
tharamthiricha sthaanam

ganithashaasthrathil, kompinettoriksinte shaakhayil, oru graded poosatt ennathu "p" ll ninnu ellah swaabhaavika samkhyakaludeyum settu n vare "ρ" enna rank function sajjeekarichirikkunna bhaagikamaayi oordar cheytha settaanu (poosatt) "p". "ρ" thaazhepparayunna rund gunangal thripthippeduthanam
posattinte oru gadakathinte rank functionte moolyathe athinte rank ennu vilikkunnu. chilappol oru graded posattine rank cheytha poosatt ennu vilikkunnu, ennaal aaa vaakyathin mattu ardhangalundu; rank cheytha poosatt kaanuka. oru graded posattinte rank allengil rank level ennathu oru nischitha rank moolyamulla posattinte ellah gadakangaludeyum upavibhagamaanu.
kompinettoriksil graducheytha posattukal oru pradhana punk vahikkunnu, oru haash diagram vazhi drishyavalkkarikkaam.
udaaharanagal.
graducheytha posattukalude chila udaaharanagal (paraantheesukalil rank functionullava) ivayaanu
badal swabhaavasavisheshathakal.
p" leeii ellah maximal cheyinukalkkum ore neelam undengil maathram oru bounded poosatt oru grading amgeekarikkunnu ettavum kuranja moolakatthinte rank 0 aayi sajjeekarikkukayum rank function poornnamaayum nirnnayikkukayum cheiyunnu. ithu thaalpparyamulla niravadhi parimitha kesukal ulkkollunnu; oru negatteevu udaaharanathinaayi chithram kaanuka. ennirunnaalum, paridhiyillaattha sthaanangal kooduthal sankeernamaayekkaam.
oraalkk "x" <"z" umm "n" + 1 "raankinte" z "umm ullappozhellaam, rankil" n "ennathinte" y "enna ghadakam" x "≤" y "<" z "ennathumaayi kandethaan kazhiyumenkil maathrame kramappeduthalumaayi poruthappedunna oru candidate rank function oru posattine graducheytha posattilekku maattukayulloo. eee vyavastha mathiyakum, kaaranam "z" ennathu "x" nte oru kavarayi edukkukayaanengil, "x", "z" ennivayude raankukal 1 kondu vyathyaasappedunnuvennu kaanikkunna "y" = "x" maathramaan saadhyamaaya choys, ithu aavasyamaanu, kaaranam oru graded posettil oraalkk "x" ≤ "y" <"z" ulla maximal raankile eathu gadakavum "y" aayi edukkaam, athu ellaayppozhum nilanilkkukayum "z" kondu moodappedukayum cheiyunnu.
palappozhum oru posatt oru rank functionaayi oru swaabhaavika sthaanaarthiyumaayi varunnu; udaaharanathinu, athinte gadakangal chila adisthaana settu "bii" yude parimithamaaya upavibhaagangalaanenkil, aaa upavibhaagangalude gadakangalude ennam edukkaam. appol nalkiyirikkunna maanadandam nirvachanathekkaal kooduthal praayogikamayirikkum, kaaranam athu kavarukalekkurichu paraamarshikkunnath ozhivaakkunnu. udaaharanathinu, "bii" thanne oru posittaanengil, "pii" athinte parimithamaaya thaazhnna settukal ulkkollunnundengil (athinte ellah gadakangalodum koodiya ellah cheria gadakangalum upavibhaagathilundaakunna), appol maanadandam svayameva thripthippedum, kaaranam thaazhathe settukalkku "x" ′ "z" ennathil ninnu ellaayppozhum "x" ennathil ninnu illaathirikkunna "z" nte paramavadhi ghadakam undaayirikkukayum athu "z" ll ninnu neekkam cheyth "y" roopappeduthukayum cheyyaam.
oru convex polytoppinte phase laattis polulla chila saadhaarana sthaanangalil alavanusarichu oru swaabhaavika grading undu, ithu rank functionaayi upayogikkukayaanenkil ettavum kuranja ghadakam, shoonyamaaya mukham, rank-1 nalkum. atharam sandarbhangalil mukalil paranja nirvachanam −1 enna moolyam rank functionu anuvadichirikkunna moolyangalude koottavumaayi cherthu valaykkunnathu soukaryapradamaayirikkum. ekapaksheeyamaaya poornnasamkhyakale rankayi anuvadikkunnathu adisthaanaparamaayi vyathyasthamaaya oru dhaarana nalkum; udaaharanathinu, oru minimum moolakatthinte nilanilpp inii urappilla.
graducheytha oru sthaanathinu (positteev poornnasankhya raankukalulla) "x" enna moolakangalonnum undaakilla, athinaayi ettavum valia moolakamaaya "x" ulla ekapaksheeyamaayi neelamulla shringalakal nilanilkkunnu, allaathapaksham athinu ekapaksheeyamaayi cheria (oduvil negatteevu) raankilulla gadakangal undaayirikkanam. udaaharanathinu, poornnasamkhyakal (saadhaarana kramatthil) oru graded poosatt aakaan kazhiyilla, allengil yukthisahamo yaadhaarthyamo aaya samkhyakalude edavela (onniladhikam gadakangalulla) aakaan kazhiyilla. (prathyekichum, graducheytha posattukal nannayi sthaapikkappettavayaanu, athaayath avaa avarohana shringalayude avasthaye (dii. sii. sii) thripthippeduthunnu avayil ananthamaaya avarohana shringalakalonnumilla.) athinaal inii muthal ithu sambhavikkatha avasthakal maathrame njangal pariganikkoo. ithu soochippikkunnathu, "x" <"y" aayirikkumbozhellaam namukk oru cover aavarthichu thiranjedukkunnathiloode "x" ll ninnu "y" lake ethaan kazhiyum, parimithamayi niravadhi thavana. (positteev poornnasankhya rank functionukalkku) kramavumaayi "ρ"-nte anuyojyatha kavarukalekkurichulla aavasyakathayil ninnu pinthudarunnu ennum ithinartham. graded posattinte nirvachanatthinte oru vakabhedamenna nilayil, rank functionukalkku ekapaksheeyamaaya (noonn negatteevu) poornnasankhya moolyangal undaayirikkaan birkhoff anuvadikkunnu. eee vakabhedathil, poornnasamkhyakale (aidantitti function vazhi) avante krameekaranatthil tharamthirikkaam, koodaathe kramavumaayi raankukalude anuyojyatha anaavashyamalla. moonnaamathe vakabhedam enna nilayil, braitt velum vesttum oru rank functione poornnasankhya moolyamullathaayi nirvachikkunnu, pakshe oordaringumaayi athinte anuyojyatha aavasyamilla; athinaal eee vakabhedathinu iii grade cheyyaan kazhiyum.jii. eee udaaharanathinu kavarukalekkurichulla aavasyakatha shoonyamaayathinaal eathu function anusarichu yadhaartha samkhyakal.
graducheytha posettukal aarohana shringalayude avastha (essy) thripthippeduthendathil ennathu shradhikkuka udaaharanathinu, swaabhaavika samkhyakalil ananthamaaya aarohana shringhala soothravaakyam adangiyirikkunnu.
oru poosit athinte thaarathamya graphile bandhippichirikkunna oro gadakavum tharamthirikkukayaanenkil maathrame tharamthirikkukayulloo, athinaal kooduthal swabhaavasavisheshathakal eee thaarathamya graph bandhippichirikkumenna karuthunnu. bandhippichirikkunna oro gadakathilum rank function oru ekeekritha shift vare maathrame ulloo (athinaal rank function ellaayppozhum thiranjedukkaam, angane avayude bandhippicha gadakathile kuranja raankile gadakangalkku rank 0 undaayirikkum).
p" nuu ettavum kuranja ghadakam ω undengil, eathenkilum moolakathinu "x" ennathinu [ω, "x"] edavelayile ellah paramavadhi shringalakalkkum ore neelam undaayirikkumenna vyavasthaykku thulyamaanu graducheyyunnathu. oru maximal cheyinile oro ghattavum oru covering relation aayathinal eee vyavastha aavasyamaanu, athu rank 1 kondu mattanam. vyavasthayum mathiyakum, kaaranam athu nilanilkkumbol, oraalkk "x" nte rank nirvachikkaan soochippicha neelam upayogikkaam (oru parimitha shringalayude neelam athinte "steppukalude" ennamaanu, athinaal athinte gadakangalude ennathekkaal onnu kuravaanu), koodaathe "x" "y" cover cheiyumbozhellaam, [ω, "y"] leeii oru maximal cheyinumaayi "x" chernnu [ω, "x"] ll oru maximal cheyin nalkunnu.
p"-nuu ettavum valia ghadakam ι undengil (athu oru bounded poosatt aayathinal), "p"-leeii ellah maximal cheyinukalkkum ore (parimithamaaya) neelam undaayirikkanamenna aavashyakathayilekku munbathe vyavastha lalithamakkaam. ithu mathiyakum, kaaranam [ω, "x"] leeii eathu jodi maximal cheyinukalum ["x", ι] leeii oru maximal cheyin vazhi "p" ll oru jodi maximal cheyinukal nalkaan kazhiyum.
saadhaarana cases.
kompinettoriksile pala rachayithaakkalum "p" yude ellah minimum gadakangalkkum 0 rank undaayirikkenda vidhathil graded posattukale nirvachikkunnu, koodaathe eathenkilum maximal moolakatthinte rankaya maximal rank "r" undu. appol graducheyyunnathu ardhamaakkunnathu mukalil soochippichathupole ellah paramavadhi shringalakalkkum "r" neelamundu ennaanu. eee saahacharyathil oraal parayunnath "p" nuu "r" rank undennaanu.
koodaathe, eee saahacharyathil, rank levalukalumaayi rank nambarukal allengil vitni nambarukal formula 3 bandhappettirikkunnu. eee samkhyakale formula _ 4 = rank "i" ulla "p" yude gadakangalude ennam kondu nirvachikkunnu.
vaitni samkhyakal niravadhi pradhana kompinettoriyal sidhaanthangalumaayi bandhappettirikkunnu. sperner sidhaantham thaazhepparayunna reethiyil roopappeduthaavunna oru mikacha udaaharanamaanu
ithinartham 

