2016丘成桐大学生数学竞赛

S.-T.YauCollegeStudentMathematicsContests2016AnalysisandDifferentialEquationsIndividualPleasesolve5outofthefollowing6problems.1.SupposethatFiscontinuouson[a,b],F′(x)existsforeveryx2(a,b),F′(x)isintegrable.ProvethatFisabsolutelycontinuousandF(b)F(a)=∫baF′(x)dx.2.SupposethatfisintegrableonRn,letK(x)=δn2e−|x|2foreachδ0.Provethattheconvolution(fK)(x)=∫Rnf(xy)K(y)dyisintegrableandjj(fK)fjjL1(Rn)!0,asδ!0.3.ProvethataboundedfunctiononintervalI=[a,b]isRiemannintegrableifandonlyifitssetofdiscontinuitieshasmeasurezero.Youmayprovethisbythefollowingsteps.DeneI(c,r)=(cr,c+r),osc(f,c,r)=supx;y2J\I(c;r)jf(x)f(y)j,osc(f,c)=limr!0osc(f,r,c).1)fiscontinuousatc2Jifandonlyifosc(f,c)=0.2)Forarbitraryϵ0,fc2Jjosc(f,c)ϵgiscompact.3)Ifthesetofdiscontinuitiesoffhasmeasure0,thenfisRiemannintegrable.4.1)LetfbetheRukowskimap:w=12(z+1z).Showthatitmapsfz2Cjjzj1gtoC/[1,1],C=C[f1g.2)Computetheintegral:∫10logxx21dx.5.Letfbeadoublyperiodicmeromorphicfunctionoverthecomplexplane,i.e.f(z+1)=f(z),f(z+i)=f(z),z2C,provethatthenumberofzerosandthenumberofpolesareequal.6.LetAbeaboundedself-adjointoperatoroveracomplexHilbertspace.ProvethatthespectrumofAisaboundedclosedsubsetofthereallineR.1