matematika

Obor matematiky

1.Historie matematiky

2.Matematická logika a základy matematiky

a:deduktivní logická (také nazývaná symbolická logika),b:důkazová teorie (také nazývaná metamatematika),c:rekurzní teorie,d:teorie modelu,e:teorie axiomových množin,f:matematický základ,g:matematická logika a další předměty ze základů matematiky.

3.Teorie čísel

a:teorie elementárních čísel,b:teorie analytických čísel,c:teorie algebraických čísel,d:teorie transcendentních čísel,e:diofantická aproximace,F:geometrie čísel,g:teorie čísel pravděpodobnosti,h:teorie výpočtových čísel,i:teorie číseladalší předměty.

4.Algebra

a:lineární algebra,b:teorie grup,c:teorie pole,d:Liegrupa,e:Liealgebra,f:Kac-Moodyalgebra,g:teorie prstence (včetně komutativní prstencové a komutativní algebry, asociativní prstencové a asociativní algebry, neasociativní prstencové a neasociativní algebry::artheory), atd. teorie mříží,j:univerzální algebrateorie,K:teorie kategorií,l:homologická algebra,m:algebraKteorie,n:diferenciální algebra,o:algebraická teorie kódování,p:jinépředmětyfalgebra.

5.Algebraická geometrie

6.Geometrie

a:Základní geometrie,b:Euklidovangeometrie,c:neeuklidovskoangeometrie (včetně Riemanniangeometrie atd.),d:sférická geometrie,e:vektorová a tenzorová analýza,f:afingeometrie,g:projektivnígeometrie,h:Diferenciální geometrie,k:komplexní geometrie,k:komplexní geometrie geometrie jiné předměty.

7.Topologie

a:bodová množina,b:algebraiktopologie,c:homotopie,d:nízkorozměrná topologie,E:homologická teorie,f:teorie dimenzionality,g:topologie na mřížce,h:teorie svazků vláken,i:geometrická topologie,j:teorie singularity,k:diferenciálnítopologie,l:topologie.a dalšísubjekty

8.Matematická analýza

a:Diferenciace,b:Integrální,c:Teorie řad,d:Další předměty matematické analýzy.

9.Nestandardní analýzy

10.Teorie funkce

a:teorie funkce reálných proměnných,B:teorie funkcíjednoduchýchakomplexníchproměnných,c:teorie funkcívícekomplexních proměnných,d:teorie aproximace funkcí,e:harmonická analýza,f:komplexní varieta,g:teorie speciálních funkcí,h:teorie funkcíadalší předměty.

11.Obyčejné diferenciální rovnice

a:kvalitativní teorie,b:teorie stability.c:analytická teorie,d:ostatní předměty běžných diferenciálních rovnic.

12.Parciální diferenciální rovnice

a:eliptické parciální diferenciální rovnice,b:hyperbolické parciální diferenciální rovnice,c:parabolické parciální diferenciální rovnice,D:Nelineární parciální diferenciální rovnice,e:Parciální diferenciální rovnice a další obory.

13.Dynamický systém

a:Diferenciálnídynamickýsystém,b:Topologickýdynamickýsystém,c:Komplexnídynamickýsystém,d:Ostatní oborydynamickéhosystému.

14.Integrace

15.Funkční analýza

a:Teorie lineárního operátoru,B:variační metoda,c:topologický lineární prostor,d:Hilbertův prostor,e:funkční prostor,f:Banachův prostor,g:operátorální gebrah:měřicí a integrální,i:zobecněná funkční teorie,j:Nelineární funkční analýza,k:Další disciplíny funkční analýzy.

16.Výpočetní matematika

a:interpolační a aproximační teorie,b:číselnéřešeníobyčejnýchdiferenciálníchrovnic,c:číselnéřešenídílčíchdiferenciálníchrovnic,d:Číselnéřešeníintegračnírovnice,e:Číselná algebra,f:Diskretizační metoda pro kontinuální problémy,g:Náhodný dílčí numerickýexperimentavýpočet,a další:

17.Teorie pravděpodobnosti

a:geometrická pravděpodobnost,b:rozdělení pravděpodobnosti,c:teorie limitů,d:náhodný proces (včetně normálního procesu a stacionárního procesu, bodového procesu atd.),e:Markovův proces,f:náhodná analýza,g:teorie martingale,h:aplikovaná teorie pravděpodobnosti (konkrétně aplikovaná na související disciplíny), teorie i.

18.MathematicalStatistics

a:teorie vzorkování (včetně distribuce vzorkování, průzkumu vzorkování atd.),b:test hypotézy,c:neparametrická statistika,d:analýza rozptylu,e:analýza korelace regrese,f:statistická reference,g:Bayesovská statistika (včetně odhadu parametrů atd.),h:experimentální návrh,h:experimentální vícerozměrná analýza,j:statistikaTeorie úsudku,k:analýza časových řad,l:ostatní předměty matematické statistiky.

19.Aplikovaná statistická matematika

a:statistická kontrola kvality,b:spolehlivá matematika,c:pojistná matematika,d:statistická simulace.

20.AppliedstatisticalmathematicsOtherdisciplines

21.OperaceVýzkum

a:lineární programování,b:nelineární programování,c:dynamické programování,d:kombinátorová optimalizace,e:parametrové programování,f:celočíselné programování,g:stochastické programování,h:teorie řazení do front,i:teorie her (také nazývaná teorie her),j:teorie inventáře,k:teorie rozhodování,l:hledání ,m:teorie grafů,n:teorie celkového plánování,o:optimalizace,p:ostatní disciplíny operačního výzkumu.

22.Kombinatorika

23.FuzzyMatematika

24.Kvantová matematika

25.Appliedmathematics(specificapplicationintorelatedsubjects)

26.Matematikaostatnípředměty

Historie vývoje

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).

Ve starověké Číně se matematika nazývala aritmetika, nazývaná také aritmetika, a nakonec se změnila na matematiku. Aritmetika starověká Čína, jedna z šesti umění (v šesti uměních nazývaná „číslo“).

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

Theknowledgeandapplicationofbasicmathematicsisanindispensablepartofthelifeofindividualsandgroups.TherefinementofitsbasicconceptscanbevidětninancientmathematicstextsinancientEgypt,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.Strukturaisadeductivesystembasedoninitialconceptsandaxioms.Theybelievethatmathematicshasthreebasicparentstructures:algebraicstructure(group,ring,field,lattice,...),orderstructure(partialorder,totalorder,...),topologicalstructure(neighborhood,limit, konektivita, dimenze,……).

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

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

Intermsofverticality,theexplorationintherespectivefieldsofmathematicshasbecomemoreandmorein-depth.

Definice

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

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

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.

Struktura

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

Prostor

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

Základy

Surfaceofrevolution (10 listů)

Hlavní článek:Matematické základy

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

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

Logika

Mainarticle:Mathematicallogic

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

Symboly

Mainarticle:Mathematicssymbols

MaybetheancientChinesecalculatoristheworld’sOneoftheearliestusedsymbolsoriginatedfromdivinationintheShangDynasty.

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

Přísnost

Mathematicslanguageisalsodifficultforbeginners.Howtomakethesewordshavemoreprecisemeaningsthandailyexpressionsalsotroublesbeginners.Wordssuchasopenanddomainhavespecialmeaningsinmathematics.Mathematicaltermsalsoincludepropernounssuchasembryoandintegrability.Butthereisareasonforusingthesespecialsymbolsandproprietaryterms:mathematicsrequiresmoreprecisionthaneverydaylanguage.Matematikscallthisrequirementfortheaccuracyoflanguageandlogic"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.Matematikshopethattheirtheoremscanbededucedbasedonaxiomswithsystematicreasoning.Thisistoavoidrelyingonunreliableintuitionstoarriveatwrong"theorems"or"proofs,"andthissituationhasvidětnmanyexamplesinhistory.Thedegreeofrigorexpectedinmathematicsvarieswithtime:theGreeksexpectedcarefularguments,butinNewton’stimethemethodsusedwerelessrigorous.Newton'sdefinitiontosolvetheproblemwasnotproperlyhandleduntilthe19thcenturybymathematicianswithrigorousanalysisandformalproof.Matematikscontinuetoargueabouttherigorofcomputer-aidedproofs.Whenalargenumberofcalculationsaredifficulttoverify,theproofcanhardlybesaidtobeeffectiveandrigorous.

Množství

Thestudyofquantitystartswithnumbers,startingwithfamiliarnaturalnumbersandintegersandrationalandirrationalnumbersdescribedinarithmetic.

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

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

Stručná historie

Stručná historieofWesternMathematics

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.

ABriefHistorie čínské matematiky

Hlavní článek:Historie čínské matematiky

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

Příbuzný

ManyoftheresearchresultsofancientChinesearithmetichavealreadyconceivedideasandmethodsthatwereonlyinvolvedinWesternmathematics.Inmoderntimes,therearealsomanyworld-leadingmathematicsresearchresultsbasedonChineseNamedbythemathematician:

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

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

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

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

[Representatives]TheresearchresultsofthemathematicianChenXingshenonindicativecategoriesareinternationallyknownas"Presentationalcategory".

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

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

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

[Korot'sTheorem】MatematikKeZhao'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]MatematikLuQikeng’sresearchresultsonmanifoldswithconstantcurvatureareknowninternationally"Lu’sConjecture".

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

[Jiang’sspace]MatematikJiangBoju’sresearchresultsonthecalculationofNielsennumbershavebeenrecognizedinternationally.Theaboveisnamed"Jiang'sProstor";thereisalso"Jiang'sSubgroup"namedafterhim.

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

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

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

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

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

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

Matematické citáty

Cizí předměty

Všechno se počítá.——Pythagoras

Geometryhasnoking'sway.——Euklidovská

MathematicsisthewordsusedbyGodtowritetheuniverse.——Galileo

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

Matematiksarealltryingtodiscoversomeorderoftheprimenumbersequenceonthisday.Wehavereasontobelievethatthisisamystery,andthehumanmindcanneverinfiltrate.——Euler

Somebeautifultheoremsinmathematicshavesuchcharacteristics:Theyareeasytogeneralizefromfacts,buttheproofsareextremelyhidden.Mathematicsisthekingofscience.——Gauss

Tato výhoda dobře strukturovaného jazyka a jeho zjednodušená notace jsou zdrojem esoterických teorií.——Laplace(PierreSimonLaplace,1749~1827)

Itwouldbeaseriousmistaketothinkthatthereisnecessityonlyingeometricproofsorsensoryevidence.——AugustinLouisCauchy(1789~1857)

Podstata matematiky spočívá ve svobodě.——Cantor(GeorgFerdinandLudwigPhilippCantor,1845~1918)

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

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

Problém je srdcem matematiky.——PaulHalmos (PaulHalmos, 1916~2006)

Čas je stálý, ale pro pilný, je "proměnný".

Čínské postavy

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

Terapie zpožděných nemocí není výstřední, hmatatelná a detekovatelná a dochází k několika dotykům.——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.

Interpunkce

Mathematicsisaninternationalsubjectthatrequiresrigorousnessinallaspects.

Mathematicsofelementarylevelandaboveinmycountrycanberegardedasscientificandtechnologicalliterature.

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

V otázce důkazu,","měl by se použít po∵(protože)a"."by se měl použít po ∴(takže).Pokud je ve velké otázce několik malých otázek, každá otázka končíPřipojit";",použijte"."k ukončení poslední otázky a použijte";"k připojení za pořadovým číslem①④.

Disciplína distribuce

Universitieswithfirst-levelmathematicsdisciplinesofnationalkeydisciplines:

(Note:1Thesecond-leveldisciplinescoveredbythenationalkeydisciplinesareallnationalkeydisciplines.)

Universitieswiththesecond-levelnationalkeydisciplinesofmathematics(notincludingtheabovelist)

b>:

Vzorec

Vzorec je důležitou součástí matematiky. Například...

vidět

• PureMathematics,Aplikovaná matematika

• Elementární matematika, pokročilá matematika

• Moderní matematika, moderní matematika

• Matematické metody

• Matematické problémy

• Matematik

• Matematické citáty

• Historie matematiky

• Historie čínské matematiky

• MatematikaKultura

• Matematické vzorce

• Matematické pojmy

• Konstantní

Osm problémů

Thefirstsevenproblemsarerecognizedasthesevenproblems,andtheeighthproblemisoneoftheworld’sthreemajorconjectures.

1.P(polynomiální algoritmus)problém vs.NP(nepolynomiálníalgoritmus)problém

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

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

Za druhé, Hodgeova domněnka

Matematiksofthe20thcenturydiscoveredapowerfulwaytostudytheshapeofcomplexobjects.Thebasicideaistoaskhowfarwecanformtheshapeofagivenobjectbygluingtogethersimplegeometricbuildingblockswithincreasingdimensions.Thistechniquehasbecomesousefulthatitcanbepromotedinmanydifferentways;iteventuallyleadstosomepowerfultoolsthatenablemathematicianstoachievegreatresultsinclassifyingthevariousobjectsencounteredintheirresearch.progress.Unfortunately,inthispromotion,thegeometricstartingpointoftheprogrambecomesblurred.Inacertainsense,certainpartswithoutanygeometricexplanationmustbeadded.Hodge'sconjectureassertsthatforaparticularlyperfecttypeofspacesuchastheso-calledprojectivealgebraicvariety,thecomponentscalledHodgeclosedchainsareactually(rationallinear)combinationsofgeometriccomponentscalledalgebraicclosedchains.

3. Poincareho domněnka (bylo prokázáno)

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

Čtyři. Riemannova hypotéza

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.Existence Yang-Millse a hmotnostní mezera

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.MatematiksandphysicistsareconvincedthatbothbreezeandturbulencecanbeexplainedandpredictedbyunderstandingthesolutionoftheNavier-Stokesequation.Althoughthevidětquationswerewritteninthe19thcentury,westillhaveverylittleunderstandingofthem.ThechallengeistomakesubstantialprogressinmathematicaltheorysothatwecansolvethemysteryhiddenintheNavier-Stokesequation.

Seven.BirchandSwinnerton-Dyerova domněnka

Matematiksarealwaysreferredtoasx^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.

Osm.Goldbachova domněnka

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

V roce 1966 ChenJingrun prokázal zavedení "1+2", to znamená, "Jakékoli velké sudé číslo lze vyjádřit jako součet prvočísla a jiného primárního faktoru, který nepřekračuje 2".

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