Izifundo ngesiFulentshi
Okungahleliwe: Isethulo samathuba - Ingxenye 1 (POLYTECHNIQUE PARIS)
I-École Polytechnique, isikhungo esaziwayo, sinikeza isifundo esithakazelisayo se-Coursera esinesihloko esithi “Okungahleliwe: isingeniso samathuba – Ingxenye 1”. Lesi sifundo, esithatha cishe amahora angama-27 esatshalaliswa emasontweni amathathu, siyithuba eliyingqayizivele kunoma ubani onentshisekelo kuzisekelo zamathuba. Idizayinelwe ukuguquguquka futhi ivumelane nejubane lomfundi ngamunye, lesi sifundo sinikeza indlela ejulile nefinyelelekayo kuthiyori yamathuba.
Uhlelo luqukethe amamojula abandakanyayo angu-8, ngalinye likhuluma ngezici ezibalulekile zesikhala esingaba khona, imithetho efanayo yamathuba, isimo, ukuzimela, nokuguquguquka okungahleliwe. Imojula ngayinye inothiswa ngamavidiyo achazayo, ukufundwa okwengeziwe kanye nemibuzo ukuze kuhlolwe futhi kuhlanganiswe ulwazi olutholiwe. Abafundi baphinde babe nethuba lokuthola isitifiketi abangabelana ngaso lapho beqeda isifundo, okwengeza inani elibalulekile ohambweni lwabo lobungcweti noma lwezemfundo.
Othisha, uSylvie Méléard, uJean-René Chazottes noCarl Graham, bonke abaxhumene ne-École Polytechnique, baletha ubungcweti nothando lwabo lwezibalo, okwenza lesi sifundo singabi semfundo kuphela, kodwa futhi sikhuthaze. Kungakhathaliseki ukuthi ungumfundi wezibalo, uchwepheshe ofuna ukujulisa ulwazi lwakho, noma umane ungumuntu othanda isayensi, lesi sifundo sikunikeza ithuba eliyingqayizivele lokuhlolisisa umhlaba othakazelisayo wamathuba, uqondiswa izingqondo ezithile ezihamba phambili e-École Polytechnique.
Okungahleliwe: Isethulo samathuba - Ingxenye 2 (POLYTECHNIQUE PARIS)
Iqhubeka nokwenza kahle kwezemfundo kwe-École Polytechnique, isifundo esithi “Okungahleliwe: isingeniso emathubeni – Ingxenye 2” ku-Coursera siwukuqhubeka okuqondile nokucebisayo kwengxenye yokuqala. Lesi sifundo, esilinganiselwa kumahora angu-17 esatshalaliswa emasontweni amathathu, sigxilisa abafundi emicabangweni ethuthuke kakhulu yethiyori yamathuba, sinikeze ukuqonda okujulile nokusebenzisa okubanzi kwalesi siyalo esithakazelisayo.
Ngamamojula angu-6 ahlelwe kahle, isifundo sihlanganisa izihloko ezinjengama-vector angahleliwe, ukuhlanganiswa kwezibalo zomthetho, umthetho we-theorem yezinombolo ezinkulu, indlela ye-Monte Carlo, kanye ne-central limit theorem. Imojuli ngayinye ifaka amavidiyo okufundisa, ukufundwa kanye nemibuzo, ukuze uthole ulwazi olujulile lokufunda. Le fomethi ivumela abafundi ukuthi bahlanganyele ngokugcwele nezinto ezisetshenziswayo futhi basebenzise imiqondo efundiwe ngendlela engokoqobo.
Othisha, uSylvie Méléard, uJean-René Chazottes noCarl Graham bayaqhubeka nokuqondisa abafundi kulolu hambo lwemfundo ngobuchule babo nothando lwabo lwezibalo. Indlela yabo yokufundisa isiza ukuqonda imiqondo eyinkimbinkimbi futhi ikhuthaze ukuhlola okujulile kwamathuba.
Lesi sifundo silungele labo asebevele banesisekelo esiqinile emathubeni futhi abafuna ukwandisa ukuqonda kwabo kanye nekhono lokusebenzisa le mibono ezinkingeni eziyinkimbinkimbi. Ngokuqeda lesi sifundo, abafundi bangakwazi futhi ukuthola isitifiketi abangabelana ngaso, okukhombisa ukuzibophezela kwabo kanye nekhono labo kule ndawo ekhethekile.
Isingeniso kuthiyori yokusabalalisa (POLYTECHNIQUE PARIS)
Isifundo "Isingeniso sethiyori yokusabalalisa", esinikezwa yi-École Polytechnique ku-Coursera, simele ukuhlola okuyingqayizivele nokujulile kwenkambu yezibalo ethuthukisiwe. Lesi sifundo, esithatha cishe amahora ayi-15 esisabalele emasontweni amathathu, siklanyelwe labo abafuna ukuqonda ukusabalalisa, umqondo oyisisekelo kuzibalo ezisetshenziswayo nokuhlaziya.
Lolu hlelo lunamamojula ayi-9, ngalinye linikeza ingxube yamavidiyo okufundisa, ukufundwa kanye nemibuzo. Lawa mamojuli ahlanganisa izici ezihlukahlukene zetiyori yokusabalalisa, okuhlanganisa nezindaba eziyinkimbinkimbi njengokuchaza okuphuma kokunye komsebenzi ongaqhubeki nokusebenzisa imisebenzi engaqhubeki njengezixazululo kuzibalo ezihlukene. Le ndlela ehlelekile ivumela abafundi ukuthi bajwayele kancane kancane imiqondo engase ibonakale yethusa ekuqaleni.
Osolwazi uFrançois Golse kanye no-Yvan Martel, bobabili abangamalungu avelele e-École Polytechnique, baletha ubungcweti obukhulu kulesi sifundo. Ukufundisa kwabo kuhlanganisa ukuqina kwezemfundo nezindlela ezintsha zokufundisa, okwenza okuqukethwe kufinyeleleke futhi kuhehe abafundi.
Lesi sifundo silungele kakhulu abafundi bezibalo, ubunjiniyela, noma imikhakha ehlobene abafuna ukujulisa ukuqonda kwabo kwezibalo eziyinkimbinkimbi. Ngokuqeda lesi sifundo, ababambiqhaza ngeke bagcine nje ngokuzuza ulwazi olubalulekile, kodwa futhi bayoba nethuba lokuzuza isitifiketi ekwabelwana ngaso, okwengeza inani elibalulekile kuphrofayela yabo yobungcweti noma yezemfundo.
Isingeniso sethiyori yeGalois (SUPERIOR NORMAL SCHOOL PARIS)
Ihlinzekwa yi-École Normale Supérieure ku-Coursera, isifundo esithi "Isingeniso Sethiyori ye-Galois" siwukuhlola okuthakazelisayo kwelinye lamagatsha ajule nanamandla ezibalo zanamuhla.Ihlala cishe amahora ayi-12, lesi sifundo sicwilisa abafundi emhlabeni oyinkimbinkimbi futhi ohehayo wethiyori ye-Galois, isiyalo esiguqule ukuqonda kobudlelwano phakathi kwezibalo ze-polynomial kanye nezakhiwo ze-algebra.
Isifundo sigxile ocwaningweni lwezimpande ze-polynomials kanye nokuvezwa kwazo kusuka kuma-coefficients, umbuzo omaphakathi ku-algebra. Ihlola umbono weqembu le-Galois, elethulwe ngu-Évariste Galois, elihlobanisa i-polynomial ngayinye neqembu lokuvunyelwa kwezimpande zayo. Le ndlela yokwenza isivumela ukuthi siqonde ukuthi kungani kungenakwenzeka ukuveza izimpande zezibalo ze-polynomial ezithile ngamafomula e-algebra, ikakhulukazi kuma-polynomials weziqu ezinkulu kunezine.
Ukuxhumana kwe-Galois, into eyinhloko yesifundo, ixhumanisa ithiyori yenkundla nethiyori yeqembu, ihlinzeka ngombono ohlukile wokuxazululeka kwezibalo ezinkulu. Isifundo sisebenzisa imiqondo eyisisekelo ku-algebra yomugqa ukuze sisondele kumbono wemizimba futhi sethule umbono wenombolo ye-algebraic, kuyilapho sihlola amaqembu ezimvume ezidingekayo ocwaningweni lwamaqembu e-Galois.
Lesi sifundo siphawuleka kakhulu ngekhono laso lokwethula imiqondo ye-algebra eyinkimbinkimbi ngendlela efinyelelekayo nelula, okuvumela abafundi ukuthi bathole imiphumela ephusile ngokushesha ngobuncane be-formalism engabonakali. Ilungele abafundi bezibalo, i-physics, noma ubunjiniyela, kanye nabafundi abathanda izibalo abafuna ukujulisa ukuqonda kwabo kwezakhiwo ze-algebraic kanye nokusebenza kwazo.
Ngokuqeda lesi sifundo, ababambiqhaza ngeke nje bazuze ukuqonda okujulile kwethiyori ye-Galois, kodwa futhi bayoba nethuba lokuzuza isitifiketi esabelane ngaso, okwengeze inani elibalulekile kuphrofayela yabo yobungcweti noma yezemfundo.
Ukuhlaziya I (ingxenye 1): Isandulelo, imibono eyisisekelo, izinombolo zangempela (ISIKOLE I-POLYTECHNIQUE FEDERALE DE LAUSANNE)
Isifundo esithi “Ukuhlaziya I (ingxenye 1): Isethulo, imibono eyisisekelo, izinombolo zangempela”, esinikezwa i-École Polytechnique Fédérale de Lausanne ku-edX, siyisethulo esijulile semiqondo eyisisekelo yokuhlaziya kwangempela. Lesi sifundo samasonto ama-5, esidinga cishe amahora angama-4-5 okufunda ngesonto, siklanyelwe ukuthi siqedelwe ngejubane lakho.
Okuqukethwe kwesifundo kuqala ngesandulelo esiphinde sivakashele futhi sijule imibono yezibalo ebalulekile njengemisebenzi ye-trigonometric (sin, cos, tan), imisebenzi ehambisanayo (exp, ln), kanye nemithetho yokubala yamandla, ama-logarithms nezimpande. Iphinde ihlanganise amasethi ayisisekelo nemisebenzi.
Umongo wesifundo ugxile ezinhlelweni zezinombolo. Kusukela kumbono onembile wezinombolo zemvelo, isifundo sichaza izinombolo ezinengqondo futhi sihlola izici zazo. Ukunakwa okukhethekile kukhokhwa ezinombolweni zangempela, ezethulwa ukuze kugcwaliswe izikhala ngezinombolo ezinengqondo. Isifundo sethula incazelo ye-axiomatic yezinombolo zangempela futhi sicwaninga izakhiwo zazo ngokuningiliziwe, kufaka phakathi imiqondo efana ne-infimum, i-supremum, inani eliphelele kanye nezinye izici ezengeziwe zezinombolo zangempela.
Lesi sifundo silungele labo abanolwazi oluyisisekelo lwezibalo futhi abafuna ukujulisa ukuqonda kwabo ngokuhlaziywa komhlaba wangempela. Iwusizo ikakhulukazi kubafundi bezibalo, i-physics, noma ubunjiniyela, kanye nanoma ubani onentshisekelo yokuqonda kahle izisekelo zezibalo.
Ngokuqeda lesi sifundo, ababambiqhaza bazothola ukuqonda okuqinile kwezinombolo zangempela nokubaluleka kwazo ekuhlaziyeni, kanye nethuba lokuzuza isitifiketi ekwabelwana ngaso, okwengeza inani elibalulekile kuphrofayela yabo yobungcweti noma yezemfundo.
Ukuhlaziya I (ingxenye 2): Isingeniso sezinombolo eziyinkimbinkimbi (ISIKOLE I-POLYTECHNIQUE FEDERALE DE LAUSANNE)
Isifundo esithi “Ukuhlaziya I (ingxenye 2): Isingeniso sezinombolo eziyinkimbinkimbi”, esinikezwa yi-École Polytechnique Fédérale de Lausanne ku-edX, siyisingeniso esikhangayo emhlabeni wezinombolo eziyinkimbinkimbi.Lesi sifundo samasonto ama-2, esidinga cishe amahora angama-4-5 okufunda ngesonto, siklanyelwe ukuthi siqedelwe ngejubane lakho.
Isifundo siqala ngokubhekana ne-equation z^2 = -1, engenaso isixazululo kusethi yezinombolo zangempela, R. Le nkinga iholela ekwethulweni kwezinombolo eziyinkimbinkimbi, C, inkambu equkethe u-R futhi esivumela ukuba sixazulule lezi zinombolo. zibalo. Isifundo sihlola izindlela ezihlukene zokumela inombolo eyinkimbinkimbi futhi sidingida izixazululo zezibalo zefomu elithi z^n = w, lapho u-n engokuka-N* kanye no-w ukuya ku-C.
Okugqamile kulesi sifundo ukutadisha i-theorem eyisisekelo ye-algebra, okuwumphumela oyinhloko wezibalo. Isifundo siphinde sihlanganise izihloko ezifana nokumelwa kweCartesian kwezinombolo eziyinkimbinkimbi, izakhiwo zazo eziyisisekelo, isici esiphambene sokuphindaphinda, ifomula ye-Euler kanye ne-de Moivre, kanye nesimo se-polar senombolo eyinkimbinkimbi.
Lesi sifundo silungele labo asebenolwazi oluthile lwezinombolo zangempela futhi abafuna ukwelula ukuqonda kwabo kube izinombolo eziyinkimbinkimbi. Iwusizo ikakhulukazi kubafundi bezibalo, i-physics, noma ubunjiniyela, kanye nanoma ubani onentshisekelo yokuqonda okujulile kwe-algebra kanye nokusetshenziswa kwayo.
Ngokuqeda lesi sifundo, ababambiqhaza bazothola ukuqonda okuqinile kwezinombolo eziyinkimbinkimbi neqhaza labo elibalulekile kwizibalo, kanye nethuba lokuzuza isitifiketi ekwabelwana ngaso, okwengeza inani elibalulekile kuphrofayela yabo yobungcweti noma yezemfundo.
Ukuhlaziya I (ingxenye 3): Ukulandelana kwezinombolo zangempela I no-II (ISIKOLE I-POLYTECHNIQUE FEDERALE DE LAUSANNE)
Isifundo esithi “Ukuhlaziya I (ingxenye 3): Ukulandelana kwezinombolo zangempela I no-II”, esinikezwa yi-École Polytechnique Fédérale de Lausanne ku-edX, sigxile ekulandelaneni kwezinombolo zangempela. Lesi sifundo samasonto ama-4, esidinga cishe amahora angama-4-5 okufunda ngesonto, siklanyelwe ukuthi siqedelwe ngejubane lakho.
Umqondo omaphakathi walesi sifundo umkhawulo wokulandelana kwezinombolo zangempela. Iqala ngokuchaza ukulandelana kwezinombolo zangempela njengomsebenzi osuka ku-N ukuya ku-R. Isibonelo, ukulandelana okuthi a_n = 1/2^n kuyahlolisiswa, okubonisa ukuthi isondela kanjani kuziro. Isifundo sibhekana ngokuqinile nencazelo yomkhawulo wokulandelana futhi sithuthukisa izindlela zokuthola ubukhona bomkhawulo.
Ngaphezu kwalokho, isifundo sisungula ukuxhumana phakathi komqondo womkhawulo kanye nalowo we-infimum kanye nenani eliphakeme lesethi. Ukusetshenziswa okubalulekile kokulandelana kwezinombolo zangempela kuboniswa yiqiniso lokuthi inombolo yangempela ngayinye ingabhekwa njengomkhawulo wokulandelana kwezinombolo ezinengqondo. Isifundo siphinde sihlole ukulandelana kwe-Cauchy nokulandelana okuchazwe ngokungeniswa komugqa, kanye nethiyori ye-Bolzano-Weierstrass.
Ababambiqhaza bazophinda bafunde ngochungechunge lwezinombolo, ngesingeniso sezibonelo ezihlukene kanye nemibandela yokuhlangana, efana nombandela we-d'Alembert, umbandela we-Cauchy, kanye nombandela we-Leibniz. Isifundo siphetha ngocwaningo lochungechunge lwezinombolo ngepharamitha.
Lesi sifundo silungele labo abanolwazi oluyisisekelo lwezibalo futhi abafuna ukujulisa ukuqonda kwabo ukulandelana kwezinombolo zangempela. Iwusizo ikakhulukazi kubafundi bezibalo, i-physics noma ubunjiniyela. Ngokuqeda lesi sifundo, ababambiqhaza bazocebisa ukuqonda kwabo izibalo futhi bangase bathole isitifiketi abangabelana ngaso, impahla yokuthuthuka kwabo emsebenzini noma ezifundweni zabo.
Ukutholwa Kwemisebenzi Yangempela Neqhubekayo: Ukuhlaziywa I (ingxenye 4) (ISIKOLE I-POLYTECHNIQUE FEDERALE DE LAUSANNE)
Kokuthi “Ukuhlaziya I (ingxenye 4): Umkhawulo womsebenzi, imisebenzi eqhubekayo”, i-École Polytechnique Fédérale de Lausanne inikezela ngohambo oluhehayo oluya ocwaningweni lwemisebenzi yangempela yokuguquguquka kwangempela.Lesi sifundo, esithatha amasonto ama-4 ngamahora ama-4 kuye kwayi-5 okufunda masonto onke, siyatholakala ku-edX futhi sivumela ukuqhubeka ngejubane lakho.
Le ngxenye yesifundo iqala ngokwethulwa kwemisebenzi yangempela, igcizelela izici zayo ezifana ne-monotonicity, i-parity, kanye ne-periodicity. Iphinde ihlole ukusebenza phakathi kwemisebenzi futhi yethule imisebenzi ethile efana nemisebenzi ye-hyperbolic. Ukunakwa okukhethekile kunikezwa imisebenzi echazwe ngokwezinyathelo, okuhlanganisa imisebenzi ye-Signum ne-Heaviside, kanye nokuguqulwa kwe-affine.
Umnyombo wesifundo ugxile emkhawulweni obukhali womsebenzi endaweni ethile, unikeza izibonelo eziphathekayo zemikhawulo yemisebenzi. Iphinde ihlanganise imiqondo yemikhawulo yesokunxele nesokudla. Okulandelayo, isifundo sibheka imikhawulo engapheli yemisebenzi futhi sinikeza amathuluzi abalulekile okubala imikhawulo, njenge-cop theorem.
Isici esibalulekile sesifundo ukwethulwa komqondo wokuqhubeka, ochazwa ngezindlela ezimbili ezahlukene, kanye nokusetshenziswa kwawo ukuze kunwetshwe imisebenzi ethile. Isifundo siphetha ngocwaningo lokuqhubeka ngezikhathi ezivulekile.
Lesi sifundo siyithuba elicebisayo kulabo abafuna ukujulisa ukuqonda kwabo imisebenzi yangempela neqhubekayo. Ilungele abafundi bezibalo, i-physics noma ubunjiniyela. Ngokuqeda lesi sifundo, ababambiqhaza ngeke bagcine ngokunweba ama-horizons abo ezibalo, kodwa bayoba nethuba lokuthola isitifiketi esivuzayo, okuvula umnyango wemibono emisha yezemfundo noma yobungcweti.
Ukuhlola Imisebenzi Ehlukene: Ukuhlaziya I (ingxenye 5) (ISIKOLE I-POLYTECHNIQUE FEDERALE DE LAUSANNE)
I-École Polytechnique Fédérale de Lausanne, ekunikezeni kwayo imfundo ku-edX, yethula “Ukuhlaziya I (ingxenye 5): Imisebenzi eqhubekayo nemisebenzi ehlukanisekayo, umsebenzi ophuma kokunye”. Lesi sifundo samasonto amane, esidinga cishe amahora angama-4-5 okufunda ngesonto, siwukuhlola okujulile kwemiqondo yokuhlukahluka nokuqhubeka kwemisebenzi.
Isifundo siqala ngocwaningo olujulile lwemisebenzi eqhubekayo, egxile ezakhiweni zabo ngezikhathi ezivaliwe. Lesi sigaba sisiza abafundi baqonde ubuningi kanye nobuncane bemisebenzi eqhubekayo. Isifundo sibe sesithula indlela yokuhlukanisa kabili futhi sethule ama-theoremu abalulekile njenge-theoremu yenani elimaphakathi kanye ne-fixed point theorem.
Ingxenye emaphakathi yesifundo inikezelwe ekwehlukaniseni nasekuhlukaniseni imisebenzi. Abafundi bafunda ukuhumusha le mibono futhi baqonde ukulingana kwayo. Isifundo sibe sesibheka ukwakhiwa komsebenzi wokuphuma kokunye futhi sihlola izici zayo ngokuningiliziwe, okuhlanganisa ukusebenza kwe-algebraic emisebenzini ephuma kokunye.
Isici esibalulekile sesifundo ukucwaninga kwezakhiwo zemisebenzi ehlukanisekayo, njengokuphuma kokubunjwa kwemisebenzi, ithiyori kaRolle, kanye nethiyori yokunyuka okulinganiselwe. Isifundo siphinde sihlole ukuqhubeka komsebenzi wokuphuma kokunye kanye nemithelela yawo ku-monotonicity yomsebenzi ohlukanisekayo.
Lesi sifundo siyithuba elihle kakhulu kulabo abafuna ukujulisa ukuqonda kwabo ngemisebenzi ehlukanisekayo neqhubekayo. Ilungele abafundi bezibalo, i-physics noma ubunjiniyela. Ngokuphothula lesi sifundo, ababambiqhaza ngeke bagcine nje ngokunweba ukuqonda kwabo imiqondo eyisisekelo yezibalo, kodwa bazothola nethuba lokuthola isitifiketi esivuzayo, okuvula umnyango wamathuba amasha emfundo noma ochwepheshe.
Ukujula Ekuhlaziyeni Kwezibalo: Ukuhlaziya I (ingxenye 6) (ISIKOLE I-POLYTECHNIQUE FEDERALE DE LAUSANNE)
Isifundo esithi “Ukuhlaziya I (ingxenye 6): Izifundo zemisebenzi, ukuthuthukiswa okulinganiselwe”, esinikezwa yi-École Polytechnique Fédérale de Lausanne ku-edX, siwukuhlola okujulile kwemisebenzi nokuthuthuka kwayo okulinganiselwe. Lesi sifundo samasonto amane, esinomthwalo wokusebenza wamahora ama-4 kuya kwayi-5 ngesonto, sivumela abafundi ukuthi bathuthuke ngejubane labo.
Lesi sahluko sesifundo sigxile ocwaningweni olunzulu lwemisebenzi, kusetshenziswa amathiyori ukuhlola ukuhlukahluka kwawo. Ngemva kokubhekana ne-finite increment theorem, isifundo sibheka ukujwayela kwayo. Isici esibalulekile sokufunda imisebenzi ukuqonda ukuziphatha kwayo ngokungapheli. Ukwenza lokhu, isifundo sethula umthetho weBernoulli-l'Hospital, ithuluzi elibalulekile lokunquma imikhawulo eyinkimbinkimbi yama-quotients athile.
Isifundo siphinde sihlole ukumelelwa kwezithombe zemisebenzi, sihlole imibuzo efana nokuba khona kwenani lendawo noma lembulunga yonke noma i-minima, kanye ne-convexity noma ukufingqa kwemisebenzi. Abafundi bazofunda ukukhomba ama-asymptote ahlukene omsebenzi.
Elinye iphuzu eliqinile lesifundo ukwethulwa kokunwetshwa okukhawulelwe komsebenzi, okunikeza ukulinganiselwa kwe-polynomial eduze kwephoyinti elinikeziwe. Lokhu kuthuthukiswa kubalulekile ukuze kube lula ukubalwa kwemikhawulo kanye nocwaningo lwezakhiwo zemisebenzi. Isifundo siphinde sihlanganise uchungechunge oluphelele kanye ne-radius yokuhlangana, kanye nochungechunge luka-Taylor, ithuluzi elinamandla lokumelela imisebenzi ehlukaniseka unomphela.
Lesi sifundo siwumthombo obalulekile walabo abafuna ukujulisa ukuqonda kwabo imisebenzi kanye nezindlela zabo zokusebenza kuzibalo. Inikeza umbono ocebisayo nonemininingwane emiqondweni ebalulekile ekuhlaziyeni kwezibalo.
Ubuchule Bokuhlanganisa: Ukuhlaziya I (ingxenye 7) (ISIKOLE I-POLYTECHNIQUE FEDERALE DE LAUSANNE)
Isifundo esithi "Ukuhlaziya I (ingxenye 7): Okuhlanganisiwe okungapheli nokuqinisekile, ukuhlanganiswa (izahluko ezikhethiwe)", okunikezwa yi-École Polytechnique Fédérale de Lausanne ku-edX, ukuhlola okuningiliziwe kokuhlanganiswa kwemisebenzi. Le mojula, ethatha amasonto amane ngokubandakanyeka kwamahora ama-4 kuya kwayi-5 ngesonto, ivumela abafundi ukuthi bathole ubuqili bokuhlanganisa ngejubane labo.
Isifundo siqala ngencazelo yengqikithi engapheli kanye nengqikithi eqondile, yethula okubalulekile okubalulekile ngezibalo ze-Riemann kanye nezibalo eziphezulu neziphansi. Bese idingida izici ezintathu ezibalulekile zokuhlanganisa okuqondile: umugqa wokuhlanganisa, ukuhlukaniswa iziqephu kwesizinda sokuhlanganisa, kanye ne-monotonicity ye-integral.
Iphuzu eliyinhloko lesifundo ithiyomu emaphakathi yemisebenzi eqhubekayo kusegimenti, eboniswa ngokuningiliziwe. Isifundo sifinyelela umvuthwandaba nge-theorem eyisisekelo yokubala okubalulekile, yethula umbono we-antiderivative yomsebenzi. Abafundi bafunda amasu ahlukene okuhlanganisa, njengokuhlanganisa izingxenye, ukushintsha okuguquguqukayo, nokuhlanganiswa ngokungeniswa.
Isifundo siphetha ngocwaningo lokuhlanganiswa kwemisebenzi ethile, okuhlanganisa ukuhlanganiswa kokunwetshwa okulinganiselwe komsebenzi, ukuhlanganiswa kochungechunge oluphelele, kanye nokuhlanganiswa kwemisebenzi eqhubekayo ye-piecewise. Lawa masu avumela okuhlanganisiwe kwemisebenzi enamafomu akhethekile ukuthi ibalwe ngokuphumelelayo. Okokugcina, isifundo sihlola okubalulekile okujwayelekile, okuchazwa ngokudlulela emkhawulweni kuma-integral, futhi silethe izibonelo eziphathekayo.
Lesi sifundo siyisisetshenziswa esibalulekile salabo abafuna ukufunda ukuhlanganisa, ithuluzi elibalulekile kwizibalo. Inikeza umbono ophelele nongokoqobo wokuhlanganisa, inothisa amakhono abafundi ezibalo.
Izifundo ngesiNgisi
Isingeniso samamodeli alayini kanye ne-Matrix Algebra (I-Harvard)
I-Harvard University, ngenkundla yayo ye-HarvardX ku-edX, inikeza isifundo “Isingeniso Kumamodeli Awumugqa kanye ne-Matrix Algebra”. Nakuba isifundo sifundiswa ngesiNgisi, sinikeza ithuba eliyingqayizivele lokufunda izisekelo ze-matrix algebra namamodeli aqondile, amakhono abalulekile emikhakheni eminingi yesayensi.
Lesi sifundo samasonto amane, esidinga amahora amabili kuya kwamane okufunda ngesonto, siklanyelwe ukuthi siqedelwe ngejubane lakho. Igxile ekusebenziseni ulimi lokuhlela lwe-R ukusebenzisa amamodeli alayini ekuhlaziyeni idatha, ikakhulukazi kusayensi yezempilo. Abafundi bazofunda ukukhohlisa i-algebra ye-matrix futhi baqonde ukusetshenziswa kwayo ekwakhiweni kokuhlola nokuhlaziya idatha yobukhulu obuphezulu.
Uhlelo luhlanganisa i-matrix algebra notation, ukusebenza kwe-matrix, ukusetshenziswa kwe-matrix algebra ekuhlaziyeni idatha, amamodeli alayini, kanye nesingeniso sokubola kwe-QR. Lesi sifundo siyingxenye yochungechunge lwezifundo eziyisikhombisa, ezingathathwa ngazinye noma njengengxenye yezitifiketi ezimbili zobungcweti ku-Data Analysis for the Life Sciences and Genomic Data Analysis.
Lesi sifundo silungele labo abafuna ukuzuza amakhono ekumodeleni izibalo nokuhlaziya idatha, ikakhulukazi kumongo wesayensi yezempilo. Inikeza isisekelo esiqinile salabo abafisa ukuqhubeka nokuhlola i-algebra ye-matrix kanye nokusebenza kwayo emikhakheni eyahlukene yesayensi nocwaningo.
I-Master Probability (i-Harvard)
LUhlu lwadlalwayo oluthi “Izibalo 110: Amathuba” ku-YouTube, olufundiswa ngesiNgisi nguJoe Blitzstein waseHarvard University, luwumthombo oyigugu walabo abafuna ukujulisa ulwazi lwabo lwamathuba.. Uhlu lwadlalwayo luhlanganisa amavidiyo ezifundo, izinto zokubuyekeza, kanye nezivivinyo ezingaphezu kuka-250 ezinezixazululo ezinemininingwane.
Lesi sifundo sesiNgisi siyisingeniso esibanzi samathuba, ethulwa njengolimi olubalulekile kanye nesethi yamathuluzi okuqonda izibalo, isayensi, ubungozi kanye nokungahleliwe. Imiqondo efundiswayo iyasebenza emikhakheni eyahlukene njengezibalo, isayensi, ubunjiniyela, ezomnotho, ezezimali kanye nempilo yansuku zonke.
Izihloko ezihlanganisiwe zihlanganisa okuyisisekelo kwamathuba, okuguquguqukayo okungahleliwe nokusabalalisa kwakho, ukusabalalisa okungaguquki nokuhlukahluka, ama-theorems omkhawulo, namaketanga e-Markov. Isifundo sidinga ulwazi lwangaphambili lwesibalo esiguquguqukayo esisodwa kanye nokujwayelana nomatikuletsheni.
Kulabo abakhululekile ngesiNgisi nabamagange ukuhlola umhlaba wamathuba ngokujula, lolu chungechunge lwezifundo zeHarvard lunikeza ithuba lokufunda elicebisayo. Ungafinyelela ohlwini lwadlalwayo nokuqukethwe kwalo okuningiliziwe ngokuqondile ku-YouTube.
Amathuba Achaziwe. Isifundo esinemibhalo engezansi yesiFulentshi (i-Harvard)
Isifundo esithi “Fat Chance: Probability from the Ground Up,” esinikezwa i-HarvardX ku-edX, siyisingeniso esithakazelisayo samathuba nezibalo. Nakuba isifundo sifundiswa ngesiNgisi, sifinyeleleka kuzithameli ezikhuluma isiFulentshi ngenxa yemibhalo engezansi yesiFulentshi etholakalayo.
Lesi sifundo samasonto ayisikhombisa, esidinga amahora angu-3 kuya kwangu-5 okufunda ngesonto, siklanyelwe labo abasanda kufunda ngokungenzeka noma abafuna ukubuyekezwa okufinyelelekayo kwemiqondo eyinhloko ngaphambi kokubhalisa esifundweni sezibalo. I-“Fat Chance” igcizelela ukuthuthukisa ukucabanga kwezibalo kunokubamba ngekhanda amagama namafomula.
Amamojula okuqala ethula amakhono okubala ayisisekelo, abese esetshenziswa ezinkingeni ezilula zamathuba. Amamojula alandelayo ahlola ukuthi le mibono namasu angashintshwa kanjani ukuze kubhekwane nohlu olubanzi lwezinkinga ezingenzeka. Isifundo siphetha ngesethulo sezibalo ngemibono yenani elilindelekile, ukuhluka nokusabalalisa okuvamile.
Lesi sifundo silungele labo abafuna ukukhulisa amakhono abo okucabanga futhi baqonde izisekelo zamathuba kanye nezibalo. Ihlinzeka ngombono ocebisayo mayelana nemvelo eqoqwayo yezibalo nokuthi isebenza kanjani ekuqondeni ubungozi nokungahleliwe.
Ukuchazwa Kwezibalo Nokumodela Kwemilingo Ye-High-Throughput (Harvard)
Isifundo "se-Statistical Inference and Modelling for High-throughput Experiments" ngesiNgisi sigxile kumasu asetshenziselwa ukwenza ukucatshangelwa kwezibalo kudatha yomphumela ophezulu. Lesi sifundo samasonto amane, esidinga amahora okufunda angama-2-4 ngesonto, siwumthombo obalulekile walabo abafuna ukuqonda nokusebenzisa izindlela zezibalo ezithuthukisiwe ezilungiselelweni zocwaningo olusebenzisa idatha.
Uhlelo luhlanganisa izihloko ezihlukahlukene, okuhlanganisa inkinga yokuqhathanisa okuningi, amanani amaphutha, izinqubo zokulawula izinga lamaphutha, amanani okutholwa okungamanga, amanani we-q, nokuhlaziywa kwedatha yokuhlola. Iphinde yethule ukumodela kwezibalo kanye nokusebenza kwayo kudatha yomphumela ophezulu, idingida ngokusatshalaliswa kwepharamethri efana ne-binomial, exponential, ne-gamma, futhi ichaza isilinganiso esiphezulu sokungenzeka.
Abafundi bazofunda ukuthi le miqondo isetshenziswa kanjani kuzimo ezinjengokulandelana kwesizukulwane esilandelayo kanye nedatha ye-microarray. Lesi sifundo siphinde sihlanganise amamodeli e-hierarchical kanye ne-Bayesian empirics, nezibonelo ezingokoqobo zokusebenzisa kwazo.
Lesi sifundo silungele labo abafuna ukujulisa ukuqonda kwabo kwezibalo kanye nokumodela ocwaningweni lwesayensi lwesimanje. Inikeza umbono ojulile wokuhlaziywa kwezibalo zedatha eyinkimbinkimbi futhi iwumthombo omuhle kakhulu wabacwaningi, abafundi nochwepheshe emikhakheni yesayensi yezempilo, i-bioinformatics kanye nezibalo.
Isingeniso ku-Probability (Harvard)
Isifundo esithi “Isingeniso Sokungenzeka”, esinikezwa i-HarvardX ku-edX, siwukuhlola okujulile kwamathuba, ulimi olubalulekile nesethi yamathuluzi okuqonda idatha, ithuba, nokungaqiniseki. Nakuba isifundo sifundiswa ngesiNgisi, sifinyeleleka kuzithameli ezikhuluma isiFulentshi ngenxa yemibhalo engezansi yesiFulentshi etholakalayo.
Lesi sifundo samasonto ayishumi, esidinga amahora okufunda angama-5-10 ngesonto, sihlose ukuletha ingqondo emhlabeni ogcwele amathuba nokungaqiniseki. Izohlinzeka ngamathuluzi adingekayo ukuze kuqondwe idatha, isayensi, ifilosofi, ubunjiniyela, ezomnotho kanye nezezimali. Ngeke ufunde kuphela ukuthi ungaxazulula kanjani izinkinga zobuchwepheshe eziyinkimbinkimbi, kodwa nokuthi ungasebenzisa kanjani lezi zixazululo ekuphileni kwansuku zonke.
Ngezibonelo ezisukela ekuhlolweni kwezokwelashwa kuye ekuqaguleni kwezemidlalo, uzothola isisekelo esiqinile socwaningo lokuqagela kwezibalo, izinqubo ze-stochastic, ama-algorithms angahleliwe, nezinye izihloko lapho amathuba adingekayo.
Lesi sifundo silungele labo abafuna ukukhulisa ukuqonda kwabo kokungaqiniseki kanye nethuba, ukwenza izibikezelo ezinhle, nokuqonda okuguquguqukayo okungahleliwe. Ihlinzeka ngombono ocebisayo mayelana nokusatshalaliswa kwamathuba avamile asetshenziswa kuzibalo nesayensi yedatha.
I-Applied Calculus (Harvard)
Isifundo esithi “Calculus Applied!”, esinikezwa i-Harvard ku-edX, siwukuhlola okujulile kokusetshenziswa kwezibalo eziguquguqukayo olulodwa kwezenhlalo, impilo, nesayensi yemvelo. Lesi sifundo, ngesiNgisi ngokuphelele, siyithuba elihle kakhulu lalabo abafuna ukuqonda ukuthi i-calculus isetshenziswa kanjani kuzimo zochwepheshe zomhlaba wangempela.
Ihlala amasonto ayishumi futhi idinga phakathi kwamahora ama-3 nama-6 okufunda ngesonto, lesi sifundo sidlula izincwadi zendabuko. Usebenzisana nochwepheshe abavela emikhakheni eyahlukene ukuze abonise ukuthi i-calculus isetshenziswa kanjani ukuze kuhlaziywe futhi kuxazululwe izinkinga zomhlaba wangempela. Abafundi bazohlola izinhlelo zokusebenza ezihlukahlukene, kusukela ekuhlaziyweni komnotho kuye ekumodeleni kwebhayoloji.
Uhlelo luhlanganisa ukusetshenziswa kokuphuma kokunye, okuhlanganisayo, izilinganiso ezihlukanisayo, futhi lugcizelela ukubaluleka kwamamodeli nemingcele yezibalo. Idizayinelwe labo abanokuqonda okuyisisekelo kokubala okuguquguqukayo okukodwa futhi abanentshisekelo ekusetshenzisweni kwayo okungokoqobo emikhakheni ehlukahlukene.
Lesi sifundo silungele abafundi, othisha, kanye nochwepheshe abafuna ukujulisa ukuqonda kwabo ukubala nokuthola izinhlelo zayo zomhlaba wangempela.
Isingeniso sokucabanga kwezibalo (Stanford)
Isifundo "Isingeniso Sokucabanga Kwezibalo", esinikezwa yiNyuvesi yaseStanford ku-Coursera, singena emhlabeni wokucabanga ngezibalo. Nakuba isifundo sifundiswa ngesiNgisi, sifinyeleleka kuzithameli ezikhuluma isiFulentshi ngenxa yemibhalo engezansi yesiFulentshi etholakalayo.
Lesi sifundo samasonto ayisikhombisa, esidinga cishe amahora angama-38 esewonke, noma cishe amahora ayi-12 ngesonto, siklanyelwe labo abafisa ukuthuthukisa ukucabanga kwezibalo, okuhlukile ekuzilolongeni nje izibalo njengoba kuvame ukwethulwa ohlelweni lwesikole. Isifundo sigxile ekuthuthukiseni indlela yokucabanga “engaphandle kwebhokisi,” okuyikhono elibalulekile ezweni lanamuhla.
Abafundi bazohlola ukuthi ochwepheshe bezibalo bacabanga kanjani ukuxazulula izinkinga zomhlaba wangempela, noma ngabe zivela emhlabeni wansuku zonke, kwisayensi, noma kwizibalo ngokwayo. Izifundo zisiza ukuthuthukisa le ndlela yokucabanga ebalulekile, idlulele ngalé kwezinqubo zokufunda ukuze kuxazululwe izinkinga ezingokwengqondo.
Lesi sifundo silungele labo abafuna ukuqinisa ukucabanga kwabo komthamo futhi baqonde izisekelo zokucabanga kwezibalo. Ihlinzeka ngombono ocebisayo mayelana nemvelo eqoqwayo yezibalo kanye nokusebenza kwayo ekuqondeni izinkinga eziyinkimbinkimbi.
Ukufunda ngezibalo no-R (Stanford)
Isifundo "sokufunda ngezibalo nge-R", esinikezwa u-Stanford, siyisingeniso sezinga eliphakathi sokufunda okugadiwe, esigxile ezindleleni zokuhlehla nezindlela zokuhlukanisa. Lesi sifundo, ngesiNgisi ngokuphelele, siyisisetshenziswa esibalulekile salabo abafuna ukuqonda nokusebenzisa izindlela zezibalo emkhakheni wesayensi yedatha.
Ihlala amasonto ayishumi nanye futhi idinga amahora angama-3-5 okufunda ngesonto, isifundo sihlanganisa izindlela ezintsha zendabuko nezijabulisayo ekumodeleni izibalo, nokuthi zisetshenziswa kanjani ngolimi lohlelo luka-R. incwadi yezifundo.
Izihloko zifaka ukuhlehla komugqa kanye ne-polynomial, ukuhlehla kwempahla kanye nokuhlaziya okubandlululayo okulandelanayo, ukuqinisekiswa okuphambanayo kanye ne-bootstrapping, ukukhethwa kwemodeli nezindlela ezijwayelekile (i-ridge ne-lasso), amamodeli angaqondile, ama-splines kanye namamodeli angeziwe ajwayelekile, izindlela ezisekelwe esihlahleni, amahlathi angahleliwe kanye nokukhulisa, ukusekela imishini ye-vector, amanethiwekhi e-neural nokufunda okujulile, amamodeli okusinda, nokuhlola okuningi.
Lesi sifundo silungele labo abanolwazi oluyisisekelo lwezibalo, i-algebra yomugqa, nesayensi yekhompiyutha, futhi abafuna ukujulisa ukuqonda kwabo kokufunda kwezibalo kanye nokusebenza kwayo kusayensi yedatha.
Uzifunda Kanjani Izibalo: Isifundo Sawo Wonke Umuntu (Stanford)
Isifundo esithi “Indlela Yokufunda Izibalo: Zabafundi”, esinikezwa uStanford. Isifundo samahhala se-inthanethi sabafundi bawo wonke amazinga ezibalo. Iphelele ngesiNgisi, ihlanganisa ulwazi olubalulekile mayelana nobuchopho nobufakazi obusha mayelana nezindlela ezingcono kakhulu zokufunda izibalo.
Ihlala amasonto ayisithupha futhi idinga ihora eli-1 kuye ku-3 lokufunda ngesonto. Lesi sifundo siklanyelwe ukuguqula ubudlelwano babafundi nezibalo. Abantu abaningi baye baba nokuhlangenwe nakho okungekuhle ngezibalo, okuholela ekungathandini noma ekuhlulekeni. Lesi sifundo sihlose ukunika abafundi ulwazi abaludingayo ukuze bajabulele izibalo.
Okumboziwe yizihloko ezifana nobuchopho nokufunda izibalo. Izinganekwane mayelana nezibalo, umqondo, amaphutha kanye nesivinini nazo zihlanganisiwe. Ukuguquguquka kwezinombolo, ukucabanga kwezibalo, ukuxhumana, amamodeli ezinombolo nakho kuyingxenye yohlelo. Izethulo zezibalo empilweni, kodwa nasemvelweni nasemsebenzini azikhohliwe. Isifundo siklanywe nge-pedagogy yokuzibandakanya, okwenza ukufunda kuhlanganyele futhi kube namandla.
Kuyinsiza ebalulekile kunoma ubani ofuna ukubona izibalo ngendlela ehlukile. Thuthukisa ukuqonda okujulile nokuhle kwalesi siyalo. Ilungele ikakhulukazi labo abake babhekana nezibalo ezingezinhle phambilini futhi ababheke ukushintsha lo mbono.
Ukuphathwa Kwamathuba (eStanford)
Izifundo "zesethulo sokuphathwa kwamathuba", ezinikezwa u-Stanford, ziyisingeniso sesiyalo sokuphathwa kwamathuba. Lo mkhakha ugxile ekukhulumeni nasekubaleni ukungaqiniseki ngendlela yamathebula edatha ahlolekayo abizwa ngokuthi ama-Stochastic Information Packets (SIPs). Lesi sifundo samasonto ayishumi sidinga ihora eli-1 kuye kwayi-5 ngesonto. Akungabazeki ukuthi iyinsiza ebalulekile kulabo abafuna ukuqonda nokusebenzisa izindlela zezibalo emkhakheni wesayensi yedatha.
Ikharikhulamu yezifundo ihlanganisa izihloko ezinjengokubona "Iphutha Lezilinganiso," isethi yamaphutha esistimu avela lapho ukungaqiniseki kumelelwa izinombolo ezizodwa, ngokuvamile isilinganiso. Ichaza ukuthi kungani amaphrojekthi amaningi ephuzile, ngaphezu kwesabelomali futhi ngaphansi kwesabelomali. Lesi sifundo siphinde sifundise i-Uncertainty Arithmetic, eyenza izibalo ngokufaka okungaqinisekile, okuholela emiphumeleni engaqinisekile ongakwazi ukubala kuyo imiphumela emaphakathi yangempela kanye namathuba okufinyelela imigomo ethile.
Abafundi bazofunda ukwenza ukulingiswa okusebenzisanayo okungabiwa nanoma yimuphi umsebenzisi we-Excel ngaphandle kokudinga izengezo noma amamakhro. Le ndlela ifaneleka ngokulinganayo iPython noma iyiphi indawo yokuhlela esekela ama-array.
Lesi sifundo silungele labo abanethezekile nge-Microsoft Excel futhi abafuna ukujulisa ukuqonda kwabo kokuphathwa kwamathuba kanye nokusetshenziswa kwayo kwisayensi yedatha.
Isayensi Yokungaqiniseki Nedatha (MIT)
Isifundo esithi "Amathuba - Isayensi Yokungaqiniseki Nedatha", esinikezwa yi-Massachusetts Institute of Technology (MIT). Isingeniso esiyisisekelo sesayensi yedatha ngamamodeli angenzeka. Lesi sifundo samasonto ayishumi nesithupha, esidinga amahora ayi-10 kuya kwayi-14 okufunda ngesonto. Ihambisana nengxenye yohlelo lwe-MIT MicroMasters kwizibalo nesayensi yedatha.
Lesi sifundo sihlola umhlaba wokungaqiniseki: kusukela ezingozini ezimakethe zezimali ezingalindelekile kuya kwezokuxhumana. Ukumodela okungenzeka kanye nenkambu ehlobene yokucatshangelwa kwezibalo. Izikhiye ezimbili zokuhlaziya le datha nokwenza izibikezelo ezinengqondo ngokwesayensi.
Abafundi bazothola ukwakheka nezinto eziyisisekelo zamamodeli angenzeka. Kubandakanya okuguquguqukayo okungahleliwe, ukusatshalaliswa kwazo, izindlela nokuhluka. Isifundo sihlanganisa nezindlela zokukhomba. Imithetho yezinombolo ezinkulu kanye nokusetshenziswa kwazo, kanye nezinqubo ezingahleliwe.
Lesi sifundo sifanelekile kulabo abafuna ulwazi oluyisisekelo kwisayensi yedatha. Ihlinzeka ngombono obanzi mayelana namamodeli angenzeka. Kusukela kuma-elementi ayisisekelo kuya kuzinqubo ezingahleliwe kanye nokuqagela kwezibalo. Konke lokhu kuwusizo ikakhulukazi kochwepheshe kanye nabafundi. Ikakhulukazi emikhakheni yesayensi yedatha, ubunjiniyela kanye nezibalo.
Amathuba eComputational and Inference (MIT)
I-Massachusetts Institute of Technology (MIT) yethula isifundo esithi “Computational Probability and Inference” ngesiNgisi. Kuhlelo, isingeniso sezinga elimaphakathi ekuhlaziyeni okungenzeka kanye nokucatshangelwa. Lesi sifundo samasonto ayishumi nambili, esidinga amahora okufunda angu-4 kuya kwangu-6 ngeviki, ukuhlola okuthakazelisayo kokuthi amathuba nokucabangela kusetshenziswa kanjani ezindaweni ezihlukahlukene njengokuhlunga ogaxekile, ukuzulazula kwe-bot yeselula, noma ngisho nasemidlalweni yamasu efana ne-Jeopardy ne-Go.
Kulesi sifundo, uzofunda imigomo yamathuba kanye nencazelo nokuthi ungayisebenzisa kanjani ezinhlelweni zekhompiyutha ezibonisa ukungaqiniseki futhi wenze izibikezelo. Uzofunda mayelana nezinhlaka zedatha ezihlukene zokugcina ukusatshalaliswa kwamathuba, njengamamodeli wesithombe esingaba khona, futhi uthuthukise ama-algorithms asebenza kahle wokubonisana nalezi zakhiwo zedatha.
Ekupheleni kwalesi sifundo, uzokwazi ukuthi ungamodela kanjani izinkinga zomhlaba wangempela ngamathuba kanye nendlela yokusebenzisa amamodeli avelayo ukuze uqonde. Awudingi ukuba nolwazi lwangaphambili emathubeni noma ekuqondeni, kodwa kufanele ukhululeke ngohlelo oluyisisekelo lwePython nokubala.
Lesi sifundo siwumthombo obalulekile walabo abafuna ukuqonda nokusebenzisa izindlela zezibalo emkhakheni wesayensi yedatha, ehlinzeka ngombono obanzi mayelana namamodeli angenzeka kanye nokusho kwezibalo.
Enhliziyweni Yokungaqiniseki: I-MIT Ihlukanisa Amathuba
Esifundweni esithi “Isingeniso Sengxenye II: Izinqubo Zokucabanga”, I-Massachusetts Institute of Technology (MIT) inikeza ukucwiliswa okuthuthukisiwe emhlabeni wamathuba kanye nencazelo. Lesi sifundo, ngesiNgisi ngokuphelele, siwukuqhubeka okunengqondo kwengxenye yokuqala, ukungena kujule ekuhlaziyeni idatha kanye nesayensi yokungaqiniseki.
Esikhathini esingamaviki ayishumi nesithupha, ngokuzibophezela kwamahora angu-6 ngesonto, lesi sifundo sihlola imithetho yezinombolo ezinkulu, izindlela ze-Bayesian inference, izibalo zakudala, nezinqubo ezingahleliwe ezifana nezinqubo ze-Poisson namaketanga e-Markov. Lokhu ukuhlola okuqinile, okuhloselwe labo asebevele banesisekelo esiqinile okungenzeka.
Lesi sifundo sigqama ngendlela yaso enembile, kuyilapho sigcina ukuqina kwezibalo. Ayethuli nje amathiyori nobufakazi, kodwa ihlose ukuthuthukisa ukuqonda okujulile kwemiqondo ngokusebenzisa ukusetshenziswa okuphathekayo. Abafundi bazofunda ukwenza imodeli yezenzakalo eziyinkimbinkimbi futhi bahumushe idatha yomhlaba wangempela.
Ilungele ochwepheshe besayensi yedatha, abacwaningi, nabafundi, lesi sifundo sinikeza umbono ohlukile wokuthi amathuba nokucabanga kukulolonga kanjani ukuqonda kwethu umhlaba. Iphelele kulabo abafuna ukujulisa ukuqonda kwabo kwesayensi yedatha nokuhlaziywa kwezibalo.
I-Analytical Combinatorics: Isifundo se-Princeton Sokucacisa Izakhiwo Eziyinkimbinkimbi (Princeton)
Isifundo se-Analytic Combinatorics, esinikezwa yiNyuvesi yase-Princeton, siwukuhlola okuthakazelisayo kwama-analytical combinatorics, isiyalo esivumela ukubikezela okunembayo kwenani lezakhiwo ezihlanganisiwe eziyinkimbinkimbi. Lesi sifundo, ngesiNgisi ngokuphelele, siyisisetshenziswa esibalulekile salabo abafuna ukuqonda nokusebenzisa izindlela ezithuthukile emkhakheni wama-combinatoric.
Ihlala amasonto amathathu futhi idinga cishe amahora angu-16 esewonke, noma cishe amahora angu-5 ngeviki, lesi sifundo sethula indlela engokomfanekiso yokuthola ubudlelwano obusebenzayo phakathi kwemisebenzi evamile, evezayo, kanye nemisebenzi eminingi ekhiqizayo. Iphinde ihlole izindlela zokuhlaziya okuyinkimbinkimbi ukuze kutholwe ama-asymptotics anemba kusukela kuzibalo zemisebenzi ekhiqizayo.
Abafundi bazothola ukuthi ama-combinatorics okuhlaziya angasetshenziswa kanjani ukubikezela amanani anembayo ezakhiweni ezinkulu zokuhlanganisa. Bazofunda ukuphatha izakhiwo zokuhlanganisa futhi basebenzise amasu okuhlaziya ayinkimbinkimbi ukuze bahlaziye lezi zakhiwo.
Lesi sifundo silungele labo abafuna ukujulisa ukuqonda kwabo ama-combinatorics kanye nokusebenza kwawo ekuxazululeni izinkinga eziyinkimbinkimbi. Inikeza umbono oyingqayizivele wokuthi ama-analytical combinatorics alolonga kanjani ukuqonda kwethu kwezibalo nezakhiwo ezihlangene.
