美國數(shù)學(xué)建模輔導(dǎo)課

一、美國數(shù)學(xué)建模競賽介紹

MCM:MathematicalContestinModelingICM:InterdisciplinaryContestinModeling主辦：美國運籌學(xué)和管理科學(xué)協(xié)會、工業(yè)及應(yīng)用數(shù)學(xué)協(xié)會、美國數(shù)學(xué)協(xié)會以及國家安全局贊助，美國數(shù)學(xué)及應(yīng)用協(xié)會承辦時間（美國東部）：每年二月份的第二個周末，中國時間：周四晚上8：00開始，周一晚上8：00結(jié)束

題目：MCM競賽一般有兩個問題，都是根據(jù)各行各業(yè)的專家建議精選出來的實際問題。

問題的答案不是唯一的，完全是開放的，

主要考察參賽者處理問題的巧妙性，靈活性，新穎性。ICM只有一道題，涵蓋數(shù)學(xué)、環(huán)境科學(xué)、環(huán)境工程及資源管理等概念的建模問題。評審標(biāo)準(zhǔn)及評審程序四個等級：Outstanding(特等獎），Meritorious(一等獎)，HonorableMention(二等獎)，SuccessfulParticipation(成功參賽獎).三輪審稿Initialround：至少兩人審稿重點看summary：文字通順，語言精練、準(zhǔn)確；包括解題的方法和結(jié)論；摘要和全文結(jié)論要一致。不合格者被淘汰。Screeninground：由第二組人審稿

重點檢查是否滿足進(jìn)入下一輪的最低要求：文字通順，語法正確仍然是必須的；文章前后的一致性更加重要；清楚地說明哪些是自己做的，哪些是引用的，引用資源的出處要明確。Finalrounds：由第三組人審稿，每篇文章多人閱讀，每次審閱時間延長；按照加分元素篩選進(jìn)入下一輪的參賽隊，被留下時間越長，得最高獎的可能性越高。

重點除了summary以外，全篇的一致性更加重要，每一解題部分都要經(jīng)過審查。可以加分的元素1、文字優(yōu)美，流暢，語言豐富，表達(dá)清晰，方程、標(biāo)點引用格式正確。2、文摘的要求：包括問題的簡單概括，建模使用的模型和方法，所得的結(jié)論。3、合理的假設(shè)以及必要的解釋。4、多個模型由淺入深地對問題求解5、對模型使用的方法條件及結(jié)論進(jìn)行分析6、模型的敏感度和穩(wěn)定度分析7、當(dāng)可引用的資源很多時，要對你引用的資源進(jìn)行分析8、理論分析和數(shù)字模擬相結(jié)合，對于理論結(jié)果要有數(shù)字模擬加以說明9、圖表的合理使用，讓閱卷人對圖表的解釋和討論一目了然10、模型的優(yōu)勢、弱勢分析二、論文的組成部分*Summary/Abstract1、簡單的背景介紹2、建立和使用的模型由淺入深地介紹；求解模型的方法，包括解析解、數(shù)值解或者圖解。3、主要的結(jié)論和建議*Overview/Analysisoftheproblem主要是對問題的理解做一解釋，對問題的解答建立在哪些前提和背景下*Assumptions要對假設(shè)做些合理性的解釋*Designofthemodels1、Effectsof……2、Determinationof……3、Equationsof……*Sensitivityandstabilityanalysis1、Sensitivityoftheequation2、Stabilityofthemodeltotheparameter*Conclusions總結(jié)建立模型的條件、方法；使用模型的注意事項；模型推廣的可能性，模型存在的問題。*Strengthsandweaknesses1、參數(shù)不是任意給出的，而是用….方法得到的2、模型運算時間很短，具有使用性*References順序與文章中引用的順序要一致三、論文目錄舉例Example1Contents

Introduction................................................................22.ThePlan........................................................................2Objectives....................................................................34.Definingthe“SweetSpot”........................................45.ModelA-CenterofPercussion:Physics’SweetSpot.......55-b.ExampleI.......................................................................56.ModelB:BrodyPowerModel..........................................66-a.FactsandAssumptions..................................................66-b.GeneralForm.................................................................76-c.ExampleII........................................................................96-d.Torque.............................................................................97.Discussion..........................................................................107-a.“Corking”........................................................……….107-b.CorkModelAugmentation..........................................117-c.Aluminumvs.Wood.....................................................137-d.AluminumModelAugmentation.................................138.SensitivityAnalysis.............................................................159.Conclusion..........................................................................1910.References........................................................................22Example2ContentsAbstract.............................................................................22.Statementoftheproblemandapproach.............……….52.1SurveyofPreviousResearch:EnvironmentalCriminology....................................................................62.2Assumptions.................................................................62.3PropositionsandFoundation.....................................8Methods........................................................................113.1ConstructionoftheMap–GeographicalMethod....113.2StaticandDynamic–RiskIntensityMethod..........18SimulationResultsandDiscussion................................304.1ResultsoftheGeographicalMethod...........................314.2ResultsoftheRiskIntensity,“Static”and“Dynamic”Method............................................................................364.3Discussion................................................................374.3.1SensitivityandRobustnessTesting.....................374.3.2AccuracyofthePrediction.................................404.3.3CombinationoftheTwoMethods.....................40StrengthandWeaknessoftheModel...........................416.ConclusionandRecommendation..............................41Appendices..........................................43ABibliography........................................43BData.......................................................45CCode......................................................46

Example3Contents1Introduction………..………12TheModels…………………32.1ASimpli?edModel...................………....32.2AnIntermediateModel...................……...32.3ACongestionModel..................…...…….42.4ExtendingtheModelUsingComputerSimulation....42.4.1SimulationAssumptions........................52.4.2Limitations....................……….....73AnalyzingtheModels……..………….83.1TheSimplestModel........................83.2IntermediateModel........................83.3CongestionModel.........................93.4SimulationResults........................124Conclusion…………...15四、摘要舉例Abstract1Theadventofcomputerandtechnologicalprogresshasintroducedanewstageinthedevelopmentofcriminology.Investigatorscannowusecomputationaltechniquesofgeographicprofilinginordertodeterminethepatternsofmovementoftheirsuspects.

Weproposeamodelthataimstopredictareaswithhighprobabilityofbeingthenextonthecriminal‘stargetlist.Wehaveassumedthatserialcrimesareinstrumentalratherthanexpressive,thusensuringthatthecriminalfollowsapredictablepatternofmovement.

Wealsoassumethatthispatternischaracterizedbyacertainstabilityandcontinuitywhichfacilitatesacorrelationwiththeactionsofothercriminalsinthatarea.Ourmodelfirstusesaninitial“geographicalmethod”whichreducestheareasunderconsiderationbasedonparameterssuchaslocationcoordinates,area,populationandcriminalrate,aswellasthehistoryandpsychologicalvalueThisinputisusedtodeterminetheshapeofaGaussian2Dfunctionshowingthedistributionoftheareaswiththehighestprobabilityofbecomingthelocationofafuturecrime.Weimprovetheseresultsbyusingtheriskintensitymethod,acombinationoftwoschemes,a“static”anda“dynamic”.Thestaticmethodconsistsoffirstgeneratingtheriskintensitiesofdifferentlocationsbasedonvariablessuchascrimeratesanddistancesfromtheanchorpoint,byusingtoolssuchasthedistance-decayfunction.Wethenassigncrimecoefficients,whichindicatetheextenttowhichthecrimecanbecategorizedasmurder,rape,arsonorrobbery.Inthedynamicmodel,wecategorizethestaticparametersintohomotypic,heterotypicandcumulativetypesbycomputingthemeanandcovariancematrixoftheseparameters.Weapplydifferentalgorithms:logisticregression,linearregressionandnearestneighboralgorithmrespectivelytothesetypesandthenweighthemdifferentlytoobtainaparameterprobability.Thisisthencombinedwiththeresultsofthestaticprocesstogeneratetheprobabilityofacrimeatacertainlocation.Wetestedourmodelusingexamplesfromdifferentcategoriesofserialcrimes:robbery,murder,arson,whichdemonstrateddistinctcriminalpatterns.Thesurfacesgeneratedusingthegeographicmethodandthefinalpredictedprobabilitiesgenerallyagreedwithourexpectationsofareaswherethecriminalwillattackagain.Thetestforsensitivitysuggestedthatparameterssuchascrimerateorpopulationdensity(areaandlocation)arewelltakenintoconsiderationbyourmodel.Smallchangesinlocation,however,affectedtoasignificantextentourresults,probablybecausethedifferencesincoordinatesofthelocationswerenotlargetobeginwith.Summary2Ourgoalwastodesignamodelthatcouldaccountforthedynamicsofvehiclesinatraf?ccircle.Wemainlyfocusedontherateofentryintothecircletodeterminethebestwaytoregulatetraf?c.Weassumedthatvehiclescirculateinasinglelaneandthatonlyincomingtraf?ccanberegulated.Forourmodel,theadjustableparametersweretherateofentryintothequeue,therateofentryintothecircle(servicerate),themaximumcapacityofthetraf?ccircle,andtherateofdeparturefromthecircle(departurerate).Vehiclesfirstenterthequeuefromtheoutsideworld,thenenterthetrafficcirclefromthequeue,andlastlyexitthetrafficcircletotheoutsideworld.Wemodeledboththeservicerateandthedeparturerateasdependentonthenumberofvehiclesinsidethetrafficcircle.Inaddition,werancomputersimulationstohaveavisualrepresentationofwhathappensintrafficcirclesduringdifferentsituations.Inmanycases,wefoundthatafastserviceratewastheoptimalwaytomaintaintrafficflow.However,whenthecirclebecamemoreHeavilytraf?cked,aslowerserviceratebetteraccommodatedthetraf?c,indicatingthatatraf?clightshouldbeused.withvariabletimingdependingontheexpectedamountoftraf?c.Themainadvantageofourapproachwasthatthemodelwasverysimpleandallowedustoclearlyseethedynamicsofthesystem.Also,thecomputersimulationsthatweranprovidedmorein-depthinformationabouttrafficflowunderconditionsthatthemodelcouldnoteasilyshowvisualobservationofthetraffic.Somedisadvantagestoourapproachwerethatwecouldnotanalyzetheaffectsofmultiplelanesnorstoplightsthatcontrolledtheflowoftrafficinsidethecircle.Inaddition,wehadnowayofanalyzingsingularitiesinthesituation,suchasvehiclesthatdrivefasterorslowerthantherestofthetrafficcircleandpedestrians.Summary3Ourbasicmodelcanbedividedintotwoparts.Thefirstpartistofindahalf-pipeshapewhichcanmaximize“VerticalAir”;inthesecondpart,weadapttheshapetomaximizethepossibletotalangleofrotation.Intheextendedmodel,weanalyzethesnowboarder’ssubjectiveinfluenceon“VerticalAir”andthetotaldegreeofrotation.Finally,wediscussthefeasibilityandthetrade-offofbuildingapracticalcourse.Inthefirstpartofourbasicmodel,weobtainadifferentialequationofElostbasedonforceanalysisandEnergyConservation.ThenwederivetherepresentationofElostbysolvingtheequation,whichisafunctional.Inthesecondpartofthebasicmodel,wederivetherepresentationoftheinitialangularmomentumbeforethefly,anddiscussthefactorsinfluencingit.Inourextendedmodel,wecalculatethe“VerticalAir”whentakingthesnowboarder’ssubjectiveinfluenceintoaccount.Weusenumericalmethodtosolvethemodelandcompareanalyticalresultswithrealityandvalidateourmethodtobecorrectandrobust.Weanalyzedtheeffectsoffactorssuchaswidth,heightandgradientangleofhalf-pipeon“verticalAir”bycontrollingotherfactors.UsingGeneAlgorithm,wegloballyoptimizedthecourseshapetoprovidelargest“verticalair”andtotaldegreeofrotation.Implementingahybridscoringsystemastheobjectivefunction,weoptimizethecourseshapetoa“half-blood”shape.Yet,themodelhasn'tprovidedanalyticsolutionofoptimalcourse.五、寫作練習(xí)1、本文是