丘成桐大学生数学竞赛,应用数学与概率统计,考纲

2023年12月10日发(作者：能刷数学试卷的app)

ComputationalMathematics,AppliedMathematics,ProbabilityandStatistics(theseconddraft)ComputationalMathematicsInterpolationandapproximationPolynomialinterpolationandleastsquareapproximation;trigonometricinterpolationandapproximation,fastFouriertransform;approximationsbyrationalfunctions;earequationsolversConvergenceofiterativemethods(bisection,secantmethod,Newtonmethod,otheriterativemethods)forbothscalarequationsandsystems;systemsandeigenvalueproblemsDirectsolvers(Gausselimination,LUdecomposition,pivoting,operationcount,bandedmatrices,round-offerroraccumulation);iterativesolvers(Jacobi,Gauss-Seidel,successiveover-relaxation,conjugategradientmethod,multi-gridmethod,Krylovmethods);numericalsolutionsforeigenvaluesandeigenvectorsNumericalsolutionsofordinarydifferentialequationsOnestepmethods(TaylorseriesmethodandRunge-Kuttamethod);stability,accuracyandconvergence;absolutestability,longtimebehavior;multi-stepmethodsNumericalsolutionsofpartialdifferentialequationsFinitedifferencemethod;stability,accuracyandconvergence,Laxequivalencetheorem;finiteelementmethod,boundaryvalueproblemsReferences:[1],ElementaryNumericalAnalysis,analgorithmicapproach,McGraw-Hill,2000.[2]n,MatrixComputations,thirdedition,JohnsHopkinsUniversityPress,1996.[3],,SolvingOrdinaryDifferentialEquations,Springer,1993.[4]sson,H.-,TimeDependentProblemsandDifferenceMethods,JohnWileySons,1995.[5],AnAnalysisoftheFiniteElementMethod,secondedition,Wellesley-CambridgePress,dMathematicsODEwithconstantcoefficients;NonlinearODE:criticalpoints,phasespace&stabilityanalysis;Hamiltonian,gradient,conservativeODE’usofVariations:Euler-LagrangeEquations;BoundaryConditions,parametricformulation;optimalcontrolandHamiltonian,rderpartialdifferentialequations(PDE)andmethodofcharacteristics;Heat,wave,andLaplace’sequation;Separationofvariablesandeigen-functionexpansions;Stationaryphasemethod;HomogenizationmethodforellipticandlinearhyperbolicPDEs;HomogenizationandfrontpropagationofHamilton-Jacobiequations;nces:a,ElementaryDifferentialEquations,Wiley,,IntroductiontoCalculusofVariationsandItsApplications,Chapman&Hall,m,“LinearandNonlinearWaves”,John-WileyandSons,,“PrinciplesofAppliedMathematics”,Addison-Wesley,usan,P-LLions,colaou,“AsymptoticAnalysisforPeriodicStructures”,North-HollandPublishingCo,,,k,“Homogenizationofdifferentialoperatorsandintegralfunctions”,Springer,,“AnIntroductiontoFrontsinRandomMedia”,SurveysandTutorialsinAppliedMathSciences,No.5,Springer,ilityRandomVariables;ConditionalProbabilityandConditionalExpectation;MarkovChains;TheExponentialDistributionandthePoissonProcess;Continuous-TimeMarkovChains;RenewalTheoryandItsApplications;QueueingTheory;ReliabilityTheory;BrownianMotionandStationaryProcesses;nce:,IntroductiontoProbabilityModelsStatisticsDistributionTheoryandBasicStatisticsFamiliesofcontinuousdistributions:Chi-sq,t,F,gamma,beta;Familiesofdiscretedistributions:Multinomial,Poisson,negativebinomial;Basicstatistics:Mean,median,quantiles,orderstatisticsLikelihoodprincipleLikelihoodfunction,parametricinference,sufficiency,factorizationtheorem,ancillarystatistic,conditionallikelihood,gNeyman-Pearsonparadigm,nullandalternativehypotheses,simpleandcompositehypotheses,typeIandtypeIIerrors,power,mostpowerfultest,likelihoodratiotest,Neyman-PearsonTheorem,monotonelikelihoodratio,uniformlymostpowerfultest,tionParameterestimation,methodofmoments,maximumlikelihoodestimation,unbiasedness,quadraticandothercriterionfunctions,Rao-BlackwellTheorem,Fisherinformation,Cramer-Raobound,confidenceinterval,anStatisticsPrior,posterior,conjugatepriors,BayesianlossNonparametricstatisticsPermutationtest,permutationdistribution,rankstatistics,Wilcoxon-Mann-Whitneytest,log-ranktest,sionLinearregression,leastsquaresmethod,Gauss-MarkovTheorem,logisticregression,maximumlikelihoodLargesampletheoryConsistency,asymptoticnormality,chi-sqapproximationtolikelihoodratiostatistic,large-samplebasedconfidenceinterval,ncesCasella,ger,R.L.(2002).StatisticalInference(2ndEd.)DuxburyPress.茆诗松，程依明，濮晓龙，概率论与数理统计教程（第二版），高等教育出版社，2008.陈家鼎，孙山泽，李东风，刘力平，数理统计学讲义，高等教育出版社，2006.郑明，陈子毅，汪嘉冈，数理统计讲义，复旦大学出版社，2006.陈希孺，倪国熙，数理统计学教程，中国科学技术大学出版社，2009.