1、OXFORD PAPERBACK REFERENCEThe Concise Oxford Dictionary of MathematicsSomeentriesinthisdictionaryhaverecommendedweblinks.When you see the symbol at the end of an entry go to thedictionarys web page at http:/ Web links intheResourcessection and click straight through to the relevant websites.Christop
2、her Clapham was until 1993 Senior Lecturer inMathematics at the University of Aberdeen and has alsotaughtatuniversitiesinNigeria,Lesotho,andMalawi.Heisthe author of Introduction to Abstract Algebra andIntroduction to Mathematical Analysis. He lives in Exeter.James NicholsonhasamathematicsdegreefromC
3、ambridge,and taught at Harrow School for twelve years beforebecomingHeadofMathematicsatBelfastRoyalAcademyin1990. He lives in Belfast, but now works mostly with theSchoolofEducationatDurhamUniversity.Heisco-authorofStatistics GCSE for AQA.2The most authoritative and up-to-date reference books forbot
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9、nary of MathematicsFOURTH EDITIONCHRISTOPHER CLAPHAMJAMES NICHOLSON10Great Clarendon Street, OxfordOX2 6DPOxfordUniversityPressisadepartmentoftheUniversityofOxford.ItfurtherstheUniversitysobjectiveofexcellenceinresearch,scholarship,andeducationbypublishingworldwideinOxford New YorkAuckland Cape Town
10、 Dar es Salaam Hong Kong KarachiKualaLumpurMadridMelbourneMexicoCityNairobiNewDelhi Shanghai Taipei TorontoWith offices inArgentina Austria Brazil Chile Czech Republic FranceGreece Guatemala Hungary Italy Japan Poland PortugalSingaporeSouthKoreaSwitzerlandThailandTurkeyUkraineVietnamOxfordisaregiste
11、redtrademarkofOxfordUniversityPressin the UK and in certain other countriesPublished in the United Statesby Oxford University Press Inc., New York Christopher Clapham 1990, 1996 Christopher Clapham and James Nicholson 2005, 2009First edition 1990Second edition 199611Third edition 2005Fourth edition
12、2009All rights reserved. No part of this publication may bereproduced,storedinaretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthepriorpermissioninwritingofOxfordUniversityPress,orasexpresslypermittedbylaw,or under terms agreed with the appropriate reprographicsrights organization. Enquir
13、ies concerning reproductionoutside the scope of the above should be sent to the RightsDepartment, Oxford University Press, at the address aboveYou must not circulate this book in any other binding orcover and you must impose this same condition on anyacquirerBritish Library Cataloguing in Publicatio
14、n DataData availableLibrary of Congress Cataloging in Publication DataData availableTypeset by SPI Publisher Services, Pondicherry, IndiaPrinted in Great Britainon acid-free paper byClays Ltd., St Ives plcISBN 97801992359401 3 5 7 9 10 8 6 4 212ContentsPrefaceDictionaryAPPENDICES1 Table of areas and
15、 volumes2 Table of centres of mass3 Table of moments of inertia4 Table of derivatives5 Table of integrals6 Table of common ordinary differential equations andsolutions7 Table of series8 Table of convergence tests for series9 Table of common inequalities10 Table of trigonometric formulae11 Table of s
16、ymbols12 Table of Greek letters1313 Table of Roman numerals14 Table of Fields Medal Winners14ContributorsC. Chatfield, BSc, PhDR. Cheal, BScJ. B. Gavin, BSc, MScUniversity of BathJ. R. Pulham, BSc, PhDUniversity of AberdeenD. P. Thomas, BSc, PhDUniversity of Dundee15Preface to Second Edition*Thisdic
17、tionaryisintendedtobeareferencebookthatgivesreliable definitions or clear and precise explanations ofmathematicalterms.Thelevelissuchthatitwillsuit,amongothers, sixth-form pupils, college students and first-yearuniversity students who are taking mathematics as one oftheircourses.Suchstudentswillbeab
18、letolookupanytermtheymaymeetandbeledontootherentriesbyfollowingupcross-references or by browsing more generally.Theconceptsandterminologyofallthosetopicsthatfeatureinpureandappliedmathematicsandstatisticscoursesatthislevel today are covered. There are also entries onmathematicians of the past and im
19、portant mathematics ofmore general interest. Computing is not included. Thereaders attention is drawn to the appendices which giveuseful tables for ready reference.Someentriesgiveastraightdefinitioninanopeningphrase.Othersgivethedefinitionintheformofacompletesentence,sometimes following an explanati
20、on of the context. Anasterisk is used to indicate words with their own entry, towhich cross-reference can be made if required.This edition is more than half as large again as the firstedition.Asignificantchangehasbeentheinclusionofentriescoveringappliedmathematicsandstatistics.Intheseareas,Iamverymu
21、chindebtedtothecontributors,whosenamesaregiven on page v. I am most grateful to these colleagues for16their specialist advice and drafting work. They are not,however, to be held responsible for the final form of theentries on their subjects. There has also been a considerableincrease in the number o
22、f short biographies, so that all themajor names are included. Other additional entries havegreatly increased the comprehensiveness of the dictionary.Thetexthasbenefitedfromthecommentsofcolleagueswhohavereaddifferentpartsofit.Eventhoughthenamesofallofthem will not be given, I should like to acknowled
23、ge heretheir help and express my thanks.Christopher Clapham17Preface to Third EditionSince the second edition was published the content andemphasisofappliedmathematicsandstatisticsatsixth-form,college and first-year university levels has changedconsiderably. This edition includes many more appliedst
24、atisticsentriesaswellasdealingcomprehensivelywiththenew decision and discrete mathematics courses, and a largenumberofnewbiographieson20th-centurymathematicians.Iam grateful to the Headmaster and Governors of BelfastRoyalAcademyfortheirsupportandencouragementtotakeonthistask,andtoLouise,Joanne,andLa
25、urafortranscribingmy notes.James Nicholson18Preface to Fourth EditionSince the third edition was published there has been adramatic increase in both access to the internet and theamount of information available. The major change to thisedition is the introduction of a substantial number ofweblinks,
26、many of which contain dynamic or interactiveillustrations related to the definition.James Nicholson19a-Prefixmeaningnot.Forexample,anasymmetricfigureisone which possesses no symmetry, which is not symmetrical.AThe number 10 in hexadecimal notation.abacus Acountingdeviceconsistingofrodsonwhichbeadsca
27、n be moved so as to represent numbers. A description of how one abacus works.abelian groupSupposethat G isa*groupwiththeoperation.Then G isabelianiftheoperation iscommutative;thatis,if, for all elementsaandbinG, a b=b a.Abel, Niels Henrik (180229) Norwegian mathematicianwho, at the age of 19, proved
28、 that the general equation ofdegreegreaterthan4cannotbesolvedalgebraically.Inotherwords, there can be no formula for the roots of such anequation similar to the familiar formula for a quadraticequation. He was also responsible for fundamentaldevelopmentsinthetheoryofalgebraicfunctions.Hediedinsome p
29、overty at the age of 26, just a few days before hewouldhavereceivedaletterannouncinghisappointmenttoaprofessorship in Berlin.Abels test A test for the convergence of an infinite serieswhichstatesthatifan isaconvergentsequence,andbnismonoticallydecreasing,i.e. bn+1 bn forall n,thenanbn isalso converg
30、ent.20aboveGreaterthan.Thelimitofafunctionat a fromaboveisthelimitof f(x)as x a forvaluesof x a.Itisofparticularimportancewhen f(x)hasadiscontinuityat a,i.e.wherethelimitsfromaboveandfrombelowdonotcoincide.Itcanbewritten asf(a+) or .abscissaThe x-coordinateinaCartesiancoordinatesysteminthe plane.abs
31、olute address In spreadsheets a formula which is toappearinanumberofcellsmaywishtousethecontentsofanothercellorcells.Sincetherelativepositionofthosecellswill be different each time the formulaappears in a new location, the spreadsheet syntax allows anabsolute address to be specified, identifying the
32、 actual rowand column for each cell. When a formula is copied andpasted to another cell, a cell reference using an absoluteaddress will remain unchanged. A formula can contain amixture of absolute and *relative addresses.absolute error See ERROR.absolute frequency Thenumberofoccurrencesofanevent.For
33、 example, if a die is rolled 20 times and 4 sixes areobservedtheabsolutefrequencyofsixesis4andthe*relativefrequency is 4/20.absolute measure of dispersion=MEASURE OF DISPERSION.absolutely convergent series A series an is said to beabsolutely convergent if is *convergent. For21example,if thentheserie
34、sisconvergentbut not absolutely convergent, whereasis absolutely convergent.absolutely summable=ABSOLUTELY CONVERGENT.absolute value For any real number a, the absolute value(alsocalledthe*modulus)of a,denotedby|a|,is a itselfif a0,anda if a 0.Thus|a|ispositiveexceptwhen a =0.The following propertie
35、s hold:(i) |ab| = |a|b|.(ii) |a+b| |a| + |b|.(iii) |ab| |a| |b|.(iv) Fora 0, |x| aif and only if axa.absorbing state See RANDOM WALK.22absorption laws For all sets A and B (subsets of some*universalset), A (A B)= A and A (A B)= A.Theseare the absorption laws.abstract algebra The area of mathematics
36、concerned withalgebraic structures, such as *groups, *rings and *fields,involving sets of elements with particular operationssatisfyingcertainaxioms.Thepurposeistoderive,fromtheset of axioms, general results that are then applicable toanyparticularexampleofthealgebraicstructureinquestion.Thetheoryof
37、certainalgebraicstructuresishighlydeveloped;inparticular,thetheoryofvectorspacesissoextensivethatitsstudy,knownas*linearalgebra,wouldprobablynolongerbe classified as abstract algebra.abstractionTheprocessofmakingageneralstatementwhichsummarizeswhatcanbeobservedinparticularinstances.Forexample,wecans
38、aythat x2 x for x1. Mathematical theorems are essentiallyabstraction of concepts to a higher level.abundant numberAnintegerthatissmallerthanthesumofits positive divisors, not including itself, For example, 12 isdivisible by 1, 2, 3, 4 and 6, and 1 + 2 + 3 + 4 + 6 = 16 12.acceleration Suppose that a
39、particle is moving in a straightline, with a point O on the line taken as origin and onedirectiontakenaspositive.Let x bethe*displacementoftheparticleattime t.Theaccelerationoftheparticleisequaltoor d2x/dt2,the*rateofchangeofthe*velocitywithrespecttot.Ifthevelocityispositive(thatis,iftheparticleismo
40、vinginthe positive direction), the acceleration is positive when theparticleisspeedingupandnegativewhenitisslowingdown.23However, if the velocity is negative, a positive accelerationmeans that the particle is slowing down and a negativeacceleration means that it is speeding up.In the preceding parag
41、raph, a common convention has beenfollowed, in which the unit vector i in the positive directionalong the line has been suppressed. Acceleration is in fact avector quantity, and in the one-dimensional case above it isequal to i.When the motion is in two or three dimensions, vectors areused explicitl
42、y. The acceleration a of a particle is a vectorequaltotherateofchangeofthevelocity vwithrespectto t.Thus a = dv/dt. If the particle has *position vector r, then.WhenCartesiancoordinatesareused, r = xi+yj+zk, and then .Ifaparticleistravellinginacirclewithconstantspeed,itstillhas an acceleration becau
43、se of the changing direction of thevelocity.Thisaccelerationistowardsthecentreofthecircleandhasmagnitude where v isthespeedoftheparticleandris the radius of the circle.Acceleration has the dimensions LT2, and the SI unit ofmeasurementisthemetrepersecondpersecond,abbreviatedto ms2.accelerationtime gr
44、aph A graph that shows accelerationplotted against time for a particle moving in a straight line.Let v(t)anda(t)bethevelocityandacceleration,respectively,of the particle at time t. The accelerationtime graph is thegraph y = a(t),wherethe t-axisishorizontalandthe y-axisis24vertical with the positive
45、direction upwards. With theconventionthatanyareabelowthehorizontalaxisisnegative,the areaunderthegraphbetween t = t1 and t = t2 isequalto v(t2)v(t1). (Here a common convention has been followed, inwhichtheunitvectoriinthepositivedirectionalongthelinehas been suppressed. The velocity and acceleration
46、 of theparticle are in fact vector quantities equal to v(t)i and a(t)i,respectively.)acceptance region See HYPOTHESIS TESTING.acceptance sampling A method of quality control where asampleistakenfromabatchandadecisionwhethertoacceptthe batch is made on the basis of the quality of the sample.The most
47、simple method is to have a straight accept/rejectcriterion,butamoresophisticatedapproachistotakeanothersample if the evidence from the existing sample, or a set ofsamples,isnotclearlyindicatingwhetherthebatchshouldbeaccepted or rejected. One of the main advantages of thisapproach is reducing the cos
48、t of taking samples to satisfyquality control criteria.accuracyAmeasureoftheprecisionofanumericalquantity,usuallygivento n *significantfigures(wheretheproportionalaccuracyistheimportantaspect)or n *decimalplaces(wherethe absolute accuracy is more important).accurate (correct) to n decimal places Rou
49、ndinganumberwiththeaccuracyspecifiedbythenumberof*decimalplacesgiven in the rounded value. So e = 2.71828 = 2.718 tothree decimal places and = 2.72 to two decimal places.25is9.30correcttotwodecimalplaces.Whereanumberofquantitiesarebeingmeasuredandaddedor subtracted, using values correct to the same number ofdecimal places ensures that they have the same degree ofaccuracy. However, if the units are changed, for examplebetween centimetres and metres, then the accuracy of themeasurement
