
wonn neuman virodhabhas

ganitat, johnn wonn newmanchya navavarun naav dilela wonn neuman virodhabhas hii kalpana aahe qee ekhadi vyakti ekak chauras yasarkhya samtol aakrutiche gunanchya sanchanmadhye vibhajan karoo shakate aanee pratyek sanchala kshetra-sanrakshn karnare ephene parivartan asey karoo shakate qee parinam mool aakarachya samaan aakarachya done samtol aakruti aahet. heey 1929 madhye johnn wonn neuman yanni nivadicha swasiddhant gruheet dharoon siddha kele. heey purvichya banach-tarsky virodhabhasavar aadharit aahe, joou badalat housedorf virodhabhasavar aadharit aahe.
banach aanee tarsky yanni heey siddha kele hote qee, samamiteey parivartanancha vapar karoon, dvi-aayami aakruti vegale karoon punha ekatra karnyachya parinamache kshetrafal mool aakrutisarkhech asane aavashyak aahe. yamule eka ashakya paiki done ekak chauras tayar hotil. parantu wonn neumanla samajale qee asha tathakathit virodhabhasi vighatananchi yukti mhanje parivartananchya gatacha vapar jyamadhye upasamuh mhanoon done generators mukt gut samavisht aahe. kshetra-sanrakshn parivartananchya gatamadhye (mug too vishesh resheey gut asso kinwa vishesh sanlagn gut) asey upasamuh asatat aanee yamule tyancha vapar karoon virodhabhasi vighatan karanyachi shakyata ughadate.
paddhatiche rekhachitra.
wonn newmanne sapadalelya paddhatiche anaupacharik varnan khalilpramane aahe. asey gruheet dhara qee σ aanee τ yaa done parivartanandvare nirman jhalelya kshetra-sanrakshn resheey parivartanancha eka mukt gut "h" aahe, jeey olakh ghatakapasun phaar door naheet. mukt gut asanyacha arth assa aahe qee tyache serve ghatak kahee "n" sathee sootra _ 1 madhye advitiyapane vyakt kele jaau shakatat, jethe sootra _ 2s aanee sootra _ 3s heey serve shunyetar purnank aahet, bahudha pahile sootra _ 2 aanee shevatche sootra _ 3 vagalata. aapan yaa gatala done bhaganmadhye vibhagoo shakatoh jeey davikadoon σ teey kahee shunyetar shaktine suru hotat (aapan yaa sanchala "a" mhanato) aanee jeey τ teey kahee shaktine suru hotat (mhanje, sootra _ 6 shoonya aahe-aapan yaa sanchala "b" mhanato aanee tyat olakh samavisht aahe).
jarr aapan euclidean 2-spacemadhil konatyahi binduvar "etch" chya vividh ghatkandware karya kele tarr aaplyala tyaa binduchi kaksha milate. asha prakare samatalatil serve bindunche kakshanmadhye vargikaran kele jaau shakate, jyapaiki satatyachya mukhytesah eka anant sankhya aahe. nivadicha svayamsiddha siddhant vaparoon, aapan pratyek kakshetoon eka bindu nivadu shakato aanee yaa bindunchya sanchala 'm' mhanoo shakato. aapan mool vagalato, joou "etch" madheel eka nishchit bindu aahe. jarr aapan nantar 'etch' chya serve ghatkandware 'emm' varr karya kele, tarr aapan pratalacha pratyek bindu (mool vagalata) achook ekada nirman karato. jarr aapan 'aaa' kinwa 'buub' chya serve ghatkandware 'emm' varr karya kele tarr aaplyala done vichchhedit sanch midatat jyanche serve guna parantu mool aahet.
aata aapan unit square kinwa unit disksarkhi kahee aakruti gheto. tyanantar aapan tyachya aat purnapane dusari aakruti nivadato, jase qee moodavar kendrit asalela eka lahan chauras. jaree kahee guna done kinwa adhik pratinni jhakalele asale, taree aapan lahan aakrutichya anek prateensah mothee aakruti zaku shakato. tyanantar aapan mothya aakruticha pratyek bindu chhotya aakrutichya eka pratila niyukt karoo shakato. pratyek prat sutrashi sambandhit sanchanna aapan 7 mhanuya. aata aapan keval kshetra-sanrakshn parivartan vaparoon, mothya aakrutitil pratyek binduche tyachya aateel binduparyant eka-eka karoon nakasha banavoo. aapan sootra _ 8 chey bindu gheto aanee tyanche bhashantar karato jenekarun sootra _ 8 chaurasache kendra moolsthani asel. tyanantar aapan tyat teey bindu gheto jeey varr paribhashit kelelya 'aaa' sanchat aahet aanee kshetra-sanrakshn operation στ dware tyanchyavar karya karato. heey tyanna 'buub' sanchat thevate. tyanantar aapan 'buub' chey guna gheto aanee tyanchyavar 'σ2' neey karya karato. teey aata 'buub' madhye asateel, parantu yaa binduncha sanch mageel sanchapasun vegala asel. aapan asha prakare pudhe jaato, 'c' 2 pasoon 'a' bindunvar 'σ3τ' aanee tyachya 'b' bindunvar 'σ4' vaparoon. asha prakare, aamhi mothya aakadyapasun (kahee nishchit bindu vagalata) serve guna eka-eka paddhatine, kendrapasoon phaar door nasalelya 'buub' prakarchya bindunparyant aanee mothya aakadyachya aat mapp kele aahet. tyanantar aapan 'aaa' prakarchya bindunche dusare mapping karoo shakato.
yaa tappyavar aapan cantor-bernstein-schroeder pramey paddhat lagoo karoo shakato. heey pramey aaplyala sangate qee jarr 'eae' seet karnyasathi 'dii' sanchatoon injection (jase qee mothya aakrutipasoon 'aii' prakarchya bindunparyant) aanee 'eae' pasoon 'dii' paryantche injection (jase qee aakrutitil 'aii' prakarchya bindunmadhun swatahsathi olakh mapping) asel tarr 'dii' aanee 'eae' darmyan eka-eka sanvad aahe. dusya shabdant, mothya aakrutipasoon tyateel 'aaa' bindunchya upasanchaparyantche mapping karoon, aapan mothya aakrutipasoon tyateel 'serve' aaa 'bindunparyantche mapping (eka dwibhajan) karoo shakato. (kahee pradeshanmadhye bindu swatahsathi mapp kele jaatat, tarr itaranmadhye mageel parichhedat varnan kelelya mappingcha vapar karoon mapp kele jaatat.) tyachpramane aapan mothya aakrutipasoon tyateel serve 'buub' bindunparyantche mapping karoo shakato. tarr heey dusya bajoone pahata, aapan aakrutila tyachya "aaa" aanee "buub" gunanmadhye vegale karoo shakato aanee nantar yaa pratyekacha sampoorn aakrutimadhye (mhanje donhi prakarche guna asalele) nakasha banavoo shakato!
heey rekhachitra kahee goshtinvar prakash takte, jase qee nishchit bindu kase hatalayache. heey asey disun yete qee hyabhovati kaam karnyasathi adhik nakashe aanee adhik sanch aavashyak aahet.
parinam.
chaurasasathi virodhabhas khalilpramane majboot keela jaau shakatoh
yache upayanchya samasyeshi sambandhit parinam aahet. wonn newmanne namood kelyapramane,
heey thode adhik spasht karnyasathi, vishisht parivartanankhali jatan kelele maryaditpane yogatmak mojmap astitvat aahe qaa haah prashna konatya parivartananna parvanagi aahe yavar avalambun asato. pratalatil sanchanche banach mojmap, jeey anuvad aanee paribhramanadvare jatan kele jaate, teey bahubhuj kshetra jatan karat asatanahi samamiteey nasalelya parivartanandvare jatan kele jaat nahee. varr spasht kelyapramane, pratalache bindu (mool vyatirikt) done ghan sanchanmadhye vibhagale jaau shakatat jyanna aapan "aaa" aanee "buub" mhanoo shakato. jarr dilelya bahubhujatil "aaa" bindu vishisht kshetra-sanrakshn parivartanane aanee "buub" bindu dusaryane rupantarit kele tarr donhi sanch done naveen bahubhujanmadhye "buub" bindunche upasanch banoo shakatat. naveen bahubhujanche kshetrafal junya bahubhujasarkhech aahe, parantu done rupantarit sanchanche mojmap purvisarkhe asoo shakat nahee (kaaran tyat "buub" binduncha keval eka bhag asato) aanee mhanoon "karya karate" asey konatehi mojmap nahee.
banach-tarsky ghatanechya abhyasadarmyan wonn newmanne vegale kelele gut ganitachya anek kshetransathi khoop mahatvache tharleh heey anukool gut aahet, kinwa aparivartaniy sarasari asalele gut aahet aanee tyat serve maryadit aanee serve nirakaran karnyayogya gut samavisht aahet. sarvasadharanpane sangayache tarr, jevha samatulyatechya vyakhyemadhye samatulyatesathi vaparla jaanara gut 'anukool' nasato tevha virodhabhasatmak vighatan udbhavate.
alikadachi pragati.
wonn newmanchya peparne resheey gut "ess. ell". (2, aar) (wagon, prashna 7.4) chya sandarbhat ekak chaurasachya aateel bhagachya virodhabhasi vighatan honyachi shakyata khulee thewali. 2000 madhye, miklos laxkovichne asey siddha kele qee asey vighatan astitvat aahe. adhik tantotant, 'aaa' heey rikt nasalelya aateel aanee moodapasoon sakaratmak antaravar asalelya pratalachya serve baddh upasanchanche kutumb asoo dya aanee 'buub' heey serve samtol sanchanche kutumb asoo dya, jyache gunavishesh 'slu' (2, aar) chya kahee ghatakankhali maryaditpane anuvadit kelelya anekanchya sanyogat moolcha eka chhidrit parisar asato. mug 'aaa' kutumbatil serve sanch 'slu' (2, aar)-samatulya aahet aanee tyachpramane 'buub' madheel sanchansathi. asey aahe qee donhi kutumbanmadhye virodhabhasi sanch asatat.