gre math sub material 1.docx
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gre math sub material 1.docx
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gremathsubmaterial1
MathematicsMaterials
Radical
1.Radicalsymbol(√),asymbolusedtoindicatethesquarerootornthroot
2.Radicalofanideal,animportantconceptinabstractalgebra
Incommutativeringtheory,abranchofmathematics,theradicalofanidealIisanidealsuchthatanelementxisintheradicalifsomepowerofxisinI.Aradicalideal(orsemiprimeideal)isanidealthatisitsownradical(thiscanbephrasedasbeingafixedpointofanoperationonidealscalled'radicalization').Theradicalofaprimaryidealisprime.
RadicalidealsdefinedherearegeneralizedtononcommutativeringsintheSemiprimeringarticle.
Definition
TheradicalofanidealIinacommutativeringR,denotedbyRad(I)or
isdefinedas
Intuitively,onecanthinkoftheradicalofIasobtainedbytakingallthepossiblerootsofelementsofI.Rad(I)turnsouttobeanidealitself,containingI.TheeasiestwaytoprovethattheradicalofIofaringAisanidealistonotethatitisthepre-imageoftheidealofnilpotentelementsinA/I.Thisissometimestakenasadefinitionofradical.
IfanidealIcoincideswithitsownradical,thenIiscalledaradicalidealorsemiprimeideal.
Examples
ConsidertheringZofintegers.
Theradicaloftheideal4Zofintegermultiplesof4is2Z.
Theradicalof5Zis5Z.
Theradicalof12Zis6Z.
Ingeneral,theradicalofmZisrZ,whereristheproductofallprimefactorsofm(seeradicalofaninteger).
Theradicalofaprimaryidealisprime.
Thenilradicalofaring
ConsiderthesetofallnilpotentelementsofR,whichwillbecalledthenilradicalofR(andwillbedenotedbyN(R)).OnecaneasilyseethatthenilradicalofRisjusttheradicalofthezeroideal(0).Thispermitsanalternativedefinitionforthe(general)radicalofanidealIinR.DefineRad(I)asthepreimageofN(R/I),thenilradicalofR/I,undertheprojectionmapR→R/I.
ToseethatthetwodefinitionsfortheradicalofIareequivalent,notefirstthatifrisinthepreimageofN(R/I),thenforsomen,rniszeroinR/I,andhencernisinI.Second,ifrnisinIforsomen,thentheimageofrninR/Iiszero,andhencernisinthepreimageofN(R/I).
Thisalternativedefinitioncanbeveryuseful,asweshallseerightbelow.See#Propertiesbelowforanothercharacterizationofthenilradical.
Properties
ThissectionwillcontinuetheconventionthatIisanidealofacommutativeringR:
ItisalwaystruethatRad(Rad(I))=Rad(I).Inwords,thissaysthatRad(I)isaradicalideal.
ForanyidealI,Rad(I)isthesmallestradicalidealcontainingI.
Rad(I)istheintersectionofalltheprimeidealsofRthatcontainI.Ononehand,everyprimeidealisradical,andsotheintersectionJoftheprimeidealscontainingIcontainsRad(I).SupposerisanelementofRwhichisnotinRad(I),andletSbetheset{rn|nisanonnegativeinteger}.BythedefinitionofRad(I),SmustbedisjointfromI.SinceSismultiplicativelyclosedandRhasidentity,anargumentwithZorn'slemmashowsthatthereexistsanidealPinthisringwhichcontainsIandismaximalwithrespecttobeingdisjointfromS.ItiswellknownthatPisnecessarilyaprimeideal.SincePcontainsI,butnotr,thisshowsthatrisnotintheintersectionofprimeidealscontainingI.Thus,theintersectionofprimeidealscontainingIiscontainedinRad(I),provingequality.
Specializingthelastpoint,thenilradicalisequaltotheintersectionofallprimeidealsofR.ThisshowsthatthenilradicalofRcanalternativelybedefinedastheintersectionoftheprimeidealsofR.
AnidealIinaringRisradicalifandonlyifthequotientringR/Iisreduced.
3.Radicalofaninteger,innumbertheory,theradicalofanintegeristheproductoftheprimeswhichdividethatinteger
Innumbertheory,theradicalofapositiveintegernisdefinedastheproductoftheprimenumbersdividingn:
Continuousfunction
Inmathematics,acontinuousfunctionisafunctionforwhich,intuitively,"small"changesintheinputresultin"small"changesintheoutput.Otherwise,afunctionissaidtobea"discontinuousfunction".Acontinuousfunctionwithacontinuousinversefunctioniscalled"bicontinuous".
Continuityoffunctionsisoneofthecoreconceptsoftopology,whichistreatedinfullgeneralitybelow.Theintroductoryportionofthisarticlefocusesonthespecialcasewheretheinputsandoutputsoffunctionsarerealnumbers.Inaddition,thisarticlediscussesthedefinitionforthemoregeneralcaseoffunctionsbetweentwometricspaces.Inordertheory,especiallyindomaintheory,oneconsidersanotionofcontinuityknownasScottcontinuity.Otherformsofcontinuitydoexistbuttheyarenotdiscussedinthisarticle.
Asanexample,considerthefunctionh(t),whichdescribestheheightofagrowingflowerattimet.Thisfunctioniscontinuous.Infact,adictumofclassicalphysicsstatesthatinnatureeverythingiscontinuous.Bycontrast,ifM(t)denotestheamountofmoneyinabankaccountattimet,thenthefunctionjumpswhenevermoneyisdepositedorwithdrawn,sothefunctionM(t)isdiscontinuous.
Real-valuedcontinuousfunctions
Definition
AfunctionfromthesetofrealnumberstotherealnumberscanberepresentedbyagraphintheCartesianplane;thefunctioniscontinuousif,roughlyspeaking,thegraphisasingleunbrokencurvewithno"holes"or"jumps".
Thereareseveralwaystomakethisintuitionmathematicallyrigorous.Thesedefinitionsareequivalenttooneanother,sothemostconvenientdefinitioncanbeusedtodeterminewhetheragivenfunctioniscontinuousornot.Inthedefinitionsbelow,
isafunctiondefinedonasubsetIofthesetRofrealnumbers.ThissubsetIisreferredtoasthedomainoff.PossiblechoicesincludeI=R,thewholesetofrealnumbers,anopeninterval
oraclosedinterval
Here,aandbarerealnumbers.
[edit]Definitionintermsoflimitsoffunctions
Thefunctionfiscontinuousatsomepointcofitsdomainifthelimitoff(x)asxapproachescthroughthedomainoffexistsandisequaltof(c).[3]Inmathematicalnotation,thisiswrittenas
Indetailthismeansthreeconditions:
first,fhastobedefinedatc.Second,thelimitonthelefthandsideofthatequationhastoexist.Third,thevalueofthislimitmustequalf(c).
Thefunctionfissaidtobecontinuousifitiscontinuousateverypointofitsdomain.Ifthepointcinthedomainoffisnotalimitpointofthedomain,thenthisconditionisvacuouslytrue,sincexcannotapproachcthroughvaluesnotequalc.Thus,forexample,everyfunctionwhosedomainisthesetofallintegersiscontinuous.
Definitionintermsoflimitsofsequences
Onecaninsteadrequirethatforanysequence
ofpointsinthedomainwhichconvergestoc,thecorrespondingsequence
convergestof(c).Inmathematicalnotation,
Weierstrassdefinition(epsilon-delta)ofcontinuousfunctions
Illustrationoftheε-δ-definition:
forε=0.5,thevalueδ=0.5satisfiestheconditionofthedefinition.
Explicitlyincludingthedefinitionofthelimitofafunction,weobtainaself-containeddefinition:
GivenafunctionfasaboveandanelementcofthedomainI,fissaidtobecontinuousatthepointcifthefollowingholds:
Foranynumberε>0,howeversmall,thereexistssomenumberδ>0suchthatforallxinthedomainoffwithc−δ Alternativelywritten,continuityoff: I→Ratc∈Imeansthatforeveryε>0thereexistsaδ>0suchthatforallx∈I,: Moreintuitively,wecansaythatifwewanttogetallthef(x)valuestostayinsomesmallneighborhoodaroundf(c),wesimplyneedtochooseasmallenoughneighborhoodforthexvaluesaroundc,andwecandothatnomatterhowsmallthef(x)neighborhoodis;fisthencontinuousatc. Inmodernterms,thisisgeneralizedbythedefinitionofcontinuityofafunctionwithrespecttoabasisforthetopology,herethemetrictopology. Definitionusingoscillation Thefailureofafunctiontobecontinuousatapointisquantifiedbyitsoscillation. Continuitycanalsobedefinedintermsofoscillation: afunctionfiscontinuousatapointx0ifandonlyifitsoscillationatthatpointiszero;[4]insymbols, Abenefitofthisdefinitionisthatitquantifiesdiscontinuity: theoscillationgiveshowmuchthefunctionisdiscontinuousatapoint. Thisdefinitionisusefulindescriptivesettheorytostudythesetofdiscontinuitiesandcontinuouspoints–thecontinuouspointsaretheintersectionofthesetswheretheoscillationislessthanε(henceaGδset)–andgivesaveryquickproofofonedirectionoftheLebesgueintegrabilitycondition.[5] Theoscillationisequivalenttotheε-δdefinitionbyasimplere-arrangement,andbyusingalimit(limsup,liminf)todefineoscillation: if(atagivenpoint)foragivenε0thereisnoδthatsatisfiestheε-δdefinition,thentheoscillationisatleastε0,andconverselyifforeveryεthereisadesiredδ,theoscillationis0.Theoscillationdefinitioncanbenaturallygeneralizedtomapsfromatopologicalspacetoametricspace. Definitionusingthehyperreals Cauchydefinedcontinuityofafunctioninthefollowingintuitiveterms: aninfinitesimalchangeintheindependentvariablecorrespondstoaninfinitesimalchangeofthedependentvariable(seeCoursd'analyse,page34).Non-standardanalysisisawayofmakingthismathematicallyrigorous.Thereallineisaugmentedbytheadditionofinfiniteandinfinitesimalnumberstoformthehyperrealnumbers.Innonstandardanalysis,continuitycanbedefinedasfollows. Afunctionƒfromtherealstotherealsiscontinuousifitsnatur
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