id sid tid token lemma pos cana-4553 1 1 communications communication NOUN cana-4553 1 2 on on ADP cana-4553 1 3 applied apply VERB cana-4553 1 4 nonlinear nonlinear ADJ cana-4553 1 5 analysis analysis NOUN cana-4553 1 6 issn issn NOUN cana-4553 1 7 : : PUNCT cana-4553 1 8 1074 1074 NUM cana-4553 1 9 - - PUNCT cana-4553 1 10 133x 133x NUM cana-4553 1 11 vol vol NOUN cana-4553 1 12 32 32 NUM cana-4553 1 13 no no NOUN cana-4553 1 14 . . PUNCT cana-4553 2 1 9s 9s NUM cana-4553 2 2 ( ( PUNCT cana-4553 2 3 2025 2025 NUM cana-4553 2 4 ) ) PUNCT cana-4553 2 5 2754 2754 NUM cana-4553 3 1 https://internationalpubls.com https://internationalpubls.com X cana-4553 3 2 probabilistic probabilistic ADJ cana-4553 3 3 perspective perspective NOUN cana-4553 3 4 of of ADP cana-4553 3 5 jungck jungck PROPN cana-4553 3 6 contractive contractive ADJ cana-4553 3 7 situations situation NOUN cana-4553 3 8 in in ADP cana-4553 3 9 machine machine NOUN cana-4553 3 10 learning learn VERB cana-4553 3 11 ashutosh ashutosh PROPN cana-4553 3 12 mishra1 mishra1 PROPN cana-4553 3 13 , , PUNCT cana-4553 3 14 piyush piyush PROPN cana-4553 3 15 kumar kumar PROPN cana-4553 3 16 tripathi2 tripathi2 PROPN cana-4553 3 17 , , PUNCT cana-4553 3 18 mukesh mukesh PROPN cana-4553 3 19 kumar kumar PROPN cana-4553 3 20 singh3 singh3 PROPN cana-4553 3 21 , , PUNCT cana-4553 3 22 manisha manisha PROPN cana-4553 3 23 gupta4 gupta4 PROPN cana-4553 3 24 , , PUNCT cana-4553 3 25 alok alok PROPN cana-4553 3 26 kumar kumar PROPN cana-4553 3 27 agrawal5 agrawal5 PROPN cana-4553 3 28 1,2,5department 1,2,5department NUM cana-4553 3 29 of of ADP cana-4553 3 30 mathematics mathematic NOUN cana-4553 3 31 , , PUNCT cana-4553 3 32 asas asa NOUN cana-4553 3 33 , , PUNCT cana-4553 3 34 amity amity NOUN cana-4553 3 35 university university NOUN cana-4553 3 36 uttar uttar PROPN cana-4553 3 37 pradesh pradesh PROPN cana-4553 3 38 ( ( PUNCT cana-4553 3 39 lucknow lucknow PROPN cana-4553 3 40 campus campus PROPN cana-4553 3 41 ) ) PUNCT cana-4553 3 42 , , PUNCT cana-4553 3 43 india india PROPN cana-4553 3 44 3armapore 3armapore NUM cana-4553 3 45 p. p. NOUN cana-4553 3 46 g. g. PROPN cana-4553 3 47 college college PROPN cana-4553 3 48 , , PUNCT cana-4553 3 49 kanpur kanpur PROPN cana-4553 3 50 , , PUNCT cana-4553 3 51 india india PROPN cana-4553 3 52 4university 4university PROPN cana-4553 3 53 of of ADP cana-4553 3 54 technology technology NOUN cana-4553 3 55 and and CCONJ cana-4553 3 56 applied apply VERB cana-4553 3 57 science science NOUN cana-4553 3 58 , , PUNCT cana-4553 3 59 muscat muscat NOUN cana-4553 3 60 email email NOUN cana-4553 3 61 : : PUNCT cana-4553 3 62 1amishra94154@gmail.com 1amishra94154@gmail.com NUM cana-4553 3 63 , , PUNCT cana-4553 3 64 1ashutosh.mishra8@s.amity.edu 1ashutosh.mishra8@s.amity.edu NUM cana-4553 3 65 ; ; PUNCT cana-4553 3 66 2pktripathi@amity.edu 2pktripathi@amity.edu NUM cana-4553 3 67 ; ; PUNCT cana-4553 3 68 3mksinghuptti@gmail.com 3mksinghuptti@gmail.com NUM cana-4553 3 69 ; ; PUNCT cana-4553 3 70 4manisha.gupta@utas.edu.om 4manisha.gupta@utas.edu.om NUM cana-4553 3 71 ; ; PUNCT cana-4553 3 72 5akagrawal@lko.amity.edu 5akagrawal@lko.amity.edu NUM cana-4553 3 73 article article NOUN cana-4553 3 74 history history NOUN cana-4553 3 75 : : PUNCT cana-4553 3 76 received receive VERB cana-4553 3 77 : : PUNCT cana-4553 3 78 12 12 NUM cana-4553 3 79 - - SYM cana-4553 3 80 01 01 NUM cana-4553 3 81 - - PUNCT cana-4553 3 82 2025 2025 NUM cana-4553 3 83 revised revise VERB cana-4553 3 84 : : PUNCT cana-4553 3 85 15 15 NUM cana-4553 3 86 - - NUM cana-4553 3 87 02 02 NUM cana-4553 3 88 - - PUNCT cana-4553 3 89 2025 2025 NUM cana-4553 3 90 accepted accept VERB cana-4553 3 91 : : PUNCT cana-4553 3 92 01 01 NUM cana-4553 3 93 - - SYM cana-4553 3 94 03 03 NUM cana-4553 3 95 - - PUNCT cana-4553 3 96 2025 2025 NUM cana-4553 3 97 abstract abstract NOUN cana-4553 3 98 : : PUNCT cana-4553 3 99 in in ADP cana-4553 3 100 the the DET cana-4553 3 101 realm realm NOUN cana-4553 3 102 of of ADP cana-4553 3 103 machine machine NOUN cana-4553 3 104 learning learning NOUN cana-4553 3 105 ( ( PUNCT cana-4553 3 106 ml ml NOUN cana-4553 3 107 ) ) PUNCT cana-4553 3 108 , , PUNCT cana-4553 3 109 the the DET cana-4553 3 110 underlying underlying ADJ cana-4553 3 111 mathematical mathematical ADJ cana-4553 3 112 frameworks framework NOUN cana-4553 3 113 significantly significantly ADV cana-4553 3 114 influence influence VERB cana-4553 3 115 the the DET cana-4553 3 116 development development NOUN cana-4553 3 117 and and CCONJ cana-4553 3 118 efficacy efficacy NOUN cana-4553 3 119 of of ADP cana-4553 3 120 algorithms algorithm NOUN cana-4553 3 121 . . PUNCT cana-4553 4 1 one one NUM cana-4553 4 2 such such ADJ cana-4553 4 3 framework framework NOUN cana-4553 4 4 is be AUX cana-4553 4 5 the the DET cana-4553 4 6 probabilistic probabilistic ADJ cana-4553 4 7 metric metric ADJ cana-4553 4 8 space space NOUN cana-4553 4 9 , , PUNCT cana-4553 4 10 which which PRON cana-4553 4 11 provides provide VERB cana-4553 4 12 a a DET cana-4553 4 13 robust robust ADJ cana-4553 4 14 foundation foundation NOUN cana-4553 4 15 for for ADP cana-4553 4 16 handling handle VERB cana-4553 4 17 uncertainty uncertainty NOUN cana-4553 4 18 and and CCONJ cana-4553 4 19 variability variability NOUN cana-4553 4 20 inherent inherent ADJ cana-4553 4 21 in in ADP cana-4553 4 22 real real ADJ cana-4553 4 23 - - PUNCT cana-4553 4 24 world world NOUN cana-4553 4 25 problems problem NOUN cana-4553 4 26 and and CCONJ cana-4553 4 27 data datum NOUN cana-4553 4 28 . . PUNCT cana-4553 5 1 this this DET cana-4553 5 2 work work NOUN cana-4553 5 3 explores explore VERB cana-4553 5 4 the the DET cana-4553 5 5 applicability applicability NOUN cana-4553 5 6 of of ADP cana-4553 5 7 probabilistic probabilistic ADJ cana-4553 5 8 metric metric ADJ cana-4553 5 9 spaces space NOUN cana-4553 5 10 in in ADP cana-4553 5 11 machine machine NOUN cana-4553 5 12 learning learning NOUN cana-4553 5 13 , , PUNCT cana-4553 5 14 with with ADP cana-4553 5 15 a a DET cana-4553 5 16 particular particular ADJ cana-4553 5 17 focus focus NOUN cana-4553 5 18 on on ADP cana-4553 5 19 the the DET cana-4553 5 20 jungck jungck NOUN cana-4553 5 21 contractive contractive ADJ cana-4553 5 22 notion notion NOUN cana-4553 5 23 . . PUNCT cana-4553 6 1 the the DET cana-4553 6 2 jungck jungck PROPN cana-4553 6 3 contraction contraction PROPN cana-4553 6 4 theorem theorem PROPN cana-4553 6 5 extends extend VERB cana-4553 6 6 the the DET cana-4553 6 7 classical classical ADJ cana-4553 6 8 banach banach NOUN cana-4553 6 9 contraction contraction NOUN cana-4553 6 10 principle principle NOUN cana-4553 6 11 to to ADP cana-4553 6 12 the the DET cana-4553 6 13 setting setting NOUN cana-4553 6 14 of of ADP cana-4553 6 15 probabilistic probabilistic ADJ cana-4553 6 16 metric metric ADJ cana-4553 6 17 spaces space NOUN cana-4553 6 18 . . PUNCT cana-4553 7 1 this this DET cana-4553 7 2 theorem theorem NOUN cana-4553 7 3 is be AUX cana-4553 7 4 pivotal pivotal ADJ cana-4553 7 5 for for ADP cana-4553 7 6 proving prove VERB cana-4553 7 7 the the DET cana-4553 7 8 existence existence NOUN cana-4553 7 9 and and CCONJ cana-4553 7 10 uniqueness uniqueness NOUN cana-4553 7 11 of of ADP cana-4553 7 12 fixed fix VERB cana-4553 7 13 points point NOUN cana-4553 7 14 in in ADP cana-4553 7 15 probabilistic probabilistic ADJ cana-4553 7 16 settings setting NOUN cana-4553 7 17 , , PUNCT cana-4553 7 18 which which PRON cana-4553 7 19 is be AUX cana-4553 7 20 crucial crucial ADJ cana-4553 7 21 for for ADP cana-4553 7 22 the the DET cana-4553 7 23 convergence convergence NOUN cana-4553 7 24 analysis analysis NOUN cana-4553 7 25 of of ADP cana-4553 7 26 ml ml ADP cana-4553 7 27 algorithms algorithm NOUN cana-4553 7 28 . . PUNCT cana-4553 8 1 this this PRON cana-4553 8 2 generalises generalise VERB cana-4553 8 3 invariant invariant ADJ cana-4553 8 4 point point NOUN cana-4553 8 5 propositions proposition NOUN cana-4553 8 6 to to ADP cana-4553 8 7 probabilistic probabilistic ADJ cana-4553 8 8 settings setting NOUN cana-4553 8 9 , , PUNCT cana-4553 8 10 offers offer VERB cana-4553 8 11 powerful powerful ADJ cana-4553 8 12 tools tool NOUN cana-4553 8 13 for for ADP cana-4553 8 14 the the DET cana-4553 8 15 convergence convergence NOUN cana-4553 8 16 analysis analysis NOUN cana-4553 8 17 of of ADP cana-4553 8 18 iterative iterative NOUN cana-4553 8 19 ml ml NOUN cana-4553 8 20 algorithms algorithm NOUN cana-4553 8 21 . . PUNCT cana-4553 9 1 we we PRON cana-4553 9 2 delve delve VERB cana-4553 9 3 into into ADP cana-4553 9 4 the the DET cana-4553 9 5 theoretical theoretical ADJ cana-4553 9 6 underpinnings underpinning NOUN cana-4553 9 7 of of ADP cana-4553 9 8 probabilistic probabilistic ADJ cana-4553 9 9 metric metric ADJ cana-4553 9 10 spaces space NOUN cana-4553 9 11 , , PUNCT cana-4553 9 12 explain explain VERB cana-4553 9 13 the the DET cana-4553 9 14 jungck jungck NOUN cana-4553 9 15 contractive contractive ADJ cana-4553 9 16 notion notion NOUN cana-4553 9 17 , , PUNCT cana-4553 9 18 and and CCONJ cana-4553 9 19 bespeak bespeak VERB cana-4553 9 20 its its PRON cana-4553 9 21 relevance relevance NOUN cana-4553 9 22 and and CCONJ cana-4553 9 23 application application NOUN cana-4553 9 24 in in ADP cana-4553 9 25 ml ml ADP cana-4553 9 26 algorithms algorithm NOUN cana-4553 9 27 . . PUNCT cana-4553 10 1 keywords keyword NOUN cana-4553 10 2 : : PUNCT cana-4553 10 3 probabilistic probabilistic ADJ cana-4553 10 4 space space NOUN cana-4553 10 5 , , PUNCT cana-4553 10 6 contractive contractive ADJ cana-4553 10 7 conditions condition NOUN cana-4553 10 8 , , PUNCT cana-4553 10 9 machine machine NOUN cana-4553 10 10 learning learning NOUN cana-4553 10 11 , , PUNCT cana-4553 10 12 convergence convergence NOUN cana-4553 10 13 analysis analysis NOUN cana-4553 10 14 . . PUNCT cana-4553 11 1 1 1 X cana-4553 11 2 . . X cana-4553 11 3 introduction introduction NOUN cana-4553 11 4 machine machine NOUN cana-4553 11 5 learning learning NOUN cana-4553 11 6 is be AUX cana-4553 11 7 an an DET cana-4553 11 8 underlying underlying ADJ cana-4553 11 9 collection collection NOUN cana-4553 11 10 as as ADP cana-4553 11 11 a a DET cana-4553 11 12 subset subset NOUN cana-4553 11 13 of of ADP cana-4553 11 14 ai ai PROPN cana-4553 11 15 ( ( PUNCT cana-4553 11 16 artificial artificial ADJ cana-4553 11 17 intelligence intelligence NOUN cana-4553 11 18 ) ) PUNCT cana-4553 11 19 that that PRON cana-4553 11 20 concentrates concentrate VERB cana-4553 11 21 the the DET cana-4553 11 22 advancement advancement NOUN cana-4553 11 23 of of ADP cana-4553 11 24 algorithms algorithm NOUN cana-4553 11 25 , , PUNCT cana-4553 11 26 as as ADV cana-4553 11 27 well well ADV cana-4553 11 28 as as ADP cana-4553 11 29 statistical statistical ADJ cana-4553 11 30 model model NOUN cana-4553 11 31 which which PRON cana-4553 11 32 enables enable VERB cana-4553 11 33 computers computer NOUN cana-4553 11 34 to to PART cana-4553 11 35 execute execute VERB cana-4553 11 36 certain certain ADJ cana-4553 11 37 tasks task NOUN cana-4553 11 38 without without ADP cana-4553 11 39 distinct distinct ADJ cana-4553 11 40 instructions instruction NOUN cana-4553 11 41 , , PUNCT cana-4553 11 42 depending depend VERB cana-4553 11 43 on on ADP cana-4553 11 44 patterns pattern NOUN cana-4553 11 45 and and CCONJ cana-4553 11 46 inference inference NOUN cana-4553 11 47 instead instead ADV cana-4553 11 48 . . PUNCT cana-4553 12 1 in in ADP cana-4553 12 2 other other ADJ cana-4553 12 3 words word NOUN cana-4553 12 4 , , PUNCT cana-4553 12 5 it it PRON cana-4553 12 6 is be AUX cana-4553 12 7 about about ADP cana-4553 12 8 creating create VERB cana-4553 12 9 systems system NOUN cana-4553 12 10 that that PRON cana-4553 12 11 can can AUX cana-4553 12 12 learn learn VERB cana-4553 12 13 from from ADP cana-4553 12 14 data datum NOUN cana-4553 12 15 . . PUNCT cana-4553 13 1 the the DET cana-4553 13 2 crux crux NOUN cana-4553 13 3 behind behind ADP cana-4553 13 4 such such ADJ cana-4553 13 5 learning learning NOUN cana-4553 13 6 is be AUX cana-4553 13 7 to to PART cana-4553 13 8 empower empower VERB cana-4553 13 9 computer computer NOUN cana-4553 13 10 related relate VERB cana-4553 13 11 programmes programme NOUN cana-4553 13 12 to to PART cana-4553 13 13 train train VERB cana-4553 13 14 automatically automatically ADV cana-4553 13 15 from from ADP cana-4553 13 16 given give VERB cana-4553 13 17 data datum NOUN cana-4553 13 18 and and CCONJ cana-4553 13 19 to to PART cana-4553 13 20 improve improve VERB cana-4553 13 21 its its PRON cana-4553 13 22 effectiveness effectiveness NOUN cana-4553 13 23 over over ADP cana-4553 13 24 time time NOUN cana-4553 13 25 , , PUNCT cana-4553 13 26 without without ADP cana-4553 13 27 being be AUX cana-4553 13 28 distinctively distinctively ADV cana-4553 13 29 programmed program VERB cana-4553 13 30 for for ADP cana-4553 13 31 every every DET cana-4553 13 32 task task NOUN cana-4553 13 33 . . PUNCT cana-4553 14 1 this this PRON cana-4553 14 2 is be AUX cana-4553 14 3 brought bring VERB cana-4553 14 4 about about ADP cana-4553 14 5 through through ADP cana-4553 14 6 various various ADJ cana-4553 14 7 learning learning NOUN cana-4553 14 8 techniques technique NOUN cana-4553 14 9 viz viz PROPN cana-4553 14 10 . . PUNCT cana-4553 15 1 supervised supervised ADJ cana-4553 15 2 , , PUNCT cana-4553 15 3 unsupervised unsupervised ADJ cana-4553 15 4 , , PUNCT cana-4553 15 5 reinforced reinforce VERB cana-4553 15 6 , , PUNCT cana-4553 15 7 and and CCONJ cana-4553 15 8 deep deep ADJ cana-4553 15 9 . . PUNCT cana-4553 16 1 supervised supervised ADJ cana-4553 16 2 learning learn VERB cana-4553 16 3 in in ADP cana-4553 16 4 this this DET cana-4553 16 5 approach approach NOUN cana-4553 16 6 , , PUNCT cana-4553 16 7 the the DET cana-4553 16 8 algorithm algorithm NOUN cana-4553 16 9 retains retain VERB cana-4553 16 10 information information NOUN cana-4553 16 11 from from ADP cana-4553 16 12 some some DET cana-4553 16 13 specific specific ADJ cana-4553 16 14 labelled label VERB cana-4553 16 15 data datum NOUN cana-4553 16 16 , , PUNCT cana-4553 16 17 meaning mean VERB cana-4553 16 18 there there ADV cana-4553 16 19 by by ADP cana-4553 16 20 that that SCONJ cana-4553 16 21 it it PRON cana-4553 16 22 is be AUX cana-4553 16 23 given give VERB cana-4553 16 24 input input NOUN cana-4553 16 25 - - PUNCT cana-4553 16 26 output output NOUN cana-4553 16 27 pairs pair NOUN cana-4553 16 28 and and CCONJ cana-4553 16 29 learns learn VERB cana-4553 16 30 to to PART cana-4553 16 31 map map VERB cana-4553 16 32 inputs input NOUN cana-4553 16 33 to to ADP cana-4553 16 34 outputs output NOUN cana-4553 16 35 . . PUNCT cana-4553 17 1 for for ADP cana-4553 17 2 example example NOUN cana-4553 17 3 , , PUNCT cana-4553 17 4 let let VERB cana-4553 17 5 us we PRON cana-4553 17 6 have have VERB cana-4553 17 7 a a DET cana-4553 17 8 given give VERB cana-4553 17 9 set set NOUN cana-4553 17 10 of of ADP cana-4553 17 11 labelled label VERB cana-4553 17 12 impressions impression NOUN cana-4553 17 13 ( ( PUNCT cana-4553 17 14 images image NOUN cana-4553 17 15 ) ) PUNCT cana-4553 17 16 of of ADP cana-4553 17 17 cats cat NOUN cana-4553 17 18 and and CCONJ cana-4553 17 19 dogs dog NOUN cana-4553 17 20 , , PUNCT cana-4553 17 21 a a DET cana-4553 17 22 supervised supervised ADJ cana-4553 17 23 learning learning NOUN cana-4553 17 24 algorithm algorithm NOUN cana-4553 17 25 would would AUX cana-4553 17 26 learn learn VERB cana-4553 17 27 to to PART cana-4553 17 28 classify classify VERB cana-4553 17 29 new new ADJ cana-4553 17 30 impressions impression NOUN cana-4553 17 31 ( ( PUNCT cana-4553 17 32 images image NOUN cana-4553 17 33 ) ) PUNCT cana-4553 17 34 as as ADP cana-4553 17 35 either either PRON cana-4553 17 36 cats cat NOUN cana-4553 17 37 or or CCONJ cana-4553 17 38 dogs dog NOUN cana-4553 17 39 [ [ X cana-4553 17 40 8 8 NUM cana-4553 17 41 ] ] PUNCT cana-4553 17 42 . . PUNCT cana-4553 18 1 mailto:amishra94154@gmail.com mailto:amishra94154@gmail.com X cana-4553 18 2 mailto:1ashutosh.mishra8@s.amity.edu mailto:1ashutosh.mishra8@s.amity.edu SYM cana-4553 18 3 mailto:2pktripathi@amity.edu mailto:2pktripathi@amity.edu PROPN cana-4553 18 4 mailto mailto NOUN cana-4553 18 5 : : PUNCT cana-4553 18 6 mksinghuptti@gmail.com3 mksinghuptti@gmail.com3 PROPN cana-4553 18 7 mailto:manisha.gupta@utas.edu.om mailto:manisha.gupta@utas.edu.om PROPN cana-4553 18 8 communications communication NOUN cana-4553 18 9 on on ADP cana-4553 18 10 applied apply VERB cana-4553 18 11 nonlinear nonlinear ADJ cana-4553 18 12 analysis analysis NOUN cana-4553 18 13 issn issn NOUN cana-4553 18 14 : : PUNCT cana-4553 18 15 1074 1074 NUM cana-4553 18 16 - - PUNCT cana-4553 18 17 133x 133x NUM cana-4553 18 18 vol vol NOUN cana-4553 18 19 32 32 NUM cana-4553 18 20 no no NOUN cana-4553 18 21 . . PUNCT cana-4553 19 1 9s 9s NUM cana-4553 19 2 ( ( PUNCT cana-4553 19 3 2025 2025 NUM cana-4553 19 4 ) ) PUNCT cana-4553 19 5 2755 2755 NUM cana-4553 20 1 https://internationalpubls.com https://internationalpubls.com X cana-4553 20 2 unsupervised unsupervised ADJ cana-4553 20 3 learning learning NOUN cana-4553 20 4 in in ADP cana-4553 20 5 this this DET cana-4553 20 6 stage stage NOUN cana-4553 20 7 , , PUNCT cana-4553 20 8 the the DET cana-4553 20 9 algorithm algorithm NOUN cana-4553 20 10 is be AUX cana-4553 20 11 given give VERB cana-4553 20 12 input input NOUN cana-4553 20 13 data datum NOUN cana-4553 20 14 without without ADP cana-4553 20 15 explicit explicit ADJ cana-4553 20 16 labels label NOUN cana-4553 20 17 , , PUNCT cana-4553 20 18 and and CCONJ cana-4553 20 19 it it PRON cana-4553 20 20 tries try VERB cana-4553 20 21 to to PART cana-4553 20 22 find find VERB cana-4553 20 23 patterns pattern NOUN cana-4553 20 24 or or CCONJ cana-4553 20 25 structure structure NOUN cana-4553 20 26 in in ADP cana-4553 20 27 the the DET cana-4553 20 28 data datum NOUN cana-4553 20 29 . . PUNCT cana-4553 21 1 clustering clustering NOUN cana-4553 21 2 is be AUX cana-4553 21 3 a a DET cana-4553 21 4 common common ADJ cana-4553 21 5 task task NOUN cana-4553 21 6 in in ADP cana-4553 21 7 unsupervised unsupervised ADJ cana-4553 21 8 learning learning NOUN cana-4553 21 9 , , PUNCT cana-4553 21 10 where where SCONJ cana-4553 21 11 the the DET cana-4553 21 12 algorithm algorithm NOUN cana-4553 21 13 groups group NOUN cana-4553 21 14 similar similar ADJ cana-4553 21 15 data datum NOUN cana-4553 21 16 points point VERB cana-4553 21 17 together together ADV cana-4553 21 18 . . PUNCT cana-4553 22 1 an an DET cana-4553 22 2 example example NOUN cana-4553 22 3 could could AUX cana-4553 22 4 be be AUX cana-4553 22 5 grouping group VERB cana-4553 22 6 similar similar ADJ cana-4553 22 7 customer customer NOUN cana-4553 22 8 purchase purchase NOUN cana-4553 22 9 behaviour behaviour NOUN cana-4553 22 10 for for ADP cana-4553 22 11 market market NOUN cana-4553 22 12 segmentation segmentation NOUN cana-4553 22 13 [ [ X cana-4553 22 14 8 8 NUM cana-4553 22 15 ] ] PUNCT cana-4553 22 16 . . PUNCT cana-4553 23 1 reinforced reinforce VERB cana-4553 23 2 learning learning NOUN cana-4553 23 3 this this PRON cana-4553 23 4 is be AUX cana-4553 23 5 decision decision NOUN cana-4553 23 6 making make VERB cana-4553 23 7 training training NOUN cana-4553 23 8 algorithms algorithm NOUN cana-4553 23 9 that that PRON cana-4553 23 10 gets get AUX cana-4553 23 11 learning learn VERB cana-4553 23 12 from from ADP cana-4553 23 13 feedback feedback NOUN cana-4553 23 14 . . PUNCT cana-4553 24 1 the the DET cana-4553 24 2 algorithm algorithm NOUN cana-4553 24 3 trains train VERB cana-4553 24 4 itself itself PRON cana-4553 24 5 to to PART cana-4553 24 6 achieve achieve VERB cana-4553 24 7 a a DET cana-4553 24 8 outcome outcome NOUN cana-4553 24 9 or or CCONJ cana-4553 24 10 goal goal NOUN cana-4553 24 11 in in ADP cana-4553 24 12 an an DET cana-4553 24 13 uncertain uncertain ADJ cana-4553 24 14 , , PUNCT cana-4553 24 15 potentially potentially ADV cana-4553 24 16 complex complex ADJ cana-4553 24 17 environment environment NOUN cana-4553 24 18 by by ADP cana-4553 24 19 taking take VERB cana-4553 24 20 actions action NOUN cana-4553 24 21 and and CCONJ cana-4553 24 22 receiving receive VERB cana-4553 24 23 rewards reward NOUN cana-4553 24 24 or or CCONJ cana-4553 24 25 penalties penalty NOUN cana-4553 24 26 . . PUNCT cana-4553 25 1 it it PRON cana-4553 25 2 is be AUX cana-4553 25 3 often often ADV cana-4553 25 4 used use VERB cana-4553 25 5 in in ADP cana-4553 25 6 areas area NOUN cana-4553 25 7 like like ADP cana-4553 25 8 gamming gamming NOUN cana-4553 25 9 , , PUNCT cana-4553 25 10 robotics robotic NOUN cana-4553 25 11 , , PUNCT cana-4553 25 12 and and CCONJ cana-4553 25 13 auto auto NOUN cana-4553 25 14 driving driving NOUN cana-4553 25 15 [ [ X cana-4553 25 16 10 10 NUM cana-4553 25 17 ] ] PUNCT cana-4553 25 18 . . PUNCT cana-4553 26 1 deep deep ADJ cana-4553 26 2 learning learn VERB cana-4553 26 3 it it PRON cana-4553 26 4 is be AUX cana-4553 26 5 an an DET cana-4553 26 6 underlying underlying ADJ cana-4553 26 7 subset subset NOUN cana-4553 26 8 of of ADP cana-4553 26 9 machine machine NOUN cana-4553 26 10 learning learning NOUN cana-4553 26 11 which which PRON cana-4553 26 12 uses use VERB cana-4553 26 13 artificial artificial ADJ cana-4553 26 14 neural neural ADJ cana-4553 26 15 networks network NOUN cana-4553 26 16 with with ADP cana-4553 26 17 many many ADJ cana-4553 26 18 blankets blanket NOUN cana-4553 26 19 of of ADP cana-4553 26 20 layers layer NOUN cana-4553 26 21 , , PUNCT cana-4553 26 22 hence hence ADV cana-4553 26 23 proposed propose VERB cana-4553 26 24 as as ADP cana-4553 26 25 β€œ " PUNCT cana-4553 26 26 deep deep ADJ cana-4553 26 27 ” " PUNCT cana-4553 26 28 , , PUNCT cana-4553 26 29 to to PART cana-4553 26 30 model model VERB cana-4553 26 31 and and CCONJ cana-4553 26 32 process process NOUN cana-4553 26 33 complex complex ADJ cana-4553 26 34 patterns pattern NOUN cana-4553 26 35 in in ADP cana-4553 26 36 large large ADJ cana-4553 26 37 amounts amount NOUN cana-4553 26 38 of of ADP cana-4553 26 39 data datum NOUN cana-4553 26 40 . . PUNCT cana-4553 27 1 it it PRON cana-4553 27 2 's be AUX cana-4553 27 3 particularly particularly ADV cana-4553 27 4 effective effective ADJ cana-4553 27 5 for for ADP cana-4553 27 6 tasks task NOUN cana-4553 27 7 such such ADJ cana-4553 27 8 as as ADP cana-4553 27 9 image image NOUN cana-4553 27 10 and and CCONJ cana-4553 27 11 speech speech NOUN cana-4553 27 12 recognition recognition NOUN cana-4553 27 13 , , PUNCT cana-4553 27 14 natural natural ADJ cana-4553 27 15 language language NOUN cana-4553 27 16 processing processing NOUN cana-4553 27 17 , , PUNCT cana-4553 27 18 and and CCONJ cana-4553 27 19 playing play VERB cana-4553 27 20 strategic strategic ADJ cana-4553 27 21 games game NOUN cana-4553 27 22 like like ADP cana-4553 27 23 go go VERB cana-4553 27 24 [ [ X cana-4553 27 25 9 9 NUM cana-4553 27 26 ] ] PUNCT cana-4553 27 27 . . PUNCT cana-4553 28 1 overall overall ADJ cana-4553 28 2 , , PUNCT cana-4553 28 3 machine machine NOUN cana-4553 28 4 learning learning NOUN cana-4553 28 5 has have VERB cana-4553 28 6 numerous numerous ADJ cana-4553 28 7 applications application NOUN cana-4553 28 8 across across ADP cana-4553 28 9 various various ADJ cana-4553 28 10 industries industry NOUN cana-4553 28 11 , , PUNCT cana-4553 28 12 including include VERB cana-4553 28 13 finance finance NOUN cana-4553 28 14 , , PUNCT cana-4553 28 15 ecommerce ecommerce PROPN cana-4553 28 16 , , PUNCT cana-4553 28 17 healthcare healthcare PROPN cana-4553 28 18 , , PUNCT cana-4553 28 19 cyber cyber ADJ cana-4553 28 20 security security NOUN cana-4553 28 21 , , PUNCT cana-4553 28 22 and and CCONJ cana-4553 28 23 many many ADJ cana-4553 28 24 more more ADJ cana-4553 28 25 . . PUNCT cana-4553 29 1 its its PRON cana-4553 29 2 ability ability NOUN cana-4553 29 3 to to PART cana-4553 29 4 find find VERB cana-4553 29 5 patterns pattern NOUN cana-4553 29 6 in in ADP cana-4553 29 7 data datum NOUN cana-4553 29 8 and and CCONJ cana-4553 29 9 make make VERB cana-4553 29 10 predictions prediction NOUN cana-4553 29 11 or or CCONJ cana-4553 29 12 decisions decision NOUN cana-4553 29 13 based base VERB cana-4553 29 14 on on ADP cana-4553 29 15 those those DET cana-4553 29 16 patterns pattern NOUN cana-4553 29 17 has have AUX cana-4553 29 18 made make VERB cana-4553 29 19 it it PRON cana-4553 29 20 a a DET cana-4553 29 21 powerful powerful ADJ cana-4553 29 22 tool tool NOUN cana-4553 29 23 for for ADP cana-4553 29 24 solving solve VERB cana-4553 29 25 a a DET cana-4553 29 26 wide wide ADJ cana-4553 29 27 range range NOUN cana-4553 29 28 of of ADP cana-4553 29 29 real real ADJ cana-4553 29 30 - - PUNCT cana-4553 29 31 world world NOUN cana-4553 29 32 problems problem NOUN cana-4553 29 33 . . PUNCT cana-4553 30 1 machine machine NOUN cana-4553 30 2 learning learn VERB cana-4553 30 3 algorithms algorithm NOUN cana-4553 30 4 often often ADV cana-4553 30 5 rely rely VERB cana-4553 30 6 on on ADP cana-4553 30 7 the the DET cana-4553 30 8 convergence convergence NOUN cana-4553 30 9 properties property NOUN cana-4553 30 10 of of ADP cana-4553 30 11 iterative iterative NOUN cana-4553 30 12 processes process NOUN cana-4553 30 13 . . PUNCT cana-4553 31 1 classical classical ADJ cana-4553 31 2 metric metric ADJ cana-4553 31 3 spaces space NOUN cana-4553 31 4 and and CCONJ cana-4553 31 5 their their PRON cana-4553 31 6 associated associate VERB cana-4553 31 7 invariant invariant ADJ cana-4553 31 8 point point NOUN cana-4553 31 9 theorems theorem NOUN cana-4553 31 10 have have AUX cana-4553 31 11 provided provide VERB cana-4553 31 12 substantial substantial ADJ cana-4553 31 13 insights insight NOUN cana-4553 31 14 into into ADP cana-4553 31 15 these these DET cana-4553 31 16 processes process NOUN cana-4553 31 17 . . PUNCT cana-4553 32 1 however however ADV cana-4553 32 2 , , PUNCT cana-4553 32 3 real real ADJ cana-4553 32 4 - - PUNCT cana-4553 32 5 world world NOUN cana-4553 32 6 data datum NOUN cana-4553 32 7 is be AUX cana-4553 32 8 frequently frequently ADV cana-4553 32 9 stochastic stochastic ADJ cana-4553 32 10 in in ADP cana-4553 32 11 nature nature NOUN cana-4553 32 12 , , PUNCT cana-4553 32 13 demanding demand VERB cana-4553 32 14 a a DET cana-4553 32 15 probabilistic probabilistic ADJ cana-4553 32 16 approach approach NOUN cana-4553 32 17 to to AUX cana-4553 32 18 better well ADV cana-4553 32 19 model model VERB cana-4553 32 20 the the DET cana-4553 32 21 uncertainty uncertainty NOUN cana-4553 32 22 and and CCONJ cana-4553 32 23 variability variability NOUN cana-4553 32 24 . . PUNCT cana-4553 33 1 probabilistic probabilistic ADJ cana-4553 33 2 metric metric ADJ cana-4553 33 3 spaces space NOUN cana-4553 33 4 , , PUNCT cana-4553 33 5 introduced introduce VERB cana-4553 33 6 by by ADP cana-4553 33 7 menger[2]in menger[2]in PROPN cana-4553 33 8 the the DET cana-4553 33 9 year year NOUN cana-4553 33 10 1940 1940 NUM cana-4553 33 11 , , PUNCT cana-4553 33 12 extend extend VERB cana-4553 33 13 the the DET cana-4553 33 14 concept concept NOUN cana-4553 33 15 of of ADP cana-4553 33 16 metric metric ADJ cana-4553 33 17 spaces space NOUN cana-4553 33 18 by by ADP cana-4553 33 19 incorporating incorporate VERB cana-4553 33 20 probability probability NOUN cana-4553 33 21 distributions distribution NOUN cana-4553 33 22 instead instead ADV cana-4553 33 23 of of ADP cana-4553 33 24 deterministic deterministic ADJ cana-4553 33 25 distances distance NOUN cana-4553 33 26 . . PUNCT cana-4553 34 1 since since SCONJ cana-4553 34 2 then then ADV cana-4553 34 3 , , PUNCT cana-4553 34 4 a a DET cana-4553 34 5 variety variety NOUN cana-4553 34 6 of of ADP cana-4553 34 7 propositions proposition NOUN cana-4553 34 8 were be AUX cana-4553 34 9 introduced introduce VERB cana-4553 34 10 using use VERB cana-4553 34 11 menger menger PROPN cana-4553 34 12 theory theory NOUN cana-4553 34 13 . . PUNCT cana-4553 35 1 some some DET cana-4553 35 2 implications implication NOUN cana-4553 35 3 of of ADP cana-4553 35 4 this this DET cana-4553 35 5 notion notion NOUN cana-4553 35 6 are be AUX cana-4553 35 7 illustrated illustrate VERB cana-4553 35 8 in in ADP cana-4553 35 9 population population NOUN cana-4553 35 10 dynamics dynamic NOUN cana-4553 35 11 of of ADP cana-4553 35 12 cells cell NOUN cana-4553 35 13 [ [ X cana-4553 35 14 11 11 NUM cana-4553 35 15 , , PUNCT cana-4553 35 16 14 14 NUM cana-4553 35 17 ] ] PUNCT cana-4553 35 18 . . PUNCT cana-4553 36 1 this this DET cana-4553 36 2 probabilistic probabilistic ADJ cana-4553 36 3 framework framework NOUN cana-4553 36 4 is be AUX cana-4553 36 5 particularly particularly ADV cana-4553 36 6 useful useful ADJ cana-4553 36 7 in in ADP cana-4553 36 8 ml ml ADP cana-4553 36 9 , , PUNCT cana-4553 36 10 where where SCONJ cana-4553 36 11 data datum NOUN cana-4553 36 12 and and CCONJ cana-4553 36 13 models model NOUN cana-4553 36 14 are be AUX cana-4553 36 15 inherently inherently ADV cana-4553 36 16 noisy noisy ADJ cana-4553 36 17 and and CCONJ cana-4553 36 18 probabilistic probabilistic ADJ cana-4553 36 19 . . PUNCT cana-4553 37 1 the the DET cana-4553 37 2 jungck jungck NOUN cana-4553 37 3 theorem theorem VERB cana-4553 37 4 on on ADP cana-4553 37 5 contraction contraction NOUN cana-4553 37 6 mapping[4 mapping[4 PROPN cana-4553 37 7 ] ] PUNCT cana-4553 37 8 , , PUNCT cana-4553 37 9 a a DET cana-4553 37 10 generalization generalization NOUN cana-4553 37 11 of of ADP cana-4553 37 12 banach banach NOUN cana-4553 37 13 's 's PART cana-4553 37 14 fixed fix VERB cana-4553 37 15 point point NOUN cana-4553 37 16 result result VERB cana-4553 37 17 to to PART cana-4553 37 18 probabilistic probabilistic VERB cana-4553 37 19 metric metric ADJ cana-4553 37 20 spaces space NOUN cana-4553 37 21 , , PUNCT cana-4553 37 22 offers offer VERB cana-4553 37 23 a a DET cana-4553 37 24 robust robust ADJ cana-4553 37 25 mechanism mechanism NOUN cana-4553 37 26 to to PART cana-4553 37 27 analyze analyze VERB cana-4553 37 28 the the DET cana-4553 37 29 convergence convergence NOUN cana-4553 37 30 of of ADP cana-4553 37 31 sequences sequence NOUN cana-4553 37 32 in in ADP cana-4553 37 33 these these DET cana-4553 37 34 spaces[1 spaces[1 NOUN cana-4553 37 35 ] ] PUNCT cana-4553 37 36 . . PUNCT cana-4553 38 1 this this DET cana-4553 38 2 theorem theorem NOUN cana-4553 38 3 is be AUX cana-4553 38 4 instrumental instrumental ADJ cana-4553 38 5 in in ADP cana-4553 38 6 ensuring ensure VERB cana-4553 38 7 the the DET cana-4553 38 8 stability stability NOUN cana-4553 38 9 and and CCONJ cana-4553 38 10 reliability reliability NOUN cana-4553 38 11 of of ADP cana-4553 38 12 iterative iterative NOUN cana-4553 38 13 ml ml NOUN cana-4553 38 14 algorithms algorithm NOUN cana-4553 38 15 . . PUNCT cana-4553 39 1 by by ADP cana-4553 39 2 leveraging leverage VERB cana-4553 39 3 the the DET cana-4553 39 4 jungck jungck NOUN cana-4553 39 5 contraction contraction NOUN cana-4553 39 6 theorem theorem VERB cana-4553 39 7 , , PUNCT cana-4553 39 8 we we PRON cana-4553 39 9 can can AUX cana-4553 39 10 develop develop VERB cana-4553 39 11 and and CCONJ cana-4553 39 12 analyze analyze VERB cana-4553 39 13 ml ml NOUN cana-4553 39 14 algorithms algorithm NOUN cana-4553 39 15 that that PRON cana-4553 39 16 are be AUX cana-4553 39 17 more more ADV cana-4553 39 18 resilient resilient ADJ cana-4553 39 19 to to ADP cana-4553 39 20 the the DET cana-4553 39 21 uncertainties uncertainty NOUN cana-4553 39 22 of of ADP cana-4553 39 23 real real ADJ cana-4553 39 24 - - PUNCT cana-4553 39 25 world world NOUN cana-4553 39 26 data datum NOUN cana-4553 39 27 . . PUNCT cana-4553 40 1 the the DET cana-4553 40 2 k k NOUN cana-4553 40 3 - - PUNCT cana-4553 40 4 means mean VERB cana-4553 40 5 clustering clustering ADJ cana-4553 40 6 algorithm algorithm NOUN cana-4553 40 7 is be AUX cana-4553 40 8 a a DET cana-4553 40 9 fundamental fundamental ADJ cana-4553 40 10 technique technique NOUN cana-4553 40 11 in in ADP cana-4553 40 12 machine machine NOUN cana-4553 40 13 learning learn VERB cana-4553 40 14 for for ADP cana-4553 40 15 partitioning partition VERB cana-4553 40 16 data datum NOUN cana-4553 40 17 into into ADP cana-4553 40 18 clusters cluster NOUN cana-4553 40 19 based base VERB cana-4553 40 20 on on ADP cana-4553 40 21 similarity[7 similarity[7 NOUN cana-4553 40 22 ] ] PUNCT cana-4553 40 23 . . PUNCT cana-4553 41 1 in in ADP cana-4553 41 2 traditional traditional ADJ cana-4553 41 3 k k NOUN cana-4553 41 4 - - PUNCT cana-4553 41 5 means mean NOUN cana-4553 41 6 , , PUNCT cana-4553 41 7 the the DET cana-4553 41 8 distance distance NOUN cana-4553 41 9 between between ADP cana-4553 41 10 points point NOUN cana-4553 41 11 and and CCONJ cana-4553 41 12 centroids centroid NOUN cana-4553 41 13 is be AUX cana-4553 41 14 deterministic deterministic ADJ cana-4553 41 15 . . PUNCT cana-4553 42 1 however however ADV cana-4553 42 2 , , PUNCT cana-4553 42 3 real real ADJ cana-4553 42 4 - - PUNCT cana-4553 42 5 world world NOUN cana-4553 42 6 data datum NOUN cana-4553 42 7 often often ADV cana-4553 42 8 contains contain VERB cana-4553 42 9 uncertainty uncertainty NOUN cana-4553 42 10 and and CCONJ cana-4553 42 11 noise noise NOUN cana-4553 42 12 , , PUNCT cana-4553 42 13 which which PRON cana-4553 42 14 can can AUX cana-4553 42 15 lead lead VERB cana-4553 42 16 to to ADP cana-4553 42 17 instability instability NOUN cana-4553 42 18 in in ADP cana-4553 42 19 the the DET cana-4553 42 20 clustering clustering ADJ cana-4553 42 21 results result NOUN cana-4553 42 22 . . PUNCT cana-4553 43 1 by by ADP cana-4553 43 2 incorporating incorporate VERB cana-4553 43 3 a a DET cana-4553 43 4 probabilistic probabilistic ADJ cana-4553 43 5 approach approach NOUN cana-4553 43 6 into into ADP cana-4553 43 7 k k PROPN cana-4553 43 8 - - PUNCT cana-4553 43 9 means means NOUN cana-4553 43 10 , , PUNCT cana-4553 43 11 we we PRON cana-4553 43 12 can can AUX cana-4553 43 13 account account VERB cana-4553 43 14 for for ADP cana-4553 43 15 this this DET cana-4553 43 16 uncertainty uncertainty NOUN cana-4553 43 17 , , PUNCT cana-4553 43 18 leading lead VERB cana-4553 43 19 to to ADP cana-4553 43 20 more more ADV cana-4553 43 21 robust robust ADJ cana-4553 43 22 and and CCONJ cana-4553 43 23 reliable reliable ADJ cana-4553 43 24 clustering clustering NOUN cana-4553 43 25 . . PUNCT cana-4553 44 1 before before ADP cana-4553 44 2 delving delve VERB cana-4553 44 3 into into ADP cana-4553 44 4 the the DET cana-4553 44 5 probabilistic probabilistic ADJ cana-4553 44 6 k k ADJ cana-4553 44 7 - - PUNCT cana-4553 44 8 means means NOUN cana-4553 44 9 algorithm algorithm NOUN cana-4553 44 10 , , PUNCT cana-4553 44 11 it it PRON cana-4553 44 12 's be AUX cana-4553 44 13 essential essential ADJ cana-4553 44 14 to to PART cana-4553 44 15 understand understand VERB cana-4553 44 16 probabilistic probabilistic ADJ cana-4553 44 17 metric metric ADJ cana-4553 44 18 spaces space NOUN cana-4553 44 19 ( ( PUNCT cana-4553 44 20 pms pm NOUN cana-4553 44 21 ) ) PUNCT cana-4553 44 22 . . PUNCT cana-4553 45 1 communications communication NOUN cana-4553 45 2 on on ADP cana-4553 45 3 applied apply VERB cana-4553 45 4 nonlinear nonlinear ADJ cana-4553 45 5 analysis analysis NOUN cana-4553 45 6 issn issn NOUN cana-4553 45 7 : : PUNCT cana-4553 45 8 1074 1074 NUM cana-4553 45 9 - - PUNCT cana-4553 45 10 133x 133x NUM cana-4553 45 11 vol vol NOUN cana-4553 45 12 32 32 NUM cana-4553 45 13 no no NOUN cana-4553 45 14 . . PUNCT cana-4553 46 1 9s 9s NUM cana-4553 46 2 ( ( PUNCT cana-4553 46 3 2025 2025 NUM cana-4553 46 4 ) ) PUNCT cana-4553 46 5 2756 2756 NUM cana-4553 47 1 https://internationalpubls.com https://internationalpubls.com X cana-4553 47 2 2 2 X cana-4553 47 3 . . X cana-4553 47 4 objectives objective NOUN cana-4553 47 5 by by ADP cana-4553 47 6 employing employ VERB cana-4553 47 7 the the DET cana-4553 47 8 jungck jungck NOUN cana-4553 47 9 contractive contractive ADJ cana-4553 47 10 statement statement NOUN cana-4553 47 11 , , PUNCT cana-4553 47 12 we we PRON cana-4553 47 13 ensure ensure VERB cana-4553 47 14 that that SCONJ cana-4553 47 15 the the DET cana-4553 47 16 probabilistic probabilistic ADJ cana-4553 47 17 gradient gradient ADJ cana-4553 47 18 descent descent NOUN cana-4553 47 19 algorithm algorithm NOUN cana-4553 47 20 converges converge VERB cana-4553 47 21 to to ADP cana-4553 47 22 an an DET cana-4553 47 23 optimal optimal ADJ cana-4553 47 24 solution solution NOUN cana-4553 47 25 , , PUNCT cana-4553 47 26 providing provide VERB cana-4553 47 27 robustness robustness NOUN cana-4553 47 28 against against ADP cana-4553 47 29 the the DET cana-4553 47 30 uncertainties uncertainty NOUN cana-4553 47 31 in in ADP cana-4553 47 32 the the DET cana-4553 47 33 training training NOUN cana-4553 47 34 data datum NOUN cana-4553 47 35 . . PUNCT cana-4553 48 1 3 3 X cana-4553 48 2 . . X cana-4553 48 3 methods method NOUN cana-4553 48 4 and and CCONJ cana-4553 48 5 preliminaries preliminary NOUN cana-4553 48 6 a. a. NOUN cana-4553 48 7 probabilistic probabilistic ADJ cana-4553 48 8 space space NOUN cana-4553 48 9 : : PUNCT cana-4553 48 10 a a DET cana-4553 48 11 probabilistic probabilistic ADJ cana-4553 48 12 distance distance NOUN cana-4553 48 13 ( ( PUNCT cana-4553 48 14 or or CCONJ cana-4553 48 15 metric metric ADJ cana-4553 48 16 ) ) PUNCT cana-4553 48 17 space space NOUN cana-4553 48 18 ( ( PUNCT cana-4553 48 19 pms pms PROPN cana-4553 48 20 ) ) PUNCT cana-4553 48 21 is be AUX cana-4553 48 22 a a DET cana-4553 48 23 generalization generalization NOUN cana-4553 48 24 of of ADP cana-4553 48 25 a a DET cana-4553 48 26 metric metric ADJ cana-4553 48 27 space space NOUN cana-4553 48 28 where where SCONJ cana-4553 48 29 the the DET cana-4553 48 30 distance distance NOUN cana-4553 48 31 between between ADP cana-4553 48 32 any any DET cana-4553 48 33 two two NUM cana-4553 48 34 points point NOUN cana-4553 48 35 is be AUX cana-4553 48 36 not not PART cana-4553 48 37 a a DET cana-4553 48 38 single single ADJ cana-4553 48 39 real real ADJ cana-4553 48 40 number number NOUN cana-4553 48 41 but but CCONJ cana-4553 48 42 a a DET cana-4553 48 43 distribution distribution NOUN cana-4553 48 44 function function NOUN cana-4553 48 45 . . PUNCT cana-4553 49 1 formally formally ADV cana-4553 49 2 , , PUNCT cana-4553 49 3 a a DET cana-4553 49 4 pms pms NOUN cana-4553 49 5 is be AUX cana-4553 49 6 a a DET cana-4553 49 7 triplet triplet NOUN cana-4553 49 8 ( ( PUNCT cana-4553 49 9 𝑃 𝑃 NOUN cana-4553 49 10 , , PUNCT cana-4553 49 11 𝐹 𝐹 PROPN cana-4553 49 12 , , PUNCT cana-4553 49 13 𝑇 𝑇 PROPN cana-4553 49 14 ) ) PUNCT cana-4553 49 15 , , PUNCT cana-4553 49 16 where where SCONJ cana-4553 49 17 𝑃 𝑃 NOUN cana-4553 49 18 is be AUX cana-4553 49 19 a a DET cana-4553 49 20 set set NOUN cana-4553 49 21 , , PUNCT cana-4553 49 22 𝐹 𝐹 PROPN cana-4553 49 23 is be AUX cana-4553 49 24 the the DET cana-4553 49 25 set set NOUN cana-4553 49 26 of of ADP cana-4553 49 27 distribution distribution NOUN cana-4553 49 28 functions function NOUN cana-4553 49 29 on on ADP cana-4553 49 30 [ [ X cana-4553 49 31 0 0 NUM cana-4553 49 32 , , PUNCT cana-4553 49 33 ∞ ∞ PROPN cana-4553 49 34 ) ) PUNCT cana-4553 49 35 , , PUNCT cana-4553 49 36 and and CCONJ cana-4553 49 37 𝑇 𝑇 PROPN cana-4553 49 38 : : PUNCT cana-4553 49 39 𝑃 𝑃 VERB cana-4553 49 40 Γ— Γ— NOUN cana-4553 49 41 𝑃 𝑃 NOUN cana-4553 49 42 β†’ β†’ SYM cana-4553 49 43 𝐹is 𝐹is PROPN cana-4553 49 44 a a DET cana-4553 49 45 mapping mapping NOUN cana-4553 49 46 that that PRON cana-4553 49 47 satisfies satisfy VERB cana-4553 49 48 certain certain ADJ cana-4553 49 49 axioms axiom NOUN cana-4553 49 50 analogous analogous ADJ cana-4553 49 51 to to ADP cana-4553 49 52 those those PRON cana-4553 49 53 of of ADP cana-4553 49 54 a a DET cana-4553 49 55 metric metric ADJ cana-4553 49 56 space space NOUN cana-4553 49 57 . . PUNCT cana-4553 50 1 definition definition NOUN cana-4553 50 2 3.1 3.1 NUM cana-4553 50 3 : : PUNCT cana-4553 51 1 [ [ X cana-4553 51 2 11 11 NUM cana-4553 51 3 ] ] PUNCT cana-4553 51 4 a a DET cana-4553 51 5 distribution distribution NOUN cana-4553 51 6 function function NOUN cana-4553 51 7 𝐹 𝐹 PROPN cana-4553 51 8 on on ADP cana-4553 51 9 [ [ X cana-4553 51 10 0 0 NUM cana-4553 51 11 , , PUNCT cana-4553 51 12 ∞ ∞ NUM cana-4553 51 13 ) ) PUNCT cana-4553 51 14 is be AUX cana-4553 51 15 a a DET cana-4553 51 16 non non ADJ cana-4553 51 17 - - ADJ cana-4553 51 18 decreasing decrease VERB cana-4553 51 19 , , PUNCT cana-4553 51 20 left left ADJ cana-4553 51 21 - - PUNCT cana-4553 51 22 continuous continuous ADJ cana-4553 51 23 function function NOUN cana-4553 51 24 , , PUNCT cana-4553 51 25 such such ADJ cana-4553 51 26 that that SCONJ cana-4553 51 27 , , PUNCT cana-4553 51 28 𝐹(0 𝐹(0 X cana-4553 51 29 ) ) PUNCT cana-4553 51 30 = = SYM cana-4553 51 31 0 0 NUM cana-4553 51 32 , , PUNCT cana-4553 51 33 and and CCONJ cana-4553 51 34 π‘™π‘–π‘š π‘™π‘–π‘š VERB cana-4553 51 35 π‘›β†’βˆž π‘›β†’βˆž NUM cana-4553 51 36 𝐹(π‘₯ 𝐹(π‘₯ NUM cana-4553 51 37 ) ) PUNCT cana-4553 51 38 = = SYM cana-4553 52 1 1 1 X cana-4553 52 2 . . PUNCT cana-4553 53 1 the the DET cana-4553 53 2 function function NOUN cana-4553 53 3 𝑇(π‘₯ 𝑇(π‘₯ PROPN cana-4553 53 4 , , PUNCT cana-4553 53 5 𝑦 𝑦 NUM cana-4553 53 6 ) ) PUNCT cana-4553 53 7 gives give VERB cana-4553 53 8 the the DET cana-4553 53 9 probability probability NOUN cana-4553 53 10 that that SCONJ cana-4553 53 11 the the DET cana-4553 53 12 distance distance NOUN cana-4553 53 13 between between ADP cana-4553 53 14 π‘₯ π‘₯ PROPN cana-4553 53 15 and and CCONJ cana-4553 53 16 𝑦is 𝑦is PRON cana-4553 53 17 less less ADJ cana-4553 53 18 than than ADP cana-4553 53 19 or or CCONJ cana-4553 53 20 equal equal ADJ cana-4553 53 21 to to ADP cana-4553 53 22 a a DET cana-4553 53 23 given give VERB cana-4553 53 24 value value NOUN cana-4553 53 25 . . PUNCT cana-4553 54 1 this this DET cana-4553 54 2 probabilistic probabilistic ADJ cana-4553 54 3 approach approach NOUN cana-4553 54 4 is be AUX cana-4553 54 5 particularly particularly ADV cana-4553 54 6 useful useful ADJ cana-4553 54 7 in in ADP cana-4553 54 8 modelling model VERB cana-4553 54 9 uncertainties uncertainty NOUN cana-4553 54 10 in in ADP cana-4553 54 11 data datum NOUN cana-4553 54 12 points point NOUN cana-4553 54 13 . . PUNCT cana-4553 55 1 definition definition NOUN cana-4553 55 2 3.2 3.2 NUM cana-4553 55 3 : : PUNCT cana-4553 56 1 [ [ X cana-4553 56 2 11 11 NUM cana-4553 56 3 ] ] PUNCT cana-4553 56 4 a a DET cana-4553 56 5 triangular triangular NOUN cana-4553 56 6 norm norm NOUN cana-4553 56 7 ( ( PUNCT cana-4553 56 8 t t NOUN cana-4553 56 9 - - PUNCT cana-4553 56 10 norm norm NOUN cana-4553 56 11 ) ) PUNCT cana-4553 56 12 can can AUX cana-4553 56 13 be be AUX cana-4553 56 14 represented represent VERB cana-4553 56 15 as as ADP cana-4553 56 16 a a DET cana-4553 56 17 mapping mapping NOUN cana-4553 56 18 𝑑 𝑑 NOUN cana-4553 56 19 : : PUNCT cana-4553 57 1 [ [ X cana-4553 57 2 0,1 0,1 NUM cana-4553 57 3 ] ] X cana-4553 57 4 Γ— Γ— NOUN cana-4553 58 1 [ [ X cana-4553 58 2 0,1 0,1 NUM cana-4553 58 3 ] ] PUNCT cana-4553 58 4 β†’ β†’ PUNCT cana-4553 58 5 [ [ X cana-4553 58 6 0,1 0,1 NUM cana-4553 58 7 ] ] X cana-4553 58 8 satisfying satisfying NOUN cana-4553 58 9 : : PUNCT cana-4553 58 10 i. i. PROPN cana-4553 58 11 t(π‘Ž t(π‘Ž PROPN cana-4553 58 12 , , PUNCT cana-4553 58 13 1 1 NUM cana-4553 58 14 ) ) PUNCT cana-4553 58 15 = = NOUN cana-4553 59 1 π‘Ž π‘Ž NOUN cana-4553 59 2 for for ADP cana-4553 59 3 every every DET cana-4553 59 4 π‘Ž π‘Ž PRON cana-4553 59 5 ∈ ∈ NOUN cana-4553 59 6 [ [ X cana-4553 59 7 0,1 0,1 NUM cana-4553 59 8 ] ] X cana-4553 59 9 ii ii NOUN cana-4553 59 10 . . PUNCT cana-4553 60 1 t(π‘Ž t(π‘Ž NOUN cana-4553 60 2 , , PUNCT cana-4553 60 3 𝑏 𝑏 NOUN cana-4553 60 4 ) ) PUNCT cana-4553 60 5 = = PUNCT cana-4553 60 6 t(𝑏 t(𝑏 NOUN cana-4553 60 7 , , PUNCT cana-4553 60 8 π‘Ž π‘Ž NOUN cana-4553 60 9 ) ) PUNCT cana-4553 60 10 for for ADP cana-4553 60 11 every every DET cana-4553 60 12 π‘Ž π‘Ž NOUN cana-4553 60 13 , , PUNCT cana-4553 60 14 𝑏 𝑏 PROPN cana-4553 60 15 ∈ ∈ PROPN cana-4553 60 16 [ [ X cana-4553 60 17 0,1 0,1 NUM cana-4553 60 18 ] ] X cana-4553 60 19 iii iii X cana-4553 60 20 . . PUNCT cana-4553 60 21 t(π‘Ž t(π‘Ž NOUN cana-4553 60 22 , , PUNCT cana-4553 60 23 𝑐 𝑐 NOUN cana-4553 60 24 ) ) PUNCT cana-4553 60 25 = = SYM cana-4553 61 1 t(𝑏 t(𝑏 NOUN cana-4553 61 2 , , PUNCT cana-4553 61 3 𝑑 𝑑 NOUN cana-4553 61 4 ) ) PUNCT cana-4553 61 5 if if SCONJ cana-4553 61 6 π‘Ž π‘Ž PRON cana-4553 61 7 β‰₯ β‰₯ NOUN cana-4553 61 8 𝑏 𝑏 NOUN cana-4553 61 9 , , PUNCT cana-4553 61 10 𝑐 𝑐 PROPN cana-4553 61 11 β‰₯ β‰₯ NUM cana-4553 61 12 𝑑 𝑑 NOUN cana-4553 61 13 iv iv NOUN cana-4553 61 14 . . PUNCT cana-4553 62 1 t(π‘Ž t(π‘Ž NOUN cana-4553 62 2 , , PUNCT cana-4553 62 3 t(𝑏 t(𝑏 NOUN cana-4553 62 4 , , PUNCT cana-4553 62 5 𝑐 𝑐 NOUN cana-4553 62 6 ) ) PUNCT cana-4553 62 7 ) ) PUNCT cana-4553 63 1 = = SYM cana-4553 63 2 t(t(π‘Ž t(t(π‘Ž NUM cana-4553 63 3 , , PUNCT cana-4553 63 4 𝑏 𝑏 NOUN cana-4553 63 5 ) ) PUNCT cana-4553 63 6 , , PUNCT cana-4553 63 7 𝑐 𝑐 NOUN cana-4553 63 8 ) ) PUNCT cana-4553 63 9 ; ; PUNCT cana-4553 63 10 π‘Ž π‘Ž X cana-4553 63 11 , , PUNCT cana-4553 63 12 𝑏,𝑐 𝑏,𝑐 NOUN cana-4553 63 13 ∈ ∈ PROPN cana-4553 64 1 [ [ X cana-4553 64 2 0,1 0,1 NUM cana-4553 64 3 ] ] PUNCT cana-4553 64 4 . . PUNCT cana-4553 65 1 b. b. PROPN cana-4553 65 2 axioms axioms PROPN cana-4553 65 3 of of ADP cana-4553 65 4 pms pms PROPN cana-4553 65 5 : : PUNCT cana-4553 65 6 i. i. NOUN cana-4553 65 7 non non PROPN cana-4553 65 8 - - NOUN cana-4553 65 9 negativity negativity ADJ cana-4553 65 10 : : PUNCT cana-4553 65 11 𝑇(π‘₯ 𝑇(π‘₯ NUM cana-4553 65 12 , , PUNCT cana-4553 65 13 𝑦 𝑦 NOUN cana-4553 65 14 ) ) PUNCT cana-4553 65 15 = = SYM cana-4553 65 16 πœ–0 πœ–0 PROPN cana-4553 65 17 ( ( PUNCT cana-4553 65 18 where where SCONJ cana-4553 65 19 πœ–0 πœ–0 PROPN cana-4553 65 20 is be AUX cana-4553 65 21 the the DET cana-4553 65 22 degenerate degenerate ADJ cana-4553 65 23 distribution distribution NOUN cana-4553 65 24 at at ADP cana-4553 65 25 0 0 NUM cana-4553 65 26 ) ) PUNCT cana-4553 65 27 , , PUNCT cana-4553 65 28 iff iff PROPN cana-4553 65 29 , , PUNCT cana-4553 65 30 π‘₯ π‘₯ PROPN cana-4553 66 1 = = PROPN cana-4553 66 2 𝑦. 𝑦. PROPN cana-4553 66 3 ii ii PROPN cana-4553 66 4 . . PUNCT cana-4553 66 5 symmetry symmetry PROPN cana-4553 66 6 : : PUNCT cana-4553 66 7 𝑇(π‘₯ 𝑇(π‘₯ NUM cana-4553 66 8 , , PUNCT cana-4553 66 9 𝑦 𝑦 NOUN cana-4553 66 10 ) ) PUNCT cana-4553 66 11 = = SYM cana-4553 66 12 𝑇(𝑦 𝑇(𝑦 PROPN cana-4553 66 13 , , PUNCT cana-4553 66 14 π‘₯ π‘₯ NOUN cana-4553 66 15 ) ) PUNCT cana-4553 66 16 , , PUNCT cana-4553 66 17 π‘“π‘œπ‘Ÿ π‘“π‘œπ‘Ÿ ADV cana-4553 66 18 π‘Žπ‘™π‘™ π‘Žπ‘™π‘™ VERB cana-4553 66 19 π‘₯ π‘₯ PRON cana-4553 66 20 ∈ ∈ PROPN cana-4553 67 1 π‘ƒπ‘Žπ‘›π‘‘ π‘ƒπ‘Žπ‘›π‘‘ PROPN cana-4553 67 2 𝑦 𝑦 NOUN cana-4553 67 3 ∈ ∈ PROPN cana-4553 67 4 𝑃. 𝑃. PROPN cana-4553 67 5 iii iii PROPN cana-4553 67 6 . . PUNCT cana-4553 68 1 triangle triangle PROPN cana-4553 68 2 inequality inequality NOUN cana-4553 68 3 : : PUNCT cana-4553 68 4 for for ADP cana-4553 68 5 all all PRON cana-4553 68 6 π‘₯ π‘₯ PROPN cana-4553 68 7 , , PUNCT cana-4553 68 8 𝑦 𝑦 NOUN cana-4553 68 9 , , PUNCT cana-4553 68 10 𝑧 𝑧 X cana-4553 68 11 in in ADP cana-4553 68 12 𝑃 𝑃 NOUN cana-4553 68 13 and and CCONJ cana-4553 68 14 for for ADP cana-4553 68 15 all all DET cana-4553 68 16 𝑑 𝑑 PROPN cana-4553 68 17 β‰₯ β‰₯ NOUN cana-4553 68 18 0 0 NUM cana-4553 68 19 , , PUNCT cana-4553 68 20 𝑇(π‘₯ 𝑇(π‘₯ NOUN cana-4553 68 21 , , PUNCT cana-4553 68 22 𝑧)(𝑑 𝑧)(𝑑 ADJ cana-4553 68 23 ) ) PUNCT cana-4553 68 24 β‰₯ β‰₯ NOUN cana-4553 68 25 sup𝑒+𝑣=𝑑 sup𝑒+𝑣=𝑑 NOUN cana-4553 68 26 π‘šπ‘–π‘› π‘šπ‘–π‘› PROPN cana-4553 68 27 { { PUNCT cana-4553 68 28 𝑇(π‘₯ 𝑇(π‘₯ PROPN cana-4553 68 29 , , PUNCT cana-4553 68 30 𝑦)(𝑒 𝑦)(𝑒 NOUN cana-4553 68 31 ) ) PUNCT cana-4553 68 32 , , PUNCT cana-4553 68 33 𝑇(𝑦 𝑇(𝑦 PROPN cana-4553 68 34 , , PUNCT cana-4553 68 35 𝑧)(𝑣 𝑧)(𝑣 NOUN cana-4553 68 36 ) ) PUNCT cana-4553 68 37 } } PUNCT cana-4553 68 38 . . PUNCT cana-4553 69 1 these these DET cana-4553 69 2 axioms axiom NOUN cana-4553 69 3 ensure ensure VERB cana-4553 69 4 that that SCONJ cana-4553 69 5 the the DET cana-4553 69 6 probabilistic probabilistic ADJ cana-4553 69 7 metric metric NOUN cana-4553 69 8 retains retain VERB cana-4553 69 9 the the DET cana-4553 69 10 essential essential ADJ cana-4553 69 11 properties property NOUN cana-4553 69 12 of of ADP cana-4553 69 13 a a DET cana-4553 69 14 classical classical ADJ cana-4553 69 15 metric metric NOUN cana-4553 69 16 , , PUNCT cana-4553 69 17 allowing allow VERB cana-4553 69 18 us we PRON cana-4553 69 19 to to PART cana-4553 69 20 extend extend VERB cana-4553 69 21 various various ADJ cana-4553 69 22 analytical analytical ADJ cana-4553 69 23 techniques technique NOUN cana-4553 69 24 to to ADP cana-4553 69 25 the the DET cana-4553 69 26 probabilistic probabilistic ADJ cana-4553 69 27 domain domain NOUN cana-4553 69 28 . . PUNCT cana-4553 70 1 c. c. PROPN cana-4553 70 2 jungck jungck PROPN cana-4553 70 3 contractive contractive ADJ cana-4553 70 4 condition condition NOUN cana-4553 70 5 : : PUNCT cana-4553 70 6 the the DET cana-4553 70 7 jungck jungck PROPN cana-4553 70 8 contraction contraction NOUN cana-4553 70 9 theorem theorem PROPN cana-4553 70 10 extends extend VERB cana-4553 70 11 the the DET cana-4553 70 12 classical classical ADJ cana-4553 70 13 banach banach NOUN cana-4553 70 14 contraction contraction NOUN cana-4553 70 15 principle principle NOUN cana-4553 70 16 to to ADP cana-4553 70 17 the the DET cana-4553 70 18 setting setting NOUN cana-4553 70 19 of of ADP cana-4553 70 20 probabilistic probabilistic ADJ cana-4553 70 21 metric metric ADJ cana-4553 70 22 spaces space NOUN cana-4553 70 23 . . PUNCT cana-4553 71 1 this this DET cana-4553 71 2 theorem theorem NOUN cana-4553 71 3 is be AUX cana-4553 71 4 pivotal pivotal ADJ cana-4553 71 5 for for ADP cana-4553 71 6 proving prove VERB cana-4553 71 7 the the DET cana-4553 71 8 existence existence NOUN cana-4553 71 9 and and CCONJ cana-4553 71 10 uniqueness uniqueness NOUN cana-4553 71 11 of of ADP cana-4553 71 12 fixed fix VERB cana-4553 71 13 points point NOUN cana-4553 71 14 in in ADP cana-4553 71 15 probabilistic probabilistic ADJ cana-4553 71 16 settings setting NOUN cana-4553 71 17 , , PUNCT cana-4553 71 18 which which PRON cana-4553 71 19 is be AUX cana-4553 71 20 crucial crucial ADJ cana-4553 71 21 for for ADP cana-4553 71 22 the the DET cana-4553 71 23 convergence convergence NOUN cana-4553 71 24 analysis analysis NOUN cana-4553 71 25 of of ADP cana-4553 71 26 ml ml ADP cana-4553 71 27 algorithms algorithm NOUN cana-4553 71 28 . . PUNCT cana-4553 72 1 theorem theorem VERB cana-4553 72 2 3.1 3.1 NUM cana-4553 72 3 ( ( PUNCT cana-4553 72 4 jungck jungck NOUN cana-4553 72 5 contraction contraction NOUN cana-4553 72 6 theorem theorem VERB cana-4553 72 7 ): ): PUNCT cana-4553 72 8 [ [ X cana-4553 72 9 4 4 X cana-4553 72 10 ] ] X cana-4553 72 11 let let VERB cana-4553 72 12 ( ( PUNCT cana-4553 72 13 𝑃 𝑃 VERB cana-4553 72 14 , , PUNCT cana-4553 72 15 𝐹 𝐹 PROPN cana-4553 72 16 , , PUNCT cana-4553 72 17 𝑇 𝑇 PROPN cana-4553 72 18 ) ) PUNCT cana-4553 72 19 be be VERB cana-4553 72 20 a a DET cana-4553 72 21 complete complete ADJ cana-4553 72 22 probabilistic probabilistic ADJ cana-4553 72 23 metric metric ADJ cana-4553 72 24 space space NOUN cana-4553 72 25 . . PUNCT cana-4553 73 1 suppose suppose VERB cana-4553 73 2 𝑓 𝑓 DET cana-4553 73 3 , , PUNCT cana-4553 73 4 𝑔 𝑔 ADJ cana-4553 73 5 : : PUNCT cana-4553 73 6 𝑃 𝑃 PROPN cana-4553 73 7 β†’ β†’ SYM cana-4553 73 8 𝑃are 𝑃are NOUN cana-4553 73 9 two two NUM cana-4553 73 10 mappings mapping NOUN cana-4553 73 11 satisfying satisfying ADJ cana-4553 73 12 : : PUNCT cana-4553 73 13 𝑇(𝑓(π‘₯ 𝑇(𝑓(π‘₯ NOUN cana-4553 73 14 ) ) PUNCT cana-4553 73 15 , , PUNCT cana-4553 73 16 𝑓(𝑦))(𝑑 𝑓(𝑦))(𝑑 PROPN cana-4553 73 17 ) ) PUNCT cana-4553 73 18 β‰₯ β‰₯ NOUN cana-4553 73 19 𝛽 𝛽 NOUN cana-4553 73 20 𝑇(𝑔(π‘₯ 𝑇(𝑔(π‘₯ NUM cana-4553 73 21 ) ) PUNCT cana-4553 73 22 , , PUNCT cana-4553 73 23 𝑔(𝑦))(𝑑 𝑔(𝑦))(𝑑 PROPN cana-4553 73 24 ) ) PUNCT cana-4553 73 25 , , PUNCT cana-4553 73 26 for for ADP cana-4553 73 27 all all DET cana-4553 73 28 π‘₯ π‘₯ PROPN cana-4553 73 29 , , PUNCT cana-4553 73 30 𝑦 𝑦 PRON cana-4553 73 31 ∈ ∈ NOUN cana-4553 73 32 𝑃 𝑃 NOUN cana-4553 73 33 and and CCONJ cana-4553 73 34 for for ADP cana-4553 73 35 some some DET cana-4553 73 36 constant constant ADJ cana-4553 73 37 𝛽 𝛽 PROPN cana-4553 73 38 ∈ ∈ PROPN cana-4553 73 39 ( ( PUNCT cana-4553 73 40 0,1).then,𝑓 0,1).then,𝑓 NOUN cana-4553 73 41 and and CCONJ cana-4553 73 42 𝑔 𝑔 PROPN cana-4553 73 43 have have VERB cana-4553 73 44 a a DET cana-4553 73 45 unique unique ADJ cana-4553 73 46 common common ADJ cana-4553 73 47 invariant invariant ADJ cana-4553 73 48 point point NOUN cana-4553 73 49 in in ADP cana-4553 73 50 𝑃. 𝑃. PROPN cana-4553 73 51 this this DET cana-4553 73 52 result result NOUN cana-4553 73 53 ensures ensure VERB cana-4553 73 54 that that SCONJ cana-4553 73 55 under under ADP cana-4553 73 56 a a DET cana-4553 73 57 contraction contraction NOUN cana-4553 73 58 condition condition NOUN cana-4553 73 59 , , PUNCT cana-4553 73 60 iterative iterative ADJ cana-4553 73 61 applications application NOUN cana-4553 73 62 of of ADP cana-4553 73 63 the the DET cana-4553 73 64 mappings mapping NOUN cana-4553 73 65 𝑓 𝑓 PRON cana-4553 73 66 and and CCONJ cana-4553 73 67 𝑔 𝑔 PROPN cana-4553 73 68 will will AUX cana-4553 73 69 converge converge VERB cana-4553 73 70 to to ADP cana-4553 73 71 a a DET cana-4553 73 72 common common ADJ cana-4553 73 73 invariant invariant ADJ cana-4553 73 74 point point NOUN cana-4553 73 75 . . PUNCT cana-4553 74 1 this this DET cana-4553 74 2 result result NOUN cana-4553 74 3 is be AUX cana-4553 74 4 useful useful ADJ cana-4553 74 5 in in ADP cana-4553 74 6 analyzing analyze VERB cana-4553 74 7 machine machine NOUN cana-4553 74 8 learning learning NOUN cana-4553 74 9 based base VERB cana-4553 74 10 algorithms algorithm NOUN cana-4553 74 11 where where SCONJ cana-4553 74 12 such such ADJ cana-4553 74 13 mappings mapping NOUN cana-4553 74 14 often often ADV cana-4553 74 15 represent represent VERB cana-4553 74 16 iterative iterative NOUN cana-4553 74 17 update update NOUN cana-4553 74 18 steps step NOUN cana-4553 74 19 , , PUNCT cana-4553 74 20 for for ADP cana-4553 74 21 example example NOUN cana-4553 74 22 , , PUNCT cana-4553 74 23 in in ADP cana-4553 74 24 analysing analyse VERB cana-4553 74 25 the the DET cana-4553 74 26 convergence convergence NOUN cana-4553 74 27 of of ADP cana-4553 74 28 probabilistic probabilistic ADJ cana-4553 74 29 k k PROPN cana-4553 74 30 - - PUNCT cana-4553 74 31 means means NOUN cana-4553 74 32 clustering clustering NOUN cana-4553 74 33 . . PUNCT cana-4553 75 1 communications communication NOUN cana-4553 75 2 on on ADP cana-4553 75 3 applied apply VERB cana-4553 75 4 nonlinear nonlinear ADJ cana-4553 75 5 analysis analysis NOUN cana-4553 75 6 issn issn NOUN cana-4553 75 7 : : PUNCT cana-4553 75 8 1074 1074 NUM cana-4553 75 9 - - PUNCT cana-4553 75 10 133x 133x NUM cana-4553 75 11 vol vol NOUN cana-4553 75 12 32 32 NUM cana-4553 75 13 no no NOUN cana-4553 75 14 . . PUNCT cana-4553 76 1 9s 9s NUM cana-4553 76 2 ( ( PUNCT cana-4553 76 3 2025 2025 NUM cana-4553 76 4 ) ) PUNCT cana-4553 76 5 2757 2757 NUM cana-4553 76 6 https://internationalpubls.com https://internationalpubls.com X cana-4553 76 7 application application NOUN cana-4553 76 8 in in ADP cana-4553 76 9 machine machine NOUN cana-4553 76 10 learning learning NOUN cana-4553 76 11 : : PUNCT cana-4553 76 12 many many ADJ cana-4553 76 13 algorithms algorithm NOUN cana-4553 76 14 , , PUNCT cana-4553 76 15 in in ADP cana-4553 76 16 machine machine NOUN cana-4553 76 17 learning learning NOUN cana-4553 76 18 , , PUNCT cana-4553 76 19 can can AUX cana-4553 76 20 be be AUX cana-4553 76 21 viewed view VERB cana-4553 76 22 as as ADP cana-4553 76 23 iterative iterative NOUN cana-4553 76 24 processes process NOUN cana-4553 76 25 that that PRON cana-4553 76 26 seek seek VERB cana-4553 76 27 to to PART cana-4553 76 28 minimize minimize VERB cana-4553 76 29 an an DET cana-4553 76 30 objective objective ADJ cana-4553 76 31 function function NOUN cana-4553 76 32 or or CCONJ cana-4553 76 33 optimize optimize VERB cana-4553 76 34 a a DET cana-4553 76 35 model model NOUN cana-4553 76 36 . . PUNCT cana-4553 77 1 the the DET cana-4553 77 2 convergence convergence NOUN cana-4553 77 3 of of ADP cana-4553 77 4 these these DET cana-4553 77 5 processes process NOUN cana-4553 77 6 is be AUX cana-4553 77 7 crucial crucial ADJ cana-4553 77 8 for for ADP cana-4553 77 9 the the DET cana-4553 77 10 reliability reliability NOUN cana-4553 77 11 and and CCONJ cana-4553 77 12 stability stability NOUN cana-4553 77 13 of of ADP cana-4553 77 14 the the DET cana-4553 77 15 algorithms algorithm NOUN cana-4553 77 16 . . PUNCT cana-4553 78 1 by by ADP cana-4553 78 2 framing frame VERB cana-4553 78 3 these these DET cana-4553 78 4 iterative iterative NOUN cana-4553 78 5 processes process NOUN cana-4553 78 6 within within ADP cana-4553 78 7 the the DET cana-4553 78 8 context context NOUN cana-4553 78 9 of of ADP cana-4553 78 10 probabilistic probabilistic ADJ cana-4553 78 11 metric metric ADJ cana-4553 78 12 spaces space NOUN cana-4553 78 13 , , PUNCT cana-4553 78 14 we we PRON cana-4553 78 15 can can AUX cana-4553 78 16 leverage leverage VERB cana-4553 78 17 the the DET cana-4553 78 18 jungck jungck NOUN cana-4553 78 19 contractive contractive ADJ cana-4553 78 20 condition condition NOUN cana-4553 78 21 to to PART cana-4553 78 22 assure assure VERB cana-4553 78 23 convergence convergence NOUN cana-4553 78 24 even even ADV cana-4553 78 25 in in ADP cana-4553 78 26 the the DET cana-4553 78 27 presence presence NOUN cana-4553 78 28 of of ADP cana-4553 78 29 uncertainty uncertainty NOUN cana-4553 78 30 . . PUNCT cana-4553 79 1 k k X cana-4553 79 2 - - PUNCT cana-4553 79 3 means mean VERB cana-4553 79 4 clustering cluster VERB cana-4553 79 5 in in ADP cana-4553 79 6 probabilistic probabilistic ADJ cana-4553 79 7 space space NOUN cana-4553 79 8 - - PUNCT cana-4553 79 9 a a DET cana-4553 79 10 case case NOUN cana-4553 79 11 study study NOUN cana-4553 79 12 : : PUNCT cana-4553 79 13 the the DET cana-4553 79 14 k k NOUN cana-4553 79 15 - - PUNCT cana-4553 79 16 means mean VERB cana-4553 79 17 clustering clustering ADJ cana-4553 79 18 algorithm algorithm NOUN cana-4553 79 19 is be AUX cana-4553 79 20 a a DET cana-4553 79 21 widely widely ADV cana-4553 79 22 used use VERB cana-4553 79 23 method method NOUN cana-4553 79 24 for for ADP cana-4553 79 25 partitioning partition VERB cana-4553 79 26 data datum NOUN cana-4553 79 27 into into ADP cana-4553 79 28 clusters cluster NOUN cana-4553 79 29 . . PUNCT cana-4553 80 1 however however ADV cana-4553 80 2 , , PUNCT cana-4553 80 3 in in ADP cana-4553 80 4 the the DET cana-4553 80 5 presence presence NOUN cana-4553 80 6 of of ADP cana-4553 80 7 uncertainty uncertainty NOUN cana-4553 80 8 and and CCONJ cana-4553 80 9 noise noise NOUN cana-4553 80 10 , , PUNCT cana-4553 80 11 the the DET cana-4553 80 12 classical classical ADJ cana-4553 80 13 k k ADJ cana-4553 80 14 - - PUNCT cana-4553 80 15 means means NOUN cana-4553 80 16 algorithm algorithm NOUN cana-4553 80 17 may may AUX cana-4553 80 18 struggle struggle VERB cana-4553 80 19 to to PART cana-4553 80 20 find find VERB cana-4553 80 21 stable stable ADJ cana-4553 80 22 clusters cluster NOUN cana-4553 80 23 . . PUNCT cana-4553 81 1 by by ADP cana-4553 81 2 adopting adopt VERB cana-4553 81 3 a a DET cana-4553 81 4 probabilistic probabilistic ADJ cana-4553 81 5 approach approach NOUN cana-4553 81 6 , , PUNCT cana-4553 81 7 we we PRON cana-4553 81 8 can can AUX cana-4553 81 9 enhance enhance VERB cana-4553 81 10 the the DET cana-4553 81 11 robustness robustness NOUN cana-4553 81 12 of of ADP cana-4553 81 13 k k NOUN cana-4553 81 14 - - PUNCT cana-4553 81 15 means means NOUN cana-4553 81 16 clustering clustering NOUN cana-4553 81 17 . . PUNCT cana-4553 82 1 probabilistic probabilistic ADJ cana-4553 82 2 distance distance NOUN cana-4553 82 3 metric metric NOUN cana-4553 82 4 : : PUNCT cana-4553 82 5 instead instead ADV cana-4553 82 6 of of ADP cana-4553 82 7 using use VERB cana-4553 82 8 a a DET cana-4553 82 9 deterministic deterministic ADJ cana-4553 82 10 distance distance NOUN cana-4553 82 11 , , PUNCT cana-4553 82 12 we we PRON cana-4553 82 13 define define VERB cana-4553 82 14 a a DET cana-4553 82 15 probabilistic probabilistic ADJ cana-4553 82 16 distance distance NOUN cana-4553 82 17 between between ADP cana-4553 82 18 points point NOUN cana-4553 82 19 . . PUNCT cana-4553 83 1 this this DET cana-4553 83 2 distance distance NOUN cana-4553 83 3 is be AUX cana-4553 83 4 modelled model VERB cana-4553 83 5 as as ADP cana-4553 83 6 a a DET cana-4553 83 7 distribution distribution NOUN cana-4553 83 8 function function NOUN cana-4553 83 9 capturing capture VERB cana-4553 83 10 the the DET cana-4553 83 11 uncertainty uncertainty NOUN cana-4553 83 12 in in ADP cana-4553 83 13 the the DET cana-4553 83 14 data datum NOUN cana-4553 83 15 . . PUNCT cana-4553 84 1 probabilistic probabilistic ADJ cana-4553 84 2 centroid centroid NOUN cana-4553 84 3 update update NOUN cana-4553 84 4 : : PUNCT cana-4553 84 5 the the DET cana-4553 84 6 update update NOUN cana-4553 84 7 step step NOUN cana-4553 84 8 for for ADP cana-4553 84 9 centroids centroid NOUN cana-4553 84 10 in in ADP cana-4553 84 11 the the DET cana-4553 84 12 classical classical ADJ cana-4553 84 13 k k NOUN cana-4553 84 14 - - PUNCT cana-4553 84 15 means means NOUN cana-4553 84 16 is be AUX cana-4553 84 17 replaced replace VERB cana-4553 84 18 by by ADP cana-4553 84 19 a a DET cana-4553 84 20 probabilistic probabilistic ADJ cana-4553 84 21 update update NOUN cana-4553 84 22 . . PUNCT cana-4553 85 1 given give VERB cana-4553 85 2 a a DET cana-4553 85 3 cluster cluster NOUN cana-4553 85 4 , , PUNCT cana-4553 85 5 the the DET cana-4553 85 6 new new ADJ cana-4553 85 7 centroid centroid NOUN cana-4553 85 8 is be AUX cana-4553 85 9 chosen choose VERB cana-4553 85 10 such such ADJ cana-4553 85 11 that that SCONJ cana-4553 85 12 it it PRON cana-4553 85 13 minimizes minimize VERB cana-4553 85 14 the the DET cana-4553 85 15 expected expect VERB cana-4553 85 16 probabilistic probabilistic ADJ cana-4553 85 17 distance distance NOUN cana-4553 85 18 to to ADP cana-4553 85 19 the the DET cana-4553 85 20 points point NOUN cana-4553 85 21 in in ADP cana-4553 85 22 the the DET cana-4553 85 23 cluster cluster NOUN cana-4553 85 24 . . PUNCT cana-4553 86 1 convergence convergence NOUN cana-4553 86 2 analysis analysis NOUN cana-4553 86 3 : : PUNCT cana-4553 86 4 using use VERB cana-4553 86 5 the the DET cana-4553 86 6 jungck jungck NOUN cana-4553 86 7 contraction contraction NOUN cana-4553 86 8 theorem theorem VERB cana-4553 86 9 , , PUNCT cana-4553 86 10 we we PRON cana-4553 86 11 can can AUX cana-4553 86 12 analyse analyse VERB cana-4553 86 13 the the DET cana-4553 86 14 convergence convergence NOUN cana-4553 86 15 of of ADP cana-4553 86 16 the the DET cana-4553 86 17 probabilistic probabilistic ADJ cana-4553 86 18 k k ADJ cana-4553 86 19 - - PUNCT cana-4553 86 20 means means NOUN cana-4553 86 21 algorithm algorithm NOUN cana-4553 86 22 . . PUNCT cana-4553 87 1 by by ADP cana-4553 87 2 ensuring ensure VERB cana-4553 87 3 that that SCONJ cana-4553 87 4 the the DET cana-4553 87 5 probabilistic probabilistic ADJ cana-4553 87 6 distance distance NOUN cana-4553 87 7 satisfies satisfy VERB cana-4553 87 8 the the DET cana-4553 87 9 contraction contraction NOUN cana-4553 87 10 condition condition NOUN cana-4553 87 11 , , PUNCT cana-4553 87 12 we we PRON cana-4553 87 13 can can AUX cana-4553 87 14 guarantee guarantee VERB cana-4553 87 15 that that SCONJ cana-4553 87 16 the the DET cana-4553 87 17 iterative iterative NOUN cana-4553 87 18 updates update NOUN cana-4553 87 19 of of ADP cana-4553 87 20 centroids centroid NOUN cana-4553 87 21 will will AUX cana-4553 87 22 converge converge VERB cana-4553 87 23 to to ADP cana-4553 87 24 a a DET cana-4553 87 25 stable stable ADJ cana-4553 87 26 configuration configuration NOUN cana-4553 87 27 . . PUNCT cana-4553 88 1 k k X cana-4553 88 2 - - PUNCT cana-4553 88 3 means mean VERB cana-4553 88 4 clustering cluster VERB cana-4553 88 5 algorithm algorithm NOUN cana-4553 88 6 in in ADP cana-4553 88 7 probabilistic probabilistic ADJ cana-4553 88 8 space space NOUN cana-4553 88 9 : : PUNCT cana-4553 88 10 the the DET cana-4553 88 11 probabilistic probabilistic ADJ cana-4553 88 12 k k NOUN cana-4553 88 13 - - PUNCT cana-4553 88 14 means means NOUN cana-4553 88 15 algorithm algorithm NOUN cana-4553 89 1 [ [ X cana-4553 89 2 7 7 X cana-4553 89 3 ] ] PUNCT cana-4553 89 4 extends extend VERB cana-4553 89 5 the the DET cana-4553 89 6 classical classical ADJ cana-4553 89 7 k k NOUN cana-4553 89 8 - - PUNCT cana-4553 89 9 means means NOUN cana-4553 89 10 by by ADP cana-4553 89 11 incorporating incorporate VERB cana-4553 89 12 probabilistic probabilistic ADJ cana-4553 89 13 distances distance NOUN cana-4553 89 14 , , PUNCT cana-4553 89 15 which which PRON cana-4553 89 16 are be AUX cana-4553 89 17 represented represent VERB cana-4553 89 18 by by ADP cana-4553 89 19 distribution distribution NOUN cana-4553 89 20 functions function NOUN cana-4553 89 21 . . PUNCT cana-4553 90 1 this this DET cana-4553 90 2 approach approach NOUN cana-4553 90 3 provides provide VERB cana-4553 90 4 a a DET cana-4553 90 5 more more ADV cana-4553 90 6 robust robust ADJ cana-4553 90 7 clustering clustering NOUN cana-4553 90 8 method method NOUN cana-4553 90 9 in in ADP cana-4553 90 10 the the DET cana-4553 90 11 presence presence NOUN cana-4553 90 12 of of ADP cana-4553 90 13 uncertainty uncertainty NOUN cana-4553 90 14 and and CCONJ cana-4553 90 15 noise noise NOUN cana-4553 90 16 . . PUNCT cana-4553 91 1 algorithm algorithm NOUN cana-4553 91 2 of of ADP cana-4553 91 3 k k X cana-4553 91 4 - - PUNCT cana-4553 91 5 means means NOUN cana-4553 91 6 in in ADP cana-4553 91 7 probabilistic probabilistic ADJ cana-4553 91 8 space space NOUN cana-4553 91 9 is be AUX cana-4553 91 10 as as SCONJ cana-4553 91 11 follows follow VERB cana-4553 91 12 : : PUNCT cana-4553 91 13 1 1 X cana-4553 91 14 . . X cana-4553 91 15 initialization initialization NOUN cana-4553 91 16 : : PUNCT cana-4553 91 17 randomly randomly ADV cana-4553 91 18 initialize initialize VERB cana-4553 91 19 π‘˜ π‘˜ DET cana-4553 91 20 probabilistic probabilistic ADJ cana-4553 91 21 centroids centroid NOUN cana-4553 91 22 , , PUNCT cana-4553 91 23 where where SCONJ cana-4553 91 24 each each DET cana-4553 91 25 centroid centroid NOUN cana-4553 91 26 is be AUX cana-4553 91 27 represented represent VERB cana-4553 91 28 by by ADP cana-4553 91 29 a a DET cana-4553 91 30 distribution distribution NOUN cana-4553 91 31 function function NOUN cana-4553 91 32 . . PUNCT cana-4553 92 1 2 2 X cana-4553 92 2 . . X cana-4553 92 3 assignment assignment NOUN cana-4553 92 4 step step NOUN cana-4553 92 5 : : PUNCT cana-4553 92 6 here here ADV cana-4553 92 7 , , PUNCT cana-4553 92 8 we we PRON cana-4553 92 9 assign assign VERB cana-4553 92 10 each each DET cana-4553 92 11 data data NOUN cana-4553 92 12 point point NOUN cana-4553 92 13 to to ADP cana-4553 92 14 the the DET cana-4553 92 15 cluster cluster NOUN cana-4553 92 16 whose whose DET cana-4553 92 17 centroid centroid NOUN cana-4553 92 18 minimizes minimize VERB cana-4553 92 19 the the DET cana-4553 92 20 probabilistic probabilistic ADJ cana-4553 92 21 distance distance NOUN cana-4553 92 22 . . PUNCT cana-4553 93 1 3 3 X cana-4553 93 2 . . X cana-4553 93 3 update update NOUN cana-4553 93 4 step step NOUN cana-4553 93 5 : : PUNCT cana-4553 93 6 for for ADP cana-4553 93 7 each each DET cana-4553 93 8 and and CCONJ cana-4553 93 9 every every PRON cana-4553 93 10 cluster cluster NOUN cana-4553 93 11 , , PUNCT cana-4553 93 12 update update VERB cana-4553 93 13 the the DET cana-4553 93 14 centroid centroid NOUN cana-4553 93 15 to to PART cana-4553 93 16 minimize minimize VERB cana-4553 93 17 the the DET cana-4553 93 18 expected expect VERB cana-4553 93 19 probabilistic probabilistic ADJ cana-4553 93 20 distance distance NOUN cana-4553 93 21 to to ADP cana-4553 93 22 the the DET cana-4553 93 23 points point NOUN cana-4553 93 24 in in ADP cana-4553 93 25 the the DET cana-4553 93 26 cluster cluster NOUN cana-4553 93 27 . . PUNCT cana-4553 94 1 4 4 X cana-4553 94 2 . . X cana-4553 94 3 convergence convergence NOUN cana-4553 94 4 check check NOUN cana-4553 94 5 : : PUNCT cana-4553 94 6 repeat repeat VERB cana-4553 94 7 the the DET cana-4553 94 8 assignment assignment NOUN cana-4553 94 9 and and CCONJ cana-4553 94 10 update update NOUN cana-4553 94 11 steps step NOUN cana-4553 94 12 until until SCONJ cana-4553 94 13 the the DET cana-4553 94 14 centroids centroid NOUN cana-4553 94 15 converge converge VERB cana-4553 94 16 . . PUNCT cana-4553 95 1 step step NOUN cana-4553 95 2 - - PUNCT cana-4553 95 3 by by ADP cana-4553 95 4 - - PUNCT cana-4553 95 5 step step NOUN cana-4553 95 6 explanation explanation NOUN cana-4553 95 7 : : PUNCT cana-4553 95 8 1 1 X cana-4553 95 9 . . X cana-4553 95 10 initialization initialization NOUN cana-4553 95 11 : : PUNCT cana-4553 95 12 randomly randomly ADV cana-4553 95 13 select select VERB cana-4553 95 14 π‘˜ π‘˜ DET cana-4553 95 15 points point NOUN cana-4553 95 16 from from ADP cana-4553 95 17 the the DET cana-4553 95 18 dataset dataset NOUN cana-4553 95 19 as as ADP cana-4553 95 20 initial initial ADJ cana-4553 95 21 centroids centroid NOUN cana-4553 95 22 . . PUNCT cana-4553 96 1 represent represent VERB cana-4553 96 2 each each DET cana-4553 96 3 centroid centroid NOUN cana-4553 96 4 as as ADP cana-4553 96 5 a a DET cana-4553 96 6 distribution distribution NOUN cana-4553 96 7 function function NOUN cana-4553 96 8 capturing capture VERB cana-4553 96 9 the the DET cana-4553 96 10 uncertainty uncertainty NOUN cana-4553 96 11 . . PUNCT cana-4553 97 1 2 2 X cana-4553 97 2 . . X cana-4553 97 3 assignment assignment NOUN cana-4553 97 4 step step NOUN cana-4553 97 5 : : PUNCT cana-4553 97 6 communications communication NOUN cana-4553 97 7 on on ADP cana-4553 97 8 applied apply VERB cana-4553 97 9 nonlinear nonlinear ADJ cana-4553 97 10 analysis analysis NOUN cana-4553 97 11 issn issn NOUN cana-4553 97 12 : : PUNCT cana-4553 97 13 1074 1074 NUM cana-4553 97 14 - - PUNCT cana-4553 97 15 133x 133x NUM cana-4553 97 16 vol vol NOUN cana-4553 97 17 32 32 NUM cana-4553 97 18 no no NOUN cana-4553 97 19 . . PUNCT cana-4553 98 1 9s 9s NUM cana-4553 98 2 ( ( PUNCT cana-4553 98 3 2025 2025 NUM cana-4553 98 4 ) ) PUNCT cana-4553 98 5 2758 2758 NUM cana-4553 98 6 https://internationalpubls.com https://internationalpubls.com X cana-4553 98 7 for for ADP cana-4553 98 8 each each DET cana-4553 98 9 data datum NOUN cana-4553 98 10 point point NOUN cana-4553 98 11 π‘₯𝑖 π‘₯𝑖 PROPN cana-4553 98 12 , , PUNCT cana-4553 98 13 compute compute VERB cana-4553 98 14 the the DET cana-4553 98 15 probabilistic probabilistic ADJ cana-4553 98 16 distance distance NOUN cana-4553 98 17 to to ADP cana-4553 98 18 each each PRON cana-4553 98 19 centroid centroid NOUN cana-4553 99 1 𝐢𝑗 𝐢𝑗 NOUN cana-4553 99 2 using use VERB cana-4553 99 3 the the DET cana-4553 99 4 distribution distribution NOUN cana-4553 99 5 function function NOUN cana-4553 99 6 𝑇(π‘₯𝑖 𝑇(π‘₯𝑖 VERB cana-4553 99 7 , , PUNCT cana-4553 99 8 𝐢𝑗 𝐢𝑗 PROPN cana-4553 99 9 ) ) PUNCT cana-4553 99 10 . . PUNCT cana-4553 100 1 assign assign NOUN cana-4553 100 2 π‘₯𝑖to π‘₯𝑖to PROPN cana-4553 101 1 the the DET cana-4553 101 2 cluster cluster NOUN cana-4553 101 3 with with ADP cana-4553 101 4 the the DET cana-4553 101 5 centroid centroid NOUN cana-4553 101 6 that that PRON cana-4553 101 7 has have VERB cana-4553 101 8 the the DET cana-4553 101 9 smallest small ADJ cana-4553 101 10 expected expect VERB cana-4553 101 11 probabilistic probabilistic ADJ cana-4553 101 12 distance distance NOUN cana-4553 101 13 . . PUNCT cana-4553 102 1 3 3 X cana-4553 102 2 . . X cana-4553 102 3 update update NOUN cana-4553 102 4 step step NOUN cana-4553 102 5 : : PUNCT cana-4553 102 6 for for ADP cana-4553 102 7 each each DET cana-4553 102 8 cluster cluster NOUN cana-4553 102 9 , , PUNCT cana-4553 102 10 𝐢𝑗 𝐢𝑗 PROPN cana-4553 102 11 , , PUNCT cana-4553 102 12 update update VERB cana-4553 102 13 the the DET cana-4553 102 14 centroid centroid NOUN cana-4553 102 15 to to ADP cana-4553 102 16 the the DET cana-4553 102 17 point point NOUN cana-4553 102 18 that that PRON cana-4553 102 19 minimizes minimize VERB cana-4553 102 20 the the DET cana-4553 102 21 expected expect VERB cana-4553 102 22 probabilistic probabilistic ADJ cana-4553 102 23 distance distance NOUN cana-4553 102 24 to to ADP cana-4553 102 25 all all DET cana-4553 102 26 points point NOUN cana-4553 102 27 in in ADP cana-4553 102 28 the the DET cana-4553 102 29 cluster cluster NOUN cana-4553 102 30 . . PUNCT cana-4553 103 1 this this PRON cana-4553 103 2 involves involve VERB cana-4553 103 3 finding find VERB cana-4553 103 4 a a DET cana-4553 103 5 new new ADJ cana-4553 103 6 distribution distribution NOUN cana-4553 103 7 function function NOUN cana-4553 103 8 that that PRON cana-4553 103 9 best well ADV cana-4553 103 10 represents represent VERB cana-4553 103 11 the the DET cana-4553 103 12 central central ADJ cana-4553 103 13 tendency tendency NOUN cana-4553 103 14 of of ADP cana-4553 103 15 the the DET cana-4553 103 16 points point NOUN cana-4553 103 17 in in ADP cana-4553 103 18 the the DET cana-4553 103 19 cluster cluster NOUN cana-4553 103 20 . . PUNCT cana-4553 104 1 4 4 X cana-4553 104 2 . . X cana-4553 104 3 convergence convergence NOUN cana-4553 104 4 check check NOUN cana-4553 104 5 : : PUNCT cana-4553 104 6 we we PRON cana-4553 104 7 check check VERB cana-4553 104 8 , , PUNCT cana-4553 104 9 if if SCONJ cana-4553 104 10 the the DET cana-4553 104 11 centroids centroid NOUN cana-4553 104 12 have have AUX cana-4553 104 13 stabilized stabilize VERB cana-4553 104 14 ( ( PUNCT cana-4553 104 15 i.e. i.e. X cana-4553 104 16 , , PUNCT cana-4553 104 17 there there PRON cana-4553 104 18 is be VERB cana-4553 104 19 little little ADJ cana-4553 104 20 to to ADP cana-4553 104 21 no no DET cana-4553 104 22 change change NOUN cana-4553 104 23 in in ADP cana-4553 104 24 the the DET cana-4553 104 25 distribution distribution NOUN cana-4553 104 26 functions function NOUN cana-4553 104 27 of of ADP cana-4553 104 28 the the DET cana-4553 104 29 centroids centroid NOUN cana-4553 104 30 ) ) PUNCT cana-4553 104 31 . . PUNCT cana-4553 105 1 if if SCONJ cana-4553 105 2 not not PART cana-4553 105 3 so so ADV cana-4553 105 4 , , PUNCT cana-4553 105 5 we we PRON cana-4553 105 6 repeat repeat VERB cana-4553 105 7 the the DET cana-4553 105 8 assignment assignment NOUN cana-4553 105 9 and and CCONJ cana-4553 105 10 update update NOUN cana-4553 105 11 steps step NOUN cana-4553 105 12 until until SCONJ cana-4553 105 13 the the DET cana-4553 105 14 convergence convergence NOUN cana-4553 105 15 is be AUX cana-4553 105 16 obtained obtain VERB cana-4553 105 17 , , PUNCT cana-4553 105 18 e.g. e.g. ADV cana-4553 105 19 consider consider VERB cana-4553 105 20 a a DET cana-4553 105 21 simple simple ADJ cana-4553 105 22 2d 2d NUM cana-4553 105 23 dataset dataset VERB cana-4553 105 24 with with ADP cana-4553 105 25 three three NUM cana-4553 105 26 clusters cluster NOUN cana-4553 105 27 . . PUNCT cana-4553 106 1 the the DET cana-4553 106 2 classical classical ADJ cana-4553 106 3 k k ADJ cana-4553 106 4 - - PUNCT cana-4553 106 5 means means NOUN cana-4553 106 6 algorithm algorithm NOUN cana-4553 106 7 may may AUX cana-4553 106 8 struggle struggle VERB cana-4553 106 9 to to PART cana-4553 106 10 identify identify VERB cana-4553 106 11 the the DET cana-4553 106 12 clusters cluster NOUN cana-4553 106 13 accurately accurately ADV cana-4553 106 14 due due ADJ cana-4553 106 15 to to ADP cana-4553 106 16 noise noise NOUN cana-4553 106 17 and and CCONJ cana-4553 106 18 uncertainty uncertainty NOUN cana-4553 106 19 in in ADP cana-4553 106 20 the the DET cana-4553 106 21 data data NOUN cana-4553 106 22 points point NOUN cana-4553 106 23 . . PUNCT cana-4553 107 1 we we PRON cana-4553 107 2 apply apply VERB cana-4553 107 3 the the DET cana-4553 107 4 probabilistic probabilistic ADJ cana-4553 107 5 k k ADJ cana-4553 107 6 - - PUNCT cana-4553 107 7 means means NOUN cana-4553 107 8 algorithm algorithm NOUN cana-4553 107 9 to to PART cana-4553 107 10 handle handle VERB cana-4553 107 11 this this DET cana-4553 107 12 uncertainty uncertainty NOUN cana-4553 107 13 . . PUNCT cana-4553 108 1 dataset dataset NOUN cana-4553 108 2 : : PUNCT cana-4553 108 3 points point NOUN cana-4553 108 4 : : PUNCT cana-4553 109 1 [ [ X cana-4553 109 2 ( ( PUNCT cana-4553 109 3 1.0 1.0 NUM cana-4553 109 4 , , PUNCT cana-4553 109 5 2.0 2.0 NUM cana-4553 109 6 ) ) PUNCT cana-4553 109 7 , , PUNCT cana-4553 109 8 ( ( PUNCT cana-4553 109 9 1.5 1.5 NUM cana-4553 109 10 , , PUNCT cana-4553 109 11 1.8 1.8 NUM cana-4553 109 12 ) ) PUNCT cana-4553 109 13 , , PUNCT cana-4553 109 14 ( ( PUNCT cana-4553 109 15 5.0 5.0 NUM cana-4553 109 16 , , PUNCT cana-4553 109 17 8.0 8.0 NUM cana-4553 109 18 ) ) PUNCT cana-4553 109 19 , , PUNCT cana-4553 109 20 ( ( PUNCT cana-4553 109 21 8.0 8.0 NUM cana-4553 109 22 , , PUNCT cana-4553 109 23 8.0 8.0 NUM cana-4553 109 24 ) ) PUNCT cana-4553 109 25 , , PUNCT cana-4553 109 26 ( ( PUNCT cana-4553 109 27 1.0 1.0 NUM cana-4553 109 28 , , PUNCT cana-4553 109 29 0.6 0.6 NUM cana-4553 109 30 ) ) PUNCT cana-4553 109 31 , , PUNCT cana-4553 109 32 ( ( PUNCT cana-4553 109 33 9.0 9.0 NUM cana-4553 109 34 , , PUNCT cana-4553 109 35 11.0 11.0 NUM cana-4553 109 36 ) ) PUNCT cana-4553 109 37 ] ] PUNCT cana-4553 109 38 initialization initialization NOUN cana-4553 109 39 : : PUNCT cana-4553 109 40 randomly randomly ADV cana-4553 109 41 initialize initialize VERB cana-4553 109 42 π‘˜ π‘˜ PRON cana-4553 109 43 = = SYM cana-4553 109 44 2 2 NUM cana-4553 109 45 probabilistic probabilistic ADJ cana-4553 109 46 centroids centroid NOUN cana-4553 109 47 . . PUNCT cana-4553 110 1 assume assume VERB cana-4553 110 2 initial initial ADJ cana-4553 110 3 centroids centroid NOUN cana-4553 110 4 are be AUX cana-4553 110 5 ( ( PUNCT cana-4553 110 6 1.0 1.0 NUM cana-4553 110 7 , , PUNCT cana-4553 110 8 2.0 2.0 NUM cana-4553 110 9 ) ) PUNCT cana-4553 110 10 and and CCONJ cana-4553 110 11 ( ( PUNCT cana-4553 110 12 5.0 5.0 NUM cana-4553 110 13 , , PUNCT cana-4553 110 14 8.0 8.0 NUM cana-4553 110 15 ) ) PUNCT cana-4553 110 16 , , PUNCT cana-4553 110 17 represented represent VERB cana-4553 110 18 by by ADP cana-4553 110 19 distribution distribution NOUN cana-4553 110 20 functions function NOUN cana-4553 110 21 with with ADP cana-4553 110 22 small small ADJ cana-4553 110 23 variances variance NOUN cana-4553 110 24 . . PUNCT cana-4553 111 1 iteration iteration NOUN cana-4553 111 2 1 1 NUM cana-4553 111 3 : : PUNCT cana-4553 111 4 assignment assignment NOUN cana-4553 111 5 step step NOUN cana-4553 111 6 compute compute NOUN cana-4553 111 7 probabilistic probabilistic ADJ cana-4553 111 8 distances distance NOUN cana-4553 111 9 between between ADP cana-4553 111 10 each each DET cana-4553 111 11 point point NOUN cana-4553 111 12 and and CCONJ cana-4553 111 13 the the DET cana-4553 111 14 centroids centroid NOUN cana-4553 111 15 using use VERB cana-4553 111 16 a a DET cana-4553 111 17 gaussian gaussian ADJ cana-4553 111 18 distribution distribution NOUN cana-4553 111 19 function function NOUN cana-4553 111 20 . . PUNCT cana-4553 112 1 assign assign VERB cana-4553 112 2 points point NOUN cana-4553 112 3 to to ADP cana-4553 112 4 the the DET cana-4553 112 5 nearest near ADJ cana-4553 112 6 centroid centroid NOUN cana-4553 112 7 based base VERB cana-4553 112 8 on on ADP cana-4553 112 9 expected expect VERB cana-4553 112 10 probabilistic probabilistic ADJ cana-4553 112 11 distance distance NOUN cana-4553 112 12 . . PUNCT cana-4553 113 1 iteration iteration NOUN cana-4553 113 2 1 1 NUM cana-4553 113 3 : : PUNCT cana-4553 113 4 update update NOUN cana-4553 113 5 step step NOUN cana-4553 113 6 update update NOUN cana-4553 113 7 centroids centroid NOUN cana-4553 113 8 by by ADP cana-4553 113 9 recalculating recalculate VERB cana-4553 113 10 the the DET cana-4553 113 11 distribution distribution NOUN cana-4553 113 12 functions function NOUN cana-4553 113 13 that that PRON cana-4553 113 14 minimize minimize VERB cana-4553 113 15 the the DET cana-4553 113 16 expected expect VERB cana-4553 113 17 probabilistic probabilistic ADJ cana-4553 113 18 distance distance NOUN cana-4553 113 19 within within ADP cana-4553 113 20 each each DET cana-4553 113 21 cluster cluster NOUN cana-4553 113 22 . . PUNCT cana-4553 114 1 iteration iteration NOUN cana-4553 114 2 1 1 NUM cana-4553 114 3 : : PUNCT cana-4553 114 4 results result NOUN cana-4553 114 5 cluster cluster VERB cana-4553 114 6 1 1 NUM cana-4553 114 7 : : PUNCT cana-4553 115 1 [ [ X cana-4553 115 2 ( ( PUNCT cana-4553 115 3 1.0 1.0 NUM cana-4553 115 4 , , PUNCT cana-4553 115 5 2.0 2.0 NUM cana-4553 115 6 ) ) PUNCT cana-4553 115 7 , , PUNCT cana-4553 115 8 ( ( PUNCT cana-4553 115 9 1.5 1.5 NUM cana-4553 115 10 , , PUNCT cana-4553 115 11 1.8 1.8 NUM cana-4553 115 12 ) ) PUNCT cana-4553 115 13 , , PUNCT cana-4553 115 14 ( ( PUNCT cana-4553 115 15 1.0 1.0 NUM cana-4553 115 16 , , PUNCT cana-4553 115 17 0.6 0.6 NUM cana-4553 115 18 ) ) PUNCT cana-4553 115 19 ] ] PUNCT cana-4553 115 20 cluster cluster NOUN cana-4553 115 21 2 2 NUM cana-4553 115 22 : : PUNCT cana-4553 115 23 [ [ X cana-4553 115 24 ( ( PUNCT cana-4553 115 25 5.0 5.0 NUM cana-4553 115 26 , , PUNCT cana-4553 115 27 8.0 8.0 NUM cana-4553 115 28 ) ) PUNCT cana-4553 115 29 , , PUNCT cana-4553 115 30 ( ( PUNCT cana-4553 115 31 8.0 8.0 NUM cana-4553 115 32 , , PUNCT cana-4553 115 33 8.0 8.0 NUM cana-4553 115 34 ) ) PUNCT cana-4553 115 35 , , PUNCT cana-4553 115 36 ( ( PUNCT cana-4553 115 37 9.0 9.0 NUM cana-4553 115 38 , , PUNCT cana-4553 115 39 11.0 11.0 NUM cana-4553 115 40 ) ) PUNCT cana-4553 115 41 ] ] PUNCT cana-4553 115 42 new new ADJ cana-4553 115 43 centroids centroid NOUN cana-4553 115 44 ( ( PUNCT cana-4553 115 45 distribution distribution NOUN cana-4553 115 46 functions function NOUN cana-4553 115 47 ): ): PUNCT cana-4553 115 48 cluster cluster NOUN cana-4553 115 49 1 1 NUM cana-4553 115 50 : : PUNCT cana-4553 115 51 mean mean VERB cana-4553 115 52 = = SYM cana-4553 115 53 ( ( PUNCT cana-4553 115 54 1.17 1.17 NUM cana-4553 115 55 , , PUNCT cana-4553 115 56 1.47 1.47 NUM cana-4553 115 57 ) ) PUNCT cana-4553 115 58 , , PUNCT cana-4553 115 59 variance variance NOUN cana-4553 115 60 = = SYM cana-4553 115 61 small small ADJ cana-4553 115 62 communications communication NOUN cana-4553 115 63 on on ADP cana-4553 115 64 applied apply VERB cana-4553 115 65 nonlinear nonlinear ADJ cana-4553 115 66 analysis analysis NOUN cana-4553 115 67 issn issn NOUN cana-4553 115 68 : : PUNCT cana-4553 115 69 1074 1074 NUM cana-4553 115 70 - - PUNCT cana-4553 115 71 133x 133x NUM cana-4553 115 72 vol vol NOUN cana-4553 115 73 32 32 NUM cana-4553 115 74 no no NOUN cana-4553 115 75 . . PUNCT cana-4553 116 1 9s 9s NUM cana-4553 116 2 ( ( PUNCT cana-4553 116 3 2025 2025 NUM cana-4553 116 4 ) ) PUNCT cana-4553 116 5 2759 2759 NUM cana-4553 116 6 https://internationalpubls.com https://internationalpubls.com X cana-4553 116 7 cluster cluster NOUN cana-4553 116 8 2 2 NUM cana-4553 116 9 : : PUNCT cana-4553 116 10 mean mean VERB cana-4553 116 11 = = SYM cana-4553 116 12 ( ( PUNCT cana-4553 116 13 7.33 7.33 NUM cana-4553 116 14 , , PUNCT cana-4553 116 15 9.0 9.0 NUM cana-4553 116 16 ) ) PUNCT cana-4553 116 17 , , PUNCT cana-4553 116 18 variance variance NOUN cana-4553 116 19 = = SYM cana-4553 116 20 small small ADJ cana-4553 116 21 iteration iteration NOUN cana-4553 116 22 2 2 NUM cana-4553 116 23 : : PUNCT cana-4553 116 24 assignment assignment NOUN cana-4553 116 25 step step NOUN cana-4553 116 26 recompute recompute NOUN cana-4553 116 27 probabilistic probabilistic ADJ cana-4553 116 28 distances distance NOUN cana-4553 116 29 using use VERB cana-4553 116 30 updated update VERB cana-4553 116 31 centroids centroid NOUN cana-4553 116 32 . . PUNCT cana-4553 117 1 reassign reassign ADJ cana-4553 117 2 points point NOUN cana-4553 117 3 to to ADP cana-4553 117 4 the the DET cana-4553 117 5 nearest near ADJ cana-4553 117 6 centroid centroid NOUN cana-4553 117 7 . . PUNCT cana-4553 118 1 iteration iteration NOUN cana-4553 118 2 2 2 NUM cana-4553 118 3 : : PUNCT cana-4553 118 4 update update NOUN cana-4553 118 5 step step NOUN cana-4553 118 6 recalculate recalculate NOUN cana-4553 118 7 centroids centroid NOUN cana-4553 118 8 for for ADP cana-4553 118 9 the the DET cana-4553 118 10 new new ADJ cana-4553 118 11 clusters cluster NOUN cana-4553 118 12 . . PUNCT cana-4553 119 1 iteration iteration NOUN cana-4553 119 2 2 2 NUM cana-4553 119 3 : : PUNCT cana-4553 119 4 results result NOUN cana-4553 119 5 cluster cluster VERB cana-4553 119 6 1 1 NUM cana-4553 119 7 : : PUNCT cana-4553 120 1 [ [ X cana-4553 120 2 ( ( PUNCT cana-4553 120 3 1.0 1.0 NUM cana-4553 120 4 , , PUNCT cana-4553 120 5 2.0 2.0 NUM cana-4553 120 6 ) ) PUNCT cana-4553 120 7 , , PUNCT cana-4553 120 8 ( ( PUNCT cana-4553 120 9 1.5 1.5 NUM cana-4553 120 10 , , PUNCT cana-4553 120 11 1.8 1.8 NUM cana-4553 120 12 ) ) PUNCT cana-4553 120 13 , , PUNCT cana-4553 120 14 ( ( PUNCT cana-4553 120 15 1.0 1.0 NUM cana-4553 120 16 , , PUNCT cana-4553 120 17 0.6 0.6 NUM cana-4553 120 18 ) ) PUNCT cana-4553 120 19 ] ] PUNCT cana-4553 120 20 cluster cluster NOUN cana-4553 120 21 2 2 NUM cana-4553 120 22 : : PUNCT cana-4553 120 23 [ [ X cana-4553 120 24 ( ( PUNCT cana-4553 120 25 5.0 5.0 NUM cana-4553 120 26 , , PUNCT cana-4553 120 27 8.0 8.0 NUM cana-4553 120 28 ) ) PUNCT cana-4553 120 29 , , PUNCT cana-4553 120 30 ( ( PUNCT cana-4553 120 31 8.0 8.0 NUM cana-4553 120 32 , , PUNCT cana-4553 120 33 8.0 8.0 NUM cana-4553 120 34 ) ) PUNCT cana-4553 120 35 , , PUNCT cana-4553 120 36 ( ( PUNCT cana-4553 120 37 9.0 9.0 NUM cana-4553 120 38 , , PUNCT cana-4553 120 39 11.0 11.0 NUM cana-4553 120 40 ) ) PUNCT cana-4553 120 41 ] ] PUNCT cana-4553 120 42 new new ADJ cana-4553 120 43 centroids centroid NOUN cana-4553 120 44 ( ( PUNCT cana-4553 120 45 distribution distribution NOUN cana-4553 120 46 functions function NOUN cana-4553 120 47 ): ): PUNCT cana-4553 120 48 cluster cluster NOUN cana-4553 120 49 1 1 NUM cana-4553 120 50 : : PUNCT cana-4553 120 51 mean mean VERB cana-4553 120 52 = = SYM cana-4553 120 53 ( ( PUNCT cana-4553 120 54 1.17 1.17 NUM cana-4553 120 55 , , PUNCT cana-4553 120 56 1.47 1.47 NUM cana-4553 120 57 ) ) PUNCT cana-4553 120 58 , , PUNCT cana-4553 120 59 variance variance NOUN cana-4553 120 60 = = PUNCT cana-4553 120 61 very very ADV cana-4553 120 62 small small ADJ cana-4553 120 63 cluster cluster NOUN cana-4553 120 64 2 2 NUM cana-4553 120 65 : : PUNCT cana-4553 120 66 mean mean VERB cana-4553 120 67 = = SYM cana-4553 120 68 ( ( PUNCT cana-4553 120 69 7.33 7.33 NUM cana-4553 120 70 , , PUNCT cana-4553 120 71 9.0 9.0 NUM cana-4553 120 72 ) ) PUNCT cana-4553 120 73 , , PUNCT cana-4553 120 74 variance variance NOUN cana-4553 120 75 = = PUNCT cana-4553 120 76 very very ADV cana-4553 120 77 small small ADJ cana-4553 120 78 convergence convergence NOUN cana-4553 120 79 check check NOUN cana-4553 120 80 : : PUNCT cana-4553 120 81 centroids centroid NOUN cana-4553 120 82 have have AUX cana-4553 120 83 stabilized stabilize VERB cana-4553 120 84 ( ( PUNCT cana-4553 120 85 no no DET cana-4553 120 86 significant significant ADJ cana-4553 120 87 change change NOUN cana-4553 120 88 in in ADP cana-4553 120 89 mean mean NOUN cana-4553 120 90 and and CCONJ cana-4553 120 91 variance variance NOUN cana-4553 120 92 of of ADP cana-4553 120 93 distribution distribution NOUN cana-4553 120 94 functions function NOUN cana-4553 120 95 ) ) PUNCT cana-4553 120 96 . . PUNCT cana-4553 121 1 algorithm algorithm PROPN cana-4553 121 2 converges converge VERB cana-4553 121 3 . . PUNCT cana-4553 122 1 final final ADJ cana-4553 122 2 clusters cluster NOUN cana-4553 122 3 : : PUNCT cana-4553 122 4 cluster cluster NOUN cana-4553 122 5 1 1 NUM cana-4553 122 6 : : PUNCT cana-4553 123 1 [ [ X cana-4553 123 2 ( ( PUNCT cana-4553 123 3 1.0 1.0 NUM cana-4553 123 4 , , PUNCT cana-4553 123 5 2.0 2.0 NUM cana-4553 123 6 ) ) PUNCT cana-4553 123 7 , , PUNCT cana-4553 123 8 ( ( PUNCT cana-4553 123 9 1.5 1.5 NUM cana-4553 123 10 , , PUNCT cana-4553 123 11 1.8 1.8 NUM cana-4553 123 12 ) ) PUNCT cana-4553 123 13 , , PUNCT cana-4553 123 14 ( ( PUNCT cana-4553 123 15 1.0 1.0 NUM cana-4553 123 16 , , PUNCT cana-4553 123 17 0.6 0.6 NUM cana-4553 123 18 ) ) PUNCT cana-4553 123 19 ] ] PUNCT cana-4553 123 20 cluster cluster NOUN cana-4553 123 21 2 2 NUM cana-4553 123 22 : : PUNCT cana-4553 123 23 [ [ X cana-4553 123 24 ( ( PUNCT cana-4553 123 25 5.0 5.0 NUM cana-4553 123 26 , , PUNCT cana-4553 123 27 8.0 8.0 NUM cana-4553 123 28 ) ) PUNCT cana-4553 123 29 , , PUNCT cana-4553 123 30 ( ( PUNCT cana-4553 123 31 8.0 8.0 NUM cana-4553 123 32 , , PUNCT cana-4553 123 33 8.0 8.0 NUM cana-4553 123 34 ) ) PUNCT cana-4553 123 35 , , PUNCT cana-4553 123 36 ( ( PUNCT cana-4553 123 37 9.0 9.0 NUM cana-4553 123 38 , , PUNCT cana-4553 123 39 11.0 11.0 NUM cana-4553 123 40 ) ) PUNCT cana-4553 123 41 ] ] PUNCT cana-4553 123 42 by by ADP cana-4553 123 43 applying apply VERB cana-4553 123 44 the the DET cana-4553 123 45 jungck jungck NOUN cana-4553 123 46 contraction contraction NOUN cana-4553 123 47 theorem theorem VERB cana-4553 123 48 , , PUNCT cana-4553 123 49 we we PRON cana-4553 123 50 ensure ensure VERB cana-4553 123 51 that that SCONJ cana-4553 123 52 the the DET cana-4553 123 53 probabilistic probabilistic ADJ cana-4553 123 54 k k NOUN cana-4553 123 55 - - PUNCT cana-4553 123 56 means means NOUN cana-4553 123 57 algorithm algorithm NOUN cana-4553 123 58 converges converge VERB cana-4553 123 59 to to ADP cana-4553 123 60 a a DET cana-4553 123 61 stable stable ADJ cana-4553 123 62 clustering clustering ADJ cana-4553 123 63 configuration configuration NOUN cana-4553 123 64 , , PUNCT cana-4553 123 65 providing provide VERB cana-4553 123 66 robustness robustness NOUN cana-4553 123 67 against against ADP cana-4553 123 68 the the DET cana-4553 123 69 inherent inherent ADJ cana-4553 123 70 uncertainties uncertainty NOUN cana-4553 123 71 in in ADP cana-4553 123 72 the the DET cana-4553 123 73 data datum NOUN cana-4553 123 74 . . PUNCT cana-4553 124 1 probabilistic probabilistic ADJ cana-4553 124 2 neural neural ADJ cana-4553 124 3 networks network NOUN cana-4553 124 4 ( ( PUNCT cana-4553 124 5 pnn)-a pnn)-a PROPN cana-4553 124 6 case case NOUN cana-4553 124 7 study study NOUN cana-4553 124 8 : : PUNCT cana-4553 124 9 neural neural ADJ cana-4553 124 10 networks network NOUN cana-4553 124 11 ( ( PUNCT cana-4553 124 12 nns nn NOUN cana-4553 124 13 ) ) PUNCT cana-4553 124 14 are be AUX cana-4553 124 15 a a DET cana-4553 124 16 cornerstone cornerstone NOUN cana-4553 124 17 of of ADP cana-4553 124 18 modern modern ADJ cana-4553 124 19 ml ml NOUN cana-4553 124 20 . . PUNCT cana-4553 125 1 training train VERB cana-4553 125 2 a a DET cana-4553 125 3 nn nn NOUN cana-4553 125 4 involves involve VERB cana-4553 125 5 iteratively iteratively ADV cana-4553 125 6 updating update VERB cana-4553 125 7 weights weight NOUN cana-4553 125 8 to to PART cana-4553 125 9 minimize minimize VERB cana-4553 125 10 a a DET cana-4553 125 11 loss loss NOUN cana-4553 125 12 function function NOUN cana-4553 125 13 . . PUNCT cana-4553 126 1 in in ADP cana-4553 126 2 the the DET cana-4553 126 3 presence presence NOUN cana-4553 126 4 of of ADP cana-4553 126 5 noisy noisy ADJ cana-4553 126 6 data datum NOUN cana-4553 126 7 , , PUNCT cana-4553 126 8 a a DET cana-4553 126 9 probabilistic probabilistic ADJ cana-4553 126 10 framework framework NOUN cana-4553 126 11 can can AUX cana-4553 126 12 enhance enhance VERB cana-4553 126 13 the the DET cana-4553 126 14 training training NOUN cana-4553 126 15 process[13].pnns process[13].pnn NOUN cana-4553 126 16 and and CCONJ cana-4553 126 17 related relate VERB cana-4553 126 18 probabilistic probabilistic ADJ cana-4553 126 19 models model NOUN cana-4553 126 20 are be AUX cana-4553 126 21 being be AUX cana-4553 126 22 applied apply VERB cana-4553 126 23 to to ADP cana-4553 126 24 various various ADJ cana-4553 126 25 domains domain NOUN cana-4553 126 26 such such ADJ cana-4553 126 27 as as ADP cana-4553 126 28 healthcare healthcare NOUN cana-4553 126 29 ( ( PUNCT cana-4553 126 30 medical medical ADJ cana-4553 126 31 diagnosis diagnosis NOUN cana-4553 126 32 and and CCONJ cana-4553 126 33 prognosis prognosis NOUN cana-4553 126 34 ) ) PUNCT cana-4553 126 35 and and CCONJ cana-4553 126 36 finance finance NOUN cana-4553 126 37 ( ( PUNCT cana-4553 126 38 risk risk NOUN cana-4553 126 39 assessment assessment NOUN cana-4553 126 40 and and CCONJ cana-4553 126 41 fraud fraud NOUN cana-4553 126 42 detection detection NOUN cana-4553 126 43 ) ) PUNCT cana-4553 126 44 . . PUNCT cana-4553 127 1 these these DET cana-4553 127 2 applications application NOUN cana-4553 127 3 benefit benefit VERB cana-4553 127 4 from from ADP cana-4553 127 5 the the DET cana-4553 127 6 ability ability NOUN cana-4553 127 7 of of ADP cana-4553 127 8 pnns pnn NOUN cana-4553 127 9 to to PART cana-4553 127 10 handle handle VERB cana-4553 127 11 uncertainty uncertainty NOUN cana-4553 127 12 and and CCONJ cana-4553 127 13 complex complex ADJ cana-4553 127 14 patterns pattern NOUN cana-4553 127 15 in in ADP cana-4553 127 16 data[13 data[13 PROPN cana-4553 127 17 ] ] PUNCT cana-4553 127 18 . . PUNCT cana-4553 128 1 1 1 X cana-4553 128 2 . . X cana-4553 128 3 probabilistic probabilistic ADJ cana-4553 128 4 weight weight NOUN cana-4553 128 5 updates update VERB cana-4553 128 6 : : PUNCT cana-4553 128 7 the the DET cana-4553 128 8 weight weight NOUN cana-4553 128 9 updates update NOUN cana-4553 128 10 are be AUX cana-4553 128 11 modelled model VERB cana-4553 128 12 probabilistically probabilistically ADV cana-4553 128 13 , , PUNCT cana-4553 128 14 capturing capture VERB cana-4553 128 15 the the DET cana-4553 128 16 uncertainty uncertainty NOUN cana-4553 128 17 in in ADP cana-4553 128 18 the the DET cana-4553 128 19 gradient gradient NOUN cana-4553 128 20 estimates estimate NOUN cana-4553 128 21 due due ADP cana-4553 128 22 to to ADP cana-4553 128 23 noisy noisy ADJ cana-4553 128 24 data datum NOUN cana-4553 128 25 . . PUNCT cana-4553 129 1 2 2 X cana-4553 129 2 . . X cana-4553 129 3 probabilistic probabilistic ADJ cana-4553 129 4 loss loss NOUN cana-4553 129 5 function function NOUN cana-4553 129 6 : : PUNCT cana-4553 129 7 the the DET cana-4553 129 8 loss loss NOUN cana-4553 129 9 function function NOUN cana-4553 129 10 is be AUX cana-4553 129 11 defined define VERB cana-4553 129 12 in in ADP cana-4553 129 13 terms term NOUN cana-4553 129 14 of of ADP cana-4553 129 15 probabilistic probabilistic ADJ cana-4553 129 16 distances distance NOUN cana-4553 129 17 , , PUNCT cana-4553 129 18 providing provide VERB cana-4553 129 19 a a DET cana-4553 129 20 more more ADV cana-4553 129 21 robust robust ADJ cana-4553 129 22 objective objective NOUN cana-4553 129 23 for for ADP cana-4553 129 24 optimization optimization NOUN cana-4553 129 25 [ [ X cana-4553 129 26 16 16 NUM cana-4553 129 27 ] ] PUNCT cana-4553 129 28 . . PUNCT cana-4553 130 1 communications communication NOUN cana-4553 130 2 on on ADP cana-4553 130 3 applied apply VERB cana-4553 130 4 nonlinear nonlinear ADJ cana-4553 130 5 analysis analysis NOUN cana-4553 130 6 issn issn NOUN cana-4553 130 7 : : PUNCT cana-4553 130 8 1074 1074 NUM cana-4553 130 9 - - PUNCT cana-4553 130 10 133x 133x NUM cana-4553 130 11 vol vol NOUN cana-4553 130 12 32 32 NUM cana-4553 130 13 no no NOUN cana-4553 130 14 . . PUNCT cana-4553 131 1 9s 9s NUM cana-4553 131 2 ( ( PUNCT cana-4553 131 3 2025 2025 NUM cana-4553 131 4 ) ) PUNCT cana-4553 131 5 2760 2760 NUM cana-4553 131 6 https://internationalpubls.com https://internationalpubls.com X cana-4553 131 7 3 3 X cana-4553 131 8 . . PUNCT cana-4553 131 9 convergence convergence NOUN cana-4553 131 10 analysis analysis NOUN cana-4553 131 11 : : PUNCT cana-4553 131 12 using use VERB cana-4553 131 13 the the DET cana-4553 131 14 jungck jungck NOUN cana-4553 131 15 contraction contraction NOUN cana-4553 131 16 theorem theorem VERB cana-4553 131 17 , , PUNCT cana-4553 131 18 we we PRON cana-4553 131 19 analyse analyse VERB cana-4553 131 20 the the DET cana-4553 131 21 convergence convergence NOUN cana-4553 131 22 of of ADP cana-4553 131 23 the the DET cana-4553 131 24 probabilistic probabilistic ADJ cana-4553 131 25 weight weight NOUN cana-4553 131 26 updates update VERB cana-4553 131 27 [ [ X cana-4553 131 28 16 16 NUM cana-4553 131 29 ] ] PUNCT cana-4553 131 30 . . PUNCT cana-4553 132 1 by by ADP cana-4553 132 2 ensuring ensure VERB cana-4553 132 3 that that SCONJ cana-4553 132 4 the the DET cana-4553 132 5 probabilistic probabilistic ADJ cana-4553 132 6 distance distance NOUN cana-4553 132 7 satisfies satisfy VERB cana-4553 132 8 the the DET cana-4553 132 9 contraction contraction NOUN cana-4553 132 10 condition condition NOUN cana-4553 132 11 , , PUNCT cana-4553 132 12 we we PRON cana-4553 132 13 can can AUX cana-4553 132 14 assure assure VERB cana-4553 132 15 convergence convergence NOUN cana-4553 132 16 to to ADP cana-4553 132 17 an an DET cana-4553 132 18 optimal optimal ADJ cana-4553 132 19 set set NOUN cana-4553 132 20 of of ADP cana-4553 132 21 weights weight NOUN cana-4553 132 22 . . PUNCT cana-4553 133 1 probabilistic probabilistic ADJ cana-4553 133 2 gradient gradient ADJ cana-4553 133 3 descent descent NOUN cana-4553 133 4 algorithm algorithm NOUN cana-4553 133 5 : : PUNCT cana-4553 133 6 1 1 X cana-4553 133 7 . . X cana-4553 133 8 initialization initialization NOUN cana-4553 133 9 : : PUNCT cana-4553 133 10 randomly randomly ADV cana-4553 133 11 initialize initialize VERB cana-4553 133 12 weights weight NOUN cana-4553 133 13 probabilistically probabilistically ADV cana-4553 133 14 . . PUNCT cana-4553 134 1 2 2 X cana-4553 134 2 . . X cana-4553 134 3 forward forward ADV cana-4553 134 4 pass pass PROPN cana-4553 134 5 : : PUNCT cana-4553 134 6 compute compute VERB cana-4553 134 7 the the DET cana-4553 134 8 output output NOUN cana-4553 134 9 of of ADP cana-4553 134 10 the the DET cana-4553 134 11 nn nn NOUN cana-4553 134 12 using use VERB cana-4553 134 13 the the DET cana-4553 134 14 current current ADJ cana-4553 134 15 probabilistic probabilistic ADJ cana-4553 134 16 weights weight NOUN cana-4553 134 17 . . PUNCT cana-4553 135 1 3 3 X cana-4553 135 2 . . X cana-4553 135 3 loss loss NOUN cana-4553 135 4 computation computation NOUN cana-4553 135 5 : : PUNCT cana-4553 135 6 compute compute VERB cana-4553 135 7 the the DET cana-4553 135 8 probabilistic probabilistic ADJ cana-4553 135 9 loss loss NOUN cana-4553 135 10 based base VERB cana-4553 135 11 on on ADP cana-4553 135 12 the the DET cana-4553 135 13 output output NOUN cana-4553 135 14 and and CCONJ cana-4553 135 15 the the DET cana-4553 135 16 true true ADJ cana-4553 135 17 labels label NOUN cana-4553 135 18 . . PUNCT cana-4553 136 1 4 4 X cana-4553 136 2 . . X cana-4553 136 3 backward backward ADJ cana-4553 136 4 pass pass NOUN cana-4553 136 5 : : PUNCT cana-4553 136 6 compute compute VERB cana-4553 136 7 the the DET cana-4553 136 8 probabilistic probabilistic ADJ cana-4553 136 9 gradient gradient NOUN cana-4553 136 10 of of ADP cana-4553 136 11 the the DET cana-4553 136 12 loss loss NOUN cana-4553 136 13 with with ADP cana-4553 136 14 respect respect NOUN cana-4553 136 15 to to ADP cana-4553 136 16 the the DET cana-4553 136 17 weights weight NOUN cana-4553 136 18 . . PUNCT cana-4553 137 1 5 5 NUM cana-4553 137 2 . . X cana-4553 137 3 weight weight NOUN cana-4553 137 4 update update NOUN cana-4553 137 5 : : PUNCT cana-4553 137 6 update update VERB cana-4553 137 7 the the DET cana-4553 137 8 weights weight NOUN cana-4553 137 9 probabilistically probabilistically ADV cana-4553 137 10 based base VERB cana-4553 137 11 on on ADP cana-4553 137 12 the the DET cana-4553 137 13 computed computed ADJ cana-4553 137 14 gradient gradient NOUN cana-4553 137 15 . . PUNCT cana-4553 138 1 6 6 X cana-4553 138 2 . . X cana-4553 138 3 convergence convergence NOUN cana-4553 138 4 check check NOUN cana-4553 138 5 : : PUNCT cana-4553 138 6 we we PRON cana-4553 138 7 repeat repeat VERB cana-4553 138 8 steps step NOUN cana-4553 138 9 2 2 NUM cana-4553 138 10 - - SYM cana-4553 138 11 5 5 NUM cana-4553 138 12 , , PUNCT cana-4553 138 13 until until SCONJ cana-4553 138 14 the the DET cana-4553 138 15 loss loss NOUN cana-4553 138 16 converges converge VERB cana-4553 138 17 . . PUNCT cana-4553 139 1 by by ADP cana-4553 139 2 employing employ VERB cana-4553 139 3 the the DET cana-4553 139 4 jungck jungck NOUN cana-4553 139 5 contractive contractive ADJ cana-4553 139 6 statement statement NOUN cana-4553 139 7 and and CCONJ cana-4553 139 8 commutativity commutativity NOUN cana-4553 139 9 [ [ X cana-4553 139 10 15 15 NUM cana-4553 139 11 ] ] PUNCT cana-4553 139 12 , , PUNCT cana-4553 139 13 we we PRON cana-4553 139 14 ensure ensure VERB cana-4553 139 15 that that SCONJ cana-4553 139 16 the the DET cana-4553 139 17 probabilistic probabilistic ADJ cana-4553 139 18 gradient gradient ADJ cana-4553 139 19 descent descent NOUN cana-4553 139 20 algorithm algorithm NOUN cana-4553 139 21 converges converge VERB cana-4553 139 22 to to ADP cana-4553 139 23 an an DET cana-4553 139 24 optimal optimal ADJ cana-4553 139 25 solution solution NOUN cana-4553 139 26 , , PUNCT cana-4553 139 27 providing provide VERB cana-4553 139 28 robustness robustness NOUN cana-4553 139 29 against against ADP cana-4553 139 30 the the DET cana-4553 139 31 uncertainties uncertainty NOUN cana-4553 139 32 in in ADP cana-4553 139 33 the the DET cana-4553 139 34 training training NOUN cana-4553 139 35 data datum NOUN cana-4553 139 36 . . PUNCT cana-4553 140 1 4 4 X cana-4553 140 2 . . X cana-4553 140 3 main main ADJ cana-4553 140 4 results result NOUN cana-4553 140 5 & & CCONJ cana-4553 140 6 statements statement NOUN cana-4553 140 7 we we PRON cana-4553 140 8 use use VERB cana-4553 140 9 the the DET cana-4553 140 10 following following NOUN cana-4553 140 11 as as ADP cana-4553 140 12 a a DET cana-4553 140 13 result result NOUN cana-4553 140 14 in in ADP cana-4553 140 15 harnessing harness VERB cana-4553 140 16 the the DET cana-4553 140 17 situation situation NOUN cana-4553 140 18 as as SCONJ cana-4553 140 19 stated state VERB cana-4553 140 20 above above ADV cana-4553 140 21 . . PUNCT cana-4553 141 1 lemma lemma PROPN cana-4553 141 2 4.1 4.1 NUM cana-4553 141 3 : : PUNCT cana-4553 142 1 [ [ X cana-4553 142 2 6 6 NUM cana-4553 142 3 ] ] X cana-4553 142 4 if if SCONJ cana-4553 142 5 ( ( PUNCT cana-4553 142 6 𝑃 𝑃 NOUN cana-4553 142 7 , , PUNCT cana-4553 142 8 𝐹 𝐹 PROPN cana-4553 142 9 , , PUNCT cana-4553 142 10 𝑇 𝑇 PROPN cana-4553 142 11 ) ) PUNCT cana-4553 142 12 be be VERB cana-4553 142 13 a a DET cana-4553 142 14 menger menger NOUN cana-4553 142 15 generalized generalize VERB cana-4553 142 16 pm pm NOUN cana-4553 142 17 - - PUNCT cana-4553 142 18 space space NOUN cana-4553 142 19 with with ADP cana-4553 142 20 a a DET cana-4553 142 21 cauchy cauchy ADJ cana-4553 142 22 sequence sequence NOUN cana-4553 142 23 < < X cana-4553 142 24 𝛼𝑛 𝛼𝑛 ADP cana-4553 142 25 > > PUNCT cana-4553 142 26 in in ADP cana-4553 142 27 𝑃along 𝑃along PROPN cana-4553 142 28 with with ADP cana-4553 142 29 some some DET cana-4553 142 30 𝛽 𝛽 NOUN cana-4553 142 31 ∈ ∈ PROPN cana-4553 142 32 ( ( PUNCT cana-4553 142 33 0,1 0,1 NOUN cana-4553 142 34 ) ) PUNCT cana-4553 142 35 such such ADJ cana-4553 142 36 that that DET cana-4553 142 37 𝐹𝛼𝑛,𝛼𝑛+1 𝐹𝛼𝑛,𝛼𝑛+1 PROPN cana-4553 142 38 ( ( PUNCT cana-4553 142 39 𝛽π‘₯ 𝛽π‘₯ ADJ cana-4553 142 40 ) ) PUNCT cana-4553 142 41 β‰₯ β‰₯ NOUN cana-4553 142 42 πΉπ›Όπ‘›βˆ’1,𝛼𝑛 πΉπ›Όπ‘›βˆ’1,𝛼𝑛 PROPN cana-4553 142 43 ( ( PUNCT cana-4553 142 44 π‘₯ π‘₯ NOUN cana-4553 142 45 ) ) PUNCT cana-4553 142 46 for for ADP cana-4553 142 47 all all DET cana-4553 142 48 positive positive ADJ cana-4553 142 49 values value NOUN cana-4553 142 50 of of ADP cana-4553 142 51 π‘₯. π‘₯. NOUN cana-4553 142 52 also also ADV cana-4553 142 53 , , PUNCT cana-4553 142 54 if if SCONJ cana-4553 142 55 for for ADP cana-4553 142 56 π‘Ÿ π‘Ÿ NOUN cana-4553 142 57 > > SYM cana-4553 142 58 1 1 NUM cana-4553 142 59 and and CCONJ cana-4553 142 60 𝑖 𝑖 SYM cana-4553 142 61 = = SYM cana-4553 142 62 𝑛 𝑛 PRON cana-4553 142 63 π‘‘π‘œ π‘‘π‘œ NOUN cana-4553 142 64 ∞ ∞ PROPN cana-4553 142 65 , , PUNCT cana-4553 142 66 we we PRON cana-4553 142 67 have have VERB cana-4553 142 68 lim lim PROPN cana-4553 142 69 π‘›β†’βˆž π‘›β†’βˆž NUM cana-4553 142 70 𝑇𝑖 𝑇𝑖 PROPN cana-4553 142 71 ( ( PUNCT cana-4553 142 72 𝐹𝛼0,𝛼1 𝐹𝛼0,𝛼1 PROPN cana-4553 142 73 ( ( PUNCT cana-4553 142 74 π‘Ÿπ‘– π‘Ÿπ‘– NOUN cana-4553 142 75 ) ) PUNCT cana-4553 142 76 ) ) PUNCT cana-4553 143 1 = = SYM cana-4553 143 2 1 1 NUM cana-4553 143 3 therefore therefore ADV cana-4553 143 4 , , PUNCT cana-4553 143 5 we we PRON cana-4553 143 6 say say VERB cana-4553 143 7 the the DET cana-4553 143 8 sequence sequence NOUN cana-4553 143 9 < < X cana-4553 143 10 𝛼𝑛 𝛼𝑛 X cana-4553 143 11 > > X cana-4553 143 12 is be AUX cana-4553 143 13 𝐹-cauchy 𝐹-cauchy PROPN cana-4553 143 14 . . PUNCT cana-4553 144 1 to to PART cana-4553 144 2 get get VERB cana-4553 144 3 the the DET cana-4553 144 4 common common ADJ cana-4553 144 5 coincidence coincidence NOUN cana-4553 144 6 point point NOUN cana-4553 144 7 for for ADP cana-4553 144 8 mappings mapping NOUN cana-4553 144 9 under under ADP cana-4553 144 10 generalized generalized ADJ cana-4553 144 11 menger menger NOUN cana-4553 144 12 space space NOUN cana-4553 144 13 for for ADP cana-4553 144 14 producing produce VERB cana-4553 144 15 some some DET cana-4553 144 16 machine machine NOUN cana-4553 144 17 learning learning NOUN cana-4553 144 18 results result NOUN cana-4553 144 19 , , PUNCT cana-4553 144 20 we we PRON cana-4553 144 21 can can AUX cana-4553 144 22 apply apply VERB cana-4553 144 23 the the DET cana-4553 144 24 following following ADJ cana-4553 144 25 statements statement NOUN cana-4553 144 26 with with ADP cana-4553 144 27 some some DET cana-4553 144 28 specific specific ADJ cana-4553 144 29 constraints constraint NOUN cana-4553 144 30 . . PUNCT cana-4553 145 1 theorem theorem VERB cana-4553 145 2 4.1 4.1 NUM cana-4553 145 3 : : PUNCT cana-4553 145 4 with with ADP cana-4553 145 5 ( ( PUNCT cana-4553 145 6 𝑃 𝑃 PROPN cana-4553 145 7 , , PUNCT cana-4553 145 8 𝐹 𝐹 PROPN cana-4553 145 9 , , PUNCT cana-4553 145 10 𝑇)a 𝑇)a VERB cana-4553 145 11 generalized generalize VERB cana-4553 145 12 pm pm NOUN cana-4553 145 13 - - PUNCT cana-4553 145 14 space space NOUN cana-4553 145 15 and and CCONJ cana-4553 145 16 a a DET cana-4553 145 17 continuous continuous ADJ cana-4553 145 18 𝑑-norm𝑇 𝑑-norm𝑇 PROPN cana-4553 145 19 in in ADP cana-4553 145 20 ( ( PUNCT cana-4553 145 21 π‘Ž π‘Ž X cana-4553 145 22 , , PUNCT cana-4553 145 23 1 1 NUM cana-4553 145 24 ) ) PUNCT cana-4553 145 25 for for ADP cana-4553 145 26 all all PRON cana-4553 145 27 0 0 NUM cana-4553 145 28 < < X cana-4553 146 1 π‘Ž π‘Ž X cana-4553 146 2 < < X cana-4553 146 3 1 1 NUM cana-4553 146 4 ; ; PUNCT cana-4553 146 5 0 0 NUM cana-4553 146 6 < < X cana-4553 146 7 πœƒ πœƒ X cana-4553 146 8 < < X cana-4553 146 9 1 1 NUM cana-4553 146 10 ; ; PUNCT cana-4553 146 11 𝛿 𝛿 DET cana-4553 146 12 ∈ ∈ NOUN cana-4553 146 13 βˆ† βˆ† PROPN cana-4553 146 14 and and CCONJ cana-4553 146 15 𝑓 𝑓 PRON cana-4553 146 16 , , PUNCT cana-4553 146 17 𝑔 𝑔 NOUN cana-4553 146 18 : : PUNCT cana-4553 146 19 𝑄 𝑄 PROPN cana-4553 146 20 β†’ β†’ SYM cana-4553 146 21 𝑃 𝑃 PROPN cana-4553 146 22 , , PUNCT cana-4553 146 23 we we PRON cana-4553 146 24 have have VERB cana-4553 146 25 ( ( PUNCT cana-4553 146 26 i i NOUN cana-4553 146 27 ) ) PUNCT cana-4553 146 28 . . PUNCT cana-4553 147 1 𝛿(𝐹𝑓𝛼,𝑓𝛽(πœƒπ‘₯ 𝛿(𝐹𝑓𝛼,𝑓𝛽(πœƒπ‘₯ NOUN cana-4553 147 2 ) ) PUNCT cana-4553 147 3 , , PUNCT cana-4553 147 4 𝐹𝑔𝛼,𝑔𝛽(π‘₯ 𝐹𝑔𝛼,𝑔𝛽(π‘₯ NUM cana-4553 147 5 ) ) PUNCT cana-4553 147 6 , , PUNCT cana-4553 147 7 𝐹𝑓𝛼,𝑔𝛼(π‘₯ 𝐹𝑓𝛼,𝑔𝛼(π‘₯ PROPN cana-4553 147 8 ) ) PUNCT cana-4553 147 9 , , PUNCT cana-4553 147 10 𝐹𝑓𝛽,𝑔𝛽(πœƒπ‘₯ 𝐹𝑓𝛽,𝑔𝛽(πœƒπ‘₯ NOUN cana-4553 147 11 ) ) PUNCT cana-4553 147 12 ) ) PUNCT cana-4553 148 1 β‰₯ β‰₯ NOUN cana-4553 148 2 0 0 NUM cana-4553 148 3 for for ADP cana-4553 148 4 every every DET cana-4553 148 5 𝛼 𝛼 NOUN cana-4553 148 6 , , PUNCT cana-4553 148 7 𝛽 𝛽 PROPN cana-4553 148 8 ∈ ∈ PROPN cana-4553 148 9 𝑄 𝑄 PROPN cana-4553 148 10 and and CCONJ cana-4553 148 11 positive positive ADJ cana-4553 148 12 values value NOUN cana-4553 148 13 of of ADP cana-4553 148 14 π‘₯ π‘₯ PROPN cana-4553 148 15 ( ( PUNCT cana-4553 148 16 ii ii NOUN cana-4553 148 17 ) ) PUNCT cana-4553 148 18 . . PUNCT cana-4553 149 1 𝑓(𝑄 𝑓(𝑄 NOUN cana-4553 149 2 ) ) PUNCT cana-4553 150 1 βŠ‚ βŠ‚ PROPN cana-4553 150 2 𝑔(𝑄 𝑔(𝑄 PROPN cana-4553 150 3 ) ) PUNCT cana-4553 150 4 ( ( PUNCT cana-4553 150 5 iii).𝛼0 iii).𝛼0 PROPN cana-4553 150 6 , , PUNCT cana-4553 150 7 𝛼1 𝛼1 NOUN cana-4553 150 8 ∈ ∈ NOUN cana-4553 150 9 𝑄 𝑄 NOUN cana-4553 151 1 so so SCONJ cana-4553 151 2 that that PRON cana-4553 151 3 𝑓𝛼0 𝑓𝛼0 NOUN cana-4553 151 4 = = SYM cana-4553 151 5 𝑔𝛼1 𝑔𝛼1 NOUN cana-4553 151 6 and and CCONJ cana-4553 151 7 lim lim PROPN cana-4553 151 8 π‘›β†’βˆž π‘›β†’βˆž NUM cana-4553 151 9 𝑇𝑖 𝑇𝑖 PROPN cana-4553 151 10 ( ( PUNCT cana-4553 151 11 𝐹𝑓𝛼0,𝑓𝛼1 𝐹𝑓𝛼0,𝑓𝛼1 ADJ cana-4553 151 12 ( ( PUNCT cana-4553 151 13 π‘Ÿπ‘– π‘Ÿπ‘– NOUN cana-4553 151 14 ) ) PUNCT cana-4553 151 15 ) ) PUNCT cana-4553 152 1 = = PUNCT cana-4553 152 2 1for 1for NUM cana-4553 152 3 π‘Ÿ π‘Ÿ X cana-4553 152 4 > > SYM cana-4553 152 5 1 1 NUM cana-4553 152 6 and and CCONJ cana-4553 152 7 𝑖 𝑖 SYM cana-4553 152 8 = = SYM cana-4553 152 9 𝑛 𝑛 DET cana-4553 152 10 π‘‘π‘œ π‘‘π‘œ NOUN cana-4553 152 11 ∞ ∞ NUM cana-4553 152 12 then then ADV cana-4553 152 13 , , PUNCT cana-4553 152 14 we we PRON cana-4553 152 15 can can AUX cana-4553 152 16 say say VERB cana-4553 152 17 to to PART cana-4553 152 18 have have VERB cana-4553 152 19 common common ADJ cana-4553 152 20 coincidence coincidence NOUN cana-4553 152 21 value value NOUN cana-4553 152 22 for𝑓and for𝑓and NOUN cana-4553 152 23 𝑔. 𝑔. PROPN cana-4553 152 24 proof proof PROPN cana-4553 152 25 : : PUNCT cana-4553 152 26 suppose suppose VERB cana-4553 152 27 , , PUNCT cana-4553 152 28 𝛼0 𝛼0 PROPN cana-4553 152 29 , , PUNCT cana-4553 152 30 𝛼1 𝛼1 NOUN cana-4553 152 31 ∈ ∈ NOUN cana-4553 152 32 𝑄 𝑄 NOUN cana-4553 152 33 so so SCONJ cana-4553 152 34 that that PRON cana-4553 152 35 𝑓𝛼0 𝑓𝛼0 NOUN cana-4553 152 36 = = SYM cana-4553 152 37 𝑔𝛼1 𝑔𝛼1 NOUN cana-4553 152 38 and and CCONJ cana-4553 152 39 lim lim PROPN cana-4553 152 40 π‘›β†’βˆž π‘›β†’βˆž NUM cana-4553 152 41 𝑇𝑖 𝑇𝑖 PROPN cana-4553 152 42 ( ( PUNCT cana-4553 152 43 𝐹𝑓𝛼0,𝑓𝛼1 𝐹𝑓𝛼0,𝑓𝛼1 ADJ cana-4553 152 44 ( ( PUNCT cana-4553 152 45 π‘Ÿπ‘– π‘Ÿπ‘– NOUN cana-4553 152 46 ) ) PUNCT cana-4553 152 47 ) ) PUNCT cana-4553 153 1 = = SYM cana-4553 153 2 1 1 NUM cana-4553 153 3 ( ( PUNCT cana-4553 153 4 1 1 NUM cana-4553 153 5 ) ) PUNCT cana-4553 153 6 as as ADP cana-4553 153 7 𝑓(𝑄 𝑓(𝑄 NOUN cana-4553 153 8 ) ) PUNCT cana-4553 153 9 βŠ‚ βŠ‚ PROPN cana-4553 153 10 𝑔(𝑄 𝑔(𝑄 PROPN cana-4553 153 11 ) ) PUNCT cana-4553 153 12 and and CCONJ cana-4553 153 13 𝑓𝛼0 𝑓𝛼0 NOUN cana-4553 153 14 = = SYM cana-4553 153 15 𝑔𝛼1 𝑔𝛼1 NOUN cana-4553 153 16 , , PUNCT cana-4553 153 17 so so SCONJ cana-4553 153 18 a a DET cana-4553 153 19 sequence sequence NOUN cana-4553 153 20 < < X cana-4553 153 21 𝛼𝑛 𝛼𝑛 X cana-4553 153 22 > > X cana-4553 153 23 can can AUX cana-4553 153 24 be be AUX cana-4553 153 25 constructed construct VERB cana-4553 153 26 to to PART cana-4553 153 27 have have VERB cana-4553 153 28 𝑓𝛼𝑛 𝑓𝛼𝑛 ADJ cana-4553 153 29 = = SYM cana-4553 153 30 𝑔𝛼𝑛+1 𝑔𝛼𝑛+1 NOUN cana-4553 153 31 consider,𝑧𝑛 consider,𝑧𝑛 PUNCT cana-4553 154 1 = = PUNCT cana-4553 154 2 𝑓𝛼𝑛 𝑓𝛼𝑛 VERB cana-4553 154 3 communications communication NOUN cana-4553 154 4 on on ADP cana-4553 154 5 applied apply VERB cana-4553 154 6 nonlinear nonlinear ADJ cana-4553 154 7 analysis analysis NOUN cana-4553 154 8 issn issn NOUN cana-4553 154 9 : : PUNCT cana-4553 154 10 1074 1074 NUM cana-4553 154 11 - - PUNCT cana-4553 154 12 133x 133x NUM cana-4553 154 13 vol vol NOUN cana-4553 154 14 32 32 NUM cana-4553 154 15 no no NOUN cana-4553 154 16 . . PUNCT cana-4553 155 1 9s 9s NUM cana-4553 155 2 ( ( PUNCT cana-4553 155 3 2025 2025 NUM cana-4553 155 4 ) ) PUNCT cana-4553 155 5 2761 2761 NUM cana-4553 155 6 https://internationalpubls.com https://internationalpubls.com X cana-4553 155 7 then then ADV cana-4553 155 8 , , PUNCT cana-4553 155 9 ( ( PUNCT cana-4553 155 10 1 1 X cana-4553 155 11 ) ) PUNCT cana-4553 155 12 gives give VERB cana-4553 155 13 , , PUNCT cana-4553 155 14 lim lim PROPN cana-4553 155 15 π‘›β†’βˆž π‘›β†’βˆž NUM cana-4553 156 1 𝑇𝑖 𝑇𝑖 PROPN cana-4553 156 2 ( ( PUNCT cana-4553 156 3 𝐹𝑧0,𝑧1 𝐹𝑧0,𝑧1 PROPN cana-4553 156 4 ( ( PUNCT cana-4553 156 5 π‘Ÿπ‘– π‘Ÿπ‘– NUM cana-4553 156 6 ) ) PUNCT cana-4553 156 7 ) ) PUNCT cana-4553 157 1 = = SYM cana-4553 157 2 1 1 NUM cana-4553 157 3 ; ; PUNCT cana-4553 157 4 𝑖 𝑖 SYM cana-4553 157 5 = = VERB cana-4553 157 6 𝑛 𝑛 DET cana-4553 157 7 π‘‘π‘œ π‘‘π‘œ PROPN cana-4553 157 8 ∞. ∞. PROPN cana-4553 157 9 substituting substitute VERB cana-4553 157 10 𝛼 𝛼 PROPN cana-4553 157 11 = = SYM cana-4553 157 12 𝛼𝑛 𝛼𝑛 PROPN cana-4553 157 13 and and CCONJ cana-4553 157 14 𝛽 𝛽 NOUN cana-4553 157 15 = = SYM cana-4553 157 16 𝛽𝑛+1 𝛽𝑛+1 PROPN cana-4553 157 17 in in ADP cana-4553 157 18 ( ( PUNCT cana-4553 157 19 𝑖 𝑖 NOUN cana-4553 157 20 ) ) PUNCT cana-4553 157 21 , , PUNCT cana-4553 157 22 we we PRON cana-4553 157 23 get get VERB cana-4553 157 24 𝛿(𝐹𝑓𝛼𝑛,𝑓𝛼𝑛+1 𝛿(𝐹𝑓𝛼𝑛,𝑓𝛼𝑛+1 PROPN cana-4553 157 25 ( ( PUNCT cana-4553 157 26 πœƒπ‘₯ πœƒπ‘₯ NOUN cana-4553 157 27 ) ) PUNCT cana-4553 157 28 , , PUNCT cana-4553 157 29 𝐹𝑔𝛼𝑛,𝑔𝛼𝑛+1 𝐹𝑔𝛼𝑛,𝑔𝛼𝑛+1 X cana-4553 157 30 ( ( PUNCT cana-4553 157 31 π‘₯ π‘₯ NOUN cana-4553 157 32 ) ) PUNCT cana-4553 157 33 , , PUNCT cana-4553 157 34 𝐹𝑓𝛼𝑛,𝑔𝛼𝑛 𝐹𝑓𝛼𝑛,𝑔𝛼𝑛 PROPN cana-4553 157 35 ( ( PUNCT cana-4553 157 36 π‘₯ π‘₯ NOUN cana-4553 157 37 ) ) PUNCT cana-4553 157 38 , , PUNCT cana-4553 157 39 𝐹𝑓𝛼𝑛+1,𝑔𝛼𝑛+1 𝐹𝑓𝛼𝑛+1,𝑔𝛼𝑛+1 PROPN cana-4553 157 40 ( ( PUNCT cana-4553 157 41 πœƒπ‘₯ πœƒπ‘₯ NOUN cana-4553 157 42 ) ) PUNCT cana-4553 157 43 ) ) PUNCT cana-4553 157 44 β‰₯ β‰₯ NOUN cana-4553 157 45 0 0 NUM cana-4553 157 46 , , PUNCT cana-4553 157 47 𝑖. 𝑖. ADV cana-4553 157 48 𝑒. 𝑒. PUNCT cana-4553 158 1 𝛿(𝐹𝑧𝑛,𝑧𝑛+1 𝛿(𝐹𝑧𝑛,𝑧𝑛+1 PROPN cana-4553 158 2 ( ( PUNCT cana-4553 158 3 πœƒπ‘₯ πœƒπ‘₯ NOUN cana-4553 158 4 ) ) PUNCT cana-4553 158 5 , , PUNCT cana-4553 158 6 πΉπ‘§π‘›βˆ’1,𝑧𝑛 πΉπ‘§π‘›βˆ’1,𝑧𝑛 PROPN cana-4553 158 7 ( ( PUNCT cana-4553 158 8 π‘₯ π‘₯ NOUN cana-4553 158 9 ) ) PUNCT cana-4553 158 10 , , PUNCT cana-4553 158 11 πΉπ‘§π‘›βˆ’1,𝑧𝑛 πΉπ‘§π‘›βˆ’1,𝑧𝑛 PROPN cana-4553 158 12 ( ( PUNCT cana-4553 158 13 π‘₯ π‘₯ NOUN cana-4553 158 14 ) ) PUNCT cana-4553 158 15 , , PUNCT cana-4553 158 16 𝐹𝑧𝑛,𝑧𝑛+1 𝐹𝑧𝑛,𝑧𝑛+1 PROPN cana-4553 158 17 ( ( PUNCT cana-4553 158 18 πœƒπ‘₯ πœƒπ‘₯ NOUN cana-4553 158 19 ) ) PUNCT cana-4553 158 20 ) ) PUNCT cana-4553 158 21 β‰₯ β‰₯ NOUN cana-4553 158 22 0 0 NUM cana-4553 158 23 and and CCONJ cana-4553 158 24 using use VERB cana-4553 158 25 virtue virtue NOUN cana-4553 158 26 of of ADP cana-4553 158 27 𝛿 𝛿 NOUN cana-4553 158 28 , , PUNCT cana-4553 158 29 for for ADP cana-4553 158 30 all all DET cana-4553 158 31 𝑛 𝑛 ADJ cana-4553 158 32 , , PUNCT cana-4553 158 33 we we PRON cana-4553 158 34 obtain obtain VERB cana-4553 158 35 𝐹𝑧𝑛,𝑧𝑛+1 𝐹𝑧𝑛,𝑧𝑛+1 PROPN cana-4553 158 36 ( ( PUNCT cana-4553 158 37 πœƒπ‘₯ πœƒπ‘₯ NOUN cana-4553 158 38 ) ) PUNCT cana-4553 158 39 β‰₯ β‰₯ NOUN cana-4553 159 1 πΉπ‘§π‘›βˆ’1,𝑧𝑛 πΉπ‘§π‘›βˆ’1,𝑧𝑛 PROPN cana-4553 159 2 ( ( PUNCT cana-4553 159 3 π‘₯ π‘₯ NOUN cana-4553 159 4 ) ) PUNCT cana-4553 159 5 so so ADV cana-4553 159 6 , , PUNCT cana-4553 159 7 by by ADP cana-4553 159 8 above above ADP cana-4553 159 9 lemma lemma PROPN cana-4553 159 10 , , PUNCT cana-4553 159 11 the the DET cana-4553 159 12 sequence sequence NOUN cana-4553 159 13 < < X cana-4553 159 14 𝑧𝑛 𝑧𝑛 X cana-4553 159 15 > > PUNCT cana-4553 159 16 = = X cana-4553 159 17 < < X cana-4553 159 18 𝑓𝛼𝑛 𝑓𝛼𝑛 X cana-4553 159 19 > > X cana-4553 159 20 is be AUX cana-4553 159 21 cauchy cauchy NOUN cana-4553 159 22 . . PUNCT cana-4553 160 1 if if SCONJ cana-4553 160 2 𝑔(𝑄 𝑔(𝑄 PROPN cana-4553 160 3 ) ) PUNCT cana-4553 160 4 is be AUX cana-4553 160 5 𝐹complete 𝐹complete PROPN cana-4553 160 6 , , PUNCT cana-4553 160 7 then then ADV cana-4553 160 8 there there PRON cana-4553 160 9 must must AUX cana-4553 160 10 be be AUX cana-4553 160 11 𝛼 𝛼 PRON cana-4553 160 12 ∈ ∈ PROPN cana-4553 160 13 𝑔(𝑄 𝑔(𝑄 NUM cana-4553 160 14 ) ) PUNCT cana-4553 160 15 , , PUNCT cana-4553 160 16 so so SCONJ cana-4553 160 17 that that SCONJ cana-4553 160 18 , , PUNCT cana-4553 160 19 < < X cana-4553 160 20 𝑧𝑛 𝑧𝑛 X cana-4553 160 21 > > X cana-4553 160 22 β†’ β†’ SYM cana-4553 160 23 𝛼 𝛼 PROPN cana-4553 160 24 , , PUNCT cana-4553 160 25 and and CCONJ cana-4553 160 26 for for ADP cana-4553 160 27 any any DET cana-4553 160 28 𝑧 𝑧 DET cana-4553 160 29 ∈ ∈ NOUN cana-4553 160 30 𝑄 𝑄 NOUN cana-4553 160 31 so so SCONJ cana-4553 160 32 that that SCONJ cana-4553 160 33 𝑔𝑧 𝑔𝑧 X cana-4553 160 34 = = SYM cana-4553 160 35 𝛼. 𝛼. NOUN cana-4553 160 36 substituting substitute VERB cana-4553 160 37 𝛼 𝛼 PROPN cana-4553 160 38 = = SYM cana-4553 160 39 𝛼𝑛 𝛼𝑛 PROPN cana-4553 160 40 and and CCONJ cana-4553 160 41 𝑦 𝑦 NOUN cana-4553 160 42 = = SYM cana-4553 160 43 𝑧 𝑧 ADJ cana-4553 160 44 , , PUNCT cana-4553 160 45 in in ADP cana-4553 160 46 ( ( PUNCT cana-4553 160 47 𝑖 𝑖 X cana-4553 160 48 ) ) PUNCT cana-4553 160 49 , , PUNCT cana-4553 160 50 we we PRON cana-4553 160 51 have have VERB cana-4553 160 52 𝛿(𝐹𝑓𝛼𝑛,𝑓𝑧(πœƒπ‘₯ 𝛿(𝐹𝑓𝛼𝑛,𝑓𝑧(πœƒπ‘₯ NOUN cana-4553 160 53 ) ) PUNCT cana-4553 160 54 , , PUNCT cana-4553 160 55 𝐹𝑔𝛼𝑛,𝑔𝑧(π‘₯ 𝐹𝑔𝛼𝑛,𝑔𝑧(π‘₯ NUM cana-4553 160 56 ) ) PUNCT cana-4553 160 57 , , PUNCT cana-4553 160 58 𝐹𝑓𝛼𝑛,𝑔𝛼𝑛 𝐹𝑓𝛼𝑛,𝑔𝛼𝑛 X cana-4553 160 59 ( ( PUNCT cana-4553 160 60 π‘₯ π‘₯ NOUN cana-4553 160 61 ) ) PUNCT cana-4553 160 62 , , PUNCT cana-4553 160 63 𝐹𝑓𝑧,𝑔𝑧(πœƒπ‘₯ 𝐹𝑓𝑧,𝑔𝑧(πœƒπ‘₯ PROPN cana-4553 160 64 ) ) PUNCT cana-4553 160 65 ) ) PUNCT cana-4553 160 66 β‰₯ β‰₯ X cana-4553 160 67 0 0 PUNCT cana-4553 161 1 𝑖. 𝑖. PROPN cana-4553 161 2 𝑒. 𝑒. PROPN cana-4553 161 3 𝛿(𝐹𝑧𝑛,𝑓𝑧(πœƒπ‘₯ 𝛿(𝐹𝑧𝑛,𝑓𝑧(πœƒπ‘₯ NUM cana-4553 161 4 ) ) PUNCT cana-4553 161 5 , , PUNCT cana-4553 161 6 πΉπ‘§π‘›βˆ’1,𝑔𝑧(π‘₯ πΉπ‘§π‘›βˆ’1,𝑔𝑧(π‘₯ PROPN cana-4553 161 7 ) ) PUNCT cana-4553 161 8 , , PUNCT cana-4553 161 9 πΉπ‘§π‘›βˆ’1,𝑓𝑧(π‘₯ πΉπ‘§π‘›βˆ’1,𝑓𝑧(π‘₯ NUM cana-4553 161 10 ) ) PUNCT cana-4553 161 11 , , PUNCT cana-4553 161 12 𝐹𝑔𝑧,𝑓𝑧(πœƒπ‘₯ 𝐹𝑔𝑧,𝑓𝑧(πœƒπ‘₯ PROPN cana-4553 161 13 ) ) PUNCT cana-4553 161 14 ) ) PUNCT cana-4553 162 1 β‰₯ β‰₯ NOUN cana-4553 162 2 0 0 PUNCT cana-4553 162 3 as as ADP cana-4553 162 4 𝑛 𝑛 PROPN cana-4553 162 5 β†’ β†’ SYM cana-4553 162 6 ∞ ∞ PROPN cana-4553 162 7 , , PUNCT cana-4553 162 8 we we PRON cana-4553 162 9 get get VERB cana-4553 162 10 , , PUNCT cana-4553 162 11 𝛿(𝐹𝑔𝑧,𝑓𝑧(πœƒπ‘₯ 𝛿(𝐹𝑔𝑧,𝑓𝑧(πœƒπ‘₯ PROPN cana-4553 162 12 ) ) PUNCT cana-4553 162 13 , , PUNCT cana-4553 162 14 𝐹𝑔𝑧,𝑔𝑧(π‘₯ 𝐹𝑔𝑧,𝑔𝑧(π‘₯ NOUN cana-4553 162 15 ) ) PUNCT cana-4553 162 16 , , PUNCT cana-4553 162 17 𝐹𝑔𝑧,𝑓𝑧(π‘₯ 𝐹𝑔𝑧,𝑓𝑧(π‘₯ PROPN cana-4553 162 18 ) ) PUNCT cana-4553 162 19 , , PUNCT cana-4553 162 20 𝐹𝑓𝑧,𝑔𝑧(πœƒπ‘₯ 𝐹𝑓𝑧,𝑔𝑧(πœƒπ‘₯ PROPN cana-4553 162 21 ) ) PUNCT cana-4553 162 22 ) ) PUNCT cana-4553 163 1 β‰₯ β‰₯ X cana-4553 163 2 0 0 PUNCT cana-4553 164 1 𝑖. 𝑖. PROPN cana-4553 164 2 𝑒. 𝑒. PROPN cana-4553 164 3 𝛿(𝐹𝑔𝑧,𝑓𝑧(πœƒπ‘₯ 𝛿(𝐹𝑔𝑧,𝑓𝑧(πœƒπ‘₯ NOUN cana-4553 164 4 ) ) PUNCT cana-4553 164 5 , , PUNCT cana-4553 164 6 1 1 NUM cana-4553 164 7 , , PUNCT cana-4553 164 8 𝐹𝑔𝑧,𝑓𝑧(π‘₯ 𝐹𝑔𝑧,𝑓𝑧(π‘₯ NOUN cana-4553 164 9 ) ) PUNCT cana-4553 164 10 , , PUNCT cana-4553 164 11 𝐹𝑓𝑧,𝑔𝑧(πœƒπ‘₯ 𝐹𝑓𝑧,𝑔𝑧(πœƒπ‘₯ PROPN cana-4553 164 12 ) ) PUNCT cana-4553 164 13 ) ) PUNCT cana-4553 165 1 β‰₯ β‰₯ NOUN cana-4553 165 2 0 0 NUM cana-4553 165 3 again again ADV cana-4553 165 4 , , PUNCT cana-4553 165 5 as as ADP cana-4553 165 6 𝐹𝑔𝑧,𝑓𝑧(πœƒπ‘₯ 𝐹𝑔𝑧,𝑓𝑧(πœƒπ‘₯ NOUN cana-4553 165 7 ) ) PUNCT cana-4553 165 8 β‰₯ β‰₯ NOUN cana-4553 165 9 1 1 NUM cana-4553 165 10 , , PUNCT cana-4553 165 11 so so ADV cana-4553 165 12 𝑓𝑧 𝑓𝑧 NOUN cana-4553 165 13 = = SYM cana-4553 165 14 𝑔𝑧 𝑔𝑧 PROPN cana-4553 165 15 = = NOUN cana-4553 165 16 𝛼 𝛼 PRON cana-4553 165 17 taking take VERB cana-4553 165 18 limit limit NOUN cana-4553 165 19 as as ADP cana-4553 165 20 nβ†’ο‚₯ nβ†’ο‚₯ NOUN cana-4553 165 21 , , PUNCT cana-4553 165 22 we we PRON cana-4553 165 23 have have VERB cana-4553 165 24 , , PUNCT cana-4553 165 25 hence hence ADV cana-4553 165 26 , , PUNCT cana-4553 165 27 𝑓 𝑓 PRON cana-4553 165 28 and and CCONJ cana-4553 165 29 𝑔 𝑔 PROPN cana-4553 165 30 have𝑧 have𝑧 PROPN cana-4553 165 31 as as ADP cana-4553 165 32 a a DET cana-4553 165 33 coincidence coincidence NOUN cana-4553 165 34 point point NOUN cana-4553 165 35 and and CCONJ cana-4553 165 36 thus thus ADV cana-4553 165 37 , , PUNCT cana-4553 165 38 for for ADP cana-4553 165 39 𝑓(𝑄 𝑓(𝑄 NOUN cana-4553 165 40 ) ) PUNCT cana-4553 165 41 to to PART cana-4553 165 42 be be AUX cana-4553 165 43 complete complete ADJ cana-4553 165 44 , , PUNCT cana-4553 165 45 we we PRON cana-4553 165 46 have have VERB cana-4553 165 47 < < X cana-4553 165 48 𝑧𝑛 𝑧𝑛 ADP cana-4553 165 49 > > X cana-4553 165 50 β†’ β†’ SYM cana-4553 165 51 𝛼 𝛼 PROPN cana-4553 165 52 ∈ ∈ PROPN cana-4553 165 53 𝑓(𝑄 𝑓(𝑄 PROPN cana-4553 165 54 ) ) PUNCT cana-4553 166 1 βŠ‚ βŠ‚ PROPN cana-4553 166 2 𝑔(𝑄 𝑔(𝑄 PROPN cana-4553 166 3 ) ) PUNCT cana-4553 167 1 so so ADV cana-4553 167 2 , , PUNCT cana-4553 167 3 𝑧 𝑧 PRON cana-4553 167 4 is be AUX cana-4553 167 5 a a DET cana-4553 167 6 common common ADJ cana-4553 167 7 coincidence coincidence NOUN cana-4553 167 8 point point NOUN cana-4553 167 9 . . PUNCT cana-4553 168 1 theorem theorem VERB cana-4553 168 2 4.2 4.2 NUM cana-4553 168 3 : : PUNCT cana-4553 168 4 with with ADP cana-4553 168 5 ( ( PUNCT cana-4553 168 6 𝑃 𝑃 PROPN cana-4553 168 7 , , PUNCT cana-4553 168 8 𝐹 𝐹 PROPN cana-4553 168 9 , , PUNCT cana-4553 168 10 𝑇)a 𝑇)a VERB cana-4553 168 11 generalized generalize VERB cana-4553 168 12 pm pm NOUN cana-4553 168 13 - - PUNCT cana-4553 168 14 space space NOUN cana-4553 168 15 and and CCONJ cana-4553 168 16 a a DET cana-4553 168 17 continuous continuous ADJ cana-4553 168 18 𝑑-norm𝑇 𝑑-norm𝑇 PROPN cana-4553 168 19 in in ADP cana-4553 168 20 ( ( PUNCT cana-4553 168 21 π‘Ž π‘Ž X cana-4553 168 22 , , PUNCT cana-4553 168 23 1 1 NUM cana-4553 168 24 ) ) PUNCT cana-4553 168 25 for for ADP cana-4553 168 26 all all PRON cana-4553 168 27 0 0 NUM cana-4553 168 28 < < X cana-4553 169 1 π‘Ž π‘Ž X cana-4553 169 2 < < X cana-4553 169 3 1 1 NUM cana-4553 169 4 ; ; PUNCT cana-4553 169 5 0 0 NUM cana-4553 169 6 < < X cana-4553 169 7 πœƒ πœƒ X cana-4553 169 8 < < X cana-4553 169 9 1 1 NUM cana-4553 169 10 ; ; PUNCT cana-4553 169 11 𝛿 𝛿 DET cana-4553 169 12 ∈ ∈ NOUN cana-4553 169 13 βˆ† βˆ† PROPN cana-4553 169 14 and and CCONJ cana-4553 169 15 𝑓 𝑓 PRON cana-4553 169 16 , , PUNCT cana-4553 169 17 𝑔 𝑔 NOUN cana-4553 169 18 : : PUNCT cana-4553 169 19 𝑄 𝑄 PROPN cana-4553 169 20 β†’ β†’ SYM cana-4553 169 21 𝑃 𝑃 PROPN cana-4553 169 22 , , PUNCT cana-4553 169 23 we we PRON cana-4553 169 24 have have VERB cana-4553 169 25 ( ( PUNCT cana-4553 169 26 i i NOUN cana-4553 169 27 ) ) PUNCT cana-4553 169 28 . . PUNCT cana-4553 170 1 𝛿(𝐹𝑓𝛼,𝑓𝛽(πœƒπ‘₯ 𝛿(𝐹𝑓𝛼,𝑓𝛽(πœƒπ‘₯ NOUN cana-4553 170 2 ) ) PUNCT cana-4553 170 3 , , PUNCT cana-4553 170 4 𝐹𝑔𝛼,𝑔𝛽(π‘₯ 𝐹𝑔𝛼,𝑔𝛽(π‘₯ NUM cana-4553 170 5 ) ) PUNCT cana-4553 170 6 , , PUNCT cana-4553 170 7 𝐹𝑓𝛼,𝑔𝛼(π‘₯ 𝐹𝑓𝛼,𝑔𝛼(π‘₯ PROPN cana-4553 170 8 ) ) PUNCT cana-4553 170 9 , , PUNCT cana-4553 170 10 𝐹𝑓𝛽,𝑔𝛽(πœƒπ‘₯ 𝐹𝑓𝛽,𝑔𝛽(πœƒπ‘₯ NOUN cana-4553 170 11 ) ) PUNCT cana-4553 170 12 ) ) PUNCT cana-4553 171 1 β‰₯ β‰₯ NOUN cana-4553 171 2 0 0 NUM cana-4553 171 3 for for ADP cana-4553 171 4 every every DET cana-4553 171 5 𝛼 𝛼 NOUN cana-4553 171 6 , , PUNCT cana-4553 171 7 𝛽 𝛽 PROPN cana-4553 171 8 ∈ ∈ PROPN cana-4553 171 9 𝑄 𝑄 PROPN cana-4553 171 10 and and CCONJ cana-4553 171 11 positive positive ADJ cana-4553 171 12 values value NOUN cana-4553 171 13 of of ADP cana-4553 171 14 π‘₯ π‘₯ PROPN cana-4553 171 15 ( ( PUNCT cana-4553 171 16 ii ii NOUN cana-4553 171 17 ) ) PUNCT cana-4553 171 18 . . PUNCT cana-4553 172 1 𝑓(𝑃 𝑓(𝑃 PROPN cana-4553 172 2 ) ) PUNCT cana-4553 173 1 βŠ‚ βŠ‚ PROPN cana-4553 173 2 𝑔(𝑃 𝑔(𝑃 PROPN cana-4553 173 3 ) ) PUNCT cana-4553 173 4 ( ( PUNCT cana-4553 173 5 iii iii NOUN cana-4553 173 6 ) ) PUNCT cana-4553 173 7 . . PUNCT cana-4553 174 1 βˆƒ βˆƒ PROPN cana-4553 174 2 𝛼0 𝛼0 PROPN cana-4553 174 3 , , PUNCT cana-4553 174 4 𝛼1 𝛼1 NOUN cana-4553 174 5 so so SCONJ cana-4553 174 6 that that DET cana-4553 174 7 𝑓𝛼0 𝑓𝛼0 NOUN cana-4553 174 8 = = SYM cana-4553 174 9 𝑔𝛼1 𝑔𝛼1 NOUN cana-4553 174 10 and and CCONJ cana-4553 174 11 lim lim PROPN cana-4553 174 12 π‘›β†’βˆž π‘›β†’βˆž NUM cana-4553 174 13 𝑇𝑖 𝑇𝑖 PROPN cana-4553 174 14 ( ( PUNCT cana-4553 174 15 𝐹𝑓𝛼0,𝑓𝛼1 𝐹𝑓𝛼0,𝑓𝛼1 ADJ cana-4553 174 16 ( ( PUNCT cana-4553 174 17 π‘Ÿπ‘– π‘Ÿπ‘– NOUN cana-4553 174 18 ) ) PUNCT cana-4553 174 19 ) ) PUNCT cana-4553 175 1 = = PUNCT cana-4553 175 2 1for 1for NUM cana-4553 175 3 π‘Ÿ π‘Ÿ X cana-4553 175 4 > > SYM cana-4553 175 5 1 1 NUM cana-4553 175 6 and and CCONJ cana-4553 175 7 𝑖 𝑖 SYM cana-4553 175 8 = = SYM cana-4553 175 9 𝑛 𝑛 DET cana-4553 175 10 π‘‘π‘œ π‘‘π‘œ NOUN cana-4553 175 11 ∞ ∞ NUM cana-4553 175 12 ( ( PUNCT cana-4553 175 13 iv iv NOUN cana-4553 175 14 ) ) PUNCT cana-4553 175 15 . . PUNCT cana-4553 176 1 either either CCONJ cana-4553 176 2 𝑓(𝑃 𝑓(𝑃 PROPN cana-4553 176 3 ) ) PUNCT cana-4553 176 4 or or CCONJ cana-4553 176 5 𝑔(𝑃 𝑔(𝑃 PROPN cana-4553 176 6 ) ) PUNCT cana-4553 176 7 is be AUX cana-4553 176 8 𝐹complete 𝐹complete PROPN cana-4553 176 9 ( ( PUNCT cana-4553 176 10 v v NOUN cana-4553 176 11 ) ) PUNCT cana-4553 176 12 . . PUNCT cana-4553 177 1 𝑓 𝑓 X cana-4553 177 2 and and CCONJ cana-4553 177 3 𝑔 𝑔 PROPN cana-4553 177 4 are be AUX cana-4553 177 5 commuting commute VERB cana-4553 177 6 functions function NOUN cana-4553 177 7 that that PRON cana-4553 177 8 commute commute VERB cana-4553 177 9 at at ADP cana-4553 177 10 their their PRON cana-4553 177 11 coincidence coincidence NOUN cana-4553 177 12 point point NOUN cana-4553 177 13 . . PUNCT cana-4553 178 1 then then ADV cana-4553 178 2 , , PUNCT cana-4553 178 3 𝑓 𝑓 PRON cana-4553 178 4 and and CCONJ cana-4553 178 5 𝑔 𝑔 AUX cana-4553 178 6 have have VERB cana-4553 178 7 a a DET cana-4553 178 8 common common ADJ cana-4553 178 9 invariant invariant ADJ cana-4553 178 10 point point NOUN cana-4553 178 11 that that PRON cana-4553 178 12 is be AUX cana-4553 178 13 unique unique ADJ cana-4553 178 14 in in ADP cana-4553 178 15 , , PUNCT cana-4553 178 16 𝑃. 𝑃. PROPN cana-4553 178 17 proof proof NOUN cana-4553 178 18 : : PUNCT cana-4553 178 19 let let VERB cana-4553 178 20 us we PRON cana-4553 178 21 choose choose VERB cana-4553 178 22 𝑃 𝑃 NOUN cana-4553 178 23 = = PUNCT cana-4553 178 24 𝑄 𝑄 PROPN cana-4553 178 25 in in ADP cana-4553 178 26 previous previous ADJ cana-4553 178 27 theorem theorem NOUN cana-4553 178 28 , , PUNCT cana-4553 178 29 then then ADV cana-4553 178 30 we we PRON cana-4553 178 31 obtain obtain VERB cana-4553 178 32 𝑧𝑛 𝑧𝑛 SCONJ cana-4553 178 33 = = PUNCT cana-4553 178 34 𝑓𝛼𝑛 𝑓𝛼𝑛 ADJ cana-4553 178 35 and and CCONJ cana-4553 178 36 sequence sequence NOUN cana-4553 178 37 < < X cana-4553 178 38 𝑧𝑛 𝑧𝑛 X cana-4553 178 39 > > X cana-4553 178 40 is be AUX cana-4553 178 41 cauchy cauchy PROPN cana-4553 178 42 . . PUNCT cana-4553 179 1 if if SCONJ cana-4553 179 2 𝑔(𝑃 𝑔(𝑃 PROPN cana-4553 179 3 ) ) PUNCT cana-4553 179 4 is be AUX cana-4553 179 5 𝐹complete 𝐹complete PROPN cana-4553 179 6 , , PUNCT cana-4553 179 7 then then ADV cana-4553 179 8 𝑧𝑛 𝑧𝑛 ADP cana-4553 179 9 β†’ β†’ SYM cana-4553 179 10 𝛼 𝛼 PROPN cana-4553 179 11 ∈ ∈ PROPN cana-4553 179 12 𝑔(𝑃 𝑔(𝑃 PROPN cana-4553 179 13 ) ) PUNCT cana-4553 179 14 , , PUNCT cana-4553 179 15 and and CCONJ cana-4553 179 16 we we PRON cana-4553 179 17 have have VERB cana-4553 179 18 𝑧 𝑧 PRON cana-4553 179 19 ∈ ∈ NOUN cana-4553 179 20 𝑃 𝑃 NOUN cana-4553 179 21 so so SCONJ cana-4553 179 22 that that SCONJ cana-4553 179 23 𝑔(𝑧 𝑔(𝑧 NOUN cana-4553 179 24 ) ) PUNCT cana-4553 180 1 = = SYM cana-4553 180 2 𝛼. 𝛼. NOUN cana-4553 180 3 replacing replace VERB cana-4553 180 4 𝛼 𝛼 PRON cana-4553 180 5 by by ADP cana-4553 180 6 𝛼𝑛 𝛼𝑛 ADP cana-4553 180 7 and and CCONJ cana-4553 180 8 𝛽 𝛽 NOUN cana-4553 180 9 by by ADP cana-4553 180 10 𝑧 𝑧 PROPN cana-4553 180 11 in in ADP cana-4553 180 12 inequality(𝑖 inequality(𝑖 PROPN cana-4553 180 13 ) ) PUNCT cana-4553 180 14 we we PRON cana-4553 180 15 have have VERB cana-4553 180 16 ; ; PUNCT cana-4553 180 17 𝑓𝑧 𝑓𝑧 ADP cana-4553 180 18 = = SYM cana-4553 180 19 𝑔𝑧 𝑔𝑧 PROPN cana-4553 180 20 = = SYM cana-4553 180 21 𝛼. 𝛼. NOUN cana-4553 180 22 as as SCONJ cana-4553 180 23 𝑓 𝑓 PROPN cana-4553 180 24 and and CCONJ cana-4553 180 25 𝑔 𝑔 PROPN cana-4553 180 26 are be AUX cana-4553 180 27 commuting commute VERB cana-4553 180 28 , , PUNCT cana-4553 180 29 therefore therefore ADV cana-4553 180 30 , , PUNCT cana-4553 180 31 we we PRON cana-4553 180 32 must must AUX cana-4553 180 33 have have VERB cana-4553 180 34 , , PUNCT cana-4553 180 35 𝑓𝑔𝑧 𝑓𝑔𝑧 PROPN cana-4553 180 36 = = SYM cana-4553 180 37 𝑔𝑓𝑧 𝑔𝑓𝑧 PROPN cana-4553 180 38 , , PUNCT cana-4553 180 39 so so ADV cana-4553 180 40 that,𝑓𝛼 that,𝑓𝛼 NOUN cana-4553 181 1 = = PRON cana-4553 181 2 𝑔𝛼. 𝑔𝛼. PROPN cana-4553 181 3 again again ADV cana-4553 181 4 , , PUNCT cana-4553 181 5 using use VERB cana-4553 181 6 𝛼 𝛼 PROPN cana-4553 181 7 = = PUNCT cana-4553 181 8 𝑧 𝑧 PROPN cana-4553 181 9 and and CCONJ cana-4553 181 10 𝛽 𝛽 NOUN cana-4553 181 11 = = SYM cana-4553 181 12 𝑓𝑧 𝑓𝑧 NOUN cana-4553 181 13 in in ADP cana-4553 181 14 inequality inequality NOUN cana-4553 181 15 ( ( PUNCT cana-4553 181 16 𝑖 𝑖 X cana-4553 181 17 ) ) PUNCT cana-4553 181 18 , , PUNCT cana-4553 181 19 we we PRON cana-4553 181 20 get get VERB cana-4553 181 21 𝛿(𝐹𝑓𝑧,𝑓𝑓𝑧(πœƒπ‘₯ 𝛿(𝐹𝑓𝑧,𝑓𝑓𝑧(πœƒπ‘₯ NOUN cana-4553 181 22 ) ) PUNCT cana-4553 181 23 , , PUNCT cana-4553 181 24 𝐹𝑔𝑧,𝑔𝑓𝑧(π‘₯ 𝐹𝑔𝑧,𝑔𝑓𝑧(π‘₯ PROPN cana-4553 181 25 ) ) PUNCT cana-4553 181 26 , , PUNCT cana-4553 181 27 𝐹𝑔𝑧,𝑓𝑧(π‘₯ 𝐹𝑔𝑧,𝑓𝑧(π‘₯ NOUN cana-4553 181 28 ) ) PUNCT cana-4553 181 29 , , PUNCT cana-4553 181 30 𝐹𝑓𝑓𝑧,𝑔𝑓𝑧(πœƒπ‘₯ 𝐹𝑓𝑓𝑧,𝑔𝑓𝑧(πœƒπ‘₯ NUM cana-4553 181 31 ) ) PUNCT cana-4553 181 32 ) ) PUNCT cana-4553 182 1 β‰₯ β‰₯ NOUN cana-4553 182 2 0 0 NUM cana-4553 182 3 communications communication NOUN cana-4553 182 4 on on ADP cana-4553 182 5 applied apply VERB cana-4553 182 6 nonlinear nonlinear ADJ cana-4553 182 7 analysis analysis NOUN cana-4553 182 8 issn issn NOUN cana-4553 182 9 : : PUNCT cana-4553 182 10 1074 1074 NUM cana-4553 182 11 - - PUNCT cana-4553 182 12 133x 133x NUM cana-4553 182 13 vol vol NOUN cana-4553 182 14 32 32 NUM cana-4553 182 15 no no NOUN cana-4553 182 16 . . PUNCT cana-4553 183 1 9s 9s NUM cana-4553 183 2 ( ( PUNCT cana-4553 183 3 2025 2025 NUM cana-4553 183 4 ) ) PUNCT cana-4553 183 5 2762 2762 NUM cana-4553 183 6 https://internationalpubls.com https://internationalpubls.com X cana-4553 183 7 𝑖. 𝑖. PROPN cana-4553 183 8 𝑒. 𝑒. PROPN cana-4553 183 9 𝛿(𝐹𝛼,𝑓𝛼(πœƒπ‘₯ 𝛿(𝐹𝛼,𝑓𝛼(πœƒπ‘₯ PROPN cana-4553 183 10 ) ) PUNCT cana-4553 183 11 , , PUNCT cana-4553 183 12 𝐹𝛼,𝑓𝛼(π‘₯ 𝐹𝛼,𝑓𝛼(π‘₯ NUM cana-4553 183 13 ) ) PUNCT cana-4553 183 14 , , PUNCT cana-4553 183 15 1 1 NUM cana-4553 183 16 , , PUNCT cana-4553 183 17 𝐹𝑓𝛼,𝑔𝛼(πœƒπ‘₯ 𝐹𝑓𝛼,𝑔𝛼(πœƒπ‘₯ PROPN cana-4553 183 18 ) ) PUNCT cana-4553 183 19 ) ) PUNCT cana-4553 184 1 β‰₯ β‰₯ NOUN cana-4553 184 2 0 0 PUNCT cana-4553 185 1 using use VERB cana-4553 185 2 the the DET cana-4553 185 3 properties property NOUN cana-4553 185 4 of of ADP cana-4553 185 5 distance distance NOUN cana-4553 185 6 metric(𝛿 metric(𝛿 NOUN cana-4553 185 7 ) ) PUNCT cana-4553 185 8 for for ADP cana-4553 185 9 all all DET cana-4553 185 10 positive,π‘₯ positive,π‘₯ PROPN cana-4553 185 11 we we PRON cana-4553 185 12 obtain obtain VERB cana-4553 185 13 , , PUNCT cana-4553 185 14 𝐹𝛼,𝑓𝛼(πœƒπ‘₯ 𝐹𝛼,𝑓𝛼(πœƒπ‘₯ NOUN cana-4553 185 15 ) ) PUNCT cana-4553 185 16 β‰₯ β‰₯ NOUN cana-4553 185 17 𝐹𝛼,𝑓𝛼(π‘₯ 𝐹𝛼,𝑓𝛼(π‘₯ X cana-4553 185 18 ) ) PUNCT cana-4553 185 19 which which PRON cana-4553 185 20 is be AUX cana-4553 185 21 true true ADJ cana-4553 185 22 only only ADV cana-4553 185 23 if if SCONJ cana-4553 185 24 𝛼 𝛼 PROPN cana-4553 185 25 = = SYM cana-4553 185 26 𝑓𝛼 𝑓𝛼 ADP cana-4553 185 27 , , PUNCT cana-4553 185 28 otherwise otherwise ADV cana-4553 185 29 we we PRON cana-4553 185 30 reach reach VERB cana-4553 185 31 to to ADP cana-4553 185 32 a a DET cana-4553 185 33 contradiction contradiction NOUN cana-4553 185 34 with with ADP cana-4553 185 35 respect respect NOUN cana-4553 185 36 to to ADP cana-4553 185 37 lemma lemma PROPN cana-4553 185 38 4.1 4.1 NUM cana-4553 185 39 . . PUNCT cana-4553 186 1 this this PRON cana-4553 186 2 implies imply VERB cana-4553 186 3 that that SCONJ cana-4553 186 4 , , PUNCT cana-4553 186 5 𝛼 𝛼 PRON cana-4553 186 6 is be AUX cana-4553 186 7 a a DET cana-4553 186 8 common common ADJ cana-4553 186 9 invariant invariant ADJ cana-4553 186 10 point point NOUN cana-4553 186 11 of of ADP cana-4553 186 12 𝑓 𝑓 PRON cana-4553 186 13 and and CCONJ cana-4553 186 14 𝑔. 𝑔. NOUN cana-4553 186 15 to to PART cana-4553 186 16 prove prove VERB cana-4553 186 17 the the DET cana-4553 186 18 uniqueness uniqueness NOUN cana-4553 186 19 , , PUNCT cana-4553 186 20 we we PRON cana-4553 186 21 suppose suppose VERB cana-4553 186 22 𝛼 𝛼 PRON cana-4553 186 23 and and CCONJ cana-4553 186 24 𝛽 𝛽 NOUN cana-4553 186 25 are be AUX cana-4553 186 26 two two NUM cana-4553 186 27 common common ADJ cana-4553 186 28 invariant invariant ADJ cana-4553 186 29 points point NOUN cana-4553 186 30 of of ADP cana-4553 186 31 𝑓 𝑓 PRON cana-4553 186 32 and and CCONJ cana-4553 186 33 𝑔. 𝑔. NOUN cana-4553 186 34 using use VERB cana-4553 186 35 ( ( PUNCT cana-4553 186 36 𝑖 𝑖 X cana-4553 186 37 ) ) PUNCT cana-4553 186 38 , , PUNCT cana-4553 186 39 we we PRON cana-4553 186 40 get get VERB cana-4553 186 41 𝛿(𝐹𝛼,𝛽(πœƒπ‘₯ 𝛿(𝐹𝛼,𝛽(πœƒπ‘₯ NOUN cana-4553 186 42 ) ) PUNCT cana-4553 186 43 , , PUNCT cana-4553 186 44 𝐹𝛼,𝛽(π‘₯ 𝐹𝛼,𝛽(π‘₯ NUM cana-4553 186 45 ) ) PUNCT cana-4553 186 46 , , PUNCT cana-4553 186 47 𝐹𝛼,𝛼(π‘₯ 𝐹𝛼,𝛼(π‘₯ NOUN cana-4553 186 48 ) ) PUNCT cana-4553 186 49 , , PUNCT cana-4553 186 50 𝐹𝛽,𝛽(πœƒπ‘₯ 𝐹𝛽,𝛽(πœƒπ‘₯ NUM cana-4553 186 51 ) ) PUNCT cana-4553 186 52 ) ) PUNCT cana-4553 187 1 β‰₯ β‰₯ NOUN cana-4553 187 2 0β‡’ 0β‡’ NUM cana-4553 187 3 𝐹𝛼,𝛽(πœƒπ‘₯ 𝐹𝛼,𝛽(πœƒπ‘₯ PROPN cana-4553 187 4 ) ) PUNCT cana-4553 187 5 β‰₯ β‰₯ NOUN cana-4553 187 6 1 1 NUM cana-4553 187 7 β‡’ β‡’ NOUN cana-4553 187 8 𝛼 𝛼 NOUN cana-4553 187 9 = = NOUN cana-4553 187 10 𝛽. 𝛽. NOUN cana-4553 187 11 this this PRON cana-4553 187 12 shows show VERB cana-4553 187 13 the the DET cana-4553 187 14 uniqueness uniqueness NOUN cana-4553 187 15 of of ADP cana-4553 187 16 the the DET cana-4553 187 17 result result NOUN cana-4553 187 18 . . PUNCT cana-4553 188 1 definition definition NOUN cana-4553 188 2 4.1 4.1 NUM cana-4553 188 3 : : PUNCT cana-4553 189 1 [ [ X cana-4553 189 2 6 6 NUM cana-4553 189 3 ] ] PUNCT cana-4553 189 4 aforementioned aforementione VERB cana-4553 189 5 t t PROPN cana-4553 189 6 – – PUNCT cana-4553 189 7 norm norm NOUN cana-4553 189 8 t t PROPN cana-4553 189 9 is be AUX cana-4553 189 10 said say VERB cana-4553 189 11 hadΕΎiΔ‡ hadΕΎiΔ‡ NOUN cana-4553 189 12 type type NOUN cana-4553 189 13 or or CCONJ cana-4553 189 14 𝐻type 𝐻type PROPN cana-4553 189 15 , , PUNCT cana-4553 189 16 denoted denote VERB cana-4553 189 17 as as ADP cana-4553 189 18 𝑇 𝑇 PROPN cana-4553 189 19 ∈ ∈ PROPN cana-4553 189 20 𝐻 𝐻 PROPN cana-4553 189 21 , , PUNCT cana-4553 189 22 if if SCONJ cana-4553 189 23 the the DET cana-4553 189 24 group group NOUN cana-4553 189 25 { { PUNCT cana-4553 189 26 𝑇𝑛}π‘›βˆˆπ‘ 𝑇𝑛}π‘›βˆˆπ‘ PROPN cana-4553 189 27 of of ADP cana-4553 189 28 it it PRON cana-4553 189 29 iterates iterate VERB cana-4553 189 30 is be AUX cana-4553 189 31 inductively inductively ADV cana-4553 189 32 depicted depict VERB cana-4553 189 33 , , PUNCT cana-4553 189 34 βˆ€π‘₯ βˆ€π‘₯ PROPN cana-4553 189 35 ∈ ∈ PROPN cana-4553 190 1 [ [ X cana-4553 190 2 0,1 0,1 NUM cana-4553 190 3 ] ] PUNCT cana-4553 190 4 by by ADP cana-4553 190 5 , , PUNCT cana-4553 190 6 𝑇0(π‘₯ 𝑇0(π‘₯ NUM cana-4553 190 7 ) ) PUNCT cana-4553 191 1 = = SYM cana-4553 191 2 1 1 NUM cana-4553 191 3 , , PUNCT cana-4553 191 4 𝑇𝑛+1(π‘₯ 𝑇𝑛+1(π‘₯ NOUN cana-4553 191 5 ) ) PUNCT cana-4553 191 6 = = SYM cana-4553 191 7 𝑇(𝑇𝑛(π‘₯ 𝑇(𝑇𝑛(π‘₯ PROPN cana-4553 191 8 ) ) PUNCT cana-4553 191 9 , , PUNCT cana-4553 191 10 π‘₯ π‘₯ NOUN cana-4553 191 11 ) ) PUNCT cana-4553 191 12 , , PUNCT cana-4553 191 13 for for ADP cana-4553 191 14 all all DET cana-4553 191 15 non non ADJ cana-4553 191 16 - - ADJ cana-4553 191 17 negative negative ADJ cana-4553 191 18 π‘₯ π‘₯ NOUN cana-4553 191 19 and and CCONJ cana-4553 191 20 as as SCONJ cana-4553 191 21 𝑇𝑛 𝑇𝑛 PROPN cana-4553 191 22 is be AUX cana-4553 191 23 defined define VERB cana-4553 191 24 , , PUNCT cana-4553 191 25 it it PRON cana-4553 191 26 is be AUX cana-4553 191 27 equi equi NOUN cana-4553 191 28 - - PUNCT cana-4553 191 29 continuous continuous ADJ cana-4553 191 30 at at ADP cana-4553 191 31 π‘₯ π‘₯ NOUN cana-4553 191 32 = = SYM cana-4553 191 33 1 1 X cana-4553 191 34 . . PUNCT cana-4553 192 1 this this PRON cana-4553 192 2 states state VERB cana-4553 192 3 the the DET cana-4553 192 4 following following NOUN cana-4553 192 5 , , PUNCT cana-4553 192 6 νœ€ νœ€ PROPN cana-4553 192 7 ∈ ∈ PROPN cana-4553 192 8 ( ( PUNCT cana-4553 192 9 0,1)βˆƒ 0,1)βˆƒ NUM cana-4553 192 10 a a DET cana-4553 192 11 πœ— πœ— NOUN cana-4553 192 12 ∈ ∈ NOUN cana-4553 192 13 ( ( PUNCT cana-4553 192 14 0,1 0,1 NOUN cana-4553 192 15 ) ) PUNCT cana-4553 192 16 , , PUNCT cana-4553 192 17 so so SCONJ cana-4553 192 18 that that SCONJ cana-4553 192 19 , , PUNCT cana-4553 192 20 π‘₯ π‘₯ PROPN cana-4553 192 21 > > X cana-4553 192 22 1 1 NUM cana-4553 192 23 βˆ’ βˆ’ PROPN cana-4553 192 24 πœ—β‡’ πœ—β‡’ INTJ cana-4553 192 25 𝑇𝑛(π‘₯ 𝑇𝑛(π‘₯ PROPN cana-4553 192 26 ) ) PUNCT cana-4553 192 27 > > X cana-4553 193 1 1 1 NUM cana-4553 193 2 βˆ’ βˆ’ NOUN cana-4553 193 3 νœ€ νœ€ NOUN cana-4553 193 4 , , PUNCT cana-4553 193 5 for for ADP cana-4553 193 6 all all DET cana-4553 193 7 𝑛 𝑛 PRON cana-4553 193 8 β‰₯ β‰₯ NUM cana-4553 193 9 1 1 NUM cana-4553 193 10 . . PUNCT cana-4553 194 1 the the DET cana-4553 194 2 statement statement NOUN cana-4553 194 3 is be AUX cana-4553 194 4 followed follow VERB cana-4553 194 5 by by ADP cana-4553 194 6 hadΕΎiΔ‡ hadΕΎiΔ‡ NOUN cana-4553 194 7 : : PUNCT cana-4553 194 8 statement statement NOUN cana-4553 194 9 4.1 4.1 NUM cana-4553 194 10 : : PUNCT cana-4553 195 1 [ [ X cana-4553 195 2 6 6 NUM cana-4553 195 3 ] ] PUNCT cana-4553 195 4 ( ( PUNCT cana-4553 195 5 i i NOUN cana-4553 195 6 ) ) PUNCT cana-4553 195 7 . . PUNCT cana-4553 196 1 if if SCONJ cana-4553 196 2 we we PRON cana-4553 196 3 take take VERB cana-4553 196 4 𝑇 𝑇 PROPN cana-4553 196 5 β‰₯ β‰₯ NUM cana-4553 196 6 𝑇𝐿 𝑇𝐿 PROPN cana-4553 196 7 , , PUNCT cana-4553 196 8 then then ADV cana-4553 196 9 for for ADP cana-4553 196 10 𝑖 𝑖 PRON cana-4553 196 11 = = SYM cana-4553 196 12 𝑛 𝑛 PRON cana-4553 196 13 π‘‘π‘œ π‘‘π‘œ NOUN cana-4553 196 14 ∞ ∞ PROPN cana-4553 196 15 , , PUNCT cana-4553 196 16 lim lim PROPN cana-4553 196 17 π‘›β†’βˆž π‘›β†’βˆž NUM cana-4553 196 18 𝑇𝑖π‘₯𝑛+1 𝑇𝑖π‘₯𝑛+1 NOUN cana-4553 196 19 = = SYM cana-4553 196 20 1 1 NUM cana-4553 196 21 , , PUNCT cana-4553 196 22 both both DET cana-4553 196 23 sides side NOUN cana-4553 196 24 imply imply VERB cana-4553 196 25 βˆ‘ βˆ‘ PROPN cana-4553 196 26 1 1 NUM cana-4553 196 27 βˆ’ βˆ’ NOUN cana-4553 196 28 π‘₯𝑛 π‘₯𝑛 PROPN cana-4553 196 29 < < X cana-4553 196 30 ∞ ∞ PROPN cana-4553 196 31 and and CCONJ cana-4553 196 32 𝑇 𝑇 PROPN cana-4553 196 33 = = SYM cana-4553 196 34 𝑇¡ 𝑇¡ PROPN cana-4553 196 35 π‘†π‘Š π‘†π‘Š NOUN cana-4553 196 36 ( ( PUNCT cana-4553 196 37 by by ADP cana-4553 196 38 𝑇 𝑇 PROPN cana-4553 196 39 β‰₯ β‰₯ NUM cana-4553 196 40 𝑇𝐿 𝑇𝐿 PROPN cana-4553 196 41 𝑖. 𝑖. ADV cana-4553 196 42 𝑒. 𝑒. PROPN cana-4553 197 1 π‘Ž π‘Ž DET cana-4553 197 2 β‰₯ β‰₯ NOUN cana-4553 197 3 𝑐 𝑐 NOUN cana-4553 197 4 , , PUNCT cana-4553 197 5 𝑏 𝑏 PRON cana-4553 197 6 β‰₯ β‰₯ NOUN cana-4553 197 7 𝑑⇒ 𝑑⇒ PROPN cana-4553 197 8 𝑇(π‘Ž 𝑇(π‘Ž PROPN cana-4553 197 9 , , PUNCT cana-4553 197 10 𝑏 𝑏 NOUN cana-4553 197 11 ) ) PUNCT cana-4553 197 12 = = SYM cana-4553 198 1 𝑇(𝑐 𝑇(𝑐 PROPN cana-4553 198 2 , , PUNCT cana-4553 198 3 𝑑 𝑑 NOUN cana-4553 198 4 ) ) PUNCT cana-4553 198 5 ) ) PUNCT cana-4553 198 6 . . PUNCT cana-4553 199 1 ( ( PUNCT cana-4553 199 2 ii ii NOUN cana-4553 199 3 ) ) PUNCT cana-4553 199 4 . . PUNCT cana-4553 200 1 when when SCONJ cana-4553 200 2 , , PUNCT cana-4553 200 3 𝑇 𝑇 PROPN cana-4553 200 4 ∈ ∈ PROPN cana-4553 200 5 𝐻 𝐻 PROPN cana-4553 200 6 , , PUNCT cana-4553 200 7 then then ADV cana-4553 200 8 for for SCONJ cana-4553 200 9 each each PRON cana-4553 200 10 ( ( PUNCT cana-4553 200 11 π‘₯𝑛)π‘›βˆˆπ‘ π‘₯𝑛)π‘›βˆˆπ‘ PROPN cana-4553 200 12 in in ADP cana-4553 200 13 [ [ X cana-4553 200 14 0,1 0,1 NUM cana-4553 200 15 ] ] PUNCT cana-4553 200 16 , , PUNCT cana-4553 200 17 lim lim PROPN cana-4553 200 18 π‘›β†’βˆž π‘›β†’βˆž NUM cana-4553 200 19 π‘₯𝑛 π‘₯𝑛 PROPN cana-4553 200 20 = = SYM cana-4553 200 21 1β‡’ 1β‡’ PROPN cana-4553 200 22 lim lim PROPN cana-4553 200 23 π‘›β†’βˆž π‘›β†’βˆž NUM cana-4553 200 24 𝑇𝑖π‘₯𝑛+𝑖 𝑇𝑖π‘₯𝑛+𝑖 PROPN cana-4553 200 25 = = PUNCT cana-4553 201 1 1,for 1,for NUM cana-4553 201 2 𝑖 𝑖 SYM cana-4553 201 3 = = VERB cana-4553 201 4 𝑛 𝑛 DET cana-4553 201 5 π‘‘π‘œ π‘‘π‘œ PROPN cana-4553 201 6 ∞. ∞. PROPN cana-4553 201 7 ( ( PUNCT cana-4553 201 8 iii iii NOUN cana-4553 201 9 ) ) PUNCT cana-4553 201 10 . . PUNCT cana-4553 202 1 when when SCONJ cana-4553 202 2 , , PUNCT cana-4553 202 3 𝑇 𝑇 PROPN cana-4553 202 4 ∈ ∈ PROPN cana-4553 202 5 π‘‡πœ‡ π‘‡πœ‡ PROPN cana-4553 202 6 𝐷 𝐷 PROPN cana-4553 202 7 , , PUNCT cana-4553 202 8 π‘‡πœ‡ π‘‡πœ‡ PROPN cana-4553 202 9 𝐴𝐴𝑇 𝐴𝐴𝑇 PROPN cana-4553 202 10 , , PUNCT cana-4553 202 11 then then ADV cana-4553 202 12 for for ADP cana-4553 202 13 𝑖 𝑖 PRON cana-4553 202 14 = = SYM cana-4553 202 15 𝑛 𝑛 PRON cana-4553 202 16 π‘‘π‘œ π‘‘π‘œ NOUN cana-4553 202 17 ∞ ∞ PROPN cana-4553 202 18 , , PUNCT cana-4553 202 19 lim lim PROPN cana-4553 202 20 π‘›β†’βˆž π‘›β†’βˆž AUX cana-4553 202 21 𝑇𝑖π‘₯𝑛+𝑖 𝑇𝑖π‘₯𝑛+𝑖 PROPN cana-4553 202 22 = = SYM cana-4553 202 23 1 1 NUM cana-4553 202 24 , , PUNCT cana-4553 202 25 both both DET cana-4553 202 26 sides side NOUN cana-4553 202 27 imply imply VERB cana-4553 202 28 βˆ‘(1 βˆ‘(1 ADJ cana-4553 202 29 βˆ’ βˆ’ PROPN cana-4553 202 30 π‘₯𝑛)πœ‡ π‘₯𝑛)πœ‡ PROPN cana-4553 202 31 < < X cana-4553 202 32 ∞. ∞. PROPN cana-4553 202 33 we we PRON cana-4553 202 34 wish wish VERB cana-4553 202 35 to to PART cana-4553 202 36 repeat repeat VERB cana-4553 202 37 the the DET cana-4553 202 38 notion notion NOUN cana-4553 202 39 of of ADP cana-4553 202 40 distribution distribution NOUN cana-4553 202 41 functions function NOUN cana-4553 202 42 and and CCONJ cana-4553 202 43 pms pm NOUN cana-4553 202 44 for for ADP cana-4553 202 45 the the DET cana-4553 202 46 interval[0 interval[0 NOUN cana-4553 202 47 , , PUNCT cana-4553 202 48 ] ] PUNCT cana-4553 202 49 ο‚₯ ο‚₯ X cana-4553 202 50 . . PUNCT cana-4553 203 1 definition definition NOUN cana-4553 203 2 4.2 4.2 NUM cana-4553 203 3 : : PUNCT cana-4553 204 1 [ [ X cana-4553 204 2 6 6 NUM cana-4553 204 3 ] ] PUNCT cana-4553 204 4 consider consider VERB cana-4553 204 5 a a DET cana-4553 204 6 class class NOUN cana-4553 204 7 of of ADP cana-4553 204 8 distribution distribution NOUN cana-4553 204 9 map𝐿+as map𝐿+as NOUN cana-4553 204 10 𝐹 𝐹 PROPN cana-4553 204 11 : : PUNCT cana-4553 205 1 [ [ X cana-4553 205 2 0 0 NUM cana-4553 205 3 , , PUNCT cana-4553 205 4 ∞ ∞ PROPN cana-4553 205 5 ] ] PUNCT cana-4553 205 6 β†’ β†’ PUNCT cana-4553 205 7 [ [ X cana-4553 205 8 0 0 NUM cana-4553 205 9 , , PUNCT cana-4553 205 10 ∞ ∞ PROPN cana-4553 205 11 ] ] PUNCT cana-4553 205 12 with with ADP cana-4553 205 13 , , PUNCT cana-4553 205 14 ( ( PUNCT cana-4553 205 15 i i NOUN cana-4553 205 16 ) ) PUNCT cana-4553 205 17 . . PUNCT cana-4553 206 1 𝐹 𝐹 PROPN cana-4553 206 2 has have VERB cana-4553 206 3 zero zero NUM cana-4553 206 4 value value NOUN cana-4553 206 5 at at ADP cana-4553 206 6 π‘₯ π‘₯ NOUN cana-4553 206 7 = = SYM cana-4553 206 8 0 0 NUM cana-4553 206 9 ( ( PUNCT cana-4553 206 10 ii ii NOUN cana-4553 206 11 ) ) PUNCT cana-4553 206 12 . . PUNCT cana-4553 207 1 𝐹 𝐹 PROPN cana-4553 207 2 is be AUX cana-4553 207 3 a a DET cana-4553 207 4 non non ADJ cana-4553 207 5 - - ADJ cana-4553 207 6 decreasing decrease VERB cana-4553 207 7 map map NOUN cana-4553 207 8 and and CCONJ cana-4553 207 9 ( ( PUNCT cana-4553 207 10 iii iii NOUN cana-4553 207 11 ) ) PUNCT cana-4553 207 12 . . PUNCT cana-4553 208 1 𝐹 𝐹 PROPN cana-4553 208 2 is be AUX cana-4553 208 3 left leave VERB cana-4553 208 4 continuous continuous ADJ cana-4553 208 5 for for ADP cana-4553 208 6 0 0 NUM cana-4553 208 7 < < X cana-4553 208 8 π‘₯ π‘₯ X cana-4553 208 9 < < X cana-4553 208 10 ∞. ∞. PROPN cana-4553 208 11 let let VERB cana-4553 208 12 𝐷+ 𝐷+ VERB cana-4553 208 13 βŠ‚ βŠ‚ PROPN cana-4553 208 14 𝐿+ 𝐿+ PROPN cana-4553 208 15 contains𝐹 contains𝐹 PROPN cana-4553 208 16 , , PUNCT cana-4553 208 17 so so SCONJ cana-4553 208 18 that that SCONJ cana-4553 208 19 , , PUNCT cana-4553 208 20 lim lim PROPN cana-4553 208 21 π‘›β†’βˆž π‘›β†’βˆž NUM cana-4553 208 22 𝐹(π‘₯ 𝐹(π‘₯ NUM cana-4553 208 23 ) ) PUNCT cana-4553 208 24 = = SYM cana-4553 208 25 1 1 X cana-4553 208 26 . . PUNCT cana-4553 208 27 then then ADV cana-4553 208 28 , , PUNCT cana-4553 208 29 the the DET cana-4553 208 30 distribution distribution NOUN cana-4553 208 31 function function NOUN cana-4553 208 32 , , PUNCT cana-4553 208 33 νœ€0 νœ€0 NOUN cana-4553 208 34 is be AUX cana-4553 208 35 defined define VERB cana-4553 208 36 as as ADP cana-4553 208 37 νœ€0 νœ€0 NOUN cana-4553 208 38 νœ€0 νœ€0 NOUN cana-4553 208 39 = = SYM cana-4553 208 40 { { PUNCT cana-4553 208 41 0 0 NUM cana-4553 208 42 , , PUNCT cana-4553 208 43 π‘₯ π‘₯ NOUN cana-4553 208 44 = = SYM cana-4553 208 45 0 0 NUM cana-4553 208 46 1 1 NUM cana-4553 208 47 , , PUNCT cana-4553 208 48 π‘₯ π‘₯ PROPN cana-4553 208 49 > > X cana-4553 208 50 0 0 PUNCT cana-4553 208 51 lies lie NOUN cana-4553 208 52 in in ADP cana-4553 208 53 𝐷+ 𝐷+ NOUN cana-4553 208 54 . . PUNCT cana-4553 209 1 statement statement NOUN cana-4553 209 2 4.2 4.2 NUM cana-4553 209 3 : : PUNCT cana-4553 209 4 consider consider VERB cana-4553 209 5 ( ( PUNCT cana-4553 209 6 𝑃 𝑃 NOUN cana-4553 209 7 , , PUNCT cana-4553 209 8 𝐹 𝐹 PROPN cana-4553 209 9 , , PUNCT cana-4553 209 10 𝑇 𝑇 PROPN cana-4553 209 11 ) ) PUNCT cana-4553 209 12 asa asa PROPN cana-4553 209 13 generalized generalize VERB cana-4553 209 14 pm pm NOUN cana-4553 209 15 - - PUNCT cana-4553 209 16 space space NOUN cana-4553 209 17 and and CCONJ cana-4553 209 18 a a DET cana-4553 209 19 continuous continuous ADJ cana-4553 209 20 𝑑-norm𝑇 𝑑-norm𝑇 PROPN cana-4553 209 21 ∈ ∈ PROPN cana-4553 209 22 𝐻 𝐻 NOUN cana-4553 209 23 with with ADP cana-4553 209 24 0 0 NUM cana-4553 209 25 < < X cana-4553 209 26 πœƒ πœƒ X cana-4553 209 27 < < X cana-4553 209 28 1 1 NUM cana-4553 209 29 ; ; PUNCT cana-4553 209 30 𝛿 𝛿 DET cana-4553 209 31 ∈ ∈ NOUN cana-4553 209 32 βˆ† βˆ† PROPN cana-4553 209 33 and and CCONJ cana-4553 209 34 𝑓 𝑓 PRON cana-4553 209 35 , , PUNCT cana-4553 209 36 𝑔 𝑔 ADJ cana-4553 209 37 : : PUNCT cana-4553 209 38 𝑃 𝑃 NOUN cana-4553 209 39 β†’ β†’ SYM cana-4553 209 40 𝑃 𝑃 NOUN cana-4553 209 41 , , PUNCT cana-4553 209 42 so so SCONJ cana-4553 209 43 that that SCONJ cana-4553 209 44 , , PUNCT cana-4553 209 45 ( ( PUNCT cana-4553 209 46 i i NOUN cana-4553 209 47 ) ) PUNCT cana-4553 209 48 . . PUNCT cana-4553 210 1 𝛿(𝐹𝑓𝛼,𝑓𝛽(πœƒπ‘₯ 𝛿(𝐹𝑓𝛼,𝑓𝛽(πœƒπ‘₯ NOUN cana-4553 210 2 ) ) PUNCT cana-4553 210 3 , , PUNCT cana-4553 210 4 𝐹𝑔𝛼,𝑔𝛽(π‘₯ 𝐹𝑔𝛼,𝑔𝛽(π‘₯ NUM cana-4553 210 5 ) ) PUNCT cana-4553 210 6 , , PUNCT cana-4553 210 7 𝐹𝑓𝛼,𝑔𝛼(π‘₯ 𝐹𝑓𝛼,𝑔𝛼(π‘₯ PROPN cana-4553 210 8 ) ) PUNCT cana-4553 210 9 , , PUNCT cana-4553 210 10 𝐹𝑓𝛽,𝑔𝛽(πœƒπ‘₯ 𝐹𝑓𝛽,𝑔𝛽(πœƒπ‘₯ NOUN cana-4553 210 11 ) ) PUNCT cana-4553 210 12 ) ) PUNCT cana-4553 211 1 β‰₯ β‰₯ NOUN cana-4553 211 2 0 0 NUM cana-4553 211 3 for for ADP cana-4553 211 4 every every DET cana-4553 211 5 𝛼 𝛼 NOUN cana-4553 211 6 , , PUNCT cana-4553 211 7 𝛽 𝛽 PROPN cana-4553 211 8 ∈ ∈ NOUN cana-4553 211 9 𝑃 𝑃 NOUN cana-4553 211 10 and and CCONJ cana-4553 211 11 positive positive ADJ cana-4553 211 12 values value NOUN cana-4553 211 13 of of ADP cana-4553 211 14 π‘₯ π‘₯ PROPN cana-4553 211 15 ( ( PUNCT cana-4553 211 16 ii ii NOUN cana-4553 211 17 ) ) PUNCT cana-4553 211 18 . . PUNCT cana-4553 212 1 𝑓(𝑃 𝑓(𝑃 PROPN cana-4553 212 2 ) ) PUNCT cana-4553 213 1 βŠ‚ βŠ‚ PROPN cana-4553 213 2 𝑔(𝑃 𝑔(𝑃 PROPN cana-4553 213 3 ) ) PUNCT cana-4553 213 4 ( ( PUNCT cana-4553 213 5 iii iii NOUN cana-4553 213 6 ) ) PUNCT cana-4553 213 7 . . PUNCT cana-4553 214 1 βˆƒ βˆƒ PROPN cana-4553 214 2 𝛼0 𝛼0 PROPN cana-4553 214 3 , , PUNCT cana-4553 214 4 𝛼1 𝛼1 NOUN cana-4553 214 5 so so SCONJ cana-4553 214 6 that that PRON cana-4553 214 7 𝑓𝛼0 𝑓𝛼0 NOUN cana-4553 214 8 = = SYM cana-4553 214 9 𝑔𝛼1 𝑔𝛼1 NOUN cana-4553 214 10 for for ADP cana-4553 214 11 which which PRON cana-4553 214 12 𝐹𝑓𝛼0,𝑓𝛼1 𝐹𝑓𝛼0,𝑓𝛼1 ADJ cana-4553 214 13 ∈ ∈ PROPN cana-4553 214 14 𝐷+ 𝐷+ X cana-4553 214 15 ( ( PUNCT cana-4553 214 16 iv iv NUM cana-4553 214 17 ) ) PUNCT cana-4553 214 18 . . PUNCT cana-4553 215 1 either either CCONJ cana-4553 215 2 𝑓(𝑃 𝑓(𝑃 PROPN cana-4553 215 3 ) ) PUNCT cana-4553 215 4 or or CCONJ cana-4553 215 5 𝑔(𝑃 𝑔(𝑃 PROPN cana-4553 215 6 ) ) PUNCT cana-4553 215 7 is be AUX cana-4553 215 8 𝐹complete 𝐹complete PROPN cana-4553 215 9 communications communication NOUN cana-4553 215 10 on on ADP cana-4553 215 11 applied apply VERB cana-4553 215 12 nonlinear nonlinear ADJ cana-4553 215 13 analysis analysis NOUN cana-4553 215 14 issn issn NOUN cana-4553 215 15 : : PUNCT cana-4553 215 16 1074 1074 NUM cana-4553 215 17 - - PUNCT cana-4553 215 18 133x 133x NUM cana-4553 215 19 vol vol NOUN cana-4553 215 20 32 32 NUM cana-4553 215 21 no no NOUN cana-4553 215 22 . . PUNCT cana-4553 216 1 9s 9s NUM cana-4553 216 2 ( ( PUNCT cana-4553 216 3 2025 2025 NUM cana-4553 216 4 ) ) PUNCT cana-4553 216 5 2763 2763 NUM cana-4553 216 6 https://internationalpubls.com https://internationalpubls.com X cana-4553 216 7 ( ( PUNCT cana-4553 216 8 vi vi NOUN cana-4553 216 9 ) ) PUNCT cana-4553 216 10 . . PUNCT cana-4553 217 1 𝑓 𝑓 X cana-4553 217 2 and and CCONJ cana-4553 217 3 𝑔 𝑔 PROPN cana-4553 217 4 are be AUX cana-4553 217 5 commuting commute VERB cana-4553 217 6 functions function NOUN cana-4553 217 7 that that PRON cana-4553 217 8 commute commute VERB cana-4553 217 9 at at ADP cana-4553 217 10 their their PRON cana-4553 217 11 coincidence coincidence NOUN cana-4553 217 12 point point NOUN cana-4553 217 13 . . PUNCT cana-4553 218 1 then then ADV cana-4553 218 2 , , PUNCT cana-4553 218 3 𝑓 𝑓 PRON cana-4553 218 4 and and CCONJ cana-4553 218 5 𝑔 𝑔 AUX cana-4553 218 6 have have VERB cana-4553 218 7 a a DET cana-4553 218 8 unique unique ADJ cana-4553 218 9 common common ADJ cana-4553 218 10 invariant invariant ADJ cana-4553 218 11 point point NOUN cana-4553 218 12 in in ADP cana-4553 218 13 𝑃. 𝑃. PROPN cana-4553 218 14 proof proof NOUN cana-4553 218 15 : : PUNCT cana-4553 218 16 in in ADP cana-4553 218 17 theorem theorem NOUN cana-4553 218 18 4.1 4.1 NUM cana-4553 218 19 , , PUNCT cana-4553 218 20 if if SCONJ cana-4553 218 21 we we PRON cana-4553 218 22 consider consider VERB cana-4553 218 23 a a DET cana-4553 218 24 πœ‡ πœ‡ X cana-4553 218 25 > > X cana-4553 218 26 1 1 NUM cana-4553 218 27 with with ADP cana-4553 218 28 a a DET cana-4553 218 29 sequence sequence NOUN cana-4553 218 30 < < X cana-4553 218 31 𝑧𝑛 𝑧𝑛 X cana-4553 218 32 > > X cana-4553 218 33 along along ADP cana-4553 218 34 with with ADP cana-4553 218 35 a a DET cana-4553 218 36 mapping mapping NOUN cana-4553 218 37 , , PUNCT cana-4553 218 38 𝑓𝑧0,𝑧1 𝑓𝑧0,𝑧1 PROPN cana-4553 218 39 ∈ ∈ PROPN cana-4553 218 40 𝐷+ 𝐷+ NOUN cana-4553 218 41 , , PUNCT cana-4553 218 42 so so SCONJ cana-4553 218 43 that that SCONJ cana-4553 218 44 , , PUNCT cana-4553 218 45 lim lim PROPN cana-4553 218 46 π‘›β†’βˆž π‘›β†’βˆž NUM cana-4553 218 47 𝐹𝑧0,𝑧1 𝐹𝑧0,𝑧1 PROPN cana-4553 218 48 ( ( PUNCT cana-4553 218 49 πœ‡π‘› πœ‡π‘› NOUN cana-4553 218 50 ) ) PUNCT cana-4553 218 51 = = SYM cana-4553 218 52 1 1 X cana-4553 218 53 . . PUNCT cana-4553 218 54 therefore therefore ADV cana-4553 218 55 , , PUNCT cana-4553 218 56 using use VERB cana-4553 218 57 ( ( PUNCT cana-4553 218 58 𝑖𝑖 𝑖𝑖 NOUN cana-4553 218 59 ) ) PUNCT cana-4553 218 60 of of ADP cana-4553 218 61 statement statement NOUN cana-4553 218 62 4.1 4.1 NUM cana-4553 218 63 , , PUNCT cana-4553 218 64 we we PRON cana-4553 218 65 get get VERB cana-4553 218 66 lim lim PROPN cana-4553 218 67 π‘›β†’βˆž π‘›β†’βˆž NUM cana-4553 219 1 𝑇𝑖 𝑇𝑖 PROPN cana-4553 219 2 ( ( PUNCT cana-4553 219 3 𝐹𝑧0,𝑧1 𝐹𝑧0,𝑧1 PROPN cana-4553 219 4 ( ( PUNCT cana-4553 219 5 πœ‡π‘– πœ‡π‘– ADP cana-4553 219 6 ) ) PUNCT cana-4553 219 7 ) ) PUNCT cana-4553 219 8 = = PUNCT cana-4553 220 1 1 1 NUM cana-4553 220 2 ; ; PUNCT cana-4553 220 3 𝑖 𝑖 SYM cana-4553 220 4 = = SYM cana-4553 220 5 𝑛 𝑛 DET cana-4553 220 6 π‘‘π‘œ π‘‘π‘œ NOUN cana-4553 220 7 ∞. ∞. PROPN cana-4553 220 8 hence hence ADV cana-4553 220 9 , , PUNCT cana-4553 220 10 theorem theorem VERB cana-4553 220 11 4.2 4.2 NUM cana-4553 220 12 follows follow VERB cana-4553 220 13 the the DET cana-4553 220 14 required require VERB cana-4553 220 15 results result NOUN cana-4553 220 16 . . PUNCT cana-4553 221 1 5 5 X cana-4553 221 2 . . X cana-4553 221 3 discussion discussion NOUN cana-4553 221 4 probabilistic probabilistic ADJ cana-4553 221 5 spaces space NOUN cana-4553 221 6 offer offer VERB cana-4553 221 7 a a DET cana-4553 221 8 powerful powerful ADJ cana-4553 221 9 framework framework NOUN cana-4553 221 10 for for ADP cana-4553 221 11 handling handle VERB cana-4553 221 12 the the DET cana-4553 221 13 uncertainties uncertainty NOUN cana-4553 221 14 and and CCONJ cana-4553 221 15 variability variability NOUN cana-4553 221 16 inherent inherent ADJ cana-4553 221 17 in in ADP cana-4553 221 18 real real ADJ cana-4553 221 19 - - PUNCT cana-4553 221 20 world world NOUN cana-4553 221 21 data datum NOUN cana-4553 221 22 [ [ X cana-4553 221 23 12 12 NUM cana-4553 221 24 ] ] PUNCT cana-4553 221 25 . . PUNCT cana-4553 222 1 by by ADP cana-4553 222 2 incorporating incorporate VERB cana-4553 222 3 probability probability NOUN cana-4553 222 4 distributions distribution NOUN cana-4553 222 5 into into ADP cana-4553 222 6 the the DET cana-4553 222 7 metric metric ADJ cana-4553 222 8 space space NOUN cana-4553 222 9 , , PUNCT cana-4553 222 10 we we PRON cana-4553 222 11 can can AUX cana-4553 222 12 model model VERB cana-4553 222 13 the the DET cana-4553 222 14 stochastic stochastic ADJ cana-4553 222 15 nature nature NOUN cana-4553 222 16 of of ADP cana-4553 222 17 data datum NOUN cana-4553 222 18 more more ADV cana-4553 222 19 effectively effectively ADV cana-4553 222 20 . . PUNCT cana-4553 223 1 the the DET cana-4553 223 2 jungck jungck PROPN cana-4553 223 3 contraction contraction NOUN cana-4553 223 4 theorem theorem VERB cana-4553 223 5 provides provide VERB cana-4553 223 6 a a DET cana-4553 223 7 robust robust ADJ cana-4553 223 8 tool tool NOUN cana-4553 223 9 for for ADP cana-4553 223 10 analysing analyse VERB cana-4553 223 11 the the DET cana-4553 223 12 convergence convergence NOUN cana-4553 223 13 of of ADP cana-4553 223 14 iterative iterative NOUN cana-4553 223 15 processes process NOUN cana-4553 223 16 in in ADP cana-4553 223 17 probabilistic probabilistic ADJ cana-4553 223 18 metric metric ADJ cana-4553 223 19 spaces space NOUN cana-4553 223 20 , , PUNCT cana-4553 223 21 which which PRON cana-4553 223 22 is be AUX cana-4553 223 23 crucial crucial ADJ cana-4553 223 24 for for ADP cana-4553 223 25 the the DET cana-4553 223 26 stability stability NOUN cana-4553 223 27 and and CCONJ cana-4553 223 28 reliability reliability NOUN cana-4553 223 29 of of ADP cana-4553 223 30 ml ml ADP cana-4553 223 31 algorithms algorithm NOUN cana-4553 223 32 . . PUNCT cana-4553 224 1 this this DET cana-4553 224 2 work work NOUN cana-4553 224 3 has have AUX cana-4553 224 4 demonstrated demonstrate VERB cana-4553 224 5 the the DET cana-4553 224 6 application application NOUN cana-4553 224 7 of of ADP cana-4553 224 8 probabilistic probabilistic ADJ cana-4553 224 9 metric metric ADJ cana-4553 224 10 spaces space NOUN cana-4553 224 11 and and CCONJ cana-4553 224 12 the the DET cana-4553 224 13 jungck jungck NOUN cana-4553 224 14 contraction contraction NOUN cana-4553 224 15 theorem theorem VERB cana-4553 224 16 in in ADP cana-4553 224 17 enhancing enhance VERB cana-4553 224 18 the the DET cana-4553 224 19 robustness robustness NOUN cana-4553 224 20 and and CCONJ cana-4553 224 21 convergence convergence NOUN cana-4553 224 22 properties property NOUN cana-4553 224 23 of of ADP cana-4553 224 24 ml ml NOUN cana-4553 224 25 algorithms algorithm NOUN cana-4553 224 26 . . PUNCT cana-4553 225 1 through through ADP cana-4553 225 2 case case NOUN cana-4553 225 3 studies study NOUN cana-4553 225 4 on on ADP cana-4553 225 5 probabilistic probabilistic ADJ cana-4553 225 6 k k PROPN cana-4553 225 7 - - PUNCT cana-4553 225 8 means mean VERB cana-4553 225 9 clustering cluster VERB cana-4553 225 10 and and CCONJ cana-4553 225 11 probabilistic probabilistic ADJ cana-4553 225 12 neural neural ADJ cana-4553 225 13 networks network NOUN cana-4553 225 14 , , PUNCT cana-4553 225 15 we we PRON cana-4553 225 16 have have AUX cana-4553 225 17 illustrated illustrate VERB cana-4553 225 18 the the DET cana-4553 225 19 practical practical ADJ cana-4553 225 20 benefits benefit NOUN cana-4553 225 21 of of ADP cana-4553 225 22 this this DET cana-4553 225 23 approach approach NOUN cana-4553 225 24 . . PUNCT cana-4553 226 1 future future ADJ cana-4553 226 2 research research NOUN cana-4553 226 3 can can AUX cana-4553 226 4 extend extend VERB cana-4553 226 5 these these DET cana-4553 226 6 concepts concept NOUN cana-4553 226 7 to to ADP cana-4553 226 8 other other ADJ cana-4553 226 9 ml ml NOUN cana-4553 226 10 algorithms algorithm NOUN cana-4553 226 11 and and CCONJ cana-4553 226 12 explore explore VERB cana-4553 226 13 the the DET cana-4553 226 14 integration integration NOUN cana-4553 226 15 of of ADP cana-4553 226 16 probabilistic probabilistic ADJ cana-4553 226 17 metric metric ADJ cana-4553 226 18 spaces space NOUN cana-4553 226 19 with with ADP cana-4553 226 20 other other ADJ cana-4553 226 21 probabilistic probabilistic ADJ cana-4553 226 22 modelling modelling NOUN cana-4553 226 23 techniques technique NOUN cana-4553 226 24 . . PUNCT cana-4553 227 1 in in ADP cana-4553 227 2 a a DET cana-4553 227 3 nutshell nutshell NOUN cana-4553 227 4 , , PUNCT cana-4553 227 5 the the DET cana-4553 227 6 probabilistic probabilistic ADJ cana-4553 227 7 metric metric ADJ cana-4553 227 8 space space NOUN cana-4553 227 9 approach approach NOUN cana-4553 227 10 , , PUNCT cana-4553 227 11 underpinned underpin VERB cana-4553 227 12 by by ADP cana-4553 227 13 the the DET cana-4553 227 14 jungck jungck PROPN cana-4553 227 15 contraction contraction PROPN cana-4553 227 16 theorem theorem VERB cana-4553 227 17 , , PUNCT cana-4553 227 18 offers offer VERB cana-4553 227 19 a a DET cana-4553 227 20 promising promising ADJ cana-4553 227 21 direction direction NOUN cana-4553 227 22 for for ADP cana-4553 227 23 developing develop VERB cana-4553 227 24 more more ADV cana-4553 227 25 robust robust ADJ cana-4553 227 26 and and CCONJ cana-4553 227 27 reliable reliable ADJ cana-4553 227 28 ml ml ADP cana-4553 227 29 algorithms algorithm NOUN cana-4553 227 30 in in ADP cana-4553 227 31 the the DET cana-4553 227 32 face face NOUN cana-4553 227 33 of of ADP cana-4553 227 34 uncertainty uncertainty NOUN cana-4553 227 35 and and CCONJ cana-4553 227 36 variability variability NOUN cana-4553 227 37 . . PUNCT cana-4553 228 1 references reference NOUN cana-4553 228 2 [ [ X cana-4553 228 3 1 1 X cana-4553 228 4 ] ] PUNCT cana-4553 228 5 s. s. PROPN cana-4553 228 6 banach banach PROPN cana-4553 228 7 , , PUNCT cana-4553 228 8 β€œ " PUNCT cana-4553 228 9 sur sur X cana-4553 228 10 les les X cana-4553 228 11 opΓ©rations opΓ©ration NOUN cana-4553 228 12 dans dan NOUN cana-4553 228 13 les les X cana-4553 228 14 ensembles ensemble NOUN cana-4553 228 15 abstraits abstrait NOUN cana-4553 228 16 et et PROPN cana-4553 228 17 leur leur X cana-4553 228 18 application application PROPN cana-4553 228 19 aux aux PROPN cana-4553 228 20 Γ©quationsintΓ©grales Γ©quationsintΓ©grales PROPN cana-4553 228 21 , , PUNCT cana-4553 228 22 ” " PUNCT cana-4553 228 23 fundamenta fundamenta PROPN cana-4553 228 24 mathematicae mathematicae PROPN cana-4553 228 25 , , PUNCT cana-4553 228 26 3(1 3(1 NUM cana-4553 228 27 ) ) PUNCT cana-4553 228 28 , , PUNCT cana-4553 228 29 pp.133 pp.133 NOUN cana-4553 228 30 - - ADJ cana-4553 228 31 181,1922 181,1922 NUM cana-4553 228 32 . . PUNCT cana-4553 229 1 [ [ X cana-4553 229 2 2 2 NUM cana-4553 229 3 ] ] PUNCT cana-4553 229 4 k. k. PROPN cana-4553 229 5 menger menger PROPN cana-4553 229 6 , , PUNCT cana-4553 229 7 β€œ " PUNCT cana-4553 229 8 statistical statistical ADJ cana-4553 229 9 metrics metric NOUN cana-4553 229 10 , , PUNCT cana-4553 229 11 ” " PUNCT cana-4553 229 12 proceedings proceeding NOUN cana-4553 229 13 of of ADP cana-4553 229 14 the the DET cana-4553 229 15 national national PROPN cana-4553 229 16 academy academy PROPN cana-4553 229 17 of of ADP cana-4553 229 18 sciences sciences PROPN cana-4553 229 19 of of ADP cana-4553 229 20 the the DET cana-4553 229 21 united united PROPN cana-4553 229 22 states states PROPN cana-4553 229 23 of of ADP cana-4553 229 24 america america PROPN cana-4553 229 25 , , PUNCT cana-4553 229 26 28(12 28(12 NUM cana-4553 229 27 ) ) PUNCT cana-4553 229 28 , , PUNCT cana-4553 229 29 pp.535 pp.535 PROPN cana-4553 229 30 - - PUNCT cana-4553 229 31 537 537 NUM cana-4553 229 32 , , PUNCT cana-4553 229 33 1942 1942 NUM cana-4553 229 34 . . PUNCT cana-4553 230 1 [ [ X cana-4553 230 2 3 3 X cana-4553 230 3 ] ] PUNCT cana-4553 230 4 e. e. PROPN cana-4553 230 5 fix fix PROPN cana-4553 230 6 , , PUNCT cana-4553 230 7 and and CCONJ cana-4553 230 8 j. j. PROPN cana-4553 230 9 l. l. PROPN cana-4553 230 10 hodges hodges PROPN cana-4553 230 11 , , PUNCT cana-4553 230 12 β€œ " PUNCT cana-4553 230 13 discriminatory discriminatory ADJ cana-4553 230 14 analysis analysis NOUN cana-4553 230 15 - - PUNCT cana-4553 230 16 nonparametric nonparametric NOUN cana-4553 230 17 discrimination discrimination NOUN cana-4553 230 18 : : PUNCT cana-4553 230 19 consistency consistency NOUN cana-4553 230 20 properties property NOUN cana-4553 230 21 . . PUNCT cana-4553 231 1 report report NOUN cana-4553 231 2 number number NOUN cana-4553 231 3 4 4 NUM cana-4553 231 4 , , PUNCT cana-4553 231 5 project project NOUN cana-4553 231 6 21 21 NUM cana-4553 231 7 - - SYM cana-4553 231 8 49 49 NUM cana-4553 231 9 - - PUNCT cana-4553 231 10 004 004 NOUN cana-4553 231 11 , , PUNCT cana-4553 231 12 ” " PUNCT cana-4553 231 13 usaf usaf ADJ cana-4553 231 14 school school NOUN cana-4553 231 15 of of ADP cana-4553 231 16 aviation aviation NOUN cana-4553 231 17 medicine medicine NOUN cana-4553 231 18 , , PUNCT cana-4553 231 19 randolph randolph PROPN cana-4553 231 20 field field NOUN cana-4553 231 21 , , PUNCT cana-4553 231 22 texas,1951 texas,1951 NOUN cana-4553 231 23 . . PUNCT cana-4553 232 1 [ [ X cana-4553 232 2 4 4 X cana-4553 232 3 ] ] X cana-4553 232 4 g. g. PROPN cana-4553 232 5 jungck jungck PROPN cana-4553 232 6 , , PUNCT cana-4553 232 7 β€œ " PUNCT cana-4553 232 8 compatible compatible ADJ cana-4553 232 9 mappings mapping NOUN cana-4553 232 10 and and CCONJ cana-4553 232 11 common common ADJ cana-4553 232 12 fixed fix VERB cana-4553 232 13 points point NOUN cana-4553 232 14 , , PUNCT cana-4553 232 15 ” " PUNCT cana-4553 232 16 international international ADJ cana-4553 232 17 journal journal NOUN cana-4553 232 18 of of ADP cana-4553 232 19 mathematics mathematics PROPN cana-4553 232 20 and and CCONJ cana-4553 232 21 mathematical mathematical ADJ cana-4553 232 22 sciences science NOUN cana-4553 232 23 , , PUNCT cana-4553 232 24 1(4 1(4 NUM cana-4553 232 25 ) ) PUNCT cana-4553 232 26 , , PUNCT cana-4553 232 27 pp.215 pp.215 NOUN cana-4553 232 28 - - PUNCT cana-4553 232 29 222,1976 222,1976 NOUN cana-4553 232 30 . . PUNCT cana-4553 233 1 [ [ X cana-4553 233 2 5 5 NUM cana-4553 233 3 ] ] PUNCT cana-4553 233 4 c. c. PROPN cana-4553 233 5 c. c. PROPN cana-4553 233 6 y. y. PROPN cana-4553 233 7 dorea dorea PROPN cana-4553 233 8 , , PUNCT cana-4553 233 9 β€œ " PUNCT cana-4553 233 10 on on ADP cana-4553 233 11 probabilistic probabilistic ADJ cana-4553 233 12 metric metric ADJ cana-4553 233 13 spaces space NOUN cana-4553 233 14 , , PUNCT cana-4553 233 15 ” " PUNCT cana-4553 233 16 bulletin bulletin NOUN cana-4553 233 17 of of ADP cana-4553 233 18 the the DET cana-4553 233 19 brazilian brazilian ADJ cana-4553 233 20 mathematical mathematical ADJ cana-4553 233 21 society society NOUN cana-4553 233 22 , , PUNCT cana-4553 233 23 14(2 14(2 NUM cana-4553 233 24 ) ) PUNCT cana-4553 233 25 , , PUNCT cana-4553 233 26 pp.99 pp.99 NOUN cana-4553 233 27 - - PUNCT cana-4553 233 28 118,1983 118,1983 NUM cana-4553 233 29 . . PUNCT cana-4553 234 1 [ [ X cana-4553 234 2 6 6 NUM cana-4553 234 3 ] ] X cana-4553 234 4 o. o. ADJ cana-4553 234 5 hadΕΎiΔ‡ hadΕΎiΔ‡ PROPN cana-4553 234 6 and and CCONJ cana-4553 234 7 e. e. PROPN cana-4553 234 8 pap pap PROPN cana-4553 234 9 , , PUNCT cana-4553 234 10 β€œ " PUNCT cana-4553 234 11 probabilistic probabilistic ADJ cana-4553 234 12 metric metric ADJ cana-4553 234 13 spaces space NOUN cana-4553 234 14 . . PUNCT cana-4553 235 1 in in ADP cana-4553 235 2 : : PUNCT cana-4553 235 3 fixed fixed ADJ cana-4553 235 4 point point NOUN cana-4553 235 5 theory theory NOUN cana-4553 235 6 in in ADP cana-4553 235 7 probabilistic probabilistic ADJ cana-4553 235 8 metric metric ADJ cana-4553 235 9 spaces space NOUN cana-4553 235 10 , , PUNCT cana-4553 235 11 ” " PUNCT cana-4553 235 12 mathematics mathematic NOUN cana-4553 235 13 and and CCONJ cana-4553 235 14 its its PRON cana-4553 235 15 appl appl NOUN cana-4553 235 16 . . PROPN cana-4553 236 1 vol vol NOUN cana-4553 236 2 . . PROPN cana-4553 237 1 536 536 NUM cana-4553 237 2 , , PUNCT cana-4553 237 3 springer springer NOUN cana-4553 237 4 , , PUNCT cana-4553 237 5 dordrecht dordrecht PROPN cana-4553 237 6 , , PUNCT cana-4553 237 7 pp.47 pp.47 PROPN cana-4553 237 8 - - PUNCT cana-4553 237 9 94 94 NUM cana-4553 237 10 , , PUNCT cana-4553 237 11 2001.https://doi.org/10.1007/978-94-017-1560-7_2 2001.https://doi.org/10.1007/978-94-017-1560-7_2 NUM cana-4553 238 1 [ [ X cana-4553 238 2 7 7 NUM cana-4553 238 3 ] ] X cana-4553 238 4 o. o. PROPN cana-4553 238 5 banerjee banerjee PROPN cana-4553 238 6 , , PUNCT cana-4553 238 7 s. s. PROPN cana-4553 238 8 merugu merugu PROPN cana-4553 238 9 , , PUNCT cana-4553 238 10 i. i. PROPN cana-4553 238 11 s. s. PROPN cana-4553 238 12 dhillon dhillon PROPN cana-4553 238 13 , , PUNCT cana-4553 238 14 and and CCONJ cana-4553 238 15 j. j. PROPN cana-4553 238 16 ghosh ghosh PROPN cana-4553 238 17 , , PUNCT cana-4553 238 18 " " PUNCT cana-4553 238 19 clustering cluster VERB cana-4553 238 20 with with ADP cana-4553 238 21 bregman bregman NOUN cana-4553 238 22 divergences divergence NOUN cana-4553 238 23 . . PUNCT cana-4553 238 24 " " PUNCT cana-4553 239 1 journal journal NOUN cana-4553 239 2 of of ADP cana-4553 239 3 machine machine NOUN cana-4553 239 4 learning learn VERB cana-4553 239 5 research research NOUN cana-4553 239 6 , , PUNCT cana-4553 239 7 vol vol NOUN cana-4553 239 8 . . PROPN cana-4553 239 9 6 6 NUM cana-4553 239 10 , , PUNCT cana-4553 239 11 pp pp ADJ cana-4553 239 12 . . PUNCT cana-4553 240 1 1705 1705 NUM cana-4553 240 2 - - SYM cana-4553 240 3 1749 1749 NUM cana-4553 240 4 , , PUNCT cana-4553 240 5 2005 2005 NUM cana-4553 240 6 . . PUNCT cana-4553 241 1 [ [ X cana-4553 241 2 8 8 X cana-4553 241 3 ] ] X cana-4553 241 4 t. t. PROPN cana-4553 241 5 hastie hastie PROPN cana-4553 241 6 , , PUNCT cana-4553 241 7 r. r. PROPN cana-4553 241 8 tibshirani tibshirani PROPN cana-4553 241 9 , , PUNCT cana-4553 241 10 j. j. PROPN cana-4553 241 11 friedman friedman PROPN cana-4553 241 12 , , PUNCT cana-4553 241 13 β€œ " PUNCT cana-4553 241 14 the the DET cana-4553 241 15 elements element NOUN cana-4553 241 16 of of ADP cana-4553 241 17 statistical statistical ADJ cana-4553 241 18 learning learning NOUN cana-4553 241 19 : : PUNCT cana-4553 241 20 data datum NOUN cana-4553 241 21 mining mining NOUN cana-4553 241 22 , , PUNCT cana-4553 241 23 inference inference NOUN cana-4553 241 24 , , PUNCT cana-4553 241 25 and and CCONJ cana-4553 241 26 prediction prediction NOUN cana-4553 241 27 ( ( PUNCT cana-4553 241 28 2nd 2nd ADJ cana-4553 241 29 ed ed NOUN cana-4553 241 30 . . PUNCT cana-4553 241 31 ) ) PUNCT cana-4553 241 32 , , PUNCT cana-4553 241 33 ” " PUNCT cana-4553 241 34 springer springer NOUN cana-4553 241 35 , , PUNCT cana-4553 241 36 2009 2009 NUM cana-4553 241 37 . . PUNCT cana-4553 242 1 communications communication NOUN cana-4553 242 2 on on ADP cana-4553 242 3 applied apply VERB cana-4553 242 4 nonlinear nonlinear ADJ cana-4553 242 5 analysis analysis NOUN cana-4553 242 6 issn issn NOUN cana-4553 242 7 : : PUNCT cana-4553 242 8 1074 1074 NUM cana-4553 242 9 - - PUNCT cana-4553 242 10 133x 133x NUM cana-4553 242 11 vol vol NOUN cana-4553 242 12 32 32 NUM cana-4553 242 13 no no NOUN cana-4553 242 14 . . PUNCT cana-4553 243 1 9s 9s NUM cana-4553 243 2 ( ( PUNCT cana-4553 243 3 2025 2025 NUM cana-4553 243 4 ) ) PUNCT cana-4553 243 5 2764 2764 NUM cana-4553 243 6 https://internationalpubls.com https://internationalpubls.com X cana-4553 244 1 [ [ X cana-4553 244 2 9 9 NUM cana-4553 244 3 ] ] X cana-4553 244 4 i. i. PROPN cana-4553 244 5 goodfellow goodfellow PROPN cana-4553 244 6 , , PUNCT cana-4553 244 7 y. y. PROPN cana-4553 244 8 bengio bengio PROPN cana-4553 244 9 , , PUNCT cana-4553 244 10 & & CCONJ cana-4553 244 11 a. a. PROPN cana-4553 244 12 courville courville PROPN cana-4553 244 13 , , PUNCT cana-4553 244 14 β€œ " PUNCT cana-4553 244 15 deep deep ADJ cana-4553 244 16 learning learning NOUN cana-4553 244 17 , , PUNCT cana-4553 244 18 ” " PUNCT cana-4553 244 19 mit mit PROPN cana-4553 244 20 press press NOUN cana-4553 244 21 , , PUNCT cana-4553 244 22 2016 2016 NUM cana-4553 244 23 . . PUNCT cana-4553 245 1 [ [ X cana-4553 245 2 10 10 NUM cana-4553 245 3 ] ] X cana-4553 245 4 r. r. PROPN cana-4553 245 5 s. s. PROPN cana-4553 245 6 suttonand suttonand PROPN cana-4553 245 7 a. a. PROPN cana-4553 245 8 g. g. PROPN cana-4553 245 9 barto barto PROPN cana-4553 245 10 , , PUNCT cana-4553 245 11 β€œ " PUNCT cana-4553 245 12 reinforcement reinforcement NOUN cana-4553 245 13 learning learning NOUN cana-4553 245 14 : : PUNCT cana-4553 245 15 an an DET cana-4553 245 16 introduction introduction NOUN cana-4553 245 17 ( ( PUNCT cana-4553 245 18 2nd 2nd ADJ cana-4553 245 19 ed ed NOUN cana-4553 245 20 . . PUNCT cana-4553 245 21 ) ) PUNCT cana-4553 245 22 ” " PUNCT cana-4553 245 23 . . PUNCT cana-4553 246 1 mit mit PROPN cana-4553 246 2 press press NOUN cana-4553 246 3 , , PUNCT cana-4553 246 4 2018 2018 NUM cana-4553 246 5 . . PUNCT cana-4553 247 1 [ [ X cana-4553 247 2 11 11 NUM cana-4553 247 3 ] ] PUNCT cana-4553 247 4 a. a. PROPN cana-4553 247 5 mishra mishra PROPN cana-4553 247 6 , , PUNCT cana-4553 247 7 p. p. PROPN cana-4553 247 8 k. k. PROPN cana-4553 247 9 tripathi tripathi PROPN cana-4553 247 10 , , PUNCT cana-4553 247 11 a. a. PROPN cana-4553 247 12 k. k. PROPN cana-4553 247 13 agrawal agrawal PROPN cana-4553 247 14 , , PUNCT cana-4553 247 15 d. d. PROPN cana-4553 247 16 r. r. PROPN cana-4553 247 17 joshi joshi PROPN cana-4553 247 18 , , PUNCT cana-4553 247 19 β€œ " PUNCT cana-4553 247 20 a a DET cana-4553 247 21 probabilistic probabilistic ADJ cana-4553 247 22 approach approach NOUN cana-4553 247 23 in in ADP cana-4553 247 24 modeling model VERB cana-4553 247 25 the the DET cana-4553 247 26 growth growth NOUN cana-4553 247 27 of of ADP cana-4553 247 28 cell cell NOUN cana-4553 247 29 population population NOUN cana-4553 247 30 , , PUNCT cana-4553 247 31 ” " PUNCT cana-4553 247 32 test test NOUN cana-4553 247 33 engineering engineering NOUN cana-4553 247 34 and and CCONJ cana-4553 247 35 management management NOUN cana-4553 247 36 , , PUNCT cana-4553 247 37 vol vol NOUN cana-4553 247 38 . . PROPN cana-4553 247 39 83 83 NUM cana-4553 247 40 , , PUNCT cana-4553 247 41 pp pp ADJ cana-4553 247 42 . . PUNCT cana-4553 248 1 737 737 NUM cana-4553 248 2 - - SYM cana-4553 248 3 742 742 NUM cana-4553 248 4 , , PUNCT cana-4553 248 5 2020 2020 NUM cana-4553 248 6 . . PUNCT cana-4553 249 1 [ [ X cana-4553 249 2 12 12 NUM cana-4553 249 3 ] ] PUNCT cana-4553 249 4 a. a. PROPN cana-4553 249 5 mishra mishra PROPN cana-4553 249 6 , , PUNCT cana-4553 249 7 achieving achieve VERB cana-4553 249 8 environmental environmental ADJ cana-4553 249 9 sustainability sustainability NOUN cana-4553 249 10 through through ADP cana-4553 249 11 contraction contraction NOUN cana-4553 249 12 mapping mapping NOUN cana-4553 249 13 in in ADP cana-4553 249 14 growth growth NOUN cana-4553 249 15 of of ADP cana-4553 249 16 cell cell NOUN cana-4553 249 17 population population NOUN cana-4553 249 18 , , PUNCT cana-4553 249 19 international international ADJ cana-4553 249 20 journal journal NOUN cana-4553 249 21 of of ADP cana-4553 249 22 engineering engineering NOUN cana-4553 249 23 research research NOUN cana-4553 249 24 and and CCONJ cana-4553 249 25 management management NOUN cana-4553 249 26 technology technology NOUN cana-4553 249 27 , , PUNCT cana-4553 249 28 issn:2348 issn:2348 NOUN cana-4553 249 29 - - SYM cana-4553 249 30 403 403 NUM cana-4553 249 31 , , PUNCT cana-4553 249 32 international international ADJ cana-4553 249 33 conference conference NOUN cana-4553 249 34 on on ADP cana-4553 249 35 mathematical mathematical ADJ cana-4553 249 36 sciences science NOUN cana-4553 249 37 and and CCONJ cana-4553 249 38 computational computational ADJ cana-4553 249 39 intelligence intelligence NOUN cana-4553 249 40 ( ( PUNCT cana-4553 249 41 icmsci-2020 icmsci-2020 NOUN cana-4553 249 42 ) ) PUNCT cana-4553 249 43 , , PUNCT cana-4553 249 44 ( ( PUNCT cana-4553 249 45 2020 2020 NUM cana-4553 249 46 ) ) PUNCT cana-4553 249 47 , , PUNCT cana-4553 249 48 35 35 NUM cana-4553 249 49 - - SYM cana-4553 249 50 36 36 NUM cana-4553 249 51 . . PUNCT cana-4553 250 1 [ [ X cana-4553 250 2 13 13 NUM cana-4553 250 3 ] ] PUNCT cana-4553 250 4 v. v. CCONJ cana-4553 250 5 fortuin fortuin NOUN cana-4553 250 6 , , PUNCT cana-4553 250 7 o. o. PROPN cana-4553 250 8 e. e. PROPN cana-4553 250 9 ganeaand ganeaand PROPN cana-4553 250 10 m. m. PROPN cana-4553 250 11 welling well VERB cana-4553 250 12 , , PUNCT cana-4553 250 13 β€œ " PUNCT cana-4553 250 14 deep deep ADJ cana-4553 250 15 generative generative ADJ cana-4553 250 16 models model NOUN cana-4553 250 17 with with ADP cana-4553 250 18 learnable learnable ADJ cana-4553 250 19 knowledge knowledge NOUN cana-4553 250 20 priors prior NOUN cana-4553 250 21 , , PUNCT cana-4553 250 22 ” " PUNCT cana-4553 250 23 in in ADP cana-4553 250 24 international international ADJ cana-4553 250 25 conference conference NOUN cana-4553 250 26 on on ADP cana-4553 250 27 machine machine NOUN cana-4553 250 28 learning learning NOUN cana-4553 250 29 ( ( PUNCT cana-4553 250 30 icml icml PROPN cana-4553 250 31 ) ) PUNCT cana-4553 250 32 , , PUNCT cana-4553 250 33 2021 2021 NUM cana-4553 250 34 . . PUNCT cana-4553 251 1 [ [ X cana-4553 251 2 14 14 NUM cana-4553 251 3 ] ] X cana-4553 251 4 a. a. PROPN cana-4553 251 5 mishra mishra PROPN cana-4553 251 6 and and CCONJ cana-4553 251 7 p. p. PROPN cana-4553 251 8 k. k. PROPN cana-4553 252 1 tripathi tripathi PROPN cana-4553 252 2 , , PUNCT cana-4553 252 3 β€œ " PUNCT cana-4553 252 4 iteration iteration NOUN cana-4553 252 5 - - PUNCT cana-4553 252 6 based base VERB cana-4553 252 7 reduction reduction NOUN cana-4553 252 8 in in ADP cana-4553 252 9 cell cell NOUN cana-4553 252 10 population population NOUN cana-4553 252 11 for for ADP cana-4553 252 12 biomedical biomedical ADJ cana-4553 252 13 applications application NOUN cana-4553 252 14 ” " PUNCT cana-4553 252 15 , , PUNCT cana-4553 252 16 proceedings proceeding NOUN cana-4553 252 17 of of ADP cana-4553 252 18 advances advance NOUN cana-4553 252 19 in in ADP cana-4553 252 20 ai ai NOUN cana-4553 252 21 for for ADP cana-4553 252 22 biomedical biomedical ADJ cana-4553 252 23 instrumentation instrumentation NOUN cana-4553 252 24 , , PUNCT cana-4553 252 25 electronics electronic NOUN cana-4553 252 26 and and CCONJ cana-4553 252 27 computing computing NOUN cana-4553 252 28 , , PUNCT cana-4553 252 29 taylor taylor PROPN cana-4553 252 30 and and CCONJ cana-4553 252 31 francis francis PROPN cana-4553 252 32 group group PROPN cana-4553 252 33 , , PUNCT cana-4553 252 34 crc crc NOUN cana-4553 252 35 press press PROPN cana-4553 252 36 , , PUNCT cana-4553 252 37 uk uk PROPN cana-4553 252 38 , , PUNCT cana-4553 252 39 pp.466 pp.466 NOUN cana-4553 252 40 - - SYM cana-4553 252 41 72 72 NUM cana-4553 252 42 , , PUNCT cana-4553 252 43 2024 2024 NUM cana-4553 252 44 . . PUNCT cana-4553 253 1 http://dx.doi.org/10.1201/9781032644752-86 http://dx.doi.org/10.1201/9781032644752-86 ADJ cana-4553 253 2 [ [ X cana-4553 253 3 15 15 NUM cana-4553 253 4 ] ] X cana-4553 253 5 a. a. PROPN cana-4553 253 6 mishra mishra PROPN cana-4553 253 7 and and CCONJ cana-4553 253 8 p. p. PROPN cana-4553 253 9 k. k. PROPN cana-4553 254 1 tripathi tripathi PROPN cana-4553 254 2 , , PUNCT cana-4553 254 3 β€œ " PUNCT cana-4553 254 4 applicability applicability NOUN cana-4553 254 5 of of ADP cana-4553 254 6 intimate intimate ADJ cana-4553 254 7 mapping mapping NOUN cana-4553 254 8 in in ADP cana-4553 254 9 digital digital ADJ cana-4553 254 10 image image NOUN cana-4553 254 11 sources source NOUN cana-4553 254 12 ” " PUNCT cana-4553 254 13 , , PUNCT cana-4553 254 14 grenze grenze PROPN cana-4553 254 15 international international PROPN cana-4553 254 16 journal journal PROPN cana-4553 254 17 of of ADP cana-4553 254 18 engineering engineering NOUN cana-4553 254 19 and and CCONJ cana-4553 254 20 technology technology NOUN cana-4553 254 21 ( ( PUNCT cana-4553 254 22 grenze grenze NOUN cana-4553 254 23 scientific scientific ADJ cana-4553 254 24 society society NOUN cana-4553 254 25 ) ) PUNCT cana-4553 254 26 , , PUNCT cana-4553 254 27 10 10 NUM cana-4553 254 28 ( ( PUNCT cana-4553 254 29 1 1 NUM cana-4553 254 30 ) ) PUNCT cana-4553 254 31 , , PUNCT cana-4553 254 32 711 711 NOUN cana-4553 254 33 - - SYM cana-4553 254 34 715 715 NUM cana-4553 254 35 , , PUNCT cana-4553 254 36 2024 2024 NUM cana-4553 254 37 . . PUNCT cana-4553 255 1 [ [ X cana-4553 255 2 16 16 NUM cana-4553 255 3 ] ] X cana-4553 255 4 a. a. PROPN cana-4553 255 5 mishra mishra PROPN cana-4553 255 6 and and CCONJ cana-4553 255 7 p. p. PROPN cana-4553 255 8 k. k. PROPN cana-4553 256 1 tripathi tripathi PROPN cana-4553 256 2 , , PUNCT cana-4553 256 3 β€œ " PUNCT cana-4553 256 4 iterative iterative ADJ cana-4553 256 5 methods method NOUN cana-4553 256 6 under under ADP cana-4553 256 7 jungck jungck PROPN cana-4553 256 8 contraction contraction NOUN cana-4553 256 9 theorem theorem VERB cana-4553 256 10 for for ADP cana-4553 256 11 root root NOUN cana-4553 256 12 finding finding NOUN cana-4553 256 13 in in ADP cana-4553 256 14 numerical numerical ADJ cana-4553 256 15 analysis analysis NOUN cana-4553 256 16 for for ADP cana-4553 256 17 ai ai PROPN cana-4553 256 18 applications application NOUN cana-4553 256 19 ” " PUNCT cana-4553 256 20 , , PUNCT cana-4553 256 21 grenze grenze PROPN cana-4553 256 22 international international PROPN cana-4553 256 23 journal journal PROPN cana-4553 256 24 of of ADP cana-4553 256 25 engineering engineering NOUN cana-4553 256 26 and and CCONJ cana-4553 256 27 technology technology NOUN cana-4553 256 28 ( ( PUNCT cana-4553 256 29 grenze grenze NOUN cana-4553 256 30 scientific scientific ADJ cana-4553 256 31 society society NOUN cana-4553 256 32 ) ) PUNCT cana-4553 256 33 , , PUNCT cana-4553 256 34 11 11 NUM cana-4553 256 35 ( ( PUNCT cana-4553 256 36 1 1 NUM cana-4553 256 37 ) ) PUNCT cana-4553 256 38 , , PUNCT cana-4553 256 39 4703 4703 NUM cana-4553 256 40 - - SYM cana-4553 256 41 4708 4708 NUM cana-4553 256 42 , , PUNCT cana-4553 256 43 2025 2025 NUM cana-4553 256 44 . . PUNCT cana-4553 257 1 http://dx.doi.org/10.1201/9781032644752-86 http://dx.doi.org/10.1201/9781032644752-86 ADJ