
poison seema

ganit mein, poison seema ack yaadrichhik chalane seey judee ack maap sthaan haye. yeh ack vastu haye jise yaadrichhik chalane key asymptotic vyavhaar koo encode karane key liye design kiya gayaa haye.ee. jaba charanon kii sankhya anant taka jaati haye tou prakshepavakra kaise alag hoe jaate hain. seema kahalaane key bavjood yeh samanya roop seey ack vishuddh maap-saiddhaantik vastu haye aur sthalvaigyaanik arth mein ack seema naheen haye. haalaanki, usa mamle mein jahaan yaadrichhik paidal yatra ack topological sthaan parr haye, poison seema martin seema seey sambandhit hoe sakati haye, joe ack vaastavik topological seema dene vaalaa ack vishleshanatmak nirmaan haye. donon seemaaen pyson sootra key saamaanyikaran key madhyam seey sthaan parr harmonic karyon seey sambandhit hain.
atee-javic tala kaa mamla.
poison sootra bataata haye kii ikai disk sootra _ 2 parr ack sakaratmak harmonic falan sootra _ 1 diyaa gayaa haye (yaani, sootra _ 3 jahaan sootra _ 4 sootra _ 5 parr pointcare metrick seey judaa lapless-beltramy operator haye) seema sootra _ 7 parr ack advitiya maap sootra _ 6 maujood haye jaise kii samaantaa 
sabhi sootra 10 key liye dhaaran kartaa haye. isaki vyakhya karane kaa ack tarika yeh haye kii sootra 12 key liye falan sootra 11 gair-nakaratmak harmonic karyon key shanku mein sabhi charam binduon koo maapane taka hain. sett sootra _ 13 kii yeh vishleshanatmak vyakhya "nyuntam martin seema" kii adhik samanya dhaarana kii or leey jaati haye (joe iss mamle mein poorna "martin seema" haye).
iss tathya kii vyakhya sambhaavit tareeke seey bhee kii jaa sakati haye. yadi sootra _ 14 sootra _ 4 (i) seey judee markov prakriya haye.ee. disk parr poincar remanian metrick key saath brounian gati), phir prakriya sootra _ 16 ack nirantar-samay martingale haye, aur iss tarah lagbhag harr jagah sootra _ 14 key liye sambhaavit (anant) prakshepavakra key winer sthaan parr ack kaary mein parivartit hotaa haye. iss prakaar poison sootra iss maapi gaee jagah kii pehchaan oopar nirmit martin seema key saath kartaa haye, aur antatah sootra _ 13 koo leabs maap key varg key saath sampann kartaa haye (dhyaan dein kii yeh pehchaan seedhe kiya jaa sakata haye kyonki winer sthaan mein ack path lagbhag nishchit roop seey sootra _ 13 parr ack bindu parr parivartit hotaa haye). markov prakriya key liye prakshepavakra key sthaan key roop mein sootra _ 13 kii yeh vyakhya poison seema key nirmaan kaa ack vishesh mamla haye.
anta mein, uparokt nirmaanon koo alag kiya jaa sakata haye, i.ee. sootra _ 5 parr kaary karane vaale ack fusian samooh kii kakshaon parr yaadrichhik chalane taka seemit. yeh samooh parr charam sakaratmak harmonic karyon kii pehchaan detaa haye, aur samooh parr yaadrichhik chalane key prakshepavakra key sthaan (donon diye gaye sambhaavyata maap key sambandh mein), sthalvaigyaanik/maapa gayaa sthaan sootra _ 5 key saath.
paribhasha.
ack asatat samooh parr ack yaadrichhik chalane kii poison seema.
sootra _ 23 koo ack asatat samooh aur sootra _ 6 koo sootra _ 23 parr ack sambhaavyata maap maanaa jaae, jiska upayog sootra _ 23 parr ack yaadrichhik chalane key sootra _ 26 koo paribhaashit karane key liye kiya jaayega (ack asatat-samay markov prakriya jisaki sankraman sambhavanaen sootra _ 28 hain); maap sootra _ 6 koo yaadrichhik chalane key liye "charan vitaran" kahaa jaataa haye. sootra _ 30 koo sootra _ 23 parr ack aur maap honey dein, joe yaadrichhik chalane key liye praarambhik sthiti hogi. sootra _ 26 key liye prakshepavakra kaa antariksh sootra _ 32 ack maap sootra _ 34 key saath sampann haye jisake seemant sootra _ 35 hain (jahaan sootra _ 36 upaayon key convolution koo darshaata haye; yeh sootra _ 37 charanon key baad yaadrichhik chalane kaa vitaran haye). sootra _ 32 parr ack samatulya sambandh sootra _ 38 bhee haye, joe sootra _ 40 seey sootra _ 41 kii pehchaan kartaa haye yadi sootra _ 42 maujood haye tou sabhi sootra _ 44 key liye sootra _ 43 (donon prakshepavakron mein samaan "poonchh" haye). sootra _ 45 kii "poison seema" taba maapa gayaa sthaan sootra _ 46 haye joe samatulya sambandh sootra _ 38 dwara sootra _ 47 key bhagfal key roop mein praapt kiya jaataa haye.
yadi formula _ 49 charan vitaran formula _ 6 key saath ack yaadrichhik chalane kaa praarambhik vitaran haye tou formula _ 46 parr maap formula _ 53 key aage badhane key roop mein praapt kiya jaataa haye. yeh formula _ 45 key liye ack sthir maap haye, jiska arth haye kii 
poison seema kii ack antarnihit paribhasha denaa sambhav haye kyonki ack sootra _ 45-sthir maap sootra _ 58 key saath adhikatam sootra _ 23-sett key roop mein, atirikt sthiti koo santusht kartaa haye kii sootra _ 59 lagbhag nishchit roop seey ack dirac dravyamaan mein kamjor roop seey parivartit hoe jaataa haye.
poison sootra.
sootra _ 1 koo sootra _ 23 parr ack sootra _ 6-harmonic falan honey dein, jiska arth haye kii sootra _ 63. phir yaadrichhik charr sootra _ 64 ack asatat-samay martingale haye aur isaliye yeh lagbhag nishchit roop seey abhisaran kartaa haye. sootra _ 65 dwara niroopit sootra _ 46 parr falan koo ack prakshepavakra key saath sootra _ 1 key moolyon kii seema laker praapt kiya jaataa haye (yeh lagbhag harr jagah sootra _ 32 aur shift-invariant parr paribhaashit kiya jaataa haye). sootra _ 69 koo sootra _ 71 (sootra _ 72 parr diarock dravyamaan) key saath oopar key sankuchan dwara praapt maap sootra _ 70 honey dein. yadi sootra _ 1 yaa tou sakaratmak haye yaa bandha huaa haye tou sootra _ 65 bhee haye aur hamaare paas "poison sootra" haih
yeh sootra _ 6-harmonic bounded functions aur anivarya roop seey sootra _ 46 parr bounded maapane yogya functions key beach ack bijection sthaapit kartaa haye. vishesh roop seey sootra _ 45 kii poison seema tuchchh haye, joe ack bindu taka cumm hoe jaati haye, aggar aur kewal tabhi jaba sootra _ 23 parr kewal bounded sootra _ 6-harmonic functions sthir houn.
samanya paribhasha.
samanya setting ack maapa sthaan parr ack "markov operator" kii haye, ack dhaarana joe ack yaadrichhik chalane seey judey markov operator sootra _ 81 koo samanya karati haye. adhikansh siddhant koo iss amoort aur bahut samanya setting mein vikasit kiya jaa sakata haye.
martin seema.
ack asatat samooh kii martin seema.
sootra _ 82 koo ack asatat samooh parr yaadrichhik chalane dein. sootra _ 83 koo sootra _ 72 seey sootra _ 85 taka sootra _ 37 charanon mein praapt karane kii sambhavana maanaa jaae, i.ee. sootra _ 87. hurry kernell paribhasha key anusaar haih
yadi chalana kshanik haye tou yeh shrinkhala sabhi sootra 89 key liye abhisari haye. ack bindu sootra 90 taya karein aur martin kernell koo iss prakaar paribhaashit karein - 
sootra _ 91. 
embedding sootra _ 92 mein bindu kii drishti seey abhisaran kii topology key liye ack apekshaakrit compact chhavi haye, aur martin compactification iss chhavi kaa samaapan haye. ack bindu sootra _ 93 koo aamtaur parr sanketan sootra _ 94 dwara darshaaya jaataa haye.
martin kernell sakaratmak harmonic kaary hain aur pratyek sakaratmak harmonic kaary koo seema parr karyon key abhinn anga key roop mein vyakt kiya jaa sakata haye, arthaat pratyek sakaratmak harmonic kaary key liye sootra 46 parr ack maap sootra _ 95 haye joe ack poison-jaisa sootra dhaaran kartaa haih
maap sootra 95 "nyuntam" martin seema parr samarthit hain, jinake tatvon koo nyuntam honey kii visheshta bhee dee jaa sakati haye. ack sakaratmak harmonic falan sootra 99 koo "nyuntam" kahaa jaataa haye yadi sootra 101 key saath kissy bhee harmonic falan sootra 100 key liye sootra 102 maujood haye jaise kii sootra 103.
vaastav mein martin compactfication kaa ack puuraa parivaar haye. harit utpaadan shrinkhala koo iss prakaar paribhaashit karein
sootra 105 dwara iss shakti shrinkhala key abhisaran kii trijya koo darshaata haye
aur sootra _ 106 key liye sootra _ 107-martin kernell koo paribhaashit karein
sootra _ 108.
embedding sootra 109 key bund honey koo sootra 107-martin compactfication kahaa jaataa haye.
ack remanian manifold kii martin seema.
ack remanian manifold key liye martin seema kaa nirmaan kiya jaataa haye, jaba yeh maujood hotaa haye, usee tarah jaise oopar, lapless-beltramy operator sootra _ 4 key harey falan kaa upayog karate hue. iss mamle mein phir seey sootra _ 113 key liye operator sootra _ 112 seey judey martin compactification kaa ack puuraa parivaar haye jahaan sootra _ 114 spectrum kaa nichala hissa haye. udaaharan jahan iss nirmaan kaa upayog ack sampeedan koo paribhaashit karane key liye kiya jaa sakata haye, way samtal mein bandhe hue kshetra aur gair-sampeedit prakaar key samamit sthaan hain.
martin aur poison seemaon key beach sambandh.
sthir falan key anuroop maap sootra _ 115 koo martin seema parr "harmonic maap" kahaa jaataa haye. iss maap key saath martin seema poison seema key liye samroop haye.
udaaharan.
neelpotent samooh.
neelpotent samoohon mein samamit yaadrichhik chalane key liye poison aur martin seemaaen tuchchh hain. doosari or, jaba yaadrichhik chalana gair-kendrit hotaa haye, tou nyuntam karyon sahit poorna martin seema kaa adhyayan, bahut cumm nirnaayak hotaa haye.
jhooth bolane vaale samooh aur asatat upsamooh.
ack ardh-saral jhooth samooh parr yaadrichhik chalane key liye (haar maap key sambandh mein bilkul nirantar charan vitaran key saath) poison seema firstenberg seema key barabar hoti haye. sambandhit samamit sthaan parr brounian gati kii poison seema bhee firstenberg seema haye. inn mamalon mein poorna martin seema kaa bhee achchi tarah seey adhyayan kiya jaataa haye aur hamesha jyaamitiya tareeke seey varnit kiya jaa sakata haye. udaaharan key liye, shreni ack key samoohon key liye (udaaharan key liye hyperbolic rikt sthaan key samamit samooh) poorna martin seema nyuntam martin seema key samaan haye (uchch shreni key samoohon mein sthiti adhik jatil haye).
ack ardh-saral jhooth samooh key ack zariski-ghaney upsamooh kii poison seema, udaaharan key liye ack jaali, bhee samooh kii firstenberg seema key barabar haye.
atee-maukhik samooh.
ack atee-maukhik samooh parr yaadrichhik chalane key liye, charan vitaran parr kamjor dhaaranaon key tahat joe hamesha ack saral chalane key liye pakad mein rahati haye (ack adhik samanya sthiti yeh haye kii pahala kshan seemit hoe) poison seema hamesha gromov seema key barabar hoti haye. udaaharan key liye, ack mukt samooh kii poison seema isake keyali pedh key siron kaa sthaan haye. poorna martin seema kii pehchaan adhik shamil haye; yadi yaadrichhik chalane kii seema seemit haye (charan vitaran ack seemit sett parr samarthit haye) tou martin seema nyuntam martin seema key saath mell khaati haye aur donon gromov seema key saath mell khaati haye.

