id sid tid token lemma pos cana-756 1 1 communications communication NOUN cana-756 1 2 on on ADP cana-756 1 3 applied apply VERB cana-756 1 4 nonlinear nonlinear ADJ cana-756 1 5 analysis analysis NOUN cana-756 1 6 issn issn NOUN cana-756 1 7 : : PUNCT cana-756 1 8 1074 1074 NUM cana-756 1 9 - - PUNCT cana-756 1 10 133x 133x NUM cana-756 1 11 vol vol NOUN cana-756 1 12 31 31 NUM cana-756 1 13 no no NOUN cana-756 1 14 . . PUNCT cana-756 2 1 3s 3s NUM cana-756 2 2 ( ( PUNCT cana-756 2 3 2024 2024 NUM cana-756 2 4 ) ) PUNCT cana-756 2 5 156 156 NUM cana-756 2 6 https://internationalpubls.com https://internationalpubls.com X cana-756 2 7 orthogonal orthogonal ADJ cana-756 2 8 generalized generalize VERB cana-756 2 9 symmetric symmetric ADJ cana-756 2 10 reverse reverse NOUN cana-756 2 11 bi-(𝜎 bi-(𝜎 NOUN cana-756 2 12 , , PUNCT cana-756 2 13 𝝉)-derivations 𝝉)-derivation NOUN cana-756 2 14 of of ADP cana-756 2 15 semi semi ADJ cana-756 2 16 prime prime ADJ cana-756 2 17 ring ring NOUN cana-756 2 18 v.s.v v.s.v NOUN cana-756 2 19 . . PUNCT cana-756 3 1 krishna krishna PROPN cana-756 3 2 murty1 murty1 ADV cana-756 3 3 , , PUNCT cana-756 3 4 k.chennakesavulu2 k.chennakesavulu2 PROPN cana-756 3 5 , , PUNCT cana-756 3 6 c. c. PROPN cana-756 3 7 jaya jaya PROPN cana-756 3 8 subba subba PROPN cana-756 3 9 reddy3 reddy3 PROPN cana-756 3 10 krishnamurty.vadrevu@gmail.com1 krishnamurty.vadrevu@gmail.com1 PROPN cana-756 3 11 , , PUNCT cana-756 3 12 intell.chenna@gmail.com2 intell.chenna@gmail.com2 PROPN cana-756 3 13 , , PUNCT cana-756 3 14 cjsreddysvu@gmail.com3 cjsreddysvu@gmail.com3 NOUN cana-756 3 15 1research 1research NUM cana-756 3 16 scholar scholar NOUN cana-756 3 17 , , PUNCT cana-756 3 18 department department NOUN cana-756 3 19 of of ADP cana-756 3 20 mathematics mathematics PROPN cana-756 3 21 , , PUNCT cana-756 3 22 s.v.university s.v.university NOUN cana-756 3 23 , , PUNCT cana-756 3 24 tirupati tirupati PROPN cana-756 3 25 , , PUNCT cana-756 3 26 andhra andhra PROPN cana-756 3 27 pradesh pradesh PROPN cana-756 3 28 , , PUNCT cana-756 3 29 india india PROPN cana-756 3 30 . . PUNCT cana-756 4 1 2professor 2professor NUM cana-756 4 2 , , PUNCT cana-756 4 3 department department NOUN cana-756 4 4 of of ADP cana-756 4 5 mathematics mathematic NOUN cana-756 4 6 , , PUNCT cana-756 4 7 pvkkit pvkkit NOUN cana-756 4 8 , , PUNCT cana-756 4 9 anantapur anantapur PROPN cana-756 4 10 , , PUNCT cana-756 4 11 andhra andhra PROPN cana-756 4 12 pradesh pradesh PROPN cana-756 4 13 , , PUNCT cana-756 4 14 india india PROPN cana-756 4 15 . . PUNCT cana-756 5 1 3professor 3professor NUM cana-756 5 2 , , PUNCT cana-756 5 3 department department NOUN cana-756 5 4 of of ADP cana-756 5 5 mathematics mathematics PROPN cana-756 5 6 , , PUNCT cana-756 5 7 s. s. PROPN cana-756 5 8 v. v. PROPN cana-756 5 9 university university PROPN cana-756 5 10 , , PUNCT cana-756 5 11 tirupati tirupati PROPN cana-756 5 12 , , PUNCT cana-756 5 13 andhra andhra PROPN cana-756 5 14 pradesh pradesh PROPN cana-756 5 15 , , PUNCT cana-756 5 16 india india PROPN cana-756 5 17 . . PUNCT cana-756 6 1 article article PROPN cana-756 6 2 history history NOUN cana-756 6 3 : : PUNCT cana-756 6 4 received receive VERB cana-756 6 5 : : PUNCT cana-756 6 6 05 05 NUM cana-756 6 7 - - PUNCT cana-756 6 8 04 04 NUM cana-756 6 9 - - PUNCT cana-756 6 10 2024 2024 NUM cana-756 6 11 revised revise VERB cana-756 6 12 : : PUNCT cana-756 6 13 25 25 NUM cana-756 6 14 - - PUNCT cana-756 6 15 05 05 NUM cana-756 6 16 - - PUNCT cana-756 6 17 2024 2024 NUM cana-756 6 18 accepted accept VERB cana-756 6 19 : : PUNCT cana-756 6 20 10 10 NUM cana-756 6 21 - - SYM cana-756 6 22 06 06 NUM cana-756 6 23 - - PUNCT cana-756 6 24 2024 2024 NUM cana-756 6 25 abstract abstract NOUN cana-756 6 26 : : PUNCT cana-756 6 27 let let VERB cana-756 6 28 r r PRON cana-756 6 29 be be AUX cana-756 6 30 a a DET cana-756 6 31 semi semi ADJ cana-756 6 32 prime prime ADJ cana-756 6 33 ring ring NOUN cana-756 6 34 . . PUNCT cana-756 7 1 suppose suppose VERB cana-756 7 2 that that SCONJ cana-756 7 3 𝜎 𝜎 PROPN cana-756 7 4 , , PUNCT cana-756 7 5 𝜏 𝜏 NOUN cana-756 7 6 are be AUX cana-756 7 7 automorphisms automorphism NOUN cana-756 7 8 on on ADP cana-756 7 9 r. r. PROPN cana-756 7 10 a a DET cana-756 7 11 symmetric symmetric ADJ cana-756 7 12 biadditive biadditive ADJ cana-756 7 13 mapping mapping NOUN cana-756 7 14 𝛿1 𝛿1 NOUN cana-756 7 15 : : PUNCT cana-756 7 16 𝑅𝑋 𝑅𝑋 PROPN cana-756 7 17 𝑅 𝑅 PROPN cana-756 7 18 β†’ β†’ SYM cana-756 7 19 𝑅 𝑅 PROPN cana-756 7 20 is be AUX cana-756 7 21 said say VERB cana-756 7 22 to to PART cana-756 7 23 be be AUX cana-756 7 24 a a DET cana-756 7 25 generalized generalized ADJ cana-756 7 26 symmetric symmetric ADJ cana-756 7 27 reverse reverse NOUN cana-756 7 28 bi-(𝜎,𝜏)derivation bi-(𝜎,𝜏)derivation PROPN cana-756 7 29 on on ADP cana-756 7 30 r r NOUN cana-756 7 31 if if SCONJ cana-756 7 32 there there PRON cana-756 7 33 exists exist VERB cana-756 7 34 a a DET cana-756 7 35 symmetric symmetric ADJ cana-756 7 36 reverse reverse NOUN cana-756 7 37 bi-(𝜎,𝜏)-derivation bi-(𝜎,𝜏)-derivation PROPN cana-756 7 38 d d NOUN cana-756 7 39 on on ADP cana-756 7 40 r r NOUN cana-756 7 41 such such ADJ cana-756 7 42 that that SCONJ cana-756 7 43 𝛿1(𝑒𝑣 𝛿1(𝑒𝑣 PROPN cana-756 7 44 , , PUNCT cana-756 7 45 𝑀 𝑀 ADP cana-756 7 46 ) ) PUNCT cana-756 7 47 = = SYM cana-756 7 48 𝛿1(𝑣 𝛿1(𝑣 PROPN cana-756 7 49 , , PUNCT cana-756 7 50 𝑀)𝜎(𝑒 𝑀)𝜎(𝑒 NOUN cana-756 7 51 ) ) PUNCT cana-756 7 52 + + NUM cana-756 7 53 𝜏(v)d(𝑒 𝜏(v)d(𝑒 NOUN cana-756 7 54 , , PUNCT cana-756 7 55 𝑀 𝑀 X cana-756 7 56 ) ) PUNCT cana-756 7 57 holds hold VERB cana-756 7 58 βˆ€ βˆ€ NOUN cana-756 7 59 𝑒 𝑒 NOUN cana-756 7 60 , , PUNCT cana-756 7 61 𝑣 𝑣 NOUN cana-756 7 62 , , PUNCT cana-756 7 63 𝑀 𝑀 ADP cana-756 7 64 ∈ ∈ NOUN cana-756 7 65 𝑅. 𝑅. NOUN cana-756 7 66 let let VERB cana-756 7 67 [ [ PUNCT cana-756 7 68 𝛿1 𝛿1 ADJ cana-756 7 69 , , PUNCT cana-756 7 70 𝐷1 𝐷1 NOUN cana-756 7 71 ] ] PUNCT cana-756 7 72 and and CCONJ cana-756 7 73 [ [ X cana-756 7 74 𝛿2 𝛿2 NOUN cana-756 7 75 , , PUNCT cana-756 7 76 𝐷2 𝐷2 PROPN cana-756 7 77 ] ] PUNCT cana-756 7 78 be be VERB cana-756 7 79 two two NUM cana-756 7 80 generalized generalized ADJ cana-756 7 81 symmetric symmetric ADJ cana-756 7 82 reverse reverse ADJ cana-756 7 83 bi-(𝜎,𝜏)-derivations bi-(𝜎,𝜏)-derivation NOUN cana-756 7 84 of of ADP cana-756 7 85 r r NOUN cana-756 7 86 with with ADP cana-756 7 87 associated associated ADJ cana-756 7 88 reverse reverse NOUN cana-756 7 89 bi-(𝜎 bi-(𝜎 NOUN cana-756 7 90 , , PUNCT cana-756 7 91 𝜏)-derivations 𝜏)-derivation NOUN cana-756 7 92 𝐷1 𝐷1 NOUN cana-756 7 93 , , PUNCT cana-756 7 94 𝐷2 𝐷2 PROPN cana-756 7 95 . . PUNCT cana-756 8 1 in in ADP cana-756 8 2 this this DET cana-756 8 3 paper paper NOUN cana-756 8 4 , , PUNCT cana-756 8 5 we we PRON cana-756 8 6 establish establish VERB cana-756 8 7 some some DET cana-756 8 8 equivalent equivalent ADJ cana-756 8 9 conditions condition NOUN cana-756 8 10 for for ADP cana-756 8 11 the the DET cana-756 8 12 orthogonality orthogonality NOUN cana-756 8 13 between between ADP cana-756 8 14 two two NUM cana-756 8 15 symmetric symmetric ADJ cana-756 8 16 generalized generalize VERB cana-756 8 17 reverse reverse ADJ cana-756 8 18 bi-(𝜎,𝜏)-derivations bi-(𝜎,𝜏)-derivation NOUN cana-756 8 19 of of ADP cana-756 8 20 semiprime semiprime NOUN cana-756 8 21 ring ring PROPN cana-756 8 22 r. r. PROPN cana-756 8 23 keywords keywords PROPN cana-756 8 24 : : PUNCT cana-756 8 25 semiprime semiprime NOUN cana-756 8 26 ring ring NOUN cana-756 8 27 , , PUNCT cana-756 8 28 generalized generalized ADJ cana-756 8 29 reverse reverse ADJ cana-756 8 30 biderivation biderivation NOUN cana-756 8 31 , , PUNCT cana-756 8 32 generalized generalized ADJ cana-756 8 33 reverse reverse NOUN cana-756 8 34 bi(𝜎,𝜏)-derivation bi(𝜎,𝜏)-derivation PROPN cana-756 8 35 , , PUNCT cana-756 8 36 orthogonal orthogonal ADJ cana-756 8 37 biderivation biderivation NOUN cana-756 8 38 . . PUNCT cana-756 9 1 1 1 X cana-756 9 2 . . X cana-756 9 3 introduction introduction NOUN cana-756 9 4 : : PUNCT cana-756 9 5 the the DET cana-756 9 6 concept concept NOUN cana-756 9 7 of of ADP cana-756 9 8 orthogonal orthogonal ADJ cana-756 9 9 derivation derivation NOUN cana-756 9 10 was be AUX cana-756 9 11 introduced introduce VERB cana-756 9 12 by by ADP cana-756 9 13 m. m. NOUN cana-756 9 14 bresar bresar VERB cana-756 9 15 and and CCONJ cana-756 9 16 j. j. PROPN cana-756 9 17 vukman vukman PROPN cana-756 10 1 [ [ X cana-756 10 2 14 14 NUM cana-756 10 3 ] ] PUNCT cana-756 11 1 and and CCONJ cana-756 11 2 proved prove VERB cana-756 11 3 some some DET cana-756 11 4 results result NOUN cana-756 11 5 on on ADP cana-756 11 6 the the DET cana-756 11 7 orthogonal orthogonal ADJ cana-756 11 8 derivations derivation NOUN cana-756 11 9 of of ADP cana-756 11 10 semiprime semiprime NOUN cana-756 11 11 rings ring NOUN cana-756 11 12 which which PRON cana-756 11 13 were be AUX cana-756 11 14 related relate VERB cana-756 11 15 to to ADP cana-756 11 16 posner posner NOUN cana-756 11 17 ’s ’s PART cana-756 11 18 first first ADJ cana-756 11 19 theorem theorem NOUN cana-756 12 1 [ [ X cana-756 12 2 9 9 NUM cana-756 12 3 ] ] PUNCT cana-756 12 4 . . PUNCT cana-756 13 1 some some DET cana-756 13 2 results result NOUN cana-756 13 3 on on ADP cana-756 13 4 ( ( PUNCT cana-756 13 5 𝜎 𝜎 INTJ cana-756 13 6 , , PUNCT cana-756 13 7 𝜏)-derivations 𝜏)-derivation NOUN cana-756 13 8 in in ADP cana-756 13 9 prime prime ADJ cana-756 13 10 rings ring NOUN cana-756 13 11 were be AUX cana-756 13 12 studied study VERB cana-756 13 13 by by ADP cana-756 13 14 m. m. NOUN cana-756 13 15 ashraf ashraf PROPN cana-756 14 1 [ [ X cana-756 14 2 13 13 NUM cana-756 14 3 ] ] PUNCT cana-756 14 4 and and CCONJ cana-756 14 5 k. k. PROPN cana-756 14 6 kaya kaya PROPN cana-756 14 7 et et PROPN cana-756 14 8 al al PROPN cana-756 14 9 . . PUNCT cana-756 15 1 [ [ X cana-756 15 2 12 12 NUM cana-756 15 3 ] ] PUNCT cana-756 15 4 . . PUNCT cana-756 16 1 j.c j.c PROPN cana-756 16 2 . . PROPN cana-756 16 3 chang chang PROPN cana-756 17 1 [ [ X cana-756 17 2 11 11 NUM cana-756 17 3 ] ] PUNCT cana-756 17 4 introduced introduce VERB cana-756 17 5 the the DET cana-756 17 6 notion notion NOUN cana-756 17 7 of of ADP cana-756 17 8 a a DET cana-756 17 9 generalized generalize VERB cana-756 17 10 ( ( PUNCT cana-756 17 11 𝛼 𝛼 NOUN cana-756 17 12 , , PUNCT cana-756 17 13 𝛽)derivation 𝛽)derivation NOUN cana-756 17 14 of of ADP cana-756 17 15 a a DET cana-756 17 16 ring ring NOUN cana-756 17 17 r r NOUN cana-756 17 18 and and CCONJ cana-756 17 19 investigated investigate VERB cana-756 17 20 some some DET cana-756 17 21 properties property NOUN cana-756 17 22 of of ADP cana-756 17 23 such such ADJ cana-756 17 24 derivations derivation NOUN cana-756 17 25 . . PUNCT cana-756 18 1 argac argac PROPN cana-756 18 2 et et PROPN cana-756 18 3 al al PROPN cana-756 18 4 . . PUNCT cana-756 19 1 [ [ X cana-756 19 2 17 17 NUM cana-756 19 3 ] ] PUNCT cana-756 19 4 introduced introduce VERB cana-756 19 5 the the DET cana-756 19 6 notion notion NOUN cana-756 19 7 of of ADP cana-756 19 8 orthogonality orthogonality NOUN cana-756 19 9 for for ADP cana-756 19 10 a a DET cana-756 19 11 pair pair NOUN cana-756 19 12 ( ( PUNCT cana-756 19 13 d d NOUN cana-756 19 14 , , PUNCT cana-756 19 15 d d NOUN cana-756 19 16 ) ) PUNCT cana-756 19 17 , , PUNCT cana-756 19 18 ( ( PUNCT cana-756 19 19 g g NOUN cana-756 19 20 , , PUNCT cana-756 19 21 g g NOUN cana-756 19 22 ) ) PUNCT cana-756 19 23 of of ADP cana-756 19 24 generalized generalized ADJ cana-756 19 25 derivations derivation NOUN cana-756 19 26 on on ADP cana-756 19 27 semiprime semiprime NOUN cana-756 19 28 rings ring NOUN cana-756 19 29 and and CCONJ cana-756 19 30 gave give VERB cana-756 19 31 several several ADJ cana-756 19 32 necessary necessary ADJ cana-756 19 33 and and CCONJ cana-756 19 34 sufficient sufficient ADJ cana-756 19 35 conditions condition NOUN cana-756 19 36 for for ADP cana-756 19 37 ( ( PUNCT cana-756 19 38 d d NOUN cana-756 19 39 , , PUNCT cana-756 19 40 d d NOUN cana-756 19 41 ) ) PUNCT cana-756 19 42 and and CCONJ cana-756 19 43 ( ( PUNCT cana-756 19 44 g g NOUN cana-756 19 45 , , PUNCT cana-756 19 46 g g NOUN cana-756 19 47 ) ) PUNCT cana-756 19 48 to to PART cana-756 19 49 be be AUX cana-756 19 50 orthogonal orthogonal ADJ cana-756 19 51 . . PUNCT cana-756 20 1 o.golbasi o.golbasi NOUN cana-756 20 2 and and CCONJ cana-756 20 3 n. n. NOUN cana-756 20 4 aydin aydin NOUN cana-756 20 5 [ [ X cana-756 20 6 18 18 NUM cana-756 20 7 ] ] PUNCT cana-756 20 8 extended extend VERB cana-756 20 9 the the DET cana-756 20 10 results result NOUN cana-756 20 11 of of ADP cana-756 20 12 argac argac PROPN cana-756 20 13 to to PART cana-756 20 14 orthogonal orthogonal VERB cana-756 20 15 generalized generalized ADJ cana-756 20 16 ( ( PUNCT cana-756 20 17 𝜎 𝜎 INTJ cana-756 20 18 , , PUNCT cana-756 20 19 𝜏)-derivations 𝜏)-derivation NOUN cana-756 20 20 . . PUNCT cana-756 21 1 orthogonality orthogonality NOUN cana-756 21 2 of of ADP cana-756 21 3 generalized generalized ADJ cana-756 21 4 ( ( PUNCT cana-756 21 5 𝜎 𝜎 INTJ cana-756 21 6 , , PUNCT cana-756 21 7 𝜏)-derivations 𝜏)-derivation NOUN cana-756 21 8 on on ADP cana-756 21 9 ideals ideal NOUN cana-756 21 10 of of ADP cana-756 21 11 semiprime semiprime NOUN cana-756 21 12 rings ring NOUN cana-756 21 13 was be AUX cana-756 21 14 studied study VERB cana-756 21 15 in in ADP cana-756 21 16 [ [ X cana-756 21 17 10 10 NUM cana-756 21 18 ] ] PUNCT cana-756 21 19 . . PUNCT cana-756 22 1 several several ADJ cana-756 22 2 studies study NOUN cana-756 22 3 were be AUX cana-756 22 4 established establish VERB cana-756 22 5 on on ADP cana-756 22 6 the the DET cana-756 22 7 orthogonality orthogonality NOUN cana-756 22 8 of of ADP cana-756 22 9 derivations derivation NOUN cana-756 22 10 , , PUNCT cana-756 22 11 biderivations biderivation NOUN cana-756 22 12 by by ADP cana-756 22 13 m.n m.n PROPN cana-756 22 14 . . PROPN cana-756 22 15 daif daif PROPN cana-756 22 16 et et PROPN cana-756 22 17 al al PROPN cana-756 22 18 . . PUNCT cana-756 23 1 [ [ X cana-756 23 2 15 15 NUM cana-756 23 3 ] ] PUNCT cana-756 23 4 and and CCONJ cana-756 23 5 c. c. PROPN cana-756 23 6 jaya jaya PROPN cana-756 23 7 subba subba PROPN cana-756 23 8 reddy reddy PROPN cana-756 23 9 et et PROPN cana-756 23 10 al al PROPN cana-756 23 11 . . PUNCT cana-756 24 1 [ [ X cana-756 24 2 2,4 2,4 NUM cana-756 24 3 , , PUNCT cana-756 24 4 5 5 NUM cana-756 24 5 ] ] PUNCT cana-756 24 6 . . PUNCT cana-756 25 1 a.ali a.ali PROPN cana-756 25 2 et et PROPN cana-756 25 3 al al PROPN cana-756 25 4 . . PUNCT cana-756 26 1 [ [ X cana-756 26 2 1 1 X cana-756 26 3 ] ] PUNCT cana-756 26 4 and and CCONJ cana-756 26 5 m.n.daif m.n.daif PROPN cana-756 26 6 et et PROPN cana-756 26 7 al al PROPN cana-756 26 8 . . PUNCT cana-756 27 1 [ [ X cana-756 27 2 16 16 NUM cana-756 27 3 ] ] PUNCT cana-756 27 4 established establish VERB cana-756 27 5 some some DET cana-756 27 6 results result NOUN cana-756 27 7 on on ADP cana-756 27 8 biderivations biderivation NOUN cana-756 27 9 of of ADP cana-756 27 10 prime prime ADJ cana-756 27 11 and and CCONJ cana-756 27 12 semiprime semiprime NOUN cana-756 27 13 rings ring NOUN cana-756 27 14 and and CCONJ cana-756 27 15 the the DET cana-756 27 16 study study NOUN cana-756 27 17 of of ADP cana-756 27 18 orthogonality orthogonality NOUN cana-756 27 19 of of ADP cana-756 27 20 symmetric symmetric ADJ cana-756 27 21 bi-(𝜎 bi-(𝜎 NOUN cana-756 27 22 , , PUNCT cana-756 27 23 𝜏)-derivations 𝜏)-derivation NOUN cana-756 27 24 in in ADP cana-756 27 25 semi semi ADJ cana-756 27 26 prime prime ADJ cana-756 27 27 rings ring NOUN cana-756 27 28 was be AUX cana-756 27 29 carried carry VERB cana-756 27 30 out out ADP cana-756 27 31 in in ADP cana-756 27 32 [ [ X cana-756 27 33 3,6 3,6 NUM cana-756 27 34 ] ] PUNCT cana-756 27 35 . . PUNCT cana-756 28 1 recently recently ADV cana-756 28 2 , , PUNCT cana-756 28 3 c. c. PROPN cana-756 28 4 jaya jaya PROPN cana-756 28 5 subba subba PROPN cana-756 28 6 reddy reddy PROPN cana-756 28 7 et et PROPN cana-756 28 8 al al PROPN cana-756 28 9 . . PUNCT cana-756 29 1 [ [ X cana-756 29 2 7 7 NUM cana-756 29 3 , , PUNCT cana-756 29 4 8 8 NUM cana-756 29 5 ] ] PUNCT cana-756 29 6 have have AUX cana-756 29 7 studied study VERB cana-756 29 8 orthogonal orthogonal ADJ cana-756 29 9 symmetric symmetric ADJ cana-756 29 10 reverse reverse ADJ cana-756 29 11 bi-(𝜎,𝜏)derivations bi-(𝜎,𝜏)derivations PROPN cana-756 29 12 in in ADP cana-756 29 13 semi semi ADJ cana-756 29 14 prime prime ADJ cana-756 29 15 rings ring NOUN cana-756 29 16 and and CCONJ cana-756 29 17 orthogonal orthogonal ADJ cana-756 29 18 generalized generalized ADJ cana-756 29 19 reverse reverse NOUN cana-756 29 20 ( ( PUNCT cana-756 29 21 Οƒ Οƒ NOUN cana-756 29 22 , , PUNCT cana-756 29 23 Ο„)derivations Ο„)derivations PROPN cana-756 29 24 in in ADP cana-756 29 25 semiprime semiprime NOUN cana-756 29 26 Ξ³ Ξ³ PROPN cana-756 29 27 - - PUNCT cana-756 29 28 rings ring NOUN cana-756 29 29 . . PUNCT cana-756 30 1 in in ADP cana-756 30 2 the the DET cana-756 30 3 present present ADJ cana-756 30 4 paper paper NOUN cana-756 30 5 , , PUNCT cana-756 30 6 we we PRON cana-756 30 7 extended extend VERB cana-756 30 8 the the DET cana-756 30 9 results result NOUN cana-756 30 10 of of ADP cana-756 30 11 orthogonality orthogonality NOUN cana-756 30 12 on on ADP cana-756 30 13 generalized generalized ADJ cana-756 30 14 symmetric symmetric ADJ cana-756 30 15 bi(𝜎 bi(𝜎 NOUN cana-756 30 16 , , PUNCT cana-756 30 17 𝜏)-derivations 𝜏)-derivation NOUN cana-756 30 18 established establish VERB cana-756 30 19 in in ADP cana-756 30 20 [ [ X cana-756 30 21 6 6 NUM cana-756 30 22 ] ] PUNCT cana-756 30 23 to to PART cana-756 30 24 generalized generalize VERB cana-756 30 25 symmetric symmetric ADJ cana-756 30 26 reverse reverse NOUN cana-756 30 27 bi-(𝜎 bi-(𝜎 NOUN cana-756 30 28 , , PUNCT cana-756 30 29 𝜏)-derivations 𝜏)-derivation NOUN cana-756 30 30 . . PUNCT cana-756 31 1 communications communication NOUN cana-756 31 2 on on ADP cana-756 31 3 applied apply VERB cana-756 31 4 nonlinear nonlinear ADJ cana-756 31 5 analysis analysis NOUN cana-756 31 6 issn issn NOUN cana-756 31 7 : : PUNCT cana-756 31 8 1074 1074 NUM cana-756 31 9 - - PUNCT cana-756 31 10 133x 133x NUM cana-756 31 11 vol vol NOUN cana-756 31 12 31 31 NUM cana-756 31 13 no no NOUN cana-756 31 14 . . PUNCT cana-756 32 1 3s 3s NUM cana-756 32 2 ( ( PUNCT cana-756 32 3 2024 2024 NUM cana-756 32 4 ) ) PUNCT cana-756 32 5 157 157 NUM cana-756 32 6 https://internationalpubls.com https://internationalpubls.com X cana-756 32 7 2 2 NUM cana-756 32 8 . . PUNCT cana-756 32 9 preliminaries preliminary NOUN cana-756 32 10 : : PUNCT cana-756 32 11 throughout throughout ADP cana-756 32 12 this this DET cana-756 32 13 paper paper NOUN cana-756 32 14 , , PUNCT cana-756 32 15 r r NOUN cana-756 32 16 will will AUX cana-756 32 17 denote denote VERB cana-756 32 18 an an DET cana-756 32 19 associative associative ADJ cana-756 32 20 ring ring NOUN cana-756 32 21 with with ADP cana-756 32 22 center center NOUN cana-756 32 23 z(r z(r NOUN cana-756 32 24 ) ) PUNCT cana-756 32 25 . . PUNCT cana-756 33 1 a a DET cana-756 33 2 ring ring NOUN cana-756 33 3 r r NOUN cana-756 33 4 is be AUX cana-756 33 5 known know VERB cana-756 33 6 to to PART cana-756 33 7 be be AUX cana-756 33 8 semiprime semiprime NOUN cana-756 33 9 if if SCONJ cana-756 33 10 𝑒𝑅𝑒 𝑒𝑅𝑒 NOUN cana-756 33 11 = = SYM cana-756 33 12 { { PUNCT cana-756 33 13 0 0 NUM cana-756 33 14 } } PUNCT cana-756 33 15 implies imply VERB cana-756 33 16 𝑒 𝑒 NOUN cana-756 33 17 = = SYM cana-756 33 18 0 0 NUM cana-756 33 19 , , PUNCT cana-756 33 20 βˆ€ βˆ€ X cana-756 33 21 𝑒 𝑒 NOUN cana-756 33 22 ∈ ∈ NOUN cana-756 33 23 𝑅. 𝑅. NOUN cana-756 33 24 we we PRON cana-756 33 25 say say VERB cana-756 33 26 that that SCONJ cana-756 33 27 r r NOUN cana-756 33 28 is be AUX cana-756 33 29 2 2 NUM cana-756 33 30 - - PUNCT cana-756 33 31 torsion torsion NOUN cana-756 33 32 - - PUNCT cana-756 33 33 free free ADJ cana-756 33 34 if if SCONJ cana-756 33 35 2𝑒 2𝑒 NUM cana-756 34 1 = = SYM cana-756 34 2 0 0 NUM cana-756 34 3 implies imply VERB cana-756 34 4 𝑒 𝑒 NOUN cana-756 34 5 = = SYM cana-756 34 6 0 0 NUM cana-756 34 7 , , PUNCT cana-756 34 8 βˆ€ βˆ€ X cana-756 34 9 𝑒 𝑒 NOUN cana-756 34 10 ∈ ∈ NOUN cana-756 34 11 𝑅. 𝑅. VERB cana-756 34 12 an an DET cana-756 34 13 additive additive ADJ cana-756 34 14 mapping mapping NOUN cana-756 34 15 𝑑 𝑑 NOUN cana-756 34 16 : : PUNCT cana-756 34 17 𝑅 𝑅 PROPN cana-756 34 18 β†’ β†’ SYM cana-756 34 19 𝑅 𝑅 PROPN cana-756 34 20 is be AUX cana-756 34 21 said say VERB cana-756 34 22 to to PART cana-756 34 23 be be AUX cana-756 34 24 a a DET cana-756 34 25 derivation derivation NOUN cana-756 34 26 ( ( PUNCT cana-756 34 27 respectively respectively ADV cana-756 34 28 , , PUNCT cana-756 34 29 reverse reverse ADJ cana-756 34 30 derivation derivation NOUN cana-756 34 31 ) ) PUNCT cana-756 34 32 on on ADP cana-756 34 33 r r NOUN cana-756 34 34 if if SCONJ cana-756 34 35 𝑑(𝑒𝑣 𝑑(𝑒𝑣 VERB cana-756 34 36 ) ) PUNCT cana-756 34 37 = = SYM cana-756 34 38 𝑑(𝑒)𝑣 𝑑(𝑒)𝑣 PROPN cana-756 34 39 + + CCONJ cana-756 34 40 𝑒𝑑(𝑣 𝑒𝑑(𝑣 NOUN cana-756 34 41 ) ) PUNCT cana-756 34 42 ( ( PUNCT cana-756 34 43 respectively respectively ADV cana-756 34 44 , , PUNCT cana-756 34 45 𝑑(𝑒𝑣 𝑑(𝑒𝑣 PROPN cana-756 34 46 ) ) PUNCT cana-756 34 47 = = SYM cana-756 34 48 𝑑(𝑣)𝑒 𝑑(𝑣)𝑒 NOUN cana-756 34 49 + + NUM cana-756 34 50 𝑣𝑑(𝑒 𝑣𝑑(𝑒 NUM cana-756 34 51 ) ) PUNCT cana-756 34 52 holds hold VERB cana-756 34 53 for for ADP cana-756 34 54 all all DET cana-756 34 55 𝑒 𝑒 NOUN cana-756 34 56 , , PUNCT cana-756 34 57 𝑣 𝑣 PRON cana-756 34 58 ∈ ∈ NOUN cana-756 34 59 𝑅. 𝑅. NOUN cana-756 34 60 suppose suppose VERB cana-756 34 61 that that SCONJ cana-756 34 62 𝜎 𝜎 PROPN cana-756 34 63 and and CCONJ cana-756 34 64 𝜏 𝜏 NOUN cana-756 34 65 are be AUX cana-756 34 66 automorphisms automorphism NOUN cana-756 34 67 of of ADP cana-756 34 68 r. r. PROPN cana-756 34 69 an an DET cana-756 34 70 additive additive ADJ cana-756 34 71 mapping mapping NOUN cana-756 34 72 𝑑 𝑑 NOUN cana-756 34 73 : : PUNCT cana-756 34 74 𝑅 𝑅 PROPN cana-756 34 75 β†’ β†’ SYM cana-756 34 76 𝑅 𝑅 PROPN cana-756 34 77 is be AUX cana-756 34 78 said say VERB cana-756 34 79 to to PART cana-756 34 80 be be AUX cana-756 34 81 a a DET cana-756 34 82 ( ( PUNCT cana-756 34 83 𝜎,𝜏)-derivation 𝜎,𝜏)-derivation PROPN cana-756 34 84 ( ( PUNCT cana-756 34 85 respectively respectively ADV cana-756 34 86 , , PUNCT cana-756 34 87 reverse reverse NOUN cana-756 34 88 ( ( PUNCT cana-756 34 89 𝜎,𝜏)-derivation 𝜎,𝜏)-derivation NUM cana-756 34 90 ) ) PUNCT cana-756 34 91 on on ADP cana-756 34 92 r r NOUN cana-756 34 93 if if SCONJ cana-756 34 94 𝑑(𝑒𝑣 𝑑(𝑒𝑣 VERB cana-756 34 95 ) ) PUNCT cana-756 34 96 = = SYM cana-756 34 97 𝑑(𝑒)𝜎(𝑣 𝑑(𝑒)𝜎(𝑣 NOUN cana-756 34 98 ) ) PUNCT cana-756 34 99 + + NUM cana-756 34 100 𝜏(𝑒)𝑑(𝑣 𝜏(𝑒)𝑑(𝑣 NOUN cana-756 34 101 ) ) PUNCT cana-756 34 102 ( ( PUNCT cana-756 34 103 respectively respectively ADV cana-756 34 104 , , PUNCT cana-756 34 105 𝑑(𝑒𝑣 𝑑(𝑒𝑣 PROPN cana-756 34 106 ) ) PUNCT cana-756 34 107 = = SYM cana-756 34 108 𝑑(𝑣)𝜎(𝑒 𝑑(𝑣)𝜎(𝑒 X cana-756 34 109 ) ) PUNCT cana-756 34 110 + + CCONJ cana-756 34 111 𝜏(𝑣)𝑑(𝑒 𝜏(𝑣)𝑑(𝑒 NOUN cana-756 34 112 ) ) PUNCT cana-756 34 113 ) ) PUNCT cana-756 35 1 holds hold VERB cana-756 35 2 , , PUNCT cana-756 35 3 βˆ€ βˆ€ X cana-756 35 4 𝑒 𝑒 NOUN cana-756 35 5 , , PUNCT cana-756 35 6 𝑣 𝑣 PRON cana-756 35 7 ∈ ∈ NOUN cana-756 35 8 𝑅. 𝑅. VERB cana-756 35 9 an an DET cana-756 35 10 additive additive ADJ cana-756 35 11 mapping mapping NOUN cana-756 35 12 𝐷1 𝐷1 NOUN cana-756 35 13 : : PUNCT cana-756 35 14 𝑅 𝑅 PROPN cana-756 35 15 β†’ β†’ SYM cana-756 35 16 𝑅 𝑅 PROPN cana-756 35 17 is be AUX cana-756 35 18 called call VERB cana-756 35 19 a a DET cana-756 35 20 generalized generalized ADJ cana-756 35 21 derivation derivation NOUN cana-756 35 22 ( ( PUNCT cana-756 35 23 respectively respectively ADV cana-756 35 24 , , PUNCT cana-756 35 25 generalized generalized ADJ cana-756 35 26 reverse reverse ADJ cana-756 35 27 derivation derivation NOUN cana-756 35 28 ) ) PUNCT cana-756 35 29 if if SCONJ cana-756 35 30 there there PRON cana-756 35 31 exists exist VERB cana-756 35 32 a a DET cana-756 35 33 derivation derivation NOUN cana-756 35 34 ( ( PUNCT cana-756 35 35 respectively respectively ADV cana-756 35 36 , , PUNCT cana-756 35 37 reverse reverse ADJ cana-756 35 38 derivation derivation NOUN cana-756 35 39 ) ) PUNCT cana-756 35 40 β€˜ ' PUNCT cana-756 35 41 d d NOUN cana-756 35 42 ’ ' PUNCT cana-756 35 43 such such ADJ cana-756 35 44 that that DET cana-756 35 45 𝐷1(𝑒𝑣 𝐷1(𝑒𝑣 NOUN cana-756 35 46 ) ) PUNCT cana-756 35 47 = = SYM cana-756 36 1 𝐷1(𝑒)𝑣 𝐷1(𝑒)𝑣 NOUN cana-756 36 2 + + CCONJ cana-756 36 3 𝑒𝑑(𝑣 𝑒𝑑(𝑣 NOUN cana-756 36 4 ) ) PUNCT cana-756 36 5 ( ( PUNCT cana-756 36 6 respectively,𝐷1(𝑒𝑣 respectively,𝐷1(𝑒𝑣 PROPN cana-756 36 7 ) ) PUNCT cana-756 36 8 = = PUNCT cana-756 37 1 𝐷1(𝑣)𝑒 𝐷1(𝑣)𝑒 X cana-756 37 2 + + NOUN cana-756 37 3 𝑣𝑑(𝑒 𝑣𝑑(𝑒 NUM cana-756 37 4 ) ) PUNCT cana-756 37 5 holds hold VERB cana-756 37 6 βˆ€ βˆ€ NOUN cana-756 37 7 𝑒 𝑒 NOUN cana-756 37 8 , , PUNCT cana-756 37 9 𝑣 𝑣 PRON cana-756 37 10 ∈ ∈ NOUN cana-756 37 11 𝑅. 𝑅. VERB cana-756 37 12 an an DET cana-756 37 13 additive additive ADJ cana-756 37 14 mapping mapping NOUN cana-756 37 15 𝐷1 𝐷1 NOUN cana-756 37 16 : : PUNCT cana-756 37 17 𝑅 𝑅 PROPN cana-756 37 18 β†’ β†’ SYM cana-756 37 19 𝑅 𝑅 PROPN cana-756 37 20 is be AUX cana-756 37 21 called call VERB cana-756 37 22 a a DET cana-756 37 23 generalized generalized ADJ cana-756 37 24 ( ( PUNCT cana-756 37 25 𝜎,𝜏)-derivation 𝜎,𝜏)-derivation NUM cana-756 37 26 ( ( PUNCT cana-756 37 27 respectively respectively ADV cana-756 37 28 , , PUNCT cana-756 37 29 generalized generalized ADJ cana-756 37 30 reverse reverse NOUN cana-756 37 31 ( ( PUNCT cana-756 37 32 𝜎 𝜎 NOUN cana-756 37 33 , , PUNCT cana-756 37 34 𝜏)derivation 𝜏)derivation PROPN cana-756 37 35 ) ) PUNCT cana-756 37 36 if if SCONJ cana-756 37 37 there there PRON cana-756 37 38 exists exist VERB cana-756 37 39 a a DET cana-756 37 40 ( ( PUNCT cana-756 37 41 𝜎,𝜏)-derivation 𝜎,𝜏)-derivation PROPN cana-756 37 42 ( ( PUNCT cana-756 37 43 respectively respectively ADV cana-756 37 44 , , PUNCT cana-756 37 45 reverse reverse NOUN cana-756 37 46 ( ( PUNCT cana-756 37 47 𝜎,𝜏)-derivation 𝜎,𝜏)-derivation NUM cana-756 37 48 ) ) PUNCT cana-756 37 49 β€˜ ' PUNCT cana-756 37 50 d d NOUN cana-756 37 51 ’ ' PUNCT cana-756 37 52 such such ADJ cana-756 37 53 that that DET cana-756 37 54 𝐷1(𝑒𝑣 𝐷1(𝑒𝑣 NOUN cana-756 37 55 ) ) PUNCT cana-756 37 56 = = SYM cana-756 37 57 𝐷1(𝑒)𝜎(𝑣 𝐷1(𝑒)𝜎(𝑣 NOUN cana-756 37 58 ) ) PUNCT cana-756 38 1 + + NUM cana-756 38 2 𝜏(𝑒)𝑑(𝑣 𝜏(𝑒)𝑑(𝑣 NOUN cana-756 38 3 ) ) PUNCT cana-756 38 4 ( ( PUNCT cana-756 38 5 respectively respectively ADV cana-756 38 6 , , PUNCT cana-756 38 7 𝐷1(𝑒𝑣 𝐷1(𝑒𝑣 PROPN cana-756 38 8 ) ) PUNCT cana-756 38 9 = = PUNCT cana-756 38 10 𝐷1(𝑣)𝜎(𝑒 𝐷1(𝑣)𝜎(𝑒 X cana-756 38 11 ) ) PUNCT cana-756 39 1 + + CCONJ cana-756 39 2 𝜏(𝑣)𝑑(𝑒 𝜏(𝑣)𝑑(𝑒 NOUN cana-756 39 3 ) ) PUNCT cana-756 39 4 holds hold VERB cana-756 39 5 βˆ€ βˆ€ NOUN cana-756 39 6 𝑒 𝑒 NOUN cana-756 39 7 , , PUNCT cana-756 39 8 𝑣 𝑣 PRON cana-756 39 9 ∈ ∈ NOUN cana-756 39 10 𝑅. 𝑅. ADV cana-756 39 11 thus thus ADV cana-756 39 12 , , PUNCT cana-756 39 13 the the DET cana-756 39 14 concept concept NOUN cana-756 39 15 of of ADP cana-756 39 16 generalized generalized ADJ cana-756 39 17 ( ( PUNCT cana-756 39 18 𝜎 𝜎 INTJ cana-756 39 19 , , PUNCT cana-756 39 20 𝜏)-derivation 𝜏)-derivation NOUN cana-756 39 21 covers cover VERB cana-756 39 22 the the DET cana-756 39 23 concept concept NOUN cana-756 39 24 of of ADP cana-756 39 25 ( ( PUNCT cana-756 39 26 𝜎,𝜏)-derivation 𝜎,𝜏)-derivation NUM cana-756 39 27 . . PUNCT cana-756 40 1 a a DET cana-756 40 2 bi bi ADJ cana-756 40 3 additive additive NOUN cana-756 40 4 mapping mapping NOUN cana-756 40 5 d1 d1 NOUN cana-756 40 6 : : PUNCT cana-756 40 7 rx rx VERB cana-756 40 8 r r NOUN cana-756 40 9 β†’ β†’ SYM cana-756 40 10 r r NOUN cana-756 40 11 is be AUX cana-756 40 12 said say VERB cana-756 40 13 to to PART cana-756 40 14 be be AUX cana-756 40 15 symmetric symmetric ADJ cana-756 40 16 if if SCONJ cana-756 40 17 d1(u d1(u PROPN cana-756 40 18 , , PUNCT cana-756 40 19 v v NOUN cana-756 40 20 ) ) PUNCT cana-756 40 21 = = SYM cana-756 40 22 d1(v d1(v PROPN cana-756 40 23 , , PUNCT cana-756 40 24 u u NOUN cana-756 40 25 ) ) PUNCT cana-756 40 26 . . PUNCT cana-756 41 1 a a DET cana-756 41 2 symmetric symmetric ADJ cana-756 41 3 bi bi NOUN cana-756 41 4 additive additive NOUN cana-756 41 5 mapping mapping NOUN cana-756 41 6 d1 d1 NOUN cana-756 41 7 : : PUNCT cana-756 41 8 rx rx VERB cana-756 41 9 r r NOUN cana-756 41 10 β†’ β†’ SYM cana-756 41 11 r r NOUN cana-756 41 12 is be AUX cana-756 41 13 said say VERB cana-756 41 14 to to PART cana-756 41 15 be be AUX cana-756 41 16 a a DET cana-756 41 17 symmetric symmetric ADJ cana-756 41 18 biderivation biderivation NOUN cana-756 41 19 on on ADP cana-756 41 20 r r PROPN cana-756 41 21 if if SCONJ cana-756 41 22 d1(uv d1(uv PROPN cana-756 41 23 , , PUNCT cana-756 41 24 w w NOUN cana-756 41 25 ) ) PUNCT cana-756 41 26 = = SYM cana-756 41 27 ud1(v ud1(v PROPN cana-756 41 28 , , PUNCT cana-756 41 29 w w NOUN cana-756 41 30 ) ) PUNCT cana-756 41 31 + + CCONJ cana-756 41 32 d1(u d1(u PROPN cana-756 41 33 , , PUNCT cana-756 41 34 w)v w)v X cana-756 41 35 holds hold VERB cana-756 41 36 βˆ€ βˆ€ NOUN cana-756 41 37 u u NOUN cana-756 41 38 , , PUNCT cana-756 41 39 v v INTJ cana-756 41 40 , , PUNCT cana-756 41 41 w w PROPN cana-756 41 42 ∈ ∈ PROPN cana-756 41 43 r. r. NOUN cana-756 41 44 a a DET cana-756 41 45 symmetric symmetric ADJ cana-756 41 46 biadditive biadditive ADJ cana-756 41 47 mapping mapping NOUN cana-756 41 48 d1 d1 NOUN cana-756 41 49 : : PUNCT cana-756 41 50 rx rx VERB cana-756 41 51 r r NOUN cana-756 41 52 β†’ β†’ SYM cana-756 41 53 r r NOUN cana-756 41 54 is be AUX cana-756 41 55 said say VERB cana-756 41 56 to to PART cana-756 41 57 be be AUX cana-756 41 58 a a DET cana-756 41 59 symmetric symmetric ADJ cana-756 41 60 bi-(𝜎,Ο„)-derivation bi-(𝜎,Ο„)-derivation NOUN cana-756 41 61 ( ( PUNCT cana-756 41 62 respectively respectively ADV cana-756 41 63 , , PUNCT cana-756 41 64 symmetric symmetric ADJ cana-756 41 65 reverse reverse ADJ cana-756 41 66 bi-(𝜎,Ο„)derivation bi-(𝜎,Ο„)derivation NOUN cana-756 41 67 ) ) PUNCT cana-756 41 68 on on ADP cana-756 41 69 r r NOUN cana-756 41 70 if if SCONJ cana-756 41 71 d1 d1 PROPN cana-756 41 72 ( ( PUNCT cana-756 41 73 uv uv INTJ cana-756 41 74 , , PUNCT cana-756 41 75 w)= w)= PROPN cana-756 41 76 d1(u d1(u PROPN cana-756 41 77 , , PUNCT cana-756 41 78 w)Οƒ(v w)Οƒ(v NOUN cana-756 41 79 ) ) PUNCT cana-756 42 1 + + CCONJ cana-756 42 2 Ο„(u Ο„(u PROPN cana-756 42 3 ) ) PUNCT cana-756 42 4 d1(v d1(v PROPN cana-756 42 5 , , PUNCT cana-756 42 6 w w NOUN cana-756 42 7 ) ) PUNCT cana-756 42 8 ( ( PUNCT cana-756 42 9 respectively respectively ADV cana-756 42 10 , , PUNCT cana-756 42 11 d1(uv d1(uv PROPN cana-756 42 12 , , PUNCT cana-756 42 13 w w NOUN cana-756 42 14 ) ) PUNCT cana-756 43 1 = = SYM cana-756 43 2 d1 d1 PROPN cana-756 43 3 ( ( PUNCT cana-756 43 4 v v NOUN cana-756 43 5 , , PUNCT cana-756 43 6 w)Οƒ(u w)Οƒ(u NOUN cana-756 43 7 ) ) PUNCT cana-756 43 8 + + SYM cana-756 43 9 Ο„(v Ο„(v NOUN cana-756 43 10 ) ) PUNCT cana-756 43 11 d1(u d1(u PROPN cana-756 43 12 , , PUNCT cana-756 43 13 w w NOUN cana-756 43 14 ) ) PUNCT cana-756 43 15 holds hold VERB cana-756 43 16 βˆ€ βˆ€ NOUN cana-756 43 17 u u NOUN cana-756 43 18 , , PUNCT cana-756 43 19 v v INTJ cana-756 43 20 , , PUNCT cana-756 43 21 w w PROPN cana-756 43 22 ∈ ∈ PROPN cana-756 43 23 r. r. NOUN cana-756 43 24 a a DET cana-756 43 25 symmetric symmetric ADJ cana-756 43 26 biadditive biadditive ADJ cana-756 43 27 mapping mapping NOUN cana-756 43 28 Ξ΄ Ξ΄ NOUN cana-756 43 29 1 1 NUM cana-756 43 30 : : PUNCT cana-756 43 31 rx rx VERB cana-756 43 32 r r NOUN cana-756 43 33 β†’ β†’ SYM cana-756 43 34 r r NOUN cana-756 43 35 is be AUX cana-756 43 36 said say VERB cana-756 43 37 to to PART cana-756 43 38 be be AUX cana-756 43 39 a a DET cana-756 43 40 generalized generalized ADJ cana-756 43 41 symmetric symmetric ADJ cana-756 43 42 biderivation biderivation NOUN cana-756 43 43 ( ( PUNCT cana-756 43 44 respectively respectively ADV cana-756 43 45 , , PUNCT cana-756 43 46 generalized generalize VERB cana-756 43 47 symmetric symmetric ADJ cana-756 43 48 reverse reverse NOUN cana-756 43 49 biderivation biderivation NOUN cana-756 43 50 ) ) PUNCT cana-756 43 51 on on ADP cana-756 43 52 r r NOUN cana-756 43 53 if if SCONJ cana-756 43 54 there there PRON cana-756 43 55 exists exist VERB cana-756 43 56 a a DET cana-756 43 57 symmetric symmetric ADJ cana-756 43 58 biderivation biderivation NOUN cana-756 43 59 ( ( PUNCT cana-756 43 60 respectively respectively ADV cana-756 43 61 , , PUNCT cana-756 43 62 symmetric symmetric ADJ cana-756 43 63 reverse reverse NOUN cana-756 43 64 biderivation biderivation NOUN cana-756 43 65 ) ) PUNCT cana-756 43 66 d1on d1on PUNCT cana-756 44 1 r r NOUN cana-756 44 2 such such ADJ cana-756 44 3 that that SCONJ cana-756 44 4 Ξ΄1(uv Ξ΄1(uv PROPN cana-756 44 5 , , PUNCT cana-756 44 6 w w NOUN cana-756 44 7 ) ) PUNCT cana-756 44 8 = = SYM cana-756 44 9 Ξ΄1(u Ξ΄1(u NOUN cana-756 44 10 , , PUNCT cana-756 44 11 w)v w)v X cana-756 44 12 + + CCONJ cana-756 44 13 ud1(v ud1(v PROPN cana-756 44 14 , , PUNCT cana-756 44 15 w)(respectively w)(respectively ADV cana-756 44 16 , , PUNCT cana-756 44 17 Ξ΄1(uv Ξ΄1(uv PROPN cana-756 44 18 , , PUNCT cana-756 44 19 w)= w)= NOUN cana-756 44 20 Ξ΄1(v Ξ΄1(v SYM cana-756 44 21 , , PUNCT cana-756 44 22 w)u w)u X cana-756 44 23 + + CCONJ cana-756 44 24 vd1(u vd1(u PROPN cana-756 44 25 , , PUNCT cana-756 44 26 w w NOUN cana-756 44 27 ) ) PUNCT cana-756 44 28 , , PUNCT cana-756 44 29 βˆ€ βˆ€ X cana-756 44 30 u u NOUN cana-756 44 31 , , PUNCT cana-756 44 32 v v INTJ cana-756 44 33 , , PUNCT cana-756 44 34 w w PROPN cana-756 44 35 ∈ ∈ PROPN cana-756 44 36 r. r. NOUN cana-756 44 37 a a DET cana-756 44 38 symmetric symmetric ADJ cana-756 44 39 biadditive biadditive ADJ cana-756 44 40 mapping mapping NOUN cana-756 44 41 Ξ΄1 Ξ΄1 NOUN cana-756 44 42 : : PUNCT cana-756 44 43 rx rx VERB cana-756 44 44 r r NOUN cana-756 44 45 β†’ β†’ SYM cana-756 44 46 r r NOUN cana-756 44 47 is be AUX cana-756 44 48 said say VERB cana-756 44 49 to to PART cana-756 44 50 be be AUX cana-756 44 51 a a DET cana-756 44 52 generalized generalized ADJ cana-756 44 53 symmetric symmetric ADJ cana-756 44 54 bi-(𝜎,Ο„)-derivation bi-(𝜎,Ο„)-derivation NOUN cana-756 44 55 ( ( PUNCT cana-756 44 56 respectively respectively ADV cana-756 44 57 , , PUNCT cana-756 44 58 generalized generalize VERB cana-756 44 59 symmetric symmetric ADJ cana-756 44 60 reverse reverse NOUN cana-756 44 61 bi-(𝜎,Ο„)derivation bi-(𝜎,Ο„)derivation NOUN cana-756 44 62 ) ) PUNCT cana-756 44 63 on on ADP cana-756 44 64 r r NOUN cana-756 44 65 if if SCONJ cana-756 44 66 there there PRON cana-756 44 67 exists exist VERB cana-756 44 68 a a DET cana-756 44 69 symmetric symmetric ADJ cana-756 44 70 bi-(𝜎,Ο„)-derivation bi-(𝜎,Ο„)-derivation NOUN cana-756 44 71 ( ( PUNCT cana-756 44 72 respectively respectively ADV cana-756 44 73 , , PUNCT cana-756 44 74 symmetric symmetric ADJ cana-756 44 75 reverse reverse NOUN cana-756 44 76 bi(𝜎,Ο„)-derivation bi(𝜎,Ο„)-derivation NOUN cana-756 44 77 ) ) PUNCT cana-756 44 78 d1 d1 PROPN cana-756 44 79 on on ADP cana-756 44 80 r r NOUN cana-756 44 81 such such ADJ cana-756 44 82 that that SCONJ cana-756 44 83 Ξ΄1(uv Ξ΄1(uv PROPN cana-756 44 84 , , PUNCT cana-756 44 85 w w NOUN cana-756 44 86 ) ) PUNCT cana-756 44 87 = = SYM cana-756 44 88 Ξ΄1(u Ξ΄1(u PROPN cana-756 44 89 , , PUNCT cana-756 44 90 w w NOUN cana-756 44 91 ) ) PUNCT cana-756 44 92 Οƒ(v Οƒ(v NOUN cana-756 44 93 ) ) PUNCT cana-756 44 94 + + CCONJ cana-756 44 95 Ο„(u Ο„(u PROPN cana-756 44 96 ) ) PUNCT cana-756 44 97 d1(v d1(v PROPN cana-756 44 98 , , PUNCT cana-756 44 99 w w NOUN cana-756 44 100 ) ) PUNCT cana-756 44 101 ( ( PUNCT cana-756 44 102 respectively respectively ADV cana-756 44 103 , , PUNCT cana-756 44 104 Ξ΄1(uv Ξ΄1(uv PROPN cana-756 44 105 , , PUNCT cana-756 44 106 w w NOUN cana-756 44 107 ) ) PUNCT cana-756 44 108 = = SYM cana-756 44 109 Ξ΄1(v Ξ΄1(v PROPN cana-756 44 110 , , PUNCT cana-756 44 111 w)Οƒ(u w)Οƒ(u NOUN cana-756 44 112 ) ) PUNCT cana-756 44 113 + + CCONJ cana-756 44 114 Ο„(v)d1(u Ο„(v)d1(u ADJ cana-756 44 115 , , PUNCT cana-756 44 116 w w NOUN cana-756 44 117 ) ) PUNCT cana-756 44 118 holds hold VERB cana-756 44 119 βˆ€ βˆ€ NOUN cana-756 44 120 u u NOUN cana-756 44 121 , , PUNCT cana-756 44 122 v v INTJ cana-756 44 123 , , PUNCT cana-756 44 124 w w PROPN cana-756 44 125 ∈ ∈ PROPN cana-756 44 126 r. r. NOUN cana-756 44 127 two two NUM cana-756 44 128 symmetric symmetric ADJ cana-756 44 129 reverse reverse NOUN cana-756 44 130 bi-(Οƒ bi-(Οƒ NOUN cana-756 44 131 , , PUNCT cana-756 44 132 Ο„)derivations Ο„)derivation NOUN cana-756 44 133 d1 d1 NOUN cana-756 44 134 , , PUNCT cana-756 44 135 d2 d2 PROPN cana-756 44 136 are be AUX cana-756 44 137 said say VERB cana-756 44 138 to to PART cana-756 44 139 be be AUX cana-756 44 140 orthogonal orthogonal ADJ cana-756 44 141 if if SCONJ cana-756 44 142 d1(u d1(u PROPN cana-756 44 143 , , PUNCT cana-756 44 144 v)rd2(v v)rd2(v PROPN cana-756 44 145 , , PUNCT cana-756 44 146 w w NOUN cana-756 44 147 ) ) PUNCT cana-756 44 148 = = SYM cana-756 44 149 { { PUNCT cana-756 44 150 0 0 NUM cana-756 44 151 } } PUNCT cana-756 44 152 = = SYM cana-756 44 153 d2(v d2(v PROPN cana-756 44 154 , , PUNCT cana-756 44 155 w)rd1(u w)rd1(u PROPN cana-756 44 156 , , PUNCT cana-756 44 157 v v NOUN cana-756 44 158 ) ) PUNCT cana-756 44 159 , , PUNCT cana-756 44 160 for for ADP cana-756 44 161 all all DET cana-756 44 162 u u NOUN cana-756 44 163 , , PUNCT cana-756 44 164 v v NOUN cana-756 44 165 , , PUNCT cana-756 44 166 w w PROPN cana-756 44 167 ∈ ∈ PROPN cana-756 44 168 r. r. PROPN cana-756 44 169 two two NUM cana-756 44 170 generalized generalize VERB cana-756 44 171 symmetric symmetric ADJ cana-756 44 172 reverse reverse NOUN cana-756 44 173 bi-(Οƒ bi-(Οƒ NOUN cana-756 44 174 , , PUNCT cana-756 44 175 Ο„)-derivations Ο„)-derivation NOUN cana-756 44 176 Ξ΄1 Ξ΄1 NOUN cana-756 44 177 , , PUNCT cana-756 44 178 Ξ΄2 Ξ΄2 PROPN cana-756 44 179 are be AUX cana-756 44 180 said say VERB cana-756 44 181 to to PART cana-756 44 182 be be AUX cana-756 44 183 orthogonal orthogonal ADJ cana-756 44 184 if if SCONJ cana-756 44 185 Ξ΄1(u Ξ΄1(u PROPN cana-756 44 186 , , PUNCT cana-756 44 187 v)rΞ΄2(v v)rΞ΄2(v PROPN cana-756 44 188 , , PUNCT cana-756 44 189 w w PROPN cana-756 44 190 ) ) PUNCT cana-756 44 191 = = SYM cana-756 44 192 { { PUNCT cana-756 44 193 0 0 NUM cana-756 44 194 } } PUNCT cana-756 44 195 = = SYM cana-756 44 196 Ξ΄2(v Ξ΄2(v PROPN cana-756 44 197 , , PUNCT cana-756 44 198 w)rΞ΄1(u w)rΞ΄1(u PROPN cana-756 44 199 , , PUNCT cana-756 44 200 v v NOUN cana-756 44 201 ) ) PUNCT cana-756 44 202 , , PUNCT cana-756 44 203 for for ADP cana-756 44 204 all all DET cana-756 44 205 u u NOUN cana-756 44 206 , , PUNCT cana-756 44 207 v v NOUN cana-756 44 208 , , PUNCT cana-756 44 209 w w PROPN cana-756 44 210 ∈ ∈ PROPN cana-756 44 211 r. r. NOUN cana-756 44 212 we we PRON cana-756 44 213 assume assume VERB cana-756 44 214 throughout throughout ADP cana-756 44 215 the the DET cana-756 44 216 paper paper NOUN cana-756 44 217 that that PRON cana-756 44 218 r r NOUN cana-756 44 219 is be AUX cana-756 44 220 a a DET cana-756 44 221 2 2 NUM cana-756 44 222 - - PUNCT cana-756 44 223 torsion torsion NOUN cana-756 44 224 - - PUNCT cana-756 44 225 free free ADJ cana-756 44 226 semiprime semiprime NOUN cana-756 44 227 ring ring NOUN cana-756 44 228 , , PUNCT cana-756 44 229 while while SCONJ cana-756 44 230 Οƒ Οƒ PROPN cana-756 44 231 and and CCONJ cana-756 44 232 Ο„ Ο„ PROPN cana-756 44 233 are be AUX cana-756 44 234 automorphisms automorphism NOUN cana-756 44 235 of of ADP cana-756 44 236 r. r. PROPN cana-756 44 237 also also ADV cana-756 44 238 d1 d1 PROPN cana-756 44 239 , , PUNCT cana-756 44 240 d2 d2 PROPN cana-756 44 241 are be AUX cana-756 44 242 reverse reverse ADJ cana-756 44 243 bi-(Οƒ bi-(Οƒ NOUN cana-756 44 244 , , PUNCT cana-756 44 245 Ο„)-derivations Ο„)-derivation NOUN cana-756 44 246 of of ADP cana-756 44 247 r r NOUN cana-756 44 248 such such ADJ cana-756 44 249 that that SCONJ cana-756 44 250 d1Ο„ d1Ο„ PROPN cana-756 44 251 = = SYM cana-756 44 252 Ο„d1 Ο„d1 PROPN cana-756 44 253 , , PUNCT cana-756 44 254 d2Ο„ d2Ο„ PROPN cana-756 44 255 = = PUNCT cana-756 44 256 Ο„d2 Ο„d2 PROPN cana-756 44 257 , , PUNCT cana-756 44 258 Οƒd1 Οƒd1 PROPN cana-756 44 259 = = SYM cana-756 44 260 d1Οƒ d1Οƒ PROPN cana-756 44 261 , , PUNCT cana-756 44 262 Οƒd2 Οƒd2 X cana-756 44 263 = = PUNCT cana-756 45 1 d2Οƒ d2Οƒ NOUN cana-756 45 2 . . PUNCT cana-756 46 1 we we PRON cana-756 46 2 denote denote VERB cana-756 46 3 two two NUM cana-756 46 4 generalized generalized ADJ cana-756 46 5 reverse reverse NOUN cana-756 46 6 bi-(𝜎 bi-(𝜎 NOUN cana-756 46 7 , , PUNCT cana-756 46 8 𝜏)-derivations 𝜏)-derivation NOUN cana-756 46 9 Ξ΄1 Ξ΄1 NOUN cana-756 46 10 : : PUNCT cana-756 46 11 rx rx VERB cana-756 46 12 r r NOUN cana-756 46 13 β†’ β†’ SYM cana-756 46 14 r r NOUN cana-756 46 15 and and CCONJ cana-756 46 16 Ξ΄2 Ξ΄2 VERB cana-756 46 17 : : PUNCT cana-756 46 18 rx rx VERB cana-756 46 19 r r NOUN cana-756 46 20 β†’ β†’ SYM cana-756 46 21 r r NOUN cana-756 46 22 determined determine VERB cana-756 46 23 by by ADP cana-756 46 24 reverse reverse NOUN cana-756 46 25 bi-(Οƒ bi-(Οƒ NOUN cana-756 46 26 , , PUNCT cana-756 46 27 Ο„)-derivations Ο„)-derivation VERB cana-756 46 28 d1 d1 NOUN cana-756 46 29 , , PUNCT cana-756 46 30 d2 d2 PROPN cana-756 46 31 of of ADP cana-756 46 32 r r NOUN cana-756 46 33 be be VERB cana-756 46 34 such such ADJ cana-756 46 35 that that SCONJ cana-756 46 36 Ξ΄1Ο„ Ξ΄1Ο„ NOUN cana-756 46 37 = = SYM cana-756 46 38 τδ1 τδ1 NOUN cana-756 46 39 , , PUNCT cana-756 46 40 Ξ΄2Ο„ Ξ΄2Ο„ PROPN cana-756 46 41 = = PUNCT cana-756 46 42 τδ2 τδ2 NOUN cana-756 46 43 , , PUNCT cana-756 46 44 σδ1 σδ1 NOUN cana-756 46 45 = = SYM cana-756 46 46 Ξ΄1Οƒ Ξ΄1Οƒ PROPN cana-756 46 47 , , PUNCT cana-756 46 48 σδ2 σδ2 NOUN cana-756 46 49 = = SYM cana-756 46 50 Ξ΄2Οƒ Ξ΄2Οƒ PROPN cana-756 46 51 . . PUNCT cana-756 47 1 lemma lemma PROPN cana-756 47 2 1 1 NUM cana-756 47 3 : : PUNCT cana-756 48 1 [ [ X cana-756 48 2 lemma lemma PROPN cana-756 48 3 1,[14 1,[14 NUM cana-756 48 4 ] ] X cana-756 48 5 ] ] X cana-756 48 6 if if SCONJ cana-756 48 7 r r NOUN cana-756 48 8 is be AUX cana-756 48 9 a a DET cana-756 48 10 2 2 NUM cana-756 48 11 - - PUNCT cana-756 48 12 torsion torsion NOUN cana-756 48 13 free free ADJ cana-756 48 14 semi semi ADJ cana-756 48 15 prime prime ADJ cana-756 48 16 ring ring NOUN cana-756 48 17 and and CCONJ cana-756 48 18 𝑒 𝑒 NOUN cana-756 48 19 , , PUNCT cana-756 48 20 𝑣 𝑣 PRON cana-756 48 21 ∈ ∈ PROPN cana-756 48 22 𝑅 𝑅 PROPN cana-756 48 23 , , PUNCT cana-756 48 24 then then ADV cana-756 48 25 the the DET cana-756 48 26 following follow VERB cana-756 48 27 conditions condition NOUN cana-756 48 28 are be AUX cana-756 48 29 equivalent equivalent ADJ cana-756 48 30 : : PUNCT cana-756 48 31 communications communication NOUN cana-756 48 32 on on ADP cana-756 48 33 applied apply VERB cana-756 48 34 nonlinear nonlinear ADJ cana-756 48 35 analysis analysis NOUN cana-756 48 36 issn issn NOUN cana-756 48 37 : : PUNCT cana-756 48 38 1074 1074 NUM cana-756 48 39 - - PUNCT cana-756 48 40 133x 133x NUM cana-756 48 41 vol vol NOUN cana-756 48 42 31 31 NUM cana-756 48 43 no no NOUN cana-756 48 44 . . PUNCT cana-756 49 1 3s 3s NUM cana-756 49 2 ( ( PUNCT cana-756 49 3 2024 2024 NUM cana-756 49 4 ) ) PUNCT cana-756 49 5 158 158 NUM cana-756 49 6 https://internationalpubls.com https://internationalpubls.com X cana-756 49 7 1.π‘’π‘Ÿπ‘£ 1.π‘’π‘Ÿπ‘£ NUM cana-756 49 8 = = SYM cana-756 49 9 0 0 NUM cana-756 49 10 , , PUNCT cana-756 49 11 for for ADP cana-756 49 12 all all DET cana-756 49 13 π‘Ÿ π‘Ÿ PRON cana-756 49 14 ∈ ∈ NOUN cana-756 49 15 𝑅. 𝑅. NOUN cana-756 49 16 2.π‘£π‘Ÿπ‘’ 2.π‘£π‘Ÿπ‘’ NUM cana-756 49 17 = = SYM cana-756 49 18 0 0 NUM cana-756 49 19 , , PUNCT cana-756 49 20 for for ADP cana-756 49 21 all all DET cana-756 49 22 π‘Ÿ π‘Ÿ PRON cana-756 49 23 ∈ ∈ NOUN cana-756 49 24 𝑅. 𝑅. ADP cana-756 49 25 3 3 NUM cana-756 49 26 . . PUNCT cana-756 49 27 π‘’π‘Ÿπ‘£ π‘’π‘Ÿπ‘£ ADJ cana-756 49 28 + + CCONJ cana-756 49 29 π‘£π‘Ÿπ‘’ π‘£π‘Ÿπ‘’ NOUN cana-756 49 30 = = SYM cana-756 49 31 0 0 NUM cana-756 49 32 , , PUNCT cana-756 49 33 for for ADP cana-756 49 34 all all DET cana-756 49 35 π‘Ÿ π‘Ÿ PRON cana-756 49 36 ∈ ∈ NOUN cana-756 49 37 𝑅. 𝑅. ADV cana-756 49 38 if if SCONJ cana-756 49 39 anyone anyone PRON cana-756 49 40 of of ADP cana-756 49 41 the the DET cana-756 49 42 above above ADJ cana-756 49 43 conditions condition NOUN cana-756 49 44 is be AUX cana-756 49 45 fulfilled fulfil VERB cana-756 49 46 , , PUNCT cana-756 49 47 then then ADV cana-756 49 48 𝑒𝑣 𝑒𝑣 NOUN cana-756 49 49 = = NOUN cana-756 49 50 𝑣𝑒 𝑣𝑒 NOUN cana-756 49 51 = = SYM cana-756 49 52 0 0 PROPN cana-756 49 53 . . PUNCT cana-756 50 1 lemma lemma PROPN cana-756 50 2 2 2 NUM cana-756 50 3 : : PUNCT cana-756 50 4 [ [ PUNCT cana-756 50 5 lemma lemma PROPN cana-756 50 6 2 2 NUM cana-756 50 7 , , PUNCT cana-756 50 8 [ [ X cana-756 50 9 5 5 NUM cana-756 50 10 ] ] PUNCT cana-756 50 11 ] ] X cana-756 50 12 let let VERB cana-756 50 13 r r PRON cana-756 50 14 be be AUX cana-756 50 15 a a DET cana-756 50 16 semiprime semiprime NOUN cana-756 50 17 ring ring NOUN cana-756 50 18 . . PUNCT cana-756 51 1 suppose suppose VERB cana-756 51 2 that that SCONJ cana-756 51 3 two two NUM cana-756 51 4 bi bi ADJ cana-756 51 5 - - ADJ cana-756 51 6 additive additive ADJ cana-756 51 7 mappings mapping NOUN cana-756 51 8 d1 d1 NOUN cana-756 51 9 : : PUNCT cana-756 51 10 r r NOUN cana-756 51 11 x x SYM cana-756 51 12 r r NOUN cana-756 51 13 β†’ β†’ SYM cana-756 51 14 r r NOUN cana-756 51 15 and and CCONJ cana-756 51 16 d2 d2 PROPN cana-756 51 17 : : PUNCT cana-756 51 18 r r NOUN cana-756 51 19 x x SYM cana-756 51 20 r r NOUN cana-756 51 21 β†’ β†’ SYM cana-756 51 22 r r NOUN cana-756 51 23 satisfies satisfie NOUN cana-756 51 24 d1(u d1(u PROPN cana-756 51 25 , , PUNCT cana-756 51 26 v)rd2(v v)rd2(v PROPN cana-756 51 27 , , PUNCT cana-756 51 28 u u NOUN cana-756 51 29 ) ) PUNCT cana-756 51 30 = = PUNCT cana-756 51 31 { { PUNCT cana-756 51 32 0 0 NUM cana-756 51 33 } } PUNCT cana-756 51 34 , , PUNCT cana-756 51 35 βˆ€ βˆ€ X cana-756 51 36 u u NOUN cana-756 51 37 , , PUNCT cana-756 51 38 v v NOUN cana-756 51 39 ∈ ∈ NOUN cana-756 51 40 r r NOUN cana-756 51 41 , , PUNCT cana-756 51 42 then then ADV cana-756 51 43 d1(u d1(u PROPN cana-756 51 44 , , PUNCT cana-756 51 45 v)rd2(v v)rd2(v PROPN cana-756 51 46 , , PUNCT cana-756 51 47 w w NOUN cana-756 51 48 ) ) PUNCT cana-756 51 49 = = SYM cana-756 51 50 { { PUNCT cana-756 51 51 0 0 NUM cana-756 51 52 } } PUNCT cana-756 51 53 , , PUNCT cana-756 51 54 βˆ€ βˆ€ X cana-756 51 55 u u NOUN cana-756 51 56 , , PUNCT cana-756 51 57 v v INTJ cana-756 51 58 , , PUNCT cana-756 51 59 w w PROPN cana-756 51 60 ∈ ∈ PROPN cana-756 51 61 r. r. PROPN cana-756 51 62 lemma lemma PROPN cana-756 51 63 3 3 NUM cana-756 51 64 : : PUNCT cana-756 52 1 [ [ X cana-756 52 2 theorem theorem ADJ cana-756 52 3 1 1 NUM cana-756 52 4 , , PUNCT cana-756 52 5 [ [ X cana-756 52 6 7 7 NUM cana-756 52 7 ] ] X cana-756 52 8 ] ] PUNCT cana-756 52 9 let let VERB cana-756 52 10 r r PRON cana-756 52 11 be be AUX cana-756 52 12 a a DET cana-756 52 13 2 2 NUM cana-756 52 14 torsion torsion NOUN cana-756 52 15 free free ADJ cana-756 52 16 semi semi ADJ cana-756 52 17 prime prime ADJ cana-756 52 18 ring ring NOUN cana-756 52 19 . . PUNCT cana-756 53 1 then then ADV cana-756 53 2 the the DET cana-756 53 3 following follow VERB cana-756 53 4 conditions condition NOUN cana-756 53 5 are be AUX cana-756 53 6 equivalent equivalent ADJ cana-756 53 7 : : PUNCT cana-756 53 8 1.two 1.two NUM cana-756 53 9 symmetric symmetric ADJ cana-756 53 10 reverse reverse NOUN cana-756 53 11 bi-(Οƒ bi-(Οƒ NOUN cana-756 53 12 , , PUNCT cana-756 53 13 Ο„)-derivations Ο„)-derivations PUNCT cana-756 53 14 d1 d1 PROPN cana-756 53 15 and and CCONJ cana-756 53 16 d2 d2 PROPN cana-756 53 17 are be AUX cana-756 53 18 orthogonal orthogonal ADJ cana-756 53 19 . . PUNCT cana-756 54 1 2.d1(u 2.d1(u NUM cana-756 54 2 , , PUNCT cana-756 54 3 v v NOUN cana-756 54 4 ) ) PUNCT cana-756 54 5 d2(v d2(v NOUN cana-756 54 6 , , PUNCT cana-756 54 7 w w NOUN cana-756 54 8 ) ) PUNCT cana-756 54 9 + + CCONJ cana-756 55 1 d2(u d2(u NOUN cana-756 55 2 , , PUNCT cana-756 55 3 v)d1(v v)d1(v PROPN cana-756 55 4 , , PUNCT cana-756 55 5 w w NOUN cana-756 55 6 ) ) PUNCT cana-756 55 7 = = SYM cana-756 55 8 0 0 NUM cana-756 55 9 , , PUNCT cana-756 55 10 βˆ€ βˆ€ X cana-756 55 11 u u NOUN cana-756 55 12 , , PUNCT cana-756 55 13 v v INTJ cana-756 55 14 , , PUNCT cana-756 55 15 w w PROPN cana-756 55 16 ∈ ∈ PROPN cana-756 55 17 r. r. PROPN cana-756 55 18 lemma lemma PROPN cana-756 55 19 4 4 NUM cana-756 55 20 : : PUNCT cana-756 55 21 let let VERB cana-756 55 22 r r PRON cana-756 55 23 be be AUX cana-756 55 24 a a DET cana-756 55 25 2 2 NUM cana-756 55 26 torsion torsion NOUN cana-756 55 27 free free ADJ cana-756 55 28 semi semi ADJ cana-756 55 29 prime prime ADJ cana-756 55 30 ring ring NOUN cana-756 55 31 . . PUNCT cana-756 56 1 then then ADV cana-756 56 2 two two NUM cana-756 56 3 symmetric symmetric ADJ cana-756 56 4 reverse reverse NOUN cana-756 56 5 bi-(𝜎 bi-(𝜎 NOUN cana-756 56 6 , , PUNCT cana-756 56 7 𝜏)derivations 𝜏)derivations PROPN cana-756 56 8 d1 d1 PROPN cana-756 56 9 and and CCONJ cana-756 56 10 d2 d2 PROPN cana-756 56 11 are be AUX cana-756 56 12 orthogonal orthogonal ADJ cana-756 56 13 if if SCONJ cana-756 56 14 and and CCONJ cana-756 56 15 only only ADV cana-756 56 16 if if SCONJ cana-756 56 17 d1d2 d1d2 X cana-756 56 18 = = SYM cana-756 56 19 0 0 X cana-756 56 20 . . PUNCT cana-756 57 1 proof proof NOUN cana-756 57 2 : : PUNCT cana-756 57 3 suppose suppose VERB cana-756 57 4 that that SCONJ cana-756 57 5 𝐷1 𝐷1 PROPN cana-756 57 6 and and CCONJ cana-756 57 7 𝐷2 𝐷2 NOUN cana-756 57 8 are be AUX cana-756 57 9 orthogonal orthogonal ADJ cana-756 57 10 . . PUNCT cana-756 58 1 since since SCONJ cana-756 58 2 𝐷1 𝐷1 PROPN cana-756 58 3 , , PUNCT cana-756 58 4 𝐷2 𝐷2 PROPN cana-756 58 5 are be AUX cana-756 58 6 orthogonal orthogonal ADJ cana-756 58 7 , , PUNCT cana-756 58 8 we we PRON cana-756 58 9 can can AUX cana-756 58 10 have have VERB cana-756 58 11 𝐷1(𝑒 𝐷1(𝑒 PROPN cana-756 58 12 , , PUNCT cana-756 58 13 𝑣)π‘Ÿπ·2(𝑣 𝑣)π‘Ÿπ·2(𝑣 NOUN cana-756 58 14 , , PUNCT cana-756 58 15 𝑀 𝑀 X cana-756 58 16 ) ) PUNCT cana-756 58 17 = = SYM cana-756 58 18 0 0 NUM cana-756 58 19 , , PUNCT cana-756 58 20 βˆ€ βˆ€ X cana-756 58 21 𝑒 𝑒 NOUN cana-756 58 22 , , PUNCT cana-756 58 23 𝑣 𝑣 X cana-756 58 24 , , PUNCT cana-756 58 25 𝑀 𝑀 ADP cana-756 58 26 , , PUNCT cana-756 58 27 π‘Ÿ π‘Ÿ X cana-756 58 28 ∈ ∈ PROPN cana-756 58 29 𝑅 𝑅 PROPN cana-756 58 30 𝐷1(𝐷1(𝑒 𝐷1(𝐷1(𝑒 PROPN cana-756 58 31 , , PUNCT cana-756 58 32 𝑣 𝑣 NOUN cana-756 58 33 ) ) PUNCT cana-756 58 34 π‘Ÿπ·2(𝑣 π‘Ÿπ·2(𝑣 NOUN cana-756 58 35 , , PUNCT cana-756 58 36 𝑀 𝑀 ADP cana-756 58 37 ) ) PUNCT cana-756 58 38 , , PUNCT cana-756 58 39 π‘š π‘š NOUN cana-756 58 40 ) ) PUNCT cana-756 58 41 = = SYM cana-756 58 42 0 0 NUM cana-756 58 43 , , PUNCT cana-756 58 44 βˆ€ βˆ€ X cana-756 58 45 𝑒 𝑒 NOUN cana-756 58 46 , , PUNCT cana-756 58 47 𝑣 𝑣 INTJ cana-756 58 48 , , PUNCT cana-756 58 49 𝑀 𝑀 ADP cana-756 58 50 , , PUNCT cana-756 58 51 π‘Ÿ π‘Ÿ AUX cana-756 58 52 , , PUNCT cana-756 58 53 π‘š π‘š PROPN cana-756 58 54 ∈ ∈ PROPN cana-756 58 55 𝑅 𝑅 PROPN cana-756 58 56 𝐷1(𝐷2(𝑣 𝐷1(𝐷2(𝑣 PROPN cana-756 58 57 , , PUNCT cana-756 58 58 𝑀 𝑀 ADP cana-756 58 59 ) ) PUNCT cana-756 58 60 , , PUNCT cana-756 58 61 π‘š)𝜎(π‘Ÿ)𝜎(𝐷1(𝑒 π‘š)𝜎(π‘Ÿ)𝜎(𝐷1(𝑒 NUM cana-756 58 62 , , PUNCT cana-756 58 63 𝑣 𝑣 NOUN cana-756 58 64 ) ) PUNCT cana-756 59 1 + + CCONJ cana-756 59 2 𝜏(𝐷2(𝑣 𝜏(𝐷2(𝑣 PROPN cana-756 59 3 , , PUNCT cana-756 59 4 𝑀))𝐷1(π‘Ÿ 𝑀))𝐷1(π‘Ÿ PRON cana-756 59 5 , , PUNCT cana-756 59 6 π‘š)𝜎(𝐷1(𝑒 π‘š)𝜎(𝐷1(𝑒 PROPN cana-756 59 7 , , PUNCT cana-756 59 8 𝑣 𝑣 NOUN cana-756 59 9 ) ) PUNCT cana-756 59 10 ) ) PUNCT cana-756 60 1 + + CCONJ cana-756 60 2 𝜏(π‘Ÿπ·2(𝑣 𝜏(π‘Ÿπ·2(𝑣 PROPN cana-756 60 3 , , PUNCT cana-756 60 4 𝑀 𝑀 ADP cana-756 60 5 ) ) PUNCT cana-756 60 6 𝐷1(𝐷1(𝑒 𝐷1(𝐷1(𝑒 PROPN cana-756 60 7 , , PUNCT cana-756 60 8 𝑣 𝑣 NOUN cana-756 60 9 ) ) PUNCT cana-756 60 10 , , PUNCT cana-756 60 11 π‘š)=0 π‘š)=0 PROPN cana-756 60 12 . . PUNCT cana-756 60 13 using use VERB cana-756 60 14 𝐷1𝜎 𝐷1𝜎 PROPN cana-756 60 15 = = SYM cana-756 60 16 𝜎𝐷1 𝜎𝐷1 PROPN cana-756 60 17 , , PUNCT cana-756 60 18 𝜏𝐷2 𝜏𝐷2 PROPN cana-756 60 19 = = SYM cana-756 60 20 𝐷2𝜏 𝐷2𝜏 PROPN cana-756 60 21 and and CCONJ cana-756 60 22 𝜎 𝜎 PROPN cana-756 60 23 and and CCONJ cana-756 60 24 𝜏 𝜏 NOUN cana-756 60 25 are be AUX cana-756 60 26 automorphisms automorphism NOUN cana-756 60 27 of of ADP cana-756 60 28 r r NOUN cana-756 60 29 , , PUNCT cana-756 60 30 we we PRON cana-756 60 31 have have VERB cana-756 60 32 𝐷1(𝐷2(𝑣 𝐷1(𝐷2(𝑣 PROPN cana-756 60 33 , , PUNCT cana-756 60 34 𝑀 𝑀 ADP cana-756 60 35 ) ) PUNCT cana-756 60 36 , , PUNCT cana-756 60 37 π‘š)π‘Ÿπ·1(𝑒 π‘š)π‘Ÿπ·1(𝑒 PUNCT cana-756 60 38 , , PUNCT cana-756 60 39 𝑣 𝑣 NOUN cana-756 60 40 ) ) PUNCT cana-756 61 1 + + CCONJ cana-756 61 2 𝐷2(𝑣 𝐷2(𝑣 PROPN cana-756 61 3 , , PUNCT cana-756 61 4 𝑀 𝑀 PROPN cana-756 61 5 ) ) PUNCT cana-756 61 6 𝐷1(π‘Ÿ 𝐷1(π‘Ÿ PROPN cana-756 61 7 , , PUNCT cana-756 61 8 π‘š π‘š NOUN cana-756 61 9 ) ) PUNCT cana-756 61 10 𝐷1(𝑒 𝐷1(𝑒 PROPN cana-756 61 11 , , PUNCT cana-756 61 12 𝑣 𝑣 NOUN cana-756 61 13 ) ) PUNCT cana-756 61 14 + + NUM cana-756 61 15 π‘Ÿπ·2(𝑣 π‘Ÿπ·2(𝑣 NOUN cana-756 61 16 , , PUNCT cana-756 61 17 𝑀 𝑀 NOUN cana-756 61 18 ) ) PUNCT cana-756 61 19 𝐷1(𝐷1(𝑒 𝐷1(𝐷1(𝑒 PROPN cana-756 61 20 , , PUNCT cana-756 61 21 𝑣 𝑣 NOUN cana-756 61 22 ) ) PUNCT cana-756 61 23 , , PUNCT cana-756 61 24 π‘š)=0 π‘š)=0 PROPN cana-756 61 25 . . PUNCT cana-756 62 1 using use VERB cana-756 62 2 the the DET cana-756 62 3 condition condition NOUN cana-756 62 4 of of ADP cana-756 62 5 orthogonality orthogonality NOUN cana-756 62 6 of of ADP cana-756 62 7 𝐷1 𝐷1 PROPN cana-756 62 8 , , PUNCT cana-756 62 9 𝐷2 𝐷2 PROPN cana-756 62 10 , , PUNCT cana-756 62 11 we we PRON cana-756 62 12 get get VERB cana-756 62 13 𝐷1𝐷2(𝑣 𝐷1𝐷2(𝑣 NOUN cana-756 62 14 , , PUNCT cana-756 62 15 𝑀)π‘Ÿπ·1(𝑒 𝑀)π‘Ÿπ·1(𝑒 NUM cana-756 62 16 , , PUNCT cana-756 62 17 𝑣)=0 𝑣)=0 PROPN cana-756 62 18 . . PUNCT cana-756 63 1 in in ADP cana-756 63 2 particular particular ADJ cana-756 63 3 if if SCONJ cana-756 63 4 we we PRON cana-756 63 5 put put VERB cana-756 63 6 𝑒 𝑒 NOUN cana-756 63 7 = = SYM cana-756 63 8 𝐷2(𝑣 𝐷2(𝑣 PROPN cana-756 63 9 , , PUNCT cana-756 63 10 𝑀 𝑀 ADP cana-756 63 11 ) ) PUNCT cana-756 63 12 in in ADP cana-756 63 13 the the DET cana-756 63 14 above above ADJ cana-756 63 15 equation equation NOUN cana-756 63 16 , , PUNCT cana-756 63 17 we we PRON cana-756 63 18 get get VERB cana-756 63 19 𝐷1𝐷2(𝑣 𝐷1𝐷2(𝑣 NOUN cana-756 63 20 , , PUNCT cana-756 63 21 𝑀)π‘Ÿπ·1(𝐷2(𝑣 𝑀)π‘Ÿπ·1(𝐷2(𝑣 NOUN cana-756 63 22 , , PUNCT cana-756 63 23 𝑀 𝑀 ADP cana-756 63 24 ) ) PUNCT cana-756 63 25 , , PUNCT cana-756 63 26 𝑣 𝑣 X cana-756 63 27 ) ) PUNCT cana-756 63 28 = = SYM cana-756 63 29 0 0 NUM cana-756 63 30 𝐷1𝐷2(𝑣 𝐷1𝐷2(𝑣 NOUN cana-756 63 31 , , PUNCT cana-756 63 32 𝑀)π‘Ÿπ·1𝐷2(𝑣 𝑀)π‘Ÿπ·1𝐷2(𝑣 NOUN cana-756 63 33 , , PUNCT cana-756 63 34 𝑀 𝑀 ADP cana-756 63 35 ) ) PUNCT cana-756 63 36 = = SYM cana-756 63 37 0 0 NUM cana-756 63 38 𝐷1𝐷2(𝑣 𝐷1𝐷2(𝑣 NOUN cana-756 63 39 , , PUNCT cana-756 63 40 𝑀 𝑀 X cana-756 63 41 ) ) PUNCT cana-756 63 42 = = SYM cana-756 63 43 0 0 PUNCT cana-756 64 1 ( ( PUNCT cana-756 64 2 by by ADP cana-756 64 3 semiprimeness semiprimeness NOUN cana-756 64 4 of of ADP cana-756 64 5 r. r. PROPN cana-756 64 6 ) ) PUNCT cana-756 64 7 𝐷1𝐷2 𝐷1𝐷2 PROPN cana-756 64 8 = = SYM cana-756 64 9 0 0 X cana-756 64 10 . . PUNCT cana-756 65 1 conversely conversely ADV cana-756 65 2 , , PUNCT cana-756 65 3 let let VERB cana-756 65 4 𝐷1 𝐷1 NOUN cana-756 65 5 and and CCONJ cana-756 65 6 𝐷2 𝐷2 NOUN cana-756 65 7 be be AUX cana-756 65 8 two two NUM cana-756 65 9 reverse reverse ADJ cana-756 65 10 bi-(𝜎 bi-(𝜎 NOUN cana-756 65 11 , , PUNCT cana-756 65 12 𝜏)-derivations 𝜏)-derivation NOUN cana-756 65 13 such such ADJ cana-756 65 14 that that DET cana-756 65 15 𝐷1𝐷2 𝐷1𝐷2 NOUN cana-756 65 16 = = SYM cana-756 65 17 0 0 X cana-756 65 18 . . PUNCT cana-756 66 1 𝐷1𝐷2(𝑒𝑣 𝐷1𝐷2(𝑒𝑣 INTJ cana-756 66 2 , , PUNCT cana-756 66 3 𝑀 𝑀 ADJ cana-756 66 4 ) ) PUNCT cana-756 66 5 = = SYM cana-756 66 6 𝐷1(𝐷2(𝑒𝑣 𝐷1(𝐷2(𝑒𝑣 NOUN cana-756 66 7 , , PUNCT cana-756 66 8 𝑀 𝑀 ADP cana-756 66 9 ) ) PUNCT cana-756 66 10 , , PUNCT cana-756 66 11 π‘š π‘š NOUN cana-756 66 12 ) ) PUNCT cana-756 67 1 = = NOUN cana-756 67 2 𝐷1(𝐷2 𝐷1(𝐷2 X cana-756 67 3 ( ( PUNCT cana-756 67 4 𝑣 𝑣 NOUN cana-756 67 5 , , PUNCT cana-756 67 6 𝑀)𝜎(𝑒 𝑀)𝜎(𝑒 NOUN cana-756 67 7 ) ) PUNCT cana-756 67 8 + + CCONJ cana-756 67 9 𝜏(𝑣 𝜏(𝑣 PROPN cana-756 67 10 ) ) PUNCT cana-756 67 11 𝐷2(𝑒 𝐷2(𝑒 PROPN cana-756 67 12 , , PUNCT cana-756 67 13 𝑀 𝑀 ADP cana-756 67 14 ) ) PUNCT cana-756 67 15 , , PUNCT cana-756 67 16 π‘š π‘š NOUN cana-756 67 17 ) ) PUNCT cana-756 67 18 communications communication NOUN cana-756 67 19 on on ADP cana-756 67 20 applied apply VERB cana-756 67 21 nonlinear nonlinear ADJ cana-756 67 22 analysis analysis NOUN cana-756 67 23 issn issn NOUN cana-756 67 24 : : PUNCT cana-756 67 25 1074 1074 NUM cana-756 67 26 - - PUNCT cana-756 67 27 133x 133x NUM cana-756 67 28 vol vol NOUN cana-756 67 29 31 31 NUM cana-756 67 30 no no NOUN cana-756 67 31 . . PUNCT cana-756 68 1 3s 3s NUM cana-756 68 2 ( ( PUNCT cana-756 68 3 2024 2024 NUM cana-756 68 4 ) ) PUNCT cana-756 68 5 159 159 NUM cana-756 68 6 https://internationalpubls.com https://internationalpubls.com X cana-756 68 7 = = X cana-756 68 8 𝐷1(𝐷2 𝐷1(𝐷2 X cana-756 68 9 ( ( PUNCT cana-756 68 10 𝑣 𝑣 NOUN cana-756 68 11 , , PUNCT cana-756 68 12 𝑀)𝜎(𝑒 𝑀)𝜎(𝑒 NOUN cana-756 68 13 ) ) PUNCT cana-756 68 14 , , PUNCT cana-756 68 15 π‘š π‘š PROPN cana-756 68 16 ) ) PUNCT cana-756 69 1 + + PUNCT cana-756 69 2 𝐷1(𝜏(𝑣)𝐷2(𝑒 𝐷1(𝜏(𝑣)𝐷2(𝑒 PROPN cana-756 69 3 , , PUNCT cana-756 69 4 𝑀 𝑀 ADP cana-756 69 5 ) ) PUNCT cana-756 69 6 , , PUNCT cana-756 69 7 π‘š π‘š NOUN cana-756 69 8 ) ) PUNCT cana-756 69 9 = = NOUN cana-756 69 10 𝐷1(𝜎(𝑒 𝐷1(𝜎(𝑒 NOUN cana-756 69 11 ) ) PUNCT cana-756 69 12 , , PUNCT cana-756 69 13 π‘š)𝜎(𝐷2(𝑣 π‘š)𝜎(𝐷2(𝑣 NOUN cana-756 69 14 , , PUNCT cana-756 69 15 𝑀 𝑀 NOUN cana-756 69 16 ) ) PUNCT cana-756 69 17 ) ) PUNCT cana-756 70 1 + + NUM cana-756 70 2 𝜏(𝜎(𝑒))𝐷1(𝐷2(𝑣 𝜏(𝜎(𝑒))𝐷1(𝐷2(𝑣 NOUN cana-756 70 3 , , PUNCT cana-756 70 4 𝑀 𝑀 NOUN cana-756 70 5 ) ) PUNCT cana-756 70 6 , , PUNCT cana-756 70 7 π‘š)+𝐷1(𝐷2(𝑒 π‘š)+𝐷1(𝐷2(𝑒 NOUN cana-756 70 8 , , PUNCT cana-756 70 9 𝑀 𝑀 ADP cana-756 70 10 ) ) PUNCT cana-756 70 11 , , PUNCT cana-756 70 12 π‘š)𝜎(𝜏(𝑣 π‘š)𝜎(𝜏(𝑣 NOUN cana-756 70 13 ) ) PUNCT cana-756 70 14 ) ) PUNCT cana-756 71 1 + + CCONJ cana-756 71 2 𝜏(𝐷2(𝑒 𝜏(𝐷2(𝑒 X cana-756 71 3 , , PUNCT cana-756 71 4 𝑀))𝐷1(𝜏(𝑣 𝑀))𝐷1(𝜏(𝑣 PROPN cana-756 71 5 ) ) PUNCT cana-756 71 6 , , PUNCT cana-756 71 7 π‘š π‘š NOUN cana-756 71 8 ) ) PUNCT cana-756 71 9 . . PUNCT cana-756 72 1 using use VERB cana-756 72 2 𝜎𝐷2 𝜎𝐷2 PROPN cana-756 72 3 = = SYM cana-756 72 4 𝐷2𝜎 𝐷2𝜎 PROPN cana-756 72 5 , , PUNCT cana-756 72 6 𝜏𝐷2 𝜏𝐷2 PROPN cana-756 72 7 = = SYM cana-756 72 8 𝐷2𝜏 𝐷2𝜏 PROPN cana-756 72 9 ; ; PUNCT cana-756 72 10 𝜎 𝜎 PROPN cana-756 72 11 and and CCONJ cana-756 72 12 𝜏 𝜏 NOUN cana-756 72 13 are be AUX cana-756 72 14 automorphisms automorphism NOUN cana-756 72 15 of of ADP cana-756 72 16 r r NOUN cana-756 72 17 and and CCONJ cana-756 72 18 𝐷1𝐷2 𝐷1𝐷2 NOUN cana-756 72 19 = = SYM cana-756 72 20 0 0 NUM cana-756 72 21 , , PUNCT cana-756 72 22 we we PRON cana-756 72 23 obtain obtain VERB cana-756 72 24 0 0 NUM cana-756 72 25 = = SYM cana-756 72 26 𝐷1(𝑒 𝐷1(𝑒 PROPN cana-756 72 27 , , PUNCT cana-756 72 28 π‘š π‘š NOUN cana-756 72 29 ) ) PUNCT cana-756 72 30 𝐷2 𝐷2 NOUN cana-756 72 31 ( ( PUNCT cana-756 72 32 𝑣 𝑣 NOUN cana-756 72 33 , , PUNCT cana-756 72 34 𝑀 𝑀 ADP cana-756 72 35 ) ) PUNCT cana-756 72 36 + + CCONJ cana-756 72 37 𝐷2(𝑒 𝐷2(𝑒 PROPN cana-756 72 38 , , PUNCT cana-756 72 39 𝑀)𝐷1(𝑣 𝑀)𝐷1(𝑣 NUM cana-756 72 40 , , PUNCT cana-756 72 41 π‘š π‘š NOUN cana-756 72 42 ) ) PUNCT cana-756 72 43 . . PUNCT cana-756 73 1 in in ADP cana-756 73 2 particular particular ADJ cana-756 73 3 , , PUNCT cana-756 73 4 𝐷1(𝑒 𝐷1(𝑒 PROPN cana-756 73 5 , , PUNCT cana-756 73 6 𝑀 𝑀 NOUN cana-756 73 7 ) ) PUNCT cana-756 73 8 𝐷2 𝐷2 NOUN cana-756 73 9 ( ( PUNCT cana-756 73 10 𝑣 𝑣 NOUN cana-756 73 11 , , PUNCT cana-756 73 12 𝑀 𝑀 ADP cana-756 73 13 ) ) PUNCT cana-756 73 14 + + CCONJ cana-756 73 15 𝐷2(𝑒 𝐷2(𝑒 PROPN cana-756 73 16 , , PUNCT cana-756 73 17 𝑀)𝐷1(𝑣 𝑀)𝐷1(𝑣 NUM cana-756 73 18 , , PUNCT cana-756 73 19 𝑀 𝑀 ADP cana-756 73 20 ) ) PUNCT cana-756 73 21 = = SYM cana-756 73 22 0 0 X cana-756 73 23 . . PUNCT cana-756 74 1 therefore therefore ADV cana-756 74 2 𝐷1(𝑒 𝐷1(𝑒 PROPN cana-756 74 3 , , PUNCT cana-756 74 4 𝑀 𝑀 NOUN cana-756 74 5 ) ) PUNCT cana-756 74 6 𝐷2(𝑀 𝐷2(𝑀 NOUN cana-756 74 7 , , PUNCT cana-756 74 8 𝑣 𝑣 NOUN cana-756 74 9 ) ) PUNCT cana-756 74 10 + + CCONJ cana-756 74 11 𝐷2(𝑒 𝐷2(𝑒 PROPN cana-756 74 12 , , PUNCT cana-756 74 13 𝑀)𝐷.1 𝑀)𝐷.1 PROPN cana-756 74 14 ( ( PUNCT cana-756 74 15 𝑀 𝑀 ADP cana-756 74 16 , , PUNCT cana-756 74 17 𝑣 𝑣 NOUN cana-756 74 18 ) ) PUNCT cana-756 74 19 = = SYM cana-756 74 20 0 0 NUM cana-756 74 21 , , PUNCT cana-756 74 22 βˆ€ βˆ€ X cana-756 74 23 𝑒 𝑒 NOUN cana-756 74 24 , , PUNCT cana-756 74 25 𝑣 𝑣 X cana-756 74 26 , , PUNCT cana-756 74 27 𝑀 𝑀 ADP cana-756 74 28 ∈ ∈ NOUN cana-756 74 29 𝑅. 𝑅. NOUN cana-756 74 30 ( ( PUNCT cana-756 74 31 since since SCONJ cana-756 74 32 𝐷1,𝐷2 𝐷1,𝐷2 NOUN cana-756 74 33 are be AUX cana-756 74 34 symmetric symmetric ADJ cana-756 74 35 ) ) PUNCT cana-756 74 36 by by ADP cana-756 74 37 lemma lemma PROPN cana-756 74 38 3 3 NUM cana-756 74 39 , , PUNCT cana-756 74 40 we we PRON cana-756 74 41 can can AUX cana-756 74 42 conclude conclude VERB cana-756 74 43 that that PRON cana-756 74 44 𝐷1 𝐷1 PROPN cana-756 74 45 and and CCONJ cana-756 74 46 𝐷2 𝐷2 NOUN cana-756 74 47 are be AUX cana-756 74 48 orthogonal orthogonal ADJ cana-756 74 49 . . PUNCT cana-756 75 1 3 3 X cana-756 75 2 . . X cana-756 75 3 main main ADJ cana-756 75 4 results result NOUN cana-756 75 5 : : PUNCT cana-756 75 6 theorem theorem NOUN cana-756 75 7 1 1 NUM cana-756 75 8 : : PUNCT cana-756 75 9 if if SCONJ cana-756 75 10 ( ( PUNCT cana-756 75 11 Ξ΄1 Ξ΄1 NOUN cana-756 75 12 , , PUNCT cana-756 75 13 d1 d1 PROPN cana-756 75 14 ) ) PUNCT cana-756 75 15 and and CCONJ cana-756 75 16 ( ( PUNCT cana-756 75 17 Ξ΄2 Ξ΄2 ADJ cana-756 75 18 , , PUNCT cana-756 75 19 d2 d2 PROPN cana-756 75 20 ) ) PUNCT cana-756 75 21 are be AUX cana-756 75 22 two two NUM cana-756 75 23 orthogonal orthogonal ADJ cana-756 75 24 generalized generalize VERB cana-756 75 25 symmetric symmetric ADJ cana-756 75 26 reverse reverse NOUN cana-756 75 27 bi-(Οƒ bi-(Οƒ NOUN cana-756 75 28 , , PUNCT cana-756 75 29 Ο„)-derivations Ο„)-derivation NOUN cana-756 75 30 of of ADP cana-756 75 31 r r NOUN cana-756 75 32 , , PUNCT cana-756 75 33 then then ADV cana-756 75 34 ( ( PUNCT cana-756 75 35 Ξ΄1 Ξ΄1 NOUN cana-756 75 36 , , PUNCT cana-756 75 37 d1 d1 PROPN cana-756 75 38 ) ) PUNCT cana-756 75 39 and and CCONJ cana-756 75 40 ( ( PUNCT cana-756 75 41 Ξ΄2 Ξ΄2 ADJ cana-756 75 42 , , PUNCT cana-756 75 43 d2 d2 PROPN cana-756 75 44 ) ) PUNCT cana-756 75 45 are be AUX cana-756 75 46 orthogonal orthogonal ADJ cana-756 75 47 if if SCONJ cana-756 75 48 and and CCONJ cana-756 75 49 only only ADV cana-756 75 50 if if SCONJ cana-756 75 51 the the DET cana-756 75 52 following follow VERB cana-756 75 53 conditions condition NOUN cana-756 75 54 are be AUX cana-756 75 55 satisfied satisfied ADJ cana-756 75 56 : : PUNCT cana-756 75 57 ( ( PUNCT cana-756 75 58 i i NOUN cana-756 75 59 ) ) PUNCT cana-756 75 60 Ξ΄1(u Ξ΄1(u PROPN cana-756 75 61 , , PUNCT cana-756 75 62 v)Ξ΄2(v v)Ξ΄2(v PROPN cana-756 75 63 , , PUNCT cana-756 75 64 w w NOUN cana-756 75 65 ) ) PUNCT cana-756 76 1 + + CCONJ cana-756 76 2 Ξ΄2 Ξ΄2 VERB cana-756 76 3 ( ( PUNCT cana-756 76 4 u u NOUN cana-756 76 5 , , PUNCT cana-756 76 6 v v NOUN cana-756 76 7 ) ) PUNCT cana-756 76 8 Ξ΄1(v Ξ΄1(v PROPN cana-756 76 9 , , PUNCT cana-756 76 10 w w NOUN cana-756 76 11 ) ) PUNCT cana-756 76 12 = = SYM cana-756 76 13 0 0 NUM cana-756 76 14 , , PUNCT cana-756 76 15 βˆ€ βˆ€ X cana-756 76 16 u u NOUN cana-756 76 17 , , PUNCT cana-756 76 18 v v INTJ cana-756 76 19 , , PUNCT cana-756 76 20 w w PROPN cana-756 76 21 , , PUNCT cana-756 76 22 r r PROPN cana-756 76 23 ∈ ∈ PROPN cana-756 76 24 r. r. PROPN cana-756 76 25 ( ( PUNCT cana-756 76 26 ii ii PROPN cana-756 76 27 ) ) PUNCT cana-756 76 28 d1(u d1(u PROPN cana-756 76 29 , , PUNCT cana-756 76 30 v)Ξ΄2(v v)Ξ΄2(v PROPN cana-756 76 31 , , PUNCT cana-756 76 32 w w NOUN cana-756 76 33 ) ) PUNCT cana-756 76 34 + + CCONJ cana-756 76 35 d2 d2 PROPN cana-756 76 36 ( ( PUNCT cana-756 76 37 u u NOUN cana-756 76 38 , , PUNCT cana-756 76 39 v v NOUN cana-756 76 40 ) ) PUNCT cana-756 76 41 Ξ΄1(v Ξ΄1(v PROPN cana-756 76 42 , , PUNCT cana-756 76 43 w w NOUN cana-756 76 44 ) ) PUNCT cana-756 76 45 = = SYM cana-756 76 46 0 0 NUM cana-756 76 47 , , PUNCT cana-756 76 48 βˆ€ βˆ€ X cana-756 76 49 u u NOUN cana-756 76 50 , , PUNCT cana-756 76 51 v v INTJ cana-756 76 52 , , PUNCT cana-756 76 53 w w PROPN cana-756 76 54 , , PUNCT cana-756 76 55 r r PROPN cana-756 76 56 ∈ ∈ PROPN cana-756 76 57 r. r. NOUN cana-756 76 58 proof proof NOUN cana-756 76 59 : : PUNCT cana-756 76 60 suppose suppose VERB cana-756 76 61 that that SCONJ cana-756 76 62 ( ( PUNCT cana-756 76 63 Ξ΄1 Ξ΄1 NOUN cana-756 76 64 , , PUNCT cana-756 76 65 d1 d1 PROPN cana-756 76 66 ) ) PUNCT cana-756 76 67 and and CCONJ cana-756 76 68 ( ( PUNCT cana-756 76 69 Ξ΄2 Ξ΄2 ADJ cana-756 76 70 , , PUNCT cana-756 76 71 d2 d2 PROPN cana-756 76 72 ) ) PUNCT cana-756 76 73 are be AUX cana-756 76 74 orthogonal orthogonal ADJ cana-756 76 75 generalized generalize VERB cana-756 76 76 symmetric symmetric ADJ cana-756 76 77 reverse reverse NOUN cana-756 76 78 bi-(Οƒ bi-(Οƒ PROPN cana-756 76 79 , , PUNCT cana-756 76 80 Ο„)derivations Ο„)derivations PROPN cana-756 76 81 of of ADP cana-756 76 82 r. r. PROPN cana-756 76 83 by by ADP cana-756 76 84 the the DET cana-756 76 85 definition definition NOUN cana-756 76 86 of of ADP cana-756 76 87 orthogonality orthogonality NOUN cana-756 76 88 Ξ΄1 Ξ΄1 NOUN cana-756 76 89 and and CCONJ cana-756 76 90 Ξ΄2 Ξ΄2 VERB cana-756 76 91 , , PUNCT cana-756 76 92 we we PRON cana-756 76 93 have have VERB cana-756 76 94 Ξ΄1(u Ξ΄1(u NUM cana-756 76 95 , , PUNCT cana-756 76 96 v)rΞ΄2(v v)rΞ΄2(v PROPN cana-756 76 97 , , PUNCT cana-756 76 98 w w PROPN cana-756 76 99 ) ) PUNCT cana-756 76 100 = = SYM cana-756 76 101 0 0 PUNCT cana-756 77 1 = = SYM cana-756 77 2 Ξ΄2(u Ξ΄2(u PROPN cana-756 77 3 , , PUNCT cana-756 77 4 v)rΞ΄1(v v)rΞ΄1(v PROPN cana-756 77 5 , , PUNCT cana-756 77 6 w w NOUN cana-756 77 7 ) ) PUNCT cana-756 77 8 ( ( PUNCT cana-756 77 9 3.1 3.1 NUM cana-756 77 10 ) ) PUNCT cana-756 77 11 hence hence ADV cana-756 77 12 , , PUNCT cana-756 77 13 Ξ΄1(u Ξ΄1(u NUM cana-756 77 14 , , PUNCT cana-756 77 15 v)Ξ΄2(v v)Ξ΄2(v PROPN cana-756 77 16 , , PUNCT cana-756 77 17 w w NOUN cana-756 77 18 ) ) PUNCT cana-756 77 19 = = SYM cana-756 77 20 0 0 PUNCT cana-756 78 1 = = SYM cana-756 78 2 Ξ΄2 Ξ΄2 VERB cana-756 78 3 ( ( PUNCT cana-756 78 4 v v NOUN cana-756 78 5 , , PUNCT cana-756 78 6 w)Ξ΄1(u w)Ξ΄1(u NOUN cana-756 78 7 , , PUNCT cana-756 78 8 v v NOUN cana-756 78 9 ) ) PUNCT cana-756 78 10 ( ( PUNCT cana-756 78 11 by by ADP cana-756 78 12 lemma lemma PROPN cana-756 78 13 1 1 NUM cana-756 78 14 ) ) PUNCT cana-756 78 15 ( ( PUNCT cana-756 78 16 3.2 3.2 NUM cana-756 78 17 ) ) PUNCT cana-756 78 18 and and CCONJ cana-756 78 19 so so ADV cana-756 78 20 Ξ΄1(u Ξ΄1(u ADJ cana-756 78 21 , , PUNCT cana-756 78 22 v)Ξ΄2(v v)Ξ΄2(v PROPN cana-756 78 23 , , PUNCT cana-756 78 24 w w NOUN cana-756 78 25 ) ) PUNCT cana-756 78 26 + + CCONJ cana-756 78 27 Ξ΄2(v Ξ΄2(v PROPN cana-756 78 28 , , PUNCT cana-756 78 29 w)Ξ΄1(u w)Ξ΄1(u PROPN cana-756 78 30 , , PUNCT cana-756 78 31 v v NOUN cana-756 78 32 ) ) PUNCT cana-756 78 33 = = SYM cana-756 78 34 0 0 NUM cana-756 78 35 Ξ΄1(u Ξ΄1(u NUM cana-756 78 36 , , PUNCT cana-756 78 37 v)Ξ΄2(v v)Ξ΄2(v PROPN cana-756 78 38 , , PUNCT cana-756 78 39 w w NOUN cana-756 78 40 ) ) PUNCT cana-756 78 41 + + CCONJ cana-756 78 42 Ξ΄2(w Ξ΄2(w PROPN cana-756 78 43 , , PUNCT cana-756 78 44 v)Ξ΄1(v v)Ξ΄1(v PRON cana-756 78 45 , , PUNCT cana-756 78 46 u u NOUN cana-756 78 47 ) ) PUNCT cana-756 78 48 = = SYM cana-756 78 49 0 0 PUNCT cana-756 78 50 ( ( PUNCT cana-756 78 51 since since SCONJ cana-756 78 52 Ξ΄1 Ξ΄1 NOUN cana-756 78 53 , , PUNCT cana-756 78 54 Ξ΄2 Ξ΄2 PROPN cana-756 78 55 are be AUX cana-756 78 56 symmetric symmetric ADJ cana-756 78 57 ) ) PUNCT cana-756 78 58 Ξ΄1(u Ξ΄1(u NUM cana-756 78 59 , , PUNCT cana-756 78 60 v)Ξ΄2(v v)Ξ΄2(v PROPN cana-756 78 61 , , PUNCT cana-756 78 62 w w NOUN cana-756 78 63 ) ) PUNCT cana-756 78 64 + + CCONJ cana-756 79 1 Ξ΄2(u Ξ΄2(u NOUN cana-756 79 2 , , PUNCT cana-756 79 3 v)Ξ΄1(v v)Ξ΄1(v PRON cana-756 79 4 , , PUNCT cana-756 79 5 w w PROPN cana-756 79 6 ) ) PUNCT cana-756 79 7 = = SYM cana-756 79 8 0 0 X cana-756 79 9 . . PUNCT cana-756 80 1 hence hence ADV cana-756 80 2 condition condition NOUN cana-756 80 3 ( ( PUNCT cana-756 80 4 i i NOUN cana-756 80 5 ) ) PUNCT cana-756 80 6 is be AUX cana-756 80 7 proved prove VERB cana-756 80 8 . . PUNCT cana-756 81 1 now now ADV cana-756 81 2 , , PUNCT cana-756 81 3 suppose suppose VERB cana-756 81 4 that that SCONJ cana-756 81 5 Ξ΄1(u Ξ΄1(u NOUN cana-756 81 6 , , PUNCT cana-756 81 7 v)Ξ΄2(v v)Ξ΄2(v PROPN cana-756 81 8 , , PUNCT cana-756 81 9 w w NOUN cana-756 81 10 ) ) PUNCT cana-756 81 11 = = SYM cana-756 82 1 0 0 X cana-756 82 2 . . PUNCT cana-756 83 1 ( ( PUNCT cana-756 83 2 from from ADP cana-756 83 3 3.2 3.2 NUM cana-756 83 4 ) ) PUNCT cana-756 83 5 again again ADV cana-756 83 6 replacing replace VERB cana-756 83 7 u u NOUN cana-756 83 8 by by ADP cana-756 83 9 ur ur INTJ cana-756 83 10 , , PUNCT cana-756 83 11 r r NOUN cana-756 83 12 ∈ ∈ NOUN cana-756 83 13 r r NOUN cana-756 83 14 in in ADP cana-756 83 15 the the DET cana-756 83 16 above above ADJ cana-756 83 17 equation equation NOUN cana-756 83 18 and and CCONJ cana-756 83 19 using use VERB cana-756 83 20 ( ( PUNCT cana-756 83 21 3.2 3.2 NUM cana-756 83 22 ) ) PUNCT cana-756 83 23 , , PUNCT cana-756 83 24 we we PRON cana-756 83 25 get get VERB cana-756 83 26 Ξ΄1(r Ξ΄1(r NOUN cana-756 83 27 , , PUNCT cana-756 83 28 v)Οƒ(u v)Οƒ(u PROPN cana-756 83 29 ) ) PUNCT cana-756 83 30 Ξ΄2(v Ξ΄2(v PROPN cana-756 83 31 , , PUNCT cana-756 83 32 w w NOUN cana-756 83 33 ) ) PUNCT cana-756 83 34 + + CCONJ cana-756 84 1 Ο„(r)d1(u Ο„(r)d1(u NOUN cana-756 84 2 , , PUNCT cana-756 84 3 v)Ξ΄2(v v)Ξ΄2(v PROPN cana-756 84 4 , , PUNCT cana-756 84 5 w w NOUN cana-756 84 6 ) ) PUNCT cana-756 84 7 = = SYM cana-756 84 8 0 0 X cana-756 84 9 . . PUNCT cana-756 85 1 since since SCONJ cana-756 85 2 Οƒ Οƒ PROPN cana-756 85 3 , , PUNCT cana-756 85 4 Ο„ Ο„ PROPN cana-756 85 5 are be AUX cana-756 85 6 automorphisms automorphism NOUN cana-756 85 7 , , PUNCT cana-756 85 8 we we PRON cana-756 85 9 get get VERB cana-756 85 10 Ξ΄1(r Ξ΄1(r NOUN cana-756 85 11 , , PUNCT cana-756 85 12 v)uΞ΄2(v v)uΞ΄2(v PROPN cana-756 85 13 , , PUNCT cana-756 85 14 w w NOUN cana-756 85 15 ) ) PUNCT cana-756 86 1 + + CCONJ cana-756 86 2 r r NOUN cana-756 86 3 d1(u d1(u PROPN cana-756 86 4 , , PUNCT cana-756 86 5 v)Ξ΄2(v v)Ξ΄2(v PROPN cana-756 86 6 , , PUNCT cana-756 86 7 w w NOUN cana-756 86 8 ) ) PUNCT cana-756 86 9 = = SYM cana-756 86 10 0 0 NUM cana-756 86 11 , , PUNCT cana-756 86 12 βˆ€ βˆ€ X cana-756 86 13 u u NOUN cana-756 86 14 , , PUNCT cana-756 86 15 v v NOUN cana-756 86 16 , , PUNCT cana-756 86 17 r r NOUN cana-756 86 18 , , PUNCT cana-756 86 19 w w PROPN cana-756 86 20 ∈ ∈ PROPN cana-756 86 21 r. r. NOUN cana-756 86 22 using use VERB cana-756 86 23 the the DET cana-756 86 24 equation equation NOUN cana-756 86 25 ( ( PUNCT cana-756 86 26 3.1 3.1 NUM cana-756 86 27 ) ) PUNCT cana-756 86 28 , , PUNCT cana-756 86 29 we we PRON cana-756 86 30 get get VERB cana-756 86 31 rd1(u rd1(u NOUN cana-756 86 32 , , PUNCT cana-756 86 33 v v NOUN cana-756 86 34 ) ) PUNCT cana-756 86 35 Ξ΄2(v Ξ΄2(v PROPN cana-756 86 36 , , PUNCT cana-756 86 37 w w NOUN cana-756 86 38 ) ) PUNCT cana-756 86 39 = = SYM cana-756 86 40 0 0 NUM cana-756 86 41 , , PUNCT cana-756 86 42 βˆ€ βˆ€ X cana-756 86 43 u u NOUN cana-756 86 44 , , PUNCT cana-756 86 45 v v NOUN cana-756 86 46 , , PUNCT cana-756 86 47 r r NOUN cana-756 86 48 , , PUNCT cana-756 86 49 w w PROPN cana-756 86 50 ∈ ∈ PROPN cana-756 86 51 r. r. PROPN cana-756 86 52 left leave VERB cana-756 86 53 multiplying multiply VERB cana-756 86 54 the the DET cana-756 86 55 above above ADJ cana-756 86 56 equation equation NOUN cana-756 86 57 by by ADP cana-756 86 58 d1(u d1(u PROPN cana-756 86 59 , , PUNCT cana-756 86 60 v)Ξ΄2(v v)Ξ΄2(v PROPN cana-756 86 61 , , PUNCT cana-756 86 62 w w NOUN cana-756 86 63 ) ) PUNCT cana-756 86 64 and and CCONJ cana-756 86 65 using use VERB cana-756 86 66 the the DET cana-756 86 67 semiprimeness semiprimeness NOUN cana-756 86 68 of of ADP cana-756 86 69 r r NOUN cana-756 86 70 , , PUNCT cana-756 86 71 communications communication NOUN cana-756 86 72 on on ADP cana-756 86 73 applied apply VERB cana-756 86 74 nonlinear nonlinear ADJ cana-756 86 75 analysis analysis NOUN cana-756 86 76 issn issn NOUN cana-756 86 77 : : PUNCT cana-756 86 78 1074 1074 NUM cana-756 86 79 - - PUNCT cana-756 86 80 133x 133x NUM cana-756 86 81 vol vol NOUN cana-756 86 82 31 31 NUM cana-756 86 83 no no NOUN cana-756 86 84 . . PUNCT cana-756 87 1 3s 3s NUM cana-756 87 2 ( ( PUNCT cana-756 87 3 2024 2024 NUM cana-756 87 4 ) ) PUNCT cana-756 87 5 160 160 NUM cana-756 87 6 https://internationalpubls.com https://internationalpubls.com X cana-756 87 7 we we PRON cana-756 87 8 get get VERB cana-756 87 9 d1(u d1(u PROPN cana-756 87 10 , , PUNCT cana-756 87 11 v)Ξ΄2(v v)Ξ΄2(v PROPN cana-756 87 12 , , PUNCT cana-756 87 13 w w NOUN cana-756 87 14 ) ) PUNCT cana-756 87 15 = = SYM cana-756 87 16 0 0 X cana-756 87 17 . . PUNCT cana-756 87 18 ( ( PUNCT cana-756 87 19 3.3 3.3 NUM cana-756 87 20 ) ) PUNCT cana-756 87 21 replacing replace VERB cana-756 87 22 u u NOUN cana-756 87 23 by by ADP cana-756 87 24 ru ru NOUN cana-756 87 25 , , PUNCT cana-756 87 26 r r NOUN cana-756 87 27 ∈ ∈ NOUN cana-756 87 28 r r NOUN cana-756 87 29 in in ADP cana-756 87 30 ( ( PUNCT cana-756 87 31 3.3 3.3 NUM cana-756 87 32 ) ) PUNCT cana-756 87 33 , , PUNCT cana-756 87 34 we we PRON cana-756 87 35 obtain obtain VERB cana-756 87 36 d1(u d1(u NOUN cana-756 87 37 , , PUNCT cana-756 87 38 v)Οƒ(r v)Οƒ(r NUM cana-756 87 39 ) ) PUNCT cana-756 87 40 Ξ΄2(v Ξ΄2(v PROPN cana-756 87 41 , , PUNCT cana-756 87 42 w w NOUN cana-756 87 43 ) ) PUNCT cana-756 87 44 + + CCONJ cana-756 88 1 Ο„(u)d1(r Ο„(u)d1(r ADJ cana-756 88 2 , , PUNCT cana-756 88 3 v)Ξ΄2(v v)Ξ΄2(v PROPN cana-756 88 4 , , PUNCT cana-756 88 5 w w NOUN cana-756 88 6 ) ) PUNCT cana-756 88 7 = = SYM cana-756 89 1 0 0 X cana-756 89 2 . . X cana-756 90 1 using use VERB cana-756 90 2 ( ( PUNCT cana-756 90 3 3.3 3.3 NUM cana-756 90 4 ) ) PUNCT cana-756 90 5 and and CCONJ cana-756 90 6 Οƒ Οƒ PROPN cana-756 90 7 is be AUX cana-756 90 8 an an DET cana-756 90 9 automorphism automorphism NOUN cana-756 90 10 of of ADP cana-756 90 11 r r NOUN cana-756 90 12 , , PUNCT cana-756 90 13 we we PRON cana-756 90 14 get get VERB cana-756 90 15 d1(u d1(u NOUN cana-756 90 16 , , PUNCT cana-756 90 17 v)r v)r PUNCT cana-756 90 18 Ξ΄2(v Ξ΄2(v PROPN cana-756 90 19 , , PUNCT cana-756 90 20 w w NOUN cana-756 90 21 ) ) PUNCT cana-756 90 22 = = SYM cana-756 90 23 0 0 NUM cana-756 90 24 , , PUNCT cana-756 90 25 βˆ€ βˆ€ X cana-756 90 26 u u NOUN cana-756 90 27 , , PUNCT cana-756 90 28 v v NOUN cana-756 90 29 , , PUNCT cana-756 90 30 r r NOUN cana-756 90 31 , , PUNCT cana-756 90 32 w w PROPN cana-756 90 33 ∈ ∈ PROPN cana-756 90 34 r r NOUN cana-756 90 35 . . PUNCT cana-756 91 1 in in ADP cana-756 91 2 view view NOUN cana-756 91 3 of of ADP cana-756 91 4 lemma lemma PROPN cana-756 91 5 1 1 NUM cana-756 91 6 , , PUNCT cana-756 91 7 we we PRON cana-756 91 8 can can AUX cana-756 91 9 have have VERB cana-756 91 10 d1(u d1(u PROPN cana-756 91 11 , , PUNCT cana-756 91 12 v)rΞ΄2(v v)rΞ΄2(v PROPN cana-756 91 13 , , PUNCT cana-756 91 14 w w PROPN cana-756 91 15 ) ) PUNCT cana-756 91 16 = = SYM cana-756 91 17 0 0 PUNCT cana-756 91 18 = = SYM cana-756 91 19 Ξ΄2(v Ξ΄2(v PROPN cana-756 91 20 , , PUNCT cana-756 91 21 w)rd1(u w)rd1(u PROPN cana-756 91 22 , , PUNCT cana-756 91 23 v v NOUN cana-756 91 24 ) ) PUNCT cana-756 91 25 ( ( PUNCT cana-756 91 26 3.4 3.4 NUM cana-756 91 27 ) ) PUNCT cana-756 91 28 and and CCONJ cana-756 91 29 d1(u d1(u PROPN cana-756 91 30 , , PUNCT cana-756 91 31 v)Ξ΄2(v v)Ξ΄2(v PROPN cana-756 91 32 , , PUNCT cana-756 91 33 w w NOUN cana-756 91 34 ) ) PUNCT cana-756 91 35 = = SYM cana-756 91 36 0 0 PUNCT cana-756 91 37 = = SYM cana-756 91 38 Ξ΄2(v Ξ΄2(v PROPN cana-756 91 39 , , PUNCT cana-756 91 40 w)d1(u w)d1(u PROPN cana-756 91 41 , , PUNCT cana-756 91 42 v v NOUN cana-756 91 43 ) ) PUNCT cana-756 91 44 . . PUNCT cana-756 92 1 ( ( PUNCT cana-756 92 2 3.5 3.5 NUM cana-756 92 3 ) ) PUNCT cana-756 92 4 again again ADV cana-756 92 5 by by ADP cana-756 92 6 the the DET cana-756 92 7 definition definition NOUN cana-756 92 8 of of ADP cana-756 92 9 orthogonality orthogonality NOUN cana-756 92 10 of of ADP cana-756 92 11 Ξ΄1 Ξ΄1 NOUN cana-756 92 12 and and CCONJ cana-756 92 13 Ξ΄2 Ξ΄2 PROPN cana-756 92 14 , , PUNCT cana-756 92 15 we we PRON cana-756 92 16 have have VERB cana-756 92 17 Ξ΄1(u Ξ΄1(u NUM cana-756 92 18 , , PUNCT cana-756 92 19 v)rΞ΄2(v v)rΞ΄2(v PROPN cana-756 92 20 , , PUNCT cana-756 92 21 w w PROPN cana-756 92 22 ) ) PUNCT cana-756 92 23 = = SYM cana-756 92 24 0 0 PUNCT cana-756 93 1 = = SYM cana-756 93 2 Ξ΄2(u Ξ΄2(u PROPN cana-756 93 3 , , PUNCT cana-756 93 4 v)rΞ΄1(v v)rΞ΄1(v PROPN cana-756 93 5 , , PUNCT cana-756 93 6 w w NOUN cana-756 93 7 ) ) PUNCT cana-756 93 8 , , PUNCT cana-756 93 9 for for ADP cana-756 93 10 all all DET cana-756 93 11 u u NOUN cana-756 93 12 , , PUNCT cana-756 93 13 v v NOUN cana-756 93 14 , , PUNCT cana-756 93 15 w w PROPN cana-756 93 16 ∈ ∈ PROPN cana-756 93 17 r. r. NOUN cana-756 93 18 ( ( PUNCT cana-756 93 19 3.6 3.6 NUM cana-756 93 20 ) ) PUNCT cana-756 93 21 by by ADP cana-756 93 22 lemma lemma PROPN cana-756 93 23 ( ( PUNCT cana-756 93 24 1 1 NUM cana-756 93 25 ) ) PUNCT cana-756 93 26 , , PUNCT cana-756 93 27 we we PRON cana-756 93 28 can can AUX cana-756 93 29 have have VERB cana-756 93 30 Ξ΄1(u Ξ΄1(u NUM cana-756 93 31 , , PUNCT cana-756 93 32 v)Ξ΄2(v v)Ξ΄2(v PROPN cana-756 93 33 , , PUNCT cana-756 93 34 w w NOUN cana-756 93 35 ) ) PUNCT cana-756 93 36 = = SYM cana-756 93 37 Ξ΄2(v Ξ΄2(v PROPN cana-756 93 38 , , PUNCT cana-756 93 39 w)Ξ΄1(u w)Ξ΄1(u PROPN cana-756 93 40 , , PUNCT cana-756 93 41 v v NOUN cana-756 93 42 ) ) PUNCT cana-756 93 43 = = SYM cana-756 93 44 0 0 NUM cana-756 93 45 and and CCONJ cana-756 93 46 also also ADV cana-756 93 47 Ξ΄2(u Ξ΄2(u PROPN cana-756 93 48 , , PUNCT cana-756 93 49 v)Ξ΄1(v v)Ξ΄1(v PRON cana-756 93 50 , , PUNCT cana-756 93 51 w w PROPN cana-756 93 52 ) ) PUNCT cana-756 93 53 = = SYM cana-756 94 1 0 0 X cana-756 94 2 . . X cana-756 94 3 consider consider VERB cana-756 94 4 Ξ΄2(u Ξ΄2(u NOUN cana-756 94 5 , , PUNCT cana-756 94 6 v)Ξ΄1(v v)Ξ΄1(v PRON cana-756 94 7 , , PUNCT cana-756 94 8 w)=0 w)=0 PROPN cana-756 94 9 , , PUNCT cana-756 94 10 βˆ€ βˆ€ X cana-756 94 11 u u NOUN cana-756 94 12 , , PUNCT cana-756 94 13 v v NOUN cana-756 94 14 , , PUNCT cana-756 94 15 w w PROPN cana-756 94 16 ∈ ∈ PROPN cana-756 94 17 r r NOUN cana-756 94 18 . . PUNCT cana-756 95 1 ( ( PUNCT cana-756 95 2 3.7 3.7 NUM cana-756 95 3 ) ) PUNCT cana-756 95 4 replacing replace VERB cana-756 95 5 u u NOUN cana-756 95 6 by by ADP cana-756 95 7 ur ur INTJ cana-756 95 8 , , PUNCT cana-756 95 9 r r NOUN cana-756 95 10 ∈ ∈ NOUN cana-756 95 11 r r NOUN cana-756 95 12 in in ADP cana-756 95 13 the the DET cana-756 95 14 equation equation NOUN cana-756 95 15 ( ( PUNCT cana-756 95 16 3.7 3.7 NUM cana-756 95 17 ) ) PUNCT cana-756 95 18 and and CCONJ cana-756 95 19 using use VERB cana-756 95 20 ( ( PUNCT cana-756 95 21 3.6 3.6 NUM cana-756 95 22 ) ) PUNCT cana-756 95 23 , , PUNCT cana-756 95 24 we we PRON cana-756 95 25 obtain obtain VERB cana-756 95 26 Ο„(r)d2(u Ο„(r)d2(u PROPN cana-756 95 27 , , PUNCT cana-756 95 28 v)Ξ΄1(v v)Ξ΄1(v PRON cana-756 95 29 , , PUNCT cana-756 95 30 w w PROPN cana-756 95 31 ) ) PUNCT cana-756 95 32 = = SYM cana-756 95 33 0 0 NUM cana-756 95 34 , , PUNCT cana-756 95 35 βˆ€ βˆ€ X cana-756 95 36 u u NOUN cana-756 95 37 , , PUNCT cana-756 95 38 v v NOUN cana-756 95 39 , , PUNCT cana-756 95 40 w w PROPN cana-756 95 41 , , PUNCT cana-756 95 42 r r PROPN cana-756 95 43 ∈ ∈ PROPN cana-756 95 44 r. r. NOUN cana-756 95 45 left leave VERB cana-756 95 46 multiplying multiply VERB cana-756 95 47 the the DET cana-756 95 48 above above ADJ cana-756 95 49 equation equation NOUN cana-756 95 50 by by ADP cana-756 95 51 d2(u d2(u PROPN cana-756 95 52 , , PUNCT cana-756 95 53 v)Ξ΄1(v v)Ξ΄1(v PRON cana-756 95 54 , , PUNCT cana-756 95 55 w w PROPN cana-756 95 56 ) ) PUNCT cana-756 95 57 and and CCONJ cana-756 95 58 using use VERB cana-756 95 59 the the DET cana-756 95 60 semiprimeness semiprimeness NOUN cana-756 95 61 of of ADP cana-756 95 62 r r NOUN cana-756 95 63 , , PUNCT cana-756 95 64 we we PRON cana-756 95 65 get get VERB cana-756 95 66 d2(u d2(u NOUN cana-756 95 67 , , PUNCT cana-756 95 68 v)Ξ΄1(v v)Ξ΄1(v PRON cana-756 95 69 , , PUNCT cana-756 95 70 w w PROPN cana-756 95 71 ) ) PUNCT cana-756 95 72 = = SYM cana-756 96 1 0 0 X cana-756 96 2 . . PUNCT cana-756 97 1 ( ( PUNCT cana-756 97 2 3.8 3.8 NUM cana-756 97 3 ) ) PUNCT cana-756 97 4 replacing replace VERB cana-756 97 5 u u NOUN cana-756 97 6 by by ADP cana-756 97 7 ru,π‘Ÿ ru,π‘Ÿ PROPN cana-756 97 8 ∈ ∈ PROPN cana-756 97 9 𝑅 𝑅 PROPN cana-756 97 10 in in ADP cana-756 97 11 ( ( PUNCT cana-756 97 12 3.8 3.8 NUM cana-756 97 13 ) ) PUNCT cana-756 97 14 and and CCONJ cana-756 97 15 using use VERB cana-756 97 16 ( ( PUNCT cana-756 97 17 3.8 3.8 NUM cana-756 97 18 ) ) PUNCT cana-756 97 19 , , PUNCT cana-756 97 20 we we PRON cana-756 97 21 get get VERB cana-756 97 22 d2(u d2(u NOUN cana-756 97 23 , , PUNCT cana-756 97 24 v)Οƒ(r)Ξ΄1(v v)Οƒ(r)Ξ΄1(v NOUN cana-756 97 25 , , PUNCT cana-756 97 26 w w PROPN cana-756 97 27 ) ) PUNCT cana-756 97 28 = = SYM cana-756 97 29 0 0 NUM cana-756 97 30 , , PUNCT cana-756 97 31 for for ADP cana-756 97 32 all all DET cana-756 97 33 u u NOUN cana-756 97 34 , , PUNCT cana-756 97 35 v v NOUN cana-756 97 36 , , PUNCT cana-756 97 37 w w PROPN cana-756 97 38 , , PUNCT cana-756 97 39 r r PROPN cana-756 97 40 ∈ ∈ PROPN cana-756 97 41 r. r. PROPN cana-756 97 42 d2(u d2(u PROPN cana-756 97 43 , , PUNCT cana-756 97 44 v)Οƒ(r)Ξ΄1(v v)Οƒ(r)Ξ΄1(v NOUN cana-756 97 45 , , PUNCT cana-756 97 46 w w PROPN cana-756 97 47 ) ) PUNCT cana-756 97 48 = = SYM cana-756 97 49 0 0 X cana-756 97 50 . . PUNCT cana-756 98 1 since since SCONJ cana-756 98 2 Οƒ Οƒ PROPN cana-756 98 3 is be AUX cana-756 98 4 an an DET cana-756 98 5 automorphism automorphism NOUN cana-756 98 6 , , PUNCT cana-756 98 7 we we PRON cana-756 98 8 obtain obtain VERB cana-756 98 9 d2(u d2(u NOUN cana-756 98 10 , , PUNCT cana-756 98 11 v v NOUN cana-756 98 12 ) ) PUNCT cana-756 98 13 rΞ΄1(v rΞ΄1(v PROPN cana-756 98 14 , , PUNCT cana-756 98 15 w w NOUN cana-756 98 16 ) ) PUNCT cana-756 98 17 = = SYM cana-756 98 18 0 0 PUNCT cana-756 99 1 = = SYM cana-756 99 2 Ξ΄1(v Ξ΄1(v PROPN cana-756 99 3 , , PUNCT cana-756 99 4 w w NOUN cana-756 99 5 ) ) PUNCT cana-756 99 6 rd2(u rd2(u PROPN cana-756 99 7 , , PUNCT cana-756 99 8 v v NOUN cana-756 99 9 ) ) PUNCT cana-756 99 10 , , PUNCT cana-756 99 11 βˆ€ βˆ€ X cana-756 99 12 u u NOUN cana-756 99 13 , , PUNCT cana-756 99 14 v v NOUN cana-756 99 15 , , PUNCT cana-756 99 16 w w PROPN cana-756 99 17 , , PUNCT cana-756 99 18 r r PROPN cana-756 99 19 ∈ ∈ PROPN cana-756 99 20 r. r. NOUN cana-756 99 21 ( ( PUNCT cana-756 99 22 3.9 3.9 NUM cana-756 99 23 ) ) PUNCT cana-756 99 24 by by ADP cana-756 99 25 lemma lemma PROPN cana-756 99 26 1 1 NUM cana-756 99 27 , , PUNCT cana-756 99 28 d2(u d2(u NOUN cana-756 99 29 , , PUNCT cana-756 99 30 v)Ξ΄1(v v)Ξ΄1(v PRON cana-756 99 31 , , PUNCT cana-756 99 32 w w PROPN cana-756 99 33 ) ) PUNCT cana-756 99 34 = = SYM cana-756 99 35 0 0 PUNCT cana-756 99 36 = = SYM cana-756 99 37 Ξ΄1(v Ξ΄1(v PROPN cana-756 99 38 , , PUNCT cana-756 99 39 w)d2(u w)d2(u PROPN cana-756 99 40 , , PUNCT cana-756 99 41 v v NOUN cana-756 99 42 ) ) PUNCT cana-756 99 43 , , PUNCT cana-756 99 44 for for ADP cana-756 99 45 all all DET cana-756 99 46 u u NOUN cana-756 99 47 , , PUNCT cana-756 99 48 v v NOUN cana-756 99 49 , , PUNCT cana-756 99 50 w w PROPN cana-756 99 51 ∈ ∈ PROPN cana-756 99 52 r. r. NOUN cana-756 99 53 ( ( PUNCT cana-756 99 54 3.10 3.10 NUM cana-756 99 55 ) ) PUNCT cana-756 99 56 from from ADP cana-756 99 57 ( ( PUNCT cana-756 99 58 3.5 3.5 NUM cana-756 99 59 ) ) PUNCT cana-756 99 60 and and CCONJ cana-756 99 61 ( ( PUNCT cana-756 99 62 3.10 3.10 NUM cana-756 99 63 ) ) PUNCT cana-756 99 64 , , PUNCT cana-756 99 65 we we PRON cana-756 99 66 can can AUX cana-756 99 67 have have VERB cana-756 99 68 d1(u d1(u PROPN cana-756 99 69 , , PUNCT cana-756 99 70 v)Ξ΄2(v v)Ξ΄2(v PROPN cana-756 99 71 , , PUNCT cana-756 99 72 w w NOUN cana-756 99 73 ) ) PUNCT cana-756 99 74 + + CCONJ cana-756 100 1 d2(u d2(u NOUN cana-756 100 2 , , PUNCT cana-756 100 3 v)Ξ΄1(v v)Ξ΄1(v PRON cana-756 100 4 , , PUNCT cana-756 100 5 w w PROPN cana-756 100 6 ) ) PUNCT cana-756 100 7 = = SYM cana-756 101 1 0 0 X cana-756 101 2 . . PUNCT cana-756 102 1 hence hence ADV cana-756 102 2 condition condition NOUN cana-756 102 3 ( ( PUNCT cana-756 102 4 ii ii NOUN cana-756 102 5 ) ) PUNCT cana-756 102 6 is be AUX cana-756 102 7 proved prove VERB cana-756 102 8 . . PUNCT cana-756 103 1 conversely conversely ADV cana-756 103 2 , , PUNCT cana-756 103 3 suppose suppose VERB cana-756 103 4 the the DET cana-756 103 5 conditions condition NOUN cana-756 103 6 ( ( PUNCT cana-756 103 7 i i NOUN cana-756 103 8 ) ) PUNCT cana-756 103 9 Ξ΄1(u Ξ΄1(u PROPN cana-756 103 10 , , PUNCT cana-756 103 11 v)Ξ΄2 v)Ξ΄2 PROPN cana-756 103 12 ( ( PUNCT cana-756 103 13 v v NOUN cana-756 103 14 , , PUNCT cana-756 103 15 w w NOUN cana-756 103 16 ) ) PUNCT cana-756 103 17 + + CCONJ cana-756 103 18 Ξ΄2 Ξ΄2 VERB cana-756 103 19 ( ( PUNCT cana-756 103 20 u u NOUN cana-756 103 21 , , PUNCT cana-756 103 22 v v NOUN cana-756 103 23 ) ) PUNCT cana-756 103 24 Ξ΄1(v Ξ΄1(v PROPN cana-756 103 25 , , PUNCT cana-756 103 26 w w NOUN cana-756 103 27 ) ) PUNCT cana-756 103 28 = = SYM cana-756 103 29 0 0 NUM cana-756 103 30 and and CCONJ cana-756 103 31 ( ( PUNCT cana-756 103 32 3.11 3.11 NUM cana-756 103 33 ) ) PUNCT cana-756 103 34 ( ( PUNCT cana-756 103 35 ii ii NOUN cana-756 103 36 ) ) PUNCT cana-756 103 37 d1(u d1(u PROPN cana-756 103 38 , , PUNCT cana-756 103 39 v)Ξ΄2 v)Ξ΄2 PROPN cana-756 103 40 ( ( PUNCT cana-756 103 41 v v NOUN cana-756 103 42 , , PUNCT cana-756 103 43 w w NOUN cana-756 103 44 ) ) PUNCT cana-756 103 45 + + CCONJ cana-756 103 46 d2 d2 PROPN cana-756 103 47 ( ( PUNCT cana-756 103 48 u u NOUN cana-756 103 49 , , PUNCT cana-756 103 50 v v NOUN cana-756 103 51 ) ) PUNCT cana-756 103 52 Ξ΄1(v Ξ΄1(v PROPN cana-756 103 53 , , PUNCT cana-756 103 54 w w NOUN cana-756 103 55 ) ) PUNCT cana-756 103 56 = = SYM cana-756 103 57 0 0 NUM cana-756 103 58 holds hold VERB cana-756 103 59 good good ADJ cana-756 103 60 . . PUNCT cana-756 104 1 ( ( PUNCT cana-756 104 2 3.12 3.12 NUM cana-756 104 3 ) ) PUNCT cana-756 104 4 we we PRON cana-756 104 5 prove prove VERB cana-756 104 6 that that SCONJ cana-756 104 7 ( ( PUNCT cana-756 104 8 Ξ΄1 Ξ΄1 NOUN cana-756 104 9 , , PUNCT cana-756 104 10 d1 d1 PROPN cana-756 104 11 ) ) PUNCT cana-756 104 12 and and CCONJ cana-756 104 13 ( ( PUNCT cana-756 104 14 Ξ΄2 Ξ΄2 ADJ cana-756 104 15 , , PUNCT cana-756 104 16 d2 d2 PROPN cana-756 104 17 ) ) PUNCT cana-756 104 18 are be AUX cana-756 104 19 orthogonal orthogonal ADJ cana-756 104 20 generalized generalize VERB cana-756 104 21 symmetric symmetric ADJ cana-756 104 22 reverse reverse NOUN cana-756 104 23 bi-(Οƒ bi-(Οƒ PROPN cana-756 104 24 , , PUNCT cana-756 104 25 Ο„)derivations Ο„)derivations PROPN cana-756 104 26 of of ADP cana-756 104 27 r. r. PROPN cana-756 104 28 replacing replace VERB cana-756 104 29 u u NOUN cana-756 104 30 by by ADP cana-756 104 31 ru ru NOUN cana-756 104 32 , , PUNCT cana-756 104 33 r r NOUN cana-756 104 34 ∈ ∈ NOUN cana-756 104 35 r r NOUN cana-756 104 36 in in ADP cana-756 104 37 ( ( PUNCT cana-756 104 38 3.11 3.11 NUM cana-756 104 39 ) ) PUNCT cana-756 104 40 and and CCONJ cana-756 104 41 using use VERB cana-756 104 42 ( ( PUNCT cana-756 104 43 3.12 3.12 NUM cana-756 104 44 ) ) PUNCT cana-756 104 45 , , PUNCT cana-756 104 46 we we PRON cana-756 104 47 get get VERB cana-756 104 48 communications communication NOUN cana-756 104 49 on on ADP cana-756 104 50 applied apply VERB cana-756 104 51 nonlinear nonlinear ADJ cana-756 104 52 analysis analysis NOUN cana-756 104 53 issn issn NOUN cana-756 104 54 : : PUNCT cana-756 104 55 1074 1074 NUM cana-756 104 56 - - PUNCT cana-756 104 57 133x 133x NUM cana-756 104 58 vol vol NOUN cana-756 104 59 31 31 NUM cana-756 104 60 no no NOUN cana-756 104 61 . . PUNCT cana-756 105 1 3s 3s NUM cana-756 105 2 ( ( PUNCT cana-756 105 3 2024 2024 NUM cana-756 105 4 ) ) PUNCT cana-756 105 5 161 161 NUM cana-756 105 6 https://internationalpubls.com https://internationalpubls.com X cana-756 105 7 Ξ΄1(u Ξ΄1(u NUM cana-756 105 8 , , PUNCT cana-756 105 9 v)Οƒ(r)Ξ΄2(v v)Οƒ(r)Ξ΄2(v NOUN cana-756 105 10 , , PUNCT cana-756 105 11 w w NOUN cana-756 105 12 ) ) PUNCT cana-756 105 13 + + CCONJ cana-756 105 14 Ξ΄2 Ξ΄2 VERB cana-756 105 15 ( ( PUNCT cana-756 105 16 u u NOUN cana-756 105 17 , , PUNCT cana-756 105 18 v)Οƒ(r)Ξ΄1(v v)Οƒ(r)Ξ΄1(v NOUN cana-756 105 19 , , PUNCT cana-756 105 20 w)= w)= NOUN cana-756 105 21 0 0 NUM cana-756 105 22 , , PUNCT cana-756 105 23 for for ADP cana-756 105 24 all all DET cana-756 105 25 u u NOUN cana-756 105 26 , , PUNCT cana-756 105 27 v v NOUN cana-756 105 28 , , PUNCT cana-756 105 29 w w PROPN cana-756 105 30 , , PUNCT cana-756 105 31 r r PROPN cana-756 105 32 ∈ ∈ PROPN cana-756 105 33 r. r. NOUN cana-756 105 34 since since SCONJ cana-756 105 35 Οƒ Οƒ PROPN cana-756 105 36 is be AUX cana-756 105 37 an an DET cana-756 105 38 automorphism automorphism NOUN cana-756 105 39 , , PUNCT cana-756 105 40 we we PRON cana-756 105 41 get get VERB cana-756 105 42 Ξ΄1(u Ξ΄1(u NOUN cana-756 105 43 , , PUNCT cana-756 105 44 v)rΞ΄2 v)rΞ΄2 PROPN cana-756 105 45 ( ( PUNCT cana-756 105 46 v v NOUN cana-756 105 47 , , PUNCT cana-756 105 48 w w NOUN cana-756 105 49 ) ) PUNCT cana-756 105 50 + + CCONJ cana-756 105 51 Ξ΄2 Ξ΄2 VERB cana-756 105 52 ( ( PUNCT cana-756 105 53 u u NOUN cana-756 105 54 , , PUNCT cana-756 105 55 v)rΞ΄1(v v)rΞ΄1(v PROPN cana-756 105 56 , , PUNCT cana-756 105 57 w w NOUN cana-756 105 58 ) ) PUNCT cana-756 105 59 = = SYM cana-756 105 60 0 0 NUM cana-756 105 61 , , PUNCT cana-756 105 62 for for ADP cana-756 105 63 all all DET cana-756 105 64 u u NOUN cana-756 105 65 , , PUNCT cana-756 105 66 v v NOUN cana-756 105 67 , , PUNCT cana-756 105 68 w w PROPN cana-756 105 69 , , PUNCT cana-756 105 70 r r PROPN cana-756 105 71 ∈ ∈ PROPN cana-756 105 72 r. r. NOUN cana-756 105 73 by by ADP cana-756 105 74 lemma lemma PROPN cana-756 105 75 1 1 NUM cana-756 105 76 , , PUNCT cana-756 105 77 we we PRON cana-756 105 78 can can AUX cana-756 105 79 conclude conclude VERB cana-756 105 80 that that SCONJ cana-756 105 81 Ξ΄1and Ξ΄1and PROPN cana-756 105 82 Ξ΄2 Ξ΄2 VERB cana-756 105 83 are be AUX cana-756 105 84 orthogonal orthogonal ADJ cana-756 105 85 . . PUNCT cana-756 106 1 theorem theorem NOUN cana-756 106 2 2 2 NUM cana-756 106 3 : : PUNCT cana-756 106 4 if if SCONJ cana-756 106 5 ( ( PUNCT cana-756 106 6 Ξ΄1 Ξ΄1 NOUN cana-756 106 7 , , PUNCT cana-756 106 8 d1 d1 PROPN cana-756 106 9 ) ) PUNCT cana-756 106 10 and and CCONJ cana-756 106 11 ( ( PUNCT cana-756 106 12 Ξ΄2 Ξ΄2 ADJ cana-756 106 13 , , PUNCT cana-756 106 14 d2 d2 PROPN cana-756 106 15 ) ) PUNCT cana-756 106 16 are be AUX cana-756 106 17 two two NUM cana-756 106 18 orthogonal orthogonal ADJ cana-756 106 19 generalized generalize VERB cana-756 106 20 symmetric symmetric ADJ cana-756 106 21 reverse reverse NOUN cana-756 106 22 bi-(Οƒ bi-(Οƒ NOUN cana-756 106 23 , , PUNCT cana-756 106 24 Ο„)-derivations Ο„)-derivation NOUN cana-756 106 25 of of ADP cana-756 106 26 r r NOUN cana-756 106 27 , , PUNCT cana-756 106 28 then then ADV cana-756 106 29 ( ( PUNCT cana-756 106 30 Ξ΄1 Ξ΄1 NOUN cana-756 106 31 , , PUNCT cana-756 106 32 d1 d1 PROPN cana-756 106 33 ) ) PUNCT cana-756 106 34 and and CCONJ cana-756 106 35 ( ( PUNCT cana-756 106 36 Ξ΄2 Ξ΄2 ADJ cana-756 106 37 , , PUNCT cana-756 106 38 d2 d2 PROPN cana-756 106 39 ) ) PUNCT cana-756 106 40 are be AUX cana-756 106 41 orthogonal orthogonal ADJ cana-756 106 42 iff iff PROPN cana-756 106 43 Ξ΄1(u Ξ΄1(u PROPN cana-756 106 44 , , PUNCT cana-756 106 45 v)Ξ΄2 v)Ξ΄2 PROPN cana-756 106 46 ( ( PUNCT cana-756 106 47 v v NOUN cana-756 106 48 , , PUNCT cana-756 106 49 w w NOUN cana-756 106 50 ) ) PUNCT cana-756 106 51 = = SYM cana-756 106 52 0 0 PUNCT cana-756 107 1 = = SYM cana-756 107 2 d1 d1 PROPN cana-756 107 3 ( ( PUNCT cana-756 107 4 u u NOUN cana-756 107 5 , , PUNCT cana-756 107 6 v v NOUN cana-756 107 7 ) ) PUNCT cana-756 107 8 Ξ΄2(v Ξ΄2(v PROPN cana-756 107 9 , , PUNCT cana-756 107 10 w w NOUN cana-756 107 11 ) ) PUNCT cana-756 107 12 . . PUNCT cana-756 108 1 proof proof NOUN cana-756 108 2 : : PUNCT cana-756 108 3 suppose suppose VERB cana-756 108 4 that that SCONJ cana-756 108 5 ( ( PUNCT cana-756 108 6 Ξ΄1 Ξ΄1 NOUN cana-756 108 7 , , PUNCT cana-756 108 8 d1 d1 PROPN cana-756 108 9 ) ) PUNCT cana-756 108 10 and and CCONJ cana-756 108 11 ( ( PUNCT cana-756 108 12 Ξ΄2 Ξ΄2 ADJ cana-756 108 13 , , PUNCT cana-756 108 14 d2 d2 PROPN cana-756 108 15 ) ) PUNCT cana-756 108 16 are be AUX cana-756 108 17 orthogonal orthogonal ADJ cana-756 108 18 generalized generalize VERB cana-756 108 19 symmetric symmetric ADJ cana-756 108 20 reverse reverse NOUN cana-756 108 21 bi-(Οƒ bi-(Οƒ PROPN cana-756 108 22 , , PUNCT cana-756 108 23 Ο„)derivations Ο„)derivations PROPN cana-756 108 24 of of ADP cana-756 108 25 r. r. PROPN cana-756 108 26 by by ADP cana-756 108 27 the the DET cana-756 108 28 definition definition NOUN cana-756 108 29 of of ADP cana-756 108 30 orthogonality orthogonality NOUN cana-756 108 31 , , PUNCT cana-756 108 32 we we PRON cana-756 108 33 have have VERB cana-756 108 34 Ξ΄1(u Ξ΄1(u NUM cana-756 108 35 , , PUNCT cana-756 108 36 v)rΞ΄2(v v)rΞ΄2(v PROPN cana-756 108 37 , , PUNCT cana-756 108 38 w w PROPN cana-756 108 39 ) ) PUNCT cana-756 108 40 = = SYM cana-756 108 41 { { PUNCT cana-756 108 42 0 0 NUM cana-756 108 43 } } PUNCT cana-756 108 44 . . PUNCT cana-756 109 1 Ξ΄1(u Ξ΄1(u NUM cana-756 109 2 , , PUNCT cana-756 109 3 v)rΞ΄2(v v)rΞ΄2(v PROPN cana-756 109 4 , , PUNCT cana-756 109 5 w w PROPN cana-756 109 6 ) ) PUNCT cana-756 109 7 = = SYM cana-756 109 8 0 0 NUM cana-756 110 1 and and CCONJ cana-756 110 2 so so ADV cana-756 110 3 Ξ΄1(u Ξ΄1(u ADJ cana-756 110 4 , , PUNCT cana-756 110 5 v)Ξ΄2(v v)Ξ΄2(v PROPN cana-756 110 6 , , PUNCT cana-756 110 7 w w NOUN cana-756 110 8 ) ) PUNCT cana-756 111 1 = = SYM cana-756 111 2 0 0 NUM cana-756 111 3 , , PUNCT cana-756 111 4 for for ADP cana-756 111 5 all all DET cana-756 111 6 u u NOUN cana-756 111 7 , , PUNCT cana-756 111 8 v v NOUN cana-756 111 9 , , PUNCT cana-756 111 10 w w PROPN cana-756 111 11 ∈ ∈ PROPN cana-756 111 12 r. r. PROPN cana-756 111 13 ( ( PUNCT cana-756 111 14 by by ADP cana-756 111 15 lemma1 lemma1 PROPN cana-756 111 16 ) ) PUNCT cana-756 111 17 ( ( PUNCT cana-756 111 18 3.13 3.13 NUM cana-756 111 19 ) ) PUNCT cana-756 111 20 from from ADP cana-756 111 21 the the DET cana-756 111 22 equation equation NOUN cana-756 111 23 ( ( PUNCT cana-756 111 24 3.5 3.5 NUM cana-756 111 25 ) ) PUNCT cana-756 111 26 of of ADP cana-756 111 27 theorem theorem NOUN cana-756 111 28 1 1 NUM cana-756 111 29 , , PUNCT cana-756 111 30 we we PRON cana-756 111 31 have have VERB cana-756 111 32 d1(u d1(u PROPN cana-756 111 33 , , PUNCT cana-756 111 34 v)Ξ΄2(v v)Ξ΄2(v PROPN cana-756 111 35 , , PUNCT cana-756 111 36 w w NOUN cana-756 111 37 ) ) PUNCT cana-756 111 38 = = SYM cana-756 112 1 0 0 X cana-756 112 2 . . PUNCT cana-756 113 1 hence hence ADV cana-756 113 2 , , PUNCT cana-756 113 3 we we PRON cana-756 113 4 conclude conclude VERB cana-756 113 5 that that PRON cana-756 113 6 Ξ΄1(u Ξ΄1(u NOUN cana-756 113 7 , , PUNCT cana-756 113 8 v)Ξ΄2(v v)Ξ΄2(v PROPN cana-756 113 9 , , PUNCT cana-756 113 10 w w NOUN cana-756 113 11 ) ) PUNCT cana-756 113 12 = = SYM cana-756 113 13 0 0 PUNCT cana-756 114 1 = = SYM cana-756 114 2 d1(u d1(u PROPN cana-756 114 3 , , PUNCT cana-756 114 4 v)Ξ΄2(v v)Ξ΄2(v PROPN cana-756 114 5 , , PUNCT cana-756 114 6 w w NOUN cana-756 114 7 ) ) PUNCT cana-756 114 8 , , PUNCT cana-756 114 9 for for ADP cana-756 114 10 all all DET cana-756 114 11 u u NOUN cana-756 114 12 , , PUNCT cana-756 114 13 v v NOUN cana-756 114 14 , , PUNCT cana-756 114 15 w w PROPN cana-756 114 16 ∈ ∈ PROPN cana-756 114 17 r. r. NOUN cana-756 114 18 conversely conversely ADV cana-756 114 19 , , PUNCT cana-756 114 20 suppose suppose VERB cana-756 114 21 that that SCONJ cana-756 114 22 Ξ΄1(u Ξ΄1(u PROPN cana-756 114 23 , , PUNCT cana-756 114 24 v)Ξ΄2 v)Ξ΄2 PROPN cana-756 114 25 ( ( PUNCT cana-756 114 26 v v NOUN cana-756 114 27 , , PUNCT cana-756 114 28 w w NOUN cana-756 114 29 ) ) PUNCT cana-756 114 30 = = SYM cana-756 114 31 d1 d1 PROPN cana-756 114 32 ( ( PUNCT cana-756 114 33 u u NOUN cana-756 114 34 , , PUNCT cana-756 114 35 v v NOUN cana-756 114 36 ) ) PUNCT cana-756 114 37 Ξ΄2(v Ξ΄2(v PROPN cana-756 114 38 , , PUNCT cana-756 114 39 w w NOUN cana-756 114 40 ) ) PUNCT cana-756 114 41 = = SYM cana-756 114 42 0 0 NUM cana-756 114 43 , , PUNCT cana-756 114 44 for for ADP cana-756 114 45 all all DET cana-756 114 46 u u NOUN cana-756 114 47 , , PUNCT cana-756 114 48 v v NOUN cana-756 114 49 , , PUNCT cana-756 114 50 w w PROPN cana-756 114 51 ∈ ∈ PROPN cana-756 114 52 r. r. NOUN cana-756 114 53 ( ( PUNCT cana-756 114 54 3.14 3.14 NUM cana-756 114 55 ) ) PUNCT cana-756 114 56 we we PRON cana-756 114 57 have have VERB cana-756 114 58 to to PART cana-756 114 59 prove prove VERB cana-756 114 60 that that SCONJ cana-756 114 61 Ξ΄1and Ξ΄1and PRON cana-756 114 62 Ξ΄2 Ξ΄2 VERB cana-756 114 63 are be AUX cana-756 114 64 orthogonal orthogonal ADJ cana-756 114 65 consider consider NOUN cana-756 114 66 Ξ΄1(u Ξ΄1(u NUM cana-756 114 67 , , PUNCT cana-756 114 68 v)Ξ΄2 v)Ξ΄2 PROPN cana-756 114 69 ( ( PUNCT cana-756 114 70 v v NOUN cana-756 114 71 , , PUNCT cana-756 114 72 w w NOUN cana-756 114 73 ) ) PUNCT cana-756 114 74 = = SYM cana-756 115 1 0 0 X cana-756 115 2 . . PUNCT cana-756 116 1 ( ( PUNCT cana-756 116 2 from from ADP cana-756 116 3 ( ( PUNCT cana-756 116 4 3.14 3.14 NUM cana-756 116 5 ) ) PUNCT cana-756 116 6 ) ) PUNCT cana-756 116 7 ( ( PUNCT cana-756 116 8 3.15 3.15 NUM cana-756 116 9 ) ) PUNCT cana-756 116 10 replacing replace VERB cana-756 116 11 u u NOUN cana-756 116 12 by by ADP cana-756 116 13 ru ru PROPN cana-756 116 14 , , PUNCT cana-756 116 15 π‘Ÿ π‘Ÿ NOUN cana-756 116 16 ∈ ∈ PROPN cana-756 116 17 𝑅 𝑅 PROPN cana-756 116 18 in in ADP cana-756 116 19 ( ( PUNCT cana-756 116 20 3.15 3.15 NUM cana-756 116 21 ) ) PUNCT cana-756 116 22 and and CCONJ cana-756 116 23 using use VERB cana-756 116 24 ( ( PUNCT cana-756 116 25 3.14 3.14 NUM cana-756 116 26 ) ) PUNCT cana-756 116 27 , , PUNCT cana-756 116 28 we we PRON cana-756 116 29 get get VERB cana-756 116 30 Ξ΄1(u Ξ΄1(u NOUN cana-756 116 31 , , PUNCT cana-756 116 32 v)Οƒ(r v)Οƒ(r NUM cana-756 116 33 ) ) PUNCT cana-756 116 34 Ξ΄2(v Ξ΄2(v PROPN cana-756 116 35 , , PUNCT cana-756 116 36 w w NOUN cana-756 116 37 ) ) PUNCT cana-756 116 38 = = SYM cana-756 117 1 0 0 X cana-756 117 2 . . PUNCT cana-756 118 1 since since SCONJ cana-756 118 2 Οƒ Οƒ PROPN cana-756 118 3 is be AUX cana-756 118 4 an an DET cana-756 118 5 automorphism automorphism NOUN cana-756 118 6 , , PUNCT cana-756 118 7 we we PRON cana-756 118 8 get get VERB cana-756 118 9 Ξ΄1(u Ξ΄1(u NUM cana-756 118 10 , , PUNCT cana-756 118 11 v v NOUN cana-756 118 12 ) ) PUNCT cana-756 118 13 rΞ΄2(v rΞ΄2(v NOUN cana-756 118 14 , , PUNCT cana-756 118 15 w w NOUN cana-756 118 16 ) ) PUNCT cana-756 118 17 = = SYM cana-756 118 18 0 0 NUM cana-756 118 19 , , PUNCT cana-756 118 20 for for ADP cana-756 118 21 all all DET cana-756 118 22 u u NOUN cana-756 118 23 , , PUNCT cana-756 118 24 v v NOUN cana-756 118 25 , , PUNCT cana-756 118 26 w w PROPN cana-756 118 27 , , PUNCT cana-756 118 28 r r PROPN cana-756 118 29 ∈ ∈ PROPN cana-756 118 30 r. r. PROPN cana-756 118 31 therefore therefore ADV cana-756 118 32 , , PUNCT cana-756 118 33 Ξ΄1and Ξ΄1and PRON cana-756 118 34 Ξ΄2 Ξ΄2 VERB cana-756 118 35 are be AUX cana-756 118 36 orthogonal orthogonal ADJ cana-756 118 37 . . PUNCT cana-756 119 1 theorem theorem VERB cana-756 119 2 3 3 NUM cana-756 119 3 : : PUNCT cana-756 119 4 if if SCONJ cana-756 119 5 ( ( PUNCT cana-756 119 6 Ξ΄1 Ξ΄1 NOUN cana-756 119 7 , , PUNCT cana-756 119 8 d1 d1 PROPN cana-756 119 9 ) ) PUNCT cana-756 119 10 and and CCONJ cana-756 119 11 ( ( PUNCT cana-756 119 12 Ξ΄2 Ξ΄2 ADJ cana-756 119 13 , , PUNCT cana-756 119 14 d2 d2 PROPN cana-756 119 15 ) ) PUNCT cana-756 119 16 are be AUX cana-756 119 17 two two NUM cana-756 119 18 orthogonal orthogonal ADJ cana-756 119 19 generalized generalize VERB cana-756 119 20 symmetric symmetric ADJ cana-756 119 21 reverse reverse NOUN cana-756 119 22 bi-(Οƒ bi-(Οƒ NOUN cana-756 119 23 , , PUNCT cana-756 119 24 Ο„)-derivations Ο„)-derivation NOUN cana-756 119 25 of of ADP cana-756 119 26 r r NOUN cana-756 119 27 , , PUNCT cana-756 119 28 then then ADV cana-756 119 29 ( ( PUNCT cana-756 119 30 Ξ΄1 Ξ΄1 NOUN cana-756 119 31 , , PUNCT cana-756 119 32 d1 d1 PROPN cana-756 119 33 ) ) PUNCT cana-756 119 34 and and CCONJ cana-756 119 35 ( ( PUNCT cana-756 119 36 Ξ΄2 Ξ΄2 ADJ cana-756 119 37 , , PUNCT cana-756 119 38 d2 d2 PROPN cana-756 119 39 ) ) PUNCT cana-756 119 40 are be AUX cana-756 119 41 orthogonal orthogonal ADJ cana-756 119 42 if if SCONJ cana-756 119 43 and and CCONJ cana-756 119 44 only only ADV cana-756 119 45 if if SCONJ cana-756 119 46 Ξ΄1(u Ξ΄1(u PROPN cana-756 119 47 , , PUNCT cana-756 119 48 v)Ξ΄2 v)Ξ΄2 PROPN cana-756 119 49 ( ( PUNCT cana-756 119 50 v v NOUN cana-756 119 51 , , PUNCT cana-756 119 52 w w NOUN cana-756 119 53 ) ) PUNCT cana-756 119 54 = = SYM cana-756 119 55 0 0 NUM cana-756 119 56 and and CCONJ cana-756 119 57 d1Ξ΄2 d1Ξ΄2 NOUN cana-756 120 1 = = SYM cana-756 120 2 0 0 SYM cana-756 120 3 = = SYM cana-756 120 4 d1d2 d1d2 NOUN cana-756 120 5 . . PUNCT cana-756 120 6 proof proof NOUN cana-756 120 7 : : PUNCT cana-756 120 8 suppose suppose VERB cana-756 120 9 that that SCONJ cana-756 120 10 ( ( PUNCT cana-756 120 11 Ξ΄1 Ξ΄1 NOUN cana-756 120 12 , , PUNCT cana-756 120 13 d1 d1 PROPN cana-756 120 14 ) ) PUNCT cana-756 120 15 and and CCONJ cana-756 120 16 ( ( PUNCT cana-756 120 17 Ξ΄2 Ξ΄2 ADJ cana-756 120 18 , , PUNCT cana-756 120 19 d2 d2 PROPN cana-756 120 20 ) ) PUNCT cana-756 120 21 are be AUX cana-756 120 22 orthogonal orthogonal ADJ cana-756 120 23 generalized generalize VERB cana-756 120 24 symmetric symmetric ADJ cana-756 120 25 reverse reverse NOUN cana-756 120 26 bi-(Οƒ bi-(Οƒ PROPN cana-756 120 27 , , PUNCT cana-756 120 28 Ο„)derivations Ο„)derivations PROPN cana-756 120 29 of of ADP cana-756 120 30 r. r. PROPN cana-756 120 31 by by ADP cana-756 120 32 the the DET cana-756 120 33 definition definition NOUN cana-756 120 34 of of ADP cana-756 120 35 orthogonality orthogonality NOUN cana-756 120 36 , , PUNCT cana-756 120 37 it it PRON cana-756 120 38 is be AUX cana-756 120 39 evident evident ADJ cana-756 120 40 that that SCONJ cana-756 120 41 Ξ΄1(u Ξ΄1(u NOUN cana-756 120 42 , , PUNCT cana-756 120 43 v)rΞ΄2(v v)rΞ΄2(v PROPN cana-756 120 44 , , PUNCT cana-756 120 45 w w PROPN cana-756 120 46 ) ) PUNCT cana-756 120 47 = = SYM cana-756 120 48 0 0 NUM cana-756 120 49 and and CCONJ cana-756 120 50 so so ADV cana-756 120 51 Ξ΄1(u Ξ΄1(u ADJ cana-756 120 52 , , PUNCT cana-756 120 53 v)Ξ΄2(v v)Ξ΄2(v PROPN cana-756 120 54 , , PUNCT cana-756 120 55 w w NOUN cana-756 120 56 ) ) PUNCT cana-756 120 57 = = SYM cana-756 120 58 0 0 NUM cana-756 120 59 . . PUNCT cana-756 121 1 ( ( PUNCT cana-756 121 2 by by ADP cana-756 121 3 using use VERB cana-756 121 4 lemma lemma PROPN cana-756 121 5 1 1 NUM cana-756 121 6 ) ) PUNCT cana-756 121 7 to to PART cana-756 121 8 prove prove VERB cana-756 121 9 that that PRON cana-756 121 10 d1Ξ΄2 d1Ξ΄2 VERB cana-756 121 11 = = SYM cana-756 121 12 0 0 NUM cana-756 121 13 : : PUNCT cana-756 121 14 consider consider VERB cana-756 121 15 Ξ΄2(v Ξ΄2(v PROPN cana-756 121 16 , , PUNCT cana-756 121 17 w)rd1(u w)rd1(u PROPN cana-756 121 18 , , PUNCT cana-756 121 19 v)=0 v)=0 PROPN cana-756 121 20 ( ( PUNCT cana-756 121 21 by by ADP cana-756 121 22 the the DET cana-756 121 23 equation equation NOUN cana-756 121 24 ( ( PUNCT cana-756 121 25 3.4 3.4 NUM cana-756 121 26 ) ) PUNCT cana-756 121 27 of of ADP cana-756 121 28 theorem1 theorem1 NOUN cana-756 121 29 ) ) PUNCT cana-756 121 30 communications communication NOUN cana-756 121 31 on on ADP cana-756 121 32 applied apply VERB cana-756 121 33 nonlinear nonlinear ADJ cana-756 121 34 analysis analysis NOUN cana-756 121 35 issn issn NOUN cana-756 121 36 : : PUNCT cana-756 121 37 1074 1074 NUM cana-756 121 38 - - PUNCT cana-756 121 39 133x 133x NUM cana-756 121 40 vol vol NOUN cana-756 121 41 31 31 NUM cana-756 121 42 no no NOUN cana-756 121 43 . . PUNCT cana-756 122 1 3s 3s NUM cana-756 122 2 ( ( PUNCT cana-756 122 3 2024 2024 NUM cana-756 122 4 ) ) PUNCT cana-756 122 5 162 162 NUM cana-756 122 6 https://internationalpubls.com https://internationalpubls.com X cana-756 122 7 Ξ΄1(Ξ΄2(v Ξ΄1(Ξ΄2(v PROPN cana-756 122 8 , , PUNCT cana-756 122 9 w)rd1(u w)rd1(u PROPN cana-756 122 10 , , PUNCT cana-756 122 11 v v NOUN cana-756 122 12 ) ) PUNCT cana-756 122 13 , , PUNCT cana-756 122 14 m m NOUN cana-756 122 15 ) ) PUNCT cana-756 122 16 = = SYM cana-756 122 17 0 0 NUM cana-756 123 1 Ξ΄1(rd1(u Ξ΄1(rd1(u NOUN cana-756 123 2 , , PUNCT cana-756 123 3 v v NOUN cana-756 123 4 ) ) PUNCT cana-756 123 5 , , PUNCT cana-756 123 6 m m NOUN cana-756 123 7 ) ) PUNCT cana-756 123 8 Οƒ(Ξ΄2(v Οƒ(Ξ΄2(v PROPN cana-756 123 9 , , PUNCT cana-756 123 10 w w NOUN cana-756 123 11 ) ) PUNCT cana-756 123 12 ) ) PUNCT cana-756 124 1 + + CCONJ cana-756 124 2 Ο„(rd1(u Ο„(rd1(u PROPN cana-756 124 3 , , PUNCT cana-756 124 4 v))d1(Ξ΄2(v v))d1(Ξ΄2(v NOUN cana-756 124 5 , , PUNCT cana-756 124 6 w w NOUN cana-756 124 7 ) ) PUNCT cana-756 124 8 , , PUNCT cana-756 124 9 m)= m)= NOUN cana-756 124 10 0 0 NUM cana-756 124 11 ( ( PUNCT cana-756 124 12 Ξ΄1(d1(u Ξ΄1(d1(u NOUN cana-756 124 13 , , PUNCT cana-756 124 14 v v NOUN cana-756 124 15 ) ) PUNCT cana-756 124 16 , , PUNCT cana-756 124 17 m)Οƒ(r m)Οƒ(r PROPN cana-756 124 18 ) ) PUNCT cana-756 124 19 + + SYM cana-756 124 20 Ο„(d1(u Ο„(d1(u ADJ cana-756 124 21 , , PUNCT cana-756 124 22 v))d1(r v))d1(r NOUN cana-756 124 23 , , PUNCT cana-756 124 24 m m NOUN cana-756 124 25 ) ) PUNCT cana-756 124 26 ) ) PUNCT cana-756 125 1 Οƒ(Ξ΄2(v Οƒ(Ξ΄2(v PROPN cana-756 125 2 , , PUNCT cana-756 125 3 w w NOUN cana-756 125 4 ) ) PUNCT cana-756 125 5 + + CCONJ cana-756 125 6 Ο„(rd1(u Ο„(rd1(u PROPN cana-756 125 7 , , PUNCT cana-756 125 8 v))d1(Ξ΄2(v v))d1(Ξ΄2(v NOUN cana-756 125 9 , , PUNCT cana-756 125 10 w w NOUN cana-756 125 11 ) ) PUNCT cana-756 125 12 , , PUNCT cana-756 125 13 m)= m)= NOUN cana-756 125 14 0 0 NUM cana-756 125 15 . . PUNCT cana-756 126 1 using use VERB cana-756 126 2 d1Ο„ d1Ο„ PROPN cana-756 126 3 = = PUNCT cana-756 126 4 Ο„d1 Ο„d1 NOUN cana-756 126 5 , , PUNCT cana-756 126 6 σδ2 σδ2 INTJ cana-756 126 7 = = SYM cana-756 126 8 Ξ΄2Οƒ Ξ΄2Οƒ PROPN cana-756 126 9 and and CCONJ cana-756 126 10 Οƒ Οƒ PROPN cana-756 126 11 and and CCONJ cana-756 126 12 Ο„ Ο„ PROPN cana-756 126 13 are be AUX cana-756 126 14 automorphisms automorphism NOUN cana-756 126 15 , , PUNCT cana-756 126 16 we we PRON cana-756 126 17 can can AUX cana-756 126 18 have have VERB cana-756 126 19 Ξ΄1(d1(u Ξ΄1(d1(u NOUN cana-756 126 20 , , PUNCT cana-756 126 21 v v NOUN cana-756 126 22 ) ) PUNCT cana-756 126 23 , , PUNCT cana-756 126 24 m m NOUN cana-756 126 25 ) ) PUNCT cana-756 126 26 rΞ΄2(v rΞ΄2(v NOUN cana-756 126 27 , , PUNCT cana-756 126 28 w w NOUN cana-756 126 29 ) ) PUNCT cana-756 126 30 + + CCONJ cana-756 127 1 d1(u d1(u PROPN cana-756 127 2 , , PUNCT cana-756 127 3 v)d1(r v)d1(r PROPN cana-756 127 4 , , PUNCT cana-756 127 5 m)Ξ΄2(v m)Ξ΄2(v PROPN cana-756 127 6 , , PUNCT cana-756 127 7 w w PROPN cana-756 127 8 ) ) PUNCT cana-756 128 1 + + X cana-756 128 2 rd1(u rd1(u PROPN cana-756 128 3 , , PUNCT cana-756 128 4 v)d1(Ξ΄2(v v)d1(Ξ΄2(v NOUN cana-756 128 5 , , PUNCT cana-756 128 6 w w NOUN cana-756 128 7 ) ) PUNCT cana-756 128 8 , , PUNCT cana-756 128 9 m m NOUN cana-756 128 10 ) ) PUNCT cana-756 128 11 = = SYM cana-756 128 12 0 0 X cana-756 128 13 . . PUNCT cana-756 129 1 ( ( PUNCT cana-756 129 2 3.16 3.16 NUM cana-756 129 3 ) ) PUNCT cana-756 129 4 using use VERB cana-756 129 5 the the DET cana-756 129 6 condition condition NOUN cana-756 129 7 of of ADP cana-756 129 8 orthogonality orthogonality NOUN cana-756 129 9 of of ADP cana-756 129 10 Ξ΄1 Ξ΄1 NOUN cana-756 129 11 and and CCONJ cana-756 129 12 Ξ΄2 Ξ΄2 ADV cana-756 129 13 , , PUNCT cana-756 129 14 we we PRON cana-756 129 15 can can AUX cana-756 129 16 have have VERB cana-756 129 17 Ξ΄1(d1(u Ξ΄1(d1(u NOUN cana-756 129 18 , , PUNCT cana-756 129 19 v v NOUN cana-756 129 20 ) ) PUNCT cana-756 129 21 , , PUNCT cana-756 129 22 m m NOUN cana-756 129 23 ) ) PUNCT cana-756 129 24 rΞ΄2(v rΞ΄2(v NOUN cana-756 129 25 , , PUNCT cana-756 129 26 w w NOUN cana-756 129 27 ) ) PUNCT cana-756 129 28 = = SYM cana-756 129 29 0 0 NUM cana-756 129 30 and and CCONJ cana-756 129 31 by by ADP cana-756 129 32 theorem theorem NOUN cana-756 129 33 2 2 NUM cana-756 129 34 , , PUNCT cana-756 129 35 we we PRON cana-756 129 36 can can AUX cana-756 129 37 have have VERB cana-756 129 38 d1(r d1(r PROPN cana-756 129 39 , , PUNCT cana-756 129 40 m m NOUN cana-756 129 41 ) ) PUNCT cana-756 129 42 Ξ΄2(v Ξ΄2(v PROPN cana-756 129 43 , , PUNCT cana-756 129 44 w w NOUN cana-756 129 45 ) ) PUNCT cana-756 129 46 = = SYM cana-756 129 47 0 0 NUM cana-756 129 48 , , PUNCT cana-756 129 49 for for ADP cana-756 129 50 all all DET cana-756 129 51 v v NOUN cana-756 129 52 , , PUNCT cana-756 129 53 w w NOUN cana-756 129 54 , , PUNCT cana-756 129 55 r r NOUN cana-756 129 56 , , PUNCT cana-756 129 57 m m PROPN cana-756 129 58 ∈ ∈ PROPN cana-756 129 59 r. r. NOUN cana-756 129 60 by by ADP cana-756 129 61 applying apply VERB cana-756 129 62 the the DET cana-756 129 63 above above ADJ cana-756 129 64 conditions condition NOUN cana-756 129 65 in in ADP cana-756 129 66 ( ( PUNCT cana-756 129 67 3.16 3.16 NUM cana-756 129 68 ) ) PUNCT cana-756 129 69 , , PUNCT cana-756 129 70 we we PRON cana-756 129 71 get get VERB cana-756 129 72 rd1(u rd1(u PROPN cana-756 129 73 , , PUNCT cana-756 129 74 v)d1(Ξ΄2(v v)d1(Ξ΄2(v NOUN cana-756 129 75 , , PUNCT cana-756 129 76 w w NOUN cana-756 129 77 ) ) PUNCT cana-756 129 78 , , PUNCT cana-756 129 79 m m NOUN cana-756 129 80 ) ) PUNCT cana-756 129 81 = = SYM cana-756 130 1 0 0 NUM cana-756 130 2 , , PUNCT cana-756 130 3 for for ADP cana-756 130 4 all all DET cana-756 130 5 r r NOUN cana-756 130 6 , , PUNCT cana-756 130 7 u u NOUN cana-756 130 8 , , PUNCT cana-756 130 9 v v NOUN cana-756 130 10 , , PUNCT cana-756 130 11 w w PROPN cana-756 130 12 ∈ ∈ PROPN cana-756 130 13 r. r. PROPN cana-756 130 14 left leave VERB cana-756 130 15 multiplying multiply VERB cana-756 130 16 the the DET cana-756 130 17 above above ADJ cana-756 130 18 equation equation NOUN cana-756 130 19 by by ADP cana-756 130 20 d1(u d1(u PROPN cana-756 130 21 , , PUNCT cana-756 130 22 v)d1(Ξ΄2(v v)d1(Ξ΄2(v NOUN cana-756 130 23 , , PUNCT cana-756 130 24 w w NOUN cana-756 130 25 ) ) PUNCT cana-756 130 26 , , PUNCT cana-756 130 27 m m PROPN cana-756 130 28 ) ) PUNCT cana-756 130 29 and and CCONJ cana-756 130 30 using use VERB cana-756 130 31 the the DET cana-756 130 32 semiprimeness semiprimeness NOUN cana-756 130 33 of of ADP cana-756 130 34 r r NOUN cana-756 130 35 , , PUNCT cana-756 130 36 we we PRON cana-756 130 37 get get VERB cana-756 130 38 d1(u d1(u NOUN cana-756 130 39 , , PUNCT cana-756 130 40 v)d1(Ξ΄2(v v)d1(Ξ΄2(v NOUN cana-756 130 41 , , PUNCT cana-756 130 42 w w NOUN cana-756 130 43 ) ) PUNCT cana-756 130 44 , , PUNCT cana-756 130 45 m m NOUN cana-756 130 46 ) ) PUNCT cana-756 130 47 = = SYM cana-756 130 48 0 0 NUM cana-756 130 49 d1(u d1(u PROPN cana-756 130 50 , , PUNCT cana-756 130 51 v)d1Ξ΄2(v v)d1Ξ΄2(v NOUN cana-756 130 52 , , PUNCT cana-756 130 53 w w NOUN cana-756 130 54 ) ) PUNCT cana-756 130 55 = = SYM cana-756 130 56 0 0 X cana-756 130 57 . . PUNCT cana-756 131 1 ( ( PUNCT cana-756 131 2 3.17 3.17 NUM cana-756 131 3 ) ) PUNCT cana-756 131 4 replacing replace VERB cana-756 131 5 u u NOUN cana-756 131 6 by by ADP cana-756 131 7 uΞ΄2(v uΞ΄2(v PROPN cana-756 131 8 , , PUNCT cana-756 131 9 w w PROPN cana-756 131 10 ) ) PUNCT cana-756 131 11 in in ADP cana-756 131 12 ( ( PUNCT cana-756 131 13 3.17 3.17 NUM cana-756 131 14 ) ) PUNCT cana-756 131 15 , , PUNCT cana-756 131 16 we we PRON cana-756 131 17 get get VERB cana-756 131 18 ( ( PUNCT cana-756 131 19 d1(Ξ΄2(v d1(Ξ΄2(v PROPN cana-756 131 20 , , PUNCT cana-756 131 21 w w NOUN cana-756 131 22 ) ) PUNCT cana-756 131 23 , , PUNCT cana-756 131 24 v)Οƒ(u v)Οƒ(u PROPN cana-756 131 25 ) ) PUNCT cana-756 131 26 + + SYM cana-756 131 27 Ο„(Ξ΄2(v Ο„(Ξ΄2(v PROPN cana-756 131 28 , , PUNCT cana-756 131 29 w w NOUN cana-756 131 30 ) ) PUNCT cana-756 131 31 ) ) PUNCT cana-756 132 1 d1(u d1(u PROPN cana-756 132 2 , , PUNCT cana-756 132 3 v))d1Ξ΄2(v v))d1Ξ΄2(v NOUN cana-756 132 4 , , PUNCT cana-756 132 5 w w NOUN cana-756 132 6 ) ) PUNCT cana-756 132 7 = = SYM cana-756 132 8 0 0 X cana-756 132 9 . . X cana-756 133 1 using use VERB cana-756 133 2 ( ( PUNCT cana-756 133 3 3.17 3.17 NUM cana-756 133 4 ) ) PUNCT cana-756 133 5 and and CCONJ cana-756 133 6 Οƒ Οƒ PROPN cana-756 133 7 is be AUX cana-756 133 8 an an DET cana-756 133 9 automorphism automorphism NOUN cana-756 133 10 of of ADP cana-756 133 11 r r NOUN cana-756 133 12 , , PUNCT cana-756 133 13 we we PRON cana-756 133 14 obtain obtain VERB cana-756 133 15 d1(Ξ΄2(v d1(Ξ΄2(v PROPN cana-756 133 16 , , PUNCT cana-756 133 17 w w NOUN cana-756 133 18 ) ) PUNCT cana-756 133 19 , , PUNCT cana-756 133 20 v)ud1Ξ΄2(v v)ud1Ξ΄2(v X cana-756 133 21 , , PUNCT cana-756 133 22 w w NOUN cana-756 133 23 ) ) PUNCT cana-756 133 24 = = SYM cana-756 133 25 0 0 NUM cana-756 133 26 , , PUNCT cana-756 133 27 for for ADP cana-756 133 28 all all DET cana-756 133 29 u u NOUN cana-756 133 30 , , PUNCT cana-756 133 31 v v NOUN cana-756 133 32 , , PUNCT cana-756 133 33 w w PROPN cana-756 133 34 ∈ ∈ PROPN cana-756 133 35 r r NOUN cana-756 133 36 d1Ξ΄2(v d1Ξ΄2(v PROPN cana-756 133 37 , , PUNCT cana-756 133 38 w w NOUN cana-756 133 39 ) ) PUNCT cana-756 133 40 u u NOUN cana-756 133 41 d1Ξ΄2(v d1Ξ΄2(v PROPN cana-756 133 42 , , PUNCT cana-756 133 43 w w NOUN cana-756 133 44 ) ) PUNCT cana-756 133 45 = = SYM cana-756 133 46 0 0 NUM cana-756 133 47 d1Ξ΄2(v d1Ξ΄2(v PROPN cana-756 133 48 , , PUNCT cana-756 133 49 w)rd1Ξ΄2(v w)rd1Ξ΄2(v NOUN cana-756 133 50 , , PUNCT cana-756 133 51 w w NOUN cana-756 133 52 ) ) PUNCT cana-756 133 53 = = SYM cana-756 133 54 0 0 NUM cana-756 133 55 d1 d1 PROPN cana-756 133 56 Ξ΄2 Ξ΄2 VERB cana-756 133 57 = = SYM cana-756 133 58 0 0 NUM cana-756 133 59 . . PUNCT cana-756 134 1 ( ( PUNCT cana-756 134 2 by by ADP cana-756 134 3 the the DET cana-756 134 4 semiprime semiprime NOUN cana-756 134 5 ness ness NOUN cana-756 134 6 of of ADP cana-756 134 7 r r NOUN cana-756 134 8 ) ) PUNCT cana-756 135 1 to to PART cana-756 135 2 prove prove VERB cana-756 135 3 that that SCONJ cana-756 135 4 d1d2 d1d2 VERB cana-756 135 5 = = SYM cana-756 135 6 0 0 NUM cana-756 135 7 : : PUNCT cana-756 135 8 let let VERB cana-756 135 9 ( ( PUNCT cana-756 135 10 Ξ΄1 Ξ΄1 NOUN cana-756 135 11 , , PUNCT cana-756 135 12 d1 d1 PROPN cana-756 135 13 ) ) PUNCT cana-756 135 14 and and CCONJ cana-756 135 15 ( ( PUNCT cana-756 135 16 Ξ΄1 Ξ΄1 NOUN cana-756 135 17 , , PUNCT cana-756 135 18 d2 d2 PROPN cana-756 135 19 ) ) PUNCT cana-756 135 20 are be AUX cana-756 135 21 two two NUM cana-756 135 22 orthogonal orthogonal ADJ cana-756 135 23 generalized generalize VERB cana-756 135 24 symmetric symmetric ADJ cana-756 135 25 reverse reverse NOUN cana-756 135 26 bi-(Οƒ bi-(Οƒ NOUN cana-756 135 27 , , PUNCT cana-756 135 28 Ο„)-derivations Ο„)-derivation NOUN cana-756 135 29 of of ADP cana-756 135 30 a a DET cana-756 135 31 semi semi ADJ cana-756 135 32 prime prime PROPN cana-756 135 33 ring ring NOUN cana-756 135 34 r. r. PROPN cana-756 135 35 first first ADV cana-756 135 36 we we PRON cana-756 135 37 prove prove VERB cana-756 135 38 d1 d1 PROPN cana-756 135 39 and and CCONJ cana-756 135 40 d2 d2 PROPN cana-756 135 41 are be AUX cana-756 135 42 orthogonal orthogonal ADJ cana-756 135 43 . . PUNCT cana-756 136 1 by by ADP cana-756 136 2 the the DET cana-756 136 3 definition definition NOUN cana-756 136 4 of of ADP cana-756 136 5 orthogonality orthogonality NOUN cana-756 136 6 of of ADP cana-756 136 7 Ξ΄1 Ξ΄1 NOUN cana-756 136 8 and and CCONJ cana-756 136 9 Ξ΄2 Ξ΄2 ADV cana-756 136 10 , , PUNCT cana-756 136 11 we we PRON cana-756 136 12 have have VERB cana-756 136 13 Ξ΄1(u Ξ΄1(u NUM cana-756 136 14 , , PUNCT cana-756 136 15 v)rΞ΄2(v v)rΞ΄2(v PROPN cana-756 136 16 , , PUNCT cana-756 136 17 w w PROPN cana-756 136 18 ) ) PUNCT cana-756 136 19 = = SYM cana-756 136 20 0 0 NUM cana-756 136 21 , , PUNCT cana-756 136 22 for for ADP cana-756 136 23 all all DET cana-756 136 24 u u NOUN cana-756 136 25 , , PUNCT cana-756 136 26 v v NOUN cana-756 136 27 , , PUNCT cana-756 136 28 w w PROPN cana-756 136 29 ∈ ∈ PROPN cana-756 136 30 r. r. NOUN cana-756 136 31 by by ADP cana-756 136 32 lemma lemma PROPN cana-756 136 33 1 1 NUM cana-756 136 34 , , PUNCT cana-756 136 35 we we PRON cana-756 136 36 have have VERB cana-756 136 37 Ξ΄1(u Ξ΄1(u NUM cana-756 136 38 , , PUNCT cana-756 136 39 v)Ξ΄2(v v)Ξ΄2(v PROPN cana-756 136 40 , , PUNCT cana-756 136 41 w w NOUN cana-756 136 42 ) ) PUNCT cana-756 136 43 = = SYM cana-756 136 44 0 0 X cana-756 136 45 . . PUNCT cana-756 137 1 ( ( PUNCT cana-756 137 2 3.18 3.18 NUM cana-756 137 3 ) ) PUNCT cana-756 137 4 replacing replace VERB cana-756 137 5 u u NOUN cana-756 137 6 by by ADP cana-756 137 7 ru ru NOUN cana-756 137 8 , , PUNCT cana-756 137 9 r r NOUN cana-756 137 10 ∈ ∈ NOUN cana-756 137 11 r r NOUN cana-756 137 12 in in ADP cana-756 137 13 the the DET cana-756 137 14 equation equation NOUN cana-756 137 15 ( ( PUNCT cana-756 137 16 3.18 3.18 NUM cana-756 137 17 ) ) PUNCT cana-756 137 18 , , PUNCT cana-756 137 19 we we PRON cana-756 137 20 get get VERB cana-756 137 21 Ξ΄1(u Ξ΄1(u NUM cana-756 137 22 , , PUNCT cana-756 137 23 v)Οƒ(r)Ξ΄2(v v)Οƒ(r)Ξ΄2(v NOUN cana-756 137 24 , , PUNCT cana-756 137 25 w w NOUN cana-756 137 26 ) ) PUNCT cana-756 137 27 + + CCONJ cana-756 137 28 Ο„(u)d1(r Ο„(u)d1(r ADJ cana-756 137 29 , , PUNCT cana-756 137 30 v)Ξ΄2(v v)Ξ΄2(v PROPN cana-756 137 31 , , PUNCT cana-756 137 32 w w NOUN cana-756 137 33 ) ) PUNCT cana-756 137 34 = = SYM cana-756 137 35 0 0 NUM cana-756 137 36 , , PUNCT cana-756 137 37 for for ADP cana-756 137 38 all all DET cana-756 137 39 u u NOUN cana-756 137 40 , , PUNCT cana-756 137 41 v v NOUN cana-756 137 42 , , PUNCT cana-756 137 43 w w PROPN cana-756 137 44 , , PUNCT cana-756 137 45 r r PROPN cana-756 137 46 ∈ ∈ PROPN cana-756 137 47 r. r. PROPN cana-756 137 48 since since SCONJ cana-756 137 49 Οƒ Οƒ PROPN cana-756 137 50 and and CCONJ cana-756 137 51 Ο„ Ο„ PROPN cana-756 137 52 are be AUX cana-756 137 53 automorphisms automorphism NOUN cana-756 137 54 of of ADP cana-756 137 55 r r NOUN cana-756 137 56 , , PUNCT cana-756 137 57 we we PRON cana-756 137 58 can can AUX cana-756 137 59 have have VERB cana-756 137 60 Ξ΄1(u Ξ΄1(u NUM cana-756 137 61 , , PUNCT cana-756 137 62 v)r v)r PUNCT cana-756 137 63 Ξ΄2(v Ξ΄2(v PROPN cana-756 137 64 , , PUNCT cana-756 137 65 w w NOUN cana-756 137 66 ) ) PUNCT cana-756 137 67 + + CCONJ cana-756 137 68 ud1(r ud1(r PROPN cana-756 137 69 , , PUNCT cana-756 137 70 v)Ξ΄2(v v)Ξ΄2(v PROPN cana-756 137 71 , , PUNCT cana-756 137 72 w w NOUN cana-756 137 73 ) ) PUNCT cana-756 137 74 = = SYM cana-756 137 75 0 0 NUM cana-756 137 76 , , PUNCT cana-756 137 77 for for ADP cana-756 137 78 all all DET cana-756 137 79 u u NOUN cana-756 137 80 , , PUNCT cana-756 137 81 v v NOUN cana-756 137 82 , , PUNCT cana-756 137 83 w w PROPN cana-756 137 84 , , PUNCT cana-756 137 85 r r PROPN cana-756 137 86 ∈ ∈ PROPN cana-756 137 87 r. r. NOUN cana-756 137 88 ( ( PUNCT cana-756 137 89 3.19 3.19 NUM cana-756 137 90 ) ) PUNCT cana-756 137 91 replacing replace VERB cana-756 137 92 w w NOUN cana-756 137 93 by by ADP cana-756 137 94 rw rw NOUN cana-756 137 95 , , PUNCT cana-756 137 96 r r NOUN cana-756 137 97 ∈ ∈ PROPN cana-756 137 98 r r NOUN cana-756 137 99 in in ADP cana-756 137 100 equation equation NOUN cana-756 137 101 ( ( PUNCT cana-756 137 102 3.19 3.19 NUM cana-756 137 103 ) ) PUNCT cana-756 137 104 and and CCONJ cana-756 137 105 using use VERB cana-756 137 106 the the DET cana-756 137 107 fact fact NOUN cana-756 137 108 that that SCONJ cana-756 137 109 Οƒ Οƒ PROPN cana-756 137 110 and and CCONJ cana-756 137 111 Ο„ Ο„ PROPN cana-756 137 112 are be AUX cana-756 137 113 automorphisms automorphism NOUN cana-756 137 114 , , PUNCT cana-756 137 115 communications communication NOUN cana-756 137 116 on on ADP cana-756 137 117 applied apply VERB cana-756 137 118 nonlinear nonlinear ADJ cana-756 137 119 analysis analysis NOUN cana-756 137 120 issn issn NOUN cana-756 137 121 : : PUNCT cana-756 137 122 1074 1074 NUM cana-756 137 123 - - PUNCT cana-756 137 124 133x 133x NUM cana-756 137 125 vol vol NOUN cana-756 137 126 31 31 NUM cana-756 137 127 no no NOUN cana-756 137 128 . . PUNCT cana-756 138 1 3s 3s NUM cana-756 138 2 ( ( PUNCT cana-756 138 3 2024 2024 NUM cana-756 138 4 ) ) PUNCT cana-756 138 5 163 163 NUM cana-756 138 6 https://internationalpubls.com https://internationalpubls.com X cana-756 138 7 we we PRON cana-756 138 8 get get VERB cana-756 138 9 Ξ΄1(u Ξ΄1(u NOUN cana-756 138 10 , , PUNCT cana-756 138 11 v)r v)r X cana-756 138 12 Ξ΄2(v Ξ΄2(v NUM cana-756 138 13 , , PUNCT cana-756 138 14 w)r w)r PUNCT cana-756 138 15 + + X cana-756 138 16 Ξ΄1(u Ξ΄1(u NUM cana-756 138 17 , , PUNCT cana-756 138 18 v)rwd2(v v)rwd2(v PROPN cana-756 138 19 , , PUNCT cana-756 138 20 r r NOUN cana-756 138 21 ) ) PUNCT cana-756 138 22 + + CCONJ cana-756 138 23 ud1(r ud1(r PROPN cana-756 138 24 , , PUNCT cana-756 138 25 v)Ξ΄2(v v)Ξ΄2(v PROPN cana-756 138 26 , , PUNCT cana-756 138 27 w)r w)r PUNCT cana-756 138 28 + + CCONJ cana-756 138 29 ud1(r ud1(r ADJ cana-756 138 30 , , PUNCT cana-756 138 31 v v NOUN cana-756 138 32 ) ) PUNCT cana-756 138 33 wd2(v wd2(v PROPN cana-756 138 34 , , PUNCT cana-756 138 35 r r NOUN cana-756 138 36 ) ) PUNCT cana-756 138 37 = = SYM cana-756 138 38 0 0 X cana-756 138 39 . . PUNCT cana-756 139 1 using use VERB cana-756 139 2 the the DET cana-756 139 3 condition condition NOUN cana-756 139 4 of of ADP cana-756 139 5 orthogonality orthogonality NOUN cana-756 139 6 of of ADP cana-756 139 7 Ξ΄1 Ξ΄1 NOUN cana-756 139 8 and and CCONJ cana-756 139 9 Ξ΄2 Ξ΄2 VERB cana-756 139 10 , , PUNCT cana-756 139 11 equation equation NOUN cana-756 139 12 ( ( PUNCT cana-756 139 13 3.9 3.9 NUM cana-756 139 14 ) ) PUNCT cana-756 139 15 of of ADP cana-756 139 16 theorem theorem ADJ cana-756 139 17 2 2 NUM cana-756 139 18 and and CCONJ cana-756 139 19 also also ADV cana-756 139 20 by by ADP cana-756 139 21 theorem theorem NOUN cana-756 139 22 2 2 NUM cana-756 139 23 , , PUNCT cana-756 139 24 the the DET cana-756 139 25 first first ADJ cana-756 139 26 three three NUM cana-756 139 27 terms term NOUN cana-756 139 28 of of ADP cana-756 139 29 the the DET cana-756 139 30 above above ADJ cana-756 139 31 equation equation NOUN cana-756 139 32 are be AUX cana-756 139 33 zero zero NUM cana-756 139 34 . . PUNCT cana-756 140 1 the the DET cana-756 140 2 above above ADJ cana-756 140 3 equation equation NOUN cana-756 140 4 reduces reduce VERB cana-756 140 5 to to PART cana-756 140 6 ud1(r ud1(r VERB cana-756 140 7 , , PUNCT cana-756 140 8 v v NOUN cana-756 140 9 ) ) PUNCT cana-756 140 10 wd2(v wd2(v PROPN cana-756 140 11 , , PUNCT cana-756 140 12 r r NOUN cana-756 140 13 ) ) PUNCT cana-756 140 14 = = SYM cana-756 140 15 0 0 NUM cana-756 140 16 , , PUNCT cana-756 140 17 for for ADP cana-756 140 18 all all DET cana-756 140 19 u u NOUN cana-756 140 20 , , PUNCT cana-756 140 21 v v NOUN cana-756 140 22 , , PUNCT cana-756 140 23 r r PROPN cana-756 140 24 ∈ ∈ PROPN cana-756 140 25 r. r. NOUN cana-756 140 26 left leave VERB cana-756 140 27 multiplying multiply VERB cana-756 140 28 the the DET cana-756 140 29 above above ADJ cana-756 140 30 equation equation NOUN cana-756 140 31 by by ADP cana-756 140 32 d1(r d1(r PROPN cana-756 140 33 , , PUNCT cana-756 140 34 v v NOUN cana-756 140 35 ) ) PUNCT cana-756 140 36 wd2(v wd2(v PROPN cana-756 140 37 , , PUNCT cana-756 140 38 r r NOUN cana-756 140 39 ) ) PUNCT cana-756 140 40 and and CCONJ cana-756 140 41 using use VERB cana-756 140 42 the the DET cana-756 140 43 semipriness semipriness NOUN cana-756 140 44 of of ADP cana-756 140 45 r r NOUN cana-756 140 46 , , PUNCT cana-756 140 47 we we PRON cana-756 140 48 obtain obtain VERB cana-756 140 49 d1(r d1(r PROPN cana-756 140 50 , , PUNCT cana-756 140 51 v)wd2(v v)wd2(v PROPN cana-756 140 52 , , PUNCT cana-756 140 53 r r NOUN cana-756 140 54 ) ) PUNCT cana-756 140 55 = = SYM cana-756 140 56 0 0 PUNCT cana-756 140 57 . . PUNCT cana-756 141 1 in in ADP cana-756 141 2 particular particular ADJ cana-756 141 3 , , PUNCT cana-756 141 4 d1(u d1(u PROPN cana-756 141 5 , , PUNCT cana-756 141 6 v v NOUN cana-756 141 7 ) ) PUNCT cana-756 141 8 wd2(v wd2(v PROPN cana-756 141 9 , , PUNCT cana-756 141 10 u u NOUN cana-756 141 11 ) ) PUNCT cana-756 141 12 = = SYM cana-756 141 13 0 0 NUM cana-756 141 14 d1(u d1(u PROPN cana-756 141 15 , , PUNCT cana-756 141 16 v v NOUN cana-756 141 17 ) ) PUNCT cana-756 141 18 wd2(u wd2(u PROPN cana-756 141 19 , , PUNCT cana-756 141 20 v v NOUN cana-756 141 21 ) ) PUNCT cana-756 141 22 = = SYM cana-756 141 23 0 0 NUM cana-756 141 24 d1(u d1(u PROPN cana-756 141 25 , , PUNCT cana-756 141 26 v v NOUN cana-756 141 27 ) ) PUNCT cana-756 141 28 wd2(v wd2(v PROPN cana-756 141 29 , , PUNCT cana-756 141 30 w w PROPN cana-756 141 31 ) ) PUNCT cana-756 141 32 = = SYM cana-756 142 1 0 0 X cana-756 142 2 . . PUNCT cana-756 142 3 ( ( PUNCT cana-756 142 4 by by ADP cana-756 142 5 lemma lemma PROPN cana-756 142 6 2 2 NUM cana-756 142 7 ) ) PUNCT cana-756 142 8 therefore therefore ADV cana-756 142 9 , , PUNCT cana-756 142 10 d1 d1 PROPN cana-756 142 11 and and CCONJ cana-756 142 12 d2 d2 PROPN cana-756 142 13 are be AUX cana-756 142 14 orthogonal orthogonal ADJ cana-756 142 15 . . PUNCT cana-756 143 1 by by ADP cana-756 143 2 lemma lemma PROPN cana-756 143 3 4 4 NUM cana-756 143 4 , , PUNCT cana-756 143 5 we we PRON cana-756 143 6 can can AUX cana-756 143 7 have have VERB cana-756 143 8 d1d2 d1d2 NOUN cana-756 143 9 = = SYM cana-756 143 10 0 0 NUM cana-756 143 11 . . PUNCT cana-756 144 1 hence hence ADV cana-756 144 2 , , PUNCT cana-756 144 3 the the DET cana-756 144 4 two two NUM cana-756 144 5 conditions condition NOUN cana-756 144 6 are be AUX cana-756 144 7 proved prove VERB cana-756 144 8 . . PUNCT cana-756 145 1 conversely conversely ADV cana-756 145 2 suppose suppose VERB cana-756 145 3 that that SCONJ cana-756 145 4 Ξ΄1(u Ξ΄1(u PROPN cana-756 145 5 , , PUNCT cana-756 145 6 v)Ξ΄2 v)Ξ΄2 PROPN cana-756 145 7 ( ( PUNCT cana-756 145 8 v v NOUN cana-756 145 9 , , PUNCT cana-756 145 10 w w NOUN cana-756 145 11 ) ) PUNCT cana-756 145 12 = = SYM cana-756 145 13 0 0 NUM cana-756 145 14 and and CCONJ cana-756 145 15 d1Ξ΄2 d1Ξ΄2 NOUN cana-756 146 1 = = SYM cana-756 146 2 0 0 PUNCT cana-756 146 3 = = SYM cana-756 146 4 d1d2 d1d2 X cana-756 146 5 . . PUNCT cana-756 147 1 ( ( PUNCT cana-756 147 2 3.20 3.20 NUM cana-756 147 3 ) ) PUNCT cana-756 147 4 let let VERB cana-756 147 5 d1Ξ΄2 d1Ξ΄2 NOUN cana-756 147 6 = = SYM cana-756 147 7 0 0 X cana-756 147 8 . . PUNCT cana-756 148 1 d1Ξ΄2(uv d1Ξ΄2(uv NOUN cana-756 148 2 , , PUNCT cana-756 148 3 w w NOUN cana-756 148 4 ) ) PUNCT cana-756 148 5 = = SYM cana-756 148 6 d1(Ξ΄2(uv d1(Ξ΄2(uv NUM cana-756 148 7 , , PUNCT cana-756 148 8 w w NOUN cana-756 148 9 ) ) PUNCT cana-756 148 10 , , PUNCT cana-756 148 11 m m PROPN cana-756 148 12 ) ) PUNCT cana-756 148 13 , , PUNCT cana-756 148 14 for for ADP cana-756 148 15 all all DET cana-756 148 16 u u NOUN cana-756 148 17 , , PUNCT cana-756 148 18 v v NOUN cana-756 148 19 , , PUNCT cana-756 148 20 w w PROPN cana-756 148 21 , , PUNCT cana-756 148 22 m m VERB cana-756 148 23 ∈ ∈ NOUN cana-756 148 24 r r NOUN cana-756 148 25 = = SYM cana-756 148 26 d1(Ξ΄2(v d1(Ξ΄2(v PROPN cana-756 148 27 , , PUNCT cana-756 148 28 w)Οƒ(u w)Οƒ(u NOUN cana-756 148 29 ) ) PUNCT cana-756 148 30 + + CCONJ cana-756 148 31 Ο„(v)d2(u Ο„(v)d2(u PROPN cana-756 148 32 , , PUNCT cana-756 148 33 w w NOUN cana-756 148 34 ) ) PUNCT cana-756 148 35 , , PUNCT cana-756 148 36 m m PROPN cana-756 148 37 ) ) PUNCT cana-756 148 38 . . PUNCT cana-756 149 1 since since SCONJ cana-756 149 2 Οƒ Οƒ PROPN cana-756 149 3 and and CCONJ cana-756 149 4 Ο„ Ο„ PROPN cana-756 149 5 are be AUX cana-756 149 6 automorphisms automorphism NOUN cana-756 149 7 , , PUNCT cana-756 149 8 we we PRON cana-756 149 9 get get VERB cana-756 149 10 = = NOUN cana-756 149 11 d1(Ξ΄2(v d1(Ξ΄2(v X cana-756 149 12 , , PUNCT cana-756 149 13 w)u w)u X cana-756 150 1 + + CCONJ cana-756 150 2 vd2(u vd2(u X cana-756 150 3 , , PUNCT cana-756 150 4 w w NOUN cana-756 150 5 ) ) PUNCT cana-756 150 6 , , PUNCT cana-756 150 7 m m NOUN cana-756 150 8 ) ) PUNCT cana-756 150 9 = = SYM cana-756 150 10 d1(u d1(u PROPN cana-756 150 11 , , PUNCT cana-756 150 12 m)Οƒ(Ξ΄2(v m)Οƒ(Ξ΄2(v PROPN cana-756 150 13 , , PUNCT cana-756 150 14 w w NOUN cana-756 150 15 ) ) PUNCT cana-756 150 16 ) ) PUNCT cana-756 151 1 + + CCONJ cana-756 151 2 Ο„ Ο„ PROPN cana-756 151 3 ( ( PUNCT cana-756 151 4 u)d1(Ξ΄2(v u)d1(Ξ΄2(v PROPN cana-756 151 5 , , PUNCT cana-756 151 6 w w NOUN cana-756 151 7 ) ) PUNCT cana-756 151 8 , , PUNCT cana-756 151 9 m m PROPN cana-756 151 10 ) ) PUNCT cana-756 151 11 + + CCONJ cana-756 151 12 d1(d2(u d1(d2(u PROPN cana-756 151 13 , , PUNCT cana-756 151 14 w w NOUN cana-756 151 15 ) ) PUNCT cana-756 151 16 , , PUNCT cana-756 151 17 m)Οƒ(v m)Οƒ(v NOUN cana-756 151 18 ) ) PUNCT cana-756 151 19 + + CCONJ cana-756 151 20 Ο„(d2(u Ο„(d2(u PROPN cana-756 151 21 , , PUNCT cana-756 151 22 w))d1(v w))d1(v NOUN cana-756 151 23 , , PUNCT cana-756 151 24 m m NOUN cana-756 151 25 ) ) PUNCT cana-756 151 26 , , PUNCT cana-756 151 27 for for ADP cana-756 151 28 all all DET cana-756 151 29 u u NOUN cana-756 151 30 , , PUNCT cana-756 151 31 v v NOUN cana-756 151 32 , , PUNCT cana-756 151 33 w w PROPN cana-756 151 34 , , PUNCT cana-756 151 35 m m VERB cana-756 151 36 ∈ ∈ NOUN cana-756 151 37 r r NOUN cana-756 151 38 . . PUNCT cana-756 151 39 again again ADV cana-756 151 40 using use VERB cana-756 151 41 the the DET cana-756 151 42 fact fact NOUN cana-756 151 43 that that SCONJ cana-756 151 44 Οƒ Οƒ PROPN cana-756 151 45 and and CCONJ cana-756 151 46 Ο„ Ο„ PROPN cana-756 151 47 are be AUX cana-756 151 48 automorphisms automorphism NOUN cana-756 151 49 , , PUNCT cana-756 151 50 Ξ΄2Οƒ Ξ΄2Οƒ PROPN cana-756 151 51 = = PUNCT cana-756 151 52 Οƒ Οƒ PROPN cana-756 151 53 Ξ΄2 Ξ΄2 PROPN cana-756 151 54 , , PUNCT cana-756 151 55 Ο„d2 Ο„d2 NOUN cana-756 152 1 = = PUNCT cana-756 152 2 d2Ο„ d2Ο„ NOUN cana-756 152 3 we we PRON cana-756 152 4 get get VERB cana-756 152 5 = = NOUN cana-756 152 6 d1(u d1(u PROPN cana-756 152 7 , , PUNCT cana-756 152 8 m)Ξ΄2(v m)Ξ΄2(v NOUN cana-756 152 9 , , PUNCT cana-756 152 10 w w PROPN cana-756 152 11 ) ) PUNCT cana-756 152 12 + + CCONJ cana-756 152 13 ud1(Ξ΄2(v ud1(Ξ΄2(v PROPN cana-756 152 14 , , PUNCT cana-756 152 15 w w NOUN cana-756 152 16 ) ) PUNCT cana-756 152 17 , , PUNCT cana-756 152 18 m m PROPN cana-756 152 19 ) ) PUNCT cana-756 153 1 + + CCONJ cana-756 153 2 d1(d2(u d1(d2(u PROPN cana-756 153 3 , , PUNCT cana-756 153 4 w w NOUN cana-756 153 5 ) ) PUNCT cana-756 153 6 , , PUNCT cana-756 153 7 m)v m)v X cana-756 154 1 + + CCONJ cana-756 154 2 d2(u d2(u NOUN cana-756 154 3 , , PUNCT cana-756 154 4 w)d1(v w)d1(v PROPN cana-756 154 5 , , PUNCT cana-756 154 6 m m NOUN cana-756 154 7 ) ) PUNCT cana-756 154 8 = = SYM cana-756 154 9 d1(u d1(u PROPN cana-756 154 10 , , PUNCT cana-756 154 11 m)Ξ΄2(v m)Ξ΄2(v NOUN cana-756 154 12 , , PUNCT cana-756 154 13 w w PROPN cana-756 154 14 ) ) PUNCT cana-756 154 15 + + CCONJ cana-756 154 16 ud1Ξ΄2(v ud1Ξ΄2(v PROPN cana-756 154 17 , , PUNCT cana-756 154 18 w w NOUN cana-756 154 19 ) ) PUNCT cana-756 154 20 + + CCONJ cana-756 154 21 d1d2(u d1d2(u NOUN cana-756 154 22 , , PUNCT cana-756 154 23 w)v w)v X cana-756 154 24 + + CCONJ cana-756 154 25 d2(u d2(u NOUN cana-756 154 26 , , PUNCT cana-756 154 27 w)d1(v w)d1(v PROPN cana-756 154 28 , , PUNCT cana-756 154 29 m m PROPN cana-756 154 30 ) ) PUNCT cana-756 154 31 , , PUNCT cana-756 154 32 βˆ€ βˆ€ X cana-756 154 33 u u NOUN cana-756 154 34 , , PUNCT cana-756 154 35 v v NOUN cana-756 154 36 , , PUNCT cana-756 154 37 w w PROPN cana-756 154 38 , , PUNCT cana-756 154 39 m m PROPN cana-756 154 40 ∈ ∈ PROPN cana-756 154 41 r. r. NOUN cana-756 154 42 ( ( PUNCT cana-756 154 43 3.21 3.21 NUM cana-756 154 44 ) ) PUNCT cana-756 154 45 by by ADP cana-756 154 46 hypothesis hypothesis NOUN cana-756 154 47 , , PUNCT cana-756 154 48 we we PRON cana-756 154 49 have have VERB cana-756 154 50 d1Ξ΄2 d1Ξ΄2 NOUN cana-756 155 1 = = SYM cana-756 155 2 0 0 NUM cana-756 155 3 = = SYM cana-756 155 4 d1d2 d1d2 NOUN cana-756 155 5 . . NOUN cana-756 155 6 by by ADP cana-756 155 7 lemma lemma PROPN cana-756 155 8 4 4 NUM cana-756 155 9 , , PUNCT cana-756 155 10 we we PRON cana-756 155 11 have have VERB cana-756 155 12 d1d2 d1d2 NOUN cana-756 155 13 = = SYM cana-756 155 14 0 0 NUM cana-756 155 15 implies imply VERB cana-756 155 16 d1 d1 PROPN cana-756 155 17 , , PUNCT cana-756 155 18 d2 d2 PROPN cana-756 155 19 are be AUX cana-756 155 20 orthogonal orthogonal ADJ cana-756 155 21 and and CCONJ cana-756 155 22 hence hence ADV cana-756 155 23 d2(u d2(u NUM cana-756 155 24 , , PUNCT cana-756 155 25 w)d1(v w)d1(v PROPN cana-756 155 26 , , PUNCT cana-756 155 27 m m PROPN cana-756 155 28 ) ) PUNCT cana-756 156 1 = = SYM cana-756 156 2 0 0 NUM cana-756 156 3 , , PUNCT cana-756 156 4 βˆ€ βˆ€ X cana-756 156 5 u u NOUN cana-756 156 6 , , PUNCT cana-756 156 7 v v NOUN cana-756 156 8 , , PUNCT cana-756 156 9 w w PROPN cana-756 156 10 , , PUNCT cana-756 156 11 m m PROPN cana-756 156 12 ∈ ∈ PROPN cana-756 156 13 r. r. PROPN cana-756 156 14 therefore therefore ADV cana-756 156 15 , , PUNCT cana-756 156 16 equation equation NOUN cana-756 156 17 ( ( PUNCT cana-756 156 18 3.21 3.21 NUM cana-756 156 19 ) ) PUNCT cana-756 156 20 becomes become VERB cana-756 156 21 d1Ξ΄2(uv d1Ξ΄2(uv NOUN cana-756 156 22 , , PUNCT cana-756 156 23 w w NOUN cana-756 156 24 ) ) PUNCT cana-756 156 25 = = SYM cana-756 156 26 d1(u d1(u PROPN cana-756 156 27 , , PUNCT cana-756 156 28 m)Ξ΄2(v m)Ξ΄2(v PROPN cana-756 156 29 , , PUNCT cana-756 156 30 w w NOUN cana-756 156 31 ) ) PUNCT cana-756 156 32 , , PUNCT cana-756 156 33 for for ADP cana-756 156 34 all all DET cana-756 156 35 u u NOUN cana-756 156 36 , , PUNCT cana-756 156 37 v v NOUN cana-756 156 38 , , PUNCT cana-756 156 39 w w PROPN cana-756 156 40 , , PUNCT cana-756 156 41 m m VERB cana-756 156 42 ∈ ∈ NOUN cana-756 156 43 r r NOUN cana-756 156 44 but but CCONJ cana-756 156 45 d1Ξ΄2 d1Ξ΄2 NOUN cana-756 156 46 = = SYM cana-756 156 47 0 0 PUNCT cana-756 157 1 and and CCONJ cana-756 157 2 hence hence ADV cana-756 157 3 we we PRON cana-756 157 4 have have VERB cana-756 157 5 d1(u d1(u PROPN cana-756 157 6 , , PUNCT cana-756 157 7 m)Ξ΄2(v m)Ξ΄2(v NOUN cana-756 157 8 , , PUNCT cana-756 157 9 w)=0 w)=0 PROPN cana-756 157 10 . . PUNCT cana-756 158 1 ( ( PUNCT cana-756 158 2 3.22 3.22 NUM cana-756 158 3 ) ) PUNCT cana-756 158 4 communications communication NOUN cana-756 158 5 on on ADP cana-756 158 6 applied apply VERB cana-756 158 7 nonlinear nonlinear ADJ cana-756 158 8 analysis analysis NOUN cana-756 158 9 issn issn NOUN cana-756 158 10 : : PUNCT cana-756 158 11 1074 1074 NUM cana-756 158 12 - - PUNCT cana-756 158 13 133x 133x NUM cana-756 158 14 vol vol NOUN cana-756 158 15 31 31 NUM cana-756 158 16 no no NOUN cana-756 158 17 . . PUNCT cana-756 159 1 3s 3s NUM cana-756 159 2 ( ( PUNCT cana-756 159 3 2024 2024 NUM cana-756 159 4 ) ) PUNCT cana-756 159 5 164 164 NUM cana-756 159 6 https://internationalpubls.com https://internationalpubls.com X cana-756 159 7 replacing replace VERB cana-756 159 8 u u NOUN cana-756 159 9 by by ADP cana-756 159 10 ru ru NOUN cana-756 159 11 , , PUNCT cana-756 159 12 r r NOUN cana-756 159 13 ∈ ∈ NOUN cana-756 159 14 r r NOUN cana-756 159 15 in in ADP cana-756 159 16 the the DET cana-756 159 17 equation equation NOUN cana-756 159 18 ( ( PUNCT cana-756 159 19 3.22 3.22 NUM cana-756 159 20 ) ) PUNCT cana-756 159 21 and and CCONJ cana-756 159 22 using use VERB cana-756 159 23 the the DET cana-756 159 24 fact fact NOUN cana-756 159 25 that that SCONJ cana-756 159 26 Οƒ Οƒ PROPN cana-756 159 27 and and CCONJ cana-756 159 28 Ο„ Ο„ PROPN cana-756 159 29 are be AUX cana-756 159 30 automorphisms automorphism NOUN cana-756 159 31 of of ADP cana-756 159 32 r r NOUN cana-756 159 33 , , PUNCT cana-756 159 34 we we PRON cana-756 159 35 obtain obtain VERB cana-756 159 36 , , PUNCT cana-756 159 37 d1(u d1(u PROPN cana-756 159 38 , , PUNCT cana-756 159 39 m m NOUN cana-756 159 40 ) ) PUNCT cana-756 159 41 rΞ΄2(v rΞ΄2(v NOUN cana-756 159 42 , , PUNCT cana-756 159 43 w w NOUN cana-756 159 44 ) ) PUNCT cana-756 160 1 + + CCONJ cana-756 160 2 u1d1(r u1d1(r ADJ cana-756 160 3 , , PUNCT cana-756 160 4 m)Ξ΄2(v m)Ξ΄2(v NOUN cana-756 160 5 , , PUNCT cana-756 160 6 w w NOUN cana-756 160 7 ) ) PUNCT cana-756 160 8 = = SYM cana-756 160 9 0 0 NUM cana-756 160 10 d1(u d1(u PROPN cana-756 160 11 , , PUNCT cana-756 160 12 m m NOUN cana-756 160 13 ) ) PUNCT cana-756 160 14 r r PROPN cana-756 160 15 Ξ΄2(v Ξ΄2(v PROPN cana-756 160 16 , , PUNCT cana-756 160 17 w w NOUN cana-756 160 18 ) ) PUNCT cana-756 160 19 = = SYM cana-756 160 20 0 0 X cana-756 160 21 . . PUNCT cana-756 161 1 ( ( PUNCT cana-756 161 2 by by ADP cana-756 161 3 the the DET cana-756 161 4 equation equation NOUN cana-756 161 5 ( ( PUNCT cana-756 161 6 3.22 3.22 NUM cana-756 161 7 ) ) PUNCT cana-756 161 8 , , PUNCT cana-756 161 9 βˆ€ βˆ€ X cana-756 161 10 u u NOUN cana-756 161 11 , , PUNCT cana-756 161 12 m m PROPN cana-756 161 13 , , PUNCT cana-756 161 14 r r NOUN cana-756 161 15 ∈ ∈ NOUN cana-756 161 16 r r NOUN cana-756 161 17 by by ADP cana-756 161 18 lemma lemma PROPN cana-756 161 19 ( ( PUNCT cana-756 161 20 1 1 NUM cana-756 161 21 ) ) PUNCT cana-756 161 22 , , PUNCT cana-756 161 23 we we PRON cana-756 161 24 have have VERB cana-756 161 25 d1(u d1(u PROPN cana-756 161 26 , , PUNCT cana-756 161 27 m m NOUN cana-756 161 28 ) ) PUNCT cana-756 161 29 Ξ΄2(v1 Ξ΄2(v1 NOUN cana-756 161 30 , , PUNCT cana-756 161 31 w w NOUN cana-756 161 32 ) ) PUNCT cana-756 161 33 = = SYM cana-756 161 34 0 0 PUNCT cana-756 161 35 = = SYM cana-756 161 36 Ξ΄2(v Ξ΄2(v PROPN cana-756 161 37 , , PUNCT cana-756 161 38 w)d1(u w)d1(u PROPN cana-756 161 39 , , PUNCT cana-756 161 40 m m NOUN cana-756 161 41 ) ) PUNCT cana-756 161 42 . . PUNCT cana-756 162 1 in in ADP cana-756 162 2 particular particular ADJ cana-756 162 3 , , PUNCT cana-756 162 4 d1(u d1(u PROPN cana-756 162 5 , , PUNCT cana-756 162 6 v v NOUN cana-756 162 7 ) ) PUNCT cana-756 162 8 Ξ΄2(v Ξ΄2(v PROPN cana-756 162 9 , , PUNCT cana-756 162 10 w w NOUN cana-756 162 11 ) ) PUNCT cana-756 162 12 = = SYM cana-756 162 13 0 0 PUNCT cana-756 162 14 = = SYM cana-756 162 15 Ξ΄2(v Ξ΄2(v PROPN cana-756 162 16 , , PUNCT cana-756 162 17 w w NOUN cana-756 162 18 ) ) PUNCT cana-756 162 19 d1(u d1(u PROPN cana-756 162 20 , , PUNCT cana-756 162 21 v v NOUN cana-756 162 22 ) ) PUNCT cana-756 162 23 . . PUNCT cana-756 163 1 ( ( PUNCT cana-756 163 2 3.23 3.23 NUM cana-756 163 3 ) ) PUNCT cana-756 163 4 from from ADP cana-756 163 5 ( ( PUNCT cana-756 163 6 3.20 3.20 NUM cana-756 163 7 ) ) PUNCT cana-756 163 8 and and CCONJ cana-756 163 9 ( ( PUNCT cana-756 163 10 3.23 3.23 NUM cana-756 163 11 ) ) PUNCT cana-756 163 12 , , PUNCT cana-756 163 13 we we PRON cana-756 163 14 have have VERB cana-756 163 15 Ξ΄1(u Ξ΄1(u NUM cana-756 163 16 , , PUNCT cana-756 163 17 v)Ξ΄2(v v)Ξ΄2(v PROPN cana-756 163 18 , , PUNCT cana-756 163 19 w w NOUN cana-756 163 20 ) ) PUNCT cana-756 163 21 = = SYM cana-756 163 22 0 0 PUNCT cana-756 163 23 = = SYM cana-756 163 24 d1(u d1(u PROPN cana-756 163 25 , , PUNCT cana-756 163 26 v v NOUN cana-756 163 27 ) ) PUNCT cana-756 163 28 Ξ΄2(v Ξ΄2(v PROPN cana-756 163 29 , , PUNCT cana-756 163 30 w w NOUN cana-756 163 31 ) ) PUNCT cana-756 163 32 . . PUNCT cana-756 164 1 by by ADP cana-756 164 2 theorem theorem NOUN cana-756 164 3 2 2 NUM cana-756 164 4 , , PUNCT cana-756 164 5 we we PRON cana-756 164 6 have have VERB cana-756 164 7 Ξ΄1 Ξ΄1 NOUN cana-756 164 8 and and CCONJ cana-756 164 9 Ξ΄2 Ξ΄2 PROPN cana-756 164 10 are be AUX cana-756 164 11 orthogonal orthogonal ADJ cana-756 164 12 . . PUNCT cana-756 165 1 theorem theorem VERB cana-756 165 2 4 4 NUM cana-756 165 3 : : PUNCT cana-756 165 4 if if SCONJ cana-756 165 5 ( ( PUNCT cana-756 165 6 Ξ΄ Ξ΄ NOUN cana-756 165 7 1 1 NUM cana-756 165 8 , , PUNCT cana-756 165 9 d1 d1 PROPN cana-756 165 10 ) ) PUNCT cana-756 165 11 and and CCONJ cana-756 165 12 ( ( PUNCT cana-756 165 13 Ξ΄ Ξ΄ PROPN cana-756 165 14 2 2 NUM cana-756 165 15 , , PUNCT cana-756 165 16 d2 d2 PROPN cana-756 165 17 ) ) PUNCT cana-756 165 18 are be AUX cana-756 165 19 two two NUM cana-756 165 20 orthogonal orthogonal ADJ cana-756 165 21 generalized generalize VERB cana-756 165 22 symmetric symmetric ADJ cana-756 165 23 reverse reverse NOUN cana-756 165 24 bi-(Οƒ bi-(Οƒ PROPN cana-756 165 25 , , PUNCT cana-756 165 26 Ο„)derivations Ο„)derivations PROPN cana-756 165 27 of of ADP cana-756 165 28 r r NOUN cana-756 165 29 , , PUNCT cana-756 165 30 then then ADV cana-756 165 31 ( ( PUNCT cana-756 165 32 i i NOUN cana-756 165 33 ) ) PUNCT cana-756 165 34 d1 d1 PROPN cana-756 165 35 and and CCONJ cana-756 165 36 d2 d2 PROPN cana-756 165 37 are be AUX cana-756 165 38 orthogonal orthogonal ADJ cana-756 165 39 reverse reverse NOUN cana-756 165 40 bi-(Οƒ bi-(Οƒ NOUN cana-756 165 41 , , PUNCT cana-756 165 42 Ο„)-derivations Ο„)-derivations PROPN cana-756 165 43 ( ( PUNCT cana-756 165 44 ii ii NOUN cana-756 165 45 ) ) PUNCT cana-756 166 1 Ξ΄2d1 Ξ΄2d1 NOUN cana-756 167 1 = = SYM cana-756 167 2 0 0 NUM cana-756 167 3 ( ( PUNCT cana-756 167 4 iii iii NOUN cana-756 167 5 ) ) PUNCT cana-756 167 6 Ξ΄1d2 Ξ΄1d2 NOUN cana-756 167 7 = = SYM cana-756 167 8 0 0 NUM cana-756 167 9 ( ( PUNCT cana-756 167 10 iv iv X cana-756 167 11 ) ) PUNCT cana-756 167 12 d2Ξ΄ d2Ξ΄ PROPN cana-756 167 13 1 1 NUM cana-756 167 14 = = SYM cana-756 167 15 0 0 NUM cana-756 167 16 ( ( PUNCT cana-756 167 17 v v NOUN cana-756 167 18 ) ) PUNCT cana-756 167 19 Ξ΄1Ξ΄ Ξ΄1Ξ΄ NOUN cana-756 167 20 2 2 NUM cana-756 167 21 = = SYM cana-756 167 22 0 0 NUM cana-756 167 23 ( ( PUNCT cana-756 167 24 vi vi NOUN cana-756 167 25 ) ) PUNCT cana-756 167 26 Ξ΄2Ξ΄ Ξ΄2Ξ΄ NOUN cana-756 167 27 1 1 NUM cana-756 167 28 = = SYM cana-756 167 29 0 0 NUM cana-756 167 30 . . PUNCT cana-756 168 1 proof proof NOUN cana-756 168 2 : : PUNCT cana-756 168 3 ( ( PUNCT cana-756 168 4 i i NOUN cana-756 168 5 ) ) PUNCT cana-756 168 6 to to PART cana-756 168 7 prove prove VERB cana-756 168 8 Ξ΄2d1 Ξ΄2d1 NOUN cana-756 168 9 = = SYM cana-756 168 10 0 0 NUM cana-756 168 11 : : PUNCT cana-756 168 12 suppose suppose VERB cana-756 168 13 that that SCONJ cana-756 168 14 Ξ΄1 Ξ΄1 NOUN cana-756 168 15 , , PUNCT cana-756 168 16 Ξ΄2 Ξ΄2 VERB cana-756 168 17 are be AUX cana-756 168 18 two two NUM cana-756 168 19 orthogonal orthogonal ADJ cana-756 168 20 generalized generalize VERB cana-756 168 21 symmetric symmetric ADJ cana-756 168 22 reverse reverse NOUN cana-756 168 23 bi-(Οƒ bi-(Οƒ NOUN cana-756 168 24 , , PUNCT cana-756 168 25 Ο„)-derivations Ο„)-derivations PROPN cana-756 168 26 of of ADP cana-756 168 27 r. r. PROPN cana-756 168 28 then then ADV cana-756 168 29 , , PUNCT cana-756 168 30 by by ADP cana-756 168 31 the the DET cana-756 168 32 definition definition NOUN cana-756 168 33 of of ADP cana-756 168 34 orthogonality orthogonality NOUN cana-756 168 35 , , PUNCT cana-756 168 36 we we PRON cana-756 168 37 have have VERB cana-756 168 38 Ξ΄1(u Ξ΄1(u NUM cana-756 168 39 , , PUNCT cana-756 168 40 v)rΞ΄2 v)rΞ΄2 PROPN cana-756 168 41 ( ( PUNCT cana-756 168 42 v v NOUN cana-756 168 43 , , PUNCT cana-756 168 44 w w NOUN cana-756 168 45 ) ) PUNCT cana-756 168 46 = = SYM cana-756 168 47 0 0 NUM cana-756 168 48 , , PUNCT cana-756 168 49 βˆ€ βˆ€ X cana-756 168 50 u u NOUN cana-756 168 51 , , PUNCT cana-756 168 52 v v NOUN cana-756 168 53 , , PUNCT cana-756 168 54 w w PROPN cana-756 168 55 , , PUNCT cana-756 168 56 r r PROPN cana-756 168 57 ∈ ∈ PROPN cana-756 168 58 r. r. NOUN cana-756 168 59 by by ADP cana-756 168 60 lemma lemma PROPN cana-756 168 61 1 1 NUM cana-756 168 62 , , PUNCT cana-756 168 63 we we PRON cana-756 168 64 can can AUX cana-756 168 65 write write VERB cana-756 168 66 Ξ΄1(u Ξ΄1(u NUM cana-756 168 67 , , PUNCT cana-756 168 68 v)Ξ΄2(v v)Ξ΄2(v PROPN cana-756 168 69 , , PUNCT cana-756 168 70 w w NOUN cana-756 168 71 ) ) PUNCT cana-756 168 72 = = SYM cana-756 168 73 0 0 X cana-756 168 74 . . PUNCT cana-756 168 75 replacing replace VERB cana-756 168 76 u u NOUN cana-756 168 77 by by ADP cana-756 168 78 ru ru NOUN cana-756 168 79 , , PUNCT cana-756 168 80 r r NOUN cana-756 168 81 ∈ ∈ NOUN cana-756 168 82 r r NOUN cana-756 168 83 in in ADP cana-756 168 84 the the DET cana-756 168 85 above above ADJ cana-756 168 86 equation equation NOUN cana-756 168 87 , , PUNCT cana-756 168 88 we we PRON cana-756 168 89 get get VERB cana-756 168 90 Ξ΄1(ru Ξ΄1(ru PROPN cana-756 168 91 , , PUNCT cana-756 168 92 v)Ξ΄2 v)Ξ΄2 PROPN cana-756 168 93 ( ( PUNCT cana-756 168 94 v v NOUN cana-756 168 95 , , PUNCT cana-756 168 96 w w NOUN cana-756 168 97 ) ) PUNCT cana-756 168 98 = = SYM cana-756 168 99 0 0 NUM cana-756 168 100 Ξ΄1(u Ξ΄1(u NUM cana-756 168 101 , , PUNCT cana-756 168 102 v)Οƒ(r)Ξ΄2 v)Οƒ(r)Ξ΄2 NOUN cana-756 168 103 ( ( PUNCT cana-756 168 104 v v NOUN cana-756 168 105 , , PUNCT cana-756 168 106 w w NOUN cana-756 168 107 ) ) PUNCT cana-756 168 108 + + CCONJ cana-756 168 109 Ο„(u)d1(r Ο„(u)d1(r ADJ cana-756 168 110 , , PUNCT cana-756 168 111 v)Ξ΄2 v)Ξ΄2 PROPN cana-756 168 112 ( ( PUNCT cana-756 168 113 v v NOUN cana-756 168 114 , , PUNCT cana-756 168 115 w w NOUN cana-756 168 116 ) ) PUNCT cana-756 168 117 = = SYM cana-756 169 1 0 0 X cana-756 169 2 . . PUNCT cana-756 170 1 ( ( PUNCT cana-756 170 2 3.24 3.24 NUM cana-756 170 3 ) ) PUNCT cana-756 170 4 replacing replace VERB cana-756 170 5 w w NOUN cana-756 170 6 by by ADP cana-756 170 7 rw rw NOUN cana-756 170 8 , , PUNCT cana-756 170 9 r r NOUN cana-756 170 10 ∈ ∈ PROPN cana-756 170 11 r r NOUN cana-756 170 12 in in ADP cana-756 170 13 equation equation NOUN cana-756 170 14 ( ( PUNCT cana-756 170 15 3.24 3.24 NUM cana-756 170 16 ) ) PUNCT cana-756 170 17 , , PUNCT cana-756 170 18 Ξ΄1(u Ξ΄1(u NOUN cana-756 170 19 , , PUNCT cana-756 170 20 v)Οƒ(r)Ξ΄2 v)Οƒ(r)Ξ΄2 NOUN cana-756 170 21 ( ( PUNCT cana-756 170 22 v v NOUN cana-756 170 23 , , PUNCT cana-756 170 24 rw rw NOUN cana-756 170 25 ) ) PUNCT cana-756 170 26 + + CCONJ cana-756 170 27 Ο„(u)d1(r Ο„(u)d1(r ADJ cana-756 170 28 , , PUNCT cana-756 170 29 v)Ξ΄2(v v)Ξ΄2(v PROPN cana-756 170 30 , , PUNCT cana-756 170 31 rw rw NOUN cana-756 170 32 ) ) PUNCT cana-756 170 33 = = SYM cana-756 170 34 0 0 NUM cana-756 170 35 Ξ΄1(u Ξ΄1(u NUM cana-756 170 36 , , PUNCT cana-756 170 37 v)Οƒ(r)Ξ΄2(v v)Οƒ(r)Ξ΄2(v NOUN cana-756 170 38 , , PUNCT cana-756 170 39 w)Οƒ(r w)Οƒ(r NOUN cana-756 170 40 ) ) PUNCT cana-756 170 41 + + SYM cana-756 170 42 Ξ΄1(u Ξ΄1(u NUM cana-756 170 43 , , PUNCT cana-756 170 44 v)Οƒ(r)Ο„(w)d2(v v)Οƒ(r)Ο„(w)d2(v PROPN cana-756 170 45 , , PUNCT cana-756 170 46 r r NOUN cana-756 170 47 ) ) PUNCT cana-756 170 48 + + CCONJ cana-756 170 49 Ο„(u)d1(r Ο„(u)d1(r ADJ cana-756 170 50 , , PUNCT cana-756 170 51 v)Ξ΄2 v)Ξ΄2 PROPN cana-756 170 52 ( ( PUNCT cana-756 170 53 v v NOUN cana-756 170 54 , , PUNCT cana-756 170 55 w)Οƒ(r w)Οƒ(r NOUN cana-756 170 56 ) ) PUNCT cana-756 170 57 + + SYM cana-756 170 58 Ο„(u)d1(r Ο„(u)d1(r ADJ cana-756 170 59 , , PUNCT cana-756 170 60 v)Ο„(w)d2(v v)Ο„(w)d2(v NOUN cana-756 170 61 , , PUNCT cana-756 170 62 r r NOUN cana-756 170 63 ) ) PUNCT cana-756 170 64 = = SYM cana-756 170 65 0 0 NUM cana-756 170 66 , , PUNCT cana-756 170 67 βˆ€ βˆ€ X cana-756 170 68 u u NOUN cana-756 170 69 , , PUNCT cana-756 170 70 v v NOUN cana-756 170 71 , , PUNCT cana-756 170 72 w w PROPN cana-756 170 73 , , PUNCT cana-756 170 74 r r PROPN cana-756 170 75 ∈ ∈ PROPN cana-756 170 76 r. r. NOUN cana-756 170 77 ( ( PUNCT cana-756 170 78 3.25 3.25 NUM cana-756 170 79 ) ) PUNCT cana-756 170 80 since since SCONJ cana-756 170 81 Οƒ Οƒ PROPN cana-756 170 82 , , PUNCT cana-756 170 83 Ο„ Ο„ PROPN cana-756 170 84 are be AUX cana-756 170 85 automorphisms automorphism NOUN cana-756 170 86 of of ADP cana-756 170 87 r r NOUN cana-756 170 88 , , PUNCT cana-756 170 89 using use VERB cana-756 170 90 equations equation NOUN cana-756 170 91 ( ( PUNCT cana-756 170 92 3.1 3.1 NUM cana-756 170 93 ) ) PUNCT cana-756 170 94 , , PUNCT cana-756 170 95 ( ( PUNCT cana-756 170 96 3.9 3.9 NUM cana-756 170 97 ) ) PUNCT cana-756 170 98 and and CCONJ cana-756 170 99 ( ( PUNCT cana-756 170 100 3.5 3.5 NUM cana-756 170 101 ) ) PUNCT cana-756 170 102 , , PUNCT cana-756 170 103 the the DET cana-756 170 104 first first ADJ cana-756 170 105 three three NUM cana-756 170 106 terms term NOUN cana-756 170 107 are be AUX cana-756 170 108 zero zero NUM cana-756 170 109 , , PUNCT cana-756 170 110 then then ADV cana-756 170 111 equation equation NOUN cana-756 170 112 ( ( PUNCT cana-756 170 113 3.25 3.25 NUM cana-756 170 114 ) ) PUNCT cana-756 170 115 reduces reduce VERB cana-756 170 116 to to PART cana-756 170 117 ud1(r ud1(r VERB cana-756 170 118 , , PUNCT cana-756 170 119 v)wd2(v v)wd2(v PROPN cana-756 170 120 , , PUNCT cana-756 170 121 r r NOUN cana-756 170 122 ) ) PUNCT cana-756 170 123 = = SYM cana-756 170 124 0 0 NUM cana-756 170 125 , , PUNCT cana-756 170 126 βˆ€ βˆ€ X cana-756 170 127 u u NOUN cana-756 170 128 , , PUNCT cana-756 170 129 v v NOUN cana-756 170 130 , , PUNCT cana-756 170 131 w w PROPN cana-756 170 132 , , PUNCT cana-756 170 133 r r NOUN cana-756 170 134 ∈ ∈ NOUN cana-756 170 135 r r NOUN cana-756 170 136 then then ADV cana-756 170 137 , , PUNCT cana-756 170 138 d1(r d1(r PROPN cana-756 170 139 , , PUNCT cana-756 170 140 v)wd2(v v)wd2(v PROPN cana-756 170 141 , , PUNCT cana-756 170 142 r)ud1(r r)ud1(r PROPN cana-756 170 143 , , PUNCT cana-756 170 144 v)wd2(v v)wd2(v PROPN cana-756 170 145 , , PUNCT cana-756 170 146 r r NOUN cana-756 170 147 ) ) PUNCT cana-756 170 148 = = SYM cana-756 170 149 0 0 NUM cana-756 170 150 , , PUNCT cana-756 170 151 βˆ€ βˆ€ X cana-756 170 152 u u NOUN cana-756 170 153 , , PUNCT cana-756 170 154 v v NOUN cana-756 170 155 , , PUNCT cana-756 170 156 w w PROPN cana-756 170 157 , , PUNCT cana-756 170 158 r r PROPN cana-756 170 159 ∈ ∈ PROPN cana-756 170 160 r. r. NOUN cana-756 170 161 by by ADP cana-756 170 162 the the DET cana-756 170 163 semiprimeness semiprimeness NOUN cana-756 170 164 of of ADP cana-756 170 165 r r NOUN cana-756 170 166 , , PUNCT cana-756 170 167 we we PRON cana-756 170 168 get get VERB cana-756 170 169 d1(r d1(r NOUN cana-756 170 170 , , PUNCT cana-756 170 171 v)wd2(v v)wd2(v PROPN cana-756 170 172 , , PUNCT cana-756 170 173 r r NOUN cana-756 170 174 ) ) PUNCT cana-756 171 1 = = SYM cana-756 171 2 0 0 NUM cana-756 171 3 which which PRON cana-756 171 4 is be AUX cana-756 171 5 same same ADJ cana-756 171 6 as as ADP cana-756 171 7 d1(r d1(r PROPN cana-756 171 8 , , PUNCT cana-756 171 9 v)wd2(r v)wd2(r NOUN cana-756 171 10 , , PUNCT cana-756 171 11 v v NOUN cana-756 171 12 ) ) PUNCT cana-756 171 13 = = SYM cana-756 171 14 0 0 X cana-756 171 15 . . X cana-756 171 16 using use VERB cana-756 171 17 lemma lemma PROPN cana-756 171 18 2 2 NUM cana-756 171 19 , , PUNCT cana-756 171 20 we we PRON cana-756 171 21 can can AUX cana-756 171 22 write write VERB cana-756 171 23 d1(r d1(r PROPN cana-756 171 24 , , PUNCT cana-756 171 25 v)wd2(v v)wd2(v PROPN cana-756 171 26 , , PUNCT cana-756 171 27 u u NOUN cana-756 171 28 ) ) PUNCT cana-756 171 29 = = SYM cana-756 171 30 0 0 NUM cana-756 171 31 , , PUNCT cana-756 171 32 βˆ€ βˆ€ X cana-756 171 33 u u NOUN cana-756 171 34 , , PUNCT cana-756 171 35 v v NOUN cana-756 171 36 , , PUNCT cana-756 171 37 w w PROPN cana-756 171 38 , , PUNCT cana-756 171 39 r r PROPN cana-756 171 40 ∈ ∈ PROPN cana-756 171 41 r. r. NOUN cana-756 171 42 by by ADP cana-756 171 43 lemma1 lemma1 PROPN cana-756 171 44 , , PUNCT cana-756 171 45 d1(r d1(r PROPN cana-756 171 46 , , PUNCT cana-756 171 47 v)d2(v v)d2(v NOUN cana-756 171 48 , , PUNCT cana-756 171 49 u u NOUN cana-756 171 50 ) ) PUNCT cana-756 171 51 = = SYM cana-756 171 52 0 0 PUNCT cana-756 172 1 = = SYM cana-756 172 2 d2(v d2(v PROPN cana-756 172 3 , , PUNCT cana-756 172 4 u)d1(r u)d1(r NOUN cana-756 172 5 , , PUNCT cana-756 172 6 v v NOUN cana-756 172 7 ) ) PUNCT cana-756 172 8 = = SYM cana-756 172 9 0 0 NUM cana-756 172 10 , , PUNCT cana-756 172 11 βˆ€ βˆ€ X cana-756 172 12 u u NOUN cana-756 172 13 , , PUNCT cana-756 172 14 v v NOUN cana-756 172 15 , , PUNCT cana-756 172 16 r r PROPN cana-756 172 17 ∈ ∈ PROPN cana-756 172 18 r. r. NOUN cana-756 172 19 ( ( PUNCT cana-756 172 20 3.26 3.26 NUM cana-756 172 21 ) ) PUNCT cana-756 172 22 which which PRON cana-756 172 23 shows show VERB cana-756 172 24 that that SCONJ cana-756 172 25 d1 d1 PROPN cana-756 172 26 , , PUNCT cana-756 172 27 d2 d2 PROPN cana-756 172 28 are be AUX cana-756 172 29 orthogonal orthogonal ADJ cana-756 172 30 . . PUNCT cana-756 173 1 communications communication NOUN cana-756 173 2 on on ADP cana-756 173 3 applied apply VERB cana-756 173 4 nonlinear nonlinear ADJ cana-756 173 5 analysis analysis NOUN cana-756 173 6 issn issn NOUN cana-756 173 7 : : PUNCT cana-756 173 8 1074 1074 NUM cana-756 173 9 - - PUNCT cana-756 173 10 133x 133x NUM cana-756 173 11 vol vol NOUN cana-756 173 12 31 31 NUM cana-756 173 13 no no NOUN cana-756 173 14 . . PUNCT cana-756 174 1 3s 3s NUM cana-756 174 2 ( ( PUNCT cana-756 174 3 2024 2024 NUM cana-756 174 4 ) ) PUNCT cana-756 174 5 165 165 NUM cana-756 174 6 https://internationalpubls.com https://internationalpubls.com X cana-756 174 7 ( ( PUNCT cana-756 174 8 ii)to ii)to NOUN cana-756 174 9 prove prove VERB cana-756 174 10 Ξ΄2d1 Ξ΄2d1 NOUN cana-756 174 11 = = NOUN cana-756 174 12 0 0 NUM cana-756 174 13 : : PUNCT cana-756 174 14 since since SCONJ cana-756 174 15 , , PUNCT cana-756 174 16 Ξ΄1 Ξ΄1 NOUN cana-756 174 17 , , PUNCT cana-756 174 18 Ξ΄2 Ξ΄2 VERB cana-756 174 19 are be AUX cana-756 174 20 two two NUM cana-756 174 21 orthogonal orthogonal ADJ cana-756 174 22 generalized generalize VERB cana-756 174 23 symmetric symmetric ADJ cana-756 174 24 reverse reverse NOUN cana-756 174 25 bi-(Οƒ bi-(Οƒ NOUN cana-756 174 26 , , PUNCT cana-756 174 27 Ο„)-derivation Ο„)-derivation NOUN cana-756 174 28 of of ADP cana-756 174 29 r. r. NOUN cana-756 174 30 by by ADP cana-756 174 31 equation equation NOUN cana-756 174 32 ( ( PUNCT cana-756 174 33 3.4 3.4 NUM cana-756 174 34 ) ) PUNCT cana-756 174 35 of of ADP cana-756 174 36 theorem theorem NOUN cana-756 174 37 1 1 NUM cana-756 174 38 , , PUNCT cana-756 174 39 we we PRON cana-756 174 40 have have VERB cana-756 174 41 d1(u d1(u NOUN cana-756 174 42 , , PUNCT cana-756 174 43 v)r v)r PUNCT cana-756 174 44 Ξ΄2(v Ξ΄2(v PROPN cana-756 174 45 , , PUNCT cana-756 174 46 w w NOUN cana-756 174 47 ) ) PUNCT cana-756 174 48 = = SYM cana-756 174 49 0 0 PUNCT cana-756 174 50 = = SYM cana-756 174 51 Ξ΄2(v Ξ΄2(v PROPN cana-756 174 52 , , PUNCT cana-756 174 53 w)rd1(u w)rd1(u PROPN cana-756 174 54 , , PUNCT cana-756 174 55 v v NOUN cana-756 174 56 ) ) PUNCT cana-756 174 57 . . PUNCT cana-756 175 1 consider consider VERB cana-756 175 2 , , PUNCT cana-756 175 3 Ξ΄2(v Ξ΄2(v PROPN cana-756 175 4 , , PUNCT cana-756 175 5 w)rd1(u w)rd1(u PROPN cana-756 175 6 , , PUNCT cana-756 175 7 v v NOUN cana-756 175 8 ) ) PUNCT cana-756 175 9 = = SYM cana-756 175 10 0 0 NUM cana-756 175 11 , , PUNCT cana-756 175 12 βˆ€ βˆ€ X cana-756 175 13 u u NOUN cana-756 175 14 , , PUNCT cana-756 175 15 v v NOUN cana-756 175 16 , , PUNCT cana-756 175 17 w w PROPN cana-756 175 18 , , PUNCT cana-756 175 19 r r NOUN cana-756 175 20 ∈ ∈ NOUN cana-756 175 21 r r NOUN cana-756 175 22 then then ADV cana-756 175 23 , , PUNCT cana-756 175 24 Ξ΄2(Ξ΄ Ξ΄2(Ξ΄ PROPN cana-756 175 25 2 2 NUM cana-756 175 26 ( ( PUNCT cana-756 175 27 v v NOUN cana-756 175 28 , , PUNCT cana-756 175 29 w)rd1 w)rd1 X cana-756 175 30 ( ( PUNCT cana-756 175 31 u u NOUN cana-756 175 32 , , PUNCT cana-756 175 33 v v NOUN cana-756 175 34 ) ) PUNCT cana-756 175 35 , , PUNCT cana-756 175 36 m m PROPN cana-756 175 37 ) ) PUNCT cana-756 176 1 = = SYM cana-756 176 2 0 0 NUM cana-756 176 3 , , PUNCT cana-756 176 4 βˆ€ βˆ€ X cana-756 176 5 u u NOUN cana-756 176 6 , , PUNCT cana-756 176 7 v v NOUN cana-756 176 8 , , PUNCT cana-756 176 9 w w NOUN cana-756 176 10 , , PUNCT cana-756 176 11 r r NOUN cana-756 176 12 , , PUNCT cana-756 176 13 m m VERB cana-756 176 14 ∈ ∈ NOUN cana-756 176 15 r r NOUN cana-756 176 16 ( ( PUNCT cana-756 176 17 Ξ΄ Ξ΄ PROPN cana-756 176 18 2 2 NUM cana-756 176 19 ( ( PUNCT cana-756 176 20 rd1(u rd1(u PROPN cana-756 176 21 , , PUNCT cana-756 176 22 v v NOUN cana-756 176 23 ) ) PUNCT cana-756 176 24 , , PUNCT cana-756 176 25 m)Οƒ(Ξ΄2(v m)Οƒ(Ξ΄2(v PROPN cana-756 176 26 , , PUNCT cana-756 176 27 w w NOUN cana-756 176 28 ) ) PUNCT cana-756 176 29 ) ) PUNCT cana-756 177 1 + + CCONJ cana-756 178 1 Ο„(rd1(u Ο„(rd1(u PROPN cana-756 178 2 , , PUNCT cana-756 178 3 v)d2(Ξ΄2(v v)d2(Ξ΄2(v NOUN cana-756 178 4 , , PUNCT cana-756 178 5 w w NOUN cana-756 178 6 ) ) PUNCT cana-756 178 7 , , PUNCT cana-756 178 8 m m NOUN cana-756 178 9 ) ) PUNCT cana-756 178 10 = = SYM cana-756 178 11 0 0 NUM cana-756 178 12 Ξ΄2(d1(u Ξ΄2(d1(u NOUN cana-756 178 13 , , PUNCT cana-756 178 14 v v NOUN cana-756 178 15 ) ) PUNCT cana-756 178 16 , , PUNCT cana-756 178 17 m)Οƒ(r)Οƒ(Ξ΄2(v m)Οƒ(r)Οƒ(Ξ΄2(v NOUN cana-756 178 18 , , PUNCT cana-756 178 19 w w NOUN cana-756 178 20 ) ) PUNCT cana-756 178 21 ) ) PUNCT cana-756 179 1 + + CCONJ cana-756 179 2 Ο„(d1(u Ο„(d1(u ADJ cana-756 179 3 , , PUNCT cana-756 179 4 v))d2(r v))d2(r NOUN cana-756 179 5 , , PUNCT cana-756 179 6 m)Οƒ(Ξ΄2(v m)Οƒ(Ξ΄2(v PROPN cana-756 179 7 , , PUNCT cana-756 179 8 w w NOUN cana-756 179 9 ) ) PUNCT cana-756 179 10 ) ) PUNCT cana-756 180 1 + + CCONJ cana-756 180 2 Ο„(rd1 Ο„(rd1 NOUN cana-756 180 3 ( ( PUNCT cana-756 180 4 u u NOUN cana-756 180 5 , , PUNCT cana-756 180 6 v)d2 v)d2 PROPN cana-756 180 7 ( ( PUNCT cana-756 180 8 Ξ΄2(v Ξ΄2(v PROPN cana-756 180 9 , , PUNCT cana-756 180 10 w w NOUN cana-756 180 11 ) ) PUNCT cana-756 180 12 , , PUNCT cana-756 180 13 m m NOUN cana-756 180 14 ) ) PUNCT cana-756 180 15 = = SYM cana-756 181 1 0 0 X cana-756 181 2 . . X cana-756 181 3 using use VERB cana-756 181 4 σδ2 σδ2 NOUN cana-756 181 5 = = SYM cana-756 181 6 Ξ΄2Οƒ Ξ΄2Οƒ PROPN cana-756 181 7 ; ; PUNCT cana-756 181 8 Ο„d1 Ο„d1 X cana-756 182 1 = = SYM cana-756 182 2 d1Ο„ d1Ο„ PROPN cana-756 182 3 ; ; PUNCT cana-756 182 4 Οƒ Οƒ PROPN cana-756 182 5 , , PUNCT cana-756 182 6 Ο„ Ο„ PROPN cana-756 182 7 are be AUX cana-756 182 8 automorphisms automorphism NOUN cana-756 182 9 of of ADP cana-756 182 10 r r NOUN cana-756 182 11 and and CCONJ cana-756 182 12 using use VERB cana-756 182 13 the the DET cana-756 182 14 fact fact NOUN cana-756 182 15 that that SCONJ cana-756 182 16 d1,d2 d1,d2 PROPN cana-756 182 17 are be AUX cana-756 182 18 orthogonal orthogonal ADJ cana-756 182 19 , , PUNCT cana-756 182 20 we we PRON cana-756 182 21 have have VERB cana-756 182 22 Ξ΄2(d1(u Ξ΄2(d1(u NOUN cana-756 182 23 , , PUNCT cana-756 182 24 v v NOUN cana-756 182 25 ) ) PUNCT cana-756 182 26 , , PUNCT cana-756 182 27 m)r m)r ADJ cana-756 182 28 Ξ΄2(v Ξ΄2(v PROPN cana-756 182 29 , , PUNCT cana-756 182 30 w w NOUN cana-756 182 31 ) ) PUNCT cana-756 182 32 = = SYM cana-756 182 33 0 0 NUM cana-756 182 34 , , PUNCT cana-756 182 35 βˆ€ βˆ€ X cana-756 182 36 u u NOUN cana-756 182 37 , , PUNCT cana-756 182 38 v v NOUN cana-756 182 39 , , PUNCT cana-756 182 40 w w PROPN cana-756 182 41 , , PUNCT cana-756 182 42 r r NOUN cana-756 182 43 ∈ ∈ PROPN cana-756 182 44 r r NOUN cana-756 182 45 Ξ΄2d1(u Ξ΄2d1(u NOUN cana-756 182 46 , , PUNCT cana-756 182 47 v)r v)r PUNCT cana-756 182 48 Ξ΄2(v Ξ΄2(v PROPN cana-756 182 49 , , PUNCT cana-756 182 50 w w NOUN cana-756 182 51 ) ) PUNCT cana-756 182 52 = = SYM cana-756 183 1 0 0 X cana-756 183 2 . . X cana-756 183 3 replacing replace VERB cana-756 183 4 w w NOUN cana-756 183 5 by by ADP cana-756 183 6 wΞ΄2(v wΞ΄2(v PROPN cana-756 183 7 , , PUNCT cana-756 183 8 w w NOUN cana-756 183 9 ) ) PUNCT cana-756 183 10 in in ADP cana-756 183 11 the the DET cana-756 183 12 above above ADJ cana-756 183 13 equation equation NOUN cana-756 183 14 , , PUNCT cana-756 183 15 we we PRON cana-756 183 16 obtain obtain VERB cana-756 183 17 Ξ΄2d1(u Ξ΄2d1(u NOUN cana-756 183 18 , , PUNCT cana-756 183 19 v)r v)r PUNCT cana-756 183 20 Ξ΄2(d1(u Ξ΄2(d1(u NOUN cana-756 183 21 , , PUNCT cana-756 183 22 v v NOUN cana-756 183 23 ) ) PUNCT cana-756 183 24 , , PUNCT cana-756 183 25 v)w v)w ADJ cana-756 184 1 + + CCONJ cana-756 184 2 Ξ΄2d1(u Ξ΄2d1(u NOUN cana-756 184 3 , , PUNCT cana-756 184 4 v)r v)r X cana-756 184 5 d1(u d1(u PROPN cana-756 184 6 , , PUNCT cana-756 184 7 v)d2(v v)d2(v PROPN cana-756 184 8 , , PUNCT cana-756 184 9 w w PROPN cana-756 184 10 ) ) PUNCT cana-756 184 11 = = SYM cana-756 185 1 0 0 X cana-756 185 2 . . PUNCT cana-756 186 1 since since SCONJ cana-756 186 2 d1 d1 PROPN cana-756 186 3 , , PUNCT cana-756 186 4 d2 d2 PROPN cana-756 186 5 are be AUX cana-756 186 6 orthogonal orthogonal ADJ cana-756 186 7 are be AUX cana-756 186 8 orthogonal orthogonal ADJ cana-756 186 9 , , PUNCT cana-756 186 10 we we PRON cana-756 186 11 get get VERB cana-756 186 12 Ξ΄2d1(u Ξ΄2d1(u NOUN cana-756 186 13 , , PUNCT cana-756 186 14 v)r v)r X cana-756 186 15 Ξ΄2d1(u Ξ΄2d1(u NOUN cana-756 186 16 , , PUNCT cana-756 186 17 v)w v)w ADJ cana-756 186 18 = = SYM cana-756 186 19 0 0 NUM cana-756 186 20 , , PUNCT cana-756 186 21 βˆ€ βˆ€ X cana-756 186 22 u u NOUN cana-756 186 23 , , PUNCT cana-756 186 24 v v NOUN cana-756 186 25 , , PUNCT cana-756 186 26 w w PROPN cana-756 186 27 , , PUNCT cana-756 186 28 r r NOUN cana-756 186 29 ∈ ∈ PROPN cana-756 186 30 r r NOUN cana-756 186 31 Ξ΄2d1(u Ξ΄2d1(u NOUN cana-756 186 32 , , PUNCT cana-756 186 33 v)r v)r X cana-756 186 34 Ξ΄2d1(u Ξ΄2d1(u NOUN cana-756 186 35 , , PUNCT cana-756 186 36 v)wΞ΄2d1(u v)wΞ΄2d1(u SYM cana-756 186 37 , , PUNCT cana-756 186 38 v)r v)r X cana-756 186 39 Ξ΄2d1(u Ξ΄2d1(u NOUN cana-756 186 40 , , PUNCT cana-756 186 41 v v NOUN cana-756 186 42 ) ) PUNCT cana-756 186 43 = = SYM cana-756 187 1 0 0 X cana-756 187 2 . . PUNCT cana-756 188 1 using use VERB cana-756 188 2 the the DET cana-756 188 3 semiprimeness semiprimeness NOUN cana-756 188 4 of of ADP cana-756 188 5 r r NOUN cana-756 188 6 , , PUNCT cana-756 188 7 we we PRON cana-756 188 8 get get VERB cana-756 188 9 Ξ΄2d1 Ξ΄2d1 INTJ cana-756 188 10 ( ( PUNCT cana-756 188 11 u u NOUN cana-756 188 12 , , PUNCT cana-756 188 13 v)r v)r PUNCT cana-756 188 14 Ξ΄2d1 Ξ΄2d1 X cana-756 188 15 ( ( PUNCT cana-756 188 16 u u NOUN cana-756 188 17 , , PUNCT cana-756 188 18 v v NOUN cana-756 188 19 ) ) PUNCT cana-756 188 20 = = SYM cana-756 188 21 0 0 PUNCT cana-756 189 1 ( ( PUNCT cana-756 189 2 since since SCONJ cana-756 189 3 r r NOUN cana-756 189 4 is be AUX cana-756 189 5 semiprime semiprime NOUN cana-756 189 6 ) ) PUNCT cana-756 190 1 Ξ΄2d1 Ξ΄2d1 X cana-756 190 2 ( ( PUNCT cana-756 190 3 u u NOUN cana-756 190 4 , , PUNCT cana-756 190 5 v v NOUN cana-756 190 6 ) ) PUNCT cana-756 190 7 = = PUNCT cana-756 190 8 0 0 PUNCT cana-756 191 1 Ξ΄2d1 Ξ΄2d1 NOUN cana-756 191 2 = = NOUN cana-756 191 3 0 0 X cana-756 191 4 . . PUNCT cana-756 192 1 ( ( PUNCT cana-756 192 2 iii)to iii)to NOUN cana-756 192 3 prove prove VERB cana-756 192 4 Ξ΄1d2 Ξ΄1d2 PROPN cana-756 192 5 = = SYM cana-756 192 6 0 0 NUM cana-756 192 7 : : PUNCT cana-756 192 8 by by ADP cana-756 192 9 ( ( PUNCT cana-756 192 10 3.9 3.9 NUM cana-756 192 11 ) ) PUNCT cana-756 192 12 of of ADP cana-756 192 13 theorem theorem NOUN cana-756 192 14 1 1 NUM cana-756 192 15 , , PUNCT cana-756 192 16 we we PRON cana-756 192 17 have have VERB cana-756 192 18 Ξ΄1(v Ξ΄1(v PROPN cana-756 192 19 , , PUNCT cana-756 192 20 w w NOUN cana-756 192 21 ) ) PUNCT cana-756 192 22 rd2(u rd2(u PROPN cana-756 192 23 , , PUNCT cana-756 192 24 v v NOUN cana-756 192 25 ) ) PUNCT cana-756 192 26 = = SYM cana-756 192 27 0 0 NUM cana-756 192 28 , , PUNCT cana-756 192 29 βˆ€ βˆ€ X cana-756 192 30 u u NOUN cana-756 192 31 , , PUNCT cana-756 192 32 v v NOUN cana-756 192 33 , , PUNCT cana-756 192 34 w w PROPN cana-756 192 35 , , PUNCT cana-756 192 36 r r NOUN cana-756 192 37 ∈ ∈ NOUN cana-756 192 38 r r NOUN cana-756 192 39 Ξ΄1(Ξ΄1(v Ξ΄1(Ξ΄1(v NOUN cana-756 192 40 , , PUNCT cana-756 192 41 w)rd2(u w)rd2(u PROPN cana-756 192 42 , , PUNCT cana-756 192 43 v v NOUN cana-756 192 44 ) ) PUNCT cana-756 192 45 , , PUNCT cana-756 192 46 m m PROPN cana-756 192 47 ) ) PUNCT cana-756 192 48 = = SYM cana-756 192 49 0 0 NUM cana-756 192 50 , , PUNCT cana-756 192 51 βˆ€ βˆ€ X cana-756 192 52 u u NOUN cana-756 192 53 , , PUNCT cana-756 192 54 v v NOUN cana-756 192 55 , , PUNCT cana-756 192 56 w w PROPN cana-756 192 57 , , PUNCT cana-756 192 58 m m PROPN cana-756 192 59 , , PUNCT cana-756 192 60 r r NOUN cana-756 192 61 ∈ ∈ NOUN cana-756 192 62 r r NOUN cana-756 192 63 Ξ΄1(rd2(u Ξ΄1(rd2(u NOUN cana-756 192 64 , , PUNCT cana-756 192 65 v v NOUN cana-756 192 66 ) ) PUNCT cana-756 192 67 , , PUNCT cana-756 192 68 m)Οƒ(Ξ΄1(v m)Οƒ(Ξ΄1(v PROPN cana-756 192 69 , , PUNCT cana-756 192 70 w w NOUN cana-756 192 71 ) ) PUNCT cana-756 192 72 ) ) PUNCT cana-756 193 1 + + CCONJ cana-756 194 1 Ο„(rd2(u Ο„(rd2(u PROPN cana-756 194 2 , , PUNCT cana-756 194 3 v))d1(Ξ΄1(v v))d1(Ξ΄1(v NOUN cana-756 194 4 , , PUNCT cana-756 194 5 w w NOUN cana-756 194 6 ) ) PUNCT cana-756 194 7 , , PUNCT cana-756 194 8 m m NOUN cana-756 194 9 ) ) PUNCT cana-756 194 10 = = SYM cana-756 194 11 0 0 NUM cana-756 194 12 Ξ΄1(d2(u Ξ΄1(d2(u PROPN cana-756 194 13 , , PUNCT cana-756 194 14 v v NOUN cana-756 194 15 ) ) PUNCT cana-756 194 16 , , PUNCT cana-756 194 17 m)Οƒ(r)Οƒ(Ξ΄1(v m)Οƒ(r)Οƒ(Ξ΄1(v PROPN cana-756 194 18 , , PUNCT cana-756 194 19 w w NOUN cana-756 194 20 ) ) PUNCT cana-756 194 21 ) ) PUNCT cana-756 195 1 + + CCONJ cana-756 195 2 Ο„(d2(u Ο„(d2(u PROPN cana-756 195 3 , , PUNCT cana-756 195 4 v))d1(r v))d1(r NOUN cana-756 195 5 , , PUNCT cana-756 195 6 m)Οƒ(Ξ΄1(v m)Οƒ(Ξ΄1(v PROPN cana-756 195 7 , , PUNCT cana-756 195 8 w w NOUN cana-756 195 9 ) ) PUNCT cana-756 195 10 ) ) PUNCT cana-756 196 1 + + CCONJ cana-756 196 2 Ο„(rd2(u Ο„(rd2(u PROPN cana-756 196 3 , , PUNCT cana-756 196 4 v))d1(Ξ΄1(v v))d1(Ξ΄1(v NOUN cana-756 196 5 , , PUNCT cana-756 196 6 w w NOUN cana-756 196 7 ) ) PUNCT cana-756 196 8 , , PUNCT cana-756 196 9 m m NOUN cana-756 196 10 ) ) PUNCT cana-756 196 11 = = SYM cana-756 196 12 0 0 NUM cana-756 196 13 , , PUNCT cana-756 196 14 βˆ€ βˆ€ X cana-756 196 15 u u NOUN cana-756 196 16 , , PUNCT cana-756 196 17 v v NOUN cana-756 196 18 , , PUNCT cana-756 196 19 w w PROPN cana-756 196 20 , , PUNCT cana-756 196 21 r r PROPN cana-756 196 22 ∈ ∈ PROPN cana-756 196 23 r. r. NOUN cana-756 196 24 using use VERB cana-756 196 25 σδ1 σδ1 PROPN cana-756 196 26 = = SYM cana-756 196 27 Ξ΄1Οƒ Ξ΄1Οƒ NOUN cana-756 196 28 ; ; PUNCT cana-756 196 29 Ο„d2 Ο„d2 NOUN cana-756 196 30 = = PUNCT cana-756 196 31 d2Ο„ d2Ο„ PROPN cana-756 196 32 ; ; PUNCT cana-756 196 33 and and CCONJ cana-756 196 34 Οƒ Οƒ PROPN cana-756 196 35 and and CCONJ cana-756 196 36 Ο„ Ο„ PROPN cana-756 196 37 are be AUX cana-756 196 38 automorphisms automorphism NOUN cana-756 196 39 of of ADP cana-756 196 40 r r NOUN cana-756 196 41 , , PUNCT cana-756 196 42 we we PRON cana-756 196 43 get get VERB cana-756 196 44 Ξ΄1(d2(u Ξ΄1(d2(u NOUN cana-756 196 45 , , PUNCT cana-756 196 46 v v NOUN cana-756 196 47 ) ) PUNCT cana-756 196 48 , , PUNCT cana-756 196 49 m)rΞ΄1(v m)rΞ΄1(v PROPN cana-756 196 50 , , PUNCT cana-756 196 51 w w PROPN cana-756 196 52 ) ) PUNCT cana-756 196 53 + + CCONJ cana-756 196 54 d2(u d2(u NOUN cana-756 196 55 , , PUNCT cana-756 196 56 v)d1(r v)d1(r NOUN cana-756 196 57 , , PUNCT cana-756 196 58 m)Ξ΄1(v m)Ξ΄1(v ADJ cana-756 196 59 , , PUNCT cana-756 196 60 w w NOUN cana-756 196 61 ) ) PUNCT cana-756 196 62 + + X cana-756 197 1 rd2(u rd2(u PROPN cana-756 197 2 , , PUNCT cana-756 197 3 v)d1(Ξ΄1(v v)d1(Ξ΄1(v PROPN cana-756 197 4 , , PUNCT cana-756 197 5 w w PROPN cana-756 197 6 ) ) PUNCT cana-756 197 7 , , PUNCT cana-756 197 8 m m NOUN cana-756 197 9 ) ) PUNCT cana-756 197 10 = = SYM cana-756 197 11 0 0 NUM cana-756 197 12 , , PUNCT cana-756 197 13 βˆ€ βˆ€ X cana-756 197 14 u u NOUN cana-756 197 15 , , PUNCT cana-756 197 16 v v NOUN cana-756 197 17 , , PUNCT cana-756 197 18 w w PROPN cana-756 197 19 , , PUNCT cana-756 197 20 r r PROPN cana-756 197 21 ∈ ∈ PROPN cana-756 197 22 r. r. NOUN cana-756 197 23 using use VERB cana-756 197 24 ( ( PUNCT cana-756 197 25 3.26 3.26 NUM cana-756 197 26 ) ) PUNCT cana-756 197 27 , , PUNCT cana-756 197 28 the the DET cana-756 197 29 above above ADJ cana-756 197 30 equation equation NOUN cana-756 197 31 reduces reduce VERB cana-756 197 32 to to ADP cana-756 197 33 Ξ΄1(d2(u Ξ΄1(d2(u PROPN cana-756 197 34 , , PUNCT cana-756 197 35 v v NOUN cana-756 197 36 ) ) PUNCT cana-756 197 37 , , PUNCT cana-756 197 38 m)rΞ΄1(v m)rΞ΄1(v PROPN cana-756 197 39 , , PUNCT cana-756 197 40 w w PROPN cana-756 197 41 ) ) PUNCT cana-756 197 42 = = SYM cana-756 197 43 0 0 NUM cana-756 197 44 communications communication NOUN cana-756 197 45 on on ADP cana-756 197 46 applied apply VERB cana-756 197 47 nonlinear nonlinear ADJ cana-756 197 48 analysis analysis NOUN cana-756 197 49 issn issn NOUN cana-756 197 50 : : PUNCT cana-756 197 51 1074 1074 NUM cana-756 197 52 - - PUNCT cana-756 197 53 133x 133x NUM cana-756 197 54 vol vol NOUN cana-756 197 55 31 31 NUM cana-756 197 56 no no NOUN cana-756 197 57 . . PUNCT cana-756 198 1 3s 3s NUM cana-756 198 2 ( ( PUNCT cana-756 198 3 2024 2024 NUM cana-756 198 4 ) ) PUNCT cana-756 198 5 166 166 NUM cana-756 198 6 https://internationalpubls.com https://internationalpubls.com X cana-756 198 7 Ξ΄1d2(u Ξ΄1d2(u PROPN cana-756 198 8 , , PUNCT cana-756 198 9 v)rΞ΄1(v v)rΞ΄1(v PROPN cana-756 198 10 , , PUNCT cana-756 198 11 w w NOUN cana-756 198 12 ) ) PUNCT cana-756 198 13 = = SYM cana-756 198 14 0 0 NUM cana-756 198 15 , , PUNCT cana-756 198 16 βˆ€ βˆ€ X cana-756 198 17 u u NOUN cana-756 198 18 , , PUNCT cana-756 198 19 v v NOUN cana-756 198 20 , , PUNCT cana-756 198 21 w w PROPN cana-756 198 22 , , PUNCT cana-756 198 23 r r PROPN cana-756 198 24 ∈ ∈ PROPN cana-756 198 25 r. r. NOUN cana-756 198 26 replacing replacing NOUN cana-756 198 27 w w NOUN cana-756 198 28 by by ADP cana-756 198 29 wd2(u wd2(u PROPN cana-756 198 30 , , PUNCT cana-756 198 31 v v NOUN cana-756 198 32 ) ) PUNCT cana-756 198 33 in in ADP cana-756 198 34 the the DET cana-756 198 35 above above ADJ cana-756 198 36 equation equation NOUN cana-756 198 37 and and CCONJ cana-756 198 38 using use VERB cana-756 198 39 the the DET cana-756 198 40 fact fact NOUN cana-756 198 41 that that SCONJ cana-756 198 42 Οƒ Οƒ PROPN cana-756 198 43 and and CCONJ cana-756 198 44 Ο„ Ο„ PROPN cana-756 198 45 are be AUX cana-756 198 46 automorphisms automorphism NOUN cana-756 198 47 of of ADP cana-756 198 48 r r NOUN cana-756 198 49 , , PUNCT cana-756 198 50 we we PRON cana-756 198 51 obtain obtain VERB cana-756 198 52 Ξ΄1d2(u Ξ΄1d2(u PROPN cana-756 198 53 , , PUNCT cana-756 198 54 v)rΞ΄1(v v)rΞ΄1(v PROPN cana-756 198 55 , , PUNCT cana-756 198 56 d2(u d2(u NOUN cana-756 198 57 , , PUNCT cana-756 198 58 v))w v))w NOUN cana-756 198 59 + + CCONJ cana-756 199 1 Ξ΄1d2(u Ξ΄1d2(u PROPN cana-756 199 2 , , PUNCT cana-756 199 3 v)rd2(u v)rd2(u PROPN cana-756 199 4 , , PUNCT cana-756 199 5 v)d v)d X cana-756 199 6 1 1 NUM cana-756 199 7 ( ( PUNCT cana-756 199 8 v v NOUN cana-756 199 9 , , PUNCT cana-756 199 10 w w NOUN cana-756 199 11 ) ) PUNCT cana-756 199 12 = = SYM cana-756 199 13 0 0 NUM cana-756 199 14 , , PUNCT cana-756 199 15 βˆ€ βˆ€ X cana-756 199 16 u u NOUN cana-756 199 17 , , PUNCT cana-756 199 18 v v NOUN cana-756 199 19 , , PUNCT cana-756 199 20 w w PROPN cana-756 199 21 , , PUNCT cana-756 199 22 r r PROPN cana-756 199 23 ∈ ∈ PROPN cana-756 199 24 r. r. NOUN cana-756 199 25 using use VERB cana-756 199 26 ( ( PUNCT cana-756 199 27 3.26 3.26 NUM cana-756 199 28 ) ) PUNCT cana-756 199 29 , , PUNCT cana-756 199 30 the the DET cana-756 199 31 above above ADJ cana-756 199 32 equation equation NOUN cana-756 199 33 reduces reduce VERB cana-756 199 34 to to ADP cana-756 199 35 Ξ΄1d2(u Ξ΄1d2(u PROPN cana-756 199 36 , , PUNCT cana-756 199 37 v)rΞ΄1(v v)rΞ΄1(v PROPN cana-756 199 38 , , PUNCT cana-756 199 39 d2(u d2(u NOUN cana-756 199 40 , , PUNCT cana-756 199 41 v))w v))w NOUN cana-756 199 42 = = NOUN cana-756 199 43 0 0 NUM cana-756 199 44 Ξ΄1d2(u Ξ΄1d2(u PROPN cana-756 199 45 , , PUNCT cana-756 199 46 v)rΞ΄1(v v)rΞ΄1(v PROPN cana-756 199 47 , , PUNCT cana-756 199 48 d2(u d2(u NOUN cana-756 199 49 , , PUNCT cana-756 199 50 v))wΞ΄1d2(u v))wΞ΄1d2(u NOUN cana-756 199 51 , , PUNCT cana-756 199 52 v)rΞ΄1(v v)rΞ΄1(v PROPN cana-756 199 53 , , PUNCT cana-756 199 54 d2(u d2(u NOUN cana-756 199 55 , , PUNCT cana-756 199 56 v v NOUN cana-756 199 57 ) ) PUNCT cana-756 199 58 ) ) PUNCT cana-756 200 1 = = PUNCT cana-756 200 2 0 0 X cana-756 200 3 . . PUNCT cana-756 200 4 using use VERB cana-756 200 5 the the DET cana-756 200 6 semiprimeness semiprimeness NOUN cana-756 200 7 of of ADP cana-756 200 8 r r NOUN cana-756 200 9 , , PUNCT cana-756 200 10 we we PRON cana-756 200 11 get get VERB cana-756 200 12 Ξ΄1d2(u Ξ΄1d2(u PROPN cana-756 200 13 , , PUNCT cana-756 200 14 v)rΞ΄1(v v)rΞ΄1(v PROPN cana-756 200 15 , , PUNCT cana-756 200 16 d2(u d2(u NOUN cana-756 200 17 , , PUNCT cana-756 200 18 v v NOUN cana-756 200 19 ) ) PUNCT cana-756 200 20 ) ) PUNCT cana-756 201 1 = = SYM cana-756 201 2 0 0 NUM cana-756 201 3 Ξ΄1d2(u Ξ΄1d2(u NOUN cana-756 201 4 , , PUNCT cana-756 201 5 v)rΞ΄1(d2(u v)rΞ΄1(d2(u NOUN cana-756 201 6 , , PUNCT cana-756 201 7 v v NOUN cana-756 201 8 ) ) PUNCT cana-756 201 9 , , PUNCT cana-756 201 10 v v NOUN cana-756 201 11 ) ) PUNCT cana-756 201 12 = = SYM cana-756 201 13 0 0 NUM cana-756 201 14 Ξ΄1d2(u Ξ΄1d2(u NOUN cana-756 201 15 , , PUNCT cana-756 201 16 v)rΞ΄1d2(u v)rΞ΄1d2(u NOUN cana-756 201 17 , , PUNCT cana-756 201 18 v v NOUN cana-756 201 19 ) ) PUNCT cana-756 201 20 = = SYM cana-756 201 21 0 0 NUM cana-756 201 22 again again ADV cana-756 201 23 using use VERB cana-756 201 24 the the DET cana-756 201 25 semiprimeness semiprimeness NOUN cana-756 201 26 of of ADP cana-756 201 27 r r NOUN cana-756 201 28 , , PUNCT cana-756 201 29 we we PRON cana-756 201 30 get get VERB cana-756 201 31 Ξ΄1d2 Ξ΄1d2 NOUN cana-756 201 32 = = SYM cana-756 201 33 0 0 NUM cana-756 201 34 . . PUNCT cana-756 202 1 ( ( PUNCT cana-756 202 2 iv)to iv)to AUX cana-756 202 3 prove prove VERB cana-756 202 4 d2Ξ΄ d2Ξ΄ PROPN cana-756 202 5 1 1 NUM cana-756 202 6 = = SYM cana-756 202 7 0 0 NUM cana-756 202 8 : : PUNCT cana-756 202 9 since since SCONJ cana-756 202 10 , , PUNCT cana-756 202 11 Ξ΄1 Ξ΄1 NOUN cana-756 202 12 , , PUNCT cana-756 202 13 Ξ΄2 Ξ΄2 VERB cana-756 202 14 are be AUX cana-756 202 15 two two NUM cana-756 202 16 orthogonal orthogonal ADJ cana-756 202 17 generalized generalize VERB cana-756 202 18 symmetric symmetric ADJ cana-756 202 19 reverse(Οƒ reverse(Οƒ PROPN cana-756 202 20 , , PUNCT cana-756 202 21 Ο„ Ο„ PROPN cana-756 202 22 ) ) PUNCT cana-756 202 23 biderivations biderivation NOUN cana-756 202 24 of of ADP cana-756 202 25 r. r. PROPN cana-756 202 26 by by ADP cana-756 202 27 ( ( PUNCT cana-756 202 28 3.9 3.9 NUM cana-756 202 29 ) ) PUNCT cana-756 202 30 of of ADP cana-756 202 31 theorem theorem NOUN cana-756 202 32 1 1 NUM cana-756 202 33 , , PUNCT cana-756 202 34 we we PRON cana-756 202 35 can can AUX cana-756 202 36 have have VERB cana-756 202 37 Ξ΄1(v Ξ΄1(v PROPN cana-756 202 38 , , PUNCT cana-756 202 39 w w NOUN cana-756 202 40 ) ) PUNCT cana-756 202 41 rd2(u rd2(u PROPN cana-756 202 42 , , PUNCT cana-756 202 43 v v NOUN cana-756 202 44 ) ) PUNCT cana-756 202 45 = = SYM cana-756 202 46 0 0 NUM cana-756 202 47 Ξ΄2(Ξ΄1(v Ξ΄2(Ξ΄1(v PROPN cana-756 202 48 , , PUNCT cana-756 202 49 w w NOUN cana-756 202 50 ) ) PUNCT cana-756 202 51 rd2(u rd2(u PROPN cana-756 202 52 , , PUNCT cana-756 202 53 v v NOUN cana-756 202 54 ) ) PUNCT cana-756 202 55 , , PUNCT cana-756 202 56 m m PROPN cana-756 202 57 ) ) PUNCT cana-756 202 58 = = SYM cana-756 203 1 0 0 X cana-756 203 2 . . PUNCT cana-756 203 3 by by ADP cana-756 203 4 expanding expand VERB cana-756 203 5 the the DET cana-756 203 6 above above ADJ cana-756 203 7 equation equation NOUN cana-756 203 8 and and CCONJ cana-756 203 9 using use VERB cana-756 203 10 the the DET cana-756 203 11 fact fact NOUN cana-756 203 12 that that SCONJ cana-756 203 13 Οƒ Οƒ PROPN cana-756 203 14 and and CCONJ cana-756 203 15 Ο„ Ο„ PROPN cana-756 203 16 are be AUX cana-756 203 17 automorphisms automorphism NOUN cana-756 203 18 of of ADP cana-756 203 19 r r NOUN cana-756 203 20 , , PUNCT cana-756 203 21 we we PRON cana-756 203 22 obtain obtain VERB cana-756 203 23 Ξ΄2(d2(u Ξ΄2(d2(u NOUN cana-756 203 24 , , PUNCT cana-756 203 25 v v NOUN cana-756 203 26 ) ) PUNCT cana-756 203 27 , , PUNCT cana-756 203 28 m)rΞ΄1(v m)rΞ΄1(v PROPN cana-756 203 29 , , PUNCT cana-756 203 30 w w PROPN cana-756 203 31 ) ) PUNCT cana-756 203 32 + + CCONJ cana-756 203 33 d2(u d2(u NOUN cana-756 203 34 , , PUNCT cana-756 203 35 v)d2(r v)d2(r NOUN cana-756 203 36 , , PUNCT cana-756 203 37 m)Ξ΄1(v m)Ξ΄1(v ADJ cana-756 203 38 , , PUNCT cana-756 203 39 w w NOUN cana-756 203 40 ) ) PUNCT cana-756 203 41 + + X cana-756 204 1 rd2(u rd2(u NOUN cana-756 204 2 , , PUNCT cana-756 204 3 v)d2Ξ΄1(v v)d2Ξ΄1(v NOUN cana-756 204 4 , , PUNCT cana-756 204 5 w w PROPN cana-756 204 6 ) ) PUNCT cana-756 204 7 = = SYM cana-756 204 8 0 0 X cana-756 204 9 . . X cana-756 205 1 using use VERB cana-756 205 2 ( ( PUNCT cana-756 205 3 3.1 3.1 NUM cana-756 205 4 ) ) PUNCT cana-756 205 5 and and CCONJ cana-756 205 6 ( ( PUNCT cana-756 205 7 3.10 3.10 NUM cana-756 205 8 ) ) PUNCT cana-756 205 9 , , PUNCT cana-756 205 10 the the DET cana-756 205 11 first first ADJ cana-756 205 12 two two NUM cana-756 205 13 terms term NOUN cana-756 205 14 are be AUX cana-756 205 15 zero zero NUM cana-756 205 16 and and CCONJ cana-756 205 17 hence hence ADV cana-756 205 18 we we PRON cana-756 205 19 get get VERB cana-756 205 20 rd2(u rd2(u PROPN cana-756 205 21 , , PUNCT cana-756 205 22 v)d2Ξ΄1(v v)d2Ξ΄1(v NOUN cana-756 205 23 , , PUNCT cana-756 205 24 w w PROPN cana-756 205 25 ) ) PUNCT cana-756 205 26 = = SYM cana-756 206 1 0 0 NUM cana-756 206 2 d2(u d2(u NOUN cana-756 206 3 , , PUNCT cana-756 206 4 v)d2Ξ΄1(v v)d2Ξ΄1(v INTJ cana-756 206 5 , , PUNCT cana-756 206 6 w)rd2(u w)rd2(u PROPN cana-756 206 7 , , PUNCT cana-756 206 8 v)d2Ξ΄1(v v)d2Ξ΄1(v NOUN cana-756 206 9 , , PUNCT cana-756 206 10 w w PROPN cana-756 206 11 ) ) PUNCT cana-756 206 12 = = SYM cana-756 206 13 0 0 NUM cana-756 206 14 , , PUNCT cana-756 206 15 βˆ€ βˆ€ X cana-756 206 16 u u NOUN cana-756 206 17 , , PUNCT cana-756 206 18 v v NOUN cana-756 206 19 , , PUNCT cana-756 206 20 w w PROPN cana-756 206 21 , , PUNCT cana-756 206 22 r r PROPN cana-756 206 23 ∈ ∈ PROPN cana-756 206 24 r. r. NOUN cana-756 206 25 by by ADP cana-756 206 26 the the DET cana-756 206 27 semiprimeness semiprimeness NOUN cana-756 206 28 of of ADP cana-756 206 29 r r NOUN cana-756 206 30 , , PUNCT cana-756 206 31 we we PRON cana-756 206 32 get get VERB cana-756 206 33 d2(u d2(u NOUN cana-756 206 34 , , PUNCT cana-756 206 35 v)d2Ξ΄1(v v)d2Ξ΄1(v NOUN cana-756 206 36 , , PUNCT cana-756 206 37 w w PROPN cana-756 206 38 ) ) PUNCT cana-756 206 39 = = SYM cana-756 207 1 0,βˆ€ 0,βˆ€ NUM cana-756 207 2 u u NOUN cana-756 207 3 , , PUNCT cana-756 207 4 v v NOUN cana-756 207 5 , , PUNCT cana-756 207 6 w w PROPN cana-756 207 7 ∈ ∈ PROPN cana-756 207 8 r. r. NOUN cana-756 207 9 ( ( PUNCT cana-756 207 10 3.27 3.27 NUM cana-756 207 11 ) ) PUNCT cana-756 207 12 replacing replace VERB cana-756 207 13 v v NOUN cana-756 207 14 = = SYM cana-756 207 15 vΞ΄1(v vΞ΄1(v PROPN cana-756 207 16 , , PUNCT cana-756 207 17 w w PROPN cana-756 207 18 ) ) PUNCT cana-756 207 19 in in ADP cana-756 207 20 the the DET cana-756 207 21 above above ADJ cana-756 207 22 equation equation NOUN cana-756 207 23 , , PUNCT cana-756 207 24 we we PRON cana-756 207 25 get get VERB cana-756 207 26 d2(u d2(u NOUN cana-756 207 27 , , PUNCT cana-756 207 28 vΞ΄1(v vΞ΄1(v NOUN cana-756 207 29 , , PUNCT cana-756 207 30 w))d2Ξ΄1(v w))d2Ξ΄1(v X cana-756 207 31 , , PUNCT cana-756 207 32 w w NOUN cana-756 207 33 ) ) PUNCT cana-756 207 34 = = SYM cana-756 207 35 0 0 NUM cana-756 207 36 d2(Ξ΄1(v d2(Ξ΄1(v NOUN cana-756 207 37 , , PUNCT cana-756 207 38 w w NOUN cana-756 207 39 ) ) PUNCT cana-756 207 40 , , PUNCT cana-756 207 41 u)vd2Ξ΄1(v u)vd2Ξ΄1(v SYM cana-756 207 42 , , PUNCT cana-756 207 43 w w NOUN cana-756 207 44 ) ) PUNCT cana-756 207 45 + + NUM cana-756 207 46 Ξ΄1(v Ξ΄1(v PROPN cana-756 207 47 , , PUNCT cana-756 207 48 w)d2(u w)d2(u PROPN cana-756 207 49 , , PUNCT cana-756 207 50 v)d2Ξ΄1(v v)d2Ξ΄1(v NOUN cana-756 207 51 , , PUNCT cana-756 207 52 w w PROPN cana-756 207 53 ) ) PUNCT cana-756 207 54 = = SYM cana-756 208 1 0 0 X cana-756 208 2 . . X cana-756 209 1 using use VERB cana-756 209 2 ( ( PUNCT cana-756 209 3 3.27 3.27 NUM cana-756 209 4 ) ) PUNCT cana-756 209 5 , , PUNCT cana-756 209 6 we we PRON cana-756 209 7 get get VERB cana-756 209 8 d2(Ξ΄1(v d2(Ξ΄1(v NOUN cana-756 209 9 , , PUNCT cana-756 209 10 w w NOUN cana-756 209 11 ) ) PUNCT cana-756 209 12 , , PUNCT cana-756 209 13 u)vd2Ξ΄1(v u)vd2Ξ΄1(v SYM cana-756 209 14 , , PUNCT cana-756 209 15 w w NOUN cana-756 209 16 ) ) PUNCT cana-756 209 17 = = SYM cana-756 210 1 0 0 NUM cana-756 210 2 d2Ξ΄1(v d2Ξ΄1(v PROPN cana-756 210 3 , , PUNCT cana-756 210 4 w)vd2Ξ΄1(v w)vd2Ξ΄1(v NOUN cana-756 210 5 , , PUNCT cana-756 210 6 w w PROPN cana-756 210 7 ) ) PUNCT cana-756 210 8 = = SYM cana-756 210 9 0 0 NUM cana-756 210 10 , , PUNCT cana-756 210 11 βˆ€ βˆ€ NOUN cana-756 210 12 v v NOUN cana-756 210 13 , , PUNCT cana-756 210 14 w w PROPN cana-756 210 15 ∈ ∈ PROPN cana-756 210 16 r. r. PROPN cana-756 210 17 hence hence ADV cana-756 210 18 , , PUNCT cana-756 210 19 d2Ξ΄ d2Ξ΄ PROPN cana-756 210 20 1 1 NUM cana-756 210 21 = = SYM cana-756 210 22 0 0 NUM cana-756 210 23 . . PUNCT cana-756 211 1 ( ( PUNCT cana-756 211 2 v)to v)to PROPN cana-756 211 3 prove prove VERB cana-756 211 4 Ξ΄1Ξ΄ Ξ΄1Ξ΄ NOUN cana-756 211 5 2 2 NUM cana-756 211 6 = = SYM cana-756 211 7 0 0 NUM cana-756 211 8 : : PUNCT cana-756 211 9 communications communication NOUN cana-756 211 10 on on ADP cana-756 211 11 applied apply VERB cana-756 211 12 nonlinear nonlinear ADJ cana-756 211 13 analysis analysis NOUN cana-756 211 14 issn issn NOUN cana-756 211 15 : : PUNCT cana-756 211 16 1074 1074 NUM cana-756 211 17 - - PUNCT cana-756 211 18 133x 133x NUM cana-756 211 19 vol vol NOUN cana-756 211 20 31 31 NUM cana-756 211 21 no no NOUN cana-756 211 22 . . PUNCT cana-756 212 1 3s 3s NUM cana-756 212 2 ( ( PUNCT cana-756 212 3 2024 2024 NUM cana-756 212 4 ) ) PUNCT cana-756 212 5 167 167 NUM cana-756 212 6 https://internationalpubls.com https://internationalpubls.com X cana-756 212 7 since since SCONJ cana-756 212 8 Ξ΄1 Ξ΄1 NOUN cana-756 212 9 , , PUNCT cana-756 212 10 Ξ΄2 Ξ΄2 PROPN cana-756 212 11 are be AUX cana-756 212 12 orthogonal orthogonal ADJ cana-756 212 13 , , PUNCT cana-756 212 14 we we PRON cana-756 212 15 can can AUX cana-756 212 16 have have VERB cana-756 212 17 Ξ΄1(u Ξ΄1(u NUM cana-756 212 18 , , PUNCT cana-756 212 19 v v NOUN cana-756 212 20 ) ) PUNCT cana-756 212 21 rΞ΄2(v rΞ΄2(v NOUN cana-756 212 22 , , PUNCT cana-756 212 23 w w NOUN cana-756 212 24 ) ) PUNCT cana-756 212 25 = = SYM cana-756 212 26 0 0 NUM cana-756 212 27 , , PUNCT cana-756 212 28 βˆ€ βˆ€ X cana-756 212 29 u u NOUN cana-756 212 30 , , PUNCT cana-756 212 31 v v INTJ cana-756 212 32 , , PUNCT cana-756 212 33 w w PROPN cana-756 212 34 , , PUNCT cana-756 212 35 r r NOUN cana-756 212 36 ∈ ∈ PROPN cana-756 212 37 r r NOUN cana-756 212 38 Ξ΄1(Ξ΄1(u Ξ΄1(Ξ΄1(u PROPN cana-756 212 39 , , PUNCT cana-756 212 40 v v NOUN cana-756 212 41 ) ) PUNCT cana-756 212 42 rΞ΄2(v rΞ΄2(v NOUN cana-756 212 43 , , PUNCT cana-756 212 44 w w NOUN cana-756 212 45 ) ) PUNCT cana-756 212 46 , , PUNCT cana-756 212 47 m m NOUN cana-756 212 48 ) ) PUNCT cana-756 213 1 = = SYM cana-756 213 2 0 0 NUM cana-756 213 3 , , PUNCT cana-756 213 4 βˆ€ βˆ€ X cana-756 213 5 u u NOUN cana-756 213 6 , , PUNCT cana-756 213 7 v v INTJ cana-756 213 8 , , PUNCT cana-756 213 9 w w PROPN cana-756 213 10 , , PUNCT cana-756 213 11 r r NOUN cana-756 213 12 , , PUNCT cana-756 213 13 m m PROPN cana-756 213 14 ∈ ∈ PROPN cana-756 213 15 r r NOUN cana-756 213 16 Ξ΄1(Ξ΄2(v Ξ΄1(Ξ΄2(v PROPN cana-756 213 17 , , PUNCT cana-756 213 18 w w NOUN cana-756 213 19 ) ) PUNCT cana-756 213 20 , , PUNCT cana-756 213 21 m)Οƒ(r)Οƒ(Ξ΄1(u m)Οƒ(r)Οƒ(Ξ΄1(u PROPN cana-756 213 22 , , PUNCT cana-756 213 23 v v NOUN cana-756 213 24 ) ) PUNCT cana-756 213 25 + + CCONJ cana-756 213 26 Ο„(Ξ΄2(v Ο„(Ξ΄2(v PROPN cana-756 213 27 , , PUNCT cana-756 213 28 w))d1(r w))d1(r NOUN cana-756 213 29 , , PUNCT cana-756 213 30 m)Οƒ(Ξ΄ m)Οƒ(Ξ΄ PROPN cana-756 213 31 1 1 NUM cana-756 213 32 ( ( PUNCT cana-756 213 33 u u NOUN cana-756 213 34 , , PUNCT cana-756 213 35 v v NOUN cana-756 213 36 ) ) PUNCT cana-756 213 37 ) ) PUNCT cana-756 214 1 + + CCONJ cana-756 214 2 Ο„(rΞ΄2(v Ο„(rΞ΄2(v PROPN cana-756 214 3 , , PUNCT cana-756 214 4 w w NOUN cana-756 214 5 ) ) PUNCT cana-756 214 6 d1(Ξ΄1(u d1(Ξ΄1(u PROPN cana-756 214 7 , , PUNCT cana-756 214 8 v v NOUN cana-756 214 9 ) ) PUNCT cana-756 214 10 , , PUNCT cana-756 214 11 m m PROPN cana-756 214 12 ) ) PUNCT cana-756 214 13 = = SYM cana-756 214 14 0 0 X cana-756 214 15 . . PUNCT cana-756 214 16 using use VERB cana-756 214 17 σδ1 σδ1 NOUN cana-756 214 18 = = SYM cana-756 214 19 Ξ΄1Οƒ Ξ΄1Οƒ NOUN cana-756 214 20 ; ; PUNCT cana-756 214 21 τδ2 τδ2 NOUN cana-756 214 22 = = SYM cana-756 214 23 Ξ΄2Ο„ Ξ΄2Ο„ PROPN cana-756 214 24 ; ; PUNCT cana-756 214 25 and and CCONJ cana-756 214 26 Οƒ Οƒ PROPN cana-756 214 27 and and CCONJ cana-756 214 28 Ο„ Ο„ PROPN cana-756 214 29 are be AUX cana-756 214 30 automorphisms automorphism NOUN cana-756 214 31 of of ADP cana-756 214 32 r. r. PROPN cana-756 214 33 Ξ΄1(Ξ΄2(v Ξ΄1(Ξ΄2(v PROPN cana-756 214 34 , , PUNCT cana-756 214 35 w w NOUN cana-756 214 36 ) ) PUNCT cana-756 214 37 , , PUNCT cana-756 214 38 m)rΞ΄1(u m)rΞ΄1(u PROPN cana-756 214 39 , , PUNCT cana-756 214 40 v)+ v)+ NOUN cana-756 214 41 Ξ΄2 Ξ΄2 VERB cana-756 214 42 ( ( PUNCT cana-756 214 43 v v NOUN cana-756 214 44 , , PUNCT cana-756 214 45 w)d1(r w)d1(r NOUN cana-756 214 46 , , PUNCT cana-756 214 47 m m PROPN cana-756 214 48 ) ) PUNCT cana-756 214 49 Ξ΄1(u Ξ΄1(u PROPN cana-756 214 50 , , PUNCT cana-756 214 51 v v NOUN cana-756 214 52 ) ) PUNCT cana-756 214 53 + + CCONJ cana-756 214 54 rΞ΄2(v rΞ΄2(v NOUN cana-756 214 55 , , PUNCT cana-756 214 56 w w NOUN cana-756 214 57 ) ) PUNCT cana-756 214 58 d1(Ξ΄1(u d1(Ξ΄1(u PROPN cana-756 214 59 , , PUNCT cana-756 214 60 v v NOUN cana-756 214 61 ) ) PUNCT cana-756 214 62 , , PUNCT cana-756 214 63 m)=0 m)=0 PROPN cana-756 214 64 . . NOUN cana-756 215 1 using use VERB cana-756 215 2 ( ( PUNCT cana-756 215 3 3.5 3.5 NUM cana-756 215 4 ) ) PUNCT cana-756 215 5 of of ADP cana-756 215 6 theorem theorem NOUN cana-756 215 7 1 1 NUM cana-756 215 8 , , PUNCT cana-756 215 9 the the DET cana-756 215 10 last last ADJ cana-756 215 11 two two NUM cana-756 215 12 terms term NOUN cana-756 215 13 of of ADP cana-756 215 14 the the DET cana-756 215 15 above above ADJ cana-756 215 16 equation equation NOUN cana-756 215 17 becomes become VERB cana-756 215 18 zero zero NUM cana-756 215 19 , , PUNCT cana-756 215 20 then then ADV cana-756 215 21 we we PRON cana-756 215 22 get get VERB cana-756 215 23 Ξ΄1Ξ΄2(v Ξ΄1Ξ΄2(v NOUN cana-756 215 24 , , PUNCT cana-756 215 25 w)r w)r X cana-756 215 26 Ξ΄1(u Ξ΄1(u NUM cana-756 215 27 , , PUNCT cana-756 215 28 v v NOUN cana-756 215 29 ) ) PUNCT cana-756 215 30 = = SYM cana-756 216 1 0 0 X cana-756 216 2 . . PUNCT cana-756 217 1 in in ADP cana-756 217 2 particular particular ADJ cana-756 217 3 if if SCONJ cana-756 217 4 we we PRON cana-756 217 5 put put VERB cana-756 217 6 u u NOUN cana-756 217 7 = = PROPN cana-756 217 8 Ξ΄2(v Ξ΄2(v PROPN cana-756 217 9 , , PUNCT cana-756 217 10 w w NOUN cana-756 217 11 ) ) PUNCT cana-756 217 12 in in ADP cana-756 217 13 the the DET cana-756 217 14 above above ADJ cana-756 217 15 equation equation NOUN cana-756 217 16 , , PUNCT cana-756 217 17 we we PRON cana-756 217 18 get get VERB cana-756 217 19 Ξ΄1Ξ΄2(v Ξ΄1Ξ΄2(v NOUN cana-756 217 20 , , PUNCT cana-756 217 21 w)rΞ΄1(Ξ΄2(v w)rΞ΄1(Ξ΄2(v PROPN cana-756 217 22 , , PUNCT cana-756 217 23 w w NOUN cana-756 217 24 ) ) PUNCT cana-756 217 25 , , PUNCT cana-756 217 26 v)=0 v)=0 PROPN cana-756 217 27 Ξ΄1Ξ΄2(v Ξ΄1Ξ΄2(v PROPN cana-756 217 28 , , PUNCT cana-756 217 29 w)rΞ΄1Ξ΄2(v w)rΞ΄1Ξ΄2(v NOUN cana-756 217 30 , , PUNCT cana-756 217 31 w w NOUN cana-756 217 32 ) ) PUNCT cana-756 217 33 = = SYM cana-756 217 34 0 0 NUM cana-756 217 35 Ξ΄1Ξ΄2(v Ξ΄1Ξ΄2(v NOUN cana-756 217 36 , , PUNCT cana-756 217 37 w w NOUN cana-756 217 38 ) ) PUNCT cana-756 217 39 = = SYM cana-756 217 40 0 0 PUNCT cana-756 218 1 ( ( PUNCT cana-756 218 2 by by ADP cana-756 218 3 semiprimeness semiprimeness NOUN cana-756 218 4 of of ADP cana-756 218 5 r. r. PROPN cana-756 218 6 ) ) PUNCT cana-756 219 1 and and CCONJ cana-756 219 2 so so ADV cana-756 219 3 Ξ΄1Ξ΄2 Ξ΄1Ξ΄2 ADP cana-756 219 4 = = SYM cana-756 219 5 0 0 PROPN cana-756 219 6 . . PUNCT cana-756 220 1 ( ( PUNCT cana-756 220 2 vi)to vi)to PART cana-756 220 3 prove prove VERB cana-756 220 4 Ξ΄2Ξ΄ Ξ΄2Ξ΄ ADJ cana-756 220 5 1 1 NUM cana-756 220 6 = = SYM cana-756 220 7 0 0 NUM cana-756 220 8 : : PUNCT cana-756 220 9 since since SCONJ cana-756 220 10 Ξ΄1 Ξ΄1 NOUN cana-756 220 11 , , PUNCT cana-756 220 12 Ξ΄2 Ξ΄2 PROPN cana-756 220 13 are be AUX cana-756 220 14 orthogonal orthogonal ADJ cana-756 220 15 , , PUNCT cana-756 220 16 we we PRON cana-756 220 17 can can AUX cana-756 220 18 have have VERB cana-756 220 19 Ξ΄2(u Ξ΄2(u NOUN cana-756 220 20 , , PUNCT cana-756 220 21 v v NOUN cana-756 220 22 ) ) PUNCT cana-756 220 23 rΞ΄1(v rΞ΄1(v PROPN cana-756 220 24 , , PUNCT cana-756 220 25 w w NOUN cana-756 220 26 ) ) PUNCT cana-756 220 27 = = SYM cana-756 220 28 0 0 NUM cana-756 220 29 , , PUNCT cana-756 220 30 βˆ€ βˆ€ X cana-756 220 31 u u NOUN cana-756 220 32 , , PUNCT cana-756 220 33 v v INTJ cana-756 220 34 , , PUNCT cana-756 220 35 w w PROPN cana-756 220 36 , , PUNCT cana-756 220 37 r r NOUN cana-756 220 38 ∈ ∈ PROPN cana-756 220 39 r r NOUN cana-756 220 40 Ξ΄2(Ξ΄2(u Ξ΄2(Ξ΄2(u PROPN cana-756 220 41 , , PUNCT cana-756 220 42 v v NOUN cana-756 220 43 ) ) PUNCT cana-756 220 44 rΞ΄1(v rΞ΄1(v PROPN cana-756 220 45 , , PUNCT cana-756 220 46 w w NOUN cana-756 220 47 ) ) PUNCT cana-756 220 48 , , PUNCT cana-756 220 49 m m NOUN cana-756 220 50 ) ) PUNCT cana-756 221 1 = = SYM cana-756 221 2 0 0 NUM cana-756 221 3 , , PUNCT cana-756 221 4 βˆ€ βˆ€ X cana-756 221 5 u u NOUN cana-756 221 6 , , PUNCT cana-756 221 7 v v INTJ cana-756 221 8 , , PUNCT cana-756 221 9 w w PROPN cana-756 221 10 , , PUNCT cana-756 221 11 r r NOUN cana-756 221 12 , , PUNCT cana-756 221 13 m m PROPN cana-756 221 14 ∈ ∈ PROPN cana-756 221 15 r. r. NOUN cana-756 221 16 by by ADP cana-756 221 17 following follow VERB cana-756 221 18 the the DET cana-756 221 19 similar similar ADJ cana-756 221 20 procedure procedure NOUN cana-756 221 21 as as SCONJ cana-756 221 22 we we PRON cana-756 221 23 adopted adopt VERB cana-756 221 24 in in ADP cana-756 221 25 the the DET cana-756 221 26 previous previous ADJ cana-756 221 27 case case NOUN cana-756 221 28 , , PUNCT cana-756 221 29 we we PRON cana-756 221 30 can can AUX cana-756 221 31 easily easily ADV cana-756 221 32 obtain obtain VERB cana-756 221 33 the the DET cana-756 221 34 result result NOUN cana-756 221 35 . . PUNCT cana-756 222 1 4 4 X cana-756 222 2 . . X cana-756 222 3 discussion discussion NOUN cana-756 222 4 : : PUNCT cana-756 222 5 the the DET cana-756 222 6 study study NOUN cana-756 222 7 of of ADP cana-756 222 8 derivations derivation NOUN cana-756 222 9 and and CCONJ cana-756 222 10 their their PRON cana-756 222 11 generalizations generalization NOUN cana-756 222 12 plays play VERB cana-756 222 13 a a DET cana-756 222 14 significant significant ADJ cana-756 222 15 role role NOUN cana-756 222 16 in in ADP cana-756 222 17 ring ring NOUN cana-756 222 18 theory theory NOUN cana-756 222 19 . . PUNCT cana-756 223 1 in in ADP cana-756 223 2 this this DET cana-756 223 3 manuscript manuscript NOUN cana-756 223 4 , , PUNCT cana-756 223 5 we we PRON cana-756 223 6 focus focus VERB cana-756 223 7 on on ADP cana-756 223 8 the the DET cana-756 223 9 concept concept NOUN cana-756 223 10 of of ADP cana-756 223 11 generalized generalized ADJ cana-756 223 12 symmetric symmetric ADJ cana-756 223 13 reverse reverse NOUN cana-756 223 14 bi-(𝜎 bi-(𝜎 NOUN cana-756 223 15 , , PUNCT cana-756 223 16 𝜏)-derivations 𝜏)-derivation NOUN cana-756 223 17 and and CCONJ cana-756 223 18 explore explore VERB cana-756 223 19 the the DET cana-756 223 20 conditions condition NOUN cana-756 223 21 for for ADP cana-756 223 22 their their PRON cana-756 223 23 orthogonality orthogonality NOUN cana-756 223 24 within within ADP cana-756 223 25 the the DET cana-756 223 26 framework framework NOUN cana-756 223 27 of of ADP cana-756 223 28 semi semi ADJ cana-756 223 29 prime prime ADJ cana-756 223 30 rings ring NOUN cana-756 223 31 . . PUNCT cana-756 224 1 let let VERB cana-756 224 2 r r PRON cana-756 224 3 be be AUX cana-756 224 4 a a DET cana-756 224 5 semi semi ADJ cana-756 224 6 prime prime ADJ cana-756 224 7 ring ring NOUN cana-756 224 8 . . PUNCT cana-756 225 1 a a DET cana-756 225 2 symmetric symmetric ADJ cana-756 225 3 bi bi ADJ cana-756 225 4 - - ADJ cana-756 225 5 additive additive ADJ cana-756 225 6 mapping mapping NOUN cana-756 225 7 Ξ΄1 Ξ΄1 NOUN cana-756 225 8 : : PUNCT cana-756 225 9 rΓ—rβ†’r rΓ—rβ†’r PROPN cana-756 225 10 is be AUX cana-756 225 11 termed term VERB cana-756 225 12 a a DET cana-756 225 13 generalized generalize VERB cana-756 225 14 symmetric symmetric ADJ cana-756 225 15 reverse reverse NOUN cana-756 225 16 bi-(Οƒ bi-(Οƒ NOUN cana-756 225 17 , , PUNCT cana-756 225 18 Ο„)-derivation Ο„)-derivation ADP cana-756 225 19 if if SCONJ cana-756 225 20 there there PRON cana-756 225 21 exists exist VERB cana-756 225 22 a a DET cana-756 225 23 symmetric symmetric ADJ cana-756 225 24 reverse reverse NOUN cana-756 225 25 bi-(𝜎 bi-(𝜎 NOUN cana-756 225 26 , , PUNCT cana-756 225 27 𝜏)-derivation 𝜏)-derivation NOUN cana-756 225 28 𝐷1 𝐷1 NOUN cana-756 225 29 on on ADP cana-756 225 30 r r NOUN cana-756 225 31 such such ADJ cana-756 225 32 that that PRON cana-756 225 33 for for ADP cana-756 225 34 all all DET cana-756 225 35 𝑒 𝑒 PROPN cana-756 225 36 , , PUNCT cana-756 225 37 𝑣 𝑣 NOUN cana-756 225 38 , , PUNCT cana-756 225 39 𝑀 𝑀 ADP cana-756 225 40 ∈ ∈ PROPN cana-756 225 41 𝑅 𝑅 PROPN cana-756 225 42 , , PUNCT cana-756 225 43 𝛿1(uv 𝛿1(uv PROPN cana-756 225 44 , , PUNCT cana-756 225 45 w)= w)= NOUN cana-756 225 46 𝛿1(v 𝛿1(v NOUN cana-756 225 47 , , PUNCT cana-756 225 48 w)Οƒ(u w)Οƒ(u NOUN cana-756 225 49 ) ) PUNCT cana-756 226 1 + + SYM cana-756 226 2 Ο„(v Ο„(v NOUN cana-756 226 3 ) ) PUNCT cana-756 226 4 𝐷1 𝐷1 NOUN cana-756 226 5 ( ( PUNCT cana-756 226 6 u u NOUN cana-756 226 7 , , PUNCT cana-756 226 8 w w NOUN cana-756 226 9 ) ) PUNCT cana-756 226 10 . . PUNCT cana-756 227 1 this this DET cana-756 227 2 definition definition NOUN cana-756 227 3 extends extend VERB cana-756 227 4 the the DET cana-756 227 5 classical classical ADJ cana-756 227 6 notion notion NOUN cana-756 227 7 of of ADP cana-756 227 8 generalized generalized ADJ cana-756 227 9 symmetric symmetric ADJ cana-756 227 10 reverse reverse ADJ cana-756 227 11 bi bi NOUN cana-756 227 12 - - NOUN cana-756 227 13 derivations derivation NOUN cana-756 227 14 by by ADP cana-756 227 15 incorporating incorporate VERB cana-756 227 16 the the DET cana-756 227 17 actions action NOUN cana-756 227 18 of of ADP cana-756 227 19 two two NUM cana-756 227 20 automorphisms automorphism NOUN cana-756 227 21 , , PUNCT cana-756 227 22 Οƒ Οƒ PROPN cana-756 227 23 and and CCONJ cana-756 227 24 Ο„ Ο„ PROPN cana-756 227 25 , , PUNCT cana-756 227 26 thus thus ADV cana-756 227 27 providing provide VERB cana-756 227 28 a a DET cana-756 227 29 richer rich ADJ cana-756 227 30 and and CCONJ cana-756 227 31 more more ADV cana-756 227 32 flexible flexible ADJ cana-756 227 33 structure structure NOUN cana-756 227 34 for for ADP cana-756 227 35 analysis analysis NOUN cana-756 227 36 . . PUNCT cana-756 228 1 we we PRON cana-756 228 2 consider consider VERB cana-756 228 3 two two NUM cana-756 228 4 generalized generalized ADJ cana-756 228 5 symmetric symmetric ADJ cana-756 228 6 reverse reverse NOUN cana-756 228 7 bi-(𝜎 bi-(𝜎 NOUN cana-756 228 8 , , PUNCT cana-756 228 9 𝜏)-derivations 𝜏)-derivation VERB cana-756 228 10 [ [ X cana-756 228 11 Ξ΄1,d1 Ξ΄1,d1 X cana-756 228 12 ] ] X cana-756 228 13 and and CCONJ cana-756 228 14 [ [ X cana-756 228 15 Ξ΄2,d2 Ξ΄2,d2 NOUN cana-756 228 16 ] ] PUNCT cana-756 228 17 of of ADP cana-756 228 18 r r NOUN cana-756 228 19 with with ADP cana-756 228 20 associated associated ADJ cana-756 228 21 reverse reverse NOUN cana-756 228 22 bi-(𝜎 bi-(𝜎 NOUN cana-756 228 23 , , PUNCT cana-756 228 24 𝜏)-derivations 𝜏)-derivation NOUN cana-756 228 25 𝐷1 𝐷1 VERB cana-756 228 26 an an DET cana-756 228 27 𝐷2 𝐷2 NOUN cana-756 228 28 the the DET cana-756 228 29 primary primary ADJ cana-756 228 30 goal goal NOUN cana-756 228 31 of of ADP cana-756 228 32 this this DET cana-756 228 33 paper paper NOUN cana-756 228 34 is be AUX cana-756 228 35 to to PART cana-756 228 36 establish establish VERB cana-756 228 37 equivalent equivalent ADJ cana-756 228 38 conditions condition NOUN cana-756 228 39 for for ADP cana-756 228 40 the the DET cana-756 228 41 orthogonality orthogonality NOUN cana-756 228 42 between between ADP cana-756 228 43 these these DET cana-756 228 44 two two NUM cana-756 228 45 mappings mapping NOUN cana-756 228 46 . . PUNCT cana-756 229 1 orthogonality orthogonality NOUN cana-756 229 2 in in ADP cana-756 229 3 this this DET cana-756 229 4 context context NOUN cana-756 229 5 refers refer VERB cana-756 229 6 to to ADP cana-756 229 7 the the DET cana-756 229 8 condition condition NOUN cana-756 229 9 where where SCONJ cana-756 229 10 the the DET cana-756 229 11 products product NOUN cana-756 229 12 of of ADP cana-756 229 13 the the DET cana-756 229 14 mappings mapping NOUN cana-756 229 15 and and CCONJ cana-756 229 16 their their PRON cana-756 229 17 associated associated ADJ cana-756 229 18 derivations derivation NOUN cana-756 229 19 satisfy satisfy VERB cana-756 229 20 specific specific ADJ cana-756 229 21 nullity nullity NOUN cana-756 229 22 conditions condition NOUN cana-756 229 23 . . PUNCT cana-756 230 1 through through ADP cana-756 230 2 our our PRON cana-756 230 3 analysis analysis NOUN cana-756 230 4 , , PUNCT cana-756 230 5 we we PRON cana-756 230 6 derive derive VERB cana-756 230 7 several several ADJ cana-756 230 8 equivalent equivalent ADJ cana-756 230 9 conditions condition NOUN cana-756 230 10 that that PRON cana-756 230 11 characterize characterize VERB cana-756 230 12 the the DET cana-756 230 13 orthogonality orthogonality NOUN cana-756 230 14 of of ADP cana-756 230 15 generalized generalized ADJ cana-756 230 16 symmetric symmetric ADJ cana-756 230 17 reverse reverse NOUN cana-756 230 18 bi-(𝜎 bi-(𝜎 NOUN cana-756 230 19 , , PUNCT cana-756 230 20 𝜏)-derivations 𝜏)-derivation NOUN cana-756 230 21 in in ADP cana-756 230 22 semi semi ADJ cana-756 230 23 prime prime ADJ cana-756 230 24 rings ring NOUN cana-756 230 25 . . PUNCT cana-756 231 1 these these DET cana-756 231 2 conditions condition NOUN cana-756 231 3 provide provide VERB cana-756 231 4 insights insight NOUN cana-756 231 5 into into ADP cana-756 231 6 the the DET cana-756 231 7 underlying underlie VERB cana-756 231 8 algebraic algebraic ADJ cana-756 231 9 structures structure NOUN cana-756 231 10 and and CCONJ cana-756 231 11 their their PRON cana-756 231 12 inter inter ADJ cana-756 231 13 relationships relationship NOUN cana-756 231 14 . . PUNCT cana-756 232 1 specifically specifically ADV cana-756 232 2 , , PUNCT cana-756 232 3 we we PRON cana-756 232 4 show show VERB cana-756 232 5 that that SCONJ cana-756 232 6 orthogonality orthogonality NOUN cana-756 232 7 can can AUX cana-756 232 8 be be AUX cana-756 232 9 characterized characterize VERB cana-756 232 10 in in ADP cana-756 232 11 terms term NOUN cana-756 232 12 of of ADP cana-756 232 13 commutativity commutativity NOUN cana-756 232 14 and and CCONJ cana-756 232 15 specific specific ADJ cana-756 232 16 interaction interaction NOUN cana-756 232 17 properties property NOUN cana-756 232 18 between between ADP cana-756 232 19 the the DET cana-756 232 20 mappings mapping NOUN cana-756 232 21 and and CCONJ cana-756 232 22 their their PRON cana-756 232 23 associated associated ADJ cana-756 232 24 derivations derivation NOUN cana-756 232 25 . . PUNCT cana-756 233 1 communications communication NOUN cana-756 233 2 on on ADP cana-756 233 3 applied apply VERB cana-756 233 4 nonlinear nonlinear ADJ cana-756 233 5 analysis analysis NOUN cana-756 233 6 issn issn NOUN cana-756 233 7 : : PUNCT cana-756 233 8 1074 1074 NUM cana-756 233 9 - - PUNCT cana-756 233 10 133x 133x NUM cana-756 233 11 vol vol NOUN cana-756 233 12 31 31 NUM cana-756 233 13 no no NOUN cana-756 233 14 . . PUNCT cana-756 234 1 3s 3s NUM cana-756 234 2 ( ( PUNCT cana-756 234 3 2024 2024 NUM cana-756 234 4 ) ) PUNCT cana-756 234 5 168 168 NUM cana-756 234 6 https://internationalpubls.com https://internationalpubls.com X cana-756 234 7 the the DET cana-756 234 8 exploration exploration NOUN cana-756 234 9 of of ADP cana-756 234 10 generalized generalized ADJ cana-756 234 11 symmetric symmetric ADJ cana-756 234 12 reverse reverse NOUN cana-756 234 13 bi-(Οƒ bi-(Οƒ PROPN cana-756 234 14 , , PUNCT cana-756 234 15 Ο„)-derivations Ο„)-derivation NOUN cana-756 234 16 opens open VERB cana-756 234 17 several several ADJ cana-756 234 18 avenues avenue NOUN cana-756 234 19 for for ADP cana-756 234 20 future future ADJ cana-756 234 21 research research NOUN cana-756 234 22 . . PUNCT cana-756 235 1 one one NUM cana-756 235 2 potential potential ADJ cana-756 235 3 direction direction NOUN cana-756 235 4 is be AUX cana-756 235 5 the the DET cana-756 235 6 investigation investigation NOUN cana-756 235 7 of of ADP cana-756 235 8 these these DET cana-756 235 9 derivations derivation NOUN cana-756 235 10 in in ADP cana-756 235 11 the the DET cana-756 235 12 context context NOUN cana-756 235 13 of of ADP cana-756 235 14 non non ADJ cana-756 235 15 - - ADJ cana-756 235 16 associative associative ADJ cana-756 235 17 algebras algebra NOUN cana-756 235 18 or or CCONJ cana-756 235 19 rings ring NOUN cana-756 235 20 with with ADP cana-756 235 21 additional additional ADJ cana-756 235 22 structures structure NOUN cana-756 235 23 or or CCONJ cana-756 235 24 other other ADJ cana-756 235 25 ring ring NOUN cana-756 235 26 - - PUNCT cana-756 235 27 theoretic theoretic NOUN cana-756 235 28 properties property NOUN cana-756 235 29 , , PUNCT cana-756 235 30 such such ADJ cana-756 235 31 as as ADP cana-756 235 32 ideals ideal NOUN cana-756 235 33 and and CCONJ cana-756 235 34 radicals radical NOUN cana-756 235 35 etc etc X cana-756 235 36 . . X cana-756 236 1 while while SCONJ cana-756 236 2 this this DET cana-756 236 3 study study NOUN cana-756 236 4 focuses focus VERB cana-756 236 5 on on ADP cana-756 236 6 semiprime semiprime NOUN cana-756 236 7 rings ring NOUN cana-756 236 8 , , PUNCT cana-756 236 9 the the DET cana-756 236 10 concepts concept NOUN cana-756 236 11 can can AUX cana-756 236 12 be be AUX cana-756 236 13 extended extend VERB cana-756 236 14 to to ADP cana-756 236 15 other other ADJ cana-756 236 16 classes class NOUN cana-756 236 17 of of ADP cana-756 236 18 rings ring NOUN cana-756 236 19 offering offer VERB cana-756 236 20 a a DET cana-756 236 21 broader broad ADJ cana-756 236 22 applicability applicability NOUN cana-756 236 23 of of ADP cana-756 236 24 the the DET cana-756 236 25 results result NOUN cana-756 236 26 . . PUNCT cana-756 237 1 5 5 X cana-756 237 2 . . X cana-756 237 3 conclusion conclusion NOUN cana-756 237 4 : : PUNCT cana-756 237 5 in in ADP cana-756 237 6 conclusion conclusion NOUN cana-756 237 7 , , PUNCT cana-756 237 8 this this DET cana-756 237 9 manuscript manuscript NOUN cana-756 237 10 contributes contribute VERB cana-756 237 11 to to ADP cana-756 237 12 the the DET cana-756 237 13 field field NOUN cana-756 237 14 of of ADP cana-756 237 15 ring ring NOUN cana-756 237 16 theory theory NOUN cana-756 237 17 by by ADP cana-756 237 18 providing provide VERB cana-756 237 19 a a DET cana-756 237 20 detailed detailed ADJ cana-756 237 21 analysis analysis NOUN cana-756 237 22 of of ADP cana-756 237 23 the the DET cana-756 237 24 orthogonality orthogonality NOUN cana-756 237 25 conditions condition NOUN cana-756 237 26 for for ADP cana-756 237 27 generalized generalize VERB cana-756 237 28 symmetric symmetric ADJ cana-756 237 29 reverse reverse NOUN cana-756 237 30 bi-(𝜎 bi-(𝜎 NOUN cana-756 237 31 , , PUNCT cana-756 237 32 𝜏)-derivations 𝜏)-derivation NOUN cana-756 237 33 in in ADP cana-756 237 34 semi semi ADJ cana-756 237 35 prime prime ADJ cana-756 237 36 rings ring NOUN cana-756 237 37 . . PUNCT cana-756 238 1 the the DET cana-756 238 2 equivalence equivalence NOUN cana-756 238 3 conditions condition NOUN cana-756 238 4 established establish VERB cana-756 238 5 here here ADV cana-756 238 6 deepen deepen VERB cana-756 238 7 our our PRON cana-756 238 8 understanding understanding NOUN cana-756 238 9 of of ADP cana-756 238 10 these these DET cana-756 238 11 mappings mapping NOUN cana-756 238 12 and and CCONJ cana-756 238 13 pave pave VERB cana-756 238 14 the the DET cana-756 238 15 way way NOUN cana-756 238 16 for for ADP cana-756 238 17 further further ADJ cana-756 238 18 explorations exploration NOUN cana-756 238 19 into into ADP cana-756 238 20 the the DET cana-756 238 21 rich rich ADJ cana-756 238 22 landscape landscape NOUN cana-756 238 23 of of ADP cana-756 238 24 derivations derivation NOUN cana-756 238 25 and and CCONJ cana-756 238 26 their their PRON cana-756 238 27 generalizations generalization NOUN cana-756 238 28 in in ADP cana-756 238 29 other other ADJ cana-756 238 30 algebraic algebraic ADJ cana-756 238 31 structures structure NOUN cana-756 238 32 . . PUNCT cana-756 239 1 references reference NOUN cana-756 239 2 : : PUNCT cana-756 240 1 [ [ X cana-756 240 2 1 1 NUM cana-756 240 3 ] ] PUNCT cana-756 240 4 a. a. PROPN cana-756 240 5 ali ali PROPN cana-756 240 6 , , PUNCT cana-756 240 7 d. d. PROPN cana-756 240 8 filippis filippis PROPN cana-756 240 9 , , PUNCT cana-756 240 10 and and CCONJ cana-756 240 11 f. f. PROPN cana-756 240 12 shujat shujat PROPN cana-756 240 13 , , PUNCT cana-756 240 14 β€œ " PUNCT cana-756 240 15 results result NOUN cana-756 240 16 concerning concern VERB cana-756 240 17 symmetric symmetric ADJ cana-756 240 18 generalized generalize VERB cana-756 240 19 biderivations biderivation NOUN cana-756 240 20 of of ADP cana-756 240 21 prime prime ADJ cana-756 240 22 and and CCONJ cana-756 240 23 semiprime semiprime NOUN cana-756 240 24 rings ring NOUN cana-756 240 25 , , PUNCT cana-756 240 26 ” " PUNCT cana-756 240 27 mathematical mathematical ADJ cana-756 240 28 bechak bechak NOUN cana-756 240 29 , , PUNCT cana-756 240 30 66(4 66(4 NUM cana-756 240 31 ) ) PUNCT cana-756 240 32 ( ( PUNCT cana-756 240 33 2014 2014 NUM cana-756 240 34 ) ) PUNCT cana-756 240 35 , , PUNCT cana-756 240 36 410–417 410–417 NUM cana-756 240 37 . . PUNCT cana-756 241 1 [ [ X cana-756 241 2 2 2 NUM cana-756 241 3 ] ] PUNCT cana-756 241 4 c. c. PROPN cana-756 241 5 jaya jaya PROPN cana-756 241 6 subba subba PROPN cana-756 241 7 reddy reddy PROPN cana-756 241 8 and and CCONJ cana-756 241 9 b.ramoorthy b.ramoorthy ADJ cana-756 241 10 reddy,β€œcommutativity reddy,β€œcommutativity NOUN cana-756 241 11 of of ADP cana-756 241 12 prime prime ADJ cana-756 241 13 ring ring NOUN cana-756 241 14 witorthogonal witorthogonal ADJ cana-756 241 15 symmetric symmetric ADJ cana-756 241 16 biderivations biderivation NOUN cana-756 241 17 , , PUNCT cana-756 241 18 ” " PUNCT cana-756 241 19 mathematical mathematical ADJ cana-756 241 20 journal journal NOUN cana-756 241 21 of of ADP cana-756 241 22 interdisciplinary interdisciplinary ADJ cana-756 241 23 sciences science NOUN cana-756 241 24 , , PUNCT cana-756 241 25 7(2 7(2 NUM cana-756 241 26 ) ) PUNCT cana-756 241 27 ( ( PUNCT cana-756 241 28 2019),117–120 2019),117–120 X cana-756 241 29 . . PUNCT cana-756 242 1 [ [ X cana-756 242 2 3 3 X cana-756 242 3 ] ] X cana-756 242 4 c. c. PROPN cana-756 242 5 jaya jaya PROPN cana-756 242 6 subba subba PROPN cana-756 242 7 reddy reddy PROPN cana-756 242 8 and and CCONJ cana-756 242 9 b. b. PROPN cana-756 242 10 ramoorthy ramoorthy PROPN cana-756 242 11 reddy,β€œorthogonal reddy,β€œorthogonal ADJ cana-756 242 12 symmetric symmetric ADJ cana-756 242 13 bi(Οƒ bi(Οƒ NOUN cana-756 242 14 , , PUNCT cana-756 242 15 Ο„)-derivations Ο„)-derivations PUNCT cana-756 242 16 in in ADP cana-756 242 17 semiprime semiprime NOUN cana-756 242 18 rings ring NOUN cana-756 242 19 , , PUNCT cana-756 242 20 ” " PUNCT cana-756 242 21 international international ADJ cana-756 242 22 journal journal NOUN cana-756 242 23 of of ADP cana-756 242 24 algebra algebra PROPN cana-756 242 25 , , PUNCT cana-756 242 26 10(9 10(9 NUM cana-756 242 27 ) ) PUNCT cana-756 242 28 ( ( PUNCT cana-756 242 29 2016),423–428 2016),423–428 X cana-756 242 30 . . PUNCT cana-756 243 1 [ [ X cana-756 243 2 4 4 NUM cana-756 243 3 ] ] X cana-756 243 4 c. c. PROPN cana-756 243 5 jaya jaya PROPN cana-756 243 6 subba subba PROPN cana-756 243 7 reddy reddy PROPN cana-756 243 8 and and CCONJ cana-756 243 9 b. b. PROPN cana-756 243 10 ramoorthy ramoorthy PROPN cana-756 243 11 reddy reddy PROPN cana-756 243 12 , , PUNCT cana-756 243 13 β€œ " PUNCT cana-756 243 14 orthogonal orthogonal ADJ cana-756 243 15 generalized generalize VERB cana-756 243 16 symmetric symmetric ADJ cana-756 243 17 bi bi NOUN cana-756 243 18 - - NOUN cana-756 243 19 derivations derivation NOUN cana-756 243 20 of of ADP cana-756 243 21 semiprime semiprime NOUN cana-756 243 22 rings ring NOUN cana-756 243 23 , , PUNCT cana-756 243 24 ” " PUNCT cana-756 243 25 contemporary contemporary ADJ cana-756 243 26 mathematics mathematic NOUN cana-756 243 27 , , PUNCT cana-756 243 28 4(1)(2017 4(1)(2017 NUM cana-756 243 29 ) ) PUNCT cana-756 243 30 , , PUNCT cana-756 243 31 21–27 21–27 NUM cana-756 243 32 . . PUNCT cana-756 244 1 [ [ X cana-756 244 2 5 5 NUM cana-756 244 3 ] ] PUNCT cana-756 244 4 c.jaya c.jaya PROPN cana-756 244 5 subba subba PROPN cana-756 244 6 reddy reddy PROPN cana-756 244 7 , , PUNCT cana-756 244 8 b.ramoorthy b.ramoorthy ADJ cana-756 244 9 reddy,”orthogonal reddy,”orthogonal ADJ cana-756 244 10 symmetric symmetric ADJ cana-756 244 11 biderivations biderivation NOUN cana-756 244 12 in in ADP cana-756 244 13 semiprime semiprime NOUN cana-756 244 14 rings ring NOUN cana-756 244 15 ” " PUNCT cana-756 244 16 , , PUNCT cana-756 244 17 international international ADJ cana-756 244 18 journal journal NOUN cana-756 244 19 of of ADP cana-756 244 20 mathematics mathematics PROPN cana-756 244 21 and and CCONJ cana-756 244 22 satistics satistic NOUN cana-756 244 23 studies,4(1 studies,4(1 PROPN cana-756 244 24 ) ) PUNCT cana-756 244 25 ( ( PUNCT cana-756 244 26 2016 2016 NUM cana-756 244 27 ) ) PUNCT cana-756 244 28 , , PUNCT cana-756 244 29 22 22 NUM cana-756 244 30 - - SYM cana-756 244 31 29 29 NUM cana-756 244 32 . . PUNCT cana-756 245 1 [ [ X cana-756 245 2 6 6 NUM cana-756 245 3 ] ] PUNCT cana-756 245 4 c.jaya c.jaya NOUN cana-756 245 5 subba subba PROPN cana-756 245 6 reddy reddy PROPN cana-756 245 7 and and CCONJ cana-756 245 8 b.ramoorthy b.ramoorthy ADJ cana-756 245 9 reddy reddy NOUN cana-756 245 10 , , PUNCT cana-756 245 11 k.chennakesavulu k.chennakesavulu ADJ cana-756 245 12 , , PUNCT cana-756 245 13 β€œ " PUNCT cana-756 245 14 orthogonality orthogonality NOUN cana-756 245 15 of of ADP cana-756 245 16 generalized generalized ADJ cana-756 245 17 ( ( PUNCT cana-756 245 18 Οƒ Οƒ PROPN cana-756 245 19 , , PUNCT cana-756 245 20 Ο„ Ο„ PROPN cana-756 245 21 ) ) PUNCT cana-756 245 22 symmetric symmetric ADJ cana-756 245 23 biderivations biderivation NOUN cana-756 245 24 in in ADP cana-756 245 25 semiprime semiprime NOUN cana-756 245 26 rings ring NOUN cana-756 245 27 ” " PUNCT cana-756 245 28 , , PUNCT cana-756 245 29 international international ADJ cana-756 245 30 journal journal NOUN cana-756 245 31 of of ADP cana-756 245 32 educational educational ADJ cana-756 245 33 science science NOUN cana-756 245 34 and and CCONJ cana-756 245 35 research research NOUN cana-756 245 36 , , PUNCT cana-756 245 37 8(6 8(6 NUM cana-756 245 38 ) ) PUNCT cana-756 245 39 ( ( PUNCT cana-756 245 40 2018 2018 NUM cana-756 245 41 ) ) PUNCT cana-756 245 42 , , PUNCT cana-756 245 43 45 45 NUM cana-756 245 44 - - SYM cana-756 245 45 52 52 NUM cana-756 245 46 . . PUNCT cana-756 246 1 [ [ X cana-756 246 2 7 7 X cana-756 246 3 ] ] PUNCT cana-756 246 4 c.jaya c.jaya NOUN cana-756 246 5 subba subba PROPN cana-756 246 6 reddy reddy PROPN cana-756 246 7 and and CCONJ cana-756 246 8 v.s.v.krishna v.s.v.krishna PROPN cana-756 246 9 murty murty NOUN cana-756 246 10 , , PUNCT cana-756 246 11 β€œ " PUNCT cana-756 246 12 orthogonal orthogonal ADJ cana-756 246 13 symmetric symmetric ADJ cana-756 246 14 reverse reverse NOUN cana-756 246 15 bi-(Οƒ bi-(Οƒ NOUN cana-756 246 16 , , PUNCT cana-756 246 17 Ο„)-derivations Ο„)-derivation VERB cana-756 246 18 in in ADP cana-756 246 19 semi semi ADJ cana-756 246 20 prime prime ADJ cana-756 246 21 rings ring NOUN cana-756 246 22 ” " PUNCT cana-756 246 23 , , PUNCT cana-756 246 24 tuijin tuijin NOUN cana-756 246 25 jishu jishu NOUN cana-756 246 26 /journal /journal ADJ cana-756 246 27 of of ADP cana-756 246 28 propulsion propulsion NOUN cana-756 246 29 technology technology NOUN cana-756 246 30 , , PUNCT cana-756 246 31 45(1 45(1 NOUN cana-756 246 32 ) ) PUNCT cana-756 246 33 ( ( PUNCT cana-756 246 34 2024 2024 NUM cana-756 246 35 ) ) PUNCT cana-756 246 36 , , PUNCT cana-756 246 37 5133 5133 NUM cana-756 246 38 - - SYM cana-756 246 39 5138 5138 NUM cana-756 246 40 . . PUNCT cana-756 247 1 [ [ X cana-756 247 2 8 8 NUM cana-756 247 3 ] ] PUNCT cana-756 247 4 c.jaya c.jaya NOUN cana-756 247 5 subba subba PROPN cana-756 247 6 reddy reddy PROPN cana-756 247 7 and and CCONJ cana-756 247 8 v.s.v v.s.v ADJ cana-756 247 9 . . PUNCT cana-756 248 1 krishna krishna PROPN cana-756 248 2 murty murty NOUN cana-756 248 3 , , PUNCT cana-756 248 4 β€œ " PUNCT cana-756 248 5 orthogonality orthogonality NOUN cana-756 248 6 of of ADP cana-756 248 7 generalized generalized ADJ cana-756 248 8 reverse reverse NOUN cana-756 248 9 ( ( PUNCT cana-756 248 10 Οƒ Οƒ NOUN cana-756 248 11 , , PUNCT cana-756 248 12 Ο„)-derivations Ο„)-derivations PUNCT cana-756 248 13 in in ADP cana-756 248 14 semiprime semiprime NOUN cana-756 248 15 Ξ³ Ξ³ PROPN cana-756 248 16 - - PUNCT cana-756 248 17 rings ring NOUN cana-756 248 18 ” " PUNCT cana-756 248 19 , , PUNCT cana-756 248 20 journal journal NOUN cana-756 248 21 of of ADP cana-756 248 22 nonlinear nonlinear ADJ cana-756 248 23 analysis analysis NOUN cana-756 248 24 and and CCONJ cana-756 248 25 optimization optimization NOUN cana-756 248 26 : : PUNCT cana-756 248 27 theory theory NOUN cana-756 248 28 and and CCONJ cana-756 248 29 applications application NOUN cana-756 248 30 , , PUNCT cana-756 248 31 15 15 NUM cana-756 248 32 ( ( PUNCT cana-756 248 33 4)1,(2024),2028 4)1,(2024),2028 NUM cana-756 248 34 . . PUNCT cana-756 249 1 [ [ X cana-756 249 2 9 9 NUM cana-756 249 3 ] ] SYM cana-756 249 4 e.c.posner e.c.posner NOUN cana-756 249 5 , , PUNCT cana-756 249 6 β€œ " PUNCT cana-756 249 7 derivations derivation NOUN cana-756 249 8 in in ADP cana-756 249 9 prime prime ADJ cana-756 249 10 rings ring NOUN cana-756 249 11 ” " PUNCT cana-756 249 12 , , PUNCT cana-756 249 13 proc.amer.mth.soc proc.amer.mth.soc NOUN cana-756 249 14 . . PROPN cana-756 249 15 , , PUNCT cana-756 249 16 8(1957 8(1957 NUM cana-756 249 17 ) ) PUNCT cana-756 249 18 , , PUNCT cana-756 249 19 1093 1093 NUM cana-756 249 20 - - SYM cana-756 249 21 1100 1100 NUM cana-756 249 22 . . PUNCT cana-756 250 1 [ [ X cana-756 250 2 10 10 NUM cana-756 250 3 ] ] X cana-756 250 4 e. e. PROPN cana-756 250 5 koc koc PROPN cana-756 250 6 , , PUNCT cana-756 250 7 β€œ " PUNCT cana-756 250 8 notes note NOUN cana-756 250 9 on on ADP cana-756 250 10 ideals ideal NOUN cana-756 250 11 and and CCONJ cana-756 250 12 orthogonal orthogonal ADJ cana-756 250 13 generalized generalize VERB cana-756 250 14 ( ( PUNCT cana-756 250 15 Οƒ Οƒ PROPN cana-756 250 16 , , PUNCT cana-756 250 17 Ο„ Ο„ PROPN cana-756 250 18 ) ) PUNCT cana-756 250 19 derivations derivation NOUN cana-756 250 20 ” " PUNCT cana-756 250 21 , , PUNCT cana-756 250 22 east east ADJ cana-756 250 23 asian asian PROPN cana-756 250 24 mathematical mathematical ADJ cana-756 250 25 journal journal NOUN cana-756 250 26 , , PUNCT cana-756 250 27 24(4 24(4 NUM cana-756 250 28 ) ) PUNCT cana-756 250 29 ( ( PUNCT cana-756 250 30 2008 2008 NUM cana-756 250 31 ) ) PUNCT cana-756 250 32 , , PUNCT cana-756 250 33 389–398 389–398 NUM cana-756 250 34 . . PUNCT cana-756 251 1 [ [ X cana-756 251 2 11 11 NUM cana-756 251 3 ] ] X cana-756 251 4 j.c.chang j.c.chang PROPN cana-756 251 5 , , PUNCT cana-756 251 6 β€œ " PUNCT cana-756 251 7 on on ADP cana-756 251 8 the the DET cana-756 251 9 identity identity NOUN cana-756 251 10 h(x h(x PROPN cana-756 251 11 ) ) PUNCT cana-756 251 12 = = PUNCT cana-756 251 13 af(x af(x PRON cana-756 251 14 ) ) PUNCT cana-756 252 1 + + CCONJ cana-756 252 2 g(x)b g(x)b PROPN cana-756 252 3 ” " PUNCT cana-756 252 4 , , PUNCT cana-756 252 5 taiwanese taiwanese ADJ cana-756 252 6 j. j. PROPN cana-756 252 7 math.7(1)(2003),103 math.7(1)(2003),103 PROPN cana-756 252 8 - - PROPN cana-756 252 9 113 113 NUM cana-756 252 10 . . PUNCT cana-756 253 1 [ [ X cana-756 253 2 12 12 NUM cana-756 253 3 ] ] PUNCT cana-756 253 4 k. k. PROPN cana-756 253 5 kaya kaya PROPN cana-756 253 6 , , PUNCT cana-756 253 7 e. e. PROPN cana-756 253 8 guven guven PROPN cana-756 253 9 , , PUNCT cana-756 253 10 and and CCONJ cana-756 253 11 m. m. NOUN cana-756 253 12 soyturk soyturk PROPN cana-756 253 13 , , PUNCT cana-756 253 14 β€œ " PUNCT cana-756 253 15 on on ADP cana-756 253 16 ( ( PUNCT cana-756 253 17 Οƒ Οƒ PROPN cana-756 253 18 , , PUNCT cana-756 253 19 Ο„ Ο„ PROPN cana-756 253 20 ) ) PUNCT cana-756 253 21 derivations derivation NOUN cana-756 253 22 of of ADP cana-756 253 23 prime prime ADJ cana-756 253 24 rings ring NOUN cana-756 253 25 ” " PUNCT cana-756 253 26 , , PUNCT cana-756 253 27 the the DET cana-756 253 28 pure pure ADJ cana-756 253 29 and and CCONJ cana-756 253 30 applied apply VERB cana-756 253 31 mathematics,13(3 mathematics,13(3 PROPN cana-756 253 32 ) ) PUNCT cana-756 253 33 ( ( PUNCT cana-756 253 34 2006 2006 NUM cana-756 253 35 ) ) PUNCT cana-756 253 36 189–195 189–195 NUM cana-756 253 37 . . PUNCT cana-756 254 1 [ [ X cana-756 254 2 13 13 NUM cana-756 254 3 ] ] PUNCT cana-756 254 4 m. m. NOUN cana-756 254 5 ashraf ashraf PROPN cana-756 254 6 , , PUNCT cana-756 254 7 β€œ " PUNCT cana-756 254 8 on on ADP cana-756 254 9 ( ( PUNCT cana-756 254 10 Οƒ Οƒ NOUN cana-756 254 11 , , PUNCT cana-756 254 12 Ο„)-derivations Ο„)-derivation NOUN cana-756 254 13 in in ADP cana-756 254 14 prime prime ADJ cana-756 254 15 rings ring NOUN cana-756 254 16 , , PUNCT cana-756 254 17 ” " PUNCT cana-756 254 18 archivum archivum NOUN cana-756 254 19 mathematicum mathematicum NOUN cana-756 254 20 , , PUNCT cana-756 254 21 38(4 38(4 NUM cana-756 254 22 ) ) PUNCT cana-756 254 23 ( ( PUNCT cana-756 254 24 2002 2002 NUM cana-756 254 25 ) ) PUNCT cana-756 254 26 , , PUNCT cana-756 254 27 259–264 259–264 NUM cana-756 254 28 . . PUNCT cana-756 255 1 [ [ X cana-756 255 2 14 14 NUM cana-756 255 3 ] ] X cana-756 255 4 m. m. NOUN cana-756 255 5 bresar bresar VERB cana-756 255 6 and and CCONJ cana-756 255 7 j.vukman,β€œorthogonal j.vukman,β€œorthogonal ADJ cana-756 255 8 derivations derivation NOUN cana-756 255 9 and and CCONJ cana-756 255 10 an an DET cana-756 255 11 extension extension NOUN cana-756 255 12 of of ADP cana-756 255 13 a a DET cana-756 255 14 theorem theorem NOUN cana-756 255 15 of of ADP cana-756 255 16 posner posner NOUN cana-756 255 17 ” " PUNCT cana-756 255 18 , , PUNCT cana-756 255 19 radovi radovi PROPN cana-756 255 20 mathematicki mathematicki PROPN cana-756 255 21 , , PUNCT cana-756 255 22 ” " PUNCT cana-756 255 23 5 5 NUM cana-756 255 24 ( ( PUNCT cana-756 255 25 1991 1991 NUM cana-756 255 26 ) ) PUNCT cana-756 255 27 , , PUNCT cana-756 255 28 237 237 NUM cana-756 255 29 - - SYM cana-756 255 30 246 246 NUM cana-756 255 31 . . PUNCT cana-756 256 1 [ [ X cana-756 256 2 15 15 NUM cana-756 256 3 ] ] X cana-756 256 4 m. m. NOUN cana-756 256 5 n. n. PROPN cana-756 256 6 daif daif PROPN cana-756 256 7 , , PUNCT cana-756 256 8 m. m. NOUN cana-756 256 9 t. t. PROPN cana-756 256 10 el el PROPN cana-756 256 11 - - PUNCT cana-756 256 12 sayiad sayiad PROPN cana-756 256 13 , , PUNCT cana-756 256 14 and and CCONJ cana-756 256 15 c. c. PROPN cana-756 256 16 haetinger haetinger NOUN cana-756 256 17 , , PUNCT cana-756 256 18 β€œ " PUNCT cana-756 256 19 orthogonal orthogonal ADJ cana-756 256 20 derivations derivation NOUN cana-756 256 21 and and CCONJ cana-756 256 22 biderivations biderivation NOUN cana-756 256 23 ” " PUNCT cana-756 256 24 , , PUNCT cana-756 256 25 jmi jmi PROPN cana-756 256 26 international international PROPN cana-756 256 27 journal journal PROPN cana-756 256 28 of of ADP cana-756 256 29 mathematical mathematical ADJ cana-756 256 30 sciences science NOUN cana-756 256 31 , , PUNCT cana-756 256 32 1(1 1(1 NUM cana-756 256 33 ) ) PUNCT cana-756 256 34 ( ( PUNCT cana-756 256 35 2010 2010 NUM cana-756 256 36 ) ) PUNCT cana-756 256 37 , , PUNCT cana-756 256 38 23–34 23–34 NUM cana-756 256 39 . . PUNCT cana-756 257 1 [ [ X cana-756 257 2 16 16 NUM cana-756 257 3 ] ] PUNCT cana-756 257 4 m. m. NOUN cana-756 257 5 n. n. PROPN cana-756 257 6 daif daif PROPN cana-756 257 7 , , PUNCT cana-756 257 8 m. m. NOUN cana-756 257 9 s. s. PROPN cana-756 257 10 t. t. PROPN cana-756 257 11 el el PROPN cana-756 257 12 - - PUNCT cana-756 257 13 sayiad sayiad PROPN cana-756 257 14 , , PUNCT cana-756 257 15 and and CCONJ cana-756 257 16 c. c. PROPN cana-756 257 17 haetinger haetinger NOUN cana-756 257 18 , , PUNCT cana-756 257 19 β€œ " PUNCT cana-756 257 20 reverse reverse VERB cana-756 257 21 , , PUNCT cana-756 257 22 jordan jordan PROPN cana-756 257 23 and and CCONJ cana-756 257 24 left left ADJ cana-756 257 25 biderivations biderivation NOUN cana-756 257 26 ” " PUNCT cana-756 257 27 , , PUNCT cana-756 257 28 oriental oriental ADJ cana-756 257 29 journal journal NOUN cana-756 257 30 of of ADP cana-756 257 31 mathematics mathematic NOUN cana-756 257 32 , , PUNCT cana-756 257 33 2(2 2(2 NUM cana-756 257 34 ) ) PUNCT cana-756 257 35 ( ( PUNCT cana-756 257 36 2010 2010 NUM cana-756 257 37 ) ) PUNCT cana-756 257 38 65–81 65–81 NUM cana-756 257 39 . . PUNCT cana-756 258 1 [ [ X cana-756 258 2 17 17 NUM cana-756 258 3 ] ] X cana-756 258 4 n. n. NOUN cana-756 258 5 argaΓ§ argaΓ§ PROPN cana-756 258 6 , , PUNCT cana-756 258 7 a. a. PROPN cana-756 258 8 nakajima nakajima PROPN cana-756 258 9 , , PUNCT cana-756 258 10 and and CCONJ cana-756 258 11 e. e. PROPN cana-756 258 12 albaş albaş PROPN cana-756 258 13 , , PUNCT cana-756 258 14 β€œ " PUNCT cana-756 258 15 on on ADP cana-756 258 16 orthogonal orthogonal ADJ cana-756 258 17 generalized generalize VERB cana-756 258 18 derivations derivation NOUN cana-756 258 19 of of ADP cana-756 258 20 semiprime semiprime NOUN cana-756 258 21 rings ring NOUN cana-756 258 22 ” " PUNCT cana-756 258 23 , , PUNCT cana-756 258 24 turkish turkish ADJ cana-756 258 25 journal journal NOUN cana-756 258 26 of of ADP cana-756 258 27 mathematics mathematic NOUN cana-756 258 28 , , PUNCT cana-756 258 29 28(2 28(2 NUM cana-756 258 30 ) ) PUNCT cana-756 258 31 ( ( PUNCT cana-756 258 32 2004 2004 NUM cana-756 258 33 ) ) PUNCT cana-756 259 1 185–194 185–194 NUM cana-756 259 2 . . PUNCT cana-756 260 1 [ [ X cana-756 260 2 18 18 NUM cana-756 260 3 ] ] X cana-756 260 4 o. o. PROPN cana-756 260 5 golbaşi golbaşi PROPN cana-756 260 6 and and CCONJ cana-756 260 7 n. n. PROPN cana-756 260 8 aydin aydin NOUN cana-756 260 9 , , PUNCT cana-756 260 10 β€œ " PUNCT cana-756 260 11 orthogonal orthogonal ADJ cana-756 260 12 generalized generalize VERB cana-756 260 13 ( ( PUNCT cana-756 260 14 Οƒ Οƒ PROPN cana-756 260 15 , , PUNCT cana-756 260 16 Ο„ Ο„ NOUN cana-756 260 17 ) ) PUNCT cana-756 260 18 -derivations -derivation NOUN cana-756 260 19 of of ADP cana-756 260 20 semiprime semiprime NOUN cana-756 260 21 rings ring NOUN cana-756 260 22 ” " PUNCT cana-756 260 23 , , PUNCT cana-756 260 24 siberian siberian ADJ cana-756 260 25 mathematical mathematical ADJ cana-756 260 26 journal journal NOUN cana-756 260 27 , , PUNCT cana-756 260 28 48(6 48(6 NOUN cana-756 260 29 ) ) PUNCT cana-756 260 30 ( ( PUNCT cana-756 260 31 2007 2007 NUM cana-756 260 32 ) ) PUNCT cana-756 260 33 , , PUNCT cana-756 260 34 979 979 NUM cana-756 260 35 - - SYM cana-756 260 36 983 983 NUM cana-756 260 37 . . PUNCT