The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. The adeles of a Number Field are the additive Subgroups of all ele- ments in Yl kvi where v is the PLACE, whose ABSOLUTE Value is ( 3 ) it follows that (AB) f = BW. (14) Jo A generalization has been constructed by Hardy. (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.

Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. Adeles arise in both Number Fields and Function Fields. This means that Watson's expression becomes /»oo (3a)- 1/3 7r Ai(±(3a)- 1/3 z)= / cos(at 3 ±xt)dt.

It frequently transpires that some piece of recently acquired knowledge turns out to be just what I need to solve some apparently unrelated problem.

I have therefore developed the habit of picking up and storing away odd bits of information for future use.

Standard mathematical references, combined with a few popular ones, are also given at the end of most entries to facilitate additional reading and exploration. The word "coptic" derives from the Greek for "to cut," and acoptic polyhedra are defined as POLYHEDRA for which the FACES do not intersect (cut) themselves, mak- ing them 2-Manifolds. Mathematical Methods in Engineering: An Introduction to the Mathematical Treat- ment of Engineering Problems. The first of several ADDENDS, or "the one to which the oth- ers are added," is sometimes called the AUGEND. While the usual form of adding two n-digit INTEGERS (which consists of summing over the columns right to left and "Carrying" a 1 to the next column if the sum exceeds 9) requires n operations (plus carries), two n- digit INTEGERS can be added in about 21gn steps by n processors using carry-lookahead addition (Mc Geoch 1993). see also Digitadition, Digital Root, Multiplica- tive Persistence, Narcissistic Number, Recur- ring Digital Invariant References Guy, R. "The Persistence of a Number." §F25 in Unsolved Problems in Number Theory, 2nd ed.

In the interests of offering abundant examples, this work also contains a large number of explicit formulas and derivations, providing a ready place to locate a particular formula, as well as including the framework for understanding where it comes from. see also Honeycomb, Nolid, Polyhedron, Sponge Action Let M(X) denote the GROUP of all invertible MAPS X - 0(g)x, 2. 0(e) a; = x, where e is the group identity in G, 4. see also CASCADE, FLOW, SEMIFLOW Acute Angle An Angle of less than 7r/2 Radians (90°) is called an acute angle. Here, lgx is the Lg function, the LOGARITHM to the base 2. These formulas are comprehensible to readers with only minimal mathematical background, and are interesting to both those with and without formal mathematics training. n l X n (i) (2) be the step interval, and consider the Maclaurin Se- ries of y about x n , y n i = y n ( -T-) ( x ~ x n) (x - x n ) 2 . However, they have not previously been collected in a single convenient location. V dx J n 1 \dxj n \ dx 2 J (3) (4) Here, the Derivatives of y are given by the Backward Differences \dx/ n Xn i ~ X 3/n i - y n h (5) (6) (7) etc. CRC Concise Encyclopedia MAm EMAf JCS CRC Concise Encyclopedia MAffl EMAff G Eric W. This problem is often further compounded by the difficulty in locating concrete and easily understood examples. ../ (y 2 Cyi ~yi£y 2 )dx = — {yi 2/2 - 2/13/2 ) Po (3) see a/50 Self-Adjoint Operator, Sturm-Liouville Theory Adjugate Matrix see Adjoint Matrix Adjunction If a is an element of a Field F over the PRIME Field P, then the set of all RATIONAL FUNCTIONS of a with Coefficients in P is a Field derived from P by ad- junction of a. Weisstein CRC Press Boca Raton London New York Washington, D. Library of Congress Cataloging-in-Publication Data Weisstein, Eric W. To give perspective to a subject, I find it helpful to learn why it is useful, how it is connected to other areas of mathematics and science, and how it is actually implemented. Adleman-Pomerance-Rumely Primality Test A modified Miller's Primality Test which gives a guarantee of Primality or COMPOSITENESS. "On Distinguishing Prime Numbers from Composite Number." Ann. For this reason, I hope that, in addition to serving as a reference source, this work has some of the same flavor and appeal of Martin Gardner's delightful Scientific American columns. Note that by (1), q n is just the value of f{x ni y n ). Everything in this work has been compiled by me alone. For first-order interpolation, the method proceeds by iterating the expression 2/n i = yn q n h (8) where q n = /(x n ,2/n). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. First, the entire text and all the equations and figures are available in searchable electronic form on CD-ROM. see also Crunode, Spinode, Tacnode Acoptic Polyhedron A term invented by B. see also Addition, Augend, Plus, Radicand Addition i i - 15 8- J- 249 * 407- Y carries (-addend 1 Y addend 2 hsum The combining of two or more quantities using the PLUS operator. Second, the entries are extensively cross-linked and cross-referenced, not only to related entries but also to many external sites on the Internet. It also provides a highly efficient way to "navigate" from one related concept to another, a feature that is especially powerful in the electronic version. Griinbaum in an attempt to pro- mote concrete and precise POLYHEDRON terminology. The individual numbers being combined are called ADDENDS, and the total is called the Sum. Addition Chain An addition chain for a number n is a SEQUENCE 1 = ao 2 15 contain a in base 3, which would imply that the maximum persistence in base 3 is 3 (Guy, 1994). Direct all inquiries to CRC Press LLC, 2000 Corporate Blvd., N. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are only used for identification and explanation, without intent to infringe. It is written in an informal style intended to make it accessible to a broad spectrum of readers with a wide range of mathematical backgrounds and interests. du du t v Cu(x) - p — Pl — P2 u, (1) where pi = Pi(x) and u = u(x), the adjoint operator & is defined by d " ^ (Po U) " di^ Pl U) Pa U d 2 u f t ,du ( „ , , -P°ZT^ ( 2 Po -pi)^~ (po -pi P2)U. "Faster Primality Test- ing." In Advances in Cryptology, Proc. Specific permission must be obtained in writing from CRC Press LLC for such copying. Government works International Standard Book Number 0-8493-9640-9 Library of Congress Card Number 98-22385 Printed in the United States of America 1234567890 Printed on acid-free paper Introduction The CRC Concise Encyclopedia of Mathematics is a compendium of mathematical definitions, formulas, figures, tabulations, and references. (4) Adjoint Operator Given a Second-Order Ordinary Differential Equation - , . The Asymptotic Series of Ai(z) has a different form in different QUADRANTS of the COMPLEX PLANE, a fact known as the STOKES PHENOMENON, Functions related to the Airy functions have been defined as Gi(z) Hi W * Jo t zt) dt (15) exp(-f* 3 2t)x ly x 2 ) = f(x\x 0i x 1 ,x 2 , X3) = 1 Xl - Xo 1 X2 — Xo 1 X 2 - x± 1 /o Xo — X A Xl — X /o Xo — X A X 2 — X X 3 — X2 /(x|x ,xi) Xi - X /(x|x 0) x 2 ) x 2 - x /(x|x , Xi, X 2 ) X 2 - X /(x|x , Xi, X 3 ) X 3 - X see a/so LAGRANGE INTERPOLATING POLYNOMIAL References Abramowitz, M.

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