Maphunziro mu French
Mwachisawawa: Chiyambi cha Kuthekera - Gawo 1 (POLYTECHNIQUE PARIS)
École Polytechnique, bungwe lodziwika bwino, limapereka maphunziro osangalatsa a Coursera omwe ali ndi mutu wakuti "Mwachisawawa: mawu oyambitsa zotheka - Gawo 1". Maphunzirowa, omwe amakhala pafupifupi maola 27 omwe afalikira kwa milungu itatu, ndi mwayi wapadera kwa aliyense amene ali ndi chidwi ndi maziko a kuthekera. Zopangidwa kuti zikhale zosinthika komanso zogwirizana ndi mayendedwe a wophunzira aliyense, maphunzirowa amapereka njira yozama komanso yofikira ku chiphunzitso chotheka.
Pulogalamuyi ili ndi ma module a 8 ochita nawo, iliyonse imayang'ana mbali zazikulu za mwayi wopezeka, malamulo ofananirako, mawonekedwe, kudziyimira pawokha, komanso zosintha mwachisawawa. Mutu uliwonse umalemeretsedwa ndi makanema ofotokozera, zowerengera zowonjezera ndi mafunso kuti muyese ndikuphatikiza zomwe mwapeza. Ophunzira alinso ndi mwayi wopeza satifiketi yogawana nawo akamaliza maphunzirowo, zomwe zimawonjezera phindu paulendo wawo wamaluso kapena maphunziro.
Alangizi, Sylvie Méléard, Jean-René Chazottes ndi Carl Graham, onse ogwirizana ndi École Polytechnique, amabweretsa ukatswiri wawo ndi chilakolako cha masamu, kupangitsa maphunzirowa kukhala ophunzitsa, komanso olimbikitsa. Kaya ndinu wophunzira wa masamu, katswiri wofuna kukulitsa chidziwitso chanu, kapena mumangokonda zasayansi, maphunzirowa amapereka mwayi wapadera wofufuza dziko lochititsa chidwi la zotheka, motsogozedwa ndi ena mwa malingaliro abwino kwambiri ku École Polytechnique.
Mwachisawawa: Chiyambi cha Kuthekera - Gawo 2 (POLYTECHNIQUE PARIS)
Kupitiliza kupambana kwamaphunziro a École Polytechnique, maphunziro akuti "Mwachisawawa: mawu oyambitsa zotheka - Gawo 2" pa Coursera ndikupitilira gawo loyamba komanso lolimbikitsa. Maphunzirowa, omwe akuti atha maola 17 afalikira kwa milungu itatu, amamiza ophunzira m'malingaliro apamwamba kwambiri a chiphunzitso cha kuthekera, ndikumvetsetsa mozama komanso kugwiritsa ntchito bwino mwambo wosangalatsawu.
Ndi ma modules 6 opangidwa bwino, maphunzirowa amakhudza mitu monga ma vectors osasintha, generalization of the calculation of law, theory of theorem of large numbers, njira ya Monte Carlo, ndi theorem yapakati. Gawo lililonse limaphatikizapo mavidiyo ophunzirira, kuwerenga ndi mafunso, kuti muphunzire mozama. Fomu iyi imalola ophunzira kuti azitha kugwiritsa ntchito zomwe aphunzira komanso kugwiritsa ntchito mfundo zomwe aphunzira m'njira yothandiza.
Alangizi, Sylvie Méléard, Jean-René Chazottes ndi Carl Graham akupitiriza kutsogolera ophunzira paulendo wophunzitsawu ndi ukatswiri wawo komanso chidwi chawo pa masamu. Njira yawo yophunzitsira imathandizira kumvetsetsa mfundo zovuta komanso imalimbikitsa kufufuza mozama za kuthekera.
Maphunzirowa ndi abwino kwa iwo omwe ali ndi maziko olimba zotheka ndipo akufuna kukulitsa kumvetsetsa kwawo komanso kuthekera kogwiritsa ntchito mfundozi pamavuto ovuta kwambiri. Pomaliza maphunzirowa, ophunzira athanso kupeza satifiketi yogawana nawo, kuwonetsa kudzipereka kwawo komanso luso lawo pagawo lapaderali.
Chiyambi cha chiphunzitso chogawa (POLYTECHNIQUE PARIS)
Kosi ya "Introduction to theory of distributions", yoperekedwa ndi École Polytechnique pa Coursera, ikuyimira kufufuza kwapadera komanso mozama kwa masamu apamwamba. Maphunzirowa, omwe amakhala pafupifupi maola 15 omwe amafalikira kwa milungu itatu, adapangidwira omwe akufuna kumvetsetsa kugawa, lingaliro lofunikira pamasamu ogwiritsidwa ntchito ndi kusanthula.
Pulogalamuyi ili ndi ma module 9, iliyonse yopereka mavidiyo ophunzirira, kuwerenga ndi mafunso. Ma modules amakhudza mbali zosiyanasiyana za chiphunzitso cha kugawa, kuphatikizapo zovuta monga kufotokozera zomwe zimachokera ku ntchito yosiya komanso kugwiritsa ntchito ntchito zosiya monga njira zothetsera ma equation osiyana. Njira yokonzedwayi imalola ophunzira kuti adziŵe pang'onopang'ono mfundo zomwe zingawoneke ngati zowopsya poyamba.
Pulofesa François Golse ndi Yvan Martel, onse odziwika a École Polytechnique, amabweretsa ukadaulo wochuluka pamaphunzirowa. Kuphunzitsa kwawo kumaphatikiza kulimbikira kwamaphunziro ndi njira zophunzitsira zatsopano, kupangitsa kuti zokhutira zikhale zopezeka komanso zokopa kwa ophunzira.
Maphunzirowa ndi oyenera makamaka kwa ophunzira a masamu, uinjiniya, kapena magawo ena okhudzana nawo omwe akufuna kukulitsa kumvetsetsa kwawo masamu ovuta. Pomaliza maphunzirowa, otenga nawo mbali sadzapeza chidziwitso chofunikira, komanso adzakhala ndi mwayi wopeza satifiketi yogawana nawo, zomwe zimawonjezera phindu lalikulu paukadaulo wawo kapena maphunziro awo.
Chiyambi cha chiphunzitso cha Galois (SUPERIOR NORMAL SCHOOL PARIS)
Kuperekedwa ndi École Normale Supérieure pa Coursera, kosi ya "Introduction to Galois Theory" ndikufufuza kochititsa chidwi kwa imodzi mwanthambi zozama komanso zokopa zamasamu amakono.Maphunzirowa amatenga pafupifupi maola 12, akumiza ophunzira m'dziko lovuta komanso lopatsa chidwi la chiphunzitso cha Galois, maphunziro omwe asintha kumvetsetsa kwa ubale pakati pa ma polynomial equations ndi algebraic.
Maphunzirowa amayang'ana pa kuphunzira za mizu ya polynomials ndi kufotokozera kwawo kuchokera ku ma coefficients, funso lapakati mu algebra. Imafufuza lingaliro la gulu la Galois, loyambitsidwa ndi Évariste Galois, lomwe limagwirizanitsa polynomial iliyonse ndi gulu la zololeza za mizu yake. Njira imeneyi imatithandiza kumvetsa chifukwa chake n'kosatheka kufotokoza magwero a ma polynomial equations pogwiritsa ntchito zilembo za algebra, makamaka ma polynomial a digirii yaikulu kuposa inayi.
Kulemberana makalata a Galois, chinthu chofunika kwambiri pa maphunzirowa, kumagwirizanitsa chiphunzitso cha m'munda ndi chiphunzitso chamagulu, kupereka malingaliro apadera pa kusungunuka kwa ma equation akuluakulu. Maphunzirowa amagwiritsa ntchito mfundo zoyambira mu linear algebra kuti afikire chiphunzitso cha matupi ndikuwonetsa lingaliro la nambala ya algebra, ndikuwunika magulu ovomerezeka ofunikira pophunzira magulu a Galois.
Maphunzirowa ndi odziwika kwambiri chifukwa cha luso lake lopereka malingaliro ovuta a algebra m'njira yofikirika komanso yosavuta, zomwe zimalola ophunzira kuti azitha kupeza zotsatira zomveka mwachangu ndi njira yodziwika bwino. Ndi yabwino kwa ophunzira a masamu, physics, kapena engineering, komanso okonda masamu omwe akufuna kukulitsa kumvetsetsa kwawo kwa algebraic komanso kagwiritsidwe ntchito kake.
Pomaliza maphunzirowa, otenga nawo mbali sangangomvetsetsa mwakuya chiphunzitso cha Galois, komanso adzakhala ndi mwayi wopeza satifiketi yogawana nawo, ndikuwonjezera phindu lalikulu ku mbiri yawo yaukadaulo kapena maphunziro.
Kusanthula I (gawo 1): Chiyambi, malingaliro oyambira, manambala enieni (SUKULU POLYTECHNIQUE FEDERALE DE LAUSANNE)
Maphunziro a "Analysis I (gawo 1): Prelude, mfundo zoyambira, manambala enieni", operekedwa ndi École Polytechnique Fédérale de Lausanne pa edX, ndi chidziwitso chozama cha mfundo zazikuluzikulu za kusanthula kwenikweni. Maphunziro a milungu 5 awa, omwe amafunikira pafupifupi maola 4-5 akuphunzira pa sabata, apangidwa kuti amalize pa liwiro lanu.
Zomwe zili mumaphunzirowa zimayamba ndi mawu oyamba omwe amabwereranso ndikuzama malingaliro ofunikira a masamu monga ntchito za trigonometric (sin, cos, tan), ntchito zofananira (exp, ln), komanso malamulo owerengera mphamvu, ma logarithms ndi mizu. Zimakhudzanso magawo oyambira ndi ntchito.
Pakatikati pa maphunzirowa amayang'ana pa kachitidwe ka manambala. Kuyambira pamalingaliro achilengedwe a manambala achilengedwe, maphunzirowa amatanthauzira mozama manambala omveka ndikuwunika mawonekedwe awo. Chisamaliro chapadera chimaperekedwa ku manambala enieni, omwe amayambitsidwa kuti akwaniritse mipata mu ziwerengero zomveka. Maphunzirowa amapereka tanthauzo la axiomatic la manambala enieni ndikuphunziranso katundu wawo mwatsatanetsatane, kuphatikizapo mfundo monga infimum, supremum, mtengo weniweni ndi zina zowonjezera za manambala enieni.
Maphunzirowa ndi abwino kwa iwo omwe ali ndi chidziwitso choyambirira cha masamu ndipo akufuna kukulitsa kumvetsetsa kwawo pakuwunika kwenikweni. Ndizothandiza makamaka kwa ophunzira a masamu, physics, kapena engineering, komanso aliyense amene ali ndi chidwi chofuna kumvetsetsa bwino masamu.
Pomaliza maphunzirowa, otenga nawo mbali amvetsetsa bwino manambala enieni komanso kufunika kwawo pakuwunika, komanso mwayi wopeza satifiketi yogawana nawo, zomwe zimawonjezera phindu lalikulu paukadaulo wawo kapena maphunziro awo.
Kusanthula I (gawo 2): Chiyambi cha manambala ovuta (SUKULU POLYTECHNIQUE FEDERALE DE LAUSANNE)
Maphunziro a "Analysis I (gawo 2): Mau oyamba a manambala ovuta", operekedwa ndi École Polytechnique Fédérale de Lausanne pa edX, ndikuyambitsa kochititsa chidwi kudziko la manambala ovuta.Maphunziro a milungu 2 awa, omwe amafunikira pafupifupi maola 4-5 akuphunzira pa sabata, apangidwa kuti amalize pa liwiro lanu.
Maphunzirowa akuyamba poyankhula ndi equation z ^ 2 = -1, yomwe ilibe yankho mu chiwerengero cha manambala enieni, R. Vutoli limayambitsa kukhazikitsidwa kwa manambala ovuta, C, gawo lomwe lili ndi R ndipo limatithandiza kuthetsa zoterezi. equations. Maphunzirowa amawunikira njira zosiyanasiyana zoyimira nambala yovuta ndikukambirana njira zothetsera ma equation a mawonekedwe z^n = w, pomwe n ndi ya N* ndi w ku C.
Chochititsa chidwi kwambiri pamaphunzirowa ndi kuphunzira kwa chiphunzitso cha algebra, chomwe ndi chotsatira chachikulu cha masamu. Maphunzirowa amaphatikizanso mitu monga chiwonetsero cha Cartesian cha manambala ovuta, zoyambira zawo, chosinthira chochulukitsa, formula ya Euler ndi de Moivre, ndi mawonekedwe a polar a nambala yovuta.
Maphunzirowa ndi abwino kwa iwo omwe ali ndi chidziwitso cha manambala enieni ndipo akufuna kukulitsa kumvetsetsa kwawo mpaka manambala ovuta. Ndizothandiza makamaka kwa ophunzira a masamu, physics, kapena engineering, komanso aliyense amene ali ndi chidwi chofuna kumvetsetsa mozama za algebra ndi momwe amagwiritsira ntchito.
Pomaliza maphunzirowa, otenga nawo mbali amvetsetsa bwino manambala ovuta komanso gawo lawo lofunikira pa masamu, komanso mwayi wopeza satifiketi yogawana nawo, zomwe zimawonjezera phindu lalikulu paukadaulo wawo kapena maphunziro awo.
Kusanthula I (gawo 3): Kutsatizana kwa manambala enieni I ndi II (SUKULU POLYTECHNIQUE FEDERALE DE LAUSANNE)
Maphunziro a "Analysis I (gawo 3): Kutsatizana kwa manambala enieni I ndi II ", operekedwa ndi École Polytechnique Fédérale de Lausanne pa edX, amayang'ana mndandanda wa manambala enieni. Maphunziro a milungu 4 awa, omwe amafunikira pafupifupi maola 4-5 akuphunzira pa sabata, apangidwa kuti amalize pa liwiro lanu.
Lingaliro lapakati la maphunzirowa ndi malire a mndandanda wa manambala enieni. Zimayamba ndi kufotokozera ndondomeko ya manambala enieni monga ntchito kuchokera ku N mpaka R. Mwachitsanzo, ndondomeko a_n = 1/2 ^ n ikufufuzidwa, kusonyeza momwe ikuyandikira zero. Maphunzirowa amalankhula mwamphamvu tanthawuzo la malire a ndondomeko ndikukhazikitsa njira zowonetsera kukhalapo kwa malire.
Kuphatikiza apo, maphunzirowa amakhazikitsa mgwirizano pakati pa lingaliro la malire ndi la infimum ndi supremum ya seti. Kugwiritsiridwa ntchito kofunika kwa kutsatizana kwa manambala enieni kumasonyezedwa ndi mfundo yakuti nambala yeniyeni iliyonse ingalingaliridwe monga malire a ndondomeko ya manambala omveka. Maphunzirowa amawunikiranso kutsatizana kwa Cauchy ndi kutsatizana komwe kumatanthauzidwa ndi kulowetsa mizere, komanso theorem ya Bolzano-Weierstrass.
Otenga nawo mbali aphunziranso za mndandanda wa manambala, ndi mawu oyamba a zitsanzo zosiyanasiyana ndi njira zolumikizirana, monga d'Alembert criterion, Cauchy criterion, ndi Leibniz criterion. Maphunzirowa amatha ndi kuphunzira manambala angapo okhala ndi parameter.
Maphunzirowa ndi abwino kwa iwo omwe ali ndi chidziwitso choyambirira cha masamu ndipo akufuna kukulitsa kumvetsetsa kwawo kwamawerengero enieni. Ndizofunikira makamaka kwa ophunzira a masamu, physics kapena engineering. Pomaliza maphunzirowa, otenga nawo mbali akulitsa kumvetsetsa kwawo masamu ndipo atha kupeza satifiketi yogawana nawo, chuma chaukadaulo kapena maphunziro awo.
Kupeza Ntchito Zenizeni ndi Zopitilira: Kusanthula I (gawo 4) (SUKULU POLYTECHNIQUE FEDERALE DE LAUSANNE)
Mu "Analysis I (gawo 4): Malire a ntchito, ntchito zopitiliza", École Polytechnique Fédérale de Lausanne imapereka ulendo wosangalatsa wophunzirira ntchito zenizeni zakusintha kwenikweni.Maphunzirowa, opitilira masabata a 4 ndi 4 mpaka maola 5 akuphunzira mlungu uliwonse, amapezeka pa edX ndipo amalola kupita patsogolo pa liwiro lanu.
Gawo ili la maphunzirowa limayamba ndikuyambitsa ntchito zenizeni, kugogomezera katundu wawo monga monotonicity, parity, ndi periodicity. Imawunikanso magwiridwe antchito pakati pa ntchito ndikuyambitsa ntchito zina monga hyperbolic function. Chisamaliro chapadera chimaperekedwa ku ntchito zomwe zimafotokozedwa pang'onopang'ono, kuphatikiza ntchito za Signum ndi Heaviside, komanso kusintha kogwirizana.
Pakatikati pa maphunzirowa amayang'ana malire akuthwa a ntchito pamfundo, kupereka zitsanzo zenizeni za malire a ntchito. Ikufotokozanso mfundo za malire a kumanzere ndi kumanja. Kenaka, maphunzirowa amayang'ana malire a ntchito zopanda malire ndipo amapereka zida zofunika zowerengera malire, monga chiphunzitso cha apolisi.
Mbali yofunika kwambiri ya maphunzirowa ndikuyambitsa lingaliro la kupitiriza, lomwe limatanthauzidwa m'njira ziwiri zosiyana, ndikugwiritsa ntchito kuwonjezera ntchito zina. Maphunzirowa amatha ndi phunziro la kupitiriza pa nthawi zotseguka.
Maphunzirowa ndi mwayi wolemeretsa kwa iwo omwe akufuna kukulitsa kumvetsetsa kwawo kwa ntchito zenizeni komanso zopitilira. Ndi yabwino kwa ophunzira a masamu, physics kapena engineering. Pomaliza maphunzirowa, otenga nawo mbali sadzangokulitsa luso lawo la masamu, komanso adzakhala ndi mwayi wopeza satifiketi yopindulitsa, kutsegulira chitseko chamalingaliro atsopano amaphunziro kapena akatswiri.
Kuwona Ntchito Zosiyanasiyana: Kusanthula I (gawo 5) (SUKULU POLYTECHNIQUE FEDERALE DE LAUSANNE)
École Polytechnique Fédérale de Lausanne, popereka maphunziro ake pa edX, ikupereka "Analysis I (gawo 5): Ntchito zopitilira ndi zosiyanitsidwa, ntchito yochokera". Maphunzirowa a masabata anayi, omwe amafunikira pafupifupi maola 4-5 pa sabata, ndikufufuza mozama malingaliro a kusiyanitsa ndi kupitiriza kwa ntchito.
Maphunzirowa amayamba ndi kuphunzira mozama za ntchito zopitirira, kuyang'ana pa katundu wawo pa nthawi zotsekedwa. Gawoli limathandiza ophunzira kumvetsetsa kuchuluka komanso kuchepera kwa ntchito zopitilira. Maphunzirowa amayambitsa njira ya magawo awiri ndikuwonetsa malingaliro ofunikira monga theorem yamtengo wapakatikati ndi theorem yokhazikika.
Gawo lapakati la maphunzirowa limaperekedwa ku kusiyanitsa ndi kusiyanitsa kwa ntchito. Ophunzira amaphunzira kutanthauzira mfundozi ndikumvetsetsa kufanana kwawo. Maphunzirowa amayang'ananso kumangidwa kwa zotumphukira ndikuyang'ana mwatsatanetsatane zinthu zake, kuphatikiza ma algebraic ntchito zotumphukira.
Mbali yofunika ya maphunzirowa ndi kuphunzira za katundu wa ntchito zosiyanitsidwa, monga zochokera ku ntchito zikuchokera, theorem Rolle, ndi finite increment theorem. Maphunzirowa amafufuzanso kupitiriza kwa ntchito yotengedwa ndi zotsatira zake pa monotonicity ya ntchito yosiyana.
Maphunzirowa ndi mwayi wabwino kwambiri kwa iwo omwe akufuna kukulitsa kumvetsetsa kwawo kwa ntchito zosiyanitsidwa komanso zopitilira. Ndi yabwino kwa ophunzira a masamu, physics kapena engineering. Pomaliza maphunzirowa, otenga nawo mbali sadzangowonjezera kumvetsetsa kwawo mfundo zazikuluzikulu zamasamu, komanso adzakhala ndi mwayi wopeza satifiketi yopindulitsa, kutsegulira khomo la mwayi watsopano wamaphunziro kapena akatswiri.
Kuzama mu Kusanthula Masamu: Kusanthula I (gawo 6) (SUKULU POLYTECHNIQUE FEDERALE DE LAUSANNE)
Maphunziro a "Analysis I (gawo 6): Maphunziro a ntchito, zochitika zochepa", zoperekedwa ndi École Polytechnique Fédérale de Lausanne pa edX, ndikufufuza mozama za ntchito ndi chitukuko chawo chochepa. Maphunzirowa a milungu inayi, okhala ndi ntchito zambiri za maola 4 mpaka 5 pamlungu, amalola ophunzira kupita patsogolo pa liwiro lawo.
Mutu uwu wa maphunzirowa ukuyang'ana pa kafukufuku wozama wa ntchito, pogwiritsa ntchito malingaliro kuti awone kusiyana kwawo. Pambuyo pothana ndi theorem yomaliza yowonjezereka, maphunzirowo amayang'ana zonse zake. Chofunika kwambiri pakuwerenga ntchito ndikumvetsetsa machitidwe awo mopanda malire. Kuti tichite izi, maphunzirowa amayambitsa lamulo la Bernoulli-l'Hospital, chida chofunikira chodziwira malire ovuta a quotients.
Maphunzirowa amawunikiranso mawonekedwe azithunzi za ntchito, ndikuwunika mafunso monga kukhalapo kwa maxima am'deralo kapena apadziko lonse lapansi kapena minima, komanso kukhazikika kapena kukhazikika kwa magwiridwe antchito. Ophunzira aphunzira kuzindikira ma asymptotes osiyanasiyana a ntchito.
Mfundo ina yamphamvu ya maphunzirowa ndikuyambitsa kukulitsa kochepa kwa ntchito, yomwe imapereka kuyerekezera kwapolynomial pafupi ndi mfundo yoperekedwa. Zochitikazi ndizofunikira kuti muchepetse kuwerengera malire komanso kuphunzira za magwiridwe antchito. Maphunzirowa amaphatikizanso mndandanda wamagulu onse ndi ma radius awo osinthika, komanso mndandanda wa Taylor, chida champhamvu choyimira ntchito zosiyanitsidwa mpaka kalekale.
Maphunzirowa ndi othandiza kwambiri kwa iwo omwe akufuna kukulitsa kumvetsetsa kwawo ntchito ndi momwe amagwiritsira ntchito masamu. Imapereka malingaliro olemeretsa komanso atsatanetsatane pamalingaliro ofunikira pakusanthula masamu.
Kudziwa Kuphatikiza: Kusanthula I (gawo 7) (SUKULU POLYTECHNIQUE FEDERALE DE LAUSANNE)
Maphunziro "Analysis I (gawo la 7): Zosasinthika komanso zotsimikizika, kuphatikiza (mitu yosankhidwa)", yoperekedwa ndi École Polytechnique Fédérale de Lausanne pa edX, ndikufufuza mwatsatanetsatane kusakanikirana kwa ntchito. Gawoli, lomwe limatenga milungu inayi ndikutengapo gawo kwa maola 4 mpaka 5 pa sabata, limalola ophunzira kuzindikira zidziwitso za kuphatikiza pa liwiro lawo.
Maphunzirowa akuyamba ndi tanthauzo la chophatikizika chosatha komanso chotsimikizika, kuwonetsa zotsimikizika kudzera mu mawerengero a Riemann ndi ndalama zapamwamba ndi zotsika. Kenako imakambilana za zinthu zitatu zofunika kwambiri zophatikizira zotsimikizika: mzere wophatikizika, kugawikana kwa gawo lophatikizika, ndi monotonicity ya chophatikiza.
Mfundo yapakati pa maphunzirowa ndi tanthauzo lachidziwitso cha ntchito zopitirira pa gawo, zomwe zikuwonetsedwa mwatsatanetsatane. Maphunzirowa afika pachimake ndi mfundo yofunikira ya integral calculus, ndikuyambitsa lingaliro la antiderivative of a function. Ophunzira amaphunzira njira zosiyanasiyana zophatikizira, monga kuphatikiza ndi magawo, kusintha masinthidwe, ndi kuphatikiza ndi induction.
Maphunzirowa amamaliza ndi phunziro la kuphatikizika kwa ntchito zinazake, kuphatikizapo kuphatikizika kwa kukulitsa kochepa kwa ntchito, kusakanikirana kwa mndandanda wamagulu onse, ndi kuphatikiza kwa piecewise ntchito zopitirirabe. Njirazi zimalola kuti zophatikizana za ntchito zomwe zili ndi mawonekedwe apadera ziwerengedwe bwino. Pomaliza, maphunzirowa amawunikira zophatikizika, zomwe zimatanthauzidwa podutsa malire pazophatikizika, ndikupereka zitsanzo zenizeni.
Maphunzirowa ndi othandiza kwambiri kwa iwo omwe akufuna kudziwa kuphatikiza, chida chofunikira kwambiri pa masamu. Limapereka chidziwitso chokwanira komanso chothandiza pakuphatikiza, kukulitsa luso la masamu a ophunzira.
Maphunziro mu Chingerezi
Chiyambi cha Linear Models ndi Matrix Algebra (Harvard)
Yunivesite ya Harvard, kudzera pa nsanja yake ya HarvardX pa edX, imapereka maphunziro "Introduction to Linear Models and Matrix Algebra". Ngakhale maphunzirowa amaphunzitsidwa mu Chingerezi, amapereka mwayi wapadera wophunzira maziko a matrix algebra ndi mitundu yofananira, maluso ofunikira m'magawo ambiri asayansi.
Maphunziro a milungu inayi awa, omwe amafunikira maola a 2 mpaka 4 pa sabata, apangidwa kuti amalize pa liwiro lanu. Imayang'ana kwambiri kugwiritsa ntchito chilankhulo cha pulogalamu ya R kugwiritsa ntchito mizere mizere pakusanthula deta, makamaka mu sayansi ya moyo. Ophunzira aphunzira kugwiritsa ntchito matrix algebra ndikumvetsetsa momwe angagwiritsire ntchito pamapangidwe oyesera komanso kusanthula kwa data kwapamwamba kwambiri.
Pulogalamuyi imakhudza zolemba za matrix algebra, magwiridwe antchito a matrix, kugwiritsa ntchito matrix algebra pakusanthula deta, zitsanzo zam'mizere, komanso mawu oyamba pakuwonongeka kwa QR. Maphunzirowa ndi gawo la maphunziro asanu ndi awiri, omwe amatha kuchitidwa payekha kapena ngati gawo la satifiketi ziwiri zaukadaulo mu Data Analysis for Life Sciences and Genomic Data Analysis.
Maphunzirowa ndi abwino kwa iwo omwe akufuna kupeza maluso owerengera komanso kusanthula deta, makamaka pankhani ya sayansi ya moyo. Zimapereka maziko olimba kwa iwo omwe akufuna kupitilirabe kufufuza matrix algebra ndikugwiritsa ntchito kwake m'magawo osiyanasiyana asayansi ndi kafukufuku.
Master Probability (Harvard)
LMndandanda wa "Statistics 110: Probability" pa YouTube, wophunzitsidwa m'Chingerezi ndi Joe Blitzstein wa ku Harvard University, ndiwothandiza kwambiri kwa iwo omwe akufuna kukulitsa chidziwitso chawo cha kuthekera.. Seweroli lili ndi makanema ophunzirira, zida zowunikira, komanso zoyeserera zopitilira 250 zokhala ndi mayankho atsatanetsatane.
Maphunziro a Chingelezi awa ndi mawu oyamba okhudzana ndi kuthekera, omwe amaperekedwa ngati chilankhulo chofunikira komanso zida zomvetsetsa ziwerengero, sayansi, zoopsa komanso kusakhazikika. Mfundo zophunzitsidwa zimagwira ntchito m'magawo osiyanasiyana monga ziwerengero, sayansi, uinjiniya, zachuma, zachuma ndi moyo watsiku ndi tsiku.
Mitu yomwe imakhudzidwa imaphatikizapo zoyambira za kuthekera, zosinthika mwachisawawa ndi kugawa kwawo, kugawa kwaunivariate ndi multivariate, theorems malire, ndi unyolo wa Markov. Maphunzirowa amafunikira chidziwitso choyambirira cha calculus yosinthika imodzi komanso kuzolowera matrices.
Kwa iwo omwe ali omasuka ndi Chingerezi komanso ofunitsitsa kufufuza dziko mozama, maphunziro a Harvard awa amapereka mwayi wophunzirira bwino. Mukhoza kulumikiza playlist ndi nkhani zake mwatsatanetsatane mwachindunji YouTube.
Kutheka Kufotokozedwa. Maphunziro a French Subtitles (Harvard)
Maphunziro a "Fat Chance: Probability from the Ground Up," operekedwa ndi HarvardX pa edX, ndi mawu oyamba ochititsa chidwi a kuthekera ndi ziwerengero. Ngakhale maphunzirowa amaphunzitsidwa m'Chingerezi, ndi omvera olankhula Chifalansa chifukwa cha mawu am'munsi achi French omwe alipo.
Maphunzirowa a masabata asanu ndi awiri, omwe amafunikira maola 3 mpaka 5 pa sabata, adapangidwira omwe angoyamba kumene kuphunzira za kuthekera kapena kufunafuna kuwunikiranso mfundo zazikuluzikulu asanalembetse maphunziro a ziwerengero. “Fat Chance” imagogomezera kukulitsa kuganiza kwa masamu m’malo moloweza mawu ndi ziganizo.
Ma module oyambilira amayambitsa luso lowerengera, lomwe limagwiritsidwa ntchito pazovuta zosavuta. Ma module otsatirawa amafufuza momwe malingaliro ndi njirazi zingasinthire kuti athetse mavuto ambiri omwe angakhalepo. Maphunzirowa amatha ndi mawu oyamba a ziwerengero kudzera mu malingaliro a mtengo woyembekezeredwa, kusiyana ndi kugawa kwachibadwa.
Maphunzirowa ndi abwino kwa iwo omwe akufuna kuwonjezera luso lawo la kulingalira komanso kumvetsetsa maziko a kuthekera ndi ziwerengero. Limapereka malingaliro olemeretsa pa kuchuluka kwa masamu ndi momwe zimagwirira ntchito pakumvetsetsa zoopsa ndi kusakhazikika.
Statistical Inference and Modelling for High-throughput Experiments (Harvard)
Maphunziro a "Statistical Inference and Modelling for High-throughput Experiments" m'Chingelezi amayang'ana kwambiri njira zomwe zimagwiritsidwa ntchito popanga ziwerengero pa data yapamwamba kwambiri. Maphunziro a masabata anayiwa, omwe amafunikira maola a 2-4 pa sabata, ndi chithandizo chofunikira kwa iwo omwe akufuna kumvetsetsa ndi kugwiritsa ntchito njira zowerengera zapamwamba pa kafukufuku wozama kwambiri.
Pulogalamuyi imakhala ndi mitu yosiyanasiyana, kuphatikiza vuto lofananiza kangapo, kuchuluka kwa zolakwika, njira zowongolera zolakwika, mitengo yotulukira zabodza, ma q-values, ndi kusanthula kwa data. Imayambitsanso masanjidwe a ziwerengero ndikugwiritsa ntchito kwake pazambiri zotsogola kwambiri, kukambirana za magawo a parametric monga binomial, exponential, ndi gamma, ndikufotokozera kuyerekezera kokwanira.
Ophunzira aphunzira momwe malingalirowa amagwiritsidwira ntchito pazinthu monga kutsatizana kwa mibadwo yotsatira ndi data ya microarray. Maphunzirowa amaphatikizanso zitsanzo zotsogola ndi maulamuliro a Bayesian, ndi zitsanzo zothandiza zakugwiritsa ntchito kwawo.
Maphunzirowa ndi abwino kwa iwo omwe akuyang'ana kukulitsa kumvetsetsa kwawo kwazomwe amawerengera komanso kupanga zitsanzo mu kafukufuku wamakono wa sayansi. Amapereka chidziwitso chozama pa kusanthula kwa chiwerengero cha deta yovuta ndipo ndi chida chabwino kwambiri kwa ofufuza, ophunzira ndi akatswiri a sayansi ya moyo, bioinformatics ndi statistics.
Chiyambi cha Kuthekera (Harvard)
Maphunziro a "Introduction to Probability", operekedwa ndi HarvardX pa edX, ndikufufuza mozama za kuthekera, chilankhulo chofunikira komanso zida zowunikira deta, mwayi, komanso kusatsimikizika. Ngakhale maphunzirowa amaphunzitsidwa m'Chingerezi, ndi omvera olankhula Chifalansa chifukwa cha mawu am'munsi achi French omwe alipo.
Maphunziro a milungu khumi awa, omwe amafunikira maola a 5-10 pa sabata, akufuna kubweretsa malingaliro kudziko lodzaza ndi mwayi komanso kusatsimikizika. Idzapereka zida zofunika kumvetsetsa deta, sayansi, filosofi, uinjiniya, zachuma ndi zachuma. Simudzangophunzira momwe mungathetsere zovuta zaukadaulo, komanso momwe mungagwiritsire ntchito mayankho awa m'moyo watsiku ndi tsiku.
Ndi zitsanzo kuyambira kuyezetsa zachipatala mpaka kulosera zamasewera, mupeza maziko olimba owerengera zowerengera, ma stochastic process, ma aligorivimu mwachisawawa, ndi mitu ina komwe kuli kofunika.
Maphunzirowa ndi abwino kwa iwo omwe akufuna kuwonjezera kumvetsetsa kwawo kusatsimikizika ndi mwayi, kulosera zabwino, komanso kumvetsetsa zosintha mwachisawawa. Zimapereka malingaliro olemeretsa pazagawidwe zomwe zingagwiritsidwe ntchito powerengera ndi sayansi ya data.
Calculus Yogwiritsidwa Ntchito (Harvard)
Maphunziro a "Calculus Applied!", operekedwa ndi Harvard pa edX, ndikufufuza mozama pakugwiritsa ntchito kawerengedwe kosinthika kamodzi mu sayansi ya chikhalidwe, moyo, ndi thupi. Maphunzirowa, onse mu Chingerezi, ndi mwayi wabwino kwambiri kwa iwo omwe akufuna kumvetsetsa momwe calculus imagwiritsidwira ntchito muzochitika zenizeni zapadziko lonse lapansi.
Maphunzirowa amatha masabata khumi ndipo amafuna pakati pa 3 ndi 6 maola ophunzirira pa sabata, maphunzirowa amapitirira mabuku achikhalidwe. Amathandizana ndi akatswiri ochokera m'magawo osiyanasiyana kuti awonetse momwe mawerengedwe amagwiritsidwira ntchito posanthula ndi kuthetsa mavuto enieni padziko lapansi. Ophunzira adzafufuza ntchito zosiyanasiyana, kuyambira kusanthula zachuma mpaka kutengera zamoyo.
Pulogalamuyi imakhudza kugwiritsa ntchito zotumphukira, zophatikizika, ma equation osiyana, ndikugogomezera kufunikira kwa masamu ndi magawo. Zapangidwira iwo omwe ali ndi chidziwitso choyambira cha calculus imodzi ndipo ali ndi chidwi ndi momwe angagwiritsire ntchito m'magawo osiyanasiyana.
Maphunzirowa ndi abwino kwa ophunzira, aphunzitsi, ndi akatswiri omwe akufuna kukulitsa kumvetsetsa kwawo kawerengedwe ndikupeza momwe angagwiritsire ntchito zenizeni padziko lapansi.
Chiyambi cha kulingalira kwa masamu (Stanford)
Maphunziro a "Introduction to Mathematical Thinking", operekedwa ndi yunivesite ya Stanford pa Coursera, akulowa m'dziko la kulingalira kwa masamu. Ngakhale maphunzirowa amaphunzitsidwa m'Chingerezi, ndi omvera olankhula Chifalansa chifukwa cha mawu am'munsi achi French omwe alipo.
Maphunziro a masabata asanu ndi awiriwa, omwe amafunikira pafupifupi maola 38, kapena pafupifupi maola 12 pa sabata, apangidwira omwe akufuna kukulitsa malingaliro a masamu, mosiyana ndi kungoyeserera masamu momwe zimachitikira nthawi zambiri m'sukulu. Maphunzirowa amayang'ana kwambiri kukulitsa malingaliro a "kunja kwa bokosi", luso lofunika kwambiri masiku ano.
Ophunzira adzafufuza momwe akatswiri a masamu amaganizira kuti athetse mavuto enieni, kaya amachokera ku zochitika za tsiku ndi tsiku, kuchokera ku sayansi, kapena masamu enieniwo. Maphunzirowa amathandizira kukhazikitsa njira yofunikirayi yoganizira, kupitilira njira zophunzirira kuti athetse zovuta zomwe sizingachitike.
Maphunzirowa ndi abwino kwa iwo omwe akufuna kulimbitsa malingaliro awo ochulukira ndikumvetsetsa maziko amalingaliro a masamu. Imapereka malingaliro olemeretsa pa kuchuluka kwa masamu ndikugwiritsa ntchito kwake kumvetsetsa zovuta zovuta.
Kuphunzira Zowerengera ndi R (Stanford)
Maphunziro a "Statistical Learning with R", operekedwa ndi Stanford, ndi chiyambi chapakatikati pa kuphunzira koyang'aniridwa, kuyang'ana kwambiri njira zobwerera m'mbuyo ndi m'magulu. Maphunzirowa, onse mu Chingerezi, ndi chida chofunikira kwambiri kwa iwo omwe akufuna kumvetsetsa ndikugwiritsa ntchito njira zowerengera pazasayansi ya data.
Masabata khumi ndi limodzi omwe amafunikira maola a 3-5 pa sabata, maphunzirowa amakhudza njira zatsopano zachikhalidwe komanso zosangalatsa pakupanga masamu, komanso momwe angagwiritsire ntchito chilankhulo cha pulogalamu ya R. buku la maphunziro.
Mitu imaphatikizapo kusinthika kwa mzere ndi polynomial regression, logistic regression and linear discriminant analysis, cross-validation and bootstrapping, modelling model and regularization njira (ridge and lasso), nonlinear model, splines and generalized additive models, tree-based procedures, random nkhalango ndi kulimbikitsa, makina othandizira ma vector, ma neural network ndi kuphunzira mozama, mitundu yopulumuka, ndi kuyesa kangapo.
Maphunzirowa ndi abwino kwa iwo omwe ali ndi chidziwitso choyambirira cha ziwerengero, linear algebra, ndi sayansi ya makompyuta, ndipo omwe akufuna kukulitsa kumvetsetsa kwawo kwa masamu ndikugwiritsa ntchito kwake mu sayansi ya data.
Momwe Mungaphunzirire Masamu: Maphunziro a Aliyense (Stanford)
Maphunziro a "Momwe Mungaphunzirire Masamu: Kwa Ophunzira", operekedwa ndi Stanford. Ndi maphunziro aulere pa intaneti a ophunzira amitundu yonse ya masamu. Konse mu Chingerezi, imaphatikiza chidziwitso chofunikira chokhudza ubongo ndi umboni watsopano wokhudza njira zabwino zofikira masamu.
Kutha milungu isanu ndi umodzi ndipo kumafuna 1 kwa maola a 3 ophunzirira pa sabata. Maphunzirowa apangidwa kuti asinthe ubale wa ophunzira ndi masamu. Anthu ambiri akhala ndi zokumana nazo zoipa pa masamu, zomwe zimawapangitsa kuipidwa kapena kulephera. Maphunzirowa akufuna kupatsa ophunzira chidziwitso chomwe akufunikira kuti asangalale ndi masamu.
Nkhani zophimbidwa ndi nkhani monga ubongo ndi masamu ophunzirira. Nthano zokhuza masamu, malingaliro, zolakwa ndi liwiro zimaphimbidwanso. Kusinthasintha kwa manambala, kulingalira masamu, kulumikizana, zitsanzo zamawerengero ndi gawo la pulogalamuyi. Kuyimira masamu m'moyo, komanso m'chilengedwe komanso kuntchito sikuyiwalika. Maphunzirowa adapangidwa ndi njira yophunzitsira yogwira ntchito, yomwe imapangitsa kuphunzira kukhala kolumikizana komanso kolimbikitsa.
Ndi chida chofunikira kwa aliyense amene akufuna kuwona masamu mosiyana. Kulitsani kumvetsetsa mozama komanso kwabwino kwa mwambowu. Ndizoyenera makamaka kwa iwo omwe adakumana ndi zovuta ndi masamu m'mbuyomu ndipo akufuna kusintha malingaliro awa.
Probability Management (Stanford)
Maphunziro a "Introduction to Probability Management", operekedwa ndi Stanford, ndi mawu oyambira pakuwongolera kuthekera. Gawoli limayang'ana kwambiri kulankhulana ndi kuwerengera zosatsimikizika monga matebulo owerengeka otchedwa Stochastic Information Packets (SIPs). Maphunzirowa a masabata khumi amafunikira 1 kwa maola a 5 pa sabata. Mosakayikira ndi chithandizo chamtengo wapatali kwa iwo omwe akufuna kumvetsetsa ndi kugwiritsa ntchito njira zowerengera m'munda wa sayansi ya deta.
Maphunzirowa ali ndi mitu monga kuzindikira "Flaw of Averages," zolakwika zadongosolo zomwe zimachitika pamene kusatsimikizika kumayimiridwa ndi manambala amodzi, nthawi zambiri avareji. Ikufotokoza chifukwa chake ma projekiti ambiri akuchedwa, pa bajeti komanso pansi pa bajeti. Maphunzirowa amaphunzitsanso Kusatsimikizika Arithmetic, yomwe imawerengera ndi zolowa zosatsimikizika, zomwe zimapangitsa kuti pakhale zotulukapo zosatsimikizika zomwe mutha kuwerengera zotsatira zenizeni zenizeni komanso mwayi wokwaniritsa zolinga zomwe zafotokozedwa.
Ophunzira aphunzira kupanga zofananira zomwe zitha kugawidwa ndi wogwiritsa ntchito aliyense wa Excel osafuna zowonjezera kapena ma macros. Njirayi ndiyoyeneranso ku Python kapena malo aliwonse opangira mapulogalamu omwe amathandizira masanjidwe.
Maphunzirowa ndi abwino kwa iwo omwe ali omasuka ndi Microsoft Excel ndipo akuyang'ana kukulitsa kumvetsetsa kwawo za kuthekera kwa kasamalidwe ndi kagwiritsidwe ntchito ka sayansi ya data.
Sayansi ya Kukayikakayika ndi Deta (MIT)
Maphunzirowa "Kutheka - Sayansi ya Kusatsimikizika ndi Zambiri", operekedwa ndi Massachusetts Institute of Technology (MIT). Ndichidziwitso chofunikira cha sayansi ya data kudzera mu zitsanzo za probabilistic. Maphunzirowa a masabata khumi ndi asanu ndi limodzi, ofunikira maola 10 mpaka 14 akuphunzira pa sabata. Imafanana ndi gawo la pulogalamu ya MIT MicroMasters mu ziwerengero ndi sayansi ya data.
Maphunzirowa amawunikira dziko losatsimikizika: kuyambira ngozi zamisika yazachuma yosayembekezereka mpaka kulumikizana. Probabilistic modelling ndi gawo logwirizana la zowerengera. Ndi makiyi awiri owunikira deta iyi ndikupanga maulosi omveka mwasayansi.
Ophunzira apeza kapangidwe kake ndi zinthu zoyambira zamatsanzo a probabilistic. Kuphatikizira mitundu yosiyanasiyana, kugawa kwawo, njira ndi zosiyana. Maphunzirowa amakhudzanso njira zofotokozera. Malamulo a ziwerengero zazikulu ndi ntchito zawo, komanso njira zowonongeka.
Maphunzirowa ndi abwino kwa iwo omwe akufuna chidziwitso chofunikira mu sayansi ya data. Zimapereka chidziwitso chokwanira pa zitsanzo za probabilistic. Kuchokera kuzinthu zoyambira kupita kuzinthu zosasintha komanso zowerengera. Zonsezi ndizothandiza makamaka kwa akatswiri ndi ophunzira. Makamaka m'magawo a sayansi ya data, uinjiniya ndi ziwerengero.
Computational Probability and Inference (MIT)
Massachusetts Institute of Technology (MIT) imapereka maphunziro a "Computational Probability and Inference" mu Chingerezi. Pa pulogalamuyi, mawu oyambira apakati pakuwunika kwa kuthekera komanso kutanthauzira. Maphunziro a masabata khumi ndi awiriwa, omwe amafunikira maola 4-6 pa sabata, ndikufufuza kochititsa chidwi kwa momwe mwayi ndi malingaliro amagwiritsidwira ntchito m'madera osiyanasiyana monga kusefa sipamu, kuyenda kwa bot, kapena ngakhale masewera anzeru ngati Jeopardy ndi Go.
M'maphunzirowa, muphunzira mfundo zakuthekera ndi kulingalira komanso momwe mungawagwiritsire ntchito pamapulogalamu apakompyuta omwe amalingalira mosatsimikizika ndikulosera. Muphunzira zamitundu yosiyanasiyana ya data yosungirako kugawa, monga ma probabilistic graphical model, ndikupanga ma aligorivimu aluso pokambirana ndi ma data awa.
Pamapeto pa maphunzirowa, mudzadziwa momwe mungatsanzire zovuta zapadziko lonse lapansi ndizotheka komanso momwe mungagwiritsire ntchito zitsanzo zomwe zatsatiridwa pakungoyerekeza. Simufunikanso kukhala ndi chidziwitso cham'mbuyomu mwina kapena kungoyerekeza, koma muyenera kukhala omasuka ndi mapulogalamu oyambira a Python ndi ma calculus.
Maphunzirowa ndi othandiza kwambiri kwa iwo omwe akufuna kumvetsetsa ndi kugwiritsa ntchito njira zowerengera m'munda wa sayansi ya data, zomwe zimapereka chidziwitso chokwanira pamitundu yotheka komanso kuwerengera mawerengero.
Pamtima Wosatsimikizika: MIT Imasokoneza Kuthekera
M'maphunzirowa "Introduction to Probability Part II: Inference processes", Massachusetts Institute of Technology (MIT) imapereka kumizidwa kwapamwamba kwambiri padziko lapansi pakuthekera komanso kulingalira. Maphunzirowa, kwathunthu mu Chingerezi, ndikupitilira gawo loyamba, kulowa mozama pakusanthula deta komanso sayansi yokayikitsa.
Kwa nthawi ya masabata khumi ndi asanu ndi limodzi, ndi kudzipereka kwa maola 6 pa sabata, maphunzirowa amafufuza malamulo a anthu ambiri, njira zowonetsera za Bayesian, ziwerengero zakale, ndi njira zowonongeka monga njira za Poisson ndi unyolo wa Markov. Uku ndikufufuza mozama, komwe kumapangidwira iwo omwe ali ndi maziko olimba mwina.
Maphunzirowa ndi odziwika bwino chifukwa cha njira yake yodziwikiratu, kwinaku akusunga masamu okhwima. Sichimangopereka malingaliro ndi maumboni, koma cholinga chake ndikukulitsa kumvetsetsa kwamalingaliro pogwiritsa ntchito konkriti. Ophunzira aphunzira kutengera zochitika zovuta ndikutanthauzira zenizeni zenizeni.
Ndioyenera kwa akatswiri a sayansi ya data, ofufuza, ndi ophunzira, maphunzirowa amapereka malingaliro apadera a momwe kuthekera ndi malingaliro amapangira kumvetsetsa kwathu dziko lapansi. Zabwino kwa iwo omwe akufuna kukulitsa kumvetsetsa kwawo kwa sayansi ya data ndi kusanthula mawerengero.
Analytical Combinatorics: A Princeton Course for Deciphering Complex Structures (Princeton)
Maphunziro a Analytic Combinatorics, operekedwa ndi Yunivesite ya Princeton, ndikufufuza kochititsa chidwi kwa ma analytical combinatorics, njira yomwe imathandizira kuneneratu kwachulukidwe kwazinthu zovuta kuphatikiza. Maphunzirowa, onse mu Chingerezi, ndi chida chofunikira kwa iwo omwe akufuna kumvetsetsa ndikugwiritsa ntchito njira zapamwamba pankhani ya ma combinatorics.
Kutenga masabata atatu ndipo kumafuna pafupifupi maola a 16, kapena pafupifupi maola 5 pa sabata, maphunzirowa amayambitsa njira yophiphiritsira yopezera maubwenzi ogwira ntchito pakati pa ntchito wamba, zowonetsera, ndi zopanga zosiyanasiyana. Imafufuzanso njira zowunikira zovuta kuti zipeze ma asymptotics enieni kuchokera ku ma equation a ntchito zopangira.
Ophunzira azindikira momwe ma combinatorics owunikira angagwiritsire ntchito kulosera kuchuluka kwake m'magulu akuluakulu ophatikiza. Aphunzira kugwiritsa ntchito zida zophatikizira ndikugwiritsa ntchito njira zowunikira zovuta kuti azisanthula izi.
Maphunzirowa ndi abwino kwa iwo omwe akufuna kukulitsa kumvetsetsa kwawo kwa ma combinatorics ndikugwiritsa ntchito kwake pakuthana ndi zovuta zovuta. Imapereka mawonekedwe apadera amomwe ma analytical combinatorics amapangira kumvetsetsa kwathu masamu ndi ma combinatorial.
