ISSN 0002-9920 of the American Mathematical Society May 2001 Volume 48, Number 5 A New Solution to the Three-Body Problem page 471 Thomas H. Wolff (1954-2000) page 482 Choreography for Six Suns (see page 481) It All Adds Up At Maple® There's more to Waterloo Maple than the world's leading mathematics Great Multi-User Pricing From pre-calculus to research ... on all primary software. The renowned Maple 6 computational system is backed by an systems ... network or standalone ... we have the perfect solution for any institution. extensive range of flexible support options and resources designed to work together to offer the best total solution for educators, students, Feature Rich Maple LabPacks The ideal solution for smaller community colleges and technical professionals. and high schools and complete with a suite of educational resources. Maple Student Edition With special student pricing for the next generation of mathematicians. 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Maple is a registered trademark of Waterloo Maple Inc. AMERICAN MATHEMATICAL SOCIETY New Titles from the AMS Recommended Text Supplementary Read1ng Problems in Mathematical Analysis II Plateau's Problem Continuity and Differentiation An Invitation to Varifold Geometry W. J. Kaczor and M. T. Nowak, Maria Curie-Sklodowska Frederick J. Almgren, Jr. University, Lublin, Poland There have been many wonderful developments in the theory of This is the sequel to Problems in Mathematical Analysis 1: Real minimal surfaces and geometric measure theory in the past 25 to Numbers, Sequences and Series (Volume 4 in the AMS series, 30 years. Many of the researchers who have produced these the Student Mathematical Library). As in the first volume, this excellent results were inspired by this little book-or by Fred book is divided into two parts. The first is a collection of exercises Almgren himself. and problems, and the second contains their solutions. The book functions of one real variable. Topics The book is indeed a delightful invitation to the world of variational mainly deals with real functions, intermediate value geometry. A central topic is Plateau's Problem, which is concerned include: properties of continuous mean value theorems, Taylor's with surfaces that model the behavior of soap films. When trying property, uniform continuity, series of functions. to resolve the problem, however, one soon finds that smooth formula, convex functions, sequences and surfaces are insufficient: Varifolds are needed. With varifolds, one The book is mainly geared toward students studying the basic can obtain geometrically meaningful solutions without having to principles of analysis. However, given its selection of problems, know in advance all their possible singularities. This new tool organization, and level, it would be an ideal choice for tutorial or makes possible much exciting new analysis and many new problem-solving seminars, particularly those geared toward the results. Putnam exam. It is also suitable for self-study. The presentation of to encourage Plateau's problem and varifolds live in the world of geometric material is designed to help student comprehension, research. measure theory, where differential geometry and measure theory them to ask their own questions, and to start combine to solve problems which have variational aspects. The Student Mathematical Library, Volume 12; 2001 ; approximately 416 pages; was to encourage young mathe­ Soltcover; ISBN 0-8218-2051-6; List $49; All AMS members $39; Order code author's hope in writing this book STMU12NT105 maticians to study this fascinating subject further. Judging from the success of his students, it achieves this exceedingly well. Also available ... Student Mathematical Library; 2001 ; approximately 88 pages; Soltcover; ISBN 0-8218-2747-2; List $19; All AMS members $15; Order code Problems in Mathematical Analysis I STML-ALMGRENNT1 05 W. J. Kaczor and M. T. Nowak, Maria Curie-Sklodowska University, Supplementary Readmg Lublin, Poland Student Mathematical Library; Volume 4, 2001 ; 380 pages; Softcover; ISBN 0· Essays in the History of Lie Groups 8218-2050·8; List $39; All AMS members $31; Order code STMU4NT1 05 Set Student Mathematical Library; 2001 ; 796 pages; Softcover; ISBN 0·8218· and Algebraic Groups 2849·5; List $78; All AMS members $62; Order code STMLSETNT1 05 Armand Borel, Institute for Advanced Study, Princeton, NJ Algebraic groups and Lie groups are important in most major Recommended Text areas of mathematics, occurring in diverse roles such as the symmetries of differential equations and as central figures in the Introduction to Topology Langlands program for number theory. In this book, Professor V. A. Vassiliev, Independent University of Moscow, Russia Borel looks at the development of the theory of Lie groups and From a review for the Russian edition ... algebraic groups, highlighting the evolution from the almost purely The book is based on a course given by the author in 1996 to first theory at the start to the global theory that we know today. local and second year students at Independent Moscow University 00. As the starting point of this passage from local to global, the the emphasis is on illustrating what is happening in topology, and author takes Lie's theory of local analytic transformation groups the proofs (or their ideas) covered are those which either have algebras. He then follows the globalization of the process and Lie important generalizations or are useful in explaining important in its two most important frameworks: (transcendental) differential concepts 00. This is an excellent book and one can gain a great geometry and algebraic geometry. Chapters II to IV are devoted to deal by reading it. The material, normally requiring several the former, Chapters V to VIII, to the latter. volumes, is covered in 123 pages, allowing the reader to appre­ The author brings a unique perspective to this study. As an impor­ ciate the interaction between basic concepts of algebraic and tant developer of some of the modern elements of both the differential topology without being buried in minutiae. differential geometric and the algebraic geometric sides of the -Mathematical Reviews has a particularly deep appreciation of the underlying theory, he This English translation of a Russian book presents the basic and his historical research in mathematics. His lifelong involvement notions of differential and algebraic topology, which are indispens­ the subject give him a special appreciation of the story of its devel­ able for specialists and useful for research mathematicians and opment. theoretical physicists. In particular, ideas and results are introduced Copublished with the London Mathematical Society. Members of the LMS related to manifolds, cell spaces, coverings and fibrations, homo­ may order directly from the AMS at the AMS member price. The LMS is registered with the Charity Commissioners. topy groups, homology and cohomology, intersection index, etc. , Volume 14; 2001; 149 pages; Soltcover; History of Mathematics; 2001; approximately 184 pages; Hardcover; Student Mathematical Library ; All AMS members $20; Order code ISBN 0-8218-0288-7; List $39; All AMS members $31; Order code ISBN 0-8218-2162-8; List $25 HMATH-BORELNT1 05 STMU14NT105 To order, call: 1-800-321-4AMS (4267), in the U.S. and Canada, or 1-401-455-4000; fax: 1-401-455-4046; email : [email protected]. Visit the AMS Bookstore and order online at www.ams.org/bookstore. Or write to: American Mathematical Society, P. 0. Box 6248, Providence, Rl 02940-6248. Prices subject to change without notice. ci&AMSL'NDEO \,; AMERICAN MATHEMATICAL SOCIETY Forthcoming! Automata Theory and its Applications B. Khoussalnov, University ofAuckland, New Zealand & Anll Nerode, Cornell University, Ithaca, NY Uniform treatment of the theory of finite state machines on finite and infinite strings and trees. Many books deal with automata on finite strings, but there are very few . expositions that prove the fundamental results of automa­ ta on infinite strings and trees. Beginning with coverage of .. Forthco,uilg! · _ , . all standard fundamental results regarding finite automa­ ta, this book deals in great detail with Biichi and Rabin Sp•llatPaitern~ ", . automata and their applications to various logical theories Higher Order.. ModelsJn Pbysics·and Mechanics ,. such as SIS and S?S, and describes game-theoretic models LA. P~letler, l.eiden Universtiy, me'Netherlimds & w.c;: TrOy, UniverSity of Pittsburgh, PA ' I . of concurrent operating and communication systems. Self-contained with numerous examples, illustrations, offers stud). model are. t? the descriptiort a.nd . c ;: Sp(ai~ Pat;~~ a. ~f nofili~ea~ . h~her o~der ;quati~ns th~t cent~al exercises. Suitable for a two-semester undergraduate · 11naiysis pf in the sciences. In a ;esults spa~o-temporal pa~ern formati~n n~tur?l _u~Ique co~bi~ation . o~ ob~ned course for computer science or mathematics majors, or by rigorous mathematical ~nalysis andcompu~tion~~ studies; tlie ~ext e~blts ~e prmCJpal famihes .of soluti?ns, ~ for a one-semester graduate course/seminar. No such as kinks,·pulses and periodic solutions, and their dependence on cntical eigenvalue -IJ?rameters,.and ~mnts ,_ ' .