
viralanaaya bharanadhikaari

chila doorachihnangal illaathaayekkaavunna oru bharanadhikaariyaanu viralamaaya bharanadhikaarikal. kooduthal amoorthamaayi paranjaal, formula 2 maarkkukalulla dairkhya formula 1-nte viralamaaya bharanadhikaarikal poornnasankhya formula 3-nte oru sreniyaanu, avide formula 4. adayaala formula 5 umm formula 6 umm bharanaadhikaariyude attangalumayi poruthappedunnu. dhoora soothravaakyam _ 7 alakkunnathinu, soothravaakyam _ 8 upayogichu formula _ 9 umm formula _ 10 umm aayirikkanam.
oru "poornna" viralamaaya bharanadhikaari, eathu poornnasankhya dooravum athinte poornna dairkhyam vare alakkaan oraale anuvadikkunnu. 13 maarkk formula dairkhya formula _ 1-nte poornnamaaya viralamaaya bharanaadhikaariyillenkil oru poornna viralamaaya bharanadhikariye "minimum" ennu vilikkunnu. mattoru vidhathil paranjaal, eathenkilum adayaalangal neekkam cheythaal adayaalangal punakrameekarikkaan kazhinjaalum oraalkk ellah doorangalum alakkaan kazhiyilla. formula 2 maarkkukalulla neelam formula _ 14-nte poornnamaaya viralamaaya bharanaadhikaariyillenkil oru poornna viralamaaya bharanadhikariye "maximal" ennu vilikkunnu. kuranjathum paramavadhi ullathumaaya oru bharanadhikariye "opttimal" ennu vilikkunnu.
vyathyastha jody adayaalangalude ennam formula _ 16 aayathinal, formula _ 2 adayaalangalulla eathenkilum maximal spezhs roolarinte dairkhya formula _ 1-leeii mukalile bandhamaanith. eee uyarnna paridhi 2,3 allengil 4 markinu maathrame nedaan kazhiyoo. kooduthal maarkkukalkk, opttimal neelavum boundum thammilulla vyathyasam kramena asamamaayum valarunnu.
udaaharanathinu, 6 markinu mukalile paridhi 15 aanu, ennaal paramavadhi neelam 13 aanu. 6 maarkkukalulla 13 neelamulla viralamaaya bharanaadhikaarikalude 3 vyathyastha konfigareshanukal undu. onnu {0,1,2,6,10,13} aanu. 7 enna neelam alakkaan, eee bharanadhikariye upayogichu 6 umm 13 umm enna adayaalangal thammilulla dooram edukkum.
oru golambu bharanadhikaariyaanu oru virala bharanadhikaariyaayathinaal ellah vyathyaasangalum formula 19 vyathyasthamayirikkanam. saadhaaranayaayi, formula 2 adayaalangalulla oru golambu bharanadhikariye formula 2 adayaalangalulla opttimal spezhs bharanaadhikaariyekkaal ganyamaayi neelamullathaayirikkum, kaaranam formula 16 oru golambu bharanaadhikaariyude neelathinu oru thaazhnna bandhamaan. oru neenda golambu bharanaadhikaariykk vidavukal undaakum, athaayath, athinu alakkaan kazhiyaatha doorangal undaakum. udaaharanathinu, opttimal golomb rular (0,1,4,10,12,17) nuu 17 neelamundu, pakshe 14 allengil 15 neelam alakkaan kazhiyilla.
vichmaan bharanadhikaarikal.
formula _ 24 neela formula _ 26-nte formula _ 25 segmentukale prathinidheekarikkunna formula _ 23 roopathilulla niravadhi opttimal bharanadhikaarikalaanu. athinaal, formula _ 27 umm formula _ 28 umm aanenkil, formula _ 29 nuu (kramatthil) undu 
1 neelamulla bhagam 1, 
1 bhagam neelam 2, 
1 bhagam neelam 3, 
7 neelamulla 2 bhagangal, 
neelamulla 2 bhagangal 4, 
1 neelatthinte bhagam 1
athu bharanadhikariye {0,1,3,6,13,20,24,28,29} nalkunnu. oru vichmaan bharanaadhikaariyude neelam formula 30 umm adayaalangalude ennam formula 31 umm aanu. ellah vichmaan bharanaadhikaarikalum opttimal allennum ellah opttimal bharanadhikaarikaleyum eee reethiyil srishtikkaan kazhiyillennum shradhikkuka. 1, 13, 17, 23, 58 ennee neelamulla opttimal bharanadhikaarikalaarum eee maathruka pinthudarunnilla. peettar lushniyude opttimal rular anumaanam shariyaanenkil aaa kramam 58 ll avasaanikkunnu. eee anumaanam neelatthil 233 vare sariyaanennu ariyappedunnu.
asymptotics.
oro soothravaakyathinum formula 33 ennathu dairkhya soothravaakyathinte oru bharanaadhikaariyude ettavum cheria samkhyayaayirikkatte. udaaharanathinu, formula 35. function formulayude asymptotic erdos, gall (1948) padikkukayum leach (1956) thudarukayum cheythu, avar paridhi soothravaakyam 37 nilavilundennum athu <br> formula 38 kondu thaazhnnathum mukalilullathumaanennum theliyichu.
32-thikanja bharanaadhikaarikalude formulakku valare mikacha uyarnna athirukal nilavilundu. avaa formula 41-nte 40-aamm upasettukalaanu, athaayath oro positteev nambar formula _ 42-neey eathenkilum formula 44-nuu oru deference formula _ 43 aayi ezhuthaam. oro nambar formulakkum _ 32-nuu formula _ 46 oru formula _ 32-perfect roolarinte ettavum cheria cardinalityaayirikte. formula _ 48. sequence formula _ 46-nte asymptotics ready, renee (1949), thudarnnu leach (1956), gole (1972) enniva padichuvennu vyakthamaan. avarude parisramangal kaaranam thaazhepparayunna mukalilum thaazheyumulla athirukal labhichu <br> soothravaakyam _ 50
adhikathe formula 51 ennu nirvachikkuka. 2020-ll, ellah neela formulakkum formula _ 52 ≤ 1 ennu nirmmaanatthiloode pegg theliyichu. opttimal rular anumaanam shariyaanenkil, ellah formulakalkkum formula _ 54, nirakalil krameekarikkumbol 'dark millukal' paattenilekk nayikkunnu, athaayath a326499. ettavum ariyappedunna spars rular formulakalonnum 2020 septambar vare ettavum kuranjathaayi theliyikkappettittilla. formula _ 56-nulla nilavile ettavum ariyappedunna formula _ 57 nirmmithikalil palathum minimal allathava, prathyekichu "claude" moolyangal ennu vishwasikkappedunnu.
udaaharanagal.
thaazhepparayunnava kuranja viralamaaya bharanaadhikaarikalude udaaharanangalaan. opttimal bharanaadhikaarikale hailait cheiyunnu. pattikappeduthaan valareyadhikam ullappol, ellaam ulppeduthiyitla. mirror emejukal kanikkunnilla.
apoornnamaaya virala bharanadhikaarikal.
apoornnaraaya chila bharanaadhikaarikalkk ore ennam adayaalangalulla opttimal spares bharanaadhikaariyekkaal kooduthal dooram vare poornnamaayum alakkaan kazhiyum. formula _ 59, formula _ 60, formula _ 61, formula _ 62 ennivaykku 18 vare alakkaan kazhiyum, athesamayam 7 maarkkukalulla opttimal spezhs roolarinu 17 vare maathrame alakkaan kazhiyoo. chuvadeyulla pattikayil eee bharanaadhikaarikale pattikappedutthiyittundu, 13 maarkkukalulla bharanadhikaarikal vare. mirror emejukal kanikkunnilla. ore ennam adayaalangalulla eathoru cheria bharanaadhikaariyekkaalum kooduthal dooram vare alakkaan kazhiyunna bharanaadhikaarikale hailait cheiyunnu.

