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DICTIONARY OF ALGEBRA, ARITHMETIC, AND TRIGONOMETRY (cid:1)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 (cid:1)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. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. 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. All rights reserved. Authorization to photocopy items for internal or personal use, or the personal or internal use of specific clients, may be granted by CRC Press LLC, provided that $.50 per page photocopied is paid directly to Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923 USA. The fee code for users of the Transactional Reporting Service is ISBN 1-58488-052-X/01/$0.00+$.50. The fee is subject to change without notice. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. 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 (cid:1)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 (cid:1)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,C anada 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 CCoorrvvaalllliiss,, Oregon Boston, Massachusetts Gunnar Stefansson Kehe Zhu Pennsylvania State University State University of New York at Albany Altoona, Pennsylvania Albany, New York (cid:1)c 2001 by CRC Press LLC itsGaloisgroupisanAbeliangroup. SeeGalois group. SeealsoAbeliangroup. A Abelian extension A Galois extension of a fieldiscalledanAbelianextensionifitsGalois group is Abelian. See Galois extension. See alsoAbeliangroup. A-balancedmapping LetMbearightmod- uleovertheringA,andletN bealeftmodule Abelian function A function f(z ,z ,z , 1 2 3 overthesameringA. AmappingφfromM×N ...,z )meromorphiconCnforwhichthereex- n toanAbeliangroupGissaidtobeA-balanced ist 2n vectors ω ∈ Cn, k = 1,2,3,...,2n, k ifφ(x,·)isagrouphomomorphismfromN to calledperiodvectors,thatarelinearlyindepen- Gforeachx ∈ M,ifφ(·,y)isagrouphomo- dentoverRandaresuchthat morphism from M to G for each y ∈ N, and if f (z+ωk)=f(z) φ(xa,y)=φ(x,ay) holdsfork =1,2,3,...,2nandz∈Cn. holdsforallx ∈M,y ∈N,anda ∈A. Abelian function field The set of Abelian A-B-bimodule AnAbeliangroupGthatisa functionsonCncorrespondingtoagivensetof leftmoduleovertheringAandarightmodule period vectors forms a field called an Abelian overtheringB andsatisfiestheassociativelaw functionfield. (ax)b = a(xb) for all a ∈ A, b ∈ B, and all x ∈G. Abeliangroup Briefly,acommutativegroup. Morecompletely,asetG,togetherwithabinary Abeliancohomology Theusualcohomology operation,usuallydenoted“+,”aunaryopera- with coefficients in an Abelian group; used if tion usually denoted “−,” and a distinguished thecontextrequiresonetodistinguishbetween elementusuallydenoted“0”satisfyingthefol- theusualcohomologyandthemoreexoticnon- lowingaxioms: Abeliancohomology. Seecohomology. (i.) a+(b+c)=(a+b)+cforall a,b,c∈G, Abeliandifferentialofthefirstkind Aholo- (ii.) a+0=aforalla ∈G, morphic differential on a closed Riemann sur- (iii.) a+(−a)=0foralla ∈G, face; that is, a differential of the form ω = (iv.) a+b=b+aforalla,b∈G. a(z)dz,wherea(z)isaholomorphicfunction. The element 0 is called the identity, −a is calledtheinverseofa, axiom(i.) iscalledthe Abeliandifferentialofthesecondkind A associativeaxiom,andaxiom(iv.) iscalledthe meromorphicdifferentialonaclosedRiemann commutativeaxiom. surface,thesingularitiesofwhicharealloforder greaterthanorequalto2;thatis,adifferential Abelianideal AnidealinaLiealgebrawhich oftheformω = a(z)dzwherea(z)isamero- formsacommutativesubalgebra. morphicfunctionwithonly0residues. Abelianintegralofthe(cid:1)firstkind Anindef- Abelian differential of the third kind A inite integral W(p) = p a(z)dz on a closed p0 differentialonaclosedRiemannsurfacethatis 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 ondkind;thatis,adifferentialoftheformω = is said to be an Abelian differential of the first a(z)dz where a(z) is meromorphic and has at kind). leastonenon-zeroresidue. Abelianintegralofthes(cid:1)econdkind Anin- Abelian equation A polynomial equation definiteintegralW(p)= p a(z)dzonaclosed f(X) = 0 is said to be an Abelian equation if Riemann surface in whichp0the function a(z) is (cid:1)c 2001 by CRC Press LLC meromorphic with all its singularities of order astheproductofprimeidealsofK ofabsolute atleast2(thedifferentiala(z)dzissaidtobean degree1ifandonlyifpisaprincipalideal. Abeliandifferentialofthesecondkind). Theterm“absoluteclassfield”isusedtodis- tinguishtheGaloisextensionsdescribedabove, Abelian integral of the(cid:1)third kind An in- whichwereintroducedbyHilbert,fromamore definiteintegralW(p)= p a(z)dzonaclosed general concept of “class field” defined by p0 Riemann surface in which the function a(z) is Tagaki. Seealsoclassfield. meromorphicandhasatleastonenon-zeroresidue(thedifferentiala(z)dzissaidtobeanAbel- absolutecovariant Acovariantofweight0. iandifferentialofthethirdkind). Seealsocovariant. Abelian Lie group A Lie group for which absoluteinequality Aninequalityinvolving theassociatedLiealgebraisAbelian. Seealso variables that is valid for all possible substitu- Liealgebra. tionsofrealnumbersforthevariables. Abelian projection operator A non-zero absoluteinvariant Anyquantityorproperty projectionoperatorEinavonNeumannalgebra of an algebraic variety that is preserved under MsuchthatthereducedvonNeumannalgebra birationaltransformations. M =EMEisAbelian. E absolutelyirreduciblecharacter Thechar- Abelian subvariety A subvariety of an acterofanabsolutelyirreduciblerepresentation. Abelianvarietythatisalsoasubgroup. Seealso Arepresentationisabsolutelyirreducibleifitis Abelianvariety. irreducibleandiftherepresentationobtainedby makinganextensionofthegroundfieldremains Abeliansurface Atwo-dimensionalAbelian irreducible. variety. SeealsoAbelianvariety. absolutely irreducible representation A Abelianvariety Acompletealgebraicvari- representation is absolutely irreducible if it is ety G that also forms a commutative algebraic irreducibleandiftherepresentationobtainedby group. Thatis,Gisagroupundergroupoper- makinganextensionofthegroundfieldremains ations that are regular functions. The fact that irreducible. an algebraic group is complete as an algebraic variety implies that the group is commutative. Seealsoregularfunction. absolutelysimplegroup Agroupthatcon- tains no serial subgroup. The notion of an absolutely simple group is a strengthening of the Abel’s Theorem Niels Henrik Abel (1802- conceptofasimplegroupthatisappropriatefor 1829) proved several results now known as infinitegroups. Seeserialsubgroup. “Abel’s Theorem,” but perhaps preeminent among these is Abel’s proof that the general quinticequationcannotbesolvedalgebraically. absolutelyuniserialalgebra LetAbeanal- Other theorems that may be found under the gebraoverthefieldK,andletLbeanextension heading “Abel’s Theorem” concern power se- field of K. Then L⊗K A can be regarded as ries, Dirichlet series, and divisors on Riemann an algebra over L. If, for every choice of L, surfaces. L⊗K A can be decomposed into a direct sum ofidealswhichareprimaryrings,thenAisan absolute class field Let k be an algebraic absolutelyuniserialalgebra. numberfield. AGaloisextensionK ofk isan absolute class field if it satisfies the following absolutemultiplecovariant Amultipleco- property regarding prime ideals of k: A prime variant of weight 0. See also multiple covari- ideal p of k of absolute degree 1 decomposes ants. (cid:1)c 2001 by CRC Press LLC absolutenumber Aspecificnumberrepre- accelerationparameter Aparameterchosen sentedbynumeralssuchas2, 3,or5.67incon- in applying successive over-relaxation (which 4 trast with a literal number which is a number is an accelerated version of the Gauss-Seidel representedbyaletter. method)tosolveasystemoflinearequationsnu- merically. Morespecifically,onesolvesAx =b absolutevalueofacomplexnumber More iterativelybysetting commonlycalledthemodulus,theabsoluteval- ueofthecomplexnumberz=a+ib,wherea xn+1 =xn+R(b−Axn) , andb arereal,isdenoted√by|z|andequalsthe non-negativerealnumber a2+b2. where (cid:5) (cid:6) absolutevalueofavector Morecommonly R = L+ω−1D −1 calledthemagnitude, theabsolutevalueofthe vector withLthelowertriangularsubmatrixofA, D −→ v =(v ,v ,...,v ) the diagonal of A, and 0 < ω < 2. Here, ω 1 2 n −→ is the acceleration parameter, also called the isdenotedby(cid:2)| v |andequalsthenon-negative relaxation parameter. Analysis is required to realnumber v2+v2+···+v2. chooseanappropriatevalueofω. 1 2 n absolute value of real number For a real acyclicchaincomplex Anaugmented,pos- numberr,thenonnegativerealnumber|r|,given itivechaincomplex by (cid:3) (cid:4) |r|= −rr iiff rr ≥<00. ···−∂n→+1 Xn −∂→n Xn−1 −∂n→−1 ... ···−∂→2 X −∂→1 X →(cid:7) A→0 1 0 abstractalgebraicvariety Asetthatisanal- forminganexactsequence. Thisinturnmeans ogous to an ordinary algebraic variety, but de- finedonlylocallyandwithoutanimbedding. that the kernel of ∂n equals the image of ∂n+1 for n ≥ 1, the kernel of (cid:7) equals the image of ∂ , and (cid:7) is surjective. Here the X and A are abstract function (1) In the theory of gen- 1 i modulesoveracommutativeunitaryring. eralized almost-periodic functions, a function mapping R to a Banach space other than the addend Inarithmetic,anumberthatistobe complexnumbers. addedtoanothernumber. Ingeneral,oneofthe (2)AfunctionfromoneBanachspacetoan- operandsofanoperationofaddition. Seealso otherBanachspacethatiseverywheredifferen- addition. tiableinthesenseofFréchet. addition (1) A basic arithmetic operation abstractvariety Ageneralizationoftheno- that expresses the relationship between the tionofanalgebraicvarietyintroducedbyWeil, numberofelementsineachoftwodisjointsets inanalogywiththedefinitionofadifferentiable andthenumberofelementsintheunionofthose manifold. An abstract variety (also called an twosets. abstract algebraic variety) consists of (i.) a family {Vα}α∈A of affine algebraic sets over a (2) The name of the binary operation in an given field k, (ii.) for each α ∈ A a family of Abelian group, when the notation “+” is used opensubsets{Wαβ}β∈AofVα,and(iii.) foreach forthatoperation. SeealsoAbeliangroup. pairαandβinAabirationaltransformationbe- (3) The name of the binary operation in a tweenW andW suchthatthecomposition ring,underwhichtheelementsformanAbelian αβ αβ of the birational transformations between subgroup. SeealsoAbeliangroup. sets of V and V and between subsets of V (4)Sometimes,thenameofoneoftheopera- α β β and V are consistent with those between sub- tionsinamulti-operatorgroup,eventhoughthe γ setsofV andV . operationisnotcommutative. α γ (cid:1)c 2001 by CRC Press LLC