第三届丘成桐大学生数学竞赛试题-纯数学.pdf

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1、INDIVIDUALTESTS.-TYAUCOLLEGEMATHCONTESTS2012AnalysisandDi®erentialEquationsPleasesolve5outofthefollowing6problems,orhighestscoresof5problemswillbecounted.1.ComputetheintegralZ1pxdx;¡1

2、44ontotheunitdisk.3.Letf(x)beaC2functiononR:Showthat0200supjf(x)j·4supjf(x)jsupjf(x)j:4.Letf(x)bearealmeasurablefunctionde¯nedon[a;b].Letn(y)bethenumberofsolutionsoftheequationf(x)=y.Provethatn(y)isameasurablefunctiononR.115.For1

3、ctionalFonLpgivenby:F(f)isequaltotheintegraloffg.ShowthatjjFjj=jjgjjq:6.LetRn=fx=(x;x;:::;x)2Rnjx>0g.Showthattheformula+12nnZ2xng(y)nu(x)=dy;x2R+nn®n@Rnjx¡yj+isasolutionoftheproblemnn¢u=0;inR+;u=gon@R+;where®nisthevolumeoftheunitndimensionalsphere.1INDIVIDUALTESTS.-TYAUC

4、OLLEGEMATHCONTESTS2012GeometryandTopologyPleasesolve5outofthefollowing6problems,orhighestscoresof5problemswillbecounted.1.Showthat¼(S2)6=0.32.LetMbeasmoothmanifoldofdimensionn,andX1;¢¢¢;Xkbekeverywherelinearlyindependentsmoothvector¯eldsonanopensetU½Msatisfyingthat[Xi;Xj

5、]=0for1·i;j·k.Provethatforanypointp2Uthereisacoordinatechart(V;yi)withp2VµUandcoordinatesfy1;¢¢¢;yngsuchthatX=@onVforeach1·i·k.i@yi23.ShowthatanyselfhomeomorphismofCPisorientationpreserving.4.Provethefollowingversionoftheisoperimetricinequality:SupposeCisasimple(thatis,w

6、ithoutself-intersection),smooth,closedcurveintheEuclideanplane,withlengthL.ShowthattheareaenclosedbyL2Cislessthanorequalto,andtheequalityoccurswhenandonly4¼whenCisaroundcircle.5.Letx:M!R3beaclosedsurfacein3-dimensionalEuclideanspace.ItsGaussiancurvatureandmeancurvaturear

7、edenotedbyKandHrespectively.Provethat:ZZZZZZZZHdA+pKdA=0;pHdA+dA=0;MMMMwherep=~x¢~nisthesupportfunctionofM,~xdenotesthepositionvectorofM,~ndenotestheunitnormaltoM,anddAistheareaelementofM.6.WritethestructureequationofanorthonormalframeonaRiemann-ianmanifold.Provethefollo

8、wingRiemannianmetricghasconstantsectionalcurvaturecusingthestructureequation:Pni2(dx)g=iP=1[1+cn(xi)2]2