математика

Клон на математиката

1.История на математиката

2.Математическа логика и основи на математиката

a: дедуктивна логика (наричана още символична логика), b: теория на доказателството (наричана още метаматематика), c: теория на рекурсията, d: теория на моделите, e: теория на множеството аксиоми, f: математическа основа, g: математическа логика и други предмети на основите на математиката.

3. Теория на числата

a:елементарна теория на числата,b:аналитична теория на числата,c:алгебрична теория на числата,d:трансцендентна теория на числата,e:диофантинова апроксимация,F:геометрия на числата,g:теория на вероятностните числа,h:теория на изчислителните числа,i:теория на числата и други предмети.

4.Алгебра

a:линейнаалгебра,b:теория на групите,c:теория на полето,d:Liegroup,e:Liealgebra,f:Kac-Moodyalgebra,g:теория на пръстена (включително комутативен пръстен и комутативна алгебра, асоциативен пръстен и асоциативна алгебра, неасоциативен пръстен и неасоциативна алгебра и др.),h:модулна теория,i: теория на решетките, j: теория на универсалната алгебра, K: теория на категориите, l: алгебра на хомологията, m: теория на алгебрата K, n: диференциална алгебра, o: теория на алгебричното кодиране, p: други предмети на фалгебрата.

5.Алгебричнагеометрия

6.Геометрия

a:Основна геометрия,b:Евклидова геометрия,c:неевклидова геометрия (включително риманова геометрия и др.),d:сферична геометрия,e:векторен и тензоанализ,f:афинна геометрия,g:проективна геометрия,h:Диференциална геометрия,i:фрактална геометрия,j:изчислителна:k, геометриядруги предмети.

7.Топология

a:топология на точките,b:алгебричнатопология,c:теория на хомотопията,d:теория на нискомерността,E:теория на хомологията,f:теория на размерността,g:топология на решетката,h:теория на влакнестия сноп,i:геометричнатопология,j:теория на сингулярността,k:диференциална топология,l:топологияи други теми.

8.Математически анализ

a: Диференциране, b: Интеграл, c: Теория на редовете, d: Други предмети на математическия анализ.

9.Нестандартен анализ

10. Теория на функциите

a: Теория на функциите на реалната променлива, B: теория на функциите на единични и комплексни променливи, c: теория на функциите на множество комплексни променливи, d: теория на апроксимацията на функциите, e: хармоничен анализ, f: комплексно многообразие, g: теория на специалните функции, h: теория на функциите и други предмети.

11.Обикновени диференциални уравнения

a:качествена теория,b:теория на стабилността.c:Аналитична теория,d:други теми за обикновени диференциални уравнения.

12. Частични диференциални уравнения

a: елиптични частни диференциални уравнения, b: хиперболични частни диференциални уравнения, c: параболични частни диференциални уравнения, D: Нелинейни частни диференциални уравнения, e: Частни диференциални уравнения и други дисциплини.

13. Динамична система

a: Диференциална динамична система, b: Топологична динамична система, c: Комплексна динамична система, d: Други дисциплини на динамичната система.

14.Интегрално уравнение

15.Функционален анализ

a: Теория на линейния оператор, B: вариационен метод, c: топологично линейно пространство, d: Хилбертово пространство, e: функционално пространство, f: Банахово пространство, g: операторна алгебра: мярка и интеграл, i: теория на обобщената функция, j: Нелинеен функционален анализ, k: Други дисциплини на функционалния анализ.

16.Изчислителна математика

a: теория на интерполацията и апроксимацията, b: числено решение на обикновени диференциални уравнения, c: числено решение на частични диференциални уравнения, d: Числено решение на интегрално уравнение, e: Числова алгебра, f: Метод на дискретизация за непрекъснати проблеми, g: Произволен числен експеримент, h: Анализ на грешки, i: Изчислителна математика и други предмети

17. Теория на вероятностите

a: геометрична вероятност, b: разпределение на вероятностите, c: теория на границите, d: случаен процес (включително нормален процес и стационарен процес, точков процес и т.н.), e: процес на Марков, f: случаен анализ, g: теория на мартингала, h: приложна теория на вероятностите (специфично приложена към свързани дисциплини), i: теория на вероятностите и други дисциплини.

18.MathematicalStatistics

a:теория на извадката (включително разпределение на извадката,извадково проучване и т.н.),b:тест на хипотезата,c:непараметрична статистика,d:анализ на дисперсията,e:корелационен регресионен анализ,f:статистическа интерференция,g:Байесова статистика (включително оценка на параметри и т.н.),h:експериментален дизайн,i: многовариантен анализ,j:статистикаТеория на преценката,k:анализ на времеви редове,l:други предмети на математическа статистика.

19.Приложна статистическа математика

a:статистически контрол на качеството,b:математика на надеждността,c:математика на застраховането,d:статистическа симулация.

20.AppliedstatisticalmathematicsOtherdisciplines

21. Операции Изследване

a:линейно програмиране,b:нелинейно програмиране,c:динамично програмиране,d:комбинаторна оптимизация,e:програмиране на параметри,f:цялочислено програмиране,g:стохастично програмиране,h:теория на опашката,i:теория на игрите (наричана още теория на игрите),j:теория на инвентара,k:теория на решенията,l:теория на търсенето ,m:теория на графите,n:теория на цялостното планиране,o:оптимизация,p:други дисциплини на изследване на операциите.

22.Комбинаторика

23.Размита математика

24.Квантова математика

25.Appliedmathematics(specificapplicationintorelatedsubjects)

26.Математикадруги предмети

История на развитието

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).

В древен Китай математиката е наречена аритметика, наричана още аритметика и накрая е променена на математика.

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

Theknowledgeandapplicationofbasicmathematicsisanindispensablepartofthelifeofindividualsandgroups.TherefinementofitsbasicconceptscanbeвижninancientmathematicstextsinancientEgypt,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.Структураisadeductivesystembasedoninitialconceptsandaxioms.Theybelievethatmathematicshasthreebasicparentstructures:algebraicstructure(group,ring,field,lattice,...),orderstructure(partialorder,totalorder,...),topologicalstructure(neighborhood,limit, свързаност, измерение,……).

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

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

Intermsofverticality,theexplorationintherespectivefieldsofmathematicshasbecomemoreandmorein-depth.

Определение

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

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

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.

Структура

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

пространство

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

Основи

Повърхност на въртене (10 листа)

Основна статия:Математически основи

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

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

Логика

Mainarticle:Mathematicallogic

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

Символи

Mainarticle:Mathematicssymbols

MaybetheancientChinesecalculatoristheworld’sOneoftheearliestusedsymbolsoriginatedfromdivinationintheShangDynasty.

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

Строгост

Mathematicslanguageisalsodifficultforbeginners.Howtomakethesewordshavemoreprecisemeaningsthandailyexpressionsalsotroublesbeginners.Wordssuchasopenanddomainhavespecialmeaningsinmathematics.Mathematicaltermsalsoincludepropernounssuchasembryoandintegrability.Butthereisareasonforusingthesespecialsymbolsandproprietaryterms:mathematicsrequiresmoreprecisionthaneverydaylanguage.Математикscallthisrequirementfortheaccuracyoflanguageandlogic"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.Математикshopethattheirtheoremscanbededucedbasedonaxiomswithsystematicreasoning.Thisistoavoidrelyingonunreliableintuitionstoarriveatwrong"theorems"or"proofs,"andthissituationhasвижnmanyexamplesinhistory.Thedegreeofrigorexpectedinmathematicsvarieswithtime:theGreeksexpectedcarefularguments,butinNewton’stimethemethodsusedwerelessrigorous.Newton'sdefinitiontosolvetheproblemwasnotproperlyhandleduntilthe19thcenturybymathematicianswithrigorousanalysisandformalproof.Математикscontinuetoargueabouttherigorofcomputer-aidedproofs.Whenalargenumberofcalculationsaredifficulttoverify,theproofcanhardlybesaidtobeeffectiveandrigorous.

Количество

Thestudyofquantitystartswithnumbers,startingwithfamiliarnaturalnumbersandintegersandrationalandirrationalnumbersdescribedinarithmetic.

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

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

Кратка история

Кратка историяofWesternMathematics

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.

ABriefИстория на китайската математика

Основна статия: История на китайската математика

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

Свързани

ManyoftheresearchresultsofancientChinesearithmetichavealreadyconceivedideasandmethodsthatwereonlyinvolvedinWesternmathematics.Inmoderntimes,therearealsomanyworld-leadingmathematicsresearchresultsbasedonChineseNamedbythemathematician:

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

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

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

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

[Representatives]TheresearchresultsofthemathematicianChenXingshenonindicativecategoriesareinternationallyknownas"Presentationalcategory".

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

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

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

[Korot'sTheorem】МатематикKeZhao'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]МатематикLuQikeng’sresearchresultsonmanifoldswithconstantcurvatureareknowninternationally"Lu’sConjecture".

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

[Jiang’sspace]МатематикJiangBoju’sresearchresultsonthecalculationofNielsennumbershavebeenrecognizedinternationally.Theaboveisnamed"Jiang'sпространство";thereisalso"Jiang'sSubgroup"namedafterhim.

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

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

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

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

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

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

Математически цитати

Чужди обекти

Всичко се брои.——Питагор

Geometryhasnoking'sway.——Евклидов

MathematicsisthewordsusedbyGodtowritetheuniverse.——Galileo

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

Математикsarealltryingtodiscoversomeorderoftheprimenumbersequenceonthisday.Wehavereasontobelievethatthisisamystery,andthehumanmindcanneverinfiltrate.——Euler

Somebeautifultheoremsinmathematicshavesuchcharacteristics:Theyareeasytogeneralizefromfacts,buttheproofsareextremelyhidden.Mathematicsisthekingofscience.——Gauss

Това е главното предимство на добре структуриран език и неговата опростена нотация често е източник на фезотерични теории.—— Лаплас (Пиер Симон Лаплас, 1749~1827)

Itwouldbeaseriousmistaketothinkthatthereisnecessityonlyingeometricproofsorsensoryevidence.——AugustinLouisCauchy(1789~1857)

Същността на математиката се крие в нейната свобода.—— Кантор (Георг Фердинанд Лудвиг Филип Кантор, 1845~1918)

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

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

Проблемът е сърцето на математиката.——Пол Халмос (PaulHalmos,1916~2006)

Времето е постоянно, но за усърдния, то е "променливо". "Минутите" на "хората" за изчисляване на времето имат 59 пъти повече време от тези "часове" на къщата за изчисляване на времето.—— Рибаков

Китайски йероглифи

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

Степента на забавяне на болестта не е ексцентрична, осезаема и откриваема и има няколко натиска.——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.

Пунктуация

Mathematicsisaninternationalsubjectthatrequiresrigorousnessinallaspects.

Mathematicsofelementarylevelandaboveinmycountrycanberegardedasscientificandtechnologicalliterature.

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

В доказателствения въпрос,", "трябва да се използва след∵(защото), и"."трябва да се използва след∴(така).Ако има няколко малки въпроса в голям въпрос, всеки въпрос завършва Connect";",използвайте"."за край напоследния въпрос,и използвайте";"за свързване след номер на последователност①②③④,и завършвайте с"."следпоследния номер на последователност.

Разпределение на дисциплината

Universitieswithfirst-levelmathematicsdisciplinesofnationalkeydisciplines:

(Note:1Thesecond-leveldisciplinescoveredbythenationalkeydisciplinesareallnationalkeydisciplines.)

Universitieswiththesecond-levelnationalkeydisciplinesofmathematics(notincludingtheabovelist)

b>:

Формула

Формулата е важна част от математиката. Например...

виж

• PureMathematics,Приложна математика

• Елементарна математика, математика за напреднали

• ModernMathematics, ModernMathematics

• Математически методи

• Математически задачи

• Математик

• Цитати по математика

• История на математиката

• История на китайската математика

• МатематикаКултура

• Математически формули

• Математически термини

• Константа

Осем проблема

Thefirstsevenproblemsarerecognizedasthesevenproblems,andtheeighthproblemisoneoftheworld’sthreemajorconjectures.

1.P(полиномен алгоритъм)проблем срещу.NP(неполиномен алгоритъм)проблем

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

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

Второ, хипотеза на Ходж

Математикsofthe20thcenturydiscoveredapowerfulwaytostudytheshapeofcomplexobjects.Thebasicideaistoaskhowfarwecanformtheshapeofagivenobjectbygluingtogethersimplegeometricbuildingblockswithincreasingdimensions.Thistechniquehasbecomesousefulthatitcanbepromotedinmanydifferentways;iteventuallyleadstosomepowerfultoolsthatenablemathematicianstoachievegreatresultsinclassifyingthevariousobjectsencounteredintheirresearch.progress.Unfortunately,inthispromotion,thegeometricstartingpointoftheprogrambecomesblurred.Inacertainsense,certainpartswithoutanygeometricexplanationmustbeadded.Hodge'sconjectureassertsthatforaparticularlyperfecttypeofspacesuchastheso-calledprojectivealgebraicvariety,thecomponentscalledHodgeclosedchainsareactually(rationallinear)combinationsofgeometriccomponentscalledalgebraicclosedchains.

3. Хипотезата на Поанкаре (е доказана)

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

Четири. Хипотеза на Риман

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. Съществуване и масова празнина на Ян-Милс

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.МатематикsandphysicistsareconvincedthatbothbreezeandturbulencecanbeexplainedandpredictedbyunderstandingthesolutionoftheNavier-Stokesequation.Althoughtheвижquationswerewritteninthe19thcentury,westillhaveverylittleunderstandingofthem.ThechallengeistomakesubstantialprogressinmathematicaltheorysothatwecansolvethemysteryhiddenintheNavier-Stokesequation.

Седем. Хипотеза на Бърчанд Суинертън-Дайър

Математикsarealwaysreferredtoasx^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.

Осем. Хипотезата на Голдбах

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

През 1966 г. ChenJingruн доказва установяването на "1+2", тоест, "Всяко голямо девет число може да бъде изразено като сумаfa основно число и друг прост множител, който не надвишава 2".

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