id sid tid token lemma pos cana-1536 1 1 communications communication NOUN cana-1536 1 2 on on ADP cana-1536 1 3 applied apply VERB cana-1536 1 4 nonlinear nonlinear ADJ cana-1536 1 5 analysis analysis NOUN cana-1536 1 6 issn issn NOUN cana-1536 1 7 : : PUNCT cana-1536 1 8 1074 1074 NUM cana-1536 1 9 - - PUNCT cana-1536 1 10 133x 133x NUM cana-1536 1 11 vol vol NOUN cana-1536 1 12 31 31 NUM cana-1536 1 13 no no NOUN cana-1536 1 14 . . PUNCT cana-1536 2 1 8s 8s PROPN cana-1536 2 2 ( ( PUNCT cana-1536 2 3 2024 2024 NUM cana-1536 2 4 ) ) PUNCT cana-1536 2 5 433 433 NUM cana-1536 2 6 https://internationalpubls.com https://internationalpubls.com NUM cana-1536 2 7 tripled triple VERB cana-1536 2 8 fixed fix VERB cana-1536 2 9 point point NOUN cana-1536 2 10 results result NOUN cana-1536 2 11 in in ADP cana-1536 2 12 𝐆𝐛-metric 𝐆𝐛-metric NOUN cana-1536 2 13 spaces space NOUN cana-1536 2 14 v. v. ADP cana-1536 2 15 rajitha1 rajitha1 PROPN cana-1536 2 16 , , PUNCT cana-1536 2 17 g. g. PROPN cana-1536 2 18 upender upender PROPN cana-1536 2 19 reddy2 reddy2 PROPN cana-1536 3 1 1research 1research NUM cana-1536 3 2 scholar scholar NOUN cana-1536 3 3 , , PUNCT cana-1536 3 4 department department NOUN cana-1536 3 5 of of ADP cana-1536 3 6 mathematics mathematics PROPN cana-1536 3 7 , , PUNCT cana-1536 3 8 osmania osmania PROPN cana-1536 3 9 university university PROPN cana-1536 3 10 , , PUNCT cana-1536 3 11 telangana telangana PROPN cana-1536 3 12 , , PUNCT cana-1536 3 13 india india PROPN cana-1536 3 14 . . PUNCT cana-1536 4 1 e e X cana-1536 4 2 - - NOUN cana-1536 4 3 mail mail NOUN cana-1536 4 4 address address NOUN cana-1536 4 5 : : PUNCT cana-1536 4 6 rajivan07@gmail.com rajivan07@gmail.com X cana-1536 4 7 2associate 2associate NUM cana-1536 4 8 professor professor NOUN cana-1536 4 9 of of ADP cana-1536 4 10 mathematics mathematic NOUN cana-1536 4 11 , , PUNCT cana-1536 4 12 nizam nizam PROPN cana-1536 4 13 college college PROPN cana-1536 4 14 ( ( PUNCT cana-1536 4 15 a a NOUN cana-1536 4 16 ) ) PUNCT cana-1536 4 17 , , PUNCT cana-1536 4 18 osmania osmania PROPN cana-1536 4 19 university university PROPN cana-1536 4 20 , , PUNCT cana-1536 4 21 telangana telangana PROPN cana-1536 4 22 , , PUNCT cana-1536 4 23 india india PROPN cana-1536 4 24 . . PUNCT cana-1536 5 1 e e X cana-1536 5 2 - - NOUN cana-1536 5 3 mail mail NOUN cana-1536 5 4 address address NOUN cana-1536 5 5 : : PUNCT cana-1536 6 1 yuviganga@gmail.com yuviganga@gmail.com PROPN cana-1536 6 2 article article NOUN cana-1536 6 3 history history NOUN cana-1536 6 4 : : PUNCT cana-1536 6 5 received receive VERB cana-1536 6 6 : : PUNCT cana-1536 6 7 07 07 NUM cana-1536 6 8 - - PUNCT cana-1536 6 9 05 05 NUM cana-1536 6 10 - - PUNCT cana-1536 6 11 2024 2024 NUM cana-1536 6 12 revised revise VERB cana-1536 6 13 : : PUNCT cana-1536 6 14 25 25 NUM cana-1536 6 15 - - PUNCT cana-1536 6 16 06 06 NUM cana-1536 6 17 - - PUNCT cana-1536 6 18 2024 2024 NUM cana-1536 6 19 accepted accept VERB cana-1536 6 20 : : PUNCT cana-1536 6 21 09 09 NUM cana-1536 6 22 - - SYM cana-1536 6 23 07 07 NUM cana-1536 6 24 - - PUNCT cana-1536 6 25 2024 2024 NUM cana-1536 6 26 abstract abstract NOUN cana-1536 6 27 in in ADP cana-1536 6 28 recent recent ADJ cana-1536 6 29 work work NOUN cana-1536 6 30 authors author NOUN cana-1536 6 31 were be AUX cana-1536 6 32 discussed discuss VERB cana-1536 6 33 fixed fix VERB cana-1536 6 34 point point NOUN cana-1536 6 35 results result NOUN cana-1536 6 36 with with ADP cana-1536 6 37 various various ADJ cana-1536 6 38 contractions contraction NOUN cana-1536 6 39 like like ADP cana-1536 6 40 ( ( PUNCT cana-1536 6 41 ψ,Ο•)-weakly ψ,Ο•)-weakly ADV cana-1536 6 42 contractive contractive ADJ cana-1536 6 43 mappings mapping NOUN cana-1536 6 44 , , PUNCT cana-1536 6 45 cyclic cyclic ADJ cana-1536 6 46 contraction contraction NOUN cana-1536 6 47 , , PUNCT cana-1536 6 48 e.a e.a PROPN cana-1536 6 49 property property PROPN cana-1536 6 50 , , PUNCT cana-1536 6 51 suzuki suzuki NOUN cana-1536 6 52 - - PUNCT cana-1536 6 53 type type NOUN cana-1536 6 54 contraction contraction NOUN cana-1536 6 55 etc etc X cana-1536 6 56 . . X cana-1536 7 1 in in ADP cana-1536 7 2 complete complete ADJ cana-1536 7 3 gb gb ADV cana-1536 7 4 - - PUNCT cana-1536 7 5 metric metric ADJ cana-1536 7 6 spaces space NOUN cana-1536 7 7 , , PUNCT cana-1536 7 8 with with ADP cana-1536 7 9 the the DET cana-1536 7 10 help help NOUN cana-1536 7 11 of of ADP cana-1536 7 12 completeness completeness NOUN cana-1536 7 13 property property NOUN cana-1536 7 14 and and CCONJ cana-1536 7 15 continuous continuous ADJ cana-1536 7 16 function function NOUN cana-1536 7 17 we we PRON cana-1536 7 18 obtained obtain VERB cana-1536 7 19 unique unique ADJ cana-1536 7 20 tripled triple VERB cana-1536 7 21 fixed fix VERB cana-1536 7 22 point point NOUN cana-1536 7 23 in in ADP cana-1536 7 24 gb gb ADV cana-1536 7 25 - - PUNCT cana-1536 7 26 metric metric ADJ cana-1536 7 27 spaces space NOUN cana-1536 7 28 . . PUNCT cana-1536 8 1 objectives objective NOUN cana-1536 8 2 : : PUNCT cana-1536 8 3 to to PART cana-1536 8 4 show show VERB cana-1536 8 5 tripled triple VERB cana-1536 8 6 fixed fix VERB cana-1536 8 7 point point NOUN cana-1536 8 8 theorems theorem NOUN cana-1536 8 9 in in ADP cana-1536 8 10 gb gb ADV cana-1536 8 11 - - PUNCT cana-1536 8 12 metric metric ADJ cana-1536 8 13 spaces space NOUN cana-1536 8 14 via via ADP cana-1536 8 15 new new ADJ cana-1536 8 16 type type NOUN cana-1536 8 17 of of ADP cana-1536 8 18 contraction contraction NOUN cana-1536 8 19 and and CCONJ cana-1536 8 20 shown show VERB cana-1536 8 21 illustrate illustrate VERB cana-1536 8 22 an an DET cana-1536 8 23 example example NOUN cana-1536 8 24 which which PRON cana-1536 8 25 supports support VERB cana-1536 8 26 the the DET cana-1536 8 27 main main ADJ cana-1536 8 28 result result NOUN cana-1536 8 29 . . PUNCT cana-1536 9 1 methods method NOUN cana-1536 9 2 : : PUNCT cana-1536 9 3 in in ADP cana-1536 9 4 recent recent ADJ cana-1536 9 5 work work NOUN cana-1536 9 6 authors author NOUN cana-1536 9 7 were be AUX cana-1536 9 8 discussed discuss VERB cana-1536 9 9 fixed fix VERB cana-1536 9 10 point point NOUN cana-1536 9 11 results result NOUN cana-1536 9 12 with with ADP cana-1536 9 13 various various ADJ cana-1536 9 14 contractions contraction NOUN cana-1536 9 15 like like ADP cana-1536 9 16 ( ( PUNCT cana-1536 9 17 ψ ψ NOUN cana-1536 9 18 , , PUNCT cana-1536 9 19 Ο•)-weakly Ο•)-weakly PUNCT cana-1536 9 20 contractive contractive ADJ cana-1536 9 21 mappings mapping NOUN cana-1536 9 22 , , PUNCT cana-1536 9 23 cyclic cyclic ADJ cana-1536 9 24 contraction contraction NOUN cana-1536 9 25 , , PUNCT cana-1536 9 26 e.a e.a PROPN cana-1536 9 27 property property PROPN cana-1536 9 28 , , PUNCT cana-1536 9 29 suzuki suzuki NOUN cana-1536 9 30 - - PUNCT cana-1536 9 31 type type NOUN cana-1536 9 32 contraction contraction NOUN cana-1536 9 33 etc etc X cana-1536 9 34 . . X cana-1536 10 1 in in ADP cana-1536 10 2 complete complete ADJ cana-1536 10 3 gb gb ADV cana-1536 10 4 - - PUNCT cana-1536 10 5 metric metric ADJ cana-1536 10 6 spaces space NOUN cana-1536 10 7 , , PUNCT cana-1536 10 8 here here ADV cana-1536 10 9 we we PRON cana-1536 10 10 have have AUX cana-1536 10 11 showed showed AUX cana-1536 10 12 tripled triple VERB cana-1536 10 13 fixed fix VERB cana-1536 10 14 point point NOUN cana-1536 10 15 results result NOUN cana-1536 10 16 by by ADP cana-1536 10 17 using use VERB cana-1536 10 18 new new ADJ cana-1536 10 19 type type NOUN cana-1536 10 20 of of ADP cana-1536 10 21 contraction contraction NOUN cana-1536 10 22 . . PUNCT cana-1536 11 1 results result NOUN cana-1536 11 2 : : PUNCT cana-1536 11 3 unique unique ADJ cana-1536 11 4 tripled triple VERB cana-1536 11 5 fixed fix VERB cana-1536 11 6 points point NOUN cana-1536 11 7 with with ADP cana-1536 11 8 new new ADJ cana-1536 11 9 type type NOUN cana-1536 11 10 of of ADP cana-1536 11 11 contraction contraction NOUN cana-1536 11 12 in in ADP cana-1536 11 13 gb gb ADV cana-1536 11 14 - - PUNCT cana-1536 11 15 metric metric ADJ cana-1536 11 16 spaces space NOUN cana-1536 11 17 . . PUNCT cana-1536 12 1 conclusions conclusion NOUN cana-1536 12 2 : : PUNCT cana-1536 12 3 in in ADP cana-1536 12 4 this this DET cana-1536 12 5 work work NOUN cana-1536 12 6 we we PRON cana-1536 12 7 have have AUX cana-1536 12 8 obtained obtain VERB cana-1536 12 9 tfp tfp PROPN cana-1536 12 10 results result NOUN cana-1536 12 11 by by ADP cana-1536 12 12 using use VERB cana-1536 12 13 a a DET cana-1536 12 14 new new ADJ cana-1536 12 15 type type NOUN cana-1536 12 16 of of ADP cana-1536 12 17 contraction contraction NOUN cana-1536 12 18 and and CCONJ cana-1536 12 19 discussed discuss VERB cana-1536 12 20 corollary corollary NOUN cana-1536 12 21 also also ADV cana-1536 12 22 an an DET cana-1536 12 23 example example NOUN cana-1536 12 24 which which PRON cana-1536 12 25 supports support VERB cana-1536 12 26 the the DET cana-1536 12 27 main main ADJ cana-1536 12 28 result result NOUN cana-1536 12 29 . . PUNCT cana-1536 13 1 keywords keyword NOUN cana-1536 13 2 : : PUNCT cana-1536 13 3 tripled triple VERB cana-1536 13 4 coincident coincident ADJ cana-1536 13 5 point point NOUN cana-1536 13 6 ( ( PUNCT cana-1536 13 7 tcip tcip PROPN cana-1536 13 8 ) ) PUNCT cana-1536 13 9 ; ; PUNCT cana-1536 13 10 tripled triple VERB cana-1536 13 11 fixed fix VERB cana-1536 13 12 point point NOUN cana-1536 13 13 ( ( PUNCT cana-1536 13 14 tfp tfp PROPN cana-1536 13 15 ) ) PUNCT cana-1536 13 16 ; ; PUNCT cana-1536 13 17 gb gb ADJ cana-1536 13 18 - - PUNCT cana-1536 13 19 metric metric ADJ cana-1536 13 20 space space NOUN cana-1536 13 21 ( ( PUNCT cana-1536 13 22 gb gb NOUN cana-1536 13 23 - - PUNCT cana-1536 13 24 ms ms NOUN cana-1536 13 25 ) ) PUNCT cana-1536 13 26 ; ; PUNCT cana-1536 13 27 cauchy cauchy NOUN cana-1536 13 28 sequence sequence NOUN cana-1536 13 29 ( ( PUNCT cana-1536 13 30 cs cs PROPN cana-1536 13 31 ) ) PUNCT cana-1536 13 32 ; ; PUNCT cana-1536 13 33 continuous continuous ADJ cana-1536 13 34 function function NOUN cana-1536 13 35 ; ; PUNCT cana-1536 13 36 completeness completeness NOUN cana-1536 13 37 property property NOUN cana-1536 13 38 . . PUNCT cana-1536 14 1 1 1 X cana-1536 14 2 . . X cana-1536 14 3 introduction introduction NOUN cana-1536 14 4 fixed fix VERB cana-1536 14 5 point point NOUN cana-1536 14 6 theory theory NOUN cana-1536 14 7 is be AUX cana-1536 14 8 a a DET cana-1536 14 9 key key ADJ cana-1536 14 10 tool tool NOUN cana-1536 14 11 in in ADP cana-1536 14 12 nonlinear nonlinear ADJ cana-1536 14 13 functional functional ADJ cana-1536 14 14 analysis analysis NOUN cana-1536 14 15 , , PUNCT cana-1536 14 16 with with ADP cana-1536 14 17 applications application NOUN cana-1536 14 18 in in ADP cana-1536 14 19 computer computer NOUN cana-1536 14 20 science science NOUN cana-1536 14 21 , , PUNCT cana-1536 14 22 chemistry chemistry NOUN cana-1536 14 23 , , PUNCT cana-1536 14 24 biology biology NOUN cana-1536 14 25 , , PUNCT cana-1536 14 26 and and CCONJ cana-1536 14 27 engineering engineering NOUN cana-1536 14 28 . . PUNCT cana-1536 15 1 the the DET cana-1536 15 2 banach banach NOUN cana-1536 15 3 contraction contraction NOUN cana-1536 15 4 principle principle NOUN cana-1536 15 5 , , PUNCT cana-1536 15 6 which which PRON cana-1536 15 7 asserts assert VERB cana-1536 15 8 that that SCONJ cana-1536 15 9 each each DET cana-1536 15 10 contraction contraction NOUN cana-1536 15 11 has have VERB cana-1536 15 12 a a DET cana-1536 15 13 unique unique ADJ cana-1536 15 14 fixed fix VERB cana-1536 15 15 point point NOUN cana-1536 15 16 in in ADP cana-1536 15 17 complete complete ADJ cana-1536 15 18 metric metric ADJ cana-1536 15 19 spaces space NOUN cana-1536 15 20 , , PUNCT cana-1536 15 21 is be AUX cana-1536 15 22 a a DET cana-1536 15 23 cornerstone cornerstone NOUN cana-1536 15 24 of of ADP cana-1536 15 25 this this DET cana-1536 15 26 study study NOUN cana-1536 15 27 . . PUNCT cana-1536 16 1 many many ADJ cana-1536 16 2 authors author NOUN cana-1536 16 3 are be AUX cana-1536 16 4 interested interested ADJ cana-1536 16 5 in in ADP cana-1536 16 6 fixed fix VERB cana-1536 16 7 point point NOUN cana-1536 16 8 theory theory NOUN cana-1536 16 9 , , PUNCT cana-1536 16 10 particularly particularly ADV cana-1536 16 11 the the DET cana-1536 16 12 banach banach NOUN cana-1536 16 13 contraction contraction NOUN cana-1536 16 14 principle principle NOUN cana-1536 16 15 , , PUNCT cana-1536 16 16 due due ADP cana-1536 16 17 to to ADP cana-1536 16 18 its its PRON cana-1536 16 19 possible possible ADJ cana-1536 16 20 applications application NOUN cana-1536 16 21 in in ADP cana-1536 16 22 the the DET cana-1536 16 23 above above ADJ cana-1536 16 24 domains domain NOUN cana-1536 16 25 . . PUNCT cana-1536 17 1 investigating investigate VERB cana-1536 17 2 the the DET cana-1536 17 3 presence presence NOUN cana-1536 17 4 and and CCONJ cana-1536 17 5 uniqueness uniqueness NOUN cana-1536 17 6 of of ADP cana-1536 17 7 a a DET cana-1536 17 8 fixed fix VERB cana-1536 17 9 point point NOUN cana-1536 17 10 for for ADP cana-1536 17 11 many many ADJ cana-1536 17 12 contraction contraction NOUN cana-1536 17 13 type type NOUN cana-1536 17 14 mappings mapping NOUN cana-1536 17 15 in in ADP cana-1536 17 16 diverse diverse ADJ cana-1536 17 17 metric metric ADJ cana-1536 17 18 spaces space NOUN cana-1536 17 19 is be AUX cana-1536 17 20 especially especially ADV cana-1536 17 21 natural natural ADJ cana-1536 17 22 and and CCONJ cana-1536 17 23 intriguing intriguing ADJ cana-1536 17 24 . . PUNCT cana-1536 18 1 in in ADP cana-1536 18 2 2008 2008 NUM cana-1536 18 3 , , PUNCT cana-1536 18 4 mustafa mustafa NOUN cana-1536 18 5 and and CCONJ cana-1536 18 6 sims sim NOUN cana-1536 18 7 introduced introduce VERB cana-1536 18 8 g g NOUN cana-1536 18 9 - - PUNCT cana-1536 18 10 metric metric ADJ cana-1536 18 11 space space NOUN cana-1536 18 12 as as ADP cana-1536 18 13 new new ADJ cana-1536 18 14 generalizations generalization NOUN cana-1536 18 15 of of ADP cana-1536 18 16 metric metric ADJ cana-1536 18 17 spaces space NOUN cana-1536 18 18 . . PUNCT cana-1536 19 1 in in ADP cana-1536 19 2 2014 2014 NUM cana-1536 19 3 , , PUNCT cana-1536 19 4 aghajani aghajani PROPN cana-1536 19 5 et et PROPN cana-1536 19 6 al al PROPN cana-1536 19 7 . . PROPN cana-1536 19 8 generalised generalise VERB cana-1536 19 9 metric metric ADJ cana-1536 19 10 spaces space NOUN cana-1536 19 11 . . PUNCT cana-1536 20 1 they they PRON cana-1536 20 2 created create VERB cana-1536 20 3 a a DET cana-1536 20 4 gb gb ADV cana-1536 20 5 - - PUNCT cana-1536 20 6 metric metric ADJ cana-1536 20 7 space space NOUN cana-1536 20 8 by by ADP cana-1536 20 9 combining combine VERB cana-1536 20 10 gmetric gmetric ADJ cana-1536 20 11 and and CCONJ cana-1536 20 12 b b X cana-1536 20 13 - - PUNCT cana-1536 20 14 metric metric ADJ cana-1536 20 15 definitions definition NOUN cana-1536 20 16 . . PUNCT cana-1536 21 1 they they PRON cana-1536 21 2 also also ADV cana-1536 21 3 noted note VERB cana-1536 21 4 that that SCONJ cana-1536 21 5 gb gb NOUN cana-1536 21 6 - - PUNCT cana-1536 21 7 ms ms NOUN cana-1536 21 8 are be AUX cana-1536 21 9 effectively effectively ADV cana-1536 21 10 larger large ADJ cana-1536 21 11 than than ADP cana-1536 21 12 g g NOUN cana-1536 21 13 - - PUNCT cana-1536 21 14 ms ms NOUN cana-1536 21 15 . . PUNCT cana-1536 22 1 the the DET cana-1536 22 2 g g PROPN cana-1536 22 3 - - PUNCT cana-1536 22 4 ms ms NOUN cana-1536 22 5 is be AUX cana-1536 22 6 a a DET cana-1536 22 7 subset subset NOUN cana-1536 22 8 of of ADP cana-1536 22 9 the the DET cana-1536 22 10 gb gb NOUN cana-1536 22 11 - - PUNCT cana-1536 22 12 ms ms NOUN cana-1536 22 13 when when SCONJ cana-1536 22 14 s s VERB cana-1536 22 15 = = NOUN cana-1536 22 16 1 1 X cana-1536 22 17 . . PUNCT cana-1536 23 1 they they PRON cana-1536 23 2 also also ADV cana-1536 23 3 proved prove VERB cana-1536 23 4 that that SCONJ cana-1536 23 5 every every DET cana-1536 23 6 gb gb NOUN cana-1536 23 7 - - PUNCT cana-1536 23 8 ms ms NOUN cana-1536 23 9 topologically topologically ADV cana-1536 23 10 equals equal VERB cana-1536 23 11 a a DET cana-1536 23 12 b b PROPN cana-1536 23 13 - - PUNCT cana-1536 23 14 ms ms NOUN cana-1536 23 15 . . PROPN cana-1536 24 1 many many ADJ cana-1536 24 2 articles article NOUN cana-1536 24 3 have have AUX cana-1536 24 4 published publish VERB cana-1536 24 5 on on ADP cana-1536 24 6 this this DET cana-1536 24 7 ms ms NOUN cana-1536 24 8 ( ( PUNCT cana-1536 24 9 see see VERB cana-1536 24 10 [ [ X cana-1536 24 11 1]-[15 1]-[15 NUM cana-1536 24 12 ] ] PUNCT cana-1536 24 13 ) ) PUNCT cana-1536 24 14 . . PUNCT cana-1536 25 1 in in ADP cana-1536 25 2 this this DET cana-1536 25 3 current current ADJ cana-1536 25 4 work work NOUN cana-1536 25 5 we we PRON cana-1536 25 6 have have AUX cana-1536 25 7 discussed discuss VERB cana-1536 25 8 with with ADP cana-1536 25 9 new new ADJ cana-1536 25 10 type type NOUN cana-1536 25 11 of of ADP cana-1536 25 12 contraction contraction NOUN cana-1536 25 13 on on ADP cana-1536 25 14 𝐺𝑏-ms 𝐺𝑏-ms PROPN cana-1536 25 15 , , PUNCT cana-1536 25 16 unique unique ADJ cana-1536 25 17 tfp tfp PROPN cana-1536 25 18 results result NOUN cana-1536 25 19 and and CCONJ cana-1536 25 20 some some DET cana-1536 25 21 corollary corollary NOUN cana-1536 25 22 also also ADV cana-1536 25 23 shown show VERB cana-1536 25 24 an an DET cana-1536 25 25 example example NOUN cana-1536 25 26 of of ADP cana-1536 25 27 our our PRON cana-1536 25 28 work work NOUN cana-1536 25 29 . . PUNCT cana-1536 26 1 mailto:rajivan07@gmail.com mailto:rajivan07@gmail.com PROPN cana-1536 26 2 communications communication NOUN cana-1536 26 3 on on ADP cana-1536 26 4 applied apply VERB cana-1536 26 5 nonlinear nonlinear ADJ cana-1536 26 6 analysis analysis NOUN cana-1536 26 7 issn issn NOUN cana-1536 26 8 : : PUNCT cana-1536 26 9 1074 1074 NUM cana-1536 26 10 - - PUNCT cana-1536 26 11 133x 133x NUM cana-1536 26 12 vol vol NOUN cana-1536 26 13 31 31 NUM cana-1536 26 14 no no NOUN cana-1536 26 15 . . PUNCT cana-1536 27 1 8s 8s PROPN cana-1536 27 2 ( ( PUNCT cana-1536 27 3 2024 2024 NUM cana-1536 27 4 ) ) PUNCT cana-1536 27 5 434 434 NUM cana-1536 27 6 https://internationalpubls.com https://internationalpubls.com X cana-1536 27 7 before before ADP cana-1536 27 8 the the DET cana-1536 27 9 main main ADJ cana-1536 27 10 work work NOUN cana-1536 27 11 we we PRON cana-1536 27 12 will will AUX cana-1536 27 13 discuss discuss VERB cana-1536 27 14 some some DET cana-1536 27 15 basic basic ADJ cana-1536 27 16 definitions definition NOUN cana-1536 27 17 . . PUNCT cana-1536 28 1 2 2 X cana-1536 28 2 . . X cana-1536 28 3 methodology methodology NOUN cana-1536 28 4 definition definition NOUN cana-1536 28 5 2.1 2.1 NUM cana-1536 28 6 let let VERB cana-1536 28 7 α½£ α½£ NOUN cana-1536 28 8 is be AUX cana-1536 28 9 a a DET cana-1536 28 10 nonempty nonempty ADJ cana-1536 28 11 set set NOUN cana-1536 28 12 , , PUNCT cana-1536 28 13 β‚’ β‚’ ADP cana-1536 28 14 𝑏 𝑏 NOUN cana-1536 28 15 : : PUNCT cana-1536 28 16 α½£3β†’ α½£3β†’ X cana-1536 28 17 ℝ+ ℝ+ PUNCT cana-1536 28 18 is be AUX cana-1536 28 19 a a DET cana-1536 28 20 function function NOUN cana-1536 28 21 satisfying satisfy VERB cana-1536 28 22 the the DET cana-1536 28 23 following follow VERB cana-1536 28 24 properties property NOUN cana-1536 28 25 : : PUNCT cana-1536 28 26 ( ( PUNCT cana-1536 28 27 β‚’ β‚’ ADP cana-1536 28 28 𝑏1 𝑏1 NOUN cana-1536 28 29 ) ) PUNCT cana-1536 28 30 β‚’ β‚’ ADP cana-1536 28 31 𝑏 𝑏 X cana-1536 28 32 ( ( PUNCT cana-1536 28 33 α΄‚ α΄‚ PROPN cana-1536 28 34 , , PUNCT cana-1536 28 35 α΄” α΄” NOUN cana-1536 28 36 , , PUNCT cana-1536 28 37 𝔣 𝔣 ADJ cana-1536 28 38 ) ) PUNCT cana-1536 28 39 = = SYM cana-1536 28 40 0 0 NUM cana-1536 28 41 , , PUNCT cana-1536 28 42 if if SCONJ cana-1536 28 43 α΄‚ α΄‚ PROPN cana-1536 28 44 = = NOUN cana-1536 28 45 α΄” α΄” NOUN cana-1536 28 46 = = SYM cana-1536 28 47 𝔣 𝔣 X cana-1536 28 48 ; ; PUNCT cana-1536 28 49 ( ( PUNCT cana-1536 28 50 β‚’ β‚’ ADP cana-1536 28 51 𝑏2 𝑏2 NOUN cana-1536 28 52 ) ) PUNCT cana-1536 28 53 0<ₒ𝑏 0<ₒ𝑏 NOUN cana-1536 28 54 ( ( PUNCT cana-1536 28 55 α΄‚ α΄‚ X cana-1536 28 56 , , PUNCT cana-1536 28 57 α΄‚ α΄‚ X cana-1536 28 58 , , PUNCT cana-1536 28 59 𝔣 𝔣 ADJ cana-1536 28 60 ) ) PUNCT cana-1536 28 61 = = SYM cana-1536 28 62 0 0 NUM cana-1536 28 63 , , PUNCT cana-1536 28 64 for for ADP cana-1536 28 65 α΄‚ α΄‚ PROPN cana-1536 28 66 , , PUNCT cana-1536 28 67 𝔣 𝔣 PROPN cana-1536 28 68 Ο΅ Ο΅ PROPN cana-1536 28 69 α½£ α½£ NOUN cana-1536 28 70 and and CCONJ cana-1536 28 71 α΄‚ α΄‚ PRON cana-1536 28 72 β‰  β‰  PROPN cana-1536 28 73 𝔣 𝔣 NOUN cana-1536 28 74 ; ; PUNCT cana-1536 28 75 ( ( PUNCT cana-1536 28 76 β‚’ β‚’ ADP cana-1536 28 77 𝑏3 𝑏3 NOUN cana-1536 28 78 ) ) PUNCT cana-1536 28 79 β‚’ β‚’ ADP cana-1536 28 80 𝑏 𝑏 X cana-1536 28 81 ( ( PUNCT cana-1536 28 82 α΄‚ α΄‚ PROPN cana-1536 28 83 , , PUNCT cana-1536 28 84 α΄‚ α΄‚ X cana-1536 28 85 , , PUNCT cana-1536 28 86 𝔣 𝔣 ADJ cana-1536 28 87 ) ) PUNCT cana-1536 28 88 ≀ ≀ NOUN cana-1536 28 89 β‚’ β‚’ ADP cana-1536 28 90 𝑏 𝑏 X cana-1536 28 91 ( ( PUNCT cana-1536 28 92 α΄‚ α΄‚ X cana-1536 28 93 , , PUNCT cana-1536 28 94 𝔣 𝔣 ADJ cana-1536 28 95 , , PUNCT cana-1536 28 96 α΄”)for α΄”)for ADP cana-1536 28 97 all all DET cana-1536 28 98 α΄‚ α΄‚ PROPN cana-1536 28 99 , , PUNCT cana-1536 28 100 α΄‚ α΄‚ NOUN cana-1536 28 101 , , PUNCT cana-1536 28 102 𝔣 𝔣 PROPN cana-1536 28 103 ∈ ∈ PROPN cana-1536 29 1 α½£ α½£ NOUN cana-1536 29 2 with with ADP cana-1536 29 3 𝔣 𝔣 PROPN cana-1536 29 4 β‰  β‰  PROPN cana-1536 29 5 α΄” α΄” NOUN cana-1536 29 6 ; ; PUNCT cana-1536 29 7 ( ( PUNCT cana-1536 29 8 β‚’ β‚’ X cana-1536 29 9 𝑏4 𝑏4 NOUN cana-1536 29 10 ) ) PUNCT cana-1536 29 11 β‚’ β‚’ ADP cana-1536 29 12 𝑏 𝑏 X cana-1536 29 13 ( ( PUNCT cana-1536 29 14 α΄‚ α΄‚ PROPN cana-1536 29 15 , , PUNCT cana-1536 29 16 α΄” α΄” NOUN cana-1536 29 17 , , PUNCT cana-1536 29 18 𝔣 𝔣 ADJ cana-1536 29 19 ) ) PUNCT cana-1536 29 20 = = PUNCT cana-1536 29 21 β‚’ β‚’ ADP cana-1536 29 22 𝑏 𝑏 PROPN cana-1536 29 23 ( ( PUNCT cana-1536 29 24 α΄” α΄” NOUN cana-1536 29 25 , , PUNCT cana-1536 29 26 𝔣 𝔣 ADJ cana-1536 29 27 , , PUNCT cana-1536 29 28 α΄‚ α΄‚ NOUN cana-1536 29 29 ) ) PUNCT cana-1536 29 30 = = PUNCT cana-1536 29 31 β‚’ β‚’ ADP cana-1536 29 32 𝑏 𝑏 NOUN cana-1536 29 33 ( ( PUNCT cana-1536 29 34 𝔣 𝔣 ADJ cana-1536 29 35 , , PUNCT cana-1536 29 36 α΄‚ α΄‚ NOUN cana-1536 29 37 , , PUNCT cana-1536 29 38 α΄” α΄” NOUN cana-1536 29 39 ) ) PUNCT cana-1536 29 40 = = SYM cana-1536 29 41 … … PUNCT cana-1536 29 42 … … PUNCT cana-1536 29 43 .. .. PUNCT cana-1536 29 44 ( ( PUNCT cana-1536 29 45 symmetry symmetry NOUN cana-1536 29 46 of of ADP cana-1536 29 47 three three NUM cana-1536 29 48 variables variable NOUN cana-1536 29 49 ) ) PUNCT cana-1536 29 50 ; ; PUNCT cana-1536 29 51 ( ( PUNCT cana-1536 29 52 β‚’ β‚’ ADP cana-1536 29 53 𝑏5 𝑏5 NOUN cana-1536 29 54 ) ) PUNCT cana-1536 29 55 β‚’ β‚’ ADP cana-1536 29 56 𝑏 𝑏 X cana-1536 29 57 ( ( PUNCT cana-1536 29 58 α΄‚ α΄‚ PROPN cana-1536 29 59 , , PUNCT cana-1536 29 60 α΄” α΄” NOUN cana-1536 29 61 , , PUNCT cana-1536 29 62 𝔣 𝔣 ADJ cana-1536 29 63 ) ) PUNCT cana-1536 29 64 ≀ ≀ NOUN cana-1536 29 65 s[ₒ𝑏 s[ₒ𝑏 NUM cana-1536 29 66 ( ( PUNCT cana-1536 29 67 α΄‚ α΄‚ X cana-1536 29 68 , , PUNCT cana-1536 29 69 π“Š π“Š PROPN cana-1536 29 70 , , PUNCT cana-1536 29 71 π“Š π“Š PROPN cana-1536 29 72 ) ) PUNCT cana-1536 30 1 + + CCONJ cana-1536 30 2 β‚’ β‚’ ADP cana-1536 30 3 𝑏 𝑏 PRON cana-1536 30 4 ( ( PUNCT cana-1536 30 5 π“Š π“Š PROPN cana-1536 30 6 , , PUNCT cana-1536 30 7 α΄” α΄” NOUN cana-1536 30 8 , , PUNCT cana-1536 30 9 𝔣 𝔣 ADJ cana-1536 30 10 ) ) PUNCT cana-1536 30 11 ] ] PUNCT cana-1536 30 12 for for ADP cana-1536 30 13 all all DET cana-1536 30 14 α΄‚ α΄‚ PROPN cana-1536 30 15 , , PUNCT cana-1536 30 16 α΄” α΄” NOUN cana-1536 30 17 , , PUNCT cana-1536 30 18 𝔣 𝔣 ADJ cana-1536 30 19 , , PUNCT cana-1536 30 20 π“Š π“Š PROPN cana-1536 30 21 ∈ ∈ PROPN cana-1536 30 22 α½£ α½£ X cana-1536 30 23 ( ( PUNCT cana-1536 30 24 rectangular rectangular ADJ cana-1536 30 25 inequality inequality NOUN cana-1536 30 26 ) ) PUNCT cana-1536 30 27 . . PUNCT cana-1536 31 1 the the DET cana-1536 31 2 function function NOUN cana-1536 31 3 β‚’ β‚’ ADP cana-1536 31 4 𝑏 𝑏 PROPN cana-1536 31 5 is be AUX cana-1536 31 6 called call VERB cana-1536 31 7 metric metric ADJ cana-1536 31 8 on on ADP cana-1536 31 9 α½£ α½£ PROPN cana-1536 31 10 and and CCONJ cana-1536 31 11 the the DET cana-1536 31 12 pair pair NOUN cana-1536 31 13 ( ( PUNCT cana-1536 31 14 α½£,ₒ𝑏 α½£,ₒ𝑏 NOUN cana-1536 31 15 ) ) PUNCT cana-1536 31 16 is be AUX cana-1536 31 17 called call VERB cana-1536 31 18 β‚’ β‚’ ADP cana-1536 31 19 𝑏-ms 𝑏-ms PROPN cana-1536 31 20 . . PROPN cana-1536 31 21 example example NOUN cana-1536 31 22 2.2 2.2 NUM cana-1536 31 23 let let VERB cana-1536 31 24 ( ( PUNCT cana-1536 31 25 α½£ α½£ NUM cana-1536 31 26 , , PUNCT cana-1536 31 27 d d NOUN cana-1536 31 28 ) ) PUNCT cana-1536 31 29 is be AUX cana-1536 31 30 a a DET cana-1536 31 31 ms ms PROPN cana-1536 31 32 . . PROPN cana-1536 32 1 the the DET cana-1536 32 2 function function NOUN cana-1536 32 3 β‚’ β‚’ ADP cana-1536 32 4 𝑏 𝑏 NOUN cana-1536 32 5 : : PUNCT cana-1536 32 6 α½£3β†’ α½£3β†’ X cana-1536 32 7 ℝ+ ℝ+ PUNCT cana-1536 32 8 defined define VERB cana-1536 32 9 by by ADP cana-1536 32 10 β‚’ β‚’ ADP cana-1536 32 11 𝑏 𝑏 PROPN cana-1536 32 12 ( ( PUNCT cana-1536 32 13 α΄‚ α΄‚ PROPN cana-1536 32 14 , , PUNCT cana-1536 32 15 α΄” α΄” NOUN cana-1536 32 16 , , PUNCT cana-1536 32 17 𝔣 𝔣 ADJ cana-1536 32 18 ) ) PUNCT cana-1536 32 19 = = SYM cana-1536 32 20 max{d(α΄‚ max{d(α΄‚ PROPN cana-1536 32 21 , , PUNCT cana-1536 32 22 α΄”),d(α΄” α΄”),d(α΄” PROPN cana-1536 32 23 , , PUNCT cana-1536 32 24 𝔣),d(𝔣,α΄‚ 𝔣),d(𝔣,α΄‚ PROPN cana-1536 32 25 ) ) PUNCT cana-1536 32 26 } } PUNCT cana-1536 32 27 and and CCONJ cana-1536 32 28 β‚’ β‚’ ADP cana-1536 32 29 𝑏 𝑏 PRON cana-1536 32 30 ( ( PUNCT cana-1536 32 31 α΄‚ α΄‚ PROPN cana-1536 32 32 , , PUNCT cana-1536 32 33 α΄” α΄” NOUN cana-1536 32 34 , , PUNCT cana-1536 32 35 𝔣 𝔣 ADJ cana-1536 32 36 ) ) PUNCT cana-1536 32 37 = = SYM cana-1536 32 38 d(α΄‚ d(α΄‚ PROPN cana-1536 32 39 , , PUNCT cana-1536 32 40 α΄”)+d(α΄” α΄”)+d(α΄” NOUN cana-1536 32 41 , , PUNCT cana-1536 32 42 𝔣)+d(𝔣,α΄‚ 𝔣)+d(𝔣,α΄‚ PROPN cana-1536 32 43 ) ) PUNCT cana-1536 32 44 for for ADP cana-1536 32 45 all all DET cana-1536 32 46 α΄‚ α΄‚ PROPN cana-1536 32 47 , , PUNCT cana-1536 32 48 α΄” α΄” NOUN cana-1536 32 49 , , PUNCT cana-1536 32 50 𝔣 𝔣 PROPN cana-1536 32 51 ∈ ∈ PROPN cana-1536 32 52 α½£. α½£. NOUN cana-1536 32 53 then then ADV cana-1536 32 54 ( ( PUNCT cana-1536 32 55 α½£ α½£ NOUN cana-1536 32 56 , , PUNCT cana-1536 32 57 β‚’ β‚’ ADP cana-1536 32 58 𝑏 𝑏 NOUN cana-1536 32 59 ) ) PUNCT cana-1536 32 60 is be AUX cana-1536 32 61 a a DET cana-1536 32 62 β‚’ β‚’ ADP cana-1536 32 63 𝑏-ms 𝑏-ms NOUN cana-1536 32 64 . . PUNCT cana-1536 33 1 definition definition NOUN cana-1536 33 2 2.3 2.3 NUM cana-1536 33 3 let let VERB cana-1536 33 4 ( ( PUNCT cana-1536 33 5 α½£,ₒ𝑏 α½£,ₒ𝑏 NOUN cana-1536 33 6 ) ) PUNCT cana-1536 33 7 be be VERB cana-1536 33 8 β‚’ β‚’ ADP cana-1536 33 9 𝑏-ms 𝑏-ms NOUN cana-1536 33 10 , , PUNCT cana-1536 33 11 for for ADP cana-1536 33 12 any any DET cana-1536 33 13 Ξ΅>0 Ξ΅>0 PROPN cana-1536 33 14 , , PUNCT cana-1536 33 15 a a DET cana-1536 33 16 sequence sequence NOUN cana-1536 33 17 { { PUNCT cana-1536 33 18 ℐ𝑛 ℐ𝑛 NOUN cana-1536 33 19 } } PUNCT cana-1536 33 20 is be AUX cana-1536 33 21 a a DET cana-1536 33 22 β‚’ β‚’ ADP cana-1536 33 23 𝑏-cs 𝑏-cs PROPN cana-1536 33 24 , , PUNCT cana-1536 33 25 if if SCONJ cana-1536 33 26 β‚’ β‚’ NOUN cana-1536 33 27 𝑏(ℐ𝑛 𝑏(ℐ𝑛 X cana-1536 33 28 , , PUNCT cana-1536 33 29 β„π‘š β„π‘š PROPN cana-1536 33 30 , , PUNCT cana-1536 33 31 ℐ𝑙 ℐ𝑙 PROPN cana-1536 33 32 ) ) PUNCT cana-1536 33 33 < < X cana-1536 33 34 Ξ΅ Ξ΅ PROPN cana-1536 33 35 for for ADP cana-1536 33 36 all all DET cana-1536 33 37 n n CCONJ cana-1536 33 38 , , PUNCT cana-1536 33 39 m m PROPN cana-1536 33 40 , , PUNCT cana-1536 33 41 l l PROPN cana-1536 33 42 β‰₯ β‰₯ NOUN cana-1536 33 43 n. n. NOUN cana-1536 33 44 definition definition NOUN cana-1536 33 45 2.4 2.4 NUM cana-1536 33 46 let let NOUN cana-1536 33 47 ( ( PUNCT cana-1536 33 48 α½£,ₒ𝑏 α½£,ₒ𝑏 NOUN cana-1536 33 49 ) ) PUNCT cana-1536 33 50 is be AUX cana-1536 33 51 β‚’ β‚’ ADP cana-1536 33 52 𝑏-ms 𝑏-ms NOUN cana-1536 33 53 . . PUNCT cana-1536 34 1 a a DET cana-1536 34 2 sequence sequence NOUN cana-1536 34 3 { { PUNCT cana-1536 34 4 ℐ𝑛 ℐ𝑛 NOUN cana-1536 34 5 } } PUNCT cana-1536 34 6 is be AUX cana-1536 34 7 called call VERB cana-1536 34 8 a a DET cana-1536 34 9 β‚’ β‚’ NOUN cana-1536 34 10 𝑏-convergent 𝑏-convergent NOUN cana-1536 34 11 to𝔖 to𝔖 PROPN cana-1536 34 12 ∈ ∈ PROPN cana-1536 34 13 α½£ α½£ NOUN cana-1536 34 14 , , PUNCT cana-1536 34 15 if if SCONJ cana-1536 34 16 for for ADP cana-1536 34 17 anyΞ΅ anyΞ΅ NOUN cana-1536 34 18 > > X cana-1536 34 19 0 0 NUM cana-1536 34 20 , , PUNCT cana-1536 34 21 there there PRON cana-1536 34 22 is be VERB cana-1536 34 23 𝑁 𝑁 PROPN cana-1536 34 24 ∈ ∈ PROPN cana-1536 34 25 β„• β„• NOUN cana-1536 34 26 such such ADJ cana-1536 34 27 that that SCONJ cana-1536 34 28 β‚’ β‚’ ADP cana-1536 34 29 𝑏(𝔖 𝑏(𝔖 PROPN cana-1536 34 30 , , PUNCT cana-1536 34 31 ℐ𝑛 ℐ𝑛 PROPN cana-1536 34 32 , , PUNCT cana-1536 34 33 ℐ𝑛 ℐ𝑛 PROPN cana-1536 34 34 ) ) PUNCT cana-1536 34 35 < < X cana-1536 34 36 Ξ΅ Ξ΅ PROPN cana-1536 34 37 for for ADP cana-1536 34 38 all all DET cana-1536 34 39 n n PRON cana-1536 34 40 β‰₯ β‰₯ NOUN cana-1536 34 41 𝑁. 𝑁. PROPN cana-1536 34 42 definition definition NOUN cana-1536 34 43 2.5 2.5 NUM cana-1536 34 44 a a DET cana-1536 34 45 β‚’ β‚’ ADP cana-1536 34 46 𝑏-ms 𝑏-ms PROPN cana-1536 34 47 ( ( PUNCT cana-1536 34 48 α½£,ₒ𝑏 α½£,ₒ𝑏 PROPN cana-1536 34 49 ) ) PUNCT cana-1536 34 50 is be AUX cana-1536 34 51 called call VERB cana-1536 34 52 β‚’ β‚’ ADP cana-1536 34 53 𝑏-complete 𝑏-complete PROPN cana-1536 34 54 . . PUNCT cana-1536 35 1 if if SCONJ cana-1536 35 2 every every DET cana-1536 35 3 β‚’ β‚’ ADP cana-1536 35 4 𝑏-cs 𝑏-cs PROPN cana-1536 35 5 in in ADP cana-1536 35 6 α½£ α½£ PROPN cana-1536 35 7 is be AUX cana-1536 35 8 β‚’ β‚’ ADP cana-1536 35 9 𝑏-convergent 𝑏-convergent NOUN cana-1536 35 10 in in ADP cana-1536 35 11 α½£. α½£. PROPN cana-1536 35 12 definition definition NOUN cana-1536 35 13 2.6 2.6 NUM cana-1536 35 14 let let VERB cana-1536 35 15 ( ( PUNCT cana-1536 35 16 α½£,ₒ𝑏 α½£,ₒ𝑏 NOUN cana-1536 35 17 ) ) PUNCT cana-1536 35 18 is be AUX cana-1536 35 19 β‚’ β‚’ ADP cana-1536 35 20 𝑏-ms 𝑏-ms NOUN cana-1536 35 21 . . PUNCT cana-1536 36 1 a a DET cana-1536 36 2 mapping mapping NOUN cana-1536 36 3 β„‹ β„‹ NOUN cana-1536 36 4 : : PUNCT cana-1536 36 5 α½£3 α½£3 PROPN cana-1536 36 6 β†’ β†’ SYM cana-1536 36 7 α½£ α½£ PRON cana-1536 36 8 is be AUX cana-1536 36 9 said say VERB cana-1536 36 10 to to PART cana-1536 36 11 be be AUX cana-1536 36 12 continuous continuous ADJ cana-1536 36 13 if if SCONJ cana-1536 36 14 for for ADP cana-1536 36 15 any any DET cana-1536 36 16 { { PUNCT cana-1536 36 17 ℐ𝑛 ℐ𝑛 NOUN cana-1536 36 18 } } PUNCT cana-1536 36 19 , , PUNCT cana-1536 36 20 { { PUNCT cana-1536 36 21 ὡ𝑛 ὡ𝑛 NOUN cana-1536 36 22 } } PUNCT cana-1536 36 23 , , PUNCT cana-1536 36 24 { { PUNCT cana-1536 36 25 ϱ𝑛}β‚’-convergent ϱ𝑛}β‚’-convergent ADJ cana-1536 36 26 sequences sequence NOUN cana-1536 36 27 to to ADP cana-1536 36 28 Ο° Ο° NOUN cana-1536 36 29 , , PUNCT cana-1536 36 30 β„“ β„“ PROPN cana-1536 36 31 , , PUNCT cana-1536 36 32 β„˜ β„˜ PROPN cana-1536 36 33 then then ADV cana-1536 36 34 { { PUNCT cana-1536 36 35 β„‹(ℐ𝑛 β„‹(ℐ𝑛 NOUN cana-1536 36 36 , , PUNCT cana-1536 36 37 ὡ𝑛 ὡ𝑛 NOUN cana-1536 36 38 , , PUNCT cana-1536 36 39 ϱ𝑛 ϱ𝑛 PROPN cana-1536 36 40 ) ) PUNCT cana-1536 36 41 } } PUNCT cana-1536 36 42 is be AUX cana-1536 36 43 β‚’ β‚’ ADP cana-1536 36 44 𝑏 𝑏 DET cana-1536 36 45 convergent convergent NOUN cana-1536 36 46 to to ADP cana-1536 36 47 β„‹(Ο° β„‹(Ο° NUM cana-1536 36 48 , , PUNCT cana-1536 36 49 β„“ β„“ PROPN cana-1536 36 50 , , PUNCT cana-1536 36 51 β„˜ β„˜ PROPN cana-1536 36 52 ) ) PUNCT cana-1536 36 53 . . PUNCT cana-1536 37 1 definition definition NOUN cana-1536 37 2 2.7 2.7 NUM cana-1536 37 3 let let VERB cana-1536 37 4 β„‹ β„‹ NOUN cana-1536 37 5 : : PUNCT cana-1536 37 6 α½£3 α½£3 PROPN cana-1536 37 7 β†’ β†’ SYM cana-1536 37 8 α½£ α½£ X cana-1536 37 9 and and CCONJ cana-1536 37 10 β„± β„± PROPN cana-1536 37 11 : : PUNCT cana-1536 37 12 α½£ α½£ X cana-1536 37 13 β†’ β†’ X cana-1536 37 14 α½£ α½£ PRON cana-1536 37 15 be be AUX cana-1536 37 16 any any DET cana-1536 37 17 two two NUM cana-1536 37 18 mappings mapping NOUN cana-1536 37 19 , , PUNCT cana-1536 37 20 then then ADV cana-1536 37 21 the the DET cana-1536 37 22 mappings mapping NOUN cana-1536 37 23 are be AUX cana-1536 37 24 said say VERB cana-1536 37 25 to to PART cana-1536 37 26 be be AUX cana-1536 37 27 commute commute VERB cana-1536 37 28 if if SCONJ cana-1536 37 29 β„±(β„‹(Ο° β„±(β„‹(Ο° PROPN cana-1536 37 30 , , PUNCT cana-1536 37 31 β„“ β„“ PROPN cana-1536 37 32 , , PUNCT cana-1536 37 33 β„˜ β„˜ NOUN cana-1536 37 34 ) ) PUNCT cana-1536 37 35 ) ) PUNCT cana-1536 38 1 = = SYM cana-1536 38 2 β„‹(β„±(Ο° β„‹(β„±(Ο° NOUN cana-1536 38 3 ) ) PUNCT cana-1536 38 4 , , PUNCT cana-1536 38 5 β„±(β„“ β„±(β„“ NUM cana-1536 38 6 ) ) PUNCT cana-1536 38 7 , , PUNCT cana-1536 38 8 β„±(β„˜ β„±(β„˜ NOUN cana-1536 38 9 ) ) PUNCT cana-1536 38 10 ) ) PUNCT cana-1536 38 11 for for ADP cana-1536 38 12 allΟ° allΟ° PROPN cana-1536 38 13 , , PUNCT cana-1536 38 14 β„“ β„“ PROPN cana-1536 38 15 , , PUNCT cana-1536 38 16 β„˜ β„˜ PROPN cana-1536 38 17 ∈ ∈ NOUN cana-1536 38 18 α½£. α½£. NOUN cana-1536 38 19 3 3 X cana-1536 38 20 . . NOUN cana-1536 38 21 results result NOUN cana-1536 38 22 and and CCONJ cana-1536 38 23 discussion discussion NOUN cana-1536 38 24 definition definition NOUN cana-1536 38 25 3.1 3.1 NUM cana-1536 38 26 let let VERB cana-1536 38 27 β„‹ β„‹ NOUN cana-1536 38 28 : : PUNCT cana-1536 38 29 α½£3 α½£3 PROPN cana-1536 38 30 β†’ β†’ SYM cana-1536 38 31 α½£ α½£ PRON cana-1536 38 32 be be AUX cana-1536 38 33 a a DET cana-1536 38 34 mapping mapping NOUN cana-1536 38 35 . . PUNCT cana-1536 39 1 an an DET cana-1536 39 2 element element NOUN cana-1536 39 3 ( ( PUNCT cana-1536 39 4 α΄‚ α΄‚ PROPN cana-1536 39 5 , , PUNCT cana-1536 39 6 α΄” α΄” NOUN cana-1536 39 7 , , PUNCT cana-1536 39 8 𝔣 𝔣 ADJ cana-1536 39 9 ) ) PUNCT cana-1536 39 10 is be AUX cana-1536 39 11 called call VERB cana-1536 39 12 tfp tfp PROPN cana-1536 39 13 of of ADP cana-1536 39 14 the the DET cana-1536 39 15 mappingβ„‹. mappingβ„‹. PROPN cana-1536 39 16 if if SCONJ cana-1536 39 17 β„‹(α΄‚ β„‹(α΄‚ NUM cana-1536 39 18 , , PUNCT cana-1536 39 19 α΄” α΄” NOUN cana-1536 39 20 , , PUNCT cana-1536 39 21 𝔣 𝔣 ADJ cana-1536 39 22 ) ) PUNCT cana-1536 39 23 = = SYM cana-1536 39 24 α΄‚ α΄‚ PROPN cana-1536 39 25 , , PUNCT cana-1536 39 26 β„‹ β„‹ PROPN cana-1536 39 27 ( ( PUNCT cana-1536 39 28 α΄” α΄” NOUN cana-1536 39 29 , , PUNCT cana-1536 39 30 𝔣 𝔣 ADJ cana-1536 39 31 , , PUNCT cana-1536 39 32 α΄‚ α΄‚ NOUN cana-1536 39 33 ) ) PUNCT cana-1536 39 34 = = SYM cana-1536 39 35 α΄” α΄” PROPN cana-1536 39 36 and and CCONJ cana-1536 39 37 β„‹(𝔣 β„‹(𝔣 PUNCT cana-1536 39 38 , , PUNCT cana-1536 39 39 α΄‚ α΄‚ NOUN cana-1536 39 40 , , PUNCT cana-1536 39 41 α΄” α΄” NOUN cana-1536 39 42 ) ) PUNCT cana-1536 39 43 = = SYM cana-1536 39 44 𝔣 𝔣 ADJ cana-1536 39 45 definition definition NOUN cana-1536 39 46 3.2 3.2 NUM cana-1536 39 47 let let VERB cana-1536 39 48 β„‹ β„‹ NOUN cana-1536 39 49 : : PUNCT cana-1536 39 50 α½£3 α½£3 PROPN cana-1536 39 51 β†’ β†’ SYM cana-1536 39 52 α½£ α½£ PRON cana-1536 39 53 be be AUX cana-1536 39 54 a a DET cana-1536 39 55 mapping mapping NOUN cana-1536 39 56 and and CCONJ cana-1536 39 57 β„± β„± PROPN cana-1536 39 58 : : PUNCT cana-1536 39 59 α½£ α½£ X cana-1536 39 60 β†’ β†’ X cana-1536 39 61 α½£ α½£ X cana-1536 39 62 be be AUX cana-1536 39 63 two two NUM cana-1536 39 64 mappings mapping NOUN cana-1536 39 65 , , PUNCT cana-1536 39 66 then then ADV cana-1536 39 67 an an DET cana-1536 39 68 element element NOUN cana-1536 39 69 ( ( PUNCT cana-1536 39 70 α΄‚ α΄‚ PROPN cana-1536 39 71 , , PUNCT cana-1536 39 72 α΄” α΄” NOUN cana-1536 39 73 , , PUNCT cana-1536 39 74 𝔣 𝔣 ADJ cana-1536 39 75 ) ) PUNCT cana-1536 39 76 is be AUX cana-1536 39 77 tcip tcip NOUN cana-1536 39 78 of of ADP cana-1536 39 79 the the DET cana-1536 39 80 mappings mapping NOUN cana-1536 39 81 β„± β„± PROPN cana-1536 39 82 and and CCONJ cana-1536 39 83 β„‹. β„‹. PROPN cana-1536 39 84 if if SCONJ cana-1536 39 85 β„‹(α΄‚ β„‹(α΄‚ NUM cana-1536 39 86 , , PUNCT cana-1536 39 87 α΄” α΄” NOUN cana-1536 39 88 , , PUNCT cana-1536 39 89 𝔣 𝔣 ADJ cana-1536 39 90 ) ) PUNCT cana-1536 39 91 = = SYM cana-1536 40 1 β„±α΄‚ β„±α΄‚ PROPN cana-1536 40 2 , , PUNCT cana-1536 40 3 β„‹ β„‹ PROPN cana-1536 40 4 ( ( PUNCT cana-1536 40 5 α΄” α΄” NOUN cana-1536 40 6 , , PUNCT cana-1536 40 7 𝔣 𝔣 ADJ cana-1536 40 8 , , PUNCT cana-1536 40 9 α΄‚ α΄‚ NOUN cana-1536 40 10 ) ) PUNCT cana-1536 40 11 = = PUNCT cana-1536 41 1 β„±α΄” β„±α΄” PROPN cana-1536 41 2 and and CCONJ cana-1536 41 3 β„‹(𝔣 β„‹(𝔣 PUNCT cana-1536 41 4 , , PUNCT cana-1536 41 5 α΄‚ α΄‚ NOUN cana-1536 41 6 , , PUNCT cana-1536 41 7 α΄” α΄” NOUN cana-1536 41 8 ) ) PUNCT cana-1536 41 9 = = PUNCT cana-1536 42 1 ℱ𝔣 ℱ𝔣 PROPN cana-1536 42 2 definition definition NOUN cana-1536 42 3 3.3 3.3 NUM cana-1536 42 4 let let VERB cana-1536 42 5 β„‹ β„‹ NOUN cana-1536 42 6 : : PUNCT cana-1536 42 7 α½£3 α½£3 PROPN cana-1536 42 8 β†’ β†’ SYM cana-1536 42 9 α½£ α½£ PRON cana-1536 42 10 be be AUX cana-1536 42 11 a a DET cana-1536 42 12 mapping mapping NOUN cana-1536 42 13 and and CCONJ cana-1536 42 14 β„± β„± PROPN cana-1536 42 15 : : PUNCT cana-1536 42 16 α½£ α½£ X cana-1536 42 17 β†’ β†’ X cana-1536 42 18 α½£ α½£ X cana-1536 42 19 be be AUX cana-1536 42 20 two two NUM cana-1536 42 21 mappings mapping NOUN cana-1536 42 22 , , PUNCT cana-1536 42 23 then then ADV cana-1536 42 24 an an DET cana-1536 42 25 element element NOUN cana-1536 42 26 ( ( PUNCT cana-1536 42 27 α΄‚ α΄‚ PROPN cana-1536 42 28 , , PUNCT cana-1536 42 29 α΄” α΄” NOUN cana-1536 42 30 , , PUNCT cana-1536 42 31 𝔣 𝔣 ADJ cana-1536 42 32 ) ) PUNCT cana-1536 42 33 is be AUX cana-1536 42 34 common common ADJ cana-1536 42 35 tfp tfp NOUN cana-1536 42 36 of of ADP cana-1536 42 37 the the DET cana-1536 42 38 mappings mapping NOUN cana-1536 42 39 β„± β„± PROPN cana-1536 42 40 and and CCONJ cana-1536 42 41 β„‹. β„‹. PROPN cana-1536 42 42 if if SCONJ cana-1536 42 43 β„‹(α΄‚ β„‹(α΄‚ NUM cana-1536 42 44 , , PUNCT cana-1536 42 45 α΄” α΄” NOUN cana-1536 42 46 , , PUNCT cana-1536 42 47 𝔣 𝔣 ADJ cana-1536 42 48 ) ) PUNCT cana-1536 42 49 = = PUNCT cana-1536 43 1 β„±α΄‚ β„±α΄‚ NOUN cana-1536 43 2 = = SYM cana-1536 43 3 α΄‚ α΄‚ PROPN cana-1536 43 4 , , PUNCT cana-1536 43 5 β„‹ β„‹ PROPN cana-1536 43 6 ( ( PUNCT cana-1536 43 7 α΄” α΄” NOUN cana-1536 43 8 , , PUNCT cana-1536 43 9 𝔣 𝔣 ADJ cana-1536 43 10 , , PUNCT cana-1536 43 11 α΄‚ α΄‚ NOUN cana-1536 43 12 ) ) PUNCT cana-1536 43 13 = = PUNCT cana-1536 44 1 β„±α΄” β„±α΄” NUM cana-1536 44 2 = = SYM cana-1536 44 3 α΄” α΄” PROPN cana-1536 44 4 and and CCONJ cana-1536 44 5 β„‹(𝔣 β„‹(𝔣 PUNCT cana-1536 44 6 , , PUNCT cana-1536 44 7 α΄‚ α΄‚ NOUN cana-1536 44 8 , , PUNCT cana-1536 44 9 α΄” α΄” NOUN cana-1536 44 10 ) ) PUNCT cana-1536 45 1 = = PUNCT cana-1536 46 1 ℱ𝔣 ℱ𝔣 PROPN cana-1536 46 2 = = SYM cana-1536 46 3 𝔣 𝔣 PROPN cana-1536 46 4 communications communication NOUN cana-1536 46 5 on on ADP cana-1536 46 6 applied apply VERB cana-1536 46 7 nonlinear nonlinear ADJ cana-1536 46 8 analysis analysis NOUN cana-1536 46 9 issn issn NOUN cana-1536 46 10 : : PUNCT cana-1536 46 11 1074 1074 NUM cana-1536 46 12 - - PUNCT cana-1536 46 13 133x 133x NUM cana-1536 46 14 vol vol NOUN cana-1536 46 15 31 31 NUM cana-1536 46 16 no no NOUN cana-1536 46 17 . . PUNCT cana-1536 47 1 8s 8s PROPN cana-1536 47 2 ( ( PUNCT cana-1536 47 3 2024 2024 NUM cana-1536 47 4 ) ) PUNCT cana-1536 47 5 435 435 NUM cana-1536 47 6 https://internationalpubls.com https://internationalpubls.com X cana-1536 47 7 theorem theorem VERB cana-1536 47 8 3.4let 3.4let NUM cana-1536 47 9 ( ( PUNCT cana-1536 47 10 α½£,ₒ𝑏 α½£,ₒ𝑏 PROPN cana-1536 47 11 ) ) PUNCT cana-1536 47 12 is be AUX cana-1536 47 13 β‚’ β‚’ ADP cana-1536 47 14 𝑏-ms 𝑏-ms NOUN cana-1536 47 15 . . PUNCT cana-1536 48 1 let let VERB cana-1536 48 2 β„‹ β„‹ NOUN cana-1536 48 3 : : PUNCT cana-1536 48 4 α½£3 α½£3 PROPN cana-1536 48 5 β†’ β†’ SYM cana-1536 48 6 α½£ α½£ X cana-1536 48 7 and and CCONJ cana-1536 48 8 β„± β„± PROPN cana-1536 48 9 : : PUNCT cana-1536 48 10 α½£ α½£ X cana-1536 48 11 β†’ β†’ X cana-1536 48 12 α½£ α½£ X cana-1536 48 13 be be VERB cana-1536 48 14 two two NUM cana-1536 48 15 mappings mapping NOUN cana-1536 48 16 such such ADJ cana-1536 48 17 that that SCONJ cana-1536 49 1 [ [ X cana-1536 49 2 β‚’ β‚’ ADP cana-1536 49 3 𝑏(β„‹(α΄‚ 𝑏(β„‹(α΄‚ PROPN cana-1536 49 4 , , PUNCT cana-1536 49 5 α΄” α΄” NOUN cana-1536 49 6 , , PUNCT cana-1536 49 7 𝔣 𝔣 ADJ cana-1536 49 8 ) ) PUNCT cana-1536 49 9 , , PUNCT cana-1536 49 10 β„‹(Ο° β„‹(Ο° PROPN cana-1536 49 11 , , PUNCT cana-1536 49 12 β„“ β„“ NOUN cana-1536 49 13 , , PUNCT cana-1536 49 14 β„˜ β„˜ PROPN cana-1536 49 15 ) ) PUNCT cana-1536 49 16 , , PUNCT cana-1536 49 17 β„‹(Ο° β„‹(Ο° PROPN cana-1536 49 18 , , PUNCT cana-1536 49 19 β„“ β„“ NOUN cana-1536 49 20 , , PUNCT cana-1536 49 21 β„˜ β„˜ PROPN cana-1536 49 22 ) ) PUNCT cana-1536 49 23 ) ) PUNCT cana-1536 50 1 + + CCONJ cana-1536 50 2 β‚’ β‚’ ADP cana-1536 50 3 𝑏(β„‹ 𝑏(β„‹ PROPN cana-1536 50 4 ( ( PUNCT cana-1536 50 5 α΄” α΄” NOUN cana-1536 50 6 , , PUNCT cana-1536 50 7 𝔣 𝔣 ADJ cana-1536 50 8 , , PUNCT cana-1536 50 9 α΄‚ α΄‚ NOUN cana-1536 50 10 ) ) PUNCT cana-1536 50 11 , , PUNCT cana-1536 50 12 β„‹(β„“ β„‹(β„“ NUM cana-1536 50 13 , , PUNCT cana-1536 50 14 β„˜ β„˜ PROPN cana-1536 50 15 , , PUNCT cana-1536 50 16 Ο° Ο° NOUN cana-1536 50 17 ) ) PUNCT cana-1536 50 18 , , PUNCT cana-1536 50 19 β„‹(β„“ β„‹(β„“ NUM cana-1536 50 20 , , PUNCT cana-1536 50 21 β„˜ β„˜ PROPN cana-1536 50 22 , , PUNCT cana-1536 50 23 Ο° Ο° NOUN cana-1536 50 24 ) ) PUNCT cana-1536 50 25 ) ) PUNCT cana-1536 51 1 + + CCONJ cana-1536 51 2 β‚’ β‚’ ADP cana-1536 51 3 𝑏(β„‹(𝔣 𝑏(β„‹(𝔣 NOUN cana-1536 51 4 , , PUNCT cana-1536 51 5 α΄‚ α΄‚ NOUN cana-1536 51 6 , , PUNCT cana-1536 51 7 α΄” α΄” NOUN cana-1536 51 8 ) ) PUNCT cana-1536 51 9 , , PUNCT cana-1536 51 10 β„‹(β„˜ β„‹(β„˜ NOUN cana-1536 51 11 , , PUNCT cana-1536 51 12 Ο° Ο° NOUN cana-1536 51 13 , , PUNCT cana-1536 51 14 β„“ β„“ NOUN cana-1536 51 15 ) ) PUNCT cana-1536 51 16 , , PUNCT cana-1536 51 17 β„‹(β„˜ β„‹(β„˜ NOUN cana-1536 51 18 , , PUNCT cana-1536 51 19 Ο° Ο° NOUN cana-1536 51 20 , , PUNCT cana-1536 51 21 β„“ β„“ NOUN cana-1536 51 22 ) ) PUNCT cana-1536 51 23 ) ) PUNCT cana-1536 51 24 ] ] PUNCT cana-1536 51 25 ≀ ≀ NUM cana-1536 51 26 πœƒ[ₒ𝑏(β„±α΄‚ πœƒ[ₒ𝑏(β„±α΄‚ NOUN cana-1536 51 27 , , PUNCT cana-1536 51 28 β„±Ο° β„±Ο° PROPN cana-1536 51 29 , , PUNCT cana-1536 51 30 β„±Ο° β„±Ο° ADJ cana-1536 51 31 ) ) PUNCT cana-1536 51 32 + + CCONJ cana-1536 51 33 β‚’ β‚’ ADP cana-1536 51 34 𝑏(β„±α΄” 𝑏(β„±α΄” PROPN cana-1536 51 35 , , PUNCT cana-1536 51 36 β„±β„“ β„±β„“ PROPN cana-1536 51 37 , , PUNCT cana-1536 51 38 β„±β„“ β„±β„“ PROPN cana-1536 51 39 ) ) PUNCT cana-1536 51 40 + + CCONJ cana-1536 51 41 β‚’ β‚’ ADP cana-1536 51 42 𝑏(ℱ𝔣 𝑏(ℱ𝔣 NUM cana-1536 51 43 , , PUNCT cana-1536 51 44 β„±β„˜ β„±β„˜ NUM cana-1536 51 45 , , PUNCT cana-1536 51 46 β„±β„˜)]------------------(3.4.1 β„±β„˜)]------------------(3.4.1 NUM cana-1536 51 47 ) ) PUNCT cana-1536 51 48 for for ADP cana-1536 51 49 all all DET cana-1536 51 50 α΄‚ α΄‚ PROPN cana-1536 51 51 , , PUNCT cana-1536 51 52 α΄” α΄” NOUN cana-1536 51 53 , , PUNCT cana-1536 51 54 𝔣 𝔣 ADJ cana-1536 51 55 , , PUNCT cana-1536 51 56 Ο° Ο° NOUN cana-1536 51 57 , , PUNCT cana-1536 51 58 β„“ β„“ PROPN cana-1536 51 59 , , PUNCT cana-1536 51 60 β„˜ β„˜ PROPN cana-1536 51 61 ∈ ∈ NOUN cana-1536 51 62 α½£. α½£. NOUN cana-1536 51 63 if if SCONJ cana-1536 51 64 β„‹and β„‹and PROPN cana-1536 51 65 β„± β„± PROPN cana-1536 51 66 satisfies satisfy VERB cana-1536 51 67 the the DET cana-1536 51 68 following follow VERB cana-1536 51 69 conditions condition NOUN cana-1536 51 70 i i PRON cana-1536 51 71 ) ) PUNCT cana-1536 51 72 β„‹(α½£3 β„‹(α½£3 PROPN cana-1536 51 73 ) ) PUNCT cana-1536 51 74 βŠ‚ βŠ‚ PROPN cana-1536 52 1 β„±(α½£ β„±(α½£ NUM cana-1536 52 2 ) ) PUNCT cana-1536 52 3 ; ; PUNCT cana-1536 52 4 ii ii X cana-1536 52 5 ) ) PUNCT cana-1536 52 6 β„±(α½£ β„±(α½£ NUM cana-1536 52 7 ) ) PUNCT cana-1536 52 8 is be AUX cana-1536 52 9 β‚’ β‚’ ADP cana-1536 52 10 𝑏-complete 𝑏-complete PROPN cana-1536 52 11 ; ; PUNCT cana-1536 52 12 iii iii X cana-1536 52 13 ) ) PUNCT cana-1536 52 14 β„± β„± PROPN cana-1536 52 15 is be AUX cana-1536 52 16 β‚’ β‚’ ADP cana-1536 52 17 𝑏-continuous 𝑏-continuous ADJ cana-1536 52 18 and and CCONJ cana-1536 52 19 commutes commute NOUN cana-1536 52 20 with with ADP cana-1536 52 21 β„‹ β„‹ PROPN cana-1536 52 22 ; ; PUNCT cana-1536 52 23 if if SCONJ cana-1536 52 24 πœƒ πœƒ PRON cana-1536 52 25 ∈ ∈ X cana-1536 52 26 [ [ X cana-1536 52 27 0,1 0,1 NUM cana-1536 52 28 ) ) PUNCT cana-1536 52 29 then then ADV cana-1536 52 30 there there PRON cana-1536 52 31 exists exist VERB cana-1536 52 32 a a DET cana-1536 52 33 unique unique ADJ cana-1536 52 34 common common ADJ cana-1536 52 35 tfp tfp NOUN cana-1536 52 36 for for ADP cana-1536 52 37 β„‹ β„‹ PROPN cana-1536 52 38 and and CCONJ cana-1536 52 39 β„± β„± PROPN cana-1536 52 40 inα½£. inα½£. VERB cana-1536 52 41 proof.letℐ0 proof.letℐ0 NOUN cana-1536 52 42 , , PUNCT cana-1536 52 43 ὡ0 ὡ0 NOUN cana-1536 52 44 , , PUNCT cana-1536 52 45 Ο±0be Ο±0be ADV cana-1536 52 46 arbitrary arbitrary ADJ cana-1536 52 47 in in ADP cana-1536 52 48 α½£ α½£ PROPN cana-1536 52 49 then then ADV cana-1536 52 50 from from ADP cana-1536 52 51 ( ( PUNCT cana-1536 52 52 i i NOUN cana-1536 52 53 ) ) PUNCT cana-1536 52 54 we we PRON cana-1536 52 55 can can AUX cana-1536 52 56 construct construct VERB cana-1536 52 57 sequences sequence NOUN cana-1536 52 58 { { PUNCT cana-1536 52 59 ℐ𝑛 ℐ𝑛 NOUN cana-1536 52 60 } } PUNCT cana-1536 52 61 , , PUNCT cana-1536 52 62 { { PUNCT cana-1536 52 63 ὡ𝑛 ὡ𝑛 NOUN cana-1536 52 64 } } PUNCT cana-1536 52 65 and and CCONJ cana-1536 52 66 { { PUNCT cana-1536 52 67 ϱ𝑛 ϱ𝑛 PRON cana-1536 52 68 } } PUNCT cana-1536 52 69 inα½£ inα½£ VERB cana-1536 52 70 such such ADJ cana-1536 52 71 that that SCONJ cana-1536 52 72 ℱℐ𝑝+1 ℱℐ𝑝+1 NOUN cana-1536 52 73 = = SYM cana-1536 52 74 β„‹(ℐ𝑝 β„‹(ℐ𝑝 NUM cana-1536 52 75 , , PUNCT cana-1536 52 76 ὡ𝑝 ὡ𝑝 X cana-1536 52 77 , , PUNCT cana-1536 52 78 ϱ𝑝 ϱ𝑝 NOUN cana-1536 52 79 ) ) PUNCT cana-1536 52 80 , , PUNCT cana-1536 52 81 ℱὡ𝑝+1 ℱὡ𝑝+1 PROPN cana-1536 52 82 = = SYM cana-1536 52 83 β„‹(ὡ𝑝 β„‹(ὡ𝑝 NOUN cana-1536 52 84 , , PUNCT cana-1536 52 85 ϱ𝑝 ϱ𝑝 ADJ cana-1536 52 86 , , PUNCT cana-1536 52 87 ℐ𝑝 ℐ𝑝 PROPN cana-1536 52 88 ) ) PUNCT cana-1536 52 89 and and CCONJ cana-1536 52 90 ℱϱ𝑝+1 ℱϱ𝑝+1 PRON cana-1536 52 91 = = SYM cana-1536 52 92 β„‹(ϱ𝑝 β„‹(ϱ𝑝 ADJ cana-1536 52 93 , , PUNCT cana-1536 52 94 ℐ𝑝 ℐ𝑝 NOUN cana-1536 52 95 , , PUNCT cana-1536 52 96 ὡ𝑝)-------(3.4.2 ὡ𝑝)-------(3.4.2 NOUN cana-1536 52 97 ) ) PUNCT cana-1536 52 98 let let VERB cana-1536 52 99 ℑ𝑝 ℑ𝑝 PROPN cana-1536 52 100 = = PUNCT cana-1536 52 101 β‚’ β‚’ ADP cana-1536 52 102 𝑏(ℱℐ𝑝 𝑏(ℱℐ𝑝 PROPN cana-1536 52 103 , , PUNCT cana-1536 52 104 ℱℐ𝑝+1 ℱℐ𝑝+1 NOUN cana-1536 52 105 , , PUNCT cana-1536 52 106 ℱℐ𝑝+1 ℱℐ𝑝+1 NOUN cana-1536 52 107 ) ) PUNCT cana-1536 53 1 + + CCONJ cana-1536 53 2 β‚’ β‚’ ADP cana-1536 53 3 𝑏(ℱὡ𝑝 𝑏(ℱὡ𝑝 PROPN cana-1536 53 4 , , PUNCT cana-1536 53 5 ℱὡ𝑝+1 ℱὡ𝑝+1 NUM cana-1536 53 6 , , PUNCT cana-1536 53 7 ℱὡ𝑝+1 ℱὡ𝑝+1 NUM cana-1536 53 8 ) ) PUNCT cana-1536 53 9 + + CCONJ cana-1536 53 10 β‚’ β‚’ ADP cana-1536 53 11 𝑏(ℱϱ𝑝 𝑏(ℱϱ𝑝 NUM cana-1536 53 12 , , PUNCT cana-1536 53 13 ℱϱ𝑝+1 ℱϱ𝑝+1 NOUN cana-1536 53 14 , , PUNCT cana-1536 53 15 ℱϱ𝑝+1 ℱϱ𝑝+1 NOUN cana-1536 53 16 ) ) PUNCT cana-1536 53 17 for for ADP cana-1536 53 18 all all DET cana-1536 53 19 𝑝 𝑝 PROPN cana-1536 53 20 ∈ ∈ PROPN cana-1536 53 21 β„•. β„•. PROPN cana-1536 53 22 now now ADV cana-1536 53 23 by by ADP cana-1536 53 24 using use VERB cana-1536 53 25 equation equation NOUN cana-1536 53 26 ( ( PUNCT cana-1536 53 27 3.4.1 3.4.1 NUM cana-1536 53 28 ) ) PUNCT cana-1536 53 29 , , PUNCT cana-1536 53 30 we we PRON cana-1536 53 31 have have VERB cana-1536 53 32 ℑ𝑝 ℑ𝑝 PROPN cana-1536 53 33 = = PUNCT cana-1536 53 34 β‚’ β‚’ ADP cana-1536 53 35 𝑏(ℱℐ𝑝 𝑏(ℱℐ𝑝 PROPN cana-1536 53 36 , , PUNCT cana-1536 53 37 ℱℐ𝑝+1 ℱℐ𝑝+1 NOUN cana-1536 53 38 , , PUNCT cana-1536 53 39 ℱℐ𝑝+1 ℱℐ𝑝+1 NOUN cana-1536 53 40 ) ) PUNCT cana-1536 53 41 + + CCONJ cana-1536 53 42 β‚’ β‚’ ADP cana-1536 53 43 𝑏(ℱὡ𝑝 𝑏(ℱὡ𝑝 PROPN cana-1536 53 44 , , PUNCT cana-1536 53 45 ℱὡ𝑝+1 ℱὡ𝑝+1 NUM cana-1536 53 46 , , PUNCT cana-1536 53 47 ℱὡ𝑝+1 ℱὡ𝑝+1 NUM cana-1536 53 48 ) ) PUNCT cana-1536 53 49 + + CCONJ cana-1536 53 50 β‚’ β‚’ ADP cana-1536 53 51 𝑏(ℱϱ𝑝 𝑏(ℱϱ𝑝 NUM cana-1536 53 52 , , PUNCT cana-1536 53 53 ℱϱ𝑝+1 ℱϱ𝑝+1 NOUN cana-1536 53 54 , , PUNCT cana-1536 53 55 ℱϱ𝑝+1 ℱϱ𝑝+1 NOUN cana-1536 53 56 ) ) PUNCT cana-1536 54 1 = = NOUN cana-1536 54 2 β‚’ β‚’ ADP cana-1536 54 3 𝑏 𝑏 PRON cana-1536 54 4 ( ( PUNCT cana-1536 54 5 β„‹(β„π‘βˆ’1 β„‹(β„π‘βˆ’1 X cana-1536 54 6 , , PUNCT cana-1536 54 7 α½΅π‘βˆ’1 α½΅π‘βˆ’1 PROPN cana-1536 54 8 , , PUNCT cana-1536 54 9 Ο±π‘βˆ’1 Ο±π‘βˆ’1 PROPN cana-1536 54 10 ) ) PUNCT cana-1536 54 11 , , PUNCT cana-1536 54 12 β„‹(ℐ𝑝 β„‹(ℐ𝑝 NUM cana-1536 54 13 , , PUNCT cana-1536 54 14 ὡ𝑝 ὡ𝑝 X cana-1536 54 15 , , PUNCT cana-1536 54 16 ϱ𝑝 ϱ𝑝 NOUN cana-1536 54 17 ) ) PUNCT cana-1536 54 18 , , PUNCT cana-1536 54 19 β„‹(ℐ𝑝 β„‹(ℐ𝑝 NUM cana-1536 54 20 , , PUNCT cana-1536 54 21 ὡ𝑝 ὡ𝑝 X cana-1536 54 22 , , PUNCT cana-1536 54 23 ϱ𝑝 ϱ𝑝 NOUN cana-1536 54 24 ) ) PUNCT cana-1536 54 25 ) ) PUNCT cana-1536 55 1 + + CCONJ cana-1536 55 2 β‚’ β‚’ ADP cana-1536 55 3 𝑏 𝑏 PRON cana-1536 55 4 ( ( PUNCT cana-1536 55 5 β„‹(α½΅π‘βˆ’1 β„‹(α½΅π‘βˆ’1 PROPN cana-1536 55 6 , , PUNCT cana-1536 55 7 Ο±π‘βˆ’1 Ο±π‘βˆ’1 PROPN cana-1536 55 8 , , PUNCT cana-1536 55 9 β„π‘βˆ’1 β„π‘βˆ’1 PROPN cana-1536 55 10 ) ) PUNCT cana-1536 55 11 , , PUNCT cana-1536 55 12 β„‹(ὡ𝑝 β„‹(ὡ𝑝 NOUN cana-1536 55 13 , , PUNCT cana-1536 55 14 ϱ𝑝 ϱ𝑝 ADJ cana-1536 55 15 , , PUNCT cana-1536 55 16 ℐ𝑝 ℐ𝑝 PROPN cana-1536 55 17 ) ) PUNCT cana-1536 55 18 , , PUNCT cana-1536 55 19 β„‹(ὡ𝑝 β„‹(ὡ𝑝 NOUN cana-1536 55 20 , , PUNCT cana-1536 55 21 ϱ𝑝 ϱ𝑝 ADP cana-1536 55 22 , , PUNCT cana-1536 55 23 ℐ𝑝 ℐ𝑝 NOUN cana-1536 55 24 ) ) PUNCT cana-1536 55 25 ) ) PUNCT cana-1536 56 1 + + CCONJ cana-1536 56 2 β‚’ β‚’ ADP cana-1536 56 3 𝑏 𝑏 PRON cana-1536 56 4 ( ( PUNCT cana-1536 56 5 β„‹(Ο±π‘βˆ’1 β„‹(Ο±π‘βˆ’1 PROPN cana-1536 56 6 , , PUNCT cana-1536 56 7 β„π‘βˆ’1 β„π‘βˆ’1 PROPN cana-1536 56 8 , , PUNCT cana-1536 56 9 α½΅π‘βˆ’1 α½΅π‘βˆ’1 PROPN cana-1536 56 10 ) ) PUNCT cana-1536 56 11 , , PUNCT cana-1536 56 12 β„‹(ϱ𝑝 β„‹(ϱ𝑝 NUM cana-1536 56 13 , , PUNCT cana-1536 56 14 ℐ𝑝 ℐ𝑝 PROPN cana-1536 56 15 , , PUNCT cana-1536 56 16 ὡ𝑝 ὡ𝑝 X cana-1536 56 17 ) ) PUNCT cana-1536 56 18 , , PUNCT cana-1536 56 19 β„‹(ϱ𝑝 β„‹(ϱ𝑝 ADV cana-1536 56 20 , , PUNCT cana-1536 56 21 ℐ𝑝 ℐ𝑝 PROPN cana-1536 56 22 , , PUNCT cana-1536 56 23 ὡ𝑝 ὡ𝑝 X cana-1536 56 24 ) ) PUNCT cana-1536 56 25 ) ) PUNCT cana-1536 56 26 ≀ ≀ NUM cana-1536 56 27 πœƒ[ₒ𝑏(β„±β„π‘βˆ’1 πœƒ[ₒ𝑏(β„±β„π‘βˆ’1 PROPN cana-1536 56 28 , , PUNCT cana-1536 56 29 ℱℐ𝑝 ℱℐ𝑝 NOUN cana-1536 56 30 , , PUNCT cana-1536 56 31 ℱℐ𝑝 ℱℐ𝑝 NOUN cana-1536 56 32 ) ) PUNCT cana-1536 57 1 + + CCONJ cana-1536 57 2 β‚’ β‚’ NUM cana-1536 57 3 𝑏(β„±α½΅π‘βˆ’1 𝑏(β„±α½΅π‘βˆ’1 PROPN cana-1536 57 4 , , PUNCT cana-1536 57 5 ℱὡ𝑝 ℱὡ𝑝 PROPN cana-1536 57 6 , , PUNCT cana-1536 57 7 ℱὡ𝑝 ℱὡ𝑝 PROPN cana-1536 57 8 ) ) PUNCT cana-1536 58 1 + + CCONJ cana-1536 58 2 β‚’ β‚’ ADP cana-1536 58 3 𝑏(β„±Ο±π‘βˆ’1 𝑏(β„±Ο±π‘βˆ’1 PROPN cana-1536 58 4 , , PUNCT cana-1536 58 5 ℱϱ𝑝 ℱϱ𝑝 PROPN cana-1536 58 6 , , PUNCT cana-1536 58 7 ℱϱ𝑝 ℱϱ𝑝 PROPN cana-1536 58 8 ) ) PUNCT cana-1536 58 9 ] ] PUNCT cana-1536 59 1 = = PUNCT cana-1536 59 2 πœƒβ„‘π‘βˆ’1 πœƒβ„‘π‘βˆ’1 PROPN cana-1536 59 3 . . PUNCT cana-1536 60 1 which which PRON cana-1536 60 2 yields yield VERB cana-1536 60 3 that that PRON cana-1536 60 4 ℑ𝑝 ℑ𝑝 PROPN cana-1536 60 5 ≀ ≀ ADJ cana-1536 60 6 πœƒπ‘β„‘0 πœƒπ‘β„‘0 NOUN cana-1536 60 7 , , PUNCT cana-1536 60 8 βˆ€π‘ βˆ€π‘ X cana-1536 60 9 ∈ ∈ PROPN cana-1536 60 10 β„•. β„•. PROPN cana-1536 60 11 now now ADV cana-1536 60 12 for for ADP cana-1536 60 13 all all DET cana-1536 60 14 π‘š π‘š NOUN cana-1536 60 15 , , PUNCT cana-1536 60 16 𝑛 𝑛 DET cana-1536 60 17 ∈ ∈ PROPN cana-1536 60 18 β„• β„• PROPN cana-1536 60 19 with with ADP cana-1536 60 20 π‘š π‘š PROPN cana-1536 60 21 > > X cana-1536 60 22 𝑛 𝑛 PROPN cana-1536 60 23 and and CCONJ cana-1536 60 24 by by ADP cana-1536 60 25 using use VERB cana-1536 60 26 ( ( PUNCT cana-1536 60 27 β‚’ β‚’ ADP cana-1536 60 28 𝑏5 𝑏5 NOUN cana-1536 60 29 ) ) PUNCT cana-1536 60 30 , , PUNCT cana-1536 60 31 we we PRON cana-1536 60 32 get get VERB cana-1536 60 33 β‚’ β‚’ ADP cana-1536 60 34 𝑏(ℱℐ𝑛 𝑏(ℱℐ𝑛 PROPN cana-1536 60 35 , , PUNCT cana-1536 60 36 β„±β„π‘š β„±β„π‘š NOUN cana-1536 60 37 , , PUNCT cana-1536 60 38 β„±β„π‘š β„±β„π‘š NOUN cana-1536 60 39 ) ) PUNCT cana-1536 61 1 + + CCONJ cana-1536 61 2 β‚’ β‚’ ADP cana-1536 61 3 𝑏(ℱὡ𝑛 𝑏(ℱὡ𝑛 PROPN cana-1536 61 4 , , PUNCT cana-1536 61 5 β„±α½΅π‘š β„±α½΅π‘š PROPN cana-1536 61 6 , , PUNCT cana-1536 61 7 β„±α½΅π‘š β„±α½΅π‘š PROPN cana-1536 61 8 ) ) PUNCT cana-1536 61 9 + + CCONJ cana-1536 61 10 β‚’ β‚’ ADP cana-1536 61 11 𝑏(ℱϱ𝑛 𝑏(ℱϱ𝑛 PROPN cana-1536 61 12 , , PUNCT cana-1536 61 13 β„±Ο±π‘š β„±Ο±π‘š PROPN cana-1536 61 14 , , PUNCT cana-1536 61 15 β„±Ο±π‘š β„±Ο±π‘š PROPN cana-1536 61 16 ) ) PUNCT cana-1536 61 17 ≀ ≀ NOUN cana-1536 61 18 𝑠[ₒ𝑏(ℱℐ𝑛 𝑠[ₒ𝑏(ℱℐ𝑛 NOUN cana-1536 61 19 , , PUNCT cana-1536 61 20 ℱℐ𝑛+1 ℱℐ𝑛+1 NOUN cana-1536 61 21 , , PUNCT cana-1536 61 22 ℱℐ𝑛+1 ℱℐ𝑛+1 PROPN cana-1536 61 23 ) ) PUNCT cana-1536 61 24 + + CCONJ cana-1536 61 25 β‚’ β‚’ ADP cana-1536 61 26 𝑏(ℱℐ𝑛+1 𝑏(ℱℐ𝑛+1 NUM cana-1536 61 27 , , PUNCT cana-1536 61 28 β„±β„π‘š β„±β„π‘š NOUN cana-1536 61 29 , , PUNCT cana-1536 61 30 β„±β„π‘š β„±β„π‘š NOUN cana-1536 61 31 ) ) PUNCT cana-1536 61 32 ] ] PUNCT cana-1536 62 1 + + CCONJ cana-1536 62 2 𝑠[ₒ𝑏(ℱὡ𝑛 𝑠[ₒ𝑏(ℱὡ𝑛 ADJ cana-1536 62 3 , , PUNCT cana-1536 62 4 ℱὡ𝑛+1 ℱὡ𝑛+1 NOUN cana-1536 62 5 , , PUNCT cana-1536 62 6 ℱὡ𝑛+1 ℱὡ𝑛+1 NOUN cana-1536 62 7 ) ) PUNCT cana-1536 62 8 + + CCONJ cana-1536 62 9 β‚’ β‚’ ADP cana-1536 62 10 𝑏(ℱὡ𝑛+1 𝑏(ℱὡ𝑛+1 NOUN cana-1536 62 11 , , PUNCT cana-1536 62 12 β„±α½΅π‘š β„±α½΅π‘š PROPN cana-1536 62 13 , , PUNCT cana-1536 62 14 β„±α½΅π‘š β„±α½΅π‘š PROPN cana-1536 62 15 ) ) PUNCT cana-1536 62 16 ] ] PUNCT cana-1536 63 1 + + CCONJ cana-1536 63 2 𝑠[ₒ𝑏(ℱϱ𝑛 𝑠[ₒ𝑏(ℱϱ𝑛 NOUN cana-1536 63 3 , , PUNCT cana-1536 63 4 ℱϱ𝑛+1 ℱϱ𝑛+1 NOUN cana-1536 63 5 , , PUNCT cana-1536 63 6 ℱϱ𝑛+1 ℱϱ𝑛+1 NOUN cana-1536 63 7 ) ) PUNCT cana-1536 63 8 + + CCONJ cana-1536 63 9 β‚’ β‚’ ADP cana-1536 63 10 𝑏(ℱϱ𝑛+1 𝑏(ℱϱ𝑛+1 PROPN cana-1536 63 11 , , PUNCT cana-1536 63 12 β„±Ο±π‘š β„±Ο±π‘š PROPN cana-1536 63 13 , , PUNCT cana-1536 63 14 β„±Ο±π‘š β„±Ο±π‘š PROPN cana-1536 63 15 ) ) PUNCT cana-1536 63 16 ] ] PUNCT cana-1536 63 17 ≀ ≀ NOUN cana-1536 64 1 [ [ X cana-1536 64 2 𝑠ₒ𝑏(ℱℐ𝑛 𝑠ₒ𝑏(ℱℐ𝑛 NOUN cana-1536 64 3 , , PUNCT cana-1536 64 4 ℱℐ𝑛+1 ℱℐ𝑛+1 NOUN cana-1536 64 5 , , PUNCT cana-1536 64 6 ℱℐ𝑛+1 ℱℐ𝑛+1 PROPN cana-1536 64 7 ) ) PUNCT cana-1536 64 8 + + CCONJ cana-1536 64 9 𝑠2ₒ𝑏(ℱℐ𝑛+1 𝑠2ₒ𝑏(ℱℐ𝑛+1 PROPN cana-1536 64 10 , , PUNCT cana-1536 64 11 ℱℐ𝑛+2 ℱℐ𝑛+2 NOUN cana-1536 64 12 , , PUNCT cana-1536 64 13 ℱℐ𝑛+2 ℱℐ𝑛+2 NOUN cana-1536 64 14 ) ) PUNCT cana-1536 65 1 + + CCONJ cana-1536 65 2 𝑠2ₒ𝑏(ℱℐ𝑛+2 𝑠2ₒ𝑏(ℱℐ𝑛+2 ADJ cana-1536 65 3 , , PUNCT cana-1536 65 4 β„±β„π‘š β„±β„π‘š NOUN cana-1536 65 5 , , PUNCT cana-1536 65 6 β„±β„π‘š β„±β„π‘š NOUN cana-1536 65 7 ) ) PUNCT cana-1536 65 8 ] ] PUNCT cana-1536 66 1 + + CCONJ cana-1536 67 1 [ [ X cana-1536 67 2 𝑠ₒ𝑏(ℱὡ𝑛 𝑠ₒ𝑏(ℱὡ𝑛 X cana-1536 67 3 , , PUNCT cana-1536 67 4 ℱὡ𝑛+1 ℱὡ𝑛+1 NOUN cana-1536 67 5 , , PUNCT cana-1536 67 6 ℱὡ𝑛+1 ℱὡ𝑛+1 NOUN cana-1536 67 7 ) ) PUNCT cana-1536 67 8 + + CCONJ cana-1536 67 9 𝑠2ₒ𝑏(ℱὡ𝑛+1 𝑠2ₒ𝑏(ℱὡ𝑛+1 ADJ cana-1536 67 10 , , PUNCT cana-1536 67 11 ℱὡ𝑛+2 ℱὡ𝑛+2 PROPN cana-1536 67 12 , , PUNCT cana-1536 67 13 ℱὡ𝑛+2 ℱὡ𝑛+2 ADJ cana-1536 67 14 ) ) PUNCT cana-1536 67 15 + + X cana-1536 67 16 𝑠2ₒ𝑏(ℱὡ𝑛+2 𝑠2ₒ𝑏(ℱὡ𝑛+2 PROPN cana-1536 67 17 , , PUNCT cana-1536 67 18 β„±α½΅π‘š β„±α½΅π‘š PROPN cana-1536 67 19 , , PUNCT cana-1536 67 20 β„±α½΅π‘š β„±α½΅π‘š PROPN cana-1536 67 21 ) ) PUNCT cana-1536 67 22 ] ] PUNCT cana-1536 68 1 + + CCONJ cana-1536 69 1 [ [ X cana-1536 69 2 𝑠ₒ𝑏(ℱϱ𝑛 𝑠ₒ𝑏(ℱϱ𝑛 NOUN cana-1536 69 3 , , PUNCT cana-1536 69 4 ℱϱ𝑛+1 ℱϱ𝑛+1 NOUN cana-1536 69 5 , , PUNCT cana-1536 69 6 ℱϱ𝑛+1 ℱϱ𝑛+1 NOUN cana-1536 69 7 ) ) PUNCT cana-1536 69 8 + + CCONJ cana-1536 69 9 𝑠2ₒ𝑏(ℱϱ𝑛+1 𝑠2ₒ𝑏(ℱϱ𝑛+1 NOUN cana-1536 69 10 , , PUNCT cana-1536 69 11 ℱϱ𝑛+2 ℱϱ𝑛+2 PROPN cana-1536 69 12 , , PUNCT cana-1536 69 13 ℱϱ𝑛+2 ℱϱ𝑛+2 NOUN cana-1536 69 14 ) ) PUNCT cana-1536 69 15 + + X cana-1536 69 16 𝑠2ₒ𝑏(ℱϱ𝑛+2 𝑠2ₒ𝑏(ℱϱ𝑛+2 ADJ cana-1536 69 17 , , PUNCT cana-1536 69 18 β„±Ο±π‘š β„±Ο±π‘š PROPN cana-1536 69 19 , , PUNCT cana-1536 69 20 β„±Ο±π‘š β„±Ο±π‘š PROPN cana-1536 69 21 ) ) PUNCT cana-1536 69 22 ] ] PUNCT cana-1536 69 23 … … PUNCT cana-1536 69 24 … … PUNCT cana-1536 69 25 … … PUNCT cana-1536 69 26 … … PUNCT cana-1536 69 27 … … PUNCT cana-1536 69 28 … … PUNCT cana-1536 69 29 … … PUNCT cana-1536 69 30 … … PUNCT cana-1536 69 31 … … PUNCT cana-1536 69 32 … … PUNCT cana-1536 69 33 … … PUNCT cana-1536 69 34 … … PUNCT cana-1536 69 35 … … PUNCT cana-1536 69 36 …… …… NOUN cana-1536 69 37 …… …… NOUN cana-1536 69 38 …… …… NOUN cana-1536 69 39 …… …… NOUN cana-1536 69 40 …… …… NOUN cana-1536 69 41 …… …… NOUN cana-1536 69 42 …… …… NOUN cana-1536 69 43 …… …… NOUN cana-1536 69 44 …… …… NOUN cana-1536 69 45 …… …… NOUN cana-1536 69 46 …… …… NOUN cana-1536 69 47 . . PUNCT cana-1536 70 1 communications communication NOUN cana-1536 70 2 on on ADP cana-1536 70 3 applied apply VERB cana-1536 70 4 nonlinear nonlinear ADJ cana-1536 70 5 analysis analysis NOUN cana-1536 70 6 issn issn NOUN cana-1536 70 7 : : PUNCT cana-1536 70 8 1074 1074 NUM cana-1536 70 9 - - PUNCT cana-1536 70 10 133x 133x NUM cana-1536 70 11 vol vol NOUN cana-1536 70 12 31 31 NUM cana-1536 70 13 no no NOUN cana-1536 70 14 . . PUNCT cana-1536 71 1 8s 8s PROPN cana-1536 71 2 ( ( PUNCT cana-1536 71 3 2024 2024 NUM cana-1536 71 4 ) ) PUNCT cana-1536 71 5 436 436 NUM cana-1536 71 6 https://internationalpubls.com https://internationalpubls.com X cana-1536 71 7 ≀ ≀ NOUN cana-1536 72 1 [ [ X cana-1536 72 2 sₒ𝑏(ℱℐ𝑛 sₒ𝑏(ℱℐ𝑛 NOUN cana-1536 72 3 , , PUNCT cana-1536 72 4 ℱℐ𝑛+1 ℱℐ𝑛+1 NOUN cana-1536 72 5 , , PUNCT cana-1536 72 6 ℱℐ𝑛+1 ℱℐ𝑛+1 PROPN cana-1536 72 7 ) ) PUNCT cana-1536 72 8 + + CCONJ cana-1536 72 9 𝑠2ₒ𝑏(ℱℐ𝑛+1 𝑠2ₒ𝑏(ℱℐ𝑛+1 PROPN cana-1536 72 10 , , PUNCT cana-1536 72 11 ℱℐ𝑛+2 ℱℐ𝑛+2 NOUN cana-1536 72 12 , , PUNCT cana-1536 72 13 ℱℐ𝑛+2 ℱℐ𝑛+2 NOUN cana-1536 72 14 ) ) PUNCT cana-1536 72 15 + + CCONJ cana-1536 73 1 𝑠3ₒ𝑏(ℱℐ𝑛+2 𝑠3ₒ𝑏(ℱℐ𝑛+2 PROPN cana-1536 73 2 , , PUNCT cana-1536 73 3 ℱℐ𝑛+3 ℱℐ𝑛+3 NOUN cana-1536 73 4 , , PUNCT cana-1536 73 5 ℱℐ𝑛+3 ℱℐ𝑛+3 PROPN cana-1536 73 6 ) ) PUNCT cana-1536 74 1 + + CCONJ cana-1536 74 2 β‹― β‹― X cana-1536 74 3 + + NOUN cana-1536 74 4 π‘ π‘šβˆ’1ₒ𝑏(β„±β„π‘šβˆ’2 π‘ π‘šβˆ’1ₒ𝑏(β„±β„π‘šβˆ’2 NOUN cana-1536 74 5 , , PUNCT cana-1536 74 6 β„±β„π‘šβˆ’1 β„±β„π‘šβˆ’1 NUM cana-1536 74 7 , , PUNCT cana-1536 74 8 β„±β„π‘šβˆ’1 β„±β„π‘šβˆ’1 NUM cana-1536 74 9 ) ) PUNCT cana-1536 75 1 + + CCONJ cana-1536 75 2 π‘ π‘šβ‚’π‘(β„±β„π‘šβˆ’1 π‘ π‘šβ‚’π‘(β„±β„π‘šβˆ’1 PROPN cana-1536 75 3 , , PUNCT cana-1536 75 4 β„±β„π‘š β„±β„π‘š NOUN cana-1536 75 5 , , PUNCT cana-1536 75 6 β„±β„π‘š β„±β„π‘š NOUN cana-1536 75 7 ) ) PUNCT cana-1536 75 8 ] ] PUNCT cana-1536 76 1 + + CCONJ cana-1536 77 1 [ [ X cana-1536 77 2 sₒ𝑏(ℱὡ𝑛 sₒ𝑏(ℱὡ𝑛 X cana-1536 77 3 , , PUNCT cana-1536 77 4 ℱὡ𝑛+1 ℱὡ𝑛+1 PROPN cana-1536 77 5 , , PUNCT cana-1536 77 6 ℱὡ𝑛+1 ℱὡ𝑛+1 NOUN cana-1536 77 7 ) ) PUNCT cana-1536 77 8 + + CCONJ cana-1536 77 9 𝑠2ₒ𝑏(ℱὡ𝑛+1 𝑠2ₒ𝑏(ℱὡ𝑛+1 ADJ cana-1536 77 10 , , PUNCT cana-1536 77 11 ℱὡ𝑛+2 ℱὡ𝑛+2 PROPN cana-1536 77 12 , , PUNCT cana-1536 77 13 ℱὡ𝑛+2 ℱὡ𝑛+2 ADJ cana-1536 77 14 ) ) PUNCT cana-1536 77 15 + + CCONJ cana-1536 77 16 𝑠3ₒ𝑏(ℱὡ𝑛+2 𝑠3ₒ𝑏(ℱὡ𝑛+2 ADJ cana-1536 77 17 , , PUNCT cana-1536 77 18 ℱὡ𝑛+3 ℱὡ𝑛+3 NOUN cana-1536 77 19 , , PUNCT cana-1536 77 20 ℱὡ𝑛+3 ℱὡ𝑛+3 NOUN cana-1536 77 21 ) ) PUNCT cana-1536 77 22 + + CCONJ cana-1536 77 23 β‹― β‹― X cana-1536 77 24 + + NUM cana-1536 77 25 π‘ π‘šβˆ’1ₒ𝑏(β„±α½΅π‘šβˆ’2 π‘ π‘šβˆ’1ₒ𝑏(β„±α½΅π‘šβˆ’2 NOUN cana-1536 77 26 , , PUNCT cana-1536 77 27 β„±α½΅π‘šβˆ’2 β„±α½΅π‘šβˆ’2 PROPN cana-1536 77 28 , , PUNCT cana-1536 77 29 β„±α½΅π‘šβˆ’1 β„±α½΅π‘šβˆ’1 NUM cana-1536 77 30 ) ) PUNCT cana-1536 78 1 + + CCONJ cana-1536 78 2 π‘ π‘šβ‚’π‘(β„±α½΅π‘šβˆ’1 π‘ π‘šβ‚’π‘(β„±α½΅π‘šβˆ’1 NUM cana-1536 78 3 , , PUNCT cana-1536 78 4 β„±α½΅π‘š β„±α½΅π‘š PROPN cana-1536 78 5 , , PUNCT cana-1536 78 6 β„±α½΅π‘š β„±α½΅π‘š PROPN cana-1536 78 7 ) ) PUNCT cana-1536 78 8 ] ] PUNCT cana-1536 79 1 + + CCONJ cana-1536 80 1 [ [ X cana-1536 80 2 𝑠ₒ𝑏(ℱϱ𝑛 𝑠ₒ𝑏(ℱϱ𝑛 NOUN cana-1536 80 3 , , PUNCT cana-1536 80 4 ℱϱ𝑛+1 ℱϱ𝑛+1 NOUN cana-1536 80 5 , , PUNCT cana-1536 80 6 ℱϱ𝑛+1 ℱϱ𝑛+1 NOUN cana-1536 80 7 ) ) PUNCT cana-1536 80 8 + + CCONJ cana-1536 80 9 𝑠2ₒ𝑏(ℱϱ𝑛+1 𝑠2ₒ𝑏(ℱϱ𝑛+1 NOUN cana-1536 80 10 , , PUNCT cana-1536 80 11 ℱϱ𝑛+2 ℱϱ𝑛+2 PROPN cana-1536 80 12 , , PUNCT cana-1536 80 13 ℱϱ𝑛+2 ℱϱ𝑛+2 NOUN cana-1536 80 14 ) ) PUNCT cana-1536 80 15 + + X cana-1536 80 16 𝑠3ₒ𝑏(ℱϱ𝑛+2 𝑠3ₒ𝑏(ℱϱ𝑛+2 ADJ cana-1536 80 17 , , PUNCT cana-1536 80 18 ℱϱ𝑛+3 ℱϱ𝑛+3 NOUN cana-1536 80 19 , , PUNCT cana-1536 80 20 ℱϱ𝑛+3 ℱϱ𝑛+3 NOUN cana-1536 80 21 ) ) PUNCT cana-1536 81 1 + + CCONJ cana-1536 81 2 β‹― β‹― VERB cana-1536 81 3 + + CCONJ cana-1536 81 4 π‘ π‘šβˆ’1ₒ𝑏(β„±Ο±π‘šβˆ’2 π‘ π‘šβˆ’1ₒ𝑏(β„±Ο±π‘šβˆ’2 NOUN cana-1536 81 5 , , PUNCT cana-1536 81 6 β„±Ο±π‘šβˆ’1 β„±Ο±π‘šβˆ’1 PROPN cana-1536 81 7 , , PUNCT cana-1536 81 8 β„±Ο±π‘šβˆ’1 β„±Ο±π‘šβˆ’1 NUM cana-1536 81 9 ) ) PUNCT cana-1536 81 10 + + CCONJ cana-1536 81 11 π‘ π‘šβ‚’π‘(β„±Ο±π‘šβˆ’1 π‘ π‘šβ‚’π‘(β„±Ο±π‘šβˆ’1 X cana-1536 81 12 , , PUNCT cana-1536 81 13 β„±Ο±π‘š β„±Ο±π‘š PROPN cana-1536 81 14 , , PUNCT cana-1536 81 15 β„±Ο±π‘š β„±Ο±π‘š PROPN cana-1536 81 16 ) ) PUNCT cana-1536 81 17 ] ] PUNCT cana-1536 81 18 ≀ ≀ NUM cana-1536 81 19 𝑠[ₒ𝑏(ℱℐ𝑛 𝑠[ₒ𝑏(ℱℐ𝑛 NOUN cana-1536 81 20 , , PUNCT cana-1536 81 21 ℱℐ𝑛+1 ℱℐ𝑛+1 NOUN cana-1536 81 22 , , PUNCT cana-1536 81 23 ℱℐ𝑛+1 ℱℐ𝑛+1 PROPN cana-1536 81 24 ) ) PUNCT cana-1536 82 1 + + CCONJ cana-1536 82 2 β‚’ β‚’ ADP cana-1536 82 3 𝑏(ℱὡ𝑛 𝑏(ℱὡ𝑛 PROPN cana-1536 82 4 , , PUNCT cana-1536 82 5 ℱὡ𝑛+1 ℱὡ𝑛+1 NOUN cana-1536 82 6 , , PUNCT cana-1536 82 7 ℱὡ𝑛+1 ℱὡ𝑛+1 NOUN cana-1536 82 8 ) ) PUNCT cana-1536 82 9 + + CCONJ cana-1536 82 10 β‚’ β‚’ ADP cana-1536 82 11 𝑏(ℱϱ𝑛 𝑏(ℱϱ𝑛 PROPN cana-1536 82 12 , , PUNCT cana-1536 82 13 ℱϱ𝑛+1 ℱϱ𝑛+1 NOUN cana-1536 82 14 , , PUNCT cana-1536 82 15 ℱϱ𝑛+1 ℱϱ𝑛+1 NOUN cana-1536 82 16 ) ) PUNCT cana-1536 82 17 ] ] PUNCT cana-1536 83 1 + + CCONJ cana-1536 83 2 𝑠2[ₒ𝑏(ℱℐ𝑛+1 𝑠2[ₒ𝑏(ℱℐ𝑛+1 NUM cana-1536 83 3 , , PUNCT cana-1536 83 4 ℱℐ𝑛+2 ℱℐ𝑛+2 NOUN cana-1536 83 5 , , PUNCT cana-1536 83 6 ℱℐ𝑛+2 ℱℐ𝑛+2 NOUN cana-1536 83 7 ) ) PUNCT cana-1536 83 8 + + CCONJ cana-1536 83 9 β‚’ β‚’ ADP cana-1536 83 10 𝑏(ℱὡ𝑛+1 𝑏(ℱὡ𝑛+1 NOUN cana-1536 83 11 , , PUNCT cana-1536 83 12 ℱὡ𝑛+2 ℱὡ𝑛+2 PROPN cana-1536 83 13 , , PUNCT cana-1536 83 14 ℱὡ𝑛+2 ℱὡ𝑛+2 NOUN cana-1536 83 15 ) ) PUNCT cana-1536 83 16 + + CCONJ cana-1536 83 17 β‚’ β‚’ ADP cana-1536 83 18 𝑏(ℱϱ𝑛+1 𝑏(ℱϱ𝑛+1 PROPN cana-1536 83 19 , , PUNCT cana-1536 83 20 ℱϱ𝑛+2 ℱϱ𝑛+2 PROPN cana-1536 83 21 , , PUNCT cana-1536 83 22 ℱϱ𝑛+2 ℱϱ𝑛+2 NOUN cana-1536 83 23 ) ) PUNCT cana-1536 83 24 ] ] PUNCT cana-1536 84 1 + + CCONJ cana-1536 84 2 β‹― β‹― VERB cana-1536 84 3 + + CCONJ cana-1536 84 4 π‘ π‘šβˆ’1[ₒ𝑏(β„±β„π‘šβˆ’2 π‘ π‘šβˆ’1[ₒ𝑏(β„±β„π‘šβˆ’2 NOUN cana-1536 84 5 , , PUNCT cana-1536 84 6 β„±β„π‘šβˆ’1 β„±β„π‘šβˆ’1 NUM cana-1536 84 7 , , PUNCT cana-1536 84 8 β„±β„π‘šβˆ’1 β„±β„π‘šβˆ’1 NUM cana-1536 84 9 ) ) PUNCT cana-1536 84 10 + + CCONJ cana-1536 84 11 β‚’ β‚’ ADP cana-1536 84 12 𝑏(β„±α½΅π‘šβˆ’2 𝑏(β„±α½΅π‘šβˆ’2 NOUN cana-1536 84 13 , , PUNCT cana-1536 84 14 β„±α½΅π‘šβˆ’2 β„±α½΅π‘šβˆ’2 PROPN cana-1536 84 15 , , PUNCT cana-1536 84 16 β„±α½΅π‘šβˆ’1 β„±α½΅π‘šβˆ’1 NUM cana-1536 84 17 ) ) PUNCT cana-1536 85 1 + + CCONJ cana-1536 85 2 β‚’ β‚’ ADP cana-1536 85 3 𝑏(β„±Ο±π‘šβˆ’2 𝑏(β„±Ο±π‘šβˆ’2 NUM cana-1536 85 4 , , PUNCT cana-1536 85 5 β„±Ο±π‘šβˆ’1 β„±Ο±π‘šβˆ’1 PROPN cana-1536 85 6 , , PUNCT cana-1536 85 7 β„±Ο±π‘šβˆ’1 β„±Ο±π‘šβˆ’1 NUM cana-1536 85 8 ) ) PUNCT cana-1536 85 9 ] ] PUNCT cana-1536 86 1 + + CCONJ cana-1536 86 2 π‘ π‘š[ₒ𝑏(β„±β„π‘šβˆ’1 π‘ π‘š[ₒ𝑏(β„±β„π‘šβˆ’1 PROPN cana-1536 86 3 , , PUNCT cana-1536 86 4 β„±β„π‘š β„±β„π‘š NOUN cana-1536 86 5 , , PUNCT cana-1536 86 6 β„±β„π‘š β„±β„π‘š NOUN cana-1536 86 7 ) ) PUNCT cana-1536 87 1 + + CCONJ cana-1536 87 2 β‚’ β‚’ ADP cana-1536 87 3 𝑏(β„±α½΅π‘šβˆ’1 𝑏(β„±α½΅π‘šβˆ’1 PROPN cana-1536 87 4 , , PUNCT cana-1536 87 5 β„±α½΅π‘š β„±α½΅π‘š PROPN cana-1536 87 6 , , PUNCT cana-1536 87 7 β„±α½΅π‘š β„±α½΅π‘š PROPN cana-1536 87 8 ) ) PUNCT cana-1536 87 9 + + NUM cana-1536 87 10 β‚’ β‚’ ADP cana-1536 87 11 𝑏(β„±Ο±π‘šβˆ’1 𝑏(β„±Ο±π‘šβˆ’1 PROPN cana-1536 87 12 , , PUNCT cana-1536 87 13 β„±Ο±π‘š β„±Ο±π‘š PROPN cana-1536 87 14 , , PUNCT cana-1536 87 15 β„±Ο±π‘š β„±Ο±π‘š PROPN cana-1536 87 16 ) ) PUNCT cana-1536 87 17 ] ] PUNCT cana-1536 87 18 ≀ ≀ NUM cana-1536 87 19 𝑠ℑ𝑛 𝑠ℑ𝑛 ADJ cana-1536 87 20 + + CCONJ cana-1536 87 21 𝑠2ℑ𝑛+1 𝑠2ℑ𝑛+1 NUM cana-1536 87 22 + + SYM cana-1536 87 23 β‹― β‹― PROPN cana-1536 87 24 … … PUNCT cana-1536 87 25 … … PUNCT cana-1536 87 26 … … PUNCT cana-1536 87 27 … … PUNCT cana-1536 87 28 … … PUNCT cana-1536 87 29 … … PUNCT cana-1536 87 30 + + NUM cana-1536 87 31 π‘ π‘šβˆ’1β„‘π‘šβˆ’2 π‘ π‘šβˆ’1β„‘π‘šβˆ’2 NOUN cana-1536 87 32 + + CCONJ cana-1536 87 33 π‘ π‘šβ„‘π‘šβˆ’1 π‘ π‘šβ„‘π‘šβˆ’1 X cana-1536 87 34 ≀ ≀ NUM cana-1536 87 35 π‘ πœƒπ‘›β„‘0 π‘ πœƒπ‘›β„‘0 NUM cana-1536 87 36 + + NUM cana-1536 87 37 𝑠2πœƒπ‘›+1β„‘0 𝑠2πœƒπ‘›+1β„‘0 PROPN cana-1536 87 38 + + CCONJ cana-1536 87 39 β‹― β‹― PROPN cana-1536 87 40 … … PUNCT cana-1536 87 41 … … PUNCT cana-1536 87 42 … … PUNCT cana-1536 87 43 … … PUNCT cana-1536 87 44 … … PUNCT cana-1536 87 45 … … PUNCT cana-1536 87 46 + + CCONJ cana-1536 87 47 π‘ π‘šβˆ’1πœƒπ‘šβˆ’2β„‘0 π‘ π‘šβˆ’1πœƒπ‘šβˆ’2β„‘0 PROPN cana-1536 87 48 + + PROPN cana-1536 87 49 π‘ π‘šπœƒπ‘šβˆ’1β„‘0 π‘ π‘šπœƒπ‘šβˆ’1β„‘0 PROPN cana-1536 87 50 ≀ ≀ NOUN cana-1536 88 1 [ [ PRON cana-1536 88 2 π‘ πœƒπ‘› π‘ πœƒπ‘› NOUN cana-1536 88 3 + + X cana-1536 88 4 𝑠2πœƒπ‘›+1 𝑠2πœƒπ‘›+1 NOUN cana-1536 88 5 + + CCONJ cana-1536 88 6 β‹― β‹― PROPN cana-1536 88 7 … … PUNCT cana-1536 88 8 … … PUNCT cana-1536 88 9 … … PUNCT cana-1536 88 10 … … PUNCT cana-1536 88 11 … … PUNCT cana-1536 88 12 … … PUNCT cana-1536 88 13 + + NUM cana-1536 88 14 π‘ π‘šβˆ’1πœƒπ‘šβˆ’2 π‘ π‘šβˆ’1πœƒπ‘šβˆ’2 NUM cana-1536 88 15 + + NUM cana-1536 88 16 π‘ π‘šπœƒπ‘šβˆ’1]β„‘0 π‘ π‘šπœƒπ‘šβˆ’1]β„‘0 SYM cana-1536 88 17 ≀ ≀ NOUN cana-1536 89 1 [ [ X cana-1536 89 2 π‘ πœƒπ‘› π‘ πœƒπ‘› NOUN cana-1536 89 3 + + X cana-1536 89 4 𝑠2πœƒπ‘›+1 𝑠2πœƒπ‘›+1 NOUN cana-1536 89 5 + + CCONJ cana-1536 89 6 β‹― β‹― PROPN cana-1536 89 7 … … PUNCT cana-1536 89 8 … … PUNCT cana-1536 89 9 … … PUNCT cana-1536 89 10 … … PUNCT cana-1536 89 11 … … PUNCT cana-1536 89 12 … … PUNCT cana-1536 89 13 ] ] X cana-1536 89 14 β„‘0 β„‘0 X cana-1536 89 15 ≀ ≀ NOUN cana-1536 89 16 π‘ πœƒπ‘›[1 π‘ πœƒπ‘›[1 NUM cana-1536 89 17 + + CCONJ cana-1536 89 18 π‘ πœƒ π‘ πœƒ NOUN cana-1536 89 19 + + CCONJ cana-1536 89 20 π‘ πœƒ2 π‘ πœƒ2 NOUN cana-1536 89 21 + + SYM cana-1536 89 22 β‹― β‹― PROPN cana-1536 89 23 … … PUNCT cana-1536 89 24 … … PUNCT cana-1536 89 25 … … PUNCT cana-1536 89 26 … … PUNCT cana-1536 89 27 … … PUNCT cana-1536 89 28 ] ] X cana-1536 89 29 β„‘0 β„‘0 ADJ cana-1536 89 30 ≀ ≀ ADJ cana-1536 89 31 π‘ πœƒπ‘› π‘ πœƒπ‘› NOUN cana-1536 89 32 1βˆ’π‘ πœƒ 1βˆ’π‘ πœƒ NOUN cana-1536 89 33 β„‘0 β„‘0 PROPN cana-1536 89 34 β†’ β†’ SYM cana-1536 89 35 0 0 NUM cana-1536 89 36 as as ADP cana-1536 89 37 𝑛 𝑛 PROPN cana-1536 89 38 β†’ β†’ SYM cana-1536 89 39 ∞. ∞. PROPN cana-1536 89 40 thereforeβ‚’(ℱℐ𝑛 thereforeβ‚’(ℱℐ𝑛 PROPN cana-1536 89 41 , , PUNCT cana-1536 89 42 β„±β„π‘š β„±β„π‘š NOUN cana-1536 89 43 , , PUNCT cana-1536 89 44 β„±β„π‘š β„±β„π‘š NOUN cana-1536 89 45 ) ) PUNCT cana-1536 89 46 + + CCONJ cana-1536 89 47 β‚’ β‚’ ADP cana-1536 89 48 ( ( PUNCT cana-1536 89 49 ℱὡ𝑛 ℱὡ𝑛 PROPN cana-1536 89 50 , , PUNCT cana-1536 89 51 β„±α½΅π‘š β„±α½΅π‘š PROPN cana-1536 89 52 , , PUNCT cana-1536 89 53 β„±α½΅π‘š β„±α½΅π‘š PROPN cana-1536 89 54 ) ) PUNCT cana-1536 89 55 + + NUM cana-1536 89 56 β‚’ β‚’ ADP cana-1536 89 57 ( ( PUNCT cana-1536 89 58 ℱϱ𝑛 ℱϱ𝑛 PROPN cana-1536 89 59 , , PUNCT cana-1536 89 60 β„±Ο±π‘š β„±Ο±π‘š PROPN cana-1536 89 61 , , PUNCT cana-1536 89 62 β„±Ο±π‘š β„±Ο±π‘š PROPN cana-1536 89 63 ) ) PUNCT cana-1536 89 64 β†’ β†’ SYM cana-1536 89 65 0 0 NUM cana-1536 89 66 as as ADP cana-1536 89 67 𝑛 𝑛 PROPN cana-1536 89 68 , , PUNCT cana-1536 89 69 π‘š π‘š X cana-1536 89 70 β†’ β†’ SYM cana-1536 89 71 ∞. ∞. PROPN cana-1536 89 72 so so ADV cana-1536 89 73 , , PUNCT cana-1536 89 74 we we PRON cana-1536 89 75 can can AUX cana-1536 89 76 conclude conclude VERB cana-1536 89 77 that that SCONJ cana-1536 89 78 the the DET cana-1536 89 79 sequences sequence NOUN cana-1536 89 80 { { PUNCT cana-1536 89 81 ℱℐ𝑛 ℱℐ𝑛 NOUN cana-1536 89 82 } } PUNCT cana-1536 89 83 , , PUNCT cana-1536 89 84 { { PUNCT cana-1536 89 85 ℱὡ𝑛 ℱὡ𝑛 NOUN cana-1536 89 86 } } PUNCT cana-1536 89 87 , , PUNCT cana-1536 89 88 { { PUNCT cana-1536 89 89 ℱϱ𝑛 ℱϱ𝑛 NOUN cana-1536 89 90 } } PUNCT cana-1536 89 91 are be AUX cana-1536 89 92 cs cs PROPN cana-1536 89 93 in in ADP cana-1536 89 94 α½£. α½£. PROPN cana-1536 89 95 but but CCONJ cana-1536 89 96 , , PUNCT cana-1536 89 97 β„±(α½£ β„±(α½£ NUM cana-1536 89 98 ) ) PUNCT cana-1536 89 99 is be AUX cana-1536 89 100 β‚’ β‚’ ADP cana-1536 89 101 𝑏-complete 𝑏-complete PROPN cana-1536 89 102 , , PUNCT cana-1536 89 103 there there PRON cana-1536 89 104 exists exist VERB cana-1536 89 105 𝜌 𝜌 ADP cana-1536 89 106 , , PUNCT cana-1536 89 107 𝜎 𝜎 NOUN cana-1536 89 108 , , PUNCT cana-1536 89 109 𝔑 𝔑 X cana-1536 89 110 ∈ ∈ PROPN cana-1536 90 1 β„±(α½£ β„±(α½£ NUM cana-1536 90 2 ) ) PUNCT cana-1536 91 1 such such ADJ cana-1536 91 2 that that SCONJ cana-1536 91 3 lim lim PROPN cana-1536 91 4 π‘›β†’βˆž π‘›β†’βˆž NUM cana-1536 91 5 ℱℐ𝑛 ℱℐ𝑛 NOUN cana-1536 91 6 β†’ β†’ SYM cana-1536 91 7 𝜌 𝜌 X cana-1536 91 8 , , PUNCT cana-1536 91 9 lim lim NOUN cana-1536 91 10 π‘›β†’βˆž π‘›β†’βˆž NUM cana-1536 91 11 ℱὡ𝑛 ℱὡ𝑛 PROPN cana-1536 91 12 β†’ β†’ SYM cana-1536 91 13 𝜎 𝜎 PROPN cana-1536 91 14 and and CCONJ cana-1536 91 15 lim lim PROPN cana-1536 91 16 π‘›β†’βˆž π‘›β†’βˆž NUM cana-1536 91 17 ℱϱ𝑛 ℱϱ𝑛 PROPN cana-1536 91 18 β†’ β†’ PUNCT cana-1536 91 19 𝔑 𝔑 VERB cana-1536 91 20 𝑖. 𝑖. ADJ cana-1536 91 21 𝑒. 𝑒. NOUN cana-1536 91 22 , , PUNCT cana-1536 91 23 lim lim PROPN cana-1536 91 24 π‘›β†’βˆž π‘›β†’βˆž NUM cana-1536 91 25 β‚’ β‚’ ADP cana-1536 91 26 𝑏(ℱℐ𝑛 𝑏(ℱℐ𝑛 PROPN cana-1536 91 27 , , PUNCT cana-1536 91 28 𝜌 𝜌 X cana-1536 91 29 , , PUNCT cana-1536 91 30 𝜌 𝜌 ADP cana-1536 91 31 ) ) PUNCT cana-1536 91 32 = = SYM cana-1536 91 33 0 0 NUM cana-1536 91 34 , , PUNCT cana-1536 91 35 lim lim NOUN cana-1536 91 36 π‘›β†’βˆž π‘›β†’βˆž NUM cana-1536 91 37 β‚’ β‚’ ADP cana-1536 91 38 𝑏(ℱὡ𝑛 𝑏(ℱὡ𝑛 PROPN cana-1536 91 39 , , PUNCT cana-1536 91 40 𝜎 𝜎 PROPN cana-1536 91 41 , , PUNCT cana-1536 91 42 𝜎 𝜎 PROPN cana-1536 91 43 ) ) PUNCT cana-1536 91 44 = = SYM cana-1536 91 45 0 0 NUM cana-1536 91 46 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ PROPN cana-1536 91 47 lim lim PROPN cana-1536 91 48 π‘›β†’βˆž π‘›β†’βˆž NUM cana-1536 91 49 β‚’ β‚’ ADP cana-1536 91 50 𝑏(ℱϱ𝑛 𝑏(ℱϱ𝑛 PROPN cana-1536 91 51 , , PUNCT cana-1536 91 52 𝔑 𝔑 ADV cana-1536 91 53 , , PUNCT cana-1536 91 54 𝔑 𝔑 ADV cana-1536 91 55 ) ) PUNCT cana-1536 91 56 = = SYM cana-1536 91 57 0 0 X cana-1536 91 58 . . PUNCT cana-1536 92 1 βˆ’ βˆ’ NOUN cana-1536 93 1 βˆ’ βˆ’ PROPN cana-1536 93 2 ( ( PUNCT cana-1536 93 3 3.4.3 3.4.3 NUM cana-1536 93 4 ) ) PUNCT cana-1536 93 5 since since SCONJ cana-1536 93 6 β„± β„± PROPN cana-1536 93 7 is be AUX cana-1536 93 8 continuous continuous ADJ cana-1536 93 9 , , PUNCT cana-1536 93 10 lim lim PROPN cana-1536 93 11 π‘›β†’βˆž π‘›β†’βˆž NUM cana-1536 93 12 β‚’ β‚’ ADP cana-1536 93 13 𝑏(ℱℱℐ𝑛 𝑏(ℱℱℐ𝑛 PROPN cana-1536 93 14 , , PUNCT cana-1536 93 15 β„±πœŒ β„±πœŒ PROPN cana-1536 93 16 , , PUNCT cana-1536 93 17 β„±πœŒ β„±πœŒ PROPN cana-1536 93 18 ) ) PUNCT cana-1536 93 19 = = SYM cana-1536 93 20 0 0 PROPN cana-1536 93 21 , , PUNCT cana-1536 93 22 lim lim NOUN cana-1536 93 23 π‘›β†’βˆž π‘›β†’βˆž NUM cana-1536 93 24 β‚’ β‚’ ADP cana-1536 93 25 𝑏(ℱℱὡ𝑛 𝑏(ℱℱὡ𝑛 NOUN cana-1536 93 26 , , PUNCT cana-1536 93 27 β„±πœŽ β„±πœŽ NOUN cana-1536 93 28 , , PUNCT cana-1536 93 29 β„±πœŽ β„±πœŽ NOUN cana-1536 93 30 ) ) PUNCT cana-1536 93 31 = = SYM cana-1536 93 32 0 0 NUM cana-1536 93 33 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ PROPN cana-1536 93 34 lim lim PROPN cana-1536 93 35 π‘›β†’βˆž π‘›β†’βˆž NUM cana-1536 93 36 β‚’ β‚’ ADP cana-1536 93 37 𝑏(ℱℱϱ𝑛 𝑏(ℱℱϱ𝑛 NOUN cana-1536 93 38 , , PUNCT cana-1536 93 39 ℱ𝔑 ℱ𝔑 PROPN cana-1536 93 40 , , PUNCT cana-1536 93 41 ℱ𝔑 ℱ𝔑 NOUN cana-1536 93 42 ) ) PUNCT cana-1536 93 43 = = SYM cana-1536 93 44 0 0 X cana-1536 93 45 . . PUNCT cana-1536 94 1 ---(3.4.4 ---(3.4.4 NUM cana-1536 94 2 ) ) PUNCT cana-1536 94 3 since since SCONJ cana-1536 94 4 β„± β„± PROPN cana-1536 94 5 commutes commute VERB cana-1536 94 6 with with ADP cana-1536 94 7 β„‹ β„‹ PROPN cana-1536 94 8 and and CCONJ cana-1536 94 9 by by ADP cana-1536 94 10 equation equation NOUN cana-1536 94 11 ( ( PUNCT cana-1536 94 12 3.4.1 3.4.1 NUM cana-1536 94 13 ) ) PUNCT cana-1536 94 14 β‚’ β‚’ ADP cana-1536 94 15 𝑏(ℱℱℐ𝑛+1 𝑏(ℱℱℐ𝑛+1 PROPN cana-1536 94 16 , , PUNCT cana-1536 94 17 β„‹(𝜌 β„‹(𝜌 NUM cana-1536 94 18 , , PUNCT cana-1536 94 19 𝜎 𝜎 NOUN cana-1536 94 20 , , PUNCT cana-1536 94 21 𝔑 𝔑 CCONJ cana-1536 94 22 ) ) PUNCT cana-1536 94 23 , , PUNCT cana-1536 94 24 β„‹(𝜌 β„‹(𝜌 NUM cana-1536 94 25 , , PUNCT cana-1536 94 26 𝜎 𝜎 NOUN cana-1536 94 27 , , PUNCT cana-1536 94 28 𝔑 𝔑 NOUN cana-1536 94 29 ) ) PUNCT cana-1536 94 30 ) ) PUNCT cana-1536 95 1 + + CCONJ cana-1536 95 2 β‚’ β‚’ ADP cana-1536 95 3 𝑏(ℱℱὡ𝑛 𝑏(ℱℱὡ𝑛 NOUN cana-1536 95 4 , , PUNCT cana-1536 95 5 β„‹(𝜎 β„‹(𝜎 PROPN cana-1536 95 6 , , PUNCT cana-1536 95 7 𝔑 𝔑 PROPN cana-1536 95 8 , , PUNCT cana-1536 95 9 𝜌 𝜌 ADP cana-1536 95 10 ) ) PUNCT cana-1536 95 11 , , PUNCT cana-1536 95 12 β„‹(𝜎 β„‹(𝜎 PROPN cana-1536 95 13 , , PUNCT cana-1536 95 14 𝔑 𝔑 PROPN cana-1536 95 15 , , PUNCT cana-1536 95 16 𝜌 𝜌 X cana-1536 95 17 ) ) PUNCT cana-1536 95 18 ) ) PUNCT cana-1536 96 1 + + CCONJ cana-1536 96 2 β‚’ β‚’ ADP cana-1536 96 3 𝑏(ℱℱϱ𝑛 𝑏(ℱℱϱ𝑛 NOUN cana-1536 96 4 , , PUNCT cana-1536 96 5 β„‹(𝔑 β„‹(𝔑 PROPN cana-1536 96 6 , , PUNCT cana-1536 96 7 𝜌 𝜌 X cana-1536 96 8 , , PUNCT cana-1536 96 9 𝜎 𝜎 NOUN cana-1536 96 10 ) ) PUNCT cana-1536 96 11 , , PUNCT cana-1536 96 12 β„‹(𝔑 β„‹(𝔑 PROPN cana-1536 96 13 , , PUNCT cana-1536 96 14 𝜌 𝜌 X cana-1536 96 15 , , PUNCT cana-1536 96 16 𝜎 𝜎 NOUN cana-1536 96 17 ) ) PUNCT cana-1536 96 18 ) ) PUNCT cana-1536 97 1 = = PUNCT cana-1536 97 2 β‚’ β‚’ ADP cana-1536 97 3 𝑏 𝑏 X cana-1536 97 4 ( ( PUNCT cana-1536 97 5 β„±(β„‹(ℐ𝑛 β„±(β„‹(ℐ𝑛 PROPN cana-1536 97 6 , , PUNCT cana-1536 97 7 ὡ𝑛 ὡ𝑛 NOUN cana-1536 97 8 , , PUNCT cana-1536 97 9 ϱ𝑛 ϱ𝑛 PROPN cana-1536 97 10 ) ) PUNCT cana-1536 97 11 ) ) PUNCT cana-1536 97 12 , , PUNCT cana-1536 97 13 β„‹(𝜌 β„‹(𝜌 NUM cana-1536 97 14 , , PUNCT cana-1536 97 15 𝜎 𝜎 NOUN cana-1536 97 16 , , PUNCT cana-1536 97 17 𝔑 𝔑 CCONJ cana-1536 97 18 ) ) PUNCT cana-1536 97 19 , , PUNCT cana-1536 97 20 β„‹(𝜌 β„‹(𝜌 NUM cana-1536 97 21 , , PUNCT cana-1536 97 22 𝜎 𝜎 NOUN cana-1536 97 23 , , PUNCT cana-1536 97 24 𝔑 𝔑 NOUN cana-1536 97 25 ) ) PUNCT cana-1536 97 26 ) ) PUNCT cana-1536 98 1 + + CCONJ cana-1536 98 2 β‚’ β‚’ ADP cana-1536 98 3 𝑏 𝑏 PRON cana-1536 98 4 ( ( PUNCT cana-1536 98 5 β„±(β„‹(ὡ𝑛 β„±(β„‹(ὡ𝑛 PROPN cana-1536 98 6 , , PUNCT cana-1536 98 7 ϱ𝑛 ϱ𝑛 PROPN cana-1536 98 8 , , PUNCT cana-1536 98 9 ℐ𝑛 ℐ𝑛 PROPN cana-1536 98 10 ) ) PUNCT cana-1536 98 11 ) ) PUNCT cana-1536 98 12 , , PUNCT cana-1536 98 13 β„‹(𝜎 β„‹(𝜎 PROPN cana-1536 98 14 , , PUNCT cana-1536 98 15 𝔑 𝔑 PROPN cana-1536 98 16 , , PUNCT cana-1536 98 17 𝜌 𝜌 ADP cana-1536 98 18 ) ) PUNCT cana-1536 98 19 , , PUNCT cana-1536 98 20 β„‹(𝜎 β„‹(𝜎 PROPN cana-1536 98 21 , , PUNCT cana-1536 98 22 𝔑 𝔑 PROPN cana-1536 98 23 , , PUNCT cana-1536 98 24 𝜌 𝜌 X cana-1536 98 25 ) ) PUNCT cana-1536 98 26 ) ) PUNCT cana-1536 98 27 + + CCONJ cana-1536 98 28 β‚’ β‚’ ADP cana-1536 98 29 𝑏 𝑏 PRON cana-1536 98 30 ( ( PUNCT cana-1536 98 31 β„±(β„‹(ϱ𝑛 β„±(β„‹(ϱ𝑛 NOUN cana-1536 98 32 , , PUNCT cana-1536 98 33 ℐ𝑛 ℐ𝑛 PROPN cana-1536 98 34 , , PUNCT cana-1536 98 35 ὡ𝑛 ὡ𝑛 NOUN cana-1536 98 36 ) ) PUNCT cana-1536 98 37 ) ) PUNCT cana-1536 98 38 , , PUNCT cana-1536 98 39 β„‹(𝔑 β„‹(𝔑 PROPN cana-1536 98 40 , , PUNCT cana-1536 98 41 𝜌 𝜌 X cana-1536 98 42 , , PUNCT cana-1536 98 43 𝜎 𝜎 NOUN cana-1536 98 44 ) ) PUNCT cana-1536 98 45 , , PUNCT cana-1536 98 46 β„‹(𝔑 β„‹(𝔑 PROPN cana-1536 98 47 , , PUNCT cana-1536 98 48 𝜌 𝜌 X cana-1536 98 49 , , PUNCT cana-1536 98 50 𝜎 𝜎 NOUN cana-1536 98 51 ) ) PUNCT cana-1536 98 52 ) ) PUNCT cana-1536 98 53 communications communication NOUN cana-1536 98 54 on on ADP cana-1536 98 55 applied apply VERB cana-1536 98 56 nonlinear nonlinear ADJ cana-1536 98 57 analysis analysis NOUN cana-1536 98 58 issn issn NOUN cana-1536 98 59 : : PUNCT cana-1536 98 60 1074 1074 NUM cana-1536 98 61 - - PUNCT cana-1536 98 62 133x 133x NUM cana-1536 98 63 vol vol NOUN cana-1536 98 64 31 31 NUM cana-1536 98 65 no no NOUN cana-1536 98 66 . . PUNCT cana-1536 99 1 8s 8s PROPN cana-1536 99 2 ( ( PUNCT cana-1536 99 3 2024 2024 NUM cana-1536 99 4 ) ) PUNCT cana-1536 99 5 437 437 NUM cana-1536 99 6 https://internationalpubls.com https://internationalpubls.com X cana-1536 99 7 = = PUNCT cana-1536 99 8 β‚’ β‚’ ADP cana-1536 99 9 𝑏 𝑏 PROPN cana-1536 99 10 ( ( PUNCT cana-1536 99 11 β„‹((ℱℐ𝑛 β„‹((ℱℐ𝑛 NUM cana-1536 99 12 , , PUNCT cana-1536 99 13 ℱὡ𝑛 ℱὡ𝑛 PROPN cana-1536 99 14 , , PUNCT cana-1536 99 15 ℱϱ𝑛 ℱϱ𝑛 PROPN cana-1536 99 16 ) ) PUNCT cana-1536 99 17 ) ) PUNCT cana-1536 99 18 , , PUNCT cana-1536 99 19 β„‹(𝜌 β„‹(𝜌 NUM cana-1536 99 20 , , PUNCT cana-1536 99 21 𝜎 𝜎 NOUN cana-1536 99 22 , , PUNCT cana-1536 99 23 𝔑 𝔑 CCONJ cana-1536 99 24 ) ) PUNCT cana-1536 99 25 , , PUNCT cana-1536 99 26 β„‹(𝜌 β„‹(𝜌 NUM cana-1536 99 27 , , PUNCT cana-1536 99 28 𝜎 𝜎 NOUN cana-1536 99 29 , , PUNCT cana-1536 99 30 𝔑 𝔑 NOUN cana-1536 99 31 ) ) PUNCT cana-1536 99 32 ) ) PUNCT cana-1536 100 1 + + CCONJ cana-1536 100 2 β‚’ β‚’ ADP cana-1536 100 3 𝑏 𝑏 PRON cana-1536 100 4 ( ( PUNCT cana-1536 100 5 β„‹((ℱὡ𝑛 β„‹((ℱὡ𝑛 NOUN cana-1536 100 6 , , PUNCT cana-1536 100 7 ℱϱ𝑛 ℱϱ𝑛 PROPN cana-1536 100 8 , , PUNCT cana-1536 100 9 ℱℐ𝑛 ℱℐ𝑛 NOUN cana-1536 100 10 ) ) PUNCT cana-1536 100 11 ) ) PUNCT cana-1536 100 12 , , PUNCT cana-1536 100 13 β„‹(𝜎 β„‹(𝜎 PROPN cana-1536 100 14 , , PUNCT cana-1536 100 15 𝔑 𝔑 PROPN cana-1536 100 16 , , PUNCT cana-1536 100 17 𝜌 𝜌 ADP cana-1536 100 18 ) ) PUNCT cana-1536 100 19 , , PUNCT cana-1536 100 20 β„‹(𝜎 β„‹(𝜎 PROPN cana-1536 100 21 , , PUNCT cana-1536 100 22 𝔑 𝔑 PROPN cana-1536 100 23 , , PUNCT cana-1536 100 24 𝜌 𝜌 X cana-1536 100 25 ) ) PUNCT cana-1536 100 26 ) ) PUNCT cana-1536 101 1 + + CCONJ cana-1536 101 2 β‚’ β‚’ ADP cana-1536 101 3 𝑏 𝑏 PRON cana-1536 101 4 ( ( PUNCT cana-1536 101 5 β„‹((ℱϱ𝑛 β„‹((ℱϱ𝑛 NOUN cana-1536 101 6 , , PUNCT cana-1536 101 7 ℱℐ𝑛 ℱℐ𝑛 NOUN cana-1536 101 8 , , PUNCT cana-1536 101 9 ℱὡ𝑛 ℱὡ𝑛 NOUN cana-1536 101 10 ) ) PUNCT cana-1536 101 11 ) ) PUNCT cana-1536 101 12 , , PUNCT cana-1536 101 13 β„‹(𝔑 β„‹(𝔑 PROPN cana-1536 101 14 , , PUNCT cana-1536 101 15 𝜌 𝜌 X cana-1536 101 16 , , PUNCT cana-1536 101 17 𝜎 𝜎 NOUN cana-1536 101 18 ) ) PUNCT cana-1536 101 19 , , PUNCT cana-1536 101 20 β„‹(𝔑 β„‹(𝔑 PROPN cana-1536 101 21 , , PUNCT cana-1536 101 22 𝜌 𝜌 X cana-1536 101 23 , , PUNCT cana-1536 101 24 𝜎 𝜎 NOUN cana-1536 101 25 ) ) PUNCT cana-1536 101 26 ) ) PUNCT cana-1536 101 27 ≀ ≀ NOUN cana-1536 101 28 πœƒ[ₒ𝑏(ℱℱℐ𝑛 πœƒ[ₒ𝑏(ℱℱℐ𝑛 PUNCT cana-1536 101 29 , , PUNCT cana-1536 101 30 β„±πœŒ β„±πœŒ PROPN cana-1536 101 31 , , PUNCT cana-1536 101 32 β„±πœŒ β„±πœŒ PROPN cana-1536 101 33 ) ) PUNCT cana-1536 101 34 + + NUM cana-1536 101 35 β‚’ β‚’ ADP cana-1536 101 36 𝑏(ℱℱὡ𝑛 𝑏(ℱℱὡ𝑛 NOUN cana-1536 101 37 , , PUNCT cana-1536 101 38 β„±πœŽ β„±πœŽ NOUN cana-1536 101 39 , , PUNCT cana-1536 101 40 β„±πœŽ β„±πœŽ NOUN cana-1536 101 41 ) ) PUNCT cana-1536 101 42 + + CCONJ cana-1536 101 43 β‚’ β‚’ ADP cana-1536 101 44 𝑏(ℱℱϱ𝑛 𝑏(ℱℱϱ𝑛 NOUN cana-1536 101 45 , , PUNCT cana-1536 101 46 ℱ𝔑 ℱ𝔑 PROPN cana-1536 101 47 , , PUNCT cana-1536 101 48 ℱ𝔑 ℱ𝔑 PROPN cana-1536 101 49 ) ) PUNCT cana-1536 101 50 ] ] PUNCT cana-1536 101 51 . . PUNCT cana-1536 102 1 from from ADP cana-1536 102 2 ( ( PUNCT cana-1536 102 3 3.4.4)we 3.4.4)we NUM cana-1536 102 4 get get VERB cana-1536 102 5 β‚’ β‚’ ADP cana-1536 102 6 𝑏(ℱℱℐ𝑛+1 𝑏(ℱℱℐ𝑛+1 NOUN cana-1536 102 7 , , PUNCT cana-1536 102 8 β„‹(𝜌 β„‹(𝜌 NUM cana-1536 102 9 , , PUNCT cana-1536 102 10 𝜎 𝜎 NOUN cana-1536 102 11 , , PUNCT cana-1536 102 12 𝔑 𝔑 CCONJ cana-1536 102 13 ) ) PUNCT cana-1536 102 14 , , PUNCT cana-1536 102 15 β„‹(𝜌 β„‹(𝜌 NUM cana-1536 102 16 , , PUNCT cana-1536 102 17 𝜎 𝜎 NOUN cana-1536 102 18 , , PUNCT cana-1536 102 19 𝔑 𝔑 NOUN cana-1536 102 20 ) ) PUNCT cana-1536 102 21 ) ) PUNCT cana-1536 103 1 + + CCONJ cana-1536 103 2 β‚’ β‚’ ADP cana-1536 103 3 𝑏(ℱℱὡ𝑛 𝑏(ℱℱὡ𝑛 NOUN cana-1536 103 4 , , PUNCT cana-1536 103 5 β„‹(𝜎 β„‹(𝜎 PROPN cana-1536 103 6 , , PUNCT cana-1536 103 7 𝔑 𝔑 PROPN cana-1536 103 8 , , PUNCT cana-1536 103 9 𝜌 𝜌 ADP cana-1536 103 10 ) ) PUNCT cana-1536 103 11 , , PUNCT cana-1536 103 12 β„‹(𝜎 β„‹(𝜎 PROPN cana-1536 103 13 , , PUNCT cana-1536 103 14 𝔑 𝔑 PROPN cana-1536 103 15 , , PUNCT cana-1536 103 16 𝜌 𝜌 X cana-1536 103 17 ) ) PUNCT cana-1536 103 18 ) ) PUNCT cana-1536 104 1 + + CCONJ cana-1536 104 2 β‚’ β‚’ ADP cana-1536 104 3 𝑏(ℱℱϱ𝑛 𝑏(ℱℱϱ𝑛 NOUN cana-1536 104 4 , , PUNCT cana-1536 104 5 β„‹(𝔑 β„‹(𝔑 PROPN cana-1536 104 6 , , PUNCT cana-1536 104 7 𝜌 𝜌 X cana-1536 104 8 , , PUNCT cana-1536 104 9 𝜎 𝜎 NOUN cana-1536 104 10 ) ) PUNCT cana-1536 104 11 , , PUNCT cana-1536 104 12 β„‹(𝔑 β„‹(𝔑 PROPN cana-1536 104 13 , , PUNCT cana-1536 104 14 𝜌 𝜌 X cana-1536 104 15 , , PUNCT cana-1536 104 16 𝜎 𝜎 NOUN cana-1536 104 17 ) ) PUNCT cana-1536 104 18 ) ) PUNCT cana-1536 104 19 β†’ β†’ SYM cana-1536 104 20 0 0 NUM cana-1536 104 21 π‘Žπ‘  π‘Žπ‘  PROPN cana-1536 104 22 𝑛 𝑛 PROPN cana-1536 104 23 β†’ β†’ PUNCT cana-1536 104 24 ∞. ∞. PROPN cana-1536 104 25 on on ADP cana-1536 104 26 the the DET cana-1536 104 27 other other ADJ cana-1536 104 28 hand hand NOUN cana-1536 104 29 , , PUNCT cana-1536 104 30 β„± β„± PROPN cana-1536 104 31 is be AUX cana-1536 104 32 continuous continuous ADJ cana-1536 104 33 , , PUNCT cana-1536 104 34 β‚’ β‚’ ADP cana-1536 104 35 𝑏(ℱℱℐ𝑛+1 𝑏(ℱℱℐ𝑛+1 PROPN cana-1536 104 36 , , PUNCT cana-1536 104 37 β„‹(𝜌 β„‹(𝜌 NUM cana-1536 104 38 , , PUNCT cana-1536 104 39 𝜎 𝜎 NOUN cana-1536 104 40 , , PUNCT cana-1536 104 41 𝔑 𝔑 CCONJ cana-1536 104 42 ) ) PUNCT cana-1536 104 43 , , PUNCT cana-1536 104 44 β„‹(𝜌 β„‹(𝜌 NUM cana-1536 104 45 , , PUNCT cana-1536 104 46 𝜎 𝜎 NOUN cana-1536 104 47 , , PUNCT cana-1536 104 48 𝔑 𝔑 NOUN cana-1536 104 49 ) ) PUNCT cana-1536 104 50 ) ) PUNCT cana-1536 105 1 + + CCONJ cana-1536 105 2 β‚’ β‚’ ADP cana-1536 105 3 𝑏(ℱℱὡ𝑛 𝑏(ℱℱὡ𝑛 NOUN cana-1536 105 4 , , PUNCT cana-1536 105 5 β„‹(𝜎 β„‹(𝜎 PROPN cana-1536 105 6 , , PUNCT cana-1536 105 7 𝔑 𝔑 PROPN cana-1536 105 8 , , PUNCT cana-1536 105 9 𝜌 𝜌 ADP cana-1536 105 10 ) ) PUNCT cana-1536 105 11 , , PUNCT cana-1536 105 12 β„‹(𝜎 β„‹(𝜎 PROPN cana-1536 105 13 , , PUNCT cana-1536 105 14 𝔑 𝔑 PROPN cana-1536 105 15 , , PUNCT cana-1536 105 16 𝜌 𝜌 X cana-1536 105 17 ) ) PUNCT cana-1536 105 18 ) ) PUNCT cana-1536 106 1 + + CCONJ cana-1536 106 2 β‚’ β‚’ ADP cana-1536 106 3 𝑏(ℱℱϱ𝑛 𝑏(ℱℱϱ𝑛 NOUN cana-1536 106 4 , , PUNCT cana-1536 106 5 β„‹(𝔑 β„‹(𝔑 ADP cana-1536 106 6 , , PUNCT cana-1536 106 7 𝜌 𝜌 X cana-1536 106 8 , , PUNCT cana-1536 106 9 𝜎 𝜎 NOUN cana-1536 106 10 ) ) PUNCT cana-1536 106 11 , , PUNCT cana-1536 106 12 β„‹(𝔑 β„‹(𝔑 PROPN cana-1536 106 13 , , PUNCT cana-1536 106 14 𝜌 𝜌 X cana-1536 106 15 , , PUNCT cana-1536 106 16 𝜎 𝜎 NOUN cana-1536 106 17 ) ) PUNCT cana-1536 106 18 ) ) PUNCT cana-1536 106 19 β†’ β†’ SYM cana-1536 106 20 β‚’ β‚’ ADP cana-1536 106 21 𝑏(β„±πœŒ 𝑏(β„±πœŒ NOUN cana-1536 106 22 , , PUNCT cana-1536 106 23 β„‹(𝜌 β„‹(𝜌 ADP cana-1536 106 24 , , PUNCT cana-1536 106 25 𝜎 𝜎 NOUN cana-1536 106 26 , , PUNCT cana-1536 106 27 𝔑 𝔑 CCONJ cana-1536 106 28 ) ) PUNCT cana-1536 106 29 , , PUNCT cana-1536 106 30 β„‹(𝜌 β„‹(𝜌 NUM cana-1536 106 31 , , PUNCT cana-1536 106 32 𝜎 𝜎 NOUN cana-1536 106 33 , , PUNCT cana-1536 106 34 𝔑 𝔑 NOUN cana-1536 106 35 ) ) PUNCT cana-1536 106 36 ) ) PUNCT cana-1536 107 1 + + CCONJ cana-1536 107 2 β‚’ β‚’ ADP cana-1536 107 3 𝑏(β„±πœŽ 𝑏(β„±πœŽ NOUN cana-1536 107 4 , , PUNCT cana-1536 107 5 β„‹(𝜎 β„‹(𝜎 PROPN cana-1536 107 6 , , PUNCT cana-1536 107 7 𝔑 𝔑 PROPN cana-1536 107 8 , , PUNCT cana-1536 107 9 𝜌 𝜌 ADP cana-1536 107 10 ) ) PUNCT cana-1536 107 11 , , PUNCT cana-1536 107 12 β„‹(𝜎 β„‹(𝜎 PROPN cana-1536 107 13 , , PUNCT cana-1536 107 14 𝔑 𝔑 PROPN cana-1536 107 15 , , PUNCT cana-1536 107 16 𝜌 𝜌 X cana-1536 107 17 ) ) PUNCT cana-1536 107 18 ) ) PUNCT cana-1536 107 19 + + CCONJ cana-1536 107 20 β‚’ β‚’ ADP cana-1536 107 21 𝑏(ℱ𝔑 𝑏(ℱ𝔑 ADJ cana-1536 107 22 , , PUNCT cana-1536 107 23 β„‹(𝔑 β„‹(𝔑 NUM cana-1536 107 24 , , PUNCT cana-1536 107 25 𝜌 𝜌 X cana-1536 107 26 , , PUNCT cana-1536 107 27 𝜎 𝜎 NOUN cana-1536 107 28 ) ) PUNCT cana-1536 107 29 , , PUNCT cana-1536 107 30 β„‹(𝔑 β„‹(𝔑 PROPN cana-1536 107 31 , , PUNCT cana-1536 107 32 𝜌 𝜌 X cana-1536 107 33 , , PUNCT cana-1536 107 34 𝜎 𝜎 NOUN cana-1536 107 35 ) ) PUNCT cana-1536 107 36 ) ) PUNCT cana-1536 107 37 𝑛 𝑛 PROPN cana-1536 107 38 β†’ β†’ SYM cana-1536 107 39 ∞. ∞. PROPN cana-1536 107 40 it it PRON cana-1536 107 41 gives give VERB cana-1536 107 42 β‚’ β‚’ ADP cana-1536 107 43 𝑏(β„±πœŒ 𝑏(β„±πœŒ NOUN cana-1536 107 44 , , PUNCT cana-1536 107 45 β„‹(𝜌 β„‹(𝜌 NUM cana-1536 107 46 , , PUNCT cana-1536 107 47 𝜎 𝜎 NOUN cana-1536 107 48 , , PUNCT cana-1536 107 49 𝔑 𝔑 CCONJ cana-1536 107 50 ) ) PUNCT cana-1536 107 51 , , PUNCT cana-1536 107 52 β„‹(𝜌 β„‹(𝜌 NUM cana-1536 107 53 , , PUNCT cana-1536 107 54 𝜎 𝜎 NOUN cana-1536 107 55 , , PUNCT cana-1536 107 56 𝔑 𝔑 NOUN cana-1536 107 57 ) ) PUNCT cana-1536 107 58 ) ) PUNCT cana-1536 108 1 + + CCONJ cana-1536 108 2 β‚’ β‚’ ADP cana-1536 108 3 𝑏(β„±πœŽ 𝑏(β„±πœŽ NOUN cana-1536 108 4 , , PUNCT cana-1536 108 5 β„‹(𝜎 β„‹(𝜎 PROPN cana-1536 108 6 , , PUNCT cana-1536 108 7 𝔑 𝔑 PROPN cana-1536 108 8 , , PUNCT cana-1536 108 9 𝜌 𝜌 ADP cana-1536 108 10 ) ) PUNCT cana-1536 108 11 , , PUNCT cana-1536 108 12 β„‹(𝜎 β„‹(𝜎 PROPN cana-1536 108 13 , , PUNCT cana-1536 108 14 𝔑 𝔑 PROPN cana-1536 108 15 , , PUNCT cana-1536 108 16 𝜌 𝜌 X cana-1536 108 17 ) ) PUNCT cana-1536 108 18 ) ) PUNCT cana-1536 108 19 + + CCONJ cana-1536 108 20 β‚’ β‚’ ADP cana-1536 108 21 𝑏(ℱ𝔑 𝑏(ℱ𝔑 ADJ cana-1536 108 22 , , PUNCT cana-1536 108 23 β„‹(𝔑 β„‹(𝔑 NUM cana-1536 108 24 , , PUNCT cana-1536 108 25 𝜌 𝜌 X cana-1536 108 26 , , PUNCT cana-1536 108 27 𝜎 𝜎 NOUN cana-1536 108 28 ) ) PUNCT cana-1536 108 29 , , PUNCT cana-1536 108 30 β„‹(𝔑 β„‹(𝔑 PROPN cana-1536 108 31 , , PUNCT cana-1536 108 32 𝜌 𝜌 X cana-1536 108 33 , , PUNCT cana-1536 108 34 𝜎 𝜎 NOUN cana-1536 108 35 ) ) PUNCT cana-1536 108 36 ) ) PUNCT cana-1536 108 37 = = SYM cana-1536 108 38 0 0 NUM cana-1536 108 39 β‚’ β‚’ ADP cana-1536 108 40 𝑏(β„±πœŒ 𝑏(β„±πœŒ NOUN cana-1536 108 41 , , PUNCT cana-1536 108 42 β„‹(𝜌 β„‹(𝜌 ADP cana-1536 108 43 , , PUNCT cana-1536 108 44 𝜎 𝜎 NOUN cana-1536 108 45 , , PUNCT cana-1536 108 46 𝔑 𝔑 CCONJ cana-1536 108 47 ) ) PUNCT cana-1536 108 48 , , PUNCT cana-1536 108 49 β„‹(𝜌 β„‹(𝜌 NUM cana-1536 108 50 , , PUNCT cana-1536 108 51 𝜎 𝜎 NOUN cana-1536 108 52 , , PUNCT cana-1536 108 53 𝔑 𝔑 NOUN cana-1536 108 54 ) ) PUNCT cana-1536 108 55 ) ) PUNCT cana-1536 108 56 = = PUNCT cana-1536 108 57 β‚’ β‚’ ADP cana-1536 108 58 𝑏(β„±πœŽ 𝑏(β„±πœŽ PROPN cana-1536 108 59 , , PUNCT cana-1536 108 60 β„‹(𝜎 β„‹(𝜎 PROPN cana-1536 108 61 , , PUNCT cana-1536 108 62 𝔑 𝔑 PROPN cana-1536 108 63 , , PUNCT cana-1536 108 64 𝜌 𝜌 ADP cana-1536 108 65 ) ) PUNCT cana-1536 108 66 , , PUNCT cana-1536 108 67 β„‹(𝜎 β„‹(𝜎 PROPN cana-1536 108 68 , , PUNCT cana-1536 108 69 𝔑 𝔑 PROPN cana-1536 108 70 , , PUNCT cana-1536 108 71 𝜌 𝜌 NOUN cana-1536 108 72 ) ) PUNCT cana-1536 108 73 ) ) PUNCT cana-1536 108 74 = = PUNCT cana-1536 108 75 β‚’ β‚’ ADP cana-1536 108 76 𝑏(ℱ𝔑 𝑏(ℱ𝔑 ADJ cana-1536 108 77 , , PUNCT cana-1536 108 78 β„‹(𝔑 β„‹(𝔑 NUM cana-1536 108 79 , , PUNCT cana-1536 108 80 𝜌 𝜌 X cana-1536 108 81 , , PUNCT cana-1536 108 82 𝜎 𝜎 NOUN cana-1536 108 83 ) ) PUNCT cana-1536 108 84 , , PUNCT cana-1536 108 85 β„‹(𝔑 β„‹(𝔑 PROPN cana-1536 108 86 , , PUNCT cana-1536 108 87 𝜌 𝜌 X cana-1536 108 88 , , PUNCT cana-1536 108 89 𝜎 𝜎 NOUN cana-1536 108 90 ) ) PUNCT cana-1536 108 91 ) ) PUNCT cana-1536 109 1 = = SYM cana-1536 109 2 0 0 NUM cana-1536 109 3 β‡’ β‡’ NOUN cana-1536 109 4 β„±πœŒ β„±πœŒ PROPN cana-1536 109 5 = = SYM cana-1536 109 6 β„‹(𝜌 β„‹(𝜌 NUM cana-1536 109 7 , , PUNCT cana-1536 109 8 𝜎 𝜎 NOUN cana-1536 109 9 , , PUNCT cana-1536 109 10 𝔑 𝔑 CCONJ cana-1536 109 11 ) ) PUNCT cana-1536 109 12 , , PUNCT cana-1536 109 13 β„±πœŽ β„±πœŽ PROPN cana-1536 109 14 = = SYM cana-1536 109 15 β„‹(𝜎 β„‹(𝜎 PROPN cana-1536 109 16 , , PUNCT cana-1536 109 17 𝔑 𝔑 PROPN cana-1536 109 18 , , PUNCT cana-1536 109 19 𝜌 𝜌 ADP cana-1536 109 20 ) ) PUNCT cana-1536 109 21 and and CCONJ cana-1536 109 22 ℱ𝔑 ℱ𝔑 PROPN cana-1536 109 23 = = SYM cana-1536 109 24 β„‹(𝔑 β„‹(𝔑 PROPN cana-1536 109 25 , , PUNCT cana-1536 109 26 𝜌 𝜌 X cana-1536 109 27 , , PUNCT cana-1536 109 28 𝜎)------------(3.4.5 𝜎)------------(3.4.5 ADJ cana-1536 109 29 ) ) PUNCT cana-1536 109 30 ∴ ∴ NOUN cana-1536 109 31 ( ( PUNCT cana-1536 109 32 𝜌 𝜌 X cana-1536 109 33 , , PUNCT cana-1536 109 34 𝜎 𝜎 NOUN cana-1536 109 35 , , PUNCT cana-1536 109 36 𝔑 𝔑 CCONJ cana-1536 109 37 ) ) PUNCT cana-1536 109 38 is be AUX cana-1536 109 39 the the DET cana-1536 109 40 tcip tcip NOUN cana-1536 109 41 of of ADP cana-1536 109 42 β„± β„± PROPN cana-1536 109 43 and and CCONJ cana-1536 109 44 β„‹. β„‹. PROPN cana-1536 109 45 now now ADV cana-1536 109 46 , , PUNCT cana-1536 109 47 β‚’ β‚’ ADP cana-1536 109 48 𝑏(β„±πœŒ 𝑏(β„±πœŒ X cana-1536 109 49 , , PUNCT cana-1536 109 50 β„±πœŽ β„±πœŽ NOUN cana-1536 109 51 , , PUNCT cana-1536 109 52 β„±πœŽ β„±πœŽ NOUN cana-1536 109 53 ) ) PUNCT cana-1536 109 54 + + CCONJ cana-1536 109 55 β‚’ β‚’ ADP cana-1536 109 56 𝑏(β„±πœŽ 𝑏(β„±πœŽ NOUN cana-1536 109 57 , , PUNCT cana-1536 109 58 ℱ𝔑 ℱ𝔑 PROPN cana-1536 109 59 , , PUNCT cana-1536 109 60 ℱ𝔑 ℱ𝔑 NOUN cana-1536 109 61 ) ) PUNCT cana-1536 110 1 + + CCONJ cana-1536 110 2 β‚’ β‚’ ADP cana-1536 110 3 𝑏 𝑏 PRON cana-1536 110 4 ( ( PUNCT cana-1536 110 5 , , PUNCT cana-1536 110 6 ℱ𝔑 ℱ𝔑 PROPN cana-1536 110 7 , , PUNCT cana-1536 110 8 β„±πœŒ β„±πœŒ PROPN cana-1536 110 9 , , PUNCT cana-1536 110 10 β„±πœŒ β„±πœŒ PROPN cana-1536 110 11 ) ) PUNCT cana-1536 110 12 = = PUNCT cana-1536 110 13 β‚’ β‚’ ADP cana-1536 110 14 𝑏(β„‹(𝜌 𝑏(β„‹(𝜌 NUM cana-1536 110 15 , , PUNCT cana-1536 110 16 𝜎 𝜎 NOUN cana-1536 110 17 , , PUNCT cana-1536 110 18 𝔑 𝔑 CCONJ cana-1536 110 19 ) ) PUNCT cana-1536 110 20 , , PUNCT cana-1536 110 21 β„‹(𝜎 β„‹(𝜎 PROPN cana-1536 110 22 , , PUNCT cana-1536 110 23 𝔑 𝔑 PROPN cana-1536 110 24 , , PUNCT cana-1536 110 25 𝜌 𝜌 ADP cana-1536 110 26 ) ) PUNCT cana-1536 110 27 , , PUNCT cana-1536 110 28 β„‹(𝜎 β„‹(𝜎 PROPN cana-1536 110 29 , , PUNCT cana-1536 110 30 𝔑 𝔑 PROPN cana-1536 110 31 , , PUNCT cana-1536 110 32 𝜌))+ₒ𝑏(β„‹(𝜎 𝜌))+ₒ𝑏(β„‹(𝜎 NOUN cana-1536 110 33 , , PUNCT cana-1536 110 34 𝔑 𝔑 PROPN cana-1536 110 35 , , PUNCT cana-1536 110 36 𝜌 𝜌 ADP cana-1536 110 37 ) ) PUNCT cana-1536 110 38 , , PUNCT cana-1536 110 39 β„‹(𝔑 β„‹(𝔑 PROPN cana-1536 110 40 , , PUNCT cana-1536 110 41 𝜌 𝜌 X cana-1536 110 42 , , PUNCT cana-1536 110 43 𝜎 𝜎 NOUN cana-1536 110 44 ) ) PUNCT cana-1536 110 45 , , PUNCT cana-1536 110 46 β„‹(𝔑 β„‹(𝔑 PROPN cana-1536 110 47 , , PUNCT cana-1536 110 48 𝜌 𝜌 X cana-1536 110 49 , , PUNCT cana-1536 110 50 𝜎 𝜎 NOUN cana-1536 110 51 ) ) PUNCT cana-1536 110 52 ) ) PUNCT cana-1536 111 1 + + ADP cana-1536 111 2 β‚’ β‚’ X cana-1536 111 3 𝑏(β„‹(𝔑 𝑏(β„‹(𝔑 NOUN cana-1536 111 4 , , PUNCT cana-1536 111 5 𝜌 𝜌 X cana-1536 111 6 , , PUNCT cana-1536 111 7 𝜎 𝜎 NOUN cana-1536 111 8 ) ) PUNCT cana-1536 111 9 , , PUNCT cana-1536 111 10 β„‹(𝜌 β„‹(𝜌 NUM cana-1536 111 11 , , PUNCT cana-1536 111 12 𝜎 𝜎 NOUN cana-1536 111 13 , , PUNCT cana-1536 111 14 𝔑 𝔑 CCONJ cana-1536 111 15 ) ) PUNCT cana-1536 111 16 , , PUNCT cana-1536 111 17 β„‹(𝜌 β„‹(𝜌 NUM cana-1536 111 18 , , PUNCT cana-1536 111 19 𝜎 𝜎 NOUN cana-1536 111 20 , , PUNCT cana-1536 111 21 𝔑 𝔑 NOUN cana-1536 111 22 ) ) PUNCT cana-1536 111 23 ) ) PUNCT cana-1536 111 24 then then ADV cana-1536 111 25 from from ADP cana-1536 111 26 equation equation NOUN cana-1536 111 27 ( ( PUNCT cana-1536 111 28 3.4.1 3.4.1 NUM cana-1536 111 29 ) ) PUNCT cana-1536 111 30 ≀ ≀ NUM cana-1536 111 31 πœƒ[ₒ𝑏(β„±πœŒ πœƒ[ₒ𝑏(β„±πœŒ NOUN cana-1536 111 32 , , PUNCT cana-1536 111 33 β„±πœŽ β„±πœŽ NOUN cana-1536 111 34 , , PUNCT cana-1536 111 35 β„±πœŽ β„±πœŽ NOUN cana-1536 111 36 ) ) PUNCT cana-1536 111 37 + + CCONJ cana-1536 111 38 β‚’ β‚’ ADP cana-1536 111 39 𝑏(β„±πœŽ 𝑏(β„±πœŽ NOUN cana-1536 111 40 , , PUNCT cana-1536 111 41 ℱ𝔑 ℱ𝔑 PROPN cana-1536 111 42 , , PUNCT cana-1536 111 43 ℱ𝔑 ℱ𝔑 NOUN cana-1536 111 44 ) ) PUNCT cana-1536 111 45 + + CCONJ cana-1536 111 46 β‚’ β‚’ ADP cana-1536 111 47 𝑏 𝑏 PRON cana-1536 111 48 ( ( PUNCT cana-1536 111 49 , , PUNCT cana-1536 111 50 ℱ𝔑 ℱ𝔑 PROPN cana-1536 111 51 , , PUNCT cana-1536 111 52 β„±πœŒ β„±πœŒ PROPN cana-1536 111 53 , , PUNCT cana-1536 111 54 β„±πœŒ β„±πœŒ PROPN cana-1536 111 55 ) ) PUNCT cana-1536 111 56 ] ] PUNCT cana-1536 111 57 which which PRON cana-1536 111 58 gives give VERB cana-1536 111 59 β‚’ β‚’ ADP cana-1536 111 60 𝑏(β„±πœŒ 𝑏(β„±πœŒ NOUN cana-1536 111 61 , , PUNCT cana-1536 111 62 β„±πœŽ β„±πœŽ NOUN cana-1536 111 63 , , PUNCT cana-1536 111 64 β„±πœŽ β„±πœŽ NOUN cana-1536 111 65 ) ) PUNCT cana-1536 111 66 + + CCONJ cana-1536 111 67 β‚’ β‚’ ADP cana-1536 111 68 𝑏(β„±πœŽ 𝑏(β„±πœŽ NOUN cana-1536 111 69 , , PUNCT cana-1536 111 70 ℱ𝔑 ℱ𝔑 PROPN cana-1536 111 71 , , PUNCT cana-1536 111 72 ℱ𝔑 ℱ𝔑 NOUN cana-1536 111 73 ) ) PUNCT cana-1536 111 74 + + CCONJ cana-1536 111 75 β‚’ β‚’ ADP cana-1536 111 76 𝑏(ℱ𝔑 𝑏(ℱ𝔑 ADJ cana-1536 111 77 , , PUNCT cana-1536 111 78 β„±πœŒ β„±πœŒ PROPN cana-1536 111 79 , , PUNCT cana-1536 111 80 β„±πœŒ β„±πœŒ PROPN cana-1536 111 81 ) ) PUNCT cana-1536 111 82 = = SYM cana-1536 111 83 0 0 NUM cana-1536 111 84 β‡’ β‡’ NOUN cana-1536 111 85 β„±πœŒ β„±πœŒ PROPN cana-1536 111 86 = = SYM cana-1536 111 87 β„±πœŽ β„±πœŽ PROPN cana-1536 111 88 , , PUNCT cana-1536 111 89 β„±πœŽ β„±πœŽ PROPN cana-1536 111 90 = = SYM cana-1536 111 91 ℱ𝔑 ℱ𝔑 PROPN cana-1536 111 92 and and CCONJ cana-1536 111 93 ℱ𝔑 ℱ𝔑 PROPN cana-1536 111 94 = = SYM cana-1536 111 95 β„±πœŒ.--------------------------------------(3.4.6 β„±πœŒ.--------------------------------------(3.4.6 NUM cana-1536 111 96 ) ) PUNCT cana-1536 111 97 by by ADP cana-1536 111 98 using use VERB cana-1536 111 99 ( ( PUNCT cana-1536 111 100 β‚’ β‚’ ADP cana-1536 111 101 𝑏5 𝑏5 NOUN cana-1536 111 102 ) ) PUNCT cana-1536 111 103 and and CCONJ cana-1536 111 104 equation equation NOUN cana-1536 111 105 ( ( PUNCT cana-1536 111 106 3.4.1 3.4.1 NUM cana-1536 111 107 ) ) PUNCT cana-1536 111 108 β‚’ β‚’ ADP cana-1536 111 109 𝑏(𝜌 𝑏(𝜌 PROPN cana-1536 111 110 , , PUNCT cana-1536 111 111 β„±πœŒ β„±πœŒ PROPN cana-1536 111 112 , , PUNCT cana-1536 111 113 β„±πœŒ β„±πœŒ PROPN cana-1536 111 114 ) ) PUNCT cana-1536 111 115 + + CCONJ cana-1536 111 116 β‚’ β‚’ ADP cana-1536 111 117 𝑏(𝜎 𝑏(𝜎 NOUN cana-1536 111 118 , , PUNCT cana-1536 111 119 β„±πœŽ β„±πœŽ NOUN cana-1536 111 120 , , PUNCT cana-1536 111 121 β„±πœŽ β„±πœŽ NOUN cana-1536 111 122 ) ) PUNCT cana-1536 111 123 + + CCONJ cana-1536 111 124 β‚’ β‚’ ADP cana-1536 111 125 𝑏(𝔑 𝑏(𝔑 NOUN cana-1536 111 126 , , PUNCT cana-1536 111 127 ℱ𝔑 ℱ𝔑 PROPN cana-1536 111 128 , , PUNCT cana-1536 111 129 ℱ𝔑 ℱ𝔑 NOUN cana-1536 111 130 ) ) PUNCT cana-1536 111 131 ≀ ≀ NOUN cana-1536 111 132 𝑠[ₒ𝑏(𝜌 𝑠[ₒ𝑏(𝜌 NOUN cana-1536 111 133 , , PUNCT cana-1536 111 134 ℱℐ𝑝+1 ℱℐ𝑝+1 NOUN cana-1536 111 135 , , PUNCT cana-1536 111 136 ℱℐ𝑝+1 ℱℐ𝑝+1 NOUN cana-1536 111 137 ) ) PUNCT cana-1536 111 138 + + CCONJ cana-1536 111 139 β‚’ β‚’ ADP cana-1536 111 140 𝑏(ℱℐ𝑝+1 𝑏(ℱℐ𝑝+1 PROPN cana-1536 111 141 , , PUNCT cana-1536 111 142 β„±πœŒ β„±πœŒ PROPN cana-1536 111 143 , , PUNCT cana-1536 111 144 β„±πœŒ)]+ β„±πœŒ)]+ PROPN cana-1536 111 145 s[ₒ𝑏(𝜎 s[ₒ𝑏(𝜎 NOUN cana-1536 111 146 , , PUNCT cana-1536 111 147 ℱὡ𝑝+1 ℱὡ𝑝+1 NUM cana-1536 111 148 , , PUNCT cana-1536 111 149 ℱὡ𝑝+1 ℱὡ𝑝+1 NUM cana-1536 111 150 ) ) PUNCT cana-1536 111 151 + + NUM cana-1536 111 152 β‚’ β‚’ ADP cana-1536 111 153 𝑏(ℱὡ𝑝+1 𝑏(ℱὡ𝑝+1 NOUN cana-1536 111 154 , , PUNCT cana-1536 111 155 β„±πœŽ β„±πœŽ PROPN cana-1536 111 156 , , PUNCT cana-1536 111 157 β„±πœŽ)]+𝑠[ₒ𝑏(𝔑 β„±πœŽ)]+𝑠[ₒ𝑏(𝔑 PROPN cana-1536 111 158 , , PUNCT cana-1536 111 159 ℱϱ𝑝+1 ℱϱ𝑝+1 PROPN cana-1536 111 160 , , PUNCT cana-1536 111 161 ℱϱ𝑝+1 ℱϱ𝑝+1 NOUN cana-1536 111 162 ) ) PUNCT cana-1536 111 163 + + CCONJ cana-1536 111 164 β‚’ β‚’ ADP cana-1536 111 165 𝑏(ℱϱ𝑝+1 𝑏(ℱϱ𝑝+1 NOUN cana-1536 111 166 , , PUNCT cana-1536 111 167 ℱ𝔑 ℱ𝔑 PROPN cana-1536 111 168 , , PUNCT cana-1536 111 169 ℱ𝔑 ℱ𝔑 NOUN cana-1536 111 170 ) ) PUNCT cana-1536 111 171 ] ] PUNCT cana-1536 111 172 communications communication NOUN cana-1536 111 173 on on ADP cana-1536 111 174 applied apply VERB cana-1536 111 175 nonlinear nonlinear ADJ cana-1536 111 176 analysis analysis NOUN cana-1536 111 177 issn issn NOUN cana-1536 111 178 : : PUNCT cana-1536 111 179 1074 1074 NUM cana-1536 111 180 - - PUNCT cana-1536 111 181 133x 133x NUM cana-1536 111 182 vol vol NOUN cana-1536 111 183 31 31 NUM cana-1536 111 184 no no NOUN cana-1536 111 185 . . PUNCT cana-1536 112 1 8s 8s PROPN cana-1536 112 2 ( ( PUNCT cana-1536 112 3 2024 2024 NUM cana-1536 112 4 ) ) PUNCT cana-1536 112 5 438 438 NUM cana-1536 112 6 https://internationalpubls.com https://internationalpubls.com X cana-1536 112 7 ≀ ≀ NUM cana-1536 112 8 𝑠[ₒ𝑏(𝜌 𝑠[ₒ𝑏(𝜌 NOUN cana-1536 112 9 , , PUNCT cana-1536 112 10 ℱℐ𝑝+1 ℱℐ𝑝+1 NOUN cana-1536 112 11 , , PUNCT cana-1536 112 12 ℱℐ𝑝+1 ℱℐ𝑝+1 NOUN cana-1536 112 13 ) ) PUNCT cana-1536 113 1 + + CCONJ cana-1536 113 2 β‚’ β‚’ ADP cana-1536 113 3 𝑏(𝜎 𝑏(𝜎 NOUN cana-1536 113 4 , , PUNCT cana-1536 113 5 ℱὡ𝑝+1 ℱὡ𝑝+1 NUM cana-1536 113 6 , , PUNCT cana-1536 113 7 ℱὡ𝑝+1)+ₒ𝑏(𝔑 ℱὡ𝑝+1)+ₒ𝑏(𝔑 NOUN cana-1536 113 8 , , PUNCT cana-1536 113 9 ℱϱ𝑝+1 ℱϱ𝑝+1 NOUN cana-1536 113 10 , , PUNCT cana-1536 113 11 ℱϱ𝑝+1 ℱϱ𝑝+1 NOUN cana-1536 113 12 ) ) PUNCT cana-1536 113 13 ] ] PUNCT cana-1536 114 1 + + PUNCT cana-1536 114 2 𝑠[ₒ𝑏(ℱℐ𝑝+1 𝑠[ₒ𝑏(ℱℐ𝑝+1 NOUN cana-1536 114 3 , , PUNCT cana-1536 114 4 β„±πœŒ β„±πœŒ PROPN cana-1536 114 5 , , PUNCT cana-1536 114 6 β„±πœŒ β„±πœŒ PROPN cana-1536 114 7 ) ) PUNCT cana-1536 114 8 + + NUM cana-1536 114 9 β‚’ β‚’ ADP cana-1536 114 10 𝑏(ℱὡ𝑝+1 𝑏(ℱὡ𝑝+1 NOUN cana-1536 114 11 , , PUNCT cana-1536 114 12 β„±πœŽ β„±πœŽ PROPN cana-1536 114 13 , , PUNCT cana-1536 114 14 β„±πœŽ)+ₒ𝑏(ℱϱ𝑝+1 β„±πœŽ)+ₒ𝑏(ℱϱ𝑝+1 NOUN cana-1536 114 15 , , PUNCT cana-1536 114 16 ℱ𝔑 ℱ𝔑 PROPN cana-1536 114 17 , , PUNCT cana-1536 114 18 ℱ𝔑 ℱ𝔑 NOUN cana-1536 114 19 ) ) PUNCT cana-1536 114 20 ] ] PUNCT cana-1536 114 21 = = SYM cana-1536 114 22 𝑠[ₒ𝑏(𝜌 𝑠[ₒ𝑏(𝜌 NUM cana-1536 114 23 , , PUNCT cana-1536 114 24 ℱℐ𝑝+1 ℱℐ𝑝+1 NOUN cana-1536 114 25 , , PUNCT cana-1536 114 26 ℱℐ𝑝+1 ℱℐ𝑝+1 NOUN cana-1536 114 27 ) ) PUNCT cana-1536 114 28 + + CCONJ cana-1536 114 29 β‚’ β‚’ ADP cana-1536 114 30 𝑏(𝜎 𝑏(𝜎 NOUN cana-1536 114 31 , , PUNCT cana-1536 114 32 ℱὡ𝑝+1 ℱὡ𝑝+1 NUM cana-1536 114 33 , , PUNCT cana-1536 114 34 ℱὡ𝑝+1)+ₒ𝑏(𝔑 ℱὡ𝑝+1)+ₒ𝑏(𝔑 NOUN cana-1536 114 35 , , PUNCT cana-1536 114 36 ℱϱ𝑝+1 ℱϱ𝑝+1 NOUN cana-1536 114 37 , , PUNCT cana-1536 114 38 ℱϱ𝑝+1 ℱϱ𝑝+1 NOUN cana-1536 114 39 ) ) PUNCT cana-1536 114 40 ] ] PUNCT cana-1536 114 41 + + NOUN cana-1536 114 42 𝑠[ₒ𝑏 𝑠[ₒ𝑏 NOUN cana-1536 114 43 ( ( PUNCT cana-1536 114 44 β„‹(ℐ𝑝 β„‹(ℐ𝑝 NUM cana-1536 114 45 , , PUNCT cana-1536 114 46 ὡ𝑝 ὡ𝑝 X cana-1536 114 47 , , PUNCT cana-1536 114 48 ϱ𝑝 ϱ𝑝 NOUN cana-1536 114 49 ) ) PUNCT cana-1536 114 50 , , PUNCT cana-1536 114 51 β„‹(𝜌 β„‹(𝜌 NUM cana-1536 114 52 , , PUNCT cana-1536 114 53 𝜎 𝜎 NOUN cana-1536 114 54 , , PUNCT cana-1536 114 55 𝔑 𝔑 CCONJ cana-1536 114 56 ) ) PUNCT cana-1536 114 57 , , PUNCT cana-1536 114 58 β„‹(𝜌 β„‹(𝜌 NUM cana-1536 114 59 , , PUNCT cana-1536 114 60 𝜎 𝜎 NOUN cana-1536 114 61 , , PUNCT cana-1536 114 62 𝔑))+ₒ𝑏 𝔑))+ₒ𝑏 NUM cana-1536 114 63 ( ( PUNCT cana-1536 114 64 β„‹(ὡ𝑝 β„‹(ὡ𝑝 NOUN cana-1536 114 65 , , PUNCT cana-1536 114 66 ϱ𝑝 ϱ𝑝 ADJ cana-1536 114 67 , , PUNCT cana-1536 114 68 ℐ𝑝 ℐ𝑝 PROPN cana-1536 114 69 ) ) PUNCT cana-1536 114 70 , , PUNCT cana-1536 114 71 β„‹(𝜎 β„‹(𝜎 PROPN cana-1536 114 72 , , PUNCT cana-1536 114 73 𝔑 𝔑 PROPN cana-1536 114 74 , , PUNCT cana-1536 114 75 𝜌 𝜌 ADP cana-1536 114 76 ) ) PUNCT cana-1536 114 77 , , PUNCT cana-1536 114 78 β„‹(𝜎 β„‹(𝜎 PROPN cana-1536 114 79 , , PUNCT cana-1536 114 80 𝔑 𝔑 PROPN cana-1536 114 81 , , PUNCT cana-1536 114 82 𝜌))+ 𝜌))+ NOUN cana-1536 114 83 β‚’ β‚’ ADP cana-1536 114 84 𝑏 𝑏 NOUN cana-1536 114 85 ( ( PUNCT cana-1536 114 86 β„‹(ϱ𝑝 β„‹(ϱ𝑝 ADV cana-1536 114 87 , , PUNCT cana-1536 114 88 ℐ𝑝 ℐ𝑝 PROPN cana-1536 114 89 , , PUNCT cana-1536 114 90 ὡ𝑝 ὡ𝑝 X cana-1536 114 91 ) ) PUNCT cana-1536 114 92 , , PUNCT cana-1536 114 93 β„‹(𝔑 β„‹(𝔑 PROPN cana-1536 114 94 , , PUNCT cana-1536 114 95 𝜌 𝜌 X cana-1536 114 96 , , PUNCT cana-1536 114 97 𝜎 𝜎 NOUN cana-1536 114 98 ) ) PUNCT cana-1536 114 99 , , PUNCT cana-1536 114 100 β„‹(𝔑 β„‹(𝔑 PROPN cana-1536 114 101 , , PUNCT cana-1536 114 102 𝜌 𝜌 X cana-1536 114 103 , , PUNCT cana-1536 114 104 𝜎 𝜎 NOUN cana-1536 114 105 ) ) PUNCT cana-1536 114 106 ) ) PUNCT cana-1536 114 107 ] ] PUNCT cana-1536 114 108 ≀ ≀ NUM cana-1536 114 109 𝑠[ₒ𝑏(𝜌 𝑠[ₒ𝑏(𝜌 NOUN cana-1536 114 110 , , PUNCT cana-1536 114 111 ℱℐ𝑝+1 ℱℐ𝑝+1 NOUN cana-1536 114 112 , , PUNCT cana-1536 114 113 ℱℐ𝑝+1 ℱℐ𝑝+1 NOUN cana-1536 114 114 ) ) PUNCT cana-1536 114 115 + + CCONJ cana-1536 114 116 β‚’ β‚’ ADP cana-1536 114 117 𝑏(𝜎 𝑏(𝜎 NOUN cana-1536 114 118 , , PUNCT cana-1536 114 119 ℱὡ𝑝+1 ℱὡ𝑝+1 NUM cana-1536 114 120 , , PUNCT cana-1536 114 121 ℱὡ𝑝+1)+ₒ𝑏(𝔑 ℱὡ𝑝+1)+ₒ𝑏(𝔑 NOUN cana-1536 114 122 , , PUNCT cana-1536 114 123 ℱϱ𝑝+1 ℱϱ𝑝+1 NOUN cana-1536 114 124 , , PUNCT cana-1536 114 125 ℱϱ𝑝+1 ℱϱ𝑝+1 NOUN cana-1536 114 126 ) ) PUNCT cana-1536 114 127 ] ] PUNCT cana-1536 115 1 + + PUNCT cana-1536 115 2 π‘ πœƒ[ₒ𝑏(ℱℐ𝑝 π‘ πœƒ[ₒ𝑏(ℱℐ𝑝 NOUN cana-1536 115 3 , , PUNCT cana-1536 115 4 β„±πœŒ β„±πœŒ PROPN cana-1536 115 5 , , PUNCT cana-1536 115 6 β„±πœŒ β„±πœŒ PROPN cana-1536 115 7 ) ) PUNCT cana-1536 115 8 + + NUM cana-1536 115 9 β‚’ β‚’ ADP cana-1536 115 10 𝑏(ℱὡ𝑝 𝑏(ℱὡ𝑝 PROPN cana-1536 115 11 , , PUNCT cana-1536 115 12 β„±πœŽ β„±πœŽ NOUN cana-1536 115 13 , , PUNCT cana-1536 115 14 β„±πœŽ β„±πœŽ NOUN cana-1536 115 15 ) ) PUNCT cana-1536 115 16 + + CCONJ cana-1536 115 17 β‚’ β‚’ ADP cana-1536 115 18 𝑏(ℱϱ𝑝 𝑏(ℱϱ𝑝 PROPN cana-1536 115 19 , , PUNCT cana-1536 115 20 ℱ𝔑 ℱ𝔑 PROPN cana-1536 115 21 , , PUNCT cana-1536 115 22 ℱ𝔑 ℱ𝔑 NOUN cana-1536 115 23 ) ) PUNCT cana-1536 115 24 ] ] PUNCT cana-1536 115 25 π‘Žπ‘  π‘Žπ‘  PROPN cana-1536 115 26 𝑝 𝑝 PROPN cana-1536 115 27 β†’ β†’ SYM cana-1536 115 28 ∞ ∞ PROPN cana-1536 115 29 , , PUNCT cana-1536 115 30 we we PRON cana-1536 115 31 get get VERB cana-1536 115 32 β‚’ β‚’ ADP cana-1536 115 33 𝑏(𝜌 𝑏(𝜌 PROPN cana-1536 115 34 , , PUNCT cana-1536 115 35 β„±πœŒ β„±πœŒ PROPN cana-1536 115 36 , , PUNCT cana-1536 115 37 β„±πœŒ β„±πœŒ PROPN cana-1536 115 38 ) ) PUNCT cana-1536 115 39 + + CCONJ cana-1536 115 40 β‚’ β‚’ ADP cana-1536 115 41 𝑏(𝜎 𝑏(𝜎 NOUN cana-1536 115 42 , , PUNCT cana-1536 115 43 β„±πœŽ β„±πœŽ NOUN cana-1536 115 44 , , PUNCT cana-1536 115 45 β„±πœŽ β„±πœŽ NOUN cana-1536 115 46 ) ) PUNCT cana-1536 115 47 + + CCONJ cana-1536 115 48 β‚’ β‚’ ADP cana-1536 115 49 𝑏(𝔑 𝑏(𝔑 NOUN cana-1536 115 50 , , PUNCT cana-1536 115 51 ℱ𝔑 ℱ𝔑 PROPN cana-1536 115 52 , , PUNCT cana-1536 115 53 ℱ𝔑 ℱ𝔑 NOUN cana-1536 115 54 ) ) PUNCT cana-1536 115 55 ≀ ≀ NOUN cana-1536 116 1 [ [ X cana-1536 116 2 β‚’ β‚’ ADP cana-1536 116 3 𝑏(𝜌 𝑏(𝜌 PROPN cana-1536 116 4 , , PUNCT cana-1536 116 5 𝜌 𝜌 X cana-1536 116 6 , , PUNCT cana-1536 116 7 𝜌 𝜌 ADP cana-1536 116 8 ) ) PUNCT cana-1536 116 9 + + CCONJ cana-1536 116 10 β‚’ β‚’ ADP cana-1536 116 11 𝑏(𝜎 𝑏(𝜎 PROPN cana-1536 116 12 , , PUNCT cana-1536 116 13 𝜎 𝜎 PROPN cana-1536 116 14 , , PUNCT cana-1536 116 15 𝜎)+ₒ𝑏(𝔑 𝜎)+ₒ𝑏(𝔑 NOUN cana-1536 116 16 , , PUNCT cana-1536 116 17 𝔑 𝔑 ADV cana-1536 116 18 , , PUNCT cana-1536 116 19 𝔑)]+ 𝔑)]+ NOUN cana-1536 116 20 + + ADJ cana-1536 116 21 π‘ πœƒ[ₒ𝑏(𝜌 π‘ πœƒ[ₒ𝑏(𝜌 PROPN cana-1536 116 22 , , PUNCT cana-1536 116 23 β„±πœŒ β„±πœŒ PROPN cana-1536 116 24 , , PUNCT cana-1536 116 25 β„±πœŒ β„±πœŒ PROPN cana-1536 116 26 ) ) PUNCT cana-1536 116 27 + + CCONJ cana-1536 116 28 β‚’ β‚’ ADP cana-1536 116 29 𝑏(𝜎 𝑏(𝜎 NOUN cana-1536 116 30 , , PUNCT cana-1536 116 31 β„±πœŽ β„±πœŽ NOUN cana-1536 116 32 , , PUNCT cana-1536 116 33 β„±πœŽ)+ₒ𝑏(𝔑 β„±πœŽ)+ₒ𝑏(𝔑 NUM cana-1536 116 34 , , PUNCT cana-1536 116 35 ℱ𝔑 ℱ𝔑 PROPN cana-1536 116 36 , , PUNCT cana-1536 116 37 ℱ𝔑 ℱ𝔑 PROPN cana-1536 116 38 ) ) PUNCT cana-1536 116 39 ] ] PUNCT cana-1536 116 40 ≀ ≀ NUM cana-1536 116 41 π‘ πœƒ[ₒ𝑏(𝜌 π‘ πœƒ[ₒ𝑏(𝜌 PROPN cana-1536 116 42 , , PUNCT cana-1536 116 43 β„±πœŒ β„±πœŒ PROPN cana-1536 116 44 , , PUNCT cana-1536 116 45 β„±πœŒ β„±πœŒ PROPN cana-1536 116 46 ) ) PUNCT cana-1536 116 47 + + CCONJ cana-1536 116 48 β‚’ β‚’ ADP cana-1536 116 49 𝑏(𝜎 𝑏(𝜎 NOUN cana-1536 116 50 , , PUNCT cana-1536 116 51 β„±πœŽ β„±πœŽ NOUN cana-1536 116 52 , , PUNCT cana-1536 116 53 β„±πœŽ)+ₒ𝑏(𝔑 β„±πœŽ)+ₒ𝑏(𝔑 NUM cana-1536 116 54 , , PUNCT cana-1536 116 55 ℱ𝔑 ℱ𝔑 PROPN cana-1536 116 56 , , PUNCT cana-1536 116 57 ℱ𝔑 ℱ𝔑 PROPN cana-1536 116 58 ) ) PUNCT cana-1536 116 59 ] ] PUNCT cana-1536 116 60 . . PUNCT cana-1536 117 1 which which PRON cana-1536 117 2 gives give VERB cana-1536 117 3 𝜌 𝜌 PRON cana-1536 117 4 = = SYM cana-1536 117 5 β„±πœŒ β„±πœŒ PROPN cana-1536 117 6 , , PUNCT cana-1536 117 7 𝜎 𝜎 NOUN cana-1536 118 1 = = SYM cana-1536 118 2 β„±πœŽ β„±πœŽ PROPN cana-1536 118 3 and and CCONJ cana-1536 118 4 𝔑 𝔑 NOUN cana-1536 118 5 = = SYM cana-1536 118 6 ℱ𝔑.-------------------------------------(3.4.7 ℱ𝔑.-------------------------------------(3.4.7 NUM cana-1536 118 7 ) ) PUNCT cana-1536 118 8 from from ADP cana-1536 118 9 equations equation NOUN cana-1536 118 10 ( ( PUNCT cana-1536 118 11 3.4.5 3.4.5 NUM cana-1536 118 12 ) ) PUNCT cana-1536 118 13 , , PUNCT cana-1536 118 14 ( ( PUNCT cana-1536 118 15 3.4.6 3.4.6 NUM cana-1536 118 16 ) ) PUNCT cana-1536 118 17 and and CCONJ cana-1536 118 18 ( ( PUNCT cana-1536 118 19 3.4.7 3.4.7 NUM cana-1536 118 20 ) ) PUNCT cana-1536 118 21 , , PUNCT cana-1536 118 22 we we PRON cana-1536 118 23 conclude conclude VERB cana-1536 118 24 that that PRON cana-1536 118 25 ( ( PUNCT cana-1536 118 26 𝜌 𝜌 X cana-1536 118 27 , , PUNCT cana-1536 118 28 𝜌 𝜌 X cana-1536 118 29 , , PUNCT cana-1536 118 30 𝜌 𝜌 X cana-1536 118 31 ) ) PUNCT cana-1536 118 32 is be AUX cana-1536 118 33 common common ADJ cana-1536 118 34 tfp tfp NOUN cana-1536 118 35 of of ADP cana-1536 118 36 β„± β„± PROPN cana-1536 118 37 and and CCONJ cana-1536 118 38 β„‹. β„‹. PROPN cana-1536 118 39 now now ADV cana-1536 118 40 we we PRON cana-1536 118 41 will will AUX cana-1536 118 42 prove prove VERB cana-1536 118 43 that that SCONJ cana-1536 118 44 uniqueness uniqueness NOUN cana-1536 118 45 property property NOUN cana-1536 118 46 , , PUNCT cana-1536 118 47 if if SCONJ cana-1536 118 48 possible possible ADJ cana-1536 118 49 let let VERB cana-1536 118 50 us we PRON cana-1536 118 51 assume assume VERB cana-1536 118 52 that that SCONJ cana-1536 118 53 𝔏 𝔏 PROPN cana-1536 118 54 = = PUNCT cana-1536 118 55 ℱ𝔏 ℱ𝔏 PROPN cana-1536 118 56 = = SYM cana-1536 118 57 β„‹(𝔏 β„‹(𝔏 PROPN cana-1536 118 58 , , PUNCT cana-1536 118 59 𝔏 𝔏 PROPN cana-1536 118 60 , , PUNCT cana-1536 118 61 𝔏)is 𝔏)is PROPN cana-1536 118 62 another another DET cana-1536 118 63 tfp tfp NOUN cana-1536 118 64 of of ADP cana-1536 118 65 β„± β„± PROPN cana-1536 118 66 and and CCONJ cana-1536 118 67 β„‹. β„‹. PROPN cana-1536 118 68 now now ADV cana-1536 118 69 consider consider VERB cana-1536 118 70 , , PUNCT cana-1536 118 71 3ₒ𝑏(𝜌 3ₒ𝑏(𝜌 NUM cana-1536 118 72 , , PUNCT cana-1536 118 73 𝔏 𝔏 PROPN cana-1536 118 74 , , PUNCT cana-1536 118 75 𝔏 𝔏 PROPN cana-1536 118 76 ) ) PUNCT cana-1536 118 77 = = SYM cana-1536 118 78 β‚’ β‚’ ADP cana-1536 118 79 𝑏 𝑏 PROPN cana-1536 118 80 ( ( PUNCT cana-1536 118 81 β„‹((𝜌 β„‹((𝜌 X cana-1536 118 82 , , PUNCT cana-1536 118 83 𝜌 𝜌 X cana-1536 118 84 , , PUNCT cana-1536 118 85 𝜌 𝜌 NOUN cana-1536 118 86 ) ) PUNCT cana-1536 118 87 ) ) PUNCT cana-1536 118 88 , , PUNCT cana-1536 118 89 β„‹(𝔏 β„‹(𝔏 NOUN cana-1536 118 90 , , PUNCT cana-1536 118 91 𝔏 𝔏 PROPN cana-1536 118 92 , , PUNCT cana-1536 118 93 𝔏 𝔏 PROPN cana-1536 118 94 ) ) PUNCT cana-1536 118 95 , , PUNCT cana-1536 118 96 β„‹(𝔏 β„‹(𝔏 NOUN cana-1536 118 97 , , PUNCT cana-1536 118 98 𝔏 𝔏 PROPN cana-1536 118 99 , , PUNCT cana-1536 118 100 𝔏 𝔏 PROPN cana-1536 118 101 ) ) PUNCT cana-1536 118 102 ) ) PUNCT cana-1536 119 1 ≀ ≀ NOUN cana-1536 119 2 3π‘ πœƒ[β‚’(β„±πœŒ 3π‘ πœƒ[β‚’(β„±πœŒ NUM cana-1536 119 3 , , PUNCT cana-1536 119 4 ℱ𝔏 ℱ𝔏 PROPN cana-1536 119 5 , , PUNCT cana-1536 119 6 ℱ𝔏 ℱ𝔏 PROPN cana-1536 119 7 ) ) PUNCT cana-1536 119 8 ] ] PUNCT cana-1536 119 9 ≀ ≀ NUM cana-1536 119 10 3π‘ πœƒβ‚’(𝜌 3π‘ πœƒβ‚’(𝜌 NUM cana-1536 119 11 , , PUNCT cana-1536 119 12 𝔏 𝔏 PROPN cana-1536 119 13 , , PUNCT cana-1536 119 14 𝔏 𝔏 PROPN cana-1536 119 15 ) ) PUNCT cana-1536 119 16 it it PRON cana-1536 119 17 is be AUX cana-1536 119 18 contradiction contradiction NOUN cana-1536 119 19 , , PUNCT cana-1536 119 20 therefore therefore ADV cana-1536 119 21 ( ( PUNCT cana-1536 119 22 𝜌 𝜌 X cana-1536 119 23 , , PUNCT cana-1536 119 24 𝜌 𝜌 X cana-1536 119 25 , , PUNCT cana-1536 119 26 𝜌 𝜌 X cana-1536 119 27 ) ) PUNCT cana-1536 119 28 is be AUX cana-1536 119 29 unique unique ADJ cana-1536 119 30 common common ADJ cana-1536 119 31 tfp tfp NOUN cana-1536 119 32 of of ADP cana-1536 119 33 β„± β„± PROPN cana-1536 119 34 and and CCONJ cana-1536 119 35 β„‹. β„‹. PROPN cana-1536 119 36 corollary corollary NOUN cana-1536 119 37 3.5let 3.5let PROPN cana-1536 119 38 ( ( PUNCT cana-1536 119 39 α½£ α½£ NOUN cana-1536 119 40 , , PUNCT cana-1536 119 41 β‚’ β‚’ ADP cana-1536 119 42 𝑏 𝑏 NOUN cana-1536 119 43 ) ) PUNCT cana-1536 119 44 is be AUX cana-1536 119 45 complete complete ADJ cana-1536 119 46 β‚’ β‚’ ADP cana-1536 119 47 b b NOUN cana-1536 119 48 - - PUNCT cana-1536 119 49 ms ms PROPN cana-1536 119 50 . . PROPN cana-1536 120 1 let let VERB cana-1536 120 2 β„‹ β„‹ NOUN cana-1536 120 3 : : PUNCT cana-1536 120 4 α½£3 α½£3 PROPN cana-1536 120 5 β†’ β†’ SYM cana-1536 120 6 α½£ α½£ PRON cana-1536 120 7 be be AUX cana-1536 120 8 a a DET cana-1536 120 9 mapping mapping NOUN cana-1536 120 10 such such ADJ cana-1536 120 11 that that SCONJ cana-1536 120 12 β‚’ β‚’ ADP cana-1536 120 13 𝑏(β„‹(α΄‚ 𝑏(β„‹(α΄‚ PROPN cana-1536 120 14 , , PUNCT cana-1536 120 15 α΄” α΄” NOUN cana-1536 120 16 , , PUNCT cana-1536 120 17 𝔣 𝔣 ADJ cana-1536 120 18 ) ) PUNCT cana-1536 120 19 , , PUNCT cana-1536 120 20 β„‹(Ο° β„‹(Ο° PROPN cana-1536 120 21 , , PUNCT cana-1536 120 22 β„“ β„“ NOUN cana-1536 120 23 , , PUNCT cana-1536 120 24 β„˜ β„˜ PROPN cana-1536 120 25 ) ) PUNCT cana-1536 120 26 , , PUNCT cana-1536 120 27 β„‹(Ο° β„‹(Ο° PROPN cana-1536 120 28 , , PUNCT cana-1536 120 29 β„“ β„“ NOUN cana-1536 120 30 , , PUNCT cana-1536 120 31 β„˜ β„˜ PROPN cana-1536 120 32 ) ) PUNCT cana-1536 120 33 ) ) PUNCT cana-1536 121 1 + + CCONJ cana-1536 121 2 β‚’ β‚’ ADP cana-1536 121 3 𝑏(β„‹ 𝑏(β„‹ PROPN cana-1536 121 4 ( ( PUNCT cana-1536 121 5 α΄” α΄” NOUN cana-1536 121 6 , , PUNCT cana-1536 121 7 𝔣 𝔣 ADJ cana-1536 121 8 , , PUNCT cana-1536 121 9 α΄‚ α΄‚ NOUN cana-1536 121 10 ) ) PUNCT cana-1536 121 11 , , PUNCT cana-1536 121 12 β„‹(β„“ β„‹(β„“ NUM cana-1536 121 13 , , PUNCT cana-1536 121 14 β„˜ β„˜ PROPN cana-1536 121 15 , , PUNCT cana-1536 121 16 Ο° Ο° NOUN cana-1536 121 17 ) ) PUNCT cana-1536 121 18 , , PUNCT cana-1536 121 19 β„‹(β„“ β„‹(β„“ NUM cana-1536 121 20 , , PUNCT cana-1536 121 21 β„˜ β„˜ NOUN cana-1536 121 22 , , PUNCT cana-1536 121 23 Ο°))+ₒ𝑏(β„‹(𝔣 Ο°))+ₒ𝑏(β„‹(𝔣 NUM cana-1536 121 24 , , PUNCT cana-1536 121 25 α΄‚ α΄‚ NOUN cana-1536 121 26 , , PUNCT cana-1536 121 27 α΄” α΄” NOUN cana-1536 121 28 ) ) PUNCT cana-1536 121 29 , , PUNCT cana-1536 121 30 β„‹(β„˜ β„‹(β„˜ NOUN cana-1536 121 31 , , PUNCT cana-1536 121 32 Ο° Ο° NOUN cana-1536 121 33 , , PUNCT cana-1536 121 34 β„“ β„“ NOUN cana-1536 121 35 ) ) PUNCT cana-1536 121 36 , , PUNCT cana-1536 121 37 β„‹(β„˜ β„‹(β„˜ NOUN cana-1536 121 38 , , PUNCT cana-1536 121 39 Ο° Ο° NOUN cana-1536 121 40 , , PUNCT cana-1536 121 41 β„“ β„“ NOUN cana-1536 121 42 ) ) PUNCT cana-1536 121 43 ) ) PUNCT cana-1536 121 44 ≀ ≀ PROPN cana-1536 121 45 πœƒ[ₒ𝑏(α΄‚ πœƒ[ₒ𝑏(α΄‚ NOUN cana-1536 121 46 , , PUNCT cana-1536 121 47 Ο° Ο° NOUN cana-1536 121 48 , , PUNCT cana-1536 121 49 Ο° Ο° NOUN cana-1536 121 50 ) ) PUNCT cana-1536 121 51 + + CCONJ cana-1536 121 52 β‚’ β‚’ ADP cana-1536 121 53 𝑏(α΄” 𝑏(α΄” PROPN cana-1536 121 54 , , PUNCT cana-1536 121 55 β„“ β„“ PROPN cana-1536 121 56 , , PUNCT cana-1536 121 57 β„“ β„“ NOUN cana-1536 121 58 ) ) PUNCT cana-1536 121 59 + + CCONJ cana-1536 121 60 β‚’ β‚’ ADP cana-1536 121 61 𝑏(𝔣 𝑏(𝔣 PROPN cana-1536 121 62 , , PUNCT cana-1536 121 63 β„˜ β„˜ NOUN cana-1536 121 64 , , PUNCT cana-1536 121 65 β„˜ β„˜ PROPN cana-1536 121 66 ) ) PUNCT cana-1536 121 67 ] ] PUNCT cana-1536 121 68 for for ADP cana-1536 121 69 all all DET cana-1536 121 70 all all PRON cana-1536 121 71 α΄‚ α΄‚ PROPN cana-1536 121 72 , , PUNCT cana-1536 121 73 α΄” α΄” NOUN cana-1536 121 74 , , PUNCT cana-1536 121 75 𝔣 𝔣 ADJ cana-1536 121 76 , , PUNCT cana-1536 121 77 Ο° Ο° NOUN cana-1536 121 78 , , PUNCT cana-1536 121 79 β„“ β„“ PROPN cana-1536 121 80 , , PUNCT cana-1536 121 81 β„˜ β„˜ PROPN cana-1536 121 82 ∈ ∈ NOUN cana-1536 121 83 α½£ α½£ NOUN cana-1536 121 84 and and CCONJ cana-1536 121 85 if if SCONJ cana-1536 121 86 πœƒ πœƒ X cana-1536 121 87 ∈ ∈ X cana-1536 121 88 [ [ X cana-1536 121 89 0,1 0,1 NUM cana-1536 121 90 ) ) PUNCT cana-1536 121 91 then then ADV cana-1536 121 92 there there PRON cana-1536 121 93 exists exist VERB cana-1536 121 94 a a DET cana-1536 121 95 unique unique ADJ cana-1536 121 96 tfp tfp NOUN cana-1536 121 97 for for ADP cana-1536 121 98 β„‹ β„‹ PROPN cana-1536 121 99 and and CCONJ cana-1536 121 100 inα½£. inα½£. NOUN cana-1536 121 101 example example NOUN cana-1536 121 102 3.6 3.6 NUM cana-1536 121 103 letα½£ letα½£ NOUN cana-1536 121 104 = = PUNCT cana-1536 122 1 [ [ X cana-1536 122 2 0,1 0,1 NUM cana-1536 122 3 ] ] PUNCT cana-1536 122 4 , , PUNCT cana-1536 122 5 define define VERB cana-1536 122 6 a a DET cana-1536 122 7 mappingₒ𝑏:α½£3 mappingₒ𝑏:α½£3 NOUN cana-1536 122 8 β†’ β†’ SYM cana-1536 122 9 α½£ α½£ NOUN cana-1536 122 10 as as ADP cana-1536 122 11 β‚’ β‚’ ADP cana-1536 122 12 𝑏(α΄‚ 𝑏(α΄‚ PROPN cana-1536 122 13 , , PUNCT cana-1536 122 14 α΄” α΄” NOUN cana-1536 122 15 , , PUNCT cana-1536 122 16 𝔣 𝔣 ADJ cana-1536 122 17 ) ) PUNCT cana-1536 122 18 = = SYM cana-1536 122 19 1 1 NUM cana-1536 122 20 9 9 NUM cana-1536 122 21 { { PUNCT cana-1536 122 22 |α΄‚ |α΄‚ NOUN cana-1536 122 23 βˆ’ βˆ’ PROPN cana-1536 122 24 α΄”| α΄”| PROPN cana-1536 122 25 + + CCONJ cana-1536 122 26 |α΄” |α΄” ADJ cana-1536 122 27 βˆ’ βˆ’ PROPN cana-1536 122 28 𝔣| 𝔣| NOUN cana-1536 122 29 + + NUM cana-1536 122 30 |α΄‚ |α΄‚ NOUN cana-1536 122 31 βˆ’ βˆ’ NOUN cana-1536 122 32 𝔣|}2 𝔣|}2 PUNCT cana-1536 122 33 for for ADP cana-1536 122 34 allα΄‚ allα΄‚ NOUN cana-1536 122 35 , , PUNCT cana-1536 122 36 α΄” α΄” NOUN cana-1536 122 37 , , PUNCT cana-1536 122 38 𝔣 𝔣 PROPN cana-1536 122 39 ∈ ∈ PROPN cana-1536 122 40 α½£. α½£. NOUN cana-1536 122 41 then then ADV cana-1536 122 42 it it PRON cana-1536 122 43 is be AUX cana-1536 122 44 clear clear ADJ cana-1536 122 45 that that SCONJ cana-1536 122 46 ( ( PUNCT cana-1536 122 47 α½£ α½£ NOUN cana-1536 122 48 , , PUNCT cana-1536 122 49 β‚’ β‚’ ADP cana-1536 122 50 𝑏 𝑏 NOUN cana-1536 122 51 ) ) PUNCT cana-1536 122 52 is be AUX cana-1536 122 53 β‚’ β‚’ ADP cana-1536 122 54 𝑏-ms 𝑏-ms NOUN cana-1536 122 55 . . PUNCT cana-1536 123 1 now now ADV cana-1536 123 2 define define VERB cana-1536 123 3 β„‹ β„‹ NOUN cana-1536 123 4 : : PUNCT cana-1536 123 5 α½£3 α½£3 PROPN cana-1536 123 6 β†’ β†’ SYM cana-1536 123 7 α½£ α½£ PRON cana-1536 123 8 byβ„‹(α΄‚ byβ„‹(α΄‚ PROPN cana-1536 123 9 , , PUNCT cana-1536 123 10 α΄” α΄” NOUN cana-1536 123 11 , , PUNCT cana-1536 123 12 𝔣)=1 𝔣)=1 PROPN cana-1536 123 13 βˆ’ βˆ’ PROPN cana-1536 124 1 α΄‚2 α΄‚2 VERB cana-1536 124 2 32 32 NUM cana-1536 124 3 βˆ’ βˆ’ PROPN cana-1536 124 4 3 3 NUM cana-1536 124 5 α΄”2 α΄”2 NOUN cana-1536 124 6 32 32 NUM cana-1536 124 7 βˆ’ βˆ’ NOUN cana-1536 124 8 5 5 NUM cana-1536 124 9 𝔣2 𝔣2 ADP cana-1536 124 10 32 32 NUM cana-1536 124 11 and and CCONJ cana-1536 124 12 β„± β„± PROPN cana-1536 124 13 : : PUNCT cana-1536 124 14 α½£ α½£ X cana-1536 124 15 β†’ β†’ SYM cana-1536 124 16 α½£ α½£ X cana-1536 124 17 by by ADP cana-1536 124 18 β„±(α΄‚ β„±(α΄‚ PRON cana-1536 124 19 ) ) PUNCT cana-1536 124 20 = = PUNCT cana-1536 124 21 α΄‚ α΄‚ X cana-1536 124 22 4 4 NUM cana-1536 124 23 for for ADP cana-1536 124 24 all all DET cana-1536 124 25 α΄‚ α΄‚ PROPN cana-1536 124 26 , , PUNCT cana-1536 124 27 α΄” α΄” NOUN cana-1536 124 28 , , PUNCT cana-1536 124 29 𝔣 𝔣 ADJ cana-1536 124 30 , , PUNCT cana-1536 124 31 Ο° Ο° NOUN cana-1536 124 32 , , PUNCT cana-1536 124 33 β„“ β„“ PROPN cana-1536 124 34 , , PUNCT cana-1536 124 35 β„˜ β„˜ PROPN cana-1536 124 36 ∈ ∈ PROPN cana-1536 124 37 α½£ α½£ NOUN cana-1536 124 38 , , PUNCT cana-1536 124 39 we we PRON cana-1536 124 40 have have VERB cana-1536 124 41 communications communication NOUN cana-1536 124 42 on on ADP cana-1536 124 43 applied apply VERB cana-1536 124 44 nonlinear nonlinear ADJ cana-1536 124 45 analysis analysis NOUN cana-1536 124 46 issn issn NOUN cana-1536 124 47 : : PUNCT cana-1536 124 48 1074 1074 NUM cana-1536 124 49 - - PUNCT cana-1536 124 50 133x 133x NUM cana-1536 124 51 vol vol NOUN cana-1536 124 52 31 31 NUM cana-1536 124 53 no no NOUN cana-1536 124 54 . . PUNCT cana-1536 125 1 8s 8s PROPN cana-1536 125 2 ( ( PUNCT cana-1536 125 3 2024 2024 NUM cana-1536 125 4 ) ) PUNCT cana-1536 125 5 439 439 NUM cana-1536 125 6 https://internationalpubls.com https://internationalpubls.com X cana-1536 125 7 β‚’ β‚’ ADP cana-1536 125 8 𝑏(β„‹(α΄‚ 𝑏(β„‹(α΄‚ PROPN cana-1536 125 9 , , PUNCT cana-1536 125 10 α΄” α΄” NOUN cana-1536 125 11 , , PUNCT cana-1536 125 12 𝔣 𝔣 ADJ cana-1536 125 13 ) ) PUNCT cana-1536 125 14 , , PUNCT cana-1536 125 15 β„‹(Ο° β„‹(Ο° PROPN cana-1536 125 16 , , PUNCT cana-1536 125 17 β„“ β„“ NOUN cana-1536 125 18 , , PUNCT cana-1536 125 19 β„˜ β„˜ PROPN cana-1536 125 20 ) ) PUNCT cana-1536 125 21 , , PUNCT cana-1536 125 22 β„‹(Ο° β„‹(Ο° PROPN cana-1536 125 23 , , PUNCT cana-1536 125 24 β„“ β„“ NOUN cana-1536 125 25 , , PUNCT cana-1536 125 26 β„˜ β„˜ PROPN cana-1536 125 27 ) ) PUNCT cana-1536 125 28 ) ) PUNCT cana-1536 126 1 + + CCONJ cana-1536 126 2 β‚’ β‚’ ADP cana-1536 126 3 𝑏(β„‹ 𝑏(β„‹ PROPN cana-1536 126 4 ( ( PUNCT cana-1536 126 5 α΄” α΄” NOUN cana-1536 126 6 , , PUNCT cana-1536 126 7 𝔣 𝔣 ADJ cana-1536 126 8 , , PUNCT cana-1536 126 9 α΄‚ α΄‚ NOUN cana-1536 126 10 ) ) PUNCT cana-1536 126 11 , , PUNCT cana-1536 126 12 β„‹(β„“ β„‹(β„“ NUM cana-1536 126 13 , , PUNCT cana-1536 126 14 β„˜ β„˜ PROPN cana-1536 126 15 , , PUNCT cana-1536 126 16 Ο° Ο° NOUN cana-1536 126 17 ) ) PUNCT cana-1536 126 18 , , PUNCT cana-1536 126 19 β„‹(β„“ β„‹(β„“ NUM cana-1536 126 20 , , PUNCT cana-1536 126 21 β„˜ β„˜ NOUN cana-1536 126 22 , , PUNCT cana-1536 126 23 Ο°))+ₒ𝑏(β„‹(𝔣 Ο°))+ₒ𝑏(β„‹(𝔣 NUM cana-1536 126 24 , , PUNCT cana-1536 126 25 α΄‚ α΄‚ NOUN cana-1536 126 26 , , PUNCT cana-1536 126 27 α΄” α΄” NOUN cana-1536 126 28 ) ) PUNCT cana-1536 126 29 , , PUNCT cana-1536 126 30 β„‹(β„˜ β„‹(β„˜ NOUN cana-1536 126 31 , , PUNCT cana-1536 126 32 Ο° Ο° NOUN cana-1536 126 33 , , PUNCT cana-1536 126 34 β„“ β„“ NOUN cana-1536 126 35 ) ) PUNCT cana-1536 126 36 , , PUNCT cana-1536 126 37 β„‹(β„˜ β„‹(β„˜ NOUN cana-1536 126 38 , , PUNCT cana-1536 126 39 Ο° Ο° NOUN cana-1536 126 40 , , PUNCT cana-1536 126 41 β„“ β„“ NOUN cana-1536 126 42 ) ) PUNCT cana-1536 126 43 ) ) PUNCT cana-1536 126 44 = = PUNCT cana-1536 126 45 β‚’ β‚’ ADP cana-1536 126 46 𝑏 𝑏 PROPN cana-1536 126 47 ( ( PUNCT cana-1536 126 48 1 1 NUM cana-1536 126 49 βˆ’ βˆ’ NOUN cana-1536 126 50 α΄‚2 α΄‚2 VERB cana-1536 126 51 32 32 NUM cana-1536 126 52 βˆ’ βˆ’ PROPN cana-1536 126 53 3 3 NUM cana-1536 126 54 α΄”2 α΄”2 NOUN cana-1536 126 55 32 32 NUM cana-1536 126 56 βˆ’ βˆ’ NOUN cana-1536 126 57 5 5 NUM cana-1536 126 58 𝔣2 𝔣2 ADP cana-1536 126 59 32 32 NUM cana-1536 126 60 , , PUNCT cana-1536 126 61 1 1 NUM cana-1536 126 62 βˆ’ βˆ’ PROPN cana-1536 126 63 Ο°2 Ο°2 PROPN cana-1536 126 64 32 32 NUM cana-1536 126 65 βˆ’ βˆ’ NOUN cana-1536 126 66 3 3 NUM cana-1536 126 67 β„“2 β„“2 NOUN cana-1536 126 68 32 32 NUM cana-1536 126 69 βˆ’ βˆ’ PROPN cana-1536 126 70 5 5 NUM cana-1536 126 71 β„˜2 β„˜2 NOUN cana-1536 126 72 32 32 NUM cana-1536 126 73 , , PUNCT cana-1536 126 74 1 1 NUM cana-1536 126 75 βˆ’ βˆ’ PROPN cana-1536 126 76 Ο°2 Ο°2 PROPN cana-1536 126 77 32 32 NUM cana-1536 126 78 βˆ’ βˆ’ NOUN cana-1536 126 79 3 3 NUM cana-1536 126 80 β„“2 β„“2 NOUN cana-1536 126 81 32 32 NUM cana-1536 126 82 βˆ’ βˆ’ PROPN cana-1536 126 83 5 5 NUM cana-1536 126 84 β„˜2 β„˜2 NOUN cana-1536 126 85 32 32 NUM cana-1536 126 86 ) ) PUNCT cana-1536 127 1 + + ADP cana-1536 127 2 β‚’ β‚’ X cana-1536 127 3 𝑏 𝑏 NOUN cana-1536 127 4 ( ( PUNCT cana-1536 127 5 1 1 NUM cana-1536 127 6 βˆ’ βˆ’ PROPN cana-1536 127 7 α΄”2 α΄”2 NOUN cana-1536 127 8 32 32 NUM cana-1536 127 9 βˆ’ βˆ’ NOUN cana-1536 127 10 3 3 NUM cana-1536 127 11 𝔣2 𝔣2 ADP cana-1536 127 12 32 32 NUM cana-1536 127 13 βˆ’ βˆ’ NUM cana-1536 127 14 5 5 NUM cana-1536 127 15 α΄‚2 α΄‚2 VERB cana-1536 127 16 32 32 NUM cana-1536 127 17 , , PUNCT cana-1536 127 18 1 1 NUM cana-1536 127 19 βˆ’ βˆ’ NOUN cana-1536 127 20 β„“2 β„“2 NOUN cana-1536 127 21 32 32 NUM cana-1536 127 22 βˆ’ βˆ’ NOUN cana-1536 127 23 3 3 NUM cana-1536 127 24 β„˜2 β„˜2 PROPN cana-1536 127 25 32 32 NUM cana-1536 127 26 βˆ’ βˆ’ SYM cana-1536 127 27 5 5 NUM cana-1536 127 28 Ο°2 Ο°2 PROPN cana-1536 127 29 32 32 NUM cana-1536 127 30 , , PUNCT cana-1536 127 31 1 1 NUM cana-1536 127 32 βˆ’ βˆ’ NOUN cana-1536 127 33 β„“2 β„“2 NOUN cana-1536 127 34 32 32 NUM cana-1536 127 35 βˆ’ βˆ’ NOUN cana-1536 127 36 3 3 NUM cana-1536 127 37 β„˜2 β„˜2 PROPN cana-1536 127 38 32 32 NUM cana-1536 127 39 βˆ’ βˆ’ SYM cana-1536 127 40 5 5 NUM cana-1536 127 41 Ο°2 Ο°2 PROPN cana-1536 127 42 32 32 NUM cana-1536 127 43 ) ) PUNCT cana-1536 128 1 + + ADP cana-1536 128 2 β‚’ β‚’ X cana-1536 128 3 𝑏 𝑏 PRON cana-1536 128 4 ( ( PUNCT cana-1536 128 5 1 1 NUM cana-1536 128 6 βˆ’ βˆ’ NOUN cana-1536 128 7 𝔣2 𝔣2 ADP cana-1536 128 8 32 32 NUM cana-1536 128 9 βˆ’ βˆ’ SYM cana-1536 128 10 3 3 NUM cana-1536 128 11 α΄‚2 α΄‚2 VERB cana-1536 128 12 32 32 NUM cana-1536 128 13 βˆ’ βˆ’ PROPN cana-1536 128 14 5 5 NUM cana-1536 128 15 α΄”2 α΄”2 NOUN cana-1536 128 16 32 32 NUM cana-1536 128 17 , , PUNCT cana-1536 128 18 1 1 NUM cana-1536 128 19 βˆ’ βˆ’ NOUN cana-1536 128 20 β„˜2 β„˜2 NOUN cana-1536 128 21 32 32 NUM cana-1536 128 22 βˆ’ βˆ’ PROPN cana-1536 128 23 3 3 NUM cana-1536 128 24 Ο°2 Ο°2 PROPN cana-1536 128 25 32 32 NUM cana-1536 128 26 βˆ’ βˆ’ SYM cana-1536 128 27 5 5 NUM cana-1536 128 28 β„“2 β„“2 PROPN cana-1536 128 29 32 32 NUM cana-1536 128 30 , , PUNCT cana-1536 128 31 1 1 NUM cana-1536 128 32 βˆ’ βˆ’ NOUN cana-1536 128 33 β„˜2 β„˜2 NOUN cana-1536 128 34 32 32 NUM cana-1536 128 35 βˆ’ βˆ’ PROPN cana-1536 128 36 3 3 NUM cana-1536 128 37 Ο°2 Ο°2 PROPN cana-1536 128 38 32 32 NUM cana-1536 128 39 βˆ’ βˆ’ SYM cana-1536 128 40 5 5 NUM cana-1536 128 41 β„“2 β„“2 PROPN cana-1536 128 42 32 32 NUM cana-1536 128 43 ) ) PUNCT cana-1536 128 44 ≀ ≀ NUM cana-1536 128 45 1 1 NUM cana-1536 128 46 8 8 NUM cana-1536 128 47 { { PUNCT cana-1536 128 48 |α΄‚2 |α΄‚2 NOUN cana-1536 128 49 βˆ’ βˆ’ NOUN cana-1536 128 50 Ο°2|2+|α΄”2 Ο°2|2+|α΄”2 NOUN cana-1536 128 51 βˆ’ βˆ’ PROPN cana-1536 128 52 β„“2|2 β„“2|2 X cana-1536 128 53 + + CCONJ cana-1536 128 54 |𝔣2 |𝔣2 PROPN cana-1536 128 55 βˆ’ βˆ’ PROPN cana-1536 128 56 β„˜2|2 β„˜2|2 PROPN cana-1536 128 57 } } PUNCT cana-1536 128 58 ≀ ≀ NUM cana-1536 128 59 1 1 NUM cana-1536 128 60 8 8 NUM cana-1536 128 61 { { PUNCT cana-1536 128 62 |α΄‚ |α΄‚ NOUN cana-1536 128 63 βˆ’ βˆ’ PROPN cana-1536 128 64 Ο°| Ο°| PROPN cana-1536 128 65 + + CCONJ cana-1536 128 66 |α΄” |α΄” PROPN cana-1536 128 67 βˆ’ βˆ’ PROPN cana-1536 128 68 β„“| β„“| PROPN cana-1536 128 69 + + CCONJ cana-1536 128 70 |𝔣 |𝔣 NOUN cana-1536 128 71 βˆ’ βˆ’ PROPN cana-1536 128 72 β„˜|}2 β„˜|}2 PROPN cana-1536 128 73 . . PROPN cana-1536 129 1 and and CCONJ cana-1536 129 2 if if SCONJ cana-1536 129 3 we we PRON cana-1536 129 4 consider consider VERB cana-1536 129 5 , , PUNCT cana-1536 129 6 β‚’ β‚’ ADP cana-1536 129 7 𝑏(α΄‚ 𝑏(α΄‚ PROPN cana-1536 129 8 , , PUNCT cana-1536 129 9 Ο° Ο° NOUN cana-1536 129 10 , , PUNCT cana-1536 129 11 Ο° Ο° NOUN cana-1536 129 12 ) ) PUNCT cana-1536 129 13 + + CCONJ cana-1536 129 14 β‚’ β‚’ ADP cana-1536 129 15 𝑏(α΄” 𝑏(α΄” PROPN cana-1536 129 16 , , PUNCT cana-1536 129 17 β„“ β„“ PROPN cana-1536 129 18 , , PUNCT cana-1536 129 19 β„“ β„“ NOUN cana-1536 129 20 ) ) PUNCT cana-1536 129 21 + + CCONJ cana-1536 129 22 β‚’ β‚’ ADP cana-1536 129 23 𝑏(𝔣 𝑏(𝔣 PROPN cana-1536 129 24 , , PUNCT cana-1536 129 25 β„˜ β„˜ NOUN cana-1536 129 26 , , PUNCT cana-1536 129 27 β„˜ β„˜ NUM cana-1536 129 28 ) ) PUNCT cana-1536 129 29 = = SYM cana-1536 129 30 2 2 NUM cana-1536 129 31 9 9 NUM cana-1536 129 32 { { PUNCT cana-1536 129 33 |α΄‚ |α΄‚ NOUN cana-1536 129 34 βˆ’ βˆ’ PROPN cana-1536 129 35 Ο°|2 Ο°|2 PROPN cana-1536 129 36 + + CCONJ cana-1536 129 37 |α΄” |α΄” ADJ cana-1536 129 38 βˆ’ βˆ’ PROPN cana-1536 129 39 β„“|2 β„“|2 NOUN cana-1536 129 40 + + X cana-1536 129 41 |βˆ’β„˜|2 |βˆ’β„˜|2 ADJ cana-1536 129 42 } } PUNCT cana-1536 129 43 = = SYM cana-1536 129 44 2 2 NUM cana-1536 129 45 9 9 NUM cana-1536 129 46 [ [ X cana-1536 129 47 |α΄‚ |α΄‚ NOUN cana-1536 130 1 βˆ’ βˆ’ PROPN cana-1536 130 2 Ο°| Ο°| PROPN cana-1536 130 3 + + CCONJ cana-1536 130 4 |α΄” |α΄” PROPN cana-1536 130 5 βˆ’ βˆ’ PROPN cana-1536 130 6 β„“| β„“| PROPN cana-1536 130 7 + + CCONJ cana-1536 130 8 |𝔣 |𝔣 NOUN cana-1536 130 9 βˆ’ βˆ’ PROPN cana-1536 130 10 β„˜|]2 β„˜|]2 NOUN cana-1536 130 11 . . PUNCT cana-1536 131 1 from from ADP cana-1536 131 2 we we PRON cana-1536 131 3 conclude conclude VERB cana-1536 131 4 that that SCONJ cana-1536 131 5 β‚’ β‚’ ADP cana-1536 131 6 𝑏(β„‹(α΄‚ 𝑏(β„‹(α΄‚ PROPN cana-1536 131 7 , , PUNCT cana-1536 131 8 α΄” α΄” NOUN cana-1536 131 9 , , PUNCT cana-1536 131 10 𝔣 𝔣 ADJ cana-1536 131 11 ) ) PUNCT cana-1536 131 12 , , PUNCT cana-1536 131 13 β„‹(Ο° β„‹(Ο° PROPN cana-1536 131 14 , , PUNCT cana-1536 131 15 β„“ β„“ NOUN cana-1536 131 16 , , PUNCT cana-1536 131 17 β„˜ β„˜ PROPN cana-1536 131 18 ) ) PUNCT cana-1536 131 19 , , PUNCT cana-1536 131 20 β„‹(Ο° β„‹(Ο° PROPN cana-1536 131 21 , , PUNCT cana-1536 131 22 β„“ β„“ NOUN cana-1536 131 23 , , PUNCT cana-1536 131 24 β„˜ β„˜ PROPN cana-1536 131 25 ) ) PUNCT cana-1536 131 26 ) ) PUNCT cana-1536 132 1 + + CCONJ cana-1536 132 2 β‚’ β‚’ ADP cana-1536 132 3 𝑏(β„‹ 𝑏(β„‹ PROPN cana-1536 132 4 ( ( PUNCT cana-1536 132 5 α΄” α΄” NOUN cana-1536 132 6 , , PUNCT cana-1536 132 7 𝔣 𝔣 ADJ cana-1536 132 8 , , PUNCT cana-1536 132 9 α΄‚ α΄‚ NOUN cana-1536 132 10 ) ) PUNCT cana-1536 132 11 , , PUNCT cana-1536 132 12 β„‹(β„“ β„‹(β„“ NUM cana-1536 132 13 , , PUNCT cana-1536 132 14 β„˜ β„˜ PROPN cana-1536 132 15 , , PUNCT cana-1536 132 16 Ο° Ο° NOUN cana-1536 132 17 ) ) PUNCT cana-1536 132 18 , , PUNCT cana-1536 132 19 β„‹(β„“ β„‹(β„“ NUM cana-1536 132 20 , , PUNCT cana-1536 132 21 β„˜ β„˜ NOUN cana-1536 132 22 , , PUNCT cana-1536 132 23 Ο°))+ₒ𝑏(β„‹(𝔣 Ο°))+ₒ𝑏(β„‹(𝔣 NUM cana-1536 132 24 , , PUNCT cana-1536 132 25 α΄‚ α΄‚ NOUN cana-1536 132 26 , , PUNCT cana-1536 132 27 α΄” α΄” NOUN cana-1536 132 28 ) ) PUNCT cana-1536 132 29 , , PUNCT cana-1536 132 30 β„‹(β„˜ β„‹(β„˜ NOUN cana-1536 132 31 , , PUNCT cana-1536 132 32 Ο° Ο° NOUN cana-1536 132 33 , , PUNCT cana-1536 132 34 β„“ β„“ NOUN cana-1536 132 35 ) ) PUNCT cana-1536 132 36 , , PUNCT cana-1536 132 37 β„‹(β„˜ β„‹(β„˜ NOUN cana-1536 132 38 , , PUNCT cana-1536 132 39 Ο° Ο° NOUN cana-1536 132 40 , , PUNCT cana-1536 132 41 β„“ β„“ NOUN cana-1536 132 42 ) ) PUNCT cana-1536 132 43 ) ) PUNCT cana-1536 132 44 ≀ ≀ PROPN cana-1536 132 45 πœƒ[ₒ𝑏(α΄‚ πœƒ[ₒ𝑏(α΄‚ NOUN cana-1536 132 46 , , PUNCT cana-1536 132 47 Ο° Ο° NOUN cana-1536 132 48 , , PUNCT cana-1536 132 49 Ο° Ο° NOUN cana-1536 132 50 ) ) PUNCT cana-1536 132 51 + + CCONJ cana-1536 132 52 β‚’ β‚’ ADP cana-1536 132 53 𝑏(α΄” 𝑏(α΄” PROPN cana-1536 132 54 , , PUNCT cana-1536 132 55 β„“ β„“ PROPN cana-1536 132 56 , , PUNCT cana-1536 132 57 β„“ β„“ NOUN cana-1536 132 58 ) ) PUNCT cana-1536 132 59 + + CCONJ cana-1536 132 60 β‚’ β‚’ ADP cana-1536 132 61 𝑏(𝔣 𝑏(𝔣 PROPN cana-1536 132 62 , , PUNCT cana-1536 132 63 β„˜ β„˜ NOUN cana-1536 132 64 , , PUNCT cana-1536 132 65 β„˜ β„˜ PROPN cana-1536 132 66 ) ) PUNCT cana-1536 132 67 ] ] PUNCT cana-1536 132 68 . . PUNCT cana-1536 133 1 then then ADV cana-1536 133 2 from from ADP cana-1536 133 3 corollary corollary ADJ cana-1536 133 4 , , PUNCT cana-1536 133 5 we we PRON cana-1536 133 6 can can AUX cana-1536 133 7 conclude conclude VERB cana-1536 133 8 ( ( PUNCT cana-1536 133 9 0 0 NUM cana-1536 133 10 , , PUNCT cana-1536 133 11 0 0 NUM cana-1536 133 12 , , PUNCT cana-1536 133 13 0 0 NUM cana-1536 133 14 ) ) PUNCT cana-1536 133 15 unique unique ADJ cana-1536 133 16 tfp tfp NOUN cana-1536 133 17 of of ADP cana-1536 133 18 β„‹ β„‹ PROPN cana-1536 133 19 . . PUNCT cana-1536 134 1 4 4 X cana-1536 134 2 . . X cana-1536 134 3 conclusion conclusion NOUN cana-1536 134 4 in in ADP cana-1536 134 5 this this DET cana-1536 134 6 work work NOUN cana-1536 134 7 we we PRON cana-1536 134 8 have have AUX cana-1536 134 9 obtained obtain VERB cana-1536 134 10 tfp tfp PROPN cana-1536 134 11 results result NOUN cana-1536 134 12 by by ADP cana-1536 134 13 using use VERB cana-1536 134 14 a a DET cana-1536 134 15 new new ADJ cana-1536 134 16 type type NOUN cana-1536 134 17 of of ADP cana-1536 134 18 contraction contraction NOUN cana-1536 134 19 and and CCONJ cana-1536 134 20 discussed discuss VERB cana-1536 134 21 some some DET cana-1536 134 22 corollary corollary NOUN cana-1536 134 23 also also ADV cana-1536 134 24 an an DET cana-1536 134 25 example example NOUN cana-1536 134 26 which which PRON cana-1536 134 27 supports support VERB cana-1536 134 28 the the DET cana-1536 134 29 main main ADJ cana-1536 134 30 result result NOUN cana-1536 134 31 . . PUNCT cana-1536 135 1 refrences refrence VERB cana-1536 135 2 [ [ X cana-1536 135 3 1 1 X cana-1536 135 4 ] ] X cana-1536 135 5 d. d. PROPN cana-1536 135 6 srilatha srilatha PROPN cana-1536 135 7 , , PUNCT cana-1536 135 8 v. v. PROPN cana-1536 135 9 kiran kiran PROPN cana-1536 135 10 . . PUNCT cana-1536 136 1 ( ( PUNCT cana-1536 136 2 2023 2023 NUM cana-1536 136 3 ) ) PUNCT cana-1536 136 4 . . PUNCT cana-1536 137 1 a a DET cana-1536 137 2 study study NOUN cana-1536 137 3 on on ADP cana-1536 137 4 tripled triple VERB cana-1536 137 5 fixed fix VERB cana-1536 137 6 point point NOUN cana-1536 137 7 results result NOUN cana-1536 137 8 in in ADP cana-1536 137 9 gjs gjs NOUN cana-1536 137 10 metric metric ADJ cana-1536 137 11 space space NOUN cana-1536 137 12 . . PUNCT cana-1536 138 1 mathematics mathematic NOUN cana-1536 138 2 and and CCONJ cana-1536 138 3 statistics statistic NOUN cana-1536 138 4 11(5 11(5 NUM cana-1536 138 5 ): ): PUNCT cana-1536 138 6 767 767 NUM cana-1536 138 7 - - SYM cana-1536 138 8 777 777 NUM cana-1536 138 9 . . PUNCT cana-1536 139 1 https://doi.org/10.13189/ms.2023.110502 https://doi.org/10.13189/ms.2023.110502 X cana-1536 140 1 [ [ X cana-1536 140 2 2 2 X cana-1536 140 3 ] ] PUNCT cana-1536 140 4 jitender jitender PROPN cana-1536 140 5 kumar kumar PROPN cana-1536 140 6 , , PUNCT cana-1536 140 7 sachin sachin PROPN cana-1536 140 8 vashistha.(2023).coupled vashistha.(2023).couple VERB cana-1536 140 9 fixed fix VERB cana-1536 140 10 point point NOUN cana-1536 140 11 theorems theorem NOUN cana-1536 140 12 for for ADP cana-1536 140 13 two two NUM cana-1536 140 14 maps map NOUN cana-1536 140 15 in in ADP cana-1536 140 16 complex complex ADJ cana-1536 140 17 valued value VERB cana-1536 140 18 𝐺𝑏metric 𝐺𝑏metric PROPN cana-1536 140 19 spaces space NOUN cana-1536 140 20 , , PUNCT cana-1536 140 21 research research NOUN cana-1536 140 22 and and CCONJ cana-1536 140 23 applications application NOUN cana-1536 140 24 towards towards ADP cana-1536 140 25 mathematics mathematic NOUN cana-1536 140 26 and and CCONJ cana-1536 140 27 computer computer NOUN cana-1536 140 28 science.1 science.1 NOUN cana-1536 140 29 : : PUNCT cana-1536 140 30 54 54 NUM cana-1536 140 31 - - SYM cana-1536 140 32 65 65 NUM cana-1536 140 33 . . PUNCT cana-1536 141 1 https://doi.org/10.9734/bpi/ratmcs/v1/18818d https://doi.org/10.9734/bpi/ratmcs/v1/18818d PROPN cana-1536 142 1 [ [ X cana-1536 142 2 3 3 NUM cana-1536 142 3 ] ] X cana-1536 142 4 v. v. CCONJ cana-1536 142 5 gupta gupta PROPN cana-1536 142 6 , , PUNCT cana-1536 142 7 o. o. PROPN cana-1536 142 8 ege ege PROPN cana-1536 142 9 , , PUNCT cana-1536 142 10 r. r. PROPN cana-1536 142 11 saini saini PROPN cana-1536 142 12 . . PUNCT cana-1536 143 1 ( ( PUNCT cana-1536 143 2 2022 2022 NUM cana-1536 143 3 ) ) PUNCT cana-1536 143 4 . . PUNCT cana-1536 144 1 extended extend VERB cana-1536 144 2 gb gb ADV cana-1536 144 3 - - PUNCT cana-1536 144 4 metric metric ADJ cana-1536 144 5 spaces space NOUN cana-1536 144 6 and and CCONJ cana-1536 144 7 some some DET cana-1536 144 8 fixed fix VERB cana-1536 144 9 pointresults pointresult NOUN cana-1536 144 10 with with ADP cana-1536 144 11 an an DET cana-1536 144 12 application application NOUN cana-1536 144 13 to to PART cana-1536 144 14 fredholm fredholm VERB cana-1536 144 15 integral integral ADJ cana-1536 144 16 equation equation NOUN cana-1536 144 17 . . PUNCT cana-1536 145 1 electronic electronic ADJ cana-1536 145 2 journal journal NOUN cana-1536 145 3 of of ADP cana-1536 145 4 mathematical mathematical ADJ cana-1536 145 5 analysis analysis NOUN cana-1536 145 6 and and CCONJ cana-1536 145 7 applications application NOUN cana-1536 145 8 . . PUNCT cana-1536 146 1 10(2):219 10(2):219 NUM cana-1536 146 2 - - SYM cana-1536 146 3 227 227 NUM cana-1536 146 4 . . PUNCT cana-1536 147 1 https://ejmaa.journals.ekb.eg/article_292053_e4bb5b4724b1f42723774131e5165b9f.pdf https://ejmaa.journals.ekb.eg/article_292053_e4bb5b4724b1f42723774131e5165b9f.pdf ADJ cana-1536 147 2 [ [ X cana-1536 147 3 4 4 NUM cana-1536 147 4 ] ] X cana-1536 147 5 ΓΆ. ΓΆ. NOUN cana-1536 147 6 acar acar PROPN cana-1536 147 7 , , PUNCT cana-1536 147 8 e. e. PROPN cana-1536 147 9 erdoğan erdoğan PROPN cana-1536 147 10 , , PUNCT cana-1536 147 11 a.s a.s PROPN cana-1536 147 12 . . PROPN cana-1536 147 13 ΓΆzkapu ΓΆzkapu PROPN cana-1536 147 14 . . PUNCT cana-1536 148 1 ( ( PUNCT cana-1536 148 2 2022 2022 NUM cana-1536 148 3 ) ) PUNCT cana-1536 148 4 . . PUNCT cana-1536 149 1 generalized generalize VERB cana-1536 149 2 integral integral ADJ cana-1536 149 3 type type NOUN cana-1536 149 4 mappings mapping NOUN cana-1536 149 5 on on ADP cana-1536 149 6 orthogonal orthogonal ADJ cana-1536 149 7 metric metric ADJ cana-1536 149 8 spaces space NOUN cana-1536 149 9 , , PUNCT cana-1536 149 10 carpathian carpathian ADJ cana-1536 149 11 mathematical mathematical ADJ cana-1536 149 12 publications publication NOUN cana-1536 149 13 . . PUNCT cana-1536 150 1 14 14 NUM cana-1536 150 2 ( ( PUNCT cana-1536 150 3 2 2 NUM cana-1536 150 4 ): ): PUNCT cana-1536 150 5 485 485 NUM cana-1536 150 6 – – PUNCT cana-1536 150 7 492 492 NUM cana-1536 150 8 . . PUNCT cana-1536 151 1 https://doi.org/10.15330/cmp.14.2.485-492 https://doi.org/10.15330/cmp.14.2.485-492 X cana-1536 152 1 [ [ X cana-1536 152 2 5 5 X cana-1536 152 3 ] ] PUNCT cana-1536 152 4 b. b. PROPN cana-1536 152 5 s. s. PROPN cana-1536 152 6 rao rao PROPN cana-1536 152 7 , , PUNCT cana-1536 152 8 l. l. PROPN cana-1536 152 9 v. v. PROPN cana-1536 152 10 kumar kumar PROPN cana-1536 152 11 , , PUNCT cana-1536 152 12 c. c. PROPN cana-1536 152 13 maheswari maheswari PROPN cana-1536 152 14 , , PUNCT cana-1536 152 15 s. s. PROPN cana-1536 152 16 m. m. PROPN cana-1536 152 17 shaw shaw PROPN cana-1536 152 18 . . PUNCT cana-1536 153 1 ( ( PUNCT cana-1536 153 2 2022).on 2022).on NUM cana-1536 153 3 certain certain ADJ cana-1536 153 4 quadruple quadruple ADJ cana-1536 153 5 fixed fix VERB cana-1536 153 6 points point NOUN cana-1536 153 7 of of ADP cana-1536 153 8 integral integral ADJ cana-1536 153 9 type type NOUN cana-1536 153 10 contraction contraction NOUN cana-1536 153 11 mappings mapping NOUN cana-1536 153 12 in in ADP cana-1536 153 13 g g NOUN cana-1536 153 14 - - PUNCT cana-1536 153 15 metric metric ADJ cana-1536 153 16 spaces space NOUN cana-1536 153 17 and and CCONJ cana-1536 153 18 its its PRON cana-1536 153 19 applications application NOUN cana-1536 153 20 , , PUNCT cana-1536 153 21 journal journal NOUN cana-1536 153 22 of of ADP cana-1536 153 23 scientific scientific ADJ cana-1536 153 24 research.14(2 research.14(2 NOUN cana-1536 153 25 ): ): PUNCT cana-1536 153 26 501 501 NUM cana-1536 153 27 - - SYM cana-1536 153 28 512 512 NUM cana-1536 153 29 . . PUNCT cana-1536 154 1 https://doi.org/10.3329/jsr.v14i2.56189 https://doi.org/10.3329/jsr.v14i2.56189 NOUN cana-1536 154 2 https://doi.org/10.13189/ms.2023.110502 https://doi.org/10.13189/ms.2023.110502 NOUN cana-1536 154 3 https://stm.bookpi.org/ratmcs-v1/article/view/10688 https://stm.bookpi.org/ratmcs-v1/article/view/10688 PROPN cana-1536 154 4 https://stm.bookpi.org/ratmcs-v1/article/view/10688 https://stm.bookpi.org/ratmcs-v1/article/view/10688 NOUN cana-1536 154 5 https://doi.org/10.9734/bpi/ratmcs/v1/18818d https://doi.org/10.9734/bpi/ratmcs/v1/18818d PROPN cana-1536 154 6 https://ejmaa.journals.ekb.eg/article_292053_e4bb5b4724b1f42723774131e5165b9f.pdf https://ejmaa.journals.ekb.eg/article_292053_e4bb5b4724b1f42723774131e5165b9f.pdf ADJ cana-1536 154 7 https://doi.org/10.15330/cmp.14.2.485-492 https://doi.org/10.15330/cmp.14.2.485-492 PROPN cana-1536 154 8 https://doi.org/10.3329/jsr.v14i2.56189 https://doi.org/10.3329/jsr.v14i2.56189 PROPN cana-1536 154 9 communications communication NOUN cana-1536 154 10 on on ADP cana-1536 154 11 applied apply VERB cana-1536 154 12 nonlinear nonlinear ADJ cana-1536 154 13 analysis analysis NOUN cana-1536 154 14 issn issn NOUN cana-1536 154 15 : : PUNCT cana-1536 154 16 1074 1074 NUM cana-1536 154 17 - - PUNCT cana-1536 154 18 133x 133x NUM cana-1536 154 19 vol vol NOUN cana-1536 154 20 31 31 NUM cana-1536 154 21 no no NOUN cana-1536 154 22 . . PUNCT cana-1536 155 1 8s 8s PROPN cana-1536 155 2 ( ( PUNCT cana-1536 155 3 2024 2024 NUM cana-1536 155 4 ) ) PUNCT cana-1536 155 5 440 440 NUM cana-1536 155 6 https://internationalpubls.com https://internationalpubls.com X cana-1536 156 1 [ [ X cana-1536 156 2 6 6 NUM cana-1536 156 3 ] ] X cana-1536 156 4 gupta gupta PROPN cana-1536 156 5 , , PUNCT cana-1536 156 6 dr dr PROPN cana-1536 156 7 . . PROPN cana-1536 156 8 vishal vishal PROPN cana-1536 156 9 & & CCONJ cana-1536 156 10 ege ege PROPN cana-1536 156 11 , , PUNCT cana-1536 156 12 ozgur ozgur PROPN cana-1536 156 13 & & CCONJ cana-1536 156 14 saini saini PROPN cana-1536 156 15 , , PUNCT cana-1536 156 16 rajani rajani PROPN cana-1536 156 17 & & CCONJ cana-1536 156 18 de de PROPN cana-1536 156 19 la la PROPN cana-1536 156 20 sen sen PROPN cana-1536 156 21 , , PUNCT cana-1536 156 22 manuel manuel NOUN cana-1536 156 23 . . PUNCT cana-1536 157 1 ( ( PUNCT cana-1536 157 2 2021 2021 NUM cana-1536 157 3 ) ) PUNCT cana-1536 157 4 . . PUNCT cana-1536 158 1 various various ADJ cana-1536 158 2 fixed fix VERB cana-1536 158 3 point point NOUN cana-1536 158 4 results result NOUN cana-1536 158 5 in in ADP cana-1536 158 6 complete complete ADJ cana-1536 158 7 gb gb ADV cana-1536 158 8 - - PUNCT cana-1536 158 9 metric metric ADJ cana-1536 158 10 spaces space NOUN cana-1536 158 11 . . PUNCT cana-1536 159 1 dynamic dynamic ADJ cana-1536 159 2 systems system NOUN cana-1536 159 3 and and CCONJ cana-1536 159 4 applications application NOUN cana-1536 159 5 . . PUNCT cana-1536 160 1 30 30 NUM cana-1536 160 2 . . X cana-1536 161 1 277 277 NUM cana-1536 161 2 - - SYM cana-1536 161 3 293 293 NUM cana-1536 161 4 . . PUNCT cana-1536 162 1 https://doi.org/10.46719/dsa20213028 https://doi.org/10.46719/dsa20213028 NOUN cana-1536 163 1 [ [ X cana-1536 163 2 7 7 X cana-1536 163 3 ] ] X cana-1536 163 4 v. v. CCONJ cana-1536 163 5 gupta gupta PROPN cana-1536 163 6 , , PUNCT cana-1536 163 7 o. o. PROPN cana-1536 163 8 ege ege PROPN cana-1536 163 9 , , PUNCT cana-1536 163 10 r. r. PROPN cana-1536 163 11 saini saini PROPN cana-1536 163 12 and and CCONJ cana-1536 163 13 m. m. PROPN cana-1536 163 14 de de PROPN cana-1536 163 15 la la PROPN cana-1536 163 16 sen sen PROPN cana-1536 163 17 . . PROPN cana-1536 163 18 ( ( PUNCT cana-1536 163 19 2021 2021 NUM cana-1536 163 20 ) ) PUNCT cana-1536 163 21 . . PUNCT cana-1536 164 1 various various ADJ cana-1536 164 2 fixed fix VERB cana-1536 164 3 point point NOUN cana-1536 164 4 results result NOUN cana-1536 164 5 in in ADP cana-1536 164 6 complete complete ADJ cana-1536 164 7 gb gb ADV cana-1536 164 8 - - PUNCT cana-1536 164 9 metric metric ADJ cana-1536 164 10 spaces space NOUN cana-1536 164 11 , , PUNCT cana-1536 164 12 dynamic dynamic ADJ cana-1536 164 13 systems system NOUN cana-1536 164 14 and and CCONJ cana-1536 164 15 applications application NOUN cana-1536 164 16 . . PUNCT cana-1536 165 1 30 30 NUM cana-1536 165 2 , , PUNCT cana-1536 165 3 2 2 NUM cana-1536 165 4 , , PUNCT cana-1536 165 5 277 277 NUM cana-1536 165 6 - - SYM cana-1536 165 7 293 293 NUM cana-1536 165 8 . . PUNCT cana-1536 166 1 doi:10.46719 doi:10.46719 PROPN cana-1536 166 2 / / SYM cana-1536 166 3 dsa20213028 dsa20213028 NOUN cana-1536 167 1 [ [ X cana-1536 167 2 8 8 NUM cana-1536 167 3 ] ] SYM cana-1536 167 4 mebawondu mebawondu NOUN cana-1536 167 5 aa aa PROPN cana-1536 167 6 , , PUNCT cana-1536 167 7 mewomo mewomo PROPN cana-1536 167 8 ot ot INTJ cana-1536 167 9 . . PUNCT cana-1536 167 10 ( ( PUNCT cana-1536 167 11 2021 2021 NUM cana-1536 167 12 ) ) PUNCT cana-1536 167 13 . . PUNCT cana-1536 168 1 suzuki suzuki NOUN cana-1536 168 2 - - PUNCT cana-1536 168 3 type type NOUN cana-1536 168 4 fixed fix VERB cana-1536 168 5 point point NOUN cana-1536 168 6 results result NOUN cana-1536 168 7 in in ADP cana-1536 168 8 gb gb ADV cana-1536 168 9 - - PUNCT cana-1536 168 10 metric metric ADJ cana-1536 168 11 spaces space NOUN cana-1536 168 12 . . PUNCT cana-1536 169 1 asian asian ADJ cana-1536 169 2 - - PUNCT cana-1536 169 3 european european ADJ cana-1536 169 4 journal journal NOUN cana-1536 169 5 of of ADP cana-1536 169 6 mathematics mathematic NOUN cana-1536 169 7 . . PUNCT cana-1536 170 1 14 14 NUM cana-1536 170 2 ( ( PUNCT cana-1536 170 3 05 05 NUM cana-1536 170 4 ) ) PUNCT cana-1536 170 5 . . PUNCT cana-1536 171 1 https://dx.doi.org/10.1142/s1793557121500704 https://dx.doi.org/10.1142/s1793557121500704 PROPN cana-1536 172 1 [ [ X cana-1536 172 2 9 9 NUM cana-1536 172 3 ] ] X cana-1536 172 4 aziz aziz PROPN cana-1536 172 5 khan khan PROPN cana-1536 172 6 , , PUNCT cana-1536 172 7 thabet thabet ADJ cana-1536 172 8 abdeljawad abdeljawad NOUN cana-1536 172 9 , , PUNCT cana-1536 172 10 wasfi wasfi NOUN cana-1536 172 11 shatanawi shatanawi PROPN cana-1536 172 12 , , PUNCT cana-1536 172 13 hasib hasib PROPN cana-1536 172 14 khan khan PROPN cana-1536 172 15 . . PUNCT cana-1536 173 1 ( ( PUNCT cana-1536 173 2 2020 2020 NUM cana-1536 173 3 ) ) PUNCT cana-1536 173 4 . . PUNCT cana-1536 174 1 fixed fix VERB cana-1536 174 2 point point NOUN cana-1536 174 3 theorems theorem NOUN cana-1536 174 4 for for ADP cana-1536 174 5 quadruple quadruple NOUN cana-1536 174 6 selfmappings selfmapping NOUN cana-1536 174 7 satisfying satisfy VERB cana-1536 174 8 integral integral ADJ cana-1536 174 9 type type NOUN cana-1536 174 10 inequalities inequality NOUN cana-1536 174 11 , , PUNCT cana-1536 174 12 filomat.34(3):905 filomat.34(3):905 NOUN cana-1536 174 13 - - PUNCT cana-1536 174 14 917 917 NUM cana-1536 174 15 . . PUNCT cana-1536 175 1 https://doi.org/10.2298/fil2003905k https://doi.org/10.2298/fil2003905k NOUN cana-1536 175 2 . . PUNCT cana-1536 176 1 [ [ X cana-1536 176 2 10 10 NUM cana-1536 176 3 ] ] X cana-1536 176 4 e. e. PROPN cana-1536 176 5 l. l. PROPN cana-1536 176 6 ghasaba ghasaba PROPN cana-1536 176 7 , , PUNCT cana-1536 176 8 h. h. PROPN cana-1536 176 9 majania majania PROPN cana-1536 176 10 , , PUNCT cana-1536 176 11 g. g. PROPN cana-1536 176 12 soleimani soleimani PROPN cana-1536 176 13 rada rada PROPN cana-1536 176 14 . . PUNCT cana-1536 177 1 ( ( PUNCT cana-1536 177 2 2020).integral 2020).integral NUM cana-1536 177 3 type type NOUN cana-1536 177 4 contraction contraction NOUN cana-1536 177 5 and and CCONJ cana-1536 177 6 coupled couple VERB cana-1536 177 7 fixed fix VERB cana-1536 177 8 point point NOUN cana-1536 177 9 theorems theorem NOUN cana-1536 177 10 in in ADP cana-1536 177 11 ordered order VERB cana-1536 177 12 g g NOUN cana-1536 177 13 - - PUNCT cana-1536 177 14 metric metric ADJ cana-1536 177 15 spaces space NOUN cana-1536 177 16 , , PUNCT cana-1536 177 17 journal journal NOUN cana-1536 177 18 of of ADP cana-1536 177 19 linear linear PROPN cana-1536 177 20 and and CCONJ cana-1536 177 21 topological topological PROPN cana-1536 177 22 algebra.9(2):113120 algebra.9(2):113120 PROPN cana-1536 177 23 . . PUNCT cana-1536 177 24 https://sanad.iau.ir/journal/jlta/article/673785?jid=673785 https://sanad.iau.ir/journal/jlta/article/673785?jid=673785 PROPN cana-1536 178 1 [ [ X cana-1536 178 2 11 11 NUM cana-1536 178 3 ] ] PUNCT cana-1536 178 4 a. a. NOUN cana-1536 178 5 hassen hassen PROPN cana-1536 178 6 , , PUNCT cana-1536 178 7 r. r. PROPN cana-1536 178 8 dusan dusan PROPN cana-1536 178 9 , , PUNCT cana-1536 178 10 a. a. NOUN cana-1536 178 11 aghajani aghajani PROPN cana-1536 178 12 , , PUNCT cana-1536 178 13 t. t. PROPN cana-1536 178 14 dosenovic dosenovic ADJ cana-1536 178 15 , , PUNCT cana-1536 178 16 m. m. NOUN cana-1536 178 17 salmi salmi PROPN cana-1536 178 18 , , PUNCT cana-1536 178 19 m. m. NOUN cana-1536 178 20 nooraniand nooraniand NOUN cana-1536 178 21 and and CCONJ cana-1536 178 22 h. h. PROPN cana-1536 178 23 qawaqneh qawaqneh PROPN cana-1536 178 24 . . PUNCT cana-1536 179 1 ( ( PUNCT cana-1536 179 2 2019 2019 NUM cana-1536 179 3 ) ) PUNCT cana-1536 179 4 . . PUNCT cana-1536 180 1 on on ADP cana-1536 180 2 fixed fix VERB cana-1536 180 3 point point NOUN cana-1536 180 4 results result NOUN cana-1536 180 5 in in ADP cana-1536 180 6 gb gb ADV cana-1536 180 7 - - PUNCT cana-1536 180 8 metric metric ADJ cana-1536 180 9 spaces space NOUN cana-1536 180 10 , , PUNCT cana-1536 180 11 mathematics mathematic NOUN cana-1536 180 12 , , PUNCT cana-1536 180 13 7 7 NUM cana-1536 180 14 , , PUNCT cana-1536 180 15 617 617 NUM cana-1536 180 16 . . PUNCT cana-1536 180 17 https://doi.org/10.3390/math7070617 https://doi.org/10.3390/math7070617 ADJ cana-1536 180 18 . . PUNCT cana-1536 181 1 [ [ X cana-1536 181 2 12 12 NUM cana-1536 181 3 ] ] X cana-1536 181 4 d. d. PROPN cana-1536 181 5 dhamodharan dhamodharan PROPN cana-1536 181 6 , , PUNCT cana-1536 181 7 rajendran rajendran PROPN cana-1536 181 8 krishna krishna PROPN cana-1536 181 9 kumar kumar PROPN cana-1536 181 10 , , PUNCT cana-1536 181 11 stojan stojan ADJ cana-1536 181 12 n n PRON cana-1536 181 13 radenovic radenovic NOUN cana-1536 181 14 . . PUNCT cana-1536 182 1 ( ( PUNCT cana-1536 182 2 2019 2019 NUM cana-1536 182 3 ) ) PUNCT cana-1536 182 4 . . PUNCT cana-1536 183 1 coupled couple VERB cana-1536 183 2 fixed fix VERB cana-1536 183 3 point point NOUN cana-1536 183 4 theorems theorem NOUN cana-1536 183 5 of of ADP cana-1536 183 6 integral integral ADJ cana-1536 183 7 type type NOUN cana-1536 183 8 contraction contraction NOUN cana-1536 183 9 in in ADP cana-1536 183 10 sb sb PROPN cana-1536 183 11 -metric -metric PROPN cana-1536 183 12 space space NOUN cana-1536 183 13 , , PUNCT cana-1536 183 14 results result NOUN cana-1536 183 15 in in ADP cana-1536 183 16 fixed fix VERB cana-1536 183 17 point point NOUN cana-1536 183 18 theory theory NOUN cana-1536 183 19 and and CCONJ cana-1536 183 20 applications application NOUN cana-1536 183 21 . . PUNCT cana-1536 184 1 25 25 NUM cana-1536 184 2 pages page NOUN cana-1536 184 3 : : PUNCT cana-1536 184 4 doi:10.30697 doi:10.30697 VERB cana-1536 184 5 / / SYM cana-1536 184 6 rfpta-2018 rfpta-2018 NOUN cana-1536 184 7 - - PUNCT cana-1536 184 8 032 032 NUM cana-1536 184 9 . . PUNCT cana-1536 185 1 [ [ X cana-1536 185 2 13 13 NUM cana-1536 185 3 ] ] SYM cana-1536 185 4 nikbakhtsarvestani nikbakhtsarvestani PROPN cana-1536 185 5 f f PROPN cana-1536 185 6 , , PUNCT cana-1536 185 7 vaezpour vaezpour NOUN cana-1536 185 8 sm sm PROPN cana-1536 185 9 , , PUNCT cana-1536 185 10 asadi asadi ADJ cana-1536 185 11 m. m. NOUN cana-1536 185 12 ( ( PUNCT cana-1536 185 13 2019).common 2019).common NUM cana-1536 185 14 fixed fix VERB cana-1536 185 15 point point NOUN cana-1536 185 16 theorems theorem NOUN cana-1536 185 17 for for ADP cana-1536 185 18 weakly weakly ADJ cana-1536 185 19 compatible compatible ADJ cana-1536 185 20 mappings mapping NOUN cana-1536 185 21 by by ADP cana-1536 185 22 ( ( PUNCT cana-1536 185 23 clr clr NOUN cana-1536 185 24 ) ) PUNCT cana-1536 185 25 property property NOUN cana-1536 185 26 on on ADP cana-1536 185 27 partial partial ADJ cana-1536 185 28 metric metric ADJ cana-1536 185 29 space space NOUN cana-1536 185 30 . . PUNCT cana-1536 186 1 iranian iranian ADJ cana-1536 186 2 journal journal PROPN cana-1536 186 3 of of ADP cana-1536 186 4 mathematical mathematical ADJ cana-1536 186 5 sciences sciences PROPN cana-1536 186 6 and and CCONJ cana-1536 186 7 informatics.14:19–32 informatics.14:19–32 PROPN cana-1536 186 8 . . PUNCT cana-1536 187 1 https://doi.org/10.7508/ijmsi.2019.02.003.5 https://doi.org/10.7508/ijmsi.2019.02.003.5 PROPN cana-1536 188 1 [ [ X cana-1536 188 2 14 14 NUM cana-1536 188 3 ] ] X cana-1536 188 4 o. o. PROPN cana-1536 188 5 ege ege PROPN cana-1536 188 6 and and CCONJ cana-1536 188 7 i. i. PROPN cana-1536 188 8 karaca karaca PROPN cana-1536 188 9 . . PUNCT cana-1536 189 1 ( ( PUNCT cana-1536 189 2 2018 2018 NUM cana-1536 189 3 ) ) PUNCT cana-1536 189 4 common common ADJ cana-1536 189 5 fixed fix VERB cana-1536 189 6 point point NOUN cana-1536 189 7 results result NOUN cana-1536 189 8 on on ADP cana-1536 189 9 complex complex ADJ cana-1536 189 10 valued value VERB cana-1536 189 11 gb gb ADV cana-1536 189 12 - - PUNCT cana-1536 189 13 metric metric ADJ cana-1536 189 14 spaces space NOUN cana-1536 189 15 , , PUNCT cana-1536 189 16 thai thai PROPN cana-1536 189 17 journal journal NOUN cana-1536 189 18 of of ADP cana-1536 189 19 mathematics mathematic NOUN cana-1536 189 20 . . PUNCT cana-1536 190 1 16 16 NUM cana-1536 190 2 , , PUNCT cana-1536 190 3 3 3 NUM cana-1536 190 4 , , PUNCT cana-1536 190 5 775-787.doi 775-787.doi NUM cana-1536 190 6 : : PUNCT cana-1536 190 7 10.1007 10.1007 NUM cana-1536 190 8 / / SYM cana-1536 190 9 s13398 s13398 NOUN cana-1536 190 10 - - PUNCT cana-1536 190 11 017 017 NUM cana-1536 190 12 - - PUNCT cana-1536 190 13 0391 0391 NUM cana-1536 190 14 - - PUNCT cana-1536 190 15 x x SYM cana-1536 191 1 [ [ X cana-1536 191 2 15 15 NUM cana-1536 191 3 ] ] X cana-1536 191 4 m. m. NOUN cana-1536 191 5 liang liang PROPN cana-1536 191 6 , , PUNCT cana-1536 191 7 c. c. PROPN cana-1536 191 8 zhu zhu PROPN cana-1536 191 9 , , PUNCT cana-1536 191 10 c. c. PROPN cana-1536 191 11 chen chen PROPN cana-1536 191 12 and and CCONJ cana-1536 191 13 z. z. PROPN cana-1536 191 14 wu wu PROPN cana-1536 191 15 . . PUNCT cana-1536 192 1 ( ( PUNCT cana-1536 192 2 2018 2018 NUM cana-1536 192 3 ) ) PUNCT cana-1536 192 4 . . PUNCT cana-1536 193 1 some some DET cana-1536 193 2 new new ADJ cana-1536 193 3 theorems theorem NOUN cana-1536 193 4 for for ADP cana-1536 193 5 cyclic cyclic ADJ cana-1536 193 6 contraction contraction NOUN cana-1536 193 7 in in ADP cana-1536 193 8 gb gb ADV cana-1536 193 9 - - PUNCT cana-1536 193 10 metric metric ADJ cana-1536 193 11 spaces space NOUN cana-1536 193 12 and and CCONJ cana-1536 193 13 some some DET cana-1536 193 14 applications application NOUN cana-1536 193 15 , , PUNCT cana-1536 193 16 applied apply VERB cana-1536 193 17 mathematics mathematic NOUN cana-1536 193 18 and and CCONJ cana-1536 193 19 computation computation NOUN cana-1536 193 20 . . 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