丘成桐大学生数学竞赛试卷

S.-T.YauCollegeStudentMathematicsContests2010

AnalysisandDifferentialEquations

Team

(Pleaseselect5problemstosolve)

1.

a)Letf(z)beholomorphicinD:|z|<1and|f(z)|≤1(z∈D).

Provethat

|f(0)|−|z|

1+|f(0)||z|≤|f(z)|≤|f(0)|+|z|

1−|f(0)||z|.(z∈D)

b)Foranyfinitecomplexvaluea,provethat1

2π󰀄2π

0log|a−eiθ|dθ=max{log|a|,0}.

2.Letf∈C1(R),f(x+1)=f(x),forallx,thenwehave

||f||∞≤󰀄1

0|f(t)|dt+󰀄1

0|f󰀁(t)|dt.

3.Considertheequation

¨x+(1+f(t))x=0.

Weassumethat󰀂∞|f(t)|dt<∞.StudytheLyapunovstabilityofthe

solution(x,˙x)=(0,0).

4.Supposef:[a,b]→RbeaL1-integrablefunction.Extendftobe

0outsidetheinterval[a,b].Let

φ(x)=1

2h󰀄x+h

x−hf

Showthat󰀄b

a|φ|≤󰀄b

a|f|.

5.Supposef∈L1[0,2π],ˆf(n)=12π󰀂2π0f(x)e−inxdx,provethat

1)∞󰀃

|n|=0|ˆf(n)|2<∞impliesf∈L2[0,2π],

2)󰀃

n|nˆf(n)|<∞impliesthatf=f0,a.e.,f0∈C1[0,2π],

whereC1[0,2π]isthespaceoffunctionsfover[0,1]such

thatbothfanditsderivativef󰀁arecontinuousfunctions.

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6.SupposeΩ⊂R3tobeasimplyconnecteddomainandΩ1⊂Ωwith

boundaryΓ.LetubeaharmonicfunctioninΩandM0=(x0,y0,z0)∈

Ω1.Calculatetheintegral:

II=−󰀄󰀄

Γ󰀆u∂

∂n(1

r)−1

r∂u

∂n󰀁dS,

where1

r=1󰀅(x−x0)2+(y−x0)2+(z−x0)2and∂

∂ndenotesthe

outnormalderivativewithrespecttoboundaryΓofthedomainΩ1.

(Hint:usetheformula∂v∂ndS=∂v∂xdy∧dz+∂v∂ydz∧dx+∂v∂zdx∧dy.)S.-T.YauCollegeStudentMathematicsContests2010

AppliedMath.,ComputationalMath.,

ProbabilityandStatistics

Team

(Pleaseselect5problemstosolve)

1.LetX1,···,Xnbeindependentandidenticallydistributedrandom

variableswithcontinuousdistributionfunctionsF(x1),···,F(xn),re-

spectively.LetY1<···

ProvethatZj=F(Yj)hasthebeta(j,n−j+1)distribution(j=

1,···,n).

2.LetX1,···,Xnbei.i.d.randomvariablewithacontinuousdensity

fatpoint0.Let

Yn,i=3

4bn(1−X2i/b2n)I(|Xi|≤bn).

Showthat󰀂ni=1(Yn,i−EYn,i)

(bn󰀂ni=1Yn,i)1/2L−→N(0,3/5),

providedbn→0andnbn→∞.

3.LetX1,···,Xnbeindependentlyandindenticallydistributedran-

domvariableswithXi∼N(θ,1).Supposethatitisknownthat|θ|≤τ,

whereτisgiven.Show

mina1,···,an+1sup

|θ|≤τE(n󰀁

i=1aiXi+an+1−θ)2=τ2n−1

τ2+n−1.

Hint:Carefullyusethesufficiencyprinciple.

4.Therulesfor“1and1”foulshootinginbasketballareasfollows.

Theshootergetstotrytomakeabasketfromthefoulline.Ifhe

succeeds,hegetsanothertry.Moreprecisely,hemake0basketsby

missingthefirsttime,1basketbymakingthefirstshotandxsmissing

thesecondone,or2basketsbymakingbothshots.

Letnbeafixedinteger,andsupposeaplayergetsntriesat“1and

1”shooting.LetN0,N1,andN2betherandomvariablesrecording

thenumberoftimeshemakes0,1,or2baskets,respectively.Note

thatN0+N1+N2=n.SupposethatshotsareindependentBernoulli

trailswithprobabilitypformakingabasket.

(a)Writedownthelikelihoodfor(N0,N1,N2).

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(b)Showthatthemaximumlikelihoodestimatorofpis

ˆp=N1+2N2N0+2N1+2N2.

(c)Isˆpanunbiasedestimatorforp?Proveordisprove.(Hint:Eˆp

isapolynomialinp,whoseorderishigherthan1forp∈(0,1).)

(d)Findtheasymptoticdistributionofˆpasntendsto∞.

5.Whenconsideringfinitedifferenceschemesapproximatingpartial

differentialequations(PDEs),forexample,thescheme

(1)un+1j=unj−λ(unj−unj−1)

whereλ=∆t

∆x,approximatingthePDE

(2)ut+ux=0,

weareofteninterestedinstability,namely

(3)||un||≤C||u0||,n∆t≤T

foraconstantC=C(T)independentofthetimestep∆tandthespa-

tialmeshsize∆x.Here||·||isagivennorm,forexampletheL2normor

theL∞norm,ofthenumericalsolutionvectorun=(un1,un2,···,unN).

Themeshpointsarexj=j∆x,tn=n∆t,andthenumericalsolution

unjapproximatestheexactsolutionu(xj,tn)ofthePDE(2)witha

periodicboundarycondition.

(i)Provethatthescheme(1)isstableinthesenseof(3)forboth

theL2normandtheL∞normunderthetimesteprestriction

λ≤1.

(ii)Sincethenumericalsolutionunisinafinitedimensionalspace,

StudentAarguesthatthestability(3),onceprovedforaspe-

cificnorm||·||a,wouldalsoautomaticallyholdforanyother

norm||·||b.Hisargumentisbasedontheequivalencyofall

normsinafinitedimensionalspace,namelyforanytwonorms

||·||aand||·||bonafinitedimensionalspaceW,thereexistsa

constantδ>0suchthat

δ||u||b≤||u||a≤1

δ||u||b.

Doyouagreewithhisargument?Ifyes,pleasegiveadetailed

proofofthefollowingtheorem:Ifaschemeisstable,namely(3)

holdsforoneparticularnorm(e.g.theL2norm),thenitisalso

stableforanyothernorm.Ifnot,pleaseexplainthemistake

madebyStudentA.

6.Wehavethefollowing3PDEs

(4)ut+Aux=0,

(5)ut+Bux=0,