Dictionary of Algebra, Arithmetic, and Trigonometry

326 Pages·2000·3.72 MB·English

by  Daniel E. Levy

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-zeroresi-
due(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 ab-
solutely 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 sub- group. 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