matematiikka

Matematiikan ala

1.Matematiikan historia

2.Matematiikan logiikka ja matematiikan perusteet

a:deduktiivinen logiikka (kutsutaan myös symbolilogiikaksi),b:todistusteoria (kutsutaan myös metamatematiikaksi),c:rekursioteoria,d:malliteoria,e:aksiomien joukkoteoria,f:matemaattinen perusta,g:matemaattinen logiikka ja muut matematiikan perusteet.

3. Lukuteoria

a:alkeislukuteoria,b:analyyttinen lukuteoria,c:algebrallinen lukuteoria,d:transsendenttilukuteoria,e:diofantiineapproksimaatio,F:lukujen geometria,g:todennäköisyyslukuteoria,h:laskennallinen lukuteoria,i:lukuteoriaja muut kohteet.

4. Algebra

a:linearalgebra,b:ryhmäteoria,c:kenttäteoria,d:lieryhmä,e:liealgebra,f:Kac-Moodyalgebra,g:rengasteoria(sisältää kommutatiivisen rengas-ja kommutatiivisen agebra,assosiatiivisen ja assosiatiivisen agebra,ei-assosiatiivisen ja ei-assosiatiivisen agebra, ei-assosiatiivisena:,etc-assosiatiivisen br. hilateoria,j:universaalialgebrateoria,K:kategoriateoria,l:homologiaalgebra,m:algebraKteoria,n:differentiaalialgebra,o:algebrallinen koodausteoria,p:muut aiheet.

5. Algebrallinen geometria

6. Geometria

a:Perusgeometria,b:euklidinen geometria,c:ei-euklidinen geometria (mukaan lukien riemanniangeometria jne.),d:pallogeometria,e:vektori ja tensorinalyysi,f:affinegeometry,g:projektiivegeometry,h:differential,frajktalgeometry:k. geometria muut aiheet.

7.Topologia

a:pisteasetustopologia,b:algebrallinen topologia,c:homotopyteoria,d:pieniulotteinen topologia,E:homologiateoria,f:ulotteisuusteoria,g:topologia hilassa,h:kuitukimpputeoria,i:geometrinen topologia,j:singulaarisuusteoria,k:differentiaalitopologia ja muut.

8. Matemaattinen analyysi

a:Differentiointi,b:Integraal,c:Sarjateoria,d:Muut matemaattisen analyysin aiheet.

9.Epästandardianalyysi

10. Funktionteoria

a:Reaalimuuttujafunktioteoria,B:yksittäisten ja monimutkaisten muuttujien funktioteoria,c:monikompleksimuuttujien funktioteoria,d:funktion lähentämisen teoria,e:harmoninen analyysi,f:kompleksimanifold,g:erikoisfunktioteoria,h:funktioteoriaja muut kohteet.

11. Tavalliset differentiaaliyhtälöt

a:laadullinen teoria,b:vakausteoria.c:Analyyttinen teoria,d:muut kohteetinäärisiin differentiaaliyhtälöihin.

12.Osittaiset differentiaaliyhtälöt

a:ellipsiosadifferentiaaliyhtälöt,b:hyperboliset osadifferentiaaliyhtälöt,c:paraboliset osadifferentiaaliyhtälöt,D:epälineaariset osadifferentiaaliyhtälöt,e:osittajadifferentiaaliyhtälötja muut tieteenalat.

13.Dynaaminen järjestelmä

a:Differentiaalidynaaminen järjestelmä,b:Topologinendynaaminen järjestelmä,c:Monimutkainen dynaaminen järjestelmä,d: Muut dynaamisen järjestelmän alat.

14.Integra-yhtälö

15. Toiminnallinen analyysi

a:Lineaarioperaattoriteoria,B:variaatiomenetelmä,c:topologinen lineaariavaruus,d:Hilbertspace,e:funktiotila,f:Banachspace,g:operatoralgebra:mitta ja integraali,i:yleisfunktioteoria,j:Epälineaarinen funktion analyysi,k:muut funktionaalisen analyysin osa-alueet.

16. Laskennallinen matematiikka

a:interpolaatio- ja likiarvoteoria,b:numeerinen ratkaisu,c:osiadifferentiaaliyhtälöiden numeerinen ratkaisu,d:integraaliyhtälön numeerinen ratkaisu,e:numeroalgebra,f:diskretisointimenetelmä jatkuville ongelmille,g:satunnainen.

17. Todennäköisyysteoria

a:geometrinen todennäköisyys,b:todennäköisyysjakauma,c:rajateoria,d:satunnaisprosessi (mukaan lukien normaaliprosessi ja kiinteä prosessi, pisteprosessi jne.), e:Markovprosessi,f:satunnaisanalyysi,g:martingaleteoria,h:sovellettu todennäköisyysteoria (erityisesti sovellettu tieteenaloihin).

18.MathematicalStatistics

a:näytteenottoteoria(mukaan lukien otosjakauma,otostutkimus jne.),b:hypoteesitesti,c:ei-parametriset tilastot,d:varianssianalyysi,e:korrelaatioregressioanalyysi,f:tilastollinen päätelmä,g:Bayesin tilastot (mukaan lukien kokeellinen arvio), monimuuttujaanalyysi,j:tilastotTuomioteoria,k:aikasarjaanalyysi,l:muut matemaattisten tilastojen aiheet.

19.Soveltava tilastomatematiikka

a:tilastollisenlaadunvalvonta,b:luotettavuusmatematiikka,c:vakuutusmatematiikka,d:tilastosimulaatio.

20.AppliedstatisticalmathematicsOtherdisciplines

21.Toimintatutkimus

a:lineaarinen ohjelmointi,b:epälineaarinen ohjelmointi,c:dynaaminen ohjelmointi,d:kombinatorinen optimointi,e:parametriohjelmointi,f:kokonaislukuohjelmointi,g:stokastinen ohjelmointi,h:jonoteoria,i:peliteoria (kutsutaan myös peliteoriaksi),j:inventorytheory,k:päätösteoria,l:searchtheory ,m:graafiteoria,n:yleinen suunnitteluteoria,o:optimointi,p:muut toimintatutkimuksen tieteenalat.

22. Kombinatoriikka

23.FuzzyMathematics

24. Kvanttimatematiikka

25.Appliedmathematics(specificapplicationintorelatedsubjects)

26.Matematiikka ja muut oppiaineet

Kehityshistoria

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).

MuinainenKiina matematiikkaa kutsuttiin aritmetiikkaksi, jota kutsuttiin myös aritmetiikkaksi ja lopulta muutettiin matematiikaksi. Muinaisen Kiinan aritmetiikka on yksi kuudesta taiteesta (kutsuttiin "numeroksi" kuuden taiteen sisällä).

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

Theknowledgeandapplicationofbasicmathematicsisanindispensablepartofthelifeofindividualsandgroups.TherefinementofitsbasicconceptscanbekatsoninancientmathematicstextsinancientEgypt,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.Rakenneisadeductivesystembasedoninitialconceptsandaxioms.Theybelievethatmathematicshasthreebasicparentstructures:algebraicstructure(group,ring,field,lattice,...),orderstructure(partialorder,totalorder,...),topologicalstructure(neighborhood,limit, liitettävyys, mitat,……).

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

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

Intermsofverticality,theexplorationintherespectivefieldsofmathematicshasbecomemoreandmorein-depth.

Määritelmä

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

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

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.

Rakenne

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

Avaruus

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

Perusasiat

Vallankumouksen pinta (10 arkkia)

Pääartikkeli: Matemaattiset perusteet

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

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

Logiikka

Mainarticle:Mathematicallogic

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

Symbolit

Mainarticle:Mathematicssymbols

MaybetheancientChinesecalculatoristheworld’sOneoftheearliestusedsymbolsoriginatedfromdivinationintheShangDynasty.

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

Tiukkaus

Mathematicslanguageisalsodifficultforbeginners.Howtomakethesewordshavemoreprecisemeaningsthandailyexpressionsalsotroublesbeginners.Wordssuchasopenanddomainhavespecialmeaningsinmathematics.Mathematicaltermsalsoincludepropernounssuchasembryoandintegrability.Butthereisareasonforusingthesespecialsymbolsandproprietaryterms:mathematicsrequiresmoreprecisionthaneverydaylanguage.Matemaatikkoscallthisrequirementfortheaccuracyoflanguageandlogic"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.Matemaatikkoshopethattheirtheoremscanbededucedbasedonaxiomswithsystematicreasoning.Thisistoavoidrelyingonunreliableintuitionstoarriveatwrong"theorems"or"proofs,"andthissituationhaskatsonmanyexamplesinhistory.Thedegreeofrigorexpectedinmathematicsvarieswithtime:theGreeksexpectedcarefularguments,butinNewton’stimethemethodsusedwerelessrigorous.Newton'sdefinitiontosolvetheproblemwasnotproperlyhandleduntilthe19thcenturybymathematicianswithrigorousanalysisandformalproof.Matemaatikkoscontinuetoargueabouttherigorofcomputer-aidedproofs.Whenalargenumberofcalculationsaredifficulttoverify,theproofcanhardlybesaidtobeeffectiveandrigorous.

Määrä

Thestudyofquantitystartswithnumbers,startingwithfamiliarnaturalnumbersandintegersandrationalandirrationalnumbersdescribedinarithmetic.

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

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

Lyhyt historia

Lyhyt historiaofWesternMathematics

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.

ABriefKiinan matematiikan historia

Pääartikkeli:Kiinan matematiikan historia

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

Liittyvät

ManyoftheresearchresultsofancientChinesearithmetichavealreadyconceivedideasandmethodsthatwereonlyinvolvedinWesternmathematics.Inmoderntimes,therearealsomanyworld-leadingmathematicsresearchresultsbasedonChineseNamedbythemathematician:

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

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

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

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

[Representatives]TheresearchresultsofthemathematicianChenXingshenonindicativecategoriesareinternationallyknownas"Presentationalcategory".

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

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

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

[Korot'sTheorem】MatemaatikkoKeZhao'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]MatemaatikkoLuQikeng’sresearchresultsonmanifoldswithconstantcurvatureareknowninternationally"Lu’sConjecture".

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

[Jiang’sspace]MatemaatikkoJiangBoju’sresearchresultsonthecalculationofNielsennumbershavebeenrecognizedinternationally.Theaboveisnamed"Jiang'sAvaruus";thereisalso"Jiang'sSubgroup"namedafterhim.

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

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

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

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

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

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

Matemaattiset lainaukset

Ulkomaiset esineet

Kaikki laskettiin. -- Pythagoras

Geometryhasnoking'sway.——Euklidinen

MathematicsisthewordsusedbyGodtowritetheuniverse.——Galileo

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

Matemaatikkosarealltryingtodiscoversomeorderoftheprimenumbersequenceonthisday.Wehavereasontobelievethatthisisamystery,andthehumanmindcanneverinfiltrate.——Euler

Somebeautifultheoremsinmathematicshavesuchcharacteristics:Theyareeasytogeneralizefromfacts,buttheproofsareextremelyhidden.Mathematicsisthekingofscience.——Gauss

Tämä on hyvin jäsennellyn kielen etu, ja sen yksinkertaistetut huomautukset pehmentävät esoteeristen teorioiden lähdettä.—Laplace (PierreSimonLaplace, 1749~1827)

Itwouldbeaseriousmistaketothinkthatthereisnecessityonlyingeometricproofsorsensoryevidence.——AugustinLouisCauchy(1789~1857)

Matematiikan ydin on vapaudessa. — Cantor (GeorgFerdin ja LudwigPhilippCantor, 1845–1918)

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

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

Matematiikan sydämen ongelma.——PaulHalmos (PaulHalmos,1916~2006)

Aika on vakio, mutta ahkeralle, on "muuttuva". Ihmistalon "minuutteilla"aikalaskennassa on 59 kertaa enemmän aikaa kuin omalla talolla "tunteja"ajan laskemikatson.——Rybakov

Kiinalaiset merkit

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

Sairauden viivästymisen nopeus on epäkeskinen, käsin kosketeltava ja havaittavissa, ja siinä on useita paineita.——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.

Välimerkit

Mathematicsisaninternationalsubjectthatrequiresrigorousnessinallaspects.

Mathematicsofelementarylevelandaboveinmycountrycanberegardedasscientificandtechnologicalliterature.

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

Intheproofquestion,""shouldbeusedafter∵(koska),and"."shouldbeusedafter∴(niin).Jos suuressa kysymyksessä on useita pieniä kysymyksiä, jokainen kysymys päättyyConnect";",käytä"."lopettaaksesi viimeisen kysymyksen⑑loppunumeron,ja käyttää";"kysyntää"

Kurinalaisuus

Universitieswithfirst-levelmathematicsdisciplinesofnationalkeydisciplines:

(Note:1Thesecond-leveldisciplinescoveredbythenationalkeydisciplinesareallnationalkeydisciplines.)

Universitieswiththesecond-levelnationalkeydisciplinesofmathematics(notincludingtheabovelist)

b>:

Kaava

Kaava on tärkeä osa matematiikkaa.Esimerkki...

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• PureMathematics,Soveltava matematiikka

• Elementary Mathematics, Advanced Mathematics

• Moderni matematiikka, moderni matematiikka

• Matemaattiset menetelmät

• Matemaattiset ongelmat

• Matemaatikko

• Matematiikan lainaukset

• Matematiikan historia

• Kiinan matematiikan historia

• MatematiikkaKulttuuri

• Matemaattiset kaavat

• Matematiikan termit

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Kahdeksan ongelmaa

Thefirstsevenproblemsarerecognizedasthesevenproblems,andtheeighthproblemisoneoftheworld’sthreemajorconjectures.

1.P(polynomialgoritmi)ongelmavs.NP(ei-polynomialgoritmi)ongelma

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

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

Toiseksi, Hodge-arvelu

Matemaatikkosofthe20thcenturydiscoveredapowerfulwaytostudytheshapeofcomplexobjects.Thebasicideaistoaskhowfarwecanformtheshapeofagivenobjectbygluingtogethersimplegeometricbuildingblockswithincreasingdimensions.Thistechniquehasbecomesousefulthatitcanbepromotedinmanydifferentways;iteventuallyleadstosomepowerfultoolsthatenablemathematicianstoachievegreatresultsinclassifyingthevariousobjectsencounteredintheirresearch.progress.Unfortunately,inthispromotion,thegeometricstartingpointoftheprogrambecomesblurred.Inacertainsense,certainpartswithoutanygeometricexplanationmustbeadded.Hodge'sconjectureassertsthatforaparticularlyperfecttypeofspacesuchastheso-calledprojectivealgebraicvariety,thecomponentscalledHodgeclosedchainsareactually(rationallinear)combinationsofgeometriccomponentscalledalgebraicclosedchains.

3. Poincaren olettamus (on todistettu)

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

Neljä.Riemannnin hypoteesi

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-Millsexistenceand masssgap

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.MatemaatikkosandphysicistsareconvincedthatbothbreezeandturbulencecanbeexplainedandpredictedbyunderstandingthesolutionoftheNavier-Stokesequation.Althoughthekatsoquationswerewritteninthe19thcentury,westillhaveverylittleunderstandingofthem.ThechallengeistomakesubstantialprogressinmathematicaltheorysothatwecansolvethemysteryhiddenintheNavier-Stokesequation.

Seitsemän.BirchandSwinnerton-Dyer-arvelu

Matemaatikkosarealwaysreferredtoasx^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.

Kahdeksan. Goldbachin arvelu

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

Vuonna 1966 ChenJingrun osoitti "1+2":n perustamisen, toisin sanoen "Jokainen iso pariluku voidaan ilmaistaalkuluvun ja toisen päätekijän summana, joka ei ylitä2".

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