id sid tid token lemma pos cana-3488 1 1 communications communication NOUN cana-3488 1 2 on on ADP cana-3488 1 3 applied apply VERB cana-3488 1 4 nonlinear nonlinear ADJ cana-3488 1 5 analysis analysis NOUN cana-3488 1 6 issn issn NOUN cana-3488 1 7 : : PUNCT cana-3488 1 8 1074 1074 NUM cana-3488 1 9 - - PUNCT cana-3488 1 10 133x 133x NUM cana-3488 1 11 vol vol NOUN cana-3488 1 12 32 32 NUM cana-3488 1 13 no no NOUN cana-3488 1 14 . . PUNCT cana-3488 2 1 7s 7 NOUN cana-3488 2 2 ( ( PUNCT cana-3488 2 3 2025 2025 NUM cana-3488 2 4 ) ) PUNCT cana-3488 2 5 806 806 NUM cana-3488 2 6 https://internationalpubls.com https://internationalpubls.com X cana-3488 2 7 exploring explore VERB cana-3488 2 8 advanced advanced ADJ cana-3488 2 9 stability stability NOUN cana-3488 2 10 of of ADP cana-3488 2 11 higher high ADJ cana-3488 2 12 - - PUNCT cana-3488 2 13 order order NOUN cana-3488 2 14 functional functional ADJ cana-3488 2 15 equations equation NOUN cana-3488 2 16 in in ADP cana-3488 2 17 neutrosophic neutrosophic ADJ cana-3488 2 18 normed norme VERB cana-3488 2 19 spaces space NOUN cana-3488 2 20 via via ADP cana-3488 2 21 hyers hyer NOUN cana-3488 2 22 - - PUNCT cana-3488 2 23 ulam ulam PROPN cana-3488 2 24 methodologies methodology NOUN cana-3488 3 1 p. p. NOUN cana-3488 3 2 agilan agilan PROPN cana-3488 4 1 πŸβˆ— πŸβˆ— NUM cana-3488 4 2 , , PUNCT cana-3488 4 3 v. v. CCONJ cana-3488 4 4 vijayan vijayan PROPN cana-3488 4 5 𝟐 𝟐 NUM cana-3488 4 6 , , PUNCT cana-3488 4 7 m. m. NOUN cana-3488 4 8 sophia sophia PROPN cana-3488 4 9 πŸ‘ πŸ‘ PROPN cana-3488 4 10 g. g. PROPN cana-3488 4 11 ganapathy ganapathy PROPN cana-3488 4 12 πŸ’ πŸ’ PROPN cana-3488 4 13 1department 1department NUM cana-3488 4 14 of of ADP cana-3488 4 15 mathematics mathematic NOUN cana-3488 4 16 , , PUNCT cana-3488 4 17 st.joseph st.joseph X cana-3488 4 18 ’s ’s PART cana-3488 4 19 college college NOUN cana-3488 4 20 of of ADP cana-3488 4 21 engineering engineering PROPN cana-3488 4 22 , , PUNCT cana-3488 4 23 omr omr PROPN cana-3488 4 24 , , PUNCT cana-3488 4 25 chennai chennai VERB cana-3488 4 26 600 600 NUM cana-3488 4 27 119 119 NUM cana-3488 4 28 , , PUNCT cana-3488 4 29 tamilnadu tamilnadu NOUN cana-3488 4 30 , , PUNCT cana-3488 4 31 india india PROPN cana-3488 4 32 . . PUNCT cana-3488 5 1 2department 2department NUM cana-3488 5 2 of of ADP cana-3488 5 3 electronics electronic NOUN cana-3488 5 4 and and CCONJ cana-3488 5 5 instrumentation instrumentation NOUN cana-3488 5 6 engineering engineering NOUN cana-3488 5 7 , , PUNCT cana-3488 5 8 st.joseph st.joseph X cana-3488 5 9 ’s ’s PART cana-3488 5 10 college college NOUN cana-3488 5 11 of of ADP cana-3488 5 12 engineering engineering PROPN cana-3488 5 13 , , PUNCT cana-3488 5 14 omr omr PROPN cana-3488 5 15 , , PUNCT cana-3488 5 16 chennai chennai VERB cana-3488 5 17 600 600 NUM cana-3488 5 18 119 119 NUM cana-3488 5 19 , , PUNCT cana-3488 5 20 tamilnadu tamilnadu ADJ cana-3488 5 21 , , PUNCT cana-3488 5 22 india india PROPN cana-3488 5 23 . . PUNCT cana-3488 6 1 e e X cana-3488 6 2 - - NOUN cana-3488 6 3 mail mail NOUN cana-3488 6 4 : : PUNCT cana-3488 6 5 vinvpn@gmail.com vinvpn@gmail.com PROPN cana-3488 6 6 . . PROPN cana-3488 6 7 3 3 NUM cana-3488 6 8 department department NOUN cana-3488 6 9 of of ADP cana-3488 6 10 mathematical mathematical ADJ cana-3488 6 11 studies study NOUN cana-3488 6 12 , , PUNCT cana-3488 6 13 simats simat NOUN cana-3488 6 14 engineering engineering NOUN cana-3488 6 15 , , PUNCT cana-3488 6 16 chennai chennai VERB cana-3488 6 17 600 600 NUM cana-3488 6 18 077 077 NUM cana-3488 6 19 , , PUNCT cana-3488 6 20 tamilnadu tamilnadu PROPN cana-3488 6 21 , , PUNCT cana-3488 6 22 india india PROPN cana-3488 6 23 . . PUNCT cana-3488 7 1 e e X cana-3488 7 2 - - NOUN cana-3488 7 3 mail mail NOUN cana-3488 7 4 : : PUNCT cana-3488 7 5 sophia.raj2005@gmail.com sophia.raj2005@gmail.com PROPN cana-3488 7 6 . . PUNCT cana-3488 8 1 4department 4department NUM cana-3488 8 2 of of ADP cana-3488 8 3 mathematics mathematic NOUN cana-3488 8 4 , , PUNCT cana-3488 8 5 r.m.d r.m.d PROPN cana-3488 8 6 engineering engineering PROPN cana-3488 8 7 college college PROPN cana-3488 8 8 , , PUNCT cana-3488 8 9 kavaraipettai kavaraipettai VERB cana-3488 8 10 601 601 NUM cana-3488 8 11 206 206 NUM cana-3488 8 12 , , PUNCT cana-3488 8 13 tamilnadu tamilnadu NOUN cana-3488 8 14 , , PUNCT cana-3488 8 15 india india PROPN cana-3488 8 16 . . PUNCT cana-3488 9 1 e e X cana-3488 9 2 - - NOUN cana-3488 9 3 mail mail NOUN cana-3488 9 4 : : PUNCT cana-3488 9 5 barathganagandhi@gmail.com barathganagandhi@gmail.com X cana-3488 9 6 . . PUNCT cana-3488 10 1 * * PUNCT cana-3488 10 2 corresponding correspond VERB cana-3488 10 3 author author NOUN cana-3488 10 4 : : PUNCT cana-3488 11 1 agilram@gmail.com agilram@gmail.com PROPN cana-3488 11 2 . . PUNCT cana-3488 11 3 article article PROPN cana-3488 11 4 history history NOUN cana-3488 11 5 : : PUNCT cana-3488 11 6 received receive VERB cana-3488 11 7 : : PUNCT cana-3488 11 8 28 28 NUM cana-3488 11 9 - - SYM cana-3488 11 10 10 10 NUM cana-3488 11 11 - - PUNCT cana-3488 11 12 2024 2024 NUM cana-3488 11 13 revised:12 revised:12 NOUN cana-3488 11 14 - - PUNCT cana-3488 11 15 11 11 NUM cana-3488 11 16 - - PUNCT cana-3488 11 17 2024 2024 NUM cana-3488 11 18 accepted:19 accepted:19 VERB cana-3488 11 19 - - PUNCT cana-3488 11 20 12 12 NUM cana-3488 11 21 - - PUNCT cana-3488 11 22 2024 2024 NUM cana-3488 11 23 abstract abstract NOUN cana-3488 11 24 : : PUNCT cana-3488 11 25 in in ADP cana-3488 11 26 this this DET cana-3488 11 27 article article NOUN cana-3488 11 28 , , PUNCT cana-3488 11 29 focuses focus VERB cana-3488 11 30 on on ADP cana-3488 11 31 examining examine VERB cana-3488 11 32 the the DET cana-3488 11 33 stability stability NOUN cana-3488 11 34 of of ADP cana-3488 11 35 higher high ADJ cana-3488 11 36 - - PUNCT cana-3488 11 37 order order NOUN cana-3488 11 38 functional functional ADJ cana-3488 11 39 equations equation NOUN cana-3488 11 40 within within ADP cana-3488 11 41 the the DET cana-3488 11 42 framework framework NOUN cana-3488 11 43 of of ADP cana-3488 11 44 neutrosophic neutrosophic ADJ cana-3488 11 45 normed norme VERB cana-3488 11 46 spaces space NOUN cana-3488 11 47 , , PUNCT cana-3488 11 48 which which PRON cana-3488 11 49 incorporate incorporate VERB cana-3488 11 50 elements element NOUN cana-3488 11 51 of of ADP cana-3488 11 52 uncertainty uncertainty NOUN cana-3488 11 53 and and CCONJ cana-3488 11 54 indeterminacy indeterminacy NOUN cana-3488 11 55 . . PUNCT cana-3488 12 1 by by ADP cana-3488 12 2 utilizing utilize VERB cana-3488 12 3 the the DET cana-3488 12 4 hyers hyer NOUN cana-3488 12 5 - - PUNCT cana-3488 12 6 ulam ulam PROPN cana-3488 12 7 method method NOUN cana-3488 12 8 , , PUNCT cana-3488 12 9 the the DET cana-3488 12 10 study study NOUN cana-3488 12 11 investigates investigate VERB cana-3488 12 12 how how SCONJ cana-3488 12 13 small small ADJ cana-3488 12 14 perturbations perturbation NOUN cana-3488 12 15 in in ADP cana-3488 12 16 the the DET cana-3488 12 17 functional functional ADJ cana-3488 12 18 equations equation NOUN cana-3488 12 19 impact impact VERB cana-3488 12 20 their their PRON cana-3488 12 21 solutions solution NOUN cana-3488 12 22 . . PUNCT cana-3488 13 1 the the DET cana-3488 13 2 research research NOUN cana-3488 13 3 extends extend VERB cana-3488 13 4 classical classical ADJ cana-3488 13 5 stability stability NOUN cana-3488 13 6 theories theory NOUN cana-3488 13 7 , , PUNCT cana-3488 13 8 such such ADJ cana-3488 13 9 as as ADP cana-3488 13 10 hyers hyer NOUN cana-3488 13 11 - - PUNCT cana-3488 13 12 ulam ulam PROPN cana-3488 13 13 stability stability NOUN cana-3488 13 14 , , PUNCT cana-3488 13 15 into into ADP cana-3488 13 16 the the DET cana-3488 13 17 neutrosophic neutrosophic ADJ cana-3488 13 18 normed normed ADJ cana-3488 13 19 space space NOUN cana-3488 13 20 context context NOUN cana-3488 13 21 , , PUNCT cana-3488 13 22 providing provide VERB cana-3488 13 23 a a DET cana-3488 13 24 broader broad ADJ cana-3488 13 25 understanding understanding NOUN cana-3488 13 26 of of ADP cana-3488 13 27 how how SCONJ cana-3488 13 28 functional functional ADJ cana-3488 13 29 equations equation NOUN cana-3488 13 30 behave behave VERB cana-3488 13 31 under under ADP cana-3488 13 32 uncertainty uncertainty NOUN cana-3488 13 33 . . PUNCT cana-3488 14 1 the the DET cana-3488 14 2 findings finding NOUN cana-3488 14 3 offer offer VERB cana-3488 14 4 significant significant ADJ cana-3488 14 5 contributions contribution NOUN cana-3488 14 6 to to ADP cana-3488 14 7 the the DET cana-3488 14 8 field field NOUN cana-3488 14 9 of of ADP cana-3488 14 10 functional functional ADJ cana-3488 14 11 equations equation NOUN cana-3488 14 12 and and CCONJ cana-3488 14 13 neutrosophic neutrosophic ADJ cana-3488 14 14 mathematics mathematic NOUN cana-3488 14 15 , , PUNCT cana-3488 14 16 opening open VERB cana-3488 14 17 up up ADP cana-3488 14 18 new new ADJ cana-3488 14 19 pathways pathway NOUN cana-3488 14 20 for for ADP cana-3488 14 21 applications application NOUN cana-3488 14 22 in in ADP cana-3488 14 23 areas area NOUN cana-3488 14 24 that that PRON cana-3488 14 25 require require VERB cana-3488 14 26 the the DET cana-3488 14 27 handling handling NOUN cana-3488 14 28 of of ADP cana-3488 14 29 imprecise imprecise ADJ cana-3488 14 30 data datum NOUN cana-3488 14 31 . . PUNCT cana-3488 15 1 keywords keyword NOUN cana-3488 15 2 : : PUNCT cana-3488 15 3 duodecic duodecic NOUN cana-3488 15 4 , , PUNCT cana-3488 15 5 tridecic tridecic NOUN cana-3488 15 6 functional functional ADJ cana-3488 15 7 equations equation NOUN cana-3488 15 8 , , PUNCT cana-3488 15 9 generalized generalize VERB cana-3488 15 10 hyers hyer NOUN cana-3488 15 11 ulam ulam PROPN cana-3488 15 12 stability stability NOUN cana-3488 15 13 . . PUNCT cana-3488 16 1 1 1 X cana-3488 16 2 . . X cana-3488 16 3 introduction introduction NOUN cana-3488 16 4 the the DET cana-3488 16 5 ulam ulam NOUN cana-3488 16 6 - - PUNCT cana-3488 16 7 hyers hyer NOUN cana-3488 16 8 stability stability NOUN cana-3488 16 9 , , PUNCT cana-3488 16 10 introduced introduce VERB cana-3488 16 11 by by ADP cana-3488 16 12 stanislaw stanislaw PROPN cana-3488 16 13 ulam ulam PROPN cana-3488 16 14 and and CCONJ cana-3488 16 15 donald donald PROPN cana-3488 16 16 hyers hyer NOUN cana-3488 16 17 in in ADP cana-3488 16 18 the the DET cana-3488 16 19 1940s 1940 NOUN cana-3488 16 20 and and CCONJ cana-3488 16 21 1941s 1941 NOUN cana-3488 16 22 respectively respectively ADV cana-3488 16 23 [ [ X cana-3488 16 24 1 1 NUM cana-3488 16 25 , , PUNCT cana-3488 16 26 2 2 NUM cana-3488 16 27 ] ] PUNCT cana-3488 16 28 , , PUNCT cana-3488 16 29 is be AUX cana-3488 16 30 a a DET cana-3488 16 31 fundamental fundamental ADJ cana-3488 16 32 concept concept NOUN cana-3488 16 33 in in ADP cana-3488 16 34 functional functional ADJ cana-3488 16 35 analysis analysis NOUN cana-3488 16 36 and and CCONJ cana-3488 16 37 mathematical mathematical ADJ cana-3488 16 38 stability stability NOUN cana-3488 16 39 theory theory NOUN cana-3488 16 40 . . PUNCT cana-3488 17 1 it it PRON cana-3488 17 2 examines examine VERB cana-3488 17 3 the the DET cana-3488 17 4 behavior behavior NOUN cana-3488 17 5 of of ADP cana-3488 17 6 solutions solution NOUN cana-3488 17 7 to to ADP cana-3488 17 8 functional functional ADJ cana-3488 17 9 equations equation NOUN cana-3488 17 10 when when SCONJ cana-3488 17 11 subjected subject VERB cana-3488 17 12 to to ADP cana-3488 17 13 small small ADJ cana-3488 17 14 perturbations perturbation NOUN cana-3488 17 15 . . PUNCT cana-3488 18 1 functional functional ADJ cana-3488 18 2 equations equation NOUN cana-3488 18 3 , , PUNCT cana-3488 18 4 which which PRON cana-3488 18 5 establish establish VERB cana-3488 18 6 specific specific ADJ cana-3488 18 7 relationships relationship NOUN cana-3488 18 8 between between ADP cana-3488 18 9 the the DET cana-3488 18 10 values value NOUN cana-3488 18 11 of of ADP cana-3488 18 12 functions function NOUN cana-3488 18 13 , , PUNCT cana-3488 18 14 are be AUX cana-3488 18 15 prevalent prevalent ADJ cana-3488 18 16 across across ADP cana-3488 18 17 various various ADJ cana-3488 18 18 fields field NOUN cana-3488 18 19 , , PUNCT cana-3488 18 20 including include VERB cana-3488 18 21 mathematics mathematic NOUN cana-3488 18 22 , , PUNCT cana-3488 18 23 physics physics NOUN cana-3488 18 24 , , PUNCT cana-3488 18 25 engineering engineering NOUN cana-3488 18 26 , , PUNCT cana-3488 18 27 and and CCONJ cana-3488 18 28 economics economic NOUN cana-3488 18 29 . . PUNCT cana-3488 19 1 the the DET cana-3488 19 2 stability stability NOUN cana-3488 19 3 of of ADP cana-3488 19 4 such such ADJ cana-3488 19 5 equations equation NOUN cana-3488 19 6 focuses focus VERB cana-3488 19 7 on on ADP cana-3488 19 8 how how SCONJ cana-3488 19 9 slight slight ADJ cana-3488 19 10 changes change NOUN cana-3488 19 11 in in ADP cana-3488 19 12 inputs input NOUN cana-3488 19 13 or or CCONJ cana-3488 19 14 parameters parameter NOUN cana-3488 19 15 influence influence VERB cana-3488 19 16 their their PRON cana-3488 19 17 solutions solution NOUN cana-3488 19 18 . . PUNCT cana-3488 20 1 the the DET cana-3488 20 2 ulam ulam PROPN cana-3488 20 3 - - PUNCT cana-3488 20 4 hyers hyer NOUN cana-3488 20 5 stability stability NOUN cana-3488 20 6 theory theory NOUN cana-3488 20 7 establishes establish VERB cana-3488 20 8 conditions condition NOUN cana-3488 20 9 under under ADP cana-3488 20 10 which which PRON cana-3488 20 11 solutions solution NOUN cana-3488 20 12 to to ADP cana-3488 20 13 functional functional ADJ cana-3488 20 14 equations equation NOUN cana-3488 20 15 remain remain VERB cana-3488 20 16 close close ADJ cana-3488 20 17 to to ADP cana-3488 20 18 the the DET cana-3488 20 19 original original ADJ cana-3488 20 20 solutions solution NOUN cana-3488 20 21 despite despite SCONJ cana-3488 20 22 small small ADJ cana-3488 20 23 perturbations perturbation NOUN cana-3488 20 24 . . PUNCT cana-3488 21 1 it it PRON cana-3488 21 2 addresses address VERB cana-3488 21 3 the the DET cana-3488 21 4 existence existence NOUN cana-3488 21 5 and and CCONJ cana-3488 21 6 uniqueness uniqueness NOUN cana-3488 21 7 of of ADP cana-3488 21 8 solutions solution NOUN cana-3488 21 9 while while SCONJ cana-3488 21 10 analyzing analyze VERB cana-3488 21 11 their their PRON cana-3488 21 12 dependence dependence NOUN cana-3488 21 13 on on ADP cana-3488 21 14 initial initial ADJ cana-3488 21 15 conditions condition NOUN cana-3488 21 16 and and CCONJ cana-3488 21 17 parameters parameter NOUN cana-3488 21 18 . . PUNCT cana-3488 22 1 this this DET cana-3488 22 2 concept concept NOUN cana-3488 22 3 is be AUX cana-3488 22 4 significant significant ADJ cana-3488 22 5 due due ADP cana-3488 22 6 to to ADP cana-3488 22 7 its its PRON cana-3488 22 8 extensive extensive ADJ cana-3488 22 9 applicability applicability NOUN cana-3488 22 10 and and CCONJ cana-3488 22 11 relevance relevance NOUN cana-3488 22 12 to to ADP cana-3488 22 13 practical practical ADJ cana-3488 22 14 problems problem NOUN cana-3488 22 15 . . PUNCT cana-3488 23 1 it it PRON cana-3488 23 2 enhances enhance VERB cana-3488 23 3 the the DET cana-3488 23 4 robustness robustness NOUN cana-3488 23 5 and and CCONJ cana-3488 23 6 reliability reliability NOUN cana-3488 23 7 of of ADP cana-3488 23 8 mathematical mathematical ADJ cana-3488 23 9 models model NOUN cana-3488 23 10 and and CCONJ cana-3488 23 11 numerical numerical ADJ cana-3488 23 12 algorithms algorithm NOUN cana-3488 23 13 , , PUNCT cana-3488 23 14 offering offer VERB cana-3488 23 15 valuable valuable ADJ cana-3488 23 16 insights insight NOUN cana-3488 23 17 into into ADP cana-3488 23 18 the the DET cana-3488 23 19 behavior behavior NOUN cana-3488 23 20 and and CCONJ cana-3488 23 21 predictability predictability NOUN cana-3488 23 22 of of ADP cana-3488 23 23 dynamic dynamic ADJ cana-3488 23 24 systems system NOUN cana-3488 23 25 . . PUNCT cana-3488 24 1 ongoing ongoing ADJ cana-3488 24 2 research research NOUN cana-3488 24 3 continues continue VERB cana-3488 24 4 to to PART cana-3488 24 5 expand expand VERB cana-3488 24 6 the the DET cana-3488 24 7 theory theory NOUN cana-3488 24 8 , , PUNCT cana-3488 24 9 uncovering uncover VERB cana-3488 24 10 connections connection NOUN cana-3488 24 11 with with ADP cana-3488 24 12 other other ADJ cana-3488 24 13 mathematical mathematical ADJ cana-3488 24 14 domains domain NOUN cana-3488 24 15 and and CCONJ cana-3488 24 16 exploring explore VERB cana-3488 24 17 its its PRON cana-3488 24 18 potential potential ADJ cana-3488 24 19 applications application NOUN cana-3488 24 20 in in ADP cana-3488 24 21 various various ADJ cana-3488 24 22 disciplines discipline NOUN cana-3488 24 23 ( ( PUNCT cana-3488 24 24 [ [ X cana-3488 24 25 3 3 NUM cana-3488 24 26 , , PUNCT cana-3488 24 27 4 4 NUM cana-3488 24 28 , , PUNCT cana-3488 24 29 5 5 NUM cana-3488 24 30 , , PUNCT cana-3488 24 31 6 6 NUM cana-3488 24 32 , , PUNCT cana-3488 24 33 7 7 NUM cana-3488 24 34 ] ] NUM cana-3488 24 35 ) ) PUNCT cana-3488 24 36 . . PUNCT cana-3488 25 1 communications communication NOUN cana-3488 25 2 on on ADP cana-3488 25 3 applied apply VERB cana-3488 25 4 nonlinear nonlinear ADJ cana-3488 25 5 analysis analysis NOUN cana-3488 25 6 issn issn NOUN cana-3488 25 7 : : PUNCT cana-3488 25 8 1074 1074 NUM cana-3488 25 9 - - PUNCT cana-3488 25 10 133x 133x NUM cana-3488 25 11 vol vol NOUN cana-3488 25 12 32 32 NUM cana-3488 25 13 no no NOUN cana-3488 25 14 . . PUNCT cana-3488 26 1 7s 7 NOUN cana-3488 26 2 ( ( PUNCT cana-3488 26 3 2025 2025 NUM cana-3488 26 4 ) ) PUNCT cana-3488 26 5 807 807 NUM cana-3488 26 6 https://internationalpubls.com https://internationalpubls.com X cana-3488 26 7 lotfi lotfi PROPN cana-3488 26 8 a. a. PROPN cana-3488 26 9 zadeh zadeh PROPN cana-3488 26 10 , , PUNCT cana-3488 26 11 a a DET cana-3488 26 12 renowned renowned ADJ cana-3488 26 13 mathematician mathematician ADJ cana-3488 26 14 and and CCONJ cana-3488 26 15 computer computer NOUN cana-3488 26 16 scientist scientist NOUN cana-3488 26 17 , , PUNCT cana-3488 26 18 introduced introduce VERB cana-3488 26 19 the the DET cana-3488 26 20 concept concept NOUN cana-3488 26 21 of of ADP cana-3488 26 22 fuzzy fuzzy ADJ cana-3488 26 23 sets set NOUN cana-3488 26 24 in in ADP cana-3488 26 25 his his PRON cana-3488 26 26 seminal seminal ADJ cana-3488 26 27 1965 1965 NUM cana-3488 26 28 paper paper NOUN cana-3488 26 29 titled title VERB cana-3488 26 30 " " PUNCT cana-3488 26 31 fuzzy fuzzy ADJ cana-3488 26 32 sets set NOUN cana-3488 26 33 " " PUNCT cana-3488 26 34 [ [ X cana-3488 26 35 8 8 NUM cana-3488 26 36 ] ] PUNCT cana-3488 26 37 . . PUNCT cana-3488 27 1 zadeh zadeh PROPN cana-3488 27 2 developed develop VERB cana-3488 27 3 this this DET cana-3488 27 4 concept concept NOUN cana-3488 27 5 to to PART cana-3488 27 6 overcome overcome VERB cana-3488 27 7 the the DET cana-3488 27 8 constraints constraint NOUN cana-3488 27 9 of of ADP cana-3488 27 10 classical classical ADJ cana-3488 27 11 set set NOUN cana-3488 27 12 theory theory NOUN cana-3488 27 13 , , PUNCT cana-3488 27 14 which which PRON cana-3488 27 15 depends depend VERB cana-3488 27 16 on on ADP cana-3488 27 17 sharp sharp ADJ cana-3488 27 18 , , PUNCT cana-3488 27 19 well well ADV cana-3488 27 20 - - PUNCT cana-3488 27 21 defined define VERB cana-3488 27 22 boundaries boundary NOUN cana-3488 27 23 for for ADP cana-3488 27 24 membership membership NOUN cana-3488 27 25 . . PUNCT cana-3488 28 1 building build VERB cana-3488 28 2 upon upon SCONJ cana-3488 28 3 this this DET cana-3488 28 4 foundation foundation NOUN cana-3488 28 5 , , PUNCT cana-3488 28 6 atanassov atanassov PROPN cana-3488 28 7 proposed propose VERB cana-3488 28 8 the the DET cana-3488 28 9 notion notion NOUN cana-3488 28 10 of of ADP cana-3488 28 11 intuitionistic intuitionistic ADJ cana-3488 28 12 fuzzy fuzzy ADJ cana-3488 28 13 sets set NOUN cana-3488 28 14 ( ( PUNCT cana-3488 28 15 ifs ifs PROPN cana-3488 28 16 ) ) PUNCT cana-3488 28 17 in in ADP cana-3488 28 18 1983 1983 NUM cana-3488 28 19 [ [ X cana-3488 28 20 9 9 NUM cana-3488 28 21 , , PUNCT cana-3488 28 22 10 10 NUM cana-3488 28 23 ] ] PUNCT cana-3488 28 24 . . PUNCT cana-3488 29 1 ifs ifs PROPN cana-3488 29 2 extended extend VERB cana-3488 29 3 fuzzy fuzzy ADJ cana-3488 29 4 sets set NOUN cana-3488 29 5 to to PART cana-3488 29 6 incorporate incorporate VERB cana-3488 29 7 a a DET cana-3488 29 8 more more ADV cana-3488 29 9 nuanced nuanced ADJ cana-3488 29 10 framework framework NOUN cana-3488 29 11 for for ADP cana-3488 29 12 addressing address VERB cana-3488 29 13 uncertainty uncertainty NOUN cana-3488 29 14 , , PUNCT cana-3488 29 15 vagueness vagueness NOUN cana-3488 29 16 , , PUNCT cana-3488 29 17 and and CCONJ cana-3488 29 18 hesitation hesitation NOUN cana-3488 29 19 . . PUNCT cana-3488 30 1 further far ADV cana-3488 30 2 advancing advance VERB cana-3488 30 3 these these DET cana-3488 30 4 ideas idea NOUN cana-3488 30 5 , , PUNCT cana-3488 30 6 neutrosophic neutrosophic ADJ cana-3488 30 7 sets set NOUN cana-3488 30 8 were be AUX cana-3488 30 9 introduced introduce VERB cana-3488 30 10 as as ADP cana-3488 30 11 an an DET cana-3488 30 12 extension extension NOUN cana-3488 30 13 of of ADP cana-3488 30 14 classical classical ADJ cana-3488 30 15 set set NOUN cana-3488 30 16 theory theory NOUN cana-3488 30 17 , , PUNCT cana-3488 30 18 providing provide VERB cana-3488 30 19 a a DET cana-3488 30 20 robust robust ADJ cana-3488 30 21 mechanism mechanism NOUN cana-3488 30 22 for for ADP cana-3488 30 23 managing manage VERB cana-3488 30 24 indeterminacy indeterminacy NOUN cana-3488 30 25 , , PUNCT cana-3488 30 26 uncertainty uncertainty NOUN cana-3488 30 27 , , PUNCT cana-3488 30 28 and and CCONJ cana-3488 30 29 incomplete incomplete ADJ cana-3488 30 30 information information NOUN cana-3488 31 1 [ [ X cana-3488 31 2 11 11 NUM cana-3488 31 3 , , PUNCT cana-3488 31 4 12 12 NUM cana-3488 31 5 ] ] PUNCT cana-3488 31 6 . . PUNCT cana-3488 32 1 fuzzy fuzzy ADJ cana-3488 32 2 normed normed ADJ cana-3488 32 3 spaces space NOUN cana-3488 32 4 ( ( PUNCT cana-3488 32 5 fns fns PROPN cana-3488 32 6 ) ) PUNCT cana-3488 32 7 are be AUX cana-3488 32 8 mathematical mathematical ADJ cana-3488 32 9 structures structure NOUN cana-3488 32 10 that that PRON cana-3488 32 11 generalize generalize VERB cana-3488 32 12 classical classical ADJ cana-3488 32 13 normed norme VERB cana-3488 32 14 spaces space NOUN cana-3488 32 15 by by ADP cana-3488 32 16 incorporating incorporate VERB cana-3488 32 17 fuzzy fuzzy ADJ cana-3488 32 18 numbers number NOUN cana-3488 32 19 . . PUNCT cana-3488 33 1 katsaras katsaras PROPN cana-3488 33 2 first first ADV cana-3488 33 3 introduced introduce VERB cana-3488 33 4 the the DET cana-3488 33 5 concept concept NOUN cana-3488 33 6 of of ADP cana-3488 33 7 fns fns PROPN cana-3488 33 8 , , PUNCT cana-3488 33 9 defining define VERB cana-3488 33 10 them they PRON cana-3488 33 11 as as ADP cana-3488 33 12 vector vector NOUN cana-3488 33 13 spaces space NOUN cana-3488 33 14 equipped equip VERB cana-3488 33 15 with with ADP cana-3488 33 16 a a DET cana-3488 33 17 fuzzy fuzzy ADJ cana-3488 33 18 norm norm NOUN cana-3488 33 19 , , PUNCT cana-3488 33 20 where where SCONJ cana-3488 33 21 the the DET cana-3488 33 22 norm norm NOUN cana-3488 33 23 values value NOUN cana-3488 33 24 are be AUX cana-3488 33 25 represented represent VERB cana-3488 33 26 as as ADP cana-3488 33 27 fuzzy fuzzy ADJ cana-3488 33 28 numbers number NOUN cana-3488 33 29 rather rather ADV cana-3488 33 30 than than ADP cana-3488 33 31 real real ADJ cana-3488 33 32 numbers number NOUN cana-3488 33 33 [ [ X cana-3488 33 34 13 13 NUM cana-3488 33 35 ] ] PUNCT cana-3488 33 36 . . PUNCT cana-3488 34 1 in in ADP cana-3488 34 2 2006 2006 NUM cana-3488 34 3 , , PUNCT cana-3488 34 4 the the DET cana-3488 34 5 concept concept NOUN cana-3488 34 6 of of ADP cana-3488 34 7 intuitionistic intuitionistic ADJ cana-3488 34 8 fuzzy fuzzy ADJ cana-3488 34 9 normed norme VERB cana-3488 34 10 spaces space NOUN cana-3488 34 11 ( ( PUNCT cana-3488 34 12 ifns ifns NOUN cana-3488 34 13 ) ) PUNCT cana-3488 34 14 was be AUX cana-3488 34 15 introduced introduce VERB cana-3488 34 16 [ [ X cana-3488 34 17 14 14 NUM cana-3488 34 18 ] ] PUNCT cana-3488 34 19 . . PUNCT cana-3488 35 1 ifns ifns PROPN cana-3488 35 2 merge merge VERB cana-3488 35 3 elements element NOUN cana-3488 35 4 of of ADP cana-3488 35 5 fuzzy fuzzy ADJ cana-3488 35 6 mathematics mathematic NOUN cana-3488 35 7 , , PUNCT cana-3488 35 8 intuitionistic intuitionistic ADJ cana-3488 35 9 fuzzy fuzzy ADJ cana-3488 35 10 sets set NOUN cana-3488 35 11 , , PUNCT cana-3488 35 12 and and CCONJ cana-3488 35 13 normed normed ADJ cana-3488 35 14 spaces space NOUN cana-3488 35 15 , , PUNCT cana-3488 35 16 offering offer VERB cana-3488 35 17 a a DET cana-3488 35 18 flexible flexible ADJ cana-3488 35 19 framework framework NOUN cana-3488 35 20 for for ADP cana-3488 35 21 addressing address VERB cana-3488 35 22 uncertainty uncertainty NOUN cana-3488 35 23 and and CCONJ cana-3488 35 24 imprecision imprecision NOUN cana-3488 35 25 in in ADP cana-3488 35 26 mathematical mathematical ADJ cana-3488 35 27 modeling modeling NOUN cana-3488 35 28 and and CCONJ cana-3488 35 29 analysis analysis NOUN cana-3488 35 30 . . PUNCT cana-3488 36 1 neutrosophic neutrosophic PROPN cana-3488 36 2 normed normed PROPN cana-3488 36 3 linear linear PROPN cana-3488 36 4 spaces space NOUN cana-3488 36 5 [ [ X cana-3488 36 6 15 15 NUM cana-3488 36 7 , , PUNCT cana-3488 36 8 16 16 NUM cana-3488 36 9 ] ] PUNCT cana-3488 36 10 extend extend VERB cana-3488 36 11 neutrosophic neutrosophic ADJ cana-3488 36 12 set set NOUN cana-3488 36 13 theory theory NOUN cana-3488 36 14 to to PART cana-3488 36 15 linear linear VERB cana-3488 36 16 algebraic algebraic ADJ cana-3488 36 17 structures structure NOUN cana-3488 36 18 , , PUNCT cana-3488 36 19 enabling enable VERB cana-3488 36 20 a a DET cana-3488 36 21 more more ADV cana-3488 36 22 comprehensive comprehensive ADJ cana-3488 36 23 representation representation NOUN cana-3488 36 24 of of ADP cana-3488 36 25 uncertainty uncertainty NOUN cana-3488 36 26 within within ADP cana-3488 36 27 vector vector NOUN cana-3488 36 28 spaces space NOUN cana-3488 36 29 . . PUNCT cana-3488 37 1 neutrosophic neutrosophic ADJ cana-3488 37 2 concepts concept NOUN cana-3488 37 3 have have AUX cana-3488 37 4 been be AUX cana-3488 37 5 widely widely ADV cana-3488 37 6 applied apply VERB cana-3488 37 7 in in ADP cana-3488 37 8 various various ADJ cana-3488 37 9 branches branch NOUN cana-3488 37 10 of of ADP cana-3488 37 11 mathematics mathematic NOUN cana-3488 37 12 , , PUNCT cana-3488 37 13 including include VERB cana-3488 37 14 groups group NOUN cana-3488 37 15 and and CCONJ cana-3488 37 16 subgroups subgroup NOUN cana-3488 37 17 [ [ X cana-3488 37 18 17 17 NUM cana-3488 37 19 ] ] PUNCT cana-3488 37 20 , , PUNCT cana-3488 37 21 vector vector NOUN cana-3488 37 22 spaces space NOUN cana-3488 37 23 [ [ X cana-3488 37 24 18 18 NUM cana-3488 37 25 ] ] PUNCT cana-3488 37 26 , , PUNCT cana-3488 37 27 ring ring NOUN cana-3488 37 28 homomorphisms homomorphism NOUN cana-3488 37 29 [ [ X cana-3488 37 30 19 19 NUM cana-3488 37 31 , , PUNCT cana-3488 37 32 20 20 NUM cana-3488 37 33 ] ] PUNCT cana-3488 37 34 , , PUNCT cana-3488 37 35 linear linear ADJ cana-3488 37 36 transformations transformation NOUN cana-3488 37 37 [ [ X cana-3488 37 38 21 21 NUM cana-3488 37 39 ] ] PUNCT cana-3488 37 40 , , PUNCT cana-3488 37 41 number number NOUN cana-3488 37 42 theory theory NOUN cana-3488 37 43 [ [ X cana-3488 37 44 22 22 NUM cana-3488 37 45 ] ] PUNCT cana-3488 37 46 , , PUNCT cana-3488 37 47 graph graph NOUN cana-3488 37 48 theory theory NOUN cana-3488 37 49 [ [ X cana-3488 37 50 23 23 NUM cana-3488 37 51 ] ] PUNCT cana-3488 37 52 , , PUNCT cana-3488 37 53 measure measure NOUN cana-3488 37 54 theory theory NOUN cana-3488 37 55 , , PUNCT cana-3488 37 56 integral integral ADJ cana-3488 37 57 theory theory NOUN cana-3488 37 58 , , PUNCT cana-3488 37 59 and and CCONJ cana-3488 37 60 probability probability NOUN cana-3488 37 61 theory theory NOUN cana-3488 37 62 [ [ X cana-3488 37 63 24 24 NUM cana-3488 37 64 ] ] PUNCT cana-3488 37 65 . . PUNCT cana-3488 38 1 additionally additionally ADV cana-3488 38 2 , , PUNCT cana-3488 38 3 neutrosophic neutrosophic PROPN cana-3488 38 4 normed norme VERB cana-3488 38 5 linear linear PROPN cana-3488 38 6 spaces space NOUN cana-3488 38 7 find find VERB cana-3488 38 8 practical practical ADJ cana-3488 38 9 applications application NOUN cana-3488 38 10 in in ADP cana-3488 38 11 diverse diverse ADJ cana-3488 38 12 fields field NOUN cana-3488 38 13 such such ADJ cana-3488 38 14 as as ADP cana-3488 38 15 decision decision NOUN cana-3488 38 16 - - PUNCT cana-3488 38 17 making making NOUN cana-3488 38 18 , , PUNCT cana-3488 38 19 control control NOUN cana-3488 38 20 systems system NOUN cana-3488 38 21 , , PUNCT cana-3488 38 22 optimization optimization NOUN cana-3488 38 23 , , PUNCT cana-3488 38 24 image image NOUN cana-3488 38 25 processing processing NOUN cana-3488 38 26 , , PUNCT cana-3488 38 27 pattern pattern NOUN cana-3488 38 28 recognition recognition NOUN cana-3488 38 29 , , PUNCT cana-3488 38 30 medical medical ADJ cana-3488 38 31 diagnosis diagnosis NOUN cana-3488 38 32 , , PUNCT cana-3488 38 33 finance finance NOUN cana-3488 38 34 and and CCONJ cana-3488 38 35 risk risk NOUN cana-3488 38 36 management management NOUN cana-3488 38 37 , , PUNCT cana-3488 38 38 information information NOUN cana-3488 38 39 retrieval retrieval NOUN cana-3488 38 40 , , PUNCT cana-3488 38 41 and and CCONJ cana-3488 38 42 artificial artificial ADJ cana-3488 38 43 intelligence intelligence NOUN cana-3488 38 44 [ [ X cana-3488 38 45 25 25 NUM cana-3488 38 46 , , PUNCT cana-3488 38 47 26 26 NUM cana-3488 38 48 , , PUNCT cana-3488 38 49 27 27 NUM cana-3488 38 50 , , PUNCT cana-3488 38 51 28 28 NUM cana-3488 38 52 , , PUNCT cana-3488 38 53 29 29 NUM cana-3488 38 54 , , PUNCT cana-3488 38 55 30 30 NUM cana-3488 38 56 , , PUNCT cana-3488 38 57 31 31 NUM cana-3488 38 58 ] ] PUNCT cana-3488 38 59 . . PUNCT cana-3488 39 1 agilan agilan PROPN cana-3488 39 2 et et PROPN cana-3488 39 3 al al PROPN cana-3488 39 4 . . PROPN cana-3488 39 5 have have AUX cana-3488 39 6 introduced introduce VERB cana-3488 39 7 novel novel ADJ cana-3488 39 8 functional functional ADJ cana-3488 39 9 equations equation NOUN cana-3488 39 10 and and CCONJ cana-3488 39 11 established establish VERB cana-3488 39 12 the the DET cana-3488 39 13 hyersulam hyersulam PROPN cana-3488 39 14 stability stability NOUN cana-3488 39 15 of of ADP cana-3488 39 16 these these DET cana-3488 39 17 equations equation NOUN cana-3488 39 18 across across ADP cana-3488 39 19 various various ADJ cana-3488 39 20 normed normed ADJ cana-3488 39 21 spaces space NOUN cana-3488 39 22 ( ( PUNCT cana-3488 39 23 [ [ X cana-3488 39 24 32 32 NUM cana-3488 39 25 , , PUNCT cana-3488 39 26 33 33 NUM cana-3488 39 27 , , PUNCT cana-3488 39 28 34 34 NUM cana-3488 39 29 , , PUNCT cana-3488 39 30 35 35 NUM cana-3488 39 31 , , PUNCT cana-3488 39 32 36 36 NUM cana-3488 39 33 , , PUNCT cana-3488 39 34 37 37 NUM cana-3488 39 35 , , PUNCT cana-3488 39 36 38 38 NUM cana-3488 39 37 ] ] PUNCT cana-3488 39 38 ) ) PUNCT cana-3488 39 39 . . PUNCT cana-3488 40 1 this this DET cana-3488 40 2 article article NOUN cana-3488 40 3 introduces introduce VERB cana-3488 40 4 a a DET cana-3488 40 5 novel novel ADJ cana-3488 40 6 mixed mixed ADJ cana-3488 40 7 duodecic duodecic NOUN cana-3488 40 8 - - PUNCT cana-3488 40 9 tridecic tridecic NOUN cana-3488 40 10 functional functional ADJ cana-3488 40 11 equation equation NOUN cana-3488 40 12 and and CCONJ cana-3488 40 13 investigates investigate VERB cana-3488 40 14 its its PRON cana-3488 40 15 ulamhyers ulamhyer NOUN cana-3488 40 16 stability stability NOUN cana-3488 40 17 within within ADP cana-3488 40 18 neutrosophic neutrosophic ADJ cana-3488 40 19 normed normed PROPN cana-3488 40 20 linear linear PROPN cana-3488 40 21 spaces space NOUN cana-3488 40 22 ( ( PUNCT cana-3488 40 23 nnls nnls NOUN cana-3488 40 24 ) ) PUNCT cana-3488 40 25 . . PUNCT cana-3488 41 1 classical classical ADJ cana-3488 41 2 approaches approach NOUN cana-3488 41 3 are be AUX cana-3488 41 4 employed employ VERB cana-3488 41 5 for for ADP cana-3488 41 6 the the DET cana-3488 41 7 stability stability NOUN cana-3488 41 8 analysis analysis NOUN cana-3488 41 9 in in ADP cana-3488 41 10 the the DET cana-3488 41 11 newly newly ADV cana-3488 41 12 proposed propose VERB cana-3488 41 13 equation equation NOUN cana-3488 41 14 . . PUNCT cana-3488 42 1 given give VERB cana-3488 42 2 the the DET cana-3488 42 3 unique unique ADJ cana-3488 42 4 properties property NOUN cana-3488 42 5 of of ADP cana-3488 42 6 neutrosophic neutrosophic ADJ cana-3488 42 7 normed norme VERB cana-3488 42 8 spaces space NOUN cana-3488 42 9 and and CCONJ cana-3488 42 10 their their PRON cana-3488 42 11 broad broad ADJ cana-3488 42 12 potential potential ADJ cana-3488 42 13 applications application NOUN cana-3488 42 14 , , PUNCT cana-3488 42 15 the the DET cana-3488 42 16 stability stability NOUN cana-3488 42 17 analysis analysis NOUN cana-3488 42 18 of of ADP cana-3488 42 19 this this DET cana-3488 42 20 equation equation NOUN cana-3488 42 21 is be AUX cana-3488 42 22 of of ADP cana-3488 42 23 considerable considerable ADJ cana-3488 42 24 importance importance NOUN cana-3488 42 25 . . PUNCT cana-3488 43 1 notably notably ADV cana-3488 43 2 , , PUNCT cana-3488 43 3 this this DET cana-3488 43 4 study study NOUN cana-3488 43 5 marks mark VERB cana-3488 43 6 the the DET cana-3488 43 7 first first ADJ cana-3488 43 8 instance instance NOUN cana-3488 43 9 in in ADP cana-3488 43 10 the the DET cana-3488 43 11 literature literature NOUN cana-3488 43 12 where where SCONJ cana-3488 43 13 the the DET cana-3488 43 14 stability stability NOUN cana-3488 43 15 of of ADP cana-3488 43 16 a a DET cana-3488 43 17 functional functional ADJ cana-3488 43 18 equation equation NOUN cana-3488 43 19 is be AUX cana-3488 43 20 examined examine VERB cana-3488 43 21 within within ADP cana-3488 43 22 the the DET cana-3488 43 23 framework framework NOUN cana-3488 43 24 of of ADP cana-3488 43 25 neutrosophic neutrosophic ADJ cana-3488 43 26 normed norme VERB cana-3488 43 27 spaces space NOUN cana-3488 43 28 , , PUNCT cana-3488 43 29 underscoring underscore VERB cana-3488 43 30 the the DET cana-3488 43 31 distinctiveness distinctiveness NOUN cana-3488 43 32 and and CCONJ cana-3488 43 33 significance significance NOUN cana-3488 43 34 of of ADP cana-3488 43 35 the the DET cana-3488 43 36 research research NOUN cana-3488 43 37 . . PUNCT cana-3488 44 1 this this DET cana-3488 44 2 study study NOUN cana-3488 44 3 sets set VERB cana-3488 44 4 out out ADP cana-3488 44 5 to to PART cana-3488 44 6 accomplish accomplish VERB cana-3488 44 7 the the DET cana-3488 44 8 following follow VERB cana-3488 44 9 key key ADJ cana-3488 44 10 objectives objective NOUN cana-3488 44 11 : : PUNCT cana-3488 44 12 ( ( PUNCT cana-3488 44 13 i i NOUN cana-3488 44 14 ) ) PUNCT cana-3488 44 15 to to PART cana-3488 44 16 expand expand VERB cana-3488 44 17 and and CCONJ cana-3488 44 18 advance advance VERB cana-3488 44 19 the the DET cana-3488 44 20 existing exist VERB cana-3488 44 21 research research NOUN cana-3488 44 22 on on ADP cana-3488 44 23 neutrosophic neutrosophic ADJ cana-3488 44 24 normed normed PROPN cana-3488 44 25 linear linear PROPN cana-3488 44 26 spaces space NOUN cana-3488 44 27 . . PUNCT cana-3488 45 1 ( ( PUNCT cana-3488 45 2 ii ii NOUN cana-3488 45 3 ) ) PUNCT cana-3488 45 4 to to PART cana-3488 45 5 establish establish VERB cana-3488 45 6 the the DET cana-3488 45 7 uniqueness uniqueness NOUN cana-3488 45 8 of of ADP cana-3488 45 9 solutions solution NOUN cana-3488 45 10 for for ADP cana-3488 45 11 the the DET cana-3488 45 12 newly newly ADV cana-3488 45 13 introduced introduce VERB cana-3488 45 14 higher high ADJ cana-3488 45 15 - - PUNCT cana-3488 45 16 order order NOUN cana-3488 45 17 functional functional ADJ cana-3488 45 18 equation equation NOUN cana-3488 45 19 . . PUNCT cana-3488 46 1 ( ( PUNCT cana-3488 46 2 ii ii NOUN cana-3488 46 3 ) ) PUNCT cana-3488 46 4 to to PART cana-3488 46 5 investigate investigate VERB cana-3488 46 6 the the DET cana-3488 46 7 hyers hyer NOUN cana-3488 46 8 - - PUNCT cana-3488 46 9 ulam ulam ADJ cana-3488 46 10 stability stability NOUN cana-3488 46 11 of of ADP cana-3488 46 12 the the DET cana-3488 46 13 proposed propose VERB cana-3488 46 14 equation equation NOUN cana-3488 46 15 within within ADP cana-3488 46 16 neutrosophic neutrosophic ADJ cana-3488 46 17 normed normed PROPN cana-3488 46 18 linear linear PROPN cana-3488 46 19 spaces space NOUN cana-3488 46 20 , , PUNCT cana-3488 46 21 employing employ VERB cana-3488 46 22 the the DET cana-3488 46 23 direct direct ADJ cana-3488 46 24 method method NOUN cana-3488 46 25 . . PUNCT cana-3488 47 1 𝔉(11ℨ 𝔉(11ℨ NOUN cana-3488 47 2 ) ) PUNCT cana-3488 48 1 = = SYM cana-3488 48 2 1,88,30,57,02,60,326𝔉(ℨ 1,88,30,57,02,60,326𝔉(ℨ NUM cana-3488 48 3 ) ) PUNCT cana-3488 48 4 βˆ’ βˆ’ PROPN cana-3488 49 1 1,56,92,14,18,83,605𝔉(βˆ’β„¨ 1,56,92,14,18,83,605𝔉(βˆ’β„¨ PROPN cana-3488 49 2 ) ) PUNCT cana-3488 49 3 ( ( PUNCT cana-3488 49 4 1 1 X cana-3488 49 5 ) ) PUNCT cana-3488 49 6 the the DET cana-3488 49 7 above above ADJ cana-3488 49 8 equation equation NOUN cana-3488 49 9 having have VERB cana-3488 49 10 solution solution NOUN cana-3488 49 11 𝔉(ℨ 𝔉(ℨ NOUN cana-3488 49 12 ) ) PUNCT cana-3488 49 13 = = PUNCT cana-3488 49 14 π’œ1 π’œ1 NOUN cana-3488 49 15 ℨ ℨ NOUN cana-3488 49 16 13 13 NUM cana-3488 50 1 + + NOUN cana-3488 50 2 π’œ2 π’œ2 NOUN cana-3488 50 3 ℨ ℨ PROPN cana-3488 50 4 12 12 NUM cana-3488 50 5 . . PUNCT cana-3488 51 1 communications communication NOUN cana-3488 51 2 on on ADP cana-3488 51 3 applied apply VERB cana-3488 51 4 nonlinear nonlinear ADJ cana-3488 51 5 analysis analysis NOUN cana-3488 51 6 issn issn NOUN cana-3488 51 7 : : PUNCT cana-3488 51 8 1074 1074 NUM cana-3488 51 9 - - PUNCT cana-3488 51 10 133x 133x NUM cana-3488 51 11 vol vol NOUN cana-3488 51 12 32 32 NUM cana-3488 51 13 no no NOUN cana-3488 51 14 . . PUNCT cana-3488 52 1 7s 7 NOUN cana-3488 52 2 ( ( PUNCT cana-3488 52 3 2025 2025 NUM cana-3488 52 4 ) ) PUNCT cana-3488 52 5 808 808 NUM cana-3488 52 6 https://internationalpubls.com https://internationalpubls.com X cana-3488 52 7 remark remark NOUN cana-3488 52 8 1.1 1.1 NUM cana-3488 52 9 let let VERB cana-3488 52 10 us we PRON cana-3488 52 11 take take VERB cana-3488 52 12 an an DET cana-3488 52 13 odd odd ADJ cana-3488 52 14 mapping mapping NOUN cana-3488 52 15 𝔉:𝒳𝑀 𝔉:𝒳𝑀 NOUN cana-3488 52 16 β†’ β†’ SYM cana-3488 52 17 𝒴𝑀 𝒴𝑀 NOUN cana-3488 52 18 which which PRON cana-3488 52 19 satisfies satisfy VERB cana-3488 52 20 the the DET cana-3488 52 21 fe fe X cana-3488 52 22 ( ( PUNCT cana-3488 52 23 1 1 NUM cana-3488 52 24 ) ) PUNCT cana-3488 52 25 then then ADV cana-3488 52 26 it it PRON cana-3488 52 27 is be AUX cana-3488 52 28 tridecic tridecic NOUN cana-3488 52 29 𝔉(11ℨ 𝔉(11ℨ NOUN cana-3488 52 30 ) ) PUNCT cana-3488 53 1 = = SYM cana-3488 53 2 96,88,90,10,407𝔉(ℨ 96,88,90,10,407𝔉(ℨ X cana-3488 53 3 ) ) PUNCT cana-3488 53 4 = = SYM cana-3488 53 5 1113𝔉(ℨ 1113𝔉(ℨ NUM cana-3488 53 6 ) ) PUNCT cana-3488 53 7 for for ADP cana-3488 53 8 all all DET cana-3488 53 9 ℨ ℨ PROPN cana-3488 53 10 ∈ ∈ NOUN cana-3488 53 11 𝒳𝑀 𝒳𝑀 NOUN cana-3488 53 12 remark remark NOUN cana-3488 53 13 1.2 1.2 NUM cana-3488 53 14 let let VERB cana-3488 53 15 us we PRON cana-3488 53 16 take take VERB cana-3488 53 17 an an DET cana-3488 53 18 even even ADV cana-3488 53 19 mapping mapping NOUN cana-3488 53 20 𝔉:𝒳𝑀 𝔉:𝒳𝑀 NOUN cana-3488 53 21 β†’ β†’ SYM cana-3488 53 22 𝒴𝑀 𝒴𝑀 NOUN cana-3488 53 23 which which PRON cana-3488 53 24 satisfies satisfy VERB cana-3488 53 25 the the DET cana-3488 53 26 fe fe X cana-3488 53 27 ( ( PUNCT cana-3488 53 28 1 1 NUM cana-3488 53 29 ) ) PUNCT cana-3488 53 30 then then ADV cana-3488 53 31 it it PRON cana-3488 53 32 is be AUX cana-3488 53 33 duodecic duodecic NOUN cana-3488 53 34 𝔉(11ℨ 𝔉(11ℨ NOUN cana-3488 53 35 ) ) PUNCT cana-3488 54 1 = = SYM cana-3488 54 2 13,84,12,87,201𝔉(ℨ 13,84,12,87,201𝔉(ℨ X cana-3488 54 3 ) ) PUNCT cana-3488 54 4 = = SYM cana-3488 54 5 1112𝔉(ℨ 1112𝔉(ℨ NUM cana-3488 54 6 ) ) PUNCT cana-3488 54 7 for for ADP cana-3488 54 8 all all DET cana-3488 54 9 ℨ ℨ NOUN cana-3488 54 10 ∈ ∈ NOUN cana-3488 54 11 𝒳𝑀 𝒳𝑀 NOUN cana-3488 54 12 2 2 NUM cana-3488 54 13 definition definition NOUN cana-3488 54 14 of of ADP cana-3488 54 15 neutrosophic neutrosophic ADJ cana-3488 54 16 normed norme VERB cana-3488 54 17 spaces space NOUN cana-3488 54 18 definition definition NOUN cana-3488 54 19 2.1 2.1 NUM cana-3488 54 20 the the DET cana-3488 54 21 seven seven NUM cana-3488 54 22 - - PUNCT cana-3488 54 23 tuple tuple NOUN cana-3488 54 24 ( ( PUNCT cana-3488 54 25 𝔸,π’œ1,π’œ2 𝔸,π’œ1,π’œ2 PROPN cana-3488 54 26 , , PUNCT cana-3488 54 27 π’œ3 π’œ3 NOUN cana-3488 54 28 βˆ—,β‹„,⊘ βˆ—,β‹„,⊘ NOUN cana-3488 54 29 ) ) PUNCT cana-3488 54 30 is be AUX cana-3488 54 31 said say VERB cana-3488 54 32 to to PART cana-3488 54 33 be be AUX cana-3488 54 34 a a DET cana-3488 54 35 neutrosophic neutrosophic ADJ cana-3488 54 36 normed normed ADJ cana-3488 54 37 space space NOUN cana-3488 54 38 ( ( PUNCT cana-3488 54 39 for for ADP cana-3488 54 40 short short ADJ cana-3488 54 41 , , PUNCT cana-3488 54 42 nns nn NOUN cana-3488 54 43 ) ) PUNCT cana-3488 54 44 if if SCONJ cana-3488 54 45 𝔸 𝔸 PROPN cana-3488 54 46 is be AUX cana-3488 54 47 a a DET cana-3488 54 48 vector vector NOUN cana-3488 54 49 space space NOUN cana-3488 54 50 , , PUNCT cana-3488 54 51 βˆ— βˆ— PROPN cana-3488 54 52 is be AUX cana-3488 54 53 a a DET cana-3488 54 54 continuous continuous ADJ cana-3488 54 55 πœ…-norm πœ…-norm NOUN cana-3488 54 56 , , PUNCT cana-3488 54 57 β‹„ β‹„ PROPN cana-3488 54 58 and and CCONJ cana-3488 54 59 ⊘ ⊘ PRON cana-3488 54 60 is be AUX cana-3488 54 61 a a DET cana-3488 54 62 continuous continuous ADJ cana-3488 54 63 πœ… πœ… PRON cana-3488 54 64 βˆ’ βˆ’ NOUN cana-3488 54 65 conorm conorm NOUN cana-3488 54 66 , , PUNCT cana-3488 54 67 and and CCONJ cana-3488 54 68 π’œ1 π’œ1 NOUN cana-3488 54 69 , , PUNCT cana-3488 54 70 π’œ2,π’œ3 π’œ2,π’œ3 NOUN cana-3488 54 71 are be AUX cana-3488 54 72 fuzzy fuzzy ADJ cana-3488 54 73 sets set NOUN cana-3488 54 74 on on ADP cana-3488 54 75 𝔸 𝔸 PROPN cana-3488 54 76 Γ— Γ— NOUN cana-3488 54 77 ( ( PUNCT cana-3488 54 78 0,∞ 0,∞ NOUN cana-3488 54 79 ) ) PUNCT cana-3488 54 80 satisfying satisfy VERB cana-3488 54 81 the the DET cana-3488 54 82 following follow VERB cana-3488 54 83 conditions condition NOUN cana-3488 54 84 . . PUNCT cana-3488 55 1 for for ADP cana-3488 55 2 every every DET cana-3488 55 3 𝑝 𝑝 NOUN cana-3488 55 4 , , PUNCT cana-3488 55 5 π‘ž π‘ž PROPN cana-3488 55 6 ∈ ∈ PROPN cana-3488 55 7 𝔸 𝔸 PROPN cana-3488 55 8 and and CCONJ cana-3488 55 9 𝑠 𝑠 PROPN cana-3488 55 10 , , PUNCT cana-3488 55 11 πœ… πœ… X cana-3488 55 12 > > X cana-3488 55 13 0 0 NUM cana-3488 55 14 , , PUNCT cana-3488 55 15 ( ( PUNCT cana-3488 55 16 πœ™1 πœ™1 NOUN cana-3488 55 17 ) ) PUNCT cana-3488 55 18 π’œ1(𝑝 π’œ1(𝑝 PROPN cana-3488 55 19 , , PUNCT cana-3488 55 20 πœ… πœ… PROPN cana-3488 55 21 ) ) PUNCT cana-3488 55 22 + + NUM cana-3488 55 23 π’œ2(𝑝 π’œ2(𝑝 NUM cana-3488 55 24 , , PUNCT cana-3488 55 25 πœ… πœ… X cana-3488 55 26 ) ) PUNCT cana-3488 56 1 + + NOUN cana-3488 56 2 π’œ3(𝑝 π’œ3(𝑝 PROPN cana-3488 56 3 , , PUNCT cana-3488 56 4 πœ… πœ… PROPN cana-3488 56 5 ) ) PUNCT cana-3488 56 6 ≀ ≀ NOUN cana-3488 56 7 3 3 NUM cana-3488 56 8 , , PUNCT cana-3488 56 9 ( ( PUNCT cana-3488 56 10 πœ™2 πœ™2 NOUN cana-3488 56 11 ) ) PUNCT cana-3488 56 12 0 0 NUM cana-3488 56 13 ≀ ≀ PROPN cana-3488 56 14 π’œ1(𝑝 π’œ1(𝑝 NOUN cana-3488 56 15 , , PUNCT cana-3488 56 16 πœ… πœ… NOUN cana-3488 56 17 ) ) PUNCT cana-3488 56 18 ≀ ≀ NUM cana-3488 56 19 1,0 1,0 NUM cana-3488 56 20 ≀ ≀ NUM cana-3488 56 21 π’œ2(𝑝 π’œ2(𝑝 PUNCT cana-3488 56 22 , , PUNCT cana-3488 56 23 πœ… πœ… NOUN cana-3488 56 24 ) ) PUNCT cana-3488 56 25 ≀ ≀ NUM cana-3488 56 26 1,0 1,0 NUM cana-3488 56 27 ≀ ≀ NUM cana-3488 56 28 π’œ3(𝑝 π’œ3(𝑝 PROPN cana-3488 56 29 , , PUNCT cana-3488 56 30 πœ… πœ… PROPN cana-3488 56 31 ) ) PUNCT cana-3488 56 32 ≀ ≀ NUM cana-3488 56 33 1 1 NUM cana-3488 56 34 , , PUNCT cana-3488 56 35 ( ( PUNCT cana-3488 56 36 πœ™3 πœ™3 NOUN cana-3488 56 37 ) ) PUNCT cana-3488 56 38 π’œ1(𝑝 π’œ1(𝑝 PROPN cana-3488 56 39 , , PUNCT cana-3488 56 40 πœ… πœ… PROPN cana-3488 56 41 ) ) PUNCT cana-3488 56 42 > > X cana-3488 56 43 0 0 NUM cana-3488 56 44 , , PUNCT cana-3488 56 45 ( ( PUNCT cana-3488 56 46 πœ™4 πœ™4 NOUN cana-3488 56 47 ) ) PUNCT cana-3488 56 48 π’œ1(𝑝 π’œ1(𝑝 PROPN cana-3488 56 49 , , PUNCT cana-3488 56 50 πœ… πœ… PROPN cana-3488 56 51 ) ) PUNCT cana-3488 56 52 = = SYM cana-3488 56 53 1 1 NUM cana-3488 56 54 , , PUNCT cana-3488 56 55 if if SCONJ cana-3488 56 56 and and CCONJ cana-3488 56 57 only only ADV cana-3488 56 58 if if SCONJ cana-3488 56 59 𝑝 𝑝 ADP cana-3488 56 60 = = SYM cana-3488 56 61 0 0 NUM cana-3488 56 62 . . PUNCT cana-3488 57 1 ( ( PUNCT cana-3488 57 2 πœ™5 πœ™5 PROPN cana-3488 57 3 ) ) PUNCT cana-3488 57 4 π’œ1(𝛼𝑝 π’œ1(𝛼𝑝 PROPN cana-3488 57 5 , , PUNCT cana-3488 57 6 πœ… πœ… X cana-3488 57 7 ) ) PUNCT cana-3488 57 8 = = PUNCT cana-3488 57 9 π’œ1 π’œ1 NOUN cana-3488 57 10 ( ( PUNCT cana-3488 57 11 𝑝 𝑝 PROPN cana-3488 57 12 , , PUNCT cana-3488 57 13 πœ… πœ… X cana-3488 57 14 |𝛼| |𝛼| PROPN cana-3488 57 15 ) ) PUNCT cana-3488 57 16 for for ADP cana-3488 57 17 each each DET cana-3488 57 18 𝛼 𝛼 PROPN cana-3488 57 19 β‰  β‰  PROPN cana-3488 57 20 0 0 NUM cana-3488 57 21 , , PUNCT cana-3488 57 22 ( ( PUNCT cana-3488 57 23 πœ™6 πœ™6 NOUN cana-3488 57 24 ) ) PUNCT cana-3488 57 25 π’œ1(𝑝 π’œ1(𝑝 PROPN cana-3488 57 26 , , PUNCT cana-3488 57 27 πœ… πœ… NOUN cana-3488 57 28 ) ) PUNCT cana-3488 57 29 βˆ— βˆ— NOUN cana-3488 57 30 π’œ1(π‘ž π’œ1(π‘ž PROPN cana-3488 57 31 , , PUNCT cana-3488 57 32 𝑠 𝑠 NOUN cana-3488 57 33 ) ) PUNCT cana-3488 57 34 ≀ ≀ PROPN cana-3488 57 35 π’œ1(𝑝 π’œ1(𝑝 PUNCT cana-3488 58 1 + + CCONJ cana-3488 58 2 π‘ž π‘ž PROPN cana-3488 58 3 , , PUNCT cana-3488 58 4 πœ… πœ… PROPN cana-3488 58 5 + + X cana-3488 58 6 𝑠 𝑠 NOUN cana-3488 58 7 ) ) PUNCT cana-3488 58 8 , , PUNCT cana-3488 58 9 ( ( PUNCT cana-3488 58 10 πœ™7 πœ™7 NOUN cana-3488 58 11 ) ) PUNCT cana-3488 58 12 π’œ1(𝑝,β‹… π’œ1(𝑝,β‹… PROPN cana-3488 58 13 ): ): PUNCT cana-3488 58 14 ( ( PUNCT cana-3488 58 15 0,∞ 0,∞ NOUN cana-3488 58 16 ) ) PUNCT cana-3488 58 17 β†’ β†’ PUNCT cana-3488 59 1 [ [ X cana-3488 59 2 0,1 0,1 NUM cana-3488 59 3 ] ] PUNCT cana-3488 59 4 is be AUX cana-3488 59 5 continuous continuous ADJ cana-3488 59 6 , , PUNCT cana-3488 59 7 ( ( PUNCT cana-3488 59 8 πœ™8 πœ™8 PROPN cana-3488 59 9 ) ) PUNCT cana-3488 59 10 lim lim PROPN cana-3488 59 11 πœ…β†’βˆž πœ…β†’βˆž NUM cana-3488 59 12 π’œ1(𝑝 π’œ1(𝑝 PROPN cana-3488 59 13 , , PUNCT cana-3488 59 14 πœ… πœ… PROPN cana-3488 59 15 ) ) PUNCT cana-3488 59 16 = = SYM cana-3488 59 17 1 1 NUM cana-3488 59 18 and and CCONJ cana-3488 59 19 lim lim PROPN cana-3488 59 20 πœ…β†’0 πœ…β†’0 PUNCT cana-3488 59 21 π’œ1(𝑝 π’œ1(𝑝 PROPN cana-3488 59 22 , , PUNCT cana-3488 59 23 πœ… πœ… PROPN cana-3488 59 24 ) ) PUNCT cana-3488 59 25 = = SYM cana-3488 59 26 0 0 NUM cana-3488 59 27 , , PUNCT cana-3488 59 28 ( ( PUNCT cana-3488 59 29 πœ™9 πœ™9 PROPN cana-3488 59 30 ) ) PUNCT cana-3488 59 31 π’œ2(𝑝 π’œ2(𝑝 PROPN cana-3488 59 32 , , PUNCT cana-3488 59 33 πœ… πœ… X cana-3488 59 34 ) ) PUNCT cana-3488 59 35 < < X cana-3488 59 36 1 1 NUM cana-3488 59 37 , , PUNCT cana-3488 59 38 ( ( PUNCT cana-3488 59 39 πœ™10 πœ™10 NOUN cana-3488 59 40 ) ) PUNCT cana-3488 59 41 π’œ2(𝑝 π’œ2(𝑝 PUNCT cana-3488 59 42 , , PUNCT cana-3488 59 43 πœ… πœ… NOUN cana-3488 59 44 ) ) PUNCT cana-3488 59 45 = = SYM cana-3488 59 46 0 0 NUM cana-3488 59 47 , , PUNCT cana-3488 59 48 if if SCONJ cana-3488 59 49 and and CCONJ cana-3488 59 50 only only ADV cana-3488 59 51 if if SCONJ cana-3488 59 52 𝑝 𝑝 ADP cana-3488 59 53 = = SYM cana-3488 59 54 0 0 PROPN cana-3488 59 55 . . PUNCT cana-3488 60 1 ( ( PUNCT cana-3488 60 2 πœ™11 πœ™11 ADJ cana-3488 60 3 ) ) PUNCT cana-3488 60 4 π’œ2(𝛼𝑝 π’œ2(𝛼𝑝 PROPN cana-3488 60 5 , , PUNCT cana-3488 60 6 πœ… πœ… NOUN cana-3488 60 7 ) ) PUNCT cana-3488 60 8 = = SYM cana-3488 60 9 π’œ2 π’œ2 NOUN cana-3488 60 10 ( ( PUNCT cana-3488 60 11 𝑝 𝑝 PROPN cana-3488 60 12 , , PUNCT cana-3488 60 13 πœ… πœ… X cana-3488 60 14 |𝛼| |𝛼| PROPN cana-3488 60 15 ) ) PUNCT cana-3488 60 16 for for ADP cana-3488 60 17 each each DET cana-3488 60 18 𝛼 𝛼 PROPN cana-3488 60 19 β‰  β‰  PROPN cana-3488 60 20 0 0 NUM cana-3488 60 21 , , PUNCT cana-3488 60 22 ( ( PUNCT cana-3488 60 23 πœ™12 πœ™12 NOUN cana-3488 60 24 ) ) PUNCT cana-3488 60 25 π’œ2(𝑝 π’œ2(𝑝 PROPN cana-3488 60 26 , , PUNCT cana-3488 60 27 πœ… πœ… X cana-3488 60 28 ) ) PUNCT cana-3488 60 29 β‹„ β‹„ PROPN cana-3488 60 30 π’œ2(π‘ž π’œ2(π‘ž PROPN cana-3488 60 31 , , PUNCT cana-3488 60 32 𝑠 𝑠 NOUN cana-3488 60 33 ) ) PUNCT cana-3488 60 34 β‰₯ β‰₯ NOUN cana-3488 60 35 π’œ2(𝑝 π’œ2(𝑝 PUNCT cana-3488 61 1 + + CCONJ cana-3488 61 2 π‘ž π‘ž X cana-3488 61 3 , , PUNCT cana-3488 61 4 πœ… πœ… PROPN cana-3488 61 5 + + X cana-3488 61 6 𝑠 𝑠 NOUN cana-3488 61 7 ) ) PUNCT cana-3488 61 8 , , PUNCT cana-3488 61 9 ( ( PUNCT cana-3488 61 10 πœ™13 πœ™13 NOUN cana-3488 61 11 ) ) PUNCT cana-3488 61 12 π’œ2(𝑝,β‹… π’œ2(𝑝,β‹… PROPN cana-3488 61 13 ): ): PUNCT cana-3488 61 14 ( ( PUNCT cana-3488 61 15 0,∞ 0,∞ NOUN cana-3488 61 16 ) ) PUNCT cana-3488 61 17 β†’ β†’ PUNCT cana-3488 62 1 [ [ X cana-3488 62 2 0,1 0,1 NUM cana-3488 62 3 ] ] PUNCT cana-3488 62 4 is be AUX cana-3488 62 5 continuous continuous ADJ cana-3488 62 6 , , PUNCT cana-3488 62 7 ( ( PUNCT cana-3488 62 8 πœ™14 πœ™14 NOUN cana-3488 62 9 ) ) PUNCT cana-3488 62 10 lim lim PROPN cana-3488 62 11 πœ…β†’βˆž πœ…β†’βˆž NUM cana-3488 62 12 π’œ2(𝑝 π’œ2(𝑝 PUNCT cana-3488 62 13 , , PUNCT cana-3488 62 14 πœ… πœ… NOUN cana-3488 62 15 ) ) PUNCT cana-3488 62 16 = = SYM cana-3488 62 17 0 0 NUM cana-3488 62 18 and and CCONJ cana-3488 62 19 lim lim PROPN cana-3488 62 20 πœ…β†’0 πœ…β†’0 PUNCT cana-3488 62 21 π’œ2(𝑝 π’œ2(𝑝 PROPN cana-3488 62 22 , , PUNCT cana-3488 62 23 πœ… πœ… NOUN cana-3488 62 24 ) ) PUNCT cana-3488 62 25 = = SYM cana-3488 62 26 1 1 NUM cana-3488 62 27 ( ( PUNCT cana-3488 62 28 πœ™15 πœ™15 NOUN cana-3488 62 29 ) ) PUNCT cana-3488 62 30 π’œ3(𝑝 π’œ3(𝑝 NOUN cana-3488 62 31 , , PUNCT cana-3488 62 32 πœ… πœ… PROPN cana-3488 62 33 ) ) PUNCT cana-3488 62 34 < < X cana-3488 62 35 1 1 NUM cana-3488 62 36 , , PUNCT cana-3488 62 37 ( ( PUNCT cana-3488 62 38 πœ™16 πœ™16 NOUN cana-3488 62 39 ) ) PUNCT cana-3488 62 40 π’œ3(𝑝 π’œ3(𝑝 PROPN cana-3488 62 41 , , PUNCT cana-3488 62 42 πœ… πœ… PROPN cana-3488 62 43 ) ) PUNCT cana-3488 62 44 = = SYM cana-3488 62 45 0 0 NUM cana-3488 62 46 , , PUNCT cana-3488 62 47 if if SCONJ cana-3488 62 48 and and CCONJ cana-3488 62 49 only only ADV cana-3488 62 50 if if SCONJ cana-3488 62 51 𝑝 𝑝 ADP cana-3488 62 52 = = SYM cana-3488 62 53 0 0 PROPN cana-3488 62 54 . . PUNCT cana-3488 63 1 ( ( PUNCT cana-3488 63 2 πœ™17 πœ™17 NOUN cana-3488 63 3 ) ) PUNCT cana-3488 63 4 π’œ3(𝛼𝑝 π’œ3(𝛼𝑝 PROPN cana-3488 63 5 , , PUNCT cana-3488 63 6 πœ… πœ… NOUN cana-3488 63 7 ) ) PUNCT cana-3488 63 8 = = SYM cana-3488 63 9 π’œ3 π’œ3 PROPN cana-3488 63 10 ( ( PUNCT cana-3488 63 11 𝑝 𝑝 PROPN cana-3488 63 12 , , PUNCT cana-3488 63 13 πœ… πœ… X cana-3488 63 14 |𝛼| |𝛼| PROPN cana-3488 63 15 ) ) PUNCT cana-3488 63 16 for for ADP cana-3488 63 17 each each DET cana-3488 63 18 𝛼 𝛼 PROPN cana-3488 63 19 β‰  β‰  PROPN cana-3488 63 20 0 0 NUM cana-3488 63 21 , , PUNCT cana-3488 63 22 ( ( PUNCT cana-3488 63 23 πœ™18 πœ™18 NOUN cana-3488 63 24 ) ) PUNCT cana-3488 63 25 π’œ3(𝑝 π’œ3(𝑝 NOUN cana-3488 63 26 , , PUNCT cana-3488 63 27 πœ… πœ… PROPN cana-3488 63 28 ) ) PUNCT cana-3488 63 29 βŠ˜π’œ3(π‘ž βŠ˜π’œ3(π‘ž NUM cana-3488 63 30 , , PUNCT cana-3488 63 31 𝑠 𝑠 NOUN cana-3488 63 32 ) ) PUNCT cana-3488 63 33 β‰₯ β‰₯ NUM cana-3488 63 34 π’œ3(𝑝 π’œ3(𝑝 NOUN cana-3488 64 1 + + CCONJ cana-3488 64 2 π‘ž π‘ž PROPN cana-3488 64 3 , , PUNCT cana-3488 64 4 πœ… πœ… PROPN cana-3488 64 5 + + X cana-3488 64 6 𝑠 𝑠 NOUN cana-3488 64 7 ) ) PUNCT cana-3488 64 8 , , PUNCT cana-3488 64 9 communications communication NOUN cana-3488 64 10 on on ADP cana-3488 64 11 applied apply VERB cana-3488 64 12 nonlinear nonlinear ADJ cana-3488 64 13 analysis analysis NOUN cana-3488 64 14 issn issn NOUN cana-3488 64 15 : : PUNCT cana-3488 64 16 1074 1074 NUM cana-3488 64 17 - - PUNCT cana-3488 64 18 133x 133x NUM cana-3488 64 19 vol vol NOUN cana-3488 64 20 32 32 NUM cana-3488 64 21 no no NOUN cana-3488 64 22 . . PUNCT cana-3488 65 1 7s 7 NOUN cana-3488 65 2 ( ( PUNCT cana-3488 65 3 2025 2025 NUM cana-3488 65 4 ) ) PUNCT cana-3488 65 5 809 809 NUM cana-3488 66 1 https://internationalpubls.com https://internationalpubls.com X cana-3488 66 2 ( ( PUNCT cana-3488 66 3 πœ™19 πœ™19 PROPN cana-3488 66 4 ) ) PUNCT cana-3488 66 5 π’œ3(𝑝,β‹… π’œ3(𝑝,β‹… NUM cana-3488 66 6 ): ): PUNCT cana-3488 66 7 ( ( PUNCT cana-3488 66 8 0,∞ 0,∞ NOUN cana-3488 66 9 ) ) PUNCT cana-3488 66 10 β†’ β†’ PUNCT cana-3488 67 1 [ [ X cana-3488 67 2 0,1 0,1 NUM cana-3488 67 3 ] ] PUNCT cana-3488 67 4 is be AUX cana-3488 67 5 continuous continuous ADJ cana-3488 67 6 , , PUNCT cana-3488 67 7 ( ( PUNCT cana-3488 67 8 πœ™20 πœ™20 NOUN cana-3488 67 9 ) ) PUNCT cana-3488 67 10 lim lim PROPN cana-3488 67 11 πœ…β†’βˆž πœ…β†’βˆž NUM cana-3488 67 12 π’œ3(𝑝 π’œ3(𝑝 PROPN cana-3488 67 13 , , PUNCT cana-3488 67 14 πœ… πœ… PROPN cana-3488 67 15 ) ) PUNCT cana-3488 67 16 = = SYM cana-3488 67 17 0 0 NUM cana-3488 67 18 and and CCONJ cana-3488 67 19 lim lim PROPN cana-3488 67 20 πœ…β†’0 πœ…β†’0 PUNCT cana-3488 67 21 π’œ3(𝑝 π’œ3(𝑝 PROPN cana-3488 67 22 , , PUNCT cana-3488 67 23 πœ… πœ… PROPN cana-3488 67 24 ) ) PUNCT cana-3488 67 25 = = SYM cana-3488 67 26 1 1 NUM cana-3488 67 27 . . SYM cana-3488 67 28 3 3 NUM cana-3488 67 29 stability stability NOUN cana-3488 67 30 results result NOUN cana-3488 67 31 in in ADP cana-3488 67 32 neutrosophic neutrosophic ADJ cana-3488 67 33 normed normed ADJ cana-3488 67 34 space space NOUN cana-3488 67 35 : : PUNCT cana-3488 67 36 hyers hyer NOUN cana-3488 67 37 classical classical ADJ cana-3488 67 38 direct direct ADJ cana-3488 67 39 method method NOUN cana-3488 67 40 theorem theorem VERB cana-3488 67 41 3.1 3.1 NUM cana-3488 67 42 assume assume VERB cana-3488 67 43 that that SCONJ cana-3488 67 44 𝒳𝑀 𝒳𝑀 PROPN cana-3488 67 45 is be AUX cana-3488 67 46 a a DET cana-3488 67 47 ls ls PROPN cana-3488 67 48 , , PUNCT cana-3488 67 49 ( ( PUNCT cana-3488 67 50 π’΅π‘š π’΅π‘š ADJ cana-3488 67 51 , , PUNCT cana-3488 67 52 π’œ1β€²,π’œ2β€² π’œ1β€²,π’œ2β€² ADJ cana-3488 67 53 , , PUNCT cana-3488 67 54 π’œ3β€² π’œ3β€² PRON cana-3488 67 55 ) ) PUNCT cana-3488 67 56 is be AUX cana-3488 67 57 a a DET cana-3488 67 58 nns nns NOUN cana-3488 67 59 and and CCONJ cana-3488 67 60 ( ( PUNCT cana-3488 67 61 𝒴𝑀 𝒴𝑀 PROPN cana-3488 67 62 , , PUNCT cana-3488 67 63 π’œ1,π’œ2 π’œ1,π’œ2 NUM cana-3488 67 64 , , PUNCT cana-3488 67 65 π’œ3 π’œ3 PROPN cana-3488 67 66 ) ) PUNCT cana-3488 67 67 an an DET cana-3488 67 68 nbs nbs PROPN cana-3488 67 69 . . PUNCT cana-3488 68 1 let let VERB cana-3488 68 2 Γ°:𝒳𝑀 Γ°:𝒳𝑀 NOUN cana-3488 68 3 ⟢ ⟢ NOUN cana-3488 68 4 π’΅π‘š π’΅π‘š NOUN cana-3488 68 5 be be AUX cana-3488 68 6 a a DET cana-3488 68 7 function function NOUN cana-3488 68 8 such such ADJ cana-3488 68 9 that that PRON cana-3488 68 10 for for ADP cana-3488 68 11 some some DET cana-3488 68 12 0 0 NUM cana-3488 68 13 < < X cana-3488 68 14 ( ( PUNCT cana-3488 68 15 𝔙 𝔙 PROPN cana-3488 68 16 1113 1113 NUM cana-3488 68 17 ) ) PUNCT cana-3488 69 1 𝐹 𝐹 PROPN cana-3488 69 2 < < X cana-3488 69 3 1 1 NUM cana-3488 69 4 with with ADP cana-3488 69 5 𝐹 𝐹 PROPN cana-3488 69 6 ∈ ∈ PROPN cana-3488 69 7 { { PUNCT cana-3488 69 8 1 1 NUM cana-3488 69 9 , , PUNCT cana-3488 69 10 βˆ’1 βˆ’1 NOUN cana-3488 69 11 } } PUNCT cana-3488 69 12 . . PUNCT cana-3488 70 1 π’œ1β€²(Γ°(11 π’œ1β€²(Γ°(11 PROPN cana-3488 70 2 𝑛𝐹ℨ),β„³ 𝑛𝐹ℨ),β„³ NOUN cana-3488 70 3 ) ) PUNCT cana-3488 70 4 β‰₯ β‰₯ NOUN cana-3488 70 5 π’œ1β€²(𝔙 π’œ1β€²(𝔙 NOUN cana-3488 70 6 𝑛𝐹ð(ℨ),β„³ 𝑛𝐹ð(ℨ),β„³ PROPN cana-3488 70 7 ) ) PUNCT cana-3488 70 8 π’œ2β€²(Γ°(11 π’œ2β€²(Γ°(11 NOUN cana-3488 70 9 𝑛𝐹ℨ),β„³ 𝑛𝐹ℨ),β„³ NOUN cana-3488 70 10 ) ) PUNCT cana-3488 70 11 ≀ ≀ NUM cana-3488 70 12 π’œ2β€²(𝔙 π’œ2β€²(𝔙 SYM cana-3488 70 13 𝑛𝐹ð(ℨ),β„³ 𝑛𝐹ð(ℨ),β„³ PROPN cana-3488 70 14 ) ) PUNCT cana-3488 70 15 π’œ3β€²(Γ°(11 π’œ3β€²(Γ°(11 NOUN cana-3488 70 16 𝑛𝐹ℨ),β„³ 𝑛𝐹ℨ),β„³ NOUN cana-3488 70 17 ) ) PUNCT cana-3488 70 18 ≀ ≀ NUM cana-3488 70 19 π’œ3β€²(𝔙 π’œ3β€²(𝔙 NOUN cana-3488 70 20 𝑛𝐹ð(ℨ),β„³ 𝑛𝐹ð(ℨ),β„³ NOUN cana-3488 70 21 ) ) PUNCT cana-3488 70 22 } } PUNCT cana-3488 70 23 ( ( PUNCT cana-3488 70 24 1 1 X cana-3488 70 25 ) ) PUNCT cana-3488 70 26 for for ADP cana-3488 70 27 all all PRON cana-3488 70 28 ℨ ℨ NOUN cana-3488 70 29 ∈ ∈ NOUN cana-3488 70 30 𝒳𝑀 𝒳𝑀 NOUN cana-3488 70 31 and and CCONJ cana-3488 70 32 all all DET cana-3488 70 33 β„³ β„³ PROPN cana-3488 70 34 > > X cana-3488 70 35 0 0 PUNCT cana-3488 71 1 and and CCONJ cana-3488 71 2 lim lim PROPN cana-3488 71 3 π‘›β†’βˆž π‘›β†’βˆž NUM cana-3488 71 4 π’œ1β€²(Γ°(11 π’œ1β€²(Γ°(11 PROPN cana-3488 71 5 𝐹𝑛ℨ 𝐹𝑛ℨ PROPN cana-3488 71 6 ) ) PUNCT cana-3488 71 7 , , PUNCT cana-3488 71 8 1113𝐹𝑛ℳ 1113𝐹𝑛ℳ NUM cana-3488 71 9 ) ) PUNCT cana-3488 71 10 = = SYM cana-3488 71 11 1 1 NUM cana-3488 71 12 lim lim NOUN cana-3488 71 13 π‘›β†’βˆž π‘›β†’βˆž NUM cana-3488 71 14 π’œ2β€²(Γ°(11 π’œ2β€²(Γ°(11 PROPN cana-3488 71 15 𝐹𝑛ℨ 𝐹𝑛ℨ PROPN cana-3488 71 16 ) ) PUNCT cana-3488 71 17 , , PUNCT cana-3488 71 18 1113𝐹𝑛ℳ 1113𝐹𝑛ℳ NUM cana-3488 71 19 ) ) PUNCT cana-3488 71 20 = = SYM cana-3488 71 21 0 0 NUM cana-3488 72 1 lim lim NOUN cana-3488 72 2 π‘›β†’βˆž π‘›β†’βˆž NUM cana-3488 72 3 π’œ3β€²(Γ°(11 π’œ3β€²(Γ°(11 PROPN cana-3488 72 4 𝐹𝑛ℨ 𝐹𝑛ℨ PROPN cana-3488 72 5 ) ) PUNCT cana-3488 72 6 , , PUNCT cana-3488 72 7 1113𝐹𝑛ℳ 1113𝐹𝑛ℳ NUM cana-3488 72 8 ) ) PUNCT cana-3488 72 9 = = SYM cana-3488 72 10 0 0 X cana-3488 72 11 } } PUNCT cana-3488 72 12 ( ( PUNCT cana-3488 72 13 2 2 X cana-3488 72 14 ) ) PUNCT cana-3488 72 15 for for ADP cana-3488 72 16 all all DET cana-3488 72 17 ℨ ℨ NOUN cana-3488 72 18 ∈ ∈ NOUN cana-3488 72 19 𝒳𝑀 𝒳𝑀 NOUN cana-3488 72 20 and and CCONJ cana-3488 72 21 all all DET cana-3488 72 22 β„³ β„³ PROPN cana-3488 72 23 > > X cana-3488 72 24 0 0 PUNCT cana-3488 72 25 . . PUNCT cana-3488 73 1 let let VERB cana-3488 73 2 an an DET cana-3488 73 3 odd odd ADJ cana-3488 73 4 function function NOUN cana-3488 73 5 𝔉:𝒳𝑀 𝔉:𝒳𝑀 NOUN cana-3488 73 6 βŸΆπ’΄π‘€ βŸΆπ’΄π‘€ NOUN cana-3488 73 7 satisfying satisfy VERB cana-3488 73 8 π’œ1(𝔉(11ℨ π’œ1(𝔉(11ℨ PUNCT cana-3488 73 9 ) ) PUNCT cana-3488 74 1 βˆ’ βˆ’ ADP cana-3488 74 2 1,88,30,57,02,60,326𝔉(ℨ 1,88,30,57,02,60,326𝔉(ℨ NUM cana-3488 74 3 ) ) PUNCT cana-3488 74 4 + + CCONJ cana-3488 74 5 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ X cana-3488 74 6 ) ) PUNCT cana-3488 74 7 β‰₯ β‰₯ NOUN cana-3488 74 8 π’œ1β€²(Γ°(ℨ),β„³ π’œ1β€²(Γ°(ℨ),β„³ NOUN cana-3488 74 9 ) ) PUNCT cana-3488 74 10 π’œ2(𝔉(11ℨ π’œ2(𝔉(11ℨ PUNCT cana-3488 74 11 ) ) PUNCT cana-3488 74 12 βˆ’ βˆ’ PROPN cana-3488 74 13 1,88,30,57,02,60,326𝔉(ℨ 1,88,30,57,02,60,326𝔉(ℨ NUM cana-3488 74 14 ) ) PUNCT cana-3488 74 15 + + CCONJ cana-3488 74 16 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ X cana-3488 74 17 ) ) PUNCT cana-3488 74 18 ≀ ≀ NOUN cana-3488 74 19 π’œ2β€²(Γ°(ℨ),β„³ π’œ2β€²(Γ°(ℨ),β„³ NOUN cana-3488 74 20 ) ) PUNCT cana-3488 74 21 π’œ3(𝔉(11ℨ π’œ3(𝔉(11ℨ X cana-3488 74 22 ) ) PUNCT cana-3488 74 23 βˆ’ βˆ’ PROPN cana-3488 74 24 1,88,30,57,02,60,326𝔉(ℨ 1,88,30,57,02,60,326𝔉(ℨ NUM cana-3488 74 25 ) ) PUNCT cana-3488 74 26 + + CCONJ cana-3488 74 27 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ X cana-3488 74 28 ) ) PUNCT cana-3488 74 29 ≀ ≀ NOUN cana-3488 75 1 π’œ3β€²(Γ°(ℨ),β„³ π’œ3β€²(Γ°(ℨ),β„³ NOUN cana-3488 75 2 ) ) PUNCT cana-3488 75 3 } } PUNCT cana-3488 75 4 ( ( PUNCT cana-3488 75 5 3 3 X cana-3488 75 6 ) ) PUNCT cana-3488 75 7 for for ADP cana-3488 75 8 all all PRON cana-3488 75 9 ℨ ℨ NOUN cana-3488 75 10 ∈ ∈ NOUN cana-3488 75 11 𝒳𝑀 𝒳𝑀 NOUN cana-3488 75 12 and and CCONJ cana-3488 75 13 all all DET cana-3488 75 14 β„³ β„³ PROPN cana-3488 75 15 > > X cana-3488 75 16 0 0 NUM cana-3488 75 17 . . PUNCT cana-3488 76 1 then then ADV cana-3488 76 2 there there PRON cana-3488 76 3 exists exist VERB cana-3488 76 4 a a DET cana-3488 76 5 unique unique ADJ cana-3488 76 6 tridecic tridecic NOUN cana-3488 76 7 mapping mapping NOUN cana-3488 76 8 𝒯:𝒳𝑀 𝒯:𝒳𝑀 CCONJ cana-3488 76 9 βŸΆπ’΄π‘€ βŸΆπ’΄π‘€ PROPN cana-3488 76 10 satisfying satisfying NOUN cana-3488 76 11 ( ( PUNCT cana-3488 76 12 1 1 NUM cana-3488 76 13 ) ) PUNCT cana-3488 76 14 and and CCONJ cana-3488 76 15 π’œ1(𝔉(ℨ π’œ1(𝔉(ℨ NOUN cana-3488 76 16 ) ) PUNCT cana-3488 76 17 βˆ’ βˆ’ PRON cana-3488 76 18 𝒯(ℨ),β„³ 𝒯(ℨ),β„³ NOUN cana-3488 76 19 ) ) PUNCT cana-3488 76 20 β‰₯ β‰₯ NOUN cana-3488 76 21 π’œ1β€²(Γ°(ℨ),β„³|1113 π’œ1β€²(Γ°(ℨ),β„³|1113 NOUN cana-3488 76 22 βˆ’ βˆ’ PROPN cana-3488 76 23 𝔙| 𝔙| PROPN cana-3488 76 24 ) ) PUNCT cana-3488 76 25 π’œ2(𝔉(ℨ π’œ2(𝔉(ℨ PROPN cana-3488 76 26 ) ) PUNCT cana-3488 76 27 βˆ’ βˆ’ NOUN cana-3488 76 28 𝒯(ℨ),β„³ 𝒯(ℨ),β„³ NOUN cana-3488 76 29 ) ) PUNCT cana-3488 76 30 ≀ ≀ NUM cana-3488 76 31 π’œ2β€²(Γ°(ℨ),β„³|1113 π’œ2β€²(Γ°(ℨ),β„³|1113 PROPN cana-3488 76 32 βˆ’π”™| βˆ’π”™| PROPN cana-3488 76 33 ) ) PUNCT cana-3488 76 34 π’œ3(𝔉(ℨ π’œ3(𝔉(ℨ PROPN cana-3488 76 35 ) ) PUNCT cana-3488 76 36 βˆ’ βˆ’ NOUN cana-3488 76 37 𝒯(ℨ),β„³ 𝒯(ℨ),β„³ NOUN cana-3488 76 38 ) ) PUNCT cana-3488 76 39 ≀ ≀ NUM cana-3488 76 40 π’œ3β€²(Γ°(ℨ),β„³|1113 π’œ3β€²(Γ°(ℨ),β„³|1113 PROPN cana-3488 76 41 βˆ’π”™| βˆ’π”™| PROPN cana-3488 76 42 ) ) PUNCT cana-3488 76 43 } } PUNCT cana-3488 76 44 ( ( PUNCT cana-3488 76 45 4 4 X cana-3488 76 46 ) ) PUNCT cana-3488 76 47 for for ADP cana-3488 76 48 all all PRON cana-3488 76 49 ℨ ℨ NOUN cana-3488 76 50 ∈ ∈ NOUN cana-3488 76 51 𝒳𝑀 𝒳𝑀 NOUN cana-3488 76 52 and and CCONJ cana-3488 76 53 all all DET cana-3488 76 54 β„³ β„³ PROPN cana-3488 76 55 > > X cana-3488 76 56 0 0 PROPN cana-3488 76 57 . . PUNCT cana-3488 77 1 proof proof NOUN cana-3488 77 2 . . PUNCT cana-3488 78 1 for for ADP cana-3488 78 2 the the DET cana-3488 78 3 first first ADJ cana-3488 78 4 case case NOUN cana-3488 78 5 𝐹 𝐹 PROPN cana-3488 78 6 = = NOUN cana-3488 78 7 1 1 X cana-3488 78 8 . . PUNCT cana-3488 78 9 using use VERB cana-3488 78 10 oddness oddness NOUN cana-3488 78 11 of of ADP cana-3488 78 12 𝔉 𝔉 PROPN cana-3488 78 13 in in ADP cana-3488 78 14 in in ADP cana-3488 78 15 ( ( PUNCT cana-3488 78 16 3 3 NUM cana-3488 78 17 ) ) PUNCT cana-3488 78 18 , , PUNCT cana-3488 78 19 we we PRON cana-3488 78 20 obtain obtain VERB cana-3488 78 21 communications communication NOUN cana-3488 78 22 on on ADP cana-3488 78 23 applied apply VERB cana-3488 78 24 nonlinear nonlinear ADJ cana-3488 78 25 analysis analysis NOUN cana-3488 78 26 issn issn NOUN cana-3488 78 27 : : PUNCT cana-3488 78 28 1074 1074 NUM cana-3488 78 29 - - PUNCT cana-3488 78 30 133x 133x NUM cana-3488 78 31 vol vol NOUN cana-3488 78 32 32 32 NUM cana-3488 78 33 no no NOUN cana-3488 78 34 . . PUNCT cana-3488 79 1 7s 7 NOUN cana-3488 79 2 ( ( PUNCT cana-3488 79 3 2025 2025 NUM cana-3488 79 4 ) ) PUNCT cana-3488 79 5 810 810 NUM cana-3488 79 6 https://internationalpubls.com https://internationalpubls.com X cana-3488 79 7 π’œ1(𝔉(11ℨ π’œ1(𝔉(11ℨ PUNCT cana-3488 79 8 ) ) PUNCT cana-3488 80 1 βˆ’ βˆ’ PROPN cana-3488 80 2 11 11 NUM cana-3488 80 3 13𝔉(ℨ),β„³ 13𝔉(ℨ),β„³ NOUN cana-3488 80 4 ) ) PUNCT cana-3488 80 5 β‰₯ β‰₯ NOUN cana-3488 81 1 π’œ1β€²(Γ°(ℨ),β„³ π’œ1β€²(Γ°(ℨ),β„³ NOUN cana-3488 81 2 ) ) PUNCT cana-3488 81 3 π’œ2(𝔉(11ℨ π’œ2(𝔉(11ℨ PUNCT cana-3488 81 4 ) ) PUNCT cana-3488 82 1 βˆ’ βˆ’ PROPN cana-3488 82 2 11 11 NUM cana-3488 82 3 13𝔉(ℨ),β„³ 13𝔉(ℨ),β„³ NOUN cana-3488 82 4 ) ) PUNCT cana-3488 82 5 ≀ ≀ NOUN cana-3488 83 1 π’œ2β€²(Γ°(ℨ),β„³ π’œ2β€²(Γ°(ℨ),β„³ NOUN cana-3488 83 2 ) ) PUNCT cana-3488 83 3 π’œ3(𝔉(11ℨ π’œ3(𝔉(11ℨ X cana-3488 83 4 ) ) PUNCT cana-3488 83 5 βˆ’ βˆ’ PROPN cana-3488 83 6 11 11 NUM cana-3488 83 7 13𝔉(ℨ),β„³ 13𝔉(ℨ),β„³ NOUN cana-3488 83 8 ) ) PUNCT cana-3488 83 9 ≀ ≀ NOUN cana-3488 84 1 π’œ3β€²(Γ°(ℨ),β„³ π’œ3β€²(Γ°(ℨ),β„³ NOUN cana-3488 84 2 ) ) PUNCT cana-3488 84 3 } } PUNCT cana-3488 84 4 ( ( PUNCT cana-3488 84 5 5 5 NUM cana-3488 84 6 ) ) PUNCT cana-3488 84 7 for for ADP cana-3488 84 8 all all PRON cana-3488 84 9 ℨ ℨ NOUN cana-3488 84 10 ∈ ∈ NOUN cana-3488 84 11 𝒳𝑀 𝒳𝑀 NOUN cana-3488 84 12 and and CCONJ cana-3488 84 13 all all DET cana-3488 84 14 β„³ β„³ PROPN cana-3488 84 15 > > X cana-3488 84 16 0 0 NUM cana-3488 84 17 . . PUNCT cana-3488 85 1 using use VERB cana-3488 85 2 ( ( PUNCT cana-3488 85 3 nns5 nns5 PROPN cana-3488 85 4 ) ) PUNCT cana-3488 85 5 , , PUNCT cana-3488 85 6 ( ( PUNCT cana-3488 85 7 nns11 nns11 PROPN cana-3488 85 8 ) ) PUNCT cana-3488 85 9 and and CCONJ cana-3488 85 10 ( ( PUNCT cana-3488 85 11 nns17 nns17 PROPN cana-3488 85 12 ) ) PUNCT cana-3488 85 13 in in ADP cana-3488 85 14 ( ( PUNCT cana-3488 85 15 5 5 NUM cana-3488 85 16 ) ) PUNCT cana-3488 85 17 , , PUNCT cana-3488 85 18 we we PRON cana-3488 85 19 have have VERB cana-3488 85 20 π’œ1 π’œ1 NOUN cana-3488 85 21 ( ( PUNCT cana-3488 85 22 𝔉(11ℨ 𝔉(11ℨ NOUN cana-3488 85 23 ) ) PUNCT cana-3488 86 1 1113 1113 NUM cana-3488 86 2 βˆ’ βˆ’ PROPN cana-3488 86 3 𝔉(ℨ 𝔉(ℨ NOUN cana-3488 86 4 ) ) PUNCT cana-3488 86 5 , , PUNCT cana-3488 86 6 β„³ β„³ PROPN cana-3488 86 7 1113 1113 NUM cana-3488 86 8 ) ) PUNCT cana-3488 86 9 β‰₯ β‰₯ NUM cana-3488 86 10 π’œ1β€²(Γ°(ℨ),β„³ π’œ1β€²(Γ°(ℨ),β„³ NOUN cana-3488 86 11 ) ) PUNCT cana-3488 86 12 π’œ2 π’œ2 PROPN cana-3488 86 13 ( ( PUNCT cana-3488 86 14 𝔉(11ℨ 𝔉(11ℨ NOUN cana-3488 86 15 ) ) PUNCT cana-3488 86 16 1113 1113 NUM cana-3488 86 17 βˆ’ βˆ’ PROPN cana-3488 86 18 𝔉(ℨ 𝔉(ℨ NOUN cana-3488 86 19 ) ) PUNCT cana-3488 86 20 , , PUNCT cana-3488 86 21 β„³ β„³ PROPN cana-3488 86 22 1113 1113 NUM cana-3488 86 23 ) ) PUNCT cana-3488 86 24 ≀ ≀ NUM cana-3488 87 1 π’œ2β€²(Γ°(ℨ),β„³ π’œ2β€²(Γ°(ℨ),β„³ NOUN cana-3488 87 2 ) ) PUNCT cana-3488 87 3 π’œ3 π’œ3 PROPN cana-3488 87 4 ( ( PUNCT cana-3488 87 5 𝔉(11ℨ 𝔉(11ℨ NOUN cana-3488 87 6 ) ) PUNCT cana-3488 87 7 1113 1113 NUM cana-3488 87 8 βˆ’ βˆ’ PROPN cana-3488 87 9 𝔉(ℨ 𝔉(ℨ NOUN cana-3488 87 10 ) ) PUNCT cana-3488 87 11 , , PUNCT cana-3488 87 12 β„³ β„³ PROPN cana-3488 87 13 1113 1113 NUM cana-3488 87 14 ) ) PUNCT cana-3488 87 15 ≀ ≀ NUM cana-3488 88 1 π’œ3β€²(Γ°(ℨ),β„³ π’œ3β€²(Γ°(ℨ),β„³ NOUN cana-3488 88 2 ) ) PUNCT cana-3488 88 3 } } PUNCT cana-3488 88 4 ( ( PUNCT cana-3488 88 5 6 6 NUM cana-3488 88 6 ) ) PUNCT cana-3488 88 7 for for ADP cana-3488 88 8 all all PRON cana-3488 88 9 ℨ ℨ NOUN cana-3488 88 10 ∈ ∈ NOUN cana-3488 88 11 𝒳𝑀 𝒳𝑀 NOUN cana-3488 88 12 and and CCONJ cana-3488 88 13 all all DET cana-3488 88 14 β„³ β„³ PROPN cana-3488 88 15 > > X cana-3488 88 16 0 0 X cana-3488 88 17 . . PUNCT cana-3488 89 1 let let VERB cana-3488 89 2 us we PRON cana-3488 89 3 take take VERB cana-3488 89 4 ℨ ℨ NOUN cana-3488 89 5 by by ADP cana-3488 89 6 11𝑛ℨ 11𝑛ℨ NUM cana-3488 89 7 in in ADP cana-3488 89 8 ( ( PUNCT cana-3488 89 9 6 6 NUM cana-3488 89 10 ) ) PUNCT cana-3488 89 11 , , PUNCT cana-3488 89 12 we we PRON cana-3488 89 13 arrive arrive VERB cana-3488 89 14 π’œ1 π’œ1 PROPN cana-3488 89 15 ( ( PUNCT cana-3488 89 16 𝔉(11(𝑛+1)ℨ 𝔉(11(𝑛+1)ℨ PROPN cana-3488 89 17 ) ) PUNCT cana-3488 89 18 1113 1113 NUM cana-3488 89 19 βˆ’ βˆ’ NOUN cana-3488 89 20 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 89 21 ) ) PUNCT cana-3488 89 22 , , PUNCT cana-3488 89 23 β„³ β„³ PROPN cana-3488 89 24 1113 1113 NUM cana-3488 89 25 ) ) PUNCT cana-3488 89 26 β‰₯ β‰₯ NOUN cana-3488 89 27 π’œ1β€²(Γ°(11 π’œ1β€²(Γ°(11 PROPN cana-3488 89 28 𝑛ℨ),β„³ 𝑛ℨ),β„³ NOUN cana-3488 89 29 ) ) PUNCT cana-3488 89 30 π’œ2 π’œ2 PROPN cana-3488 89 31 ( ( PUNCT cana-3488 89 32 𝔉(11(𝑛+1)ℨ 𝔉(11(𝑛+1)ℨ PROPN cana-3488 89 33 ) ) PUNCT cana-3488 89 34 1113 1113 NUM cana-3488 89 35 βˆ’ βˆ’ NOUN cana-3488 89 36 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 89 37 ) ) PUNCT cana-3488 89 38 , , PUNCT cana-3488 89 39 β„³ β„³ PROPN cana-3488 89 40 1113 1113 NUM cana-3488 89 41 ) ) PUNCT cana-3488 90 1 ≀ ≀ NUM cana-3488 90 2 π’œ2β€²(Γ°(11 π’œ2β€²(Γ°(11 X cana-3488 90 3 𝑛ℨ),β„³ 𝑛ℨ),β„³ NOUN cana-3488 90 4 ) ) PUNCT cana-3488 90 5 π’œ3 π’œ3 PROPN cana-3488 90 6 ( ( PUNCT cana-3488 90 7 𝔉(11(𝑛+1)ℨ 𝔉(11(𝑛+1)ℨ PROPN cana-3488 90 8 ) ) PUNCT cana-3488 90 9 1113 1113 NUM cana-3488 90 10 βˆ’ βˆ’ NOUN cana-3488 90 11 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 90 12 ) ) PUNCT cana-3488 90 13 , , PUNCT cana-3488 90 14 β„³ β„³ PROPN cana-3488 90 15 1113 1113 NUM cana-3488 90 16 ) ) PUNCT cana-3488 90 17 ≀ ≀ PUNCT cana-3488 91 1 π’œ3β€²(Γ°(11 π’œ3β€²(Γ°(11 PROPN cana-3488 91 2 𝑛ℨ),β„³ 𝑛ℨ),β„³ NOUN cana-3488 91 3 ) ) PUNCT cana-3488 91 4 } } PUNCT cana-3488 91 5 ( ( PUNCT cana-3488 91 6 7 7 X cana-3488 91 7 ) ) PUNCT cana-3488 91 8 for for ADP cana-3488 91 9 all all DET cana-3488 91 10 ℨ ℨ NOUN cana-3488 91 11 ∈ ∈ NOUN cana-3488 91 12 𝒳𝑀 𝒳𝑀 NOUN cana-3488 91 13 and and CCONJ cana-3488 91 14 all all DET cana-3488 91 15 β„³ β„³ PROPN cana-3488 91 16 > > X cana-3488 91 17 0 0 NUM cana-3488 91 18 . . PUNCT cana-3488 92 1 it it PRON cana-3488 92 2 is be AUX cana-3488 92 3 simple simple ADJ cana-3488 92 4 to to PART cana-3488 92 5 confirm confirm VERB cana-3488 92 6 that that SCONJ cana-3488 92 7 ( ( PUNCT cana-3488 92 8 7 7 NUM cana-3488 92 9 ) ) PUNCT cana-3488 92 10 and and CCONJ cana-3488 92 11 using use VERB cana-3488 92 12 ( ( PUNCT cana-3488 92 13 1 1 NUM cana-3488 92 14 ) ) PUNCT cana-3488 92 15 , , PUNCT cana-3488 92 16 ( ( PUNCT cana-3488 92 17 nns5 nns5 PROPN cana-3488 92 18 ) ) PUNCT cana-3488 92 19 , , PUNCT cana-3488 92 20 ( ( PUNCT cana-3488 92 21 nns11 nns11 PROPN cana-3488 92 22 ) ) PUNCT cana-3488 92 23 and and CCONJ cana-3488 92 24 ( ( PUNCT cana-3488 92 25 nns17 nns17 PROPN cana-3488 92 26 ) ) PUNCT cana-3488 92 27 that that SCONJ cana-3488 92 28 π’œ1 π’œ1 PROPN cana-3488 92 29 ( ( PUNCT cana-3488 92 30 𝔉(11(𝑛+1)ℨ 𝔉(11(𝑛+1)ℨ PROPN cana-3488 92 31 ) ) PUNCT cana-3488 92 32 1113(𝑛+1 1113(𝑛+1 PROPN cana-3488 92 33 ) ) PUNCT cana-3488 92 34 βˆ’ βˆ’ NOUN cana-3488 92 35 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 92 36 ) ) PUNCT cana-3488 92 37 1113𝑛 1113𝑛 NOUN cana-3488 92 38 , , PUNCT cana-3488 92 39 β„³ β„³ PROPN cana-3488 92 40 1113β‹…1113𝑛 1113β‹…1113𝑛 NUM cana-3488 92 41 ) ) PUNCT cana-3488 92 42 β‰₯ β‰₯ NOUN cana-3488 92 43 π’œ1β€² π’œ1β€² NOUN cana-3488 92 44 ( ( PUNCT cana-3488 92 45 Γ°(ℨ Γ°(ℨ PROPN cana-3488 92 46 ) ) PUNCT cana-3488 92 47 , , PUNCT cana-3488 92 48 β„³ β„³ PROPN cana-3488 92 49 𝔙𝑛 𝔙𝑛 PROPN cana-3488 92 50 ) ) PUNCT cana-3488 92 51 π’œ2 π’œ2 PROPN cana-3488 92 52 ( ( PUNCT cana-3488 92 53 𝔉(11𝑛+1ℨ 𝔉(11𝑛+1ℨ PROPN cana-3488 92 54 ) ) PUNCT cana-3488 92 55 1113(𝑛+1 1113(𝑛+1 NOUN cana-3488 92 56 ) ) PUNCT cana-3488 92 57 βˆ’ βˆ’ NOUN cana-3488 92 58 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 92 59 ) ) PUNCT cana-3488 92 60 1113𝑛 1113𝑛 NOUN cana-3488 92 61 , , PUNCT cana-3488 92 62 β„³ β„³ PROPN cana-3488 92 63 1113β‹…1113𝑛 1113β‹…1113𝑛 NUM cana-3488 92 64 ) ) PUNCT cana-3488 92 65 ≀ ≀ NUM cana-3488 92 66 π’œ2β€² π’œ2β€² ADV cana-3488 92 67 ( ( PUNCT cana-3488 92 68 Γ°(ℨ Γ°(ℨ NOUN cana-3488 92 69 ) ) PUNCT cana-3488 92 70 , , PUNCT cana-3488 92 71 β„³ β„³ PROPN cana-3488 92 72 𝔙𝑛 𝔙𝑛 PROPN cana-3488 92 73 ) ) PUNCT cana-3488 92 74 π’œ3 π’œ3 PROPN cana-3488 92 75 ( ( PUNCT cana-3488 92 76 𝔉(11(𝑛+1)ℨ 𝔉(11(𝑛+1)ℨ PROPN cana-3488 92 77 ) ) PUNCT cana-3488 92 78 1113(𝑛+1 1113(𝑛+1 PROPN cana-3488 92 79 ) ) PUNCT cana-3488 92 80 βˆ’ βˆ’ NOUN cana-3488 92 81 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 92 82 ) ) PUNCT cana-3488 92 83 1113𝑛 1113𝑛 NOUN cana-3488 92 84 , , PUNCT cana-3488 92 85 β„³ β„³ PROPN cana-3488 92 86 1113β‹…1113𝑛 1113β‹…1113𝑛 NUM cana-3488 92 87 ) ) PUNCT cana-3488 93 1 ≀ ≀ NUM cana-3488 93 2 π’œ3β€² π’œ3β€² NOUN cana-3488 93 3 ( ( PUNCT cana-3488 93 4 Γ°(ℨ Γ°(ℨ PROPN cana-3488 93 5 ) ) PUNCT cana-3488 93 6 , , PUNCT cana-3488 93 7 β„³ β„³ PROPN cana-3488 93 8 𝔙𝑛 𝔙𝑛 PROPN cana-3488 93 9 ) ) PUNCT cana-3488 93 10 } } PUNCT cana-3488 93 11 ( ( PUNCT cana-3488 93 12 8) 8) NUM cana-3488 93 13 for for ADP cana-3488 93 14 all all DET cana-3488 93 15 ℨ ℨ NOUN cana-3488 93 16 ∈ ∈ NOUN cana-3488 93 17 𝒳𝑀 𝒳𝑀 NOUN cana-3488 93 18 and and CCONJ cana-3488 93 19 all all DET cana-3488 93 20 β„³ β„³ PROPN cana-3488 93 21 > > X cana-3488 93 22 0 0 X cana-3488 93 23 . . PUNCT cana-3488 93 24 swapping swap VERB cana-3488 93 25 β„³ β„³ NOUN cana-3488 93 26 into into ADP cana-3488 93 27 𝔙𝑛ℳ 𝔙𝑛ℳ NOUN cana-3488 93 28 in in ADP cana-3488 93 29 ( ( PUNCT cana-3488 93 30 8) 8) NUM cana-3488 93 31 , , PUNCT cana-3488 93 32 we we PRON cana-3488 93 33 have have VERB cana-3488 93 34 π’œ1 π’œ1 NOUN cana-3488 93 35 ( ( PUNCT cana-3488 93 36 𝔉(11𝑛+1ℨ 𝔉(11𝑛+1ℨ PROPN cana-3488 93 37 ) ) PUNCT cana-3488 93 38 1113(𝑛+1 1113(𝑛+1 NOUN cana-3488 93 39 ) ) PUNCT cana-3488 93 40 βˆ’ βˆ’ NOUN cana-3488 93 41 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 93 42 ) ) PUNCT cana-3488 93 43 1113𝑛 1113𝑛 NOUN cana-3488 93 44 , , PUNCT cana-3488 93 45 ℳ⋅𝔙𝑛 ℳ⋅𝔙𝑛 PROPN cana-3488 93 46 1113β‹…1113𝑛 1113β‹…1113𝑛 NUM cana-3488 93 47 ) ) PUNCT cana-3488 93 48 β‰₯ β‰₯ NOUN cana-3488 94 1 π’œ1β€²(Γ°(ℨ),β„³ π’œ1β€²(Γ°(ℨ),β„³ NOUN cana-3488 94 2 ) ) PUNCT cana-3488 94 3 π’œ2 π’œ2 PROPN cana-3488 94 4 ( ( PUNCT cana-3488 94 5 𝔉(11𝑛+1ℨ 𝔉(11𝑛+1ℨ PROPN cana-3488 94 6 ) ) PUNCT cana-3488 94 7 1113(𝑛+1 1113(𝑛+1 NOUN cana-3488 94 8 ) ) PUNCT cana-3488 94 9 βˆ’ βˆ’ NOUN cana-3488 94 10 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 94 11 ) ) PUNCT cana-3488 94 12 1113𝑛 1113𝑛 NOUN cana-3488 94 13 , , PUNCT cana-3488 94 14 ℳ⋅𝔙𝑛 ℳ⋅𝔙𝑛 PROPN cana-3488 94 15 1113β‹…1113𝑛 1113β‹…1113𝑛 NUM cana-3488 94 16 ) ) PUNCT cana-3488 94 17 ≀ ≀ NOUN cana-3488 95 1 π’œ2β€²(Γ°(ℨ),β„³ π’œ2β€²(Γ°(ℨ),β„³ NOUN cana-3488 95 2 ) ) PUNCT cana-3488 95 3 π’œ3 π’œ3 PROPN cana-3488 95 4 ( ( PUNCT cana-3488 95 5 𝔉(11𝑛+1ℨ 𝔉(11𝑛+1ℨ PROPN cana-3488 95 6 ) ) PUNCT cana-3488 95 7 1113(𝑛+1 1113(𝑛+1 NOUN cana-3488 95 8 ) ) PUNCT cana-3488 95 9 βˆ’ βˆ’ NOUN cana-3488 95 10 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 95 11 ) ) PUNCT cana-3488 95 12 1113𝑛 1113𝑛 NOUN cana-3488 95 13 , , PUNCT cana-3488 95 14 ℳ⋅𝔙𝑛 ℳ⋅𝔙𝑛 PROPN cana-3488 95 15 1113β‹…1113𝑛 1113β‹…1113𝑛 NUM cana-3488 95 16 ) ) PUNCT cana-3488 95 17 ≀ ≀ NUM cana-3488 96 1 π’œ3β€²(Γ°(ℨ),β„³ π’œ3β€²(Γ°(ℨ),β„³ NOUN cana-3488 96 2 ) ) PUNCT cana-3488 96 3 } } PUNCT cana-3488 96 4 ( ( PUNCT cana-3488 96 5 9 9 X cana-3488 96 6 ) ) PUNCT cana-3488 96 7 communications communication NOUN cana-3488 96 8 on on ADP cana-3488 96 9 applied apply VERB cana-3488 96 10 nonlinear nonlinear ADJ cana-3488 96 11 analysis analysis NOUN cana-3488 96 12 issn issn NOUN cana-3488 96 13 : : PUNCT cana-3488 96 14 1074 1074 NUM cana-3488 96 15 - - PUNCT cana-3488 96 16 133x 133x NUM cana-3488 96 17 vol vol NOUN cana-3488 96 18 32 32 NUM cana-3488 96 19 no no NOUN cana-3488 96 20 . . PUNCT cana-3488 97 1 7s 7 NOUN cana-3488 97 2 ( ( PUNCT cana-3488 97 3 2025 2025 NUM cana-3488 97 4 ) ) PUNCT cana-3488 97 5 811 811 NUM cana-3488 97 6 https://internationalpubls.com https://internationalpubls.com X cana-3488 97 7 for for ADP cana-3488 97 8 all all DET cana-3488 97 9 ℨ ℨ NOUN cana-3488 97 10 ∈ ∈ NOUN cana-3488 97 11 𝒳𝑀 𝒳𝑀 NOUN cana-3488 97 12 and and CCONJ cana-3488 97 13 all all DET cana-3488 97 14 β„³ β„³ PROPN cana-3488 97 15 > > X cana-3488 97 16 0 0 NUM cana-3488 97 17 . . PUNCT cana-3488 98 1 it it PRON cana-3488 98 2 is be AUX cana-3488 98 3 simple simple ADJ cana-3488 98 4 to to PART cana-3488 98 5 observe observe VERB cana-3488 98 6 that that SCONJ cana-3488 98 7 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 98 8 ) ) PUNCT cana-3488 98 9 1113𝑛 1113𝑛 NOUN cana-3488 98 10 βˆ’ βˆ’ NOUN cana-3488 98 11 𝔉(ℨ 𝔉(ℨ NOUN cana-3488 98 12 ) ) PUNCT cana-3488 98 13 = = SYM cana-3488 98 14 βˆ‘π‘›βˆ’1𝑖=0 βˆ‘π‘›βˆ’1𝑖=0 PROPN cana-3488 98 15 𝔉(11𝑖+1ℨ 𝔉(11𝑖+1ℨ PROPN cana-3488 98 16 ) ) PUNCT cana-3488 98 17 1113(𝑖+1 1113(𝑖+1 NUM cana-3488 98 18 ) ) PUNCT cana-3488 98 19 βˆ’ βˆ’ ADP cana-3488 98 20 𝔉(11𝑖ℨ 𝔉(11𝑖ℨ NOUN cana-3488 98 21 ) ) PUNCT cana-3488 98 22 1113𝑖 1113𝑖 NUM cana-3488 98 23 ( ( PUNCT cana-3488 98 24 10 10 NUM cana-3488 98 25 ) ) PUNCT cana-3488 98 26 for for ADP cana-3488 98 27 all all PRON cana-3488 98 28 ℨ ℨ NOUN cana-3488 98 29 ∈ ∈ NOUN cana-3488 98 30 𝒳𝑀. 𝒳𝑀. PUNCT cana-3488 99 1 it it PRON cana-3488 99 2 follows follow VERB cana-3488 99 3 from from ADP cana-3488 99 4 ( ( PUNCT cana-3488 99 5 9 9 NUM cana-3488 99 6 ) ) PUNCT cana-3488 99 7 and and CCONJ cana-3488 99 8 ( ( PUNCT cana-3488 99 9 10 10 NUM cana-3488 99 10 ) ) PUNCT cana-3488 99 11 , , PUNCT cana-3488 99 12 we we PRON cana-3488 99 13 get get VERB cana-3488 99 14 π’œ1 π’œ1 NOUN cana-3488 99 15 ( ( PUNCT cana-3488 99 16 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 99 17 ) ) PUNCT cana-3488 99 18 1113𝑛 1113𝑛 NOUN cana-3488 99 19 βˆ’ βˆ’ PROPN cana-3488 99 20 𝔉(ℨ),βˆ‘π‘›βˆ’1𝑖=0 𝔉(ℨ),βˆ‘π‘›βˆ’1𝑖=0 ADJ cana-3488 99 21 𝔙𝑖 𝔙𝑖 PROPN cana-3488 99 22 β„³ β„³ PROPN cana-3488 99 23 1113β‹…1113𝑖 1113β‹…1113𝑖 PROPN cana-3488 99 24 ) ) PUNCT cana-3488 100 1 = = PUNCT cana-3488 100 2 π’œ1 π’œ1 NOUN cana-3488 100 3 ( ( PUNCT cana-3488 100 4 βˆ‘ βˆ‘ PROPN cana-3488 100 5 π‘›βˆ’1 π‘›βˆ’1 PROPN cana-3488 100 6 𝑖=0 𝑖=0 PROPN cana-3488 100 7 𝔉(11𝑖+1ℨ 𝔉(11𝑖+1ℨ NOUN cana-3488 100 8 ) ) PUNCT cana-3488 100 9 1113(𝑖+1 1113(𝑖+1 NUM cana-3488 100 10 ) ) PUNCT cana-3488 100 11 βˆ’ βˆ’ ADP cana-3488 100 12 𝔉(11𝑖ℨ 𝔉(11𝑖ℨ NUM cana-3488 100 13 ) ) PUNCT cana-3488 100 14 1113𝑖 1113𝑖 NUM cana-3488 100 15 , , PUNCT cana-3488 100 16 βˆ‘π‘›βˆ’1𝑖=0 βˆ‘π‘›βˆ’1𝑖=0 PROPN cana-3488 100 17 𝔙𝑖 𝔙𝑖 PROPN cana-3488 100 18 β„³ β„³ PROPN cana-3488 100 19 1113β‹…1113𝑖 1113β‹…1113𝑖 PROPN cana-3488 100 20 ) ) PUNCT cana-3488 100 21 π’œ2 π’œ2 NOUN cana-3488 100 22 ( ( PUNCT cana-3488 100 23 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 100 24 ) ) PUNCT cana-3488 100 25 1113𝑛 1113𝑛 NOUN cana-3488 100 26 βˆ’ βˆ’ PROPN cana-3488 100 27 𝔉(ℨ),βˆ‘π‘›βˆ’1𝑖=0 𝔉(ℨ),βˆ‘π‘›βˆ’1𝑖=0 ADJ cana-3488 100 28 𝔙𝑖 𝔙𝑖 PROPN cana-3488 100 29 β„³ β„³ PROPN cana-3488 100 30 1113β‹…1113𝑖 1113β‹…1113𝑖 PROPN cana-3488 100 31 ) ) PUNCT cana-3488 101 1 = = SYM cana-3488 101 2 π’œ2 π’œ2 NOUN cana-3488 101 3 ( ( PUNCT cana-3488 101 4 βˆ‘ βˆ‘ PROPN cana-3488 101 5 π‘›βˆ’1 π‘›βˆ’1 PROPN cana-3488 101 6 𝑖=0 𝑖=0 PROPN cana-3488 101 7 𝔉(11𝑖+1ℨ 𝔉(11𝑖+1ℨ NOUN cana-3488 101 8 ) ) PUNCT cana-3488 101 9 1113(𝑖+1 1113(𝑖+1 NUM cana-3488 101 10 ) ) PUNCT cana-3488 101 11 βˆ’ βˆ’ ADP cana-3488 101 12 𝔉(11𝑖ℨ 𝔉(11𝑖ℨ NUM cana-3488 101 13 ) ) PUNCT cana-3488 101 14 11𝑖 11𝑖 NOUN cana-3488 101 15 , , PUNCT cana-3488 101 16 βˆ‘π‘›βˆ’1𝑖=0 βˆ‘π‘›βˆ’1𝑖=0 PROPN cana-3488 101 17 𝔙𝑖 𝔙𝑖 PROPN cana-3488 101 18 β„³ β„³ PROPN cana-3488 101 19 1113β‹…1113𝑖 1113β‹…1113𝑖 PROPN cana-3488 101 20 ) ) PUNCT cana-3488 101 21 π’œ3 π’œ3 NOUN cana-3488 101 22 ( ( PUNCT cana-3488 101 23 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 101 24 ) ) PUNCT cana-3488 101 25 1113𝑛 1113𝑛 NOUN cana-3488 101 26 βˆ’ βˆ’ PROPN cana-3488 101 27 𝔉(ℨ),βˆ‘π‘›βˆ’1𝑖=0 𝔉(ℨ),βˆ‘π‘›βˆ’1𝑖=0 ADJ cana-3488 101 28 𝔙𝑖 𝔙𝑖 PROPN cana-3488 101 29 β„³ β„³ PROPN cana-3488 101 30 1113β‹…1113𝑖 1113β‹…1113𝑖 PROPN cana-3488 101 31 ) ) PUNCT cana-3488 102 1 = = SYM cana-3488 102 2 π’œ3 π’œ3 PROPN cana-3488 102 3 ( ( PUNCT cana-3488 102 4 βˆ‘ βˆ‘ PROPN cana-3488 102 5 π‘›βˆ’1 π‘›βˆ’1 PROPN cana-3488 102 6 𝑖=0 𝑖=0 PROPN cana-3488 102 7 𝔉(11𝑖+1ℨ 𝔉(11𝑖+1ℨ NOUN cana-3488 102 8 ) ) PUNCT cana-3488 102 9 1113(𝑖+1 1113(𝑖+1 NUM cana-3488 102 10 ) ) PUNCT cana-3488 102 11 βˆ’ βˆ’ ADP cana-3488 102 12 𝔉(11𝑖ℨ 𝔉(11𝑖ℨ NUM cana-3488 102 13 ) ) PUNCT cana-3488 102 14 1113𝑖 1113𝑖 NUM cana-3488 102 15 , , PUNCT cana-3488 102 16 βˆ‘π‘›βˆ’1𝑖=0 βˆ‘π‘›βˆ’1𝑖=0 PROPN cana-3488 102 17 𝔙𝑖 𝔙𝑖 PROPN cana-3488 102 18 β„³ β„³ PROPN cana-3488 102 19 1113β‹…1113𝑖 1113β‹…1113𝑖 PROPN cana-3488 102 20 ) ) PUNCT cana-3488 102 21 } } PUNCT cana-3488 102 22 ( ( PUNCT cana-3488 102 23 11 11 NUM cana-3488 102 24 ) ) PUNCT cana-3488 102 25 for for ADP cana-3488 102 26 all all DET cana-3488 102 27 ℨ ℨ NOUN cana-3488 102 28 ∈ ∈ NOUN cana-3488 102 29 𝒳𝑀 𝒳𝑀 NOUN cana-3488 102 30 and and CCONJ cana-3488 102 31 all all DET cana-3488 102 32 β„³ β„³ PROPN cana-3488 102 33 > > X cana-3488 102 34 0 0 NUM cana-3488 102 35 . . PUNCT cana-3488 103 1 using use VERB cana-3488 103 2 ( ( PUNCT cana-3488 103 3 nns5 nns5 PROPN cana-3488 103 4 ) ) PUNCT cana-3488 103 5 , , PUNCT cana-3488 103 6 ( ( PUNCT cana-3488 103 7 nns11 nns11 PROPN cana-3488 103 8 ) ) PUNCT cana-3488 103 9 and and CCONJ cana-3488 103 10 ( ( PUNCT cana-3488 103 11 nns17 nns17 PROPN cana-3488 103 12 ) ) PUNCT cana-3488 103 13 in in ADP cana-3488 103 14 ( ( PUNCT cana-3488 103 15 11 11 NUM cana-3488 103 16 ) ) PUNCT cana-3488 103 17 , , PUNCT cana-3488 103 18 we we PRON cana-3488 103 19 have have VERB cana-3488 103 20 π’œ1 π’œ1 NOUN cana-3488 103 21 ( ( PUNCT cana-3488 103 22 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 103 23 ) ) PUNCT cana-3488 103 24 1113𝑛 1113𝑛 NOUN cana-3488 103 25 βˆ’ βˆ’ NOUN cana-3488 103 26 𝔉(ℨ 𝔉(ℨ NOUN cana-3488 103 27 ) ) PUNCT cana-3488 103 28 , , PUNCT cana-3488 103 29 βˆ‘π‘›βˆ’1𝑖=0 βˆ‘π‘›βˆ’1𝑖=0 PROPN cana-3488 103 30 𝔙𝑖ℳ 𝔙𝑖ℳ PROPN cana-3488 103 31 1113β‹…1113𝑖 1113β‹…1113𝑖 NUM cana-3488 103 32 ) ) PUNCT cana-3488 103 33 β‰₯ β‰₯ PROPN cana-3488 103 34 βˆπ‘›βˆ’1 βˆπ‘›βˆ’1 PROPN cana-3488 103 35 𝑖=0 𝑖=0 PROPN cana-3488 103 36 π’œ1 π’œ1 NOUN cana-3488 103 37 ( ( PUNCT cana-3488 103 38 𝔉(11𝑖+1ℨ 𝔉(11𝑖+1ℨ PROPN cana-3488 103 39 ) ) PUNCT cana-3488 103 40 1113(𝑖+1 1113(𝑖+1 NUM cana-3488 103 41 ) ) PUNCT cana-3488 103 42 βˆ’ βˆ’ ADP cana-3488 104 1 𝔉(11𝑖ℨ 𝔉(11𝑖ℨ NOUN cana-3488 104 2 ) ) PUNCT cana-3488 104 3 1113𝑖 1113𝑖 NUM cana-3488 104 4 , , PUNCT cana-3488 104 5 𝔙𝑖 𝔙𝑖 PROPN cana-3488 104 6 β„³ β„³ PROPN cana-3488 104 7 1113β‹…1113𝑖 1113β‹…1113𝑖 PROPN cana-3488 104 8 ) ) PUNCT cana-3488 104 9 π’œ2 π’œ2 NOUN cana-3488 104 10 ( ( PUNCT cana-3488 104 11 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 104 12 ) ) PUNCT cana-3488 104 13 1113𝑛 1113𝑛 NOUN cana-3488 104 14 βˆ’ βˆ’ NOUN cana-3488 104 15 𝔉(ℨ 𝔉(ℨ NOUN cana-3488 104 16 ) ) PUNCT cana-3488 104 17 , , PUNCT cana-3488 104 18 βˆ‘π‘›βˆ’1𝑖=0 βˆ‘π‘›βˆ’1𝑖=0 PROPN cana-3488 104 19 𝔙𝑖ℳ 𝔙𝑖ℳ PROPN cana-3488 104 20 1113β‹…1113𝑖 1113β‹…1113𝑖 NUM cana-3488 104 21 ) ) PUNCT cana-3488 104 22 ≀ ≀ PUNCT cana-3488 105 1 βˆπ‘›βˆ’1 βˆπ‘›βˆ’1 PROPN cana-3488 105 2 𝑖=0 𝑖=0 PROPN cana-3488 105 3 π’œ2 π’œ2 PROPN cana-3488 105 4 ( ( PUNCT cana-3488 105 5 𝔉(11𝑖+1ℨ 𝔉(11𝑖+1ℨ PROPN cana-3488 105 6 ) ) PUNCT cana-3488 105 7 1113(𝑖+1 1113(𝑖+1 NUM cana-3488 105 8 ) ) PUNCT cana-3488 105 9 βˆ’ βˆ’ ADP cana-3488 105 10 𝔉(11𝑖ℨ 𝔉(11𝑖ℨ NOUN cana-3488 105 11 ) ) PUNCT cana-3488 105 12 1113𝑖 1113𝑖 NUM cana-3488 105 13 , , PUNCT cana-3488 105 14 𝔙𝑖 𝔙𝑖 PROPN cana-3488 105 15 β„³ β„³ PROPN cana-3488 105 16 1113β‹…1113𝑖 1113β‹…1113𝑖 PROPN cana-3488 105 17 ) ) PUNCT cana-3488 105 18 π’œ3 π’œ3 NOUN cana-3488 105 19 ( ( PUNCT cana-3488 105 20 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 105 21 ) ) PUNCT cana-3488 105 22 1113𝑛 1113𝑛 NOUN cana-3488 105 23 βˆ’ βˆ’ NOUN cana-3488 105 24 𝔉(ℨ 𝔉(ℨ NOUN cana-3488 105 25 ) ) PUNCT cana-3488 105 26 , , PUNCT cana-3488 105 27 βˆ‘π‘›βˆ’1𝑖=0 βˆ‘π‘›βˆ’1𝑖=0 PROPN cana-3488 105 28 𝔙𝑖ℳ 𝔙𝑖ℳ PROPN cana-3488 105 29 1113β‹…1113𝑖 1113β‹…1113𝑖 NUM cana-3488 105 30 ) ) PUNCT cana-3488 105 31 ≀ ≀ PROPN cana-3488 106 1 βˆπ‘›βˆ’1 βˆπ‘›βˆ’1 ADP cana-3488 106 2 𝑖=0 𝑖=0 PROPN cana-3488 106 3 π’œ3 π’œ3 PROPN cana-3488 106 4 ( ( PUNCT cana-3488 106 5 𝔉(11𝑖+1ℨ 𝔉(11𝑖+1ℨ PROPN cana-3488 106 6 ) ) PUNCT cana-3488 106 7 1113(𝑖+1 1113(𝑖+1 NUM cana-3488 106 8 ) ) PUNCT cana-3488 106 9 βˆ’ βˆ’ ADP cana-3488 106 10 𝔉(11𝑖ℨ 𝔉(11𝑖ℨ NOUN cana-3488 106 11 ) ) PUNCT cana-3488 106 12 1113𝑖 1113𝑖 NUM cana-3488 106 13 , , PUNCT cana-3488 106 14 𝔙𝑖 𝔙𝑖 PROPN cana-3488 106 15 β„³ β„³ PROPN cana-3488 106 16 1113β‹…1113𝑖 1113β‹…1113𝑖 PROPN cana-3488 106 17 ) ) PUNCT cana-3488 106 18 } } PUNCT cana-3488 106 19 ( ( PUNCT cana-3488 106 20 12 12 NUM cana-3488 106 21 ) ) PUNCT cana-3488 106 22 where where SCONJ cana-3488 106 23 ∏ ∏ PROPN cana-3488 106 24 π‘›βˆ’1 π‘›βˆ’1 PROPN cana-3488 106 25 π’₯=0 π’₯=0 PUNCT cana-3488 107 1 𝑄𝑗 𝑄𝑗 PROPN cana-3488 107 2 = = SYM cana-3488 107 3 𝑄1 𝑄1 PROPN cana-3488 107 4 βˆ— βˆ— PROPN cana-3488 107 5 𝑄2 𝑄2 PROPN cana-3488 107 6 βˆ— βˆ— PROPN cana-3488 107 7 β‹― β‹― PROPN cana-3488 107 8 βˆ— βˆ— NOUN cana-3488 107 9 𝑄𝑛 𝑄𝑛 PROPN cana-3488 107 10 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ cana-3488 107 11 ∐ ∐ X cana-3488 107 12 π‘›βˆ’1 π‘›βˆ’1 PROPN cana-3488 107 13 π’₯=0 π’₯=0 PUNCT cana-3488 107 14 𝑅𝑗 𝑅𝑗 PROPN cana-3488 107 15 = = PUNCT cana-3488 107 16 𝑅1 𝑅1 PROPN cana-3488 107 17 β‹„ β‹„ PROPN cana-3488 107 18 𝑅2 𝑅2 VERB cana-3488 107 19 β‹„ β‹„ PROPN cana-3488 108 1 β‹―β‹„ β‹―β‹„ PROPN cana-3488 108 2 π‘…π‘›π‘Žπ‘›π‘‘ π‘…π‘›π‘Žπ‘›π‘‘ PROPN cana-3488 108 3 ∐ ∐ ADV cana-3488 108 4 π‘›βˆ’1 π‘›βˆ’1 NUM cana-3488 108 5 π’₯=0 π’₯=0 NUM cana-3488 109 1 𝑆𝑗 𝑆𝑗 PROPN cana-3488 109 2 = = PUNCT cana-3488 109 3 𝑆1βŠ˜π‘†2βŠ˜β‹―βŠ˜ 𝑆1βŠ˜π‘†2βŠ˜β‹―βŠ˜ NUM cana-3488 109 4 𝑆𝑛 𝑆𝑛 ADJ cana-3488 109 5 for for ADP cana-3488 109 6 all all DET cana-3488 109 7 ℨ ℨ NOUN cana-3488 109 8 ∈ ∈ NOUN cana-3488 109 9 𝒳𝑀 𝒳𝑀 NOUN cana-3488 109 10 and and CCONJ cana-3488 109 11 all all DET cana-3488 109 12 β„³ β„³ PROPN cana-3488 109 13 > > X cana-3488 109 14 0 0 NUM cana-3488 109 15 . . PUNCT cana-3488 110 1 hence hence ADV cana-3488 110 2 , , PUNCT cana-3488 110 3 from from ADP cana-3488 110 4 ( ( PUNCT cana-3488 110 5 12 12 NUM cana-3488 110 6 ) ) PUNCT cana-3488 110 7 and and CCONJ cana-3488 110 8 ( ( PUNCT cana-3488 110 9 9 9 NUM cana-3488 110 10 ) ) PUNCT cana-3488 110 11 , , PUNCT cana-3488 110 12 we we PRON cana-3488 110 13 arrive arrive VERB cana-3488 110 14 π’œ1 π’œ1 NOUN cana-3488 110 15 ( ( PUNCT cana-3488 110 16 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 110 17 ) ) PUNCT cana-3488 110 18 1113𝑛 1113𝑛 NOUN cana-3488 110 19 βˆ’ βˆ’ NOUN cana-3488 110 20 𝔉(ℨ 𝔉(ℨ NOUN cana-3488 110 21 ) ) PUNCT cana-3488 110 22 , , PUNCT cana-3488 110 23 βˆ‘π‘›βˆ’1𝑖=0 βˆ‘π‘›βˆ’1𝑖=0 PROPN cana-3488 110 24 𝔙𝑖 𝔙𝑖 PROPN cana-3488 110 25 β„³ β„³ PROPN cana-3488 110 26 1113β‹…1113𝑖 1113β‹…1113𝑖 PROPN cana-3488 110 27 ) ) PUNCT cana-3488 110 28 β‰₯ β‰₯ NOUN cana-3488 111 1 βˆπ‘›βˆ’1 βˆπ‘›βˆ’1 PROPN cana-3488 111 2 𝑖=0 𝑖=0 PROPN cana-3488 111 3 π’œ1β€²(Γ°(ℨ),β„³ π’œ1β€²(Γ°(ℨ),β„³ PROPN cana-3488 111 4 ) ) PUNCT cana-3488 111 5 = = SYM cana-3488 111 6 π’œ1β€²(Γ°(ℨ),β„³ π’œ1β€²(Γ°(ℨ),β„³ PROPN cana-3488 111 7 ) ) PUNCT cana-3488 111 8 π’œ2 π’œ2 NOUN cana-3488 111 9 ( ( PUNCT cana-3488 111 10 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 111 11 ) ) PUNCT cana-3488 111 12 1113𝑛 1113𝑛 NOUN cana-3488 111 13 βˆ’ βˆ’ NOUN cana-3488 111 14 𝔉(ℨ 𝔉(ℨ NOUN cana-3488 111 15 ) ) PUNCT cana-3488 111 16 , , PUNCT cana-3488 111 17 βˆ‘π‘›βˆ’1𝑖=0 βˆ‘π‘›βˆ’1𝑖=0 PROPN cana-3488 111 18 𝔙𝑖 𝔙𝑖 PROPN cana-3488 111 19 β„³ β„³ PROPN cana-3488 111 20 1113β‹…1113𝑖 1113β‹…1113𝑖 PROPN cana-3488 111 21 ) ) PUNCT cana-3488 111 22 ≀ ≀ NOUN cana-3488 112 1 βˆπ‘›βˆ’1 βˆπ‘›βˆ’1 PROPN cana-3488 112 2 𝑖=0 𝑖=0 PUNCT cana-3488 112 3 π’œ2β€²(Γ°(ℨ),β„³ π’œ2β€²(Γ°(ℨ),β„³ NOUN cana-3488 112 4 ) ) PUNCT cana-3488 112 5 = = SYM cana-3488 112 6 π’œ2β€²(Γ°(ℨ),β„³ π’œ2β€²(Γ°(ℨ),β„³ PROPN cana-3488 112 7 ) ) PUNCT cana-3488 112 8 π’œ3 π’œ3 NOUN cana-3488 112 9 ( ( PUNCT cana-3488 112 10 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 112 11 ) ) PUNCT cana-3488 112 12 1113𝑛 1113𝑛 NOUN cana-3488 112 13 βˆ’ βˆ’ NOUN cana-3488 112 14 𝔉(ℨ 𝔉(ℨ NOUN cana-3488 112 15 ) ) PUNCT cana-3488 112 16 , , PUNCT cana-3488 112 17 βˆ‘π‘›βˆ’1𝑖=0 βˆ‘π‘›βˆ’1𝑖=0 PROPN cana-3488 112 18 𝔙𝑖 𝔙𝑖 PROPN cana-3488 112 19 β„³ β„³ PROPN cana-3488 112 20 1113β‹…1113𝑖 1113β‹…1113𝑖 PROPN cana-3488 112 21 ) ) PUNCT cana-3488 112 22 ≀ ≀ PUNCT cana-3488 113 1 βˆπ‘›βˆ’1 βˆπ‘›βˆ’1 PROPN cana-3488 113 2 𝑖=0 𝑖=0 PUNCT cana-3488 113 3 π’œ1β€²(Γ°(ℨ),β„³ π’œ1β€²(Γ°(ℨ),β„³ PROPN cana-3488 113 4 ) ) PUNCT cana-3488 113 5 = = PUNCT cana-3488 113 6 π’œ3β€²(Γ°(ℨ),β„³ π’œ3β€²(Γ°(ℨ),β„³ NOUN cana-3488 113 7 ) ) PUNCT cana-3488 113 8 } } PUNCT cana-3488 113 9 ( ( PUNCT cana-3488 113 10 13 13 NUM cana-3488 113 11 ) ) PUNCT cana-3488 113 12 for for ADP cana-3488 113 13 all all DET cana-3488 113 14 ℨ ℨ NOUN cana-3488 113 15 ∈ ∈ NOUN cana-3488 113 16 𝒳𝑀 𝒳𝑀 NOUN cana-3488 113 17 and and CCONJ cana-3488 113 18 all all DET cana-3488 113 19 β„³ β„³ PROPN cana-3488 113 20 > > X cana-3488 113 21 0 0 NUM cana-3488 113 22 . . PUNCT cana-3488 114 1 replacing replace VERB cana-3488 114 2 ℨ ℨ NOUN cana-3488 114 3 by by ADP cana-3488 114 4 11π‘šβ„¨ 11π‘šβ„¨ NUM cana-3488 114 5 in in ADP cana-3488 114 6 ( ( PUNCT cana-3488 114 7 13 13 NUM cana-3488 114 8 ) ) PUNCT cana-3488 114 9 and and CCONJ cana-3488 114 10 using use VERB cana-3488 114 11 ( ( PUNCT cana-3488 114 12 1 1 NUM cana-3488 114 13 ) ) PUNCT cana-3488 114 14 , , PUNCT cana-3488 114 15 ( ( PUNCT cana-3488 114 16 nns5 nns5 PROPN cana-3488 114 17 ) ) PUNCT cana-3488 114 18 , , PUNCT cana-3488 114 19 ( ( PUNCT cana-3488 114 20 nns11 nns11 PROPN cana-3488 114 21 ) ) PUNCT cana-3488 114 22 and and CCONJ cana-3488 114 23 ( ( PUNCT cana-3488 114 24 nns17 nns17 PROPN cana-3488 114 25 ) ) PUNCT cana-3488 114 26 , , PUNCT cana-3488 114 27 we we PRON cana-3488 114 28 obtain obtain VERB cana-3488 114 29 communications communication NOUN cana-3488 114 30 on on ADP cana-3488 114 31 applied apply VERB cana-3488 114 32 nonlinear nonlinear ADJ cana-3488 114 33 analysis analysis NOUN cana-3488 114 34 issn issn NOUN cana-3488 114 35 : : PUNCT cana-3488 114 36 1074 1074 NUM cana-3488 114 37 - - PUNCT cana-3488 114 38 133x 133x NUM cana-3488 114 39 vol vol NOUN cana-3488 114 40 32 32 NUM cana-3488 114 41 no no NOUN cana-3488 114 42 . . PUNCT cana-3488 115 1 7s 7 NOUN cana-3488 115 2 ( ( PUNCT cana-3488 115 3 2025 2025 NUM cana-3488 115 4 ) ) PUNCT cana-3488 115 5 812 812 NUM cana-3488 115 6 https://internationalpubls.com https://internationalpubls.com X cana-3488 115 7 π’œ1 π’œ1 NOUN cana-3488 115 8 ( ( PUNCT cana-3488 115 9 𝔉(11𝑛+π‘šβ„¨ 𝔉(11𝑛+π‘šβ„¨ PROPN cana-3488 115 10 ) ) PUNCT cana-3488 115 11 1113(𝑛+π‘š 1113(𝑛+π‘š NUM cana-3488 115 12 ) ) PUNCT cana-3488 116 1 βˆ’ βˆ’ PROPN cana-3488 116 2 𝔉(11π‘šβ„¨ 𝔉(11π‘šβ„¨ NUM cana-3488 116 3 ) ) PUNCT cana-3488 116 4 1113π‘š 1113π‘š NOUN cana-3488 116 5 , , PUNCT cana-3488 116 6 βˆ‘π‘›βˆ’1𝑖=0 βˆ‘π‘›βˆ’1𝑖=0 PROPN cana-3488 116 7 𝔙𝑖 𝔙𝑖 PROPN cana-3488 116 8 β„³ β„³ PROPN cana-3488 116 9 1113β‹…1113(𝑖+π‘š 1113β‹…1113(𝑖+π‘š NUM cana-3488 116 10 ) ) PUNCT cana-3488 116 11 ) ) PUNCT cana-3488 117 1 β‰₯ β‰₯ NOUN cana-3488 117 2 π’œ1β€²(Γ°(11 π’œ1β€²(Γ°(11 PROPN cana-3488 117 3 π‘šβ„¨),β„³ π‘šβ„¨),β„³ PROPN cana-3488 117 4 ) ) PUNCT cana-3488 118 1 = = SYM cana-3488 118 2 π’œ1β€² π’œ1β€² NOUN cana-3488 118 3 ( ( PUNCT cana-3488 118 4 Γ°(ℨ Γ°(ℨ PROPN cana-3488 118 5 ) ) PUNCT cana-3488 118 6 , , PUNCT cana-3488 118 7 β„³ β„³ PROPN cana-3488 118 8 π”™π‘š π”™π‘š PROPN cana-3488 118 9 ) ) PUNCT cana-3488 118 10 π’œ2 π’œ2 NOUN cana-3488 118 11 ( ( PUNCT cana-3488 118 12 𝔉(11𝑛+π‘šβ„¨ 𝔉(11𝑛+π‘šβ„¨ PROPN cana-3488 118 13 ) ) PUNCT cana-3488 118 14 1113(𝑛+π‘š 1113(𝑛+π‘š NUM cana-3488 118 15 ) ) PUNCT cana-3488 118 16 βˆ’ βˆ’ PROPN cana-3488 119 1 𝔉(11π‘šβ„¨ 𝔉(11π‘šβ„¨ NUM cana-3488 119 2 ) ) PUNCT cana-3488 119 3 1113π‘š 1113π‘š NOUN cana-3488 119 4 , , PUNCT cana-3488 119 5 βˆ‘π‘›βˆ’1𝑖=0 βˆ‘π‘›βˆ’1𝑖=0 PROPN cana-3488 119 6 𝔙𝑖 𝔙𝑖 PROPN cana-3488 119 7 β„³ β„³ PROPN cana-3488 119 8 1113β‹…1113(𝑖+π‘š 1113β‹…1113(𝑖+π‘š NUM cana-3488 119 9 ) ) PUNCT cana-3488 119 10 ) ) PUNCT cana-3488 120 1 ≀ ≀ PROPN cana-3488 120 2 π’œ2β€²(Γ°(11 π’œ2β€²(Γ°(11 X cana-3488 120 3 π‘šβ„¨),β„³ π‘šβ„¨),β„³ PROPN cana-3488 120 4 ) ) PUNCT cana-3488 120 5 = = SYM cana-3488 120 6 π’œ2β€² π’œ2β€² X cana-3488 120 7 ( ( PUNCT cana-3488 120 8 Γ°(ℨ Γ°(ℨ NOUN cana-3488 120 9 ) ) PUNCT cana-3488 120 10 , , PUNCT cana-3488 120 11 β„³ β„³ PROPN cana-3488 120 12 π”™π‘š π”™π‘š PROPN cana-3488 120 13 ) ) PUNCT cana-3488 120 14 π’œ3 π’œ3 NOUN cana-3488 120 15 ( ( PUNCT cana-3488 120 16 𝔉(11𝑛+π‘šβ„¨ 𝔉(11𝑛+π‘šβ„¨ PROPN cana-3488 120 17 ) ) PUNCT cana-3488 120 18 1113(𝑛+π‘š 1113(𝑛+π‘š NUM cana-3488 120 19 ) ) PUNCT cana-3488 120 20 βˆ’ βˆ’ PROPN cana-3488 120 21 𝔉(11π‘šβ„¨ 𝔉(11π‘šβ„¨ NUM cana-3488 120 22 ) ) PUNCT cana-3488 120 23 1113π‘š 1113π‘š NOUN cana-3488 120 24 , , PUNCT cana-3488 120 25 βˆ‘π‘›βˆ’1𝑖=0 βˆ‘π‘›βˆ’1𝑖=0 PROPN cana-3488 120 26 𝔙𝑖 𝔙𝑖 PROPN cana-3488 120 27 β„³ β„³ PROPN cana-3488 120 28 1113β‹…1113(𝑖+π‘š 1113β‹…1113(𝑖+π‘š NUM cana-3488 120 29 ) ) PUNCT cana-3488 120 30 ) ) PUNCT cana-3488 121 1 ≀ ≀ PROPN cana-3488 121 2 π’œ3β€²(Γ°(11 π’œ3β€²(Γ°(11 PROPN cana-3488 121 3 π‘šβ„¨),β„³ π‘šβ„¨),β„³ NOUN cana-3488 121 4 ) ) PUNCT cana-3488 122 1 = = SYM cana-3488 122 2 π’œ1β€² π’œ1β€² NOUN cana-3488 122 3 ( ( PUNCT cana-3488 122 4 Γ°(ℨ Γ°(ℨ PROPN cana-3488 122 5 ) ) PUNCT cana-3488 122 6 , , PUNCT cana-3488 122 7 β„³ β„³ PROPN cana-3488 122 8 π”™π‘š π”™π‘š PROPN cana-3488 122 9 ) ) PUNCT cana-3488 122 10 } } PUNCT cana-3488 122 11 ( ( PUNCT cana-3488 122 12 14 14 NUM cana-3488 122 13 ) ) PUNCT cana-3488 122 14 for for ADP cana-3488 122 15 all all DET cana-3488 122 16 ℨ ℨ NOUN cana-3488 122 17 ∈ ∈ NOUN cana-3488 122 18 𝒳𝑀 𝒳𝑀 NOUN cana-3488 122 19 and and CCONJ cana-3488 122 20 all all DET cana-3488 122 21 β„³ β„³ PROPN cana-3488 122 22 > > X cana-3488 122 23 0 0 NUM cana-3488 123 1 also also ADV cana-3488 123 2 π‘š,𝑛 π‘š,𝑛 PROPN cana-3488 123 3 are be AUX cana-3488 123 4 positive positive ADJ cana-3488 123 5 numbers number NOUN cana-3488 123 6 . . PUNCT cana-3488 124 1 changing change VERB cana-3488 124 2 β„³ β„³ NOUN cana-3488 124 3 by by ADP cana-3488 124 4 π”™π‘šβ„³ π”™π‘šβ„³ PROPN cana-3488 124 5 in in ADP cana-3488 124 6 ( ( PUNCT cana-3488 124 7 14 14 NUM cana-3488 124 8 ) ) PUNCT cana-3488 124 9 , , PUNCT cana-3488 124 10 we we PRON cana-3488 124 11 get get VERB cana-3488 124 12 π’œ1 π’œ1 NOUN cana-3488 124 13 ( ( PUNCT cana-3488 124 14 𝔉(11𝑛+π‘šβ„¨ 𝔉(11𝑛+π‘šβ„¨ PROPN cana-3488 124 15 ) ) PUNCT cana-3488 124 16 1113(𝑛+π‘š 1113(𝑛+π‘š NUM cana-3488 124 17 ) ) PUNCT cana-3488 125 1 βˆ’ βˆ’ PROPN cana-3488 126 1 𝔉(11π‘šβ„¨ 𝔉(11π‘šβ„¨ NUM cana-3488 126 2 ) ) PUNCT cana-3488 126 3 1113π‘š 1113π‘š NUM cana-3488 126 4 , , PUNCT cana-3488 126 5 βˆ‘π‘›βˆ’1𝑖=0 βˆ‘π‘›βˆ’1𝑖=0 PROPN cana-3488 126 6 𝔙𝑖+π‘š 𝔙𝑖+π‘š PROPN cana-3488 126 7 β„³ β„³ PROPN cana-3488 126 8 1113β‹…1113(𝑖+π‘š 1113β‹…1113(𝑖+π‘š NUM cana-3488 126 9 ) ) PUNCT cana-3488 126 10 ) ) PUNCT cana-3488 127 1 β‰₯ β‰₯ NUM cana-3488 127 2 π’œ1β€²(Γ°(ℨ),β„³ π’œ1β€²(Γ°(ℨ),β„³ NOUN cana-3488 127 3 ) ) PUNCT cana-3488 128 1 π’œ2 π’œ2 NOUN cana-3488 128 2 ( ( PUNCT cana-3488 128 3 𝔉(11𝑛+π‘šβ„¨ 𝔉(11𝑛+π‘šβ„¨ PROPN cana-3488 128 4 ) ) PUNCT cana-3488 128 5 1113(𝑛+π‘š 1113(𝑛+π‘š NUM cana-3488 128 6 ) ) PUNCT cana-3488 128 7 βˆ’ βˆ’ PROPN cana-3488 128 8 𝔉(11π‘šβ„¨ 𝔉(11π‘šβ„¨ NUM cana-3488 128 9 ) ) PUNCT cana-3488 128 10 1113π‘š 1113π‘š NUM cana-3488 128 11 , , PUNCT cana-3488 128 12 βˆ‘π‘›βˆ’1𝑖=0 βˆ‘π‘›βˆ’1𝑖=0 PROPN cana-3488 128 13 𝔙𝑖+π‘š 𝔙𝑖+π‘š PROPN cana-3488 128 14 β„³ β„³ PROPN cana-3488 128 15 1113β‹…1113(𝑖+π‘š 1113β‹…1113(𝑖+π‘š NUM cana-3488 128 16 ) ) PUNCT cana-3488 128 17 ) ) PUNCT cana-3488 128 18 ≀ ≀ NUM cana-3488 129 1 π’œ2β€²(Γ°(ℨ),β„³ π’œ2β€²(Γ°(ℨ),β„³ NOUN cana-3488 129 2 ) ) PUNCT cana-3488 129 3 π’œ3 π’œ3 NOUN cana-3488 129 4 ( ( PUNCT cana-3488 129 5 𝔉(11𝑛+π‘šβ„¨ 𝔉(11𝑛+π‘šβ„¨ PROPN cana-3488 129 6 ) ) PUNCT cana-3488 129 7 1113(𝑛+π‘š 1113(𝑛+π‘š NUM cana-3488 129 8 ) ) PUNCT cana-3488 129 9 βˆ’ βˆ’ PROPN cana-3488 130 1 𝔉(11π‘šβ„¨ 𝔉(11π‘šβ„¨ NUM cana-3488 130 2 ) ) PUNCT cana-3488 130 3 1113π‘š 1113π‘š NUM cana-3488 130 4 , , PUNCT cana-3488 130 5 βˆ‘π‘›βˆ’1𝑖=0 βˆ‘π‘›βˆ’1𝑖=0 PROPN cana-3488 130 6 𝔙𝑖+π‘š 𝔙𝑖+π‘š PROPN cana-3488 130 7 β„³ β„³ PROPN cana-3488 130 8 1113β‹…1113(𝑖+π‘š 1113β‹…1113(𝑖+π‘š NUM cana-3488 130 9 ) ) PUNCT cana-3488 130 10 ) ) PUNCT cana-3488 131 1 ≀ ≀ NUM cana-3488 132 1 π’œ3β€²(Γ°(ℨ),β„³ π’œ3β€²(Γ°(ℨ),β„³ X cana-3488 132 2 ) ) PUNCT cana-3488 132 3 } } PUNCT cana-3488 132 4 ( ( PUNCT cana-3488 132 5 15 15 X cana-3488 132 6 ) ) PUNCT cana-3488 132 7 which which PRON cana-3488 132 8 implies imply VERB cana-3488 132 9 π’œ1 π’œ1 PROPN cana-3488 132 10 ( ( PUNCT cana-3488 132 11 𝔉(11𝑛+π‘šβ„¨ 𝔉(11𝑛+π‘šβ„¨ PROPN cana-3488 132 12 ) ) PUNCT cana-3488 132 13 1113(𝑛+π‘š 1113(𝑛+π‘š NUM cana-3488 132 14 ) ) PUNCT cana-3488 132 15 βˆ’ βˆ’ PROPN cana-3488 132 16 𝔉(11π‘šβ„¨ 𝔉(11π‘šβ„¨ NUM cana-3488 132 17 ) ) PUNCT cana-3488 132 18 1113π‘š 1113π‘š NUM cana-3488 132 19 , , PUNCT cana-3488 132 20 β„³ β„³ PROPN cana-3488 132 21 ) ) PUNCT cana-3488 132 22 β‰₯ β‰₯ NOUN cana-3488 132 23 π’œ1β€² π’œ1β€² NOUN cana-3488 132 24 ( ( PUNCT cana-3488 132 25 Γ°(ℨ Γ°(ℨ PROPN cana-3488 132 26 ) ) PUNCT cana-3488 132 27 , , PUNCT cana-3488 132 28 β„³ β„³ NOUN cana-3488 132 29 βˆ‘π‘›βˆ’1𝑖=π‘š βˆ‘π‘›βˆ’1𝑖=π‘š PUNCT cana-3488 133 1 𝔙𝑖 𝔙𝑖 ADP cana-3488 133 2 1113β‹…1113𝑖 1113β‹…1113𝑖 NUM cana-3488 133 3 ) ) PUNCT cana-3488 133 4 π’œ2 π’œ2 NOUN cana-3488 133 5 ( ( PUNCT cana-3488 133 6 𝔉(11𝑛+π‘šβ„¨ 𝔉(11𝑛+π‘šβ„¨ PROPN cana-3488 133 7 ) ) PUNCT cana-3488 133 8 1113(𝑛+π‘š 1113(𝑛+π‘š NUM cana-3488 133 9 ) ) PUNCT cana-3488 133 10 βˆ’ βˆ’ PROPN cana-3488 133 11 𝔉(11π‘šβ„¨ 𝔉(11π‘šβ„¨ NUM cana-3488 133 12 ) ) PUNCT cana-3488 133 13 1113π‘š 1113π‘š NUM cana-3488 133 14 , , PUNCT cana-3488 133 15 β„³ β„³ NOUN cana-3488 133 16 ) ) PUNCT cana-3488 133 17 ≀ ≀ NUM cana-3488 133 18 π’œ2β€² π’œ2β€² ADV cana-3488 133 19 ( ( PUNCT cana-3488 133 20 Γ°(ℨ Γ°(ℨ NOUN cana-3488 133 21 ) ) PUNCT cana-3488 133 22 , , PUNCT cana-3488 133 23 β„³ β„³ NOUN cana-3488 133 24 βˆ‘π‘›βˆ’1𝑖=π‘š βˆ‘π‘›βˆ’1𝑖=π‘š PUNCT cana-3488 134 1 𝔙𝑖 𝔙𝑖 ADP cana-3488 134 2 1113β‹…1113𝑖 1113β‹…1113𝑖 NUM cana-3488 134 3 ) ) PUNCT cana-3488 134 4 π’œ3 π’œ3 NOUN cana-3488 134 5 ( ( PUNCT cana-3488 134 6 𝔉(11𝑛+π‘šβ„¨ 𝔉(11𝑛+π‘šβ„¨ PROPN cana-3488 134 7 ) ) PUNCT cana-3488 134 8 1113(𝑛+π‘š 1113(𝑛+π‘š NUM cana-3488 134 9 ) ) PUNCT cana-3488 135 1 βˆ’ βˆ’ PROPN cana-3488 135 2 𝔉(11π‘šβ„¨ 𝔉(11π‘šβ„¨ NUM cana-3488 135 3 ) ) PUNCT cana-3488 135 4 1113π‘š 1113π‘š NUM cana-3488 135 5 , , PUNCT cana-3488 135 6 β„³ β„³ NOUN cana-3488 135 7 ) ) PUNCT cana-3488 135 8 ≀ ≀ NUM cana-3488 135 9 π’œ3β€² π’œ3β€² NOUN cana-3488 135 10 ( ( PUNCT cana-3488 135 11 Γ°(ℨ Γ°(ℨ PROPN cana-3488 135 12 ) ) PUNCT cana-3488 135 13 , , PUNCT cana-3488 135 14 β„³ β„³ NOUN cana-3488 135 15 βˆ‘π‘›βˆ’1𝑖=π‘š βˆ‘π‘›βˆ’1𝑖=π‘š PUNCT cana-3488 136 1 𝔙𝑖 𝔙𝑖 ADP cana-3488 136 2 1113β‹…1113𝑖 1113β‹…1113𝑖 NUM cana-3488 136 3 ) ) PUNCT cana-3488 136 4 } } PUNCT cana-3488 136 5 ( ( PUNCT cana-3488 136 6 16 16 NUM cana-3488 136 7 ) ) PUNCT cana-3488 136 8 here here ADV cana-3488 136 9 { { PUNCT cana-3488 136 10 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 136 11 ) ) PUNCT cana-3488 136 12 1113𝑛 1113𝑛 NOUN cana-3488 136 13 } } PUNCT cana-3488 136 14 is be AUX cana-3488 136 15 a a DET cana-3488 136 16 cauchy cauchy ADJ cana-3488 136 17 sequence sequence NOUN cana-3488 136 18 in in ADP cana-3488 136 19 ( ( PUNCT cana-3488 136 20 𝒴𝑀 𝒴𝑀 NOUN cana-3488 136 21 , , PUNCT cana-3488 136 22 π’œ1 π’œ1 NOUN cana-3488 136 23 , , PUNCT cana-3488 136 24 π’œ2,π’œ3 π’œ2,π’œ3 NOUN cana-3488 136 25 ) ) PUNCT cana-3488 136 26 also also ADV cana-3488 136 27 a a DET cana-3488 136 28 complete complete ADJ cana-3488 136 29 nns nn NOUN cana-3488 136 30 - - PUNCT cana-3488 136 31 space space NOUN cana-3488 136 32 is be AUX cana-3488 136 33 ( ( PUNCT cana-3488 136 34 𝒴𝑀 𝒴𝑀 NOUN cana-3488 136 35 , , PUNCT cana-3488 136 36 π’œ1,π’œ2 π’œ1,π’œ2 NUM cana-3488 136 37 , , PUNCT cana-3488 136 38 π’œ3 π’œ3 PROPN cana-3488 136 39 ) ) PUNCT cana-3488 136 40 then then ADV cana-3488 136 41 this this DET cana-3488 136 42 sequence sequence NOUN cana-3488 136 43 converges converge VERB cana-3488 136 44 to to ADP cana-3488 136 45 a a DET cana-3488 136 46 particular particular ADJ cana-3488 136 47 point point NOUN cana-3488 136 48 𝒯(ℨ 𝒯(ℨ NUM cana-3488 136 49 ) ) PUNCT cana-3488 137 1 ∈ ∈ PROPN cana-3488 137 2 π‘Œ. π‘Œ. PROPN cana-3488 137 3 lim lim PROPN cana-3488 137 4 π‘›β†’βˆž π‘›β†’βˆž NUM cana-3488 137 5 π’œ1 π’œ1 PROPN cana-3488 137 6 ( ( PUNCT cana-3488 137 7 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 137 8 ) ) PUNCT cana-3488 137 9 1113𝑛 1113𝑛 NOUN cana-3488 137 10 βˆ’ βˆ’ NOUN cana-3488 137 11 𝒯(ℨ),β„³ 𝒯(ℨ),β„³ NOUN cana-3488 137 12 ) ) PUNCT cana-3488 138 1 = = SYM cana-3488 138 2 1 1 NUM cana-3488 138 3 , , PUNCT cana-3488 138 4 lim lim PROPN cana-3488 138 5 π‘›β†’βˆž π‘›β†’βˆž NUM cana-3488 138 6 π’œ2 π’œ2 PROPN cana-3488 138 7 ( ( PUNCT cana-3488 138 8 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 138 9 ) ) PUNCT cana-3488 138 10 1113𝑛 1113𝑛 NOUN cana-3488 138 11 βˆ’ βˆ’ NOUN cana-3488 138 12 𝒯(ℨ),β„³ 𝒯(ℨ),β„³ NOUN cana-3488 138 13 ) ) PUNCT cana-3488 138 14 = = SYM cana-3488 138 15 0 0 NUM cana-3488 139 1 lim lim NOUN cana-3488 139 2 π‘›β†’βˆž π‘›β†’βˆž NUM cana-3488 139 3 π’œ3 π’œ3 PROPN cana-3488 139 4 ( ( PUNCT cana-3488 139 5 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 139 6 ) ) PUNCT cana-3488 139 7 1113𝑛 1113𝑛 NOUN cana-3488 139 8 βˆ’ βˆ’ NOUN cana-3488 139 9 𝒯(ℨ),β„³ 𝒯(ℨ),β„³ NOUN cana-3488 139 10 ) ) PUNCT cana-3488 139 11 = = SYM cana-3488 139 12 0 0 NUM cana-3488 139 13 and and CCONJ cana-3488 139 14 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 139 15 ) ) PUNCT cana-3488 139 16 1113𝑛 1113𝑛 NUM cana-3488 139 17 ⟢ ⟢ NOUN cana-3488 139 18 𝑁𝑁𝑆 𝑁𝑁𝑆 NOUN cana-3488 139 19 𝒯(ℨ 𝒯(ℨ NUM cana-3488 139 20 ) ) PUNCT cana-3488 139 21 , , PUNCT cana-3488 139 22 π‘Žπ‘  π‘Žπ‘  PROPN cana-3488 139 23 𝑛 𝑛 PROPN cana-3488 139 24 β†’ β†’ PUNCT cana-3488 139 25 ∞. ∞. PROPN cana-3488 139 26 taking take VERB cana-3488 139 27 π‘š π‘š NOUN cana-3488 139 28 = = SYM cana-3488 139 29 0 0 NUM cana-3488 139 30 in in ADP cana-3488 139 31 ( ( PUNCT cana-3488 139 32 15 15 NUM cana-3488 139 33 ) ) PUNCT cana-3488 139 34 , , PUNCT cana-3488 139 35 we we PRON cana-3488 139 36 reach reach VERB cana-3488 139 37 communications communication NOUN cana-3488 139 38 on on ADP cana-3488 139 39 applied apply VERB cana-3488 139 40 nonlinear nonlinear ADJ cana-3488 139 41 analysis analysis NOUN cana-3488 139 42 issn issn NOUN cana-3488 139 43 : : PUNCT cana-3488 139 44 1074 1074 NUM cana-3488 139 45 - - PUNCT cana-3488 139 46 133x 133x NUM cana-3488 139 47 vol vol NOUN cana-3488 139 48 32 32 NUM cana-3488 139 49 no no NOUN cana-3488 139 50 . . PUNCT cana-3488 140 1 7s 7 NOUN cana-3488 140 2 ( ( PUNCT cana-3488 140 3 2025 2025 NUM cana-3488 140 4 ) ) PUNCT cana-3488 140 5 813 813 NUM cana-3488 140 6 https://internationalpubls.com https://internationalpubls.com X cana-3488 140 7 π’œ1 π’œ1 NOUN cana-3488 140 8 ( ( PUNCT cana-3488 140 9 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 140 10 ) ) PUNCT cana-3488 140 11 1113𝑛 1113𝑛 NOUN cana-3488 140 12 βˆ’ βˆ’ ADP cana-3488 140 13 𝔉(ℨ),β„³ 𝔉(ℨ),β„³ NOUN cana-3488 140 14 ) ) PUNCT cana-3488 140 15 β‰₯ β‰₯ NOUN cana-3488 140 16 π’œ1β€² π’œ1β€² NOUN cana-3488 140 17 ( ( PUNCT cana-3488 140 18 ψ(ℨ ψ(ℨ PROPN cana-3488 140 19 ) ) PUNCT cana-3488 140 20 , , PUNCT cana-3488 140 21 β„³ β„³ PROPN cana-3488 140 22 βˆ‘π‘›βˆ’1𝑖=0 βˆ‘π‘›βˆ’1𝑖=0 NOUN cana-3488 140 23 𝔙𝑖 𝔙𝑖 PROPN cana-3488 140 24 1113β‹…1113𝑖 1113β‹…1113𝑖 PROPN cana-3488 140 25 ) ) PUNCT cana-3488 140 26 π’œ2 π’œ2 NOUN cana-3488 140 27 ( ( PUNCT cana-3488 140 28 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 140 29 ) ) PUNCT cana-3488 140 30 1113𝑛 1113𝑛 NOUN cana-3488 140 31 βˆ’ βˆ’ ADP cana-3488 140 32 𝔉(ℨ),β„³ 𝔉(ℨ),β„³ NOUN cana-3488 140 33 ) ) PUNCT cana-3488 140 34 ≀ ≀ NUM cana-3488 140 35 π’œ2β€² π’œ2β€² NOUN cana-3488 140 36 ( ( PUNCT cana-3488 140 37 ψ(ℨ ψ(ℨ NOUN cana-3488 140 38 ) ) PUNCT cana-3488 140 39 , , PUNCT cana-3488 140 40 β„³ β„³ PROPN cana-3488 140 41 βˆ‘π‘›βˆ’1𝑖=0 βˆ‘π‘›βˆ’1𝑖=0 NOUN cana-3488 140 42 𝔙𝑖 𝔙𝑖 PROPN cana-3488 140 43 1113β‹…1113𝑖 1113β‹…1113𝑖 PROPN cana-3488 140 44 ) ) PUNCT cana-3488 140 45 π’œ3 π’œ3 NOUN cana-3488 140 46 ( ( PUNCT cana-3488 140 47 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 140 48 ) ) PUNCT cana-3488 140 49 1113𝑛 1113𝑛 NOUN cana-3488 140 50 βˆ’ βˆ’ ADP cana-3488 140 51 𝔉(ℨ),β„³ 𝔉(ℨ),β„³ NOUN cana-3488 140 52 ) ) PUNCT cana-3488 140 53 ≀ ≀ NUM cana-3488 140 54 π’œ3β€² π’œ3β€² NOUN cana-3488 140 55 ( ( PUNCT cana-3488 140 56 ψ(ℨ ψ(ℨ NOUN cana-3488 140 57 ) ) PUNCT cana-3488 140 58 , , PUNCT cana-3488 140 59 β„³ β„³ PROPN cana-3488 140 60 βˆ‘π‘›βˆ’1𝑖=0 βˆ‘π‘›βˆ’1𝑖=0 NOUN cana-3488 140 61 𝔙𝑖 𝔙𝑖 PROPN cana-3488 140 62 1113β‹…1113𝑖 1113β‹…1113𝑖 PROPN cana-3488 140 63 ) ) PUNCT cana-3488 140 64 } } PUNCT cana-3488 140 65 ( ( PUNCT cana-3488 140 66 17 17 NUM cana-3488 140 67 ) ) PUNCT cana-3488 140 68 for for ADP cana-3488 140 69 all all DET cana-3488 140 70 ℨ ℨ NOUN cana-3488 140 71 ∈ ∈ NOUN cana-3488 140 72 𝒳𝑀 𝒳𝑀 NOUN cana-3488 140 73 and and CCONJ cana-3488 140 74 all all DET cana-3488 140 75 β„³ β„³ PROPN cana-3488 140 76 > > X cana-3488 140 77 0 0 NUM cana-3488 140 78 . . PUNCT cana-3488 141 1 considering consider VERB cana-3488 141 2 𝑛 𝑛 PROPN cana-3488 141 3 β†’ β†’ SYM cana-3488 141 4 ∞ ∞ PROPN cana-3488 141 5 in in ADP cana-3488 141 6 ( ( PUNCT cana-3488 141 7 17 17 NUM cana-3488 141 8 ) ) PUNCT cana-3488 141 9 , , PUNCT cana-3488 141 10 we we PRON cana-3488 141 11 arrive arrive VERB cana-3488 141 12 π’œ1(𝒯(ℨ π’œ1(𝒯(ℨ NOUN cana-3488 141 13 ) ) PUNCT cana-3488 142 1 βˆ’ βˆ’ NUM cana-3488 142 2 𝔉(ℨ),β„³ 𝔉(ℨ),β„³ SYM cana-3488 142 3 ) ) PUNCT cana-3488 142 4 β‰₯ β‰₯ PROPN cana-3488 142 5 π’œ1β€²(Γ°(ℨ),β„³|1113 π’œ1β€²(Γ°(ℨ),β„³|1113 NOUN cana-3488 142 6 βˆ’ βˆ’ PROPN cana-3488 142 7 𝔙| 𝔙| PROPN cana-3488 142 8 ) ) PUNCT cana-3488 142 9 π’œ2(𝒯(ℨ π’œ2(𝒯(ℨ NUM cana-3488 142 10 ) ) PUNCT cana-3488 142 11 βˆ’ βˆ’ NUM cana-3488 142 12 𝔉(ℨ),β„³ 𝔉(ℨ),β„³ NOUN cana-3488 142 13 ) ) PUNCT cana-3488 142 14 ≀ ≀ NUM cana-3488 142 15 π’œ2β€²(Γ°(ℨ),β„³|1113 π’œ2β€²(Γ°(ℨ),β„³|1113 PROPN cana-3488 142 16 βˆ’π”™| βˆ’π”™| PROPN cana-3488 142 17 ) ) PUNCT cana-3488 142 18 π’œ3(𝒯(ℨ π’œ3(𝒯(ℨ PUNCT cana-3488 142 19 ) ) PUNCT cana-3488 143 1 βˆ’ βˆ’ NUM cana-3488 143 2 𝔉(ℨ),β„³ 𝔉(ℨ),β„³ NOUN cana-3488 143 3 ) ) PUNCT cana-3488 143 4 ≀ ≀ NUM cana-3488 143 5 π’œ3β€²(Γ°(ℨ),β„³|1113 π’œ3β€²(Γ°(ℨ),β„³|1113 PROPN cana-3488 143 6 βˆ’π”™| βˆ’π”™| PROPN cana-3488 143 7 ) ) PUNCT cana-3488 143 8 } } PUNCT cana-3488 143 9 ( ( PUNCT cana-3488 143 10 18 18 NUM cana-3488 143 11 ) ) PUNCT cana-3488 143 12 for for ADP cana-3488 143 13 all all PRON cana-3488 143 14 ℨ ℨ NOUN cana-3488 143 15 ∈ ∈ NOUN cana-3488 143 16 𝒳𝑀 𝒳𝑀 NOUN cana-3488 143 17 and and CCONJ cana-3488 143 18 all all DET cana-3488 143 19 β„³ β„³ PROPN cana-3488 143 20 > > X cana-3488 143 21 0 0 NUM cana-3488 143 22 . . PUNCT cana-3488 144 1 next next ADV cana-3488 144 2 , , PUNCT cana-3488 144 3 we we PRON cana-3488 144 4 have have VERB cana-3488 144 5 to to PART cana-3488 144 6 show show VERB cana-3488 144 7 𝔉 𝔉 PROPN cana-3488 144 8 satisfies satisfie NOUN cana-3488 144 9 ( ( PUNCT cana-3488 144 10 1 1 NUM cana-3488 144 11 ) ) PUNCT cana-3488 144 12 , , PUNCT cana-3488 144 13 letting let VERB cana-3488 144 14 ℨ ℨ PRON cana-3488 144 15 by by ADP cana-3488 144 16 11𝑛ℨ 11𝑛ℨ NUM cana-3488 144 17 in in ADP cana-3488 144 18 ( ( PUNCT cana-3488 144 19 3 3 NUM cana-3488 144 20 ) ) PUNCT cana-3488 144 21 respectively respectively ADV cana-3488 144 22 , , PUNCT cana-3488 144 23 we we PRON cana-3488 144 24 have have VERB cana-3488 144 25 π’œ1 π’œ1 NOUN cana-3488 144 26 ( ( PUNCT cana-3488 144 27 1 1 NUM cana-3488 144 28 1113𝑛 1113𝑛 NOUN cana-3488 145 1 [ [ X cana-3488 145 2 𝔉(3 𝔉(3 NUM cana-3488 145 3 β‹… β‹… PROPN cana-3488 145 4 11𝑛ℨ 11𝑛ℨ NUM cana-3488 145 5 ) ) PUNCT cana-3488 145 6 βˆ’ βˆ’ PROPN cana-3488 145 7 1,88,30,57,02,60,326𝔉(11𝑛ℨ 1,88,30,57,02,60,326𝔉(11𝑛ℨ NUM cana-3488 145 8 ) ) PUNCT cana-3488 145 9 + + NUM cana-3488 145 10 1,56,92,14,18,83,605𝔉(βˆ’11𝑛ℨ)],β„³ 1,56,92,14,18,83,605𝔉(βˆ’11𝑛ℨ)],β„³ NUM cana-3488 145 11 ) ) PUNCT cana-3488 145 12 β‰₯ β‰₯ NOUN cana-3488 145 13 π’œ1β€²(Γ°(11 π’œ1β€²(Γ°(11 PROPN cana-3488 145 14 𝑛ℨ),1113𝑛ℳ 𝑛ℨ),1113𝑛ℳ PROPN cana-3488 145 15 ) ) PUNCT cana-3488 145 16 π’œ2 π’œ2 PROPN cana-3488 145 17 ( ( PUNCT cana-3488 145 18 1 1 NUM cana-3488 145 19 1113𝑛 1113𝑛 NOUN cana-3488 145 20 [ [ X cana-3488 145 21 𝔉(3 𝔉(3 NUM cana-3488 145 22 β‹… β‹… PROPN cana-3488 145 23 11𝑛ℨ 11𝑛ℨ NUM cana-3488 145 24 ) ) PUNCT cana-3488 145 25 βˆ’ βˆ’ PROPN cana-3488 145 26 1,88,30,57,02,60,326𝔉(11𝑛ℨ 1,88,30,57,02,60,326𝔉(11𝑛ℨ NUM cana-3488 145 27 ) ) PUNCT cana-3488 145 28 + + NUM cana-3488 145 29 1,56,92,14,18,83,605𝔉(βˆ’11𝑛ℨ)],β„³ 1,56,92,14,18,83,605𝔉(βˆ’11𝑛ℨ)],β„³ NUM cana-3488 145 30 ) ) PUNCT cana-3488 145 31 ≀ ≀ NUM cana-3488 145 32 π’œ2β€²(Γ°(11 π’œ2β€²(Γ°(11 PROPN cana-3488 145 33 𝑛ℨ),1113𝑛ℳ 𝑛ℨ),1113𝑛ℳ PROPN cana-3488 145 34 ) ) PUNCT cana-3488 145 35 π’œ3 π’œ3 PROPN cana-3488 145 36 ( ( PUNCT cana-3488 145 37 1 1 NUM cana-3488 145 38 1113𝑛 1113𝑛 NOUN cana-3488 146 1 [ [ X cana-3488 146 2 𝔉(3 𝔉(3 NUM cana-3488 146 3 β‹… β‹… PROPN cana-3488 146 4 11𝑛ℨ 11𝑛ℨ NUM cana-3488 146 5 ) ) PUNCT cana-3488 146 6 βˆ’ βˆ’ PROPN cana-3488 146 7 1,88,30,57,02,60,326𝔉(11𝑛ℨ 1,88,30,57,02,60,326𝔉(11𝑛ℨ NUM cana-3488 146 8 ) ) PUNCT cana-3488 146 9 + + NUM cana-3488 146 10 1,56,92,14,18,83,605𝔉(βˆ’11𝑛ℨ)],β„³ 1,56,92,14,18,83,605𝔉(βˆ’11𝑛ℨ)],β„³ NUM cana-3488 146 11 ) ) PUNCT cana-3488 146 12 ≀ ≀ NOUN cana-3488 146 13 π’œ3β€²(Γ°(11 π’œ3β€²(Γ°(11 PROPN cana-3488 146 14 𝑛ℨ),1113𝑛ℳ 𝑛ℨ),1113𝑛ℳ PROPN cana-3488 146 15 ) ) PUNCT cana-3488 146 16 } } PUNCT cana-3488 146 17 ( ( PUNCT cana-3488 146 18 19 19 NUM cana-3488 146 19 ) ) PUNCT cana-3488 146 20 for for ADP cana-3488 146 21 all all PRON cana-3488 146 22 ℨ ℨ NOUN cana-3488 146 23 ∈ ∈ NOUN cana-3488 146 24 𝒳𝑀 𝒳𝑀 NOUN cana-3488 146 25 and and CCONJ cana-3488 146 26 all all DET cana-3488 146 27 β„³ β„³ PROPN cana-3488 146 28 > > X cana-3488 146 29 0 0 PUNCT cana-3488 146 30 . . PUNCT cana-3488 147 1 now now ADV cana-3488 147 2 , , PUNCT cana-3488 147 3 π’œ1(𝒯(11ℨ π’œ1(𝒯(11ℨ NUM cana-3488 147 4 ) ) PUNCT cana-3488 147 5 βˆ’ βˆ’ ADP cana-3488 147 6 1,88,30,57,02,60,326𝒯(ℨ 1,88,30,57,02,60,326𝒯(ℨ NUM cana-3488 147 7 ) ) PUNCT cana-3488 147 8 + + NUM cana-3488 147 9 1,56,92,14,18,83,605𝒯(βˆ’β„¨),β„³ 1,56,92,14,18,83,605𝒯(βˆ’β„¨),β„³ X cana-3488 147 10 ) ) PUNCT cana-3488 147 11 β‰₯ β‰₯ NOUN cana-3488 147 12 π’œ1(𝒯(11ℨ π’œ1(𝒯(11ℨ NUM cana-3488 147 13 ) ) PUNCT cana-3488 148 1 βˆ’ βˆ’ PROPN cana-3488 148 2 1 1 NUM cana-3488 148 3 1113𝑛 1113𝑛 NOUN cana-3488 148 4 𝔉(11ℨ 𝔉(11ℨ NOUN cana-3488 148 5 ) ) PUNCT cana-3488 148 6 , , PUNCT cana-3488 148 7 β„³ β„³ PROPN cana-3488 148 8 4 4 NUM cana-3488 148 9 ) ) PUNCT cana-3488 148 10 βˆ— βˆ— NOUN cana-3488 148 11 π’œ1(βˆ’1,88,30,57,02,60,326𝒯(ℨ π’œ1(βˆ’1,88,30,57,02,60,326𝒯(ℨ NOUN cana-3488 148 12 ) ) PUNCT cana-3488 149 1 + + CCONJ cana-3488 149 2 1,88,30,57,02,60,326 1,88,30,57,02,60,326 NUM cana-3488 149 3 1 1 NUM cana-3488 149 4 1113𝑛 1113𝑛 NUM cana-3488 149 5 𝔉(ℨ 𝔉(ℨ NOUN cana-3488 149 6 ) ) PUNCT cana-3488 149 7 , , PUNCT cana-3488 149 8 β„³ β„³ PROPN cana-3488 149 9 4 4 NUM cana-3488 149 10 ) ) PUNCT cana-3488 149 11 βˆ— βˆ— NOUN cana-3488 149 12 π’œ1(1,56,92,14,18,83,605𝒯(βˆ’β„¨ π’œ1(1,56,92,14,18,83,605𝒯(βˆ’β„¨ NUM cana-3488 149 13 ) ) PUNCT cana-3488 150 1 + + CCONJ cana-3488 151 1 1,56,92,14,18,83,605 1,56,92,14,18,83,605 NUM cana-3488 151 2 1 1 NUM cana-3488 151 3 1113𝑛 1113𝑛 NOUN cana-3488 151 4 𝔉(βˆ’β„¨ 𝔉(βˆ’β„¨ NOUN cana-3488 151 5 ) ) PUNCT cana-3488 151 6 , , PUNCT cana-3488 151 7 β„³ β„³ PROPN cana-3488 151 8 4 4 NUM cana-3488 151 9 ) ) PUNCT cana-3488 151 10 βˆ— βˆ— NOUN cana-3488 151 11 π’œ1 π’œ1 NOUN cana-3488 151 12 ( ( PUNCT cana-3488 151 13 1 1 NUM cana-3488 151 14 1113𝑛 1113𝑛 NOUN cana-3488 151 15 𝔉(11ℨ 𝔉(11ℨ NOUN cana-3488 151 16 ) ) PUNCT cana-3488 152 1 βˆ’ βˆ’ PROPN cana-3488 152 2 1,88,30,57,02,60,326 1,88,30,57,02,60,326 NUM cana-3488 152 3 1 1 NUM cana-3488 152 4 1113𝑛 1113𝑛 NOUN cana-3488 152 5 𝔉(ℨ 𝔉(ℨ NOUN cana-3488 152 6 ) ) PUNCT cana-3488 153 1 + + CCONJ cana-3488 153 2 1,56,92,14,18,83,605 1,56,92,14,18,83,605 NUM cana-3488 153 3 1 1 NUM cana-3488 153 4 1113𝑛 1113𝑛 NOUN cana-3488 153 5 𝔉(βˆ’β„¨ 𝔉(βˆ’β„¨ NOUN cana-3488 153 6 ) ) PUNCT cana-3488 153 7 , , PUNCT cana-3488 153 8 β„³ β„³ PROPN cana-3488 153 9 4 4 NUM cana-3488 153 10 ) ) PUNCT cana-3488 153 11 ( ( PUNCT cana-3488 153 12 20 20 X cana-3488 153 13 ) ) PUNCT cana-3488 153 14 communications communication NOUN cana-3488 153 15 on on ADP cana-3488 153 16 applied apply VERB cana-3488 153 17 nonlinear nonlinear ADJ cana-3488 153 18 analysis analysis NOUN cana-3488 153 19 issn issn NOUN cana-3488 153 20 : : PUNCT cana-3488 153 21 1074 1074 NUM cana-3488 153 22 - - PUNCT cana-3488 153 23 133x 133x NUM cana-3488 153 24 vol vol NOUN cana-3488 153 25 32 32 NUM cana-3488 153 26 no no NOUN cana-3488 153 27 . . PUNCT cana-3488 154 1 7s 7 NOUN cana-3488 154 2 ( ( PUNCT cana-3488 154 3 2025 2025 NUM cana-3488 154 4 ) ) PUNCT cana-3488 154 5 814 814 NUM cana-3488 154 6 https://internationalpubls.com https://internationalpubls.com X cana-3488 154 7 π’œ2(𝒯(11ℨ π’œ2(𝒯(11ℨ NOUN cana-3488 154 8 ) ) PUNCT cana-3488 154 9 βˆ’ βˆ’ ADP cana-3488 154 10 1,88,30,57,02,60,326𝒯(ℨ 1,88,30,57,02,60,326𝒯(ℨ NUM cana-3488 154 11 ) ) PUNCT cana-3488 155 1 + + NUM cana-3488 155 2 1,56,92,14,18,83,605𝒯(βˆ’β„¨),β„³ 1,56,92,14,18,83,605𝒯(βˆ’β„¨),β„³ X cana-3488 155 3 ) ) PUNCT cana-3488 155 4 β‰₯ β‰₯ NOUN cana-3488 155 5 π’œ1(𝒯(11ℨ π’œ1(𝒯(11ℨ NUM cana-3488 155 6 ) ) PUNCT cana-3488 156 1 βˆ’ βˆ’ PROPN cana-3488 156 2 1 1 NUM cana-3488 156 3 1113𝑛 1113𝑛 NOUN cana-3488 156 4 𝔉(11ℨ 𝔉(11ℨ NOUN cana-3488 156 5 ) ) PUNCT cana-3488 156 6 , , PUNCT cana-3488 156 7 β„³ β„³ PROPN cana-3488 156 8 4 4 NUM cana-3488 156 9 ) ) PUNCT cana-3488 156 10 β‹„ β‹„ NOUN cana-3488 156 11 π’œ2(βˆ’1,88,30,57,02,60,326𝒯(ℨ π’œ2(βˆ’1,88,30,57,02,60,326𝒯(ℨ NOUN cana-3488 156 12 ) ) PUNCT cana-3488 157 1 + + CCONJ cana-3488 157 2 1,88,30,57,02,60,326 1,88,30,57,02,60,326 NUM cana-3488 157 3 1 1 NUM cana-3488 157 4 1113𝑛 1113𝑛 NUM cana-3488 157 5 𝔉(ℨ 𝔉(ℨ NOUN cana-3488 157 6 ) ) PUNCT cana-3488 157 7 , , PUNCT cana-3488 157 8 β„³ β„³ PROPN cana-3488 157 9 4 4 NUM cana-3488 157 10 ) ) PUNCT cana-3488 157 11 β‹„ β‹„ NOUN cana-3488 157 12 π’œ2(βˆ’1,56,92,14,18,83,605𝒯(βˆ’β„¨ π’œ2(βˆ’1,56,92,14,18,83,605𝒯(βˆ’β„¨ PRON cana-3488 157 13 ) ) PUNCT cana-3488 158 1 + + CCONJ cana-3488 158 2 1,56,92,14,18,83,605 1,56,92,14,18,83,605 NUM cana-3488 158 3 1 1 NUM cana-3488 158 4 1113𝑛 1113𝑛 NOUN cana-3488 158 5 𝔉(βˆ’β„¨ 𝔉(βˆ’β„¨ NOUN cana-3488 158 6 ) ) PUNCT cana-3488 158 7 , , PUNCT cana-3488 158 8 β„³ β„³ PROPN cana-3488 158 9 4 4 NUM cana-3488 158 10 ) ) PUNCT cana-3488 158 11 β‹„ β‹„ PROPN cana-3488 158 12 π’œ2 π’œ2 PROPN cana-3488 158 13 ( ( PUNCT cana-3488 158 14 1 1 NUM cana-3488 158 15 1113𝑛 1113𝑛 NOUN cana-3488 158 16 𝔉(11ℨ 𝔉(11ℨ NOUN cana-3488 158 17 ) ) PUNCT cana-3488 158 18 βˆ’ βˆ’ PROPN cana-3488 159 1 1,88,30,57,02,60,326 1,88,30,57,02,60,326 NUM cana-3488 159 2 1 1 NUM cana-3488 159 3 1113𝑛 1113𝑛 NOUN cana-3488 159 4 𝔉(ℨ 𝔉(ℨ NOUN cana-3488 159 5 ) ) PUNCT cana-3488 160 1 + + CCONJ cana-3488 160 2 1,56,92,14,18,83,605 1,56,92,14,18,83,605 NUM cana-3488 160 3 1 1 NUM cana-3488 160 4 1113𝑛 1113𝑛 NOUN cana-3488 160 5 𝔉(βˆ’β„¨ 𝔉(βˆ’β„¨ NOUN cana-3488 160 6 ) ) PUNCT cana-3488 160 7 , , PUNCT cana-3488 160 8 β„³ β„³ PROPN cana-3488 160 9 4 4 NUM cana-3488 160 10 ) ) PUNCT cana-3488 160 11 ( ( PUNCT cana-3488 160 12 21 21 NUM cana-3488 160 13 ) ) PUNCT cana-3488 160 14 and and CCONJ cana-3488 160 15 π’œ3(𝒯(11ℨ π’œ3(𝒯(11ℨ X cana-3488 160 16 ) ) PUNCT cana-3488 160 17 βˆ’ βˆ’ PROPN cana-3488 160 18 1,88,30,57,02,60,326𝒯(ℨ 1,88,30,57,02,60,326𝒯(ℨ NUM cana-3488 160 19 ) ) PUNCT cana-3488 160 20 + + NUM cana-3488 160 21 1,56,92,14,18,83,605𝒯(βˆ’β„¨),β„³ 1,56,92,14,18,83,605𝒯(βˆ’β„¨),β„³ X cana-3488 160 22 ) ) PUNCT cana-3488 160 23 β‰₯ β‰₯ NOUN cana-3488 160 24 π’œ3(𝒯(11ℨ π’œ3(𝒯(11ℨ PUNCT cana-3488 160 25 ) ) PUNCT cana-3488 160 26 βˆ’ βˆ’ PROPN cana-3488 160 27 1 1 NUM cana-3488 160 28 1113𝑛 1113𝑛 NOUN cana-3488 160 29 𝔉(11ℨ 𝔉(11ℨ NOUN cana-3488 160 30 ) ) PUNCT cana-3488 160 31 , , PUNCT cana-3488 160 32 β„³ β„³ PROPN cana-3488 160 33 4 4 NUM cana-3488 160 34 ) ) PUNCT cana-3488 160 35 βŠ˜π’œ3(βˆ’1,88,30,57,02,60,326𝒯(ℨ βŠ˜π’œ3(βˆ’1,88,30,57,02,60,326𝒯(ℨ PROPN cana-3488 160 36 ) ) PUNCT cana-3488 160 37 + + CCONJ cana-3488 160 38 1,88,30,57,02,60,326 1,88,30,57,02,60,326 NUM cana-3488 160 39 1 1 NUM cana-3488 160 40 1113𝑛 1113𝑛 NUM cana-3488 160 41 𝔉(ℨ 𝔉(ℨ NOUN cana-3488 160 42 ) ) PUNCT cana-3488 160 43 , , PUNCT cana-3488 160 44 β„³ β„³ PROPN cana-3488 160 45 4 4 NUM cana-3488 160 46 ) ) PUNCT cana-3488 160 47 βŠ˜π’œ3(1,56,92,14,18,83,605𝒯(βˆ’β„¨ βŠ˜π’œ3(1,56,92,14,18,83,605𝒯(βˆ’β„¨ NOUN cana-3488 160 48 ) ) PUNCT cana-3488 161 1 + + CCONJ cana-3488 162 1 1,56,92,14,18,83,605 1,56,92,14,18,83,605 NUM cana-3488 162 2 1 1 NUM cana-3488 162 3 1113𝑛 1113𝑛 NOUN cana-3488 162 4 𝔉(βˆ’β„¨ 𝔉(βˆ’β„¨ NOUN cana-3488 162 5 ) ) PUNCT cana-3488 162 6 , , PUNCT cana-3488 162 7 β„³ β„³ PROPN cana-3488 162 8 4 4 NUM cana-3488 162 9 ) ) PUNCT cana-3488 162 10 βŠ˜π’œ1 βŠ˜π’œ1 NOUN cana-3488 162 11 ( ( PUNCT cana-3488 162 12 1 1 NUM cana-3488 162 13 1113𝑛 1113𝑛 NOUN cana-3488 162 14 𝔉(11ℨ 𝔉(11ℨ NOUN cana-3488 162 15 ) ) PUNCT cana-3488 162 16 βˆ’ βˆ’ PROPN cana-3488 163 1 1,88,30,57,02,60,326 1,88,30,57,02,60,326 NUM cana-3488 163 2 1 1 NUM cana-3488 163 3 1113𝑛 1113𝑛 NOUN cana-3488 163 4 𝔉(ℨ 𝔉(ℨ NOUN cana-3488 163 5 ) ) PUNCT cana-3488 164 1 + + CCONJ cana-3488 164 2 1,56,92,14,18,83,605 1,56,92,14,18,83,605 NUM cana-3488 164 3 1 1 NUM cana-3488 164 4 1113𝑛 1113𝑛 NOUN cana-3488 164 5 𝔉(βˆ’β„¨ 𝔉(βˆ’β„¨ NOUN cana-3488 164 6 ) ) PUNCT cana-3488 164 7 , , PUNCT cana-3488 164 8 β„³ β„³ PROPN cana-3488 164 9 4 4 NUM cana-3488 164 10 ) ) PUNCT cana-3488 164 11 ( ( PUNCT cana-3488 164 12 22 22 NUM cana-3488 164 13 ) ) PUNCT cana-3488 164 14 for for ADP cana-3488 164 15 all all DET cana-3488 164 16 ℨ ℨ NOUN cana-3488 164 17 ∈ ∈ NOUN cana-3488 164 18 𝒳𝑀 𝒳𝑀 NOUN cana-3488 164 19 and and CCONJ cana-3488 164 20 all all DET cana-3488 164 21 β„³ β„³ PROPN cana-3488 164 22 > > X cana-3488 164 23 0 0 NUM cana-3488 164 24 . . PUNCT cana-3488 165 1 also also ADV cana-3488 165 2 , , PUNCT cana-3488 165 3 lim lim PROPN cana-3488 165 4 π‘›β†’βˆž π‘›β†’βˆž NUM cana-3488 165 5 π’œ1 π’œ1 PROPN cana-3488 165 6 ( ( PUNCT cana-3488 165 7 1 1 NUM cana-3488 165 8 1113𝑛 1113𝑛 NOUN cana-3488 166 1 [ [ X cana-3488 166 2 𝔉(3 𝔉(3 NUM cana-3488 166 3 β‹… β‹… PROPN cana-3488 166 4 11𝑛ℨ 11𝑛ℨ NUM cana-3488 166 5 ) ) PUNCT cana-3488 166 6 βˆ’ βˆ’ PROPN cana-3488 166 7 1,88,30,57,02,60,326𝔉(11𝑛ℨ 1,88,30,57,02,60,326𝔉(11𝑛ℨ NUM cana-3488 166 8 ) ) PUNCT cana-3488 166 9 + + NUM cana-3488 166 10 1,56,92,14,18,83,605𝔉(βˆ’11𝑛ℨ 1,56,92,14,18,83,605𝔉(βˆ’11𝑛ℨ PROPN cana-3488 166 11 ) ) PUNCT cana-3488 167 1 ] ] PUNCT cana-3488 167 2 , , PUNCT cana-3488 167 3 β„³ β„³ PROPN cana-3488 167 4 4 4 NUM cana-3488 167 5 ) ) PUNCT cana-3488 167 6 = = SYM cana-3488 167 7 1 1 NUM cana-3488 167 8 lim lim NOUN cana-3488 167 9 π‘›β†’βˆž π‘›β†’βˆž NUM cana-3488 167 10 π’œ2 π’œ2 PROPN cana-3488 167 11 ( ( PUNCT cana-3488 167 12 1 1 NUM cana-3488 167 13 1113𝑛 1113𝑛 NOUN cana-3488 167 14 [ [ X cana-3488 167 15 𝔉(3 𝔉(3 NUM cana-3488 167 16 β‹… β‹… PROPN cana-3488 167 17 11𝑛ℨ 11𝑛ℨ NUM cana-3488 167 18 ) ) PUNCT cana-3488 167 19 βˆ’ βˆ’ PROPN cana-3488 167 20 1,88,30,57,02,60,326𝔉(11𝑛ℨ 1,88,30,57,02,60,326𝔉(11𝑛ℨ NUM cana-3488 167 21 ) ) PUNCT cana-3488 167 22 + + NUM cana-3488 167 23 1,56,92,14,18,83,605𝔉(βˆ’11𝑛ℨ 1,56,92,14,18,83,605𝔉(βˆ’11𝑛ℨ PROPN cana-3488 167 24 ) ) PUNCT cana-3488 167 25 ] ] PUNCT cana-3488 167 26 , , PUNCT cana-3488 167 27 β„³ β„³ PROPN cana-3488 167 28 4 4 NUM cana-3488 167 29 ) ) PUNCT cana-3488 167 30 = = SYM cana-3488 167 31 0 0 NUM cana-3488 167 32 lim lim NOUN cana-3488 167 33 π‘›β†’βˆž π‘›β†’βˆž NUM cana-3488 167 34 π’œ3 π’œ3 PROPN cana-3488 167 35 ( ( PUNCT cana-3488 167 36 1 1 NUM cana-3488 167 37 1113𝑛 1113𝑛 NOUN cana-3488 167 38 [ [ X cana-3488 167 39 𝔉(3 𝔉(3 NUM cana-3488 167 40 β‹… β‹… PROPN cana-3488 167 41 11𝑛ℨ 11𝑛ℨ NUM cana-3488 167 42 ) ) PUNCT cana-3488 167 43 βˆ’ βˆ’ PROPN cana-3488 167 44 1,88,30,57,02,60,326𝔉(11𝑛ℨ 1,88,30,57,02,60,326𝔉(11𝑛ℨ NUM cana-3488 167 45 ) ) PUNCT cana-3488 167 46 + + NUM cana-3488 167 47 1,56,92,14,18,83,605𝔉(βˆ’11𝑛ℨ 1,56,92,14,18,83,605𝔉(βˆ’11𝑛ℨ PROPN cana-3488 167 48 ) ) PUNCT cana-3488 167 49 ] ] PUNCT cana-3488 167 50 , , PUNCT cana-3488 167 51 β„³ β„³ PROPN cana-3488 167 52 4 4 NUM cana-3488 167 53 ) ) PUNCT cana-3488 167 54 = = SYM cana-3488 167 55 0 0 X cana-3488 167 56 } } PUNCT cana-3488 167 57 ( ( PUNCT cana-3488 167 58 23 23 NUM cana-3488 167 59 ) ) PUNCT cana-3488 167 60 for for ADP cana-3488 167 61 all all DET cana-3488 167 62 ℨ ℨ NOUN cana-3488 167 63 ∈ ∈ NOUN cana-3488 167 64 𝒳𝑀 𝒳𝑀 NOUN cana-3488 167 65 and and CCONJ cana-3488 167 66 all all DET cana-3488 167 67 β„³ β„³ PROPN cana-3488 167 68 > > X cana-3488 167 69 0 0 NUM cana-3488 167 70 . . PUNCT cana-3488 168 1 by by ADP cana-3488 168 2 taking take VERB cana-3488 168 3 𝑛 𝑛 PRON cana-3488 168 4 β†’ β†’ SYM cana-3488 168 5 ∞ ∞ NUM cana-3488 168 6 in in ADP cana-3488 168 7 ( ( PUNCT cana-3488 168 8 21 21 NUM cana-3488 168 9 ) ) PUNCT cana-3488 168 10 , , PUNCT cana-3488 168 11 ( ( PUNCT cana-3488 168 12 22 22 NUM cana-3488 168 13 ) ) PUNCT cana-3488 168 14 and and CCONJ cana-3488 168 15 using use VERB cana-3488 168 16 ( ( PUNCT cana-3488 168 17 23 23 NUM cana-3488 168 18 ) ) PUNCT cana-3488 168 19 , , PUNCT cana-3488 168 20 we we PRON cana-3488 168 21 proved prove VERB cana-3488 168 22 that that SCONJ cana-3488 168 23 𝒯 𝒯 PROPN cana-3488 168 24 satisfies satisfie NOUN cana-3488 168 25 ( ( PUNCT cana-3488 168 26 1 1 NUM cana-3488 168 27 ) ) PUNCT cana-3488 168 28 . . PUNCT cana-3488 169 1 therefore therefore ADV cana-3488 169 2 , , PUNCT cana-3488 169 3 𝒯 𝒯 PROPN cana-3488 169 4 is be AUX cana-3488 169 5 a a DET cana-3488 169 6 tridecic tridecic NOUN cana-3488 169 7 mapping mapping NOUN cana-3488 169 8 . . PUNCT cana-3488 170 1 next next ADV cana-3488 170 2 , , PUNCT cana-3488 170 3 we we PRON cana-3488 170 4 need need VERB cana-3488 170 5 to to PART cana-3488 170 6 prove prove VERB cana-3488 170 7 𝒯(ℨ 𝒯(ℨ NOUN cana-3488 170 8 ) ) PUNCT cana-3488 170 9 is be AUX cana-3488 170 10 unique unique ADJ cana-3488 170 11 , , PUNCT cana-3488 170 12 let let VERB cana-3488 170 13 𝒯′(ℨ 𝒯′(ℨ NOUN cana-3488 170 14 ) ) PUNCT cana-3488 170 15 be be AUX cana-3488 170 16 another another DET cana-3488 170 17 tridecic tridecic NOUN cana-3488 170 18 fe fe NOUN cana-3488 170 19 satisfying satisfying NOUN cana-3488 170 20 ( ( PUNCT cana-3488 170 21 1 1 NUM cana-3488 170 22 ) ) PUNCT cana-3488 170 23 and and CCONJ cana-3488 170 24 ( ( PUNCT cana-3488 170 25 4 4 NUM cana-3488 170 26 ) ) PUNCT cana-3488 170 27 , , PUNCT cana-3488 170 28 then then ADV cana-3488 170 29 π’œ1(𝒯(ℨ π’œ1(𝒯(ℨ NOUN cana-3488 170 30 ) ) PUNCT cana-3488 171 1 βˆ’ βˆ’ PROPN cana-3488 171 2 𝒯′(ℨ),β„³ 𝒯′(ℨ),β„³ PROPN cana-3488 171 3 ) ) PUNCT cana-3488 171 4 β‰₯ β‰₯ AUX cana-3488 171 5 π’œ1 π’œ1 NOUN cana-3488 171 6 ( ( PUNCT cana-3488 171 7 𝒯(11 𝒯(11 PROPN cana-3488 171 8 𝑛ℨ 𝑛ℨ NOUN cana-3488 171 9 ) ) PUNCT cana-3488 171 10 βˆ’ βˆ’ PROPN cana-3488 171 11 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 171 12 ) ) PUNCT cana-3488 171 13 , , PUNCT cana-3488 171 14 β„³.1113𝑛 β„³.1113𝑛 NUM cana-3488 171 15 2 2 X cana-3488 171 16 ) ) PUNCT cana-3488 171 17 βˆ— βˆ— NOUN cana-3488 171 18 π’œ1 π’œ1 NOUN cana-3488 171 19 ( ( PUNCT cana-3488 171 20 𝔉(11 𝔉(11 PROPN cana-3488 171 21 𝑛ℨ 𝑛ℨ NOUN cana-3488 171 22 ) ) PUNCT cana-3488 172 1 βˆ’ βˆ’ PROPN cana-3488 172 2 𝒯′(11𝑛ℨ 𝒯′(11𝑛ℨ PROPN cana-3488 172 3 ) ) PUNCT cana-3488 172 4 , , PUNCT cana-3488 172 5 β„³.1113𝑛 β„³.1113𝑛 NUM cana-3488 172 6 2 2 X cana-3488 172 7 ) ) PUNCT cana-3488 172 8 β‰₯ β‰₯ NOUN cana-3488 172 9 π’œ1β€² π’œ1β€² X cana-3488 172 10 ( ( PUNCT cana-3488 172 11 Γ°(11 Γ°(11 NOUN cana-3488 172 12 𝑛ℨ 𝑛ℨ NOUN cana-3488 172 13 ) ) PUNCT cana-3488 172 14 , , PUNCT cana-3488 172 15 1113𝑛 1113𝑛 NUM cana-3488 172 16 β„³ β„³ PROPN cana-3488 172 17 |1113βˆ’π”™| |1113βˆ’π”™| PROPN cana-3488 172 18 2 2 NUM cana-3488 172 19 ) ) PUNCT cana-3488 172 20 β‰₯ β‰₯ NOUN cana-3488 172 21 π’œ1β€² π’œ1β€² NOUN cana-3488 172 22 ( ( PUNCT cana-3488 172 23 Γ°(ℨ Γ°(ℨ PROPN cana-3488 172 24 ) ) PUNCT cana-3488 172 25 , , PUNCT cana-3488 172 26 1113𝑛 1113𝑛 NUM cana-3488 172 27 β„³ β„³ PROPN cana-3488 172 28 |1113βˆ’π”™| |1113βˆ’π”™| PROPN cana-3488 172 29 2⋅𝔙𝑛 2⋅𝔙𝑛 NUM cana-3488 172 30 ) ) PUNCT cana-3488 172 31 π’œ2(𝒯(ℨ π’œ2(𝒯(ℨ NUM cana-3488 172 32 ) ) PUNCT cana-3488 172 33 βˆ’ βˆ’ PROPN cana-3488 172 34 𝒯′(ℨ),β„³ 𝒯′(ℨ),β„³ PROPN cana-3488 172 35 ) ) PUNCT cana-3488 172 36 ≀ ≀ NUM cana-3488 172 37 π’œ2 π’œ2 NOUN cana-3488 172 38 ( ( PUNCT cana-3488 172 39 𝒯(11 𝒯(11 PROPN cana-3488 172 40 𝑛ℨ 𝑛ℨ NOUN cana-3488 172 41 ) ) PUNCT cana-3488 172 42 βˆ’ βˆ’ PROPN cana-3488 172 43 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 172 44 ) ) PUNCT cana-3488 172 45 , , PUNCT cana-3488 172 46 β„³.1113𝑛 β„³.1113𝑛 NUM cana-3488 172 47 2 2 X cana-3488 172 48 ) ) PUNCT cana-3488 172 49 β‹„ β‹„ PROPN cana-3488 172 50 π’œ2 π’œ2 PROPN cana-3488 172 51 ( ( PUNCT cana-3488 172 52 𝔉(11 𝔉(11 PROPN cana-3488 172 53 𝑛ℨ 𝑛ℨ NOUN cana-3488 172 54 ) ) PUNCT cana-3488 172 55 βˆ’ βˆ’ PROPN cana-3488 172 56 𝒯′(11𝑛ℨ 𝒯′(11𝑛ℨ PROPN cana-3488 172 57 ) ) PUNCT cana-3488 172 58 , , PUNCT cana-3488 172 59 β„³.1113𝑛 β„³.1113𝑛 NUM cana-3488 172 60 2 2 X cana-3488 172 61 ) ) PUNCT cana-3488 172 62 ≀ ≀ NUM cana-3488 172 63 π’œ2β€² π’œ2β€² ADV cana-3488 172 64 ( ( PUNCT cana-3488 172 65 Γ°(11 Γ°(11 NOUN cana-3488 172 66 𝑛ℨ 𝑛ℨ NOUN cana-3488 172 67 ) ) PUNCT cana-3488 172 68 , , PUNCT cana-3488 172 69 1113𝑛ℳ 1113𝑛ℳ NUM cana-3488 172 70 |1113βˆ’π”™| |1113βˆ’π”™| NUM cana-3488 172 71 2 2 NUM cana-3488 172 72 ) ) PUNCT cana-3488 172 73 ≀ ≀ NUM cana-3488 172 74 π’œ2β€² π’œ2β€² ADV cana-3488 172 75 ( ( PUNCT cana-3488 172 76 Γ°(ℨ Γ°(ℨ NOUN cana-3488 172 77 ) ) PUNCT cana-3488 172 78 , , PUNCT cana-3488 172 79 1113𝑛 1113𝑛 NUM cana-3488 172 80 β„³ β„³ PROPN cana-3488 172 81 |1113βˆ’π”™| |1113βˆ’π”™| PROPN cana-3488 172 82 2⋅𝔙𝑛 2⋅𝔙𝑛 NUM cana-3488 172 83 ) ) PUNCT cana-3488 172 84 communications communication NOUN cana-3488 172 85 on on ADP cana-3488 172 86 applied apply VERB cana-3488 172 87 nonlinear nonlinear ADJ cana-3488 172 88 analysis analysis NOUN cana-3488 172 89 issn issn NOUN cana-3488 172 90 : : PUNCT cana-3488 172 91 1074 1074 NUM cana-3488 172 92 - - PUNCT cana-3488 172 93 133x 133x NUM cana-3488 172 94 vol vol NOUN cana-3488 172 95 32 32 NUM cana-3488 172 96 no no NOUN cana-3488 172 97 . . PUNCT cana-3488 173 1 7s 7 NOUN cana-3488 173 2 ( ( PUNCT cana-3488 173 3 2025 2025 NUM cana-3488 173 4 ) ) PUNCT cana-3488 173 5 815 815 NUM cana-3488 173 6 https://internationalpubls.com https://internationalpubls.com X cana-3488 173 7 π’œ3(𝒯(ℨ π’œ3(𝒯(ℨ NUM cana-3488 173 8 ) ) PUNCT cana-3488 173 9 βˆ’ βˆ’ PROPN cana-3488 174 1 𝒯′(ℨ),β„³ 𝒯′(ℨ),β„³ PROPN cana-3488 174 2 ) ) PUNCT cana-3488 174 3 ≀ ≀ PROPN cana-3488 174 4 π’œ3 π’œ3 NOUN cana-3488 174 5 ( ( PUNCT cana-3488 174 6 𝒯(11 𝒯(11 PROPN cana-3488 174 7 𝑛ℨ 𝑛ℨ NOUN cana-3488 174 8 ) ) PUNCT cana-3488 174 9 βˆ’ βˆ’ PROPN cana-3488 174 10 𝔉(11𝑛ℨ 𝔉(11𝑛ℨ NUM cana-3488 174 11 ) ) PUNCT cana-3488 174 12 , , PUNCT cana-3488 174 13 β„³.1113𝑛 β„³.1113𝑛 NUM cana-3488 174 14 2 2 X cana-3488 174 15 ) ) PUNCT cana-3488 174 16 βˆ— βˆ— NOUN cana-3488 174 17 π’œ3 π’œ3 PROPN cana-3488 174 18 ( ( PUNCT cana-3488 174 19 𝔉(11 𝔉(11 PROPN cana-3488 174 20 𝑛ℨ 𝑛ℨ NOUN cana-3488 174 21 ) ) PUNCT cana-3488 174 22 βˆ’ βˆ’ PROPN cana-3488 174 23 𝒯′(11𝑛ℨ 𝒯′(11𝑛ℨ PROPN cana-3488 174 24 ) ) PUNCT cana-3488 174 25 , , PUNCT cana-3488 174 26 β„³.1113𝑛 β„³.1113𝑛 NUM cana-3488 174 27 2 2 X cana-3488 174 28 ) ) PUNCT cana-3488 174 29 ≀ ≀ NUM cana-3488 174 30 π’œ3β€² π’œ3β€² NOUN cana-3488 174 31 ( ( PUNCT cana-3488 174 32 Γ°(11 Γ°(11 NOUN cana-3488 174 33 𝑛ℨ 𝑛ℨ NOUN cana-3488 174 34 ) ) PUNCT cana-3488 174 35 , , PUNCT cana-3488 174 36 1113𝑛 1113𝑛 NUM cana-3488 174 37 β„³ β„³ PROPN cana-3488 174 38 |1113βˆ’π”™| |1113βˆ’π”™| PROPN cana-3488 174 39 2 2 NUM cana-3488 174 40 ) ) PUNCT cana-3488 174 41 ≀ ≀ NUM cana-3488 175 1 π’œ3β€² π’œ3β€² NOUN cana-3488 175 2 ( ( PUNCT cana-3488 175 3 Γ°(ℨ Γ°(ℨ PROPN cana-3488 175 4 ) ) PUNCT cana-3488 175 5 , , PUNCT cana-3488 175 6 1113𝑛 1113𝑛 NUM cana-3488 175 7 β„³ β„³ PROPN cana-3488 175 8 |1113βˆ’π”™| |1113βˆ’π”™| PROPN cana-3488 175 9 2⋅𝔙𝑛 2⋅𝔙𝑛 NUM cana-3488 175 10 ) ) PUNCT cana-3488 175 11 for for ADP cana-3488 175 12 all all DET cana-3488 175 13 ℨ ℨ NOUN cana-3488 175 14 ∈ ∈ NOUN cana-3488 175 15 𝒳𝑀 𝒳𝑀 NOUN cana-3488 175 16 and and CCONJ cana-3488 175 17 all all DET cana-3488 175 18 β„³ β„³ PROPN cana-3488 175 19 > > X cana-3488 175 20 0 0 PUNCT cana-3488 175 21 . . PUNCT cana-3488 176 1 since since SCONJ cana-3488 176 2 lim lim PROPN cana-3488 176 3 π‘›β†’βˆž π‘›β†’βˆž NUM cana-3488 176 4 1113𝑛 1113𝑛 NOUN cana-3488 176 5 β„³ β„³ PROPN cana-3488 176 6 |1113βˆ’π”™| |1113βˆ’π”™| NOUN cana-3488 176 7 2 2 NUM cana-3488 176 8 𝔙𝑛 𝔙𝑛 PROPN cana-3488 176 9 = = SYM cana-3488 176 10 ∞ ∞ PROPN cana-3488 176 11 , , PUNCT cana-3488 176 12 we we PRON cana-3488 176 13 obtain obtain VERB cana-3488 176 14 lim lim PROPN cana-3488 176 15 π‘›β†’βˆž π‘›β†’βˆž NUM cana-3488 176 16 π’œ1β€² π’œ1β€² PROPN cana-3488 176 17 ( ( PUNCT cana-3488 176 18 Γ°(ℨ Γ°(ℨ PROPN cana-3488 176 19 ) ) PUNCT cana-3488 176 20 , , PUNCT cana-3488 176 21 1113𝑛 1113𝑛 NUM cana-3488 176 22 β„³ β„³ PROPN cana-3488 176 23 |1113βˆ’π”™| |1113βˆ’π”™| PROPN cana-3488 176 24 2⋅𝔙𝑛 2⋅𝔙𝑛 NUM cana-3488 176 25 ) ) PUNCT cana-3488 177 1 = = SYM cana-3488 177 2 1 1 NUM cana-3488 177 3 lim lim NOUN cana-3488 177 4 π‘›β†’βˆž π‘›β†’βˆž NUM cana-3488 177 5 π’œ2β€² π’œ2β€² PROPN cana-3488 177 6 ( ( PUNCT cana-3488 177 7 Γ°(ℨ Γ°(ℨ NOUN cana-3488 177 8 ) ) PUNCT cana-3488 177 9 , , PUNCT cana-3488 177 10 1113𝑛 1113𝑛 NUM cana-3488 177 11 β„³ β„³ PROPN cana-3488 177 12 |1113βˆ’π”™| |1113βˆ’π”™| PROPN cana-3488 177 13 2⋅𝔙𝑛 2⋅𝔙𝑛 NUM cana-3488 177 14 ) ) PUNCT cana-3488 177 15 = = SYM cana-3488 178 1 0 0 NUM cana-3488 178 2 lim lim NOUN cana-3488 178 3 π‘›β†’βˆž π‘›β†’βˆž NUM cana-3488 178 4 π’œ3β€² π’œ3β€² NOUN cana-3488 178 5 ( ( PUNCT cana-3488 178 6 Γ°(ℨ Γ°(ℨ PROPN cana-3488 178 7 ) ) PUNCT cana-3488 178 8 , , PUNCT cana-3488 178 9 1113𝑛 1113𝑛 NUM cana-3488 178 10 β„³ β„³ PROPN cana-3488 178 11 |1113βˆ’π”™| |1113βˆ’π”™| PROPN cana-3488 178 12 2⋅𝔙𝑛 2⋅𝔙𝑛 NUM cana-3488 178 13 ) ) PUNCT cana-3488 179 1 = = PUNCT cana-3488 179 2 0 0 X cana-3488 179 3 } } PUNCT cana-3488 179 4 for for ADP cana-3488 179 5 all all DET cana-3488 179 6 ℨ ℨ NOUN cana-3488 179 7 ∈ ∈ NOUN cana-3488 179 8 𝒳𝑀 𝒳𝑀 NOUN cana-3488 179 9 and and CCONJ cana-3488 179 10 all all DET cana-3488 179 11 β„³ β„³ PROPN cana-3488 179 12 > > X cana-3488 179 13 0 0 NUM cana-3488 179 14 . . PUNCT cana-3488 180 1 thus thus ADV cana-3488 180 2 π’œ1(𝒯(ℨ π’œ1(𝒯(ℨ NOUN cana-3488 180 3 ) ) PUNCT cana-3488 181 1 βˆ’ βˆ’ PROPN cana-3488 181 2 𝒯′(ℨ),β„³ 𝒯′(ℨ),β„³ PROPN cana-3488 181 3 ) ) PUNCT cana-3488 181 4 = = SYM cana-3488 181 5 1 1 NUM cana-3488 181 6 π’œ2(𝒯(ℨ π’œ2(𝒯(ℨ NUM cana-3488 181 7 ) ) PUNCT cana-3488 181 8 βˆ’ βˆ’ PROPN cana-3488 182 1 𝒯′(ℨ),β„³ 𝒯′(ℨ),β„³ PROPN cana-3488 182 2 ) ) PUNCT cana-3488 182 3 = = NOUN cana-3488 182 4 0 0 NUM cana-3488 182 5 π’œ3(𝒯(ℨ π’œ3(𝒯(ℨ NUM cana-3488 182 6 ) ) PUNCT cana-3488 183 1 βˆ’ βˆ’ PROPN cana-3488 183 2 𝒯′(ℨ),β„³ 𝒯′(ℨ),β„³ PROPN cana-3488 183 3 ) ) PUNCT cana-3488 183 4 = = SYM cana-3488 183 5 0 0 X cana-3488 183 6 } } PUNCT cana-3488 183 7 for for ADP cana-3488 183 8 all all DET cana-3488 183 9 ℨ ℨ NOUN cana-3488 183 10 ∈ ∈ NOUN cana-3488 183 11 𝒳𝑀 𝒳𝑀 NOUN cana-3488 183 12 and and CCONJ cana-3488 183 13 all all DET cana-3488 183 14 β„³ β„³ PROPN cana-3488 183 15 > > X cana-3488 183 16 0 0 NUM cana-3488 183 17 . . PUNCT cana-3488 184 1 hence hence ADV cana-3488 184 2 , , PUNCT cana-3488 184 3 𝒯(ℨ 𝒯(ℨ NUM cana-3488 184 4 ) ) PUNCT cana-3488 184 5 = = SYM cana-3488 184 6 𝒯′(ℨ 𝒯′(ℨ NOUN cana-3488 184 7 ) ) PUNCT cana-3488 184 8 . . PUNCT cana-3488 185 1 therefore therefore ADV cana-3488 185 2 , , PUNCT cana-3488 185 3 𝒯(ℨ 𝒯(ℨ NUM cana-3488 185 4 ) ) PUNCT cana-3488 185 5 is be AUX cana-3488 185 6 unique unique ADJ cana-3488 185 7 . . PUNCT cana-3488 186 1 for for ADP cana-3488 186 2 second second ADJ cana-3488 186 3 case case NOUN cana-3488 186 4 , , PUNCT cana-3488 186 5 we we PRON cana-3488 186 6 have have VERB cana-3488 186 7 to to PART cana-3488 186 8 take take VERB cana-3488 186 9 𝐹 𝐹 PROPN cana-3488 186 10 = = PUNCT cana-3488 186 11 βˆ’1 βˆ’1 NOUN cana-3488 186 12 . . PUNCT cana-3488 187 1 considering consider VERB cana-3488 187 2 ℨ ℨ NOUN cana-3488 187 3 by by ADP cana-3488 187 4 ℨ ℨ PROPN cana-3488 187 5 13 13 NUM cana-3488 187 6 in in ADP cana-3488 187 7 ( ( PUNCT cana-3488 187 8 5 5 NUM cana-3488 187 9 ) ) PUNCT cana-3488 187 10 , , PUNCT cana-3488 187 11 we we PRON cana-3488 187 12 have have VERB cana-3488 187 13 π’œ1 π’œ1 NOUN cana-3488 187 14 ( ( PUNCT cana-3488 187 15 𝔉(ℨ 𝔉(ℨ X cana-3488 187 16 ) ) PUNCT cana-3488 187 17 βˆ’ βˆ’ PROPN cana-3488 187 18 11 11 NUM cana-3488 188 1 13𝔉 13𝔉 NUM cana-3488 188 2 ( ( PUNCT cana-3488 188 3 ℨ ℨ PROPN cana-3488 188 4 13 13 NUM cana-3488 188 5 ) ) PUNCT cana-3488 188 6 , , PUNCT cana-3488 188 7 β„³ β„³ PROPN cana-3488 188 8 ) ) PUNCT cana-3488 188 9 β‰₯ β‰₯ NOUN cana-3488 188 10 π’œ1β€² π’œ1β€² NOUN cana-3488 188 11 ( ( PUNCT cana-3488 188 12 Γ° Γ° PROPN cana-3488 188 13 ( ( PUNCT cana-3488 188 14 ℨ ℨ NOUN cana-3488 188 15 13 13 NUM cana-3488 188 16 ) ) PUNCT cana-3488 188 17 , , PUNCT cana-3488 188 18 β„³ β„³ PROPN cana-3488 188 19 ) ) PUNCT cana-3488 188 20 π’œ2 π’œ2 NOUN cana-3488 188 21 ( ( PUNCT cana-3488 188 22 𝔉(ℨ 𝔉(ℨ X cana-3488 188 23 ) ) PUNCT cana-3488 188 24 βˆ’ βˆ’ PROPN cana-3488 188 25 11 11 NUM cana-3488 188 26 13𝔉 13𝔉 NUM cana-3488 188 27 ( ( PUNCT cana-3488 188 28 ℨ ℨ PROPN cana-3488 188 29 13 13 NUM cana-3488 188 30 ) ) PUNCT cana-3488 188 31 , , PUNCT cana-3488 188 32 β„³ β„³ NOUN cana-3488 188 33 ) ) PUNCT cana-3488 188 34 ≀ ≀ NUM cana-3488 188 35 π’œ2β€² π’œ2β€² NOUN cana-3488 188 36 ( ( PUNCT cana-3488 188 37 Γ° Γ° X cana-3488 188 38 ( ( PUNCT cana-3488 188 39 ℨ ℨ NOUN cana-3488 188 40 13 13 NUM cana-3488 188 41 ) ) PUNCT cana-3488 188 42 , , PUNCT cana-3488 188 43 β„³ β„³ PROPN cana-3488 188 44 ) ) PUNCT cana-3488 188 45 π’œ3 π’œ3 NOUN cana-3488 188 46 ( ( PUNCT cana-3488 188 47 𝔉(ℨ 𝔉(ℨ X cana-3488 188 48 ) ) PUNCT cana-3488 188 49 βˆ’ βˆ’ PROPN cana-3488 188 50 11 11 NUM cana-3488 188 51 13𝔉 13𝔉 NUM cana-3488 188 52 ( ( PUNCT cana-3488 188 53 ℨ ℨ PROPN cana-3488 188 54 13 13 NUM cana-3488 188 55 ) ) PUNCT cana-3488 188 56 , , PUNCT cana-3488 188 57 β„³ β„³ NOUN cana-3488 188 58 ) ) PUNCT cana-3488 188 59 ≀ ≀ NUM cana-3488 188 60 π’œ3β€² π’œ3β€² NOUN cana-3488 189 1 ( ( PUNCT cana-3488 189 2 Γ° Γ° PROPN cana-3488 189 3 ( ( PUNCT cana-3488 189 4 ℨ ℨ NOUN cana-3488 189 5 13 13 NUM cana-3488 189 6 ) ) PUNCT cana-3488 189 7 , , PUNCT cana-3488 189 8 β„³ β„³ PROPN cana-3488 189 9 ) ) PUNCT cana-3488 189 10 } } PUNCT cana-3488 189 11 ( ( PUNCT cana-3488 189 12 24 24 NUM cana-3488 189 13 ) ) PUNCT cana-3488 189 14 for for ADP cana-3488 189 15 all all DET cana-3488 189 16 ℨ ℨ NOUN cana-3488 189 17 ∈ ∈ NOUN cana-3488 189 18 𝒳𝑀 𝒳𝑀 NOUN cana-3488 189 19 and and CCONJ cana-3488 189 20 all all DET cana-3488 189 21 β„³ β„³ PROPN cana-3488 189 22 > > X cana-3488 189 23 0 0 X cana-3488 189 24 . . PUNCT cana-3488 189 25 corollary corollary ADJ cana-3488 189 26 3.2 3.2 NUM cana-3488 189 27 let let VERB cana-3488 189 28 β„’ β„’ NOUN cana-3488 189 29 , , PUNCT cana-3488 189 30 π‘Ÿ π‘Ÿ NOUN cana-3488 189 31 are be AUX cana-3488 189 32 constants constant NOUN cana-3488 189 33 , , PUNCT cana-3488 189 34 with with ADP cana-3488 189 35 β„’ β„’ PROPN cana-3488 189 36 > > SYM cana-3488 189 37 0 0 NUM cana-3488 189 38 and and CCONJ cana-3488 189 39 π‘Ÿ π‘Ÿ NOUN cana-3488 189 40 β‰  β‰  PROPN cana-3488 189 41 12 12 NUM cana-3488 189 42 and and CCONJ cana-3488 189 43 let let VERB cana-3488 189 44 an an DET cana-3488 189 45 odd odd ADJ cana-3488 189 46 function function NOUN cana-3488 189 47 𝔉:𝒳𝑀 𝔉:𝒳𝑀 NOUN cana-3488 189 48 βŸΆπ’΄π‘€ βŸΆπ’΄π‘€ NOUN cana-3488 189 49 satisfies satisfie NOUN cana-3488 189 50 π’œ1(𝔉(11ℨ π’œ1(𝔉(11ℨ PUNCT cana-3488 189 51 ) ) PUNCT cana-3488 190 1 βˆ’ βˆ’ ADP cana-3488 190 2 1,88,30,57,02,60,326𝔉(ℨ 1,88,30,57,02,60,326𝔉(ℨ NUM cana-3488 190 3 ) ) PUNCT cana-3488 190 4 + + CCONJ cana-3488 190 5 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ X cana-3488 190 6 ) ) PUNCT cana-3488 190 7 β‰₯ β‰₯ NOUN cana-3488 190 8 { { PUNCT cana-3488 190 9 π’œ1β€²(β„’,β„³ π’œ1β€²(β„’,β„³ PROPN cana-3488 190 10 ) ) PUNCT cana-3488 190 11 , , PUNCT cana-3488 190 12 π’œ1β€²(β„’(||ℨ|| π’œ1β€²(β„’(||ℨ|| PROPN cana-3488 190 13 π‘Ÿ),β„³ π‘Ÿ),β„³ NUM cana-3488 190 14 ) ) PUNCT cana-3488 190 15 , , PUNCT cana-3488 190 16 π’œ2(𝔉(11ℨ π’œ2(𝔉(11ℨ PUNCT cana-3488 190 17 ) ) PUNCT cana-3488 190 18 βˆ’ βˆ’ PROPN cana-3488 190 19 1,88,30,57,02,60,326𝔉(ℨ 1,88,30,57,02,60,326𝔉(ℨ NUM cana-3488 190 20 ) ) PUNCT cana-3488 190 21 + + CCONJ cana-3488 190 22 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ X cana-3488 190 23 ) ) PUNCT cana-3488 190 24 ≀ ≀ NOUN cana-3488 190 25 { { PUNCT cana-3488 190 26 π’œ2β€²(β„’,β„³ π’œ2β€²(β„’,β„³ NOUN cana-3488 190 27 ) ) PUNCT cana-3488 190 28 , , PUNCT cana-3488 190 29 π’œ2β€²(β„’(||ℨ|| π’œ2β€²(β„’(||ℨ|| PROPN cana-3488 190 30 π‘Ÿ),β„³ π‘Ÿ),β„³ PROPN cana-3488 190 31 ) ) PUNCT cana-3488 190 32 , , PUNCT cana-3488 190 33 π’œ3(𝔉(11ℨ π’œ3(𝔉(11ℨ X cana-3488 190 34 ) ) PUNCT cana-3488 190 35 βˆ’ βˆ’ PROPN cana-3488 190 36 1,88,30,57,02,60,326𝔉(ℨ 1,88,30,57,02,60,326𝔉(ℨ NUM cana-3488 190 37 ) ) PUNCT cana-3488 190 38 + + CCONJ cana-3488 190 39 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ X cana-3488 190 40 ) ) PUNCT cana-3488 190 41 ≀ ≀ NOUN cana-3488 190 42 { { PUNCT cana-3488 190 43 π’œ3β€²(β„’,β„³ π’œ3β€²(β„’,β„³ NOUN cana-3488 190 44 ) ) PUNCT cana-3488 190 45 , , PUNCT cana-3488 190 46 π’œ3β€²(β„’(||ℨ|| π’œ3β€²(β„’(||ℨ|| PROPN cana-3488 190 47 π‘Ÿ),β„³ π‘Ÿ),β„³ PROPN cana-3488 190 48 ) ) PUNCT cana-3488 190 49 , , PUNCT cana-3488 190 50 } } PUNCT cana-3488 190 51 ( ( PUNCT cana-3488 190 52 25 25 NUM cana-3488 190 53 ) ) PUNCT cana-3488 190 54 for for ADP cana-3488 190 55 all all PRON cana-3488 190 56 ℨ ℨ NOUN cana-3488 190 57 ∈ ∈ NOUN cana-3488 190 58 𝒳𝑀 𝒳𝑀 NOUN cana-3488 190 59 and and CCONJ cana-3488 190 60 all all DET cana-3488 190 61 β„³ β„³ PROPN cana-3488 190 62 > > X cana-3488 190 63 0 0 NUM cana-3488 190 64 , , PUNCT cana-3488 190 65 then then ADV cana-3488 190 66 there there PRON cana-3488 190 67 exists exist VERB cana-3488 190 68 a a DET cana-3488 190 69 unique unique ADJ cana-3488 190 70 tridecic tridecic NOUN cana-3488 190 71 function function NOUN cana-3488 190 72 𝒯:𝒳𝑀 𝒯:𝒳𝑀 PROPN cana-3488 190 73 βŸΆπ’΄π‘€ βŸΆπ’΄π‘€ NOUN cana-3488 190 74 such such ADJ cana-3488 190 75 that that SCONJ cana-3488 190 76 communications communication NOUN cana-3488 190 77 on on ADP cana-3488 190 78 applied apply VERB cana-3488 190 79 nonlinear nonlinear ADJ cana-3488 190 80 analysis analysis NOUN cana-3488 190 81 issn issn NOUN cana-3488 190 82 : : PUNCT cana-3488 190 83 1074 1074 NUM cana-3488 190 84 - - PUNCT cana-3488 190 85 133x 133x NUM cana-3488 190 86 vol vol NOUN cana-3488 190 87 32 32 NUM cana-3488 190 88 no no NOUN cana-3488 190 89 . . PUNCT cana-3488 191 1 7s 7 NOUN cana-3488 191 2 ( ( PUNCT cana-3488 191 3 2025 2025 NUM cana-3488 191 4 ) ) PUNCT cana-3488 191 5 816 816 NUM cana-3488 191 6 https://internationalpubls.com https://internationalpubls.com X cana-3488 191 7 π’œ1(𝔉(ℨ π’œ1(𝔉(ℨ NOUN cana-3488 191 8 ) ) PUNCT cana-3488 191 9 βˆ’ βˆ’ NUM cana-3488 191 10 𝒯(ℨ),β„³ 𝒯(ℨ),β„³ NOUN cana-3488 191 11 ) ) PUNCT cana-3488 191 12 β‰₯ β‰₯ NOUN cana-3488 191 13 { { PUNCT cana-3488 191 14 π’œ1β€²(β„’ π’œ1β€²(β„’ NUM cana-3488 191 15 , , PUNCT cana-3488 191 16 |11 |11 PROPN cana-3488 191 17 13 13 NUM cana-3488 191 18 βˆ’ βˆ’ PROPN cana-3488 191 19 1|β„³ 1|β„³ NUM cana-3488 191 20 ) ) PUNCT cana-3488 191 21 , , PUNCT cana-3488 191 22 π’œ1β€²(β„’||ℨ|| π’œ1β€²(β„’||ℨ|| VERB cana-3488 191 23 π‘Ÿ π‘Ÿ NOUN cana-3488 191 24 , , PUNCT cana-3488 191 25 |1113 |1113 PROPN cana-3488 191 26 βˆ’ βˆ’ PROPN cana-3488 191 27 11π‘Ÿ|β„³ 11π‘Ÿ|β„³ NUM cana-3488 191 28 ) ) PUNCT cana-3488 191 29 , , PUNCT cana-3488 191 30 π’œ2(𝔉(ℨ π’œ2(𝔉(ℨ X cana-3488 191 31 ) ) PUNCT cana-3488 191 32 βˆ’ βˆ’ ADP cana-3488 191 33 𝒯(ℨ),β„³ 𝒯(ℨ),β„³ NOUN cana-3488 191 34 ) ) PUNCT cana-3488 191 35 ≀ ≀ NOUN cana-3488 191 36 { { PUNCT cana-3488 191 37 π’œ2β€²(β„’ π’œ2β€²(β„’ X cana-3488 191 38 , , PUNCT cana-3488 191 39 |11 |11 PROPN cana-3488 191 40 13 13 NUM cana-3488 191 41 βˆ’ βˆ’ PROPN cana-3488 191 42 1|β„³ 1|β„³ NUM cana-3488 191 43 ) ) PUNCT cana-3488 191 44 , , PUNCT cana-3488 191 45 π’œ2β€²(β„’||ℨ|| π’œ2β€²(β„’||ℨ|| NOUN cana-3488 191 46 π‘Ÿ π‘Ÿ NOUN cana-3488 191 47 , , PUNCT cana-3488 191 48 |1113 |1113 PROPN cana-3488 191 49 βˆ’ βˆ’ PROPN cana-3488 191 50 11π‘Ÿ|β„³ 11π‘Ÿ|β„³ NUM cana-3488 191 51 ) ) PUNCT cana-3488 191 52 , , PUNCT cana-3488 191 53 π’œ3(𝔉(ℨ π’œ3(𝔉(ℨ X cana-3488 191 54 ) ) PUNCT cana-3488 191 55 βˆ’ βˆ’ NOUN cana-3488 191 56 𝒯(ℨ),β„³ 𝒯(ℨ),β„³ NOUN cana-3488 191 57 ) ) PUNCT cana-3488 191 58 ≀ ≀ NOUN cana-3488 191 59 { { PUNCT cana-3488 191 60 π’œ3β€²(β„’ π’œ3β€²(β„’ NUM cana-3488 191 61 , , PUNCT cana-3488 191 62 |11 |11 PROPN cana-3488 191 63 13 13 NUM cana-3488 191 64 βˆ’ βˆ’ PROPN cana-3488 191 65 1|β„³ 1|β„³ NUM cana-3488 191 66 ) ) PUNCT cana-3488 191 67 , , PUNCT cana-3488 191 68 π’œ3β€²(β„’||ℨ|| π’œ3β€²(β„’||ℨ|| ADJ cana-3488 191 69 π‘Ÿ π‘Ÿ NOUN cana-3488 191 70 , , PUNCT cana-3488 191 71 |1113 |1113 PROPN cana-3488 191 72 βˆ’ βˆ’ PROPN cana-3488 191 73 11π‘Ÿ|β„³ 11π‘Ÿ|β„³ NUM cana-3488 191 74 ) ) PUNCT cana-3488 191 75 , , PUNCT cana-3488 191 76 } } PUNCT cana-3488 191 77 ( ( PUNCT cana-3488 191 78 26 26 NUM cana-3488 191 79 ) ) PUNCT cana-3488 191 80 for for ADP cana-3488 191 81 all all PRON cana-3488 191 82 ℨ ℨ NOUN cana-3488 191 83 ∈ ∈ NOUN cana-3488 191 84 𝒳𝑀 𝒳𝑀 NOUN cana-3488 191 85 and and CCONJ cana-3488 191 86 all all DET cana-3488 191 87 β„³ β„³ PROPN cana-3488 191 88 > > X cana-3488 191 89 0 0 PUNCT cana-3488 191 90 . . PUNCT cana-3488 192 1 theorem theorem VERB cana-3488 192 2 3.3 3.3 NUM cana-3488 192 3 assume assume VERB cana-3488 192 4 that that SCONJ cana-3488 192 5 𝒳𝑀 𝒳𝑀 PROPN cana-3488 192 6 is be AUX cana-3488 192 7 a a DET cana-3488 192 8 ls ls PROPN cana-3488 192 9 , , PUNCT cana-3488 192 10 ( ( PUNCT cana-3488 192 11 π’΅π‘š π’΅π‘š ADJ cana-3488 192 12 , , PUNCT cana-3488 192 13 π’œ1β€²,π’œ2β€² π’œ1β€²,π’œ2β€² ADJ cana-3488 192 14 , , PUNCT cana-3488 192 15 π’œ3β€² π’œ3β€² PRON cana-3488 192 16 ) ) PUNCT cana-3488 192 17 is be AUX cana-3488 192 18 a a DET cana-3488 192 19 nns nns NOUN cana-3488 192 20 and and CCONJ cana-3488 192 21 ( ( PUNCT cana-3488 192 22 𝒴𝑀 𝒴𝑀 PROPN cana-3488 192 23 , , PUNCT cana-3488 192 24 π’œ1,π’œ2 π’œ1,π’œ2 NUM cana-3488 192 25 , , PUNCT cana-3488 192 26 π’œ3 π’œ3 PROPN cana-3488 192 27 ) ) PUNCT cana-3488 192 28 an an DET cana-3488 192 29 nbs nbs PROPN cana-3488 192 30 . . PUNCT cana-3488 193 1 let let VERB cana-3488 193 2 Γ°:𝒳𝑀 Γ°:𝒳𝑀 NOUN cana-3488 193 3 ⟢ ⟢ NOUN cana-3488 193 4 π’΅π‘š π’΅π‘š NOUN cana-3488 193 5 be be AUX cana-3488 193 6 a a DET cana-3488 193 7 function function NOUN cana-3488 193 8 such such ADJ cana-3488 193 9 that that PRON cana-3488 193 10 for for ADP cana-3488 193 11 some some DET cana-3488 193 12 0 0 NUM cana-3488 193 13 < < X cana-3488 193 14 ( ( PUNCT cana-3488 193 15 𝔙 𝔙 PROPN cana-3488 193 16 1112 1112 NUM cana-3488 193 17 ) ) PUNCT cana-3488 194 1 𝐹 𝐹 PROPN cana-3488 194 2 < < X cana-3488 194 3 1 1 NUM cana-3488 194 4 with with ADP cana-3488 194 5 𝐹 𝐹 PROPN cana-3488 194 6 ∈ ∈ PROPN cana-3488 194 7 { { PUNCT cana-3488 194 8 1 1 NUM cana-3488 194 9 , , PUNCT cana-3488 194 10 βˆ’1 βˆ’1 NOUN cana-3488 194 11 } } PUNCT cana-3488 194 12 . . PUNCT cana-3488 195 1 π’œ1β€²(Γ°(11 π’œ1β€²(Γ°(11 PROPN cana-3488 195 2 𝑛𝐹ℨ),β„³ 𝑛𝐹ℨ),β„³ NOUN cana-3488 195 3 ) ) PUNCT cana-3488 195 4 β‰₯ β‰₯ NOUN cana-3488 195 5 π’œ1β€²(𝔙 π’œ1β€²(𝔙 NOUN cana-3488 195 6 𝑛𝐹ð(ℨ),β„³ 𝑛𝐹ð(ℨ),β„³ PROPN cana-3488 195 7 ) ) PUNCT cana-3488 195 8 π’œ2β€²(Γ°(11 π’œ2β€²(Γ°(11 NOUN cana-3488 195 9 𝑛𝐹ℨ),β„³ 𝑛𝐹ℨ),β„³ NOUN cana-3488 195 10 ) ) PUNCT cana-3488 195 11 ≀ ≀ NUM cana-3488 195 12 π’œ2β€²(𝔙 π’œ2β€²(𝔙 SYM cana-3488 195 13 𝑛𝐹ð(ℨ),β„³ 𝑛𝐹ð(ℨ),β„³ PROPN cana-3488 195 14 ) ) PUNCT cana-3488 195 15 π’œ3β€²(Γ°(11 π’œ3β€²(Γ°(11 NOUN cana-3488 195 16 𝑛𝐹ℨ),β„³ 𝑛𝐹ℨ),β„³ NOUN cana-3488 195 17 ) ) PUNCT cana-3488 195 18 ≀ ≀ NUM cana-3488 195 19 π’œ3β€²(𝔙 π’œ3β€²(𝔙 NOUN cana-3488 195 20 𝑛𝐹ð(ℨ),β„³ 𝑛𝐹ð(ℨ),β„³ NOUN cana-3488 195 21 ) ) PUNCT cana-3488 195 22 } } PUNCT cana-3488 195 23 ( ( PUNCT cana-3488 195 24 27 27 NUM cana-3488 195 25 ) ) PUNCT cana-3488 195 26 for for ADP cana-3488 195 27 all all DET cana-3488 195 28 ℨ ℨ NOUN cana-3488 195 29 ∈ ∈ NOUN cana-3488 195 30 𝒳𝑀 𝒳𝑀 NOUN cana-3488 195 31 and and CCONJ cana-3488 195 32 all all DET cana-3488 195 33 β„³ β„³ PROPN cana-3488 195 34 > > X cana-3488 195 35 0 0 PUNCT cana-3488 196 1 and and CCONJ cana-3488 196 2 lim lim PROPN cana-3488 196 3 π‘›β†’βˆž π‘›β†’βˆž NUM cana-3488 196 4 π’œ1β€²(Γ°(11 π’œ1β€²(Γ°(11 PROPN cana-3488 196 5 𝐹𝑛ℨ 𝐹𝑛ℨ PROPN cana-3488 196 6 ) ) PUNCT cana-3488 196 7 , , PUNCT cana-3488 196 8 1112𝐹𝑛ℳ 1112𝐹𝑛ℳ NUM cana-3488 196 9 ) ) PUNCT cana-3488 196 10 = = SYM cana-3488 197 1 1 1 NUM cana-3488 197 2 lim lim NOUN cana-3488 197 3 π‘›β†’βˆž π‘›β†’βˆž NUM cana-3488 197 4 π’œ2β€²(Γ°(11 π’œ2β€²(Γ°(11 PROPN cana-3488 197 5 𝐹𝑛ℨ 𝐹𝑛ℨ PROPN cana-3488 197 6 ) ) PUNCT cana-3488 197 7 , , PUNCT cana-3488 197 8 1112𝐹𝑛ℳ 1112𝐹𝑛ℳ NUM cana-3488 197 9 ) ) PUNCT cana-3488 197 10 = = SYM cana-3488 197 11 0 0 NUM cana-3488 197 12 lim lim NOUN cana-3488 197 13 π‘›β†’βˆž π‘›β†’βˆž NUM cana-3488 197 14 π’œ3β€²(Γ°(11 π’œ3β€²(Γ°(11 PROPN cana-3488 197 15 𝐹𝑛ℨ 𝐹𝑛ℨ PROPN cana-3488 197 16 ) ) PUNCT cana-3488 197 17 , , PUNCT cana-3488 197 18 1112𝐹𝑛ℳ 1112𝐹𝑛ℳ NUM cana-3488 197 19 ) ) PUNCT cana-3488 197 20 = = SYM cana-3488 197 21 0 0 X cana-3488 197 22 } } PUNCT cana-3488 197 23 ( ( PUNCT cana-3488 197 24 28 28 NUM cana-3488 197 25 ) ) PUNCT cana-3488 197 26 for for ADP cana-3488 197 27 all all DET cana-3488 197 28 ℨ ℨ NOUN cana-3488 197 29 ∈ ∈ NOUN cana-3488 197 30 𝒳𝑀 𝒳𝑀 NOUN cana-3488 197 31 and and CCONJ cana-3488 197 32 all all DET cana-3488 197 33 β„³ β„³ PROPN cana-3488 197 34 > > X cana-3488 197 35 0 0 PUNCT cana-3488 197 36 . . PUNCT cana-3488 198 1 let let VERB cana-3488 198 2 an an DET cana-3488 198 3 even even ADV cana-3488 198 4 function function NOUN cana-3488 198 5 𝔉:𝒳𝑀 𝔉:𝒳𝑀 NOUN cana-3488 198 6 βŸΆπ’΄π‘€ βŸΆπ’΄π‘€ NOUN cana-3488 198 7 satisfying satisfy VERB cana-3488 198 8 π’œ1(𝔉(11ℨ π’œ1(𝔉(11ℨ PUNCT cana-3488 198 9 ) ) PUNCT cana-3488 199 1 βˆ’ βˆ’ ADP cana-3488 199 2 1,88,30,57,02,60,326𝔉(ℨ 1,88,30,57,02,60,326𝔉(ℨ NUM cana-3488 199 3 ) ) PUNCT cana-3488 199 4 + + CCONJ cana-3488 199 5 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ X cana-3488 199 6 ) ) PUNCT cana-3488 199 7 β‰₯ β‰₯ NOUN cana-3488 199 8 π’œ1β€²(Γ°(ℨ),β„³ π’œ1β€²(Γ°(ℨ),β„³ NOUN cana-3488 199 9 ) ) PUNCT cana-3488 199 10 π’œ2(𝔉(11ℨ π’œ2(𝔉(11ℨ PUNCT cana-3488 199 11 ) ) PUNCT cana-3488 199 12 βˆ’ βˆ’ PROPN cana-3488 199 13 1,88,30,57,02,60,326𝔉(ℨ 1,88,30,57,02,60,326𝔉(ℨ NUM cana-3488 199 14 ) ) PUNCT cana-3488 199 15 + + CCONJ cana-3488 199 16 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ X cana-3488 199 17 ) ) PUNCT cana-3488 199 18 ≀ ≀ NOUN cana-3488 199 19 π’œ2β€²(Γ°(ℨ),β„³ π’œ2β€²(Γ°(ℨ),β„³ NOUN cana-3488 199 20 ) ) PUNCT cana-3488 199 21 π’œ3(𝔉(11ℨ π’œ3(𝔉(11ℨ X cana-3488 199 22 ) ) PUNCT cana-3488 199 23 βˆ’ βˆ’ PROPN cana-3488 199 24 1,88,30,57,02,60,326𝔉(ℨ 1,88,30,57,02,60,326𝔉(ℨ NUM cana-3488 199 25 ) ) PUNCT cana-3488 199 26 + + CCONJ cana-3488 199 27 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ X cana-3488 199 28 ) ) PUNCT cana-3488 199 29 ≀ ≀ NOUN cana-3488 200 1 π’œ3β€²(Γ°(ℨ),β„³ π’œ3β€²(Γ°(ℨ),β„³ NOUN cana-3488 200 2 ) ) PUNCT cana-3488 200 3 } } PUNCT cana-3488 200 4 ( ( PUNCT cana-3488 200 5 29 29 NUM cana-3488 200 6 ) ) PUNCT cana-3488 200 7 for for ADP cana-3488 200 8 all all DET cana-3488 200 9 ℨ ℨ NOUN cana-3488 200 10 ∈ ∈ NOUN cana-3488 200 11 𝒳𝑀 𝒳𝑀 NOUN cana-3488 200 12 and and CCONJ cana-3488 200 13 all all DET cana-3488 200 14 β„³ β„³ PROPN cana-3488 200 15 > > X cana-3488 200 16 0 0 NUM cana-3488 200 17 . . PUNCT cana-3488 201 1 then then ADV cana-3488 201 2 there there PRON cana-3488 201 3 exists exist VERB cana-3488 201 4 a a DET cana-3488 201 5 unique unique ADJ cana-3488 201 6 duodecic duodecic NOUN cana-3488 201 7 mapping mapping NOUN cana-3488 201 8 π’Ÿ:𝒳𝑀 π’Ÿ:𝒳𝑀 NOUN cana-3488 201 9 βŸΆπ’΄π‘€ βŸΆπ’΄π‘€ NOUN cana-3488 201 10 satisfying satisfying NOUN cana-3488 201 11 ( ( PUNCT cana-3488 201 12 1 1 NUM cana-3488 201 13 ) ) PUNCT cana-3488 201 14 and and CCONJ cana-3488 201 15 communications communication NOUN cana-3488 201 16 on on ADP cana-3488 201 17 applied apply VERB cana-3488 201 18 nonlinear nonlinear ADJ cana-3488 201 19 analysis analysis NOUN cana-3488 201 20 issn issn NOUN cana-3488 201 21 : : PUNCT cana-3488 201 22 1074 1074 NUM cana-3488 201 23 - - PUNCT cana-3488 201 24 133x 133x NUM cana-3488 201 25 vol vol NOUN cana-3488 201 26 32 32 NUM cana-3488 201 27 no no NOUN cana-3488 201 28 . . PUNCT cana-3488 202 1 7s 7 NOUN cana-3488 202 2 ( ( PUNCT cana-3488 202 3 2025 2025 NUM cana-3488 202 4 ) ) PUNCT cana-3488 202 5 817 817 NUM cana-3488 202 6 https://internationalpubls.com https://internationalpubls.com X cana-3488 202 7 π’œ1(𝔉(ℨ π’œ1(𝔉(ℨ NOUN cana-3488 202 8 ) ) PUNCT cana-3488 202 9 βˆ’ βˆ’ ADP cana-3488 202 10 π’Ÿ(ℨ),β„³ π’Ÿ(ℨ),β„³ SYM cana-3488 202 11 ) ) PUNCT cana-3488 202 12 β‰₯ β‰₯ NOUN cana-3488 202 13 π’œ1β€²(Γ°(ℨ π’œ1β€²(Γ°(ℨ NOUN cana-3488 202 14 ) ) PUNCT cana-3488 202 15 , , PUNCT cana-3488 202 16 |11 |11 PROPN cana-3488 202 17 12 12 NUM cana-3488 202 18 βˆ’ βˆ’ NOUN cana-3488 202 19 𝔙|β„³ 𝔙|β„³ NOUN cana-3488 202 20 ) ) PUNCT cana-3488 202 21 π’œ2(𝔉(ℨ π’œ2(𝔉(ℨ NOUN cana-3488 202 22 ) ) PUNCT cana-3488 202 23 βˆ’ βˆ’ ADP cana-3488 202 24 𝒯(ℨ),β„³ 𝒯(ℨ),β„³ NOUN cana-3488 202 25 ) ) PUNCT cana-3488 202 26 ≀ ≀ NUM cana-3488 202 27 π’œ2β€²(Γ°(ℨ π’œ2β€²(Γ°(ℨ NOUN cana-3488 202 28 ) ) PUNCT cana-3488 202 29 , , PUNCT cana-3488 202 30 |11 |11 PROPN cana-3488 202 31 12 12 NUM cana-3488 202 32 βˆ’ βˆ’ NOUN cana-3488 202 33 𝔙|β„³ 𝔙|β„³ NOUN cana-3488 202 34 ) ) PUNCT cana-3488 202 35 π’œ3(𝔉(ℨ π’œ3(𝔉(ℨ PROPN cana-3488 202 36 ) ) PUNCT cana-3488 202 37 βˆ’ βˆ’ NOUN cana-3488 202 38 π’Ÿ(ℨ),β„³ π’Ÿ(ℨ),β„³ NOUN cana-3488 202 39 ) ) PUNCT cana-3488 202 40 ≀ ≀ NOUN cana-3488 202 41 π’œ3β€²(Γ°(ℨ π’œ3β€²(Γ°(ℨ NOUN cana-3488 202 42 ) ) PUNCT cana-3488 202 43 , , PUNCT cana-3488 202 44 |11 |11 PROPN cana-3488 202 45 12 12 NUM cana-3488 202 46 βˆ’π”™|β„³ βˆ’π”™|β„³ NOUN cana-3488 202 47 ) ) PUNCT cana-3488 202 48 } } PUNCT cana-3488 202 49 ( ( PUNCT cana-3488 202 50 30 30 NUM cana-3488 202 51 ) ) PUNCT cana-3488 202 52 for for ADP cana-3488 202 53 all all DET cana-3488 202 54 ℨ ℨ NOUN cana-3488 202 55 ∈ ∈ NOUN cana-3488 202 56 𝒳𝑀 𝒳𝑀 NOUN cana-3488 202 57 and and CCONJ cana-3488 202 58 all all DET cana-3488 202 59 β„³ β„³ PROPN cana-3488 202 60 > > X cana-3488 202 61 0 0 PROPN cana-3488 202 62 . . PUNCT cana-3488 203 1 proof proof NOUN cana-3488 203 2 . . PUNCT cana-3488 204 1 for for ADP cana-3488 204 2 the the DET cana-3488 204 3 first first ADJ cana-3488 204 4 case case NOUN cana-3488 204 5 𝐹 𝐹 PROPN cana-3488 204 6 = = NOUN cana-3488 204 7 1 1 X cana-3488 204 8 . . PUNCT cana-3488 204 9 by by ADP cana-3488 204 10 applying apply VERB cana-3488 204 11 the the DET cana-3488 204 12 evenness evenness ADJ cana-3488 204 13 condition condition NOUN cana-3488 204 14 of of ADP cana-3488 204 15 𝔉 𝔉 PROPN cana-3488 204 16 in in ADP cana-3488 204 17 in in ADP cana-3488 204 18 ( ( PUNCT cana-3488 204 19 29 29 NUM cana-3488 204 20 ) ) PUNCT cana-3488 204 21 , , PUNCT cana-3488 204 22 we we PRON cana-3488 204 23 arrive arrive VERB cana-3488 204 24 π’œ1(𝔉(11ℨ π’œ1(𝔉(11ℨ PUNCT cana-3488 204 25 ) ) PUNCT cana-3488 205 1 βˆ’ βˆ’ ADP cana-3488 205 2 1,56,92,14,18,83,605𝔉(ℨ),β„³ 1,56,92,14,18,83,605𝔉(ℨ),β„³ NUM cana-3488 205 3 ) ) PUNCT cana-3488 205 4 β‰₯ β‰₯ NOUN cana-3488 205 5 π’œ1β€²(Γ°(ℨ),β„³ π’œ1β€²(Γ°(ℨ),β„³ NOUN cana-3488 205 6 ) ) PUNCT cana-3488 205 7 π’œ2(𝔉(11ℨ π’œ2(𝔉(11ℨ PUNCT cana-3488 205 8 ) ) PUNCT cana-3488 206 1 βˆ’ βˆ’ ADP cana-3488 206 2 1,56,92,14,18,83,605𝔉(ℨ),β„³ 1,56,92,14,18,83,605𝔉(ℨ),β„³ NUM cana-3488 206 3 ) ) PUNCT cana-3488 206 4 ≀ ≀ NOUN cana-3488 207 1 π’œ2β€²(Γ°(ℨ),β„³ π’œ2β€²(Γ°(ℨ),β„³ NOUN cana-3488 207 2 ) ) PUNCT cana-3488 207 3 π’œ3(𝔉(11ℨ π’œ3(𝔉(11ℨ X cana-3488 207 4 ) ) PUNCT cana-3488 207 5 βˆ’ βˆ’ ADP cana-3488 207 6 1,56,92,14,18,83,605𝔉(ℨ),β„³ 1,56,92,14,18,83,605𝔉(ℨ),β„³ NUM cana-3488 207 7 ) ) PUNCT cana-3488 207 8 ≀ ≀ NOUN cana-3488 208 1 π’œ3β€²(Γ°(ℨ),β„³ π’œ3β€²(Γ°(ℨ),β„³ NOUN cana-3488 208 2 ) ) PUNCT cana-3488 208 3 } } PUNCT cana-3488 208 4 ( ( PUNCT cana-3488 208 5 31 31 NUM cana-3488 208 6 ) ) PUNCT cana-3488 208 7 for for ADP cana-3488 208 8 all all DET cana-3488 208 9 ℨ ℨ NOUN cana-3488 208 10 ∈ ∈ NOUN cana-3488 208 11 𝒳𝑀 𝒳𝑀 NOUN cana-3488 208 12 and and CCONJ cana-3488 208 13 all all DET cana-3488 208 14 β„³ β„³ PROPN cana-3488 208 15 > > X cana-3488 208 16 0 0 NUM cana-3488 208 17 . . PUNCT cana-3488 209 1 from from ADP cana-3488 209 2 ( ( PUNCT cana-3488 209 3 31 31 NUM cana-3488 209 4 ) ) PUNCT cana-3488 209 5 we we PRON cana-3488 209 6 have have AUX cana-3488 209 7 π’œ1(𝔉(11ℨ π’œ1(𝔉(11ℨ PUNCT cana-3488 209 8 ) ) PUNCT cana-3488 209 9 βˆ’ βˆ’ PROPN cana-3488 209 10 11 11 NUM cana-3488 209 11 12𝔉(ℨ),β„³ 12𝔉(ℨ),β„³ NOUN cana-3488 209 12 ) ) PUNCT cana-3488 209 13 β‰₯ β‰₯ NOUN cana-3488 209 14 π’œ1β€²(Γ°(ℨ),β„³ π’œ1β€²(Γ°(ℨ),β„³ NOUN cana-3488 209 15 ) ) PUNCT cana-3488 209 16 π’œ2(𝔉(11ℨ π’œ2(𝔉(11ℨ PUNCT cana-3488 209 17 ) ) PUNCT cana-3488 209 18 βˆ’ βˆ’ PROPN cana-3488 209 19 11 11 NUM cana-3488 209 20 12𝔉(ℨ),β„³ 12𝔉(ℨ),β„³ NOUN cana-3488 209 21 ) ) PUNCT cana-3488 209 22 ≀ ≀ NOUN cana-3488 209 23 π’œ2β€²(Γ°(ℨ),β„³ π’œ2β€²(Γ°(ℨ),β„³ NOUN cana-3488 209 24 ) ) PUNCT cana-3488 209 25 π’œ3(𝔉(11ℨ π’œ3(𝔉(11ℨ X cana-3488 209 26 ) ) PUNCT cana-3488 209 27 βˆ’ βˆ’ PROPN cana-3488 209 28 11 11 NUM cana-3488 209 29 12𝔉(ℨ),β„³ 12𝔉(ℨ),β„³ NOUN cana-3488 209 30 ) ) PUNCT cana-3488 209 31 ≀ ≀ NUM cana-3488 210 1 π’œ3β€²(Γ°(ℨ),β„³ π’œ3β€²(Γ°(ℨ),β„³ NOUN cana-3488 210 2 ) ) PUNCT cana-3488 210 3 } } PUNCT cana-3488 210 4 ( ( PUNCT cana-3488 210 5 32 32 NUM cana-3488 210 6 ) ) PUNCT cana-3488 210 7 for for ADP cana-3488 210 8 all all DET cana-3488 210 9 ℨ ℨ NOUN cana-3488 210 10 ∈ ∈ NOUN cana-3488 210 11 𝒳𝑀 𝒳𝑀 NOUN cana-3488 210 12 and and CCONJ cana-3488 210 13 all all DET cana-3488 210 14 β„³ β„³ PROPN cana-3488 210 15 > > X cana-3488 210 16 0 0 X cana-3488 210 17 . . PUNCT cana-3488 211 1 corollary corollary NOUN cana-3488 211 2 3.4 3.4 NUM cana-3488 211 3 let let VERB cana-3488 211 4 β„’ β„’ NOUN cana-3488 211 5 , , PUNCT cana-3488 211 6 π‘Ÿ π‘Ÿ NOUN cana-3488 211 7 are be AUX cana-3488 211 8 constants constant NOUN cana-3488 211 9 , , PUNCT cana-3488 211 10 with with ADP cana-3488 211 11 β„’ β„’ PROPN cana-3488 211 12 > > SYM cana-3488 211 13 0 0 NUM cana-3488 211 14 and and CCONJ cana-3488 211 15 π‘Ÿ π‘Ÿ NOUN cana-3488 211 16 β‰  β‰  PROPN cana-3488 211 17 12 12 NUM cana-3488 211 18 and and CCONJ cana-3488 211 19 let let VERB cana-3488 211 20 an an DET cana-3488 211 21 even even ADV cana-3488 211 22 mapping mapping NOUN cana-3488 211 23 𝔉:𝒳𝑀 𝔉:𝒳𝑀 NOUN cana-3488 211 24 βŸΆπ’΄π‘€ βŸΆπ’΄π‘€ NOUN cana-3488 211 25 satisfies satisfie NOUN cana-3488 211 26 π’œ1(𝔉(11ℨ π’œ1(𝔉(11ℨ PUNCT cana-3488 211 27 ) ) PUNCT cana-3488 212 1 βˆ’ βˆ’ ADP cana-3488 212 2 1,88,30,57,02,60,326𝔉(ℨ 1,88,30,57,02,60,326𝔉(ℨ NUM cana-3488 212 3 ) ) PUNCT cana-3488 212 4 + + CCONJ cana-3488 212 5 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ X cana-3488 212 6 ) ) PUNCT cana-3488 212 7 β‰₯ β‰₯ NOUN cana-3488 212 8 { { PUNCT cana-3488 212 9 π’œ1β€²(β„’,β„³ π’œ1β€²(β„’,β„³ PROPN cana-3488 212 10 ) ) PUNCT cana-3488 212 11 , , PUNCT cana-3488 212 12 π’œ1β€²(β„’(||ℨ|| π’œ1β€²(β„’(||ℨ|| PROPN cana-3488 212 13 π‘Ÿ),β„³ π‘Ÿ),β„³ NUM cana-3488 212 14 ) ) PUNCT cana-3488 212 15 , , PUNCT cana-3488 212 16 π’œ2(𝔉(11ℨ π’œ2(𝔉(11ℨ PUNCT cana-3488 212 17 ) ) PUNCT cana-3488 212 18 βˆ’ βˆ’ PROPN cana-3488 212 19 1,88,30,57,02,60,326𝔉(ℨ 1,88,30,57,02,60,326𝔉(ℨ NUM cana-3488 212 20 ) ) PUNCT cana-3488 212 21 + + CCONJ cana-3488 212 22 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ X cana-3488 212 23 ) ) PUNCT cana-3488 212 24 ≀ ≀ NOUN cana-3488 212 25 { { PUNCT cana-3488 212 26 π’œ2β€²(β„’,β„³ π’œ2β€²(β„’,β„³ NOUN cana-3488 212 27 ) ) PUNCT cana-3488 212 28 , , PUNCT cana-3488 212 29 π’œ2β€²(β„’(||ℨ|| π’œ2β€²(β„’(||ℨ|| PROPN cana-3488 212 30 π‘Ÿ),β„³ π‘Ÿ),β„³ PROPN cana-3488 212 31 ) ) PUNCT cana-3488 212 32 , , PUNCT cana-3488 212 33 π’œ3(𝔉(11ℨ π’œ3(𝔉(11ℨ X cana-3488 212 34 ) ) PUNCT cana-3488 212 35 βˆ’ βˆ’ PROPN cana-3488 212 36 1,88,30,57,02,60,326𝔉(ℨ 1,88,30,57,02,60,326𝔉(ℨ NUM cana-3488 212 37 ) ) PUNCT cana-3488 212 38 + + CCONJ cana-3488 212 39 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ X cana-3488 212 40 ) ) PUNCT cana-3488 212 41 ≀ ≀ NOUN cana-3488 212 42 { { PUNCT cana-3488 212 43 π’œ3β€²(β„’,β„³ π’œ3β€²(β„’,β„³ NOUN cana-3488 212 44 ) ) PUNCT cana-3488 212 45 , , PUNCT cana-3488 212 46 π’œ3β€²(β„’(||ℨ|| π’œ3β€²(β„’(||ℨ|| PROPN cana-3488 212 47 π‘Ÿ),β„³ π‘Ÿ),β„³ PROPN cana-3488 212 48 ) ) PUNCT cana-3488 212 49 , , PUNCT cana-3488 212 50 } } PUNCT cana-3488 212 51 ( ( PUNCT cana-3488 212 52 33 33 NUM cana-3488 212 53 ) ) PUNCT cana-3488 212 54 for for ADP cana-3488 212 55 all all DET cana-3488 212 56 ℨ ℨ NOUN cana-3488 212 57 ∈ ∈ NOUN cana-3488 212 58 𝒳𝑀 𝒳𝑀 NOUN cana-3488 212 59 and and CCONJ cana-3488 212 60 all all DET cana-3488 212 61 β„³ β„³ PROPN cana-3488 212 62 > > X cana-3488 212 63 0 0 NUM cana-3488 212 64 , , PUNCT cana-3488 212 65 then then ADV cana-3488 212 66 there there PRON cana-3488 212 67 exists exist VERB cana-3488 212 68 a a DET cana-3488 212 69 unique unique ADJ cana-3488 212 70 duodecic duodecic NOUN cana-3488 212 71 function function VERB cana-3488 212 72 π’Ÿ:𝒳𝑀 π’Ÿ:𝒳𝑀 NOUN cana-3488 212 73 βŸΆπ’΄π‘€ βŸΆπ’΄π‘€ INTJ cana-3488 213 1 such such ADJ cana-3488 213 2 that that SCONJ cana-3488 213 3 communications communication NOUN cana-3488 213 4 on on ADP cana-3488 213 5 applied apply VERB cana-3488 213 6 nonlinear nonlinear ADJ cana-3488 213 7 analysis analysis NOUN cana-3488 213 8 issn issn NOUN cana-3488 213 9 : : PUNCT cana-3488 213 10 1074 1074 NUM cana-3488 213 11 - - PUNCT cana-3488 213 12 133x 133x NUM cana-3488 213 13 vol vol NOUN cana-3488 213 14 32 32 NUM cana-3488 213 15 no no NOUN cana-3488 213 16 . . PUNCT cana-3488 214 1 7s 7 NOUN cana-3488 214 2 ( ( PUNCT cana-3488 214 3 2025 2025 NUM cana-3488 214 4 ) ) PUNCT cana-3488 214 5 818 818 NUM cana-3488 214 6 https://internationalpubls.com https://internationalpubls.com X cana-3488 214 7 π’œ1(𝔉(ℨ π’œ1(𝔉(ℨ NOUN cana-3488 214 8 ) ) PUNCT cana-3488 214 9 βˆ’ βˆ’ ADP cana-3488 214 10 π’Ÿ(ℨ),β„³ π’Ÿ(ℨ),β„³ SYM cana-3488 214 11 ) ) PUNCT cana-3488 214 12 β‰₯ β‰₯ NOUN cana-3488 214 13 { { PUNCT cana-3488 214 14 π’œ1β€²(β„’ π’œ1β€²(β„’ NUM cana-3488 214 15 , , PUNCT cana-3488 214 16 |11 |11 PROPN cana-3488 214 17 12 12 NUM cana-3488 214 18 βˆ’ βˆ’ PROPN cana-3488 214 19 1|β„³ 1|β„³ NUM cana-3488 214 20 ) ) PUNCT cana-3488 214 21 , , PUNCT cana-3488 214 22 π’œ1β€²(β„’||ℨ|| π’œ1β€²(β„’||ℨ|| VERB cana-3488 214 23 π‘Ÿ π‘Ÿ NOUN cana-3488 214 24 , , PUNCT cana-3488 214 25 |1112 |1112 NOUN cana-3488 214 26 βˆ’ βˆ’ PROPN cana-3488 214 27 11π‘Ÿ|β„³ 11π‘Ÿ|β„³ NUM cana-3488 214 28 ) ) PUNCT cana-3488 214 29 , , PUNCT cana-3488 214 30 π’œ2(𝔉(ℨ π’œ2(𝔉(ℨ PROPN cana-3488 214 31 ) ) PUNCT cana-3488 214 32 βˆ’ βˆ’ PROPN cana-3488 214 33 π’Ÿ(ℨ),β„³ π’Ÿ(ℨ),β„³ SYM cana-3488 214 34 ) ) PUNCT cana-3488 214 35 ≀ ≀ NOUN cana-3488 214 36 { { PUNCT cana-3488 214 37 π’œ2β€²(β„’ π’œ2β€²(β„’ X cana-3488 214 38 , , PUNCT cana-3488 214 39 |11 |11 PROPN cana-3488 214 40 12 12 NUM cana-3488 214 41 βˆ’ βˆ’ PROPN cana-3488 214 42 1|β„³ 1|β„³ NUM cana-3488 214 43 ) ) PUNCT cana-3488 214 44 , , PUNCT cana-3488 214 45 π’œ2β€²(β„’||ℨ|| π’œ2β€²(β„’||ℨ|| PROPN cana-3488 214 46 π‘Ÿ π‘Ÿ X cana-3488 214 47 , , PUNCT cana-3488 214 48 |1112 |1112 NOUN cana-3488 214 49 βˆ’ βˆ’ PROPN cana-3488 214 50 11π‘Ÿ|β„³ 11π‘Ÿ|β„³ NUM cana-3488 214 51 ) ) PUNCT cana-3488 214 52 , , PUNCT cana-3488 214 53 π’œ3(𝔉(ℨ π’œ3(𝔉(ℨ X cana-3488 214 54 ) ) PUNCT cana-3488 214 55 βˆ’ βˆ’ NOUN cana-3488 214 56 π’Ÿ(ℨ),β„³ π’Ÿ(ℨ),β„³ SYM cana-3488 214 57 ) ) PUNCT cana-3488 214 58 ≀ ≀ NOUN cana-3488 214 59 { { PUNCT cana-3488 214 60 π’œ3β€²(β„’ π’œ3β€²(β„’ NUM cana-3488 214 61 , , PUNCT cana-3488 214 62 |11 |11 PROPN cana-3488 214 63 12 12 NUM cana-3488 214 64 βˆ’ βˆ’ PROPN cana-3488 214 65 1|β„³ 1|β„³ NUM cana-3488 214 66 ) ) PUNCT cana-3488 214 67 , , PUNCT cana-3488 214 68 π’œ3β€²(β„’||ℨ|| π’œ3β€²(β„’||ℨ|| NOUN cana-3488 214 69 π‘Ÿ π‘Ÿ NOUN cana-3488 214 70 , , PUNCT cana-3488 214 71 |1112 |1112 NOUN cana-3488 214 72 βˆ’ βˆ’ PROPN cana-3488 214 73 11π‘Ÿ|β„³ 11π‘Ÿ|β„³ NUM cana-3488 214 74 ) ) PUNCT cana-3488 214 75 , , PUNCT cana-3488 214 76 } } PUNCT cana-3488 214 77 ( ( PUNCT cana-3488 214 78 34 34 NUM cana-3488 214 79 ) ) PUNCT cana-3488 214 80 for for ADP cana-3488 214 81 all all DET cana-3488 214 82 ℨ ℨ NOUN cana-3488 214 83 ∈ ∈ NOUN cana-3488 214 84 𝒳𝑀 𝒳𝑀 NOUN cana-3488 214 85 and and CCONJ cana-3488 214 86 all all DET cana-3488 214 87 β„³ β„³ PROPN cana-3488 214 88 > > X cana-3488 214 89 0 0 PUNCT cana-3488 214 90 . . PUNCT cana-3488 215 1 theorem theorem ADJ cana-3488 215 2 3.5 3.5 NUM cana-3488 215 3 assume assume VERB cana-3488 215 4 that that SCONJ cana-3488 215 5 𝒳𝑀 𝒳𝑀 PROPN cana-3488 215 6 is be AUX cana-3488 215 7 a a DET cana-3488 215 8 ls ls PROPN cana-3488 215 9 , , PUNCT cana-3488 215 10 ( ( PUNCT cana-3488 215 11 π’΅π‘š π’΅π‘š ADJ cana-3488 215 12 , , PUNCT cana-3488 215 13 π’œ1β€²,π’œ2β€² π’œ1β€²,π’œ2β€² ADJ cana-3488 215 14 , , PUNCT cana-3488 215 15 π’œ3β€² π’œ3β€² PRON cana-3488 215 16 ) ) PUNCT cana-3488 215 17 is be AUX cana-3488 215 18 a a DET cana-3488 215 19 nns nns NOUN cana-3488 215 20 and and CCONJ cana-3488 215 21 ( ( PUNCT cana-3488 215 22 𝒴𝑀 𝒴𝑀 PROPN cana-3488 215 23 , , PUNCT cana-3488 215 24 π’œ1,π’œ2 π’œ1,π’œ2 NUM cana-3488 215 25 , , PUNCT cana-3488 215 26 π’œ3 π’œ3 PROPN cana-3488 215 27 ) ) PUNCT cana-3488 215 28 an an DET cana-3488 215 29 nbs nbs PROPN cana-3488 215 30 . . PUNCT cana-3488 216 1 let let VERB cana-3488 216 2 Γ°:𝒳𝑀 Γ°:𝒳𝑀 NOUN cana-3488 216 3 ⟢ ⟢ NOUN cana-3488 216 4 π’΅π‘š π’΅π‘š NOUN cana-3488 216 5 be be AUX cana-3488 216 6 a a DET cana-3488 216 7 function function NOUN cana-3488 216 8 such such ADJ cana-3488 216 9 that that PRON cana-3488 216 10 for for ADP cana-3488 216 11 some some DET cana-3488 216 12 0 0 NUM cana-3488 216 13 < < X cana-3488 216 14 ( ( PUNCT cana-3488 216 15 𝔙 𝔙 PROPN cana-3488 216 16 1112 1112 NUM cana-3488 216 17 ) ) PUNCT cana-3488 217 1 𝐹 𝐹 PROPN cana-3488 217 2 < < X cana-3488 217 3 1,0 1,0 NUM cana-3488 217 4 < < X cana-3488 217 5 ( ( PUNCT cana-3488 217 6 𝔙 𝔙 PROPN cana-3488 217 7 1113 1113 NUM cana-3488 217 8 ) ) PUNCT cana-3488 218 1 𝐹 𝐹 PROPN cana-3488 218 2 < < X cana-3488 218 3 1 1 NUM cana-3488 218 4 with with ADP cana-3488 218 5 𝐹 𝐹 PROPN cana-3488 218 6 ∈ ∈ PROPN cana-3488 218 7 { { PUNCT cana-3488 218 8 1,βˆ’1 1,βˆ’1 NUM cana-3488 218 9 } } PUNCT cana-3488 218 10 . . PUNCT cana-3488 219 1 let let VERB cana-3488 219 2 𝔉:𝒳𝑀 𝔉:𝒳𝑀 ADJ cana-3488 219 3 βŸΆπ’΄π‘€ βŸΆπ’΄π‘€ NOUN cana-3488 219 4 be be AUX cana-3488 219 5 a a DET cana-3488 219 6 function function NOUN cana-3488 219 7 satisfying satisfy VERB cana-3488 219 8 the the DET cana-3488 219 9 inequality inequality NOUN cana-3488 219 10 with with ADP cana-3488 219 11 conditions condition NOUN cana-3488 219 12 ( ( PUNCT cana-3488 219 13 1 1 NUM cana-3488 219 14 ) ) PUNCT cana-3488 219 15 , , PUNCT cana-3488 219 16 ( ( PUNCT cana-3488 219 17 27 27 NUM cana-3488 219 18 ) ) PUNCT cana-3488 219 19 , , PUNCT cana-3488 219 20 ( ( PUNCT cana-3488 219 21 2 2 X cana-3488 219 22 ) ) PUNCT cana-3488 219 23 and and CCONJ cana-3488 219 24 ( ( PUNCT cana-3488 219 25 28 28 NUM cana-3488 219 26 ) ) PUNCT cana-3488 219 27 . . PUNCT cana-3488 220 1 then then ADV cana-3488 220 2 . . PUNCT cana-3488 221 1 π’œ1(𝔉(11ℨ π’œ1(𝔉(11ℨ X cana-3488 221 2 ) ) PUNCT cana-3488 222 1 βˆ’ βˆ’ ADP cana-3488 222 2 1,88,30,57,02,60,326𝔉(ℨ 1,88,30,57,02,60,326𝔉(ℨ NUM cana-3488 222 3 ) ) PUNCT cana-3488 222 4 + + CCONJ cana-3488 222 5 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ X cana-3488 222 6 ) ) PUNCT cana-3488 222 7 β‰₯ β‰₯ NOUN cana-3488 222 8 π’œ1β€²(Γ°(ℨ),β„³ π’œ1β€²(Γ°(ℨ),β„³ NOUN cana-3488 222 9 ) ) PUNCT cana-3488 222 10 π’œ2(𝔉(11ℨ π’œ2(𝔉(11ℨ PUNCT cana-3488 222 11 ) ) PUNCT cana-3488 222 12 βˆ’ βˆ’ PROPN cana-3488 222 13 1,88,30,57,02,60,326𝔉(ℨ 1,88,30,57,02,60,326𝔉(ℨ NUM cana-3488 222 14 ) ) PUNCT cana-3488 222 15 + + CCONJ cana-3488 222 16 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ X cana-3488 222 17 ) ) PUNCT cana-3488 222 18 ≀ ≀ NOUN cana-3488 222 19 π’œ2β€²(Γ°(ℨ),β„³ π’œ2β€²(Γ°(ℨ),β„³ NOUN cana-3488 222 20 ) ) PUNCT cana-3488 222 21 π’œ3(𝔉(11ℨ π’œ3(𝔉(11ℨ X cana-3488 222 22 ) ) PUNCT cana-3488 222 23 βˆ’ βˆ’ PROPN cana-3488 222 24 1,88,30,57,02,60,326𝔉(ℨ 1,88,30,57,02,60,326𝔉(ℨ NUM cana-3488 222 25 ) ) PUNCT cana-3488 222 26 + + CCONJ cana-3488 222 27 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ X cana-3488 222 28 ) ) PUNCT cana-3488 222 29 ≀ ≀ NOUN cana-3488 223 1 π’œ3β€²(Γ°(ℨ),β„³ π’œ3β€²(Γ°(ℨ),β„³ NOUN cana-3488 223 2 ) ) PUNCT cana-3488 223 3 } } PUNCT cana-3488 223 4 ( ( PUNCT cana-3488 223 5 35 35 NUM cana-3488 223 6 ) ) PUNCT cana-3488 223 7 for for ADP cana-3488 223 8 all all DET cana-3488 223 9 ℨ ℨ NOUN cana-3488 223 10 ∈ ∈ NOUN cana-3488 223 11 𝒳𝑀 𝒳𝑀 NOUN cana-3488 223 12 and and CCONJ cana-3488 223 13 all all DET cana-3488 223 14 β„³ β„³ PROPN cana-3488 223 15 > > X cana-3488 223 16 0 0 NUM cana-3488 223 17 . . PUNCT cana-3488 224 1 then then ADV cana-3488 224 2 there there PRON cana-3488 224 3 exists exist VERB cana-3488 224 4 a a DET cana-3488 224 5 unique unique ADJ cana-3488 224 6 tridecic tridecic NOUN cana-3488 224 7 function function NOUN cana-3488 224 8 𝒯:𝒳𝑀 𝒯:𝒳𝑀 PROPN cana-3488 224 9 βŸΆπ’΄π‘€ βŸΆπ’΄π‘€ NOUN cana-3488 224 10 and and CCONJ cana-3488 224 11 a a DET cana-3488 224 12 unique unique ADJ cana-3488 224 13 duodecic duodecic NOUN cana-3488 224 14 function function VERB cana-3488 224 15 π’Ÿ:𝒳𝑀 π’Ÿ:𝒳𝑀 NOUN cana-3488 224 16 βŸΆπ’΄π‘€ βŸΆπ’΄π‘€ PROPN cana-3488 225 1 satisfying satisfy VERB cana-3488 225 2 ( ( PUNCT cana-3488 225 3 1 1 NUM cana-3488 225 4 ) ) PUNCT cana-3488 225 5 and and CCONJ cana-3488 225 6 π’œ1(𝑔(ℨ π’œ1(𝑔(ℨ NOUN cana-3488 225 7 ) ) PUNCT cana-3488 225 8 βˆ’ βˆ’ PROPN cana-3488 225 9 𝒯(ℨ 𝒯(ℨ NUM cana-3488 225 10 ) ) PUNCT cana-3488 225 11 βˆ’ βˆ’ ADP cana-3488 225 12 π’Ÿ(ℨ),β„³ π’Ÿ(ℨ),β„³ SYM cana-3488 225 13 ) ) PUNCT cana-3488 225 14 β‰₯ β‰₯ NOUN cana-3488 225 15 π’œ1β€²(Γ°(ℨ π’œ1β€²(Γ°(ℨ NOUN cana-3488 225 16 ) ) PUNCT cana-3488 225 17 , , PUNCT cana-3488 225 18 |11 |11 PROPN cana-3488 225 19 13 13 NUM cana-3488 225 20 βˆ’π”™|β„³ βˆ’π”™|β„³ NOUN cana-3488 225 21 ) ) PUNCT cana-3488 225 22 βˆ— βˆ— NOUN cana-3488 225 23 π’œ1β€²(Γ°(βˆ’β„¨ π’œ1β€²(Γ°(βˆ’β„¨ NUM cana-3488 225 24 ) ) PUNCT cana-3488 225 25 , , PUNCT cana-3488 225 26 |11 |11 PROPN cana-3488 225 27 13 13 NUM cana-3488 225 28 βˆ’ βˆ’ NOUN cana-3488 225 29 𝔙|β„³ 𝔙|β„³ NOUN cana-3488 225 30 ) ) PUNCT cana-3488 225 31 βˆ— βˆ— NOUN cana-3488 225 32 π’œ1β€²(Γ°(ℨ π’œ1β€²(Γ°(ℨ NOUN cana-3488 225 33 ) ) PUNCT cana-3488 225 34 , , PUNCT cana-3488 225 35 |11 |11 PROPN cana-3488 225 36 12 12 NUM cana-3488 225 37 βˆ’π”™|β„³ βˆ’π”™|β„³ NOUN cana-3488 225 38 ) ) PUNCT cana-3488 225 39 βˆ— βˆ— NOUN cana-3488 225 40 π’œ1β€²(Γ°(βˆ’β„¨ π’œ1β€²(Γ°(βˆ’β„¨ NUM cana-3488 225 41 ) ) PUNCT cana-3488 225 42 , , PUNCT cana-3488 225 43 |11 |11 PROPN cana-3488 225 44 12 12 NUM cana-3488 225 45 βˆ’ βˆ’ NOUN cana-3488 225 46 𝔙|β„³ 𝔙|β„³ NOUN cana-3488 225 47 ) ) PUNCT cana-3488 225 48 π’œ2(𝑔(ℨ π’œ2(𝑔(ℨ PROPN cana-3488 225 49 ) ) PUNCT cana-3488 226 1 βˆ’ βˆ’ PROPN cana-3488 226 2 𝒯(ℨ 𝒯(ℨ NUM cana-3488 226 3 ) ) PUNCT cana-3488 226 4 βˆ’ βˆ’ ADP cana-3488 226 5 π’Ÿ(ℨ),β„³ π’Ÿ(ℨ),β„³ SYM cana-3488 226 6 ) ) PUNCT cana-3488 226 7 ≀ ≀ NUM cana-3488 226 8 π’œ2β€²(Γ°(ℨ π’œ2β€²(Γ°(ℨ NOUN cana-3488 226 9 ) ) PUNCT cana-3488 226 10 , , PUNCT cana-3488 226 11 |11 |11 PROPN cana-3488 226 12 13 13 NUM cana-3488 226 13 βˆ’π”™|β„³ βˆ’π”™|β„³ NOUN cana-3488 226 14 ) ) PUNCT cana-3488 226 15 β‹„ β‹„ NOUN cana-3488 226 16 π’œ2β€²(Γ°(βˆ’β„¨ π’œ2β€²(Γ°(βˆ’β„¨ PUNCT cana-3488 226 17 ) ) PUNCT cana-3488 226 18 , , PUNCT cana-3488 226 19 |11 |11 PROPN cana-3488 226 20 13 13 NUM cana-3488 226 21 βˆ’π”™|β„³ βˆ’π”™|β„³ NOUN cana-3488 226 22 ) ) PUNCT cana-3488 226 23 β‹„ β‹„ PROPN cana-3488 226 24 π’œ2β€²(Γ°(ℨ π’œ2β€²(Γ°(ℨ NOUN cana-3488 226 25 ) ) PUNCT cana-3488 226 26 , , PUNCT cana-3488 226 27 |11 |11 PROPN cana-3488 226 28 12 12 NUM cana-3488 226 29 βˆ’π”™|β„³ βˆ’π”™|β„³ PROPN cana-3488 226 30 ) ) PUNCT cana-3488 226 31 β‹„ β‹„ NOUN cana-3488 226 32 π’œ2β€²(Γ°(βˆ’β„¨ π’œ2β€²(Γ°(βˆ’β„¨ PUNCT cana-3488 226 33 ) ) PUNCT cana-3488 226 34 , , PUNCT cana-3488 226 35 |11 |11 PROPN cana-3488 226 36 12 12 NUM cana-3488 226 37 βˆ’ βˆ’ NOUN cana-3488 226 38 𝔙|β„³ 𝔙|β„³ NOUN cana-3488 226 39 ) ) PUNCT cana-3488 226 40 π’œ3(𝑔(ℨ π’œ3(𝑔(ℨ NUM cana-3488 226 41 ) ) PUNCT cana-3488 227 1 βˆ’ βˆ’ PROPN cana-3488 227 2 𝒯(ℨ 𝒯(ℨ NUM cana-3488 227 3 ) ) PUNCT cana-3488 227 4 βˆ’ βˆ’ ADP cana-3488 227 5 π’Ÿ(ℨ),β„³ π’Ÿ(ℨ),β„³ NOUN cana-3488 227 6 ) ) PUNCT cana-3488 227 7 ≀ ≀ NOUN cana-3488 227 8 π’œ3β€²(Γ°(ℨ π’œ3β€²(Γ°(ℨ NOUN cana-3488 227 9 ) ) PUNCT cana-3488 227 10 , , PUNCT cana-3488 227 11 |11 |11 PROPN cana-3488 227 12 13 13 NUM cana-3488 227 13 βˆ’π”™|β„³)βŠ˜π’œ3β€²(Γ°(βˆ’β„¨ βˆ’π”™|β„³)βŠ˜π’œ3β€²(Γ°(βˆ’β„¨ NOUN cana-3488 227 14 ) ) PUNCT cana-3488 227 15 , , PUNCT cana-3488 227 16 |11 |11 PROPN cana-3488 227 17 13 13 NUM cana-3488 227 18 βˆ’ βˆ’ NOUN cana-3488 227 19 𝔙|β„³ 𝔙|β„³ NOUN cana-3488 227 20 ) ) PUNCT cana-3488 227 21 βŠ˜π’œ3β€²(Γ°(ℨ βŠ˜π’œ3β€²(Γ°(ℨ NUM cana-3488 227 22 ) ) PUNCT cana-3488 227 23 , , PUNCT cana-3488 227 24 |11 |11 PROPN cana-3488 227 25 12 12 NUM cana-3488 227 26 βˆ’ βˆ’ NOUN cana-3488 227 27 𝔙|β„³)βŠ˜π’œ3β€²(Γ°(βˆ’β„¨ 𝔙|β„³)βŠ˜π’œ3β€²(Γ°(βˆ’β„¨ NUM cana-3488 227 28 ) ) PUNCT cana-3488 227 29 , , PUNCT cana-3488 227 30 |11 |11 PROPN cana-3488 227 31 12 12 NUM cana-3488 227 32 βˆ’π”™|β„³ βˆ’π”™|β„³ NOUN cana-3488 227 33 ) ) PUNCT cana-3488 227 34 } } PUNCT cana-3488 227 35 ( ( PUNCT cana-3488 227 36 36 36 NUM cana-3488 227 37 ) ) PUNCT cana-3488 227 38 for for ADP cana-3488 227 39 all all DET cana-3488 227 40 ℨ ℨ NOUN cana-3488 227 41 ∈ ∈ NOUN cana-3488 227 42 𝒳𝑀 𝒳𝑀 NOUN cana-3488 227 43 and and CCONJ cana-3488 227 44 all all DET cana-3488 227 45 β„³ β„³ PROPN cana-3488 227 46 > > X cana-3488 227 47 0 0 PROPN cana-3488 227 48 . . PUNCT cana-3488 227 49 proof proof NOUN cana-3488 227 50 . . PUNCT cana-3488 228 1 let let VERB cana-3488 228 2 π”‰π‘œ(ℨ π”‰π‘œ(ℨ VERB cana-3488 228 3 ) ) PUNCT cana-3488 229 1 = = SYM cana-3488 229 2 𝔉(ℨ)βˆ’π”‰(βˆ’β„¨ 𝔉(ℨ)βˆ’π”‰(βˆ’β„¨ X cana-3488 229 3 ) ) PUNCT cana-3488 229 4 2 2 NUM cana-3488 229 5 for for ADP cana-3488 229 6 all all DET cana-3488 229 7 ℨ ℨ NOUN cana-3488 229 8 ∈ ∈ NOUN cana-3488 229 9 𝒳𝑀. 𝒳𝑀. PUNCT cana-3488 229 10 then then ADV cana-3488 229 11 π”‰π‘œ(0 π”‰π‘œ(0 NOUN cana-3488 229 12 ) ) PUNCT cana-3488 230 1 = = SYM cana-3488 230 2 0 0 NUM cana-3488 230 3 and and CCONJ cana-3488 230 4 π”‰π‘œ(βˆ’β„¨ π”‰π‘œ(βˆ’β„¨ NOUN cana-3488 230 5 ) ) PUNCT cana-3488 230 6 = = SYM cana-3488 230 7 βˆ’π”‰π‘œ(ℨ βˆ’π”‰π‘œ(ℨ NOUN cana-3488 230 8 ) ) PUNCT cana-3488 230 9 for for ADP cana-3488 230 10 all all PRON cana-3488 230 11 ℨ ℨ NOUN cana-3488 230 12 ∈ ∈ NOUN cana-3488 230 13 𝒳𝑀. 𝒳𝑀. PUNCT cana-3488 230 14 hence hence ADV cana-3488 230 15 by by ADP cana-3488 230 16 theorem theorem NOUN cana-3488 230 17 3.1 3.1 NUM cana-3488 230 18 , , PUNCT cana-3488 230 19 we we PRON cana-3488 230 20 have have VERB cana-3488 230 21 communications communication NOUN cana-3488 230 22 on on ADP cana-3488 230 23 applied apply VERB cana-3488 230 24 nonlinear nonlinear ADJ cana-3488 230 25 analysis analysis NOUN cana-3488 230 26 issn issn NOUN cana-3488 230 27 : : PUNCT cana-3488 230 28 1074 1074 NUM cana-3488 230 29 - - PUNCT cana-3488 230 30 133x 133x NUM cana-3488 230 31 vol vol NOUN cana-3488 230 32 32 32 NUM cana-3488 231 1 no no NOUN cana-3488 231 2 . . PUNCT cana-3488 232 1 7s 7 NOUN cana-3488 232 2 ( ( PUNCT cana-3488 232 3 2025 2025 NUM cana-3488 232 4 ) ) PUNCT cana-3488 232 5 819 819 NUM cana-3488 232 6 https://internationalpubls.com https://internationalpubls.com X cana-3488 232 7 π’œ1(π”‰π‘œ(ℨ π’œ1(π”‰π‘œ(ℨ NOUN cana-3488 232 8 ) ) PUNCT cana-3488 233 1 βˆ’ βˆ’ ADP cana-3488 233 2 𝒯(ℨ),β„³ 𝒯(ℨ),β„³ NOUN cana-3488 233 3 ) ) PUNCT cana-3488 233 4 β‰₯ β‰₯ NOUN cana-3488 233 5 π’œ1β€²(Γ°(ℨ π’œ1β€²(Γ°(ℨ NOUN cana-3488 233 6 ) ) PUNCT cana-3488 233 7 , , PUNCT cana-3488 233 8 |11 |11 PROPN cana-3488 233 9 13 13 NUM cana-3488 233 10 βˆ’ βˆ’ NOUN cana-3488 233 11 𝔙|β„³ 𝔙|β„³ NOUN cana-3488 233 12 ) ) PUNCT cana-3488 233 13 βˆ— βˆ— NOUN cana-3488 233 14 π’œ1β€²(Γ°(βˆ’β„¨ π’œ1β€²(Γ°(βˆ’β„¨ NUM cana-3488 233 15 ) ) PUNCT cana-3488 233 16 , , PUNCT cana-3488 233 17 |11 |11 PROPN cana-3488 233 18 13 13 NUM cana-3488 233 19 βˆ’ βˆ’ PROPN cana-3488 233 20 𝔙|β„³ 𝔙|β„³ PROPN cana-3488 233 21 ) ) PUNCT cana-3488 233 22 π’œ2(π”‰π‘œ(ℨ π’œ2(π”‰π‘œ(ℨ NOUN cana-3488 233 23 ) ) PUNCT cana-3488 233 24 βˆ’ βˆ’ ADP cana-3488 234 1 𝒯(ℨ),β„³ 𝒯(ℨ),β„³ NOUN cana-3488 234 2 ) ) PUNCT cana-3488 234 3 ≀ ≀ NUM cana-3488 234 4 π’œ2β€²(Γ°(ℨ π’œ2β€²(Γ°(ℨ NOUN cana-3488 234 5 ) ) PUNCT cana-3488 234 6 , , PUNCT cana-3488 234 7 |11 |11 PROPN cana-3488 234 8 13 13 NUM cana-3488 234 9 βˆ’π”™|β„³ βˆ’π”™|β„³ NOUN cana-3488 234 10 ) ) PUNCT cana-3488 234 11 β‹„ β‹„ NOUN cana-3488 234 12 π’œ2β€²(Γ°(βˆ’β„¨ π’œ2β€²(Γ°(βˆ’β„¨ PUNCT cana-3488 234 13 ) ) PUNCT cana-3488 234 14 , , PUNCT cana-3488 234 15 |11 |11 PROPN cana-3488 234 16 13 13 NUM cana-3488 234 17 βˆ’π”™|β„³ βˆ’π”™|β„³ NOUN cana-3488 234 18 ) ) PUNCT cana-3488 234 19 π’œ3(π”‰π‘œ(ℨ π’œ3(π”‰π‘œ(ℨ NOUN cana-3488 234 20 ) ) PUNCT cana-3488 234 21 βˆ’ βˆ’ PRON cana-3488 234 22 𝒯(ℨ),β„³ 𝒯(ℨ),β„³ NOUN cana-3488 234 23 ) ) PUNCT cana-3488 234 24 ≀ ≀ NOUN cana-3488 234 25 π’œ3β€²(Γ°(ℨ π’œ3β€²(Γ°(ℨ NOUN cana-3488 234 26 ) ) PUNCT cana-3488 234 27 , , PUNCT cana-3488 234 28 |11 |11 PROPN cana-3488 234 29 13 13 NUM cana-3488 234 30 βˆ’π”™|β„³)βŠ˜π’œ3β€²(Γ°(βˆ’β„¨ βˆ’π”™|β„³)βŠ˜π’œ3β€²(Γ°(βˆ’β„¨ NOUN cana-3488 234 31 ) ) PUNCT cana-3488 234 32 , , PUNCT cana-3488 234 33 |11 |11 PROPN cana-3488 234 34 13 13 NUM cana-3488 234 35 βˆ’ βˆ’ NOUN cana-3488 234 36 𝔙|β„³ 𝔙|β„³ NOUN cana-3488 234 37 ) ) PUNCT cana-3488 234 38 } } PUNCT cana-3488 234 39 ( ( PUNCT cana-3488 234 40 37 37 NUM cana-3488 234 41 ) ) PUNCT cana-3488 234 42 for for ADP cana-3488 234 43 all all PRON cana-3488 234 44 ℨ ℨ NOUN cana-3488 234 45 ∈ ∈ NOUN cana-3488 234 46 𝒳𝑀 𝒳𝑀 NOUN cana-3488 234 47 and and CCONJ cana-3488 234 48 all all DET cana-3488 234 49 β„³ β„³ PROPN cana-3488 234 50 > > X cana-3488 234 51 0 0 NUM cana-3488 234 52 . . PUNCT cana-3488 235 1 also also ADV cana-3488 235 2 , , PUNCT cana-3488 235 3 let let VERB cana-3488 235 4 𝔉𝑒(ℨ 𝔉𝑒(ℨ PRON cana-3488 235 5 ) ) PUNCT cana-3488 235 6 = = SYM cana-3488 235 7 𝔉(ℨ)+𝔉(βˆ’β„¨ 𝔉(ℨ)+𝔉(βˆ’β„¨ X cana-3488 235 8 ) ) PUNCT cana-3488 235 9 2 2 NUM cana-3488 235 10 for for ADP cana-3488 235 11 all all DET cana-3488 235 12 ℨ ℨ NOUN cana-3488 235 13 ∈ ∈ NOUN cana-3488 235 14 𝒳𝑀. 𝒳𝑀. PUNCT cana-3488 235 15 then then ADV cana-3488 235 16 𝔉𝑒(0 𝔉𝑒(0 PROPN cana-3488 235 17 ) ) PUNCT cana-3488 236 1 = = SYM cana-3488 236 2 0 0 NUM cana-3488 236 3 and and CCONJ cana-3488 236 4 𝔉𝑒(βˆ’β„¨ 𝔉𝑒(βˆ’β„¨ ADV cana-3488 236 5 ) ) PUNCT cana-3488 236 6 = = SYM cana-3488 236 7 𝔉𝑒(ℨ 𝔉𝑒(ℨ PROPN cana-3488 236 8 ) ) PUNCT cana-3488 236 9 for for ADP cana-3488 236 10 all all DET cana-3488 236 11 ℨ ℨ NOUN cana-3488 236 12 ∈ ∈ NOUN cana-3488 236 13 𝒳𝑀. 𝒳𝑀. PUNCT cana-3488 236 14 hence hence ADV cana-3488 236 15 by by ADP cana-3488 236 16 theorem theorem NOUN cana-3488 236 17 3.3 3.3 NUM cana-3488 236 18 , , PUNCT cana-3488 236 19 we we PRON cana-3488 236 20 have have VERB cana-3488 236 21 π’œ1(𝔉𝑒(ℨ π’œ1(𝔉𝑒(ℨ PRON cana-3488 236 22 ) ) PUNCT cana-3488 237 1 βˆ’ βˆ’ NUM cana-3488 237 2 π’Ÿ(ℨ),β„³ π’Ÿ(ℨ),β„³ SYM cana-3488 237 3 ) ) PUNCT cana-3488 237 4 β‰₯ β‰₯ NOUN cana-3488 237 5 π’œ1β€²(Γ°(ℨ π’œ1β€²(Γ°(ℨ NOUN cana-3488 237 6 ) ) PUNCT cana-3488 237 7 , , PUNCT cana-3488 237 8 |11 |11 PROPN cana-3488 237 9 12 12 NUM cana-3488 237 10 βˆ’ βˆ’ NOUN cana-3488 237 11 𝔙|β„³ 𝔙|β„³ NOUN cana-3488 237 12 ) ) PUNCT cana-3488 237 13 βˆ— βˆ— NOUN cana-3488 237 14 π’œ1β€²(Γ°(βˆ’β„¨ π’œ1β€²(Γ°(βˆ’β„¨ NUM cana-3488 237 15 ) ) PUNCT cana-3488 237 16 , , PUNCT cana-3488 237 17 |11 |11 PROPN cana-3488 237 18 12 12 NUM cana-3488 237 19 βˆ’ βˆ’ NOUN cana-3488 237 20 𝔙|β„³ 𝔙|β„³ NOUN cana-3488 237 21 ) ) PUNCT cana-3488 237 22 π’œ2(𝔉𝑒(ℨ π’œ2(𝔉𝑒(ℨ NOUN cana-3488 237 23 ) ) PUNCT cana-3488 237 24 βˆ’ βˆ’ NUM cana-3488 237 25 π’Ÿ(ℨ),β„³ π’Ÿ(ℨ),β„³ NOUN cana-3488 237 26 ) ) PUNCT cana-3488 237 27 ≀ ≀ NUM cana-3488 237 28 π’œ2β€²(Γ°(ℨ π’œ2β€²(Γ°(ℨ NOUN cana-3488 237 29 ) ) PUNCT cana-3488 237 30 , , PUNCT cana-3488 237 31 |11 |11 PROPN cana-3488 237 32 12 12 NUM cana-3488 237 33 βˆ’π”™|β„³ βˆ’π”™|β„³ PROPN cana-3488 237 34 ) ) PUNCT cana-3488 237 35 β‹„ β‹„ NOUN cana-3488 237 36 π’œ2β€²(Γ°(βˆ’β„¨ π’œ2β€²(Γ°(βˆ’β„¨ PUNCT cana-3488 237 37 ) ) PUNCT cana-3488 237 38 , , PUNCT cana-3488 237 39 |11 |11 PROPN cana-3488 237 40 12 12 NUM cana-3488 237 41 βˆ’π”™|β„³ βˆ’π”™|β„³ NOUN cana-3488 237 42 ) ) PUNCT cana-3488 237 43 π’œ3(𝔉𝑒(ℨ π’œ3(𝔉𝑒(ℨ NOUN cana-3488 237 44 ) ) PUNCT cana-3488 237 45 βˆ’ βˆ’ NUM cana-3488 237 46 π’Ÿ(ℨ),β„³ π’Ÿ(ℨ),β„³ NOUN cana-3488 237 47 ) ) PUNCT cana-3488 237 48 ≀ ≀ NOUN cana-3488 237 49 π’œ3β€²(Γ°(ℨ π’œ3β€²(Γ°(ℨ NOUN cana-3488 237 50 ) ) PUNCT cana-3488 237 51 , , PUNCT cana-3488 237 52 |11 |11 PROPN cana-3488 237 53 12 12 NUM cana-3488 237 54 βˆ’π”™|β„³)βŠ˜π’œ1β€²(Γ°(βˆ’β„¨ βˆ’π”™|β„³)βŠ˜π’œ1β€²(Γ°(βˆ’β„¨ NUM cana-3488 237 55 ) ) PUNCT cana-3488 237 56 , , PUNCT cana-3488 237 57 |11 |11 PROPN cana-3488 237 58 12 12 NUM cana-3488 237 59 βˆ’π”™|β„³ βˆ’π”™|β„³ PROPN cana-3488 237 60 ) ) PUNCT cana-3488 237 61 } } PUNCT cana-3488 237 62 ( ( PUNCT cana-3488 237 63 38 38 NUM cana-3488 237 64 ) ) PUNCT cana-3488 237 65 for for ADP cana-3488 237 66 all all DET cana-3488 237 67 ℨ ℨ NOUN cana-3488 237 68 ∈ ∈ NOUN cana-3488 237 69 𝒳𝑀 𝒳𝑀 NOUN cana-3488 237 70 and and CCONJ cana-3488 237 71 all all DET cana-3488 237 72 β„³ β„³ PROPN cana-3488 237 73 > > X cana-3488 237 74 0 0 X cana-3488 237 75 . . PUNCT cana-3488 237 76 define define VERB cana-3488 237 77 𝑔(ℨ 𝑔(ℨ NOUN cana-3488 237 78 ) ) PUNCT cana-3488 237 79 = = SYM cana-3488 237 80 π”‰π‘œ(ℨ π”‰π‘œ(ℨ NOUN cana-3488 237 81 ) ) PUNCT cana-3488 237 82 + + CCONJ cana-3488 237 83 𝔉𝑒(ℨ 𝔉𝑒(ℨ NOUN cana-3488 237 84 ) ) PUNCT cana-3488 237 85 ( ( PUNCT cana-3488 237 86 39 39 NUM cana-3488 237 87 ) ) PUNCT cana-3488 237 88 for for ADP cana-3488 237 89 all all PRON cana-3488 237 90 ℨ ℨ NOUN cana-3488 237 91 ∈ ∈ NOUN cana-3488 237 92 𝒳𝑀. 𝒳𝑀. PUNCT cana-3488 237 93 from from ADP cana-3488 237 94 ( ( PUNCT cana-3488 237 95 37),(38 37),(38 NUM cana-3488 237 96 ) ) PUNCT cana-3488 237 97 and and CCONJ cana-3488 237 98 ( ( PUNCT cana-3488 237 99 39 39 NUM cana-3488 237 100 ) ) PUNCT cana-3488 237 101 , , PUNCT cana-3488 237 102 we we PRON cana-3488 237 103 arrive arrive VERB cana-3488 237 104 π’œ1(𝔉(ℨ π’œ1(𝔉(ℨ NOUN cana-3488 237 105 ) ) PUNCT cana-3488 237 106 βˆ’ βˆ’ PROPN cana-3488 237 107 𝒯(ℨ 𝒯(ℨ NUM cana-3488 237 108 ) ) PUNCT cana-3488 237 109 βˆ’ βˆ’ PROPN cana-3488 237 110 π’Ÿ(ℨ),2β„³ π’Ÿ(ℨ),2β„³ NOUN cana-3488 237 111 ) ) PUNCT cana-3488 237 112 = = PUNCT cana-3488 237 113 π’œ1(π”‰π‘œ(ℨ π’œ1(π”‰π‘œ(ℨ NOUN cana-3488 237 114 ) ) PUNCT cana-3488 237 115 + + SYM cana-3488 237 116 𝔉𝑒(ℨ 𝔉𝑒(ℨ PROPN cana-3488 237 117 ) ) PUNCT cana-3488 237 118 βˆ’ βˆ’ PROPN cana-3488 237 119 𝒯(ℨ 𝒯(ℨ NUM cana-3488 237 120 ) ) PUNCT cana-3488 237 121 βˆ’ βˆ’ PROPN cana-3488 237 122 π’Ÿ(ℨ),2β„³ π’Ÿ(ℨ),2β„³ NOUN cana-3488 237 123 ) ) PUNCT cana-3488 237 124 β‰₯ β‰₯ NOUN cana-3488 237 125 π’œ1(π”‰π‘œ(ℨ π’œ1(π”‰π‘œ(ℨ NOUN cana-3488 237 126 ) ) PUNCT cana-3488 237 127 βˆ’ βˆ’ NUM cana-3488 237 128 𝒯(ℨ),β„³ 𝒯(ℨ),β„³ NOUN cana-3488 237 129 ) ) PUNCT cana-3488 237 130 βˆ— βˆ— NOUN cana-3488 237 131 π’œ1(𝔉𝑒(ℨ π’œ1(𝔉𝑒(ℨ NOUN cana-3488 237 132 ) ) PUNCT cana-3488 238 1 βˆ’ βˆ’ ADP cana-3488 238 2 π’Ÿ(ℨ),β„³ π’Ÿ(ℨ),β„³ SYM cana-3488 238 3 ) ) PUNCT cana-3488 238 4 β‰₯ β‰₯ NOUN cana-3488 238 5 π’œ1β€²(Γ°(ℨ π’œ1β€²(Γ°(ℨ NOUN cana-3488 238 6 ) ) PUNCT cana-3488 238 7 , , PUNCT cana-3488 238 8 |11 |11 PROPN cana-3488 238 9 13 13 NUM cana-3488 238 10 βˆ’ βˆ’ NOUN cana-3488 238 11 𝔙|β„³ 𝔙|β„³ NOUN cana-3488 238 12 ) ) PUNCT cana-3488 238 13 βˆ— βˆ— NOUN cana-3488 238 14 π’œ1β€²(Γ°(βˆ’β„¨ π’œ1β€²(Γ°(βˆ’β„¨ NUM cana-3488 238 15 ) ) PUNCT cana-3488 238 16 , , PUNCT cana-3488 238 17 |11 |11 PROPN cana-3488 238 18 13 13 NUM cana-3488 238 19 βˆ’π”™|β„³ βˆ’π”™|β„³ NOUN cana-3488 238 20 ) ) PUNCT cana-3488 238 21 βˆ— βˆ— NOUN cana-3488 238 22 π’œ1β€²(Γ°(ℨ π’œ1β€²(Γ°(ℨ NOUN cana-3488 238 23 ) ) PUNCT cana-3488 238 24 , , PUNCT cana-3488 238 25 |11 |11 PROPN cana-3488 238 26 12 12 NUM cana-3488 238 27 βˆ’π”™|β„³ βˆ’π”™|β„³ NOUN cana-3488 238 28 ) ) PUNCT cana-3488 238 29 βˆ— βˆ— NOUN cana-3488 238 30 π’œ1β€²(Γ°(βˆ’β„¨ π’œ1β€²(Γ°(βˆ’β„¨ NUM cana-3488 238 31 ) ) PUNCT cana-3488 238 32 , , PUNCT cana-3488 238 33 |11 |11 PROPN cana-3488 238 34 12 12 NUM cana-3488 238 35 βˆ’π”™|β„³ βˆ’π”™|β„³ PROPN cana-3488 238 36 ) ) PUNCT cana-3488 238 37 and and CCONJ cana-3488 238 38 π’œ2(𝔉(ℨ π’œ2(𝔉(ℨ NOUN cana-3488 238 39 ) ) PUNCT cana-3488 238 40 βˆ’ βˆ’ PROPN cana-3488 238 41 𝒯(ℨ 𝒯(ℨ NUM cana-3488 238 42 ) ) PUNCT cana-3488 238 43 βˆ’ βˆ’ PROPN cana-3488 238 44 π’Ÿ(ℨ),2β„³ π’Ÿ(ℨ),2β„³ NOUN cana-3488 238 45 ) ) PUNCT cana-3488 238 46 = = SYM cana-3488 238 47 π’œ2(π”‰π‘œ(ℨ π’œ2(π”‰π‘œ(ℨ NOUN cana-3488 238 48 ) ) PUNCT cana-3488 238 49 + + SYM cana-3488 238 50 𝔉𝑒(ℨ 𝔉𝑒(ℨ PROPN cana-3488 238 51 ) ) PUNCT cana-3488 238 52 βˆ’ βˆ’ PROPN cana-3488 238 53 𝒯(ℨ 𝒯(ℨ NUM cana-3488 238 54 ) ) PUNCT cana-3488 238 55 βˆ’ βˆ’ ADP cana-3488 238 56 π’Ÿ(ℨ),2β„³ π’Ÿ(ℨ),2β„³ NOUN cana-3488 238 57 ) ) PUNCT cana-3488 238 58 ≀ ≀ NUM cana-3488 238 59 π’œ2(π”‰π‘œ(ℨ π’œ2(π”‰π‘œ(ℨ NOUN cana-3488 238 60 ) ) PUNCT cana-3488 238 61 βˆ’ βˆ’ ADP cana-3488 238 62 𝒯(ℨ),β„³ 𝒯(ℨ),β„³ NOUN cana-3488 238 63 ) ) PUNCT cana-3488 238 64 β‹„ β‹„ PROPN cana-3488 238 65 π’œ2(𝔉𝑒(ℨ π’œ2(𝔉𝑒(ℨ NOUN cana-3488 238 66 ) ) PUNCT cana-3488 238 67 βˆ’ βˆ’ NUM cana-3488 238 68 π’Ÿ(ℨ),β„³ π’Ÿ(ℨ),β„³ NOUN cana-3488 238 69 ) ) PUNCT cana-3488 238 70 ≀ ≀ NUM cana-3488 238 71 π’œ2β€²(Γ°(ℨ π’œ2β€²(Γ°(ℨ NOUN cana-3488 238 72 ) ) PUNCT cana-3488 238 73 , , PUNCT cana-3488 238 74 |11 |11 PROPN cana-3488 238 75 13 13 NUM cana-3488 238 76 βˆ’ βˆ’ PROPN cana-3488 238 77 𝔙|β„³ 𝔙|β„³ NOUN cana-3488 238 78 ) ) PUNCT cana-3488 238 79 β‹„π’œ2β€²(Γ°(βˆ’β„¨ β‹„π’œ2β€²(Γ°(βˆ’β„¨ NOUN cana-3488 238 80 ) ) PUNCT cana-3488 238 81 , , PUNCT cana-3488 238 82 |11 |11 PROPN cana-3488 238 83 13 13 NUM cana-3488 238 84 βˆ’ βˆ’ PROPN cana-3488 238 85 𝔙|β„³ 𝔙|β„³ NOUN cana-3488 238 86 ) ) PUNCT cana-3488 238 87 β‹„ β‹„ PROPN cana-3488 238 88 π’œ2β€²(Γ°(ℨ π’œ2β€²(Γ°(ℨ NOUN cana-3488 238 89 ) ) PUNCT cana-3488 238 90 , , PUNCT cana-3488 238 91 |11 |11 PROPN cana-3488 238 92 12 12 NUM cana-3488 238 93 βˆ’ βˆ’ NOUN cana-3488 238 94 𝔙|β„³ 𝔙|β„³ NOUN cana-3488 238 95 ) ) PUNCT cana-3488 238 96 β‹„π’œ2β€²(Γ°(βˆ’β„¨ β‹„π’œ2β€²(Γ°(βˆ’β„¨ NOUN cana-3488 238 97 ) ) PUNCT cana-3488 238 98 , , PUNCT cana-3488 238 99 |11 |11 PROPN cana-3488 238 100 12 12 NUM cana-3488 238 101 βˆ’π”™|β„³ βˆ’π”™|β„³ NOUN cana-3488 238 102 ) ) PUNCT cana-3488 238 103 and and CCONJ cana-3488 238 104 π’œ3(𝔉(ℨ π’œ3(𝔉(ℨ NOUN cana-3488 238 105 ) ) PUNCT cana-3488 238 106 βˆ’ βˆ’ PROPN cana-3488 238 107 𝒯(ℨ 𝒯(ℨ NUM cana-3488 238 108 ) ) PUNCT cana-3488 238 109 βˆ’ βˆ’ PROPN cana-3488 238 110 π’Ÿ(ℨ),2β„³ π’Ÿ(ℨ),2β„³ NOUN cana-3488 238 111 ) ) PUNCT cana-3488 238 112 = = PUNCT cana-3488 238 113 π’œ3(π”‰π‘œ(ℨ π’œ3(π”‰π‘œ(ℨ NOUN cana-3488 238 114 ) ) PUNCT cana-3488 238 115 + + SYM cana-3488 238 116 𝔉𝑒(ℨ 𝔉𝑒(ℨ PROPN cana-3488 238 117 ) ) PUNCT cana-3488 238 118 βˆ’ βˆ’ PROPN cana-3488 238 119 𝒯(ℨ 𝒯(ℨ NUM cana-3488 238 120 ) ) PUNCT cana-3488 238 121 βˆ’ βˆ’ ADP cana-3488 238 122 π’Ÿ(ℨ),2β„³ π’Ÿ(ℨ),2β„³ NOUN cana-3488 238 123 ) ) PUNCT cana-3488 238 124 ≀ ≀ NUM cana-3488 238 125 π’œ2(π”‰π‘œ(ℨ π’œ2(π”‰π‘œ(ℨ NOUN cana-3488 238 126 ) ) PUNCT cana-3488 238 127 βˆ’ βˆ’ ADP cana-3488 239 1 𝒯(ℨ),β„³)βŠ˜π’œ3(𝔉𝑒(ℨ 𝒯(ℨ),β„³)βŠ˜π’œ3(𝔉𝑒(ℨ NOUN cana-3488 239 2 ) ) PUNCT cana-3488 239 3 βˆ’ βˆ’ PROPN cana-3488 239 4 π’Ÿ(ℨ),β„³ π’Ÿ(ℨ),β„³ NOUN cana-3488 239 5 ) ) PUNCT cana-3488 239 6 ≀ ≀ NOUN cana-3488 239 7 π’œ3β€²(Γ°(ℨ π’œ3β€²(Γ°(ℨ NOUN cana-3488 239 8 ) ) PUNCT cana-3488 239 9 , , PUNCT cana-3488 239 10 |11 |11 PROPN cana-3488 239 11 13 13 NUM cana-3488 239 12 βˆ’π”™|β„³)βŠ˜π’œ3β€²(Γ°(βˆ’β„¨ βˆ’π”™|β„³)βŠ˜π’œ3β€²(Γ°(βˆ’β„¨ NOUN cana-3488 239 13 ) ) PUNCT cana-3488 239 14 , , PUNCT cana-3488 239 15 |11 |11 PROPN cana-3488 239 16 13 13 NUM cana-3488 239 17 βˆ’ βˆ’ NOUN cana-3488 239 18 𝔙|β„³ 𝔙|β„³ NOUN cana-3488 239 19 ) ) PUNCT cana-3488 239 20 ⊘ ⊘ X cana-3488 239 21 π’œ3β€²(Γ°(ℨ π’œ3β€²(Γ°(ℨ NOUN cana-3488 239 22 ) ) PUNCT cana-3488 239 23 , , PUNCT cana-3488 239 24 |11 |11 PROPN cana-3488 239 25 12 12 NUM cana-3488 239 26 βˆ’ βˆ’ NOUN cana-3488 239 27 𝔙|β„³)βŠ˜π’œ3β€²(Γ°(βˆ’β„¨ 𝔙|β„³)βŠ˜π’œ3β€²(Γ°(βˆ’β„¨ NUM cana-3488 239 28 ) ) PUNCT cana-3488 239 29 , , PUNCT cana-3488 239 30 |11 |11 PROPN cana-3488 239 31 12 12 NUM cana-3488 239 32 βˆ’π”™|β„³ βˆ’π”™|β„³ NOUN cana-3488 239 33 ) ) PUNCT cana-3488 239 34 for for ADP cana-3488 239 35 all all DET cana-3488 239 36 ℨ ℨ NOUN cana-3488 239 37 ∈ ∈ NOUN cana-3488 239 38 𝒳𝑀 𝒳𝑀 NOUN cana-3488 239 39 and and CCONJ cana-3488 239 40 all all DET cana-3488 239 41 β„³ β„³ PROPN cana-3488 239 42 > > X cana-3488 239 43 0 0 X cana-3488 239 44 . . PUNCT cana-3488 240 1 corollary corollary ADJ cana-3488 240 2 3.6 3.6 NUM cana-3488 240 3 let let VERB cana-3488 240 4 β„’ β„’ NOUN cana-3488 240 5 , , PUNCT cana-3488 240 6 π‘Ÿ π‘Ÿ NOUN cana-3488 240 7 are be AUX cana-3488 240 8 constants constant NOUN cana-3488 240 9 with with ADP cana-3488 240 10 β„’ β„’ PROPN cana-3488 240 11 > > SYM cana-3488 240 12 0 0 NUM cana-3488 240 13 and and CCONJ cana-3488 240 14 π‘Ÿ π‘Ÿ NOUN cana-3488 240 15 β‰  β‰  PROPN cana-3488 240 16 13,12 13,12 NUM cana-3488 240 17 and and CCONJ cana-3488 240 18 let let VERB cana-3488 240 19 the the DET cana-3488 240 20 function function NOUN cana-3488 240 21 𝔉:𝒳𝑀 𝔉:𝒳𝑀 NOUN cana-3488 240 22 βŸΆπ’΄π‘€ βŸΆπ’΄π‘€ NOUN cana-3488 240 23 satisfies satisfy VERB cana-3488 240 24 communications communication NOUN cana-3488 240 25 on on ADP cana-3488 240 26 applied apply VERB cana-3488 240 27 nonlinear nonlinear ADJ cana-3488 240 28 analysis analysis NOUN cana-3488 240 29 issn issn NOUN cana-3488 240 30 : : PUNCT cana-3488 240 31 1074 1074 NUM cana-3488 240 32 - - PUNCT cana-3488 240 33 133x 133x NUM cana-3488 240 34 vol vol NOUN cana-3488 240 35 32 32 NUM cana-3488 240 36 no no NOUN cana-3488 240 37 . . PUNCT cana-3488 241 1 7s 7 NOUN cana-3488 241 2 ( ( PUNCT cana-3488 241 3 2025 2025 NUM cana-3488 241 4 ) ) PUNCT cana-3488 241 5 820 820 NUM cana-3488 241 6 https://internationalpubls.com https://internationalpubls.com X cana-3488 241 7 π’œ1(𝔉(11ℨ π’œ1(𝔉(11ℨ PUNCT cana-3488 241 8 ) ) PUNCT cana-3488 242 1 βˆ’ βˆ’ ADP cana-3488 242 2 1,88,30,57,02,60,326𝔉(ℨ 1,88,30,57,02,60,326𝔉(ℨ NUM cana-3488 242 3 ) ) PUNCT cana-3488 242 4 + + CCONJ cana-3488 242 5 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ X cana-3488 242 6 ) ) PUNCT cana-3488 242 7 β‰₯ β‰₯ NOUN cana-3488 242 8 { { PUNCT cana-3488 242 9 π’œ1β€²(β„’,β„³ π’œ1β€²(β„’,β„³ PROPN cana-3488 242 10 ) ) PUNCT cana-3488 242 11 , , PUNCT cana-3488 242 12 π’œ1β€²(β„’(||ℨ|| π’œ1β€²(β„’(||ℨ|| PROPN cana-3488 242 13 π‘Ÿ),β„³ π‘Ÿ),β„³ NUM cana-3488 242 14 ) ) PUNCT cana-3488 242 15 , , PUNCT cana-3488 242 16 π’œ2(𝔉(11ℨ π’œ2(𝔉(11ℨ PUNCT cana-3488 242 17 ) ) PUNCT cana-3488 242 18 βˆ’ βˆ’ PROPN cana-3488 242 19 1,88,30,57,02,60,326𝔉(ℨ 1,88,30,57,02,60,326𝔉(ℨ NUM cana-3488 242 20 ) ) PUNCT cana-3488 242 21 + + CCONJ cana-3488 242 22 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ X cana-3488 242 23 ) ) PUNCT cana-3488 242 24 ≀ ≀ NOUN cana-3488 242 25 { { PUNCT cana-3488 242 26 π’œ2β€²(β„’,β„³ π’œ2β€²(β„’,β„³ NOUN cana-3488 242 27 ) ) PUNCT cana-3488 242 28 , , PUNCT cana-3488 242 29 π’œ2β€²(β„’(||ℨ|| π’œ2β€²(β„’(||ℨ|| PROPN cana-3488 242 30 π‘Ÿ),β„³ π‘Ÿ),β„³ PROPN cana-3488 242 31 ) ) PUNCT cana-3488 242 32 , , PUNCT cana-3488 242 33 π’œ3(𝔉(11ℨ π’œ3(𝔉(11ℨ X cana-3488 242 34 ) ) PUNCT cana-3488 242 35 βˆ’ βˆ’ PROPN cana-3488 242 36 1,88,30,57,02,60,326𝔉(ℨ 1,88,30,57,02,60,326𝔉(ℨ NUM cana-3488 242 37 ) ) PUNCT cana-3488 242 38 + + CCONJ cana-3488 242 39 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ 1,56,92,14,18,83,605𝔉(βˆ’β„¨),β„³ X cana-3488 242 40 ) ) PUNCT cana-3488 242 41 ≀ ≀ NOUN cana-3488 242 42 { { PUNCT cana-3488 242 43 π’œ3β€²(β„’,β„³ π’œ3β€²(β„’,β„³ NOUN cana-3488 242 44 ) ) PUNCT cana-3488 242 45 , , PUNCT cana-3488 242 46 π’œ3β€²(β„’(||ℨ|| π’œ3β€²(β„’(||ℨ|| PROPN cana-3488 242 47 π‘Ÿ),β„³ π‘Ÿ),β„³ PROPN cana-3488 242 48 ) ) PUNCT cana-3488 242 49 , , PUNCT cana-3488 242 50 } } PUNCT cana-3488 242 51 ( ( PUNCT cana-3488 242 52 40 40 NUM cana-3488 242 53 ) ) PUNCT cana-3488 242 54 for for ADP cana-3488 242 55 all all DET cana-3488 242 56 ℨ ℨ NOUN cana-3488 242 57 ∈ ∈ NOUN cana-3488 242 58 𝒳𝑀 𝒳𝑀 NOUN cana-3488 242 59 and and CCONJ cana-3488 242 60 all all DET cana-3488 242 61 β„³ β„³ PROPN cana-3488 242 62 > > X cana-3488 242 63 0 0 NUM cana-3488 242 64 . . PUNCT cana-3488 243 1 then then ADV cana-3488 243 2 there there PRON cana-3488 243 3 exists exist VERB cana-3488 243 4 a a DET cana-3488 243 5 unique unique ADJ cana-3488 243 6 tridecic tridecic NOUN cana-3488 243 7 function function NOUN cana-3488 243 8 𝒯:𝒳𝑀 𝒯:𝒳𝑀 PROPN cana-3488 243 9 βŸΆπ’΄π‘€ βŸΆπ’΄π‘€ NOUN cana-3488 243 10 and and CCONJ cana-3488 243 11 a a DET cana-3488 243 12 unique unique ADJ cana-3488 243 13 duodecic duodecic NOUN cana-3488 243 14 function function VERB cana-3488 243 15 π’Ÿ:𝒳𝑀 π’Ÿ:𝒳𝑀 NOUN cana-3488 243 16 βŸΆπ’΄π‘€ βŸΆπ’΄π‘€ INTJ cana-3488 243 17 such such ADJ cana-3488 243 18 that that PRON cana-3488 243 19 π’œ1(𝔉(ℨ π’œ1(𝔉(ℨ NOUN cana-3488 243 20 ) ) PUNCT cana-3488 243 21 βˆ’ βˆ’ PROPN cana-3488 243 22 𝒯(ℨ 𝒯(ℨ NUM cana-3488 243 23 ) ) PUNCT cana-3488 243 24 βˆ’ βˆ’ ADP cana-3488 243 25 π’Ÿ(ℨ),β„³ π’Ÿ(ℨ),β„³ SYM cana-3488 243 26 ) ) PUNCT cana-3488 243 27 β‰₯ β‰₯ NOUN cana-3488 243 28 { { PUNCT cana-3488 243 29 π’œ1β€²(β„’ π’œ1β€²(β„’ NUM cana-3488 243 30 , , PUNCT cana-3488 243 31 |11 |11 PROPN cana-3488 243 32 13 13 NUM cana-3488 243 33 βˆ’ βˆ’ PROPN cana-3488 243 34 1|β„³ 1|β„³ NUM cana-3488 243 35 ) ) PUNCT cana-3488 243 36 βˆ— βˆ— NOUN cana-3488 243 37 π’œ1β€²(β„’ π’œ1β€²(β„’ NUM cana-3488 243 38 , , PUNCT cana-3488 243 39 |11 |11 PROPN cana-3488 243 40 12 12 NUM cana-3488 243 41 βˆ’ βˆ’ PROPN cana-3488 243 42 1|β„³ 1|β„³ NUM cana-3488 243 43 ) ) PUNCT cana-3488 243 44 , , PUNCT cana-3488 243 45 π’œ1β€²(β„’||ℨ|| π’œ1β€²(β„’||ℨ|| VERB cana-3488 243 46 π‘Ÿ π‘Ÿ NOUN cana-3488 243 47 , , PUNCT cana-3488 243 48 |1113 |1113 PROPN cana-3488 243 49 βˆ’ βˆ’ PROPN cana-3488 243 50 11π‘Ÿ|β„³ 11π‘Ÿ|β„³ NUM cana-3488 243 51 ) ) PUNCT cana-3488 243 52 βˆ— βˆ— NOUN cana-3488 243 53 π’œ1β€²(β„’||ℨ|| π’œ1β€²(β„’||ℨ|| NOUN cana-3488 243 54 π‘Ÿ π‘Ÿ NOUN cana-3488 243 55 , , PUNCT cana-3488 243 56 |1112 |1112 NOUN cana-3488 243 57 βˆ’ βˆ’ PROPN cana-3488 243 58 11π‘Ÿ|β„³ 11π‘Ÿ|β„³ NUM cana-3488 243 59 ) ) PUNCT cana-3488 243 60 , , PUNCT cana-3488 243 61 π’œ2(𝔉(ℨ π’œ2(𝔉(ℨ PROPN cana-3488 243 62 ) ) PUNCT cana-3488 243 63 βˆ’ βˆ’ PROPN cana-3488 243 64 𝒯(ℨ 𝒯(ℨ NUM cana-3488 243 65 ) ) PUNCT cana-3488 243 66 βˆ’ βˆ’ ADP cana-3488 243 67 π’Ÿ(ℨ),β„³ π’Ÿ(ℨ),β„³ SYM cana-3488 243 68 ) ) PUNCT cana-3488 243 69 ≀ ≀ NOUN cana-3488 243 70 { { PUNCT cana-3488 243 71 π’œ2β€²(β„’ π’œ2β€²(β„’ X cana-3488 243 72 , , PUNCT cana-3488 243 73 |11 |11 PROPN cana-3488 243 74 13 13 NUM cana-3488 243 75 βˆ’ βˆ’ NUM cana-3488 243 76 1|β„³ 1|β„³ NUM cana-3488 243 77 ) ) PUNCT cana-3488 243 78 β‹„ β‹„ PROPN cana-3488 243 79 π’œ2β€²(β„’ π’œ2β€²(β„’ NOUN cana-3488 243 80 , , PUNCT cana-3488 243 81 |11 |11 PROPN cana-3488 243 82 12 12 NUM cana-3488 243 83 βˆ’ βˆ’ PROPN cana-3488 243 84 1|β„³ 1|β„³ NUM cana-3488 243 85 ) ) PUNCT cana-3488 243 86 , , PUNCT cana-3488 243 87 π’œ2β€²(β„’||ℨ|| π’œ2β€²(β„’||ℨ|| NOUN cana-3488 243 88 π‘Ÿ π‘Ÿ NOUN cana-3488 243 89 , , PUNCT cana-3488 243 90 |1113 |1113 PROPN cana-3488 243 91 βˆ’ βˆ’ PROPN cana-3488 243 92 11π‘Ÿ|β„³ 11π‘Ÿ|β„³ NUM cana-3488 243 93 ) ) PUNCT cana-3488 243 94 β‹„ β‹„ PROPN cana-3488 243 95 π’œ2β€²(β„’||ℨ|| π’œ2β€²(β„’||ℨ|| NOUN cana-3488 243 96 π‘Ÿ π‘Ÿ X cana-3488 243 97 , , PUNCT cana-3488 243 98 |1112 |1112 NOUN cana-3488 243 99 βˆ’ βˆ’ PROPN cana-3488 243 100 11π‘Ÿ|β„³ 11π‘Ÿ|β„³ NUM cana-3488 243 101 ) ) PUNCT cana-3488 243 102 , , PUNCT cana-3488 243 103 π’œ3(𝔉(ℨ π’œ3(𝔉(ℨ X cana-3488 243 104 ) ) PUNCT cana-3488 243 105 βˆ’ βˆ’ PROPN cana-3488 243 106 𝒯(ℨ 𝒯(ℨ NUM cana-3488 243 107 ) ) PUNCT cana-3488 244 1 βˆ’ βˆ’ ADP cana-3488 244 2 π’Ÿ(ℨ),β„³ π’Ÿ(ℨ),β„³ SYM cana-3488 244 3 ) ) PUNCT cana-3488 244 4 ≀ ≀ NOUN cana-3488 244 5 { { PUNCT cana-3488 244 6 π’œ3β€²(β„’ π’œ3β€²(β„’ NUM cana-3488 244 7 , , PUNCT cana-3488 244 8 |11 |11 PROPN cana-3488 244 9 13 13 NUM cana-3488 244 10 βˆ’ βˆ’ NOUN cana-3488 244 11 1|β„³)βŠ˜π’œ3β€²(β„’ 1|β„³)βŠ˜π’œ3β€²(β„’ NUM cana-3488 244 12 , , PUNCT cana-3488 244 13 |11 |11 PROPN cana-3488 244 14 12 12 NUM cana-3488 244 15 βˆ’ βˆ’ PROPN cana-3488 244 16 1|β„³ 1|β„³ NUM cana-3488 244 17 ) ) PUNCT cana-3488 244 18 , , PUNCT cana-3488 244 19 π’œ3β€²(β„’||ℨ|| π’œ3β€²(β„’||ℨ|| ADJ cana-3488 244 20 π‘Ÿ π‘Ÿ NOUN cana-3488 244 21 , , PUNCT cana-3488 244 22 |1113 |1113 NOUN cana-3488 244 23 βˆ’ βˆ’ PROPN cana-3488 244 24 11π‘Ÿ|β„³)βŠ˜π’œ3β€²(β„’||ℨ|| 11π‘Ÿ|β„³)βŠ˜π’œ3β€²(β„’||ℨ|| SYM cana-3488 244 25 π‘Ÿ π‘Ÿ X cana-3488 244 26 , , PUNCT cana-3488 244 27 |1112 |1112 NOUN cana-3488 244 28 βˆ’ βˆ’ PROPN cana-3488 244 29 11π‘Ÿ|β„³ 11π‘Ÿ|β„³ NUM cana-3488 244 30 ) ) PUNCT cana-3488 244 31 , , PUNCT cana-3488 244 32 } } PUNCT cana-3488 244 33 ( ( PUNCT cana-3488 244 34 41 41 NUM cana-3488 244 35 ) ) PUNCT cana-3488 244 36 for for ADP cana-3488 244 37 all all DET cana-3488 244 38 ℨ ℨ NOUN cana-3488 244 39 ∈ ∈ NOUN cana-3488 244 40 𝒳𝑀 𝒳𝑀 NOUN cana-3488 244 41 and and CCONJ cana-3488 244 42 all all DET cana-3488 244 43 β„³ β„³ PROPN cana-3488 244 44 > > NOUN cana-3488 244 45 0 0 NUM cana-3488 244 46 . . PUNCT cana-3488 245 1 4 4 NUM cana-3488 245 2 conclusion conclusion NOUN cana-3488 245 3 the the DET cana-3488 245 4 numerical numerical ADJ cana-3488 245 5 stability stability PROPN cana-3488 245 6 analysis analysis NOUN cana-3488 245 7 conducted conduct VERB cana-3488 245 8 in in ADP cana-3488 245 9 this this DET cana-3488 245 10 study study NOUN cana-3488 245 11 underscores underscore VERB cana-3488 245 12 the the DET cana-3488 245 13 viability viability NOUN cana-3488 245 14 and and CCONJ cana-3488 245 15 robustness robustness NOUN cana-3488 245 16 of of ADP cana-3488 245 17 mixed mixed ADJ cana-3488 245 18 - - PUNCT cana-3488 245 19 type type NOUN cana-3488 245 20 duodecic duodecic NOUN cana-3488 245 21 and and CCONJ cana-3488 245 22 tridecic tridecic NOUN cana-3488 245 23 functional functional ADJ cana-3488 245 24 equations equation NOUN cana-3488 245 25 in in ADP cana-3488 245 26 neutrosophic neutrosophic ADJ cana-3488 245 27 normed norme VERB cana-3488 245 28 spaces space NOUN cana-3488 245 29 . . PUNCT cana-3488 246 1 by by ADP cana-3488 246 2 employing employ VERB cana-3488 246 3 the the DET cana-3488 246 4 hyers hyer NOUN cana-3488 246 5 classical classical ADJ cana-3488 246 6 direct direct ADJ cana-3488 246 7 method method NOUN cana-3488 246 8 , , PUNCT cana-3488 246 9 we we PRON cana-3488 246 10 provided provide VERB cana-3488 246 11 a a DET cana-3488 246 12 thorough thorough ADJ cana-3488 246 13 examination examination NOUN cana-3488 246 14 of of ADP cana-3488 246 15 stability stability NOUN cana-3488 246 16 characteristics characteristic NOUN cana-3488 246 17 , , PUNCT cana-3488 246 18 ensuring ensure VERB cana-3488 246 19 that that SCONJ cana-3488 246 20 these these DET cana-3488 246 21 functional functional ADJ cana-3488 246 22 equations equation NOUN cana-3488 246 23 can can AUX cana-3488 246 24 be be AUX cana-3488 246 25 confidently confidently ADV cana-3488 246 26 applied apply VERB cana-3488 246 27 in in ADP cana-3488 246 28 various various ADJ cana-3488 246 29 scientific scientific ADJ cana-3488 246 30 and and CCONJ cana-3488 246 31 engineering engineering NOUN cana-3488 246 32 disciplines discipline NOUN cana-3488 246 33 . . PUNCT cana-3488 247 1 this this DET cana-3488 247 2 methodology methodology NOUN cana-3488 247 3 not not PART cana-3488 247 4 only only ADV cana-3488 247 5 validated validate VERB cana-3488 247 6 the the DET cana-3488 247 7 stability stability NOUN cana-3488 247 8 but but CCONJ cana-3488 247 9 also also ADV cana-3488 247 10 enriched enrich VERB cana-3488 247 11 the the DET cana-3488 247 12 theoretical theoretical ADJ cana-3488 247 13 landscape landscape NOUN cana-3488 247 14 , , PUNCT cana-3488 247 15 paving pave VERB cana-3488 247 16 the the DET cana-3488 247 17 way way NOUN cana-3488 247 18 for for ADP cana-3488 247 19 future future ADJ cana-3488 247 20 research research NOUN cana-3488 247 21 in in ADP cana-3488 247 22 this this DET cana-3488 247 23 domain domain NOUN cana-3488 247 24 . . PUNCT cana-3488 248 1 conflicts conflict NOUN cana-3488 248 2 of of ADP cana-3488 248 3 interest interest NOUN cana-3488 248 4 : : PUNCT cana-3488 248 5 the the DET cana-3488 248 6 authors author NOUN cana-3488 248 7 declare declare VERB cana-3488 248 8 no no DET cana-3488 248 9 conflicts conflict NOUN cana-3488 248 10 of of ADP cana-3488 248 11 interest interest NOUN cana-3488 248 12 . . PUNCT cana-3488 249 1 references reference NOUN cana-3488 249 2 [ [ X cana-3488 249 3 1 1 NUM cana-3488 249 4 ] ] X cana-3488 249 5 s.m s.m PROPN cana-3488 249 6 . . PROPN cana-3488 249 7 ulam ulam PROPN cana-3488 249 8 , , PUNCT cana-3488 249 9 problems problem NOUN cana-3488 249 10 in in ADP cana-3488 249 11 modern modern ADJ cana-3488 249 12 mathematics mathematic NOUN cana-3488 249 13 , , PUNCT cana-3488 249 14 science science NOUN cana-3488 249 15 editions edition NOUN cana-3488 249 16 , , PUNCT cana-3488 249 17 wiley wiley NOUN cana-3488 249 18 , , PUNCT cana-3488 249 19 newyork newyork PROPN cana-3488 249 20 , , PUNCT cana-3488 249 21 1964 1964 NUM cana-3488 249 22 . . PUNCT cana-3488 250 1 [ [ X cana-3488 250 2 2 2 NUM cana-3488 250 3 ] ] X cana-3488 250 4 d.h d.h PROPN cana-3488 250 5 . . PROPN cana-3488 250 6 hyers hyer NOUN cana-3488 250 7 , , PUNCT cana-3488 250 8 on on ADP cana-3488 250 9 the the DET cana-3488 250 10 stability stability NOUN cana-3488 250 11 of of ADP cana-3488 250 12 the the DET cana-3488 250 13 linear linear ADJ cana-3488 250 14 functional functional ADJ cana-3488 250 15 equation equation NOUN cana-3488 250 16 , , PUNCT cana-3488 250 17 proc.nat proc.nat PROPN cana-3488 250 18 . . PUNCT cana-3488 251 1 acad.sci acad.sci X cana-3488 251 2 . . PUNCT cana-3488 251 3 ,u.s.a ,u.s.a PROPN cana-3488 251 4 . . PUNCT cana-3488 251 5 ,27 ,27 PUNCT cana-3488 251 6 ( ( PUNCT cana-3488 251 7 1941 1941 NUM cana-3488 251 8 ) ) PUNCT cana-3488 251 9 222 222 NUM cana-3488 251 10 - - SYM cana-3488 251 11 224 224 NUM cana-3488 251 12 . . PUNCT cana-3488 252 1 [ [ X cana-3488 252 2 3 3 NUM cana-3488 252 3 ] ] X cana-3488 252 4 pl pl PROPN cana-3488 252 5 . . PROPN cana-3488 252 6 kannappan kannappan PROPN cana-3488 252 7 , , PUNCT cana-3488 252 8 functional functional ADJ cana-3488 252 9 equations equation NOUN cana-3488 252 10 and and CCONJ cana-3488 252 11 inequalities inequality NOUN cana-3488 252 12 with with ADP cana-3488 252 13 applications application NOUN cana-3488 252 14 , , PUNCT cana-3488 252 15 springer springer NOUN cana-3488 252 16 monographs monograph NOUN cana-3488 252 17 in in ADP cana-3488 252 18 mathematics mathematic NOUN cana-3488 252 19 , , PUNCT cana-3488 252 20 2009 2009 NUM cana-3488 252 21 . . PUNCT cana-3488 253 1 [ [ X cana-3488 253 2 4 4 NUM cana-3488 253 3 ] ] X cana-3488 253 4 j.m j.m PROPN cana-3488 253 5 . . PROPN cana-3488 253 6 rassias rassias PROPN cana-3488 253 7 , , PUNCT cana-3488 253 8 on on ADP cana-3488 253 9 approximately approximately ADV cana-3488 253 10 of of ADP cana-3488 253 11 approximately approximately ADV cana-3488 253 12 linear linear ADJ cana-3488 253 13 mappings mapping NOUN cana-3488 253 14 by by ADP cana-3488 253 15 linear linear PROPN cana-3488 253 16 mappings mapping NOUN cana-3488 253 17 , , PUNCT cana-3488 253 18 j. j. PROPN cana-3488 253 19 funct funct PROPN cana-3488 253 20 . . PUNCT cana-3488 254 1 anal anal PROPN cana-3488 254 2 . . PUNCT cana-3488 255 1 usa usa PROPN cana-3488 255 2 , , PUNCT cana-3488 255 3 46 46 NUM cana-3488 255 4 , , PUNCT cana-3488 255 5 ( ( PUNCT cana-3488 255 6 1982 1982 NUM cana-3488 255 7 ) ) PUNCT cana-3488 255 8 126 126 NUM cana-3488 255 9 - - SYM cana-3488 255 10 130 130 NUM cana-3488 255 11 . . PUNCT cana-3488 256 1 [ [ X cana-3488 256 2 5 5 NUM cana-3488 256 3 ] ] PUNCT cana-3488 256 4 th.m th.m NOUN cana-3488 256 5 . . PUNCT cana-3488 257 1 rassias rassias PROPN cana-3488 257 2 , , PUNCT cana-3488 257 3 on on ADP cana-3488 257 4 the the DET cana-3488 257 5 stability stability NOUN cana-3488 257 6 of of ADP cana-3488 257 7 the the DET cana-3488 257 8 linear linear ADJ cana-3488 257 9 mapping mapping NOUN cana-3488 257 10 in in ADP cana-3488 257 11 banach banach NOUN cana-3488 257 12 spaces space NOUN cana-3488 257 13 , , PUNCT cana-3488 257 14 proc.amer.math proc.amer.math PROPN cana-3488 257 15 . . PUNCT cana-3488 258 1 soc soc PROPN cana-3488 258 2 . . PUNCT cana-3488 258 3 , , PUNCT cana-3488 258 4 72 72 NUM cana-3488 258 5 ( ( PUNCT cana-3488 258 6 1978 1978 NUM cana-3488 258 7 ) ) PUNCT cana-3488 258 8 , , PUNCT cana-3488 258 9 297300 297300 NUM cana-3488 258 10 . . PUNCT cana-3488 259 1 communications communication NOUN cana-3488 259 2 on on ADP cana-3488 259 3 applied apply VERB cana-3488 259 4 nonlinear nonlinear ADJ cana-3488 259 5 analysis analysis NOUN cana-3488 259 6 issn issn NOUN cana-3488 259 7 : : PUNCT cana-3488 259 8 1074 1074 NUM cana-3488 259 9 - - PUNCT cana-3488 259 10 133x 133x NUM cana-3488 259 11 vol vol NOUN cana-3488 259 12 32 32 NUM cana-3488 259 13 no no NOUN cana-3488 259 14 . . PUNCT cana-3488 260 1 7s 7 NOUN cana-3488 260 2 ( ( PUNCT cana-3488 260 3 2025 2025 NUM cana-3488 260 4 ) ) PUNCT cana-3488 260 5 821 821 NUM cana-3488 260 6 https://internationalpubls.com https://internationalpubls.com X cana-3488 261 1 [ [ X cana-3488 261 2 6 6 NUM cana-3488 261 3 ] ] PUNCT cana-3488 261 4 th.m th.m PROPN cana-3488 261 5 . . PUNCT cana-3488 262 1 rassias rassias PROPN cana-3488 262 2 , , PUNCT cana-3488 262 3 functional functional ADJ cana-3488 262 4 equations equation NOUN cana-3488 262 5 , , PUNCT cana-3488 262 6 inequalities inequality NOUN cana-3488 262 7 and and CCONJ cana-3488 262 8 applications application NOUN cana-3488 262 9 , , PUNCT cana-3488 262 10 kluwer kluwer NOUN cana-3488 262 11 acedamic acedamic ADJ cana-3488 262 12 publishers publisher NOUN cana-3488 262 13 , , PUNCT cana-3488 262 14 dordrecht dordrecht PROPN cana-3488 262 15 , , PUNCT cana-3488 262 16 bostan bostan PROPN cana-3488 262 17 london london PROPN cana-3488 262 18 , , PUNCT cana-3488 262 19 2003 2003 NUM cana-3488 262 20 . . PUNCT cana-3488 263 1 [ [ X cana-3488 263 2 7 7 X cana-3488 263 3 ] ] X cana-3488 263 4 p. p. NOUN cana-3488 263 5 gavruta gavruta PROPN cana-3488 263 6 , , PUNCT cana-3488 263 7 a a DET cana-3488 263 8 generalization generalization NOUN cana-3488 263 9 of of ADP cana-3488 263 10 the the DET cana-3488 263 11 hyers hyers PROPN cana-3488 263 12 - - PUNCT cana-3488 263 13 ulam ulam ADJ cana-3488 263 14 - - PUNCT cana-3488 263 15 rassias rassias PROPN cana-3488 263 16 stability stability NOUN cana-3488 263 17 of of ADP cana-3488 263 18 approximately approximately ADV cana-3488 263 19 additive additive ADJ cana-3488 263 20 mappings mapping NOUN cana-3488 263 21 , , PUNCT cana-3488 263 22 j. j. PROPN cana-3488 263 23 math math PROPN cana-3488 263 24 . . PUNCT cana-3488 264 1 anal anal PROPN cana-3488 264 2 . . PUNCT cana-3488 264 3 appl appl PROPN cana-3488 264 4 . . PROPN cana-3488 265 1 , , PUNCT cana-3488 265 2 184 184 NUM cana-3488 265 3 ( ( PUNCT cana-3488 265 4 1994 1994 NUM cana-3488 265 5 ) ) PUNCT cana-3488 265 6 , , PUNCT cana-3488 265 7 431 431 NUM cana-3488 265 8 - - SYM cana-3488 265 9 436 436 NUM cana-3488 265 10 . . PUNCT cana-3488 266 1 [ [ X cana-3488 266 2 8 8 NUM cana-3488 266 3 ] ] X cana-3488 266 4 zadeh zadeh PROPN cana-3488 266 5 , , PUNCT cana-3488 266 6 l. l. PROPN cana-3488 266 7 a. a. PROPN cana-3488 266 8 fuzzy fuzzy PROPN cana-3488 266 9 sets set NOUN cana-3488 266 10 . . PUNCT cana-3488 267 1 information information NOUN cana-3488 267 2 and and CCONJ cana-3488 267 3 control control NOUN cana-3488 267 4 , , PUNCT cana-3488 267 5 1965 1965 NUM cana-3488 267 6 , , PUNCT cana-3488 267 7 8(3 8(3 NUM cana-3488 267 8 ) ) PUNCT cana-3488 267 9 , , PUNCT cana-3488 267 10 338 338 NUM cana-3488 267 11 - - SYM cana-3488 267 12 353 353 NUM cana-3488 267 13 . . PUNCT cana-3488 268 1 [ [ X cana-3488 268 2 9 9 NUM cana-3488 268 3 ] ] PUNCT cana-3488 268 4 atanassov atanassov NOUN cana-3488 268 5 , , PUNCT cana-3488 268 6 k. k. PROPN cana-3488 268 7 intuitionistic intuitionistic ADJ cana-3488 268 8 fuzzy fuzzy ADJ cana-3488 268 9 sets set NOUN cana-3488 268 10 . . PUNCT cana-3488 269 1 fuzzy fuzzy ADJ cana-3488 269 2 sets set NOUN cana-3488 269 3 and and CCONJ cana-3488 269 4 systems system NOUN cana-3488 269 5 , , PUNCT cana-3488 269 6 1986 1986 NUM cana-3488 269 7 20 20 NUM cana-3488 269 8 , , PUNCT cana-3488 269 9 87 87 NUM cana-3488 269 10 - - SYM cana-3488 269 11 96 96 NUM cana-3488 269 12 . . PUNCT cana-3488 270 1 [ [ X cana-3488 270 2 10 10 NUM cana-3488 270 3 ] ] X cana-3488 270 4 atanassov atanassov NOUN cana-3488 270 5 , , PUNCT cana-3488 270 6 k. k. PROPN cana-3488 270 7 t. t. PROPN cana-3488 271 1 more more ADJ cana-3488 271 2 on on ADP cana-3488 271 3 intuitionistic intuitionistic ADJ cana-3488 271 4 fuzzy fuzzy ADJ cana-3488 271 5 sets set NOUN cana-3488 271 6 . . PUNCT cana-3488 272 1 fuzzy fuzzy ADJ cana-3488 272 2 sets set NOUN cana-3488 272 3 and and CCONJ cana-3488 272 4 systems,1989 systems,1989 VERB cana-3488 272 5 , , PUNCT cana-3488 272 6 33(1 33(1 NUM cana-3488 272 7 ) ) PUNCT cana-3488 272 8 , , PUNCT cana-3488 272 9 37 37 NUM cana-3488 272 10 - - SYM cana-3488 272 11 45 45 NUM cana-3488 272 12 . . PUNCT cana-3488 273 1 [ [ X cana-3488 273 2 11 11 NUM cana-3488 273 3 ] ] X cana-3488 273 4 smarandache smarandache NOUN cana-3488 273 5 , , PUNCT cana-3488 273 6 f. f. PROPN cana-3488 273 7 neutrosophy neutrosophy PROPN cana-3488 273 8 : : PUNCT cana-3488 273 9 neutrosophic neutrosophic ADJ cana-3488 273 10 probability probability NOUN cana-3488 273 11 set set NOUN cana-3488 273 12 and and CCONJ cana-3488 273 13 logic logic NOUN cana-3488 273 14 : : PUNCT cana-3488 273 15 analytic analytic ADJ cana-3488 273 16 synthesis synthesis NOUN cana-3488 273 17 and and CCONJ cana-3488 273 18 synthetic synthetic ADJ cana-3488 273 19 analysis analysis NOUN cana-3488 273 20 . . PUNCT cana-3488 274 1 1998 1998 NUM cana-3488 274 2 . . PUNCT cana-3488 275 1 [ [ X cana-3488 275 2 12 12 NUM cana-3488 275 3 ] ] X cana-3488 275 4 smarandache smarandache NOUN cana-3488 275 5 , , PUNCT cana-3488 275 6 f. f. PROPN cana-3488 275 7 a a DET cana-3488 275 8 unifying unifying ADJ cana-3488 275 9 field field NOUN cana-3488 275 10 in in ADP cana-3488 275 11 logics logic NOUN cana-3488 275 12 : : PUNCT cana-3488 275 13 neutrosophic neutrosophic ADJ cana-3488 275 14 logic logic NOUN cana-3488 275 15 . . PUNCT cana-3488 276 1 in in ADP cana-3488 276 2 philosophy philosophy NOUN cana-3488 276 3 , , PUNCT cana-3488 276 4 american american PROPN cana-3488 276 5 research research NOUN cana-3488 276 6 press press NOUN cana-3488 276 7 . . PUNCT cana-3488 277 1 1999,(1 1999,(1 NUM cana-3488 277 2 - - SYM cana-3488 277 3 141 141 NUM cana-3488 277 4 ) ) PUNCT cana-3488 277 5 . . PUNCT cana-3488 278 1 [ [ X cana-3488 278 2 13 13 NUM cana-3488 278 3 ] ] PUNCT cana-3488 278 4 katsaras katsara NOUN cana-3488 278 5 , , PUNCT cana-3488 278 6 a.k a.k PROPN cana-3488 278 7 . . PROPN cana-3488 278 8 fuzzy fuzzy ADJ cana-3488 278 9 topological topological ADJ cana-3488 278 10 vector vector NOUN cana-3488 278 11 spaces space NOUN cana-3488 278 12 . . PUNCT cana-3488 279 1 fuzzy fuzzy ADJ cana-3488 279 2 sets set NOUN cana-3488 279 3 syst syst PROPN cana-3488 279 4 . . PUNCT cana-3488 279 5 1984 1984 NUM cana-3488 279 6 , , PUNCT cana-3488 279 7 12 12 NUM cana-3488 279 8 , , PUNCT cana-3488 279 9 143–154 143–154 NUM cana-3488 279 10 . . PUNCT cana-3488 280 1 [ [ X cana-3488 280 2 14 14 NUM cana-3488 280 3 ] ] X cana-3488 280 4 saadati.r saadati.r NUM cana-3488 280 5 , , PUNCT cana-3488 280 6 park.j.h park.j.h ADJ cana-3488 280 7 , , PUNCT cana-3488 280 8 on on ADP cana-3488 280 9 the the DET cana-3488 280 10 intuitionistic intuitionistic ADJ cana-3488 280 11 fuzzy fuzzy ADJ cana-3488 280 12 topological topological ADJ cana-3488 280 13 spaces space NOUN cana-3488 280 14 , , PUNCT cana-3488 280 15 chaos chaos NOUN cana-3488 280 16 solitons soliton NOUN cana-3488 280 17 fractals fractal VERB cana-3488 280 18 27 27 NUM cana-3488 280 19 , , PUNCT cana-3488 280 20 2006 2006 NUM cana-3488 280 21 , , PUNCT cana-3488 280 22 331–344 331–344 NUM cana-3488 280 23 . . PUNCT cana-3488 281 1 [ [ X cana-3488 281 2 15 15 NUM cana-3488 281 3 ] ] X cana-3488 281 4 bera bera NOUN cana-3488 281 5 . . PUNCT cana-3488 282 1 t t PROPN cana-3488 282 2 , , PUNCT cana-3488 282 3 mahapatra mahapatra PROPN cana-3488 282 4 . . PUNCT cana-3488 283 1 n.k n.k PROPN cana-3488 283 2 , , PUNCT cana-3488 283 3 on on ADP cana-3488 283 4 neutrosophic neutrosophic ADJ cana-3488 283 5 soft soft ADJ cana-3488 283 6 linear linear ADJ cana-3488 283 7 spaces space NOUN cana-3488 283 8 . . PUNCT cana-3488 284 1 fuzzy fuzzy PROPN cana-3488 284 2 inf inf PROPN cana-3488 284 3 . . PUNCT cana-3488 285 1 eng eng PROPN cana-3488 285 2 . . PROPN cana-3488 285 3 9(3 9(3 NUM cana-3488 285 4 ) ) PUNCT cana-3488 285 5 , , PUNCT cana-3488 285 6 2017 2017 NUM cana-3488 285 7 , , PUNCT cana-3488 285 8 299–324 299–324 NUM cana-3488 285 9 . . PUNCT cana-3488 286 1 [ [ X cana-3488 286 2 16 16 NUM cana-3488 286 3 ] ] PUNCT cana-3488 286 4 bera.t bera.t NOUN cana-3488 286 5 , , PUNCT cana-3488 286 6 mahapatra.n.k mahapatra.n.k PROPN cana-3488 286 7 , , PUNCT cana-3488 286 8 neutrosophic neutrosophic ADJ cana-3488 286 9 soft soft ADJ cana-3488 286 10 normed norme VERB cana-3488 286 11 linear linear PROPN cana-3488 286 12 spaces space NOUN cana-3488 286 13 , , PUNCT cana-3488 286 14 neutrosophic neutrosophic ADJ cana-3488 286 15 sets set NOUN cana-3488 286 16 and and CCONJ cana-3488 286 17 system system NOUN cana-3488 286 18 , , PUNCT cana-3488 286 19 23 23 NUM cana-3488 286 20 , , PUNCT cana-3488 286 21 2018 2018 NUM cana-3488 286 22 , , PUNCT cana-3488 286 23 52–71 52–71 NUM cana-3488 286 24 . . PUNCT cana-3488 287 1 [ [ X cana-3488 287 2 17 17 NUM cana-3488 287 3 ] ] X cana-3488 287 4 agboola agboola PROPN cana-3488 287 5 , , PUNCT cana-3488 287 6 a.a.a a.a.a PROPN cana-3488 287 7 , , PUNCT cana-3488 287 8 . . PUNCT cana-3488 287 9 akwu akwu PROPN cana-3488 287 10 , , PUNCT cana-3488 287 11 a.d a.d PROPN cana-3488 287 12 , , PUNCT cana-3488 287 13 . . PUNCT cana-3488 288 1 and and CCONJ cana-3488 288 2 oyebo oyebo NOUN cana-3488 288 3 , , PUNCT cana-3488 288 4 y.t y.t PROPN cana-3488 288 5 , , PUNCT cana-3488 288 6 . . PUNCT cana-3488 289 1 neutrosophic neutrosophic ADJ cana-3488 289 2 groups group NOUN cana-3488 289 3 and and CCONJ cana-3488 289 4 subgroups subgroup NOUN cana-3488 289 5 , , PUNCT cana-3488 289 6 international international PROPN cana-3488 289 7 .j .j PROPN cana-3488 289 8 .math .math PROPN cana-3488 289 9 . . PUNCT cana-3488 290 1 combin combin NOUN cana-3488 290 2 , , PUNCT cana-3488 290 3 vol vol NOUN cana-3488 290 4 . . PROPN cana-3488 291 1 3 3 NUM cana-3488 291 2 , , PUNCT cana-3488 291 3 pp pp ADJ cana-3488 291 4 . . PUNCT cana-3488 292 1 1 1 NUM cana-3488 292 2 - - SYM cana-3488 292 3 9 9 NUM cana-3488 292 4 , , PUNCT cana-3488 292 5 2012 2012 NUM cana-3488 292 6 . . PUNCT cana-3488 293 1 [ [ X cana-3488 293 2 18 18 NUM cana-3488 293 3 ] ] X cana-3488 293 4 agboola agboola PROPN cana-3488 293 5 , , PUNCT cana-3488 293 6 a.a.a a.a.a PROPN cana-3488 293 7 , , PUNCT cana-3488 293 8 . . PUNCT cana-3488 294 1 and and CCONJ cana-3488 294 2 akinleye akinleye NOUN cana-3488 294 3 , , PUNCT cana-3488 294 4 s.a s.a PROPN cana-3488 294 5 , , PUNCT cana-3488 294 6 . . PUNCT cana-3488 295 1 neutrosophic neutrosophic ADJ cana-3488 295 2 vector vector NOUN cana-3488 295 3 spaces space NOUN cana-3488 295 4 , , PUNCT cana-3488 295 5 neutrosophic neutrosophic ADJ cana-3488 295 6 sets set NOUN cana-3488 295 7 and and CCONJ cana-3488 295 8 systems system NOUN cana-3488 295 9 , , PUNCT cana-3488 295 10 vol vol NOUN cana-3488 295 11 . . PROPN cana-3488 295 12 4 4 NUM cana-3488 295 13 , , PUNCT cana-3488 295 14 pp pp ADJ cana-3488 295 15 . . PUNCT cana-3488 295 16 917 917 NUM cana-3488 295 17 , , PUNCT cana-3488 295 18 2014 2014 NUM cana-3488 295 19 . . PUNCT cana-3488 296 1 [ [ X cana-3488 296 2 19 19 NUM cana-3488 296 3 ] ] PUNCT cana-3488 296 4 vasantha vasantha NOUN cana-3488 296 5 , , PUNCT cana-3488 296 6 w.b w.b PROPN cana-3488 296 7 . . PROPN cana-3488 296 8 ; ; PUNCT cana-3488 296 9 smarandache smarandache PROPN cana-3488 296 10 , , PUNCT cana-3488 296 11 f. f. PROPN cana-3488 296 12 neutrosophic neutrosophic PROPN cana-3488 296 13 rings ring NOUN cana-3488 296 14 ; ; PUNCT cana-3488 296 15 hexis hexis ADJ cana-3488 296 16 : : PUNCT cana-3488 296 17 phoenix phoenix PROPN cana-3488 296 18 , , PUNCT cana-3488 296 19 az az PROPN cana-3488 296 20 , , PUNCT cana-3488 296 21 usa usa PROPN cana-3488 296 22 , , PUNCT cana-3488 296 23 2006 2006 NUM cana-3488 296 24 ; ; PUNCT cana-3488 296 25 isbn isbn ADJ cana-3488 296 26 978 978 NUM cana-3488 296 27 - - SYM cana-3488 296 28 1 1 NUM cana-3488 296 29 - - PUNCT cana-3488 296 30 931233 931233 NUM cana-3488 296 31 - - SYM cana-3488 296 32 209 209 NUM cana-3488 296 33 [ [ X cana-3488 296 34 20 20 NUM cana-3488 296 35 ] ] PUNCT cana-3488 296 36 abobala abobala NOUN cana-3488 296 37 , , PUNCT cana-3488 296 38 m m PROPN cana-3488 296 39 , , PUNCT cana-3488 296 40 . . PUNCT cana-3488 297 1 classical classical ADJ cana-3488 297 2 homomorphisms homomorphism NOUN cana-3488 297 3 between between ADP cana-3488 297 4 refined refined ADJ cana-3488 297 5 neutrosophic neutrosophic ADJ cana-3488 297 6 rings ring NOUN cana-3488 297 7 and and CCONJ cana-3488 297 8 neutrosophic neutrosophic ADJ cana-3488 297 9 rings ring NOUN cana-3488 297 10 , , PUNCT cana-3488 297 11 international international ADJ cana-3488 297 12 journal journal NOUN cana-3488 297 13 of of ADP cana-3488 297 14 neutrosophic neutrosophic ADJ cana-3488 297 15 science science NOUN cana-3488 297 16 , , PUNCT cana-3488 297 17 vol vol NOUN cana-3488 297 18 . . PROPN cana-3488 297 19 5 5 NUM cana-3488 297 20 , , PUNCT cana-3488 297 21 pp pp ADJ cana-3488 297 22 . . PUNCT cana-3488 298 1 72 72 NUM cana-3488 298 2 - - SYM cana-3488 298 3 75 75 NUM cana-3488 298 4 , , PUNCT cana-3488 298 5 2020 2020 NUM cana-3488 298 6 . . PUNCT cana-3488 299 1 [ [ X cana-3488 299 2 21 21 NUM cana-3488 299 3 ] ] PUNCT cana-3488 299 4 abobala abobala NOUN cana-3488 299 5 , , PUNCT cana-3488 299 6 m. m. NOUN cana-3488 299 7 , , PUNCT cana-3488 299 8 on on ADP cana-3488 299 9 the the DET cana-3488 299 10 representation representation NOUN cana-3488 299 11 of of ADP cana-3488 299 12 neutrosophic neutrosophic ADJ cana-3488 299 13 matrices matrix NOUN cana-3488 299 14 by by ADP cana-3488 299 15 neutrosophic neutrosophic ADJ cana-3488 299 16 linear linear ADJ cana-3488 299 17 transformations transformation NOUN cana-3488 299 18 , , PUNCT cana-3488 299 19 journal journal NOUN cana-3488 299 20 of of ADP cana-3488 299 21 mathematics mathematic NOUN cana-3488 299 22 , , PUNCT cana-3488 299 23 hindawi hindawi ADJ cana-3488 299 24 , , PUNCT cana-3488 299 25 2021 2021 NUM cana-3488 299 26 . . PUNCT cana-3488 300 1 [ [ X cana-3488 300 2 22 22 NUM cana-3488 300 3 ] ] PUNCT cana-3488 300 4 abobala abobala NOUN cana-3488 300 5 , , PUNCT cana-3488 300 6 m. m. NOUN cana-3488 300 7 , , PUNCT cana-3488 300 8 partial partial ADJ cana-3488 300 9 foundation foundation NOUN cana-3488 300 10 of of ADP cana-3488 300 11 neutrosophic neutrosophic ADJ cana-3488 300 12 number number NOUN cana-3488 300 13 theory theory NOUN cana-3488 300 14 , , PUNCT cana-3488 300 15 neutrosophic neutrosophic ADJ cana-3488 300 16 sets set NOUN cana-3488 300 17 and and CCONJ cana-3488 300 18 systems system NOUN cana-3488 300 19 , , PUNCT cana-3488 300 20 vol vol NOUN cana-3488 300 21 . . PROPN cana-3488 300 22 39 39 NUM cana-3488 300 23 , , PUNCT cana-3488 300 24 2021 2021 NUM cana-3488 300 25 . . PUNCT cana-3488 301 1 [ [ X cana-3488 301 2 23 23 NUM cana-3488 301 3 ] ] SYM cana-3488 301 4 kandasamy kandasamy ADJ cana-3488 301 5 vasantha vasantha NOUN cana-3488 301 6 , , PUNCT cana-3488 301 7 k. k. PROPN cana-3488 301 8 ilanthenral ilanthenral ADJ cana-3488 301 9 , , PUNCT cana-3488 301 10 and and CCONJ cana-3488 301 11 florentin florentin PROPN cana-3488 301 12 smarandache smarandache PROPN cana-3488 301 13 . . PUNCT cana-3488 302 1 neutrosophic neutrosophic ADJ cana-3488 302 2 graphs graph NOUN cana-3488 302 3 : : PUNCT cana-3488 302 4 a a DET cana-3488 302 5 new new ADJ cana-3488 302 6 dimension dimension NOUN cana-3488 302 7 to to AUX cana-3488 302 8 graph graph VERB cana-3488 302 9 theory theory NOUN cana-3488 302 10 . . PUNCT cana-3488 303 1 infinite infinite ADJ cana-3488 303 2 study study NOUN cana-3488 303 3 , , PUNCT cana-3488 303 4 2015 2015 NUM cana-3488 303 5 . . PUNCT cana-3488 304 1 [ [ X cana-3488 304 2 24 24 NUM cana-3488 304 3 ] ] X cana-3488 304 4 smarandache smarandache NOUN cana-3488 304 5 , , PUNCT cana-3488 304 6 florentin florentin PROPN cana-3488 304 7 . . PUNCT cana-3488 304 8 introduction introduction NOUN cana-3488 304 9 to to ADP cana-3488 304 10 neutrosophic neutrosophic ADJ cana-3488 304 11 measure measure NOUN cana-3488 304 12 , , PUNCT cana-3488 304 13 neutrosophic neutrosophic ADJ cana-3488 304 14 integral integral ADJ cana-3488 304 15 , , PUNCT cana-3488 304 16 and and CCONJ cana-3488 304 17 neutrosophic neutrosophic ADJ cana-3488 304 18 probability probability NOUN cana-3488 304 19 . . PUNCT cana-3488 305 1 infinite infinite ADJ cana-3488 305 2 study study NOUN cana-3488 305 3 , , PUNCT cana-3488 305 4 2013 2013 NUM cana-3488 305 5 . . PUNCT cana-3488 306 1 [ [ X cana-3488 306 2 25 25 NUM cana-3488 306 3 ] ] SYM cana-3488 306 4 broumi broumi PROPN cana-3488 306 5 , , PUNCT cana-3488 306 6 s. s. PROPN cana-3488 306 7 ; ; PUNCT cana-3488 306 8 sundareswaran sundareswaran VERB cana-3488 306 9 , , PUNCT cana-3488 306 10 r. r. PROPN cana-3488 306 11 ; ; PUNCT cana-3488 306 12 shanmugapriya shanmugapriya PROPN cana-3488 306 13 , , PUNCT cana-3488 306 14 m. m. NOUN cana-3488 306 15 ; ; PUNCT cana-3488 306 16 singh singh PROPN cana-3488 306 17 , , PUNCT cana-3488 306 18 p.k p.k PROPN cana-3488 306 19 . . PROPN cana-3488 306 20 ; ; PUNCT cana-3488 306 21 voskoglou voskoglou PROPN cana-3488 306 22 , , PUNCT cana-3488 306 23 m. m. NOUN cana-3488 306 24 ; ; PUNCT cana-3488 306 25 talea talea ADJ cana-3488 306 26 , , PUNCT cana-3488 306 27 m. m. NOUN cana-3488 306 28 faculty faculty NOUN cana-3488 306 29 performance performance NOUN cana-3488 306 30 evaluation evaluation NOUN cana-3488 306 31 through through ADP cana-3488 306 32 multi multi ADJ cana-3488 306 33 - - ADJ cana-3488 306 34 criteria criterion NOUN cana-3488 306 35 decision decision NOUN cana-3488 306 36 analysis analysis NOUN cana-3488 306 37 using use VERB cana-3488 306 38 interval interval NOUN cana-3488 306 39 - - PUNCT cana-3488 306 40 valued value VERB cana-3488 306 41 fermatean fermatean NOUN cana-3488 306 42 neutrosophic neutrosophic ADJ cana-3488 306 43 sets set NOUN cana-3488 306 44 . . PUNCT cana-3488 307 1 mathematics mathematic NOUN cana-3488 307 2 , , PUNCT cana-3488 307 3 11 11 NUM cana-3488 307 4 , , PUNCT cana-3488 307 5 3817 3817 NUM cana-3488 307 6 , , PUNCT cana-3488 307 7 2023 2023 NUM cana-3488 307 8 . . PUNCT cana-3488 308 1 [ [ X cana-3488 308 2 26 26 NUM cana-3488 308 3 ] ] X cana-3488 308 4 riaz riaz PROPN cana-3488 308 5 , , PUNCT cana-3488 308 6 m. m. NOUN cana-3488 308 7 ; ; PUNCT cana-3488 308 8 farid farid PROPN cana-3488 308 9 , , PUNCT cana-3488 308 10 h.m.a h.m.a NOUN cana-3488 308 11 . . PUNCT cana-3488 308 12 ; ; PUNCT cana-3488 308 13 antucheviciene antucheviciene PROPN cana-3488 308 14 , , PUNCT cana-3488 308 15 j. j. PROPN cana-3488 308 16 ; ; PUNCT cana-3488 308 17 demir demir PROPN cana-3488 308 18 , , PUNCT cana-3488 308 19 g. g. PROPN cana-3488 308 20 efficient efficient ADJ cana-3488 308 21 decision decision NOUN cana-3488 308 22 making make VERB cana-3488 308 23 for for ADP cana-3488 308 24 sustainable sustainable ADJ cana-3488 308 25 energy energy NOUN cana-3488 308 26 using use VERB cana-3488 308 27 single single ADV cana-3488 308 28 - - PUNCT cana-3488 308 29 valued value VERB cana-3488 308 30 neutrosophic neutrosophic ADJ cana-3488 308 31 prioritized prioritize VERB cana-3488 308 32 interactive interactive ADJ cana-3488 308 33 aggregation aggregation NOUN cana-3488 308 34 operators operator NOUN cana-3488 308 35 . . PUNCT cana-3488 309 1 mathematics mathematic NOUN cana-3488 309 2 , , PUNCT cana-3488 309 3 11 11 NUM cana-3488 309 4 , , PUNCT cana-3488 309 5 2186 2186 NUM cana-3488 309 6 . . PUNCT cana-3488 310 1 2023.[27 2023.[27 NUM cana-3488 310 2 ] ] X cana-3488 311 1 boloΘ™ boloΘ™ PROPN cana-3488 311 2 , , PUNCT cana-3488 311 3 m.-i m.-i PROPN cana-3488 311 4 . . PUNCT cana-3488 311 5 ; ; PUNCT cana-3488 312 1 bradea bradea NOUN cana-3488 312 2 , , PUNCT cana-3488 312 3 i.-a i.-a NOUN cana-3488 312 4 . . PUNCT cana-3488 312 5 ; ; PUNCT cana-3488 312 6 delcea delcea PROPN cana-3488 312 7 , , PUNCT cana-3488 312 8 c. c. PROPN cana-3488 312 9 optimization optimization NOUN cana-3488 312 10 of of ADP cana-3488 312 11 financial financial ADJ cana-3488 312 12 asset asset NOUN cana-3488 312 13 neutrosophic neutrosophic ADJ cana-3488 312 14 portfolios portfolio NOUN cana-3488 312 15 . . PUNCT cana-3488 313 1 mathematics mathematic NOUN cana-3488 313 2 , , PUNCT cana-3488 313 3 9 9 NUM cana-3488 313 4 , , PUNCT cana-3488 313 5 1162 1162 NUM cana-3488 313 6 , , PUNCT cana-3488 313 7 2021 2021 NUM cana-3488 313 8 . . PUNCT cana-3488 314 1 [ [ X cana-3488 314 2 28 28 NUM cana-3488 314 3 ] ] X cana-3488 314 4 zavadskas zavadskas NOUN cana-3488 314 5 , , PUNCT cana-3488 314 6 e.k e.k PROPN cana-3488 314 7 . . PROPN cana-3488 314 8 ; ; PUNCT cana-3488 314 9 bausys bausy NOUN cana-3488 314 10 , , PUNCT cana-3488 314 11 r. r. PROPN cana-3488 314 12 ; ; PUNCT cana-3488 314 13 lescauskiene lescauskiene PROPN cana-3488 314 14 , , PUNCT cana-3488 314 15 i. i. NOUN cana-3488 314 16 ; ; PUNCT cana-3488 314 17 usovaite usovaite PROPN cana-3488 314 18 , , PUNCT cana-3488 314 19 a. a. NOUN cana-3488 314 20 multimoora multimoora PROPN cana-3488 314 21 under under ADP cana-3488 314 22 interval interval NOUN cana-3488 314 23 - - PUNCT cana-3488 314 24 valued value VERB cana-3488 314 25 neutrosophic neutrosophic ADJ cana-3488 314 26 sets set NOUN cana-3488 314 27 as as ADP cana-3488 314 28 the the DET cana-3488 314 29 basis basis NOUN cana-3488 314 30 for for ADP cana-3488 314 31 the the DET cana-3488 314 32 quantitative quantitative ADJ cana-3488 314 33 heuristic heuristic ADJ cana-3488 314 34 evaluation evaluation NOUN cana-3488 314 35 methodology methodology NOUN cana-3488 314 36 hebin hebin NOUN cana-3488 314 37 . . PUNCT cana-3488 315 1 mathematics mathematic NOUN cana-3488 315 2 , , PUNCT cana-3488 315 3 9 9 NUM cana-3488 315 4 , , PUNCT cana-3488 315 5 66 66 NUM cana-3488 315 6 2021 2021 NUM cana-3488 315 7 . . PUNCT cana-3488 316 1 [ [ X cana-3488 316 2 29 29 NUM cana-3488 316 3 ] ] X cana-3488 316 4 belal belal PROPN cana-3488 316 5 amin amin PROPN cana-3488 316 6 , , PUNCT cana-3488 316 7 a. a. NOUN cana-3488 316 8 a. a. NOUN cana-3488 316 9 salama salama PROPN cana-3488 316 10 , , PUNCT cana-3488 316 11 i. i. PROPN cana-3488 316 12 m. m. PROPN cana-3488 316 13 el el PROPN cana-3488 316 14 - - PUNCT cana-3488 316 15 henawy henawy PROPN cana-3488 316 16 , , PUNCT cana-3488 316 17 khaled khaled PROPN cana-3488 316 18 mahfouz mahfouz PROPN cana-3488 316 19 , , PUNCT cana-3488 316 20 mona mona PROPN cana-3488 316 21 g. g. PROPN cana-3488 316 22 gafar gafar PROPN cana-3488 316 23 , , PUNCT cana-3488 316 24 intelligent intelligent ADJ cana-3488 316 25 neutrosophic neutrosophic ADJ cana-3488 316 26 diagnostic diagnostic ADJ cana-3488 316 27 system system NOUN cana-3488 316 28 for for ADP cana-3488 316 29 cardiotocography cardiotocography NOUN cana-3488 316 30 data datum NOUN cana-3488 316 31 , , PUNCT cana-3488 316 32 computational computational ADJ cana-3488 316 33 intelligence intelligence NOUN cana-3488 316 34 and and CCONJ cana-3488 316 35 neuroscience neuroscience NOUN cana-3488 316 36 , , PUNCT cana-3488 316 37 vol vol NOUN cana-3488 316 38 . . NOUN cana-3488 316 39 2021 2021 NUM cana-3488 316 40 , , PUNCT cana-3488 316 41 article article NOUN cana-3488 316 42 i i PROPN cana-3488 316 43 d d PROPN cana-3488 316 44 6656770 6656770 NUM cana-3488 316 45 , , PUNCT cana-3488 316 46 12 12 NUM cana-3488 316 47 pages page NOUN cana-3488 316 48 , , PUNCT cana-3488 316 49 2021 2021 NUM cana-3488 316 50 . . PUNCT cana-3488 317 1 [ [ X cana-3488 317 2 30 30 NUM cana-3488 317 3 ] ] X cana-3488 317 4 sumin sumin PROPN cana-3488 317 5 zhang zhang PROPN cana-3488 317 6 , , PUNCT cana-3488 317 7 jun jun PROPN cana-3488 317 8 ye ye PROPN cana-3488 317 9 , , PUNCT cana-3488 317 10 multiple multiple ADJ cana-3488 317 11 attribute attribute NOUN cana-3488 317 12 group group NOUN cana-3488 317 13 decision decision NOUN cana-3488 317 14 - - PUNCT cana-3488 317 15 making make VERB cana-3488 317 16 models model NOUN cana-3488 317 17 using use VERB cana-3488 317 18 single single ADV cana-3488 317 19 - - PUNCT cana-3488 317 20 valued value VERB cana-3488 317 21 neutrosophic neutrosophic ADJ cana-3488 317 22 and and CCONJ cana-3488 317 23 linguistic linguistic ADJ cana-3488 317 24 neutrosophic neutrosophic ADJ cana-3488 317 25 hybrid hybrid ADJ cana-3488 317 26 element element NOUN cana-3488 317 27 aggregation aggregation NOUN cana-3488 317 28 algorithms algorithm NOUN cana-3488 317 29 , , PUNCT cana-3488 317 30 journal journal NOUN cana-3488 317 31 of of ADP cana-3488 317 32 mathematics mathematic NOUN cana-3488 317 33 , , PUNCT cana-3488 317 34 vol vol NOUN cana-3488 317 35 . . NOUN cana-3488 317 36 2022 2022 NUM cana-3488 317 37 , , PUNCT cana-3488 317 38 article article NOUN cana-3488 317 39 i i PROPN cana-3488 317 40 d d PROPN cana-3488 317 41 1021280 1021280 NUM cana-3488 317 42 , , PUNCT cana-3488 317 43 10 10 NUM cana-3488 317 44 pages page NOUN cana-3488 317 45 , , PUNCT cana-3488 317 46 2022 2022 NUM cana-3488 317 47 . . PUNCT cana-3488 318 1 [ [ X cana-3488 318 2 31 31 NUM cana-3488 318 3 ] ] PUNCT cana-3488 318 4 zhongliang zhongliang PROPN cana-3488 318 5 fu fu PROPN cana-3488 318 6 , , PUNCT cana-3488 318 7 chunping chunpe VERB cana-3488 318 8 liu liu PROPN cana-3488 318 9 , , PUNCT cana-3488 318 10 shengyi shengyi PROPN cana-3488 318 11 ruan ruan PROPN cana-3488 318 12 , , PUNCT cana-3488 318 13 kun kun PROPN cana-3488 318 14 chen chen PROPN cana-3488 318 15 , , PUNCT cana-3488 318 16 design design NOUN cana-3488 318 17 of of ADP cana-3488 318 18 neutrosophic neutrosophic ADJ cana-3488 318 19 self self NOUN cana-3488 318 20 - - PUNCT cana-3488 318 21 tuning tune VERB cana-3488 318 22 pid pid NOUN cana-3488 318 23 controller controller NOUN cana-3488 318 24 for for ADP cana-3488 318 25 ac ac PROPN cana-3488 318 26 permanent permanent ADJ cana-3488 318 27 magnet magnet NOUN cana-3488 318 28 synchronous synchronous ADJ cana-3488 318 29 motor motor NOUN cana-3488 318 30 based base VERB cana-3488 318 31 on on ADP cana-3488 318 32 neutrosophic neutrosophic ADJ cana-3488 318 33 theory theory NOUN cana-3488 318 34 , , PUNCT cana-3488 318 35 mathematical mathematical ADJ cana-3488 318 36 problems problem NOUN cana-3488 318 37 in in ADP cana-3488 318 38 engineering engineering NOUN cana-3488 318 39 , , PUNCT cana-3488 318 40 vol vol NOUN cana-3488 318 41 . . PROPN cana-3488 318 42 2021 2021 NUM cana-3488 318 43 , , PUNCT cana-3488 318 44 article article NOUN cana-3488 318 45 i i PROPN cana-3488 318 46 d d PROPN cana-3488 318 47 5548184 5548184 NUM cana-3488 318 48 , , PUNCT cana-3488 318 49 14 14 NUM cana-3488 318 50 pages page NOUN cana-3488 318 51 , , PUNCT cana-3488 318 52 2021 2021 NUM cana-3488 318 53 . . PUNCT cana-3488 319 1 [ [ X cana-3488 319 2 32 32 NUM cana-3488 319 3 ] ] X cana-3488 319 4 pasupathi pasupathi NOUN cana-3488 319 5 , , PUNCT cana-3488 319 6 a. a. NOUN cana-3488 319 7 ; ; PUNCT cana-3488 319 8 konsalraj konsalraj PROPN cana-3488 319 9 , , PUNCT cana-3488 319 10 j. j. PROPN cana-3488 319 11 ; ; PUNCT cana-3488 319 12 fatima fatima PROPN cana-3488 319 13 , , PUNCT cana-3488 319 14 n. n. NOUN cana-3488 319 15 ; ; PUNCT cana-3488 319 16 velusamy velusamy PROPN cana-3488 319 17 , , PUNCT cana-3488 319 18 v. v. PROPN cana-3488 319 19 ; ; PUNCT cana-3488 319 20 mlaiki mlaiki PROPN cana-3488 319 21 , , PUNCT cana-3488 319 22 n. n. NOUN cana-3488 319 23 ; ; PUNCT cana-3488 319 24 souayah souayah NOUN cana-3488 319 25 , , PUNCT cana-3488 319 26 n. n. NOUN cana-3488 319 27 direct direct ADJ cana-3488 319 28 and and CCONJ cana-3488 319 29 fixed fix VERB cana-3488 319 30 - - PUNCT cana-3488 319 31 point point NOUN cana-3488 319 32 stability stability NOUN cana-3488 319 33 – – PUNCT cana-3488 319 34 instability instability NOUN cana-3488 319 35 of of ADP cana-3488 319 36 additive additive ADJ cana-3488 319 37 functional functional ADJ cana-3488 319 38 equation equation NOUN cana-3488 319 39 in in ADP cana-3488 319 40 banach banach NOUN cana-3488 319 41 and and CCONJ cana-3488 319 42 quasi quasi ADJ cana-3488 319 43 - - ADJ cana-3488 319 44 beta beta ADJ cana-3488 319 45 normed norme VERB cana-3488 319 46 spaces space NOUN cana-3488 319 47 . . PUNCT cana-3488 320 1 symmetry symmetry NOUN cana-3488 320 2 2022 2022 NUM cana-3488 320 3 , , PUNCT cana-3488 320 4 14 14 NUM cana-3488 320 5 , , PUNCT cana-3488 320 6 1700 1700 NUM cana-3488 320 7 . . PUNCT cana-3488 321 1 [ [ X cana-3488 321 2 33 33 NUM cana-3488 321 3 ] ] SYM cana-3488 321 4 agilan agilan ADJ cana-3488 321 5 , , PUNCT cana-3488 321 6 p. p. NOUN cana-3488 321 7 ; ; PUNCT cana-3488 321 8 julietraja julietraja PROPN cana-3488 321 9 , , PUNCT cana-3488 321 10 k. k. PROPN cana-3488 321 11 ; ; PUNCT cana-3488 321 12 mlaiki mlaiki PROPN cana-3488 321 13 , , PUNCT cana-3488 321 14 n. n. NOUN cana-3488 321 15 ; ; PUNCT cana-3488 321 16 mukheimer mukheimer NOUN cana-3488 321 17 , , PUNCT cana-3488 321 18 a. a. NOUN cana-3488 321 19 intuitionistic intuitionistic ADJ cana-3488 321 20 fuzzy fuzzy ADJ cana-3488 321 21 stability stability NOUN cana-3488 321 22 of of ADP cana-3488 321 23 an an DET cana-3488 321 24 euler euler ADJ cana-3488 321 25 – – PUNCT cana-3488 321 26 lagrange lagrange ADJ cana-3488 321 27 symmetry symmetry NOUN cana-3488 321 28 additive additive ADJ cana-3488 321 29 functional functional ADJ cana-3488 321 30 equation equation NOUN cana-3488 321 31 via via ADP cana-3488 321 32 direct direct ADJ cana-3488 321 33 and and CCONJ cana-3488 321 34 fixed fix VERB cana-3488 321 35 point point NOUN cana-3488 321 36 technique technique NOUN cana-3488 321 37 ( ( PUNCT cana-3488 321 38 fpt fpt PROPN cana-3488 321 39 ) ) PUNCT cana-3488 321 40 . . PUNCT cana-3488 322 1 symmetry symmetry PROPN cana-3488 322 2 2022 2022 NUM cana-3488 322 3 , , PUNCT cana-3488 322 4 14 14 NUM cana-3488 322 5 , , PUNCT cana-3488 322 6 2454 2454 NUM cana-3488 322 7 . . PUNCT cana-3488 323 1 communications communication NOUN cana-3488 323 2 on on ADP cana-3488 323 3 applied apply VERB cana-3488 323 4 nonlinear nonlinear ADJ cana-3488 323 5 analysis analysis NOUN cana-3488 323 6 issn issn NOUN cana-3488 323 7 : : PUNCT cana-3488 323 8 1074 1074 NUM cana-3488 323 9 - - PUNCT cana-3488 323 10 133x 133x NUM cana-3488 323 11 vol vol NOUN cana-3488 323 12 32 32 NUM cana-3488 323 13 no no NOUN cana-3488 323 14 . . PUNCT cana-3488 324 1 7s 7 NOUN cana-3488 324 2 ( ( PUNCT cana-3488 324 3 2025 2025 NUM cana-3488 324 4 ) ) PUNCT cana-3488 324 5 822 822 NUM cana-3488 324 6 https://internationalpubls.com https://internationalpubls.com X cana-3488 325 1 [ [ X cana-3488 325 2 34 34 NUM cana-3488 325 3 ] ] SYM cana-3488 325 4 agilan agilan ADJ cana-3488 325 5 , , PUNCT cana-3488 325 6 p. p. NOUN cana-3488 325 7 ; ; PUNCT cana-3488 325 8 almazah almazah PROPN cana-3488 325 9 , , PUNCT cana-3488 325 10 m.a.a m.a.a PROPN cana-3488 325 11 . . PUNCT cana-3488 325 12 ; ; PUNCT cana-3488 326 1 julietraja julietraja PROPN cana-3488 326 2 , , PUNCT cana-3488 326 3 k. k. PROPN cana-3488 326 4 ; ; PUNCT cana-3488 326 5 alsinai alsinai PROPN cana-3488 326 6 , , PUNCT cana-3488 326 7 a. a. NOUN cana-3488 326 8 classical classical NOUN cana-3488 326 9 and and CCONJ cana-3488 326 10 fixed fix VERB cana-3488 326 11 point point NOUN cana-3488 326 12 approach approach NOUN cana-3488 326 13 to to ADP cana-3488 326 14 the the DET cana-3488 326 15 stability stability NOUN cana-3488 326 16 analysis analysis NOUN cana-3488 326 17 of of ADP cana-3488 326 18 a a DET cana-3488 326 19 bilateral bilateral ADJ cana-3488 326 20 symmetric symmetric ADJ cana-3488 326 21 additive additive ADJ cana-3488 326 22 functional functional ADJ cana-3488 326 23 equation equation NOUN cana-3488 326 24 in in ADP cana-3488 326 25 fuzzy fuzzy ADJ cana-3488 326 26 and and CCONJ cana-3488 326 27 random random ADJ cana-3488 326 28 normed normed ADJ cana-3488 326 29 spaces space NOUN cana-3488 326 30 . . PUNCT cana-3488 327 1 mathematics mathematic NOUN cana-3488 327 2 2023 2023 NUM cana-3488 327 3 , , PUNCT cana-3488 327 4 11 11 NUM cana-3488 327 5 , , PUNCT cana-3488 327 6 681 681 NUM cana-3488 327 7 . . PUNCT cana-3488 328 1 [ [ X cana-3488 328 2 35 35 NUM cana-3488 328 3 ] ] SYM cana-3488 328 4 agilan agilan ADJ cana-3488 328 5 , , PUNCT cana-3488 328 6 p. p. NOUN cana-3488 328 7 ; ; PUNCT cana-3488 328 8 julietraja julietraja PROPN cana-3488 328 9 . . PUNCT cana-3488 328 10 ; ; PUNCT cana-3488 328 11 k k PROPN cana-3488 328 12 almazah almazah PROPN cana-3488 328 13 , , PUNCT cana-3488 328 14 m.a.a m.a.a PROPN cana-3488 328 15 . . PUNCT cana-3488 328 16 ; ; PUNCT cana-3488 329 1 alsinai alsinai PROPN cana-3488 329 2 , , PUNCT cana-3488 329 3 a. a. NOUN cana-3488 329 4 stability stability NOUN cana-3488 329 5 analysis analysis NOUN cana-3488 329 6 of of ADP cana-3488 329 7 a a DET cana-3488 329 8 new new ADJ cana-3488 329 9 class class NOUN cana-3488 329 10 of of ADP cana-3488 329 11 series series NOUN cana-3488 329 12 type type NOUN cana-3488 329 13 additive additive ADJ cana-3488 329 14 functional functional ADJ cana-3488 329 15 equation equation NOUN cana-3488 329 16 in in ADP cana-3488 329 17 banach banach NOUN cana-3488 329 18 spaces space NOUN cana-3488 329 19 : : PUNCT cana-3488 329 20 direct direct ADJ cana-3488 329 21 and and CCONJ cana-3488 329 22 fixed fix VERB cana-3488 329 23 point point NOUN cana-3488 329 24 techniques technique NOUN cana-3488 329 25 mathematics mathematic NOUN cana-3488 329 26 2023 2023 NUM cana-3488 329 27 , , PUNCT cana-3488 329 28 11 11 NUM cana-3488 329 29 , , PUNCT cana-3488 329 30 887 887 NUM cana-3488 329 31 . . PUNCT cana-3488 329 32 doi.org/10.3390/math11040887 doi.org/10.3390/math11040887 VERB cana-3488 329 33 . . PUNCT cana-3488 330 1 [ [ X cana-3488 330 2 36 36 NUM cana-3488 330 3 ] ] PUNCT cana-3488 330 4 aloqaily aloqaily ADV cana-3488 330 5 , , PUNCT cana-3488 330 6 ahmad ahmad PROPN cana-3488 330 7 , , PUNCT cana-3488 330 8 p. p. PROPN cana-3488 330 9 agilan agilan PROPN cana-3488 330 10 , , PUNCT cana-3488 330 11 k. k. PROPN cana-3488 330 12 julietraja julietraja PROPN cana-3488 330 13 , , PUNCT cana-3488 330 14 s. s. PROPN cana-3488 330 15 annadurai annadurai PROPN cana-3488 330 16 , , PUNCT cana-3488 330 17 and and CCONJ cana-3488 330 18 nabil nabil PROPN cana-3488 330 19 mlaiki mlaiki PROPN cana-3488 330 20 . . PUNCT cana-3488 331 1 a a DET cana-3488 331 2 novel novel ADJ cana-3488 331 3 stability stability NOUN cana-3488 331 4 analysis analysis NOUN cana-3488 331 5 of of ADP cana-3488 331 6 functional functional ADJ cana-3488 331 7 equation equation NOUN cana-3488 331 8 in in ADP cana-3488 331 9 neutrosophic neutrosophic ADJ cana-3488 331 10 normed norme VERB cana-3488 331 11 spaces space NOUN cana-3488 331 12 . . PUNCT cana-3488 332 1 boundary boundary ADJ cana-3488 332 2 value value NOUN cana-3488 332 3 problems problem NOUN cana-3488 332 4 , , PUNCT cana-3488 332 5 2024 2024 NUM cana-3488 332 6 , , PUNCT cana-3488 332 7 no no INTJ cana-3488 332 8 . . NOUN cana-3488 332 9 1 1 NUM cana-3488 332 10 ( ( PUNCT cana-3488 332 11 2024 2024 NUM cana-3488 332 12 ): ): PUNCT cana-3488 332 13 47 47 NUM cana-3488 332 14 . . PUNCT cana-3488 333 1 [ [ X cana-3488 333 2 37 37 NUM cana-3488 333 3 ] ] PUNCT cana-3488 333 4 p. p. NOUN cana-3488 333 5 agilan agilan PROPN cana-3488 333 6 , , PUNCT cana-3488 333 7 k. k. PROPN cana-3488 333 8 julietraja julietraja PROPN cana-3488 333 9 , , PUNCT cana-3488 333 10 b. b. PROPN cana-3488 333 11 kanimozhi kanimozhi PROPN cana-3488 333 12 , , PUNCT cana-3488 333 13 a. a. PROPN cana-3488 333 14 alsinai alsinai PROPN cana-3488 333 15 , , PUNCT cana-3488 333 16 hyers hyer NOUN cana-3488 333 17 stability stability NOUN cana-3488 333 18 of of ADP cana-3488 333 19 aqc aqc PROPN cana-3488 333 20 functional functional ADJ cana-3488 333 21 equation equation NOUN cana-3488 333 22 , , PUNCT cana-3488 333 23 dyn dyn NOUN cana-3488 333 24 . . PUNCT cana-3488 334 1 contin contin NOUN cana-3488 334 2 . . PUNCT cana-3488 335 1 discr discr PROPN cana-3488 335 2 . . PUNCT cana-3488 336 1 impuls impuls PROPN cana-3488 336 2 . . PUNCT cana-3488 337 1 syst syst PROPN cana-3488 337 2 . . PUNCT cana-3488 337 3 ser ser PROPN cana-3488 337 4 . . PUNCT cana-3488 338 1 b b X cana-3488 338 2 : : PUNCT cana-3488 338 3 appl appl PROPN cana-3488 338 4 . . PROPN cana-3488 338 5 algor algor PROPN cana-3488 338 6 . . PUNCT cana-3488 339 1 31 31 NUM cana-3488 339 2 ( ( PUNCT cana-3488 339 3 2024 2024 NUM cana-3488 339 4 ) ) PUNCT cana-3488 339 5 , , PUNCT cana-3488 339 6 63–75 63–75 NUM cana-3488 339 7 . . PUNCT cana-3488 340 1 [ [ X cana-3488 340 2 38 38 NUM cana-3488 340 3 ] ] PUNCT cana-3488 340 4 p. p. NOUN cana-3488 340 5 agilan agilan PROPN cana-3488 340 6 , , PUNCT cana-3488 340 7 k. k. PROPN cana-3488 340 8 julietraja julietraja PROPN cana-3488 340 9 , , PUNCT cana-3488 340 10 sarah sarah PROPN cana-3488 340 11 aljohani aljohani PROPN cana-3488 340 12 , , PUNCT cana-3488 340 13 nabil nabil PROPN cana-3488 340 14 mlaiki mlaiki PROPN cana-3488 340 15 , , PUNCT cana-3488 340 16 generalised generalise VERB cana-3488 340 17 ulam ulam PROPN cana-3488 340 18 - - PUNCT cana-3488 340 19 hyers hyer NOUN cana-3488 340 20 stability stability NOUN cana-3488 340 21 analysis analysis NOUN cana-3488 340 22 for for ADP cana-3488 340 23 system system NOUN cana-3488 340 24 of of ADP cana-3488 340 25 additive additive ADJ cana-3488 340 26 functional functional ADJ cana-3488 340 27 equation equation NOUN cana-3488 340 28 in in ADP cana-3488 340 29 fuzzy fuzzy ADJ cana-3488 340 30 and and CCONJ cana-3488 340 31 random random ADJ cana-3488 340 32 normed normed ADJ cana-3488 340 33 spaces space NOUN cana-3488 340 34 : : PUNCT cana-3488 340 35 direct direct ADJ cana-3488 340 36 and and CCONJ cana-3488 340 37 fixed fix VERB cana-3488 340 38 point point NOUN cana-3488 340 39 approach approach NOUN cana-3488 340 40 , , PUNCT cana-3488 340 41 int int NOUN cana-3488 340 42 . . PUNCT cana-3488 341 1 j. j. PROPN cana-3488 341 2 anal anal PROPN cana-3488 341 3 . . PUNCT cana-3488 342 1 appl appl PROPN cana-3488 342 2 . . PROPN cana-3488 342 3 , , PUNCT cana-3488 342 4 22 22 NUM cana-3488 342 5 ( ( PUNCT cana-3488 342 6 2024 2024 NUM cana-3488 342 7 ) ) PUNCT cana-3488 342 8 , , PUNCT cana-3488 342 9 201 201 NUM cana-3488 342 10 . . PUNCT