Manyan Darussan Lissafi na Kyauta daga Jami'o'in Elite

Abubuwan da ke cikin shafi

Darussa a cikin Faransanci

Random: Gabatarwa zuwa Yiwuwa – Kashi na 1 (POLYTECHNIQUE PARIS)

École Polytechnique, sanannen cibiya, yana ba da kwas mai ban sha'awa akan Coursera mai taken "Random: gabatarwa ga yuwuwar - Sashe na 1". Wannan darasi, wanda ke ɗaukar kusan sa'o'i 27 yana bazu cikin makonni uku, wata dama ce ta musamman ga duk mai sha'awar tushen yuwuwar. An ƙera shi don zama mai sassauƙa da daidaitawa da taki na kowane ɗalibi, wannan kwas ɗin yana ba da zurfi da isa ga ka'idar yiwuwar.

Shirin ya ƙunshi nau'ikan haɗaka guda 8, kowanne yana magana da mahimman abubuwan sararin yuwuwar, dokokin yuwuwar iri ɗaya, daidaitawa, 'yancin kai, da masu canjin bazuwar. Kowane tsarin yana wadatar da bidiyoyin bayani, ƙarin karatu da tambayoyi don gwadawa da haɓaka ilimin da aka samu. Har ila yau, ɗalibai suna da damar samun takardar shaidar da za a iya raba su bayan kammala karatun, suna ƙara ƙima mai mahimmanci ga ƙwararrunsu ko tafiyar ilimi.

Masu koyarwa, Sylvie Méléard, Jean-René Chazottes da Carl Graham, duk suna da alaƙa da École Polytechnique, suna kawo ƙwarewarsu da sha'awar ilimin lissafi, suna yin wannan kwas ɗin ba kawai ilimi ba, har ma da ban sha'awa. Ko kai ɗalibin lissafi ne, ƙwararren mai neman zurfafa ilimin ku, ko kuma ƙwararren masanin kimiyya ne kawai, wannan kwas ɗin yana ba da dama ta musamman don shiga cikin duniyar yuwuwar yuwuwar, wanda wasu mafi kyawun tunani a École Polytechnique ke jagoranta.

Random: Gabatarwa zuwa Yiwuwa – Kashi na 2 (POLYTECHNIQUE PARIS)

Ci gaba da ingantaccen ilimi na École Polytechnique, kwas ɗin "Random: Gabatarwa ga yuwuwar - Kashi na 2" akan Coursera ci gaba ne kai tsaye da haɓaka ɓangaren farko. Wannan darasi, wanda aka kiyasta zai wuce sa'o'i 17 yana bazuwa cikin makonni uku, yana nutsar da ɗalibai cikin ƙarin haɓakar ra'ayoyin ka'idar yuwuwar, samar da zurfin fahimta da fa'ida aikace-aikace na wannan horo mai ban sha'awa.

Tare da ingantattun kayayyaki guda 6, kwas ɗin ya ƙunshi batutuwa irin su bazuwar vectors, ƙaddamar da lissafin doka, dokar manyan lambobi, hanyar Monte Carlo, da ka'idar iyaka ta tsakiya. Kowane tsarin ya ƙunshi bidiyoyi na ilimi, karatu da tambayoyi, don ƙwarewar koyo mai zurfi. Wannan tsari yana bawa ɗalibai damar yin aiki tare da kayan kuma suyi amfani da abubuwan da aka koya ta hanya mai amfani.

Masu koyarwa, Sylvie Méléard, Jean-René Chazottes da Carl Graham sun ci gaba da jagorantar ɗalibai ta wannan tafiya ta ilimi tare da ƙwarewarsu da sha'awar ilimin lissafi. Hanyar koyarwarsu tana sauƙaƙe fahimtar ra'ayoyi masu rikitarwa kuma suna ƙarfafa zurfin bincike mai yiwuwa.

Wannan karatun yana da kyau ga waɗanda suka riga sun sami tushe mai ƙarfi a cikin yuwuwar kuma suna son faɗaɗa fahimtarsu da ikon amfani da waɗannan ra'ayoyin zuwa matsaloli masu rikitarwa. Ta hanyar kammala wannan kwas, ɗalibai kuma za su iya samun takardar shaidar da za a iya raba su, suna nuna himma da ƙwarewarsu a wannan yanki na musamman.

Gabatarwa ga ka'idar rarraba (POLYTECHNIQUE PARIS)

Kos ɗin "Gabatarwa ga ka'idar rarrabawa", wanda École Polytechnique akan Coursera ya bayar, yana wakiltar bincike na musamman da zurfafa bincike na babban filin lissafi. Wannan darasi, wanda ke ɗaukar kusan sa'o'i 15 yana bazu cikin makonni uku, an tsara shi don waɗanda ke neman fahimtar rarrabawa, mahimman ra'ayi a cikin lissafi da bincike.

Shirin ya ƙunshi matakai 9, kowannensu yana ba da haɗin bidiyon ilimi, karatu da quizzes. Waɗannan samfuran sun ƙunshi bangarori daban-daban na ka'idar rarraba, gami da al'amurra masu sarƙaƙƙiya kamar ma'anar abin da aka samo asali na aikin da aka daina aiki da kuma amfani da ayyukan dakatarwa azaman mafita ga daidaitattun daidaito. Wannan tsarin da aka tsara yana bawa ɗalibai damar a hankali su saba da ra'ayoyi waɗanda ka iya kama da ban tsoro da farko.

Furofesa François Golse da Yvan Martel, manyan membobin École Polytechnique, sun kawo ƙware sosai ga wannan kwas. Koyarwarsu ta haɗu da ƙwaƙƙwaran ilimi da sabbin hanyoyin koyarwa, sa abun ciki ya sami dama da jan hankali ga ɗalibai.

Wannan kwas ɗin ya dace musamman ga ɗalibai a fannin lissafi, injiniyanci, ko fannonin da ke da alaƙa waɗanda ke neman zurfafa fahimtar aikace-aikacen lissafi masu rikitarwa. Ta hanyar kammala wannan kwas, mahalarta ba kawai za su sami ilimi mai mahimmanci ba, har ma za su sami damar samun takardar shedar raba gardama, suna ƙara ƙima ga ƙwararrun su ko bayanan ilimi.

Gabatarwa ga ka'idar Galois (MAFITA NORMAL SCHOOL PARIS)

Wanda École Normale Supérieure ya bayar akan Coursera, darasin "Gabatarwa ga Ka'idar Galois" bincike ne mai ban sha'awa na ɗayan manyan rassa masu zurfi da tasiri na lissafin zamani.Dadewa kusan awanni 12, wannan kwas ɗin tana nutsar da ɗalibai cikin haɗaɗɗiyar duniyar ka'idar Galois, horon da ya kawo sauyi ga fahimtar alakar da ke tsakanin ma'auni mai yawa da tsarin algebraic.

Kwas ɗin yana mai da hankali kan nazarin tushen polynomials da maganganunsu daga ƙididdiga, tambaya ta tsakiya a cikin algebra. Ya binciko ra'ayin ƙungiyar Galois, wanda Évariste Galois ya gabatar, wanda ke danganta kowane ɗabi'a da rukuni na ruɗaɗɗen tushen sa. Wannan hanya tana ba mu damar fahimtar dalilin da ya sa ba zai yiwu a bayyana tushen wasu ma'auni masu yawa ta tsarin algebraic ba, musamman ga nau'ikan digiri fiye da hudu.

Wasiƙun Galois, wani mahimmin ɓangaren kwas ɗin, yana haɗa ka'idar filin zuwa ka'idar rukuni, yana ba da hangen nesa na musamman kan warware ma'auni masu tsattsauran ra'ayi. Kwas ɗin yana amfani da mahimman ra'ayoyi a cikin algebra na layi don kusanci ka'idar jiki da gabatar da ra'ayi na lambar algebra, yayin da ake bincika ƙungiyoyin abubuwan da suka dace don nazarin ƙungiyoyin Galois.

Wannan kwas ɗin sananne ne musamman don ikonsa na gabatar da ƙayyadaddun ra'ayoyin algebra cikin sauƙi da sauƙi, kyale ɗalibai su sami sakamako mai ma'ana cikin sauri tare da ƙaramin ƙa'idar tsari. Yana da kyau ga ɗaliban lissafi, kimiyyar lissafi, ko injiniyanci, da masu sha'awar lissafi da ke neman zurfafa fahimtar tsarin algebra da aikace-aikacen su.

Ta hanyar kammala wannan kwas, mahalarta ba kawai za su sami zurfin fahimtar ka'idar Galois ba, amma kuma za su sami damar samun takardar shaidar da za a iya raba su, suna ƙara ƙima ga ƙwararrun su ko bayanan ilimi.

Analysis I (Kashi na 1): Gabatarwa, ra'ayi na asali, ainihin lambobi (MAKARANTA POLYTECHNIQUE FEDERALE DE LAUSANNE)

Kos din "Bincike I (Kashi na 1): Gabatarwa, ra'ayi na asali, lambobi na gaske", wanda École Polytechnique Fédérale de Lausanne ke bayarwa akan edX, gabatarwa ne mai zurfi ga mahimman ra'ayoyin bincike na gaske. Wannan darasi na mako 5, yana buƙatar kusan sa'o'i 4-5 na nazari a kowane mako, an tsara shi don kammala shi cikin takun ku.

Abubuwan da ke cikin kwas ɗin suna farawa da share fage wanda ke sake dubawa da zurfafa mahimman ra'ayoyin ilimin lissafi kamar ayyukan trigonometric (zunubi, cos, tan), ayyuka masu daidaitawa (exp, ln), da kuma ƙa'idodin lissafi don iko, logarithms da tushen. Hakanan ya ƙunshi saiti da ayyuka na asali.

Jigon kwas ɗin yana mai da hankali kan tsarin lamba. An fara daga fahimtar lambobi na dabi'a, kwas ɗin yana ƙayyadaddun ma'anar lambobi masu ma'ana da bincika abubuwan su. An biya kulawa ta musamman ga lambobi na ainihi, an gabatar da su don cike giɓi a lambobi masu ma'ana. Kwas ɗin yana ba da ma'anar axiomatic na ainihin lambobi kuma yana nazarin kaddarorin su daki-daki, gami da ra'ayoyi kamar rashin ƙarfi, mafi girman ƙima, ƙima da sauran ƙarin kaddarorin lambobi na gaske.

Wannan karatun yana da kyau ga waɗanda ke da ilimin lissafi na asali kuma suna son zurfafa fahimtar bincike na zahiri. Yana da amfani musamman ga ɗaliban lissafi, kimiyyar lissafi, ko injiniyanci, da kuma duk mai sha'awar fahimtar tushen ilimin lissafi.

Ta hanyar kammala wannan kwas, mahalarta za su sami cikakkiyar fahimtar lambobi na ainihi da mahimmancin su a cikin bincike, da kuma damar samun takardar shaidar da za a iya raba su, suna ƙara ƙima ga ƙwararrun su ko bayanan ilimi.

Analysis I (Kashi na 2): Gabatarwa zuwa hadaddun lambobi (MAKARANTA POLYTECHNIQUE FEDERALE DE LAUSANNE)

Kos din "Bincike I (Kashi na 2): Gabatarwa ga lambobi masu rikitarwa", wanda École Polytechnique Fédérale de Lausanne ke bayarwa akan edX, gabatarwa ce mai jan hankali ga duniyar lambobi masu rikitarwa.Wannan darasi na mako 2, yana buƙatar kusan sa'o'i 4-5 na nazari a kowane mako, an tsara shi don kammala shi cikin takun ku.

Kwas ɗin yana farawa ne ta hanyar magance ma'aunin z^2 = -1, wanda ba shi da mafita a cikin saitin lambobi na ainihi, R. Wannan matsala ta haifar da ƙaddamar da lambobi masu rikitarwa, C, filin da ya ƙunshi R kuma yana ba mu damar magance irin wannan. daidaito. Kwas ɗin ya bincika hanyoyi daban-daban na wakiltar hadadden lamba kuma ya tattauna mafita ga daidaiton nau'in z^n = w, inda n ya kasance na N* da w zuwa C.

Babban mahimmancin kwas ɗin shine nazarin ainihin ka'idar algebra, wanda shine babban sakamako a cikin lissafi. Har ila yau, kwas ɗin ya ƙunshi batutuwa irin su wakilcin Cartesian na lambobi masu rikitarwa, abubuwan da suke da su na farko, abubuwan da suka bambanta don ninkawa, tsarin Euler da de Moivre, da nau'in polar nau'i mai rikitarwa.

Wannan hanya ita ce manufa ga waɗanda suka riga sun sami wasu ilimin lambobi na ainihi kuma suna so su fadada fahimtar su zuwa lambobi masu rikitarwa. Yana da amfani musamman ga ɗaliban lissafi, kimiyyar lissafi, ko injiniyanci, da kuma duk mai sha'awar zurfin fahimtar algebra da aikace-aikacen sa.

Ta hanyar kammala wannan kwas, mahalarta za su sami ƙwararrun fahimtar lambobi masu rikitarwa da muhimmiyar rawar da suke takawa a cikin lissafi, da kuma damar samun takardar shaidar da za a iya raba su, suna ƙara ƙima ga ƙwararrunsu ko bayanan ilimi.

Nazarin I (Kashi na 3): Jerin lambobi na ainihi I da II (MAKARANTA POLYTECHNIQUE FEDERALE DE LAUSANNE)

Kos din "Bincike I (Kashi na 3): Jerin lambobi na ainihi I da II", wanda École Polytechnique Fédérale de Lausanne ke bayarwa akan edX, yana mai da hankali kan jeri na ainihin lambobi. Wannan darasi na mako 4, yana buƙatar kusan sa'o'i 4-5 na nazari a kowane mako, an tsara shi don kammala shi cikin takun ku.

Babban manufar wannan kwas shine iyakar jerin lambobi na ainihi. Yana farawa ta hanyar ayyana jerin lambobi na ainihi a matsayin aiki daga N zuwa R. Misali, ana binciko jerin a_n = 1/2^n, yana nuna yadda yake tunkarar sifili. Kwas ɗin yana magana da ƙarfi sosai akan ma'anar iyakar jeri kuma yana haɓaka hanyoyin tabbatar da wanzuwar iyaka.

Bugu da ƙari, kwas ɗin yana kafa hanyar haɗi tsakanin ra'ayi na iyaka da na marasa lafiya da mafi girman saiti. Wani muhimmin aikace-aikacen jerin lambobi na ainihi ana kwatanta shi ta gaskiyar cewa kowace lamba ta gaske za a iya la'akari da iyakar jerin lambobi masu hankali. Har ila yau, kwas ɗin yana bincika jerin Cauchy da jerin abubuwan da aka ayyana ta hanyar shigar da layi, da kuma ka'idar Bolzano-Weierstrass.

Mahalarta kuma za su koyi game da jerin lambobi, tare da gabatarwa ga misalai daban-daban da ma'auni, kamar ma'aunin d'Alembert, ma'aunin Cauchy, da ma'aunin Leibniz. Kwas ɗin ya ƙare tare da nazarin jerin lambobi tare da ma'auni.

Wannan karatun yana da kyau ga waɗanda ke da ilimin lissafi na asali kuma suna son zurfafa fahimtar jerin lambobi na ainihi. Yana da amfani musamman ga ɗaliban lissafi, kimiyyar lissafi ko injiniyanci. Ta hanyar kammala wannan kwas, mahalarta za su haɓaka fahimtar ilimin lissafi kuma suna iya samun takardar shaidar da za a iya raba su, wata kadara don ƙwararrun su ko ci gaban ilimi.

Gano Haƙiƙanin Ayyuka da Ci gaba: Bincike na I (Kashi na 4) (MAKARANTA POLYTECHNIQUE FEDERALE DE LAUSANNE)

A cikin "Bincike I (Kashi na 4): Iyakar aiki, ci gaba da ayyuka", École Polytechnique Fédérale de Lausanne yana ba da tafiya mai ban sha'awa cikin nazarin ainihin ayyuka na ainihin canji.Wannan hanya, mai dorewa na makonni 4 tare da sa'o'i 4 zuwa 5 na nazarin mako-mako, yana samuwa akan edX kuma yana ba da damar ci gaba a kan ku.

Wannan ɓangaren kwas ɗin yana farawa tare da gabatar da ayyuka na gaske, yana mai da hankali ga kaddarorin su kamar monotonicity, daidaito, da lokaci-lokaci. Hakanan yana bincika ayyuka tsakanin ayyuka da gabatar da takamaiman ayyuka kamar ayyukan hyperbolic. An ba da kulawa ta musamman ga ayyuka da aka ayyana mataki-mataki, gami da ayyukan Signum da Heaviside, da kuma canjin affine.

Jigon kwas ɗin yana mai da hankali kan ƙayyadaddun ƙayyadaddun aiki a wani wuri, yana ba da misalai na ƙayyadaddun iyakoki na ayyuka. Hakanan ya ƙunshi ra'ayoyin iyakokin hagu da dama. Na gaba, kwas ɗin yana duba iyakokin ayyuka marasa iyaka kuma yana ba da kayan aiki masu mahimmanci don ƙididdige iyaka, kamar ka'idar ɗan sanda.

Wani muhimmin al'amari na kwas ɗin shine gabatar da manufar ci gaba, wanda aka ayyana ta hanyoyi guda biyu, da kuma amfani da shi don faɗaɗa wasu ayyuka. Kwas ɗin ya ƙare tare da nazarin ci gaba a kan buɗaɗɗen tazara.

Wannan kwas ɗin wata dama ce mai wadatarwa ga waɗanda ke neman zurfafa fahimtarsu game da ayyuka na gaske da ci gaba. Ya dace da ɗaliban lissafi, kimiyyar lissafi ko injiniyanci. Ta hanyar kammala wannan kwas, mahalarta ba kawai za su faɗaɗa ilimin lissafin su ba, har ma za su sami damar samun takardar shaida mai lada, buɗe kofa ga sabbin hanyoyin ilimi ko ƙwarewa.

Binciko Ayyuka Daban-daban: Bincike na I (Kashi na 5) (MAKARANTA POLYTECHNIQUE FEDERALE DE LAUSANNE)

École Polytechnique Fédérale de Lausanne, a cikin tayin ilimantarwa akan edX, yana gabatar da "Bincike I (sashe na 5): Ci gaba da ayyuka da ayyuka daban-daban, aikin haɓaka". Wannan darasi na mako hudu, yana buƙatar kimanin sa'o'i 4-5 na nazari a kowane mako, bincike ne mai zurfi na ra'ayoyin bambanci da ci gaba da ayyuka.

Kwas ɗin yana farawa tare da zurfafa nazarin ayyukan ci gaba, yana mai da hankali kan kaddarorin su akan rufaffiyar tazara. Wannan sashe yana taimaka wa ɗalibai su fahimci matsakaicin da mafi ƙarancin ayyukan ci gaba. Sa'an nan kwas ɗin ya gabatar da hanyar bisection kuma yana gabatar da mahimman ka'idoji kamar matsakaicin ƙimar ƙimar da ƙayyadaddun ka'idar batu.

Babban ɓangaren kwas ɗin ya keɓe ga bambance-bambance da bambancin ayyuka. Dalibai suna koyon fassara waɗannan ra'ayoyin kuma su fahimci daidaicinsu. Sa'an nan kwas ɗin ya dubi ginin aikin haɓaka kuma yayi nazarin kaddarorinsa dalla-dalla, gami da ayyukan algebra akan ayyukan da aka samo asali.

Wani muhimmin al'amari na kwas ɗin shine nazarin kaddarorin ayyuka daban-daban, kamar abubuwan da suka samo asali na abubuwan da ke tattare da ayyuka, ka'idar Rolle's theorem, da ƙayyadaddun ƙa'idodin haɓaka. Har ila yau, kwas ɗin ya bincika ci gaba da aikin da aka samo asali da kuma abubuwan da ke tattare da shi a kan kadaitacce na wani aiki mai ban mamaki.

Wannan hanya ita ce kyakkyawar dama ga waɗanda suke so su zurfafa fahimtar ayyuka daban-daban da ci gaba. Ya dace da ɗaliban lissafi, kimiyyar lissafi ko injiniyanci. Ta hanyar kammala wannan kwas, mahalarta ba wai kawai za su faɗaɗa fahimtar mahimman ra'ayoyin ilimin lissafi ba, har ma za su sami damar samun takardar shedar lada, buɗe kofa ga sababbin damar ilimi ko ƙwarewa.

Zurfafa cikin Nazarin Lissafi: Nazari na I (Kashi na 6) (MAKARANTA POLYTECHNIQUE FEDERALE DE LAUSANNE)

Kos din "Bincike I (Sashe na 6): Nazarin ayyuka, iyakancewar ci gaba", wanda École Polytechnique Fédérale de Lausanne ke bayarwa akan edX, zurfin bincike ne na ayyuka da iyakancewar ci gaban su. Wannan kwas na mako huɗu, tare da nauyin aiki na sa'o'i 4 zuwa 5 a kowane mako, yana bawa ɗalibai damar ci gaba a cikin matakan da suka dace.

Wannan babi na kwas ɗin yana mai da hankali kan zurfin nazarin ayyuka, ta yin amfani da ka'idoji don bincika bambance-bambancen su. Bayan magance ƙayyadaddun ka'idar haɓaka haɓaka, kwas ɗin yana duba gaba ɗaya. Wani muhimmin al'amari na nazarin ayyuka shine fahimtar halayensu a iyaka. Don yin wannan, kwas ɗin yana gabatar da ƙa'idar Bernoulli-l'Asibiti, kayan aiki mai mahimmanci don ƙayyade ƙayyadaddun iyaka na wasu ƙididdiga.

Har ila yau, kwas ɗin yana bincika yanayin wakilcin ayyuka, nazarin tambayoyi kamar kasancewar maxima na gida ko na duniya ko minima, da madaidaicin ko madaidaicin ayyuka. Dalibai za su koyi gano daban-daban asymptotes na wani aiki.

Wani mahimmin batu na kwas ɗin shine ƙaddamar da ƙayyadaddun fadada aiki, wanda ke ba da ƙima mai yawa a kusa da wani batu. Wadannan ci gaba suna da mahimmanci don sauƙaƙe lissafin iyaka da nazarin kaddarorin ayyuka. Har ila yau, kwas ɗin ya ƙunshi jerin integer da radius na haɗin kai, da kuma jerin jerin Taylor, kayan aiki mai ƙarfi don wakiltar ayyuka masu banƙyama.

Wannan kwas wata hanya ce mai mahimmanci ga waɗanda ke neman zurfafa fahimtar ayyuka da aikace-aikacen su a cikin lissafi. Yana ba da ingantacciyar fahimta da cikakken hangen nesa kan mahimman ra'ayoyi a cikin binciken lissafi.

Jagoran Haɗin Kai: Nazari na I (Kashi na 7) (MAKARANTA POLYTECHNIQUE FEDERALE DE LAUSANNE)

Kwas ɗin yana farawa tare da ma'anar ma'anar ma'auni marar iyaka da ƙayyadaddun ƙayyadaddun ƙayyadaddun, yana gabatar da ƙayyadaddun mahimmanci ta hanyar jimlar Riemann da babba da ƙananan kuɗi. Sa'an nan kuma ya tattauna mahimman kaddarorin guda uku na ƙayyadaddun ƙayyadaddun ƙayyadaddun ƙayyadaddun ƙayyadaddun ƙayyadaddun ƙayyadaddun ƙayyadaddun ƙayyadaddun abubuwa: layi na haɗin kai, rabe-raben yanki na haɗin kai, da monotonicity na haɗin kai.

Babban mahimmin darasi shine ma'anar ma'anar don ci gaba da ayyuka akan sashe, wanda aka nuna dalla-dalla. Kwas ɗin ya kai kololuwar sa tare da ainihin ka'idar lissafi na haɗin kai, yana gabatar da ra'ayi na hana haɓakar wani aiki. Dalibai suna koyon dabarun haɗin kai iri-iri, kamar haɗakarwa ta sassa, canza masu canji, da haɗawa ta hanyar ƙaddamarwa.

Kos ɗin ya ƙare tare da nazarin haɗakar ayyuka na musamman, gami da haɗaɗɗen ƙayyadaddun faɗaɗa aiki, haɗar jerin ƙididdiga, da haɗin ayyukan ci gaba na yanki. Waɗannan fasahohin suna ba da damar haɗaɗɗun ayyuka tare da siffofi na musamman don ƙididdige su da inganci. A ƙarshe, kwas ɗin yana bincika ƙayyadaddun abubuwan haɗin kai, wanda aka ayyana ta hanyar wuce iyaka a cikin abubuwan haɗin gwiwa, kuma yana gabatar da misalai na gaske.

Wannan kwas wata hanya ce mai mahimmanci ga waɗanda ke neman ƙwarewar haɗin kai, kayan aiki na asali a cikin lissafi. Yana ba da cikakkiyar hangen nesa mai amfani game da haɗin kai, wadatar da ƙwarewar lissafin ɗalibai.

Darussa a Turanci

Gabatarwa zuwa Samfuran Lissafi da Matrix Algebra (Harvard)

Jami'ar Harvard, ta hanyar dandalinta na HarvardX akan edX, tana ba da kwas ɗin "Gabatarwa zuwa Model Linear da Matrix Algebra". Kodayake ana koyar da kwas ɗin a cikin Ingilishi, yana ba da dama ta musamman don koyan tushe na algebra na matrix da ƙirar layi, ƙwarewa masu mahimmanci a fannonin kimiyya da yawa.

Wannan darasi na mako hudu, yana buƙatar sa'o'i 2 zuwa 4 na nazari a kowane mako, an tsara shi don kammala shi da sauri. Yana mai da hankali kan yin amfani da yaren shirye-shirye na R don amfani da ƙirar layi a cikin nazarin bayanai, musamman a cikin ilimin kimiyyar rayuwa. Dalibai za su koyi sarrafa algebra na matrix kuma su fahimci aikace-aikacen sa a cikin ƙirar gwaji da ƙididdigar bayanai masu girma.

Shirin ya ƙunshi bayanin algebra na matrix, ayyukan matrix, aikace-aikacen matrix algebra zuwa nazarin bayanai, ƙirar layi, da gabatarwa ga bazuwar QR. Wannan kwas na cikin jerin darussa guda bakwai, waɗanda za a iya ɗauka ɗaya ɗaya ko a matsayin wani ɓangare na takaddun ƙwararru guda biyu a cikin Binciken Bayanai don Kimiyyar Rayuwa da Nazarin Bayanan Halittar Halittar Halitta.

Wannan kwas ɗin ya dace da waɗanda ke neman samun ƙwarewa a ƙirar ƙididdiga da nazarin bayanai, musamman a mahallin kimiyyar rayuwa. Yana ba da tushe mai ƙarfi ga waɗanda ke son ƙara bincika algebra na matrix da aikace-aikacen sa a fannonin kimiyya da bincike daban-daban.

Yiwuwar Jagora (Harvard)

LJerin waƙa na "Kididdiga 110: Yiwuwa" akan YouTube, wanda Joe Blitzstein na Jami'ar Harvard ya koyar da Ingilishi, hanya ce mai kima ga waɗanda ke neman zurfafa iliminsu na yuwuwar.. Lissafin waƙa ya haɗa da bidiyon darasi, kayan bita, da motsa jiki sama da 250 tare da cikakkun bayanai.

Wannan darasi na Ingilishi cikakkiyar gabatarwa ce ga yuwuwar, wanda aka gabatar a matsayin harshe mai mahimmanci da saitin kayan aikin don fahimtar ƙididdiga, kimiyya, haɗari da bazuwar. Abubuwan da ake koyarwa suna aiki a fannoni daban-daban kamar kididdiga, kimiyya, injiniyanci, tattalin arziki, kuɗi da rayuwar yau da kullun.

Batutuwan da aka rufe sun haɗa da tushen yuwuwar, masu canjin bazuwar da rarrabawar su, rabe-raben rabe-rabe iri-iri, ƙayyadaddun ka'idoji, da sarƙoƙin Markov. Kwas ɗin yana buƙatar sanin farko na ƙididdiga mai canzawa ɗaya da sanin matrices.

Ga waɗanda ke jin daɗin Ingilishi kuma suna sha'awar bincika duniyar yuwuwar zurfafa, wannan jerin kwas na Harvard yana ba da damar koyo mai wadatarwa. Kuna iya samun damar lissafin waƙa da cikakkun abubuwan da ke cikinsa kai tsaye akan YouTube.

Yiwuwar Bayyana. Darasi tare da Fassarar Faransanci (Harvard)

Hanya "Fat Chance: Probability from the Ground Up," wanda HarvardX ya bayar akan edX, gabatarwa ne mai ban sha'awa ga yiwuwar da ƙididdiga. Kodayake ana koyar da kwas ɗin a cikin Ingilishi, ana samun dama ga masu sauraron Faransanci godiya ga fassarar Faransanci da ke akwai.

Wannan kwas na mako bakwai, yana buƙatar sa'o'i 3 zuwa 5 na karatu a kowane mako, an tsara shi don waɗanda suka kasance sababbi ga nazarin yiwuwar ko neman damar yin bitar mahimman ra'ayoyin kafin shiga cikin kwas ɗin kididdiga. matakin Jami'a. "Fat Chance" yana jaddada haɓaka tunanin lissafi maimakon haddar kalmomi da dabaru.

Na'urorin farko suna gabatar da ƙwarewar ƙidayar ƙidayar, waɗanda sannan a yi amfani da su ga matsalolin yuwuwar sauƙi. Modules na gaba suna bincika yadda za'a iya daidaita waɗannan ra'ayoyi da dabaru don magance ɗimbin matsalolin yuwuwar. Kwas ɗin ya ƙare tare da gabatarwa ga ƙididdiga ta hanyar ra'ayi na ƙimar da ake tsammani, bambance-bambance da rarraba na al'ada.

Wannan karatun yana da kyau ga waɗanda ke neman haɓaka ƙwarewar tunani mai ƙididdigewa da fahimtar tushen yuwuwar da ƙididdiga. Yana ba da ingantaccen hangen nesa game da tarin yanayin lissafi da kuma yadda ya shafi fahimtar haɗari da bazuwar.

Ƙididdigar Ƙididdiga da Ƙirƙira don Ƙwararrun Gwaji (Harvard)

Kwas ɗin “Inference Inference and Modeling for High-throughput Experiments” a cikin Ingilishi yana mai da hankali kan dabarun da ake amfani da su don yin ƙididdige ƙididdiga akan manyan bayanai. Wannan darasi na mako hudu, yana buƙatar sa'o'i 2-4 na nazari a kowane mako, hanya ce mai mahimmanci ga waɗanda ke neman fahimta da amfani da hanyoyin ƙididdiga masu ci gaba a cikin saitunan bincike mai zurfi.

Shirin ya ƙunshi batutuwa iri-iri, gami da matsalar kwatantawa da yawa, ƙimar kuskure, hanyoyin sarrafa ƙimar kuskure, ƙimar gano ƙarya, ƙima-q, da binciken bayanan bincike. Hakanan yana gabatar da ƙirar ƙididdiga da aikace-aikacen sa zuwa manyan bayanan da aka samar, suna tattaunawa akan rarraba madaidaici kamar binomial, exponential, da gamma, da kuma kwatanta ƙimar ƙima.

Dalibai za su koyi yadda ake amfani da waɗannan ra'ayoyin a cikin mahallin kamar jerin tsararru na gaba da bayanan ƙananan bayanai. Har ila yau, kwas ɗin ya ƙunshi nau'ikan tsari da ƙa'idodin Bayesian, tare da misalai masu amfani na amfani da su.

Wannan kwas ɗin yana da kyau ga waɗanda ke neman zurfafa fahimtar ƙididdigar ƙididdiga da ƙirar ƙira a cikin binciken kimiyya na zamani. Yana ba da hangen nesa mai zurfi game da ƙididdigar ƙididdiga na bayanai masu rikitarwa kuma yana da kyakkyawar hanya ga masu bincike, ɗalibai da masu sana'a a fannin ilimin kimiyyar rayuwa, bioinformatics da kididdiga.

Gabatarwa zuwa Yiwuwa (Harvard)

Kos ɗin "Gabatarwa ga Yiwuwa", wanda HarvardX ke bayarwa akan edX, bincike ne mai zurfi na yuwuwar, mahimman harshe da kayan aiki don fahimtar bayanai, dama, da rashin tabbas. Kodayake ana koyar da kwas ɗin a cikin Ingilishi, ana samun dama ga masu sauraron Faransanci godiya ga fassarar Faransanci da ke akwai.

Wannan darasi na mako goma, yana buƙatar sa'o'i 5-10 na nazari a kowane mako, yana da nufin kawo tunani zuwa duniyar da ke cike da dama da rashin tabbas. Zai samar da kayan aikin da ake buƙata don fahimtar bayanai, kimiyya, falsafa, injiniyanci, tattalin arziki da kuɗi. Ba wai kawai za ku koyi yadda ake warware matsalolin fasaha masu rikitarwa ba, har ma yadda ake amfani da waɗannan mafita a rayuwar yau da kullun.

Tare da misalan da suka kama daga gwajin likita zuwa tsinkayar wasanni, za ku sami ingantaccen tushe don nazarin ƙididdiga na ƙididdiga, matakai na stochastics, algorithms na bazuwar, da sauran batutuwa inda yiwuwar ya zama dole.

Wannan karatun yana da kyau ga waɗanda ke neman ƙara fahimtar rashin tabbas da dama, yin tsinkaya mai kyau, da fahimtar sauye-sauyen bazuwar. Yana ba da ingantaccen hangen nesa kan rabon yuwuwar gama gari da aka yi amfani da shi a cikin ƙididdiga da kimiyyar bayanai.

Ƙididdigar Ƙirar (Harvard)

Kwas ɗin "Kalli da Aka Aiwatar!", wanda Harvard ke bayarwa akan edX, bincike ne mai zurfi na aikace-aikacen lissafin ƙididdiga guda ɗaya a cikin zamantakewa, rayuwa, da kimiyyar jiki. Wannan darasi, gabaɗaya a cikin Ingilishi, kyakkyawar dama ce ga waɗanda ke neman fahimtar yadda ake amfani da ƙididdiga a cikin mahallin ƙwararrun ƙwararrun duniya.

Tsawon makonni goma kuma yana buƙatar tsakanin awanni 3 zuwa 6 na karatu a kowane mako, wannan kwas ɗin ya wuce litattafan gargajiya. Yana aiki tare da ƙwararru daga fannoni daban-daban don nuna yadda ake amfani da ƙididdiga don tantancewa da magance matsalolin gaske. Dalibai za su bincika aikace-aikace iri-iri, kama daga nazarin tattalin arziki zuwa ƙirar halitta.

Shirin ya ƙunshi amfani da abubuwan da aka samo asali, abubuwan haɗin kai, ma'auni daban-daban, kuma yana jaddada mahimmancin ƙirar lissafi da sigogi. An ƙirƙira shi don waɗanda ke da ainihin fahimtar ƙididdiga mai canzawa ɗaya kuma suna sha'awar aikace-aikacen sa a fagage daban-daban.

Wannan kwas ɗin cikakke ne ga ɗalibai, malamai, da ƙwararru waɗanda ke neman zurfafa fahimtar lissafi da gano aikace-aikacen sa na zahiri.

Gabatarwa ga tunanin ilimin lissafi (Stanford)

Kos ɗin "Gabatarwa ga Tunanin Lissafi", wanda Jami'ar Stanford ta bayar akan Coursera, nutsewa ne cikin duniyar tunanin ilimin lissafi. Kodayake ana koyar da kwas ɗin a cikin Ingilishi, ana samun dama ga masu sauraron Faransanci godiya ga fassarar Faransanci da ke akwai.

Wannan kwas na mako bakwai, wanda ke buƙatar kusan sa'o'i 38 gabaɗaya, ko kuma kusan awanni 12 a kowane mako, an tsara shi ne don waɗanda ke son haɓaka tunanin ilimin lissafi, wanda ya bambanta da yin lissafin kawai kamar yadda ake gabatar da shi a cikin tsarin makarantu. Kwas ɗin yana mai da hankali kan haɓaka hanyar tunani “a wajen akwatin”, fasaha mai mahimmanci a duniyar yau.

Dalibai za su binciko yadda ƙwararrun masu ilimin lissafin ke tunani don magance matsalolin duniya, ko sun taso daga duniyar yau da kullun, daga kimiyya, ko kuma daga lissafin kanta. Kwas ɗin yana taimakawa haɓaka wannan mahimmancin hanyar tunani, wuce hanyoyin ilmantarwa don warware matsalolin da ba a sani ba.

Wannan kwas ɗin ya dace ga waɗanda ke neman ƙarfafa ƙididdigar ƙima da fahimtar tushen tunanin ilimin lissafi. Yana ba da kyakkyawar hangen nesa game da tarin nau'ikan lissafi da aikace-aikacensa don fahimtar matsaloli masu rikitarwa.

Koyon Ƙididdiga tare da R (Stanford)

Koyarwar “Ƙididdiga tare da R”, wanda Stanford ke bayarwa, gabatarwa ce ta matsakaicin matakin ga ilmantarwa mai kulawa, mai da hankali kan koma baya da hanyoyin rarrabuwa. Wannan kwas, gabaɗaya a cikin Ingilishi, hanya ce mai mahimmanci ga waɗanda ke neman fahimta da amfani da hanyoyin ƙididdiga a fagen kimiyyar bayanai.

Tsawon makonni goma sha ɗaya kuma yana buƙatar sa'o'i 3-5 na karatu a kowane mako, kwas ɗin ya ƙunshi sabbin hanyoyin gargajiya da masu ban sha'awa a cikin ƙirar ƙididdiga, da yadda ake amfani da su cikin yaren shirye-shiryen R. An sabunta kwas ɗin a cikin 2021 don bugu na biyu na littafin koyarwa.

Batutuwa sun haɗa da juzu'i na layi da nau'i-nau'i, sauye-sauye na logistic da bincike na nuna wariya na madaidaiciya, tabbatarwa-giciye da bootstrapping, zaɓin ƙirar ƙira da hanyoyin daidaitawa (ridge da lasso), samfuran marasa tsari, splines da ƙirar ƙari gabaɗaya, hanyoyin tushen itace, dazuzzukan dazuzzuka da haɓakawa, goyan bayan injunan vector, hanyoyin sadarwar jijiyoyi da zurfin koyo, ƙirar tsira, da gwaji da yawa.

Wannan kwas ɗin ya dace da waɗanda ke da ainihin ilimin ƙididdiga, algebra na layi, da kimiyyar kwamfuta, kuma waɗanda ke neman zurfafa fahimtar ilimin ƙididdiga da aikace-aikacen sa a cikin ilimin kimiyyar bayanai.

Yadda ake Koyan Lissafi: Darasin Ga Kowa (Stanford)

Kos ɗin "Yadda ake Koyan Lissafi: Ga ɗalibai", wanda Stanford ke bayarwa. Darussan kan layi kyauta ne ga masu koyan duk matakan lissafi. Gabaɗaya a cikin Ingilishi, yana haɗa mahimman bayanai game da ƙwaƙwalwa tare da sabbin shaidu game da mafi kyawun hanyoyin kusanci ilimin lissafi.

Tsawan makonni shida kuma yana buƙatar 1 zuwa 3 hours na nazari a kowane mako. An tsara kwas ɗin don canza dangantakar ɗalibai da lissafi. Mutane da yawa sun sami mummunan gogewa game da lissafi, wanda ke haifar da ƙiyayya ko gazawa. Wannan kwas na nufin baiwa xaliban bayanan da suke buƙata don jin daɗin ilimin lissafi.

An rufe batutuwa kamar su kwakwalwa da koyon lissafi. Hakanan an rufe tatsuniyoyi game da lissafi, tunani, kuskure da sauri. Sassaucin lambobi, tunanin lissafi, haɗin kai, ƙirar ƙididdiga suma wani ɓangare ne na shirin. Ba a manta da wakilcin ilimin lissafi a rayuwa, amma kuma a cikin yanayi da kuma aiki. An tsara wannan kwas ɗin tare da koyar da haɗin kai mai aiki, yin ilmantarwa mai ma'amala da kuzari.

Hanya ce mai mahimmanci ga duk wanda ke son ganin ilimin lissafi daban. Haɓaka zurfin fahimta da kyakkyawar fahimta game da wannan horo. Ya dace musamman ga waɗanda suka sami mummunan gogewa tare da lissafi a baya kuma suna neman canza wannan fahimta.

Gudanar da Yiwuwar (Stanford)

Kwas ɗin "Gabatarwa ga Gudanar da Yiwuwa", wanda Stanford ya bayar, gabatarwa ce ga horo na sarrafa yiwuwar. Wannan filin yana mai da hankali kan sadarwa da ƙididdige rashin tabbas a cikin nau'in tebur ɗin bayanan da ake iya tantancewa da ake kira Fakitin Bayanin Stochastic (SIPs). Wannan kwas na mako goma yana buƙatar sa'o'i 1 zuwa 5 na nazari a kowane mako. Babu shakka yana da amfani mai mahimmanci ga masu neman fahimta da kuma amfani da hanyoyin ƙididdiga a fannin kimiyyar bayanai.

Tsarin karatun ya ƙunshi batutuwa kamar sanin “Aibi na Matsakaici,” saitin kurakurai na tsari waɗanda ke tasowa lokacin da rashin tabbas ke wakilta ta lambobi ɗaya, yawanci matsakaita. Yana bayyana dalilin da yasa yawancin ayyuka suka makara, fiye da kasafin kuɗi da kuma ƙarƙashin kasafin kuɗi. Har ila yau, kwas ɗin yana koyar da Ƙimar rashin tabbas, wanda ke yin ƙididdiga tare da abubuwan da ba su da tabbas, wanda ke haifar da rashin tabbas daga abin da za ku iya ƙididdige matsakaicin sakamako na gaskiya da kuma damar cimma takamaiman manufofi.

Dalibai za su koyi yadda ake ƙirƙira abubuwan kwaikwayo masu ma'amala waɗanda za a iya rabawa tare da kowane mai amfani da Excel ba tare da buƙatar ƙara-ins ko macros ba. Wannan hanyar ta dace daidai da Python ko kowane yanayin shirye-shirye wanda ke goyan bayan tsararraki.

Wannan karatun yana da kyau ga waɗanda ke jin daɗin Microsoft Excel kuma suna neman zurfafa fahimtarsu game da sarrafa yuwuwar da aikace-aikacen sa a cikin ilimin kimiyyar bayanai.

Kimiyyar Rashin tabbas da Bayanai (MIT)

Kwas ɗin " Yiwuwar - Kimiyyar Rashin tabbas da Bayanai ", wanda Cibiyar Fasaha ta Massachusetts (MIT) ta bayar. Gabatarwa ce ta asali ga kimiyyar bayanai ta hanyar samfuri mai yiwuwa. Wannan kwas ɗin yana ɗaukar makonni goma sha shida, yana buƙatar awanni 10 zuwa 14 na karatu a kowane mako. Ya dace da wani ɓangare na shirin MIT MicroMasters a cikin ƙididdiga da kimiyyar bayanai.

Wannan kwas ɗin yana bincika duniyar rashin tabbas: daga hatsarori a cikin kasuwannin kuɗi marasa tabbas zuwa sadarwa. Samfura mai yiwuwa da kuma filin da ya danganci ƙididdiga. Maɓallai biyu ne don nazarin wannan bayanai da yin hasashe masu inganci a kimiyyance.

Dalibai za su gano tsari da abubuwan asali na ƙirar ƙima. Ciki har da masu canjin bazuwar, rabon su, hanyoyin da bambance-bambance. Har ila yau, kwas ɗin ya ƙunshi hanyoyin bincike. Dokokin manyan lambobi da aikace-aikacen su, da kuma matakan bazuwar.

Wannan kwas ɗin cikakke ne ga waɗanda ke son ilimin asali a kimiyyar bayanai. Yana ba da cikakkiyar hangen nesa akan samfuran yuwuwar. Daga asali abubuwa zuwa matakai bazuwar da ƙididdiga ƙididdiga. Duk wannan yana da amfani musamman ga ƙwararru da ɗalibai. Musamman a fannonin kimiyyar bayanai, injiniyanci da kididdiga.

Yiwuwar Ƙididdigar Ƙididdigar Ƙididdiga da Ƙarfafawa (MIT)

Cibiyar Fasaha ta Massachusetts (MIT) tana gabatar da kwas na "Iwuwar Kwamfuta da Ƙaddamarwa" a cikin Turanci. Shirin ya ƙunshi gabatarwar matsakaici-mataki zuwa bincike mai yuwuwa da ƙima. Wannan darasi na mako goma sha biyu, yana buƙatar sa'o'i 4-6 na karatu a kowane mako, bincike ne mai ban sha'awa na yadda ake amfani da yuwuwar da ƙima a cikin yankuna daban-daban kamar tace spam, kewayawa ta hannu, ko ma a cikin dabarun dabarun kamar Jeopardy da Go.

A cikin wannan kwas ɗin, zaku koyi ƙa'idodin yuwuwa da ƙima da yadda ake aiwatar da su a cikin shirye-shiryen kwamfuta waɗanda ke yin tunani tare da rashin tabbas da yin tsinkaya. Za ku koyi game da tsarin bayanai daban-daban don adana rabe-raben yuwuwar, kamar ƙirar zane mai yiwuwa, da haɓaka ingantaccen algorithms don yin tunani tare da waɗannan tsarin bayanan.

A ƙarshen wannan kwas, za ku san yadda ake ƙirƙira matsalolin duniyar gaske tare da yuwuwar da yadda ake amfani da samfuran da aka samo don yin tunani. Ba kwa buƙatar samun gogewa ta farko a cikin yuwuwar ko fa'ida, amma yakamata ku kasance cikin kwanciyar hankali da ainihin shirye-shiryen Python da ƙididdiga.

Wannan kwas wata hanya ce mai mahimmanci ga waɗanda ke neman fahimta da amfani da hanyoyin ƙididdiga a fagen ilimin kimiyyar bayanai, suna ba da cikakkiyar hangen nesa kan ƙirar yuwuwar da ƙididdige ƙididdiga.

A Zuciyar Rashin tabbas: MIT Demystifies Yiwuwa

A cikin kwas ɗin "Gabatarwa ga Yiwuwar Sashe na II: Tsarukan Inference", Cibiyar Fasaha ta Massachusetts (MIT) tana ba da ci gaba mai zurfi a cikin duniyar yuwuwar da ƙima. Wannan darasi, gabaɗaya a cikin Ingilishi, ci gaba ne mai ma'ana na ɓangaren farko, mai zurfi cikin nazarin bayanai da kuma kimiyyar rashin tabbas.

A cikin tsawon makonni goma sha shida, tare da ƙaddamar da sa'o'i 6 a kowane mako, wannan kwas ɗin yana bincika dokokin manyan lambobi, hanyoyin ƙididdigewa na Bayesian, ƙididdiga na gargajiya, da tsarin bazuwar irin su hanyoyin Poisson da sarƙoƙi na Markov. Wannan bincike ne mai tsauri, wanda aka yi niyya ga waɗanda suka riga sun sami ingantaccen tushe cikin yuwuwar.

Wannan kwas ɗin ya yi fice don dabarar sa ta fahimta, yayin da yake riƙe da ƙarfin lissafi. Ba wai kawai yana gabatar da ka'idoji da hujjoji ba, amma yana da nufin haɓaka zurfin fahimtar ra'ayoyi ta hanyar aikace-aikacen kankare. Dalibai za su koyi yin ƙira ga hadaddun al'amura da fassara ainihin bayanan duniya.

Mafi dacewa ga ƙwararrun ƙwararrun kimiyyar bayanai, masu bincike, da ɗalibai, wannan kwas ɗin yana ba da hangen nesa na musamman kan yadda yuwuwar da ƙima ke haifar da fahimtarmu game da duniya. Cikakke ga waɗanda ke neman zurfafa fahimtar ilimin kimiyyar bayanai da ƙididdigar ƙididdiga.

Haɗuwa Na Nazari: Koyarwar Princeton don Fasa Tsarukan Maɗaukaki (Princeton)

Kos ɗin Combinatorics na Analytic, wanda Jami'ar Princeton ke bayarwa, bincike ne mai ban sha'awa na haɗaɗɗen ƙididdiga, horon da ke ba da takamaiman ƙididdiga masu ƙididdigewa na hadaddun tsarin haɗin kai. Wannan darasi, gabaɗaya a cikin Ingilishi, hanya ce mai mahimmanci ga waɗanda ke neman fahimta da amfani da hanyoyin ci-gaba a fagen haɗin gwiwa.

Tsawon makonni uku kuma yana buƙatar kusan sa'o'i 16 gabaɗaya, ko kusan sa'o'i 5 a kowane mako, wannan kwas ɗin yana gabatar da hanyar alama don samun alaƙar aiki tsakanin talakawa, ma'auni, da ayyuka masu haɓaka iri-iri. Hakanan yana bincika hanyoyin bincike mai rikitarwa don samo madaidaicin asymptotics daga ma'auni na samar da ayyuka.

Dalibai za su gano yadda za a iya amfani da na'urori na nazari don yin tsinkaya madaidaicin adadi a cikin manyan tsarin haɗin gwiwa. Za su koyi sarrafa tsarin haɗin gwiwa da amfani da dabarun bincike masu rikitarwa don nazarin waɗannan sifofin.

Wannan karatun yana da kyau ga waɗanda ke neman zurfafa fahimtar haɗin gwiwa da aikace-aikacen sa wajen magance matsaloli masu rikitarwa. Yana ba da hangen nesa na musamman kan yadda mahaɗar ƙididdiga ke tsara fahimtar mu na tsarin lissafi da tsarin haɗin kai.