丘成桐大学生数学竞赛试卷.pdf

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1、S.-T.YauCollegeStudentMathematicsContests2010AnalysisandDi®erentialEquationsTeam(Pleaseselect5problemstosolve)1.a)Letf(z)beholomorphicinD:jzj<1andjf(z)j·1(z2D).Provethatjf(0)j¡jzjjf(0)j+jzj·jf(z)j·:(z2D)1+jf(0)jjzj1¡jf(0)jjzjb)Forany¯nitecomplexvaluea,provethatZ2¼1iµlogj

2、a¡ejdµ=maxflogjaj;0g:2¼02.Letf2C1(R);f(x+1)=f(x);forallx,thenwehaveZ1Z10jjfjj1·jf(t)jdt+jf(t)jdt:003.ConsidertheequationxÄ+(1+f(t))x=0:R1Weassumethatjf(t)jdt<1.StudytheLyapunovstabilityofthesolution(x;x_)=(0;0).4.Supposef:[a;b]!RbeaL1-integrablefunction.Extendftobe0outsi

3、detheinterval[a;b].LetZx+h1Á(x)=f2hx¡hShowthatZbZbjÁj·jfj:aa11R2¼¡inx5.Supposef2L[0;2¼];f^(n)=f(x)edx,provethat2¼0X11)jf^(n)j2<1impliesf2L2[0;2¼];jnj=0X2)jnf^(n)j<1impliesthatf=f;a:e:;f2C1[0;2¼];00nwhereC1[0;2¼]isthespaceoffunctionsfover[0;1]such0thatbothfanditsderivativ

4、efarecontinuousfunctions.1236.Suppose•½Rtobeasimplyconnecteddomainand•1½•withboundary¡.Letubeaharmonicfunctionin•andM0=(x0;y0;z0)2•1.Calculatetheintegral:ZZ¡@11@u¢II=¡u()¡dS;¡@nrr@n11@where=panddenotesther(x¡x0)2+(y¡x0)2+(z¡x0)2@noutnormalderivativewithrespecttoboundary¡

5、ofthedomain•1.(Hint:usetheformula@vdS=@vdy^dz+@vdz^dx+@vdx^dy.)@n@x@y@zS.-T.YauCollegeStudentMathematicsContests2010AppliedMath.,ComputationalMath.,ProbabilityandStatisticsTeam(Pleaseselect5problemstosolve)1.LetX1;¢¢¢;Xnbeindependentandidenticallydistributedrandomvariabl

6、eswithcontinuousdistributionfunctionsF(x1);¢¢¢;F(xn),re-spectively.LetY1<¢¢¢

7、bnShowthatPni=1P(Yn;i¡EYn;i)Ln¡!N(0;3=5);(bnYn;i)1=2i=1providedbn!0andnbn!1.3.LetX1;¢¢¢;Xnbeindependentlyandindenticallydistributedran-domvariableswithXi»N(µ;1).Supposethatitisknownthatjµj·¿,where¿isgiven.ShowXn¿2n¡12aminsupE(aiXi+an+1¡µ)=2¡1:1;¢¢¢;an+1jµj·¿¿+ni=1Hint:Ca

8、refullyusethesu±ciencyprinciple.4.Therulesfor1and1"foulshootinginbasketballareasfollows.Theshootergets