math

BranchofMathematics

1.HistoryofMathematics

2.MathematicalLogicandFundamentalsofMathematics

a:deductivelogic(alsocalledsymboliclogic),b:prooftheory(alsocalledmetamathematics),c:recursiontheory,d:modeltheory,e:axiomsettheory,f:mathematicalfoundation,g:Mathematicallogicandothersubjectsoffoundationsofmathematics.

3.Numbertheory

a:elementarynumbertheory,b:analyticnumbertheory,c:algebraicnumbertheory,d:transcendentnumbertheory,e:Diophantineapproximation,F:geometryofnumbers,g:probabilitynumbertheory,h:computationalnumbertheory,i:numbertheoryandothersubjects.

4.Algebra

a:linearalgebra,b:grouptheory,c:fieldtheory,d:Liegroup,e:Liealgebra,f:Kac-Moodyalgebra,g:ringtheory(includingcommutativeringandcommutativealgebra,associativeringandassociativealgebra,non-associativeringandnon-associativealgebra,etc.),h:modulartheory,i:latticetheory,j:universalalgebratheory,K:categorytheory,l:homologyalgebra,m:algebraKtheory,n:differentialalgebra,o:algebraiccodingtheory,p:othersubjectsofalgebra.

5.AlgebraicGeometry

6.Geometry

a:BasicGeometry,b:Euclideangeometry,c:non-Euclideangeometry(includingRiemanniangeometry,etc.),d:sphericalgeometry,e:vectorandtensoranalysis,f:affinegeometry,g:projectivegeometry,h:Differentialgeometry,i:fractalgeometry,j:computationalgeometry,k:geometryothersubjects.

7.Topology

a:pointsettopology,b:algebraictopology,c:homotopytheory,d:low-dimensionaltopology,E:homologytheory,f:dimensionalitytheory,g:topologyonlattice,h:fiberbundletheory,i:geometrictopology,j:singularitytheory,k:differentialtopology,l:topologyandothersubjects.

8.Mathematicalanalysis

a:Differentiation,b:Integral,c:Seriestheory,d:Othersubjectsofmathematicalanalysis.

9.Non-standardanalysis

10.Functiontheory

a:Realvariablefunctiontheory,B:theoryoffunctionsofsingleandcomplexvariables,c:theoryoffunctionsofmultiplecomplexvariables,d:theoryoffunctionapproximation,e:harmonicanalysis,f:complexmanifold,g:specialfunctiontheory,h:functiontheoryandothersubjects.

11.Ordinarydifferentialequations

a:qualitativetheory,b:stabilitytheory.c:Analyticaltheory,d:othersubjectsofordinarydifferentialequations.

12.Partialdifferentialequations

a:ellipticpartialdifferentialequations,b:hyperbolicpartialdifferentialequations,c:parabolicpartialdifferentialequations,D:Non-linearpartialdifferentialequations,e:Partialdifferentialequationsandotherdisciplines.

13.Dynamicalsystem

a:Differentialdynamicalsystem,b:Topologicaldynamicalsystem,c:Complexdynamicalsystem,d:Otherdisciplinesofdynamicalsystem.

14.Integralequation

15.Functionalanalysis

a:Linearoperatortheory,B:variationalmethod,c:topologicallinearspace,d:Hilbertspace,e:functionspace,f:Banachspace,g:operatoralgebrah:measureandintegral,i:generalizedfunctiontheory,j:Non-linearfunctionalanalysis,k:Otherdisciplinesoffunctionalanalysis.

16.ComputationalMathematics

a:interpolationandapproximationtheory,b:numericalsolutionofordinarydifferentialequations,c:numericalsolutionofpartialdifferentialequations,d:Numericalsolutionofintegralequation,e:Numericalalgebra,f:Discretizationmethodforcontinuousproblems,g:Randomnumericalexperiment,h:Erroranalysis,i:Computationalmathematicsandothersubjects.

17.ProbabilityTheory

a:geometricprobability,b:probabilitydistribution,c:limittheory,d:randomprocess(includingnormalprocessandStationaryprocess,pointprocess,etc.),e:Markovprocess,f:randomanalysis,g:martingaletheory,h:appliedprobabilitytheory(specificallyappliedtorelateddisciplines),i:probabilitytheoryandotherdisciplines.

18.MathematicalStatistics

a:samplingtheory(includingsamplingdistribution,samplingsurvey,etc.),b:hypothesisTest,c:non-parametricstatistics,d:analysisofvariance,e:correlationregressionanalysis,f:statisticalinference,g:Bayesianstatistics(includingparameterestimation,etc.),h:experimentaldesign,i:multivariateanalysis,j:statisticsJudgmenttheory,k:timeseriesanalysis,l:othersubjectsofmathematicalstatistics.

19.Appliedstatisticalmathematics

a:statisticalqualitycontrol,b:reliabilitymathematics,c:insurancemathematics,d:statisticalsimulation.

20.AppliedstatisticalmathematicsOtherdisciplines

21.OperationsResearch

a:linearprogramming,b:nonlinearprogramming,c:dynamicprogramming,d:combinatorialoptimization,e:parameterprogramming,f:integerprogramming,g:stochasticprogramming,h:queuingTheory,i:gametheory(alsocalledgametheory),j:inventorytheory,k:decisiontheory,l:searchtheory,m:graphtheory,n:overallplanningtheory,o:optimization,p:otherdisciplinesofoperationsresearch.

22.Combinatorics

23.FuzzyMathematics

24.QuantumMathematics

25.Appliedmathematics(specificapplicationintorelatedsubjects)

26.Mathematicsothersubjects

DevelopmentHistory

Mathematics(Chinesepinyin:shùxué;Greek:μαθηματικ;English:mathematicsormaths),itsEnglishisderivedfromtheancientGreekμθημα(máthēma),withlearning,Knowledge,andscience.AncientGreekscholarsregardeditasthestartingpointofphilosophy,the"foundationoflearning."Inaddition,thereisamorenarrowandtechnicalmeaning-"mathematicsresearch".Evenwithinitsetymology,themeaningofitsadjectives,whichisrelatedtolearning,isalsousedforindexlearning.

ThepluralforminEnglish,andthepluralforminFrenchwith-es,formmathématiques,whichcanbetracedbacktotheneutralpluralinLatin(mathematica),translatedbyCicerofromthepluralinGreekταμαθηματικά(tamathēmatiká).

InancientChina,mathematicswascalledarithmetic,alsocalledarithmetic,andfinallychangedtomathematics.ArithmeticinancientChinaisoneofthesixarts(called"number"inthesixarts).

Mathematicsoriginatedfromtheearlyproductionactivitiesofmankind.TheBabylonianshaveaccumulatedcertainmathematicalknowledgesinceancienttimesandcanapplypracticalproblems.Fromtheperspectiveofmathematics,theirmathematicalknowledgeisonlyobtainedfromobservationandexperience,withoutcomprehensiveconclusionsandproofs,buttheymustfullyaffirmtheircontributionstomathematics.

Theknowledgeandapplicationofbasicmathematicsisanindispensablepartofthelifeofindividualsandgroups.TherefinementofitsbasicconceptscanbeseeninancientmathematicstextsinancientEgypt,MesopotamiaandancientIndia.Sincethen,itsdevelopmenthascontinuedtomakesmallprogress.Butthealgebraandgeometryatthattimehaveremainedindependentforalongtime.

Algebraisarguablythemostwidelyaccepted"mathematics".Itcanbesaidthatthefirstmathematicsthateveryonecomesintocontactwithisalgebrasincetheylearntocountwhentheyareyoung.Andmathematicsisasubjectthatstudies"number",andalgebraisalsooneofthemostimportantcomponentsofmathematics.Geometryisthebranchofmathematicsthatwasfirststudiedbypeople.

UntiltheRenaissanceinthe16thcentury,DescartesfoundedAnalyticGeometry,linkingthealgebraandgeometrythatwerecompletelyseparateatthattime.Sincethen,wecanfinallyusecalculationstoprovegeometrictheorems;atthesametime,wecanalsousegraphicstovisuallyrepresentabstractalgebraicequationsandtrigonometricfunctions.Later,moresubtlecalculuswasdeveloped.

Currently,mathematicsincludesmultiplebranches.TheBourbakiSchoolofFrance,foundedinthe1930s,believesthatmathematics,atleastpuremathematics,isthetheoryofstudyingabstractstructures.Structureisadeductivesystembasedoninitialconceptsandaxioms.Theybelievethatmathematicshasthreebasicparentstructures:algebraicstructure(group,ring,field,lattice,...),orderstructure(partialorder,totalorder,...),topologicalstructure(neighborhood,limit,connectivity,dimension,……).

Mathematicsisusedinmanydifferentfields,includingscience,engineering,medicineandeconomics.Theapplicationofmathematicsinthesefieldsisgenerallyreferredtoasappliedmathematics,andsometimesitwillalsostimulatenewmathematicaldiscoveriesandpromotethedevelopmentofnewmathematicsdisciplines.Mathematiciansalsostudypuremathematics,thatis,mathematicsitself,withoutanypracticalapplicationasthegoal.Althoughthereisalotofworkstartingwiththestudyofpuremathematics,youmayfindsuitableapplicationslater.

Specifically,therearesub-fieldsusedtoexplorethelinksbetweenthecoreofmathematicsandotherfields:fromlogic,settheory(mathematicsfoundation),tomathematicsbasedondifferentscientificexperiences(appliedmathematics),Withmorerecentresearchonuncertainty(chaos,fuzzymathematics).

Intermsofverticality,theexplorationintherespectivefieldsofmathematicshasbecomemoreandmorein-depth.

Definition

Aristotledefinedmathematicsas"quantitativemathematics",andthisdefinitionwasuntilthe18thcentury.Sincethe19thcentury,mathematicalresearchhasbecomemoreandmorerigorous,beginningtoinvolveabstracttopicssuchasgrouptheoryandprojectiongeometrythathavenoclearrelationshipwithquantityandmeasurement.Mathematiciansandphilosophershavebeguntoproposevariousnewdefinitions.Someofthesedefinitionsemphasizethedeductivenatureofalotofmathematics,someemphasizeitsabstractness,andsomeemphasizecertaintopicsinmathematics.Evenamongprofessionals,thereisnoconsensusonthedefinitionofmathematics.Thereisevennoconsensusonwhethermathematicsisartorscience.[8]Manyprofessionalmathematiciansarenotinterestedinthedefinitionofmathematics,orthinkitisundefinable.Somejustsay,"Mathematicsisdonebymathematicians."

Thethreemaintypesdefinedbymathematicsarecalledlogicians,intuitionists,andformalists,eachofwhichreflectsadifferentphilosophySchoolofThought.Thereareseriousproblems,noonegenerallyacceptsit,andnoreconciliationseemsfeasible.

TheearlydefinitionofmathematicallogicwasBenjaminPeirce's"SciencethatDrawsNecessaryConclusions"(1870).InPrincipiaMathematica,BertrandRussellandAlfredNorthWhiteheadproposedaphilosophicalprogramcalledlogicismandtriedtoprovethatallmathematicalconcepts,statements,andprinciplescanbedefinedandprovedbysymboliclogic.ThelogicaldefinitionofmathematicsisRussell's"Allmathematicsissymboliclogic"(1903).

Thedefinitionofintuitionism,fromthemathematicianL.E.J.Brouwer,toidentifymathematicswithcertainspiritualphenomena.Anexampleoftheintuitionisticdefinitionis"mathematicsismentalactivityconstructedoneaftertheother."Thecharacteristicofintuitionismisthatitrejectssomemathematicalideasthatareconsideredvalidaccordingtootherdefinitions.Inparticular,althoughothermathematicalphilosophiesallowobjectsthatcanbeproventoexist,eveniftheycannotbeconstructed,intuitionismonlyallowsmathematicalobjectsthatcanbeactuallyconstructed.

Formalismdefinesmathematicswithitssymbolsandoperatingrules.HaskellCurrysimplydefinesmathematicsas"thescienceofformalsystems."[33]Theformalsystemisasetofsymbols,ortokens,andtherearerulesthattellhowtokensarecombinedintoformulas.Intheformalsystem,thewordaxiomhasaspecialmeaning,whichisdifferentfromtheordinarymeaningof"self-evidenttruth".Inaformalsystem,anaxiomisacombinationoftokenscontainedinagivenformalsystem,withouttheneedtousetherulesofthesystemtoderive.

Structure

Manymathematicalobjectssuchasnumbers,functions,andgeometryreflecttheinternalstructureofcontinuousoperationsorrelationsdefinedinthem.Mathematicsstudiesthepropertiesofthesestructures.Forexample,numbertheorystudieshowintegersarerepresentedinarithmeticoperations.Inaddition,thingswithsimilarpropertiesindifferentstructuresoftenhappen.Thismakesitpossibletodescribetheirstatethroughfurtherabstractionandthenuseaxiomsforatypeofstructure.Whatneedstobestudiedistofindoutwhatsatisfiestheseinallstructures.Thestructureofaxioms.Therefore,wecanlearnaboutgroups,rings,domains,andotherabstractsystems.Thesestudies(throughstructuresdefinedbyalgebraicoperations)canformthefieldofabstractalgebra.Becauseabstractalgebrahasgreatversatility,itcanoftenbeappliedtosomeseeminglyirrelevantproblems.Forexample,someancientrulerdrawingproblemsarefinallysolvedusingGaloistheory,whichinvolvesdomaintheoryandgroups.s.Anotherexampleofalgebratheoryislinearalgebra,whichmakesageneralstudyofvectorspacesinwhichtheelementshavequantityanddirectionality.Thesephenomenashowthatgeometryandalgebra,whichwereoriginallyconsideredtobeunrelated,areactuallystronglycorrelated.Combinatorialmathematicsstudiesenumeratethemethodsthatsatisfythenumericalobjectsofagivenstructure.

Space

ThestudyofspacecomesfromEuropeangeometry.Trigonometrycombinesspaceandnumber,andincludesthefamousPythagoreantheorem,trigonometricfunctions,etc.Today'sresearchonspacehasbeenextendedtohigher-dimensionalgeometry,non-Euclideangeometryandtopology.Numberandspaceplayimportantrolesinanalyticgeometry,differentialgeometryandalgebraicgeometry.Indifferentialgeometry,thereareconceptssuchasfiberbundlesandcalculationsonmanifolds.Inalgebraicgeometry,therearedescriptionsofgeometricobjectssuchasthesolutionsetofpolynomialequations,combiningtheconceptsofnumberandspace;thereisalsothestudyoftopologicalgroups,combiningstructureandspace.LiQunisusedtostudyspace,structureandchange.

Basics

Surfaceofrevolution(10sheets)

Mainarticle:Mathematicalbasics

TomakeitclearThefieldsofmathematicalfoundations,mathematicallogicandsettheoryweredeveloped.TheGermanmathematicianKantor(1845~1918)pioneeredsettheoryandboldlymarchedtowards"infinity",inordertoprovideasolidfoundationforallbranchesofmathematics,anditscontentisalsoquiterich.Thoughthasmadeaninestimablecontributiontothefuturedevelopmentofmathematics.

Settheoryhasgraduallypenetratedintovariousbranchesofmathematicsintheearly20thcenturyandhasbecomeanindispensabletoolinanalysistheory,measurementtheory,topologyandmathematicalsciences.Atthebeginningofthe20thcentury,themathematicianHilbertspreadCantor'sideasinGermany,callingsettheorythe"mathematician'sparadise"and"themostamazingproductofmathematicalthought."TheBritishphilosopherRussellpraisedCantor'sworkas"thegreatestworkthatthiseracanboast."

Logic

Mainarticle:Mathematicallogic

MathematicallogicfocusesonputtingmathematicsinoneOnasolidaxiomaticframework,andstudytheresultsofthisframework.Foritspart,itistheoriginofGödel'ssecondincompletenesstheorem,andthisisperhapsthemostwidelyspreadresultinlogic.Modernlogicisdividedintorecursiontheory,modeltheory,andprooftheory,anditiscloselyrelatedtotheoreticalcomputerscience.

Symbols

Mainarticle:Mathematicssymbols

MaybetheancientChinesecalculatoristheworld’sOneoftheearliestusedsymbolsoriginatedfromdivinationintheShangDynasty.

Mostofthemathematicalsymbolsweusetodaywerenotinventeduntilthe16thcentury.Priortothis,mathematicswaswritteninwords,whichwasanassiduousprogramthatwouldlimitthedevelopmentofmathematics.Today'ssymbolsmakemathematicseasierforpeopletooperate,butbeginnersoftenfeeltimidaboutthis.Itisextremelycompressed:afewsymbolscontainalotofinformation.Likemusicalnotation,today'smathematicalnotationhasacleargrammarandinformationcodesthataredifficulttowriteinotherways.

Rigorousness

Mathematicslanguageisalsodifficultforbeginners.Howtomakethesewordshavemoreprecisemeaningsthandailyexpressionsalsotroublesbeginners.Wordssuchasopenanddomainhavespecialmeaningsinmathematics.Mathematicaltermsalsoincludepropernounssuchasembryoandintegrability.Butthereisareasonforusingthesespecialsymbolsandproprietaryterms:mathematicsrequiresmoreprecisionthaneverydaylanguage.Mathematicianscallthisrequirementfortheaccuracyoflanguageandlogic"rigorous."

Mathematicsisauniversalmethodforhumanstostrictlydescribetheabstractstructureandpatternsofthings,anditcanbeappliedtoanyproblemintherealworld.Inthissense,mathematicsisaformalscience,notanaturalscience.Allmathematicalobjectsareessentiallyartificiallydefined.Theydonotexistinnature,butonlyinhumanthinkingandconcepts.Therefore,thecorrectnessofmathematicalpropositionscannotbeverifiedbyrepeatableexperiments,observations,ormeasurementslikenaturalsciences,suchasphysicsandchemistry,whosegoalistostudynaturalphenomena.Instead,theycanbedirectlyprovedbyrigorouslogicalreasoning.Oncetheconclusionisprovedthroughlogicalreasoning,thentheconclusioniscorrect.

Theaxiomatizationmethodofmathematicsisessentiallythedirectapplicationoflogicmethodsinmathematics.Intheaxiomsystem,allpropositionsareconnectedbyrigorouslogic.Startingfromtheoriginalconceptthatisdirectlyadoptedwithoutdefinition,otherderivedconceptsaregraduallyestablishedthroughthemeansoflogicaldefinition;startingfromtheaxiomthatisdirectlyadoptedasthepremisewithoutproof,andthelogicaldeductionmethodisusedtograduallyobtainfurtherTheconclusionisthetheorem;thenalltheconceptsandtheoremsarecombinedintoawholewithinternallogicalconnection,whichconstitutestheaxiomsystem.

Strictnessisaveryimportantandbasicpartofmathematicalproof.Mathematicianshopethattheirtheoremscanbededucedbasedonaxiomswithsystematicreasoning.Thisistoavoidrelyingonunreliableintuitionstoarriveatwrong"theorems"or"proofs,"andthissituationhasseenmanyexamplesinhistory.Thedegreeofrigorexpectedinmathematicsvarieswithtime:theGreeksexpectedcarefularguments,butinNewton’stimethemethodsusedwerelessrigorous.Newton'sdefinitiontosolvetheproblemwasnotproperlyhandleduntilthe19thcenturybymathematicianswithrigorousanalysisandformalproof.Mathematicianscontinuetoargueabouttherigorofcomputer-aidedproofs.Whenalargenumberofcalculationsaredifficulttoverify,theproofcanhardlybesaidtobeeffectiveandrigorous.

Quantity

Thestudyofquantitystartswithnumbers,startingwithfamiliarnaturalnumbersandintegersandrationalandirrationalnumbersdescribedinarithmetic.

Tobespecific:Duetotheneedofcounting,humansabstractnaturalnumbersfromrealthings,whicharethestartingpointofall"numbers"inmathematics.Naturalnumbersarenotclosedtosubtraction.Inordertoclosetosubtraction,weexpandthenumbersystemtointegers;tonotclosetodivision,andtoclosetodivision,weexpandthenumbersystemtorationalnumbers;forsquarerootoperations,weexpandthenumbersystemtorationalnumbers.Thesystemisextendedtoalgebraicnumbers(infact,algebraicnumbersareabroaderconcept).Ontheotherhand,thelimitoperationisnotclosed,andweextendthenumbersystemtorealnumbers.Finally,inordertopreventnegativenumbersfrombeingunabletoraiseevenpowersintherealnumberrange,weextendthenumbersystemtocomplexnumbers.Complexnumbersarethesmallestalgebraiccloseddomainscontainingrealnumbers.Weperformfourarithmeticoperationsonanycomplexnumber,andthesimplificationresultsareallcomplexnumbers.

Anotherconceptrelatedto"quantity"isthe"potential"ofinfinitesets,whichleadstothecardinalnumberandanotherconceptofinfinityafterwards:theAlephnumber,whichallowstheinfinitesetbetweenThesizecanbemeaningfullycompared.

ABriefHistory

ABriefHistoryofWesternMathematics

Theevolutionofmathematicscanberegardedasthecontinuousdevelopmentofabstraction,ortheextensionofthesubjectmatter,whiletheEastandWestCulturehasalsoadoptedadifferentperspective.Europeancivilizationhasdevelopedgeometry,whileChinahasdevelopedarithmetic.Thefirstabstractedconceptisprobablythenumber(Chinesecomputingchip).Therecognitionofsomethingsimilarbetweentwoapplesandtwoorangesisabreakthroughinhumanthinking.Inadditiontoknowinghowtocountthenumberofactualobjects,prehistorichumansalsoknowhowtocountthenumberofabstractconcepts,suchastime—days,seasons,andyears.Arithmetic(addition,subtraction,multiplication,anddivision)alsoarisesnaturally.

Furthermore,youneedwritingorothersystemsthatcanrecordnumbers,suchasFumuortheChipusedbytheIncas.Therehavebeenmanydifferentcountingsystemsinhistory.

Inancienttimes,themainprincipleinmathematicswastostudyastronomy,thereasonabledistributionoflandandfoodcrops,taxationandtraderelatedcalculations.Mathematicsisformedtounderstandtherelationshipbetweennumbers,tomeasuretheland,andtopredictastronomicalevents.Theseneedscanbesimplysummarizedasmathematicalresearchonquantity,structure,spaceandtime.

WesternEuropewentthroughtheRenaissanceerafromancientGreecetothe16thcentury.Elementaryalgebraandtrigonometryaregenerallycomplete,buttheconceptoflimithasnotyetappeared.

TheemergenceoftheconceptofvariablesinEuropeinthe17thcenturymadepeoplebegintostudytheinterrelationshipsbetweenchangingquantitiesandthemutualtransformationsbetweenfigures.Duringtheestablishmentofclassicalmechanics,themethodofcalculuscombinedwithgeometricprecisionwasinvented.Withthefurtherdevelopmentofnaturalscienceandtechnology,thefieldsofsettheoryandmathematicallogic,whichareproducedtostudythefoundationofmathematics,havealsobeguntodevelopslowly.

ABriefHistoryofChineseMathematics

Mainarticle:HistoryofChineseMathematics

Theancientnameofmathematicsisarithmetic.ItisanimportantsubjectinancientChinesescience.AccordingtoancientChinesemathematicsThecharacteristicsofdevelopmentcanbedividedintofiveperiods:budding;formationofthesystem;development;prosperityandtheintegrationofChineseandWesternmathematics.

Related

ManyoftheresearchresultsofancientChinesearithmetichavealreadyconceivedideasandmethodsthatwereonlyinvolvedinWesternmathematics.Inmoderntimes,therearealsomanyworld-leadingmathematicsresearchresultsbasedonChineseNamedbythemathematician:

[LiShanlan'sIdenticalEquation]TheresearchresultsofthemathematicianLiShanlanonthesumofseries,Itisnamed"LiShanlan'sidentity"(orLi'sidentity)intheworld.

[FahrenheitTheorem]MathematicianHuaLuogeng’sresearchresultsoncompletetrigonometricsumsarecalled"FahrenheitTheorem”;inaddition,themethodheproposedwithmathematicianWangYuanfortheapproximatecalculationofmultipleintegralsisknowninternationallyasthe“Hua-WangMethod”.

[Su’sCone]MathematicianSuBuqing’sresearchachievementsinaffinedifferentialgeometryareinternationallyItwasnamed"Su'sCone".

[Xiong'sinfiniteorder]MathematicianXiongQinglai'sresearchresultsonwholefunctionsandmeromorphicfunctionsofinfiniteorderItishailedas"Xiong'sInfiniteClass"bytheinternationalmathematicscircle.

[Representatives]TheresearchresultsofthemathematicianChenXingshenonindicativecategoriesareinternationallyknownas"Presentationalcategory".

[Zhou'sCoordinates】MathematicianZhouWeiliang’sresearchinalgebraicgeometryTheresultiscalled"Zhou'sCoordinates"bytheinternationalmathematicscircle;therearealso"Zhou'sTheorem"and"Zhou'sRing"namedafterhim.

[WuThemethod]ThemethodofthemathematicianWuWenjunonthemechanicalproofofgeometrictheoremsisinternationallyknownasthe"Wu'smethod";thereisalsothe"Wu'sformula"namedafterhim.p>

[Wang’sParadox】MathematicianWangHao’spropositiononmathematicallogicwasInternationallydefinedas"Wang’sParadox."

[Korot'sTheorem】MathematicianKeZhao'squestionaboutCarterTheresearchresultsofLan’sproblemarecalled"Kot'stheorem"bytheinternationalmathematicscommunity;inaddition,theresearchresultsofhisandmathematicianSunQiinnumbertheoryarecalled"Ke-SunConjecture"intheworld.

[Chen’sTheorem]ThepropositionputforwardbymathematicianChenJingruninthestudyofGoldbach’sconjectureishailedas"Chen’sTheorem"bytheinternationalmathematicscommunity.

[Yang-ZhangTheorem]TheresearchresultsofmathematiciansYangLeandZhangGuanghouinfunctiontheoryarecalled"Yang-ZhangTheorem"internationally.".

[Lu’sConjecture]MathematicianLuQikeng’sresearchresultsonmanifoldswithconstantcurvatureareknowninternationally"Lu’sConjecture".

[Xia’sInequality]MathematicianXiaDaoxing’sresearchresultsonfunctionalintegralsandinvariantmeasuretheoryarecalled"Xia’sinequality".

[Jiang’sspace]MathematicianJiangBoju’sresearchresultsonthecalculationofNielsennumbershavebeenrecognizedinternationally.Theaboveisnamed"Jiang'sSpace";thereisalso"Jiang'sSubgroup"namedafterhim.

[Hou'sTheorem】MathematicianHouZhenting’sresearchresultsonMarkovprocesseshavebeennamed"Hou'sTheorem"internationally.

[Zhou'sguessb>]MathematicianZhouHaizhong’sresearchresultsonthedistributionofMersenneprimenumbersareinternationallynamed"Zhou’sConjecture".

[Wang’sTheorem]MathematicianWangXutang’sresearchresultsonpointsettopologyarehailedas"Wang’sTheorem"bytheinternationalmathematicscommunity.

[Yuan"Yuan'sLemma"]MathematicianYuanYaxiang’sresearchresultsinnonlinearprogramminghavebeennamed"Yuan'sLemma"internationally.

【Jing’soperator】MathematicianJingNaihuan’sresearchachievementsinsymmetricfunctionsGuoisnamed"Jing'sOperator"internationally.

[Chen’sGrammar]TheresearchresultsofmathematicianChenYongchuanincombinatoricswerenamed"Chen'sGrammar".

MathematicalQuotes

ForeignObjects

Everythingiscounted.——Pythagoras

Geometryhasnoking'sway.——Euclidean

MathematicsisthewordsusedbyGodtowritetheuniverse.——Galileo

Iamdeterminedtogiveupthatmereabstractgeometry.Thatistosay,nolongerconsiderquestionsthatareonlyusedtopracticethinking.Ididthistostudyanotherkindofgeometry,thatis,geometrythataimstoexplainnaturalphenomena.——Descartes(ReneDescartes,1596~1650)

Mathematiciansarealltryingtodiscoversomeorderoftheprimenumbersequenceonthisday.Wehavereasontobelievethatthisisamystery,andthehumanmindcanneverinfiltrate.——Euler

Somebeautifultheoremsinmathematicshavesuchcharacteristics:Theyareeasytogeneralizefromfacts,buttheproofsareextremelyhidden.Mathematicsisthekingofscience.——Gauss

Thisistheadvantageofawell-structuredlanguage,anditssimplifiednotationisoftenthesourceofesoterictheories.——Laplace(PierreSimonLaplace,1749~1827)

Itwouldbeaseriousmistaketothinkthatthereisnecessityonlyingeometricproofsorsensoryevidence.——AugustinLouisCauchy(1789~1857)

Theessenceofmathematicsliesinitsfreedom.——Cantor(GeorgFerdinandLudwigPhilippCantor,1845~1918)

Musiccaninspireorsoothefeelings,paintingcanmakepeoplepleasingtotheeye,poetrycanmovetheheartstrings,philosophycangivepeoplewisdom,andsciencecanimprovemateriallife,Butmathematicscangivealloftheabove.——Klein(ChristianFelixKlein,1849-1925)

Aslongasabranchofsciencecanaskalotofquestions,itisfullofvitality,andthelackofproblemsheraldstheendordeclineofindependentdevelopment.——Hilbert(DavidHilbert,1862~1943)

Theproblemistheheartofmathematics.——PaulHalmos(PaulHalmos,1916~2006)

Timeisaconstant,butforthediligent,itisa"variable".Peoplewhouse"minutes"tocalculatetimehave59timesmoretimethanthosewhouse"hours"tocalculatetime.——Rybakov

ChineseCharacters

Thingsareanalogous,eachhasitsownmerits,soalthoughthebranchesaredivided,theysharethesameknowledge,butonlyoneend.Thereasonisanalyzedwithwords,andpicturesareusedfordisintegration.Theconcubinealsomakesappointmentsandcanbecircumscribed.——LiuHui

Therateofdelayedillnessisnoteccentric,tangibleanddetectable,andthereareseveraltopush.——ZuChongzhi(429~500)

Newmathematicalmethodsandconceptsareoftenmoreimportantthansolvingmathematicalproblemsthemselves.——HuaLuogeng

Themathematicalexpressionisaccurateandconcise,thelogicisabstractanduniversal,andtheformisflexibleandchangeable.Itisanidealtoolforcosmiccommunication.——ZhouHaizhong

Scienceneedsexperimentation.Buttheexperimentcannotbeabsolutelyaccurate.Ifthereisamathematicaltheory,itisentirelycorrectbyrelyingoninference.Thereasonwhysciencecannotleavemathematics.

Manybasicscientificconceptsoftenneedmathematicalconceptstoexpress.Somathematicianshavefoodtoeat,butitisnaturalthattheycannotwintheNobelPrize.ThereisnoNobelPrizeinmathematics,whichmaybeagoodthing.TheNobelPrizeistoocompellingandpreventsmathematiciansfromfocusingontheirownresearch.——ChenXingshen

Aftermodernhigh-energyphysicsarrivedatquantumphysics,thereweremanyexperimentsthatcouldn’tbedoneatall.Usingpenandpapertocalculateathome,thisisnotfarfromwhatmathematiciansthought,somathematicsisinphysics.Hasincrediblepower.——QiuChengtong

Payattentiontotheorderofreadingandwritinghomework.Wemustdevelopgoodlearningmethods,trytoreviewtheknowledgelearnedthatdaywhenwegohome,especiallythenoteswetake,andthenwritehomework,sothattheeffectwillbebetter.

Punctuation

Mathematicsisaninternationalsubjectthatrequiresrigorousnessinallaspects.

Mathematicsofelementarylevelandaboveinmycountrycanberegardedasscientificandtechnologicalliterature.

mycountrystipulatesthatthefullstopofbibliographicarticlesmustuse".".Mathematicsisusedforthispurpose,secondlytoavoidconfusionwithsubscripts,andthirdlybecausemycountryhassubmittedinternationalresearchreportsonmathematics,Buttheydonotuseit,becausemostforeignperiodsarenot".".

Intheproofquestion,","shouldbeusedafter∵(because),and"."shouldbeusedafter∴(so).Ifthereareseveralsmallquestionsinabigquestion,eachquestionendsConnect";",use"."toendthelastquestion,anduse";"toconnectafterthesequencenumber①②③④,andendwith"."afterthelastsequencenumber.

Disciplinedistribution

Universitieswithfirst-levelmathematicsdisciplinesofnationalkeydisciplines:

(Note:1Thesecond-leveldisciplinescoveredbythenationalkeydisciplinesareallnationalkeydisciplines.)

Universitieswiththesecond-levelnationalkeydisciplinesofmathematics(notincludingtheabovelist)

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Formula

Theformulaisanimportantpartofmathematics.Forexample...

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EightProblems

Thefirstsevenproblemsarerecognizedasthesevenproblems,andtheeighthproblemisoneoftheworld’sthreemajorconjectures.

1.P(polynomialalgorithm)problemvs.NP(non-polynomialalgorithm)problem

OnaSaturdaynight,youparticipatedinagrandParty.Feelingembarrassed,youwanttoknowiftherearepeopleyoualreadyknowinthishall.YourhostsuggestedtoyouthatyoumustknowtheladyRosewhoisnearthecornerofthedessertplate.Withinasecond,youcanscanthereandfindthatyourmasteriscorrect.However,ifthereisnosuchhint,youhavetolookaroundtheentirehallandexamineeveryoneonebyonetoseeifthereisanyoneyouknow.Generatingasolutiontoaproblemusuallytakesmuchmoretimethanverifyingagivensolution.Thisisanexampleofthisgeneralphenomenon.

Similarly,ifsomeonetellsyouthatthenumber13,717,421canbewrittenastheproductoftwosmallernumbers,youmaynotknowwhetheryoushouldtrusthim,butifhetellsyouItcanbefactoredinto3607times3803,soyoucaneasilyverifythatthisiscorrectwithapocketcalculator.Regardlessofwhetherwewriteprogramsdexterously,determiningwhetherananswercanbequicklyverifiedusinginternalknowledge,orwhetherittakesalotoftimetosolvewithoutsuchhintsisregardedasoneofthemostprominentproblemsinlogicandcomputerscience.ItwasstatedbyStephenCookin1971.

Second,Hodgeconjecture

Mathematiciansofthe20thcenturydiscoveredapowerfulwaytostudytheshapeofcomplexobjects.Thebasicideaistoaskhowfarwecanformtheshapeofagivenobjectbygluingtogethersimplegeometricbuildingblockswithincreasingdimensions.Thistechniquehasbecomesousefulthatitcanbepromotedinmanydifferentways;iteventuallyleadstosomepowerfultoolsthatenablemathematicianstoachievegreatresultsinclassifyingthevariousobjectsencounteredintheirresearch.progress.Unfortunately,inthispromotion,thegeometricstartingpointoftheprogrambecomesblurred.Inacertainsense,certainpartswithoutanygeometricexplanationmustbeadded.Hodge'sconjectureassertsthatforaparticularlyperfecttypeofspacesuchastheso-calledprojectivealgebraicvariety,thecomponentscalledHodgeclosedchainsareactually(rationallinear)combinationsofgeometriccomponentscalledalgebraicclosedchains.

3.Poincare'sconjecture(hasbeenproven)

Ifwestretcharubberbandaroundthesurfaceofanapple,thenwecanbothDon'ttearitoff,don'tletitleavethesurface,makeitmoveslowlyandshrinktoapoint.Ontheotherhand,ifweimaginethatthesamerubberbandisstretchedonatiresurfaceintheproperdirection,thereisnowaytoshrinkittoapointwithoutbreakingtherubberbandorthetiresurface.WesaythatthesurfaceoftheAppleis"singlyconnected",butthetiresurfaceisnot.Aboutahundredyearsago,Poincaréalreadyknewthatatwo-dimensionalspherecanessentiallybedescribedbysimpleconnectivity.Heproposedthecorrespondenceproblemofathree-dimensionalsphere(afour-dimensionalspace)thathasaunitdistancefromtheorigin.Thisproblemimmediatelybecameextremelydifficult,andsincethen,mathematicianshavebeenstrugglingwithit.

Four.Riemannhypothesis

Somenumbershavespecialpropertiesthatcannotbeexpressedastheproductoftwosmallernumbers,forexample:2,3,5,7andsoon.Suchnumbersarecalledprimenumbers;theyplayanimportantroleinpuremathematicsanditsapplications.Inallnaturalnumbers,thedistributionofthisprimenumberdoesnotfollowanyregularpattern;however,theGermanmathematicianRiemann(1826~1866)observedthatthefrequencyofprimenumbersiscloselyrelatedtoacarefullyconstructedso-calledRiemannZetafunctionThebehaviorofz(s).ThefamousRiemannhypothesisassertsthatallmeaningfulsolutionstotheequationz(s)=0areonastraightline.Thishasbeenverifiedforthefirst1,500,000,000solutions.Provingthatitistrueforeverymeaningfulsolutionwillbringlighttomanymysteriessurroundingthedistributionofprimenumbers.

5.Yang-Millsexistenceandmassgap

Thewayoftheworldisestablishedfortheworldofelementaryparticles.Abouthalfacenturyago,YangZhenningandMillsdiscoveredthatquantumphysicsrevealedastrikingrelationshipbetweenelementaryparticlephysicsandthemathematicsofgeometricobjects.ThepredictionbasedontheYoung-Millsequationhasbeenconfirmedinthefollowinghigh-energyexperimentsperformedinlaboratoriesaroundtheworld:Brockhaven,Stanford,EuropeanInstituteofParticlePhysics,andTsukuba.Nevertheless,theirequationsthatdescribeheavyparticlesandaremathematicallyrigoroushavenoknownsolutions.Inparticular,the"massgap"hypothesis,whichisconfirmedbymostphysicistsandappliedintheirexplanationoftheinvisibilityof"quarks",hasneverreceivedamathematicallysatisfactoryconfirmation.Progressonthisissuerequirestheintroductionoffundamentallynewconceptsinbothphysicsandmathematics.

6.TheexistenceandsmoothnessoftheNavier-Stokesequation

TheundulatingwavesfollowusTheboatiswindingthroughthelake,andtheturbulentaircurrentfollowstheflightofourmodernjetplane.MathematiciansandphysicistsareconvincedthatbothbreezeandturbulencecanbeexplainedandpredictedbyunderstandingthesolutionoftheNavier-Stokesequation.Althoughtheseequationswerewritteninthe19thcentury,westillhaveverylittleunderstandingofthem.ThechallengeistomakesubstantialprogressinmathematicaltheorysothatwecansolvethemysteryhiddenintheNavier-Stokesequation.

Seven.BirchandSwinnerton-Dyerconjecture

Mathematiciansarealwaysreferredtoasx^2+y^2=z^2andthecharacterizationofallintegersolutionsofalgebraicequationsisfascinating.Euclidoncegaveacompletesolutiontothisequation,butformorecomplexequations,thisbecomesextremelydifficult.Infact,asYu.V.Matiyasevichpointedout,Hilbert’stenthproblemisunsolvable,thatis,thereisnogeneralmethodtodeterminewhethersuchamethodhasanintegersolution.WhenthesolutionisapointofanAbeliancluster,BechandSwinnerton-DellconjecturethatthesizeofthegroupofrationalpointsisrelatedtothebehaviorofaZetafunctionz(s)nearthepoints=1.Inparticular,thisinterestingconjectureholdsthatifz(1)isequalto0,thenthereareaninfinitenumberofrationalpoints(solutions);onthecontrary,ifz(1)isnotequalto0,thenthereareonlyafinitenumberofsuchpoints.

Eight.Goldbach’sConjecture

InalettertoEuleronJune7,1742,Goldbachproposedthefollowingconjecture:a)Anyevennumbernotlessthan6canbeexpressedasthesumoftwooddprimenumbers;b)Anyoddnumbernotlessthan9canbeexpressedasthesumofthreeoddprimenumbers.Euleralsoproposedanotherequivalentversioninhisreply,thatis,anyevennumbergreaterthan2canbewrittenasthesumoftwoprimenumbers.UsuallythesetwopropositionsarecollectivelyreferredtoasGoldbach'sconjecture.Theproposition"Anybigevennumbercanbeexpressedasthesumofanumberwithnomorethanaprimefactorandanothernumberwithnomorethanbprimefactors"isrecordedas"a+b",theCorinthiansconjectureistoprove"1+1"isestablished.

In1966,ChenJingrunprovedtheestablishmentof"1+2",thatis,"Anybigevennumbercanbeexpressedasthesumofaprimenumberandanotherprimefactornotexceeding2".

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