матх

БранцхофМатхематицс

1.ХисториофМатхематицс

2.МатхематицалЛогицаандФундаменталсофМатхематицс

а:дедуктивна логика(такођезвана симболичка),б:теорија доказа (такођезвана метаматематика),ц:теорија рекурзије,д:теорија модела,е:теорија скупова аксиома,ф:математичка основа,г:математички логички други предмети засновани на математици.

3.Теорија бројева

а:елементарнатеорија бројева,б:аналитичкатеорија бројева,ц:алгебарска теорија бројева,д:теорија трансцендентног броја,е:Диофантова апроксимација,Ф:геометрија бројева,г:теорија броја вероватноће,х:теорија рачунарских бројева,и:теорија бројева.и други подобјекти.

4.Алгебра

а:линеаралгебра,б:гроуптхеори,ц:фиелдтхеори,д:Лиегроуп,е:Лиеалгебра,ф:Кац-Моодиалгебра,г:рингтхеори(укључујући комутативно прстенасто и комутативну алгебру,асоцијативно прстенасто и асоцијативну алгебру,неасоцијативно прстенасто и небра,етцд. латтицетхеори,ј:универзалнаалгебратхеори,К:цатегоритхеори,л:хомологиалгебра,м:алгебраКтхеори,н:диференцијална алгебра,о:алгебраиццодингтхеори,п:отхерсубјецтсофалгебра.

5.АлгебраицГеометри

6.Геометрија

а:БасицГеометри,б:Еуцлидеангеометри,ц:не-Еуцлидеангеометри(укључујући Риеманиангеометри, итд.),д:спхерицалгеометри,е:вецторандтенсораналисис,ф:аффинегеометри,г:пројецтивегеометри:пројецтивегеометри:цометри:аффинегеометри,цомметриацометриацометри геометријадруги предмети.

7.Топологија

а:топологија скупа тачака,б:алгебарскатопологија,ц:теорија хомотопије,д:нискодимензионална топологија,Е:теорија хомологије,ф:теорија димензионалности,г:топологија на решетки,х:теорија снопа влакана,и:геометријскатопологија,ј:теорија сингуларности,к:диференцијалнатопологија иубтопологија,.

8.Матхематицаланалисис

а:Диференцијација,б:Интеграл,ц:Теорија серија,д:Други предмети математичке анализе.

9.Нестандардна анализа

10.Теорија функција

а:Теорија функција реалних променљивих,Б:теорија функција једне и комплексне променљиве,ц:теорија функција вишеструких комплексних променљивих,д:теорија апроксимације функција,е:хармоничка анализа,ф:комплексна многострукост,г:теорија специјалних функција,х:теорија функција и други субјекти.

11.Обичне диференцијалне једначине

а:куалитативетхеори,б:стаблетхеори.ц:Аналитицалтхеори,д:отхерсубјецтс софординаридифферентиалекуатионс.

12.Парцијалне диференцијалне једначине

а: елиптичне парцијалне диференцијалне једначине, б: хиперболичке парцијалне диференцијалне једначине, ц: параболичке парцијалне диференцијалне једначине, Д: нелинеарне парцијалне диференцијалне једначине, е: делимичне диференцијалне једначине и друге дисциплине.

13.Динамицалсистем

а:Диференцијалнидинамичкисистем,б:Тополошкидинамичкисистем,ц:Сложенидинамичкисистем,д:Друге дисциплинединамичког система.

14.Интегрална једначина

15.Функционалнаанализа

а:Теорија линеарног оператора,Б:варијационаметода,ц:тополошкилинеарни простор,д:Хилбертов простор,е:функционални простор,ф:банахов простор,г:операторалгебра:мера и интеграл,и:теорија генерализованих функција,ј:Нелинеарна функционалнаанализа,к:Друге дисциплине функционалне анализе.

16.ЦомпутатионалМатхематицс

а:интерполацијаитеорија апроксимације,б:нумеричко решење за редне диференцијалне једначине,ц:нумеричко решење делимичних диференцијалних једначина,д:Нумеричко решење интегралне једначине,е:Нумеричка алгебра,ф:Метода дискретизације за континуалне проблеме,г:Насумичног броја,насумични број м.

17.Теорија вероватноће

а:геометријска вероватноћа,б:дистрибуција вероватноће,ц:теорија ограничења,д:случајни процес(укључујући нормалан процес и стационарни процес,процес тачке итд.),е:Марковпроцес,ф:рандоманализа,г:мартингалетеорија,х:примењена теорија вероватноће (специфичнопримењена теорија вероватноће),и:бованепримењена дистореија.

18.MathematicalStatistics

а:теорија узорковања(укључујући дистрибуцију узорковања,испитивање узорковања итд.),б:тест хипотезе,ц:непараметријска статистика,д:анализа варијансе,е:корелацијарегресионаанализа,ф:статистикалинференце,г:Бајесова статистика(укључујући,екетпараметерест.десигн:х: мултиваријантнаанализа,ј:статистицсЈудгменттхеори,к:тимесериесаналисис,л:отхерсубјецтсофматхематицалстатистицс.

19.Примењена статистичкаматематика

а:статистицтицалкуалитицонтрол,б:релиабилитиматхематицс,ц:инсуранцематхематицс,д:статистицалсимулатион.

20.AppliedstatisticalmathematicsOtherdisciplines

21.ОператионсРесеарцх

а:линеарно програмирање,б:нелинеарно програмирање,ц:динамичкопрограмирање,д:комбинаторнаоптимизација,е:програмирање параметара,ф:целобројно програмирање,г:стохастичко програмирање,х:теорија чекања,и:теорија игара (такође названа теорија игара),ј:теорија инвентара,к:одлука:теорија игара ,м:теорија графова,н:општа теорија планирања,о:оптимизација,п:друге дисциплине истраживања операција.

22.Комбинаторика

23.ФуззиМатхематицс

24.Квантна математика

25.Appliedmathematics(specificapplicationintorelatedsubjects)

26.Математикадруги предмети

ДевелопментХистори

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).

У Древној Кини, математика се звала аритметика, такође названа аритметика, и коначно промењена у математику. Аритметика древне Кине је једна од шест уметности (названа „број“ у шест уметности).

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

Theknowledgeandapplicationofbasicmathematicsisanindispensablepartofthelifeofindividualsandgroups.TherefinementofitsbasicconceptscanbeвидиninancientmathematicstextsinancientEgypt,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, повезаност, димензија,……).

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

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

Intermsofverticality,theexplorationintherespectivefieldsofmathematicshasbecomemoreandmorein-depth.

Дефиниција

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

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

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,itcanoftenbeappliedtosomeвидиminglyirrelevantproblems.Forexample,someancientrulerdrawingproblemsarefinallysolvedusingGaloistheory,whichinvolvesdomaintheoryandgroups.s.Anotherexampleofalgebratheoryislinearalgebra,whichmakesageneralstudyofvectorspacesinwhichtheelementshavequantityanddirectionality.Thesephenomenashowthatgeometryandalgebra,whichwereoriginallyconsideredtobeunrelated,areactuallystronglycorrelated.Combinatorialmathematicsstudiesenumeratethemethodsthatsatisfythenumericalobjectsofagivenstructure.

Спаце

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

Основе

Сурфацеофреволутион (10 листова)

Главни чланак:Математичке основе

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.

Симболи

Mainarticle:Mathematicssymbols

MaybetheancientChinesecalculatoristheworld’sOneoftheearliestusedsymbolsoriginatedfromdivinationintheShangDynasty.

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

Ригорозност

Mathematicslanguageisalsodifficultforbeginners.Howtomakethesewordshavemoreprecisemeaningsthandailyexpressionsalsotroublesbeginners.Wordssuchasopenanddomainhavespecialmeaningsinmathematics.Mathematicaltermsalsoincludepropernounssuchasembryoandintegrability.Butthereisareasonforusingthesespecialsymbolsandproprietaryterms:mathematicsrequiresmoreprecisionthaneverydaylanguage.математичарscallthisrequirementfortheaccuracyoflanguageandlogic"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.математичарshopethattheirtheoremscanbededucedbasedonaxiomswithsystematicreasoning.Thisistoavoidrelyingonunreliableintuitionstoarriveatwrong"theorems"or"proofs,"andthissituationhasвидиnmanyexamplesinhistory.Thedegreeofrigorexpectedinmathematicsvarieswithtime:theGreeksexpectedcarefularguments,butinNewton’stimethemethodsusedwerelessrigorous.Newton'sdefinitiontosolvetheproblemwasnotproperlyhandleduntilthe19thcenturybymathematicianswithrigorousanalysisandformalproof.математичарscontinuetoargueabouttherigorofcomputer-aidedproofs.Whenalargenumberofcalculationsaredifficulttoverify,theproofcanhardlybesaidtobeeffectiveandrigorous.

Количина

Thestudyofquantitystartswithnumbers,startingwithfamiliarnaturalnumbersandintegersandrationalandirrationalnumbersdescribedinarithmetic.

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

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

Кратка историја

Кратка историјаofWesternMathematics

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.

ABriefХисториофЦхинесеМатхематицс

Главни чланак: Историја кинеске математике

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

Повезан

ManyoftheresearchresultsofancientChinesearithmetichavealreadyconceivedideasandmethodsthatwereonlyinvolvedinWesternmathematics.Inmoderntimes,therearealsomanyworld-leadingmathematicsresearchresultsbasedonChineseNamedbythemathematician:

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

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

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

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

[Representatives]TheresearchresultsofthemathematicianChenXingshenonindicativecategoriesareinternationallyknownas"Presentationalcategory".

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

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

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

[Korot'sTheorem】математичарKeZhao'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]математичарLuQikeng’sresearchresultsonmanifoldswithconstantcurvatureareknowninternationally"Lu’sConjecture".

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

[Jiang’sspace]математичарJiangBoju’sresearchresultsonthecalculationofNielsennumbershavebeenrecognizedinternationally.Theaboveisnamed"Jiang'sСпаце";thereisalso"Jiang'sSubgroup"namedafterhim.

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

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

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

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

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

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

МатхематицалКуотес

ФореигнОбјецтс

Све се рачуна.——Питагора

Геометрихаснокинг'сваи.——Еуклидски

MathematicsisthewordsusedbyGodtowritetheuniverse.——Galileo

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

математичарsarealltryingtodiscoversomeorderoftheprimenumbersequenceonthisday.Wehavereasontobelievethatthisisamystery,andthehumanmindcanneverinfiltrate.——Euler

Somebeautifultheoremsinmathematicshavesuchcharacteristics:Theyareeasytogeneralizefromfacts,buttheproofsareextremelyhidden.Mathematicsisthekingofscience.——Gauss

Ово је предност добро структурираног језика, а његова поједностављена нотација умекшава извор фезотеричних теорија.——Лаплас (ПиерреСимонЛаплаце,1749~1827)

Itwouldbeaseriousmistaketothinkthatthereisnecessityonlyingeometricproofsorsensoryevidence.——AugustinLouisCauchy(1789~1857)

Суштина математике лежи у слободи.——Кантор (ГеоргФердинандЛудвигПхилиппЦантор,1845~1918)

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

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

Проблем је срце математике.——Паул Халмос(ПаулХалмос,1916~2006)

Време је константно, али за марљиво, је „променљиво“. „Минути“ за израчунавање времена имају 59 пута више времена од оних „сати“ у кући за израчунавање времена.——Рибаков

ЦхинесеЦхарацтерс

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

Стопа кашњења болести није ексцентрична, опипљива и уочљива, и постоји неколико притисака.——ЗуЦхонгзхи(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.

Интерпункција

Mathematicsisaninternationalsubjectthatrequiresrigorousnessinallaspects.

Mathematicsofelementarylevelandaboveinmycountrycanberegardedasscientificandtechnologicalliterature.

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

Интхепроофкуестион,","треба да се користи после∵(зато),и"."треба да се користи после∴(тако).Ако постоји неколико малих питања у великом питању,свакопитање завршава повезивање";",користите"."завршитепоследње питање,и користите";"за повезивање после②секвенца③након".

Дисциплинедистрибутион

Universitieswithfirst-levelmathematicsdisciplinesofnationalkeydisciplines:

(Note:1Thesecond-leveldisciplinescoveredbythenationalkeydisciplinesareallnationalkeydisciplines.)

Universitieswiththesecond-levelnationalkeydisciplinesofmathematics(notincludingtheabovelist)

b><б>:

Формула

Формула је важан део математике. На пример...

види

• PureMathematics,Примењена математика

• Основна математика, напредна математика

• МодернаМатхематицс,МодернМатхематицс

• МатхематицалМетходс

• Математички проблеми

• математичар

• Матхематицскуотес

• ХисториофМатхематицс

• ХисториофЦхинесеМатхематицс

• МатхематицсЦултуре

• Математичке формуле

• Матхематицстермс

• Константно

ЕигхтПроблемс

Thefirstsevenproblemsarerecognizedasthesevenproblems,andtheeighthproblemisoneoftheworld’sthreemajorconjectures.

1.П(полиномски алгоритам)проблемвс.НП(неполиномски алгоритам)проблем

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

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

Друго, Хоџова претпоставка

математичарsofthe20thcenturydiscoveredapowerfulwaytostudytheshapeofcomplexobjects.Thebasicideaistoaskhowfarwecanformtheshapeofagivenobjectbygluingtogethersimplegeometricbuildingblockswithincreasingdimensions.Thistechniquehasbecomesousefulthatitcanbepromotedinmanydifferentways;iteventuallyleadstosomepowerfultoolsthatenablemathematicianstoachievegreatresultsinclassifyingthevariousobjectsencounteredintheirresearch.progress.Unfortunately,inthispromotion,thegeometricstartingpointoftheprogrambecomesblurred.Inacertainsense,certainpartswithoutanygeometricexplanationmustbeadded.Hodge'sconjectureassertsthatforaparticularlyperfecttypeofspacesuchastheso-calledprojectivealgebraicvariety,thecomponentscalledHodgeclosedchainsareactually(rationallinear)combinationsofgeometriccomponentscalledalgebraicclosedchains.

3. Поинкареова хипотека (доказана)

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

Фоур.Риеманнхипотхесис

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.Ианг-Миллсекистенцеандмассгап

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.математичарsandphysicistsareconvincedthatbothbreezeandturbulencecanbeexplainedandpredictedbyunderstandingthesolutionoftheNavier-Stokesequation.Althoughtheвидиquationswerewritteninthe19thcentury,westillhaveverylittleunderstandingofthem.ThechallengeistomakesubstantialprogressinmathematicaltheorysothatwecansolvethemysteryhiddenintheNavier-Stokesequation.

Седам.БирцхандСвиннертон-Диерцоњецтуре

математичарsarealwaysreferredtoasx^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.

Еигхт.Голдбацх'с Цоњецтуре

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

Године 1966, ЦхенЈингру је доказао успостављање „1+2“, односно „било који велики број може бити изражен као сумофактор првог менија и други примарни фактор који не прелази 2“.

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