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ISSN 0002-9920 of the American Mathematical Society May 2003 Volume 50, Number 5 The Mathematics of Learning: Dealing with Data page 537 PRIMES Is in P: A Breakthrough for "Everyman" page 54 5 CQD Input: integer n > 1 ~::' 1. If n = ab for a E N and b > 1 ro q 1otli · ai output COMPOSITE. calcu 2. Find the smallest r tiarri e. such that Or ( n) > 4 log2 n. I ad n: % Or ( n) is the order of n modulo r 1ute1n · 3. If 1 < (a, n) < n for some a ::::; r illae output COMPOSITE. 4. If n ::::; r, output PRIME. tntur, 5. For a = 1 to l2..j¢(r:) lognJ do If (X+ a)n f= xn +a (mod xr - 1, n) ' l iD id ca" output COMPOSITE. ·,' )' occ % ¢(r) is Euler's totient ta Iatori 6. output PRI ME. se e n lit~Ollt res ri:uigna /emolumenta c~l>ie.,a ' ·'. qua~ t , dispendiui1i mediocre. Iargi~r .... cbmpe~ ~ ereaque sciehtia.e 'dignitas req.Uirere ''vid :rinia subsidici ad solutionern probleriichis aitis ac celehris sedulci excoiaritur. .J?ro ·dtiori.es nori dubihimus', ~ quirt , . 'd.utie ' metl ~ntes' quariim' .e fficaciari1. ac_brciuitatern . Is n prime? (see p age 553) Distance Asynchronous Online On Ct.1r11pus Education is no longer just about classrooms and labs. With the growing diversity and complexity of educational programs. you need a software system that lets you efficiently deliver effective learning tools to literally, the world. Maple· now offers yo u a choice to add ress the reality of today's mathematics education . Maple 8 - the standard Perfect for students in mathematics, sciences, and engineering. Maple 8 offers all the power, flexibility, and resources your technical students need to manage even the most complex mathematical concepts. TM ~••!"! _:"!!!"!_:!".:_~------ -. Maple T.A. - automated assessment Online math skills assessment without resorting to multiple choice •ft 4--~r ..._.,._.,.,..._,.._ ......... -.... .. _ ___ _ or true or false. Maple T.A. supports natural entry and intelligent mapkiD -- Ill, ·-------r-.--.._ ,.. ____u __ ·----r-- .,.,. .__,..... __ ~ testing of answers. A complete question-set management ---__ --r- infrastructure lets you manage even the largest class sizes. - ... - r MapleNET - online education A complete standards-based solution for authoring , • OS!!!.,..""2r _.,.._ delivering , and managing interactive learning modules • A~~ ·~ through browsers. Derived from the legendary Maple •l'loi;IIJ'I!Irr~ engine, MapleNet is the only comprehensive solution for Web based education in mathematics. Give your institution and students the cornpetitive edge. For a FREE 30-day Maple 8 Trial CD , or to register for a FREE MapleNet or Maple T.A. Online Seminar call 800.267.6583 or e-mail [email protected]. ~ Maple soft command the brIlliance www.maplesoft.com info@maplesoft .com , www.mapleapps.com , US & Canada 800.267.6583 ~f. !fJ t > ·t l rJ I' I 'I (,' v'il'Prl(,n ~.'d!J p It I 2(J•J,j M lpit; s I '"qh'f'P'fi tr ldt''l'dfk o! \'\i11t>• Ov M1ple llr Vl<lpiB:-..Oit M<ipleNPt diHJ MaJ•Ir l A Clrt:' lr1rlem;~rks ot w tlf rloo ~\lp p 111\ All Olth fr,vif'llla k::, I'< rr(Jl'\ tl, ('I 'It I fl',,1•'l J, <' d . w ' I OUTSTANDING NEW MATHEMATICS BOOKS Second Order Partial Complex Polynomials The Changing Shape of Differential Equations in Terry Shei/-Sma/1 Geometry Hilbert Spaces Srudies the geometric theory of polynomials Celebrating a Century of Geometry G. Da Prato and j. Zabczyk and rational functions in the plane. Chapters and Geometry Teaching build the foundations of complex variable Second order linear parabolic and elliptic Edited by Chris Pritchard theory, melding together ideas from algebra, equations arise frequently in mathematical Contains popular articles including sections on topology and analysis. Presents several physics, biology and finance. The authors Pythagoras, the golden ratio and recreational unsolved problems and the full solurions present a state of the art treatment of the geometry. Thirty 'Desert Island Theorems' to some well-known conjectures. subject from a new perspective, and discuss from distinguished mathematicians and 0-521 -40068-6 Hb $95.00 how the results can be applied to control educationalists give light to some surprising Cambridge Studies in Advanced Mathematics 75 theory. and beautiful results. Contributors include a 0-521-77729-1 Pb $45.00 Nobel Laureate and a Pulitzer Prizewinner. London Mathematical Society Lecture Note Series 293 Abstract Regular Polytopes 0-521-82451-6 Hb $90.00 0-521-53162-4 Pb $40.00 Peter McMullen and Egon Schulte Co-published with The Mathematical Association of A Panorama of Number Theory This book is the first comprehensive, America or The View from Baker•s up-to-date account of this subject and its Garden ramifications. Over the past rwenry years, a rich new theory featuring an attractive Graph Directed Edited by Gisbert Wiistholz interplay of mathematical areas, including Markov Systems The papers presented here cover a broad geometry, combinatorics, group theory and Geometry and Dynamics of Limit Sets spectrum of number theory including topology has arisen. Dan Mauldin and Mariusz Urbanski geometric, algebrao-geometric and analytic 0-521-81496-0 Hb $110.00 This far-reaching generalization of the theory aspects, appealing to number theorists, Encyclopedia of Mathematics and its Applications 92 algebraic geometers, and geometers with of conformal iterated systems can be applied a number theoretic background. in many situations, including the theory of 0-521-80799-9 Hb $80.00 Hodge Theory and Complex dynamical systems. Includes much of the Algebraic Geometry necessary background material and an extensive list of references suggesting further reading. Volume I A Second Course in 0-521-82538-5 Hb $55.00* Mathematical Analysis Claire Voisin Cambridge Tracts in Mathematics 1.48 Second Edition Translated by Leila Schneps }.C. Burkill and H. Burki/1 A modern introduction to Kaehlerian Quasi-Frobenius Rings geometry and Hodge structure, this book This straightforward course, based on the starts with basic material on complex WK. Nicholson and M.F.Yousif idea of a limit, is intended for students of variables, complex manifolds, holomorphic This book provides an elementary, complete mathematics and physics who have acquired vector bundles, sheaves and cohomology account of quasi-Frobenius rings at a level a working knowledge of calculus and are theory, and culminates with the Hodge allowing researchers and graduate students ready for a more systematic treatment. decomposition theorem. Includes exercises. to gain entry to the field. 0-521-52343-5 Pb $45.00 0-521-80260-1 Hb $80.00 0-521-81593-2 Hb $70.00* Cambridge Mathematical Library Cambridge Studies in Advanced Mathematics 76 Cambridge TractS in Mathematics !54 *Prices subject to change. A Panorama ofNumber Theory or The Vieo.v from taker's Garden CAMBRIDGE 800-872-7423 Available in bookstores or from UNIVERSITY PRESS us.cambridge.org/mathematics NEW f6 NOTEWORTHY from Birkhiiuser The Topos of Music Mathematical Finance Partially Observable Geometric Logic of Concepts, and Probability Linear Systems Under Theory, and Performance Discrete Introduction Dependent Noises G. MAZZOLA, University of Zurich, Switzerland P. KOCH MEDINA, Swiss Re; and S. MERINO, UBS, both, In Collaboration with S. GOLLER and S. MULLER A. E. BASHIROV, Eastern Mediterranean University, Zurich, Switzerland Famagusta, Cypras The Tapas of Music is an updated, in-depth English This work gives a self-contained presentation of the This book provides a mathematical view of specific extension of the German Geometrie der TOne. It reflects theory underlying the valuation of derivative financial the dramatic progress of mathematical music theory engineering problems with known and new methods of instruments, a standard part of the toolbox of finance control and estimation in noisy media. The main feature and its use of information technology. The conceptual professionals. The material will be accessible to basis has been generalized to tapas-theoretic founda­ is the examination of stochastic optimal control and students and practitioners who have a worki ng know­ estimation problems with the noise processes acting tions, including geometric musical logic. The theoretical ledge of linear algebra and calculus. The book also models and results include topologies for rhythm, dependently on the state (or signal) and observation off ers an introduction to modern probability theory, systems. melody, and harmony, as well as a classification theory primarily within the context of finite sample spaces. of musical objects that comprises the framework of the 2003/3 64 PP./HARDCOVER/$99 .00 2002/ 336 PP./SOFTCOVER/$49. 95 ISBN 3-7643-6999-X tapas-theoretic concept. The book's formal language ISBN 3-7643-6921-3 and models are currently being used by leading SYSTEMS AND CONTROL: FOUNDATIONS AND APPLICATIONS researchers in Europe and Northern America and have become a foundation of music software design. Adaptive Finite Element Operator Theory: 2002/ 1,3 68 PP./HARDCOVER WITH CD-ROM/$ 139.00 Methods for Differential Advances and Applications ISBN 3-7643-5731-2 Equations Series Ed iter: l. GOHBERG, Tel Aviv University, W. BAN GERTH, The University of Texas at Austin, TX; and RamatAviv, Isreal Cryptographic Applications R. RANNACHER, University of Heidelberg, Germany This series is devoted to the publication of current of Analytical Number This work discusses the concept of "self-adaptivity" in research in operator theory, with particular emphasis Theory the numerical solution of differential equations, with on applications to classical analysis and the theory of emphasis on Galerkin finite element methods. The key integral equations, as well as to numerical analysis, Complexity Lower Bounds and mathematical physics and mathematical methods in Pseudorandomness issues are a posteriori error estimation and automatic mesh adaptation. Besides the traditional approach of electrical engineering. I. SHPARLINSKI, Macquarie University, Sydney, Australia energy-norm error control, a new duality-based tech­ This book introduces new applications for analytic nique, the Dual Weighted Residual method for goal­ Hilbert Space, Boundary Value number theory in cryptography and related areas, such oriented error estimation, is discussed in detail.
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