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amoremathematicallanguagefisonetoone graphoffxinmorethanonepointthenfxisnotonetooneThereasonfxwouldnotbeonetoone 3and43Thatwouldmeanthatf2andf4bothequal3andonetoone mostonc. 4SHANKARAPAILOORCorollary2.7.LetMbecompactandf:M!Nanalytic.Iffisnotconstant,thenf(M)=N.Asasmoothmanifold,Mcarriesk-formswhichwedenoteby k(M;C)(see[3]or[5])fordenition.Thatbeingsaid, 0(M;C)aresimplyC1 H// YIev0 Xf// Ywheref:X!Yisanarbitrarymap.ThespaceYdoesnotcomewithaspeciedbasepoint,sowechoosey0:=f(x0)asbasepoint.Thenf:X!Yisapointedmap,thoughH:A!YIis(usually)not.ChangingHtoapointedmap.Notethat 4KEITHCONRADProof.Letf(T)2OK[T]beEisensteinatsomeprimeideal.Iff(T)isreducibleinK[T]thenf(T)=g(T)h(T)forsomenonconstantg(T)andh(T)inK[T].WerstshowthatgandhcanbechoseninOK[T].Asfismonic,wecanassumegand Weusethefollowingbasicnotations.xTdenotesthetransposeofvectorx.Giventwofunctionsfandg,f=O(g)ifsupnjf(n)=g(n)j1.f= (g)ifg=O(f).Ifbothf=O(g)andf= (g),thenf=(g).Duetopageconstraints,wehavetoomitproofsof 3AnuninterpretedfunctionFofaritynsatisesonlyoneaxiom:Ifei=e0ifor1in,thenF(e1;::;en)=F(e01;::;e0n).Uninterpretedfunctionsarecommonlyusedtoabstractprogramminglanguageoperatorsthatareotherwisehardtore 2];n=7,xi=i 2n.DerivativeApproximations:iff(x)2Cn+1[a;b],P(x)interpolatesf(x)atdistinctx0;x1;:::;xn2[a;b]andx[a;b],thenf[x0;x1;:::;xn]=f(n)()=n!forsomepoint2[minfx;xig;maxfx;xig]:2 INTERPOLATIONER "#"e andf(#( andhence .If ,1},thendifferentiationyields(f%'f)f%$1=cf%$corf%(f%'f$c)=1$c.Sincec) (p)=!,thenf(p)=p;ifg(p)=0,thenf(f (p)|1,thenp&J(f).InordertoshowthatJ( beasubsequencethatconvergestoano x;sincef1(f(x))=3p x3=xandf(f1(x))=(3p x)3=x: 3.Letf(x)=2x;thenf1(x)=1 2x;sincef1(f(x))=1 2(2x)=xandf(f1(x))=21 2x=x: 5.Letf(x)=7x+2;thenf1(x)=x2 7;sincef1(f(x))=7x+22 7=xandf(f1(x))=7x2 E(w)=Mr=M (e)=Hr Fr1 Fr1(w)=Mr1 G(Mr=Mr1)=Hr1KFFFFFFFFFE=F1 F1(w)=M1 G(M1=M1)=H1 F=F0 F0(w)=M0 G(Mr=M0)=H0Letwbeaprimitiventhrootofunity.ThenF(w)hasprimitiventhirootofun 70L.C.Ciungu(2)IfX=xwewrite[x)insteadof[fxg)and[x)=fy2Ajyxnforsomen1g.[x)iscalledprincipallter.(3)IfXisalterofAandx2A,thenF(x)=[F[fxg)=fy2Ajy(f1 x)n1 (f2 x)n2 ::: (fm x)nmforsomem1;n1;n2;:::;nm simple(acl:lab,x:intfaclg)=if(memberuseracl)thenfgxelse1ThisfunctiontakesalabellikeACL(Alice,Bob)asitsrstargument,andanintegerprotectedbythatACLasitssec-ondargument.Ifthecurrentuser(representedbyva 644T.OSTROGORSKIInsections2and3wereviewsomepropertiesofthehomogeneousconesfollowingmostlyVinberg[4].Asanexampleofanapplicationofthistheorytoanalysiswestudyintegraltransformsandtheirasymptoticbehaviour 644T.OSTROGORSKIInsections2and3wereviewsomepropertiesofthehomogeneousconesfollowingmostlyVinberg[4].Asanexampleofanapplicationofthistheorytoanalysiswestudyintegraltransformsandtheirasymptoticbehaviour 308OKadaWTakahashiWegivetworesultswhichareusedinSections34and5Lemma2234Let22beaninvariantsubmeanonXThenlimsfs2022f20limsfsforeveryf2Xwherelimsfssupsinft21sftandlimsfsinfssupt21sftLemma23Letubeabounded