id sid tid token lemma pos cana-3839 1 1 communications communication NOUN cana-3839 1 2 on on ADP cana-3839 1 3 applied apply VERB cana-3839 1 4 nonlinear nonlinear ADJ cana-3839 1 5 analysis analysis NOUN cana-3839 1 6 issn issn NOUN cana-3839 1 7 : : PUNCT cana-3839 1 8 1074 1074 NUM cana-3839 1 9 - - PUNCT cana-3839 1 10 133x 133x NUM cana-3839 1 11 vol vol NOUN cana-3839 1 12 32 32 NUM cana-3839 1 13 no no NOUN cana-3839 1 14 . . PUNCT cana-3839 2 1 9s 9s NUM cana-3839 2 2 ( ( PUNCT cana-3839 2 3 2025 2025 NUM cana-3839 2 4 ) ) PUNCT cana-3839 2 5 77 77 NUM cana-3839 3 1 https://internationalpubls.com https://internationalpubls.com X cana-3839 3 2 results result VERB cana-3839 3 3 on on ADP cana-3839 3 4 fixed fix VERB cana-3839 3 5 points point NOUN cana-3839 3 6 in in ADP cana-3839 3 7 vector vector NOUN cana-3839 3 8 valued value VERB cana-3839 4 1 ๐‘ฎ-metric ๐‘ฎ-metric PROPN cana-3839 4 2 spaces space VERB cana-3839 4 3 pooja pooja PROPN cana-3839 4 4 yadav1 yadav1 PROPN cana-3839 4 5 , , PUNCT cana-3839 4 6 mamta mamta PROPN cana-3839 4 7 kamra2 kamra2 PROPN cana-3839 5 1 [ [ X cana-3839 5 2 1 1 NUM cana-3839 5 3 , , PUNCT cana-3839 5 4 2 2 NUM cana-3839 5 5 ] ] PUNCT cana-3839 5 6 " " PUNCT cana-3839 5 7 department department NOUN cana-3839 5 8 of of ADP cana-3839 5 9 mathematics mathematic NOUN cana-3839 5 10 , , PUNCT cana-3839 5 11 indira indira PROPN cana-3839 5 12 gandhi gandhi PROPN cana-3839 5 13 university university PROPN cana-3839 5 14 meerpur(rewari meerpur(rewari PROPN cana-3839 5 15 ) ) PUNCT cana-3839 5 16 , , PUNCT cana-3839 5 17 haryana-122502 haryana-122502 NOUN cana-3839 5 18 , , PUNCT cana-3839 5 19 india india PROPN cana-3839 5 20 . . PUNCT cana-3839 5 21 " " PUNCT cana-3839 6 1 article article NOUN cana-3839 6 2 history history NOUN cana-3839 6 3 : : PUNCT cana-3839 6 4 received receive VERB cana-3839 6 5 : : PUNCT cana-3839 6 6 12 12 NUM cana-3839 6 7 - - SYM cana-3839 6 8 11 11 NUM cana-3839 6 9 - - PUNCT cana-3839 6 10 2024 2024 NUM cana-3839 6 11 revised:24 revised:24 X cana-3839 6 12 - - PUNCT cana-3839 6 13 12 12 NUM cana-3839 6 14 - - PUNCT cana-3839 6 15 2024 2024 NUM cana-3839 6 16 accepted:09 accepted:09 NOUN cana-3839 6 17 - - PUNCT cana-3839 6 18 01 01 NUM cana-3839 6 19 - - PUNCT cana-3839 6 20 2025 2025 NUM cana-3839 6 21 abstract abstract NOUN cana-3839 6 22 : : PUNCT cana-3839 6 23 introduction introduction NOUN cana-3839 6 24 : : PUNCT cana-3839 6 25 fixed fix VERB cana-3839 6 26 point point NOUN cana-3839 6 27 theory theory NOUN cana-3839 6 28 furnishes furnish VERB cana-3839 6 29 a a DET cana-3839 6 30 robust robust ADJ cana-3839 6 31 framework framework NOUN cana-3839 6 32 in in ADP cana-3839 6 33 mathematics mathematic NOUN cana-3839 6 34 for for ADP cana-3839 6 35 examining examine VERB cana-3839 6 36 the the DET cana-3839 6 37 function function NOUN cana-3839 6 38 and and CCONJ cana-3839 6 39 system system NOUN cana-3839 6 40 behavior behavior NOUN cana-3839 6 41 by by ADP cana-3839 6 42 centering center VERB cana-3839 6 43 on on ADP cana-3839 6 44 fixed fix VERB cana-3839 6 45 points point NOUN cana-3839 6 46 . . PUNCT cana-3839 7 1 kirk kirk PROPN cana-3839 7 2 et et PROPN cana-3839 7 3 al al PROPN cana-3839 7 4 . . PROPN cana-3839 7 5 brought bring VERB cana-3839 7 6 forth forth ADP cana-3839 7 7 the the DET cana-3839 7 8 concept concept NOUN cana-3839 7 9 of of ADP cana-3839 7 10 cyclic cyclic ADJ cana-3839 7 11 transformation transformation NOUN cana-3839 7 12 in in ADP cana-3839 7 13 2003 2003 NUM cana-3839 7 14 . . PUNCT cana-3839 8 1 cyclic cyclic ADJ cana-3839 8 2 contraction contraction NOUN cana-3839 8 3 broadens broaden VERB cana-3839 8 4 the the DET cana-3839 8 5 standard standard ADJ cana-3839 8 6 contraction contraction NOUN cana-3839 8 7 transformations transformation NOUN cana-3839 8 8 in in ADP cana-3839 8 9 metric metric ADJ cana-3839 8 10 space space NOUN cana-3839 8 11 . . PUNCT cana-3839 9 1 by by ADP cana-3839 9 2 relaxing relax VERB cana-3839 9 3 the the DET cana-3839 9 4 requirement requirement NOUN cana-3839 9 5 of of ADP cana-3839 9 6 the the DET cana-3839 9 7 triangle triangle NOUN cana-3839 9 8 inequality inequality NOUN cana-3839 9 9 , , PUNCT cana-3839 9 10 ๐บ-metric ๐บ-metric ADJ cana-3839 9 11 space space NOUN cana-3839 9 12 extends extend VERB cana-3839 9 13 the the DET cana-3839 9 14 scope scope NOUN cana-3839 9 15 of of ADP cana-3839 9 16 metric metric ADJ cana-3839 9 17 spaces space NOUN cana-3839 9 18 . . PUNCT cana-3839 10 1 vector vector NOUN cana-3839 10 2 ๐บ-metric ๐บ-metric ADJ cana-3839 10 3 space space NOUN cana-3839 10 4 is be AUX cana-3839 10 5 gms gm VERB cana-3839 10 6 in in ADP cana-3839 10 7 which which PRON cana-3839 10 8 the the DET cana-3839 10 9 metric metric NOUN cana-3839 10 10 is be AUX cana-3839 10 11 lattice lattice NOUN cana-3839 10 12 valued value VERB cana-3839 10 13 . . PUNCT cana-3839 11 1 objectives objective NOUN cana-3839 11 2 : : PUNCT cana-3839 11 3 we we PRON cana-3839 11 4 give give VERB cana-3839 11 5 the the DET cana-3839 11 6 concept concept NOUN cana-3839 11 7 of of ADP cana-3839 11 8 a a DET cana-3839 11 9 vector vector NOUN cana-3839 11 10 ๐บ ๐บ NOUN cana-3839 11 11 metric metric ADJ cana-3839 11 12 space space NOUN cana-3839 11 13 and and CCONJ cana-3839 11 14 define define VERB cana-3839 11 15 some some DET cana-3839 11 16 related relate VERB cana-3839 11 17 concepts concept NOUN cana-3839 11 18 . . PUNCT cana-3839 12 1 we we PRON cana-3839 12 2 establish establish VERB cana-3839 12 3 some some DET cana-3839 12 4 fixed fix VERB cana-3839 12 5 point point NOUN cana-3839 12 6 results result NOUN cana-3839 12 7 on on ADP cana-3839 12 8 vector vector NOUN cana-3839 12 9 ๐บ-metric ๐บ-metric ADJ cana-3839 12 10 space space NOUN cana-3839 12 11 by by ADP cana-3839 12 12 using use VERB cana-3839 12 13 cyclic cyclic ADJ cana-3839 12 14 contraction contraction NOUN cana-3839 12 15 . . PUNCT cana-3839 13 1 to to PART cana-3839 13 2 underscore underscore VERB cana-3839 13 3 the the DET cana-3839 13 4 practical practical ADJ cana-3839 13 5 implications implication NOUN cana-3839 13 6 and and CCONJ cana-3839 13 7 importance importance NOUN cana-3839 13 8 of of ADP cana-3839 13 9 our our PRON cana-3839 13 10 findings finding NOUN cana-3839 13 11 , , PUNCT cana-3839 13 12 we we PRON cana-3839 13 13 offer offer VERB cana-3839 13 14 a a DET cana-3839 13 15 range range NOUN cana-3839 13 16 of of ADP cana-3839 13 17 examples example NOUN cana-3839 13 18 and and CCONJ cana-3839 13 19 corollaries corollary NOUN cana-3839 13 20 . . PUNCT cana-3839 14 1 methods method NOUN cana-3839 14 2 : : PUNCT cana-3839 14 3 this this DET cana-3839 14 4 research research NOUN cana-3839 14 5 paper paper NOUN cana-3839 14 6 outlines outline VERB cana-3839 14 7 fixed fix VERB cana-3839 14 8 point point NOUN cana-3839 14 9 theorems theorem NOUN cana-3839 14 10 for for ADP cana-3839 14 11 selftransformations selftransformation NOUN cana-3839 14 12 in in ADP cana-3839 14 13 vector vector NOUN cana-3839 14 14 ๐บ-metric ๐บ-metric ADJ cana-3839 14 15 space space NOUN cana-3839 14 16 , , PUNCT cana-3839 14 17 with with ADP cana-3839 14 18 the the DET cana-3839 14 19 help help NOUN cana-3839 14 20 of of ADP cana-3839 14 21 cyclic cyclic ADJ cana-3839 14 22 contraction contraction NOUN cana-3839 14 23 . . PUNCT cana-3839 15 1 conclusions conclusion NOUN cana-3839 15 2 : : PUNCT cana-3839 15 3 within within ADP cana-3839 15 4 this this DET cana-3839 15 5 manuscript manuscript NOUN cana-3839 15 6 , , PUNCT cana-3839 15 7 we we PRON cana-3839 15 8 illustrate illustrate VERB cana-3839 15 9 fixed fix VERB cana-3839 15 10 point point NOUN cana-3839 15 11 theorems theorem NOUN cana-3839 15 12 for for ADP cana-3839 15 13 selftransformations selftransformation NOUN cana-3839 15 14 in in ADP cana-3839 15 15 v v NOUN cana-3839 15 16 - - PUNCT cana-3839 15 17 complete complete ADJ cana-3839 15 18 vector vector NOUN cana-3839 15 19 ๐บ-metric ๐บ-metric ADJ cana-3839 15 20 space space NOUN cana-3839 15 21 by by ADP cana-3839 15 22 utilizing utilize VERB cana-3839 15 23 cyclic cyclic ADJ cana-3839 15 24 contraction contraction NOUN cana-3839 15 25 . . PUNCT cana-3839 16 1 these these DET cana-3839 16 2 outcomes outcome NOUN cana-3839 16 3 are be AUX cana-3839 16 4 expected expect VERB cana-3839 16 5 to to PART cana-3839 16 6 inspire inspire VERB cana-3839 16 7 researchers researcher NOUN cana-3839 16 8 to to PART cana-3839 16 9 explore explore VERB cana-3839 16 10 problem problem NOUN cana-3839 16 11 - - PUNCT cana-3839 16 12 solving solve VERB cana-3839 16 13 opportunities opportunity NOUN cana-3839 16 14 in in ADP cana-3839 16 15 diverse diverse ADJ cana-3839 16 16 areas area NOUN cana-3839 16 17 such such ADJ cana-3839 16 18 as as ADP cana-3839 16 19 differential differential ADJ cana-3839 16 20 equations equation NOUN cana-3839 16 21 and and CCONJ cana-3839 16 22 functional functional ADJ cana-3839 16 23 analysis analysis NOUN cana-3839 16 24 . . PUNCT cana-3839 17 1 keywords keyword NOUN cana-3839 17 2 : : PUNCT cana-3839 17 3 cyclic cyclic ADJ cana-3839 17 4 contraction contraction NOUN cana-3839 17 5 , , PUNCT cana-3839 17 6 vector vector NOUN cana-3839 17 7 ๐บ-metric ๐บ-metric ADJ cana-3839 17 8 space space NOUN cana-3839 17 9 , , PUNCT cana-3839 17 10 vector vector NOUN cana-3839 17 11 lattice lattice PROPN cana-3839 17 12 . . PUNCT cana-3839 18 1 subject subject PROPN cana-3839 18 2 classification:(2010 classification:(2010 PROPN cana-3839 18 3 ) ) PUNCT cana-3839 18 4 47h10 47h10 NUM cana-3839 18 5 , , PUNCT cana-3839 18 6 47h07 47h07 NOUN cana-3839 18 7 1 1 X cana-3839 18 8 . . PUNCT cana-3839 19 1 introduction introduction NOUN cana-3839 19 2 . . PUNCT cana-3839 20 1 fixed fix VERB cana-3839 20 2 point point NOUN cana-3839 20 3 theory theory NOUN cana-3839 20 4 furnishes furnish VERB cana-3839 20 5 a a DET cana-3839 20 6 robust robust ADJ cana-3839 20 7 framework framework NOUN cana-3839 20 8 in in ADP cana-3839 20 9 mathematics mathematic NOUN cana-3839 20 10 for for ADP cana-3839 20 11 examining examine VERB cana-3839 20 12 the the DET cana-3839 20 13 function function NOUN cana-3839 20 14 and and CCONJ cana-3839 20 15 system system NOUN cana-3839 20 16 behavior behavior NOUN cana-3839 20 17 by by ADP cana-3839 20 18 centering center VERB cana-3839 20 19 on on ADP cana-3839 20 20 fixed fix VERB cana-3839 20 21 points point NOUN cana-3839 20 22 . . PUNCT cana-3839 21 1 kirk kirk PROPN cana-3839 21 2 et et PROPN cana-3839 21 3 al al PROPN cana-3839 21 4 . . PUNCT cana-3839 21 5 error error PROPN cana-3839 21 6 ! ! PUNCT cana-3839 22 1 reference reference NOUN cana-3839 22 2 source source NOUN cana-3839 22 3 not not PART cana-3839 22 4 found find VERB cana-3839 22 5 . . PUNCT cana-3839 23 1 brought bring VERB cana-3839 23 2 forth forth ADP cana-3839 23 3 the the DET cana-3839 23 4 concept concept NOUN cana-3839 23 5 of of ADP cana-3839 23 6 cyclic cyclic ADJ cana-3839 23 7 transformation transformation NOUN cana-3839 23 8 in in ADP cana-3839 23 9 2003 2003 NUM cana-3839 23 10 . . PUNCT cana-3839 24 1 cyclic cyclic PROPN cana-3839 24 2 contraction(cc contraction(cc PROPN cana-3839 24 3 ) ) PUNCT cana-3839 24 4 broadens broaden VERB cana-3839 24 5 the the DET cana-3839 24 6 standard standard ADJ cana-3839 24 7 contraction contraction NOUN cana-3839 24 8 transformations transformation NOUN cana-3839 24 9 in in ADP cana-3839 24 10 metric metric ADJ cana-3839 24 11 space space NOUN cana-3839 24 12 . . PUNCT cana-3839 25 1 it it PRON cana-3839 25 2 is be AUX cana-3839 25 3 utilized utilize VERB cana-3839 25 4 for for ADP cana-3839 25 5 exploring explore VERB cana-3839 25 6 transformations transformation NOUN cana-3839 25 7 and and CCONJ cana-3839 25 8 fixed fix VERB cana-3839 25 9 points.the points.the DET cana-3839 25 10 exploration exploration NOUN cana-3839 25 11 of of ADP cana-3839 25 12 fixed fix VERB cana-3839 25 13 point point NOUN cana-3839 25 14 theorems(fpt theorems(fpt ADJ cana-3839 25 15 ) ) PUNCT cana-3839 25 16 for for SCONJ cana-3839 25 17 cyclic cyclic ADJ cana-3839 25 18 transformations transformation NOUN cana-3839 25 19 have have AUX cana-3839 25 20 been be AUX cana-3839 25 21 widely widely ADV cana-3839 25 22 studied study VERB cana-3839 25 23 ( ( PUNCT cana-3839 25 24 defined define VERB cana-3839 25 25 by by ADP cana-3839 25 26 error error NOUN cana-3839 25 27 ! ! PUNCT cana-3839 26 1 reference reference NOUN cana-3839 26 2 source source NOUN cana-3839 26 3 not not PART cana-3839 26 4 found find VERB cana-3839 26 5 . . PUNCT cana-3839 27 1 , , PUNCT cana-3839 27 2 error error NOUN cana-3839 27 3 ! ! PUNCT cana-3839 28 1 reference reference NOUN cana-3839 28 2 source source NOUN cana-3839 28 3 not not PART cana-3839 28 4 found find VERB cana-3839 28 5 . . PUNCT cana-3839 29 1 , , PUNCT cana-3839 29 2 error error NOUN cana-3839 29 3 ! ! PUNCT cana-3839 30 1 reference reference NOUN cana-3839 30 2 source source NOUN cana-3839 30 3 not not PART cana-3839 30 4 found find VERB cana-3839 30 5 . . PUNCT cana-3839 31 1 , , PUNCT cana-3839 31 2 error error NOUN cana-3839 31 3 ! ! PUNCT cana-3839 32 1 reference reference NOUN cana-3839 32 2 source source NOUN cana-3839 32 3 not not PART cana-3839 32 4 found find VERB cana-3839 32 5 . . PUNCT cana-3839 32 6 ) ) PUNCT cana-3839 32 7 . . PUNCT cana-3839 33 1 by by ADP cana-3839 33 2 relaxing relax VERB cana-3839 33 3 the the DET cana-3839 33 4 requirement requirement NOUN cana-3839 33 5 of of ADP cana-3839 33 6 the the DET cana-3839 33 7 triangle triangle NOUN cana-3839 33 8 inequality inequality NOUN cana-3839 33 9 , , PUNCT cana-3839 33 10 ๐บ-metric ๐บ-metric ADJ cana-3839 33 11 space(gms space(gms NOUN cana-3839 33 12 ) ) PUNCT cana-3839 33 13 extends extend VERB cana-3839 33 14 the the DET cana-3839 33 15 scope scope NOUN cana-3839 33 16 of of ADP cana-3839 33 17 metric metric ADJ cana-3839 33 18 1 1 NUM cana-3839 33 19 poojayadav.math.rs@igu.ac.in poojayadav.math.rs@igu.ac.in NOUN cana-3839 33 20 1 1 NUM cana-3839 33 21 mkhaneja15@gmail.com mkhaneja15@gmail.com NOUN cana-3839 33 22 communications communication NOUN cana-3839 33 23 on on ADP cana-3839 33 24 applied apply VERB cana-3839 33 25 nonlinear nonlinear ADJ cana-3839 33 26 analysis analysis NOUN cana-3839 33 27 issn issn NOUN cana-3839 33 28 : : PUNCT cana-3839 33 29 1074 1074 NUM cana-3839 33 30 - - PUNCT cana-3839 33 31 133x 133x NUM cana-3839 33 32 vol vol NOUN cana-3839 33 33 32 32 NUM cana-3839 33 34 no no NOUN cana-3839 33 35 . . PUNCT cana-3839 34 1 9s 9s NUM cana-3839 34 2 ( ( PUNCT cana-3839 34 3 2025 2025 NUM cana-3839 34 4 ) ) PUNCT cana-3839 34 5 78 78 NUM cana-3839 35 1 https://internationalpubls.com https://internationalpubls.com X cana-3839 35 2 spaces space VERB cana-3839 35 3 . . PUNCT cana-3839 36 1 vector vector NOUN cana-3839 36 2 ๐บ-metric ๐บ-metric ADJ cana-3839 36 3 space(vgms space(vgms PROPN cana-3839 36 4 ) ) PUNCT cana-3839 36 5 is be AUX cana-3839 36 6 gms gm VERB cana-3839 36 7 in in ADP cana-3839 36 8 which which PRON cana-3839 36 9 the the DET cana-3839 36 10 metric metric NOUN cana-3839 36 11 is be AUX cana-3839 36 12 lattice lattice NOUN cana-3839 36 13 valued value VERB cana-3839 36 14 . . PUNCT cana-3839 37 1 this this DET cana-3839 37 2 research research NOUN cana-3839 37 3 paper paper NOUN cana-3839 37 4 outlines outline VERB cana-3839 37 5 fpt fpt PROPN cana-3839 37 6 for for ADP cana-3839 37 7 self self NOUN cana-3839 37 8 - - PUNCT cana-3839 37 9 transformations transformation NOUN cana-3839 37 10 in in ADP cana-3839 37 11 vgms vgms NOUN cana-3839 37 12 , , PUNCT cana-3839 37 13 with with ADP cana-3839 37 14 the the DET cana-3839 37 15 help help NOUN cana-3839 37 16 of of ADP cana-3839 37 17 cc cc PROPN cana-3839 37 18 . . PUNCT cana-3839 38 1 we we PRON cana-3839 38 2 lay lie VERB cana-3839 38 3 out out ADP cana-3839 38 4 a a DET cana-3839 38 5 series series NOUN cana-3839 38 6 of of ADP cana-3839 38 7 definitions definition NOUN cana-3839 38 8 and and CCONJ cana-3839 38 9 examples example NOUN cana-3839 38 10 that that PRON cana-3839 38 11 will will AUX cana-3839 38 12 be be AUX cana-3839 38 13 applicable applicable ADJ cana-3839 38 14 in in ADP cana-3839 38 15 the the DET cana-3839 38 16 forthcoming forthcoming ADJ cana-3839 38 17 portion portion NOUN cana-3839 38 18 . . PUNCT cana-3839 39 1 for for ADP cana-3839 39 2 an an DET cana-3839 39 3 additional additional ADJ cana-3839 39 4 comprehensive comprehensive ADJ cana-3839 39 5 analysis analysis NOUN cana-3839 39 6 of of ADP cana-3839 39 7 the the DET cana-3839 39 8 results result NOUN cana-3839 39 9 regarding regard VERB cana-3839 39 10 the the DET cana-3839 39 11 vector vector NOUN cana-3839 39 12 lattice lattice NOUN cana-3839 39 13 and and CCONJ cana-3839 39 14 gms gms NOUN cana-3839 39 15 , , PUNCT cana-3839 39 16 we we PRON cana-3839 39 17 may may AUX cana-3839 39 18 refer refer VERB cana-3839 39 19 to to ADP cana-3839 39 20 ( ( PUNCT cana-3839 39 21 error error NOUN cana-3839 39 22 ! ! PUNCT cana-3839 40 1 reference reference NOUN cana-3839 40 2 source source NOUN cana-3839 40 3 not not PART cana-3839 40 4 found find VERB cana-3839 40 5 . . PUNCT cana-3839 41 1 , , PUNCT cana-3839 41 2 error error NOUN cana-3839 41 3 ! ! PUNCT cana-3839 42 1 reference reference NOUN cana-3839 42 2 source source NOUN cana-3839 42 3 not not PART cana-3839 42 4 found find VERB cana-3839 42 5 . . PUNCT cana-3839 43 1 , , PUNCT cana-3839 43 2 error error NOUN cana-3839 43 3 ! ! PUNCT cana-3839 44 1 reference reference NOUN cana-3839 44 2 source source NOUN cana-3839 44 3 not not PART cana-3839 44 4 found find VERB cana-3839 44 5 . . PUNCT cana-3839 45 1 , , PUNCT cana-3839 45 2 error error NOUN cana-3839 45 3 ! ! PUNCT cana-3839 46 1 reference reference NOUN cana-3839 46 2 source source NOUN cana-3839 46 3 not not PART cana-3839 46 4 found find VERB cana-3839 46 5 . . PUNCT cana-3839 47 1 , , PUNCT cana-3839 47 2 error error NOUN cana-3839 47 3 ! ! PUNCT cana-3839 48 1 reference reference NOUN cana-3839 48 2 source source NOUN cana-3839 48 3 not not PART cana-3839 48 4 found find VERB cana-3839 48 5 . . PUNCT cana-3839 49 1 , , PUNCT cana-3839 49 2 error error NOUN cana-3839 49 3 ! ! PUNCT cana-3839 50 1 reference reference NOUN cana-3839 50 2 source source NOUN cana-3839 50 3 not not PART cana-3839 50 4 found find VERB cana-3839 50 5 . . PUNCT cana-3839 51 1 , , PUNCT cana-3839 51 2 error error NOUN cana-3839 51 3 ! ! PUNCT cana-3839 52 1 reference reference NOUN cana-3839 52 2 source source NOUN cana-3839 52 3 not not PART cana-3839 52 4 found find VERB cana-3839 52 5 . . PUNCT cana-3839 53 1 , , PUNCT cana-3839 53 2 error error NOUN cana-3839 53 3 ! ! PUNCT cana-3839 54 1 reference reference NOUN cana-3839 54 2 source source NOUN cana-3839 54 3 not not PART cana-3839 54 4 found find VERB cana-3839 54 5 . . PUNCT cana-3839 55 1 , , PUNCT cana-3839 55 2 error error NOUN cana-3839 55 3 ! ! PUNCT cana-3839 56 1 reference reference NOUN cana-3839 56 2 source source NOUN cana-3839 56 3 not not PART cana-3839 56 4 found find VERB cana-3839 56 5 . . PUNCT cana-3839 57 1 , , PUNCT cana-3839 57 2 error error NOUN cana-3839 57 3 ! ! PUNCT cana-3839 58 1 reference reference NOUN cana-3839 58 2 source source NOUN cana-3839 58 3 not not PART cana-3839 58 4 found find VERB cana-3839 58 5 . . PUNCT cana-3839 59 1 , , PUNCT cana-3839 59 2 error error NOUN cana-3839 59 3 ! ! PUNCT cana-3839 60 1 reference reference NOUN cana-3839 60 2 source source NOUN cana-3839 60 3 not not PART cana-3839 60 4 found find VERB cana-3839 60 5 . . PUNCT cana-3839 61 1 , , PUNCT cana-3839 61 2 error error NOUN cana-3839 61 3 ! ! PUNCT cana-3839 62 1 reference reference NOUN cana-3839 62 2 source source NOUN cana-3839 62 3 not not PART cana-3839 62 4 found find VERB cana-3839 62 5 . . PUNCT cana-3839 63 1 , , PUNCT cana-3839 63 2 error error NOUN cana-3839 63 3 ! ! PUNCT cana-3839 64 1 reference reference NOUN cana-3839 64 2 source source NOUN cana-3839 64 3 not not PART cana-3839 64 4 found find VERB cana-3839 64 5 . . PUNCT cana-3839 64 6 ) ) PUNCT cana-3839 64 7 . . PUNCT cana-3839 65 1 definition definition NOUN cana-3839 65 2 1.1[5 1.1[5 NUM cana-3839 65 3 ] ] PUNCT cana-3839 65 4 let let VERB cana-3839 65 5 โ„œ โ„œ PROPN cana-3839 65 6 be be AUX cana-3839 65 7 a a DET cana-3839 65 8 non non ADJ cana-3839 65 9 - - ADJ cana-3839 65 10 null null ADJ cana-3839 65 11 set set NOUN cana-3839 65 12 and and CCONJ cana-3839 65 13 consider consider VERB cana-3839 65 14 { { PUNCT cana-3839 65 15 ๐œ’โ„}๐œ‡=1 ๐œ’โ„}๐œ‡=1 NOUN cana-3839 65 16 ๐‘š ๐‘š PROPN cana-3839 65 17 as as ADP cana-3839 65 18 a a DET cana-3839 65 19 collection collection NOUN cana-3839 65 20 of of ADP cana-3839 65 21 non non ADJ cana-3839 65 22 - - ADJ cana-3839 65 23 void void ADJ cana-3839 65 24 subsets subset NOUN cana-3839 65 25 of of ADP cana-3839 65 26 โ„œ โ„œ PROPN cana-3839 65 27 with with ADP cana-3839 65 28 ๐œ‡ ๐œ‡ ADP cana-3839 65 29 = = NOUN cana-3839 65 30 โˆชโ„=1 โˆชโ„=1 X cana-3839 65 31 ๐‘š ๐‘š X cana-3839 65 32 ๐œ’๐‘—. ๐œ’๐‘—. ADP cana-3839 65 33 a a DET cana-3839 65 34 transformation transformation NOUN cana-3839 65 35 โˆ โˆ PROPN cana-3839 65 36 : : PUNCT cana-3839 65 37 ๐œ‡ ๐œ‡ X cana-3839 65 38 โ†’ โ†’ X cana-3839 65 39 ๐œ‡ ๐œ‡ X cana-3839 65 40 is be AUX cana-3839 65 41 termed term VERB cana-3839 65 42 a a DET cana-3839 65 43 cyclic cyclic ADJ cana-3839 65 44 transformation transformation NOUN cana-3839 65 45 if if SCONJ cana-3839 65 46 โˆ(๐œ’โ„ โˆ(๐œ’โ„ NUM cana-3839 65 47 ) ) PUNCT cana-3839 65 48 โŠ† โŠ† NUM cana-3839 65 49 ๐œ’โ„+1 ๐œ’โ„+1 NOUN cana-3839 65 50 โ„ โ„ NOUN cana-3839 65 51 = = SYM cana-3839 65 52 1,2 1,2 NUM cana-3839 65 53 , , PUNCT cana-3839 65 54 โ€ฆ โ€ฆ PUNCT cana-3839 65 55 , , PUNCT cana-3839 65 56 ๐‘š ๐‘š NOUN cana-3839 65 57 , , PUNCT cana-3839 65 58 ๐‘คโ„Ž๐‘’๐‘Ÿ๐‘’ ๐‘คโ„Ž๐‘’๐‘Ÿ๐‘’ ADJ cana-3839 65 59 ๐œ’๐‘š+1 ๐œ’๐‘š+1 NUM cana-3839 65 60 = = SYM cana-3839 65 61 ๐œ’1 ๐œ’1 PROPN cana-3839 65 62 . . PUNCT cana-3839 66 1 definition1.2error definition1.2error NOUN cana-3839 66 2 ! ! PROPN cana-3839 66 3 reference reference NOUN cana-3839 66 4 source source NOUN cana-3839 66 5 not not PART cana-3839 66 6 found find VERB cana-3839 66 7 . . PUNCT cana-3839 67 1 let let VERB cana-3839 67 2 โ„œ โ„œ PROPN cana-3839 67 3 be be AUX cana-3839 67 4 a a DET cana-3839 67 5 non non ADJ cana-3839 67 6 - - ADJ cana-3839 67 7 null null ADJ cana-3839 67 8 set set NOUN cana-3839 67 9 and and CCONJ cana-3839 67 10 ๐‘‰ ๐‘‰ PROPN cana-3839 67 11 be be VERB cana-3839 67 12 a a DET cana-3839 67 13 vector vector NOUN cana-3839 67 14 lattice lattice NOUN cana-3839 67 15 , , PUNCT cana-3839 67 16 consider consider VERB cana-3839 67 17 a a DET cana-3839 67 18 function function NOUN cana-3839 67 19 ๐บ ๐บ NOUN cana-3839 67 20 : : PUNCT cana-3839 67 21 โ„œ โ„œ PROPN cana-3839 67 22 ร— ร— NOUN cana-3839 67 23 โ„œ โ„œ PROPN cana-3839 67 24 ร— ร— NOUN cana-3839 67 25 โ„œ โ„œ NOUN cana-3839 67 26 โ†’ โ†’ SYM cana-3839 67 27 ๐‘‰ ๐‘‰ PROPN cana-3839 67 28 that that PRON cana-3839 67 29 obeys obey VERB cana-3839 67 30 the the DET cana-3839 67 31 properties property NOUN cana-3839 67 32 mentioned mention VERB cana-3839 67 33 below below ADV cana-3839 67 34 : : PUNCT cana-3839 67 35 ( ( PUNCT cana-3839 67 36 i i NOUN cana-3839 67 37 ) ) PUNCT cana-3839 67 38 ๐บ( ๐บ( NOUN cana-3839 67 39 โ™ญ โ™ญ PROPN cana-3839 67 40 1 1 NUM cana-3839 67 41 , , PUNCT cana-3839 67 42 โ™ญ โ™ญ PROPN cana-3839 67 43 2 2 NUM cana-3839 67 44 , , PUNCT cana-3839 67 45 โ™ญ โ™ญ PROPN cana-3839 67 46 3 3 NUM cana-3839 67 47 ) ) PUNCT cana-3839 67 48 = = SYM cana-3839 67 49 0 0 PUNCT cana-3839 68 1 if if SCONJ cana-3839 68 2 โ™ญ โ™ญ PROPN cana-3839 68 3 1 1 NUM cana-3839 68 4 = = SYM cana-3839 68 5 โ™ญ โ™ญ PROPN cana-3839 68 6 2 2 X cana-3839 68 7 = = SYM cana-3839 68 8 โ™ญ โ™ญ PROPN cana-3839 68 9 3 3 NUM cana-3839 68 10 , , PUNCT cana-3839 68 11 ( ( PUNCT cana-3839 68 12 ii ii NOUN cana-3839 68 13 ) ) PUNCT cana-3839 68 14 0 0 PUNCT cana-3839 68 15 < < X cana-3839 68 16 ๐บ( ๐บ( X cana-3839 68 17 โ™ญ โ™ญ PROPN cana-3839 68 18 1 1 NUM cana-3839 68 19 , , PUNCT cana-3839 68 20 โ™ญ โ™ญ PROPN cana-3839 68 21 1 1 NUM cana-3839 68 22 , , PUNCT cana-3839 68 23 โ™ญ โ™ญ PROPN cana-3839 68 24 2 2 NUM cana-3839 68 25 ) ) PUNCT cana-3839 68 26 โˆ€ โˆ€ NOUN cana-3839 69 1 โ™ญ โ™ญ PROPN cana-3839 69 2 1 1 NUM cana-3839 69 3 , , PUNCT cana-3839 69 4 โ™ญ โ™ญ PROPN cana-3839 69 5 2 2 NUM cana-3839 69 6 โˆˆ โˆˆ PROPN cana-3839 69 7 โ„œ โ„œ PROPN cana-3839 69 8 with with ADP cana-3839 69 9 โ™ญ โ™ญ PROPN cana-3839 69 10 1 1 NUM cana-3839 69 11 โ‰  โ‰  PROPN cana-3839 69 12 โ™ญ โ™ญ PROPN cana-3839 69 13 2 2 NUM cana-3839 69 14 , , PUNCT cana-3839 69 15 ( ( PUNCT cana-3839 69 16 iii iii NOUN cana-3839 69 17 ) ) PUNCT cana-3839 69 18 ๐บ( ๐บ( NOUN cana-3839 69 19 โ™ญ โ™ญ PROPN cana-3839 69 20 1 1 NUM cana-3839 69 21 , , PUNCT cana-3839 69 22 โ™ญ โ™ญ PROPN cana-3839 69 23 1 1 NUM cana-3839 69 24 , , PUNCT cana-3839 69 25 โ™ญ โ™ญ PROPN cana-3839 69 26 2 2 X cana-3839 69 27 ) ) PUNCT cana-3839 69 28 โชฏ โชฏ NOUN cana-3839 69 29 ๐บ( ๐บ( X cana-3839 69 30 โ™ญ โ™ญ PROPN cana-3839 69 31 1 1 NUM cana-3839 69 32 , , PUNCT cana-3839 69 33 โ™ญ โ™ญ PROPN cana-3839 69 34 2 2 NUM cana-3839 69 35 , , PUNCT cana-3839 69 36 โ™ญ โ™ญ PROPN cana-3839 69 37 3 3 NUM cana-3839 69 38 ) ) PUNCT cana-3839 69 39 โˆ€ โˆ€ NOUN cana-3839 70 1 โ™ญ โ™ญ PROPN cana-3839 70 2 1 1 NUM cana-3839 70 3 , , PUNCT cana-3839 70 4 โ™ญ โ™ญ PROPN cana-3839 70 5 2 2 NUM cana-3839 70 6 , , PUNCT cana-3839 70 7 โ™ญ โ™ญ PROPN cana-3839 70 8 3 3 NUM cana-3839 70 9 โˆˆ โˆˆ PROPN cana-3839 70 10 โ„œ โ„œ PROPN cana-3839 70 11 with with ADP cana-3839 70 12 โ™ญ โ™ญ PROPN cana-3839 70 13 3 3 NUM cana-3839 70 14 โ‰  โ‰  PROPN cana-3839 70 15 โ™ญ โ™ญ PROPN cana-3839 70 16 2 2 NUM cana-3839 70 17 , , PUNCT cana-3839 70 18 ( ( PUNCT cana-3839 70 19 iv iv X cana-3839 70 20 ) ) PUNCT cana-3839 70 21 ๐บ( ๐บ( NOUN cana-3839 70 22 โ™ญ โ™ญ PROPN cana-3839 70 23 1 1 NUM cana-3839 70 24 , , PUNCT cana-3839 70 25 โ™ญ โ™ญ PROPN cana-3839 70 26 2 2 NUM cana-3839 70 27 , , PUNCT cana-3839 70 28 โ™ญ โ™ญ PROPN cana-3839 70 29 3 3 NUM cana-3839 70 30 ) ) PUNCT cana-3839 70 31 = = NOUN cana-3839 70 32 ๐บ( ๐บ( NOUN cana-3839 70 33 โ™ญ โ™ญ PROPN cana-3839 70 34 1 1 NUM cana-3839 70 35 , , PUNCT cana-3839 70 36 โ™ญ โ™ญ PROPN cana-3839 70 37 3 3 NUM cana-3839 70 38 , , PUNCT cana-3839 70 39 โ™ญ โ™ญ PROPN cana-3839 70 40 2 2 NUM cana-3839 70 41 ) ) PUNCT cana-3839 70 42 = = NOUN cana-3839 70 43 ๐บ( ๐บ( NOUN cana-3839 70 44 โ™ญ โ™ญ PROPN cana-3839 70 45 2 2 NUM cana-3839 70 46 , , PUNCT cana-3839 70 47 โ™ญ โ™ญ PROPN cana-3839 70 48 3 3 NUM cana-3839 70 49 , , PUNCT cana-3839 70 50 โ™ญ โ™ญ PROPN cana-3839 70 51 1 1 NUM cana-3839 70 52 ) ) PUNCT cana-3839 70 53 = = SYM cana-3839 70 54 โ‹ฏ โ‹ฏ PROPN cana-3839 70 55 ( ( PUNCT cana-3839 70 56 tri tri ADJ cana-3839 70 57 - - ADJ cana-3839 70 58 variable variable ADJ cana-3839 70 59 symmetry symmetry NOUN cana-3839 70 60 ) ) PUNCT cana-3839 70 61 ( ( PUNCT cana-3839 70 62 ๐‘ฃ)๐บ( ๐‘ฃ)๐บ( NOUN cana-3839 70 63 โ™ญ โ™ญ NOUN cana-3839 70 64 1 1 NUM cana-3839 70 65 , , PUNCT cana-3839 70 66 โ™ญ โ™ญ PROPN cana-3839 70 67 2 2 NUM cana-3839 70 68 , , PUNCT cana-3839 70 69 โ™ญ โ™ญ PROPN cana-3839 70 70 3 3 X cana-3839 70 71 ) ) PUNCT cana-3839 70 72 โชฏ โชฏ NOUN cana-3839 70 73 ๐บ( ๐บ( X cana-3839 70 74 โ™ญ โ™ญ PROPN cana-3839 70 75 1 1 NUM cana-3839 70 76 , , PUNCT cana-3839 70 77 ๐œŽ ๐œŽ PROPN cana-3839 70 78 , , PUNCT cana-3839 70 79 ๐œŽ ๐œŽ PROPN cana-3839 70 80 ) ) PUNCT cana-3839 70 81 + + SYM cana-3839 70 82 ๐บ(๐œŽ ๐บ(๐œŽ PROPN cana-3839 70 83 , , PUNCT cana-3839 70 84 โ™ญ โ™ญ PROPN cana-3839 70 85 2 2 NUM cana-3839 70 86 , , PUNCT cana-3839 70 87 โ™ญ โ™ญ PROPN cana-3839 70 88 3 3 NUM cana-3839 70 89 ) ) PUNCT cana-3839 70 90 โˆ€ โˆ€ NOUN cana-3839 71 1 โ™ญ โ™ญ PROPN cana-3839 71 2 1 1 NUM cana-3839 71 3 , , PUNCT cana-3839 71 4 โ™ญ โ™ญ PROPN cana-3839 71 5 2 2 NUM cana-3839 71 6 , , PUNCT cana-3839 71 7 โ™ญ โ™ญ PROPN cana-3839 71 8 3 3 NUM cana-3839 71 9 , , PUNCT cana-3839 71 10 ๐œŽ ๐œŽ PROPN cana-3839 71 11 โˆˆ โˆˆ PROPN cana-3839 71 12 โ„œ. โ„œ. VERB cana-3839 71 13 the the DET cana-3839 71 14 triplet triplet NOUN cana-3839 71 15 ( ( PUNCT cana-3839 71 16 โ„œ โ„œ PROPN cana-3839 71 17 , , PUNCT cana-3839 71 18 ๐บ ๐บ PROPN cana-3839 71 19 , , PUNCT cana-3839 71 20 ๐‘‰ ๐‘‰ PROPN cana-3839 71 21 ) ) PUNCT cana-3839 71 22 is be AUX cana-3839 71 23 defined define VERB cana-3839 71 24 as as ADP cana-3839 71 25 vector vector NOUN cana-3839 71 26 ๐บ-metric ๐บ-metric ADJ cana-3839 71 27 space(vgms space(vgms PROPN cana-3839 71 28 ) ) PUNCT cana-3839 71 29 . . PUNCT cana-3839 72 1 definition definition NOUN cana-3839 72 2 1.3error 1.3error NUM cana-3839 72 3 ! ! PUNCT cana-3839 73 1 reference reference NOUN cana-3839 73 2 source source NOUN cana-3839 73 3 not not PART cana-3839 73 4 found find VERB cana-3839 73 5 . . PUNCT cana-3839 74 1 a a DET cana-3839 74 2 vgms vgms NOUN cana-3839 74 3 ( ( PUNCT cana-3839 74 4 โ„œ โ„œ PROPN cana-3839 74 5 , , PUNCT cana-3839 74 6 ๐บ ๐บ PROPN cana-3839 74 7 , , PUNCT cana-3839 74 8 ๐‘‰ ๐‘‰ PROPN cana-3839 74 9 ) ) PUNCT cana-3839 74 10 is be AUX cana-3839 74 11 called call VERB cana-3839 74 12 symmetric symmetric ADJ cana-3839 74 13 vgms vgms NOUN cana-3839 74 14 if if SCONJ cana-3839 74 15 ๐บ(๐œ ๐บ(๐œ PROPN cana-3839 74 16 , , PUNCT cana-3839 74 17 โ™ญ โ™ญ PROPN cana-3839 74 18 , , PUNCT cana-3839 74 19 โ™ญ โ™ญ PROPN cana-3839 74 20 ) ) PUNCT cana-3839 75 1 = = SYM cana-3839 75 2 ๐บ ๐บ PROPN cana-3839 75 3 ( ( PUNCT cana-3839 75 4 โ™ญ โ™ญ PROPN cana-3839 75 5 , , PUNCT cana-3839 75 6 ๐œ ๐œ PROPN cana-3839 75 7 , , PUNCT cana-3839 75 8 ๐œ ๐œ NOUN cana-3839 75 9 ) ) PUNCT cana-3839 75 10 โˆ€๐œ โˆ€๐œ NOUN cana-3839 75 11 , , PUNCT cana-3839 75 12 โ™ญ โ™ญ PROPN cana-3839 75 13 โˆˆ โˆˆ PROPN cana-3839 75 14 โ„œ. โ„œ. PROPN cana-3839 75 15 example example NOUN cana-3839 75 16 1.4 1.4 NUM cana-3839 75 17 let let VERB cana-3839 75 18 โ„œ โ„œ PROPN cana-3839 75 19 be be AUX cana-3839 75 20 a a DET cana-3839 75 21 non non ADJ cana-3839 75 22 - - ADJ cana-3839 75 23 null null ADJ cana-3839 75 24 set set NOUN cana-3839 75 25 and and CCONJ cana-3839 75 26 take take VERB cana-3839 75 27 ๐‘‰ ๐‘‰ PROPN cana-3839 75 28 as as ADP cana-3839 75 29 a a DET cana-3839 75 30 vector vector NOUN cana-3839 75 31 lattice lattice NOUN cana-3839 75 32 . . PUNCT cana-3839 76 1 then then ADV cana-3839 76 2 the the DET cana-3839 76 3 function function PROPN cana-3839 76 4 ๐บ ๐บ NOUN cana-3839 76 5 : : PUNCT cana-3839 76 6 โ„œ โ„œ PROPN cana-3839 76 7 ร— ร— NOUN cana-3839 76 8 โ„œ โ„œ PROPN cana-3839 76 9 ร— ร— NOUN cana-3839 76 10 โ„œ โ„œ NOUN cana-3839 76 11 โ†’ โ†’ PUNCT cana-3839 76 12 ๐‘‰ ๐‘‰ PROPN cana-3839 76 13 is be AUX cana-3839 76 14 determined determine VERB cana-3839 76 15 by by ADP cana-3839 76 16 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 76 17 , , PUNCT cana-3839 76 18 โ™ญ โ™ญ INTJ cana-3839 76 19 , , PUNCT cana-3839 76 20 ๐œ ๐œ PROPN cana-3839 76 21 ) ) PUNCT cana-3839 76 22 = = SYM cana-3839 76 23 1 1 NUM cana-3839 76 24 2๐‘› 2๐‘› NUM cana-3839 76 25 ( ( PUNCT cana-3839 76 26 |๐œ‡ |๐œ‡ NOUN cana-3839 76 27 โˆ’ โˆ’ PROPN cana-3839 77 1 โ™ญ โ™ญ INTJ cana-3839 77 2 | | PROPN cana-3839 77 3 , , PUNCT cana-3839 77 4 | | ADV cana-3839 77 5 โ™ญ โ™ญ PROPN cana-3839 78 1 โˆ’ โˆ’ PROPN cana-3839 78 2 ๐œ| ๐œ| PROPN cana-3839 78 3 , , PUNCT cana-3839 78 4 |๐œ |๐œ NOUN cana-3839 78 5 โˆ’ โˆ’ PROPN cana-3839 78 6 ๐œ‡| ๐œ‡| PROPN cana-3839 78 7 ) ) PUNCT cana-3839 78 8 โˆ€๐œ‡ โˆ€๐œ‡ NOUN cana-3839 78 9 , , PUNCT cana-3839 78 10 โ™ญ โ™ญ PROPN cana-3839 78 11 , , PUNCT cana-3839 78 12 ๐œ ๐œ PROPN cana-3839 78 13 โˆˆ โˆˆ PROPN cana-3839 78 14 โ„œ โ„œ PROPN cana-3839 78 15 and and CCONJ cana-3839 78 16 โˆ€๐‘›. โˆ€๐‘›. PROPN cana-3839 78 17 then then ADV cana-3839 78 18 ( ( PUNCT cana-3839 78 19 โ„œ โ„œ PROPN cana-3839 78 20 , , PUNCT cana-3839 78 21 ๐บ ๐บ PROPN cana-3839 78 22 , , PUNCT cana-3839 78 23 ๐‘‰ ๐‘‰ PROPN cana-3839 78 24 ) ) PUNCT cana-3839 78 25 is be AUX cana-3839 78 26 a a DET cana-3839 78 27 vgms vgms NOUN cana-3839 78 28 on on ADP cana-3839 78 29 โ„œ. โ„œ. PROPN cana-3839 78 30 example example NOUN cana-3839 78 31 1.5 1.5 NUM cana-3839 78 32 let let VERB cana-3839 78 33 โ„œ โ„œ NOUN cana-3839 78 34 = = PUNCT cana-3839 79 1 [ [ X cana-3839 79 2 1 1 NUM cana-3839 79 3 , , PUNCT cana-3839 79 4 โˆž โˆž PROPN cana-3839 79 5 ) ) PUNCT cana-3839 79 6 , , PUNCT cana-3839 79 7 ๐‘‰ ๐‘‰ PROPN cana-3839 79 8 be be VERB cana-3839 79 9 a a DET cana-3839 79 10 vector vector NOUN cana-3839 79 11 lattice lattice NOUN cana-3839 79 12 , , PUNCT cana-3839 79 13 ๐บ1 ๐บ1 NOUN cana-3839 79 14 and and CCONJ cana-3839 79 15 ๐บ2 ๐บ2 NOUN cana-3839 79 16 be be AUX cana-3839 79 17 two two NUM cana-3839 79 18 vector vector NOUN cana-3839 79 19 ๐บ-metrics ๐บ-metrics PROPN cana-3839 79 20 on on ADP cana-3839 79 21 โ„œ. โ„œ. PROPN cana-3839 79 22 the the DET cana-3839 79 23 transformation transformation NOUN cana-3839 79 24 ๐บ ๐บ PROPN cana-3839 79 25 : : PUNCT cana-3839 79 26 โ„œ โ„œ PROPN cana-3839 79 27 ร— ร— NOUN cana-3839 79 28 โ„œ โ„œ PROPN cana-3839 79 29 ร— ร— NOUN cana-3839 79 30 โ„œ โ„œ NOUN cana-3839 79 31 โ†’ โ†’ PUNCT cana-3839 79 32 ๐‘‰ ๐‘‰ PROPN cana-3839 79 33 is be AUX cana-3839 79 34 determined determine VERB cana-3839 79 35 by by ADP cana-3839 79 36 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 79 37 , , PUNCT cana-3839 79 38 โ™ญ โ™ญ INTJ cana-3839 79 39 , , PUNCT cana-3839 79 40 ๐œŽ ๐œŽ NOUN cana-3839 79 41 ) ) PUNCT cana-3839 79 42 = = SYM cana-3839 79 43 ( ( PUNCT cana-3839 79 44 ๐บ1(๐œ‡ ๐บ1(๐œ‡ PROPN cana-3839 79 45 , , PUNCT cana-3839 79 46 โ™ญ โ™ญ PROPN cana-3839 79 47 , , PUNCT cana-3839 79 48 ๐œŽ ๐œŽ NOUN cana-3839 79 49 ) ) PUNCT cana-3839 80 1 + + CCONJ cana-3839 80 2 ๐บ2(๐œ‡ ๐บ2(๐œ‡ ADJ cana-3839 80 3 , , PUNCT cana-3839 80 4 โ™ญ โ™ญ PROPN cana-3839 80 5 , , PUNCT cana-3839 80 6 ๐œŽ ๐œŽ NOUN cana-3839 80 7 ) ) PUNCT cana-3839 80 8 ) ) PUNCT cana-3839 80 9 1 1 NUM cana-3839 80 10 2 2 NUM cana-3839 80 11 โˆ€ โˆ€ NOUN cana-3839 80 12 ๐œ‡ ๐œ‡ ADP cana-3839 80 13 , , PUNCT cana-3839 80 14 โ™ญ โ™ญ INTJ cana-3839 80 15 , , PUNCT cana-3839 80 16 ๐œŽ ๐œŽ PROPN cana-3839 81 1 โˆˆ โˆˆ PROPN cana-3839 81 2 โ„œ. โ„œ. PROPN cana-3839 81 3 then then ADV cana-3839 81 4 ( ( PUNCT cana-3839 81 5 โ„œ โ„œ PROPN cana-3839 81 6 , , PUNCT cana-3839 81 7 ๐บ ๐บ PROPN cana-3839 81 8 , , PUNCT cana-3839 81 9 ๐‘‰ ๐‘‰ PROPN cana-3839 81 10 ) ) PUNCT cana-3839 81 11 is be AUX cana-3839 81 12 a a DET cana-3839 81 13 vgms vgms NOUN cana-3839 81 14 on on ADP cana-3839 81 15 โ„œ. โ„œ. PROPN cana-3839 81 16 communications communication NOUN cana-3839 81 17 on on ADP cana-3839 81 18 applied apply VERB cana-3839 81 19 nonlinear nonlinear ADJ cana-3839 81 20 analysis analysis NOUN cana-3839 81 21 issn issn NOUN cana-3839 81 22 : : PUNCT cana-3839 81 23 1074 1074 NUM cana-3839 81 24 - - PUNCT cana-3839 81 25 133x 133x NUM cana-3839 81 26 vol vol NOUN cana-3839 81 27 32 32 NUM cana-3839 81 28 no no NOUN cana-3839 81 29 . . PUNCT cana-3839 82 1 9s 9s NUM cana-3839 82 2 ( ( PUNCT cana-3839 82 3 2025 2025 NUM cana-3839 82 4 ) ) PUNCT cana-3839 82 5 79 79 NUM cana-3839 82 6 https://internationalpubls.com https://internationalpubls.com X cana-3839 82 7 definition definition NOUN cana-3839 82 8 1.6error 1.6error PROPN cana-3839 82 9 ! ! PUNCT cana-3839 82 10 reference reference NOUN cana-3839 82 11 source source NOUN cana-3839 82 12 not not PART cana-3839 82 13 found find VERB cana-3839 82 14 . . PUNCT cana-3839 83 1 in in ADP cana-3839 83 2 a a DET cana-3839 83 3 vgms vgms NOUN cana-3839 83 4 ( ( PUNCT cana-3839 83 5 โ„œ โ„œ PROPN cana-3839 83 6 , , PUNCT cana-3839 83 7 ๐บ ๐บ PROPN cana-3839 83 8 , , PUNCT cana-3839 83 9 ๐‘‰ ๐‘‰ PROPN cana-3839 83 10 ) ) PUNCT cana-3839 83 11 , , PUNCT cana-3839 83 12 โŒฉ โŒฉ PROPN cana-3839 83 13 ๐œ๐‘› ๐œ๐‘› NOUN cana-3839 83 14 โŒช โŒช NOUN cana-3839 83 15 โˆˆ โˆˆ PROPN cana-3839 83 16 โ„œ โ„œ PROPN cana-3839 83 17 is be AUX cana-3839 83 18 called call VERB cana-3839 83 19 ๐‘‰ ๐‘‰ PROPN cana-3839 83 20 convergent convergent NOUN cana-3839 83 21 to to ADP cana-3839 83 22 some some PRON cana-3839 83 23 ๐œ ๐œ ADP cana-3839 83 24 โˆˆ โˆˆ PROPN cana-3839 83 25 ๐‘‰ ๐‘‰ PROPN cana-3839 83 26 if if SCONJ cana-3839 83 27 โˆƒ โˆƒ PROPN cana-3839 83 28 โŒฉ โŒฉ PROPN cana-3839 83 29 ๐œ‡๐‘› ๐œ‡๐‘› NOUN cana-3839 83 30 โŒช โŒช NOUN cana-3839 83 31 โˆˆ โˆˆ NOUN cana-3839 83 32 ๐‘‰ ๐‘‰ PROPN cana-3839 83 33 satisfying satisfy VERB cana-3839 83 34 ๐œ‡๐‘› ๐œ‡๐‘› NOUN cana-3839 83 35 โ†“ โ†“ NOUN cana-3839 83 36 0 0 NUM cana-3839 83 37 and and CCONJ cana-3839 83 38 ๐บ(๐œ ๐บ(๐œ PROPN cana-3839 83 39 , , PUNCT cana-3839 83 40 ๐œ๐‘› ๐œ๐‘› PRON cana-3839 83 41 , , PUNCT cana-3839 83 42 ๐œ๐‘š ๐œ๐‘š NOUN cana-3839 83 43 ) ) PUNCT cana-3839 83 44 โ‰ค โ‰ค NOUN cana-3839 83 45 ๐œ‡๐‘› ๐œ‡๐‘› ADP cana-3839 83 46 โˆ€๐‘› โˆ€๐‘› NOUN cana-3839 83 47 , , PUNCT cana-3839 83 48 ๐‘š ๐‘š PROPN cana-3839 83 49 โˆˆ โˆˆ PROPN cana-3839 83 50 โ„ค+ โ„ค+ NOUN cana-3839 83 51 . . PUNCT cana-3839 83 52 definition definition NOUN cana-3839 83 53 1.7error 1.7error NUM cana-3839 83 54 ! ! PUNCT cana-3839 84 1 reference reference NOUN cana-3839 84 2 source source NOUN cana-3839 84 3 not not PART cana-3839 84 4 found find VERB cana-3839 84 5 . . PUNCT cana-3839 85 1 in in ADP cana-3839 85 2 a a DET cana-3839 85 3 vgms vgms NOUN cana-3839 85 4 ( ( PUNCT cana-3839 85 5 โ„œ โ„œ PROPN cana-3839 85 6 , , PUNCT cana-3839 85 7 ๐บ ๐บ PROPN cana-3839 85 8 , , PUNCT cana-3839 85 9 ๐‘‰ ๐‘‰ PROPN cana-3839 85 10 ) ) PUNCT cana-3839 85 11 , , PUNCT cana-3839 85 12 โŒฉ โŒฉ PROPN cana-3839 85 13 ๐œ๐‘› ๐œ๐‘› NOUN cana-3839 85 14 โŒช โŒช NOUN cana-3839 85 15 โˆˆ โˆˆ PROPN cana-3839 85 16 โ„œ โ„œ PROPN cana-3839 85 17 is be AUX cana-3839 85 18 called call VERB cana-3839 85 19 ๐‘‰cauchy ๐‘‰cauchy PROPN cana-3839 85 20 sequence(vcs sequence(vcs ADV cana-3839 85 21 ) ) PUNCT cana-3839 85 22 if if SCONJ cana-3839 85 23 โˆƒ โˆƒ PROPN cana-3839 85 24 โŒฉ โŒฉ PROPN cana-3839 85 25 ๐œ‡๐‘› ๐œ‡๐‘› NOUN cana-3839 85 26 โŒช โŒช NOUN cana-3839 85 27 โˆˆ โˆˆ NOUN cana-3839 85 28 ๐‘‰ ๐‘‰ PROPN cana-3839 85 29 satisfying satisfy VERB cana-3839 85 30 ๐œ‡๐‘› ๐œ‡๐‘› NOUN cana-3839 85 31 โ†“ โ†“ NOUN cana-3839 85 32 0 0 NUM cana-3839 85 33 and and CCONJ cana-3839 85 34 ๐บ(๐œ๐‘› ๐บ(๐œ๐‘› ADV cana-3839 85 35 , , PUNCT cana-3839 85 36 ๐œ๐‘š ๐œ๐‘š NOUN cana-3839 85 37 , , PUNCT cana-3839 85 38 ๐œ๐œ‡ ๐œ๐œ‡ NOUN cana-3839 85 39 ) ) PUNCT cana-3839 85 40 โ‰ค โ‰ค NUM cana-3839 85 41 ๐œ‡๐‘› ๐œ‡๐‘› NOUN cana-3839 85 42 holds hold VERB cana-3839 85 43 for for ADP cana-3839 85 44 all all DET cana-3839 85 45 ๐‘› ๐‘› PROPN cana-3839 85 46 , , PUNCT cana-3839 85 47 ๐‘š ๐‘š NOUN cana-3839 85 48 , , PUNCT cana-3839 85 49 ๐œ‡ ๐œ‡ ADP cana-3839 85 50 โˆˆ โˆˆ PROPN cana-3839 85 51 โ„ค+ โ„ค+ PUNCT cana-3839 85 52 . . PUNCT cana-3839 86 1 definition definition NOUN cana-3839 86 2 1.8error 1.8error NUM cana-3839 86 3 ! ! PUNCT cana-3839 86 4 reference reference NOUN cana-3839 86 5 source source NOUN cana-3839 86 6 not not PART cana-3839 86 7 found find VERB cana-3839 86 8 . . PUNCT cana-3839 87 1 in in ADP cana-3839 87 2 a a DET cana-3839 87 3 vgms vgms NOUN cana-3839 87 4 ( ( PUNCT cana-3839 87 5 โ„œ โ„œ PROPN cana-3839 87 6 , , PUNCT cana-3839 87 7 ๐บ ๐บ PROPN cana-3839 87 8 , , PUNCT cana-3839 87 9 ๐‘‰ ๐‘‰ PROPN cana-3839 87 10 ) ) PUNCT cana-3839 87 11 , , PUNCT cana-3839 87 12 the the DET cana-3839 87 13 term term NOUN cana-3839 87 14 ๐‘‰-complete ๐‘‰-complete PROPN cana-3839 87 15 implies imply VERB cana-3839 87 16 that that SCONJ cana-3839 87 17 every every DET cana-3839 87 18 vcs vcs NOUN cana-3839 87 19 in in ADP cana-3839 87 20 โ„œ โ„œ PROPN cana-3839 87 21 is be AUX cana-3839 87 22 ๐‘‰-convergence ๐‘‰-convergence PROPN cana-3839 87 23 to to ADP cana-3839 87 24 a a DET cana-3839 87 25 limit limit NOUN cana-3839 87 26 in in ADP cana-3839 87 27 โ„œ. โ„œ. PROPN cana-3839 87 28 definition definition NOUN cana-3839 87 29 1.9[5 1.9[5 NUM cana-3839 87 30 ] ] X cana-3839 87 31 a a DET cana-3839 87 32 cyclic cyclic ADJ cana-3839 87 33 transformation transformation NOUN cana-3839 87 34 โˆ โˆ PROPN cana-3839 87 35 on on ADP cana-3839 87 36 ๐œ‡ ๐œ‡ ADP cana-3839 87 37 is be AUX cana-3839 87 38 said say VERB cana-3839 87 39 to to PART cana-3839 87 40 be be AUX cana-3839 87 41 cyclic cyclic ADJ cana-3839 87 42 contraction contraction NOUN cana-3839 87 43 if if SCONJ cana-3839 87 44 โˆƒ โˆƒ PROPN cana-3839 87 45 0 0 NUM cana-3839 87 46 โ‰ค โ‰ค NOUN cana-3839 87 47 ๐œŒ ๐œŒ ADP cana-3839 87 48 < < X cana-3839 87 49 1 1 NUM cana-3839 87 50 such such ADJ cana-3839 87 51 that that DET cana-3839 87 52 ๐บ(โˆ(๐œ‡ ๐บ(โˆ(๐œ‡ NOUN cana-3839 87 53 ) ) PUNCT cana-3839 87 54 , , PUNCT cana-3839 87 55 โˆ โˆ PROPN cana-3839 87 56 ( ( PUNCT cana-3839 87 57 โ™ญ โ™ญ PROPN cana-3839 87 58 ) ) PUNCT cana-3839 87 59 , , PUNCT cana-3839 87 60 โˆ โˆ PROPN cana-3839 87 61 ( ( PUNCT cana-3839 87 62 โ™ญ โ™ญ PROPN cana-3839 87 63 ) ) PUNCT cana-3839 87 64 ) ) PUNCT cana-3839 87 65 โชฏ โชฏ NOUN cana-3839 87 66 ๐œŒ๐บ(๐œ‡ ๐œŒ๐บ(๐œ‡ ADP cana-3839 87 67 , , PUNCT cana-3839 87 68 โ™ญ โ™ญ INTJ cana-3839 87 69 , , PUNCT cana-3839 87 70 โ™ญ โ™ญ PROPN cana-3839 87 71 ) ) PUNCT cana-3839 87 72 โˆ€๐œ‡ โˆ€๐œ‡ X cana-3839 87 73 โˆˆ โˆˆ PROPN cana-3839 87 74 ๐œ’๐‘ž ๐œ’๐‘ž PROPN cana-3839 87 75 and and CCONJ cana-3839 87 76 โ™ญ โ™ญ PROPN cana-3839 87 77 โˆˆ โˆˆ PROPN cana-3839 87 78 ๐œ’๐‘ž+1 ๐œ’๐‘ž+1 PROPN cana-3839 87 79 , , PUNCT cana-3839 87 80 ๐‘ž ๐‘ž X cana-3839 87 81 = = SYM cana-3839 87 82 1,2 1,2 NUM cana-3839 87 83 , , PUNCT cana-3839 87 84 โ€ฆ โ€ฆ PUNCT cana-3839 87 85 ๐‘š. ๐‘š. ADJ cana-3839 87 86 2 2 NUM cana-3839 87 87 main main ADJ cana-3839 87 88 results result NOUN cana-3839 87 89 in in ADP cana-3839 87 90 this this DET cana-3839 87 91 section section NOUN cana-3839 87 92 , , PUNCT cana-3839 87 93 we we PRON cana-3839 87 94 establish establish VERB cana-3839 87 95 some some DET cana-3839 87 96 fpt fpt NOUN cana-3839 87 97 and and CCONJ cana-3839 87 98 corollaries corollary NOUN cana-3839 87 99 for for ADP cana-3839 87 100 self self NOUN cana-3839 87 101 - - PUNCT cana-3839 87 102 transformations transformation NOUN cana-3839 87 103 in in ADP cana-3839 87 104 vgms vgms NOUN cana-3839 87 105 ( ( PUNCT cana-3839 87 106 โ„œ โ„œ PROPN cana-3839 87 107 , , PUNCT cana-3839 87 108 ๐บ ๐บ PROPN cana-3839 87 109 , , PUNCT cana-3839 87 110 ๐‘‰ ๐‘‰ PROPN cana-3839 87 111 ) ) PUNCT cana-3839 87 112 by by ADP cana-3839 87 113 employing employ VERB cana-3839 87 114 the the DET cana-3839 87 115 methodology methodology NOUN cana-3839 87 116 of of ADP cana-3839 87 117 cc cc PROPN cana-3839 87 118 . . PROPN cana-3839 87 119 theorem theorem VERB cana-3839 87 120 2.1 2.1 NUM cana-3839 87 121 let let NOUN cana-3839 87 122 ( ( PUNCT cana-3839 87 123 โ„œ โ„œ PROPN cana-3839 87 124 , , PUNCT cana-3839 87 125 ๐บ ๐บ PROPN cana-3839 87 126 , , PUNCT cana-3839 87 127 ๐‘‰ ๐‘‰ PROPN cana-3839 87 128 ) ) PUNCT cana-3839 87 129 be be VERB cana-3839 87 130 symmetric symmetric ADJ cana-3839 87 131 and and CCONJ cana-3839 87 132 ๐บcomplete ๐บcomplete ADJ cana-3839 87 133 vgms vgms NOUN cana-3839 87 134 and and CCONJ cana-3839 87 135 { { PUNCT cana-3839 87 136 ๐œ’๐‘ž}๐‘ž=1 ๐œ’๐‘ž}๐‘ž=1 X cana-3839 87 137 ๐‘š ๐‘š PROPN cana-3839 87 138 as as ADP cana-3839 87 139 a a DET cana-3839 87 140 collection collection NOUN cana-3839 87 141 of of ADP cana-3839 87 142 non non ADJ cana-3839 87 143 - - ADJ cana-3839 87 144 empty empty ADJ cana-3839 87 145 subset subset NOUN cana-3839 87 146 of of ADP cana-3839 87 147 โ„œ โ„œ PROPN cana-3839 87 148 with with ADP cana-3839 87 149 ๐œ‡ ๐œ‡ ADP cana-3839 87 150 = = VERB cana-3839 87 151 โˆช๐‘ž=1 โˆช๐‘ž=1 PROPN cana-3839 87 152 ๐‘š ๐‘š X cana-3839 87 153 ๐œ’๐‘ž.let ๐œ’๐‘ž.let PROPN cana-3839 87 154 โˆ โˆ NOUN cana-3839 87 155 : : PUNCT cana-3839 87 156 ๐œ‡ ๐œ‡ X cana-3839 87 157 โ†’ โ†’ X cana-3839 87 158 ๐œ‡ ๐œ‡ SCONJ cana-3839 87 159 be be AUX cana-3839 87 160 cyclic cyclic ADJ cana-3839 87 161 transformation transformation NOUN cana-3839 87 162 satisfying satisfying NOUN cana-3839 87 163 โˆ(๐œ’๐‘ž โˆ(๐œ’๐‘ž NOUN cana-3839 87 164 ) ) PUNCT cana-3839 87 165 โŠ† โŠ† NUM cana-3839 87 166 ๐œ’๐‘ž+1 ๐œ’๐‘ž+1 NUM cana-3839 87 167 ๐‘ž ๐‘ž NOUN cana-3839 87 168 = = SYM cana-3839 87 169 1,2 1,2 NUM cana-3839 87 170 , , PUNCT cana-3839 87 171 โ€ฆ โ€ฆ PUNCT cana-3839 87 172 , , PUNCT cana-3839 87 173 ๐‘š ๐‘š NOUN cana-3839 87 174 , , PUNCT cana-3839 87 175 ๐‘คโ„Ž๐‘’๐‘Ÿ๐‘’ ๐‘คโ„Ž๐‘’๐‘Ÿ๐‘’ ADJ cana-3839 87 176 ๐œ’๐‘š+1 ๐œ’๐‘š+1 NUM cana-3839 87 177 = = SYM cana-3839 87 178 ๐œ’1 ๐œ’1 PROPN cana-3839 87 179 . . PUNCT cana-3839 88 1 suppose suppose VERB cana-3839 88 2 that that SCONJ cana-3839 88 3 โˆƒ โˆƒ PROPN cana-3839 88 4 constant constant ADJ cana-3839 88 5 ๐œ•1 ๐œ•1 NOUN cana-3839 88 6 , , PUNCT cana-3839 88 7 ๐œ•2 ๐œ•2 NOUN cana-3839 88 8 , , PUNCT cana-3839 88 9 ๐œ•3 ๐œ•3 NOUN cana-3839 88 10 , , PUNCT cana-3839 88 11 ๐œ•4 ๐œ•4 PROPN cana-3839 88 12 , , PUNCT cana-3839 88 13 ๐œ•5 ๐œ•5 NUM cana-3839 88 14 , , PUNCT cana-3839 88 15 ๐œ•6 ๐œ•6 PROPN cana-3839 88 16 and and CCONJ cana-3839 88 17 ๐œ•7 ๐œ•7 VERB cana-3839 88 18 with with ADP cana-3839 88 19 0 0 NUM cana-3839 88 20 โ‰ค โ‰ค NUM cana-3839 88 21 ๐œ•1 ๐œ•1 NOUN cana-3839 88 22 + + SYM cana-3839 88 23 ๐œ•2 ๐œ•2 NOUN cana-3839 89 1 + + SYM cana-3839 89 2 ๐œ•3 ๐œ•3 NOUN cana-3839 89 3 + + SYM cana-3839 89 4 ๐œ•4 ๐œ•4 NOUN cana-3839 89 5 + + CCONJ cana-3839 89 6 ๐œ•5 ๐œ•5 NUM cana-3839 89 7 + + CCONJ cana-3839 89 8 ๐œ•6 ๐œ•6 NUM cana-3839 89 9 + + CCONJ cana-3839 89 10 ๐œ•7 ๐œ•7 VERB cana-3839 89 11 < < X cana-3839 89 12 1 1 NUM cana-3839 89 13 such such ADJ cana-3839 89 14 that that SCONJ cana-3839 89 15 the the DET cana-3839 89 16 transformation transformation NOUN cana-3839 89 17 โˆ โˆ PROPN cana-3839 89 18 satisfies satisfy VERB cana-3839 89 19 ๐บ(โˆ(๐œ‡ ๐บ(โˆ(๐œ‡ NUM cana-3839 89 20 ) ) PUNCT cana-3839 89 21 , , PUNCT cana-3839 89 22 โˆ โˆ PROPN cana-3839 89 23 ( ( PUNCT cana-3839 89 24 โ™ญ โ™ญ PROPN cana-3839 89 25 ) ) PUNCT cana-3839 89 26 , , PUNCT cana-3839 89 27 โˆ(๐‘Ÿ โˆ(๐‘Ÿ NOUN cana-3839 89 28 ) ) PUNCT cana-3839 89 29 ) ) PUNCT cana-3839 90 1 โชฏ โชฏ PROPN cana-3839 90 2 ๐œ•1๐บ(๐œ‡ ๐œ•1๐บ(๐œ‡ NUM cana-3839 90 3 , , PUNCT cana-3839 90 4 โ™ญ โ™ญ PROPN cana-3839 90 5 , , PUNCT cana-3839 90 6 ๐‘Ÿ ๐‘Ÿ NOUN cana-3839 90 7 ) ) PUNCT cana-3839 91 1 + + CCONJ cana-3839 91 2 ๐œ•2๐บ(๐œ‡ ๐œ•2๐บ(๐œ‡ PROPN cana-3839 91 3 , , PUNCT cana-3839 91 4 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 91 5 ) ) PUNCT cana-3839 91 6 , , PUNCT cana-3839 91 7 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 91 8 ) ) PUNCT cana-3839 91 9 ) ) PUNCT cana-3839 92 1 + + CCONJ cana-3839 92 2 ๐œ•3๐บ ๐œ•3๐บ PROPN cana-3839 92 3 ( ( PUNCT cana-3839 92 4 โ™ญ โ™ญ INTJ cana-3839 92 5 , , PUNCT cana-3839 92 6 โˆ โˆ PROPN cana-3839 92 7 ( ( PUNCT cana-3839 92 8 โ™ญ โ™ญ PROPN cana-3839 92 9 ) ) PUNCT cana-3839 92 10 , , PUNCT cana-3839 92 11 โˆ โˆ PROPN cana-3839 92 12 ( ( PUNCT cana-3839 92 13 โ™ญ โ™ญ PROPN cana-3839 92 14 ) ) PUNCT cana-3839 92 15 ) ) PUNCT cana-3839 93 1 + + PROPN cana-3839 93 2 ๐œ•4๐บ(๐‘Ÿ ๐œ•4๐บ(๐‘Ÿ PROPN cana-3839 93 3 , , PUNCT cana-3839 93 4 โˆ(๐‘Ÿ โˆ(๐‘Ÿ ADV cana-3839 93 5 ) ) PUNCT cana-3839 93 6 , , PUNCT cana-3839 93 7 โˆ(๐‘Ÿ โˆ(๐‘Ÿ NOUN cana-3839 93 8 ) ) PUNCT cana-3839 93 9 ) ) PUNCT cana-3839 94 1 + + CCONJ cana-3839 94 2 ๐œ•5๐บ(๐œ‡ ๐œ•5๐บ(๐œ‡ PROPN cana-3839 94 3 , , PUNCT cana-3839 94 4 โˆ โˆ PROPN cana-3839 94 5 ( ( PUNCT cana-3839 94 6 โ™ญ โ™ญ PROPN cana-3839 94 7 ) ) PUNCT cana-3839 94 8 , , PUNCT cana-3839 94 9 โˆ โˆ PROPN cana-3839 94 10 ( ( PUNCT cana-3839 94 11 โ™ญ โ™ญ PROPN cana-3839 94 12 ) ) PUNCT cana-3839 94 13 ) ) PUNCT cana-3839 95 1 + + CCONJ cana-3839 95 2 ๐œ•6๐บ ๐œ•6๐บ PROPN cana-3839 95 3 ( ( PUNCT cana-3839 95 4 โ™ญ โ™ญ INTJ cana-3839 95 5 , , PUNCT cana-3839 95 6 โˆ(๐‘Ÿ โˆ(๐‘Ÿ ADJ cana-3839 95 7 ) ) PUNCT cana-3839 95 8 , , PUNCT cana-3839 95 9 โˆ(๐‘Ÿ โˆ(๐‘Ÿ NOUN cana-3839 95 10 ) ) PUNCT cana-3839 95 11 ) ) PUNCT cana-3839 96 1 + + ADP cana-3839 96 2 ๐œ•7๐บ(๐‘Ÿ ๐œ•7๐บ(๐‘Ÿ PROPN cana-3839 96 3 , , PUNCT cana-3839 96 4 โˆ(๐œ‡ โˆ(๐œ‡ ADJ cana-3839 96 5 ) ) PUNCT cana-3839 96 6 , , PUNCT cana-3839 96 7 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 96 8 ) ) PUNCT cana-3839 96 9 ) ) PUNCT cana-3839 96 10 ( ( PUNCT cana-3839 96 11 1 1 X cana-3839 96 12 ) ) PUNCT cana-3839 96 13 โˆ€๐œ‡ โˆ€๐œ‡ X cana-3839 96 14 โˆˆ โˆˆ PROPN cana-3839 96 15 ๐œ’๐‘ž ๐œ’๐‘ž PROPN cana-3839 96 16 and and CCONJ cana-3839 96 17 โ™ญ โ™ญ PROPN cana-3839 96 18 , , PUNCT cana-3839 96 19 ๐‘Ÿ ๐‘Ÿ X cana-3839 96 20 โˆˆ โˆˆ PROPN cana-3839 96 21 ๐œ’๐‘ž+1 ๐œ’๐‘ž+1 PROPN cana-3839 96 22 , , PUNCT cana-3839 96 23 ๐‘ž ๐‘ž X cana-3839 96 24 = = SYM cana-3839 96 25 1,2 1,2 NUM cana-3839 96 26 , , PUNCT cana-3839 96 27 โ€ฆ โ€ฆ PUNCT cana-3839 96 28 ๐‘š. ๐‘š. ADV cana-3839 96 29 then then ADV cana-3839 96 30 ๐œ’ ๐œ’ X cana-3839 96 31 has have VERB cana-3839 96 32 fp fp NOUN cana-3839 96 33 in in ADP cana-3839 96 34 ๐œ‡ ๐œ‡ ADP cana-3839 96 35 = = NOUN cana-3839 96 36 โˆฉ๐‘ž=1 โˆฉ๐‘ž=1 ADJ cana-3839 96 37 ๐‘š ๐‘š X cana-3839 96 38 ๐œ’๐‘ž ๐œ’๐‘ž NOUN cana-3839 96 39 which which PRON cana-3839 96 40 is be AUX cana-3839 96 41 unique unique ADJ cana-3839 96 42 . . PUNCT cana-3839 97 1 proof proof NOUN cana-3839 97 2 : : PUNCT cana-3839 97 3 we we PRON cana-3839 97 4 can can AUX cana-3839 97 5 write write VERB cana-3839 97 6 eq eq NOUN cana-3839 97 7 ( ( PUNCT cana-3839 97 8 1 1 NUM cana-3839 97 9 ) ) PUNCT cana-3839 97 10 as as ADP cana-3839 97 11 ๐บ(โˆ(๐œ‡ ๐บ(โˆ(๐œ‡ NOUN cana-3839 97 12 ) ) PUNCT cana-3839 97 13 , , PUNCT cana-3839 97 14 โˆ โˆ PROPN cana-3839 97 15 ( ( PUNCT cana-3839 97 16 โ™ญ โ™ญ PROPN cana-3839 97 17 ) ) PUNCT cana-3839 97 18 , , PUNCT cana-3839 97 19 โˆ โˆ PROPN cana-3839 97 20 ( ( PUNCT cana-3839 97 21 โ™ญ โ™ญ PROPN cana-3839 97 22 ) ) PUNCT cana-3839 97 23 ) ) PUNCT cana-3839 97 24 โชฏ โชฏ PROPN cana-3839 97 25 ๐œ•1๐บ(๐œ‡ ๐œ•1๐บ(๐œ‡ NUM cana-3839 97 26 , , PUNCT cana-3839 97 27 โ™ญ โ™ญ PROPN cana-3839 97 28 , , PUNCT cana-3839 97 29 โ™ญ โ™ญ PROPN cana-3839 97 30 ) ) PUNCT cana-3839 98 1 + + CCONJ cana-3839 98 2 ๐œ•2๐บ(๐œ‡ ๐œ•2๐บ(๐œ‡ PROPN cana-3839 98 3 , , PUNCT cana-3839 98 4 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 98 5 ) ) PUNCT cana-3839 98 6 , , PUNCT cana-3839 98 7 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 98 8 ) ) PUNCT cana-3839 98 9 ) ) PUNCT cana-3839 99 1 + + CCONJ cana-3839 99 2 ๐œ•3๐บ ๐œ•3๐บ PROPN cana-3839 99 3 ( ( PUNCT cana-3839 99 4 โ™ญ โ™ญ INTJ cana-3839 99 5 , , PUNCT cana-3839 99 6 โˆ โˆ PROPN cana-3839 99 7 ( ( PUNCT cana-3839 99 8 โ™ญ โ™ญ PROPN cana-3839 99 9 ) ) PUNCT cana-3839 99 10 , , PUNCT cana-3839 99 11 โˆ โˆ PROPN cana-3839 99 12 ( ( PUNCT cana-3839 99 13 โ™ญ โ™ญ PROPN cana-3839 99 14 ) ) PUNCT cana-3839 99 15 ) ) PUNCT cana-3839 99 16 communications communication NOUN cana-3839 99 17 on on ADP cana-3839 99 18 applied apply VERB cana-3839 99 19 nonlinear nonlinear ADJ cana-3839 99 20 analysis analysis NOUN cana-3839 99 21 issn issn NOUN cana-3839 99 22 : : PUNCT cana-3839 99 23 1074 1074 NUM cana-3839 99 24 - - PUNCT cana-3839 99 25 133x 133x NUM cana-3839 99 26 vol vol NOUN cana-3839 99 27 32 32 NUM cana-3839 99 28 no no NOUN cana-3839 99 29 . . PUNCT cana-3839 100 1 9s 9s NUM cana-3839 100 2 ( ( PUNCT cana-3839 100 3 2025 2025 NUM cana-3839 100 4 ) ) PUNCT cana-3839 100 5 80 80 NUM cana-3839 100 6 https://internationalpubls.com https://internationalpubls.com X cana-3839 101 1 + + PROPN cana-3839 101 2 ๐œ•4๐บ ๐œ•4๐บ PROPN cana-3839 101 3 ( ( PUNCT cana-3839 101 4 โ™ญ โ™ญ INTJ cana-3839 101 5 , , PUNCT cana-3839 101 6 โˆ โˆ PROPN cana-3839 101 7 ( ( PUNCT cana-3839 101 8 โ™ญ โ™ญ PROPN cana-3839 101 9 ) ) PUNCT cana-3839 101 10 , , PUNCT cana-3839 101 11 โˆ โˆ PROPN cana-3839 101 12 ( ( PUNCT cana-3839 101 13 โ™ญ โ™ญ PROPN cana-3839 101 14 ) ) PUNCT cana-3839 101 15 ) ) PUNCT cana-3839 102 1 + + CCONJ cana-3839 102 2 ๐œ•5๐บ(๐œ‡ ๐œ•5๐บ(๐œ‡ PROPN cana-3839 102 3 , , PUNCT cana-3839 102 4 โˆ โˆ PROPN cana-3839 102 5 ( ( PUNCT cana-3839 102 6 โ™ญ โ™ญ PROPN cana-3839 102 7 ) ) PUNCT cana-3839 102 8 , , PUNCT cana-3839 102 9 โˆ โˆ PROPN cana-3839 102 10 ( ( PUNCT cana-3839 102 11 โ™ญ โ™ญ PROPN cana-3839 102 12 ) ) PUNCT cana-3839 102 13 ) ) PUNCT cana-3839 103 1 + + CCONJ cana-3839 103 2 ๐œ•6๐บ ๐œ•6๐บ PROPN cana-3839 103 3 ( ( PUNCT cana-3839 103 4 โ™ญ โ™ญ INTJ cana-3839 103 5 , , PUNCT cana-3839 103 6 โˆ โˆ PROPN cana-3839 103 7 ( ( PUNCT cana-3839 103 8 โ™ญ โ™ญ PROPN cana-3839 103 9 ) ) PUNCT cana-3839 103 10 , , PUNCT cana-3839 103 11 โˆ โˆ PROPN cana-3839 103 12 ( ( PUNCT cana-3839 103 13 โ™ญ โ™ญ PROPN cana-3839 103 14 ) ) PUNCT cana-3839 103 15 ) ) PUNCT cana-3839 104 1 + + CCONJ cana-3839 104 2 ๐œ•7๐บ ๐œ•7๐บ PROPN cana-3839 104 3 ( ( PUNCT cana-3839 104 4 โ™ญ โ™ญ INTJ cana-3839 104 5 , , PUNCT cana-3839 104 6 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 104 7 ) ) PUNCT cana-3839 104 8 , , PUNCT cana-3839 104 9 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 104 10 ) ) PUNCT cana-3839 104 11 ) ) PUNCT cana-3839 104 12 ๐บ(โˆ(๐œ‡ ๐บ(โˆ(๐œ‡ NUM cana-3839 104 13 ) ) PUNCT cana-3839 104 14 , , PUNCT cana-3839 104 15 โˆ โˆ PROPN cana-3839 104 16 ( ( PUNCT cana-3839 104 17 โ™ญ โ™ญ PROPN cana-3839 104 18 ) ) PUNCT cana-3839 104 19 , , PUNCT cana-3839 104 20 โˆ โˆ PROPN cana-3839 104 21 ( ( PUNCT cana-3839 104 22 โ™ญ โ™ญ PROPN cana-3839 104 23 ) ) PUNCT cana-3839 104 24 ) ) PUNCT cana-3839 104 25 โชฏ โชฏ PROPN cana-3839 104 26 ๐œ•1๐บ(๐œ‡ ๐œ•1๐บ(๐œ‡ NUM cana-3839 104 27 , , PUNCT cana-3839 104 28 โ™ญ โ™ญ PROPN cana-3839 104 29 , , PUNCT cana-3839 104 30 โ™ญ โ™ญ PROPN cana-3839 104 31 ) ) PUNCT cana-3839 105 1 + + CCONJ cana-3839 105 2 ๐œ•2๐บ(๐œ‡ ๐œ•2๐บ(๐œ‡ PROPN cana-3839 105 3 , , PUNCT cana-3839 105 4 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 105 5 ) ) PUNCT cana-3839 105 6 , , PUNCT cana-3839 105 7 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 105 8 ) ) PUNCT cana-3839 105 9 ) ) PUNCT cana-3839 106 1 + + CCONJ cana-3839 106 2 ( ( PUNCT cana-3839 106 3 ๐œ•3 ๐œ•3 NOUN cana-3839 106 4 + + SYM cana-3839 106 5 ๐œ•4 ๐œ•4 NOUN cana-3839 106 6 + + CCONJ cana-3839 106 7 ๐œ•6)๐บ ๐œ•6)๐บ PROPN cana-3839 106 8 ( ( PUNCT cana-3839 106 9 โ™ญ โ™ญ INTJ cana-3839 106 10 , , PUNCT cana-3839 106 11 โˆ โˆ PROPN cana-3839 106 12 ( ( PUNCT cana-3839 106 13 โ™ญ โ™ญ PROPN cana-3839 106 14 ) ) PUNCT cana-3839 106 15 , , PUNCT cana-3839 106 16 โˆ โˆ PROPN cana-3839 106 17 ( ( PUNCT cana-3839 106 18 โ™ญ โ™ญ PROPN cana-3839 106 19 ) ) PUNCT cana-3839 106 20 ) ) PUNCT cana-3839 107 1 + + ADV cana-3839 107 2 ๐œ•5๐บ(๐œ‡ ๐œ•5๐บ(๐œ‡ PROPN cana-3839 107 3 , , PUNCT cana-3839 107 4 โˆ โˆ PROPN cana-3839 107 5 ( ( PUNCT cana-3839 107 6 โ™ญ โ™ญ PROPN cana-3839 107 7 ) ) PUNCT cana-3839 107 8 , , PUNCT cana-3839 107 9 โˆ โˆ PROPN cana-3839 107 10 ( ( PUNCT cana-3839 107 11 โ™ญ โ™ญ PROPN cana-3839 107 12 ) ) PUNCT cana-3839 107 13 ) ) PUNCT cana-3839 108 1 + + CCONJ cana-3839 108 2 ๐œ•7๐บ ๐œ•7๐บ PROPN cana-3839 108 3 ( ( PUNCT cana-3839 108 4 โ™ญ โ™ญ INTJ cana-3839 108 5 , , PUNCT cana-3839 108 6 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 108 7 ) ) PUNCT cana-3839 108 8 , , PUNCT cana-3839 108 9 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 108 10 ) ) PUNCT cana-3839 108 11 ) ) PUNCT cana-3839 108 12 ( ( PUNCT cana-3839 108 13 2 2 X cana-3839 108 14 ) ) PUNCT cana-3839 108 15 interchanging interchange VERB cana-3839 108 16 the the DET cana-3839 108 17 role role NOUN cana-3839 108 18 of of ADP cana-3839 108 19 ๐œ‡ ๐œ‡ ADP cana-3839 108 20 and and CCONJ cana-3839 108 21 โ™ญ โ™ญ INTJ cana-3839 108 22 , , PUNCT cana-3839 108 23 we we PRON cana-3839 108 24 get get VERB cana-3839 108 25 ๐บ(โˆ ๐บ(โˆ NOUN cana-3839 108 26 ( ( PUNCT cana-3839 108 27 โ™ญ โ™ญ PROPN cana-3839 108 28 ) ) PUNCT cana-3839 108 29 , , PUNCT cana-3839 108 30 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 108 31 ) ) PUNCT cana-3839 108 32 , , PUNCT cana-3839 108 33 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 108 34 ) ) PUNCT cana-3839 108 35 ) ) PUNCT cana-3839 108 36 โชฏ โชฏ PROPN cana-3839 108 37 ๐œ•1๐บ ๐œ•1๐บ PROPN cana-3839 108 38 ( ( PUNCT cana-3839 108 39 โ™ญ โ™ญ PROPN cana-3839 108 40 , , PUNCT cana-3839 108 41 ๐œ‡ ๐œ‡ ADP cana-3839 108 42 , , PUNCT cana-3839 108 43 ๐œ‡ ๐œ‡ ADP cana-3839 108 44 ) ) PUNCT cana-3839 108 45 + + CCONJ cana-3839 108 46 ๐œ•2๐บ ๐œ•2๐บ PROPN cana-3839 108 47 ( ( PUNCT cana-3839 108 48 โ™ญ โ™ญ INTJ cana-3839 108 49 , , PUNCT cana-3839 108 50 โˆ โˆ PROPN cana-3839 108 51 ( ( PUNCT cana-3839 108 52 โ™ญ โ™ญ PROPN cana-3839 108 53 ) ) PUNCT cana-3839 108 54 , , PUNCT cana-3839 108 55 โˆ โˆ PROPN cana-3839 108 56 ( ( PUNCT cana-3839 108 57 โ™ญ โ™ญ PROPN cana-3839 108 58 ) ) PUNCT cana-3839 108 59 ) ) PUNCT cana-3839 109 1 + + CCONJ cana-3839 109 2 ( ( PUNCT cana-3839 109 3 ๐œ•3 ๐œ•3 NOUN cana-3839 109 4 + + SYM cana-3839 109 5 ๐œ•4 ๐œ•4 NOUN cana-3839 109 6 + + CCONJ cana-3839 109 7 ๐œ•6)๐บ(๐œ‡ ๐œ•6)๐บ(๐œ‡ NOUN cana-3839 109 8 , , PUNCT cana-3839 109 9 โˆ(๐œ‡ โˆ(๐œ‡ ADV cana-3839 109 10 ) ) PUNCT cana-3839 109 11 , , PUNCT cana-3839 109 12 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 109 13 ) ) PUNCT cana-3839 109 14 ) ) PUNCT cana-3839 110 1 + + VERB cana-3839 110 2 ๐œ•5๐บ ๐œ•5๐บ PROPN cana-3839 110 3 ( ( PUNCT cana-3839 110 4 โ™ญ โ™ญ INTJ cana-3839 110 5 , , PUNCT cana-3839 110 6 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 110 7 ) ) PUNCT cana-3839 110 8 , , PUNCT cana-3839 110 9 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 110 10 ) ) PUNCT cana-3839 110 11 ) ) PUNCT cana-3839 110 12 + + CCONJ cana-3839 110 13 ๐œ•7๐บ(๐œ‡ ๐œ•7๐บ(๐œ‡ VERB cana-3839 110 14 , , PUNCT cana-3839 110 15 โˆ โˆ PROPN cana-3839 110 16 ( ( PUNCT cana-3839 110 17 โ™ญ โ™ญ PROPN cana-3839 110 18 ) ) PUNCT cana-3839 110 19 , , PUNCT cana-3839 110 20 โˆ โˆ PROPN cana-3839 110 21 ( ( PUNCT cana-3839 110 22 โ™ญ โ™ญ PROPN cana-3839 110 23 ) ) PUNCT cana-3839 110 24 ) ) PUNCT cana-3839 110 25 ๐บ(โˆ(๐œ‡ ๐บ(โˆ(๐œ‡ NUM cana-3839 110 26 ) ) PUNCT cana-3839 110 27 , , PUNCT cana-3839 110 28 โˆ โˆ PROPN cana-3839 110 29 ( ( PUNCT cana-3839 110 30 โ™ญ โ™ญ PROPN cana-3839 110 31 ) ) PUNCT cana-3839 110 32 , , PUNCT cana-3839 110 33 โˆ โˆ PROPN cana-3839 110 34 ( ( PUNCT cana-3839 110 35 โ™ญ โ™ญ PROPN cana-3839 110 36 ) ) PUNCT cana-3839 110 37 ) ) PUNCT cana-3839 110 38 โชฏ โชฏ PROPN cana-3839 110 39 ๐œ•1๐บ(๐œ‡ ๐œ•1๐บ(๐œ‡ NUM cana-3839 110 40 , , PUNCT cana-3839 110 41 โ™ญ โ™ญ PROPN cana-3839 110 42 , , PUNCT cana-3839 110 43 โ™ญ โ™ญ PROPN cana-3839 110 44 ) ) PUNCT cana-3839 111 1 + + CCONJ cana-3839 111 2 ๐œ•2๐บ ๐œ•2๐บ PROPN cana-3839 111 3 ( ( PUNCT cana-3839 111 4 โ™ญ โ™ญ INTJ cana-3839 111 5 , , PUNCT cana-3839 111 6 โˆ โˆ PROPN cana-3839 111 7 ( ( PUNCT cana-3839 111 8 โ™ญ โ™ญ PROPN cana-3839 111 9 ) ) PUNCT cana-3839 111 10 , , PUNCT cana-3839 111 11 โˆ โˆ PROPN cana-3839 111 12 ( ( PUNCT cana-3839 111 13 โ™ญ โ™ญ PROPN cana-3839 111 14 ) ) PUNCT cana-3839 111 15 ) ) PUNCT cana-3839 112 1 + + CCONJ cana-3839 112 2 ( ( PUNCT cana-3839 112 3 ๐œ•3 ๐œ•3 NOUN cana-3839 112 4 + + SYM cana-3839 112 5 ๐œ•4 ๐œ•4 NOUN cana-3839 112 6 + + CCONJ cana-3839 112 7 ๐œ•6)๐บ(๐œ‡ ๐œ•6)๐บ(๐œ‡ NOUN cana-3839 112 8 , , PUNCT cana-3839 112 9 โˆ(๐œ‡ โˆ(๐œ‡ ADV cana-3839 112 10 ) ) PUNCT cana-3839 112 11 , , PUNCT cana-3839 112 12 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 112 13 ) ) PUNCT cana-3839 112 14 ) ) PUNCT cana-3839 113 1 + + VERB cana-3839 113 2 ๐œ•5๐บ ๐œ•5๐บ PROPN cana-3839 113 3 ( ( PUNCT cana-3839 113 4 โ™ญ โ™ญ INTJ cana-3839 113 5 , , PUNCT cana-3839 113 6 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 113 7 ) ) PUNCT cana-3839 113 8 , , PUNCT cana-3839 113 9 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 113 10 ) ) PUNCT cana-3839 113 11 ) ) PUNCT cana-3839 113 12 + + CCONJ cana-3839 113 13 ๐œ•7๐บ(๐œ‡ ๐œ•7๐บ(๐œ‡ VERB cana-3839 113 14 , , PUNCT cana-3839 113 15 โˆ โˆ PROPN cana-3839 113 16 ( ( PUNCT cana-3839 113 17 โ™ญ โ™ญ PROPN cana-3839 113 18 ) ) PUNCT cana-3839 113 19 , , PUNCT cana-3839 113 20 โˆ โˆ PROPN cana-3839 113 21 ( ( PUNCT cana-3839 113 22 โ™ญ โ™ญ PROPN cana-3839 113 23 ) ) PUNCT cana-3839 113 24 ) ) PUNCT cana-3839 113 25 ( ( PUNCT cana-3839 113 26 3 3 X cana-3839 113 27 ) ) PUNCT cana-3839 113 28 adding add VERB cana-3839 113 29 eq(2 eq(2 PROPN cana-3839 113 30 ) ) PUNCT cana-3839 113 31 and and CCONJ cana-3839 113 32 eq eq NOUN cana-3839 113 33 ( ( PUNCT cana-3839 113 34 3 3 NUM cana-3839 113 35 ) ) PUNCT cana-3839 113 36 , , PUNCT cana-3839 113 37 we we PRON cana-3839 113 38 get get VERB cana-3839 113 39 2๐บ(โˆ(๐œ‡ 2๐บ(โˆ(๐œ‡ NUM cana-3839 113 40 ) ) PUNCT cana-3839 113 41 , , PUNCT cana-3839 113 42 โˆ โˆ PROPN cana-3839 113 43 ( ( PUNCT cana-3839 113 44 โ™ญ โ™ญ PROPN cana-3839 113 45 ) ) PUNCT cana-3839 113 46 , , PUNCT cana-3839 113 47 โˆ โˆ PROPN cana-3839 113 48 ( ( PUNCT cana-3839 113 49 โ™ญ โ™ญ PROPN cana-3839 113 50 ) ) PUNCT cana-3839 113 51 ) ) PUNCT cana-3839 113 52 โชฏ โชฏ NOUN cana-3839 113 53 2๐œ•1๐บ(๐œ‡ 2๐œ•1๐บ(๐œ‡ NUM cana-3839 113 54 , , PUNCT cana-3839 113 55 โ™ญ โ™ญ INTJ cana-3839 113 56 , , PUNCT cana-3839 113 57 โ™ญ โ™ญ PROPN cana-3839 113 58 ) ) PUNCT cana-3839 114 1 + + CCONJ cana-3839 114 2 ( ( PUNCT cana-3839 114 3 ๐œ•2 ๐œ•2 NOUN cana-3839 114 4 + + CCONJ cana-3839 114 5 ๐œ•3 ๐œ•3 NOUN cana-3839 114 6 + + SYM cana-3839 114 7 ๐œ•4 ๐œ•4 NOUN cana-3839 114 8 + + CCONJ cana-3839 114 9 ๐œ•6)๐บ(๐œ‡ ๐œ•6)๐บ(๐œ‡ NOUN cana-3839 114 10 , , PUNCT cana-3839 114 11 โˆ(๐œ‡ โˆ(๐œ‡ ADV cana-3839 114 12 ) ) PUNCT cana-3839 114 13 , , PUNCT cana-3839 114 14 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 114 15 ) ) PUNCT cana-3839 114 16 ) ) PUNCT cana-3839 115 1 + + CCONJ cana-3839 115 2 ( ( PUNCT cana-3839 115 3 ๐œ•2 ๐œ•2 NOUN cana-3839 115 4 + + CCONJ cana-3839 115 5 ๐œ•3 ๐œ•3 NOUN cana-3839 115 6 + + SYM cana-3839 115 7 ๐œ•4 ๐œ•4 NOUN cana-3839 115 8 + + CCONJ cana-3839 115 9 ๐œ•6)๐บ ๐œ•6)๐บ PROPN cana-3839 115 10 ( ( PUNCT cana-3839 115 11 โ™ญ โ™ญ INTJ cana-3839 115 12 , , PUNCT cana-3839 115 13 โˆ โˆ PROPN cana-3839 115 14 ( ( PUNCT cana-3839 115 15 โ™ญ โ™ญ PROPN cana-3839 115 16 ) ) PUNCT cana-3839 115 17 , , PUNCT cana-3839 115 18 โˆ โˆ PROPN cana-3839 115 19 ( ( PUNCT cana-3839 115 20 โ™ญ โ™ญ PROPN cana-3839 115 21 ) ) PUNCT cana-3839 115 22 ) ) PUNCT cana-3839 116 1 + + CCONJ cana-3839 116 2 ( ( PUNCT cana-3839 116 3 ๐œ•5 ๐œ•5 NOUN cana-3839 116 4 + + NUM cana-3839 116 5 ๐œ•7)๐บ(๐œ‡ ๐œ•7)๐บ(๐œ‡ NOUN cana-3839 116 6 , , PUNCT cana-3839 116 7 โˆ โˆ PROPN cana-3839 116 8 ( ( PUNCT cana-3839 116 9 โ™ญ โ™ญ PROPN cana-3839 116 10 ) ) PUNCT cana-3839 116 11 , , PUNCT cana-3839 116 12 โˆ โˆ PROPN cana-3839 116 13 ( ( PUNCT cana-3839 116 14 โ™ญ โ™ญ PROPN cana-3839 116 15 ) ) PUNCT cana-3839 116 16 ) ) PUNCT cana-3839 117 1 + + CCONJ cana-3839 117 2 ( ( PUNCT cana-3839 117 3 ๐œ•5 ๐œ•5 PRON cana-3839 117 4 + + SYM cana-3839 117 5 ๐œ•7)๐บ ๐œ•7)๐บ PROPN cana-3839 117 6 ( ( PUNCT cana-3839 117 7 โ™ญ โ™ญ PROPN cana-3839 117 8 , , PUNCT cana-3839 117 9 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 117 10 ) ) PUNCT cana-3839 117 11 , , PUNCT cana-3839 117 12 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 117 13 ) ) PUNCT cana-3839 117 14 ) ) PUNCT cana-3839 117 15 ๐บ(โˆ(๐œ‡ ๐บ(โˆ(๐œ‡ NUM cana-3839 117 16 ) ) PUNCT cana-3839 117 17 , , PUNCT cana-3839 117 18 โˆ โˆ PROPN cana-3839 117 19 ( ( PUNCT cana-3839 117 20 โ™ญ โ™ญ PROPN cana-3839 117 21 ) ) PUNCT cana-3839 117 22 , , PUNCT cana-3839 117 23 โˆ โˆ PROPN cana-3839 117 24 ( ( PUNCT cana-3839 117 25 โ™ญ โ™ญ PROPN cana-3839 117 26 ) ) PUNCT cana-3839 117 27 ) ) PUNCT cana-3839 117 28 โชฏ โชฏ PROPN cana-3839 117 29 ๐œ•1๐บ(๐œ‡ ๐œ•1๐บ(๐œ‡ NUM cana-3839 117 30 , , PUNCT cana-3839 117 31 โ™ญ โ™ญ PROPN cana-3839 117 32 , , PUNCT cana-3839 117 33 โ™ญ โ™ญ PROPN cana-3839 117 34 ) ) PUNCT cana-3839 118 1 + + NUM cana-3839 118 2 ๐œ•2 ๐œ•2 NUM cana-3839 118 3 + + CCONJ cana-3839 118 4 ๐œ•3 ๐œ•3 NOUN cana-3839 118 5 + + SYM cana-3839 118 6 ๐œ•4 ๐œ•4 X cana-3839 118 7 + + CCONJ cana-3839 118 8 ๐œ•6 ๐œ•6 NUM cana-3839 118 9 2 2 NUM cana-3839 119 1 [ [ X cana-3839 119 2 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 119 3 , , PUNCT cana-3839 119 4 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 119 5 ) ) PUNCT cana-3839 119 6 , , PUNCT cana-3839 119 7 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 119 8 ) ) PUNCT cana-3839 119 9 ) ) PUNCT cana-3839 120 1 + + CCONJ cana-3839 120 2 ๐บ ๐บ PROPN cana-3839 120 3 ( ( PUNCT cana-3839 120 4 โ™ญ โ™ญ PROPN cana-3839 120 5 , , PUNCT cana-3839 120 6 โˆ โˆ PROPN cana-3839 120 7 ( ( PUNCT cana-3839 120 8 โ™ญ โ™ญ PROPN cana-3839 120 9 ) ) PUNCT cana-3839 120 10 , , PUNCT cana-3839 120 11 โˆ โˆ PROPN cana-3839 120 12 ( ( PUNCT cana-3839 120 13 โ™ญ โ™ญ PROPN cana-3839 120 14 ) ) PUNCT cana-3839 120 15 ) ) PUNCT cana-3839 120 16 ] ] PUNCT cana-3839 121 1 + + CCONJ cana-3839 121 2 ๐œ•5+๐œ•7 ๐œ•5+๐œ•7 NUM cana-3839 121 3 2 2 NUM cana-3839 122 1 [ [ X cana-3839 122 2 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 122 3 , , PUNCT cana-3839 122 4 โˆ โˆ NUM cana-3839 122 5 ( ( PUNCT cana-3839 122 6 โ™ญ โ™ญ PROPN cana-3839 122 7 ) ) PUNCT cana-3839 122 8 , , PUNCT cana-3839 122 9 โˆ โˆ PROPN cana-3839 122 10 ( ( PUNCT cana-3839 122 11 โ™ญ โ™ญ PROPN cana-3839 122 12 ) ) PUNCT cana-3839 122 13 ) ) PUNCT cana-3839 123 1 + + CCONJ cana-3839 123 2 ๐บ ๐บ PROPN cana-3839 123 3 ( ( PUNCT cana-3839 123 4 โ™ญ โ™ญ PROPN cana-3839 123 5 , , PUNCT cana-3839 123 6 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 123 7 ) ) PUNCT cana-3839 123 8 , , PUNCT cana-3839 123 9 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 123 10 ) ) PUNCT cana-3839 123 11 ) ) PUNCT cana-3839 123 12 ] ] PUNCT cana-3839 123 13 ( ( PUNCT cana-3839 123 14 4 4 X cana-3839 123 15 ) ) PUNCT cana-3839 123 16 putting put VERB cana-3839 123 17 โ™ญ โ™ญ X cana-3839 123 18 = = SYM cana-3839 123 19 โˆ(๐œ‡ โˆ(๐œ‡ X cana-3839 123 20 ) ) PUNCT cana-3839 123 21 in in ADP cana-3839 123 22 eq eq NOUN cana-3839 123 23 ( ( PUNCT cana-3839 123 24 4 4 NUM cana-3839 123 25 ) ) PUNCT cana-3839 123 26 ๐บ(โˆ(๐œ‡ ๐บ(โˆ(๐œ‡ NUM cana-3839 123 27 ) ) PUNCT cana-3839 123 28 , , PUNCT cana-3839 123 29 โˆ2(๐œ‡ โˆ2(๐œ‡ NUM cana-3839 123 30 ) ) PUNCT cana-3839 123 31 , , PUNCT cana-3839 123 32 โˆ2(๐œ‡ โˆ2(๐œ‡ NUM cana-3839 123 33 ) ) PUNCT cana-3839 123 34 ) ) PUNCT cana-3839 123 35 โชฏ โชฏ PROPN cana-3839 123 36 ๐œ•1๐บ(๐œ‡ ๐œ•1๐บ(๐œ‡ VERB cana-3839 123 37 , , PUNCT cana-3839 123 38 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 123 39 ) ) PUNCT cana-3839 123 40 , , PUNCT cana-3839 123 41 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 123 42 ) ) PUNCT cana-3839 123 43 ) ) PUNCT cana-3839 124 1 + + CCONJ cana-3839 124 2 ๐œ•2 ๐œ•2 NUM cana-3839 125 1 + + CCONJ cana-3839 125 2 ๐œ•3 ๐œ•3 NOUN cana-3839 125 3 + + SYM cana-3839 125 4 ๐œ•4 ๐œ•4 X cana-3839 125 5 + + CCONJ cana-3839 125 6 ๐œ•6 ๐œ•6 NUM cana-3839 125 7 2 2 NUM cana-3839 126 1 [ [ X cana-3839 126 2 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 126 3 , , PUNCT cana-3839 126 4 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 126 5 ) ) PUNCT cana-3839 126 6 , , PUNCT cana-3839 126 7 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 126 8 ) ) PUNCT cana-3839 126 9 ) ) PUNCT cana-3839 127 1 + + CCONJ cana-3839 127 2 ๐บ(โˆ(๐œ‡ ๐บ(โˆ(๐œ‡ NUM cana-3839 127 3 ) ) PUNCT cana-3839 127 4 , , PUNCT cana-3839 127 5 โˆ2(๐œ‡ โˆ2(๐œ‡ NUM cana-3839 127 6 ) ) PUNCT cana-3839 127 7 , , PUNCT cana-3839 127 8 โˆ2(๐œ‡ โˆ2(๐œ‡ NUM cana-3839 127 9 ) ) PUNCT cana-3839 127 10 ) ) PUNCT cana-3839 127 11 ] ] PUNCT cana-3839 128 1 + + CCONJ cana-3839 128 2 ๐œ•5+๐œ•7 ๐œ•5+๐œ•7 NUM cana-3839 128 3 2 2 NUM cana-3839 129 1 [ [ X cana-3839 129 2 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 129 3 , , PUNCT cana-3839 129 4 โˆ2(๐œ‡ โˆ2(๐œ‡ NUM cana-3839 129 5 ) ) PUNCT cana-3839 129 6 , , PUNCT cana-3839 129 7 โˆ2(๐œ‡ โˆ2(๐œ‡ NUM cana-3839 129 8 ) ) PUNCT cana-3839 129 9 ) ) PUNCT cana-3839 130 1 + + CCONJ cana-3839 130 2 ๐บ(โˆ(๐œ‡ ๐บ(โˆ(๐œ‡ NUM cana-3839 130 3 ) ) PUNCT cana-3839 130 4 , , PUNCT cana-3839 130 5 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 130 6 ) ) PUNCT cana-3839 130 7 , , PUNCT cana-3839 130 8 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 130 9 ) ) PUNCT cana-3839 130 10 ) ) PUNCT cana-3839 130 11 ] ] PUNCT cana-3839 131 1 ( ( PUNCT cana-3839 131 2 5 5 NUM cana-3839 131 3 ) ) PUNCT cana-3839 131 4 by by ADP cana-3839 131 5 using use VERB cana-3839 131 6 definition definition NOUN cana-3839 131 7 of of ADP cana-3839 131 8 vgms vgms NOUN cana-3839 131 9 , , PUNCT cana-3839 131 10 we we PRON cana-3839 131 11 get get VERB cana-3839 131 12 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 131 13 , , PUNCT cana-3839 131 14 โˆ2(๐œ‡ โˆ2(๐œ‡ NUM cana-3839 131 15 ) ) PUNCT cana-3839 131 16 , , PUNCT cana-3839 131 17 โˆ2(๐œ‡ โˆ2(๐œ‡ NUM cana-3839 131 18 ) ) PUNCT cana-3839 131 19 ) ) PUNCT cana-3839 132 1 โชฏ โชฏ NOUN cana-3839 132 2 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 132 3 , , PUNCT cana-3839 132 4 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 132 5 ) ) PUNCT cana-3839 132 6 , , PUNCT cana-3839 132 7 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 132 8 ) ) PUNCT cana-3839 132 9 ) ) PUNCT cana-3839 133 1 + + CCONJ cana-3839 133 2 ๐บ(โˆ(๐œ‡ ๐บ(โˆ(๐œ‡ NUM cana-3839 133 3 ) ) PUNCT cana-3839 133 4 , , PUNCT cana-3839 133 5 โˆ2(๐œ‡ โˆ2(๐œ‡ NUM cana-3839 133 6 ) ) PUNCT cana-3839 133 7 , , PUNCT cana-3839 133 8 โˆ2(๐œ‡ โˆ2(๐œ‡ NUM cana-3839 133 9 ) ) PUNCT cana-3839 133 10 ) ) PUNCT cana-3839 134 1 ( ( PUNCT cana-3839 134 2 6 6 NUM cana-3839 134 3 ) ) PUNCT cana-3839 134 4 then then ADV cana-3839 134 5 eq(5 eq(5 NOUN cana-3839 134 6 ) ) PUNCT cana-3839 134 7 becomes become VERB cana-3839 134 8 ๐บ(โˆ(๐œ‡ ๐บ(โˆ(๐œ‡ NOUN cana-3839 134 9 ) ) PUNCT cana-3839 134 10 , , PUNCT cana-3839 134 11 โˆ2(๐œ‡ โˆ2(๐œ‡ NUM cana-3839 134 12 ) ) PUNCT cana-3839 134 13 , , PUNCT cana-3839 134 14 โˆ2(๐œ‡ โˆ2(๐œ‡ NUM cana-3839 134 15 ) ) PUNCT cana-3839 134 16 ) ) PUNCT cana-3839 135 1 โชฏ โชฏ PROPN cana-3839 135 2 ๐œ•1๐บ(๐œ‡ ๐œ•1๐บ(๐œ‡ VERB cana-3839 135 3 , , PUNCT cana-3839 135 4 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 135 5 ) ) PUNCT cana-3839 135 6 , , PUNCT cana-3839 135 7 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 135 8 ) ) PUNCT cana-3839 135 9 ) ) PUNCT cana-3839 136 1 + + CCONJ cana-3839 136 2 ๐œ•2 ๐œ•2 NUM cana-3839 137 1 + + CCONJ cana-3839 137 2 ๐œ•3 ๐œ•3 NOUN cana-3839 137 3 + + SYM cana-3839 137 4 ๐œ•4 ๐œ•4 X cana-3839 137 5 + + CCONJ cana-3839 137 6 ๐œ•6 ๐œ•6 NUM cana-3839 137 7 2 2 NUM cana-3839 138 1 [ [ X cana-3839 138 2 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 138 3 , , PUNCT cana-3839 138 4 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 138 5 ) ) PUNCT cana-3839 138 6 , , PUNCT cana-3839 138 7 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 138 8 ) ) PUNCT cana-3839 138 9 ) ) PUNCT cana-3839 139 1 + + CCONJ cana-3839 139 2 ๐บ(โˆ(๐œ‡ ๐บ(โˆ(๐œ‡ NUM cana-3839 139 3 ) ) PUNCT cana-3839 139 4 , , PUNCT cana-3839 139 5 โˆ2(๐œ‡ โˆ2(๐œ‡ NUM cana-3839 139 6 ) ) PUNCT cana-3839 139 7 , , PUNCT cana-3839 139 8 โˆ2(๐œ‡ โˆ2(๐œ‡ NUM cana-3839 139 9 ) ) PUNCT cana-3839 139 10 ) ) PUNCT cana-3839 139 11 ] ] PUNCT cana-3839 140 1 + + CCONJ cana-3839 140 2 ๐œ•5+๐œ•7 ๐œ•5+๐œ•7 NUM cana-3839 140 3 2 2 NUM cana-3839 141 1 [ [ X cana-3839 141 2 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 141 3 , , PUNCT cana-3839 141 4 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 141 5 ) ) PUNCT cana-3839 141 6 , , PUNCT cana-3839 141 7 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 141 8 ) ) PUNCT cana-3839 141 9 ) ) PUNCT cana-3839 142 1 + + CCONJ cana-3839 142 2 communications communication NOUN cana-3839 142 3 on on ADP cana-3839 142 4 applied apply VERB cana-3839 142 5 nonlinear nonlinear ADJ cana-3839 142 6 analysis analysis NOUN cana-3839 142 7 issn issn NOUN cana-3839 142 8 : : PUNCT cana-3839 142 9 1074 1074 NUM cana-3839 142 10 - - PUNCT cana-3839 142 11 133x 133x NUM cana-3839 142 12 vol vol NOUN cana-3839 142 13 32 32 NUM cana-3839 142 14 no no NOUN cana-3839 142 15 . . PUNCT cana-3839 143 1 9s 9s NUM cana-3839 143 2 ( ( PUNCT cana-3839 143 3 2025 2025 NUM cana-3839 143 4 ) ) PUNCT cana-3839 143 5 81 81 NUM cana-3839 143 6 https://internationalpubls.com https://internationalpubls.com X cana-3839 143 7 ๐บ(โˆ(๐œ‡ ๐บ(โˆ(๐œ‡ NUM cana-3839 143 8 ) ) PUNCT cana-3839 143 9 , , PUNCT cana-3839 143 10 โˆ2(๐œ‡ โˆ2(๐œ‡ NUM cana-3839 143 11 ) ) PUNCT cana-3839 143 12 , , PUNCT cana-3839 143 13 โˆ2(๐œ‡ โˆ2(๐œ‡ NUM cana-3839 143 14 ) ) PUNCT cana-3839 143 15 ) ) PUNCT cana-3839 143 16 ] ] PUNCT cana-3839 144 1 ( ( PUNCT cana-3839 144 2 1 1 NUM cana-3839 144 3 โˆ’ โˆ’ PROPN cana-3839 144 4 โˆ‘7 โˆ‘7 PROPN cana-3839 144 5 ๐‘–=2 ๐‘–=2 PROPN cana-3839 144 6 ๐œ•๐‘– ๐œ•๐‘– NOUN cana-3839 144 7 2 2 NUM cana-3839 144 8 ) ) PUNCT cana-3839 144 9 ๐บ(โˆ(๐œ‡ ๐บ(โˆ(๐œ‡ NUM cana-3839 144 10 ) ) PUNCT cana-3839 144 11 , , PUNCT cana-3839 144 12 โˆ2(๐œ‡ โˆ2(๐œ‡ NUM cana-3839 144 13 ) ) PUNCT cana-3839 144 14 , , PUNCT cana-3839 144 15 โˆ2(๐œ‡ โˆ2(๐œ‡ NUM cana-3839 144 16 ) ) PUNCT cana-3839 144 17 ) ) PUNCT cana-3839 144 18 โชฏ โชฏ NOUN cana-3839 144 19 ( ( PUNCT cana-3839 144 20 ๐œ•1 ๐œ•1 VERB cana-3839 144 21 + + CCONJ cana-3839 144 22 โˆ‘7 โˆ‘7 PROPN cana-3839 144 23 ๐‘–=2 ๐‘–=2 PROPN cana-3839 144 24 ๐œ•๐‘– ๐œ•๐‘– NOUN cana-3839 144 25 2 2 NUM cana-3839 144 26 ) ) PUNCT cana-3839 144 27 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 144 28 , , PUNCT cana-3839 144 29 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 144 30 ) ) PUNCT cana-3839 144 31 , , PUNCT cana-3839 144 32 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 144 33 ) ) PUNCT cana-3839 144 34 ) ) PUNCT cana-3839 144 35 ๐บ(โˆ(๐œ‡ ๐บ(โˆ(๐œ‡ NUM cana-3839 144 36 ) ) PUNCT cana-3839 144 37 , , PUNCT cana-3839 144 38 โˆ2(๐œ‡ โˆ2(๐œ‡ NUM cana-3839 144 39 ) ) PUNCT cana-3839 144 40 , , PUNCT cana-3839 144 41 โˆ2(๐œ‡ โˆ2(๐œ‡ NUM cana-3839 144 42 ) ) PUNCT cana-3839 144 43 ) ) PUNCT cana-3839 144 44 โชฏ โชฏ NOUN cana-3839 144 45 ๐œ•1 ๐œ•1 VERB cana-3839 144 46 + + CCONJ cana-3839 144 47 โˆ‘7 โˆ‘7 PROPN cana-3839 144 48 ๐‘–=2 ๐‘–=2 PROPN cana-3839 144 49 ๐œ•๐‘– ๐œ•๐‘– NOUN cana-3839 144 50 2 2 NUM cana-3839 144 51 1โˆ’ 1โˆ’ NUM cana-3839 144 52 โˆ‘7 โˆ‘7 PROPN cana-3839 144 53 ๐‘–=2 ๐‘–=2 PROPN cana-3839 144 54 ๐œ•๐‘– ๐œ•๐‘– PROPN cana-3839 144 55 2 2 NUM cana-3839 144 56 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 144 57 , , PUNCT cana-3839 144 58 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 144 59 ) ) PUNCT cana-3839 144 60 , , PUNCT cana-3839 144 61 โˆ(๐œ‡ โˆ(๐œ‡ NOUN cana-3839 144 62 ) ) PUNCT cana-3839 144 63 ) ) PUNCT cana-3839 144 64 ( ( PUNCT cana-3839 144 65 7 7 X cana-3839 144 66 ) ) PUNCT cana-3839 144 67 putting put VERB cana-3839 144 68 ๐‘ž ๐‘ž X cana-3839 144 69 = = NOUN cana-3839 144 70 ๐œ•1 ๐œ•1 VERB cana-3839 144 71 + + CCONJ cana-3839 144 72 โˆ‘7 โˆ‘7 PROPN cana-3839 144 73 ๐‘–=2 ๐‘–=2 PROPN cana-3839 144 74 ๐œ•๐‘– ๐œ•๐‘– NOUN cana-3839 144 75 2 2 NUM cana-3839 144 76 1โˆ’ 1โˆ’ NUM cana-3839 144 77 โˆ‘7 โˆ‘7 PROPN cana-3839 144 78 ๐‘–=2 ๐‘–=2 PROPN cana-3839 144 79 ๐œ•๐‘– ๐œ•๐‘– PROPN cana-3839 144 80 2 2 NUM cana-3839 144 81 . . PUNCT cana-3839 145 1 since since SCONJ cana-3839 145 2 0 0 NUM cana-3839 145 3 โ‰ค โ‰ค NUM cana-3839 145 4 โˆ‘7 โˆ‘7 PROPN cana-3839 145 5 ๐‘–=1 ๐‘–=1 PROPN cana-3839 145 6 ๐œ•๐‘– ๐œ•๐‘– PROPN cana-3839 145 7 < < X cana-3839 145 8 1 1 NUM cana-3839 145 9 , , PUNCT cana-3839 145 10 so so ADV cana-3839 145 11 0 0 NUM cana-3839 145 12 โ‰ค โ‰ค NOUN cana-3839 145 13 ๐‘ž ๐‘ž X cana-3839 145 14 < < X cana-3839 145 15 1 1 NUM cana-3839 145 16 . . PUNCT cana-3839 145 17 then then ADV cana-3839 145 18 eq(7 eq(7 NOUN cana-3839 145 19 ) ) PUNCT cana-3839 145 20 becomes become VERB cana-3839 145 21 ๐บ(โˆ(๐œ‡ ๐บ(โˆ(๐œ‡ NOUN cana-3839 145 22 ) ) PUNCT cana-3839 145 23 , , PUNCT cana-3839 145 24 โˆ2(๐œ‡ โˆ2(๐œ‡ NUM cana-3839 145 25 ) ) PUNCT cana-3839 145 26 , , PUNCT cana-3839 145 27 โˆ2(๐œ‡ โˆ2(๐œ‡ NUM cana-3839 145 28 ) ) PUNCT cana-3839 145 29 ) ) PUNCT cana-3839 145 30 โชฏ โชฏ NOUN cana-3839 145 31 ๐‘ž๐บ(๐œ‡ ๐‘ž๐บ(๐œ‡ PRON cana-3839 145 32 , , PUNCT cana-3839 145 33 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 145 34 , , PUNCT cana-3839 145 35 โˆ๐œ‡ โˆ๐œ‡ PROPN cana-3839 145 36 ) ) PUNCT cana-3839 145 37 ( ( PUNCT cana-3839 145 38 8) 8) NUM cana-3839 145 39 โˆ€๐œ‡ โˆ€๐œ‡ NUM cana-3839 145 40 โˆˆ โˆˆ PROPN cana-3839 145 41 ๐œ’๐‘ž. ๐œ’๐‘ž. NOUN cana-3839 145 42 further far ADV cana-3839 145 43 , , PUNCT cana-3839 145 44 ๐บ(โˆ๐œŒ(๐œ‡ ๐บ(โˆ๐œŒ(๐œ‡ NOUN cana-3839 145 45 ) ) PUNCT cana-3839 145 46 , , PUNCT cana-3839 145 47 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 145 48 ) ) PUNCT cana-3839 145 49 , , PUNCT cana-3839 145 50 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 145 51 ) ) PUNCT cana-3839 145 52 ) ) PUNCT cana-3839 146 1 โชฏ โชฏ PROPN cana-3839 146 2 ๐‘ž๐œŒ๐บ(๐œ‡ ๐‘ž๐œŒ๐บ(๐œ‡ PROPN cana-3839 146 3 , , PUNCT cana-3839 146 4 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 146 5 , , PUNCT cana-3839 146 6 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 146 7 ) ) PUNCT cana-3839 146 8 โˆ€๐œŒ โˆ€๐œŒ PUNCT cana-3839 146 9 โˆˆ โˆˆ PROPN cana-3839 146 10 โ„• โ„• PROPN cana-3839 146 11 ๐œ‡ ๐œ‡ ADP cana-3839 146 12 โˆˆ โˆˆ NOUN cana-3839 146 13 ๐œ’๐‘ž ๐œ’๐‘ž NOUN cana-3839 146 14 . . PUNCT cana-3839 147 1 for for ADP cana-3839 147 2 ๐œŒ ๐œŒ ADP cana-3839 147 3 , , PUNCT cana-3839 147 4 ๐‘— ๐‘— PROPN cana-3839 147 5 โˆˆ โˆˆ PROPN cana-3839 147 6 โ„• โ„• PROPN cana-3839 147 7 and and CCONJ cana-3839 147 8 ๐œ‡ ๐œ‡ ADP cana-3839 147 9 โˆˆ โˆˆ PROPN cana-3839 147 10 ๐œ’๐‘ž ๐œ’๐‘ž NOUN cana-3839 147 11 ๐บ(โˆ๐œŒ(๐œ‡ ๐บ(โˆ๐œŒ(๐œ‡ NOUN cana-3839 147 12 ) ) PUNCT cana-3839 147 13 , , PUNCT cana-3839 147 14 โˆ๐œŒ+๐‘—(๐œ‡ โˆ๐œŒ+๐‘—(๐œ‡ ADP cana-3839 147 15 ) ) PUNCT cana-3839 147 16 , , PUNCT cana-3839 147 17 โˆ๐œŒ+๐‘—(๐œ‡ โˆ๐œŒ+๐‘—(๐œ‡ NUM cana-3839 147 18 ) ) PUNCT cana-3839 147 19 ) ) PUNCT cana-3839 147 20 โชฏ โชฏ NOUN cana-3839 147 21 ๐บ(โˆ๐œŒ(๐œ‡ ๐บ(โˆ๐œŒ(๐œ‡ NOUN cana-3839 147 22 ) ) PUNCT cana-3839 147 23 , , PUNCT cana-3839 147 24 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 147 25 ) ) PUNCT cana-3839 147 26 , , PUNCT cana-3839 147 27 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 147 28 ) ) PUNCT cana-3839 147 29 ) ) PUNCT cana-3839 148 1 + + CCONJ cana-3839 148 2 ๐บ(โˆ๐œŒ+1(๐œ‡ ๐บ(โˆ๐œŒ+1(๐œ‡ NUM cana-3839 148 3 ) ) PUNCT cana-3839 148 4 , , PUNCT cana-3839 148 5 โˆ๐œŒ+๐‘—(๐œ‡ โˆ๐œŒ+๐‘—(๐œ‡ ADP cana-3839 148 6 ) ) PUNCT cana-3839 148 7 , , PUNCT cana-3839 148 8 โˆ๐œŒ+๐‘—(๐œ‡ โˆ๐œŒ+๐‘—(๐œ‡ NUM cana-3839 148 9 ) ) PUNCT cana-3839 148 10 ) ) PUNCT cana-3839 149 1 โชฏ โชฏ NOUN cana-3839 149 2 ๐บ(โˆ๐œŒ(๐œ‡ ๐บ(โˆ๐œŒ(๐œ‡ NOUN cana-3839 149 3 ) ) PUNCT cana-3839 149 4 , , PUNCT cana-3839 149 5 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 149 6 ) ) PUNCT cana-3839 149 7 , , PUNCT cana-3839 149 8 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 149 9 ) ) PUNCT cana-3839 149 10 ) ) PUNCT cana-3839 150 1 + + CCONJ cana-3839 150 2 ๐บ(โˆ๐œŒ+1(๐œ‡ ๐บ(โˆ๐œŒ+1(๐œ‡ NUM cana-3839 150 3 ) ) PUNCT cana-3839 150 4 , , PUNCT cana-3839 150 5 โˆ๐œŒ+2(๐œ‡ โˆ๐œŒ+2(๐œ‡ ADJ cana-3839 150 6 ) ) PUNCT cana-3839 150 7 , , PUNCT cana-3839 150 8 โˆ๐œŒ+2(๐œ‡ โˆ๐œŒ+2(๐œ‡ ADJ cana-3839 150 9 ) ) PUNCT cana-3839 150 10 ) ) PUNCT cana-3839 151 1 + + VERB cana-3839 151 2 ๐บ(โˆ๐œŒ+2(๐œ‡ ๐บ(โˆ๐œŒ+2(๐œ‡ ADJ cana-3839 151 3 ) ) PUNCT cana-3839 151 4 , , PUNCT cana-3839 151 5 โˆ๐œŒ+๐‘—(๐œ‡ โˆ๐œŒ+๐‘—(๐œ‡ ADP cana-3839 151 6 ) ) PUNCT cana-3839 151 7 , , PUNCT cana-3839 151 8 โˆ๐œŒ+๐‘—(๐œ‡ โˆ๐œŒ+๐‘—(๐œ‡ NUM cana-3839 151 9 ) ) PUNCT cana-3839 151 10 โ‹ฎ โ‹ฎ NOUN cana-3839 151 11 โชฏ โชฏ VERB cana-3839 151 12 โˆ‘๐‘— โˆ‘๐‘— PROPN cana-3839 151 13 ๐‘–=1 ๐‘–=1 PROPN cana-3839 151 14 ๐บ(โˆ๐œŒ+๐‘–โˆ’1(๐œ‡ ๐บ(โˆ๐œŒ+๐‘–โˆ’1(๐œ‡ NOUN cana-3839 151 15 ) ) PUNCT cana-3839 151 16 , , PUNCT cana-3839 151 17 โˆ๐œŒ+๐‘–(๐œ‡ โˆ๐œŒ+๐‘–(๐œ‡ NUM cana-3839 151 18 ) ) PUNCT cana-3839 151 19 , , PUNCT cana-3839 151 20 โˆ๐œŒ+๐‘–(๐œ‡ โˆ๐œŒ+๐‘–(๐œ‡ NUM cana-3839 151 21 ) ) PUNCT cana-3839 151 22 ) ) PUNCT cana-3839 152 1 โชฏ โชฏ VERB cana-3839 152 2 โˆ‘๐‘— โˆ‘๐‘— PROPN cana-3839 152 3 ๐‘–=1 ๐‘–=1 PROPN cana-3839 152 4 ๐‘ž๐œŒ+๐‘–โˆ’1๐บ(๐œ‡ ๐‘ž๐œŒ+๐‘–โˆ’1๐บ(๐œ‡ X cana-3839 152 5 , , PUNCT cana-3839 152 6 โˆ๐œ‡ โˆ๐œ‡ VERB cana-3839 152 7 , , PUNCT cana-3839 152 8 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 152 9 ) ) PUNCT cana-3839 152 10 ๐บ(โˆ๐œŒ(๐œ‡ ๐บ(โˆ๐œŒ(๐œ‡ NUM cana-3839 152 11 ) ) PUNCT cana-3839 152 12 , , PUNCT cana-3839 152 13 โˆ๐œŒ+๐‘—(๐œ‡ โˆ๐œŒ+๐‘—(๐œ‡ ADP cana-3839 152 14 ) ) PUNCT cana-3839 152 15 , , PUNCT cana-3839 152 16 โˆ๐œŒ+๐‘—(๐œ‡ โˆ๐œŒ+๐‘—(๐œ‡ NUM cana-3839 152 17 ) ) PUNCT cana-3839 152 18 ) ) PUNCT cana-3839 152 19 โชฏ โชฏ NOUN cana-3839 152 20 ๐‘ž๐œŒ ๐‘ž๐œŒ PROPN cana-3839 152 21 1โˆ’๐‘ž 1โˆ’๐‘ž NUM cana-3839 152 22 ๐บ(๐œ‡ ๐บ(๐œ‡ ADP cana-3839 152 23 , , PUNCT cana-3839 152 24 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 152 25 , , PUNCT cana-3839 152 26 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 152 27 ) ) PUNCT cana-3839 152 28 ( ( PUNCT cana-3839 152 29 9 9 X cana-3839 152 30 ) ) PUNCT cana-3839 152 31 this this PRON cana-3839 152 32 implies imply VERB cana-3839 152 33 โŒฉ โŒฉ ADV cana-3839 152 34 โˆ๐œŒ๐œ‡ โˆ๐œŒ๐œ‡ NOUN cana-3839 152 35 โŒช โŒช NOUN cana-3839 152 36 is be AUX cana-3839 152 37 a a DET cana-3839 152 38 vcs vcs NOUN cana-3839 152 39 in in ADP cana-3839 152 40 g g NOUN cana-3839 152 41 - - PUNCT cana-3839 152 42 complete complete ADJ cana-3839 152 43 vgms vgms NOUN cana-3839 152 44 and and CCONJ cana-3839 152 45 it it PRON cana-3839 152 46 converges converge VERB cana-3839 152 47 to to ADP cana-3839 152 48 ๐œ‡1 ๐œ‡1 PROPN cana-3839 152 49 โˆˆ โˆˆ PROPN cana-3839 152 50 โ„œ. โ„œ. PROPN cana-3839 153 1 so so ADV cana-3839 153 2 , , PUNCT cana-3839 153 3 โˆƒ โˆƒ PROPN cana-3839 153 4 a a DET cana-3839 153 5 sequence sequence NOUN cana-3839 153 6 ๐œ‡๐œŒ ๐œ‡๐œŒ NOUN cana-3839 153 7 โˆˆ โˆˆ NOUN cana-3839 153 8 ๐‘‰ ๐‘‰ PROPN cana-3839 153 9 such such ADJ cana-3839 153 10 that that SCONJ cana-3839 153 11 ๐œ‡๐œŒ ๐œ‡๐œŒ VERB cana-3839 153 12 โ†“ โ†“ NOUN cana-3839 153 13 0 0 NUM cana-3839 153 14 and and CCONJ cana-3839 153 15 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADJ cana-3839 153 16 , , PUNCT cana-3839 153 17 โˆ๐œŒ(๐œ‡ โˆ๐œŒ(๐œ‡ NOUN cana-3839 153 18 ) ) PUNCT cana-3839 153 19 , , PUNCT cana-3839 153 20 โˆ๐œŒ(๐œ‡ โˆ๐œŒ(๐œ‡ NOUN cana-3839 153 21 ) ) PUNCT cana-3839 153 22 ) ) PUNCT cana-3839 154 1 โชฏ โชฏ NOUN cana-3839 154 2 ๐œ‡๐œŒ. ๐œ‡๐œŒ. VERB cana-3839 154 3 now now ADV cana-3839 155 1 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADJ cana-3839 155 2 , , PUNCT cana-3839 155 3 โˆ((๐œ‡1 โˆ((๐œ‡1 NUM cana-3839 155 4 ) ) PUNCT cana-3839 155 5 ) ) PUNCT cana-3839 155 6 , , PUNCT cana-3839 155 7 โˆ(๐œ‡1 โˆ(๐œ‡1 PROPN cana-3839 155 8 ) ) PUNCT cana-3839 155 9 ) ) PUNCT cana-3839 156 1 โชฏ โชฏ PROPN cana-3839 156 2 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADV cana-3839 156 3 , , PUNCT cana-3839 156 4 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 156 5 ) ) PUNCT cana-3839 156 6 , , PUNCT cana-3839 156 7 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 156 8 ) ) PUNCT cana-3839 156 9 ) ) PUNCT cana-3839 157 1 + + CCONJ cana-3839 157 2 ๐บ(โˆ๐œŒ+1(๐œ‡ ๐บ(โˆ๐œŒ+1(๐œ‡ NUM cana-3839 157 3 ) ) PUNCT cana-3839 157 4 , , PUNCT cana-3839 157 5 โˆ(๐œ‡1 โˆ(๐œ‡1 PROPN cana-3839 157 6 ) ) PUNCT cana-3839 157 7 , , PUNCT cana-3839 157 8 โˆ(๐œ‡1 โˆ(๐œ‡1 PROPN cana-3839 157 9 ) ) PUNCT cana-3839 157 10 ) ) PUNCT cana-3839 158 1 โชฏ โชฏ PROPN cana-3839 158 2 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADV cana-3839 158 3 , , PUNCT cana-3839 158 4 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 158 5 ) ) PUNCT cana-3839 158 6 , , PUNCT cana-3839 158 7 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 158 8 ) ) PUNCT cana-3839 158 9 ) ) PUNCT cana-3839 159 1 + + CCONJ cana-3839 159 2 ๐บ(โˆ(๐œ‡1 ๐บ(โˆ(๐œ‡1 NOUN cana-3839 159 3 ) ) PUNCT cana-3839 159 4 , , PUNCT cana-3839 159 5 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 159 6 ) ) PUNCT cana-3839 159 7 , , PUNCT cana-3839 159 8 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 159 9 ) ) PUNCT cana-3839 159 10 ) ) PUNCT cana-3839 159 11 by by ADP cana-3839 159 12 using use VERB cana-3839 159 13 ( ( PUNCT cana-3839 159 14 4 4 NUM cana-3839 159 15 ) ) PUNCT cana-3839 159 16 , , PUNCT cana-3839 159 17 we we PRON cana-3839 159 18 get get VERB cana-3839 159 19 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADJ cana-3839 159 20 , , PUNCT cana-3839 159 21 โˆ(๐œ‡1 โˆ(๐œ‡1 PROPN cana-3839 159 22 ) ) PUNCT cana-3839 159 23 , , PUNCT cana-3839 159 24 โˆ(๐œ‡1 โˆ(๐œ‡1 PROPN cana-3839 159 25 ) ) PUNCT cana-3839 159 26 ) ) PUNCT cana-3839 160 1 โชฏ โชฏ PROPN cana-3839 160 2 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADV cana-3839 160 3 , , PUNCT cana-3839 160 4 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 160 5 ) ) PUNCT cana-3839 160 6 , , PUNCT cana-3839 160 7 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 160 8 ) ) PUNCT cana-3839 160 9 ) ) PUNCT cana-3839 161 1 + + CCONJ cana-3839 161 2 ๐œ•1๐บ(๐œ‡1 ๐œ•1๐บ(๐œ‡1 PROPN cana-3839 161 3 , , PUNCT cana-3839 161 4 โˆ๐œŒ(๐œ‡ โˆ๐œŒ(๐œ‡ NOUN cana-3839 161 5 ) ) PUNCT cana-3839 161 6 , , PUNCT cana-3839 161 7 โˆ๐œŒ(๐œ‡ โˆ๐œŒ(๐œ‡ NOUN cana-3839 161 8 ) ) PUNCT cana-3839 161 9 ) ) PUNCT cana-3839 162 1 + + CCONJ cana-3839 162 2 communications communication NOUN cana-3839 162 3 on on ADP cana-3839 162 4 applied apply VERB cana-3839 162 5 nonlinear nonlinear ADJ cana-3839 162 6 analysis analysis NOUN cana-3839 162 7 issn issn NOUN cana-3839 162 8 : : PUNCT cana-3839 162 9 1074 1074 NUM cana-3839 162 10 - - PUNCT cana-3839 162 11 133x 133x NUM cana-3839 162 12 vol vol NOUN cana-3839 162 13 32 32 NUM cana-3839 162 14 no no NOUN cana-3839 162 15 . . PUNCT cana-3839 163 1 9s 9s NUM cana-3839 163 2 ( ( PUNCT cana-3839 163 3 2025 2025 NUM cana-3839 163 4 ) ) PUNCT cana-3839 163 5 82 82 NUM cana-3839 164 1 https://internationalpubls.com https://internationalpubls.com X cana-3839 164 2 ๐œ•2+๐œ•3+๐œ•4+๐œ•6 ๐œ•2+๐œ•3+๐œ•4+๐œ•6 X cana-3839 164 3 2 2 NUM cana-3839 164 4 [ [ X cana-3839 164 5 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADJ cana-3839 164 6 , , PUNCT cana-3839 164 7 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 164 8 , , PUNCT cana-3839 164 9 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 164 10 ) ) PUNCT cana-3839 164 11 + + CCONJ cana-3839 164 12 ๐บ(โˆ๐œŒ(๐œ‡ ๐บ(โˆ๐œŒ(๐œ‡ NUM cana-3839 164 13 ) ) PUNCT cana-3839 164 14 , , PUNCT cana-3839 164 15 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 164 16 ) ) PUNCT cana-3839 164 17 , , PUNCT cana-3839 164 18 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 164 19 ) ) PUNCT cana-3839 164 20 ) ) PUNCT cana-3839 164 21 ] ] PUNCT cana-3839 165 1 + + CCONJ cana-3839 165 2 ๐œ•5+๐œ•7 ๐œ•5+๐œ•7 NUM cana-3839 165 3 2 2 NUM cana-3839 165 4 [ [ X cana-3839 165 5 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADJ cana-3839 165 6 , , PUNCT cana-3839 165 7 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 165 8 ) ) PUNCT cana-3839 165 9 , , PUNCT cana-3839 165 10 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 165 11 ) ) PUNCT cana-3839 165 12 ) ) PUNCT cana-3839 166 1 + + CCONJ cana-3839 166 2 ๐บ(โˆ๐œŒ(๐œ‡ ๐บ(โˆ๐œŒ(๐œ‡ NUM cana-3839 166 3 ) ) PUNCT cana-3839 166 4 , , PUNCT cana-3839 166 5 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 166 6 , , PUNCT cana-3839 166 7 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 166 8 ) ) PUNCT cana-3839 166 9 ] ] PUNCT cana-3839 167 1 โชฏ โชฏ X cana-3839 167 2 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADV cana-3839 167 3 , , PUNCT cana-3839 167 4 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 167 5 ) ) PUNCT cana-3839 167 6 , , PUNCT cana-3839 167 7 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 167 8 ) ) PUNCT cana-3839 167 9 ) ) PUNCT cana-3839 168 1 + + CCONJ cana-3839 168 2 ๐œ•1๐บ(๐œ‡1 ๐œ•1๐บ(๐œ‡1 PROPN cana-3839 168 3 , , PUNCT cana-3839 168 4 โˆ๐œŒ(๐œ‡ โˆ๐œŒ(๐œ‡ NOUN cana-3839 168 5 ) ) PUNCT cana-3839 168 6 , , PUNCT cana-3839 168 7 โˆ๐œŒ(๐œ‡ โˆ๐œŒ(๐œ‡ NOUN cana-3839 168 8 ) ) PUNCT cana-3839 168 9 ) ) PUNCT cana-3839 169 1 + + CCONJ cana-3839 169 2 ๐œ•2+๐œ•3+๐œ•4+๐œ•6 ๐œ•2+๐œ•3+๐œ•4+๐œ•6 NOUN cana-3839 169 3 2 2 NUM cana-3839 169 4 [ [ X cana-3839 169 5 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADJ cana-3839 169 6 , , PUNCT cana-3839 169 7 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 169 8 , , PUNCT cana-3839 169 9 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 169 10 ) ) PUNCT cana-3839 169 11 + + CCONJ cana-3839 169 12 ๐บ(โˆ๐œŒ(๐œ‡ ๐บ(โˆ๐œŒ(๐œ‡ NUM cana-3839 169 13 ) ) PUNCT cana-3839 169 14 , , PUNCT cana-3839 169 15 ๐œ‡1 ๐œ‡1 ADJ cana-3839 169 16 , , PUNCT cana-3839 169 17 ๐œ‡1 ๐œ‡1 NOUN cana-3839 169 18 ) ) PUNCT cana-3839 169 19 + + CCONJ cana-3839 169 20 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADJ cana-3839 169 21 , , PUNCT cana-3839 169 22 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 169 23 ) ) PUNCT cana-3839 169 24 , , PUNCT cana-3839 169 25 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 169 26 ) ) PUNCT cana-3839 169 27 ) ) PUNCT cana-3839 169 28 ] ] PUNCT cana-3839 170 1 + + CCONJ cana-3839 170 2 ๐œ•5+๐œ•7 ๐œ•5+๐œ•7 NUM cana-3839 170 3 2 2 NUM cana-3839 170 4 [ [ X cana-3839 170 5 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADJ cana-3839 170 6 , , PUNCT cana-3839 170 7 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 170 8 ) ) PUNCT cana-3839 170 9 , , PUNCT cana-3839 170 10 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 170 11 ) ) PUNCT cana-3839 170 12 ) ) PUNCT cana-3839 171 1 + + CCONJ cana-3839 171 2 ๐บ(โˆ๐œŒ(๐œ‡ ๐บ(โˆ๐œŒ(๐œ‡ NUM cana-3839 171 3 ) ) PUNCT cana-3839 171 4 , , PUNCT cana-3839 171 5 ๐œ‡1 ๐œ‡1 ADJ cana-3839 171 6 , , PUNCT cana-3839 171 7 ๐œ‡1 ๐œ‡1 NOUN cana-3839 171 8 ) ) PUNCT cana-3839 172 1 + + CCONJ cana-3839 172 2 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADJ cana-3839 172 3 , , PUNCT cana-3839 172 4 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 172 5 , , PUNCT cana-3839 172 6 โˆ๐œ‡1 โˆ๐œ‡1 NOUN cana-3839 172 7 ) ) PUNCT cana-3839 172 8 ] ] PUNCT cana-3839 173 1 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADJ cana-3839 173 2 , , PUNCT cana-3839 173 3 โˆ(๐œ‡1 โˆ(๐œ‡1 PROPN cana-3839 173 4 ) ) PUNCT cana-3839 173 5 , , PUNCT cana-3839 173 6 โˆ(๐œ‡1 โˆ(๐œ‡1 PROPN cana-3839 173 7 ) ) PUNCT cana-3839 173 8 ) ) PUNCT cana-3839 173 9 โชฏ โชฏ PROPN cana-3839 174 1 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADV cana-3839 174 2 , , PUNCT cana-3839 174 3 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 174 4 ) ) PUNCT cana-3839 174 5 , , PUNCT cana-3839 174 6 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 174 7 ) ) PUNCT cana-3839 174 8 ) ) PUNCT cana-3839 175 1 + + CCONJ cana-3839 175 2 ๐œ•1๐บ(๐œ‡1 ๐œ•1๐บ(๐œ‡1 PROPN cana-3839 175 3 , , PUNCT cana-3839 175 4 โˆ๐œŒ(๐œ‡ โˆ๐œŒ(๐œ‡ NOUN cana-3839 175 5 ) ) PUNCT cana-3839 175 6 , , PUNCT cana-3839 175 7 โˆ๐œŒ(๐œ‡ โˆ๐œŒ(๐œ‡ NOUN cana-3839 175 8 ) ) PUNCT cana-3839 175 9 ) ) PUNCT cana-3839 176 1 + + CCONJ cana-3839 176 2 ๐œ•2+๐œ•3+๐œ•4+๐œ•6 ๐œ•2+๐œ•3+๐œ•4+๐œ•6 NOUN cana-3839 176 3 2 2 NUM cana-3839 176 4 [ [ X cana-3839 176 5 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADJ cana-3839 176 6 , , PUNCT cana-3839 176 7 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 176 8 , , PUNCT cana-3839 176 9 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 176 10 ) ) PUNCT cana-3839 176 11 + + CCONJ cana-3839 177 1 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADJ cana-3839 177 2 , , PUNCT cana-3839 177 3 โˆ๐œŒ(๐œ‡ โˆ๐œŒ(๐œ‡ NOUN cana-3839 177 4 ) ) PUNCT cana-3839 177 5 , , PUNCT cana-3839 177 6 โˆ๐œŒ(๐œ‡ โˆ๐œŒ(๐œ‡ NOUN cana-3839 177 7 ) ) PUNCT cana-3839 177 8 ) ) PUNCT cana-3839 178 1 + + CCONJ cana-3839 178 2 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADJ cana-3839 178 3 , , PUNCT cana-3839 178 4 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 178 5 ) ) PUNCT cana-3839 178 6 , , PUNCT cana-3839 178 7 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 178 8 ) ) PUNCT cana-3839 178 9 ) ) PUNCT cana-3839 178 10 ] ] PUNCT cana-3839 179 1 + + CCONJ cana-3839 179 2 ๐œ•5+๐œ•7 ๐œ•5+๐œ•7 NUM cana-3839 179 3 2 2 NUM cana-3839 179 4 [ [ X cana-3839 179 5 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADJ cana-3839 179 6 , , PUNCT cana-3839 179 7 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 179 8 ) ) PUNCT cana-3839 179 9 , , PUNCT cana-3839 179 10 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 179 11 ) ) PUNCT cana-3839 179 12 ) ) PUNCT cana-3839 180 1 + + CCONJ cana-3839 181 1 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADJ cana-3839 181 2 , , PUNCT cana-3839 181 3 โˆ๐œŒ(๐œ‡ โˆ๐œŒ(๐œ‡ NOUN cana-3839 181 4 ) ) PUNCT cana-3839 181 5 , , PUNCT cana-3839 181 6 โˆ๐œŒ(๐œ‡ โˆ๐œŒ(๐œ‡ NOUN cana-3839 181 7 ) ) PUNCT cana-3839 181 8 ) ) PUNCT cana-3839 182 1 + + CCONJ cana-3839 182 2 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADJ cana-3839 182 3 , , PUNCT cana-3839 182 4 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 182 5 , , PUNCT cana-3839 182 6 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 182 7 ) ) PUNCT cana-3839 182 8 ] ] PUNCT cana-3839 183 1 โชฏ โชฏ NOUN cana-3839 183 2 ๐œ‡๐œŒ+1 ๐œ‡๐œŒ+1 NOUN cana-3839 184 1 + + CCONJ cana-3839 184 2 ๐œ•1๐œ‡๐œŒ ๐œ•1๐œ‡๐œŒ PROPN cana-3839 184 3 + + CCONJ cana-3839 184 4 ๐œ•2+๐œ•3+๐œ•4+๐œ•6 ๐œ•2+๐œ•3+๐œ•4+๐œ•6 NOUN cana-3839 184 5 2 2 NUM cana-3839 184 6 [ [ X cana-3839 184 7 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADJ cana-3839 184 8 , , PUNCT cana-3839 184 9 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 184 10 , , PUNCT cana-3839 184 11 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 184 12 ) ) PUNCT cana-3839 185 1 + + NUM cana-3839 185 2 ๐œ‡๐œŒ ๐œ‡๐œŒ NOUN cana-3839 185 3 + + ADJ cana-3839 185 4 ๐œ‡๐œŒ+1 ๐œ‡๐œŒ+1 NOUN cana-3839 185 5 ] ] X cana-3839 185 6 + + CCONJ cana-3839 185 7 ๐œ•5+๐œ•7 ๐œ•5+๐œ•7 NUM cana-3839 185 8 2 2 NUM cana-3839 185 9 [ [ X cana-3839 185 10 ๐œ‡๐œŒ+1 ๐œ‡๐œŒ+1 NOUN cana-3839 185 11 + + X cana-3839 185 12 ๐œ‡๐œŒ ๐œ‡๐œŒ NOUN cana-3839 185 13 + + CCONJ cana-3839 185 14 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADJ cana-3839 185 15 , , PUNCT cana-3839 185 16 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 185 17 , , PUNCT cana-3839 185 18 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 185 19 ) ) PUNCT cana-3839 185 20 ] ] PUNCT cana-3839 186 1 โชฏ โชฏ NOUN cana-3839 186 2 ๐œ‡๐œŒ ๐œ‡๐œŒ NOUN cana-3839 187 1 + + CCONJ cana-3839 187 2 ๐œ•1๐œ‡๐œŒ ๐œ•1๐œ‡๐œŒ PROPN cana-3839 187 3 + + CCONJ cana-3839 187 4 ๐œ•2+๐œ•3+๐œ•4+๐œ•6 ๐œ•2+๐œ•3+๐œ•4+๐œ•6 NOUN cana-3839 187 5 2 2 NUM cana-3839 188 1 [ [ X cana-3839 188 2 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADJ cana-3839 188 3 , , PUNCT cana-3839 188 4 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 188 5 , , PUNCT cana-3839 188 6 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 188 7 ) ) PUNCT cana-3839 189 1 + + CCONJ cana-3839 189 2 ๐œ‡๐œŒ ๐œ‡๐œŒ NOUN cana-3839 189 3 + + PRON cana-3839 189 4 ๐œ‡๐œŒ ๐œ‡๐œŒ NOUN cana-3839 189 5 ] ] X cana-3839 189 6 + + CCONJ cana-3839 189 7 ๐œ•5+๐œ•7 ๐œ•5+๐œ•7 NUM cana-3839 189 8 2 2 NUM cana-3839 190 1 [ [ NOUN cana-3839 190 2 ๐œ‡๐œŒ ๐œ‡๐œŒ NOUN cana-3839 190 3 + + PRON cana-3839 190 4 ๐œ‡๐œŒ ๐œ‡๐œŒ NOUN cana-3839 190 5 + + CCONJ cana-3839 190 6 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADJ cana-3839 190 7 , , PUNCT cana-3839 190 8 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 190 9 , , PUNCT cana-3839 190 10 โˆ๐œ‡1 โˆ๐œ‡1 NOUN cana-3839 190 11 ) ) PUNCT cana-3839 190 12 ] ] PUNCT cana-3839 191 1 ( ( PUNCT cana-3839 191 2 10 10 NUM cana-3839 191 3 ) ) PUNCT cana-3839 191 4 ( ( PUNCT cana-3839 191 5 1 1 NUM cana-3839 191 6 โˆ’ โˆ’ PROPN cana-3839 191 7 โˆ‘7 โˆ‘7 PROPN cana-3839 191 8 ๐‘–=2 ๐‘–=2 PROPN cana-3839 191 9 ๐œ•๐‘– ๐œ•๐‘– PROPN cana-3839 191 10 2 2 NUM cana-3839 191 11 ) ) PUNCT cana-3839 191 12 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADJ cana-3839 191 13 , , PUNCT cana-3839 191 14 โˆ(๐œ‡1 โˆ(๐œ‡1 PROPN cana-3839 191 15 ) ) PUNCT cana-3839 191 16 , , PUNCT cana-3839 191 17 โˆ(๐œ‡1 โˆ(๐œ‡1 PROPN cana-3839 191 18 ) ) PUNCT cana-3839 191 19 ) ) PUNCT cana-3839 191 20 โชฏ โชฏ VERB cana-3839 191 21 2๐œ•1+โˆ‘7 2๐œ•1+โˆ‘7 NUM cana-3839 191 22 ๐‘–=2 ๐‘–=2 PUNCT cana-3839 191 23 ๐œ•๐‘– ๐œ•๐‘– NOUN cana-3839 191 24 2 2 NUM cana-3839 191 25 ๐œ‡๐œŒ ๐œ‡๐œŒ NOUN cana-3839 191 26 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADJ cana-3839 191 27 , , PUNCT cana-3839 191 28 โˆ(๐œ‡1 โˆ(๐œ‡1 PROPN cana-3839 191 29 ) ) PUNCT cana-3839 191 30 , , PUNCT cana-3839 191 31 โˆ(๐œ‡1 โˆ(๐œ‡1 PROPN cana-3839 191 32 ) ) PUNCT cana-3839 191 33 ) ) PUNCT cana-3839 192 1 โชฏ โชฏ VERB cana-3839 192 2 2๐œ•1+โˆ‘7 2๐œ•1+โˆ‘7 NUM cana-3839 192 3 ๐‘–=2 ๐‘–=2 PUNCT cana-3839 192 4 ๐œ•๐‘– ๐œ•๐‘– NOUN cana-3839 192 5 2โˆ’โˆ‘7 2โˆ’โˆ‘7 PROPN cana-3839 192 6 ๐‘–=2 ๐‘–=2 PROPN cana-3839 192 7 ๐œ•๐‘– ๐œ•๐‘– NOUN cana-3839 192 8 ๐œ‡๐œŒ ๐œ‡๐œŒ VERB cana-3839 192 9 โ†“ โ†“ NOUN cana-3839 192 10 0 0 NUM cana-3839 193 1 it it PRON cana-3839 193 2 follows follow VERB cana-3839 193 3 that that DET cana-3839 193 4 ๐œ‡1 ๐œ‡1 NOUN cana-3839 193 5 = = SYM cana-3839 193 6 โˆ๐œ‡1 โˆ๐œ‡1 NOUN cana-3839 193 7 . . PUNCT cana-3839 194 1 now now ADV cana-3839 194 2 , , PUNCT cana-3839 194 3 we we PRON cana-3839 194 4 assert assert VERB cana-3839 194 5 that that SCONJ cana-3839 194 6 ๐œ‡1 ๐œ‡1 PROPN cana-3839 194 7 is be AUX cana-3839 194 8 unique unique ADJ cana-3839 194 9 fp fp NOUN cana-3839 194 10 . . PUNCT cana-3839 195 1 if if SCONJ cana-3839 195 2 ๐œ•1 ๐œ•1 NOUN cana-3839 195 3 is be AUX cana-3839 195 4 another another DET cana-3839 195 5 fp fp NOUN cana-3839 195 6 of of ADP cana-3839 195 7 โˆ โˆ PROPN cana-3839 195 8 , , PUNCT cana-3839 195 9 then then ADV cana-3839 195 10 โˆ€๐œ‡1 โˆ€๐œ‡1 ADP cana-3839 195 11 โˆˆ โˆˆ PROPN cana-3839 195 12 ๐œ’๐‘ž ๐œ’๐‘ž NOUN cana-3839 195 13 and and CCONJ cana-3839 195 14 ๐œ•1 ๐œ•1 VERB cana-3839 195 15 โˆˆ โˆˆ PROPN cana-3839 195 16 ๐œ’๐‘ž+1 ๐œ’๐‘ž+1 NOUN cana-3839 195 17 , , PUNCT cana-3839 195 18 ๐‘— ๐‘— NOUN cana-3839 195 19 = = SYM cana-3839 195 20 1,2 1,2 NUM cana-3839 195 21 , , PUNCT cana-3839 195 22 โ€ฆ โ€ฆ PUNCT cana-3839 195 23 ๐‘š ๐‘š X cana-3839 195 24 , , PUNCT cana-3839 195 25 we we PRON cana-3839 195 26 have have AUX cana-3839 195 27 ๐บ(๐œ‡1 ๐บ(๐œ‡1 PROPN cana-3839 195 28 , , PUNCT cana-3839 195 29 ๐œ•1 ๐œ•1 NOUN cana-3839 195 30 , , PUNCT cana-3839 195 31 ๐œ•1 ๐œ•1 NUM cana-3839 195 32 ) ) PUNCT cana-3839 195 33 = = PUNCT cana-3839 195 34 ๐บ(โˆ๐œ‡1 ๐บ(โˆ๐œ‡1 NUM cana-3839 195 35 , , PUNCT cana-3839 195 36 โˆ๐œ•1 โˆ๐œ•1 NUM cana-3839 195 37 , , PUNCT cana-3839 195 38 โˆ๐œ•1 โˆ๐œ•1 NUM cana-3839 195 39 ) ) PUNCT cana-3839 195 40 โชฏ โชฏ PROPN cana-3839 195 41 ๐œ•1๐บ(๐œ‡1 ๐œ•1๐บ(๐œ‡1 PROPN cana-3839 195 42 , , PUNCT cana-3839 195 43 ๐œ•1 ๐œ•1 NOUN cana-3839 195 44 , , PUNCT cana-3839 195 45 ๐œ•1 ๐œ•1 NOUN cana-3839 195 46 ) ) PUNCT cana-3839 196 1 + + CCONJ cana-3839 196 2 ๐œ•2๐บ(๐œ‡1 ๐œ•2๐บ(๐œ‡1 ADJ cana-3839 196 3 , , PUNCT cana-3839 196 4 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 196 5 , , PUNCT cana-3839 196 6 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 196 7 ) ) PUNCT cana-3839 197 1 + + CCONJ cana-3839 197 2 ๐œ•3๐บ(๐œ•1 ๐œ•3๐บ(๐œ•1 PROPN cana-3839 197 3 , , PUNCT cana-3839 197 4 โˆ๐œ•1 โˆ๐œ•1 NUM cana-3839 197 5 , , PUNCT cana-3839 197 6 โˆ๐œ•1 โˆ๐œ•1 NUM cana-3839 197 7 ) ) PUNCT cana-3839 197 8 communications communication NOUN cana-3839 197 9 on on ADP cana-3839 197 10 applied apply VERB cana-3839 197 11 nonlinear nonlinear ADJ cana-3839 197 12 analysis analysis NOUN cana-3839 197 13 issn issn NOUN cana-3839 197 14 : : PUNCT cana-3839 197 15 1074 1074 NUM cana-3839 197 16 - - PUNCT cana-3839 197 17 133x 133x NUM cana-3839 197 18 vol vol NOUN cana-3839 197 19 32 32 NUM cana-3839 197 20 no no NOUN cana-3839 197 21 . . PUNCT cana-3839 198 1 9s 9s NUM cana-3839 198 2 ( ( PUNCT cana-3839 198 3 2025 2025 NUM cana-3839 198 4 ) ) PUNCT cana-3839 198 5 83 83 NUM cana-3839 198 6 https://internationalpubls.com https://internationalpubls.com X cana-3839 198 7 + + PROPN cana-3839 198 8 ๐œ•4๐บ(๐œ•1 ๐œ•4๐บ(๐œ•1 PROPN cana-3839 198 9 , , PUNCT cana-3839 198 10 โˆ๐œ•1 โˆ๐œ•1 NUM cana-3839 198 11 , , PUNCT cana-3839 198 12 โˆ๐œ•1 โˆ๐œ•1 NUM cana-3839 198 13 ) ) PUNCT cana-3839 198 14 + + CCONJ cana-3839 198 15 ๐œ•5๐บ(๐œ‡1 ๐œ•5๐บ(๐œ‡1 ADJ cana-3839 198 16 , , PUNCT cana-3839 198 17 โˆ๐œ•1 โˆ๐œ•1 NUM cana-3839 198 18 , , PUNCT cana-3839 198 19 โˆ๐œ•1 โˆ๐œ•1 NUM cana-3839 198 20 ) ) PUNCT cana-3839 199 1 + + CCONJ cana-3839 199 2 ๐œ•6๐บ(๐œ•1 ๐œ•6๐บ(๐œ•1 PROPN cana-3839 199 3 , , PUNCT cana-3839 199 4 โˆ๐œ•1 โˆ๐œ•1 NUM cana-3839 199 5 , , PUNCT cana-3839 199 6 โˆ๐œ•1 โˆ๐œ•1 NUM cana-3839 199 7 ) ) PUNCT cana-3839 200 1 + + NOUN cana-3839 200 2 ๐œ•7๐บ(๐œ•1 ๐œ•7๐บ(๐œ•1 PROPN cana-3839 200 3 , , PUNCT cana-3839 200 4 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 200 5 , , PUNCT cana-3839 200 6 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 200 7 ) ) PUNCT cana-3839 200 8 โชฏ โชฏ PROPN cana-3839 200 9 ๐œ•1๐บ(๐œ‡1 ๐œ•1๐บ(๐œ‡1 PROPN cana-3839 200 10 , , PUNCT cana-3839 200 11 ๐œ•1 ๐œ•1 NOUN cana-3839 200 12 , , PUNCT cana-3839 200 13 ๐œ•1 ๐œ•1 NOUN cana-3839 200 14 ) ) PUNCT cana-3839 200 15 + + CCONJ cana-3839 200 16 ๐œ•5๐บ(๐œ‡1 ๐œ•5๐บ(๐œ‡1 ADJ cana-3839 200 17 , , PUNCT cana-3839 200 18 ๐œ•1 ๐œ•1 NOUN cana-3839 200 19 , , PUNCT cana-3839 200 20 ๐œ•1 ๐œ•1 NUM cana-3839 200 21 ) ) PUNCT cana-3839 201 1 + + CCONJ cana-3839 201 2 ๐œ•7๐บ(๐œ•1 ๐œ•7๐บ(๐œ•1 PROPN cana-3839 201 3 , , PUNCT cana-3839 201 4 ๐œ‡1 ๐œ‡1 PROPN cana-3839 201 5 , , PUNCT cana-3839 201 6 ๐œ‡1 ๐œ‡1 NOUN cana-3839 201 7 ) ) PUNCT cana-3839 201 8 โชฏ โชฏ PROPN cana-3839 201 9 ๐œ•1๐บ(๐œ‡1 ๐œ•1๐บ(๐œ‡1 PROPN cana-3839 201 10 , , PUNCT cana-3839 201 11 ๐œ•1 ๐œ•1 NOUN cana-3839 201 12 , , PUNCT cana-3839 201 13 ๐œ•1 ๐œ•1 NOUN cana-3839 201 14 ) ) PUNCT cana-3839 202 1 + + CCONJ cana-3839 202 2 ๐œ•5๐บ(๐œ‡1 ๐œ•5๐บ(๐œ‡1 ADJ cana-3839 202 3 , , PUNCT cana-3839 202 4 ๐œ•1 ๐œ•1 NOUN cana-3839 202 5 , , PUNCT cana-3839 202 6 ๐œ•1 ๐œ•1 NUM cana-3839 202 7 ) ) PUNCT cana-3839 202 8 + + SYM cana-3839 202 9 ๐œ•7๐บ(๐œ‡1 ๐œ•7๐บ(๐œ‡1 NOUN cana-3839 202 10 , , PUNCT cana-3839 202 11 ๐œ•1 ๐œ•1 NOUN cana-3839 202 12 , , PUNCT cana-3839 202 13 ๐œ•1 ๐œ•1 NOUN cana-3839 202 14 ) ) PUNCT cana-3839 202 15 โชฏ โชฏ NOUN cana-3839 202 16 ( ( PUNCT cana-3839 202 17 ๐œ•1 ๐œ•1 VERB cana-3839 202 18 + + SYM cana-3839 202 19 ๐œ•5 ๐œ•5 PRON cana-3839 203 1 + + CCONJ cana-3839 203 2 ๐œ•7)๐บ(๐œ‡1 ๐œ•7)๐บ(๐œ‡1 ADJ cana-3839 203 3 , , PUNCT cana-3839 203 4 ๐œ•1 ๐œ•1 NOUN cana-3839 203 5 , , PUNCT cana-3839 203 6 ๐œ•1 ๐œ•1 NOUN cana-3839 203 7 ) ) PUNCT cana-3839 203 8 . . PUNCT cana-3839 204 1 since since SCONJ cana-3839 204 2 0 0 NUM cana-3839 204 3 โ‰ค โ‰ค NUM cana-3839 204 4 ๐œ•1 ๐œ•1 NOUN cana-3839 204 5 + + SYM cana-3839 204 6 ๐œ•5 ๐œ•5 PRON cana-3839 204 7 + + CCONJ cana-3839 204 8 ๐œ•7 ๐œ•7 VERB cana-3839 204 9 < < X cana-3839 204 10 โˆ‘7 โˆ‘7 PROPN cana-3839 204 11 ๐‘–=1 ๐‘–=1 PROPN cana-3839 204 12 ๐œ•๐‘– ๐œ•๐‘– PROPN cana-3839 204 13 < < X cana-3839 204 14 1 1 X cana-3839 204 15 . . PUNCT cana-3839 205 1 so so ADV cana-3839 205 2 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADJ cana-3839 205 3 , , PUNCT cana-3839 205 4 ๐œ•1 ๐œ•1 NOUN cana-3839 205 5 , , PUNCT cana-3839 205 6 ๐œ•1 ๐œ•1 NUM cana-3839 205 7 ) ) PUNCT cana-3839 205 8 = = SYM cana-3839 205 9 0 0 PUNCT cana-3839 206 1 this this PRON cana-3839 206 2 implies imply VERB cana-3839 206 3 ๐œ‡1 ๐œ‡1 NOUN cana-3839 206 4 = = SYM cana-3839 206 5 ๐œ•1 ๐œ•1 NOUN cana-3839 206 6 . . PUNCT cana-3839 207 1 thus thus ADV cana-3839 207 2 โˆ โˆ PROPN cana-3839 207 3 has have VERB cana-3839 207 4 fp fp NOUN cana-3839 207 5 in in ADP cana-3839 207 6 ๐œ‡ ๐œ‡ ADP cana-3839 207 7 which which PRON cana-3839 207 8 is be AUX cana-3839 207 9 unique unique ADJ cana-3839 207 10 . . PUNCT cana-3839 208 1 setting set VERB cana-3839 208 2 ๐œ•2 ๐œ•2 PUNCT cana-3839 209 1 = = SYM cana-3839 209 2 ๐œ• ๐œ• PROPN cana-3839 209 3 , , PUNCT cana-3839 209 4 ๐œ•3 ๐œ•3 NOUN cana-3839 209 5 = = SYM cana-3839 209 6 ๐œŒ ๐œŒ X cana-3839 209 7 and and CCONJ cana-3839 209 8 ๐œ•๐‘– ๐œ•๐‘– NOUN cana-3839 209 9 = = NOUN cana-3839 209 10 0 0 NUM cana-3839 209 11 for for ADP cana-3839 209 12 ๐‘– ๐‘– PRON cana-3839 209 13 = = SYM cana-3839 209 14 1,3 1,3 NUM cana-3839 209 15 , , PUNCT cana-3839 209 16 โ€ฆ โ€ฆ PUNCT cana-3839 209 17 , , PUNCT cana-3839 209 18 7 7 NUM cana-3839 209 19 in in ADP cana-3839 209 20 the the DET cana-3839 209 21 previous previous ADJ cana-3839 209 22 theorem theorem NOUN cana-3839 209 23 , , PUNCT cana-3839 209 24 we we PRON cana-3839 209 25 yield yield VERB cana-3839 209 26 the the DET cana-3839 209 27 subsequent subsequent ADJ cana-3839 209 28 corollary corollary NOUN cana-3839 209 29 . . PUNCT cana-3839 210 1 corollary corollary ADJ cana-3839 210 2 2.2 2.2 NUM cana-3839 210 3 : : PUNCT cana-3839 210 4 let let VERB cana-3839 210 5 ( ( PUNCT cana-3839 210 6 โ„œ โ„œ PROPN cana-3839 210 7 , , PUNCT cana-3839 210 8 ๐บ ๐บ PROPN cana-3839 210 9 , , PUNCT cana-3839 210 10 ๐‘‰ ๐‘‰ PROPN cana-3839 210 11 ) ) PUNCT cana-3839 210 12 be be VERB cana-3839 210 13 ๐บ ๐บ NOUN cana-3839 210 14 โˆ’ โˆ’ NOUN cana-3839 210 15 complete complete ADJ cana-3839 210 16 and and CCONJ cana-3839 210 17 symmetric symmetric ADJ cana-3839 210 18 vgms vgms NOUN cana-3839 210 19 and and CCONJ cana-3839 210 20 { { PUNCT cana-3839 210 21 ๐œ’๐‘ž}๐‘ž=1 ๐œ’๐‘ž}๐‘ž=1 X cana-3839 210 22 ๐‘š ๐‘š PROPN cana-3839 210 23 be be AUX cana-3839 210 24 family family NOUN cana-3839 210 25 of of ADP cana-3839 210 26 the the DET cana-3839 210 27 non non ADJ cana-3839 210 28 - - ADJ cana-3839 210 29 empty empty ADJ cana-3839 210 30 subset subset NOUN cana-3839 210 31 of of ADP cana-3839 210 32 ๐œ‡ ๐œ‡ ADP cana-3839 210 33 with with ADP cana-3839 210 34 ๐œ‡ ๐œ‡ SCONJ cana-3839 210 35 = = NOUN cana-3839 210 36 โˆช๐‘ž=1 โˆช๐‘ž=1 X cana-3839 210 37 ๐‘š ๐‘š X cana-3839 210 38 ๐œ’๐‘ž. ๐œ’๐‘ž. NOUN cana-3839 210 39 let let VERB cana-3839 210 40 โˆ โˆ NOUN cana-3839 210 41 : : PUNCT cana-3839 210 42 ๐œ‡ ๐œ‡ X cana-3839 210 43 โ†’ โ†’ X cana-3839 210 44 ๐œ‡ ๐œ‡ SCONJ cana-3839 210 45 be be AUX cana-3839 210 46 a a DET cana-3839 210 47 transformation transformation NOUN cana-3839 210 48 satisfying satisfy VERB cana-3839 210 49 โˆ(๐œ’๐‘ž โˆ(๐œ’๐‘ž NOUN cana-3839 210 50 ) ) PUNCT cana-3839 211 1 โŠ† โŠ† NUM cana-3839 211 2 ๐œ’๐‘ž+1 ๐œ’๐‘ž+1 NUM cana-3839 211 3 ๐‘ž ๐‘ž NOUN cana-3839 211 4 = = SYM cana-3839 211 5 1,2 1,2 NUM cana-3839 211 6 , , PUNCT cana-3839 211 7 โ€ฆ โ€ฆ PUNCT cana-3839 211 8 , , PUNCT cana-3839 211 9 ๐‘š ๐‘š NOUN cana-3839 211 10 , , PUNCT cana-3839 211 11 ๐‘คโ„Ž๐‘’๐‘Ÿ๐‘’ ๐‘คโ„Ž๐‘’๐‘Ÿ๐‘’ ADJ cana-3839 211 12 ๐œ’๐‘š+1 ๐œ’๐‘š+1 NUM cana-3839 211 13 = = SYM cana-3839 211 14 ๐œ’1 ๐œ’1 PROPN cana-3839 211 15 . . PUNCT cana-3839 211 16 suppose suppose VERB cana-3839 211 17 that that SCONJ cana-3839 211 18 โˆƒ โˆƒ PROPN cana-3839 211 19 constant constant ADJ cana-3839 211 20 ๐œ• ๐œ• PROPN cana-3839 211 21 , , PUNCT cana-3839 211 22 โ„œ โ„œ NOUN cana-3839 211 23 with with ADP cana-3839 211 24 0 0 NUM cana-3839 211 25 < < X cana-3839 211 26 ๐œ• ๐œ• PROPN cana-3839 211 27 + + ADP cana-3839 211 28 ๐œŒ ๐œŒ X cana-3839 211 29 < < X cana-3839 211 30 1 1 NUM cana-3839 211 31 such such ADJ cana-3839 211 32 that that SCONJ cana-3839 211 33 the the DET cana-3839 211 34 transformation transformation NOUN cana-3839 211 35 โˆ โˆ PROPN cana-3839 211 36 satisfies satisfy VERB cana-3839 211 37 ๐บ(โˆ๐œ‡ ๐บ(โˆ๐œ‡ NOUN cana-3839 211 38 , , PUNCT cana-3839 211 39 โˆ โˆ PROPN cana-3839 211 40 โ™ญ โ™ญ INTJ cana-3839 211 41 , , PUNCT cana-3839 211 42 โˆ โˆ NUM cana-3839 211 43 โ™ญ โ™ญ PRON cana-3839 211 44 ) ) PUNCT cana-3839 211 45 โชฏ โชฏ NOUN cana-3839 211 46 ๐œ•๐บ(๐œ‡ ๐œ•๐บ(๐œ‡ NOUN cana-3839 211 47 , , PUNCT cana-3839 211 48 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 211 49 , , PUNCT cana-3839 211 50 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 211 51 ) ) PUNCT cana-3839 211 52 + + CCONJ cana-3839 211 53 ๐œŒ๐บ ๐œŒ๐บ NOUN cana-3839 211 54 ( ( PUNCT cana-3839 211 55 โ™ญ โ™ญ INTJ cana-3839 211 56 , , PUNCT cana-3839 211 57 โˆ โˆ PROPN cana-3839 211 58 โ™ญ โ™ญ INTJ cana-3839 211 59 , , PUNCT cana-3839 211 60 โˆ โˆ NUM cana-3839 211 61 โ™ญ โ™ญ NUM cana-3839 211 62 ) ) PUNCT cana-3839 211 63 โˆ€๐œ‡ โˆ€๐œ‡ X cana-3839 211 64 โˆˆ โˆˆ PROPN cana-3839 211 65 ๐œ’๐‘ž ๐œ’๐‘ž PROPN cana-3839 211 66 and and CCONJ cana-3839 211 67 โ™ญ โ™ญ PROPN cana-3839 211 68 , , PUNCT cana-3839 211 69 ๐‘Ÿ ๐‘Ÿ X cana-3839 211 70 โˆˆ โˆˆ PROPN cana-3839 211 71 ๐œ’๐‘ž+1 ๐œ’๐‘ž+1 PROPN cana-3839 211 72 , , PUNCT cana-3839 211 73 ๐‘ž ๐‘ž X cana-3839 211 74 = = SYM cana-3839 211 75 1,2 1,2 NUM cana-3839 211 76 , , PUNCT cana-3839 211 77 โ€ฆ โ€ฆ PUNCT cana-3839 211 78 ๐‘š. ๐‘š. ADV cana-3839 211 79 then then ADV cana-3839 211 80 ๐œ’ ๐œ’ X cana-3839 211 81 has have VERB cana-3839 211 82 fp fp NOUN cana-3839 211 83 in in ADP cana-3839 211 84 ๐œ‡ ๐œ‡ ADP cana-3839 211 85 = = NOUN cana-3839 211 86 โˆฉ๐‘ž=1 โˆฉ๐‘ž=1 ADJ cana-3839 211 87 ๐‘š ๐‘š X cana-3839 211 88 ๐œ’๐‘ž ๐œ’๐‘ž NOUN cana-3839 211 89 which which PRON cana-3839 211 90 is be AUX cana-3839 211 91 unique unique ADJ cana-3839 211 92 . . PUNCT cana-3839 212 1 example example NOUN cana-3839 212 2 2.3 2.3 NUM cana-3839 212 3 : : PUNCT cana-3839 212 4 the the DET cana-3839 212 5 transformation transformation NOUN cana-3839 212 6 โˆ โˆ PROPN cana-3839 212 7 : : PUNCT cana-3839 213 1 [ [ X cana-3839 213 2 0,1 0,1 NUM cana-3839 213 3 ] ] PUNCT cana-3839 213 4 โ†’ โ†’ X cana-3839 214 1 [ [ X cana-3839 214 2 0,1 0,1 NUM cana-3839 214 3 ] ] PUNCT cana-3839 214 4 defined define VERB cana-3839 214 5 by by ADP cana-3839 214 6 โˆ๐œ‡ โˆ๐œ‡ PROPN cana-3839 214 7 = = NOUN cana-3839 214 8 { { PUNCT cana-3839 214 9 ๐œ‡ ๐œ‡ ADP cana-3839 214 10 4 4 NUM cana-3839 214 11 0 0 NUM cana-3839 214 12 โชฏ โชฏ NOUN cana-3839 214 13 ๐œ‡ ๐œ‡ ADP cana-3839 214 14 < < X cana-3839 214 15 1 1 NUM cana-3839 214 16 2 2 NUM cana-3839 214 17 ๐œ‡ ๐œ‡ ADP cana-3839 214 18 5 5 NUM cana-3839 214 19 1 1 NUM cana-3839 214 20 2 2 NUM cana-3839 214 21 โชฏ โชฏ NOUN cana-3839 214 22 ๐œ‡ ๐œ‡ ADP cana-3839 214 23 < < X cana-3839 214 24 1 1 NUM cana-3839 214 25 is be AUX cana-3839 214 26 a a DET cana-3839 214 27 discontinuous discontinuous ADJ cana-3839 214 28 transformation transformation NOUN cana-3839 214 29 with with ADP cana-3839 214 30 0 0 NUM cana-3839 214 31 as as ADV cana-3839 214 32 unique unique ADJ cana-3839 214 33 fp fp X cana-3839 214 34 . . NOUN cana-3839 214 35 setting set VERB cana-3839 214 36 ๐œ•1 ๐œ•1 PUNCT cana-3839 214 37 = = SYM cana-3839 214 38 ๐œ• ๐œ• NOUN cana-3839 214 39 and and CCONJ cana-3839 214 40 ๐œ•๐‘– ๐œ•๐‘– PROPN cana-3839 214 41 = = PROPN cana-3839 214 42 0 0 NUM cana-3839 214 43 for for ADP cana-3839 214 44 ๐‘– ๐‘– NOUN cana-3839 214 45 = = SYM cana-3839 214 46 2 2 NUM cana-3839 214 47 , , PUNCT cana-3839 214 48 โ€ฆ โ€ฆ PUNCT cana-3839 214 49 , , PUNCT cana-3839 214 50 7 7 NUM cana-3839 214 51 in in ADP cana-3839 214 52 the the DET cana-3839 214 53 theorem theorem NOUN cana-3839 214 54 2.1 2.1 NUM cana-3839 214 55 , , PUNCT cana-3839 214 56 we we PRON cana-3839 214 57 yield yield VERB cana-3839 214 58 the the DET cana-3839 214 59 subsequent subsequent ADJ cana-3839 214 60 corollary corollary NOUN cana-3839 214 61 . . PUNCT cana-3839 215 1 corollary corollary ADJ cana-3839 215 2 2.4:(banach 2.4:(banach NUM cana-3839 215 3 contraction contraction NOUN cana-3839 215 4 principle principle NOUN cana-3839 215 5 ) ) PUNCT cana-3839 215 6 let let VERB cana-3839 215 7 ( ( PUNCT cana-3839 215 8 โ„œ โ„œ PROPN cana-3839 215 9 , , PUNCT cana-3839 215 10 ๐บ ๐บ PROPN cana-3839 215 11 , , PUNCT cana-3839 215 12 ๐‘‰ ๐‘‰ PROPN cana-3839 215 13 ) ) PUNCT cana-3839 215 14 be be VERB cana-3839 215 15 ๐บ ๐บ NOUN cana-3839 215 16 โˆ’ โˆ’ NOUN cana-3839 215 17 complete complete ADJ cana-3839 215 18 and and CCONJ cana-3839 215 19 symmetric symmetric ADJ cana-3839 215 20 vgms vgms NOUN cana-3839 215 21 and and CCONJ cana-3839 215 22 { { PUNCT cana-3839 215 23 ๐ด๐‘—}๐‘—=1 ๐ด๐‘—}๐‘—=1 NOUN cana-3839 215 24 ๐‘š ๐‘š VERB cana-3839 215 25 be be AUX cana-3839 215 26 family family NOUN cana-3839 215 27 of of ADP cana-3839 215 28 non non ADJ cana-3839 215 29 - - ADJ cana-3839 215 30 empty empty ADJ cana-3839 215 31 subset subset NOUN cana-3839 215 32 of of ADP cana-3839 215 33 ๐œ‡ ๐œ‡ ADP cana-3839 215 34 with with ADP cana-3839 215 35 ๐œ‡ ๐œ‡ SCONJ cana-3839 215 36 = = NOUN cana-3839 215 37 โˆช๐‘ž=1 โˆช๐‘ž=1 X cana-3839 215 38 ๐‘š ๐‘š X cana-3839 215 39 ๐œ’๐‘ž. ๐œ’๐‘ž. NOUN cana-3839 215 40 let let VERB cana-3839 215 41 โˆ โˆ NOUN cana-3839 215 42 : : PUNCT cana-3839 215 43 ๐œ‡ ๐œ‡ X cana-3839 215 44 โ†’ โ†’ X cana-3839 215 45 ๐œ‡ ๐œ‡ SCONJ cana-3839 215 46 be be AUX cana-3839 215 47 a a DET cana-3839 215 48 transformation transformation NOUN cana-3839 215 49 satisfying satisfy VERB cana-3839 215 50 โˆ(๐œ’๐‘ž โˆ(๐œ’๐‘ž NOUN cana-3839 215 51 ) ) PUNCT cana-3839 215 52 โŠ† โŠ† NUM cana-3839 215 53 ๐œ’๐‘ž+1 ๐œ’๐‘ž+1 NUM cana-3839 215 54 ๐‘ž ๐‘ž NOUN cana-3839 215 55 = = SYM cana-3839 215 56 1,2 1,2 NUM cana-3839 215 57 , , PUNCT cana-3839 215 58 โ€ฆ โ€ฆ PUNCT cana-3839 215 59 , , PUNCT cana-3839 215 60 ๐‘š ๐‘š NOUN cana-3839 215 61 , , PUNCT cana-3839 215 62 ๐‘คโ„Ž๐‘’๐‘Ÿ๐‘’ ๐‘คโ„Ž๐‘’๐‘Ÿ๐‘’ ADJ cana-3839 215 63 ๐œ’๐‘š+1 ๐œ’๐‘š+1 NUM cana-3839 215 64 = = SYM cana-3839 215 65 ๐œ’1 ๐œ’1 PROPN cana-3839 215 66 . . PUNCT cana-3839 216 1 suppose suppose VERB cana-3839 216 2 that that SCONJ cana-3839 216 3 โˆƒ โˆƒ PROPN cana-3839 216 4 constant constant ADJ cana-3839 216 5 ๐œ• ๐œ• NOUN cana-3839 216 6 with with ADP cana-3839 216 7 0 0 NUM cana-3839 216 8 < < X cana-3839 216 9 ๐œ• ๐œ• NOUN cana-3839 216 10 < < X cana-3839 216 11 1 1 NUM cana-3839 216 12 such such ADJ cana-3839 216 13 that that SCONJ cana-3839 216 14 the the DET cana-3839 216 15 transformation transformation NOUN cana-3839 216 16 โˆ โˆ PROPN cana-3839 216 17 satisfies satisfy VERB cana-3839 216 18 ๐บ(โˆ๐œ‡ ๐บ(โˆ๐œ‡ NOUN cana-3839 216 19 , , PUNCT cana-3839 216 20 โˆ โˆ PROPN cana-3839 216 21 โ™ญ โ™ญ PROPN cana-3839 216 22 , , PUNCT cana-3839 216 23 โˆ๐‘Ÿ โˆ๐‘Ÿ NOUN cana-3839 216 24 ) ) PUNCT cana-3839 216 25 โชฏ โชฏ NOUN cana-3839 216 26 ๐œ•๐บ(๐œ‡ ๐œ•๐บ(๐œ‡ NOUN cana-3839 216 27 , , PUNCT cana-3839 216 28 โ™ญ โ™ญ INTJ cana-3839 216 29 , , PUNCT cana-3839 216 30 ๐‘Ÿ ๐‘Ÿ NOUN cana-3839 216 31 ) ) PUNCT cana-3839 216 32 โˆ€๐œ‡ โˆ€๐œ‡ NUM cana-3839 216 33 โˆˆ โˆˆ PROPN cana-3839 216 34 ๐œ’๐‘ž ๐œ’๐‘ž PROPN cana-3839 216 35 and and CCONJ cana-3839 216 36 โ™ญ โ™ญ PROPN cana-3839 216 37 , , PUNCT cana-3839 216 38 ๐‘Ÿ ๐‘Ÿ X cana-3839 216 39 โˆˆ โˆˆ PROPN cana-3839 216 40 ๐œ’๐‘ž+1 ๐œ’๐‘ž+1 PROPN cana-3839 216 41 , , PUNCT cana-3839 216 42 ๐‘ž ๐‘ž X cana-3839 216 43 = = SYM cana-3839 216 44 1,2 1,2 NUM cana-3839 216 45 , , PUNCT cana-3839 216 46 โ€ฆ โ€ฆ PUNCT cana-3839 216 47 ๐‘š. ๐‘š. ADV cana-3839 216 48 then then ADV cana-3839 216 49 ๐œ’ ๐œ’ X cana-3839 216 50 has have VERB cana-3839 216 51 fp fp NOUN cana-3839 216 52 in in ADP cana-3839 216 53 ๐œ‡ ๐œ‡ ADP cana-3839 216 54 = = NOUN cana-3839 216 55 โˆฉ๐‘ž=1 โˆฉ๐‘ž=1 ADJ cana-3839 216 56 ๐‘š ๐‘š X cana-3839 216 57 ๐œ’๐‘ž ๐œ’๐‘ž NOUN cana-3839 216 58 which which PRON cana-3839 216 59 is be AUX cana-3839 216 60 unique unique ADJ cana-3839 216 61 . . PUNCT cana-3839 217 1 theorem theorem VERB cana-3839 217 2 2.5 2.5 NUM cana-3839 217 3 let let VERB cana-3839 217 4 ( ( PUNCT cana-3839 217 5 โ„œ โ„œ PROPN cana-3839 217 6 , , PUNCT cana-3839 217 7 ๐บ ๐บ PROPN cana-3839 217 8 , , PUNCT cana-3839 217 9 ๐‘‰ ๐‘‰ PROPN cana-3839 217 10 ) ) PUNCT cana-3839 217 11 be be VERB cana-3839 217 12 ๐บ ๐บ NOUN cana-3839 217 13 โˆ’ โˆ’ NOUN cana-3839 217 14 complete complete ADJ cana-3839 217 15 and and CCONJ cana-3839 217 16 symmetric symmetric ADJ cana-3839 217 17 vgms vgms NOUN cana-3839 217 18 and and CCONJ cana-3839 217 19 { { PUNCT cana-3839 217 20 ๐œ’๐‘ž}๐‘ž=1 ๐œ’๐‘ž}๐‘ž=1 X cana-3839 217 21 ๐‘š ๐‘š PROPN cana-3839 217 22 be be AUX cana-3839 217 23 family family NOUN cana-3839 217 24 of of ADP cana-3839 217 25 non non ADJ cana-3839 217 26 - - ADJ cana-3839 217 27 empty empty ADJ cana-3839 217 28 subset subset NOUN cana-3839 217 29 of of ADP cana-3839 217 30 ๐œ‡ ๐œ‡ ADP cana-3839 217 31 with with ADP cana-3839 217 32 ๐œ‡ ๐œ‡ SCONJ cana-3839 217 33 = = NOUN cana-3839 217 34 โˆช๐‘ž=1 โˆช๐‘ž=1 X cana-3839 217 35 ๐‘š ๐‘š X cana-3839 217 36 ๐œ’๐‘ž. ๐œ’๐‘ž. NOUN cana-3839 217 37 let let VERB cana-3839 217 38 โˆ โˆ NOUN cana-3839 217 39 : : PUNCT cana-3839 217 40 ๐œ‡ ๐œ‡ X cana-3839 217 41 โ†’ โ†’ X cana-3839 217 42 ๐œ‡ ๐œ‡ SCONJ cana-3839 217 43 be be AUX cana-3839 217 44 a a DET cana-3839 217 45 transformation transformation NOUN cana-3839 217 46 satisfying satisfy VERB cana-3839 217 47 communications communication NOUN cana-3839 217 48 on on ADP cana-3839 217 49 applied apply VERB cana-3839 217 50 nonlinear nonlinear ADJ cana-3839 217 51 analysis analysis NOUN cana-3839 217 52 issn issn NOUN cana-3839 217 53 : : PUNCT cana-3839 217 54 1074 1074 NUM cana-3839 217 55 - - PUNCT cana-3839 217 56 133x 133x NUM cana-3839 217 57 vol vol NOUN cana-3839 217 58 32 32 NUM cana-3839 217 59 no no NOUN cana-3839 217 60 . . PUNCT cana-3839 218 1 9s 9s NUM cana-3839 218 2 ( ( PUNCT cana-3839 218 3 2025 2025 NUM cana-3839 218 4 ) ) PUNCT cana-3839 218 5 84 84 NUM cana-3839 218 6 https://internationalpubls.com https://internationalpubls.com X cana-3839 218 7 โˆ(๐œ’๐‘ž โˆ(๐œ’๐‘ž NOUN cana-3839 218 8 ) ) PUNCT cana-3839 218 9 โŠ† โŠ† NUM cana-3839 218 10 ๐œ’๐‘ž+1 ๐œ’๐‘ž+1 NUM cana-3839 218 11 ๐‘ž ๐‘ž NOUN cana-3839 218 12 = = SYM cana-3839 218 13 1,2 1,2 NUM cana-3839 218 14 , , PUNCT cana-3839 218 15 โ€ฆ โ€ฆ PUNCT cana-3839 218 16 , , PUNCT cana-3839 218 17 ๐‘š ๐‘š NOUN cana-3839 218 18 , , PUNCT cana-3839 218 19 ๐‘คโ„Ž๐‘’๐‘Ÿ๐‘’ ๐‘คโ„Ž๐‘’๐‘Ÿ๐‘’ ADJ cana-3839 218 20 ๐œ’๐‘š+1 ๐œ’๐‘š+1 NUM cana-3839 218 21 = = SYM cana-3839 218 22 ๐œ’1 ๐œ’1 PROPN cana-3839 218 23 . . PUNCT cana-3839 218 24 suppose suppose VERB cana-3839 218 25 that that SCONJ cana-3839 218 26 there there PRON cana-3839 218 27 exist exist VERB cana-3839 218 28 constant constant ADJ cana-3839 218 29 ๐œ•1 ๐œ•1 NOUN cana-3839 218 30 , , PUNCT cana-3839 218 31 ๐œ•2 ๐œ•2 NOUN cana-3839 218 32 , , PUNCT cana-3839 218 33 ๐œ•3 ๐œ•3 NOUN cana-3839 218 34 , , PUNCT cana-3839 218 35 ๐œ•4 ๐œ•4 NOUN cana-3839 218 36 with with ADP cana-3839 218 37 0 0 NUM cana-3839 218 38 < < X cana-3839 218 39 ๐œ•1 ๐œ•1 NOUN cana-3839 218 40 + + SYM cana-3839 218 41 ๐œ•2 ๐œ•2 NOUN cana-3839 219 1 + + CCONJ cana-3839 219 2 ๐œ•3 ๐œ•3 NOUN cana-3839 219 3 + + CCONJ cana-3839 219 4 ๐œ•4 ๐œ•4 X cana-3839 219 5 < < X cana-3839 219 6 1 1 NUM cana-3839 219 7 2 2 NUM cana-3839 219 8 such such ADJ cana-3839 219 9 that that SCONJ cana-3839 219 10 the the DET cana-3839 219 11 transformation transformation NOUN cana-3839 219 12 โˆ โˆ PROPN cana-3839 219 13 satisfies satisfy VERB cana-3839 219 14 ๐บ(โˆ๐œ‡ ๐บ(โˆ๐œ‡ NOUN cana-3839 219 15 , , PUNCT cana-3839 219 16 โˆ โˆ PROPN cana-3839 219 17 โ™ญ โ™ญ INTJ cana-3839 219 18 , , PUNCT cana-3839 219 19 โˆ โˆ NUM cana-3839 219 20 โ™ญ โ™ญ PRON cana-3839 219 21 ) ) PUNCT cana-3839 219 22 โชฏ โชฏ NOUN cana-3839 219 23 { { PUNCT cana-3839 219 24 ๐œ•1[๐บ(๐œ‡ ๐œ•1[๐บ(๐œ‡ ADJ cana-3839 219 25 , , PUNCT cana-3839 219 26 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 219 27 , , PUNCT cana-3839 219 28 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 219 29 ) ) PUNCT cana-3839 220 1 + + CCONJ cana-3839 220 2 ๐บ ๐บ PROPN cana-3839 220 3 ( ( PUNCT cana-3839 220 4 โ™ญ โ™ญ PROPN cana-3839 220 5 , , PUNCT cana-3839 220 6 โˆ โˆ PROPN cana-3839 220 7 โ™ญ โ™ญ INTJ cana-3839 220 8 , , PUNCT cana-3839 220 9 โˆ โˆ PROPN cana-3839 220 10 โ™ญ โ™ญ NUM cana-3839 220 11 ) ) PUNCT cana-3839 220 12 ] ] PUNCT cana-3839 220 13 , , PUNCT cana-3839 220 14 ๐œ•2[๐บ(๐œ‡ ๐œ•2[๐บ(๐œ‡ ADP cana-3839 220 15 , , PUNCT cana-3839 220 16 โˆ โˆ PROPN cana-3839 220 17 โ™ญ โ™ญ INTJ cana-3839 220 18 , , PUNCT cana-3839 220 19 โˆ โˆ NUM cana-3839 220 20 โ™ญ โ™ญ NUM cana-3839 220 21 ) ) PUNCT cana-3839 221 1 + + CCONJ cana-3839 221 2 ๐บ ๐บ PROPN cana-3839 221 3 ( ( PUNCT cana-3839 221 4 โ™ญ โ™ญ INTJ cana-3839 221 5 , , PUNCT cana-3839 221 6 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 221 7 , , PUNCT cana-3839 221 8 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 221 9 ) ) PUNCT cana-3839 221 10 ] ] PUNCT cana-3839 221 11 , , PUNCT cana-3839 221 12 ๐œ•3[๐บ(๐œ‡ ๐œ•3[๐บ(๐œ‡ X cana-3839 221 13 , , PUNCT cana-3839 221 14 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 221 15 , , PUNCT cana-3839 221 16 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 221 17 ) ) PUNCT cana-3839 222 1 + + CCONJ cana-3839 222 2 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 222 3 , , PUNCT cana-3839 222 4 โˆ โˆ PROPN cana-3839 222 5 โ™ญ โ™ญ INTJ cana-3839 222 6 , , PUNCT cana-3839 222 7 โˆ โˆ PROPN cana-3839 222 8 โ™ญ โ™ญ NUM cana-3839 222 9 ) ) PUNCT cana-3839 222 10 ] ] PUNCT cana-3839 222 11 , , PUNCT cana-3839 222 12 ๐œ•4[๐บ ๐œ•4[๐บ PROPN cana-3839 222 13 ( ( PUNCT cana-3839 222 14 โ™ญ โ™ญ PROPN cana-3839 222 15 , , PUNCT cana-3839 222 16 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 222 17 , , PUNCT cana-3839 222 18 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 222 19 ) ) PUNCT cana-3839 222 20 + + CCONJ cana-3839 222 21 ๐บ ๐บ PROPN cana-3839 222 22 ( ( PUNCT cana-3839 222 23 โ™ญ โ™ญ PROPN cana-3839 222 24 , , PUNCT cana-3839 222 25 โˆ โˆ PROPN cana-3839 222 26 โ™ญ โ™ญ INTJ cana-3839 222 27 , , PUNCT cana-3839 222 28 โˆ โˆ PROPN cana-3839 222 29 โ™ญ โ™ญ NUM cana-3839 222 30 ) ) PUNCT cana-3839 222 31 ] ] PUNCT cana-3839 222 32 } } PUNCT cana-3839 222 33 ( ( PUNCT cana-3839 222 34 11 11 NUM cana-3839 222 35 ) ) PUNCT cana-3839 222 36 โˆ€๐œ‡ โˆ€๐œ‡ X cana-3839 222 37 โˆˆ โˆˆ PROPN cana-3839 222 38 ๐œ’๐‘ž ๐œ’๐‘ž PROPN cana-3839 222 39 and and CCONJ cana-3839 222 40 โ™ญ โ™ญ PROPN cana-3839 222 41 , , PUNCT cana-3839 222 42 ๐‘Ÿ ๐‘Ÿ X cana-3839 222 43 โˆˆ โˆˆ PROPN cana-3839 222 44 ๐œ’๐‘ž+1 ๐œ’๐‘ž+1 PROPN cana-3839 222 45 , , PUNCT cana-3839 222 46 ๐‘ž ๐‘ž X cana-3839 222 47 = = SYM cana-3839 222 48 1,2 1,2 NUM cana-3839 222 49 , , PUNCT cana-3839 222 50 โ€ฆ โ€ฆ PUNCT cana-3839 222 51 ๐‘š. ๐‘š. ADV cana-3839 222 52 then then ADV cana-3839 222 53 ๐œ’ ๐œ’ X cana-3839 222 54 has have VERB cana-3839 222 55 fp fp NOUN cana-3839 222 56 in in ADP cana-3839 222 57 ๐œ‡ ๐œ‡ ADP cana-3839 222 58 = = NOUN cana-3839 222 59 โˆฉ๐‘ž=1 โˆฉ๐‘ž=1 ADJ cana-3839 222 60 ๐‘š ๐‘š X cana-3839 222 61 ๐œ’๐‘ž ๐œ’๐‘ž NOUN cana-3839 222 62 which which PRON cana-3839 222 63 is be AUX cana-3839 222 64 unique unique ADJ cana-3839 222 65 . . PUNCT cana-3839 223 1 proof proof NOUN cana-3839 223 2 : : PUNCT cana-3839 223 3 interchanging interchange VERB cana-3839 223 4 the the DET cana-3839 223 5 role role NOUN cana-3839 223 6 of of ADP cana-3839 223 7 ๐œ‡ ๐œ‡ ADP cana-3839 223 8 and and CCONJ cana-3839 223 9 โ™ญ โ™ญ PROPN cana-3839 223 10 in in ADP cana-3839 223 11 ( ( PUNCT cana-3839 223 12 11 11 NUM cana-3839 223 13 ) ) PUNCT cana-3839 223 14 , , PUNCT cana-3839 223 15 we we PRON cana-3839 223 16 get get VERB cana-3839 223 17 ๐บ(โˆ ๐บ(โˆ NOUN cana-3839 223 18 โ™ญ โ™ญ NOUN cana-3839 223 19 , , PUNCT cana-3839 223 20 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 223 21 , , PUNCT cana-3839 223 22 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 223 23 ) ) PUNCT cana-3839 223 24 โชฏ โชฏ NOUN cana-3839 223 25 { { PUNCT cana-3839 223 26 ๐œ•1[๐บ ๐œ•1[๐บ PROPN cana-3839 223 27 ( ( PUNCT cana-3839 223 28 โ™ญ โ™ญ INTJ cana-3839 223 29 , , PUNCT cana-3839 223 30 โˆ โˆ PROPN cana-3839 223 31 โ™ญ โ™ญ INTJ cana-3839 223 32 , , PUNCT cana-3839 223 33 โˆ โˆ NUM cana-3839 223 34 โ™ญ โ™ญ X cana-3839 223 35 ) ) PUNCT cana-3839 224 1 + + CCONJ cana-3839 224 2 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 224 3 , , PUNCT cana-3839 224 4 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 224 5 , , PUNCT cana-3839 224 6 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 224 7 ) ) PUNCT cana-3839 224 8 ] ] PUNCT cana-3839 224 9 , , PUNCT cana-3839 224 10 ๐œ•2[๐บ ๐œ•2[๐บ NOUN cana-3839 224 11 ( ( PUNCT cana-3839 224 12 โ™ญ โ™ญ INTJ cana-3839 224 13 , , PUNCT cana-3839 224 14 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 224 15 , , PUNCT cana-3839 224 16 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 224 17 ) ) PUNCT cana-3839 224 18 + + CCONJ cana-3839 224 19 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 224 20 , , PUNCT cana-3839 224 21 โˆ โˆ PROPN cana-3839 224 22 โ™ญ โ™ญ INTJ cana-3839 224 23 , , PUNCT cana-3839 224 24 โˆ โˆ PROPN cana-3839 224 25 โ™ญ โ™ญ NUM cana-3839 224 26 ) ) PUNCT cana-3839 224 27 ] ] PUNCT cana-3839 224 28 , , PUNCT cana-3839 224 29 ๐œ•3[๐บ ๐œ•3[๐บ PROPN cana-3839 224 30 ( ( PUNCT cana-3839 224 31 โ™ญ โ™ญ INTJ cana-3839 224 32 , , PUNCT cana-3839 224 33 โˆ โˆ PROPN cana-3839 224 34 โ™ญ โ™ญ INTJ cana-3839 224 35 , , PUNCT cana-3839 224 36 โˆ โˆ NUM cana-3839 224 37 โ™ญ โ™ญ NUM cana-3839 224 38 ) ) PUNCT cana-3839 225 1 + + CCONJ cana-3839 225 2 ๐บ ๐บ PROPN cana-3839 225 3 ( ( PUNCT cana-3839 225 4 โ™ญ โ™ญ INTJ cana-3839 225 5 , , PUNCT cana-3839 225 6 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 225 7 , , PUNCT cana-3839 225 8 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 225 9 ) ) PUNCT cana-3839 225 10 ] ] PUNCT cana-3839 225 11 , , PUNCT cana-3839 225 12 ๐œ•4[๐บ((๐œ‡ ๐œ•4[๐บ((๐œ‡ PROPN cana-3839 225 13 , , PUNCT cana-3839 225 14 โˆ โˆ PROPN cana-3839 225 15 โ™ญ โ™ญ INTJ cana-3839 225 16 , , PUNCT cana-3839 225 17 โˆ โˆ NUM cana-3839 225 18 โ™ญ โ™ญ X cana-3839 225 19 ) ) PUNCT cana-3839 225 20 + + CCONJ cana-3839 225 21 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 225 22 , , PUNCT cana-3839 225 23 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 225 24 , , PUNCT cana-3839 225 25 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 225 26 ) ) PUNCT cana-3839 225 27 ] ] PUNCT cana-3839 225 28 } } PUNCT cana-3839 225 29 ( ( PUNCT cana-3839 225 30 12 12 X cana-3839 225 31 ) ) PUNCT cana-3839 225 32 adding add VERB cana-3839 225 33 eq(11 eq(11 PROPN cana-3839 225 34 ) ) PUNCT cana-3839 225 35 and and CCONJ cana-3839 225 36 eq(12 eq(12 PROPN cana-3839 225 37 ) ) PUNCT cana-3839 225 38 , , PUNCT cana-3839 225 39 we we PRON cana-3839 225 40 get get VERB cana-3839 225 41 2๐บ(โˆ๐œ‡ 2๐บ(โˆ๐œ‡ NOUN cana-3839 225 42 , , PUNCT cana-3839 225 43 โˆ โˆ PROPN cana-3839 225 44 โ™ญ โ™ญ INTJ cana-3839 225 45 , , PUNCT cana-3839 225 46 โˆ โˆ NUM cana-3839 225 47 โ™ญ โ™ญ PRON cana-3839 225 48 ) ) PUNCT cana-3839 225 49 โชฏ โชฏ NOUN cana-3839 225 50 { { PUNCT cana-3839 225 51 2๐œ•1[๐บ(๐œ‡ 2๐œ•1[๐บ(๐œ‡ NUM cana-3839 225 52 , , PUNCT cana-3839 225 53 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 225 54 , , PUNCT cana-3839 225 55 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 225 56 ) ) PUNCT cana-3839 226 1 + + CCONJ cana-3839 226 2 ๐บ ๐บ PROPN cana-3839 226 3 ( ( PUNCT cana-3839 226 4 โ™ญ โ™ญ PROPN cana-3839 226 5 , , PUNCT cana-3839 226 6 โˆ โˆ PROPN cana-3839 226 7 โ™ญ โ™ญ INTJ cana-3839 226 8 , , PUNCT cana-3839 226 9 โˆ โˆ PUNCT cana-3839 226 10 โ™ญ โ™ญ NUM cana-3839 226 11 )],2๐œ•2[๐บ(๐œ‡ )],2๐œ•2[๐บ(๐œ‡ PROPN cana-3839 226 12 , , PUNCT cana-3839 226 13 โˆ โˆ PROPN cana-3839 226 14 โ™ญ โ™ญ INTJ cana-3839 226 15 , , PUNCT cana-3839 226 16 โˆ โˆ NUM cana-3839 226 17 โ™ญ โ™ญ NUM cana-3839 226 18 ) ) PUNCT cana-3839 227 1 + + CCONJ cana-3839 227 2 ๐บ ๐บ PROPN cana-3839 227 3 ( ( PUNCT cana-3839 227 4 โ™ญ โ™ญ INTJ cana-3839 227 5 , , PUNCT cana-3839 227 6 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 227 7 , , PUNCT cana-3839 227 8 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 227 9 ) ) PUNCT cana-3839 227 10 ] ] PUNCT cana-3839 227 11 , , PUNCT cana-3839 227 12 ( ( PUNCT cana-3839 227 13 ๐œ•3 ๐œ•3 X cana-3839 227 14 + + CCONJ cana-3839 227 15 ๐œ•4)[๐บ(๐œ‡ ๐œ•4)[๐บ(๐œ‡ ADJ cana-3839 227 16 , , PUNCT cana-3839 227 17 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 227 18 , , PUNCT cana-3839 227 19 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 227 20 ) ) PUNCT cana-3839 228 1 + + CCONJ cana-3839 228 2 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 228 3 , , PUNCT cana-3839 228 4 โˆ โˆ PROPN cana-3839 228 5 โ™ญ โ™ญ INTJ cana-3839 228 6 , , PUNCT cana-3839 228 7 โˆ โˆ PROPN cana-3839 228 8 โ™ญ โ™ญ NUM cana-3839 228 9 ) ) PUNCT cana-3839 228 10 ] ] PUNCT cana-3839 228 11 , , PUNCT cana-3839 228 12 ๐œ•3 ๐œ•3 X cana-3839 228 13 + + SYM cana-3839 228 14 ๐œ•4)([๐บ ๐œ•4)([๐บ PROPN cana-3839 228 15 ( ( PUNCT cana-3839 228 16 โ™ญ โ™ญ INTJ cana-3839 228 17 , , PUNCT cana-3839 228 18 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 228 19 , , PUNCT cana-3839 228 20 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 228 21 ) ) PUNCT cana-3839 229 1 + + CCONJ cana-3839 229 2 ๐บ ๐บ PROPN cana-3839 229 3 ( ( PUNCT cana-3839 229 4 โ™ญ โ™ญ PROPN cana-3839 229 5 , , PUNCT cana-3839 229 6 โˆ โˆ PROPN cana-3839 229 7 โ™ญ โ™ญ INTJ cana-3839 229 8 , , PUNCT cana-3839 229 9 โˆ โˆ PROPN cana-3839 229 10 โ™ญ โ™ญ NUM cana-3839 229 11 ) ) PUNCT cana-3839 229 12 ] ] PUNCT cana-3839 229 13 } } PUNCT cana-3839 229 14 ๐บ(โˆ๐œ‡ ๐บ(โˆ๐œ‡ NOUN cana-3839 229 15 , , PUNCT cana-3839 229 16 โˆ โˆ PROPN cana-3839 229 17 โ™ญ โ™ญ INTJ cana-3839 229 18 , , PUNCT cana-3839 229 19 โˆ โˆ NUM cana-3839 229 20 โ™ญ โ™ญ PRON cana-3839 229 21 ) ) PUNCT cana-3839 229 22 โชฏ โชฏ NOUN cana-3839 229 23 { { PUNCT cana-3839 229 24 ๐œ•1[๐บ(๐œ‡ ๐œ•1[๐บ(๐œ‡ ADJ cana-3839 229 25 , , PUNCT cana-3839 229 26 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 229 27 , , PUNCT cana-3839 229 28 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 229 29 ) ) PUNCT cana-3839 230 1 + + CCONJ cana-3839 230 2 ๐บ ๐บ PROPN cana-3839 230 3 ( ( PUNCT cana-3839 230 4 โ™ญ โ™ญ PROPN cana-3839 230 5 , , PUNCT cana-3839 230 6 โˆ โˆ PROPN cana-3839 230 7 โ™ญ โ™ญ INTJ cana-3839 230 8 , , PUNCT cana-3839 230 9 โˆ โˆ PROPN cana-3839 230 10 โ™ญ โ™ญ NUM cana-3839 230 11 ) ) PUNCT cana-3839 230 12 ] ] PUNCT cana-3839 230 13 , , PUNCT cana-3839 230 14 ๐œ•2[๐บ(๐œ‡ ๐œ•2[๐บ(๐œ‡ ADP cana-3839 230 15 , , PUNCT cana-3839 230 16 โˆ โˆ PROPN cana-3839 230 17 โ™ญ โ™ญ INTJ cana-3839 230 18 , , PUNCT cana-3839 230 19 โˆ โˆ NUM cana-3839 230 20 โ™ญ โ™ญ NUM cana-3839 230 21 ) ) PUNCT cana-3839 231 1 + + CCONJ cana-3839 231 2 ๐บ ๐บ PROPN cana-3839 231 3 ( ( PUNCT cana-3839 231 4 โ™ญ โ™ญ INTJ cana-3839 231 5 , , PUNCT cana-3839 231 6 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 231 7 , , PUNCT cana-3839 231 8 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 231 9 ) ) PUNCT cana-3839 231 10 ] ] PUNCT cana-3839 231 11 , , PUNCT cana-3839 231 12 ๐œ•3+๐œ•4 ๐œ•3+๐œ•4 NOUN cana-3839 231 13 2 2 NUM cana-3839 231 14 [ [ X cana-3839 231 15 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 231 16 , , PUNCT cana-3839 231 17 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 231 18 , , PUNCT cana-3839 231 19 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 231 20 ) ) PUNCT cana-3839 232 1 + + CCONJ cana-3839 232 2 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 232 3 , , PUNCT cana-3839 232 4 โˆ โˆ PROPN cana-3839 232 5 โ™ญ โ™ญ INTJ cana-3839 232 6 , , PUNCT cana-3839 232 7 โˆ โˆ PROPN cana-3839 232 8 โ™ญ โ™ญ NUM cana-3839 232 9 ) ) PUNCT cana-3839 232 10 ] ] PUNCT cana-3839 232 11 , , PUNCT cana-3839 232 12 ๐œ•3+๐œ•4 ๐œ•3+๐œ•4 NOUN cana-3839 232 13 2 2 NUM cana-3839 232 14 [ [ X cana-3839 232 15 ๐บ ๐บ PROPN cana-3839 232 16 ( ( PUNCT cana-3839 232 17 โ™ญ โ™ญ PROPN cana-3839 232 18 , , PUNCT cana-3839 232 19 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 232 20 , , PUNCT cana-3839 232 21 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 232 22 ) ) PUNCT cana-3839 233 1 + + CCONJ cana-3839 233 2 ๐บ ๐บ PROPN cana-3839 233 3 ( ( PUNCT cana-3839 233 4 โ™ญ โ™ญ PROPN cana-3839 233 5 , , PUNCT cana-3839 233 6 โˆ โˆ PROPN cana-3839 233 7 โ™ญ โ™ญ INTJ cana-3839 233 8 , , PUNCT cana-3839 233 9 โˆ โˆ PROPN cana-3839 233 10 โ™ญ โ™ญ NUM cana-3839 233 11 ) ) PUNCT cana-3839 233 12 ] ] PUNCT cana-3839 233 13 } } PUNCT cana-3839 233 14 โชฏ โชฏ NOUN cana-3839 233 15 { { PUNCT cana-3839 233 16 2๐œ•1+๐œ•3+๐œ•4 2๐œ•1+๐œ•3+๐œ•4 NUM cana-3839 233 17 2 2 NUM cana-3839 233 18 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 233 19 , , PUNCT cana-3839 233 20 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 233 21 , , PUNCT cana-3839 233 22 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 233 23 ) ) PUNCT cana-3839 234 1 + + CCONJ cana-3839 234 2 2๐œ•1+๐œ•3+๐œ•4 2๐œ•1+๐œ•3+๐œ•4 NUM cana-3839 234 3 2 2 NUM cana-3839 234 4 ๐บ ๐บ PROPN cana-3839 234 5 ( ( PUNCT cana-3839 234 6 โ™ญ โ™ญ PROPN cana-3839 234 7 , , PUNCT cana-3839 234 8 โˆ โˆ PROPN cana-3839 234 9 โ™ญ โ™ญ INTJ cana-3839 234 10 , , PUNCT cana-3839 234 11 โˆ โˆ NUM cana-3839 234 12 โ™ญ โ™ญ NUM cana-3839 234 13 ) ) PUNCT cana-3839 234 14 + + CCONJ cana-3839 234 15 2๐œ•2+๐œ•3+๐œ•4 2๐œ•2+๐œ•3+๐œ•4 NUM cana-3839 234 16 2 2 NUM cana-3839 234 17 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 234 18 , , PUNCT cana-3839 234 19 โˆ โˆ PROPN cana-3839 234 20 โ™ญ โ™ญ INTJ cana-3839 234 21 , , PUNCT cana-3839 234 22 โˆ โˆ NUM cana-3839 234 23 โ™ญ โ™ญ NUM cana-3839 234 24 ) ) PUNCT cana-3839 235 1 + + CCONJ cana-3839 235 2 2๐œ•2+๐œ•3+๐œ•4 2๐œ•2+๐œ•3+๐œ•4 NUM cana-3839 235 3 2 2 NUM cana-3839 235 4 ๐บ ๐บ PROPN cana-3839 235 5 ( ( PUNCT cana-3839 235 6 โ™ญ โ™ญ PROPN cana-3839 235 7 , , PUNCT cana-3839 235 8 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 235 9 , , PUNCT cana-3839 235 10 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 235 11 ) ) PUNCT cana-3839 235 12 } } PUNCT cana-3839 235 13 โชฏ โชฏ NOUN cana-3839 235 14 { { PUNCT cana-3839 235 15 2๐œ•1+๐œ•3+๐œ•4 2๐œ•1+๐œ•3+๐œ•4 NUM cana-3839 235 16 2 2 NUM cana-3839 235 17 [ [ X cana-3839 235 18 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 235 19 , , PUNCT cana-3839 235 20 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 235 21 , , PUNCT cana-3839 235 22 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 235 23 ) ) PUNCT cana-3839 236 1 + + CCONJ cana-3839 236 2 ๐บ ๐บ PROPN cana-3839 236 3 ( ( PUNCT cana-3839 236 4 โ™ญ โ™ญ PROPN cana-3839 236 5 , , PUNCT cana-3839 236 6 โˆ โˆ PROPN cana-3839 236 7 โ™ญ โ™ญ INTJ cana-3839 236 8 , , PUNCT cana-3839 236 9 โˆ โˆ PROPN cana-3839 236 10 โ™ญ โ™ญ NUM cana-3839 236 11 ) ) PUNCT cana-3839 236 12 ] ] PUNCT cana-3839 237 1 + + CCONJ cana-3839 237 2 2๐œ•2+๐œ•3+๐œ•4 2๐œ•2+๐œ•3+๐œ•4 NUM cana-3839 237 3 2 2 NUM cana-3839 237 4 [ [ X cana-3839 237 5 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 237 6 , , PUNCT cana-3839 237 7 โˆ โˆ PROPN cana-3839 237 8 โ™ญ โ™ญ INTJ cana-3839 237 9 , , PUNCT cana-3839 237 10 โˆ โˆ NUM cana-3839 237 11 โ™ญ โ™ญ NUM cana-3839 237 12 ) ) PUNCT cana-3839 238 1 + + CCONJ cana-3839 238 2 ๐บ ๐บ PROPN cana-3839 238 3 ( ( PUNCT cana-3839 238 4 โ™ญ โ™ญ INTJ cana-3839 238 5 , , PUNCT cana-3839 238 6 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 238 7 , , PUNCT cana-3839 238 8 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 238 9 ) ) PUNCT cana-3839 238 10 ] ] PUNCT cana-3839 238 11 } } PUNCT cana-3839 238 12 ( ( PUNCT cana-3839 238 13 13 13 X cana-3839 238 14 ) ) PUNCT cana-3839 238 15 putting put VERB cana-3839 238 16 โ™ญ โ™ญ X cana-3839 238 17 = = PUNCT cana-3839 239 1 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 239 2 in in ADP cana-3839 239 3 eq(13 eq(13 NOUN cana-3839 239 4 ) ) PUNCT cana-3839 239 5 , , PUNCT cana-3839 239 6 we we PRON cana-3839 239 7 get get VERB cana-3839 239 8 ๐บ(โˆ๐œ‡ ๐บ(โˆ๐œ‡ NOUN cana-3839 239 9 , , PUNCT cana-3839 239 10 โˆ2๐œ‡ โˆ2๐œ‡ NOUN cana-3839 239 11 , , PUNCT cana-3839 239 12 โˆ2๐œ‡ โˆ2๐œ‡ NUM cana-3839 239 13 ) ) PUNCT cana-3839 239 14 โชฏ โชฏ NOUN cana-3839 239 15 { { PUNCT cana-3839 239 16 2๐œ•1 2๐œ•1 NUM cana-3839 239 17 + + NUM cana-3839 239 18 ๐œ•3 ๐œ•3 NOUN cana-3839 239 19 + + SYM cana-3839 239 20 ๐œ•4 ๐œ•4 NOUN cana-3839 239 21 2 2 NUM cana-3839 239 22 [ [ X cana-3839 239 23 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 239 24 , , PUNCT cana-3839 239 25 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 239 26 , , PUNCT cana-3839 239 27 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 239 28 ) ) PUNCT cana-3839 240 1 + + NUM cana-3839 240 2 ๐บ(โˆ๐œ‡ ๐บ(โˆ๐œ‡ NOUN cana-3839 240 3 , , PUNCT cana-3839 240 4 โˆ2๐œ‡ โˆ2๐œ‡ NOUN cana-3839 240 5 , , PUNCT cana-3839 240 6 โˆ2๐œ‡ โˆ2๐œ‡ NUM cana-3839 240 7 ) ) PUNCT cana-3839 240 8 ] ] PUNCT cana-3839 241 1 + + CCONJ cana-3839 241 2 2๐œ•2 2๐œ•2 NUM cana-3839 241 3 + + CCONJ cana-3839 241 4 ๐œ•3 ๐œ•3 NOUN cana-3839 241 5 + + SYM cana-3839 241 6 ๐œ•4 ๐œ•4 NOUN cana-3839 241 7 2 2 NUM cana-3839 241 8 [ [ X cana-3839 241 9 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 241 10 , , PUNCT cana-3839 241 11 โˆ2๐œ‡ โˆ2๐œ‡ NOUN cana-3839 241 12 , , PUNCT cana-3839 241 13 โˆ2๐œ‡ โˆ2๐œ‡ NUM cana-3839 241 14 ) ) PUNCT cana-3839 241 15 + + NUM cana-3839 241 16 ๐บ(โˆ๐œ‡ ๐บ(โˆ๐œ‡ NOUN cana-3839 241 17 , , PUNCT cana-3839 241 18 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 241 19 , , PUNCT cana-3839 241 20 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 241 21 ) ) PUNCT cana-3839 241 22 ] ] PUNCT cana-3839 241 23 } } PUNCT cana-3839 241 24 โชฏ โชฏ NOUN cana-3839 241 25 { { PUNCT cana-3839 241 26 2๐œ•1 2๐œ•1 NUM cana-3839 241 27 + + NUM cana-3839 241 28 ๐œ•3 ๐œ•3 NOUN cana-3839 241 29 + + SYM cana-3839 241 30 ๐œ•4 ๐œ•4 NOUN cana-3839 241 31 2 2 NUM cana-3839 241 32 [ [ X cana-3839 241 33 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 241 34 , , PUNCT cana-3839 241 35 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 241 36 , , PUNCT cana-3839 241 37 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 241 38 ) ) PUNCT cana-3839 241 39 + + NUM cana-3839 241 40 ๐บ(โˆ๐œ‡ ๐บ(โˆ๐œ‡ NOUN cana-3839 241 41 , , PUNCT cana-3839 241 42 โˆ2๐œ‡ โˆ2๐œ‡ NOUN cana-3839 241 43 , , PUNCT cana-3839 241 44 โˆ2๐œ‡ โˆ2๐œ‡ NUM cana-3839 241 45 ) ) PUNCT cana-3839 241 46 ] ] PUNCT cana-3839 242 1 + + CCONJ cana-3839 242 2 communications communication NOUN cana-3839 242 3 on on ADP cana-3839 242 4 applied apply VERB cana-3839 242 5 nonlinear nonlinear ADJ cana-3839 242 6 analysis analysis NOUN cana-3839 242 7 issn issn NOUN cana-3839 242 8 : : PUNCT cana-3839 242 9 1074 1074 NUM cana-3839 242 10 - - PUNCT cana-3839 242 11 133x 133x NUM cana-3839 242 12 vol vol NOUN cana-3839 242 13 32 32 NUM cana-3839 242 14 no no NOUN cana-3839 242 15 . . PUNCT cana-3839 243 1 9s 9s NUM cana-3839 243 2 ( ( PUNCT cana-3839 243 3 2025 2025 NUM cana-3839 243 4 ) ) PUNCT cana-3839 243 5 85 85 NUM cana-3839 243 6 https://internationalpubls.com https://internationalpubls.com X cana-3839 243 7 2๐œ•2+๐œ•3+๐œ•4 2๐œ•2+๐œ•3+๐œ•4 NUM cana-3839 243 8 2 2 NUM cana-3839 243 9 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 243 10 , , PUNCT cana-3839 243 11 โˆ2๐œ‡ โˆ2๐œ‡ NOUN cana-3839 243 12 , , PUNCT cana-3839 243 13 โˆ2๐œ‡ โˆ2๐œ‡ NUM cana-3839 243 14 ) ) PUNCT cana-3839 243 15 } } PUNCT cana-3839 243 16 ( ( PUNCT cana-3839 243 17 14 14 NUM cana-3839 243 18 ) ) PUNCT cana-3839 243 19 by by ADP cana-3839 243 20 using use VERB cana-3839 243 21 ( ( PUNCT cana-3839 243 22 vgm5 vgm5 PROPN cana-3839 243 23 ) ) PUNCT cana-3839 243 24 , , PUNCT cana-3839 243 25 we we PRON cana-3839 243 26 get get VERB cana-3839 243 27 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 243 28 , , PUNCT cana-3839 243 29 โˆ2๐œ‡ โˆ2๐œ‡ NOUN cana-3839 243 30 , , PUNCT cana-3839 243 31 โˆ2๐œ‡ โˆ2๐œ‡ NUM cana-3839 243 32 ) ) PUNCT cana-3839 243 33 โชฏ โชฏ NOUN cana-3839 243 34 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 243 35 , , PUNCT cana-3839 243 36 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 243 37 , , PUNCT cana-3839 243 38 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 243 39 ) ) PUNCT cana-3839 243 40 + + NUM cana-3839 243 41 ๐บ(โˆ๐œ‡ ๐บ(โˆ๐œ‡ NOUN cana-3839 243 42 , , PUNCT cana-3839 243 43 โˆ2๐œ‡ โˆ2๐œ‡ NOUN cana-3839 243 44 , , PUNCT cana-3839 243 45 โˆ2๐œ‡ โˆ2๐œ‡ NUM cana-3839 243 46 ) ) PUNCT cana-3839 243 47 ( ( PUNCT cana-3839 243 48 15 15 NUM cana-3839 243 49 ) ) PUNCT cana-3839 243 50 then then ADV cana-3839 243 51 eq(14 eq(14 NOUN cana-3839 243 52 ) ) PUNCT cana-3839 243 53 becomes become VERB cana-3839 243 54 ๐บ(โˆ๐œ‡ ๐บ(โˆ๐œ‡ NOUN cana-3839 243 55 , , PUNCT cana-3839 243 56 โˆ2๐œ‡ โˆ2๐œ‡ NOUN cana-3839 243 57 , , PUNCT cana-3839 243 58 โˆ2๐œ‡ โˆ2๐œ‡ NUM cana-3839 243 59 ) ) PUNCT cana-3839 243 60 โชฏ โชฏ NOUN cana-3839 243 61 { { PUNCT cana-3839 243 62 2๐œ•1 2๐œ•1 NUM cana-3839 243 63 + + NUM cana-3839 243 64 ๐œ•3 ๐œ•3 NOUN cana-3839 243 65 + + SYM cana-3839 243 66 ๐œ•4 ๐œ•4 NOUN cana-3839 243 67 2 2 NUM cana-3839 243 68 [ [ X cana-3839 243 69 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 243 70 , , PUNCT cana-3839 243 71 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 243 72 , , PUNCT cana-3839 243 73 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 243 74 ) ) PUNCT cana-3839 244 1 + + NUM cana-3839 244 2 ๐บ(โˆ๐œ‡ ๐บ(โˆ๐œ‡ NOUN cana-3839 244 3 , , PUNCT cana-3839 244 4 โˆ2๐œ‡ โˆ2๐œ‡ NOUN cana-3839 244 5 , , PUNCT cana-3839 244 6 โˆ2๐œ‡ โˆ2๐œ‡ NUM cana-3839 244 7 ) ) PUNCT cana-3839 244 8 ] ] PUNCT cana-3839 245 1 + + CCONJ cana-3839 245 2 2๐œ•2 2๐œ•2 NUM cana-3839 245 3 + + CCONJ cana-3839 245 4 ๐œ•3 ๐œ•3 NOUN cana-3839 245 5 + + SYM cana-3839 245 6 ๐œ•4 ๐œ•4 NOUN cana-3839 245 7 2 2 NUM cana-3839 245 8 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 245 9 , , PUNCT cana-3839 245 10 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 245 11 , , PUNCT cana-3839 245 12 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 245 13 ) ) PUNCT cana-3839 246 1 + + NUM cana-3839 246 2 ๐บ(โˆ๐œ‡ ๐บ(โˆ๐œ‡ NOUN cana-3839 246 3 , , PUNCT cana-3839 246 4 โˆ2๐œ‡ โˆ2๐œ‡ NOUN cana-3839 246 5 , , PUNCT cana-3839 246 6 โˆ2๐œ‡ โˆ2๐œ‡ NUM cana-3839 246 7 ) ) PUNCT cana-3839 246 8 } } PUNCT cana-3839 246 9 โชฏ โชฏ NOUN cana-3839 246 10 { { PUNCT cana-3839 246 11 โˆ‘ โˆ‘ PROPN cana-3839 246 12 4 4 NUM cana-3839 246 13 ๐‘–=1 ๐‘–=1 SYM cana-3839 246 14 ๐œ•๐‘–๐บ(๐œ‡ ๐œ•๐‘–๐บ(๐œ‡ NOUN cana-3839 246 15 , , PUNCT cana-3839 246 16 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 246 17 , , PUNCT cana-3839 246 18 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 246 19 ) ) PUNCT cana-3839 247 1 + + CCONJ cana-3839 247 2 โˆ‘ โˆ‘ PROPN cana-3839 247 3 4 4 NUM cana-3839 247 4 ๐‘–=1 ๐‘–=1 PROPN cana-3839 247 5 ๐œ•๐‘–๐บ(โˆ๐œ‡ ๐œ•๐‘–๐บ(โˆ๐œ‡ NOUN cana-3839 247 6 , , PUNCT cana-3839 247 7 โˆ2๐œ‡ โˆ2๐œ‡ NOUN cana-3839 247 8 , , PUNCT cana-3839 247 9 โˆ2๐œ‡ โˆ2๐œ‡ NUM cana-3839 247 10 ) ) PUNCT cana-3839 247 11 } } PUNCT cana-3839 247 12 ( ( PUNCT cana-3839 247 13 1 1 NUM cana-3839 247 14 โˆ’ โˆ’ NOUN cana-3839 247 15 โˆ‘ โˆ‘ PROPN cana-3839 247 16 4 4 NUM cana-3839 247 17 ๐‘–=1 ๐‘–=1 PROPN cana-3839 247 18 ๐œ•๐‘–)๐บ(โˆ๐œ‡ ๐œ•๐‘–)๐บ(โˆ๐œ‡ NOUN cana-3839 247 19 , , PUNCT cana-3839 247 20 โˆ2๐œ‡ โˆ2๐œ‡ NOUN cana-3839 247 21 , , PUNCT cana-3839 247 22 โˆ2๐œ‡ โˆ2๐œ‡ NUM cana-3839 247 23 ) ) PUNCT cana-3839 247 24 โชฏ โชฏ NOUN cana-3839 247 25 โˆ‘ โˆ‘ PROPN cana-3839 247 26 4 4 NUM cana-3839 247 27 ๐‘–=1 ๐‘–=1 PROPN cana-3839 247 28 ๐œ•๐‘–(๐บ(๐œ‡ ๐œ•๐‘–(๐บ(๐œ‡ ADV cana-3839 247 29 , , PUNCT cana-3839 247 30 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 247 31 , , PUNCT cana-3839 247 32 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 247 33 ) ) PUNCT cana-3839 247 34 ) ) PUNCT cana-3839 247 35 ๐บ(โˆ๐œ‡ ๐บ(โˆ๐œ‡ NOUN cana-3839 247 36 , , PUNCT cana-3839 247 37 โˆ2๐œ‡ โˆ2๐œ‡ NOUN cana-3839 247 38 , , PUNCT cana-3839 247 39 โˆ2๐œ‡ โˆ2๐œ‡ NUM cana-3839 247 40 ) ) PUNCT cana-3839 247 41 โชฏ โชฏ NOUN cana-3839 247 42 โˆ‘4 โˆ‘4 PROPN cana-3839 247 43 ๐‘–=1 ๐‘–=1 PROPN cana-3839 248 1 ๐œ•๐‘– ๐œ•๐‘– NOUN cana-3839 248 2 1โˆ’โˆ‘4 1โˆ’โˆ‘4 NUM cana-3839 248 3 ๐‘–=1 ๐‘–=1 PUNCT cana-3839 249 1 ๐œ•๐‘– ๐œ•๐‘– NOUN cana-3839 249 2 ๐บ(๐œ‡ ๐บ(๐œ‡ NOUN cana-3839 249 3 , , PUNCT cana-3839 249 4 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 249 5 , , PUNCT cana-3839 249 6 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 249 7 ) ) PUNCT cana-3839 250 1 ( ( PUNCT cana-3839 250 2 16 16 X cana-3839 250 3 ) ) PUNCT cana-3839 250 4 putting put VERB cana-3839 250 5 ๐‘ž ๐‘ž X cana-3839 250 6 = = PUNCT cana-3839 250 7 โˆ‘4 โˆ‘4 PROPN cana-3839 250 8 ๐‘–=1 ๐‘–=1 PROPN cana-3839 251 1 ๐œ•๐‘– ๐œ•๐‘– NOUN cana-3839 251 2 1โˆ’โˆ‘4 1โˆ’โˆ‘4 NUM cana-3839 252 1 ๐‘–=1 ๐‘–=1 PROPN cana-3839 252 2 ๐œ•๐‘– ๐œ•๐‘– NOUN cana-3839 252 3 . . PUNCT cana-3839 253 1 since since SCONJ cana-3839 253 2 0 0 NUM cana-3839 253 3 โ‰ค โ‰ค NUM cana-3839 253 4 โˆ‘4 โˆ‘4 X cana-3839 253 5 ๐‘–=1 ๐‘–=1 PROPN cana-3839 253 6 ๐œ•๐‘– ๐œ•๐‘– NOUN cana-3839 253 7 < < X cana-3839 253 8 1 1 NUM cana-3839 253 9 2 2 NUM cana-3839 253 10 , , PUNCT cana-3839 253 11 so so ADV cana-3839 253 12 0 0 NUM cana-3839 253 13 โ‰ค โ‰ค NOUN cana-3839 253 14 ๐‘ž ๐‘ž X cana-3839 253 15 < < X cana-3839 253 16 1 1 NUM cana-3839 253 17 . . PUNCT cana-3839 253 18 then then ADV cana-3839 253 19 eq(16 eq(16 PROPN cana-3839 253 20 ) ) PUNCT cana-3839 253 21 becomes become VERB cana-3839 253 22 ๐บ(โˆ๐œ‡ ๐บ(โˆ๐œ‡ NOUN cana-3839 253 23 , , PUNCT cana-3839 253 24 โˆ2๐œ‡ โˆ2๐œ‡ NOUN cana-3839 253 25 , , PUNCT cana-3839 253 26 โˆ2๐œ‡ โˆ2๐œ‡ NUM cana-3839 253 27 ) ) PUNCT cana-3839 253 28 โชฏ โชฏ NOUN cana-3839 253 29 ๐‘ž๐บ(๐œ‡ ๐‘ž๐บ(๐œ‡ NOUN cana-3839 253 30 , , PUNCT cana-3839 253 31 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 253 32 , , PUNCT cana-3839 253 33 โˆ๐œ‡ โˆ๐œ‡ PROPN cana-3839 253 34 ) ) PUNCT cana-3839 253 35 ( ( PUNCT cana-3839 254 1 17 17 NUM cana-3839 254 2 ) ) PUNCT cana-3839 254 3 โˆ€๐œ‡ โˆ€๐œ‡ NOUN cana-3839 255 1 โˆˆ โˆˆ PROPN cana-3839 255 2 ๐œ’๐‘ž. ๐œ’๐‘ž. NOUN cana-3839 255 3 further further PROPN cana-3839 255 4 ๐บ(โˆ๐œŒ๐œ‡ ๐บ(โˆ๐œŒ๐œ‡ PROPN cana-3839 255 5 , , PUNCT cana-3839 255 6 โˆ๐œŒ+1๐œ‡ โˆ๐œŒ+1๐œ‡ PROPN cana-3839 255 7 , , PUNCT cana-3839 255 8 โˆ๐œŒ+1๐œ‡ โˆ๐œŒ+1๐œ‡ NUM cana-3839 255 9 ) ) PUNCT cana-3839 255 10 โชฏ โชฏ NOUN cana-3839 255 11 ๐‘ž๐œŒ๐บ(๐œ‡ ๐‘ž๐œŒ๐บ(๐œ‡ PROPN cana-3839 255 12 , , PUNCT cana-3839 255 13 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 255 14 , , PUNCT cana-3839 255 15 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 255 16 ) ) PUNCT cana-3839 255 17 โˆ€๐œŒ โˆ€๐œŒ PUNCT cana-3839 255 18 โˆˆ โˆˆ PROPN cana-3839 255 19 โ„• โ„• PROPN cana-3839 255 20 ๐œ‡ ๐œ‡ ADP cana-3839 255 21 โˆˆ โˆˆ NOUN cana-3839 255 22 ๐œ’๐‘ž ๐œ’๐‘ž NOUN cana-3839 255 23 . . PUNCT cana-3839 256 1 for for ADP cana-3839 256 2 ๐œŒ ๐œŒ ADP cana-3839 256 3 , , PUNCT cana-3839 256 4 ๐‘— ๐‘— PROPN cana-3839 256 5 โˆˆ โˆˆ PROPN cana-3839 256 6 โ„• โ„• PROPN cana-3839 256 7 and and CCONJ cana-3839 256 8 ๐œ‡ ๐œ‡ ADP cana-3839 256 9 โˆˆ โˆˆ PROPN cana-3839 256 10 ๐œ’๐‘ž ๐œ’๐‘ž NOUN cana-3839 256 11 ๐บ(โˆ๐œŒ(๐œ‡ ๐บ(โˆ๐œŒ(๐œ‡ NOUN cana-3839 256 12 ) ) PUNCT cana-3839 256 13 , , PUNCT cana-3839 256 14 โˆ๐œŒ+๐‘—(๐œ‡ โˆ๐œŒ+๐‘—(๐œ‡ ADP cana-3839 256 15 ) ) PUNCT cana-3839 256 16 , , PUNCT cana-3839 256 17 โˆ๐œŒ+๐‘—(๐œ‡ โˆ๐œŒ+๐‘—(๐œ‡ NUM cana-3839 256 18 ) ) PUNCT cana-3839 256 19 ) ) PUNCT cana-3839 256 20 โชฏ โชฏ NOUN cana-3839 256 21 ๐บ(โˆ๐œŒ(๐œ‡ ๐บ(โˆ๐œŒ(๐œ‡ NOUN cana-3839 256 22 ) ) PUNCT cana-3839 256 23 , , PUNCT cana-3839 256 24 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 256 25 ) ) PUNCT cana-3839 256 26 , , PUNCT cana-3839 256 27 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 256 28 ) ) PUNCT cana-3839 256 29 ) ) PUNCT cana-3839 257 1 + + CCONJ cana-3839 257 2 ๐บ(โˆ๐œŒ+1(๐œ‡ ๐บ(โˆ๐œŒ+1(๐œ‡ NUM cana-3839 257 3 ) ) PUNCT cana-3839 257 4 , , PUNCT cana-3839 257 5 โˆ๐œŒ+๐‘—(๐œ‡ โˆ๐œŒ+๐‘—(๐œ‡ ADP cana-3839 257 6 ) ) PUNCT cana-3839 257 7 , , PUNCT cana-3839 257 8 โˆ๐œŒ+๐‘—(๐œ‡ โˆ๐œŒ+๐‘—(๐œ‡ NUM cana-3839 257 9 ) ) PUNCT cana-3839 257 10 ) ) PUNCT cana-3839 258 1 โชฏ โชฏ NOUN cana-3839 258 2 ๐บ(โˆ๐œŒ(๐œ‡ ๐บ(โˆ๐œŒ(๐œ‡ NOUN cana-3839 258 3 ) ) PUNCT cana-3839 258 4 , , PUNCT cana-3839 258 5 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 258 6 ) ) PUNCT cana-3839 258 7 , , PUNCT cana-3839 258 8 โˆ๐œŒ+1(๐œ‡ โˆ๐œŒ+1(๐œ‡ NOUN cana-3839 258 9 ) ) PUNCT cana-3839 258 10 ) ) PUNCT cana-3839 259 1 + + CCONJ cana-3839 259 2 ๐บ(โˆ๐œŒ+1(๐œ‡ ๐บ(โˆ๐œŒ+1(๐œ‡ NUM cana-3839 259 3 ) ) PUNCT cana-3839 259 4 , , PUNCT cana-3839 259 5 โˆ๐œŒ+2(๐œ‡ โˆ๐œŒ+2(๐œ‡ ADJ cana-3839 259 6 ) ) PUNCT cana-3839 259 7 , , PUNCT cana-3839 259 8 โˆ๐œŒ+2(๐œ‡ โˆ๐œŒ+2(๐œ‡ ADJ cana-3839 259 9 ) ) PUNCT cana-3839 259 10 ) ) PUNCT cana-3839 260 1 + + VERB cana-3839 260 2 ๐บ(โˆ๐œŒ+2(๐œ‡ ๐บ(โˆ๐œŒ+2(๐œ‡ ADJ cana-3839 260 3 ) ) PUNCT cana-3839 260 4 , , PUNCT cana-3839 260 5 โˆ๐œŒ+๐‘—(๐œ‡ โˆ๐œŒ+๐‘—(๐œ‡ ADP cana-3839 260 6 ) ) PUNCT cana-3839 260 7 , , PUNCT cana-3839 260 8 โˆ๐œŒ+๐‘—(๐œ‡ โˆ๐œŒ+๐‘—(๐œ‡ NUM cana-3839 260 9 ) ) PUNCT cana-3839 260 10 โ‹ฎ โ‹ฎ NOUN cana-3839 260 11 โชฏ โชฏ VERB cana-3839 260 12 โˆ‘๐‘— โˆ‘๐‘— PROPN cana-3839 260 13 ๐‘–=1 ๐‘–=1 PROPN cana-3839 260 14 ๐บ(โˆ๐œŒ+๐‘–โˆ’1(๐œ‡ ๐บ(โˆ๐œŒ+๐‘–โˆ’1(๐œ‡ NOUN cana-3839 260 15 ) ) PUNCT cana-3839 260 16 , , PUNCT cana-3839 260 17 โˆ๐œŒ+๐‘–(๐œ‡ โˆ๐œŒ+๐‘–(๐œ‡ NUM cana-3839 260 18 ) ) PUNCT cana-3839 260 19 , , PUNCT cana-3839 260 20 โˆ๐œŒ+๐‘–(๐œ‡ โˆ๐œŒ+๐‘–(๐œ‡ NUM cana-3839 260 21 ) ) PUNCT cana-3839 260 22 ) ) PUNCT cana-3839 260 23 communications communication NOUN cana-3839 260 24 on on ADP cana-3839 260 25 applied apply VERB cana-3839 260 26 nonlinear nonlinear ADJ cana-3839 260 27 analysis analysis NOUN cana-3839 260 28 issn issn NOUN cana-3839 260 29 : : PUNCT cana-3839 260 30 1074 1074 NUM cana-3839 260 31 - - PUNCT cana-3839 260 32 133x 133x NUM cana-3839 260 33 vol vol NOUN cana-3839 260 34 32 32 NUM cana-3839 260 35 no no NOUN cana-3839 260 36 . . PUNCT cana-3839 261 1 9s 9s NUM cana-3839 261 2 ( ( PUNCT cana-3839 261 3 2025 2025 NUM cana-3839 261 4 ) ) PUNCT cana-3839 261 5 86 86 NUM cana-3839 262 1 https://internationalpubls.com https://internationalpubls.com X cana-3839 262 2 โชฏ โชฏ NOUN cana-3839 262 3 โˆ‘๐‘— โˆ‘๐‘— PROPN cana-3839 262 4 ๐‘–=1 ๐‘–=1 PROPN cana-3839 262 5 ๐‘ž๐œŒ+๐‘–โˆ’1๐บ(๐œ‡ ๐‘ž๐œŒ+๐‘–โˆ’1๐บ(๐œ‡ X cana-3839 262 6 , , PUNCT cana-3839 262 7 โˆ๐œ‡ โˆ๐œ‡ VERB cana-3839 262 8 , , PUNCT cana-3839 262 9 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 262 10 ) ) PUNCT cana-3839 262 11 ๐บ(โˆ๐œŒ(๐œ‡ ๐บ(โˆ๐œŒ(๐œ‡ NUM cana-3839 262 12 ) ) PUNCT cana-3839 262 13 , , PUNCT cana-3839 262 14 โˆ๐œŒ+๐‘—(๐œ‡ โˆ๐œŒ+๐‘—(๐œ‡ ADP cana-3839 262 15 ) ) PUNCT cana-3839 262 16 , , PUNCT cana-3839 262 17 โˆ๐œŒ+๐‘—(๐œ‡ โˆ๐œŒ+๐‘—(๐œ‡ NUM cana-3839 262 18 ) ) PUNCT cana-3839 262 19 ) ) PUNCT cana-3839 262 20 โชฏ โชฏ NOUN cana-3839 262 21 ๐‘ž๐œŒ ๐‘ž๐œŒ PROPN cana-3839 262 22 1โˆ’๐‘ž 1โˆ’๐‘ž NUM cana-3839 262 23 ๐บ(๐œ‡ ๐บ(๐œ‡ ADP cana-3839 262 24 , , PUNCT cana-3839 262 25 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 262 26 , , PUNCT cana-3839 262 27 โˆ๐œ‡ โˆ๐œ‡ NOUN cana-3839 262 28 ) ) PUNCT cana-3839 262 29 ( ( PUNCT cana-3839 262 30 18 18 NUM cana-3839 262 31 ) ) PUNCT cana-3839 262 32 this this PRON cana-3839 262 33 implies imply VERB cana-3839 262 34 { { PUNCT cana-3839 262 35 โˆ๐œŒ๐œ‡ โˆ๐œŒ๐œ‡ NOUN cana-3839 262 36 } } PUNCT cana-3839 262 37 is be AUX cana-3839 262 38 a a DET cana-3839 262 39 vcs vcs NOUN cana-3839 262 40 in in ADP cana-3839 262 41 g g NOUN cana-3839 262 42 - - PUNCT cana-3839 262 43 complete complete ADJ cana-3839 262 44 vgms vgms NOUN cana-3839 262 45 and and CCONJ cana-3839 262 46 so so ADV cana-3839 262 47 it it PRON cana-3839 262 48 converges converge VERB cana-3839 262 49 to to ADP cana-3839 262 50 ๐œ‡1 ๐œ‡1 PROPN cana-3839 262 51 โˆˆ โˆˆ PROPN cana-3839 262 52 ๐œ•. ๐œ•. PROPN cana-3839 262 53 now now ADV cana-3839 262 54 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADJ cana-3839 262 55 , , PUNCT cana-3839 262 56 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 262 57 , , PUNCT cana-3839 262 58 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 262 59 ) ) PUNCT cana-3839 262 60 โชฏ โชฏ NOUN cana-3839 263 1 ๐บ(๐œ‡1 ๐บ(๐œ‡1 PROPN cana-3839 263 2 , , PUNCT cana-3839 263 3 โˆ๐œŒ+1๐œ‡ โˆ๐œŒ+1๐œ‡ PROPN cana-3839 263 4 , , PUNCT cana-3839 263 5 โˆ๐œŒ+1๐œ‡ โˆ๐œŒ+1๐œ‡ NUM cana-3839 263 6 ) ) PUNCT cana-3839 264 1 + + X cana-3839 264 2 ๐บ(โˆ๐œŒ+1๐œ‡ ๐บ(โˆ๐œŒ+1๐œ‡ NUM cana-3839 264 3 , , PUNCT cana-3839 264 4 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 264 5 , , PUNCT cana-3839 264 6 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 264 7 ) ) PUNCT cana-3839 264 8 โชฏ โชฏ NOUN cana-3839 264 9 ๐บ(๐œ‡1 ๐บ(๐œ‡1 PROPN cana-3839 264 10 , , PUNCT cana-3839 264 11 โˆ๐œŒ+1๐œ‡ โˆ๐œŒ+1๐œ‡ PROPN cana-3839 264 12 , , PUNCT cana-3839 264 13 โˆ๐œŒ+1๐œ‡ โˆ๐œŒ+1๐œ‡ NUM cana-3839 264 14 ) ) PUNCT cana-3839 265 1 + + NOUN cana-3839 265 2 2๐œ•1+๐œ•3+๐œ•4 2๐œ•1+๐œ•3+๐œ•4 NUM cana-3839 265 3 2 2 NUM cana-3839 265 4 [ [ X cana-3839 265 5 ๐บ(โˆ๐œŒ๐œ‡ ๐บ(โˆ๐œŒ๐œ‡ PROPN cana-3839 265 6 , , PUNCT cana-3839 265 7 โˆ๐œŒ+1๐œ‡ โˆ๐œŒ+1๐œ‡ PROPN cana-3839 265 8 , , PUNCT cana-3839 265 9 โˆ๐œŒ+1๐œ‡ โˆ๐œŒ+1๐œ‡ NUM cana-3839 265 10 ) ) PUNCT cana-3839 266 1 + + ADJ cana-3839 266 2 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADJ cana-3839 266 3 , , PUNCT cana-3839 266 4 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 266 5 , , PUNCT cana-3839 266 6 โˆ๐œ‡1 โˆ๐œ‡1 NOUN cana-3839 266 7 ) ) PUNCT cana-3839 266 8 ] ] PUNCT cana-3839 267 1 + + CCONJ cana-3839 267 2 2๐œ•2+๐œ•3+๐œ•4 2๐œ•2+๐œ•3+๐œ•4 NUM cana-3839 267 3 2 2 NUM cana-3839 267 4 [ [ X cana-3839 267 5 ๐บ(โˆ๐œŒ๐œ‡ ๐บ(โˆ๐œŒ๐œ‡ PROPN cana-3839 267 6 , , PUNCT cana-3839 267 7 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 267 8 , , PUNCT cana-3839 267 9 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 267 10 ) ) PUNCT cana-3839 268 1 + + ADJ cana-3839 268 2 ๐บ(๐œ‡1 ๐บ(๐œ‡1 PROPN cana-3839 268 3 , , PUNCT cana-3839 268 4 โˆ๐œŒ+1๐œ‡ โˆ๐œŒ+1๐œ‡ PROPN cana-3839 268 5 , , PUNCT cana-3839 268 6 โˆ๐œŒ+1๐œ‡ โˆ๐œŒ+1๐œ‡ NUM cana-3839 268 7 ) ) PUNCT cana-3839 268 8 ] ] PUNCT cana-3839 268 9 ( ( PUNCT cana-3839 268 10 19 19 NUM cana-3839 268 11 ) ) PUNCT cana-3839 268 12 by by ADP cana-3839 268 13 using use VERB cana-3839 268 14 ๐œ‡1 ๐œ‡1 PROPN cana-3839 268 15 = = SYM cana-3839 268 16 lim lim PROPN cana-3839 268 17 ๐œŒโ†’โˆž ๐œŒโ†’โˆž NUM cana-3839 268 18 โˆ๐œŒ๐œ‡ โˆ๐œŒ๐œ‡ X cana-3839 268 19 = = SYM cana-3839 268 20 lim lim NOUN cana-3839 268 21 ๐œŒโ†’โˆž ๐œŒโ†’โˆž NUM cana-3839 268 22 โˆ๐œŒ+1๐œ‡. โˆ๐œŒ+1๐œ‡. NOUN cana-3839 268 23 then then ADV cana-3839 268 24 eq(19 eq(19 PROPN cana-3839 268 25 ) ) PUNCT cana-3839 268 26 becomes become VERB cana-3839 268 27 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADJ cana-3839 268 28 , , PUNCT cana-3839 268 29 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 268 30 , , PUNCT cana-3839 268 31 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 268 32 ) ) PUNCT cana-3839 268 33 โชฏ โชฏ NOUN cana-3839 268 34 0 0 NUM cana-3839 269 1 + + CCONJ cana-3839 269 2 2๐œ•1 2๐œ•1 NUM cana-3839 269 3 + + NUM cana-3839 269 4 ๐œ•3 ๐œ•3 NOUN cana-3839 269 5 + + SYM cana-3839 269 6 ๐œ•4 ๐œ•4 NOUN cana-3839 269 7 2 2 NUM cana-3839 269 8 [ [ SYM cana-3839 269 9 0 0 X cana-3839 270 1 + + CCONJ cana-3839 270 2 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADJ cana-3839 270 3 , , PUNCT cana-3839 270 4 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 270 5 , , PUNCT cana-3839 270 6 โˆ๐œ‡1 โˆ๐œ‡1 NOUN cana-3839 270 7 ) ) PUNCT cana-3839 270 8 ] ] PUNCT cana-3839 271 1 + + CCONJ cana-3839 271 2 2๐œ•2 2๐œ•2 NUM cana-3839 271 3 + + CCONJ cana-3839 271 4 ๐œ•3 ๐œ•3 NOUN cana-3839 271 5 + + SYM cana-3839 271 6 ๐œ•4 ๐œ•4 NOUN cana-3839 271 7 2 2 NUM cana-3839 271 8 [ [ X cana-3839 271 9 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADJ cana-3839 271 10 , , PUNCT cana-3839 271 11 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 271 12 , , PUNCT cana-3839 271 13 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 271 14 ) ) PUNCT cana-3839 271 15 + + CCONJ cana-3839 271 16 0 0 X cana-3839 271 17 ] ] X cana-3839 271 18 โชฏ โชฏ X cana-3839 271 19 โˆ‘4 โˆ‘4 PROPN cana-3839 271 20 ๐‘™=1 ๐‘™=1 PROPN cana-3839 271 21 ๐œ•๐‘–๐บ(๐œ‡1 ๐œ•๐‘–๐บ(๐œ‡1 PROPN cana-3839 271 22 , , PUNCT cana-3839 271 23 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 271 24 , , PUNCT cana-3839 271 25 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 271 26 ) ) PUNCT cana-3839 271 27 ( ( PUNCT cana-3839 271 28 20 20 NUM cana-3839 271 29 ) ) PUNCT cana-3839 271 30 since since SCONJ cana-3839 271 31 0 0 NUM cana-3839 271 32 โ‰ค โ‰ค NUM cana-3839 271 33 โˆ‘4 โˆ‘4 X cana-3839 272 1 ๐‘–=1 ๐‘–=1 PROPN cana-3839 272 2 ๐œ•๐‘– ๐œ•๐‘– NOUN cana-3839 272 3 < < X cana-3839 272 4 1 1 NUM cana-3839 272 5 , , PUNCT cana-3839 272 6 it it PRON cana-3839 272 7 follows follow VERB cana-3839 272 8 that that DET cana-3839 272 9 ๐œ‡1 ๐œ‡1 NOUN cana-3839 272 10 = = SYM cana-3839 272 11 โˆ๐œ‡1 โˆ๐œ‡1 NOUN cana-3839 272 12 . . PUNCT cana-3839 273 1 now now ADV cana-3839 273 2 , , PUNCT cana-3839 273 3 we we PRON cana-3839 273 4 assert assert VERB cana-3839 273 5 that that SCONJ cana-3839 273 6 ๐œ‡1 ๐œ‡1 PROPN cana-3839 273 7 is be AUX cana-3839 273 8 unique unique ADJ cana-3839 273 9 fp fp NOUN cana-3839 273 10 . . PUNCT cana-3839 274 1 if if SCONJ cana-3839 274 2 ๐œ•1 ๐œ•1 NOUN cana-3839 274 3 is be AUX cana-3839 274 4 another another DET cana-3839 274 5 fp fp NOUN cana-3839 274 6 of of ADP cana-3839 274 7 โˆ โˆ PROPN cana-3839 274 8 , , PUNCT cana-3839 274 9 then then ADV cana-3839 274 10 โˆ€๐œ‡1 โˆ€๐œ‡1 ADP cana-3839 274 11 โˆˆ โˆˆ PROPN cana-3839 274 12 ๐œ’๐‘ž ๐œ’๐‘ž NOUN cana-3839 274 13 and and CCONJ cana-3839 274 14 ๐œ•1 ๐œ•1 VERB cana-3839 274 15 โˆˆ โˆˆ PROPN cana-3839 274 16 ๐œ’๐‘ž+1 ๐œ’๐‘ž+1 NOUN cana-3839 274 17 , , PUNCT cana-3839 274 18 ๐‘— ๐‘— NOUN cana-3839 274 19 = = SYM cana-3839 274 20 1,2 1,2 NUM cana-3839 274 21 , , PUNCT cana-3839 274 22 โ€ฆ โ€ฆ PUNCT cana-3839 274 23 ๐‘š ๐‘š X cana-3839 274 24 , , PUNCT cana-3839 274 25 we we PRON cana-3839 274 26 have have AUX cana-3839 274 27 ๐บ(๐œ‡1 ๐บ(๐œ‡1 PROPN cana-3839 274 28 , , PUNCT cana-3839 274 29 ๐œ•1 ๐œ•1 NOUN cana-3839 274 30 , , PUNCT cana-3839 274 31 ๐œ•1 ๐œ•1 NUM cana-3839 274 32 ) ) PUNCT cana-3839 274 33 = = PUNCT cana-3839 274 34 ๐บ(โˆ๐œ‡1 ๐บ(โˆ๐œ‡1 NUM cana-3839 274 35 , , PUNCT cana-3839 274 36 โˆ๐œ•1 โˆ๐œ•1 NUM cana-3839 274 37 , , PUNCT cana-3839 274 38 โˆ๐œ•1 โˆ๐œ•1 NUM cana-3839 274 39 ) ) PUNCT cana-3839 274 40 โชฏ โชฏ NOUN cana-3839 274 41 { { PUNCT cana-3839 274 42 ๐œ•1[๐บ(๐œ‡1 ๐œ•1[๐บ(๐œ‡1 PROPN cana-3839 274 43 , , PUNCT cana-3839 274 44 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 274 45 , , PUNCT cana-3839 274 46 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 274 47 ) ) PUNCT cana-3839 275 1 + + CCONJ cana-3839 275 2 ๐บ(๐œ•1 ๐บ(๐œ•1 ADJ cana-3839 275 3 , , PUNCT cana-3839 275 4 โˆ๐œ•1 โˆ๐œ•1 NUM cana-3839 275 5 , , PUNCT cana-3839 275 6 โˆ๐œ•1 โˆ๐œ•1 NUM cana-3839 275 7 ) ) PUNCT cana-3839 275 8 ] ] PUNCT cana-3839 275 9 , , PUNCT cana-3839 275 10 ๐œ•2[๐บ(๐œ‡1 ๐œ•2[๐บ(๐œ‡1 PROPN cana-3839 275 11 , , PUNCT cana-3839 275 12 โˆ๐œ•1 โˆ๐œ•1 NUM cana-3839 275 13 , , PUNCT cana-3839 275 14 โˆ๐œ•1 โˆ๐œ•1 NUM cana-3839 275 15 ) ) PUNCT cana-3839 276 1 + + CCONJ cana-3839 276 2 ๐บ(๐œ•1 ๐บ(๐œ•1 ADJ cana-3839 276 3 , , PUNCT cana-3839 276 4 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 276 5 , , PUNCT cana-3839 276 6 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 276 7 ) ) PUNCT cana-3839 277 1 ] ] PUNCT cana-3839 277 2 , , PUNCT cana-3839 277 3 ๐œ•3[๐บ(๐œ‡1 ๐œ•3[๐บ(๐œ‡1 NOUN cana-3839 277 4 , , PUNCT cana-3839 277 5 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 277 6 , , PUNCT cana-3839 277 7 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 277 8 ) ) PUNCT cana-3839 278 1 + + CCONJ cana-3839 279 1 ๐บ(๐œ‡1 ๐บ(๐œ‡1 ADJ cana-3839 279 2 , , PUNCT cana-3839 279 3 โˆ๐œ•1 โˆ๐œ•1 NUM cana-3839 279 4 , , PUNCT cana-3839 279 5 โˆ๐œ•1 โˆ๐œ•1 NUM cana-3839 279 6 ) ) PUNCT cana-3839 279 7 ] ] PUNCT cana-3839 279 8 , , PUNCT cana-3839 279 9 ๐œ•4[๐บ(๐œ•1 ๐œ•4[๐บ(๐œ•1 NOUN cana-3839 279 10 , , PUNCT cana-3839 279 11 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 279 12 , , PUNCT cana-3839 279 13 โˆ๐œ‡1 โˆ๐œ‡1 ADJ cana-3839 279 14 ) ) PUNCT cana-3839 280 1 + + CCONJ cana-3839 280 2 ๐บ(๐œ•1 ๐บ(๐œ•1 ADJ cana-3839 280 3 , , PUNCT cana-3839 280 4 โˆ๐œ•1 โˆ๐œ•1 NUM cana-3839 280 5 , , PUNCT cana-3839 280 6 โˆ๐œ•1 โˆ๐œ•1 NUM cana-3839 280 7 ) ) PUNCT cana-3839 280 8 ] ] PUNCT cana-3839 280 9 } } PUNCT cana-3839 280 10 โชฏ โชฏ NOUN cana-3839 280 11 { { PUNCT cana-3839 280 12 ๐œ•1(0 ๐œ•1(0 PROPN cana-3839 280 13 ) ) PUNCT cana-3839 280 14 , , PUNCT cana-3839 280 15 ๐œ•2[๐บ(๐œ‡1 ๐œ•2[๐บ(๐œ‡1 PROPN cana-3839 280 16 , , PUNCT cana-3839 280 17 ๐œ•1 ๐œ•1 NOUN cana-3839 280 18 , , PUNCT cana-3839 280 19 ๐œ•1 ๐œ•1 NOUN cana-3839 280 20 ) ) PUNCT cana-3839 281 1 + + CCONJ cana-3839 281 2 ๐บ(๐œ•1 ๐บ(๐œ•1 ADJ cana-3839 281 3 , , PUNCT cana-3839 281 4 ๐œ‡1 ๐œ‡1 ADJ cana-3839 281 5 , , PUNCT cana-3839 281 6 ๐œ‡1 ๐œ‡1 PROPN cana-3839 281 7 ) ) PUNCT cana-3839 281 8 ] ] PUNCT cana-3839 281 9 , , PUNCT cana-3839 281 10 ๐œ•3๐บ(๐œ‡1 ๐œ•3๐บ(๐œ‡1 PROPN cana-3839 281 11 , , PUNCT cana-3839 281 12 ๐œ•1 ๐œ•1 NOUN cana-3839 281 13 , , PUNCT cana-3839 281 14 ๐œ•1 ๐œ•1 NOUN cana-3839 281 15 ) ) PUNCT cana-3839 281 16 , , PUNCT cana-3839 281 17 ๐œ•4๐บ(๐œ‡1 ๐œ•4๐บ(๐œ‡1 NOUN cana-3839 281 18 , , PUNCT cana-3839 281 19 ๐œ•1 ๐œ•1 NOUN cana-3839 281 20 , , PUNCT cana-3839 281 21 ๐œ•1 ๐œ•1 NOUN cana-3839 281 22 ) ) PUNCT cana-3839 281 23 } } PUNCT cana-3839 281 24 โชฏ โชฏ NOUN cana-3839 281 25 { { PUNCT cana-3839 281 26 2๐œ•2๐บ(๐œ‡1 2๐œ•2๐บ(๐œ‡1 NOUN cana-3839 281 27 , , PUNCT cana-3839 281 28 ๐œ•1 ๐œ•1 NOUN cana-3839 281 29 , , PUNCT cana-3839 281 30 ๐œ•1 ๐œ•1 NOUN cana-3839 281 31 ) ) PUNCT cana-3839 281 32 , , PUNCT cana-3839 281 33 ๐œ•3๐บ(๐œ‡1 ๐œ•3๐บ(๐œ‡1 NOUN cana-3839 281 34 , , PUNCT cana-3839 281 35 ๐œ•1 ๐œ•1 NOUN cana-3839 281 36 , , PUNCT cana-3839 281 37 ๐œ•1 ๐œ•1 NOUN cana-3839 281 38 ) ) PUNCT cana-3839 281 39 , , PUNCT cana-3839 281 40 ๐œ•4๐บ(๐œ‡1 ๐œ•4๐บ(๐œ‡1 NOUN cana-3839 281 41 , , PUNCT cana-3839 281 42 ๐œ•1 ๐œ•1 NOUN cana-3839 281 43 , , PUNCT cana-3839 281 44 ๐œ•1 ๐œ•1 NOUN cana-3839 281 45 ) ) PUNCT cana-3839 281 46 } } PUNCT cana-3839 281 47 โชฏ โชฏ NOUN cana-3839 281 48 ( ( PUNCT cana-3839 281 49 2๐œ•2 2๐œ•2 NUM cana-3839 281 50 + + CCONJ cana-3839 281 51 ๐œ•3 ๐œ•3 NOUN cana-3839 281 52 + + SYM cana-3839 281 53 ๐œ•4)๐บ(๐œ‡1 ๐œ•4)๐บ(๐œ‡1 ADJ cana-3839 281 54 , , PUNCT cana-3839 281 55 ๐œ•1 ๐œ•1 NOUN cana-3839 281 56 , , PUNCT cana-3839 281 57 ๐œ•1 ๐œ•1 NOUN cana-3839 281 58 ) ) PUNCT cana-3839 281 59 . . PUNCT cana-3839 282 1 since since SCONJ cana-3839 282 2 0 0 NUM cana-3839 282 3 โ‰ค โ‰ค NUM cana-3839 282 4 2๐œ•2 2๐œ•2 NUM cana-3839 282 5 + + CCONJ cana-3839 282 6 ๐œ•3 ๐œ•3 NOUN cana-3839 282 7 + + CCONJ cana-3839 282 8 ๐œ•4 ๐œ•4 X cana-3839 282 9 < < X cana-3839 282 10 1 1 X cana-3839 282 11 . . PUNCT cana-3839 282 12 so so ADV cana-3839 282 13 โˆ โˆ PROPN cana-3839 282 14 has have VERB cana-3839 282 15 fp fp NOUN cana-3839 282 16 in in ADP cana-3839 282 17 ๐œ‡ ๐œ‡ ADP cana-3839 282 18 which which PRON cana-3839 282 19 is be AUX cana-3839 282 20 unique unique ADJ cana-3839 282 21 . . PUNCT cana-3839 283 1 communications communication NOUN cana-3839 283 2 on on ADP cana-3839 283 3 applied apply VERB cana-3839 283 4 nonlinear nonlinear ADJ cana-3839 283 5 analysis analysis NOUN cana-3839 283 6 issn issn NOUN cana-3839 283 7 : : PUNCT cana-3839 283 8 1074 1074 NUM cana-3839 283 9 - - PUNCT cana-3839 283 10 133x 133x NUM cana-3839 283 11 vol vol NOUN cana-3839 283 12 32 32 NUM cana-3839 283 13 no no NOUN cana-3839 283 14 . . PUNCT cana-3839 284 1 9s 9s NUM cana-3839 284 2 ( ( PUNCT cana-3839 284 3 2025 2025 NUM cana-3839 284 4 ) ) PUNCT cana-3839 284 5 87 87 NUM cana-3839 284 6 https://internationalpubls.com https://internationalpubls.com X cana-3839 284 7 3 3 NUM cana-3839 284 8 conclusion conclusion NOUN cana-3839 284 9 within within ADP cana-3839 284 10 this this DET cana-3839 284 11 manuscript manuscript NOUN cana-3839 284 12 , , PUNCT cana-3839 284 13 we we PRON cana-3839 284 14 illustrate illustrate VERB cana-3839 284 15 fpt fpt NOUN cana-3839 284 16 for for ADP cana-3839 284 17 self self NOUN cana-3839 284 18 - - PUNCT cana-3839 284 19 transformations transformation NOUN cana-3839 284 20 in in ADP cana-3839 284 21 v v NOUN cana-3839 284 22 - - PUNCT cana-3839 284 23 complete complete ADJ cana-3839 284 24 vgms vgms NOUN cana-3839 284 25 by by ADP cana-3839 284 26 utilizing utilize VERB cana-3839 284 27 cyclic cyclic ADJ cana-3839 284 28 contraction contraction NOUN cana-3839 284 29 . . PUNCT cana-3839 285 1 these these DET cana-3839 285 2 outcomes outcome NOUN cana-3839 285 3 are be AUX cana-3839 285 4 expected expect VERB cana-3839 285 5 to to PART cana-3839 285 6 inspire inspire VERB cana-3839 285 7 researchers researcher NOUN cana-3839 285 8 to to PART cana-3839 285 9 explore explore VERB cana-3839 285 10 problem problem NOUN cana-3839 285 11 - - PUNCT cana-3839 285 12 solving solve VERB cana-3839 285 13 opportunities opportunity NOUN cana-3839 285 14 in in ADP cana-3839 285 15 diverse diverse ADJ cana-3839 285 16 areas area NOUN cana-3839 285 17 such such ADJ cana-3839 285 18 as as ADP cana-3839 285 19 differential differential ADJ cana-3839 285 20 equations equation NOUN cana-3839 285 21 and and CCONJ cana-3839 285 22 functional functional ADJ cana-3839 285 23 analysis analysis NOUN cana-3839 285 24 . . PUNCT cana-3839 286 1 refrences refrence VERB cana-3839 286 2 [ [ X cana-3839 286 3 1 1 X cana-3839 286 4 ] ] X cana-3839 286 5 b. b. PROPN cana-3839 286 6 c. c. PROPN cana-3839 286 7 dhage dhage PROPN cana-3839 286 8 , , PUNCT cana-3839 286 9 bull.cal bull.cal PROPN cana-3839 286 10 . . PUNCT cana-3839 286 11 math math PROPN cana-3839 286 12 . . PUNCT cana-3839 287 1 soc soc PROPN cana-3839 287 2 . . PUNCT cana-3839 287 3 , , PUNCT cana-3839 287 4 84(1992 84(1992 NUM cana-3839 287 5 ) ) PUNCT cana-3839 287 6 , , PUNCT cana-3839 287 7 329 329 NUM cana-3839 287 8 - - SYM cana-3839 287 9 336 336 NUM cana-3839 287 10 . . PUNCT cana-3839 288 1 [ [ X cana-3839 288 2 2 2 NUM cana-3839 288 3 ] ] X cana-3839 288 4 b. b. PROPN cana-3839 288 5 c. c. PROPN cana-3839 288 6 dhage dhage PROPN cana-3839 288 7 , , PUNCT cana-3839 288 8 bull bull NOUN cana-3839 288 9 . . PUNCT cana-3839 289 1 cal cal PROPN cana-3839 289 2 . . PUNCT cana-3839 290 1 math math NOUN cana-3839 290 2 . . PUNCT cana-3839 291 1 soc soc PROPN cana-3839 291 2 . . PROPN cana-3839 291 3 , , PUNCT cana-3839 291 4 86(1994 86(1994 NUM cana-3839 291 5 ) ) PUNCT cana-3839 291 6 , , PUNCT cana-3839 291 7 503 503 NUM cana-3839 291 8 - - SYM cana-3839 291 9 508 508 NUM cana-3839 291 10 . . PUNCT cana-3839 292 1 [ [ X cana-3839 292 2 3 3 X cana-3839 292 3 ] ] X cana-3839 292 4 d. d. PROPN cana-3839 292 5 w. w. PROPN cana-3839 292 6 boyd boyd PROPN cana-3839 292 7 , , PUNCT cana-3839 292 8 j. j. PROPN cana-3839 292 9 s. s. PROPN cana-3839 292 10 w. w. PROPN cana-3839 292 11 wong wong PROPN cana-3839 292 12 , , PUNCT cana-3839 292 13 on on ADP cana-3839 292 14 nonlinear nonlinear ADJ cana-3839 292 15 contractions contraction NOUN cana-3839 292 16 , , PUNCT cana-3839 292 17 proc proc NOUN cana-3839 292 18 . . PUNCT cana-3839 292 19 am be AUX cana-3839 292 20 . . PUNCT cana-3839 293 1 math math NOUN cana-3839 293 2 . . PUNCT cana-3839 294 1 soc soc PROPN cana-3839 294 2 . . PROPN cana-3839 294 3 , , PUNCT cana-3839 294 4 20(1969 20(1969 NUM cana-3839 294 5 ) ) PUNCT cana-3839 294 6 , , PUNCT cana-3839 294 7 458 458 NUM cana-3839 294 8 - - SYM cana-3839 294 9 464 464 NUM cana-3839 294 10 . . PUNCT cana-3839 295 1 [ [ X cana-3839 295 2 4 4 X cana-3839 295 3 ] ] X cana-3839 295 4 h. h. PROPN cana-3839 295 5 aydi aydi PROPN cana-3839 295 6 , , PUNCT cana-3839 295 7 w. w. PROPN cana-3839 295 8 shatanawi shatanawi PROPN cana-3839 295 9 , , PUNCT cana-3839 295 10 c. c. PROPN cana-3839 295 11 vetro vetro PROPN cana-3839 295 12 , , PUNCT cana-3839 295 13 on on ADP cana-3839 295 14 generalized generalized ADJ cana-3839 295 15 weakly weakly ADJ cana-3839 295 16 g g NOUN cana-3839 295 17 - - PUNCT cana-3839 295 18 contraction contraction NOUN cana-3839 295 19 mapping mapping NOUN cana-3839 295 20 in in ADP cana-3839 295 21 g g NOUN cana-3839 295 22 - - PUNCT cana-3839 295 23 metric metric ADJ cana-3839 295 24 spaces space NOUN cana-3839 295 25 , , PUNCT cana-3839 295 26 comput comput NOUN cana-3839 295 27 . . PUNCT cana-3839 296 1 math math NOUN cana-3839 296 2 . . PUNCT cana-3839 297 1 appl appl PROPN cana-3839 297 2 . . PROPN cana-3839 297 3 , , PUNCT cana-3839 297 4 62(2011 62(2011 NUM cana-3839 297 5 ) ) PUNCT cana-3839 297 6 , , PUNCT cana-3839 297 7 4222 4222 NUM cana-3839 297 8 - - SYM cana-3839 297 9 4229 4229 NUM cana-3839 297 10 . . PUNCT cana-3839 298 1 [ [ X cana-3839 298 2 5 5 X cana-3839 298 3 ] ] PUNCT cana-3839 298 4 e. e. PROPN cana-3839 298 5 karapinar karapinar PROPN cana-3839 298 6 , , PUNCT cana-3839 298 7 i. i. PROPN cana-3839 298 8 m. m. PROPN cana-3839 298 9 erhan erhan PROPN cana-3839 298 10 , , PUNCT cana-3839 298 11 cyclic cyclic ADJ cana-3839 298 12 contraction contraction NOUN cana-3839 298 13 and and CCONJ cana-3839 298 14 fixed fix VERB cana-3839 298 15 point point NOUN cana-3839 298 16 theorems theorem NOUN cana-3839 298 17 , , PUNCT cana-3839 298 18 filomat filomat NOUN cana-3839 298 19 , , PUNCT cana-3839 298 20 26(2012 26(2012 NOUN cana-3839 298 21 ) ) PUNCT cana-3839 298 22 , , PUNCT cana-3839 298 23 777 777 NUM cana-3839 298 24 - - SYM cana-3839 298 25 782 782 NUM cana-3839 298 26 . . PUNCT cana-3839 299 1 [ [ X cana-3839 299 2 6 6 NUM cana-3839 299 3 ] ] X cana-3839 299 4 h. h. PROPN cana-3839 299 5 aydi aydi VERB cana-3839 299 6 , , PUNCT cana-3839 299 7 generalized generalized ADJ cana-3839 299 8 cyclic cyclic ADJ cana-3839 299 9 contraction contraction NOUN cana-3839 299 10 in in ADP cana-3839 299 11 g g NOUN cana-3839 299 12 - - PUNCT cana-3839 299 13 metric metric ADJ cana-3839 299 14 spaces space NOUN cana-3839 299 15 , , PUNCT cana-3839 299 16 j. j. PROPN cana-3839 299 17 nonlinear nonlinear PROPN cana-3839 299 18 sci sci PROPN cana-3839 299 19 . . PUNCT cana-3839 299 20 appl appl PROPN cana-3839 299 21 . . PUNCT cana-3839 300 1 in in ADP cana-3839 300 2 press press NOUN cana-3839 300 3 . . PUNCT cana-3839 301 1 [ [ X cana-3839 301 2 7 7 X cana-3839 301 3 ] ] PUNCT cana-3839 301 4 l. l. PROPN cana-3839 301 5 k. k. PROPN cana-3839 301 6 dey dey PROPN cana-3839 301 7 , , PUNCT cana-3839 301 8 fixed fix VERB cana-3839 301 9 point point NOUN cana-3839 301 10 theorems theorem NOUN cana-3839 301 11 in in ADP cana-3839 301 12 vector vector NOUN cana-3839 301 13 metric metric ADJ cana-3839 301 14 spaces space NOUN cana-3839 301 15 , , PUNCT cana-3839 301 16 int int NOUN cana-3839 301 17 . . PUNCT cana-3839 302 1 j. j. PROPN cana-3839 302 2 math math PROPN cana-3839 302 3 . . PUNCT cana-3839 303 1 anal anal PROPN cana-3839 303 2 . . PROPN cana-3839 303 3 , , PUNCT cana-3839 304 1 4(2010 4(2010 NOUN cana-3839 304 2 ) ) PUNCT cana-3839 304 3 , , PUNCT cana-3839 304 4 2395 2395 NUM cana-3839 304 5 - - SYM cana-3839 304 6 2400 2400 NUM cana-3839 304 7 . . PUNCT cana-3839 305 1 [ [ X cana-3839 305 2 8 8 NUM cana-3839 305 3 ] ] X cana-3839 305 4 luxemburg luxemburg PROPN cana-3839 305 5 , , PUNCT cana-3839 305 6 zannen zannen PROPN cana-3839 305 7 , , PUNCT cana-3839 305 8 riesz riesz PROPN cana-3839 305 9 spaces space NOUN cana-3839 305 10 , , PUNCT cana-3839 305 11 north north NOUN cana-3839 305 12 - - PUNCT cana-3839 305 13 holland holland NOUN cana-3839 305 14 pub pub NOUN cana-3839 305 15 . . PUNCT cana-3839 306 1 co. co. PROPN cana-3839 306 2 , , PUNCT cana-3839 306 3 amsterdam amsterdam PROPN cana-3839 306 4 , , PUNCT cana-3839 306 5 1971 1971 NUM cana-3839 306 6 . . PUNCT cana-3839 307 1 [ [ X cana-3839 307 2 9 9 NUM cana-3839 307 3 ] ] PUNCT cana-3839 307 4 m. m. NOUN cana-3839 307 5 a. a. NOUN cana-3839 307 6 alghamdi alghamdi PROPN cana-3839 307 7 , , PUNCT cana-3839 307 8 a. a. NOUN cana-3839 307 9 petrusel petrusel NOUN cana-3839 307 10 , , PUNCT cana-3839 307 11 n. n. PROPN cana-3839 307 12 shahzad shahzad PROPN cana-3839 307 13 , , PUNCT cana-3839 307 14 a a DET cana-3839 307 15 fixed fix VERB cana-3839 307 16 point point NOUN cana-3839 307 17 theorems theorem NOUN cana-3839 307 18 for for ADP cana-3839 307 19 cyclic cyclic ADJ cana-3839 307 20 generalized generalized ADJ cana-3839 307 21 contraction contraction NOUN cana-3839 307 22 in in ADP cana-3839 307 23 metric metric ADJ cana-3839 307 24 spaces space NOUN cana-3839 307 25 , , PUNCT cana-3839 307 26 fixed fix VERB cana-3839 307 27 point point NOUN cana-3839 307 28 theory theory NOUN cana-3839 307 29 appl appl PROPN cana-3839 307 30 . . PROPN cana-3839 307 31 , , PUNCT cana-3839 307 32 122(2012 122(2012 NUM cana-3839 307 33 ) ) PUNCT cana-3839 307 34 . . PUNCT cana-3839 308 1 [ [ X cana-3839 308 2 10 10 NUM cana-3839 308 3 ] ] X cana-3839 308 4 n. n. NOUN cana-3839 308 5 bilgili bilgili NOUN cana-3839 308 6 , , PUNCT cana-3839 308 7 e. e. PROPN cana-3839 308 8 karapฤฑnar karapฤฑnar PROPN cana-3839 308 9 , , PUNCT cana-3839 308 10 cyclic cyclic ADJ cana-3839 308 11 contractions contraction NOUN cana-3839 308 12 via via ADP cana-3839 308 13 auxiliary auxiliary ADJ cana-3839 308 14 functions function NOUN cana-3839 308 15 on on ADP cana-3839 308 16 g g NOUN cana-3839 308 17 - - PUNCT cana-3839 308 18 metric metric ADJ cana-3839 308 19 spaces space NOUN cana-3839 308 20 , , PUNCT cana-3839 308 21 fixed fix VERB cana-3839 308 22 point point NOUN cana-3839 308 23 theory theory NOUN cana-3839 308 24 appl appl PROPN cana-3839 308 25 . . PROPN cana-3839 308 26 , , PUNCT cana-3839 308 27 49 49 NUM cana-3839 308 28 ( ( PUNCT cana-3839 308 29 2013 2013 NUM cana-3839 308 30 ) ) PUNCT cana-3839 308 31 . . PUNCT cana-3839 309 1 [ [ X cana-3839 309 2 11 11 NUM cana-3839 309 3 ] ] PUNCT cana-3839 309 4 p. p. NOUN cana-3839 309 5 yadav yadav PROPN cana-3839 309 6 , , PUNCT cana-3839 309 7 m. m. NOUN cana-3839 309 8 kamra kamra PROPN cana-3839 309 9 , , PUNCT cana-3839 309 10 common common ADJ cana-3839 309 11 fixed fix VERB cana-3839 309 12 point point NOUN cana-3839 309 13 theorems theorem NOUN cana-3839 309 14 for for ADP cana-3839 309 15 four four NUM cana-3839 309 16 transformations transformation NOUN cana-3839 309 17 in in ADP cana-3839 309 18 vector vector NOUN cana-3839 309 19 s s NOUN cana-3839 309 20 - - ADJ cana-3839 309 21 metric metric ADJ cana-3839 309 22 spaces space NOUN cana-3839 309 23 , , PUNCT cana-3839 309 24 aip aip PROPN cana-3839 309 25 conf conf PROPN cana-3839 309 26 . . PUNCT cana-3839 310 1 proc proc PROPN cana-3839 310 2 . . PROPN cana-3839 310 3 , , PUNCT cana-3839 310 4 3081(2024 3081(2024 NUM cana-3839 310 5 ) ) PUNCT cana-3839 310 6 , , PUNCT cana-3839 310 7 020006 020006 NUM cana-3839 310 8 - - SYM cana-3839 310 9 1 1 NUM cana-3839 310 10 - - PUNCT cana-3839 310 11 020006 020006 NUM cana-3839 310 12 - - SYM cana-3839 310 13 11 11 NUM cana-3839 310 14 . . PUNCT cana-3839 311 1 doi.org/10.1063/5.0196129 doi.org/10.1063/5.0196129 ADJ cana-3839 311 2 . . PUNCT cana-3839 312 1 [ [ X cana-3839 312 2 12 12 NUM cana-3839 312 3 ] ] PUNCT cana-3839 312 4 p. p. NOUN cana-3839 312 5 yadav yadav PROPN cana-3839 312 6 , , PUNCT cana-3839 312 7 m. m. NOUN cana-3839 312 8 kamra kamra PROPN cana-3839 312 9 , , PUNCT cana-3839 312 10 some some DET cana-3839 312 11 fixed fix VERB cana-3839 312 12 point point NOUN cana-3839 312 13 results result NOUN cana-3839 312 14 on on ADP cana-3839 312 15 a a DET cana-3839 312 16 vector vector NOUN cana-3839 312 17 valued value VERB cana-3839 312 18 s s NOUN cana-3839 312 19 - - ADJ cana-3839 312 20 metric metric ADJ cana-3839 312 21 space space NOUN cana-3839 312 22 , , PUNCT cana-3839 312 23 j. j. PROPN cana-3839 312 24 tianjin tianjin PROPN cana-3839 312 25 univ univ PROPN cana-3839 312 26 . . PUNCT cana-3839 313 1 sci sci PROPN cana-3839 313 2 . . PROPN cana-3839 313 3 technol technol PROPN cana-3839 313 4 . . PROPN cana-3839 313 5 , , PUNCT cana-3839 313 6 55(2022 55(2022 NOUN cana-3839 313 7 ) ) PUNCT cana-3839 313 8 , , PUNCT cana-3839 313 9 209 209 NUM cana-3839 313 10 - - SYM cana-3839 313 11 228 228 NUM cana-3839 313 12 . . PUNCT cana-3839 314 1 doi doi PROPN cana-3839 314 2 10.17605 10.17605 NUM cana-3839 314 3 / / SYM cana-3839 314 4 osf.io osf.io PROPN cana-3839 314 5 / / SYM cana-3839 314 6 xuhgk xuhgk NOUN cana-3839 314 7 . . PUNCT cana-3839 315 1 [ [ X cana-3839 315 2 13 13 NUM cana-3839 315 3 ] ] PUNCT cana-3839 315 4 p. p. NOUN cana-3839 315 5 yadav yadav PROPN cana-3839 315 6 , , PUNCT cana-3839 315 7 m. m. NOUN cana-3839 315 8 kamra kamra PROPN cana-3839 315 9 , , PUNCT cana-3839 315 10 rajpal rajpal PROPN cana-3839 315 11 , , PUNCT cana-3839 315 12 common common ADJ cana-3839 315 13 fixed fix VERB cana-3839 315 14 point point NOUN cana-3839 315 15 theorems theorem NOUN cana-3839 315 16 for for ADP cana-3839 315 17 self self NOUN cana-3839 315 18 mappings mapping NOUN cana-3839 315 19 on on ADP cana-3839 315 20 a a DET cana-3839 315 21 complete complete ADJ cana-3839 315 22 vector vector NOUN cana-3839 315 23 s s ADJ cana-3839 315 24 - - ADJ cana-3839 315 25 metric metric ADJ cana-3839 315 26 space space NOUN cana-3839 315 27 , , PUNCT cana-3839 315 28 spec spec PROPN cana-3839 315 29 ugdym ugdym PROPN cana-3839 315 30 . . PUNCT cana-3839 315 31 , , PUNCT cana-3839 315 32 43(2022 43(2022 PROPN cana-3839 315 33 ) ) PUNCT cana-3839 315 34 , , PUNCT cana-3839 315 35 6439 6439 NUM cana-3839 315 36 - - SYM cana-3839 315 37 6450 6450 NUM cana-3839 315 38 . . PUNCT cana-3839 316 1 [ [ X cana-3839 316 2 14 14 NUM cana-3839 316 3 ] ] X cana-3839 316 4 r. r. PROPN cana-3839 316 5 p. p. PROPN cana-3839 316 6 pathak pathak PROPN cana-3839 316 7 , , PUNCT cana-3839 316 8 r. r. PROPN cana-3839 316 9 bhardwaj bhardwaj PROPN cana-3839 316 10 , , PUNCT cana-3839 316 11 fixed fix VERB cana-3839 316 12 point point NOUN cana-3839 316 13 and and CCONJ cana-3839 316 14 common common ADJ cana-3839 316 15 fixed fix VERB cana-3839 316 16 point point NOUN cana-3839 316 17 theorems theorem NOUN cana-3839 316 18 in in ADP cana-3839 316 19 vector vector NOUN cana-3839 316 20 metric metric ADJ cana-3839 316 21 spaces space NOUN cana-3839 316 22 , , PUNCT cana-3839 316 23 math math NOUN cana-3839 316 24 . . PUNCT cana-3839 317 1 theory theory NOUN cana-3839 317 2 model model PROPN cana-3839 317 3 . . PUNCT cana-3839 317 4 , , PUNCT cana-3839 317 5 3(2013 3(2013 NUM cana-3839 317 6 ) ) PUNCT cana-3839 317 7 . . PUNCT cana-3839 318 1 [ [ X cana-3839 318 2 15 15 NUM cana-3839 318 3 ] ] X cana-3839 318 4 r. r. PROPN cana-3839 318 5 saadati saadati PROPN cana-3839 318 6 , , PUNCT cana-3839 318 7 s. s. PROPN cana-3839 318 8 m. m. PROPN cana-3839 318 9 vaezpour vaezpour NOUN cana-3839 318 10 , , PUNCT cana-3839 318 11 p. p. NOUN cana-3839 318 12 vetro vetro PROPN cana-3839 318 13 , , PUNCT cana-3839 318 14 b.e b.e PROPN cana-3839 318 15 . . PROPN cana-3839 318 16 rhoades rhoade NOUN cana-3839 318 17 , , PUNCT cana-3839 318 18 fixed fix VERB cana-3839 318 19 point point NOUN cana-3839 318 20 theorems theorem NOUN cana-3839 318 21 in in ADP cana-3839 318 22 generalized generalize VERB cana-3839 318 23 partially partially ADV cana-3839 318 24 ordered order VERB cana-3839 318 25 g g NOUN cana-3839 318 26 - - PUNCT cana-3839 318 27 metric metric ADJ cana-3839 318 28 spaces space NOUN cana-3839 318 29 , , PUNCT cana-3839 318 30 math math NOUN cana-3839 318 31 . . PUNCT cana-3839 318 32 compat compat PROPN cana-3839 318 33 . . PUNCT cana-3839 319 1 modelling,52(2010 modelling,52(2010 NOUN cana-3839 319 2 ) ) PUNCT cana-3839 319 3 , , PUNCT cana-3839 319 4 797 797 NUM cana-3839 319 5 - - SYM cana-3839 319 6 801 801 NUM cana-3839 319 7 . . PUNCT cana-3839 320 1 [ [ X cana-3839 320 2 16 16 NUM cana-3839 320 3 ] ] X cana-3839 320 4 w. w. PROPN cana-3839 320 5 shatanawi shatanawi PROPN cana-3839 320 6 , , PUNCT cana-3839 320 7 some some DET cana-3839 320 8 fixed fix VERB cana-3839 320 9 point point NOUN cana-3839 320 10 theorems theorem NOUN cana-3839 320 11 in in ADP cana-3839 320 12 ordered order VERB cana-3839 320 13 g g NOUN cana-3839 320 14 - - PUNCT cana-3839 320 15 metric metric ADJ cana-3839 320 16 spaces space NOUN cana-3839 320 17 and and CCONJ cana-3839 320 18 applications application NOUN cana-3839 320 19 , , PUNCT cana-3839 320 20 abst abst PROPN cana-3839 320 21 . . PUNCT cana-3839 320 22 appl appl PROPN cana-3839 320 23 . . PUNCT cana-3839 321 1 anal anal PROPN cana-3839 321 2 . . PROPN cana-3839 321 3 , , PUNCT cana-3839 321 4 2011(2011 2011(2011 NUM cana-3839 321 5 ) ) PUNCT cana-3839 321 6 , , PUNCT cana-3839 321 7 article article NOUN cana-3839 321 8 id:126205 id:126205 X cana-3839 321 9 . . PUNCT cana-3839 322 1 [ [ X cana-3839 322 2 17 17 NUM cana-3839 322 3 ] ] X cana-3839 322 4 w. w. PROPN cana-3839 322 5 a. a. PROPN cana-3839 322 6 kirk kirk PROPN cana-3839 322 7 , , PUNCT cana-3839 322 8 p. p. PROPN cana-3839 322 9 s. s. PROPN cana-3839 322 10 srinavasan srinavasan PROPN cana-3839 322 11 , , PUNCT cana-3839 322 12 p. p. PROPN cana-3839 322 13 veeramani veeramani PROPN cana-3839 322 14 , , PUNCT cana-3839 322 15 fixed fix VERB cana-3839 322 16 point point NOUN cana-3839 322 17 for for ADP cana-3839 322 18 mappings mapping NOUN cana-3839 322 19 satisfying satisfy VERB cana-3839 322 20 cyclical cyclical ADJ cana-3839 322 21 contraction contraction NOUN cana-3839 322 22 condition condition NOUN cana-3839 322 23 , , PUNCT cana-3839 322 24 fixed fix VERB cana-3839 322 25 point point NOUN cana-3839 322 26 theory theory NOUN cana-3839 322 27 , , PUNCT cana-3839 322 28 4(2003 4(2003 NOUN cana-3839 322 29 ) ) PUNCT cana-3839 322 30 , , PUNCT cana-3839 322 31 79 79 NUM cana-3839 322 32 - - SYM cana-3839 322 33 89 89 NUM cana-3839 322 34 . . PUNCT cana-3839 323 1 [ [ X cana-3839 323 2 18 18 NUM cana-3839 323 3 ] ] PUNCT cana-3839 323 4 z. z. PROPN cana-3839 323 5 mustafa mustafa PROPN cana-3839 323 6 , , PUNCT cana-3839 323 7 b. b. PROPN cana-3839 323 8 i. i. PROPN cana-3839 323 9 sims sims PROPN cana-3839 323 10 , , PUNCT cana-3839 323 11 a a DET cana-3839 323 12 new new ADJ cana-3839 323 13 approach approach NOUN cana-3839 323 14 to to ADP cana-3839 323 15 generalized generalize VERB cana-3839 323 16 metric metric ADJ cana-3839 323 17 spaces space NOUN cana-3839 323 18 , , PUNCT cana-3839 323 19 j. j. PROPN cana-3839 323 20 nonlinear nonlinear PROPN cana-3839 323 21 convex convex PROPN cana-3839 323 22 anal anal NOUN cana-3839 323 23 . . PUNCT cana-3839 323 24 , , PUNCT cana-3839 323 25 7(2006 7(2006 NUM cana-3839 323 26 ) ) PUNCT cana-3839 323 27 , , PUNCT cana-3839 323 28 289 289 NUM cana-3839 323 29 - - SYM cana-3839 323 30 297 297 NUM cana-3839 323 31 . . PUNCT cana-3839 324 1 communications communication NOUN cana-3839 324 2 on on ADP cana-3839 324 3 applied apply VERB cana-3839 324 4 nonlinear nonlinear ADJ cana-3839 324 5 analysis analysis NOUN cana-3839 324 6 issn issn NOUN cana-3839 324 7 : : PUNCT cana-3839 324 8 1074 1074 NUM cana-3839 324 9 - - PUNCT cana-3839 324 10 133x 133x NUM cana-3839 324 11 vol vol NOUN cana-3839 324 12 32 32 NUM cana-3839 324 13 no no NOUN cana-3839 324 14 . . PUNCT cana-3839 325 1 9s 9s NUM cana-3839 325 2 ( ( PUNCT cana-3839 325 3 2025 2025 NUM cana-3839 325 4 ) ) PUNCT cana-3839 325 5 88 88 NUM cana-3839 326 1 https://internationalpubls.com https://internationalpubls.com X cana-3839 326 2 [ [ X cana-3839 326 3 19 19 NUM cana-3839 326 4 ] ] PUNCT cana-3839 326 5 z. z. PROPN cana-3839 326 6 mustafa mustafa PROPN cana-3839 326 7 , , PUNCT cana-3839 326 8 b. b. PROPN cana-3839 326 9 i. i. PROPN cana-3839 326 10 sims sims PROPN cana-3839 326 11 , , PUNCT cana-3839 326 12 fixed fix VERB cana-3839 326 13 point point NOUN cana-3839 326 14 theorems theorem NOUN cana-3839 326 15 for for ADP cana-3839 326 16 contractive contractive ADJ cana-3839 326 17 mappings mapping NOUN cana-3839 326 18 in in ADP cana-3839 326 19 complete complete ADJ cana-3839 326 20 ๐บmetric ๐บmetric ADJ cana-3839 326 21 spaces space NOUN cana-3839 326 22 , , PUNCT cana-3839 326 23 j. j. PROPN cana-3839 326 24 fixed fix VERB cana-3839 326 25 point point PROPN cana-3839 326 26 theory theory NOUN cana-3839 326 27 appl appl PROPN cana-3839 326 28 . . PROPN cana-3839 326 29 , , PUNCT cana-3839 326 30 2009 2009 NUM cana-3839 326 31 . . PUNCT cana-3839 327 1 article article NOUN cana-3839 327 2 i i PROPN cana-3839 327 3 d d PROPN cana-3839 327 4 917175 917175 NUM cana-3839 327 5 . . PUNCT