id sid tid token lemma pos cana-3901 1 1 communications communication NOUN cana-3901 1 2 on on ADP cana-3901 1 3 applied apply VERB cana-3901 1 4 nonlinear nonlinear ADJ cana-3901 1 5 analysis analysis NOUN cana-3901 1 6 issn issn NOUN cana-3901 1 7 : : PUNCT cana-3901 1 8 1074 1074 NUM cana-3901 1 9 - - PUNCT cana-3901 1 10 133x 133x NUM cana-3901 1 11 vol vol NOUN cana-3901 1 12 32 32 NUM cana-3901 1 13 no no NOUN cana-3901 1 14 . . PUNCT cana-3901 2 1 9s 9s NUM cana-3901 2 2 ( ( PUNCT cana-3901 2 3 2025 2025 NUM cana-3901 2 4 ) ) PUNCT cana-3901 2 5 353 353 NUM cana-3901 2 6 https://internationalpubls.com https://internationalpubls.com X cana-3901 2 7 on on ADP cana-3901 2 8 the the DET cana-3901 2 9 structure structure NOUN cana-3901 2 10 of of ADP cana-3901 2 11 𝝈𝟏 𝝈𝟏 PROPN cana-3901 2 12 near near ADP cana-3901 2 13 rings ring NOUN cana-3901 2 14 1v.alies 1v.alies PROPN cana-3901 2 15 anbukani anbukani ADJ cana-3901 2 16 , , PUNCT cana-3901 2 17 2g.sugantha 2g.sugantha NUM cana-3901 2 18 , , PUNCT cana-3901 2 19 3p.sivagami 3p.sivagami NUM cana-3901 2 20 1research 1research NUM cana-3901 2 21 scholar scholar NOUN cana-3901 2 22 of of ADP cana-3901 2 23 mathematics mathematic NOUN cana-3901 2 24 ( ( PUNCT cana-3901 2 25 part part NOUN cana-3901 2 26 time time NOUN cana-3901 2 27 ) ) PUNCT cana-3901 2 28 , , PUNCT cana-3901 2 29 reg reg VERB cana-3901 2 30 no no PRON cana-3901 2 31 : : PUNCT cana-3901 2 32 21122102092008 21122102092008 NUM cana-3901 2 33 , , PUNCT cana-3901 2 34 email:alieslivingston@gmail.com email:alieslivingston@gmail.com NOUN cana-3901 2 35 pg pg NOUN cana-3901 2 36 and and CCONJ cana-3901 2 37 research research PROPN cana-3901 2 38 department department PROPN cana-3901 2 39 of of ADP cana-3901 2 40 mathematics mathematic NOUN cana-3901 2 41 , , PUNCT cana-3901 2 42 kamaraj kamaraj ADJ cana-3901 2 43 college college NOUN cana-3901 2 44 , , PUNCT cana-3901 2 45 thoothukudi thoothukudi PROPN cana-3901 2 46 – – PUNCT cana-3901 2 47 628003 628003 NUM cana-3901 2 48 . . PUNCT cana-3901 3 1 ( ( PUNCT cana-3901 3 2 affiliated affiliate VERB cana-3901 3 3 to to ADP cana-3901 3 4 manonmaniam manonmaniam PROPN cana-3901 3 5 sundaranar sundaranar PROPN cana-3901 3 6 university university PROPN cana-3901 3 7 , , PUNCT cana-3901 3 8 abishekapatti abishekapatti ADJ cana-3901 3 9 , , PUNCT cana-3901 3 10 tirunelveli tirunelveli ADJ cana-3901 3 11 – – PUNCT cana-3901 3 12 627012 627012 NUM cana-3901 3 13 ) ) PUNCT cana-3901 4 1 2assistant 2assistant NUM cana-3901 4 2 professor professor NOUN cana-3901 4 3 of of ADP cana-3901 4 4 mathematics mathematics PROPN cana-3901 4 5 , , PUNCT cana-3901 4 6 pope pope PROPN cana-3901 4 7 ’s ’s PART cana-3901 4 8 college college PROPN cana-3901 4 9 ( ( PUNCT cana-3901 4 10 autonomous autonomous ADJ cana-3901 4 11 ) ) PUNCT cana-3901 4 12 , , PUNCT cana-3901 4 13 sawyerpuram sawyerpuram NOUN cana-3901 4 14 , , PUNCT cana-3901 4 15 tamil tamil PROPN cana-3901 4 16 nadu nadu NOUN cana-3901 4 17 627 627 NUM cana-3901 4 18 251 251 NUM cana-3901 4 19 , , PUNCT cana-3901 4 20 india india PROPN cana-3901 4 21 . . PUNCT cana-3901 5 1 e.mail:sugi.trini@gmail.com e.mail:sugi.trini@gmail.com PROPN cana-3901 5 2 ( ( PUNCT cana-3901 5 3 affiliated affiliate VERB cana-3901 5 4 to to ADP cana-3901 5 5 manonmaniam manonmaniam PROPN cana-3901 5 6 sundaranar sundaranar PROPN cana-3901 5 7 university university PROPN cana-3901 5 8 , , PUNCT cana-3901 5 9 abishekapatti abishekapatti ADJ cana-3901 5 10 , , PUNCT cana-3901 5 11 tirunelveli tirunelveli ADJ cana-3901 5 12 – – PUNCT cana-3901 5 13 627012 627012 NUM cana-3901 5 14 ) ) PUNCT cana-3901 5 15 3associate 3associate NUM cana-3901 5 16 professor professor NOUN cana-3901 5 17 of of ADP cana-3901 5 18 mathematics mathematic NOUN cana-3901 5 19 , , PUNCT cana-3901 5 20 pg pg NOUN cana-3901 5 21 and and CCONJ cana-3901 5 22 research research PROPN cana-3901 5 23 department department PROPN cana-3901 5 24 of of ADP cana-3901 5 25 mathematics mathematic NOUN cana-3901 5 26 , , PUNCT cana-3901 5 27 kamaraj kamaraj ADJ cana-3901 5 28 college college NOUN cana-3901 5 29 , , PUNCT cana-3901 5 30 thoothukudi thoothukudi PROPN cana-3901 5 31 – – PUNCT cana-3901 5 32 628003 628003 NUM cana-3901 5 33 . . PUNCT cana-3901 6 1 e.mail:sivagamimuthu75@gmail.com e.mail:sivagamimuthu75@gmail.com X cana-3901 6 2 ( ( PUNCT cana-3901 6 3 affiliated affiliate VERB cana-3901 6 4 to to ADP cana-3901 6 5 manonmaniam manonmaniam PROPN cana-3901 6 6 sundaranar sundaranar PROPN cana-3901 6 7 university university PROPN cana-3901 6 8 , , PUNCT cana-3901 6 9 abishekapatti abishekapatti ADJ cana-3901 6 10 , , PUNCT cana-3901 6 11 tirunelveli tirunelveli ADJ cana-3901 6 12 – – PUNCT cana-3901 6 13 627012 627012 NUM cana-3901 6 14 ) ) PUNCT cana-3901 6 15 article article NOUN cana-3901 6 16 history history NOUN cana-3901 6 17 : : PUNCT cana-3901 6 18 received receive VERB cana-3901 6 19 : : PUNCT cana-3901 6 20 15 15 NUM cana-3901 6 21 - - SYM cana-3901 6 22 11 11 NUM cana-3901 6 23 - - PUNCT cana-3901 6 24 2024 2024 NUM cana-3901 6 25 revised revise VERB cana-3901 6 26 : : PUNCT cana-3901 6 27 26 26 NUM cana-3901 6 28 - - SYM cana-3901 6 29 12 12 NUM cana-3901 6 30 - - PUNCT cana-3901 6 31 2024 2024 NUM cana-3901 6 32 accepted:10 accepted:10 PROPN cana-3901 6 33 - - PUNCT cana-3901 6 34 01 01 NUM cana-3901 6 35 - - PUNCT cana-3901 6 36 2025 2025 NUM cana-3901 6 37 abstract abstract NOUN cana-3901 6 38 : : PUNCT cana-3901 6 39 if if SCONJ cana-3901 6 40 , , PUNCT cana-3901 6 41 in in ADP cana-3901 6 42 a a DET cana-3901 6 43 ring ring NOUN cana-3901 6 44 ( ( PUNCT cana-3901 6 45 n n CCONJ cana-3901 6 46 , , PUNCT cana-3901 6 47 + + ADV cana-3901 6 48 , , PUNCT cana-3901 6 49 βˆ™ βˆ™ PROPN cana-3901 6 50 ) ) PUNCT cana-3901 6 51 we we PRON cana-3901 6 52 ignore ignore VERB cana-3901 6 53 the the DET cana-3901 6 54 commutativity commutativity NOUN cana-3901 6 55 of of ADP cana-3901 6 56 β€˜ ' PUNCT cana-3901 6 57 + + NOUN cana-3901 6 58 ’ ' PUNCT cana-3901 6 59 and and CCONJ cana-3901 6 60 one one NUM cana-3901 6 61 of of ADP cana-3901 6 62 the the DET cana-3901 6 63 distributive distributive ADJ cana-3901 6 64 laws law NOUN cana-3901 6 65 , , PUNCT cana-3901 6 66 ( ( PUNCT cana-3901 6 67 n n CCONJ cana-3901 6 68 , , PUNCT cana-3901 6 69 + + ADV cana-3901 6 70 , , PUNCT cana-3901 6 71 βˆ™ βˆ™ PROPN cana-3901 6 72 ) ) PUNCT cana-3901 6 73 becomes become VERB cana-3901 6 74 a a DET cana-3901 6 75 near near ADJ cana-3901 6 76 - - PUNCT cana-3901 6 77 ring ring NOUN cana-3901 6 78 . . PUNCT cana-3901 7 1 if if SCONJ cana-3901 7 2 we we PRON cana-3901 7 3 do do AUX cana-3901 7 4 not not PART cana-3901 7 5 stipulate stipulate VERB cana-3901 7 6 the the DET cana-3901 7 7 left left ADJ cana-3901 7 8 distributive distributive ADJ cana-3901 7 9 law law NOUN cana-3901 7 10 , , PUNCT cana-3901 7 11 ( ( PUNCT cana-3901 7 12 n n CCONJ cana-3901 7 13 , , PUNCT cana-3901 7 14 + + ADV cana-3901 7 15 , , PUNCT cana-3901 7 16 βˆ™ βˆ™ PROPN cana-3901 7 17 ) ) PUNCT cana-3901 7 18 is be AUX cana-3901 7 19 a a DET cana-3901 7 20 right right ADJ cana-3901 7 21 nearring nearring NOUN cana-3901 7 22 . . PUNCT cana-3901 8 1 this this DET cana-3901 8 2 research research NOUN cana-3901 8 3 aims aim VERB cana-3901 8 4 to to PART cana-3901 8 5 introduce introduce VERB cana-3901 8 6 the the DET cana-3901 8 7 concept concept NOUN cana-3901 8 8 of of ADP cana-3901 8 9 Οƒ_1near Οƒ_1near NOUN cana-3901 8 10 - - PUNCT cana-3901 8 11 ring ring NOUN cana-3901 8 12 . . PUNCT cana-3901 9 1 n n PRON cana-3901 9 2 is be AUX cana-3901 9 3 called call VERB cana-3901 9 4 Οƒ_1near Οƒ_1near PROPN cana-3901 9 5 ring ring NOUN cana-3901 9 6 if if SCONJ cana-3901 9 7 n n PRON cana-3901 9 8 is be AUX cana-3901 9 9 a a DET cana-3901 9 10 right right ADJ cana-3901 9 11 near near NOUN cana-3901 9 12 - - PUNCT cana-3901 9 13 ring ring NOUN cana-3901 9 14 and and CCONJ cana-3901 9 15 xy^2 xy^2 NOUN cana-3901 9 16 = = NOUN cana-3901 9 17 yxyfor yxyfor PROPN cana-3901 9 18 all all DET cana-3901 9 19 x x NOUN cana-3901 9 20 , , PUNCT cana-3901 9 21 y∈n y∈n NOUN cana-3901 9 22 .the .the DET cana-3901 9 23 element element ADJ cana-3901 9 24 wise wise ADJ cana-3901 9 25 characterization characterization NOUN cana-3901 9 26 for for ADP cana-3901 9 27 Οƒ_1near Οƒ_1near NOUN cana-3901 9 28 - - PUNCT cana-3901 9 29 ring ring NOUN cana-3901 9 30 will will AUX cana-3901 9 31 be be AUX cana-3901 9 32 investigated investigate VERB cana-3901 9 33 and and CCONJ cana-3901 9 34 shall shall AUX cana-3901 9 35 establish establish VERB cana-3901 9 36 theorems theorem NOUN cana-3901 9 37 and and CCONJ cana-3901 9 38 properties property NOUN cana-3901 9 39 in in ADP cana-3901 9 40 this this DET cana-3901 9 41 near near ADJ cana-3901 9 42 ring ring NOUN cana-3901 9 43 . . PUNCT cana-3901 10 1 mathematics mathematic NOUN cana-3901 10 2 subject subject ADJ cana-3901 10 3 classification classification NOUN cana-3901 10 4 : : PUNCT cana-3901 10 5 16y30 16y30 NUM cana-3901 10 6 . . PUNCT cana-3901 11 1 keywords keyword NOUN cana-3901 11 2 : : PUNCT cana-3901 11 3 Οƒ_1 Οƒ_1 NOUN cana-3901 11 4 near near ADP cana-3901 11 5 -ring -ring PROPN cana-3901 11 6 , , PUNCT cana-3901 11 7 commutativity commutativity NOUN cana-3901 11 8 , , PUNCT cana-3901 11 9 near near NOUN cana-3901 11 10 - - PUNCT cana-3901 11 11 field field NOUN cana-3901 11 12 . . PUNCT cana-3901 12 1 1 1 NUM cana-3901 12 2 introduction introduction NOUN cana-3901 12 3 a a DET cana-3901 12 4 right right ADJ cana-3901 12 5 near near NOUN cana-3901 12 6 - - PUNCT cana-3901 12 7 ring ring NOUN cana-3901 12 8 is be AUX cana-3901 12 9 a a DET cana-3901 12 10 non non ADJ cana-3901 12 11 - - ADJ cana-3901 12 12 empty empty ADJ cana-3901 12 13 set set NOUN cana-3901 12 14 n n CCONJ cana-3901 12 15 together together ADV cana-3901 12 16 with with ADP cana-3901 12 17 two two NUM cana-3901 12 18 binary binary ADJ cana-3901 12 19 operations operation NOUN cana-3901 12 20 β€œ " PUNCT cana-3901 12 21 + + ADJ cana-3901 12 22 ” " PUNCT cana-3901 12 23 and and CCONJ cana-3901 12 24 β€œ " PUNCT cana-3901 12 25 . . PUNCT cana-3901 12 26 ” " PUNCT cana-3901 13 1 such such ADJ cana-3901 13 2 that that SCONJ cana-3901 13 3 ( ( PUNCT cana-3901 13 4 1 1 NUM cana-3901 13 5 ) ) PUNCT cana-3901 13 6 ( ( PUNCT cana-3901 13 7 n n CCONJ cana-3901 13 8 , , PUNCT cana-3901 13 9 + + PUNCT cana-3901 13 10 ) ) PUNCT cana-3901 13 11 is be AUX cana-3901 13 12 a a DET cana-3901 13 13 group group NOUN cana-3901 13 14 . . PUNCT cana-3901 14 1 ( ( PUNCT cana-3901 14 2 2 2 NUM cana-3901 14 3 ) ) PUNCT cana-3901 14 4 ( ( PUNCT cana-3901 14 5 n n X cana-3901 14 6 , , PUNCT cana-3901 14 7 βˆ™ βˆ™ PROPN cana-3901 14 8 ) ) PUNCT cana-3901 14 9 is be AUX cana-3901 14 10 a a DET cana-3901 14 11 semi semi NOUN cana-3901 14 12 - - NOUN cana-3901 14 13 group group NOUN cana-3901 14 14 and and CCONJ cana-3901 14 15 ( ( PUNCT cana-3901 14 16 3 3 NUM cana-3901 14 17 ) ) PUNCT cana-3901 14 18 ( ( PUNCT cana-3901 14 19 𝑛1 𝑛1 NOUN cana-3901 14 20 + + NOUN cana-3901 14 21 𝑛2)𝑛3 𝑛2)𝑛3 PROPN cana-3901 14 22 = = SYM cana-3901 14 23 𝑛1𝑛3 𝑛1𝑛3 X cana-3901 15 1 + + NUM cana-3901 15 2 𝑛2𝑛3 𝑛2𝑛3 NOUN cana-3901 15 3 for for ADP cana-3901 15 4 all all DET cana-3901 15 5 𝑛1 𝑛1 NOUN cana-3901 15 6 , , PUNCT cana-3901 15 7 𝑛2 𝑛2 NOUN cana-3901 15 8 , , PUNCT cana-3901 15 9 𝑛3 𝑛3 NOUN cana-3901 15 10 ∈ ∈ PROPN cana-3901 15 11 𝑁. 𝑁. PROPN cana-3901 15 12 throughout throughout ADP cana-3901 15 13 this this DET cana-3901 15 14 paper paper NOUN cana-3901 15 15 n n PRON cana-3901 15 16 stands stand VERB cana-3901 15 17 for for ADP cana-3901 15 18 a a DET cana-3901 15 19 right right ADJ cana-3901 15 20 near near NOUN cana-3901 15 21 - - PUNCT cana-3901 15 22 ring ring NOUN cana-3901 15 23 . . PUNCT cana-3901 16 1 ( ( PUNCT cana-3901 16 2 𝑁 𝑁 PROPN cana-3901 16 3 , , PUNCT cana-3901 16 4 + + NOUN cana-3901 16 5 , , PUNCT cana-3901 16 6 . . PUNCT cana-3901 16 7 ) ) PUNCT cana-3901 17 1 with with ADP cana-3901 17 2 at at ADV cana-3901 17 3 least least ADV cana-3901 17 4 two two NUM cana-3901 17 5 elements element NOUN cana-3901 17 6 1 1 NUM cana-3901 17 7 and and CCONJ cana-3901 17 8 β€²0β€² β€²0β€² NOUN cana-3901 17 9 denotes denote VERB cana-3901 17 10 the the DET cana-3901 17 11 identity identity NOUN cana-3901 17 12 element element NOUN cana-3901 17 13 of of ADP cana-3901 17 14 the the DET cana-3901 17 15 group group NOUN cana-3901 17 16 ( ( PUNCT cana-3901 17 17 n n CCONJ cana-3901 17 18 , , PUNCT cana-3901 17 19 + + NOUN cana-3901 17 20 ) ) PUNCT cana-3901 17 21 . . PUNCT cana-3901 18 1 obviously obviously ADV cana-3901 18 2 , , PUNCT cana-3901 18 3 0n 0n NOUN cana-3901 18 4 = = SYM cana-3901 18 5 0 0 NUM cana-3901 18 6 for for ADP cana-3901 18 7 all all DET cana-3901 18 8 n n NOUN cana-3901 18 9 in in ADP cana-3901 18 10 n. n. NOUN cana-3901 18 11 n n NUM cana-3901 18 12 is be AUX cana-3901 18 13 said say VERB cana-3901 18 14 to to PART cana-3901 18 15 be be AUX cana-3901 18 16 zerosymmetric zerosymmetric ADJ cana-3901 18 17 if if SCONJ cana-3901 18 18 n0 n0 ADJ cana-3901 18 19 = = NOUN cana-3901 18 20 0 0 NUM cana-3901 19 1 for for ADP cana-3901 19 2 all all DET cana-3901 19 3 n n NOUN cana-3901 19 4 in in ADP cana-3901 19 5 n. n. NOUN cana-3901 19 6 as as ADP cana-3901 19 7 in in ADP cana-3901 19 8 [ [ X cana-3901 19 9 2 2 NUM cana-3901 19 10 ] ] PUNCT cana-3901 19 11 , , PUNCT cana-3901 19 12 a a DET cana-3901 19 13 subgroup subgroup NOUN cana-3901 19 14 of of ADP cana-3901 19 15 ( ( PUNCT cana-3901 19 16 𝑀 𝑀 PROPN cana-3901 19 17 , , PUNCT cana-3901 19 18 + + NOUN cana-3901 19 19 ) ) PUNCT cana-3901 19 20 of of ADP cana-3901 19 21 ( ( PUNCT cana-3901 19 22 𝑁 𝑁 PROPN cana-3901 19 23 , , PUNCT cana-3901 19 24 + + NOUN cana-3901 19 25 ) ) PUNCT cana-3901 19 26 is be AUX cana-3901 19 27 called call VERB cana-3901 19 28 an an DET cana-3901 19 29 n n NOUN cana-3901 19 30 - - PUNCT cana-3901 19 31 subgroup subgroup NOUN cana-3901 19 32 of of ADP cana-3901 19 33 n n PRON cana-3901 19 34 if if SCONJ cana-3901 19 35 𝑁𝑀 𝑁𝑀 PROPN cana-3901 19 36 βŠ‚ βŠ‚ PROPN cana-3901 19 37 𝑀 𝑀 PROPN cana-3901 19 38 and and CCONJ cana-3901 19 39 an an DET cana-3901 19 40 invariant invariant ADJ cana-3901 19 41 n n PRON cana-3901 19 42 subgroup subgroup NOUN cana-3901 19 43 of of ADP cana-3901 19 44 n n ADV cana-3901 19 45 if if SCONJ cana-3901 19 46 , , PUNCT cana-3901 19 47 in in ADP cana-3901 19 48 addition addition NOUN cana-3901 19 49 , , PUNCT cana-3901 19 50 𝑀𝑁 𝑀𝑁 PROPN cana-3901 19 51 βŠ‚ βŠ‚ PROPN cana-3901 19 52 𝑀. 𝑀. PROPN cana-3901 19 53 in in ADP cana-3901 19 54 [ [ X cana-3901 19 55 6 6 NUM cana-3901 19 56 ] ] PUNCT cana-3901 19 57 , , PUNCT cana-3901 19 58 n n PRON cana-3901 19 59 is be AUX cana-3901 19 60 defined define VERB cana-3901 19 61 to to PART cana-3901 19 62 be be AUX cana-3901 19 63 pseudo pseudo NOUN cana-3901 19 64 commutative commutative ADJ cana-3901 19 65 if if SCONJ cana-3901 19 66 π‘₯𝑦𝑧 π‘₯𝑦𝑧 NOUN cana-3901 19 67 = = VERB cana-3901 19 68 𝑧𝑦π‘₯ 𝑧𝑦π‘₯ NOUN cana-3901 19 69 for for ADP cana-3901 19 70 all all PRON cana-3901 19 71 π‘₯ π‘₯ PROPN cana-3901 19 72 , , PUNCT cana-3901 19 73 𝑦 𝑦 NOUN cana-3901 19 74 , , PUNCT cana-3901 19 75 𝑧 𝑧 PROPN cana-3901 19 76 in in ADP cana-3901 19 77 n. n. NOUN cana-3901 19 78 the the DET cana-3901 19 79 concept concept NOUN cana-3901 19 80 of of ADP cana-3901 19 81 a a DET cana-3901 19 82 mate mate NOUN cana-3901 19 83 function function NOUN cana-3901 19 84 in in ADP cana-3901 19 85 n n NUM cana-3901 19 86 has have AUX cana-3901 19 87 been be AUX cana-3901 19 88 introduced introduce VERB cana-3901 19 89 in in ADP cana-3901 19 90 [ [ X cana-3901 19 91 4 4 NUM cana-3901 19 92 ] ] PUNCT cana-3901 19 93 with with ADP cana-3901 19 94 a a DET cana-3901 19 95 view view NOUN cana-3901 19 96 to to ADP cana-3901 19 97 handling handle VERB cana-3901 19 98 the the DET cana-3901 19 99 regularity regularity NOUN cana-3901 19 100 structure structure NOUN cana-3901 19 101 with with ADP cana-3901 19 102 considerable considerable ADJ cana-3901 19 103 case case NOUN cana-3901 19 104 . . PUNCT cana-3901 20 1 a a DET cana-3901 20 2 map map NOUN cana-3901 20 3 β€²fβ€² β€²fβ€² PROPN cana-3901 20 4 from from ADP cana-3901 20 5 n n PROPN cana-3901 20 6 into into ADP cana-3901 20 7 n n PROPN cana-3901 20 8 is be AUX cana-3901 20 9 called call VERB cana-3901 20 10 ( ( PUNCT cana-3901 20 11 i i NOUN cana-3901 20 12 ) ) PUNCT cana-3901 20 13 a a DET cana-3901 20 14 mate mate NOUN cana-3901 20 15 function function NOUN cana-3901 20 16 for for ADP cana-3901 20 17 𝑁 𝑁 PROPN cana-3901 20 18 if if SCONJ cana-3901 20 19 π‘₯ π‘₯ NOUN cana-3901 20 20 = = PRON cana-3901 20 21 π‘₯𝑓 π‘₯𝑓 X cana-3901 20 22 ( ( PUNCT cana-3901 20 23 π‘₯)π‘₯ π‘₯)π‘₯ ADJ cana-3901 20 24 , , PUNCT cana-3901 20 25 ( ( PUNCT cana-3901 20 26 ii ii NOUN cana-3901 20 27 ) ) PUNCT cana-3901 20 28 a a DET cana-3901 20 29 p3 p3 PROPN cana-3901 20 30 mate mate NOUN cana-3901 20 31 function function NOUN cana-3901 20 32 , , PUNCT cana-3901 20 33 if if SCONJ cana-3901 20 34 , , PUNCT cana-3901 20 35 in in ADP cana-3901 20 36 addition addition NOUN cana-3901 20 37 , , PUNCT cana-3901 20 38 π‘₯𝑓 π‘₯𝑓 ADP cana-3901 20 39 ( ( PUNCT cana-3901 20 40 π‘₯ π‘₯ NOUN cana-3901 20 41 ) ) PUNCT cana-3901 20 42 = = SYM cana-3901 20 43 𝑓 𝑓 PROPN cana-3901 20 44 ( ( PUNCT cana-3901 20 45 π‘₯)π‘₯ π‘₯)π‘₯ X cana-3901 20 46 for for ADP cana-3901 20 47 all all DET cana-3901 20 48 π‘₯ π‘₯ NOUN cana-3901 20 49 in in ADP cana-3901 20 50 𝑁. 𝑁. PROPN cana-3901 20 51 by by ADP cana-3901 20 52 identity identity NOUN cana-3901 20 53 1 1 NUM cana-3901 20 54 of of ADP cana-3901 20 55 𝑁 𝑁 PROPN cana-3901 20 56 , , PUNCT cana-3901 20 57 we we PRON cana-3901 20 58 mean mean VERB cana-3901 20 59 only only ADV cana-3901 20 60 the the DET cana-3901 20 61 multiplicative multiplicative ADJ cana-3901 20 62 identity identity NOUN cana-3901 20 63 of of ADP cana-3901 20 64 𝑁. 𝑁. PROPN cana-3901 20 65 basic basic ADJ cana-3901 20 66 concepts concept NOUN cana-3901 20 67 and and CCONJ cana-3901 20 68 terms term NOUN cana-3901 20 69 used use VERB cana-3901 20 70 but but CCONJ cana-3901 20 71 left leave VERB cana-3901 20 72 undefined undefined ADJ cana-3901 20 73 in in ADP cana-3901 20 74 this this DET cana-3901 20 75 paper paper NOUN cana-3901 20 76 can can AUX cana-3901 20 77 be be AUX cana-3901 20 78 found find VERB cana-3901 20 79 in in ADP cana-3901 20 80 [ [ X cana-3901 20 81 2 2 NUM cana-3901 20 82 ] ] PUNCT cana-3901 20 83 . . PUNCT cana-3901 21 1 2 2 NUM cana-3901 21 2 notations notation NOUN cana-3901 21 3 in in ADP cana-3901 21 4 this this DET cana-3901 21 5 section section NOUN cana-3901 21 6 , , PUNCT cana-3901 21 7 we we PRON cana-3901 21 8 furnish furnish VERB cana-3901 21 9 the the DET cana-3901 21 10 notations notation NOUN cana-3901 21 11 which which PRON cana-3901 21 12 are be AUX cana-3901 21 13 used use VERB cana-3901 21 14 frequently frequently ADV cana-3901 21 15 throughout throughout ADP cana-3901 21 16 this this DET cana-3901 21 17 paper paper NOUN cana-3901 21 18 . . PUNCT cana-3901 22 1 ( ( PUNCT cana-3901 22 2 i i NOUN cana-3901 22 3 ) ) PUNCT cana-3901 22 4 e e NOUN cana-3901 22 5 denotes denote VERB cana-3901 22 6 the the DET cana-3901 22 7 set set NOUN cana-3901 22 8 of of ADP cana-3901 22 9 all all DET cana-3901 22 10 idempotent idempotent NOUN cana-3901 22 11 of of ADP cana-3901 22 12 n. n. NOUN cana-3901 22 13 ( ( PUNCT cana-3901 22 14 e e NOUN cana-3901 22 15 in in ADP cana-3901 22 16 n n PROPN cana-3901 22 17 is be AUX cana-3901 22 18 called call VERB cana-3901 22 19 an an DET cana-3901 22 20 idempotent idempotent NOUN cana-3901 22 21 if if SCONJ cana-3901 22 22 𝑒2 𝑒2 PROPN cana-3901 22 23 = = SYM cana-3901 22 24 𝑒 𝑒 NOUN cana-3901 22 25 ) ) PUNCT cana-3901 22 26 mailto:alieslivingston@gmail.com mailto:alieslivingston@gmail.com NOUN cana-3901 22 27 communications communication NOUN cana-3901 22 28 on on ADP cana-3901 22 29 applied apply VERB cana-3901 22 30 nonlinear nonlinear ADJ cana-3901 22 31 analysis analysis NOUN cana-3901 22 32 issn issn NOUN cana-3901 22 33 : : PUNCT cana-3901 22 34 1074 1074 NUM cana-3901 22 35 - - PUNCT cana-3901 22 36 133x 133x NUM cana-3901 22 37 vol vol NOUN cana-3901 22 38 32 32 NUM cana-3901 22 39 no no NOUN cana-3901 22 40 . . PUNCT cana-3901 23 1 9s 9s NUM cana-3901 23 2 ( ( PUNCT cana-3901 23 3 2025 2025 NUM cana-3901 23 4 ) ) PUNCT cana-3901 23 5 354 354 NUM cana-3901 23 6 https://internationalpubls.com https://internationalpubls.com X cana-3901 23 7 ( ( PUNCT cana-3901 23 8 ii ii NOUN cana-3901 23 9 ) ) PUNCT cana-3901 23 10 l l NOUN cana-3901 23 11 denotes denote VERB cana-3901 23 12 the the DET cana-3901 23 13 set set NOUN cana-3901 23 14 of of ADP cana-3901 23 15 all all DET cana-3901 23 16 nilpotent nilpotent NOUN cana-3901 23 17 of of ADP cana-3901 23 18 n. n. NOUN cana-3901 23 19 ( ( PUNCT cana-3901 23 20 a a DET cana-3901 23 21 in in ADP cana-3901 23 22 n n PRON cana-3901 23 23 is be AUX cana-3901 23 24 nilpotent nilpotent ADJ cana-3901 23 25 if if SCONJ cana-3901 23 26 ak ak PROPN cana-3901 23 27 = = PROPN cana-3901 23 28 0 0 PROPN cana-3901 23 29 for for ADP cana-3901 23 30 some some DET cana-3901 23 31 positive positive ADJ cana-3901 23 32 integer integer NOUN cana-3901 23 33 k. k. PROPN cana-3901 23 34 ) ) PUNCT cana-3901 24 1 and and CCONJ cana-3901 24 2 𝑁 𝑁 PROPN cana-3901 24 3 is be AUX cana-3901 24 4 said say VERB cana-3901 24 5 to to PART cana-3901 24 6 be be AUX cana-3901 24 7 reduced reduce VERB cana-3901 24 8 if if SCONJ cana-3901 24 9 l={0 l={0 PROPN cana-3901 24 10 } } PUNCT cana-3901 24 11 . . PUNCT cana-3901 25 1 ( ( PUNCT cana-3901 25 2 iii iii X cana-3901 25 3 ) ) PUNCT cana-3901 25 4 𝑁0 𝑁0 ADJ cana-3901 25 5 = = PUNCT cana-3901 25 6 { { PUNCT cana-3901 25 7 𝑛 𝑛 PRON cana-3901 25 8 ∈ ∈ NOUN cana-3901 25 9 𝑁 𝑁 PROPN cana-3901 25 10 / / SYM cana-3901 25 11 𝑛0 𝑛0 VERB cana-3901 25 12 = = NOUN cana-3901 25 13 0 0 NUM cana-3901 25 14 } } PUNCT cana-3901 25 15 zero zero NUM cana-3901 25 16 - - PUNCT cana-3901 25 17 symmetric symmetric ADJ cana-3901 25 18 part part NOUN cana-3901 25 19 of of ADP cana-3901 25 20 n n PRON cana-3901 25 21 and and CCONJ cana-3901 25 22 𝑁 𝑁 PROPN cana-3901 25 23 is be AUX cana-3901 25 24 called call VERB cana-3901 25 25 zero zero NUM cana-3901 25 26 symmetric symmetric NOUN cana-3901 25 27 if if SCONJ cana-3901 25 28 𝑁 𝑁 PROPN cana-3901 25 29 = = SYM cana-3901 25 30 𝑁0 𝑁0 PROPN cana-3901 25 31 . . PUNCT cana-3901 26 1 ( ( PUNCT cana-3901 26 2 iv iv X cana-3901 26 3 ) ) PUNCT cana-3901 26 4 𝑁𝑑 𝑁𝑑 PROPN cana-3901 26 5 = = SYM cana-3901 26 6 { { PUNCT cana-3901 26 7 𝑛 𝑛 PRON cana-3901 26 8 ∈ ∈ NOUN cana-3901 26 9 𝑁 𝑁 PROPN cana-3901 26 10 / / SYM cana-3901 26 11 𝑛(π‘₯ 𝑛(π‘₯ PROPN cana-3901 26 12 + + CCONJ cana-3901 26 13 𝑦 𝑦 X cana-3901 26 14 ) ) PUNCT cana-3901 26 15 = = SYM cana-3901 26 16 𝑛π‘₯ 𝑛π‘₯ PROPN cana-3901 27 1 + + CCONJ cana-3901 27 2 𝑛𝑦 𝑛𝑦 PROPN cana-3901 27 3 for for ADP cana-3901 27 4 all all DET cana-3901 27 5 π‘₯ π‘₯ PROPN cana-3901 27 6 , , PUNCT cana-3901 27 7 𝑦 𝑦 NOUN cana-3901 27 8 in in ADP cana-3901 27 9 𝑁 𝑁 NOUN cana-3901 27 10 } } PUNCT cana-3901 27 11 – – PUNCT cana-3901 27 12 set set NOUN cana-3901 27 13 of of ADP cana-3901 27 14 all all DET cana-3901 27 15 distributive distributive ADJ cana-3901 27 16 element element NOUN cana-3901 27 17 of of ADP cana-3901 27 18 n n PROPN cana-3901 27 19 and and CCONJ cana-3901 27 20 𝑁 𝑁 PROPN cana-3901 27 21 is be AUX cana-3901 27 22 called call VERB cana-3901 27 23 distributive distributive ADJ cana-3901 27 24 if if SCONJ cana-3901 27 25 𝑁 𝑁 PROPN cana-3901 27 26 = = SYM cana-3901 27 27 𝑁𝑑. 𝑁𝑑. PROPN cana-3901 27 28 ( ( PUNCT cana-3901 27 29 v v NOUN cana-3901 27 30 ) ) PUNCT cana-3901 27 31 𝐢(𝑁 𝐢(𝑁 NUM cana-3901 27 32 ) ) PUNCT cana-3901 28 1 = = PRON cana-3901 28 2 { { PUNCT cana-3901 28 3 𝑛 𝑛 PRON cana-3901 28 4 ∈ ∈ NOUN cana-3901 28 5 𝑁 𝑁 PROPN cana-3901 28 6 / / SYM cana-3901 28 7 𝑛π‘₯ 𝑛π‘₯ PROPN cana-3901 28 8 = = PUNCT cana-3901 28 9 π‘₯𝑛 π‘₯𝑛 PROPN cana-3901 28 10 for for ADP cana-3901 28 11 all all DET cana-3901 28 12 π‘₯ π‘₯ PRON cana-3901 28 13 in in ADP cana-3901 28 14 𝑁 𝑁 PROPN cana-3901 28 15 } } PUNCT cana-3901 28 16 centre centre NOUN cana-3901 28 17 of of ADP cana-3901 28 18 n. n. PROPN cana-3901 28 19 ( ( PUNCT cana-3901 28 20 vi vi PROPN cana-3901 28 21 ) ) PUNCT cana-3901 28 22 if if SCONJ cana-3901 28 23 a a PRON cana-3901 28 24 is be AUX cana-3901 28 25 any any DET cana-3901 28 26 non non ADJ cana-3901 28 27 – – PUNCT cana-3901 28 28 empty empty ADJ cana-3901 28 29 subset subset NOUN cana-3901 28 30 of of ADP cana-3901 28 31 𝑁 𝑁 PROPN cana-3901 28 32 , , PUNCT cana-3901 28 33 then then ADV cana-3901 28 34 i i PROPN cana-3901 28 35 ) ) PUNCT cana-3901 28 36 π΄βˆ— π΄βˆ— PROPN cana-3901 28 37 = = SYM cana-3901 28 38 𝐴 𝐴 PROPN cana-3901 28 39 βˆ’ βˆ’ PROPN cana-3901 28 40 { { PUNCT cana-3901 28 41 0 0 NUM cana-3901 28 42 } } SYM cana-3901 28 43 ii ii NOUN cana-3901 28 44 ) ) PUNCT cana-3901 28 45 𝐢(𝐴 𝐢(𝐴 PROPN cana-3901 28 46 ) ) PUNCT cana-3901 28 47 = = PRON cana-3901 28 48 { { PUNCT cana-3901 28 49 𝑛 𝑛 PRON cana-3901 28 50 ∈ ∈ NOUN cana-3901 29 1 𝑁/π‘›π‘Ž 𝑁/π‘›π‘Ž NOUN cana-3901 29 2 = = PUNCT cana-3901 29 3 π‘Žπ‘› π‘Žπ‘› VERB cana-3901 29 4 for for ADP cana-3901 29 5 all all PRON cana-3901 29 6 π‘Ž π‘Ž PRON cana-3901 29 7 ∈ ∈ PROPN cana-3901 29 8 𝐴 𝐴 PROPN cana-3901 29 9 } } PUNCT cana-3901 29 10 iii iii PROPN cana-3901 29 11 ) ) PUNCT cana-3901 29 12 when when SCONJ cana-3901 29 13 𝐴 𝐴 PROPN cana-3901 29 14 = = SYM cana-3901 29 15 𝑁 𝑁 PROPN cana-3901 29 16 , , PUNCT cana-3901 29 17 𝐢(𝑁 𝐢(𝑁 NUM cana-3901 29 18 ) ) PUNCT cana-3901 29 19 = = PRON cana-3901 29 20 { { PUNCT cana-3901 29 21 π‘›π‘Ž π‘›π‘Ž X cana-3901 29 22 = = PUNCT cana-3901 29 23 π‘Žπ‘› π‘Žπ‘› PROPN cana-3901 29 24 for for ADP cana-3901 29 25 all all PRON cana-3901 29 26 π‘Ž π‘Ž DET cana-3901 29 27 ∈ ∈ ADJ cana-3901 29 28 𝑁 𝑁 PROPN cana-3901 29 29 } } PUNCT cana-3901 29 30 βˆ’called βˆ’calle VERB cana-3901 29 31 the the DET cana-3901 29 32 centre centre NOUN cana-3901 29 33 of of ADP cana-3901 29 34 𝑁. 𝑁. PROPN cana-3901 29 35 ( ( PUNCT cana-3901 29 36 vii vii PROPN cana-3901 29 37 ) ) PUNCT cana-3901 29 38 when when SCONJ cana-3901 29 39 𝐸 𝐸 PROPN cana-3901 29 40 βŠ† βŠ† NUM cana-3901 29 41 𝐢(𝑁 𝐢(𝑁 NUM cana-3901 29 42 ) ) PUNCT cana-3901 29 43 , , PUNCT cana-3901 29 44 we we PRON cana-3901 29 45 say say VERB cana-3901 29 46 that that SCONJ cana-3901 29 47 the the DET cana-3901 29 48 idempotent idempotent NOUN cana-3901 29 49 are be AUX cana-3901 29 50 central central ADJ cana-3901 29 51 . . PUNCT cana-3901 30 1 3 3 NUM cana-3901 30 2 preliminary preliminary ADJ cana-3901 30 3 results result NOUN cana-3901 30 4 we we PRON cana-3901 30 5 freely freely ADV cana-3901 30 6 make make VERB cana-3901 30 7 use use NOUN cana-3901 30 8 of of ADP cana-3901 30 9 the the DET cana-3901 30 10 following follow VERB cana-3901 30 11 results result NOUN cana-3901 30 12 and and CCONJ cana-3901 30 13 designate designate VERB cana-3901 30 14 them they PRON cana-3901 30 15 as as ADP cana-3901 30 16 r r NOUN cana-3901 30 17 ( ( PUNCT cana-3901 30 18 1 1 NUM cana-3901 30 19 ) ) PUNCT cana-3901 30 20 , , PUNCT cana-3901 30 21 r r NOUN cana-3901 30 22 ( ( PUNCT cana-3901 30 23 2) 2) NUM cana-3901 30 24 .... .... SYM cana-3901 30 25 etc etc X cana-3901 30 26 . . X cana-3901 31 1 r r NOUN cana-3901 31 2 ( ( PUNCT cana-3901 31 3 1 1 NUM cana-3901 31 4 ) ) PUNCT cana-3901 31 5 n n VERB cana-3901 31 6 is be AUX cana-3901 31 7 sub sub NOUN cana-3901 31 8 directly directly ADV cana-3901 31 9 irreducible irreducible ADJ cana-3901 31 10 if if SCONJ cana-3901 31 11 and and CCONJ cana-3901 31 12 only only ADV cana-3901 31 13 if if SCONJ cana-3901 31 14 the the DET cana-3901 31 15 intersection intersection NOUN cana-3901 31 16 of of ADP cana-3901 31 17 any any DET cana-3901 31 18 family family NOUN cana-3901 31 19 of of ADP cana-3901 31 20 non non ADJ cana-3901 31 21 - - ADJ cana-3901 31 22 zero zero NUM cana-3901 31 23 ideals ideal NOUN cana-3901 31 24 of of ADP cana-3901 31 25 n n PRON cana-3901 31 26 is be AUX cana-3901 31 27 again again ADV cana-3901 31 28 non non ADJ cana-3901 31 29 - - ADJ cana-3901 31 30 zero zero NUM cana-3901 31 31 ( ( PUNCT cana-3901 31 32 theorem theorem VERB cana-3901 31 33 1.60 1.60 NUM cana-3901 31 34 , , PUNCT cana-3901 31 35 p.25 p.25 NOUN cana-3901 31 36 of of ADP cana-3901 31 37 [ [ X cana-3901 31 38 2 2 NUM cana-3901 31 39 ] ] SYM cana-3901 31 40 ) ) PUNCT cana-3901 31 41 r r NOUN cana-3901 31 42 ( ( PUNCT cana-3901 31 43 2 2 NUM cana-3901 31 44 ) ) PUNCT cana-3901 32 1 n n PRON cana-3901 32 2 has have VERB cana-3901 32 3 no no DET cana-3901 32 4 non non ADJ cana-3901 32 5 - - ADJ cana-3901 32 6 zero zero ADJ cana-3901 32 7 nilpotent nilpotent ADJ cana-3901 32 8 elements element NOUN cana-3901 32 9 if if SCONJ cana-3901 32 10 and and CCONJ cana-3901 32 11 only only ADV cana-3901 32 12 if if SCONJ cana-3901 32 13 π‘₯2 π‘₯2 NOUN cana-3901 32 14 = = SYM cana-3901 32 15 0 0 NUM cana-3901 32 16 β‡’ β‡’ NOUN cana-3901 32 17 π‘₯ π‘₯ X cana-3901 32 18 = = SYM cana-3901 32 19 0 0 NUM cana-3901 32 20 for for ADP cana-3901 32 21 all all DET cana-3901 32 22 π‘₯ π‘₯ NOUN cana-3901 32 23 in in ADP cana-3901 32 24 𝑁 𝑁 PROPN cana-3901 32 25 ( ( PUNCT cana-3901 32 26 problem problem NOUN cana-3901 32 27 14 14 NUM cana-3901 32 28 , , PUNCT cana-3901 32 29 p.9 p.9 NOUN cana-3901 32 30 of of ADP cana-3901 32 31 [ [ X cana-3901 32 32 3 3 NUM cana-3901 32 33 ] ] NUM cana-3901 32 34 ) ) PUNCT cana-3901 32 35 . . PUNCT cana-3901 33 1 r r NOUN cana-3901 33 2 ( ( PUNCT cana-3901 33 3 3 3 NUM cana-3901 33 4 ) ) PUNCT cana-3901 33 5 if if SCONJ cana-3901 33 6 f f PROPN cana-3901 33 7 is be AUX cana-3901 33 8 a a DET cana-3901 33 9 mate mate NOUN cana-3901 33 10 function function NOUN cana-3901 33 11 for for ADP cana-3901 33 12 n n CCONJ cana-3901 33 13 , , PUNCT cana-3901 33 14 then then ADV cana-3901 33 15 for for ADP cana-3901 33 16 every every DET cana-3901 33 17 π‘₯ π‘₯ NOUN cana-3901 33 18 in in ADP cana-3901 33 19 𝑁 𝑁 NOUN cana-3901 33 20 , , PUNCT cana-3901 33 21 π‘₯𝑓 π‘₯𝑓 ADP cana-3901 33 22 ( ( PUNCT cana-3901 33 23 π‘₯ π‘₯ NOUN cana-3901 33 24 ) ) PUNCT cana-3901 33 25 , , PUNCT cana-3901 33 26 𝑓 𝑓 DET cana-3901 33 27 ( ( PUNCT cana-3901 33 28 π‘₯)π‘₯ π‘₯)π‘₯ X cana-3901 33 29 ∈ ∈ PROPN cana-3901 33 30 𝐸 𝐸 PROPN cana-3901 33 31 and and CCONJ cana-3901 33 32 𝑁π‘₯ 𝑁π‘₯ PROPN cana-3901 33 33 = = SYM cana-3901 33 34 𝑁𝑓 𝑁𝑓 PROPN cana-3901 33 35 ( ( PUNCT cana-3901 33 36 π‘₯)π‘₯ π‘₯)π‘₯ ADJ cana-3901 33 37 , , PUNCT cana-3901 33 38 π‘₯𝑁 π‘₯𝑁 NOUN cana-3901 33 39 = = NOUN cana-3901 33 40 π‘₯𝑓 π‘₯𝑓 X cana-3901 33 41 ( ( PUNCT cana-3901 33 42 π‘₯)𝑁 π‘₯)𝑁 NUM cana-3901 33 43 ( ( PUNCT cana-3901 33 44 lemma lemma PROPN cana-3901 33 45 3.2 3.2 NUM cana-3901 33 46 of of ADP cana-3901 33 47 [ [ X cana-3901 33 48 5 5 NUM cana-3901 33 49 ] ] NUM cana-3901 33 50 ) ) PUNCT cana-3901 33 51 . . PUNCT cana-3901 34 1 r r NOUN cana-3901 34 2 ( ( PUNCT cana-3901 34 3 4 4 NUM cana-3901 34 4 ) ) PUNCT cana-3901 34 5 if if SCONJ cana-3901 34 6 𝐿 𝐿 PROPN cana-3901 34 7 = = SYM cana-3901 34 8 { { PUNCT cana-3901 34 9 0 0 NUM cana-3901 34 10 } } PUNCT cana-3901 34 11 and and CCONJ cana-3901 34 12 𝑁 𝑁 PROPN cana-3901 34 13 = = SYM cana-3901 34 14 𝑁0 𝑁0 PROPN cana-3901 34 15 , , PUNCT cana-3901 34 16 then then ADV cana-3901 34 17 ( ( PUNCT cana-3901 34 18 i i NOUN cana-3901 34 19 ) ) PUNCT cana-3901 34 20 π‘₯𝑦 π‘₯𝑦 PROPN cana-3901 34 21 = = SYM cana-3901 34 22 0 0 NUM cana-3901 34 23 β‡’ β‡’ NOUN cana-3901 34 24 𝑦π‘₯ 𝑦π‘₯ NOUN cana-3901 34 25 = = SYM cana-3901 34 26 0 0 NUM cana-3901 34 27 for for ADP cana-3901 34 28 all all DET cana-3901 34 29 π‘₯ π‘₯ PROPN cana-3901 34 30 , , PUNCT cana-3901 34 31 𝑦 𝑦 NOUN cana-3901 34 32 in in ADP cana-3901 34 33 𝑁.(ii 𝑁.(ii PROPN cana-3901 34 34 ) ) PUNCT cana-3901 34 35 n n PRON cana-3901 34 36 has have VERB cana-3901 34 37 insertion insertion NOUN cana-3901 34 38 of of ADP cana-3901 34 39 factors factor NOUN cana-3901 34 40 propertyifp propertyifp VERB cana-3901 34 41 for for ADP cana-3901 34 42 shorti.e shorti.e NOUN cana-3901 34 43 for for ADP cana-3901 34 44 π‘₯ π‘₯ PROPN cana-3901 34 45 , , PUNCT cana-3901 34 46 𝑦 𝑦 NOUN cana-3901 34 47 in in ADP cana-3901 34 48 𝑁 𝑁 NOUN cana-3901 34 49 , , PUNCT cana-3901 34 50 π‘₯𝑦 π‘₯𝑦 ADJ cana-3901 34 51 = = SYM cana-3901 34 52 0 0 NUM cana-3901 34 53 β‡’ β‡’ NOUN cana-3901 34 54 π‘₯𝑛𝑦 π‘₯𝑛𝑦 ADP cana-3901 34 55 = = SYM cana-3901 34 56 0 0 PROPN cana-3901 34 57 . . PUNCT cana-3901 35 1 for for ADP cana-3901 35 2 all all DET cana-3901 35 3 n n NOUN cana-3901 35 4 in in ADP cana-3901 35 5 n. n. PROPN cana-3901 35 6 if if SCONJ cana-3901 35 7 n n PRON cana-3901 35 8 satisfies satisfie NOUN cana-3901 35 9 ( ( PUNCT cana-3901 35 10 i i NOUN cana-3901 35 11 ) ) PUNCT cana-3901 35 12 and and CCONJ cana-3901 35 13 ( ( PUNCT cana-3901 35 14 ii ii NOUN cana-3901 35 15 ) ) PUNCT cana-3901 35 16 then then ADV cana-3901 35 17 n n PRON cana-3901 35 18 is be AUX cana-3901 35 19 said say VERB cana-3901 35 20 to to PART cana-3901 35 21 have have VERB cana-3901 35 22 ( ( PUNCT cana-3901 35 23 βˆ— βˆ— NOUN cana-3901 35 24 , , PUNCT cana-3901 35 25 ifp ifp NOUN cana-3901 35 26 ) ) PUNCT cana-3901 35 27 ( ( PUNCT cana-3901 35 28 lemma lemma PROPN cana-3901 35 29 2.3 2.3 NUM cana-3901 35 30 of of ADP cana-3901 35 31 [ [ X cana-3901 35 32 5 5 NUM cana-3901 35 33 ] ] SYM cana-3901 35 34 ) ) PUNCT cana-3901 35 35 r r NOUN cana-3901 35 36 ( ( PUNCT cana-3901 35 37 5 5 NUM cana-3901 35 38 ) ) PUNCT cana-3901 35 39 any any DET cana-3901 35 40 pseudo pseudo NOUN cana-3901 35 41 commutative commutative ADJ cana-3901 35 42 near near ADP cana-3901 35 43 - - PUNCT cana-3901 35 44 ring ring NOUN cana-3901 35 45 with with ADP cana-3901 35 46 a a DET cana-3901 35 47 right right ADJ cana-3901 35 48 identity identity NOUN cana-3901 35 49 is be AUX cana-3901 35 50 weak weak ADJ cana-3901 35 51 commutative commutative ADJ cana-3901 35 52 ( ( PUNCT cana-3901 35 53 ( ( PUNCT cana-3901 35 54 i.e i.e X cana-3901 35 55 ) ) PUNCT cana-3901 35 56 . . PUNCT cana-3901 36 1 π‘₯𝑦𝑧 π‘₯𝑦𝑧 NOUN cana-3901 36 2 = = NUM cana-3901 36 3 π‘₯𝑦𝑧 π‘₯𝑦𝑧 VERB cana-3901 36 4 for for ADP cana-3901 36 5 all all DET cana-3901 36 6 π‘₯ π‘₯ PROPN cana-3901 36 7 , , PUNCT cana-3901 36 8 𝑦 𝑦 NOUN cana-3901 36 9 , , PUNCT cana-3901 36 10 𝑧 𝑧 VERB cana-3901 36 11 in in ADP cana-3901 36 12 𝑁 𝑁 PROPN cana-3901 36 13 [ [ X cana-3901 36 14 2 2 NUM cana-3901 36 15 ] ] PUNCT cana-3901 36 16 ) ) PUNCT cana-3901 36 17 ( ( PUNCT cana-3901 36 18 proposition proposition NOUN cana-3901 36 19 2.9 2.9 NUM cana-3901 36 20 of of ADP cana-3901 36 21 [ [ X cana-3901 36 22 6 6 NUM cana-3901 36 23 ] ] SYM cana-3901 36 24 ) ) PUNCT cana-3901 36 25 r r NOUN cana-3901 36 26 ( ( PUNCT cana-3901 36 27 6 6 NUM cana-3901 36 28 ) ) PUNCT cana-3901 36 29 a a DET cana-3901 36 30 zero zero NUM cana-3901 36 31 - - PUNCT cana-3901 36 32 symmetric symmetric ADJ cana-3901 36 33 near near ADP cana-3901 36 34 - - PUNCT cana-3901 36 35 ring ring NOUN cana-3901 36 36 n n NOUN cana-3901 36 37 is be AUX cana-3901 36 38 a a DET cana-3901 36 39 nearfield nearfield NOUN cana-3901 36 40 if if SCONJ cana-3901 36 41 𝑁𝑑 𝑁𝑑 PROPN cana-3901 36 42 β‰  β‰  PROPN cana-3901 36 43 { { PUNCT cana-3901 36 44 0 0 NUM cana-3901 36 45 } } PUNCT cana-3901 36 46 and and CCONJ cana-3901 36 47 for for ADP cana-3901 36 48 all all DET cana-3901 36 49 𝑛 𝑛 DET cana-3901 36 50 ∈ ∈ NOUN cana-3901 36 51 𝑁 𝑁 PROPN cana-3901 36 52 βˆ’ βˆ’ PROPN cana-3901 36 53 { { PUNCT cana-3901 36 54 0 0 NUM cana-3901 36 55 } } PUNCT cana-3901 36 56 , , PUNCT cana-3901 36 57 𝑁𝑛 𝑁𝑛 PROPN cana-3901 36 58 = = SYM cana-3901 36 59 𝑁 𝑁 PROPN cana-3901 36 60 ( ( PUNCT cana-3901 36 61 theorem theorem NOUN cana-3901 36 62 8.3 8.3 NUM cana-3901 36 63 , , PUNCT cana-3901 36 64 p.249 p.249 NOUN cana-3901 36 65 of of ADP cana-3901 36 66 [ [ X cana-3901 36 67 2 2 NUM cana-3901 36 68 ] ] NUM cana-3901 36 69 ) ) PUNCT cana-3901 36 70 . . PUNCT cana-3901 37 1 r r NOUN cana-3901 37 2 ( ( PUNCT cana-3901 37 3 7 7 X cana-3901 37 4 ) ) PUNCT cana-3901 37 5 let let VERB cana-3901 37 6 n n PRON cana-3901 37 7 be be AUX cana-3901 37 8 a a DET cana-3901 37 9 right right ADJ cana-3901 37 10 near near NOUN cana-3901 37 11 - - PUNCT cana-3901 37 12 ring ring NOUN cana-3901 37 13 . . PUNCT cana-3901 38 1 if if SCONJ cana-3901 38 2 for for ADP cana-3901 38 3 every every DET cana-3901 38 4 π‘₯ π‘₯ PROPN cana-3901 38 5 , , PUNCT cana-3901 38 6 𝑦 𝑦 NOUN cana-3901 38 7 in in ADP cana-3901 38 8 n n CCONJ cana-3901 38 9 , , PUNCT cana-3901 38 10 π‘₯𝑁 π‘₯𝑁 PROPN cana-3901 38 11 𝑦 𝑦 NOUN cana-3901 38 12 = = PUNCT cana-3901 38 13 𝑁π‘₯𝑦 𝑁π‘₯𝑦 PROPN cana-3901 38 14 then then ADV cana-3901 38 15 we we PRON cana-3901 38 16 say say VERB cana-3901 38 17 n n PRON cana-3901 38 18 is be AUX cana-3901 38 19 a a DET cana-3901 38 20 Ξ²1 Ξ²1 NOUN cana-3901 38 21 near near ADP cana-3901 38 22 - - PUNCT cana-3901 38 23 ring ring NOUN cana-3901 38 24 . . PUNCT cana-3901 39 1 ( ( PUNCT cana-3901 39 2 definition definition NOUN cana-3901 39 3 3.1.1 3.1.1 NUM cana-3901 39 4 of of ADP cana-3901 39 5 [ [ X cana-3901 39 6 4 4 NUM cana-3901 39 7 ] ] SYM cana-3901 39 8 ) ) PUNCT cana-3901 39 9 4 4 NUM cana-3901 39 10 . . X cana-3901 39 11 definition definition NOUN cana-3901 39 12 and and CCONJ cana-3901 39 13 examples example NOUN cana-3901 39 14 of of ADP cana-3901 39 15 𝜎1 𝜎1 ADJ cana-3901 39 16 near near ADV cana-3901 39 17 – – PUNCT cana-3901 39 18 rings ring NOUN cana-3901 39 19 in in ADP cana-3901 39 20 this this DET cana-3901 39 21 section section NOUN cana-3901 39 22 we we PRON cana-3901 39 23 define define VERB cana-3901 39 24 𝜎1near 𝜎1near NUM cana-3901 39 25 - - PUNCT cana-3901 39 26 ring ring NOUN cana-3901 39 27 and and CCONJ cana-3901 39 28 give give VERB cana-3901 39 29 certain certain ADJ cana-3901 39 30 examples example NOUN cana-3901 39 31 of of ADP cana-3901 39 32 this this DET cana-3901 39 33 new new ADJ cana-3901 39 34 concept concept NOUN cana-3901 39 35 . . PUNCT cana-3901 40 1 communications communication NOUN cana-3901 40 2 on on ADP cana-3901 40 3 applied apply VERB cana-3901 40 4 nonlinear nonlinear ADJ cana-3901 40 5 analysis analysis NOUN cana-3901 40 6 issn issn NOUN cana-3901 40 7 : : PUNCT cana-3901 40 8 1074 1074 NUM cana-3901 40 9 - - PUNCT cana-3901 40 10 133x 133x NUM cana-3901 40 11 vol vol NOUN cana-3901 40 12 32 32 NUM cana-3901 40 13 no no NOUN cana-3901 40 14 . . PUNCT cana-3901 41 1 9s 9s NUM cana-3901 41 2 ( ( PUNCT cana-3901 41 3 2025 2025 NUM cana-3901 41 4 ) ) PUNCT cana-3901 41 5 355 355 NUM cana-3901 42 1 https://internationalpubls.com https://internationalpubls.com X cana-3901 42 2 definition definition NOUN cana-3901 42 3 4.1 4.1 NUM cana-3901 42 4 let let VERB cana-3901 42 5 n n PRON cana-3901 42 6 be be AUX cana-3901 42 7 a a DET cana-3901 42 8 right right ADJ cana-3901 42 9 nearring nearring NOUN cana-3901 42 10 . . PUNCT cana-3901 43 1 then then ADV cana-3901 43 2 n n ADV cana-3901 43 3 is be AUX cana-3901 43 4 said say VERB cana-3901 43 5 to to PART cana-3901 43 6 be be AUX cana-3901 43 7 an an DET cana-3901 43 8 𝜎1near 𝜎1near NOUN cana-3901 43 9 - - PUNCT cana-3901 43 10 ring ring NOUN cana-3901 43 11 if if SCONJ cana-3901 43 12 π‘₯𝑦2 π‘₯𝑦2 NOUN cana-3901 43 13 = = PUNCT cana-3901 43 14 𝑦π‘₯𝑦 𝑦π‘₯𝑦 VERB cana-3901 43 15 for for ADP cana-3901 43 16 all all DET cana-3901 43 17 π‘₯ π‘₯ PROPN cana-3901 43 18 , , PUNCT cana-3901 43 19 𝑦 𝑦 NOUN cana-3901 43 20 ∈ ∈ NOUN cana-3901 43 21 𝑁. 𝑁. PROPN cana-3901 43 22 example example NOUN cana-3901 43 23 4.2(a 4.2(a NUM cana-3901 43 24 ) ) PUNCT cana-3901 43 25 the the DET cana-3901 43 26 near near ADV cana-3901 43 27 - - PUNCT cana-3901 43 28 ring ring NOUN cana-3901 43 29 ( ( PUNCT cana-3901 43 30 𝑁 𝑁 PROPN cana-3901 43 31 , , PUNCT cana-3901 43 32 + + NOUN cana-3901 43 33 , , PUNCT cana-3901 43 34 . . PUNCT cana-3901 43 35 ) ) PUNCT cana-3901 44 1 defined define VERB cana-3901 44 2 on on ADP cana-3901 44 3 klein klein PROPN cana-3901 44 4 ’s ’s PART cana-3901 44 5 four four NUM cana-3901 44 6 group group NOUN cana-3901 44 7 ( ( PUNCT cana-3901 44 8 𝑁 𝑁 PROPN cana-3901 44 9 , , PUNCT cana-3901 44 10 + + PUNCT cana-3901 44 11 ) ) PUNCT cana-3901 44 12 with with ADP cana-3901 44 13 𝑁 𝑁 PROPN cana-3901 44 14 = = SYM cana-3901 44 15 { { PUNCT cana-3901 44 16 0 0 NUM cana-3901 44 17 , , PUNCT cana-3901 44 18 π‘Ž π‘Ž NOUN cana-3901 44 19 , , PUNCT cana-3901 44 20 𝑏 𝑏 NOUN cana-3901 44 21 , , PUNCT cana-3901 44 22 𝑐 𝑐 NOUN cana-3901 44 23 } } PUNCT cana-3901 44 24 where where SCONJ cana-3901 44 25 β€˜ ' PUNCT cana-3901 44 26 βˆ™ βˆ™ PROPN cana-3901 44 27 ’ ' PUNCT cana-3901 44 28 is be AUX cana-3901 44 29 defined define VERB cana-3901 44 30 as as ADP cana-3901 44 31 per per ADP cana-3901 44 32 scheme scheme NOUN cana-3901 44 33 12 12 NUM cana-3901 44 34 , , PUNCT cana-3901 44 35 𝑃 𝑃 VERB cana-3901 44 36 408 408 NUM cana-3901 44 37 of of ADP cana-3901 44 38 pilz pilz PROPN cana-3901 45 1 [ [ X cana-3901 45 2 2 2 NUM cana-3901 45 3 ] ] PUNCT cana-3901 45 4 . . PUNCT cana-3901 46 1 βˆ™ βˆ™ NOUN cana-3901 46 2 0 0 PUNCT cana-3901 47 1 a a DET cana-3901 47 2 b b X cana-3901 47 3 c c NOUN cana-3901 47 4 0 0 NUM cana-3901 47 5 0 0 NUM cana-3901 47 6 0 0 NUM cana-3901 47 7 0 0 NUM cana-3901 47 8 0 0 NUM cana-3901 47 9 a a DET cana-3901 47 10 0 0 NUM cana-3901 47 11 a a DET cana-3901 47 12 0 0 NUM cana-3901 47 13 a a DET cana-3901 47 14 b b NOUN cana-3901 47 15 0 0 NUM cana-3901 47 16 0 0 NUM cana-3901 47 17 0 0 NUM cana-3901 47 18 0 0 NUM cana-3901 47 19 c c NOUN cana-3901 47 20 0 0 NUM cana-3901 48 1 a a DET cana-3901 48 2 0 0 NUM cana-3901 48 3 a a PRON cana-3901 48 4 is be AUX cana-3901 48 5 a a DET cana-3901 48 6 𝜎1near 𝜎1near NOUN cana-3901 48 7 - - PUNCT cana-3901 48 8 ring ring NOUN cana-3901 48 9 b b NOUN cana-3901 48 10 ) ) PUNCT cana-3901 48 11 let let VERB cana-3901 48 12 ( ( PUNCT cana-3901 48 13 𝑁 𝑁 PROPN cana-3901 48 14 , , PUNCT cana-3901 48 15 + + NOUN cana-3901 48 16 ) ) PUNCT cana-3901 48 17 be be VERB cana-3901 48 18 the the DET cana-3901 48 19 klein klein PROPN cana-3901 48 20 ’s ’s PART cana-3901 48 21 four four NUM cana-3901 48 22 group group NOUN cana-3901 48 23 as as ADP cana-3901 48 24 in in ADP cana-3901 48 25 ( ( PUNCT cana-3901 48 26 a a NOUN cana-3901 48 27 ) ) PUNCT cana-3901 48 28 above above ADV cana-3901 48 29 . . PUNCT cana-3901 49 1 if if SCONJ cana-3901 49 2 multiplication multiplication NOUN cana-3901 49 3 is be AUX cana-3901 49 4 defined define VERB cana-3901 49 5 as as ADP cana-3901 49 6 per per ADP cana-3901 49 7 scheme scheme NOUN cana-3901 49 8 22 22 NUM cana-3901 49 9 , , PUNCT cana-3901 49 10 p.408 p.408 ADV cana-3901 49 11 of of ADP cana-3901 49 12 pilz[2 pilz[2 NOUN cana-3901 49 13 ] ] PUNCT cana-3901 49 14 βˆ™ βˆ™ PROPN cana-3901 49 15 0 0 PUNCT cana-3901 50 1 a a DET cana-3901 50 2 b b X cana-3901 50 3 c c NOUN cana-3901 50 4 0 0 NUM cana-3901 50 5 0 0 NUM cana-3901 50 6 0 0 NUM cana-3901 50 7 0 0 NUM cana-3901 50 8 0 0 NUM cana-3901 50 9 a a DET cana-3901 50 10 a a DET cana-3901 50 11 a a DET cana-3901 50 12 a a DET cana-3901 50 13 a a DET cana-3901 50 14 b b NOUN cana-3901 50 15 0 0 NUM cana-3901 50 16 0 0 NUM cana-3901 50 17 0 0 NUM cana-3901 50 18 0 0 NUM cana-3901 50 19 c c NOUN cana-3901 50 20 a a DET cana-3901 50 21 a a DET cana-3901 50 22 a a DET cana-3901 50 23 a a PRON cana-3901 50 24 then then ADV cana-3901 50 25 n n NOUN cana-3901 50 26 is be AUX cana-3901 50 27 not not PART cana-3901 50 28 a a DET cana-3901 50 29 𝜎1near 𝜎1near NOUN cana-3901 50 30 - - PUNCT cana-3901 50 31 ring ring NOUN cana-3901 50 32 since since SCONJ cana-3901 50 33 π‘Žπ‘2 π‘Žπ‘2 PROPN cana-3901 50 34 β‰  β‰  PROPN cana-3901 50 35 π‘π‘Žπ‘. π‘π‘Žπ‘. X cana-3901 50 36 5 5 NUM cana-3901 50 37 . . PUNCT cana-3901 50 38 properties property NOUN cana-3901 50 39 of of ADP cana-3901 50 40 𝜎1near 𝜎1near NOUN cana-3901 50 41 - - PUNCT cana-3901 50 42 ring ring NOUN cana-3901 50 43 in in ADP cana-3901 50 44 this this DET cana-3901 50 45 section section NOUN cana-3901 50 46 we we PRON cana-3901 50 47 prove prove VERB cana-3901 50 48 certain certain ADJ cana-3901 50 49 important important ADJ cana-3901 50 50 properties property NOUN cana-3901 50 51 of of ADP cana-3901 50 52 𝜎1 𝜎1 ADJ cana-3901 50 53 near near ADJ cana-3901 50 54 - - PUNCT cana-3901 50 55 ring ring NOUN cana-3901 50 56 and and CCONJ cana-3901 50 57 give give VERB cana-3901 50 58 a a DET cana-3901 50 59 complete complete ADJ cana-3901 50 60 characterization characterization NOUN cana-3901 50 61 of of ADP cana-3901 50 62 such such ADJ cana-3901 50 63 near near ADP cana-3901 50 64 - - PUNCT cana-3901 50 65 ring ring NOUN cana-3901 50 66 . . PUNCT cana-3901 51 1 proposition proposition NOUN cana-3901 51 2 5.1 5.1 NUM cana-3901 51 3 let let VERB cana-3901 51 4 𝑁 𝑁 PROPN cana-3901 51 5 = = SYM cana-3901 51 6 𝑁𝑑 𝑁𝑑 ADV cana-3901 51 7 be be AUX cana-3901 51 8 a a DET cana-3901 51 9 𝜎1near 𝜎1near NOUN cana-3901 51 10 - - PUNCT cana-3901 51 11 ring ring NOUN cana-3901 51 12 with with ADP cana-3901 51 13 identity identity NOUN cana-3901 51 14 . . PUNCT cana-3901 52 1 then then ADV cana-3901 52 2 n n PRON cana-3901 52 3 is be AUX cana-3901 52 4 commutative commutative ADJ cana-3901 52 5 . . PUNCT cana-3901 53 1 proof proof NOUN cana-3901 53 2 let let VERB cana-3901 53 3 n n PRON cana-3901 53 4 be be AUX cana-3901 53 5 a a DET cana-3901 53 6 𝜎1near 𝜎1near NOUN cana-3901 53 7 - - PUNCT cana-3901 53 8 ring ring NOUN cana-3901 53 9 . . PUNCT cana-3901 54 1 then then ADV cana-3901 54 2 for for ADP cana-3901 54 3 all all DET cana-3901 54 4 π‘₯ π‘₯ PROPN cana-3901 54 5 , , PUNCT cana-3901 54 6 𝑦 𝑦 NOUN cana-3901 54 7 ∈ ∈ NOUN cana-3901 54 8 𝑁 𝑁 NOUN cana-3901 54 9 , , PUNCT cana-3901 54 10 π‘₯𝑦2 π‘₯𝑦2 NOUN cana-3901 54 11 = = PUNCT cana-3901 54 12 𝑦π‘₯𝑦 𝑦π‘₯𝑦 NOUN cana-3901 54 13 …… …… NOUN cana-3901 54 14 …… …… NOUN cana-3901 54 15 ... ... PUNCT cana-3901 54 16 ( ( PUNCT cana-3901 54 17 1 1 X cana-3901 54 18 ) ) PUNCT cana-3901 54 19 replace replace VERB cana-3901 54 20 the the DET cana-3901 54 21 element element NOUN cana-3901 54 22 𝑦 𝑦 NOUN cana-3901 54 23 by by ADP cana-3901 54 24 𝑦 𝑦 PRON cana-3901 54 25 + + CCONJ cana-3901 54 26 𝑒. 𝑒. VERB cana-3901 54 27 in in ADP cana-3901 54 28 ( ( PUNCT cana-3901 54 29 1 1 X cana-3901 54 30 ) ) PUNCT cana-3901 54 31 π‘₯(𝑦 π‘₯(𝑦 NOUN cana-3901 54 32 + + NUM cana-3901 54 33 𝑒)2 𝑒)2 NOUN cana-3901 54 34 = = SYM cana-3901 54 35 ( ( PUNCT cana-3901 54 36 𝑦 𝑦 NOUN cana-3901 54 37 + + CCONJ cana-3901 54 38 𝑒)π‘₯(𝑦 𝑒)π‘₯(𝑦 NOUN cana-3901 54 39 + + CCONJ cana-3901 54 40 𝑒 𝑒 X cana-3901 54 41 ) ) PUNCT cana-3901 54 42 π‘₯(𝑦 π‘₯(𝑦 NOUN cana-3901 54 43 + + CCONJ cana-3901 54 44 𝑒)(𝑦 𝑒)(𝑦 NOUN cana-3901 54 45 + + CCONJ cana-3901 54 46 𝑒 𝑒 X cana-3901 54 47 ) ) PUNCT cana-3901 54 48 = = SYM cana-3901 54 49 ( ( PUNCT cana-3901 54 50 𝑦 𝑦 NOUN cana-3901 54 51 + + CCONJ cana-3901 54 52 𝑒)π‘₯(𝑦 𝑒)π‘₯(𝑦 NOUN cana-3901 54 53 + + CCONJ cana-3901 54 54 𝑒 𝑒 NOUN cana-3901 54 55 ) ) PUNCT cana-3901 54 56 communications communication NOUN cana-3901 54 57 on on ADP cana-3901 54 58 applied apply VERB cana-3901 54 59 nonlinear nonlinear ADJ cana-3901 54 60 analysis analysis NOUN cana-3901 54 61 issn issn NOUN cana-3901 54 62 : : PUNCT cana-3901 54 63 1074 1074 NUM cana-3901 54 64 - - PUNCT cana-3901 54 65 133x 133x NUM cana-3901 54 66 vol vol NOUN cana-3901 54 67 32 32 NUM cana-3901 54 68 no no NOUN cana-3901 54 69 . . PUNCT cana-3901 55 1 9s 9s NUM cana-3901 55 2 ( ( PUNCT cana-3901 55 3 2025 2025 NUM cana-3901 55 4 ) ) PUNCT cana-3901 55 5 356 356 NUM cana-3901 55 6 https://internationalpubls.com https://internationalpubls.com NOUN cana-3901 55 7 π‘₯[(𝑦 π‘₯[(𝑦 NOUN cana-3901 55 8 + + CCONJ cana-3901 55 9 𝑒 𝑒 NOUN cana-3901 55 10 ) ) PUNCT cana-3901 55 11 . . PUNCT cana-3901 56 1 𝑦 𝑦 PRON cana-3901 56 2 + + CCONJ cana-3901 56 3 ( ( PUNCT cana-3901 56 4 𝑦 𝑦 NOUN cana-3901 56 5 + + CCONJ cana-3901 56 6 𝑒 𝑒 NOUN cana-3901 56 7 ) ) PUNCT cana-3901 56 8 . . PUNCT cana-3901 57 1 𝑒 𝑒 X cana-3901 57 2 ] ] X cana-3901 57 3 = = SYM cana-3901 57 4 ( ( PUNCT cana-3901 57 5 𝑦 𝑦 NOUN cana-3901 57 6 + + CCONJ cana-3901 57 7 𝑒)π‘₯(𝑦 𝑒)π‘₯(𝑦 NOUN cana-3901 57 8 + + CCONJ cana-3901 57 9 𝑒 𝑒 X cana-3901 57 10 ) ) PUNCT cana-3901 58 1 π‘₯[𝑦. π‘₯[𝑦. PROPN cana-3901 58 2 𝑦 𝑦 NUM cana-3901 58 3 + + SYM cana-3901 58 4 𝑦 𝑦 PROPN cana-3901 58 5 + + SYM cana-3901 58 6 𝑦 𝑦 NOUN cana-3901 58 7 + + CCONJ cana-3901 58 8 𝑒 𝑒 X cana-3901 58 9 ] ] X cana-3901 58 10 = = SYM cana-3901 58 11 ( ( PUNCT cana-3901 58 12 𝑦π‘₯ 𝑦π‘₯ PROPN cana-3901 58 13 + + CCONJ cana-3901 58 14 π‘₯)(𝑦 π‘₯)(𝑦 X cana-3901 58 15 + + CCONJ cana-3901 58 16 𝑒 𝑒 X cana-3901 58 17 ) ) PUNCT cana-3901 58 18 ( ( PUNCT cana-3901 58 19 π‘₯𝑦)𝑦 π‘₯𝑦)𝑦 PROPN cana-3901 58 20 + + CCONJ cana-3901 58 21 π‘₯𝑦 π‘₯𝑦 VERB cana-3901 58 22 + + CCONJ cana-3901 58 23 π‘₯𝑦 π‘₯𝑦 NOUN cana-3901 58 24 + + NUM cana-3901 58 25 π‘₯ π‘₯ NOUN cana-3901 58 26 = = SYM cana-3901 58 27 ( ( PUNCT cana-3901 58 28 𝑦π‘₯)𝑦 𝑦π‘₯)𝑦 NUM cana-3901 58 29 + + NUM cana-3901 58 30 𝑦π‘₯ 𝑦π‘₯ NOUN cana-3901 58 31 + + CCONJ cana-3901 58 32 π‘₯𝑦 π‘₯𝑦 NOUN cana-3901 58 33 + + NUM cana-3901 58 34 π‘₯ π‘₯ X cana-3901 58 35 ( ( PUNCT cana-3901 58 36 π‘₯𝑦)𝑦 π‘₯𝑦)𝑦 PROPN cana-3901 58 37 + + NOUN cana-3901 58 38 π‘₯𝑦 π‘₯𝑦 NOUN cana-3901 58 39 = = SYM cana-3901 58 40 ( ( PUNCT cana-3901 58 41 𝑦π‘₯)𝑦 𝑦π‘₯)𝑦 NUM cana-3901 58 42 + + NUM cana-3901 58 43 𝑦π‘₯ 𝑦π‘₯ PROPN cana-3901 58 44 ( ( PUNCT cana-3901 58 45 by by ADP cana-3901 58 46 right right ADJ cana-3901 58 47 cancellation cancellation NOUN cana-3901 58 48 law law NOUN cana-3901 58 49 ) ) PUNCT cana-3901 58 50 π‘₯(𝑦. π‘₯(𝑦. SYM cana-3901 58 51 𝑦 𝑦 X cana-3901 58 52 ) ) PUNCT cana-3901 59 1 + + CCONJ cana-3901 59 2 π‘₯. π‘₯. PROPN cana-3901 59 3 𝑦 𝑦 NOUN cana-3901 59 4 = = X cana-3901 59 5 𝑦(π‘₯. 𝑦(π‘₯. NOUN cana-3901 59 6 𝑦 𝑦 NOUN cana-3901 59 7 ) ) PUNCT cana-3901 59 8 + + CCONJ cana-3901 59 9 𝑦. 𝑦. NOUN cana-3901 59 10 π‘₯ π‘₯ PROPN cana-3901 59 11 ( ( PUNCT cana-3901 59 12 by by ADP cana-3901 59 13 associative associative ADJ cana-3901 59 14 law law NOUN cana-3901 59 15 ) ) PUNCT cana-3901 59 16 β‡’ β‡’ VERB cana-3901 59 17 π‘₯𝑦2 π‘₯𝑦2 NOUN cana-3901 60 1 + + CCONJ cana-3901 60 2 π‘₯𝑦 π‘₯𝑦 NOUN cana-3901 60 3 = = ADJ cana-3901 60 4 π‘₯𝑦2 π‘₯𝑦2 NOUN cana-3901 60 5 + + CCONJ cana-3901 60 6 𝑦π‘₯ 𝑦π‘₯ PROPN cana-3901 61 1 [ [ X cana-3901 61 2 by by ADP cana-3901 61 3 equation equation NOUN cana-3901 61 4 1 1 NUM cana-3901 61 5 ] ] SYM cana-3901 61 6 π‘₯𝑦 π‘₯𝑦 PROPN cana-3901 61 7 = = SYM cana-3901 61 8 𝑦π‘₯ 𝑦π‘₯ NOUN cana-3901 61 9 βˆ€ βˆ€ NOUN cana-3901 61 10 π‘₯ π‘₯ NOUN cana-3901 61 11 , , PUNCT cana-3901 61 12 𝑦 𝑦 NOUN cana-3901 61 13 ∈ ∈ NOUN cana-3901 61 14 𝑁 𝑁 NOUN cana-3901 61 15 thus thus ADV cana-3901 61 16 𝑁 𝑁 PROPN cana-3901 61 17 is be AUX cana-3901 61 18 commutative commutative ADJ cana-3901 61 19 . . PUNCT cana-3901 62 1 remark remark VERB cana-3901 62 2 5.2 5.2 NUM cana-3901 62 3 a a DET cana-3901 62 4 quasi quasi NOUN cana-3901 62 5 weak weak ADJ cana-3901 62 6 commutative commutative ADJ cana-3901 62 7 near near NOUN cana-3901 62 8 - - PUNCT cana-3901 62 9 ring ring NOUN cana-3901 62 10 can can AUX cana-3901 62 11 become become VERB cana-3901 62 12 a a DET cana-3901 62 13 𝜎1near 𝜎1near ADJ cana-3901 62 14 - - PUNCT cana-3901 62 15 ring ring NOUN cana-3901 62 16 proof proof NOUN cana-3901 62 17 let let VERB cana-3901 62 18 𝑁 𝑁 PROPN cana-3901 62 19 be be AUX cana-3901 62 20 a a DET cana-3901 62 21 quasi quasi ADJ cana-3901 62 22 - - ADJ cana-3901 62 23 weak weak ADJ cana-3901 62 24 commutative commutative ADJ cana-3901 62 25 near near NOUN cana-3901 62 26 - - PUNCT cana-3901 62 27 ring ring NOUN cana-3901 62 28 then then ADV cana-3901 62 29 π‘₯𝑦𝑧 π‘₯𝑦𝑧 VERB cana-3901 62 30 = = SYM cana-3901 62 31 𝑦π‘₯𝑧 𝑦π‘₯𝑧 NOUN cana-3901 62 32 for for ADP cana-3901 62 33 all all PRON cana-3901 62 34 π‘₯ π‘₯ PROPN cana-3901 62 35 , , PUNCT cana-3901 62 36 𝑦 𝑦 NOUN cana-3901 62 37 , , PUNCT cana-3901 62 38 𝑧 𝑧 DET cana-3901 62 39 ∈ ∈ NOUN cana-3901 62 40 𝑁. 𝑁. PROPN cana-3901 62 41 if if SCONJ cana-3901 62 42 𝑦 𝑦 NOUN cana-3901 62 43 = = SYM cana-3901 62 44 𝑧 𝑧 ADJ cana-3901 62 45 , , PUNCT cana-3901 62 46 then then ADV cana-3901 62 47 π‘₯𝑦𝑦 π‘₯𝑦𝑦 PROPN cana-3901 62 48 = = SYM cana-3901 62 49 𝑦π‘₯𝑦 𝑦π‘₯𝑦 NOUN cana-3901 62 50 , , PUNCT cana-3901 62 51 ( ( PUNCT cana-3901 62 52 𝑖𝑒 𝑖𝑒 NOUN cana-3901 62 53 ) ) PUNCT cana-3901 62 54 π‘₯𝑦2 π‘₯𝑦2 NOUN cana-3901 62 55 = = SYM cana-3901 62 56 𝑦π‘₯𝑦. 𝑦π‘₯𝑦. X cana-3901 62 57 consequently consequently ADV cana-3901 62 58 𝑁 𝑁 PROPN cana-3901 62 59 becomes become VERB cana-3901 62 60 a a DET cana-3901 62 61 𝜎1near 𝜎1near NOUN cana-3901 62 62 - - PUNCT cana-3901 62 63 ring ring NOUN cana-3901 62 64 . . PUNCT cana-3901 63 1 proposition proposition NOUN cana-3901 63 2 5.3 5.3 NUM cana-3901 63 3 𝜎1near 𝜎1near NUM cana-3901 63 4 - - PUNCT cana-3901 63 5 ring ring NOUN cana-3901 63 6 is be AUX cana-3901 63 7 always always ADV cana-3901 63 8 zero zero NUM cana-3901 63 9 symmetric symmetric NOUN cana-3901 63 10 . . PUNCT cana-3901 64 1 proof proof NOUN cana-3901 64 2 suppose suppose VERB cana-3901 64 3 𝑁 𝑁 PROPN cana-3901 64 4 is be AUX cana-3901 64 5 a a DET cana-3901 64 6 𝜎1near 𝜎1near NOUN cana-3901 64 7 - - PUNCT cana-3901 64 8 ring ring NOUN cana-3901 64 9 then then ADV cana-3901 64 10 π‘₯𝑦2 π‘₯𝑦2 ADV cana-3901 64 11 = = PUNCT cana-3901 64 12 𝑦π‘₯𝑦 𝑦π‘₯𝑦 VERB cana-3901 64 13 π‘“π‘œπ‘Ÿ π‘“π‘œπ‘Ÿ ADV cana-3901 64 14 π‘Žπ‘™π‘™ π‘Žπ‘™π‘™ VERB cana-3901 64 15 π‘₯ π‘₯ PROPN cana-3901 64 16 , , PUNCT cana-3901 64 17 𝑦 𝑦 NOUN cana-3901 64 18 ∈ ∈ NOUN cana-3901 64 19 𝑁. 𝑁. PROPN cana-3901 65 1 when when SCONJ cana-3901 65 2 𝑦 𝑦 NOUN cana-3901 65 3 = = SYM cana-3901 65 4 0 0 NUM cana-3901 65 5 , , PUNCT cana-3901 65 6 π‘₯0 π‘₯0 NOUN cana-3901 65 7 = = SYM cana-3901 65 8 0 0 PUNCT cana-3901 65 9 π‘₯ π‘₯ NOUN cana-3901 65 10 0 0 NUM cana-3901 66 1 = = SYM cana-3901 66 2 0 0 NUM cana-3901 67 1 it it PRON cana-3901 67 2 follows follow VERB cana-3901 67 3 that that SCONJ cana-3901 67 4 𝑁 𝑁 PROPN cana-3901 67 5 is be AUX cana-3901 67 6 zero zero NUM cana-3901 67 7 symmetric symmetric ADJ cana-3901 67 8 . . PUNCT cana-3901 68 1 proposition proposition NOUN cana-3901 68 2 5.4 5.4 NUM cana-3901 68 3 let let VERB cana-3901 68 4 𝑁 𝑁 PROPN cana-3901 68 5 be be AUX cana-3901 68 6 a a DET cana-3901 68 7 𝜎1near 𝜎1near NOUN cana-3901 68 8 - - PUNCT cana-3901 68 9 ring ring NOUN cana-3901 68 10 . . PUNCT cana-3901 69 1 if if SCONJ cana-3901 69 2 𝑁 𝑁 PROPN cana-3901 69 3 is be AUX cana-3901 69 4 weak weak ADJ cana-3901 69 5 commutative commutative ADJ cana-3901 69 6 then then ADV cana-3901 69 7 π‘₯2𝑦 π‘₯2𝑦 PROPN cana-3901 69 8 = = PUNCT cana-3901 69 9 𝑦2π‘₯. 𝑦2π‘₯. VERB cana-3901 69 10 for for ADP cana-3901 69 11 all all DET cana-3901 69 12 π‘₯ π‘₯ PROPN cana-3901 69 13 , , PUNCT cana-3901 69 14 𝑦 𝑦 NOUN cana-3901 69 15 ∈ ∈ NOUN cana-3901 69 16 𝑁 𝑁 ADJ cana-3901 69 17 proof proof NOUN cana-3901 69 18 let let VERB cana-3901 69 19 𝑁 𝑁 PROPN cana-3901 69 20 be be AUX cana-3901 69 21 a a DET cana-3901 69 22 weak weak ADJ cana-3901 69 23 commutative commutative ADJ cana-3901 69 24 near near NOUN cana-3901 69 25 - - PUNCT cana-3901 69 26 ring ring NOUN cana-3901 69 27 then then ADV cana-3901 69 28 π‘₯𝑦𝑧 π‘₯𝑦𝑧 VERB cana-3901 69 29 = = SYM cana-3901 69 30 π‘₯𝑧𝑦 π‘₯𝑧𝑦 NOUN cana-3901 69 31 … … PUNCT cana-3901 69 32 … … PUNCT cana-3901 69 33 … … PUNCT cana-3901 69 34 . . PUNCT cana-3901 69 35 . . PUNCT cana-3901 70 1 … … PUNCT cana-3901 70 2 ( ( PUNCT cana-3901 70 3 1 1 X cana-3901 70 4 ) ) PUNCT cana-3901 70 5 let let VERB cana-3901 70 6 𝑁 𝑁 PROPN cana-3901 70 7 be be AUX cana-3901 70 8 a a DET cana-3901 70 9 𝜎1near 𝜎1near NOUN cana-3901 70 10 - - PUNCT cana-3901 70 11 ring ring NOUN cana-3901 70 12 then then ADV cana-3901 70 13 π‘₯𝑦2 π‘₯𝑦2 NOUN cana-3901 70 14 = = SYM cana-3901 70 15 𝑦π‘₯𝑦 𝑦π‘₯𝑦 NOUN cana-3901 70 16 … … PUNCT cana-3901 70 17 … … PUNCT cana-3901 70 18 … … PUNCT cana-3901 70 19 … … PUNCT cana-3901 70 20 .(2 .(2 NUM cana-3901 70 21 ) ) PUNCT cana-3901 70 22 now now ADV cana-3901 70 23 , , PUNCT cana-3901 70 24 π‘₯2𝑦 π‘₯2𝑦 PROPN cana-3901 70 25 = = PUNCT cana-3901 70 26 π‘₯π‘₯𝑦 π‘₯π‘₯𝑦 ADJ cana-3901 70 27 = = PUNCT cana-3901 70 28 π‘₯𝑦π‘₯ π‘₯𝑦π‘₯ NOUN cana-3901 71 1 [ [ PUNCT cana-3901 71 2 by by ADP cana-3901 71 3 equation equation NOUN cana-3901 71 4 ( ( PUNCT cana-3901 71 5 1 1 NUM cana-3901 71 6 ) ) PUNCT cana-3901 71 7 ] ] PUNCT cana-3901 72 1 = = PUNCT cana-3901 72 2 𝑦π‘₯2 𝑦π‘₯2 NOUN cana-3901 73 1 [ [ PUNCT cana-3901 73 2 by by ADP cana-3901 73 3 equation equation NOUN cana-3901 73 4 ( ( PUNCT cana-3901 73 5 2 2 NUM cana-3901 73 6 ) ) PUNCT cana-3901 73 7 ] ] PUNCT cana-3901 73 8 hence hence ADV cana-3901 73 9 π‘₯2𝑦 π‘₯2𝑦 PROPN cana-3901 73 10 = = SYM cana-3901 73 11 𝑦π‘₯2for 𝑦π‘₯2for PROPN cana-3901 73 12 all all DET cana-3901 73 13 π‘₯ π‘₯ PROPN cana-3901 73 14 , , PUNCT cana-3901 73 15 𝑦 𝑦 NOUN cana-3901 73 16 ∈ ∈ NOUN cana-3901 73 17 𝑁. 𝑁. PROPN cana-3901 73 18 communications communication NOUN cana-3901 73 19 on on ADP cana-3901 73 20 applied apply VERB cana-3901 73 21 nonlinear nonlinear ADJ cana-3901 73 22 analysis analysis NOUN cana-3901 73 23 issn issn NOUN cana-3901 73 24 : : PUNCT cana-3901 73 25 1074 1074 NUM cana-3901 73 26 - - PUNCT cana-3901 73 27 133x 133x NUM cana-3901 73 28 vol vol NOUN cana-3901 73 29 32 32 NUM cana-3901 73 30 no no NOUN cana-3901 73 31 . . PUNCT cana-3901 74 1 9s 9s NUM cana-3901 74 2 ( ( PUNCT cana-3901 74 3 2025 2025 NUM cana-3901 74 4 ) ) PUNCT cana-3901 74 5 357 357 NUM cana-3901 75 1 https://internationalpubls.com https://internationalpubls.com X cana-3901 75 2 proposition proposition NOUN cana-3901 75 3 5.5 5.5 NUM cana-3901 75 4 homomorphic homomorphic ADJ cana-3901 75 5 image image NOUN cana-3901 75 6 of of ADP cana-3901 75 7 a a DET cana-3901 75 8 𝜎1near 𝜎1near NOUN cana-3901 75 9 - - PUNCT cana-3901 75 10 ring ring NOUN cana-3901 75 11 is be AUX cana-3901 75 12 also also ADV cana-3901 75 13 a a DET cana-3901 75 14 𝜎1near 𝜎1near ADJ cana-3901 75 15 - - PUNCT cana-3901 75 16 ring ring NOUN cana-3901 75 17 . . PUNCT cana-3901 76 1 proof proof NOUN cana-3901 76 2 the the DET cana-3901 76 3 proof proof NOUN cana-3901 76 4 is be AUX cana-3901 76 5 straight straight ADV cana-3901 76 6 forward forward ADV cana-3901 76 7 . . PUNCT cana-3901 77 1 proposition proposition NOUN cana-3901 77 2 5.6 5.6 NUM cana-3901 77 3 if if SCONJ cana-3901 77 4 i i PRON cana-3901 77 5 is be AUX cana-3901 77 6 an an DET cana-3901 77 7 ideal ideal NOUN cana-3901 77 8 of of ADP cana-3901 77 9 the the DET cana-3901 77 10 𝜎1near 𝜎1near NOUN cana-3901 77 11 - - PUNCT cana-3901 77 12 ring ring NOUN cana-3901 77 13 n n NOUN cana-3901 77 14 then then ADV cana-3901 78 1 n n CCONJ cana-3901 78 2 / / SYM cana-3901 79 1 i i PRON cana-3901 79 2 is be AUX cana-3901 79 3 also also ADV cana-3901 79 4 an an DET cana-3901 79 5 𝜎1near 𝜎1near ADJ cana-3901 79 6 - - PUNCT cana-3901 79 7 ring ring NOUN cana-3901 79 8 proof proof NOUN cana-3901 79 9 the the DET cana-3901 79 10 function function NOUN cana-3901 79 11 βˆ… βˆ… NOUN cana-3901 79 12 : : PUNCT cana-3901 79 13 𝑁 𝑁 PROPN cana-3901 79 14 β†’ β†’ SYM cana-3901 79 15 𝑁/𝐼 𝑁/𝐼 NOUN cana-3901 79 16 defined define VERB cana-3901 79 17 by by ADP cana-3901 79 18 βˆ…(π‘₯ βˆ…(π‘₯ NOUN cana-3901 79 19 ) ) PUNCT cana-3901 80 1 = = PUNCT cana-3901 80 2 𝐼 𝐼 PROPN cana-3901 80 3 + + CCONJ cana-3901 80 4 π‘₯ π‘₯ PROPN cana-3901 80 5 is be AUX cana-3901 80 6 an an DET cana-3901 80 7 epimorphism epimorphism NOUN cana-3901 80 8 . . PUNCT cana-3901 81 1 the the DET cana-3901 81 2 rest rest NOUN cana-3901 81 3 of of ADP cana-3901 81 4 the the DET cana-3901 81 5 proof proof NOUN cana-3901 81 6 is be AUX cana-3901 81 7 taken take VERB cana-3901 81 8 care care NOUN cana-3901 81 9 of of ADP cana-3901 81 10 by by ADP cana-3901 81 11 the the DET cana-3901 81 12 above above ADJ cana-3901 81 13 proposition proposition NOUN cana-3901 81 14 5.5 5.5 NUM cana-3901 81 15 . . PUNCT cana-3901 82 1 proposition proposition NOUN cana-3901 82 2 5.7 5.7 NUM cana-3901 82 3 every every DET cana-3901 82 4 𝜎1near 𝜎1near NUM cana-3901 82 5 - - PUNCT cana-3901 82 6 ring ring NOUN cana-3901 82 7 n n NOUN cana-3901 82 8 is be AUX cana-3901 82 9 isomorphic isomorphic ADJ cana-3901 82 10 to to ADP cana-3901 82 11 a a DET cana-3901 82 12 sub sub NOUN cana-3901 82 13 direct direct ADJ cana-3901 82 14 product product NOUN cana-3901 82 15 of of ADP cana-3901 82 16 sub sub NOUN cana-3901 82 17 directly directly ADV cana-3901 82 18 irreducible irreducible ADJ cana-3901 82 19 𝜎1near 𝜎1near NUM cana-3901 82 20 - - PUNCT cana-3901 82 21 ring ring NOUN cana-3901 82 22 . . PUNCT cana-3901 83 1 proof proof NOUN cana-3901 83 2 by by ADP cana-3901 83 3 theorem theorem NOUN cana-3901 83 4 1.62 1.62 NUM cana-3901 83 5 , , PUNCT cana-3901 83 6 p p NOUN cana-3901 83 7 26 26 NUM cana-3901 83 8 of of ADP cana-3901 83 9 pilz pilz PROPN cana-3901 83 10 [ [ X cana-3901 83 11 2 2 NUM cana-3901 83 12 ] ] PUNCT cana-3901 83 13 , , PUNCT cana-3901 83 14 n n X cana-3901 83 15 is be AUX cana-3901 83 16 isomorphic isomorphic ADJ cana-3901 83 17 to to ADP cana-3901 83 18 a a DET cana-3901 83 19 sub sub NOUN cana-3901 83 20 direct direct ADJ cana-3901 83 21 product product NOUN cana-3901 83 22 of of ADP cana-3901 83 23 sub sub NOUN cana-3901 83 24 directly directly ADV cana-3901 83 25 irreducible irreducible ADJ cana-3901 83 26 𝜎1 𝜎1 ADJ cana-3901 83 27 – – PUNCT cana-3901 83 28 near near ADP cana-3901 83 29 - - PUNCT cana-3901 83 30 ring.𝑁𝑖 ring.𝑁𝑖 NOUN cana-3901 83 31 ’s ’s NOUN cana-3901 83 32 and and CCONJ cana-3901 83 33 each each DET cana-3901 83 34 𝑁𝑖 𝑁𝑖 PROPN cana-3901 83 35 is be AUX cana-3901 83 36 a a DET cana-3901 83 37 homorphic homorphic ADJ cana-3901 83 38 image image NOUN cana-3901 83 39 of of ADP cana-3901 83 40 𝑁 𝑁 PROPN cana-3901 83 41 under under ADP cana-3901 83 42 the the DET cana-3901 83 43 multiplication multiplication NOUN cana-3901 83 44 πœ‹π‘–. πœ‹π‘–. ADP cana-3901 83 45 now now ADV cana-3901 83 46 the the DET cana-3901 83 47 desired desire VERB cana-3901 83 48 result result NOUN cana-3901 83 49 follows follow VERB cana-3901 83 50 from from ADP cana-3901 83 51 the the DET cana-3901 83 52 above above ADJ cana-3901 83 53 proposition proposition NOUN cana-3901 83 54 5.5 5.5 NUM cana-3901 83 55 . . PUNCT cana-3901 84 1 proposition proposition NOUN cana-3901 84 2 5.8 5.8 NUM cana-3901 84 3 let let VERB cana-3901 84 4 n n PRON cana-3901 84 5 be be AUX cana-3901 84 6 a a DET cana-3901 84 7 𝜎1near 𝜎1near NOUN cana-3901 84 8 - - PUNCT cana-3901 84 9 ring ring NOUN cana-3901 84 10 with with ADP cana-3901 84 11 a a DET cana-3901 84 12 mate mate NOUN cana-3901 84 13 function function NOUN cana-3901 84 14 𝑓. 𝑓. NOUN cana-3901 84 15 then then ADV cana-3901 84 16 we we PRON cana-3901 84 17 have have VERB cana-3901 84 18 i i PRON cana-3901 84 19 ) ) PUNCT cana-3901 84 20 𝐿 𝐿 PROPN cana-3901 84 21 = = SYM cana-3901 84 22 { { PUNCT cana-3901 84 23 0 0 NUM cana-3901 84 24 } } SYM cana-3901 84 25 ii ii NOUN cana-3901 84 26 ) ) PUNCT cana-3901 85 1 𝑁 𝑁 PROPN cana-3901 85 2 has have VERB cana-3901 85 3 ( ( PUNCT cana-3901 85 4 βˆ— βˆ— NOUN cana-3901 85 5 , , PUNCT cana-3901 85 6 ifp ifp ADJ cana-3901 85 7 ) ) PUNCT cana-3901 85 8 iii iii PROPN cana-3901 85 9 ) ) PUNCT cana-3901 85 10 𝐸 𝐸 PROPN cana-3901 85 11 βŠ† βŠ† NUM cana-3901 85 12 𝐢(𝑁 𝐢(𝑁 NUM cana-3901 85 13 ) ) PUNCT cana-3901 85 14 proof proof NOUN cana-3901 85 15 let let VERB cana-3901 85 16 n n PRON cana-3901 85 17 be be AUX cana-3901 85 18 a a DET cana-3901 85 19 𝜎1near 𝜎1near NOUN cana-3901 85 20 - - PUNCT cana-3901 85 21 ring ring NOUN cana-3901 85 22 then then ADV cana-3901 85 23 π‘₯𝑦2 π‘₯𝑦2 NOUN cana-3901 85 24 = = PUNCT cana-3901 85 25 𝑦π‘₯𝑦 𝑦π‘₯𝑦 VERB cana-3901 85 26 βˆ€ βˆ€ NOUN cana-3901 85 27 π‘₯ π‘₯ NOUN cana-3901 85 28 , , PUNCT cana-3901 85 29 𝑦 𝑦 NOUN cana-3901 85 30 ∈ ∈ NOUN cana-3901 85 31 𝑁. 𝑁. PROPN cana-3901 85 32 … … PUNCT cana-3901 85 33 … … PUNCT cana-3901 86 1 …… …… NOUN cana-3901 86 2 …… …… NOUN cana-3901 86 3 ( ( PUNCT cana-3901 86 4 1 1 NUM cana-3901 86 5 ) ) PUNCT cana-3901 86 6 since since SCONJ cana-3901 86 7 𝑓 𝑓 PRON cana-3901 86 8 is be AUX cana-3901 86 9 a a DET cana-3901 86 10 mate mate NOUN cana-3901 86 11 function function NOUN cana-3901 86 12 for for ADP cana-3901 86 13 n n PRON cana-3901 86 14 then then ADV cana-3901 86 15 π‘₯ π‘₯ PRON cana-3901 86 16 = = PUNCT cana-3901 86 17 π‘₯𝑓(π‘₯)π‘₯ π‘₯𝑓(π‘₯)π‘₯ VERB cana-3901 86 18 ∈ ∈ PROPN cana-3901 86 19 π‘₯𝑁π‘₯ π‘₯𝑁π‘₯ NOUN cana-3901 86 20 for for ADP cana-3901 86 21 all all PRON cana-3901 86 22 π‘₯ π‘₯ DET cana-3901 86 23 ∈ ∈ NOUN cana-3901 87 1 𝑁 𝑁 PROPN cana-3901 87 2 ∴ ∴ NOUN cana-3901 87 3 π‘₯ π‘₯ X cana-3901 87 4 = = PUNCT cana-3901 87 5 π‘₯𝑛π‘₯ π‘₯𝑛π‘₯ NOUN cana-3901 87 6 for for ADP cana-3901 87 7 some some DET cana-3901 87 8 n. n. NOUN cana-3901 87 9 … … PUNCT cana-3901 87 10 … … PUNCT cana-3901 87 11 … … PUNCT cana-3901 87 12 …… …… NOUN cana-3901 87 13 …… …… NOUN cana-3901 87 14 ( ( PUNCT cana-3901 87 15 2 2 NUM cana-3901 87 16 ) ) PUNCT cana-3901 87 17 i)for i)for PROPN cana-3901 87 18 𝑛 𝑛 PROPN cana-3901 87 19 , , PUNCT cana-3901 87 20 π‘₯ π‘₯ PRON cana-3901 87 21 ∈ ∈ PROPN cana-3901 87 22 𝑁 𝑁 NOUN cana-3901 87 23 , , PUNCT cana-3901 87 24 𝑛π‘₯2 𝑛π‘₯2 VERB cana-3901 87 25 = = SYM cana-3901 87 26 π‘₯𝑛π‘₯ π‘₯𝑛π‘₯ PROPN cana-3901 88 1 [ [ PUNCT cana-3901 88 2 by by ADP cana-3901 88 3 equation equation NOUN cana-3901 88 4 ( ( PUNCT cana-3901 88 5 1 1 NUM cana-3901 88 6 ) ) PUNCT cana-3901 88 7 ] ] PUNCT cana-3901 89 1 = = PUNCT cana-3901 89 2 π‘₯ π‘₯ X cana-3901 90 1 [ [ X cana-3901 90 2 by by ADP cana-3901 90 3 equation equation NOUN cana-3901 90 4 ( ( PUNCT cana-3901 90 5 2 2 NUM cana-3901 90 6 ) ) PUNCT cana-3901 90 7 ] ] PUNCT cana-3901 90 8 suppose suppose VERB cana-3901 90 9 π‘₯2 π‘₯2 NOUN cana-3901 90 10 = = SYM cana-3901 90 11 0 0 PUNCT cana-3901 91 1 clearly clearly ADV cana-3901 91 2 then then ADV cana-3901 91 3 π‘₯ π‘₯ VERB cana-3901 91 4 = = SYM cana-3901 91 5 0 0 NUM cana-3901 91 6 . . PUNCT cana-3901 92 1 [ [ X cana-3901 92 2 since since SCONJ cana-3901 92 3 n n PROPN cana-3901 92 4 is be AUX cana-3901 92 5 zero zero NUM cana-3901 92 6 symmetric symmetric NOUN cana-3901 92 7 ] ] PUNCT cana-3901 92 8 . . PUNCT cana-3901 93 1 then then ADV cana-3901 93 2 𝑅(2 𝑅(2 NUM cana-3901 93 3 ) ) PUNCT cana-3901 93 4 guarantees guarantee VERB cana-3901 93 5 that that SCONJ cana-3901 93 6 𝐿 𝐿 PROPN cana-3901 93 7 = = SYM cana-3901 93 8 { { PUNCT cana-3901 93 9 0 0 NUM cana-3901 93 10 } } PUNCT cana-3901 93 11 ( ( PUNCT cana-3901 93 12 ii ii NOUN cana-3901 93 13 ) ) PUNCT cana-3901 93 14 by by ADP cana-3901 93 15 i i X cana-3901 93 16 ) ) PUNCT cana-3901 93 17 𝐿 𝐿 PROPN cana-3901 93 18 = = SYM cana-3901 93 19 { { PUNCT cana-3901 93 20 0 0 NUM cana-3901 93 21 } } PUNCT cana-3901 93 22 . . PUNCT cana-3901 94 1 now now ADV cana-3901 94 2 r(4 r(4 PROPN cana-3901 94 3 ) ) PUNCT cana-3901 94 4 guarantees guarantee VERB cana-3901 94 5 that that SCONJ cana-3901 94 6 n n PRON cana-3901 94 7 has have VERB cana-3901 94 8 ( ( PUNCT cana-3901 94 9 βˆ— βˆ— NOUN cana-3901 94 10 , , PUNCT cana-3901 94 11 ifp ifp NOUN cana-3901 94 12 ) ) PUNCT cana-3901 94 13 ( ( PUNCT cana-3901 95 1 iii)let iii)let NOUN cana-3901 95 2 𝑒 𝑒 PROPN cana-3901 95 3 ∈ ∈ NOUN cana-3901 95 4 𝐸. 𝐸. PROPN cana-3901 95 5 since since SCONJ cana-3901 95 6 n n NUM cana-3901 95 7 is be AUX cana-3901 95 8 a a DET cana-3901 95 9 𝜎1near 𝜎1near ADJ cana-3901 95 10 βˆ’ βˆ’ PROPN cana-3901 95 11 ring ring NOUN cana-3901 95 12 , , PUNCT cana-3901 95 13 𝑛𝑒2 𝑛𝑒2 X cana-3901 95 14 = = SYM cana-3901 95 15 𝑒𝑛𝑒 𝑒𝑛𝑒 VERB cana-3901 95 16 ⟹ ⟹ PUNCT cana-3901 95 17 𝑛𝑒 𝑛𝑒 PROPN cana-3901 96 1 = = PUNCT cana-3901 96 2 𝑒𝑛𝑒for 𝑒𝑛𝑒for ADP cana-3901 96 3 all all DET cana-3901 96 4 𝑛 𝑛 PRON cana-3901 96 5 in in ADP cana-3901 96 6 𝑁 𝑁 PROPN cana-3901 96 7 … … PUNCT cana-3901 96 8 … … PUNCT cana-3901 96 9 . . PUNCT cana-3901 96 10 . . PUNCT cana-3901 97 1 … … PUNCT cana-3901 97 2 … … PUNCT cana-3901 97 3 ( ( PUNCT cana-3901 97 4 3 3 X cana-3901 97 5 ) ) PUNCT cana-3901 97 6 communications communication NOUN cana-3901 97 7 on on ADP cana-3901 97 8 applied apply VERB cana-3901 97 9 nonlinear nonlinear ADJ cana-3901 97 10 analysis analysis NOUN cana-3901 97 11 issn issn NOUN cana-3901 97 12 : : PUNCT cana-3901 97 13 1074 1074 NUM cana-3901 97 14 - - PUNCT cana-3901 97 15 133x 133x NUM cana-3901 97 16 vol vol NOUN cana-3901 97 17 32 32 NUM cana-3901 97 18 no no NOUN cana-3901 97 19 . . PUNCT cana-3901 98 1 9s 9s NUM cana-3901 98 2 ( ( PUNCT cana-3901 98 3 2025 2025 NUM cana-3901 98 4 ) ) PUNCT cana-3901 98 5 358 358 NUM cana-3901 98 6 https://internationalpubls.com https://internationalpubls.com X cana-3901 99 1 also also ADV cana-3901 99 2 we we PRON cana-3901 99 3 have have AUX cana-3901 99 4 ( ( PUNCT cana-3901 99 5 𝑒𝑛𝑒 𝑒𝑛𝑒 VERB cana-3901 99 6 βˆ’ βˆ’ NOUN cana-3901 99 7 𝑒𝑛)𝑒 𝑒𝑛)𝑒 X cana-3901 99 8 = = SYM cana-3901 99 9 0 0 NUM cana-3901 100 1 this this PRON cana-3901 100 2 implies imply VERB cana-3901 100 3 𝑒(𝑒𝑛𝑒 𝑒(𝑒𝑛𝑒 INTJ cana-3901 100 4 βˆ’ βˆ’ X cana-3901 100 5 𝑒𝑛 𝑒𝑛 NOUN cana-3901 100 6 ) ) PUNCT cana-3901 100 7 = = SYM cana-3901 100 8 0 0 NUM cana-3901 100 9 and and CCONJ cana-3901 100 10 𝑒𝑛(𝑒𝑛𝑒 𝑒𝑛(𝑒𝑛𝑒 PROPN cana-3901 100 11 βˆ’ βˆ’ PROPN cana-3901 100 12 𝑒𝑛 𝑒𝑛 NOUN cana-3901 100 13 ) ) PUNCT cana-3901 100 14 = = PUNCT cana-3901 100 15 0 0 PUNCT cana-3901 101 1 [ [ X cana-3901 101 2 𝑏𝑦(𝑖𝑖 𝑏𝑦(𝑖𝑖 NOUN cana-3901 101 3 ) ) PUNCT cana-3901 101 4 ] ] PUNCT cana-3901 101 5 also also ADV cana-3901 101 6 𝑒𝑛𝑒(𝑒𝑛𝑒 𝑒𝑛𝑒(𝑒𝑛𝑒 VERB cana-3901 101 7 βˆ’ βˆ’ NUM cana-3901 101 8 𝑒𝑛 𝑒𝑛 NOUN cana-3901 101 9 ) ) PUNCT cana-3901 101 10 = = SYM cana-3901 101 11 𝑒𝑛. 𝑒𝑛. ADP cana-3901 101 12 0 0 NUM cana-3901 101 13 = = SYM cana-3901 101 14 0 0 PUNCT cana-3901 102 1 [ [ X cana-3901 102 2 since since SCONJ cana-3901 102 3 n n PROPN cana-3901 102 4 is be AUX cana-3901 102 5 zero zero NUM cana-3901 102 6 symmetric symmetric NOUN cana-3901 102 7 ] ] PUNCT cana-3901 102 8 now now ADV cana-3901 102 9 𝑒𝑛𝑒(𝑒𝑛𝑒 𝑒𝑛𝑒(𝑒𝑛𝑒 VERB cana-3901 102 10 βˆ’ βˆ’ NUM cana-3901 102 11 𝑒𝑛 𝑒𝑛 NOUN cana-3901 102 12 ) ) PUNCT cana-3901 102 13 βˆ’ βˆ’ PROPN cana-3901 102 14 𝑒𝑛(𝑒𝑛𝑒 𝑒𝑛(𝑒𝑛𝑒 PROPN cana-3901 102 15 βˆ’ βˆ’ X cana-3901 102 16 𝑒𝑛 𝑒𝑛 NOUN cana-3901 102 17 ) ) PUNCT cana-3901 102 18 = = SYM cana-3901 103 1 0 0 X cana-3901 103 2 . . PUNCT cana-3901 104 1 consequently consequently ADV cana-3901 104 2 , , PUNCT cana-3901 104 3 ( ( PUNCT cana-3901 104 4 𝑒𝑛𝑒 𝑒𝑛𝑒 PROPN cana-3901 104 5 βˆ’ βˆ’ PROPN cana-3901 104 6 𝑒𝑛)2 𝑒𝑛)2 PROPN cana-3901 104 7 = = PROPN cana-3901 104 8 0 0 PROPN cana-3901 105 1 and and CCONJ cana-3901 105 2 ( ( PUNCT cana-3901 105 3 i i NOUN cana-3901 105 4 ) ) PUNCT cana-3901 105 5 guarantees guarantee VERB cana-3901 105 6 𝑒𝑛𝑒 𝑒𝑛𝑒 PRON cana-3901 105 7 βˆ’ βˆ’ PUNCT cana-3901 105 8 𝑒𝑛 𝑒𝑛 NOUN cana-3901 105 9 = = NOUN cana-3901 105 10 0 0 PROPN cana-3901 105 11 . . PUNCT cana-3901 106 1 therefore therefore ADV cana-3901 106 2 𝑒𝑛𝑒 𝑒𝑛𝑒 VERB cana-3901 106 3 = = PUNCT cana-3901 106 4 𝑒𝑛 𝑒𝑛 NOUN cana-3901 106 5 for for ADP cana-3901 106 6 all all DET cana-3901 106 7 n n PRON cana-3901 106 8 in in ADP cana-3901 106 9 n n PRON cana-3901 106 10 … … PUNCT cana-3901 106 11 … … PUNCT cana-3901 106 12 … … PUNCT cana-3901 106 13 … … PUNCT cana-3901 106 14 … … SYM cana-3901 106 15 . . NUM cana-3901 106 16 … … PUNCT cana-3901 106 17 .(4 .(4 NUM cana-3901 106 18 ) ) PUNCT cana-3901 106 19 from from ADP cana-3901 106 20 equations equation NOUN cana-3901 106 21 ( ( PUNCT cana-3901 106 22 3 3 NUM cana-3901 106 23 ) ) PUNCT cana-3901 106 24 and and CCONJ cana-3901 106 25 ( ( PUNCT cana-3901 106 26 4 4 X cana-3901 106 27 ) ) PUNCT cana-3901 106 28 we we PRON cana-3901 106 29 get get VERB cana-3901 106 30 𝑒𝑛 𝑒𝑛 ADP cana-3901 106 31 = = PUNCT cana-3901 106 32 𝑛𝑒 𝑛𝑒 PROPN cana-3901 106 33 for for ADP cana-3901 106 34 all all DET cana-3901 106 35 n n NOUN cana-3901 106 36 in in ADP cana-3901 106 37 n. n. NOUN cana-3901 106 38 thus thus ADV cana-3901 106 39 𝐸 𝐸 PROPN cana-3901 106 40 βŠ† βŠ† NUM cana-3901 106 41 𝐢(𝑁 𝐢(𝑁 NUM cana-3901 106 42 ) ) PUNCT cana-3901 106 43 proposition proposition NOUN cana-3901 106 44 5.9 5.9 NUM cana-3901 106 45 let let VERB cana-3901 106 46 𝑁 𝑁 PROPN cana-3901 106 47 be be AUX cana-3901 106 48 a a DET cana-3901 106 49 pseudo pseudo NOUN cana-3901 106 50 commutative commutative ADJ cana-3901 106 51 near near ADP cana-3901 106 52 -ring -ring PROPN cana-3901 106 53 with with ADP cana-3901 106 54 right right ADJ cana-3901 106 55 identity identity NOUN cana-3901 106 56 . . PUNCT cana-3901 107 1 then then ADV cana-3901 107 2 if if SCONJ cana-3901 107 3 𝑁 𝑁 PROPN cana-3901 107 4 is be AUX cana-3901 107 5 a a DET cana-3901 107 6 𝜎1near 𝜎1near NOUN cana-3901 107 7 - - PUNCT cana-3901 107 8 ring ring NOUN cana-3901 107 9 then then ADV cana-3901 107 10 for for ADP cana-3901 107 11 any any DET cana-3901 107 12 π‘Ž π‘Ž NOUN cana-3901 107 13 , , PUNCT cana-3901 107 14 𝑏 𝑏 NOUN cana-3901 107 15 in in ADP cana-3901 107 16 𝑁 𝑁 PROPN cana-3901 107 17 , , PUNCT cana-3901 107 18 π‘Žπ‘ π‘Žπ‘ ADP cana-3901 107 19 = = SYM cana-3901 107 20 0 0 NUM cana-3901 107 21 implies imply VERB cana-3901 107 22 π‘π‘Ž π‘π‘Ž X cana-3901 107 23 = = SYM cana-3901 107 24 0 0 NUM cana-3901 107 25 proof proof NOUN cana-3901 107 26 let let VERB cana-3901 107 27 𝑁 𝑁 PROPN cana-3901 107 28 be be AUX cana-3901 107 29 a a DET cana-3901 107 30 pseudo pseudo NOUN cana-3901 107 31 commutative commutative ADJ cana-3901 107 32 near near ADP cana-3901 107 33 - - PUNCT cana-3901 107 34 ring ring NOUN cana-3901 107 35 . . PUNCT cana-3901 108 1 then then ADV cana-3901 108 2 π‘₯𝑦𝑧 π‘₯𝑦𝑧 VERB cana-3901 108 3 = = PUNCT cana-3901 108 4 𝑧𝑦π‘₯ 𝑧𝑦π‘₯ NOUN cana-3901 108 5 for for ADP cana-3901 108 6 all all PRON cana-3901 108 7 π‘₯ π‘₯ PROPN cana-3901 108 8 , , PUNCT cana-3901 108 9 𝑦 𝑦 NOUN cana-3901 108 10 , , PUNCT cana-3901 108 11 𝑧 𝑧 PRON cana-3901 108 12 ∈ ∈ NOUN cana-3901 108 13 𝑁 𝑁 PROPN cana-3901 108 14 … … PUNCT cana-3901 108 15 .. .. PUNCT cana-3901 108 16 … … PUNCT cana-3901 108 17 . . PUNCT cana-3901 109 1 ( ( PUNCT cana-3901 109 2 1 1 X cana-3901 109 3 ) ) PUNCT cana-3901 109 4 now now ADV cana-3901 109 5 r r NOUN cana-3901 109 6 ( ( PUNCT cana-3901 109 7 5 5 NUM cana-3901 109 8 ) ) PUNCT cana-3901 109 9 guarantees guarantee VERB cana-3901 109 10 that that SCONJ cana-3901 109 11 𝑁 𝑁 PROPN cana-3901 109 12 is be AUX cana-3901 109 13 weak weak ADJ cana-3901 109 14 commutative commutative ADJ cana-3901 109 15 . . PUNCT cana-3901 110 1 ∴ ∴ NOUN cana-3901 110 2 π‘₯𝑦𝑧 π‘₯𝑦𝑧 NOUN cana-3901 110 3 = = SYM cana-3901 110 4 π‘₯𝑧𝑦 π‘₯𝑧𝑦 VERB cana-3901 110 5 for for ADP cana-3901 110 6 all all DET cana-3901 110 7 π‘₯ π‘₯ PROPN cana-3901 110 8 , , PUNCT cana-3901 110 9 𝑦 𝑦 NOUN cana-3901 110 10 , , PUNCT cana-3901 110 11 𝑧 𝑧 PRON cana-3901 110 12 ∈ ∈ NOUN cana-3901 110 13 𝑁 𝑁 PROPN cana-3901 110 14 … … SYM cana-3901 110 15 … … NUM cana-3901 110 16 . . PUNCT cana-3901 110 17 … … PUNCT cana-3901 110 18 .. .. PUNCT cana-3901 110 19 (2 (2 NUM cana-3901 110 20 ) ) PUNCT cana-3901 110 21 now now ADV cana-3901 110 22 , , PUNCT cana-3901 110 23 ( ( PUNCT cana-3901 110 24 π‘₯π‘Žπ‘₯)(𝑦𝑏𝑦 π‘₯π‘Žπ‘₯)(𝑦𝑏𝑦 ADV cana-3901 110 25 ) ) PUNCT cana-3901 110 26 = = SYM cana-3901 111 1 π‘Žπ‘₯2𝑏𝑦2 π‘Žπ‘₯2𝑏𝑦2 NOUN cana-3901 111 2 π‘₯π‘Žπ‘₯ π‘₯π‘Žπ‘₯ NOUN cana-3901 111 3 𝑦𝑏𝑦 𝑦𝑏𝑦 NOUN cana-3901 111 4 = = PROPN cana-3901 111 5 π‘Žπ‘₯π‘₯𝑏𝑦𝑦 π‘Žπ‘₯π‘₯𝑏𝑦𝑦 PROPN cana-3901 111 6 π‘₯π‘Ž(π‘₯𝑦𝑏)𝑦 π‘₯π‘Ž(π‘₯𝑦𝑏)𝑦 PUNCT cana-3901 111 7 = = SYM cana-3901 111 8 π‘Ž(π‘₯π‘₯𝑏)𝑦𝑦 π‘Ž(π‘₯π‘₯𝑏)𝑦𝑦 NOUN cana-3901 111 9 π‘₯π‘Ž(𝑏𝑦π‘₯)𝑦 π‘₯π‘Ž(𝑏𝑦π‘₯)𝑦 PROPN cana-3901 111 10 = = SYM cana-3901 111 11 π‘Ž(𝑏π‘₯π‘₯)𝑦𝑦 π‘Ž(𝑏π‘₯π‘₯)𝑦𝑦 PRON cana-3901 112 1 [ [ PUNCT cana-3901 112 2 by by ADP cana-3901 112 3 equation equation NOUN cana-3901 112 4 ( ( PUNCT cana-3901 112 5 1 1 NUM cana-3901 112 6 ) ) PUNCT cana-3901 112 7 ] ] PUNCT cana-3901 112 8 ( ( PUNCT cana-3901 112 9 π‘₯π‘Žπ‘)𝑦π‘₯𝑦 π‘₯π‘Žπ‘)𝑦π‘₯𝑦 NOUN cana-3901 112 10 = = SYM cana-3901 112 11 π‘Žπ‘ π‘Žπ‘ PROPN cana-3901 112 12 π‘₯π‘₯𝑦 π‘₯π‘₯𝑦 ADJ cana-3901 112 13 𝑦 𝑦 DET cana-3901 112 14 π‘π‘Žπ‘₯𝑦π‘₯𝑦 π‘π‘Žπ‘₯𝑦π‘₯𝑦 NOUN cana-3901 112 15 = = PUNCT cana-3901 112 16 π‘Žπ‘(π‘₯π‘₯𝑦)𝑦 π‘Žπ‘(π‘₯π‘₯𝑦)𝑦 X cana-3901 113 1 [ [ X cana-3901 113 2 by by ADP cana-3901 113 3 equation equation NOUN cana-3901 113 4 ( ( PUNCT cana-3901 113 5 1 1 NUM cana-3901 113 6 ) ) PUNCT cana-3901 113 7 ] ] PUNCT cana-3901 113 8 π‘π‘Ž π‘π‘Ž X cana-3901 113 9 π‘₯𝑦 π‘₯𝑦 NOUN cana-3901 113 10 π‘₯𝑦 π‘₯𝑦 NOUN cana-3901 113 11 = = SYM cana-3901 113 12 π‘Žπ‘ π‘Žπ‘ PART cana-3901 113 13 π‘₯ π‘₯ NOUN cana-3901 113 14 𝑦π‘₯𝑦 𝑦π‘₯𝑦 NOUN cana-3901 114 1 [ [ PUNCT cana-3901 114 2 by by ADP cana-3901 114 3 equation equation NOUN cana-3901 114 4 ( ( PUNCT cana-3901 114 5 2 2 NUM cana-3901 114 6 ) ) PUNCT cana-3901 114 7 ] ] PUNCT cana-3901 114 8 π‘π‘Ž π‘π‘Ž X cana-3901 114 9 = = X cana-3901 114 10 π‘Žπ‘. π‘Žπ‘. NOUN cana-3901 114 11 since since SCONJ cana-3901 114 12 π‘Žπ‘ π‘Žπ‘ PROPN cana-3901 114 13 = = SYM cana-3901 114 14 0 0 NUM cana-3901 114 15 it it PRON cana-3901 114 16 follows follow VERB cana-3901 114 17 that that PRON cana-3901 114 18 π‘π‘Ž π‘π‘Ž PUNCT cana-3901 114 19 = = NOUN cana-3901 114 20 0 0 X cana-3901 114 21 . . PUNCT cana-3901 115 1 proposition proposition NOUN cana-3901 115 2 5.10 5.10 NUM cana-3901 115 3 n n NOUN cana-3901 115 4 is be AUX cana-3901 115 5 a a DET cana-3901 115 6 𝜎1 𝜎1 ADJ cana-3901 115 7 near near ADP cana-3901 115 8 - - PUNCT cana-3901 115 9 ring ring NOUN cana-3901 115 10 if if SCONJ cana-3901 116 1 and and CCONJ cana-3901 116 2 only only ADV cana-3901 116 3 if if SCONJ cana-3901 116 4 every every PRON cana-3901 116 5 x x NOUN cana-3901 116 6 in in ADP cana-3901 116 7 n n NUM cana-3901 116 8 can can AUX cana-3901 116 9 be be AUX cana-3901 116 10 written write VERB cana-3901 116 11 as as ADP cana-3901 116 12 π‘₯𝑦2 π‘₯𝑦2 NOUN cana-3901 116 13 = = SYM cana-3901 116 14 𝑒 𝑒 PROPN cana-3901 116 15 + + NOUN cana-3901 116 16 𝑣 𝑣 ADP cana-3901 116 17 where where SCONJ cana-3901 116 18 𝑒 𝑒 X cana-3901 116 19 πœ– πœ– PROPN cana-3901 116 20 π‘π‘œ π‘π‘œ PROPN cana-3901 116 21 and and CCONJ cana-3901 116 22 𝑣 𝑣 PRON cana-3901 116 23 πœ– πœ– X cana-3901 117 1 𝑁𝑐 𝑁𝑐 PROPN cana-3901 117 2 and and CCONJ cana-3901 117 3 𝑒 𝑒 X cana-3901 117 4 = = SYM cana-3901 117 5 𝑦0(𝑛𝑦 𝑦0(𝑛𝑦 PROPN cana-3901 117 6 + + CCONJ cana-3901 117 7 π‘š π‘š NOUN cana-3901 117 8 ) ) PUNCT cana-3901 117 9 βˆ’ βˆ’ NOUN cana-3901 117 10 𝑦0 𝑦0 PROPN cana-3901 117 11 π‘₯ π‘₯ PROPN cana-3901 117 12 𝑦𝑐 𝑦𝑐 PROPN cana-3901 117 13 , , PUNCT cana-3901 117 14 𝑣 𝑣 X cana-3901 117 15 = = PUNCT cana-3901 117 16 𝑦0π‘₯ 𝑦0π‘₯ NOUN cana-3901 117 17 𝑦𝑐 𝑦𝑐 PROPN cana-3901 117 18 + + CCONJ cana-3901 117 19 𝑦𝑐 𝑦𝑐 PROPN cana-3901 117 20 , , PUNCT cana-3901 117 21 𝑦 𝑦 NOUN cana-3901 117 22 = = X cana-3901 117 23 𝑦0 𝑦0 NOUN cana-3901 117 24 + + CCONJ cana-3901 117 25 𝑦𝑐 𝑦𝑐 PROPN cana-3901 117 26 πœ– πœ– X cana-3901 117 27 𝑁0⨁𝑁𝑐 𝑁0⨁𝑁𝑐 NOUN cana-3901 117 28 where where SCONJ cana-3901 117 29 𝑦0 𝑦0 NOUN cana-3901 117 30 , , PUNCT cana-3901 117 31 𝑛 𝑛 DET cana-3901 117 32 ∈ ∈ PROPN cana-3901 117 33 𝑁0 𝑁0 VERB cana-3901 117 34 , , PUNCT cana-3901 117 35 𝑦𝑐 𝑦𝑐 PROPN cana-3901 117 36 , , PUNCT cana-3901 117 37 π‘š π‘š PROPN cana-3901 117 38 ∈ ∈ NOUN cana-3901 117 39 𝑁𝑐. 𝑁𝑐. PROPN cana-3901 117 40 further far ADV cana-3901 117 41 more more ADV cana-3901 117 42 𝑒 𝑒 ADP cana-3901 117 43 ∈ ∈ PROPN cana-3901 117 44 𝑁0 𝑁0 ADJ cana-3901 117 45 , , PUNCT cana-3901 117 46 𝑣 𝑣 PRON cana-3901 117 47 ∈ ∈ NOUN cana-3901 117 48 𝑁𝑐. 𝑁𝑐. PROPN cana-3901 117 49 proof proof NOUN cana-3901 117 50 for for ADP cana-3901 117 51 the the DET cana-3901 117 52 β€˜ ' PUNCT cana-3901 117 53 only only ADJ cana-3901 117 54 if if SCONJ cana-3901 117 55 ’ ' PUNCT cana-3901 117 56 part part NOUN cana-3901 117 57 , , PUNCT cana-3901 117 58 let let VERB cana-3901 117 59 𝑦 𝑦 NOUN cana-3901 117 60 ∈ ∈ NOUN cana-3901 117 61 𝑁. 𝑁. PROPN cana-3901 117 62 since since SCONJ cana-3901 117 63 n n NUM cana-3901 117 64 is be AUX cana-3901 117 65 𝜎1 𝜎1 ADJ cana-3901 117 66 there there ADV cana-3901 117 67 exist exist VERB cana-3901 117 68 π‘₯ π‘₯ NOUN cana-3901 117 69 in in ADP cana-3901 117 70 n n CCONJ cana-3901 118 1 such such ADJ cana-3901 118 2 that that DET cana-3901 118 3 π‘₯𝑦2 π‘₯𝑦2 NOUN cana-3901 118 4 = = SYM cana-3901 118 5 𝑦π‘₯𝑦. 𝑦π‘₯𝑦. X cana-3901 118 6 by by ADP cana-3901 118 7 using use VERB cana-3901 118 8 pierce pierce PROPN cana-3901 118 9 decomposition decomposition NOUN cana-3901 118 10 we we PRON cana-3901 118 11 can can AUX cana-3901 118 12 write write VERB cana-3901 118 13 π‘₯ π‘₯ NOUN cana-3901 118 14 = = PUNCT cana-3901 118 15 𝑛 𝑛 PROPN cana-3901 118 16 + + CCONJ cana-3901 118 17 π‘š π‘š PROPN cana-3901 118 18 and and CCONJ cana-3901 118 19 𝑦 𝑦 NOUN cana-3901 118 20 = = X cana-3901 118 21 𝑦0 𝑦0 NOUN cana-3901 118 22 + + CCONJ cana-3901 118 23 𝑦𝑐 𝑦𝑐 ADP cana-3901 118 24 where where SCONJ cana-3901 118 25 π‘₯ π‘₯ PRON cana-3901 118 26 ∈ ∈ PROPN cana-3901 118 27 𝑁 𝑁 PROPN cana-3901 118 28 , , PUNCT cana-3901 118 29 𝑛 𝑛 PROPN cana-3901 118 30 , , PUNCT cana-3901 118 31 𝑦0 𝑦0 NOUN cana-3901 118 32 ∈ ∈ PROPN cana-3901 118 33 𝑁0 𝑁0 ADJ cana-3901 118 34 and and CCONJ cana-3901 118 35 π‘š π‘š NOUN cana-3901 118 36 , , PUNCT cana-3901 118 37 𝑦𝑐 𝑦𝑐 PROPN cana-3901 118 38 ∈ ∈ PROPN cana-3901 119 1 𝑁𝑐 𝑁𝑐 PROPN cana-3901 119 2 now now ADV cana-3901 119 3 π‘₯𝑦2 π‘₯𝑦2 ADV cana-3901 119 4 = = SYM cana-3901 119 5 ( ( PUNCT cana-3901 119 6 𝑦0 𝑦0 NOUN cana-3901 119 7 + + CCONJ cana-3901 119 8 𝑦𝑐)(𝑛 𝑦𝑐)(𝑛 NOUN cana-3901 119 9 + + CCONJ cana-3901 119 10 π‘š)𝑦 π‘š)𝑦 NOUN cana-3901 119 11 communications communication NOUN cana-3901 119 12 on on ADP cana-3901 119 13 applied apply VERB cana-3901 119 14 nonlinear nonlinear ADJ cana-3901 119 15 analysis analysis NOUN cana-3901 119 16 issn issn NOUN cana-3901 119 17 : : PUNCT cana-3901 119 18 1074 1074 NUM cana-3901 119 19 - - PUNCT cana-3901 119 20 133x 133x NUM cana-3901 119 21 vol vol NOUN cana-3901 119 22 32 32 NUM cana-3901 119 23 no no NOUN cana-3901 119 24 . . PUNCT cana-3901 120 1 9s 9s NUM cana-3901 120 2 ( ( PUNCT cana-3901 120 3 2025 2025 NUM cana-3901 120 4 ) ) PUNCT cana-3901 120 5 359 359 NUM cana-3901 121 1 https://internationalpubls.com https://internationalpubls.com X cana-3901 121 2 = = SYM cana-3901 121 3 ( ( PUNCT cana-3901 121 4 𝑦0 𝑦0 NOUN cana-3901 121 5 + + CCONJ cana-3901 121 6 𝑦𝑐)(𝑛𝑦 𝑦𝑐)(𝑛𝑦 PUNCT cana-3901 121 7 + + CCONJ cana-3901 121 8 π‘šπ‘¦ π‘šπ‘¦ X cana-3901 121 9 ) ) PUNCT cana-3901 121 10 = = SYM cana-3901 121 11 ( ( PUNCT cana-3901 121 12 𝑦0 𝑦0 NOUN cana-3901 121 13 + + CCONJ cana-3901 121 14 𝑦𝑐)(𝑛𝑦 𝑦𝑐)(𝑛𝑦 PUNCT cana-3901 121 15 + + CCONJ cana-3901 121 16 π‘š π‘š X cana-3901 121 17 ) ) PUNCT cana-3901 122 1 [ [ PUNCT cana-3901 122 2 𝑠𝑖𝑛𝑐𝑒 𝑠𝑖𝑛𝑐𝑒 X cana-3901 122 3 π‘š π‘š ADP cana-3901 122 4 ∈ ∈ PROPN cana-3901 122 5 𝑁𝑐 𝑁𝑐 PROPN cana-3901 122 6 ] ] X cana-3901 122 7 = = PUNCT cana-3901 122 8 𝑦0(𝑛𝑦 𝑦0(𝑛𝑦 PROPN cana-3901 122 9 + + CCONJ cana-3901 122 10 π‘š π‘š X cana-3901 122 11 ) ) PUNCT cana-3901 122 12 + + NUM cana-3901 122 13 𝑦𝑐(𝑛𝑦 𝑦𝑐(𝑛𝑦 NOUN cana-3901 122 14 + + CCONJ cana-3901 122 15 π‘š π‘š X cana-3901 122 16 ) ) PUNCT cana-3901 122 17 = = SYM cana-3901 122 18 𝑦0(𝑛𝑦 𝑦0(𝑛𝑦 PROPN cana-3901 122 19 + + CCONJ cana-3901 122 20 π‘š π‘š X cana-3901 122 21 ) ) PUNCT cana-3901 122 22 + + NUM cana-3901 122 23 𝑦𝑐 𝑦𝑐 PROPN cana-3901 123 1 [ [ X cana-3901 123 2 𝑠𝑖𝑛𝑐𝑒 𝑠𝑖𝑛𝑐𝑒 X cana-3901 123 3 𝑦𝑐 𝑦𝑐 ADP cana-3901 123 4 ∈ ∈ PROPN cana-3901 123 5 𝑁𝑐 𝑁𝑐 PROPN cana-3901 123 6 ] ] X cana-3901 123 7 = = PUNCT cana-3901 123 8 𝑦0(𝑛𝑦 𝑦0(𝑛𝑦 PROPN cana-3901 123 9 + + CCONJ cana-3901 123 10 π‘š π‘š NOUN cana-3901 123 11 ) ) PUNCT cana-3901 123 12 βˆ’ βˆ’ NOUN cana-3901 123 13 𝑦0 𝑦0 NOUN cana-3901 123 14 π‘₯ π‘₯ PROPN cana-3901 123 15 𝑦𝑐 𝑦𝑐 NOUN cana-3901 123 16 + + CCONJ cana-3901 123 17 𝑦0 𝑦0 NOUN cana-3901 123 18 π‘₯ π‘₯ PROPN cana-3901 123 19 𝑦𝑐 𝑦𝑐 PROPN cana-3901 123 20 + + CCONJ cana-3901 123 21 𝑦𝑐 𝑦𝑐 PROPN cana-3901 123 22 = = PUNCT cana-3901 123 23 𝑒 𝑒 PROPN cana-3901 123 24 + + NOUN cana-3901 123 25 𝑣 𝑣 ADP cana-3901 123 26 where where SCONJ cana-3901 123 27 𝑒 𝑒 X cana-3901 123 28 = = SYM cana-3901 123 29 𝑦0(𝑛𝑦 𝑦0(𝑛𝑦 PROPN cana-3901 123 30 + + CCONJ cana-3901 123 31 π‘š π‘š NOUN cana-3901 123 32 ) ) PUNCT cana-3901 123 33 βˆ’ βˆ’ PROPN cana-3901 123 34 𝑦0 𝑦0 NOUN cana-3901 123 35 π‘₯ π‘₯ PROPN cana-3901 123 36 𝑦𝑐 𝑦𝑐 PROPN cana-3901 123 37 and and CCONJ cana-3901 123 38 𝑣 𝑣 X cana-3901 123 39 = = PUNCT cana-3901 123 40 𝑦0 𝑦0 PROPN cana-3901 123 41 π‘₯ π‘₯ X cana-3901 123 42 𝑦𝑐 𝑦𝑐 PROPN cana-3901 123 43 + + CCONJ cana-3901 123 44 𝑦𝑐 𝑦𝑐 NOUN cana-3901 123 45 now now ADV cana-3901 123 46 , , PUNCT cana-3901 123 47 𝑒. 𝑒. NOUN cana-3901 123 48 π‘œ π‘œ PROPN cana-3901 124 1 = = PUNCT cana-3901 125 1 [ [ X cana-3901 125 2 𝑦0(𝑛𝑦 𝑦0(𝑛𝑦 NUM cana-3901 125 3 + + CCONJ cana-3901 125 4 π‘š π‘š NOUN cana-3901 125 5 ) ) PUNCT cana-3901 125 6 βˆ’ βˆ’ NOUN cana-3901 125 7 𝑦0 𝑦0 NOUN cana-3901 126 1 π‘₯ π‘₯ NOUN cana-3901 126 2 𝑦𝑐].0 𝑦𝑐].0 PROPN cana-3901 126 3 = = SYM cana-3901 126 4 𝑦0(𝑛𝑦 𝑦0(𝑛𝑦 PROPN cana-3901 127 1 + + CCONJ cana-3901 127 2 π‘š)0 π‘š)0 ADJ cana-3901 127 3 βˆ’ βˆ’ PROPN cana-3901 127 4 𝑦0 𝑦0 NOUN cana-3901 127 5 π‘₯ π‘₯ PROPN cana-3901 127 6 𝑦𝑐 𝑦𝑐 PROPN cana-3901 127 7 .0 .0 NUM cana-3901 127 8 = = SYM cana-3901 127 9 𝑦0(𝑛𝑦0 𝑦0(𝑛𝑦0 NOUN cana-3901 127 10 + + CCONJ cana-3901 127 11 π‘š0 π‘š0 NOUN cana-3901 127 12 ) ) PUNCT cana-3901 127 13 βˆ’ βˆ’ NOUN cana-3901 127 14 𝑦0 𝑦0 NOUN cana-3901 127 15 π‘₯ π‘₯ NOUN cana-3901 127 16 𝑦𝑐 𝑦𝑐 PROPN cana-3901 127 17 . . PUNCT cana-3901 127 18 0 0 NUM cana-3901 128 1 = = NOUN cana-3901 128 2 𝑦0(𝑛𝑦0 𝑦0(𝑛𝑦0 NOUN cana-3901 128 3 + + CCONJ cana-3901 128 4 π‘š π‘š NOUN cana-3901 128 5 ) ) PUNCT cana-3901 128 6 βˆ’ βˆ’ NOUN cana-3901 128 7 𝑦0 𝑦0 NOUN cana-3901 128 8 π‘₯ π‘₯ X cana-3901 128 9 𝑦𝑐 𝑦𝑐 PROPN cana-3901 129 1 [ [ X cana-3901 129 2 𝑠𝑖𝑛𝑐𝑒 𝑠𝑖𝑛𝑐𝑒 X cana-3901 129 3 π‘š π‘š ADP cana-3901 129 4 , , PUNCT cana-3901 129 5 𝑦𝑐 𝑦𝑐 PROPN cana-3901 129 6 ∈ ∈ PROPN cana-3901 129 7 𝑁𝑐 𝑁𝑐 PROPN cana-3901 129 8 ] ] X cana-3901 129 9 = = SYM cana-3901 129 10 𝑦0(𝑛𝑦𝑐 𝑦0(𝑛𝑦𝑐 X cana-3901 129 11 + + NUM cana-3901 129 12 π‘šπ‘¦π‘ π‘šπ‘¦π‘ ADJ cana-3901 129 13 ) ) PUNCT cana-3901 129 14 βˆ’ βˆ’ PROPN cana-3901 129 15 𝑦0 𝑦0 NOUN cana-3901 129 16 π‘₯ π‘₯ X cana-3901 129 17 𝑦𝑐 𝑦𝑐 PROPN cana-3901 130 1 [ [ X cana-3901 130 2 𝑠𝑖𝑛𝑐𝑒 𝑠𝑖𝑛𝑐𝑒 ADJ cana-3901 130 3 𝑦0 𝑦0 NOUN cana-3901 130 4 = = SYM cana-3901 130 5 𝑦𝑐 𝑦𝑐 X cana-3901 130 6 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ X cana-3901 130 7 π‘š π‘š X cana-3901 130 8 ∈ ∈ PROPN cana-3901 131 1 𝑁𝑐 𝑁𝑐 X cana-3901 131 2 ] ] X cana-3901 131 3 = = SYM cana-3901 131 4 0 0 X cana-3901 131 5 . . PUNCT cana-3901 132 1 also also ADV cana-3901 132 2 , , PUNCT cana-3901 132 3 𝑣. 𝑣. NOUN cana-3901 132 4 0 0 PUNCT cana-3901 133 1 = = PUNCT cana-3901 134 1 [ [ X cana-3901 134 2 𝑦0 𝑦0 NOUN cana-3901 134 3 π‘₯ π‘₯ X cana-3901 134 4 𝑦𝑐 𝑦𝑐 PROPN cana-3901 134 5 + + CCONJ cana-3901 134 6 𝑦𝑐 𝑦𝑐 PROPN cana-3901 134 7 ] ] PUNCT cana-3901 134 8 . . PUNCT cana-3901 134 9 0 0 PUNCT cana-3901 135 1 = = NOUN cana-3901 135 2 𝑦0 𝑦0 NOUN cana-3901 135 3 π‘₯ π‘₯ INTJ cana-3901 135 4 𝑦𝑐0 𝑦𝑐0 VERB cana-3901 135 5 + + CCONJ cana-3901 135 6 𝑦𝑐 𝑦𝑐 PROPN cana-3901 135 7 0 0 NUM cana-3901 135 8 = = SYM cana-3901 135 9 𝑦0 𝑦0 PROPN cana-3901 135 10 π‘₯ π‘₯ X cana-3901 135 11 𝑦𝑐 𝑦𝑐 PROPN cana-3901 135 12 + + CCONJ cana-3901 135 13 𝑦𝑐 𝑦𝑐 PROPN cana-3901 136 1 [ [ X cana-3901 136 2 𝑠𝑖𝑛𝑐𝑒 𝑠𝑖𝑛𝑐𝑒 X cana-3901 136 3 𝑦𝑐 𝑦𝑐 ADP cana-3901 136 4 ∈ ∈ PROPN cana-3901 136 5 𝑁𝑐 𝑁𝑐 PROPN cana-3901 136 6 ] ] X cana-3901 136 7 = = PUNCT cana-3901 136 8 𝑣 𝑣 ADP cana-3901 136 9 thus thus ADV cana-3901 136 10 π‘₯𝑦2 π‘₯𝑦2 ADJ cana-3901 136 11 = = SYM cana-3901 136 12 𝑒 𝑒 PROPN cana-3901 137 1 + + NOUN cana-3901 137 2 𝑣 𝑣 ADP cana-3901 137 3 where where SCONJ cana-3901 137 4 𝑒 𝑒 X cana-3901 137 5 ∈ ∈ PROPN cana-3901 137 6 𝑁0 𝑁0 X cana-3901 137 7 and and CCONJ cana-3901 137 8 𝑣 𝑣 DET cana-3901 137 9 ∈ ∈ PROPN cana-3901 137 10 𝑁𝑐. 𝑁𝑐. PROPN cana-3901 137 11 for for ADP cana-3901 137 12 the the DET cana-3901 137 13 β€œ " PUNCT cana-3901 137 14 if if SCONJ cana-3901 137 15 part part NOUN cana-3901 137 16 ” " PUNCT cana-3901 137 17 assume assume VERB cana-3901 137 18 for for ADP cana-3901 137 19 every every DET cana-3901 137 20 y y PROPN cana-3901 137 21 in in ADP cana-3901 137 22 n n NOUN cana-3901 137 23 with with ADP cana-3901 137 24 π‘₯𝑦2 π‘₯𝑦2 NOUN cana-3901 137 25 = = SYM cana-3901 137 26 𝑒 𝑒 PROPN cana-3901 138 1 + + NOUN cana-3901 138 2 𝑣 𝑣 ADP cana-3901 138 3 where where SCONJ cana-3901 138 4 𝑒 𝑒 X cana-3901 138 5 ∈ ∈ PROPN cana-3901 138 6 𝑁0 𝑁0 X cana-3901 138 7 and and CCONJ cana-3901 138 8 𝑣 𝑣 ADP cana-3901 138 9 ∈ ∈ PRON cana-3901 139 1 𝑁𝑐 𝑁𝑐 PROPN cana-3901 139 2 with with ADP cana-3901 139 3 𝑒 𝑒 NOUN cana-3901 139 4 = = SYM cana-3901 139 5 𝑦0(𝑛𝑦 𝑦0(𝑛𝑦 PROPN cana-3901 139 6 + + CCONJ cana-3901 139 7 π‘š π‘š X cana-3901 139 8 ) ) PUNCT cana-3901 139 9 βˆ’ βˆ’ PROPN cana-3901 139 10 𝑦0π‘₯ 𝑦0π‘₯ PROPN cana-3901 139 11 𝑦𝑐 𝑦𝑐 PROPN cana-3901 139 12 , , PUNCT cana-3901 139 13 𝑣 𝑣 X cana-3901 139 14 = = SYM cana-3901 139 15 𝑦0 𝑦0 PROPN cana-3901 139 16 π‘₯ π‘₯ X cana-3901 139 17 𝑦𝑐 𝑦𝑐 PROPN cana-3901 139 18 + + CCONJ cana-3901 139 19 𝑦𝑐 𝑦𝑐 ADP cana-3901 139 20 where where SCONJ cana-3901 139 21 𝑦 𝑦 NOUN cana-3901 139 22 = = X cana-3901 139 23 𝑦0 𝑦0 PROPN cana-3901 139 24 + + CCONJ cana-3901 139 25 𝑦𝑐 𝑦𝑐 NOUN cana-3901 139 26 , , PUNCT cana-3901 139 27 𝑦0 𝑦0 NOUN cana-3901 139 28 , , PUNCT cana-3901 139 29 𝑛 𝑛 PRON cana-3901 139 30 ∈ ∈ PROPN cana-3901 140 1 𝑁𝑐 𝑁𝑐 PROPN cana-3901 140 2 and and CCONJ cana-3901 140 3 𝑦𝑐 𝑦𝑐 PROPN cana-3901 140 4 , , PUNCT cana-3901 140 5 π‘š π‘š X cana-3901 140 6 ∈ ∈ PROPN cana-3901 141 1 𝑁𝑐 𝑁𝑐 PROPN cana-3901 141 2 we we PRON cana-3901 141 3 shall shall AUX cana-3901 141 4 show show VERB cana-3901 141 5 that that SCONJ cana-3901 141 6 n n PRON cana-3901 141 7 is be AUX cana-3901 141 8 a a DET cana-3901 141 9 𝜎1near 𝜎1near ADJ cana-3901 141 10 – – PUNCT cana-3901 141 11 ring ring NOUN cana-3901 141 12 . . PUNCT cana-3901 142 1 now now ADV cana-3901 142 2 , , PUNCT cana-3901 142 3 π‘₯𝑦2 π‘₯𝑦2 NOUN cana-3901 142 4 = = SYM cana-3901 142 5 𝑒 𝑒 PROPN cana-3901 142 6 + + X cana-3901 142 7 𝑣 𝑣 X cana-3901 142 8 = = PUNCT cana-3901 142 9 𝑦0(𝑛𝑦 𝑦0(𝑛𝑦 PROPN cana-3901 142 10 + + CCONJ cana-3901 142 11 π‘š π‘š X cana-3901 142 12 ) ) PUNCT cana-3901 142 13 βˆ’ βˆ’ PROPN cana-3901 143 1 𝑦0π‘₯ 𝑦0π‘₯ PROPN cana-3901 143 2 𝑦 𝑦 PROPN cana-3901 143 3 𝑐 𝑐 NOUN cana-3901 143 4 + + NUM cana-3901 143 5 𝑦0 𝑦0 PROPN cana-3901 143 6 π‘₯ π‘₯ PROPN cana-3901 143 7 𝑦𝑐 𝑦𝑐 PROPN cana-3901 143 8 + + CCONJ cana-3901 143 9 𝑦𝑐 𝑦𝑐 PROPN cana-3901 143 10 = = SYM cana-3901 143 11 𝑦0(𝑛𝑦 𝑦0(𝑛𝑦 PROPN cana-3901 143 12 + + CCONJ cana-3901 143 13 π‘šπ‘¦ π‘šπ‘¦ ADP cana-3901 143 14 ) ) PUNCT cana-3901 144 1 + + CCONJ cana-3901 144 2 𝑦𝑐 𝑦𝑐 PROPN cana-3901 145 1 [ [ X cana-3901 145 2 𝑠𝑖𝑛𝑐𝑒 𝑠𝑖𝑛𝑐𝑒 X cana-3901 145 3 π‘š π‘š ADP cana-3901 145 4 ∈ ∈ PRON cana-3901 145 5 𝑁𝑐 𝑁𝑐 PROPN cana-3901 145 6 ] ] X cana-3901 145 7 = = PUNCT cana-3901 146 1 𝑦0(𝑛 𝑦0(𝑛 PROPN cana-3901 146 2 + + NOUN cana-3901 146 3 π‘š)𝑦 π‘š)𝑦 PUNCT cana-3901 147 1 + + CCONJ cana-3901 147 2 𝑦𝑐 𝑦𝑐 NOUN cana-3901 147 3 = = SYM cana-3901 147 4 𝑦0 𝑦0 NOUN cana-3901 147 5 π‘₯𝑦 π‘₯𝑦 CCONJ cana-3901 147 6 + + NUM cana-3901 147 7 𝑦𝑐 𝑦𝑐 PROPN cana-3901 147 8 π‘₯𝑦 π‘₯𝑦 NOUN cana-3901 147 9 [ [ PUNCT cana-3901 147 10 𝑠𝑖𝑛𝑐𝑒 𝑠𝑖𝑛𝑐𝑒 PROPN cana-3901 147 11 𝑦𝑐 𝑦𝑐 PROPN cana-3901 147 12 , , PUNCT cana-3901 147 13 ∈ ∈ PROPN cana-3901 148 1 𝑁𝑐 𝑁𝑐 PROPN cana-3901 148 2 ] ] X cana-3901 148 3 = = SYM cana-3901 148 4 ( ( PUNCT cana-3901 148 5 𝑦0 𝑦0 NOUN cana-3901 148 6 + + CCONJ cana-3901 148 7 𝑦𝑐)π‘₯𝑦 𝑦𝑐)π‘₯𝑦 PUNCT cana-3901 148 8 = = PRON cana-3901 148 9 𝑦π‘₯𝑦 𝑦π‘₯𝑦 VERB cana-3901 148 10 thus thus ADV cana-3901 148 11 for for ADP cana-3901 148 12 every every DET cana-3901 148 13 𝑦 𝑦 PROPN cana-3901 148 14 ∈ ∈ NOUN cana-3901 148 15 𝑁 𝑁 NOUN cana-3901 148 16 , , PUNCT cana-3901 148 17 π‘₯𝑦2 π‘₯𝑦2 NOUN cana-3901 148 18 = = PUNCT cana-3901 148 19 𝑦π‘₯𝑦 𝑦π‘₯𝑦 VERB cana-3901 148 20 for for ADP cana-3901 148 21 all all DET cana-3901 148 22 π‘₯ π‘₯ PROPN cana-3901 148 23 in in ADP cana-3901 148 24 n. n. NOUN cana-3901 148 25 hence hence ADV cana-3901 148 26 n n ADV cana-3901 148 27 is be AUX cana-3901 148 28 a a DET cana-3901 148 29 𝜎1near 𝜎1near ADJ cana-3901 148 30 – – PUNCT cana-3901 148 31 ring ring NOUN cana-3901 148 32 . . PUNCT cana-3901 149 1 theorem theorem VERB cana-3901 149 2 5.11 5.11 NUM cana-3901 149 3 let let VERB cana-3901 149 4 n n PRON cana-3901 149 5 be be AUX cana-3901 149 6 a a DET cana-3901 149 7 zero zero NUM cana-3901 149 8 symmetric symmetric ADJ cana-3901 149 9 weak weak ADJ cana-3901 149 10 commutative commutative ADJ cana-3901 149 11 𝜎1 𝜎1 NOUN cana-3901 149 12 near near ADV cana-3901 149 13 – – PUNCT cana-3901 149 14 ring ring NOUN cana-3901 149 15 then then ADV cana-3901 149 16 right right ADV cana-3901 149 17 cancellative cancellative ADJ cana-3901 149 18 near near ADP cana-3901 149 19 rings ring NOUN cana-3901 149 20 are be AUX cana-3901 149 21 integral integral ADJ cana-3901 149 22 . . PUNCT cana-3901 150 1 communications communication NOUN cana-3901 150 2 on on ADP cana-3901 150 3 applied apply VERB cana-3901 150 4 nonlinear nonlinear ADJ cana-3901 150 5 analysis analysis NOUN cana-3901 150 6 issn issn NOUN cana-3901 150 7 : : PUNCT cana-3901 150 8 1074 1074 NUM cana-3901 150 9 - - PUNCT cana-3901 150 10 133x 133x NUM cana-3901 150 11 vol vol NOUN cana-3901 150 12 32 32 NUM cana-3901 150 13 no no NOUN cana-3901 150 14 . . PUNCT cana-3901 151 1 9s 9s NUM cana-3901 151 2 ( ( PUNCT cana-3901 151 3 2025 2025 NUM cana-3901 151 4 ) ) PUNCT cana-3901 151 5 360 360 NUM cana-3901 151 6 https://internationalpubls.com https://internationalpubls.com X cana-3901 151 7 proof proof NOUN cana-3901 151 8 let let VERB cana-3901 151 9 π‘Ž π‘Ž PROPN cana-3901 151 10 β‰  β‰  PROPN cana-3901 151 11 π‘œ π‘œ NOUN cana-3901 151 12 and and CCONJ cana-3901 151 13 π‘Žπ‘ π‘Žπ‘ NOUN cana-3901 151 14 = = SYM cana-3901 151 15 0 0 PROPN cana-3901 151 16 . . PUNCT cana-3901 152 1 then then ADV cana-3901 152 2 π‘π‘Ž π‘π‘Ž VERB cana-3901 152 3 = = NOUN cana-3901 152 4 0 0 PUNCT cana-3901 153 1 [ [ X cana-3901 153 2 by by ADP cana-3901 153 3 proposition proposition NOUN cana-3901 153 4 5.10 5.10 NUM cana-3901 153 5 ] ] PUNCT cana-3901 153 6 = = SYM cana-3901 153 7 0π‘Ž 0π‘Ž NOUN cana-3901 153 8 and and CCONJ cana-3901 153 9 therefore therefore ADV cana-3901 153 10 by by ADP cana-3901 153 11 right right ADJ cana-3901 153 12 cancellative cancellative ADJ cana-3901 153 13 law law NOUN cana-3901 153 14 we we PRON cana-3901 153 15 get get VERB cana-3901 153 16 𝑏 𝑏 NOUN cana-3901 153 17 = = SYM cana-3901 153 18 0 0 PROPN cana-3901 153 19 . . PUNCT cana-3901 154 1 and and CCONJ cana-3901 154 2 if if SCONJ cana-3901 154 3 𝑏 𝑏 PROPN cana-3901 154 4 β‰  β‰  PROPN cana-3901 154 5 0 0 NUM cana-3901 154 6 and and CCONJ cana-3901 154 7 π‘Žπ‘ π‘Žπ‘ NOUN cana-3901 155 1 = = SYM cana-3901 155 2 0 0 PUNCT cana-3901 155 3 then then ADV cana-3901 155 4 π‘Žπ‘ π‘Žπ‘ PROPN cana-3901 155 5 = = SYM cana-3901 155 6 0𝑏. 0𝑏. NOUN cana-3901 155 7 this this PRON cana-3901 155 8 implies imply VERB cana-3901 155 9 π‘Ž π‘Ž PROPN cana-3901 155 10 = = SYM cana-3901 155 11 0 0 NUM cana-3901 155 12 , , PUNCT cana-3901 155 13 by by ADP cana-3901 155 14 right right ADJ cana-3901 155 15 cancellative cancellative ADJ cana-3901 155 16 law law NOUN cana-3901 155 17 . . PUNCT cana-3901 156 1 hence hence ADV cana-3901 156 2 π‘Žπ‘ π‘Žπ‘ NOUN cana-3901 156 3 = = SYM cana-3901 156 4 0 0 PROPN cana-3901 156 5 ⟹ ⟹ NUM cana-3901 156 6 either either CCONJ cana-3901 156 7 π‘Ž π‘Ž X cana-3901 156 8 = = SYM cana-3901 156 9 0 0 NUM cana-3901 156 10 or or CCONJ cana-3901 156 11 𝑏 𝑏 NOUN cana-3901 156 12 = = SYM cana-3901 156 13 0 0 NUM cana-3901 156 14 and and CCONJ cana-3901 156 15 the the DET cana-3901 156 16 result result NOUN cana-3901 156 17 follows follow VERB cana-3901 156 18 . . PUNCT cana-3901 157 1 theorem theorem VERB cana-3901 157 2 5.12 5.12 NUM cana-3901 157 3 let let VERB cana-3901 157 4 n n PRON cana-3901 157 5 be be AUX cana-3901 157 6 a a DET cana-3901 157 7 𝜎1 𝜎1 ADJ cana-3901 157 8 near near ADV cana-3901 157 9 – – PUNCT cana-3901 157 10 ring ring NOUN cana-3901 157 11 with with ADP cana-3901 157 12 mate mate NOUN cana-3901 157 13 function function NOUN cana-3901 157 14 𝑓. 𝑓. NOUN cana-3901 157 15 if if SCONJ cana-3901 157 16 n n PRON cana-3901 157 17 is be AUX cana-3901 157 18 regular regular ADJ cana-3901 158 1 and and CCONJ cana-3901 158 2 i i PRON cana-3901 158 3 is be AUX cana-3901 158 4 a a DET cana-3901 158 5 proper proper ADJ cana-3901 158 6 ideal ideal NOUN cana-3901 158 7 of of ADP cana-3901 158 8 n n CCONJ cana-3901 158 9 , , PUNCT cana-3901 158 10 then then ADV cana-3901 158 11 every every DET cana-3901 158 12 element element NOUN cana-3901 158 13 of of ADP cana-3901 158 14 i i PRON cana-3901 158 15 is be AUX cana-3901 158 16 a a DET cana-3901 158 17 zero zero NUM cana-3901 158 18 divisor divisor NOUN cana-3901 158 19 . . PUNCT cana-3901 159 1 proof proof NOUN cana-3901 159 2 let let VERB cana-3901 159 3 n n PRON cana-3901 159 4 be be AUX cana-3901 159 5 an an DET cana-3901 159 6 𝜎1near 𝜎1near NOUN cana-3901 159 7 – – PUNCT cana-3901 159 8 ring ring NOUN cana-3901 159 9 then then ADV cana-3901 159 10 π‘₯𝑦2 π‘₯𝑦2 NOUN cana-3901 159 11 = = PUNCT cana-3901 159 12 𝑦π‘₯𝑦 𝑦π‘₯𝑦 VERB cana-3901 159 13 βˆ€ βˆ€ NOUN cana-3901 159 14 π‘₯ π‘₯ NOUN cana-3901 159 15 , , PUNCT cana-3901 159 16 𝑦 𝑦 NOUN cana-3901 159 17 ∈ ∈ NOUN cana-3901 159 18 𝑁 𝑁 PROPN cana-3901 159 19 … … SYM cana-3901 159 20 … … PUNCT cana-3901 159 21 … … PUNCT cana-3901 159 22 …… …… NOUN cana-3901 159 23 …… …… NOUN cana-3901 159 24 .. .. PUNCT cana-3901 159 25 ( ( PUNCT cana-3901 159 26 1 1 X cana-3901 159 27 ) ) PUNCT cana-3901 159 28 since since SCONJ cana-3901 159 29 𝑓 𝑓 PRON cana-3901 159 30 is be AUX cana-3901 159 31 a a DET cana-3901 159 32 mate mate NOUN cana-3901 159 33 function function NOUN cana-3901 159 34 for for ADP cana-3901 159 35 𝑁 𝑁 PROPN cana-3901 159 36 , , PUNCT cana-3901 159 37 then then ADV cana-3901 159 38 π‘₯ π‘₯ X cana-3901 159 39 = = PUNCT cana-3901 159 40 π‘₯𝑓(π‘₯)π‘₯ π‘₯𝑓(π‘₯)π‘₯ PROPN cana-3901 159 41 ∈ ∈ PROPN cana-3901 159 42 π‘₯𝑁π‘₯ π‘₯𝑁π‘₯ X cana-3901 159 43 ∴ ∴ NOUN cana-3901 159 44 π‘₯ π‘₯ PROPN cana-3901 159 45 = = PUNCT cana-3901 159 46 π‘₯𝑛π‘₯ π‘₯𝑛π‘₯ NOUN cana-3901 159 47 = = PUNCT cana-3901 159 48 𝑛π‘₯2 𝑛π‘₯2 VERB cana-3901 159 49 for for ADP cana-3901 159 50 some some DET cana-3901 159 51 𝑛 𝑛 PRON cana-3901 160 1 [ [ X cana-3901 160 2 by by ADP cana-3901 160 3 equation equation NOUN cana-3901 160 4 1 1 NUM cana-3901 160 5 ] ] PUNCT cana-3901 160 6 put put NOUN cana-3901 160 7 π‘₯2 π‘₯2 NOUN cana-3901 160 8 = = SYM cana-3901 160 9 0 0 PUNCT cana-3901 160 10 ⟹ ⟹ NUM cana-3901 160 11 𝑛 𝑛 PRON cana-3901 160 12 0 0 NUM cana-3901 161 1 = = SYM cana-3901 161 2 π‘₯ π‘₯ NOUN cana-3901 161 3 ⟹ ⟹ PUNCT cana-3901 162 1 π‘₯ π‘₯ X cana-3901 162 2 = = SYM cana-3901 162 3 0 0 NUM cana-3901 162 4 ∴ ∴ PROPN cana-3901 162 5 𝐿 𝐿 PROPN cana-3901 162 6 = = SYM cana-3901 162 7 { { PUNCT cana-3901 162 8 0 0 NUM cana-3901 162 9 } } PUNCT cana-3901 162 10 . . PUNCT cana-3901 163 1 let let VERB cana-3901 163 2 π‘Ž π‘Ž NOUN cana-3901 163 3 ∈ ∈ NOUN cana-3901 163 4 𝐼. 𝐼. NOUN cana-3901 163 5 then then ADV cana-3901 163 6 na na INTJ cana-3901 163 7 is be AUX cana-3901 163 8 an an DET cana-3901 163 9 n n ADJ cana-3901 163 10 – – PUNCT cana-3901 163 11 subgroup subgroup NOUN cana-3901 163 12 . . PUNCT cana-3901 164 1 since since SCONJ cana-3901 164 2 n n PRON cana-3901 164 3 is be AUX cana-3901 164 4 regular regular ADJ cana-3901 164 5 π‘Ž π‘Ž NOUN cana-3901 164 6 = = ADJ cana-3901 164 7 π‘Žπ‘₯π‘Ž π‘Žπ‘₯π‘Ž NOUN cana-3901 164 8 for for ADP cana-3901 164 9 some some DET cana-3901 164 10 π‘₯ π‘₯ DET cana-3901 164 11 ∈ ∈ PROPN cana-3901 164 12 𝑁. 𝑁. PROPN cana-3901 164 13 let let VERB cana-3901 164 14 𝑛 𝑛 DET cana-3901 164 15 π‘Ž π‘Ž NOUN cana-3901 164 16 ∈ ∈ NOUN cana-3901 164 17 π‘π‘Ž π‘π‘Ž PROPN cana-3901 164 18 for for ADP cana-3901 164 19 any any DET cana-3901 164 20 𝑛 𝑛 PRON cana-3901 164 21 ∈ ∈ PROPN cana-3901 164 22 𝑁. 𝑁. PROPN cana-3901 164 23 ⟹ ⟹ PUNCT cana-3901 164 24 π‘›π‘Ž π‘›π‘Ž ADP cana-3901 164 25 = = SYM cana-3901 164 26 𝑛(π‘Žπ‘₯π‘Ž 𝑛(π‘Žπ‘₯π‘Ž NOUN cana-3901 164 27 ) ) PUNCT cana-3901 164 28 = = VERB cana-3901 164 29 π‘›π‘Žπ‘₯π‘Ž π‘›π‘Žπ‘₯π‘Ž NOUN cana-3901 164 30 ∈ ∈ NOUN cana-3901 164 31 π‘π‘Žπ‘π‘Ž π‘π‘Žπ‘π‘Ž PROPN cana-3901 164 32 and and CCONJ cana-3901 164 33 if if SCONJ cana-3901 164 34 π‘š π‘š PROPN cana-3901 164 35 ∈ ∈ PROPN cana-3901 164 36 π‘π‘Žπ‘π‘Ž π‘π‘Žπ‘π‘Ž PROPN cana-3901 164 37 then then ADV cana-3901 164 38 for for ADP cana-3901 164 39 𝑒 𝑒 X cana-3901 164 40 , , PUNCT cana-3901 164 41 𝑣 𝑣 PRON cana-3901 164 42 ∈ ∈ NOUN cana-3901 164 43 𝑁 𝑁 PROPN cana-3901 164 44 π‘š π‘š NOUN cana-3901 164 45 = = PUNCT cana-3901 164 46 π‘’π‘Žπ‘£π‘Ž π‘’π‘Žπ‘£π‘Ž NOUN cana-3901 164 47 = = SYM cana-3901 164 48 ( ( PUNCT cana-3901 164 49 π‘’π‘Žπ‘£)π‘Ž π‘’π‘Žπ‘£)π‘Ž PROPN cana-3901 164 50 ∈ ∈ PROPN cana-3901 165 1 πΌπ‘Ž πΌπ‘Ž PROPN cana-3901 165 2 ( ( PUNCT cana-3901 165 3 𝑁𝐼𝑁 𝑁𝐼𝑁 PROPN cana-3901 165 4 βŠ† βŠ† NUM cana-3901 165 5 𝐼 𝐼 PROPN cana-3901 165 6 ) ) PUNCT cana-3901 165 7 ⟹ ⟹ PUNCT cana-3901 165 8 π‘š π‘š X cana-3901 165 9 ∈ ∈ PROPN cana-3901 166 1 π‘π‘Ž π‘π‘Ž PROPN cana-3901 166 2 consequently consequently ADV cana-3901 166 3 , , PUNCT cana-3901 166 4 π‘π‘Žπ‘π‘Ž π‘π‘Žπ‘π‘Ž PROPN cana-3901 166 5 = = PUNCT cana-3901 167 1 π‘π‘Ž π‘π‘Ž PROPN cana-3901 167 2 let let VERB cana-3901 167 3 π‘›π‘Ž π‘›π‘Ž PART cana-3901 167 4 = = PUNCT cana-3901 167 5 π‘’π‘Žπ‘£π‘Ž π‘’π‘Žπ‘£π‘Ž NOUN cana-3901 167 6 then then ADV cana-3901 167 7 π‘›π‘Ž π‘›π‘Ž ADP cana-3901 167 8 βˆ’ βˆ’ PROPN cana-3901 167 9 π‘’π‘Žπ‘£π‘Ž π‘’π‘Žπ‘£π‘Ž NOUN cana-3901 167 10 = = SYM cana-3901 167 11 0 0 NUM cana-3901 167 12 ⟹ ⟹ NUM cana-3901 167 13 ( ( PUNCT cana-3901 167 14 𝑛 𝑛 PROPN cana-3901 167 15 βˆ’ βˆ’ NOUN cana-3901 167 16 π‘’π‘Žπ‘£)π‘Ž π‘’π‘Žπ‘£)π‘Ž PROPN cana-3901 167 17 = = SYM cana-3901 167 18 0 0 NUM cana-3901 167 19 … … SYM cana-3901 167 20 … … PUNCT cana-3901 167 21 … … PUNCT cana-3901 167 22 … … PUNCT cana-3901 167 23 … … PUNCT cana-3901 167 24 … … PUNCT cana-3901 167 25 (2 (2 NUM cana-3901 167 26 ) ) PUNCT cana-3901 167 27 if if SCONJ cana-3901 167 28 a a PRON cana-3901 167 29 is be AUX cana-3901 167 30 not not PART cana-3901 167 31 a a DET cana-3901 167 32 zero zero NUM cana-3901 167 33 divisor divisor NOUN cana-3901 167 34 , , PUNCT cana-3901 167 35 then then ADV cana-3901 167 36 𝑛 𝑛 DET cana-3901 167 37 βˆ’ βˆ’ PROPN cana-3901 167 38 π‘’π‘Žπ‘£ π‘’π‘Žπ‘£ PROPN cana-3901 167 39 = = NOUN cana-3901 167 40 0 0 PROPN cana-3901 167 41 . . PUNCT cana-3901 167 42 i.e.)𝑛 i.e.)𝑛 PUNCT cana-3901 168 1 = = PUNCT cana-3901 168 2 π‘’π‘Žπ‘£ π‘’π‘Žπ‘£ PROPN cana-3901 169 1 ∈ ∈ PROPN cana-3901 169 2 𝑁𝐼𝑁 𝑁𝐼𝑁 PROPN cana-3901 169 3 βŠ† βŠ† NUM cana-3901 169 4 𝐼 𝐼 PROPN cana-3901 169 5 ⟹ ⟹ NUM cana-3901 169 6 𝑁 𝑁 PROPN cana-3901 169 7 = = SYM cana-3901 169 8 𝐼 𝐼 PROPN cana-3901 169 9 , , PUNCT cana-3901 169 10 which which PRON cana-3901 169 11 is be AUX cana-3901 169 12 a a DET cana-3901 169 13 contradiction contradiction NOUN cana-3901 169 14 to to ADP cana-3901 169 15 i i PRON cana-3901 169 16 is be AUX cana-3901 169 17 a a DET cana-3901 169 18 proper proper ADJ cana-3901 169 19 ideal ideal NOUN cana-3901 169 20 of of ADP cana-3901 169 21 n. n. NOUN cana-3901 169 22 hence hence ADV cana-3901 169 23 a a PRON cana-3901 169 24 is be AUX cana-3901 169 25 a a DET cana-3901 169 26 right right ADJ cana-3901 169 27 zero zero NUM cana-3901 169 28 divisor divisor NOUN cana-3901 169 29 . . PUNCT cana-3901 170 1 now now ADV cana-3901 170 2 equation equation NOUN cana-3901 170 3 ( ( PUNCT cana-3901 170 4 2 2 NUM cana-3901 170 5 ) ) PUNCT cana-3901 170 6 ⟹ ⟹ NUM cana-3901 170 7 π‘Ž(𝑛 π‘Ž(𝑛 PROPN cana-3901 170 8 βˆ’ βˆ’ PROPN cana-3901 170 9 π‘’π‘Žπ‘£ π‘’π‘Žπ‘£ PROPN cana-3901 170 10 ) ) PUNCT cana-3901 170 11 = = PUNCT cana-3901 171 1 0 0 X cana-3901 171 2 . . PUNCT cana-3901 172 1 [ [ X cana-3901 172 2 by by ADP cana-3901 172 3 proposition proposition NOUN cana-3901 172 4 5.9 5.9 NUM cana-3901 172 5 ] ] PUNCT cana-3901 172 6 . . PUNCT cana-3901 173 1 this this PRON cana-3901 173 2 leads lead VERB cana-3901 173 3 to to ADP cana-3901 173 4 the the DET cana-3901 173 5 result result NOUN cana-3901 173 6 that that SCONJ cana-3901 173 7 a a PRON cana-3901 173 8 is be AUX cana-3901 173 9 a a DET cana-3901 173 10 left left ADJ cana-3901 173 11 divisor divisor NOUN cana-3901 173 12 too too ADV cana-3901 173 13 . . PUNCT cana-3901 174 1 thus thus ADV cana-3901 174 2 the the DET cana-3901 174 3 result result NOUN cana-3901 174 4 follows follow VERB cana-3901 174 5 . . PUNCT cana-3901 175 1 corollary corollary ADJ cana-3901 175 2 5.12 5.12 NUM cana-3901 175 3 if if SCONJ cana-3901 175 4 n n PRON cana-3901 175 5 is be AUX cana-3901 175 6 a a DET cana-3901 175 7 𝜎1 𝜎1 ADJ cana-3901 175 8 near near ADV cana-3901 175 9 – – PUNCT cana-3901 175 10 ring ring NOUN cana-3901 175 11 with with ADP cana-3901 175 12 no no DET cana-3901 175 13 nonzero nonzero ADJ cana-3901 175 14 divisors divisor NOUN cana-3901 175 15 then then ADV cana-3901 175 16 n n PRON cana-3901 175 17 contains contain VERB cana-3901 175 18 no no DET cana-3901 175 19 proper proper ADJ cana-3901 175 20 ideals ideal NOUN cana-3901 175 21 of of ADP cana-3901 175 22 n. n. PROPN cana-3901 175 23 lemma lemma PROPN cana-3901 175 24 5.13 5.13 NUM cana-3901 175 25 if if SCONJ cana-3901 175 26 n n NOUN cana-3901 175 27 is be AUX cana-3901 175 28 𝜎1 𝜎1 ADJ cana-3901 175 29 – – PUNCT cana-3901 175 30 near near ADP cana-3901 175 31 ring ring NOUN cana-3901 175 32 with with ADP cana-3901 175 33 mate mate NOUN cana-3901 175 34 function function NOUN cana-3901 175 35 𝑓 𝑓 PRON cana-3901 175 36 then then ADV cana-3901 175 37 for for ADP cana-3901 175 38 π‘Ž π‘Ž NOUN cana-3901 175 39 , , PUNCT cana-3901 175 40 𝑏 𝑏 PROPN cana-3901 175 41 ∈ ∈ PROPN cana-3901 175 42 𝑁 𝑁 PROPN cana-3901 175 43 , , PUNCT cana-3901 175 44 π‘Žπ‘ π‘Žπ‘ PROPN cana-3901 175 45 = = PROPN cana-3901 175 46 𝑏2 𝑏2 PROPN cana-3901 175 47 and and CCONJ cana-3901 175 48 π‘π‘Ž π‘π‘Ž X cana-3901 176 1 = = NOUN cana-3901 176 2 π‘Ž2 π‘Ž2 NOUN cana-3901 176 3 imply imply VERB cana-3901 176 4 π‘Ž π‘Ž NOUN cana-3901 176 5 = = NOUN cana-3901 176 6 𝑏. 𝑏. NOUN cana-3901 176 7 communications communication NOUN cana-3901 176 8 on on ADP cana-3901 176 9 applied apply VERB cana-3901 176 10 nonlinear nonlinear ADJ cana-3901 176 11 analysis analysis NOUN cana-3901 176 12 issn issn NOUN cana-3901 176 13 : : PUNCT cana-3901 176 14 1074 1074 NUM cana-3901 176 15 - - PUNCT cana-3901 176 16 133x 133x NUM cana-3901 176 17 vol vol NOUN cana-3901 176 18 32 32 NUM cana-3901 176 19 no no NOUN cana-3901 176 20 . . PUNCT cana-3901 177 1 9s 9s NUM cana-3901 177 2 ( ( PUNCT cana-3901 177 3 2025 2025 NUM cana-3901 177 4 ) ) PUNCT cana-3901 177 5 361 361 NUM cana-3901 178 1 https://internationalpubls.com https://internationalpubls.com X cana-3901 178 2 proof proof NOUN cana-3901 178 3 : : PUNCT cana-3901 178 4 let let VERB cana-3901 178 5 n n PRON cana-3901 178 6 be be AUX cana-3901 178 7 a a DET cana-3901 178 8 𝜎1near 𝜎1near NOUN cana-3901 178 9 - - PUNCT cana-3901 178 10 ring ring NOUN cana-3901 178 11 with with ADP cana-3901 178 12 a a DET cana-3901 178 13 mate mate NOUN cana-3901 178 14 function function NOUN cana-3901 178 15 𝑓. 𝑓. NOUN cana-3901 178 16 then then ADV cana-3901 178 17 we we PRON cana-3901 178 18 have have VERB cana-3901 178 19 𝐿 𝐿 PROPN cana-3901 178 20 = = SYM cana-3901 178 21 { { PUNCT cana-3901 178 22 0} 0} NUM cana-3901 178 23 … … SYM cana-3901 178 24 … … PUNCT cana-3901 178 25 … … PUNCT cana-3901 178 26 … … PUNCT cana-3901 178 27 … … PUNCT cana-3901 178 28 … … PUNCT cana-3901 178 29 … … PUNCT cana-3901 178 30 … … PUNCT cana-3901 178 31 (1 (1 X cana-3901 178 32 ) ) PUNCT cana-3901 179 1 [ [ X cana-3901 179 2 by by ADP cana-3901 179 3 proposition proposition NOUN cana-3901 179 4 5.8 5.8 NUM cana-3901 179 5 ] ] PUNCT cana-3901 179 6 let let VERB cana-3901 179 7 𝑒1 𝑒1 PROPN cana-3901 179 8 = = PROPN cana-3901 179 9 π‘Ž π‘Ž PRON cana-3901 179 10 βˆ’ βˆ’ PROPN cana-3901 179 11 𝑏 𝑏 PROPN cana-3901 179 12 𝑒2 𝑒2 PROPN cana-3901 179 13 = = PROPN cana-3901 179 14 π‘Žπ‘’1 π‘Žπ‘’1 PROPN cana-3901 179 15 and and CCONJ cana-3901 179 16 𝑒3 𝑒3 PROPN cana-3901 179 17 = = SYM cana-3901 179 18 𝑏𝑒1 𝑏𝑒1 NOUN cana-3901 179 19 we we PRON cana-3901 179 20 can can AUX cana-3901 179 21 easily easily ADV cana-3901 179 22 obtain obtain VERB cana-3901 179 23 the the DET cana-3901 179 24 following follow VERB cana-3901 179 25 equations equation NOUN cana-3901 179 26 . . PUNCT cana-3901 180 1 𝑒1π‘Ž 𝑒1π‘Ž NOUN cana-3901 180 2 = = PUNCT cana-3901 180 3 ( ( PUNCT cana-3901 180 4 π‘Ž π‘Ž NOUN cana-3901 180 5 βˆ’ βˆ’ NOUN cana-3901 180 6 𝑏)π‘Ž 𝑏)π‘Ž ADJ cana-3901 180 7 = = SYM cana-3901 180 8 π‘Ž2 π‘Ž2 NOUN cana-3901 180 9 βˆ’ βˆ’ PROPN cana-3901 180 10 π‘π‘Ž π‘π‘Ž X cana-3901 180 11 = = NOUN cana-3901 180 12 0 0 NUM cana-3901 180 13 …… …… NOUN cana-3901 180 14 …… …… NOUN cana-3901 180 15 …… …… NOUN cana-3901 180 16 ( ( PUNCT cana-3901 180 17 2 2 NUM cana-3901 180 18 ) ) PUNCT cana-3901 180 19 𝑒1𝑏 𝑒1𝑏 NOUN cana-3901 180 20 = = PUNCT cana-3901 180 21 ( ( PUNCT cana-3901 180 22 π‘Ž π‘Ž DET cana-3901 180 23 βˆ’ βˆ’ NOUN cana-3901 180 24 𝑏)𝑏 𝑏)𝑏 NOUN cana-3901 180 25 = = PUNCT cana-3901 180 26 π‘Žπ‘ π‘Žπ‘ PROPN cana-3901 180 27 βˆ’ βˆ’ PROPN cana-3901 180 28 𝑏2 𝑏2 NOUN cana-3901 180 29 = = NOUN cana-3901 180 30 0 0 NUM cana-3901 180 31 …… …… NOUN cana-3901 180 32 …… …… NOUN cana-3901 180 33 …… …… NOUN cana-3901 180 34 . . PUNCT cana-3901 181 1 ( ( PUNCT cana-3901 181 2 3 3 X cana-3901 181 3 ) ) PUNCT cana-3901 181 4 also also ADV cana-3901 181 5 . . PUNCT cana-3901 182 1 𝑒1 𝑒1 PROPN cana-3901 182 2 2 2 NUM cana-3901 182 3 = = SYM cana-3901 182 4 ( ( PUNCT cana-3901 182 5 π‘Ž π‘Ž NOUN cana-3901 182 6 βˆ’ βˆ’ PROPN cana-3901 182 7 𝑏)𝑒1 𝑏)𝑒1 PROPN cana-3901 182 8 = = PUNCT cana-3901 182 9 π‘Žπ‘’1 π‘Žπ‘’1 PROPN cana-3901 182 10 βˆ’ βˆ’ PROPN cana-3901 182 11 𝑏𝑒1 𝑏𝑒1 NOUN cana-3901 182 12 = = PROPN cana-3901 182 13 𝑒2 𝑒2 PROPN cana-3901 182 14 βˆ’ βˆ’ PROPN cana-3901 182 15 𝑒3 𝑒3 NOUN cana-3901 182 16 … … PUNCT cana-3901 182 17 …… …… NOUN cana-3901 182 18 …… …… NOUN cana-3901 182 19 ( ( PUNCT cana-3901 182 20 4 4 NUM cana-3901 182 21 ) ) PUNCT cana-3901 182 22 as as SCONJ cana-3901 182 23 n n X cana-3901 182 24 is be AUX cana-3901 182 25 zero zero NUM cana-3901 182 26 symmetric symmetric NOUN cana-3901 182 27 . . PUNCT cana-3901 183 1 𝑒2 𝑒2 PROPN cana-3901 183 2 2 2 NUM cana-3901 183 3 = = SYM cana-3901 183 4 ( ( PUNCT cana-3901 183 5 π‘Žπ‘’1)(π‘Žπ‘’1 π‘Žπ‘’1)(π‘Žπ‘’1 PROPN cana-3901 183 6 ) ) PUNCT cana-3901 183 7 = = PUNCT cana-3901 183 8 π‘Ž(𝑒1π‘Ž)𝑒1 π‘Ž(𝑒1π‘Ž)𝑒1 NOUN cana-3901 183 9 = = SYM cana-3901 183 10 π‘Ž. π‘Ž. NOUN cana-3901 183 11 0𝑒1 0𝑒1 PUNCT cana-3901 184 1 = = SYM cana-3901 184 2 0 0 PUNCT cana-3901 185 1 [ [ X cana-3901 185 2 by by ADP cana-3901 185 3 equation equation NOUN cana-3901 185 4 ( ( PUNCT cana-3901 185 5 2 2 NUM cana-3901 185 6 ) ) PUNCT cana-3901 185 7 ] ] X cana-3901 185 8 … … PUNCT cana-3901 185 9 … … PUNCT cana-3901 185 10 … … PUNCT cana-3901 185 11 . . PUNCT cana-3901 186 1 ( ( PUNCT cana-3901 186 2 5 5 NUM cana-3901 186 3 ) ) PUNCT cana-3901 186 4 also also ADV cana-3901 186 5 , , PUNCT cana-3901 186 6 𝑒3 𝑒3 PROPN cana-3901 186 7 2 2 NUM cana-3901 186 8 = = SYM cana-3901 186 9 ( ( PUNCT cana-3901 186 10 𝑏𝑒1)(𝑏𝑒1 𝑏𝑒1)(𝑏𝑒1 PROPN cana-3901 186 11 ) ) PUNCT cana-3901 186 12 = = SYM cana-3901 187 1 𝑏(𝑒1𝑏)𝑒1 𝑏(𝑒1𝑏)𝑒1 PROPN cana-3901 187 2 = = PUNCT cana-3901 188 1 𝑏(0)𝑒1 𝑏(0)𝑒1 NOUN cana-3901 189 1 = = NOUN cana-3901 189 2 0 0 PUNCT cana-3901 190 1 [ [ X cana-3901 190 2 by by ADP cana-3901 190 3 equation equation NOUN cana-3901 190 4 ( ( PUNCT cana-3901 190 5 3 3 NUM cana-3901 190 6 ) ) PUNCT cana-3901 190 7 ] ] PUNCT cana-3901 190 8 … … PUNCT cana-3901 190 9 … … PUNCT cana-3901 190 10 … … PUNCT cana-3901 190 11 . . PUNCT cana-3901 191 1 ( ( PUNCT cana-3901 191 2 6 6 NUM cana-3901 191 3 ) ) PUNCT cana-3901 191 4 equations equation NOUN cana-3901 191 5 ( ( PUNCT cana-3901 191 6 5 5 NUM cana-3901 191 7 ) ) PUNCT cana-3901 191 8 & & CCONJ cana-3901 191 9 ( ( PUNCT cana-3901 191 10 6 6 NUM cana-3901 191 11 ) ) PUNCT cana-3901 191 12 imply imply VERB cana-3901 191 13 𝑒2 𝑒2 PROPN cana-3901 191 14 = = PROPN cana-3901 191 15 0 0 NUM cana-3901 191 16 and and CCONJ cana-3901 191 17 𝑒3 𝑒3 PROPN cana-3901 191 18 = = SYM cana-3901 191 19 0 0 NUM cana-3901 191 20 respectively respectively ADV cana-3901 191 21 . . PUNCT cana-3901 192 1 [ [ X cana-3901 192 2 by by ADP cana-3901 192 3 equation equation NOUN cana-3901 192 4 ( ( PUNCT cana-3901 192 5 1 1 NUM cana-3901 192 6 ) ) PUNCT cana-3901 192 7 ] ] PUNCT cana-3901 192 8 making make VERB cana-3901 192 9 use use NOUN cana-3901 192 10 of of ADP cana-3901 192 11 these these PRON cana-3901 192 12 in in ADP cana-3901 192 13 equation equation NOUN cana-3901 192 14 ( ( PUNCT cana-3901 192 15 4 4 NUM cana-3901 192 16 ) ) PUNCT cana-3901 192 17 , , PUNCT cana-3901 192 18 we we PRON cana-3901 192 19 get get VERB cana-3901 192 20 𝑒1 𝑒1 NOUN cana-3901 192 21 2 2 NUM cana-3901 192 22 = = SYM cana-3901 192 23 0 0 NUM cana-3901 192 24 . . PUNCT cana-3901 193 1 it it PRON cana-3901 193 2 follows follow VERB cana-3901 193 3 that that SCONJ cana-3901 193 4 𝑒1 𝑒1 PROPN cana-3901 193 5 = = PROPN cana-3901 193 6 0 0 NUM cana-3901 193 7 , , PUNCT cana-3901 193 8 [ [ X cana-3901 193 9 by by ADP cana-3901 193 10 equation equation NOUN cana-3901 193 11 ( ( PUNCT cana-3901 193 12 1 1 NUM cana-3901 193 13 ) ) PUNCT cana-3901 193 14 ] ] PUNCT cana-3901 193 15 i.e. i.e. X cana-3901 193 16 ) ) PUNCT cana-3901 193 17 π‘Ž π‘Ž PRON cana-3901 193 18 βˆ’ βˆ’ NOUN cana-3901 193 19 𝑏 𝑏 NOUN cana-3901 193 20 = = SYM cana-3901 193 21 0 0 NUM cana-3901 193 22 . . PUNCT cana-3901 194 1 thus thus ADV cana-3901 194 2 π‘Ž π‘Ž X cana-3901 194 3 = = SYM cana-3901 194 4 𝑏 𝑏 NOUN cana-3901 194 5 references reference NOUN cana-3901 194 6 : : PUNCT cana-3901 194 7 [ [ X cana-3901 194 8 1 1 NUM cana-3901 194 9 ] ] X cana-3901 194 10 j.r.clay j.r.clay NOUN cana-3901 194 11 , , PUNCT cana-3901 194 12 the the DET cana-3901 194 13 near near NOUN cana-3901 194 14 - - PUNCT cana-3901 194 15 rings ring NOUN cana-3901 194 16 on on ADP cana-3901 194 17 groups group NOUN cana-3901 194 18 of of ADP cana-3901 194 19 low low ADJ cana-3901 194 20 order order NOUN cana-3901 194 21 , , PUNCT cana-3901 194 22 math math NOUN cana-3901 194 23 z. z. PROPN cana-3901 194 24 104 104 NUM cana-3901 194 25 ( ( PUNCT cana-3901 194 26 1968 1968 NUM cana-3901 194 27 ) ) PUNCT cana-3901 194 28 , , PUNCT cana-3901 194 29 364 364 NUM cana-3901 194 30 - - SYM cana-3901 194 31 371 371 NUM cana-3901 194 32 . . PUNCT cana-3901 195 1 [ [ X cana-3901 195 2 2 2 NUM cana-3901 195 3 ] ] X cana-3901 195 4 gunter gunter NOUN cana-3901 195 5 pilz pilz PROPN cana-3901 195 6 , , PUNCT cana-3901 195 7 near near ADP cana-3901 195 8 ring ring NOUN cana-3901 195 9 , , PUNCT cana-3901 195 10 north north NOUN cana-3901 195 11 holland holland PROPN cana-3901 195 12 , , PUNCT cana-3901 195 13 amsterdam,1983 amsterdam,1983 PROPN cana-3901 195 14 . . PUNCT cana-3901 196 1 [ [ X cana-3901 196 2 3 3 NUM cana-3901 196 3 ] ] SYM cana-3901 196 4 n.h.mccoy n.h.mccoy NOUN cana-3901 196 5 , , PUNCT cana-3901 196 6 the the DET cana-3901 196 7 theory theory NOUN cana-3901 196 8 of of ADP cana-3901 196 9 rings ring NOUN cana-3901 196 10 , , PUNCT cana-3901 196 11 macmillan macmillan PROPN cana-3901 196 12 & & CCONJ cana-3901 196 13 co,1970 co,1970 PROPN cana-3901 196 14 . . PUNCT cana-3901 197 1 [ [ X cana-3901 197 2 4 4 X cana-3901 197 3 ] ] X cana-3901 197 4 sugantha sugantha NOUN cana-3901 197 5 g g PROPN cana-3901 197 6 and and CCONJ cana-3901 197 7 balakrishnan balakrishnan PROPN cana-3901 197 8 r r NOUN cana-3901 197 9 , , PUNCT cana-3901 197 10 Ξ²1 Ξ²1 PROPN cana-3901 197 11 near near ADJ cana-3901 197 12 - - PUNCT cana-3901 197 13 rings ring NOUN cana-3901 197 14 , , PUNCT cana-3901 197 15 international international ADJ cana-3901 197 16 journal journal NOUN cana-3901 197 17 of of ADP cana-3901 197 18 algebra algebra PROPN cana-3901 197 19 vol.4 vol.4 PROPN cana-3901 197 20 2010 2010 NUM cana-3901 197 21 , , PUNCT cana-3901 197 22 no no INTJ cana-3901 197 23 . . NOUN cana-3901 197 24 2 2 NUM cana-3901 197 25 , , PUNCT cana-3901 197 26 71 71 NUM cana-3901 197 27 - - SYM cana-3901 197 28 79 79 NUM cana-3901 198 1 [ [ X cana-3901 198 2 5 5 NUM cana-3901 198 3 ] ] PUNCT cana-3901 198 4 s.suryanarayanan s.suryanarayanan ADJ cana-3901 198 5 and and CCONJ cana-3901 198 6 n.ganesan n.ganesan NOUN cana-3901 198 7 , , PUNCT cana-3901 198 8 stable stable ADJ cana-3901 198 9 and and CCONJ cana-3901 198 10 pseudo pseudo NOUN cana-3901 198 11 stable stable ADJ cana-3901 198 12 near near ADP cana-3901 198 13 - - PUNCT cana-3901 198 14 rings ring NOUN cana-3901 198 15 , , PUNCT cana-3901 198 16 indian indian PROPN cana-3901 198 17 j. j. PROPN cana-3901 198 18 pure pure PROPN cana-3901 198 19 and and CCONJ cana-3901 198 20 appl.math appl.math NUM cana-3901 198 21 19 19 NUM cana-3901 198 22 ( ( PUNCT cana-3901 198 23 12 12 NUM cana-3901 198 24 ) ) PUNCT cana-3901 198 25 december december PROPN cana-3901 198 26 , , PUNCT cana-3901 198 27 1988 1988 NUM cana-3901 198 28 , , PUNCT cana-3901 198 29 1206 1206 NUM cana-3901 198 30 - - SYM cana-3901 198 31 1216 1216 NUM cana-3901 198 32 . . PUNCT cana-3901 199 1 [ [ X cana-3901 199 2 6 6 NUM cana-3901 199 3 ] ] X cana-3901 199 4 s.uma s.uma ADV cana-3901 199 5 , , PUNCT cana-3901 199 6 pseudo pseudo NOUN cana-3901 199 7 commutative commutative ADJ cana-3901 199 8 near near ADP cana-3901 199 9 - - PUNCT cana-3901 199 10 rings ring NOUN cana-3901 199 11 , , PUNCT cana-3901 199 12 scientia scientia PROPN cana-3901 199 13 magna magna PROPN cana-3901 199 14 , , PUNCT cana-3901 199 15 an an DET cana-3901 199 16 international international ADJ cana-3901 199 17 journal journal NOUN cana-3901 199 18 . . PUNCT cana-3901 200 1 vol.6 vol.6 PROPN cana-3901 200 2 ( ( PUNCT cana-3901 200 3 2010 2010 NUM cana-3901 200 4 ) ) PUNCT cana-3901 200 5 no no DET cana-3901 200 6 2 2 NUM cana-3901 200 7 , , PUNCT cana-3901 200 8 75 75 NUM cana-3901 200 9 - - SYM cana-3901 200 10 85 85 NUM cana-3901 200 11 . . PUNCT cana-3901 201 1 [ [ X cana-3901 201 2 7 7 X cana-3901 201 3 ] ] X cana-3901 201 4 dheena dheena PROPN cana-3901 201 5 p. p. NOUN cana-3901 201 6 on on ADP cana-3901 201 7 strongly strongly ADV cana-3901 201 8 regular regular ADJ cana-3901 201 9 near near ADJ cana-3901 201 10 – – PUNCT cana-3901 201 11 rings ring NOUN cana-3901 201 12 , , PUNCT cana-3901 201 13 journal journal NOUN cana-3901 201 14 of of ADP cana-3901 201 15 the the DET cana-3901 201 16 indian indian PROPN cana-3901 201 17 math.soc math.soc PROPN cana-3901 201 18 . . PROPN cana-3901 201 19 ,49 ,49 PROPN cana-3901 201 20 ( ( PUNCT cana-3901 201 21 1985),201 1985),201 NUM cana-3901 201 22 - - SYM cana-3901 201 23 206 206 NUM cana-3901 201 24 . . PUNCT cana-3901 202 1 [ [ X cana-3901 202 2 8 8 NUM cana-3901 202 3 ] ] X cana-3901 202 4 dheena dheena PROPN cana-3901 202 5 p. p. PROPN cana-3901 202 6 , , PUNCT cana-3901 202 7 a a DET cana-3901 202 8 generalization generalization NOUN cana-3901 202 9 of of ADP cana-3901 202 10 strongly strongly ADV cana-3901 202 11 regular regular ADJ cana-3901 202 12 near near ADJ cana-3901 202 13 – – PUNCT cana-3901 202 14 rings ring NOUN cana-3901 202 15 , , PUNCT cana-3901 202 16 indian indian PROPN cana-3901 202 17 j. j. PROPN cana-3901 202 18 pure pure PROPN cana-3901 202 19 appl appl PROPN cana-3901 202 20 . . PUNCT cana-3901 202 21 math math PROPN cana-3901 202 22 . . PUNCT cana-3901 202 23 , , PUNCT cana-3901 202 24 20 20 NUM cana-3901 202 25 ( ( PUNCT cana-3901 202 26 1 1 NUM cana-3901 202 27 ): ): PUNCT cana-3901 202 28 1989,58 1989,58 NOUN cana-3901 202 29 - - SYM cana-3901 202 30 63 63 NUM cana-3901 202 31 . . PUNCT cana-3901 203 1 [ [ X cana-3901 203 2 9 9 NUM cana-3901 203 3 ] ] X cana-3901 203 4 von von PROPN cana-3901 203 5 neumann neumann PROPN cana-3901 203 6 j. j. PROPN cana-3901 203 7 , , PUNCT cana-3901 203 8 on on ADP cana-3901 203 9 regular regular ADJ cana-3901 203 10 rings ring NOUN cana-3901 203 11 , , PUNCT cana-3901 203 12 proc proc NOUN cana-3901 203 13 . . PUNCT cana-3901 204 1 nat nat PROPN cana-3901 204 2 . . PUNCT cana-3901 205 1 acad acad PROPN cana-3901 205 2 . . PUNCT cana-3901 206 1 usa usa PROPN cana-3901 206 2 , , PUNCT cana-3901 206 3 22 22 NUM cana-3901 206 4 ( ( PUNCT cana-3901 206 5 1936),707 1936),707 NOUN cana-3901 206 6 - - PUNCT cana-3901 206 7 713 713 NUM cana-3901 206 8 . . PUNCT cana-3901 207 1 [ [ X cana-3901 207 2 10 10 NUM cana-3901 207 3 ] ] X cana-3901 207 4 s.uma s.uma ADV cana-3901 207 5 , , PUNCT cana-3901 207 6 r.balakrishnan r.balakrishnan ADJ cana-3901 207 7 , , PUNCT cana-3901 207 8 t.tamilzchelvan t.tamilzchelvan NOUN cana-3901 207 9 𝛼1 𝛼1 NOUN cana-3901 207 10 near near ADP cana-3901 207 11 - - PUNCT cana-3901 207 12 rings ring NOUN cana-3901 207 13 , , PUNCT cana-3901 207 14 international international ADJ cana-3901 207 15 journal journal NOUN cana-3901 207 16 of of ADP cana-3901 207 17 algebra algebra PROPN cana-3901 207 18 vol.4 vol.4 PROPN cana-3901 207 19 2010 2010 NUM cana-3901 207 20 , , PUNCT cana-3901 207 21 no.2 no.2 PROPN cana-3901 207 22 , , PUNCT cana-3901 207 23 71 71 NUM cana-3901 207 24 - - SYM cana-3901 207 25 79 79 NUM cana-3901 207 26 . . PUNCT cana-3901 208 1 [ [ X cana-3901 208 2 11 11 NUM cana-3901 208 3 ] ] X cana-3901 208 4 volety volety NOUN cana-3901 208 5 v.s. v.s. PROPN cana-3901 208 6 ramachadran ramachadran ADJ cana-3901 208 7 , , PUNCT cana-3901 208 8 commutativity commutativity NOUN cana-3901 208 9 of of ADP cana-3901 208 10 semi semi ADV cana-3901 208 11 near near ADP cana-3901 208 12 rings ring NOUN cana-3901 208 13 – – PUNCT cana-3901 208 14 journal journal NOUN cana-3901 208 15 of of ADP cana-3901 208 16 science science NOUN cana-3901 208 17 and and CCONJ cana-3901 208 18 arts art NOUN cana-3901 208 19 , , PUNCT cana-3901 208 20 2011 2011 NUM cana-3901 208 21 , , PUNCT cana-3901 208 22 no.4 no.4 PROPN cana-3901 208 23 ( ( PUNCT cana-3901 208 24 17 17 NUM cana-3901 208 25 ) ) PUNCT cana-3901 208 26 367 367 NUM cana-3901 208 27 368 368 NUM cana-3901 208 28 . . PUNCT