kaantor settu

ganithashaasthrathil, niravadhi avyakthamaaya gunangalulla orotta line vibhaagathil kidakkunna oru koottam poyintukalaanu kaantar settu. 1874ll henry joonn steefan smith ithu kandethukayum 1883ll jarmman ganithashaasthranjanaaya jorg kaantor ithu avatharippikkukayum cheythu.
eee settu pariganikkunnathiloode, kaantarum mattullavarum aadhunika pooyint settu topology adithara paakunnathin sahayichu. oru line segmentinte madhyabhaagathe moonnilonnu neekkam cheyth sheshikkunna cheria segmentukalodoppam prakriya aavarthichu nirmmicha kaantor ternery settaanu ettavum saadhaaranamaaya nirmmaanam. kaantar tritala nirmmaanatthe kadannupokunnathil maathrame paraamarshichittulloo, kooduthal pothuvaaya oru aashayathinte udaaharanamaayi, evideyum saandramallaatha oru thikanja koottatthinte.
kooduthal pothuve, topologyil, kaantor ternery settinu (athinte subspace topology sajjeekarichirikkunnu) oru topological space homomorphic aanu "eiii" kaantor space. l enna sidhaantham. iii. jeey. ennirunnaalum, ithu thikanja shoonyamallaathathum othukkamullathum alakkaanaavaathathum poojyamaayathumaaya alavinu thulyamaanu.
tritala ganathinte nirmmanavum soothravaakyavum.
oru koottam line segmentukalil ninnu "ooppan" middle therd aavarthichu illaathaakkiyaanu kaantar ternery settu formula _ 1 srishtikkunnathu. onnu aarambhikkunnath interwell formula _ 3 ll ninnu ooppan middle therd formula _ 2 illaathaakkiyaanu, rund line segmentukal avasheshikkunnu formula _ 4. aduthathaayi, eee sheshikkunna segmentukalil oronninteyum ooppan middle therd illaathaakkukayum naalu line segmentukal avasheshikkukayum cheiyunnu formula _ 5.
kaantar ternery settil eee ananthamaaya prakriyayude oru ghattathilum illaathaakkaatha interwell formula _ 6 leeii ellah pooyintukalum adangiyirikkunnu. athe vasthuthakal krameekarichukondu aavarthichu vivarikkaam.
oppam
formula _ 9-nuu, angane
eee prakriyayude aadya aarr gattangal chuvade chithreekarichirikkunnu.
svayam-samana parivartthanangalude ashayam upayogichu, formula _ 14 formula _ 15, formula _ 16 enniva kaantar settinulla vyakthamaaya clossed formulakal ivayaanu
oro madhya moonnaam bhaagavum athinu chuttumulla clossed interwell formula 19 ll ninnu ooppan interwell formula 18 aayi neekkam cheyyumbol, allengil
mukalil paranjirikkunna clossed interwell formula _ 22-nte madhyabhaagathulla moonnaamathe formula _ 21 formula _ 23 upayogichu murichukadannu neekkam cheiyunnu.
madhya moonnilonnum neekkam cheiyunna eee prakriya oru parimitha upavibhaga niyamatthinte lalithamaaya udaaharanamaanu. kaantor ternery settinte poorakam oru fractal stringinte udaaharanamaanu.
ganithashaasthrathil paranjaal, kaantar settil yoonit interwell formula _ 6-nte ellah yadhaartha samkhyakalum adangiyirikkunnu, avaa oru tritheeya (bays 3) bhinnasankhyayaayi prakadippikkunnathin akkam 1 aavasyamilla. melpparanja rekhaachithram vyakthamaakkunnathupole, kaantar settile oro pooyintum ananthamaaya aazhathilulla bainari trealoodeyulla oru paathayiloode savisheshamaayi sthithicheyyunnu, avide oro thalatthilum idathu allengil valath thirinj pooyint eathu vashathaan sthithicheyyunnathu ennathinu anusarichu oro levelilum paatha idathotto valathotto thiriyunnu. oro idathu valavineyum 0 ennum oro valath valavineyum 2 ennum prathinidheekarikkunnath oru pooyintinu tritala bhinnasankhya nalkunnu.
curling" vazhi mandelbroatinte nirmmaanam.
prakrithiyude fractal jyamithiyil, ganithashaasthranjanaaya benoit mandelbroat ganithashaasthra paatakarallaattha vaayanakkaare formula _ 1-nte nirmmaanam sangalppikkaan sahaayikkunnathin oru vichitramaaya chinta pareekshanam nalkunnu. addehathinte vivaranam aarambhikkunnath oru baar sangalppikkunnathiloode, orupakshe bharam kuranja lohathinte, athil baarinte dravyam athinte attangalilekku aavarthichu maarunnathiloode "thairu" cheiyunnu. baarinte bhagangal cheruthaayitheerumbol, avaa nertthathum idathoornnathumaaya slaggukalaayi maarunnu, avaa oduvil valare cheruthum kaanaanaakaathavidham mangukayum cheiyunnu.curlinga curling ennu njaan vilikkunna prakriyayude falamaanu kaantor baarinte nirmmaanam. oru raundu baaril ninnaan ithu aarambhikkunnath. valare kuranja saandrathayullathaayi kanakkakkunnathaanu nallath. eee baarinte madhyabhaagathu ninnu avasaanathe moonnilonnu vare dravyam "thairu" undaakkunnu, angane randaamathethinte sthaanangal maattamillathe thudarunnu. adutha dravyam oro attatthinteyum madhya moonnilonnil ninnu athinte avasana moonnilonnu vare thairu puranthallunnu, angane ananthamaaya uyarnna saandrathayulla ananthamaaya nerttha sluggukalude ananthamaaya valia ennam avasheshikkunnathuvare ananthamaayi. ulppaadana prakriyayaal prachodippikkappedunna valare nirddishta reethiyilaan eee sluggukal variyil akalam paalikkunnath. eee thithreekaranatthil, kurling (ithinu oduvil chuttika aavasyamaanu!) printarinte presses kannum pinthudarunnathu nirtthumbol nirtthunnu; avasaanathe varii avasaanathethil ninnu verthirichariyaan kazhiyaathathaan, pakshe onnu athinte oro bhaagavum rund samanthara karutha slaggukalekkaal chaaraniratthilulla sluggai kaanapedunnu."
rachana.
kaantar settine ozhivaakkaatha pooyintukalude koottamaayi nirvachichirikkunnathil, anupatham (i.iii.bakkiyulla yoonit edavelayude alavu) neekkam cheytha motham dairkhyathaal kandethanakum. eee aakethuka jyaamitheeya purogathi aanu
angane avasheshikkunna anupatham 1 − 1 = 0 aanu.
eee kanakkukoottal soochippikkunnathu kaantar settil poojyamallaatha neelamulla edavelakalonnum ulkkollaan kazhiyilla ennaanu. enthengilum avasheshikkunnuvennathu athishayakaramaanennu thonniyekkaam-ellaathinumupari, neekkam cheytha edavelakalude dairkhyathinte aakethuka yadhaartha edavelayude dairkhyathinu thulyamaanu. ennirunnaalum, prakriya sookshumamaayi parishodhikkumbol enthengilum avasheshikkunnuvennu velippeduthunnu, kaaranam oro edavelayudeyum "madhya moonnaam bhagam" neekkam cheiyunnathil ooppan settukal (avayude end pooyintukal ulppedaatha settukal) neekkam cheiyunnathu ulppedunnu. athinaal yadhaartha edavelayil ninnu line segment (,) neekkam cheiyunnathu [0,1] poyintukalkku pinnilayi avasheshikkunnu. thudarnnulla gattangal eee (allengil mattu) end pooyintukal neekkam cheiyunnilla, kaaranam neekkam cheytha edavelakal ellaayppozhum sheshikkunna edavelakalkku aantharikamaanu. athinaal kaantar settu shoonyamalla, vaasthavathil ennamatta ananthamaaya pooyintukal adangiyirikkunnu (ananthamaaya bainari treayile paathakalude kaaryatthil mukalil paranja vivaranathil ninnu thaazhepparayunnathupole).
nirmmaana vibhaagangalude anthima pooyintukal maathrame avasheshikkunnulloo ennu thonniyekkaam, pakshe anganeyalla. udaaharanathinu, eee samkhyaykku savisheshamaaya tritheeya roopamaanu 0.020202.. =. athu thaazhe moonnaamathethum aaa moonnaamathe moonnaamathe mukalile moonnaamathethum aaa mukalile moonnaamathe thaazhathe moonnaamathethum angane palathum. ithu orikkalum madhyabhaagangalilonnillath athu orikkalum neekkam cheyyappedunnilla. ennirunnaalum ithu oru madhyabhaagathinteyum avasana poyintalla, kaaranam ithu 1/3 enna oru shakthiyudeyum gunithamalla.
segmentukalude ellah end pooyintukalum trirashtra bhinnasamkhyakale "avasaanippikkunnu", avaa settil adangiyirikkunnu.
ithu ennaavunnathra ananthamaaya oru koottamaanu.
cardinalitye sambandhichidatholam, kaantar settinte mikkavarum ellah gadakangalum edavelakalude avasana poyintukaloo naalil onnu polulla yukthisahamaya poyintukaloo allah. muzhuvan kaantar settum vaasthavathil kanakkakkaanaavilla.
swathukkal.
cardinality.
eee prakriyayil thudakkathil avasheshichirunnathra pooyintukal avasheshikkunnundennum athinaal kaantar settu ennaanaavaathathaanennum kaanikkan kazhiyum. ithu kaanunnathin, kaantar settu formula _ 1 muthal clossed interwell [0,1] vare surjective (i) enna oru function "f" undennu nammal kaanikkunnu.iii. "f" formula _ 1 ll ninnu [0,1] lake mapp cheiyunnu, angane formula _ 1-nte cardinality [0,1]-ll kurravalla. formula _ 1 [0,1] nte upavibhagamaayathinaal, athinte cardinalityum valuthalla, athinaal kaantor-bernstain-schroeder sidhaantham anusarichu rund kaardinaalittikalum yadhaarthathil thulyamayirikkanam.
eee function nirmmikkunnathin, [0,1] edavelayile pooyintukal bays 3 (allengil tritheeya) notteshante adisthaanathil pariganikkuka. sariyaaya tritheeya bhinnasamkhyakal, kooduthal krithyamaayi soothravaakyathinte gadakangal _ 32, eee notteshanil onniladhikam praathinidyangal amgeekarikkunnu, udaaharanathinu, athu 0.13 = 3 ennu ezhuthaam, mathramalla 0.0222 ennum ezhuthaam.. 3 = 3, koodaathe, athu 0.13 = 3 ennu ezhuthaam, mathramalla 0.1222 ennum ezhuthaam.. 3 = 3.
madhyabhaagathe moonnaamathethu neekkam cheyyumbol, athil 0.1xxxxx enna roopathinte tritheeya akkangalulla samkhyakal adangiyirikkunnu... 3 evide XXXX... 3 ennathu 00000-nuu idayilaanu.. 3 umm 22222 umm.. 3. athinaal aadya ghattathinu sesham avasheshikkunna samkhyakal ivayaanu
radics pooyintinu sheshamulla aadya akkam 1 allatha oru tritala praathinidyamulla samkhyakal aadya ghattathinu sesham avasheshikkunnavayaanennu paranjukondu ithu sangrahikkaam.
randaamathe ghattam 0.01xxxx enna foominte nambarukal neekkam cheiyunnu... 3 umm 0.21xxxx umm... 3, (anthima bindukkalkku uchithamaaya shradhayode) aadyathe "rund" akkangalil onnumillaatha oru tritheeya samkhyayullavayaanu sheshikkunna samkhyakal ennu nigamanam cheyyaam.
eee reethiyil thudarunnathinu, oru samkhyaye "n" ghattathil ozhivaakkaathirikkaan, athinu "n" akkam 1 allatha oru tritala praathinidhyam undaayirikkanam. oru samkhya kaantar settil aayirikkanamenkil, athu oru ghattathilum ozhivaakkaruth, poornnamaayum 0s umm 2s umm adangiya oru samkhyaa praathinidhyam athu amgeekarikkanam.
1, = 0.13, = 0.213 thudangiya samkhyakal kaantar settilaanennathu oonnipparayendathaanu, kaaranam avaykku poornnamaayum 0s umm 2s umm adangiyirikkunna tritala akkangalunda 1 = 0.222.. 3 = 3, = 0.0222... 3 = 3 umm = 0.20222.. 3 = 3.
pinneedulla ellah samkhyakalum "end pooyintukal" aanu, eee udaaharanagal formula _ 1-nte sariyaaya paridhi poyintukalaanu. formula _ 1, e-yude idathu paridhi pooyintukalkkum ithu sariyaanu.jii. = 0.1222... 3 = 3 = 3 umm = 0.21222.. 3 = 3 = 3. eee anthima bindukkalellaam roopathinte "sariyaaya tritheeya" bhinnasamkhyakalaanu (soothravaakyam _ 35-nte gadakangal), ivide bhinnasankhya athinte irreducible roopathilaayirikkumbol "q" ennathu 3-nte sakthiyaanu. eee bhinnasamkhyakalude tritala praathinidhyam avasaanikkunnu (i.iii., parimithamaanu) allengil-mukalil ninnu oorkkuka, sariyaaya trirashtra bhinnasamkhyakalkku oronninum 2 praathinidyangalundu-ananthavum ananthavum aavarthikkunna 0s allengil ananthamaaya aavarthikkunna 2s-ll "avasaanikkunnu". atharam oru bhinnasankhya athinte trirashtra praathinidhyatthil ananthamaaya niravadhi aavarthichulla 0-kalil nambar 1-kalum "end" kalum adangiyittundengil formula _ 1-nte idathu paridhi poyintaanu. athupole, oru sariyaaya trirashtra bhinnasankhya ennathu formula _ 1-nte sariyaaya paridhi poyintaanu, athinte trirashtra vipuleekaranathil ananthamaaya niravadhi aavarthikkunna 2s-kalil nambar 1-kalum "end"-kalum adangiyirikkunnundel.
eee end pooyintukalude koottam formula _ 1 ll saandramaanu (ennaal [0,1] ll saandramalla) koodaathe ennaavunna ananthamaaya oru koottam undaakkunnu. allatha" end pooyintukalaaya formula _ 1 leeii samkhyakalkku avayude tritheeya praathinidhyatthil 0s umm 2s umm maathrameyulloo, pakshe avaykku 0 enna akkathinteyo 2 enna akkathinteyo ananthamaaya aavarthanathil avasanikkaan kazhiyilla, kaaranam appol athu oru end poyintaayirikkum.
poornnamaayum 0s umm 2s umm adangiyirikkunna tritheeya samkhyakal eduthu, ellah 2s-kaleyum 1s upayogichu maattisthaapikkukayum, oru yadhaartha samkhyayude bainari praathinidhyamaayi seekwansine vyaakhyaanikkukayum cheythaanu formula _ 1 muthal [0,1] vareyulla function nirvachikkappedunnathu. oru soothravaakyathil,
[0,1] leeii eathu samkhyaykkum, athinte bainari praathinidhyam ellah 1 kaleyum 2 kall upayogichu mattisthaapikkunnathiloo formula _ 1 leeii oru samkhyayaaya "x" nte tritala praathinidhyathilekku vivarthanam cheyyaan kazhiyum. ithinoppam, "f" ("x") = "y" athinaal "y" ennathu "f" nte paridhiyilaan. udaaharanathinu "y" = = 0.100110011001.. 2 =, nammal "x" = = 0.200220022002 ennu ezhuthunnu.. 3 =. thathfalamaayi, "f" ennathu surjective aanu. ennirunnaalum, "f" ennathu "injective allah"-"f" ("x") yojikkunna moolyangal neekkam cheytha "madhya moonnilonninte" ethir attathaayirikkanam. udaaharanathinu, edukkuka
angane
athinaal [0,1] enna edavelayilullathupole thanne niravadhi pooyintukal kaantar settilum undu (ithinu ennaanaavaatha curdinality undu, ennirunnaalum, neekkam cheytha edavelakalude end pooyintukalude koottam kanakkakkaavunnathaanu, athinaal kaantar settil interwell end poyintukalallaattha ennamatta niravadhi samkhyakal undaayirikkanam. mukalil soochippichathupole, atharam oru samkhyayude oru udaaharanam, athu 0.020202 ennu ezhuthaam.. 3 = trikonamithi sankethathil. vaasthavathil, eathenkilum formula _ 47 nalkiyaal, formula _ 49 polulla formula _ 48 nilavilundu. ithu 1917 ll steyinhouse aadyamaayi pradharshippichu, addeham oru jyaamitheeya vaadathiloode, oro formulakkum formula _ 47 ennathinu thulyamaaya avakaashavaadam theliyichu. eee nirmmaanam formula _ 52 muthal formula _ 53 vare oru kuthivayppu nalkunnathinaal, namukk formula _ 54 oru udanadi falamaayi undu. eathenkilum ananthamaaya settu formula _ 56-nuu (tarsky thiranjeduttha svayamsidhathinu thulyamaanennu kaanikkunna oru prasthavana) formula _ 55 ennu karuthuka, ithu formula _ 57 enna mattoru prakadanam nalkunnu.
kaantar settil athu edukkunna edavelayude athra pooyintukal adangiyirikkunnu, ennittum poojyamallaatha neelatthinte edavela illa. yukthirahithamaya samkhyakalkku ore swabhaavamundu, ennaal kaantar settinu adaykkunnathinte adhika swabhaavamundu, athinaal oro edavelayilum saandramaaya yukthirahithamaya samkhyakalil ninnu vyathyasthamayi ithu oru edavelayilum saandramalla.
ellah beejaganita yuktirahita samkhyakalum saadhaaranamaanennu anumaanikkappettittund. kaantar settile amgangal saadhaaranamallathathinaal, kaantar settile ellah amgangalum yukthisahamo ateendriyo aanennu ithu soochippikkunnu.
svayam samanatha.
kaantar settu oru fractalinte prototypaanu. oro copy 3 enna ghadakam churukki vivarthanam cheythaal athu svayam samaanamaan, kaaranam ithu rund koppikalkku thulyamaanu. kooduthal krithyamaayi paranjaal, kaantar settu rund functionukalude samyojanathinu thulyamaanu, idathu, valath svayam-samanatha parivartthanangal, formula _ 58, formula _ 15, ithu kaantar settine homeomorphism vare maattamillathe vidunnu formula _ 60
formula _ 61, formula _ 62 ennivayude aavarthichulla aavarthanathe ananthamaaya bainari trea aayi drishyavalkkarikkaam. athaayath, maratthinte oro nodilum, idathotto valathotto ulla subtree pariganikkaam. settu formula _ 63 function compositionumaayi chernnu oru monoid, diadic monoid roopappedunnu.
bainari treaude automorphisangal athinte hyperbolic bhramanangalaan, avaa modular grouppaanu nalkunnathu. athinaal, kaantar settu formula _ 1 leeii eathenkilum rund pooyint formula _ 64, formula _ 65 ennivaykku, formula _ 68 ulla oru homomorphism formula _ 67 nilavilundu enna ardhathil kaantar settu oru ekeekritha sthalamaanu. kaantar settu discreet space formula _ 70 nte ennaavunna niravadhi koppikalude uthppanna sthalamaayi kaanukayaanengil formula _ 69 nte vyakthamaaya nirmmaanam kooduthal eluppathil vivarikkan kazhiyum. appol formula _ 72 nirvachichirikkunna mapp formula _ 71 formula _ 64, formula _ 65 enniva kaimarunna oru involutive homomorphism aanu.
samrakshana niyamam.
scalingin svayam samaanathaykkum pinnil ellaayppozhum eathenkilum tharatthilulla samrakshana niyamamaanennu kandethiyittundu. kaantar settinte kaaryatthil, nirmmaana prakriyayude eathu ghattathilum nilanilkkunna ellah edavelakaludeyum formula 75th moment (formula 76 fractal dimension aanenkil) kaantar settinte kaaryatthil onninu thulyamaaya sthiraankathinu thulyamaanennu kaanaan kazhiyum.
athinte nirmmaanatthinte 79-aamm ghattathil sisttathil valuppa formula 78-nte formula 77 edavelakal undennu namukkariyam. appol athijeevicha edavelakale formula _ 80 ennu label cheythaal formula _ 75th moment formula _ 83 muthal formula _ 82 aanu.
kaantar settinte hosdorff alavu ln (2)/ln (3) ± 0.631 nuu thulyamaanu.
topological, analytical propertykal.
kaantor settu" saadhaaranayaayi mukalil vivaricha yadhaartha, madhya-moonnaam kaantor settine soochippikkunnundengi, topologistukal palappozhum "eiii" kaantor settinekkurichu samsaarikkunnu, athaayath homomorphic (topological thulyamaaya) eathu topological space.
melpparanja sangraham kaanikkunnathupole, kaantar settu ennaanaavaathathaanu, pakshe lebsg alavu 0 undu. kaantar settu ooppan settukalude oru yooniyante poorakamaayathinaal, athu thanne realukalude oru clossed upavibhagamaanu, athinaal oru samboornna metrik space aanu. ithu poornnamaayum parimithamayathinaal, heine-borrell sidhaantham parayunnath athu othukkamullathayirikkanam ennaanu.
kaantar settile eathu pooyintinum pooyintinte eathenkilum cheria ayalpakkathinum, 0s, 2s ennivayude tritheeya samkhyayulla mattethenkilum samkhyayum, tritheeya samkhyakalil 1s adangiyirikkunna samkhyakalum undu. athinaal, kaantar settile oro pooyintum kaantar settinte oru shekharana poyintaanu (klasttar pooyint allengil limitt pooyint ennum vilikkunnu), pakshe onnumilla oru interior pooyint. oro pooyintum oru shekharana pooyintaaya oru clossed settine topologyil oru perfect settu ennum vilikkunnu, athesamayam interior poyintukalillaattha edavelayude clossed upavibhagam edavelayil evideyum saandramalla.
kaantar settinte oro pooyintum kaantar settinte poorakatthinte shekharana poyintaanu.
kaantar settile eathenkilum rund poyintukalkku, avaa vyathyasamulla chila tritala akkangal undaakum-onninu 0 umm mattethinu 2 umm undaakum. eee akkatthinte moolyathe aashrayich kaantarine "pakuthikalaayi" vibhajikkunnathiloode, oraalkk kaantarinte oru vibhajanam labikkum, athu yadhaartha rund pooyintukale verthirikkunna rund clossed settukalaayi sajjeekarikkunnu. kaantar settile relative topologyil, pooyintukal oru cloppen settu upayogichu verthirichirikkunnu. thathfalamaayi, kaantar settu poornnamaayum vichedikkappedunnu. poornnamaayum vichedikkappetta oru compact housedorff space enna nilayil, kaantar settu oru kallu sthalathinte udaaharanamaanu.
oru topological space enna nilayil, kaantar settu swaabhaavikamaayum space formulayude ennaavunna niravadhi pakarppukalude uthppannathin homomorphic aanu, avide oro copy vyathirikthamaya topology vahikkunnu. rund akkangalilulla ellah sequenceukaludeyum space ithaanu 
2-adic poornnasamkhyakalude koottam upayogichu ithu thirichariyaan kazhiyum. uthppanna topology ooppan settukalude adisthaanam silindar settukalaanu; yadhaartha variyile prakrithi topologyil ninnu kaantar settu paarambaryamaayi labhikkuna subspace topologyilekku homomorphism ivaye mapp cheiyunnu. compact speysukalude uthppannamenna nilayil kaantar spacinte eee swabhaavam, tyconophinte sidhaanthathiloode kaantar space kompaakt aanennathin randaamathe thelivu nalkunnu.
melpparanja swabhaavasavisheshathakalil ninnu, kaantar settu "p"-adic poornnasamkhyakalilekkum athil ninnu oru pooyint neekkam cheythaal "p"-adic samkhyakalilekkum homomorphic aanu.
kaantar settu realsinte oru upavibhagamaanu, avaa saadhaarana dhoora metrik sambandhicha oru metrik space aanu; athinaal athe metrik upayogichu kaantar settu thanne oru metrik space aanu. pakaramaayi, oraalkk soothravaakyam _ 86-ll "p"-adic metrik upayogikkaam rund sequence formula _ 87 nalkiyaal, avaa thammilulla dooram formula _ 88 aanu, ivide formula _ 89 ettavum cheria soochikayaanu, athaayath formula _ 90; atharam soochika illengil, rund sequenceukalum onnuthanneyaanu, oraal dooram poojyamaayi nirvachikkunnu. eee rund alavukalum kaantar settil ore topology srishtikkunnu.
kaantar settu poornnamaayum vichedikkappetta thikanja compact metrik space aanennu nammal mukalil kandu. theerchayaayum, orarthathil ithu maathramaanu shoonyamaayi poornnamaayum vichedikkappetta oro compact metrik sthalavum kaantar settinu homomorphic aanu. kaantar settilekkulla homomorphic spaceukalekkurichu kooduthal arian kaantar space kaanuka.
compact metrik speysukalude vibhaagathil kaantar settine chilappol "saarvathrika" maayi kanakkakkunnu, kaaranam eathoru compact metrik spays kaantar settinte thudarchayaaya chithramaanu; ennirunnaalum eee nirmmaanam advitheeyamalla, athinaal kaantar settu krithyamaaya kattagarikkal ardhathil saarvathrikamalla. saarvathrika" propertic pravarthanaparamaaya vishakalanathil pradhana prayogangalundu, avide ithu chilappol "compact metrik speysukalkkulla praathinidhya sidhaantham" ennariyappedunnu.
q" ≤2 enna eathenkilum poornnasamkhyaykku, g = z "q" ω (kanakkakkavunna nerittulla thuka) enna groupile topology vyathirikthamaan. ponderjagin duval γ yum z "q" ω aanenkilum, γ-nte topology kompaakt aanu. γ poornnamaayum vichedikkappettathum thikanjathumaanennu oraalkk kaanaan kazhiyum-athinaal ithu kaantar settinu homomorphic aanu. q" = 2 enna kesil homomorphism vyakthamaayi ezhuthunnathu ettavum eluppamaanu. (rudin 1962 p 40 kaanuka.)
kaantar settinte jyaamitheeya sharashari ekadesham 0.274974 aanu.
alavum saadhyathayum.
kaantar settine bainari seekwansukalude compact groupaayi kaanaan kazhiyum, athinaal, ithu oru swaabhaavika haar alavu kondu sambannamaanu. settinte alavu 1 aayi normalise cheyyumbol, ithu nanayam eriyunna ananthamaaya shreniyude oru maathrkayaanu. koodaathe, edavelayile saadhaarana lebsg alavu kaantar settile haar alavinte chithramaanennu oraalkk kaanikkan kazhiyum, athesamayam ternery settilekkulla swaabhaavika kuthivayppu oru eaka alavinulla canonical udaaharanamaanu. haar alavu eathenkilum probabilityude oru chithramaanennum ithu kaantarine chila vazhikalil oru saarvathrika probability space sajjamakkunnuvennum kaanikkam.
lebsg alaveduppu sidhaanthathil, kaantar settu ennaanaavaathathum poojya alavullathumaaya oru koottatthinte udaaharanamaanu. ithinu vipareethamaayi, logue 2/logue 3 enna alavil settinu 1 enna hosdorff alavu undu.
kaantor nambarukal.
kaantor settile amgamaayi nammal oru kaantor nambarine nirvachikkukayaanenkil,
vivaranathmaka settu sidhaantham.
kaantar settu oru cheria settaanu (allengil aadya vibhaagathinte oru koottam) [0,1] nte upavibhagamaayi (athu oru bair space aayathinal svayam oru upavibhaagamaayittallengilum). cardinality, alavu, (bair) vibhaagam ennivayude kaaryatthil "valuppam" enna ashayangal yojikkendathillennu kaantar settu angane theliyikkunnu. settu formula _ 91 pole, kaantar settu formula _ 1 oru nall settu (poojyatthinte alavu) enna ardhathil "cheruthaanu", ithu [0,1] nte oru cheria upavibhagamaanu. ennirunnaalum, kanakkakkaavunnathum "cheria" cardinality ullathumaaya formula _ 91 ll ninnu vyathyasthamayi, formula _ 94, formula _ 1 nte cardinality [0,1] nuu thulyamaanu, thudar soothravaakyam _ 96, cardinality enna ardhathil "valuth" aanu. vaasthavathil, cheriyathum ennaal positteev alavullathumaaya [0,1] oru upavibhaagavum, poojya alavil poojyamallaathathum alakkavunnathumaaya oru upavibhaagavum nirmmikkanum kazhiyum alavu formula 98-nte formula 97-nte "phaatt" kaantar settukalude kanakkakkavunna union edukkunnathiloode (nirmmaanathinaayi chuvade sajjeekarichirikkunna smith-voltera-kaantar kaanuka), namukk oru settu formula 99 labikkum, athu positteev alavu (1 nuu thulyamaanu) ennaal [0,1] ll valare kuravaanu, kaaranam oro formula _ 97 evideyum saandramalla. appol 101 enna nischitha soothravaakyam pariganikkuka. soothravaakyam 102 aayathinal, soothravaakyam 103 valare kuravaayirikkilla, pakshe soothravaakyam 104 aayathinal, soothravaakyam 103 nuu poojyamaayirikkanam alavu.
vakabhedangal.
smith-voltera-kaantar settu.
kaantar settilennapole oro kashanathinteyum madhya moonnilonnu aavarthichu neekkam cheiyunnathinupakaram, madhyathil ninnu mattethenkilum nischitha sathamaanam (0 sathamaanavum 100% umm ozhike) neekkam cheiyunnathu thudaraam. edavelayude madhyabhagam neekkam cheytha saahacharyathil, namukk shradheyamaayi axus cheyyaavunna oru cases labikkum-settil [0,1] leeii ellah samkhyakalum adangiyirikkunnu, athu poornnamaayum 0s umm 9s umm ulkkollunna dashaamshamaayi ezhuthaam. oro ghattathilum oru nischitha sathamaanam neekkam cheiyukayaanenkil, limitting settinu poojyatthinte alavu undaakum, kaaranam backie formulayude neelam _ 106 formula _ 109 polulla eathenkilum formula _ 108 nuu formula _ 107 aayi aayirikkum.
maruvashathu, oro aavarthanathilum segmentinte madhyabhaagathe cheria bhinnasamkhyakal neekkam cheiyunnathiloode positteev alavilulla "phaatt kaantar settukal" srishtikkaan kazhiyum. athinaal, evideyum saandrathayillaathappol thanne positteev lebsg alavukalulla kaantar settilekku homomorphic settukal nirmmikkaan kazhiyum. n" th aavarthanathil oro segmentinteyum madhyathil ninnu dairkhya soothravaakyam _ 110 (formula _ 111) neekkam cheiyukayaanenkil, neekkam cheytha motham neelam formula _ 112 aanu, koodaathe parimithappedutthunna settinu formula _ 113 nte lebsg alavu undaakum. athinaal, oru ardhathil, madhya-moonnaam kaantar settu formula _ 114 ulla oru parimithappedutthunna kesaanu. formula _ 115 aanenkil, baakkiyullavaykk formula _ 116 ulla positteev alavu undaakum. cases formula _ 117 smith-voltera-kaantar settu ennariyappedunnu, ithinu formula _ 118 nte lebsg alavu undu.
stokkaastik kaantar settu.
thulyamaayi vibhajikkunnathinupakaram kramarahithamayi vibhajichu kaantar settinte nirmmaanam parishkkarikkam. koodaathe, samayam ulppeduthunnathinu labhyamaya ellah edavelakalum vibhajikkunnathinupakaram oro ghattathilum labhyamaya edavelakalil onnu maathrame namukk vibhajikkaan kazhiyoo. stokkaastik tridic kaantar settinte kaaryatthil thathfalamaayundaakunna prakriya inipparayunna nirakk samavaakyam upayogichu vivarikkaam.
koodaathe stokkaastik diadic kaantar settinum
ivide formula _ 121 ennathu formula _ 64 num formula _ 123 num edayilulla valippathinte edavelakalude ennamaanu. triadic kaantar settinte kaaryatthil fractal dimension formula _ 124 aanu, athaayath 
athinte nirnnayaka counderpart formulayekkaal kuravu _ 125. stokkaastik diadic kaantar settinte kaaryatthil
fractal dimension formula 126 aanu, ithu veendum athinte nirnnayaka counderpart formula 127 nekkaal kuravaanu. stokkaastik diadic kaantarinte kaaryatthil formula 128-nulla pariharam dynamic scaling pradarshippikkunnu, kaaranam deerghakaala paridhiyile pariharam formula 129 aanu, avide stokkaastik diadic kaantar settu formula 130-nte fractal dimension. rund saahacharyangalilum, triadic kaantar settu pole, stokkaastik triadic, diadic kaantar settinte formula 75-aam moment (formula 132) umm samrakshitha alavukalaanu.
kaantar podi.
kaantor settinte oru bahumukha pathippaanu kaantor dust. kaantor settinte oru parimithamaaya cartician uthppannam svayam eduthu athine oru kaantor space aakki maattunnathiloode ithu roopappeduthaam. kaantar settu pole, kaantar podikku poojya alavu undu.
kaantar settinte mattoru mattoru analog searpinsky paravathaaniyaan, avide oru chathuratthe ombathu cheria chathurangalayi vibhajichu madhyabhagam neekkam cheiyunnu. bakkiyulla chathurangal pinneedu onpathu veetham aayi vibhajikkukayum madhyabhagam neekkam cheyyukayum cheiyunnu, angane ananthamaayi. ithinte oru oru analog menchar sponjaanu.
charitraparamaaya paraamarshangal.
kaantor innu kaantor ternery settu formula _ 133 ennu vilikkunnathu oru thikanja pooyint settinte udaaharanamaayi avatharippichu, ithu ellaayidathum edavelakalil saandramalla, ethra cheruthenkilum."tribhuja vipuleekaranangalude adisthaanathil kaantar formula 133 neey" formula nalkiyirikkunna ellah yadhaartha samkhyakaludeyum koottam "ennu visheshippichu formula 135 ivide cofficient formula 136 ekapaksheeyamaayi 0 umm 2 umm enna rund moolyangal edukkunnu, koodaathe seereesil oru parimitha samkhyayo ananthamaaya samkhyakalo adangiyirikkaam."
oru topological space formula _ 137 athinte ellah pooyintukalum limitt pooyintukalaanenkil allengil, thathulyamaayi, athu athil ninnu uruthirinja settu formula _ 138-maayi poruthappedunnundel, anuyojyamaanu. formula _ 133 polulla yadhaartha variyude upavibhaagangale induced subspace topologic keezhilulla topological speysukalaayi kaanaan kazhiyum.
trikonamitheeya shreniyude savisheshathayekkurichula addehathinte phalangal kaantarine uthbhavicha settukalude padanatthilekku nayichu. ananthamaaya settukalude amoorthavum pothuvayathumaaya oru sidhaantham vikasippikkunnathinulla paathayilekku addehathe nayikkaan randaamathethu valareyadhikam cheythu.
kaantor podipadalangalekkurichum prakrithidatha fractalukalumaayum stattistikkal physixumaayum avayude bandhathekkurichu benoit mandelbroat dhaaraalam ezhuthiyittundu. ganithashaasthrathilum bhauthikashaasthrathilum ullavarkku atharam gadanakalude aashayakkuzhappamundaakkunnathu aswasthappeduthunnatho aaya swabhaavathekkurichu addeham kooduthal prathiphalippichu. prakrithiyude fractal jyamithiyil, "1962-ll njaan eee vishayathil aarambhichappol, kaantor podi kuranjathu coach, peeno valayangal pole bheemaakaaramaanennu ellaavarum sammathichirunnu" ennu addeham vivarikkukayum "oro aathmaabhimaanam pulartthunna bhauthikashaasthranjanum kantorinekkurichu paraamarshikkunnathiloode yaanthrikamaayi oof cheyyappedukayum, formula 133 shaasthrathil rasakaramaanennu avakaashappedunnavaril ninnu oru mile oodaan thayyarakukayum cheythu."

