reeman-roqa pramey

nirdharit shoonya aanee anumat dhruvansah meromorphic functionschya jagechya parimananchya gananesathi, reeman-roqa pramey heey ganitatil eka mahattvache pramey aahe, visheshtah jatil vishleshan aanee bijganitiy bhoomitimadhye. heey prushthbhagachya purnapane topological vansh "jee" shhee jodalelya sankshipt reeman prushthbhagachya jatil vishleshanacha asha prakare sambandh thevate, jeey purnapane bijganitiy settingsmadhye nele jaau shakate.
suruvatila remanchi asamanata mhanoon siddha jhaale, remanchya alpakalin vidyarthyachya kamanantar heey pramey reeman prushthbhagasathi tyachya nishchit swarupaparyant pohochale. nantar tyache bijganitiy vakranmadhye, uchch-aayami prakaranmadhye aanee tyapalikade samanyikaran karnyat aale.
prathamik kalpana.
reeman prushthbhagache sootra _ 1 hii eka topological jaaga aahe jee sthanik patalivar formula _ 2 chya khulya upasanchasathi, jatil sankhyanchya sanchasathi samaroop aahe. yavyatirikt, yaa khulya upasanchanmadhil sankraman nakashe holomorphic asane aavashyak aahe. nantarchi aut sootra _ 2 vareel holomorphic aanee meromorphic functionshee sambandhit jatil vishleshanachya kalpana aanee paddhati prushthbhagachya sootra _ 1 madhye hastantarit karnyas anumati dete. reeman-rotch pramey uddeshansathi, prushthbhagache sootra _ 1 nehmich sankshipt manale jaate. bolchalchya bhashet, reeman prushthbhagache vansh sootra _ 6 mhanje tyachya handlechi sankhya; udaharanarth ujavikade darshavilelya reeman prushthbhagachi vansh teen aahe. adhik tantotantpane, yaa vanshala pahilya betty kramankacha ardha bhag, mhanje aay. mhanoon paribhashit kele jaate.eae., jatil gunank asalelya pahilya ekvachan samarupata gatachya sootrachya _ 2-parimanachya _ 8 chya ardhyavar. vanshasankalit reeman prushthbhaganche homomorphismparyant vargikaran karato, i.eae.jarr aanee tyanchi vansh samaan asel tarach asey done prushthbhag homomorphic asatat. tyamule, hii vanshaval hii reeman prushthbhagachi eka mahattvachi topological aparivartaniy aahe. dusarikade, hodge siddhant darshavito qee vansh sootra 1 vareel holomorphic vann-formchya jagechya sootra _ 2-parimanashi julato, mhanoon vansh reeman prushthbhagabaddal jatil-vishleshanatmak mahitee dekhil encode karato.
vibhajak sootra _ 11 haah prushthbhagachya bindunvaril mukt abelian gatacha eka ghatak aahe. samatulya, vibhajak heey prushthbhagachya bindunche purnank gunankansah maryadit resheey sanyojan aahe.
konatehi meromorphic karya sootra _ 12 heey darshavilelya vibhajak sutrala janm dete _ 13 mhanoon paribhashit kele aahe
jethe sootra _ 15 haah sootra _ 12 chya serve shunyancha aanee dhruvancha sanch aahe aanee sootra _ 17 heey asey dile aaheh
sanch sootra _ 15 heey maryadit mhanoon olakhale jaate; heey sootra _ 1 sankshipt asanyacha aanee (shoonya nasalelya) holomorphic karyachya shoonyanna sanchay bindu nasalyachi vastusthiti aahe. mhanoon, sootra _ 13 changlya prakare paribhashit aahe. yaa swarupachya konatyahi vibhajkala mukhya vibhajak mhanatat. mukhya vibhajkapasun bhinn asalelya done vibhajkansathi resheey samatulya mhanatat. meromorphic 1-formcha vibhajak ashach prakare paribhashit keela jaato. jaagtik meromorphic 1-swarupachya vibhajkala canonical devisor (samanyatah sootra _ 22 darshavile jaate) mhanatat. konatehi done meromorphic 1-form resheey samatulya vibhajak deil, mhanoon canonical devisor advitiyapane resheey samatulyateparyant nirdharit keela jaato (mhanoon "canonical devisor").
chinh sootra _ 23 heey vibhajak sootra _ 11, i chey "degree" (kadhikadhi nirdeshank dekhil mhanatat) darshavate.eae. sootra _ 11 madhye vistaratoon gunankanchi berries. heey darshavile jaau shakate qee jaagtik meromorphic karyachya vibhajakachi degree nehmich 0 asate, mhanoon vibhajakachi degree keval tyachya resheey samatulya vargavar avalambun asate.
26 heey sootra heey prathamik swarasya asalele praman aaheh prushthbhagavaril meromorphic functions sootra _ 28 chya vector spacemadhil pariman (sootra _ 2 varr), mhanje sootra _ 29 chey serve gunank nakaratmak naheet. antardnyanane, aapan heey serve meromorphic functions mhanoon vichar karoo shakato jyanche pratyek binduvaril dhruv sootra _ 11 madheel sambandhit gunankapeksha white nasatat; jarr sootra _ 32 madheel sootra _ 11 madheel gunank nakaratmak asel, tarr aaplyala tyaa sootra _ 28 madhye sootra _ 32 varr kiman tyaa gunakaratil shoonya asane aavashyak aahe-sootra _ 11 madheel gunank sakaratmak asalyas, sootra _ 28 madhye jastit jaast tyaa kramacha dhruv asoo shakato. resheey samatulya vibhajkansathi vector jaaga naisargikaritya jaagtik meromorphic karyasah gunakaradvare samaroop asatat (jeey skelarparyant changlya prakare paribhashit kele jaate).
pramey vidhan.
vansh sootra _ 6 chya sankshipt reeman prushthbhagasathi reeman-roqa pramey, jyamadhye vihit vibhajak sootra _ 22 sthitee aahet
samanyatah, sankhya sootra _ 26 heey swarasyapoorn asate, tarr sootra _ 41 hii sudharana sandnya (jyala visheshatecha nirdeshank dekhil mhanatat) mhanoon manali jaate, tyamule pramey andaje asey mhanat vyakhya kelee jaau shakate
kaaran teey vector jaageche pariman aahe, sudharana sandnya sootra _ 41 nehmich nakaratmak nasate, jenekarun
yala 'remanchi asamanata' mhanatat. "yaa vidhanacha 'rochacha bhag' haah asamanatechya bajunmadhil sambhavya farkache varnan aahe. vansh sootra _ 6 chya samanya reeman prushthbhagavar, sootra _ 22 chey ansh sootra _ 46 asate, jeey vibhajakache pratinidhitva karnyasathi nivadalelya meromorphic swarupapasoon swatantra asate. heey prameyamadhye sootra _ 47 thevalyane yete. visheshtah, joparyant sootra _ 11 madhye kiman sootra _ 49 degree aahe, toparyant durustichi sandnya 0 aahe, jenekarun
heey pramey aata nimn vanshanchya prushthbhagansathi spasht kele jaeel. itar anek jawalun sambandhit pramey dekhil aaheth reshechya bandalcha vapar karoon yaa prameyala samatulya rachana aanee bijganitiy vakransathi pramey samanyikaran.
udaharane.
yaa vishayachya prushthbhagavaril bindu sootra _ 51 nivadoon aanee sankhyanchya kramababat pramey spasht kele jaeel.
aay.eae., sootra _ 51 vagalata sarvatra holomorphic asalelya karyanchya jaageche pariman jethe karyala jastit jaast sootra _ 54 varr kramacha dhruv asanyachi parvanagi aahe. sootra _ 55 sathee, karye asha prakare sampoorn asane aavashyak aahe, i.eae.sampoorn prushthbhagachya sutravar holomorphic. leuvilchya prameyanusar, asey karya apariharyapane sthir asate. mhanoon, sootra _ 57. sarvasadharanpane, anukram sootra _ 58 haah eka vadhata kram aahe.
jeane shoonya.
reeman goal (jyala jatil prakshepak resha dekhil mhanatat) sahajpane jodalela aahe aanee mhanoon tyachi pahili ekvachan samarupata shoonya aahe. visheshtah tyachi vanshasankhya shoonya aahe. goal sootra _ 2 chya done pratinni jhakala jaau shakato, sankraman nakasha yaa dware deela jaat aahe
mhanoon, sootra _ 62 chya eka prativaril formula _ 61 haah reeman golavareel meromorphic swarupaparyant vistaratoho tyala anantatevar duheri dhruv aahe, kaaran
asha prakare, tyache vibhajak sootra _ 64 (jethe sootra _ 51 haah anant bindu aahe).
mhanoon, pramey mhanato qee anukram sootra _ 58 vachate
haah kram aanshik apurnankanchya siddhantatun dekhil vachala jaau shakato. yaulat jarr haah kram asha prakare suru jhaala tarr sootra _ 6 shoonya asane aavashyak aahe.
eka vansh.
pudhil prakaran heey vansh sootra _ 68 chey reeman prushthbhag aahe, jase qee torrus sootra _ 69, jethe sootra _ 70 heey dvimiteey jaali aahe (sootra _ 71 sathee samaroop gut). tyachi vansh eka aaheh ujavikadil chitrat darshavilyapramane, tyacha pahila ekvachan samarupata gut done loopdware muktapane tayar keela jaato. sootra _ 73 vareel manak jatil samanvay sootra _ 32 heey sootra _ 1 varr eka-swaroop sootra _ 61 dete jeey sarvatra holomorphic aahe, i.eae., tyala khambhe ajibat naheet. mhanoon, sootra _ 22, sootra _ 77 chaa vibhajak shoonya aahe.
yaa prushthbhagavar, haah kram aahe
aanee heey cases sootra _ 68 chey vaishishtya darshavate. kharokhar, sootra _ 79 sathee, sootra _ 80, varr namood kelyapramane. sootra _ 82 asalelya sootra _ 81 sathee, sootra _ 83 chee degree katekorpane nakaratmak aahe, jenekarun durustichi sandnya 0 aahe. parimanancha kram dekhil lambvartulakar karyanchya siddhantavarun ghetala jaau shakato.
vansh done aanee tyahoon pudhe.
sootra _ 84 sathee, varr namood kelela kram aahe
yavarun asey disun yete qee? bindunusar, 2 vya padvichi mudat ektar 1 kinwa 2 asate. heey siddha kele jaau shakate qee konatyahi vansh 2 vakramadhye nemke sahaa bindu asatat jyanche anukram 1,1,2,2, aahet... aanee urvarit bindunmadhye samanya kram 1,1,1,2 aahet... visheshtah, janas 2 vakr haah hyperalyptic vakr asato. sootra 85 sathee heey nehmich kharey aahe qee bahutek bindunvar kram sootra 86 neey suru hoto aanee itar kramansah maryaditpane anek bindu asatat (warestrass bindu pahaa).
reshechya bandalsathi reeman-rotch.
reeman prushthbhagavaril vibhajak aanee holomorphic reshechya bandalmadhil jawalche patravyavahar vaparoon, pramey vegalya, parantu samatulya paddhatine dekhil sangitale jaau shakateh 'x' varr 'l' haah holomorphic reshecha bundle asoo dya. sootra _ 87 heey "l" chya holomorphic vibhaganchi jaaga darshavu dya. hii jaaga maryadit-aayami asel; tyache pariman sootra _ 88 darshavile jaate. 'keey' heey 'ex' varr canonical bundle darshavuya. mug, reeman-roqa pramey asey sangato qee
mageel vibhagache pramey heey jevha "l" heey bindu bundle asate, tyache vishesh udaharan aahe.
heey pramey heey darshavinyasathi lagoo kele jaau shakate qee "k" chey "g" resheey swatantra holomorphic vibhag aahet kinwa "x" varr eka-prakar aahe. l" laa kshullak bundle mhanoon ghetalyas, sootra _ 90 kaaran "x" vareel ekmev holomorphic karye sthirank aahet. l" chee degree shoonya aahe aanee sootra _ 91 heey kshullak bundle aahe. tyamule,
mhanoon, sootra _ 93, heey siddha karate qee "jee" holomorphic eka-prakar aahet.
canonical bandalchi degree.
22 yaa adhikrut bundle sutrala sootra 93 asalyane, 96 yaa sutrala reeman-rotch lagoo kelyane milate.
jyache punarlekhan kele jaau shakate
mhanoon canonical bandalchi degree sootra _ 99 aahe.
bijganitiy vakransathi reeman-roqa pramey.
reeman prushthbhagavaril vibhajkansathichya reeman-roqa prameyavaril vareel sutratil pratyek vastoocha bijganitiy bhoomitimadhye eka samanarth aahe. reeman prushthbhagache anuroop heey kshetra "keey" varr eka ekeri nasalele bijganitiy vakr "see" aahe. paribhashetil farak (vakr viruddha. prushthbhag) yache kaaran mhanje riman prushthbhagache vastavik anekpat mhanoon pariman done aahe, parantu eka jatil anekpat mhanoon aahe. reeman prushthbhagachi ghattapana hii bijganitiy vakr purn honyachya attis samantar asate, jee prakshepak asanyaitakich aahe. keey" yaa samanya kshetravar, ekvachan (saha) samarupatechi konatihi changli kalpana nahee. tathakathit bhaumitik vansh ashee paribhashit kelee jaate
aay.eae.jaagtik staravar paribhashit (bijganitiy) eka-swarupanchya jaageche pariman mhanoon (kahler vibhedak pahaa). shevati, reeman prushthbhagavaril meromorphic functions sthanik patalivar holomorphic functionsche apurnank mhanoon darshavili jaatat. mhanoon tyanchi jaaga tarksangat karye ghetat, jeey sthanik patalivar niyamit karyanche apurnank asatat. asha prakare, vakravareel tarksangat karyanchya jagechya parimanasathi ('keey' varr) sootra _ 26 lihine, jyache pratyek binduvaril dhruv 'dii' madheel sambandhit gunankapeksha white nasatat, vareel sutrapramanech aaheh
jethe "c" haah bijganitiyadrushtya bund kshetra "k" varr prakshepak ekvachanitar bijganitiy vakr aahe. kharey tarr, konatyahi kshetravaril prakshepak vakransathi tech sootra lagoo hote, apavad mhanje vibhajkachya anshane aadhar kshetrachya sambhavya vistaratoon vistaratoon gunakarancha aanee vibhajakache samarthan samarthan bindunchya avashesh kshetrancha vichar karane aavashyak aahe. shevati, artinian ringvareel yogya vakratesathi, vibhajkashi sambandhit reshechya bundleche euler vaishishtya vibhajkachya anshadvare (yogya prakare paribhashit) aanee sanrachanatmak sheef sootra _ 103 chya euler vaishishtyanusar dile jaate.
prameyatil chikatpanachi dharana dekhil shithil kelee jaau shakateh bijganitiyadrushtya bund kshetravaril (prakshepak) vakratesathi, jyanchya serve sthanik rings gorenstein rings aahet, vareel pramanech vidhan aahe, parantu varr paribhashit kelelya bhaumitik vanshala "jee" aii "yaa ankganitiy vanshane badalale aahe, jyachi vyakhya ashee aaheh
(gulgulit vakransathi, bhaumitik vanshganitashi sahmat aahe.) heey pramey samanya ekvachan vakranparyant (aanee uchch-aayami prakaranparyant) dekhil vistarit kele gele aahe.
anuprayog.
hilbert bahupadi.
reeman-rotcha eka mahattvacha parinam mhanje too vakrareshevaril reshechya bandalchya hilbert bahupadichi ganana karnyasathi eka sootra deto. jarr reshechya bundleche sootra _ 105 purese asel, tarr hilbert bahupadi pratham degreeche sootra _ 106 deil, jeey prakshepak jaaget antarbhut karel. udaharanarth, canonical sheef sootra _ 107 madhye degree sootra _ 108 aahe, jeey genus sootra _ 109 sathee puresa resha bundle dete. jarr aapan sootra _ 110 nishchit kele tarr reeman-rock sootra vachate.
sootra _ 107 chey degree sootra _ 112 hilbert bahupadi dett aahe
kaaran tri-kayanatriy aavaran sootra _ 115 heey vakr antarbhut karnyasathi vaparale jaate, hilbert bahupadi
sootra _ 116
samanyatah vakranchi hilbert yojana (aanee bijganitiy vakranchi moduliswak) tayar kartana vicharat ghetali jaate. hii bahupadi aahe
sootra _ 117
aanee tyala jee vansh vakratechi hilbert bahupadi mhanatat.
pluricanonical embedding.
yaa samikaranache aanakhi vishleshan kartana, yoolarche vaishishtya asey aahe
sootra _ 119 pasoon
sootra _ 121 sathee, tyachi degree serve sootra _ 109 sathee nakaratmak asalyane, yacha arth assa qee tyala konatehi jaagtik vibhag naheet, sootra _ 123 chya jaagtik vibhaganmadhun kahee prakshepak jaaget eka embedding aahe. visheshtah, sootra _ 115 sootra _ 125 madhye embedding dete jethe sootra _ 126 heey sootra _ 127 pasoon aahe. heey bijganitiy vakranchya modulis jagechya bandhkamat upayukt aahe kaaran hilbert bahupadi sootra _ 128 saha hilbert yojana tayar karnyasathi prakshepak jaaga mhanoon tyacha vapar keela jaau shakato.
ekatmatesah samtal vakranchi prajati.
jevha yogyaritya mojale jaate, tevha 'dii' degreechya aparivartaniy samtal bijganitiy vakramadhye ('dii' −1) ('dii' −2)/2 − 'jee' ekankakta asate. asey aahe qee, jarr vakramadhye ("d" −1) ("d" −2)/2 bhinn ekankata asateel, tarr tee eka tarksangat vakr aahe aanee asha prakare, eka tarksangat mapdand swikarte.
reeman-herwitz sootra.
reeman prushthbhag kinwa bijganitiy vakranmadhil (remified) nakashanshi sambandhit reeman-herwitz sootra heey reeman-roqa prameyacha parinam aahe.
vishesh vibhajkanvar cliffordche pramey.
vishesh vibhajkanvar cliffordche pramey dekhil reeman-rotch prameyacha parinam aahe. tyat asey mhatale aahe qee vishesh vibhajkasathi (i.eae., asey qee sootra _ 129) sootra _ 130 chey samadhan karnare khaleel asamanata aaheh
purava.
bijganitiy vakransathi purava.
bijganitiy vakransathiche vidhan sair dwaitata vaparoon siddha kele jaau shakate. purnank sootra _ 26 heey "d" ("cf") shhee sambandhit reshechya bundle sootra _ 133 chya jaagtik vibhaganchya jaageche pariman aahe."cartier devisor). sheef cohomologychya sandarbhat, mhanoon aaplyakade sootra 134 aahe aanee tyachpramane sootra 135 aahe. parantu vakratechya vishisht prakaranat ekeri nasalelya prakshepak prakaransathi sair dwaitata sangate qee sootra 136 heey duheri sootra _ 137 sathee samaroop aahe. asha prakare davya hatachi bajoo vibhajak 'dii' chya euler vaishishtyachya barobarichi aahe. jevha "d" = 0, tevha aaplyala sanrachanechya aavaranasathi euler vaishishtyapurn sootra _ 138 heey vyakhyenusar aadhalate. samanya vibhajakache pramey siddha karnyasathi, nantar konee vibhajkamadhye eka eka karoon bindu jodoon pudhe jaau shakate aanee yoolarche vaishishtya ujavya hatachya bajus tyanusar rupantarit hote yachee khatri karoo shakate.
sankshipt reeman prushthbhagansathi purava.
chauche pramey aanee gaga tattva vaparoon bijganitiy aavruttivarun sankshipt reeman prushthbhaganche pramey kadhale jaau shakateh kharey tarr, pratyek sankshipt reeman prushthbhag kahee jatil prakshepak jaaget bijganitiy samikaranandware paribhashit keela jaato. (chauche pramey asey sangate qee prakshepak jaagechi konatihi bund vishleshanatmak upprakar bijganitiy samikaranandware paribhashit kelee jaate aanee gaga tattva asey mhanate qee bijganitiy vividhateche sheef cohomology heey tyach samikaranandware paribhashit kelelya vishleshanatmak vividhatechya sheef kohomologisarkhech aahe).
bijganitiy vakranchya babatit puravyashi samaan yuktivad karoon chauchya prameyacha vapar talhata yeil, parantu sootra _ 133 chya jaagi 'etch' yaa meromorphic functionschya sheef sootra _ 140 laa asha prakare badalale jaau shakate qee vibhajak sootra _ 29 chey serve gunank nakaratmak nastil. yethe jevha konee vibhajkala bindu jodato tevha yularache vaishishtya ichchitatenusar badalate hii vastusthiti, chhotya achook kramane prerit kelelya lamb achook kramane vachata yete.
jethe sootra _ 143 heey "pii" varr gaganchumbi imaratiche aavaran aahe aanee nakasha sootra _ 144 heey sootra _ 145 vaa laurent gunank darshavite, jethe sootra _ 145 aahe.
ankganitiy reeman-roqa pramey.
ankganitiy reeman-roqa prameyatil eka aavruttit asey mhatale aahe qee jarr 'keey' heey jaagtik kshetra asel aanee 'eff' heey 'keey' chya edelesache yogyaritya swikararh karya asel tarr pratyek ideal 'aii' sathee, aekhadyache poison yogasutra sootra aaheh
vishesh prakaranat jevha "k" heey maryadit kshetravaril bijganitiy vakranche karya kshetra asate aanee "f" heey "k" varr kshullak asalele konatehi akshar asate, tevha heey bhaumitik reeman-rotch pramey punarprapt karate.
ankganitiy reeman-roqa siddhantachya itar aavrutya paramparik reeman-roqa prameyala adhik tantotant samayik karnyasathi arakelov siddhantacha vapar karatat.
reeman-roqa siddhantache samanyikaran.
1850 chya dashakat reeman aanee rotchney reeman prushthbhagasathi aanee 1931 madhye friedrich carl schmitt yanni bijganitiy vakransathi, jevha teey maryadit vaishishtyanchya paripoorn kshetranvar kaam karat hote, tevha vakransathi reeman-rotch pramey siddha keela. pieter roketne mhatalyapramane,
eff. chee pahili mukhya kamgiri. keey. schmitt haah eka shodh aahe qee sankshipt reeman prushthbhagavaril reeman-rotche shastriy pramey maryadit aadhar kshetrasah karya kshetranmadhye hastantarit kele jaau shakate. pratyakshat, tyacha reeman-rotch prameyatil purava nishchitpane maryadit nasalelya, aniyantrit paripoorn aadhar kshetransathi karya karato.
vakransathicha tyanantarcha siddhant tyatoon milanari mahitee parishkrut karnyacha prayatna karato (udaharanarth brill-nother siddhantat) yaa arthane too payabhoot aahe.
uchch parimananchya aavrutya aahet (vibhajak kinwa reshechya bandalchya yogya kalpanesathi). tyanche samanya sootra heey pramey done bhaganmadhye vibhaganyavar avalambun asate. eka, jyala aata sair dwaitata mhatale jaeel, sootra 41 sandnyecha arth pahilya sheef cohomology gatache pariman mhanoon karato; sootra 26 saha shoonya cohomology gatache pariman kinwa vibhaganchya jagesah, prameyachi davi bajoo euler vaishishtyapurn banate aanee ujavya hatachi bajoo reeman prushthbhagachya topologinusar durust kelelya "degree" mhanoon tyachi ganana karate.
dvi-aayamanchya bijganitiy bhoomitimadhye asey sootra italian shalechya bhoomitidware sapadale; prushthbhagansathi reeman-rock pramey siddha jhaale (anek aavrutya aahet, jyat pahilya bahudha kamal notharmule aahet).
eka "aine"-aayami samanyikaran, hirzebrook-riman-roqa pramey, bijganitiy topologymadhil vaishishtyapurn vargancha vapar mhanoon friedrich hirzebrookne shodhala aanee siddha keela; kunihiko kodayarachya karyane tyachyavar khoop prabhav padala. jawaljawal tyach velee jeane-piyare sair heey sair dwaitateche samanya swaroop dett hote, jase qee aata aaplyala mahit aahe.
alekzandar grothendicne 1957 madhye eka durgami samanyikaran siddha kele, jyala aata grauthendic-riman-rock pramey mhanoon olakhale jaate. tyachya karyat reeman-rotche vividhatebaddalache pramey mhanoon navhe tarr done prakaranmadhil aakarvadabaddal punarvyakhyan kele aahe. puravyancha tapasheel armond borel aanee jeane-piyare sair yanni 1958 madhye prakashit keela. nantar, grauthendic aanee tyachya tyachya purava sulabh aanee samanya keela.
shevati bijganitiy topologymadhehi eka samanya aavrutti aadhalali. yaa serve ghadamodi moolatah 1950 teey 1960 darmyan karnyat aalya. tyanantar atiyah-singer nirdeshank prameyane samanyikaranacha aanakhi eka marg khula keela. parinami, susangat aavaranache euler vaishishtya vajavipane gananiy aahe. paryayi bereejmadhye fakt eka berijsathi, lupt hott asalelya prameyasarkhya pudhil yuktivadancha vapar karane aavashyak aahe.

