id sid tid token lemma pos cana-918 1 1 communications communication NOUN cana-918 1 2 on on ADP cana-918 1 3 applied apply VERB cana-918 1 4 nonlinear nonlinear ADJ cana-918 1 5 analysis analysis NOUN cana-918 1 6 issn issn NOUN cana-918 1 7 : : PUNCT cana-918 1 8 1074 1074 NUM cana-918 1 9 - - PUNCT cana-918 1 10 133x 133x NUM cana-918 1 11 vol vol NOUN cana-918 1 12 31 31 NUM cana-918 1 13 no no NOUN cana-918 1 14 . . PUNCT cana-918 2 1 4s 4s NUM cana-918 2 2 ( ( PUNCT cana-918 2 3 2024 2024 NUM cana-918 2 4 ) ) PUNCT cana-918 2 5 392 392 NUM cana-918 2 6 https://internationalpubls.com https://internationalpubls.com X cana-918 2 7 association association NOUN cana-918 2 8 schemes scheme NOUN cana-918 2 9 for for ADP cana-918 2 10 some some DET cana-918 2 11 finite finite PROPN cana-918 2 12 group group NOUN cana-918 2 13 rings ring NOUN cana-918 2 14 ii ii PROPN cana-918 2 15 anuradha anuradha PROPN cana-918 2 16 sabharwal sabharwal PROPN cana-918 2 17 1 1 NUM cana-918 2 18 * * NOUN cana-918 2 19 , , PUNCT cana-918 2 20 pooja pooja PROPN cana-918 2 21 yadav2 yadav2 PROPN cana-918 2 22 and and CCONJ cana-918 2 23 r. r. PROPN cana-918 2 24 k. k. PROPN cana-918 2 25 sharma3 sharma3 PROPN cana-918 3 1 1department 1department NUM cana-918 3 2 of of ADP cana-918 3 3 mathematics mathematic NOUN cana-918 3 4 , , PUNCT cana-918 3 5 university university NOUN cana-918 3 6 of of ADP cana-918 3 7 delhi delhi PROPN cana-918 3 8 , , PUNCT cana-918 3 9 delhi-110007 delhi-110007 ADJ cana-918 3 10 , , PUNCT cana-918 3 11 india india PROPN cana-918 3 12 . . PUNCT cana-918 4 1 2department 2department NUM cana-918 4 2 of of ADP cana-918 4 3 mathematics mathematic NOUN cana-918 4 4 , , PUNCT cana-918 4 5 kamala kamala PROPN cana-918 4 6 nehru nehru PROPN cana-918 4 7 college college PROPN cana-918 4 8 , , PUNCT cana-918 4 9 university university PROPN cana-918 4 10 of of ADP cana-918 4 11 delhi delhi PROPN cana-918 4 12 , , PUNCT cana-918 4 13 new new ADJ cana-918 4 14 delhi-110049 delhi-110049 NOUN cana-918 4 15 , , PUNCT cana-918 4 16 india india PROPN cana-918 4 17 . . PUNCT cana-918 5 1 3department 3department NUM cana-918 5 2 of of ADP cana-918 5 3 mathematics mathematic NOUN cana-918 5 4 , , PUNCT cana-918 5 5 indian indian PROPN cana-918 5 6 institute institute PROPN cana-918 5 7 of of ADP cana-918 5 8 technology technology PROPN cana-918 5 9 delhi delhi PROPN cana-918 5 10 , , PUNCT cana-918 5 11 new new ADJ cana-918 5 12 delhi-110016 delhi-110016 PROPN cana-918 5 13 , , PUNCT cana-918 5 14 india india PROPN cana-918 5 15 . . PUNCT cana-918 6 1 * * PUNCT cana-918 6 2 corresponding correspond VERB cana-918 6 3 author author NOUN cana-918 6 4 : : PUNCT cana-918 6 5 anuradha anuradha PROPN cana-918 6 6 sabharwal sabharwal PROPN cana-918 6 7 article article NOUN cana-918 6 8 history history NOUN cana-918 6 9 : : PUNCT cana-918 6 10 received receive VERB cana-918 6 11 : : PUNCT cana-918 6 12 20 20 NUM cana-918 6 13 - - PUNCT cana-918 6 14 04 04 NUM cana-918 6 15 - - PUNCT cana-918 6 16 2024 2024 NUM cana-918 6 17 revised revise VERB cana-918 6 18 : : PUNCT cana-918 6 19 10 10 NUM cana-918 6 20 - - SYM cana-918 6 21 06 06 NUM cana-918 6 22 - - PUNCT cana-918 6 23 2024 2024 NUM cana-918 6 24 accepted accept VERB cana-918 6 25 : : PUNCT cana-918 6 26 24 24 NUM cana-918 6 27 - - PUNCT cana-918 6 28 06 06 NUM cana-918 6 29 - - PUNCT cana-918 6 30 2024 2024 NUM cana-918 6 31 abstract abstract NOUN cana-918 6 32 : : PUNCT cana-918 6 33 association association NOUN cana-918 6 34 schemes scheme NOUN cana-918 6 35 have have AUX cana-918 6 36 been be AUX cana-918 6 37 used use VERB cana-918 6 38 in in ADP cana-918 6 39 coding code VERB cana-918 6 40 theory theory NOUN cana-918 6 41 and and CCONJ cana-918 6 42 other other ADJ cana-918 6 43 combinatorial combinatorial ADJ cana-918 6 44 problems problem NOUN cana-918 6 45 . . PUNCT cana-918 7 1 in in ADP cana-918 7 2 this this DET cana-918 7 3 paper paper NOUN cana-918 7 4 , , PUNCT cana-918 7 5 we we PRON cana-918 7 6 construct construct VERB cana-918 7 7 association association NOUN cana-918 7 8 schemes scheme NOUN cana-918 7 9 for for ADP cana-918 7 10 the the DET cana-918 7 11 abelian abelian ADJ cana-918 7 12 groups group NOUN cana-918 7 13 โ„ค2 โ„ค2 VERB cana-918 7 14 ๐‘Ÿ ๐‘Ÿ NOUN cana-918 7 15 , , PUNCT cana-918 7 16 โ„ค๐‘›1 โ„ค๐‘›1 X cana-918 7 17 ร— ร— PROPN cana-918 7 18 โ„ค๐‘›2 โ„ค๐‘›2 NOUN cana-918 7 19 ร—โ‹ฏร— ร—โ‹ฏร— PROPN cana-918 7 20 โ„ค๐‘›๐‘Ÿ โ„ค๐‘›๐‘Ÿ PROPN cana-918 7 21 , , PUNCT cana-918 7 22 set set VERB cana-918 7 23 of of ADP cana-918 7 24 ๐‘› ๐‘› DET cana-918 7 25 ร— ร— NOUN cana-918 7 26 ๐‘› ๐‘› PROPN cana-918 7 27 matrices matrix NOUN cana-918 7 28 over over ADP cana-918 7 29 โ„ค๐‘š โ„ค๐‘š PROPN cana-918 7 30 and and CCONJ cana-918 7 31 for for ADP cana-918 7 32 the the DET cana-918 7 33 general general ADJ cana-918 7 34 linear linear PROPN cana-918 7 35 group group NOUN cana-918 7 36 of of ADP cana-918 7 37 order order NOUN cana-918 7 38 2 2 NUM cana-918 7 39 over over ADP cana-918 7 40 โ„ค2 โ„ค2 PROPN cana-918 7 41 , , PUNCT cana-918 7 42 โ„ค4 โ„ค4 PROPN cana-918 7 43 , , PUNCT cana-918 7 44 and and CCONJ cana-918 7 45 โ„ค6 โ„ค6 NOUN cana-918 7 46 . . PUNCT cana-918 8 1 we we PRON cana-918 8 2 also also ADV cana-918 8 3 obtain obtain VERB cana-918 8 4 association association NOUN cana-918 8 5 schemes scheme NOUN cana-918 8 6 for for ADP cana-918 8 7 symmetric symmetric ADJ cana-918 8 8 groups group NOUN cana-918 8 9 and and CCONJ cana-918 8 10 alternating alternate VERB cana-918 8 11 groups group NOUN cana-918 8 12 of of ADP cana-918 8 13 degree degree NOUN cana-918 8 14 4 4 NUM cana-918 8 15 and and CCONJ cana-918 8 16 5 5 NUM cana-918 8 17 using use VERB cana-918 8 18 canonical canonical ADJ cana-918 8 19 forms form NOUN cana-918 8 20 . . PUNCT cana-918 9 1 keywords keyword NOUN cana-918 9 2 : : PUNCT cana-918 9 3 group group NOUN cana-918 9 4 ring ring NOUN cana-918 9 5 ; ; PUNCT cana-918 9 6 association association NOUN cana-918 9 7 scheme scheme NOUN cana-918 9 8 . . PUNCT cana-918 10 1 2020 2020 NUM cana-918 10 2 mathematics mathematic NOUN cana-918 10 3 subject subject ADJ cana-918 10 4 classification classification NOUN cana-918 10 5 . . PUNCT cana-918 11 1 05e30 05e30 NOUN cana-918 11 2 . . PUNCT cana-918 12 1 1 1 X cana-918 12 2 . . X cana-918 12 3 introduction introduction NOUN cana-918 12 4 association association NOUN cana-918 12 5 schemes scheme NOUN cana-918 12 6 ( ( PUNCT cana-918 12 7 as as ADP cana-918 12 8 ) ) PUNCT cana-918 12 9 introduced introduce VERB cana-918 12 10 by by ADP cana-918 12 11 bose bose NOUN cana-918 12 12 and and CCONJ cana-918 12 13 shimamoto shimamoto NOUN cana-918 13 1 [ [ X cana-918 13 2 1 1 NUM cana-918 13 3 ] ] PUNCT cana-918 13 4 , , PUNCT cana-918 13 5 play play VERB cana-918 13 6 a a DET cana-918 13 7 key key ADJ cana-918 13 8 role role NOUN cana-918 13 9 in in ADP cana-918 13 10 the the DET cana-918 13 11 study study NOUN cana-918 13 12 of of ADP cana-918 13 13 algebraic algebraic ADJ cana-918 13 14 combinatorics combinatoric NOUN cana-918 13 15 . . PUNCT cana-918 14 1 it it PRON cana-918 14 2 has have VERB cana-918 14 3 applications application NOUN cana-918 14 4 in in ADP cana-918 14 5 graph graph NOUN cana-918 14 6 theory theory NOUN cana-918 14 7 , , PUNCT cana-918 14 8 coding code VERB cana-918 14 9 theory theory NOUN cana-918 14 10 , , PUNCT cana-918 14 11 group group NOUN cana-918 14 12 theory theory NOUN cana-918 14 13 and and CCONJ cana-918 14 14 design design NOUN cana-918 14 15 theory theory NOUN cana-918 14 16 [ [ X cana-918 14 17 2 2 NUM cana-918 14 18 , , PUNCT cana-918 14 19 3 3 NUM cana-918 14 20 , , PUNCT cana-918 14 21 4 4 NUM cana-918 14 22 , , PUNCT cana-918 14 23 5 5 NUM cana-918 14 24 , , PUNCT cana-918 14 25 6 6 NUM cana-918 14 26 , , PUNCT cana-918 14 27 7 7 NUM cana-918 14 28 , , PUNCT cana-918 14 29 8 8 NUM cana-918 14 30 , , PUNCT cana-918 14 31 9 9 NUM cana-918 14 32 , , PUNCT cana-918 14 33 10 10 NUM cana-918 14 34 , , PUNCT cana-918 14 35 11 11 NUM cana-918 14 36 ] ] PUNCT cana-918 14 37 . . PUNCT cana-918 15 1 jรธrgensen jรธrgensen PROPN cana-918 15 2 โ€™s โ€™s PART cana-918 15 3 list list NOUN cana-918 15 4 of of ADP cana-918 15 5 non non ADJ cana-918 15 6 - - ADJ cana-918 15 7 symmetric symmetric ADJ cana-918 15 8 association association NOUN cana-918 15 9 schemes scheme NOUN cana-918 15 10 with with ADP cana-918 15 11 classes class NOUN cana-918 15 12 smaller small ADJ cana-918 15 13 than than ADP cana-918 15 14 96 96 NUM cana-918 15 15 in in ADP cana-918 15 16 vertices vertex NOUN cana-918 15 17 in in ADP cana-918 15 18 [ [ X cana-918 15 19 12 12 NUM cana-918 15 20 ] ] PUNCT cana-918 15 21 , , PUNCT cana-918 15 22 inspires inspire VERB cana-918 15 23 us we PRON cana-918 15 24 to to PART cana-918 15 25 research research VERB cana-918 15 26 non non ADJ cana-918 15 27 - - ADJ cana-918 15 28 symmetric symmetric ADJ cana-918 15 29 association association NOUN cana-918 15 30 schemes scheme NOUN cana-918 15 31 for for ADP cana-918 15 32 various various ADJ cana-918 15 33 finite finite ADJ cana-918 15 34 groups group NOUN cana-918 15 35 and and CCONJ cana-918 15 36 group group NOUN cana-918 15 37 rings ring NOUN cana-918 15 38 . . PUNCT cana-918 16 1 in in ADP cana-918 16 2 our our PRON cana-918 16 3 previous previous ADJ cana-918 16 4 work work NOUN cana-918 16 5 [ [ X cana-918 16 6 13 13 NUM cana-918 16 7 ] ] PUNCT cana-918 16 8 , , PUNCT cana-918 16 9 we we PRON cana-918 16 10 have have AUX cana-918 16 11 constructed construct VERB cana-918 16 12 non non ADJ cana-918 16 13 symmetric symmetric NOUN cana-918 16 14 commutative commutative ADJ cana-918 16 15 as as ADP cana-918 16 16 for for ADP cana-918 16 17 symmetric symmetric ADJ cana-918 16 18 groups group NOUN cana-918 16 19 , , PUNCT cana-918 16 20 dihedral dihedral ADJ cana-918 16 21 groups group NOUN cana-918 16 22 , , PUNCT cana-918 16 23 abelian abelian ADJ cana-918 16 24 groups group NOUN cana-918 17 1 โ„ค๐‘ โ„ค๐‘ ADJ cana-918 17 2 ๐‘Ÿ ๐‘Ÿ NOUN cana-918 17 3 ( ( PUNCT cana-918 17 4 where where SCONJ cana-918 17 5 ๐‘ ๐‘ NOUN cana-918 17 6 is be AUX cana-918 17 7 an an DET cana-918 17 8 odd odd ADJ cana-918 17 9 prime prime NOUN cana-918 17 10 ) ) PUNCT cana-918 17 11 , , PUNCT cana-918 17 12 โ€ˆ โ€ˆ SPACE cana-918 17 13 โ„ค๐‘1 โ„ค๐‘1 X cana-918 17 14 ร— ร— NOUN cana-918 17 15 โ„ค๐‘2 โ„ค๐‘2 NOUN cana-918 17 16 ร—โ‹ฏร— ร—โ‹ฏร— PROPN cana-918 17 17 โ„ค๐‘๐‘Ÿ โ„ค๐‘๐‘Ÿ PROPN cana-918 17 18 ( ( PUNCT cana-918 17 19 ๐‘๐‘– ๐‘๐‘– NOUN cana-918 17 20 โ€ฒ๐‘  โ€ฒ๐‘  PROPN cana-918 17 21 are be AUX cana-918 17 22 distinct distinct ADJ cana-918 17 23 primes prime NOUN cana-918 17 24 ) ) PUNCT cana-918 17 25 , , PUNCT cana-918 17 26 finite finite PROPN cana-918 17 27 group group NOUN cana-918 17 28 rings ring NOUN cana-918 17 29 over over ADP cana-918 17 30 โ„ค๐‘› โ„ค๐‘› PROPN cana-918 17 31 and and CCONJ cana-918 17 32 circulant circulant ADJ cana-918 17 33 matrices matrix NOUN cana-918 17 34 over over ADP cana-918 17 35 โ„ค๐‘ โ„ค๐‘ PROPN cana-918 17 36 , , PUNCT cana-918 17 37 for for ADP cana-918 17 38 ๐‘ ๐‘ PROPN cana-918 17 39 prime prime NOUN cana-918 17 40 . . PUNCT cana-918 18 1 in in ADP cana-918 18 2 the the DET cana-918 18 3 present present ADJ cana-918 18 4 study study NOUN cana-918 18 5 , , PUNCT cana-918 18 6 we we PRON cana-918 18 7 construct construct VERB cana-918 18 8 association association NOUN cana-918 18 9 schemes scheme NOUN cana-918 18 10 for for ADP cana-918 18 11 the the DET cana-918 18 12 abelian abelian ADJ cana-918 18 13 groups group NOUN cana-918 18 14 โ„ค2 โ„ค2 VERB cana-918 18 15 ๐‘Ÿ ๐‘Ÿ NOUN cana-918 18 16 = = SYM cana-918 18 17 โ„ค2 โ„ค2 PROPN cana-918 18 18 ร— ร— PROPN cana-918 18 19 โ„ค2 โ„ค2 PROPN cana-918 18 20 ร—โ‹ฏร— ร—โ‹ฏร— PROPN cana-918 18 21 โ„ค2โŸ โ„ค2โŸ VERB cana-918 18 22 ๐‘Ÿ ๐‘Ÿ PRON cana-918 18 23 ๐‘ก๐‘–๐‘š๐‘’๐‘  ๐‘ก๐‘–๐‘š๐‘’๐‘  ADJ cana-918 18 24 and and CCONJ cana-918 18 25 โ„ค๐‘›1 โ„ค๐‘›1 X cana-918 18 26 ร— ร— PROPN cana-918 18 27 โ„ค๐‘›2 โ„ค๐‘›2 NOUN cana-918 18 28 ร—โ‹ฏร— ร—โ‹ฏร— PROPN cana-918 18 29 โ„ค๐‘›๐‘Ÿ โ„ค๐‘›๐‘Ÿ PROPN cana-918 18 30 , , PUNCT cana-918 18 31 the the DET cana-918 18 32 general general ADJ cana-918 18 33 linear linear PROPN cana-918 18 34 group group NOUN cana-918 18 35 ๐บ๐ฟ(2 ๐บ๐ฟ(2 PROPN cana-918 18 36 , , PUNCT cana-918 18 37 โ„ค2 โ„ค2 PROPN cana-918 18 38 ) ) PUNCT cana-918 18 39 , , PUNCT cana-918 18 40 ๐บ๐ฟ(2 ๐บ๐ฟ(2 PROPN cana-918 18 41 , , PUNCT cana-918 18 42 โ„ค4 โ„ค4 PROPN cana-918 18 43 ) ) PUNCT cana-918 18 44 , , PUNCT cana-918 18 45 ๐บ๐ฟ(2 ๐บ๐ฟ(2 PROPN cana-918 18 46 , , PUNCT cana-918 18 47 โ„ค6 โ„ค6 NOUN cana-918 18 48 ) ) PUNCT cana-918 18 49 , , PUNCT cana-918 18 50 symmetric symmetric ADJ cana-918 18 51 group group NOUN cana-918 18 52 and and CCONJ cana-918 18 53 alternating alternate VERB cana-918 18 54 group group NOUN cana-918 18 55 of of ADP cana-918 18 56 order order NOUN cana-918 18 57 4 4 NUM cana-918 18 58 and and CCONJ cana-918 18 59 5 5 NUM cana-918 18 60 . . PUNCT cana-918 19 1 in in ADP cana-918 19 2 this this DET cana-918 19 3 paper paper NOUN cana-918 19 4 , , PUNCT cana-918 19 5 let let VERB cana-918 19 6 ๐’ซ ๐’ซ NOUN cana-918 19 7 denote denote VERB cana-918 19 8 the the DET cana-918 19 9 partition partition NOUN cana-918 19 10 of of ADP cana-918 19 11 ๐‘Œ ๐‘Œ PROPN cana-918 19 12 ร— ร— PROPN cana-918 19 13 ๐‘Œ ๐‘Œ PROPN cana-918 19 14 , , PUNCT cana-918 19 15 where where SCONJ cana-918 19 16 ๐‘Œ ๐‘Œ PROPN cana-918 19 17 is be AUX cana-918 19 18 a a DET cana-918 19 19 finite finite ADJ cana-918 19 20 set set NOUN cana-918 19 21 , , PUNCT cana-918 19 22 and and CCONJ cana-918 19 23 let let VERB cana-918 19 24 โ€ˆ โ€ˆ SPACE cana-918 20 1 โ„ค๐‘ โ„ค๐‘ ADJ cana-918 20 2 ๐‘Ÿ ๐‘Ÿ NOUN cana-918 20 3 = = PUNCT cana-918 21 1 โ„ค๐‘ โ„ค๐‘ ADJ cana-918 21 2 ร— ร— NOUN cana-918 21 3 โ„ค๐‘ โ„ค๐‘ PROPN cana-918 21 4 ร—โ‹ฏร— ร—โ‹ฏร— NOUN cana-918 21 5 โ„ค๐‘โŸ โ„ค๐‘โŸ NOUN cana-918 21 6 ๐‘Ÿ ๐‘Ÿ NOUN cana-918 21 7 โ€ˆ โ€ˆ SPACE cana-918 21 8 ๐‘ก๐‘–๐‘š๐‘’๐‘  ๐‘ก๐‘–๐‘š๐‘’๐‘  ADJ cana-918 21 9 . . PUNCT cana-918 22 1 some some DET cana-918 22 2 basic basic ADJ cana-918 22 3 literature literature NOUN cana-918 22 4 and and CCONJ cana-918 22 5 preliminaries preliminary NOUN cana-918 22 6 on on ADP cana-918 22 7 association association NOUN cana-918 22 8 schemes scheme NOUN cana-918 22 9 are be AUX cana-918 22 10 given give VERB cana-918 22 11 below below ADV cana-918 22 12 . . PUNCT cana-918 23 1 1.1 1.1 NUM cana-918 23 2 . . PUNCT cana-918 24 1 association association NOUN cana-918 24 2 scheme scheme NOUN cana-918 24 3 definition definition NOUN cana-918 24 4 1 1 NUM cana-918 24 5 . . PUNCT cana-918 25 1 let let VERB cana-918 25 2 ๐’ซ ๐’ซ NOUN cana-918 25 3 be be AUX cana-918 25 4 a a DET cana-918 25 5 partition partition NOUN cana-918 25 6 of of ADP cana-918 25 7 ๐‘Œ ๐‘Œ PROPN cana-918 25 8 ร— ร— NOUN cana-918 25 9 ๐‘Œ ๐‘Œ PROPN cana-918 25 10 where where SCONJ cana-918 25 11 ๐‘Œ ๐‘Œ PROPN cana-918 25 12 is be AUX cana-918 25 13 a a DET cana-918 25 14 finite finite NOUN cana-918 25 15 set set NOUN cana-918 25 16 and and CCONJ cana-918 25 17 let let VERB cana-918 25 18 ๐’ฎ0 ๐’ฎ0 NOUN cana-918 25 19 , , PUNCT cana-918 25 20 ๐’ฎ1 ๐’ฎ1 NOUN cana-918 25 21 , , PUNCT cana-918 25 22 โ€ฆ โ€ฆ PUNCT cana-918 25 23 , , PUNCT cana-918 26 1 ๐’ฎ๐“ƒ ๐’ฎ๐“ƒ ADP cana-918 26 2 binary binary ADJ cana-918 26 3 relations relation NOUN cana-918 26 4 on on ADP cana-918 26 5 ๐’ซ. ๐’ซ. PROPN cana-918 26 6 then then ADV cana-918 26 7 ๐’œ ๐’œ PROPN cana-918 26 8 = = SYM cana-918 26 9 ( ( PUNCT cana-918 26 10 ๐‘Œ,๐’ซ ๐‘Œ,๐’ซ NOUN cana-918 26 11 ) ) PUNCT cana-918 26 12 forms form VERB cana-918 26 13 ๐‘›-class ๐‘›-class PROPN cana-918 26 14 association association NOUN cana-918 26 15 scheme scheme NOUN cana-918 26 16 if if SCONJ cana-918 26 17 the the DET cana-918 26 18 subsequent subsequent ADJ cana-918 26 19 conditions condition NOUN cana-918 26 20 hold hold VERB cana-918 26 21 : : PUNCT cana-918 26 22 ( ( PUNCT cana-918 26 23 1 1 X cana-918 26 24 ) ) PUNCT cana-918 26 25 identity identity NOUN cana-918 26 26 relation relation NOUN cana-918 26 27 ๐’ฎ0 ๐’ฎ0 NOUN cana-918 26 28 = = SYM cana-918 26 29 { { PUNCT cana-918 26 30 ( ( PUNCT cana-918 26 31 ๐‘Ž ๐‘Ž X cana-918 26 32 , , PUNCT cana-918 26 33 ๐‘Ž ๐‘Ž NOUN cana-918 26 34 ): ): PUNCT cana-918 26 35 ๐‘Ž ๐‘Ž PROPN cana-918 26 36 โˆˆ โˆˆ PROPN cana-918 26 37 ๐‘Œ ๐‘Œ PROPN cana-918 26 38 } } PUNCT cana-918 26 39 โˆˆ โˆˆ PROPN cana-918 26 40 ๐’ซ. ๐’ซ. NOUN cana-918 26 41 communications communication NOUN cana-918 26 42 on on ADP cana-918 26 43 applied apply VERB cana-918 26 44 nonlinear nonlinear ADJ cana-918 26 45 analysis analysis NOUN cana-918 26 46 issn issn NOUN cana-918 26 47 : : PUNCT cana-918 26 48 1074 1074 NUM cana-918 26 49 - - PUNCT cana-918 26 50 133x 133x NUM cana-918 26 51 vol vol NOUN cana-918 26 52 31 31 NUM cana-918 26 53 no no NOUN cana-918 26 54 . . PUNCT cana-918 27 1 4s 4s NUM cana-918 27 2 ( ( PUNCT cana-918 27 3 2024 2024 NUM cana-918 27 4 ) ) PUNCT cana-918 27 5 393 393 NUM cana-918 27 6 https://internationalpubls.com https://internationalpubls.com X cana-918 27 7 ( ( PUNCT cana-918 27 8 2 2 NUM cana-918 27 9 ) ) PUNCT cana-918 27 10 ๐’ฎโˆ— ๐’ฎโˆ— NOUN cana-918 28 1 = = SYM cana-918 28 2 { { PUNCT cana-918 28 3 ( ( PUNCT cana-918 28 4 ๐‘Ž ๐‘Ž X cana-918 28 5 , , PUNCT cana-918 28 6 ๐‘ ๐‘ NOUN cana-918 28 7 ): ): PUNCT cana-918 28 8 ( ( PUNCT cana-918 28 9 ๐‘ ๐‘ NOUN cana-918 28 10 , , PUNCT cana-918 28 11 ๐‘Ž ๐‘Ž NOUN cana-918 28 12 ) ) PUNCT cana-918 28 13 โˆˆ โˆˆ PROPN cana-918 28 14 ๐’ฎ ๐’ฎ PROPN cana-918 28 15 } } PUNCT cana-918 28 16 โˆˆ โˆˆ PROPN cana-918 28 17 ๐’ซ ๐’ซ NOUN cana-918 28 18 for for ADP cana-918 28 19 any any DET cana-918 28 20 relation relation NOUN cana-918 28 21 ๐’ฎ ๐’ฎ PROPN cana-918 28 22 โˆˆ โˆˆ PROPN cana-918 28 23 ๐’ซ. ๐’ซ. NOUN cana-918 28 24 ( ( PUNCT cana-918 28 25 3 3 NUM cana-918 28 26 ) ) PUNCT cana-918 28 27 if if SCONJ cana-918 28 28 ( ( PUNCT cana-918 28 29 ๐‘Ž ๐‘Ž X cana-918 28 30 , , PUNCT cana-918 28 31 ๐‘ ๐‘ NOUN cana-918 28 32 ) ) PUNCT cana-918 28 33 โˆˆ โˆˆ PROPN cana-918 29 1 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 29 2 , , PUNCT cana-918 29 3 the the DET cana-918 29 4 number number NOUN cana-918 29 5 of of ADP cana-918 29 6 elements element NOUN cana-918 29 7 ๐‘ ๐‘ PROPN cana-918 29 8 โˆˆ โˆˆ NOUN cana-918 29 9 ๐‘Œ ๐‘Œ PROPN cana-918 29 10 such such ADJ cana-918 29 11 that that SCONJ cana-918 29 12 ( ( PUNCT cana-918 29 13 ๐‘Ž ๐‘Ž X cana-918 29 14 , , PUNCT cana-918 29 15 ๐‘ ๐‘ NOUN cana-918 29 16 ) ) PUNCT cana-918 29 17 โˆˆ โˆˆ PROPN cana-918 29 18 ๐‘†๐‘™ ๐‘†๐‘™ PROPN cana-918 29 19 , , PUNCT cana-918 29 20 ( ( PUNCT cana-918 29 21 ๐‘ ๐‘ NOUN cana-918 29 22 , , PUNCT cana-918 29 23 ๐‘ ๐‘ NOUN cana-918 29 24 ) ) PUNCT cana-918 29 25 โˆˆ โˆˆ PROPN cana-918 30 1 ๐‘†๐‘š ๐‘†๐‘š PROPN cana-918 30 2 is be AUX cana-918 30 3 a a DET cana-918 30 4 constant constant ADJ cana-918 30 5 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š NOUN cana-918 30 6 ๐‘˜ ๐‘˜ X cana-918 30 7 not not PART cana-918 30 8 depending depend VERB cana-918 30 9 on on ADP cana-918 30 10 choice choice NOUN cana-918 30 11 of of ADP cana-918 30 12 ๐‘Ž ๐‘Ž NOUN cana-918 30 13 and and CCONJ cana-918 30 14 ๐‘ ๐‘ NOUN cana-918 30 15 for for ADP cana-918 30 16 all all DET cana-918 30 17 integers integer NOUN cana-918 30 18 0 0 NUM cana-918 30 19 โ‰ค โ‰ค NUM cana-918 30 20 ๐‘˜ ๐‘˜ X cana-918 30 21 , , PUNCT cana-918 30 22 ๐‘™,๐‘š ๐‘™,๐‘š NOUN cana-918 30 23 โ‰ค โ‰ค NUM cana-918 30 24 ๐‘›. ๐‘›. NOUN cana-918 30 25 the the DET cana-918 30 26 integers integer NOUN cana-918 30 27 { { PUNCT cana-918 30 28 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š NOUN cana-918 30 29 ๐‘˜ ๐‘˜ PROPN cana-918 30 30 } } PUNCT cana-918 30 31 0โ‰ค๐‘˜,๐‘™,๐‘šโ‰ค๐‘› 0โ‰ค๐‘˜,๐‘™,๐‘šโ‰ค๐‘› NUM cana-918 30 32 are be AUX cana-918 30 33 called call VERB cana-918 30 34 parameters parameter NOUN cana-918 30 35 or or CCONJ cana-918 30 36 intersection intersection NOUN cana-918 30 37 numbers number NOUN cana-918 30 38 of of ADP cana-918 30 39 ๐’œ. ๐’œ. NOUN cana-918 30 40 if if SCONJ cana-918 30 41 each each DET cana-918 30 42 relation relation NOUN cana-918 30 43 ๐’ฎ ๐’ฎ NOUN cana-918 30 44 in in ADP cana-918 30 45 ๐’ซ ๐’ซ PROPN cana-918 30 46 is be AUX cana-918 30 47 a a DET cana-918 30 48 symmetric symmetric ADJ cana-918 30 49 relation relation NOUN cana-918 30 50 , , PUNCT cana-918 30 51 that that ADV cana-918 30 52 is is ADV cana-918 30 53 , , PUNCT cana-918 30 54 ๐’ฎ ๐’ฎ PROPN cana-918 30 55 = = SYM cana-918 30 56 ๐’ฎโˆ— ๐’ฎโˆ— PROPN cana-918 30 57 , , PUNCT cana-918 30 58 then then ADV cana-918 30 59 ๐’œ ๐’œ NOUN cana-918 30 60 is be AUX cana-918 30 61 called call VERB cana-918 30 62 symmetric symmetric ADJ cana-918 30 63 association association NOUN cana-918 30 64 scheme scheme NOUN cana-918 30 65 and and CCONJ cana-918 30 66 if if SCONJ cana-918 30 67 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š PROPN cana-918 30 68 ๐‘˜ ๐‘˜ VERB cana-918 30 69 = = VERB cana-918 31 1 ๐‘๐‘š๐‘™ ๐‘๐‘š๐‘™ PROPN cana-918 31 2 ๐‘˜ ๐‘˜ DET cana-918 31 3 โˆ€ โˆ€ NOUN cana-918 31 4 0 0 NUM cana-918 31 5 โ‰ค โ‰ค NUM cana-918 31 6 ๐‘˜ ๐‘˜ X cana-918 31 7 , , PUNCT cana-918 31 8 ๐‘™,๐‘š ๐‘™,๐‘š NOUN cana-918 31 9 โ‰ค โ‰ค NUM cana-918 31 10 ๐‘› ๐‘› NOUN cana-918 31 11 , , PUNCT cana-918 31 12 then then ADV cana-918 31 13 it it PRON cana-918 31 14 is be AUX cana-918 31 15 called call VERB cana-918 31 16 commutative commutative ADJ cana-918 31 17 association association NOUN cana-918 31 18 scheme scheme NOUN cana-918 31 19 . . PUNCT cana-918 32 1 let let VERB cana-918 32 2 the the DET cana-918 32 3 set set NOUN cana-918 32 4 ๐‘Ž๐’ฎ ๐‘Ž๐’ฎ PUNCT cana-918 33 1 = = PUNCT cana-918 33 2 { { PUNCT cana-918 33 3 ๐‘ ๐‘ NOUN cana-918 33 4 โˆˆ โˆˆ PROPN cana-918 33 5 ๐‘Œ| ๐‘Œ| PROPN cana-918 33 6 ( ( PUNCT cana-918 33 7 ๐‘Ž ๐‘Ž PROPN cana-918 33 8 , , PUNCT cana-918 33 9 ๐‘ ๐‘ NOUN cana-918 33 10 ) ) PUNCT cana-918 33 11 โˆˆ โˆˆ PROPN cana-918 33 12 ๐’ฎ ๐’ฎ PROPN cana-918 33 13 } } PUNCT cana-918 33 14 for for ADP cana-918 33 15 ๐‘Ž ๐‘Ž PROPN cana-918 33 16 โˆˆ โˆˆ PROPN cana-918 33 17 ๐‘Œ ๐‘Œ PROPN cana-918 33 18 and and CCONJ cana-918 33 19 ๐’ฎ ๐’ฎ PROPN cana-918 33 20 โˆˆ โˆˆ PROPN cana-918 33 21 ๐’ซ. ๐’ซ. NOUN cana-918 33 22 the the DET cana-918 33 23 elements element NOUN cana-918 33 24 ๐‘Ž ๐‘Ž NOUN cana-918 33 25 and and CCONJ cana-918 33 26 ๐‘ ๐‘ NOUN cana-918 33 27 in in ADP cana-918 33 28 ๐‘Œ ๐‘Œ PROPN cana-918 33 29 are be AUX cana-918 33 30 called call VERB cana-918 33 31 ๐‘˜๐‘กโ„Ž ๐‘˜๐‘กโ„Ž NOUN cana-918 33 32 associates associate NOUN cana-918 33 33 if if SCONJ cana-918 33 34 ( ( PUNCT cana-918 33 35 ๐‘Ž ๐‘Ž X cana-918 33 36 , , PUNCT cana-918 33 37 ๐‘ ๐‘ NOUN cana-918 33 38 ) ) PUNCT cana-918 33 39 โˆˆ โˆˆ NOUN cana-918 33 40 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 33 41 with with ADP cana-918 33 42 ๐‘Ž ๐‘Ž PRON cana-918 33 43 โ‰  โ‰  PROPN cana-918 33 44 ๐‘. ๐‘. NOUN cana-918 33 45 note note NOUN cana-918 33 46 that that SCONJ cana-918 33 47 every every DET cana-918 33 48 symmetric symmetric ADJ cana-918 33 49 association association NOUN cana-918 33 50 scheme scheme NOUN cana-918 33 51 is be AUX cana-918 33 52 commutative commutative ADJ cana-918 33 53 . . PUNCT cana-918 34 1 with with ADP cana-918 34 2 regards regard NOUN cana-918 34 3 to to ADP cana-918 34 4 more more ADJ cana-918 34 5 basic basic ADJ cana-918 34 6 association association NOUN cana-918 34 7 schemes scheme NOUN cana-918 34 8 results result NOUN cana-918 34 9 , , PUNCT cana-918 34 10 refer refer VERB cana-918 34 11 [ [ X cana-918 34 12 7 7 NUM cana-918 34 13 , , PUNCT cana-918 34 14 14 14 NUM cana-918 34 15 ] ] PUNCT cana-918 34 16 . . PUNCT cana-918 35 1 definition definition NOUN cana-918 35 2 2 2 NUM cana-918 35 3 . . PUNCT cana-918 36 1 a a DET cana-918 36 2 finite finite ADJ cana-918 36 3 group group NOUN cana-918 36 4 ๐บ ๐บ PROPN cana-918 36 5 with with ADP cana-918 36 6 the the DET cana-918 36 7 conjugacy conjugacy PROPN cana-918 36 8 classes class NOUN cana-918 36 9 ๐ถ0 ๐ถ0 PROPN cana-918 36 10 , , PUNCT cana-918 36 11 ๐ถ1 ๐ถ1 NUM cana-918 36 12 , , PUNCT cana-918 36 13 โ€ฆ โ€ฆ PUNCT cana-918 36 14 , , PUNCT cana-918 36 15 ๐ถ๐‘‘ ๐ถ๐‘‘ PROPN cana-918 36 16 produces produce VERB cana-918 36 17 a a DET cana-918 36 18 commutative commutative ADJ cana-918 36 19 association association NOUN cana-918 36 20 scheme scheme NOUN cana-918 36 21 with with ADP cana-918 36 22 a a DET cana-918 36 23 class class NOUN cana-918 36 24 of of ADP cana-918 36 25 relations relation NOUN cana-918 36 26 on on ADP cana-918 36 27 ๐บ ๐บ PROPN cana-918 36 28 defined define VERB cana-918 36 29 by by ADP cana-918 36 30 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 36 31 = = SYM cana-918 36 32 { { PUNCT cana-918 36 33 ( ( PUNCT cana-918 36 34 ๐‘Ž ๐‘Ž X cana-918 36 35 , , PUNCT cana-918 36 36 ๐‘)| ๐‘)| NOUN cana-918 36 37 ๐‘๐‘Ž ๐‘๐‘Ž X cana-918 36 38 โˆ’1 โˆ’1 NOUN cana-918 36 39 โˆˆ โˆˆ NOUN cana-918 36 40 ๐ถ๐‘˜ ๐ถ๐‘˜ PROPN cana-918 36 41 } } PUNCT cana-918 36 42 โˆ€ โˆ€ X cana-918 36 43 0 0 NUM cana-918 36 44 โ‰ค โ‰ค NOUN cana-918 36 45 ๐‘˜ ๐‘˜ DET cana-918 36 46 โ‰ค โ‰ค ADJ cana-918 36 47 ๐‘‘. ๐‘‘. NOUN cana-918 36 48 this this DET cana-918 36 49 scheme scheme NOUN cana-918 36 50 is be AUX cana-918 36 51 called call VERB cana-918 36 52 the the DET cana-918 36 53 group group NOUN cana-918 36 54 association association NOUN cana-918 36 55 scheme scheme NOUN cana-918 36 56 of of ADP cana-918 36 57 ๐บ. ๐บ. NOUN cana-918 36 58 association association NOUN cana-918 36 59 schemes scheme NOUN cana-918 36 60 can can AUX cana-918 36 61 be be AUX cana-918 36 62 determined determine VERB cana-918 36 63 for for ADP cana-918 36 64 all all DET cana-918 36 65 those those DET cana-918 36 66 groups group NOUN cana-918 36 67 whose whose DET cana-918 36 68 conjugacy conjugacy ADJ cana-918 36 69 classes class NOUN cana-918 36 70 are be AUX cana-918 36 71 known know VERB cana-918 36 72 . . PUNCT cana-918 37 1 lemma lemma PROPN cana-918 37 2 1 1 X cana-918 37 3 . . PUNCT cana-918 38 1 let let VERB cana-918 38 2 ๐‘Œ ๐‘Œ PROPN cana-918 38 3 = = SYM cana-918 38 4 โ„ค๐‘› โ„ค๐‘› PROPN cana-918 38 5 and and CCONJ cana-918 38 6 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 38 7 defines define VERB cana-918 38 8 relations relation NOUN cana-918 38 9 on on ADP cana-918 38 10 ๐’ซ ๐’ซ NOUN cana-918 38 11 by by ADP cana-918 38 12 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 38 13 = = SYM cana-918 38 14 ( ( PUNCT cana-918 38 15 ๐‘Ž ๐‘Ž X cana-918 38 16 , , PUNCT cana-918 38 17 ๐‘)|๐‘Ž ๐‘)|๐‘Ž NOUN cana-918 38 18 = = SYM cana-918 38 19 ๐‘˜ ๐‘˜ PROPN cana-918 38 20 + + CCONJ cana-918 38 21 ๐‘|๐‘Ž ๐‘|๐‘Ž NOUN cana-918 38 22 , , PUNCT cana-918 38 23 ๐‘ ๐‘ PROPN cana-918 38 24 โˆˆ โˆˆ PROPN cana-918 38 25 โ„ค๐‘›โˆ€๐‘˜ โ„ค๐‘›โˆ€๐‘˜ PROPN cana-918 38 26 โˆˆ โˆˆ PROPN cana-918 38 27 โ„ค๐‘›. โ„ค๐‘›. PROPN cana-918 38 28 then then ADV cana-918 38 29 ( ( PUNCT cana-918 38 30 ๐‘Œ ๐‘Œ PROPN cana-918 38 31 , , PUNCT cana-918 38 32 ๐’ซ ๐’ซ NOUN cana-918 38 33 ) ) PUNCT cana-918 38 34 is be AUX cana-918 38 35 a a DET cana-918 38 36 non non ADJ cana-918 38 37 symmetric symmetric PROPN cana-918 38 38 commutative commutative PROPN cana-918 38 39 association association NOUN cana-918 38 40 scheme scheme NOUN cana-918 38 41 with with ADP cana-918 38 42 parameters parameter NOUN cana-918 38 43 given give VERB cana-918 38 44 by by ADP cana-918 38 45 : : PUNCT cana-918 38 46 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š PROPN cana-918 38 47 ๐‘˜ ๐‘˜ NOUN cana-918 38 48 = = PUNCT cana-918 38 49 { { PUNCT cana-918 38 50 1 1 NUM cana-918 38 51 ๐‘–๐‘“๐‘˜ ๐‘–๐‘“๐‘˜ NOUN cana-918 38 52 = = SYM cana-918 39 1 ๐‘™ ๐‘™ PROPN cana-918 40 1 + + NOUN cana-918 40 2 ๐‘š ๐‘š NOUN cana-918 40 3 , , PUNCT cana-918 40 4 0 0 NUM cana-918 40 5 ๐‘–๐‘“๐‘˜ ๐‘–๐‘“๐‘˜ PROPN cana-918 40 6 โ‰  โ‰  PROPN cana-918 40 7 ๐‘™ ๐‘™ PROPN cana-918 41 1 + + NUM cana-918 41 2 ๐‘š ๐‘š PROPN cana-918 41 3 where where SCONJ cana-918 41 4 ๐‘˜ ๐‘˜ ADP cana-918 41 5 , , PUNCT cana-918 41 6 ๐‘™ ๐‘™ X cana-918 41 7 , , PUNCT cana-918 41 8 ๐‘š ๐‘š PROPN cana-918 41 9 โˆˆ โˆˆ NOUN cana-918 41 10 ๐‘๐‘›. ๐‘๐‘›. NOUN cana-918 41 11 proof proof NOUN cana-918 41 12 . . PUNCT cana-918 42 1 since since SCONJ cana-918 42 2 โ„ค๐‘› โ„ค๐‘› PROPN cana-918 42 3 is be AUX cana-918 42 4 an an DET cana-918 42 5 abelian abelian ADJ cana-918 42 6 group group NOUN cana-918 42 7 , , PUNCT cana-918 42 8 ( ( PUNCT cana-918 42 9 โ„ค๐‘› โ„ค๐‘› PROPN cana-918 42 10 , , PUNCT cana-918 42 11 ๐’ซ ๐’ซ NOUN cana-918 42 12 ) ) PUNCT cana-918 42 13 under under ADP cana-918 42 14 given give VERB cana-918 42 15 relations relation NOUN cana-918 42 16 becomes become VERB cana-918 42 17 a a DET cana-918 42 18 commutative commutative ADJ cana-918 42 19 association association NOUN cana-918 42 20 scheme scheme NOUN cana-918 42 21 . . PUNCT cana-918 43 1 for for ADP cana-918 43 2 arbitrary arbitrary ADJ cana-918 43 3 relations relation NOUN cana-918 43 4 ๐’ฎ๐‘™ ๐’ฎ๐‘™ PROPN cana-918 43 5 , , PUNCT cana-918 43 6 ๐’ฎ๐‘š ๐’ฎ๐‘š NOUN cana-918 43 7 , , PUNCT cana-918 43 8 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 43 9 in in ADP cana-918 43 10 ๐’ซ ๐’ซ NOUN cana-918 43 11 , , PUNCT cana-918 43 12 we we PRON cana-918 43 13 find find VERB cana-918 43 14 cardinality cardinality PROPN cana-918 43 15 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š NOUN cana-918 43 16 ๐‘˜ ๐‘˜ NOUN cana-918 43 17 = = PUNCT cana-918 43 18 |๐‘Ž๐’ฎ๐‘™ |๐‘Ž๐’ฎ๐‘™ PROPN cana-918 43 19 โˆฉ โˆฉ NOUN cana-918 43 20 ๐‘๐’ฎ๐‘š ๐‘๐’ฎ๐‘š VERB cana-918 43 21 โˆ— โˆ— NOUN cana-918 43 22 | | ADV cana-918 43 23 whenever whenever SCONJ cana-918 43 24 ( ( PUNCT cana-918 43 25 ๐‘Ž ๐‘Ž X cana-918 43 26 , , PUNCT cana-918 43 27 ๐‘ ๐‘ NOUN cana-918 43 28 ) ) PUNCT cana-918 43 29 โˆˆ โˆˆ NOUN cana-918 44 1 ๐’ฎ๐‘˜. ๐’ฎ๐‘˜. PROPN cana-918 44 2 let let VERB cana-918 44 3 ( ( PUNCT cana-918 44 4 ๐‘Ž ๐‘Ž X cana-918 44 5 , , PUNCT cana-918 44 6 ๐‘ ๐‘ NOUN cana-918 44 7 ) ) PUNCT cana-918 44 8 be be AUX cana-918 44 9 an an DET cana-918 44 10 arbitrary arbitrary ADJ cana-918 44 11 element element NOUN cana-918 44 12 of of ADP cana-918 44 13 ๐‘Œ ๐‘Œ PROPN cana-918 44 14 in in ADP cana-918 44 15 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 44 16 and and CCONJ cana-918 44 17 let let VERB cana-918 44 18 ๐‘Ž๐’ฎ๐‘™ ๐‘Ž๐’ฎ๐‘™ NOUN cana-918 44 19 = = PRON cana-918 44 20 ๐‘Ž ๐‘Ž NOUN cana-918 44 21 โ€ฒ โ€ฒ NOUN cana-918 44 22 and and CCONJ cana-918 44 23 ๐‘๐’ฎ๐‘š ๐‘๐’ฎ๐‘š VERB cana-918 44 24 โˆ— โˆ— NOUN cana-918 44 25 = = PUNCT cana-918 44 26 ๐‘โ€ฒ. ๐‘โ€ฒ. ADP cana-918 44 27 this this PRON cana-918 44 28 implies imply VERB cana-918 44 29 , , PUNCT cana-918 44 30 ๐‘Ž ๐‘Ž NOUN cana-918 44 31 = = SYM cana-918 44 32 ๐‘Žโ€ฒ ๐‘Žโ€ฒ PRON cana-918 44 33 + + NUM cana-918 44 34 ๐‘™ ๐‘™ NUM cana-918 44 35 , , PUNCT cana-918 44 36 ๐‘โ€ฒ ๐‘โ€ฒ PUNCT cana-918 45 1 = = SYM cana-918 45 2 ๐‘ ๐‘ PRON cana-918 45 3 + + NOUN cana-918 45 4 ๐‘š ๐‘š NOUN cana-918 45 5 and and CCONJ cana-918 45 6 ๐‘Ž ๐‘Ž NOUN cana-918 45 7 = = SYM cana-918 45 8 ๐‘ ๐‘ PROPN cana-918 45 9 + + ADP cana-918 45 10 ๐‘˜ ๐‘˜ PROPN cana-918 46 1 and and CCONJ cana-918 46 2 we we PRON cana-918 46 3 get get VERB cana-918 46 4 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š PROPN cana-918 46 5 ๐‘˜ ๐‘˜ NOUN cana-918 46 6 = = NOUN cana-918 46 7 1 1 NUM cana-918 46 8 if if SCONJ cana-918 46 9 ๐‘Žโ€ฒ ๐‘Žโ€ฒ PRON cana-918 46 10 = = SYM cana-918 46 11 ๐‘โ€ฒ ๐‘โ€ฒ PROPN cana-918 46 12 which which PRON cana-918 46 13 implies imply VERB cana-918 46 14 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š PROPN cana-918 46 15 ๐‘˜ ๐‘˜ NOUN cana-918 46 16 = = NOUN cana-918 46 17 1 1 NUM cana-918 46 18 if if SCONJ cana-918 46 19 ๐‘˜ ๐‘˜ ADP cana-918 46 20 = = SYM cana-918 46 21 ๐‘™ ๐‘™ PROPN cana-918 46 22 + + CCONJ cana-918 46 23 ๐‘š. ๐‘š. ADV cana-918 46 24 further far ADV cana-918 46 25 , , PUNCT cana-918 46 26 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š VERB cana-918 46 27 ๐‘˜ ๐‘˜ X cana-918 47 1 = = NOUN cana-918 47 2 0 0 PUNCT cana-918 48 1 if if SCONJ cana-918 48 2 ๐‘Žโ€ฒ ๐‘Žโ€ฒ DET cana-918 48 3 โ‰  โ‰  PROPN cana-918 48 4 ๐‘โ€ฒ ๐‘โ€ฒ PROPN cana-918 48 5 , , PUNCT cana-918 48 6 equivalently equivalently ADV cana-918 48 7 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š VERB cana-918 48 8 ๐‘˜ ๐‘˜ PROPN cana-918 48 9 = = NOUN cana-918 48 10 0 0 NUM cana-918 48 11 , , PUNCT cana-918 48 12 whenever whenever SCONJ cana-918 48 13 ๐‘˜ ๐‘˜ PRON cana-918 48 14 โ‰  โ‰  PROPN cana-918 48 15 ๐‘™ ๐‘™ PROPN cana-918 48 16 + + CCONJ cana-918 48 17 ๐‘š. ๐‘š. ADJ cana-918 48 18 โ—ป โ—ป NOUN cana-918 48 19 in in ADP cana-918 48 20 the the DET cana-918 48 21 next next ADJ cana-918 48 22 section section NOUN cana-918 48 23 of of ADP cana-918 48 24 this this DET cana-918 48 25 paper paper NOUN cana-918 48 26 , , PUNCT cana-918 48 27 we we PRON cana-918 48 28 work work VERB cana-918 48 29 on on ADP cana-918 48 30 non non ADJ cana-918 48 31 symmetric symmetric PROPN cana-918 48 32 association association NOUN cana-918 48 33 scheme scheme NOUN cana-918 48 34 of of ADP cana-918 48 35 the the DET cana-918 48 36 cyclic cyclic ADJ cana-918 48 37 groups group NOUN cana-918 48 38 โ„ค2 โ„ค2 PROPN cana-918 48 39 ๐‘Ÿ ๐‘Ÿ NOUN cana-918 48 40 , , PUNCT cana-918 48 41 โ„ค๐‘›1 โ„ค๐‘›1 VERB cana-918 48 42 ร— ร— PROPN cana-918 48 43 โ„ค๐‘›2 โ„ค๐‘›2 NOUN cana-918 48 44 ร—โ‹ฏร— ร—โ‹ฏร— PROPN cana-918 48 45 โ„ค๐‘›๐‘Ÿ โ„ค๐‘›๐‘Ÿ PROPN cana-918 48 46 and and CCONJ cana-918 48 47 general general ADJ cana-918 48 48 linear linear PROPN cana-918 48 49 group group NOUN cana-918 48 50 of of ADP cana-918 48 51 order order NOUN cana-918 48 52 2 2 NUM cana-918 48 53 over over ADP cana-918 48 54 โ„ค2 โ„ค2 PROPN cana-918 48 55 . . PUNCT cana-918 49 1 2 2 X cana-918 49 2 . . X cana-918 49 3 association association NOUN cana-918 49 4 schemes scheme NOUN cana-918 49 5 for for ADP cana-918 49 6 some some DET cana-918 49 7 finite finite ADJ cana-918 49 8 groups group NOUN cana-918 49 9 theorem theorem VERB cana-918 49 10 1 1 X cana-918 49 11 . . PUNCT cana-918 50 1 let let VERB cana-918 50 2 ๐‘Œ ๐‘Œ PROPN cana-918 50 3 = = SYM cana-918 50 4 โ„ค2 โ„ค2 PROPN cana-918 50 5 ๐‘Ÿ ๐‘Ÿ NOUN cana-918 50 6 = = SYM cana-918 50 7 โ„ค2 โ„ค2 PROPN cana-918 50 8 ร— ร— PROPN cana-918 50 9 โ„ค2 โ„ค2 PROPN cana-918 50 10 ร—โ‹ฏร— ร—โ‹ฏร— PROPN cana-918 50 11 โ„ค2 โ„ค2 PROPN cana-918 50 12 and and CCONJ cana-918 50 13 ๐‘Ÿ ๐‘Ÿ PRON cana-918 51 1 โ‰ฅ โ‰ฅ NUM cana-918 51 2 2 2 X cana-918 51 3 . . PUNCT cana-918 52 1 we we PRON cana-918 52 2 define define VERB cana-918 52 3 relations relation NOUN cana-918 52 4 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 52 5 on on ADP cana-918 52 6 ๐’ซ ๐’ซ NOUN cana-918 52 7 by by ADP cana-918 52 8 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 52 9 = = SYM cana-918 52 10 { { PUNCT cana-918 52 11 ( ( PUNCT cana-918 52 12 ๐‘Ž ๐‘Ž X cana-918 52 13 , , PUNCT cana-918 52 14 ๐‘)| ๐‘)| NOUN cana-918 52 15 ๐‘๐‘  ๐‘๐‘  NOUN cana-918 52 16 โ‰ก โ‰ก PROPN cana-918 52 17 ( ( PUNCT cana-918 52 18 ๐‘ก๐‘  ๐‘ก๐‘  PROPN cana-918 52 19 + + CCONJ cana-918 52 20 ๐‘Ž๐‘  ๐‘Ž๐‘  PROPN cana-918 52 21 ) ) PUNCT cana-918 52 22 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 52 23 2 2 NUM cana-918 52 24 | | NOUN cana-918 52 25 ๐‘ก๐‘  ๐‘ก๐‘  ADP cana-918 52 26 โˆˆ โˆˆ NOUN cana-918 52 27 โ„ค2 โ„ค2 X cana-918 52 28 โˆ€ โˆ€ NOUN cana-918 52 29 1 1 NUM cana-918 52 30 โ‰ค โ‰ค NUM cana-918 52 31 ๐‘  ๐‘  PRON cana-918 52 32 โ‰ค โ‰ค ADJ cana-918 52 33 ๐‘Ÿ ๐‘Ÿ NOUN cana-918 52 34 , , PUNCT cana-918 52 35 ๐‘Ž ๐‘Ž X cana-918 52 36 = = SYM cana-918 52 37 ( ( PUNCT cana-918 52 38 ๐‘Ž1 ๐‘Ž1 PROPN cana-918 52 39 , , PUNCT cana-918 52 40 ๐‘Ž2 ๐‘Ž2 PROPN cana-918 52 41 , , PUNCT cana-918 52 42 โ€ฆ โ€ฆ PUNCT cana-918 52 43 , , PUNCT cana-918 52 44 ๐‘Ž๐‘Ÿ ๐‘Ž๐‘Ÿ NOUN cana-918 52 45 ) ) PUNCT cana-918 52 46 , , PUNCT cana-918 52 47 ๐‘ ๐‘ NOUN cana-918 52 48 = = PUNCT cana-918 52 49 ( ( PUNCT cana-918 52 50 ๐‘1 ๐‘1 NOUN cana-918 52 51 , , PUNCT cana-918 52 52 ๐‘2 ๐‘2 PROPN cana-918 52 53 , , PUNCT cana-918 52 54 โ€ฆ โ€ฆ PUNCT cana-918 52 55 , , PUNCT cana-918 52 56 ๐‘๐‘Ÿ ๐‘๐‘Ÿ NOUN cana-918 52 57 ) ) PUNCT cana-918 52 58 โˆˆ โˆˆ PROPN cana-918 52 59 ๐‘Œ ๐‘Œ PROPN cana-918 52 60 } } PUNCT cana-918 52 61 where where SCONJ cana-918 52 62 ๐‘˜ ๐‘˜ PRON cana-918 52 63 = = PUNCT cana-918 53 1 2๐‘Ÿโˆ’1๐‘ก1 2๐‘Ÿโˆ’1๐‘ก1 NUM cana-918 53 2 + + CCONJ cana-918 53 3 2 2 NUM cana-918 53 4 ๐‘Ÿโˆ’2๐‘ก2 ๐‘Ÿโˆ’2๐‘ก2 NUM cana-918 53 5 + + ADJ cana-918 53 6 โ‹ฏ+ โ‹ฏ+ NOUN cana-918 53 7 2๐‘ก๐‘Ÿโˆ’1 2๐‘ก๐‘Ÿโˆ’1 ADJ cana-918 53 8 + + CCONJ cana-918 53 9 ๐‘ก๐‘Ÿ ๐‘ก๐‘Ÿ VERB cana-918 53 10 . . PUNCT cana-918 54 1 then then ADV cana-918 54 2 ๐’œ ๐’œ NOUN cana-918 54 3 = = SYM cana-918 54 4 ( ( PUNCT cana-918 54 5 ๐‘Œ,๐’ซ ๐‘Œ,๐’ซ NOUN cana-918 54 6 ) ) PUNCT cana-918 54 7 is be AUX cana-918 54 8 a a DET cana-918 54 9 symmetric symmetric ADJ cana-918 54 10 and and CCONJ cana-918 54 11 commutative commutative ADJ cana-918 54 12 association association NOUN cana-918 54 13 scheme scheme NOUN cana-918 54 14 with with ADP cana-918 54 15 parameters parameter NOUN cana-918 55 1 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š VERB cana-918 55 2 ๐‘˜ ๐‘˜ X cana-918 55 3 = = PUNCT cana-918 55 4 { { PUNCT cana-918 55 5 1 1 NUM cana-918 55 6 if if SCONJ cana-918 55 7 ๐‘ก๐‘  ๐‘ก๐‘  INTJ cana-918 55 8 ( ( PUNCT cana-918 55 9 ๐‘˜ ๐‘˜ NOUN cana-918 55 10 ) ) PUNCT cana-918 55 11 โ‰ก โ‰ก PROPN cana-918 55 12 ๐‘ก๐‘  ๐‘ก๐‘  INTJ cana-918 55 13 ( ( PUNCT cana-918 55 14 ๐‘™ ๐‘™ NUM cana-918 55 15 ) ) PUNCT cana-918 55 16 + + X cana-918 56 1 ๐‘ก๐‘  ๐‘ก๐‘  X cana-918 56 2 ( ( PUNCT cana-918 56 3 ๐‘š)๐‘š๐‘œ๐‘‘ ๐‘š)๐‘š๐‘œ๐‘‘ ADJ cana-918 56 4 2 2 NUM cana-918 56 5 โˆ€ โˆ€ NOUN cana-918 56 6 1 1 NUM cana-918 56 7 โ‰ค โ‰ค NUM cana-918 56 8 ๐‘  ๐‘  PRON cana-918 56 9 โ‰ค โ‰ค NUM cana-918 56 10 ๐‘Ÿ ๐‘Ÿ NOUN cana-918 56 11 0 0 NUM cana-918 56 12 otherwise otherwise ADV cana-918 56 13 where where SCONJ cana-918 56 14 ๐‘˜ ๐‘˜ PROPN cana-918 56 15 = = PUNCT cana-918 56 16 โˆ‘ โˆ‘ PROPN cana-918 56 17 2๐‘Ÿโˆ’๐‘ ๐‘ก๐‘  2๐‘Ÿโˆ’๐‘ ๐‘ก๐‘  PROPN cana-918 56 18 ( ( PUNCT cana-918 56 19 ๐‘˜)๐‘Ÿ ๐‘˜)๐‘Ÿ X cana-918 56 20 ๐‘ =1 ๐‘ =1 PUNCT cana-918 56 21 ; ; PUNCT cana-918 56 22 ๐‘™ ๐‘™ X cana-918 56 23 = = PUNCT cana-918 56 24 โˆ‘ โˆ‘ PROPN cana-918 56 25 2๐‘Ÿโˆ’๐‘ ๐‘ก๐‘  2๐‘Ÿโˆ’๐‘ ๐‘ก๐‘  PROPN cana-918 56 26 ( ( PUNCT cana-918 56 27 ๐‘™)๐‘Ÿ ๐‘™)๐‘Ÿ X cana-918 56 28 ๐‘ =1 ๐‘ =1 PUNCT cana-918 56 29 ; ; PUNCT cana-918 56 30 ๐‘š ๐‘š X cana-918 56 31 = = PUNCT cana-918 56 32 โˆ‘ โˆ‘ PROPN cana-918 56 33 2๐‘Ÿโˆ’๐‘ ๐‘ก๐‘  2๐‘Ÿโˆ’๐‘ ๐‘ก๐‘  PROPN cana-918 56 34 ( ( PUNCT cana-918 56 35 ๐‘š)๐‘Ÿ ๐‘š)๐‘Ÿ X cana-918 56 36 ๐‘ =1 ๐‘ =1 PUNCT cana-918 56 37 for for ADP cana-918 56 38 some some DET cana-918 56 39 ๐‘ก๐‘  ๐‘ก๐‘  ADJ cana-918 56 40 ( ( PUNCT cana-918 56 41 ๐‘˜ ๐‘˜ NOUN cana-918 56 42 ) ) PUNCT cana-918 56 43 , , PUNCT cana-918 56 44 ๐‘ก๐‘  ๐‘ก๐‘  X cana-918 56 45 ( ( PUNCT cana-918 56 46 ๐‘™ ๐‘™ NOUN cana-918 56 47 ) ) PUNCT cana-918 56 48 , , PUNCT cana-918 56 49 ๐‘ก๐‘  ๐‘ก๐‘  X cana-918 56 50 ( ( PUNCT cana-918 56 51 ๐‘š ๐‘š NOUN cana-918 56 52 ) ) PUNCT cana-918 56 53 โˆˆ โˆˆ PROPN cana-918 56 54 โ„ค2 โ„ค2 PROPN cana-918 56 55 . . PUNCT cana-918 57 1 communications communication NOUN cana-918 57 2 on on ADP cana-918 57 3 applied apply VERB cana-918 57 4 nonlinear nonlinear ADJ cana-918 57 5 analysis analysis NOUN cana-918 57 6 issn issn NOUN cana-918 57 7 : : PUNCT cana-918 57 8 1074 1074 NUM cana-918 57 9 - - PUNCT cana-918 57 10 133x 133x NUM cana-918 57 11 vol vol NOUN cana-918 57 12 31 31 NUM cana-918 57 13 no no NOUN cana-918 57 14 . . PUNCT cana-918 58 1 4s 4s NUM cana-918 58 2 ( ( PUNCT cana-918 58 3 2024 2024 NUM cana-918 58 4 ) ) PUNCT cana-918 58 5 394 394 NUM cana-918 58 6 https://internationalpubls.com https://internationalpubls.com X cana-918 58 7 proof proof NOUN cana-918 58 8 . . PUNCT cana-918 59 1 observe observe VERB cana-918 59 2 that that SCONJ cana-918 59 3 |๐‘Œ| |๐‘Œ| NOUN cana-918 59 4 = = SYM cana-918 59 5 2๐‘Ÿ 2๐‘Ÿ NUM cana-918 59 6 = = PUNCT cana-918 59 7 |๐’ฎ๐‘˜| |๐’ฎ๐‘˜| ADV cana-918 59 8 for for ADP cana-918 59 9 all all PRON cana-918 59 10 0 0 NUM cana-918 59 11 โ‰ค โ‰ค NOUN cana-918 59 12 ๐‘˜ ๐‘˜ DET cana-918 59 13 โ‰ค โ‰ค ADJ cana-918 59 14 2๐‘Ÿ 2๐‘Ÿ NOUN cana-918 59 15 โˆ’ โˆ’ NOUN cana-918 59 16 1 1 NUM cana-918 59 17 . . PUNCT cana-918 60 1 the the DET cana-918 60 2 relations relation NOUN cana-918 60 3 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 60 4 are be AUX cana-918 60 5 disjoint disjoint ADJ cana-918 60 6 and and CCONJ cana-918 60 7 โ‹ƒ๐’ฎ๐‘˜:0 โ‹ƒ๐’ฎ๐‘˜:0 VERB cana-918 60 8 โ‰ค โ‰ค NOUN cana-918 60 9 ๐‘˜ ๐‘˜ DET cana-918 60 10 โ‰ค โ‰ค ADJ cana-918 60 11 2๐‘Ÿ 2๐‘Ÿ NUM cana-918 60 12 โˆ’ โˆ’ NOUN cana-918 60 13 1 1 NUM cana-918 60 14 = = SYM cana-918 60 15 ๐’ซ. ๐’ซ. PROPN cana-918 60 16 for for ADP cana-918 60 17 arbitrary arbitrary ADJ cana-918 60 18 relations relation NOUN cana-918 60 19 ๐’ฎ๐‘™ ๐’ฎ๐‘™ PROPN cana-918 60 20 , , PUNCT cana-918 60 21 ๐’ฎ๐‘š ๐’ฎ๐‘š NOUN cana-918 60 22 , , PUNCT cana-918 60 23 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 60 24 in in ADP cana-918 60 25 ๐’ซ ๐’ซ PROPN cana-918 60 26 , , PUNCT cana-918 60 27 we we PRON cana-918 60 28 prove prove VERB cana-918 60 29 that that SCONJ cana-918 60 30 the the DET cana-918 60 31 parameters parameter NOUN cana-918 60 32 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š VERB cana-918 60 33 ๐‘˜ ๐‘˜ ADV cana-918 60 34 = = PUNCT cana-918 60 35 |๐‘Ž๐’ฎ๐‘™ |๐‘Ž๐’ฎ๐‘™ PROPN cana-918 61 1 โˆฉ โˆฉ NOUN cana-918 61 2 ๐‘๐’ฎ๐‘š ๐‘๐’ฎ๐‘š PART cana-918 61 3 โˆ— โˆ— NOUN cana-918 61 4 | | ADV cana-918 61 5 is be AUX cana-918 61 6 constant constant ADJ cana-918 61 7 for for ADP cana-918 61 8 ( ( PUNCT cana-918 61 9 ๐‘Ž ๐‘Ž X cana-918 61 10 , , PUNCT cana-918 61 11 ๐‘ ๐‘ NOUN cana-918 61 12 ) ) PUNCT cana-918 61 13 โˆˆ โˆˆ NOUN cana-918 62 1 ๐’ฎ๐‘˜. ๐’ฎ๐‘˜. PROPN cana-918 62 2 let let VERB cana-918 62 3 ( ( PUNCT cana-918 62 4 ๐‘Ž ๐‘Ž X cana-918 62 5 , , PUNCT cana-918 62 6 ๐‘ ๐‘ NOUN cana-918 62 7 ) ) PUNCT cana-918 62 8 be be AUX cana-918 62 9 an an DET cana-918 62 10 arbitrary arbitrary ADJ cana-918 62 11 element element NOUN cana-918 62 12 of of ADP cana-918 62 13 ๐‘Œ ๐‘Œ PROPN cana-918 62 14 in in ADP cana-918 62 15 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 62 16 where where SCONJ cana-918 62 17 ๐‘Ž ๐‘Ž NOUN cana-918 62 18 = = PUNCT cana-918 62 19 ( ( PUNCT cana-918 62 20 ๐‘Ž1 ๐‘Ž1 PROPN cana-918 62 21 , , PUNCT cana-918 62 22 ๐‘Ž2 ๐‘Ž2 PROPN cana-918 62 23 , , PUNCT cana-918 62 24 โ€ฆ โ€ฆ PUNCT cana-918 62 25 , , PUNCT cana-918 62 26 ๐‘Ž๐‘Ÿ ๐‘Ž๐‘Ÿ NOUN cana-918 62 27 ) ) PUNCT cana-918 62 28 , , PUNCT cana-918 63 1 ๐‘ ๐‘ NOUN cana-918 63 2 = = PUNCT cana-918 63 3 ( ( PUNCT cana-918 63 4 ๐‘1 ๐‘1 NOUN cana-918 63 5 , , PUNCT cana-918 63 6 ๐‘2 ๐‘2 PROPN cana-918 63 7 , , PUNCT cana-918 63 8 โ€ฆ โ€ฆ PUNCT cana-918 63 9 , , PUNCT cana-918 63 10 ๐‘๐‘Ÿ ๐‘๐‘Ÿ NOUN cana-918 63 11 ) ) PUNCT cana-918 63 12 . . PUNCT cana-918 64 1 let let VERB cana-918 64 2 ๐‘Ž๐’ฎ๐‘™ ๐‘Ž๐’ฎ๐‘™ VERB cana-918 64 3 = = PRON cana-918 64 4 ๐‘Ž ๐‘Ž NOUN cana-918 64 5 โ€ฒ โ€ฒ NOUN cana-918 64 6 = = SYM cana-918 64 7 ( ( PUNCT cana-918 64 8 ๐‘Žโ€ฒ1 ๐‘Žโ€ฒ1 NOUN cana-918 64 9 , , PUNCT cana-918 64 10 ๐‘Žโ€ฒ2 ๐‘Žโ€ฒ2 ADJ cana-918 64 11 , , PUNCT cana-918 64 12 โ€ฆ โ€ฆ PUNCT cana-918 64 13 , , PUNCT cana-918 64 14 ๐‘Žโ€ฒ๐‘Ÿ ๐‘Žโ€ฒ๐‘Ÿ NUM cana-918 64 15 ) ) PUNCT cana-918 64 16 and and CCONJ cana-918 64 17 ๐‘๐’ฎ๐‘š ๐‘๐’ฎ๐‘š VERB cana-918 64 18 โˆ— โˆ— NOUN cana-918 64 19 = = SYM cana-918 64 20 ๐‘โ€ฒ ๐‘โ€ฒ X cana-918 64 21 = = SYM cana-918 64 22 ( ( PUNCT cana-918 64 23 ๐‘โ€ฒ1 ๐‘โ€ฒ1 INTJ cana-918 64 24 , , PUNCT cana-918 64 25 ๐‘โ€ฒ2 ๐‘โ€ฒ2 NOUN cana-918 64 26 , , PUNCT cana-918 64 27 โ€ฆ โ€ฆ PUNCT cana-918 64 28 , , PUNCT cana-918 64 29 ๐‘โ€ฒ๐‘Ÿ ๐‘โ€ฒ๐‘Ÿ NOUN cana-918 64 30 ) ) PUNCT cana-918 64 31 . . PUNCT cana-918 65 1 now now ADV cana-918 65 2 ( ( PUNCT cana-918 65 3 ๐‘Ž ๐‘Ž X cana-918 65 4 , , PUNCT cana-918 65 5 ๐‘ ๐‘ NOUN cana-918 65 6 ) ) PUNCT cana-918 65 7 โˆˆ โˆˆ PROPN cana-918 66 1 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 66 2 implies imply VERB cana-918 66 3 ๐‘๐‘  ๐‘๐‘  PROPN cana-918 66 4 โ‰ก โ‰ก PROPN cana-918 66 5 ( ( PUNCT cana-918 66 6 ๐‘ก๐‘  ๐‘ก๐‘  X cana-918 66 7 ( ( PUNCT cana-918 66 8 ๐‘˜ ๐‘˜ NOUN cana-918 66 9 ) ) PUNCT cana-918 66 10 + + CCONJ cana-918 66 11 ๐‘Ž๐‘  ๐‘Ž๐‘  PROPN cana-918 66 12 ) ) PUNCT cana-918 66 13 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 66 14 2 2 NUM cana-918 66 15 where where SCONJ cana-918 66 16 ๐‘˜ ๐‘˜ PROPN cana-918 66 17 = = PUNCT cana-918 66 18 โˆ‘ โˆ‘ PROPN cana-918 66 19 2๐‘Ÿโˆ’๐‘ ๐‘ก๐‘  2๐‘Ÿโˆ’๐‘ ๐‘ก๐‘  PROPN cana-918 66 20 ( ( PUNCT cana-918 66 21 ๐‘˜)๐‘Ÿ ๐‘˜)๐‘Ÿ X cana-918 66 22 ๐‘ =1 ๐‘ =1 PUNCT cana-918 66 23 for for ADP cana-918 66 24 some some DET cana-918 66 25 ๐‘ก๐‘  ๐‘ก๐‘  ADJ cana-918 66 26 ( ( PUNCT cana-918 66 27 ๐‘˜ ๐‘˜ NOUN cana-918 66 28 ) ) PUNCT cana-918 66 29 โˆˆ โˆˆ PROPN cana-918 66 30 โ„ค2 โ„ค2 NOUN cana-918 66 31 โˆ€ โˆ€ NOUN cana-918 66 32 1 1 NUM cana-918 66 33 โ‰ค โ‰ค NUM cana-918 66 34 ๐‘  ๐‘  PRON cana-918 66 35 โ‰ค โ‰ค NUM cana-918 66 36 ๐‘Ÿ. ๐‘Ÿ. NOUN cana-918 66 37 similarly similarly ADV cana-918 66 38 , , PUNCT cana-918 66 39 ( ( PUNCT cana-918 66 40 ๐‘Ž ๐‘Ž X cana-918 66 41 , , PUNCT cana-918 66 42 ๐‘Žโ€ฒ ๐‘Žโ€ฒ NUM cana-918 66 43 ) ) PUNCT cana-918 66 44 โˆˆ โˆˆ PROPN cana-918 66 45 ๐’ฎ๐‘™ ๐’ฎ๐‘™ PROPN cana-918 66 46 and and CCONJ cana-918 66 47 ( ( PUNCT cana-918 66 48 ๐‘โ€ฒ ๐‘โ€ฒ PROPN cana-918 66 49 , , PUNCT cana-918 66 50 ๐‘ ๐‘ NOUN cana-918 66 51 ) ) PUNCT cana-918 66 52 โˆˆ โˆˆ PROPN cana-918 66 53 ๐’ฎ๐‘š ๐’ฎ๐‘š NOUN cana-918 66 54 , , PUNCT cana-918 66 55 implies imply VERB cana-918 66 56 ๐‘Žโ€ฒ๐‘  ๐‘Žโ€ฒ๐‘  PROPN cana-918 66 57 โ‰ก โ‰ก PROPN cana-918 66 58 ( ( PUNCT cana-918 66 59 ๐‘ก๐‘  ๐‘ก๐‘  PROPN cana-918 66 60 ( ( PUNCT cana-918 66 61 ๐‘™ ๐‘™ NUM cana-918 66 62 ) ) PUNCT cana-918 67 1 + + CCONJ cana-918 67 2 ๐‘Ž๐‘  ๐‘Ž๐‘  PROPN cana-918 67 3 ) ) PUNCT cana-918 67 4 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 67 5 2 2 NUM cana-918 68 1 where where SCONJ cana-918 68 2 ๐‘™ ๐‘™ NOUN cana-918 68 3 = = PUNCT cana-918 68 4 โˆ‘ โˆ‘ PROPN cana-918 68 5 2๐‘Ÿโˆ’๐‘ ๐‘ก๐‘  2๐‘Ÿโˆ’๐‘ ๐‘ก๐‘  PROPN cana-918 68 6 ( ( PUNCT cana-918 68 7 ๐‘™)๐‘Ÿ ๐‘™)๐‘Ÿ NOUN cana-918 68 8 ๐‘ =1 ๐‘ =1 PUNCT cana-918 68 9 for for ADP cana-918 68 10 some some DET cana-918 68 11 ๐‘ก๐‘  ๐‘ก๐‘  ADJ cana-918 68 12 ( ( PUNCT cana-918 68 13 ๐‘™ ๐‘™ NUM cana-918 68 14 ) ) PUNCT cana-918 68 15 โˆˆ โˆˆ PROPN cana-918 68 16 โ„ค2 โ„ค2 NOUN cana-918 68 17 โˆ€ โˆ€ NOUN cana-918 68 18 1 1 NUM cana-918 68 19 โ‰ค โ‰ค NUM cana-918 68 20 ๐‘  ๐‘  PRON cana-918 68 21 โ‰ค โ‰ค ADJ cana-918 68 22 ๐‘Ÿ ๐‘Ÿ NOUN cana-918 68 23 , , PUNCT cana-918 68 24 and and CCONJ cana-918 68 25 ๐‘๐‘  ๐‘๐‘  PROPN cana-918 68 26 โ‰ก โ‰ก PROPN cana-918 68 27 ( ( PUNCT cana-918 68 28 ๐‘ก๐‘  ๐‘ก๐‘  INTJ cana-918 68 29 ( ( PUNCT cana-918 68 30 ๐‘™ ๐‘™ NUM cana-918 68 31 ) ) PUNCT cana-918 68 32 + + CCONJ cana-918 68 33 ๐‘โ€ฒ๐‘  ๐‘โ€ฒ๐‘  NOUN cana-918 68 34 ) ) PUNCT cana-918 68 35 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 68 36 2 2 NUM cana-918 68 37 where where SCONJ cana-918 68 38 ๐‘š ๐‘š X cana-918 68 39 = = SYM cana-918 68 40 โˆ‘ โˆ‘ PROPN cana-918 68 41 2๐‘Ÿโˆ’๐‘ ๐‘ก๐‘  2๐‘Ÿโˆ’๐‘ ๐‘ก๐‘  PROPN cana-918 68 42 ( ( PUNCT cana-918 68 43 ๐‘š)๐‘Ÿ ๐‘š)๐‘Ÿ X cana-918 68 44 ๐‘ =1 ๐‘ =1 PUNCT cana-918 68 45 for for ADP cana-918 68 46 some some DET cana-918 68 47 ๐‘ก๐‘  ๐‘ก๐‘  ADJ cana-918 68 48 ( ( PUNCT cana-918 68 49 ๐‘š ๐‘š NOUN cana-918 68 50 ) ) PUNCT cana-918 68 51 โˆˆ โˆˆ NOUN cana-918 68 52 โ„ค2 โ„ค2 NOUN cana-918 68 53 โˆ€ โˆ€ NOUN cana-918 68 54 1 1 NUM cana-918 68 55 โ‰ค โ‰ค NUM cana-918 68 56 ๐‘  ๐‘  PRON cana-918 68 57 โ‰ค โ‰ค NUM cana-918 68 58 ๐‘Ÿ. ๐‘Ÿ. NOUN cana-918 68 59 we we PRON cana-918 68 60 observe observe VERB cana-918 68 61 that that SCONJ cana-918 68 62 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š PROPN cana-918 68 63 ๐‘˜ ๐‘˜ NOUN cana-918 68 64 = = NOUN cana-918 68 65 1 1 NUM cana-918 68 66 if if SCONJ cana-918 68 67 and and CCONJ cana-918 68 68 only only ADV cana-918 68 69 if if SCONJ cana-918 68 70 ๐‘Žโ€ฒ ๐‘Žโ€ฒ PRON cana-918 68 71 = = SYM cana-918 68 72 ๐‘โ€ฒ ๐‘โ€ฒ PROPN cana-918 68 73 that that PRON cana-918 68 74 is be AUX cana-918 68 75 , , PUNCT cana-918 68 76 if if SCONJ cana-918 68 77 ๐‘ก๐‘  ๐‘ก๐‘  INTJ cana-918 68 78 ( ( PUNCT cana-918 68 79 ๐‘˜ ๐‘˜ NOUN cana-918 68 80 ) ) PUNCT cana-918 68 81 โ‰ก โ‰ก PROPN cana-918 68 82 ( ( PUNCT cana-918 68 83 ๐‘ก๐‘  ๐‘ก๐‘  PROPN cana-918 68 84 ( ( PUNCT cana-918 68 85 ๐‘™ ๐‘™ NUM cana-918 68 86 ) ) PUNCT cana-918 68 87 + + X cana-918 68 88 ๐‘ก๐‘  ๐‘ก๐‘  ADJ cana-918 68 89 ( ( PUNCT cana-918 68 90 ๐‘š ๐‘š NOUN cana-918 68 91 ) ) PUNCT cana-918 68 92 ) ) PUNCT cana-918 68 93 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 68 94 2 2 NUM cana-918 68 95 for for ADP cana-918 68 96 all all DET cana-918 68 97 1 1 NUM cana-918 68 98 โ‰ค โ‰ค NUM cana-918 69 1 ๐‘  ๐‘  PRON cana-918 69 2 โ‰ค โ‰ค NUM cana-918 69 3 ๐‘Ÿ. ๐‘Ÿ. NOUN cana-918 69 4 to to PART cana-918 69 5 show show VERB cana-918 69 6 ( ( PUNCT cana-918 69 7 ๐‘Œ ๐‘Œ PROPN cana-918 69 8 , , PUNCT cana-918 69 9 ๐’ซ ๐’ซ NOUN cana-918 69 10 ) ) PUNCT cana-918 69 11 is be AUX cana-918 69 12 a a DET cana-918 69 13 symmetric symmetric ADJ cana-918 69 14 association association NOUN cana-918 69 15 scheme scheme NOUN cana-918 69 16 , , PUNCT cana-918 69 17 we we PRON cana-918 69 18 prove prove VERB cana-918 69 19 that that SCONJ cana-918 69 20 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 69 21 โˆ— โˆ— NOUN cana-918 69 22 = = PUNCT cana-918 70 1 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 70 2 for for ADP cana-918 70 3 all all PRON cana-918 70 4 0 0 NUM cana-918 70 5 โ‰ค โ‰ค NOUN cana-918 70 6 ๐‘˜ ๐‘˜ PRON cana-918 70 7 < < X cana-918 70 8 2๐‘Ÿ. 2๐‘Ÿ. NUM cana-918 70 9 let let VERB cana-918 70 10 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 70 11 โˆ— โˆ— NOUN cana-918 70 12 = = PUNCT cana-918 70 13 ๐’ฎ๐พ ๐’ฎ๐พ NOUN cana-918 70 14 where where SCONJ cana-918 70 15 ๐พ ๐พ PROPN cana-918 70 16 = = SYM cana-918 70 17 โˆ‘ โˆ‘ PUNCT cana-918 70 18 2๐‘Ÿโˆ’๐‘ ๐‘‡๐‘  2๐‘Ÿโˆ’๐‘ ๐‘‡๐‘  NUM cana-918 70 19 ๐‘Ÿ ๐‘Ÿ NOUN cana-918 70 20 ๐‘ =1 ๐‘ =1 PUNCT cana-918 70 21 and and CCONJ cana-918 70 22 ๐‘˜ ๐‘˜ X cana-918 70 23 = = PUNCT cana-918 70 24 โˆ‘ โˆ‘ PROPN cana-918 70 25 2๐‘Ÿโˆ’๐‘ ๐‘ก๐‘  2๐‘Ÿโˆ’๐‘ ๐‘ก๐‘  NUM cana-918 70 26 ๐‘Ÿ ๐‘Ÿ NOUN cana-918 70 27 ๐‘ =1 ๐‘ =1 PUNCT cana-918 70 28 for for ADP cana-918 70 29 some some PRON cana-918 70 30 ๐‘ก๐‘  ๐‘ก๐‘  NOUN cana-918 70 31 , , PUNCT cana-918 70 32 ๐‘‡๐‘  ๐‘‡๐‘  PROPN cana-918 70 33 โˆˆ โˆˆ PROPN cana-918 70 34 โ„ค2 โ„ค2 PROPN cana-918 70 35 . . PUNCT cana-918 71 1 if if SCONJ cana-918 71 2 ( ( PUNCT cana-918 71 3 ๐‘ ๐‘ NOUN cana-918 71 4 , , PUNCT cana-918 71 5 ๐‘Ž ๐‘Ž NOUN cana-918 71 6 ) ) PUNCT cana-918 71 7 โˆˆ โˆˆ PROPN cana-918 71 8 ๐’ฎ๐พ ๐’ฎ๐พ PROPN cana-918 71 9 , , PUNCT cana-918 71 10 then then ADV cana-918 71 11 further far ADV cana-918 71 12 ๐‘Ž๐‘  ๐‘Ž๐‘  ADP cana-918 71 13 โ‰ก โ‰ก PROPN cana-918 71 14 ๐‘‡๐‘  ๐‘‡๐‘  PROPN cana-918 71 15 + + CCONJ cana-918 71 16 ๐‘๐‘  ๐‘๐‘  NOUN cana-918 71 17 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 71 18 2 2 NUM cana-918 71 19 and and CCONJ cana-918 71 20 ๐‘๐‘  ๐‘๐‘  PROPN cana-918 71 21 โ‰ก โ‰ก PROPN cana-918 71 22 ๐‘ก๐‘  ๐‘ก๐‘  VERB cana-918 71 23 + + CCONJ cana-918 71 24 ๐‘Ž๐‘  ๐‘Ž๐‘  PROPN cana-918 71 25 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ PROPN cana-918 71 26 2 2 NUM cana-918 71 27 which which PRON cana-918 71 28 implies imply VERB cana-918 71 29 that that SCONJ cana-918 71 30 ๐‘‡๐‘  ๐‘‡๐‘  PROPN cana-918 71 31 = = SYM cana-918 71 32 ๐‘ก๐‘  ๐‘ก๐‘  X cana-918 71 33 โˆ€ โˆ€ NOUN cana-918 71 34 1 1 NUM cana-918 71 35 โ‰ค โ‰ค NUM cana-918 71 36 ๐‘  ๐‘  PRON cana-918 71 37 โ‰ค โ‰ค NUM cana-918 71 38 ๐‘Ÿ. ๐‘Ÿ. NOUN cana-918 71 39 thus thus ADV cana-918 71 40 ๐พ ๐พ PROPN cana-918 71 41 = = SYM cana-918 71 42 ๐‘˜ ๐‘˜ PROPN cana-918 71 43 , , PUNCT cana-918 71 44 and and CCONJ cana-918 71 45 hence hence ADV cana-918 71 46 , , PUNCT cana-918 71 47 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 71 48 โˆ— โˆ— NOUN cana-918 71 49 = = SYM cana-918 72 1 ๐’ฎ๐‘˜. ๐’ฎ๐‘˜. ADJ cana-918 72 2 โ—ป โ—ป NOUN cana-918 72 3 theorem theorem VERB cana-918 72 4 2 2 X cana-918 72 5 . . PUNCT cana-918 73 1 let let VERB cana-918 73 2 ๐‘Œ ๐‘Œ PROPN cana-918 73 3 = = PRON cana-918 73 4 โ„ค๐‘›1 โ„ค๐‘›1 VERB cana-918 74 1 ร— ร— NOUN cana-918 74 2 โ„ค๐‘›2 โ„ค๐‘›2 NOUN cana-918 74 3 ร—โ‹ฏร— ร—โ‹ฏร— PROPN cana-918 74 4 โ„ค๐‘›๐‘Ÿ โ„ค๐‘›๐‘Ÿ PROPN cana-918 74 5 , , PUNCT cana-918 74 6 where where SCONJ cana-918 74 7 ๐‘›1 ๐‘›1 NOUN cana-918 74 8 , , PUNCT cana-918 74 9 ๐‘›2 ๐‘›2 NOUN cana-918 74 10 , , PUNCT cana-918 74 11 โ€ฆ โ€ฆ PUNCT cana-918 74 12 , , PUNCT cana-918 74 13 ๐‘›๐‘Ÿ ๐‘›๐‘Ÿ ADP cana-918 74 14 are be AUX cana-918 74 15 pairwise pairwise NOUN cana-918 74 16 co co NOUN cana-918 74 17 - - NOUN cana-918 74 18 prime prime ADJ cana-918 74 19 . . PUNCT cana-918 75 1 the the DET cana-918 75 2 relations relation NOUN cana-918 75 3 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 75 4 on on ADP cana-918 75 5 ๐’ซ ๐’ซ NOUN cana-918 75 6 defined define VERB cana-918 75 7 by by ADP cana-918 75 8 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 75 9 = = SYM cana-918 75 10 { { PUNCT cana-918 75 11 ( ( PUNCT cana-918 75 12 ๐‘Ž ๐‘Ž X cana-918 75 13 , , PUNCT cana-918 75 14 ๐‘)| ๐‘)| NOUN cana-918 75 15 ๐‘๐‘  ๐‘๐‘  NOUN cana-918 75 16 โ‰ก โ‰ก PROPN cana-918 75 17 ( ( PUNCT cana-918 75 18 ๐‘˜ ๐‘˜ PROPN cana-918 75 19 + + SYM cana-918 75 20 ๐‘Ž๐‘  ๐‘Ž๐‘  NOUN cana-918 75 21 ) ) PUNCT cana-918 75 22 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 75 23 ๐‘›๐‘  ๐‘›๐‘  VERB cana-918 75 24 โˆ€ โˆ€ X cana-918 75 25 1 1 NUM cana-918 75 26 โ‰ค โ‰ค NUM cana-918 75 27 ๐‘  ๐‘  DET cana-918 75 28 โ‰ค โ‰ค PROPN cana-918 75 29 ๐‘Ÿ| ๐‘Ÿ| VERB cana-918 75 30 ๐‘Ž ๐‘Ž X cana-918 75 31 = = SYM cana-918 75 32 ( ( PUNCT cana-918 75 33 ๐‘Ž1 ๐‘Ž1 PROPN cana-918 75 34 , , PUNCT cana-918 75 35 ๐‘Ž2 ๐‘Ž2 PROPN cana-918 75 36 , , PUNCT cana-918 75 37 โ€ฆ โ€ฆ PUNCT cana-918 75 38 , , PUNCT cana-918 75 39 ๐‘Ž๐‘Ÿ ๐‘Ž๐‘Ÿ NOUN cana-918 75 40 ) ) PUNCT cana-918 75 41 , , PUNCT cana-918 76 1 ๐‘ ๐‘ NOUN cana-918 76 2 = = PUNCT cana-918 76 3 ( ( PUNCT cana-918 76 4 ๐‘1 ๐‘1 NOUN cana-918 76 5 , , PUNCT cana-918 76 6 ๐‘2 ๐‘2 PROPN cana-918 76 7 , , PUNCT cana-918 76 8 โ€ฆ โ€ฆ PUNCT cana-918 76 9 , , PUNCT cana-918 76 10 ๐‘๐‘Ÿ ๐‘๐‘Ÿ NOUN cana-918 76 11 ) ) PUNCT cana-918 76 12 โˆˆ โˆˆ PROPN cana-918 76 13 ๐‘Œ ๐‘Œ PROPN cana-918 76 14 } } PUNCT cana-918 76 15 where where SCONJ cana-918 76 16 0 0 NUM cana-918 76 17 โ‰ค โ‰ค NOUN cana-918 76 18 ๐‘˜ ๐‘˜ ADP cana-918 76 19 < < X cana-918 76 20 ๐‘›1๐‘›2โ‹ฏ๐‘›๐‘Ÿ ๐‘›1๐‘›2โ‹ฏ๐‘›๐‘Ÿ PROPN cana-918 76 21 , , PUNCT cana-918 76 22 is be AUX cana-918 76 23 a a DET cana-918 76 24 non non ADJ cana-918 76 25 - - ADJ cana-918 76 26 symmetric symmetric ADJ cana-918 76 27 and and CCONJ cana-918 76 28 commutative commutative ADJ cana-918 76 29 association association NOUN cana-918 76 30 scheme scheme NOUN cana-918 76 31 with with ADP cana-918 76 32 parameters parameter NOUN cana-918 76 33 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š VERB cana-918 76 34 ๐‘˜ ๐‘˜ X cana-918 76 35 = = PUNCT cana-918 76 36 { { PUNCT cana-918 76 37 1 1 NUM cana-918 76 38 if if SCONJ cana-918 76 39 ๐‘˜ ๐‘˜ PRON cana-918 76 40 โ‰ก โ‰ก PROPN cana-918 76 41 ( ( PUNCT cana-918 76 42 ๐‘™ ๐‘™ PROPN cana-918 76 43 + + NUM cana-918 76 44 ๐‘š ๐‘š NOUN cana-918 76 45 ) ) PUNCT cana-918 76 46 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 76 47 ๐‘›๐‘  ๐‘›๐‘  VERB cana-918 76 48 โˆ€ โˆ€ X cana-918 76 49 1 1 NUM cana-918 76 50 โ‰ค โ‰ค NUM cana-918 76 51 ๐‘  ๐‘  PRON cana-918 76 52 โ‰ค โ‰ค NUM cana-918 76 53 ๐‘Ÿ ๐‘Ÿ NOUN cana-918 76 54 0 0 PUNCT cana-918 76 55 otherwise otherwise ADV cana-918 76 56 where where SCONJ cana-918 76 57 0 0 NUM cana-918 76 58 โ‰ค โ‰ค NUM cana-918 76 59 ๐‘˜ ๐‘˜ X cana-918 76 60 , , PUNCT cana-918 76 61 ๐‘™,๐‘š ๐‘™,๐‘š NOUN cana-918 76 62 โ‰ค โ‰ค ADJ cana-918 76 63 ๐‘›1๐‘›2โ‹ฏ๐‘›๐‘Ÿ ๐‘›1๐‘›2โ‹ฏ๐‘›๐‘Ÿ PROPN cana-918 76 64 โˆ’ โˆ’ ADP cana-918 76 65 1 1 X cana-918 76 66 . . PUNCT cana-918 77 1 proof proof NOUN cana-918 77 2 . . PUNCT cana-918 78 1 observe observe VERB cana-918 78 2 that that SCONJ cana-918 78 3 |๐‘Œ| |๐‘Œ| NOUN cana-918 78 4 = = SYM cana-918 78 5 ๐‘›1๐‘›2โ‹ฏ๐‘›๐‘Ÿ ๐‘›1๐‘›2โ‹ฏ๐‘›๐‘Ÿ PROPN cana-918 78 6 = = PUNCT cana-918 78 7 |๐’ฎ๐‘˜| |๐’ฎ๐‘˜| ADV cana-918 78 8 for for ADP cana-918 78 9 all all PRON cana-918 78 10 0 0 NUM cana-918 78 11 โ‰ค โ‰ค NOUN cana-918 78 12 ๐‘˜ ๐‘˜ DET cana-918 78 13 < < X cana-918 78 14 ๐‘›1๐‘›2โ‹ฏ๐‘›๐‘Ÿ. ๐‘›1๐‘›2โ‹ฏ๐‘›๐‘Ÿ. PUNCT cana-918 78 15 the the DET cana-918 78 16 relations relation NOUN cana-918 79 1 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 79 2 being be AUX cana-918 79 3 disjoint disjoint ADJ cana-918 79 4 , , PUNCT cana-918 79 5 form form NOUN cana-918 79 6 partition partition NOUN cana-918 79 7 of of ADP cana-918 79 8 ๐’ซ. ๐’ซ. PROPN cana-918 79 9 for for ADP cana-918 79 10 arbitrary arbitrary ADJ cana-918 79 11 relations relation NOUN cana-918 79 12 ๐’ฎ๐‘™ ๐’ฎ๐‘™ PROPN cana-918 79 13 , , PUNCT cana-918 79 14 ๐’ฎ๐‘š ๐’ฎ๐‘š NOUN cana-918 79 15 , , PUNCT cana-918 79 16 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 79 17 in in ADP cana-918 79 18 ๐’ซ ๐’ซ PROPN cana-918 79 19 , , PUNCT cana-918 79 20 we we PRON cana-918 79 21 prove prove VERB cana-918 79 22 that that SCONJ cana-918 79 23 for for ADP cana-918 79 24 each each DET cana-918 79 25 pair pair NOUN cana-918 79 26 ๐‘Ž ๐‘Ž NOUN cana-918 79 27 , , PUNCT cana-918 79 28 ๐‘ ๐‘ NOUN cana-918 79 29 with with ADP cana-918 79 30 ( ( PUNCT cana-918 79 31 ๐‘Ž ๐‘Ž NOUN cana-918 79 32 , , PUNCT cana-918 79 33 ๐‘ ๐‘ NOUN cana-918 79 34 ) ) PUNCT cana-918 79 35 โˆˆ โˆˆ PROPN cana-918 80 1 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 80 2 , , PUNCT cana-918 80 3 the the DET cana-918 80 4 number number NOUN cana-918 80 5 of of ADP cana-918 80 6 elements element NOUN cana-918 80 7 in in ADP cana-918 80 8 the the DET cana-918 80 9 set set NOUN cana-918 80 10 { { PUNCT cana-918 80 11 ๐‘ ๐‘ PROPN cana-918 80 12 โˆˆ โˆˆ PROPN cana-918 80 13 ๐‘Œ| ๐‘Œ| PROPN cana-918 80 14 ( ( PUNCT cana-918 80 15 ๐‘Ž ๐‘Ž PROPN cana-918 80 16 , , PUNCT cana-918 80 17 ๐‘ ๐‘ NOUN cana-918 80 18 ) ) PUNCT cana-918 80 19 โˆˆ โˆˆ PROPN cana-918 80 20 ๐’ฎ๐‘™ ๐’ฎ๐‘™ PROPN cana-918 80 21 , , PUNCT cana-918 80 22 ( ( PUNCT cana-918 80 23 ๐‘ ๐‘ NOUN cana-918 80 24 , , PUNCT cana-918 80 25 ๐‘ ๐‘ NOUN cana-918 80 26 ) ) PUNCT cana-918 80 27 โˆˆ โˆˆ NOUN cana-918 81 1 ๐’ฎ๐‘š ๐’ฎ๐‘š NOUN cana-918 81 2 } } PUNCT cana-918 81 3 is be AUX cana-918 81 4 invariant invariant ADJ cana-918 81 5 . . PUNCT cana-918 82 1 let let VERB cana-918 82 2 ( ( PUNCT cana-918 82 3 ๐‘Ž ๐‘Ž X cana-918 82 4 , , PUNCT cana-918 82 5 ๐‘ ๐‘ NOUN cana-918 82 6 ) ) PUNCT cana-918 82 7 โˆˆ โˆˆ PROPN cana-918 83 1 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 83 2 where where SCONJ cana-918 83 3 ๐‘Ž ๐‘Ž NOUN cana-918 83 4 = = PUNCT cana-918 83 5 ( ( PUNCT cana-918 83 6 ๐‘Ž1 ๐‘Ž1 PROPN cana-918 83 7 , , PUNCT cana-918 83 8 ๐‘Ž2 ๐‘Ž2 PROPN cana-918 83 9 , , PUNCT cana-918 83 10 โ€ฆ โ€ฆ PUNCT cana-918 83 11 , , PUNCT cana-918 83 12 ๐‘Ž๐‘Ÿ ๐‘Ž๐‘Ÿ NOUN cana-918 83 13 ) ) PUNCT cana-918 83 14 , , PUNCT cana-918 83 15 ๐‘ ๐‘ NOUN cana-918 83 16 = = PUNCT cana-918 83 17 ( ( PUNCT cana-918 83 18 ๐‘1 ๐‘1 NOUN cana-918 83 19 , , PUNCT cana-918 83 20 ๐‘2 ๐‘2 PROPN cana-918 83 21 , , PUNCT cana-918 83 22 โ€ฆ โ€ฆ PUNCT cana-918 83 23 , , PUNCT cana-918 83 24 ๐‘๐‘Ÿ ๐‘๐‘Ÿ NOUN cana-918 83 25 ) ) PUNCT cana-918 83 26 โˆˆ โˆˆ PROPN cana-918 83 27 ๐‘Œ. ๐‘Œ. PROPN cana-918 83 28 further far ADV cana-918 83 29 , , PUNCT cana-918 83 30 suppose suppose VERB cana-918 83 31 that that SCONJ cana-918 83 32 ๐‘Ž๐’ฎ๐‘™ ๐‘Ž๐’ฎ๐‘™ VERB cana-918 83 33 = = PRON cana-918 83 34 ๐‘Ž ๐‘Ž NOUN cana-918 83 35 โ€ฒ โ€ฒ NOUN cana-918 83 36 = = SYM cana-918 83 37 ( ( PUNCT cana-918 83 38 ๐‘Žโ€ฒ1 ๐‘Žโ€ฒ1 NOUN cana-918 83 39 , , PUNCT cana-918 83 40 ๐‘Žโ€ฒ2 ๐‘Žโ€ฒ2 ADJ cana-918 83 41 , , PUNCT cana-918 83 42 โ€ฆ โ€ฆ PUNCT cana-918 83 43 , , PUNCT cana-918 83 44 ๐‘Žโ€ฒ๐‘Ÿ ๐‘Žโ€ฒ๐‘Ÿ NUM cana-918 83 45 ) ) PUNCT cana-918 83 46 , , PUNCT cana-918 83 47 ๐‘๐’ฎ๐‘š ๐‘๐’ฎ๐‘š VERB cana-918 83 48 โˆ— โˆ— NOUN cana-918 83 49 = = SYM cana-918 83 50 ๐‘โ€ฒ ๐‘โ€ฒ X cana-918 83 51 = = SYM cana-918 83 52 ( ( PUNCT cana-918 83 53 ๐‘โ€ฒ1 ๐‘โ€ฒ1 INTJ cana-918 83 54 , , PUNCT cana-918 83 55 ๐‘โ€ฒ2 ๐‘โ€ฒ2 NOUN cana-918 83 56 , , PUNCT cana-918 83 57 โ€ฆ โ€ฆ PUNCT cana-918 83 58 , , PUNCT cana-918 83 59 ๐‘โ€ฒ๐‘Ÿ ๐‘โ€ฒ๐‘Ÿ NOUN cana-918 83 60 ) ) PUNCT cana-918 83 61 . . PUNCT cana-918 84 1 now now ADV cana-918 84 2 ( ( PUNCT cana-918 84 3 ๐‘Ž ๐‘Ž X cana-918 84 4 , , PUNCT cana-918 84 5 ๐‘ ๐‘ NOUN cana-918 84 6 ) ) PUNCT cana-918 84 7 โˆˆ โˆˆ PROPN cana-918 85 1 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 85 2 implies imply VERB cana-918 85 3 ๐‘๐‘  ๐‘๐‘  PROPN cana-918 85 4 โ‰ก โ‰ก PROPN cana-918 85 5 ( ( PUNCT cana-918 85 6 ๐‘˜ ๐‘˜ PROPN cana-918 85 7 + + SYM cana-918 85 8 ๐‘Ž๐‘  ๐‘Ž๐‘  NOUN cana-918 85 9 ) ) PUNCT cana-918 85 10 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 85 11 ๐‘›๐‘  ๐‘›๐‘  VERB cana-918 85 12 โˆ€ โˆ€ X cana-918 85 13 1 1 NUM cana-918 85 14 โ‰ค โ‰ค NUM cana-918 85 15 ๐‘  ๐‘  PRON cana-918 85 16 โ‰ค โ‰ค NUM cana-918 85 17 ๐‘Ÿ. ๐‘Ÿ. NOUN cana-918 85 18 similarly similarly ADV cana-918 85 19 , , PUNCT cana-918 85 20 ( ( PUNCT cana-918 86 1 ๐‘Ž ๐‘Ž X cana-918 86 2 , , PUNCT cana-918 86 3 ๐‘Žโ€ฒ ๐‘Žโ€ฒ NUM cana-918 86 4 ) ) PUNCT cana-918 86 5 โˆˆ โˆˆ PROPN cana-918 86 6 ๐’ฎ๐‘™ ๐’ฎ๐‘™ PROPN cana-918 86 7 and and CCONJ cana-918 86 8 ( ( PUNCT cana-918 86 9 ๐‘โ€ฒ ๐‘โ€ฒ PROPN cana-918 86 10 , , PUNCT cana-918 86 11 ๐‘ ๐‘ NOUN cana-918 86 12 ) ) PUNCT cana-918 86 13 โˆˆ โˆˆ PROPN cana-918 86 14 ๐’ฎ๐‘š. ๐’ฎ๐‘š. NOUN cana-918 86 15 we we PRON cana-918 86 16 get get VERB cana-918 86 17 ๐‘Žโ€ฒ๐‘  ๐‘Žโ€ฒ๐‘  NOUN cana-918 86 18 โ‰ก โ‰ก PROPN cana-918 86 19 ( ( PUNCT cana-918 86 20 ๐‘™ ๐‘™ PROPN cana-918 86 21 + + NUM cana-918 86 22 ๐‘Ž๐‘  ๐‘Ž๐‘  PROPN cana-918 86 23 ) ) PUNCT cana-918 86 24 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 86 25 ๐‘›๐‘  ๐‘›๐‘  VERB cana-918 86 26 and and CCONJ cana-918 86 27 ๐‘๐‘  ๐‘๐‘  PROPN cana-918 86 28 โ‰ก โ‰ก PROPN cana-918 86 29 ( ( PUNCT cana-918 86 30 ๐‘š ๐‘š PROPN cana-918 86 31 + + CCONJ cana-918 86 32 ๐‘โ€ฒ๐‘  ๐‘โ€ฒ๐‘  NOUN cana-918 86 33 ) ) PUNCT cana-918 86 34 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 86 35 ๐‘›๐‘  ๐‘›๐‘  VERB cana-918 86 36 โˆ€ โˆ€ X cana-918 86 37 1 1 NUM cana-918 86 38 โ‰ค โ‰ค NUM cana-918 86 39 ๐‘  ๐‘  PRON cana-918 86 40 โ‰ค โ‰ค NUM cana-918 86 41 ๐‘Ÿ. ๐‘Ÿ. NOUN cana-918 87 1 the the DET cana-918 87 2 above above ADJ cana-918 87 3 equations equation NOUN cana-918 87 4 give give VERB cana-918 87 5 that that PRON cana-918 87 6 , , PUNCT cana-918 87 7 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š PROPN cana-918 87 8 ๐‘˜ ๐‘˜ X cana-918 87 9 = = NOUN cana-918 87 10 1 1 NUM cana-918 87 11 if if SCONJ cana-918 87 12 and and CCONJ cana-918 87 13 only only ADV cana-918 87 14 if if SCONJ cana-918 87 15 ๐‘Žโ€ฒ ๐‘Žโ€ฒ NUM cana-918 87 16 = = SYM cana-918 87 17 ๐‘โ€ฒ. ๐‘โ€ฒ. NOUN cana-918 87 18 that that PRON cana-918 87 19 is be AUX cana-918 87 20 , , PUNCT cana-918 87 21 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š PROPN cana-918 87 22 ๐‘˜ ๐‘˜ X cana-918 87 23 = = NOUN cana-918 87 24 1 1 NUM cana-918 87 25 whenever whenever SCONJ cana-918 87 26 ๐‘˜ ๐‘˜ DET cana-918 87 27 โ‰ก โ‰ก PROPN cana-918 87 28 ( ( PUNCT cana-918 87 29 ๐‘™ ๐‘™ PROPN cana-918 87 30 + + NUM cana-918 87 31 ๐‘š ๐‘š NOUN cana-918 87 32 ) ) PUNCT cana-918 87 33 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 87 34 ๐‘›๐‘  ๐‘›๐‘  VERB cana-918 87 35 โˆ€ โˆ€ X cana-918 87 36 1 1 NUM cana-918 87 37 โ‰ค โ‰ค NUM cana-918 87 38 ๐‘  ๐‘  PRON cana-918 87 39 โ‰ค โ‰ค NUM cana-918 87 40 ๐‘Ÿ. ๐‘Ÿ. NOUN cana-918 87 41 hence hence ADV cana-918 87 42 , , PUNCT cana-918 87 43 ( ( PUNCT cana-918 87 44 ๐‘Œ ๐‘Œ PROPN cana-918 87 45 , , PUNCT cana-918 87 46 ๐’ซ ๐’ซ NOUN cana-918 87 47 ) ) PUNCT cana-918 87 48 is be AUX cana-918 87 49 a a DET cana-918 87 50 non non ADJ cana-918 87 51 - - ADJ cana-918 87 52 symmetric symmetric ADJ cana-918 87 53 and and CCONJ cana-918 87 54 commutative commutative ADJ cana-918 87 55 association association NOUN cana-918 87 56 scheme scheme NOUN cana-918 87 57 . . PUNCT cana-918 88 1 โ—ป โ—ป PROPN cana-918 88 2 theorem theorem VERB cana-918 88 3 3 3 X cana-918 88 4 . . PUNCT cana-918 89 1 let let VERB cana-918 89 2 ๐‘Œ ๐‘Œ PROPN cana-918 89 3 = = PRON cana-918 89 4 โ„ค๐‘›1 โ„ค๐‘›1 VERB cana-918 89 5 ร— ร— PROPN cana-918 89 6 โ„ค๐‘›2 โ„ค๐‘›2 NOUN cana-918 89 7 ร—โ‹ฏร— ร—โ‹ฏร— PROPN cana-918 89 8 โ„ค๐‘›๐‘Ÿ โ„ค๐‘›๐‘Ÿ PROPN cana-918 89 9 , , PUNCT cana-918 89 10 where where SCONJ cana-918 89 11 2 2 X cana-918 89 12 < < X cana-918 89 13 ๐‘›1 ๐‘›1 ADJ cana-918 89 14 โ‰ค โ‰ค PUNCT cana-918 89 15 ๐‘›2 ๐‘›2 NOUN cana-918 89 16 โ‰ค โ‰ค NOUN cana-918 89 17 โ‹ฏ โ‹ฏ NOUN cana-918 89 18 โ‰ค โ‰ค NOUN cana-918 89 19 ๐‘›๐‘Ÿ ๐‘›๐‘Ÿ ADP cana-918 89 20 and and CCONJ cana-918 89 21 ๐‘”๐‘๐‘‘(๐‘›1 ๐‘”๐‘๐‘‘(๐‘›1 PROPN cana-918 89 22 , , PUNCT cana-918 89 23 ๐‘›2 ๐‘›2 NOUN cana-918 89 24 , , PUNCT cana-918 89 25 โ€ฆ โ€ฆ PUNCT cana-918 89 26 , , PUNCT cana-918 89 27 ๐‘›๐‘Ÿ ๐‘›๐‘Ÿ PROPN cana-918 89 28 ) ) PUNCT cana-918 89 29 โ‰  โ‰  PROPN cana-918 89 30 1 1 NUM cana-918 89 31 . . PUNCT cana-918 90 1 the the DET cana-918 90 2 relations relation NOUN cana-918 90 3 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 90 4 on on ADP cana-918 90 5 ๐’ซ ๐’ซ NOUN cana-918 90 6 defined define VERB cana-918 90 7 by by ADP cana-918 90 8 communications communication NOUN cana-918 90 9 on on ADP cana-918 90 10 applied apply VERB cana-918 90 11 nonlinear nonlinear ADJ cana-918 90 12 analysis analysis NOUN cana-918 90 13 issn issn NOUN cana-918 90 14 : : PUNCT cana-918 90 15 1074 1074 NUM cana-918 90 16 - - PUNCT cana-918 90 17 133x 133x NUM cana-918 90 18 vol vol NOUN cana-918 90 19 31 31 NUM cana-918 90 20 no no NOUN cana-918 90 21 . . PUNCT cana-918 91 1 4s 4s NUM cana-918 91 2 ( ( PUNCT cana-918 91 3 2024 2024 NUM cana-918 91 4 ) ) PUNCT cana-918 91 5 395 395 NUM cana-918 91 6 https://internationalpubls.com https://internationalpubls.com X cana-918 92 1 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 92 2 = = SYM cana-918 92 3 { { PUNCT cana-918 92 4 ( ( PUNCT cana-918 92 5 ๐‘Ž ๐‘Ž X cana-918 92 6 , , PUNCT cana-918 92 7 ๐‘)|๐‘๐‘Ÿ ๐‘)|๐‘๐‘Ÿ PROPN cana-918 92 8 โ‰ก โ‰ก PROPN cana-918 92 9 ( ( PUNCT cana-918 92 10 ๐‘˜ ๐‘˜ X cana-918 92 11 + + PUNCT cana-918 92 12 ๐‘Ž๐‘Ÿ)๐‘š๐‘œ๐‘‘๐‘›๐‘Ÿ ๐‘Ž๐‘Ÿ)๐‘š๐‘œ๐‘‘๐‘›๐‘Ÿ NOUN cana-918 92 13 , , PUNCT cana-918 92 14 ๐‘๐‘  ๐‘๐‘  PROPN cana-918 92 15 โ‰ก โ‰ก PROPN cana-918 92 16 ( ( PUNCT cana-918 92 17 ๐‘˜ ๐‘˜ PROPN cana-918 92 18 + + CCONJ cana-918 92 19 ๐‘ก๐‘  ๐‘ก๐‘  ADJ cana-918 92 20 + + SYM cana-918 92 21 ๐‘Ž๐‘  ๐‘Ž๐‘  PROPN cana-918 92 22 ) ) PUNCT cana-918 92 23 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 92 24 ๐‘›๐‘  ๐‘›๐‘  VERB cana-918 92 25 โˆ€ โˆ€ X cana-918 92 26 1 1 NUM cana-918 92 27 โ‰ค โ‰ค NUM cana-918 92 28 ๐‘  ๐‘  ADP cana-918 92 29 < < X cana-918 92 30 ๐‘Ÿ| ๐‘Ÿ| PUNCT cana-918 92 31 ๐‘ก๐‘  ๐‘ก๐‘  ADP cana-918 92 32 โˆˆ โˆˆ PROPN cana-918 92 33 ๐‘๐‘›๐‘  ๐‘๐‘›๐‘  NOUN cana-918 92 34 , , PUNCT cana-918 92 35 ๐‘Ž ๐‘Ž NOUN cana-918 92 36 = = SYM cana-918 92 37 ( ( PUNCT cana-918 92 38 ๐‘Ž1 ๐‘Ž1 PROPN cana-918 92 39 , , PUNCT cana-918 92 40 ๐‘Ž2 ๐‘Ž2 PROPN cana-918 92 41 , , PUNCT cana-918 92 42 โ€ฆ โ€ฆ PUNCT cana-918 92 43 , , PUNCT cana-918 92 44 ๐‘Ž๐‘Ÿ ๐‘Ž๐‘Ÿ NOUN cana-918 92 45 ) ) PUNCT cana-918 92 46 , , PUNCT cana-918 92 47 ๐‘ ๐‘ NOUN cana-918 92 48 = = PUNCT cana-918 92 49 ( ( PUNCT cana-918 92 50 ๐‘1 ๐‘1 NOUN cana-918 92 51 , , PUNCT cana-918 92 52 ๐‘2 ๐‘2 PROPN cana-918 92 53 , , PUNCT cana-918 92 54 โ€ฆ โ€ฆ PUNCT cana-918 92 55 , , PUNCT cana-918 92 56 ๐‘๐‘Ÿ ๐‘๐‘Ÿ NOUN cana-918 92 57 ) ) PUNCT cana-918 92 58 โˆˆ โˆˆ PROPN cana-918 92 59 ๐‘Œ ๐‘Œ PROPN cana-918 92 60 } } PUNCT cana-918 92 61 where where SCONJ cana-918 92 62 ๐‘˜ ๐‘˜ X cana-918 92 63 = = PUNCT cana-918 92 64 ๐‘›2๐‘›3โ‹ฏ๐‘›๐‘Ÿ๐‘ก1 ๐‘›2๐‘›3โ‹ฏ๐‘›๐‘Ÿ๐‘ก1 NOUN cana-918 92 65 + + CCONJ cana-918 92 66 ๐‘›3๐‘›4โ‹ฏ๐‘›๐‘Ÿ๐‘ก2 ๐‘›3๐‘›4โ‹ฏ๐‘›๐‘Ÿ๐‘ก2 NOUN cana-918 92 67 + + NOUN cana-918 92 68 โ‹ฏ+ โ‹ฏ+ NOUN cana-918 92 69 ๐‘›๐‘Ÿ๐‘ก๐‘Ÿโˆ’1 ๐‘›๐‘Ÿ๐‘ก๐‘Ÿโˆ’1 VERB cana-918 92 70 + + CCONJ cana-918 92 71 ๐‘ก๐‘Ÿ ๐‘ก๐‘Ÿ ADJ cana-918 92 72 , , PUNCT cana-918 92 73 is be AUX cana-918 92 74 a a DET cana-918 92 75 non non ADJ cana-918 92 76 - - ADJ cana-918 92 77 symmetric symmetric ADJ cana-918 92 78 and and CCONJ cana-918 92 79 commutative commutative ADJ cana-918 92 80 association association NOUN cana-918 92 81 scheme scheme NOUN cana-918 92 82 with with ADP cana-918 92 83 parameters parameter NOUN cana-918 92 84 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š VERB cana-918 92 85 ๐‘˜ ๐‘˜ X cana-918 92 86 = = PUNCT cana-918 92 87 { { PUNCT cana-918 92 88 1 1 NUM cana-918 92 89 ๐‘–๐‘“๐‘˜ ๐‘–๐‘“๐‘˜ PROPN cana-918 92 90 โ‰ก โ‰ก PROPN cana-918 92 91 ( ( PUNCT cana-918 92 92 ๐‘™ ๐‘™ PROPN cana-918 92 93 + + NUM cana-918 92 94 ๐‘š ๐‘š NOUN cana-918 92 95 ) ) PUNCT cana-918 92 96 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 92 97 ๐‘›๐‘Ÿ ๐‘›๐‘Ÿ ADP cana-918 92 98 , , PUNCT cana-918 92 99 and and CCONJ cana-918 92 100 0 0 NUM cana-918 92 101 otherwise otherwise ADV cana-918 92 102 ๐‘˜ ๐‘˜ X cana-918 92 103 + + CCONJ cana-918 92 104 ๐‘ก๐‘  ๐‘ก๐‘  ADJ cana-918 92 105 ( ( PUNCT cana-918 92 106 ๐‘˜ ๐‘˜ NOUN cana-918 92 107 ) ) PUNCT cana-918 92 108 โ‰ก โ‰ก PROPN cana-918 92 109 ( ( PUNCT cana-918 92 110 ๐‘™ ๐‘™ PROPN cana-918 93 1 + + NUM cana-918 93 2 ๐‘š ๐‘š X cana-918 93 3 + + CCONJ cana-918 93 4 ๐‘ก๐‘  ๐‘ก๐‘  ADJ cana-918 93 5 ( ( PUNCT cana-918 93 6 ๐‘™ ๐‘™ NUM cana-918 93 7 ) ) PUNCT cana-918 93 8 + + X cana-918 93 9 ๐‘ก๐‘  ๐‘ก๐‘  ADJ cana-918 93 10 ( ( PUNCT cana-918 93 11 ๐‘š ๐‘š NOUN cana-918 93 12 ) ) PUNCT cana-918 93 13 ) ) PUNCT cana-918 93 14 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 93 15 ๐‘›๐‘  ๐‘›๐‘  VERB cana-918 93 16 โˆ€ โˆ€ X cana-918 93 17 1 1 NUM cana-918 93 18 โ‰ค โ‰ค NUM cana-918 93 19 ๐‘  ๐‘  X cana-918 93 20 < < X cana-918 93 21 ๐‘Ÿ ๐‘Ÿ X cana-918 93 22 where where SCONJ cana-918 93 23 ๐‘˜ ๐‘˜ NOUN cana-918 93 24 = = X cana-918 93 25 ๐‘›2๐‘›3โ‹ฏ๐‘›๐‘Ÿ๐‘ก1 ๐‘›2๐‘›3โ‹ฏ๐‘›๐‘Ÿ๐‘ก1 NOUN cana-918 93 26 ( ( PUNCT cana-918 93 27 ๐‘˜ ๐‘˜ NOUN cana-918 93 28 ) ) PUNCT cana-918 93 29 + + NUM cana-918 93 30 ๐‘›3๐‘›4โ‹ฏ๐‘›๐‘Ÿ๐‘ก2 ๐‘›3๐‘›4โ‹ฏ๐‘›๐‘Ÿ๐‘ก2 NOUN cana-918 93 31 ( ( PUNCT cana-918 93 32 ๐‘˜ ๐‘˜ NOUN cana-918 93 33 ) ) PUNCT cana-918 94 1 + + NOUN cana-918 94 2 โ‹ฏ+ โ‹ฏ+ NOUN cana-918 94 3 ๐‘›๐‘Ÿ๐‘‘๐‘Ÿโˆ’1 ๐‘›๐‘Ÿ๐‘‘๐‘Ÿโˆ’1 VERB cana-918 94 4 ( ( PUNCT cana-918 94 5 ๐‘˜ ๐‘˜ NOUN cana-918 94 6 ) ) PUNCT cana-918 94 7 + + CCONJ cana-918 94 8 ๐‘ก๐‘Ÿ ๐‘ก๐‘Ÿ VERB cana-918 94 9 ( ( PUNCT cana-918 94 10 ๐‘˜ ๐‘˜ NOUN cana-918 94 11 ) ) PUNCT cana-918 94 12 ; ; PUNCT cana-918 94 13 ๐‘™ ๐‘™ X cana-918 94 14 = = PUNCT cana-918 94 15 ๐‘›2๐‘›3โ‹ฏ๐‘›๐‘Ÿ๐‘ก1 ๐‘›2๐‘›3โ‹ฏ๐‘›๐‘Ÿ๐‘ก1 NOUN cana-918 94 16 ( ( PUNCT cana-918 94 17 ๐‘™ ๐‘™ NOUN cana-918 94 18 ) ) PUNCT cana-918 94 19 + + NUM cana-918 94 20 ๐‘›3๐‘›4โ‹ฏ๐‘›๐‘Ÿ๐‘ก2 ๐‘›3๐‘›4โ‹ฏ๐‘›๐‘Ÿ๐‘ก2 NOUN cana-918 94 21 ( ( PUNCT cana-918 94 22 ๐‘™ ๐‘™ X cana-918 94 23 ) ) PUNCT cana-918 94 24 + + NOUN cana-918 94 25 โ‹ฏ+ โ‹ฏ+ NOUN cana-918 94 26 ๐‘›๐‘Ÿ๐‘ก๐‘Ÿโˆ’1 ๐‘›๐‘Ÿ๐‘ก๐‘Ÿโˆ’1 VERB cana-918 94 27 ( ( PUNCT cana-918 94 28 ๐‘™ ๐‘™ NUM cana-918 94 29 ) ) PUNCT cana-918 94 30 + + CCONJ cana-918 94 31 ๐‘ก๐‘Ÿ ๐‘ก๐‘Ÿ VERB cana-918 94 32 ( ( PUNCT cana-918 94 33 ๐‘™ ๐‘™ NOUN cana-918 94 34 ) ) PUNCT cana-918 94 35 ; ; PUNCT cana-918 94 36 ๐‘š ๐‘š X cana-918 94 37 = = PUNCT cana-918 94 38 ๐‘›2๐‘›3โ‹ฏ๐‘›๐‘Ÿ๐‘ก1 ๐‘›2๐‘›3โ‹ฏ๐‘›๐‘Ÿ๐‘ก1 NOUN cana-918 94 39 ( ( PUNCT cana-918 94 40 ๐‘š ๐‘š NOUN cana-918 94 41 ) ) PUNCT cana-918 94 42 + + NUM cana-918 94 43 ๐‘›3๐‘›4โ‹ฏ๐‘›๐‘Ÿ๐‘ก2 ๐‘›3๐‘›4โ‹ฏ๐‘›๐‘Ÿ๐‘ก2 NOUN cana-918 94 44 ( ( PUNCT cana-918 94 45 ๐‘š ๐‘š NOUN cana-918 94 46 ) ) PUNCT cana-918 94 47 + + NOUN cana-918 94 48 โ‹ฏ+ โ‹ฏ+ NOUN cana-918 94 49 ๐‘›๐‘Ÿ๐‘ก๐‘Ÿโˆ’1 ๐‘›๐‘Ÿ๐‘ก๐‘Ÿโˆ’1 PRON cana-918 94 50 ( ( PUNCT cana-918 94 51 ๐‘š ๐‘š NOUN cana-918 94 52 ) ) PUNCT cana-918 94 53 + + CCONJ cana-918 94 54 ๐‘ก๐‘Ÿ ๐‘ก๐‘Ÿ VERB cana-918 94 55 ( ( PUNCT cana-918 94 56 ๐‘š ๐‘š NOUN cana-918 94 57 ) ) PUNCT cana-918 94 58 for for ADP cana-918 94 59 some some DET cana-918 94 60 ๐‘ก๐‘  ๐‘ก๐‘  ADJ cana-918 94 61 ( ( PUNCT cana-918 94 62 ๐‘˜ ๐‘˜ NOUN cana-918 94 63 ) ) PUNCT cana-918 94 64 , , PUNCT cana-918 94 65 ๐‘ก๐‘  ๐‘ก๐‘  X cana-918 94 66 ( ( PUNCT cana-918 94 67 ๐‘™ ๐‘™ NOUN cana-918 94 68 ) ) PUNCT cana-918 94 69 , , PUNCT cana-918 94 70 ๐‘ก๐‘  ๐‘ก๐‘  X cana-918 94 71 ( ( PUNCT cana-918 94 72 ๐‘š ๐‘š NOUN cana-918 94 73 ) ) PUNCT cana-918 94 74 โˆˆ โˆˆ PROPN cana-918 94 75 โ„ค๐‘›๐‘  โ„ค๐‘›๐‘  PROPN cana-918 94 76 where where SCONJ cana-918 94 77 1 1 NUM cana-918 94 78 โ‰ค โ‰ค NUM cana-918 94 79 ๐‘  ๐‘  PRON cana-918 94 80 โ‰ค โ‰ค NUM cana-918 94 81 ๐‘Ÿ. ๐‘Ÿ. NOUN cana-918 94 82 for for ADP cana-918 94 83 each each DET cana-918 94 84 0 0 NUM cana-918 94 85 โ‰ค โ‰ค NOUN cana-918 94 86 ๐‘˜ ๐‘˜ PRON cana-918 94 87 < < X cana-918 94 88 ๐‘›1๐‘›2 ๐‘›1๐‘›2 X cana-918 94 89 โ€ฆ โ€ฆ SYM cana-918 94 90 ๐‘›๐‘Ÿ ๐‘›๐‘Ÿ ADP cana-918 94 91 , , PUNCT cana-918 94 92 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 94 93 โˆ— โˆ— NOUN cana-918 94 94 = = PUNCT cana-918 94 95 ๐’ฎ๐พ ๐’ฎ๐พ NOUN cana-918 94 96 can can AUX cana-918 94 97 be be AUX cana-918 94 98 calculated calculate VERB cana-918 94 99 by by ADP cana-918 94 100 solving solve VERB cana-918 94 101 the the DET cana-918 94 102 following follow VERB cana-918 94 103 equations equation NOUN cana-918 94 104 : : PUNCT cana-918 94 105 ๐พ ๐พ NOUN cana-918 94 106 + + CCONJ cana-918 94 107 ๐‘˜ ๐‘˜ PROPN cana-918 94 108 โ‰ก โ‰ก PROPN cana-918 94 109 0 0 NUM cana-918 94 110 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 94 111 ๐‘›๐‘Ÿ ๐‘›๐‘Ÿ ADP cana-918 94 112 ๐พ ๐พ PROPN cana-918 94 113 + + CCONJ cana-918 94 114 ๐‘˜ ๐‘˜ X cana-918 94 115 + + PUNCT cana-918 94 116 ๐‘‡๐‘  ๐‘‡๐‘  NOUN cana-918 94 117 + + CCONJ cana-918 94 118 ๐‘ก๐‘  ๐‘ก๐‘  ADJ cana-918 94 119 โ‰ก โ‰ก PROPN cana-918 94 120 0 0 NUM cana-918 94 121 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 94 122 ๐‘›๐‘  ๐‘›๐‘  VERB cana-918 94 123 โˆ€ โˆ€ X cana-918 94 124 1 1 NUM cana-918 94 125 โ‰ค โ‰ค NUM cana-918 94 126 ๐‘  ๐‘  PRON cana-918 94 127 โ‰ค โ‰ค NOUN cana-918 95 1 ๐‘Ÿ ๐‘Ÿ DET cana-918 95 2 โˆ’ โˆ’ PROPN cana-918 95 3 1 1 NUM cana-918 95 4 where where SCONJ cana-918 95 5 ๐‘˜ ๐‘˜ NOUN cana-918 95 6 = = PUNCT cana-918 95 7 ๐‘›2๐‘›3โ‹ฏ๐‘›๐‘Ÿ๐‘ก1 ๐‘›2๐‘›3โ‹ฏ๐‘›๐‘Ÿ๐‘ก1 NOUN cana-918 95 8 + + CCONJ cana-918 95 9 ๐‘›3๐‘›4โ‹ฏ๐‘›๐‘Ÿ๐‘ก2 ๐‘›3๐‘›4โ‹ฏ๐‘›๐‘Ÿ๐‘ก2 NOUN cana-918 95 10 + + NOUN cana-918 95 11 โ‹ฏ+ โ‹ฏ+ NOUN cana-918 95 12 ๐‘›๐‘Ÿ๐‘ก๐‘Ÿโˆ’1 ๐‘›๐‘Ÿ๐‘ก๐‘Ÿโˆ’1 VERB cana-918 95 13 + + CCONJ cana-918 95 14 ๐‘ก๐‘Ÿ ๐‘ก๐‘Ÿ ADJ cana-918 95 15 and and CCONJ cana-918 95 16 ๐พ ๐พ NOUN cana-918 95 17 = = SYM cana-918 95 18 ๐‘›2๐‘›3โ‹ฏ๐‘›๐‘Ÿ๐‘‡1 ๐‘›2๐‘›3โ‹ฏ๐‘›๐‘Ÿ๐‘‡1 NOUN cana-918 95 19 + + NUM cana-918 95 20 ๐‘›3๐‘›4โ‹ฏ๐‘›๐‘Ÿ๐‘‡2 ๐‘›3๐‘›4โ‹ฏ๐‘›๐‘Ÿ๐‘‡2 NOUN cana-918 95 21 + + CCONJ cana-918 95 22 โ‹ฏ+ โ‹ฏ+ NOUN cana-918 95 23 ๐‘›๐‘Ÿ๐‘‡๐‘Ÿโˆ’1 ๐‘›๐‘Ÿ๐‘‡๐‘Ÿโˆ’1 PUNCT cana-918 96 1 + + CCONJ cana-918 96 2 ๐‘‡๐‘Ÿ ๐‘‡๐‘Ÿ PROPN cana-918 96 3 for for ADP cana-918 96 4 some some PRON cana-918 96 5 ๐‘ก๐‘  ๐‘ก๐‘  NOUN cana-918 96 6 , , PUNCT cana-918 96 7 ๐‘‡๐‘  ๐‘‡๐‘  PROPN cana-918 96 8 โˆˆ โˆˆ PROPN cana-918 96 9 โ„ค๐‘›๐‘ . โ„ค๐‘›๐‘ . PROPN cana-918 96 10 proof proof NOUN cana-918 96 11 . . PUNCT cana-918 97 1 again again ADV cana-918 97 2 recall recall VERB cana-918 97 3 , , PUNCT cana-918 97 4 |๐‘Œ| |๐‘Œ| NOUN cana-918 97 5 = = SYM cana-918 97 6 ๐‘›1๐‘›2โ‹ฏ๐‘›๐‘Ÿ ๐‘›1๐‘›2โ‹ฏ๐‘›๐‘Ÿ PROPN cana-918 97 7 = = PUNCT cana-918 97 8 |๐’ฎ๐‘˜| |๐’ฎ๐‘˜| ADV cana-918 97 9 for for ADP cana-918 97 10 all all PRON cana-918 97 11 0 0 NUM cana-918 97 12 โ‰ค โ‰ค NOUN cana-918 97 13 ๐‘˜ ๐‘˜ DET cana-918 97 14 < < X cana-918 97 15 ๐‘›1๐‘›2โ‹ฏ๐‘›๐‘Ÿ. ๐‘›1๐‘›2โ‹ฏ๐‘›๐‘Ÿ. CCONJ cana-918 97 16 above above ADV cana-918 97 17 defined define VERB cana-918 97 18 relations relation NOUN cana-918 98 1 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 98 2 being be AUX cana-918 98 3 disjoint disjoint ADJ cana-918 98 4 , , PUNCT cana-918 98 5 form form VERB cana-918 98 6 a a DET cana-918 98 7 partition partition NOUN cana-918 98 8 of of ADP cana-918 98 9 ๐’ซ. ๐’ซ. PROPN cana-918 98 10 for for ADP cana-918 98 11 arbitrary arbitrary ADJ cana-918 98 12 relations relation NOUN cana-918 98 13 ๐’ฎ๐‘™ ๐’ฎ๐‘™ PROPN cana-918 98 14 , , PUNCT cana-918 98 15 ๐’ฎ๐‘š ๐’ฎ๐‘š NOUN cana-918 98 16 , , PUNCT cana-918 98 17 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 98 18 in in ADP cana-918 98 19 ๐’ซ ๐’ซ NOUN cana-918 98 20 , , PUNCT cana-918 98 21 we we PRON cana-918 98 22 show show VERB cana-918 98 23 that that SCONJ cana-918 98 24 for for ADP cana-918 98 25 each each DET cana-918 98 26 ( ( PUNCT cana-918 98 27 ๐‘Ž ๐‘Ž PROPN cana-918 98 28 , , PUNCT cana-918 98 29 ๐‘ ๐‘ NOUN cana-918 98 30 ) ) PUNCT cana-918 98 31 โˆˆ โˆˆ PROPN cana-918 98 32 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 98 33 , , PUNCT cana-918 98 34 the the DET cana-918 98 35 number number NOUN cana-918 98 36 of of ADP cana-918 98 37 elements element NOUN cana-918 98 38 in in ADP cana-918 98 39 the the DET cana-918 98 40 set set NOUN cana-918 98 41 { { PUNCT cana-918 98 42 ๐‘ ๐‘ PROPN cana-918 98 43 โˆˆ โˆˆ PROPN cana-918 98 44 ๐‘Œ| ๐‘Œ| PROPN cana-918 98 45 ( ( PUNCT cana-918 98 46 ๐‘Ž ๐‘Ž PROPN cana-918 98 47 , , PUNCT cana-918 98 48 ๐‘ ๐‘ NOUN cana-918 98 49 ) ) PUNCT cana-918 98 50 โˆˆ โˆˆ PROPN cana-918 98 51 ๐’ฎ๐‘™ ๐’ฎ๐‘™ PROPN cana-918 98 52 , , PUNCT cana-918 98 53 ( ( PUNCT cana-918 98 54 ๐‘ ๐‘ NOUN cana-918 98 55 , , PUNCT cana-918 98 56 ๐‘ ๐‘ NOUN cana-918 98 57 ) ) PUNCT cana-918 98 58 โˆˆ โˆˆ NOUN cana-918 98 59 ๐’ฎ๐‘š ๐’ฎ๐‘š NOUN cana-918 98 60 } } PUNCT cana-918 98 61 is be AUX cana-918 98 62 invariant invariant ADJ cana-918 98 63 . . PUNCT cana-918 99 1 let let VERB cana-918 99 2 ( ( PUNCT cana-918 99 3 ๐‘Ž ๐‘Ž X cana-918 99 4 , , PUNCT cana-918 99 5 ๐‘ ๐‘ NOUN cana-918 99 6 ) ) PUNCT cana-918 99 7 โˆˆ โˆˆ PROPN cana-918 100 1 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 100 2 where where SCONJ cana-918 100 3 ๐‘Ž ๐‘Ž NOUN cana-918 100 4 = = PUNCT cana-918 100 5 ( ( PUNCT cana-918 100 6 ๐‘Ž1 ๐‘Ž1 PROPN cana-918 100 7 , , PUNCT cana-918 100 8 ๐‘Ž2 ๐‘Ž2 PROPN cana-918 100 9 , , PUNCT cana-918 100 10 โ€ฆ โ€ฆ PUNCT cana-918 100 11 , , PUNCT cana-918 100 12 ๐‘Ž๐‘Ÿ ๐‘Ž๐‘Ÿ NOUN cana-918 100 13 ) ) PUNCT cana-918 100 14 , , PUNCT cana-918 100 15 ๐‘ ๐‘ NOUN cana-918 100 16 = = PUNCT cana-918 100 17 ( ( PUNCT cana-918 100 18 ๐‘1 ๐‘1 NOUN cana-918 100 19 , , PUNCT cana-918 100 20 ๐‘2 ๐‘2 PROPN cana-918 100 21 , , PUNCT cana-918 100 22 โ€ฆ โ€ฆ PUNCT cana-918 100 23 , , PUNCT cana-918 100 24 ๐‘๐‘Ÿ ๐‘๐‘Ÿ NOUN cana-918 100 25 ) ) PUNCT cana-918 100 26 โˆˆ โˆˆ PROPN cana-918 100 27 ๐‘Œ. ๐‘Œ. PROPN cana-918 100 28 suppose suppose VERB cana-918 100 29 ๐‘Ž๐’ฎ๐‘™ ๐‘Ž๐’ฎ๐‘™ PROPN cana-918 100 30 = = PRON cana-918 100 31 ๐‘Ž ๐‘Ž NOUN cana-918 100 32 โ€ฒ โ€ฒ NOUN cana-918 100 33 = = SYM cana-918 100 34 ( ( PUNCT cana-918 100 35 ๐‘Žโ€ฒ1 ๐‘Žโ€ฒ1 NOUN cana-918 100 36 , , PUNCT cana-918 100 37 ๐‘Žโ€ฒ2 ๐‘Žโ€ฒ2 ADJ cana-918 100 38 , , PUNCT cana-918 100 39 โ€ฆ โ€ฆ PUNCT cana-918 100 40 , , PUNCT cana-918 100 41 ๐‘Žโ€ฒ๐‘Ÿ ๐‘Žโ€ฒ๐‘Ÿ NUM cana-918 100 42 ) ) PUNCT cana-918 100 43 , , PUNCT cana-918 100 44 ๐‘๐’ฎ๐‘š ๐‘๐’ฎ๐‘š VERB cana-918 100 45 โˆ— โˆ— NOUN cana-918 100 46 = = SYM cana-918 100 47 ๐‘โ€ฒ ๐‘โ€ฒ X cana-918 100 48 = = SYM cana-918 100 49 ( ( PUNCT cana-918 100 50 ๐‘โ€ฒ1 ๐‘โ€ฒ1 INTJ cana-918 100 51 , , PUNCT cana-918 100 52 ๐‘โ€ฒ2 ๐‘โ€ฒ2 NOUN cana-918 100 53 , , PUNCT cana-918 100 54 โ€ฆ โ€ฆ PUNCT cana-918 100 55 , , PUNCT cana-918 100 56 ๐‘โ€ฒ๐‘Ÿ ๐‘โ€ฒ๐‘Ÿ NOUN cana-918 100 57 ) ) PUNCT cana-918 100 58 . . PUNCT cana-918 101 1 then then ADV cana-918 101 2 ( ( PUNCT cana-918 101 3 ๐‘Ž ๐‘Ž X cana-918 101 4 , , PUNCT cana-918 101 5 ๐‘ ๐‘ NOUN cana-918 101 6 ) ) PUNCT cana-918 101 7 โˆˆ โˆˆ PROPN cana-918 102 1 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 102 2 implies imply VERB cana-918 102 3 ๐‘๐‘Ÿ ๐‘๐‘Ÿ ADP cana-918 102 4 โ‰ก โ‰ก PROPN cana-918 102 5 ( ( PUNCT cana-918 102 6 ๐‘˜ ๐‘˜ PROPN cana-918 102 7 + + NOUN cana-918 102 8 ๐‘Ž๐‘Ÿ ๐‘Ž๐‘Ÿ NOUN cana-918 102 9 ) ) PUNCT cana-918 102 10 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 102 11 ๐‘›๐‘Ÿ ๐‘›๐‘Ÿ ADP cana-918 102 12 , , PUNCT cana-918 102 13 ๐‘๐‘  ๐‘๐‘  PROPN cana-918 102 14 โ‰ก โ‰ก PROPN cana-918 102 15 ( ( PUNCT cana-918 102 16 ๐‘˜ ๐‘˜ PROPN cana-918 102 17 + + CCONJ cana-918 102 18 ๐‘ก๐‘  ๐‘ก๐‘  ADJ cana-918 102 19 ( ( PUNCT cana-918 102 20 ๐‘˜ ๐‘˜ NOUN cana-918 102 21 ) ) PUNCT cana-918 102 22 + + CCONJ cana-918 102 23 ๐‘Ž๐‘  ๐‘Ž๐‘  PROPN cana-918 102 24 ) ) PUNCT cana-918 102 25 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 102 26 ๐‘›๐‘  ๐‘›๐‘  VERB cana-918 102 27 โˆ€ โˆ€ X cana-918 102 28 1 1 NUM cana-918 102 29 โ‰ค โ‰ค NUM cana-918 102 30 ๐‘  ๐‘  PRON cana-918 102 31 โ‰ค โ‰ค NOUN cana-918 102 32 ๐‘Ÿ ๐‘Ÿ PRON cana-918 102 33 โˆ’ โˆ’ PROPN cana-918 102 34 1 1 NUM cana-918 102 35 where where SCONJ cana-918 102 36 ๐‘˜ ๐‘˜ NOUN cana-918 102 37 = = X cana-918 102 38 ๐‘›2๐‘›3โ‹ฏ๐‘›๐‘Ÿ๐‘ก1 ๐‘›2๐‘›3โ‹ฏ๐‘›๐‘Ÿ๐‘ก1 NOUN cana-918 102 39 ( ( PUNCT cana-918 102 40 ๐‘˜ ๐‘˜ NOUN cana-918 102 41 ) ) PUNCT cana-918 102 42 + + NUM cana-918 102 43 ๐‘›3๐‘›4โ‹ฏ๐‘›๐‘Ÿ๐‘ก2 ๐‘›3๐‘›4โ‹ฏ๐‘›๐‘Ÿ๐‘ก2 NOUN cana-918 102 44 ( ( PUNCT cana-918 102 45 ๐‘˜ ๐‘˜ NOUN cana-918 102 46 ) ) PUNCT cana-918 102 47 + + NOUN cana-918 102 48 โ‹ฏ+ โ‹ฏ+ NOUN cana-918 102 49 ๐‘›๐‘Ÿ๐‘ก๐‘Ÿโˆ’1 ๐‘›๐‘Ÿ๐‘ก๐‘Ÿโˆ’1 NOUN cana-918 102 50 ( ( PUNCT cana-918 102 51 ๐‘˜ ๐‘˜ NOUN cana-918 102 52 ) ) PUNCT cana-918 103 1 + + CCONJ cana-918 103 2 ๐‘ก๐‘Ÿ ๐‘ก๐‘Ÿ VERB cana-918 103 3 ( ( PUNCT cana-918 103 4 ๐‘˜ ๐‘˜ NOUN cana-918 103 5 ) ) PUNCT cana-918 103 6 for for ADP cana-918 103 7 some some DET cana-918 103 8 ๐‘ก๐‘  ๐‘ก๐‘  NOUN cana-918 103 9 โˆˆ โˆˆ PROPN cana-918 103 10 ๐‘๐‘›๐‘  ๐‘๐‘›๐‘  NOUN cana-918 103 11 โˆ€ โˆ€ NOUN cana-918 103 12 1 1 NUM cana-918 103 13 โ‰ค โ‰ค NUM cana-918 103 14 ๐‘  ๐‘  PRON cana-918 103 15 โ‰ค โ‰ค NUM cana-918 103 16 ๐‘Ÿ. ๐‘Ÿ. NOUN cana-918 103 17 since since SCONJ cana-918 103 18 ( ( PUNCT cana-918 103 19 ๐‘Ž ๐‘Ž X cana-918 103 20 , , PUNCT cana-918 103 21 ๐‘Žโ€ฒ ๐‘Žโ€ฒ NUM cana-918 103 22 ) ) PUNCT cana-918 103 23 โˆˆ โˆˆ PROPN cana-918 103 24 ๐’ฎ๐‘™ ๐’ฎ๐‘™ PROPN cana-918 103 25 and and CCONJ cana-918 103 26 ( ( PUNCT cana-918 103 27 ๐‘โ€ฒ ๐‘โ€ฒ PROPN cana-918 103 28 , , PUNCT cana-918 103 29 ๐‘ ๐‘ NOUN cana-918 103 30 ) ) PUNCT cana-918 103 31 โˆˆ โˆˆ PROPN cana-918 103 32 ๐‘†๐‘š ๐‘†๐‘š PROPN cana-918 103 33 , , PUNCT cana-918 103 34 we we PRON cana-918 103 35 have have VERB cana-918 103 36 ๐‘Žโ€ฒ๐‘Ÿ ๐‘Žโ€ฒ๐‘Ÿ PROPN cana-918 103 37 โ‰ก โ‰ก PROPN cana-918 103 38 ( ( PUNCT cana-918 103 39 ๐‘™ ๐‘™ PROPN cana-918 103 40 + + NUM cana-918 103 41 ๐‘Ž๐‘Ÿ ๐‘Ž๐‘Ÿ NOUN cana-918 103 42 ) ) PUNCT cana-918 103 43 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 103 44 ๐‘›๐‘Ÿ ๐‘›๐‘Ÿ ADP cana-918 103 45 , , PUNCT cana-918 103 46 ๐‘Žโ€ฒ๐‘  ๐‘Žโ€ฒ๐‘  PROPN cana-918 103 47 โ‰ก โ‰ก PROPN cana-918 103 48 ( ( PUNCT cana-918 103 49 ๐‘™ ๐‘™ X cana-918 103 50 + + CCONJ cana-918 103 51 ๐‘ก๐‘  ๐‘ก๐‘  ADJ cana-918 103 52 ( ( PUNCT cana-918 103 53 ๐‘™ ๐‘™ NOUN cana-918 103 54 ) ) PUNCT cana-918 104 1 + + CCONJ cana-918 104 2 ๐‘Ž๐‘  ๐‘Ž๐‘  PROPN cana-918 104 3 ) ) PUNCT cana-918 104 4 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 104 5 ๐‘›๐‘  ๐‘›๐‘  VERB cana-918 104 6 โˆ€ โˆ€ X cana-918 104 7 1 1 NUM cana-918 104 8 โ‰ค โ‰ค NUM cana-918 104 9 ๐‘  ๐‘  PRON cana-918 104 10 โ‰ค โ‰ค NOUN cana-918 105 1 ๐‘Ÿ ๐‘Ÿ PRON cana-918 105 2 โˆ’ โˆ’ PROPN cana-918 105 3 1 1 NUM cana-918 105 4 where where SCONJ cana-918 105 5 ๐‘™ ๐‘™ NOUN cana-918 105 6 = = SYM cana-918 105 7 ๐‘›2๐‘›3โ‹ฏ๐‘›๐‘Ÿ๐‘ก1 ๐‘›2๐‘›3โ‹ฏ๐‘›๐‘Ÿ๐‘ก1 NOUN cana-918 105 8 ( ( PUNCT cana-918 105 9 ๐‘™ ๐‘™ NOUN cana-918 105 10 ) ) PUNCT cana-918 105 11 + + NUM cana-918 105 12 ๐‘›3๐‘›4โ‹ฏ๐‘›๐‘Ÿ๐‘ก2 ๐‘›3๐‘›4โ‹ฏ๐‘›๐‘Ÿ๐‘ก2 NOUN cana-918 105 13 ( ( PUNCT cana-918 105 14 ๐‘™ ๐‘™ X cana-918 105 15 ) ) PUNCT cana-918 105 16 + + NOUN cana-918 105 17 โ‹ฏ+ โ‹ฏ+ NOUN cana-918 105 18 ๐‘›๐‘Ÿ๐‘ก๐‘Ÿโˆ’1 ๐‘›๐‘Ÿ๐‘ก๐‘Ÿโˆ’1 VERB cana-918 105 19 ( ( PUNCT cana-918 105 20 ๐‘™ ๐‘™ NUM cana-918 105 21 ) ) PUNCT cana-918 105 22 + + CCONJ cana-918 105 23 ๐‘ก๐‘Ÿ ๐‘ก๐‘Ÿ VERB cana-918 105 24 ( ( PUNCT cana-918 105 25 ๐‘™ ๐‘™ NOUN cana-918 105 26 ) ) PUNCT cana-918 105 27 for for ADP cana-918 105 28 some some DET cana-918 105 29 ๐‘ก๐‘  ๐‘ก๐‘  ADJ cana-918 105 30 ( ( PUNCT cana-918 105 31 ๐‘™ ๐‘™ NOUN cana-918 105 32 ) ) PUNCT cana-918 105 33 โˆˆ โˆˆ NOUN cana-918 105 34 โ„ค๐‘›๐‘ โˆ€1 โ„ค๐‘›๐‘ โˆ€1 VERB cana-918 105 35 โ‰ค โ‰ค NUM cana-918 105 36 ๐‘  ๐‘  PRON cana-918 105 37 โ‰ค โ‰ค ADJ cana-918 105 38 ๐‘Ÿ ๐‘Ÿ NOUN cana-918 105 39 , , PUNCT cana-918 105 40 and and CCONJ cana-918 105 41 ๐‘๐‘Ÿ ๐‘๐‘Ÿ ADP cana-918 105 42 โ‰ก โ‰ก PROPN cana-918 105 43 ( ( PUNCT cana-918 105 44 ๐‘š ๐‘š PROPN cana-918 105 45 + + NOUN cana-918 105 46 ๐‘โ€ฒ๐‘Ÿ ๐‘โ€ฒ๐‘Ÿ X cana-918 105 47 ) ) PUNCT cana-918 105 48 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 105 49 ๐‘›๐‘Ÿ ๐‘›๐‘Ÿ ADP cana-918 105 50 , , PUNCT cana-918 105 51 ๐‘๐‘  ๐‘๐‘  PROPN cana-918 105 52 โ‰ก โ‰ก PROPN cana-918 105 53 ( ( PUNCT cana-918 105 54 ๐‘š ๐‘š PROPN cana-918 105 55 + + CCONJ cana-918 105 56 ๐‘ก๐‘  ๐‘ก๐‘  ADJ cana-918 105 57 ( ( PUNCT cana-918 105 58 ๐‘š ๐‘š NOUN cana-918 105 59 ) ) PUNCT cana-918 105 60 + + CCONJ cana-918 105 61 ๐‘โ€ฒ๐‘  ๐‘โ€ฒ๐‘  NOUN cana-918 105 62 ) ) PUNCT cana-918 105 63 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 105 64 ๐‘›๐‘  ๐‘›๐‘  VERB cana-918 105 65 โˆ€ โˆ€ X cana-918 105 66 1 1 NUM cana-918 105 67 โ‰ค โ‰ค NUM cana-918 105 68 ๐‘  ๐‘  PRON cana-918 105 69 โ‰ค โ‰ค NOUN cana-918 106 1 ๐‘Ÿ ๐‘Ÿ PRON cana-918 106 2 โˆ’ โˆ’ PROPN cana-918 106 3 1 1 NUM cana-918 106 4 where where SCONJ cana-918 106 5 ๐‘š ๐‘š NOUN cana-918 106 6 = = SYM cana-918 106 7 ๐‘›2๐‘›3โ‹ฏ๐‘›๐‘Ÿ๐‘ก1 ๐‘›2๐‘›3โ‹ฏ๐‘›๐‘Ÿ๐‘ก1 NOUN cana-918 106 8 ( ( PUNCT cana-918 106 9 ๐‘š ๐‘š NOUN cana-918 106 10 ) ) PUNCT cana-918 106 11 + + NUM cana-918 106 12 ๐‘›3๐‘›4โ‹ฏ๐‘›๐‘Ÿ๐‘ก2 ๐‘›3๐‘›4โ‹ฏ๐‘›๐‘Ÿ๐‘ก2 NOUN cana-918 106 13 ( ( PUNCT cana-918 106 14 ๐‘š ๐‘š NOUN cana-918 106 15 ) ) PUNCT cana-918 106 16 + + NOUN cana-918 106 17 โ‹ฏ+ โ‹ฏ+ NOUN cana-918 106 18 ๐‘›๐‘Ÿ๐‘ก๐‘Ÿโˆ’1 ๐‘›๐‘Ÿ๐‘ก๐‘Ÿโˆ’1 PRON cana-918 106 19 ( ( PUNCT cana-918 106 20 ๐‘š ๐‘š NOUN cana-918 106 21 ) ) PUNCT cana-918 106 22 + + CCONJ cana-918 106 23 ๐‘ก๐‘Ÿ ๐‘ก๐‘Ÿ VERB cana-918 106 24 ( ( PUNCT cana-918 106 25 ๐‘š ๐‘š NOUN cana-918 106 26 ) ) PUNCT cana-918 106 27 for for ADP cana-918 106 28 some some DET cana-918 106 29 ๐‘ก๐‘  ๐‘ก๐‘  ADJ cana-918 106 30 ( ( PUNCT cana-918 106 31 ๐‘š ๐‘š NOUN cana-918 106 32 ) ) PUNCT cana-918 106 33 โˆˆ โˆˆ NOUN cana-918 106 34 โ„ค๐‘›๐‘  โ„ค๐‘›๐‘  PROPN cana-918 106 35 โˆ€ โˆ€ NOUN cana-918 106 36 1 1 NUM cana-918 106 37 โ‰ค โ‰ค NUM cana-918 106 38 ๐‘  ๐‘  PRON cana-918 106 39 โ‰ค โ‰ค NUM cana-918 106 40 ๐‘Ÿ. ๐‘Ÿ. NOUN cana-918 106 41 we we PRON cana-918 106 42 observe observe VERB cana-918 106 43 that that SCONJ cana-918 106 44 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š PROPN cana-918 106 45 ๐‘˜ ๐‘˜ NOUN cana-918 106 46 = = NOUN cana-918 106 47 1 1 NUM cana-918 106 48 if if SCONJ cana-918 106 49 and and CCONJ cana-918 106 50 only only ADV cana-918 106 51 if if SCONJ cana-918 106 52 ๐‘Žโ€ฒ ๐‘Žโ€ฒ NUM cana-918 106 53 = = SYM cana-918 106 54 ๐‘โ€ฒ. ๐‘โ€ฒ. NOUN cana-918 106 55 that that PRON cana-918 106 56 is be AUX cana-918 106 57 , , PUNCT cana-918 106 58 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š PROPN cana-918 106 59 ๐‘˜ ๐‘˜ X cana-918 106 60 = = NOUN cana-918 106 61 1 1 NUM cana-918 106 62 if if SCONJ cana-918 106 63 ๐‘˜ ๐‘˜ PRON cana-918 106 64 โ‰ก โ‰ก PROPN cana-918 106 65 ( ( PUNCT cana-918 106 66 ๐‘™ ๐‘™ PROPN cana-918 106 67 + + NUM cana-918 106 68 ๐‘š ๐‘š NOUN cana-918 106 69 ) ) PUNCT cana-918 106 70 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 106 71 ๐‘›๐‘Ÿ ๐‘›๐‘Ÿ ADP cana-918 106 72 and and CCONJ cana-918 106 73 ๐‘˜ ๐‘˜ PROPN cana-918 107 1 + + CCONJ cana-918 107 2 ๐‘ก๐‘  ๐‘ก๐‘  INTJ cana-918 107 3 ( ( PUNCT cana-918 107 4 ๐‘˜ ๐‘˜ NOUN cana-918 107 5 ) ) PUNCT cana-918 107 6 โ‰ก โ‰ก PROPN cana-918 107 7 ( ( PUNCT cana-918 107 8 ๐‘™ ๐‘™ PROPN cana-918 107 9 + + NUM cana-918 107 10 ๐‘š ๐‘š X cana-918 107 11 + + CCONJ cana-918 107 12 ๐‘ก๐‘  ๐‘ก๐‘  ADJ cana-918 107 13 ( ( PUNCT cana-918 107 14 ๐‘™ ๐‘™ NUM cana-918 107 15 ) ) PUNCT cana-918 107 16 + + X cana-918 107 17 ๐‘ก๐‘  ๐‘ก๐‘  ADJ cana-918 107 18 ( ( PUNCT cana-918 107 19 ๐‘š ๐‘š NOUN cana-918 107 20 ) ) PUNCT cana-918 107 21 ) ) PUNCT cana-918 107 22 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 107 23 ๐‘›๐‘  ๐‘›๐‘  VERB cana-918 107 24 โˆ€ โˆ€ X cana-918 107 25 1 1 NUM cana-918 107 26 โ‰ค โ‰ค NUM cana-918 107 27 ๐‘  ๐‘  PRON cana-918 107 28 โ‰ค โ‰ค NOUN cana-918 108 1 ๐‘Ÿ ๐‘Ÿ DET cana-918 108 2 โˆ’ โˆ’ PROPN cana-918 108 3 1 1 NUM cana-918 108 4 hence hence ADV cana-918 108 5 , , PUNCT cana-918 108 6 ( ( PUNCT cana-918 108 7 ๐‘Œ ๐‘Œ PROPN cana-918 108 8 , , PUNCT cana-918 108 9 ๐’ซ ๐’ซ NOUN cana-918 108 10 ) ) PUNCT cana-918 108 11 is be AUX cana-918 108 12 a a DET cana-918 108 13 non non ADJ cana-918 108 14 - - ADJ cana-918 108 15 symmetric symmetric ADJ cana-918 108 16 and and CCONJ cana-918 108 17 commutative commutative ADJ cana-918 108 18 association association NOUN cana-918 108 19 scheme scheme NOUN cana-918 108 20 . . PUNCT cana-918 109 1 next next ADV cana-918 109 2 , , PUNCT cana-918 109 3 we we PRON cana-918 109 4 find find VERB cana-918 109 5 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 109 6 โˆ—. โˆ—. NOUN cana-918 109 7 let let VERB cana-918 109 8 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 109 9 โˆ— โˆ— NOUN cana-918 109 10 = = PUNCT cana-918 109 11 ๐’ฎ๐พ ๐’ฎ๐พ NOUN cana-918 109 12 where where SCONJ cana-918 109 13 ๐พ ๐พ NOUN cana-918 109 14 = = SYM cana-918 109 15 ๐‘›2๐‘›3โ‹ฏ๐‘›๐‘Ÿ๐‘‡1 ๐‘›2๐‘›3โ‹ฏ๐‘›๐‘Ÿ๐‘‡1 NOUN cana-918 109 16 + + CCONJ cana-918 109 17 ๐‘›3๐‘›4โ‹ฏ๐‘›๐‘Ÿ๐‘‡2 ๐‘›3๐‘›4โ‹ฏ๐‘›๐‘Ÿ๐‘‡2 NOUN cana-918 109 18 + + NOUN cana-918 109 19 โ‹ฏ+ โ‹ฏ+ NOUN cana-918 109 20 ๐‘›๐‘Ÿ๐‘‡๐‘Ÿโˆ’1 ๐‘›๐‘Ÿ๐‘‡๐‘Ÿโˆ’1 PUNCT cana-918 110 1 + + CCONJ cana-918 110 2 ๐‘‡๐‘Ÿ ๐‘‡๐‘Ÿ PROPN cana-918 110 3 for for ADP cana-918 110 4 some some DET cana-918 110 5 ๐‘‡๐‘  ๐‘‡๐‘  PROPN cana-918 110 6 โˆˆ โˆˆ PROPN cana-918 110 7 โ„ค๐‘›๐‘ . โ„ค๐‘›๐‘ . PROPN cana-918 110 8 if if SCONJ cana-918 110 9 ( ( PUNCT cana-918 110 10 ๐‘ ๐‘ NOUN cana-918 110 11 , , PUNCT cana-918 110 12 ๐‘Ž ๐‘Ž NOUN cana-918 110 13 ) ) PUNCT cana-918 110 14 โˆˆ โˆˆ PROPN cana-918 110 15 ๐’ฎ๐พ ๐’ฎ๐พ PROPN cana-918 110 16 , , PUNCT cana-918 110 17 then then ADV cana-918 110 18 ๐‘Ž๐‘Ÿ ๐‘Ž๐‘Ÿ PRON cana-918 110 19 โ‰ก โ‰ก PROPN cana-918 110 20 ๐พ ๐พ PROPN cana-918 110 21 + + CCONJ cana-918 110 22 ๐‘๐‘Ÿ ๐‘๐‘Ÿ PROPN cana-918 110 23 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ PROPN cana-918 110 24 ๐‘›๐‘Ÿ ๐‘›๐‘Ÿ ADP cana-918 110 25 ; ; PUNCT cana-918 110 26 ๐‘Ž๐‘  ๐‘Ž๐‘  PROPN cana-918 110 27 โ‰ก โ‰ก PROPN cana-918 110 28 ๐พ ๐พ PROPN cana-918 110 29 + + CCONJ cana-918 110 30 ๐‘‡๐‘  ๐‘‡๐‘  PROPN cana-918 110 31 + + CCONJ cana-918 110 32 ๐‘๐‘  ๐‘๐‘  NOUN cana-918 110 33 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 110 34 ๐‘›๐‘  ๐‘›๐‘  VERB cana-918 110 35 โˆ€ โˆ€ X cana-918 110 36 1 1 NUM cana-918 110 37 โ‰ค โ‰ค NUM cana-918 110 38 ๐‘  ๐‘  PRON cana-918 110 39 โ‰ค โ‰ค NOUN cana-918 110 40 ๐‘Ÿ ๐‘Ÿ PRON cana-918 110 41 โˆ’ โˆ’ PROPN cana-918 110 42 1 1 X cana-918 110 43 . . PUNCT cana-918 111 1 also also ADV cana-918 111 2 , , PUNCT cana-918 111 3 since since SCONJ cana-918 111 4 ( ( PUNCT cana-918 111 5 ๐‘Ž ๐‘Ž X cana-918 111 6 , , PUNCT cana-918 111 7 ๐‘ ๐‘ NOUN cana-918 111 8 ) ) PUNCT cana-918 111 9 โˆˆ โˆˆ PROPN cana-918 111 10 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 111 11 , , PUNCT cana-918 111 12 we we PRON cana-918 111 13 get get VERB cana-918 111 14 ๐‘๐‘Ÿ ๐‘๐‘Ÿ ADP cana-918 111 15 โ‰ก โ‰ก PROPN cana-918 111 16 ๐‘˜ ๐‘˜ PROPN cana-918 112 1 + + CCONJ cana-918 112 2 ๐‘Ž๐‘Ÿ ๐‘Ž๐‘Ÿ DET cana-918 112 3 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 112 4 ๐‘›๐‘Ÿ ๐‘›๐‘Ÿ ADP cana-918 112 5 ; ; PUNCT cana-918 112 6 ๐‘๐‘  ๐‘๐‘  PROPN cana-918 112 7 โ‰ก โ‰ก PROPN cana-918 112 8 ๐‘˜ ๐‘˜ PROPN cana-918 113 1 + + CCONJ cana-918 113 2 ๐‘ก๐‘  ๐‘ก๐‘  ADJ cana-918 113 3 + + CCONJ cana-918 113 4 ๐‘Ž๐‘  ๐‘Ž๐‘  NOUN cana-918 113 5 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 113 6 ๐‘›๐‘  ๐‘›๐‘  VERB cana-918 113 7 โˆ€ โˆ€ X cana-918 113 8 1 1 NUM cana-918 113 9 โ‰ค โ‰ค NUM cana-918 113 10 ๐‘  ๐‘  PRON cana-918 113 11 โ‰ค โ‰ค NOUN cana-918 113 12 ๐‘Ÿ ๐‘Ÿ NOUN cana-918 113 13 โˆ’ โˆ’ PROPN cana-918 113 14 1 1 X cana-918 113 15 . . PUNCT cana-918 113 16 communications communication NOUN cana-918 113 17 on on ADP cana-918 113 18 applied apply VERB cana-918 113 19 nonlinear nonlinear ADJ cana-918 113 20 analysis analysis NOUN cana-918 113 21 issn issn NOUN cana-918 113 22 : : PUNCT cana-918 113 23 1074 1074 NUM cana-918 113 24 - - PUNCT cana-918 113 25 133x 133x NUM cana-918 113 26 vol vol NOUN cana-918 113 27 31 31 NUM cana-918 113 28 no no NOUN cana-918 113 29 . . PUNCT cana-918 114 1 4s 4s NUM cana-918 114 2 ( ( PUNCT cana-918 114 3 2024 2024 NUM cana-918 114 4 ) ) PUNCT cana-918 114 5 396 396 NUM cana-918 114 6 https://internationalpubls.com https://internationalpubls.com X cana-918 114 7 this this PRON cana-918 114 8 implies imply VERB cana-918 114 9 that that SCONJ cana-918 114 10 ๐พ ๐พ PROPN cana-918 114 11 + + PROPN cana-918 114 12 ๐‘˜ ๐‘˜ PROPN cana-918 114 13 โ‰ก โ‰ก PROPN cana-918 114 14 0 0 NUM cana-918 114 15 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ PROPN cana-918 114 16 ๐‘›๐‘Ÿ ๐‘›๐‘Ÿ ADP cana-918 114 17 and and CCONJ cana-918 114 18 ๐พ ๐พ PROPN cana-918 114 19 + + NOUN cana-918 114 20 ๐‘˜ ๐‘˜ X cana-918 114 21 + + NOUN cana-918 114 22 ๐‘‡๐‘  ๐‘‡๐‘  NOUN cana-918 114 23 + + CCONJ cana-918 114 24 ๐‘ก๐‘  ๐‘ก๐‘  ADJ cana-918 114 25 โ‰ก โ‰ก PROPN cana-918 114 26 0 0 NUM cana-918 114 27 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 114 28 ๐‘›๐‘  ๐‘›๐‘  VERB cana-918 114 29 โˆ€ โˆ€ X cana-918 114 30 1 1 NUM cana-918 114 31 โ‰ค โ‰ค NUM cana-918 114 32 ๐‘  ๐‘  PRON cana-918 114 33 โ‰ค โ‰ค NOUN cana-918 114 34 ๐‘Ÿ ๐‘Ÿ PRON cana-918 114 35 โˆ’ โˆ’ PROPN cana-918 114 36 1 1 X cana-918 114 37 . . PUNCT cana-918 114 38 solving solve VERB cana-918 114 39 these these DET cana-918 114 40 equations equation NOUN cana-918 114 41 , , PUNCT cana-918 114 42 we we PRON cana-918 114 43 get get VERB cana-918 114 44 the the DET cana-918 114 45 value value NOUN cana-918 114 46 of of ADP cana-918 114 47 ๐พ ๐พ PROPN cana-918 114 48 as as SCONJ cana-918 114 49 claimed claim VERB cana-918 114 50 . . PUNCT cana-918 115 1 โ—ป โ—ป PROPN cana-918 115 2 corollary corollary ADJ cana-918 115 3 1 1 X cana-918 115 4 . . PUNCT cana-918 116 1 let let VERB cana-918 116 2 ๐‘Œ ๐‘Œ PROPN cana-918 116 3 = = SYM cana-918 116 4 โ„ณ๐‘›(โ„ค๐‘š โ„ณ๐‘›(โ„ค๐‘š NOUN cana-918 116 5 ) ) PUNCT cana-918 116 6 be be VERB cana-918 116 7 the the DET cana-918 116 8 set set NOUN cana-918 116 9 of of ADP cana-918 116 10 all all DET cana-918 116 11 ๐‘› ๐‘› DET cana-918 116 12 ร— ร— NOUN cana-918 116 13 ๐‘› ๐‘› PRON cana-918 116 14 matrices matrix NOUN cana-918 116 15 over over ADP cana-918 116 16 โ„ค๐‘š โ„ค๐‘š PROPN cana-918 116 17 where where SCONJ cana-918 116 18 ๐‘š ๐‘š PROPN cana-918 116 19 โ‰ฅ โ‰ฅ NUM cana-918 116 20 2 2 NUM cana-918 116 21 and and CCONJ cana-918 116 22 ๐‘› ๐‘› PRON cana-918 116 23 โ‰ฅ โ‰ฅ NUM cana-918 116 24 2 2 NUM cana-918 116 25 . . PUNCT cana-918 117 1 any any DET cana-918 117 2 element element NOUN cana-918 117 3 of of ADP cana-918 117 4 โ„ณ๐‘›(โ„ค๐‘š โ„ณ๐‘›(โ„ค๐‘š NOUN cana-918 117 5 ) ) PUNCT cana-918 117 6 can can AUX cana-918 117 7 be be AUX cana-918 117 8 written write VERB cana-918 117 9 as as ADP cana-918 117 10 ๐‘›2 ๐‘›2 NOUN cana-918 117 11 - - PUNCT cana-918 117 12 tuple tuple NOUN cana-918 117 13 in in ADP cana-918 117 14 โ„ค๐‘š โ„ค๐‘š PROPN cana-918 117 15 ๐‘›2 ๐‘›2 NOUN cana-918 117 16 . . PUNCT cana-918 118 1 therefore therefore ADV cana-918 118 2 , , PUNCT cana-918 118 3 relations relation NOUN cana-918 118 4 defined define VERB cana-918 118 5 on on ADP cana-918 118 6 โ„ค๐‘š โ„ค๐‘š PROPN cana-918 118 7 ๐‘›2 ๐‘›2 NOUN cana-918 118 8 will will AUX cana-918 118 9 form form VERB cana-918 118 10 an an DET cana-918 118 11 association association NOUN cana-918 118 12 scheme scheme NOUN cana-918 118 13 over over ADP cana-918 118 14 โ„ณ๐‘›(โ„ค๐‘š โ„ณ๐‘›(โ„ค๐‘š NOUN cana-918 118 15 ) ) PUNCT cana-918 118 16 . . PUNCT cana-918 119 1 proof proof NOUN cana-918 119 2 . . PUNCT cana-918 120 1 follows follow VERB cana-918 120 2 from from ADP cana-918 120 3 theorem theorem NOUN cana-918 120 4 1 1 NUM cana-918 120 5 when when SCONJ cana-918 120 6 ๐‘š ๐‘š PROPN cana-918 120 7 = = SYM cana-918 120 8 2 2 NUM cana-918 120 9 , , PUNCT cana-918 120 10 and and CCONJ cana-918 120 11 from from ADP cana-918 120 12 theorem theorem NOUN cana-918 120 13 3 3 NUM cana-918 120 14 when when SCONJ cana-918 120 15 ๐‘š ๐‘š PROPN cana-918 120 16 > > X cana-918 120 17 2 2 NUM cana-918 120 18 . . PUNCT cana-918 121 1 โ—ป โ—ป PROPN cana-918 121 2 presentations presentation NOUN cana-918 121 3 of of ADP cana-918 121 4 some some DET cana-918 121 5 general general ADJ cana-918 121 6 linear linear ADJ cana-918 121 7 groups group NOUN cana-918 121 8 ๐บ๐ฟ(2 ๐บ๐ฟ(2 VERB cana-918 121 9 , , PUNCT cana-918 121 10 โ„ค๐‘› โ„ค๐‘› NOUN cana-918 121 11 ) ) PUNCT cana-918 121 12 are be AUX cana-918 121 13 given give VERB cana-918 121 14 in in ADP cana-918 121 15 [ [ NOUN cana-918 121 16 15 15 NUM cana-918 121 17 ] ] X cana-918 121 18 for for ADP cana-918 121 19 ๐‘› ๐‘› NOUN cana-918 121 20 = = SYM cana-918 121 21 4 4 NUM cana-918 121 22 , , PUNCT cana-918 121 23 6 6 NUM cana-918 121 24 , , PUNCT cana-918 121 25 8 8 NUM cana-918 121 26 , , PUNCT cana-918 121 27 10 10 NUM cana-918 121 28 . . PUNCT cana-918 122 1 with with ADP cana-918 122 2 the the DET cana-918 122 3 help help NOUN cana-918 122 4 of of ADP cana-918 122 5 these these DET cana-918 122 6 presentations presentation NOUN cana-918 122 7 , , PUNCT cana-918 122 8 we we PRON cana-918 122 9 can can AUX cana-918 122 10 compute compute VERB cana-918 122 11 the the DET cana-918 122 12 canonical canonical ADJ cana-918 122 13 form form NOUN cana-918 122 14 for for ADP cana-918 122 15 these these DET cana-918 122 16 groups group NOUN cana-918 122 17 ( ( PUNCT cana-918 122 18 provided provide VERB cana-918 122 19 in in ADP cana-918 122 20 table table NOUN cana-918 122 21 1 1 NUM cana-918 122 22 canonical canonical ADJ cana-918 122 23 forms form NOUN cana-918 122 24 of of ADP cana-918 122 25 some some DET cana-918 122 26 general general ADJ cana-918 122 27 linear linear ADJ cana-918 122 28 groups group NOUN cana-918 122 29 the the DET cana-918 122 30 the the DET cana-918 122 31 canonical canonical ADJ cana-918 122 32 form form NOUN cana-918 122 33 of of ADP cana-918 122 34 the the DET cana-918 122 35 presentation presentation NOUN cana-918 122 36 of of ADP cana-918 122 37 ๐บ๐ฟ(2 ๐บ๐ฟ(2 PROPN cana-918 122 38 , , PUNCT cana-918 122 39 โ„ค2 โ„ค2 PROPN cana-918 122 40 ) ) PUNCT cana-918 122 41 is be AUX cana-918 122 42 ๐ด๐‘Ž๐ต๐‘ ๐ด๐‘Ž๐ต๐‘ PROPN cana-918 122 43 : : PUNCT cana-918 122 44 0 0 NUM cana-918 122 45 โ‰ค โ‰ค NUM cana-918 122 46 ๐‘Ž ๐‘Ž DET cana-918 122 47 โ‰ค โ‰ค NUM cana-918 122 48 1,0 1,0 NUM cana-918 122 49 โ‰ค โ‰ค NOUN cana-918 123 1 ๐‘ ๐‘ DET cana-918 123 2 โ‰ค โ‰ค NUM cana-918 123 3 2 2 NUM cana-918 123 4 . . PUNCT cana-918 124 1 the the DET cana-918 124 2 group group NOUN cana-918 124 3 is be AUX cana-918 124 4 isomorphic isomorphic ADJ cana-918 124 5 onto onto ADP cana-918 124 6 ๐‘†3 ๐‘†3 PROPN cana-918 124 7 , , PUNCT cana-918 124 8 symmetric symmetric ADJ cana-918 124 9 group group NOUN cana-918 124 10 on on ADP cana-918 124 11 3 3 NUM cana-918 124 12 - - PUNCT cana-918 124 13 symbols symbol NOUN cana-918 124 14 . . PUNCT cana-918 125 1 hence hence ADV cana-918 125 2 , , PUNCT cana-918 125 3 another another DET cana-918 125 4 familiar familiar ADJ cana-918 125 5 presentation presentation NOUN cana-918 125 6 can can AUX cana-918 125 7 be be AUX cana-918 125 8 given give VERB cana-918 125 9 as as ADP cana-918 125 10 ๐บ๐ฟ(2 ๐บ๐ฟ(2 PROPN cana-918 125 11 , , PUNCT cana-918 125 12 โ„ค2 โ„ค2 NOUN cana-918 125 13 ) ) PUNCT cana-918 125 14 = = SYM cana-918 125 15 โŸจ๐ด โŸจ๐ด PROPN cana-918 125 16 , , PUNCT cana-918 125 17 ๐ต|๐ด ๐ต|๐ด PROPN cana-918 125 18 2 2 NUM cana-918 125 19 = = SYM cana-918 125 20 ๐ต3 ๐ต3 NOUN cana-918 125 21 = = SYM cana-918 125 22 ๐ผ ๐ผ PROPN cana-918 125 23 , , PUNCT cana-918 125 24 ๐ด๐ต ๐ด๐ต NOUN cana-918 125 25 = = NOUN cana-918 125 26 ๐ตโˆ’1๐ดโŸฉ. ๐ตโˆ’1๐ดโŸฉ. NOUN cana-918 125 27 using use VERB cana-918 125 28 this this DET cana-918 125 29 presentation presentation NOUN cana-918 125 30 , , PUNCT cana-918 125 31 we we PRON cana-918 125 32 get get VERB cana-918 125 33 a a DET cana-918 125 34 non non ADJ cana-918 125 35 symmetric symmetric PROPN cana-918 125 36 association association NOUN cana-918 125 37 scheme scheme NOUN cana-918 125 38 . . PUNCT cana-918 126 1 table table NOUN cana-918 126 2 1 1 NUM cana-918 126 3 canonical canonical ADJ cana-918 126 4 forms form NOUN cana-918 126 5 of of ADP cana-918 126 6 some some DET cana-918 126 7 general general ADJ cana-918 126 8 linear linear PROPN cana-918 126 9 groups group NOUN cana-918 126 10 group group NOUN cana-918 126 11 generators generator NOUN cana-918 126 12 presentation presentation VERB cana-918 126 13 canonical canonical ADJ cana-918 126 14 form form NOUN cana-918 126 15 ๐บ๐ฟ(2 ๐บ๐ฟ(2 PROPN cana-918 126 16 , , PUNCT cana-918 126 17 โ„ค2 โ„ค2 PROPN cana-918 126 18 ) ) PUNCT cana-918 126 19 ๐ด ๐ด PROPN cana-918 126 20 , , PUNCT cana-918 126 21 ๐ต ๐ต NOUN cana-918 126 22 ฯ„2 ฯ„2 NOUN cana-918 126 23 = = SYM cana-918 126 24 ฯƒ3 ฯƒ3 NOUN cana-918 126 25 = = SYM cana-918 126 26 ( ( PUNCT cana-918 126 27 ฯ„ฯƒ)3 ฯ„ฯƒ)3 NOUN cana-918 126 28 = = SYM cana-918 126 29 1 1 NUM cana-918 126 30 ๐ด๐‘Ž๐ต๐‘ ๐ด๐‘Ž๐ต๐‘ NOUN cana-918 126 31 : : PUNCT cana-918 126 32 0 0 NUM cana-918 126 33 โ‰ค โ‰ค NUM cana-918 127 1 ๐‘Ž ๐‘Ž X cana-918 127 2 โ‰ค โ‰ค NUM cana-918 127 3 1 1 NUM cana-918 127 4 , , PUNCT cana-918 127 5 0 0 NUM cana-918 127 6 โ‰ค โ‰ค NUM cana-918 127 7 ๐‘ ๐‘ PRON cana-918 127 8 โ‰ค โ‰ค NUM cana-918 127 9 2 2 NUM cana-918 127 10 ๐บ๐ฟ(2 ๐บ๐ฟ(2 NOUN cana-918 127 11 , , PUNCT cana-918 127 12 โ„ค4 โ„ค4 PROPN cana-918 127 13 ) ) PUNCT cana-918 127 14 ๐ด ๐ด PROPN cana-918 127 15 , , PUNCT cana-918 127 16 ๐ต ๐ต PROPN cana-918 127 17 , , PUNCT cana-918 127 18 ๐ถ ๐ถ PROPN cana-918 127 19 ๐ด2 ๐ด2 PROPN cana-918 127 20 = = PUNCT cana-918 127 21 ๐ต2 ๐ต2 NOUN cana-918 127 22 = = NOUN cana-918 127 23 ๐ถ4 ๐ถ4 NOUN cana-918 127 24 = = PUNCT cana-918 127 25 ๐‘‹3 ๐‘‹3 NOUN cana-918 127 26 = = PUNCT cana-918 127 27 ๐‘Œ4 ๐‘Œ4 NOUN cana-918 127 28 = = SYM cana-918 127 29 ( ( PUNCT cana-918 127 30 ๐ถ๐ต๐ด)4 ๐ถ๐ต๐ด)4 SYM cana-918 127 31 = = SYM cana-918 127 32 ๐ผ ๐ผ PROPN cana-918 127 33 , , PUNCT cana-918 127 34 ๐ถ2๐ด ๐ถ2๐ด NOUN cana-918 127 35 = = PROPN cana-918 127 36 ๐ด๐ถ2 ๐ด๐ถ2 PROPN cana-918 127 37 , , PUNCT cana-918 127 38 ๐ถ2๐ต ๐ถ2๐ต PROPN cana-918 127 39 = = SYM cana-918 127 40 ๐ต๐ถ2 ๐ต๐ถ2 PROPN cana-918 127 41 , , PUNCT cana-918 127 42 ๐ต๐ถ ๐ต๐ถ PROPN cana-918 127 43 = = SYM cana-918 127 44 ๐ถโˆ’1๐ต ๐ถโˆ’1๐ต PROPN cana-918 127 45 where where SCONJ cana-918 127 46 ๐‘‹ ๐‘‹ NOUN cana-918 127 47 = = PUNCT cana-918 127 48 ๐ด๐ต ๐ด๐ต PROPN cana-918 127 49 and and CCONJ cana-918 127 50 ๐‘Œ ๐‘Œ PROPN cana-918 127 51 = = PUNCT cana-918 127 52 ๐‘‹๐ถ๐‘‹ ๐‘‹๐ถ๐‘‹ PROPN cana-918 127 53 ๐ด๐‘Ž๐ถ๐‘๐‘Œ๐‘๐‘‹๐‘‘ ๐ด๐‘Ž๐ถ๐‘๐‘Œ๐‘๐‘‹๐‘‘ NOUN cana-918 127 54 : : PUNCT cana-918 127 55 0 0 NUM cana-918 127 56 โ‰ค โ‰ค NUM cana-918 128 1 ๐‘Ž ๐‘Ž PRON cana-918 128 2 โ‰ค โ‰ค NUM cana-918 128 3 1,0 1,0 NUM cana-918 128 4 โ‰ค โ‰ค NOUN cana-918 128 5 ๐‘ ๐‘ PROPN cana-918 128 6 , , PUNCT cana-918 128 7 ๐‘ ๐‘ PROPN cana-918 128 8 โ‰ค โ‰ค NOUN cana-918 128 9 3 3 NUM cana-918 128 10 , , PUNCT cana-918 128 11 0 0 NUM cana-918 128 12 โ‰ค โ‰ค NUM cana-918 128 13 ๐‘‘ ๐‘‘ VERB cana-918 128 14 โ‰ค โ‰ค ADJ cana-918 128 15 2 2 NUM cana-918 128 16 ๐บ๐ฟ(2 ๐บ๐ฟ(2 NOUN cana-918 128 17 , , PUNCT cana-918 128 18 โ„ค6 โ„ค6 NOUN cana-918 128 19 ) ) PUNCT cana-918 128 20 ๐ด ๐ด PROPN cana-918 128 21 , , PUNCT cana-918 128 22 ๐ต ๐ต PROPN cana-918 128 23 , , PUNCT cana-918 128 24 ๐ถ ๐ถ PROPN cana-918 128 25 ๐ด2 ๐ด2 PROPN cana-918 128 26 = = PUNCT cana-918 128 27 ๐ต2 ๐ต2 NOUN cana-918 128 28 = = NOUN cana-918 128 29 ๐ถ4 ๐ถ4 NOUN cana-918 128 30 = = PUNCT cana-918 128 31 ๐‘‹12 ๐‘‹12 PROPN cana-918 128 32 = = PUNCT cana-918 128 33 ๐‘Œ6 ๐‘Œ6 X cana-918 128 34 = = SYM cana-918 128 35 ( ( PUNCT cana-918 128 36 ๐ด๐ต)3 ๐ด๐ต)3 PROPN cana-918 128 37 = = SYM cana-918 128 38 ( ( PUNCT cana-918 128 39 ๐ต๐ถ)2 ๐ต๐ถ)2 NUM cana-918 128 40 = = SYM cana-918 128 41 ๐ผ ๐ผ PROPN cana-918 128 42 , , PUNCT cana-918 128 43 ๐ถ๐ด ๐ถ๐ด PROPN cana-918 128 44 = = SYM cana-918 128 45 ๐ด๐ถ2 ๐ด๐ถ2 PROPN cana-918 128 46 , , PUNCT cana-918 128 47 ๐ถ2๐ต ๐ถ2๐ต PROPN cana-918 128 48 = = SYM cana-918 128 49 ๐ต๐ถ2 ๐ต๐ถ2 PROPN cana-918 128 50 , , PUNCT cana-918 128 51 ( ( PUNCT cana-918 128 52 ๐ด๐ถ)12 ๐ด๐ถ)12 NOUN cana-918 128 53 = = SYM cana-918 128 54 ๐ถ2 ๐ถ2 PROPN cana-918 128 55 , , PUNCT cana-918 128 56 ( ( PUNCT cana-918 128 57 ๐ถ๐ด)2๐ต(๐ถ๐ด)2 ๐ถ๐ด)2๐ต(๐ถ๐ด)2 PROPN cana-918 128 58 = = PUNCT cana-918 128 59 ( ( PUNCT cana-918 128 60 ๐ด๐ถ)2๐ต(๐ด๐ถ)2 ๐ด๐ถ)2๐ต(๐ด๐ถ)2 NOUN cana-918 128 61 where where SCONJ cana-918 128 62 ๐‘‹ ๐‘‹ PROPN cana-918 128 63 = = SYM cana-918 128 64 ๐ต(๐ถ๐ด)2๐ต ๐ต(๐ถ๐ด)2๐ต PROPN cana-918 128 65 and and CCONJ cana-918 128 66 ๐‘Œ ๐‘Œ PROPN cana-918 128 67 = = PUNCT cana-918 128 68 ๐ถ๐ด๐ต ๐ถ๐ด๐ต PROPN cana-918 128 69 ๐ถ๐‘Ž๐‘‹๐‘๐‘Œ๐‘๐ต๐‘‘ ๐ถ๐‘Ž๐‘‹๐‘๐‘Œ๐‘๐ต๐‘‘ NOUN cana-918 128 70 : : PUNCT cana-918 128 71 0 0 NUM cana-918 128 72 โ‰ค โ‰ค NUM cana-918 128 73 ๐‘Ž ๐‘Ž ADJ cana-918 128 74 , , PUNCT cana-918 128 75 ๐‘‘ ๐‘‘ PROPN cana-918 128 76 โ‰ค โ‰ค NUM cana-918 128 77 1 1 NUM cana-918 128 78 , , PUNCT cana-918 128 79 0 0 NUM cana-918 128 80 โ‰ค โ‰ค NUM cana-918 128 81 ๐‘ ๐‘ DET cana-918 128 82 โ‰ค โ‰ค NUM cana-918 128 83 11 11 NUM cana-918 128 84 , , PUNCT cana-918 128 85 0 0 NUM cana-918 128 86 โ‰ค โ‰ค NUM cana-918 128 87 ๐‘ ๐‘ PROPN cana-918 128 88 โ‰ค โ‰ค NUM cana-918 128 89 5 5 NUM cana-918 128 90 theorem theorem NOUN cana-918 128 91 4 4 NUM cana-918 128 92 . . PUNCT cana-918 128 93 let let VERB cana-918 128 94 ๐‘Œ ๐‘Œ PROPN cana-918 128 95 = = SYM cana-918 128 96 ๐บ๐ฟ(2 ๐บ๐ฟ(2 PROPN cana-918 128 97 , , PUNCT cana-918 128 98 โ„ค2 โ„ค2 PROPN cana-918 128 99 ) ) PUNCT cana-918 128 100 . . PUNCT cana-918 129 1 define define VERB cana-918 129 2 relations relation NOUN cana-918 129 3 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 129 4 on on ADP cana-918 129 5 ๐’ซ ๐’ซ NOUN cana-918 129 6 by by ADP cana-918 129 7 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 129 8 = = SYM cana-918 129 9 { { PUNCT cana-918 129 10 ( ( PUNCT cana-918 129 11 ๐ด ๐ด PROPN cana-918 129 12 ๐‘Ž๐ต๐‘ ๐‘Ž๐ต๐‘ NOUN cana-918 129 13 , , PUNCT cana-918 129 14 ๐ด๐‘˜+๐‘Ž๐ต๐‘˜+๐‘)| ๐ด๐‘˜+๐‘Ž๐ต๐‘˜+๐‘)| X cana-918 129 15 0 0 NUM cana-918 129 16 โ‰ค โ‰ค NUM cana-918 130 1 ๐‘Ž ๐‘Ž PRON cana-918 130 2 โ‰ค โ‰ค NUM cana-918 130 3 1 1 NUM cana-918 130 4 , , PUNCT cana-918 130 5 0 0 NUM cana-918 130 6 โ‰ค โ‰ค NUM cana-918 130 7 ๐‘ ๐‘ PRON cana-918 130 8 โ‰ค โ‰ค NUM cana-918 130 9 2 2 NUM cana-918 130 10 } } PUNCT cana-918 130 11 for for ADP cana-918 130 12 all all PRON cana-918 130 13 0 0 NUM cana-918 130 14 โ‰ค โ‰ค NOUN cana-918 130 15 ๐‘˜ ๐‘˜ DET cana-918 130 16 โ‰ค โ‰ค ADJ cana-918 130 17 5 5 NUM cana-918 130 18 . . PUNCT cana-918 130 19 then then ADV cana-918 130 20 ( ( PUNCT cana-918 130 21 ๐‘Œ ๐‘Œ PROPN cana-918 130 22 , , PUNCT cana-918 130 23 ๐’ซ ๐’ซ NOUN cana-918 130 24 ) ) PUNCT cana-918 130 25 is be AUX cana-918 130 26 a a DET cana-918 130 27 non non ADJ cana-918 130 28 symmetric symmetric PROPN cana-918 130 29 association association NOUN cana-918 130 30 scheme scheme NOUN cana-918 130 31 with with ADP cana-918 130 32 parameters parameter NOUN cana-918 130 33 as as SCONJ cana-918 130 34 follows follow VERB cana-918 130 35 : : PUNCT cana-918 130 36 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š PROPN cana-918 130 37 ๐‘˜ ๐‘˜ NOUN cana-918 130 38 = = PUNCT cana-918 130 39 { { PUNCT cana-918 130 40 1 1 NUM cana-918 130 41 if if SCONJ cana-918 130 42 ๐‘˜ ๐‘˜ ADP cana-918 130 43 = = SYM cana-918 130 44 ๐‘™ ๐‘™ PROPN cana-918 130 45 + + NUM cana-918 130 46 ๐‘š ๐‘š PROPN cana-918 130 47 , , PUNCT cana-918 130 48 0 0 PUNCT cana-918 131 1 if if SCONJ cana-918 131 2 ๐‘˜ ๐‘˜ PRON cana-918 131 3 โ‰  โ‰  PROPN cana-918 131 4 ๐‘™ ๐‘™ PROPN cana-918 132 1 + + NUM cana-918 132 2 ๐‘š ๐‘š X cana-918 132 3 where where SCONJ cana-918 132 4 0 0 NUM cana-918 132 5 โ‰ค โ‰ค NUM cana-918 132 6 ๐‘˜ ๐‘˜ X cana-918 132 7 , , PUNCT cana-918 132 8 ๐‘™,๐‘š ๐‘™,๐‘š NOUN cana-918 132 9 โ‰ค โ‰ค ADV cana-918 132 10 5 5 NUM cana-918 132 11 . . PUNCT cana-918 133 1 proof proof NOUN cana-918 133 2 . . PUNCT cana-918 134 1 observe observe VERB cana-918 134 2 that that DET cana-918 134 3 ๐’ฎ0 ๐’ฎ0 NOUN cana-918 134 4 = = PRON cana-918 134 5 { { PUNCT cana-918 134 6 ( ( PUNCT cana-918 134 7 ๐‘€,๐‘€)| ๐‘€,๐‘€)| PROPN cana-918 134 8 ๐‘€ ๐‘€ PROPN cana-918 134 9 โˆˆ โˆˆ PROPN cana-918 134 10 ๐บ๐ฟ(2 ๐บ๐ฟ(2 PROPN cana-918 134 11 , , PUNCT cana-918 134 12 โ„ค2 โ„ค2 PROPN cana-918 134 13 ) ) PUNCT cana-918 134 14 } } PUNCT cana-918 134 15 is be AUX cana-918 134 16 an an DET cana-918 134 17 identity identity NOUN cana-918 134 18 relation relation NOUN cana-918 134 19 . . PUNCT cana-918 135 1 for for ADP cana-918 135 2 arbitrary arbitrary ADJ cana-918 135 3 relations relation NOUN cana-918 135 4 ๐’ฎ๐‘™ ๐’ฎ๐‘™ PROPN cana-918 135 5 , , PUNCT cana-918 135 6 ๐’ฎ๐‘š ๐’ฎ๐‘š NOUN cana-918 135 7 , , PUNCT cana-918 135 8 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 135 9 โˆˆ โˆˆ PROPN cana-918 135 10 ๐’ซ ๐’ซ NOUN cana-918 135 11 , , PUNCT cana-918 135 12 we we PRON cana-918 135 13 find find VERB cana-918 135 14 the the DET cana-918 135 15 cardinality cardinality NOUN cana-918 135 16 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š VERB cana-918 136 1 ๐‘˜ ๐‘˜ INTJ cana-918 136 2 such such ADJ cana-918 136 3 that that SCONJ cana-918 136 4 |๐‘€๐’ฎ๐‘™ |๐‘€๐’ฎ๐‘™ ADJ cana-918 136 5 โˆฉ โˆฉ NOUN cana-918 136 6 ๐‘๐’ฎ๐‘š ๐‘๐’ฎ๐‘š NOUN cana-918 136 7 โˆ— โˆ— NOUN cana-918 137 1 | | NOUN cana-918 137 2 = = NOUN cana-918 138 1 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š PROPN cana-918 138 2 ๐‘˜ ๐‘˜ NOUN cana-918 138 3 for for ADP cana-918 138 4 all all DET cana-918 138 5 ( ( PUNCT cana-918 138 6 ๐‘€,๐‘ ๐‘€,๐‘ NOUN cana-918 138 7 ) ) PUNCT cana-918 138 8 โˆˆ โˆˆ PROPN cana-918 139 1 ๐’ฎ๐‘˜. ๐’ฎ๐‘˜. PROPN cana-918 139 2 let let VERB cana-918 139 3 ( ( PUNCT cana-918 139 4 ๐‘€,๐‘ ๐‘€,๐‘ NOUN cana-918 139 5 ) ) PUNCT cana-918 139 6 โˆˆ โˆˆ PROPN cana-918 139 7 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 139 8 and and CCONJ cana-918 139 9 let let VERB cana-918 139 10 ๐‘€๐’ฎ๐‘™ ๐‘€๐’ฎ๐‘™ NOUN cana-918 139 11 = = SYM cana-918 139 12 ๐‘€ ๐‘€ PROPN cana-918 139 13 โ€ฒ โ€ฒ NOUN cana-918 139 14 and and CCONJ cana-918 139 15 ๐‘๐’ฎ๐‘š ๐‘๐’ฎ๐‘š NOUN cana-918 139 16 โˆ— โˆ— NOUN cana-918 139 17 = = PRON cana-918 139 18 ๐‘โ€ฒ. ๐‘โ€ฒ. NOUN cana-918 139 19 that that PRON cana-918 139 20 is be AUX cana-918 139 21 , , PUNCT cana-918 139 22 ๐‘€ ๐‘€ PROPN cana-918 139 23 = = PUNCT cana-918 139 24 ๐ด๐‘Ž๐ต๐‘ ๐ด๐‘Ž๐ต๐‘ PROPN cana-918 139 25 , , PUNCT cana-918 139 26 ๐‘ ๐‘ PROPN cana-918 139 27 = = SYM cana-918 139 28 ๐ด๐‘˜+๐‘Ž๐ต๐‘˜+๐‘;๐‘€ ๐ด๐‘˜+๐‘Ž๐ต๐‘˜+๐‘;๐‘€ PROPN cana-918 139 29 = = SYM cana-918 139 30 ๐ด๐‘Ž1๐ต๐‘1 ๐ด๐‘Ž1๐ต๐‘1 PROPN cana-918 139 31 , , PUNCT cana-918 139 32 ๐‘€โ€ฒ ๐‘€โ€ฒ NOUN cana-918 139 33 = = PUNCT cana-918 139 34 ๐ด๐‘™+๐‘Ž1๐ต๐‘™+๐‘1 ๐ด๐‘™+๐‘Ž1๐ต๐‘™+๐‘1 PROPN cana-918 139 35 ; ; PUNCT cana-918 139 36 ๐‘โ€ฒ ๐‘โ€ฒ X cana-918 139 37 = = SYM cana-918 139 38 ๐ด๐‘Ž2๐ต๐‘2 ๐ด๐‘Ž2๐ต๐‘2 PROPN cana-918 139 39 , , PUNCT cana-918 139 40 ๐‘ ๐‘ PROPN cana-918 139 41 = = PUNCT cana-918 139 42 ๐ด๐‘š+๐‘Ž2๐ต๐‘š+๐‘2 ๐ด๐‘š+๐‘Ž2๐ต๐‘š+๐‘2 ADJ cana-918 139 43 where where SCONJ cana-918 139 44 0 0 NUM cana-918 139 45 โ‰ค โ‰ค NUM cana-918 139 46 ๐‘Ž ๐‘Ž ADJ cana-918 139 47 , , PUNCT cana-918 139 48 ๐‘Ž1 ๐‘Ž1 ADJ cana-918 139 49 , , PUNCT cana-918 139 50 ๐‘Ž2 ๐‘Ž2 NOUN cana-918 139 51 โ‰ค โ‰ค NUM cana-918 139 52 1 1 NUM cana-918 139 53 and and CCONJ cana-918 139 54 0 0 NUM cana-918 139 55 โ‰ค โ‰ค NOUN cana-918 139 56 communications communication NOUN cana-918 139 57 on on ADP cana-918 139 58 applied apply VERB cana-918 139 59 nonlinear nonlinear ADJ cana-918 139 60 analysis analysis NOUN cana-918 139 61 issn issn NOUN cana-918 139 62 : : PUNCT cana-918 139 63 1074 1074 NUM cana-918 139 64 - - PUNCT cana-918 139 65 133x 133x NUM cana-918 139 66 vol vol NOUN cana-918 139 67 31 31 NUM cana-918 139 68 no no NOUN cana-918 139 69 . . PUNCT cana-918 140 1 4s 4s NUM cana-918 140 2 ( ( PUNCT cana-918 140 3 2024 2024 NUM cana-918 140 4 ) ) PUNCT cana-918 140 5 397 397 NUM cana-918 140 6 https://internationalpubls.com https://internationalpubls.com X cana-918 140 7 ๐‘ ๐‘ NOUN cana-918 140 8 , , PUNCT cana-918 140 9 ๐‘1 ๐‘1 NOUN cana-918 140 10 , , PUNCT cana-918 140 11 ๐‘2 ๐‘2 PROPN cana-918 140 12 โ‰ค โ‰ค NOUN cana-918 140 13 2 2 NUM cana-918 140 14 . . PUNCT cana-918 141 1 since since SCONJ cana-918 141 2 every every DET cana-918 141 3 pair pair NOUN cana-918 141 4 ( ( PUNCT cana-918 141 5 ๐‘€,๐‘ ๐‘€,๐‘ NOUN cana-918 141 6 ) ) PUNCT cana-918 141 7 in in ADP cana-918 141 8 ๐‘Œ ๐‘Œ PROPN cana-918 141 9 are be AUX cana-918 141 10 ๐‘˜๐‘กโ„Ž ๐‘˜๐‘กโ„Ž VERB cana-918 141 11 associates associate NOUN cana-918 141 12 for for ADP cana-918 141 13 exactly exactly ADV cana-918 141 14 one one NUM cana-918 141 15 ๐‘˜ ๐‘˜ NOUN cana-918 141 16 , , PUNCT cana-918 141 17 we we PRON cana-918 141 18 find find VERB cana-918 141 19 that that SCONJ cana-918 141 20 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š PROPN cana-918 141 21 ๐‘˜ ๐‘˜ PRON cana-918 141 22 can can AUX cana-918 141 23 be be AUX cana-918 141 24 either either CCONJ cana-918 141 25 0 0 NUM cana-918 141 26 or or CCONJ cana-918 141 27 1 1 NUM cana-918 141 28 . . PUNCT cana-918 142 1 hence hence ADV cana-918 142 2 , , PUNCT cana-918 142 3 with with ADP cana-918 142 4 the the DET cana-918 142 5 help help NOUN cana-918 142 6 of of ADP cana-918 142 7 the the DET cana-918 142 8 above above ADJ cana-918 142 9 equations equation NOUN cana-918 142 10 , , PUNCT cana-918 142 11 we we PRON cana-918 142 12 obtain obtain VERB cana-918 142 13 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š PROPN cana-918 142 14 ๐‘˜ ๐‘˜ NOUN cana-918 142 15 = = NOUN cana-918 142 16 1 1 NUM cana-918 142 17 if if SCONJ cana-918 142 18 ๐‘€โ€ฒ ๐‘€โ€ฒ PROPN cana-918 142 19 = = PUNCT cana-918 142 20 ๐‘โ€ฒ ๐‘โ€ฒ PROPN cana-918 142 21 equivalently equivalently ADV cana-918 142 22 , , PUNCT cana-918 142 23 if if SCONJ cana-918 142 24 ๐‘˜ ๐‘˜ PRON cana-918 142 25 = = SYM cana-918 142 26 ๐‘™ ๐‘™ PROPN cana-918 142 27 + + CCONJ cana-918 142 28 ๐‘š. ๐‘š. ADV cana-918 142 29 moreover moreover ADV cana-918 142 30 , , PUNCT cana-918 142 31 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š PROPN cana-918 142 32 ๐‘˜ ๐‘˜ X cana-918 142 33 = = NOUN cana-918 142 34 0 0 PUNCT cana-918 143 1 if if SCONJ cana-918 143 2 ๐‘€โ€ฒ ๐‘€โ€ฒ PROPN cana-918 143 3 โ‰  โ‰  PROPN cana-918 143 4 ๐‘โ€ฒ ๐‘โ€ฒ NOUN cana-918 143 5 that that PRON cana-918 143 6 is be AUX cana-918 143 7 , , PUNCT cana-918 143 8 whenever whenever SCONJ cana-918 143 9 ๐‘˜ ๐‘˜ PRON cana-918 143 10 โ‰  โ‰  PROPN cana-918 143 11 ๐‘™ ๐‘™ PROPN cana-918 143 12 + + CCONJ cana-918 143 13 ๐‘š. ๐‘š. NOUN cana-918 143 14 also also ADV cana-918 143 15 it it PRON cana-918 143 16 can can AUX cana-918 143 17 be be AUX cana-918 143 18 easily easily ADV cana-918 143 19 proved prove VERB cana-918 143 20 that that SCONJ cana-918 143 21 the the DET cana-918 143 22 relations relation NOUN cana-918 143 23 are be AUX cana-918 143 24 not not PART cana-918 143 25 symmetric symmetric ADJ cana-918 143 26 and and CCONJ cana-918 143 27 hence hence ADV cana-918 143 28 ( ( PUNCT cana-918 143 29 ๐‘Œ ๐‘Œ PROPN cana-918 143 30 , , PUNCT cana-918 143 31 ๐’ซ ๐’ซ NOUN cana-918 143 32 ) ) PUNCT cana-918 143 33 is be AUX cana-918 143 34 a a DET cana-918 143 35 non non ADJ cana-918 143 36 symmetric symmetric PROPN cana-918 143 37 association association NOUN cana-918 143 38 scheme scheme NOUN cana-918 143 39 . . PUNCT cana-918 144 1 โ—ป โ—ป PROPN cana-918 144 2 theorem theorem VERB cana-918 144 3 5 5 NUM cana-918 144 4 . . PUNCT cana-918 145 1 let let VERB cana-918 145 2 ๐‘Œ ๐‘Œ PROPN cana-918 145 3 = = SYM cana-918 145 4 ๐บ๐ฟ(2 ๐บ๐ฟ(2 PROPN cana-918 145 5 , , PUNCT cana-918 145 6 โ„ค4 โ„ค4 PROPN cana-918 145 7 ) ) PUNCT cana-918 145 8 and and CCONJ cana-918 145 9 ๐’ซ ๐’ซ NOUN cana-918 145 10 be be VERB cana-918 145 11 a a DET cana-918 145 12 partition partition NOUN cana-918 145 13 of of ADP cana-918 145 14 ๐‘Œ ๐‘Œ PROPN cana-918 145 15 ร— ร— PROPN cana-918 145 16 ๐‘Œ. ๐‘Œ. PROPN cana-918 145 17 for for ADP cana-918 145 18 all all PRON cana-918 145 19 ๐‘˜ ๐‘˜ PRON cana-918 145 20 written write VERB cana-918 145 21 in in ADP cana-918 145 22 form form NOUN cana-918 145 23 of of ADP cana-918 145 24 48๐‘  48๐‘  NOUN cana-918 145 25 + + CCONJ cana-918 145 26 16๐‘› 16๐‘› NOUN cana-918 145 27 + + NUM cana-918 145 28 4๐‘ก 4๐‘ก NOUN cana-918 145 29 + + X cana-918 145 30 ๐‘Ÿ ๐‘Ÿ NOUN cana-918 145 31 ( ( PUNCT cana-918 145 32 ๐‘  ๐‘  PROPN cana-918 145 33 โˆˆ โˆˆ PROPN cana-918 145 34 โ„ค2 โ„ค2 PROPN cana-918 145 35 ; ; PUNCT cana-918 145 36 ๐‘› ๐‘› PROPN cana-918 145 37 โˆˆ โˆˆ PROPN cana-918 145 38 โ„ค3 โ„ค3 NOUN cana-918 145 39 ; ; PUNCT cana-918 145 40 ๐‘Ÿ ๐‘Ÿ X cana-918 145 41 , , PUNCT cana-918 145 42 ๐‘ก ๐‘ก PROPN cana-918 145 43 โˆˆ โˆˆ PROPN cana-918 145 44 โ„ค4 โ„ค4 NOUN cana-918 145 45 ) ) PUNCT cana-918 145 46 , , PUNCT cana-918 145 47 the the DET cana-918 145 48 relations relation NOUN cana-918 145 49 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 145 50 in in ADP cana-918 145 51 ๐’ซ ๐’ซ NOUN cana-918 145 52 defined define VERB cana-918 145 53 by by ADP cana-918 145 54 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 145 55 = = SYM cana-918 145 56 { { PUNCT cana-918 145 57 ( ( PUNCT cana-918 145 58 ๐ด ๐ด PROPN cana-918 145 59 ๐‘Ž๐ถ๐‘๐‘Œ๐‘๐‘‹๐‘‘ ๐‘Ž๐ถ๐‘๐‘Œ๐‘๐‘‹๐‘‘ PROPN cana-918 145 60 , , PUNCT cana-918 145 61 ๐ด๐‘Ž+๐‘ ๐ถ๐‘+๐‘ก๐‘Œ๐‘+๐‘Ÿ๐‘‹๐‘‘+๐‘›)| ๐ด๐‘Ž+๐‘ ๐ถ๐‘+๐‘ก๐‘Œ๐‘+๐‘Ÿ๐‘‹๐‘‘+๐‘›)| PROPN cana-918 145 62 ๐‘Ž ๐‘Ž PROPN cana-918 145 63 โˆˆ โˆˆ PROPN cana-918 145 64 โ„ค2 โ„ค2 NOUN cana-918 145 65 ; ; PUNCT cana-918 145 66 ๐‘ ๐‘ X cana-918 145 67 , , PUNCT cana-918 145 68 ๐‘ ๐‘ PROPN cana-918 145 69 โˆˆ โˆˆ PROPN cana-918 145 70 โ„ค4 โ„ค4 NOUN cana-918 145 71 ; ; PUNCT cana-918 145 72 ๐‘‘ ๐‘‘ PROPN cana-918 145 73 โˆˆ โˆˆ PROPN cana-918 145 74 โ„ค3 โ„ค3 NOUN cana-918 145 75 } } PUNCT cana-918 145 76 is be AUX cana-918 145 77 a a DET cana-918 145 78 non non ADJ cana-918 145 79 - - ADJ cana-918 145 80 symmetric symmetric ADJ cana-918 145 81 association association NOUN cana-918 145 82 scheme scheme NOUN cana-918 145 83 with with ADP cana-918 145 84 parameters parameter NOUN cana-918 145 85 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š VERB cana-918 145 86 ๐‘˜ ๐‘˜ X cana-918 145 87 = = PUNCT cana-918 145 88 { { PUNCT cana-918 145 89 1 1 NUM cana-918 145 90 ๐‘–๐‘“ ๐‘–๐‘“ PRON cana-918 145 91 ๐‘ (๐‘˜ ๐‘ (๐‘˜ NOUN cana-918 145 92 ) ) PUNCT cana-918 145 93 โ‰ก โ‰ก PROPN cana-918 145 94 ( ( PUNCT cana-918 145 95 ๐‘ (๐‘™ ๐‘ (๐‘™ PROPN cana-918 145 96 ) ) PUNCT cana-918 146 1 + + CCONJ cana-918 146 2 ๐‘ (๐‘š ๐‘ (๐‘š NOUN cana-918 146 3 ) ) PUNCT cana-918 146 4 ) ) PUNCT cana-918 146 5 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 146 6 2 2 NUM cana-918 146 7 , , PUNCT cana-918 146 8 ๐‘›(๐‘˜ ๐‘›(๐‘˜ PROPN cana-918 146 9 ) ) PUNCT cana-918 146 10 โ‰ก โ‰ก PROPN cana-918 146 11 ( ( PUNCT cana-918 146 12 ๐‘›(๐‘™ ๐‘›(๐‘™ PROPN cana-918 146 13 ) ) PUNCT cana-918 146 14 + + SYM cana-918 146 15 ๐‘›(๐‘š ๐‘›(๐‘š NOUN cana-918 146 16 ) ) PUNCT cana-918 146 17 ) ) PUNCT cana-918 146 18 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 146 19 2 2 NUM cana-918 146 20 , , PUNCT cana-918 146 21 ๐‘ก(๐‘˜ ๐‘ก(๐‘˜ NOUN cana-918 146 22 ) ) PUNCT cana-918 146 23 โ‰ก โ‰ก PROPN cana-918 146 24 ( ( PUNCT cana-918 146 25 ๐‘ก(๐‘™ ๐‘ก(๐‘™ PROPN cana-918 146 26 ) ) PUNCT cana-918 146 27 + + NUM cana-918 146 28 ๐‘ก(๐‘š ๐‘ก(๐‘š NOUN cana-918 146 29 ) ) PUNCT cana-918 146 30 ) ) PUNCT cana-918 146 31 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 146 32 4 4 NUM cana-918 146 33 , , PUNCT cana-918 146 34 ๐‘Ž๐‘›๐‘‘ ๐‘Ž๐‘›๐‘‘ ADJ cana-918 146 35 ๐‘Ÿ(๐‘˜ ๐‘Ÿ(๐‘˜ PROPN cana-918 146 36 ) ) PUNCT cana-918 146 37 โ‰ก โ‰ก PROPN cana-918 146 38 ( ( PUNCT cana-918 146 39 ๐‘Ÿ(๐‘™ ๐‘Ÿ(๐‘™ PROPN cana-918 146 40 ) ) PUNCT cana-918 146 41 + + CCONJ cana-918 146 42 ๐‘Ÿ(๐‘š ๐‘Ÿ(๐‘š NOUN cana-918 146 43 ) ) PUNCT cana-918 146 44 ) ) PUNCT cana-918 146 45 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 146 46 4 4 NUM cana-918 146 47 0 0 NUM cana-918 146 48 ๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’ ๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’ NOUN cana-918 146 49 proof proof NOUN cana-918 146 50 . . PUNCT cana-918 147 1 observe observe VERB cana-918 147 2 that that SCONJ cana-918 147 3 |๐’ฎ๐‘˜| |๐’ฎ๐‘˜| ADV cana-918 147 4 = = PUNCT cana-918 147 5 |๐‘Œ| |๐‘Œ| VERB cana-918 147 6 for for ADP cana-918 147 7 all all PRON cana-918 147 8 0 0 NUM cana-918 147 9 โ‰ค โ‰ค NOUN cana-918 148 1 ๐‘˜ ๐‘˜ PRON cana-918 148 2 < < X cana-918 148 3 96 96 NUM cana-918 148 4 . . PUNCT cana-918 149 1 the the DET cana-918 149 2 relations relation NOUN cana-918 149 3 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 149 4 are be AUX cana-918 149 5 disjoint disjoint ADJ cana-918 149 6 and and CCONJ cana-918 149 7 โˆช โˆช ADJ cana-918 149 8 ๐’ฎ๐‘˜:0 ๐’ฎ๐‘˜:0 VERB cana-918 149 9 โ‰ค โ‰ค NOUN cana-918 149 10 ๐‘˜ ๐‘˜ PRON cana-918 149 11 < < X cana-918 149 12 96 96 NUM cana-918 149 13 = = SYM cana-918 149 14 ๐’ซ. ๐’ซ. PROPN cana-918 149 15 ๐’ฎ0 ๐’ฎ0 NOUN cana-918 149 16 = = SYM cana-918 149 17 { { PUNCT cana-918 149 18 ( ( PUNCT cana-918 149 19 ๐‘€,๐‘€):๐‘€ ๐‘€,๐‘€):๐‘€ PROPN cana-918 149 20 โˆˆ โˆˆ PROPN cana-918 149 21 ๐บ๐ฟ(2 ๐บ๐ฟ(2 NOUN cana-918 149 22 , , PUNCT cana-918 149 23 โ„ค4 โ„ค4 PROPN cana-918 149 24 ) ) PUNCT cana-918 149 25 } } PUNCT cana-918 149 26 is be AUX cana-918 149 27 an an DET cana-918 149 28 identity identity NOUN cana-918 149 29 relation relation NOUN cana-918 149 30 . . PUNCT cana-918 150 1 let let VERB cana-918 150 2 ๐’ฎ๐‘™ ๐’ฎ๐‘™ PROPN cana-918 150 3 , , PUNCT cana-918 150 4 ๐’ฎ๐‘š ๐’ฎ๐‘š NOUN cana-918 150 5 , , PUNCT cana-918 150 6 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 150 7 be be VERB cana-918 150 8 arbitrary arbitrary ADJ cana-918 150 9 relations relation NOUN cana-918 150 10 in in ADP cana-918 150 11 ๐’ซ ๐’ซ PROPN cana-918 150 12 and and CCONJ cana-918 150 13 ( ( PUNCT cana-918 150 14 ๐‘€,๐‘ ๐‘€,๐‘ NOUN cana-918 150 15 ) ) PUNCT cana-918 150 16 be be AUX cana-918 150 17 any any DET cana-918 150 18 element element NOUN cana-918 150 19 of of ADP cana-918 150 20 ๐’ฎ๐‘˜. ๐’ฎ๐‘˜. PROPN cana-918 150 21 since since SCONJ cana-918 150 22 ๐‘€ ๐‘€ PROPN cana-918 150 23 โˆˆ โˆˆ PROPN cana-918 150 24 ๐‘Œ ๐‘Œ PROPN cana-918 150 25 , , PUNCT cana-918 150 26 ๐‘€ ๐‘€ PROPN cana-918 150 27 is be AUX cana-918 150 28 of of ADP cana-918 150 29 the the DET cana-918 150 30 form form NOUN cana-918 150 31 ๐ด๐‘Ž๐ถ๐‘๐‘Œ๐‘๐‘‹๐‘‘ ๐ด๐‘Ž๐ถ๐‘๐‘Œ๐‘๐‘‹๐‘‘ ADV cana-918 150 32 for for ADP cana-918 150 33 some some DET cana-918 150 34 ๐‘Ž ๐‘Ž NOUN cana-918 150 35 , , PUNCT cana-918 150 36 ๐‘ ๐‘ NOUN cana-918 150 37 , , PUNCT cana-918 150 38 ๐‘ ๐‘ PROPN cana-918 150 39 , , PUNCT cana-918 150 40 ๐‘‘ ๐‘‘ X cana-918 150 41 ( ( PUNCT cana-918 150 42 from from ADP cana-918 150 43 table table NOUN cana-918 150 44 [ [ X cana-918 150 45 table table NOUN cana-918 150 46 : : PUNCT cana-918 150 47 canonical canonical ADJ cana-918 150 48 forms form NOUN cana-918 150 49 an an DET cana-918 150 50 , , PUNCT cana-918 150 51 sn sn NOUN cana-918 150 52 ] ] PUNCT cana-918 150 53 ) ) PUNCT cana-918 150 54 . . PUNCT cana-918 151 1 suppose suppose VERB cana-918 151 2 ๐‘€๐’ฎ๐‘™ ๐‘€๐’ฎ๐‘™ NOUN cana-918 151 3 = = SYM cana-918 151 4 ๐‘€ ๐‘€ PROPN cana-918 151 5 โ€ฒ โ€ฒ NOUN cana-918 151 6 and and CCONJ cana-918 151 7 ๐‘๐’ฎ๐‘š ๐‘๐’ฎ๐‘š NOUN cana-918 151 8 โˆ— โˆ— NOUN cana-918 151 9 = = SYM cana-918 151 10 ๐‘โ€ฒ ๐‘โ€ฒ X cana-918 151 11 where where SCONJ cana-918 151 12 ๐‘€โ€ฒ ๐‘€โ€ฒ NOUN cana-918 151 13 , , PUNCT cana-918 151 14 ๐‘โ€ฒ ๐‘โ€ฒ X cana-918 151 15 โˆˆ โˆˆ PROPN cana-918 151 16 ๐‘Œ. ๐‘Œ. PROPN cana-918 151 17 now now ADV cana-918 151 18 ( ( PUNCT cana-918 151 19 ๐‘€ ๐‘€ PROPN cana-918 151 20 , , PUNCT cana-918 151 21 ๐‘ ๐‘ PROPN cana-918 151 22 ) ) PUNCT cana-918 151 23 โˆˆ โˆˆ PROPN cana-918 151 24 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 151 25 , , PUNCT cana-918 151 26 ( ( PUNCT cana-918 151 27 ๐‘€,๐‘€โ€ฒ ๐‘€,๐‘€โ€ฒ NOUN cana-918 151 28 ) ) PUNCT cana-918 151 29 โˆˆ โˆˆ PROPN cana-918 151 30 ๐’ฎ๐‘™ ๐’ฎ๐‘™ PROPN cana-918 151 31 and and CCONJ cana-918 151 32 ( ( PUNCT cana-918 151 33 ๐‘โ€ฒ ๐‘โ€ฒ NOUN cana-918 151 34 , , PUNCT cana-918 151 35 ๐‘ ๐‘ PROPN cana-918 151 36 ) ) PUNCT cana-918 151 37 โˆˆ โˆˆ PROPN cana-918 151 38 ๐’ฎ๐‘š ๐’ฎ๐‘š NOUN cana-918 151 39 , , PUNCT cana-918 151 40 implies imply VERB cana-918 151 41 ๐‘ ๐‘ PROPN cana-918 151 42 = = SYM cana-918 151 43 ๐ด(๐‘Ž+๐‘  ๐ด(๐‘Ž+๐‘  PROPN cana-918 151 44 ( ( PUNCT cana-918 151 45 ๐‘˜ ๐‘˜ NOUN cana-918 151 46 ) ) PUNCT cana-918 151 47 ) ) PUNCT cana-918 151 48 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 151 49 2 2 NUM cana-918 151 50 ๐ถ(๐‘+๐‘ก ๐ถ(๐‘+๐‘ก X cana-918 151 51 ( ( PUNCT cana-918 151 52 ๐‘˜ ๐‘˜ NOUN cana-918 151 53 ) ) PUNCT cana-918 151 54 ) ) PUNCT cana-918 151 55 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 151 56 4 4 NUM cana-918 151 57 ๐‘Œ(๐‘+๐‘Ÿ ๐‘Œ(๐‘+๐‘Ÿ NOUN cana-918 151 58 ( ( PUNCT cana-918 151 59 ๐‘˜ ๐‘˜ NOUN cana-918 151 60 ) ) PUNCT cana-918 151 61 ) ) PUNCT cana-918 151 62 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 151 63 4 4 NUM cana-918 151 64 ๐‘‹(๐‘‘+๐‘› ๐‘‹(๐‘‘+๐‘› NOUN cana-918 151 65 ( ( PUNCT cana-918 151 66 ๐‘˜ ๐‘˜ NOUN cana-918 151 67 ) ) PUNCT cana-918 151 68 ) ) PUNCT cana-918 151 69 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 151 70 3 3 NUM cana-918 151 71 , , PUNCT cana-918 151 72 mโ€ฒ mโ€ฒ NOUN cana-918 151 73 = = SYM cana-918 151 74 โ€ˆ โ€ˆ SPACE cana-918 151 75 a(a+s a(a+s PUNCT cana-918 151 76 ( ( PUNCT cana-918 151 77 l l NOUN cana-918 151 78 ) ) PUNCT cana-918 151 79 ) ) PUNCT cana-918 151 80 mod mod PROPN cana-918 151 81 โ€‰ โ€‰ SPACE cana-918 151 82 2 2 NUM cana-918 151 83 โ€ˆ โ€ˆ SPACE cana-918 151 84 c(b+t c(b+t PROPN cana-918 151 85 ( ( PUNCT cana-918 151 86 l l NOUN cana-918 151 87 ) ) PUNCT cana-918 151 88 ) ) PUNCT cana-918 151 89 mod mod PROPN cana-918 151 90 โ€‰ โ€‰ SPACE cana-918 151 91 4 4 NUM cana-918 151 92 โ€ˆ โ€ˆ SPACE cana-918 151 93 y(c+r y(c+r PROPN cana-918 151 94 ( ( PUNCT cana-918 151 95 l l NOUN cana-918 151 96 ) ) PUNCT cana-918 151 97 ) ) PUNCT cana-918 152 1 mod mod PROPN cana-918 152 2 โ€‰ โ€‰ SPACE cana-918 152 3 4 4 NUM cana-918 152 4 โ€ˆ โ€ˆ SPACE cana-918 152 5 x(d+n x(d+n PUNCT cana-918 153 1 ( ( PUNCT cana-918 153 2 l l NOUN cana-918 153 3 ) ) PUNCT cana-918 153 4 ) ) PUNCT cana-918 154 1 mod mod PROPN cana-918 154 2 โ€‰ โ€‰ SPACE cana-918 154 3 3 3 NUM cana-918 154 4 , , PUNCT cana-918 154 5 and and CCONJ cana-918 154 6 nโ€ฒ nโ€ฒ PROPN cana-918 154 7 = = SYM cana-918 154 8 a(a+s a(a+s PROPN cana-918 154 9 ( ( PUNCT cana-918 154 10 k)โˆ’s(m k)โˆ’s(m NOUN cana-918 154 11 ) ) PUNCT cana-918 154 12 ) ) PUNCT cana-918 155 1 mod mod PROPN cana-918 155 2 โ€‰ โ€‰ SPACE cana-918 155 3 2 2 NUM cana-918 155 4 c(b+t c(b+t PROPN cana-918 155 5 ( ( PUNCT cana-918 155 6 k)โˆ’t(m k)โˆ’t(m NOUN cana-918 155 7 ) ) PUNCT cana-918 155 8 ) ) PUNCT cana-918 155 9 mod mod PROPN cana-918 155 10 โ€‰ โ€‰ SPACE cana-918 155 11 4 4 NUM cana-918 155 12 y(c+r y(c+r PROPN cana-918 155 13 ( ( PUNCT cana-918 155 14 k)โˆ’r(m k)โˆ’r(m NOUN cana-918 155 15 ) ) PUNCT cana-918 155 16 ) ) PUNCT cana-918 156 1 mod mod PROPN cana-918 156 2 โ€‰ โ€‰ SPACE cana-918 156 3 4 4 NUM cana-918 156 4 x(d+n x(d+n PROPN cana-918 156 5 ( ( PUNCT cana-918 156 6 k)โˆ’n(m k)โˆ’n(m NOUN cana-918 156 7 ) ) PUNCT cana-918 156 8 ) ) PUNCT cana-918 157 1 mod mod PROPN cana-918 157 2 โ€‰ โ€‰ SPACE cana-918 157 3 3 3 NUM cana-918 157 4 respectively respectively ADV cana-918 157 5 , , PUNCT cana-918 157 6 where where SCONJ cana-918 157 7 ๐‘˜ ๐‘˜ PRON cana-918 157 8 = = SYM cana-918 157 9 48๐‘ (๐‘˜ 48๐‘ (๐‘˜ NUM cana-918 157 10 ) ) PUNCT cana-918 157 11 + + X cana-918 157 12 16๐‘›(๐‘˜ 16๐‘›(๐‘˜ NUM cana-918 157 13 ) ) PUNCT cana-918 157 14 + + NUM cana-918 157 15 4๐‘ก(๐‘˜ 4๐‘ก(๐‘˜ NUM cana-918 157 16 ) ) PUNCT cana-918 158 1 + + CCONJ cana-918 158 2 ๐‘Ÿ(๐‘˜ ๐‘Ÿ(๐‘˜ NUM cana-918 158 3 ) ) PUNCT cana-918 158 4 , , PUNCT cana-918 158 5 ๐‘™ ๐‘™ PROPN cana-918 158 6 = = SYM cana-918 158 7 48๐‘ (๐‘™ 48๐‘ (๐‘™ NUM cana-918 158 8 ) ) PUNCT cana-918 158 9 + + NUM cana-918 158 10 16๐‘›(๐‘™ 16๐‘›(๐‘™ NUM cana-918 158 11 ) ) PUNCT cana-918 159 1 + + CCONJ cana-918 159 2 4๐‘ก(๐‘™ 4๐‘ก(๐‘™ NUM cana-918 159 3 ) ) PUNCT cana-918 160 1 + + CCONJ cana-918 160 2 ๐‘Ÿ(๐‘™ ๐‘Ÿ(๐‘™ NOUN cana-918 160 3 ) ) PUNCT cana-918 160 4 and and CCONJ cana-918 160 5 ๐‘š ๐‘š X cana-918 160 6 = = SYM cana-918 160 7 48๐‘ (๐‘š 48๐‘ (๐‘š PROPN cana-918 160 8 ) ) PUNCT cana-918 161 1 + + CCONJ cana-918 161 2 16๐‘›(๐‘š 16๐‘›(๐‘š X cana-918 161 3 ) ) PUNCT cana-918 162 1 + + CCONJ cana-918 162 2 4๐‘ก(๐‘š 4๐‘ก(๐‘š NUM cana-918 162 3 ) ) PUNCT cana-918 163 1 + + CCONJ cana-918 163 2 ๐‘Ÿ(๐‘š ๐‘Ÿ(๐‘š NOUN cana-918 163 3 ) ) PUNCT cana-918 163 4 , , PUNCT cana-918 163 5 for for ADP cana-918 163 6 some some DET cana-918 163 7 ๐‘ (๐‘˜ ๐‘ (๐‘˜ NOUN cana-918 163 8 ) ) PUNCT cana-918 163 9 , , PUNCT cana-918 163 10 ๐‘ (๐‘™ ๐‘ (๐‘™ PROPN cana-918 163 11 ) ) PUNCT cana-918 163 12 , , PUNCT cana-918 163 13 ๐‘ (๐‘š ๐‘ (๐‘š PROPN cana-918 163 14 ) ) PUNCT cana-918 163 15 โˆˆ โˆˆ PROPN cana-918 163 16 โ„ค2 โ„ค2 PROPN cana-918 163 17 ; ; PUNCT cana-918 163 18 ๐‘› ๐‘› PROPN cana-918 163 19 ( ( PUNCT cana-918 163 20 ๐‘˜ ๐‘˜ NOUN cana-918 163 21 ) ) PUNCT cana-918 163 22 , , PUNCT cana-918 163 23 ๐‘›(๐‘™ ๐‘›(๐‘™ PROPN cana-918 163 24 ) ) PUNCT cana-918 163 25 , , PUNCT cana-918 163 26 ๐‘›(๐‘š ๐‘›(๐‘š NOUN cana-918 163 27 ) ) PUNCT cana-918 163 28 โˆˆ โˆˆ PROPN cana-918 163 29 โ„ค3 โ„ค3 NOUN cana-918 163 30 and and CCONJ cana-918 163 31 ๐‘ก(๐‘˜ ๐‘ก(๐‘˜ NOUN cana-918 163 32 ) ) PUNCT cana-918 163 33 , , PUNCT cana-918 163 34 ๐‘ก(๐‘™ ๐‘ก(๐‘™ PROPN cana-918 163 35 ) ) PUNCT cana-918 163 36 , , PUNCT cana-918 163 37 ๐‘ก(๐‘š ๐‘ก(๐‘š NOUN cana-918 163 38 ) ) PUNCT cana-918 163 39 , , PUNCT cana-918 163 40 ๐‘Ÿ(๐‘˜ ๐‘Ÿ(๐‘˜ PROPN cana-918 163 41 ) ) PUNCT cana-918 163 42 , , PUNCT cana-918 163 43 ๐‘Ÿ(๐‘™ ๐‘Ÿ(๐‘™ PROPN cana-918 163 44 ) ) PUNCT cana-918 163 45 , , PUNCT cana-918 163 46 ๐‘Ÿ(๐‘š ๐‘Ÿ(๐‘š PROPN cana-918 163 47 ) ) PUNCT cana-918 163 48 โˆˆ โˆˆ PROPN cana-918 163 49 โ„ค4 โ„ค4 NOUN cana-918 163 50 . . PUNCT cana-918 164 1 since since SCONJ cana-918 164 2 every every DET cana-918 164 3 pair pair NOUN cana-918 164 4 ( ( PUNCT cana-918 164 5 ๐‘€,๐‘ ๐‘€,๐‘ NOUN cana-918 164 6 ) ) PUNCT cana-918 164 7 in in ADP cana-918 164 8 ๐’ซ ๐’ซ PROPN cana-918 164 9 are be AUX cana-918 164 10 ๐‘˜๐‘กโ„Ž ๐‘˜๐‘กโ„Ž VERB cana-918 164 11 associates associate NOUN cana-918 164 12 for for ADP cana-918 164 13 exactly exactly ADV cana-918 164 14 one one NUM cana-918 164 15 ๐‘˜ ๐‘˜ NOUN cana-918 164 16 , , PUNCT cana-918 164 17 we we PRON cana-918 164 18 find find VERB cana-918 164 19 that that SCONJ cana-918 164 20 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š PROPN cana-918 164 21 ๐‘˜ ๐‘˜ PRON cana-918 164 22 can can AUX cana-918 164 23 be be AUX cana-918 164 24 either either CCONJ cana-918 164 25 0 0 NUM cana-918 164 26 or or CCONJ cana-918 164 27 1 1 NUM cana-918 164 28 . . PUNCT cana-918 165 1 hence hence ADV cana-918 165 2 we we PRON cana-918 165 3 obtain obtain VERB cana-918 165 4 that that DET cana-918 165 5 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š PROPN cana-918 165 6 ๐‘˜ ๐‘˜ NOUN cana-918 165 7 = = NOUN cana-918 165 8 1 1 NUM cana-918 165 9 if if SCONJ cana-918 166 1 and and CCONJ cana-918 166 2 only only ADV cana-918 166 3 if if SCONJ cana-918 166 4 ๐‘€โ€ฒ ๐‘€โ€ฒ NOUN cana-918 166 5 = = PUNCT cana-918 166 6 ๐‘โ€ฒ ๐‘โ€ฒ NOUN cana-918 166 7 that that ADV cana-918 166 8 is be AUX cana-918 166 9 , , PUNCT cana-918 166 10 if if SCONJ cana-918 166 11 ๐‘ (๐‘˜ ๐‘ (๐‘˜ NOUN cana-918 166 12 ) ) PUNCT cana-918 166 13 โ‰ก โ‰ก PROPN cana-918 166 14 ( ( PUNCT cana-918 166 15 ๐‘ (๐‘™ ๐‘ (๐‘™ PROPN cana-918 166 16 ) ) PUNCT cana-918 166 17 + + CCONJ cana-918 166 18 ๐‘ (๐‘š ๐‘ (๐‘š NOUN cana-918 166 19 ) ) PUNCT cana-918 166 20 ) ) PUNCT cana-918 166 21 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 166 22 2 2 NUM cana-918 166 23 , , PUNCT cana-918 166 24 ๐‘ก(๐‘˜ ๐‘ก(๐‘˜ NOUN cana-918 166 25 ) ) PUNCT cana-918 166 26 โ‰ก โ‰ก PROPN cana-918 166 27 ( ( PUNCT cana-918 166 28 ๐‘ก(๐‘™ ๐‘ก(๐‘™ PROPN cana-918 166 29 ) ) PUNCT cana-918 166 30 + + NUM cana-918 166 31 ๐‘ก(๐‘š ๐‘ก(๐‘š NOUN cana-918 166 32 ) ) PUNCT cana-918 166 33 ) ) PUNCT cana-918 166 34 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 166 35 4 4 NUM cana-918 166 36 , , PUNCT cana-918 166 37 ๐‘Ÿ(๐‘˜ ๐‘Ÿ(๐‘˜ NUM cana-918 166 38 ) ) PUNCT cana-918 166 39 โ‰ก โ‰ก PROPN cana-918 166 40 ( ( PUNCT cana-918 166 41 ๐‘Ÿ(๐‘™ ๐‘Ÿ(๐‘™ PROPN cana-918 166 42 ) ) PUNCT cana-918 166 43 + + CCONJ cana-918 166 44 ๐‘Ÿ(๐‘š ๐‘Ÿ(๐‘š NOUN cana-918 166 45 ) ) PUNCT cana-918 166 46 ) ) PUNCT cana-918 166 47 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 166 48 4 4 NUM cana-918 166 49 and and CCONJ cana-918 166 50 ๐‘›(๐‘˜ ๐‘›(๐‘˜ PROPN cana-918 166 51 ) ) PUNCT cana-918 166 52 โ‰ก โ‰ก PROPN cana-918 166 53 ( ( PUNCT cana-918 166 54 ๐‘›(๐‘™ ๐‘›(๐‘™ PROPN cana-918 166 55 ) ) PUNCT cana-918 166 56 + + SYM cana-918 166 57 ๐‘›(๐‘š ๐‘›(๐‘š NOUN cana-918 166 58 ) ) PUNCT cana-918 166 59 ) ) PUNCT cana-918 166 60 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 166 61 3 3 X cana-918 166 62 . . PUNCT cana-918 166 63 โ—ป โ—ป PROPN cana-918 166 64 theorem theorem VERB cana-918 166 65 6 6 NUM cana-918 166 66 . . PUNCT cana-918 167 1 let let VERB cana-918 167 2 ๐‘Œ ๐‘Œ PROPN cana-918 167 3 = = SYM cana-918 167 4 ๐บ๐ฟ(2 ๐บ๐ฟ(2 PROPN cana-918 167 5 , , PUNCT cana-918 167 6 โ„ค6 โ„ค6 NOUN cana-918 167 7 ) ) PUNCT cana-918 167 8 and and CCONJ cana-918 167 9 ๐’ซ ๐’ซ NOUN cana-918 167 10 be be VERB cana-918 167 11 a a DET cana-918 167 12 partition partition NOUN cana-918 167 13 of of ADP cana-918 167 14 ๐‘Œ ๐‘Œ PROPN cana-918 167 15 ร— ร— PROPN cana-918 167 16 ๐‘Œ. ๐‘Œ. PROPN cana-918 167 17 for for ADP cana-918 167 18 all all PRON cana-918 167 19 ๐‘˜ ๐‘˜ PRON cana-918 167 20 written write VERB cana-918 167 21 in in ADP cana-918 167 22 form form NOUN cana-918 167 23 of of ADP cana-918 167 24 144๐‘  144๐‘  NOUN cana-918 167 25 + + CCONJ cana-918 167 26 72๐‘› 72๐‘› NOUN cana-918 168 1 + + CCONJ cana-918 168 2 12๐‘ก 12๐‘ก NOUN cana-918 168 3 + + SYM cana-918 168 4 ๐‘Ÿ ๐‘Ÿ NOUN cana-918 168 5 ( ( PUNCT cana-918 168 6 ๐‘  ๐‘  PROPN cana-918 168 7 , , PUNCT cana-918 168 8 ๐‘› ๐‘› PROPN cana-918 168 9 โˆˆ โˆˆ PROPN cana-918 168 10 โ„ค2 โ„ค2 PROPN cana-918 168 11 , , PUNCT cana-918 168 12 ๐‘ก ๐‘ก PROPN cana-918 168 13 โˆˆ โˆˆ PROPN cana-918 168 14 โ„ค6 โ„ค6 NOUN cana-918 168 15 , , PUNCT cana-918 168 16 ๐‘Ÿ ๐‘Ÿ X cana-918 168 17 โˆˆ โˆˆ PROPN cana-918 168 18 โ„ค12 โ„ค12 PROPN cana-918 168 19 ) ) PUNCT cana-918 168 20 , , PUNCT cana-918 168 21 the the DET cana-918 168 22 relations relation NOUN cana-918 168 23 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 168 24 in in ADP cana-918 168 25 ๐’ซ ๐’ซ NOUN cana-918 168 26 defined define VERB cana-918 168 27 by by ADP cana-918 168 28 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 168 29 = = SYM cana-918 168 30 { { PUNCT cana-918 168 31 ( ( PUNCT cana-918 168 32 ๐ถ ๐ถ PROPN cana-918 168 33 ๐‘Ž๐‘‹๐‘๐‘Œ๐‘๐ต๐‘‘ ๐‘Ž๐‘‹๐‘๐‘Œ๐‘๐ต๐‘‘ PROPN cana-918 168 34 , , PUNCT cana-918 168 35 ๐ถ๐‘Ž+๐‘ ๐‘‹๐‘+๐‘Ÿ๐‘Œ๐‘+๐‘ก๐ต๐‘‘+๐‘› ๐ถ๐‘Ž+๐‘ ๐‘‹๐‘+๐‘Ÿ๐‘Œ๐‘+๐‘ก๐ต๐‘‘+๐‘› PROPN cana-918 168 36 ) ) PUNCT cana-918 168 37 | | ADV cana-918 168 38 ๐‘Ž ๐‘Ž X cana-918 168 39 , , PUNCT cana-918 168 40 ๐‘‘ ๐‘‘ PROPN cana-918 168 41 โˆˆ โˆˆ PROPN cana-918 168 42 โ„ค2 โ„ค2 PROPN cana-918 168 43 ; ; PUNCT cana-918 168 44 ๐‘ ๐‘ PROPN cana-918 168 45 โˆˆ โˆˆ PROPN cana-918 168 46 โ„ค12 โ„ค12 PROPN cana-918 168 47 ; ; PUNCT cana-918 168 48 ๐‘ ๐‘ PROPN cana-918 168 49 โˆˆ โˆˆ PROPN cana-918 168 50 โ„ค6 โ„ค6 NOUN cana-918 168 51 } } PUNCT cana-918 168 52 is be AUX cana-918 168 53 a a DET cana-918 168 54 non non ADJ cana-918 168 55 - - ADJ cana-918 168 56 symmetric symmetric ADJ cana-918 168 57 association association NOUN cana-918 168 58 scheme scheme NOUN cana-918 168 59 with with ADP cana-918 168 60 parameters parameter NOUN cana-918 168 61 communications communication NOUN cana-918 168 62 on on ADP cana-918 168 63 applied apply VERB cana-918 168 64 nonlinear nonlinear ADJ cana-918 168 65 analysis analysis NOUN cana-918 168 66 issn issn NOUN cana-918 168 67 : : PUNCT cana-918 168 68 1074 1074 NUM cana-918 168 69 - - PUNCT cana-918 168 70 133x 133x NUM cana-918 168 71 vol vol NOUN cana-918 168 72 31 31 NUM cana-918 168 73 no no NOUN cana-918 168 74 . . PUNCT cana-918 169 1 4s 4s NUM cana-918 169 2 ( ( PUNCT cana-918 169 3 2024 2024 NUM cana-918 169 4 ) ) PUNCT cana-918 169 5 398 398 NUM cana-918 169 6 https://internationalpubls.com https://internationalpubls.com X cana-918 169 7 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š PROPN cana-918 169 8 ๐‘˜ ๐‘˜ NOUN cana-918 169 9 = = PUNCT cana-918 169 10 { { PUNCT cana-918 169 11 1 1 NUM cana-918 169 12 ๐‘–๐‘“ ๐‘–๐‘“ PRON cana-918 169 13 ๐‘ (๐‘˜ ๐‘ (๐‘˜ NOUN cana-918 169 14 ) ) PUNCT cana-918 169 15 โ‰ก โ‰ก PROPN cana-918 169 16 ( ( PUNCT cana-918 169 17 ๐‘ (๐‘™ ๐‘ (๐‘™ PROPN cana-918 169 18 ) ) PUNCT cana-918 169 19 + + CCONJ cana-918 169 20 ๐‘ (๐‘š ๐‘ (๐‘š NOUN cana-918 169 21 ) ) PUNCT cana-918 169 22 ) ) PUNCT cana-918 169 23 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 169 24 2 2 NUM cana-918 169 25 , , PUNCT cana-918 169 26 ๐‘›(๐‘˜ ๐‘›(๐‘˜ PROPN cana-918 169 27 ) ) PUNCT cana-918 169 28 โ‰ก โ‰ก PROPN cana-918 169 29 ( ( PUNCT cana-918 169 30 ๐‘›(๐‘™ ๐‘›(๐‘™ PROPN cana-918 169 31 ) ) PUNCT cana-918 169 32 + + SYM cana-918 169 33 ๐‘›(๐‘š ๐‘›(๐‘š NOUN cana-918 169 34 ) ) PUNCT cana-918 169 35 ) ) PUNCT cana-918 169 36 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 169 37 2 2 NUM cana-918 169 38 , , PUNCT cana-918 169 39 ๐‘ก(๐‘˜ ๐‘ก(๐‘˜ NOUN cana-918 169 40 ) ) PUNCT cana-918 169 41 โ‰ก โ‰ก PROPN cana-918 169 42 ( ( PUNCT cana-918 169 43 ๐‘ก(๐‘™ ๐‘ก(๐‘™ PROPN cana-918 169 44 ) ) PUNCT cana-918 169 45 + + NUM cana-918 169 46 ๐‘ก(๐‘š ๐‘ก(๐‘š NOUN cana-918 169 47 ) ) PUNCT cana-918 169 48 ) ) PUNCT cana-918 169 49 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 169 50 6 6 NUM cana-918 169 51 , , PUNCT cana-918 169 52 ๐‘Ž๐‘›๐‘‘ ๐‘Ž๐‘›๐‘‘ ADJ cana-918 169 53 ๐‘Ÿ(๐‘˜ ๐‘Ÿ(๐‘˜ PROPN cana-918 169 54 ) ) PUNCT cana-918 169 55 โ‰ก โ‰ก PROPN cana-918 169 56 ( ( PUNCT cana-918 169 57 ๐‘Ÿ(๐‘™ ๐‘Ÿ(๐‘™ PROPN cana-918 169 58 ) ) PUNCT cana-918 169 59 + + CCONJ cana-918 169 60 ๐‘Ÿ(๐‘š ๐‘Ÿ(๐‘š NOUN cana-918 169 61 ) ) PUNCT cana-918 169 62 ) ) PUNCT cana-918 170 1 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 170 2 12 12 NUM cana-918 170 3 0 0 NUM cana-918 170 4 ๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’ ๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’ NOUN cana-918 170 5 proof proof NOUN cana-918 170 6 . . PUNCT cana-918 171 1 here here ADV cana-918 171 2 , , PUNCT cana-918 171 3 |๐‘๐‘˜| |๐‘๐‘˜| X cana-918 171 4 = = NOUN cana-918 171 5 |๐‘Œ| |๐‘Œ| VERB cana-918 171 6 for for ADP cana-918 171 7 all all DET cana-918 171 8 0 0 NUM cana-918 171 9 โ‰ค โ‰ค NOUN cana-918 171 10 ๐‘˜ ๐‘˜ PRON cana-918 171 11 < < X cana-918 171 12 288 288 NUM cana-918 171 13 . . PUNCT cana-918 172 1 the the DET cana-918 172 2 relations relation NOUN cana-918 172 3 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 172 4 are be AUX cana-918 172 5 disjoint disjoint ADJ cana-918 172 6 and and CCONJ cana-918 172 7 โˆช โˆช ADJ cana-918 172 8 ๐’ฎ๐‘˜:0 ๐’ฎ๐‘˜:0 VERB cana-918 172 9 โ‰ค โ‰ค NOUN cana-918 172 10 ๐‘˜ ๐‘˜ DET cana-918 172 11 < < X cana-918 172 12 288 288 NUM cana-918 172 13 = = SYM cana-918 172 14 ๐’ซ. ๐’ซ. NOUN cana-918 172 15 proceeding proceeding NOUN cana-918 172 16 as as ADP cana-918 172 17 in in ADP cana-918 172 18 proof proof NOUN cana-918 172 19 of of ADP cana-918 172 20 theorem theorem NOUN cana-918 172 21 5 5 NUM cana-918 172 22 , , PUNCT cana-918 172 23 we we PRON cana-918 172 24 can can AUX cana-918 172 25 find find VERB cana-918 172 26 cardinality cardinality NOUN cana-918 172 27 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š PROPN cana-918 172 28 ๐‘˜ ๐‘˜ INTJ cana-918 172 29 such such ADJ cana-918 172 30 that that PRON cana-918 172 31 for for ADP cana-918 172 32 all all PRON cana-918 172 33 ( ( PUNCT cana-918 172 34 ๐‘ฅ ๐‘ฅ PROPN cana-918 172 35 , , PUNCT cana-918 172 36 ๐‘ฆ ๐‘ฆ NOUN cana-918 172 37 ) ) PUNCT cana-918 172 38 โˆˆ โˆˆ PROPN cana-918 173 1 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 173 2 , , PUNCT cana-918 173 3 |๐‘ฅ๐’ฎ๐‘™ |๐‘ฅ๐’ฎ๐‘™ ADJ cana-918 173 4 โˆฉ โˆฉ NOUN cana-918 173 5 ๐‘ฆ๐’ฎ๐‘š ๐‘ฆ๐’ฎ๐‘š NOUN cana-918 173 6 โˆ— โˆ— NOUN cana-918 173 7 | | ADV cana-918 174 1 = = NOUN cana-918 175 1 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š PROPN cana-918 176 1 ๐‘˜ ๐‘˜ PRON cana-918 176 2 is be AUX cana-918 176 3 a a DET cana-918 176 4 constant constant ADJ cana-918 176 5 . . PUNCT cana-918 177 1 โ—ป โ—ป PROPN cana-918 177 2 note note NOUN cana-918 177 3 : : PUNCT cana-918 177 4 let let VERB cana-918 177 5 ๐‘†๐ฟ(2 ๐‘†๐ฟ(2 PROPN cana-918 177 6 , , PUNCT cana-918 177 7 โ„ค๐‘ โ„ค๐‘ PROPN cana-918 177 8 ) ) PUNCT cana-918 177 9 be be AUX cana-918 177 10 the the DET cana-918 177 11 special special ADJ cana-918 177 12 linear linear NOUN cana-918 177 13 group group NOUN cana-918 177 14 over over ADP cana-918 177 15 โ„ค๐‘ โ„ค๐‘ PROPN cana-918 177 16 and and CCONJ cana-918 177 17 ๐บ๐ฟ(2 ๐บ๐ฟ(2 NOUN cana-918 177 18 , , PUNCT cana-918 177 19 โ„ค๐‘ โ„ค๐‘ PROPN cana-918 177 20 ) ) PUNCT cana-918 177 21 be be AUX cana-918 177 22 general general ADJ cana-918 177 23 linear linear ADJ cana-918 177 24 group group NOUN cana-918 177 25 over over ADP cana-918 177 26 โ„ค๐‘. โ„ค๐‘. PROPN cana-918 177 27 the the DET cana-918 177 28 structure structure NOUN cana-918 177 29 of of ADP cana-918 177 30 these these DET cana-918 177 31 conjugacy conjugacy PROPN cana-918 177 32 classes class NOUN cana-918 177 33 are be AUX cana-918 177 34 worked work VERB cana-918 177 35 out out ADP cana-918 177 36 in in ADP cana-918 177 37 detail detail NOUN cana-918 177 38 in in ADP cana-918 177 39 [ [ X cana-918 177 40 16 16 NUM cana-918 177 41 ] ] PUNCT cana-918 177 42 . . PUNCT cana-918 178 1 using use VERB cana-918 178 2 these these DET cana-918 178 3 classes class NOUN cana-918 178 4 one one NUM cana-918 178 5 can can AUX cana-918 178 6 compute compute VERB cana-918 178 7 group group NOUN cana-918 178 8 association association NOUN cana-918 178 9 scheme scheme NOUN cana-918 178 10 for for ADP cana-918 178 11 ๐‘†๐ฟ(2 ๐‘†๐ฟ(2 PROPN cana-918 178 12 , , PUNCT cana-918 178 13 โ„ค๐‘ โ„ค๐‘ PROPN cana-918 178 14 ) ) PUNCT cana-918 178 15 and and CCONJ cana-918 178 16 ๐บ๐ฟ(2 ๐บ๐ฟ(2 NOUN cana-918 178 17 , , PUNCT cana-918 178 18 โ„ค๐‘ โ„ค๐‘ PROPN cana-918 178 19 ) ) PUNCT cana-918 178 20 using use VERB cana-918 178 21 definition definition NOUN cana-918 178 22 2 2 NUM cana-918 178 23 . . PUNCT cana-918 178 24 let let VERB cana-918 178 25 us we PRON cana-918 178 26 represent represent VERB cana-918 178 27 ๐‘†๐‘› ๐‘†๐‘› PROPN cana-918 178 28 as as SCONJ cana-918 178 29 the the DET cana-918 178 30 symmetric symmetric ADJ cana-918 178 31 group group NOUN cana-918 178 32 of of ADP cana-918 178 33 degree degree NOUN cana-918 178 34 ๐‘›. ๐‘›. NOUN cana-918 178 35 group group NOUN cana-918 178 36 association association NOUN cana-918 178 37 scheme scheme NOUN cana-918 178 38 for for ADP cana-918 178 39 symmetric symmetric ADJ cana-918 178 40 groups group NOUN cana-918 178 41 have have AUX cana-918 178 42 been be AUX cana-918 178 43 described describe VERB cana-918 178 44 by by ADP cana-918 178 45 tomiyama tomiyama PROPN cana-918 178 46 and and CCONJ cana-918 178 47 yamazaki yamazaki PROPN cana-918 178 48 in in ADP cana-918 178 49 [ [ X cana-918 178 50 17 17 NUM cana-918 178 51 ] ] PUNCT cana-918 178 52 . . PUNCT cana-918 179 1 in in ADP cana-918 179 2 [ [ X cana-918 179 3 13 13 NUM cana-918 179 4 ] ] PUNCT cana-918 179 5 , , PUNCT cana-918 179 6 sabharwal sabharwal PROPN cana-918 179 7 et et PROPN cana-918 179 8 al al PROPN cana-918 179 9 . . PROPN cana-918 179 10 identified identify VERB cana-918 179 11 the the DET cana-918 179 12 association association NOUN cana-918 179 13 schemes scheme NOUN cana-918 179 14 for for SCONJ cana-918 179 15 the the DET cana-918 179 16 symmetric symmetric ADJ cana-918 179 17 groups group NOUN cana-918 179 18 ๐‘†3 ๐‘†3 PROPN cana-918 179 19 and and CCONJ cana-918 179 20 ๐‘†4 ๐‘†4 PROPN cana-918 179 21 without without ADP cana-918 179 22 using use VERB cana-918 179 23 the the DET cana-918 179 24 conjugacy conjugacy PROPN cana-918 179 25 classes class NOUN cana-918 179 26 . . PUNCT cana-918 180 1 in in ADP cana-918 180 2 next next ADJ cana-918 180 3 theorem theorem NOUN cana-918 180 4 , , PUNCT cana-918 180 5 we we PRON cana-918 180 6 have have AUX cana-918 180 7 determined determine VERB cana-918 180 8 non non ADJ cana-918 180 9 symmetric symmetric PROPN cana-918 180 10 commutative commutative PROPN cana-918 180 11 association association NOUN cana-918 180 12 scheme scheme NOUN cana-918 180 13 for for ADP cana-918 180 14 the the DET cana-918 180 15 symmetric symmetric ADJ cana-918 180 16 groups group NOUN cana-918 180 17 ๐‘†4 ๐‘†4 PROPN cana-918 180 18 and and CCONJ cana-918 180 19 ๐‘†5 ๐‘†5 PROPN cana-918 180 20 and and CCONJ cana-918 180 21 alternating alternate VERB cana-918 180 22 groups group NOUN cana-918 180 23 ๐ด3 ๐ด3 PROPN cana-918 180 24 , , PUNCT cana-918 180 25 ๐ด4 ๐ด4 PROPN cana-918 180 26 and and CCONJ cana-918 180 27 ๐ด5 ๐ด5 NOUN cana-918 180 28 without without ADP cana-918 180 29 using use VERB cana-918 180 30 conjugacy conjugacy ADJ cana-918 180 31 classes class NOUN cana-918 180 32 . . PUNCT cana-918 181 1 lemma lemma PROPN cana-918 181 2 2 2 X cana-918 181 3 . . PUNCT cana-918 181 4 let let VERB cana-918 181 5 ๐‘‹ ๐‘‹ PROPN cana-918 181 6 = = SYM cana-918 181 7 ๐ด3 ๐ด3 PROPN cana-918 181 8 = = SYM cana-918 181 9 ๐œŽ ๐œŽ PROPN cana-918 181 10 ๐‘–:0 ๐‘–:0 X cana-918 181 11 โ‰ค โ‰ค NUM cana-918 181 12 ๐‘– ๐‘– SYM cana-918 181 13 โ‰ค โ‰ค NOUN cana-918 181 14 2 2 NUM cana-918 181 15 where where SCONJ cana-918 181 16 ๐œŽ3 ๐œŽ3 NOUN cana-918 181 17 = = SYM cana-918 181 18 1 1 NUM cana-918 181 19 . . PUNCT cana-918 182 1 then then ADV cana-918 182 2 the the DET cana-918 182 3 relations relation NOUN cana-918 182 4 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 182 5 on on ADP cana-918 182 6 ๐’ซ ๐’ซ NOUN cana-918 182 7 defined define VERB cana-918 182 8 by by ADP cana-918 182 9 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 182 10 = = SYM cana-918 182 11 { { PUNCT cana-918 182 12 ( ( PUNCT cana-918 182 13 ๐œŽ ๐œŽ NOUN cana-918 182 14 ๐‘– ๐‘– X cana-918 182 15 , , PUNCT cana-918 182 16 ๐œŽ๐‘˜+๐‘– ๐œŽ๐‘˜+๐‘– X cana-918 182 17 ) ) PUNCT cana-918 182 18 | | ADV cana-918 182 19 0 0 NUM cana-918 182 20 โ‰ค โ‰ค NUM cana-918 182 21 ๐‘– ๐‘– SYM cana-918 182 22 โ‰ค โ‰ค NUM cana-918 182 23 2 2 NUM cana-918 182 24 } } PUNCT cana-918 182 25 for for ADP cana-918 182 26 all all PRON cana-918 182 27 0 0 NUM cana-918 182 28 โ‰ค โ‰ค NOUN cana-918 182 29 ๐‘˜ ๐‘˜ DET cana-918 182 30 โ‰ค โ‰ค ADJ cana-918 182 31 2 2 NUM cana-918 182 32 is be AUX cana-918 182 33 a a DET cana-918 182 34 non non ADJ cana-918 182 35 - - ADJ cana-918 182 36 symmetric symmetric ADJ cana-918 182 37 commutative commutative ADJ cana-918 182 38 association association NOUN cana-918 182 39 scheme scheme NOUN cana-918 182 40 and and CCONJ cana-918 182 41 intersection intersection NOUN cana-918 182 42 numbers number NOUN cana-918 182 43 of of ADP cana-918 182 44 this this DET cana-918 182 45 association association NOUN cana-918 182 46 scheme scheme NOUN cana-918 182 47 are be AUX cana-918 182 48 as as SCONJ cana-918 182 49 follows follow VERB cana-918 182 50 : : PUNCT cana-918 183 1 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š PROPN cana-918 183 2 ๐‘˜ ๐‘˜ NOUN cana-918 183 3 = = PUNCT cana-918 183 4 { { PUNCT cana-918 183 5 1 1 NUM cana-918 183 6 if if SCONJ cana-918 183 7 ๐‘˜ ๐‘˜ ADP cana-918 183 8 = = SYM cana-918 183 9 ๐‘™ ๐‘™ PROPN cana-918 183 10 + + NUM cana-918 183 11 ๐‘š ๐‘š PROPN cana-918 183 12 , , PUNCT cana-918 183 13 0 0 PUNCT cana-918 183 14 if if SCONJ cana-918 183 15 ๐‘˜ ๐‘˜ PRON cana-918 183 16 โ‰  โ‰  PROPN cana-918 183 17 ๐‘™ ๐‘™ PROPN cana-918 183 18 + + NUM cana-918 183 19 ๐‘š ๐‘š ADP cana-918 183 20 proof proof NOUN cana-918 183 21 . . PUNCT cana-918 184 1 since since SCONJ cana-918 184 2 ๐ด3 ๐ด3 PROPN cana-918 184 3 is be AUX cana-918 184 4 isomorphic isomorphic ADJ cana-918 184 5 to to ADP cana-918 184 6 โ„ค3 โ„ค3 NOUN cana-918 184 7 by by ADP cana-918 184 8 mapping map VERB cana-918 184 9 ๐œŽ๐‘– ๐œŽ๐‘– NUM cana-918 184 10 โ†ฆ โ†ฆ PROPN cana-918 184 11 ๐‘– ๐‘– NOUN cana-918 184 12 , , PUNCT cana-918 184 13 and and CCONJ cana-918 184 14 the the DET cana-918 184 15 set set NOUN cana-918 184 16 of of ADP cana-918 184 17 relations relation NOUN cana-918 184 18 { { PUNCT cana-918 184 19 ( ( PUNCT cana-918 184 20 ๐‘– ๐‘– PROPN cana-918 184 21 , , PUNCT cana-918 184 22 ๐‘˜ ๐‘˜ PROPN cana-918 184 23 + + NOUN cana-918 184 24 ๐‘– ๐‘– X cana-918 184 25 ) ) PUNCT cana-918 184 26 | | ADV cana-918 184 27 ๐‘– ๐‘– NOUN cana-918 184 28 = = NOUN cana-918 184 29 0,1,2 0,1,2 X cana-918 184 30 } } PUNCT cana-918 184 31 forms form NOUN cana-918 184 32 as as ADP cana-918 184 33 for for ADP cana-918 184 34 โ„ค3 โ„ค3 NOUN cana-918 184 35 , , PUNCT cana-918 184 36 we we PRON cana-918 184 37 can can AUX cana-918 184 38 conclude conclude VERB cana-918 184 39 the the DET cana-918 184 40 result result NOUN cana-918 184 41 . . PUNCT cana-918 185 1 โ—ป โ—ป PROPN cana-918 185 2 presentations presentation NOUN cana-918 185 3 of of ADP cana-918 185 4 alternating alternate VERB cana-918 185 5 and and CCONJ cana-918 185 6 symmetric symmetric ADJ cana-918 185 7 groups group NOUN cana-918 185 8 of of ADP cana-918 185 9 degree degree NOUN cana-918 185 10 less less ADJ cana-918 185 11 than than ADP cana-918 185 12 8 8 NUM cana-918 185 13 are be AUX cana-918 185 14 given give VERB cana-918 185 15 in in ADP cana-918 185 16 [ [ X cana-918 185 17 18 18 NUM cana-918 185 18 ] ] PUNCT cana-918 185 19 . . PUNCT cana-918 186 1 with with ADP cana-918 186 2 the the DET cana-918 186 3 help help NOUN cana-918 186 4 of of ADP cana-918 186 5 these these DET cana-918 186 6 presentations presentation NOUN cana-918 186 7 , , PUNCT cana-918 186 8 we we PRON cana-918 186 9 can can AUX cana-918 186 10 compute compute VERB cana-918 186 11 the the DET cana-918 186 12 canonical canonical ADJ cana-918 186 13 form form NOUN cana-918 186 14 for for ADP cana-918 186 15 these these DET cana-918 186 16 groups group NOUN cana-918 186 17 ( ( PUNCT cana-918 186 18 provided provide VERB cana-918 186 19 in in ADP cana-918 186 20 table table NOUN cana-918 186 21 2 2 NUM cana-918 186 22 canonical canonical ADJ cana-918 186 23 forms form NOUN cana-918 186 24 of of ADP cana-918 186 25 alternating alternate VERB cana-918 186 26 and and CCONJ cana-918 186 27 symmetric symmetric ADJ cana-918 186 28 groups group NOUN cana-918 186 29 of of ADP cana-918 186 30 degree degree NOUN cana-918 186 31 4 4 NUM cana-918 186 32 and and CCONJ cana-918 186 33 5 5 NUM cana-918 186 34 ) ) PUNCT cana-918 186 35 . . PUNCT cana-918 187 1 in in ADP cana-918 187 2 [ [ X cana-918 187 3 13 13 NUM cana-918 187 4 ] ] PUNCT cana-918 187 5 , , PUNCT cana-918 187 6 canonical canonical ADJ cana-918 187 7 form form NOUN cana-918 187 8 of of ADP cana-918 187 9 ๐‘†4 ๐‘†4 PROPN cana-918 187 10 is be AUX cana-918 187 11 discussed discuss VERB cana-918 187 12 . . PUNCT cana-918 188 1 table table NOUN cana-918 188 2 2 2 NUM cana-918 188 3 canonical canonical ADJ cana-918 188 4 forms form NOUN cana-918 188 5 of of ADP cana-918 188 6 alternating alternate VERB cana-918 188 7 and and CCONJ cana-918 188 8 symmetric symmetric ADJ cana-918 188 9 groups group NOUN cana-918 188 10 of of ADP cana-918 188 11 degree degree NOUN cana-918 188 12 4 4 NUM cana-918 188 13 and and CCONJ cana-918 188 14 5 5 NUM cana-918 188 15 group group NOUN cana-918 188 16 generators generator NOUN cana-918 188 17 presentation presentation NOUN cana-918 188 18 canonical canonical ADJ cana-918 188 19 form form NOUN cana-918 188 20 ๐ด4 ๐ด4 PROPN cana-918 188 21 ๐œ ๐œ PROPN cana-918 188 22 , , PUNCT cana-918 188 23 ๐œŽ ๐œŽ NOUN cana-918 188 24 ๐ด2 ๐ด2 NOUN cana-918 188 25 = = PUNCT cana-918 188 26 ๐ต3 ๐ต3 PROPN cana-918 188 27 = = SYM cana-918 188 28 ๐ผ ๐ผ PROPN cana-918 188 29 , , PUNCT cana-918 188 30 ๐ด๐ต ๐ด๐ต NOUN cana-918 188 31 = = SYM cana-918 188 32 ๐ตโˆ’1๐ด ๐ตโˆ’1๐ด PROPN cana-918 188 33 ๐œ๐‘Ž๐œŽ๐œ๐‘๐œŽ๐‘ ๐œ๐‘Ž๐œŽ๐œ๐‘๐œŽ๐‘ ADJ cana-918 188 34 : : PUNCT cana-918 188 35 0 0 NUM cana-918 188 36 โ‰ค โ‰ค NUM cana-918 188 37 ๐‘Ž ๐‘Ž ADJ cana-918 188 38 , , PUNCT cana-918 188 39 ๐‘ ๐‘ PROPN cana-918 188 40 โ‰ค โ‰ค NUM cana-918 188 41 1 1 NUM cana-918 188 42 , , PUNCT cana-918 188 43 0 0 NUM cana-918 188 44 โ‰ค โ‰ค NUM cana-918 188 45 ๐‘ ๐‘ PROPN cana-918 188 46 โ‰ค โ‰ค ADJ cana-918 188 47 2 2 NUM cana-918 188 48 ๐‘†4 ๐‘†4 PROPN cana-918 188 49 ๐œ ๐œ NUM cana-918 188 50 , , PUNCT cana-918 188 51 ๐œŽ ๐œŽ PROPN cana-918 188 52 ๐œ2 ๐œ2 NOUN cana-918 188 53 = = SYM cana-918 188 54 ๐œŽ3 ๐œŽ3 NOUN cana-918 188 55 = = SYM cana-918 188 56 ( ( PUNCT cana-918 188 57 ๐œ๐œŽ)3 ๐œ๐œŽ)3 PROPN cana-918 188 58 = = SYM cana-918 188 59 1 1 NUM cana-918 188 60 ๐ด2๐œ๐‘Ž๐œŽ๐‘๐œ๐œŽ๐‘ ๐ด2๐œ๐‘Ž๐œŽ๐‘๐œ๐œŽ๐‘ NOUN cana-918 188 61 : : PUNCT cana-918 188 62 0 0 NUM cana-918 188 63 โ‰ค โ‰ค NUM cana-918 189 1 ๐‘Ž ๐‘Ž X cana-918 189 2 โ‰ค โ‰ค NUM cana-918 189 3 1 1 NUM cana-918 189 4 , , PUNCT cana-918 189 5 0 0 NUM cana-918 189 6 โ‰ค โ‰ค NUM cana-918 189 7 ๐‘ ๐‘ DET cana-918 189 8 โ‰ค โ‰ค NUM cana-918 189 9 2 2 NUM cana-918 189 10 , , PUNCT cana-918 189 11 0 0 NUM cana-918 189 12 โ‰ค โ‰ค NUM cana-918 189 13 ๐‘ ๐‘ NOUN cana-918 189 14 โ‰ค โ‰ค NOUN cana-918 189 15 3 3 NUM cana-918 189 16 communications communication NOUN cana-918 189 17 on on ADP cana-918 189 18 applied apply VERB cana-918 189 19 nonlinear nonlinear ADJ cana-918 189 20 analysis analysis NOUN cana-918 189 21 issn issn NOUN cana-918 189 22 : : PUNCT cana-918 189 23 1074 1074 NUM cana-918 189 24 - - PUNCT cana-918 189 25 133x 133x NUM cana-918 189 26 vol vol NOUN cana-918 189 27 31 31 NUM cana-918 189 28 no no NOUN cana-918 189 29 . . PUNCT cana-918 190 1 4s 4s NUM cana-918 190 2 ( ( PUNCT cana-918 190 3 2024 2024 NUM cana-918 190 4 ) ) PUNCT cana-918 190 5 399 399 NUM cana-918 190 6 https://internationalpubls.com https://internationalpubls.com X cana-918 190 7 ๐ด5 ๐ด5 PROPN cana-918 190 8 ๐œ ๐œ PROPN cana-918 190 9 , , PUNCT cana-918 190 10 ๐œŽ ๐œŽ PROPN cana-918 190 11 ๐œ2 ๐œ2 NOUN cana-918 190 12 = = SYM cana-918 190 13 ๐œŽ3 ๐œŽ3 NOUN cana-918 190 14 = = SYM cana-918 190 15 ๐›พ5 ๐›พ5 NOUN cana-918 190 16 = = SYM cana-918 190 17 1 1 NUM cana-918 190 18 where where SCONJ cana-918 190 19 ๐›พ ๐›พ NOUN cana-918 190 20 = = SYM cana-918 190 21 ๐œ๐œŽ ๐œ๐œŽ X cana-918 190 22 ๐œ๐‘Ž๐œŽ๐œ๐œŽ2๐›พ๐‘๐œ๐‘๐œŽ๐œ๐œŽ๐‘‘ ๐œ๐‘Ž๐œŽ๐œ๐œŽ2๐›พ๐‘๐œ๐‘๐œŽ๐œ๐œŽ๐‘‘ PROPN cana-918 190 23 : : PUNCT cana-918 190 24 0 0 NUM cana-918 190 25 โ‰ค โ‰ค NUM cana-918 190 26 ๐‘Ž ๐‘Ž ADJ cana-918 190 27 , , PUNCT cana-918 190 28 ๐‘ ๐‘ PROPN cana-918 190 29 โ‰ค โ‰ค NOUN cana-918 190 30 1 1 NUM cana-918 190 31 , , PUNCT cana-918 190 32 0 0 NUM cana-918 190 33 โ‰ค โ‰ค NUM cana-918 190 34 ๐‘ ๐‘ DET cana-918 190 35 โ‰ค โ‰ค NUM cana-918 190 36 4 4 NUM cana-918 190 37 , , PUNCT cana-918 190 38 0 0 NUM cana-918 190 39 โ‰ค โ‰ค NUM cana-918 190 40 ๐‘‘ ๐‘‘ VERB cana-918 190 41 โ‰ค โ‰ค ADJ cana-918 190 42 2 2 NUM cana-918 190 43 ๐‘†5 ๐‘†5 PROPN cana-918 190 44 ๐œ ๐œ PROPN cana-918 190 45 , , PUNCT cana-918 191 1 ๐œŽ ๐œŽ NOUN cana-918 191 2 ๐œ5 ๐œ5 PROPN cana-918 191 3 = = SYM cana-918 191 4 ๐œŽ6 ๐œŽ6 PROPN cana-918 191 5 = = SYM cana-918 191 6 ( ( PUNCT cana-918 191 7 ๐›พ)2 ๐›พ)2 NOUN cana-918 191 8 = = SYM cana-918 191 9 ( ( PUNCT cana-918 191 10 ๐›ฟ)2 ๐›ฟ)2 PROPN cana-918 191 11 = = NOUN cana-918 191 12 1 1 NUM cana-918 191 13 where where SCONJ cana-918 191 14 ๐›พ ๐›พ NOUN cana-918 191 15 = = SYM cana-918 191 16 ๐œ๐œŽ ๐œ๐œŽ NOUN cana-918 191 17 and and CCONJ cana-918 191 18 ๐›ฟ ๐›ฟ PROPN cana-918 191 19 = = SYM cana-918 191 20 ๐œ2๐œŽ2 ๐œ2๐œŽ2 PROPN cana-918 191 21 ๐œ๐‘Ž๐œŽ๐›พ๐‘๐œ4๐œŽ5๐›ฟ๐‘๐œ๐œŽ๐‘‘ ๐œ๐‘Ž๐œŽ๐›พ๐‘๐œ4๐œŽ5๐›ฟ๐‘๐œ๐œŽ๐‘‘ NOUN cana-918 191 22 : : PUNCT cana-918 191 23 0 0 NUM cana-918 191 24 โ‰ค โ‰ค NUM cana-918 192 1 ๐‘Ž ๐‘Ž X cana-918 192 2 โ‰ค โ‰ค NUM cana-918 192 3 4 4 NUM cana-918 192 4 , , PUNCT cana-918 192 5 0 0 NUM cana-918 192 6 โ‰ค โ‰ค NUM cana-918 192 7 ๐‘ ๐‘ NOUN cana-918 192 8 , , PUNCT cana-918 192 9 ๐‘ ๐‘ PROPN cana-918 192 10 โ‰ค โ‰ค NOUN cana-918 192 11 1 1 NUM cana-918 192 12 , , PUNCT cana-918 192 13 0 0 NUM cana-918 192 14 โ‰ค โ‰ค NUM cana-918 192 15 ๐‘‘ ๐‘‘ VERB cana-918 192 16 โ‰ค โ‰ค ADJ cana-918 192 17 5 5 NUM cana-918 192 18 theorem theorem NOUN cana-918 192 19 7 7 NUM cana-918 192 20 . . PUNCT cana-918 193 1 let let VERB cana-918 193 2 ๐‘Œ ๐‘Œ PROPN cana-918 193 3 = = PUNCT cana-918 193 4 ๐ด4 ๐ด4 PROPN cana-918 193 5 and and CCONJ cana-918 193 6 ๐’ซ ๐’ซ NOUN cana-918 193 7 be be VERB cana-918 193 8 a a DET cana-918 193 9 partition partition NOUN cana-918 193 10 of of ADP cana-918 193 11 ๐‘Œ ๐‘Œ PROPN cana-918 193 12 ร— ร— PROPN cana-918 193 13 ๐‘Œ. ๐‘Œ. PROPN cana-918 193 14 for for ADP cana-918 193 15 all all PRON cana-918 193 16 ๐‘˜ ๐‘˜ PRON cana-918 193 17 written write VERB cana-918 193 18 in in ADP cana-918 193 19 form form NOUN cana-918 193 20 of of ADP cana-918 193 21 3๐‘› 3๐‘› NUM cana-918 193 22 + + CCONJ cana-918 193 23 ๐‘ก ๐‘ก PROPN cana-918 193 24 ( ( PUNCT cana-918 193 25 ๐‘› ๐‘› PROPN cana-918 193 26 โˆˆ โˆˆ PROPN cana-918 193 27 โ„ค4 โ„ค4 NOUN cana-918 193 28 , , PUNCT cana-918 193 29 ๐‘ก ๐‘ก PROPN cana-918 193 30 โˆˆ โˆˆ PROPN cana-918 193 31 โ„ค3 โ„ค3 NOUN cana-918 193 32 ) ) PUNCT cana-918 193 33 , , PUNCT cana-918 193 34 the the DET cana-918 193 35 relations relation NOUN cana-918 193 36 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 193 37 on on ADP cana-918 193 38 ๐’ซ ๐’ซ NOUN cana-918 193 39 defined define VERB cana-918 193 40 by by ADP cana-918 193 41 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 193 42 = = SYM cana-918 193 43 { { PUNCT cana-918 193 44 ( ( PUNCT cana-918 193 45 ๐›ผ๐‘– ๐›ผ๐‘– PROPN cana-918 193 46 ( ( PUNCT cana-918 193 47 ๐‘— ๐‘— NOUN cana-918 193 48 ) ) PUNCT cana-918 193 49 , , PUNCT cana-918 193 50 ๐›ผ(๐‘–+๐‘ก)๐‘š๐‘œ๐‘‘ ๐›ผ(๐‘–+๐‘ก)๐‘š๐‘œ๐‘‘ ADJ cana-918 193 51 3 3 NUM cana-918 193 52 ( ( PUNCT cana-918 193 53 ๐‘—+๐‘›)๐‘š๐‘œ๐‘‘ ๐‘—+๐‘›)๐‘š๐‘œ๐‘‘ NOUN cana-918 193 54 4 4 X cana-918 193 55 ) ) PUNCT cana-918 193 56 | | ADV cana-918 193 57 0 0 NUM cana-918 193 58 โ‰ค โ‰ค NUM cana-918 193 59 ๐‘– ๐‘– SYM cana-918 193 60 โ‰ค โ‰ค NOUN cana-918 193 61 2 2 NUM cana-918 193 62 , , PUNCT cana-918 193 63 0 0 NUM cana-918 193 64 โ‰ค โ‰ค NUM cana-918 194 1 ๐‘— ๐‘— PRON cana-918 194 2 โ‰ค โ‰ค NUM cana-918 194 3 3 3 NUM cana-918 194 4 } } PUNCT cana-918 194 5 where where SCONJ cana-918 194 6 ๐›ผ๐‘– ๐›ผ๐‘– PROPN cana-918 194 7 ( ( PUNCT cana-918 194 8 0 0 NUM cana-918 194 9 ) ) PUNCT cana-918 194 10 = = SYM cana-918 194 11 ๐œŽ๐‘–+1 ๐œŽ๐‘–+1 PROPN cana-918 194 12 , , PUNCT cana-918 194 13 ๐›ผ๐‘– ๐›ผ๐‘– PROPN cana-918 194 14 ( ( PUNCT cana-918 194 15 1 1 NUM cana-918 194 16 ) ) PUNCT cana-918 194 17 = = NOUN cana-918 194 18 ๐œŽ๐œ๐œŽ๐‘– ๐œŽ๐œ๐œŽ๐‘– NOUN cana-918 194 19 , , PUNCT cana-918 194 20 ๐›ผ๐‘– ๐›ผ๐‘– PROPN cana-918 194 21 ( ( PUNCT cana-918 194 22 2 2 NUM cana-918 194 23 ) ) PUNCT cana-918 194 24 = = SYM cana-918 194 25 ๐œ๐œŽ๐‘–+1 ๐œ๐œŽ๐‘–+1 NOUN cana-918 194 26 , , PUNCT cana-918 194 27 ๐›ผ๐‘– ๐›ผ๐‘– PROPN cana-918 194 28 ( ( PUNCT cana-918 194 29 3 3 NUM cana-918 194 30 ) ) PUNCT cana-918 194 31 = = PUNCT cana-918 194 32 ๐œ๐œŽ๐œ๐œŽ๐‘– ๐œ๐œŽ๐œ๐œŽ๐‘– X cana-918 194 33 is be AUX cana-918 194 34 a a DET cana-918 194 35 non non ADJ cana-918 194 36 - - ADJ cana-918 194 37 symmetric symmetric ADJ cana-918 194 38 association association NOUN cana-918 194 39 scheme scheme NOUN cana-918 194 40 with with ADP cana-918 194 41 parameters parameter NOUN cana-918 195 1 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š VERB cana-918 195 2 ๐‘˜ ๐‘˜ X cana-918 195 3 = = PUNCT cana-918 195 4 { { PUNCT cana-918 195 5 1 1 NUM cana-918 195 6 ๐‘–๐‘“ ๐‘–๐‘“ ADP cana-918 195 7 ๐‘›(๐‘˜ ๐‘›(๐‘˜ NOUN cana-918 195 8 ) ) PUNCT cana-918 195 9 โ‰ก โ‰ก PROPN cana-918 195 10 ( ( PUNCT cana-918 195 11 ๐‘›(๐‘™ ๐‘›(๐‘™ PROPN cana-918 195 12 ) ) PUNCT cana-918 195 13 + + SYM cana-918 195 14 ๐‘›(๐‘š ๐‘›(๐‘š NOUN cana-918 195 15 ) ) PUNCT cana-918 195 16 ) ) PUNCT cana-918 195 17 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 195 18 4 4 NUM cana-918 195 19 , , PUNCT cana-918 195 20 ๐‘Ž๐‘›๐‘‘ ๐‘Ž๐‘›๐‘‘ ADJ cana-918 195 21 ๐‘ก(๐‘˜ ๐‘ก(๐‘˜ NOUN cana-918 195 22 ) ) PUNCT cana-918 195 23 โ‰ก โ‰ก PROPN cana-918 195 24 ( ( PUNCT cana-918 195 25 ๐‘ก(๐‘™ ๐‘ก(๐‘™ PROPN cana-918 195 26 ) ) PUNCT cana-918 195 27 + + NUM cana-918 195 28 ๐‘ก(๐‘š ๐‘ก(๐‘š NOUN cana-918 195 29 ) ) PUNCT cana-918 195 30 ) ) PUNCT cana-918 196 1 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 196 2 3 3 NUM cana-918 196 3 0 0 NUM cana-918 196 4 ๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’ ๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’ NOUN cana-918 196 5 where where SCONJ cana-918 196 6 ๐‘˜ ๐‘˜ X cana-918 196 7 = = NOUN cana-918 196 8 3๐‘›(๐‘˜ 3๐‘›(๐‘˜ NUM cana-918 196 9 ) ) PUNCT cana-918 196 10 + + CCONJ cana-918 196 11 ๐‘ก(๐‘˜ ๐‘ก(๐‘˜ NOUN cana-918 196 12 ) ) PUNCT cana-918 196 13 ; ; PUNCT cana-918 196 14 ๐‘™ ๐‘™ X cana-918 196 15 = = SYM cana-918 196 16 3๐‘›(๐‘™ 3๐‘›(๐‘™ NUM cana-918 196 17 ) ) PUNCT cana-918 196 18 + + CCONJ cana-918 196 19 ๐‘ก(๐‘™ ๐‘ก(๐‘™ PROPN cana-918 196 20 ) ) PUNCT cana-918 196 21 ; ; PUNCT cana-918 196 22 ๐‘š ๐‘š X cana-918 196 23 = = SYM cana-918 196 24 3๐‘›(๐‘š 3๐‘›(๐‘š NUM cana-918 196 25 ) ) PUNCT cana-918 197 1 + + PUNCT cana-918 197 2 ๐‘ก(๐‘š ๐‘ก(๐‘š NOUN cana-918 197 3 ) ) PUNCT cana-918 197 4 ; ; PUNCT cana-918 197 5 for for ADP cana-918 197 6 some some DET cana-918 197 7 ๐‘ก(๐‘˜ ๐‘ก(๐‘˜ NOUN cana-918 197 8 ) ) PUNCT cana-918 197 9 , , PUNCT cana-918 197 10 ๐‘ก(๐‘™ ๐‘ก(๐‘™ PROPN cana-918 197 11 ) ) PUNCT cana-918 197 12 , , PUNCT cana-918 197 13 ๐‘ก(๐‘š ๐‘ก(๐‘š NOUN cana-918 197 14 ) ) PUNCT cana-918 197 15 โˆˆ โˆˆ PROPN cana-918 197 16 โ„ค3 โ„ค3 NOUN cana-918 197 17 and and CCONJ cana-918 197 18 ๐‘›(๐‘˜ ๐‘›(๐‘˜ NOUN cana-918 197 19 ) ) PUNCT cana-918 197 20 , , PUNCT cana-918 197 21 ๐‘›(๐‘™ ๐‘›(๐‘™ PROPN cana-918 197 22 ) ) PUNCT cana-918 197 23 , , PUNCT cana-918 197 24 ๐‘›(๐‘š ๐‘›(๐‘š NOUN cana-918 197 25 ) ) PUNCT cana-918 197 26 โˆˆ โˆˆ PROPN cana-918 197 27 โ„ค4 โ„ค4 NOUN cana-918 197 28 . . PUNCT cana-918 198 1 proof proof NOUN cana-918 198 2 . . PUNCT cana-918 199 1 observe observe VERB cana-918 199 2 that that SCONJ cana-918 199 3 |๐’ฎ๐‘˜| |๐’ฎ๐‘˜| ADV cana-918 199 4 = = PUNCT cana-918 199 5 |๐‘Œ| |๐‘Œ| VERB cana-918 199 6 for for ADP cana-918 199 7 all all DET cana-918 199 8 0 0 NUM cana-918 199 9 โ‰ค โ‰ค NOUN cana-918 200 1 ๐‘˜ ๐‘˜ PRON cana-918 200 2 < < X cana-918 200 3 12 12 NUM cana-918 200 4 . . PUNCT cana-918 201 1 the the DET cana-918 201 2 relations relation NOUN cana-918 201 3 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 201 4 are be AUX cana-918 201 5 disjoint disjoint ADJ cana-918 201 6 and and CCONJ cana-918 201 7 โˆช โˆช VERB cana-918 201 8 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ NOUN cana-918 201 9 : : SYM cana-918 201 10 0 0 NUM cana-918 201 11 โ‰ค โ‰ค NOUN cana-918 201 12 ๐‘˜ ๐‘˜ PRON cana-918 201 13 < < X cana-918 201 14 12 12 NUM cana-918 201 15 = = SYM cana-918 201 16 ๐’ซ. ๐’ซ. NOUN cana-918 201 17 ๐’ฎ0 ๐’ฎ0 NOUN cana-918 201 18 = = SYM cana-918 201 19 { { PUNCT cana-918 201 20 ( ( PUNCT cana-918 201 21 ๐›ผ๐‘– ๐›ผ๐‘– PROPN cana-918 201 22 ( ( PUNCT cana-918 201 23 ๐‘— ๐‘— NOUN cana-918 201 24 ) ) PUNCT cana-918 201 25 , , PUNCT cana-918 201 26 ๐›ผ๐‘– ๐›ผ๐‘– PROPN cana-918 201 27 ( ( PUNCT cana-918 201 28 ๐‘— ๐‘— NOUN cana-918 201 29 ) ) PUNCT cana-918 201 30 ) ) PUNCT cana-918 201 31 : : PUNCT cana-918 201 32 0 0 NUM cana-918 201 33 โ‰ค โ‰ค NUM cana-918 201 34 ๐‘– ๐‘– SYM cana-918 201 35 โ‰ค โ‰ค NOUN cana-918 201 36 2 2 NUM cana-918 201 37 , , PUNCT cana-918 201 38 0 0 NUM cana-918 201 39 โ‰ค โ‰ค NUM cana-918 201 40 ๐‘— ๐‘— PRON cana-918 201 41 โ‰ค โ‰ค NUM cana-918 201 42 3 3 NUM cana-918 201 43 } } PUNCT cana-918 201 44 is be AUX cana-918 201 45 an an DET cana-918 201 46 identity identity NOUN cana-918 201 47 relation relation NOUN cana-918 201 48 . . PUNCT cana-918 202 1 for for ADP cana-918 202 2 arbitrary arbitrary ADJ cana-918 202 3 relations relation NOUN cana-918 202 4 ๐’ฎ๐‘™ ๐’ฎ๐‘™ PROPN cana-918 202 5 , , PUNCT cana-918 202 6 ๐’ฎ๐‘š ๐’ฎ๐‘š NOUN cana-918 202 7 , , PUNCT cana-918 202 8 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 202 9 in in ADP cana-918 202 10 ๐’ซ ๐’ซ PROPN cana-918 202 11 , , PUNCT cana-918 202 12 we we PRON cana-918 202 13 prove prove VERB cana-918 202 14 that that SCONJ cana-918 202 15 for for ADP cana-918 202 16 all all PRON cana-918 202 17 ( ( PUNCT cana-918 202 18 ๐‘ฅ ๐‘ฅ PROPN cana-918 202 19 , , PUNCT cana-918 202 20 ๐‘ฆ ๐‘ฆ NOUN cana-918 202 21 ) ) PUNCT cana-918 202 22 โˆˆ โˆˆ PROPN cana-918 202 23 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 202 24 , , PUNCT cana-918 202 25 |๐‘ฅ๐’ฎ๐‘™ |๐‘ฅ๐’ฎ๐‘™ ADJ cana-918 202 26 โˆฉ โˆฉ NOUN cana-918 202 27 ๐‘ฆ๐’ฎ๐‘š ๐‘ฆ๐’ฎ๐‘š NOUN cana-918 202 28 โˆ— โˆ— NOUN cana-918 202 29 | | ADV cana-918 202 30 is be AUX cana-918 202 31 constant constant ADJ cana-918 202 32 . . PUNCT cana-918 203 1 now now ADV cana-918 203 2 let let VERB cana-918 203 3 ( ( PUNCT cana-918 203 4 ๐‘ฅ ๐‘ฅ NOUN cana-918 203 5 , , PUNCT cana-918 203 6 ๐‘ฆ ๐‘ฆ X cana-918 203 7 ) ) PUNCT cana-918 203 8 be be VERB cana-918 203 9 an an DET cana-918 203 10 arbitrary arbitrary ADJ cana-918 203 11 element element NOUN cana-918 203 12 of of ADP cana-918 203 13 ๐’ฎ๐‘˜. ๐’ฎ๐‘˜. PROPN cana-918 203 14 since since SCONJ cana-918 203 15 ๐‘ฅ ๐‘ฅ PROPN cana-918 203 16 โˆˆ โˆˆ PROPN cana-918 203 17 ๐‘Œ ๐‘Œ PROPN cana-918 203 18 , , PUNCT cana-918 203 19 ๐‘ฅ ๐‘ฅ PRON cana-918 203 20 is be AUX cana-918 203 21 of of ADP cana-918 203 22 the the DET cana-918 203 23 form form NOUN cana-918 203 24 ๐œ๐‘Ž๐œŽ๐œ๐‘๐œŽ๐‘ ๐œ๐‘Ž๐œŽ๐œ๐‘๐œŽ๐‘ ADJ cana-918 203 25 for for ADP cana-918 203 26 some some DET cana-918 203 27 ๐‘Ž ๐‘Ž NOUN cana-918 203 28 , , PUNCT cana-918 203 29 ๐‘ ๐‘ NOUN cana-918 203 30 , , PUNCT cana-918 203 31 ๐‘ ๐‘ PROPN cana-918 203 32 ( ( PUNCT cana-918 203 33 from from ADP cana-918 203 34 table table NOUN cana-918 203 35 [ [ X cana-918 203 36 table table NOUN cana-918 203 37 : : PUNCT cana-918 203 38 canonical canonical ADJ cana-918 203 39 forms form NOUN cana-918 203 40 an an DET cana-918 203 41 , , PUNCT cana-918 203 42 sn sn NOUN cana-918 203 43 ] ] PUNCT cana-918 203 44 ) ) PUNCT cana-918 203 45 and and CCONJ cana-918 203 46 further far ADV cana-918 203 47 ๐‘ฅ ๐‘ฅ PRON cana-918 203 48 = = SYM cana-918 203 49 ๐›ผ๐‘– ๐›ผ๐‘– PROPN cana-918 203 50 ( ( PUNCT cana-918 203 51 ๐‘— ๐‘— NOUN cana-918 203 52 ) ) PUNCT cana-918 203 53 for for ADP cana-918 203 54 some some PRON cana-918 203 55 ๐‘– ๐‘– ADP cana-918 203 56 , , PUNCT cana-918 203 57 ๐‘—. ๐‘—. NOUN cana-918 203 58 suppose suppose VERB cana-918 203 59 ๐‘ฅ๐’ฎ๐‘™ ๐‘ฅ๐’ฎ๐‘™ X cana-918 203 60 = = SYM cana-918 203 61 ๐‘ฅ ๐‘ฅ NOUN cana-918 203 62 โ€ฒ โ€ฒ NUM cana-918 203 63 and and CCONJ cana-918 203 64 ๐‘ฆ๐’ฎ๐‘š ๐‘ฆ๐’ฎ๐‘š NOUN cana-918 203 65 โˆ— โˆ— NOUN cana-918 203 66 = = SYM cana-918 203 67 ๐‘ฆโ€ฒ ๐‘ฆโ€ฒ PUNCT cana-918 203 68 where where SCONJ cana-918 203 69 ๐‘ฅโ€ฒ ๐‘ฅโ€ฒ PROPN cana-918 203 70 , , PUNCT cana-918 203 71 ๐‘ฆโ€ฒ ๐‘ฆโ€ฒ X cana-918 203 72 โˆˆ โˆˆ PROPN cana-918 203 73 ๐‘Œ. ๐‘Œ. PROPN cana-918 203 74 now now ADV cana-918 203 75 ( ( PUNCT cana-918 203 76 ๐‘ฅ ๐‘ฅ PROPN cana-918 203 77 , , PUNCT cana-918 203 78 ๐‘ฆ ๐‘ฆ NOUN cana-918 203 79 ) ) PUNCT cana-918 203 80 โˆˆ โˆˆ PROPN cana-918 204 1 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 204 2 implies imply VERB cana-918 204 3 ๐‘ฆ ๐‘ฆ NOUN cana-918 204 4 = = SYM cana-918 204 5 ๐›ผ ๐›ผ PROPN cana-918 204 6 ( ( PUNCT cana-918 204 7 ๐‘–+๐‘ก(๐‘˜ ๐‘–+๐‘ก(๐‘˜ NOUN cana-918 204 8 ) ) PUNCT cana-918 204 9 ) ) PUNCT cana-918 204 10 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 204 11 3 3 NUM cana-918 204 12 ( ( PUNCT cana-918 204 13 ๐‘—+๐‘›(๐‘˜ ๐‘—+๐‘›(๐‘˜ NOUN cana-918 204 14 ) ) PUNCT cana-918 204 15 ) ) PUNCT cana-918 204 16 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 204 17 4 4 NUM cana-918 204 18 where where SCONJ cana-918 204 19 ๐‘˜ ๐‘˜ X cana-918 204 20 = = NOUN cana-918 204 21 3๐‘›(๐‘˜ 3๐‘›(๐‘˜ NUM cana-918 204 22 ) ) PUNCT cana-918 204 23 + + CCONJ cana-918 204 24 ๐‘ก(๐‘˜ ๐‘ก(๐‘˜ NOUN cana-918 204 25 ) ) PUNCT cana-918 204 26 for for ADP cana-918 204 27 some some DET cana-918 204 28 ๐‘ก(๐‘˜ ๐‘ก(๐‘˜ NOUN cana-918 204 29 ) ) PUNCT cana-918 204 30 โˆˆ โˆˆ PROPN cana-918 204 31 โ„ค3 โ„ค3 NOUN cana-918 204 32 , , PUNCT cana-918 204 33 ๐‘› ๐‘› PROPN cana-918 204 34 ( ( PUNCT cana-918 204 35 ๐‘˜ ๐‘˜ NOUN cana-918 204 36 ) ) PUNCT cana-918 204 37 โˆˆ โˆˆ PROPN cana-918 204 38 โ„ค4 โ„ค4 NOUN cana-918 204 39 . . PUNCT cana-918 205 1 similarly similarly ADV cana-918 205 2 , , PUNCT cana-918 205 3 ( ( PUNCT cana-918 205 4 ๐‘ฅ ๐‘ฅ X cana-918 205 5 , , PUNCT cana-918 205 6 ๐‘ฅโ€ฒ ๐‘ฅโ€ฒ NUM cana-918 205 7 ) ) PUNCT cana-918 206 1 โˆˆ โˆˆ PROPN cana-918 206 2 ๐’ฎ๐‘™ ๐’ฎ๐‘™ PROPN cana-918 206 3 and and CCONJ cana-918 206 4 ( ( PUNCT cana-918 206 5 ๐‘ฆโ€ฒ ๐‘ฆโ€ฒ NOUN cana-918 206 6 , , PUNCT cana-918 206 7 ๐‘ฆ ๐‘ฆ X cana-918 206 8 ) ) PUNCT cana-918 206 9 โˆˆ โˆˆ PROPN cana-918 206 10 ๐’ฎ๐‘š ๐’ฎ๐‘š NOUN cana-918 206 11 , , PUNCT cana-918 206 12 implies imply VERB cana-918 206 13 ๐‘ฅโ€ฒ ๐‘ฅโ€ฒ PUNCT cana-918 206 14 = = SYM cana-918 206 15 ๐›ผ ๐›ผ X cana-918 206 16 ( ( PUNCT cana-918 206 17 ๐‘–+๐‘ก(๐‘™ ๐‘–+๐‘ก(๐‘™ NOUN cana-918 206 18 ) ) PUNCT cana-918 206 19 ) ) PUNCT cana-918 206 20 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 206 21 3 3 NUM cana-918 206 22 ( ( PUNCT cana-918 206 23 ๐‘—+๐‘›(๐‘™ ๐‘—+๐‘›(๐‘™ NOUN cana-918 206 24 ) ) PUNCT cana-918 206 25 ) ) PUNCT cana-918 206 26 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 206 27 4 4 NUM cana-918 206 28 where where SCONJ cana-918 206 29 ๐‘™ ๐‘™ NOUN cana-918 206 30 = = SYM cana-918 206 31 3๐‘›(๐‘™ 3๐‘›(๐‘™ NUM cana-918 206 32 ) ) PUNCT cana-918 206 33 + + CCONJ cana-918 206 34 ๐‘ก(๐‘™ ๐‘ก(๐‘™ PROPN cana-918 206 35 ) ) PUNCT cana-918 206 36 for for ADP cana-918 206 37 some some DET cana-918 206 38 ๐‘ก(๐‘™ ๐‘ก(๐‘™ PROPN cana-918 206 39 ) ) PUNCT cana-918 206 40 โˆˆ โˆˆ PROPN cana-918 206 41 โ„ค3 โ„ค3 NOUN cana-918 206 42 , , PUNCT cana-918 206 43 ๐‘› ๐‘› PROPN cana-918 206 44 ( ( PUNCT cana-918 206 45 ๐‘™ ๐‘™ X cana-918 206 46 ) ) PUNCT cana-918 206 47 โˆˆ โˆˆ PROPN cana-918 206 48 โ„ค4 โ„ค4 NOUN cana-918 206 49 , , PUNCT cana-918 206 50 and and CCONJ cana-918 206 51 ๐‘ฆโ€ฒ ๐‘ฆโ€ฒ X cana-918 206 52 = = SYM cana-918 206 53 ๐›ผ ๐›ผ PROPN cana-918 206 54 ( ( PUNCT cana-918 206 55 ๐‘–+๐‘ก(๐‘˜)โˆ’๐‘ก(๐‘š ๐‘–+๐‘ก(๐‘˜)โˆ’๐‘ก(๐‘š NUM cana-918 206 56 ) ) PUNCT cana-918 206 57 ) ) PUNCT cana-918 206 58 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 206 59 3 3 NUM cana-918 206 60 ( ( PUNCT cana-918 206 61 ๐‘—+๐‘›(๐‘˜)โˆ’๐‘›(๐‘š ๐‘—+๐‘›(๐‘˜)โˆ’๐‘›(๐‘š NOUN cana-918 206 62 ) ) PUNCT cana-918 206 63 ) ) PUNCT cana-918 206 64 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 206 65 4 4 NUM cana-918 206 66 where where SCONJ cana-918 206 67 ๐‘š ๐‘š NOUN cana-918 206 68 = = SYM cana-918 206 69 3๐‘›(๐‘š 3๐‘›(๐‘š NUM cana-918 206 70 ) ) PUNCT cana-918 207 1 + + PUNCT cana-918 207 2 ๐‘ก(๐‘š ๐‘ก(๐‘š NOUN cana-918 207 3 ) ) PUNCT cana-918 207 4 for for ADP cana-918 207 5 some some DET cana-918 207 6 ๐‘ก(๐‘š ๐‘ก(๐‘š NOUN cana-918 207 7 ) ) PUNCT cana-918 207 8 โˆˆ โˆˆ PROPN cana-918 207 9 โ„ค3 โ„ค3 NOUN cana-918 207 10 , , PUNCT cana-918 207 11 ๐‘› ๐‘› PROPN cana-918 207 12 ( ( PUNCT cana-918 207 13 ๐‘š ๐‘š NOUN cana-918 207 14 ) ) PUNCT cana-918 207 15 โˆˆ โˆˆ PROPN cana-918 207 16 โ„ค4 โ„ค4 NOUN cana-918 207 17 . . PUNCT cana-918 208 1 since since SCONJ cana-918 208 2 every every DET cana-918 208 3 pair pair NOUN cana-918 208 4 ( ( PUNCT cana-918 208 5 ๐‘ฅ ๐‘ฅ NOUN cana-918 208 6 , , PUNCT cana-918 208 7 ๐‘ฆ ๐‘ฆ NOUN cana-918 208 8 ) ) PUNCT cana-918 208 9 in in ADP cana-918 208 10 ๐’ซ ๐’ซ PROPN cana-918 208 11 are be AUX cana-918 208 12 ๐‘˜๐‘กโ„Ž ๐‘˜๐‘กโ„Ž VERB cana-918 208 13 associates associate NOUN cana-918 208 14 for for ADP cana-918 208 15 exactly exactly ADV cana-918 208 16 one one NUM cana-918 208 17 k k NOUN cana-918 208 18 , , PUNCT cana-918 208 19 we we PRON cana-918 208 20 find find VERB cana-918 208 21 that that SCONJ cana-918 208 22 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š PROPN cana-918 208 23 ๐‘˜ ๐‘˜ PRON cana-918 208 24 can can AUX cana-918 208 25 be be AUX cana-918 208 26 either either CCONJ cana-918 208 27 0 0 NUM cana-918 208 28 or or CCONJ cana-918 208 29 1 1 NUM cana-918 208 30 . . PUNCT cana-918 209 1 hence hence ADV cana-918 209 2 we we PRON cana-918 209 3 obtain obtain VERB cana-918 209 4 that that DET cana-918 209 5 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š PROPN cana-918 209 6 ๐‘˜ ๐‘˜ NOUN cana-918 209 7 = = NOUN cana-918 209 8 1 1 NUM cana-918 209 9 if if SCONJ cana-918 209 10 and and CCONJ cana-918 209 11 only only ADV cana-918 209 12 if if SCONJ cana-918 209 13 ๐‘ฅโ€ฒ ๐‘ฅโ€ฒ VERB cana-918 209 14 = = PUNCT cana-918 209 15 ๐‘ฆโ€ฒ ๐‘ฆโ€ฒ X cana-918 209 16 that that PRON cana-918 209 17 is be AUX cana-918 209 18 , , PUNCT cana-918 209 19 if if SCONJ cana-918 209 20 ๐‘ก(๐‘˜ ๐‘ก(๐‘˜ NOUN cana-918 209 21 ) ) PUNCT cana-918 209 22 โ‰ก โ‰ก PROPN cana-918 209 23 ( ( PUNCT cana-918 209 24 ๐‘ก(๐‘™ ๐‘ก(๐‘™ PROPN cana-918 209 25 ) ) PUNCT cana-918 209 26 + + NUM cana-918 209 27 ๐‘ก(๐‘š ๐‘ก(๐‘š NOUN cana-918 209 28 ) ) PUNCT cana-918 209 29 ) ) PUNCT cana-918 209 30 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 209 31 3 3 NUM cana-918 209 32 and and CCONJ cana-918 209 33 ๐‘›(๐‘˜ ๐‘›(๐‘˜ PROPN cana-918 209 34 ) ) PUNCT cana-918 209 35 โ‰ก โ‰ก PROPN cana-918 209 36 ( ( PUNCT cana-918 209 37 ๐‘›(๐‘™ ๐‘›(๐‘™ PROPN cana-918 209 38 ) ) PUNCT cana-918 209 39 + + SYM cana-918 209 40 ๐‘›(๐‘š ๐‘›(๐‘š NOUN cana-918 209 41 ) ) PUNCT cana-918 209 42 ) ) PUNCT cana-918 209 43 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 210 1 4 4 X cana-918 210 2 . . PUNCT cana-918 210 3 โ—ป โ—ป PROPN cana-918 210 4 theorem theorem VERB cana-918 210 5 8 8 NUM cana-918 210 6 . . PUNCT cana-918 211 1 let let VERB cana-918 211 2 ๐‘Œ ๐‘Œ PROPN cana-918 211 3 = = PUNCT cana-918 211 4 ๐ด5 ๐ด5 NOUN cana-918 211 5 and and CCONJ cana-918 211 6 ๐’ซ ๐’ซ NOUN cana-918 211 7 be be VERB cana-918 211 8 a a DET cana-918 211 9 partition partition NOUN cana-918 211 10 of of ADP cana-918 211 11 ๐‘Œ ๐‘Œ PROPN cana-918 211 12 ร— ร— PROPN cana-918 211 13 ๐‘Œ. ๐‘Œ. PROPN cana-918 211 14 for for ADP cana-918 211 15 all all PRON cana-918 211 16 ๐‘˜ ๐‘˜ PRON cana-918 211 17 written write VERB cana-918 211 18 in in ADP cana-918 211 19 form form NOUN cana-918 211 20 of of ADP cana-918 211 21 12๐‘› 12๐‘› NUM cana-918 211 22 + + CCONJ cana-918 211 23 4๐‘ก 4๐‘ก NOUN cana-918 211 24 + + X cana-918 211 25 ๐‘Ÿ ๐‘Ÿ NOUN cana-918 211 26 ( ( PUNCT cana-918 211 27 ๐‘› ๐‘› PROPN cana-918 211 28 โˆˆ โˆˆ PROPN cana-918 211 29 โ„ค5 โ„ค5 NOUN cana-918 211 30 , , PUNCT cana-918 211 31 ๐‘ก ๐‘ก PROPN cana-918 211 32 โˆˆ โˆˆ PROPN cana-918 211 33 โ„ค3 โ„ค3 NOUN cana-918 211 34 , , PUNCT cana-918 211 35 ๐‘Ÿ ๐‘Ÿ X cana-918 211 36 โˆˆ โˆˆ PROPN cana-918 211 37 โ„ค4 โ„ค4 PROPN cana-918 211 38 ) ) PUNCT cana-918 211 39 , , PUNCT cana-918 211 40 the the DET cana-918 211 41 relations relation NOUN cana-918 211 42 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 211 43 on on ADP cana-918 211 44 ๐’ซ ๐’ซ NOUN cana-918 211 45 defined define VERB cana-918 211 46 by by ADP cana-918 211 47 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 211 48 = = SYM cana-918 211 49 { { PUNCT cana-918 211 50 ( ( PUNCT cana-918 211 51 ๐›ผ๐‘–๐‘— ๐›ผ๐‘–๐‘— PROPN cana-918 211 52 ( ( PUNCT cana-918 211 53 ๐‘  ๐‘  NOUN cana-918 211 54 ) ) PUNCT cana-918 211 55 , , PUNCT cana-918 211 56 ๐›ผ(๐‘–+๐‘›)๐‘š๐‘œ๐‘‘ ๐›ผ(๐‘–+๐‘›)๐‘š๐‘œ๐‘‘ NOUN cana-918 211 57 5(๐‘—+๐‘ก)๐‘š๐‘œ๐‘‘ 5(๐‘—+๐‘ก)๐‘š๐‘œ๐‘‘ PROPN cana-918 211 58 3 3 NUM cana-918 211 59 ( ( PUNCT cana-918 211 60 ๐‘ +๐‘Ÿ)๐‘š๐‘œ๐‘‘ ๐‘ +๐‘Ÿ)๐‘š๐‘œ๐‘‘ PROPN cana-918 211 61 4 4 NUM cana-918 211 62 ) ) PUNCT cana-918 211 63 | | ADV cana-918 211 64 ๐‘– ๐‘– SYM cana-918 211 65 โˆˆ โˆˆ PROPN cana-918 211 66 โ„ค5 โ„ค5 NOUN cana-918 211 67 , , PUNCT cana-918 211 68 ๐‘— ๐‘— PROPN cana-918 211 69 โˆˆ โˆˆ PROPN cana-918 211 70 โ„ค3 โ„ค3 NOUN cana-918 211 71 , , PUNCT cana-918 211 72 ๐‘  ๐‘  PROPN cana-918 211 73 โˆˆ โˆˆ PROPN cana-918 211 74 โ„ค4 โ„ค4 PROPN cana-918 211 75 } } PUNCT cana-918 211 76 communications communication NOUN cana-918 211 77 on on ADP cana-918 211 78 applied apply VERB cana-918 211 79 nonlinear nonlinear ADJ cana-918 211 80 analysis analysis NOUN cana-918 211 81 issn issn NOUN cana-918 211 82 : : PUNCT cana-918 211 83 1074 1074 NUM cana-918 211 84 - - PUNCT cana-918 211 85 133x 133x NUM cana-918 211 86 vol vol NOUN cana-918 211 87 31 31 NUM cana-918 211 88 no no NOUN cana-918 211 89 . . PUNCT cana-918 212 1 4s 4s NUM cana-918 212 2 ( ( PUNCT cana-918 212 3 2024 2024 NUM cana-918 212 4 ) ) PUNCT cana-918 212 5 400 400 NUM cana-918 212 6 https://internationalpubls.com https://internationalpubls.com X cana-918 212 7 where where SCONJ cana-918 212 8 ฮฑ๐‘–๐‘— ฮฑ๐‘–๐‘— X cana-918 212 9 ( ( PUNCT cana-918 212 10 0 0 NUM cana-918 212 11 ) ) PUNCT cana-918 212 12 = = SYM cana-918 213 1 ฯƒฯ„ฯƒ2ฮณ๐‘–ฯƒฯ„ฯƒ๐‘— ฯƒฯ„ฯƒ2ฮณ๐‘–ฯƒฯ„ฯƒ๐‘— PROPN cana-918 213 2 , , PUNCT cana-918 213 3 ฮฑ๐‘–๐‘— ฮฑ๐‘–๐‘— X cana-918 213 4 ( ( PUNCT cana-918 213 5 1 1 NUM cana-918 213 6 ) ) PUNCT cana-918 213 7 = = SYM cana-918 214 1 ฯƒฯ„ฯƒ2ฮณ๐‘–ฯ„ฯƒฯ„ฯƒ๐‘— ฯƒฯ„ฯƒ2ฮณ๐‘–ฯ„ฯƒฯ„ฯƒ๐‘— PROPN cana-918 214 2 , , PUNCT cana-918 214 3 ฮฑ๐‘–๐‘— ฮฑ๐‘–๐‘— X cana-918 214 4 ( ( PUNCT cana-918 214 5 2 2 NUM cana-918 214 6 ) ) PUNCT cana-918 214 7 = = SYM cana-918 214 8 ฯ„ฯƒฯ„ฯƒ2ฮณ๐‘–ฯƒฯ„ฯƒ๐‘— ฯ„ฯƒฯ„ฯƒ2ฮณ๐‘–ฯƒฯ„ฯƒ๐‘— ADP cana-918 214 9 , , PUNCT cana-918 214 10 ฮฑ๐‘–๐‘— ฮฑ๐‘–๐‘— X cana-918 214 11 ( ( PUNCT cana-918 214 12 3 3 NUM cana-918 214 13 ) ) PUNCT cana-918 214 14 = = PRON cana-918 214 15 ฯ„ฯƒฯ„ฯƒ2ฮณ๐‘–ฯ„ฯƒฯ„ฯƒ๐‘— ฯ„ฯƒฯ„ฯƒ2ฮณ๐‘–ฯ„ฯƒฯ„ฯƒ๐‘— X cana-918 214 16 is be AUX cana-918 214 17 a a DET cana-918 214 18 non non ADJ cana-918 214 19 - - ADJ cana-918 214 20 symmetric symmetric ADJ cana-918 214 21 association association NOUN cana-918 214 22 scheme scheme NOUN cana-918 214 23 with with ADP cana-918 214 24 parameters parameter NOUN cana-918 214 25 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š VERB cana-918 214 26 ๐‘˜ ๐‘˜ X cana-918 214 27 = = PUNCT cana-918 214 28 { { PUNCT cana-918 214 29 1 1 NUM cana-918 214 30 if if SCONJ cana-918 214 31 ๐‘›(๐‘˜ ๐‘›(๐‘˜ NOUN cana-918 214 32 ) ) PUNCT cana-918 214 33 โ‰ก โ‰ก PROPN cana-918 214 34 ( ( PUNCT cana-918 214 35 ๐‘›(๐‘™ ๐‘›(๐‘™ PROPN cana-918 214 36 ) ) PUNCT cana-918 214 37 + + SYM cana-918 214 38 ๐‘›(๐‘š ๐‘›(๐‘š NOUN cana-918 214 39 ) ) PUNCT cana-918 214 40 ) ) PUNCT cana-918 214 41 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 214 42 5 5 NUM cana-918 214 43 , , PUNCT cana-918 214 44 ๐‘ก(๐‘˜ ๐‘ก(๐‘˜ NOUN cana-918 214 45 ) ) PUNCT cana-918 214 46 โ‰ก โ‰ก PROPN cana-918 214 47 ( ( PUNCT cana-918 214 48 ๐‘ก(๐‘™ ๐‘ก(๐‘™ PROPN cana-918 214 49 ) ) PUNCT cana-918 214 50 + + NUM cana-918 214 51 ๐‘ก(๐‘š ๐‘ก(๐‘š NOUN cana-918 214 52 ) ) PUNCT cana-918 214 53 ) ) PUNCT cana-918 214 54 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 214 55 3 3 NUM cana-918 214 56 , , PUNCT cana-918 214 57 ๐‘Ž๐‘›๐‘‘ ๐‘Ž๐‘›๐‘‘ ADJ cana-918 214 58 ๐‘Ÿ(๐‘˜ ๐‘Ÿ(๐‘˜ PROPN cana-918 214 59 ) ) PUNCT cana-918 214 60 โ‰ก โ‰ก PROPN cana-918 214 61 ( ( PUNCT cana-918 214 62 ๐‘Ÿ(๐‘™ ๐‘Ÿ(๐‘™ PROPN cana-918 214 63 ) ) PUNCT cana-918 214 64 + + CCONJ cana-918 214 65 ๐‘Ÿ(๐‘š ๐‘Ÿ(๐‘š NOUN cana-918 214 66 ) ) PUNCT cana-918 214 67 ) ) PUNCT cana-918 214 68 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 214 69 4 4 NUM cana-918 214 70 0 0 NUM cana-918 214 71 otherwise otherwise ADV cana-918 214 72 } } PUNCT cana-918 214 73 proof proof NOUN cana-918 214 74 . . PUNCT cana-918 215 1 observe observe VERB cana-918 215 2 that that SCONJ cana-918 215 3 |๐’ฎ๐‘˜| |๐’ฎ๐‘˜| ADV cana-918 215 4 = = PUNCT cana-918 215 5 |๐‘Œ| |๐‘Œ| VERB cana-918 215 6 for for ADP cana-918 215 7 all all DET cana-918 215 8 0 0 NUM cana-918 215 9 โ‰ค โ‰ค NOUN cana-918 215 10 ๐‘˜ ๐‘˜ PRON cana-918 215 11 < < X cana-918 215 12 60 60 NUM cana-918 215 13 . . PUNCT cana-918 216 1 the the DET cana-918 216 2 relations relation NOUN cana-918 216 3 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 216 4 are be AUX cana-918 216 5 disjoint disjoint ADJ cana-918 216 6 and and CCONJ cana-918 216 7 โˆช โˆช ADV cana-918 216 8 { { PUNCT cana-918 216 9 ๐’ฎ๐‘˜:0 ๐’ฎ๐‘˜:0 X cana-918 216 10 โ‰ค โ‰ค NOUN cana-918 216 11 ๐‘˜ ๐‘˜ PRON cana-918 216 12 < < X cana-918 216 13 60 60 NUM cana-918 216 14 } } PUNCT cana-918 216 15 = = SYM cana-918 216 16 ๐’ซ. ๐’ซ. NOUN cana-918 216 17 ๐’ฎ0 ๐’ฎ0 NOUN cana-918 216 18 = = SYM cana-918 216 19 { { PUNCT cana-918 216 20 ( ( PUNCT cana-918 216 21 ๐›ผ๐‘–๐‘— ๐›ผ๐‘–๐‘— PROPN cana-918 216 22 ( ( PUNCT cana-918 216 23 ๐‘  ๐‘  PROPN cana-918 216 24 ) ) PUNCT cana-918 216 25 , , PUNCT cana-918 216 26 ๐›ผ๐‘–๐‘— ๐›ผ๐‘–๐‘— PROPN cana-918 216 27 ( ( PUNCT cana-918 216 28 ๐‘  ๐‘  NOUN cana-918 216 29 ) ) PUNCT cana-918 216 30 ) ) PUNCT cana-918 216 31 : : PUNCT cana-918 217 1 0 0 NUM cana-918 217 2 โ‰ค โ‰ค NUM cana-918 217 3 ๐‘– ๐‘– SYM cana-918 217 4 โ‰ค โ‰ค NOUN cana-918 217 5 4 4 NUM cana-918 217 6 , , PUNCT cana-918 217 7 0 0 NUM cana-918 217 8 โ‰ค โ‰ค NUM cana-918 217 9 ๐‘— ๐‘— PRON cana-918 217 10 โ‰ค โ‰ค NUM cana-918 217 11 2 2 NUM cana-918 217 12 , , PUNCT cana-918 217 13 0 0 NUM cana-918 217 14 โ‰ค โ‰ค NUM cana-918 217 15 ๐‘  ๐‘  X cana-918 217 16 โ‰ค โ‰ค NUM cana-918 217 17 3 3 NUM cana-918 217 18 } } PUNCT cana-918 217 19 is be AUX cana-918 217 20 an an DET cana-918 217 21 identity identity NOUN cana-918 217 22 relation relation NOUN cana-918 217 23 . . PUNCT cana-918 218 1 let let VERB cana-918 218 2 ๐’ฎ๐‘™ ๐’ฎ๐‘™ PROPN cana-918 218 3 , , PUNCT cana-918 218 4 ๐’ฎ๐‘š ๐’ฎ๐‘š NOUN cana-918 218 5 , , PUNCT cana-918 218 6 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 218 7 be be VERB cana-918 218 8 arbitrary arbitrary ADJ cana-918 218 9 relations relation NOUN cana-918 218 10 in in ADP cana-918 218 11 ๐’ซ ๐’ซ PROPN cana-918 218 12 and and CCONJ cana-918 218 13 ( ( PUNCT cana-918 218 14 ๐‘ฅ ๐‘ฅ PROPN cana-918 218 15 , , PUNCT cana-918 218 16 ๐‘ฆ ๐‘ฆ X cana-918 218 17 ) ) PUNCT cana-918 218 18 be be VERB cana-918 218 19 any any DET cana-918 218 20 element element NOUN cana-918 218 21 of of ADP cana-918 218 22 ๐’ฎ๐‘˜. ๐’ฎ๐‘˜. PROPN cana-918 218 23 since since SCONJ cana-918 218 24 ๐‘ฅ ๐‘ฅ PROPN cana-918 218 25 โˆˆ โˆˆ PROPN cana-918 218 26 ๐‘Œ ๐‘Œ PROPN cana-918 218 27 , , PUNCT cana-918 218 28 ๐‘ฅ ๐‘ฅ PRON cana-918 218 29 is be AUX cana-918 218 30 of of ADP cana-918 218 31 the the DET cana-918 218 32 form form NOUN cana-918 218 33 ๐œ๐‘Ž๐œŽ๐œ๐œŽ2๐›พ๐‘๐œ๐‘๐œŽ๐œ๐œŽ๐‘‘ ๐œ๐‘Ž๐œŽ๐œ๐œŽ2๐›พ๐‘๐œ๐‘๐œŽ๐œ๐œŽ๐‘‘ VERB cana-918 218 34 for for ADP cana-918 218 35 some some DET cana-918 218 36 ๐‘Ž ๐‘Ž NOUN cana-918 218 37 , , PUNCT cana-918 218 38 ๐‘ ๐‘ NOUN cana-918 218 39 , , PUNCT cana-918 218 40 ๐‘ ๐‘ PROPN cana-918 218 41 , , PUNCT cana-918 218 42 ๐‘‘ ๐‘‘ X cana-918 218 43 ( ( PUNCT cana-918 218 44 from from ADP cana-918 218 45 table table NOUN cana-918 218 46 [ [ X cana-918 218 47 table table NOUN cana-918 218 48 : : PUNCT cana-918 218 49 canonical canonical ADJ cana-918 218 50 forms form NOUN cana-918 218 51 an an DET cana-918 218 52 , , PUNCT cana-918 218 53 sn sn NOUN cana-918 218 54 ] ] PUNCT cana-918 218 55 ) ) PUNCT cana-918 218 56 and and CCONJ cana-918 218 57 further far ADV cana-918 218 58 ๐‘ฅ ๐‘ฅ PRON cana-918 218 59 = = SYM cana-918 218 60 ๐›ผ๐‘–๐‘— ๐›ผ๐‘–๐‘— PROPN cana-918 218 61 ( ( PUNCT cana-918 218 62 ๐‘  ๐‘  NOUN cana-918 218 63 ) ) PUNCT cana-918 218 64 for for ADP cana-918 218 65 some some PRON cana-918 218 66 ๐‘– ๐‘– NOUN cana-918 218 67 , , PUNCT cana-918 218 68 ๐‘— ๐‘— INTJ cana-918 218 69 , , PUNCT cana-918 218 70 ๐‘ . ๐‘ . PROPN cana-918 218 71 suppose suppose VERB cana-918 218 72 ๐‘ฅ๐’ฎ๐‘™ ๐‘ฅ๐’ฎ๐‘™ ADJ cana-918 218 73 = = SYM cana-918 218 74 ๐‘ฅ ๐‘ฅ NOUN cana-918 219 1 โ€ฒ โ€ฒ NUM cana-918 219 2 and and CCONJ cana-918 219 3 ๐‘ฆ๐’ฎ๐‘š ๐‘ฆ๐’ฎ๐‘š NOUN cana-918 219 4 โˆ— โˆ— NOUN cana-918 219 5 = = SYM cana-918 219 6 ๐‘ฆโ€ฒ ๐‘ฆโ€ฒ PUNCT cana-918 219 7 where where SCONJ cana-918 219 8 ๐‘ฅโ€ฒ ๐‘ฅโ€ฒ PROPN cana-918 219 9 , , PUNCT cana-918 219 10 ๐‘ฆโ€ฒ ๐‘ฆโ€ฒ X cana-918 219 11 โˆˆ โˆˆ PROPN cana-918 219 12 ๐‘Œ. ๐‘Œ. PROPN cana-918 219 13 now now ADV cana-918 219 14 ( ( PUNCT cana-918 219 15 ๐‘ฅ ๐‘ฅ PROPN cana-918 219 16 , , PUNCT cana-918 219 17 ๐‘ฆ ๐‘ฆ NOUN cana-918 219 18 ) ) PUNCT cana-918 219 19 โˆˆ โˆˆ PROPN cana-918 220 1 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 220 2 implies imply VERB cana-918 220 3 ๐‘ฆ ๐‘ฆ NOUN cana-918 220 4 = = SYM cana-918 220 5 ๐›ผ ๐›ผ PROPN cana-918 220 6 ( ( PUNCT cana-918 220 7 ๐‘–+๐‘›(๐‘˜ ๐‘–+๐‘›(๐‘˜ NUM cana-918 220 8 ) ) PUNCT cana-918 220 9 ) ) PUNCT cana-918 220 10 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 220 11 5 5 NUM cana-918 220 12 ( ( PUNCT cana-918 220 13 ๐‘—+๐‘ก(๐‘˜ ๐‘—+๐‘ก(๐‘˜ NOUN cana-918 220 14 ) ) PUNCT cana-918 220 15 ) ) PUNCT cana-918 220 16 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 220 17 3 3 NUM cana-918 220 18 ( ( PUNCT cana-918 220 19 ๐‘ +๐‘Ÿ(๐‘˜ ๐‘ +๐‘Ÿ(๐‘˜ NOUN cana-918 220 20 ) ) PUNCT cana-918 220 21 ) ) PUNCT cana-918 220 22 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 220 23 4 4 NUM cana-918 220 24 where where SCONJ cana-918 220 25 ๐‘˜ ๐‘˜ PRON cana-918 220 26 = = SYM cana-918 220 27 12๐‘›(๐‘˜ 12๐‘›(๐‘˜ NUM cana-918 220 28 ) ) PUNCT cana-918 220 29 + + NUM cana-918 220 30 4๐‘ก(๐‘˜ 4๐‘ก(๐‘˜ NUM cana-918 220 31 ) ) PUNCT cana-918 221 1 + + CCONJ cana-918 221 2 ๐‘Ÿ(๐‘˜ ๐‘Ÿ(๐‘˜ NUM cana-918 221 3 ) ) PUNCT cana-918 221 4 for for ADP cana-918 221 5 some some DET cana-918 221 6 ๐‘ก(๐‘˜ ๐‘ก(๐‘˜ NOUN cana-918 221 7 ) ) PUNCT cana-918 221 8 โˆˆ โˆˆ PROPN cana-918 221 9 โ„ค3 โ„ค3 NOUN cana-918 221 10 , , PUNCT cana-918 221 11 ๐‘› ๐‘› PROPN cana-918 221 12 ( ( PUNCT cana-918 221 13 ๐‘˜ ๐‘˜ NOUN cana-918 221 14 ) ) PUNCT cana-918 221 15 โˆˆ โˆˆ PROPN cana-918 221 16 โ„ค5 โ„ค5 NOUN cana-918 221 17 , , PUNCT cana-918 221 18 ๐‘Ÿ ๐‘Ÿ X cana-918 221 19 ( ( PUNCT cana-918 221 20 ๐‘˜ ๐‘˜ NOUN cana-918 221 21 ) ) PUNCT cana-918 221 22 โˆˆ โˆˆ PROPN cana-918 221 23 โ„ค4 โ„ค4 NOUN cana-918 221 24 . . PUNCT cana-918 222 1 similarly similarly ADV cana-918 222 2 , , PUNCT cana-918 222 3 ( ( PUNCT cana-918 222 4 ๐‘ฅ ๐‘ฅ X cana-918 222 5 , , PUNCT cana-918 222 6 ๐‘ฅโ€ฒ ๐‘ฅโ€ฒ NUM cana-918 222 7 ) ) PUNCT cana-918 223 1 โˆˆ โˆˆ PROPN cana-918 223 2 ๐’ฎ๐‘™ ๐’ฎ๐‘™ PROPN cana-918 223 3 and and CCONJ cana-918 223 4 ( ( PUNCT cana-918 223 5 ๐‘ฆโ€ฒ ๐‘ฆโ€ฒ NOUN cana-918 223 6 , , PUNCT cana-918 223 7 ๐‘ฆ ๐‘ฆ X cana-918 223 8 ) ) PUNCT cana-918 223 9 โˆˆ โˆˆ PROPN cana-918 223 10 ๐’ฎ๐‘š ๐’ฎ๐‘š NOUN cana-918 223 11 , , PUNCT cana-918 223 12 implies imply VERB cana-918 223 13 ๐‘ฅโ€ฒ ๐‘ฅโ€ฒ PUNCT cana-918 223 14 = = SYM cana-918 223 15 ๐›ผ ๐›ผ X cana-918 223 16 ( ( PUNCT cana-918 223 17 ๐‘–+๐‘›(๐‘™ ๐‘–+๐‘›(๐‘™ NOUN cana-918 223 18 ) ) PUNCT cana-918 223 19 ) ) PUNCT cana-918 223 20 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 223 21 5 5 NUM cana-918 223 22 ( ( PUNCT cana-918 223 23 ๐‘—+๐‘ก(๐‘™ ๐‘—+๐‘ก(๐‘™ NOUN cana-918 223 24 ) ) PUNCT cana-918 223 25 ) ) PUNCT cana-918 223 26 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 223 27 3 3 NUM cana-918 223 28 ( ( PUNCT cana-918 223 29 ๐‘ +๐‘Ÿ(๐‘™ ๐‘ +๐‘Ÿ(๐‘™ NOUN cana-918 223 30 ) ) PUNCT cana-918 223 31 ) ) PUNCT cana-918 223 32 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 223 33 4 4 NUM cana-918 224 1 where where SCONJ cana-918 224 2 ๐‘™ ๐‘™ NOUN cana-918 224 3 = = SYM cana-918 224 4 12๐‘›(๐‘™ 12๐‘›(๐‘™ NUM cana-918 224 5 ) ) PUNCT cana-918 224 6 + + CCONJ cana-918 224 7 4๐‘ก(๐‘™ 4๐‘ก(๐‘™ NUM cana-918 224 8 ) ) PUNCT cana-918 225 1 + + CCONJ cana-918 225 2 ๐‘Ÿ(๐‘™ ๐‘Ÿ(๐‘™ NOUN cana-918 225 3 ) ) PUNCT cana-918 225 4 for for ADP cana-918 225 5 some some DET cana-918 225 6 ๐‘ก(๐‘™ ๐‘ก(๐‘™ PROPN cana-918 225 7 ) ) PUNCT cana-918 225 8 โˆˆ โˆˆ PROPN cana-918 225 9 โ„ค3 โ„ค3 NOUN cana-918 225 10 , , PUNCT cana-918 225 11 ๐‘› ๐‘› PROPN cana-918 225 12 ( ( PUNCT cana-918 225 13 ๐‘™ ๐‘™ X cana-918 225 14 ) ) PUNCT cana-918 225 15 โˆˆ โˆˆ PROPN cana-918 225 16 โ„ค5 โ„ค5 NOUN cana-918 225 17 , , PUNCT cana-918 225 18 ๐‘Ÿ ๐‘Ÿ X cana-918 225 19 ( ( PUNCT cana-918 225 20 ๐‘™ ๐‘™ X cana-918 225 21 ) ) PUNCT cana-918 225 22 โˆˆ โˆˆ PROPN cana-918 225 23 โ„ค4 โ„ค4 NOUN cana-918 225 24 ; ; PUNCT cana-918 225 25 and and CCONJ cana-918 225 26 ๐‘ฆโ€ฒ ๐‘ฆโ€ฒ X cana-918 225 27 = = SYM cana-918 225 28 ๐›ผ ๐›ผ PROPN cana-918 225 29 ( ( PUNCT cana-918 225 30 ๐‘–+๐‘›(๐‘˜)โˆ’๐‘›(๐‘š ๐‘–+๐‘›(๐‘˜)โˆ’๐‘›(๐‘š NOUN cana-918 225 31 ) ) PUNCT cana-918 225 32 ) ) PUNCT cana-918 225 33 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 225 34 5 5 NUM cana-918 225 35 ( ( PUNCT cana-918 225 36 ๐‘—+๐‘ก(๐‘˜)โˆ’๐‘ก(๐‘š ๐‘—+๐‘ก(๐‘˜)โˆ’๐‘ก(๐‘š NUM cana-918 225 37 ) ) PUNCT cana-918 225 38 ) ) PUNCT cana-918 225 39 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 225 40 3 3 NUM cana-918 225 41 ( ( PUNCT cana-918 225 42 ๐‘ +๐‘Ÿ(๐‘˜)โˆ’๐‘Ÿ(๐‘š ๐‘ +๐‘Ÿ(๐‘˜)โˆ’๐‘Ÿ(๐‘š PROPN cana-918 225 43 ) ) PUNCT cana-918 225 44 ) ) PUNCT cana-918 225 45 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 225 46 4 4 NUM cana-918 225 47 where where SCONJ cana-918 225 48 ๐‘š ๐‘š X cana-918 225 49 = = SYM cana-918 225 50 12๐‘›(๐‘š 12๐‘›(๐‘š NUM cana-918 225 51 ) ) PUNCT cana-918 225 52 + + CCONJ cana-918 225 53 4๐‘ก(๐‘š 4๐‘ก(๐‘š NUM cana-918 225 54 ) ) PUNCT cana-918 225 55 + + CCONJ cana-918 225 56 ๐‘Ÿ(๐‘š ๐‘Ÿ(๐‘š NOUN cana-918 225 57 ) ) PUNCT cana-918 225 58 for for ADP cana-918 225 59 some some DET cana-918 225 60 ๐‘ก(๐‘š ๐‘ก(๐‘š NOUN cana-918 225 61 ) ) PUNCT cana-918 225 62 โˆˆ โˆˆ PROPN cana-918 225 63 โ„ค3 โ„ค3 NOUN cana-918 225 64 , , PUNCT cana-918 225 65 ๐‘› ๐‘› PROPN cana-918 225 66 ( ( PUNCT cana-918 225 67 ๐‘š ๐‘š NOUN cana-918 225 68 ) ) PUNCT cana-918 225 69 โˆˆ โˆˆ PROPN cana-918 225 70 โ„ค5 โ„ค5 NOUN cana-918 225 71 , , PUNCT cana-918 225 72 ๐‘Ÿ ๐‘Ÿ X cana-918 225 73 ( ( PUNCT cana-918 225 74 ๐‘š ๐‘š NOUN cana-918 225 75 ) ) PUNCT cana-918 225 76 โˆˆ โˆˆ PROPN cana-918 225 77 โ„ค4 โ„ค4 NOUN cana-918 225 78 . . PUNCT cana-918 226 1 since since SCONJ cana-918 226 2 every every DET cana-918 226 3 pair pair NOUN cana-918 226 4 ( ( PUNCT cana-918 226 5 ๐‘ฅ ๐‘ฅ NOUN cana-918 226 6 , , PUNCT cana-918 226 7 ๐‘ฆ ๐‘ฆ NOUN cana-918 226 8 ) ) PUNCT cana-918 226 9 in in ADP cana-918 226 10 ๐’ซ ๐’ซ PROPN cana-918 226 11 are be AUX cana-918 226 12 ๐‘˜๐‘กโ„Ž ๐‘˜๐‘กโ„Ž VERB cana-918 226 13 associates associate NOUN cana-918 226 14 for for ADP cana-918 226 15 exactly exactly ADV cana-918 226 16 one one NUM cana-918 226 17 k k NOUN cana-918 226 18 , , PUNCT cana-918 226 19 we we PRON cana-918 226 20 find find VERB cana-918 226 21 that that SCONJ cana-918 226 22 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š PROPN cana-918 226 23 ๐‘˜ ๐‘˜ PRON cana-918 226 24 can can AUX cana-918 226 25 be be AUX cana-918 226 26 either either CCONJ cana-918 226 27 0 0 NUM cana-918 226 28 or or CCONJ cana-918 226 29 1 1 NUM cana-918 226 30 . . PUNCT cana-918 227 1 hence hence ADV cana-918 227 2 we we PRON cana-918 227 3 obtain obtain VERB cana-918 227 4 that that DET cana-918 227 5 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š PROPN cana-918 227 6 ๐‘˜ ๐‘˜ NOUN cana-918 227 7 = = NOUN cana-918 227 8 1 1 NUM cana-918 227 9 if if SCONJ cana-918 227 10 and and CCONJ cana-918 227 11 only only ADV cana-918 227 12 if if SCONJ cana-918 227 13 ๐‘ฅโ€ฒ ๐‘ฅโ€ฒ VERB cana-918 227 14 = = PUNCT cana-918 227 15 ๐‘ฆโ€ฒ ๐‘ฆโ€ฒ X cana-918 227 16 that that PRON cana-918 227 17 is be AUX cana-918 227 18 , , PUNCT cana-918 227 19 if if SCONJ cana-918 227 20 ๐‘ก(๐‘˜ ๐‘ก(๐‘˜ NOUN cana-918 227 21 ) ) PUNCT cana-918 227 22 โ‰ก โ‰ก PROPN cana-918 227 23 ( ( PUNCT cana-918 227 24 ๐‘ก(๐‘™ ๐‘ก(๐‘™ PROPN cana-918 227 25 ) ) PUNCT cana-918 227 26 + + NUM cana-918 227 27 ๐‘ก(๐‘š ๐‘ก(๐‘š NOUN cana-918 227 28 ) ) PUNCT cana-918 227 29 ) ) PUNCT cana-918 227 30 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 227 31 3 3 NUM cana-918 227 32 , , PUNCT cana-918 227 33 ๐‘Ÿ(๐‘˜ ๐‘Ÿ(๐‘˜ NUM cana-918 227 34 ) ) PUNCT cana-918 227 35 โ‰ก โ‰ก PROPN cana-918 227 36 ( ( PUNCT cana-918 227 37 ๐‘Ÿ(๐‘™ ๐‘Ÿ(๐‘™ PROPN cana-918 227 38 ) ) PUNCT cana-918 227 39 + + CCONJ cana-918 227 40 ๐‘Ÿ(๐‘š ๐‘Ÿ(๐‘š NOUN cana-918 227 41 ) ) PUNCT cana-918 227 42 ) ) PUNCT cana-918 227 43 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 227 44 4 4 NUM cana-918 227 45 and and CCONJ cana-918 227 46 ๐‘›(๐‘˜ ๐‘›(๐‘˜ PROPN cana-918 227 47 ) ) PUNCT cana-918 227 48 โ‰ก โ‰ก PROPN cana-918 227 49 ( ( PUNCT cana-918 227 50 ๐‘›(๐‘™ ๐‘›(๐‘™ PROPN cana-918 227 51 ) ) PUNCT cana-918 227 52 + + SYM cana-918 227 53 ๐‘›(๐‘š ๐‘›(๐‘š NOUN cana-918 227 54 ) ) PUNCT cana-918 227 55 ) ) PUNCT cana-918 227 56 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 228 1 5 5 NUM cana-918 228 2 . . PUNCT cana-918 228 3 โ—ป โ—ป PROPN cana-918 228 4 theorem theorem VERB cana-918 228 5 9 9 NUM cana-918 228 6 . . PUNCT cana-918 229 1 let let VERB cana-918 229 2 ๐‘Œ ๐‘Œ PROPN cana-918 229 3 = = PUNCT cana-918 229 4 ๐‘†4 ๐‘†4 PROPN cana-918 229 5 and and CCONJ cana-918 229 6 ๐’ซ ๐’ซ PROPN cana-918 229 7 be be VERB cana-918 229 8 a a DET cana-918 229 9 partition partition NOUN cana-918 229 10 of of ADP cana-918 229 11 ๐‘Œ ๐‘Œ PROPN cana-918 229 12 ร— ร— PROPN cana-918 229 13 ๐‘Œ. ๐‘Œ. PROPN cana-918 229 14 for for ADP cana-918 229 15 all all PRON cana-918 229 16 ๐‘˜ ๐‘˜ PRON cana-918 229 17 written write VERB cana-918 229 18 in in ADP cana-918 229 19 form form NOUN cana-918 229 20 of of ADP cana-918 229 21 4๐‘› 4๐‘› NOUN cana-918 229 22 + + CCONJ cana-918 230 1 ๐‘ก ๐‘ก PROPN cana-918 230 2 ( ( PUNCT cana-918 230 3 ๐‘› ๐‘› PROPN cana-918 230 4 โˆˆ โˆˆ PROPN cana-918 230 5 โ„ค6 โ„ค6 NOUN cana-918 230 6 , , PUNCT cana-918 230 7 ๐‘ก ๐‘ก PROPN cana-918 230 8 โˆˆ โˆˆ PROPN cana-918 230 9 โ„ค4 โ„ค4 NOUN cana-918 230 10 ) ) PUNCT cana-918 230 11 , , PUNCT cana-918 230 12 the the DET cana-918 230 13 relations relation NOUN cana-918 230 14 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 230 15 on on ADP cana-918 230 16 ๐’ซ ๐’ซ NOUN cana-918 230 17 defined define VERB cana-918 230 18 by by ADP cana-918 230 19 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 230 20 = = SYM cana-918 230 21 { { PUNCT cana-918 230 22 ( ( PUNCT cana-918 230 23 ๐›ผ๐‘– ๐›ผ๐‘– PROPN cana-918 230 24 ( ( PUNCT cana-918 230 25 ๐‘— ๐‘— NOUN cana-918 230 26 ) ) PUNCT cana-918 230 27 , , PUNCT cana-918 230 28 ๐›ผ(๐‘–+๐‘ก)๐‘š๐‘œ๐‘‘ ๐›ผ(๐‘–+๐‘ก)๐‘š๐‘œ๐‘‘ ADJ cana-918 230 29 4 4 NUM cana-918 230 30 ( ( PUNCT cana-918 230 31 ๐‘—+๐‘›)๐‘š๐‘œ๐‘‘ ๐‘—+๐‘›)๐‘š๐‘œ๐‘‘ NOUN cana-918 230 32 6 6 NUM cana-918 230 33 ) ) PUNCT cana-918 230 34 | | ADV cana-918 230 35 ๐‘– ๐‘– SYM cana-918 230 36 โˆˆ โˆˆ PROPN cana-918 230 37 โ„ค4 โ„ค4 PROPN cana-918 230 38 , , PUNCT cana-918 230 39 ๐‘— ๐‘— PROPN cana-918 230 40 โˆˆ โˆˆ PROPN cana-918 230 41 โ„ค6 โ„ค6 NOUN cana-918 230 42 } } PUNCT cana-918 230 43 where where SCONJ cana-918 230 44 ฮฑ๐‘– ฮฑ๐‘– ADP cana-918 230 45 ( ( PUNCT cana-918 230 46 0 0 NUM cana-918 230 47 ) ) PUNCT cana-918 230 48 = = SYM cana-918 230 49 ฯ„ฯƒ๐‘– ฯ„ฯƒ๐‘– PROPN cana-918 230 50 , , PUNCT cana-918 230 51 ฮฑ๐‘– ฮฑ๐‘– ADP cana-918 230 52 ( ( PUNCT cana-918 230 53 1 1 X cana-918 230 54 ) ) PUNCT cana-918 230 55 = = SYM cana-918 230 56 ฯƒฯ„ฯƒ๐‘– ฯƒฯ„ฯƒ๐‘– PROPN cana-918 230 57 , , PUNCT cana-918 230 58 ฮฑ๐‘– ฮฑ๐‘– ADP cana-918 230 59 ( ( PUNCT cana-918 230 60 2 2 NUM cana-918 230 61 ) ) PUNCT cana-918 230 62 = = SYM cana-918 230 63 ฯƒ2ฯ„ฯƒ๐‘– ฯƒ2ฯ„ฯƒ๐‘– NOUN cana-918 230 64 , , PUNCT cana-918 230 65 ฮฑ๐‘– ฮฑ๐‘– ADP cana-918 230 66 ( ( PUNCT cana-918 230 67 3 3 X cana-918 230 68 ) ) PUNCT cana-918 230 69 = = SYM cana-918 230 70 ฯƒ๐‘– ฯƒ๐‘– PROPN cana-918 230 71 , , PUNCT cana-918 230 72 ฮฑ๐‘– ฮฑ๐‘– ADP cana-918 230 73 ( ( PUNCT cana-918 230 74 4 4 NUM cana-918 230 75 ) ) PUNCT cana-918 230 76 = = NOUN cana-918 230 77 ฯ„ฯƒฯ„ฯƒ๐‘– ฯ„ฯƒฯ„ฯƒ๐‘– NOUN cana-918 230 78 , , PUNCT cana-918 230 79 ฮฑ๐‘– ฮฑ๐‘– ADP cana-918 230 80 ( ( PUNCT cana-918 230 81 5 5 NUM cana-918 230 82 ) ) PUNCT cana-918 230 83 = = NOUN cana-918 231 1 ฯ„ฯƒ2ฯ„ฯƒ๐‘– ฯ„ฯƒ2ฯ„ฯƒ๐‘– PROPN cana-918 231 2 is be AUX cana-918 231 3 a a DET cana-918 231 4 non non ADJ cana-918 231 5 - - ADJ cana-918 231 6 symmetric symmetric ADJ cana-918 231 7 association association NOUN cana-918 231 8 scheme scheme NOUN cana-918 231 9 with with ADP cana-918 231 10 parameters parameter NOUN cana-918 231 11 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š VERB cana-918 231 12 ๐‘˜ ๐‘˜ X cana-918 231 13 = = PUNCT cana-918 231 14 { { PUNCT cana-918 231 15 1 1 NUM cana-918 231 16 ๐‘–๐‘“ ๐‘–๐‘“ ADP cana-918 231 17 ๐‘›(๐‘˜ ๐‘›(๐‘˜ NOUN cana-918 231 18 ) ) PUNCT cana-918 231 19 โ‰ก โ‰ก PROPN cana-918 231 20 ( ( PUNCT cana-918 231 21 ๐‘›(๐‘™ ๐‘›(๐‘™ PROPN cana-918 231 22 ) ) PUNCT cana-918 231 23 + + SYM cana-918 231 24 ๐‘›(๐‘š ๐‘›(๐‘š NOUN cana-918 231 25 ) ) PUNCT cana-918 231 26 ) ) PUNCT cana-918 231 27 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 231 28 6 6 NUM cana-918 231 29 , , PUNCT cana-918 231 30 ๐‘Ž๐‘›๐‘‘ ๐‘Ž๐‘›๐‘‘ ADJ cana-918 231 31 ๐‘ก(๐‘˜ ๐‘ก(๐‘˜ NOUN cana-918 231 32 ) ) PUNCT cana-918 231 33 โ‰ก โ‰ก PROPN cana-918 231 34 ( ( PUNCT cana-918 231 35 ๐‘ก(๐‘™ ๐‘ก(๐‘™ PROPN cana-918 231 36 ) ) PUNCT cana-918 231 37 + + NUM cana-918 231 38 ๐‘ก(๐‘š ๐‘ก(๐‘š NOUN cana-918 231 39 ) ) PUNCT cana-918 231 40 ) ) PUNCT cana-918 231 41 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 231 42 4 4 NUM cana-918 231 43 0 0 NUM cana-918 231 44 ๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’ ๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’ NOUN cana-918 231 45 proof proof NOUN cana-918 231 46 . . PUNCT cana-918 232 1 ๐’ฎ0 ๐’ฎ0 PROPN cana-918 232 2 = = PRON cana-918 232 3 { { PUNCT cana-918 232 4 ( ( PUNCT cana-918 232 5 ๐›ผ๐‘– ๐›ผ๐‘– PROPN cana-918 232 6 ( ( PUNCT cana-918 232 7 ๐‘— ๐‘— NOUN cana-918 232 8 ) ) PUNCT cana-918 232 9 , , PUNCT cana-918 232 10 ๐›ผ๐‘– ๐›ผ๐‘– PROPN cana-918 232 11 ( ( PUNCT cana-918 232 12 ๐‘— ๐‘— NOUN cana-918 232 13 ) ) PUNCT cana-918 232 14 ) ) PUNCT cana-918 232 15 : : PUNCT cana-918 233 1 0 0 NUM cana-918 233 2 โ‰ค โ‰ค NUM cana-918 233 3 ๐‘– ๐‘– SYM cana-918 233 4 โ‰ค โ‰ค NOUN cana-918 233 5 2 2 NUM cana-918 233 6 , , PUNCT cana-918 233 7 0 0 NUM cana-918 233 8 โ‰ค โ‰ค NUM cana-918 233 9 ๐‘— ๐‘— PRON cana-918 233 10 โ‰ค โ‰ค NUM cana-918 233 11 3 3 NUM cana-918 233 12 } } PUNCT cana-918 233 13 is be AUX cana-918 233 14 an an DET cana-918 233 15 identity identity NOUN cana-918 233 16 relation relation NOUN cana-918 233 17 . . PUNCT cana-918 234 1 it it PRON cana-918 234 2 can can AUX cana-918 234 3 be be AUX cana-918 234 4 verified verify VERB cana-918 234 5 that that SCONJ cana-918 234 6 ( ( PUNCT cana-918 234 7 ๐‘Œ ๐‘Œ PROPN cana-918 234 8 , , PUNCT cana-918 234 9 ๐’ซ ๐’ซ NOUN cana-918 234 10 ) ) PUNCT cana-918 234 11 is be AUX cana-918 234 12 an an DET cana-918 234 13 association association NOUN cana-918 234 14 scheme scheme NOUN cana-918 234 15 and and CCONJ cana-918 234 16 the the DET cana-918 234 17 relations relation NOUN cana-918 234 18 are be AUX cana-918 234 19 non non ADJ cana-918 234 20 - - ADJ cana-918 234 21 symmetric symmetric ADJ cana-918 234 22 . . PUNCT cana-918 235 1 let let VERB cana-918 235 2 ๐’ฎ๐‘™ ๐’ฎ๐‘™ PROPN cana-918 235 3 , , PUNCT cana-918 235 4 ๐’ฎ๐‘š ๐’ฎ๐‘š NOUN cana-918 235 5 , , PUNCT cana-918 235 6 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 235 7 โˆˆ โˆˆ PROPN cana-918 235 8 ๐’ซ. ๐’ซ. NOUN cana-918 235 9 to to PART cana-918 235 10 find find VERB cana-918 235 11 cardinality cardinality NOUN cana-918 235 12 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š PROPN cana-918 235 13 ๐‘˜ ๐‘˜ INTJ cana-918 235 14 such such ADJ cana-918 235 15 that that PRON cana-918 235 16 for for ADP cana-918 235 17 all all PRON cana-918 235 18 ( ( PUNCT cana-918 235 19 ๐‘ฅ ๐‘ฅ PROPN cana-918 235 20 , , PUNCT cana-918 235 21 ๐‘ฆ ๐‘ฆ NOUN cana-918 235 22 ) ) PUNCT cana-918 235 23 โˆˆ โˆˆ PROPN cana-918 236 1 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 236 2 , , PUNCT cana-918 236 3 |๐‘ฅ๐’ฎ๐‘™ |๐‘ฅ๐’ฎ๐‘™ ADJ cana-918 236 4 โˆฉ โˆฉ NOUN cana-918 236 5 ๐‘ฆ๐’ฎ๐‘š ๐‘ฆ๐’ฎ๐‘š NOUN cana-918 236 6 โˆ— โˆ— NOUN cana-918 236 7 | | ADV cana-918 236 8 = = SYM cana-918 236 9 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š PROPN cana-918 236 10 ๐‘˜ ๐‘˜ INTJ cana-918 236 11 . . PUNCT cana-918 237 1 communications communication NOUN cana-918 237 2 on on ADP cana-918 237 3 applied apply VERB cana-918 237 4 nonlinear nonlinear ADJ cana-918 237 5 analysis analysis NOUN cana-918 237 6 issn issn NOUN cana-918 237 7 : : PUNCT cana-918 237 8 1074 1074 NUM cana-918 237 9 - - PUNCT cana-918 237 10 133x 133x NUM cana-918 237 11 vol vol NOUN cana-918 237 12 31 31 NUM cana-918 237 13 no no NOUN cana-918 237 14 . . PUNCT cana-918 238 1 4s 4s NUM cana-918 238 2 ( ( PUNCT cana-918 238 3 2024 2024 NUM cana-918 238 4 ) ) PUNCT cana-918 238 5 401 401 NUM cana-918 238 6 https://internationalpubls.com https://internationalpubls.com X cana-918 238 7 let let VERB cana-918 238 8 ( ( PUNCT cana-918 238 9 ๐‘ฅ ๐‘ฅ NOUN cana-918 238 10 , , PUNCT cana-918 238 11 ๐‘ฆ ๐‘ฆ NOUN cana-918 238 12 ) ) PUNCT cana-918 238 13 โˆˆ โˆˆ PROPN cana-918 239 1 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 239 2 , , PUNCT cana-918 239 3 ๐‘ฅ๐’ฎ๐‘™ ๐‘ฅ๐’ฎ๐‘™ X cana-918 239 4 = = SYM cana-918 239 5 ๐‘ฅ ๐‘ฅ PROPN cana-918 239 6 โ€ฒ โ€ฒ NUM cana-918 239 7 and and CCONJ cana-918 239 8 ๐‘ฆ๐’ฎ๐‘š ๐‘ฆ๐’ฎ๐‘š NOUN cana-918 239 9 โˆ— โˆ— NOUN cana-918 239 10 = = SYM cana-918 239 11 ๐‘ฆโ€ฒ ๐‘ฆโ€ฒ PUNCT cana-918 239 12 where where SCONJ cana-918 239 13 ๐‘ฅโ€ฒ ๐‘ฅโ€ฒ PROPN cana-918 239 14 , , PUNCT cana-918 239 15 ๐‘ฆโ€ฒ ๐‘ฆโ€ฒ X cana-918 239 16 โˆˆ โˆˆ PROPN cana-918 239 17 ๐‘Œ. ๐‘Œ. PROPN cana-918 239 18 since since SCONJ cana-918 239 19 ๐‘ฅ ๐‘ฅ PROPN cana-918 239 20 โˆˆ โˆˆ PROPN cana-918 239 21 ๐‘Œ ๐‘Œ PROPN cana-918 239 22 , , PUNCT cana-918 239 23 ๐‘ฅ ๐‘ฅ PRON cana-918 239 24 is be AUX cana-918 239 25 of of ADP cana-918 239 26 the the DET cana-918 239 27 form form NOUN cana-918 239 28 ๐œ๐‘Ž๐œŽ๐‘๐œ๐œŽ๐‘ ๐œ๐‘Ž๐œŽ๐‘๐œ๐œŽ๐‘ NOUN cana-918 239 29 for for ADP cana-918 239 30 some some DET cana-918 239 31 ๐‘Ž ๐‘Ž PROPN cana-918 239 32 , , PUNCT cana-918 239 33 ๐‘ ๐‘ NOUN cana-918 239 34 , , PUNCT cana-918 239 35 ๐‘ ๐‘ PROPN cana-918 239 36 ( ( PUNCT cana-918 239 37 from from ADP cana-918 239 38 table table NOUN cana-918 239 39 [ [ X cana-918 239 40 table table NOUN cana-918 239 41 : : PUNCT cana-918 239 42 canonical canonical ADJ cana-918 239 43 forms form NOUN cana-918 239 44 an an DET cana-918 239 45 , , PUNCT cana-918 239 46 sn sn NOUN cana-918 239 47 ] ] PUNCT cana-918 239 48 ) ) PUNCT cana-918 239 49 and and CCONJ cana-918 239 50 further far ADV cana-918 239 51 ๐‘ฅ ๐‘ฅ PRON cana-918 239 52 = = SYM cana-918 239 53 ๐›ผ๐‘– ๐›ผ๐‘– PROPN cana-918 239 54 ( ( PUNCT cana-918 239 55 ๐‘— ๐‘— NOUN cana-918 239 56 ) ) PUNCT cana-918 239 57 for for ADP cana-918 239 58 some some PRON cana-918 239 59 ๐‘– ๐‘– ADP cana-918 239 60 , , PUNCT cana-918 239 61 ๐‘—. ๐‘—. NOUN cana-918 239 62 proceeding proceeding NOUN cana-918 239 63 as as ADP cana-918 239 64 in in ADP cana-918 239 65 proof proof NOUN cana-918 239 66 of of ADP cana-918 239 67 previous previous ADJ cana-918 239 68 theorem theorem NOUN cana-918 239 69 , , PUNCT cana-918 239 70 we we PRON cana-918 239 71 get get VERB cana-918 239 72 ๐‘ฆ ๐‘ฆ NOUN cana-918 239 73 = = SYM cana-918 239 74 ๐›ผ ๐›ผ X cana-918 239 75 ( ( PUNCT cana-918 239 76 ๐‘–+๐‘ก(๐‘˜ ๐‘–+๐‘ก(๐‘˜ NOUN cana-918 239 77 ) ) PUNCT cana-918 239 78 ) ) PUNCT cana-918 239 79 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 239 80 4 4 NUM cana-918 239 81 ( ( PUNCT cana-918 239 82 ๐‘—+๐‘›(๐‘˜ ๐‘—+๐‘›(๐‘˜ NOUN cana-918 239 83 ) ) PUNCT cana-918 239 84 ) ) PUNCT cana-918 239 85 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 239 86 6 6 NUM cana-918 239 87 where where SCONJ cana-918 239 88 ๐‘˜ ๐‘˜ PROPN cana-918 239 89 = = SYM cana-918 239 90 4๐‘›(๐‘˜ 4๐‘›(๐‘˜ PROPN cana-918 239 91 ) ) PUNCT cana-918 240 1 + + CCONJ cana-918 240 2 ๐‘ก(๐‘˜ ๐‘ก(๐‘˜ NOUN cana-918 240 3 ) ) PUNCT cana-918 240 4 for for ADP cana-918 240 5 some some DET cana-918 240 6 ๐‘ก(๐‘˜ ๐‘ก(๐‘˜ NOUN cana-918 240 7 ) ) PUNCT cana-918 240 8 โˆˆ โˆˆ PROPN cana-918 240 9 โ„ค4 โ„ค4 PROPN cana-918 240 10 , , PUNCT cana-918 240 11 ๐‘› ๐‘› PROPN cana-918 240 12 ( ( PUNCT cana-918 240 13 ๐‘˜ ๐‘˜ NOUN cana-918 240 14 ) ) PUNCT cana-918 240 15 โˆˆ โˆˆ PROPN cana-918 240 16 โ„ค6 โ„ค6 NOUN cana-918 240 17 ; ; PUNCT cana-918 240 18 ๐‘ฅโ€ฒ ๐‘ฅโ€ฒ PUNCT cana-918 240 19 = = SYM cana-918 240 20 ๐›ผ ๐›ผ X cana-918 240 21 ( ( PUNCT cana-918 240 22 ๐‘–+๐‘ก(๐‘™ ๐‘–+๐‘ก(๐‘™ NOUN cana-918 240 23 ) ) PUNCT cana-918 240 24 ) ) PUNCT cana-918 240 25 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 240 26 4 4 NUM cana-918 240 27 ( ( PUNCT cana-918 240 28 ๐‘—+๐‘›(๐‘™ ๐‘—+๐‘›(๐‘™ NOUN cana-918 240 29 ) ) PUNCT cana-918 240 30 ) ) PUNCT cana-918 240 31 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 240 32 6 6 NUM cana-918 241 1 where where SCONJ cana-918 241 2 ๐‘™ ๐‘™ NOUN cana-918 241 3 = = SYM cana-918 241 4 4๐‘›(๐‘™ 4๐‘›(๐‘™ PROPN cana-918 241 5 ) ) PUNCT cana-918 241 6 + + CCONJ cana-918 241 7 ๐‘ก(๐‘™ ๐‘ก(๐‘™ PROPN cana-918 241 8 ) ) PUNCT cana-918 241 9 for for ADP cana-918 241 10 some some DET cana-918 241 11 ๐‘ก(๐‘™ ๐‘ก(๐‘™ PROPN cana-918 241 12 ) ) PUNCT cana-918 241 13 โˆˆ โˆˆ PROPN cana-918 241 14 โ„ค4 โ„ค4 PROPN cana-918 241 15 , , PUNCT cana-918 241 16 ๐‘› ๐‘› PROPN cana-918 241 17 ( ( PUNCT cana-918 241 18 ๐‘™ ๐‘™ X cana-918 241 19 ) ) PUNCT cana-918 241 20 โˆˆ โˆˆ PROPN cana-918 241 21 โ„ค6 โ„ค6 NOUN cana-918 241 22 , , PUNCT cana-918 241 23 and and CCONJ cana-918 241 24 ๐‘ฆโ€ฒ ๐‘ฆโ€ฒ X cana-918 241 25 = = SYM cana-918 241 26 ๐›ผ ๐›ผ PROPN cana-918 241 27 ( ( PUNCT cana-918 241 28 ๐‘–+๐‘ก(๐‘˜)โˆ’๐‘ก(๐‘š ๐‘–+๐‘ก(๐‘˜)โˆ’๐‘ก(๐‘š NUM cana-918 241 29 ) ) PUNCT cana-918 241 30 ) ) PUNCT cana-918 241 31 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 241 32 4 4 NUM cana-918 241 33 ( ( PUNCT cana-918 241 34 ๐‘—+๐‘›(๐‘˜)โˆ’๐‘›(๐‘š ๐‘—+๐‘›(๐‘˜)โˆ’๐‘›(๐‘š NOUN cana-918 241 35 ) ) PUNCT cana-918 241 36 ) ) PUNCT cana-918 241 37 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 241 38 6 6 NUM cana-918 241 39 where where SCONJ cana-918 241 40 ๐‘š ๐‘š NOUN cana-918 241 41 = = SYM cana-918 241 42 4๐‘›(๐‘š 4๐‘›(๐‘š NUM cana-918 241 43 ) ) PUNCT cana-918 242 1 + + CCONJ cana-918 242 2 ๐‘ก(๐‘š ๐‘ก(๐‘š NOUN cana-918 242 3 ) ) PUNCT cana-918 242 4 for for ADP cana-918 242 5 some some DET cana-918 242 6 ๐‘ก(๐‘š ๐‘ก(๐‘š NOUN cana-918 242 7 ) ) PUNCT cana-918 242 8 โˆˆ โˆˆ PROPN cana-918 242 9 โ„ค4 โ„ค4 PROPN cana-918 242 10 , , PUNCT cana-918 242 11 ๐‘› ๐‘› PROPN cana-918 242 12 ( ( PUNCT cana-918 242 13 ๐‘š ๐‘š NOUN cana-918 242 14 ) ) PUNCT cana-918 242 15 โˆˆ โˆˆ PROPN cana-918 242 16 โ„ค6 โ„ค6 NOUN cana-918 242 17 . . PUNCT cana-918 243 1 using use VERB cana-918 243 2 these these DET cana-918 243 3 equations equation NOUN cana-918 243 4 , , PUNCT cana-918 243 5 we we PRON cana-918 243 6 have have VERB cana-918 243 7 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š PROPN cana-918 243 8 ๐‘˜ ๐‘˜ NOUN cana-918 243 9 = = ADJ cana-918 243 10 1 1 NUM cana-918 243 11 if if SCONJ cana-918 243 12 and and CCONJ cana-918 243 13 only only ADV cana-918 243 14 if if SCONJ cana-918 243 15 ๐‘ฅโ€ฒ ๐‘ฅโ€ฒ VERB cana-918 243 16 = = PUNCT cana-918 243 17 ๐‘ฆโ€ฒ ๐‘ฆโ€ฒ X cana-918 243 18 that that PRON cana-918 243 19 is be AUX cana-918 243 20 , , PUNCT cana-918 243 21 if if SCONJ cana-918 243 22 ๐‘ก(๐‘˜ ๐‘ก(๐‘˜ NOUN cana-918 243 23 ) ) PUNCT cana-918 243 24 โ‰ก โ‰ก PROPN cana-918 243 25 ( ( PUNCT cana-918 243 26 ๐‘ก(๐‘™ ๐‘ก(๐‘™ PROPN cana-918 243 27 ) ) PUNCT cana-918 243 28 + + NUM cana-918 243 29 ๐‘ก(๐‘š ๐‘ก(๐‘š NOUN cana-918 243 30 ) ) PUNCT cana-918 243 31 ) ) PUNCT cana-918 243 32 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 243 33 4 4 NUM cana-918 243 34 and and CCONJ cana-918 243 35 ๐‘›(๐‘˜ ๐‘›(๐‘˜ PROPN cana-918 243 36 ) ) PUNCT cana-918 243 37 โ‰ก โ‰ก PROPN cana-918 243 38 ( ( PUNCT cana-918 243 39 ๐‘›(๐‘™ ๐‘›(๐‘™ PROPN cana-918 243 40 ) ) PUNCT cana-918 244 1 + + SYM cana-918 244 2 ๐‘›(๐‘š ๐‘›(๐‘š NOUN cana-918 244 3 ) ) PUNCT cana-918 244 4 ) ) PUNCT cana-918 244 5 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 245 1 6 6 NUM cana-918 245 2 . . PUNCT cana-918 246 1 โ—ป โ—ป PROPN cana-918 246 2 theorem theorem VERB cana-918 246 3 10 10 NUM cana-918 246 4 . . PUNCT cana-918 247 1 let let VERB cana-918 247 2 ๐‘Œ ๐‘Œ PROPN cana-918 247 3 = = PUNCT cana-918 247 4 ๐‘†5 ๐‘†5 PROPN cana-918 247 5 and and CCONJ cana-918 247 6 ๐’ซ ๐’ซ NOUN cana-918 247 7 be be VERB cana-918 247 8 a a DET cana-918 247 9 partition partition NOUN cana-918 247 10 of of ADP cana-918 247 11 ๐‘Œ ๐‘Œ PROPN cana-918 247 12 ร— ร— PROPN cana-918 247 13 ๐‘Œ. ๐‘Œ. PROPN cana-918 247 14 for for ADP cana-918 247 15 all all PRON cana-918 247 16 ๐‘˜ ๐‘˜ PRON cana-918 247 17 written write VERB cana-918 247 18 in in ADP cana-918 247 19 form form NOUN cana-918 247 20 of of ADP cana-918 247 21 24๐‘ก 24๐‘ก NOUN cana-918 247 22 + + CCONJ cana-918 247 23 4๐‘› 4๐‘› NOUN cana-918 248 1 + + CCONJ cana-918 248 2 ๐‘Ÿ ๐‘Ÿ NOUN cana-918 248 3 ( ( PUNCT cana-918 248 4 ๐‘ก ๐‘ก PROPN cana-918 248 5 โˆˆ โˆˆ PROPN cana-918 248 6 โ„ค5 โ„ค5 NOUN cana-918 248 7 , , PUNCT cana-918 248 8 ๐‘› ๐‘› DET cana-918 248 9 โˆˆ โˆˆ PROPN cana-918 248 10 โ„ค6 โ„ค6 NOUN cana-918 248 11 , , PUNCT cana-918 248 12 ๐‘Ÿ ๐‘Ÿ X cana-918 248 13 โˆˆ โˆˆ PROPN cana-918 248 14 โ„ค4 โ„ค4 PROPN cana-918 248 15 ) ) PUNCT cana-918 248 16 , , PUNCT cana-918 248 17 the the DET cana-918 248 18 relations relation NOUN cana-918 248 19 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 248 20 on on ADP cana-918 248 21 ๐’ซ ๐’ซ NOUN cana-918 248 22 defined define VERB cana-918 248 23 by by ADP cana-918 248 24 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 248 25 = = SYM cana-918 248 26 { { PUNCT cana-918 248 27 ( ( PUNCT cana-918 248 28 ๐›ผ๐‘–๐‘— ๐›ผ๐‘–๐‘— PROPN cana-918 248 29 ( ( PUNCT cana-918 248 30 ๐‘  ๐‘  PROPN cana-918 248 31 ) ) PUNCT cana-918 248 32 , , PUNCT cana-918 248 33 ๐›ผ(๐‘–+๐‘ก)๐‘š๐‘œ๐‘‘ ๐›ผ(๐‘–+๐‘ก)๐‘š๐‘œ๐‘‘ PROPN cana-918 248 34 5(๐‘—+๐‘›)๐‘š๐‘œ๐‘‘ 5(๐‘—+๐‘›)๐‘š๐‘œ๐‘‘ NUM cana-918 248 35 6 6 NUM cana-918 248 36 ( ( PUNCT cana-918 248 37 ๐‘ +๐‘Ÿ)๐‘š๐‘œ๐‘‘ ๐‘ +๐‘Ÿ)๐‘š๐‘œ๐‘‘ PROPN cana-918 248 38 4 4 NUM cana-918 248 39 ) ) PUNCT cana-918 248 40 | | ADV cana-918 248 41 ๐‘– ๐‘– SYM cana-918 248 42 โˆˆ โˆˆ PROPN cana-918 248 43 โ„ค5 โ„ค5 NOUN cana-918 248 44 , , PUNCT cana-918 248 45 ๐‘— ๐‘— PROPN cana-918 248 46 โˆˆ โˆˆ PROPN cana-918 248 47 โ„ค6 โ„ค6 NOUN cana-918 248 48 , , PUNCT cana-918 248 49 ๐‘  ๐‘  PROPN cana-918 248 50 โˆˆ โˆˆ PROPN cana-918 248 51 โ„ค4 โ„ค4 PROPN cana-918 248 52 } } PUNCT cana-918 248 53 where where SCONJ cana-918 248 54 ฮฑ๐‘–๐‘— ฮฑ๐‘–๐‘— X cana-918 248 55 ( ( PUNCT cana-918 248 56 0 0 NUM cana-918 248 57 ) ) PUNCT cana-918 248 58 = = NOUN cana-918 248 59 ฯ„๐‘–ฯƒฯ„4ฯƒ5ฯ„ฯƒ๐‘— ฯ„๐‘–ฯƒฯ„4ฯƒ5ฯ„ฯƒ๐‘— NOUN cana-918 248 60 , , PUNCT cana-918 248 61 ฮฑ๐‘–๐‘— ฮฑ๐‘–๐‘— X cana-918 248 62 ( ( PUNCT cana-918 248 63 1 1 NUM cana-918 248 64 ) ) PUNCT cana-918 248 65 = = NOUN cana-918 248 66 ฯ„๐‘–ฯƒฮณฯ„4ฯƒ5ฯ„ฯƒ๐‘— ฯ„๐‘–ฯƒฮณฯ„4ฯƒ5ฯ„ฯƒ๐‘— ADV cana-918 248 67 , , PUNCT cana-918 248 68 ฮฑ๐‘–๐‘— ฮฑ๐‘–๐‘— X cana-918 248 69 ( ( PUNCT cana-918 248 70 2 2 NUM cana-918 248 71 ) ) PUNCT cana-918 248 72 = = SYM cana-918 248 73 ฯ„๐‘–ฯƒฯ„4ฯƒ5ฮดฯ„ฯƒ๐‘— ฯ„๐‘–ฯƒฯ„4ฯƒ5ฮดฯ„ฯƒ๐‘— NOUN cana-918 248 74 , , PUNCT cana-918 248 75 ฮฑ๐‘–๐‘— ฮฑ๐‘–๐‘— X cana-918 248 76 ( ( PUNCT cana-918 248 77 3 3 NUM cana-918 248 78 ) ) PUNCT cana-918 248 79 = = PUNCT cana-918 248 80 ฯ„๐‘–ฯƒฮณฯ„4ฯƒ5ฮดฯ„ฯƒ๐‘— ฯ„๐‘–ฯƒฮณฯ„4ฯƒ5ฮดฯ„ฯƒ๐‘— X cana-918 248 81 is be AUX cana-918 248 82 a a DET cana-918 248 83 non non ADJ cana-918 248 84 - - ADJ cana-918 248 85 symmetric symmetric ADJ cana-918 248 86 association association NOUN cana-918 248 87 scheme scheme NOUN cana-918 248 88 with with ADP cana-918 248 89 parameters parameter NOUN cana-918 248 90 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š VERB cana-918 248 91 ๐‘˜ ๐‘˜ X cana-918 248 92 = = PUNCT cana-918 248 93 { { PUNCT cana-918 248 94 1 1 NUM cana-918 248 95 if if SCONJ cana-918 248 96 ๐‘›(๐‘˜ ๐‘›(๐‘˜ NOUN cana-918 248 97 ) ) PUNCT cana-918 248 98 โ‰ก โ‰ก PROPN cana-918 248 99 ( ( PUNCT cana-918 248 100 ๐‘›(๐‘™ ๐‘›(๐‘™ PROPN cana-918 248 101 ) ) PUNCT cana-918 248 102 + + SYM cana-918 248 103 ๐‘›(๐‘š ๐‘›(๐‘š NOUN cana-918 248 104 ) ) PUNCT cana-918 248 105 ) ) PUNCT cana-918 248 106 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 248 107 6 6 NUM cana-918 248 108 , , PUNCT cana-918 248 109 ๐‘ก(๐‘˜ ๐‘ก(๐‘˜ NUM cana-918 248 110 ) ) PUNCT cana-918 248 111 โ‰ก โ‰ก PROPN cana-918 248 112 ( ( PUNCT cana-918 248 113 ๐‘ก(๐‘™ ๐‘ก(๐‘™ PROPN cana-918 248 114 ) ) PUNCT cana-918 248 115 + + NUM cana-918 248 116 ๐‘ก(๐‘š ๐‘ก(๐‘š NOUN cana-918 248 117 ) ) PUNCT cana-918 248 118 ) ) PUNCT cana-918 248 119 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 248 120 5 5 NUM cana-918 248 121 , , PUNCT cana-918 248 122 ๐‘Ž๐‘›๐‘‘ ๐‘Ž๐‘›๐‘‘ ADJ cana-918 248 123 ๐‘Ÿ(๐‘˜ ๐‘Ÿ(๐‘˜ PROPN cana-918 248 124 ) ) PUNCT cana-918 248 125 โ‰ก โ‰ก PROPN cana-918 248 126 ( ( PUNCT cana-918 248 127 ๐‘Ÿ(๐‘™ ๐‘Ÿ(๐‘™ PROPN cana-918 248 128 ) ) PUNCT cana-918 248 129 + + CCONJ cana-918 248 130 ๐‘Ÿ(๐‘š ๐‘Ÿ(๐‘š NOUN cana-918 248 131 ) ) PUNCT cana-918 248 132 ) ) PUNCT cana-918 248 133 ๐‘š๐‘œ๐‘‘ ๐‘š๐‘œ๐‘‘ NOUN cana-918 248 134 4 4 NUM cana-918 248 135 0 0 NUM cana-918 248 136 otherwise otherwise ADV cana-918 248 137 } } PUNCT cana-918 248 138 proof proof NOUN cana-918 248 139 . . PUNCT cana-918 249 1 here here ADV cana-918 249 2 , , PUNCT cana-918 249 3 |๐’ฎ๐‘˜| |๐’ฎ๐‘˜| ADV cana-918 249 4 = = PUNCT cana-918 249 5 |๐‘Œ| |๐‘Œ| VERB cana-918 249 6 for for ADP cana-918 249 7 all all DET cana-918 249 8 0 0 NUM cana-918 249 9 โ‰ค โ‰ค NOUN cana-918 249 10 ๐‘˜ ๐‘˜ PRON cana-918 249 11 < < X cana-918 249 12 120 120 NUM cana-918 249 13 . . PUNCT cana-918 250 1 the the DET cana-918 250 2 relations relation NOUN cana-918 250 3 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 250 4 are be AUX cana-918 250 5 disjoint disjoint ADJ cana-918 250 6 and and CCONJ cana-918 250 7 โˆช โˆช VERB cana-918 250 8 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 250 9 : : PUNCT cana-918 250 10 0 0 NUM cana-918 250 11 โ‰ค โ‰ค NOUN cana-918 251 1 ๐‘˜ ๐‘˜ PRON cana-918 251 2 < < X cana-918 251 3 120 120 NUM cana-918 251 4 = = SYM cana-918 251 5 ๐’ซ. ๐’ซ. PROPN cana-918 251 6 proceeding proceeding NOUN cana-918 251 7 as as ADP cana-918 251 8 in in ADP cana-918 251 9 proof proof NOUN cana-918 251 10 of of ADP cana-918 251 11 theorem theorem NOUN cana-918 251 12 8 8 NUM cana-918 251 13 , , PUNCT cana-918 251 14 we we PRON cana-918 251 15 can can AUX cana-918 251 16 find find VERB cana-918 251 17 cardinality cardinality NOUN cana-918 251 18 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š PROPN cana-918 251 19 ๐‘˜ ๐‘˜ INTJ cana-918 251 20 such such ADJ cana-918 251 21 that that PRON cana-918 251 22 for for ADP cana-918 251 23 all all PRON cana-918 251 24 ( ( PUNCT cana-918 251 25 ๐‘ฅ ๐‘ฅ PROPN cana-918 251 26 , , PUNCT cana-918 251 27 ๐‘ฆ ๐‘ฆ NOUN cana-918 251 28 ) ) PUNCT cana-918 251 29 โˆˆ โˆˆ PROPN cana-918 252 1 ๐’ฎ๐‘˜ ๐’ฎ๐‘˜ PROPN cana-918 252 2 , , PUNCT cana-918 252 3 |๐‘ฅ๐’ฎ๐‘™ |๐‘ฅ๐’ฎ๐‘™ ADJ cana-918 252 4 โˆฉ โˆฉ NOUN cana-918 252 5 ๐‘ฆ๐’ฎ๐‘š ๐‘ฆ๐’ฎ๐‘š NOUN cana-918 252 6 โˆ— โˆ— NOUN cana-918 252 7 | | ADV cana-918 253 1 = = NOUN cana-918 254 1 ๐‘๐‘™๐‘š ๐‘๐‘™๐‘š PROPN cana-918 255 1 ๐‘˜ ๐‘˜ PRON cana-918 255 2 is be AUX cana-918 255 3 a a DET cana-918 255 4 constant constant ADJ cana-918 255 5 . . PUNCT cana-918 256 1 โ—ป โ—ป PROPN cana-918 256 2 references reference NOUN cana-918 256 3 [ [ X cana-918 256 4 1 1 NUM cana-918 256 5 ] ] X cana-918 256 6 r. r. PROPN cana-918 256 7 c. c. PROPN cana-918 256 8 bose bose PROPN cana-918 256 9 and and CCONJ cana-918 256 10 t. t. PROPN cana-918 256 11 shimamoto shimamoto PROPN cana-918 256 12 , , PUNCT cana-918 256 13 " " PUNCT cana-918 256 14 classification classification NOUN cana-918 256 15 and and CCONJ cana-918 256 16 analysis analysis NOUN cana-918 256 17 of of ADP cana-918 256 18 partially partially ADV cana-918 256 19 balanced balanced ADJ cana-918 256 20 incomplete incomplete ADJ cana-918 256 21 block block NOUN cana-918 256 22 designs design NOUN cana-918 256 23 with with ADP cana-918 256 24 two two NUM cana-918 256 25 associate associate ADJ cana-918 256 26 classes class NOUN cana-918 256 27 , , PUNCT cana-918 256 28 " " PUNCT cana-918 256 29 journal journal NOUN cana-918 256 30 of of ADP cana-918 256 31 the the DET cana-918 256 32 american american PROPN cana-918 256 33 statistical statistical PROPN cana-918 256 34 association association PROPN cana-918 256 35 , , PUNCT cana-918 256 36 vol vol NOUN cana-918 256 37 . . PROPN cana-918 257 1 47 47 NUM cana-918 257 2 , , PUNCT cana-918 257 3 no no INTJ cana-918 257 4 . . NOUN cana-918 257 5 258 258 NUM cana-918 257 6 , , PUNCT cana-918 257 7 pp pp ADJ cana-918 257 8 . . PUNCT cana-918 258 1 151 151 NUM cana-918 258 2 - - SYM cana-918 258 3 184 184 NUM cana-918 258 4 , , PUNCT cana-918 258 5 1952 1952 NUM cana-918 258 6 . . PUNCT cana-918 259 1 [ [ X cana-918 259 2 2 2 X cana-918 259 3 ] ] PUNCT cana-918 259 4 p. p. NOUN cana-918 259 5 delsarte delsarte PROPN cana-918 259 6 and and CCONJ cana-918 259 7 v. v. ADP cana-918 259 8 levenshtein levenshtein NUM cana-918 259 9 , , PUNCT cana-918 259 10 " " PUNCT cana-918 259 11 association association NOUN cana-918 259 12 schemes scheme NOUN cana-918 259 13 and and CCONJ cana-918 259 14 coding code VERB cana-918 259 15 theory theory NOUN cana-918 259 16 , , PUNCT cana-918 259 17 " " PUNCT cana-918 259 18 ieee ieee NOUN cana-918 259 19 transactions transaction NOUN cana-918 259 20 on on ADP cana-918 259 21 information information NOUN cana-918 259 22 theory theory NOUN cana-918 259 23 , , PUNCT cana-918 259 24 vol vol NOUN cana-918 259 25 . . PROPN cana-918 259 26 44 44 NUM cana-918 259 27 , , PUNCT cana-918 259 28 no no INTJ cana-918 259 29 . . NOUN cana-918 259 30 6 6 NUM cana-918 259 31 , , PUNCT cana-918 259 32 pp pp ADJ cana-918 259 33 . . PUNCT cana-918 260 1 2477 2477 NUM cana-918 260 2 2504 2504 NUM cana-918 260 3 , , PUNCT cana-918 260 4 1998 1998 NUM cana-918 260 5 . . PUNCT cana-918 261 1 [ [ X cana-918 261 2 3 3 X cana-918 261 3 ] ] X cana-918 261 4 t.-t t.-t NOUN cana-918 261 5 . . PUNCT cana-918 262 1 xia xia PROPN cana-918 262 2 , , PUNCT cana-918 262 3 y.-y y.-y PROPN cana-918 262 4 . . PUNCT cana-918 263 1 tan tan PROPN cana-918 263 2 , , PUNCT cana-918 263 3 x. x. PROPN cana-918 263 4 liang liang PROPN cana-918 263 5 and and CCONJ cana-918 263 6 j. j. PROPN cana-918 263 7 h. h. PROPN cana-918 263 8 koolen koolen PROPN cana-918 263 9 , , PUNCT cana-918 263 10 " " PUNCT cana-918 263 11 on on ADP cana-918 263 12 association association NOUN cana-918 263 13 schemes scheme NOUN cana-918 263 14 generated generate VERB cana-918 263 15 by by ADP cana-918 263 16 a a DET cana-918 263 17 relation relation NOUN cana-918 263 18 or or CCONJ cana-918 263 19 an an DET cana-918 263 20 idempotent idempotent NOUN cana-918 263 21 , , PUNCT cana-918 263 22 " " PUNCT cana-918 263 23 linear linear ADJ cana-918 263 24 algebra algebra NOUN cana-918 263 25 and and CCONJ cana-918 263 26 its its PRON cana-918 263 27 applications application NOUN cana-918 263 28 , , PUNCT cana-918 263 29 vol vol NOUN cana-918 263 30 . . PUNCT cana-918 264 1 670 670 NUM cana-918 264 2 , , PUNCT cana-918 264 3 pp pp ADJ cana-918 264 4 . . PUNCT cana-918 265 1 1 1 NUM cana-918 265 2 - - SYM cana-918 265 3 18 18 NUM cana-918 265 4 , , PUNCT cana-918 265 5 2023 2023 NUM cana-918 265 6 . . PUNCT cana-918 266 1 [ [ X cana-918 266 2 4 4 NUM cana-918 266 3 ] ] PUNCT cana-918 266 4 a. a. NOUN cana-918 266 5 ลกvob ลกvob PROPN cana-918 266 6 , , PUNCT cana-918 266 7 " " PUNCT cana-918 266 8 lcd lcd NOUN cana-918 266 9 codes code NOUN cana-918 266 10 from from ADP cana-918 266 11 equitable equitable ADJ cana-918 266 12 partitions partition NOUN cana-918 266 13 of of ADP cana-918 266 14 association association NOUN cana-918 266 15 schemes scheme NOUN cana-918 266 16 , , PUNCT cana-918 266 17 " " PUNCT cana-918 266 18 applicable applicable ADJ cana-918 266 19 algebra algebra NOUN cana-918 266 20 in in ADP cana-918 266 21 engineering engineering NOUN cana-918 266 22 , , PUNCT cana-918 266 23 communication communication NOUN cana-918 266 24 and and CCONJ cana-918 266 25 computing computing NOUN cana-918 266 26 , , PUNCT cana-918 266 27 vol vol NOUN cana-918 266 28 . . PROPN cana-918 266 29 34 34 NUM cana-918 266 30 , , PUNCT cana-918 266 31 p. p. NOUN cana-918 266 32 889โ€“896 889โ€“896 NUM cana-918 266 33 , , PUNCT cana-918 266 34 2021 2021 NUM cana-918 266 35 . . PUNCT cana-918 267 1 [ [ X cana-918 267 2 5 5 NUM cana-918 267 3 ] ] PUNCT cana-918 267 4 b. b. PROPN cana-918 267 5 xu xu PROPN cana-918 267 6 , , PUNCT cana-918 267 7 " " PUNCT cana-918 267 8 partial partial ADJ cana-918 267 9 geometric geometric ADJ cana-918 267 10 designs design NOUN cana-918 267 11 arising arise VERB cana-918 267 12 from from ADP cana-918 267 13 association association NOUN cana-918 267 14 schemes scheme NOUN cana-918 267 15 , , PUNCT cana-918 267 16 " " PUNCT cana-918 267 17 european european ADJ cana-918 267 18 journal journal PROPN cana-918 267 19 of of ADP cana-918 267 20 combinatorics combinatorics PROPN cana-918 267 21 , , PUNCT cana-918 267 22 vol vol NOUN cana-918 267 23 . . PROPN cana-918 267 24 109 109 NUM cana-918 267 25 , , PUNCT cana-918 267 26 2023 2023 NUM cana-918 267 27 . . PUNCT cana-918 268 1 [ [ X cana-918 268 2 6 6 NUM cana-918 268 3 ] ] X cana-918 268 4 y. y. PROPN cana-918 268 5 filmus filmus PROPN cana-918 268 6 and and CCONJ cana-918 268 7 f. f. PROPN cana-918 268 8 ihringer ihringer PROPN cana-918 268 9 , , PUNCT cana-918 268 10 " " PUNCT cana-918 268 11 boolean boolean ADJ cana-918 268 12 degree degree NOUN cana-918 268 13 1 1 NUM cana-918 268 14 functions function NOUN cana-918 268 15 on on ADP cana-918 268 16 some some DET cana-918 268 17 classical classical ADJ cana-918 268 18 association association NOUN cana-918 268 19 schemes scheme NOUN cana-918 268 20 , , PUNCT cana-918 268 21 " " PUNCT cana-918 268 22 journal journal NOUN cana-918 268 23 of of ADP cana-918 268 24 combinatorial combinatorial ADJ cana-918 268 25 theory theory NOUN cana-918 268 26 , , PUNCT cana-918 268 27 series series PROPN cana-918 268 28 a a NOUN cana-918 268 29 , , PUNCT cana-918 268 30 vol vol NOUN cana-918 268 31 . . NOUN cana-918 268 32 162 162 NUM cana-918 268 33 , , PUNCT cana-918 268 34 pp pp ADJ cana-918 268 35 . . PUNCT cana-918 269 1 241 241 NUM cana-918 269 2 - - SYM cana-918 269 3 270 270 NUM cana-918 269 4 , , PUNCT cana-918 269 5 2019 2019 NUM cana-918 269 6 . . PUNCT cana-918 270 1 [ [ X cana-918 270 2 7 7 NUM cana-918 270 3 ] ] X cana-918 270 4 p.-h p.-h NOUN cana-918 270 5 . . PUNCT cana-918 271 1 zieschan zieschan PROPN cana-918 271 2 , , PUNCT cana-918 271 3 theory theory NOUN cana-918 271 4 of of ADP cana-918 271 5 association association NOUN cana-918 271 6 schemes scheme NOUN cana-918 271 7 , , PUNCT cana-918 271 8 the the DET cana-918 271 9 netherlands netherlands PROPN cana-918 271 10 : : PUNCT cana-918 271 11 springer springer NOUN cana-918 271 12 , , PUNCT cana-918 271 13 2005 2005 NUM cana-918 271 14 . . PUNCT cana-918 272 1 [ [ X cana-918 272 2 8 8 NUM cana-918 272 3 ] ] PUNCT cana-918 272 4 m. m. NOUN cana-918 272 5 klin klin PROPN cana-918 272 6 , , PUNCT cana-918 272 7 j. j. PROPN cana-918 272 8 lauri lauri PROPN cana-918 272 9 and and CCONJ cana-918 272 10 m. m. PROPN cana-918 272 11 ziv ziv PROPN cana-918 272 12 - - PUNCT cana-918 272 13 av av PROPN cana-918 272 14 , , PUNCT cana-918 272 15 " " PUNCT cana-918 272 16 links link NOUN cana-918 272 17 between between ADP cana-918 272 18 two two NUM cana-918 272 19 semisymmetric semisymmetric ADJ cana-918 272 20 graphs graph NOUN cana-918 272 21 on on ADP cana-918 272 22 112 112 NUM cana-918 272 23 vertices vertex NOUN cana-918 272 24 via via ADP cana-918 272 25 association association NOUN cana-918 272 26 schemes scheme NOUN cana-918 272 27 , , PUNCT cana-918 272 28 " " PUNCT cana-918 272 29 journal journal NOUN cana-918 272 30 of of ADP cana-918 272 31 symbolic symbolic ADJ cana-918 272 32 computation computation NOUN cana-918 272 33 , , PUNCT cana-918 272 34 vol vol NOUN cana-918 272 35 . . PROPN cana-918 273 1 47 47 NUM cana-918 273 2 , , PUNCT cana-918 273 3 no no INTJ cana-918 273 4 . . NOUN cana-918 273 5 10 10 NUM cana-918 273 6 , , PUNCT cana-918 273 7 pp pp ADJ cana-918 273 8 . . PUNCT cana-918 274 1 1175 1175 NUM cana-918 274 2 - - SYM cana-918 274 3 1191 1191 NUM cana-918 274 4 , , PUNCT cana-918 274 5 2012 2012 NUM cana-918 274 6 . . PUNCT cana-918 275 1 communications communication NOUN cana-918 275 2 on on ADP cana-918 275 3 applied apply VERB cana-918 275 4 nonlinear nonlinear ADJ cana-918 275 5 analysis analysis NOUN cana-918 275 6 issn issn NOUN cana-918 275 7 : : PUNCT cana-918 275 8 1074 1074 NUM cana-918 275 9 - - PUNCT cana-918 275 10 133x 133x NUM cana-918 275 11 vol vol NOUN cana-918 275 12 31 31 NUM cana-918 275 13 no no NOUN cana-918 275 14 . . PUNCT cana-918 276 1 4s 4s NUM cana-918 276 2 ( ( PUNCT cana-918 276 3 2024 2024 NUM cana-918 276 4 ) ) PUNCT cana-918 276 5 402 402 NUM cana-918 276 6 https://internationalpubls.com https://internationalpubls.com X cana-918 277 1 [ [ X cana-918 277 2 9 9 NUM cana-918 277 3 ] ] X cana-918 277 4 e. e. PROPN cana-918 277 5 bannai bannai PROPN cana-918 277 6 , , PUNCT cana-918 277 7 e. e. PROPN cana-918 277 8 bannai bannai PROPN cana-918 277 9 , , PUNCT cana-918 277 10 h. h. PROPN cana-918 277 11 tanaka tanaka PROPN cana-918 277 12 and and CCONJ cana-918 277 13 y. y. PROPN cana-918 277 14 zhu zhu PROPN cana-918 277 15 , , PUNCT cana-918 277 16 " " PUNCT cana-918 277 17 design design NOUN cana-918 277 18 theory theory NOUN cana-918 277 19 from from ADP cana-918 277 20 the the DET cana-918 277 21 viewpoint viewpoint NOUN cana-918 277 22 of of ADP cana-918 277 23 algebraic algebraic ADJ cana-918 277 24 combinatorics combinatoric NOUN cana-918 277 25 . . PUNCT cana-918 278 1 graphs graph NOUN cana-918 278 2 and and CCONJ cana-918 278 3 combinatorics combinatoric NOUN cana-918 278 4 , , PUNCT cana-918 278 5 " " PUNCT cana-918 278 6 graphs graph NOUN cana-918 278 7 and and CCONJ cana-918 278 8 combinatorics combinatoric NOUN cana-918 278 9 , , PUNCT cana-918 278 10 vol vol NOUN cana-918 278 11 . . PROPN cana-918 278 12 33 33 NUM cana-918 278 13 , , PUNCT cana-918 278 14 pp pp ADJ cana-918 278 15 . . PUNCT cana-918 279 1 1 1 NUM cana-918 279 2 - - SYM cana-918 279 3 41 41 NUM cana-918 279 4 , , PUNCT cana-918 279 5 2017 2017 NUM cana-918 279 6 . . PUNCT cana-918 280 1 [ [ X cana-918 280 2 10 10 NUM cana-918 280 3 ] ] X cana-918 280 4 w. w. PROPN cana-918 280 5 j. j. PROPN cana-918 280 6 martin martin PROPN cana-918 280 7 , , PUNCT cana-918 280 8 " " PUNCT cana-918 280 9 designs design NOUN cana-918 280 10 in in ADP cana-918 280 11 product product NOUN cana-918 280 12 association association NOUN cana-918 280 13 schemes scheme NOUN cana-918 280 14 , , PUNCT cana-918 280 15 " " PUNCT cana-918 280 16 designs design NOUN cana-918 280 17 , , PUNCT cana-918 280 18 codes code NOUN cana-918 280 19 and and CCONJ cana-918 280 20 cryptography cryptography NOUN cana-918 280 21 , , PUNCT cana-918 280 22 vol vol NOUN cana-918 280 23 . . PROPN cana-918 280 24 16 16 NUM cana-918 280 25 , , PUNCT cana-918 280 26 pp pp ADJ cana-918 280 27 . . PUNCT cana-918 281 1 271 271 NUM cana-918 281 2 - - SYM cana-918 281 3 289 289 NUM cana-918 281 4 , , PUNCT cana-918 281 5 1999 1999 NUM cana-918 281 6 . . PUNCT cana-918 282 1 [ [ X cana-918 282 2 11 11 NUM cana-918 282 3 ] ] X cana-918 282 4 b. b. PROPN cana-918 282 5 xu xu PROPN cana-918 282 6 , , PUNCT cana-918 282 7 " " PUNCT cana-918 282 8 on on ADP cana-918 282 9 ip ip NOUN cana-918 282 10 - - PUNCT cana-918 282 11 graphs graph NOUN cana-918 282 12 of of ADP cana-918 282 13 association association NOUN cana-918 282 14 schemes scheme NOUN cana-918 282 15 and and CCONJ cana-918 282 16 applications application NOUN cana-918 282 17 to to ADP cana-918 282 18 group group NOUN cana-918 282 19 theory theory NOUN cana-918 282 20 , , PUNCT cana-918 282 21 " " PUNCT cana-918 282 22 journal journal NOUN cana-918 282 23 of of ADP cana-918 282 24 combinatorial combinatorial ADJ cana-918 282 25 theory theory NOUN cana-918 282 26 , , PUNCT cana-918 282 27 series series PROPN cana-918 282 28 a a NOUN cana-918 282 29 , , PUNCT cana-918 282 30 vol vol NOUN cana-918 282 31 . . PROPN cana-918 282 32 117 117 NUM cana-918 282 33 , , PUNCT cana-918 282 34 no no INTJ cana-918 282 35 . . NOUN cana-918 282 36 7 7 NUM cana-918 282 37 , , PUNCT cana-918 282 38 pp pp ADJ cana-918 282 39 . . PUNCT cana-918 283 1 981 981 NUM cana-918 283 2 - - SYM cana-918 283 3 995 995 NUM cana-918 283 4 , , PUNCT cana-918 283 5 2010 2010 NUM cana-918 283 6 . . PUNCT cana-918 284 1 [ [ X cana-918 284 2 12 12 NUM cana-918 284 3 ] ] PUNCT cana-918 284 4 l. l. PROPN cana-918 284 5 k. k. PROPN cana-918 284 6 jรธrgensen jรธrgensen PROPN cana-918 284 7 , , PUNCT cana-918 284 8 " " PUNCT cana-918 284 9 schur schur NOUN cana-918 284 10 rings ring NOUN cana-918 284 11 and and CCONJ cana-918 284 12 non non ADJ cana-918 284 13 - - ADJ cana-918 284 14 symmetric symmetric ADJ cana-918 284 15 association association NOUN cana-918 284 16 schemes scheme NOUN cana-918 284 17 on on ADP cana-918 284 18 64 64 NUM cana-918 284 19 vertices vertex NOUN cana-918 284 20 , , PUNCT cana-918 284 21 " " PUNCT cana-918 284 22 discrete discrete ADJ cana-918 284 23 mathematics mathematic NOUN cana-918 284 24 , , PUNCT cana-918 284 25 vol vol NOUN cana-918 284 26 . . PROPN cana-918 284 27 310 310 NUM cana-918 284 28 , , PUNCT cana-918 284 29 no no INTJ cana-918 284 30 . . NOUN cana-918 284 31 22 22 NUM cana-918 284 32 , , PUNCT cana-918 284 33 pp pp ADJ cana-918 284 34 . . PUNCT cana-918 285 1 3259 3259 NUM cana-918 285 2 - - SYM cana-918 285 3 3266 3266 NUM cana-918 285 4 , , PUNCT cana-918 285 5 2010 2010 NUM cana-918 285 6 . . PUNCT cana-918 286 1 [ [ X cana-918 286 2 13 13 NUM cana-918 286 3 ] ] PUNCT cana-918 286 4 a. a. NOUN cana-918 286 5 sabharwal sabharwal NOUN cana-918 286 6 and and CCONJ cana-918 286 7 p. p. PROPN cana-918 286 8 yadav yadav PROPN cana-918 286 9 , , PUNCT cana-918 286 10 " " PUNCT cana-918 286 11 association association NOUN cana-918 286 12 schemes scheme NOUN cana-918 286 13 over over ADP cana-918 286 14 some some DET cana-918 286 15 finite finite ADJ cana-918 286 16 group group NOUN cana-918 286 17 rings ring NOUN cana-918 286 18 , , PUNCT cana-918 286 19 " " PUNCT cana-918 286 20 in in ADP cana-918 286 21 mathematical mathematical ADJ cana-918 286 22 modeling modeling NOUN cana-918 286 23 , , PUNCT cana-918 286 24 computational computational ADJ cana-918 286 25 intelligence intelligence NOUN cana-918 286 26 techniques technique NOUN cana-918 286 27 and and CCONJ cana-918 286 28 renewable renewable ADJ cana-918 286 29 energy energy NOUN cana-918 286 30 , , PUNCT cana-918 286 31 2022 2022 NUM cana-918 286 32 . . PUNCT cana-918 287 1 [ [ X cana-918 287 2 14 14 NUM cana-918 287 3 ] ] PUNCT cana-918 287 4 p.-h p.-h NOUN cana-918 287 5 . . PUNCT cana-918 288 1 zieschang zieschang PROPN cana-918 288 2 , , PUNCT cana-918 288 3 an an DET cana-918 288 4 algebraic algebraic ADJ cana-918 288 5 approach approach NOUN cana-918 288 6 to to ADP cana-918 288 7 association association NOUN cana-918 288 8 schemes scheme NOUN cana-918 288 9 , , PUNCT cana-918 288 10 springer springer NOUN cana-918 288 11 , , PUNCT cana-918 288 12 2006 2006 NUM cana-918 288 13 . . PUNCT cana-918 289 1 [ [ X cana-918 289 2 15 15 NUM cana-918 289 3 ] ] X cana-918 289 4 r. r. PROPN cana-918 289 5 k. k. PROPN cana-918 289 6 sharma sharma PROPN cana-918 289 7 , , PUNCT cana-918 289 8 p. p. NOUN cana-918 289 9 yadav yadav PROPN cana-918 289 10 and and CCONJ cana-918 289 11 p. p. PROPN cana-918 289 12 kanwar kanwar PROPN cana-918 289 13 , , PUNCT cana-918 289 14 " " PUNCT cana-918 289 15 lie lie VERB cana-918 289 16 regular regular ADJ cana-918 289 17 generators generator NOUN cana-918 289 18 of of ADP cana-918 289 19 general general ADJ cana-918 289 20 linear linear ADJ cana-918 289 21 groups group NOUN cana-918 289 22 , , PUNCT cana-918 289 23 " " PUNCT cana-918 289 24 communications communication NOUN cana-918 289 25 in in ADP cana-918 289 26 algebra algebra NOUN cana-918 289 27 , , PUNCT cana-918 289 28 vol vol NOUN cana-918 289 29 . . PROPN cana-918 289 30 40 40 NUM cana-918 289 31 , , PUNCT cana-918 289 32 no no INTJ cana-918 289 33 . . NOUN cana-918 289 34 4 4 NUM cana-918 289 35 , , PUNCT cana-918 289 36 pp pp ADJ cana-918 289 37 . . PUNCT cana-918 290 1 1304 1304 NUM cana-918 290 2 - - SYM cana-918 290 3 1315 1315 NUM cana-918 290 4 , , PUNCT cana-918 290 5 2012 2012 NUM cana-918 290 6 . . PUNCT cana-918 291 1 [ [ X cana-918 291 2 16 16 NUM cana-918 291 3 ] ] X cana-918 291 4 c. c. NOUN cana-918 291 5 bonnafรฉ bonnafรฉ NOUN cana-918 291 6 , , PUNCT cana-918 291 7 representations representation NOUN cana-918 291 8 of of ADP cana-918 291 9 sl2 sl2 PROPN cana-918 291 10 ( ( PUNCT cana-918 291 11 fq fq PROPN cana-918 291 12 ) ) PUNCT cana-918 291 13 , , PUNCT cana-918 291 14 vol vol NOUN cana-918 291 15 . . PROPN cana-918 291 16 13 13 NUM cana-918 291 17 , , PUNCT cana-918 291 18 springer springer NOUN cana-918 291 19 science science PROPN cana-918 291 20 & & CCONJ cana-918 291 21 business business NOUN cana-918 291 22 media medium NOUN cana-918 291 23 . . PUNCT cana-918 291 24 , , PUNCT cana-918 291 25 2010 2010 NUM cana-918 291 26 . . PUNCT cana-918 292 1 [ [ X cana-918 292 2 17 17 NUM cana-918 292 3 ] ] PUNCT cana-918 292 4 m. m. NOUN cana-918 292 5 tomiyama tomiyama PROPN cana-918 292 6 and and CCONJ cana-918 292 7 n. n. PROPN cana-918 292 8 yamazaki yamazaki PROPN cana-918 292 9 , , PUNCT cana-918 292 10 " " PUNCT cana-918 292 11 characterization characterization NOUN cana-918 292 12 of of ADP cana-918 292 13 the the DET cana-918 292 14 group group NOUN cana-918 292 15 association association NOUN cana-918 292 16 scheme scheme NOUN cana-918 292 17 of of ADP cana-918 292 18 the the DET cana-918 292 19 symmetric symmetric ADJ cana-918 292 20 group group NOUN cana-918 292 21 , , PUNCT cana-918 292 22 " " PUNCT cana-918 292 23 european european ADJ cana-918 292 24 journal journal PROPN cana-918 292 25 of of ADP cana-918 292 26 combinatorics combinatorics PROPN cana-918 292 27 , , PUNCT cana-918 292 28 vol vol NOUN cana-918 292 29 . . PROPN cana-918 293 1 19 19 NUM cana-918 293 2 , , PUNCT cana-918 293 3 no no INTJ cana-918 293 4 . . NOUN cana-918 293 5 2 2 NUM cana-918 293 6 , , PUNCT cana-918 293 7 pp pp ADJ cana-918 293 8 . . PUNCT cana-918 294 1 237 237 NUM cana-918 294 2 - - SYM cana-918 294 3 255 255 NUM cana-918 294 4 , , PUNCT cana-918 294 5 1998 1998 NUM cana-918 294 6 . . PUNCT cana-918 295 1 [ [ X cana-918 295 2 18 18 NUM cana-918 295 3 ] ] X cana-918 295 4 h. h. PROPN cana-918 295 5 s. s. PROPN cana-918 295 6 coxeter coxeter PROPN cana-918 295 7 and and CCONJ cana-918 295 8 w. w. PROPN cana-918 295 9 o. o. PROPN cana-918 295 10 moser moser PROPN cana-918 295 11 , , PUNCT cana-918 295 12 generators generator NOUN cana-918 295 13 and and CCONJ cana-918 295 14 relations relation NOUN cana-918 295 15 for for ADP cana-918 295 16 discrete discrete ADJ cana-918 295 17 groups group NOUN cana-918 295 18 , , PUNCT cana-918 295 19 vol vol NOUN cana-918 295 20 . . PROPN cana-918 296 1 14 14 NUM cana-918 296 2 , , PUNCT cana-918 296 3 springer springer NOUN cana-918 296 4 science science PROPN cana-918 296 5 & & CCONJ cana-918 296 6 business business NOUN cana-918 296 7 media medium NOUN cana-918 296 8 , , PUNCT cana-918 296 9 2013 2013 NUM cana-918 296 10 . . PUNCT cana-918 297 1 [ [ X cana-918 297 2 19 19 NUM cana-918 297 3 ] ] X cana-918 297 4 k.-u k.-u PROPN cana-918 297 5 . . PUNCT cana-918 298 1 schmidt schmidt PROPN cana-918 298 2 , , PUNCT cana-918 298 3 " " PUNCT cana-918 298 4 quadratic quadratic ADJ cana-918 298 5 and and CCONJ cana-918 298 6 symmetric symmetric ADJ cana-918 298 7 bilinear bilinear NOUN cana-918 298 8 forms form NOUN cana-918 298 9 over over ADP cana-918 298 10 finite finite ADJ cana-918 298 11 fields field NOUN cana-918 298 12 and and CCONJ cana-918 298 13 their their PRON cana-918 298 14 association association NOUN cana-918 298 15 schemes scheme NOUN cana-918 298 16 , , PUNCT cana-918 298 17 " " PUNCT cana-918 298 18 algebraic algebraic ADJ cana-918 298 19 combinatorics combinatoric NOUN cana-918 298 20 , , PUNCT cana-918 298 21 vol vol NOUN cana-918 298 22 . . PROPN cana-918 299 1 3 3 NUM cana-918 299 2 , , PUNCT cana-918 299 3 no no INTJ cana-918 299 4 . . NOUN cana-918 299 5 1 1 NUM cana-918 299 6 , , PUNCT cana-918 299 7 pp pp ADJ cana-918 299 8 . . PUNCT cana-918 300 1 161 161 NUM cana-918 300 2 - - SYM cana-918 300 3 189 189 NUM cana-918 300 4 , , PUNCT cana-918 300 5 2020 2020 NUM cana-918 300 6 . . PUNCT cana-918 301 1 [ [ X cana-918 301 2 20 20 NUM cana-918 301 3 ] ] PUNCT cana-918 301 4 r. r. PROPN cana-918 301 5 feng feng PROPN cana-918 301 6 , , PUNCT cana-918 301 7 y. y. PROPN cana-918 301 8 wang wang PROPN cana-918 301 9 , , PUNCT cana-918 301 10 c. c. PROPN cana-918 301 11 ma ma PROPN cana-918 301 12 and and CCONJ cana-918 301 13 j. j. PROPN cana-918 301 14 ma ma PROPN cana-918 301 15 , , PUNCT cana-918 301 16 " " PUNCT cana-918 301 17 eigenvalues eigenvalue NOUN cana-918 301 18 of of ADP cana-918 301 19 association association NOUN cana-918 301 20 schemes scheme NOUN cana-918 301 21 of of ADP cana-918 301 22 quadratic quadratic ADJ cana-918 301 23 forms form NOUN cana-918 301 24 , , PUNCT cana-918 301 25 " " PUNCT cana-918 301 26 discrete discrete ADJ cana-918 301 27 mathematics mathematic NOUN cana-918 301 28 , , PUNCT cana-918 301 29 vol vol NOUN cana-918 301 30 . . PROPN cana-918 301 31 308 308 NUM cana-918 301 32 , , PUNCT cana-918 301 33 no no INTJ cana-918 301 34 . . NOUN cana-918 301 35 14 14 NUM cana-918 301 36 , , PUNCT cana-918 301 37 pp pp ADJ cana-918 301 38 . . PUNCT cana-918 302 1 3023 3023 NUM cana-918 302 2 - - SYM cana-918 302 3 3047 3047 NUM cana-918 302 4 , , PUNCT cana-918 302 5 2008 2008 NUM cana-918 302 6 . . PUNCT