ankganitiya pragati parr roth kii pramey

ankganitiya pragati parr roth kii pramey praakritik sankhyaon key upasamuchchay mein ankganitiya pragati key astitva seey sambandhit yogaatmak sanyojan kaa parinam haye. yeh pahali baar 1953 mein closs roth dwara saabit kiya gayaa thaa. roth kii pramey mamle key sootra _ 1 key liye zemready kii pramey kaa ack vishesh mamla haye.
bayaan.
praakritik sankhyaon key ack upasamuchchay "aeye" koo sakaratmak uchch ghanatva kahaa jaataa haye yadi
ankganitiya pragati parr roth kii pramey (anant sanskaran): dhanaatmak uchch ghanatva vaali praakritik sankhyaon key ack upasamuchchay mein 3-avadhi kii ankganitiya pragati hoti haye.
ack vaikalpik, adhik gunaatmak, pramey kaa sutrikaran ack selam-spensor sett key adhikatam aakar seey sambandhit haye joe sootra _ 3 kaa ack upasamuchchay haye. sootra _ 4 sootra _ 5 key sabse badey upasamuchchay kaa aakar haye jisamen koee 3-avadhi ankganitiya pragati naheen haye.
ankganitiya pragati parr roth kii pramey (parishkaran sanskaran): sootra _ 6. 
formula _ 4 parr oopari aur nichali seema mein sudhaar abhee bhee ack khuli shodh samasya haye.
itihas.
iss disha mein pahala parinam 1927 mein vaine dair wareden kii pramey thee, joe bataati haye kii paryaapt roop seey badey n key liye, sootra _ 9 rangon key saath purnank sootra _ 8 koo rangane kaa parinam ack sootra _ 10 shabd ankganitiya pragati mein hoga.
baad mein 1936 mein erdos aur turran ney ack bahut majboot parinam kaa anuman lagaaya kii sakaratmak ghanatva vaale purnankon key kissy bhee upasamuchchay mein manmaane dhang seey lambi ankganitiya pragati hoti haye. 1942 mein, rafal salem aur donald see. spensor ney 3-app-free sett (i.ee. aakar sootra _ 11 key binaa 3-avadhi ankganitiya pragati vaalaa ack sett, kuchh sootra _ 13 key liye erdós aur turán key usa sootra _ 12 key ack atirikt anuman koo galat thaharaata haye.
1953 mein, roth ney praarambhik anuman koo aanshik roop seey yeh saabit karke hull kiya kii unamen fourier vishleshanatmak vidhiyon kaa upayog karke lambai 3 kii ankganitiya pragati honey chaahiye. antatah, 1975 mein, zemready ney mool anuman koo poorna roop seey hull karate hue, sanyojan takneekon kaa upayog karke zemready kii pramey koo saabit kiya.
pramaan takneek.
roth dwara diyaa gayaa mool pramaan forier vishleshanatmak tarikon kaa upayog kartaa haye. baad mein zemready key niyamitata lemma kaa upayog karke ack aur pramaan diyaa gayaa.
forier vishleshan key madhyam seey pramaan rekhaachitra.
1953 mein, roth ney sootra _ 14 kii oopari seema saabit karane key liye fourier vishleshan kaa upayog kiya. neeche iss pramaan kaa ack rekhaachitra haye.
falan sootra _ 15 key forier parivartan koo falan sootra _ 16 santoshjanak honey key liye paribhaashit karein.
jahan sootra _ 18 haye.
sootra _ 19 koo sootra _ 20 kaa 3-app-mukt upasamuchchay honey dein. pramaan 3 charanon mein aage badhata haye.
charan 1.
karyon key liye, sootra _ 25 paribhaashit karein 
lemma kii ginati sootra _ 27 koo sootra _ 28 koo santusht kartaa haye. sootra _ 29 koo paribhaashit karein. phir sootra _ 30.
ginati kaa crum hamein bataata haye kii yadi sootra _ 31 aur sootra _ 32 key fourier parivartan "karib" hain, tou donon key beach 3-avadhi ankganitiya pragati kii sankhya bhee "karib" honey chaahiye."sootra koo sootra _ 33 sootra _ 19 kaa ghanatva honey dein. sootra _ 35 (i) koo paribhaashit karein.e sootra _ 19) kaa sanketak kaary, aur sootra _ 37. charan 1 koo phir ganana lemma koo sootra _ 31 aur sootra _ 32 parr lagu karke kaataa jaa sakata haye, joe hamein bataata haye kii kuchh sootra _ 40 maujood haye jaise kii 
charan 2.
charan 1 seey sootra _ 40 koo dekhate hue, humm pahale dikhaate hain kii sootra _ 43 koo apekshaakrit badey upa-pragati mein vibhajit karna sambhav haye jaise kii varna sootra _ 44 pratyek upa-pragati parr motey taur parr sthir haye.
lemma 1: sootra _ 45 koo maan leejiye kii ack saarvbhaumik sthir sootra _ 47 key liye sootra _ 46. phir sootra _ 43 koo ankganitiya pragati sootra _ 49 mein lambai sootra _ 50 key saath vibhajit karna sambhav haye jaise kii sabhi sootra _ 52 key liye sootra _ 51. 
isake baad, humm upaprograshan mein vibhajan praapt karane key liye lemma 1 lagu karate hain. phir humm iss tathya kaa upayog karate hain kii sootra _ 40 ney charan 1 mein ack badey gunank kaa utpaadan kiya haye yeh dikhaane key liye kii inn upa-pragati mein seey ack mein ghanatva vriddhi honey chaahieh
lemma 2: sootra _ 19 koo sootra _ 56 aur sootra _ 57 key saath sootra _ 43 kaa 3-app-mukt subset honey dein. phir, ack upa-pragati sootra _ 58 maujood haye jaise kii sootra _ 59 aur sootra _ 60.
charan 3.
abb humm charan 2 koo dohrate hain. sootra _ 61 koo sootra _ 63 veen punaravritti key baad sootra _ 19 kaa ghanatva honey dein. hamaare paas vaha sootra 64 aur sootra 65 pahale haye, usa sootra 66 koo dekhein (i.ee. sootra _ 67 taka pahunchen jaise kii sootra _ 68) adhikansh sootra _ 69 charanon key baad. humm phir seey sootra _ 66 koo dohra dete hain (i.e sootra _ 71 taka pahunchen) adhikansh sootra _ 72 charanon key baad. choonki sootra _ 73, iss prakriya koo adhikansh sootra _ 74 charanon key baad samaapt honah chaahiye.
sootra _ 63 punaraavrittiyon key baad sootra _ 75 hamari vartamaan pragati kaa aakar hoe. lemma 2 dwara, humm hamesha prakriya koo jaari rakh sakate hain jaba bhee sootra _ 77 aur iss prakaar jaba prakriya samaapt hoti haye tou hamaare paas vaha sootra _ 78 bhee hotaa haye, dhyaan dein kii jaba humm ack upapragati parr jaate hain, tou hamaare sett kaa aakar ack ghana mool seey cumm hoe jaataa haye. isaliye
isaliye sootra _ 80 tou sootra _ 81 jaisa kii vaanchhit haye. sootra _ 82
durbhagya seey, yeh takneek seedhe badey ankganitiya pragati koo samanya naheen karati haye taaki zemerady kii pramey koo saabit kiya jaa sakey. iss pramaan kaa vistaar dashakon taka ganitvidon koo 1998 taka door curr diyaa, jaba timothi govers ney uchch-crum key fourier vishleshan key kshetra koo vishesh roop seey zemready kii pramey koo saabit karane key liye uparokt pramaan koo samanya banaane key liye vikasit kiya.
graph niyamitata key madhyam seey pramaan rekhaachitra.
neeche szemerady niyamitata lemma kaa upayog karke ack pramaan kii ruprekha dee gaee haye.
sootra _ 83 koo ack graph aur sootra _ 84 honey dein. humm sootra _ 85 koo ack sootra _ 86-niyamit jodee kahate hain yadi sootra _ 88 key saath sabhi sootra _ 87 key liye, ack kaa sootra _ 89 haye.
sootra _ 91 kaa ack vibhajan sootra _ 90 ack sootra _ 86-niyamit vibhajan haye yadi
taba szemerédi niyamitata lemma kahata haye kii pratyek sootra _ 94 key liye, ack sthir sootra _ 95 maujood haye jaise kii pratyek graph mein ack sootra _ 86-niyamit vibhajan hotaa haye joe adhikansh sootra _ 95 bhaagon mein hotaa haye.
humm yeh bhee saabit curr sakate hain kii sootra 86-niyamit sheershon key beach trikon kaee anya trikonon key saath aaney chaahiye. ise trikon ginati lemma key roop mein jaanaa jaataa haye.
trikon ganana lemmah sootra _ 83 koo ack graph aur sootra _ 100 koo sootra _ 83 key sheershon key upasamuchchay honey dein jaise kii sootra _ 102 kuchh sootra _ 104 key liye sabhi sootra _ 86-niyamit jode hain. sootra _ 105 koo kramshah kinare key ghanatva sootra _ 106 koo darshaata haye. yadi sootra _ 107 haye, tou trigun sootra _ 108 kii sankhya cumm seey cumm sootra _ 83 mein tribhuj banaane key liye sootra _ 109 haye.
trikon ginati lemma aur szemerady niyamitata lemma kaa upayog karke, humm tribhuj hataane vaale lemma koo saabit curr sakate hain, joe graph hataane vaale lemma kaa ack vishesh mamla haye.
trikon hataane kaa lemmah sabhi sootra 104 key liye, sootra 13 maujood haye jaise kii sootra 114 sheershon parr kissy bhee graph koo sootra 115 trikon seey cumm yaa barabar key saath adhikansh sootra 116 kinaaron koo hataakar trikon-mukt banaaya jaa sakata haye.
sootra _ 118 sheershon parr graph sootra _ 83 seey sambandhit ack dilchasp upsamhaar haye jahan sootra _ 83 kaa pratyek kinara ack advitiya trikon mein sthit haye. vishisht roop seey, inn sabhi graphon mein sootra _ 120 kinare honey chaahiye.
binaa kissy 3-avadhi key ankganitiya pragati key ack nirdhaarit sootra _ 19 lein. abb, ack tripakshiya graph sootra _ 83 kaa nirmaan karein jiska bhaag sootra _ 100 sootra _ 124 kii sabhi pratiyaan hain. ack sheersh sootra _ 125 koo ack sheersh sootra _ 126 seey jodein yadi sootra _ 127 haye. isi tarah, sootra _ 128 koo sootra _ 126 key saath jodein yadi sootra _ 130 haye. anta mein, sootra _ 125 koo sootra _ 128 key saath jodein yadi sootra _ 133 haye.
yeh nirmaan iss prakaar seey sthaapit kiya gayaa haye kii yadi sootra 109 ack trikon banaata haye, tou hamein sootra 135 praapt hotaa haye joe sabhi sootra 19 seey sambandhit hain. yeye sankhyaaen soochibaddh crum mein ack ankganitiya pragati banaati hain. sootra _ 19 parr dhaarana taba hamein bataati haye kii yeh pragati tuchchh honey chaahieh oopar soochibaddh tatva sabhi samaan hain. lekin yeh sthiti iss daave key barabar haye kii sootra _ 109 sootra _ 124 mein ack ankganitiya pragati haye. isake parinamswarup, sootra _ 83 kaa pratyek kinara bilkul ack trikon mein sthit haye. vaanchhit nishkarsh nimnalikhit haye. sootra _ 82
vistaar aur saamaanyikaran.
zemerady kii pramey ney mool anuman koo hull kiya aur roth kii pramey koo manmaane dhang seey lambai kii ankganitiya pragati key liye saamaanyikrit kiya. taba seey ise nae aur dilchasp parinam banaane key liye kaee faishanon mein vistaarit kiya gayaa haye.
firstenberg aur katznelson ney ack bahuayami sanskaran koo saabit karane key liye ergodic siddhant kaa upayog kiya aur libman aur bergleson ney ise bahupad pragati taka bhee vistaarit kiya. haal hii mein, green aur taao ney green-taao pramey koo saabit kiya joe kahata haye kii abhaajya sankhyaon mein manmaane dhang seey lambi ankganitiya pragati hoti haye. choonki abhaajya sankhyaaen ghanatva 0 kaa ack upasamuchchay hain, isaliye unhone ack "saapeksh" zemerady pramey pesh kiya joe ghanatva 0 vaale upasamuchchay parr lagu hotaa haye joe kuchh chhadm-samroopata sthitiyon koo santusht kartaa haye. baad mein conlon, lomdi aur jhaao ney aavashyak chhadm-vikriti kii sthiti koo kamjor karke iss pramey koo majboot kiya. 2020 mein, khilane aur sisask ney saabit kiya kii kissy bhee sett sootra _ 19 mein aise sootra _ 143 vichalan mein lambai 3 kii ankganitiya pragati honey chaahiye; yeh erdos key ack anya anuman kaa pahala gair-tuchchh mamla haye joe yeh bataata haye kii iss tarah key kissy bhee sett mein vaastav mein manmaane dhang seey lambi ankganitiya pragati honey chaahiye.
seema mein sudhaar.
roth kii pramey mein seema mein sudhaar parr bhee kaam kiya gayaa haye. roth kii pramey key mool pramaan seey seema seey pataa chalaa kii
kuchh nirantar sootra key liye _ 145. varshon seey iss seema koo lagaatar zemerady, heath-braun, borgen aur sanders dwara cumm kiya gayaa haye. vartamaan (julaai 2020) sabse achchha baadhya khilane aur sisask key kaaran haye jinhone ack poorna sthiraank c> 0 kaa astitva dikhaaya haye jaise kii
farwari 2023 mein kellie aur meka dwara ack poorvamudran ney ack naee seema dee 
aur chaar dinon baad khilane aur sisask ney parinam koo saral banaa diyaa aur formula _ 148 mein thodaa sudhaar kiya.
dusare chhor parr bhee kaam kiya gayaa haye, jisamen teen-avadhi kii ankganitiya pragati key binaa sabse badey sett kaa nirmaan kiya gayaa haye. sabse achchha nirmaan 1946 key baad seey mushkil seey sudhaar kiya gayaa haye jaba selem aur spensor dwara praarambhik nirmaan mein sudhaar kiya gayaa aur saabit huaa
70 seey adhik varshon mein koee sudhaar naheen honey key kaaran, yeh anuman lagaaya jaataa haye kii bearand kaa sett binaa kissy teen-avadhi kii pragati key aakar mein sabse badey sambhav sett key bahut karib haye. yadi sahee haye, tou kellie-meka bound iss anuman koo saabit karega.
parimit kshetron mein roth kii pramey.
ack bhinnata key roop mein, humm seemit kshetron parr anuroop samasya parr vichaar curr sakate hain. parimit kshetra sootra _ 150 parr vichaar karein, aur sootra _ 150 koo sootra _ 150 key sabse badey upasamuchchay kaa aakar honey dein jisamen koee 3-avadhi ankganitiya pragati naheen haye. yeh samasya vaastav mein cap sett samasya key barabar haye, joe sootra _ 150 key sabse badey upasamuchchay key liye poochhati haye jaise kii koee 3 ank ack rekha parr naheen hotey hain. cap sett samasya koo card game sett key saamaanyikaran key roop mein dekhaa jaa sakata haye.
1982 mein, braun aur buhler ney pahali baar 1995 mein sootra 154 dikhaaya, roy mesuhalam ney roth kii pramey key fourier-vishleshanatmak pramaan key samaan takneek kaa upayog kiya yeh dikhaane key liye kii sootra 155 koo ballebaaz aur kaatz dwara 2012 mein sootra 156 mein sudhaar kiya gayaa thaa.
2016 mein, earnie crot, vsvoload lev, petter paal putch, jorden elemberg aur diane jijaswit ney usa sootra _ 157 koo saabit karane key liye bahupad vidhi key aadhar parr ack naee takneek vikasit kii.
sabse achchi gyaat nichali seema lagbhag sootra _ 158 haye, joe 2022 mein tyrell dwara dee gaee haye.
lokpriya antaron key saath roth kii pramey.
roth kii pramey key ack anya saamaanyikaran seey pataa chalata haye kii sakaratmak ghanatva upasamuchchay key liye, naa kewal 3-avadhi kii ankganitiya pragati maujood haye, balki ack hii samanya antar key saath kaee 3-aips maujood hain.
lokpriya antaron key saath roth kii prameyah sabhi sootra _ 104 key liye, kuchh sootra _ 160 maujood haye jaise kii pratyek sootra _ 161 aur sootra _ 163 key saath sootra _ 162 key liye kuchh sootra _ 164 maujood haye jaise kii sootra _ 165
yadi sootra _ 19 koo sootra _ 167 seey yaadrichhik roop seey chunaa jaataa haye tou humm sootra _ 169 key pratyek moolya key liye sootra _ 168 pragati honey kii ummeed karenge. lokpriya antar pramey iss prakaar bataata haye kii sakaratmak ghanatva key saath pratyek sootra _ 24 key liye, kuchh sootra _ 169 haye jaise kii samanya antar sootra _ 169 key saath 3-aips kii sankhya hamari apeksha key karib haye.
iss pramey koo pahali baar 2005 mein harey rung seey saabit kiya gayaa thaa, jinhone sootra _ 173 kii ack seema dee thee jahan sootra _ 174 tower kaary haye. 2019 mein, lomdi aur faim ney haal hii mein sootra _ 175 kii seema mein sudhaar kiya
3-aips aur 4-aips donon key liye sootra _ 176 mein ack sambandhit kathan bhee sahee haye. haalaanki, daavaa 5-aips key liye galat dikhaaya gayaa haye.

