vermittels Briefmarken. DieVerzeichnisse zu Literatur, Abbildungen, Per-sonen mit Lebensdaten und Sachen übertref-fen den sonst üblichen Rahmen. Konstruktions-zeichnungen veranschaulichen Vieles. Folglichist es nicht nur ein kompaktes Nachschlage-werk, denn hier handelt es sich um mehr alsum komprimiertes Allgemeinwissen. Die einzel-nen Kapitel der acht Abschnitte sind klar, zumTeil diffizil gegliedert und scharf gegen einan-der abgegrenzt. Es ist nicht einfach eine weitere“Geschichte der Mathematik“, sondern ein allge-meinbildendes Buch, das schon durch die vielenZitate, durch den jeweiligen Rundblick auf dieallgemeine Geschichte, auf die politische und aufdie wirtschaftliche Situation der Menschen eineGeschlossenheit in der historischen Abfolge er-reicht, die bei einer normalen “Geschichte derMathematik” in dieser Art und Weise wohl nichtdarstellbar wäre.Sowohl die eine als auch die andere eigene, ver-meintlich historische Tatsache erscheint somitplötzlich unter einem anderen Blickwinkel, alsman dies bislang gewohnt war; freilich gestehtVerf. hierbei nicht nur sich selbst, sondern auchanderen genügend Raum für Spekulationen zu.1.2 Ethnomathematik führt bereits aufgrundmathematikhistorischer, pädagogischer undpolitischer Aspekte – “Darum kann und darf diemoderne Historiographie der Mathematik nichtan der Ethnomathematik vorbeigehen.” (S. 17)– u. U. zu einer heute stark veränderten Ein-schätzung der Leistungen früherer Kulturvölker.In 2 Entwicklung der Mathematik in asiatis-chen Kulturen geht es um die Mathematik im

alten China: dortiges Wissen zwischen 200 v.und 300 n. d. Z. gilt als unübertrefflich; im 7.Jahrhundert: Überseehandel, Schießpulver, Kar-tographie, Druck auch mit beweglichen Lettern;beachtliche Leistungen des 13. Jahrhunderts wer-den vorgestellt. – Zur frühen Mathematik inJapan gibt es hingegen keinerlei verlässliche In-formationen. Die Abschottung Japans von frem-den Einflüssen während der sog. Edo-Epochezwischen 1603 und 1867/68 setzte zeitlich etwamit dem Beginn eines glanzvollen Zeitabschnittsvom Ende des 16. Jahrhunderts bis etwa 1675 ein:“Gestützt auf chinesische Quellen erreichte diejapanische Mathematik ein hohes Niveau, dur-chaus vergleichbar mit der Entwicklung in West-europa in der Zeit von Leibniz und Newton, bishin zur Infinitesimalrechnung.” (72)Starker hellenistischer Einfluss um die Zei-tenwende im Nordwesten Indiens hatteAuswirkungen auf die Entwicklung der dortigenMathematik. Die “Null” spielte die Hauptrolle:dies “. . . führte schließlich zur Etablierung desdezimalen Positionssystems. Es war spätestensim 7. Jahrhundert n. Chr. weit verbreitet.” (99)3 Frühzeit der Mathematik im Vorderen Ori-ent: Anhand bedeutender Zeugnisse aus demMittleren Reich vom 18./17. Jahrhundert v. d. Z.,und zwar kaum mehr als zehn Papyri, er-läutert Verf. die altägyptische Rechentechnik,geht auf die “Hau“-Rechnungen und auf die dor-tige Lösung geometrischer Probleme ein. – Dienach Thales und Pythagoras benannten Sätzewaren in Mesopotamien wohl schon um 1500v. d. Z. bekannt. “Mit der Erfindung der Buch-stabenschrift durch die Phönizier hat sich dieKeilschrift überlebt. Aus dem phönizischen gin-gen schließlich das griechische, das lateinischeund das frühslawische Alphabet hervor. Die Ent-wicklung der Keilschrift hatte auch Auswirkun-gen auf die Schreibweise mathematischer Texte.”(126)4 Mathematik in griechisch-hellenistischer Zeitund Spätantike: Die griechische Mathematik er-reichte “einen prinzipiell neuen Stand” (147),wobei “aus einer nahezu empirisch ent-standenen und oft nach Art von Rezeptenbetriebenen Mathematik eine systematische,logisch-deduktiv dargelegte, eigenständige Wis-senschaft Mathematik mit spezifischen Zielset-zungen und Methoden” (147) hervorging. Diegriechisch-hellenistische Mathematik vom 7./6.v. bis etwa zum 5. Jahrhundert n. d. Z. umfasstevier sich gegen einander abgrenzende Epochen:


1) ionische mit Seitenblick auf die Naturphiloso-phie: “In dieser Periode erfolgte schließlich dieHerausbildung der selbstständigen WissenschaftMathematik.” (150); 2) athenische; 3) hellenistis-che oder sog. alexandrinische stach ca. vierein-halb Jahrhunderte lang hervor: Euklid; Eratos-thenes; Archimedes: “Mit Archimedes erreichtedie Mathematik der Antike ihren Höhepunkt.”(194); Apollonios von Perge; Ptolemaios; Heron;Diophant; Pappos; 4) Spätantike.5 Mathematik in den Ländern des Islam: “. . . einebeispiellose Entfaltung von Kultur und Wis-senschaft . . . von Philosophie und Medizin, vonBaukunst, Gartentechnik und handwerklichenFertigkeiten” (222) bewirkte, dass sich vom 7.bis zum 15. Jahrhundert diese Kulturgüter mitdem Islam ausbreiteten. Verf. hebt die Bedeu-tung islamischer Gelehrter hervor, beschreibt al-Hwarizmi’s Verdienste um die Gleichungslehre[Algebra] und um das indische Ziffernrechnen[Algorismus]; er zeigt, wie die “Algebra” al-Hwarizmi’s von Nachfolgern weiterbetriebenund ausgebaut wurde und stellt so einen ak-tuellen Übergang zu 6 Mathematik im Eu-ropäischen Mittelalter her. – Alcuin von Yorkmachte Tours zu einem Bildungszentrum; Hra-banus Maurus als “Praeceptor Germaniae“; Ger-bert’scher Abacus. “Bei aller Anerkennung dereigenständigen Leistungen in der Mathematikdes Frühmittelalters bleibt doch die historischeTatsache, dass der eigentliche Aufschwung dereuropäischen Mathematik kausal an die Bekan-ntschaft mit der islamischen Mathematik gebun-den war” (276). Der Ruf der seinerzeitigenGelehrten wird hervorgehoben; verwiesen wirdauf die großartige damalige Architektonik, nurmöglich auf der Basis der angewandten Geome-trie, und auf die Bauhüttentradition. “Im 11./12.Jahrhundert setzte sich in der Kirche die Er-kenntnis durch, dass im Interesse der eigenenpolitischen und ökonomischen Macht des Klerusdie Geistlichen Lesen, Schreiben und Rechnenerlernen und wissenschaftliche Bildung erwer-ben und über die Klöster hinaus ins öffentlicheLeben tragen sollten. Ebenso wurden in raschaufblühenden Städten die Bürger gedrängt, sichKenntnisse und Bildung anzueignen.” (281)Drei geistige Ströme “trafen zusammen,berührten und durchdrangen sich im Laufe des12./13. Jahrhunderts” (281): griechisch-hellenis-tisches Erbe; bedeutende Teile der islamischenMathematik; christliche Denkweisen. Auf denUniversitäten tritt in der Scholastik die “Einpas-

sung von Aristoteles in die katholische Theolo-gie” (285) durch Albertus Magnus und Thomasvon Aquin hervor: “Beim Bildungsmonopol desKlerus waren fast nur Geistliche imstande, denZugang zur Wissenschaft zu finden.” (288), undso werden vorgestellt: Gerbert von Aurillac,Robert Grosseteste, Roger Bacon, Thomas Brad-wardine, Nicolaus Oresme; doch: “Das Erbeder klassischen Antike war weitgehend ver-schüttet, lebte allenfalls fort in den Dom- undKlosterschulen des sich ausbreitenden Chris-tentums.” (296) – 7 Mathematik während derRenaissance: Die Nationalsprachen traten inErscheinung. Hans Wußing beschreibt diesenZeitabschnitt sowohl insgesamt besonders ein-fühlsam – Man wollte Größe und Glanz desAlten Rom wiedererstehen lassen [Kurt Vogel] -,als auch im Detail; nach dem Hinweis auf tech-nische Neuerungen hebt er die damalige “Stag-nation des wissenschaftlichen Lebens” (304)an den Universitäten hervor. “Die Erneuerungund Fortentwicklung der Wissenschaften gingvielmehr von einer mit der Entwicklung desFrühkapitalismus eng verbundenen Gruppevon Kaufleuten, Handwerkern, darstellendenund bildenden Künstlern, Priestern, Geschütz-meistern, Rechenmeistern, Architekten undÄrzten aus” (304); “Mathematik war nicht längerbloßes Bildungselement im System kirchlich-scholastischer Gelehrsamkeit, eingegliedert indas Studium der sieben freien Künste. Nun, inder Renaissance griff die Mathematik ins täglicheLeben ein, ihre Produktionspotenz wurde deut-lich. Aufs Ganze gesehen und ein wenig grobbetrachtet, schritt die Mathematik während derRenaissance in drei Hauptrichtungen vorwärts:Rechenmeister und frühe Algebra; Geometrieund Perspektive; Astronomie und Trigonome-trie.” (309f.) Nach den Kurzbiographien vonLeonardo von Pisa, Luca Pacioli, Nicolas Chu-quet, Robert Recorde, Simon Stevin, PedroNunes, werden als sog. “deutsche Cossisten”Adam Ries und Söhne; Christoff Rudolff, dervor 1543 [!] starb; Michael Stifel mit Diskussionüber Irrationalitäten gewürdigt; Christoph Clav-ius’ Wirken, “Euklid des 16. Jahrhunderts” (346),kommt zur Sprache; Verf. zeigt, wie “die Abbil-dung der Kugeloberfläche in die Ebene” (349) zuverschiedenartigen Projektionen bei Weltkartendiente, hebt Albrecht Dürer – “. . . dürfte intu-itiv den mathematischen Begriff der Abbildungund sogar den der Funktion vor Augen gehabthaben” (357) – hervor, spricht die Bedeutung von


Johannes von Gmunden, Georg von Peuerbach,Johannes Regiomontanus und Nicolaus Coper-nicus an: “Aus historischer Sicht hat Copernicuseine Revolution des Denkens in der Astronomieausgelöst. Sie wurde zum Symbol der zu An-fang des 17. Jahrhunderts einsetzenden Wis-senschaftlichen Revolution.” (374) – 8 Mathema-tik während der Wissenschaftlichen Revolution:“Der Begriff ‘Wissenschaftliche Revolution’ hatsich in jüngerer Zeit als Strukturbegriff in derWissenschaftsgeschichte eingebürgert und zwarfür die Zeit vom ausgehenden 16. Jahrhundertbis zum Beginn des 18. Jahrhunderts, mit demBlick auf die gänzliche Umgestaltung der Natur-wissenschaften nach Inhalt, Methode, Kommu-nikationsformen und gesellschaftlicher Relevanzund auf Wechselbeziehungen zu Religion undPhilosophie. . . .Vier Generationen von Männern haben diesedurchgreifende Wandlung vollzogen . . . IhreLeistungen und die ihrer Mitstreiter bewirk-ten eine gänzliche Umgestaltung der Mathe-matik, der Astronomie und der Mechanik, ins-besondere der Dynamik . . . Mit der Fortent-wicklung von Algebra und Geometrie ent-standen eine Mathematik der Variablen unddie Infinitesimalmathematik, und damit wurdeden Naturforschern ein Instrumentarium höch-ster Leistungsfähigkeit in die Hand gegeben.”(379f.); gemäß einem Schema wird die Entwick-lung der drei Zweige: Funktionsbegriff – In-finitesimalrechnung – Analytische Geometrie“in der Historiographie der Mathematik als In-halt der Wissenschaftlichen Revolution in derMathematik bezeichnet” (380). Eigene naturwis-senschaftlich fundierte Gesellschaften wurdenbegründet: “London, Paris, Berlin und St. Peters-burg mit ihren Akademien waren während des17./18. Jahrhunderts die Zentren mathematisch-naturwissenschaftlicher Forschung.” (384) – 8.2Algebra wird zur selbstständigen mathematis-chen Disziplin: “Die entscheidende Wendungder Dinge, die den Übergang zur selbstständigenDisziplin Algebra nach sich zog, war der Schrittvon der geometrischen Lösung von Gleichun-gen dritten und vierten Grades zur rechnerischenAuflösung in Formelgestalt.” (386) Es geht umScipione del Ferro, Niccolò Tartaglia, GirolamoCardano und Rafael Bombelli; François Viète’sBedeutung wird hervorgehoben. René Descartes– er war zeitlebens scheu und zurückhaltendund lebte zurückgezogen; “Von einer entwick-elten analytischen Geometrie kann bei Descar-

tes noch keine Rede sein. Es gibt z. B. kein ex-plizit gehandhabtes Koordinatensystem.” (406)-, sowie Pierre de Fermat – “Fermats Zielstel-lung läuft . . . darauf hinaus, die Identität einesdurch eine algebraische Gleichung definiertengeometrischen Ortes mit schon bekannten Kur-ven nachzuweisen.” (409) – werden ausführlichgewürdigt; hinsichtlich der projektiven Geome-trie stehen Blaise Pascal und die Jesuiten imZentrum; Verf. rekapituliert die Herausbildungder Logarithmen; er spricht die Abhängigkeitzwischen Stifel, dem “Zauderer” Jost Bürgi,John Napier und Henry Briggs an; die erstenAnsätze zu Rechenmaschinen durch WilhelmSchickard, Pascal und Gottfried Wilhelm Leib-niz werden detailliert diskutiert. – “Die Bere-itstellung algebraischer Methoden, die Ausbil-dung einer Symbolik, die Durchbildung kalkül-haften Denkens gehören zu den unabding-baren historischen Voraussetzungen für dieAusbildung infinitesimaler Methoden, sowohlim Formalen als auch im Methodischen. Eineentsprechende Vorbedingung für das Entste-hen der Infinitesimalmathematik erfüllt auch dieGeometrie während des 16./17. Jahrhunderts.Hier stehen vielleicht sogar noch deutlicher alsbei der Algebra die Rezeption und der Aus-bau der aus der Antike übernommenen Metho-den im Vordergrund. Ebene und räumliche Ge-ometrie verloren im 15./16. Jahrhundert . . . inden Händen der Praktiker ihren esoterischenCharakter.” (427) Vorangegangene mechanisch-physikalische, mechanisch-geometrische undgeometrische Probleme werden demgemäß klas-sifiziert, denn: “. . . es musste erst erkannt wer-den, dass Tangentenproblem und Flächeninhalt-sproblem inhaltlich zueinander inverse Prob-leme sind: Der Fundamentalsatz der Differen-tial- und Integralrechnung musste erst entdecktwerden.” (429) – Diese Erkenntnis geht auf IsaacBarrow zurück. Im Folgenden werden Galilei,Kepler, Cavalieri, Torricelli, Wallis und Fer-mat ausführlich besprochen, wobei notgedrun-gen Formeln erscheinen. – Verf. bringt den ger-ade in der Epoche der Coß verlorengegangenenBezug zur Naturphilosophie schier zwangsläu-fig wieder ein: “Mit der Erfindung der Fluxions-rechnung durch Newton und schließlich mit derEntwicklung des Calculus durch Leibniz wirddie Infinitesimalmathematik nach Form und In-halt einen gewissen vorläufigen Abschluss er-reichen. Das 18. Jahrhundert sah dann denweiteren Ausbau der infinitesimalen Metho-


den, die volle Herausbildung des Funktions-begriffes und die höheren Gefilde der Analy-sis . . . ” (452), und er resümiert: “Isaac New-ton war einer der bedeutendsten Naturforscher,den die Menschheit bisher hervorgebracht hat.Er . . . fand das allgemeine Gravitationsgesetz. . . leistete als Präsident der Royal Society wirk-same Arbeit bei der Organisation der Wis-senschaften, schuf mit seiner Fluxionsrechnungeine spezifische Form der Infinitesimalmath-ematik und leistete wesentliche Beiträge zurReihenlehre und zur Algebra.” (453) – Ana-log steht für Leibniz, der seines schlechtenGedächtnisses wegen angeblich alle Gedanken-blitze und Ideen schriftlich festhielt: “Leib-niz nimmt . . . in einer Geschichte der Mathe-matik einen Ehrenplatz ein . . . Leibniz liefertewesentliche Beiträge zur theoretischen und prak-tischen Mechanik . . . Auf seine Initiative gingdie Gründung einer Akademie in Berlin zurück.. . . Leibniz war von einer fast unglaublichengeistigen Beweglichkeit und raschen Auffas-sungsgabe, von einem nie erlahmenden Arbeits-eifer und schier unerschöpflichem Ideenreich-tum.” (464) – Der sog. Prioritätsstreit wird inknapper, jedoch distanzierter Form skizziert.

W. Kaunzner (Regensburg)

1167.01013Wußing, Hans6000 years of mathematics. A cultural-historicaljourney. 2. From Euler to present. With the col-laboration of Heinz-Wilhelm Alten and HeikoWesemüller-Kock. With an outlook by Eber-hard Zeidler. (6000 Jahre Mathematik. Eine kul-turgeschichtliche Zeitreise. 2. Von Euler bis zurGegenwart.) (German)Berlin: Springer (ISBN 978-3-540-77313-9/hbk).xviii, 675 p. EUR 29.95; SFR 46.50 (2009).Nach dem begrifflichen Unterschied zwischenGeschichte der Mathematik und Historiogra-phie: “Die Historiographie hat die Aufgabe, Ent-wicklung und Entfaltung der Mathematik alshistorischen Prozess zu erfassen.” (Band 1, S.2), verdeutlichte Hans Wußing sein Vorhaben:“. . . die Idee, eine die Fächer übergreifende His-toriographie der Mathematik ins Auge zu fassen,leicht lesbar, mit wenigen Formeln, dafür abermit reichlich kulturellen, philosophischen undhistorischen Bezügen, alle Zeiten und Kulturenberührend” (Band 1, S. 2). Man kann ihm zumGelingen dieser Absicht gratulieren: In zwei Bän-den, betitelt 6000 Jahre Mathematik, ist ihm dies

wahrlich gelungen! Es war ein gewaltiges Un-terfangen, das sich naturgemäß über Jahre er-streckte, und nun mit Band 2, der relativ schnellauf Band 1 folgte, vollendet wurde. Dass essich nicht um Lehr- oder gar um Lesebücherim herkömmlichen Sinn handelt, hat man baldherausgefunden, und so wird man wohl vomAnfang an mehr nach Gemeinsamkeiten in ver-schiedenen Abschnitten bzw. Kapiteln suchen,als dass man die Seiten lediglich der Reihe nachdurchgeht.Mit diesen beiden preisgünstigen Büchern vonHans Wußing wird einem vieles geboten.Über die in Band 1 vordringlich gezeigte Be-trachtungsweise zu einzelnen Epochen bzw.zu Leben, Quellen, Wirken und Nachwirkenbedeutender Wissenschaftler hinaus, werdenhier in Band 2 vor allem auch die “persön-lichen Schicksale der Mathematiker als Forscherund Lehrer” (S. V) herausgestellt; zusätzlichwird nun freilich wesentlichen “Gedanken zurZukunft der Mathematik” (S. V) ein gebühren-der Platz eingeräumt. “Dem Autor Wußing istes gelungen, die Fülle des Stoffes soweit wiemöglich in den aufeinander folgenden Epochenjeweils nach Gebieten gegliedert darzustellenund parallel dazu die Entwicklung der Math-ematik in einzelnen Ländern zu beschreiben,. . . Als besonders problematisch erwies sich dieDarstellung der Entwicklung im 20. Jahrhundertangesichts der kaum noch überschaubaren Fülleder Forschungsergebnisse” (S. VI) heißt es weiterim Vorwort des Herausgebers. Während den vierAbschnitten – Nr. 9-12 – auch diesmal “chronolo-gisch angeordnete Tabellen mit den wichtigstenpolitischen Ereignissen der jeweiligen Epochevorangestellt” (S. VI) werden, erscheint, an-ders als in Band 1, folglich “für die zeitlichversetzten und regional verschiedenen Ausprä-gungen der vielen Stilrichtungen in Baukunst,Malerei, Musik und Literatur . . . eine textgebun-dene Darstellung angemessen” (S. VI). – Dievier Abschnitte sind wiederum stark und über-sichtlich gegliedert, so dass man ohne weiteresauch dem einzelnen Problem gezielt nachzuge-hen vermag. Sein Ziel, die organische Einheit derMathematik darzustellen, verliert Hans Wußingnatürlich auch hier nirgends aus dem Blick.Wie bereits gewohnt, lockert er die Lesbarkeitdurch eine große Anzahl von Abbildungen auf:zunächst Gemälde, dann Fotos; Briefmarken;Stemmata für quellenmäßige bzw. immanenteLängs- und Querverbindungen, so: Weiterent-


wicklung der Analysis seit Leibniz und New-ton (S. 232), Entstehung der Funktionalanalysis(S. 410), weitere (S. 493, 524, 536, 537); nur relativwenige Skizzen und Tabellen; fast 90 Seiten Liter-atur, Abbildungs-, Personen-, Sachverzeichnis. –Viele Zitate gemäß dem umfangreichen, großen-teils auf neue englisch- und deutschsprachigeWerke gestützten Literaturverzeichnis verdeut-lichen so manchen Sachverhalt.Das eigene Empfinden des Verfs., verbunden mitseinem Spürsinn für aktuelle Fragen, tritt offen-sichtlich an einigen Stellen parallel mit dem sei-nerzeitigen Fortschrittsdenken in Wissenschaft,Kunst und Kultur während so mancher Epochedeutlich zutage, vor allem dann, wenn er diezugehörige Problematik bereits in einer seinervielen Veröffentlichungen – zur Renaissance,zur Algebra usw. – untersucht hatte; folglichbeschreibt er die betreffenden Zeitabschnitte hiersowohl insgesamt, als auch im Detail, mit beson-derem Einfühlungsvermögen. – Eine Reihe vonKapiteln wurde von anderen Autoren bear-beitet. – In Abschnitt 11 Globalisierung derMathematik seit dem Ende des 19. Jahrhun-derts wird vom Verf. ein neues, geschicktes Ver-fahren gewählt, wodurch nicht mehr einzelnePersönlichkeiten, sondern entweder die aktuelleProblematik oder die durch die Zeitumständehervorgerufenen Notwendigkeiten bzw. Verän-derungen wirtschaftlicher, politischer oder son-stiger Art im Vordergrund stehen.Hans Wußing hat hier neben Band 1 ein zweitesherausragendes Werk vorgelegt, das freilicherst derjenige zweckmäßig heranziehen kann,der über die gängige Geschichte der Mathe-matik hinaus die allgemeine wissenschaftlicheSituation innerhalb eines bestimmten Zeitab-schnitts bzw. das Umfeld eines Einzelnen erken-nen möchte. Dann wird man auf diese bei-den Bücher zurückgreifen, weil sie vermut-lich auf lange Sicht hin ein bisher einmaligesUnterfangen bleiben! So handelt es sich hiergemäß “Wer vieles bringt, wird manchem et-was bringen” um eine Art Wissenschaftslexikon,in dem sowohl wichtige Daten aus vielenLebensläufen, als auch, neben obligatorischenQuellen, die Querverbindungen zu den aktuel-len Zeitgenossen herausgearbeitet wurden.Weil Hans Wußing in Band 2 die Detailsder einzelnen Fachgebiete so tiefgründig wiemöglich aufzeigt, werden die Inhalte der jeweili-gen Kapitel hierdurch immer stärker miteinan-der verwoben; er geht folglich laufend weiter

in Richtung “Grundlagen der Mathematik“; inAbschnitt 11 wird somit klar aufgezeigt, wieursprünglich getrennt erdachte und behandeltemathematische Disziplinen – hier Zahlentheorie,Wahrscheinlichkeit, Algebra, Geometrie – im 20.Jahrhundert schließlich untrennbar ineinandergriffen, wobei allerdings auch die Wahrschein-lichkeitstheorie zunächst axiomatisiert werdenmußte. Hieraus folgt, dass es wohl kaum ak-tuelle mathematische Probleme – gelöste undungelöste, so wie etwa die 23 von DavidHilbert vorgeführten – gibt, die hierbei von HansWußing nicht zumindest stichwortmäßig mitangesprochen wurden. Im 20. Jahrhundert er-hielten z. B. die Forschungen zur algebraischenGeometrie wichtige Impulse von der Zahlenthe-orie; die Bedeutung der topologischen bzw. ana-lytischen Methoden in der algebraischen Geome-trie hatte deutlich zugenommen, “so dass derenIntegration in den algebraisch-arithmetischenRahmen der algebraischen Geometrie eine wei-tere zentrale Aufgabe darstellte.” (S. 437)Wer es nur kurz haben will, der erfährt HansWußings Vorstellungen bereits in der Einleitung(S. 1-4); ansonsten jedoch, wobei die Zuhilfe-nahme eines Fach- bzw. Fremdwörterbuchs un-abdingbar erscheint, findet er sie der Reihe nachim ganzen Band. So ausführlich und begeisternddieses Buch in den meisten Passagen geschriebenist, so schockierend lesen sich allerdings mancheanderen Stellen, z. B. einige Lebensläufe.Je ein paar Splitter: 9 Mathematik im Zeital-ter des Absolutismus und der Aufklärung: Aus-gehend von der “Verbesserung der Produktionund zur Erhöhung des Gewinnes” (S. 9), ver-lagerte sich die Produktion zunehmend vomHandwerk zu den Manufakturen. “Es gingdarum, . . . den menschlichen Verstand als ober-ste Instanz für Rechtfertigung oder Abänderunggesellschaftlicher Zustände einzusetzen.” (S. 10)– “Mit Euler und Daniel Bernoulli tritt ein neuer‘Funktionentyp’ auf.” (S. 40); in Eulers Mechani-ca, sive motus scientia analytica exposita: “Stattder synthetisch-geometrischen Methode, die bisdahin vorherrschend gewesen war und noch den‘Principia’ [Newton’s] als mathematisches In-strumentarium gedient hatte, wird nun system-atisch die Infinitesimalmathematik, die Anal-ysis, zur Behandlung mechanischer Problemeherangezogen.” (S. 42); “Euler war wohl derproduktivste Wissenschaftler, den die Welt her-vorgebracht hat.” (S. 45); “Erst zu Beginn des18. Jahrhunderts fing man an, Erfindungen und


technischen Fortschritt als Wissenschaft zu be-trachten.” (S. 83)10 Mathematik während der Industriellen Revo-lution: Vervollkommnung der Dampfmaschine,Aufkommen von Werkzeugmaschinen; “. . . viele. . . Vorgänge und Erscheinungen in der belebtenwie in der unbelebten Natur mussten mathe-matisch erfasst und in ihren Abläufen modell-haft studiert werden, bevor sie in technischenVerfahren mit industriellen Prozeduren genutztwerden konnten. Dies alles gab der Entwicklungder Mathematik . . . ungeheuren Antrieb, vorallem in der Analysis und Geometrie, aber auchin der Algebra, der Wahrscheinlichkeitsrech-nung und der Statistik.” (S. 102); “. . . in Deutsch-land bzw. in Mitteleuropa ging die in den 30erJahren einsetzende Industrielle Revolution Handin Hand . . . in Wechselwirkung mit Teilen derMathematik.” (S. 109); “Im Berührungsfeld zwis-chen Mathematik und Physik . . . während des19. Jahrhunderts . . . erhielt die Physik eine an-dere, eine deduktiv aufgebaute Struktur.” (S. 128)– 10.2 Entwicklungen in der Geometrie: “DieAnerkennung der nicht-euklidischen Geometriebzw. Geometrien . . . war erst möglich, nach-dem Modelle für die Existenz solcher Geome-trien gefunden worden waren.” (S. 163); “MitHilbert erfuhr der von Euklid einst beschritteneWeg der Axiomatisierung seine Vollendung.”(S. 177) – 10.3 Wandel in der Algebra: “Dasallgemeine Auflösungsproblem für algebraischeGleichungen offenbarte . . . im 18. Jahrhundertseine wirklichen Tücken. . . . Lagrange . . . kamzu dem Schluss, dass diese Verfahren [del Ferro,Tartaglia, Cardano, Hudde, Tschirnhaus, Eu-ler, Bezout] bei den (allgemeinen) Gleichun-gen höheren als vierten Grades . . . versagenmüssen.” (S. 178) – Fundamentalsatz → reelleZahlen → einfach-komplexe Zahlen → hyper-komplexe Zahlen; Gruppe → Körper werdenvon Hans Wußing in spannender zeitlicherund inhaltlicher Folge eingebracht: “Bereits imLaufe des 19. Jahrhunderts wurden in abstrak-ter Form einige algebraische Grundstrukturenwie Gruppe, Körper, Ideal herausgearbeitet.”(S. 206); “Historisch gesehen lag mit dem ab-strakten Gruppenbegriff der früheste Fall derEmanzipation einer algebraischen Struktur vor.”(S. 209) – 10.6: Analysis in neuem Gewande:“Auf der eigentlichen Ebene [der Analysis], dermathematischen, ging es um die begrifflicheBewältigung des mathematischen Unendlich”(S. 233). “Zu Beginn des 19. Jahrhunderts trat die

Diskussion der Grundlagen der Analysis in einezweite Etappe ein, von der Konstatierung derMängel zur Bewältigung der Schwierigkeiten.”(S. 236); “Lange Zeit, bis ins 19. Jahrhundert,war das Integral als Umkehrung eines differen-zierten Ausdrucks, Integration als inverse Oper-ation zum Differenzieren verstanden worden.”(S. 263) So läßt Verf. das 19. Jahrhundert sowohlals das “Zeitalter der Strenge” (S. 278) gelten,doch er zitiert auch: “Tatsächlich ergab sich ausder Entwicklung neuer Sätze in der Analysisein wichtiger Anstoß für das wachsende In-teresse an ihren Grundlagen.” (S. 278) Als lo-gische Folge verlangte die rapide fortschreit-ende Ingenieurkunst jedoch praktische Metho-den und schnelle Verfahren, die dann freilicheine Zeit lang bisweilen als “Schmierölmathe-matik” abgetan wurden. – Anläßlich 10.7 DerWeg zur klassischen Wahrscheinlichkeitsrech-nung: Zufall? “Diese zutiefst philosophischeFrage wurde im 17. Jahrhundert von heraus-ragenden Gelehrten erörtert, . . . Die klassischeMechanik der Laplace-Zeit beschreibt alle Be-wegungen durch Differentialgleichungen 2. Ord-nung. . . . Für Zufall ist kein Platz.” (S. 281f.)In 11 Globalisierung der Mathematik seit demEnde des 19. Jahrhunderts resümiert Verf.bzgl. 11.0.10 Entwicklung nach dem ZweitenWeltkrieg: . . . : “Heute halten viele Mathe-matiker die Unterscheidung zwischen sogenan-nter ‘reiner’ und ‘angewandter’ Mathematikhöchstens noch in historischem Rückblick fürberechtigt, da viele Gebiete ehemals ‘reiner’Mathematik plötzlich, ganz unerwartet sogar,für Praxis und Anwendungen wichtig gewor-den sind.” (S. 373) – 11.1 Die Begründungder Mengenlehre: Ersichtlich liegt dem Un-endlichkeitsbegriff, “zeitlich vor der Mengen-lehre, die Vorstellung des nur potentiell Un-endlichen zugrunde. Die Mengenlehre behan-delt das aktual Unendliche.” (S. 380). – 11.3.2Entstehung der Funktionalanalysis in: 11.3 Eineneue Disziplin: Funktionalanalysis: “. . . zu An-fang der 30er Jahre erreichte sie [= die Funk-tionalanalysis] den Rang einer selbständigenmathematischen Disziplin. . . . Dem Wesen nachwar die Entstehung der Funktionalanalysiseine Übertragung einzelner oder mehrerer Be-griffe wie Kompaktheit, Beschränktheit, Konver-genz, Abstand, Stetigkeit, Vollständigkeit, Di-mension, Skalarprodukt, Linearität usw. vom n-dimensionalen euklidischen Raum Rn und denauf ihm erklärten Funktionen auf unendlichdi-


mensionale ‘Funktionenräume’ verschiedenenTyps und ihre ‘Operatoren’.” (S. 411); “Die Funk-tionalanalysis hat verschiedenartige historischeWurzeln, die einerseits dem Streben nach Vere-inheitlichung und andererseits konkreten Prob-lemen bei Anwendungen entsprangen.” (S. 412);“Hilbert hat einen großen Einfluss ausgeübt, ge-rade durch das erkennbar werdende Ziel, dieMathematik als einheitlich zu führenden Prozesszu verstehen.” (S. 413); “Den endgültigen Durch-bruch der Funktionalanalysis verdankt manJohn von Neumann. Er konnte, beginnend mit1928, die Anwendbarkeit der Hilbertschen Spek-traltheorie in der Quantenmechanik zeigen.”(S. 414) – 11.4 Algebra im 20. Jahrhundert: “DieHauptmerkmale des . . . Prozesses beim Auf-bau der sog. ‘Modernen Algebra’ bestehen ineiner verstärkten Hinwendung zum Abstrak-ten, der durchgehenden Axiomatisierung undin der sich wie in der gesamten Mathematikauch in der Algebra vollziehenden mengenthe-oretischen Durchdringung.” (S. 423); “Ende derzwanziger Jahre waren . . . drei Säulen der mo-dernen Algebra – Gruppentheorie, Körperthe-orie, Algebrentheorie (Theorie hyperkomplexerSysteme) – in ihrer abstrakten und axioma-tisierten Form entwickelt. Durch diese Theorienihrer Grundstrukturen war die Algebra zu einemklar gegliederten Zweig der Mathematik gewor-den.” (S. 427); “Da Boole die Wahrscheinlich-keitsrechnung zu den ‘reinen’ bzw. exakten Wis-senschaften zählte, unterstellte er, dass sich dieErgebnisse dieser Wissenschaft mit Mitteln derLogik erfassen lassen müssten, also ein deduk-tiver Aufbau möglich sein müsse.” (S. 441), sodass Hilbert forderte: “Das mathematische Be-weisen selbst sollte zum Gegenstand der Un-tersuchungen gemacht werden.” (S. 456); “In-haltliches Schließen wurde bei Hilbert durch eineKette rein formaler Handlungen, das heißt durchRechnen mit Zeichen nach festen Regeln ersetzt.Für Hilbert ist damit die Mathematik zur all-gemeinen Lehre von den Formalismen gewor-den.” (S. 459); “Ungebrochen blieb Hilberts Ver-trauen in die Kraft menschlichen Denkens undmenschlicher Erkenntnisfähigkeit.” (S. 463); an-dersartigen Vorstellungen setzte er sein “Wirmüssen wissen. Wir werden wissen.” entgegen.12 Gedanken zur Zukunft der Mathematik – EinAusblick von Eberhard Zeidler ist vielseitig undvielschichtig aufgebaut, besonders in 12.2 Strate-gien der Mathematik für die Zukunft und 12.5

Die philosophische Dimension der Mathematik.W. Kaunzner (Regensburg)

1167.00006Gowers, William TimothyDoes mathematics need a philosophy?Hersh, Reuben (ed.), 18 unconventional es-says on the nature of mathematics. New York,NY: Springer (ISBN 0-387-25717-9/pbk). 182-200(2006).The author discusses the title problem consid-ering some specific questions concerning theempty set, subsets of natural numbers, orderedpairs, truth and provability, the axiom of choice.He comes to the conclusion that mathematicsdoes and does not need philosophy. The author’sown philosophy of mathematics is – as he de-scribes it – naturalism, i.e., he claims that a properphilosophical account of mathematics should begrounded in the actual practice of mathemati-cians. Roman Murawski (Poznan)

1167.01010Sonar, ThomasThe death of Gottfried Wilhelm Leibniz. Truthand legend in the light of the sources. (Der Toddes Gottfried Wilhelm Leibniz. Wahrheit undLegende im Licht der Quellen.) (German)Abh. Braunschw. Wiss. Ges. 59, 161-201 (2008).It required nearly two years until T. Sonar clari-fied the circumstances of Leibniz’ death, his lasthours and funeral. Motivation of his researchwas the fact that several other authors did notknow where Leibniz was buried. T. Sonar wasacquainted with the fact, that Leibniz’ grave issituated in the “Neustädter" church in Hannover.In 1902, 1957 and 1993 the tomb was carefullyexamined and restored. G. W. Leibniz died the14th of November 1717. Today it can be as-sumed that coronary failure and a renal colicwere the causes of his death (Dissertation E. Gör-lich). Some persons were with Leibniz in his lasthours: “Secretarius" Johann Georg Eckhart, thestudent “Amanuensis" Johann Hermann Vogler,the physician Johann Philipp Seip, and one ortwo servants.J. Eckhart, J. Vogler and J. Seip later give accountof that night. Most biographers knew of Eckhartsstory, cited or embellished him (Ludovici 1737,Eberhard 1795, Guhrauer 1846). As recently asin the beginning of the twentieth century twoletters of J. Vogler were found in Copenhagen(by P. Ritter). There were similarities in the three


reports, but the stories as stated by Eckart of awritten paper with unreadable characters, lastwords or statements against last rites seems notvery plausible.T. Sonar has published parts of the arti-cle in [Math. Intell. 28, No. 2, 37-40 (2006;Zbl 1165.01014) in English and in German inthe (Festschrift for Karin Reich. Augsburg: ERVDr. Erwin Rauner Verlag. Algorismus 60, 101-110(2007)]. Silke Göbel (Berlin)

1167.03006Urbaniak, RafalLesniewski and Russell’s paradox: some prob-lems.Hist. Philos. Log. 29, No. 2, 115-146 (2008).This paper aims to “re-examine the usual claimthat Lesniewski provided a satisfactory solutionto Russell’s paradox" (p. 115). The author main-tains that such a solution need not just presenta logical system which avoids contradiction, butshould also recover the salient aspects of our no-tion of distributive class, as with the claim “Mydog’s leg is not an element of the class of dogs"(p. 132).The first part of the paper summarizesLesniewski’s system Ontology where the centralinnovations are that (i) names a, b, c, . . . includeempty names, traditional singular names andplural names and (ii) a connective ε is used to linkthese names. The problem is then to introducea list of axioms for the operator Kl which willaccord closely enough to an intuitive notion ofclass without producing contradictions. The au-thor reconstructs B. Sobocinski’s proofs that someplausible axioms lead to contradiction via theexistence of a Russell class [Methodos, Milano 1,94–107 (1949; Zbl 34.15101)]. This motivates theintroduction of a collective or mereological no-tion of class KlM. This blocks the existence of aRussell class along with the empty class. But theauthor complains that “mereological classes inmany respects behave differently than it is ex-pected of classes in the classical set theory" (p.132).The second part of the paper explores theprospects for recovering a distributive notion ofclass in the system Ontology. Several optionsare considered, but each proposal has counter-intuitive consequences. The most promisingstrategy is to introduce a new defined “higher-order varepsilon" symbol which can simulatethe usual class membership relation. The author

leaves it as an open question for future researchwhether or not any satisfactory solution alongthese lines is available.A helpful aspect of this paper is the extensivelist of references to work on Lesniewski’s logicalsystems which ranges far beyond what relatesdirectly to Russell’s Paradox.

Chris Pincock (West Lafayette)

1167.03025Edmundo, Mário J.; Peatfield, Nicholas J.o-minimal Cech cohomology.Q. J. Math. 59, No. 2, 213-220 (2008).Let N denote an arbitrary o-minimal structure;see [L. van den Dries, Tame topology and o-minimal structures. London Mathematical Soci-ety Lecture Note Series. 248. Cambridge: Cam-bridge University Press (1998; Zbl 953.03045)].Consider the category of N-definable sets withmorphisms being the N-definable continuousfunctions. For this category, the authors prove theexistence of a Cech cohomology theory H

∗

, d∗(with coefficients in an abelian group) which sat-isfies the Eilenberg-Steenrod axioms. The con-struction mainly follows the classical proof in[S. Eilenberg and N. Steenrod, Foundations ofalgebraic topology. Princeton Mathematical Se-ries No. 15. Princeton: University Press (1952;Zbl 47.41402)]. Only the homotopy axiom re-quires additional work.

Andreas Fischer (Saskatoon)

1167.03028Shioya, MasahiroA proof of Shelah’s strong covering theorem forPκλ.Asian J. Math. 12, No. 1, 83-97 (2008).A theorem of S. Shelah [“Advances in cardinalarithmetic”, in: N. W. Sauer et al. (eds.), Finiteand infinite combinatorics. Dordrecht: Kluwer.NATO ASI Ser., Ser. C, Math. Phys. Sci. 411,355–383 (1993; Zbl 844.03028)] answered a long-standing open question in a surprising way: forevery regular uncountable κ and λ > κ, thesmallest size of a stationary set in Pκλ is thesame as the smallest size of an unbounded setin Pκλ. By work of Baumgartner and Taylor, thiswas known in the situation when λ < κ+ω, butthe argument stopped there. Shelah’s proof usespcf theory and an ingenious system of elemen-tary submodels. Unfortunately, close inspectionof the details of this proof found some incorrect


ones, as noticed by Shioya and apparently con-firmed by Shelah.In this paper, Shioya gives the correct version ofShelah’s proof, with the missing details pointedout and corrected. The historical notes in the in-troduction suggest that the corrections were sug-gested by Shelah in response to Shioya’s inquiryand then further revised by Shioya. In additionto this proof, the paper gives a different, self-contained proof due to Shioya.The paper is well written and interesting to read.I was slightly annoyed by the use of ΠA in placeof ΠA/U in the section reviewing pcf theory,making some of the statements quoted formallyincorrect. Mirna Dzamonja (Norwich)

1167.03030Matet, PierreGuessing with mutually stationary sets.Can. Math. Bull. 51, No. 4, 579-583 (2008).Letκ be a regular uncountable cardinal andλ > κa cardinal. Then Pκ(λ) denotes the collection ofall subsets of λ of size< κ. For a regular uncount-able cardinal ν let Eνω denote the set of all infinitelimit ordinals α < ν such that cf α = ω.A 2-person game Gκ,λ(A) is defined for A ⊆ Pκ(λ).Players I and II alternately pick elements of Pκ(λ),thus building a sequence 〈an : n < ω〉 with thecondition that a0 ⊆ a1 ⊆ a2 ⊆ . . . Player II winsthe game just in case

⋃n<ω an ∈ A. Let NGκ,λ be

the set of all subsets B of Pκ(λ) such that Player IIhas a winning strategy in Gκ,λ(Pκ(λ)\B). Then it isproved: Suppose that κ and µn for n < ω are reg-ular cardinals such that ω1 ≤ κ ≤ µ0 < µ1 < . . .and 2<κ ≤ (supµn : n < ω)ℵ0 , and λ is a cardinalsuch that λ > µn for any n < ω. Suppose furtherthat for each n < ω, Tn is a stationary subset ofEµnω . Then, letting

S = a ∈ Pκ(λ) : ∀n < ω (sup(a ∩ µn) ∈ Tn),

there is a sequence 〈sa : a ∈ Pκ(λ)〉 with sa ⊆ asuch that for every X ⊆ λ, a ∈ S : sa = X ∩ a ∈NG+

κ,λ. Hereby the author uses the mutually sta-tionary sets of Foreman and Magidor as a toolto establish the validity of the two-cardinal ver-sion of the diamond principle in some specialcases. Egbert Harzheim (Köln)

1167.03031Hjorth, Greg; Miller, Benjamin D.Ends of graphed equivalence relations. II.Isr. J. Math. 169, 393-415 (2009).

For Part I see [B. D. Miller, Isr. J. Math. 169, 375–392 (2009; Zbl Zbl 1166.03026)].Given a graphing G of a countable Borel equiva-lence relation on a Polish space, the authors showthat if there is a Borel way of selecting a non-empty closed set of countably many ends fromeach G-component, then there is a Borel way ofselecting an end or line from each G-component.Their method yields also Glimm-Effros-style di-chotomies that characterize the circumstancesunder which: (1) there is a Borel way of selectinga point or end from each G-component; and (2)there is a Borel way of selecting a point, end orline from each G-component.

Denis I. Saveliev (Moskva)

1167.03035Minami, HiroakiSuslin forcing and parametrized ♦ principles.J. Symb. Log. 73, No. 3, 752-764 (2008).The ♦ principle, introduced by Jensen, has beenestablished as a central concept of modern settheory. It can be formulated as follows: “There isa guessing sequence (gα)α∈ω1 that matches everysequence y ⊂ ω1 at stationary many points α, i.e.,gα = y ∩ α."Several weaker variants have been introduced,amongst them K. J. Devlin and S. Shelah’sweak Diamond [Isr. J. Math. 29, 239–247 (1978;Zbl 403.03040)] and M. Hrušák’s ♦d [Fundam.Math. 167, No. 3, 277–289 (2001; Zbl 972.03046)].The combination of these two variants, to-gether with the restriction to Borel functions,led to J. T. Moore, M. Hrušák and M. Džamonja’sparametrized ♦principles [Trans. Am. Math. Soc.356, No. 6, 2281–2306 (2004; Zbl 1053.03027)]:“For every Borel F : 2<ω1 → ωω, there is a guess-ing sequence g : ω1 → ωω such that for allf : ω1 → ω the real g(α) ‘covers’ the real F( f α)stationary often."There are many notions of “covering”. In factmany of the usual cardinal invariants are definedas the minimal size of a family that covers all thereals; so for such a cardinal invariant xwe get theparametrized ♦-principle ♦(x) defined as above.For example, the bounding number b corre-sponds the following notion of “r covers s”: Thereare infinitely many n such that r(n) > s(n).Generally, ♦(x) implies x ≤ ℵ1 (in the same waythat the usual ♦ implies CH). In [Moore, Hrušákand Džamonja, loc. cit.], it is shown that certaincountable support iterations force the following:2ℵ0 = ℵ2 and ♦(x) holds iff x = ℵ1 (simultaneously


for all Borel invariants x). This fact was then ap-plied to show, e.g., the consistency of

2ℵ0 = ℵ2 + ♦(x) + ¬♦(y)

for several pairs (x, y) of cardinal characteristics.(In fact, y = ℵ2, which implies the failure of dia-mond). Note that, due to general restrictions im-posed by countable support iterations, the con-tinuum has to be ℵ2 in these models.In [Arch. Math. Logic 44, No. 4, 513–526 (2005;Zbl 1079.03036)], the author of the paper underreview introduced finite support ccc forcing con-structions to show the consistency of

CH + ♦(x) + ¬♦(y).

In the paper under review, similar constructions(finite support ccc forcings) are used to give newproofs of the results of [Moore, Hrušák and Dža-monja, loc. cit.]. Moreover, these constructionsallow large continuum and therefore results ofthe form

2ℵ0 ℵ2 + ♦(x) + ¬♦(y).

This is proven, e.g., for (x, y) = (cov(N),non(M))and (b, cov(M)), as well as several other pairs ofcharacteristics. Along the way, the author alsoproves some results about Suslin ccc forcings.

Jakob Kellner (Wien)

1166.03026Miller, Benjamin D.Ends of graphed equivalence relations. I.Isr. J. Math. 169, 375-392 (2009).Let E be a Borel equivalence relation on a Polishspace X. E is called finite (countable, aperiodic)if all of its equivalence classes are finite (count-able, infinite). E is called smooth if there exists asequence 〈Bn〉n∈N of E-invariant Borel sets suchthat xEy ⇔ ∀n ∈ N (x ∈ Bn ⇔ y ∈ Bn), for allx, y ∈ X. E is called hyperfinite if there exists anincreasing sequence 〈En〉n∈N of finite Borel equiv-alence relations such that E =

⋃n∈N En.

A Borel graph G on X is called a graphing ofE if its connected components coincide withthe equivalence classes of E. A ray throughG is an injective sequence α ∈ XN such that〈α(n), α(n+1)〉 ∈ G, for all n ∈N. Two raysα, β arecalled end equivalent if for any finite S ⊆ X, thereexists a path from α to β through G|(X\S). An endof G is an end equivalence class. A graphing iscalled endless if it has no rays. The author inves-

tigates relationships between Borel equivalencerelations and their graphings.The main results of the paper: (1) E admits anendless graphing iff E is smooth; (2) E admits alocally finite single-ended graphing iff E is ape-riodic; (3) E admits a graphing for which thereis a Borel way of selecting two ends from eachcomponent iff E is hyperfinite; and (4) E admitsa graphing for which there is a Borel way of se-lecting a finite set of at least three ends from eachcomponent iff E is smooth.For Part II see [G. Hjorth and B. D. Miller, Isr. J.Math. 169, 393–415 (2009)].

Denis I. Saveliev (Moskva)

1167.06008Olson, Jeffrey S.Free representable idempotent commutativeresiduated lattices.Int. J. Algebra Comput. 18, No. 8, 1365-1394(2008).A commutative residuated lattice is an algebraA = 〈A; ·,→,∧,∨, e〉, where 〈A;∧,∨〉 is a lattice,〈A; ·, e〉 is a commutative monoid, and A is resid-uated in the sense that a · c ≤ b iff c ≤ a→ b for alla, b, c ∈ A. The commutative residuated lattice Ais ‘idempotent’ if a · a = a for all a ∈ A, and ‘rep-resentable’ if it is a subdirect product of linearlyordered commutative residuated lattices.The variety RICRL of all representable idempo-tent commutative residuated lattices is shown tobe locally finite in [J. G. Raftery, Trans. Am. Math.Soc. 359, No. 9, 4405–4427 (2007; Zbl 1117.03070)];the main objective of this paper is to computethe precise size of the free algebras in RICRL. Toobtain the objective, many strong results on thefinite members of RICRL are derived. The overallapproach makes use of results in [J. Berman andW. J. Blok, Trans. Am. Math. Soc. 302, 427–465(1987; Zbl 633.08005)]. A formulaΦ(n) is derivedthat gives the size of the free algebra in RICRLon n generators.For example, the following are calculated:Φ(1) = 144Φ(2) ≈ 3.00362 × 1069

Φ(3) ≈ 7.01164 × 105554

Φ(4) ≈ 1.13215 × 10938926. Clint van Alten (Wits)

1167.06012Ma, Jingjing; Redfield, Robert H.Positive derivations on Archimedean lattice-ordered rings.Positivity 13, No. 1, 165-191 (2009).


A lattice-ordered ring A is called an f -ring ifx∧y = 0 and 0 ≤ z imply zx∧y = xz∧y = 0 for allx, y, z ∈ A. It is well-known that the only positivederivation (i.e., one that maps positive elementsto positive elements) on a reduced Archimedeanf -ring is the zero derivation, cf. [P. Colville, G.Davis and K. Keimel, J. Aust. Math. Soc., Ser. A 23,371–375 (1977; Zbl 376.06021)].The paper under review examines the situationfor general Archimedean lattice-ordered rings.The beginning sections focus on derivatives andderivations on certain polynomial and grouprings. For instance, it is shown that for grouprings of finite cyclic groups the only derivationvanishing on the underlying ring is the zeroderivation. For infinite cyclic groups such deriva-tions are always based on the derivative, seeCorollary 3.4 or Theorem 3.2 for a more generalstatement.In the second half of the article the authorsturn their attention to derivatives and deriva-tions on Archimedean lattice-ordered rings. Itis proved that on purely transcendental exten-sions of totally ordered fields derivations thatare at the same time lattice homomorphisms,are translations of the usual derivative (Theo-rem 9.1), and on algebraic extensions of totallyordered fields often the only positive derivationis the zero derivation (see Section 6 for detailedstatements). The authors also give some resultsfor lattice-ordered matrix rings (Section 8) andlattice-ordered rings in which all squares are pos-itive (Section 7).

The paper abounds with examples that illustratethe results and show the necessity of hypothe-ses. Igor Klep (Ljubljana)

1167.08002Bloom, Stephen L.; Ésik, Zoltan; Kuich, WernerPartial Conway and iteration semirings.Fundam. Inform. 86, No. 1-2, 19-40 (2008).In this comprehensive algebraico-computer-science investigation, the authors prove severalinteresting theorems on semirings and their gen-eralization to partial semirings, with applica-tions to languages recognizable by finite au-tomata as well as regular languages. The au-thors’ main framework comes from results of (1)J. C. Conway’s book [Regular algebra and finitemachines. London: Chapman and Hall (1971;Zbl 231.94041)] and (2) J. S. Golan’s book [Semir-ings and their applications. Dordrecht: KluwerAcademic Publishers (1999; Zbl 947.16034)].Their main results are (1) the initial develop-ment of a general theory of partial Conwaysemirings and (2) proof of a theorem of Kleene-Schützenberger type for partial Conway semir-ings. The authors are studying the axiomatiza-tion of rational power series so that they canexploit analogues of Schützenberger’s idea ofequating power series recognizable by finiteweighed automata with rational power-seies.The authors close with a useful bibliography con-taining 19 items. Albert A. Mullin (Madison)

Number Theory. Algebra. Algebraic Geometry. 21

Number theory. Algebra. Algebraic Geometry

1167.11001Gauss, Carl FriedrichInvestigations in higher arithmetic. Edited inGerman by H. Maser. (Disquisitiones Arith-meticae. Untersuchungen über höhere Arith-metik.) Facsimile reprint of the 1889 originalpublished by Julius Springer. (German)Remagen: Verlag Kessel (ISBN 978-3-941300-09-5/pbk). xiii, 695 p. EUR 24.00 (2009).There is nothing more to say about this bookby the princeps mathematicorum which waspublished in Latin in 1801 and which shapedclassical number theory. The present publica-tion is a photomechanical reproduction of thefamous German translation by H. Maser (1889;JFM 21.0166.04)]. It contains not only the textof the Disquisitiones Arithmicae but also somemathematical treasures from Gauss’ abatement.Although it is creditable that Maser’s translationis now back in print this translation itself seemsnow ripe to be re-examined and critically accom-panied by a historian of mathematics. We stillhave to wait. Thomas Sonar (Braunschweig)

1166.13001Drton, Mathias; Sturmfels, Bernd;Sullivant, SethLectures on algebraic statistics.Oberwolfach Seminars 39. Basel: Birkhäuser(ISBN 978-3-7643-8904-8/pbk). viii, 271 p.EUR 24.00/net; SFR 42.90; $ 39.95; £ 19.99 (2009).The seminar lectures give an overview of thepresent status of the research area in which tech-niques originating in algebraic geometry, com-mutative algebra and combinatorics are adaptedto solve problems of interest in statistics. Thecontents of the book is divided in seven chap-ters. The first five chapters are expanded ver-sions of lectures given at an Oberwolfach Semi-nar in May 2008: Markov Bases, Likelihood Infer-ence, Conditional Independence, Hidden Vari-ables, Bayesian Integrals. Each of them is devotedto an important model/problem of statistics andexplains ties between them and notions/resultsof algebraic geometry/computational algebra.The sixth chapter contains students’ solutionsto eight problems proposed during the Ober-wolfach Seminar. In the final chapter one finds

twelve open research problems related to alge-braic statistics, presented by lecturers and stu-dents.The authors have striven to gather as many factsas possible and to organize the large amountof material in a most profitable way for the as-tute reader. The general idea is that statisticalhypotheses can be often tested in an exact ap-proach by using algebraic tools developed tosolve (semi-)algebraic equations. Many exam-ples discussed at length in these lecture notesindicates that, in order to draw conclusions withpractical applicability from a geometric study ofthe parameter space, heavy computations are un-avoidable. Several software packages are usedto perform such demanding computations. Thenecessary pieces of code are carefully explained.The volume makes a difficult reading by thewealth of techniques and results touched upon.It is not self-contained. The reader is warned thatto study all the material, at least as much enthu-siasm and energy as that witnessed by the par-ticipants to the seminar are needed.

Mihai Cipu (Bucuresti)

1166.14001Lakshmibai, V.; Brown, JustinFlag varieties. An interplay of geometry, com-binatorics, and representation theory.Texts and Readings in Mathematics 53. NewDelhi: Hindustan Book Agency (ISBN 978-81-85931-92-0/hbk). xiv, 272 p. $ 48.00 (2009).Let K be a field (often assumed in the book to bealgebraically closed) and V be an n-dimensionalK-vector space. A flag in V is a chain of subspaces:0 = V0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂ Vn = V where the di-mension of Vi is i. The set of all flags in V is calledthe flag variety. Consider the general linear groupGLn(K). The flag variety can be identified withthe quotient of GLn(K) by the Borel subgroupof upper triangular matrices. Equivalently, onecan take the special linear group SLn(K) mod-ulo upper triangular matrices. For an arbitrarysemisimple algebraic group G and Borel sub-group B, the quotient G/B may be consideredas a generalized flag variety. The (classical) flag va-riety can also be considered as a subvariety ofthe Grassmannian variety Gd,n where 1 ≤ d ≤ n−1.The Grassmannian variety Gd,n is by definition

22 Number Theory. Algebra. Algebraic Geometry.

the set of all d-dimensional subspaces of V. Usingthe Plücker map, the variety Gd,n can be identi-fied with a subvariety of the projective space onthe dth exterior power of V; in fact, as the zero setof the so-called Plücker relations. The Grassman-nians can also be placed into a general algebraicgroup context since Gd,n can be identified withthe quotient of GLn(K) by a parabolic subgroup.For either the flag variety or a Grassmannian va-riety, an important collection of varieties are theSchubert subvarieties. For the flag variety, con-sidered as G/B, given an element w of the Weylgroup, the Schubert variety associated to w isthe Zariski closure of the B-orbit of the coset wBinside G/B. For the Grassmannian Gd,n, let Id,ndenote the set of d-tuples (i1, i2, . . . , id) such that1 ≤ i1 < i2 < · · · < id ≤ n. Classically and morecombinatorially, a Schubert variety can be associ-ated to each such tuple by using a fixed basis forV. These varieties can also be identified using B-orbits (for B being the upper triangular matricesinside GLn(K)).Flag varieties, Grassmannian varieties, and Schu-bert varieties have played a key role in the devel-opment of algebraic geometry and related sub-jects (such as algebraic groups). The goal of thisbook is to provide an introduction to these ob-jects, presented (as suggested by the title) fromthe perspective that flag varieties involve the in-terplay of algebraic geometry, algebraic groups,combinatorics, and representation theory. Thisbook stems from a series of lectures given by thefirst author at the Institute for Advanced Studyin 2007 and is aimed at an introductory level.The authors are quite thorough in presenting thenecessary background material to develop thesubject. In principle, with a solid background ingraduate algebra, a student or researcher couldlearn the basic ideas from this book, althoughsome prior knowledge of algebraic geometry andalgebraic groups would be beneficial. The dis-cussion of flag varieties is focused on the clas-sical case (working with the general or speciallinear group) in order to bring the reader morequickly to the key ideas. Sufficient references areprovided to direct the interested reader to themore general case and further developments.To help the reader gain understanding of theideas, a number of examples are given as well asproofs of most results. References are providedfor the results (typically more significant) whichare stated without proof. If used as a textbook,all but the first chapter contains a few exercises,

which could be supplemented with problemsfrom standard texts. We now briefly describe thecontents of the book, from which one can see thatit could serve as a concise reference for a numberof topics.The first chapter provides a brief overview of keyideas from commutative algebra (e.g., Noethe-rian, localization, radicals, Krull dimension, reg-ular, and Cohen-Macauley) and algebraic geom-etry (e.g., affine and projective varieties, schemes,sheaves of modules, vector and line bundles,the Picard group and tangent spaces). Chap-ter Two presents a basic discussion of semisim-ple modules and rings, including the Artin-Molien-Wedderburn Structure Theorem. Brauergroups, central simple algebras, and group al-gebras (including Maschke’s Theorem) are alsointroduced. Chapter Three presents some ba-sic ideas in the representation theory of finitegroups over fields of characteristic zero or primeto the order of the group, including characters,irreducible characters, tensor products, restric-tion and induction. Chapter Four then focuseson the representation theory of the finite sym-metric group Sn over fields of characteristic zeroor prime larger than n. Young tableaux are in-troduced, which appear throughout the remain-der of the book. The simple (or irreducible) Sn-modules correspond to partitions of n, and atableaux can be formed from a partition. Twoconstructions of the simple modules are pre-sented: Frobenius-Young modules (arising as leftideals within the group algebra) and Specht mod-ules (arising from Sn acting on a polynomial ringin n variables). The chapter ends with a discus-sion of the representation theory of the alternat-ing group An. In Chapter Five, the discussioncontinues with computations on the charactersand dimensions of Young modules. The chaptercontains some general discussion of symmetricpolynomials and various bases for them.Beginning with Chapter Six, the focus shifts to-ward algebraic groups. The chapter begins witha discussion of endomorphism algebras and thendiscusses Schur-Weyl Duality: the isomorphismof the group algebra of GLn(C) with an endo-morphism algebra over Sd and vice versa. Thisduality allows one to obtain some of the simpleGLn(C), the so-called Schur modules, from Youngmodules. The characters of the Schur modulesare discussed. Over SLn(C), all of the finite di-mensional simple modules arise as certain Schurmodules. From these, one can then give a com-


plete description of the finite dimensional sim-ple GLn(C)-modules. Before beginning a generaldiscussion of algebraic groups, Chapters Sevenand Eight discuss Lie algebras. Chapter Sevenprovides a brief introduction to Lie algebras andthe structure of semisimple Lie algebras, includ-ing root systems. Precise details are given onlyfor the special linear Lie algebra sln(C). Chap-ter Eight turns to the representation theory ofsemisimple Lie algebras (in characteristic zero).The universal enveloping algebra and weighttheory are introduced, along with the construc-tion of irreducible highest weight modules. Thechapter ends with Weyl’s character and dimen-sion formulas, and some specific discussion ofthe sln(C) case.Chapters Nine through Eleven discuss algebraicgroups. The basic concepts are presented inChapter Nine, including the associated Lie alge-bra, tori, Borel subgroups, parabolic subgroups,Jordan decomposition, semisimple groups, re-ductive groups, and group actions. The flag va-riety is introduced along with its connection toG/B (as discussed above). Finally, the collectionof Borel subgroups is studied and seen to beidentifiable with the flag variety G/B. In Chap-ter Ten, the basic structure theories of reductiveand semisimple algebraic groups are presented,including a discussion of groups of adjoint anduniversal type for a given root system. Schu-bert varieties are then introduced along with theBruhat Decomposition of G/B into a union of B-orbits. In Chapter Eleven, the representation the-ory of semisimple algebraic groups is discussed,including weight theory and the correspondencebetween simple modules and dominant weights,and the simple modules are constructed (al-gebraically and geometrically) in characteristiczero. Weyl modules are introduced as duals tosections of a line bundle on G/B, and their charac-ters and dimensions are given (analogous to theLie algebra case) by Weyl’s character formula. Incharacteristic zero, the Weyl modules are simple,but that is generally not true in prime charac-teristic. Related to the algebraic group discus-sion, the book contains an Appendix on Cheval-ley groups.In Chapter 12, the reader arrives in a sense atthe goal of the book. Grassmannian varietiesand their Schubert subvarieties are introduced,including a thorough discussion of the Plückermap, coordinates, and relations. A Plücker co-ordinate pτ for Gd,n corresponds to a d-tuple

τ ∈ Id,n. The set Idn of d-tuples is partially or-dered by (i1, i2, . . . , id) ≤ ( j1, j2, . . . , jd) if and onlyif is ≤ js for each s. A monomial pτ1 pτ2 . . . pτm

in Plücker coordinates is said to be standard ifτ1 ≥ τ2 ≥ · · · ≥ τm. For a Schubert variety X(ω)associated to a tuple ω, such a monomial is stan-dard on X(ω) if in addition ω ≥ τ1. The notion ofStandard Monomial Theory is developed, with thekey result being that the monomials of degreem standard on X(ω) give a K-basis for the de-gree m homogeneous coordinates of X(ω). Fur-ther discussion is also given on unions and in-tersections of Schubert varieties. Next, the well-known sheaf cohomology vanishing theorem ispresented. Specifically, for a Schubert variety X,the higher cohomology of powers of OX(1) van-ishes, and the zeroth cohomology (global sec-tions) has a basis given by standard monomials.The remainder of the chapter presents an analo-gous development of standard monomial theory(and the vanishing theorem) for the flag variety.Here, one works with tableaux and must con-sider standard tableaux (which extends the classi-cal notion of standard).In Chapter Thirteen, computations of the singu-lar locus (the non-smooth points) of Schubert va-rieties for SLn(K)/B are presented. Information onthe singular locus can be obtained from combi-natorial information in the Weyl group. Finally,in Chapter Fourteen, two key applications arepresented. The first application involves classi-cal invariant theory. With V as above, let X be thesum of m-copies of V along with q-copies of thedual module V∗ for m, q > n. Let GLn(K) act diag-onally on X. Of interest is the subring of GLn(K)-fixed points of K[X]. This can be identified withthe coordinate algebra of a determinantal vari-ety. The determinantal variety can be identifiedwith an open subset of a Schubert variety, whichallows one to use the standard monomial theoryfor Schubert varieties to identify generators andrelations for the ring of invariants. The secondapplication is the degeneration of Schubert va-rieties (for the Grassmannian or flag variety) totoric varieties. The toric varieties are associatedto distributive lattices within Id,n (or a union ofsuch). Christopher P. Bendel (Menomonie)


15001Higham, Nicholas J.Functions of matrices.Theory and computation.Philadelphia, PA: Society for Industrial andApplied Mathematics (SIAM) (ISBN 978-0-898716-46-7/hbk). xx, 425 p. $ 59.00 (2008).This book treats of matrix functions andboth states pure theoretical fundamentals andpresents applied numerical methods. To thereader benefit, the scope of the book is far be-yond its title and it also comprises basic matrixtheory and description of numerical packagesprocedures. Furthermore, it includes remarkablehistoric facts and also serves as a complete bibli-ographic reference. Surpassing his own style theauthor excels himself in details, presentation andstyle, providing essays, comments and a metic-ulous layout, which reflects care and dedicationof a master to his work.The book contains a preface, 14 chapters, 5 ap-pendixes and a bibliography with an impressivenumber of 625 entries. Chapters include interest-ing examples and always contain abundant notesand references, ending with a list of problemswith the solutions appearing in one of the ap-pendixes. The first chapter introduces definitionsfor matrix functions, polynomial interpolation,Cauchy integral theorem, matrix square rootsand logarithms. Chapter 2 is concerned with ap-plications to differential equations, Markov mod-els, control theory, eigenvalue problem and non-linear matrix problems. The third chapter con-siders the conditioning and sensitivity of compu-tations of matrix functions especially by meansof the Fréchet derivative. In Chapter 4 methodsfor matrix power, polynomial evaluation, Tay-lor series, rational approximation, diagonaliza-tion and others are discussed. Related theorems,algorithms, cost and stability studies are treated.The next three chapters focus on the matrix signfunction, matrix square root and matrix pth root.Chapter 8 considers the polar decomposition andits relation with the singular value decomposi-tion. Chapter 9 is devoted to Schur-Parlett al-gorithm by the use of the Taylor series expan-sion. Chapter 10 discusses the matrix exponen-tial, “the most studied matrix function", reflect-ing its importance in the solution of differentialequations. A natural continuation is the matrixlogarithm presented in Chapter 11, and matrixsine and cosine in Chapter 12. In Chapter 13 isgiven a substantial treatment of a function matrix

times a vector, considering polynomial interpo-lation, Krylov subspace methods and numericalquadrature. The concluding Chapter 14 includessome additional topics under the title of miscel-lany. In the Appendixes are the notation, defini-tions, operation counts, toolbox references, andthe solutions of the problems.As a final word, this book should be a referenceto anyone involved with matrix analysis and itappears to be a classic in the coming years, to-gether with the ones of F. R. Gantmacher [Thetheory of matrices. (1953; Zbl 50.24804)] and J.H. Wilkinson [The algebraic eigenvalue problem(1965; Zbl 258.65037)]. Edgar Pereira (Covilha)

1167.15002Knop, Larry E.Linear algebra. A first course with applications.Textbooks in Mathematics. Boca Raton, FL:CRC Press (ISBN 978-1-58488-782-9/hbk). xix,725 p. $ 99.95 (2009).This textbook investigates the classical ideasof linear algebra, including vector spaces, sub-spaces, basis, span, linear independence, lineartransformation, eigenvalues, and eigenvectors. Itincludes a variety of applications integrated intothe text, from inventories to graphics to Google’sPageRank. This classroom-tested book has anearly focus on the structure of linear algebra, giv-ing students as much of the semester as possibleto absorb the difficult parts of the material.Computational aspects provide changes of pace.Moving from the specific to the general, the au-thor raises questions, provides motivation, anddiscusses strategy before presenting answers.Discussions of motivation and strategy includecontent and context to help students learn. More-over, this book provides the option to use com-putational technology: The text offers just-in-time and just enough introductions to MapleTM,MATLAB, and T1-83 Plus for numerical calcula-tions of matrix row reduction, matrix inverses,determinants, eigenvalues, and eigenvectors.

Qing-Wen Wang (Shanghai)

1167.17004Benkart, Georgia; Gregory, Thomas; Premet,AlexanderThe Recognition Theorem for graded Lie alge-bras in prime characteristic.Mem. Am. Math. Soc. 920, 145 p. (2009).The finite dimensional simple Lie algebras overan algebraically closed field of characteristic p ≥


5 have been classified thanks to work of Block,Wilson, Strade and Premet. Firstly, we have theLie algebras of simple algebraic groups (factoringout the centre when necessary). These algebraswhere called “classical" by Seligman. Secondly,there are the algebras corresponding to the fourfamilies W, S, H, K (Witt, special, Hamiltonianand contact) of infinite dimensional complex Liealgebras arising in Cartan’s work on Lie pseu-dogroups. These algebras were called “Lie alge-bras of Cartan type" by Kostrikin and Shafare-vich. They conjectured in 1966 that any restrictedfinite dimensional simple Lie algebra over an al-gebraically closed field of characteristic p ≥ 7 iseither classical or of Cartan type. This was provedin 1984 for p > 7 by R. E. Block and R. L. Wilson[J. Algebra 114, 115–259 (1988; Zbl 644.17008)].Thirdly, there are, only in characteristic 5, the Me-likyan algebras. The Classification Theorem as-serts that every finite dimensional simple Lie al-gebra over an algebraically closed field of char-acteristic p ≥ 5 is of classical, Cartan or Melikyantype.The so-called “Recognition Theorem" states thefollowing (see page ix):Let g be a finite-dimensional graded Lie algebraover an algebraically closed field of characteristicp > 3. Assume that:(a) g0 is a direct sum of ideals, each of which isabelian, a classical simple Lie algebra, or one ofthe Lie algebras gln, sln, or pgln with p|n;(b) g0 is an irreducible g0-module;(c) If x ∈ ⊕ j≥0 g j and [x, g−1] = 0, then x = 0;(d) If x ∈ ⊕ j≥0 g− j and [x, g1] = 0, then x = 0.Then g is isomorphic as a graded Lie algebra toone of the following:(1) a classical simple Lie algebra with a standardgrading;(2) pglm for some m such that p|m with a standardgrading;(3) a Cartan type Lie algebra with the naturalgrading or its reverse;(4) a Melikyan algebra (in characteristic 5) witheither the natural grading or its reverse.This theorem plays a vital role in the extensionof the classification theory to characteristic 5 and7, since the classification theory aims to showthat any finite dimensional simple Lie algebraL admits a filtration L = L−q ⊃ · · · ⊃ L0 ⊃ · · · ⊃

Lr ⊃ Lr+1 = 0 such that the corresponding gradedLie algebra g = ⊕iLi/Li+1 satisfies the assump-tions (a)–(d) above. In the work under reviewthe authors present the first complete proof of the

Recognition Theorem. The proof contains manyimportant ideas due to V. G. Kac who first un-dertook the proof of the Recognition Theorem ina pioneering work in 1970 [Math. USSR, Izv. 4,391–413 (1970); translation from Izv. Akad. NaukSSSR, Ser. Mat. 34, 385–408 (1970; Zbl 254.17007)].

Rudolf Tange (York)

1166.20056Ganyushkin, Olexandr; Mazorchuk, Volody-myrClassical finite transformation semigroups.An introduction.Algebra and Applications 9. London: Springer(ISBN 978-1-84800-280-7/hbk; 978-1-84800-281-4/e-book). xii, 314 p. EUR 62.95/net; SFR 104.50;$ 89.95; £ 45.00 (2009).“Semigroup theory is a relatively young part ofmathematics. As a separate direction of algebrawith its own objects, formulations of problems,and methods of investigations, semigroup the-ory was formed about 60 years ago."There are only a few old books about transfor-mation semigroups. The present book fills a gapin the literature.“This monograph gives a self-contained intro-duction to the modern theory of finite trans-formation semigroups with a strong emphasison concrete examples and combinatorial appli-cations. It covers the following topics on theexamples of the three classical finite transfor-mation semigroups: transformations and semi-groups, ideals and Green’s relations, subsemi-groups, congruences, endomorphisms, nilpotentsubsemigroups, presentations, actions on sets,linear representations, cross-sections and vari-ants. The book contains many exercises and his-torical comments and is directed, first of all, toboth graduate and postgraduate students look-ing for an introduction to the theory of transfor-mation semigroups, but should also prove usefulto tutors and researchers."The titles of the chapters of the book are as fol-lows: Ordinary and Partial Transformations; TheSemigroups Tn, PTn, and ISn; Generating Sys-tems; Ideals and Green’s Relations; Subgroupsand Subsemigroups; Other Relations on Semi-groups; Endomorphisms; Nilpotent Subsemi-groups; Presentation; Transitive Actions; Lin-ear Representations; Cross-Sections; Variants;Order-Related Subsemigroups.

Sheng Chen (Harbin)


1167.20030Szabó, Sándor; Sands, Arthur D.Factoring groups into subsets.Lecture Notes in Pure and Applied Mathemat-ics 257. Boca Raton, FL: CRC Press (ISBN 978-1-4200-9046-8/pbk). xv, 269 p. $ 179.95 (2009).Let G be a finite Abelian group and let A and Bbe subsets of G. The sum A + B of the subsets Aand B is defined to be the set a+b : a ∈ A, b ∈ B.If the elements in this subset are distinct, the sumis called direct. This is a book on the factorizationtheory of Abelian groups into direct sums of itssubsets, focusing mainly on cyclic groups.Below we list the titles of the chapters: 1. Intro-duction. 2. New factorizations from old ones. 3.Non-periodic factorizations. 4. Periodic factor-izations. 5. Various factorizations. 6. Factoringby many factors. 7. Group of integers. 8. Infinitegroups. 9. Combinatorics. 10. Codes. 11. Someclassical problems.Reviewer’s remark: Some overlaps occur withanother similar book published in 2004 by thefirst author: Topics in factorization of Abeliangroups [Texts and Readings in Mathematics29. New Delhi: Hindustan Book Agency (2004;Zbl 1086.20026) and Basel: Birkhäuser (2004;Zbl 1085.20032)].

Grigore Calugareanu (Cluj-Napoca)

1166.11004Agoh, TakashiThe square and cube of Fermat quotients andvalues of Mirimanoff polynomials.JP J. Algebra Number Theory Appl. 11, No. 1,113-127 (2008).The author studies relations between the Fer-mat quotients qp(a) = (ap−1

− 1)/p and values ofthe Mirimanoff polynomials Gm(x) =

∑p−1k=1 xk/km

(m = 1, 2, . . . ). Here p is a prime ≥ 5. In partic-ular, the author proves congruences mod p forqp(a)2 and qp(a)3 with small a in terms of val-ues of Gm(x). His starting point is the congruenceqp(2)2

≡ −G2(2) (mod p) as well as a similar butmore complicated congruence for qp(2)3, whichwere proved by A. Granville [Integers 4, PaperA22, 3 p., electronic only (2004; Zbl 1083.11005)]and K. Dilcher and L. Skula [ibid. 6, Paper A24,12 p. (2006; Zbl 1103.11011)], respectively. Thepresent results contain these congruences as spe-cial cases. Also Bernoulli numbers Bn, especiallyBp−3, play a role here, due to their known connec-tion mod p to Gm(±1). Tauno Metsänkylä (Turku)

1166.11013Csikvári, PéterSubset sums avoiding quadratic nonresidues.Acta Arith. 135, No. 1, 91-98 (2008).If the set of all subset sums

FS(A) =∑

εaa : εa ∈ 0, 1,∑

εa > 0

of a set A ⊆ Fp avoids the quadratic nonresidues,what can be said about the cardinality of A? Thepresent paper establishes an upper bound of theform

|A| = O(n(p) log3 p),

where n(p) is the least quadratic nonresidue mod-ulo p, as an almost immediate consequence of thedeeper statement that if |A| is sufficiently large,then FS(A) contains a long arithmetic progres-sion. More precisely, if |A| ≥ 2000t log3 p, thenthere exists a d , 0 such that d, 2d, . . . , td ⊆FS(A), a result which was originally proved inthe integers (rather than Fp) by P. Erdo and A.Sárközy [Discrete Math. 102, No. 3, 249–264 (1992;Zbl 758.11007)].On the other hand, it is shown (non-constructively) that there exists a set A with theproperty that FS(A) avoids the quadratic non-residues which has cardinality

|A| = Ω(log log p).

The proof uses a maximality argument combinedwith the fact that if Q + B = Fp, where Q denotesthe quadratic residues, then B has to have size atleast 1

4 log p, which in turn follows from standardWeil sum estimates. Julia Wolf (Piscataway)

1166.11014Wan, ZhexianA generalization of Witt’s theorem andSylvester’s law of nullity.Algebra Colloq. 15, No. 2, 181-184 (2008).In his groundbreaking article on quadratic formsE. Witt [J. Reine Angew. Math. 176, 31–44 (1937;Zbl 15.05701)] has established among otherthings the famous cancellation theorem, whichsays that q ⊥ q1 ' q ⊥ q2 implies q1 ' q2 for non-degenerate quadratic forms q, q1, q2 over a fieldof characteristic different from 2.The paper under review gives a generalization ofWitt’s cancellation theorem by allowing q, q1, q2to be degenerate. This generalization howevercan already been found in the existing literaturewith slightly different proofs, e.g. in T. Y. Lam’s


book [Introduction to quadratic forms over fields(Graduate Studies in Mathematics 67, AmericanMathematical Society (AMS), Providence) (2005;Zbl 1068.11023)].Moreover, the author shows that the generaliza-tion of Witt’s theorem over R is equivalent toSylvester’s law of inertia. The latter is called lawof nullity in this paper, a name which is usuallyattributed to Sylvester’s result about the nullity(i.e. dimension of the kernel) of a product of twomatrices. Roland Lötscher (Basel)

1166.11017Abdón, Miriam; Bezerra, Juscelino; Quoos, Lu-cianeFurther examples of maximal curves.J. Pure Appl. Algebra 213, No. 6, 1192-1196(2009).A smooth, projective and geometrically irre-ducible curve C of genus g over a finite fieldFq2 is called maximal over Fq2 if #C(Fq2 ) attainsthe Hasse-Weil upper bound, i.e., #C(Fq2 ) =

q2 + 1 + 2gq. For such curves, one necessarilyhas g ≤ q(q − 1)/2. Further, there exists a uniquemaximal curve H over Fq2 , with affine modelyq + y = xq+1 and called the Hermitian curve, suchthat g has the maximal value q(q − 1)/2.This paper studies some classes of maximalcurves and discusses whether or not the curvesstudied are Galois-covered by H . First, the au-thors study the nonsingular models of the curves

C(q,n) : yq2− y = x

qn+1q+1 defined over Fq2n , where

n ≥ 3 is an odd integer. If ` is a prime, theF`6 -curves C(`, 3) were previously studied by A.García and H. Stichtenoth. They showed thatthese curves are maximal over F`6 of genus(`2−1)(`2

−`)/2. An interesting observation is thefollowing: the curve C(3, 3) is not Galois-coveredby the Hermitian curveH , in contrast to the caseof C(2, 3).In this paper the curves C(q,n) are shown to bemaximal for any q. Further, it is shown that thecurves C(2,n) are Galois-covered by H . The au-thors also discuss the family of maximal curvesCb defined over Fq2n by the affine equation yN =

−xb(x + 1), 1 ≤ b ≤ N − 1, where N is an odd divi-sor of qn + 1 and (N, b) = (N, b + 1) = 1.

Cristian D. Gonzales-Aviles (La Serena)

1166.11019Garcia, Arnaldo; Tafazolian, SaeedCertain maximal curves and Cartier operators.

Acta Arith. 135, No. 3, 199-218 (2008).In this article the authors use Cartier operatorC to improve several results on some particularmaximal curves overFq2 , where q = pn. The mainargument is that if C/Fq2 is a maximal curve,then Cn = 0. The first family is Fermat curvesC(m) : xm + ym + zm = 0 with m prime to p. Theyshow that C(m) is maximal over Fq2 if and onlyif m divides q + 1. This generalizes [A. Aguglia,G. Korchmáros and F. Torres, Acta Arith. 98, No.2, 165–179 (2001; Zbl 1015.11026)] and [T. Ko-dama and T. Washio, Manuscr. Math. 60, 185–195(1988; Zbl 653.14014)]. The second part deals withArtin-Schreier curves. If C : yq

− y = f (x) withf (x) ∈ Fq2 [x] of degree d prime to p is maximalover Fq2 , then C is isomorphic to the projectivecurve defined by the affine equation yd + y = xd

with d dividing q + 1. The main ideas come from[S. Irokawa and R. Sasaki, Tsukuba J. Math. 15,No. 1, 185–192 (1991; Zbl 745.14010)]. Finally theyshow that there is a unique maximal hyperellip-tic curve over Fq2 with genus (q− 1)/2 which canbe given by y2 = xq + x when p , 2 and a uniquemaximal hyperelliptic curve over Fq2 with genusq/2 which can can be given by y2 + y = xq+1 whenp = 2. The main idea is a characterization of hy-perelliptic curves with zero Hasse-Witt matrixobtained in [R. C. Valentini, Manuscr. Math. 86,No. 2, 185–194 (1995; Zbl 833.14021)].

Christophe Ritzenthaler (Marseille)

1166.11020Howe, Everett W.Supersingular genus-2 curves over fields ofcharacteristic 3.Lauter, Kristin E. (ed.) et al., Computationalarithmetic geometry. AMS special session, SanFrancisco, CA, USA, April 29–30, 2006. Prov-idence, RI: American Mathematical Society(AMS) (ISBN 978-0-8218-4320-8/pbk). Contem-porary Mathematics 463, 49-69 (2008).The main purpose of the article is to completethe result of E. W. Howe, E. Nart and the reviewer[Ann. Inst. Fourier 59, No. 1, 239–289 (2009)]by giving the Weil polynomials for supersingu-lar genus 2 curves over the finite fields Fq withq = 3d. The result is the following: if d is odd theWeil polynomial belongs to the following list1. (x2 + q)(x2

− sx + q) for all s ∈ ±√

3q;2. (x2 + q)2, if q > 3;3. x4 + q;4. x4 + qx2 + q2;


5. x4− 2qx2 + q2 if q > 3.

And if d is even it belongs to the list1) (x2

− 2sx + q)(x2 + sx + q) for all s ∈ ±√

q;2) (x2

− sx + q)2 for all s ∈ 0,±√

q;3) (x2

− 2sx + q)2 for all s ∈ ±√

q, if q > 9;4) x4 + q2;5) x4

− sx3 + qx2− sqx + q2 for all s ∈ ±

√q.

The Weil polynomials of abelian surfaces whichcannot be obtained are treated using various ar-guments. For positive results, note that for eachclass, an explicit curve can be constructed. Thisis done by using the first part of the paper whichgives a beautiful analysis of the coarse modulispaceA of triples (C,E, φ) where C is a supersin-gular genus 2 curve over a field of characteris-tic 3, E the elliptic curve with j-invariant 0 andφ : C→ E a degree 3 map. The author shows (Th.2.1) that C is supersingular if and only if its Igusainvariants J2, J4 and J8 are zero and shows thatC : y2 = x6 + Ax3 + Bx + A2 represents the invari-ant [0 : 0 : A : 0 : B]. Two other nice propertiesare shown : if C is a genus 2 triple cover of a su-persingular elliptic curve in characteristic 3 thenC is supersingular (Cor.3.3) ; if C : y2 = f (x) withf a sextic polynomial is a supersingular genus 2curve such that f can be written as the productof two cubic factors then C is a triple cover of asupersingular elliptic curve (Th.3.4).Finally the author shows that A is isomorphicto the affine line with one point removed and isa degree 20 cover of the moduli space of super-singular genus 2 curves. On the other hand, theauthor shows that A is also isomorphic to themoduli space of pairs (C,G) where C is supersin-gular and G is an order 4 subgroup of Jac(C)[2]that is not isotropic with respect to the Weil pair-ing. Christophe Ritzenthaler (Marseille)

1166.11022Bruin, NilsThe arithmetic of Prym varieties in genus 3.Compos. Math. 144, No. 2, 317-338 (2008).Let π : D→ C be an unramified finite morphismof degree two between curves over a field K. ThePrym variety of D/C is the connected componentof the identity element of the kernel of the mapπ∗ : Jac(D) → Jac(C), denoted by Prym(D/C). Inhis earlier work, the author considered this setupwhere C is a hyperelliptic curve, and had successin applying the arithmetic theory of this setup,combined with explicit Chaubauty methods, todetermining the set of rational points on suchcurves. In the paper under review, he considers

the case where C is non-hyperelliptic of genus 3,and accomplishes the following:(1) An explicit construction of a curve F of genus2 for which Jac(F) Prym(D/C);(2) a construction of a map φ : D → Jac(F) Prym(D/C) which does not require a rationalpoint on D.When K = C, the description Prym(D/C) Jac(F)can be found in [E. Arbarello, M. Cornalba, P. A.Griffiths, and J. Harris, Geometry of algebraiccurves, Vol. I, Grundlehren der mathematischenWissenschaften, 267. New York etc.: Springer-Verlag (1985; Zbl 559.14017)]. When D has a ra-tional point, we have the well-known Abel-Prymmap (id∗ − ι∗) : D → Prym(D/C) where ι is thenontrivial involution on D with no fixed points.Combined with earlier work, the construction ofa curve F described above gives a complete de-scription of how principally polarized Abeliansurfaces arise as Prym varieties with g(C) = 3over base fields of characteristic zero which isnot necessarily algebraically closed.The author also proves a case of the converse:Given an arbitrary curve F of genus 2 over a num-ber field K, there is a Prym variety Prym(D/C)which is isomorphic to Jac(F), and he remarksthat the construction of Prym(D/C) certainlyworks over any base field of odd characteristicwith sufficiently many elements.As an application the author presents threeexamples of non-hyperelliptic curves C/Q ofgenus 3: (1) A curve C with exactly one rationalpoint; (2) An unramified double cover D/Q of Csuch that both curves have points locally every-where but have no rational points; (3) A curve Cwith all 28 bitangents rational, for which he useshis general form of a non-hyperelliptic curve ofgenus 3 and a Kummer surface Jac(F)/〈±1〉.

Sungkon Chang (Savannah)

1166.11023Poulakis, DimitriosOn the rational points of the curve f (X,Y)q =h(X)g(X,Y).Can. Math. Bull. 52, No. 1, 117-126 (2009).Let φ : D → C be an unramified morphism ofprojective smooth curves defined over Q. By theChevalley-Weil theorem, there is a number fieldK such that φ−1(C(Q)) ⊂ D(K). In the paper un-der review, the author considers a morphism ψbetween two special families of affine curves Vand W, and explicitly computes a number fieldK such that


ψ−1(W(Q)) ⊂ V(K),

and uses this setup to completely determine therational solutions of three examples of curvesand two examples of families of curves. Thisexplicit computation is generalized in the re-cent works K. Draziotis and D. Poulakis [“ExplicitChevalley-Weil theorem for affine plane curves,"Rocky Mt. J. Math. 39, No. 1, 49–70 (2009) andAn effective version of Chevalley-Weil theoremfor projective plane curves, arxiv:0904.3845v1].Let F(X,Y) := f (X,Y)q

− h(X) g(X,Y) be a polyno-mial inZ[X,Y] where f , h, and g are fairly generalpolynomials. Let h(X) = h1(X) h2(X) be a certainfactorization in Z[X], which always is possible,such that q | deg(h1). See the paper for details.Consider the varieties

V : F(X,Y) = 0, Tq = h1(X),W : F(X,Y) = 0.

For q = 2 or 3, he explicitly computes a finite setS such that the number field K described above isgiven by Q(

q√b ∈ S). More specifically, he shows

that for each (x, y) ∈ W(Q), there is a b ∈ S suchthat the twist bTq = h1(X) has a solution (x, t)where t ∈ Q. Thus, it reduces the problem to thatof solving finitely many twists of a superellipticcurve. It seems to the reviewer that the restric-tion on q in the theorem is only for practical com-putability purpose and that the theorem can besomewhat explicitly stated for q being a primenumber.He gives a geometric interpretation of this re-duction of the original problem in which it isestablished that the morphism between the pro-jective desingularizations D and C of V and W,respectively, is unramified, and hence, being putinto the context of the Chevalley-Weil Theorem.He uses the crucial condition q | deg(h1) to pullthe unramified property out of the towers of thefunction fields Q(A), Q(W), and Q(Z) where Zis the affine variety Tq = h1(X), and he uses thecondition in the proof of the main theorem forthe affine varieties as well.


1166.11037Bleher, Frauke M.; Chinburg, Ted;Froelich, JenniferCapping groups and some cases of theFontaine-Mazur conjecture.Proc. Am. Math. Soc. 137, No. 5, 1551-1560(2009).

The authors prove some particular cases of theFontaine-Mazur conjecture. Let p be an oddprime and GQ,p the Galois group over Q of themaximal unramified-outside-p extension Qp ofQ. They show that, under certain hypotheses, theuniversal deformation of the action of GQ,p onthe 2-torsion of an elliptic curve defined overQ has finite image. They compute the associ-ated universal deformation ring and show thatS4 caps Q for the prime 2, where S4 is the doublecover of the symmetric group S4 whose Sylow2-subgroups are generalized quaternion groups.This article lies on elliptic curves theory, repre-sentation of groups theory and universal defor-mation ring theory, see in particular F. M. Bleherand T. Chinburg [Math. Ann. 337, No. 4, 739–767(2007)] and N. Boston [Contemp. Math. 416, 31–40(2006)].Notations-definitions: The concept of cappinggroups defined in [Bleher and Chinburg, loc.cit.] is recalled here: let l be a prime number,and suppose there is a short exact sequence1 → K → Γ

π→ G → 1, where Γ and G are

profinite groups, π is a continuous group homo-morphism and K is a closed normal subgroup ofΓ. We say that G caps Γ (via π) for l if there is noclosed normal subgroup K0 of Γ satisfying K0 < Kand for which K/K0 is a non-trivial pro-l group.The two main results of the article follow:Theorem: Let F4 be a totally real quartic field ofodd prime discriminant p ≡ 5 mod 8 (see also[Bleher and Chinburg, loc. cit.], proposition 6.2).Let N be the Galois closure of F4 over Q. ThenN is a S4-extension of Q and N has three cubicsubfields.Suppose that the following are true:1) F4 has odd class number.2) There is a unit of F4 which is not a square atsome unramified prime of F4 over p.3) The class number of one cubic subfield F3 of Nis exactly divisible by 2.4) There is a unit of F3 which is not a square atsome prime of F3 over p.Then N is contained in an S4-extension N′ of Q.The group S4 caps GQ,p for l = 2 via the resultingsurjection π : GQ,p → Gal(N′/Q).Theorem: Let p be a prime as before. There is,up to isomorphism, a unique two-dimensionalirreducible representation V of Gal(N/Q) =S4 over k = Z/2. The universal deforma-tion ring R(Gal(Qp/Q),V) R(S4,V) of V isisomorphic to Z2[[t]]/(t3

− 2t). In particular,R(Gal(Qp/Q),V) is a complete intersection ring.


Moreover, the image of the universal deforma-tion ρ : Gal(Qp/Q) → GL2(R(Gal(Qp/Q),V)) isisomorphic to S4. Roland Quême (Brax)

1166.11038Dembélé, LassinaA non-solvable Galois extension of Q ramifiedat 2 only. (English. Abridged French version)C. R., Math., Acad. Sci. Paris 347, No. 3-4, 111-116 (2009).B. H. Gross [Int. Math. Res. Not. 1998, No. 16,865–875 (1998; Zbl 978.110181)] proposed the fol-lowing conjecture: for any prime number p, thereexists a finite non-solvable Galois extension K/Qramified at p only. Via results of Serre, one knowshow to construct such an extension for primesp ≥ 11. In this note, the author settles the conjec-ture for p = 2, by using the Galois representationsattached to Hilbert modular forms over the max-imal totally real subfield F of the cyclotomic fieldQ(ζ32).Precisely: Theorem. There exist two SL2(F28 )-extensions E/F and E′/F ramified at the uniqueprime ideal above 2 only. The fields E andE′ are both Galois over Q(

√2), with Ga-

lois group SL2(F28 ).4, and are interchanged byGal(Q(

√2)/Q).

Corollary. There exists a finite, totally complex,non-solvable Galois extension K/Q that is ram-ified at 2 only, with Galois group SL2(F28 )2.8.Moreover, the root discriminant of K is < 2

478 =

58, 68 . . . . This bound has been lowered by J.-P.Serre [C. R., Math., Acad. Sci. Paris 347, No. 3–4,117–118 (2009)]. Richard Massy (Valenciennes)

1167.11007Müller, TomA generalization of a theorem by Krížek, Lucaand Somer on elite primes.Analysis, München 28, No. 4, 375-382 (2008).Let Fn = 22n

+ 1 be the nth Fermat numberwith n = 0, 1, . . .. An odd prime p is calledelite if only finitely many Fermat numbers arequadratic residues modulo p. Such numbers canbe used as basis for the Proth test to check for theprimality of all sufficiently large Fermat num-bers. In 2002, the reviewer together with M.Krížek and L. Somer [J. Number Theory 97, No.1, 95–112 (2002; Zbl 1026.11011)] proved that thecounting function of the elite primes p ≤ x isO(x/(log x)2). In particular, the sum of the re-ciprocals of the elite primes converges. In this

nicely written paper, the author proves an anal-ogous result for the b-elite primes (b > 1 an in-teger), which are odd primes p such that onlyfinitely many of the generalized Fermat num-bers Fb,n = b2n

+ 1 are quadratic residues modulop. The author shows that if Nb(x) is the num-ber of b-elite primes p ≤ x, then the inequal-ity Nb(x) < (64 + 176 log b)x/(log x)2 holds for allx > b32.The method follows closely the Krížek, Luca,Somer proof of this result for the case b = 2 withthe added value that all the constants appearingin that proof are made explicit. Writing Eb for theset of all b-elite primes, the author also showsthat E = ∪b≥2Eb has a positive relative lower den-sity of at least 1/2 as a subset of all primes byfinding for each prime p ≡ ±3 (mod 8) a numberb such that p ∈ Eb.The paper concludes with several remarks,heuristics and open questions on b-elite primesand on b-anti elite primes, which are defined asbeing those odd primes p such that only finitelymany of the numbers Fb,n are quadratic non-residues modulo p. Florian Luca (Morelia)

1167.11008Łuczak, Tomasz; Schoen, TomaszOn a problem of Konyagin.Acta Arith. 134, No. 2, 101-109 (2008).Let (G,+) be an abelian group. For A ⊆ G andt ∈ G, let ν(t) = |(a, b) ∈ A × A : t =a + b|. Let ν(A) = mint∈A+A ν(t). Sergei V. Konya-gin [see V. F. Lev, “Reconstructing integer setsfrom their representation functions", Electron. J.Comb. 11, No. 1, Research paper R78, 6 p., elec-tronic only (2004; Zbl 1068.11006)], asked the fol-lowing question:“Do there exist constants ε,C > 0 such that forevery sufficiently large prime p and each set A ⊆Z/pZwith |A| <

√p, we have ν(A) ≤ C|A|1−ε?"

In the paper under review the authors prove aresult towards this direction. The main theoremstates that there are positive constants C1,C2 suchthat if A ⊆ Z/pZ verifies |A| < C1p2−d−1

, d integer,d ≥ 3 then

ν(A) ≤ C2|A|.

More precise, but technically complicated, infor-mation connecting C1,C2 and d is given.Ingredients of the proof are Dirichlet’s approxi-mation theorem and the following result of Plün-necke and Ruzsa:


“Let C,D be finite subsets of an abelian group. If|C + D| ≤ K|D|, then for every k ≥ 1,

|kC| ≤ Kk|D|.

Georges Grekos (St Etienne)

1167.11018Mizuno, YoshinoriOn Fourier coefficients of Eisenstein series andNiebur Poincaré series of integral weight.J. Number Theory 128, No. 4, 898-909 (2008).There are several known theorems of the follow-ing basic structure: Certain averages of automor-phic forms evaluated at suitable points (e.g. CMpoints) give rise to Fourier coefficients of anotherclass of automorphic forms. The paper under re-view gives another contribution to this topic:Let K be an imaginary quadratic number field ofdiscriminant −D and ring of integers O. For anyfunction f on the three-dimensional hyperbolicspace H3 which is automorphic with respectto SL2(O) the author defines an average valueTrm( f ) over the three-dimensional analogue ofCM points depending on a natural number m. If fis an Eisenstein series onH3 for the group SL2(O)then Trm( f ) is essentially equal to the Fourier co-efficient of a linear combination of Eisenstein se-ries on the upper half-plane H2 for the groupΓ0(D) (see Theorems 1 and 5).An analogous result holds for the Poincaré se-ries constructed with the Bessel functions Iν in-stead of the Eisenstein series on H3 and H2, re-spectively. (The author calls these series NieburPoincaré series since they appear in a paper ofD. Kiebur [Nagoya Math. J. 52, 133–145 (1973;Zbl 288.10010)]. The reviewer wants to pointout that the same series came up simultane-ously in the Ph. D. Thesis of H. Neunhöffer[Über die analytische Fortsetzung von Poincaré-Reihen. Sitzungsber. Heidelberger Akad. Wiss.,Math.-Naturwiss. Kl., 1973, 2. Abhdl. 62 S. (1973;Zbl 272.10015)].)In the course of the proof of Theorem 1 a simplearithmetic expression for the zeta function of rep-resentation numbers by positive definite integralbinary Hermitian forms is given (see Theorem3). Theorem 4 gives a nice expression for a cer-tain weighted sum of all Trm( f ) (m ∈ N) for anEisenstein series f onH3 in terms of a finite sumof products of L-functions. There are results onanalogues of Salié sums which are of indepen-dent interest. Jürgen Elstrodt (Münster)

1167.11020Dummigan, Neil; Watkins, MarkCritical values of symmetric power L-functions.Pure Appl. Math. Q. 5, No. 1, 127-161 (2009).The authors study the critical values of symmet-ric power L-functions attached to elliptic curvesdefined over Q, in view of the Bloch-Kato con-jecture. They first show how to obtain a Deligneperiod, then they calculate the Tamagawa factorswhich appear in the Bloch-Kato conjecture. Moreprecisely, they compute the `-part of the Tam-agawa factor at a prime p of multiplicative re-duction (or potentially multiplicative reductionif p , 2), for any prime ` , p. Finally, they boundtorsion terms (which also appear in the Bloch-Kato conjecture), which they use, along with theircomputations for the algebraic parts of the rele-vant L-values to reach conclusions about the sizeof the Shafarevich-Tate group.For example, suppose n = 2l+1 is an odd integer,and consider the nth symmetric square SymnE.Assume that L(SymnE, l + 1) , 0. Then the Bloch-Kato conjecture reads (up to sign) as follows:Conjecture (Bloch-Kato):

L(SymnE, l + 1)c+(SymnM(l + 1))

=(∏

p≤∞ cp)|ST|

|H0(Q,A(l + 1))|2

Here, M is the motive h1(E), for a motive X, X(k)stands for the kth Tate twist, and c+(SymnM(l +1)) is the Deligne-period (which the authorscarefully compute in this paper), ST is theShafarevich-Tate group, cp is the Tamagawa fac-tor at p and |H0(Q,A(l + 1))|2 is the torsion termmentioned above.A result of M. Flach [J. Reine Angew. Math.412, 113–127 (1990; Zbl 711.14001)] implies inthis case that the order of the Shafarevich-Tategroup (if finite) is a perfect square or a twice aperfect square. The bounds obtained by the au-thors for the Tamagawa factors and the torsionterms can be used to find (large enough) primeswhich should only contribute to the order of theShafarevich-Tate group. In their computationalexamples, they use this fact to indicate how theircalculations go hand-in-hand with Flach’s afore-mentioned theorem. Kâzım Büyükboduk (Bonn)

1167.11021Jiang, Dihua; Nien, Chufeng; Qin, YujunLocal Shalika models and functoriality.Manuscr. Math. 127, No. 2, 187-217 (2008).This paper relates local Langlands functorial lifts


and Shalika models. Let F be a p-adic local field.The main result is the following: an irreduciblesupercuspidal representation of GL2n(F) is a lo-cal Langlands functorial transfer from SO2n+1(F)if and only if it has a nonzero Shalika model.Shalika models are defined for representationsof GL2n(F). In order to prove the main result, theauthors consider generalized Shalika models de-fined for representations of SO4n(F). They showthat an irreducible admissible representation ofSO4n(F) cannot have both nonzero generalizedShalika model and a nonzero Whittaker model.This is used for proving that if an irreducible uni-tary supercuspidal representation τ of GL2n(F)has a nonzero Shalika model, then the unitarilyinduced representation I(s, τ) of SO4n(F) reducesat s = 1. This is then related to various knowncharacterizations of the local Langlands transferproperty. Altogether, they are stated as follows:Theorem. Let τ be an irreducible supercuspidalrepresentation of GL2n(F). Then the following areequivalent.(1) τ has a nonzero Shalika model.(2) The local exterior square L-factor L(s, τ,∧2)has a pole at s = 0.(3) The local exterior square γ-factor γ(s, τ,∧2, ψ)has a pole at s = 1.(4) The unitarily induced representation I(1, τ) ofSO4n(F) is reducible.(5) τ is a local Langlands functorial transfer fromSO2n+1(F).If one of the above holds for τ, then τ is self-dual.As an application of their results, the authorsprove three conjectures in the theory of automor-phic forms. Dubravka Ban (Carbondale)

1167.11021Rebolledo, MarusiaSupersingular module, Gross-Kudla formulaand rational points of modular curves. (Mod-ule supersingulier, formule de Gross-Kudlaet points rationnels de courbes modulaires.)(French)Pac. J. Math. 234, No. 1, 167-184 (2008).In this well-written article, the author proves thatfor all prime number p outside a set C of analyticdensity 9/210, the quotient of the modular curveX0(pr), r > 1, by the Atkin-Lehner involution hasno rational points except cusps and CM points(theorem 0.6). The set C is determined by explicitquadratic congruences.The method consists in applying a criterion dueto P. J. R. Parent [Compos. Math. 141, No. 3, 561–

572 (2005)] to well-chosen degree zero elementsin the supersingular module annihilated by thewinding ideal.More precisely, for p > 3 any prime number letP denote the free abelian group generated bythe isomorphism classes of supersingular ellip-tic curves over Fp, let P0 denote the subgroup ofdegree zero elements in P, let Ie denote the anni-hilator in the Hecke algebra of the set of primitivecusps forms f of level Γ0(p) such that L( f , 1) , 0,and letP0[Ie] denote the subgroup ofP0 on whichIe acts trivially. Thanks to the famous formula in-volving triple products of L-functions due to B.H. Gross and S. S. Kudla [Compos. Math. 81, No.2, 143–209 (1992; Zbl 807.11027)], the author con-structs an explicit family (y0

m), m > 0, of elementsin P0[Ie] ⊗ Q. She then applies Parent’s criterionto this family. She proves in fact that the sequence(y0

m), m > 0, generatesP0[Ie]⊗Q as a module overthe Hecke algebra.Previous results were due to Parent [loc. cit.] re-placing the set C by one of analytic density 7/29.Parent used another family of P0[Ie] ⊗ Q whichalso generates this vector space. The author pro-vides the link between these two families: up tosome Eisenstein element, y0

m is a linear combina-tion of Parent’s elements (proposition 0.2).

Benoît Stroh (Paris)

1167.11024Stoll, MichaelFinite descent obstructions and rational pointson curves.Algebra Number Theory 1, No. 4, 349-391 (2007).Let k be a number field, and X be a smooth projec-tive k-variety. In this paper, the author considersa slightly modified set of adelic points

X(Ak)• :=∏v-∞

X(kv) ×∏v|∞

π0

(X(kv)

)where the factors at infinite places v in the usualset of adelic points are reduced to the set of con-nected components of X(kv), and approaches theproblem of determining the existence of a ratio-nal point on X using descent methods withinX(Ak)• via torsors under finite étale group schemes,which can be viewed as generalizations of then-Selmer group of an abelian variety. Before wediscuss the main results, I would like to pointout that the paper contains a nice overview ofthe descent via torsors under finite étale groupschemes, which is an introduction suitable for


those who are not familar with this extent of ageneralization of descent methods. One of themain results of this paper in the context of theBrauer-Manin obstruction is an improvementon S. Siksek [“The Brauer-Manin obstruction forcurves having split Jacobians”, J. Théor. NombresBordx. 16, No. 3, 773–777 (2004; Zbl 1076.14033)],which is a positive answer toward the author’sconjecture that the Brauer-Manin obstruction isthe only obstruction against rational points oncurves – Skorobogatov first formulated it as aquestion. Using the descent via torsors underall finite abelian étale group schemes, the au-thor establishes that for all curves C the subsetC(Ak)f-ab

• consisting of points in C(Ak)• whichsurvive the descent is equal to the Brauer setC(A‖)Br

• , and proves the conjecture in the descentcontext, provided that there is a nonconstant k-morphism from C into an abelian variety A suchthat the divisible part of the Shafarevich-Tategroup X(k,A) of A is trivial and A(k) is finite.The author believes that this condition is satis-fied for all curves of genus ≥ 2. Let us review thedescent via torsors under finite group schemes.An X-torsor under a finite étale group schemeG is a smooth projective variety Y with the fol-lowing commutative diagram such that Y × G isidentified with the fiber product Y ×X Y:

Y × Gµ

−−−−−→ Y

pr1

y yπY π

−−−−−→ X

where µ is a right action of G on Y, pr1 is theprojection, and π is a finite étale morphism.Note that to a point P in X(k), we can asso-ciate an element in the Galois cohomology setH1(k,G) as the group scheme G acts on the fiberπInv(P). In other words, the X-torsor induces amap φY : X(k) → H1(k,G), and hence, the com-mutative diagram:

X(k)φY

−−−−−→ H1(k,G)y yres

X(Ak)•δ

−−−−−→∏

v H1(kv,G)

whose setting is quite analogous to that of then-Selmer group of an abelian variety. The setCov(X) is defined to be the subset consisting of el-ements Q in X(Ak)• such that δ(Q) ∈ Img(res) forall X-torsors under all finite étale group schemes,

and the subsets X(Ak)f-sol• and X(Ak)f-ab

• are sim-ilarly defined with the solvable/abelian groupschemes. Denoting by X(k) the topological clo-sure of X(k) in X(Ak)•, we have

X(k) ⊂ X(k) ⊂ X(Ak)f-cov• ⊂

X(Ak)f-sol• ⊂ X(Ak)f-ab

• ⊂ X(Ak)•.

A significant part of this paper is for the develop-ment of a theory toward the occasions of equali-ties between these sets and toward the relation-ships with various Brauer sets, and as mentionedearlier, for the case of all curves, it is particularlyfruitful.For X being an abelian variety A, the authortakes a more natural generalization of n-descent;namely,

0→ A(k)→ Sel→ T Sh(k,A)→ 0

where A(k) = A(k) ⊗ Z, Sel = lim Sel(n), andT X(k,A) is the Tate module of X(k,A). In thispaper two main results for abelian varieties inthis paper are introduced:(1) For a set of finite places of k of good reduc-tion for A and of density 1, we have canonicalinjective homomorphisms

A(k)→ Sel(k,A)→∏v∈S

A(Fv)

where the last map factors through∏

v∈S A(kv),and A(k) = A(k) as images in

∏v∈S A(kv).

This is an improvement on J.-P. Serre [Theorem3, Sur les groupes de congruence des variétésabeéliennes. II, Izv. Akad. Nauk SSSR, Ser. Mat.35, 731–737 (1971; Zbl 222.14025)];(2) If Z ⊂ A is a finite subscheme of A over k, thenfor a set S of places of k of density 1 the intersec-tion Z(Ak)• ∩ Sel(k,A) in

∏v∈S A(kv)/A(kv)0 is the

image of Z(k).The second result means that for a finite sub-scheme Z of A, the intersection Z(Ak)• ∩ Sel(k,A)is the only obstruction against rational points onZ.The author also formulates an “Adelic Mordell-Lang Conjecture”, and explains its implicationon some subvarieties of A:Adelic Mordell-Lang Conjecture: Let X ⊂ A be asubvariety not containing a translate of a non-trivial subgroup of A. Then, there is a finite sub-


scheme Z ⊂ X such that X(Ak)• ∩ Sel(k,A) ⊂Z(adel)•.This conjecture together with the second resultfor A implies that X(k) = Z(k), and the chain ofadelic subsets shown above collapses to X(k) =X(Ak)f-ab

• . The author also remarks that the aboveconjecture is true when k is a global function field,A is ordinary, and X is not defined over kp wherep is the characteristic of k.


1167.11035Matomäki, KaisaOn the exceptional set in Goldbach’s problemin short intervals.Monatsh. Math. 155, No. 2, 167-189 (2008).Let E(X,H) denote the number of even integersn with X < n ≤ X + H such that n cannot bewritten as a sum of two primes, and set H = Xθ.It is known that E(X,H) = o(H) as X → ∞, if θis larger than a certain small number. In fact, G.Harman showed that if θ ≥ 11/180, then for anyfixed A, one has E(X,H) H(log X)−A (see Chap-ter 10 of [Prime-detecting sieves. London Math-ematical Society Monographs 33. Princeton, NJ:Princeton University Press (2007)]).Meanwhile, the methods of H. L. Montgomeryand R. C. Vaughan enabled us to establish asharper bound of the form E(X,H) H1−δ withsome constant δ > 0, at least when H is as largeas X. Several mathematicians have worked toshow bounds of the latter type for smaller H,and for example T. P. Peneva [Monatsh. Math.132, No. 1, 49–65 (2001; Zbl 974.11037), Corri-gendum 141, No. 3, 209–217 (2004)] and A. Lan-guasco [Monatsh. Math. 141, No. 2, 147–169 (2004;Zbl 1059.11059)] showed such bounds, respec-tively, for θ > 1/3 and θ > 7/24. And in thepaper under review, the author proves that forθ > 1/5, one has E(X,H) H1−δ with some effec-tively computable positive number δ dependingon θ, substantially improving previous results inthis direction. (The author points out, and over-comes, a flaw in the argument of Languasco (loc.cit.).) It may be said in brief that the proof is con-structed by incorporating ideas of sieve methodsinto the previous work being based on the circlemethod. Koichi Kawada (Morioka)

1167.11037Zerbes, Sarah LiviaGeneralised Euler characteristics of Selmergroups.

Proc. Lond. Math. Soc. (3) 98, No. 3, 775-796(2009).In [J. Lond. Math. Soc., II. Ser. 70, No. 3, 586–608(2004; Zbl 1065.11039)], the author computed theΣ-Euler characteristic of the compact dual of theSelmer group of an elliptic curve E over a num-ber field F. The aim of the paper under review isto generalize these results to an elliptic curve ofgeneral rank.Let Fcyc be the cyclotomicZp-extension of F, andlet Γ = Gal(Fcyc/F). Let F∞/F be an admissiblep-adic Lie extension. That is, Σ = Gal(F∞/F) isa p-adic Lie group, F∞ is unramified outside afinite set of primes of F, F∞ ⊇ Fcyc and Σ hasno elements of order p. In this case Σ has finitep-cohomological dimension and it is equal to itsdimension as p-adic Lie group.The essential change of hypothesis from her pre-vious paper [op. cit.] is that now it is no longerassumed that the group E(F) of F-rational pointson E is finite. In this situation it is no longertrue that the Selmer group Sel(E/F∞) has finiteΣ-Euler characteristic. Instead it is considered ageneralized Σ-Euler characteristic, whose defini-tion depends essentially on the hypothesis thatFcyc⊆ F∞.

The main result, under some conditions of finite-ness and for p ≥ 5, is that Sel(E/F∞) has finitegeneralized Σ-Euler characteristic if and only ifSel(E/Fcyc) has finite generalized Γ-Euler charac-teristic and

χ(Σ, Sel(E/F∞)) =

χ(Γ, Sel(E/Fcyc)) ×

∣∣∣∣∣∣∣∏v∈M Lv(E, 1)

∣∣∣∣∣∣∣p

whereM is the set of primes of F not dividing pwhose inertia group in Σ is infinite and Lv(E, s)denotes the Euler factor of E at v.The organization of the paper is as follows. InSection 2 preliminaries such as Selmer groupsand generalized Euler characteristic are given.The local and global results are given in Sections3 and 4, respectively. In the last section it is shownthat the finiteness conditions are satisfied for alarge class of p-adic Lie extensions of F.

Gabriel D. Villa-Salvador (México D.F.)

1167.12002Fornasiero, AntongiulioEmbedding Henselian fields into power series.J. Algebra 304, No. 1, 112-156 (2006).


Given any real number x ∈ R, there exists aunique integer n ∈ Z such that n ≤ x < n + 1,that is, n is the integral part of x. This propertymotivates the definition of integer part of an or-dered fieldK as a subring R ⊆ K containing 1 andsuch that for every x ∈ K there exists a uniquer ∈ R such that r ≤ x < r+1. A field of generalizedpower series k((t)) over an ordered field k ⊆ Rhas an integer part:

∑γ<0 aγtγ + n | n ∈ Z.

The main goal of this paper is to embed K intofields of generalized power series. When thecharacteristic of k is 0, the field k is contained inK and there exist a good section s : Γ → K∗ andits factor set f. If k((Γ, f)) is the field of generalizedpower series with factor set f and s′ : Γ→ k((Γ, f))is its canonical section, the author shows thatK has a truncation-closed analytic embeddingover k, φ in k(Γ, f) such that s′ φ = s (Theorem5.1). As a corollary it is obtained a generaliza-tion of Ax-Kochen-Ershov theorem [Yu. L. Ershov,Sov. Math., Dokl. 6, 1390–1393 (1965); translationfrom Dokl. Akad. Nauk SSSR 165, 21–23 (1965;Zbl 152.02403)]. It is also obtained an extensionof Mourgues-Ressayre theorem [M. H. Mourguesand J. P. Ressayre, J. Symb. Log. 58, No. 2, 641–647 (1993; Zbl 786.12005)]: every ordered field,which is Henselian in its natural valuation hasan integer part (Theorem 2.16 and Corollary 5.3).When K is of positive characteristic, the authoruses the Hypothesis A of I. Kaplansky [DukeMath. J. 9, 303–321 (1942)] to prove that ifK is analgebraically maximal valued field containing kso that there is a p-good section ofΓ intoK∗,K sat-isfies Hypothesis A and ℵ is an uncountable car-dinal such that tr deg(K/k(Γ, f)) ≤ ℵ, then thereis a truncation-closed embedding into k((Γ, f))ℵ.For the mixed characteristic case, under some ex-tra assumptions, the author shows thatK can beembedded into K((Λ,m)) in a truncation-closedway, where Λ is certain quotient of Γ and K is afield associated to K. In the last section, amongother things, it is given a simplified version ofthe examples of S. Boughattas [J. Symb. Log. 58,No. 1, 326–333 (1993; Zbl 779.12003)] of valuedfields that do not admit integer parts and hencedo not admit truncation-closed embeddings.

Gabriel D. Villa-Salvador (México D.F.)

1166.12004Nguyen An KhuongOn d-solvability for linear differential equa-tions.J. Symb. Comput. 44, No. 5, 421-434 (2009).

Let F be an ordinary differential field of char-acteristic zero, L(y) = 0 be a linear differentialequation of degree n and d be a natural numberless then n. In the paper the following question,originally posed by Fichs in 1883 is discussed.Is it possible to express all solutions of the equa-tion L(y) = 0 in terms of solutions of homoge-neous linear differential equations of order lessthan or equal to d?The author specifies the statement of the problemwithin the frames of differential Galois theory.For this purpose he introduces the concept of d-solvability for Picard–Vessiot extensions and lin-ear algebraic groups. Then he proves the equiva-lence of these concepts within differential Galoistheory.The main result of the paper is the theorem con-necting the d-solvability of a connected semisim-ple algebraic group with the minimal dimensionsof representations of simple subalgebras of its Liealgebra.The results of the paper can be useful, for exam-ple, in number theory in the proof of algebraicindependence of values of E-functions [see thereviewer, Mat. Zametki 34, No. 4, 481–484 (1983;Zbl 548.12014)].

Nikolay Vasilye Grigorenko (Kyïv)

1166.13007Brenner, HolgerLooking out for stable syzygy bundles.Adv. Math. 219, No. 2, 401-427 (2008).The motivation for this paper comes from the the-ory of tight closure, in particular from an earlierpaper of the main author which gives inclusionand exclusion criteria for tight closure in termsof the maximal and minimal slope of twistedsyzygy bundles [H. Brenner, Trans. Am. Math.Soc. 356, No. 1, 371–392 (2004; Zbl 1041.13002)].Under the condition that the syzygy bundle is(semi)stable, these criteria yield a simple numer-ical characterization of tight closure. That is whywe are now “looking out for stable syzygy bun-dles".After a comprehensive introduction, Section 1presents some restriction theorems due toMehta and Ramanathan, Flenner, Bogomolovand Langer saying that the restriction of a(semi)stable bundle on a smooth projective va-riety X to a complete intersection curve of suf-ficiently high degree is again semistable. In par-ticular, these can be applied for restrictions fromPN to a complete intersection curve.


If the restriction of a coherent torsion-free sheafSto a generic projective lineP1

⊆ PN is semistable,i.e. of the form O(a) ⊕ . . . ⊕ O(a), then S issemistable. In Section 4, this fact is used to showthat the syzygy bundle of d + 1 generic forms ofdegree d is semistable.Section 5 gives a semistability criterion for a co-herent torsion-free sheaf S on PN in terms of theexterior powers

∧rS.

In Section 6, which is the most technical part ofthe paper, Klyachko’s theory of toric bundles isused to obtain a combinatorial-numerical crite-rion for a monomial ideal to have a stable syzygybundle on PN.Section 7 presents some interesting corollariesand examples of these results. In particular,Corollary 7.1 says that the syzygy bundle of allmonomials of fixed degree in a polynomial ringis semistable. The short proof works in any char-acteristic.Section 8 presents some partial results about thesemistability of the syzygy bundle of genericforms of given degrees. Using results by Bohn-horst, Spindler and Hein, combinatorial criteriafor semistability and numerical tight closure re-sults similar to those for monomial ideals areshown.In the Appendix by G. Hein, three interestingtheorems about the (semi)stability of the syzygybundles of generic forms of constant degrees areshown. Each theorem gives instances of mor-phisms

⊕nOPN → OPN (d) whose kernel is a

semistable coherent sheaf. The semistability isshown via restriction to some curve C ⊆ PN. Thefirst theorem, Theorem A.1, implies the semista-bility criterion Theorem 8.6 and the tight closurecriterion Corollary 8.7 in the main paper.

Helena Fischbacher-Weitz (Osnabrück)

1166.13010Lopatin, A.A.Invariants of quivers under the action of classi-cal groups.J. Algebra 321, No. 4, 1079-1106 (2009).M. Artin conjectured in a paper [J. Algebra 11,532–563 (1969; Zbl 222.16007)] that the algebraof invariants of several m × m matrices underthe action of GLm by simultaneous conjugationis generated by the traces of the monomials in thematrices. This was proved for a field of character-istic 0 by Sibirskii and Procesi independently. Theappropriate version of the above conjecture in ar-bitrary characteristic is that the algebra of invari-

ants is generated by the coefficients of the char-acteristic polynomial (i.e. by the traces of exteriorpowers) of the monomials in the matrices. Thiswas first proved by S. Donkin [Comment. Math.Helv. 69, No. 1, 137–141 (1994; Zbl 816.16015)].Le Bruyn and Procesi, and Schofield proved gen-eralizations of the above result to the algebra ofinvariants of a quiver in characteristic 0. Herethe space of tuples of matrices is replaced bythe representation space of a quiver. The “clas-sical case" of invariants of several matrices canbe obtained by taking the one point quiver withn loops, where n is the number of matrices. M.Domokos and A. N. Zubkov [Transform. Groups 6,No. 1, 9–24 (2001; Zbl 984.16023)] extended theseresults to arbitrary characteristic. Their methodof proof was new and quite elementary.In the paper under review the author general-izes the notion of a quiver, to include other ba-sic spaces whose invariants have been studied:spaces of forms and spaces of (anti)symmetricmatrices. Furthermore, the action of products ofclassical groups is considered rather than just aproduct of general (or special) linear groups. Themain result of the paper is Thm. 1 in Section 4. Itgives generators for the algebra of invariants ofa “mixed quiver" in arbitrary characteristic (thecharacteristic has to be, 2 if an orthogonal groupis involved). To describe the basic invariants thenotion of a tableau with substitutions is needed, seeSection 3.2. Sections 5-9 are dedicated to the proofof Thm. 1. Partial generalizations along the linesindicated above were already in the literature.Section 2.2 contains an overview of the knownresults. Rudolf Tange (York)

1166.13025Herzog, JürgenCombinatorics and commutative algebra.Colomé-Nin, Gemma (ed.) et al., Three lec-tures on commutative algebra. Based on coursesof the winter school on commutative alge-bra and applications, Barcelona, Spain, January30–February 3, 2006. Providence, RI: AmericanMathematical Society (AMS) (ISBN 978-0-8218-4434-2/pbk). University Lecture Series 42, 73-121 (2008).This is a survey on the works on combinatorialcommutative algebra by the author and his col-laborators over the last 10 years. The author con-siders the monomial ideals I ⊂ S = k[x1, . . . , xn]of a polynomial ring S, which are interestingboth from commutative algebra and combina-


torics. In particular, when I is squarefree, it canbe regarded as the Stanley-Reisner ideal I∆ cor-responding to a simplicial complex ∆.The first chapter deals with algebraic shifting the-ory. A simplicial complex ∆ is said to be shiftedwhen I∆ is strongly stable. A translation σ of sim-plicial complexes is called a shifting operationwhen σ(∆) is shifted. Three shifting operationsare considered: combinatorial shifting, symmet-ric algebraic shifting, exterior algebraic shifting.The last two are defined with generic initial ideal(Gin) of I∆ or its analogue in the exterior algebra.The main interest here is comparison of gradedBetti numbers βi j(I∆) and βi j(Iσ(∆)). A few conjec-tures and the partial answers given by the authorare presented.In the second chapter, the set of monomials cor-responding to (the base B of ) a polymatroidsP is considered. The author considers the basering K[B] = K[tu : u ∈ B] ⊂ K[t1, . . . , tn]. Themain interest here is White’s conjecture con-cerning the defining ideal of K[B]. The authorpresents a partial answer to this conjecture to-gether with some results on a comparison of K[B]and K[P] = K[tu : u ∈ P] ⊂ K[t1, . . . , tn], whichturns out be equal to Ehrhart ring K[P] definedby the cone of P. On the other hand, one can con-sider the polymatroidal ideal 〈xu : u ∈ B〉 ⊂ S. Al-though this ideal rarely has the Cohen-Macaulayproperty, it has a special property of having lin-ear quotients and is closed under ideal product.Besides the Stanley-Reisner ideal I∆, one can con-sider a squarefree monomial ideal I(∆) called thefacet ideal corresponding to a given simplicialcomplex ∆. In the third chapter, after preparingcombinatorial theory of facet ideals such as therelation with Alexander duality, notions of vertexcover and quasi-tree, the author proves anotherversion of Dirac’s theorem on chordal graph anda characterization of powers of monomial idealswith linear resolutions. Alexander duality andHilbert-Burch theorem are used to prove Dirac’stheorem, while the theory of x-regularity and theproperty that polymatroidal ideals are closed un-der product are effectively used in handling pow-ers of monomial ideals.The edge ideal I(G) ⊂ S is the one dimensionalversion of facet ideal and the general questionto ask when S/I(G) is Cohen-Macaulay is al-most hopeless to answer, because it is equiva-lent to classify all the Cohen-Macaulay simpli-cial complexes. However, it is possible to findgood classes of graphs, in which one can classify

Cohen-Macaulay graphs. In the fourth chapter,the author gives the classification of the Cohen-Macaulay bipartite graphs, which has close re-lation with meet-semilattices. This leads to thestudy of the squarefree monomial ideal HL asso-ciated with a finite meet-semilattice L and thissurvey is closed with the proofs of several com-binatorial and commutative algebraic propertiesof this ideal. Yukihide Takayama (Shiga)

1167.13006Fasel, J.; Srinivas, V.Chow-Witt groups and Grothendieck-Wittgroups of regular schemes.Adv. Math. 221, No. 1, 302-329 (2009).Let A be a noetherian commutative ring of Krulldimension d and let P be a projective A-moduleof rank r. The well-known splitting theoremof J.-P. Serre [Algebre Theorie Nombres, Sem.P. Dubreil, M.-L. Dubreil-Jacotin et C. Pisot 11(1957/58), No. 23, 18 p. (1958; Zbl 132.41202)]asserts P splits off a free factor of rank one ifr > d. The authors try to give obstructions forsplitting off from P a free factor of rank one. Inorder to do this they use higher Grothendieck-Witt groups. The construction of Grothendieck-Witt groups was introduced by M. Schlicht-ing [“Hermitian K-theory, derived equivalencesand Karoubi’s fundamental theorem", preprint,http://www.math.lsu.edu/˜mschlich/research/prelim.html] and then Balmerand Walter defined derived Grothendieck-Witt groups [C. Walter, “Grothendieck-Wittgroups of triangulated categories", preprint,http://www.math.uiuc.edu/K-theory/0643/].The authors define Euler classes in the derivedGrothendieck-Witt groups and show the follow-ing:Theorem. Let A be a noetherian ring of dimen-sion d with 1/2 ∈ A. Let P be a projective mod-ule of rank d. If d = 2 or d = 3, then e(P) = 0 inGWd(A,det(P)∨) if and only if P ' Q⊕A for someprojective module Q. Piotr Krason (Szczecin)

1167.13007Khatami, Leila; Tousi, Massoud;Yassemi, SiamakFiniteness of Gorenstein injective dimensionof modules.Proc. Am. Math. Soc. 137, No. 7, 2201-2207(2009).This paper contains generalizations (theorems1,2,3 below) of the so-called Bass formula (the


Bass formula is: Every finitely generated mod-ule M over a noetherian ring satisfies dimR M ≤idRM and, if R is local, one has idRM =depthR [see H. Bass, Math. Z. 82, 8–28 (1963;Zbl 112.26604)]; in addition, the paper also con-tains an application (theorem 4 below) of thesegeneralizations.Concretely, there are the following results:Theorem 1. If M is a module of finite positiveGorenstein injective dimension over a noetherianring R then

GidRM =

supdepthRp −widthRp Mp|p ∈ SpecR

(Gid is Gorenstein injective dimension and thewidth of a module M over a local ring (R,m) iswidthRM := infi|TorR

i (R/m,M) , 0).Theorem 1 generalizes a result from L. G.Chouinard [Proc. Am. Math. Soc. 60, 57–60 (1976;Zbl 343.13005)] (there one has injective dimen-sion instead of Gorenstein injective dimension;note that Choinard’s result is a generalization ofBass’ formula in the sense that it ’computes’ the(finite) injective dimension in a more general sit-uation)).Theorem 2. Let M be a non-zero finite moduleof finite positive Gorenstein injective dimensionover a local ring R. Then

GidR(M) = depthR.

Theorem 3. Let a be an ideal of a local ring R,let M be a non-zero a-cofinite R-modulo of finiteGorenstein injective dimension. Then

dimR M ≤ GidRM.

There is also an application of theorem 1:Theorem 4. Let a be an ideal of a local ring Rand assume that there exists a non-zero a-cofiniteR-module M with finite Gorenstein injective di-mension and such that dimR M = dim R. Then Ris Cohen-Macaulay. Michael Hellus (Leipzig)

1166.14009Tsuda, Teruhisa; Takenawa, TomoyukiTropical representation of Weyl groups associ-ated with certain rational varieties.Adv. Math. 221, No. 3, 936-954 (2009).The theory of birational representations of Weylgroups associated with algebraic varieties hasits origins in work of Coble and Kantor and

of Du Val. They showed that certain types ofCremona transformations act on the configura-tion space Xm,n of n points in general positionin the projective space Pm−1. From this one de-duces a birational action on Xm,n of the Weylgroup W(T2,m,n−m) associated to the Dynkin di-agram T2,m,n−m [see V. G. Kac, Infinite dimen-sional Lie algebras. 3rd ed. Cambridge Univer-sity Press, Cambridge (1990; Zbl 716.17022), §4.7-4.10]. Dolgachev and Ortland showed that theCremona action of W(T2,m,n−m) induces pseudo-isomorphisms between certain rational varieties(generalized Del Pezzo varieties) blown up fromPm−1 at n points in general position.In the paper under review the authors generalizethe above results to Weyl groups correspondingto certain Dynkin diagrams Tk

l , where k and l areN-tuples of strictly positive integers.In Section 2 they construct the roots and co-roots of Tk

l in the Néron-Severi bilattice N(X) =

(H2(X,Z),H2(X,Z)) of a blowup ε : X → (P1)N

along certain subvarieties that are not only pointsets. The construction of X involves certain pa-rameters and one obtains a family of varietiesXa where a runs through tuples of so-calledmultiplicative root variables. The Weyl groupW = W(T2,m,n−m) acts on these tuples. It is shown(Thm 2.2) that the action of W on N(X) lifts to abirational action w : Xa → Xw(a) on the family Xa.To understand the action of W on defining poly-nomials of exceptional divisors the authors in-troduce in Section 3 a geometric framework ofτ-functions.If Tk

l is of affine type, then the action of the latticepart of W on N(X) provides a discrete dynamicalsystem of Painlevé type. In Section 4 the A(1)

n andD(1)

n cases are demonstrated.In Section 5 the authors indicate some relation-ships between the τ-functions and the universalcharacters S[λ,µ] which describe the irreduciblecharacter of the rational representation of GLn(C)associated to the pair of partitions [λ, µ].

Rudolf Tange (York)

1166.14010McQuillan, MichaelCanonical models of foliations.Pure Appl. Math. Q. 4, No. 3, 877-1012 (2008).The paper forms part of a special issue of thejournal in honor of F. Bogomolov. The papers [F.A. Bogomolov, Dokl. Akad. Nauk SSSR 236, 1041–1044 (1977; Zbl 415.14013)] and [M. McQuillan,Publ. Math., Inst. Hautes Étud. Sci. 87, 121–174


(1998; Zbl 1006.32020)] can be seen as a precedentof the paper.In the introduction, the author explains the lineof the paper and why he follows it instead of thatgiven in some successive preprints by him with aclose purpose. As he says, once one accepts thatits above cited paper is really the key point, thenone leads to a truly satisfactory theory of generaltype objects which is properly speaking the gen-eralization of the uniformization theorem to thisgenerality and which goes beyond the limiteddetails that algebra alone provides.The material of the paper is dense in many pointsand exhaustive in its details which constitute themost precise description of the canonical modelthat the theory allows. The basic objects of studyare pairs (X,F ), where X is a normal projec-tive variety over a field k and F an integrablefoliation. The paper contains five sections. Thefirst one entitled singularities is an exposition ofpurely algebraic properties, with a second sub-section devoted to dimension 2 and a third sub-section that discusses residues around singular-ities in an algebraic way appropriate to the min-imal model theorem. The second section treatsthe positive characteristic case and its main goalis the foliated bend and break lemma of Mikaoyawhich on surfaces yields the cone theorem. Analternative approach by Bogomolov and the au-thor is also summarized in this section. The thirdsection is devoted to minimal models althoughthe object is not exactly a minimal model theorembut to get a model with KF as positive as possible(see Definition III.3.1 of the paper). Theorem 1 ofthe paper states the canonical model theorem.The central part of this article is section IV enti-tled, classification, where Theorems 2 and 3 arestated and proved. The classification takes intoaccount the existence of minimal models andattends the numerical Kodaira dimension andthe Kodaira dimension of the model. Thus whenν(F ) = −∞, this implies that the foliation is aconic pencil, although the converse is only truewith canonical singularities. The main propertyin case ν(F ) = 2 is the lack of an algebraic de-scription since the canonical ring need not befinitely generated. A complete list of possibilitiesfor the case ν(F ) = 0 is given in IV.3.6, that isTheorem 2 of the paper. The case ν(F ) = 1 mustbe divided in two subcases: κ(F ) ≥ 0, here abun-dance holds and there is a Kodaira fibration; andκ(F ) = −∞, where the canonical models tran-spire to be the Baily-Borel compactification of

Hilbert-Modular surfaces in either of their nat-ural foliations (Theorem 3 of the paper). Finally,more or less, by definition, a choice of metriza-tion of TF , allows us to consider certain functo-rial object σ : S → [S/F ] as a varying family ofconformal structures. The papers ends with a lastsection devoted to the study of these conformalstructures. Carlos Galindo (Castellon)

1166.14018Chiodo, AlessandroTowards an enumerative geometry of the mod-uli space of twisted curves and rth roots.Compos. Math. 144, No. 6, 1461-1496 (2008).An r-spin (algebraic) curve C is a curve of genusg together with a choice of an r-th root of itscanonical bundle KC. If C has marked pointsx1, . . . , xn, then one considers the r-th root ofKC(m1x1 + · · · + mnxn), where m1, . . . ,mn are suit-able weights. The interest for the moduli spaceof such spin curves Mr

g,n (and their compactifi-

cation à la Deligne-MumfordMrg,n) arose with E.

Witten’s paper [Algebraic geometry associatedwith matrix models of two dimensional grav-ity, in Topological methods in modern mathe-matics. Proceedings of a symposium in honor ofJohn Milnor’s sixtieth birthday, SUNY at StonyBrook, 1991, Publish or Perish Inc. 235–269 (1993,Zbl 812.14017)], in which he conjectured thata suitable potential made up with intersectionnumbers of tautological classes on the modulispace of r-spin curves must satisfy the r-KdV hi-erarchy.In the classical case r = 1, D. Mumford [in: Arith-metic and geometry, Pap. dedic. I. R. Shafare-vich, Vol. II: Geometry, Prog. Math. 36, 271–328(1983; Zbl 554.14008)] applied the Grothendieck-Riemann-Roch formula to the universal familyof algebraic curves to relate characteristic classesof the Hodge bundle to the κ and the ψ classes.In this paper, the author pursues the same projectin the case r ≥ 2 and the result is a nice compactformula. Clearly, most of the work amounts todeal with the boundary ofM

rg,n. At the end, the

author gives some examples and some applica-tions to Gromov-Witten invariants.

Gabriele Mondello (Roma)

1166.14019Hacking, PaulThe moduli space of curves is rigid.Algebra Number Theory 2, No. 7, 809-818 (2008).The subject of the paper under review is the study


of the deformations of the moduli stackMg,n ofgenus g stable curves with n marked points. Themain result is thatMg,n is rigid: that is, it has nonontrivial infinitesimal deformations. Two otherrelated results are shown: namely, that the pair(Mg,n, ∂Mg,n), where ∂Mg,n is the locus of sin-gular stable curves, has no locally trivial defor-mations, and that the same holds for the coarsemoduli space Mg,n, with the possible exceptionof finitely many cases in which g + n ≤ 3.On the contrary, it is expected that moduli spacesof surfaces have non-trivial infinitesimal defor-mations in general. A simple example of this phe-nomenon is given in the last section of the paper.The motivation for this investigation of defor-mation spaces comes from the following princi-ple, proposed by M. Kapranov in an unpublishedmanuscript: Given a smooth n-dimensional va-riety X = X(0), we can associate to it a sequenceof varieties X(i) (with i ≥ 0) in which X(i + 1)is the moduli space of varieties obtained as de-formations of X(i). Then this process should stopafter n steps, i.e. X(n) should be rigid and X(n+1)should be a point. Orsola Tommasi (Hannover)

1166.14029Brion, MichelAnti-affine algebraic groups.J. Algebra 321, No. 3, 934-952 (2009).A group scheme G of finite type over a field k issaid to be anti-affine ifO (G) = k, i.e. every regularfunction on G is constant. Any anti-affine alge-braic group is smooth, connected, and commu-tative. The most well-known examples of anti-affine groups are abelian varieties, however thiscollection of group schemes includes universalvector extensions and some semi-abelian vari-eties. For any group G of finite type one hastwo subgroups Gant and Gaff, where Gant is thesmallest normal subgroup of G such than G/Gantis affine (and, as the notation suggests, Gant isanti-affine; furthermore it is central in G) andGaff is the smallest connected affine subgroupsuch that G/Gaff is an abelian variety (hence anti-affine). One then gets the Rosenlicht decompo-sition: G = GaffGant, and (Gant)aff is a subgroupof Gaff ∩ Gant of finite index. The main objectivein this paper is to classify anti-affine algebraicgroups, and for certain choices of k one obtains avery nice classification.Before giving the classification, certain proper-ties of anti-affine algebraic groups are given. Itis shown that G (always assumed to be of fi-

nite type) is anti-affine over k if and only if itis anti-affine over any extension of k (an easyresult that follows from the definition). Further-more, G is anti-affine if and only if every finite-dimensional linear representation of G is trivial– this further justifies the term “anti-affine” asaffine groups admit a faithful linear representa-tion. It is shown how the product of two anti-affine groups are anti-affine; and if an anti-affinegroup G decomposes as the product of two con-nected group schemes they must both be anti-affine. If f : G → H is an isogeny of connectedcommutative algebraic groups, then G is anti-affine if and only if H is, thereby showing that“anti-affineness” is stable under isogenies. Fi-nally, the multiplication map x 7→ nx : G → Gis an isogeny when G is anti-affine.The classification begins by identifying all con-nected group schemes with an abelian variety. AsG/Gaff is an abelian variety, say G/Gaff A via anisomorphism α,we say that G is a group schemeover A. In the case where G is a semi-abelian va-riety it is proved that G corresponds to a sub-Γk-lattice of A∨ (ks), where ks is the separable closureof k and Γk =Gal(ks/k) ; furthermore this corre-spondence is bijective. As a consequence, if G isan anti-affine algebraic group over a field of pos-itive characteristic then G is necessarily a semi-abelian variety; furthermore if k is a finite fieldthen G is an abelian variety. Thus in characteris-tic p the anti-affine algebraic groups correspondto pairs (A, Λ) , where Λ is the lattice describedabove.The classification is more complicated in char-acteristic zero – for example one has universalvector extensions which are anti-affine. For G anA-group scheme and E (A) the universal vectorextension of A there is a map E (A) → G whichinduces a map γ : H1 (A,OA)∗ = E (A)aff → Gaff. Itis proved that G is anti-affine if and only if γ issurjective, and that anti-affine groups over A ob-tained as vector extensions are classified by sub-k-vector spaces of H1 (A,OA) . Thus in character-istic zero the anti-affine algebraic groups corre-spond to triples (A, Λ,V) , where Λ is as aboveand V is a subspace of H1 (A,OA) .Some consequences of the work above are given,most notably some counterexamples to Hilbert’sfourteenth problem. Hilbert’s fourteenth prob-lem asks if the coordinate ring of every quasi-affine variety is finitely generated. Here is itshown that X, where X → G is the Gm-torsorassociated to an ample invertible sheaf on a


non-complete anti-affine algebraic group, is notnoetherian. Alan Koch (Decatur)

1166.14032Oh, Suho; Postnikov, Alexander;Yoo, HwanchulBruhat order, smooth Schubert varieties, andhyperplane arrangements.J. Comb. Theory, Ser. A 115, No. 7, 1156-1166(2008).Let w ∈ Sn be a permutation. Consider a polyno-mial Pw(q) =

∑u≤w ql(u), where the sum is over all

permutations u ∈ Sn below w in the strong Bruhatorder. The polynomial Pw(q) is the Poincaré poly-nomial of the Schubert variety Xw = BwB/B in theflag variety SL(n,C)/B.Define the inversion hyperplane arrangementAw as the collection of the hyperplanes xi−x j = 0in Rn, for all inversions 1 ≤ i < j ≤ n, w(i) > w( j).Let Rw(q) =

∑r qd(r0,r) be the generating function

that counts regions r of the arrangement Aw ac-cording to the distance d(r0, r) from the fixed ini-tial region r0 with (1, 2, . . . ,n) ∈ r0.The main result of the paper is the claim thatPw(q) = Rw(q) if and only if the Schubert vari-ety Xw is smooth. In this case, an explicit fac-torization of the polynomial Pw(q) as a productof q-numbers [e1 + 1]q . . . [en + 1]q is obtained. Thenumbers e1, . . . , en are called exponents. They canbe computed using the left-to-right maxima (akarecords) of the permutation w. Here the inversiongraph Gw, whose edges correspond to inversionsin w, is a chordal graph. The numbers e1, . . . , enare the roots of the chromatic polynomial χGw (t)of the inversion graph. The polynomial χGw (t)is also the characteristic polynomial of the in-version hyperplane arrangement Aw. Chordalgraphs and perfect elimination ordering are themain technical tools of the paper under review.In the final section, a generalization of the con-struction to other root systems is proposed. Foran element w of the Weyl group, the arrange-ment Aw is defined as the collection of hyper-planes α(x) = 0 for all roots α such that α > 0 andw(α) < 0. The authors conjecture that coincidenceof polynomials Pw(q) and Rw(q) corresponds torational smoothness of the Schubert variety Xwin the generalized flag variety G/B.

Ivan V. Arzhantsev (Moskva)

1166.14038Kishimoto, TakashiA new proof of the non-tameness of the Nagata

automorphism from the point of view of theSarkisov program.Compos. Math. 144, No. 4, 963-977 (2008).Denote by G = Aut(Cn) the group of automor-phisms of the complex affine space of dimen-sion n. Any element of G is a polynomial map(x1, . . . , xn) → ( f1(x1, . . . , xn), . . . , fn(x1, . . . , xn))which admits a polynomial inverse. Denote byA ⊂ G its subgroup of affine elements (elementswith each fi being of degree 1), and by J ⊂ G itssubgroup of de Jonquières elements (each fi is apolynomial in x1, . . . , xi).The tame problem asks whether G is generatedby A and J. If n = 1, the three groups are equal.If n = 2, the result is the famous Jung’s theo-rem. If n = 3, Nagata gave a counterexample,of the form (x, y, z) 7→ (x − 2y(xz + y2) − z(xz +y2)2, y+ z(xz+ y2), z) and only recently Shestakovand Umirbaev proved that this automorphism isnot “tame", which means that it is not generaredby A and J. In dimension at least 4, the problemis still open.The proof of the non-tameness of the Nagata au-tomorphism is algebraic, and one could ask fora geometric proof of this. In the article under re-view, the author tries to give a necessary criterionfor an element f ∈ G = Aut(C3) to be tame.Extending f to a birational map F of the projec-tive space, F has base-points only in the plane atinfinity. One can apply to F a Sarkisov program.Then, the author’s main result consists of say-ing that if f is tame, then the maximal center ofthe first link of any Sarkisov factorisation of F is apoint or a line. To the contrary, every Sarkisov de-composition of the Nagata automorphism startswith the blow-up of a smooth conic.Reviewer’s remark: Unfortunately, there are seri-ous gaps in the proof of almost every lemma usedto prove the result, that any interested readercould easily find (for example, Lemma 3.2 ad-mits trivial counterexamples). Moreover, the cri-terion given is false, as the following simple ex-ample (already found by Shestakov and Umir-baev) shows:Let f1, f2 ∈ G be the tame automorphisms of theform f1 : (x, y, z) 7→ (x + 3/2yz + z3, y + z2, z),f2 : (x, y, z) 7→ (x, y, z + x2

− y3), and let f = f2 f1be the tame automorphism of the form(x, y, z) 7→ (x + 3/2yz + z3, y + z2, z + x2 + 3xyz −y3 + z2(2xz − 3/4y2)).Then, the base locus in the plane at infinity isgiven by z2(2xz − 3/4y2), and is the union of aline (z = 0) and a smooth conic (2xz− 3/4y2 = 0),


being tangent at (1 : 0 : 0). The multiplicity ofthe line and the conic is 1, and the multiplicityof the point of intersection is 2. Their Sarkisovweight is thus 1 for all three. One can moreovercompute that no other infinitely near valuationhas a Sarkisov weight strictly larger than 1. Inconsequence, the blow-up of the smooth conicis an admissible start of the Sarkisov algorithm(and also the blow-up of the line or the point ofintersection). Jérémy Blanc (Genève)

1167.14002Bertolin, CristianaExtensions and biextensions of locally constantgroup schemes, tori and abelian schemes.Math. Z. 261, No. 4, 845-868 (2009).Let S be a scheme. By a semiabelian scheme overS we understand a group scheme over S whichis an extension of an abelian scheme A over S byan S–torus T. The objective of the paper underreview is to study biextensions of a pair of semi-abelian schemes (G1,G2) by a third semiabelianscheme G3. The definition of biextensions (intro-duced by D. Mumford in 1968) is briefly recalledand motivated.The author’s main result is that biextensions of(G1,G2) by G3 are essentially the same as biex-tensions of (A1,A2) by T3, where the semiabelianscheme Gi is an extension of the abelian schemeAi by the torus Ti. This includes for example thatbiextensions of a pair of abelian schemes (A1,A2)by a third abelian scheme A3 are trivial.In order to obtain this result, homomorphismgroups and extension groups, as well as the asso-ciated fppf-sheafs on S involving tori and abelianschemes are computed. Also locally constantgroup schemes, locally isomorphic to a finitelygenerated free commutative group (that is, char-acter groups of tori) have to be taken into ac-count. On the way several representability re-sults are obtained.Throughout the article much attention is paid tokeep the assumptions on the base scheme S min-imal. Many of the technical difficulties stem fromthis. Peter Jossen (Budapest)

1167.14011Esnault, Hélène; Hai, Phùng HôThe fundamental groupoid scheme and appli-cations.Ann. Inst. Fourier 58, No. 7, 2381-2412 (2008).The aim of the paper is to reconcile Grothen-

dick’s construction of the arithmetic fundamen-tal group of a k-scheme X [cf. A. Grothendieck,Revetêments étales et groupe fondamental,SGA 1, Lectures Notes in Mathematics (1971;Zbl 234.14002)] and Nori’s Tannaka construc-tion of the fundamental group scheme of aproper, reduced, strongly connected k-schemeX endowed wih a k-rational point x ∈ X(k)[cf. M. V. Nori, Compos. Math. 33, 29–41 (1976;Zbl 337.14016)] by using Deligne’s Tannakaformalism [cf. P. Deligne, Catégories tannaki-ennes. The Grothendieck Festschrift, Collect.Artic. in Honor of the 60th Birthday of A.Grothendieck. Vol. II, Prog. Math. 87, 111–195(1990; Zbl 727.14010)].In order to do this they give a linear structure toGrothendieck’s construction when X is a smoothscheme of finite type over a characteristic 0 fieldk and k = H0

DR(X): first the authors extend Nori’sconstruction to not necessarily proper schemesby definining over X the category FConn(X) of fi-nite flat connections; then they define the funda-mental groupoid scheme

∏(X, x) of X with base

point a geometric point x : Spec(k) → X as thek-groupoid scheme acting on Spec(k) given byAut⊗(ρx) where

ρx : FConn(X)→ Vectx

is the fiber functor assigning to each connec-tion the fiber of the underlining bundle at x; fi-nally they compare their fundamental groupoidscheme to the Grothendieck’s arithmetic funda-mental group π1(X, x). This allows the authors tolink the existence of sections of the Galois groupGal(k/k) to π1(X, x) with the existence of a neu-tral fiber functor on the category which linearizesit. Finally they apply the construction to affinecurves and neutral fiber functors coming from atangent vector at a rational point at infinity, in or-der to follow this rational point in the universalcovering of the affine curve. Marco Antei (Bonn)

1167.14014Moriwaki, AtsushiContinuity of volumes on arithmetic varieties.J. Algebr. Geom. 18, No. 3, 407-457 (2009).The notion of volume function is important inalgebraic geometry. It is a birational invariantwhich measures the asymptotic behaviour of thesectional algebra of a line bundle on a projectivevariety. In the article under review, the authorhas studied the analogue of the volume function


in Arakelov geometry, notably the continuity ofthe arithmetic volume function. Recall that thearithmetic volume function of a C∞ Hermitianline bundle L on an arithmetic variety X of totaldimension d is defined as

vol(L) := lim supm→+∞

h0(X,mL)md/d!

,

where h0(X,mL) is defined as log #s ∈

H0(X,L⊗m) : ‖s‖sup 6 1. By the equalityvol(mL) = mdvol(L), one can extend the defini-tion domain of this function to Pic(X)⊗ZQ, wherePic(X) denotes the group of smooth Hermitianline bundles on X, whose group law is writtenadditively. Namely, for all L,A1, · · · ,An in Pic(X),there exists a positive constant C such that∣∣∣vol(L + ε1A1 + · · · + εnAn) − vol(L)

∣∣∣ 66 C

(|ε1| + · · · + |εn|

).

The proof of this result uses an estimation ofdistorsion functions due to T. Bouche [Ann. Inst.Fourier 40, No. 1, 117–130 (1990; Zbl 685.32015)and G. Tian [J. Differ. Geom. 32, No. 1, 99–130(1990; Zbl 706.53036)].As an application, the author generalizes thearithmetic Hilbert-Samuel function to the nefcase, in the following sense: assume that L and Nare in Pic(X) with L nef, then

h0(X,mL + N) =c1(L)d

d!md + o(md) (m→∞).

Consequently, a nef C∞ Hermitian line bundle Lis big (i.e. vol(L) > 0) if and only if c1(L)d > 0.Similarly, for C∞ Hermitian line bundle L withnef generic fibre and semi-positive metric, if L hasmoderate growth of positive even cohomologies,then the inequality vol(L) > c1(L)d holds. This re-sult generalizes the arithmetic Hodge index the-orem due to G. Faltings [Ann. Math. (2) 119, 387–424 (1984; Zbl 559.14005)] and P. Hriljac [Am. J.Math. 107, 23–38 (1985; Zbl 593.14004)] and thearithmetic Bogomolov-Gieseker’s inequality.

Huayi Chen (Paris)

1167.14027Mourougane, Christophe; Takayama, Shige-haruHodge metrics and the curvature of higher di-

rect images.Ann. Sci. Éc. Norm. Supér. (4) 41, No. 6, 905-924(2008).In this interesting paper, the authors study higherdirect images by a smooth morphism of the rela-tive canonical bundle twisted by a semi-positivevector bundle. They prove that they are locallyfree and semi-positively curved when endowedwith a suitable Hodge type metric.The main result is the following. Let f : X → Ybe a holomorphic map of complex manifolds,which is smooth, proper, surjective and withconnected fibers. Assume that f is Kähler. Thismeans that there exists a real (1, 1) form ωX onX such that for any point y ∈ Y, there existsa local coordinate W, (t1, ..., tm) around y withω f := ωX + c f ∗(

√−1

∑j dt j ∧ dt j) is Kähler on

f−1(W) for a certain constant c. Assume that (E, h)is a holomorphic vector bundle on X with a her-mitian metric h of semi-positive curvature in thesense of Nakano. Then, the theorem states thatfor any q ≥ 0, the direct image sheaf

Rq f∗ΩnX/Y(E)

is locally free and Nakano semi-positive, wheren is the dimension of fibers.This is a generalization of a deep result of B.Berndtsson [Ann. Math. (2) 169, No. 2, 531–560(2009)] who studied the case of E a line bundleand q = 0. It also improves the main result ofthe authors in [J. Reine Angew. Math. 606, 167–178 (2007; Zbl 1128.14030)] where Griffiths semi-positivity was considered.The sheaf f∗Ω

n−qX/Y(E) has a natural hermitian met-

ric coming from the metric ω f and h. For localholomorphic sections of this sheaf, there is actu-ally an L2-inner product:

〈s1, s2〉 =

∫f−1(y)

(s1| f−1(y)) ∧ ?(s2| f−1(y))

where ? is the star operator induced by h| f−1(y)and ω f | f−1(y). By pull-back, this gives a hermitianHodge metric on Rq f∗Ωn

X/Y(E). A major difficultyof the problem is that f∗Ω

n−qX/Y(E) is not locally free

and there is no vanishing of the (1, 0)-derivativeof s ∈ H0(X,Ωn−q

X/Y(E)) on f−1(y). But, by restrict-ing to the image of Rq f∗Ωn

X/Y(E) → f∗Ωn−qX/Y(E),

considering the Hodge metric introduced before,and using the harmonic theory developed byTakegoshi for representation of relative cohomol-


ogy, the computation turns out to be a general-ization of Berndtsson’s work. The paper is wellwritten. Julien Keller (Marseille)

1167.14033Zhu, XinwenAffine Demazure modules and T-fixed pointsubschemes in the affine Grassmannian.Adv. Math. 221, No. 2, 570-600 (2009).A geometrical proof of the Frenkel-Kac-Segal iso-morphism is given. In the following, G will bealways a simple, connected algebraic group overC with Lie algebra g. Let g = g ⊗ C[t, t−1] ⊕ CKbe the associated untwisted affine Kac-Moodyalgebra. Given a positive integer k, let V(kΛ) =

Indgg⊗C[t]⊕CKC be the level k module of g and let

L(kΛ) be its unique irreducible quotient; V(kΛ)and L(kΛ) have a structure of vertex algebra. Lett ⊂ g be a Cartan algebra and let t ⊂ g be theassociated Heisenberg Lie algebras. Let VRG be⊕

λ∈RGπλ where RG is the coroot lattice of g and

πλ is the Fock module of twith highest weight ιλ.If g is simply-laced, then VRG have a structure ofvertex algebras and the Frenkel-Kac-Segal statesthat, given a simple-laced simple algebra g, (1)VRG L(Λ) as vertex algebras. In particular, theyare isomorphic as t-modules.The author consider the following geometricalinterpretation. Let GrG = GK/GO be the affineGrassmannian of G, where GK is the group ofmaps from the punctured disc to G and GO isthe group of maps from the disc to G. If Gis simply-connected then the Picard group ofGrG is generated by an ample invertible sheafLG. Then the Borel-Weyl theorem for affine Kac-Moody algebra identifies L(kΛ) (resp. VRG ) withH0(GrG,L⊗k

G )∗ (resp. H0(GrG,OGrT ⊗ LG)∗). ThusL(Λ) VRG as t-module if and only if the restric-tion morphism ϕ : LG → OGrT ⊗ LG induces anisomorphism between the spaces of global sec-tions. If G is not simple connected, then GrG is notconnected and the author define LG as the linebundle whose restriction to each connected com-ponent is the ample generator of its Picard group.The Borel-Weyl theorem hold again (see Propo-sition 1.4.4). Recall that GrG is stratified by GO-orbits, GrλG, indexed by the dominant coweights

λ. Moreover, GrG = lim→

Grλ

G, where the Schu-

bert varieties Grλ

G are defined as the closures ofthe GrλG.The maximal torus T of G acts on the Schubertvarieties and the natural embedding GrT ⊂ GrG

identifies GrT with the T-fixed point scheme of

GrG (see §1.3). Moreover GrT ×GrG Grλ

G is the T-

fixed point scheme of Grλ

G. The main theorem of

this article states that the restriction of ϕ to Grλ

Ginduces an isomorphism on the global sections,

(2) H0(Grλ

G,LG)→ H0(Grλ

G,O(GrλG)T ⊗LG), if G has

type A or D. Furthermore, the same fact holds formany coweights if G has type E. In many parts of

the proof it is used that (Grλ

G)T is a finite scheme.The difficult part of the proof is showing the in-jectivity of (2). Indeed it is proved that the restric-

tion ofLkG from Gr

λ

G to (Grλ

G)T induces a surjectivemorphism on the global sections for any simplealgebraic group G.The first step to prove the injectivity is a re-duction to the case of fundamental dominantcoweights. Given two dominant coweights λand µ, the author constructs a family of vari-

eties with generic fibre is Grλ

G × Grµ

G while the

special fibre is Grλ+µ

G , using the following resulton the Demazure affine modules (here G is the

simply connected cover of G): H0(Grλ+µ

G ,LkG)

H0(Grλ

G,LkG)⊗H0(Gr

µ

G,LkG) as G-modules. The au-

thor includes a proof of this isomorphism whichis more geometrical than the original one. Toprove the reduction step, the author uses thethis family together with the interpretation of theaffine Grassmannian as a module space over asmooth curve.Next, he prove that the Schubert variety Gr

λ

G con-tains many rational curve with known degree. Ifλ is minuscule (and G is simply laced) then thesecurves have degree one. The author proves theinjectivity by showing that an arbitrarily fixed

section of H0(Grλ

G,Iλ(1)) is zero over certain T-

invariant subvarieties Z (by induction on the di-mension of Z). Here Iλ ⊂ O

GrλG

is the ideal sheaf

defining (Grλ

G)T. Moreover, the previous subva-rieties includes all the T-invariant curves andthe whole Schubert variety. Therefore, the caseof type A is proved.Next, he prove the injectivity when λ is a longestroot. The proof is similar, but he need to con-sider also curve of degree 2. Moreover he usesthe result for G = SL2. This fact proves the caseof D4.The case of Dn is proved by induction on n andby induction on i, where λ is i-th fundamental


coweight and the weight are indexed accord-ing to [N. Bourbaki, Groupes et algébres de Lie.Chapitres 4, 5 et 6. Elements de Mathematique.(Paris) etc.: Masson. (1981; Zbl 483.22001)]. Heneed also the isomorphism (2) for the case A.Finally, the author can prove some other caseswhen the type of G is E and he conjectures thatthe map is always injective for the type E.It is necessary to note that the author can re-prove the FKS-isomorphism for all the simplylaced group. The isomorphism (1) as t-modulesclearly follows from (2) for the type A and D.To prove the case E the author show that also inthis case any connected components of GrG is thedirect limits of Schubert varieties associated toweight for which (2) is an isomorphism.To prove that (1) is an isomorphism of vertex al-gebras he uses the languages of Kac-moody fac-torization algebras. Finally, he prove an identifi-cation of the modules over VRG with the modulesover L(Λ). Alessandro Ruzzi (Roma)

1167.15007Fortier Bourque, Maxime; Ransford, ThomasSuper-identical pseudospectra.J. Lond. Math. Soc., II. Ser. 79, No. 2, 511-528(2009).Let A ∈ CN×N. Given ε > 0, its ε-pseudospectrumis defined by σε(A) = z ∈ C : ‖(A − zI)−1

‖ > ε−1

(if z is an eigenvalue of A, then ‖(A− zI)−1‖ = ∞).

Here ‖.‖ denotes the spectral norm.L. N. Trefethen and M. Embree [Spectra and pseu-dospectra. The behavior of nonnormal matricesand operators. Princeton, NJ: Princeton Univer-sity Press. (2005; Zbl 1085.15009)] ask: Do pseu-dospectra determine behaviour? Precisely, sup-pose that A,B ∈ CN×N have identical pseudospec-tra, i.e., σε(A) = σε(B) for all ε > 0, or equivalently

‖(A − zI)−1‖ = ‖(B − zI)−1

‖ for all z ∈ C. (1)

Does ‖p(A)‖ = ‖p(B)‖ hold for every polyno-mial p? The answer is negative.Denoting by s1, . . . , sN the singular values indecreasing order, (1) means that sN(A − zI) =sN(B − zI). The authors study what happens un-der the stronger condition s j(A − zI) = s j(B − zI),j = 1, . . . ,N; then A and B are said to have super-identical pseudospectra (“sips” in the sequel).Drawing on ideas from invariant theory, the au-thors prove that there exist an integer m (de-pending on N) and a closed subset E of CN×N ofmeasure zero such that if A1, . . . ,Am ∈ CN×N

\ E

have sips, then at least two of them are unitar-ily equivalent. They also study other propertiesand non-properties of matrices with sips. Forexample, they construct non-derogatory matri-ces A,B ∈ R4×4 with sips such that ‖A2

‖ , ‖B2‖.

Jorma K. Merikoski (Tampere)

1166.16006Demonet, LaurentCluster algebras and preprojective algebras: thenon-simply laced case. (Algèbres amassées etalgèbres préprojectives: le cas non simplementlacé.) (French)C. R., Math., Acad. Sci. Paris 346, No. 7-8, 379-384 (2008).This paper is about cluster algebras [S. Fomin andA. Zelevinsky, J. Am. Math. Soc. 15, No. 2, 497-529 (2002; Zbl 1021.16017)]. A cluster structure inthe algebra of rational functions C[N], N a maxi-mal unipotent subgroup of a simple Lie group G,was introduced by A. Berenstein, S. Fomin and A.Zelevinsky, [Duke Math. J. 126, No. 1, 1-52 (2005;Zbl 1135.16013)]; when the associated Dynkin di-agram is simply-laced, this cluster structure wasinterpreted in terms of preprojective algebras byC. Geiß, B. Leclerc and J. Schröer, [Invent. Math.165, No. 3, 589-632 (2006)]. In this last paper, itwas shown that the cluster monomials in C[N]are linearly independent, answering a conjectureof Fomin and Zelevinsky (in the simply-lacedcase).The author announces the extension of the resultsof Geiß et al. [loc. cit.], in particular the proof ofthe conjecture of Fomin and Zelevinsky, to thenon simply-laced case. The starting point is toconsider a non simply-laced Dynkin diagram asa quotient of a simply-laced one by the action ofa diagram automorphism. Parallel approaches tothe same question appear in [G. Dupont, J. Alge-bra 320, No. 4, 1626-1661 (2008; Zbl 1155.16011)]and [D. Yang, Algebra Colloq. 16, No. 1, 143-154(2009)]. Nicolás Andruskiewitsch (Cordoba)

1166.16010Herman, Allen; Olteanu, Gabriela; del Río, Án-gelThe Schur group of an Abelian number field.J. Pure Appl. Algebra 213, No. 1, 22-33 (2009).Let K be a field and A a central simple K-algebra.We say that A is a Schur algebra, if it is generatedby a finite group of units. The subgroup S(K)of the Brauer group Br(K), formed by similar-ity classes containing a Schur algebra, is called


the Schur group of K. By the Brauer-Witt theo-rem [see, for example, T. Yamada, The Schur sub-group of the Brauer group. (Lect. Notes Math.397), (1974; Zbl 321.20004)], each class in S(K)can be presented by a cyclotomic algebra, i.e. acrossed product of the form (L/K, α), where L/Kis a cyclotomic extension and the factor set α hasvalues in the group W(L) of roots of unity in L.When K is a finite Abelian extension of the fieldQ of rational numbers, the elements of S(K) arepartially characterized by Benard-Schacher the-ory [see M. Benard, M. M. Schacher, J. Algebra 22,378-385 (1972; Zbl 239.20007)]. In the first place,it shows that S(K) is of finite exponent which di-vides the order of W(K). Secondly, it determinesthe behaviour of the local invariants of an alge-bra A ∈ S(K) over the prime ideals of the max-imal order of K lying above an arbitrary primenumber r ∈ Q. This singles out a positive integermr(A), called an r-local invariant of A. As shownby the theory, mr(A) , 1, for finitely many r, andthe Schur index ind(A) equals the least commonmultiple of the mr(A), when r runs across the setof prime numbers. This ensures the existence ofa maximum r-local index δr(A).The paper under review characterizes δr(A) interms of global information determined by K. Itprovides explicit formulae for the powers in theprimary decomposition of δr(A). This completesand unifies earlier results of G. J. Janusz [Ann.Math. (2) 103, 253-281 (1976; Zbl 352.16004)] andJ. W. Pendergrass [J. Algebra 41, 422-438 (1976;Zbl 359.16006)]. In addition, the authors correctsome of the calculations made by Pendergrass.

Ivan D. Chipchakov (Sofia)

1167.16004Bazzoni, SilvanaWhen are definable classes tilting and cotiltingclasses?J. Algebra 320, No. 12, 4281-4299 (2008).A class of modules is ‘definable’ if it is closed un-der direct products, direct limits and pure sub-modules. By two correspondences RF

ρ→ TR and

RTσ→ FR a bijection is obtained between defin-

able classes of R-Mod and definable classes ofMod-R, such that resolving classes correspondto coresolving classes and classes of cofinite typecorrespond to classes of finite type.The asymmetry which appears between the no-tions of finite type and cofinite type for n-tiltingand n-cotilting classes is mentioned. In partic-

ular, the tilting classes are of finite type, whilecotilting classes are not always of cofinite type.To clarify this phenomenon, the following theo-rem for n-cotilting classes is proved, the analogueof which for n-tilting classes does not hold.For n ∈ N and class F ⊆ R-Mod the followingare equivalent: 1) F is an n-cotilting class; 2) F isresolving, definable and F ⊥ ⊆ I\; 3) F is resolv-ing, definable and closed under n-submodules.For n-cotilting classes the following result isproved. Let n ∈ N and T be a definable classof R-Mod. The following are equivalent: 1) T isan n-tilting class; 2) σ(T ) is an n-cotilting class ofcofinite type; 3) T is coresolving, special preen-veloping and closed under n-images.As a conclusion is mentioned that if F is an n-cotilting class of R-Mod, then ρ(F ) is a definablecoresolving class closed under n-images and inthis case ρ(F ) is an n-tilting class if and only if Fis an n-cotilting class of cofinite type.

A. I. Kashu (Kishinev)

1167.16014Mounirh, KarimNondegenerate semiramified valued andgraded division algebras.Commun. Algebra 36, No. 12, 4386-4406 (2008).The paper under review is devoted to the studyof (non)degenenerate division algebras over aHenselian valued field. The notion of a non-degenerate division algebra proposed by theauthor is coherent with the definition of S. A.Amitsur and D. Saltman [J. Algebra 51, 76-87(1978; Zbl 391.13001)] and agrees with the oneadopted by K. McKinnie [J. Algebra 320, No. 5,1887-1907 (2008; Zbl 1156.16012)] in the contextof inertially split semiramified division algebrasover a Henselian valued field (K, v). Its advan-tages are exhibited on the basis of some of hisearlier results on nicely semiramified divisionK-algebras. In addition, the paper sheds newlight on the existing relationship between thesealgebras and cyclic division p-algebras.The author proves that inertially split divisionK-algebras of prime power degrees are nonde-generate cyclic and nicely semiramified of Schurindex equal to the exponent, provided that theresidue field of (K, v) is finite. He finds criteriafor tame semiramified division K-algebras (aswell as for semiramified graded division alge-bras over graded fields) to be degenerate, anduses them for obtaining examples of division p-


algebras which are not tensor products of cyclicalgebras.The second main result of the paper shows thatnondegenerate tame semiramified division K-algebras of prime power degree are indecompos-able, and so generalizes a theorem proved (by adifferent method), for inertially split semirami-fied division p-algebras, in the above-quoted pa-per by McKinnie. Ivan D. Chipchakov (Sofia)

1167.16017Bazlov, Yuri; Berenstein, ArkadyBraided doubles and rational Cherednik alge-bras.Adv. Math. 220, No. 5, 1466-1530 (2009).In the paper under review the authors intro-duce and study a new class of associative al-gebras, which they call ‘braided doubles’. Thisclass provides a unifying framework for classi-cal and quantum universal enveloping algebrasand rational Cherednik algebras. A braided dou-ble is an algebra with triangular decompositionA = U− ⊗ H ⊗ U+ over a Hopf algebra H suchthat U− and U+ are generated by dually pairedH-modules V and V∗, whose commutator (in A)lies in H.The main result of the paper is that any ratio-nal Cherednik algebra canonically embeds in thebraided Heisenberg double attached to the corre-sponding complex reflection group. This gives anew, quantum group-like realization of rationalCherednik algebras.Free braided doubles are classified in terms ofquasi-Yetter-Drinfeld modules. Braided doublesare quotients of free braided doubles that still ad-mit triangular decomposition. In particular, thereis a natural notion of a minimal double, examplesof which are (quantum) universal enveloping al-gebras. In many cases it is shown that a mini-mal double embeds into a braided Heisenbergdouble. As a corollary the authors get the PBWtheorem for rational Cherednik algebras over anarbitrary field. Volodymyr Mazorchuk (Uppsala)

1166.15016Ran, André C.M.; Rodman, LeibaOn the index of conditional stability of stableinvariant Lagrangian subspaces.SIAM J. Matrix Anal. Appl. 29, No. 4, 1181-1190(2008).The authors continue their investigation of con-ditional α-stability initiated by them [IntegralEquations Oper. Theory 27, No. 1, 71–102 (1997;

Zbl 901.15007)]. Basic results from several of theirearlier works are used and inaccurate statementstherein concerning the index are now made pre-cise.Let the fieldF denote eitherC orR. Fix an invert-ible 2n × 2n Hermitian (resp. symmetric or skewsymmetric as the case may be) matrix H in F2n×2n

and consider matrices A that satisfy HA = A∗H(resp. HA = ±ATH). An n-dimensional subspaceM of F2n is said to be H-Lagrangian if y∗Hx = 0(resp. yTHx = 0) for all x, y ∈ M. The authorsthen seek to establish small perturbations of theoriginal matrices associated toMwhich result insmall changes inM, the index in the title quali-fying for this purpose. But the set IL(A,H) of allA-invariant H-Lagrangian subspaces need not,a priori, be non empty. The main results (The-orem 2.1, resp. 3.1) provide spectral criteria onA guaranteeing the existence of conditionally α-stable subspaces. An application to algebraic Ric-cati equations is also given.

George F. Nassopoulos (Athens)

1167.18003Palu, YannGrothendieck group and generalized mutationrule for 2-Calabi-Yau triangulated categories.J. Pure Appl. Algebra 213, No. 7, 1438-1449(2009).Let C be the stable category of a Frobenius cat-egory E. The author shows a generalized mu-tation rule for quivers of cluster-tilting subcat-egories in C. When the cluster-tilting subcate-gories are related by a single mutation, this re-sult shows that their quivers are related by theFomin-Zelevinsky mutation rule.The author computes the Grothendieck groupof the triangulated category C, and comparesthe Grothendieck group of a cluster category CA

with the group K0(CA), defined in [M. Barot, D.Kussin and H. Lenzing, J. Pure Appl. Algebra 212,No. 1, 33–46 (2008; Zbl 1148.16005)]. Here CAis any cluster category associated with a finite-dimensional hereditary algebra.

Pere Ara (Bellaterra)

1167.18003Weibel, CharlesNK0 and NK1 of the groups C4 and D4.Comment. Math. Helv. 84, No. 2, 339-349 (2009).Let R be a ring and Ki(R) denote the i-th K-theorygroup of R, (i ∈ Z). Recall that NKi(R) is de-fined as the quotient Ki(R[X])/Ki(R). The author


explicitly computes these groups in the follow-ing cases: R = Z[G] (the integral group ring),G = D2,C4 or D4 (dihedral of order 2, cyclicof order 4 and dihedral of order 8, respectively)and i = 0, 1. The computations keep track of themodule structure of these groups given by Ver-schiebung and Frobenius operators, as well asthe continuous module structure over the ring ofbig Witt vectors W(Z). Let V denote the continu-ous W(F2)-module xF2[x]: it is a countable directsum of copies of F2 on generator xi, i > 0 and themodule structure is determined by Vm(xn) = xmn,[a]xn = anxn; Fm(xn) = 0 if (m,n) = 1 (m > 1) andFd(xn) = dxn/d wif d|n.The results are as follows.Theorem 1.3: For D2 = C2 × C2 we haveNK0(Z[D2]) V,NK1(Z[D2]) ΩF2[x],Theorem 1.4: NK1(Z[C4]) ΩF2[x] andNK0(Z[C4]) V,Theorem 2.5: NK0(Z[D4]) is isomorphic to thecyclic Cartier module NK1(Z[C2], 1 − τ) (de-scribed in the paper). As a group, this is the directsum of a countable infinite freeZ/4-module anda countable infinite free Z/2-module.The author also proves that NK0(Z[D4]) surjectsonto NK1(Z[D4]).

Daniel Juan Pineda (Michoacan)

1166.18007Tabuada, GonçaloHigher K-theory via universal invariants.Duke Math. J. 145, No. 1, 121-206 (2008).Recently it was observed that differential graded(dg) categories can be helpful in better un-derstanding of triangulated categories [cf. B.Keller, “On differential graded categories”, Pro-ceedings of the international congress of math-ematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume II: Invited lectures. Zürich:European Mathematical Society (EMS). 151–190(2006; Zbl 1140.18008)]. From a noncommutativealgebro-geometric point of view (i.e., the studyof dg categories and their homological invari-ants) dg categories have been considered as non-commutative schemes [cf. V. Drinfeld, J. Algebra272, No. 2, 643–691 (2004; Zbl 1064.18009) and M.Kontsevich].In this article, using Grothendieck’s derivatorsformalism, the author contructs the universal lo-calizing invariant of dg categories, i.e., a mor-phismUl from the pointed derivator HO(dgcat),associated with Morita’s homotopy theory of dgcategories [cf. G. Tabuada, C. R., Math., Acad. Sci.

Paris 340, No. 1, 15–19 (2005; Zbl 1060.18010)]to a triangulated strong derivator Mloc

dg so thatUl commutes with filtered homotopy colimits,preserves the point, maps each exact sequenceof dg categories to a triangle, and is universalfor these properties. Also, the author constructsthe universal additive invariant of dg categories,i.e., the universal morphism of derivators Uafrom HO(dgcat) to a strong triangulated deriva-torMadd

dg that satisfies the first two properties andthe third one only for split exact sequences.In this context, Waldhausen’s K-theory spectrumappears as a spectrum of morphisms in the basecategoryMadd

dg (e) of the additive motivator.Tihomir Petrov (Northridge)

1166.20012Evans, Anthony B.The admissibility of sporadic simple groups.J. Algebra 321, No. 1, 105-116 (2009).A complete mapping of a group G is a bijectionθ : G → G for which the mapping g → gθ(g)is also a bijection; G is admissible if G admitscomplete mappings. The Cayley table of a finitegroup G is a Latin square, and this Latin squarehas an orthogonal mate if and only if G is admis-sible.A long-standing problem is that of determin-ing which groups are admissible. M. Hall andL. J. Paige [Pac. J. Math. 5, 541-549 (1955;Zbl 66.27703)] conjectured that all finite groupswith trivial or noncyclic Sylow 2-subgroups areadmissible. In an unpublished paper, S. Wilcoxproved that any minimal counterexample to thisconjecture must be simple, and further, must beeither a Tits group or a sporadic simple group.In this paper, the author improves on this resultby proving that J4 is the only possible minimalcounterexample to this conjecture. J. Bray reportshaving proved that this group is also not a coun-terexample, thus completing a proof of the Hall-Paige conjecture.

Anatoli Kondrat’ev (Ekaterinburg)

1166.20032Francaviglia, StefanoGeodesic currents and length compactness forautomorphisms of free groups.Trans. Am. Math. Soc. 361, No. 1, 161-176 (2009).Let F be a free group of finite rank k ≥ 2and ϑF its hyperbolic boundary, which can beviewed as the space of right-infinite, freely re-duced, words in an alphabet of F. The bound-


ary is endowed with the Cantor-set topology. Letϑ2F = (x, y) | x, y ∈ ϑF, x , y. ϑ2F can be identi-fied with the set of oriented bi-infinite geodesicsin the Cayley graph of F. The base-ball of ϑ2F isthe set of geodesics passing through 1.A geodesic current on F is a non-negative locallyfinite Borel measure on ϑ2F that is F-invariant.The length L(η) of a current η is the measure η(B)of the base-ball of ϑ2F.If ϕ is an automorphism of F, then it is extendedto a homeomorphism of ϑF (which is denotedagain by ϕ) [D. Cooper, J. Algebra 111, 453-456(1987; Zbl 628.20029)]. This gives rise to an ac-tion of ϕ on the space of the geodesic currents onF: Let η be a geodesic current, define ϕη by set-ting ϕη(S) = η(ϕ−1(S)) for a Borel subset S ⊆ ϑ2F.Then it was proved [by I. Kapovich, Topologicaland asymptotic aspects of group theory. Con-temp. Math. 394, 149-176 (2006; Zbl 1110.20034)]that ϕη is a geodesic current on F.Let T : ϑF→ ϑF be the shift operator deleting thefirst letter of one right-infinite word. The spaceof frequency measures is the set of finite-massT-invariant non-negative Borel measures on ϑF.There is a natural homeomorphism from thespace of frequency measures to the space ofgeodesic currents (Theorem 3.4 in the paper orin [I. Kapovich, Exp. Math. 16, No. 1, 67-76 (2007;Zbl 1158.20014)]), therefore the action of an au-tomorphism of F on the space of the geodesiccurrents on F induces an action of this automor-phism on the frequency measures.If we pass to the uniform current ηA (see Defini-tion 3.5 in the paper), then the length of the auto-morphismϕ is defined as the length of the imageof the uniform current, that is, L(ϕ) = L(ϕηA) =ηA(ϕ−1(B)).Now we can state the main result of the pa-per: For a sequence ϕn of automorphisms of F, ifthere is a word w such that the cyclically reducedlength of ϕn(w) goes to∞, then L(ϕn)→∞.

Dimitrios Varsos (Athenai)

1167.20021Tortora, AntonioSome properties of Bell groups.Commun. Algebra 37, No. 2, 431-438 (2009).Let n , 0, 1 be an integer. A group G is called n-Bell if [xn, y] = [x, yn] for all x and y in G. A groupG is said to be n-Levi (n-Abelian, resp.) wheneverit satisfies the law [xn, y] = [x, y]n ((xy)n = xnyn,resp.).

The main results of the paper under review arethe following. Theorem 2.4. Let G be a groupand let n , 0, 1 be an integer. (i) If G is n-Bell, then G/Z2(G) has finite exponent dividing3n2(n− 1)2/2. (ii) If G is n-Levi, then G/Z2(G) hasfinite exponent dividing 3n(n − 1).A group G is said to be n-soluble if it has a fi-nite n-Abelian series, that is, a series 1 = G0 CG1 C · · · C Gk = G in which each factor Gi+1/Gi isn-Abelian.Theorem 3.3. Let G be an n-Bell group. Then thefactor group G/Z2(G) is n-Abelian. In particular,G is an n-soluble group of length at most 3.Following R. Baer [Proc. Am. Math. Soc. 4, 15-26 (1953; Zbl 50.02201)], the n-center of a groupG is defined by Z(G; n) = x ∈ G : (xy)n =xnyn for all y ∈ G. A group G is called n-nilpotent if it has a finite n-central series, thatis, a normal series 1 = G0 C G1 C · · · C Gk = Gsuch that each factor Gi+1/Gi is contained in then-center of G/Gi.Theorem 3.4. Every n-Bell group is n-nilpotent ofclass at most 5.Theorem 3.5. Let G be a locally graded n-Bellgroup. Then there exists an integer f (n) , 0, 1such that G is f (n)-nilpotent of class at most 2.Theorem 4.1. Let G be a locally finite n-Bell group,let π1 and π2 be the sets of primes dividing n andn − 1, respectively, and let π = π1 ∪ π2. ThenG = A × B × C, where A is a π1-group, B is a π2-group, and C is a 2-Engel π′-group. Moreover, Ais n-Kappe and B is (n − 1)-Kappe.Some commutator calculus are used in proofs.

Alireza Abdollahi (Isfahan)

1166.22002Uspenskij, Vladimir V.On subgroups of minimal topological groups.Topology Appl. 155, No. 14, 1580-1606 (2008).In this extraordinarily interesting paper whichcontains abundant material, the author answerstwo questions, namely:Pestov, Arhangelskii 1980: What are the sub-groups of minimal topological groups?Roelcke 1990: What are the subgroups of lowerprecompact topological groups?The author shows that every topological groupG is isomorphic to a subgroup of a complete min-imal group which is Roelcke-precompact, topo-logically simple and has the same weight as G.A Hausdorff group is called minimal if there ex-ists no strictly coarser Hausdorff group topology.


The greatest bound for the left and the right uni-formity L ∧ R on a topological group is calledlower uniformity or Roelcke uniformity.The proof of this deep and fundamental theo-rem is given in two steps. In the first the authorproves that every topological group G is isomor-phic to a subgroup of Iso(M) where M is a com-plete metrizableω-homogeneous Urysohn spaceof the same weight as G.In the case that G is second countable, M isthe Urysohn universal metric space U1 whichis characterized by the following two properties:it contains an isometric copy of every separa-ble metric space of diameter ≤ 1 and it is ω-homogeneous.In the case that G has uncountable weight, theproof is much harder and makes use of Katetovfunctions and Graev metrics on free topologicalgroups.In the second step Uspenskij shows that thegroup Iso(M) where M is an ω-homogeneousUrysohn complete metric space is Roelcke-precompact, minimal, and topologically simple.The weight of Iso(M) is equal to the weight of M.This is done by describing explicitly the Roelckecompactification of this group. It can be iden-tified with the so called bi-Katetov functionsM × M → [0, 1] which form an involutive or-dered semigroup. Deep insight in the structureof this semigroup is given.

Lydia Außenhofer (Castellón)

1166.22008Hirano, Miki; Oda, TakayukiCalculus of principal series Whittaker func-tions on GL(3,C).J. Funct. Anal. 256, No. 7, 2222-2267 (2009).Let G = GL(3,C) with Lie algebra g and K a

maximal compact subgroup of G. Consider theIwasawa decomposition G = NAK of G andthe corresponding minimal parabolic subgroupP = MAN of G. A nondegenerate character η of Ndefines a bundle over N\G. Denote by C∞η (N\G)the space of its smooth sections on which thegroup G acts by right translation C∞IndG

Nη. Let πbe an irreducible admissible representation of Gand choose a K-type (τ?,V?

τ ) which occurs withmultiplicity one inπ. Here (τ?,V?

τ ) stands for thecontragredient representation of a finite dimen-sional representation (τ,Vτ) of K. Fix an injectiveK-homomorphism i ∈ HomK(τ?, π |K). Associ-ated with an element T ∈ Hom(g,K)(π,C∞IndG

Nη) isan element Ti ∈ HomK(τ?,C∞IndG

Nη) defined by:T(i(v?))(h) =< v?,Ti(h) > for all v? ∈ V?

τ , h ∈ G.The elements of the space

Wh(π, η, τ) =⋃i∈HomK(τ?,π|K)

Ti ∈ HomK

(τ?,C∞IndG

Nη) ∣∣∣∣

T ∈ Hom(g,K)

(π,C∞IndG

Nη)

are called Whittaker functions on G with respectto (π, η, τ). It is difficult to give explicit formulasfor these functions for a general G, even in lowdimensional cases.In the paper under review, the authors giveexplicit formulas for Whittaker functions onGL(3,C). They use the so-called Gelfand-Zelevinsky basis of simple K-modules and theDirac-Schmid operators. They also relate Whit-taker functions on GL(3,C) with Whittaker func-tions on GL(2,C), using some propagation for-mula. Salah Mehdi (Metz)

Real and Complex Analysis. Functional Analysis and Operator Theory. 51

Real and Complex Analysis.Functional Analysis and Operator Theory.

1166.28001Fremlin, D. H.Measure theory. Vol. 4. Topological measurespaces. Part I, II. Corrected second printing ofthe 2003 original.Colchester: Torres Fremlin (ISBN 0-9538129-4-4). 528 p./Pt. I, 439 p., 19 p. (errata)/Pt. II. £ 73.00;$ 120.00 (2006).The fourth volume of the treatise, entitled “Topo-logical measure spaces" is devoted to study mea-sure in the context of space and topology.The first chapter of this volume is “an introduc-tion to some of the most important ways in whichtopologies and measures can interact". The start-ing point is to give some definitions (e.g. innerregular measure, topological measure space, τ-additive measure, outer regular measure, quasi-Radon measure space, Borel and Baire mea-sures, support set) and some examples. Succes-sive paragraphs of this chapter develop the prob-lems connected with notions introduced in thefirst paragraph.The second chapter is some introduction to thedescriptive set theory. The intention of the au-thor connected with this part of the treatise isbest illustrated by a quotation: “The first sec-tion describes Souslin’s operation and its basicset-theoretic properties up to the theory of ‘con-stituents’, mostly steering away from topologicalideas, but with some remarks on σ-algebras andSouslin-F sets. §422 deals with usco-compact re-lations and K-analytic spaces, working throughthe topological properties which will be usefullater, and giving a version of the First Separa-tion Theorem. §423 looks at ‘analytic’ or ‘Souslin’spaces, treating them primarily as a special kindof K-analytic space, with the von Neumann-Jankow selection theorem. §424 is devoted to‘standard Borel spaces’; it is largely a series ofeasy applications of results in §423, but there isone substantial theorem on Borel measurable ac-tions of Polish groups."The chapter with number 43 (third chapter inthe fourth volume) is some continuation of con-siderations contained in the first chapter of thisvolume. The paragraph begins with the sectionconcerning Souslin operations. In the next sec-

tion the author describes basic measure-theoreticproperties of K-analytic spaces. The main resultof this section is the theorem (Aldaz, Render):Let X be a K-analytic Hausdorff space and µ alocally finite measure on X which is inner regu-lar with respect to the closed sets. Then µ has anextension to a Radon measure on X (in particularµ is τ additive).In the third paragraph the author considers someproperties of analytic spaces. The fourth para-graph the author begins with the formulationof the basic question: “What kinds of measurescan arise on what kinds of topological space?"The main idea of this paragraph (entitled “Borelmeasures") is presented in some parts of intro-duction: “In 434A I set out a crude classifica-tion of Borel measures on topological spaces. Forcompact Hausdorff spaces, at least, the first ques-tion is whether they carry Borel measures whichare not, in effect, Radon measures; this leads usto the definition of ‘Radon’ space which is alsoof interest in the context of general Hausdorffspaces. I give a brief account of the propertiesof Radon spaces. I look also at two special top-ics: ‘quasi-dyadic’ spaces and a construction ofBorel product measures by integration of sec-tions. In the study of Radon spaces we find our-selves looking at ‘universally measurable’ sub-sets of topological spaces. [. . . ] Three furtherclasses of topological space, defined in termsof the types of topological measure which theycarry, are the ‘Borel-measure-compact’, ‘Borel-measure-complete’ and ‘pre-Radon’ spaces; I dis-cuss them briefly in 434G-434J. They provideuseful methods for deciding whether Hausdorffspaces are Radon".The fourth chapter the author begins with theparagraph containing theorem on the existenceof invariant measures: a locally compact Haus-dorff topological group has left and right Haarmeasures, which are both Radon measures. Thenext two paragraphs describe properties and em-phasize the uniqueness of Haar measures. Para-graph 444 is devoted to convolution measures,function and measures and, moreover, containsgeneral results concerning continuous group ac-tions on quasi-Radon measure spaces. The suc-

52 Real and Complex Analysis. Functional Analysis and Operator Theory.

cessive part of this chapter contains a proof ofthe Pontryagin-van Kampen duality theorem. Inthe last four paragraphs the author tackles thefollowing problems: the structure of locally com-pact groups, translation-invariant liftings andlower densities; Vitali’s theorem and a densitytheorem for groups with B-sequences; invariantmeasures on Polish spaces and (locally compact)amenable groups.The chapter 45 (the last chapter in the first partof the fourth volume) is entitled “Perfect mea-sures and disintegration". The starting pointsof the considerations are the notions: compactmeasure and countably compact measures andthe Ryll-Nardzewski theorem: Any semi-finitecountably compact measure is perfect. §452 is de-voted to the problem of integration and disinte-gration of measures (special types of disintegra-tion connected with the phrase: “regular condi-tional probability"). In the next paragraph the au-thor describes some cases in which strong liftingsare known to exist. This part of the chapter is fin-ished with a note on the relation between strongliftings and a Stone space and with an exam-ple of a space with no strong lifting. Paragraph454 (entitled: “Measures on product spaces") con-tains, among other things, the Marczewski andRyll-Nardzewski Thorem. The next part is con-nected with Markov processes, which lead tothe straightforward existence theorem “depen-dent only on a natural consistency condition onthe conditional distributions". In the last parts ofthis chapter the author gives results connectedwith (universal) Gaussian distributions, exten-sions of measures (Strassen’s theorem); relativelyindependent families of σ algebras and randomvariables; relative distribution; relative productsof probability spaces; exchangeable families ofinverse-measure-preserving functions and sym-metric quasi-Radon measures.The second part of the fourth volume (chapter 46)begins with the theme: “Pointwise compact setsof measurable functions". Within the range of thistheme the following barycenters (sufficient con-ditions for existence, Krein theorem, measures onsets of extreme points); pointwise compact setsof continuous functions (the topology of point-wise convergence on C(X), weak convergence,convex hulls, separately continuous function);pointwise convergence on spaces of measurablefunctions; Talangrand’s measure; stable sets offunctions; quasi-Radon measures for weak and

strong topologies; universally measurable linearoperators and locally uniformly rotund norms.In the introduction to the Chapter 47 (entitled:“Geometric measure theory") the author writes:“The greater part of it is directed specificallyat a version of the Divergence Theorem". Thefirst paragraph is devoted to Hausdorff mea-sures on general metric spaces. The further partsof this chapter the author describes in the fol-lowing way: “§472, at least, deals with some-thing which must be central to any approach,Besicovitch’s Density Theorem for Radon mea-sures on Rr. In §473 I examine Lipschitz func-tions, and give crude forms of some fundamen-tal inequalities relating integrals

∫‖grad f ‖dµ

with other measures of the variation of a func-tion f . In §474 I introduce perimeter measuresλ∂E and outward-normal functions ψE as thosefor which the Divergence Theorem, in the form∫

E divφdµ =∫φ · ψdλ∂E will be valid, and give

the geometric description of ψe(x) as the Fed-erer exterior normal to E at x. In §475 I showthat λ∂E can be identified with normalized Haus-dorff (r− l)-dimensional measure on the essentialboundary of E. [. . . ] In §476 I turn to a differenttopic, the problem of finding the subsets ofRr onwhich Lebesgue measure is most ‘concentrated’in some sense. I present a number of classicalresults, the deepest being the Isoperimetric The-orem: among sets with a given measure, thosewith the smallest perimeters are the balls."The Chapter 48 (entitled “Gauge integrals") is de-voted to the Kurzweil - Henstock integrals. Thefirst paragraph contains the terminology and thedescription of examples. The best description ofthe content of next chapters gives the author inthe introduction to this part of the treatise: “In§482 I give a handful of general theorems show-ing what kinds of result can be expected andwhat difficulties arise. In §483, I work throughthe principal properties of the Henstock integralon the real line, showing, in particular, that itcoincides with the Perron and special Denjoy in-tegrals. Finally, in §484, I look at a very strikingintegral on Rr, due to W. F. Pfeffer."The title of the last chapter: “Further topics" ex-plains the intentions of the author. This part of thebook contains the following problems: concen-trations of measure in product space and in par-mutations group; extremely amenable groups;locally compact groups; product sets includedin given sets of positive measure; Poison distri-butions and Poisson point process.


This volume ends with appendices containingsome information relevant to some topics pre-sented in this volume. In this part of the treatisethe author gives some facts connected with settheory and general topology, topological σ alge-bras; locally convex spaces; topological groupsand Banach algebras.Each section of this treatise ends with “basic ex-ercises", “further exercises" and “Notes and com-ments". Ryszard Pawlak (Łódz)

1166.28002Fremlin, D. H.Measure theory. Vol. 5.Set-theoretic measure theory. Part I, II.Colchester: Torres Fremlin (ISBN 978-0-9538129-5-0/Pt.I; 978-0-9538129-6-7/Pt.II). 292 p.$ 30.81/pt.I; $ 34.45/pt.II (2008).The intentions of the author connected with thefifth volume (entitled: “Broad foundations") ofthe treatise “Measure theory" is best illustratedby a quotation from the introduction: “For thefinal volume of this treatise, I have collected re-sults which demand more sophisticated set the-ory than elsewhere. The line is not sharp, buttypically we are much closer to questions whichare undecidable in ZFC. Only in Chapter 55 arethese brought to the forefront of the discussion,but elsewhere much of the work depends on for-mulations carefully chosen to express, as argu-ments in ZFC, ideas which arose in contexts inwhich some special axiom – Martin’s axiom, forinstance – was being assumed".The first chapter “is centered on a study of par-tially ordered set". In this part of the book theauthor talks over the following problems: ide-als of sets, supported relations and Galois-Tukeyconnections, Stone spaces, cardinal functionsof Boolean algebras, free subalgebras, Balcar-Frenek and Pierce-Koppelberg theorems, precal-ibers of supported relations, Martin numbers andFreese-Nation numbers of partially ordered sets.The main ideas of the results contained in thechapter 52 are best conveyed by the author:“From the point of view of this book, the mostimportant cardinals are those associated withmeasures and measure algebras, especially, ofcourse, Lebesgue measure and the usual mea-sure µI of 0, 1I. In this chapter I try to coverthe principal known facts about these which aretheorems of ZFC. I start with a review of the the-ory for general measure spaces in §521, includingsome material which returns to the classification

scheme of Chapter 21, exploring relationships be-tween (strict) localizability, magnitude and Ma-haram type. §522 examines Lebesgue measureand the surprising connexions found by BAR-TOSZYNSKI 84 and RAISONNIER & STERN85 between the cardinals associated with theLebesgue null ideal and the corresponding onesbased on the ideal of meager subsets of K. §523looks at the measures µI for uncountable sets I,giving formulae for the additivities and cofinali-ties of their null ideals, and bounds for their cov-ering numbers, uniformities and shrinking num-bers. Remarkably, these cardinals are enough totell us most of what we want to know concern-ing the cardinal functions of general Radon mea-sures and semi-finite measure algebras (§524).These three sections are heavily dependent onthe Galois-Tukey connections and Tukey func-tions of §§512-513. Precalibers do not seem to fitinto this scheme, and the relatively partial infor-mation I have is in §525. The second half of thechapter deals with special topics which can beapproached with the methods so far developed.In §526 I return to the ideal of subsets ofN withasymptotic density zero, seeking to locate it inthe Tukey classification. Further σ-ideals whichare of interest in measure theory are the ‘skewproducts’ of §527. In §528 I examine some inter-esting Boolean algebras, the ‘amoeba algebras’first introduced by MARTIN & SOLOVAY 70,giving the results of TRUSS 88 on the connexionsbetween different amoeba algebras and localiza-tion posets. Finally, in §529, I look at a handful ofother structures, concentrating on results involv-ing cardinals already described".In the Chapter 53 (entitled “Topologies andmeasures") the author returns to problems con-sidered in earlier volumes. §531 is connectedwith general Radon measures (Maharam typesof Radon measures). Most notably, the authorexamines the class MahR(X) – the set of Ma-haram types of Maharam-type-homogeneousRadon probability measures on a Hausdorffspace X, while in §532 the author considers theclass Mahcr(X) – the set of Maharam types ofMaharam-type-homogeneous completion regu-lar topological probability measures on topolog-ical space X. To describe the next parts of thisChapter let us quote the author: “In §534 I setout the elementary theory of ‘strong measurezero’ ideals in uniform spaces, concentrating onaspects which can be studied in terms of con-cepts already induced. Here there are some very


natural questions which have not I think been an-swered (534Z). In the same section I run throughthe properties of Hausdorff measures when ex-amined in the light of the concepts in Chapter 52.In §535 I look at liftings and strong liftings, ex-tending the results of §§341 and 453; in particular,asking which non-complete probability spaceshave liftings. In §536 I run over what is knownabout Alexandra Bellow’s problem concerningpointwise compact sets of continuous functions.With a little help from special axioms, there aresome striking possibilities concerning repeatedintegrals, which I examine in §537. In §539, Icomplete my account of the result of B. Balcar,T. Jech and T. Pazak that it is consistent to sup-pose that every Dedekind complete ccc weakly(σ,∞)-distributive Boolean algebra is a Maharamalgebra, and work through applications of themethods of Chapter 52 to Maharam submeasuresand algebras. Moving into new territory, I devotea section to a study of special types of filter onNassociated with measure-theoretic phenomena,and to medial limits".The Chapter 54 the author begins with the ques-tion: is there a non-trivial measure space in whichevery set is measurable? The first two paragraphsof this chapter have an introducing characterand contain statements connected with: (ω1) sat-urated ideals and cardinal arithmetic. The mainresult contained in §543 is the Gitik-Shelah theo-rem: Let κ be an atomlessly-measurable cardinal,with witnessing probability ν, then the Maharamtype of ν is at least min(κ(+ω), 2κ).The mainstream of problems of the paragraph544 is well described in the introduction to thispart of this book: “As is to be expected, a wit-nessing measure on a real-valued-measurablecardinal has some striking properties, especiallyif it is normal. What is less obvious is thatthe mere existence of such a cardinal can haveimplications for apparently unrelated questionsin analysis. In 544J, for instance, we see thatif there is any atomlessly-measurable cardinalthen we have a version of Fubini’s theorem,∫ ∫

f (x, y)dxdy =∫ ∫

f (x, y)dydx, for many func-tions f on R2 which are not jointly measurable.In this section I explore results of this kind.We find that, in the presence of an atomlessly-measurable cardinal, the covering number of theLebesgue null ideal is large while its unifor-mity is small. There is a second inequality onrepeated integrals to add to the one already givenin 543C, and which tells us something about

measure-precalibers; I add a couple of variations(544I-544J). Next, I give a pair of theorems on ameasure-combinatorial property of the filter ofconegligible sets of a normal witnessing mea-sure. Revisiting the theory of Borel measures onmetrizable spaces, discussed in §438 on the as-sumption that no real-valued-measurable cardi-nal was present, we find that there are some non-trivial arguments applicable to spaces with non-measure-free weight (544K-544L)".The fifth paragraph of this chapter is devotedto “product measure extension axiom" (PMEA)and “normal measure axiom" (NMA). The mainidea of the paragraphs 546 and 547 is signaledin the introduction to these sections: “One wayof interpreting the Gitik-Shelah theorem is to saythat it shows that ‘simple’ atomless probabilityalgebras cannot be of the form PX/N(µ). Simi-larly, the results of §541–§542 show that any cccBoolean algebra expressible as the quotient of apower set by a non-trivial σ-ideal involves us indramatic complexities, though it is not clear thatthese must appear in the quotient algebra itself.In this section I give two further results of M. Gi-tik and S. Shelah showing that certain algebrascannot appear in this way. I try to present theideas in a form which leads naturally to someoutstanding questions". “Given a family Aii∈Iof sets in measure space, when can we find a dis-joint family A′i i∈I such that A′i ⊆ Ai has the sameouter measure as Ai for every i? A partial resultis in Theorem 547F. Allied questions are: whencan we find a set D such that Ai ∩ D and Ai \ Dhave the same outer measure as Ai for every i?(547G) or are just non-negligible? (5471)."The paragraphs 551–555 (the chapter 55 entitled:“Possible worlds") are well exemplified in the in-troduction written by the author: “For a measuretheorist, by far the most important forcings arethose of ‘adding random reals’. I give three sec-tions to these. Without great difficulty, we candetermine the behaviour of the cardinals in Ci-chon’s diagram, at least if many random reals areadded. Going deeper, there are things to be saidabout outer measure and Sierpinski sets, and ex-tensions of Radon measures. In the same section Igive a version of the fundamental result that sim-ple iteration of random real forcings gives ran-dom real forcings. In §553 I collect results whichare connected with other topics dealt with above(Rothberger’s property, precalibers, ultrafilters,medial limits) and in which the arguments seemto me to develop properties of measure algebras


which may be of independent interest. In prepa-ration for this work, and also for §554, I start witha section devoted to a rather technical general ac-count of forcings with quotients of σ-algebras ofsets, aiming to find effective representations ofnames for points, sets, functions, measure alge-bras and filters.Very similar ideas can also take us a long waywith Cohen real forcing. Here I give little morethan obvious parallels to the first part of §552,with an account of Freese-Nation numbers suf-ficient to support Carlson’s theorem that a Borellifting of Lebesgue measure can exist when thecontinuum hypothesis is false.One of the most remarkable applications of ran-dom reals is in Solovay’s proof that if it is con-sistent to suppose that there is a two-valued-measurable cardinal, then it is consistent to sup-pose that there is an atomlessly-measurable car-dinal. By taking a bit of trouble over the lem-mas, we can get a good deal more, includingthe corresponding theorem relating supercom-pact cardinals to the normal measure axiom; andsimilar techniques show the (possibility of inter-esting power set σ-quotient algebras". In the firstpart of the paragraph 556 the author developsthe topic “Forcing with Boolean algebra". In thefurther part one can find some theorems withproofs connected with these methods: a stronglaw with large numbers; Dye’s theorem on or-bit isomorphic measure-preserving transforma-tions; Kawada theorem on invariant measure.The last part of the book is entitled “Choice anddeterminancy". In the first part of this chapter“Analysis without choice" the author “looks atbasic facts from real analysis, functional analysisand general topology" which can be proved inZermelo–Fraenkel set theory. The starting pointof these considerations is a remark that in theabsence of choice, the union of a sequence ofcountable sets need not be countable. For the de-scription of remaining parts of this chapter wecan quote the author: “§562 deals with ‘codable’Borel sets and functions, using Borel codes tokeep track of constructions for objects, so that ifwe know a sequence of codes we can avoid hav-ing to make a sequence of choices. A ‘Borel-codedmeasure’ is now one which behaves well withrespect to codable sequences of measurable sets;for such a measure we have an integral with ver-sions of the convergence theorems, and Lebesguemeasure fits naturally into the structure. In §566,with ZF + AC(ω), we are back in familiar terri-

tory, and most of the results of Volumes 1 and2 can be proved if we are willing to re-examinesome definitions and hypotheses. Finally, in §567,I look at infinite; games and half a dozen of theconsequences of AD, with a postscript on deter-minacy in the context of ZF + AC."This volume ends with appendices containingsome information relevant to some topics pre-sented in this volume. In this part of the treatisethe author gives some facts connected with theset theory, Shelah’s pcf theory (reduced products,cofinalities); forcing; general topology (cardinalfunctions, Vietoris topology, category and Baireproperty, paracompact spaces), real-entiere func-tions.Each section of this treatise ends with “basic ex-ercises", “further exercises" and “Notes and com-ments". Ryszard Pawlak (Łódz)

1167.30001Freitag, Eberhard; Busam, RolfComplex analysis. Transl. from the German byDan Fulea. 2nd ed.Universitext. Berlin: Springer (ISBN 978-3-540-93982-5/pbk; 978-3-540-93983-2/e-book). x,532 p. EUR 44.95/net; SFR 75.00; $ 69.95; £ 35.99(2009).The first edition [Zbl 783.30001] of this originallyGerman textbook on the classic theory of func-tions of a complex variable appeared in 1993.Thereafter, in the course of the following decade,the authors have improved this popular text bysuccessive revisions and enlargements, therebyfollowing the continuously growing number ofhints, suggestions, and demands communicatedto them by colleagues, students, and other en-gaged readers. As a result, two new editions ofthe book were brought about in relatively shorttime. More precisely, the second German edition[Zbl 928.30001] was published in 1995, while thethird one [Zbl 1058.30500] became available in2000. Another five years later, in 2005, an Englishtranslation [Zbl 1085.30001] of the then forthcom-ing fourth German edition of this meanwhilewidely used primer was provided, again withfurther material added. In the meantime, thefourth edition of the German version of the texthas appeared, too. The book under review is thesecond English edition of the well-tried introduc-tion to classic complex analysis of one variable byE. Freitag and R. Busam. Having left the text ofthe foregoing first English edition largely intact,the authors have nevertheless followed further


useful suggestions to improve the text by mi-nor corrections and additions. Compared to theoriginal German edition published sixteen yearsago, the exposition appears now in substantiallyclarified and polished form, enhanced by a largenumber of illustrating applications, instructiveexamples, and accompanying related exercises,together with guiding hints as for solutions tothe latter.Now as before, the material is organized in sevenchapters covering the following topics:1. Differential calculus in the complex plane C,including complex derivatives and the Cauchy-Riemann differential equations.2. Integral calculus in the complex plane C, fo-cussing on Cauchy’s integral theorem and onCauchy’s integral formulas.3. Sequences and series of analytic functions,covering the method of uniform approxima-tion, power series, mapping properties of ana-lytic functions, singularities of analytic functions,Laurent series, the residue theorem, and concreteapplications of the residue theorem.4. Construction of analytic functions, explainingthe Gamma function, the Weierstrass productformula, the Mittag-Leffler partial fraction de-composition, and the weak version of the Rie-mann mapping theorem. Two appendices to thischapter discuss the homotopical version and thehomological version of the Cauchy integral the-orem, respectively, whilst a third appendix pro-vides characterizations of elementary plane do-mains.5. Elliptic functions, comprising Liouville’s theo-rems, double periodic functions, period lattices,one-dimensional complex tori, the Weierstrass℘-function, the field of elliptic functions, the addi-tion theorem for the℘-function, elliptic integrals,Abel’s theorem, the elliptic modular group, andthe modular j-function.6. Elliptic modular forms, providing an intro-duction to the elementary theory by discussingthe Siegel modular group and its fundamentalregion, the so-called “k/12-formula" for mero-morphic modular forms of weight k, the algebraof modular forms, theta series, modular formsfor congruence subgroups, the theta group, andrings of theta functions.7. Analytic number theory, emphasizing some ofthe fascinating applications of complex-analyticmethods in this classic area of mathematics. Thisincludes the use of Eisenstein series and theirFourier expansion for computing the number of

representations of a natural number as a sum offour or eight squares, Dirichlet series and theirrole in number theory, the Riemann zeta func-tion in the theory of distribution of prime num-bers, and finally a complex-analytic proof of theprime number theorem with a weak form for theerror term. This proof of a version of the primenumber theorem is based on rather elementaryproperties of the Riemann zeta function, uses asfew advanced methods as possible, and is there-fore particularly accessible and instructive forstudents.There is an additional Chapter 8 providing ratherdetailed solutions to the vast number of carefullyselected exercises that accompany each singlesection of the book. These exercises often leadthe reader to supplementary concepts and re-sults, thereby enhancing both the main text andthe value of the book tremendously.All together, this introduction to complex analy-sis is fairly special, and somehow unique withinthe existing related textbook literature. Its mainfeatures can be summarized as follows:The authors present, apart from the standard ma-terial, a wide range of topics that are usually notcovered by introductory texts, but certainly be-long to the basic knowledge of modern complexanalysis. From the very beginning on, the geo-metric aspects of complex analysis are stronglyemphasized and serve as driving motivation,thereby reflecting the contemporary viewpointin combination with the classical heritage ofCauchy, Abel, Jacobi, Riemann, and Weierstrassin very instructive a manner. In spite of the manyextras offered by this fine book, the authors havespent great effort to introduce as few notions aspossible in order to quickly advance to the core ofthe respective topic and, in this vein, have strivento give simplified ad-hoc proofs of general theo-rems in special contexts whenever possible. Also,many of the fundamental concepts and resultsare discussed from various viewpoints, whichhelps the reader acquire a thorough understand-ing of the matter. Finally, the entire expositionstands out by its lucidity, elegance, depth, andversatility. The numerous historical remarks cer-tainly underline the authors’ didactic mastery,just as the carefully compiled bibliography does.No doubt, this excellent, comprehensive, andnearly self-contained textbook on complex anal-ysis deserves a wide international audience ofreaders, who will profit a great deal from study-ing it. As the authors point out, a forthcoming


second volume of their treatise, with a specialemphasis placed on Riemann surfaces, is to ap-pear in the near future, which the mathematicalcommunity will appreciate just as well.

Werner Kleinert (Berlin)

1167.41001Cheney, Ward; Light, WillA course in approximation theory.Graduate Studies in Mathematics 101. Prov-idence, RI: American Mathematical Society(AMS) (ISBN 978-0-8218-4798-5/hbk). xiv, 359 p.$ 69.00 (2009).This book is an excellent course on some mod-ern chapters of approximation. It deals especiallywith approximation of functions with severalvariables, as opposed to the classical approxima-tion theory of functions of one variable. The bookcovers areas of researches of great interest in thelast decades, in connections with obtaining highperformance algorithms for computers, but alsomade for a deep study of the abstract models.The book collects and organizes a large numberof results accessible until now only in researchpapers. It contains modern methods and technicsof approximations for diverse applicative fields,like engineering, life sciences, business and eco-nomics. On the other hand it points out that aconsistent source of development of the theorylied in practical applications.The elaboration of the book is a success. The bookpoints out the beauty of theoretical constructionsand the diversity of approximation theory. It willbecome surely a reference book.The simple enumeration of the book’s 36 chap-ters gives a good idea of the treated problems: 1.Introductory discussion of interpolation; 2. Lin-ear interpolation operators; 3. Optimization ofthe Lagrange operator; 4. Multivariate polyno-mials; 5. Moving the nodes; 6. Projections; 7.Tensor-product interpolation; 8. The Boolean al-gebra of projections; 9. The Newton paradigmfor interpolation; 10. The Lagrange paradigm forinterpolation; 11. Interpolation by translates ofa single function; 12. Positive definite functions;13. Strictly positive definite functions; 14. Com-pletely monotone functions; 15. The Schoenberginterpolation theorem; 16. The Micchelli inter-polation theorem; 17. Positive definite functionson spheres; 18. Approximation by positive def-inite functions; 19. Approximate reconstructionof functions and tomography; 20. Approxima-tion by convolution; 21 The good kernels; 22.

Ridge functions; 23. Ridge function approxima-tion via convolutions; 24. Density of ridge func-tions; 25. Artificial neural networks; 26. Cheby-shev centers; 27. Optimal reconstruction of func-tions; 28. Algorithmic orthogonal projections; 29.Cardinal B-splines and the sinc function; 30. TheGolomb-Weinberger theory; 31. Hilbert functionspaces and reproducing kernels; 32. Sphericalthin-plate splines; 33. Box splines; 34. WaveletsI; 35. Wavelets II; 36. Quasi-interpolation.The approach is made from different points ofview, combining the theoretical aspects with spe-cial problems, emphasizing the strong connec-tions between them. From the very rich materialtreated in the book we shall remark here onlysome topics. A large part of the book is concernedwith interpolation theory for functions with sev-eral variables. It contains a pertinent discussionon the possible general methods and of possi-ble classes of functions which can be used forinterpolations. We note the polynomial interpo-lation and the interpolation and approximationby positive definite functions. The book presentsalso some special types of approximations. So wemention the very general and important problemof reconstruction of functions by starting fromthe known values taken by a finite number offunctionals applied on it. A particular case ofsuch problems raised in tomography. An otherspecial problem is approximation by ridge func-tions, which is a particular case of approximationby superposition functions. A main aim is to findextensions of the spline functions in the multi-dimensional case. Finally the book contains anintroduction in wavelets theory, one of the mostdynamic chapter in approximation from the lasttwo decades. Among the main general theoreti-cal tools used in the book we mention the Hilbertspaces, the Fourier transform and the convolu-tion operators.The book is constructed as a course in approxi-mation. In this order there is made a gradually in-troduction into the subject, from simple to moregeneral and special problems. Each chapter con-tains a concise and suggestive presentation ofthe main ideas and attains the big problems ofthe domain. A rich information is presented ina condensed space. The material is linked to awell - chosen bibliography which accompanieseach chapter. The whole bibliography has about600 titles, consisting in fundamental books onapproximation and important papers, many ofthem very recent. Also, each chapter contains a


weighty part of attractive problems and exerciseswhich allow the reader to have a more deep con-tact with the subject. Radu Paltanea (Brasov)

1166.42001Van Fleet, Patrick J.Discrete wavelet transformations. An elemen-tary approach with applications.Wiley-Interscience. Hoboken, NJ: John Wiley &Sons (ISBN 978-0-470-18311-3/hbk). xxiv, 535 p.EUR 80.50; £ 60.95 (2008).The book concerns the basic theory and prac-tice of wavelet computations. Several books ad-dressing wavelet theory and applications areavailable, they are however, rather sophisticatedand mainly oriented at the scientific audience.This book is in turn targeted on teachers and stu-dents. Assuming that the filter coefficients, and(fast) computational algorithms, are sufficient inpractice, especially in image and signal process-ing applications, the book is focused on the alge-braic stage based on the wavelet filter coefficientsand related to them computational algorithms.The contents of the book can be split into thefollowing general three parts. The first (Chap-ters 1-5) is preliminary, and recollects pertinentfacts about vectors, matrices and digital images,complex numbers and Fourier series, and con-volutions and filters. The second (Chapters 6-9)deals with Haar, classic Daubechies and Coifletorthogonal filters associated with the respectivecompactly supported orthogonal wavelets. Con-structions of the filters are presented in detailsalong with the following transform algorithmsfor 1D and 2D data. Filters properties are ex-pressed in terms of algebraic and Fourier seriesconditions while the transform algorithms arepresented in a matrix vein. Two explanatoryapplications are presented: edge detection andimage denoising based on Donoho shrinkageidea. In the latter, the popular VisuShrink andSureShrink algorithms are described. The lastpart (Chapters 10-12) describes biorthogonal fil-ters with the emphasis given to spline-based,LeGall 5/3, and Cohen-Daubechies-Feauveau 9/7filters. The associated transform algorithms arepresented and the practical issues related toboundary effects and filters symmetry are bothelaborated. Eventually, the JPEG2000 compres-sion algorithm is used to illustrate the poten-tial of biorthogonal filters application to imagecompression. The ’lecture’ sections are inter-twined with both exercises oriented at more gen-

eral and more fundamental relations on whichwavelet filte r constructions are based upon andexamples accompanied by software implemen-tations. In a short appendix the basic facts frommathematical statistics (to formally support theshrinkage algorithms) are reported. A pack-age/toolbox supplementing the book topics isavailable on-line, via the book’s WWW page:http://cam.mathlab.stthomas.edu/wavelets/packages.php, for use with Mathematica andMatlab. The book, delivering the theoretical back-ground, exercises to ‘built up’ proper intuitions,and ready to use software packages to ‘play’ withapplications, is very helpful and supportive inthis routine. Therefore, it can be recommendedfor teaching wavelets and as primer in this field.

Zygmunt Hasiewicz (Wrocław)

1167.47001Ammari, Habib; Kang, Hyeonbae;Lee, HyundaeLayer potential techniques in spectral analysis.Mathematical Surveys and Monographs 153.Providence, RI: American Mathematical Soci-ety (AMS) (ISBN 978-0-8218-4784-8/hbk). vi,202 p. $ 69.00 (2009).The present book deals with using the layer po-tential technique to tackle the asymptotic theoryfor eigenvalue problems. The general approachin this book is developed in detail for the Lapla-cian and the Lamé system in two situations: oneunder variation of domains or boundary condi-tions, and the other due to the presence of small-volume inclusions. The book consists of threeparts.The first part is devoted to the theory developedby E. Gohberg and E. I. Sigal. In the second part,eigenvalue perturbation problems are studied.In Chapter 2, the layer potential technique asso-ciated with the Laplacian, the Helmholtz equa-tion and the Lamé system is briefly reviewed. InChapter 3, the authors derive complete expan-sions for the eigenvalues of the Neumann Lapla-cian in bounded singularly perturbed domains.Chapter 4 designs a simple method for detect-ing small internal corrosion by vibration analy-sis. Chapter 5 studies perturbations of scatteringfrequencies of resonators with narrow slits andslots. The book provides, on the one hand, resultson the existence and localization of the scatter-ing frequencies and, on the other hand, on theleading-order terms in their asymptotic expan-sions in terms of the characteristic width of the


slits or the slots. In Chapter 6, the book developsthe asymptotic theory for eigenvalue problems tothe Lamé system with Neumann boundary con-dition. Complete asymptotic expansions of theperturbations due to the presence of an elasticinclusion are derived rigorously. Leading-orderterms in these expansions are explicitly given.The inclusion may be hard or soft. A hard inclu-sion is characterized by the boundary conditionu = 0 on its boundary, while a soft inclusion ischaracterized by the transmission condition onits boundary.The third part of the book investigates theband gap structure of the frequency spectrumfor waves in a high contrast, two-composedmedium. This provides a new tool for investi-gating photonic and phononic crystals. Chapter 7deals with the Floquet transform. Then the struc-ture of spectra of periodic elliptic operators is di-cussed. Also, quasi-periodic layer potentials forthe Helmholtz equation and the Lamé systemare investigated. A spectral and spatial represen-tation of Green’s function in periodic domains isprovided. In Chapter 8, the authors perform ananalysis of the spectral properties of high con-trast band gap materials, consisting of a back-ground medium which is perforated by a peri-odic array of holes, with respect to the index ratioand small perturbations in the geometry of holes.Chapter 9 is devoted to phononic band gaps. Thefinal Chapter 10 investigates shape optimizationproblems. Dagmar Medková (Praha)

1167.47002Chidume, CharlesGeometric properties of Banach spaces andnonlinear iterations.Lecture Notes in Mathematics 1965. Berlin:Springer (ISBN 978-1-84882-189-7/pbk; 978-1-84882-190-3/e-book). xvii, 326 p. EUR 53.45(2009).This monograph gives an introduction to andoverview of the author’s extensive work on fixedpoint iterations. It consists of three parts. Part 1(Chapters 1 to 5) is dedicated to the geometricproperties of Banach spaces, namely, convexity,smoothness and the duality map.In Part 2 (Chapters 6 to 14), the author gives manyresults about fixed points of different classes ofmappings. He focuses the main attention on theiterative processes that converge to a fixed point.The celebrated Banach contraction theorem as-sures that the Picard iteration formula

x0 ∈ K, xn+1 = Txn (n ≥ 0)

provides a sequence which converges to theunique fixed point, if K is a complete metricspace and T : K→ K is contractive. If T is nonex-pansive, other iteration processes are considered.Under certain conditions, the Mann iteration for-mula

x0 ∈ K, xn+1 = (1 − αn)xn + αnTxn (n ≥ 0),

where (αn)n≥0 ⊂]0, 1[, limn→∞ αn = 0 and∑n≥0 αn = ∞, provides a sequence convergent to

a fixed point. Throughout the book, the authorconsiders different classes of operators: contrac-tive, nonexpansive, quasi-nonexpansive, asymp-totically regular, uniformly asymptotically regu-lar, etc.In Part 3 (Chapters 15 to 22), common fixed pointsfor (finite, countable) families of mappings arestudied. The final Chapter 23 presents a lot of thesame results on set-valued mappings, plus somegeneral comments, examples and open ques-tions.Each chapter contains a section of exercises andanother of historical remarks. The book endswith 561 references, of which 100 are of the au-thor, and a short index. It contains some minormistakes: for example, the Goldstine theorem iscalled Goldstein theorem.Almost all the theorems of this monograph aredue to the author and his collaborators.

Antonio Martinón (La Laguna)

1167.26001Ambrosio, Luigi; Malý, JanVery weak notions of differentiability.Proc. R. Soc. Edinb., Sect. A, Math. 137, No. 3,447-455 (2007).The authors study some weak notions of differ-entiability arising in connection with the spatialregularity of flows associated with non-smoothvector fields. The main difference from other sim-ilar concepts is that the convergence of differencequotients has to be understood as convergence inmeasure. So, the following definitions are intro-duced.Let Ω be an open subset of RN. A measurablefunction f : Ω→ R is called:– (Fréchet-)differentiable in measure if there ex-ists a measurable function g : Ω→ RN such that

limh→0

f (x + h) + f (x) − g(x) · h|h|

= 0


locally in measure with respect to x ∈ Ω,– Gateaux differentiable in measure if there existsa measurable function g : Ω→ RN such that

limδ→0

f (x + δy) + f (x) − δg(x) · yδ

= 0

for each y ∈ RN, locally in measure with respectto x ∈ Ω, and– directionally differentiable in measure if thereexists a measurable function W : Ω × RN

→ Rsuch that

limδ→0

f (x + δy) + f (x) − δW(x, y)δ

= 0

locally in measure with respect to (x, y) ∈ Ω×RN.The main theorem states that all these notionsof differentiability are equivalent. Moreover, theauthors show that classical approximate differen-tiability is stronger than differentiability in mea-sure. Uta Freiberg (Canberra)

1167.26002Lewis, Thomas M.A probabilistic property of Katsuura’s continu-ous nowhere differentiable function.J. Math. Anal. Appl. 353, No. 1, 224-231 (2009).Let f be H. Katsuura’s continuous nowheredifferentiable function [Am. Math. Mon. 98,No. 5, 411–416 (1991; Zbl 752.26005)]. Let γ =

((2/3)(1/3)(2/3))1/3, µ = − log2 γ, σ =√

2/3, and

∆k(x, h) = log2 | f (x + 3−kh) − f (x)| + kµ,

where (x, h) ∈ [0, 1)2 and k ≥ 0. Suppose that m1and m2 denote the linear and planar Lebesguemeasures, respectively.The main result of the paper states that:(a) For almost every (x, h) ∈ [0, 1)2 with respect tom2, we have

limk→∞

∆k(x, h)k

= 0.

(b) For all numbers a < b, as k→∞we have

m2

(x, h) ∈ [0, 1)2 :

∆k(x, h)

σ√

k∈ (a, b]

→

→1√

2π

∫ b

ae−x2/2 dx. (0.0)

(c) For almost every (x, h) ∈ [0, 1)2, with respectto m2 we have

lim supk→∞

∆k(x, h)

σ√

2k log log k= 1.

In the second step of the work, the author letsν(A) = m1x ∈ [0, 1] : f (x) ∈ A for each Borel setA ⊆ R1, and proves that ν([a, b]) ≤ 6(b−a)q, whereq = ln(2/3) ln(1/3) ≈ 0.369, and [a, b] ⊂ [0, 1].

Mehdi Hassani (Zanjan)

1167.26007Cattiaux, Patrick; Guillin, Arnaud; Wang, Feng-Yu; Wu, LimingLyapunov conditions for super Poincaré in-equalities.J. Funct. Anal. 256, No. 6, 1821-1841 (2009).In this interesting paper, the authors show howto use Lyapunov functions to obtain functionalinequalities which are stronger than Poincaréinequality (for instance logarithm Sobolev orF−Sobolev. They build Lyapunov functions, ei-ther as a function of the log-density or as a func-tion of the Riemannian distance.

Raouf Ghomrasni (Muizenberg)

1166.30011Rasila, AnttiIntroduction to quasiconformal mappings in n-space.Ponnusamy, S. (ed.) et al., Proceedings of the in-ternational workshop on quasiconformal map-pings and their applications, December 27,2005–January 1, 2006. New Delhi: Narosa Pub-lishing House (ISBN 81-7319-807-1/hbk). 239-260 (2007).These lecture notes are an introduction to thetheory of quasiconformal mappings in the n-dimensional Euclidean space Rn, where n ≥ 2.The focus is on distortion results and on classicaltechniques involving the conformal modulus ofa path family. Basic notations and results alongwith examples are given. For a thorough inves-tigation of these topics, see e.g. J. Väisälä [Lec-tures on n-dimensional quasiconformal map-pings. Berlin-Heidelberg-New York: Springer-Verlag. XIV, 144 p. (1971; Zbl 221.30031)] andthe recent survey of F. W. Gehring [Kühnau, R.(ed.), Handbook of complex analysis: geomet-ric function theory. Volume 2. Amsterdam: Else-vier/North Holland. 1–29 (2005; Zbl 1078.30014)].

Antti H. Rasila (Helsinki)

1166.30016Dyakonov, Konstantin M.


Meromorphic functions and their derivatives:equivalence of norms.Indiana Univ. Math. J. 57, No. 4, 1557-1571(2008).Let Hp = Hp(C+), 0 < p ≤ ∞, denote the Hardyspace of the upper half-plane C+ = z ∈ C : =z >0. A function θ ∈ H∞(C+) is an inner function if

limy→0+|θ(x + iy)| = 1 for almost all x ∈ R.

LetKpθ = Hp

∩ θHp, 1 < p < ∞,

denote the star-invariant subspace in Hp gener-ated by an inner function θ.In the paper under review the author studies thefollowing question: When are the Lp-norms of fand f ′ comparable for all f ∈ Kp

θ?The main results of the paper are Theorems 1.1and 1.3, and Propositions 1.2 and 1.4.Theorem 1.1. Let 1 < p < ∞ and let θ be an innerfunction. The following are equivalent:(i.1) The operator d/dx : Kp

θ → Lp(R) is an iso-morphism onto its range.(ii.1) θ′ ∈ L∞(R) and infφ ‖φ′‖∞ > 0, where φranges over the nonconstant inner divisors of θ.(iii.3) θ is a finite product of interpolatingBlaschke products, and

0 < inf=z : z ∈ θ−1(0) ≤

sup=z : z ∈ θ−1(0) < ∞.(1)

Recall that a sequence of points z j inC+ is calledan interpolating sequence if the restriction mapf 7→ f (z j), going from H∞ to `∞, is surjective.By Carleson’s interpolating theorem z j is an in-terpolating sequence if and only if

infj

∏k:k, j

∣∣∣∣∣∣z j − zk

z j − zk

∣∣∣∣∣∣ > 0.

An interpolating Blaschke product is a Blaschkeproduct whose zeros are simple and form an in-terpolating sequence.As a byproduct of the proof of Theorem 1.1 theauthor obtains the comparability of the norms‖ f ′‖Y and ‖ f ‖Y under condition (iii.1), for all ra-tional functions f ∈ Kp

θ in the case when Y iseither the analytic subspace of BMO(R) or theanalytic Lipschitz-Zygmund space on C+. Thisresult is stated as Proposition 1.2.

Recalling the M. Riesz decomposition Lp(R) =

Hp⊕ Hp for 1 < p < ∞, write P+ : Lp(R) → Hp

for the canonical projection that arises, and thendefine the Toeplitz operator Tϕ with symbolϕ ∈ L∞(R) by Tϕ f = P+(ϕ f ) for f ∈ Hp. Whenϕ ∈ H∞, Tϕ is called a coanalytic Toeplitz opera-tor.Theorem 1.3. Let 1 < p < ∞ and let θ be an innerfunction. The following are equivalent:(i.2) The operator d/dx : Kp

θ → Lp(R) is bounded,and there exists a function ψ ∈ H∞ such thatTψd/dx = I on Kθ (here I is the identity map).(ii.2) θ is an interpolating Blaschke product sat-isfying (1).Theorem 1.3 is supplemented with a weightednorm estimate for Kθ functions and their deriva-tives. This result is stated as Proposition 1.4 andit reads as follows: If 1 < p < ∞, the weight wsatisfies the Muckenhoupt (Ap) condition, and ifθ satisfies (ii.2), then

‖ f ′‖p,w =

(∫R

| f ′|pwdx) 1

p

is comparable with ‖ f ‖p,w for all rational func-tions f in K2

θ. Jouni Rättyä (Joensuu)

1167.30010Kühnau, ReinerQuadratic forms in geometric function theory,quasiconformal extensions, Fredhohn eigen-values.Agranovsky, Mark (ed.) et al., Complex anal-ysis and dynamical systems III. Proceedingsof the 3rd conference in honor of the retire-ment of Dov Aharonov, Lev Aizenberg, SamuelKrushkal, and Uri Srebro, Nahariya, Israel, Jan-uary 2–6, 2006. Providence, RI: American Math-ematical Society (AMS); Ramat Gan: Bar-IlanUniversity (ISBN 978-0-8218-4150-1/pbk). Con-temporary Mathematics 455. Israel Mathemati-cal Conference Proceedings, 237-256 (2008).The author gives a survey of the role of quadraticforms as functionals for extremal problems inGeometric Function Theory. This field is closelyconnected with the names of H. Grunsky, G. M.Golusin, and M. Schiffer. Many extremal prob-lems are discussed, including the author’s ex-tensive studies on Fredholm eigenvalues. Qua-sicircles, images of the unit circle under a quasi-conformal mapping of the plane, are in the centerof this study. It turns out, for instance, that thequasiconformal reflection coefficient of a qua-


sicircle is connected with the Fredholm eigen-value. Numerous open problems are pointed out.The study of quasicircles offers many challengingproblems that have been studied very recently bymany authors. See e.g. C. Bishop [Ill. J. Math. 51,No. 4, 1243–1248 (2007; Zbl 1153.30014)] and I.Prause [Comput. Methods Funct. Theory 7, No.2, 527–541 (2007)]. Matti Vuorinen (Turku)

1167.30014Kraus, Daniela; Roth, OliverCritical points of inner functions, nonlinearpartial differential equations, and an extensionof Liouville’s theorem.J. Lond. Math. Soc., II. Ser. 77, No. 1, 183-202(2008).The authors discuss a method for construct-ing holomorphic maps with prescribed criticalpoints via a study of the Gaussian curvatureequation

∆u = 4|h′(z)|2e2u, ∗

where h is a holomorphic function on a domainΩ ⊂ C.First they find a solution of (∗) in |z| < 1 withdegenerate boundary data u = +∞ on |z| = 1,where h(z) is an infinite Blaschke product; this isa special case of the Berger-Nirenberg problemin differential geometry, namely the question ofwhich functions κ : R→ R on a Riemann surfaceR arise as the Gaussian curvature of a conformalRiemannian metric λ(z)|dz| on R (see, for exam-ple, D. Hulin and M. Troyanov [Math. Ann. 293,No. 2, 277–315 (1992; Zbl 799.53047)] for refer-ences).Next, they establish an extension of Liouville’sclassical representation theorem for the solutionsof the Liouville equation ∆u = 4e2u to the moregeneral equation (∗).They are thus able to construct holomorphicmaps with prescribed critical points and spec-ified boundary behaviour. For instance, theyshow that for every Blaschke sequence z j in|z| < 1 there is always a (not necessarily unique)Blaschke product with z j as its set of criticalpoints.They also extend earlier work of F. Fournier andSt. Ruscheweyh [Proc. Am. Math. Soc. 127, No.11, 3287–3294 (1999; Zbl 923.30006) and Math.Proc. Camb. Philos. Soc. 130, No. 2, 353–364(2001; Zbl 973.30003)] and R. Kühnau [Mitt. Math.Semin. Giessen 229, 45–53 (1997; Zbl 910.30023)]to prove the following result. Let Ω ⊆ C bea bounded simply-connected domain, z1, . . . , zn

points in Ω, and φ : ∂Ω → R+ a continuousfunction. Then there exists a holomorphic func-tion f : Ω → D with critical points precisely atz j (counted with multiplicities) such that

limz→ξ| f ′(z)|/(1 − | f (z)|2) = φ(ξ), ξ ∈ ∂Ω.

Further, if g : Ω → D is another holomorphicfunction with these properties, then g = T f forsome conformal disk automorphism T.

D. A. Brannan (Milton Keynes)

1167.30016Hanche-Olsen, HaraldOn Goursat’s proof of Cauchy’s integral theo-rem.Am. Math. Mon. 115, No. 7, 648-652 (2008).The purpose of the present article is to point outthat the homotopy version of the Cauchy inte-gral theorem is easily derived directly by Gour-sat’s proof. A homotopy in a region Ω means acontinuous mapping σ : [0, 1] × [0, 1] → Ω. Thehomotopy is Lipschitz continuous if σ satisfiesthe Lipschitz condition. A parameterized path αin [0, 1]× [0, 1] is mapped to a path inΩ by σ α.The boundary α0 of [0, 1]× [0, 1] is parameterizedby the arc length, starting at (0, 0) and proceedingin the counterclockwise direction. Theorem 2: Letf be analytic in Ω. If σ is a Lipschitz continuoushomotopy, then ∫

σα0

f (z)dz = 0.

The usual homotopy forms of the Cauchy in-tegral theorem follow immediately from Theo-rem 2. Next, the author shows that the restrictionto Lipschitz continuous homotopies is harmless.Historical remarks are added at the end of thearticle. Dmitri V. Prokhorov (Saratov)

1167.31005Deblassie, DanteThe growth of the Martin kernel in a horn-shaped domain.Indiana Univ. Math. J. 57, No. 7, 3115-3130(2008).Let a and b be positive, Lipschitz continuousfunctions on [1,∞[ such that b < a. Consider thehorn-shaped domain

Ωa,b = (ρ, z) : ρ > 1, b(ρ) < z < a(ρ) × Sd−1,


where (ρ, z, φ) ∈ [0,∞[×R × Sd−1 are cylindri-cal coordinates in Rd+1. It was proved by H.Aikawa and M. Murata [J. Anal. Math. 69, 137–152(1996; Zbl 865.31009)] that all the Martin bound-ary points of (∆Rd+1 ,Ωa,b) are minimal, and thatthe Martin boundary is homeomorphic to the Eu-clidean boundary with an adjoined set Sd−1, pro-vided that ∫

∞

1

a(ρ) − b(ρ)ρ2 dρ < ∞.

For anyφ ∈ Sd−1, the corresponding (normalized)Martin kernel is denoted by Kφ(ρ, z, θ); note thatthe maximal growth of Kφ is in the direction of φ.The goal of this paper is to determine the growthrate, under the above conditions and the addi-tional condition that

limρ2>ρ1→∞

a(ρ2) − a(ρ1)ρ2 − ρ1

= 0 =

limρ2>ρ1→∞

b(ρ2) − b(ρ1)ρ2 − ρ1

.

It is shown that, for each η ∈ ]0, 1[,

limρ→∞

[∫ ρ

1

dtθ(t)

]−1

log Kφ(ρ, z, φ) = π,

uniformly for |(z−ψ(ρ))/θ(ρ)| < (1− η)/2, whereθ(ρ) = a(ρ) − b(ρ) and ψ(ρ) = (a(ρ) + b(ρ))/2.

Neil A. Watson (Christchurch)

1166.32003Bavrin, I.I.Integral representations of functions of severalcomplex variables. (English. Russian original)Dokl. Math. 75, No. 3, 395-398 (2007); translationfrom Dokl. Akad. Nauk, Ross. Akad. Nauk 414,No. 4, 439–442 (2007).The author presents an integral representationof holomorphic functions in a special class ofconvex, complete, bicircular domains in C2. Thisrepresentation expresses the values of a holomor-phic function in a domain of this class in termsof the values of the following linear differentialoperator

(Lz f )(z) =

f (z) + (z1 − z02)∂ f∂z2

(z) + (z2 − z02)∂ f∂z2

(z)

on the boundary. The integral representation inquestion implies the well-known Temlyakov in-tegral representations of the first, second andthird kinds.

Sergey M. Ivashkovich (Villeneuve d’Ascq)

1166.32010Fornæss, John Erik; Lee, LinaKobayashi, Carathéodory and Sibony metric.Complex Var. Elliptic Equ. 54, No. 3-4, 293-301(2009).The order of growth for the Sibony differentialmetric FS

Ω on a C2-smooth bounded domainΩ inCn is determined near a pseudoconcave bound-ary point and compared with that of the differ-ential metrics of Caratheodory and Kobayashi,which will be denoted by FC

Ω and FKΩ, respec-

tively. Let AΩ(P) denote for a point P ∈ Ω the setof all plurisubharmonic functions u : Ω→ [0, 1],such that u(P) = 0, and log u is also plurisubhar-monic. Then the metric FS

Ω is given by

FSΩ(P; X) := sup

√(∂∂u(P)X,X) |u ∈ AΩ(P),

P ∈ Ω, X ∈ Cn.

The result is as follows:Theorem: Let P ∈ ∂Ω be a non-pseudoconvexpoint and Pδ the point on the inner normal to ∂Ωat P with boundary distance δ and ν the innerunit normal at P. Then

FSΩ(Pδ, ν) ≈

1√δ, FC

Ω(Pδ, ν) ≈ 1.

Together with the estimate

FKΩ(Pδ, ν) ≈

1δ3/4

which is due to S. Krantz, it follows that the or-ders of growth for these differential metrics arepairwise different in this situation.The main lemma in the proof of the above theo-rem is the following estimate on FS

Ω for the ringdomain

Ωm := 14< |z|2 + |w|m < 3, m ≥ 2

at the point pδ := ( 12 + δ, 0), namely

FSΩm

(pδ,

(10

))≈

1

δ1− 1m

.


This result is generalized to the higher dimen-sional ring domains

Ω′ :=1

4< |z1|

2 + |z2|m2 + · · · + |zn|

mn < 1

2 ≤ m2 ≤ · · · ≤ mn

which serve as suitable local domains of compar-ison at pseudoconcave boundary points.

Gregor Herbort (Wuppertal)

1166.32011Fu, SiqiThe Kobayashi metric in the normal directionand the mapping problem.Complex Var. Elliptic Equ. 54, No. 3-4, 303-316(2009).Let Ω be a bounded domain in Cn with a C1,1-boundary. The paper under review deals withthe relationship between the growth order of theKobayashi differential metric FK

Ω and the pseudo-convexity of Ω. Let r denote a defining functionfor Ω. The following results are obtainedTheorem 1: If ∂Ω ∈ C1,1, then one has for anyX ∈ Cn:

FKΩ(p,X) &

|〈∂r(z),X〉|

|r(z)|12

.

It is also shown, that the growth rate 12 cannot

be improved without the assumption of pseudo-convexity, namelyTheorem 2: a) Assume thatΩ has a C2-boundary,and one has

FKΩ(p,X) &

|〈∂r(z),X〉||r(z)|α

with someα > 12 . ThenΩmust be pseudoconvex.

b) If Ω is pseudoconvex and ∂Ω ∈ C3, then oneeven has

FKΩ(p,X) &

|〈∂r(z),X〉|

|r(z)|23

.

The proofs of these results are based uponthe method of comparison domains. For thispurpose the Kobayashi metric on the non-pseudoconvex domains

Ωm :=|z| < R | Re zn < A(|z|m + |zn||z|)

and

Ωm :=z | |z| < R,Re+

B

n∑j=2

|z j|m + |zn||z|

<A|z1|

m

are studied, where z = (z1, . . . , zn−1), and A,B,R >0, and m ≥ 2.A key lemma is this:Lemma: a) If Λ(k) denotes the cone (0 < k < 1)

Λ(k) := z | |z| < R1, −Re zn > k|z|

then one has

FKΩm

(z; X) &|Xn|

d1− 1m

Ωm(z)

on Ωm ∩ Λ(k).b) If we additionally assume, with some K > 1,that |X| ≤ K|Xn|, then with suitable constantsC2,C3 > 0, the estimate

C2|Xn|

d1− 12m

Ωm(z)≤ FK

Ωm(z; X) ≤ C3

|Xn|

d1− 12m

Ωm(z)

holds.c) The Kobayashi metric on the domain Ωm can,non-tangentially, be estimated from above by

FKΩm

(z; X) ≤ C|Xn|

d1− 12m

Ωm(z),

whenever |X| ≤ K|Xn|.The article concludes with applications of theabove estimates to the question of Lipschitz con-tinuity for proper holomorphic mappings thatpreserve the complex normal direction.

Gregor Herbort (Wuppertal)

1167.32003Arcozzi, N.; Rochberg, R.; Sawyer, E.Carleson measures for the Drury-ArvesonHardy space and other Besov-Sobolev spaceson complex balls.Adv. Math. 218, No. 4, 1107-1180 (2008).Let Bn denote the unit ball in Cn. For 0 ≤ σ < ∞,1 < p < ∞, and m ≥ 0 an integer with m+σ > n/p,the analytic Besov spaces Bσp(Bn) are the sets ofholomorphic functions f on Bn satisfying


m−1∑k=0

| f (k)(0)|p+∫Bn

∣∣∣(1 − |z|2)n+σ f (m)(z)∣∣∣p dλn(z) < ∞,

(1)

where dλn(z) = (1−|z|2)−n−1dz is the invariant vol-ume measure on Bn. In the above, f (k) is the k-thorder complex derivative of f . The definition ofBσp is independent of the choice of m. With p = 2and σ = 1

2 one obtains the Drury-Arveson Hardyspace H2

n, whereas Bn/22 gives the ordinary Hardy

space H2.A measure µ on Bn is a Carleson measure for Bσpif

∫Bn| f (z)|p dµ ≤ Cµ‖ f ‖pBσp . In [Mem. Am. Math.

Soc. 859 (2006; Zbl 1112.46027)], the authors de-scribed the Carleson measures for B0

p(Bn) with1 < p < 2 + 1

n−1 . In the present paper, the authorsfocus their attention on Bσ2(Bn) with 0 ≤ σ ≤ 1

2 .With appropriate modifications, they show thatthe results forσ = 0 extend to the range 0 ≤ σ < 1

2 .Their characterization is given in terms of an in-equality for µ on the Bergman tree Tn associatedto the ball Bn. Specifically it is proved in the pa-per that the discrete tree condition∑

β≥α

[2σd(β)I∗µ(β)

]2≤ CI∗µ(α) < ∞,

α ∈ Tn,

(2)

characterizes the Carleson measures for Bσ2(Bn)with 0 ≤ σ < 1

2 . On the other hand, by results ofC. Cascante and J. Ortega [Can. J. Math. 47, No. 6,1177–1200 (1995; Zbl 845.46027)], there exist pos-itive measures µ that are Carleson for Bσ2 , σ ≥ 1

2 ,but fail to satisfy (2).For σ = 1

2 , the authors prove that the Carlesonnorm of µ, ‖µ‖Carleson, is comparable with the bestconsta nt C in the inequality∫

Bn

∫Bn

(Re

11 − w · z

)f (z) dµ(z)g(w) dµ(w)

≤ C‖ f ‖L2(µ)‖g‖L2(µ).

It is shown that when n > 1 the finiteness of‖µ‖Carleson is equivalent neither to the tree condi-tion (2), nor to the simple condition

2d(α)I∗(α) ≤ C, α ∈ Tn. (4)

The latter condition has been shown to be nec-essary when σ < 1

2 . The case σ = 12 requires

the introduction of additional structure on theBergman tree. This additional structure is then

used to prove that the Carleson measures for B122

are characterized by the condition (4) togetherwith the “split" tree condition∑

k≥0

∑γ≥α

2d(γ)−k∑

(δ,δ′)∈G(k)(γ)

I∗µ(δ)I∗µ(δ′) ≤

CI∗µ(α), α ∈ Tm.

Manfred Stoll (Columbia)

1167.32007Lieb, IngoThe Levi problem. (Das Levische Problem.)(German)Lieb, Ingo et al., Beiträge zur komplexen Anal-ysis. Bonn: Univ. Bonn, Mathematisches Insti-tut. Bonner Mathematische Schriften 387, 1-34(2007).In this interesting paper, a thorough study of theevolution of the Levi problem over almost onehundred years is presented. From the originalquestion of E. E. Levi concerning the domains ofholomorphy, the reader may follow the resultsobtained by different great mathematicians untilthe general formulation of today, namely, how isthe theory of functions in a domain influencedby the geometry of its border. Some biographicalnotes on the greatest names who worked on thesubject are also included. Reading this paper is amust for anyone interested in this subject.

Eugen Pascu (Montréal)

1167.32023Błocki, ZbigniewA gradient estimate in the Calabi-Yau theorem.Math. Ann. 344, No. 2, 317-327 (2009).This paper is a nice contribution to the theory ofcomplex Monge-Ampère equations on compactKähler manifolds. The setting is the same as inYau’s proof of the Calabi conjecture [S.-T. Yau,Commun. Pure Appl. Math. 31, 339–411 (1978;Zbl 369.53059)]: (M, ω) is a compact Kähler man-ifold and ϕ is a real function so that ω + i∂∂ϕis positive definite and with maxM ϕ = 0. Oneassumes that

(ω + i∂∂ϕ)n = fωn,


for some positive smooth function f with∫

M( f −1)ωn = 0, which is the complex Monge-Ampèreequation that arises in the Calabi conjecture. Inhis solution of the conjecture, Yau proved thatthe functionϕ has a priori bounds in all Ck norms,depending only on M, ω, f . To show this, he firstproved that the C0 norm of ϕ is bounded uni-formly, and then used this to derive a C2 uni-form bound. Standard interpolation estimatesthen provide a C1 bound on ϕ, but the questionremained whether one could derive a C1 boundon ϕ depending only on the C0 bound.The author previously worked on this prob-lem in [Math. Z. 244, No. 1, 153–161 (2003;Zbl 1076.32036)] where he proved such a boundif (M, ω) has nonnegative bisectional curvature(a rather stringent hypothesis). In this paper, theauthor generalizes his previous work and provesthe desired C1 bound on ϕ, which depends onlyon M, ω, f and a bound on maxM |ϕ|.This result has already had some interesting ap-plications, for example by the author as wellas D. H. Phong and J. Sturm to the problem ofgeodesics in the space of Kähler potentials. Theauthor promises further applications to complexHessian equations.

Valentino Tosatti (Cambridge)

1167.33002Elbert, Árpád; Laforgia, AndreaThe zeros of the complementary error function.Numer. Algorithms 49, No. 1-4, 153-157 (2008).The well-known complementary error functionerfc (z) is defined by

erfc (z) =2√π

∫∞

ze−s2

ds.

It is shown that erfc (z) has no zeros in the sector3π/4 ≤ arg z ≤ 5π/4.The authors establish this result by considera-tion of the two sectors 3π/4 ≤ arg z ≤ π andπ < arg z ≤ 5π/4. In the first sector, they writez = −X + iY, with X ≥ 0 and 0 ≤ Y ≤ X, and de-compose the integral into integrals taken alongthe straight line paths (z,−X), (−X, 0) and (0,∞).They show that the real part of the decomposedintegral is positive. Similar considerations withz = −X − iY are applied to the second sector.

R. B. Paris (Dundee)

41006Temlyakov, V.N.Relaxation in greedy approximation.Constr. Approx. 28, No. 1, 1-25 (2008).The author studies greedy algorithms in a Ba-nach space from the point of view of conver-gence and rate of convergence. There are twowell-studied approximation methods: the WeakChebyshev Greedy Algorithm (WCGA) and theWeak Relaxed Greedy Algorithm (WRGA). TheWRGA is simpler than the WCGA in the sense ofcomputational complexity. However, the WRGAhas limited applicability. It converges only forelements of the closure of the convex hull of adictionary. In this paper he studies algorithmsthat combine good features of both algorithms,the WRGA and the WCGA. In the constructionof such algorithms he uses different forms of re-laxation. First results on such algorithms havebeen obtained in a Hilbert space by A. Barron,A. Cohen, W. Dahmen, and R. DeVore. Their pa-per was a motivation for the research reportedhere. Francisco Pérez Acosta (La Laguna)

1166.42007Iosevich, A.; Rudnev, M.Freiman theorem, Fourier transform and addi-tive structure of measures.J. Aust. Math. Soc. 86, No. 1, 97-109 (2009).This very interesting paper is written so clear andnicely that the proper way to describe it seemsto combine the most impressive points from theauthors’ text. We omit quotation marks.The main result of the paper is a structural the-orem that relates Lp, p ≥ 4, estimates for theFourier transform of measures in Rd with trans-lation invariance properties of their supports. Alot of information about microstructure of thesupports is contained in the decay properties ofthe Fourier transforms. The main result of thepaper reads as follows. Let µ be a compactlysupported measure satisfying µ[Bδ(x)] ≈ δd/2 foreach x ∈ suppµ, where Bδ(x) is a ball of radius δcentered at x. Suppose that

∫t≤|ξ|≤2t

|µ(ξ)|2ldξ &∫

t≤|ξ|≤2t|µ(ξ)|2dξ ≈ td/2

for sufficiently large t and l ≥ 2. Then µ is arith-metic, that is, its support possesses an additivestructure.


This work is motivated by and closely related todistance problems. There are several interestingexamples in the paper.

Elijah Liflyand (Ramat-Gan)

1166.42018Bechler, PawełWavelet approximation of distributions withbounded variation derivatives.J. Fourier Anal. Appl. 15, No. 1, 31-57 (2009).The topic of this paper is a wavelet approxima-tion of distributions from the spaces BVr, whichare the spaces of distributions f such that thedistributional derivatives Dα f with |α| ≤ r existas measures of bounded variation. These spacesare a generalization of the classical BV spaces tohigher orders of smoothness. The main aim ofthis paper is to generalize as many of known re-sults about the Haar approximations as possibleto higher orders of smoothness of wavelet ap-proximations, where smoothness is measured inthe sense of BV. This paper discusses estimatesfor wavelet coefficients of BVr distributions, di-rect (Jackson) and inverse (Bernstein) inequal-ities for n-term approximation of elements ofBVr in the Lp spaces using compactly supportedwavelets. The optimal rates of approximation areestablished. The linear approximation in similarcontexts is considered for comparison.

V. Leontiev (Ul’yanovsk)

1167.43006Delmonico, CédricConvolution operators and homomorphisms oflocally compact groups.J. Aust. Math. Soc. 84, No. 3, 329-344 (2008).This paper is concerned with a mapping ofp-convolution operators induced by a contin-uous homomorphism ω : G → H when Gand H are locally compact groups. The resultsunify and generalize earlier works for locallycompact abelian groups by N. Lohoué [C. R.Acad. Sci., Paris, Sér. A 270, 589–591 (1970;Zbl 188.20302)], S. Saeki [Tohoku Math. J., II.Ser. 22, 409–419 (1970; Zbl 206.12601)], F. Lust-Piquard [Ann. Inst. Fourier 39, No. 4, 969–1006(1989; Zbl 675.43001)], and for particular ho-momorphisms by J.-P. Anker [Comment. Math.Helv. 58, 622–645 (1983; Zbl 535.22006)] as wellas K. de Leeuw [Ann. Math. (2) 81, 364–379 (1965;Zbl 171.11803)] or for the special case p = 2 by E.Bédos [Proc. Am. Math. Soc. 120, No. 2, 603–608(1994; Zbl 806.22004)] and by M. B. Bekka, E. Kani-

uth, A. T. Lau and G. Schlichting [Proc. Am. Math.Soc. 124, No. 10, 3151–3158 (1996; Zbl 861.43002)].Let 1 < p < ∞, let G and H be locally com-pact groups and ω : G → H a continuous ho-momorphism. For any such group X, let CVp(X)denote the Banach algebra of p-convolution op-erators, that is, the algebra of continuous opera-tors on Lp(X) commuting with left translation,equipped with the operator norm. Let λp

X bethe map from bounded measures to the corre-sponding p-convolution operators, which asso-ciates with a bounded measure µ the canonicalconvolution operator obtained with the modu-lar function and the reflection of the measure µabout the identity.The first main theorem is that given ω, G and Has above and G amenable, then there exists a lin-ear contraction, in an abuse of notation denotedω, from CVp(G) to CVp(H), which observes theintertwining relation

ω λpG = λp

H ω

on all bounded measures. The image of an op-erator T under such a contraction ω is uniquelydetermined for any convolution operator T withcompact support.As a consequence, if ω is the inclusion of Gd inG, where Gd is the group G with the discretetopology, and if Gd is amenable, then for any dis-crete measure µ, the operator norms of λp

G(µ) andof λp

Gd(µ) can be shown to be equal. Moreover,

the author constructs a linear contraction σ fromCVp(G) to CVp(Gd) which fixes convolution op-erators corresponding to all discrete measures.If G is amenable, then σ serves as a non-abelianversion of the map of p-multipliers from G to itsBohr compactification.Furthermore, in case G and H are abelian, the au-thor shows that if the Fourier transform T of ap-convolution operator T corresponding to a p-pseudomeasure (from the dual space of the Herzalgebra) is continuous, then one recovers a re-sult by Lohoué [loc. cit.]: (ω(T)) = T ω. Underthe same assumptions, Reiter’s relativisation ofthe spectrum [Theorem 7.2.2, in: H. Reiter andJ. Stegeman, Classical harmonic analysis and lo-cally compact groups, 2nd ed., Oxford: Claren-don Press (2000; Zbl 965.43001)] can be provedfor all continuous homomorphisms.

Bernhard Bodmann (Houston, TX)


1166.46003Martín, MiguelThe group of isometries of a Banach space andduality.J. Funct. Anal. 255, No. 10, 2966-2976 (2008).A subspace X of C(K) is said to be C-rich if, forevery nonempty open subset U ⊂ K and everyε > 0, there is an f ∈ SC(K) with support in U anddist( f ,X) < ε.It is shown that for every separable Banach spaceE there is a C-rich separable subspace X(E) ofC[0, 1] such that X(E)∗ contains an isometric copyof E∗ as an L-summand. The particular case ofX = X(`2) presents an example of a real Banachspace with a “poor" group of isometries Iso(X) (ithas no nontrivial one-parameter uniformly con-tinuous subgroup), but Iso(X∗) is “very big" (asbig as Iso(`2)). Vladimir Kadets (Kharkov)

1166.46007Kalenda, Ondrej F.K.Natural examples of Valdivia compact spaces.J. Math. Anal. Appl. 350, No. 2, 464-484 (2009).A compact space K is called Valdivia if it is,for a set Γ, homeomorphic to a subset K′ of RΓ

such that the set x ∈ K′ : γ ∈ Γ : x(γ) ,0 is countable is dense in K′. With the Val-divia compacta are associated spaces C(K) andBanach spaces X for which the dual unit ball isValdivia compact in the weak* topology, witha condition of linearity. The theory of Valdiviacompacta is presented in the author’s survey pa-per [O. F. K. Kalenda, Extr. Math. 15, No. 1, 1–85(2000; Zbl 983.46021)].The paper under review is a good addendumto this survey. It collects known and unknownexamples of Valdivia compact spaces, their con-tinuous images and associated classes of Ba-nach spaces which appear naturally in vari-ous branches of mathematics. The author de-scribes topological constructions which preserveand generate Valdivia compact spaces. Linearlyordered Valdivia compact spaces and Valdiviacompact groups are described. The associatedclasses of Banach spaces are considered: ordercontinuous Banach lattices, commutative andnoncommutative L1 spaces, dual C∗ algebras.Some interesting questions are posed. For exam-ple, does C(K), where K is a compact group, havea Markushevich basis? Now, it is known (in folk-lore) that yes. Anatolij M. Plichko (Krakow)

1166.46015Mockenhaupt, G.; Ricker, W.J.Optimal extension of the Hausdorff-Young in-equality.J. Reine Angew. Math. 620, 195-211 (2008).Let 1 ≤ p ≤ 2. The Hausdorff–Young inequal-ity establishes that the Fourier transform definesa continuous injective operator from Lp(T) into`p′ (Z). The authors consider the optimal exten-sion of the Hausdorff–Young inequality in thefollowing sense: Fixing the range space `p′ (Z),is it possible to extend the Fourier transform toa larger Banach function space Fp(T) preservingcontinuity in such a way that this extension isoptimal? Here, optimal means that this spacemust satisfy the following property: If Z is anyBanach function space over (T,B(T), dt) with σ-order continuous norm so that Lp(T) is contin-uously included in Z and the Fourier transformhas a continuous extension to Z, then Z is con-tinuously included in Fp(T). The answer to thisquestion is positive (Theorem 1.1), and the au-thors give three complementary descriptions ofthe space Fp(T), which is called the optimal latticedomain of the Fourier transform (Theorem 1.2).The first one – the space ∆p(T) – is given in termsof a sort of Köthe dual of the subspace of Lp′ (T)of functions which are the inverse of the Fouriertransform of some element of `p(Z). The secondone – the space Φp(T) – is given by the functionsf in L1(T) satisfying that the Fourier transformof fχA lies in `p′ (Z) for every A ∈ B(T). Thethird one – the space Γp(T) – is given in termsof the maps S f : L∞(T) → c0(Z) with f ∈ L1(T),defined as compositions of multiplication opera-tors and the Fourier transform. Finally, Theorem1.4 establishes that, for 1 < p < 2, the inclusionsLp(T) ⊂ Fp(T) ⊂ L1(T) are proper.It should be noted that Theorem 1.1 and Theorem1.2 solve the following question that was raisedsome forty years ago by R. E. Edwards: What canbe said about the family of functions in L1(T)having the property that the Fourier transform offχA, A ∈ B(T), lies in `p′ (Z)? The answer is thatthis is exactly the space Fp(T). A lot of relevantcomments are given. It should also be pointedout that, from the technical point of view, thisinteresting paper uses a recent development ofmathematical procedures based on vector mea-sures – the characterization of the optimal do-main for operators on Banach function spaces,see the references in the paper – for obtainingimportant results regarding classical problems in


harmonic analysis.Enrique Alfonso Sánchez-Pérez (València)

1166.46034Størmer, ErlingMultiplicative properties of positive maps.Math. Scand. 100, No. 1, 184-192 (2007).In this interesting note, the author studies theproperties of a weak∗-continuous positive linearmapping from a W∗-algebra M into itself thatmaps the unit 1M in M to itself. Denoting theJordan product

(a, b) 7→ a b = 12 (ab + ba),

the definite set Mϕ of ϕ, which is a Jordan subal-gebra of M, is defined by

Mϕ = a ∈M : ϕ(a a∗) = ϕ(a) ϕ(a)∗.

The W∗-algebra M is said to be ϕ-finite if thereexists a faithful family F of ϕ-invariant weak∗-continuous states on the W∗-algebra generatedby ϕ(M). For such a W∗-algebra M, let

Mϕ = a ∈Mϕ : ϕk(a) ∈Mϕ, k = 1, 2, . . . ,

and let

Cϕ =

∞⋂n=0

ϕn(Mϕ).

Then, Mϕ is a weakly closed Jordan subalgebra ofMϕ such that ϕ(Mϕ) is contained in Mϕ and Cϕ isa Jordan subalgebra of M such that ϕ(Cϕ) and Cϕcoincide. The main results of the paper are that,for each element a in M, every weak limit point ofthe sequence (ϕn(a)) lies in Cϕ, and if, for each el-ement p inF and each element b in Cϕ, if p(ab) isequal to zero then the sequence (ϕn(a)) convergesweakly to zero. The converse of the second state-ment holds when weak convergence is replacedby convergence in the strong∗-topology.In the final section of the paper, the author de-scribes the connections between his results andthose of W. Arveson [Int. J. Math. 15, No. 3, 289–312 (2004; Zbl 1065.46047)].

C. M. Edwards (Oxford)

1166.46035Bekjan, Turdebek N.Noncommutative maximal ergodic theoremsfor positive contractions.J. Funct. Anal. 254, No. 9, 2401-2418 (2008).Two of the recent major developments in oper-

ator ergodic theory are due to M. Junge and Q.–H. Xu [J. Am. Math. Soc. 20, No. 2, 385–439 (2007;Zbl 1116.46053)] and G. Cohen [J. Funct. Anal.242, No. 2, 658–668 (2007; Zbl 1128.47011)]. Thefirst one, the proof of the noncommutative max-imal ergodic inequality for (both averages andproducts of) positive contractions T : Lp(M) →Lp(M), p > 1, where M is a semifinite von Neu-mann algebra, is a major breakthrough whichpaved the way to noncommutative ergodic the-orems for operators beyond the ones consideredin the Lance–Yeadon line of results (see T. Fackand H. Kosaki [Pac. J. Math. 123, 269–300 (1986;Zbl 617.46063)], M. Goldstein [Funkts. Anal.Prilozh. 14, No. 4, 69–70 (1980; Zbl 467.46056)],F. J. Yeadon [J. Lond. Math. Soc., II. Ser 16, 326–332 (1977; Zbl 369.46061)], E. C. Lance [Invent.Math. 37, 201–214 (1976; Zbl 338.46054)], andM. Goldstein and S. N. Litvinov [Stud. Math. 143,No. 1, 33–41 (2000; Zbl 968.46049)]). The sec-ond one, the maximal inequality for the prod-ucts of conditional expectation operators onLp(X), 1 < p ≤ ∞, settled an old conjecture ofD. L. Burkholder on the a.e. convergence of theproducts of conditional expectation operators onLp.The main result of the article under considera-tion is a generalization of the maximal ergodicinequality (product version) of M. Junge and Q.–H. ,Xu to the setting where the numerical rangeof the operator T in L2(M) is contained in a Stolzregion of the form Dδ = z ∈ C : |1− z| < δ(1− |z|)for some δ > 0 :Theorem. Let T be a positive contraction on asemifinite von Neumann algebra M with a nor-mal faithful trace τ such that τ(T(x)) ≤ τ(x) forall x ∈ L1(M)∩M+. If the numerical range of T iscontained in a Stolz region Dδ with vertex 1, then

‖ supn

+Tn(x)‖2 ≤ C2,δ‖x‖2 for all x ∈ L2(M).

This result is also extended to Lp(M), 1 < p ≤ ∞.Then, via the method of B. Delyon and F. Delyon[Bull. Soc. Math. Fr. 127, No. 1, 25–41 (1999;Zbl 937.47004)], the maximal inequality of G. Co-hen is extended to the non-commutative setting.As application, some almost uniform and bilater-ally almost uniform convergence results for suchoperators are obtained.This is a well written paper with lucid expositionand clear arguments. Dogan Çömez (Fargo)


1167.46007Petsoulas, Giorgos; Raikoftsalis, TheocharisA Gowers tree like space and the space of itsbounded linear operators.Stud. Math. 190, No. 3, 233-281 (2009).W. T. Gowers [Trans. Am. Math. Soc. 344, No. 1,407–420 (1994; Zbl 811.46014)] gave the first ex-ample of a separable infinite-dimensional Ba-nach space, GT, so that all infinite-dimensionalsubspaces have non-separable dual and yet GTdoes not contain an isomorph of `1. In par-ticular, GT contains no isomorph of c0, `1,or an infinite-dimensional reflexive subspace.Other such examples appear in [S. A. Argyros,A. D. Arvantitakis, A. G. Tolias, London Mathe-matical Society Lecture Notes Series 337, 1–90(2006; Zbl 1131.46006)].In the present paper, the authors construct a vari-ant of GT called Xgt and prove that Xgt is heredi-tarily indecomposable and every operator on Xgtcan be written as λId + W (with W being weaklycompact and strictly singular). Moreover, Xgt is adual space and its predual has the same proper-ties. Furthermore, for every infinite-dimensionalsubspace Y of Xgt, Y∗ is non-separable, `2 embedsinto Y∗, and `2(R) embeds into Y∗∗.

Edward W. Odell (Austin)

1167.46021Adams, David R.My love affair with the Sobolev inequality.Maz’ya, Vladimir (ed.), Sobolev spaces inmathematics. I: Sobolev type inequalities.New York, NY: Springer; Novosibirsk: TamaraRozhkovskaya Publisher (ISBN 978-0-387-85647-6/hbk; 978-5-901873-24-3/hbk; 978-0-387-85648-3/e-book; 978-0-387-85791-6/set). Interna-tional Mathematical Series 8, 1-23 (2009).This is a very well written survey of the efforts infinding different versions of the Sobolev inequal-ity by the author and others. Historical notes andthe author’s personal recollections, together withmathematical explanations, make the text easilyunderstandable and picturesque.

Alexei Lukashov (Istanbul)

1167.46025Shvartsman, PavelThe Whitney extension problem and Lipschitzselections of set-valued mappings in jet-spaces.Trans. Am. Math. Soc. 360, No. 10, 5529-5550(2008).Let ω be a continuous, positive concave func-

tion defined on [0,∞[ satisfying ω(0) = 0. We letCk,ω(Rn) denote the space of all real functions fon Rn with continuous derivatives up to order kfor which the seminorm

p( f ) :=∑|α|=k

supx,y

|Dα f (x) −Dα f (y)|ω(‖x − y‖)

is finite, endowed with the natural norm. Theauthor studies the following variant of a classi-cal problem known in the literature as Whitney’sextension problem: Let S be a closed subset ofRn. Given a positive integer k and an arbitraryfunction f : S → R, what are necessary and suf-ficient conditions for f to be the restriction to Sof a function F ∈ Ck,ω(Rn)? An impressive break-through in the solution of this problem has beenmade by C. Fefferman in a series of articles [see,e.g., Ann. of Math. (2) 161, No. 1, 509–577 (2005;Zbl 1102.58005)].The main result of this paper is the followingTheorem. Let G be a mapping defined on a fi-nite subset S of Rn which assignes a convex setof polynomials G(x) ⊂ Pk of degree at most k ofdimension at most ` to every point x ∈ S. Sup-pose that, for every subset S′ of S consisting ofat most 2min(`+1,dimPk) points, there is a functionFS′ ∈ Ck,ω(Rn) with norm ≤ 1 and whose Tay-lor polynomial at x of degree k belongs to G(x)for all x ∈ S′. Then there is F ∈ Ck,ω(Rn) withnorm ≤ γ (a positive constant depending onlyon k,n and the cardinality of S) such that theTaylor polynomial of F at x belongs to G(x) forall x ∈ S. This is an extension of a result due toC. Fefferman [Rev. Mat. Iberoam. 21, No. 2, 577–688 (2005; Zbl 1102.58004)].The crucial ingredient of the present approach isan isomorphism between the restrictions of ele-ments of Ck,ω(Rn) to S and a certain space of Lip-schitz mappings from S into the space of poten-tial k-jets Pk ×R

n endowed with a certain metricdω. In this way, Whitney’s problem is reformu-lated as a problem about Lipschitz selections ofset-valued maps defined on S with values in thepower set of the space of potential k-jets. Themain theorem is equivalent to a Helly-type cri-terion for the existence of a Lipschitz selection,whose proof relies on methods and ideas devel-oped by the author for the case of set-valuedmappings taking their values in Banach spaces[see, e.g., P. Shvartsman, J. Geom. Anal. 12, No. 2,289–324 (2002; Zbl 1031.52004)]. As in this case,the new proof depends on Helly’s intersection


theorem and a combinatorial result on the struc-ture of finite metric graphs. José Bonet (Valencia)

1167.46036Pask, David; Rennie, Adam; Sims, AidanThe noncommutative geometry of k-graph C∗-algebras.J. K-Theory 1, No. 2, 259-304 (2008).The authors extend the construction, due to thetwo first named authors [J. Funct. Anal. 233,No. 1, 92–134 (2006; Zbl 1099.46047)], of semifi-nite spectral triples for graph C∗-algebras to thecase of k-graph C∗-algebras.The first important step is that of analysing theexistence of faithful semifinite traces on the C∗-algebra C∗(Λ) of a higher-rank graph Λ. For tech-nical reasons, the graphs are assumed to be lo-cally convex and row-finite. Then such tracesare classified by so-called k-graph traces whichin turn are functions g from the set Λ0 of ver-tices to the positive reals satisfying the conditiong(v) =

∑λ∈vΛ≤n g(s(λ)) for v ∈ Λ0 and n ∈ Nk.

This classification is obtained via a faithful con-ditional expectationΦ from C∗(Λ) onto the maxi-mal abelian subalgebra of the fixed point algebraF under the gauge-action γ.To construct the spectral triple, first a C∗-moduleX over F is constructed, alongside with a Kas-parov module (which is even if and only if k isan even integer). Then a semifinite spectral triplefor locally convex k-graph C∗-algebras which ad-mit a faithful, semifinite, lower-semicontinuous,gauge invariant trace is constructed, see Sec-tion 6. The spectral triple is shown to be (k,∞)-summable relative to a canonical choice of a vonNeumann algebra with faithful semifinite nor-mal trace. In Theorem 7.1, the authors establishan index theorem for their spectral triple. An ex-ample is presented which illustrates that the in-formation coming from the semifinite index ismore refined than in the case of the usual Fred-holm index.In an appendix, necessary and sufficient condi-tions are presented for a k-graph to admit a faith-ful graph trace. Nadia Larsen (Oslo)

1167.46042Arveson, WilliamQuantum channels that preserve entanglement.Math. Ann. 343, No. 4, 757-771 (2009).Let M and N be full matrix algebras. A unitalcompletely positive (UCP) map φ : M → N(quantum channel) preserves entanglement if its

inflation φ ⊗ IdN : M ⊗ N → N ⊗ N maps ev-ery maximally entangled pure state ρ (or everystate of maximal Schmidt rank, which are calledmarginally cyclic in this paper) of N ⊗N, into anentangled state of M ⊗N.In Section 2, complementing a result of M.Horodecki, P. W. Shor, and M. B. Ruskai [Rev. Math.Phys. 15, No. 6, 629–641 (2003; Zbl 1080.81006)],the author shows that every UCP map that is notentanglement breaking must preserve entangle-ment. Further, the parametrization of states givenby W. Arveson [The probability of entanglement,arXiv:0712.4163 (2007)] can be appropriatelyadapted to UCP maps so as to make the spaceΦr of all UCP maps φ : B(K) → B(H) of rank≤ r into a compact probability space that carriesa unique invariant probability measure Pr, and itis shown in Section 3 that Pr is concentrated onthe set of maps of rank r. Thus, the probabilityspace (Φr,Pr) represents choosing a UCP map ofrank r at random. A zero-one law for channelsis proven which expresses in probabilistic termsthe dichotomy that a UCP map either preservesentanglement or is entanglement breaking.The main results of W. Arveson [loc. cit.] are thenapplied in Sect. 4 to show that there are plentyof entanglement preserving UCP maps of everypossible rank, and that almost surely every UCPmapΦr : B(K)→ B(H) of rank r ≤ n/2 preservesentanglement. Here n = dim H,m = dim K, andit is assumed that n ≤ m. Moreover, for everyr = 1, 2, . . . ,mn, the set of entanglement preserv-ing UCP maps of rank r is a relatively open subsetof Φr of positive measure and for the maximumrank r = mn its probability is strictly between 0and 1.Sect. 5 concludes with a discussion of extremepoints of the convex set of UCP maps that im-plies: for every integer r satisfying 1 ≤ r ≤ n, theextremals of rank r constitute a relatively opendense set having probability 1. There are no ex-tremal UCP maps φ : B(K) → B(H) of rank > n.Thus, whenever an extremal UCP map of rankr exists, then almost surely every UCP map ofrank r is extremal. Alexandr S. Holevo (Moskva)

1166.47005Gheorghe, DanaA Kato perturbation-type result for open linearrelations in normed spaces.Bull. Aust. Math. Soc. 79, No. 1, 85-101 (2009).Given two normed spaces X,Y, any linear sub-space Z ⊂ X×Y is said to be a linear relation. One


usually sets D(Z) = x ∈ X | ∃y ∈ Y : (x, y) ∈ Z,R(Z) = y ∈ Y | ∃x ∈ X : (x, y) ∈ Z, N(Z) = x ∈X | (x, 0) ∈ Z, M(Z) = y ∈ Y | (0, y) ∈ Z. Whendim(N(Z)) < ∞ and codim(R(Z)) < ∞, the num-ber ind(Z) = dim(N(Z)) − codim(R(Z)) is said tobe the topological index of Z. The relation Z issaid to be open if there exists ρ > 0 such thatρBY ⊂ y ∈ Y | ∃x ∈ BX ∩D(Z) : (x, y) ∈ Z, whereBX,BY are the closed unit balls in X,Y, respec-tively.The main result of the paper is the follow-ing Theorem. Let X,Y be normed spaces andlet Z1 ⊂ X × Y be an open relation satisfyingdim(N(Z1)) < ∞ and codim(R(Z1)) < ∞. Thenthere exists an ε > 0 such that for any linearrelation Z2 ⊂ X × Y with M(Z1) = M(Z2) andδ(Z1,Z2) < ε one has dim(N(Z2)) ≤ dim(N(Z1)),codim(R(Z2)) ≤ codim(R(Z1)) and ind(Z2) ≤ind(Z1). In addition, ind(Z2) = ind(Z1) if and onlyif Z2 is open.Here, δ(Z1,Z2) denotes the gap between theclosed subspaces Z1,Z2.

Florian-Horia Vasilescu (Villeneuve d’Ascq)

1166.47024Xia, JingboOn certain quotient modules of the Bergmanmodule.Indiana Univ. Math. J. 57, No. 1, 545-575 (2008).Let Bn denote the open unit ball of Cn. TheBergman space L2

a of the holomorphic functionsthat are square integrable for the Lebesgue mea-sure on Bn is a Hilbert module over the poly-nomial algebra C[z1, . . . , zn]. Given a subset L ofBn, the functions of L2

a which vanish on L form asubmoduleM(L) of L2

a and the quotient L2a/M(L)

is naturally identified with the Hilbert subspaceQ(L) generated by the kernel functions at thepoints of L. For f a continuous function on Bn,let SL

f denote the compression to Q(L) of the op-erator of multiplication by f on the larger spaceL2 and let S(L) denote the C∗-algebra generatedby these compressions.The author proves that, for each 1 ≤ p < n,there exists a subset L such that the commu-tators [(SL

z j)∗,SL

zi] belong to the t-Schatten class

for every 1 ≤ i, j ≤ n and t > p and the self-commutators of the SL

zido not belong to the p-

Schatten class. Moreover, the operation of restric-tion to the boundary ∂Bn on C(Bn) defines a C∗-isomorphism between the quotient of S(L) by the

ideal of the compact operators and C(∂Bn). Inthe last section, it is proved that there exists aset L such that all the commutators [(SL

z j)∗,SL

zi] be-

long to the trace class. The sets L are obtainedby choosing maximal countable subsets of Bn inwhich the kernel functions make large enoughangles with each other so that the projection ontoQ(L) can be approximated by the sum of the one-dimensional projections onto the lines they gen-erate.This interesting work is a contribution to thestudy of questions raised by W. Arveson [J. Oper.Theory 54, No. 1, 101–117 (2005; Zbl 1107.47006)]and R. Douglas [J. Oper. Theory 55, No. 1, 117–133(2006; Zbl 1108.47030)]. It also contains a discus-sion of a problem proposed by W. Arveson in theMyhill Lectures given in SUNY Buffalo in April2006. Antonio Serra (Lisboa)

1166.47031Avila, Artur; Jitomirskaya, SvetlanaThe Ten Martini problem.Ann. Math. (2) 170, No. 1, 303-342 (2009).From the Introduction: In this paper, we solvethe Ten Martini Problem as stated in [B. Simon,in: Mathematical physics 2000. Internationalcongress, London, GB, 2000 (London: ImperialCollege Press), 283–288 (2000)].Theorem. The spectrum of the almost Mathieuoperator is a Cantor set for all irrational α andfor all λ , 0.The almost Mathieu operator is the Schrödingeroperator on `2(Z),

(Hλ,α,θu)n = un+1 + un−1 + 2λ cos 2π(θ + nα)un,

where λ, α, θ ∈ R are parameters (called the cou-pling, frequency, and phase, respectively), and oneassumes that λ , 0. The interest in this partic-ular model is motivated both by its connectionsto physics and by a remarkable richness of therelated spectral theory. This has made the lat-ter a subject of intense research in the last threedecades.If α =

pq is rational, it is well-known that the spec-

trum consists of the union of q intervals calledbands, possibly touching at the endpoints. In thecase of irrational α, the spectrum Σλ,α (which inthis case does not depend on θ) has been conjec-tured for a long time to be a Cantor set. To provethis conjecture has been dubbed The Ten Mar-tini Problem by B. Simon [op. cit.]. For a historyof this problem, see [Y. Last, in: Sturm–Liouville


theory. Past and present (Basel: Birkhäuser), 99–120 (2005; Zbl 1098.39011)].

1167.47013Davidson, Kenneth R.; Paulsen, Vern I.; Raghu-pathi, Mrinal; Singh, DineshA constrained Nevanlinna–Pick interpolationproblem.Indiana Univ. Math. J. 58, No. 2, 709-732 (2009).The classical Nevanlinna-Pick problem consistsof finding a function f analytic in the unit diskD, with f (z j) = w j, j = 1, 2, . . . ,n, and ‖ f ‖∞ ≤ 1,where z1, . . . , zn are distinct points in D, andw1, . . . ,wn are complex numbers. In the presentpaper, the authors study this problem with anadditional constraint, namely, f ′(0) = 0.Two different sets of necessary and sufficient con-ditions for the problem to have a solution areobtained. Denote

φλ(z) =z − λ

1 − λz,

Kα,β(z) = (α + βz)(α + βw) +z2w2

1 − zw.

Theorem. The Nevanlinna–Pick problem withthe constraint f ′(0) = 0 has a solution if and onlyif one of the following two conditions holds.(1) There exists λ ∈ D such that the matrixz2

i z2j − φλ(wi)φλ(w j)

1 − ziz j

≥ 0 .

(2) For all α, β with |α|2 + β|2 = 1, the matrix[(1 − wiw j) Kα,β(zi, z j)

]≥ 0 .

Leonid Golinskii (Kharkov)

1167.47017Gleason, Jim; Rosentrater, C.RayXia’s analytic model of a subnormal operatorand its applications.Rocky Mt. J. Math. 38, No. 3, 849-889 (2008).The present paper gives a nice exposition onsubnormal operators, presenting in an attrac-tive form the analytic model of D.–X. Xia [Inte-gral Equations Oper. Theory 10, 258–289 (1987;Zbl 645.47020)] and its applications. New proofsof some results are given and many examples tohelp clarify the model are provided. Each sectionis concluded with a paragraph of notes and openproblems, which gives some additional back-

ground to the topics covered in the section andsome related open problems.One of the basic results in operator theory is thespectral theorem for normal operators. Charac-terizations of such a type have been given for thetheory of Hilbert space contractions. In the sec-ond section of the paper, the authors review theeffort made towards this end for subnormal oper-ators. Xia’s analytic model is presented and someof the complete unitary invariants for pure sub-normal operators are described. The third mainsection of the paper explores these unitary in-variants when the self-commutator is of finiterank. A study of the one-dimensional and thetwo-dimensional case is made, and a classifica-tion of subnormal operators of finite type is pre-sented. Ilie Valusescu (Bucuresti)

1167.47020Ueki, Sei-Ichiro; Luo, LuoCompact weighted composition operatorsand multiplication operators between Hardyspaces.Abstr. Appl. Anal. 2008, Article ID 196498, 12 p.(2008).For N ∈N, let BN be the unit ball of CN. For eachp, 1 ≤ p < ∞, the Hardy space Hp(BN) consists ofthe holomorphic functions f in the ball such that

sup0<r<1

∫∂BN

| f (rζ)|p dσ(ζ) =∫∂BN

| f ∗(ζ)|p dσ(ζ) = ‖ f ‖p < ∞,

where dσ is the normalized Lebesgue measureon the boundary of BN and f ∗ is the radial limitof f , which exists for almost every ζ ∈ ∂BN.Given a holomorphic self-map ϕ of BN and aholomorphic map u in BN, the weighted com-position operator MuCϕ is defined by MuCϕ f =u( f ϕ), where f is holomorphic.Let X and Y be Banach spaces and T be a boundedlinear operator from X into Y. The essential norm‖T‖e,X→Y is the distance from T to the set of com-pact operators from X into Y.The pull-back measure µϕ,u induced by the self-map ϕ and u ∈ Hq(BN) is the finite positive Borelmeasure on BN defined by

µϕ,u(E) =

∫ϕ∗−1(E)

|u∗|q dσ,


for all Borel sets E. Notice that ϕ∗ is a map of∂BN into BN. For each ζ ∈ ∂BN and t > 0, let theCarleson S(ζ, t) be z ∈ BN : |1 − 〈z, ζ〉| < 1.Many authors have studied weighted composi-tion operators on different holomorphic func-tion spaces. M. D. Contreras and A. Hernandez–Díaz [J. Math. Anal. Appl. 263, No. 1, 224–233(2001; Zbl 1026.47016); Integral Equations Oper.Theory 46, No. 2, 165–188 (2003; Zbl 1042.47017)]characterized the compactness of MuCϕ fromHp(B1) into Hq(B1) with 1 < p ≤ q < ∞ in terms ofthe pull-back measure, but they didn’t estimate‖MuCϕ‖e,Hp→Hq .The authors’ main result is as follows. Let 1 <p ≤ q < ∞. If MuCϕ is a bounded weighted com-position operator from Hp(BN) into Hq(BN), then

‖MuCϕ‖e,Hp→Hq ∼

lim sup|w|→1−

∫∂BN

|u∗(ζ)|q(

1 − |w|2

|1 − 〈ϕ∗(ζ),w〉|

)qN/p

dσ(ζ)

∼ lim supt→0

supζ∈∂BN

µϕ,u(S(ζ, t))tqN/p .

The notation ∼means that the ratio of two termsare bounded below and above by constants de-pendent on the dimension N and other parame-ters.They also show that a multiplication operator Mufrom Hp(BN) into Hq(BN) with 1 < p ≤ q < ∞ iscompact if and only if u = 0.

Héctor N. Salas (Mayagüez)

1167.47023Dyakonov, Konstantin M.Toeplitz operators and arguments of analyticfunctions.Math. Ann. 344, No. 2, 353-380 (2009).For a Toeplitz operator Tϕ with symbol ϕ ∈L∞(T), let KerpTϕ denote its kernel on the Hardyspace Hp, 1 ≤ p ≤ ∞, of the unit circle T. Re-lations between KerpTϕ and the smoothness ofϕ are studied. Let Λα (resp., Aα), 0 < α < ∞,denote the Lipschitz (Hölder-Zygmund) space(resp., its subspace of analytic functions). Sup-pose that β = α− 1/p and N is the integral part of1/(αp). Let us formulate two sample results forunimodular symbolsϕ ∈ Λα. If 1/p < α < ∞, thenKerpTϕ ⊂ Aβ and ‖ f ‖Λβ ≤ const ‖ϕ‖Λα‖ f ‖p for eachf ∈ KerpTϕ. If 0 < α < ∞, then KerpTϕ ⊂ Aα and‖ f ‖Λα ≤ const ‖ϕ‖N+2

Λα ‖ f ‖p for every f ∈ KerpTϕ.The constants in the above inequalities are inde-

pendent of f and ϕ. Similar and more general re-sults for various smoothness classes are obtained,and several approaches are discussed.

Alexei Yu. Karlovich (Lisboa)

1167.47026Widom, HaroldAsymptotic of a class of operator determinantswith application to the cylindrical Toda equa-tions.Baik, Jinho (ed.) et al., Integrable systems andrandom matrices. In honor of Percy Deift. Con-ference on integrable systems, random matri-ces, and applications in honor of Percy Deift’s60th birthday, New York, NY, USA, May 22–26,2006. Providence, RI: American MathematicalSociety (AMS) (ISBN 978-0-8218-4240-9/pbk).Contemporary Mathematics 458, 31-53 (2008).The paper is a continuation of results by theauthor and C. A. Tracy [Commun. Math. Phys.190, No. 3, 697–721 (1998; Zbl 907.35125)] on theasymptotics of certain solutions to the cylindricalToda equations

q′′k (t) + t−1q′k(t) = 4(expqk(t) − qk−1(t)−expqk+1(t) − qk(t)),

where k runs through Z. These solutions are ex-pressed in terms of determinants det(I + Kk(t)),where Kk(t) denotes certain integral operatorsacting on L2(R+). Generalizations to a wider classof functions arising as operator determinantshave been given in [H. Widom, Operator Theory:Advances and Applications 170, 249–256 (2006;Zbl 1123.47027)]. So far, all results were restrictedto the case where the symbol of a convolutionoperator associated to K0(t) does not vanish andhas zero index. This condition is what the authorcalls the regular case and it leads to asymptoticsof the form det(I + K0(t)) ∼ bta as t → 0+. Bothconstants a and b can be expressed by integralformulas.In the present paper, singular cases are alsotreated. Here, the symbol has a double zeroat a single point and an additional logarith-mic factor appears in the asymptotic expansion:det(I + K0(t)) ∼ bta log t−1. These results are spe-cialized to the most interesting case of the n-periodic Toda kernels in that qk+n = qk for allk. Then a and b can be calculated more explicitlyin terms of the zeros of an associated functionand Barnes G-function for both the regular andthe singular case.


The first part of the paper reviews the regularcase in a more general setting than before. Asa main tool for calculating a formula for thesmall time asymptotics of det(I + Kk(t)), the Kac–Achieser theorem together with a clever matrixdecompositions of the operators is used. Thenall expressions appearing in this formula can beobtained in a more explicit way at least in then-periodic case.In the second part, the singular situation is ana-lyzed. The proofs follow the same line as in theregular case, using approximations and the the-ory of Wiener–Hopf operators. However, a care-ful analysis is needed, the assumptions on the in-tegral operators have to be strengthened, and theL2-spaces before have to be replaced by weightedL2-spaces.As an application, the asymptotics of a solutionsto the cylindrical sinh-Gordon equation and the

Bullough–Dodd equation are discussed in theend of the paper. Wolfram Bauer (Greifswald)

1167.47056Benchohra, Mouffak; Hamidi, Naima; Ntouyas,Sotiris K.An asymptotic stable result of solutions for anintegral equation of mixed type on time scales.Commun. Appl. Nonlinear Anal. 14, No. 4, 35-45 (2007).The authors consider an integral equation ofmixed type with a Volterra integral operator anda Urysohn integral operator. They discuss the ex-istence of asymptotically stable solutions for thestudied integral equations on time scales. Newresults are obtained by using a generalization toFréchet spaces of a Krasnosel’skij type fixed pointtheorem. Lechoslaw Hacia (Poznan)

76 Differential, Difference and Integral Equations.

Differential, Difference and Integral Equations.

1167.35001Shargorodsky, E.; Toland, J.F.Bernoulli free-boundary problems.Mem. Am. Math. Soc. 914, 70 p. (2008).A Bernoulli free-boundary problem is one offinding domains in the plane on which aharmonic function simultaneously satisfies lin-ear homogeneous Dirichlet and inhomogeneousNeumann boundary conditions. The boundaryis called “free" because it is not prescribed a pri-ori, and the problem of determining free bound-aries from the given data is nonlinear. The nameBernoulli was originally associated which suchproblems in hydrodynamics. Questions of exis-tence, multiplicity or uniqueness, and regularityof free boundaries for prescribed data are veryimportant and their solutions lead to nonlinearproblems. In the booklet an equivalence is shownbetween Bernoulli free-boundary problems anda class of equations for real-valued functions ofone real variable. Since no restriction is imposedby the authors on the amplitudes or shapes offree boundaries, this equivalence is global, andvalid even for very weak solutions.Furthermore it is shown in the paper that theabove equivalence can be expressed as nonlin-ear Riemann-Hilbert problems and the theory ofHardy spaces in the unit disc plays a central role.Also, they have gradient structure and their solu-tions are critical points of a natural Lagrangian.Therefore, the Calculus of Variations can success-fully be applied as a tool for the study. Somerather natural conjectures about the regularity offree boundaries remain unresolved.Jürgen Socolowsky (Brandenburg an der Havel)

1167.35038Ludu, AndreiNonlinear waves and solitons on contours andclosed surfaces.Springer Series in Synergetics; Springer Com-plexity. Berlin: Springer (ISBN 978-3-540-72872-6/pbk). xx, 466 p. EUR 89.95/net; SFR 157.00;$ 119.00; £ 69.00 (2007).It is extremely difficult to comprise in a singlebook the study of nonlinear equations model-ing the nature surrounding us even when con-centrating on a special phenomenon. However,

by choosing models of nonlinear effects that oc-cur mostly on closed, compact curves and sur-faces, the author succeeds in writing a mono-graph which introduces the physics of solitonson compact systems to readers who may not haveany such prior knowledge. Yet, one does not re-main with the feeling that the book lacks eitherrigor or substance. The text is suitable for a grad-uate course on special topics or it can be used byreaders with various backgrounds and interestswho simply want to understand the connectionsbetween geometry and the phenomena of non-linear waves.Chapters 1–3 offer an introduction, first to the no-tion of soliton, then to a number of elementarynotions from general topology, followed by basicrepresentation formulas (Cauchy, Green, Stokes)relying on boundary values. A soliton is un-derstood from the perspective of mathematicalphysics as a solitary wave solution of a nonlin-ear evolutionary system. Asymptotically it pre-serves its shape and velocity against interactionswith other type solutions of the system or othertype of localized disturbances.The remainder of what is called Part I: “Mathe-matical prerequisites” is the most geometric sec-tion of the book. In fact, it consists of a nice in-troduction to vector fields, differential forms andderivatives (Chapter 4), followed by the differ-ential geometry of curves (Chapter 5). There isa regrettable typo in Theorem 12, section 5.2,where the well-known isoperimetric inequalityfor a simple, closed, planar curve or length Land area of the enclosed domain A should readL2≥ 4πA. In Chapter 6, we encounter first exam-

ples of solitons in various examples of nonlinearkinematics of two-dimensional curves. Chapters7 and 8 generalize the treatment of curves to sur-faces. We see elements of the differential geome-try of surfaces, surface differential operators (gra-dient, divergence, Laplacian, curl) and elementsof dynamics of moving surfaces.The most extensive part, Part II: “Solitons andnonlinear waves on closed curves and surfaces”,focuses on the fundamentals of nonlinear hy-drodynamics. Giving both frameworks of fluiddynamics (Lagrangian and Eulerian), the authorcontinues with the Eulerian approach which de-scribes the motion of a fluid from a stationary

Differential, Difference and Integral Equations. 77

lab frame by employing appropriate geometri-cal tools. The description is quite detailed andvery welcome for anybody unfamiliar with theset up. We understand why the early treatmentof the boundary is essential, as Chapter 10 looksat the dynamics of the fluid confined in a com-pact domain with free boundaries. Chapters 11and 12 presents examples of nonlinear evolutionequations (KdV and modified KdV equation forexample) in one dimension respectively two di-mensions (flattened droplets). We then pass nat-urally to nonlinear waves in three dimensions(Chapter 13) through the study of shape oscilla-tions of drops. Chapter 14 (Other Special Non-linear Compact Systems) stands a bit apart fromthe rest. It concludes, even if rather briefly, thispart of the book providing a different geometricapproach to predict compact solitons.The final part, Part III: “Physical nonlinear sys-tems at different scales”, is devoted to selectedphysical applications. Among them is the ap-plication of theory of nonlinear integrable sys-tems on free one-dimensional systems to the ex-istence of solitons on filaments (Chapter 15). Fur-ther, applications of soliton theory to nuclei orquantum Hall liquids (Chapter 16) and a descrip-tion of nonlinear modes in neutron stars (Chap-ter 17). The text concludes with a mathemati-cal annex which contains among other thingsthe one-soliton solutions of the KdV and MKdVequations and nonlinear dispersion relation ap-proach. The latter provides information on therelations between, say for example, amplitudeand speed of a soliton solution without actuallysolving the equation. Alina Stancu (Lowell)

1167.37001Gerdjikov, Vladimir Stefanov; Vilasi, Gaetano;Yanovski, Alexandar BorissovIntegrable Hamiltonian hierarchies. Spectraland geometric methods.Lecture Notes in Physics 748. Berlin: Springer(ISBN 978-3-540-77053-4/hbk). xii, 653 p.EUR 96.25 (2008).This monograph aims to discuss about theinverse scattering transform and the bi-Hamiltonian theory of soliton equations in bothanalytical and geometric approaches. All materi-als on those two important topics in soliton the-ory are divided into two corresponding parts.Generally speaking, the first part consisting ofChapters 1–9 deals with the inverse scatteringtransform – the spectral method to solve non-

linear equations, and the second part consistingof Chapters 10–16 deals with the Hamiltoniantheory laying a foundation of integrability – thegeometric theory of recursion operators. Otherrelated interesting topics discussed include theclassical r-matrix method, the so-called linearbundles of Lie algebras, and the Hamiltonian dy-namics with fermionic variables.Here is a sketchy summary of Part 1. Chap-ter 1 gives a brief historical review and funda-mental properties of soliton equations. Chap-ter 2 outlines the Ablowitz-Kuap-Newell-Segurmethod of solving soliton equations. Chapter3 is devoted to the direct scattering problemfor the Zakharov-Shabat system. Chapter 4 pri-marily outlines the classical approach based onthe Gelfand-Levitan-Marchenko equation, theRiemann-Hilbert method, and the Zakharov-Shabat dressing method for solving the inversescattering problem of the Zakharov-Shabat sys-tem. Chapter 5 mainly shows that the mappingof the potential of the spectral operator ontothe minimal sets of scattering data is one-to-one. Chapter 6 shows how the expansions overthe so-called squared eigenfunctions can be usedto analyze solvable soliton equations associatedwith the Zakharov-Shabat system. Chapter 7explains how to transform those solvable soli-ton equations into infinite-dimensional Hamil-tonian systems. Chapter 8 deals with the gauge-equivalence between the solvable soliton equa-tions. Chapter 9 discusses about a modern ap-proach to Hamiltonian systems using the classi-cal r-matrix theory.Here is a sketchy summary of Part 2. Chapter 10gives a brief introduction to the geometric the-ory of Hamiltonian systems of soliton equations.Chapter 11 lists some facts from differential ge-ometry and introduces the basic notation. Chap-ter 12 discusses how to define Poisson bracketsbased on the existences of symplectic structuresand Poisson structures on manifolds and sub-manifolds. Chapter 13 develops new mathemat-ical tools to study mixed tensor fields on Poissonmanifolds including Nijenhuis tensors. Chapter14 discusses about relations between the inte-grability problem and the Nijenhuis structure.Chapter 15 is entirely devoted to the geometrictheory of soliton equations associated with theZakharov-Shabat system. Chapter 16 considersthe linear bundles of Lie algebras, which lead tocompatible Poisson tensors in a straightforwardway.


The book is well organized and clearly written. Inparticular, the authors provide informative andhelpful comments, together with bibliographicalreview, at the end of each chapter. The interestedreaders, therefore, can easily find various rele-vant theories and their references. It is an excel-lent research monograph summarizing and eval-uating our growing body of research on solitontheory and integrable Hamiltonian systems.

Ma Wen-Xiu (Tampa)

Mentzen, Mieczysław K.Group extension of dynamical systems in er-godic theory and topological dynamics.Lectures Notes in Nonlinear Analysis 6.Torun: Nicolaus Copernicus University, JuliuszSchauder Center for Nonlinear Studies (ISBN83-231-1625-3/pbk). 193 p. (2005).This thesis describes a body of work on ergodicproperties of group extensions of dynamical sys-tems. After an overview of ergodic theory andtopological dynamics, there are detailed exposi-tions of the following works. In Chapter 2, thejoint work of the author with A. del Junco and M.Lemanczyk [Stud. Math. 112, No. 2, 141–164 (1995;Zbl 814.28007)] on ergodic properties of semisim-ple maps is described; Chapter 3 covers the jointwork with M. Lemanczyk and H. Nakada [Stud.Math. 156, No. 1, 31–57 (2003; Zbl 1019.37001)]on semisimplicity for extensions of ergodic circlerotations and the construction of weak-mixingsemisimple extensions; Chapter 4 covers the jointwork with E. Glasner and A. Siemaszko [Contemp.Math. 215, 19–42 (1998; Zbl 916.54024)] on nor-mal factors, using joinings to construct measure-theoretic analogues of the natural factors in topo-logical dynamics; Chapter 5 covers the joint workwith M. Lemanczyk [Monatsh. Math. 134, No. 3,227–246 (2002; Zbl 1002.54023)] on topologicaldynamical properties of extensions by cocyclesin locally compact groups; Chapter 6 describesthe work of the author [Colloq. Math. 95, No. 2,241–253 (2003; Zbl 1030.54031)] on the groupof essential values for continuous cocycles overan ergodic rotation taking values in a torsion-free locally compact group; Chapter 7 is basedon the joint work with A. Siemaszko [Colloq.Math. 101, No. 1, 75–88 (2004; Zbl 1058.54018)]on finding minimal sets for cylinder flows (ex-tensions taking values in the reals); finally Chap-ter 8 covers the author’s work [Topol. Meth-ods Nonlinear Anal. 23, No. 2, 357–375 (2004;

Zbl 1073.54019)] on comparisons between topo-logical and measure-theoretic dynamical proper-ties of extensions. Two appendices give the back-ground material needed on Lebesgue spaces andtopology. Thomas Ward (Norwich)

1167.37003Pinto, Alberto A.; Rand, David A.; Ferreira,FlávioFine structures of hyperbolic diffeomorphisms.Springer Monographs in Mathematics. Berlin:Springer (ISBN 978-3-540-87524-6/hbk). xvi,353 p. EUR 79.95/net; SFR 133.00; $ 129.00; £ 63.99(2009).In this book the laminations by stable and un-stable manifolds associated with a C1+ hyper-bolic diffeomorphism are studied. It is shownthat the holonomies between the 1-dimensionalleaves are C1+α for some 0 < α < 1 and that theholonomies vary Hölder continuously with re-spect to the domain and target leaves. Hence, thelaminations by stable and unstable manifolds areC1+ foliated. This result is very useful in a numberof contexts and it is used in all of the followingfourteen chapters of the book.These chapters are: Chapter 1: Introduction.Chapter 2: Hölder ratios structures. Chap-ter 3: Solenoid functions. Chapter 4: Self-renormalizable structures. Chapter 5: Rigidity.Chapter 6: Gibbs measures. Chapter 7: Measurescaling functions. Chapter 8: Measure solenoidfunctions. Chapter 9: Cocycle-gap pairs. Chapter10: Hausdorff realizations. Chapter 11: ExtendedLifšic-Sinai eigenvalue formula. Chapter 12: Arcexchange systems and renormalization. Chapter13: Golden tilings. Chapter 14: Pseudo-Anosovdiffeomorphisms in pseudo-surfaces.The book contains also five appendices: Ap-pendix A: Classifying C1+ structures on the realline. Appendix B: Classifying C1+ structures onCantor sets. Appendix C: Expanding dynamicsof the circle. Appendix D: Markov maps on train-tracks. Appendix E: Explosion of smoothness forMarkov families.This book fills a gap in the rich and enlarged lit-erature about theory of dynamical systems.

Alois Klíc (Praha)

1167.49001Hinze, M.; Pinnau, R.; Ulbrich, M.; Ulbrich, S.Optimization with PDE constraints.Mathematical Modelling: Theory and Applica-tions 23. Dordrecht: Springer (ISBN 978-1-4020-


8838-4/hbk; 978-1-4020-8839-1/e-book). xi, 270 p.EUR 85.55 (2009).The book presents a state-of-the-art of optimiza-tion problems described by partial differentialequations (PDEs) and algorithms for obtainingtheir solutions. Solving optimization problemswith constraints given in terms of PDEs is one ofthe most challenging problems appearing, e.g., inindustry, medical and economical applications.The book consists of four chapters.Chapter 1 provides an introduction to analyticalbackground and optimality theory for optimiza-tion problems with PDEs. First, necessary back-ground in functional analysis, Sobolev spaces,the theory of week solutions for elliptic andparabolic equations and Gáteaux and Fréchet dif-ferentiability are presented. Next, existence ofoptimal controls both for linear quadratic andnonlinear abstract problems in Hilbert spaces areconsidered. Further, first order optimality con-ditions for problems described by PDEs withcontrol and state constraints are derived. Ellip-tic, parabolic and Navier-Stokes optimal controlproblems are used as illustrative examples.The second chapter presents a selection of impor-tant algorithms for optimization problems withPDEs. Several variants of generalized Newtonmethods in Banach spaces are derived and anal-ysed. Problems of convergence of the consideredalgorithms are discussed in details. Elliptic andNavier-Stokes optimal control problems are usedas illustrative examples.The third chapter gives an introduction to dis-crete concepts for optimization problems withPDEs constraints. Two approaches: “First dis-cretize, then optimize” and “First optimize, thendiscretize” are compared and discussed. Severalnumerical examples together with error analysisare presented.The final fourth chapter is devoted to the studyof two industrial applications in which optimiza-tion with PDEs plays a crucial role. Two problemsfrom modern semiconductor design and glassindustry are described in details together withtheir numerical studies of related optimal con-trol problems.Every chapter of the book was written by anotherauthor. In spite of this, four chapters of the bookare logically and smoothly connected. Notationsand terminology used throughout the book areconsequent. This well-written book can be rec-ommended to scientists and graduate studentsworking in the fields of optimal control theory,

optimization algorithms and numerical solvingof optimization problems described by PDEs.

Wiesław Kotarski (Sosnowiec)

1166.34018Chen, Jian; Yi, Yingfei; Zhang, XiangFirst integrals and normal forms for germs ofanalytic vector fields.J. Differ. Equations 245, No. 5, 1167-1184 (2008).This paper deals with the existence of analytic(or formal) first integrals in a neighborhood of aconstant solution or a singularity of an analyticvector field taking into account the resonances ofthe eigenvalues associated to the linear part ofthe vector field near the considered solution.In the first main result of the paper a quasi-periodicdifferential system of the following form is consid-ered:

θ = ω +Ω(θ, x), x = Ax + f (θ, x), (1)

where θ is a 2π-periodic variable in Rm, x ∈ Rn,ω ∈ Rm are the frequencies, Ω = O (‖x‖) andf = O

(‖x‖2

)are analytic functions in their vari-

ables and 2π-periodic in θ. The results holdtrue without any modification if one considers(θ, x) ∈ Cm

×Cn. The constant solution for whichthe existence of local first integrals is investigatedis x = 0.As defined in the paper, a non-constant functionH(θ, x) is an analytic first integral of (1) if it is an-alytic in its variables, 2π-periodic in θ, and thederivative of H(θ, x) along the flow of (1) van-ishes.In order to state the first main result of the paper,we need to establish some notation:Z stands forthe set of integer numbers, N is the set of non-negative integer numbers, i =

√−1 and 〈·, ·〉 is

the usual inner product of two vectors.Theorem. For the differential system (1), letλ ∈ Cn be the n-tuple of eigenvalues of thematrix A and let γ denote the rank of the setR := (k, l) ∈ Zm

×Nn : i 〈k, ω〉 + 〈l, λ〉 = 0. Thenthe number of functionally independent analyticfirst integrals in a neighborhood of the constantsolution x = 0 is less than or equal to γ.An example that shows that the number γ isoptimal is given by completely integrable non-resonant Hamiltonian vector fields. The for-mer result gives a generalization of a result ofH. Poincaré [Rend. Circ. Mat. Palermo 5, 161–191 (1891; JFM 23.0319.01)] in which only au-tonomous differential systems are considered


and also a generalization of a result of W. Li,J. Llibre and X. Zhang [Z. Angew. Math. Phys. 54,No. 2, 235–255 (2003; Zbl 1043.37010)] in whichonly periodic systems are taken into account.The following two results in the paper investi-gate the close relation among integrability, theconvergence of the normalizing transformationnear a singularity and the resonance conditions.To end with, a classical result about planarHamiltonian systems is proved with a very orig-inal use of the Euler–Lagrange equation of calcu-lus of variations. It is shown that a planar analyticHamiltonian system has an isochronous center if,and only if, it can be analytically linearized.

Maite Grau (Lleida)

1166.34055Costin, Ovidiu; Garoufalidis, StavrosResurgence of the Euler-MacLaurin summationformula.Ann. Inst. Fourier 58, No. 3, 893-914 (2008).The Euler-MacLaurin summation formula re-lates summation to integration as follows:

N∑k=1

f (k/N) =

N∫ 1

0f (s)ds + (1/2)( f (1) − f (0)) + R( f ,N)

where the remainder term R( f ,N) has an asymp-totic expansion given in terms of Bernoulli num-bers and values of the derivatives of f at 0 and 1.For a typical analytic function this expansion isa divergent Gevrey-1 series.Under some decay assumptions of the functionin a half-plane (resp. in the vertical strip con-taining the summation interval), Hardy (resp.Abel-Plana) proved that the asymptotic expan-sion is a Borel summable series, and gave an ex-act Euler-MacLaurin summation formula. Usinga mild resurgence hypothesis for the function tobe summed, the authors give a Borel summabletransseries expression for the remainder term,as well as a Laplace integral formula, with anexplicit integrand which is a resurgent functionitself. In particular, their summation formula al-lows for resurgent functions with singularities inthe vertical strip containing the summation in-terval. Finally, they give two applications of theresults. One concerns the construction of solu-tions of linear difference equations with a smallparameter. Another concerns resurgence of 1-

dimensional sums of quantum factorials, that areassociated to knotted 3-dimensional objects.

Vladimir P. Kostov (Nice)

1167.34021Grishina, Yu.A.; Davydov, A.A.Structural stability of simplest dynamical in-equalities. (English. Russian original)Proc. Steklov Inst. Math. 256, 80-91 (2007); trans-lation from Tr. Mat. Inst. Steklova 256, 89–101(2007).Objects of interest in this paper are inequalitiesof the form

(x(x, y)−a(x, y)2 + (y(x, y)−b(x, y))2≤ f (x, y), (1)

where v = (a, b) is a fixed smooth vector fieldon the plane R2 and f : R2

→ R is a fixedsmooth function. The set of dynamical inequali-ties is identified with all triples (a, b, f ) endowedwith the fine Whitney Ck topology. A set is Ck-generic if it holds on a dense open set in the fineWhitney Ck topology.The set of points on which a2 + b2 > f > 0 holdsis said to be steep. These are the points at whichthe controlled object cannot resist drift. In thispaper it is assumed that either the set on whicha2 + b2 < f or the set on which f < 0 contains allsufficiently distant points, which in turn impliesboundedness of the steep domain.A velocity (x, y) is feasible at a point (x, y) if in-equality (1) holds there. A feasible motion is anabsolutely continuous mapping from a time in-terval into the phase space such that the velocityis feasible at each point at which the mapping isdifferentiable. A point A is reachable from a pointB is there exists a feasible motion that takes pointB to point A in finite time. The union of all pointsthat are reachable from a given point (respec-tively, from which a given point is reachable) isthe positive (respectively, negative) orbit of thispoint. A dynamical inequality is structurally sta-ble if, for any inequality in a neighborhood ofit, there exists a near-identity homeomorphismh of phase space that transforms all families ofpositive and negative orbits of the points of oneinequality into the corresponding families of theother.In this paper the authors prove the structural sta-bility of the simplest generic smooth dynamicalinequality with bounded steep domain.

Douglas S. Shafer (Charlotte)


1167.34023Diagana, TokaExistence of weighted pseudo almost periodicsolutions to some classes of hyperbolic evolu-tion equations.J. Math. Anal. Appl. 350, No. 1, 18-28 (2009).The author studies the existence and uniquenessof weighted pseudo almost periodic solutions (inthe sense of mild solutions) to evolution equa-tions of the type

ddt

[u(t) + f (t,Bu(t))

]= Au(t) + g(t,Cu(t)) ,

t ∈ R,

where A is a linear sectorial operator in a Banachspace X with domain D(A) and generates a hy-perbolic analytic semigroup, B and C are lineardensely defined closed operators in X, f mapsR × X into an interpolations space Xα betweenD(A) and X, and g maps R ×X into X. Indeed, Band C are supposed to be linear bounded oper-ators mapping Xα into X. Moreover, f and g areassumed to be weighted pseudo almost periodicin t (uniformly w.r.t. the second argument) anduniformly Lipschitz continuous in the second ar-gument.Such problems arise in control theory when con-sidering functional-differential equations withfeedback control. As an example, the author con-siders a linear partial integro-differential equa-tion of parabolic type. Etienne Emmrich (Berlin)

1167.34041Gontsov, R.R.; Poberezhnyi, V.A.Various versions of the Riemann-Hilbert prob-lem for linear differential equations. (English.Russian original)Russ. Math. Surv. 63, No. 4, 603-639 (2008); trans-lation from Usp. Mat. Nauk (0042-1316) 63, No.4, 3-42 (2008).The paper reviews some important results inthe Riemann-Hilbert (RH) problem in its varioussettings including the classical problem of exis-tence of a Fuchsian system and a Fuchsian scalarODE with given singularities and monodromy,as well as non-classical RH problems on a com-pact genus g Riemann surface and a RH problemfor a system of ODEs with irregular singulari-ties. The authors treat all the versions of the RHproblem using holomorphic vector bundles andmeromorphic connections.In the classical case, the authors show how theBirkhoff-Grothendieck’s theorem results in the

Plemelj’s theorem, i.e. why the monodromy rep-resentation of the fundamental group of thepunctured sphere can always be realized by asystem of ODEs that is Fuchsian at all but pos-sibly one point, at which the system is regular.In the case of the RH problem for a scalar Fuch-sian equation, the authors formulate a conjectureon a number of additional (apparent) singular-ities necessary for the RH problem solvabilityfor an arbitrary monodromy representation andgive necessary and sufficient conditions of theRH problem solvability in terms of existence ofa stable pair of the holomorphic vector bundleof certain splitting type and a meromorphic con-nection.It is known that the RH problem on a compactgenus g Riemann surface is solvable in the classof Pfaffian systems with regular singularities ifadditional (apparent) singularities are allowed.The authors formulate various sufficient condi-tions for the RH problem solvability and discusssome open questions.The generalized RH problem with irregular sin-gularities consists in finding of a global systemof ODEs with given singular points a1, . . . , an ofgiven Poincaré ranks r1, . . . , rn and with givenmonodromy representation of the fundamentalgroup such that, in some neighborhoods of thesingular points ai, this global system is mero-morphically equivalent to a local system witha minimal Poincaré rank ri. The authors formu-late an analog of the Plemelj’s theorem, assertingthat the generalized monodromy data (i.e. themonodromy representation of the fundamentalgroup and the local systems of minimal Poincarérank) can be realized by a system of ODEs thathas minimal Poincaré ranks at all but possiblyone singular point whose Poincaré rank can beestimated using the order of the system, numberof singular points and the sum of all assignedPoincaré ranks. Also, the authors provide sev-eral sufficient conditions for the solvability of thegeneralized RH problem and discuss the classi-cal question of existence of a Birkhoff standardform for a system with one Fuchsian and one ir-regular singular point.

Andrei A. Kapaev (St. Petersburg)

1166.35004Visintin, AugustoHomogenization of nonlinear visco-elasticcomposites.J. Math. Pures Appl. (9) 89, No. 5, 477-504 (2008).


The paper presents an homogenization resultfor nonlinear viscoelastic materials. The consti-tutive equation for a basic material is taken asσ − B(x) : ∂ε/∂t ∈ β(ε, x) , where σ (resp. ε) isthe stress (resp. the linearized strain) tensor, B(x)is a positive definite fourth-order viscosity ten-sor and β is maximal monotone mapping in thespace of second-order tensors. The problem isposed inΩ× (0,T), whereΩ is a smooth domainof R3. The author adds the quasi-static balanceequation −∇ · σ =

−→f in Ω × (0,T). The author

builds analogical discrete or continuous mod-els considering stress fields independent of theelements building the overall model and strainfields obtained as the sum or the integral of thestress fields.After a long presentation of the main tools hewill use throughout the work, the author intro-duces a weak formulation of the first problemunder consideration and proves an existence anduniqueness result. This is obtained using a timediscretization procedure and some Korn inequal-ity in order to establish estimates which allow topass to the limit with respect to the time step. In afurther section, the author moves to the compos-ite situation. This construction leads to a consti-tutive equation which depends on a highly oscil-lating parameter y. The author here first uses thetwo-scale convergence in order to describe theasymptotic behavior of the solution of the corre-sponding weak formulation. Then he proves thata single-scale homogenization may be applied,after the elimination of the variable y through anintegration on the unit cell (0, 1)3. He then com-pares the two convergence results, and also theseconvergence results to the analogical models. Inthe last part of the paper, the author applies someΓ-convergence methods to the original problem,introducing an appropriate energy functional.

Alain Brillard (Riedisheim)

1166.35005Chill, R.; Jendoubi, M.A.Convergence to steady states of solutions ofnon-autonomous heat equations in RN.J. Dyn. Differ. Equations 19, No. 3, 777-788(2007).In this paper, the authors consider the followingnon-autonomous heat equation:

ut − ∆u + f (u) = g(t, x) for (t, x) ∈ R+ ×RN,

u(0, x) = u0(x) for x ∈ RN (1)

with N ≥ 3 and f (u) = u − |u|p−1u (u ∈ R) forsome 1 < p < N+2

N−2 , g ∈ L2(R+; L2∩ Lq) for some

q > N (where Lp := Lp(RN), supp g(t, .) ⊂ K for allt ≥ 0 and some K ⊂ RN compact, g(t, x) ≥ 0 forall (t, x) ∈ R+ × RN, and u0 ∈ H1(RN) ∩ L∞(RN)has compact support.The main result of the paper states that if u ∈C(R+; H1) is a positive solution of (1) such thatsupt∈R+

‖u(t)‖H1 < +∞, and if there exists δ > 0such that

supt∈R+

t1+δ

∫∞

t‖g(s)‖2L2(RN)ds < +∞

, then limt→∞ u(t) := w exists in H1, and −∆w +f (w) = 0.This result extends previous results by C. Cor-tazar, M. del Pino and M. Elguetta [Commun. Par-tial Differ. Equations 24, No. 11–12, 2147–2172(1999; Zbl 940.35107)] and by E. Feireisl and H.Petzeltová [Differ. Integral Equ. 10, No. 1, 181–196 (1997; Zbl 879.35023)] who considered theautonomous case g = 0, as well as J. Busca,M. A. Jendoubi and P. Polácik [Commun. PartialDiffer. Equations 27, No. 9–10, 1793–1814 (2002;Zbl 1021.35013)] who considered a larger class ofnonlinearities f .

Behzad Djafari-Rouhani (El-Paso)

1166.35014Mayboroda, Svitlana; Maz’ya, VladimirBoundedness of the gradient of a solution andWiener test of order one for the biharmonicequation.Invent. Math. 175, No. 2, 287-334 (2009).The very interesting paper under review dealswith the behaviour of solutions to the biharmonicequation

∆2u = f in Omega, u ∈ W22(Ω)

over an arbitrary domain Ω ⊂ Rn, where W22(Ω)

is the completion of C∞0 (Ω) with respect to thenorm ‖∆u‖L2(Ω).The authors introduce new techniques which al-low to obtain boundedness of the gradient of thesolution to the biharmonic equation under no restric-tions on the domainΩ. In particular, the followingresults are established:(1) Boundedness of the gradient of a solution inany three-dimensional domain;(2) Pointwise estimates for the derivatives of thebiharmonic Green function;


(3) Necessary and sufficient conditions of Wienertype for continuity of the gradient of a solution.

Lubomira Softova (Bari)

1166.35038Alexandre, R.; Morimoto, Y.; Ukai, S.; Xu, C.-J.;Yang, T.Uncertainty principle and kinetic equations.J. Funct. Anal. 255, No. 8, 2013-2066 (2008).The main aim of the paper is to apply uncer-tainty principle to the study of smoothing effectsarising from the non-cutoff cross-sections for thespace inhomogeneous kinematic equations. Gen-eralized version of the uncertainty principle isproved. Based on this version results on thesmoothing effect on solutions to the kinematicequations are proved, by analyzing the interac-tion between the transport operator and the reg-ularity assumption in the microscopic velocity,together with a mild regularity assumption onthe source term. First the regularity for trans-port equation is regarded, then existence andregularity of solutions for linearized Boltzmannequation and linearized Landau equation areproved. Moreover, the regularity result for non-linear Boltzmann equation is also proved.

Leszek Skrzypczak (Poznan)DE055662704: 1166.35038

Ding, Yong; Yao, XiaohuaLp− Lq estimates for dispersive equations and

related applications.J. Math. Anal. Appl. 356, No. 2, 711-728 (2009).The authors are interested in Lp

−Lq estimates forthe solutions to the following Cauchy problem

∂tu = iP(D)u, u(0, x) = x0(x), x ∈ Rn.

Here P : Rn→ R is a real elliptic polynomial of

order m ≥ 2 with principal part Pm(ξ), withoutloss of generality Pm(ξ) > 0 for ξ , 0. There existresults for the non-degenerate case of P, that is,the Hessian matrix HPm of Pm satisfies HPm , 0for ξ , 0. This condition is equivalent to the con-dition(H1): 1/λk(ξ) = O(|ξ|−(m−2)) as |ξ| → ∞, hereλk = λk(ξ) is the k-th eigenvalue of HP.In the present paper the authors devote to thedegenerate case(Hb), b ∈ (0, 1): 1/λk(ξ) = O(|ξ|−(m−2)b).By using well-known tools of harmonic analysisthey study oscillatory integrals. The most inter-esting case is the behavior of oscillatory integrals

around stationary points of the phase function,here stationary phase method is applied. The pa-per is completed by considerations to resolventestimates for higher-order Schrödinger operatorsiP(D) + V(x,D) with a special potential V(x,D).

Michael Reissig (Freiberg)

1167.35019Vázquez, Juan Luis; Vitillaro, EnzoOn the Laplace equation with dynamicalboundary conditions of reactive-diffusive type.J. Math. Anal. Appl. 354, No. 2, 674-688 (2009).This paper deals with the Laplace equation ina bounded regular domain Ω of RN (N ≥ 2)coupled with a dynamical boundary conditionof reactive-diffusive type. In particular we studythe problem

∆u = 0 in (0,∞) ×Ω,ut = kuν + l∆Γu on (0,∞) × Γ,u(0, x) = u0(x) on Γ,

where u = u(t, x), t ≥ 0, x ∈ Ω, Γ = ∂Ω, ∆ = ∆xdenotes the Laplacian operator with respect tothe space variable, while∆Γ denotes the Laplace-Beltrami operator on Γ, ν is the outward normalto Ω, and k and l are given real constants. Well-posedness is proved for any given initial distri-bution u0 on Γ, together with the regularity ofthe solution. Moreover the Fourier method is ap-plied to represent it in term of the eigenfunctionsof a related eigenvalue problem.

1167.35028Rozenblum, Grigori; Sobolev, Alexander V.Discrete spectrum distribution of the Landauoperator pertubed by an expanding electric po-tential.Suslina, T. (ed.) et al., Spectral theory of dif-ferential operators. M. Sh. Birman 80th an-niversary collection. Providence, RI: Ameri-can Mathematical Society (AMS) (ISBN 978-0-8218-4738-1/hbk). Translations. Series 2. Amer-ican Mathematical Society 225; Advances in theMathematical Sciences 62, 169-190 (2008).The 2-dimensional Landau Hamiltonian H0 =(−i∇− a)2 describing a charged quantum particlemoving in the plane in a constant magnetic fieldB = rot a is one of the earliest explicitly solv-able models of Quantum Mechanics. Its spec-trum consists of infinitely degenerate eigenval-ues or Landau levels Λq : q = 0, 1, . . .. Under aperturbation by an electric or magnetic field de-


caying at infinity the Landau levels split beingtheir limit points.The subject of the article is to study the dis-crete spectrum of the Landau Hamiltonian H0perturbed by an expanding potential V(t)(x) =V(t−1x), t > 0. Under the condition V ∈ L1(R2) ∩L2(R2) the operator H = H(t) = H0 + V(t) is prop-erly defined as an operator sum in L2(R2), andV(t) is H0-compact. The aim of the authors isthe investigation of the number N(λ1, λ2; H(t))of eigenvalues of H(t) on the interval (λ1, λ2) b(Λν, Λν+1) with some ν = −1, 0, 1, . . . , as t → ∞.For λ1 = −∞ the designation N(λ2; H(t)) is ac-cepted. Boris V. Loginov (Ul’yanovsk)

1166.37004Giordano, Thierry; Matui, Hiroki;Putnam, Ian F.; Skau, Christian F.The absorption theorem for affable equivalencerelations.Ergodic Theory Dyn. Syst. 28, No. 5, 1509-1531(2008).A. Connes, J. Feldman and B. Weiss [Ergodic The-ory Dyn. Syst. 1, 431–450 (1981; Zbl 491.28018)]proved that any free ergodic action of anamenable group by measure class preservingtransformations is orbit equivalent to an actionof the group of integers. This paper continuesa series of results pursuing similar results inthe setting of topological dynamics. In an ear-lier work, T. Giordano, I. Putnam and C. Skau [Er-godic Theory Dyn. Syst. 24, No. 2, 441–475 (2004;Zbl 1074.37010)] developed a sophisticated ma-chinery to probe this kind of question for a count-able amenable group acting freely and minimallyon a compact metric zero-dimensional space X.An equivalence relation with countable (possi-bly finite) equivalence classes is called étale if theprojection (x, y) 7→ x is a local homeomorphismin a suitable topology on the relation R ⊂ X × X.An étale relation is called affable (AF-able, fromthe close relation to the notion of approximatelyfinite dimensional in operator algebras) if it canbe topologized so that it is the union of an in-creasing sequence of open subrelations each ofwhich has the property that the complement ofthe diagonal in R is compact (that is, so that it isan AF equivalence relation). A basic example isthat the orbit equivalence relation of an action ofa locally finite countable group on a Cantor setis an AF equivalence relation. Various propertiesof and tests for this notion were developed.

In this paper the main result is extended signif-icantly, by showing that a minimal AF equiva-lence relation R extended by modification on athin closed subset Y ⊂ X is affable. This is al-ready a substantial step forward if Y is requiredto be a finite set, and this rather technical result isa potent tool in the determination of orbit equiv-alence for Zd actions in particular, providing akey step in the recent work of the same four au-thors showing that any minimal Z2 action on aCantor set is orbit equivalent (to an AF equiva-lence relation and hence) to a minimal Z action[J. Am. Math. Soc. 21, 863–892 (2008)].

Thomas Ward (Norwich)

1166.37010Gouëzel, Sébastien; Liverani, CarlangeloCompact locally maximal hyperbolic sets forsmooth maps: Fine statistical properties.J. Differ. Geom. 79, No. 3, 433-477 (2008).The authors study a compact locally maximalhyperbolic set by means of geometrically de-fined functional spaces asserting that this pa-per provides a self-contained theory that notonly reproduces all the known classical results,but also gives the insights on the statisticalproperties of the set. We consider an open setU ⊂ X of a smooth manifold X and a mapT ∈ Cr(U,X)(r > 1), diffeomorphism on its imagesuch that Λ =

⋂TnU is non-empty and compact.

Suppose thatΛ is a hyperbolic set for T. We fix anopen neighorhood U′ ofΛ such that TU′ ⊂ U andT−1U′⊂U and small enough so that the restrictionof T to U′ is still hyperbolic. We also fix a smallneighborhood V ofΛ, compactly contained in U′.Let λ > 1 (resp. ν < 1) be two constants smaller(resp, larger) than the minimal expansion (resp.maximal contraction) of T in the unstable (resp.stable) direction. Denote by W0 the set of Cr−1

function φ associating, to each x ∈ U and eachds dimensional subspace E of the tangent spaceTxX at x, a real number φ(x,E), where ds is thedimension of the stable manifold Ws(x). Denoteby W1 the set of Cr function φ : U→ R.For x ∈ Λ, set φ(x) = φ(x,Es(x)) for φ ∈ W0 andφ(x) = φ(x) for φ ∈ W1. The authors construct afunctional spaceS and define a linear mapL de-rived from T on it in the following way. Let G bethe Grassmanian of ds-dimensional oriented sub-spaces of the tangent bundle TX to X. The set Eof all triple (x,E, ω) with (x,E) ∈ G, ω ∈ Λds E′ ⊗Cconsists a complex line bundle over G, where E′

is the dual space of E. We consider the vector


space S of all Cr−1 sections of the line bundle E.A Cr-function π on X is called a truncation func-tion if π(x) ∈ [0, 1] for x ∈ X and equal to 1 ona neighborhood of Λ and compactly supportedin T(V). Now, for each such function π, and eachφ ∈ Wι, ι ∈ 0, 1, we define a transfer operatorLπ,φ : S → S by Lπ,φα = T∗(π, eφα) for α ∈ S,where T∗ denotes the naturally defined pushfor-ward under T. It is clear that the iteration of Lπ,φwould shed light on the mixing property of T.The operator Lπ,φ does not have good asymtoticproperties on S with its Cr−1 norm, but the au-thors show that it behaves well on the spacesBp,q,ι, which are obtained by the comletion of Sin some semi-norms ‖ · ‖p,q,φ, defined for somepairs p, q ∈ N × R+. If W is a submanifold ofdimension ds contained in U, ϕ is a continuousfunction on W with compact support and α ∈ S,then we have∫

WLπ,φα =

∫T−1W

ϕ T · πeφα.

The authors consider the spectral radius ρ ofL = Lπ,φ|Bp,q,φ and proves that ρ is a simple eigen-value L, when T is orientation mixing. They givealso the description for the correlation functions(Theorem 1.2 in the Introduction) as follows: Letφ ∈ Wι for some ι ∈ (0, 1). Let p ∈ N, and q ∈ R+

satisfy p + q ≤ r−1 + ι. Let σ > max(λ−p, νq). Thenthere exists a unique measure µ on Λ, a positivenumber C, a finite-dimensional vector space F, alinear map M : F → F with simple eigenvalue 1,linear maps τ1 : Cp(U) → F and τ2 : Cq(U) → Fsuch that for every ψ1 ∈ Cp(U) and ψ2 ∈ Cq(U)and n ∈N the following holds:

|

∫(ψ1 · ψ2)Tndµ − τ2(ψ2) Mn, τ1(ψ1)| ≤

Cσn|ψ2|Cq(U)|ψ1|Cp(U)

Concerning the variational principle, the authorsprove the following: The spectral radius is equalto the topological pressure Ptop(φ) of the function.In addition, the measureµ is the unique probabil-ity measure satisfying the variational principle:

hµ(T) +

∫φ dµ = Ptop(φ),

where hµ(T) is the topological entropy of T.The measure is the so-called Gibbs measure ofT : Λ→ Λ, corresponding to the potential φ.

Finally, the authors try to give an idea of thebreadth of the obtained results by discussingsome natural examples to which they can be ap-plied, e.g., perturbation theory.

Akihiko Morimoto (Nagoya)

1166.39007Hazama, FumioDiscrete tomography through distribution the-ory.Publ. Res. Inst. Math. Sci. 44, No. 4, 1069-1095(2008).The paper is concerned with the zero-sum ar-rays, that are the C-valued functions on Zn,a = (ai) ∈ (C)Z

n, i = (i1, . . . , in) ∈ Zn, ai ∈ C, that

verify the condition dt+p(a) =∑

i∈Zn ti−pai = 0, forall p ∈ Zn and for a fixed array t = (ti) with finitesupport, called window, which in addition havea polynomial growth ak = O(|k|N) as |k| → ∞.The author studies the annihilator of the Diracdelta function with its higher derivatives, andthen he establishes a one-to-one correspondencebetween the space of zero-sum arrays of a win-dow and the space of polynomial solutions ofthe associated partial differential equation (PDE).A dimension formula for the space of polyno-mial solutions of linear PDEs with constant coef-ficients is also presented. An inductive procedurefor construction of a solution from that for PDEin a lesser number of variables and several ex-amples of windows are finally addressed.

Rodica Luca Tudorache (Iasi)

1167.39009Bohner, Martin; Došlý, Ondrej; Kratz, WernerSturmian and spectral theory for discrete sym-plectic systems.Trans. Am. Math. Soc. 361, No. 6, 3109-3123(2009).The paper is devoted to oscillation properties ofsymplectic difference systems

zk+1 = Skzk, k ∈ 0, . . . ,N, S

where zk =(xk

uk

), xk,uk ∈ R

n and the matricesSk are symplectic, that is ST

JS = J , withJ = ( 0 Ii 0 ). The authors present two Sturmiancomparison theorems for the numbers of focalpoints of conjoined bases of a pair of symplec-tic systems. Then they prove the Sturmian typeseparation theorem for conjoined bases of (S),namely they show that the difference betweenthe numbers of focal points in (0,N + 1] of any


two conjoined bases of (S) is at most n. Some spec-tral properties of (S), among which: the Rayleighprinciple for the discrete quadratic functionalassociated with eigenvalue problems associatedwith (S) and the completeness of finite eigenvec-tors in the space of the admissible sequences, arealso investigated. Rodica Luca Tudorache (Iasi)

1166.45006Troy, William C.Traveling waves and synchrony in an excitablelarge-scale neuronal network with asymmetricconnections.SIAM J. Appl. Dyn. Syst. 7, No. 4, 1247-1282,electronic only (2008).The author studies travelling waves, the spreadof synchronous oscillations and the effects ofvariations in threshold for the following systemof integro-differential equations:

∂u(x, t)∂t

= − u(x, t) − v(x, t)+∫∞

−∞

w(x − x′) f (u(x′, t)−

− θ) dx′ + ζ(x, t),∂v(x, t)∂t

=ε(βu(x, t) − v(x, t)),

which can be used to model the spread ofexcitation waves in slices of brain cortex in

which synaptic inhibition is pharmacologicallyblocked. Here, u represents the activity level ofthe population of excitatory neurons with longrange connections, v is the negative feedback re-covery variable, ζ is the external input, w is anasymmetric and strictly positive coupling func-tion, f is the Heaviside function and ε, θ and βare positive parameters.The author investigates the dynamics of this sys-tem when the eigenvalues of the linearised sys-tem around the trivial solution are complex. Theparameters β and θ are used here as a bifurcationparameters. The wave fronts, 1-pulse waves and2-pulse waves are analysed here for β larger thanthe critical value. For such values of β, at leasttwo distinct families of stable wave fronts coex-ist. Due to the asymmetry of w, these solutionspropagate in opposite directions with differentspeeds and shapes. Existence and a similar be-haviour is also found for the 1-pulse travellingwaves. Moreover, there exists a critical value forθ such that, for values of θ above it, the 1-pulseor the 2-pulse waves can propagate only in onedirection, a fact that cannot be observed for asymmetric w. A second critical value for β isfound, where spatially independent bulk oscil-lations appear and the system becomes unstable.Numerical experiments were carried outthroughout this work, illustrating the theoreti-cal results. Iulian Stoleriu (Iasi)

Discrete Mathematics. 87

Discrete Mathematics.

1166.52001Sadun, LorenzoTopology of tiling spaces.University Lecture Series 46. Providence, RI:American Mathematical Society (AMS) (ISBN978-0-8218-4727-5/pbk). x, 118 p. $ 29.00 (2008).Tiling theory has many interesting facets and ap-plications, both within mathematics and in sci-ence at large. This is particularly true of aperiodictilings, which are best described in terms of theensembles (or hulls) they define. Their combi-natorial, geometric and harmonic properties arewell studied, with many relations to “real world”quasi-crystals, but they also have an interestingtopological structure. The latter is the theme ofthis graduate level lecture series, with emphasison the cohomological structure of tiling spacesdefined by inflation (or substitution) rules.After a brief summary of basic notions in Chap-ter 1, the next chapter describes tiling spacesas inverse limits. This structure gives access totheir cohomological structure (Chap. 3), the lattesbeing modified to include rotation symmetriesin Chapter 4. Then, the text expands on equiv-ariance concepts and connections to K-theory(Chap. 5), on some more advanced (and moretechnical) aspects needed for the classic pinwheeltiling (Chap. 6), and finally on some recent (andperhaps also preliminary) extensions towardstilings without finite local complexity (Chap. 7).The text comes with exercises and selected solu-tions that help the understanding considerably.Overall, this is a nice text and a welcome addi-tion to the still rather incomplete literature onaperiodic order (as the author states, it is verydifficult to write a comprehensive account). Nev-ertheless, some hints on existing survey volumescould have strenghtened the links to other math-ematical disciplines. The interested reader mightwish to consult other references for a first ori-entation for instance “Directions in Mathemat-ical Quasicrystals”, M. Baake and R. V. Moody(eds.) [CRM Monograph Series. 13. Providence,RI: American Mathematical Society (AMS) (2000;Zbl 955.00025), and the [CRM Monogr. Ser. 13,371–373 (2000; Zbl 968.52014)] by M. Baake andU. Grimm in it. Michael Baake (Bielefeld)

1167.05003Corteel, Sylvie; Nadeau, PhilippeBijections for permutation tableaux.Eur. J. Comb. 30, No. 1, 295-310 (2009).In [J. Comb. Theory, Ser. A 114, 211-234 (2007;Zbl 1116.05003)], E. Steingrímsson and L. K.Williams introduce the concept of permutationtableaux, a class of 0/1-tableaux that are naturallyin bijection with permutations. In particular, theygive a bijection between permutation tableauxand permutations which allows to read severaldata of the permutation like exceedances, cross-ings, alignments from the corresponding tableau.In this paper, the authors propose two bijec-tions between permutation tableaux and permu-tations which carry tableau statistics to descents,right-to-left minima, and occurrences of the gen-eralized pattern 31-2 of permutations. Based onthe correspondences, subclasses of permutationtableaux are defined that are in bijection with setpartitions. Astrid Reifegerste (Magdeburg)

1167.05005Postnikov, Alex; Reiner, Victor;Williams, LaurenFaces of generalized permutohedra.Doc. Math., J. DMV 13, 207-273 (2008).Use the authors’ abstract: The aim of the pa-per is to calculate face numbers of simple gen-eralized permutohedra, and study their f -, h-,andγ-vectors. These polytopes include permuto-hedra, associahedra, graph-associahedra, simplegraphic zonotopes, nestohedra, and other inter-esting polytopes.We give several explicit formulas for h-vectorsand γ-vectors involving descent statistics. Thisincludes a combinatorial interpretation for γ-vectors of a large class of generalized permu-tohedra which are flag simple polytopes, andconfirms for them S. R. Gal’s conjecture [Dis-crete Comput. Geom. 34, No. 2, 269-284 (2005;Zbl 1085.52005)] on the nonnegativity of γ-vectors.We calculate explicit generating functions andformulae for h-polynomials of various fami-lies of graph-associahedra, including those cor-responding to all Dynkin diagrams of finiteand affine types. We also discuss relations withNarayana numbers and with Simon Newcomb’s

88 Discrete Mathematics.

problem. (Newcomb’s problem is the problem ofcounting permutations of a multiset with a givennumber of descents.)We give (and conjecture) upper and lowerbounds for f -, h-, and γ-vectors within severalclasses of generalized permutohedra.An appendix discusses the equivalence of vari-ous notions of deformations of simple polytopes.

Astrid Reifegerste (Magdeburg)

1166.05004Berger, Eli; Ziv, RanA note on the edge cover number and indepen-dence number in hypergraphs.Discrete Math. 308, No. 12, 2649-2654 (2008).The main result of this nice paper is that for eachhypergraph with rank r > 2, with m edges, inde-pendence numberα and edge cover numberρ theinequality ρ ≤ (r−2)m+α

r−1 holds. It can be used fora partial solution of the Aharoni-Berger’s con-jecture (which, in turn, a generalization of thewell-known Ryser’s conjecture on r-partite hy-pergraphs). Péter L. Erdos (Budapest)

1166.05005Dukes, Peter; Howard, LeaSmall maximally disjoint union-free families.Discrete Math. 308, No. 18, 4272-4275 (2008).This tidy paper studies the minimum possiblecardinality of maximally disjoint union-free (orDUF for short) k-graphs: a k-uniform hypergraphis DUF, if all disjoint pairs of k-edges have dis-tinct unions. Maximally DUF means that the hy-pergraph is saturated for property DUF: addingone new k-edge to the hypergraph would de-stroy the property. The quantity φ(n) denotes theminimum size of maximally DUF k-graph on ann-element underlying set. The papers deals withthe case k = 3. It is shown that number φ(n)is not strictly increasing for n ≥ 3. Furthermore— using the classical Mantel’s theorem — thelower bound φ(n) ≥ n2/12 − n/6 is proved. Thepaper constructs small size maximally DUF 3-graphs for infinitely many n which proves thatφ(n) ≤ n2/12 = O(n). Péter L. Erdos (Budapest)

1166.05006Sonntag, Martin; Teichert, Hanns-MartinCompetition hypergraphs of digraphs with cer-tain properties. I: Strong connectedness.Discuss. Math., Graph Theory 28, No. 1, 5-21(2008).For the directed graph D = (V,A) the competition

graph C(D) = (V,E) contains those vertex pairsas edges which have common out-neighbor inD. The structure was introduced by J.E. Cohenin 1968 in connection with food webs in ecosys-tems. The competition graphs have a completecharacterization if D is strongly connected. (Morepreciously if C(D) is a competition graph of somedirected graph, then it is the competition graph ofsome strongly connected directed graph as well.)A similar notion using hypergraphs describeseven better the properties of the food web. IfD = (V,A) is a directed graph, then its competitionhypergraph (or CH for short) is CH(D) = (V,E),where each edge Ev : v ∈ V contains the in-neighbors of v (if there are at least two differentones). These edges represent the hunters of thesame prey. It is known a necessary and sufficientcondition for a hypergraph to be the CH of a di-rected graph, however, there are CHs which donot emerge as the CH of some strongly connecteddigraph. The authors find those digraphs whichhave the smallest number of strongly connectedcomponents among all digraph defying the samecompetition hypergraph.

Péter L. Erdos (Budapest)

1166.05007Sonntag, Martin; Teichert, Hanns-MartinCompetition hypergraphs of digraphs with cer-tain properties. II: Hamiltonicity.Discuss. Math., Graph Theory 28, No. 1, 23-34(2008).For the directed graph D = (V,A) its competitionhypergraph (or CH for short) is CH(D) = (V,E),where Ev = w ∈ V s.t. < w, v > ∈ A (assumingthere are at least two common in-neighbors). Thishypergraph’s edges represent the predators ofthe same prey in the food web. The authors givethe full characterization of those hypergraphswhich arise as the CH of some Hamiltonian di-rected graphs, or, more generally directed graphswith τ-cycle factor. Péter L. Erdos (Budapest)

1167.05016Chia, Gek L.; Lee, Chan L.Skewness and crossing numbers of graphs.Bull. Inst. Comb. Appl. 55, 17-32 (2009).The skewness of a graph G, denoted by sk(G), isthe minimum number of edges of G whose dele-tion gives a planar graph. If G has p vertices, qedges and girth g, then sk(G) ≥ dq−g(p−2)/(g−2)e.The authors: (i) present several families of graphsfor which equality holds in the bound given; (ii)

Discrete Mathematics. 89

determine the skewness of some complete tri-partite graphs and of some generalized Petersengraphs; (iii) find a new proof for the skewness ofthe n-cube; and (iv) determine the crossing num-ber of certain generalized Petersen graphs.

Arthur T. White (Kalamazoo)

1167.05020Korzhik, Vladimir P.Exponentially many nonisomorphic orientabletriangular embeddings of K12s+3.Discrete Math. 309, No. 4, 852-866 (2009).The orientable case n ≡ (mod 12) of the Hea-wood Map Color Theorem was solved by us-ing, for each s, an index three current graphwith current group Z12s+3 (see G. Ringel ["MapColor Theorem," Springer-Verlag, Berlin (1974;Zbl 287.05102)]), to give an orientable triangu-lar imbedding of K12s+3. The present author pro-vides a new index three current graph, with thesame current group and the same result. He thenshows that, for s ≥ 11, there are at least 22s−11 non-isomorphic orientable triangular imbeddings ofK12s+3, thus completing the proof that there areconstants M, c > 0, and b ≥ 1/12 such that forevery n ≥ M there are at least c2bn nonisomor-phic orientable as well as nonorientable genusimbeddings of the complete graph Kn.

Arthur T. White (Kalamazoo)

1166.52009Gatzouras, D.; Giannopoulos, A.Threshold for the volume spanned by randompoints with independent coordinates.Isr. J. Math. 169, 125-153 (2009).Let µ be an even Borel probability measure onthe real lineRwith compact support, and defineα := supx ∈ R : µ([x,∞)) > 0 to be the rightend-point of the support of µ. Let X be a randomvariable with distribution µ, let ϕ(t) := E(etX) de-note the moment generating function of X, andletψ(t) := lnϕ(t) be its cumulant generating func-tion. Let

λ(x) := suptx − ψ(t) : t ∈ R

be the Legendre transform of ψ, and define

κ :=1

2α

∫ α

−αλ(x) dx.

Now let X1, . . . ,Xn be independent and identi-cally distributed random variables with distri-bution µ, set X = (X1, . . . ,Xn) and, for fixedN > n take N independent copies X1, . . . ,XNof X; this defines the random polytope KN :=convX1, . . . ,XN. If

limx↑α

− lnµ([x,∞))λ(x)

= 1,

then the authors show that, for every ε > 0, theexpected volume of KN satisfies

limn→∞

sup(2α)−nE(|KN |) : N ≤ exp((κ − ε)n) = 0,

limn→∞

inf(2α)−nE(|KN |) : N ≥ exp((κ + ε)n) = 1.

Peter McMullen (London)

1167.05041Boros, Endre; Gurvich, Vladimir;Zverovich, IgorNeighborhood hypergraphs of bipartitegraphs.J. Graph Theory 58, No. 1, 69-95 (2008).The matrix symmetrization problem is to decidewhether a given n × n matrix can be made sym-metric by row and column permutations. Thetopic of the paper is the computational complex-ity of this problem and related ones. The paperfirst gives a survey on previous results, partiallycorrecting errors in earlier papers. It then pro-ceeds by analyzing the complexity of severalvariants of the bipartite neighborhood recogni-tion problem. Benjamin Doerr (Saarbrücken)

90 Topology and Geometry.

Topology and Geometry.

1166.51001Weiss, Richard M.The structure of affine buildings.Annals of Mathematics Studies 168. Princeton,NJ: Princeton University Press (ISBN 978-0-691-13881-7/pbk; 978-0-691-13659-2/hbk). x, 368 p.$ 49.95, £ 29.95/pbk; $ 99.50, £ 59.95/hbk (2009).The monograph assumes some familiarity withCoxeter groups and buildings and is a sequelto the author’s book [The structure of spheri-cal buildings. Princeton, NJ: Princeton Univer-sity Press (2004; Zbl 1061.51011)], which is citedfrequently in the text, and a brief review of itsmost important definitions and results is givenin Appendix A together with some additionalsmall results on Coxeter chamber systems andbuildings. The main goal of the book under re-view is to present a complete proof of the clas-sification of affine buildings whose buildings atinfinity satisfy the Moufang property.All spherical buildings of rank at least 3 as wellas all their irreducible residues of rank at least2 satisfy the Moufang property, and the authorand J. Tits [Moufang polygons. Springer Mono-graphs in Mathematics. Berlin: Springer (2002;Zbl 1010.20017)] classified all Moufang general-ized polygons. A summary of the results on Mo-ufang spherical buildings is given in AppendixB. The proof of the classification of Bruhat-Titsbuildings largely follows the work of F. Bruhatand J. Tits [Publ. Math., Inst. Hautes Étud. Sci.No. 41, 5–251 (1972; Zbl 254.14017)] and J. Tits[Buildings and the geometry of diagrams, Lect.3rd 1984 Sess. C.I.M.E., Como/Italy 1984, Lect.Notes Math. 1181, 159–190 (1986; Zbl 611.20026)]with a greater emphasis on the role of root datain the case of rank 2.The first few chapters serve to introduce affinebuildings, present their basic properties and sub-structures, construct the building at infinity andestablish that this is a building of spherical type.Root data with valuations are already introducedin Chapter 3 although they are not used untilmuch later in the text. In Chapter 9 affine build-ings of rank 1, that is, thick trees, are investigatedand the important connection between trees anddiscrete valuations of fields is established. Thefollowing two chapters introduce two families of

trees, wall trees and panel trees, for affine build-ings of arbitrary rank and Chapter 12 shows thatan affine building is uniquely determined by itsbuilding at infinity together with its structure ofwall and panel trees.The next chapter uses the Moufang property atinfinity to obtain a valuation of the associatedroot datum and Chapter 14 shows that every rootdatum with valuation arises in this manner froman affine building. This establishes that Bruhat-Tits pairs (∆,A), whereA is the system of apart-ments of the Bruhat-Tits building∆, are classifiedby root data (of Moufang spherical buildings ofrank at least 2) with valuation, up to equipollenceof valuations. From the determination of all Mo-ufang spherical buildings one knows all possibleroot data. The aim then is to determine whensuch a root datum has a valuation. Chapters 15and 16 reduce this problem to the rank 2 caseand develop general conditions for a valuationof the underlying field (or skew field or octoniandivision algebra) of a root datum to extend toa valuation of the root datum. The problem isthen solved in chapters 19 to 25 by a case by caseanalysis for each the seven families of Moufangpolygons.Since the building at infinity also depends on thesystem of apartments, one has, for the classifica-tion of Bruhat-Tits buildings (rather than Bruhat-Tits pairs), to determine when two root data withvaluations correspond to the same Bruhat-Titsbuilding. This is done on Chapter 17 with the in-vestigation of completions. The following chap-ter then examines more closely the structure ofthe residues of a Bruhat-Tits building. Chapters26 and 27 add some comments on Bruhat-Titspairs of type F4 and algebraic Bruhat-Tits build-ings, and summarizes the classification. The lastchapter before the appendices introduces locallyfinite Bruhat-Tits buildings and outlines some oftheir principal features.

Günter F. Steinke (Christchurch)

1167.51001Deza, Michel Marie; Deza, ElenaEncyclopedia of distances.Berlin: Springer (ISBN 978-3-642-00233-5/hbk;978-3-642-00234-2/e-book). xiv, 590 p., with CD-ROM. EUR 149.95/net; SFR 249.00; $ 229.00;

Topology and Geometry. 91

£ 130.00 (2009).This ‘Encyclopedia of distances’ is a rich sourceof material on definitions in connection with thenotion “distance” in different areas of mathemat-ics and applied sciences. This includes distancesin (convex) geometry, probability, statistics, cod-ing/graph theory, data analysis, pattern recogni-tion, computer networks, computer graphics, as-tronomy, astrology, biology, physics, chemistryand also in areas like economy, philosophy andmedicine.The book is not just a collection of definitions, butin most cases it includes an introduction into thecorresponding theory. The book allows to refreshone’s own knowledge and also to learn aboutnew notions of distances in the domains listedabove.In this way, the book serves not only as a refer-ence book but also as an introduction of distancesin several other scientific fields. Altogether, thebook is recommended not only for students inthe undergraduate and graduate level, but alsofor scientists in mathematics and other relatedfields.A CD-Rom is included. Gertraud Ehrig (Berlin)

1167.51002Stahl, SaulA gateway to modern geometry. The Poincaréhalf-plane. 2nd ed.Boston, MA: Jones and Bartlett Publishers(ISBN 978-0-7637-5381-8/hbk). viii, 255 p. £ 26.99(2008).If one wants to prove theorems of hyperbolicgeometry that are not theorems of absolute ge-ometry, then there is, quite often, only one ap-proach: the computational one in some model ofhyperbolic geometry. So, if one wants to teachstudents how to answer actual questions in hy-perbolic geometry, then the time-honored tradi-tion of synthetic geometric reasoning, so fruitfulin Euclidean geometry, where almost all ques-tions of interest have found a synthetic geomet-ric answer, is little more than a useless ornament.With this in mind, the author by-passes the ax-iomatic construction of hyperbolic geometry, tointroduce – after a cursory presentation of Book Iof Euclid’s Elements, the mention of four proposi-tions each from Books III and VI, of Hilbert’s ax-ioms for Euclidean three-dimensional geometry,and a chapter on inversions – Poincaré’s upperhalf-plane model of the hyperbolic plane and itsrigid motions. The only intrusion of axiomatic

thinking comes not from Hilbert, but from Eu-clid, the validity of whose first four postulates ischecked in the model. The fact that, given threeangles that add up to less than 2π, there is a trian-gle with those angles, as well as a well-motivateddefinition of hyperbolic area, for which the factthat the area of a hyperbolic triangle is its defectbecomes a theorem, and a chapter on hyperbolictrigonometry round up the tools needed to proveresults in hyperbolic geometry. These results arelisted as exercises, the likes of which one hardlyever finds in textbooks on hyperbolic geometry,such as “the altitude to the hypotenuse of a righttriangle cannot exceed ln(1 +

√2)", “any circle

inscribed in an equilateral triangle has diameterat most ln 3", “the length of the midline of anequilateral triangle is ≤ 2 ln((1 +

√5/2)2)”.

The remaining chapters are devoted to (i) a studyof hyperbolic rigid motions, including their vi-sualization by means of “flow diagrams", whichare graphic representations of some of the or-bits, (ii) the proof that in absolute geometry overthe real numbers the sum of the angles of a tri-angle is ≤ 2π, (iii) spherical trigonometry andelliptic geometry, (iv) differential geometry withthe deduction of the formula for the Gaussiancurvature (defined in the manner of Gauss, as aratio of areas) of a surface which is the graph ofa function z = f (x, y), (v) the Poincaré unit diskmodel, (vi) the Beltrami-Klein unit disk model,(vii) a brief history of non-Euclidean geometry,and (viii) three-dimensional hyperbolic spaceH3

and its rigid motions, with proofs that the hyper-bolic geometry of any sphere inH3 is isometric tothe Euclidean geometry of some sphere, and thatevery horosphere is isometric to the Euclideanplane.While the author has succeeded in neatly in-troducing the computational aspects of hyper-bolic geometry, and offers students the neces-sary tools to actually work out problems ofhyperbolic geometry, he also deemed it neces-sary to pepper the text with disparaging re-marks about the axiomatic approach. Accord-ing to the author, Hilbert’s “axiom system isconsidered definitive", “but it is at the sametime ignored by most pedagogues because ofits complexity and subtlety" (p. 17). Such state-ments could hardly be made by someone awareof Alfred Tarski’s work (see [W. Schwabhäuser,W. Szmielew and A. Tarski [MetamathematischeMethoden in der Geometrie. Berlin: Springer-Verlag (1983; Zbl 564.51001)], A. Tarski and S. Gi-


vant [Bull. Symb. Log. 5, No. 2, 175–214 (1999;Zbl 932.01031)], or the reviewer’s [Axiomati-zations of hyperbolic and absolute geometries.Mathematics and its Applications (Springer) 581,119–153 (2006; Zbl 1106.51008)]). It is ironic that,in a text concerned only with geometry overthe real numbers, the author finds that Hilbert’saxiom of continuity “concerns some issues thathave no application to Euclidean geometry as itis understood by most geometers" (p. 20). Ratherthan emphasizing the definitive nature of the20th century achievement in setting axiomaticson the firm foundation of first-order logic, theauthor presents this achievement more as a his-torical curiosity, as “needless to say, future gen-erations may deem this century’s versions to bedeficient and opt for yet other characterizations"(p. 1).It is also doubtful whether North American un-dergraduates have had enough Euclidean ge-ometry in high school to solve the exercises inSection 1.2 (whose only theoretical underpin-ning consists of a presentation of the contentsof mostly Book I of Euclid’s Elements). Amongthem, one finds Menelaus’s and Ceva’s theorems,as well as the concurrence of the medians, anglebisectors, and of the altitudes of a triangle.

Victor V. Pambuccian (Phoenix)

1167.53001De Paris, Alessandro; Vinogradov, AlexandreFat manifolds and linear connections.Hackensack, NJ: World Scientific (ISBN 978-981-281-904-8/hbk). xii, 297 p. $ 76.00; £ 41.00(2009).By a fat manifold the authors mean a vector bun-dle, whose fiber over a point of the base manifoldis said to be a fat point over the point. At firstsight, this definition seems to be not more than a‘change of terminology’. In this case it would nothave much sense, however, the new terminologyis just a consequence of a well-motivated andcarefully thought-out philosophy. This philoso-phy is partly based on some non-classical princi-ples of modern physics, especially of gauge the-ory (and not on suspicious methaphysical spec-ulations. . . ).The authors’ terminology stresses that some spe-cific points are considered which have a non triv-ial internal structure. If this structure is classicalin the sense that single points of a fibre can beobserved (and hence distinguished) by classicalmeans, then the fat points form a smooth mani-

fold. Dramaticly new phenomena occur when itis impossible to distinguish the points of a fibreby classical observations; then we need a newconceptual framework as well as new tools fortheir satisfactory description. The fat calculus de-veloped in this volume seems to be a promisingattempt to achieve this aim.Mathematically, the first main point of the au-thors’ philosophy is that differential calculus isjust an aspect of commutative algebra. The foun-dations of calculus on manifolds were systemat-ically developed in the book “Smooth manifoldsand observables” [New York, NY: Springer (2003;Zbl 1021.58001)] by J. Nestruev in this spirit. Letme mention that Jet Nestruev is the code-name ofa group of Russian mathematicians, and Alexan-dre Vinogradov, one of the authors of this book,is a member of this group.The main ideas and tools borrowed fromNestruev’s work are summarized in detail (89pages) in Chapter 0 of this book. In Chapter 1the simplest elements of fat calculus (fat tangentvectors, fat vector fields, etc.) are discussed. Theauthors’ underlying philosophy forces that thetheory of linear connections should be developedin terms of the corresponding modules of sec-tions, i.e., without any use of the total space of thegiven vector bundle. Chapter 2 is devoted to thetheory of linear connections from this algebraicviewpoint, while Chapter 3 treats the covariantdifferential attached to a linear connection. Inthe concluding Chapter 4 important cohomolog-ical aspects of linear connections are discussed.Among some basic examples, Maxwell’s equa-tions are considered in detail.I found this nice and elegant volume very stim-ulating. József Szilasi (Debrecen)

1166.51004Bruen, Aiden A.; McQuillan, James M.Those amazing Desargues configurations.J. Comb. Math. Comb. Comput. 66, 33-41 (2008).In a projective plane PG(2,F) over a divisionring F consider two triangles P1,P3,P5 andP2,P4,P6 being in perspective from a point V,V = P1P2 ∩ P3P4 ∩ P5P6, then the points R =P1P3∩ P2P4, S = P4P6∩ P3P5, and T = P1P5∩ P2P6are on the Desargues line, and the ten pointsP1, ...,P6,V,R,S,T and the ten lines P1P2, P3P4,P5P6, P1P3, P2P4, P3P5, P4P6, P1P5, P2P6, RS forma Desargues configuration. The authors speak of aSpecial Desargues configuration, if


(i) V is on the Desargues line of the Desarguesconfiguration and

(ii) no three points of P1, ...,P6 are collinear, i.e.,P1, ...,P6 is a 6-arc.

Special Desargues configurations exist in eachprojective Desarguesian plane. Furthermore, theauthors define: A 6-arc Γ := P1, ...,P6 is calledSpecial Desarguesian arc, if there exists a pointV < Γ such that the following two conditionshold:

(i) There exist three distinct lines through Veach of which meets Γ in two points.

(ii) Of the four ways to partition P1, ...,P6 intopairs of disjoint triangles that are in per-spective from V, at least two of these pairs,together with V, yield a Special Desarguesconfiguration.

The authors prove:

1. Assume that F is a (finite or infinite) field ofcharacteristic 2, that the triangles P1,P3,P5

and P2,P4,P6 are in perspective from apoint V and that P1, . . . ,P6 =: Γ is a 6-arc.Then the resulting Desargues configurationis a Special Desargues configuration if, andonly if, Γ lies on a unique (non-degenerate)conic C. Moreover, the nucleus N of C is inci-dent with the Desargues line which is there-fore VN.

2. If a Very Special Desarguesian arc exists in aprojective plane over a (finite or infinite) fieldF, then F must have characteristic 2.

3. In a projective plane over a (finite or infinite)field F of characteristic 2 every Special De-sargues configuration yields a Very SpecialDesarguesian arc.

Rolf Riesinger (Wien)

1166.53025Brendle, Simon; Marques, Fernando C.Blow-up phenomena for the Yamabe equation.II.J. Differ. Geom. 81, No. 2, 225-250 (2009).The authors investigate the compactness comjec-ture for the sphere Sn. They consider the YamabePDE which is concerned with finding a metric ofconstant scalar curvature in the conformal classof g. They demonstrate that there exists a Rie-mannian metric g on Sn of class C∞ and a se-quence of positive functions vν ∈ C∞(Sn), ν ∈ N,so that (i) g is not conformally flat; (ii) vν is asolution of the Yamabe PDE for all ν ∈ N; (iii)

Eg(vν) < Y(Sn) for all ν ∈ N, and Eg(vν) → Y(Sn)as ν → ∞; (iv) supSn vν → ∞ as ν → ∞, whereY(Sn) denotes the Yamabe energy of the roundmetric on Sn.In comparison one could mention that for eachinteger n ≥ 52 there exists a smooth Riemannianmetric g on Sn such that a set of constant scalarcurvature metrics in the conformal class of g isnon-compact. Results on lower dimensions arealso discussed in the article in relation with theworks of others authors.[For part I, cf. J. Am. Math. Soc. 21, No. 4, 951–979(2008).] Sergey Ludkovsky (Moskva)

1166.53026Gadgil, Siddartha; Seshadri, HarishOn the topology of manifolds with positiveisotropic curvature.Proc. Am. Math. Soc. 137, No. 5, 1807-1811(2009).The authors show that a closed orientable Rie-mannian n-manifold, n ≥ 5, with positiveisotropic curvature and free fundamental groupis homeomorphic to a connected sum of copiesof Sn−1

× S1.Let (M, g) be a closed, orientable, Riemannianmanifold with positive isotropic curvature. By apaper of M. J. Micallef and J. D. Moore [Ann. Math.(2) 127, No. 1, 199–227 (1988; Zbl 661.53027)], ifM is simply connected, then M is homeomorphicto a sphere of the same dimension. The authorsgeneralize this to the case when the fundamentalgroup M is a free group.Theorem 1.1. Let M be a closed, orientable Rie-mannian n-manifold with positive isotropic cur-vature. Suppose that π1(M) is a free group onk generators. Then, if n , 4 or k = 1 (i.e.,π1(M) = Z), M is homeomorphic to the con-nected sum of k copies of Sn−1

× S1.The proof of this result is based on a theoremof M. Micaleff and J. Moore (op. cit.) whichstates that for a closed manifold M with posi-tive isotropic curvature, πi(M) = 0 for 2 ≤ i ≤ n

2and the following purely topological result es-tablished in the present paper:Theorem 1.3. Let M be a smooth, orientable,closed n-manifold such thatπ1(M) is a free groupon k generators and πi(M) = 0 for 2 ≤ i ≤ n

2 . Ifn , 4 or k = 1, then M is homeomorphic to theconnected sum of k copies of Sn−1

× S1.Ioan Pop (Iasi)


1166.53028Minerbe, VincentWeighted Sobolev inequalities and Ricci flatmanifolds.Geom. Funct. Anal. 18, No. 5, 1696-1749 (2008).Let M be a complete connected non-compact Rie-mannian manifold of dimension n ≥ 3 with non-negative Ricci curvature. If P ∈ M, let V(r,P)be the volume of the ball of radius r about P.Let ρP(t) := tnV(P, t)−1. The authors establishweighted Sobolev inequalities and a Hardy in-equality and thereby obtain rigidity results forRicci flat manifolds. The main result of the paperis the following:Theorem: Assume there is a point P ∈ M, ν > 2,and C0 > 0 so that for any t ≥ s > 0, V(P,t)

V(P,s) ≥ C0( ts )ν.

Then M satisfies a weighted Sobolev inequalityand a Hardy inequality for any smooth func-tion f with compact support of the form forS = S(n,C0, ν) and H = H(n,C0, ν):(∫

M| f |

2nn−2ρP(rP)−

2n−2

)1− 2n

≤ S∫

M|d f |2

and ∫M| f |2|r−2

P ≤ H∫

M|d f |2 .

This enables the author to draw useful geomet-ric consequences concerning Ricci flat manifoldsand to obtain some interesting results in L2 coho-mology. Peter B. Gilkey (Eugene)

1166.53034Haskins, Mark; Nikolaos, KapouleasGluing constructions of special Lagrangiancones.Ji, Lizhen (ed.) et al., Handbook of geometricanalysis. No. 1. Somerville, MA: InternationalPress; Beijing: Higher Education Press (ISBN978-1-57146-130-8/hbk). Advanced Lectures inMathematics (ALM) 7, 77-145 (2008).This is a survey of a recent work of the authorson constructing special Lagrangian cones in Cn,n ≥ 3 [Invent. Math. 167, No. 2, 223–294 (2007)],and some other preprints in preparation). Theauthors have developed a new geometric PDEgluing method to construct such cones, that con-trast with previous constructions by other au-thors, using techniques from equivariant differ-ential geometry, and integrable systems, for ex-ample. Due to the 1-1 correspondence betweenregular Lagrangian cones and Legendrian sub-

manifolds of S2n−1 (that generate the cones), themethod is in fact based on constructing specialLegendrian immersed (n − 1)-dimensional sub-manifolds. The construction follows several com-plex steps.In section 3 the building blocks are defined withsome basic SO(p)×SO(q)-invariant special Legen-drian submanifolds ofS2n−1, where p+q = n. Theysatisfy a system of first order nonlinear ODEs,and constitute a family Xτ : I×Sp−1

×Sq−1→ S2n−1

of special Legendrian cylinders, where |τ| ≤τmax < +∞, and I an interval of R. When τ → 0,Xτ approximates a necklace of equatorial (n− 1)-spheres, of different kinds depending on (p, q).The geometry of Xτ is described, namely its sym-metry groups, and its embeddedness. The casesp = 1 and p = q are special and treated withparticular attention.The image of Xτ on certain domains S[k] (almostspherical regions) are close to “approximating"spheres. S[k] connects to S[k − 1] and S[k + 1] viasome transition regions. If p = 1 these transitionregions are approaching an (n − 1)-dimensionalLagrangian catenoid of size τ1/n−1, when τ → 0.For p > 1, Xτ approaches the complement oftwo orthogonal oriented equatorial sub-spheresSp and Sq, of dimensions p − 1 and q − 1 respec-tively, of a equatorial special Legendrian (n − 1)-dimensional sphere SL ⊂ S2n−1

⊂ Cn. The transi-tion regions are contained in small tubular neigh-bourhoods of the attachment set Sp ∪ Sq and canbe of different type, where products of sphereswith Lagrangian catenoids appear.The initial almost special Legendrian immer-sions Y : M → S2n−1 are constructed using mul-tiple copies of the building blocks, fusing to-gether at a common approximating sphere. Thisis a similar gluing construction of CMC surfacesgiven by previous work of the second author,for example [Ann. Math. (2), 131, No. 2, 239–330(1990; Zbl 699.53007)] and by R. M. Schoen [Com-mun. Pure Appl. Math. 41, No. 3, 317–392 (1988;Zbl 674.35027)].Fixing a large number m ∈ N, one determines alarge number k0 of spherical regions (for exam-ple, k0 = 2(n− 1)m− 1 for p = 1, and k0 = 2pm− 1for p = q) and a small τ > 0 satisfying a certaincondition. Then Xτ factors through an embed-ding of a closed manifold which contains an oddnumber k0 of almost spherical regions. This im-mersion Y : M → S2n−1 is equivariant under asuitable large group of isometries G. M is cov-ered by a finite number of open sets M =

⋃gi=0 Mi


where M0 corresponds to a central sphere wherefusion takes place, and M0 ∩Mi = A+

i ∪ A−i arethe two connected components of the gluing re-gion, and Mi are disjoint for i ≥ 1 and consistsof g copies of the building blocks Xτ that arefusing in the central sphere. Y agrees with Xτ inM1\(A+

1∪A−1 ), and byG-equivariance, determinesY on Mi\(A+

i ∪ A−i ). On M0\⋃

i(A+i ∪ A−i ), Y is an

embedding, and on A±i , Y is determined by itsdefinition on A+

1 (a cylindrical domain for someinterval I), similarly defined as by A. Butscher[Commun. Anal. Geom. 12, No. 4, 733–791 (2004;Zbl 1070.53026)] and by Y.-I. Lee [ibid. 11, No. 3,391–423 (2003; Zbl 1099.53053)].In section 5 the authors use PDEs to correcta initial almost special Legendrian immersionX : M→ S2n−1 of Lagrangian angle θ to a specialLegendrian one. Given a function f : M → R, fcan be extended to a tubular neighbourhood Nof X(M) using the exponential map and a nor-mal vector, and then extended to a cone over Nas homogeneous of degree 2. This defines a Leg-endrian submanifold X f : M → S2n−1, the Leg-endrian graph of f on X, by taking the flow ofV = −J∇ f at time t = 1. The Lagrangian angle ofX f is θ f = θ + L f + Q f , where L f = (∆ + 2n) f ,(∆ is the Laplacian on M defined by X) and Q fis quadratic on f and its first derivatives. Theauthors study the inhomogeneous linear equa-tion L f = E and the G-equivariant kernel of L,on a initial almost special Legendrian immersionY : M→ S2n−1.In section 6 the solutions of this operator arestudied on transition regions and then on almostspherical regions with their adjacent transitionregions. For that they consider the approximatedkernel (the span of the eigenfunctions forLwithsmall eigenvalues), and then the extended sub-stitute kernel,Kextended, on regions containing analmost spherical region, and correcting the in-homogeneous term so that it becomes orthogo-nal to this kernel, they can solve the Dirichletproblem on any extended almost spherical re-gion.Kextended consists on a space of dimension mof globally defined functions that vanish on thetransition regions, and were introduced by thesecond author (see ref. above) and it is an help-ful tool that simplifies the study of the Dirichletproblem.From these solutions they obtain global approx-imated solutions that satisfy good estimates,and modify θ f of the final submanifold to beany sufficiently small prescribed element from

Ktiny extended ≡ Rm. Each parameter ξ ∈ Rm de-fines another vector ζ ∈ Rm ( m is one or two,and corresponds to a well defined subspace ofKextended), that, for ξ sufficiently small, param-eterizes a modified G-invariant special Legen-drian immersion Yζ : M → S2n−1, that is builtusing some special Legendrian modification Xτ,α

of Xτ. There is continuous dependence on ζ.Finally in section 8, estimates of the quadraticterm QV = θV − θ − LV are obtained, whereV ∈ C2,β

sym(M) (sym means it is G-equivariant), θVis the Lagrangian angle of the Legendrian graph(Yζ)V andθ that of Yζ. For sufficiently large m andsmall τ, the authors conclude that there exist ζand V satisfying certain boundedness conditions(depending on m and on τ and on an indepen-dent constant C), such that (Yζ)V is special Leg-endrian andG-equivariant. The proof consists onapplying Schauder fix point theorem on a suit-able map defined on the variables v ∈ C2,β

sym(M)and ξ. Isabel Salavessa (Lisboa)

1166.53052Albers, Peter; Hofer, HelmutOn the Weinstein conjecture in higher dimen-sions.Comment. Math. Helv. 84, No. 2, 429-436 (2009).By a contact manifold we mean a (2n − 1)-dimensional manifold M together with a 1-formλ such that λ∧ (dλ)n−1 is a volume form. It is wellknown that a contact manifold admits a uniquevector field X, called the Reeb vector field, suchthat iXdλ = 0 and iXλ = 1.In [J. Differ. Equations 33, 353–358 (1979;Zbl 388.58020)], A. Weinstein conjectured that theReeb vector field on a compact simply-connectedcontact manifold has a closed orbit. Many au-thors, including this reviewer, believe that theconjecture is true without the assumption ofsimple connectivity. In dimension 3 the conjec-ture was recently proved in full generality byC. H. Taubes [Geom. Topol. 11, 2117–2202 (2007;Zbl 1135.57015)]. The present paper considers theWeinstein conjecture in higher dimensions.In dimension 3 one has the notion of an over-twisted contact structure. Let ξ denote the con-tact structure or hyperplane field defined byλ = 0. Roughly speaking, ξ being overtwistedmeans that one has a contact embedding of adisk, tangent to ξ at the origin and such that asone moves outward radially, the plane ξ turnsover in a finite distance.


As a higher dimensional generalization of be-ing overtwisted, K. Niederkrüger [Algebr. Geom.Topol. 6, 2473–2508 (2006; Zbl 1129.53056)] in-troduced the notion of a Plastikstufe. Let D2

denote the closed unit disk with coordinates(x, y). The contact manifold contains a Plastik-stufe with singular set S if it admits a closed sub-manifold S of dimension n−2 and an embeddingι : D2

×S −→M with ι(0×S) = S having the fol-lowing properties: (1) There exists a contact formλPS inducing ξ such that the 1-form β = ι∗λPSsatisfies β ∧ dβ = 0 and β , 0 on (D2

\0) × S.Near 0 × S, β = xdy − ydx and the pull-backof β to ∂D2

× S vanishes. (2) The complementof 0 × S in (D2

\∂D2) × S is smoothly foliatedby β via an S1-family of leaves diffeomorphic to(0, 1)× S, where one of the ends converges to thesingular set 0 × S and the other is asymptoticto the leaf ∂D2

× S. The set ι(D2× S) is called the

Plastikstufe and a closed contact manifold is saidto be PS-overtwisted is it admits a contact formλPS inducing ξ containing a Plastikstufe.The main result of the present paper is that givena closed PS-overtwisted contact manifold withcontact structure ξ, every Reeb vector field as-sociated to a contact form λ inducing ξ has acontractible periodic orbit.

David E. Blair (East Lansing)

1167.53022Gilkey, Peter B.; Nikcevic, Stana Ž.Geometrical representations of equiaffine cur-vature operators.Result. Math. 52, No. 3-4, 281-287 (2008).The authors examine the way passing from thealgebraic context to the geometric setting forsome classes of equiaffine curvature operators.It is shown that any equiaffine algebraic opera-tor arises from an equiaffine connection.Also, the decomposition problem of curvatureoperators [N. Bokan, Rend. Circ. Mat. Palermo,II. Ser. 39, No. 3, 331–380 (1990; Zbl 728.53016);I. E. Hirica [Balkan J. Geom. Appl. 4, No. 1, 69–90 (1999; Zbl 982.53013)] is studied. It is shownthat the space of equiaffine algebraic curvatureoperators splits into two summands which areirreducible under the action of the general groupin dimension m > 3. This gives rise to two addi-tional geometric representation questions whichthe authors answer affirmatively:1. Is every equiaffine algebraic curvature opera-tor which is projectively flat representable by aequiaffine connection which is projectively flat?

2. Is every Ricci flat algebraic curvature operatorrepresentable by a Ricci flat torsion free connec-tion? Iulia Hirica (Bucuresti)

1167.53034Ballmann, Werner; Buyalo, SergeiPeriodic rank one geodesics in Hadamardspaces.Burns, Keith (ed.) et al., Geometric and prob-abilistic structures in dynamics. Workshop ondynamical systems and related topics in honorof Michael Brin on the occasion of his 60thbirthday, College Park, MD, USA, March 15–18,2008. Providence, RI: American MathematicalSociety (AMS) (ISBN 978-0-8218-4286-7/pbk).Contemporary Mathematics 469, 19-27 (2008).The main aim of the present article is the properformulation of problems concerning rank rigid-ity and geodesic flow of Hadamard spaces.Let X be a locally compact Hadamard space. Let∂X be the ideal boundary of X and set X = X∪∂X.The cone topology turns X into a a compactHausdorff space. There is a natural metric on∂X, the Tits metric dT, introduced by M. Gro-mov [W. Ballmann, M. Gromov, V. Schroeder, Mani-folds of nonpositive curvature. Progress in Math-ematics, 61. (Boston-Basel-Stuttgart): Birkhäuser.(1985; Zbl 591.53001)]. Let ∂TX denote the space∂X together with the Tits metric and the Titstopology induced by dT. The Tits topology isfiner than the cone topology and ∂TX is a CAT(1)-space. Typically, ∂TX is a rather wild noncompacttopological space.Let Γ be a group of isometries of X. Recall that theinduced action of Γ on ∂X and X respects the conetopology and the Tits metric, respectively. A se-quence γnn∈N ⊂ Γ converges to a point ξ ∈ ∂X(γn → ξ) if γnx → ξ for one ( and hence any)x ∈ X. Let Λ = Λ(Γ) be the limit set of Γ, that is,Λ is the set of all ξ ∈ ∂X occurring as such limits.So, Λ ⊂ X is closed and Γ-invariant.The article focuses on studying the interplay be-tween the action of Γ and the geometry of X, andfor this they assume Γ is sufficiently large.A subset F ⊂ X is k-flat if F is convex and iso-metric to the k-dimensional Euclidean space Rk.A 1-flat is also called a line, a 2-flat a flat plane. Aflat half-plane is a convex subset of X isometric tothe subset y ≥ 0 ⊂ R2. In the article it is pointedout that many questions concerning the asymp-totic geometry of X are related to the existence offlats or flat half-planes.


A line L ⊂ X has rank one if it does not bound aflat half-plane. This holds if and only if the end-points of L have Tits distance > π. A line L is Γ-periodic if there is a unit speed parametrization σof L, an element γ ∈ Γ, and a number w > 0 suchthat γσ(t) = γ(t + w), for all t ∈ R. A Γ-periodicline has rank one if and only if its endpoints haveinfinite Tits distance. The authors stress the factthat the existence of γ-periodic line of rank oneis a a central issue for algebraic properties of Γ,asymptotic properties of stochastic processes onX, and dynamical properties of the geodesic flowof X module Γ [W. Ballmann, Lectures on spacesof nonpositive curvature. DMV Seminar. Bd. 25.Basel: Birkhäuser Verlag (1995; Zbl 834.53003)].The main problems listed in the article are thefollowing conjectures:Conjecture A. (Closing lemma) Suppose that Γ isproperly discontinuous and that Λ = ∂Λ. Thendiam(∂TX) > π implies that X contains a Γ-periodic line of rank one.They claim the Conjecture A holds for the fol-lowing cases [see W. Ballmann and M. Brin, Publ.Math., Inst. Hautes Étud. Sci. 82, 169–209 (1995;Zbl 866.53029); Lê Hông Vân and K. Ono, Topol-ogy 34, No. 1, 155–176 (1995; Zbl 822.58019)]:(1) under the assumption that Γ satisfies the du-ality condition of Chen and Eberlein.(2) under the assumption that X is a piecewisesmooth complex of dimension two and Γ is aproperly discontinuous group of isometries of Xwith compact quotient.Conjecture B. (Diameter rigidity) Suppose that Xis geodesically complete and that Λ = ∂X. Thendiam(∂TX) = π implies that X is a symmetricspace or a Euclidean building of rank at leasttwo, or that X is reducible.They claim that diam(∂TX) = π if and only if alllines in X bound a flat half-plane. Conjecture Bholds for the cases quoted below:(1) when X is a Hadamard manifold and Γsatisfies a duality condition (see [W. Ballmann,Lectures on spaces of nonpositive curvature.With an appendix by Misha Brin: Ergodicity ofgeodesic flows. Basel: Birkhäuser Verlag (1995;Zbl 834.53003], Chapter IV),(2) X is a homogeneous Hadamard manifold(proved by [J. Heber, Int. J. Math. 6, No. 2, 279–296(1995; Zbl 830.53040)]).A homogeneous Hadamard manifold can be rep-resented as a simpy connected solvable groupwith a left-invariant metric [E. Heintze, Math.Ann. 211, 23–34 (1974; Zbl 273.53042)].

By the authors own words, the conjectures abovehave to be taken with a grain of salt, because itmay exist an exotic counter-example. The pointis that, except for inessentialities, they believe theconjecture’s formulations seem to be correct.The following propositions are proved by the au-thors:Proposition 1. Suppose that λ = ∂X and that,for each ξ ∈ ∂TX, there is an η ∈ ∂TX withd(ξ, η) > π. Then X contains a Γ-periodic lineof rank one.Proprosition 2. Let X be geodesically completeand Γ be uniform.(1) If ∂X contains exactly one Γ-minimal subset,then the action of Γ on ∂X is topologically transi-tive.(2) If |∂X| ≥ 3 and if ∂X contains more than one Γ-minimal subset, then (∂X)2 contains a non-emptyΓ-invariant subset P) (with respect to the diago-nal action of Γ) which does not intersect the diag-onal such that, for all x ∈ X and (ξ, η) ∈ P, thereis a flat plane containing the sector (x, ξ, η).

Celso M. Doria (Florianapolis)

1167.53036Sun, Hongwei; Wang, YushengPositive sectional curvature, symmetry andChern’s conjecture.Differ. Geom. Appl. 27, No. 1, 129-136 (2009).This paper deals with the study of the funda-mental group of a closed manifold M of positivesectional curvature. It is well-known that π1(M)is finite. The Chern conjecture, asserting that anyAbelian subgroup ofπ1(M) is cyclic, is true if M isa space form, but it is false in general cases. Un-der the hypothesis that M admits an isometriccircle T1-action, a positive answer to the men-tioned conjecture is due to Rong.The authors assume that a torus Tk acts effec-tively by isometrics on a closed n-manifold ofpositive sectional curvature. Under this hypoth-esis a classification of the group π1(M) has beenrecently obtained. This helps in proving the fol-lowing results:If n > 23 and k > n+1

8 + 1, then π1(M) containsno Zp ⊕ Zp subgroup, with p prime, p , 3 andany element of order 2 of π1(M) belongs to thecenter of π1(M). Furthermore, if there are no Tk-fixed points or if the order of π1(M) is even, then,up to isomorphisms, π1(M) is the fundamentalgroup of a rational homology 5-sphere or of a 3-dimensional space-form with constant curvatureone. Maria Falcitelli (Bari)


1167.53045Klein, SebastianReconstructing the geometric structure of a Rie-mannian symmetric space from its Satake dia-gram.Geom. Dedicata 138, 25-50 (2009).Let M be a Riemannian symmetric space of non-compact type. The local geometry of M is de-scribed completely by its Riemannian metric andits Riemannian curvature tensor. The author pro-vides algorithms for the reconstruction of thesefundamental geometric tensors from the Satakediagram of M. The algorithms are adjusted tothe use with computer algebra systems. As anexample, the author has implemented the algo-rithms as a Maple package. Moreover, as an ap-plication, he classifies the totally geodesic sub-manifolds of the Riemannian symmetric spaceSU(3)/SO(3) by classifying all Lie triple systemstherein. This result corrects the classification ofmaximal totally geodesic submanifolds by B.-Y.Chen and T. Nagano [Duke Math. J. 45, 405–425(1978; Zbl 384.53024)].The author takes advantage of the Lie algebraicdescription of the Riemannian metric and thecurvature tensor on M. These tensors are deter-mined by the inner product given by the Rieman-nian metric on a single tangent space TpM, andthe curvature tensor at p. Suppose that M = G/K,where G is the transvection group of M and Kthe isotropy group of a point in M. The geodesicsymmetry at o := eK gives rise to an involutionσ on the Lie algebra g of G. The map σ inducesthe decomposition g = k ⊕ m, where k is the 1-eigenspace of σ andm is its−1-eigenspace, whichis canonically isomorphic to ToM. Then the cur-vature tensor R of M at o is given by the formulaR(u, v)w = −[[u, v],w] for u, v,w ∈ m.Now the author proceeds as follows. In a firststep he uses the Dynkin diagram of g, which canbe read from the Satake diagram of M, to recon-struct algorithmically the root system ∆ of g. Inthe second step he gives an algorithm to con-struct a Chevalley basis for the complexificationgC of g using ∆. Then the Lie bracket of gC is de-termined by the Chevalley constants, which arecalculated by the algorithm. In the third step hecalculates the action of the complexification ofσ on this Chevalley basis of gC, and hence theaction of σ on g. The algorithm for this calcula-tion uses Cartan’s classification of Riemanniansymmetric spaces. The knowledge of σ allows toreproduce the decomposition g = k⊕m. Since the

chosen Chevalley basis was particularly suitedto the position of g within gC, he can now deter-mine the Lie bracket on g, the inner product onm, and the curvature tensor R at o.

Anke Pohl (Paderborn)

1167.53055Ecker, KlausA formula relating entropy monotonicity toHarnack inequalities.Commun. Anal. Geom. 15, No. 5, 1025-1061(2007).The author obtains an entropy formula for a fam-ily (Ωt)t∈[0,T) of bounded open subsets ofRn withsmooth boundary hypersurfaces evolving withnormal speed

βt = −∂x∂t· ν,

where x denotes the embedding map of ∂Ωt andν the normal pointing out of Ωt.The formula is an adaptation of Perelman’s en-tropy formula for the Ricci flow [G. Perelman,arXiv e-print service, Cornell University Library,Paper No. 0211159, 39 p., electronic only (2002;Zbl 1130.53001)].In the important case of the mean curvature flow,where βt is given by the mean curvature of theboundary of the subsetΩt, the author conjecturesa Harnack-type inequality which would implythe monotonicity of the entropy. Under the as-sumption that the conjecture is valid, a lower lo-cal volume ratio bound for solutions of the meancurvature flow is obtained, which rules out cer-tain eternal solutions of the mean curvature flowas rescaling limits metric, as the stationary solu-tions corresponding to a pair of hyperplanes orto the catenoid minimal surface, or as the trans-lating solution corresponding to the grim reaperhypersurface. Pierre Bayard (Morelia)

1166.54004Avilés, AntonioCompact spaces that do not map onto finiteproducts.Fundam. Math. 202, No. 1, 81-96 (2009).The author considers several problems concern-ing the possibility of mapping a finite productspace onto another product space with more fac-tors. The first problem concerns the possibility ofmapping a finite power of the closed Euclideanball B of a nonseparable Hilbert space endowedwith its weak topology onto a product space. It is


shown that for natural numbers n,m with n < m,if X1 × · · · ×Xm is a continuous image of Bn, thenXi must be metrizable for some i ≤ m. Conse-quently, B2 is not a continuous image of B. Thissettles a problem of the author and Kalenda.The second problem concerns the following con-jecture of S. Mardešic posed in [Glas. Mat., III.Ser. 5(25), 163–170 (1970; Zbl 195.52301)]: LetL1, . . . ,Ln be linearly ordered compact spaces, letX0, . . . ,Xn be infinitely compact spaces and letf :

∏ni=1 Li →

∏nj=0 X j be a continuous surjection.

Then there exist 0 ≤ i, j ≤ n, i , j, such that Xiand X j are metrizable. Mardešic showed that theanswer to the conjecture is affirmative under theassumption that all spaces Xi are separable. Al-though the author does not solve the conjecturecompletely, he proves that if there is a continu-ous surjection f :

∏ni=1 Li →

∏nj=0 X j, then there

exist 0 ≤ i, j ≤ n, i , j, such that Xi and X j areseparable.The third problem deals with σ-product spacesσn(Γ) = x ∈ 0, 1Γ : |supp(x)| ≤ n, and whethera product of such spaces can be a continuousimpage of another product of the same type.Examples of nonseparable compact spaces withthe property that any continuous image which ishomeomorphic to a finite product of spaces hasa maximal prescribed number of nonseparablefactors are provided. The main tools employedin the paper are some properties defined in termsof Knaster-disjoint families.

Jiling Cao (Auckland)

1166.54011Pol, Elzbieta; Pol, RomanA metric space with the Haver property whosesquare fails this property.Proc. Am. Math. Soc. 137, No. 2, 745-750 (2009).A metric space (X, d) has the Haver property iffor every sequence εi : i ∈ ω of positive num-bers there exists a sequence Vi : i ∈ ω of dis-joint collections of open subsets of X such that⋃Vi : i ∈ ω covers X and, for each i ∈ ω, every

element ofVi has diameter less than εi.A space X is said to have the property C if for anysequence Ui : i ∈ ω of open covers of X, thereexists a sequence Vi : i ∈ ω of disjoint collec-tions of open subsets of X such that

⋃Vi : i ∈ ω

is a cover of X and, for each i ∈ ω, every elementofVi is contained in an element ofUi. A metriz-able space X has the property C if and only if, forany metric d on X which generates the topologyof X, the space (X, d) has the Haver property.

The main result of the paper is Theorem 1.1which states that there exist separable metricspaces (X0, d0) and (X1, d1) with the property Csuch that the metrics d0 and d1 generate the sametopology on X0 ∩ X1 , ∅ while the metric space(X0 ∩ X1,maxd0, d1) does not have the Haverproperty. Therefore there exists a separable com-plete metric space with the property C whosesquare fails to have the Haver property.

Vladimir Tkachuk (Mexico)

1166.54015Holický, PetrDecompositions of Borel bimeasurable map-pings between complete metric spaces.Topology Appl. 156, No. 2, 217-226 (2008).The σ-fieldEX of extended Borel subsets of a met-ric space X is the smallest σ-field of subsets of Xwhich contains Borel sets and which is closedwith respect to the unions of discrete families. Amapping f : X → Y between metric spaces isextended Borel measurable if f−1(B) is extendedBorel in X for every Borel subset B of Y. A ex-tended Borel measurable mapping f : X → Ybetween metric spaces is extended Borel bimea-surable if f (B) is extended Borel for every subsetB of X.A family Dαα∈A of subsets of a topological spaceX is σ-discretely decomposable (σ-d.d. in brief) inX if there are Dα(n), α ∈ A, n = 1, 2, . . . such thatDα =

⋃∞

n=1 Dα(n) for every α ∈ A and Dα(n)α∈Ais discrete for each n = 1, 2, . . . A family Dαα∈Aof subsets of a topological space X is almost σ-discretely decomposable (almost σ-d.d. in brief)in X if there exists a σ-discrete set S in X such thatthe family Dα\α∈A is σ-d.d.The mains results of this paper are the followingdecomposition theorems for an extended Borelmeasurable mapping f : E → Y, where X and Yare complete metric spaces and E ∈ EX:1. (Theorem 2.1) Let f : E → Y be and ex-tended Borel bimeasurable mapping. There is aσ-discrete subset S of Y and a σ-discrete partitionD of E\ f−1(S) such that each D ⊂ EX and eachf |M is an extended Borel isomorphism for everyM ∈ D.2. (Theorems 3.1 and 3.2) The mapping f is ex-tended Borel bimeasurable and preserves almostσ-d.d. families if and only if there are pairwisedisjoint extended Borel subsets E0,E1, . . . of Xsuch that E =

⋃∞

n=0 En, f (E0) is σ-discrete andf |En is an extended Borel isomorphism for everyn = 1, 2, . . . Assuming Fleissner’s axiom (SCω2),


the mapping f is extended Borel bimeasurableif and only if there are disjoint extended Borelsubsets E0,E1, . . . of X such that E =

⋃∞

n=0 En,f (E0) is σ-discrete and f |En is an extended Borelisomorphism for every n = 1, 2, . . .Axiom SCω2 was given in [W. G. Fleissner,Trans. Am. Math. Soc. 251, 309–328 (1979;Zbl 428.03044)].In the case of separable complete metric spacesX and Y, the existence of a countable decompo-sition of Borel bimeasurable mappings followsfrom the well-known results of R. Purves [Fun-dam. Math. 58, 149–157 (1966; Zbl 143.07101)],and of N. Luzin and P. S. Novikov [see, e.g.,Theorem 18.10 in A. S. Kechris, Classical Descrip-tive Set Theory, Springer-Verlag. New York (1955;Zbl 819.04002)].3. (Theorems 3.3 and 3.4) The mapping f pre-serves almost σ-d.d. families and f (B) is extendedBorel for every closed subset B of E if and onlyif there are extended Borel subsets E0,E1, . . . ofX such that E =

⋃∞

n=0 En, f (E0) is σ-discrete, f |En

maps closed sets to extended Borel sets, and f |En

has compact fibers ( f |En )−1(y), y ∈ Y, for ev-ery n = 1, 2, . . . Assuming SCω2, the precedingequivalence holds without almost σ-d.d. condi-tion.The existence of analogical countable decompo-sition in ZFC remains open; in this case it is giventhe σ-discrete decomposition considered in 1.

Manuel Lopez Pellicer (Valencia)

1167.54006Kunen, KennethOrdered spaces, metric preimages, and functionalgebras.Topology Appl. 156, No. 7, 1199-1215 (2009).Let X be a compact Hausdorff space and C(X) de-note the algebra of all continuous complex val-ued mappings on X with the sup norm topol-ogy. Let A ⊂ C(X) be a subalgebra which sepa-rates points and contains the constant functions.We say that X has the complex Stone-Weierstrassproperty CSWP in case every such A is dense inC(X). A result of W. Rudin [Proc. Am. Math. Soc.8, 39–42 (1957; Zbl 77.31103)] says that if X is ametric space, then (*) X has the CSWP if, and onlyif, X contains no copy of the Cantor set. Further-more, the metric hypothesis can not be omitted.The present author has shown [Fundam. Math.182, No. 2, 151–167 (2004; Zbl 1063.46036)] that ifX is a linearly ordered topological space (LOTS),then (*) holds. In the present paper he shows,

among other things, that if L1,L2,L3 are LOTSesand X ⊂ L1 × L2 × L3, then (*) holds. He also con-structs from the technical apparatus of his papera compact, separable, first countable LOTS L suchthat Ln has the CSWP for each integer n ≥ 1. Anumber of open questions are also posed.

James V. Whittaker (Vancouver)

1167.54009Charatonik, J.J.; Charatonik, W.J.; Prajs, J.R.Confluent mappings and arc Kelley continua.Rocky Mt. J. Math. 38, No. 4, 1091-1115 (2008).A metric continuum X is called a Kelley contin-uum provided that for each point p, each con-tinuum K in X containing p and each sequenceof pn converging to p, there exists a sequence ofsubcontinua Kn containing pn that converges toK. A Kelley continuum X is called an arc con-tinuum provided that each subcontinuum K ofX containing p can be approximated by arcwiseconnected continua containing p. A continuumhomeomorphic to the inverse limit of locally con-nected continua with confluent bonding maps issaid to be confluently LC-representable. The au-thors study the close relationships between thearc Kelley continua and confluent mappings.The main result says that if a continuum X ad-mits, for each ε > 0, a confluent ε-mapping ontoa(n) (arc) Kelley continuum, then X itself is a(n)(arc) Kelley continuum. In particular, each con-fluently LC-representable continuum is arc Kel-ley. The authors study conditions under whichthe different hyperspaces of a continuum X areLC-representable. They also offer a number of in-teresting open problems.

Alejandro Illanes (México, D.F.)

1166.55001Rognes, JohnGalois extensions of structured ring spectra.Stably dualizable groups.Mem. Am. Math. Soc. 898, 137 p. (2008).Author’s abstract: We introduce the notion ofa Galois extension of commutative S-algebras(E∞-ring spectra), often localized with respect toa fixed homology theory. There are numerousexamples, including some involving Eilenberg-Mac Lane spectra of commutative rings, realand comple25x topological K-theory, Lubin-Tatespectra and cochain S-algebras. We establish themain theorem of Galois theory in this gener-ality. Its proof involves the notions of sepa-rable and étale extensions of commutative S-


algebras, and the Goerss-Hopkins-Miller theoryfor E∞-mapping spaces. We show that the globalsphere spectrum S is separably closed, usingMinkowski’s discriminant theorem, and we esti-mate the separable closure of its localization withrespect to each of the Morava K-theories. We alsodefine Hopf-Galois extensions of commutative S-algebras, and study the complex cobordism spec-trum MU as a common integral model for all ofthe local Lubin-Tate Galois extensions.

Sunil Chebolu (Illinois)

1167.55004Song, Yongjin; Tillmann, UlrikeBraids, mapping class groups, and categoricaldelooping.Math. Ann. 339, No. 2, 377-393 (2007).This paper studies a family of group homomor-phisms φ : β2g → Γg,1 from the braid group on 2gstrings to the mapping class group of an orientedgenus g surface with one boundary component.These arise from the fact that Dehn twists sat-isfy the braid relations. They extend to a homo-morphism β∞ → Γ∞ of the corresponding stablegroups. The main result is that the map inducedin homology by this homomorphism is trivial,proving a conjecture of Harer.Work of J. S. Maginnis [Braids and mappingclass groups. PhD thesis, Stanford University(1987)] shows that the map induced by φ from∐

g≥0 Bβ2g to∐

g≥0 BΓg,1 is not a map of doubleloop spaces. The main point here is to show thatnevertheless the induced map on group comple-tions φ : Bβ+

∞ → BΓ+∞ is a double loop map. A

straightforward lemma, using that Bβ+∞ ' Ω

2S3

is the free double loop space on S1, then showsthat any such map is trivial in homology.Categorial delooping techniques are used. Thusφ is extended to a monoidal functor Φ : T → Sbetween monoidal 2-categories. Then the in-duced map ΩBΦ : ΩBT → ΩBS is automati-cally a double loop map. Here S is the surfacecategory of [U. Tillmann, Invent. Math. 130, No.2, 257–275 (1997; Zbl 891.55019)] with ΩBS 'Z×BΓ+

∞. The categoryT is a monoidal 2-categorybuilt out of braids and it may be thought of asa special type of cobordism category. AlthoughΩBT is not homotopy equivalent to Z × Bβ+

∞, itis shown thatφ factors through the identity com-ponent Ω0BT which has a splitting as a doubleloop space asΩ2S3

×Ω2W (where the homotopytype of W is unknown).

An appendix is devoted to proving analogousresults for two geometrically defined maps, de-fined by identifying the braid group as a sub-group of the mapping class group of a subsur-face. Sarah Whitehouse (Sheffield)

1167.55006Richter, WilliamCombinatorial proofs of the lambda algebra ba-sis and EHP sequence.Proc. Am. Math. Soc. 137, No. 7, 2471-2482(2009).The Λ algebra is, for a fixed prime p, the asso-ciative differential bigraded algebra introducedby A. K. Bousfield et al. [Topology 5, 331–342(1966; Zbl 158.20502)] with the property thatH(Λ) = ExtA∗ (Z/(p),Z/(p)), the E2 term of theAdams spectral sequence of spheres. Λ has sub-complexes Λ(n) forming short exact sequencesΛ(n)→ Λ(n + 1)→ ΣnΛ(2n + 1) whose associatedlong exact sequences in cohomology correspondto the EHP sequences of homotopy groups. Inthis paper the author constructs an admissiblebasis of monomials for Λ, giving combinatorialproofs of its linear independence and spanningproperties and uses it to establish results aboutcomposition products in Λ which imply the ex-actness of the sequence above. He also gives newproofs of results about the Hopf invariant of anunstable Λ composition and about admissibleand symmetric Adem relations.

Keith Johnson (Halifax)

1166.57002Dasbach, Oliver T.; Lin, Xiao-SongA volumish theorem for the Jones polynomialof alternating knots.Pac. J. Math. 231, No. 2, 279-291 (2007).The present paper is inspired by the desire for ageometrical or topological interpretation for theJones polynomial.In this setting, the Volume Conjecture states thatthe coloured Jones polynomial of a knot K (whichis given by the Jones polynomial and the Jonespolynomial of cablings of K) determines the Gro-mov norm of the knot complement of K, andhence – for hyperbolic knots – the hyperbolicvolume: see [H. Murakami and J. Murakami, ActaMath. 186, No. 1, 85–104 (2001; Zbl 983.57009)].In the present paper the authors prove a Volu-mish Theorem for alternating knots in terms ofthe Jones polynomial, rather than the colouredJones polynomial: the ratio of the volume and


certain sums of coefficients of the Jones polyno-mial is bounded from above and from below byconstants.The proof of the Volumish Theorem makes use ofa result by M. Lackenby in [Proc. Lond. Math. Soc.,III. Ser. 88, No. 1, 204–224 (2004; Zbl 1041.57002)](see also [I. Agol, P. A. Storm, W. P. Thurston and N.Dunfield, J. Am. Math. Soc. 20, No. 4, 1053–1077(2007; Zbl 1155.58016)] for an improvement).The paper also contains some numerical data onthe relation between other coefficients and thehyperbolic volume, for both alternating and non-alternating knots. These data give some hope fora Volumish Theorem for non-alternating knots,as well. Maria Rita Casali (Modena)

1166.57003Futer, David; Ishikawa, Masaharu; Kabaya,Yuichi; Mattman, Thomas W.; Shimokawa,KoyaFinite surgeries on three-tangle Pretzel knots.Algebr. Geom. Topol. 9, No. 2, 743-771 (2009).T. W. Mattman [J. Knot Theory Ramifications 11,No. 6, 891–902 (2002; Zbl 1023.57016)] almostfinished the determination of Pretzel knots ad-mitting non-trivial finite Dehn surgery. In fact,he could not exclude the (−2, p, q)-Pretzel knotwhere p and q are odd and 5 ≤ p ≤ q. Thepaper under review concludes that this familydoes not admit non-trivial finite surgery. Thus,the only non-trivial finite Dehn surgeries on hy-perbolic Pretzel knots are 17, 18, 19-surgeries onthe (−2, 3, 7)-Pretzel knot, and 22, 23-surgeries onthe (−2, 3, 9)-Pretzel knot.The argument is as follows. First, assume 7 ≤p ≤ q or p = 5 and q ≥ 11. Then the 6-theorem by I. Agol [Geom. Topol. 4, 431–449(2000; Zbl 959.57009)] and M. Lackenby [Invent.Math. 140, No. 2, 243–282 (2000; Zbl 947.57016)]eliminates almost all slopes. The length of a slopeis calculated based on the construction of an idealtriangulation of the knot complement and a cuspcross-section. The remaining candidates are ex-cluded by Culler-Shalen norm arguments andsome group theoretic calculations.Hence, three knots, (−2, 5, 5), (−2, 5, 7) and(−2, 5, 9), remain. For the (−2, 5, 5)-Pretzel knot,there are five boundary slopes by A. E. Hatcherand U. Oertel [Topology 28, No. 4, 453–480 (1989;Zbl 686.57006)]. The detection of those bound-ary slopes except for one is shown by a similarargument to Y. Kabaya [A method to find idealpoints from ideal triangulations, preprint]. Then

the calculation of the Culler-Shalen norm impliesthat the trivial slope is the only finite surgery. Thecase for the (−2, 5, 9)-Pretzel knot is similar. How-ever, the case for the (−2, 5, 7)-Pretzel knot needsextra information for the number of ideal pointsfor some boundary slope. This is carried out bythe method of T. Ohtsuki [J. Math. Soc. Japan 46,No. 1, 51–87 (1994; Zbl 837.57006) and TopologyAppl. 93, No. 2, 131–159 (1999; Zbl 924.57004)].The same result was independently obtained byK. Ichihara and I. D. Jong [Algebr. Geom. Topol. 9,No. 2, 731–742 (2009; Zbl 1165.57006)]. The argu-ment is different, and they further determine allfinite surgeries for Montesinos knots.

Masakazu Teragaito (Hiroshima)

1166.57006van der Veen, RolandThe volume conjecture for augmented knottedtrivalent graphs.Algebr. Geom. Topol. 9, No. 2, 691-722 (2009).The volume conjecture [see R. M. Kashaev,Lett. Math. Phys. 39, No. 3, 269–275 (1997;Zbl 876.57007), H. Murakami and J. Mu-rakami, Acta Math. 186, No. 1, 85–104 (2001;Zbl 983.57009)] states that for any knot K

limn→∞

2πn

log∣∣∣∣Jn(K)

(eπi2n

)∣∣∣∣ = Vol(S3\ K),

where Jn is the n-colored Jones polynomial of Kand Vol is the simplicial volume.The SO(3)-volume conjecture is the same state-ment where n is restricted to be odd. This modi-fied version is expected to hold for links.The author proposes a generalization of theSO(3)-volume conjecture to knotted trivalentgraphs (KTGs), [see D. P. Thurston, The alge-bra of knotted trivalent graphs and Turaev’sshadow world. In: Ohtsuki, T. (ed.) et al., Invari-ants of knots and 3-manifolds. Proceedings of theworkshop, Kyoto, Japan, September 17–21, 2001.Coventry: Geometry and Topology Publications.Geom. Topol. Monogr. 4, 337–362 (2002)]. A KTGis a thickened trivalent one-dimensional complexembedded as a surface into S3. The author provesthat any KTG can be obtained from the standardtetrahedron using a combination of four certainoperations (the KTG moves). These moves canbe used to read the colored Jones invariant of theKTG.A fifth set of operations is introduced (the n-unzips) that consists of splitting (unzipping)an edge into two parallel edges and adding n


parallel rings encircling the two new unzippededges. An augmented KTG is a KTG that isobtained from the standard tetrahedron, usingthe KTG moves but replacing the unzip movesby n-unzips. The conjecture is proved for aug-mented KTGs with sufficiently many augmenta-tion rings.As a corollary it is re-proved that every linkis a sublink of an arithmetic link [see M. D.Baker, Pac. J. Math. 203, No. 2, 257–263 (2002;Zbl 1051.57007)]. Matilde Lalin (Edmonton)

1166.57014Gordon, Cameron McA.; Wu, Ying-QingToroidal Dehn fillings on hyperbolic 3-manifolds.Mem. Am. Math. Soc. 909, 140 p. (2008).For a hyperbolic 3-manifold M and a slope ron a torus boundary component T0 of M the3-manifold obtained by r-Dehn filling on M isdenoted by M(r). Thurston’s hyperbolic Dehnsurgery theorem asserts that for each boundarycomponent of M there are only finitely manyslopes r for which M(r) is not hyperbolic. Such aslope is called exceptional and by the Geometriza-tion Conjecture (recently proved by Perelman) aslope is exceptional if and only if M(r) is either re-ducible, ∂-reducible, annular, toroidal, or a smallSeifert fiber space. It is known that for two excep-tional slopes r, s on T0 the geometric intersectionnumber ∆(r, s) is small. In particular if r, s aretwo toroidal slopes (i.e., M(r), M(s) are toroidal),C. C. McA. Gordon [Trans. Am. Math. Soc. 350,No. 5, 1713–1790 (1998; Zbl 896.57011)] showedthat ∆(r, s) ≤ 8 and there are exactly two mani-folds M with∆ = 8, one with∆ = 7, and one with∆ = 6. For ∆ = 5 there are infinitely many suchmanifolds.In the paper under review the authors classify allhyperbolic 3-manifolds that admit toroidal Dehnfillings with ∆ = 5 or 4.The Main Theorem provides a list of 14 specifichyperbolic 3-manifolds Mi where ∂Mi consistsof two tori T0, T1 for i ∈ 1, 2, 3, 14 and a singletorus T0 otherwise. For each Mi there are toroidalslopes ri, si on T0 such that∆(ri, si) = 4 for seven ofthe Mi’s and ∆(ri, si) = 5 for the other seven. If Mis any hyperbolic 3-manifold admitting toroidalslopes r, s with ∆(r, s) = 4 or 5 then either (M, r, s)is equivalent to some (Mi, ri, si) or M is obtainedfrom M1, M2, M3 or M14 by attaching a solid torusto T1.

A corollary is that for a hyperbolic 3-manifold Mwith more than one boundary component andtwo toroidal slopes r, s with ∆(r, s) ≥ 4, either∆(r, s) = 4 and (M, r, s) is equivalent to (Mi, ri, si)for i ∈ 1, 2, 3, 14, or ∆(r, s) = 5 and (M, r, s) isequivalent to (M3, r3, s3).By showing that only M1, M2, and M3 can beembedded into S3, the authors obtain a completeclassification of all hyperbolic knots in S3 havingtoroidal surgeries with ∆(r, s) ≥ 4.The proofs of the theorems involve an intricate,delicate and detailed analysis of the intersectiongraphs of the two punctured tori in M that comefrom the essential tori in M(r) and M(s).

Wolfgang Heil (Tallahassee)

1166.57018Salamon, Dietmar; Wehrheim, KatrinInstanton Floer homology with Lagrangianboundary conditions.Geom. Topol. 12, No. 2, 747-918 (2008).Motivated by the authors’ programme to provethe Atiyah-Floer conjecture [M. Atiyah, Newinvariants of 3-and 4-dimensional manifolds.The mathematical heritage of Hermann Weyl,Proc. Symp., Durham/NC 1987, Proc. Symp.Pure Math. 48, 285-299 (1988; Zbl 667.57018)],the paper under review is to construct an in-stanton Floer homology group HF(Y,L) for acompact oriented 3-manifold Y with boundaryΣ := ∂Y and a gauge invariant, monotone, ir-reducible infinite-dimensional Lagrangian sub-manifold L of the moduli space A(Σ) of flatSU(2)-connections over the boundary Σ. The ac-tion of the gauge group G(Σ) = C∞(Σ,G) onA(Σ) is Hamiltonian, and the corresponding mo-ment map is the curvature. The moduli spaceMΣ := A flat(Σ)/G(Σ) of flat connections is a (sin-gular) symplectic manifold. A gauge invariant,monotone, irreducible infinite-dimensional La-grangian submanifold L of the moduli spaceA(Σ) descends to a (singular) Lagrangian sub-manifold L := L/G(Σ) ⊂MΣ.On the space A(Y,L) := A ∈ A(Y) | A|Σ ∈ Lof connections on Y with boundary values inL there is a gauge invariant Chern-Simon func-tional CSL with values inR/4π2Z, whose differ-ential is the usual Chern-Simons 1-form. The crit-ical points are the flat connections inA(Y,L). Thegradient flow lines of CSL with respect to the L2

inner product are smooth maps R→A(Y) : s→A(s) satisfying the equation: ∂sA + ∗FA = 0 andA(s)|Σ ∈ L for any s ∈ R. As in A. Floer’s original


work [Commun. Math. Phys. 118, No. 2, 215–240(1988; Zbl 684.53027)], the instanton Floer homol-ogy group HF(Y,L) is obtained by using the so-lutions of the equation to construct a boundaryoperator on the chain complex generated by thegauge equivalence classes of the nontrivial flatconnections inAflat(Y,L).For a disjoint union H of handlebodies with∂H = Σ, if Y

⋃Σ H is an integral homology 3-

sphere then the subsetLH ⊂ LH ⊂ A(Σ) of all flatconnections on Σ that extend to flat connectionson Y is a gauge invariant, monotone, irreducibleinfinite-dimensional Lagrangian submanifold ofA(Σ). In this case the authors expect that the ho-mology HF(Y,L) is naturally isomorphic to theinstanton Floer homology group HF(Y

⋃Σ H) of

a homology 3-sphere Y⋃Σ H constructed in [loc.

cit.].Given a Heegaard splitting M = H0

⋃Σ H1 of

a homology 3-sphere into two handlebodies Hiwith ∂Hi = Σ, the 3-manifold Y := [0, 1] × Σ hastwo boundary components ∂Y = Σ t H1, andattaching the disjoint union of the handlebod-ies H := H0 t H1 yields the homology 3-sphereY

⋃ΣtΣ H M. The Lagrangian submanifold is

LH0×LH1 LH ⊂ A(ΣtΣ). In this case the expec-tation is that HF([0, 1] × Σ,LH0 × LH1 ) FH(M)and that the symplectic Floer homology of thepair of Lagrangian submanifolds LH0 , LH1 of thesingular symplectic manifold MΣ is also isomor-phic to HF([0, 1] × Σ,LH0 × LH1 ).The construction of this detailed paper is basedon the foundational analysis by K. Wehrheim in[Commun. Contemp. Math. 6, No. 4, 601–635(2004; Zbl 1083.53086), Commun. Math. Phys.254, No. 1, 45–89 (2005; Zbl 1083.53029), ibid. 258,No. 2, 275–315 (2005; Zbl 1088.53013)]. Thoughthe exposition is to follow the work of A. Floer[loc. cit.] and S. K. Donaldson [Floer homologygroups in Yang-Mills theory. Cambridge: Cam-bridge University Press (2002; Zbl 998.53057)],some new phenomena and difficulties occur andare explained and overcome in details. They arealso helpful for understanding Floer’s and Don-aldson’s papers. In addition, this paper is read-able and a very good reference for people whowish to become acquainted with this field.

Guang-Cun Lu (Beijing)

1167.57002Minsky, Yair N.; Moriah, Yoav; Schleimer, SaulHigh distance knots.Algebr. Geom. Topol. 7, 1471-1483 (2007).

In the paper under review, for any pair of inte-gers g and n, the authors present a knot in the3-sphere with the exterior admitting a Heegaardsplitting of genus g having distance greater thann. The distance of a Heegaard splitting, whichis an invariant reflecting well the complexity ofa 3-manifold, was defined by J. Hempel [Topol-ogy 40, No. 3, 631–637 (2001; Zbl 985.57014)].Roughly speaking it is the distance betweenthe two disk complexes defined by D. McCul-lough [J. Differ. Geom. 33, No. 1, 1–65 (1991;Zbl 721.57008)] for compression bodies boundedby a Heegaard surface. Here we regard them assubcomplexes in the curve complex of the Hee-gaard surface. In that paper, Hempel gave ex-amples of 3-manifolds with arbitrary high dis-tance Heegaard splitting, and from these, it iseasy to find knots with exteriors having arbi-trary high distance Heegaard splitting in some3-manifolds. However, to construct such a knotin the prescribed ambient manifold is a challeng-ing problem, which is affirmatively answered inthe paper under review for the case that the man-ifold is the 3-sphere. One of the key ingredientsof the authors’ proof is to find a pseudo-Anosovgluing map for a Heegaard splitting of some knotexterior. Also, as a corollary, they give a knot inthe 3-sphere with tunnel number t so that theknot has no (t, b)-decomposition for any pair ofintegers t and b. Kazuhiro Ichihara (Nara)

1167.57005Juhász, AndrásFloer homology and surface decompositions.Geom. Topol. 12, No. 1, 299-350 (2008).The main theorem of the present article is aboutthe way Sutured Floer Homology, denoted SFH,changes under surface decomposition.(Main) Theorem 1.3. Let (M, γ) be a balanced su-tured manifold and let (M, γ) (M′, γ′) be asutured manifold decomposition. Suppose thatS is open and for every component V of R(γ) theset of closed components of S∩V consists of par-allel oriented boundary-coherent simple curves.Then

SFH(M′, γ′) =⊕s∈OS

SFH(M, γ, s).

In particular, SFH(M′, γ′) is a direct summand ofSFH(M, γ). OS stands for the set of outer Spinc

structures with respect to S.The proof of the main theorem goes along a gen-eralization of S. Sarkar’s and J. Wang’s algorithm


[“An Algorithm for Computing some HeegardFloer Homologies", arXiv:math.GT/ 0607777] tocompute SFH(M, γ) from any given balanced di-agram of (M, γ). The main applications of theapparatus developed in the article are listed asfollows:

1. a simplified proof that shows Knot Floer Ho-mology detects fibred knots, which has beenconjectured by Ozsváth-Szabó and was firstproved by Y. Ni [Invent. Math. 170, No. 3,577–608 (2007; Zbl 1138.57031)];

2. a simple proof of a theorem that Link FloerHomology detects the Thurston norm, whichwas proved for links in S3 by P. Ozsváth andZ. Szabó [Algebr. Geom. Topol. 8, No. 2, 615–692 (2008; Zbl 1144.57011)]. The author gen-eralizes this result to links in arbitrary 3-manifolds.

3. As an easy consequence of Theorem 1.3, theMurasugi sum formula is reproved (it hadalready been proved in [Y. Ni, Algebr. GeomTopol. 6, 513–537 (2006; Zbl 1103.57021)]).

The proofs of these three applications above aresimpler than the original ones because the authormakes no use of symplectic or contact geometry.Before the applications, the author obtains twoby-product theorems of the main theorem, pro-viding us with positive answers to Question 9.19and Conjecture 10.2 of [A. Juhász, Algebr. Geom.Topol. 6, 1429–1457 (2006; Zbl 1129.57039)].Theorem 1.4. Suppose that the balanced suturedmanifold (M, γ) is taut. Then

Z ≤ SFH(M, γ).

If Y is a closed connected oriented 3-manifoldand R ⊂ Y is a compact oriented surface withno closed components then we can obtain a bal-anced sutured manifold Y(R) = (M, γ), whereM = Y\Int(R × I) and γ = ∂R× [see Juhász, loc.cit., Example 2.6]. Furthermore, if K ⊂ Y is a knot,α ∈ H2(Y,K;Z), and i ∈ Z then let

HFK(Y,K, α, i) =⊕s∈Spinc(Y,K): <c1(s),α>=2i

HFK(Y,K, s). (0.-4)

Thus, the following theorem follows from themain theorem.Theorem 1.5. Let K be a null-homologous knot ina closed connected 3-manifold Y and let S ⊂ Y bea Seifert surface of K. Then

SFH(Y(S)) ' HFK(Y,K, [S], g(S)).

These two theorems together give a proof thatKnot Floer Homology detects the genus of a knot,as first proved by P. Ozsváth and Z. Szabó [Geom.Topol. 8, 311–334 (2004; Zbl 1056.57020)]. In par-ticular, if Y is a rational homology 3-sphere thenHFK(K, g(K)) , 0 and HFK(K, i) = 0 for i > g(K).The author also uses the theorems proved aboveto push forward the understanding of the ques-tion concerning the minimal n such that there isa depth n foliation on S3

\N(K), where K is a knotin S3 and N(K) ⊂ S3 is a normal neighbourhoodof K.Theorem 1.8. Let K be a null-homologous genus gknot in a rational homology 3-sphere Y. Supposethat the coefficient ag of the Alexander polyno-mial 4K(t) of K is non-zero and

rank(HFK(Y,K, g)) < 4.

Then Y\N(K) has a depth ≤ 2 taut foliation trans-verse to ∂N(K). Celso M. Doria (Florianapolis)

1166.58006Tang, Xiang; Yao, Yi-JunA universal deformation formula forH∞ with-out projectivity assumption.J. Noncommut. Geom. 3, No. 2, 151-179 (2009).The authors aim to find an universal deformationformula for Connes-Moscovici’s Hopf subalgebraH

1CM ⊂ HCM, by using a Fedosov’s quantization

type for symplectic diffeomorphisms. By univer-sal deformation formula of a Hopf algebra A,they mean an element R ∈ A[[~]]

⊗C[[~]] A[[~]]

satisfying the following conditions: (a) ((4 ⊗1)R)(R ⊗ 1) = ((1 ⊗ 4)R)(1 ⊗ R); (b) (ε ⊗ 1)(R) =1 ⊗ 1 = (1 ⊗ ε)(R). Here 4 is the coproduct andε the counit in A. The Connes-Moscovici HopfalgebraHCM is the enveloping algebra of the Liealgebra which is the linear space < X,Y, δn >n≥1,with the following brackets:

[X,Y] = X,[Y, δn] = nδn,

[δn, δm] = 0,[X, δn] = δn+1,

m,n ≥ 1.

(♣)

The coproduct 4 in HCM is defined by: 4(Y) =Y ⊗ 1 + 1 ⊗ Y, 4(X) = X ⊗ 1 + 1 ⊗ X + δ1 ⊗ Y,4(δ1) = δ1 ⊗ 1 + 1 ⊗ δ1, where 4(δn) is defined re-


cursively, using above brackets (♣) and the iden-tity 4(h1h2) = 4(h1)4(h2), ∀h1, h2 ∈ HCM. Thealgebra generated by δnn≥1 is the graded sub-Hopf algebra H1

CM ⊂ HCM, where the gradia-tion is defined by gr(δn1 , . . . , δns ) = n1 + · · · + ns.H

1CM is Hopf-isomorphic to the so-called de Fadd

di Bruno algebra, HFB ≡ R[a1, . . . , an, . . . ], wherean : G→ R are suitable functions over the groupG of so-called formal tangent diffeomorphisms. TheFedosov’s quantization of a symplectic manifold(M, ω) is obtained by a bijection f 7→ f , betweenthe commutative algebra C∞(M) and a noncom-mutative algebra WD, identified by a connec-tion D, (Fedosov’s connection), on a suitable vectorfiber bundle W → M, depending on a deforma-tion formal parameter ~. The product inWD can beconsidered a generalized “Moyal product”. Thistype of quantization can be extended to sym-plectic diffeomorphisms by extending the fiberbundle W. Then by generalizing the Fedosov’smethod they are able to obtain a quantization ofa pseudogroup Γ of a symplectic manifold M.Their main result is to prove that, by consideringM ≡ R×R+, it is possible construct a Moyal typeproduct on C∞0 (M)×r Γ[[~]] that can be expressedby an element R ∈ H1

CM

⊗C[[~]]H

1CM. (Here ×r

denotes semidirect product.)Reviewer’s remark. Hopf algebras are very use-ful structures whether in commutative geome-try or in noncommutative ones. Let us recallhere that conservation laws of PDE’s identifyextended Hopf algebras that are commutativewhen PDE’s are defined in the category of com-mutative manifolds, and result, instead, quan-tum algebras for quantum PDE’s. Then extended(quantum) Hopf algebras play a central roleto characterize global solutions in (quantum)PDE’s. (See recent works on the geometry of(quantum) PDE’s by the reviewer of this paper.)

Agostino Prástaro (Roma)

1167.58001Mitrea, Dorina; Mitrea, Marius; Shaw, Mei-ChiTraces of differential forms on Lipschitz do-mains, the boundary de Rham complex andHodge decompositions.Indiana Univ. Math. J. 57, No. 5, 2061-2095(2008).The authors are interested to study some aspectsof the de Rham theory relatively to Lipschitz sub-domains Ω of finite n-dimensional compact Rie-mannian manifolds M. They refer to the follow-ing theorems, well-known in functional analysis.

(In the paper also some related authors with theirreferences are quoted.)Theorem A. Ω is a bounded, Lipschitz domainin Rn and 1 < p < ∞, α ∈ R, then the operatorof restriction to Ω, mapping Lp

α(Rn) onto Lpα(Ω),

has a linear continuous right inverseE : Lpα(Ω)→

Lpα(Rn), such that (Eu)|Ω = u, for each u ∈ Lp

α(Ω).Theorem B. Assume that Ω is a bounded, Lip-schitz domain in Rn and that 1 < p < ∞,α ∈ (1/p, 1 + 1/p) ⊂ R. Then the operator of re-striction C∞(Ω) → Lip(∂Ω), extends to a map-ping TR : Lp

α(Ω)→ Bp,pα−1/p(∂Ω). Furthermore, this

operator is onto; indeed it has a bounded linearright inverse.The purpose of this paper is just to prove suit-able versions of Theorem A and Theorem B inthe case when differential forms are consideredin place of scalar functions.The main result is an extension theorem fromΩ to M, for suitable classes of differential p-forms, (denoted Dl(d; Lp

s (Ω))), and the existenceof a linear, bounded mapping projecting differen-tial forms of Dl(d; Lp

s (Ω)) to other ones (denotedNBp,p

s−1/p(∂Ω,Λl)), on ∂Ω.The paper, after a detailed introduction, splitsinto seven more sections. 2. The geometrical set-ting. 3. Differential forms with Sobolev-Besov co-efficients. (Dl(d; Lp

s (Ω)) is the space of l-forms uwith coefficients in Lp

s (Ω) for which du has alsocoefficients in Lp

s (Ω). NBp,ps−1/p(∂Ω,Λl) is the space

of differential forms ξ ∈ Bp,ps−1/p(∂Ω,Λl+1), such

that there exists η ∈ Bp,ps−1/p(∂Ω,Λl+2) satisfying

〈ξ, (δ f )|∂Ω〉, ∀ f ∈ C∞(M, Λl+2). In the paper, Lps de-

note Sobolev spaces, and Bp,ps Besov spaces.) 4.

Traces of differential forms. 5. The boundary deRham complex. 6. Extending differential formsfrom Ω to M. 7. Hoodge decomposition. (Here,an Hoodge-type decomposition is given on thetopologic vector space Lp

s (Ω,Λl).)Agostino Prástaro (Roma)

1167.58011Moerdijk, I.; Mrcun, J.On the developability of Lie subalgebroids.Adv. Math. 210, No. 1, 1-21 (2007).The paper under review is the second half of atwo-parts work dedicated to the question of in-tegrability of Lie algebroids (the first half is re-viewed in [Adv. Math. 204, No. 1, 101–115 (2006;Zbl 1131.58015)]).If G is a Lie groupoid with Lie algebroid g, givena Lie subalgebroid h ⊂ g, there is a Lie groupoid


H, with Lie algebroid h, and a groupoid homo-morphism H→ G which is an immersion. Unlikethe case of Lie groups, in general, we have:

1. it is not possible to choose this immersion tobe injective;

2. even when such an immersion exists, the clo-sure of its image is not a Lie subgroupoid.

In [Moerdijk (loc. cit.)], the authors have givencriteria for such properties to hold. These criteriaare stated in terms of properties of the invari-ant foliation F (h) of G associated to Lie subalge-broid h (defined by integrating the invariant vec-tor fields tangent to h). Namely, they show that (i)one can choose an injective immersion H→ G iffthe foliationF (h) has trivial holonomy; and (ii) ifthe foliation F (h) is transversely complete, thenone can choose an injective immersion H → Gfor which the closure of its image is a Lie sub-groupoid H ⊂ G. Moreover, the image is closediff this injective immersion is actually an embed-ding.In the present paper, the authors complete theirstudy of Lie subalgebroids by considering thequestion: when is a Lie subalgebroid integrableby a closed Lie subgroup? More precisely, if Gis a Lie groupoid with Lie algebroid g, call a Liesubalgebroid h ⊂ g developable if it can be inte-grated to a closed subgroupoid of the universalcovering groupoid G. The reason for this nameis the following: recall that a foliation F of amanifold M is called developable if its lift to theuniversal covering space M is given by the fibersof a submersion. Now, the authors show that aLie subalgebroid h ⊂ g is developable iff the liftof F (h) to the universal covering groupoid G is adevelopable foliation.The main theorem in this paper states that fora proper, Hausdorff, Lie groupoid G, a Lie sub-algebroid h ⊂ g for which F (h) is transverselycomplete is developable iff a certain Lie algebroidb(G, h) is integrable.Note that, contrary to the usual Lie theory, not ev-ery Lie algebroid is integrable. The first obstruc-tion to integrability was noted by R. Almeida andP. Molino [C. R. Acad. Sci., Paris, Sér. I 300, 13–15 (1985; Zbl 582.57015)], while a complete the-ory was given more recently in [M. Crainic andR. L. Fernandes, Ann. Math. (2) 157, No. 2, 575–620 (2003; Zbl 1037.22003)]. The classical resultof Almeida and Molino states that a transverselycomplete foliation F on a compact manifold Mis developable iff a certain Lie algebroid b(M,F )

is integrable. The Almeida and Molino result canbe seen as a very special case of the main theo-rem in this paper, where G is the pair groupoidM ×M and h ⊂ g is the subalgebroid TF ⊂ TM.A fundamental fact underlying the proof of themain theorem is that if G is a proper, Hausdorff,Lie groupoid, then for every Lie subgroupoidH ⊂ G for which F (h) is transversely complete,the Lie subalgebroid h ⊂ g is given by the kernelof a Lie algebroid valued Maurer-Cartan form.

Rui Loja Fernandes (Lisbon)

1167.58013Daletskii, AlexeiFamilies of Witten Laplacians associated withinteracting particle systems, and von Neumannalgebras.Albeverio, Sergio (ed) et al., Traces in num-ber theory, geometry and quantum fields. Wies-baden: Vieweg (ISBN 978-3-8348-0371-9/hbk).Aspects of Mathematics E38, 111-127 (2008).The author reports on recent results relatingvon Neumann algebras to Witten Laplacians as-sociated with Campbell measures on compactconfiguration spaces. In this framework, he de-fines traces of semigroup operators, proving ananalogue of the McKean-Singer formula, and es-tablishes the stability of the regularized index ofthe corresponding Dirac operator.The paper, after an introduction, splits into fourmore sections and one appendix, entitled 2. Fam-ily of Witten Laplacians; 3. Von Neumann al-gebras associated with configuration spaces; 4.Probabilistic representations of the heat kernels;5. Stability of the index; Appendix: Gibbs mea-sures on configuration spaces.Let X be a Riemannian manifold of infinite vol-ume. Set ΓX ≡ γ ⊂ X | |γ ∩ Λ| < ∞ for eachcompact Λ ⊂ X, where |A| denotes cardinalityof a set A. Assume that there exists an infinitegroup G of isometries of X such that X/G is acompact Riemannian manifold. Let v ∈ C2

0(R)with supp(v) ⊂ [−r, r], r > 0, and V : X × X→ R,V(x, y) = v(ρ(x, y)), where ρ is the Riemanniandistance on X. For a fixed locally finite set γ ⊂ X,σγ(dx) = e−Eγ(x)dx is a measure on X, whereEγ(x) =

∑y∈γ V(x, y) < ∞, and dx is the Rieman-

nian measure on X. Let H(p)σγ ≡ dp(dp

γ)∗ + (dpγ)∗dp

denote the Witten-Bismut-Laplacian on L2σγΩ

p,the space of p-forms on X which are square-integrable with respect to the volume measureσγ. Let U : L2

σγΩp(X) → L2Ωp(X) be the uni-

tary isomorphism defined by the multiplication


by e−12 Eγ(x). Then the operator H(p)

γ ≡ UH(p)σγ U :

L2Ωp(X) → L2Ωp(X), expressed by means of thede Rham Laplacian H(p), on p-forms on X is givenby H(p)

γ = H(p) + W(p)γ = 4(p) + R(p) + W(p)

γ (modified

Weitzenböck formula), where W(p)γ is a term de-

pending on Eγ, and G-invariant. Then, for any p,one can construct a von Neumann W∗-algebraCp

containing the semigroup T(p)γ,t ≡ e−tH(p)

γ , t ≥ 0. Forsuch W∗-algebras, one identifies a faithful nor-mal semifinite trace TR and a McKean-Singerformula:∑

0≤p≤dim X

(−1)pTRT(p)i =∑

0≤p≤dim X

(−1)pTRP(p),(♣)

where T(p)i and P(p) are, respectively, the maps

T(p)i : γ 7→ T(p)

γ,t and P(p) : γ 7→ P(p)γ , with

P(p)γ : L2Ωp(X)→ lim ker H(p)

γ the orthogonal pro-jection. Formula (♣) follows from the McKean–Singer formula in von Neumann algebra, ap-plied to the algebra C ≡

⊕p C

p and operatorsD ≡

∑p dp, D∗ ≡

∑p(dp

γ)∗ in L2µΩ ≡

⊕p L2

µΩp. The

right-hand side of formula (♣) can be understoodas a regularized index of the Dirac operator D+D∗

acting on the space L2µΩ. The McKean–Singer for-

mula states that for a Dirac-type operator D, overM, and for each t > 0, the index ind(D) of Dadmits the following representation by means

of the heat kernel 〈x|e−tD2|y〉 of the square of

D, i.e., the kernel of the heat semigroup e−tD2,

((e−tD2 s)(x) =∫

M〈x|e−tD2|y〉s(y) dy) : ind(D) =

STR(e−tD2) =

∫M STR〈x|e

−tD2|x〉 dx, where dx is

the Riemannian measure on M and STR is thesupertrace. (If E = E+

⊕E− is a Z2-graded vec-

tor space and A ∈ End(E), one defines the super-trace by STR(A) = TR(cA), where c ∈ End(E) isthe chirality operator identified with ±1 on E±.)The following properties hold for the heat ker-nel: (i) it is smooth; (ii) smooth kernels are trace-class, hence the kernel of D is finite-dimensional;(iii) there is uniform convergence of 〈x|e−tD2

|y〉 tothe projection onto ker(D) as t → ∞; (iv) thereexists an asymptotic expansion for 〈x|e−tD2

|y〉 atsmall t. Thus the McKean–Singer formula allowsto state that ind(D) is given in terms of the re-striction to the diagonal of the heat kernel of D2

at arbitrarily small times. Furthermore, by uti-lizing the asymptotic property (iv), there existslimt→0 STR〈x|e−tD2

|x〉 and this can be expressedby means geometric characteristics. Then, onecan define a supertrace on P, obtaining the im-portant formula STRP ≡

∑p(−1)pTRP(p) = χ(M),

where χ(M) is the Euler characteristic of M ≡

X/G. This proves that the regularized index ofthe Dirac operator D + D∗ does not depend onthe choice of the potential V and the measureµ on ΓX. As a byproduct, one gets also that un-der the following hypotheses: (a) the action ofG on ΓX is ergodic, (b) χ(M) , 0, it follows thatdim ker Hγ = ∞ for a.a. γ.

Agostino Prástaro (Roma)

Probability Theory. Statistics. Applications to Economics and Biology. 109

Probability Theory. Statistics.Applications to Economics and Biology.

1166.60001Gut, AllanStopped random walks. Limit theorems and ap-plications. 2nd ed.Springer Series in Operations Research and Fi-nancial Engineering. New York, NY: Springer(ISBN 978-0-387-87834-8/hbk; 978-0-387-87835-5/ebook). xiii, 263 p. EUR 34.95/net; SFR 58.50;£ 32.99; $ 49.95 (2009).This is an updated second edition of the book[A. Gut, Stopped random walks. Limit the-orems and applications. Applied Probability,Vol. 5. (New York) etc.: Springer-Verlag. (1988;Zbl 634.60061)]. The present edition contains anew Chapter 6 devoted to the perturbed randomwalks, that is processes of the type Zn = Sn + ξn,where Sn is a random walk with positive finitemean increments, and ξn is a perturbation se-quence satisfying n−1ξn → 0 a.s. Various limittheorems are proven in this chapter, in particu-lar for the case Zn = ng( 1

n∑n

k=1 Yk), with Yk be-ing i.i.d. random variables with positive finitemean, and g(·) belonging to a certain class ofnon-negative continuous functions. The book’sbibliography is also considerably extended andcontains 324 entries. Ilya Pavlyukevich (Berlin)

1167.60001Lamberton, Damien; Lapeyre, BernhardIntroduction to stochastic calculus applied tofinance. 2nd ed.Chapman & Hall/CRC Financial MathematicsSeries. Boca Raton, FL: Chapman & Hall/CRC(ISBN 978-1-58488-626-6/hbk). 253 p. £ 34.99;$ 69.95 (2008).This book is the second English edition (from2008) of the textbook whose first French editionwas among the first books that provided an ex-cellent mathematical introduction into the the-ory of mathematical finance in continuous time.Aimed at a mathematically trained readership, itis still one of the best introductions to this field.The earlier parts of the text contain a brief intro-duction to the mathematical theory of stochas-tic calculus, as needed for solving mathemat-ical problems from finance. The latter area isthe main theme of the book and is covered in

a wider scope – ranging from optimal stoppingproblems for American options and the Black-Scholes model, over martingale and PDE meth-ods for option pricing, to models for interest rateand credit derivatives. Also Monte-Carlo simu-lation methods are treated. Despite this range,the book is comparably compact, the presenta-tion being concise and to the point. Results anddefinitions are rigorously stated, and referencesare provided for results whose proofs are omit-ted. Some results are provided through guidedexercises, for instance the Dupire local volatilitymodel or the caplet pricing formula for the Li-bor market model. This serves mathematicallyadvanced readers well who want to learn aboutmathematical finance but do not need to be hand-waved through the ideas of stochastic calculus insome more informal way. On the other hand, thefocus on applications sets the book apart fromother mathematical books that mainly concen-trate on stochastic theory with some excursionsto financial applications. The new second editionhas been complemented by additional topics, forinstance the Libor market model, credit risk mod-els and the change of numeraire technique. Alsofurther exercises have been added, this includescomputer experiments with code provided on-line in Scilab, which is open source software.

Dirk Becherer (Berlin)

1166.62001Claeskens, Gerda; Hjort, Nils LidModel selection and model averaging.Cambridge Series on Statistical and Probabilis-tic Mathematics 27. Cambridge: CambridgeUniversity Press (ISBN 978-0-521-85225-8/hbk).xvii, 312 p. £ 40.00; $ 70.00 (2008).Statisticians and data analysts have to makechoices when data have been collected and amodel must be chosen to best describe and sum-marise these data. Another choice has also tobe made about whether all measured variablesare important enough to be included in a modelaimed for prediction. However, given a data set,there are numerous models that could possiblybe used, and the choice of the most suitable mod-els becomes a central and crucial task.

110 Probability Theory. Statistics. Applications to Economics and Biology.

The past two decades have seen rapid advancesboth in the ability to fit models and in the theo-retical understanding of model selection neededto harness this ability, and this book providesa synthesis of research from this active field. Inaddition, it gives practical advice to researchersconfronted with conflicting results. More specif-ically, within 10 chapters, model choice crite-ria are explained, discussed and compared, in-cluding the Akaike’s information criterion, theBayesian information criterion and the focusedinformation criterion. The uncertainties involvedwith model selection are also addressed, withdiscussions of frequentist and Bayesian meth-ods. Finally, the book presents model averagingschemes that combine the strength of several can-didate models.Worked examples on real data are complementedby derivations that provide deeper insight intothe methodology. Exercises, both theoretical anddata-based, guide the reader to familiarity withthe methods. All data analyses are compatiblewith open-source R software, and data sets andR code are available from a companion website.The methods used in the book are mostly basedon likelihoods. Therefore, to read the book itwould be helpful for the reader to have at leastsome knowledge of the likelihood functions. Thebook further assumes that the readers are famil-iar with applied regression and basic matrix com-putations. It is intended for those interested inmodel selection and model averaging, includingstudents with a background in regression mod-elling and researchers in statistically orientedfields. Christina Diakaki (Chania)

1166.62003Sarkar, DeepayanLattice. Multivariate data visualization with R.Use R!. New York, NY: Springer (ISBN 978-0-387-75968-5/pbk). xvii, 265 p. EUR 44.95/net;SFR 78.50; £ 34.50; $ 54.95 (2008).The Lattice package is a software that extendsthe R language and environment for statisticalcomputing by providing a coherent set of toolsto produce statistical graphics with emphasison multivariate data. However, Lattice is a self-contained system that is largely independent ofother graphic facilities in R and easily extend-able to handle demands of cutting edge research.Written by the developer of the Lattice system,the book describes Lattice in considerable depth.

Organised in 3 parts, the book begins with theessentials and systematically delves into specificlow level details necessary within the 14 chaptersincluding approximately 150 figures producedwith Lattice. Many of the examples emphasiseprinciples of good practical design, and almostall use real data sets that are publicly available invarious R packages. All codes and figures in thebook are also available online, along with supple-mentary material covering more advanced top-ics.No prior experience with Lattice is required toread the book, although basic familiarity withR is assumed. As such, it is intended for bothlong-time and new R users looking for a systemto produce conventional statistical graphics. It isalso intended for existing Lattice users willing tolearn more about R programming, so as to gainincreased flexibility, and developers who wish toimplement new graphical displays, building onthe infrastructure already available in Lattice.

Vangelis Grigoroudis (Chania)

1166.62049Ryan, Thomas P.Modern regression methods. 2nd ed.Wiley Series in Probability and Statistics.Hoboken, NJ: John Wiley & Sons (ISBN 978-0-470-08186-0/hbk). xvii, 642 p. EUR 82.20; £ 67.95(2009).[For the review of the first edition from 1997 seeZbl 885.62074.]This new edition has been updated and en-hanced to include all new information on thelatest advances and research in the evolvingfield of regression analysis. It covers diagnos-tics, transformations, linear regression, logisticregression, nonparametric regression, robust re-gression, ridge regression, nonlinear regression,and experimental designs for regression.There are a total of 16 chapters in this book. InChapter 1, an introduction to the fundamentalsof simple linear regression is given. Chapter 2completes the introduction to simple linear re-gression given in Chapter 1 by discussing diag-nostic procedures and additional plots. Chapter3 covers simple linear regression with matrix al-gebra. In Chapter 4, an introduction to funda-mental concepts of multiple linear regression isgiven which includes orthogonal and correlatedregressors, multicollinearity, signs of regressioncoefficients, and centering and scaling. Chapter5 discusses how to use various types of regres-


sion plots for detecting influential observationsand for detecting the possible need to transformone or more regressors. Emphasis has been givento partial residual plots, partial regression plots,and variations of each. Chapter 6 examines someof the commonly used methods for deciding on atransformation of the dependent variable and/orregressors. In Chapter 7, various methods usedfor selecting a subset of regression variables arediscussed, and Chapter 8 considers various is-sues in the use of polynomial terms and alsogives indications when trigonometric terms arelikely to be of value.Chapter 9 contains a survey of the state-of-the-art of many topics in logistic regression. Chap-ter 10 presents several nonparametric regressiontechniques, ranging from monotonic regressionto local regression. Chapter 11 contributes to ro-bust regression. Ridge regression is discussed inChapter 12. Chapter 13 provides an introductionto nonlinear regression, with two examples usedto illustrate basic concepts. Chapter 14 focuseson the statistical aspects of experimental designsfor regression. Chapter 15 contains many differ-ent regression methods that were not coveredin the preceding chapters, including quantile re-gression and Poisson regression. In Chapter 16,analyses of five challenging data sets are pre-sented.Each chapter contains a brief section for SAS,Minitab, SPSS, BMDP, and Systat users, thoughparticular software packages are not emphasizedoverall. Besides, each chapter ends with a sum-mary, references and exercises and some chap-ters have their own appendices. It is noted thatanswers to selected exercises are provided. In ad-dition, Some statistical tables are also given at theend of the book.This book can serve as a textbook for undergrad-uates and graduate courses on regression mod-els. It can also be used as a reference book forfaculty and professionals. Yuehua Wu (Toronto)

1166.62076Marques de Sá, Joaquim P.Applied statistics. Using SPSS, STATISTICA,MATLAB and R. With CD-ROM. 2nd ed.Berlin: Springer (ISBN 978-3-540-71971-7/hbl).xxiv, 505 p. EUR 59.95/net; SFR 98.50; £ 46.00;$ 79.95 (2007).[For the review of the first edition from 2003 seeZbl 1028.62082.]

This book is intended as a reference book for stu-dents, professionals and research workers whoneed to apply statistical analyses to a large vari-ety of practical problems using SPSS, MATLAB,STATISTICA and R. Within 10 chapters and 6appendices, it provides a comprehensive cover-age of the main statistical analysis topics impor-tant for practical applications, such as data de-scription, statistical inference, classification, re-gression, factor analysis, survival data and direc-tional statistics. The relevant notions and meth-ods are explained concisely and illustrated withpractical examples that use real data and are pre-sented so as to clarify sensible practical issues,while the solutions presented in the examplesare obtained with one of the aforementioned soft-ware packages.The book provides guidance on how to use SPSS,MATLAB, STATISTICA and R in statistical analy-sis applications without having to delve into themanuals. The accompanying CD-Rom includesseveral specific software tools for the topics de-scribed in the book, including sets of MATLABand R functions, as well as the data sets usedin the examples and exercises covering a broadspectrum of areas from engineering, medicine,biology, psychology, economy, geology, and as-tronomy. Vangelis Grigoroudis (Chania)

1167.62002Kowalski, Jeanne; Tu, Xin M.Modern applied U-statistics.Wiley Series in Probability and Statistics;Wiley-Interscience. Hoboken, NJ: John Wiley& Sons (ISBN 978-0-471-68227-1/hbk). xi, 378 p.EUR 69.90; £ 52.95 (2008).With focus on longitudinal data modelling, thisbook is an introduction to U-statistics and its ap-plications in genetics, biomedical and psycho-social research. After introducing in Chapter1the fundamental statistical ideas needed forthe development of U-statistics theory, paramet-ric and distribution-free regression models forcross-sectional study analyses are discussed inChapter 2. Univariate U-statistics are introducedin Chapter 3, with associated models and infer-ences. Models for clustered data arising, for ex-ample, from cross-sectional study designs in sur-vey and epidemiological research, are discussedin Chapter 4 and the models discussed includeparametric, distribution-free, missing data andstructural equations models. Chapter 5 gener-alizes the idea of univariate U-statistics to the


multivariate setting and extends the theory de-veloped in Chapter 3 to longitudinal data anal-ysis. Chapter 6 introduces functional responsemodels which integrate nonparametric statisticsand distribution-free models. Inference for thesemodels are discussed by developing U-statisticsbased on estimating equations.Examples are discussed throughout the book.Every chapter ends with Exercises com-plementing the material in the chapterand the book ends with a list of refer-ences and a subject index. The dedicatedwebsite www.cancerbiostats.onc.jhmi.edu/Kowalski/ustatsbook has been created for up-dates and data applications.Chapter 1 on Preliminaries is written in a sloppymanner and a few conceptual mistakes havecrept in. The examples discussed in every chap-ter complement the theory discussed. There areat least 70 misprints / mistakes and this reflectsthe poor style of writing. But the book is a wel-come addition to the U-statistics literature andthe new material discussed in the book will be ofinterest to practitioners in the fields of bioinfor-matics and psycho-social research.

Ravi Sreenivasan (Mysore)

1167.91001Balk, Bert M.Price and quantity index numbers. Models formeasuring aggregate change and difference.Cambridge: Cambridge University Press (ISBN978-0-521-88907-0/hbk). xv, 283 p. £ 48.00; $ 85.00(2008).The book is the first comprehensive text on in-dex number theory since Irving Fisher’s (1922)“The Making of Index Numbers". There, the ax-iomatic approach to the theory of price and quan-tity index was presented. The author presentstheir original system of axioms. Many conclu-sions implied by these axioms are studied. Thementioned conclusions are given as a mathemat-ical theorems which are proved.In my opinion, this book should be read by eacheconomist who applies price and quantity indexand by each statistician who studies the price andquantity problem. Krzysztof Piasecki (Poznan)

1167.91002Schöne, StefanAuctions in the electricity market. Biddingwhen production capacity is constrained.Lecture Notes in Economics and Mathemati-

cal Systems 617. Berlin: Springer (ISBN 978-3-540-85364-0/pbk; 978-3-540-85365-7/ebook). xvi,218 p. EUR 64.95/net; SFR 108.00; £ 59.99; $ 84.95(2009).Electricity is an essential commodity traded atpower exchanges. Its price is very volatile withina day and over the year. This raises questionsabout the efficiency of the trading rules.This work analyzes the bidding behaviour oftwo risk-neutral generators under three auctiontypes. Each of the generators owns one powerplant with the same production capacity. Pro-duction costs are affiliated. This allows for in-dependence or positive correlation. The authoranalyzes and compares a uniform-price, a dis-criminatory, and a generalized second-price auc-tion. Optimal bids, cost efficiency, profits, andconsumer prices are examined. A simple prob-ability density function of affiliated productioncosts is given and used for examples. Numericalresults are presented.The results of the analysis can help improvingthe bidding strategies of producers, selecting thebest auction type at power exchanges or detect-ing price manipulations.

Rózsa Horvàth-Bokor (Budapest)

1167.91003Anand, Paul (ed.); Pattanaik, Prasanta K. (ed.);Puppe, Clemens (ed.)The handbook of rational and social choice. Anoverview of new foundations and applications.Oxford: Oxford University Press (ISBN 978-0-19-929042-0/hbk). ix, 581 p. £ 85.00 (2009).This handbook of different authors provides anoverview of issues arising in work on the foun-dations of decision theory and social choice. Thiscollection shows how the related areas of deci-sion theory and social choice have developed intheir applications and moved well beyond therebasic models of expected utility and utilitarianapproches to welfare economics.Containing 23 contributions, the handbookshows how the normative foundations of eco-nomics have changed dramatically as more gen-eral and explicit models of utility and groupchoice have been developed. These develop-ments have been brought together in a mannerthat seeks to identify and make accessible thatrecent themes and developments. This collectionwill be of particular value to each researchers ineconomics with interests in utility or welfare butit will also be of interest to social scientists or


philosophers interested in theories of rationalityor group decision-making.

Klaus Ehemann (Karlsruhe)

1166.91001Mak, Don K.Mathematical techniques in financial markettrading.Hackensack, NJ: World Scientific (ISBN 981-256-699-6/hbk). xvi, 304 p. £ 39.00 (2006).This book aims to analyze the equipment thatprofessional traders use and to distinguish thetools from the junk. It consists of 14 Chapters.The latest development of scientific investigationin the financial market is presented. A new field,called Econophysics, has cropped up. Since thereexist evidence that the market is non-random,market crashes have been considered to be non-random events. The trending indicators used bytraders are analyzed. The exponential movingaverage is characterized using spectrum analy-sis. Other low pass filters are also considered.Causal wavelet filters, which are actually band-pass filters are described. It is demonstrated howa series of causal wavelet filters with differentfrequency ranges can be constructed. Frequencyresponse function of a causal system is studied.Several newly created causal high-pass filters arecompared. Some of the popular trading tacticsare analyzed in order to differentiate the truthsfrom the myths. The advantages and disadvan-tages of a long-term timeframe are pointed out.The market is assumed to be random in Chapters12 and 13, where, in particular, the optimizationof the gain by moving the stop-loss is proposed.Most of the mathematical derivations and severalcomputer programs are listed in the Appendices.The book will be useful both for practitioners andacademicians. Yuliya S. Mishura (Kyïv)

1166.92023Xia, XuhuaBioinformatics and the cell. Modern computa-tional approaches in genomics, proteomics andtranscriptomics.New York, NY: Springer (ISBN 978-0-387-71336-6/hbk). xv, 349 p. EUR 106.95 (2007).In this book the author makes an “effort to ren-der both mathematical equations and biology tonumbers". Following this premise, he works outa lot of illustrative examples to make biologistsunderstand the mathematics and computational

scientists understand the biology of a wide rangeof problems in bioinformatics.In the first chapters the author introduces themathematics of string-matching algorithms inFASTA and BLAST and explains pairwise andmultiple sequence alignments. There are sectionsabout aligning rRNA genes with the constraint ofsecondary structure and about alignments of nu-cleotide sequences against amino acid sequences.A whole chapter is devoted to contig assemblyalgorithms. Several chapters deal with gene andmotif predictions. Position weight matrices, per-ceptrons and hidden Markov models are intro-duced. The Gibbs sampler is used to identify reg-ulatory sequences in DNA or functional motifsin proteins.The book also covers the analysis of proteinsand proteomes, e.g., calculation of the isoelec-tric point, protein separation with 2D-PAGE,and mass spectrometry. The reader learns howto calculate expected 2D-PAGE separation pat-terns and how to use these in-silico gels to iden-tify post-translational protein modifications. Af-ter estimating the molecular mass of peptidesfrom MS outputs using charge deconvolutions,the author explains the different steps of peptidemass fingerprinting to identify proteins in spotsof 2D-SDS-PAGE gels.Two chapters introduce the reader to essentialbiological processes such as genome replication,transcription, and translation within the frame-work of molecular evolution. A long chapter isdevoted to phylogenetic methods. It is an intro-duction to the construction of branching patternsand shall prepare the readers for more advancedtopics in molecular phylogenetics. Other topicscovered by the book are the characterization oftranslation efficiency and clustering algorithmssuch as UPGMA and self-organizing maps. Thereader learns about the EM algorithm for maxi-mum likelihood calculations and about Bayesianinference including Markov chain Monte Carloalgorithms for evaluating posterior probabilities.The book is addressed to graduate students ma-joring in sciences and software engineering. Bi-ologists with a sound knowledge of computerprogramming should be able to implement thepresented algorithms in their own programs.

Wiebke Werft (Heidelberg)


60024Kallenberg, OlavSome highlights from the theory of multivari-ate symmetries.Rend. Mat. Appl., VII. Ser. 28, No. 1, 19-32(2008).This paper’s stated aim is to provide an in-formal introduction to some basic notions andresults on various probabilistic symmetries ford-dimensional arrays. The detailed theory un-derpinning these results can be found in [O.Kallenberg, Probabilistic Symmetries and In-variance Principles. Probability and Its Ap-plications. (New York), NY: Springer. (2005;Zbl 1084.60003)]. First an historical overview isgiven of characterizations of infinite sequenceswhere the distributions are invariant under con-tractions, permutations or rotations. Extensionsto separately and jointly contractable and ex-changeable arrays are provided followed by rep-resentations for rotatable arrays. The summary ofresults on representations of exchangeable ran-dom sheets shows the author’s deep insight intothe connections between representations of con-tactable or exchangeable arrays and representa-tions of rotatable random functionals in terms ofmultiple Wiener Itô integrals, which in turn canbe used to obtain representations of contactableor exchangeable random sheets. The paper con-cludes with seven open problems.

Neville Weber (Sydney)

1167.60002Rosin, Amber; Sharobiem, Mary; Swift, Ran-dallDice sums.Math. Sci. 33, No. 2, 99-109 (2008).A pair of dice with faces labeled 1, 3, 4, 5, 6, 8 and1, 2, 2, 3, 3, 4, respectively, has the same prob-abilities of sums 2, 3, . . . , 12 as a standard pair[Sicherman’s dice, M. Gårdner, Sci. Am. 238, 19–32 (1978)]. The paper reviews the literature on re-lated subjects since 1978 and discusses some newproblems: What can be done with a pair having ksides and l sides? Find pairs of relabeled n-sideddice giving the sums 2, 3, . . . , 2n with generalizedstandard probabilities. Here the discussion is vi-tiated by printing errors.A different problem: Is it possible to label thefaces of mn-sided dice so that the sums m,m +1, . . . ,mn, are equiprobable? A number of waysis given. A. J. Stam (Winsum)

1166.60045Barlow, Martin T.; Bass, Richard F.; Chen, Zhen-Qing; Kassmann, MoritzNon-local Dirichlet forms and symmetric jumpprocesses.Trans. Am. Math. Soc. 361, No. 4, 1963-1999(2009).The authors consider the non-local symmetricDirichlet form

E( f , f ) :=∫Rd

∫Rd

( f (y) − f (x))2 J(x, y)dxdy

where the jump kernel J is a symmetric function,with support in |x − y| < 1 and satisfies:

κ1|y − x|−d−α≤ J(x, y) ≤ κ2|y − x|−d−β

for |x− y| < 1 and 0 < α < β < 2. This assumptionallows the corresponding jump process to havejump intensities whose sizes depend on the posi-tion of the process and the direction of the jump.No continuity assumptions on the jump kernel amade.The main result of the paper is the constructionof a Hunt process associated with the Dirichletform. This process can be realized as a weaklimit of certain more regular jump processes.Upper and lower bounds for the (symmetric)transition density (heat kernel) are proved. Aparabolic Harnack inequality for non-negative(caloric) functions that solve the heat equationwith respect to the Dirichlet form is established.A counterexample in included, showing that thecorresponding harmonic functions need not becontinuous. Liliana Popa (Iasi)

1166.60048Adler, Mark; Delépine, Jonathan; Van Moer-beke, PierreDyson’s nonintersecting Brownian motionswith a few outliers.Commun. Pure Appl. Math. 62, No. 3, 334-395(2009).Consider n nonintersecting real Brownianbridges all starting from the origin at time t = 0and returning to the origin at time t = 1 (DysonBrownian motions). For large n, the averagemean density of the particles has its support,for each 0 < t < 1 in the interval (−c, c) wherec =

√2nt(1 − t). The Airy process A(t) (intro-

duced by M. Prähofer and H. Spohn [J. Stat. Phys.108, No. 5–6, 1071–1106 (2002; Zbl 1025.82010)])is defined as the motion of these nonintersecting


Brownian bridges for large n, but viewed froman observer on the (right-hand) edge curve

C : y =√

2nt(1 − t) > 0

with time and space appropriately rescaled. Inthis new scale the Airy process describes the fluc-tuations of the Brownian particles near the edgecurve C.The authors study the case where among thosen paths, 0 ≤ r ≤ n are forced to reach a givenfinal target a = ρ0

√n/2, say, while the remaining

n − r particles still return to the origin. The mainresult states that for large n no new process ap-pears as long as we have not yet reached the timewhere the tangent to C passing through a touchesC. At the point of tangency the fluctuations obeya new statistics, which is called the Airy processwith r outliers or r-Airy process. The r-Airy pro-cess is an extension of the Airy process, but itis in contrast to the Airy process not stationary.The logarithm of the probability that at time t thecloud does not exceed x is given by the Fredholmdeterminant of a new kernel (extending the Airykernel), and it satisfies a nonlinear PDE in x andt, from which the asymptotic behaviour of theprocess can be deduced for t → −∞. This ker-nel is closely related to one found by Baik, BenArous, and Péché in the context of multivariatestatistics. René L. Schilling (Dresden)

1167.60006Pellegrini, ClémentExistence, uniqueness and approximation of astochastic Schrödinger equation: The diffusivecase.Ann. Probab. 36, No. 6, 2332-2353 (2008).Stochastic Schödinger equations, or Belavkinequations, are perturbations of Schrödinger typeequations. They describe the evolution of anopen quantum system undergoing a continuousquantum measurement which is at the origin ofthe stochastic character of the evolution. Exis-tence and uniqueness are proven for the solutionof these equations in the diffusive case with atwo-level system. Then the model is physicallyjustified by proving that it is a continuous timelimit of a discrete physical procedure with re-peated quantum interactions.

Dominique Lepingle (Orléans)

60009Pugachev, O.V.Quasi-invariance of Poisson distributions withrespect to transformations of configurations.(English. Russian original)Dokl. Math. 77, No. 3, 420-423 (2008); translationfrom Dokl. Akad. Nauk, Ross. Akad. Nauk 420,No. 4, 455–458 (2008).Since Cameron, Martin, Prokhorov, Skorehodand Girsanov a lot of measures were studied forquasi-invariance under transformations. In thispaper the author presents conditions for the exis-tence and derives formulas for Radon-Nikodýmdensities of Poisson measures πσ under certaintranslations. The vector fields v used to constructthe translations are assumed to be Lipschitz.In the first theorem for the configuration space Γon Rd and the translation T(γ) = x + φ(γ)v(x) |x ∈ γ, where φ is a continuously differentiablefunction and v is globally Lipschitz, one of theauthor’s results is that under some additional as-sumptions πσ is quasi-invariant under T and hegives a rather involved formulae for the density.Finally, for the configuration space over aconnected noncompact smooth Riemannian d-manifold M, v being locally Lipschitz and trans-lations Tt = Ft(x) | x ∈ γ, where Ftt∈R denotesthe flow of diffeomorphism of M generated by v,another result says that under certain conditionsone has πσ T−1

t πσ with density ρt ∈ Lq(π)having an explicit bound.

Michael Röckner (Bielefeld)

1167.91004Clarke, Nancy E.A witness version of the cops and robber game.Discrete Math. 309, No. 10, 3292-3298 (2009).In the usual game of cops and robbers, playedon a graph with a loop at each vertex, there arek > 0 cops and a single robber. To begin, eachof the cops chooses a vertex and then the rob-ber chooses a vertex. The sides move alternately,a move consisting of moving along an edge toan adjacent vertex (possibly the same one, via aloop). Both sides have perfect information, andthe cops win if at some time the robber and oneof the cops occupy the same vertex. A graph onwhich a single cop can always win is called acopwin. In this paper a variation of this gameis studied in which the robber still has perfectinformation but the cops do not. They receiveinformation intermittently from witnesses whoreport sightings of the robber. This information


may come at regular intervals, every k units oftime (the k-regular case), or may come sporad-ically. A graph on which a single cop can winwith sightings every k units of time is called k-winnable. Some results are obtained for irregularinformation games, but the bulk of the paper isdevoted to k-regular games. An example is givenof a copwin graph that is not k-winnable for anyk, and several theorems are obtained for k-regulargames. Special attention is paid to triangle-freek-winnable graphs. Gerald A. Heuer (Moorhead)

1166.91010Wu, YaokunLit-only sigma game on a line graph.Eur. J. Comb. 30, No. 1, 84-95 (2009).Let Γ be a connected graph and V(Γ) its vertexset. Denote by F the set of functions from V(Γ)to the binary field F2. For any x ∈ F, a move ofthe lit-only sigma game on Γ consists of choos-ing some vertex v with x(v) = 1 and changing

the values of x at all those neighbors of v. An or-bit is a maximal subset of F satisfying the prop-erty that any two elements can reach each otherby a series of moves. The minimum light num-ber of Γ is maxK minx∈K #supp(x), where K runsthrough all orbits. It is shown that the minimumlight number of the line graph of Γ is equal tomaxb(Γ), ρ(σ1(Γ)), where b(Γ) is the isoperimet-ric number of Γ and ρ(σ1(Γ)) is the covering ra-dius of the subspace of set of functions from E(Γ)(the set of edges of Γ) to the binary fieldF2 gener-ated by the rows of the adjacency matrix of L(Γ).The sizes of all orbits of the lit-only sigma gameon L(Γ) are also determined. It is also shown thatwhen Γ is a tree, the minimum light number ofL(Γ) is b(Γ). Furthermore, it is shown that, if thetree has n ≥ 3 vertices, the group formed by themoves of the lit-only sigma game on L(Γ) is thesymmetric group on n elements.

Giacomo Bonanno (Davis)

Numerical Analysis. Modelling. Computer Science. Algorithms. 117

Numerical Analysis. Modelling.Computer Science. Algorithms.

1167.65001Oldham, Keith B.; Myland, Jan; Spanier, JeromeAn atlas of functions. With Equator, the atlasfunction calculator. With CD-ROM. 2nd ed.New York, NY: Springer (ISBN 978-0-387-48806-6/hbk). xi, 748 p. EUR 85.95/net; SFR 144.40;£ 68.99; $ 129.00 (2008).This book provides elementary and basic infor-mation on several hundred functions and comeswith a CD containing a piece of software, calledEquator, for numerical evaluation, tabulationand visualization of these functions. Such a com-bination of a book and software is useful for sci-entists concerned with the quantitative aspectsof their field. The CD is also sold separately.The book is organized into 64 chapters and sev-eral appendices one of which describes the useof Equator. This software is designed to run un-der the Windows Vista operating system. Thevisual appearance and layout of the book is mod-ern with very nice colorful graphs or surfaces offunctions. Each chapter contains a lot of infor-mation about the functions studied in that chap-ter. Compared to the first edition of the book,authored by the first and third author, the sec-ond edition contains more material, the graphicshas been updated, the textual algorithm descrip-tions of function evaluation has been replacedby the Equator program (for a review of the firstedition, see G. Tee: Computing Reviews, April1988, 203–204 and also J. Spanier and K. B. Old-ham [An atlas of functions. (Washington - NewYork - London): Hemisphere Publishing Corpo-ration, a subsidiary of Harper & Row, Publishers,Inc.; distr. outside North America by Springer-Verlag, Berlin etc. (1987; Zbl 618.65007)]). Thisbook defends well its place in the mathematicalliterature.Reviewer’s remarks. Numerical evaluation ofspecial functions is discussed in the followingbooks:A. Gil, J. Segura and N. M. Temme [Numer-ical methods for special functions. (Philadel-phia), PA: Society for Industrial and AppliedMathematics (SIAM). (2007; Zbl 1144.65016)], A.Cuyt, V. B. Petersen, B. Verdonk, H. Waadelandand W. B. Jones [Handbook of continued frac-

tions for special functions. With contributionsby Franky Backeljauw and Catherine Bonan-Hamada. Verified numerical output by StefanBecuwe and Cuyt. (Dordrecht): Springer. (2008;Zbl 1150.30003)], W. J. Thompson, Atlas for com-puting mathematical functions: an illustratedguide for practitioners, with program in Fortran90 and Mathematica. Incl. 1 CD-ROM. (Chich-ester): John Wiley & Sons. (1997; Zbl 885.68089)].

Matti Vuorinen (Turku)

1167.65002Grigoryan, Artyom M.;Grigoryan, Merughan M.Brief notes in advanced DSP.Fourier analysis mit Matlab.Boca Raton, FL: CRC Press (ISBN 978-1-4398-0137-6/hbk). xiii, 354 p. $ 99.95 (2009).The discrete Fourier transform (DFT) is the mostused tool for solving problems in the frameworkof linear systems. In digital signal processing(DSP), DFT gives the push for developing otherdiscrete transforms, such as the Hadamard, co-sine, sine, Hartley and Haar transforms.Based on the authors’ research in applied Fourieranalysis, this book addresses many concepts andapplications of discrete unitary transforms inDSP. Numerous examples and included MAT-LAB codes illustrate how to apply the ideas inpractice.This book is divided into 6 chapters. Chapter 1covers the basic properties of DFT. The concept ofsplitting of DFT is described. The authors discussthe discrete paired transform that yields to an ef-fective splitting of DFT. Fast Fourier transforms(FFT) are based on the splitting by the pairedtransform. In Chapter 2, the method of the liftingschemes and integer transforms with control bitsare described and applied to integer approxima-tions of DFT. Chapter 3 is devoted to the discretecosine transforms (DCT). The method of pairedtransforms for splitting the DCT is handled. In-teger approximations of DCT are presented bymethods of lifting schemes. In Chapter 4, the dis-crete Hadamard transform is discussed. Chapter5 examines the the decomposition of a signal byso-called section basis signals as well as a de-

118 Numerical Analysis. Modelling. Computer Science. Algorithms.

composition of images by direction images. Inthe last chapter, the authors consider the signalmultiresolution which is based on Fourier trans-form wavelets.Each chapter closes with a list of problems.This well-written book presents many interest-ing problems and concepts of various discreteunitary transforms. This book will be useful forstudents and researchers in electrical and com-puter engineering. Manfred Tasche (Rostock)

1167.65041Butcher, J. C.Numerical methods for ordinary differentialequations. 2nd revised ed.Hoboken, NJ: John Wiley & Sons (ISBN 978-0-470-72335-7/hbk). xix, 463 p. (2008).This book is the second edition of the book, firstpublished in 2003 by J.Wiley and sons, 435 pp[Zbl 1040.65057]. It consists of 5 chapters, each ofwhich consists of several sections. It has a bibli-ography pp.453-458, and an Index, pp.459-463.Chapter 1. Differential and difference equations,consists of 5 sections. It contains some introduc-tory background material.Chapter 2. Numerical differential equation meth-ods, consists of 8 sections. It contains a generaloverview of the numerical methods for ordinarydifferential equations (ODE). These methods in-clude the Euler method, the Runge-Kutta meth-ods, multistep and hybrid methods, and an intro-duction to the implementation of these methods.Chapter 3. Runge-Kutta methods, consists of10 sections. It contains a detailed analysis ofthese methods, explicit and implicit Runge-Kuttamethods, stability and implementation of thesemethods, and their algebraic properties. A sep-arate section deals with the implementation is-sues.Chapter 4. Linear multistep methods, consists of7 sections. The order of the linear multistep meth-ods, their error estimates, stability analysis, andimplementation issues are discussed in detail.Chapter 5. General linear methods, consists of 6sections. It treats some of the known methods, inparticular, the Runge-Kutta methods, the linearmultistep methods, some known unconventialmethods, and some recently discovered meth-ods as general linear methods. The stability anal-ysis of general linear methods, the order of thesemethods and the error analysis of these meth-ods are discussed, and design criteria for generallinear methods are proposed. The author is one

of the well-known contributors to the numeri-cal methods for solving ODE. Part of the results,presented in this book, belong to the author.The book is a useful contribution to the literature.It is written both to the students and specialistsin the field of numerical solution of ODE.

Alexander G. Ramm (Manhattan)

1166.68005Ho, Tracey; Lun, Desmond S.Network coding. An introduction.Cambridge: Cambridge University Press (ISBN978-0-521-87310-9/hbk). xi, 170 p. £ 30.00; $ 60.00(2008).The book under review explains the ideas andprinciples of network coding, a relatively youngfield of study. The authors propose some defi-nitions of the coding of networks. One of themgives a coding at a network node in a packetnetwork, applied to the content of packets. Theauthors show and analyze distinctions betweennetwork and channel coding for noisy links aswell as source coding applied to compression al-gorithms for source processes. They discuss theutility of network coding in so-called butterflynetworks and characterize the network capacityin intra-session coding. Then they extend theirdiscussion to non-multipath problems in loss-less networks, referring to inter-session coding.Next, they consider lossy networks and study thecapacity results for random linear network cod-ing. In the following chapters they analyze dif-ferent strategies connected with network codingand selection of appropriate subgraphs. Finally,security aspects of network coding are consid-ered. The book can be an interesting source ofknowledge about network coding for both theo-reticians and practitioners.

Jozef Wozniak (Gdansk)

1166.68012Brass, PeterAdvanced data structures.Cambridge: Cambridge University Press (ISBN978-0-521-88037-4/hbk). xvi, 456 p. £ 39.99;$ 75.00 (2009).This book represents a work of highly scientificvalue, accessible for graduate students interestedin acquiring knowledge about fundamental datastructures, and for researchers or practitionerseager to find a deep insight into the subjectstreated here.


The book is composed of nine chapters (Ele-mentary structures, Search trees, Balanced searchtrees, Tree structures for sets of intervals, Heaps,Union-find and related structures, Data struc-ture transformations, Data structures for strings,Hash tables), a bibliography and an appendix.Through his rigorous, clear, precise, and almostexhaustive treatment of the topics exposed, theauthor succeeds to bring the focus back to datastructures as a fundamental subtopic of algo-rithms. The reader can notice the intensive doc-umentation given by the author, and the com-prehensive bibliography consisting of more than500 references.Although the author provides a formal analysisof the data structures and algorithms exposed,his approach is also a practical one – the opera-tions available for each data structure are illus-trated through C code or described in naturallanguage. Mirel Cosulschi (Craiova)

1166.68037Kolman, Eyal; Margaliot, MichaelKnowledge-based neurocomputing: a fuzzylogic approach.Studies in Fuzziness and Soft Computing 234.Berlin: Springer (ISBN 978-3-540-88076-9/hbk;978-3-540-88077-6/ebook). xv, 100 p. EUR 106.95;SFR 166.00 (2009).This book is part of the series “Studies in Fuzzi-ness and Soft Computing". It introduces a novelapproach to building fuzzy rule-bases, calledFuzzy All-permutations Rule-Bases (FARB). Thebook starts with a broad introduction to thefield of connectionism, namely artificial neu-ral networks, fuzzy rule-bases, and neurocom-puting, before it introduces the FARB concept.The authors show that inferring the FARB us-ing standard tools from fuzzy logic theory yieldsan input-output relation that is mathematicallyequivalent to that of standard neural networks.Chapter 2 formally defines the FARB approachand shows that its special structure implies aclosed-form formula for its input-output map-ping.Some special cases of the FARB are discussed inChapter 3, where mathematical equivalence be-tween these cases and various types of artificialneural networks is discussed. Example applica-tions are presented to illustrate the main ideas.Since FARB may include either a large number ofrules or complicated rules which may obstruct itscomprehensibility, Chapter 4 presents a system-

atic procedure for rule reduction and simplifica-tion to minimize the effect of this. Applicationof FARB to large-scale problems is discussed inChapters 5 and 6, where FARB is used for knowl-edge extraction from artificial neural networkstrained to solve several benchmark problems.New knowledge-based design methods are de-scribed and their usefulness is demonstrated bydesigning networks to solve language recogni-tion problems. Future research is summarized inChapter 7, where potential applications based ontransferring concepts and ideas between fuzzylogic and networks are discussed.

Neli Zlatareva (New Britain)

1166.68039Dompere, Kofi KissiFuzzy rationality. A critique and methodologi-cal unity of classical, bounded and other ratio-nalities.Studies in Fuzziness and Soft Computing235. Berlin: Springer (ISBN 978-3-540-88082-0/hbk; 978-3-540-88083-7/e-book). xxi, 283 p.EUR 106.95; SFR 166.00 (2009).The present volume belongs to a series of mono-graphs on fuzzy aspects of intelligent behaviour.This one regards the concept of rationality re-lated to information, knowledge and optimalchoice in an uncertain environment. The text ofthe volume is divided into four main chaptersdealing with rationality regarding general epis-temics, information and knowledge, decision-making and also ideological aspects of rationaldecision-choice. Approximately forty per cent ofthe volume’s extent is devoted to a large list ofreferences. The presentation of the topics dealtwith in the volume uses a minimum of formalmathematics. The preferred method used by theauthor is structural analysis and detailed heuris-tic discussion of particular topics.

Milan Mareš (Praha)

1167.68008Cardoso, João M. P.; Diniz, Pedro C.Compilation techniques for reconfigurable ar-chitectures.New York, NY: Springer (ISBN 978-0-387-09670-4/hbk). xii, 223 p. EUR 86.95/net; SFR 144.50;£ 68.99; $ 129.00 (2009).This is a research monograph treating mappingof applications, typically written in high-levelimperative programming languages, such as Cor MATLAB, to efficient reconfigurable hardware


architecture consisting of multiple processing el-ements and storage structures. The results ofsuch mappings could allow to respond to specificdomain requirements such as input data ratesor stringent real-time constraints. Although thisresearch domain is far from mature now, manyresearch efforts have been made and they are de-scribed in this book.The book consists of 7 chapters. After a shortintroduction, reconfigurable architectures are re-viewed, and early works by authors such as Es-trin, Miller, Cocker, Reddick and Feustel andcompanies like Altera and Xilinx are quoted. Keycharacteristics of reconfigurable architectures arediscussed. Granularity is one of the key aspectsthat differentiate reconfigurable architectures:

– In fine-grained architectures the configurablecells include logic gates, thus allowing theimplementation of arbitrary and specializeddata-path hardware designs,

– in coarse-grained architectures the config-urable cells, often designated as Field-Programmable ALU Arrays (FPAAs), includeALUs and distributed memories,

– in mix-coarse grained architectures the config-urable cells include microprocessor cores com-bined with very fine-grained reconfigurablelogic.

The three subsequent chapters present phases ofthe overall compilation process in the followingsequence:

– At first generic compilation and synthesisflow is discussed. The compiler must not onlytake into account known instruction set ar-chitecture, but also synthesize an application-specific architecture implemented with recon-figurable hardware resources. Moreover, com-pilers must deal with the many aspects ofcompilation techniques for parallel comput-ing, like processor synchronization, data par-titioning and code generation. The programcode resulting after this phase of compilationis not optimal – therefore the next phase iscode optimization.

– Code transformations may be deployed atbit level (bit-width narrowing, conversion be-tween floating-point and fixed-point data for-mat), at instruction level (operator strengthand height reduction) or at loop-level (looptiling, loop strip-mining, loop merging).Transformations may be data-oriented orfunction-oriented. An attempt is made to give

advice which code transformation is to be cho-sen.

– Mapping and execution optimizations are re-viewed. The complexity of this problem is in-creased as compared with traditional archi-tectures by the spatial nature of reconfigurablearchitectures. Compilers must judiciously bal-ance the use of different kinds of resources inspace and time. Very important here is looppipelining.

In the 6th chapter, plenty of compilers for recon-figurable architectures are compared. The bookdistinguishes between compilers for fine-grainedreconfigurable devices and coarse-grained archi-tectures. Several compilers in each group areshortly characterized and a table comparisonis included. Chapter 7 discusses perspectiveson programming reconfigurable computing plat-forms. The book contains 350 items in its bibliog-raphy.The authors finish the book with the follow-ing statements: “We believe compilation tech-niques for reconfigurable computing platformsoffer many exciting research and developmentopportunities. We hope this book, to our knowl-edge the first book completely dedicated to thetopic of compilation for reconfigurable architec-tures, will motivate further research efforts in thisdomain and serve as a base for a deeper under-standing of the overall compilation and synthesisproblems, current solutions, and open issues."

Antoni Michalski (Warszawa)

1167.68009Wang, JieComputer network security. Theory and prac-tice.Beijing: Higher Education Press; Berlin:Springer (ISBN 978-7-04-024162-4; 978-3-540-79697-8/hbk;). xviii, 384 p. EUR 119.95/net;SFR 199.50; £ 108.00; $ 169.00 (2009).The book under review provides the reader withan interesting and well-written overview of avariety of aspects of network security. The ti-tle recalls a popular book by Bruce Schneier[Applied cryptography. New York: Wiley (1993;Zbl 789.94001)]. So does the table of contents.However, among classical algorithms one canalso find recent proposals, such as the latest WiFicryptography instruments, including AES stan-dard or Bluetooth 2.1 EDR security extensions.The book not only presents a theoretical de-scription of security mechanisms, but also prac-


tical aspects related to them. For example, thevery interesting part concerning Bluetooth secu-rity can be found in Chapter 6.5. What one canfind a bit disturbing is an uneven detail levelamong the variety of topics covered. For exam-ple SSH is described on about one page, whileSSL/TLS, serving similar services and complex-ity, is presented in much more detail. The booklacks the description of TACACS+ and Diam-eter authentication protocols similar to Radiusand, to some extent, Kerberos protocols. Also,modern WEP attack, published in the year 2007(PTW – from the names of its authors Pychkine,Tews and Weinmann) is not mentioned. The au-thor does not present anonymizing issues, whichrecently gained popularity in network cryptog-raphy. Nevertheless, the book is very interestingand covers current security issues not only fromthe theoretical but, at least in some cases, alsofrom the practical point of view. The book is rec-ommended to all theoreticians and practitionersthat would like to have available a comprehen-sive compendium treating modern network se-curity. Jozef Wozniak (Gdansk)

1167.68010Magoulés, Frédéric; Pan, Jie; Tan, Kiat-An; Ku-mar, AbhinitIntroduction to grid computing.Chapman & Hall/CRC Numerical Analysis andScientific Computing. Boca Raton, FL: CRCPress (ISBN 978-1-4200-7406-2/hbk). xxiii, 310 p.$ 79.95 (2009).The book illustrates the state-of-the-art technolo-gies and research activities in the field of Gridcomputing. The classical topics of Grid com-puting being covered are: virtual organizations,scheduling algorithms, fault tolerance, workflowmanagement, data management, security, stan-dards, monitoring, as well as currently availablemiddleware. In particular, a number of Grid ap-plications are carefully treated. The Monte Carlomethod is presented not only with its gridifi-cation, but also within the context of concreteproblem solving for computational mechanicsand computational finance. The same approachis applied to partial differential equations comingfrom options pricing in computational finance.Written in a tutorial style, the book can be suc-cessfully used as support for Grid computinglectures. Particularly useful are in this directionthe three annexes giving details about the two

most popular Grid middleware systems, GlobusToolkit and gLite. Dana Petcu (Timisoara)

1167.68018Große, Daniel (ed.); Sülflow, André (ed.);Drechsler, Nicole (ed.)EXplayN – strategy optimization and selectedgame problems. (EXplayN – Strategieopti-mierung und Analyse ausgewählter Spielprob-leme.)Technische Informatik. Aachen: Shaker Ver-lag (ISBN 978-3-8322-7059-9/pbk). xii, 170 p.EUR 28.80 (2008).The book represents the project report and is de-voted to computer description and simulation ofdifferent types of game problems. Computer sim-ulation of game problems, their detailed analysisand algorithmic description became the edge ofinterests for mathematics, applied mathematics,informatics, and computer science. The biggestproblem in computer simulation of game prob-lems is their NP–property. The last means thatwhen the description of the problem is increasedlinearly number of algorithmic steps is increasedexponentially.Within years there were number of attempts tosolve these problems using different approaches.It is worth to mention such well known com-puter systems as Deep Thought, Deep Blue, andDeep Fritz. The reviewed project EXplayN wasperformed in 2005–2007 in the University of Bre-men.The book contains introduction, 5 chapters, con-clusion, and description of the work of computersystem EXplayN. The book has formal descrip-tion of some of the game problems in the the-ory of sets form. The problems considered aremainly related to one- and two person games.Main stress is made on combinatorial problemswhere the number of calculations leading to theoptimal strategy is vital.The computational model is described in the cor-responding computing language. The interest-ing part of this project is the implementation ofheuristic algorithms inbuilt in algorithmic form.The book contains a number of simple gameproblems which are used to demonstrate ap-proaches and techniques.

Andreiy Kondrat’yev (Red Level)

1167.90001Bartholomew-Biggs, MichaelNonlinear optimization with engineering ap-


plications.Springer Optimization and Its Applications 19.New York, NY: Springer (ISBN 978-0-387-78722-0/hbk). xvi, 280 p. EUR 46.95/net; SFR 78.00;£ 35.50; $ 69.95 (2008).This book gives on 280 pages a broad overviewof nonlinear optimization. The presented meth-ods include direct search techniques, the steepestdescend approach, trust region methods as wellas Newton and quasi-Newton techniques withglobalization strategies for the unconstrainedcase. For constrained optimization problemspenalty approaches, SQP methods, barrier tech-niques and interior point methods are discussed.Additionally, the book covers several topics asline search approaches, filter methods and thecomputation of derivatives. The presented op-timization approaches are compared with eachother by means of several examples with up to200 variables. The numerical results are mostlyobtained by the software package OPTIMA writ-ten by the author. The book contains 25 chapterswith an average of 11 pages. Due to the varietyof considered solution techniques, the presenta-tion of each method and its theoretical analysis israther brief. Nevertheless, the introduction of thedifferent techniques is written in a very compre-hensible way. Furthermore, each section containsexercises to verify and deepen the understandingof the material. Andrea Walther (Paderborn)

1167.90002Nakayama, Hirotaka; Yun, Yeboon; Yoon, MinSequential approximate multiobjective opti-mization using computational intelligence.Vector Optimization. Berlin: Springer(ISBN 978-3-540-88909-0/hbk; 978-3-540-88910-6/ebook). xvi, 197 p. EUR 96.25; SFR 149.50(2009).The problem of multiobjective optimization isconsidered emphasizing the cases where objec-tive functions are “expensive", and value judg-ments of decision makers can change in decisionmaking process. To reduce the occurring in suchcases computational burden metamodels of ob-jective functions are used. The aspiration levelapproach is used to cope with changes in atti-tudes of decision makers. The basic concepts ofmultiobjective optimization are included mak-ing the book self-contained. The performance ofthe presented methods is demonstrated by theresults of applications to real world problems.

In Chapter 1 basic concepts of multiobjective op-timization are introduced and some scalariza-tion methods are described. In the next chap-ter the following interactive methods are de-scribed: goal programming, weighting, and sat-isficing trade-off. In Chapter 3 genetic algorithmsfor Pareto frontier generation are presented. Theperformance of genetic algorithms essentiallydepends on fitness evaluation of the individu-als of the current generation; thus a half of thethird chapter is devoted to the applicability ofthe data envelopment method to the fitness eval-uation and to the approximation of the Paretofrontier. In Chapter 4 computational intelligencemethods and their applicability in multiobjec-tive optimization are discussed. Machine learn-ing is explained along pattern classification andregression problems. Radial basis function net-works are briefly introduced. Support vector ma-chines are presented in a more detailed way. Theclaim on the back cover “This book highlightsa new direction of multiobjective optimization,which has never been treated in previous pub-lications" most likely is related to the materialof the chapters 5 and 6. In Chapter 5 (Sequen-tial Approximate Optimization) the methods foroptimization of “expensive" functions are con-sidered using an auxiliary approximate functionwhose minimum point is chosen as a point of cur-rent observation (either computation or experi-mental measurement) of the objective functionvalue. In different publications such auxiliaryfunctions are called models, metamodels, surro-gates, and response surfaces. Regression modelsare introduced to explain some methods of opti-mal design of experiments. For global optimiza-tion of “expensive" functions methods based onstatistical models (random functions) and radialbasis functions are recommended. Chapter 6 isdevoted to apply the theory and methods intro-duced in previous chapters to multiobjective op-timization. The efficiency of the method basedon integration of the sequential approximate op-timization and the aspiration level approach isdemonstrated. In Chapter 7 two engineering ap-plications are described: reinforcement of cable-stayed bridges, and startup scheduling of powerplants. Several application examples are consid-ered also in other chapters of the book.The book is aimed to researchers, practitioners inindustries and students of graduate course andhigh grade of undergraduate course.

Antanas Žilinskas (Vilnius)


1166.90001Perret, Francis-Luc (ed.); Jaffeux, Corynne (ed.);Fender, Michel (ed.); Wieser, Philippe (ed.)Essentials of logistics and management. 2nd re-vised ed.Management of Technology Series. Lausanne:EPFL Press (ISBN 978-2-940222-16-2; 978-1-4200-4619-9/hbk). $ 94.95 (2007).The book touches almost all topics, which areconnected with logistics and management. Theauthors give an overview of key notions ofthe above-mentioned professional areas startingfrom financial accounting and logistical strat-egy and their connection with globalisation pro-cesses. They go through customer service man-agement and logistics including transport, mar-keting and purchasing management. The bookcomprehends also some issues of operationsresearch, modelling and simulation and otherauxiliary tools as information systems and in-formation technology for logistics. Topics likeworld-class management, world-class logisticsand managing human resources conclude thebook. The terms related to these topics are ver-bally explained, but almost no quantitative meth-ods or models are suggested here for reader tobe offered by some tool, which could be used forobtaining a decision in a practical situation. Theauthors avoid any mean of management scienceand this way the book represents only a sort ofintroductory textbook for students.

Jaroslav Janácek (Žilina)

1166.90004Ozcan, Yasar A.Health care benchmarking and performanceevaluation. An assessment using data envelop-ment analysis (DEA). With CD-ROM.International Series in Operations Research& Management Science 120. New York, NY:Springer (ISBN 978-0-387-75447-5/hbk). xxvii,214 p. EUR 79.95/net; SFR 133.00; £ 63.99; $ 99.00(2008).The present book by Doctor Yasar Ozcan is clearand very well structured. It starts with an inter-esting overview on efficiency and effectivenessperformance measurement, through Data Envel-opment Analysis (DEA). Then, using the exposedmodels and methods, the author addresses anumber of appealing health care applications,from hospital management to medical measur-ing of performance, passing through staff perfor-mance evaluation. This is an important text book

for post-graduate students and professionals thatdevote their work and research on improving theefficiency of heath care services. It also providesan interesting introduction and overview on effi-ciency analysis, performance measurement andDEA for those graduated in Management Sci-ence, Operations Research or Applied Mathe-matics that are intended to follow their researchwork and activity on heath system issues.

Pedro Martins (Coimbra)

1167.93001Lévine, JeanAnalysis and control of nonlinear systems. Aflatness-based approach.Mathematical Engineering. Berlin: Springer(ISBN 978-3-642-00838-2/hbk; 978-3-642-00839-9/ebook). xiii, 319 p. EUR 79.95/net; SFR 133.00;£ 71.99; $ 109.00 (2009).This book is focused mainly on the analysis of so-called nonlinear flat systems. Loosely speaking anonlinear system is said to be flat when thereis a (flat) output which exhibits the followingproperties: this output and its derivatives can beused to express all the system variables. For a flatsystem, the trajectories can be mapped in a one-to-one correspondence onto geometrical curveswhich are not necessarily defined by differen-tial equations. It follows that if one can obtain anumerical approximation for these flat outputs,one will be able to define the state parameter ofthe system, in a straightforward way, by usingnumerical techniques. The mathematical frame-work of this approach is (elementary) differentialgeometry.The book comprises two parts: the first partwhich displays the essential background on di-mensional geometry and the theory of flat sys-tems, and the second part which provides someapplications in the real world. The book is pub-lished in the mathematical engineering series,and as a consequence the differential geometrycontent is presented in a very accessible appliedmathematics way. Guy Jumarie (Montréal)

1167.93002Stokey, Nancy L.The economics of inaction. Stochastic controlwith fixed costs.Princeton, NJ: Princeton University Press(ISBN 978-0-691-13505-2/hbk). ix, 308 p. £ 29.00;$ 49.50 (2009).The aim of the book is to provide an account of


applications of optimal stochastic control tech-niques to economic problems where control isexercised only occasionally. This happens in sit-uations where control actions entail a fixed costand as a consequence large occasional changesare preferable to a sequence of smaller, morefrequent changes. This behavior is typical in anumber of important economic settings and theinterest in this kind of models had a sensibleincrease in recent years. Examples of this typeof problems are connected to price adjustment,investment behavior in manifacturing plants,job creation and destruction, individual portfo-lio behavior and attitude of consumers towardsdurable goods such as houses and automobiles.The reading of the book requires rather advancedmathematical preliminaries, which are presentedat a level that is acceptable for economists andrigorous enough to be useful in model build-ing. All background material is given in the ini-tial Chapters 2 to 5 while a few special topicsare provided in two appendixes. Such materialdeals with basic results in stochastic analysis andstochastic control, namely Brownian motion anddiffusions, stochastic calculus and martingalesand Hamilton-Jacobi-Bellman equation. The ap-plications to economic problems then follow.Chapter 6 deals with the simplest example inthe presence of a fixed cost, namely the prob-lem of exercising a one time option of infiniteduration. In this case action is taken only onceand the action itself is fixed so that the only is-sue is timing. Two approaches are studied. Thefirst is a direct approach similar to the one used tosolve the analogous deterministic problem whilethe second approach is based on the use of theHamilton-Jacobi-Bellman equation. A numericalexample is also studied in detail.Chapter 7 is concerned with more general mod-els with fixed costs. Here, in particular, actionscan be taken many times but there is an explicitfixed cost of adjustment. The size of the adjust-ment must also be chosen and decisions must beforward looking. A typical example is given bythe so called menu cost model, where the profitof a firm depends on the price of its own prod-uct relative to a general price index, the latterbeing modeled as a geometric Brownian motion.Changing the price is assumed to entail a fixedcost but no variable costs. The problem consistsin the maximization of the expected discountedreturns net of adjustment costs and it is shownthat its solution is characterized by an inaction in-

terval (where the firm does nothing) and a returnpoint inside the inaction interval. When the rela-tive price leaves this interval the firm adjusts thenominal price so that the relative price is equalto the return price. Long run averages under theoptimal policy can also be described, e.g., frac-tion of adjustments at each threshold, averagetime between adjustments, long-run density forrelative price. The solution to the problem is ob-tained both using a direct approach, which char-acterizes the optimal policy and value function,and using the Hamilton-Jacobi-Bellman equationwhich characterizes also the long-run averages.Exogenous opportunities for costless adjustmentare also discussed and finally a numerical exam-ple is developed in detail. Chapter 8 considersmodels with fixed as well as variable costs of ad-justment and the problem is slightly more com-plicated than in the menu cost model, but withan optimal policy that still involves exercisingcontrol only occasionally. As a typical example astandard inventory model is examined in whicha plant produces output, customers place orderand the difference between supply and demandis the net inflow into a buffer stock (with negativestocks interpreted as back-orders). Control is ex-ercised by a manager who can add to the stock bypurchasing the good elsewhere or can reduce itby selling on a secondary market. The associatedcosts are holding costs depending on the size ofthe stock, fixed costs of adjustment and a variablecost proportional to the size of adjustment andrepresenting the unit cost of purchasing goodsfrom an outside source or the unit revenue fromdisposing of goods on a secondary market. Themanager’s problem is to choose a policy for min-imizing the expected discounted value of totalcosts: holding costs plus fixed and variable ad-justment costs. It is shown that also in this casethere exists an inaction interval where no controlis applied. When the boundaries (lower and up-per) of the inaction region are reached the stockis adjusted to a suitable (lower or upper) returnpoint. As in the menu cost model the optimal pol-icy and associated value function is derived bothusing a direct approach and using the Hamilton-Jacobi-Bellman equation. Chapter 9 deals mainlywith the so called model of housing consumptionand portfolio choice, that provides an example ofmodels with continuous-time control variables.It is characterized by a consumer whose only con-sumption is the service flow from a house andwhose only income is the return of a portfolio


of two financial assets, a riskless asset paying afixed interest rate and a risky asset with a stochas-tic return. Housing has interest and maintainancecosts and the consumer is required to hold a min-imum equity level in the house that is a fixed frac-tion of its value, but he can also hold additionalwealth as housing equity, holding a mortgageon the remaining balance. Portfolio adjustmentis costless and is carried out continuously, whilethe consumer can adjust the housing consump-tion flow only by selling the old house and buy-ing a new one and this involves paying a cost thatis proportional to the size of the old house. Thisproblem is studied using the Hamilton-Jacobi-Bellman equation and it is shown that the opti-mal policy for housing transactions is character-ized, as in the previous chapters, by upper andlower thresholds for the ratio of total wealth tohousing wealth and a return point between thethresholds, while the optimal portfolio selectionis derived as a function of the ratio in the in-action region. Several extensions and numericalexamples are examined in detail.Chapter 10 deals with a model similar to the in-ventory model studied in Chapter 8 but with nofixed costs of adjustment. Here the difference be-tween inflows and outflows is described by aBrownian motion and it is first studied the con-trol problem consisting in keeping it inside theinaction interval. On the basis of the results ob-tained in this case the optimal control problemconsisting in the minimization of the expecteddiscounted sum of various costs is studied. Threetypes of costs are considered: a unit cost foradding to the stock at the lower threshold, a unitcost for decreasing the stock at the upper thresh-old and a strictly convex carrying cost. As a resultof the absence of fixed costs the size of the op-timal upward and downward adjustments arezero and the regulation is just sufficient to avoidthe exiting of the process from the inaction inter-val. A numerical example concludes the chapter.In Chapter 11 several investment problems arestudied using models similar to the inventorymodel of Chapter 10. They are characterized bya firm whose net revenue flow depends on itscapital stock and a geometric Brownian motiondescribing demand. The problem is to choose theinvestment policy that maximizes the expecteddiscounted value of the net revenue minus in-vestment costs. Various assumptions about theinvestment costs are examined. The first consid-ers a different purchase and sale price for capital

and in this case the optimal policy has the samestructure as in the inventory model in Chapter10 with the thresholds being functions of the de-mand. If in addition to the capital price there isalso a strictly convex cost of adjustment then thecontrol consists in a continuous flow when thecapital stock is outside the inaction interval. Spe-cial cases and further investment models are alsoconsidered in this chapter.The final Chapter 12 considers aggregate ver-sions of the menu cost model of Chapter 7. Sincein this model the exogenous shocks experiencedby individual firms are shocks to an economywide price index, then in the aggregate ver-sion of that model shocks cannot be modeled asindependent and identically distributed acrossagents, so that arguments based on a law of largenumbers cannot be used for the derivation of astationary cross sectional distribution and ana-lytical results are much harder to obtain than inthe case of agents subject to idiosyncratic shocks.The goal of the analysis of these models is thestudy of the hypothesis that monetary policyhas real effects in the short run. Two economiesare examined, identical except for the monetarypolicy adopted. In the first economy the moneysupply process is monotone and in this settingshocks to the money supply have no effect. In thesecond the log money supply follows a Brown-ian motion with zero drift and effects of shocksto the money supply follow and depend on thestate of the economy when the shock arrives. Afew numerical examples are provided and somevariations of the above model are also studied.

Giovanni Di Masi (Padova)

1166.93001Lozovanu, Dmitrii; Pickl, StefanOptimization and multiobjective control oftime-discrete systems. Dynamic networks andmultilayered structures. With foreword byGeorge Leitmann.Berlin: Springer (ISBN 978-3-540-85024-3/hbk;978-3-540-85025-0/ebook). xvi, 285 p. EUR 96.25(2009).The main idea of this book is to apply theory ofcooperative and noncooperative dynamic gamesin multiobjective optimization of discrete-timedynamic systems. The authors introduce a readerinto the most important results of dynamicgames including characterization of Pareto op-tima and Nash and von Stackelberg equilib-ria. Regarding control strategies for discrete-time


systems the first results are related to zero sumdynamic games on networks and conditions ofsaddle-points are formulated. Moreover station-ary strategies for cyclic games are proposed.Then the authors study multiobjective controlproblems for discrete time systems with varyingtime of states transition. Both cooperative andnoncooperative problems are considered. More-over the authors combine cooperative and non-cooperative approaches by introducing Pareto-Nash and Pareto-Stackelberg solutions and ap-ply them in the control design context. The re-sults for discrete-time control problems are gen-eralized onto optimal dynamic flow problems in-cluding multicommodity flow tasks. Once morea game theoretic approach is proposed. The sameframework is used to model a general multilay-ered decision process on networks. As examplestechnology emission means model and Kyotogames are analyzed. A. Swierniak (Gliwice)

1166.93002Suh, Suk-Hwan; Kang, Seong-Kuoon; Chung,Dae-Hyuk; Stroud, IanTheory and design of CNC systems.Springer Series in Advanced Manufactur-ing. London: Springer (ISBN 978-1-84800-335-4/hbk). xx, 455 p. EUR 129.95/net; SFR 216.00;£ 85.00; $ 149.00 (2008).Based on numerous new references and on ownexperience, the authors present this referencework in the field of Computerized NumericalControl (CNC). The book was originally pub-lished in Korean and represents a good guidefor all specialists and students interested in thedesign and exploitation of CNC systems.Part one is devoted to the main component of theCNC systems. Special attention is given to thecode interpreter (which translates the programparts into internal commands) and to the inter-polator (which has the role of generator of axismovement data from the block data generatedby the interpreter).Due to the importance of the acceleration anddeceleration (Acc/Dec) for the movement of themachine axis, in chapters 4, 5 and 6, the authorsdescribe the Acc/Dec control methods and therole of the PID position controller. The NumericalControl Kernel (NCK) system implemented forthe Acc/Dec-After-Interpolation and Acc/Dec-Before-Interpolation based on algorithms men-tioned in the previous chapters is presented, too.

Part two of the book starts with the descriptionof the control of mechanical behaviour of the ma-chine with the exception of axis control. This Pro-grammable Logic Control (PLC) is defined as asoftware-based control system which has moreadvantages in comparison with a hardware sys-tem. Then the authors present the main-machineinterface. Designing of an operation panel of thatkind takes into consideration a lot of factors: ma-chine information, operation node, etc.Chapter 9 is devoted to the design of the CNCarchitecture. In this context the main restrictionand performance indices are analyzed.The modern orientation in the design of CNCsystems is discussed in chapters 10 and 11.Here the authors show that the conventionalcontrollers based on closed hardware structureshave been replaced by controllers based on PChardwares, and, on the other hand, a new modelof data transfer between CAD/CAM systems andCNC machines (known as STEP-NC) reflects thechanges in the manufacturing environement.

Sergiu T. Chiriacescu (Brasov)

1166.94001Chernov, V. M.Arithmetical methods of the synthesis of fastalgorithms of discrete orthogonal transforms.(Arifmetiqeskie medody sinteza bystryhalgoritmov dinskretnyh ortogonal~nyhpreobrazovani$i.) (Russian)Matematika, Prikladnaya Matematika.Moskva: Fizmatlit (ISBN 978-5-9221-0940-6/hbk). 261 p. (2007).The book under review represents an exami-nation of the boundary domain between infor-mation science, in particular, processing manydimensional numerical systems, and mathemat-ics, as number theory and algebra.The analyses described touch upon the com-plicated problems synthesis theory of so-calledfast algorithms of discrete orthogonal transfor-mations. The researcher’s intention is to developon this base efficient methods for discrete infor-mation analyses. The fields of numerical signaland transformations processing, applied mathe-matics are presented in nine chapters of the book.It should say, that each chapter of the book pro-vided by special commentary and citations list.

Vladimir N. Karpushkin (Moskva)

1166.94002Steinbach, Bernd; Posthoff, Christian


Logic functions and equations. Examples andexercises.Dordrecht: Springer (ISBN 978-1-4020-9594-8/hbk; 978-1-4020-9595-5/e-book). xxii, 231 p.EUR 85.55 (2009).This book is an extension, or better, a com-plement to the book “Logic Functions andEquations–Binary Models for Computer Sci-ence" written by the same authors [(Dordrecht):Springer. (2004; Zbl 1087.94030)]. It has two parts:Basic Software (XBOOLE Monitor, Basics andlogic functions, Logic equations, Boolean differ-ential calculus, The solution of logic equations)and Applications (Logics and arithmetics, Com-binatorial circuits, Finite-state machines), lists offigures, tables, an Index of terms and a list ofreferences with 24 titles. The authors have devel-oped a logic system called XBOOLE which per-forms logical operations on the given functions,helping the readers to solve the problems givenin the book. Many examples and complete solu-tions to the problems are shown, so the readerscan study at home. This book is more focused onthe solutions of Boolean equations, rather thanthe classical minimization of logical expressions.Its use can increase the quality of learning con-siderably through the use of examples and ex-ercises which can be handled on a much higherlevel when computer support is provided. Thisbook, allowing to learn from examples is veryuseful not only for the students, but also the pro-fessors. Ioan Tomescu (Bucuresti)

1166.94003Yan, Song Y.Cryptography attacks on RSA.New York, NY: Springer (ISBN 978-0-387-48741-0/hbk). xx, 255 p. EUR 69.95/net; SFR 122.00;£ 54.00; $ 89.95 (2008).This book comprises of ten chapters: Chapter 1presents the theory and practice of tractable andintractable computations in number theory. Thetheory of RSA cryptographic system and its vari-ants are discussed in Chapter 2. The attacks basedon solutions to the Integer Factorization Prob-lem (IFP) and the attacks based on solutions toDiscrete Logarithm Problem (DLP) are discussedin Chapter 3 and Chapter 4, respectively. Chap-ter 5 discusses some quantum attacks on RSA,whereas some number-theoretic attacks on RSAare discussed in Chapter 6. Some crytoanalyticattacks on the short RSA public exponent arediscussed in Chapter 7, and some attacks on the

short RSA private exponent are the subject ofChapter 8. Completely different type of attacksviz. the side-channel attacks on RSA are pre-sented in Chapter 9. Chapter 10 presents somequantum resistant, non-factoring based crypto-graphic systems as an alternative to RSA whichprovides an immediate replacement once RSA isproved to be insecure.The book covers almost all major known crypto-analytic attacks and defenses of the RSA crypto-graphic attacks and defenses of the RSA cryp-tographic system and its variants. It is a self-contained book and the materials presented havebeen classroom tested for various courses incryptography and cryptoanalysis, and has beendesigned for a professional audience of practi-tioners and researchers. Of course, the book issuitable as a graduate text in the field.

Bal Kishan Dass (Delhi)

1166.94004Pun, Man-On; Morelli, Michele; Kuo, C.-C. JayMulti-carrier techniques for broadband wire-less communications. A signal processing per-spective.Communications and Signal Processing 3. Lon-don: Imperial College Press (ISBN 978-1-86094-946-3/hbk). xiv, 257 p. £ 41.00; $ 75.00$ (2007).This book is intended to provide an accessibleintroduction to orthogonal frequency-divisionmultiplexing based systems from a signal pro-cessing perspective. The first part provides aconcise treatment of some fundamental conceptsrelated to wireless communications and multi-carrier systems. The second part offers a compre-hensive survey of resent developments on a va-riety of critical design issues including synchro-nization techniques, channel estimation meth-ods, adaptive resource allocation and practicalschemes for reducing peak-to-average power ra-tio of the transmitted waveform.

Tiit Riismaa (Tallinn)

1166.65011Jenkinson, O.; Pollicott, M.A dynamical approach to accelerating numeri-cal integration with equidistributed points.Proc. Steklov Inst. Math. 256, 275-289 (2007) andTr. Mat. Inst. Steklova 256 (2007).The paper is interesting and well written.The authors show how the rate of conver-gence of the approximate integral with respectto Lebesgue measure can be significantly accel-


erated for the case of a real analytic function byconsidering linear combinations of some equidis-tributions (Theorem 1); the rate of convergenceis better than that of any Newton-Cotes rule.An analogue of Theorem 1 for the d-dimensionalLebesgue measure is also presented (Theorem2). Afrodita Iorgulescu (Bucharest)

1166.65014Hari, V.; Matejaš, J.Accuracy of two SVD algorithms for 2×2 trian-gular matrices.Appl. Math. Comput. 210, No. 1, 232-257 (2009).The paper proves and compares accuracy esti-mates of two algorithms for computing the sin-gular value decomposition (SVD) of 2 × 2 trian-gular matrices. The first algorithm is a new one.It is composed of four known algorithms. Two ofthem are used for the upper- and the other twofor the lower-triangular 2 × 2 matrix. The sec-ond algorithm is the existing algorithm whichis coded as LAPACK auxiliary routine xLASV2.For simplicity, it is referred to it as the LAPACKalgorithm.In the beginning of the paper, the formulas whichdefine the new algorithm are briefly derived.They are based on the Voevodin formulas forcomputing the SVD of general 2 × 2 matrices.These formulas have already been used as partof the Kogbetliantz method for computing theSVD of n × n triangular matrices. However, forthe completeness of the exposition, the formulasare derived in a new simple way, which addi-tionally delivers several essential details whichare used later. A 2 × 2 triangular matrix can beeither upper- or lower-triangular. For the upper-triangular case, two algorithms are derived, de-pending on whether the left or the right rota-tion is applied first. They are called UL and URalgorithm. By a proper selection between thesetwo algorithms, a new accurate algorithm for theupper-triangular case is defined. Similarly, forthe lower-triangular case, two algorithms are ob-tained, and selection is made to obtain an accu-rate algorithm. A nice fact is that instead of usingfour pairs of formulas (used in UL, UR, LL andLR algorithms) one can use just one.A subtle error analysis of the new algorithmis provided. Some natural assumptions for theanalysis are given. First, basic error analysis as-sumptions are accomplished, which are basedon the IEEE standard. In the analysis, the non-linear parts of the errors are not neglected, and

the linear and the nonlinear part of each errorare estimated separately. In addition, the signs ofthe errors, especially of the intermediate quan-tities are kept on the track. Finally, the errors ofthe output data are expressed in terms of somequantities which depend on the off-diagonally ofthe matrix. The proofs are somewhat lengthy, sothey are extracted and placed in the AppendixA.The same analysis is applied to the LAPACK al-gorithm. The error bounds for the two algorithmsare compared and the results of simple numeri-cal tests are presented. In general, the errors ob-tained for the new algorithm are somewhat betterthan those obtained for the LAPACK algorithm.More importantly, in the case of an almost diag-onal 2× 2 triangular matrix, the estimates for thenew algorithm are notably better than those ofthe LAPACK algorithm.This fact is important since each of these two al-gorithms can be used as the core algorithm of theKogbetliantz method for n × n triangular matri-ces. Since after some iteration, the Kogbetliantzmethod works with an almost diagonal matrixthe new algorithm seems to be a more appropri-ate candidate to be used as the core of the Kog-betliantz method. Tzvetan Semerdjiev (Sofia)

1166.65021Cucker, Felipe; Krick, Teresa; Malajovich, Gre-gorio; Wschebor, MarioA numerical algorithm for zero counting. I:Complexity and accuracy.J. Complexity 24, No. 5-6, 582-605 (2008).The authors present an algorithm that countsthe number of distinct real zeros of a polyno-mial square system f . The algorithm performsO(log(n · D · k( f ))), where n is the number ofpolynomials, D is a bound on the polynomials’degree, and k( f ) is a condition number of thesystem. The algorithm uses finite-precision arith-metic and a major feature of the results is a boundfor the precision required to ensure that the re-turned output is correct which is polynomial inn,D, and logaritmic in k( f ). Each iteration, thatuses an exponential number of operations, canbe computed in parallel polynomial time in n,log(D), and log(k( f )). Sonia Pérez Díaz (Madrid)

1166.65045Sousa, ErcíliaOn the edge of stability analysis.Appl. Numer. Math. 59, No. 6, 1322-1336 (2009).


The application of high order finite differenceschemes to solve numerically partial differentialequations (PDEs) with boundary conditions re-quires a special treatment in a neighbourhood ofthe boundary, since in many cases additional nu-merical schemes have to be introduced, and thestability of the overall scheme can be altered. Inthis respect, with the von Neumann analysis it isnot always possible to establish accurately the in-fluence of the boundary conditions, whereas thespectral analysis may fail to capture transient ef-fects in certain time-dependent PDEs.On the other hand, the Lax stability analysisprovides a more accurate characterisation of theinfluence of numerical schemes dealing withboundary conditions, but it is not always easy toget the corresponding stability conditions. It istherefore of the maximal interest to obtain prop-erties relating the von Neumann and spectralanalyses with the Lax procedure and thus pro-vide efficient ways of testing when a particularfinite difference scheme is stable.This is precisely the aim of the present paper,where it is shown under which circumstances thevon Neumann analysis together with the spectralanalysis provide sufficient conditions to achievestability according with the Lax criterion. Thetreatment is carried out for a general class of fi-nite difference schemes and is illustrated in prac-tice on a pair of examples.

Fernando Casas (Castellon)

1166.65059Kondratyuk, Yaroslav; Stevenson, RobAn optimal adaptive finite element method forthe Stokes problem.SIAM J. Numer. Anal. 46, No. 2, 747-775 (2008).A new adaptive finite element method for solv-ing the Stokes equations is developed, which isshown to converge with the best possible rate [cf.P. Binev, W. Dahmen and R. DeVore, Numer. Math.97, No. 2, 219–268 (2004; Zbl 1063.65120); and R.Stevenson, Found. Comput. Math. 7, No. 2, 245–269 (2007; Zbl 1136.65109)]. The method consistsof 3 nested loops. The outermost loop consists ofan adaptive finite element method for solving thepressure from the (elliptic) Schur complementsystem that arises by eliminating the velocity.Each of the arising finite element problems isa Stokes-type problem, with the pressure be-ing sought in the current trial space and thedivergence-free constraint being reduced to or-thogonality of the divergence to this trial space.

Such a problem is still continuous in the velocityfield. In the middle loop, its solution is approxi-mated using the Uzawa scheme. In the innermostloop, the solution of the elliptic system for the ve-locity field that has to be solved, in each Uzawaiteration is approximated by an adaptive finiteelement method.Finally, the authors present numerical resultsand compare them with those obtained with themethod from E. Bänsch, P. Morin and R. H. No-chetto [SIAM J. Numer. Anal. 40, No. 4, 1207–1229(2002; Zbl 1027.65148)]. H. P. Dikshit (Bhopal)

1167.65018la Cruz, William; Raydan, MarcosResidual iterative schemes for large-scale non-symmetric positive definite linear systems.Comput. Appl. Math. 27, No. 2, 151-173 (2008).A new iterative scheme for the solution of large-scale nonsymmetric linear systems, whose ma-trix has a positive (or negative) definite symmet-ric part is presented and analyzed. The proposedscheme uses the residual vector as search direc-tion. The stepsize and up to date literature fea-tures for residual methods for nonlinear systemsused, are differentiating the proposed schemefrom the method of Richardson.Numerical experiments are included to showthat without preconditioning, the proposedscheme outperforms some recently proposedvariations of Richardson’s method, and com-pletes with well known and well estab-lished Krylov subspace methods: GMRES andBiCGSTAB. The computational experimentsshow that in the presence of suitable precondi-tioning strategies, residual iterative methods canbe competitive and some times advantageouswhen compared with Krylov subspace methods.

Vasilis Dimitriou (Chania)

1167.65050Ben-Artzi, Matania; Falcovitz, Joseph; Li,JiequanThe convergence of the GRP scheme.Discrete Contin. Dyn. Syst. 23, No. 1-2, 1-27(2009).This paper deals with the convergence of thesecond-order GRP (Generalized Riemann Prob-lem) numerical scheme to the entropy solutionfor scalar conservation laws with strictly convexfluxes. The approximate profiles at each time stepare linear in each cell, with possible jump discon-tinuities (of functional values and slopes) across


cell boundaries. The basic observation is that thediscrete values produced by the scheme are ex-act averages of an approximate conservation law, which enables the use of properties of such so-lutions in the proof.In particular, the “total-variation" of the schemecan be controlled, using analytic properties. Inpractice, the GRP code allows “sawteeth" pro-files (i.e., the piecewise linear approximation isnot monotone even if the sequences of averagesis such). The “reconstruction" procedure consid-ered here also allows the formation of “sawteeth"profiles, with an hypothesis of “Godunov Com-patibility", which limits the slopes in cases ofnon-monotone profiles. The scheme is proved toconverge to a weak solution of the conservationlaw. In the case of a monotone initial profile it isshown (under a further hypothesis on the slopes)that the limit solution is indeed the entropy so-lution.The constructed solution satisfies the “finitepropagation speed", so that no rarefaction shockscan appear in intervals such that the initial func-tion is monotone in their domain of dependence.However, the characterization of the limit solu-tion as the unique entropy solution, for generalinitial data, is still an open problem.

Yasar Sözen (Istanbul)

1166.90006Garavello, Mauro; Piccoli, BenedettoTime-varying Riemann solvers for conserva-tion laws on networks.J. Differ. Equations 247, No. 2, 447-464 (2009).This paper deals with a scalar partial differen-tial equation in conservation form on a network,composed by a finite collection of arcs connectedtogether at nodes. To describe the dynamics, itis sufficient to define solutions to Riemann prob-lems at nodes, which are Cauchy problems withconstant initial conditions on the arcs meeting atthe node. Using the same terminology of conser-vation laws on Euclidean spaces, the maps pro-viding such solutions are called Riemann solversat nodes. The authors prove existence of solu-tions to Cauchy problems on the whole network.

Nicolae Pop (Baia Mare)

1167.90003Erbao, Cao; Mingyong, LaiA hybrid differential evolution algorithm to ve-hicle routing problem with fuzzy demands.J. Comput. Appl. Math. 231, No. 1, 302-310

(2009).The authors consider a non-deterministic versionof the vehicle routing problem. The deterministicversion of the problem consists in finding routesof minimal total cost, beginning and ending ina central depot for a fleet of vehicles to servea number of customers. Each customer must bevisited exactly once by one vehicle possibly un-der some additional constraints. The authors de-scribe a fuzzy version of credibility measure the-ory. Using this theory, a vehicle routing problemwith fuzzy demand is introduced. A chance con-strained program model is presented. After that,stochastic simulation and differential evolutionalgorithms are integrated to design a hybrid al-gorithm, which focuses on minimizing the totaltraveled distance. A dispatcher preference indexexpressing dispatcher’s attitude to risk is intro-duced and its in a certain sense best value is ob-tained by the hybrid algorithm. The effectivenessof the hybrid algorithm is illustrated by numeri-cal examples in the concluding part of the paper.

Karel Zimmermann (Praha)

1166.94005Montoya Zegarra, Javier A.; Leite, Neucimar J.;Da Silva Torres, RicardoWavelet-based fingerprint image retrieval.J. Comput. Appl. Math. 227, No. 2, 294-307(2009).The authors present a method for fingerprintbased identification based on feature vector com-parison in which a basic component of the fea-ture vector consists of Gabor coefficients, or othertypes of wavelet coefficients.After providing an expository introduction tometric access methods for image retrieval anda comprehensive overview of the specific sys-tem architecture proposed here, in addition toa discussion of normalization techniques whenthe images in question are fingerprints, the au-thors outline several specific possible feature de-scriptors, including quadtree wavelets, “Gaborwavelets”, and steerable wavelets. The featurevector components and corresponding metric ac-cessibility properties are described precisely ineach case.Numerical comparisons are made based on im-age ensembles coming from the Bologna FVC2002 and 2004 fingerprint databases. Based onseveral numerical experiments the authors con-clude that the Gabor wavelets outperform sta-tistically the approaches using other types of


wavelets in terms of retrieval accuracy, basedon the specific metric accessibility methods em-ployed. No direct comparisons are made withother methods for fingerprint identification andcomparison. Joseph Lakey (Las Cruces)

1166.94011Gabidulin, Ernst M.; Loidreau, PierreProperties of subspace subcodes of Gabidulincodes.Adv. Math. Commun. 2, No. 2, 147-157 (2008).This paper extends the results previously ob-tained and presented by E. M. Gabidulin, P. Loin-dreau [in: 2005 IEEE International Symposium onInformation Theory, 121–123 (2005)] and showsthat in some cases questions remaining open forsubspace subcodes of Reed-Solomon codes inHamming metric can be answered for Gabidulincodes in rank metric: When the length of the par-ent Gabidulin code is equal to the degree of thealphabet field, there exists a rank-preserving iso-morphism between the subspace subcode and aGabidulin code with smaller parameters. In thatcase. the authors design a systematic procedureencoding all the possible information as well asa decoding algorithm correcting up to the capa-bility of the subspace subcode. Then, they gen-eralise the results to the direct sum of subspacesubcodes and show that the number of decod-able error-patterns is larger than what is theoret-icalIy possible for a code with the same parame-ters, but without this additional structure. Finallythey prove that subfield subcodes of Gabidulincodes can be seen, modulo the action of the gen-

erallinear group, as the direct sum of Gabidulincodes with smaller parameters.

Giorgio Faina (Perugia)

1167.94006Charters, P.Generalizing binary quadratic residue codes tohigher power residues over larger fields.Finite Fields Appl. 15, No. 3, 404-413 (2009).The binary quadratic residue codes are well-known codes in coding theory. Their construc-tion starts from p ≡ ±1 (mod 8) so that 2 is aquadratic residue modulo p. This article investi-gates a generalization of these quadratic residuecodes. Let p and q be two distinct prime numberssuch that q|(p− 1) and such that q is a q-th powerresidue modulo p. Then there exists an elementβ ∈ Fp such that βq

≡ q (mod p), and it is possi-ble to divide Fp into q cosets. These cosets are thebasis to define the q-th power residue codes. Theauthor presents generating polynomials for thesecodes, and defines a new notion correspondingto the binary concept of an idempotent. The con-struction of these q-th power residue codes isalso briefly explained in Section 15.2 of E. W.Berlekamp [Algebraic Coding Theory (McGraw-Hill, New York, NY) (1968)]. Using different tech-niques from the book of Berlekamp, the authorpresents a lower bound on the codeword weightof the dual of these q-th power residue codes.This lower bound then leads to a lower boundon the weight of the codewords of the q-th powerresidue codes. Leo Storme (Gent)

132 Mathematical Physics.

Mathematical Physics.

1166.70004O’Reilly, Oliver M.Intermediate dynamics for engineers. A unifiedtreatment of Newton-Euler and Lagrangian me-chanics.Cambridge: Cambridge University Press (ISBN978-0-521-87483-0/hbk). xiv, 392 p. £ 55.00;$ 99.00/hbk; $ 79.00/ebook (2008).The book attracts by excellent design and sim-plicity of presentation. The theoretical mechanicsfor engineers is explained with sufficient com-pleteness. At the same time, a good selectedbibliography gives the possibility to become ac-quainted with modern results and methods. Thisis one of a few books in which, along with famoustextbooks of J. L. Lagrange, W. R. Hamilton, E. T.Whittaker, L. A. Pars, the popular Russian au-thors A. I. Lur’e, F. Gantmacher, V. I. Arnol’d arealso cited. The use of tensor symbolics from thefirst pages of the book is very useful for presenta-tion and will help students in their future studyof continuum mechanics and other subjects. Thesubject matter is illustrated by numerous highlystructured examples and exercises, featuring awide range of applications and numerical simu-lations.This book has sufficient material for twosemester-length courses in intermediate engi-neering dynamics. For the first course, a Newton-Euler approach can be used, followed by a La-grangian approach in the second course. Un-doubtedly, this is an excellent textbook on me-chanics for engineers, creating a reliable base forfollowing training.

Alexander Mikhailovich Kovalev (Donetsk)

1167.70001Klein, Felix; Sommerfeld, ArnoldThe theory of the top. Volume I. Introduction tothe kinematics and kinetics of the top. Prefaceby Michael Eckert. Translated from the Germanoriginal by Raymond J. Nagem and Guido San-dri.Basel: Birkhäuser (ISBN 978-0-8176-4720-9/hbk). xviii, 279 p. EUR 64.90/net; SFR 115.00;£ 49.00; $ 79.95 (2008).Without doubt, this book is a classic in physics(mechanics), written at the end of the nineteenth

century by Felix Klein, a mathematician of worldfame and by Arnold Sommerfeld, a rising star inphysics at that time. Felix Klein was, by the way,also responsible for the ”golden age” of mathe-matics in Göttingen in the first third of the twenti-eth cenntury, because he convinced the Prussianminister of culture at that time to establish a cen-ter of excellence in mathematics in Göttingen, forwhich the next two new hired professors were D.Hilbert and H. Minkowski.Klein taught besides the usual classes in puremathematics also classes on very applied sub-jects, like “’elementary geometry”, “the theoryof top” or “’technical mechanics” for students inmathematics who wished to become high schoolteachers. In these classes usually one of his assis-tents was charged in elaborating the manuscriptinto a booklet. In the case of the theory of top itwas Sommerfeld’s duty, who at this time (1895)was assistent in mathematics. Indeed the out-come was not a booklet but a book of about thou-sand pages edited in four volumes, where the lastvolume appeared about 15 years after the firstone, when Sommerfeld was already professor,now of physics, and not in Göttingen anymore.It is interesting to note that Klein had the impres-sion that the whole project derailed, because inthe third and fourth volume, which dealt withapplications, and in particular in the fourth vol-ume, where technical applications were treated,almost no use was made of the theoretical frame-work developed in the first two volumes, thatis, the advanced function-theoretic methods andthe exact representation of the motion by ellip-tic functions. For the technical applications, theconcepts of momentum and rigid body dynam-ics were sufficient. Also the representation of thesingularity-free rotation, for which four param-eters instead of three were necessary and theirquaternion geometric meaning was elaboratedin the first volume, was not relevant to the ap-plications, even if it was a masterpiece of thepresentation.In the opinion of the reviewer, to give in the for-mulation of the kinetic equations preference tothe method of impact forces over the now usualmethod of continuous forces is certainly a weakpoint in this book. Today we know that this for-mulation used by Klein in this book completely

Mathematical Physics. 133

disappeared from the literature on classical me-chanics. Although the difference between thesetwo formulations – impact forces versus contin-uous forces – is not really substantial, it requiressome effort to get acquainted with these equa-tions.The key equations in applications of the the-ory of top are Euler’s equations describing therotations of a rigid body. Here V. I. Arnol’din his book [Mathematical methods of classicalmechanics: Textbook. 3rd ed., rev. and compl.(Russian). Moskva: Nauka (1989; Zbl 692.70003)]gives an extremely elegant and easily compre-hensible derivation. It is based on a clever nota-tion giving the same letter in lower case or uppercase to a quantity depending whether it is repre-sented in the inertial or in the body fixed frame,respectively. It is interesting to note that Kleinalso uses this notation and may have been insome sense influencial for Arnol’d’s approach.Another item should be mentioned here in admi-ration of Klein’s work, and this is his definitionof the concept of stability, which also meets allrequirements which nowadays are made for thisimportant subject. Some examples, like the force-free motion and the stabilty of the rotation axisare worked out in detail.It is very positive that, by producing the Englishedition of this book, a large group of contempo-rary scientists and especially mathematicians canadmire and appreciate what a broad understand-ing mathematicians like Felix Klein not only oftheir field but also of physics had.Very interesting ample notes of the translators(about 50 pages), where also various motionsare worked out, physically interpreted and ex-cellently illustrated by numerous figures, and anice preface by Michael Eckert make this volumeeven more interesting to read than the Germanoriginal. Hans Troger (Wien)

1166.74002Sampson, William W.Modelling stochastic fibrous materials withMathematica.Engineering Materials and Processes. London:Springer (ISBN 978-1-84800-990-5/hbk; 978-1-84800-991-2/ebook). xi, 277 p. EUR 89.95/net;SFR 149.50; £ 60.00; $ 139.00 (2009).This book provides an overview of the structureof stochastic fibrous materials, and the use ofMathematica code to develop models describ-ing their structure and performance. The author

classifies paper and similar composite materialsas stochastic. He considers random structures tobe a special class of stochastic fibrous materi-als. These materials possess the following prop-erties: i) the fibres are deposited independently ofone another; ii) they have an equal probability oflanding at all points in the network and of mak-ing all possible angles with any arbitrary chosen,fixed axis. “Departures from randomness" canbe caused by preferential orientation of fibres toa given direction, fibre clumping and/or disper-sion.The book is divided into 7 chapters: 1. Introduc-tion; 2. Statistical tools and terminology; 3. PlanarPoisson point and line processes; 4. Poisson fibreprocesses I: fibre phase; 5. Poisson fibre processesII: void phase; 6. Stochastic departures from ran-domness; 7. Three-dimensional networks.This well-written book is a reader-friendly andgood-organised manual in the field of compos-ite materials. It can be highly recommended toexperts in mechanics of solids, engineers, and tograduate, postgraduate and doctoral students.

Igor Andrianov (Köln)

1167.74002Löbach, DominiqueInterior stress regularity for the Prandtl-Reussand Hencky model of perfect plasticity usingthe Perzyna approximation.Bonner Mathematische Schriften 386. Bonn:Univ. Bonn, Mathematisches Institut. 53 p.(2007).The author gives a brief introduction into the the-ory of perfect plasticity and Perzyna viscoplas-ticity used for the approximation of Hencky andPrandtl-Reus model. All models introduced havein common that the material behavior is sep-arated into two kinds: elastic and plastic. Thegeometrical linear theory is used. The authorshows the convergence of the sequence σµ of thePerzyna models to the stress solution σ of theHencky model. The local regularity of the stresstensor σµ is studied, and the local differentiabilityof the stress tensor σ is established in the case ofvon Mises yield criterion. The author proves theregularity of the quasi-static Perzyna viscoplas-ticity model, and, using finite differences, showsthe existence of the time derivative σµ. The localdifferentiability of the stress tensor is establishedin the Prandtl-Reus model with von Mises yieldcriterion. Georg V. Jaiani (Tbilisi)


1166.76001Pringle, Jim E.; King, Andrew R.Astrophysical flows.Cambridge: Cambridge University Press (ISBN978-0-521-86936-2/hbk). x, 206 p. £ 45.00; $ 87.00(2007).Almost all of the baryonic Universe is fluid, andthe study how these fluids move is central to as-trophysics. This new graduate textbook providesa basic understanding of the fluid dynamic pro-cesses relevant to astrophysics. The mathemat-ics used to describe these processes is simpli-fied to bring out the underlying physics. To keepthe book to a manageable size, the authors havehad to be selective. In particular, they omittedall discussion of dissipative fluid processes suchas viscosity and magnetic diffusivity. The workcontains 14 chapters. The problems at the ends ofthe chapters are intended to illustrate the coursematerial further and also to introduce additionalideas. Thus they are an integral part of the book,and the reader will benefit from working throughthem.The book starts with a brief derivation and dis-cussion of the equations of fluid dynamics. Thisis aimed mainly at establishing the notation, aswell as at stressing those properties of fluidsrelevant to astrophysics which may be less fa-miliar to fluid dynamicists from other fields. Inthe second chapter, topics are considered whichare basic to understanding of flows in compress-ible media, that means pressure waves and mag-netic waves in uniform media, and nonlinearflows in one dimension including shocks areexplained. As many astrophysical phenomenaare approximately spherically symmetric, chap-ter 3 deals with spherically symmetric flows,steady ones and explosive ones. In the next fewchapters, it is considered what happens if thestationary fluid configuration is perturbed. Theunperturbed configuration is assumed to be afluid at rest in a stationary gravitational po-tential well. Stellar models, Eulerian and La-grangian perturbations, and adiabatic perturba-tions are introduced in chapter 4. As an exam-ple, the Schwarzschild stability criterion of afluid against convection is derived. Chapter 5discusses waves in stratified media, in a plane-parallel atmosphere and in a polytropic one. Inchapter 6 a brief look at damping and excitationprocesses in stars is taken. Influences of thermalconductivity, opacity, and energy production bynuclear fusion are considered. In chapter 7 again

the stability of a static atmosphere is discussed,but with the complication of an added magneticfield. Again a variational principle is derived. Es-pecially the buoyancy and Parker instabilities aretreated. It is found that a magnetic field can ei-ther stabilize or destabilize the fluid. Instabilitiesgenerated by the effects of heating and cooling,coupled with the effect of thermal conductivity,are explained in chapter 8. Instabilities driven bythe self-gravity of the fluid are discussed in chap-ter 9. Such instabilities are extremely importantprocesses concerning the formation of astrophys-ical objects such as stars and planets, and moregenerally, large-scale structures such as galaxiesand clusters. Chapter 10 deals with linear shearflows in incompressible fluids. A condition forinstability of a stratified shear-flow in a gravita-tional field is derived. Having in mind mainlythe effect of rotation on stellar objects, chapter 11discusses rotating flows. Chapter 12 deals withthe stability of a differentially rotating fluid. Herethe Rayleigh’s criterion for axisymmetric pertur-bations is derived. Describing the star simply asa circular cylinder of incompressible fluid, chap-ter 13 discusses modes in rotating stars. Finally,chapter 14 shows that instabilities in cylindricalshear flows are possible, even if the Rayleigh cri-terion is satisfied, if there exist non-axisymmetricperturbations.

Claudia-Veronika Meister (Darmstadt)

1166.76002Aye, Khin MyoComplexity regulations and the inversion ofenvironmental hydrological distributed mod-els.Göttingen: Cuvillier; Bonn: Univ. Bonn,Naturwissenschaftlich-Mathematische Fakul-tät (Diss.) (ISBN 3-86727-028-7). 198 p.EUR 27.00 (2006).Publisher’s description: For the rainfall-runoffprocess, a nonlinear rainfall-runoff model is de-veloped based on the water balance equation.Observed monthly precipitation and potentialevapotranspiration, unit area of the reservoirsand selected maximum storage values are usedto simulate the models using MATLAB. Thecontinuity, monotonicity and uniqueness of thismodel are described. It was found that the down-hill simplex method is suitable for the inversionof the rainfall-runoff model. The results from thenumerical experiments for model validation in-dicate that a system of two or three reservoirs


can be fitted with another system of up to sevenreservoirs.

1166.76003Prosperetti, Andrea (ed.);Tryggvason, Grétar (ed.)Computational methods for multiple flow.Cambridge: Cambridge University Press (ISBN0-521-84764-8/hbk). xxiii, 470 p. £ 55.00; $ 95.00(2007).The rough contents of this graduate-level text-book are as follows: Ch. 1, Introduction: A com-putational approach to multiphase flow, by A.Prosperetti and G. Tryggvason; Ch. 2, Direct nu-merical simulations of finite Reynolds numberflows, by G. Tryggvason and S. Balachandar; Ch.3, Immersed boundary methods for fluid in-terfaces, by G. Tryggvason, M. Sussman and M.Y. Hussaini; Ch. 4, Structured grid methods forsolid particles, by S. Balachandar; Ch. 5, Finite el-ement methods for particulate flows, by H. Hu;Ch. 6, Lattice Boltzmann models for multiphaseflows, by S. Chen, X. He and L.-S. Luo; Ch. 7,Boundary integral methods for Stokes flows, byJ. Blawzdziewicz; Ch. 8, Averaged equations formultiphase flow, by A. Prosperetti; Ch. 9, Point-particle methods for disperse flows, by K. Squires;Ch. 10, Segregated methods for two-fluid mod-els, by A. Prosperetti, S. Sundaresan, S. Pannala andD. Z. Zhang; Ch. 11, Coupled methods for multi-fluid models, by A. Prosperetti.The main aim of this multi-author volume is toprovide a detailed description of computationalmethods used in the simulation of multiphaseflows. It is well known that it is quite difficultto set up fully controlled physical experimentsfor such flows. Consequently, the authors, whichare outstanding researchers, carry out a lot ofcomputational experiments in order to controlthe behavior of such fairly complicated systems.Their computational effort is an important stepto a better understanding of very complex andlarge-scale multiphase systems.

Titus Petrila (Cluj-Napoca)

1167.78001Rebhan, EckhardTheoretical physics. Electrodynamics.(Theoretische Physik: Elektrodynamik.)(German)Heidelberg: Elsevier/Spektrum AkademischerVerlag (ISBN 978-3-8274-1717-6/pbk). xi, 406 p.EUR 26.00; SFR 40.00 (2007).

Das Kapitel “Mathematische Vorbereitung” istsehr gut dargestellt auf dem Niveau, das für dieStudierenden der Physik und der Elektrotech-nik verständlich sein sollte. Vom Umfang her istes vollkommen ausreichend, um die später imBuch eingeführten physikalischen Zusammen-hänge zu beschreiben. Darüber hinaus könnendiese Grundlagen der Vektoranalysis später alsNachschlagewerk benutzt werden.Im Kapitel 3 wird die Gesamtheit der Maxwell-Gleichungen für Vakuumfelder aufgestellt. Dochbevor der Autor die Maxwell-Gleichungen fürzeitabhängige Felder in voller Allgemeinheit ein-führt, erklärt er die Maxwell-Gleichungen derElektrostatik und Magnetostatik. Dabei werdendie dazugehörigen Grundbegriffe wie Ladun-gen, Ströme, Felder und die entsprechendenKräfte erläutert. Die Einführung folgt der his-torischen Entwicklung der Elektrodynamik, in-dem die Grundprinzipien wie das der Super-position nach der Einführung des Coulomb-Gesetzes erläutert werden. Die Reihenfolge: sta-tionäre Ströme – Biot-Savart-Gesetz – LorentzKraft gibt die Chronologie der einzelnen Schrittein der Geschichte der Elektrodynamik wider.In den Kapiteln 4 und 5 werden die Elektro-und Magnetostatik ausführlich behandelt undmit vielen Beispielen erläutert.Das Kapitel 6 “Stromkreise mit stationären undzeitlich langsam veränderlichen Strömen” ist einKapitel, das jeder Elektroingenieur irgendwannwieder aufschlagen würde. Es enthält die the-oretische Vorbereitung für die Kurse über elek-trische Maschinen.Das Buch schließt mit einem Kapitel über schnellveränderliche elektromagnetische Felder. DasKapitel beginnt klassisch mit der Einführung derWellengleichung und wird mit der Behandlungvon elektromagnetischen Feldern in Leitern undHohlleitern sowie in Materie fortgesetzt.Darüber hinaus gibt es in jedem Kapitel Ab-schnitte, die nicht in den zeitlichen Rahmen einerVorlesung über “Theoretische Elektrotechnik”hineinpassen, dafür aber im Buch ihren Platz ein-nehmen konnten.Das Buch eignet sich ebenso gut als vorlesungs-begleitendes Material wie zum Selbststudium.

Ursula van Rienen (Rostock)

1167.80001Bergheau, Jean-Michel; Fortunier, RolandFinite element simulation of heat transfer.Translated from the French original by Robert


Meillier.Hoboken, NJ: John Wiley & Sons; London:ISTE (ISBN 978-1-84821-053-0/hbk). 279 p.EUR 95.00; £ 70.00; $ 140.00 (2008).This book is suitable for graduate students in en-gineering and also for use in senior-level course.Some knowledge of use of ordinary, and par-tial differential equations to describe problemsrelated to engineering analysis is assumed. Theaim of this book is to present the oasis and ap-plication of the FEM to the solution of industrialthermal problems. It consists of three parts: Part1, dedicated to the solution of steady state heatconduction problems, introduces the FEM. Dif-ferent steps of physical modeling leading to aBVP are described in Chapter 1. In Chapter 2are introduced the major principles of finite ele-ment approximation leading to an approximatesolution of continuous problems. Chapter 3 con-cerns the description of isoparametric elementswhich are by far the most widely used elementsin most computer codes. Part 2 extends the fieldof application of the method to transient stateconduction problems (Chapter 4), the most com-mon nonlinearities (Chapter 5), and transportphenomena (diffusion convection problems) inChapter 6. In this last chapter, either a liquidmedium or a mobile solid medium is considered.The velocity field in the material is assumed to beknown. The last part, Part 3, deals with coupledproblems. In Chapters 7 and 6 by boundary con-ditions coupled problems are treated: radiationproblems, fluid and structure coupling in a pip-ing system, and in Chapter 9, thermometallurgi-cal coupling including additional state variables.The last three chapters, Chapter 10, 11, 12 presentby partial differential equations thermochemical,electrothermal, and magnetothermal coupling.This book is a survey of various thermal prob-lems which professional engineers may have tosimulate. Pavol Chocholatý (Bratislava)

1166.81001Jaeger, GreggQuantum information. An overview. With fore-word by Tommaso Toffoli.New York, NY: Springer (ISBN 978-0-387-35725-6/hbk). xviii, 284 p. EUR 44.95/net; SFR 74.00;£ 34.50; $ 49.95 (2007).As the title states, this book provides a conciseoverview over quantum information theory (in-cluding some short glimpses into experiments).Since on only 284 pages basically all important

subjects are treated, the author mostly concen-trates on fundamental concepts, while techni-cal details, or more sophisticated mathematicalbackground are added only when necessary. Thefirst chapter starts with a discussion of the prop-erties of qubits and their state space. It is fol-lowed by a survey on observables and measure-ments (Chapter 2), and a first look onto non-classical correlations between two quantum sys-tems. This includes in particular a more detaileddiscussion of hidden variable models and Bellinequalities (Chapter 3). After introducing themost important concepts of classical informationtheory (Chapter 4), the book turns in Chapter5 to quantum information and looks at mea-sures for the amount of information which is con-tained in a given quantum system (e.g. entropy).This is followed by two chapters about entan-gled states, i.e. non-classical correlations alreadyvisited in Chapter 3. Now the treatment is moreformal and mainly devoted to quantitative as-pects like entanglement measures. The bipartitecase is treated in Chapter 6 the multipartite casein Chapter 7. Chapter 8 then discusses the esti-mation of quantum states and processes, whilechapter 9 deals with channels, channel capaci-ties and other topics from quantum communi-cation. Chapter 10 discusses quantum decoher-ence and quantum error correction. Basically itis a nice albeit unusual idea to put both subjectsinto one chapter, because decoherence is the mostimportant source of noise in quantum informa-tion processing and quantum error correction isthe default strategy to cope with it. On the otherhand, however, this organization of the subjectsbreaks somehow the close connection betweenerror correction and channel capacities (in par-ticular because channel capacities are discussedfirst). After this chapter the book turns to quan-tum broadcasting, copying and deletion. Theseare considered to be impossible in quantum me-chanics, since they can not be performed withouterror on a single input system in an arbitrary un-known state (but the error can be made arbitrarysmall if enough independent input systems in thesame state are available). It is a little bit unfortu-nate that this subject is not discussed in closerconnection to state estimation which is closelyrelated in particular to optimal cloning. The lastthree chapters are devoted to the main tasks ofquantum information processing: quantum keydistribution (Chapter 12), quantum computing(Chapter 13), and quantum algorithm (Chapter


14). All three chapter are self contained in thesense that they contain all material from clas-sical informatics which are necessary to followthe presentation (e.g. a short introduction to theconcept of complexity classes). Finally the bookis complemented by two appendices providinga very brief overview over mathematical foun-dations and physical postulates of quantum me-chanics. This is actually a nice book which canbe recommended to researchers in other fieldswho wants to get a fast and sound overview overthe subject. It is also suitable for students learn-ing quantum information theory, provided it issupplemented by additional publications fromwhich technical details not given in this bookcan be taken. For researchers working in quan-tum information theory it does not contain newmaterial, but can serve as a compact referencebook. Michael Keyl (Braunschweig)

1166.81002Schwabl, FranzQuantum mechanics. 4th ed. Translated from7th German ed.Berlin: Springer (ISBN 978-3-540-71932-8/hbk).xv, 424 p. EUR 54.95/net; SFR 96.00; £ 42.50;$ 69.95 (2007).This is the translation of the seventh revised edi-tion of the well known textbook in German. It isan excellent introduction for students of physicsor mathematics into the fundamentals of quan-tum mechanics covering the methods used in ap-plications. It is written in a language well suitedfor the course in theoretical physics omittingdeeper mathematical details but explaining themethods extensively and in a very instructivemanner. There are only few points to criticize.Introducing the Schrödinger equation the authorargues that it must be first order in time becausethe initial condition should be the wave functionat a given instant of time. But the evidence of thelatter condition is not clear. Schrödinger himself,trying to represent atomic spectra as the solutionof an eigenvalue problem, gave the striking ar-gument that the energy spectra must be boundedfrom below. At the beginning of the final chapter,when the author explains the difference betweenthe classical particle concept in Newtonian orrelativistic mechanics and that in quantum me-chanics, one reads ‘coordinate system’ instead of‘frame of reference’. Another point is why thecomplicated concept of the delta function mustbe used to explain the momentum representation

when the elementary concept of Fourier transfor-mation suffices for the explanation. These defec-tive appearances show up some bounds but theseshould not be stressed over. The main point isthat the fundamentals and methods of quantummechanics are mediated very well and guide thereader to apply them successfully. This book canbe best recommended to students and lecturers.

K.-E. Hellwig (Berlin)

1166.81004Teschl, GeraldMathematical methods in quantum mechanics.With applications to Schrödinger operators.Graduate Studies in Mathematics 99. Prov-idence, RI: American Mathematical Society(AMS) (ISBN 978-0-8218-4660-5/hbk). xiv, 305 p.$ 59.00 (2009).This book is a self-contained brief introductionto the mathematical methods of quantum me-chanics, with a view towards applications toSchrödinger operators. It is intended for begin-ning graduate students in both mathematics andphysics and provides a solid foundation for read-ing more advanced books and current researchliterature.The book consists of two main parts.Part 1, “Mathematical foundations of quantummechanics”, is of six chapters: Hilbert spaces;Self-adjointness and spectrum; The spectral theo-rem; Applications of the spectral theorem; Quan-tum dynamics; Perturbation theory for self-adjoint operators.This part is a concise introduction to the spectraltheory of unbounded operators. Only those top-ics that will be needed for later applications arecovered. The spectral theorem is a central topicin this approach and is introduced at an earlystage. From the start the author considers thecase of unbounded operators. The existence ofspectral measures is established via the Herglotztheorem rather than the Riesz representation the-orem. Section “Quantum dynamics” along withthe Stone classical theorem contains the state-ment and proof of the RAGE theorem (which pro-vides the connection between long time behav-ior and spectral types) and the Trotter product-formula. These results are not included in mostof classical textbooks. The final section of thispart contains the basic results of the perturba-tion theory for self-adjoint operators (relativelybounded perturbations, form-bounded pertur-bations and KLNM theorem, Hilbert-Schmidt


and trace class operators, relatively compact per-turbations, strong and norm resolvent conver-gence).Part 2, “Schrödinger operators”, contains sixchapters: The free Schrödinger operator; Al-gebraic methods; One-dimensional Schrödingeroperators; One-particle Schrödinger operators;Atomic Schrödinger operators; Scattering theory.This part starts with a review of some basicfacts concerning the Fourier transform and thefree Schrödinger equation (the free resolvent andtime evolution are computed). The position, mo-mentum and angular momentum are discussedvia algebraic methods (all these methods canbe found in almost any physics textbook onquantum mechanics, and according to the au-thor “. . . the only contribution is to provide somemathematical details”). In sections concerningthe Schrödinger operators the problems of self-adjointness and the structure of the spectrum (thestructure of an essential spectrum, absence of sin-gular continuous spectrum, . . . ) are discussed.One-dimensional Schrödinger operators are con-sidered in more detail. In particular, in the sectionon the inverse spectral theory a simple proof ofthe Borg-Marchenko theorem is presented. Thelast chapter of this part contains an introductionto the mathematical scattering theory (abstracttheory, decomposition into an incoming and anoutgoing part, scattering theory of Schrödingeroperators with short range potentials based onthe Enss approach).The required background material on mea-sure theory and integration, Hilbert spaces, andbounded linear operators is included in the pre-liminary chapter. In addition, there is an ap-pendix (again with proofs) providing all neces-sary results from measure theory and integra-tion. The only prerequisite is a solid knowledgeof advanced calculus and an introduction to com-plex analysis are required. The book is written ina very clear and compact style. It is well suitedfor self-study and includes numerous exercises(many with hints). Michael Perelmuter (Kyïv)

1166.81005Weiss, UlrichQuantum dissipative systems. 3rd ed.Series in Modern Condensed Matter Physics13. Hackensack, NJ: World Scientific (ISBN 978-981-279-162-7/pbk). xviii, 507 p. £ 31.00; $ 58.00(2008).The book provides the third edition of an impor-

tant monograph on the theoretical descriptionof dissipative effects in quantum systems, thusdealing with the treatment open quantum sys-tems. The style of the book is appropriate foradvanced graduate students and researchers inthe field.The main emphasis is on phenomena relevantfor condensed matter physics and chemistry,thus providing a complementary standpointwith respect to other monographs addressingopen quantum systems such as [H.-P. Breuerand F. Petruccione, The theory of open quan-tum systems. Oxford: Oxford University Press(2002; Zbl 1053.81001) and C. W. Gardiner andP. Zoller, Quantum noise. Berlin: Springer (2004;Zbl 1072.81002)].The basic tool used for the study of quantumsystems interacting with an environment is thefunctional integral. The path integral formal-ism leads to formally exact results, when con-sidering a bilinear coupling between the sys-tem and a harmonic bath, which is the typicalframework considered in the book. The referencesystem-plus-reservoir models considered are infact the Caldeira Leggett model and the spin-boson model.The book is written in a concise style, discussingfrom a theoretical-mathematical standpoint awealth of system relevant in physics and chem-istry. The book is divided in five parts.Part I, “General theory of open quantum sys-tems”, provides a quite brief account of other ap-proaches to open quantum systems, the introduc-tion of system-plus-reservoir models addressedin the book and a thorough presentation of pathintegrals techniques applied to open quantumsystems.Part II, “Few simple applications”, considers ex-actly solvable damped linear systems.Part III, “Quantum statistical decay”, studies de-cay of a meta stable state, thus addressing classi-cal and quantum rate theory.Part IV, “The dissipative two-state system”, dealswith the spin-Boson model, in which a two-levelsystem is linearly coupled to the coordinates of aset of harmonic oscillators.Part V, “The dissipative multi-state system”,extends the previous analysis to a multi-dimensional system coupled to an environment.Each part has an accurate organization in smallersections and paragraphs, and a rich bibliographyis provided at the end of the book. Moreover inthe preface the author provides a detailed com-


parison of the contents of the present edition withthe previous ones. Bassano Vacchini (Milano)

1167.81001Cancès, Eric; Le Bris, Claude; Maday, YvonMathematical methods in quantum chemistry.An introduction. (Méthodes mathématiques enchimie quantique. Une introduction.) (French)Mathématiques & Applications (Berlin) 53.Berlin: Springer (ISBN 3-540-30996-9/pbk). xvi,409 p. EUR 82.10 (2006).The reviewing book “Méthodes Mathématiquesen Chimie Quantique" by Eric Cancès, Claude LeBris, and Yvon Maday presents the mathemati-cal foundations of quantum chemistry. The lay-out of this book consists of eleven Chapters andtwo Appendices - the latter are dedicated to thefoundations of quantum mechanics and of thetheory of operators. The Introductory Chapter 1discusses the different levels of treating or mod-eling of many-electron systems, such as atoms,ions, molecules, and solids: the empirical, semi-empirical, and ab initio levels and provides thereader the preliminaries of the N-electron vari-ational calculus: the N-representability problem,the Hartree-Fock (HF) variational principle, thatarises on a class N-electron wave functions cho-sen as Slater determinants, and the correspond-ing Hartree-Fock self-consistent field equations,including the spin, the concept of one-electrondensity and the density functional theory (DFT)that is given via the Thomas-Fermi (TF) modeland via the Kohn-Sham (KS) self-consistent fieldmethod (see e.g., in this regard, the book [E. S.Kryachko and E. V. Ludeña, Energy Density Func-tional Theory of Many-Electron Systems. Dor-drecht: Kluwer (1990)]).Chapter 2 introduces the reader to the math-ematical analysis linked to many-body quan-tum theory and particularly includes the Hölder,Sobolev-Gagliardo-Nirenberg, and Hardy in-equalities and the derivation of the Euler-Lagrange equations of motion. Chapter 3 is thekey Introductory Chapter to the Thomas-Fermi-von Weizsäcker (TFW) model. It also deals partic-ularly with the Thomas-Fermi model with Fermi-Amaldi correction term. Chapter 4 develops theDFT and focuses on the concept of the energydensity functional, especially on that of the TFWmodel.The next Chapters are partly focusing on themost significant problems of quantum chemistry:Chapter 5 examines the Hartree-Fock model;

Chapter 6 - the hydrogen atom, the concepts ofatomic orbitals and their linear combinations, theAufbau principle, and the Roothaan approach,the Møller-Plesset perturbation method, and themethod of configuration interaction; Chapter 7discusses different quantum chemical basis sets,and finally, Chapter 8 the convergence problemof self-consistent field approaches.The next Chapters, 9 and 10, are dedicated tothe methods of study of condensed phases andperiodic systems, such as the HF, TF, TFW, andKS methods. The last Chapter 11 concludes thisbook. To summarize, this book is actually a ratherbroad reflection of the current state of mathemat-ical art of quantum chemistry that deserves to bestudied by the readers interested in the mathe-matical grounds of quantum chemistry and theirdirect applications in the research.

Eugene Kryachko (Liège)

1167.81002Gaitan, FrankQuantum error correction and fault tolerantquantum computing.Boca Raton, FL: CRC Press (ISBN 978-0-8493-7199-8/hbk). xviii, 292 p. $ 99.95 (2008).This is an excellent textbook introducing to andrepresenting the title problem. It is written ina very clear language which motivates eachstep of the formal developments which are pre-sented with mathematical rigor, formulating def-initions, theorems, and proofs. Exercises help thereader to realize the material just presented. Eachchapter ends with a collection of problems anda list of references. After the preface the intro-duction begins with a section on the histori-cal background followed by sections on classi-cal error correcting codes, how information canbe stored in quantum systems, and the prob-lem to stabilize quantum information process-ing in spite of the oo-cloning theorem. Here alsoa first pass through the methods of quantumerror correcting codes is given which containsimportant fundamentals frequently referred tolater on in the book. Chapter two, entitled ‘quan-tum error correcting codes’, begins introducingthe concepts of operations, measurements, andthe general channel concept as a operator-sumrepresentation (replacing the concept of com-plete positive map which seems to be omitted).The depolarizing channel as the simplest errormodel and generalizations are considered. Af-ter some additional definitions to quantum er-


ror correcting codes introduced in section 1.4.2the chapter closes describing the Calderbank-Shore-Steane code. Chapter three contains a de-tailed description of quantum stabilizer codes,their mathematical background, examples anddifferent realizations of such codes. In chapterfour the standard form of stabilizer codes is in-troduced. On this basis efficient encoding anddecoding and the respective circuits are consid-ered. The subject of chapter five is error correc-tion with fault-tolerant quantum computing. Thedevelopments culminate in a set of fault tolerantencoded quantum gates sufficient for universalquantum computation using any quantum sta-bilizer code. Two examples are given. The useof quantum error correcting codes enlarges thealgorithm n consideration and so new sourcesof errors arise. The question whether it is stillpossible to restrict remaining errors arbitrarilyis answered by the accuracy threshold theoremwhich is subject to chapter six. Under certain as-sumptions it is proved that remaining errors canbe omitted efficiently if the error probability dueto storage registers and a universal set of gatesis smaller than a certain value, called the accu-racy threshold. In chapter seven several construc-tions of bounds for quantum error correctingcodes are considered. Eventually, entanglementpurification protocols for teleportation througha noisy channel are considered and comparedwith quantum error correcting codes. Three ap-pendices concern the fundamentals of the theoryof(discrete) groups, quantum mechanics, and thecircuit model of quantum algorithms. - It shall bestated again that this book is an excellent intro-duction and representation of the material andcan be best recommended. K.-E. Hellwig (Berlin)

1167.81003Home, Dipankar; Whitaker, AndrewEinstein’s struggle with quantum theory. Areappraisal. With a foreword by Sir Roger Pen-rose.New York, NY: Springer (ISBN 978-0-387-71519-3/hbk). xxi, 370 p. EUR 123.00 (2007).In his foreword Roger Penrose welcomes thisbook as being a historical work which recognizesEinstein’s merits when he was criticizing state-ments that quantum theory is being a completeand final physical theory. Clearly, Einstein’s crit-icism was based on the classical particle conceptand therefore not accepted by the younger gener-ation of physicists. Actually, his critics has caused

people to think with the result of deeper insightsand progress in quantum theory. A prominentexample is the Einstein-Podolski-Rosen paperwhich shows up deep understanding of quan-tum theory. Originally considered as paradox butafter fifty years of contemplation it has been un-derstood as a phenomenon, called quantum cor-relation, verified by experimental proofs. Quan-tum correlations became a fundament of quan-tum information processing. This stands in con-trast to Bohr’s complementarity principle whichfifty years after the Solvay conferences is merelysubject of philosophical discussions and nevergot a mathematically rigorous formulation.The book is organized in four parts. The firstone, ‘Setting the scene’, begins describing the re-lation of Einstein to E. Mach, before and after1921, showing up a philosophical background. Inchapter two the development of quantum theoryuntil 1925 is reported. A summary of Einstein’scontributions is given. After a sketch of quantummechanics in chapter three features of quantumreality which contrast classical pictures are con-sidered. Chapter four is entitled ‘The standardinterpretation of quantum mechanics’. Whethersingle particle interpretations and ensemble in-terpretations may be considered as different ver-sions of a standard one is, however, anyone’sguess.Part two concerns Einstein’s approaches to quan-tum theory after 1925. Chapter five begins withan overview of 1925–1935. The history of an un-published manuscript attempting to introducehidden variables by Einstein, written 1927, to-gether with several comments is given. Thestatements and discussions at the Como andthe Solvay conferences in 1927 as well as atthe Solvay conference in 1930 are described intwo sections. A detailed discussion of the Bohr-Einstein debate is given in the last two sectionsof this chapter. The subject of chapter six is theEPR argument of 1935 together with later publi-cations by Einstein, interpretations, reactions aswell as the answer of Bohr. In the first sectionit is attempted to explain the fact that the EPRpaper was not recognized for a long time. Onereason is seen in the use of the concept ‘to bean element of reality’ which was given a pre-cise meaning by EPR and which plays a centralpoint in the in the argumentation. Clearly, say-ing that the formulation of the paper was un-sympathetic for the majority of physicists comesto saying that the time was not ripe to under-


stand EPR. W.r.t. the second section it should beremarked that there is no difference between Ein-stein and Bell locality! Einstein obviously omit-ted the use of technical terms in his short essay for‘Dialetica’, a philosophical magazine. Moreoverit should be noted, that the statement about theequations (6.1) and (6.2) are wrong: The Schmidtcoefficients are uniquely determined. The statedfree choice of an orthonormal basis ψ1,n is onlypossible, when the Schmidt coefficients mutu-ally are equal as in formula (6.3). A series of foursections is devoted to the EPR argument againstthe completeness of quantum theory. It may berather confusing for readers who are not aware ofthe content of the EPR 1935 paper since it beginswith Einstein’s 1948 paper in Dialectica and endswith his 1935 paper. So the reader misses the de-velopment of thoughts leading to the 1948 paper.It would be helpful when it would begin with thefamous 1935 paper. The awkward technical fea-tures of this paper could be surrounded replacingthe consideration of position and momentum ob-servables by spin observables (Bohm’s version).In the section following that containing Bohr’sanswer to EPR, conclusions are drawn from thestatement quantum mechanics is either incom-plete or non-local. These conclusions presupposethat quantum mechanics is accepted, a statementnot proposed by EPR. The final section reviewsthe correspondence of Einstein with Schrdingerafter 1935 including Schrödinger’s cat problem.The short chapter seven reports about Einstein’sthoughts on macroscopic limits of quantum the-ory. The sections of chapter eight are a collectionof Einstein’s views on different topics of natu-ral sciences and quantum theory as far as thisis possible. The first section asks: Did Einsteinreject quantum theory, another is entitled ‘Ein-stein’s approach to the Copenhagen interpreta-tion’, a third one ‘Einstein and ensembles’ and soon. It is useful giving answers and references tofrequently posed questions.In the third part, called denouement, more re-cent developments and problems are described.Chapter nine gives a rather detailed review ofBell’s contributions and quantum non-locality,chapter 10 is concerned with non-standard inter-pretations, e.g., the many worlds interpretation,stochastic interpretation etc.. Chapter eleven isentitled ‘Einstein and quantum information the-ory’. It gives an overview and, clearly, mentionsthe contribution of the EPR situation. The chap-

ter twelve is devoted to bridging the quantumclassical divide.The fourth an final part is an outlook. Chapterthirteen gives outlooks to general foundationsof quantum theory. Here one misses a chapteron the operational approach which arose around60 years ago, has developed remarkably and be-came an important tool for quantum informationprocessing. Chapter fourteen is an assessment ofEinstein’s view and contribution.This book presents a lot of commented historicalfacts together with the respective references. Theauthors preferably describe these facts in collo-quial language. A broader use of formal descrip-tions of the backgrounds would be at some placesadvantageous, but this may be a matter of taste.Altogether, this book is a precious and usefulcontribution to the history of quantum mechan-ics which especially gives insight into the think-ing and influence to the development of quan-tum theory by Albert Einstein.

K.-E. Hellwig (Berlin)

1166.82001Haug, Hartmut; Koch, Stephan W.Quantum theory of the optical and electronicproperties of semiconductors. 5th ed.Hackensack, NJ: World Scientific (ISBN 978-981-283-883-4/hbk; 978-981-283-884-1/pbk). xiii,469 p. £ 46.00; $ 86.00; £ 26.00; $ 48.00 (2009).See the review of the 4th edition (2005) inZbl 1094.82003This fifth edition includes an additional chap-ter on ‘Quantum Optical Effects’ where the the-ory of quantum optical effects in semiconductorsis detailed. Besides deriving the ‘semiconductorluminescence equations’ and the expression forthe stationary luminescence spectrum, results arepresented to show the importance of Coulombiceffects on the semiconductor luminescence andto elucidate the role of excitonic populations.

1167.82001Brun, RaymondIntroduction to reactive gas dynamics.Oxford: Oxford University Press (ISBN 978-0-19-955268-9/hbk). xix, 408 p. £ 49.95 (2009).The particles which compose a gas may be sub-ject to elastic collisions, or inelastic collisions orreactive collisions, that is to say collisions whichare strong enough to create new species, like inionization, for instance. Whilst elastic and inelas-tic collisions can be encompassed in unified mod-


els, reactive collisions are very complex and re-quires lengthy studies, and it is exactly the pur-pose of the present book. Loosely speaking, re-active collisions are described by various exten-sions of the Boltzmann equation.The book comprises two parts which deal withmicroscopic and mesoscopic aspects on the onehand, and the macroscopic (thermodynamic) as-pect on the other hand. The first part displaysthe fundamental statistics of the problem (equi-librium, non-equilibrium and quasi-equilibriumcollisional regimes, transport phenomena, relax-ation, Chapman-Enskog method) and the secondpart analyzes gas flows, gas dynamics, and reac-tive flows with and without dissipative regime.To read the book you need the basic of thermo-dynamics statistical physics and fluid mechanics.We shall mention that the book price is quite rea-sonable. Guy Jumarie (Montréal)

1167.82002Eschrig, HelmutThe particle world of condensed matter. An in-troduction to the notion of quasi-particle.EAGLE 24. Leipzig: Edition am GutenbergplatzLeipzig (EAGLE) (ISBN 3-937219-24-2/pbk).172 p. EUR 24.50; SFR 43.50 (2005).The book is devoted to the notion of quasi-particles in the world of condensed matter. Suchparticles has become one of the most profoundand fundamental ideas in the quantum theoryof condensed matter. Here the author tries to ex-plain how the quasi-particles appear as an in-dispensable element of the description of matter.The book consists of 8 chapters:Chapter 1 (Technical tools) has an introductorycharacter, i.e. there the fundamental notions fora mathematical description of a quantum many-body system are presented.In Chapter 2 (Macroscopic quantum systems) theimplications of the thermodynamic limit and ofmacroscopic states are provided, i.e., the thermo-dynamic limit, pure and mixed quantum states,thermodynamics states are given.In Chapter 3 (Quasi-stationary excitations) theGreen’s functions technique for the descriptionof quasi-particles and of collective excitations ispresented.In Chapter 4 (Model Hamiltonians) transforma-tions of the many-body simple Hamiltonians intomodel Hamiltonians acting on the appropriateFock spaces are given.

In Chapter 5 (Quasi-particles) quasi-particles ofthe solid state theory, the Bloch eletrons and thelattice phonons are considered.In Chapter 6 (The nature of the vacuum) the ap-propriate Fock space of a solid is determined bythe nature of its symmetry-broken vacuum.Chapter 7 (What matter consists of) deals witha review of the hierarchy of Hamiltonians and“from the first principles" in the solid state the-ory.In Chapter 8 (Epilogue: the unifying picture ofphysics) is given an epilogue on highlightsof theunified picture of physics.In Appendices the self-energy operator and thedensity functional theory are presented.The book is presented in a simple way for a gen-eral audience of students with a little physicalknowledge. Note that derivations of many for-mulas are omitted.

Farruh Mukhamedov (Kuantan)

1167.82003Kardar, MehranStatistical physics of fields.Cambridge: Cambridge University Press (ISBN978-0-521-87341-3/hbk). x, 359 p. £ 40.00; $ 75.00(2007).This nice graduate-level text covers the secondpart of author’s course on Statistical Physicsgiven at MIT. It extends the first book in theseries [M. Kardar, Statistical physics of parti-cles. (Cambridge): Cambridge University Press.(2007; Zbl 1148.82001)] which covers more stan-dard subjects of statistical mechanics. The secondbook treats the collective behaviour of interact-ing particle systems, including phase transitionsand critical phenomena.Chapter 1 “Collective behavior, from particles tofields” explains how interacting particle systemscan be described using coarse-grained fields andintroduces the notions of phase transition andcritical phenomena.In Chapter 2 “Statistical fields” the author in-troduces the Landau-Ginzburg Hamiltonian andderives some results using its mean-field approx-imation. The second part of the chapter dealswith problems of symmetry breaking, Goldstonemodes and domain walls.Chapter 3 “Fluctuations” treats fluctuations, cor-relations and their relation to susceptibilities. Theauthor introduces both lower and upper criticaldimension and gives the Ginzburg criterion for


the importance of fluctuations near a phase tran-sition.Chapter 4 “The scaling hypothesis” starts witha discussion of homogeneity of free energy anddivergence of correlation length near the phasetransition. It further introduces to the concept ofthe Renormalization Group (RG). This concept isthen applied to a simple Gaussian model.Chapter 5 “Perturbative renormalization group”explains how complicated systems can be treatedas perturbation of simple ones. It then introducesa diagrammatic representation of a perturbationtheory. The second half of this chapter deals withperturbations in the RG approach.In Chapter 6 “Lattice systems” various latticespin models are defined, comprising Ising, O(n),Potts models. Transfer matrix and RG are appliedon Ising models in dimension one and two.Chapter 7 “Series expansions” covers bothlow- and high-temperature expansions of lat-tice models. It includes the exact solution ofone-dimensional and the self-duality of two-dimensional Ising models and explains how aGaussian model appears as approximation ofhigh-temperature expansions. Further, it givesthe exact free energy of the Ising model on squarelattice and describes its critical behaviour.In Chapter 8 “Beyond spin waves” the authorreturns to the continuous setting. He defines thenon-linear σ-model and studies it using the per-turbative RG approach. Further parts of the chap-ter cover topological defects in the XY model, theRG for the Coulomb gas and the melting of 2Dsolids.Chapter 9 “Dissipative dynamics” starts with anintroduction to the theory of Brownian motion. Itcontinues with the description of the equilibriumdynamics of a field using the Landau-Ginzburgequation of dynamics of a conserved field. It dis-cusses generic scale invariance of equilibriumsystems and non-equilibrium dynamics of opensystems.Chapter 10 exposes “Directed paths in randommedia” and their importance for various appli-cations. It gives, among others, high-temperatureexpansions for random-bond Ising models, stud-ies one-dimensional chains using the transfermatrix, and shortly discusses a connection tospin-glass models. Finally, quantum interferenceof strongly localized electrons is exposed.Each chapter of the book contains many instruc-tive problems. Solutions to selected problems are

provided on almost one hundred last pages of thebook. Jirí Cerný (Zürich)

1166.83001Carmeli, MosheCosmological relativity. The special and gen-eral theories for the structure of the universe.Hackensack, NJ: World Scientific (ISBN 978-981-270-075-9/hbk). xi, 138 p. £ 22.00; $ 37.00(2006).In the present monograph the author mainly de-velops his ideas which he had presented in earlierbooks [see Cosmological special relativity: Thelarge scale structure of space, time and velocity.Singapore: World Scientific (1997; Zbl 946.83001),2nd ed. (2002; Zbl 1011.83001)]. Several exampleshave been worked out in much details with theaim that it be readable for a wider audience.Cosmological as well as black hole solutions forgeneral relativity belong to this set of topicsas well as the equation of motion in the EIH-approximation, where EIH abbreviates for Ein-stein, Infeld and Hoffmann.

Hans-Jürgen Schmidt (Potsdam)

1166.85001Giovannini, MassimoA primer on the physics of the cosmic mi-crowave background.Hackensack, NJ: World Scientific (ISBN 978-981-279-142-9/hbk). xiv, 474 p. £ 48.00; $ 89.00(2008).This book grew out of various series of lectures,so its level of presentation is not homogeneous:in some parts, very detailed special backgroundknowledge is necessary to understand the for-mulas, whereas in others, quite elementary ba-sic material is outlined in detail. Apart from thisreservation, one can say, that this book covers al-most everything, which one needs to know aboutthe cosmic microwave background radiation.A mathematician might wish to start readingthe appendices: Appendix A is on the conceptof distance in cosmology, it covers many differ-ent measures like luminosity distance, redshiftdistance, the many different possible coordinatedistances, and their interrelation. Appendix Bpresents the kinetic description of hot plasmas,starting with generalities on thermodynamic sys-tems, fermions and bosons, and their applicationto the big-bang nucleosynthesis.Appendix C “Scalar modes of the geometry"presents the fluctuations of the Einstein tensor


and their classification, and appendix D givesthe gauge-independent treatment of metric fluc-tuations.The main body of the book contains exactly whatone would expect, so we may give here the pub-lisher’s description: “In the last fifteen years, var-ious areas of high energy physics, astrophysicsand theoretical physics have converged on thestudy of cosmology so that any graduate studentin these disciplines today needs a reasonably self-contained introduction to the Cosmic MicrowaveBackground (CMB). This book presents the es-sential theoretical tools necessary to acquire amodern working knowledge of CMB physics.The style of the book, falling somewhere betweena monograph and a set of lecture notes, is peda-gogical and the author uses the typical approachof theoretical physics to explain the main prob-lems in detail, touching on the main assumptionsand derivations of a fascinating subject."

Hans-Jürgen Schmidt (Potsdam)

1166.86001Nolet, GuustA breviary of seismic tomography. Imaging theinterior of the earth and sun.Cambridge: Cambridge University Press (ISBN978-0-521-88244-6/hbk). xiv, 344 p. £ 32.99; $ 65.00(2008).This book is a good textbook covering all themajor aspects of seismic tomography. The bookpresent a comprehensive introduction to seismictomography including the basic theory of wavepropagation and practical recommendation forimplementing numerical model using publiclyavailable software. This book is appropriate foradvanced undergraduate and graduate courses.It is also an invaluable guide for seismology re-search practitioners in geophysics and astron-omy. Chengshu Wang (Denver)

1166.86002Klein, RupertMathematics in the climate of global change.(Mathematik im Klima des globalen Wandels.)(German)Jahresber. Dtsch. Math.-Ver. 109, No. 1, 3-29(2007).The investigators of the climate and its conse-quences meet the problem that their object ofstudy - the Earth system, or essential parts of thissystem - cannot be modeled and studied in detailin the laboratory. It is too large and too complex.

Thus, observations and direct measurements, aswell as modeling and computer simulations, areof great importance. The author of the article, firstdiscusses two essential sources of the complexityof the Earth’s system, the multitude of differentphysical processes and the interrelation of nu-merous - small and large - spatial and temporalscales. Modern mathematics, above all numericalmathematics and asymptotic analyses, may es-sentially contribute to the solution of both prob-lems. The author concentrates in the article on themany-scales problem only. Using the exampleof the harmonic oscillator with small mass anddamping, a two-scale problem with respect ofthe time is analysed. The sublinear growth con-dition is explained. It is shown that multi-scaleapproaches help to identify not only small-scaleand large-scale parts of solutions, but also therelated physical processes. Finally, a multi-scalemodel of the synoptic-planetary interactions inthe tropes is presented. This model possesses amulti-scale ansatz of the zonal spatial variable.


1167.86001Chen, ZhangxinReservoir simulation. Mathematical tech-niques in oil recovery.CBMS-NSF Regional Conference Series in Ap-plied Mathematics 77. Philadelphia, PA: So-ciety for Industrial and Applied Mathematics(SIAM) (ISBN 978-0-898716-40-5/pbk). xxviii,219 p. $ 86.00 (2007).This book’s methodological approach familiar-izes readers with the mathematical tools requiredin simulations of oil recovery processes. It be-gins with an overview of classical reservoir en-gineering and basic reservoir simulation meth-ods and then progresses through a discussion oftypes of flows – single-phase, two-phase, blackoil (three-phase), single phase with multicompo-nents, compositional, and thermal. The text orig-inates from the material presented by the authorat a CBMS-NSF Regional Conference during aten-lecture series on multiphase flows in porousmedia and their simulations.The author provides a thorough glossary ofpetrolium engineering terms and their units,along with basic flow and transport equationsand their unusual features, and correspondingrock and fluid properties. The practical aspectsof reservoir simulation, such as data gatheringand analysis, selection of a simulation model, his-


tory matching, and reservoir perfomance predic-tion, are summarized. The book is aimed at ad-vanced undergraduates and first-year graduatestudents in geology, petroleum engineering, andapplied mathematics; as a refence book for geol-ogists, petroleum engineers, and applied mathe-maticians; or as a handbook for practitioners inthe oil industry. Prerequisites are calculus, basicphysics, and some knowledge of partial differen-tial equations and matrix algebra.

Felix Kaplanski (Tallinn)

1167.86002Freeden, Willi; Schreiner, MichaelSpherical functions of mathematical geo-sciences. A scalar, vectorial, and tensorial setup.Advances in Geophysical and EnvironmentalMechanics and Mathematics. Berlin: Springer(ISBN 978-3-540-85111-0/hbk; 978-3-540-85112-7/ebook). xv, 602 p. EUR 181.85 (2009).In geosciences, following the developing of tech-nology and computer, mathematics concernedwith geoscientific problems is becoming moreand more important. From modern satellite-positioning, the Earth’s surface deviates from asphere, therefore, spherical functions and con-cepts play an essential part in all geosciences.This book concentrates the introduction of math-ematical representation of spherical vector andtensor fields. Vector/tensor spherical harmonicsare used throughout mathematics. This book is avaluable reference for scientists and practitionerswhen facing spherical problems.

Chengshu Wang (Denver)

1166.78003Marengo, Edwin A.; Khodja, Mohamed R.;Boucherif, AbdelkaderInverse source problem in nonhomogeneousbackground media. Part II: Vector formulationand antenna substrate performance characteri-zation.SIAM J. Appl. Math. 69, No. 1, 81-110 (2008).This paper solves analytically and illustrates nu-merically the full-vector, electromagnetic inversesource problem of synthesizing an unknownsource embedded in a given substrate mediumof volume V and radiating a prescribed exte-rior field. The derived formulation and resultsgeneralize previous work on the scalar versionof the problem, especially the recent Part I ofthis paper [A. J. Devaney, E. A. Marengo and M.Li, SIAM J. Appl. Math. 67, No. 5, 1353–1378

(2007; Zbl 1130.45011)]. Emphasis is put on sub-strates having constant constitutive propertieswithin the source volume V, which, for formaltractability, is taken to be a spherical shape. Theadopted approach is one of constrained opti-mization which also relies on spherical wave-function theory. We find that the observed peaksin the spectrum of the singular values are pri-marily due to the phenomenon of Mie resonance.Therefore, for a given antenna radiating at a pre-scribed frequency, the set of solutions to the Mieresonance conditions corresponds to a set of con-stitutive parameters that maximize the radiatedelectromagnetic fields. The derived theory andassociated implications for antenna substratesare illustrated numerically.

Xavier Antoine (Vandœuvre-lès-Nancy)

1166.81015Mora, Carlos M.Heisenberg evolution of quantum observablesrepresented by unbounded operators.J. Funct. Anal. 255, No. 12, 3249-3273 (2008).Consider a small quantum system with statespace h weakly coupled to a heat bath. In aMarkovian setting, an observable A of the smallsystem evolves in accordance with the adjointquantum master equation

ddtTt(A) =

Tt(A)G + G∗Tt(A) +

∞∑k=1

L∗kTt(A)Lk,

T0(A) = A,

where G,L1,L2, . . . are linear operators in h satis-fying some certain conditions.For bounded observables A (i.e. A is a boundedoperator in h), the existence and uniqueness ofsolutions for the equations are established in A.M. Chebotarev [Theor. Math. Phys. 80, No. 2, 804–818 (1989); translation from Teor. Mat. Fiz. 80,No. 2, 192–211 (1989; Zbl 694.47022)] by semi-group methods. This paper deals with the adjointquantum master equations with initial condi-tions given by unbounded operators, such as theposition and momentum operators of quantumoscillators and the occupation number operatorin many-body particle systems. It shows the exis-tence and uniqueness of solutions of the operatorequations governing the motion of unboundedobservables by developing the relation between


operator evolution equations arising in quantummechanics and stochastic evolution equations ofSchrödinger type. It also explores quantum dy-namical semigroup properties of the Heisenbergevolutions of unbounded observables in a gen-eral setting. Miyeon Kwon (Platteville, WI)

1166.81028Cipriani, FabioDirichlet forms on noncommutative spaces.Franz, Uwe (ed.) et al., Quantum potential the-ory. Lectures given at the school ‘Quantum po-tential theory: Structure and applications tophysics’, Greifswald, Germany, February 26 toMarch 9, 2007. Berlin: Springer (ISBN 978-3-540-69364-2/pbk). Lecture Notes in Mathemat-ics 1954, 161-276 (2008).The lecture notes under review present a detailedintroduction to the theory of noncommutativesymmetric Dirichlet forms and describe rich con-nections of this theory to quantum dynamics and(noncommutative) geometry. Both in the classi-cal and quantum contexts Dirichlet forms area crucial tool for studying properties of evolu-tions modelled by positivity-preserving contrac-tive semigroups. The author presents both thegeneral theory and many specific examples ofquantum Markovian semigroups; although most(not all!) of the results presented in the notes arenot new, so far they have been available only inscattered form in research papers.The context of the notes is as follows: in Sec-tion 1 the author discusses the development ofthe classical theory of Dirichlet forms on Hilbertspaces of the form L2(X,m) and the necessity toinvestigate the analogue of this theory for Dirich-let forms on Hilbert spaces of the type L2(A, φ),where A is a von Neumann algebra (or a C∗-algebra) and φ is a state on A, playing the role ofa reference measure. In Section 2 the crucial inter-dependence between maps on A (viewed as thenoncommutative L∞-space), on L2(A, φ), and onthe predual of A (noncommutative L1-space) isdiscussed and the properties of symmetric em-beddings between these spaces established. Asthe reference state is in general not assumed to betracial, the crucial role is played by the so-calledKMS-symmetry. It is shown that there is a 1-1correspondence between the class of (symmetric)Dirichlet forms on L2(A, φ) and positive Marko-vian semigroups of KMS-symmetric maps on A.Many concrete examples of such semigroups areconstructed, in particular on CAR and CCR alge-

bras, and their dynamical properties such as er-godicity investigated. Section 3 continues the dis-cussion of examples focusing on quantum spinsystems arising in quantum statistical mechan-ics. In Section 4 the author associates to any com-pletely Dirichlet form on a C∗-algebra A with afaithful semifinite lower semicontinuous trace aHilbert A-bimodule H and a derivation ∂ map-ping a dense subalgebra of A into H. This corre-sponds in the classical context to the constructionof a gradient (derivation) from the energy func-tional (Dirichlet form) and is further used to ob-tain natural decompositions of a Dirichlet forminto nondegenerate and ‘killing’ parts. The nexttwo sections contain applications of the theorydeveloped earlier to classical questions such asthe positivity of the curvature on a given Rieman-nian manifold and quasi-equivalence of classicalDirichlet forms with identical domains.The author is one of the leading experts involvedin the development of the field and the text is en-riched by many valuable side remarks and bibli-ographic pointers. Unfortunately the notes con-tain several typographic errors, but in most casesthey do not affect the mathematical content.

Adam Skalski (Lancaster)

1166.82016Burkhardt, Theodore W.First-passage and extreme-value statistics of aparticle subject to a constant force plus a ran-dom force.J. Stat. Phys. 133, No. 2, 217-230 (2008).The authors consider a particle which movesalong the x-axis and is subject to a constant force,such as gravity, plus a random force in the formof Gaussian white noise. They analyse the statis-tics of first arrival at point x1 of a particle whichstarts at xo with velocity vo. The probability dis-tribution of the changes of position and veloc-ity of the particle satisfies the time-dependentFokker-Planck equation. The probability that theparticle has not yet arrived at x1 after a time t,the mean time of first arrival, and the velocitydistribution at first arrival are considered. Alsothe statistics of the first return of the particle to itsstarting point is studied. Finally, it is pointed outthat the extreme-value statistics of the particleand the first passage statistics are closely related.The distribution of the maximum displacementm =maxt[x(t)] is found.



1166.82021Val’kov, V.V.; Dzebisashvili, D.M.Effective interactions in the periodic Andersonmodel in the regime of mixed valency withstrong correlations. (English. Russian original)Theor. Math. Phys. 157, No. 2, 1565-1576 (2008);translation from Teor. Mat. Fiz. 157, No. 2, 235–249 (2008).The paper is devoted to construction of effectiveHamiltonian for the periodic Anderson model(PAM) in the intermediate valence regime, whichdescribes antiferromegnetic interactions that in-fluence the formation of the superconductingphase with the order parameter having the d-type symmetry. First, for the PAM Hamiltonianis used the atomic representation and splittingthe hybridization interaction operator into twoterms. The first term reflects processes resultingin mixing collectivized and localized electronswithout changing the number of states called thepairs in the localized subsystem (the low energysector of the Hilbert space). The second termis responsible for hybridization processes withsimultaneous pair creation or annihilation, cor-responding to processes inducing transitions tothe high-energy sector of the Hilbert space. Dueto the intra-atom Coulomb repulsion strength islarge these processes are taken into account per-turbatively. Thus, the application the sequenceof unitary transformations to obtain the effectiveHamiltonian allows one to obtain an expansionin powers of the small parameter. The small pa-rameter is determined by only those hybridiza-tion processes that involve high-energy states ofthe localized subsystem. Correspondingly onlythese processes are excluded in the first and sec-ond order by using the unitary transformations.The hybridization processes, then remains in thelower Hubbard subband. As a result, the authorsobtain the effective Hamiltonian describing asexchange interactions as interactions explicitlyindicating a possibility of the Cooper instabil-ity and the creation of a superconducting phasewith developed antiferromagnetic fluctuations.The obtained effective Hamiltonian is used fornumerical study of the magnitudes of exchangeinteractions and their dependences on the dis-tance necessary for understanding the nature ofthe magnetically ordered phase. The concrete nu-merical results present the dependences of theeffective f-electron jump integral and the super-exchange interaction parameter on the coordi-nate sphere number at various values of the hy-

bridization constant. Moreover, there are the re-sults of calculation of the indirect interaction ex-change integral at various distances between rareearth ions and also its dependence on the chemi-cal potential at different values of the hybridiza-tion constant. I. A. Parinov (Rostov-na-Donu)

1167.82005Bostan, A.; Boukraa, S.; Hassani, S.; Maillard,J-M; Weil, J-A; Zenine, N.Globally nilpotent differential operators andthe square Ising model.J. Phys. A, Math. Theor. 42, No. 12, Article ID125206, 50 p. (2009).Various multiple integrals with one parameterare recalled. Examples are the n-particle con-tribution of the magnetic susceptibility of theisotropic square Ising model, lattice form factorsand two-point correlation functions, or examplesfrom enumerative combinatorics. The univari-ate analytic functions defined by these integralsare holonomic and even G-functions: they satisfyFuchsian linear differential equations with poly-nomial coefficients and have some arithmeticgrowth property on the coefficients of a solutionseries. These differential operators are selectedFuchsian linear differential operators, and theirremarkable properties have a deep geometricalorigin: they are globally nilpotent, or, sometimes,even have zero p-curvature. In some examplesfrom enumerative combinatorics, quantities arenot naturally expressed as n-fold integrals of analgebraic integrand. The discovery of the globalnilpotence of the corresponding minimal orderFuchsian linear differential operator has to beseen as a strong indication that they can be ex-pressed as n-fold integrals of an algebraic inte-grand. Other quantities, like the n-particle con-tributions to the magnetic susceptibility of thesquare Ising model are defined as such n-fold in-tegrals. The integrand of the algebraic functioncan be chosen continuous and single valued onthe torus of integration: they are a family of pe-riods. In this last case, the purpose of the paperis not to give another proof of global nilpotence,but to understand, how these differential factors’succeed’ to be globally nilpotent. Throughout allthe examples displayed in the paper the (mini-mal order) linear differential operators of quitelarge order (for instance order 23, 33, 50) actuallyfactorize into products, or direct sums and prod-ucts, of linear differential operators of smallerorders (up to four). Focusing on the factorized

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