DICTIONARY OF ALGEBRA, ARITHMETIC, AND TRIGONOMETRY c 2001 by CRC Press LLC Comprehensive Dictionary of Mathematics Douglas N. Clark Editor-in-Chief Stan Gibilisco Editorial Advisor PUBLISHED VOLUMES Analysis, Calculus, and Differential Equations Douglas N. Clark Algebra, Arithmetic and Trigonometry Steven G. Krantz FORTHCOMING VOLUMES Classical & Theoretical Mathematics Catherine Cavagnaro and Will Haight Applied Mathematics for Engineers and Scientists Emma Previato The Comprehensive Dictionary of Mathematics Douglas N. Clark c 2001 by CRC Press LLC a Volume in the Comprehensive Dictionary of Mathematics DICTIONARY OF ALGEBRA, ARITHMETIC, AND TRIGONOMETRY Edited by Steven G. Krantz CRC Press Boca Raton London New York Washington, D.C. Library of Congress Cataloging-in-Publication Data Catalog record is available from the Library of Congress. This book contains information obtained from authentic and highly regarded sources. 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Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. © 2001 by CRC Press LLC No claim to original U.S. Government works International Standard Book Number 1-58488-052-X Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper PREFACE The second volume of the CRC Press Comprehensive Dictionary of Mathematics covers algebra, arithmetic and trigonometry broadly, with an overlap into differential geometry, algebraic geometry, topology and other related fields. The authorship is by well over 30 mathematicians, active in teaching and research, including the editor. Because it is a dictionary and not an encyclopedia, definitions are only occasionally accompanied by a discussion or example. In a dictionary of mathematics, the primary goal is to define each term rigorously. The derivation of a term is almost never attempted. The dictionary is written to be a useful reference for a readership that includes students, scientists, and engineers with a wide range of backgrounds, as well as specialists in areas of analysis and differential equations and mathematicians in related fields. Therefore, the definitions are intended to be accessible, as well as rigorous. To be sure, the degree of accessibility may depend upon the individual term, in a dictionary with terms ranging from Abelian cohomology to z intercept. Occasionally a term must be omitted because it is archaic. Care was taken when such circum- stances arose to ensure that the term was obsolete. An example of an archaic term deemed to be obsolete, and hence not included, is “right line”. This term was used throughout a turn-of-the-century analytic geometry textbook we needed to consult, but it was not defined there. Finally, reference to a contemporary English language dictionary yielded “straight line” as a synonym for “right line”. The authors are grateful to the series editor, Stanley Gibilisco, for dealing with our seemingly endless procedural questions and to Nora Konopka, for always acting efficiently and cheerfully with CRC Press liaison matters. Douglas N. Clark Editor-in-Chief c 2001 by CRC Press LLC CONTRIBUTORS Edward Aboufadel Neil K. Dickson Grand Valley State University University of Glasgow Allendale, Michigan Glasgow, United Kingdom Gerardo Aladro David E. Dobbs Florida International University University of Tennessee Miami, Florida Knoxville, Tennessee Mohammad Azarian Marcus Feldman University of Evansville Washington University Evansville. Indiana St. Louis, Missouri Susan Barton Stephen Humphries West Virginia Institute of Technology Brigham Young University Montgomery, West Virginia Provo, Utah Albert Boggess Shanyu Ji Texas A&M University University of Houston College Station, Texas Houston, Texas Robert Borrelli Kenneth D. Johnson Harvey Mudd College University of Georgia Claremont, California Athens, Georgia Stephen W. Brady Bao Qin Li Wichita State University Florida International University Wichita, Kansas Miami, Florida Der Chen Chang Robert E. MacRae Georgetown University University of Colorado Washington, D.C. Boulder, Colorado Stephen A. Chiappari Charles N. Moore Santa Clara University Kansas State University Santa Clara. California Manhattan, Kansas Joseph A. Cima Hossein Movahedi-Lankarani The University of North Carolina at Chapel Hill Pennsylvania State University Chapel Hill, North Carolina Altoona, Pennsylvania Courtney S. Coleman Shashikant B. Mulay Harvey Mudd College University of Tennessee Claremont, California Knoxville, Tennessee John B. Conway Judy Kenney Munshower University of Tennessee Avila College Knoxville, Tennessee Kansas City, Missouri c 2001 by CRC Press LLC Charles W. Neville Anthony D. Thomas CWN Research University of Wisconsin Berlin, Connecticut Platteville. Wisconsin Daniel E. Otero Michael Tsatsomeros Xavier University University of Regina Cincinnati, Ohio Regina, Saskatchewan,Canada Josef Paldus James S. Walker University of Waterloo University of Wisconsin at Eau Claire Waterloo, Ontario, Canada Eau Claire, Wisconsin Harold R. Parks C. Eugene Wayne Oregon State University Boston University Corvallis, Oregon Boston, Massachusetts Gunnar Stefansson Kehe Zhu Pennsylvania State University State University of New York at Albany Altoona, Pennsylvania Albany, New York c 2001 by CRC Press LLC its Galois group is an Abelian group. See Galois group. See also Abelian group. Abelian extension A Galois extension of a A field is called an Abelian extension if its Galois group is Abelian. See Galois extension. See also Abelian group. A-balanced mapping Let M be a right mod- ule over the ring A, and let N be a left module Abelian function A function f(z1,z2,z3, n over the same ring A. A mapping φ from M ×N ...,zn) meromorphic on C for which there ex- n to an Abelian group G is said to be A-balanced ist 2n vectors ωk ∈ C , k = 1, 2, 3,...,2n, if φ(x,·) is a group homomorphism from N to called period vectors, that are linearly indepen- G for each x ∈ M,ifφ(·,y)is a group homo- dent over R and are such that morphism from M to G for each y ∈ N, and + = if f (z ωk) f(z) φ(xa,y) = φ(x,ay) holds for k = 1, 2, 3,...,2n and z ∈ Cn. holds for all x ∈ M, y ∈ N, and a ∈ A. Abelian function field The set of Abelian A-B-bimodule An Abelian group G that is a functions on Cn corresponding to a given set of left module over the ring A and a right module period vectors forms a field called an Abelian over the ring B and satisfies the associative law function field. (ax)b = a(xb) for all a ∈ A, b ∈ B, and all x ∈ G. Abelian group Briefly, a commutative group. More completely, a set G, together with a binary Abelian cohomology The usual cohomology operation, usually denoted “+,” a unary opera- with coefficients in an Abelian group; used if tion usually denoted “−,” and a distinguished the context requires one to distinguish between element usually denoted “0” satisfying the fol- the usual cohomology and the more exotic non- lowing axioms: Abelian cohomology. See cohomology. (i.) a + (b + c) = (a + b) + c for all a,b,c ∈ G, Abelian differential of the first kind A holo- (ii.) a + 0 = a for all a ∈ G, morphic differential on a closed Riemann sur- (iii.) a + (−a) = 0 for all a ∈ G, face; that is, a differential of the form ω = (iv.) a + b = b + a for all a,b ∈ G. a(z)dz, where a(z) is a holomorphic function. The element 0 is called the identity, −a is called the inverse of a, axiom (i.) is called the Abelian differential of the second kind A associative axiom, and axiom (iv.) is called the meromorphic differential on a closed Riemann commutative axiom. surface, the singularities of which are all of order greater than or equal to 2; that is, a differential Abelian ideal An ideal in a Lie algebra which of the form ω = a(z)dz where a(z) is a mero- forms a commutative subalgebra. morphic function with only 0 residues. Abelian integral of the first kind An indef- Abelian differential of the third kind A inite integral W(p) = p a(z)dz on a closed p0 differential on a closed Riemann surface that is Riemann surface in which the function a(z) is not an Abelian differential of the first or sec- holomorphic (the differential ω(z) = a(z)dz ond kind; that is, a differential of the form ω = is said to be an Abelian differential of the first a(z)dz where a(z) is meromorphic and has at kind).