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 , . 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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. 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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. 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