简介：Leta,b,cberelativelyprimepositiveintegerssuchthata2+b2=c2.Je′smanowicz’conjectureonPythagoreannumbersstatesthatforanypositiveintegerN,theDiophantineequation(aN)x+(bN)y=(cN)zhasnopositivesolution(x,y,z)otherthanx=y=z=2.Inthispaper,weprovethisconjectureforthecasethataorbisapowerof2.

简介：Generalizedcase-cohortdesignhasbeenprovedtobeacost-effectivewaytoenhancetheefficiencyoflargeepidemiologicalcohort.Inthisarticle,weproposeaninferenceprocedureforestimatingtheunknownparametersinCox’sproportionalhazardsmodelingeneralizedcase-cohortdesignandestablishanoptimalsamplesizeallocationtoachievethemaximumpoweratagivenbudget.Thefinitesampleperformanceoftheproposedmethodisevaluatedthroughsimulationstudies.TheproposedmethodisappliedtoarealdatasetfromtheNationalWilm’sTumorStudyGroup.

简介：Inthispaper,westudyoptimalrecovery(reconstruction)offunctionsonthesphereintheaveragecasesetting.WeobtaintheasymptoticordersofaveragesamplingnumbersofaSobolevspaceonthespherewithaGaussianmeasureintheLd-1q(S)metricfor1≤q≤∞,andshowthatsomeworst-caseasymptoticallyoptimalalgorithmsarealsoasymptoticallyoptimalintheaveragecasesettingintheLdq(S-1)metricfor1≤q≤∞.

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