id sid tid token lemma pos cana-1962 1 1 communications communication NOUN cana-1962 1 2 on on ADP cana-1962 1 3 applied apply VERB cana-1962 1 4 nonlinear nonlinear ADJ cana-1962 1 5 analysis analysis NOUN cana-1962 1 6 issn issn NOUN cana-1962 1 7 : : PUNCT cana-1962 1 8 1074 1074 NUM cana-1962 1 9 - - PUNCT cana-1962 1 10 133x 133x NUM cana-1962 1 11 vol vol NOUN cana-1962 1 12 32 32 NUM cana-1962 1 13 no no NOUN cana-1962 1 14 . . NOUN cana-1962 1 15 3 3 NUM cana-1962 1 16 ( ( PUNCT cana-1962 1 17 2025 2025 NUM cana-1962 1 18 ) ) PUNCT cana-1962 1 19 315 315 NUM cana-1962 1 20 https://internationalpubls.com https://internationalpubls.com X cana-1962 1 21 algebraic algebraic ADJ cana-1962 1 22 identities identity NOUN cana-1962 1 23 on on ADP cana-1962 1 24 generalized generalized ADJ cana-1962 1 25 derivations derivation NOUN cana-1962 1 26 in in ADP cana-1962 1 27 prime prime ADJ cana-1962 1 28 rings ring NOUN cana-1962 1 29 with with ADP cana-1962 1 30 involution involution NOUN cana-1962 1 31 m. m. NOUN cana-1962 1 32 el el PROPN cana-1962 1 33 hamdaoui1 hamdaoui1 PROPN cana-1962 1 34 , , PUNCT cana-1962 1 35 nawal nawal PROPN cana-1962 1 36 m. m. PROPN cana-1962 1 37 noureldeen2,3 noureldeen2,3 PROPN cana-1962 1 38 and and CCONJ cana-1962 1 39 ahmed ahme VERB cana-1962 1 40 aboubakr2,4 aboubakr2,4 PROPN cana-1962 1 41 , , PUNCT cana-1962 1 42 * * PROPN cana-1962 1 43 1department 1department NUM cana-1962 1 44 of of ADP cana-1962 1 45 mathematics mathematic NOUN cana-1962 1 46 , , PUNCT cana-1962 1 47 polydisciplinary polydisciplinary ADJ cana-1962 1 48 faculty faculty NOUN cana-1962 1 49 , , PUNCT cana-1962 1 50 lsi lsi PROPN cana-1962 1 51 , , PUNCT cana-1962 1 52 taza taza PROPN cana-1962 1 53 sidi sidi PROPN cana-1962 1 54 mohammed mohammed PROPN cana-1962 1 55 ben ben PROPN cana-1962 1 56 abdellah abdellah PROPN cana-1962 1 57 university university PROPN cana-1962 1 58 , , PUNCT cana-1962 1 59 morocco morocco PROPN cana-1962 1 60 . . PUNCT cana-1962 1 61 mathsup2011@gmail.com mathsup2011@gmail.com X cana-1962 2 1 2department 2department NUM cana-1962 2 2 of of ADP cana-1962 2 3 mathematics mathematic NOUN cana-1962 2 4 , , PUNCT cana-1962 2 5 taibah taibah PROPN cana-1962 2 6 university university PROPN cana-1962 2 7 , , PUNCT cana-1962 2 8 medina medina PROPN cana-1962 2 9 , , PUNCT cana-1962 2 10 saudi saudi PROPN cana-1962 2 11 arabia arabia PROPN cana-1962 2 12 . . PUNCT cana-1962 3 1 3department 3department NUM cana-1962 3 2 of of ADP cana-1962 3 3 mathematics mathematic NOUN cana-1962 3 4 , , PUNCT cana-1962 3 5 women woman NOUN cana-1962 3 6 ’s ’s PART cana-1962 3 7 college college PROPN cana-1962 3 8 of of ADP cana-1962 3 9 arts art NOUN cana-1962 3 10 , , PUNCT cana-1962 3 11 sciences science NOUN cana-1962 3 12 and and CCONJ cana-1962 3 13 education education NOUN cana-1962 3 14 , , PUNCT cana-1962 3 15 ain ain PROPN cana-1962 3 16 shams shams PROPN cana-1962 3 17 university university PROPN cana-1962 3 18 , , PUNCT cana-1962 3 19 egypt egypt PROPN cana-1962 3 20 . . PUNCT cana-1962 4 1 neldeen@taibahu.edu.sa neldeen@taibahu.edu.sa PROPN cana-1962 4 2 4department 4department NUM cana-1962 4 3 of of ADP cana-1962 4 4 mathematics mathematic NOUN cana-1962 4 5 , , PUNCT cana-1962 4 6 fayoum fayoum PROPN cana-1962 4 7 university university PROPN cana-1962 4 8 , , PUNCT cana-1962 4 9 63514 63514 NUM cana-1962 4 10 fayoum fayoum PROPN cana-1962 4 11 , , PUNCT cana-1962 4 12 egypt egypt PROPN cana-1962 4 13 , , PUNCT cana-1962 4 14 aaboubakr aaboubakr PROPN cana-1962 4 15 @taibahu.edu.sa @taibahu.edu.sa PROPN cana-1962 4 16 & & CCONJ cana-1962 4 17 afs00@fayoum.edu.eg afs00@fayoum.edu.eg PROPN cana-1962 4 18 * * PUNCT cana-1962 4 19 corresponding correspond VERB cana-1962 4 20 author author NOUN cana-1962 4 21 . . PUNCT cana-1962 5 1 article article NOUN cana-1962 5 2 history history NOUN cana-1962 5 3 : : PUNCT cana-1962 5 4 received receive VERB cana-1962 5 5 : : PUNCT cana-1962 5 6 01 01 NUM cana-1962 5 7 - - SYM cana-1962 5 8 08 08 NUM cana-1962 5 9 - - PUNCT cana-1962 5 10 2024 2024 NUM cana-1962 5 11 revised revise VERB cana-1962 5 12 : : PUNCT cana-1962 5 13 21 21 NUM cana-1962 5 14 - - SYM cana-1962 5 15 09 09 NUM cana-1962 5 16 - - PUNCT cana-1962 5 17 2024 2024 NUM cana-1962 5 18 accepted accept VERB cana-1962 5 19 : : PUNCT cana-1962 5 20 02 02 NUM cana-1962 5 21 - - PUNCT cana-1962 5 22 10 10 NUM cana-1962 5 23 - - PUNCT cana-1962 5 24 2024 2024 NUM cana-1962 5 25 abstract abstract NOUN cana-1962 5 26 : : PUNCT cana-1962 5 27 this this DET cana-1962 5 28 research research NOUN cana-1962 5 29 paper paper NOUN cana-1962 5 30 aims aim VERB cana-1962 5 31 to to PART cana-1962 5 32 investigate investigate VERB cana-1962 5 33 the the DET cana-1962 5 34 commutativity commutativity NOUN cana-1962 5 35 properties property NOUN cana-1962 5 36 of of ADP cana-1962 5 37 prime prime ADJ cana-1962 5 38 rings ring NOUN cana-1962 5 39 in in ADP cana-1962 5 40 relation relation NOUN cana-1962 5 41 to to ADP cana-1962 5 42 generalized generalize VERB cana-1962 5 43 derivations derivation NOUN cana-1962 5 44 and and CCONJ cana-1962 5 45 a a DET cana-1962 5 46 left left ADJ cana-1962 5 47 multiplier multiplier ADV cana-1962 5 48 that that PRON cana-1962 5 49 fulfil fulfil VERB cana-1962 5 50 specific specific ADJ cana-1962 5 51 algebraic algebraic ADJ cana-1962 5 52 conditions condition NOUN cana-1962 5 53 involving involve VERB cana-1962 5 54 involution involution NOUN cana-1962 5 55 . . PUNCT cana-1962 6 1 additionally additionally ADV cana-1962 6 2 , , PUNCT cana-1962 6 3 we we PRON cana-1962 6 4 present present VERB cana-1962 6 5 various various ADJ cana-1962 6 6 examples example NOUN cana-1962 6 7 studies study NOUN cana-1962 6 8 to to PART cana-1962 6 9 illustrate illustrate VERB cana-1962 6 10 that that SCONJ cana-1962 6 11 the the DET cana-1962 6 12 constraints constraint NOUN cana-1962 6 13 imposed impose VERB cana-1962 6 14 in in ADP cana-1962 6 15 our our PRON cana-1962 6 16 theorems theorem NOUN cana-1962 6 17 are be AUX cana-1962 6 18 indeed indeed ADV cana-1962 6 19 necessary necessary ADJ cana-1962 6 20 and and CCONJ cana-1962 6 21 can can AUX cana-1962 6 22 not not PART cana-1962 6 23 be be AUX cana-1962 6 24 omitted omit VERB cana-1962 6 25 without without ADP cana-1962 6 26 compromising compromise VERB cana-1962 6 27 the the DET cana-1962 6 28 validity validity NOUN cana-1962 6 29 of of ADP cana-1962 6 30 our our PRON cana-1962 6 31 results result NOUN cana-1962 6 32 . . PUNCT cana-1962 7 1 introduction introduction NOUN cana-1962 7 2 : : PUNCT cana-1962 7 3 consider consider VERB cana-1962 7 4 a a DET cana-1962 7 5 ring ring NOUN cana-1962 7 6 𝔖 𝔖 NOUN cana-1962 7 7 that that PRON cana-1962 7 8 satisfies satisfy VERB cana-1962 7 9 the the DET cana-1962 7 10 associative associative ADJ cana-1962 7 11 property property NOUN cana-1962 7 12 , , PUNCT cana-1962 7 13 and and CCONJ cana-1962 7 14 let let VERB cana-1962 7 15 𝑍(𝔖 𝑍(𝔖 PRON cana-1962 7 16 ) ) PUNCT cana-1962 7 17 represent represent VERB cana-1962 7 18 its its PRON cana-1962 7 19 center center NOUN cana-1962 7 20 . . PUNCT cana-1962 8 1 an an DET cana-1962 8 2 involution involution NOUN cana-1962 8 3 denoted denote VERB cana-1962 8 4 by by ADP cana-1962 8 5 βˆ— βˆ— NOUN cana-1962 8 6 , , PUNCT cana-1962 8 7 which which PRON cana-1962 8 8 is be AUX cana-1962 8 9 an an DET cana-1962 8 10 additive additive ADJ cana-1962 8 11 function function NOUN cana-1962 8 12 mapping mapping NOUN cana-1962 8 13 𝔖 𝔖 PROPN cana-1962 8 14 to to ADP cana-1962 8 15 itself itself PRON cana-1962 8 16 . . PUNCT cana-1962 9 1 this this DET cana-1962 9 2 involution involution NOUN cana-1962 9 3 has have VERB cana-1962 9 4 specific specific ADJ cana-1962 9 5 properties property NOUN cana-1962 9 6 : : PUNCT cana-1962 9 7 for for ADP cana-1962 9 8 any any DET cana-1962 9 9 𝛼 𝛼 NOUN cana-1962 9 10 and and CCONJ cana-1962 9 11 𝛽 𝛽 NOUN cana-1962 9 12 in in ADP cana-1962 9 13 𝔖 𝔖 PROPN cana-1962 9 14 , , PUNCT cana-1962 9 15 applying apply VERB cana-1962 9 16 the the DET cana-1962 9 17 involution involution NOUN cana-1962 9 18 twice twice ADV cana-1962 9 19 returns return VERB cana-1962 9 20 the the DET cana-1962 9 21 original original ADJ cana-1962 9 22 element element NOUN cana-1962 9 23 ( ( PUNCT cana-1962 9 24 ( ( PUNCT cana-1962 9 25 π›Όβˆ—)βˆ— π›Όβˆ—)βˆ— X cana-1962 9 26 = = SYM cana-1962 9 27 𝛼 𝛼 X cana-1962 9 28 ) ) PUNCT cana-1962 9 29 , , PUNCT cana-1962 9 30 it it PRON cana-1962 9 31 distributes distribute VERB cana-1962 9 32 over over ADP cana-1962 9 33 addition addition NOUN cana-1962 9 34 ( ( PUNCT cana-1962 9 35 ( ( PUNCT cana-1962 9 36 𝛼 𝛼 X cana-1962 9 37 + + X cana-1962 9 38 𝛽)βˆ— 𝛽)βˆ— PROPN cana-1962 9 39 = = SYM cana-1962 9 40 π›Όβˆ— π›Όβˆ— NOUN cana-1962 10 1 + + CCONJ cana-1962 10 2 π›½βˆ— π›½βˆ— NOUN cana-1962 10 3 ) ) PUNCT cana-1962 10 4 , , PUNCT cana-1962 10 5 and and CCONJ cana-1962 10 6 it it PRON cana-1962 10 7 reverses reverse VERB cana-1962 10 8 the the DET cana-1962 10 9 order order NOUN cana-1962 10 10 of of ADP cana-1962 10 11 multiplication multiplication NOUN cana-1962 10 12 ( ( PUNCT cana-1962 10 13 ( ( PUNCT cana-1962 10 14 𝛼𝛽)βˆ— 𝛼𝛽)βˆ— PROPN cana-1962 10 15 = = SYM cana-1962 10 16 π›½βˆ—π›Όβˆ— π›½βˆ—π›Όβˆ— X cana-1962 10 17 ) ) PUNCT cana-1962 10 18 . . PUNCT cana-1962 11 1 we we PRON cana-1962 11 2 categorize categorize VERB cana-1962 11 3 elements element NOUN cana-1962 11 4 as as ADP cana-1962 11 5 hermitian hermitian ADJ cana-1962 11 6 when when SCONJ cana-1962 11 7 they they PRON cana-1962 11 8 remain remain VERB cana-1962 11 9 unchanged unchanged ADJ cana-1962 11 10 under under ADP cana-1962 11 11 the the DET cana-1962 11 12 involution involution NOUN cana-1962 11 13 ( ( PUNCT cana-1962 11 14 π›Όβˆ— π›Όβˆ— NOUN cana-1962 11 15 = = SYM cana-1962 11 16 𝛼 𝛼 NOUN cana-1962 11 17 ) ) PUNCT cana-1962 11 18 , , PUNCT cana-1962 11 19 and and CCONJ cana-1962 11 20 as as ADP cana-1962 11 21 skew skew NOUN cana-1962 11 22 - - PUNCT cana-1962 11 23 hermitian hermitian ADJ cana-1962 11 24 when when SCONJ cana-1962 11 25 they they PRON cana-1962 11 26 change change VERB cana-1962 11 27 sign sign NOUN cana-1962 11 28 ( ( PUNCT cana-1962 11 29 π›Όβˆ— π›Όβˆ— NOUN cana-1962 11 30 = = SYM cana-1962 11 31 βˆ’π›Ό βˆ’π›Ό PROPN cana-1962 11 32 ) ) PUNCT cana-1962 11 33 . . PUNCT cana-1962 12 1 we we PRON cana-1962 12 2 use use VERB cana-1962 12 3 β„‹(𝔖 β„‹(𝔖 NOUN cana-1962 12 4 ) ) PUNCT cana-1962 12 5 to to PART cana-1962 12 6 represent represent VERB cana-1962 12 7 the the DET cana-1962 12 8 collection collection NOUN cana-1962 12 9 of of ADP cana-1962 12 10 all all DET cana-1962 12 11 hermitian hermitian ADJ cana-1962 12 12 elements element NOUN cana-1962 12 13 in in ADP cana-1962 12 14 𝔖 𝔖 PROPN cana-1962 12 15 , , PUNCT cana-1962 12 16 and and CCONJ cana-1962 12 17 𝑆(𝔖 𝑆(𝔖 PROPN cana-1962 12 18 ) ) PUNCT cana-1962 12 19 for for ADP cana-1962 12 20 all all DET cana-1962 12 21 skew skew ADJ cana-1962 12 22 - - PUNCT cana-1962 12 23 hermitian hermitian ADJ cana-1962 12 24 elements element NOUN cana-1962 12 25 . . PUNCT cana-1962 13 1 the the DET cana-1962 13 2 involution involution NOUN cana-1962 13 3 is be AUX cana-1962 13 4 classified classify VERB cana-1962 13 5 as as ADP cana-1962 13 6 first first ADJ cana-1962 13 7 kind kind NOUN cana-1962 13 8 if if SCONJ cana-1962 13 9 β„‹(𝔖 β„‹(𝔖 NUM cana-1962 13 10 ) ) PUNCT cana-1962 13 11 is be AUX cana-1962 13 12 a a DET cana-1962 13 13 subset subset NOUN cana-1962 13 14 of of ADP cana-1962 13 15 the the DET cana-1962 13 16 center center NOUN cana-1962 13 17 of of ADP cana-1962 13 18 𝔖 𝔖 PROPN cana-1962 13 19 , , PUNCT cana-1962 13 20 denoted denote VERB cana-1962 13 21 as as ADP cana-1962 13 22 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 13 23 ) ) PUNCT cana-1962 13 24 . . PUNCT cana-1962 14 1 if if SCONJ cana-1962 14 2 this this PRON cana-1962 14 3 is be AUX cana-1962 14 4 not not PART cana-1962 14 5 the the DET cana-1962 14 6 case case NOUN cana-1962 14 7 , , PUNCT cana-1962 14 8 it it PRON cana-1962 14 9 ’s ’s AUX cana-1962 14 10 considered consider VERB cana-1962 14 11 second second ADJ cana-1962 14 12 kind kind NOUN cana-1962 14 13 , , PUNCT cana-1962 14 14 and and CCONJ cana-1962 14 15 in in ADP cana-1962 14 16 this this DET cana-1962 14 17 scenario scenario NOUN cana-1962 14 18 , , PUNCT cana-1962 14 19 the the DET cana-1962 14 20 intersection intersection NOUN cana-1962 14 21 of of ADP cana-1962 14 22 β„‹(𝔖 β„‹(𝔖 NOUN cana-1962 14 23 ) ) PUNCT cana-1962 14 24 and and CCONJ cana-1962 14 25 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 14 26 ) ) PUNCT cana-1962 14 27 contains contain VERB cana-1962 14 28 more more ADJ cana-1962 14 29 than than ADP cana-1962 14 30 just just ADV cana-1962 14 31 the the DET cana-1962 14 32 zero zero NUM cana-1962 14 33 element element NOUN cana-1962 14 34 . . PUNCT cana-1962 15 1 we we PRON cana-1962 15 2 also also ADV cana-1962 15 3 define define VERB cana-1962 15 4 several several ADJ cana-1962 15 5 types type NOUN cana-1962 15 6 of of ADP cana-1962 15 7 mappings mapping NOUN cana-1962 15 8 on on ADP cana-1962 15 9 ring ring NOUN cana-1962 15 10 𝔖. 𝔖. PROPN cana-1962 15 11 a a DET cana-1962 15 12 left left ADJ cana-1962 15 13 multiplier multiplier ADV cana-1962 15 14 , , PUNCT cana-1962 15 15 β–³ β–³ PROPN cana-1962 15 16 , , PUNCT cana-1962 15 17 is be AUX cana-1962 15 18 an an DET cana-1962 15 19 additive additive ADJ cana-1962 15 20 map map NOUN cana-1962 15 21 where where SCONJ cana-1962 15 22 β–³ β–³ X cana-1962 15 23 ( ( PUNCT cana-1962 15 24 πœπœ” πœπœ” NOUN cana-1962 15 25 ) ) PUNCT cana-1962 15 26 = = NOUN cana-1962 15 27 β–³ β–³ X cana-1962 15 28 ( ( PUNCT cana-1962 15 29 𝜐)πœ” 𝜐)πœ” ADJ cana-1962 15 30 , , PUNCT cana-1962 15 31 βˆ€πœ βˆ€πœ NOUN cana-1962 15 32 , , PUNCT cana-1962 15 33 πœ” πœ” PART cana-1962 15 34 ∈ ∈ NOUN cana-1962 15 35 𝔖. 𝔖. PROPN cana-1962 15 36 a a DET cana-1962 15 37 derivation derivation NOUN cana-1962 15 38 , , PUNCT cana-1962 15 39 πœ“ πœ“ PROPN cana-1962 15 40 , , PUNCT cana-1962 15 41 is be AUX cana-1962 15 42 an an DET cana-1962 15 43 additive additive ADJ cana-1962 15 44 mapping mapping NOUN cana-1962 15 45 that that PRON cana-1962 15 46 satisfies satisfy VERB cana-1962 15 47 πœ“(πœπœ” πœ“(πœπœ” NOUN cana-1962 15 48 ) ) PUNCT cana-1962 15 49 = = SYM cana-1962 16 1 πœ“(𝜐)πœ” πœ“(𝜐)πœ” X cana-1962 16 2 + + CCONJ cana-1962 16 3 πœπœ“(πœ” πœπœ“(πœ” NOUN cana-1962 16 4 ) ) PUNCT cana-1962 16 5 , , PUNCT cana-1962 16 6 βˆ€πœ βˆ€πœ NOUN cana-1962 16 7 , , PUNCT cana-1962 16 8 πœ” πœ” PART cana-1962 16 9 ∈ ∈ NOUN cana-1962 16 10 𝔖. 𝔖. NOUN cana-1962 16 11 extending extend VERB cana-1962 16 12 this this DET cana-1962 16 13 concept concept NOUN cana-1962 16 14 , , PUNCT cana-1962 16 15 we we PRON cana-1962 16 16 define define VERB cana-1962 16 17 a a DET cana-1962 16 18 generalized generalized ADJ cana-1962 16 19 derivation derivation NOUN cana-1962 16 20 , , PUNCT cana-1962 16 21 Ξ³ Ξ³ PROPN cana-1962 16 22 , , PUNCT cana-1962 16 23 which which PRON cana-1962 16 24 is be AUX cana-1962 16 25 linked link VERB cana-1962 16 26 to to ADP cana-1962 16 27 a a DET cana-1962 16 28 derivation derivation NOUN cana-1962 16 29 πœ“. πœ“. NOUN cana-1962 16 30 this this DET cana-1962 16 31 function function NOUN cana-1962 16 32 satisfies satisfy VERB cana-1962 16 33 Ξ³(πœπœ” Ξ³(πœπœ” NOUN cana-1962 16 34 ) ) PUNCT cana-1962 16 35 = = SYM cana-1962 16 36 Ξ³(𝜐)πœ” Ξ³(𝜐)πœ” PROPN cana-1962 16 37 + + CCONJ cana-1962 16 38 πœπœ“(πœ” πœπœ“(πœ” NOUN cana-1962 16 39 ) ) PUNCT cana-1962 16 40 , , PUNCT cana-1962 16 41 βˆ€πœ βˆ€πœ NOUN cana-1962 16 42 , , PUNCT cana-1962 16 43 πœ” πœ” PART cana-1962 16 44 ∈ ∈ NOUN cana-1962 16 45 𝔖. 𝔖. NOUN cana-1962 16 46 it it PRON cana-1962 16 47 ’s ’ VERB cana-1962 16 48 worth worth ADJ cana-1962 16 49 noting note VERB cana-1962 16 50 that that SCONJ cana-1962 16 51 any any DET cana-1962 16 52 derivation derivation NOUN cana-1962 16 53 can can AUX cana-1962 16 54 be be AUX cana-1962 16 55 considered consider VERB cana-1962 16 56 a a DET cana-1962 16 57 generalized generalized ADJ cana-1962 16 58 derivation derivation NOUN cana-1962 16 59 associated associate VERB cana-1962 16 60 with with ADP cana-1962 16 61 itself itself PRON cana-1962 16 62 . . PUNCT cana-1962 17 1 objectives objective NOUN cana-1962 17 2 : : PUNCT cana-1962 17 3 in in ADP cana-1962 17 4 this this DET cana-1962 17 5 study study NOUN cana-1962 17 6 , , PUNCT cana-1962 17 7 we we PRON cana-1962 17 8 intend intend VERB cana-1962 17 9 to to PART cana-1962 17 10 examine examine VERB cana-1962 17 11 the the DET cana-1962 17 12 commutativity commutativity NOUN cana-1962 17 13 of of ADP cana-1962 17 14 a a DET cana-1962 17 15 prime prime ADJ cana-1962 17 16 ring ring NOUN cana-1962 17 17 𝔖 𝔖 NOUN cana-1962 17 18 by by ADP cana-1962 17 19 utilizing utilize VERB cana-1962 17 20 generalized generalized ADJ cana-1962 17 21 derivations derivation NOUN cana-1962 17 22 Ξ³1 Ξ³1 NOUN cana-1962 17 23 , , PUNCT cana-1962 17 24 Ξ³2 Ξ³2 PROPN cana-1962 17 25 , , PUNCT cana-1962 17 26 and and CCONJ cana-1962 17 27 a a DET cana-1962 17 28 left leave VERB cana-1962 17 29 multiplier multipli ADJ cana-1962 17 30 β–³ β–³ NOUN cana-1962 17 31 , , PUNCT cana-1962 17 32 while while SCONJ cana-1962 17 33 adhering adhere VERB cana-1962 17 34 to to ADP cana-1962 17 35 specific specific ADJ cana-1962 17 36 algebraic algebraic ADJ cana-1962 17 37 identities identity NOUN cana-1962 17 38 that that PRON cana-1962 17 39 involve involve VERB cana-1962 17 40 involution involution NOUN cana-1962 17 41 . . PUNCT cana-1962 18 1 specifically specifically ADV cana-1962 18 2 , , PUNCT cana-1962 18 3 we we PRON cana-1962 18 4 will will AUX cana-1962 18 5 delve delve VERB cana-1962 18 6 into into ADP cana-1962 18 7 the the DET cana-1962 18 8 commutativity commutativity NOUN cana-1962 18 9 of of ADP cana-1962 18 10 rings ring NOUN cana-1962 18 11 𝔖 𝔖 PROPN cana-1962 18 12 that that PRON cana-1962 18 13 fulfill fulfill VERB cana-1962 18 14 the the DET cana-1962 18 15 following follow VERB cana-1962 18 16 algebraic algebraic ADJ cana-1962 18 17 conditions condition NOUN cana-1962 18 18 : : PUNCT cana-1962 19 1 β€’ β€’ ADP cana-1962 19 2 [ [ X cana-1962 19 3 Ξ³1(𝜐 Ξ³1(𝜐 X cana-1962 19 4 ) ) PUNCT cana-1962 19 5 , , PUNCT cana-1962 19 6 Ξ³2(πœβˆ— Ξ³2(πœβˆ— PROPN cana-1962 19 7 ) ) PUNCT cana-1962 19 8 ] ] PUNCT cana-1962 20 1 + + X cana-1962 20 2 β–³ β–³ X cana-1962 20 3 ( ( PUNCT cana-1962 20 4 [ [ X cana-1962 20 5 𝜐 𝜐 X cana-1962 20 6 , , PUNCT cana-1962 20 7 πœβˆ— πœβˆ— PROPN cana-1962 20 8 ] ] PUNCT cana-1962 20 9 ) ) PUNCT cana-1962 20 10 ∈ ∈ PROPN cana-1962 20 11 𝑍(𝔖 𝑍(𝔖 PROPN cana-1962 20 12 ) ) PUNCT cana-1962 20 13 , , PUNCT cana-1962 20 14 for for ADP cana-1962 20 15 all all PRON cana-1962 20 16 𝜐 𝜐 PROPN cana-1962 20 17 ∈ ∈ PROPN cana-1962 20 18 𝔖 𝔖 PROPN cana-1962 20 19 ; ; PUNCT cana-1962 20 20 β€’ β€’ PRON cana-1962 20 21 Ξ³1(𝜐 Ξ³1(𝜐 PROPN cana-1962 20 22 ) ) PUNCT cana-1962 20 23 ∘ ∘ NOUN cana-1962 20 24 Ξ³2(πœβˆ— Ξ³2(πœβˆ— PUNCT cana-1962 20 25 ) ) PUNCT cana-1962 21 1 + + NOUN cana-1962 21 2 β–³ β–³ X cana-1962 21 3 ( ( PUNCT cana-1962 21 4 𝜐 𝜐 PROPN cana-1962 21 5 ∘ ∘ PROPN cana-1962 21 6 πœβˆ— πœβˆ— PROPN cana-1962 21 7 ) ) PUNCT cana-1962 21 8 ∈ ∈ PROPN cana-1962 21 9 𝑍(𝔖 𝑍(𝔖 PROPN cana-1962 21 10 ) ) PUNCT cana-1962 21 11 , , PUNCT cana-1962 21 12 for for ADP cana-1962 21 13 all all PRON cana-1962 21 14 𝜐 𝜐 PROPN cana-1962 21 15 ∈ ∈ PROPN cana-1962 21 16 𝔖 𝔖 PROPN cana-1962 21 17 ; ; PUNCT cana-1962 21 18 β€’ β€’ ADP cana-1962 22 1 [ [ X cana-1962 22 2 Ξ³1(𝜐 Ξ³1(𝜐 X cana-1962 22 3 ) ) PUNCT cana-1962 22 4 , , PUNCT cana-1962 22 5 Ξ³2(πœβˆ— Ξ³2(πœβˆ— PROPN cana-1962 22 6 ) ) PUNCT cana-1962 22 7 ] ] PUNCT cana-1962 23 1 + + X cana-1962 23 2 β–³ β–³ X cana-1962 23 3 ( ( PUNCT cana-1962 23 4 𝜐 𝜐 PROPN cana-1962 23 5 ∘ ∘ PROPN cana-1962 23 6 πœβˆ— πœβˆ— PROPN cana-1962 23 7 ) ) PUNCT cana-1962 23 8 ∈ ∈ PROPN cana-1962 23 9 𝑍(𝔖 𝑍(𝔖 PROPN cana-1962 23 10 ) ) PUNCT cana-1962 23 11 , , PUNCT cana-1962 23 12 for for ADP cana-1962 23 13 all all PRON cana-1962 23 14 𝜐 𝜐 PROPN cana-1962 23 15 ∈ ∈ PROPN cana-1962 23 16 𝔖 𝔖 PROPN cana-1962 23 17 ; ; PUNCT cana-1962 23 18 β€’ β€’ PRON cana-1962 23 19 Ξ³1(𝜐 Ξ³1(𝜐 PROPN cana-1962 23 20 ) ) PUNCT cana-1962 23 21 ∘ ∘ NOUN cana-1962 23 22 Ξ³2(πœβˆ— Ξ³2(πœβˆ— PUNCT cana-1962 23 23 ) ) PUNCT cana-1962 24 1 + + NOUN cana-1962 24 2 β–³ β–³ X cana-1962 24 3 ( ( PUNCT cana-1962 24 4 [ [ X cana-1962 24 5 𝜐 𝜐 X cana-1962 24 6 , , PUNCT cana-1962 24 7 πœβˆ— πœβˆ— PROPN cana-1962 24 8 ] ] PUNCT cana-1962 24 9 ) ) PUNCT cana-1962 24 10 ∈ ∈ PROPN cana-1962 24 11 𝑍(𝔖 𝑍(𝔖 PROPN cana-1962 24 12 ) ) PUNCT cana-1962 24 13 , , PUNCT cana-1962 24 14 for for ADP cana-1962 24 15 all all PRON cana-1962 24 16 𝜐 𝜐 PROPN cana-1962 24 17 ∈ ∈ PROPN cana-1962 24 18 𝔖 𝔖 NOUN cana-1962 24 19 ; ; PUNCT cana-1962 24 20 β€’ β€’ NUM cana-1962 24 21 Ξ³(πœπœβˆ— Ξ³(πœπœβˆ— NOUN cana-1962 24 22 ) ) PUNCT cana-1962 24 23 Β± Β± NUM cana-1962 24 24 Ξ³(𝜐)Ξ³(πœβˆ— Ξ³(𝜐)Ξ³(πœβˆ— NUM cana-1962 24 25 ) ) PUNCT cana-1962 25 1 + + NOUN cana-1962 25 2 β–³ β–³ X cana-1962 25 3 ( ( PUNCT cana-1962 25 4 [ [ X cana-1962 25 5 𝜐 𝜐 X cana-1962 25 6 , , PUNCT cana-1962 25 7 πœβˆ— πœβˆ— PROPN cana-1962 25 8 ] ] PUNCT cana-1962 25 9 ) ) PUNCT cana-1962 25 10 ∈ ∈ PROPN cana-1962 25 11 𝑍(𝔖 𝑍(𝔖 PROPN cana-1962 25 12 ) ) PUNCT cana-1962 25 13 , , PUNCT cana-1962 25 14 for for ADP cana-1962 25 15 all all PRON cana-1962 25 16 𝜐 𝜐 PROPN cana-1962 25 17 ∈ ∈ PROPN cana-1962 25 18 𝔖 𝔖 PROPN cana-1962 25 19 ; ; PUNCT cana-1962 25 20 mailto:afs00@fayoum.edu.eg mailto:afs00@fayoum.edu.eg NOUN cana-1962 25 21 communications communication NOUN cana-1962 25 22 on on ADP cana-1962 25 23 applied apply VERB cana-1962 25 24 nonlinear nonlinear ADJ cana-1962 25 25 analysis analysis NOUN cana-1962 25 26 issn issn NOUN cana-1962 25 27 : : PUNCT cana-1962 25 28 1074 1074 NUM cana-1962 25 29 - - PUNCT cana-1962 25 30 133x 133x NUM cana-1962 25 31 vol vol NOUN cana-1962 25 32 32 32 NUM cana-1962 25 33 no no NOUN cana-1962 25 34 . . NOUN cana-1962 25 35 3 3 NUM cana-1962 25 36 ( ( PUNCT cana-1962 25 37 2025 2025 NUM cana-1962 25 38 ) ) PUNCT cana-1962 26 1 316 316 NUM cana-1962 26 2 https://internationalpubls.com https://internationalpubls.com SYM cana-1962 26 3 β€’ β€’ NUM cana-1962 26 4 Ξ³(πœπœβˆ— Ξ³(πœπœβˆ— NOUN cana-1962 26 5 ) ) PUNCT cana-1962 26 6 Β± Β± NOUN cana-1962 26 7 Ξ³(πœβˆ—)Ξ³(𝜐 Ξ³(πœβˆ—)Ξ³(𝜐 NOUN cana-1962 26 8 ) ) PUNCT cana-1962 27 1 + + NOUN cana-1962 27 2 β–³ β–³ X cana-1962 27 3 ( ( PUNCT cana-1962 27 4 𝜐 𝜐 PROPN cana-1962 27 5 ∘ ∘ PROPN cana-1962 27 6 πœβˆ— πœβˆ— PROPN cana-1962 27 7 ) ) PUNCT cana-1962 27 8 ∈ ∈ PROPN cana-1962 27 9 𝑍(𝔖 𝑍(𝔖 PROPN cana-1962 27 10 ) ) PUNCT cana-1962 27 11 , , PUNCT cana-1962 27 12 for for ADP cana-1962 27 13 all all PRON cana-1962 27 14 𝜐 𝜐 PROPN cana-1962 27 15 ∈ ∈ NOUN cana-1962 27 16 𝔖. 𝔖. PROPN cana-1962 27 17 lastly lastly ADV cana-1962 27 18 , , PUNCT cana-1962 27 19 we we PRON cana-1962 27 20 offer offer VERB cana-1962 27 21 examples example NOUN cana-1962 27 22 to to PART cana-1962 27 23 illustrate illustrate VERB cana-1962 27 24 that that SCONJ cana-1962 27 25 the the DET cana-1962 27 26 constraints constraint NOUN cana-1962 27 27 applied apply VERB cana-1962 27 28 to to ADP cana-1962 27 29 our our PRON cana-1962 27 30 hypotheses hypothesis NOUN cana-1962 27 31 are be AUX cana-1962 27 32 necessary necessary ADJ cana-1962 27 33 and and CCONJ cana-1962 27 34 not not PART cana-1962 27 35 redundant redundant ADJ cana-1962 27 36 . . PUNCT cana-1962 28 1 results result NOUN cana-1962 28 2 : : PUNCT cana-1962 28 3 building build VERB cana-1962 28 4 on on ADP cana-1962 28 5 the the DET cana-1962 28 6 work work NOUN cana-1962 28 7 of of ADP cana-1962 28 8 nejjar nejjar NOUN cana-1962 28 9 ( ( PUNCT cana-1962 28 10 [ [ X cana-1962 28 11 8 8 NUM cana-1962 28 12 ] ] PUNCT cana-1962 28 13 , , PUNCT cana-1962 28 14 theorems theorem NOUN cana-1962 28 15 3.5 3.5 NUM cana-1962 28 16 , , PUNCT cana-1962 28 17 3.8 3.8 NUM cana-1962 28 18 ) ) PUNCT cana-1962 28 19 , , PUNCT cana-1962 28 20 who who PRON cana-1962 28 21 demonstrated demonstrate VERB cana-1962 28 22 that that SCONJ cana-1962 28 23 a a DET cana-1962 28 24 2 2 NUM cana-1962 28 25 - - PUNCT cana-1962 28 26 torsion torsion NOUN cana-1962 28 27 - - PUNCT cana-1962 28 28 free free ADJ cana-1962 28 29 prime prime ADJ cana-1962 28 30 ring ring NOUN cana-1962 28 31 with with ADP cana-1962 28 32 involution involution NOUN cana-1962 28 33 and and CCONJ cana-1962 28 34 a a DET cana-1962 28 35 derivation derivation NOUN cana-1962 28 36 πœ“ πœ“ ADP cana-1962 28 37 satisfying satisfy VERB cana-1962 28 38 certain certain ADJ cana-1962 28 39 conditions condition NOUN cana-1962 28 40 must must AUX cana-1962 28 41 be be AUX cana-1962 28 42 commutative commutative ADJ cana-1962 28 43 , , PUNCT cana-1962 28 44 we we PRON cana-1962 28 45 explore explore VERB cana-1962 28 46 broader broad ADJ cana-1962 28 47 generalizations generalization NOUN cana-1962 28 48 of of ADP cana-1962 28 49 these these DET cana-1962 28 50 conditions condition NOUN cana-1962 28 51 . . PUNCT cana-1962 29 1 specifically specifically ADV cana-1962 29 2 , , PUNCT cana-1962 29 3 nejjar nejjar PROPN cana-1962 29 4 showed show VERB cana-1962 29 5 that that SCONJ cana-1962 29 6 if if SCONJ cana-1962 29 7 the the DET cana-1962 29 8 derivation derivation NOUN cana-1962 29 9 πœ“ πœ“ PROPN cana-1962 29 10 meets meet VERB cana-1962 29 11 either either PRON cana-1962 29 12 of of ADP cana-1962 29 13 the the DET cana-1962 29 14 following follow VERB cana-1962 29 15 criteria criterion NOUN cana-1962 29 16 : : PUNCT cana-1962 30 1 [ [ X cana-1962 30 2 πœ“(𝜐 πœ“(𝜐 X cana-1962 30 3 ) ) PUNCT cana-1962 30 4 , , PUNCT cana-1962 30 5 πœ“(𝜐 πœ“(𝜐 X cana-1962 30 6 ) ) PUNCT cana-1962 30 7 ] ] PUNCT cana-1962 31 1 Β± Β± NOUN cana-1962 32 1 𝜐 𝜐 INTJ cana-1962 32 2 ∘ ∘ NOUN cana-1962 32 3 𝜐 𝜐 X cana-1962 32 4 ∈ ∈ PROPN cana-1962 32 5 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 32 6 ) ) PUNCT cana-1962 32 7 , , PUNCT cana-1962 32 8 βˆ€πœ βˆ€πœ NOUN cana-1962 32 9 ∈ ∈ PROPN cana-1962 32 10 𝔖 𝔖 PROPN cana-1962 32 11 or or CCONJ cana-1962 32 12 πœ“(𝜐 πœ“(𝜐 NOUN cana-1962 32 13 ) ) PUNCT cana-1962 32 14 ∘ ∘ X cana-1962 32 15 πœ“(𝜐 πœ“(𝜐 NOUN cana-1962 32 16 ) ) PUNCT cana-1962 32 17 Β± Β± NOUN cana-1962 33 1 𝜐 𝜐 NOUN cana-1962 33 2 ∘ ∘ NOUN cana-1962 33 3 𝜐 𝜐 X cana-1962 33 4 ∈ ∈ PROPN cana-1962 33 5 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 33 6 ) ) PUNCT cana-1962 33 7 , , PUNCT cana-1962 33 8 βˆ€πœ βˆ€πœ X cana-1962 33 9 ∈ ∈ PROPN cana-1962 33 10 𝔖 𝔖 PROPN cana-1962 33 11 , , PUNCT cana-1962 33 12 then then ADV cana-1962 33 13 𝔖 𝔖 PROPN cana-1962 33 14 is be AUX cana-1962 33 15 necessarily necessarily ADV cana-1962 33 16 commutative commutative ADJ cana-1962 33 17 . . PUNCT cana-1962 34 1 our our PRON cana-1962 34 2 work work NOUN cana-1962 34 3 extends extend VERB cana-1962 34 4 these these DET cana-1962 34 5 findings finding NOUN cana-1962 34 6 by by ADP cana-1962 34 7 introducing introduce VERB cana-1962 34 8 new new ADJ cana-1962 34 9 identities identity NOUN cana-1962 34 10 for for ADP cana-1962 34 11 pairs pair NOUN cana-1962 34 12 of of ADP cana-1962 34 13 generalized generalized ADJ cana-1962 34 14 derivations derivation NOUN cana-1962 34 15 that that PRON cana-1962 34 16 are be AUX cana-1962 34 17 connected connect VERB cana-1962 34 18 to to ADP cana-1962 34 19 a a DET cana-1962 34 20 left left ADJ cana-1962 34 21 multiplier multipli ADJ cana-1962 34 22 β–³ β–³ PROPN cana-1962 34 23 . . PUNCT cana-1962 35 1 finally finally ADV cana-1962 35 2 , , PUNCT cana-1962 35 3 we we PRON cana-1962 35 4 offer offer VERB cana-1962 35 5 examples example NOUN cana-1962 35 6 to to PART cana-1962 35 7 illustrate illustrate VERB cana-1962 35 8 that that SCONJ cana-1962 35 9 the the DET cana-1962 35 10 constraints constraint NOUN cana-1962 35 11 applied apply VERB cana-1962 35 12 to to ADP cana-1962 35 13 our our PRON cana-1962 35 14 hypotheses hypothesis NOUN cana-1962 35 15 are be AUX cana-1962 35 16 necessary necessary ADJ cana-1962 35 17 and and CCONJ cana-1962 35 18 not not PART cana-1962 35 19 redundant redundant ADJ cana-1962 35 20 . . PUNCT cana-1962 36 1 conclusions conclusion NOUN cana-1962 36 2 : : PUNCT cana-1962 36 3 in in ADP cana-1962 36 4 this this DET cana-1962 36 5 research research NOUN cana-1962 36 6 , , PUNCT cana-1962 36 7 we we PRON cana-1962 36 8 investigate investigate VERB cana-1962 36 9 the the DET cana-1962 36 10 commutativity commutativity NOUN cana-1962 36 11 of of ADP cana-1962 36 12 prime prime ADJ cana-1962 36 13 rings ring NOUN cana-1962 36 14 𝔖 𝔖 PROPN cana-1962 36 15 admitting admit VERB cana-1962 36 16 an an DET cana-1962 36 17 involution involution NOUN cana-1962 36 18 and and CCONJ cana-1962 36 19 generalized generalized ADJ cana-1962 36 20 derivations derivation NOUN cana-1962 36 21 satisfying satisfy VERB cana-1962 36 22 some some DET cana-1962 36 23 algebraic algebraic ADJ cana-1962 36 24 identities identity NOUN cana-1962 36 25 . . PUNCT cana-1962 37 1 we we PRON cana-1962 37 2 can can AUX cana-1962 37 3 conclude conclude VERB cana-1962 37 4 our our PRON cana-1962 37 5 paper paper NOUN cana-1962 37 6 with with ADP cana-1962 37 7 an an DET cana-1962 37 8 open open ADJ cana-1962 37 9 question question NOUN cana-1962 37 10 . . PUNCT cana-1962 38 1 open open ADJ cana-1962 38 2 question question NOUN cana-1962 38 3 : : PUNCT cana-1962 38 4 are be AUX cana-1962 38 5 these these DET cana-1962 38 6 results result NOUN cana-1962 38 7 correct correct ADJ cana-1962 38 8 if if SCONJ cana-1962 38 9 we we PRON cana-1962 38 10 replace replace VERB cana-1962 38 11 the the DET cana-1962 38 12 generalized generalized ADJ cana-1962 38 13 derivation derivation NOUN cana-1962 38 14 by by ADP cana-1962 38 15 the the DET cana-1962 38 16 generalized generalized ADJ cana-1962 38 17 ( ( PUNCT cana-1962 38 18 𝛼 𝛼 NOUN cana-1962 38 19 , , PUNCT cana-1962 38 20 𝛽 𝛽 NOUN cana-1962 38 21 ) ) PUNCT cana-1962 38 22 βˆ’derivation βˆ’derivation NOUN cana-1962 38 23 , , PUNCT cana-1962 38 24 where where SCONJ cana-1962 38 25 𝛼 𝛼 X cana-1962 38 26 and and CCONJ cana-1962 38 27 𝛽 𝛽 PROPN cana-1962 38 28 are be AUX cana-1962 38 29 automorphisms automorphism NOUN cana-1962 38 30 of of ADP cana-1962 38 31 ring ring NOUN cana-1962 38 32 𝔖 𝔖 PROPN cana-1962 38 33 ? ? PUNCT cana-1962 39 1 keywords keyword NOUN cana-1962 39 2 : : PUNCT cana-1962 39 3 generalized generalized ADJ cana-1962 39 4 derivation derivation NOUN cana-1962 39 5 , , PUNCT cana-1962 39 6 prime prime ADJ cana-1962 39 7 ring ring NOUN cana-1962 39 8 with with ADP cana-1962 39 9 involution involution NOUN cana-1962 39 10 , , PUNCT cana-1962 39 11 integral integral ADJ cana-1962 39 12 domain domain NOUN cana-1962 39 13 . . PUNCT cana-1962 40 1 1 1 X cana-1962 40 2 . . X cana-1962 40 3 introduction introduction NOUN cana-1962 40 4 consider consider VERB cana-1962 40 5 a a DET cana-1962 40 6 ring ring NOUN cana-1962 40 7 𝔖 𝔖 NOUN cana-1962 40 8 that that PRON cana-1962 40 9 satisfies satisfy VERB cana-1962 40 10 the the DET cana-1962 40 11 associative associative ADJ cana-1962 40 12 property property NOUN cana-1962 40 13 , , PUNCT cana-1962 40 14 and and CCONJ cana-1962 40 15 let let VERB cana-1962 40 16 𝑍(𝔖 𝑍(𝔖 PRON cana-1962 40 17 ) ) PUNCT cana-1962 40 18 represent represent VERB cana-1962 40 19 its its PRON cana-1962 40 20 center center NOUN cana-1962 40 21 . . PUNCT cana-1962 41 1 we we PRON cana-1962 41 2 ’ll ’ll AUX cana-1962 41 3 use use VERB cana-1962 41 4 the the DET cana-1962 41 5 notation notation NOUN cana-1962 41 6 [ [ X cana-1962 41 7 𝛼 𝛼 X cana-1962 41 8 , , PUNCT cana-1962 41 9 𝛽 𝛽 NOUN cana-1962 41 10 ] ] PUNCT cana-1962 41 11 to to PART cana-1962 41 12 represent represent VERB cana-1962 41 13 the the DET cana-1962 41 14 commutator commutator NOUN cana-1962 41 15 𝛼𝛽 𝛼𝛽 NOUN cana-1962 41 16 βˆ’ βˆ’ PROPN cana-1962 41 17 𝛽𝛼 𝛽𝛼 PROPN cana-1962 41 18 , , PUNCT cana-1962 41 19 βˆ€π›Ό βˆ€π›Ό PROPN cana-1962 41 20 , , PUNCT cana-1962 41 21 𝛽 𝛽 PROPN cana-1962 41 22 ∈ ∈ PROPN cana-1962 41 23 𝔖. 𝔖. PROPN cana-1962 41 24 similarly similarly ADV cana-1962 41 25 , , PUNCT cana-1962 41 26 we we PRON cana-1962 41 27 ’ll ’ll AUX cana-1962 41 28 denote denote VERB cana-1962 41 29 the the DET cana-1962 41 30 anticommutator anticommutator NOUN cana-1962 41 31 𝛼 𝛼 PROPN cana-1962 41 32 ∘ ∘ PROPN cana-1962 41 33 𝛽 𝛽 PROPN cana-1962 41 34 , , PUNCT cana-1962 41 35 defined define VERB cana-1962 41 36 as as ADP cana-1962 41 37 𝛼𝛽 𝛼𝛽 NOUN cana-1962 41 38 + + CCONJ cana-1962 41 39 𝛽𝛼. 𝛽𝛼. PROPN cana-1962 41 40 a a DET cana-1962 41 41 ring ring NOUN cana-1962 41 42 𝔖 𝔖 PROPN cana-1962 41 43 is be AUX cana-1962 41 44 considered consider VERB cana-1962 41 45 prime prime ADJ cana-1962 41 46 if if SCONJ cana-1962 41 47 , , PUNCT cana-1962 41 48 for for ADP cana-1962 41 49 any any DET cana-1962 41 50 two two NUM cana-1962 41 51 elements element NOUN cana-1962 41 52 𝛼 𝛼 VERB cana-1962 41 53 , , PUNCT cana-1962 41 54 𝛽 𝛽 PROPN cana-1962 41 55 ∈ ∈ PROPN cana-1962 41 56 𝔖 𝔖 PROPN cana-1962 41 57 , , PUNCT cana-1962 41 58 the the DET cana-1962 41 59 condition condition NOUN cana-1962 41 60 𝛼𝔖𝛽 𝛼𝔖𝛽 NUM cana-1962 41 61 = = SYM cana-1962 41 62 0 0 NUM cana-1962 41 63 necessitates necessitate VERB cana-1962 41 64 that that SCONJ cana-1962 41 65 either either CCONJ cana-1962 41 66 𝛼 𝛼 X cana-1962 41 67 = = SYM cana-1962 41 68 0 0 NUM cana-1962 41 69 or or CCONJ cana-1962 41 70 𝛽 𝛽 NOUN cana-1962 41 71 = = NOUN cana-1962 41 72 0 0 PROPN cana-1962 41 73 . . PUNCT cana-1962 42 1 in in ADP cana-1962 42 2 the the DET cana-1962 42 3 context context NOUN cana-1962 42 4 of of ADP cana-1962 42 5 𝔖 𝔖 PROPN cana-1962 42 6 , , PUNCT cana-1962 42 7 an an DET cana-1962 42 8 involution involution NOUN cana-1962 42 9 denoted denote VERB cana-1962 42 10 by by ADP cana-1962 42 11 βˆ— βˆ— NOUN cana-1962 42 12 , , PUNCT cana-1962 42 13 which which PRON cana-1962 42 14 is be AUX cana-1962 42 15 an an DET cana-1962 42 16 additive additive ADJ cana-1962 42 17 function function NOUN cana-1962 42 18 mapping mapping NOUN cana-1962 42 19 𝔖 𝔖 PROPN cana-1962 42 20 to to ADP cana-1962 42 21 itself itself PRON cana-1962 42 22 . . PUNCT cana-1962 43 1 this this DET cana-1962 43 2 involution involution NOUN cana-1962 43 3 has have VERB cana-1962 43 4 specific specific ADJ cana-1962 43 5 properties property NOUN cana-1962 43 6 : : PUNCT cana-1962 43 7 for for ADP cana-1962 43 8 any any DET cana-1962 43 9 𝛼 𝛼 NOUN cana-1962 43 10 and and CCONJ cana-1962 43 11 𝛽 𝛽 NOUN cana-1962 43 12 in in ADP cana-1962 43 13 𝔖 𝔖 PROPN cana-1962 43 14 , , PUNCT cana-1962 43 15 applying apply VERB cana-1962 43 16 the the DET cana-1962 43 17 involution involution NOUN cana-1962 43 18 twice twice ADV cana-1962 43 19 returns return VERB cana-1962 43 20 the the DET cana-1962 43 21 original original ADJ cana-1962 43 22 element element NOUN cana-1962 43 23 ( ( PUNCT cana-1962 43 24 ( ( PUNCT cana-1962 43 25 π›Όβˆ—)βˆ— π›Όβˆ—)βˆ— X cana-1962 43 26 = = SYM cana-1962 43 27 𝛼 𝛼 X cana-1962 43 28 ) ) PUNCT cana-1962 43 29 , , PUNCT cana-1962 43 30 it it PRON cana-1962 43 31 distributes distribute VERB cana-1962 43 32 over over ADP cana-1962 43 33 addition addition NOUN cana-1962 43 34 ( ( PUNCT cana-1962 43 35 ( ( PUNCT cana-1962 43 36 𝛼 𝛼 X cana-1962 43 37 + + X cana-1962 43 38 𝛽)βˆ— 𝛽)βˆ— PROPN cana-1962 43 39 = = SYM cana-1962 43 40 π›Όβˆ— π›Όβˆ— NOUN cana-1962 44 1 + + CCONJ cana-1962 44 2 π›½βˆ— π›½βˆ— NOUN cana-1962 44 3 ) ) PUNCT cana-1962 44 4 , , PUNCT cana-1962 44 5 and and CCONJ cana-1962 44 6 it it PRON cana-1962 44 7 reverses reverse VERB cana-1962 44 8 the the DET cana-1962 44 9 order order NOUN cana-1962 44 10 of of ADP cana-1962 44 11 multiplication multiplication NOUN cana-1962 44 12 ( ( PUNCT cana-1962 44 13 ( ( PUNCT cana-1962 44 14 𝛼𝛽)βˆ— 𝛼𝛽)βˆ— PROPN cana-1962 44 15 = = SYM cana-1962 44 16 π›½βˆ—π›Όβˆ— π›½βˆ—π›Όβˆ— X cana-1962 44 17 ) ) PUNCT cana-1962 44 18 . . PUNCT cana-1962 45 1 we we PRON cana-1962 45 2 categorize categorize VERB cana-1962 45 3 elements element NOUN cana-1962 45 4 as as ADP cana-1962 45 5 hermitian hermitian ADJ cana-1962 45 6 when when SCONJ cana-1962 45 7 they they PRON cana-1962 45 8 remain remain VERB cana-1962 45 9 unchanged unchanged ADJ cana-1962 45 10 under under ADP cana-1962 45 11 the the DET cana-1962 45 12 involution involution NOUN cana-1962 45 13 ( ( PUNCT cana-1962 45 14 π›Όβˆ— π›Όβˆ— NOUN cana-1962 45 15 = = SYM cana-1962 45 16 𝛼 𝛼 NOUN cana-1962 45 17 ) ) PUNCT cana-1962 45 18 , , PUNCT cana-1962 45 19 and and CCONJ cana-1962 45 20 as as ADP cana-1962 45 21 skew skew NOUN cana-1962 45 22 - - PUNCT cana-1962 45 23 hermitian hermitian ADJ cana-1962 45 24 when when SCONJ cana-1962 45 25 they they PRON cana-1962 45 26 change change VERB cana-1962 45 27 sign sign NOUN cana-1962 45 28 ( ( PUNCT cana-1962 45 29 π›Όβˆ— π›Όβˆ— NOUN cana-1962 45 30 = = SYM cana-1962 45 31 βˆ’π›Ό βˆ’π›Ό PROPN cana-1962 45 32 ) ) PUNCT cana-1962 45 33 . . PUNCT cana-1962 46 1 we we PRON cana-1962 46 2 use use VERB cana-1962 46 3 β–³ β–³ PROPN cana-1962 46 4 ( ( PUNCT cana-1962 46 5 𝔖 𝔖 NOUN cana-1962 46 6 ) ) PUNCT cana-1962 46 7 to to PART cana-1962 46 8 represent represent VERB cana-1962 46 9 the the DET cana-1962 46 10 collection collection NOUN cana-1962 46 11 of of ADP cana-1962 46 12 all all DET cana-1962 46 13 hermitian hermitian ADJ cana-1962 46 14 elements element NOUN cana-1962 46 15 in in ADP cana-1962 46 16 𝔖 𝔖 PROPN cana-1962 46 17 , , PUNCT cana-1962 46 18 and and CCONJ cana-1962 46 19 𝑆(𝔖 𝑆(𝔖 PROPN cana-1962 46 20 ) ) PUNCT cana-1962 46 21 for for ADP cana-1962 46 22 all all DET cana-1962 46 23 skewhermitian skewhermitian ADJ cana-1962 46 24 elements element NOUN cana-1962 46 25 . . PUNCT cana-1962 47 1 the the DET cana-1962 47 2 involution involution NOUN cana-1962 47 3 is be AUX cana-1962 47 4 classified classify VERB cana-1962 47 5 as as ADP cana-1962 47 6 first first ADJ cana-1962 47 7 kind kind NOUN cana-1962 47 8 if if SCONJ cana-1962 47 9 β–³ β–³ X cana-1962 47 10 ( ( PUNCT cana-1962 47 11 𝔖 𝔖 NOUN cana-1962 47 12 ) ) PUNCT cana-1962 47 13 is be AUX cana-1962 47 14 a a DET cana-1962 47 15 subset subset NOUN cana-1962 47 16 of of ADP cana-1962 47 17 the the DET cana-1962 47 18 center center NOUN cana-1962 47 19 of of ADP cana-1962 47 20 𝔖 𝔖 PROPN cana-1962 47 21 , , PUNCT cana-1962 47 22 denoted denote VERB cana-1962 47 23 as as ADP cana-1962 47 24 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 47 25 ) ) PUNCT cana-1962 47 26 . . PUNCT cana-1962 48 1 if if SCONJ cana-1962 48 2 this this PRON cana-1962 48 3 is be AUX cana-1962 48 4 not not PART cana-1962 48 5 the the DET cana-1962 48 6 case case NOUN cana-1962 48 7 , , PUNCT cana-1962 48 8 it it PRON cana-1962 48 9 ’s ’s AUX cana-1962 48 10 considered consider VERB cana-1962 48 11 second second ADJ cana-1962 48 12 kind kind NOUN cana-1962 48 13 , , PUNCT cana-1962 48 14 and and CCONJ cana-1962 48 15 in in ADP cana-1962 48 16 this this DET cana-1962 48 17 scenario scenario NOUN cana-1962 48 18 , , PUNCT cana-1962 48 19 the the DET cana-1962 48 20 intersection intersection NOUN cana-1962 48 21 of of ADP cana-1962 48 22 β–³ β–³ X cana-1962 48 23 ( ( PUNCT cana-1962 48 24 𝔖 𝔖 NOUN cana-1962 48 25 ) ) PUNCT cana-1962 48 26 and and CCONJ cana-1962 48 27 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 48 28 ) ) PUNCT cana-1962 48 29 contains contain VERB cana-1962 48 30 more more ADJ cana-1962 48 31 than than ADP cana-1962 48 32 just just ADV cana-1962 48 33 the the DET cana-1962 48 34 zero zero NUM cana-1962 48 35 element element NOUN cana-1962 48 36 . . PUNCT cana-1962 49 1 we we PRON cana-1962 49 2 also also ADV cana-1962 49 3 define define VERB cana-1962 49 4 several several ADJ cana-1962 49 5 types type NOUN cana-1962 49 6 of of ADP cana-1962 49 7 mappings mapping NOUN cana-1962 49 8 on on ADP cana-1962 49 9 ring ring NOUN cana-1962 49 10 𝔖. 𝔖. PROPN cana-1962 49 11 a a DET cana-1962 49 12 left left ADJ cana-1962 49 13 multiplier multiplier ADV cana-1962 49 14 , , PUNCT cana-1962 49 15 β–³ β–³ PROPN cana-1962 49 16 , , PUNCT cana-1962 49 17 is be AUX cana-1962 49 18 an an DET cana-1962 49 19 additive additive ADJ cana-1962 49 20 map map NOUN cana-1962 49 21 where where SCONJ cana-1962 49 22 β–³ β–³ X cana-1962 49 23 ( ( PUNCT cana-1962 49 24 πœπœ” πœπœ” NOUN cana-1962 49 25 ) ) PUNCT cana-1962 49 26 = = NOUN cana-1962 49 27 β–³ β–³ X cana-1962 49 28 ( ( PUNCT cana-1962 49 29 𝜐)πœ” 𝜐)πœ” ADJ cana-1962 49 30 , , PUNCT cana-1962 49 31 βˆ€πœ βˆ€πœ NOUN cana-1962 49 32 , , PUNCT cana-1962 49 33 πœ” πœ” PART cana-1962 49 34 ∈ ∈ NOUN cana-1962 49 35 𝔖. 𝔖. PROPN cana-1962 49 36 a a DET cana-1962 49 37 derivation derivation NOUN cana-1962 49 38 , , PUNCT cana-1962 49 39 πœ“ πœ“ PROPN cana-1962 49 40 , , PUNCT cana-1962 49 41 is be AUX cana-1962 49 42 an an DET cana-1962 49 43 additive additive ADJ cana-1962 49 44 mapping mapping NOUN cana-1962 49 45 that that PRON cana-1962 49 46 satisfies satisfy VERB cana-1962 49 47 πœ“(πœπœ” πœ“(πœπœ” NOUN cana-1962 49 48 ) ) PUNCT cana-1962 49 49 = = SYM cana-1962 50 1 πœ“(𝜐)πœ” πœ“(𝜐)πœ” X cana-1962 50 2 + + CCONJ cana-1962 50 3 πœπœ“(πœ” πœπœ“(πœ” NOUN cana-1962 50 4 ) ) PUNCT cana-1962 50 5 , , PUNCT cana-1962 50 6 βˆ€πœ βˆ€πœ NOUN cana-1962 50 7 , , PUNCT cana-1962 50 8 πœ” πœ” PART cana-1962 50 9 ∈ ∈ NOUN cana-1962 50 10 𝔖. 𝔖. NOUN cana-1962 50 11 extending extend VERB cana-1962 50 12 this this DET cana-1962 50 13 concept concept NOUN cana-1962 50 14 , , PUNCT cana-1962 50 15 we we PRON cana-1962 50 16 define define VERB cana-1962 50 17 a a DET cana-1962 50 18 generalized generalized ADJ cana-1962 50 19 derivation derivation NOUN cana-1962 50 20 , , PUNCT cana-1962 50 21 Ξ³ Ξ³ PROPN cana-1962 50 22 , , PUNCT cana-1962 50 23 which which PRON cana-1962 50 24 is be AUX cana-1962 50 25 linked link VERB cana-1962 50 26 to to ADP cana-1962 50 27 a a DET cana-1962 50 28 derivation derivation NOUN cana-1962 50 29 πœ“. πœ“. NOUN cana-1962 50 30 this this DET cana-1962 50 31 function function NOUN cana-1962 50 32 satisfies satisfy VERB cana-1962 50 33 Ξ³(πœπœ” Ξ³(πœπœ” NOUN cana-1962 50 34 ) ) PUNCT cana-1962 50 35 = = SYM cana-1962 50 36 Ξ³(𝜐)πœ” Ξ³(𝜐)πœ” PROPN cana-1962 50 37 + + CCONJ cana-1962 50 38 πœπœ“(πœ” πœπœ“(πœ” NOUN cana-1962 50 39 ) ) PUNCT cana-1962 50 40 , , PUNCT cana-1962 50 41 βˆ€πœ βˆ€πœ NOUN cana-1962 50 42 , , PUNCT cana-1962 50 43 πœ” πœ” PART cana-1962 50 44 ∈ ∈ NOUN cana-1962 50 45 𝔖. 𝔖. NOUN cana-1962 50 46 it it PRON cana-1962 50 47 ’s ’ VERB cana-1962 50 48 worth worth ADJ cana-1962 50 49 noting note VERB cana-1962 50 50 that that SCONJ cana-1962 50 51 any any DET cana-1962 50 52 derivation derivation NOUN cana-1962 50 53 can can AUX cana-1962 50 54 be be AUX cana-1962 50 55 considered consider VERB cana-1962 50 56 a a DET cana-1962 50 57 generalized generalized ADJ cana-1962 50 58 derivation derivation NOUN cana-1962 50 59 associated associate VERB cana-1962 50 60 with with ADP cana-1962 50 61 itself itself PRON cana-1962 50 62 . . PUNCT cana-1962 51 1 in in ADP cana-1962 51 2 recent recent ADJ cana-1962 51 3 decades decade NOUN cana-1962 51 4 , , PUNCT cana-1962 51 5 numerous numerous ADJ cana-1962 51 6 mathematicians mathematician NOUN cana-1962 51 7 have have AUX cana-1962 51 8 explored explore VERB cana-1962 51 9 the the DET cana-1962 51 10 relationship relationship NOUN cana-1962 51 11 between between ADP cana-1962 51 12 the the DET cana-1962 51 13 commutativity commutativity NOUN cana-1962 51 14 of of ADP cana-1962 51 15 a a DET cana-1962 51 16 ring ring NOUN cana-1962 51 17 𝔖 𝔖 NOUN cana-1962 51 18 and and CCONJ cana-1962 51 19 certain certain ADJ cana-1962 51 20 types type NOUN cana-1962 51 21 of of ADP cana-1962 51 22 additive additive ADJ cana-1962 51 23 mappings mapping NOUN cana-1962 51 24 , , PUNCT cana-1962 51 25 such such ADJ cana-1962 51 26 as as ADP cana-1962 51 27 automorphisms automorphism NOUN cana-1962 51 28 and and CCONJ cana-1962 51 29 generalized generalized ADJ cana-1962 51 30 derivations derivation NOUN cana-1962 51 31 acting act VERB cana-1962 51 32 on on ADP cana-1962 51 33 subsets subset NOUN cana-1962 51 34 of of ADP cana-1962 51 35 rings ring NOUN cana-1962 51 36 . . PUNCT cana-1962 52 1 the the DET cana-1962 52 2 seminal seminal ADJ cana-1962 52 3 work work NOUN cana-1962 52 4 of of ADP cana-1962 52 5 posner posner NOUN cana-1962 52 6 established establish VERB cana-1962 52 7 the the DET cana-1962 52 8 most most ADV cana-1962 52 9 significant significant ADJ cana-1962 52 10 theorem theorem NOUN cana-1962 52 11 on on ADP cana-1962 52 12 commuting commute VERB cana-1962 52 13 and and CCONJ cana-1962 52 14 related related ADJ cana-1962 52 15 mappings mapping NOUN cana-1962 52 16 . . PUNCT cana-1962 53 1 posner posner NOUN cana-1962 53 2 demonstrated demonstrate VERB cana-1962 53 3 that that SCONJ cana-1962 53 4 a a DET cana-1962 53 5 prime prime ADJ cana-1962 53 6 ring ring NOUN cana-1962 53 7 𝔖 𝔖 PROPN cana-1962 53 8 is be AUX cana-1962 53 9 commutative commutative ADJ cana-1962 53 10 if if SCONJ cana-1962 53 11 it it PRON cana-1962 53 12 possesses possess VERB cana-1962 53 13 a a DET cana-1962 53 14 nonzero nonzero ADJ cana-1962 53 15 derivation derivation NOUN cana-1962 53 16 πœ“ πœ“ ADP cana-1962 53 17 such such ADJ cana-1962 53 18 that that SCONJ cana-1962 53 19 [ [ X cana-1962 53 20 πœ“(𝜐 πœ“(𝜐 X cana-1962 53 21 ) ) PUNCT cana-1962 53 22 , , PUNCT cana-1962 53 23 𝜐 𝜐 X cana-1962 53 24 ] ] X cana-1962 53 25 ∈ ∈ PROPN cana-1962 53 26 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 53 27 ) ) PUNCT cana-1962 53 28 , , PUNCT cana-1962 53 29 βˆ€πœ βˆ€πœ X cana-1962 53 30 ∈ ∈ PROPN cana-1962 53 31 𝔖. 𝔖. PROPN cana-1962 53 32 this this DET cana-1962 53 33 foundational foundational ADJ cana-1962 53 34 result result NOUN cana-1962 53 35 has have AUX cana-1962 53 36 been be AUX cana-1962 53 37 further far ADV cana-1962 53 38 refined refine VERB cana-1962 53 39 and and CCONJ cana-1962 53 40 extended extend VERB cana-1962 53 41 by by ADP cana-1962 53 42 various various ADJ cana-1962 53 43 researchers researcher NOUN cana-1962 53 44 , , PUNCT cana-1962 53 45 as as SCONJ cana-1962 53 46 documented document VERB cana-1962 53 47 in in ADP cana-1962 53 48 the the DET cana-1962 53 49 comprehensive comprehensive ADJ cana-1962 53 50 bibliography bibliography NOUN cana-1962 53 51 provided provide VERB cana-1962 53 52 in in ADP cana-1962 53 53 [ [ X cana-1962 53 54 2 2 NUM cana-1962 53 55 ] ] PUNCT cana-1962 53 56 , , PUNCT cana-1962 53 57 [ [ X cana-1962 53 58 3 3 NUM cana-1962 53 59 ] ] PUNCT cana-1962 53 60 , , PUNCT cana-1962 53 61 [ [ X cana-1962 53 62 13 13 NUM cana-1962 53 63 ] ] PUNCT cana-1962 53 64 , , PUNCT cana-1962 53 65 and and CCONJ cana-1962 53 66 [ [ X cana-1962 53 67 7 7 NUM cana-1962 53 68 ] ] PUNCT cana-1962 53 69 . . PUNCT cana-1962 54 1 more more ADV cana-1962 54 2 recently recently ADV cana-1962 54 3 , , PUNCT cana-1962 54 4 some some DET cana-1962 54 5 authors author NOUN cana-1962 54 6 have have VERB cana-1962 54 7 communications communication NOUN cana-1962 54 8 on on ADP cana-1962 54 9 applied apply VERB cana-1962 54 10 nonlinear nonlinear ADJ cana-1962 54 11 analysis analysis NOUN cana-1962 54 12 issn issn NOUN cana-1962 54 13 : : PUNCT cana-1962 54 14 1074 1074 NUM cana-1962 54 15 - - PUNCT cana-1962 54 16 133x 133x NUM cana-1962 54 17 vol vol NOUN cana-1962 54 18 32 32 NUM cana-1962 55 1 no no NOUN cana-1962 55 2 . . NOUN cana-1962 55 3 3 3 NUM cana-1962 55 4 ( ( PUNCT cana-1962 55 5 2025 2025 NUM cana-1962 55 6 ) ) PUNCT cana-1962 55 7 317 317 NUM cana-1962 55 8 https://internationalpubls.com https://internationalpubls.com X cana-1962 55 9 investigated investigate VERB cana-1962 55 10 these these DET cana-1962 55 11 concepts concept NOUN cana-1962 55 12 in in ADP cana-1962 55 13 the the DET cana-1962 55 14 context context NOUN cana-1962 55 15 of of ADP cana-1962 55 16 rings ring NOUN cana-1962 55 17 with with ADP cana-1962 55 18 involution involution NOUN cana-1962 55 19 . . PUNCT cana-1962 56 1 for for ADP cana-1962 56 2 instance instance NOUN cana-1962 56 3 in in ADP cana-1962 56 4 [ [ X cana-1962 56 5 1 1 NUM cana-1962 56 6 ] ] PUNCT cana-1962 56 7 , , PUNCT cana-1962 56 8 ali ali PROPN cana-1962 56 9 and and CCONJ cana-1962 56 10 dar dar PROPN cana-1962 56 11 obtained obtain VERB cana-1962 56 12 a a DET cana-1962 56 13 βˆ—-version βˆ—-version NOUN cana-1962 56 14 of of ADP cana-1962 56 15 posner posner NOUN cana-1962 56 16 ’s ’s PART cana-1962 56 17 second second ADJ cana-1962 56 18 theorem theorem NOUN cana-1962 56 19 . . PUNCT cana-1962 57 1 specifically specifically ADV cana-1962 57 2 , , PUNCT cana-1962 57 3 they they PRON cana-1962 57 4 showed show VERB cana-1962 57 5 that that SCONJ cana-1962 57 6 a a DET cana-1962 57 7 ring ring NOUN cana-1962 57 8 𝔖 𝔖 PROPN cana-1962 57 9 is be AUX cana-1962 57 10 commutative commutative ADJ cana-1962 57 11 if if SCONJ cana-1962 57 12 it it PRON cana-1962 57 13 possesses possess VERB cana-1962 57 14 a a DET cana-1962 57 15 derivation derivation NOUN cana-1962 57 16 πœ“ πœ“ NOUN cana-1962 57 17 with with ADP cana-1962 57 18 an an DET cana-1962 57 19 involution involution NOUN cana-1962 57 20 βˆ— βˆ— NOUN cana-1962 57 21 , , PUNCT cana-1962 57 22 where where SCONJ cana-1962 57 23 π‘β„Žπ‘Žπ‘Ÿ(𝔖 π‘β„Žπ‘Žπ‘Ÿ(𝔖 NOUN cana-1962 57 24 ) ) PUNCT cana-1962 57 25 β‰  β‰  PROPN cana-1962 57 26 2 2 NUM cana-1962 57 27 , , PUNCT cana-1962 57 28 πœ“(𝑆(𝔖 πœ“(𝑆(𝔖 NOUN cana-1962 57 29 ) ) PUNCT cana-1962 57 30 ∩ ∩ PROPN cana-1962 57 31 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 57 32 ) ) PUNCT cana-1962 57 33 ) ) PUNCT cana-1962 58 1 β‰  β‰  PROPN cana-1962 58 2 0 0 NUM cana-1962 58 3 , , PUNCT cana-1962 58 4 and and CCONJ cana-1962 58 5 [ [ X cana-1962 58 6 πœ“(𝜐 πœ“(𝜐 X cana-1962 58 7 ) ) PUNCT cana-1962 58 8 , , PUNCT cana-1962 58 9 πœβˆ— πœβˆ— PROPN cana-1962 58 10 ] ] X cana-1962 58 11 ∈ ∈ PROPN cana-1962 58 12 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 58 13 ) ) PUNCT cana-1962 58 14 , , PUNCT cana-1962 58 15 βˆ€πœ βˆ€πœ X cana-1962 58 16 ∈ ∈ PROPN cana-1962 58 17 𝔖. 𝔖. PROPN cana-1962 58 18 in in ADP cana-1962 58 19 this this DET cana-1962 58 20 study study NOUN cana-1962 58 21 , , PUNCT cana-1962 58 22 we we PRON cana-1962 58 23 intend intend VERB cana-1962 58 24 to to PART cana-1962 58 25 examine examine VERB cana-1962 58 26 the the DET cana-1962 58 27 commutativity commutativity NOUN cana-1962 58 28 of of ADP cana-1962 58 29 a a DET cana-1962 58 30 prime prime ADJ cana-1962 58 31 ring ring NOUN cana-1962 58 32 𝔖 𝔖 NOUN cana-1962 58 33 by by ADP cana-1962 58 34 utilizing utilize VERB cana-1962 58 35 generalized generalized ADJ cana-1962 58 36 derivations derivation NOUN cana-1962 58 37 Ξ³1 Ξ³1 NOUN cana-1962 58 38 , , PUNCT cana-1962 58 39 Ξ³2 Ξ³2 PROPN cana-1962 58 40 , , PUNCT cana-1962 58 41 and and CCONJ cana-1962 58 42 a a DET cana-1962 58 43 left leave VERB cana-1962 58 44 multiplier multipli ADJ cana-1962 58 45 β–³ β–³ NOUN cana-1962 58 46 , , PUNCT cana-1962 58 47 while while SCONJ cana-1962 58 48 adhering adhere VERB cana-1962 58 49 to to ADP cana-1962 58 50 specific specific ADJ cana-1962 58 51 algebraic algebraic ADJ cana-1962 58 52 identities identity NOUN cana-1962 58 53 that that PRON cana-1962 58 54 involve involve VERB cana-1962 58 55 involution involution NOUN cana-1962 58 56 . . PUNCT cana-1962 59 1 specifically specifically ADV cana-1962 59 2 , , PUNCT cana-1962 59 3 we we PRON cana-1962 59 4 will will AUX cana-1962 59 5 delve delve VERB cana-1962 59 6 into into ADP cana-1962 59 7 the the DET cana-1962 59 8 commutativity commutativity NOUN cana-1962 59 9 of of ADP cana-1962 59 10 rings ring NOUN cana-1962 59 11 𝔖 𝔖 PROPN cana-1962 59 12 that that PRON cana-1962 59 13 fulfill fulfill VERB cana-1962 59 14 the the DET cana-1962 59 15 following follow VERB cana-1962 59 16 algebraic algebraic ADJ cana-1962 59 17 conditions condition NOUN cana-1962 59 18 : : PUNCT cana-1962 60 1 β€’ β€’ ADP cana-1962 60 2 [ [ X cana-1962 60 3 Ξ³1(𝜐 Ξ³1(𝜐 X cana-1962 60 4 ) ) PUNCT cana-1962 60 5 , , PUNCT cana-1962 60 6 Ξ³2(πœβˆ— Ξ³2(πœβˆ— PROPN cana-1962 60 7 ) ) PUNCT cana-1962 60 8 ] ] PUNCT cana-1962 61 1 + + X cana-1962 61 2 β–³ β–³ X cana-1962 61 3 ( ( PUNCT cana-1962 61 4 [ [ X cana-1962 61 5 𝜐 𝜐 X cana-1962 61 6 , , PUNCT cana-1962 61 7 πœβˆ— πœβˆ— PROPN cana-1962 61 8 ] ] PUNCT cana-1962 61 9 ) ) PUNCT cana-1962 61 10 ∈ ∈ PROPN cana-1962 61 11 𝑍(𝔖 𝑍(𝔖 PROPN cana-1962 61 12 ) ) PUNCT cana-1962 61 13 , , PUNCT cana-1962 61 14 for for ADP cana-1962 61 15 all all PRON cana-1962 61 16 𝜐 𝜐 PROPN cana-1962 61 17 ∈ ∈ PROPN cana-1962 61 18 𝔖 𝔖 PROPN cana-1962 61 19 ; ; PUNCT cana-1962 61 20 β€’ β€’ PRON cana-1962 61 21 Ξ³1(𝜐 Ξ³1(𝜐 PROPN cana-1962 61 22 ) ) PUNCT cana-1962 61 23 ∘ ∘ NOUN cana-1962 61 24 Ξ³2(πœβˆ— Ξ³2(πœβˆ— PUNCT cana-1962 61 25 ) ) PUNCT cana-1962 62 1 + + NOUN cana-1962 62 2 β–³ β–³ X cana-1962 62 3 ( ( PUNCT cana-1962 62 4 𝜐 𝜐 PROPN cana-1962 62 5 ∘ ∘ PROPN cana-1962 62 6 πœβˆ— πœβˆ— PROPN cana-1962 62 7 ) ) PUNCT cana-1962 62 8 ∈ ∈ PROPN cana-1962 62 9 𝑍(𝔖 𝑍(𝔖 PROPN cana-1962 62 10 ) ) PUNCT cana-1962 62 11 , , PUNCT cana-1962 62 12 for for ADP cana-1962 62 13 all all PRON cana-1962 62 14 𝜐 𝜐 PROPN cana-1962 62 15 ∈ ∈ PROPN cana-1962 62 16 𝔖 𝔖 PROPN cana-1962 62 17 ; ; PUNCT cana-1962 62 18 β€’ β€’ ADP cana-1962 63 1 [ [ X cana-1962 63 2 Ξ³1(𝜐 Ξ³1(𝜐 X cana-1962 63 3 ) ) PUNCT cana-1962 63 4 , , PUNCT cana-1962 63 5 Ξ³2(πœβˆ— Ξ³2(πœβˆ— PROPN cana-1962 63 6 ) ) PUNCT cana-1962 63 7 ] ] PUNCT cana-1962 64 1 + + X cana-1962 64 2 β–³ β–³ X cana-1962 64 3 ( ( PUNCT cana-1962 64 4 𝜐 𝜐 PROPN cana-1962 64 5 ∘ ∘ PROPN cana-1962 64 6 πœβˆ— πœβˆ— PROPN cana-1962 64 7 ) ) PUNCT cana-1962 64 8 ∈ ∈ PROPN cana-1962 64 9 𝑍(𝔖 𝑍(𝔖 PROPN cana-1962 64 10 ) ) PUNCT cana-1962 64 11 , , PUNCT cana-1962 64 12 for for ADP cana-1962 64 13 all all PRON cana-1962 64 14 𝜐 𝜐 PROPN cana-1962 64 15 ∈ ∈ PROPN cana-1962 64 16 𝔖 𝔖 PROPN cana-1962 64 17 ; ; PUNCT cana-1962 64 18 β€’ β€’ PRON cana-1962 64 19 Ξ³1(𝜐 Ξ³1(𝜐 PROPN cana-1962 64 20 ) ) PUNCT cana-1962 64 21 ∘ ∘ NOUN cana-1962 64 22 Ξ³2(πœβˆ— Ξ³2(πœβˆ— PUNCT cana-1962 64 23 ) ) PUNCT cana-1962 65 1 + + NOUN cana-1962 65 2 β–³ β–³ X cana-1962 65 3 ( ( PUNCT cana-1962 65 4 [ [ X cana-1962 65 5 𝜐 𝜐 X cana-1962 65 6 , , PUNCT cana-1962 65 7 πœβˆ— πœβˆ— PROPN cana-1962 65 8 ] ] PUNCT cana-1962 65 9 ) ) PUNCT cana-1962 65 10 ∈ ∈ PROPN cana-1962 65 11 𝑍(𝔖 𝑍(𝔖 PROPN cana-1962 65 12 ) ) PUNCT cana-1962 65 13 , , PUNCT cana-1962 65 14 for for ADP cana-1962 65 15 all all PRON cana-1962 65 16 𝜐 𝜐 PROPN cana-1962 65 17 ∈ ∈ PROPN cana-1962 65 18 𝔖 𝔖 NOUN cana-1962 65 19 ; ; PUNCT cana-1962 65 20 β€’ β€’ NUM cana-1962 65 21 Ξ³(πœπœβˆ— Ξ³(πœπœβˆ— NOUN cana-1962 65 22 ) ) PUNCT cana-1962 65 23 Β± Β± NUM cana-1962 65 24 Ξ³(𝜐)Ξ³(πœβˆ— Ξ³(𝜐)Ξ³(πœβˆ— NUM cana-1962 65 25 ) ) PUNCT cana-1962 66 1 + + NOUN cana-1962 66 2 β–³ β–³ X cana-1962 66 3 ( ( PUNCT cana-1962 66 4 [ [ X cana-1962 66 5 𝜐 𝜐 X cana-1962 66 6 , , PUNCT cana-1962 66 7 πœβˆ— πœβˆ— PROPN cana-1962 66 8 ] ] PUNCT cana-1962 66 9 ) ) PUNCT cana-1962 66 10 ∈ ∈ PROPN cana-1962 66 11 𝑍(𝔖 𝑍(𝔖 PROPN cana-1962 66 12 ) ) PUNCT cana-1962 66 13 , , PUNCT cana-1962 66 14 for for ADP cana-1962 66 15 all all PRON cana-1962 66 16 𝜐 𝜐 PROPN cana-1962 66 17 ∈ ∈ PROPN cana-1962 66 18 𝔖 𝔖 NOUN cana-1962 66 19 ; ; PUNCT cana-1962 66 20 β€’ β€’ NUM cana-1962 66 21 Ξ³(πœπœβˆ— Ξ³(πœπœβˆ— NOUN cana-1962 66 22 ) ) PUNCT cana-1962 66 23 Β± Β± NOUN cana-1962 66 24 Ξ³(πœβˆ—)Ξ³(𝜐 Ξ³(πœβˆ—)Ξ³(𝜐 NOUN cana-1962 66 25 ) ) PUNCT cana-1962 67 1 + + NOUN cana-1962 67 2 β–³ β–³ X cana-1962 67 3 ( ( PUNCT cana-1962 67 4 𝜐 𝜐 PROPN cana-1962 67 5 ∘ ∘ PROPN cana-1962 67 6 πœβˆ— πœβˆ— PROPN cana-1962 67 7 ) ) PUNCT cana-1962 67 8 ∈ ∈ PROPN cana-1962 67 9 𝑍(𝔖 𝑍(𝔖 PROPN cana-1962 67 10 ) ) PUNCT cana-1962 67 11 , , PUNCT cana-1962 67 12 for for ADP cana-1962 67 13 all all PRON cana-1962 67 14 𝜐 𝜐 PROPN cana-1962 67 15 ∈ ∈ NOUN cana-1962 67 16 𝔖. 𝔖. PROPN cana-1962 67 17 lastly lastly ADV cana-1962 67 18 , , PUNCT cana-1962 67 19 we we PRON cana-1962 67 20 offer offer VERB cana-1962 67 21 examples example NOUN cana-1962 67 22 to to PART cana-1962 67 23 illustrate illustrate VERB cana-1962 67 24 that that SCONJ cana-1962 67 25 the the DET cana-1962 67 26 constraints constraint NOUN cana-1962 67 27 applied apply VERB cana-1962 67 28 to to ADP cana-1962 67 29 our our PRON cana-1962 67 30 hypotheses hypothesis NOUN cana-1962 67 31 are be AUX cana-1962 67 32 necessary necessary ADJ cana-1962 67 33 and and CCONJ cana-1962 67 34 not not PART cana-1962 67 35 redundant redundant ADJ cana-1962 67 36 . . PUNCT cana-1962 68 1 2 2 X cana-1962 68 2 . . X cana-1962 68 3 preliminaries preliminary NOUN cana-1962 68 4 in in ADP cana-1962 68 5 this this DET cana-1962 68 6 section section NOUN cana-1962 68 7 , , PUNCT cana-1962 68 8 we we PRON cana-1962 68 9 commence commence VERB cana-1962 68 10 our our PRON cana-1962 68 11 analysis analysis NOUN cana-1962 68 12 by by ADP cana-1962 68 13 introducing introduce VERB cana-1962 68 14 several several ADJ cana-1962 68 15 well well ADV cana-1962 68 16 - - PUNCT cana-1962 68 17 established establish VERB cana-1962 68 18 results result NOUN cana-1962 68 19 that that PRON cana-1962 68 20 will will AUX cana-1962 68 21 be be AUX cana-1962 68 22 extensively extensively ADV cana-1962 68 23 leveraged leverage VERB cana-1962 68 24 in in ADP cana-1962 68 25 the the DET cana-1962 68 26 proof proof NOUN cana-1962 68 27 of of ADP cana-1962 68 28 our our PRON cana-1962 68 29 theorems theorem NOUN cana-1962 68 30 . . PUNCT cana-1962 69 1 specifically specifically ADV cana-1962 69 2 , , PUNCT cana-1962 69 3 for for ADP cana-1962 69 4 all all DET cana-1962 69 5 𝜐 𝜐 PROPN cana-1962 69 6 , , PUNCT cana-1962 69 7 πœ” πœ” VERB cana-1962 69 8 , , PUNCT cana-1962 69 9 𝑧 𝑧 DET cana-1962 69 10 ∈ ∈ PROPN cana-1962 69 11 𝔖 𝔖 PROPN cana-1962 69 12 , , PUNCT cana-1962 69 13 the the DET cana-1962 69 14 following follow VERB cana-1962 69 15 identities identity NOUN cana-1962 69 16 hold hold VERB cana-1962 69 17 : : PUNCT cana-1962 70 1 1 1 X cana-1962 70 2 . . PUNCT cana-1962 71 1 [ [ X cana-1962 71 2 𝜐 𝜐 X cana-1962 71 3 , , PUNCT cana-1962 71 4 , , PUNCT cana-1962 71 5 πœ”π‘§ πœ”π‘§ PROPN cana-1962 71 6 ] ] X cana-1962 71 7 = = SYM cana-1962 71 8 πœ”[𝜐 πœ”[𝜐 PROPN cana-1962 71 9 , , PUNCT cana-1962 71 10 𝑧 𝑧 X cana-1962 71 11 ] ] PUNCT cana-1962 72 1 + + CCONJ cana-1962 72 2 [ [ X cana-1962 72 3 𝜐 𝜐 NOUN cana-1962 72 4 , , PUNCT cana-1962 72 5 πœ”]𝑧 πœ”]𝑧 NOUN cana-1962 72 6 ; ; PUNCT cana-1962 73 1 [ [ X cana-1962 73 2 [ [ X cana-1962 73 3 πœπœ” πœπœ” X cana-1962 73 4 , , PUNCT cana-1962 73 5 𝑧 𝑧 X cana-1962 73 6 ] ] X cana-1962 73 7 = = PUNCT cana-1962 74 1 [ [ X cana-1962 74 2 𝜐 𝜐 NOUN cana-1962 74 3 , , PUNCT cana-1962 74 4 𝑧]πœ” 𝑧]πœ” ADJ cana-1962 74 5 + + CCONJ cana-1962 74 6 𝜐[πœ” 𝜐[πœ” NOUN cana-1962 74 7 , , PUNCT cana-1962 74 8 𝑧 𝑧 NOUN cana-1962 74 9 ] ] X cana-1962 74 10 . . PUNCT cana-1962 75 1 2 2 X cana-1962 75 2 . . X cana-1962 76 1 πœπœ” πœπœ” NOUN cana-1962 76 2 ∘ ∘ NOUN cana-1962 76 3 𝑧 𝑧 PROPN cana-1962 76 4 = = X cana-1962 76 5 ( ( PUNCT cana-1962 76 6 𝜐 𝜐 X cana-1962 76 7 ∘ ∘ X cana-1962 76 8 𝑧)πœ” 𝑧)πœ” PUNCT cana-1962 77 1 + + NUM cana-1962 77 2 𝜐[πœ” 𝜐[πœ” NOUN cana-1962 77 3 , , PUNCT cana-1962 77 4 𝑧 𝑧 X cana-1962 77 5 ] ] X cana-1962 77 6 = = SYM cana-1962 77 7 𝜐(πœ” 𝜐(πœ” ADJ cana-1962 77 8 ∘ ∘ NUM cana-1962 77 9 𝑧 𝑧 NOUN cana-1962 77 10 ) ) PUNCT cana-1962 77 11 βˆ’ βˆ’ PROPN cana-1962 78 1 [ [ X cana-1962 78 2 𝜐 𝜐 NOUN cana-1962 78 3 , , PUNCT cana-1962 78 4 𝑧]πœ” 𝑧]πœ” ADJ cana-1962 78 5 ; ; PUNCT cana-1962 78 6 𝜐 𝜐 PRON cana-1962 78 7 ∘ ∘ NOUN cana-1962 78 8 , , PUNCT cana-1962 78 9 πœ”π‘§ πœ”π‘§ PRON cana-1962 78 10 = = PUNCT cana-1962 78 11 πœ”(𝜐 πœ”(𝜐 PROPN cana-1962 78 12 ∘ ∘ NUM cana-1962 78 13 𝑧 𝑧 X cana-1962 78 14 ) ) PUNCT cana-1962 78 15 + + PROPN cana-1962 79 1 [ [ X cana-1962 79 2 𝜐 𝜐 NOUN cana-1962 79 3 , , PUNCT cana-1962 79 4 πœ”]𝑧 πœ”]𝑧 NOUN cana-1962 79 5 = = SYM cana-1962 79 6 ( ( PUNCT cana-1962 79 7 𝜐 𝜐 X cana-1962 79 8 ∘ ∘ X cana-1962 79 9 πœ”)𝑧 πœ”)𝑧 PUNCT cana-1962 79 10 + + CCONJ cana-1962 79 11 πœ”[𝑧 πœ”[𝑧 PROPN cana-1962 79 12 , , PUNCT cana-1962 79 13 𝜐 𝜐 X cana-1962 79 14 ] ] X cana-1962 79 15 . . PUNCT cana-1962 80 1 lemma lemma PROPN cana-1962 80 2 1 1 NUM cana-1962 81 1 [ [ X cana-1962 81 2 [ [ X cana-1962 81 3 5 5 NUM cana-1962 81 4 ] ] PUNCT cana-1962 81 5 lemmas lemma VERB cana-1962 81 6 2.1 2.1 NUM cana-1962 81 7 and and CCONJ cana-1962 81 8 2.2 2.2 NUM cana-1962 81 9 ] ] PUNCT cana-1962 81 10 " " PUNCT cana-1962 81 11 let let VERB cana-1962 81 12 𝔖 𝔖 PRON cana-1962 81 13 be be AUX cana-1962 81 14 a a DET cana-1962 81 15 prime prime ADJ cana-1962 81 16 ring ring NOUN cana-1962 81 17 with with ADP cana-1962 81 18 involution involution NOUN cana-1962 81 19 βˆ— βˆ— NOUN cana-1962 81 20 of of ADP cana-1962 81 21 the the DET cana-1962 81 22 second second ADJ cana-1962 81 23 kind kind NOUN cana-1962 81 24 such such ADJ cana-1962 81 25 that that DET cana-1962 81 26 π‘β„Žπ‘Žπ‘Ÿ(𝔖 π‘β„Žπ‘Žπ‘Ÿ(𝔖 NOUN cana-1962 81 27 ) ) PUNCT cana-1962 81 28 β‰  β‰  PROPN cana-1962 81 29 2 2 NUM cana-1962 81 30 , , PUNCT cana-1962 81 31 the the DET cana-1962 81 32 following follow VERB cana-1962 81 33 assertions assertion NOUN cana-1962 81 34 are be AUX cana-1962 81 35 equivalents equivalent NOUN cana-1962 81 36 : : PUNCT cana-1962 81 37 " " PUNCT cana-1962 82 1 1 1 X cana-1962 82 2 . . PUNCT cana-1962 83 1 [ [ X cana-1962 83 2 𝜐 𝜐 X cana-1962 83 3 , , PUNCT cana-1962 83 4 πœβˆ— πœβˆ— PROPN cana-1962 83 5 ] ] X cana-1962 83 6 ∈ ∈ PROPN cana-1962 83 7 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 83 8 ) ) PUNCT cana-1962 83 9 βˆ€πœ βˆ€πœ NOUN cana-1962 84 1 ∈ ∈ PROPN cana-1962 84 2 𝔖. 𝔖. PROPN cana-1962 84 3 2 2 NUM cana-1962 84 4 . . PUNCT cana-1962 85 1 𝜐 𝜐 PROPN cana-1962 85 2 ∘ ∘ PROPN cana-1962 85 3 πœβˆ— πœβˆ— PROPN cana-1962 85 4 ∈ ∈ PROPN cana-1962 85 5 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 85 6 ) ) PUNCT cana-1962 85 7 βˆ€πœ βˆ€πœ NOUN cana-1962 86 1 ∈ ∈ PROPN cana-1962 86 2 𝔖. 𝔖. PROPN cana-1962 86 3 3 3 NUM cana-1962 86 4 . . PUNCT cana-1962 87 1 𝔖 𝔖 PROPN cana-1962 87 2 is be AUX cana-1962 87 3 an an DET cana-1962 87 4 integral integral ADJ cana-1962 87 5 domain domain NOUN cana-1962 87 6 . . PUNCT cana-1962 88 1 lemma lemma PROPN cana-1962 88 2 2 2 NUM cana-1962 89 1 [ [ X cana-1962 89 2 [ [ X cana-1962 89 3 11 11 NUM cana-1962 89 4 ] ] X cana-1962 89 5 lemma lemma PROPN cana-1962 89 6 2.5 2.5 NUM cana-1962 89 7 ] ] PUNCT cana-1962 89 8 " " PUNCT cana-1962 89 9 let let VERB cana-1962 89 10 𝔖 𝔖 PRON cana-1962 89 11 be be AUX cana-1962 89 12 a a DET cana-1962 89 13 prime prime ADJ cana-1962 89 14 ring ring NOUN cana-1962 89 15 with with ADP cana-1962 89 16 involution involution NOUN cana-1962 89 17 of of ADP cana-1962 89 18 the the DET cana-1962 89 19 second second ADJ cana-1962 89 20 kind kind NOUN cana-1962 89 21 such such ADJ cana-1962 89 22 that that DET cana-1962 89 23 π‘β„Žπ‘Žπ‘Ÿ(𝔖 π‘β„Žπ‘Žπ‘Ÿ(𝔖 NOUN cana-1962 89 24 ) ) PUNCT cana-1962 89 25 β‰  β‰  PROPN cana-1962 89 26 2 2 X cana-1962 89 27 . . PUNCT cana-1962 90 1 let let VERB cana-1962 90 2 πœ“ πœ“ PART cana-1962 90 3 be be AUX cana-1962 90 4 a a DET cana-1962 90 5 derivation derivation NOUN cana-1962 90 6 of of ADP cana-1962 90 7 𝔖 𝔖 PROPN cana-1962 90 8 such such ADJ cana-1962 90 9 that that SCONJ cana-1962 90 10 πœ“(β„Ž πœ“(β„Ž NOUN cana-1962 90 11 ) ) PUNCT cana-1962 90 12 = = SYM cana-1962 90 13 0 0 NUM cana-1962 90 14 for for ADP cana-1962 90 15 all all PRON cana-1962 90 16 β„Ž β„Ž ADP cana-1962 90 17 ∈ ∈ PROPN cana-1962 90 18 β–³ β–³ X cana-1962 90 19 ( ( PUNCT cana-1962 90 20 𝔖 𝔖 NOUN cana-1962 90 21 ) ) PUNCT cana-1962 90 22 ∩ ∩ NOUN cana-1962 90 23 𝑍(𝔖 𝑍(𝔖 ADV cana-1962 90 24 ) ) PUNCT cana-1962 90 25 . . PUNCT cana-1962 91 1 then then ADV cana-1962 91 2 πœ“(𝜐 πœ“(𝜐 X cana-1962 91 3 ) ) PUNCT cana-1962 91 4 = = SYM cana-1962 91 5 0 0 NUM cana-1962 91 6 βˆ€πœ βˆ€πœ NOUN cana-1962 91 7 ∈ ∈ PROPN cana-1962 91 8 𝔖 𝔖 PROPN cana-1962 91 9 " " PUNCT cana-1962 91 10 . . PUNCT cana-1962 92 1 lemma lemma PROPN cana-1962 92 2 3 3 NUM cana-1962 93 1 [ [ X cana-1962 93 2 [ [ X cana-1962 93 3 4 4 X cana-1962 93 4 ] ] X cana-1962 93 5 lemma lemma PROPN cana-1962 93 6 2 2 NUM cana-1962 93 7 ] ] PUNCT cana-1962 93 8 " " PUNCT cana-1962 93 9 let let VERB cana-1962 93 10 𝔖 𝔖 PRON cana-1962 93 11 be be AUX cana-1962 93 12 a a DET cana-1962 93 13 prime prime ADJ cana-1962 93 14 ring ring NOUN cana-1962 93 15 . . PUNCT cana-1962 94 1 if if SCONJ cana-1962 94 2 π‘Žπ‘ π‘Žπ‘ PROPN cana-1962 94 3 ∈ ∈ PROPN cana-1962 94 4 𝑍(𝔖 𝑍(𝔖 PROPN cana-1962 94 5 ) ) PUNCT cana-1962 94 6 for for ADP cana-1962 94 7 some some PRON cana-1962 94 8 0 0 NUM cana-1962 94 9 β‰  β‰  PROPN cana-1962 94 10 π‘Ž π‘Ž PROPN cana-1962 94 11 ∈ ∈ PROPN cana-1962 94 12 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 94 13 ) ) PUNCT cana-1962 94 14 , , PUNCT cana-1962 94 15 then then ADV cana-1962 94 16 𝑏 𝑏 PROPN cana-1962 94 17 ∈ ∈ PROPN cana-1962 94 18 𝑍(𝔖 𝑍(𝔖 PROPN cana-1962 94 19 ) ) PUNCT cana-1962 94 20 . . PUNCT cana-1962 95 1 in in ADP cana-1962 95 2 particular particular ADJ cana-1962 95 3 , , PUNCT cana-1962 95 4 if if SCONJ cana-1962 95 5 π‘Žπ‘ π‘Žπ‘ PROPN cana-1962 95 6 = = SYM cana-1962 95 7 0 0 PROPN cana-1962 95 8 , , PUNCT cana-1962 95 9 then then ADV cana-1962 95 10 𝑏 𝑏 PROPN cana-1962 95 11 = = SYM cana-1962 95 12 0 0 NUM cana-1962 95 13 " " PUNCT cana-1962 95 14 . . PUNCT cana-1962 96 1 lemma lemma PROPN cana-1962 96 2 4 4 NUM cana-1962 97 1 [ [ PUNCT cana-1962 97 2 [ [ X cana-1962 97 3 12 12 NUM cana-1962 97 4 ] ] X cana-1962 97 5 lemma lemma PROPN cana-1962 97 6 2.2 2.2 NUM cana-1962 97 7 ] ] PUNCT cana-1962 97 8 " " PUNCT cana-1962 97 9 let let VERB cana-1962 97 10 𝔖 𝔖 PRON cana-1962 97 11 be be AUX cana-1962 97 12 a a DET cana-1962 97 13 ring ring NOUN cana-1962 97 14 and and CCONJ cana-1962 97 15 πœ“ πœ“ NOUN cana-1962 97 16 be be AUX cana-1962 97 17 a a DET cana-1962 97 18 multiplicative multiplicative ADJ cana-1962 97 19 derivation derivation NOUN cana-1962 97 20 of of ADP cana-1962 97 21 𝔖. 𝔖. PROPN cana-1962 97 22 then then ADV cana-1962 97 23 πœ“(𝑍(𝔖 πœ“(𝑍(𝔖 PROPN cana-1962 97 24 ) ) PUNCT cana-1962 97 25 ) ) PUNCT cana-1962 98 1 βŠ† βŠ† NUM cana-1962 98 2 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 98 3 ) ) PUNCT cana-1962 98 4 . . PUNCT cana-1962 98 5 " " PUNCT cana-1962 99 1 3 3 X cana-1962 99 2 . . X cana-1962 99 3 main main ADJ cana-1962 99 4 results result NOUN cana-1962 99 5 building build VERB cana-1962 99 6 on on ADP cana-1962 99 7 the the DET cana-1962 99 8 work work NOUN cana-1962 99 9 of of ADP cana-1962 99 10 nejjar nejjar NOUN cana-1962 99 11 ( ( PUNCT cana-1962 99 12 [ [ X cana-1962 99 13 8 8 NUM cana-1962 99 14 ] ] PUNCT cana-1962 99 15 , , PUNCT cana-1962 99 16 theorems theorem NOUN cana-1962 99 17 3.5 3.5 NUM cana-1962 99 18 , , PUNCT cana-1962 99 19 3.8 3.8 NUM cana-1962 99 20 ) ) PUNCT cana-1962 99 21 , , PUNCT cana-1962 99 22 who who PRON cana-1962 99 23 demonstrated demonstrate VERB cana-1962 99 24 that that SCONJ cana-1962 99 25 a a DET cana-1962 99 26 2 2 NUM cana-1962 99 27 - - PUNCT cana-1962 99 28 torsion torsion NOUN cana-1962 99 29 - - PUNCT cana-1962 99 30 free free ADJ cana-1962 99 31 prime prime ADJ cana-1962 99 32 ring ring NOUN cana-1962 99 33 with with ADP cana-1962 99 34 involution involution NOUN cana-1962 99 35 and and CCONJ cana-1962 99 36 a a DET cana-1962 99 37 derivation derivation NOUN cana-1962 99 38 πœ“ πœ“ ADP cana-1962 99 39 satisfying satisfy VERB cana-1962 99 40 certain certain ADJ cana-1962 99 41 conditions condition NOUN cana-1962 99 42 must must AUX cana-1962 99 43 be be AUX cana-1962 99 44 commutative commutative ADJ cana-1962 99 45 , , PUNCT cana-1962 99 46 we we PRON cana-1962 99 47 explore explore VERB cana-1962 99 48 broader broad ADJ cana-1962 99 49 generalizations generalization NOUN cana-1962 99 50 of of ADP cana-1962 99 51 these these DET cana-1962 99 52 conditions condition NOUN cana-1962 99 53 . . PUNCT cana-1962 100 1 specifically specifically ADV cana-1962 100 2 , , PUNCT cana-1962 100 3 nejjar nejjar PROPN cana-1962 100 4 showed show VERB cana-1962 100 5 that that SCONJ cana-1962 100 6 if if SCONJ cana-1962 100 7 the the DET cana-1962 100 8 derivation derivation NOUN cana-1962 100 9 πœ“ πœ“ PROPN cana-1962 100 10 meets meet VERB cana-1962 100 11 communications communication NOUN cana-1962 100 12 on on ADP cana-1962 100 13 applied apply VERB cana-1962 100 14 nonlinear nonlinear ADJ cana-1962 100 15 analysis analysis NOUN cana-1962 100 16 issn issn NOUN cana-1962 100 17 : : PUNCT cana-1962 100 18 1074 1074 NUM cana-1962 100 19 - - PUNCT cana-1962 100 20 133x 133x NUM cana-1962 100 21 vol vol NOUN cana-1962 100 22 32 32 NUM cana-1962 100 23 no no NOUN cana-1962 100 24 . . NOUN cana-1962 100 25 3 3 NUM cana-1962 100 26 ( ( PUNCT cana-1962 100 27 2025 2025 NUM cana-1962 100 28 ) ) PUNCT cana-1962 100 29 318 318 NUM cana-1962 100 30 https://internationalpubls.com https://internationalpubls.com X cana-1962 100 31 either either PRON cana-1962 100 32 of of ADP cana-1962 100 33 the the DET cana-1962 100 34 following follow VERB cana-1962 100 35 criteria criterion NOUN cana-1962 100 36 : : PUNCT cana-1962 100 37 [ [ X cana-1962 100 38 πœ“(𝜐 πœ“(𝜐 X cana-1962 100 39 ) ) PUNCT cana-1962 100 40 , , PUNCT cana-1962 100 41 πœ“(𝜐 πœ“(𝜐 X cana-1962 100 42 ) ) PUNCT cana-1962 100 43 ] ] PUNCT cana-1962 101 1 Β± Β± NOUN cana-1962 102 1 𝜐 𝜐 INTJ cana-1962 102 2 ∘ ∘ NOUN cana-1962 102 3 𝜐 𝜐 X cana-1962 102 4 ∈ ∈ PROPN cana-1962 102 5 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 102 6 ) ) PUNCT cana-1962 102 7 , , PUNCT cana-1962 102 8 βˆ€πœ βˆ€πœ NOUN cana-1962 102 9 ∈ ∈ PROPN cana-1962 102 10 𝔖 𝔖 PROPN cana-1962 102 11 or or CCONJ cana-1962 102 12 πœ“(𝜐 πœ“(𝜐 NOUN cana-1962 102 13 ) ) PUNCT cana-1962 102 14 ∘ ∘ X cana-1962 102 15 πœ“(𝜐 πœ“(𝜐 NOUN cana-1962 102 16 ) ) PUNCT cana-1962 102 17 Β± Β± NOUN cana-1962 103 1 𝜐 𝜐 NOUN cana-1962 103 2 ∘ ∘ NOUN cana-1962 103 3 𝜐 𝜐 X cana-1962 103 4 ∈ ∈ PROPN cana-1962 103 5 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 103 6 ) ) PUNCT cana-1962 103 7 , , PUNCT cana-1962 103 8 βˆ€πœ βˆ€πœ X cana-1962 103 9 ∈ ∈ PROPN cana-1962 103 10 𝔖 𝔖 PROPN cana-1962 103 11 , , PUNCT cana-1962 103 12 then then ADV cana-1962 103 13 𝔖 𝔖 PROPN cana-1962 103 14 is be AUX cana-1962 103 15 necessarily necessarily ADV cana-1962 103 16 commutative commutative ADJ cana-1962 103 17 . . PUNCT cana-1962 104 1 our our PRON cana-1962 104 2 work work NOUN cana-1962 104 3 extends extend VERB cana-1962 104 4 these these DET cana-1962 104 5 findings finding NOUN cana-1962 104 6 by by ADP cana-1962 104 7 introducing introduce VERB cana-1962 104 8 new new ADJ cana-1962 104 9 identities identity NOUN cana-1962 104 10 for for ADP cana-1962 104 11 pairs pair NOUN cana-1962 104 12 of of ADP cana-1962 104 13 generalized generalized ADJ cana-1962 104 14 derivations derivation NOUN cana-1962 104 15 that that PRON cana-1962 104 16 are be AUX cana-1962 104 17 connected connect VERB cana-1962 104 18 to to ADP cana-1962 104 19 a a DET cana-1962 104 20 left left ADJ cana-1962 104 21 multiplier multipli ADJ cana-1962 104 22 β–³ β–³ PROPN cana-1962 104 23 . . PUNCT cana-1962 105 1 throughout throughout ADP cana-1962 105 2 the the DET cana-1962 105 3 following follow VERB cana-1962 105 4 results result NOUN cana-1962 105 5 , , PUNCT cana-1962 105 6 let let VERB cana-1962 105 7 𝔖 𝔖 PRON cana-1962 105 8 be be AUX cana-1962 105 9 a a DET cana-1962 105 10 prime prime ADJ cana-1962 105 11 ring ring NOUN cana-1962 105 12 with with ADP cana-1962 105 13 involution involution NOUN cana-1962 105 14 , , PUNCT cana-1962 105 15 β–³ β–³ X cana-1962 105 16 be be AUX cana-1962 105 17 a a DET cana-1962 105 18 left left ADJ cana-1962 105 19 multiplier multiplier ADV cana-1962 105 20 , , PUNCT cana-1962 105 21 and and CCONJ cana-1962 105 22 ( ( PUNCT cana-1962 105 23 Ξ³1 Ξ³1 PROPN cana-1962 105 24 , , PUNCT cana-1962 105 25 Ξ³2 Ξ³2 PROPN cana-1962 105 26 ) ) PUNCT cana-1962 105 27 be be VERB cana-1962 105 28 two two NUM cana-1962 105 29 generalized generalized ADJ cana-1962 105 30 derivations derivation NOUN cana-1962 105 31 , , PUNCT cana-1962 105 32 which which PRON cana-1962 105 33 are be AUX cana-1962 105 34 linked link VERB cana-1962 105 35 to to ADP cana-1962 105 36 nonzero nonzero PROPN cana-1962 105 37 derivations derivation NOUN cana-1962 105 38 ( ( PUNCT cana-1962 105 39 πœ“1 πœ“1 NOUN cana-1962 105 40 , , PUNCT cana-1962 105 41 πœ“2 πœ“2 NOUN cana-1962 105 42 ) ) PUNCT cana-1962 105 43 , , PUNCT cana-1962 105 44 respectively respectively ADV cana-1962 105 45 . . PUNCT cana-1962 106 1 theorem theorem VERB cana-1962 106 2 1 1 NUM cana-1962 106 3 the the DET cana-1962 106 4 following following ADJ cana-1962 106 5 statements statement NOUN cana-1962 106 6 are be AUX cana-1962 106 7 equivalent equivalent ADJ cana-1962 106 8 : : PUNCT cana-1962 106 9 ( ( PUNCT cana-1962 106 10 i i NOUN cana-1962 106 11 ) ) PUNCT cana-1962 107 1 [ [ X cana-1962 107 2 Ξ³1(𝜐 Ξ³1(𝜐 X cana-1962 107 3 ) ) PUNCT cana-1962 107 4 , , PUNCT cana-1962 107 5 Ξ³2(πœβˆ— Ξ³2(πœβˆ— PROPN cana-1962 107 6 ) ) PUNCT cana-1962 107 7 ] ] PUNCT cana-1962 108 1 + + X cana-1962 108 2 β–³ β–³ X cana-1962 108 3 ( ( PUNCT cana-1962 108 4 [ [ X cana-1962 108 5 𝜐 𝜐 X cana-1962 108 6 , , PUNCT cana-1962 108 7 πœβˆ— πœβˆ— PROPN cana-1962 108 8 ] ] PUNCT cana-1962 108 9 ) ) PUNCT cana-1962 108 10 ∈ ∈ PROPN cana-1962 108 11 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 108 12 ) ) PUNCT cana-1962 108 13 , , PUNCT cana-1962 108 14 βˆ€πœ βˆ€πœ X cana-1962 108 15 ∈ ∈ PROPN cana-1962 108 16 𝔖. 𝔖. PROPN cana-1962 108 17 ( ( PUNCT cana-1962 108 18 ii ii NOUN cana-1962 108 19 ) ) PUNCT cana-1962 108 20 Ξ³1(𝜐 Ξ³1(𝜐 NOUN cana-1962 108 21 ) ) PUNCT cana-1962 108 22 ∘ ∘ NOUN cana-1962 108 23 Ξ³2(πœβˆ— Ξ³2(πœβˆ— PUNCT cana-1962 108 24 ) ) PUNCT cana-1962 109 1 + + NOUN cana-1962 109 2 β–³ β–³ X cana-1962 109 3 ( ( PUNCT cana-1962 109 4 𝜐 𝜐 PROPN cana-1962 109 5 ∘ ∘ PROPN cana-1962 109 6 πœβˆ— πœβˆ— PROPN cana-1962 109 7 ) ) PUNCT cana-1962 109 8 ∈ ∈ PROPN cana-1962 109 9 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 109 10 ) ) PUNCT cana-1962 109 11 βˆ€πœ βˆ€πœ NOUN cana-1962 109 12 ∈ ∈ PROPN cana-1962 109 13 𝔖. 𝔖. PROPN cana-1962 109 14 ( ( PUNCT cana-1962 109 15 iii iii NOUN cana-1962 109 16 ) ) PUNCT cana-1962 110 1 [ [ X cana-1962 110 2 Ξ³1(𝜐 Ξ³1(𝜐 X cana-1962 110 3 ) ) PUNCT cana-1962 110 4 , , PUNCT cana-1962 110 5 Ξ³2(πœβˆ— Ξ³2(πœβˆ— PROPN cana-1962 110 6 ) ) PUNCT cana-1962 110 7 ] ] PUNCT cana-1962 111 1 + + X cana-1962 111 2 β–³ β–³ X cana-1962 111 3 ( ( PUNCT cana-1962 111 4 [ [ X cana-1962 111 5 𝜐 𝜐 X cana-1962 111 6 ∘ ∘ NOUN cana-1962 111 7 πœβˆ— πœβˆ— PROPN cana-1962 111 8 ] ] PUNCT cana-1962 111 9 ) ) PUNCT cana-1962 111 10 ∈ ∈ PROPN cana-1962 111 11 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 111 12 ) ) PUNCT cana-1962 111 13 βˆ€πœ βˆ€πœ NOUN cana-1962 111 14 ∈ ∈ PROPN cana-1962 111 15 𝔖. 𝔖. PROPN cana-1962 111 16 ( ( PUNCT cana-1962 111 17 iv iv X cana-1962 111 18 ) ) PUNCT cana-1962 111 19 Ξ³1(𝜐 Ξ³1(𝜐 NOUN cana-1962 111 20 ) ) PUNCT cana-1962 111 21 ∘ ∘ NOUN cana-1962 111 22 Ξ³2(πœβˆ— Ξ³2(πœβˆ— PUNCT cana-1962 111 23 ) ) PUNCT cana-1962 112 1 + + NOUN cana-1962 112 2 β–³ β–³ X cana-1962 112 3 ( ( PUNCT cana-1962 112 4 [ [ X cana-1962 112 5 𝜐 𝜐 X cana-1962 112 6 , , PUNCT cana-1962 112 7 πœβˆ— πœβˆ— PROPN cana-1962 112 8 ] ] PUNCT cana-1962 112 9 ) ) PUNCT cana-1962 112 10 ∈ ∈ PROPN cana-1962 112 11 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 112 12 ) ) PUNCT cana-1962 112 13 βˆ€πœ βˆ€πœ NOUN cana-1962 112 14 ∈ ∈ PROPN cana-1962 112 15 𝔖. 𝔖. PROPN cana-1962 112 16 ( ( PUNCT cana-1962 112 17 v v NOUN cana-1962 112 18 ) ) PUNCT cana-1962 112 19 𝔖 𝔖 NOUN cana-1962 112 20 is be AUX cana-1962 112 21 an an DET cana-1962 112 22 integral integral ADJ cana-1962 112 23 domain domain NOUN cana-1962 112 24 . . PUNCT cana-1962 113 1 proof proof NOUN cana-1962 113 2 . . PUNCT cana-1962 114 1 it it PRON cana-1962 114 2 is be AUX cana-1962 114 3 necessary necessary ADJ cana-1962 114 4 to to PART cana-1962 114 5 demonstrate demonstrate VERB cana-1962 114 6 that that SCONJ cana-1962 114 7 ( ( PUNCT cana-1962 114 8 𝑖 𝑖 X cana-1962 114 9 ) ) PUNCT cana-1962 114 10 , , PUNCT cana-1962 114 11 ( ( PUNCT cana-1962 114 12 𝑖𝑖 𝑖𝑖 NOUN cana-1962 114 13 ) ) PUNCT cana-1962 114 14 , , PUNCT cana-1962 114 15 ( ( PUNCT cana-1962 114 16 𝑖𝑖𝑖 𝑖𝑖𝑖 NOUN cana-1962 114 17 ) ) PUNCT cana-1962 114 18 and and CCONJ cana-1962 114 19 ( ( PUNCT cana-1962 114 20 𝑖𝑣 𝑖𝑣 X cana-1962 114 21 ) ) PUNCT cana-1962 114 22 ⟹ ⟹ VERB cana-1962 115 1 ( ( PUNCT cana-1962 115 2 𝑣 𝑣 NOUN cana-1962 115 3 ) ) PUNCT cana-1962 115 4 . . PUNCT cana-1962 116 1 ( ( PUNCT cana-1962 116 2 𝑖 𝑖 X cana-1962 116 3 ) ) PUNCT cana-1962 116 4 ⟹ ⟹ VERB cana-1962 117 1 ( ( PUNCT cana-1962 117 2 𝑣 𝑣 X cana-1962 117 3 ) ) PUNCT cana-1962 117 4 suppose suppose VERB cana-1962 117 5 that that SCONJ cana-1962 117 6 [ [ X cana-1962 117 7 Ξ³1(𝜐 Ξ³1(𝜐 X cana-1962 117 8 ) ) PUNCT cana-1962 117 9 , , PUNCT cana-1962 117 10 Ξ³2(πœβˆ— Ξ³2(πœβˆ— PROPN cana-1962 117 11 ) ) PUNCT cana-1962 117 12 ] ] PUNCT cana-1962 118 1 + + X cana-1962 118 2 β–³ β–³ X cana-1962 118 3 ( ( PUNCT cana-1962 118 4 [ [ X cana-1962 118 5 𝜐 𝜐 X cana-1962 118 6 , , PUNCT cana-1962 118 7 πœβˆ— πœβˆ— PROPN cana-1962 118 8 ] ] PUNCT cana-1962 118 9 ) ) PUNCT cana-1962 118 10 ∈ ∈ PROPN cana-1962 118 11 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 118 12 ) ) PUNCT cana-1962 118 13 βˆ€πœ βˆ€πœ NOUN cana-1962 118 14 ∈ ∈ PROPN cana-1962 118 15 𝔖. 𝔖. PROPN cana-1962 118 16 ( ( PUNCT cana-1962 118 17 1 1 NUM cana-1962 118 18 ) ) PUNCT cana-1962 118 19 replacing replace VERB cana-1962 118 20 𝜐 𝜐 PRON cana-1962 118 21 by by ADP cana-1962 118 22 𝜐 𝜐 PROPN cana-1962 118 23 + + CCONJ cana-1962 118 24 πœ” πœ” X cana-1962 118 25 in in ADV cana-1962 118 26 ( ( PUNCT cana-1962 118 27 1 1 NUM cana-1962 118 28 ) ) PUNCT cana-1962 118 29 and and CCONJ cana-1962 118 30 using use VERB cana-1962 118 31 it it PRON cana-1962 118 32 , , PUNCT cana-1962 118 33 we we PRON cana-1962 118 34 obtain obtain VERB cana-1962 118 35 [ [ X cana-1962 118 36 Ξ³1(𝜐 Ξ³1(𝜐 NOUN cana-1962 118 37 ) ) PUNCT cana-1962 118 38 , , PUNCT cana-1962 118 39 Ξ³2(πœ”βˆ— Ξ³2(πœ”βˆ— X cana-1962 118 40 ) ) PUNCT cana-1962 118 41 ] ] PUNCT cana-1962 119 1 + + CCONJ cana-1962 120 1 [ [ X cana-1962 120 2 Ξ³1(πœ” Ξ³1(πœ” PROPN cana-1962 120 3 ) ) PUNCT cana-1962 120 4 , , PUNCT cana-1962 120 5 Ξ³2(πœβˆ— Ξ³2(πœβˆ— PROPN cana-1962 120 6 ) ) PUNCT cana-1962 120 7 ] ] PUNCT cana-1962 121 1 + + X cana-1962 121 2 β–³ β–³ X cana-1962 121 3 ( ( PUNCT cana-1962 121 4 [ [ X cana-1962 121 5 𝜐 𝜐 X cana-1962 121 6 , , PUNCT cana-1962 121 7 πœ”βˆ— πœ”βˆ— PROPN cana-1962 121 8 ] ] PUNCT cana-1962 121 9 ) ) PUNCT cana-1962 121 10 + + NOUN cana-1962 121 11 β–³ β–³ X cana-1962 121 12 ( ( PUNCT cana-1962 121 13 [ [ X cana-1962 121 14 πœ” πœ” X cana-1962 121 15 , , PUNCT cana-1962 121 16 πœβˆ— πœβˆ— PROPN cana-1962 121 17 ] ] PUNCT cana-1962 121 18 ) ) PUNCT cana-1962 121 19 ∈ ∈ PROPN cana-1962 121 20 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 121 21 ) ) PUNCT cana-1962 121 22 βˆ€πœ βˆ€πœ NOUN cana-1962 121 23 , , PUNCT cana-1962 121 24 πœ” πœ” PART cana-1962 121 25 ∈ ∈ PROPN cana-1962 121 26 𝔖. 𝔖. PROPN cana-1962 121 27 ( ( PUNCT cana-1962 121 28 2 2 NUM cana-1962 121 29 ) ) PUNCT cana-1962 121 30 replacing replace VERB cana-1962 121 31 πœ” πœ” X cana-1962 121 32 by by ADP cana-1962 121 33 πœ”β„Ž πœ”β„Ž INTJ cana-1962 121 34 in in ADP cana-1962 121 35 ( ( PUNCT cana-1962 121 36 2 2 NUM cana-1962 121 37 ) ) PUNCT cana-1962 121 38 where where SCONJ cana-1962 121 39 β„Ž β„Ž X cana-1962 121 40 ∈ ∈ PROPN cana-1962 121 41 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 121 42 ) ) PUNCT cana-1962 121 43 ∩ ∩ NOUN cana-1962 121 44 β–³ β–³ X cana-1962 121 45 ( ( PUNCT cana-1962 121 46 𝔖 𝔖 PROPN cana-1962 121 47 ) ) PUNCT cana-1962 121 48 and and CCONJ cana-1962 121 49 using use VERB cana-1962 121 50 it it PRON cana-1962 121 51 again again ADV cana-1962 121 52 , , PUNCT cana-1962 121 53 we we PRON cana-1962 121 54 get get VERB cana-1962 121 55 [ [ X cana-1962 121 56 Ξ³1(𝜐 Ξ³1(𝜐 NOUN cana-1962 121 57 ) ) PUNCT cana-1962 121 58 , , PUNCT cana-1962 121 59 πœ”βˆ—]πœ“2(β„Ž πœ”βˆ—]πœ“2(β„Ž NOUN cana-1962 121 60 ) ) PUNCT cana-1962 121 61 + + CCONJ cana-1962 122 1 [ [ X cana-1962 122 2 πœ” πœ” X cana-1962 122 3 , , PUNCT cana-1962 122 4 Ξ³2(πœβˆ—)]πœ“1(β„Ž Ξ³2(πœβˆ—)]πœ“1(β„Ž PUNCT cana-1962 122 5 ) ) PUNCT cana-1962 122 6 ∈ ∈ PROPN cana-1962 122 7 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 122 8 ) ) PUNCT cana-1962 122 9 βˆ€πœ βˆ€πœ NOUN cana-1962 122 10 , , PUNCT cana-1962 122 11 πœ” πœ” PART cana-1962 122 12 ∈ ∈ PROPN cana-1962 122 13 𝔖. 𝔖. PROPN cana-1962 122 14 ( ( PUNCT cana-1962 122 15 3 3 NUM cana-1962 122 16 ) ) PUNCT cana-1962 122 17 replacing replace VERB cana-1962 122 18 𝜐 𝜐 PRON cana-1962 122 19 by by ADP cana-1962 122 20 πœβ„Ž πœβ„Ž PRON cana-1962 122 21 in in ADP cana-1962 122 22 ( ( PUNCT cana-1962 122 23 3 3 NUM cana-1962 122 24 ) ) PUNCT cana-1962 122 25 where where SCONJ cana-1962 122 26 β„Ž β„Ž X cana-1962 122 27 ∈ ∈ PROPN cana-1962 122 28 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 122 29 ) ) PUNCT cana-1962 122 30 ∩ ∩ NOUN cana-1962 122 31 β–³ β–³ X cana-1962 122 32 ( ( PUNCT cana-1962 122 33 𝔖 𝔖 PROPN cana-1962 122 34 ) ) PUNCT cana-1962 122 35 and and CCONJ cana-1962 122 36 using use VERB cana-1962 122 37 it it PRON cana-1962 122 38 again again ADV cana-1962 122 39 , , PUNCT cana-1962 122 40 we we PRON cana-1962 122 41 obtain obtain VERB cana-1962 122 42 [ [ X cana-1962 122 43 𝜐 𝜐 NOUN cana-1962 122 44 , , PUNCT cana-1962 122 45 πœ”βˆ—]πœ“1(β„Ž)πœ“2(β„Ž πœ”βˆ—]πœ“1(β„Ž)πœ“2(β„Ž ADJ cana-1962 122 46 ) ) PUNCT cana-1962 123 1 + + CCONJ cana-1962 124 1 [ [ X cana-1962 124 2 πœ” πœ” X cana-1962 124 3 , , PUNCT cana-1962 124 4 πœβˆ—]πœ“1(β„Ž)πœ“2(β„Ž πœβˆ—]πœ“1(β„Ž)πœ“2(β„Ž ADJ cana-1962 124 5 ) ) PUNCT cana-1962 124 6 ∈ ∈ PROPN cana-1962 124 7 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 124 8 ) ) PUNCT cana-1962 124 9 βˆ€πœ βˆ€πœ NOUN cana-1962 124 10 , , PUNCT cana-1962 124 11 πœ” πœ” PART cana-1962 124 12 ∈ ∈ PROPN cana-1962 124 13 𝔖. 𝔖. PROPN cana-1962 124 14 ( ( PUNCT cana-1962 124 15 4 4 X cana-1962 124 16 ) ) PUNCT cana-1962 124 17 taking take VERB cana-1962 124 18 πœ” πœ” PRON cana-1962 124 19 = = NUM cana-1962 124 20 𝜐 𝜐 X cana-1962 124 21 in in ADP cana-1962 124 22 ( ( PUNCT cana-1962 124 23 4 4 NUM cana-1962 124 24 ) ) PUNCT cana-1962 124 25 , , PUNCT cana-1962 124 26 we we PRON cana-1962 124 27 arrive arrive VERB cana-1962 124 28 at at ADP cana-1962 124 29 2[𝜐 2[𝜐 NUM cana-1962 124 30 , , PUNCT cana-1962 124 31 πœβˆ—]πœ“1(β„Ž)πœ“2(β„Ž πœβˆ—]πœ“1(β„Ž)πœ“2(β„Ž ADJ cana-1962 124 32 ) ) PUNCT cana-1962 124 33 ∈ ∈ PROPN cana-1962 124 34 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 124 35 ) ) PUNCT cana-1962 124 36 βˆ€πœ βˆ€πœ NOUN cana-1962 125 1 ∈ ∈ PROPN cana-1962 125 2 𝔖. 𝔖. PROPN cana-1962 125 3 ( ( PUNCT cana-1962 125 4 5 5 NUM cana-1962 125 5 ) ) PUNCT cana-1962 125 6 since since SCONJ cana-1962 125 7 π‘β„Žπ‘Žπ‘Ÿ(𝔖 π‘β„Žπ‘Žπ‘Ÿ(𝔖 NOUN cana-1962 125 8 ) ) PUNCT cana-1962 125 9 β‰  β‰  PROPN cana-1962 125 10 2 2 NUM cana-1962 125 11 , , PUNCT cana-1962 125 12 and and CCONJ cana-1962 125 13 by by ADP cana-1962 125 14 using use VERB cana-1962 125 15 lemme lemme PROPN cana-1962 125 16 1 1 NUM cana-1962 125 17 , , PUNCT cana-1962 125 18 we we PRON cana-1962 125 19 get get VERB cana-1962 125 20 [ [ X cana-1962 125 21 𝜐 𝜐 NOUN cana-1962 125 22 , , PUNCT cana-1962 125 23 πœβˆ— πœβˆ— PROPN cana-1962 125 24 ] ] X cana-1962 125 25 ∈ ∈ PROPN cana-1962 125 26 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 125 27 ) ) PUNCT cana-1962 125 28 or or CCONJ cana-1962 125 29 πœ“1(β„Ž)πœ“2(β„Ž πœ“1(β„Ž)πœ“2(β„Ž X cana-1962 125 30 ) ) PUNCT cana-1962 126 1 = = SYM cana-1962 126 2 0 0 NUM cana-1962 126 3 , , PUNCT cana-1962 126 4 by by ADP cana-1962 126 5 primness primness NOUN cana-1962 126 6 that that PRON cana-1962 126 7 leads lead VERB cana-1962 126 8 to to ADP cana-1962 126 9 𝔖 𝔖 PROPN cana-1962 126 10 is be AUX cana-1962 126 11 an an DET cana-1962 126 12 integral integral ADJ cana-1962 126 13 domain domain NOUN cana-1962 126 14 or or CCONJ cana-1962 126 15 πœ“1(β„Ž πœ“1(β„Ž ADJ cana-1962 126 16 ) ) PUNCT cana-1962 126 17 = = SYM cana-1962 126 18 0 0 NUM cana-1962 126 19 or or CCONJ cana-1962 126 20 πœ“2(β„Ž πœ“2(β„Ž NOUN cana-1962 126 21 ) ) PUNCT cana-1962 126 22 = = SYM cana-1962 126 23 0 0 NUM cana-1962 126 24 for for ADP cana-1962 126 25 all all PRON cana-1962 126 26 β„Ž β„Ž ADP cana-1962 126 27 ∈ ∈ PROPN cana-1962 126 28 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 126 29 ) ) PUNCT cana-1962 126 30 . . PUNCT cana-1962 127 1 if if SCONJ cana-1962 127 2 πœ“1(β„Ž πœ“1(β„Ž ADJ cana-1962 127 3 ) ) PUNCT cana-1962 127 4 = = SYM cana-1962 127 5 0 0 NUM cana-1962 127 6 for for ADP cana-1962 127 7 all all DET cana-1962 127 8 β„Ž β„Ž PART cana-1962 127 9 ∈ ∈ PROPN cana-1962 127 10 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 127 11 ) ) PUNCT cana-1962 127 12 ∩ ∩ NOUN cana-1962 127 13 β–³ β–³ X cana-1962 127 14 ( ( PUNCT cana-1962 127 15 𝔖 𝔖 PROPN cana-1962 127 16 ) ) PUNCT cana-1962 127 17 , , PUNCT cana-1962 127 18 lemma lemma PROPN cana-1962 127 19 2 2 NUM cana-1962 127 20 leads lead VERB cana-1962 127 21 us we PRON cana-1962 127 22 to to ADP cana-1962 127 23 πœ“1(𝜐 πœ“1(𝜐 NOUN cana-1962 127 24 ) ) PUNCT cana-1962 127 25 = = SYM cana-1962 127 26 0 0 NUM cana-1962 127 27 , , PUNCT cana-1962 127 28 βˆ€πœ βˆ€πœ NOUN cana-1962 127 29 ∈ ∈ PROPN cana-1962 127 30 𝔖 𝔖 PROPN cana-1962 127 31 , , PUNCT cana-1962 127 32 a a DET cana-1962 127 33 contradiction contradiction NOUN cana-1962 127 34 . . PUNCT cana-1962 128 1 using use VERB cana-1962 128 2 the the DET cana-1962 128 3 same same ADJ cana-1962 128 4 technique technique NOUN cana-1962 128 5 , , PUNCT cana-1962 128 6 we we PRON cana-1962 128 7 obtain obtain VERB cana-1962 128 8 that that DET cana-1962 128 9 πœ“2(𝑍(𝔖 πœ“2(𝑍(𝔖 NUM cana-1962 128 10 ) ) PUNCT cana-1962 128 11 ∩ ∩ NOUN cana-1962 128 12 β–³ β–³ X cana-1962 128 13 ( ( PUNCT cana-1962 128 14 𝔖 𝔖 NOUN cana-1962 128 15 ) ) PUNCT cana-1962 128 16 ) ) PUNCT cana-1962 129 1 β‰  β‰  PROPN cana-1962 129 2 0 0 X cana-1962 129 3 . . PUNCT cana-1962 129 4 ( ( PUNCT cana-1962 129 5 𝑖𝑖 𝑖𝑖 NOUN cana-1962 129 6 ) ) PUNCT cana-1962 129 7 ⟹ ⟹ VERB cana-1962 129 8 ( ( PUNCT cana-1962 129 9 𝑣 𝑣 NOUN cana-1962 129 10 ) ) PUNCT cana-1962 129 11 . . PUNCT cana-1962 130 1 suppose suppose VERB cana-1962 130 2 that that SCONJ cana-1962 130 3 Ξ³1(𝜐 Ξ³1(𝜐 PROPN cana-1962 130 4 ) ) PUNCT cana-1962 130 5 ∘ ∘ NOUN cana-1962 130 6 Ξ³2(πœβˆ— Ξ³2(πœβˆ— PUNCT cana-1962 130 7 ) ) PUNCT cana-1962 131 1 + + NOUN cana-1962 131 2 β–³ β–³ X cana-1962 131 3 ( ( PUNCT cana-1962 131 4 𝜐 𝜐 PROPN cana-1962 131 5 ∘ ∘ PROPN cana-1962 131 6 πœβˆ— πœβˆ— PROPN cana-1962 131 7 ) ) PUNCT cana-1962 131 8 ∈ ∈ PROPN cana-1962 131 9 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 131 10 ) ) PUNCT cana-1962 132 1 βˆ€πœ βˆ€πœ NOUN cana-1962 132 2 ∈ ∈ PROPN cana-1962 132 3 𝔖. 𝔖. PROPN cana-1962 132 4 ( ( PUNCT cana-1962 132 5 6 6 NUM cana-1962 132 6 ) ) PUNCT cana-1962 132 7 replacing replace VERB cana-1962 132 8 𝜐 𝜐 PRON cana-1962 132 9 by by ADP cana-1962 132 10 𝜐 𝜐 PROPN cana-1962 132 11 + + CCONJ cana-1962 132 12 πœ” πœ” X cana-1962 132 13 in in ADP cana-1962 132 14 ( ( PUNCT cana-1962 132 15 6 6 NUM cana-1962 132 16 ) ) PUNCT cana-1962 132 17 and and CCONJ cana-1962 132 18 using use VERB cana-1962 132 19 it it PRON cana-1962 132 20 , , PUNCT cana-1962 132 21 we we PRON cana-1962 132 22 obtain obtain VERB cana-1962 132 23 Ξ³1(𝜐 Ξ³1(𝜐 X cana-1962 132 24 ) ) PUNCT cana-1962 132 25 ∘ ∘ NUM cana-1962 132 26 Ξ³2(πœ”βˆ— Ξ³2(πœ”βˆ— NOUN cana-1962 132 27 ) ) PUNCT cana-1962 133 1 + + CCONJ cana-1962 133 2 Ξ³1(πœ” Ξ³1(πœ” PROPN cana-1962 133 3 ) ) PUNCT cana-1962 133 4 ∘ ∘ NOUN cana-1962 133 5 Ξ³2(πœβˆ— Ξ³2(πœβˆ— PUNCT cana-1962 133 6 ) ) PUNCT cana-1962 134 1 + + NOUN cana-1962 134 2 β–³ β–³ X cana-1962 134 3 ( ( PUNCT cana-1962 134 4 𝜐 𝜐 PROPN cana-1962 134 5 ∘ ∘ X cana-1962 134 6 πœ”βˆ— πœ”βˆ— PROPN cana-1962 134 7 ) ) PUNCT cana-1962 134 8 + + NOUN cana-1962 134 9 β–³ β–³ X cana-1962 134 10 ( ( PUNCT cana-1962 134 11 πœ” πœ” SYM cana-1962 134 12 ∘ ∘ NUM cana-1962 134 13 πœβˆ— πœβˆ— PROPN cana-1962 134 14 ) ) PUNCT cana-1962 134 15 ∈ ∈ PROPN cana-1962 134 16 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 134 17 ) ) PUNCT cana-1962 134 18 βˆ€πœ βˆ€πœ NOUN cana-1962 134 19 , , PUNCT cana-1962 134 20 πœ” πœ” PART cana-1962 134 21 ∈ ∈ PROPN cana-1962 134 22 𝔖. 𝔖. PROPN cana-1962 134 23 ( ( PUNCT cana-1962 134 24 7 7 X cana-1962 134 25 ) ) PUNCT cana-1962 134 26 replacing replace VERB cana-1962 134 27 πœ” πœ” X cana-1962 134 28 by by ADP cana-1962 134 29 πœ”πœ πœ”πœ NOUN cana-1962 134 30 in in ADP cana-1962 134 31 ( ( PUNCT cana-1962 134 32 7 7 NUM cana-1962 134 33 ) ) PUNCT cana-1962 134 34 where where SCONJ cana-1962 134 35 𝜏 𝜏 PRON cana-1962 134 36 ∈ ∈ PROPN cana-1962 134 37 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 134 38 ) ) PUNCT cana-1962 134 39 ∩ ∩ NOUN cana-1962 134 40 β–³ β–³ X cana-1962 134 41 ( ( PUNCT cana-1962 134 42 𝔖 𝔖 NOUN cana-1962 134 43 ) ) PUNCT cana-1962 134 44 and and CCONJ cana-1962 134 45 we we PRON cana-1962 134 46 using use VERB cana-1962 134 47 it it PRON cana-1962 134 48 again again ADV cana-1962 134 49 , , PUNCT cana-1962 134 50 we we PRON cana-1962 134 51 get get VERB cana-1962 134 52 ( ( PUNCT cana-1962 134 53 Ξ³1(𝜐 Ξ³1(𝜐 NOUN cana-1962 134 54 ) ) PUNCT cana-1962 134 55 ∘ ∘ NOUN cana-1962 134 56 πœ”βˆ—)πœ“2(𝜏 πœ”βˆ—)πœ“2(𝜏 PUNCT cana-1962 134 57 ) ) PUNCT cana-1962 135 1 + + CCONJ cana-1962 135 2 ( ( PUNCT cana-1962 135 3 πœ” πœ” NOUN cana-1962 135 4 , , PUNCT cana-1962 135 5 Ξ³2(πœβˆ—))πœ“1(𝜏 Ξ³2(πœβˆ—))πœ“1(𝜏 NOUN cana-1962 135 6 ) ) PUNCT cana-1962 135 7 ∈ ∈ PROPN cana-1962 135 8 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 135 9 ) ) PUNCT cana-1962 135 10 βˆ€πœ βˆ€πœ NOUN cana-1962 135 11 , , PUNCT cana-1962 135 12 πœ” πœ” PART cana-1962 135 13 ∈ ∈ PROPN cana-1962 135 14 𝔖. 𝔖. PROPN cana-1962 135 15 ( ( PUNCT cana-1962 135 16 8) 8) NUM cana-1962 135 17 replacing replace VERB cana-1962 135 18 𝜐 𝜐 PRON cana-1962 135 19 by by ADP cana-1962 135 20 𝜐𝜏 𝜐𝜏 NOUN cana-1962 135 21 in in ADP cana-1962 135 22 ( ( PUNCT cana-1962 135 23 8) 8) NUM cana-1962 135 24 where where SCONJ cana-1962 135 25 𝜏 𝜏 PRON cana-1962 135 26 ∈ ∈ PROPN cana-1962 135 27 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 135 28 ) ) PUNCT cana-1962 135 29 ∩ ∩ NOUN cana-1962 135 30 β–³ β–³ X cana-1962 135 31 ( ( PUNCT cana-1962 135 32 𝔖 𝔖 NOUN cana-1962 135 33 ) ) PUNCT cana-1962 135 34 and and CCONJ cana-1962 135 35 we we PRON cana-1962 135 36 using use VERB cana-1962 135 37 it it PRON cana-1962 135 38 again again ADV cana-1962 135 39 , , PUNCT cana-1962 135 40 we we PRON cana-1962 135 41 have have VERB cana-1962 135 42 ( ( PUNCT cana-1962 135 43 𝜐 𝜐 X cana-1962 135 44 ∘ ∘ X cana-1962 135 45 πœ”βˆ—)πœ“1(𝜏)πœ“2(𝜏 πœ”βˆ—)πœ“1(𝜏)πœ“2(𝜏 NOUN cana-1962 135 46 ) ) PUNCT cana-1962 136 1 + + CCONJ cana-1962 136 2 ( ( PUNCT cana-1962 136 3 πœ” πœ” PRON cana-1962 136 4 ∘ ∘ NUM cana-1962 136 5 πœβˆ—)πœ“1(𝜏)πœ“2(𝜏 πœβˆ—)πœ“1(𝜏)πœ“2(𝜏 NOUN cana-1962 136 6 ) ) PUNCT cana-1962 136 7 ∈ ∈ PROPN cana-1962 136 8 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 136 9 ) ) PUNCT cana-1962 136 10 βˆ€πœ βˆ€πœ NOUN cana-1962 136 11 , , PUNCT cana-1962 136 12 πœ” πœ” PART cana-1962 136 13 ∈ ∈ PROPN cana-1962 136 14 𝔖. 𝔖. PROPN cana-1962 136 15 ( ( PUNCT cana-1962 136 16 9 9 X cana-1962 136 17 ) ) PUNCT cana-1962 136 18 taking take VERB cana-1962 136 19 πœ” πœ” PRON cana-1962 136 20 = = NUM cana-1962 136 21 𝜐 𝜐 X cana-1962 136 22 in in ADP cana-1962 136 23 ( ( PUNCT cana-1962 136 24 9 9 NUM cana-1962 136 25 ) ) PUNCT cana-1962 136 26 , , PUNCT cana-1962 136 27 we we PRON cana-1962 136 28 arrive arrive VERB cana-1962 136 29 at at ADP cana-1962 136 30 communications communication NOUN cana-1962 136 31 on on ADP cana-1962 136 32 applied apply VERB cana-1962 136 33 nonlinear nonlinear ADJ cana-1962 136 34 analysis analysis NOUN cana-1962 136 35 issn issn NOUN cana-1962 136 36 : : PUNCT cana-1962 136 37 1074 1074 NUM cana-1962 136 38 - - PUNCT cana-1962 136 39 133x 133x NUM cana-1962 136 40 vol vol NOUN cana-1962 136 41 32 32 NUM cana-1962 136 42 no no NOUN cana-1962 136 43 . . NOUN cana-1962 136 44 3 3 NUM cana-1962 136 45 ( ( PUNCT cana-1962 136 46 2025 2025 NUM cana-1962 136 47 ) ) PUNCT cana-1962 136 48 319 319 NUM cana-1962 136 49 https://internationalpubls.com https://internationalpubls.com X cana-1962 136 50 2(𝜐 2(𝜐 NUM cana-1962 136 51 ∘ ∘ PROPN cana-1962 136 52 πœβˆ—)πœ“1(𝜏)πœ“2(𝜏 πœβˆ—)πœ“1(𝜏)πœ“2(𝜏 NOUN cana-1962 136 53 ) ) PUNCT cana-1962 136 54 ∈ ∈ PROPN cana-1962 136 55 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 136 56 ) ) PUNCT cana-1962 136 57 βˆ€πœ βˆ€πœ NOUN cana-1962 137 1 ∈ ∈ PROPN cana-1962 137 2 𝔖. 𝔖. PROPN cana-1962 137 3 ( ( PUNCT cana-1962 137 4 10 10 NUM cana-1962 137 5 ) ) PUNCT cana-1962 137 6 since since SCONJ cana-1962 137 7 π‘β„Žπ‘Žπ‘Ÿ(𝔖 π‘β„Žπ‘Žπ‘Ÿ(𝔖 NOUN cana-1962 137 8 ) ) PUNCT cana-1962 137 9 β‰  β‰  PROPN cana-1962 137 10 2 2 NUM cana-1962 137 11 , , PUNCT cana-1962 137 12 and and CCONJ cana-1962 137 13 the the DET cana-1962 137 14 primeness primeness NOUN cana-1962 137 15 of of ADP cana-1962 137 16 𝔖 𝔖 PROPN cana-1962 137 17 , , PUNCT cana-1962 137 18 then then ADV cana-1962 137 19 𝜐 𝜐 PROPN cana-1962 137 20 ∘ ∘ NOUN cana-1962 137 21 πœβˆ— πœβˆ— PROPN cana-1962 137 22 ∈ ∈ PROPN cana-1962 137 23 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 137 24 ) ) PUNCT cana-1962 137 25 or or CCONJ cana-1962 137 26 πœ“1(𝜏)πœ“2(𝜏 πœ“1(𝜏)πœ“2(𝜏 PUNCT cana-1962 137 27 ) ) PUNCT cana-1962 137 28 = = SYM cana-1962 137 29 0 0 NUM cana-1962 137 30 , , PUNCT cana-1962 137 31 by by ADP cana-1962 137 32 lemme lemme PROPN cana-1962 137 33 1 1 NUM cana-1962 137 34 in in ADP cana-1962 137 35 case case NOUN cana-1962 137 36 𝜐 𝜐 PROPN cana-1962 137 37 ∘ ∘ PROPN cana-1962 137 38 πœβˆ— πœβˆ— PROPN cana-1962 137 39 ∈ ∈ PROPN cana-1962 137 40 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 137 41 ) ) PUNCT cana-1962 137 42 , , PUNCT cana-1962 137 43 then then ADV cana-1962 137 44 𝔖 𝔖 PROPN cana-1962 137 45 is be AUX cana-1962 137 46 an an DET cana-1962 137 47 integral integral ADJ cana-1962 137 48 domain domain NOUN cana-1962 137 49 or or CCONJ cana-1962 137 50 πœ“1(𝜏 πœ“1(𝜏 NOUN cana-1962 137 51 ) ) PUNCT cana-1962 137 52 = = SYM cana-1962 137 53 0 0 NUM cana-1962 137 54 or or CCONJ cana-1962 137 55 πœ“2(𝜏 πœ“2(𝜏 NUM cana-1962 137 56 ) ) PUNCT cana-1962 137 57 = = SYM cana-1962 137 58 0 0 NUM cana-1962 137 59 βˆ€πœ βˆ€πœ PROPN cana-1962 137 60 ∈ ∈ PROPN cana-1962 137 61 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 137 62 ) ) PUNCT cana-1962 137 63 . . PUNCT cana-1962 138 1 continue continue VERB cana-1962 138 2 using use VERB cana-1962 138 3 the the DET cana-1962 138 4 same same ADJ cana-1962 138 5 technique technique NOUN cana-1962 138 6 , , PUNCT cana-1962 138 7 using use VERB cana-1962 138 8 the the DET cana-1962 138 9 first first ADJ cana-1962 138 10 case case NOUN cana-1962 138 11 to to PART cana-1962 138 12 arrive arrive VERB cana-1962 138 13 at at ADP cana-1962 138 14 𝔖 𝔖 PROPN cana-1962 138 15 is be AUX cana-1962 138 16 an an DET cana-1962 138 17 integral integral ADJ cana-1962 138 18 domain domain NOUN cana-1962 138 19 . . PUNCT cana-1962 139 1 using use VERB cana-1962 139 2 the the DET cana-1962 139 3 same same ADJ cana-1962 139 4 technique technique NOUN cana-1962 139 5 , , PUNCT cana-1962 139 6 we we PRON cana-1962 139 7 discover discover VERB cana-1962 139 8 that that SCONJ cana-1962 139 9 the the DET cana-1962 139 10 other other ADJ cana-1962 139 11 identities identity NOUN cana-1962 139 12 lead lead VERB cana-1962 139 13 to to ADP cana-1962 139 14 𝔖 𝔖 PROPN cana-1962 139 15 being be AUX cana-1962 139 16 an an DET cana-1962 139 17 integral integral ADJ cana-1962 139 18 domain domain NOUN cana-1962 139 19 . . PUNCT cana-1962 140 1 it it PRON cana-1962 140 2 ’s ’ VERB cana-1962 140 3 obvious obvious ADJ cana-1962 140 4 that that SCONJ cana-1962 140 5 ±𝐼𝔖 ±𝐼𝔖 PROPN cana-1962 140 6 and and CCONJ cana-1962 140 7 0𝔖 0𝔖 NOUN cana-1962 140 8 are be AUX cana-1962 140 9 multipliers multiplier NOUN cana-1962 140 10 of of ADP cana-1962 140 11 𝔖. 𝔖. PROPN cana-1962 140 12 therefore therefore ADV cana-1962 140 13 , , PUNCT cana-1962 140 14 the the DET cana-1962 140 15 theorem theorem ADJ cana-1962 140 16 1 1 NUM cana-1962 140 17 leads lead VERB cana-1962 140 18 to to ADP cana-1962 140 19 the the DET cana-1962 140 20 following follow VERB cana-1962 140 21 corollary corollary NOUN cana-1962 140 22 : : PUNCT cana-1962 140 23 . . PUNCT cana-1962 141 1 corollary corollary ADJ cana-1962 141 2 1 1 NUM cana-1962 141 3 the the DET cana-1962 141 4 following following ADJ cana-1962 141 5 statements statement NOUN cana-1962 141 6 are be AUX cana-1962 141 7 equivalent equivalent ADJ cana-1962 141 8 : : PUNCT cana-1962 141 9 ( ( PUNCT cana-1962 141 10 i i NOUN cana-1962 141 11 ) ) PUNCT cana-1962 142 1 [ [ X cana-1962 142 2 Ξ³1(𝜐 Ξ³1(𝜐 X cana-1962 142 3 ) ) PUNCT cana-1962 142 4 , , PUNCT cana-1962 142 5 Ξ³2(πœβˆ— Ξ³2(πœβˆ— PROPN cana-1962 142 6 ) ) PUNCT cana-1962 142 7 ] ] PUNCT cana-1962 142 8 Β± Β± NOUN cana-1962 143 1 [ [ X cana-1962 143 2 𝜐 𝜐 X cana-1962 143 3 , , PUNCT cana-1962 143 4 πœβˆ— πœβˆ— PROPN cana-1962 143 5 ] ] X cana-1962 143 6 ∈ ∈ PROPN cana-1962 143 7 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 143 8 ) ) PUNCT cana-1962 143 9 βˆ€ βˆ€ X cana-1962 144 1 𝜐 𝜐 X cana-1962 144 2 ∈ ∈ NOUN cana-1962 144 3 𝔖. 𝔖. PROPN cana-1962 144 4 ( ( PUNCT cana-1962 144 5 in in ADP cana-1962 144 6 particular particular ADJ cana-1962 144 7 , , PUNCT cana-1962 144 8 [ [ X cana-1962 144 9 Ξ³1(𝜐 Ξ³1(𝜐 X cana-1962 144 10 ) ) PUNCT cana-1962 144 11 , , PUNCT cana-1962 144 12 Ξ³2(πœβˆ— Ξ³2(πœβˆ— PROPN cana-1962 144 13 ) ) PUNCT cana-1962 144 14 ] ] PUNCT cana-1962 144 15 Β± Β± NOUN cana-1962 145 1 [ [ X cana-1962 145 2 𝜐 𝜐 X cana-1962 145 3 , , PUNCT cana-1962 145 4 πœβˆ— πœβˆ— PROPN cana-1962 145 5 ] ] X cana-1962 145 6 = = SYM cana-1962 145 7 0 0 NUM cana-1962 145 8 βˆ€ βˆ€ NOUN cana-1962 145 9 𝜐 𝜐 NOUN cana-1962 145 10 ∈ ∈ PROPN cana-1962 145 11 𝔖. 𝔖. PROPN cana-1962 145 12 ) ) PUNCT cana-1962 145 13 ( ( PUNCT cana-1962 145 14 ii ii NOUN cana-1962 145 15 ) ) PUNCT cana-1962 145 16 Ξ³1(𝜐 Ξ³1(𝜐 NOUN cana-1962 145 17 ) ) PUNCT cana-1962 145 18 ∘ ∘ NOUN cana-1962 145 19 Ξ³2(πœβˆ— Ξ³2(πœβˆ— SYM cana-1962 145 20 ) ) PUNCT cana-1962 145 21 Β± Β± NOUN cana-1962 146 1 𝜐 𝜐 NOUN cana-1962 146 2 ∘ ∘ NOUN cana-1962 146 3 πœβˆ— πœβˆ— PROPN cana-1962 146 4 ∈ ∈ PROPN cana-1962 146 5 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 146 6 ) ) PUNCT cana-1962 146 7 βˆ€ βˆ€ X cana-1962 147 1 𝜐 𝜐 X cana-1962 147 2 ∈ ∈ NOUN cana-1962 147 3 𝔖. 𝔖. PROPN cana-1962 147 4 ( ( PUNCT cana-1962 147 5 in in ADP cana-1962 147 6 particular particular ADJ cana-1962 147 7 , , PUNCT cana-1962 147 8 Ξ³1(𝜐 Ξ³1(𝜐 PROPN cana-1962 147 9 ) ) PUNCT cana-1962 147 10 ∘ ∘ NOUN cana-1962 147 11 Ξ³2(πœβˆ— Ξ³2(πœβˆ— SYM cana-1962 147 12 ) ) PUNCT cana-1962 147 13 Β± Β± NOUN cana-1962 148 1 𝜐 𝜐 NOUN cana-1962 148 2 ∘ ∘ NOUN cana-1962 148 3 πœβˆ— πœβˆ— PROPN cana-1962 148 4 = = SYM cana-1962 148 5 0 0 NUM cana-1962 148 6 βˆ€ βˆ€ NOUN cana-1962 148 7 𝜐 𝜐 NOUN cana-1962 148 8 ∈ ∈ PROPN cana-1962 148 9 𝔖. 𝔖. PROPN cana-1962 148 10 ) ) PUNCT cana-1962 148 11 ( ( PUNCT cana-1962 148 12 iii iii NOUN cana-1962 148 13 ) ) PUNCT cana-1962 149 1 [ [ X cana-1962 149 2 Ξ³1(𝜐 Ξ³1(𝜐 X cana-1962 149 3 ) ) PUNCT cana-1962 149 4 , , PUNCT cana-1962 149 5 Ξ³2(πœβˆ— Ξ³2(πœβˆ— PROPN cana-1962 149 6 ) ) PUNCT cana-1962 149 7 ] ] PUNCT cana-1962 149 8 Β± Β± NOUN cana-1962 150 1 [ [ X cana-1962 150 2 𝜐 𝜐 NOUN cana-1962 150 3 ∘ ∘ NOUN cana-1962 150 4 πœβˆ— πœβˆ— PROPN cana-1962 150 5 ] ] X cana-1962 150 6 ∈ ∈ PROPN cana-1962 150 7 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 150 8 ) ) PUNCT cana-1962 150 9 βˆ€ βˆ€ X cana-1962 151 1 𝜐 𝜐 X cana-1962 151 2 ∈ ∈ NOUN cana-1962 151 3 𝔖. 𝔖. PROPN cana-1962 151 4 ( ( PUNCT cana-1962 151 5 in in ADP cana-1962 151 6 particular particular ADJ cana-1962 151 7 , , PUNCT cana-1962 151 8 [ [ X cana-1962 151 9 Ξ³1(𝜐 Ξ³1(𝜐 X cana-1962 151 10 ) ) PUNCT cana-1962 151 11 , , PUNCT cana-1962 151 12 Ξ³2(πœβˆ— Ξ³2(πœβˆ— PROPN cana-1962 151 13 ) ) PUNCT cana-1962 151 14 ] ] PUNCT cana-1962 151 15 Β± Β± NOUN cana-1962 152 1 [ [ X cana-1962 152 2 𝜐 𝜐 NOUN cana-1962 152 3 ∘ ∘ NOUN cana-1962 152 4 πœβˆ— πœβˆ— PROPN cana-1962 152 5 ] ] X cana-1962 152 6 = = SYM cana-1962 152 7 0 0 NUM cana-1962 152 8 βˆ€ βˆ€ NOUN cana-1962 152 9 𝜐 𝜐 NOUN cana-1962 152 10 ∈ ∈ PROPN cana-1962 152 11 𝔖. 𝔖. PROPN cana-1962 152 12 ) ) PUNCT cana-1962 152 13 ( ( PUNCT cana-1962 152 14 iv iv X cana-1962 152 15 ) ) PUNCT cana-1962 152 16 Ξ³1(𝜐 Ξ³1(𝜐 NOUN cana-1962 152 17 ) ) PUNCT cana-1962 152 18 ∘ ∘ NOUN cana-1962 152 19 Ξ³2(πœβˆ— Ξ³2(πœβˆ— SYM cana-1962 152 20 ) ) PUNCT cana-1962 152 21 Β± Β± NOUN cana-1962 153 1 [ [ X cana-1962 153 2 𝜐 𝜐 X cana-1962 153 3 , , PUNCT cana-1962 153 4 πœβˆ— πœβˆ— PROPN cana-1962 153 5 ] ] X cana-1962 153 6 ∈ ∈ PROPN cana-1962 153 7 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 153 8 ) ) PUNCT cana-1962 153 9 βˆ€ βˆ€ X cana-1962 154 1 𝜐 𝜐 X cana-1962 154 2 ∈ ∈ NOUN cana-1962 154 3 𝔖. 𝔖. PROPN cana-1962 154 4 ( ( PUNCT cana-1962 154 5 in in ADP cana-1962 154 6 particular particular ADJ cana-1962 154 7 , , PUNCT cana-1962 154 8 Ξ³1(𝜐 Ξ³1(𝜐 PROPN cana-1962 154 9 ) ) PUNCT cana-1962 154 10 ∘ ∘ NOUN cana-1962 154 11 Ξ³2(πœβˆ— Ξ³2(πœβˆ— SYM cana-1962 154 12 ) ) PUNCT cana-1962 154 13 Β± Β± NOUN cana-1962 155 1 [ [ X cana-1962 155 2 𝜐 𝜐 X cana-1962 155 3 , , PUNCT cana-1962 155 4 πœβˆ— πœβˆ— PROPN cana-1962 155 5 ] ] X cana-1962 155 6 = = SYM cana-1962 155 7 0 0 NUM cana-1962 155 8 βˆ€ βˆ€ NOUN cana-1962 155 9 𝜐 𝜐 NOUN cana-1962 155 10 ∈ ∈ PROPN cana-1962 155 11 𝔖. 𝔖. PROPN cana-1962 155 12 ) ) PUNCT cana-1962 155 13 ( ( PUNCT cana-1962 155 14 v v NOUN cana-1962 155 15 ) ) PUNCT cana-1962 156 1 [ [ X cana-1962 156 2 Ξ³1(𝜐 Ξ³1(𝜐 X cana-1962 156 3 ) ) PUNCT cana-1962 156 4 , , PUNCT cana-1962 156 5 Ξ³2(πœβˆ— Ξ³2(πœβˆ— PROPN cana-1962 156 6 ) ) PUNCT cana-1962 156 7 ] ] PUNCT cana-1962 157 1 ∈ ∈ PROPN cana-1962 157 2 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 157 3 ) ) PUNCT cana-1962 157 4 βˆ€ βˆ€ X cana-1962 158 1 𝜐 𝜐 X cana-1962 158 2 ∈ ∈ NOUN cana-1962 158 3 𝔖. 𝔖. PROPN cana-1962 158 4 ( ( PUNCT cana-1962 158 5 in in ADP cana-1962 158 6 particular particular ADJ cana-1962 158 7 , , PUNCT cana-1962 158 8 [ [ X cana-1962 158 9 Ξ³1(𝜐 Ξ³1(𝜐 X cana-1962 158 10 ) ) PUNCT cana-1962 158 11 , , PUNCT cana-1962 158 12 Ξ³2(πœβˆ— Ξ³2(πœβˆ— PROPN cana-1962 158 13 ) ) PUNCT cana-1962 158 14 ] ] PUNCT cana-1962 159 1 = = SYM cana-1962 159 2 0 0 NUM cana-1962 159 3 βˆ€ βˆ€ NOUN cana-1962 159 4 𝜐 𝜐 NOUN cana-1962 159 5 ∈ ∈ PROPN cana-1962 159 6 𝔖. 𝔖. PROPN cana-1962 159 7 ) ) PUNCT cana-1962 159 8 ( ( PUNCT cana-1962 159 9 vi vi NOUN cana-1962 159 10 ) ) PUNCT cana-1962 159 11 Ξ³1(𝜐 Ξ³1(𝜐 NOUN cana-1962 159 12 ) ) PUNCT cana-1962 159 13 ∘ ∘ NOUN cana-1962 159 14 Ξ³2(πœβˆ— Ξ³2(πœβˆ— SYM cana-1962 159 15 ) ) PUNCT cana-1962 159 16 ∈ ∈ PROPN cana-1962 159 17 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 159 18 ) ) PUNCT cana-1962 159 19 βˆ€ βˆ€ X cana-1962 160 1 𝜐 𝜐 X cana-1962 160 2 ∈ ∈ NOUN cana-1962 160 3 𝔖. 𝔖. PROPN cana-1962 160 4 ( ( PUNCT cana-1962 160 5 in in ADP cana-1962 160 6 particular particular ADJ cana-1962 160 7 , , PUNCT cana-1962 160 8 Ξ³1(𝜐 Ξ³1(𝜐 PROPN cana-1962 160 9 ) ) PUNCT cana-1962 160 10 ∘ ∘ NOUN cana-1962 160 11 Ξ³2(πœβˆ— Ξ³2(πœβˆ— PRON cana-1962 160 12 ) ) PUNCT cana-1962 160 13 = = SYM cana-1962 160 14 0 0 NUM cana-1962 160 15 βˆ€ βˆ€ NOUN cana-1962 160 16 𝜐 𝜐 NOUN cana-1962 160 17 ∈ ∈ PROPN cana-1962 160 18 𝔖. 𝔖. PROPN cana-1962 160 19 ) ) PUNCT cana-1962 160 20 ( ( PUNCT cana-1962 160 21 vii vii PROPN cana-1962 160 22 ) ) PUNCT cana-1962 160 23 𝔖 𝔖 PROPN cana-1962 160 24 is be AUX cana-1962 160 25 an an DET cana-1962 160 26 integral integral ADJ cana-1962 160 27 domain domain NOUN cana-1962 160 28 . . PUNCT cana-1962 161 1 a a DET cana-1962 161 2 derivation derivation NOUN cana-1962 161 3 πœ“ πœ“ NOUN cana-1962 161 4 is be AUX cana-1962 161 5 classified classify VERB cana-1962 161 6 as as ADP cana-1962 161 7 a a DET cana-1962 161 8 generalized generalized ADJ cana-1962 161 9 derivation derivation NOUN cana-1962 161 10 . . PUNCT cana-1962 162 1 by by ADP cana-1962 162 2 setting set VERB cana-1962 162 3 Ξ³ Ξ³ PROPN cana-1962 162 4 = = SYM cana-1962 162 5 πœ“ πœ“ PROPN cana-1962 162 6 and and CCONJ cana-1962 162 7 β–³ β–³ NOUN cana-1962 162 8 = = SYM cana-1962 162 9 ±𝐼𝔖 ±𝐼𝔖 NOUN cana-1962 162 10 in in ADP cana-1962 162 11 the the DET cana-1962 162 12 preceding precede VERB cana-1962 162 13 theorem theorem NOUN cana-1962 162 14 , , PUNCT cana-1962 162 15 we we PRON cana-1962 162 16 arrive arrive VERB cana-1962 162 17 at at ADP cana-1962 162 18 the the DET cana-1962 162 19 following follow VERB cana-1962 162 20 corollary corollary NOUN cana-1962 162 21 , , PUNCT cana-1962 162 22 which which PRON cana-1962 162 23 holds hold VERB cana-1962 162 24 significant significant ADJ cana-1962 162 25 importance importance NOUN cana-1962 162 26 in in ADP cana-1962 162 27 the the DET cana-1962 162 28 work work NOUN cana-1962 162 29 of of ADP cana-1962 162 30 [ [ X cana-1962 162 31 [ [ X cana-1962 162 32 8 8 NUM cana-1962 162 33 ] ] SYM cana-1962 162 34 ] ] PUNCT cana-1962 162 35 . . PUNCT cana-1962 163 1 corollary corollary ADJ cana-1962 163 2 2 2 NUM cana-1962 164 1 [ [ X cana-1962 164 2 [ [ X cana-1962 164 3 8 8 NUM cana-1962 164 4 ] ] X cana-1962 164 5 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘šπ‘  π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘šπ‘  PROPN cana-1962 164 6 3.5 3.5 NUM cana-1962 164 7 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ cana-1962 164 8 3.8 3.8 NUM cana-1962 164 9 ] ] PUNCT cana-1962 164 10 let let VERB cana-1962 164 11 𝔖 𝔖 PRON cana-1962 164 12 be be AUX cana-1962 164 13 a a DET cana-1962 164 14 prime prime ADJ cana-1962 164 15 ring ring NOUN cana-1962 164 16 with with ADP cana-1962 164 17 involution involution NOUN cana-1962 164 18 . . PUNCT cana-1962 165 1 if if SCONJ cana-1962 165 2 𝔖 𝔖 PROPN cana-1962 165 3 admits admit VERB cana-1962 165 4 a a DET cana-1962 165 5 derivation derivation NOUN cana-1962 165 6 πœ“ πœ“ NOUN cana-1962 165 7 , , PUNCT cana-1962 165 8 then then ADV cana-1962 165 9 the the DET cana-1962 165 10 following follow VERB cana-1962 165 11 assertions assertion NOUN cana-1962 165 12 are be AUX cana-1962 165 13 equivalents equivalent NOUN cana-1962 165 14 : : PUNCT cana-1962 165 15 ( ( PUNCT cana-1962 165 16 i i NOUN cana-1962 165 17 ) ) PUNCT cana-1962 166 1 [ [ X cana-1962 166 2 πœ“(𝜐 πœ“(𝜐 X cana-1962 166 3 ) ) PUNCT cana-1962 166 4 , , PUNCT cana-1962 166 5 πœ“(πœβˆ— πœ“(πœβˆ— NUM cana-1962 166 6 ) ) PUNCT cana-1962 166 7 ] ] PUNCT cana-1962 167 1 Β± Β± NOUN cana-1962 168 1 [ [ X cana-1962 168 2 𝜐 𝜐 X cana-1962 168 3 , , PUNCT cana-1962 168 4 πœβˆ— πœβˆ— PROPN cana-1962 168 5 ] ] X cana-1962 168 6 ∈ ∈ PROPN cana-1962 168 7 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 168 8 ) ) PUNCT cana-1962 168 9 βˆ€ βˆ€ X cana-1962 169 1 𝜐 𝜐 X cana-1962 169 2 ∈ ∈ PROPN cana-1962 170 1 𝔖. 𝔖. PROPN cana-1962 170 2 ( ( PUNCT cana-1962 170 3 ii ii NOUN cana-1962 170 4 ) ) PUNCT cana-1962 170 5 πœ“(𝜐 πœ“(𝜐 NOUN cana-1962 170 6 ) ) PUNCT cana-1962 170 7 ∘ ∘ NOUN cana-1962 170 8 πœ“(πœβˆ— πœ“(πœβˆ— NUM cana-1962 170 9 ) ) PUNCT cana-1962 170 10 Β± Β± NOUN cana-1962 171 1 𝜐 𝜐 NOUN cana-1962 171 2 ∘ ∘ NOUN cana-1962 171 3 πœβˆ— πœβˆ— PROPN cana-1962 171 4 ∈ ∈ PROPN cana-1962 171 5 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 171 6 ) ) PUNCT cana-1962 171 7 βˆ€ βˆ€ X cana-1962 172 1 𝜐 𝜐 X cana-1962 172 2 ∈ ∈ PROPN cana-1962 173 1 𝔖. 𝔖. PROPN cana-1962 173 2 ( ( PUNCT cana-1962 173 3 iii iii NOUN cana-1962 173 4 ) ) PUNCT cana-1962 174 1 [ [ X cana-1962 174 2 πœ“(𝜐 πœ“(𝜐 X cana-1962 174 3 ) ) PUNCT cana-1962 174 4 , , PUNCT cana-1962 174 5 πœ“(πœβˆ— πœ“(πœβˆ— NUM cana-1962 174 6 ) ) PUNCT cana-1962 174 7 ] ] PUNCT cana-1962 174 8 Β± Β± NOUN cana-1962 175 1 [ [ X cana-1962 175 2 𝜐 𝜐 NOUN cana-1962 175 3 ∘ ∘ NOUN cana-1962 175 4 πœβˆ— πœβˆ— PROPN cana-1962 175 5 ] ] X cana-1962 175 6 ∈ ∈ PROPN cana-1962 175 7 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 175 8 ) ) PUNCT cana-1962 175 9 βˆ€ βˆ€ X cana-1962 176 1 𝜐 𝜐 X cana-1962 176 2 ∈ ∈ PROPN cana-1962 176 3 𝔖. 𝔖. PROPN cana-1962 176 4 ( ( PUNCT cana-1962 176 5 iv iv X cana-1962 176 6 ) ) PUNCT cana-1962 176 7 πœ“(𝜐 πœ“(𝜐 NOUN cana-1962 176 8 ) ) PUNCT cana-1962 176 9 ∘ ∘ NOUN cana-1962 176 10 πœ“(πœβˆ— πœ“(πœβˆ— NUM cana-1962 176 11 ) ) PUNCT cana-1962 176 12 Β± Β± NOUN cana-1962 177 1 [ [ X cana-1962 177 2 𝜐 𝜐 X cana-1962 177 3 , , PUNCT cana-1962 177 4 πœβˆ— πœβˆ— PROPN cana-1962 177 5 ] ] X cana-1962 177 6 ∈ ∈ PROPN cana-1962 177 7 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 177 8 ) ) PUNCT cana-1962 177 9 βˆ€ βˆ€ X cana-1962 178 1 𝜐 𝜐 X cana-1962 178 2 ∈ ∈ PROPN cana-1962 178 3 𝔖. 𝔖. PROPN cana-1962 178 4 ( ( PUNCT cana-1962 178 5 v v NOUN cana-1962 178 6 ) ) PUNCT cana-1962 178 7 𝔖 𝔖 NOUN cana-1962 178 8 is be AUX cana-1962 178 9 an an DET cana-1962 178 10 integral integral ADJ cana-1962 178 11 domain domain NOUN cana-1962 178 12 . . PUNCT cana-1962 179 1 theorem theorem VERB cana-1962 179 2 2 2 NUM cana-1962 179 3 the the DET cana-1962 179 4 following following ADJ cana-1962 179 5 statements statement NOUN cana-1962 179 6 are be AUX cana-1962 179 7 equivalent equivalent ADJ cana-1962 179 8 : : PUNCT cana-1962 179 9 ( ( PUNCT cana-1962 179 10 i i NOUN cana-1962 179 11 ) ) PUNCT cana-1962 179 12 Ξ³1(πœπœβˆ— Ξ³1(πœπœβˆ— PROPN cana-1962 179 13 ) ) PUNCT cana-1962 179 14 Β± Β± NUM cana-1962 179 15 Ξ³2(𝜐)Ξ³2(πœβˆ— Ξ³2(𝜐)Ξ³2(πœβˆ— PROPN cana-1962 179 16 ) ) PUNCT cana-1962 180 1 + + NOUN cana-1962 180 2 β–³ β–³ X cana-1962 180 3 ( ( PUNCT cana-1962 180 4 [ [ X cana-1962 180 5 𝜐 𝜐 X cana-1962 180 6 , , PUNCT cana-1962 180 7 πœβˆ— πœβˆ— PROPN cana-1962 180 8 ] ] PUNCT cana-1962 180 9 ) ) PUNCT cana-1962 180 10 ∈ ∈ PROPN cana-1962 180 11 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 180 12 ) ) PUNCT cana-1962 180 13 βˆ€ βˆ€ X cana-1962 181 1 𝜐 𝜐 X cana-1962 181 2 ∈ ∈ PROPN cana-1962 181 3 𝔖. 𝔖. PROPN cana-1962 181 4 ( ( PUNCT cana-1962 181 5 ii ii NOUN cana-1962 181 6 ) ) PUNCT cana-1962 181 7 Ξ³1(πœπœβˆ— Ξ³1(πœπœβˆ— NOUN cana-1962 181 8 ) ) PUNCT cana-1962 181 9 Β± Β± NUM cana-1962 181 10 Ξ³2(𝜐)Ξ³2(πœβˆ— Ξ³2(𝜐)Ξ³2(πœβˆ— PROPN cana-1962 181 11 ) ) PUNCT cana-1962 182 1 + + NOUN cana-1962 182 2 β–³ β–³ X cana-1962 182 3 ( ( PUNCT cana-1962 182 4 𝜐 𝜐 PROPN cana-1962 182 5 ∘ ∘ PROPN cana-1962 182 6 πœβˆ— πœβˆ— PROPN cana-1962 182 7 ) ) PUNCT cana-1962 182 8 ∈ ∈ PROPN cana-1962 182 9 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 182 10 ) ) PUNCT cana-1962 182 11 βˆ€ βˆ€ X cana-1962 183 1 𝜐 𝜐 X cana-1962 183 2 ∈ ∈ PROPN cana-1962 183 3 𝔖. 𝔖. PROPN cana-1962 183 4 ( ( PUNCT cana-1962 183 5 iii iii NOUN cana-1962 183 6 ) ) PUNCT cana-1962 183 7 Ξ³1(πœπœβˆ— Ξ³1(πœπœβˆ— NOUN cana-1962 183 8 ) ) PUNCT cana-1962 183 9 Β± Β± NOUN cana-1962 183 10 Ξ³2(πœβˆ—)Ξ³2(𝜐 Ξ³2(πœβˆ—)Ξ³2(𝜐 PUNCT cana-1962 183 11 ) ) PUNCT cana-1962 184 1 + + NOUN cana-1962 184 2 β–³ β–³ X cana-1962 184 3 ( ( PUNCT cana-1962 184 4 [ [ X cana-1962 184 5 𝜐 𝜐 X cana-1962 184 6 , , PUNCT cana-1962 184 7 πœβˆ— πœβˆ— PROPN cana-1962 184 8 ] ] PUNCT cana-1962 184 9 ) ) PUNCT cana-1962 184 10 ∈ ∈ PROPN cana-1962 184 11 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 184 12 ) ) PUNCT cana-1962 184 13 βˆ€ βˆ€ X cana-1962 185 1 𝜐 𝜐 X cana-1962 185 2 ∈ ∈ PROPN cana-1962 185 3 𝔖. 𝔖. PROPN cana-1962 185 4 ( ( PUNCT cana-1962 185 5 iv iv NOUN cana-1962 185 6 ) ) PUNCT cana-1962 185 7 Ξ³1(πœπœβˆ— Ξ³1(πœπœβˆ— NOUN cana-1962 185 8 ) ) PUNCT cana-1962 185 9 Β± Β± NOUN cana-1962 185 10 Ξ³2(πœβˆ—)Ξ³2(𝜐 Ξ³2(πœβˆ—)Ξ³2(𝜐 PUNCT cana-1962 185 11 ) ) PUNCT cana-1962 186 1 + + NOUN cana-1962 186 2 β–³ β–³ X cana-1962 186 3 ( ( PUNCT cana-1962 186 4 𝜐 𝜐 PROPN cana-1962 186 5 ∘ ∘ PROPN cana-1962 186 6 πœβˆ— πœβˆ— PROPN cana-1962 186 7 ) ) PUNCT cana-1962 186 8 ∈ ∈ PROPN cana-1962 186 9 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 186 10 ) ) PUNCT cana-1962 186 11 βˆ€ βˆ€ X cana-1962 187 1 𝜐 𝜐 X cana-1962 187 2 ∈ ∈ PROPN cana-1962 187 3 𝔖. 𝔖. PROPN cana-1962 187 4 ( ( PUNCT cana-1962 187 5 v v NOUN cana-1962 187 6 ) ) PUNCT cana-1962 187 7 𝔖 𝔖 NOUN cana-1962 187 8 is be AUX cana-1962 187 9 an an DET cana-1962 187 10 integral integral ADJ cana-1962 187 11 domain domain NOUN cana-1962 187 12 . . PUNCT cana-1962 188 1 proof proof NOUN cana-1962 188 2 . . PUNCT cana-1962 189 1 it it PRON cana-1962 189 2 is be AUX cana-1962 189 3 sufficient sufficient ADJ cana-1962 189 4 for for SCONJ cana-1962 189 5 us we PRON cana-1962 189 6 to to PART cana-1962 189 7 demonstrate demonstrate VERB cana-1962 189 8 that that SCONJ cana-1962 189 9 ( ( PUNCT cana-1962 189 10 𝑖 𝑖 X cana-1962 189 11 ) ) PUNCT cana-1962 189 12 , , PUNCT cana-1962 189 13 ( ( PUNCT cana-1962 189 14 𝑖𝑖 𝑖𝑖 NOUN cana-1962 189 15 ) ) PUNCT cana-1962 189 16 , , PUNCT cana-1962 189 17 ( ( PUNCT cana-1962 189 18 𝑖𝑖𝑖 𝑖𝑖𝑖 NOUN cana-1962 189 19 ) ) PUNCT cana-1962 189 20 , , PUNCT cana-1962 189 21 and and CCONJ cana-1962 189 22 ( ( PUNCT cana-1962 189 23 𝑖𝑣 𝑖𝑣 X cana-1962 189 24 ) ) PUNCT cana-1962 189 25 ⟹ ⟹ VERB cana-1962 190 1 ( ( PUNCT cana-1962 190 2 𝑣 𝑣 NOUN cana-1962 190 3 ) ) PUNCT cana-1962 190 4 . . PUNCT cana-1962 191 1 we we PRON cana-1962 191 2 begin begin VERB cana-1962 191 3 with with ADP cana-1962 191 4 ( ( PUNCT cana-1962 191 5 𝑖 𝑖 X cana-1962 191 6 ) ) PUNCT cana-1962 191 7 ⟹ ⟹ VERB cana-1962 191 8 ( ( PUNCT cana-1962 191 9 𝑣 𝑣 NOUN cana-1962 191 10 ) ) PUNCT cana-1962 191 11 and and CCONJ cana-1962 191 12 suppose suppose VERB cana-1962 191 13 that that SCONJ cana-1962 191 14 Ξ³1(πœπœβˆ— Ξ³1(πœπœβˆ— NOUN cana-1962 191 15 ) ) PUNCT cana-1962 191 16 + + CCONJ cana-1962 191 17 Ξ³2(𝜐)Ξ³2(πœβˆ— Ξ³2(𝜐)Ξ³2(πœβˆ— VERB cana-1962 191 18 ) ) PUNCT cana-1962 192 1 + + NOUN cana-1962 192 2 β–³ β–³ X cana-1962 192 3 ( ( PUNCT cana-1962 192 4 [ [ X cana-1962 192 5 𝜐 𝜐 X cana-1962 192 6 , , PUNCT cana-1962 192 7 πœβˆ— πœβˆ— PROPN cana-1962 192 8 ] ] PUNCT cana-1962 192 9 ) ) PUNCT cana-1962 192 10 ∈ ∈ PROPN cana-1962 192 11 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 192 12 ) ) PUNCT cana-1962 192 13 βˆ€πœ βˆ€πœ NOUN cana-1962 192 14 ∈ ∈ PROPN cana-1962 192 15 𝔖. 𝔖. PROPN cana-1962 192 16 ( ( PUNCT cana-1962 192 17 11 11 NUM cana-1962 192 18 ) ) PUNCT cana-1962 192 19 replacing replace VERB cana-1962 192 20 𝜐 𝜐 PRON cana-1962 192 21 by by ADP cana-1962 192 22 𝜐 𝜐 PROPN cana-1962 192 23 + + CCONJ cana-1962 192 24 πœ” πœ” X cana-1962 192 25 in in ADP cana-1962 192 26 ( ( PUNCT cana-1962 192 27 11 11 NUM cana-1962 192 28 ) ) PUNCT cana-1962 192 29 and and CCONJ cana-1962 192 30 using use VERB cana-1962 192 31 it it PRON cana-1962 192 32 , , PUNCT cana-1962 192 33 we we PRON cana-1962 192 34 obtain obtain VERB cana-1962 192 35 Ξ³1(πœπœ”βˆ— Ξ³1(πœπœ”βˆ— PUNCT cana-1962 192 36 ) ) PUNCT cana-1962 193 1 + + NUM cana-1962 193 2 Ξ³1(πœ”πœβˆ— Ξ³1(πœ”πœβˆ— NOUN cana-1962 193 3 ) ) PUNCT cana-1962 194 1 + + CCONJ cana-1962 194 2 Ξ³2(𝜐)Ξ³2(πœ”βˆ— Ξ³2(𝜐)Ξ³2(πœ”βˆ— X cana-1962 194 3 ) ) PUNCT cana-1962 194 4 + + CCONJ cana-1962 194 5 Ξ³2(πœ”)Ξ³2(πœβˆ— Ξ³2(πœ”)Ξ³2(πœβˆ— X cana-1962 194 6 ) ) PUNCT cana-1962 195 1 + + NOUN cana-1962 195 2 β–³ β–³ X cana-1962 195 3 ( ( PUNCT cana-1962 195 4 [ [ X cana-1962 195 5 𝜐 𝜐 X cana-1962 195 6 , , PUNCT cana-1962 195 7 πœ”βˆ— πœ”βˆ— PROPN cana-1962 195 8 ] ] PUNCT cana-1962 195 9 ) ) PUNCT cana-1962 196 1 + + NOUN cana-1962 196 2 β–³ β–³ X cana-1962 196 3 ( ( PUNCT cana-1962 196 4 [ [ X cana-1962 196 5 πœ” πœ” X cana-1962 196 6 , , PUNCT cana-1962 196 7 πœβˆ— πœβˆ— PROPN cana-1962 196 8 ] ] PUNCT cana-1962 196 9 ) ) PUNCT cana-1962 196 10 ∈ ∈ PROPN cana-1962 196 11 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 196 12 ) ) PUNCT cana-1962 196 13 βˆ€πœ βˆ€πœ NOUN cana-1962 196 14 , , PUNCT cana-1962 196 15 πœ” πœ” PART cana-1962 196 16 ∈ ∈ PROPN cana-1962 196 17 𝔖. 𝔖. PROPN cana-1962 196 18 ( ( PUNCT cana-1962 196 19 12 12 NUM cana-1962 196 20 ) ) PUNCT cana-1962 196 21 replacing replace VERB cana-1962 196 22 πœ” πœ” X cana-1962 196 23 by by ADP cana-1962 196 24 πœ”πœ πœ”πœ NOUN cana-1962 196 25 in in ADP cana-1962 196 26 ( ( PUNCT cana-1962 196 27 12 12 NUM cana-1962 196 28 ) ) PUNCT cana-1962 196 29 where where SCONJ cana-1962 196 30 𝜏 𝜏 PRON cana-1962 196 31 ∈ ∈ PROPN cana-1962 196 32 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 196 33 ) ) PUNCT cana-1962 196 34 ∩ ∩ NOUN cana-1962 196 35 β–³ β–³ X cana-1962 196 36 ( ( PUNCT cana-1962 196 37 𝔖 𝔖 PROPN cana-1962 196 38 ) ) PUNCT cana-1962 196 39 and and CCONJ cana-1962 196 40 using use VERB cana-1962 196 41 it it PRON cana-1962 196 42 , , PUNCT cana-1962 196 43 we we PRON cana-1962 196 44 get get VERB cana-1962 196 45 communications communication NOUN cana-1962 196 46 on on ADP cana-1962 196 47 applied apply VERB cana-1962 196 48 nonlinear nonlinear ADJ cana-1962 196 49 analysis analysis NOUN cana-1962 196 50 issn issn NOUN cana-1962 196 51 : : PUNCT cana-1962 196 52 1074 1074 NUM cana-1962 196 53 - - PUNCT cana-1962 196 54 133x 133x NUM cana-1962 196 55 vol vol NOUN cana-1962 196 56 32 32 NUM cana-1962 196 57 no no NOUN cana-1962 196 58 . . NOUN cana-1962 196 59 3 3 NUM cana-1962 196 60 ( ( PUNCT cana-1962 196 61 2025 2025 NUM cana-1962 196 62 ) ) PUNCT cana-1962 196 63 320 320 NUM cana-1962 196 64 https://internationalpubls.com https://internationalpubls.com X cana-1962 196 65 ( ( PUNCT cana-1962 196 66 πœπœ”βˆ— πœπœ”βˆ— NOUN cana-1962 196 67 + + CCONJ cana-1962 196 68 πœ”πœβˆ—)πœ“1(𝜏 πœ”πœβˆ—)πœ“1(𝜏 NOUN cana-1962 196 69 ) ) PUNCT cana-1962 197 1 + + CCONJ cana-1962 197 2 Ξ³2(𝜐)πœ”βˆ—πœ“2(𝜏 Ξ³2(𝜐)πœ”βˆ—πœ“2(𝜏 ADJ cana-1962 197 3 ) ) PUNCT cana-1962 198 1 + + SYM cana-1962 198 2 πœ”Ξ³2(πœβˆ—)πœ“2(𝜏 πœ”Ξ³2(πœβˆ—)πœ“2(𝜏 X cana-1962 198 3 ) ) PUNCT cana-1962 198 4 ∈ ∈ PROPN cana-1962 198 5 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 198 6 ) ) PUNCT cana-1962 198 7 βˆ€πœ βˆ€πœ NOUN cana-1962 198 8 , , PUNCT cana-1962 198 9 πœ” πœ” PART cana-1962 198 10 ∈ ∈ PROPN cana-1962 198 11 𝔖. 𝔖. PROPN cana-1962 198 12 ( ( PUNCT cana-1962 198 13 13 13 NUM cana-1962 198 14 ) ) PUNCT cana-1962 198 15 replacing replace VERB cana-1962 198 16 𝜐 𝜐 PRON cana-1962 198 17 by by ADP cana-1962 198 18 πœπœ„ πœπœ„ NOUN cana-1962 198 19 in in ADP cana-1962 198 20 ( ( PUNCT cana-1962 198 21 13 13 NUM cana-1962 198 22 ) ) PUNCT cana-1962 198 23 where where SCONJ cana-1962 198 24 πœ„ πœ„ PROPN cana-1962 198 25 ∈ ∈ PROPN cana-1962 198 26 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 198 27 ) ) PUNCT cana-1962 198 28 ∩ ∩ NOUN cana-1962 198 29 β–³ β–³ X cana-1962 198 30 ( ( PUNCT cana-1962 198 31 𝔖 𝔖 PROPN cana-1962 198 32 ) ) PUNCT cana-1962 198 33 and and CCONJ cana-1962 198 34 using use VERB cana-1962 198 35 it it PRON cana-1962 198 36 , , PUNCT cana-1962 198 37 we we PRON cana-1962 198 38 obtain obtain VERB cana-1962 198 39 ( ( PUNCT cana-1962 198 40 πœπœ”βˆ— πœπœ”βˆ— NOUN cana-1962 198 41 + + CCONJ cana-1962 198 42 πœ”πœβˆ—)πœ“2(πœ„)πœ“2(𝜏 πœ”πœβˆ—)πœ“2(πœ„)πœ“2(𝜏 PROPN cana-1962 198 43 ) ) PUNCT cana-1962 198 44 ∈ ∈ PROPN cana-1962 198 45 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 198 46 ) ) PUNCT cana-1962 198 47 βˆ€πœ βˆ€πœ NOUN cana-1962 198 48 , , PUNCT cana-1962 198 49 πœ” πœ” PART cana-1962 198 50 ∈ ∈ PROPN cana-1962 198 51 𝔖. 𝔖. PROPN cana-1962 198 52 ( ( PUNCT cana-1962 198 53 14 14 NUM cana-1962 198 54 ) ) PUNCT cana-1962 198 55 taking take VERB cana-1962 198 56 πœ” πœ” PRON cana-1962 198 57 = = NUM cana-1962 198 58 𝜐 𝜐 X cana-1962 198 59 in in ADP cana-1962 198 60 ( ( PUNCT cana-1962 198 61 14 14 NUM cana-1962 198 62 ) ) PUNCT cana-1962 198 63 , , PUNCT cana-1962 198 64 we we PRON cana-1962 198 65 get get VERB cana-1962 198 66 2(𝜐 2(𝜐 NUM cana-1962 198 67 ∘ ∘ NUM cana-1962 198 68 πœβˆ—)πœ“2(πœ„)πœ“2(𝜏 πœβˆ—)πœ“2(πœ„)πœ“2(𝜏 X cana-1962 198 69 ) ) PUNCT cana-1962 198 70 ∈ ∈ PROPN cana-1962 198 71 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 198 72 ) ) PUNCT cana-1962 198 73 βˆ€πœ βˆ€πœ NOUN cana-1962 198 74 ∈ ∈ PROPN cana-1962 198 75 𝔖. 𝔖. PROPN cana-1962 198 76 ( ( PUNCT cana-1962 198 77 15 15 NUM cana-1962 198 78 ) ) PUNCT cana-1962 198 79 since since SCONJ cana-1962 198 80 π‘β„Žπ‘Žπ‘Ÿ(𝔖 π‘β„Žπ‘Žπ‘Ÿ(𝔖 NOUN cana-1962 198 81 ) ) PUNCT cana-1962 199 1 β‰  β‰  PROPN cana-1962 199 2 2 2 NUM cana-1962 199 3 , , PUNCT cana-1962 199 4 and and CCONJ cana-1962 199 5 by by ADP cana-1962 199 6 lemme lemme PROPN cana-1962 199 7 1 1 NUM cana-1962 199 8 , , PUNCT cana-1962 199 9 we we PRON cana-1962 199 10 get get VERB cana-1962 199 11 𝜐 𝜐 PRON cana-1962 199 12 ∘ ∘ NOUN cana-1962 199 13 πœβˆ— πœβˆ— PROPN cana-1962 199 14 ∈ ∈ PROPN cana-1962 199 15 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 199 16 ) ) PUNCT cana-1962 199 17 or or CCONJ cana-1962 199 18 πœ“2(πœ„)πœ“2(𝜏 πœ“2(πœ„)πœ“2(𝜏 PUNCT cana-1962 199 19 ) ) PUNCT cana-1962 199 20 = = SYM cana-1962 199 21 0 0 NUM cana-1962 199 22 , , PUNCT cana-1962 199 23 by by ADP cana-1962 199 24 primness primness NOUN cana-1962 199 25 that that PRON cana-1962 199 26 leads lead VERB cana-1962 199 27 to to ADP cana-1962 199 28 𝔖 𝔖 PROPN cana-1962 199 29 is be AUX cana-1962 199 30 an an DET cana-1962 199 31 integral integral ADJ cana-1962 199 32 domain domain NOUN cana-1962 199 33 or or CCONJ cana-1962 199 34 πœ“2(𝜏 πœ“2(𝜏 SYM cana-1962 199 35 ) ) PUNCT cana-1962 199 36 = = SYM cana-1962 200 1 0 0 NUM cana-1962 200 2 βˆ€πœ βˆ€πœ PROPN cana-1962 200 3 ∈ ∈ PROPN cana-1962 200 4 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 200 5 ) ) PUNCT cana-1962 200 6 , , PUNCT cana-1962 200 7 then then ADV cana-1962 200 8 πœ“2 πœ“2 NOUN cana-1962 200 9 = = SYM cana-1962 200 10 0 0 NUM cana-1962 200 11 , , PUNCT cana-1962 200 12 it it PRON cana-1962 200 13 ’s ’ VERB cana-1962 200 14 a a DET cana-1962 200 15 contradiction contradiction NOUN cana-1962 200 16 . . PUNCT cana-1962 201 1 using use VERB cana-1962 201 2 the the DET cana-1962 201 3 same same ADJ cana-1962 201 4 technique technique NOUN cana-1962 201 5 , , PUNCT cana-1962 201 6 we we PRON cana-1962 201 7 discover discover VERB cana-1962 201 8 that that SCONJ cana-1962 201 9 the the DET cana-1962 201 10 other other ADJ cana-1962 201 11 identities identity NOUN cana-1962 201 12 lead lead VERB cana-1962 201 13 to to ADP cana-1962 201 14 𝔖 𝔖 PROPN cana-1962 201 15 being be AUX cana-1962 201 16 an an DET cana-1962 201 17 integral integral ADJ cana-1962 201 18 domain domain NOUN cana-1962 201 19 . . PUNCT cana-1962 202 1 replacing replace VERB cana-1962 202 2 β–³ β–³ PUNCT cana-1962 202 3 by by ADP cana-1962 202 4 ±𝐼𝔖 ±𝐼𝔖 PROPN cana-1962 202 5 and and CCONJ cana-1962 202 6 0𝔖 0𝔖 ADJ cana-1962 202 7 in in ADP cana-1962 202 8 above above ADP cana-1962 202 9 theorem theorem VERB cana-1962 202 10 , , PUNCT cana-1962 202 11 we we PRON cana-1962 202 12 get get VERB cana-1962 202 13 the the DET cana-1962 202 14 following follow VERB cana-1962 202 15 corollary corollary NOUN cana-1962 202 16 . . PUNCT cana-1962 203 1 corollary corollary ADJ cana-1962 203 2 3 3 NUM cana-1962 203 3 the the DET cana-1962 203 4 following follow VERB cana-1962 203 5 assertions assertion NOUN cana-1962 203 6 are be AUX cana-1962 203 7 equivalents equivalent NOUN cana-1962 203 8 : : PUNCT cana-1962 203 9 ( ( PUNCT cana-1962 203 10 i i NOUN cana-1962 203 11 ) ) PUNCT cana-1962 203 12 Ξ³1(πœπœβˆ— Ξ³1(πœπœβˆ— PROPN cana-1962 203 13 ) ) PUNCT cana-1962 203 14 Β± Β± NUM cana-1962 203 15 Ξ³2(𝜐)Ξ³2(πœβˆ— Ξ³2(𝜐)Ξ³2(πœβˆ— PROPN cana-1962 203 16 ) ) PUNCT cana-1962 203 17 Β± Β± NOUN cana-1962 204 1 [ [ X cana-1962 204 2 𝜐 𝜐 X cana-1962 204 3 , , PUNCT cana-1962 204 4 πœβˆ— πœβˆ— PROPN cana-1962 204 5 ] ] X cana-1962 204 6 ∈ ∈ PROPN cana-1962 204 7 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 204 8 ) ) PUNCT cana-1962 204 9 βˆ€ βˆ€ X cana-1962 205 1 𝜐 𝜐 X cana-1962 205 2 ∈ ∈ PROPN cana-1962 205 3 𝔖. 𝔖. PROPN cana-1962 205 4 ( ( PUNCT cana-1962 205 5 ii ii NOUN cana-1962 205 6 ) ) PUNCT cana-1962 205 7 Ξ³1(πœπœβˆ— Ξ³1(πœπœβˆ— NOUN cana-1962 205 8 ) ) PUNCT cana-1962 205 9 Β± Β± NUM cana-1962 205 10 Ξ³2(𝜐)Ξ³2(πœβˆ— Ξ³2(𝜐)Ξ³2(πœβˆ— PROPN cana-1962 205 11 ) ) PUNCT cana-1962 205 12 Β± Β± NOUN cana-1962 206 1 𝜐 𝜐 NOUN cana-1962 206 2 ∘ ∘ NOUN cana-1962 206 3 πœβˆ— πœβˆ— PROPN cana-1962 206 4 ∈ ∈ PROPN cana-1962 206 5 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 206 6 ) ) PUNCT cana-1962 206 7 βˆ€ βˆ€ X cana-1962 207 1 𝜐 𝜐 X cana-1962 207 2 ∈ ∈ PROPN cana-1962 207 3 𝔖. 𝔖. PROPN cana-1962 207 4 ( ( PUNCT cana-1962 207 5 iii iii NOUN cana-1962 207 6 ) ) PUNCT cana-1962 207 7 Ξ³1(πœπœβˆ— Ξ³1(πœπœβˆ— NOUN cana-1962 207 8 ) ) PUNCT cana-1962 207 9 Β± Β± NOUN cana-1962 207 10 Ξ³2(πœβˆ—)Ξ³2(𝜐 Ξ³2(πœβˆ—)Ξ³2(𝜐 NOUN cana-1962 207 11 ) ) PUNCT cana-1962 207 12 Β± Β± NOUN cana-1962 208 1 [ [ X cana-1962 208 2 𝜐 𝜐 X cana-1962 208 3 , , PUNCT cana-1962 208 4 πœβˆ— πœβˆ— PROPN cana-1962 208 5 ] ] X cana-1962 208 6 ∈ ∈ PROPN cana-1962 208 7 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 208 8 ) ) PUNCT cana-1962 208 9 βˆ€ βˆ€ X cana-1962 209 1 𝜐 𝜐 X cana-1962 209 2 ∈ ∈ PROPN cana-1962 209 3 𝔖. 𝔖. PROPN cana-1962 209 4 ( ( PUNCT cana-1962 209 5 iv iv NOUN cana-1962 209 6 ) ) PUNCT cana-1962 209 7 Ξ³1(πœπœβˆ— Ξ³1(πœπœβˆ— NOUN cana-1962 209 8 ) ) PUNCT cana-1962 209 9 Β± Β± NOUN cana-1962 209 10 Ξ³2(πœβˆ—)Ξ³2(𝜐 Ξ³2(πœβˆ—)Ξ³2(𝜐 NOUN cana-1962 209 11 ) ) PUNCT cana-1962 209 12 Β± Β± NOUN cana-1962 210 1 𝜐 𝜐 NOUN cana-1962 210 2 ∘ ∘ NOUN cana-1962 210 3 πœβˆ— πœβˆ— PROPN cana-1962 210 4 ∈ ∈ PROPN cana-1962 210 5 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 210 6 ) ) PUNCT cana-1962 210 7 βˆ€ βˆ€ X cana-1962 211 1 𝜐 𝜐 X cana-1962 211 2 ∈ ∈ PROPN cana-1962 211 3 𝔖. 𝔖. PROPN cana-1962 211 4 ( ( PUNCT cana-1962 211 5 v v NOUN cana-1962 211 6 ) ) PUNCT cana-1962 211 7 Ξ³1(πœπœβˆ— Ξ³1(πœπœβˆ— NOUN cana-1962 211 8 ) ) PUNCT cana-1962 211 9 Β± Β± NUM cana-1962 211 10 Ξ³2(𝜐)Ξ³2(πœβˆ— Ξ³2(𝜐)Ξ³2(πœβˆ— PROPN cana-1962 211 11 ) ) PUNCT cana-1962 211 12 ∈ ∈ PROPN cana-1962 211 13 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 211 14 ) ) PUNCT cana-1962 211 15 βˆ€ βˆ€ X cana-1962 212 1 𝜐 𝜐 X cana-1962 212 2 ∈ ∈ PROPN cana-1962 212 3 𝔖. 𝔖. PROPN cana-1962 212 4 ( ( PUNCT cana-1962 212 5 vi vi NOUN cana-1962 212 6 ) ) PUNCT cana-1962 212 7 Ξ³1(πœπœβˆ— Ξ³1(πœπœβˆ— NOUN cana-1962 212 8 ) ) PUNCT cana-1962 212 9 Β± Β± NOUN cana-1962 212 10 Ξ³2(πœβˆ—)Ξ³2(𝜐 Ξ³2(πœβˆ—)Ξ³2(𝜐 PUNCT cana-1962 212 11 ) ) PUNCT cana-1962 213 1 ∈ ∈ PROPN cana-1962 213 2 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 213 3 ) ) PUNCT cana-1962 213 4 βˆ€ βˆ€ X cana-1962 214 1 𝜐 𝜐 X cana-1962 214 2 ∈ ∈ PROPN cana-1962 214 3 𝔖. 𝔖. PROPN cana-1962 214 4 ( ( PUNCT cana-1962 214 5 vii vii PROPN cana-1962 214 6 ) ) PUNCT cana-1962 214 7 𝔖 𝔖 PROPN cana-1962 214 8 is be AUX cana-1962 214 9 an an DET cana-1962 214 10 integral integral ADJ cana-1962 214 11 domain domain NOUN cana-1962 214 12 . . PUNCT cana-1962 215 1 corollary corollary ADJ cana-1962 215 2 4 4 NUM cana-1962 216 1 [ [ X cana-1962 216 2 [ [ X cana-1962 216 3 6 6 NUM cana-1962 216 4 ] ] PUNCT cana-1962 216 5 , , PUNCT cana-1962 216 6 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘šπ‘  π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘šπ‘  PROPN cana-1962 216 7 1 1 NUM cana-1962 216 8 , , PUNCT cana-1962 216 9 2 2 NUM cana-1962 216 10 , , PUNCT cana-1962 216 11 3 3 NUM cana-1962 216 12 ] ] PUNCT cana-1962 216 13 the the DET cana-1962 216 14 following follow VERB cana-1962 216 15 assertions assertion NOUN cana-1962 216 16 are be AUX cana-1962 216 17 equivalents equivalent NOUN cana-1962 216 18 : : PUNCT cana-1962 216 19 ( ( PUNCT cana-1962 216 20 i i NOUN cana-1962 216 21 ) ) PUNCT cana-1962 216 22 Ξ³(πœπœβˆ— Ξ³(πœπœβˆ— NOUN cana-1962 216 23 ) ) PUNCT cana-1962 216 24 Β± Β± NUM cana-1962 216 25 Ξ³(𝜐)Ξ³(πœβˆ— Ξ³(𝜐)Ξ³(πœβˆ— NUM cana-1962 216 26 ) ) PUNCT cana-1962 216 27 ] ] PUNCT cana-1962 217 1 Β± Β± NOUN cana-1962 218 1 [ [ X cana-1962 218 2 𝜐 𝜐 X cana-1962 218 3 , , PUNCT cana-1962 218 4 πœβˆ— πœβˆ— PROPN cana-1962 218 5 ] ] X cana-1962 218 6 ∈ ∈ PROPN cana-1962 218 7 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 218 8 ) ) PUNCT cana-1962 218 9 βˆ€ βˆ€ X cana-1962 219 1 𝜐 𝜐 X cana-1962 219 2 ∈ ∈ PROPN cana-1962 219 3 𝔖. 𝔖. PROPN cana-1962 219 4 ( ( PUNCT cana-1962 219 5 ii ii NOUN cana-1962 219 6 ) ) PUNCT cana-1962 219 7 Ξ³(πœπœβˆ— Ξ³(πœπœβˆ— NOUN cana-1962 219 8 ) ) PUNCT cana-1962 219 9 Β± Β± NOUN cana-1962 220 1 [ [ X cana-1962 220 2 𝜐 𝜐 X cana-1962 220 3 , , PUNCT cana-1962 220 4 πœβˆ— πœβˆ— PROPN cana-1962 220 5 ] ] X cana-1962 220 6 ∈ ∈ PROPN cana-1962 220 7 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 220 8 ) ) PUNCT cana-1962 220 9 βˆ€ βˆ€ X cana-1962 221 1 𝜐 𝜐 X cana-1962 221 2 ∈ ∈ PROPN cana-1962 221 3 𝔖. 𝔖. PROPN cana-1962 221 4 ( ( PUNCT cana-1962 221 5 iii iii NOUN cana-1962 221 6 ) ) PUNCT cana-1962 221 7 Ξ³(πœπœβˆ— Ξ³(πœπœβˆ— NOUN cana-1962 221 8 ) ) PUNCT cana-1962 221 9 Β± Β± NUM cana-1962 221 10 πœπœβˆ— πœπœβˆ— NOUN cana-1962 221 11 ∈ ∈ PROPN cana-1962 221 12 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 221 13 ) ) PUNCT cana-1962 221 14 βˆ€ βˆ€ X cana-1962 222 1 𝜐 𝜐 X cana-1962 222 2 ∈ ∈ PROPN cana-1962 222 3 𝔖. 𝔖. PROPN cana-1962 222 4 ( ( PUNCT cana-1962 222 5 iv iv NOUN cana-1962 222 6 ) ) PUNCT cana-1962 222 7 Ξ³(πœπœβˆ— Ξ³(πœπœβˆ— NOUN cana-1962 222 8 ) ) PUNCT cana-1962 222 9 Β± Β± NOUN cana-1962 222 10 πœβˆ—πœ πœβˆ—πœ NUM cana-1962 222 11 ∈ ∈ PROPN cana-1962 222 12 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 222 13 ) ) PUNCT cana-1962 222 14 βˆ€ βˆ€ X cana-1962 223 1 𝜐 𝜐 X cana-1962 223 2 ∈ ∈ PROPN cana-1962 223 3 𝔖. 𝔖. PROPN cana-1962 223 4 ( ( PUNCT cana-1962 223 5 v v NOUN cana-1962 223 6 ) ) PUNCT cana-1962 223 7 Ξ³(𝜐)Ξ³(πœβˆ— Ξ³(𝜐)Ξ³(πœβˆ— NUM cana-1962 223 8 ) ) PUNCT cana-1962 223 9 Β± Β± NOUN cana-1962 224 1 [ [ X cana-1962 224 2 𝜐 𝜐 X cana-1962 224 3 , , PUNCT cana-1962 224 4 πœβˆ— πœβˆ— PROPN cana-1962 224 5 ] ] X cana-1962 224 6 ∈ ∈ PROPN cana-1962 224 7 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 224 8 ) ) PUNCT cana-1962 224 9 βˆ€ βˆ€ X cana-1962 225 1 𝜐 𝜐 X cana-1962 225 2 ∈ ∈ PROPN cana-1962 225 3 𝔖. 𝔖. PROPN cana-1962 225 4 ( ( PUNCT cana-1962 225 5 vi vi NOUN cana-1962 225 6 ) ) PUNCT cana-1962 225 7 Ξ³(𝜐)Ξ³(πœβˆ— Ξ³(𝜐)Ξ³(πœβˆ— NUM cana-1962 225 8 ) ) PUNCT cana-1962 225 9 Β± Β± NOUN cana-1962 225 10 πœπœβˆ— πœπœβˆ— NOUN cana-1962 225 11 ∈ ∈ PROPN cana-1962 225 12 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 225 13 ) ) PUNCT cana-1962 225 14 βˆ€ βˆ€ X cana-1962 226 1 𝜐 𝜐 X cana-1962 226 2 ∈ ∈ PROPN cana-1962 226 3 𝔖. 𝔖. PROPN cana-1962 226 4 ( ( PUNCT cana-1962 226 5 vii vii PROPN cana-1962 226 6 ) ) PUNCT cana-1962 226 7 Ξ³(𝜐)Ξ³(πœβˆ— Ξ³(𝜐)Ξ³(πœβˆ— NUM cana-1962 226 8 ) ) PUNCT cana-1962 226 9 Β± Β± PROPN cana-1962 226 10 πœβˆ—πœ πœβˆ—πœ NUM cana-1962 226 11 ∈ ∈ PROPN cana-1962 226 12 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 226 13 ) ) PUNCT cana-1962 226 14 βˆ€ βˆ€ X cana-1962 227 1 𝜐 𝜐 X cana-1962 227 2 ∈ ∈ PROPN cana-1962 227 3 𝔖. 𝔖. PROPN cana-1962 227 4 ( ( PUNCT cana-1962 227 5 viii viii NOUN cana-1962 227 6 ) ) PUNCT cana-1962 227 7 Ξ³(πœπœβˆ— Ξ³(πœπœβˆ— NOUN cana-1962 227 8 ) ) PUNCT cana-1962 227 9 Β± Β± NOUN cana-1962 227 10 Ξ³(πœβˆ—)Ξ³(𝜐 Ξ³(πœβˆ—)Ξ³(𝜐 NOUN cana-1962 227 11 ) ) PUNCT cana-1962 227 12 Β± Β± NOUN cana-1962 228 1 𝜐 𝜐 NOUN cana-1962 228 2 ∘ ∘ NOUN cana-1962 228 3 πœβˆ— πœβˆ— PROPN cana-1962 228 4 ∈ ∈ PROPN cana-1962 228 5 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 228 6 ) ) PUNCT cana-1962 228 7 βˆ€ βˆ€ X cana-1962 229 1 𝜐 𝜐 X cana-1962 229 2 ∈ ∈ NOUN cana-1962 229 3 𝔖. 𝔖. PROPN cana-1962 229 4 ( ( PUNCT cana-1962 229 5 ix ix PROPN cana-1962 229 6 ) ) PUNCT cana-1962 229 7 Ξ³(πœβˆ—)Ξ³(𝜐 Ξ³(πœβˆ—)Ξ³(𝜐 ADJ cana-1962 229 8 ) ) PUNCT cana-1962 229 9 Β± Β± NOUN cana-1962 230 1 𝜐 𝜐 NOUN cana-1962 230 2 ∘ ∘ NOUN cana-1962 230 3 πœβˆ— πœβˆ— PROPN cana-1962 230 4 ∈ ∈ PROPN cana-1962 230 5 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 230 6 ) ) PUNCT cana-1962 230 7 βˆ€ βˆ€ X cana-1962 231 1 𝜐 𝜐 X cana-1962 231 2 ∈ ∈ NOUN cana-1962 231 3 𝔖. 𝔖. PROPN cana-1962 231 4 ( ( PUNCT cana-1962 231 5 x x NOUN cana-1962 231 6 ) ) PUNCT cana-1962 231 7 Ξ³(πœβˆ—)Ξ³(𝜐 Ξ³(πœβˆ—)Ξ³(𝜐 ADJ cana-1962 231 8 ) ) PUNCT cana-1962 231 9 Β± Β± PROPN cana-1962 231 10 πœπœβˆ— πœπœβˆ— NOUN cana-1962 231 11 ∈ ∈ PROPN cana-1962 231 12 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 231 13 ) ) PUNCT cana-1962 231 14 βˆ€ βˆ€ X cana-1962 232 1 𝜐 𝜐 X cana-1962 232 2 ∈ ∈ NOUN cana-1962 232 3 𝔖. 𝔖. PROPN cana-1962 232 4 ( ( PUNCT cana-1962 232 5 xi xi NOUN cana-1962 232 6 ) ) PUNCT cana-1962 232 7 Ξ³(πœβˆ—)Ξ³(𝜐 Ξ³(πœβˆ—)Ξ³(𝜐 ADJ cana-1962 232 8 ) ) PUNCT cana-1962 232 9 Β± Β± PROPN cana-1962 232 10 πœβˆ—πœ πœβˆ—πœ NUM cana-1962 232 11 ∈ ∈ PROPN cana-1962 232 12 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 232 13 ) ) PUNCT cana-1962 232 14 βˆ€ βˆ€ X cana-1962 233 1 𝜐 𝜐 X cana-1962 233 2 ∈ ∈ PROPN cana-1962 233 3 𝔖. 𝔖. PROPN cana-1962 233 4 ( ( PUNCT cana-1962 233 5 xii xii NOUN cana-1962 233 6 ) ) PUNCT cana-1962 233 7 𝔖 𝔖 NOUN cana-1962 233 8 is be AUX cana-1962 233 9 an an DET cana-1962 233 10 integral integral ADJ cana-1962 233 11 domain domain NOUN cana-1962 233 12 . . PUNCT cana-1962 234 1 the the DET cana-1962 234 2 following follow VERB cana-1962 234 3 example example NOUN cana-1962 234 4 shows show VERB cana-1962 234 5 that that SCONJ cana-1962 234 6 the the DET cana-1962 234 7 assumption assumption NOUN cana-1962 234 8 of of ADP cana-1962 234 9 βˆ— βˆ— NOUN cana-1962 234 10 is be AUX cana-1962 234 11 of of ADP cana-1962 234 12 the the DET cana-1962 234 13 second second ADJ cana-1962 234 14 kind kind NOUN cana-1962 234 15 in in ADP cana-1962 234 16 theorem theorem NOUN cana-1962 234 17 1 1 NUM cana-1962 234 18 and and CCONJ cana-1962 234 19 the the DET cana-1962 234 20 primness primness NOUN cana-1962 234 21 of of ADP cana-1962 234 22 𝔖 𝔖 PROPN cana-1962 234 23 in in ADP cana-1962 234 24 theorem theorem NOUN cana-1962 234 25 2 2 NUM cana-1962 234 26 , , PUNCT cana-1962 234 27 are be AUX cana-1962 234 28 not not PART cana-1962 234 29 superfluous superfluous ADJ cana-1962 234 30 . . PUNCT cana-1962 234 31 example example NOUN cana-1962 235 1 1 1 NUM cana-1962 235 2 let let VERB cana-1962 235 3 β„€ β„€ PRON cana-1962 235 4 be be AUX cana-1962 235 5 the the DET cana-1962 235 6 set set NOUN cana-1962 235 7 of of ADP cana-1962 235 8 integers integer NOUN cana-1962 235 9 . . PUNCT cana-1962 236 1 1 1 X cana-1962 236 2 . . X cana-1962 236 3 let let VERB cana-1962 236 4 us we PRON cana-1962 236 5 define define VERB cana-1962 236 6 𝔖 𝔖 PROPN cana-1962 236 7 and and CCONJ cana-1962 236 8 Ξ³ Ξ³ X cana-1962 236 9 , , PUNCT cana-1962 236 10 πœ“, πœ“, X cana-1962 236 11 β–³ β–³ X cana-1962 236 12 ,βˆ— ,βˆ— PUNCT cana-1962 236 13 : : PUNCT cana-1962 236 14 𝔖 𝔖 PROPN cana-1962 236 15 β†’ β†’ SYM cana-1962 236 16 𝔖 𝔖 PROPN cana-1962 236 17 as as SCONJ cana-1962 236 18 follows follow VERB cana-1962 236 19 : : PUNCT cana-1962 236 20 𝔖 𝔖 PROPN cana-1962 236 21 = = PRON cana-1962 236 22 { { PUNCT cana-1962 236 23 ( ( PUNCT cana-1962 236 24 𝑣 𝑣 PART cana-1962 236 25 πœ” πœ” PART cana-1962 236 26 0 0 NUM cana-1962 236 27 𝑧 𝑧 NOUN cana-1962 236 28 ) ) PUNCT cana-1962 237 1 | | ADV cana-1962 237 2 𝜐 𝜐 NOUN cana-1962 237 3 , , PUNCT cana-1962 237 4 πœ” πœ” VERB cana-1962 237 5 , , PUNCT cana-1962 237 6 𝑧 𝑧 DET cana-1962 237 7 ∈ ∈ PROPN cana-1962 237 8 β„€ β„€ PROPN cana-1962 237 9 } } PUNCT cana-1962 237 10 , , PUNCT cana-1962 237 11 ( ( PUNCT cana-1962 237 12 𝑣 𝑣 DET cana-1962 237 13 πœ” πœ” PART cana-1962 237 14 0 0 NUM cana-1962 237 15 𝑧 𝑧 NOUN cana-1962 237 16 ) ) PUNCT cana-1962 237 17 βˆ— βˆ— NOUN cana-1962 237 18 = = PUNCT cana-1962 237 19 ( ( PUNCT cana-1962 237 20 𝑣 𝑣 PRON cana-1962 237 21 βˆ’πœ” βˆ’πœ” NOUN cana-1962 237 22 0 0 PUNCT cana-1962 237 23 𝑧 𝑧 X cana-1962 237 24 ) ) PUNCT cana-1962 237 25 . . PUNCT cana-1962 238 1 taking take VERB cana-1962 238 2 Ξ³1 Ξ³1 NOUN cana-1962 238 3 = = SYM cana-1962 238 4 Ξ³2 Ξ³2 NOUN cana-1962 238 5 = = SYM cana-1962 238 6 Ξ³ Ξ³ X cana-1962 238 7 π‘ π‘’π‘β„Ž π‘ π‘’π‘β„Ž NOUN cana-1962 238 8 Ξ³ Ξ³ X cana-1962 238 9 ( ( PUNCT cana-1962 238 10 𝑣 𝑣 PART cana-1962 238 11 πœ” πœ” PART cana-1962 238 12 0 0 NUM cana-1962 238 13 𝑧 𝑧 NOUN cana-1962 238 14 ) ) PUNCT cana-1962 239 1 = = SYM cana-1962 239 2 ( ( PUNCT cana-1962 239 3 𝑣 𝑣 PRON cana-1962 239 4 βˆ’πœ” βˆ’πœ” NOUN cana-1962 239 5 0 0 PUNCT cana-1962 239 6 𝑧 𝑧 X cana-1962 239 7 ) ) PUNCT cana-1962 239 8 , , PUNCT cana-1962 239 9 πœ“1 πœ“1 NOUN cana-1962 239 10 = = SYM cana-1962 239 11 πœ“2 πœ“2 NOUN cana-1962 239 12 = = SYM cana-1962 239 13 πœ“ πœ“ PROPN cana-1962 239 14 π‘ π‘’π‘β„Ž π‘ π‘’π‘β„Ž X cana-1962 239 15 πœ“ πœ“ PROPN cana-1962 239 16 ( ( PUNCT cana-1962 239 17 𝑣 𝑣 PART cana-1962 239 18 πœ” πœ” PART cana-1962 239 19 0 0 NUM cana-1962 239 20 𝑧 𝑧 NOUN cana-1962 239 21 ) ) PUNCT cana-1962 239 22 = = SYM cana-1962 240 1 ( ( PUNCT cana-1962 240 2 0 0 NUM cana-1962 240 3 βˆ’2πœ” βˆ’2πœ” NUM cana-1962 240 4 0 0 NUM cana-1962 240 5 0 0 NUM cana-1962 240 6 ) ) PUNCT cana-1962 240 7 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ X cana-1962 240 8 β–³ β–³ X cana-1962 240 9 ( ( PUNCT cana-1962 240 10 𝑣 𝑣 PART cana-1962 240 11 πœ” πœ” PART cana-1962 240 12 0 0 NUM cana-1962 240 13 𝑧 𝑧 NOUN cana-1962 240 14 ) ) PUNCT cana-1962 241 1 = = SYM cana-1962 241 2 ( ( PUNCT cana-1962 241 3 0 0 NUM cana-1962 241 4 0 0 NUM cana-1962 241 5 0 0 NUM cana-1962 241 6 𝑧 𝑧 X cana-1962 241 7 ) ) PUNCT cana-1962 241 8 . . PUNCT cana-1962 242 1 communications communication NOUN cana-1962 242 2 on on ADP cana-1962 242 3 applied apply VERB cana-1962 242 4 nonlinear nonlinear ADJ cana-1962 242 5 analysis analysis NOUN cana-1962 242 6 issn issn NOUN cana-1962 242 7 : : PUNCT cana-1962 242 8 1074 1074 NUM cana-1962 242 9 - - PUNCT cana-1962 242 10 133x 133x NUM cana-1962 242 11 vol vol NOUN cana-1962 242 12 32 32 NUM cana-1962 242 13 no no NOUN cana-1962 242 14 . . NOUN cana-1962 242 15 3 3 NUM cana-1962 242 16 ( ( PUNCT cana-1962 242 17 2025 2025 NUM cana-1962 242 18 ) ) PUNCT cana-1962 242 19 321 321 NUM cana-1962 242 20 https://internationalpubls.com https://internationalpubls.com X cana-1962 243 1 it it PRON cana-1962 243 2 can can AUX cana-1962 243 3 be be AUX cana-1962 243 4 confirmed confirm VERB cana-1962 243 5 that that SCONJ cana-1962 243 6 Ξ³ Ξ³ PROPN cana-1962 243 7 serves serve VERB cana-1962 243 8 as as ADP cana-1962 243 9 a a DET cana-1962 243 10 generalized generalized ADJ cana-1962 243 11 derivation derivation NOUN cana-1962 243 12 linked link VERB cana-1962 243 13 to to ADP cana-1962 243 14 a a DET cana-1962 243 15 non non ADJ cana-1962 243 16 - - ADJ cana-1962 243 17 zero zero NUM cana-1962 243 18 derivation derivation NOUN cana-1962 243 19 πœ“ πœ“ NOUN cana-1962 243 20 , , PUNCT cana-1962 243 21 where where SCONJ cana-1962 243 22 β–³ β–³ PROPN cana-1962 243 23 acts act VERB cana-1962 243 24 as as ADP cana-1962 243 25 the the DET cana-1962 243 26 left left ADJ cana-1962 243 27 multiplier multiplier ADV cana-1962 243 28 . . PUNCT cana-1962 244 1 furthermore furthermore ADV cana-1962 244 2 , , PUNCT cana-1962 244 3 for for ADP cana-1962 244 4 any any DET cana-1962 244 5 two two NUM cana-1962 244 6 elements element NOUN cana-1962 244 7 𝑀 𝑀 PROPN cana-1962 244 8 and and CCONJ cana-1962 244 9 𝑁 𝑁 PROPN cana-1962 244 10 in in ADP cana-1962 244 11 𝔖 𝔖 PROPN cana-1962 244 12 , , PUNCT cana-1962 244 13 the the DET cana-1962 244 14 following follow VERB cana-1962 244 15 conditions condition NOUN cana-1962 244 16 are be AUX cana-1962 244 17 consistently consistently ADV cana-1962 244 18 satisfied satisfied ADJ cana-1962 244 19 ( ( PUNCT cana-1962 244 20 i i NOUN cana-1962 244 21 ) ) PUNCT cana-1962 245 1 [ [ X cana-1962 245 2 𝛀(𝑀 𝛀(𝑀 NOUN cana-1962 245 3 ) ) PUNCT cana-1962 245 4 , , PUNCT cana-1962 245 5 𝛀(π‘€βˆ— 𝛀(π‘€βˆ— NOUN cana-1962 245 6 ) ) PUNCT cana-1962 245 7 ] ] PUNCT cana-1962 246 1 + + X cana-1962 246 2 β–³ β–³ X cana-1962 246 3 ( ( PUNCT cana-1962 246 4 [ [ X cana-1962 246 5 𝑀 𝑀 PROPN cana-1962 246 6 , , PUNCT cana-1962 246 7 π‘€βˆ— π‘€βˆ— PROPN cana-1962 246 8 ] ] PUNCT cana-1962 246 9 ) ) PUNCT cana-1962 246 10 ∈ ∈ PROPN cana-1962 246 11 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 246 12 ) ) PUNCT cana-1962 246 13 , , PUNCT cana-1962 246 14 βˆ€π‘€ βˆ€π‘€ PROPN cana-1962 246 15 ∈ ∈ PROPN cana-1962 246 16 𝔖. 𝔖. PROPN cana-1962 246 17 ( ( PUNCT cana-1962 246 18 ii ii NOUN cana-1962 246 19 ) ) PUNCT cana-1962 246 20 𝛀(𝑀 𝛀(𝑀 NOUN cana-1962 246 21 ) ) PUNCT cana-1962 246 22 ∘ ∘ PROPN cana-1962 246 23 𝛀(π‘€βˆ— 𝛀(π‘€βˆ— PROPN cana-1962 246 24 ) ) PUNCT cana-1962 247 1 + + NOUN cana-1962 247 2 β–³ β–³ X cana-1962 247 3 ( ( PUNCT cana-1962 247 4 𝑀 𝑀 PROPN cana-1962 247 5 ∘ ∘ PROPN cana-1962 247 6 π‘€βˆ— π‘€βˆ— PROPN cana-1962 247 7 ) ) PUNCT cana-1962 247 8 ∈ ∈ PROPN cana-1962 247 9 𝑍(𝔖 𝑍(𝔖 PROPN cana-1962 247 10 ) ) PUNCT cana-1962 247 11 , , PUNCT cana-1962 247 12 βˆ€π‘€ βˆ€π‘€ PROPN cana-1962 247 13 ∈ ∈ PROPN cana-1962 247 14 𝔖. 𝔖. PROPN cana-1962 247 15 ( ( PUNCT cana-1962 247 16 iii iii NOUN cana-1962 247 17 ) ) PUNCT cana-1962 248 1 [ [ X cana-1962 248 2 𝛀(𝑀 𝛀(𝑀 NOUN cana-1962 248 3 ) ) PUNCT cana-1962 248 4 , , PUNCT cana-1962 248 5 𝛀(π‘€βˆ— 𝛀(π‘€βˆ— NOUN cana-1962 248 6 ) ) PUNCT cana-1962 248 7 ] ] PUNCT cana-1962 249 1 + + X cana-1962 249 2 β–³ β–³ X cana-1962 249 3 ( ( PUNCT cana-1962 249 4 𝑀 𝑀 PROPN cana-1962 249 5 ∘ ∘ PROPN cana-1962 249 6 π‘€βˆ— π‘€βˆ— PROPN cana-1962 249 7 ) ) PUNCT cana-1962 249 8 ∈ ∈ PROPN cana-1962 249 9 𝑍(𝔖 𝑍(𝔖 PROPN cana-1962 249 10 ) ) PUNCT cana-1962 249 11 , , PUNCT cana-1962 249 12 βˆ€π‘€ βˆ€π‘€ PROPN cana-1962 249 13 ∈ ∈ PROPN cana-1962 249 14 𝔖. 𝔖. PROPN cana-1962 249 15 ( ( PUNCT cana-1962 249 16 iv iv NOUN cana-1962 249 17 ) ) PUNCT cana-1962 249 18 𝛀(𝑀 𝛀(𝑀 NOUN cana-1962 249 19 ) ) PUNCT cana-1962 249 20 ∘ ∘ PROPN cana-1962 249 21 𝛀(π‘€βˆ— 𝛀(π‘€βˆ— PROPN cana-1962 249 22 ) ) PUNCT cana-1962 250 1 + + NOUN cana-1962 250 2 β–³ β–³ X cana-1962 250 3 ( ( PUNCT cana-1962 250 4 [ [ X cana-1962 250 5 𝑀 𝑀 PROPN cana-1962 250 6 , , PUNCT cana-1962 250 7 π‘€βˆ— π‘€βˆ— PROPN cana-1962 250 8 ] ] PUNCT cana-1962 250 9 ) ) PUNCT cana-1962 250 10 ∈ ∈ PROPN cana-1962 250 11 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 250 12 ) ) PUNCT cana-1962 250 13 , , PUNCT cana-1962 250 14 βˆ€π‘€ βˆ€π‘€ PROPN cana-1962 250 15 ∈ ∈ PROPN cana-1962 250 16 𝔖. 𝔖. PROPN cana-1962 250 17 however however ADV cana-1962 250 18 , , PUNCT cana-1962 250 19 𝔖 𝔖 PROPN cana-1962 250 20 is be AUX cana-1962 250 21 not not PART cana-1962 250 22 an an DET cana-1962 250 23 integral integral ADJ cana-1962 250 24 domain domain NOUN cana-1962 250 25 due due ADP cana-1962 250 26 to to ADP cana-1962 250 27 βˆ— βˆ— NOUN cana-1962 250 28 is be AUX cana-1962 250 29 an an DET cana-1962 250 30 involution involution NOUN cana-1962 250 31 of of ADP cana-1962 250 32 the the DET cana-1962 250 33 first first ADJ cana-1962 250 34 kind kind NOUN cana-1962 250 35 , , PUNCT cana-1962 250 36 not not PART cana-1962 250 37 the the DET cana-1962 250 38 second second ADJ cana-1962 250 39 kind kind NOUN cana-1962 250 40 . . PUNCT cana-1962 251 1 2 2 X cana-1962 251 2 . . X cana-1962 251 3 suppose suppose VERB cana-1962 251 4 that that SCONJ cana-1962 251 5 𝑇 𝑇 PROPN cana-1962 251 6 = = SYM cana-1962 251 7 𝔖 𝔖 PROPN cana-1962 251 8 Γ— Γ— NOUN cana-1962 251 9 β„‚ β„‚ PROPN cana-1962 251 10 is be AUX cana-1962 251 11 a a DET cana-1962 251 12 non non ADJ cana-1962 251 13 - - ADJ cana-1962 251 14 prime prime ADJ cana-1962 251 15 ring ring NOUN cana-1962 251 16 , , PUNCT cana-1962 251 17 where where SCONJ cana-1962 251 18 𝔖 𝔖 PROPN cana-1962 251 19 is be AUX cana-1962 251 20 the the DET cana-1962 251 21 same same ADJ cana-1962 251 22 as as ADP cana-1962 251 23 in in ADP cana-1962 251 24 part part NOUN cana-1962 251 25 1 1 NUM cana-1962 251 26 and and CCONJ cana-1962 251 27 β„‚ β„‚ PROPN cana-1962 251 28 is be AUX cana-1962 251 29 the the DET cana-1962 251 30 ring ring NOUN cana-1962 251 31 of of ADP cana-1962 251 32 complex complex ADJ cana-1962 251 33 numbers number NOUN cana-1962 251 34 with with ADP cana-1962 251 35 conjugate conjugate ADJ cana-1962 251 36 involution involution NOUN cana-1962 251 37 ⋆. ⋆. PUNCT cana-1962 251 38 we we PRON cana-1962 251 39 define define VERB cana-1962 251 40 in in ADP cana-1962 251 41 𝑇 𝑇 PROPN cana-1962 251 42 the the DET cana-1962 251 43 involution involution NOUN cana-1962 251 44 𝜏 𝜏 X cana-1962 251 45 of of ADP cana-1962 251 46 the the DET cana-1962 251 47 second second ADJ cana-1962 251 48 kind kind NOUN cana-1962 251 49 such such ADJ cana-1962 251 50 that that SCONJ cana-1962 251 51 𝜏(𝜐 𝜏(𝜐 PROPN cana-1962 251 52 , , PUNCT cana-1962 251 53 πœ” πœ” ADJ cana-1962 251 54 ) ) PUNCT cana-1962 251 55 = = SYM cana-1962 251 56 ( ( PUNCT cana-1962 251 57 πœβˆ— πœβˆ— PROPN cana-1962 251 58 , , PUNCT cana-1962 251 59 πœ†β‹† πœ†β‹† X cana-1962 251 60 ) ) PUNCT cana-1962 251 61 such such ADJ cana-1962 251 62 that that SCONJ cana-1962 251 63 βˆ— βˆ— NOUN cana-1962 251 64 is be AUX cana-1962 251 65 the the DET cana-1962 251 66 same same ADJ cana-1962 251 67 involution involution NOUN cana-1962 251 68 in in ADP cana-1962 251 69 part part NOUN cana-1962 251 70 1 1 X cana-1962 251 71 . . PUNCT cana-1962 252 1 we we PRON cana-1962 252 2 can can AUX cana-1962 252 3 easily easily ADV cana-1962 252 4 demonstrate demonstrate VERB cana-1962 252 5 that that SCONJ cana-1962 252 6 the the DET cana-1962 252 7 map map NOUN cana-1962 252 8 𝐺 𝐺 NOUN cana-1962 252 9 , , PUNCT cana-1962 252 10 defined define VERB cana-1962 252 11 in in ADP cana-1962 252 12 𝑇 𝑇 PROPN cana-1962 252 13 by by ADP cana-1962 252 14 𝐺(𝑀 𝐺(𝑀 NOUN cana-1962 252 15 , , PUNCT cana-1962 252 16 𝑁 𝑁 PROPN cana-1962 252 17 ) ) PUNCT cana-1962 252 18 = = SYM cana-1962 252 19 ( ( PUNCT cana-1962 252 20 Ξ³(𝑀),0 Ξ³(𝑀),0 PROPN cana-1962 252 21 ) ) PUNCT cana-1962 252 22 , , PUNCT cana-1962 252 23 is be AUX cana-1962 252 24 a a DET cana-1962 252 25 generalized generalized ADJ cana-1962 252 26 derivation derivation NOUN cana-1962 252 27 links link NOUN cana-1962 252 28 to to ADP cana-1962 252 29 the the DET cana-1962 252 30 derivation derivation NOUN cana-1962 252 31 𝑔(𝑀 𝑔(𝑀 PROPN cana-1962 252 32 , , PUNCT cana-1962 252 33 𝑁 𝑁 PROPN cana-1962 252 34 ) ) PUNCT cana-1962 252 35 = = SYM cana-1962 252 36 ( ( PUNCT cana-1962 252 37 πœ“(𝑀),0 πœ“(𝑀),0 PROPN cana-1962 252 38 ) ) PUNCT cana-1962 252 39 , , PUNCT cana-1962 252 40 where where SCONJ cana-1962 252 41 πœ“ πœ“ PROPN cana-1962 252 42 is be AUX cana-1962 252 43 the the DET cana-1962 252 44 same same ADJ cana-1962 252 45 in in ADP cana-1962 252 46 part part NOUN cana-1962 252 47 1 1 NUM cana-1962 252 48 . . PUNCT cana-1962 253 1 it it PRON cana-1962 253 2 can can AUX cana-1962 253 3 be be AUX cana-1962 253 4 easily easily ADV cana-1962 253 5 verified verify VERB cana-1962 253 6 that that SCONJ cana-1962 253 7 : : PUNCT cana-1962 253 8 ( ( PUNCT cana-1962 253 9 i i NOUN cana-1962 253 10 ) ) PUNCT cana-1962 253 11 𝐺(π‘€π‘€βˆ— 𝐺(π‘€π‘€βˆ— PROPN cana-1962 253 12 ) ) PUNCT cana-1962 253 13 βˆ’ βˆ’ NOUN cana-1962 253 14 𝐺(𝑀)𝐺(π‘€βˆ— 𝐺(𝑀)𝐺(π‘€βˆ— NUM cana-1962 253 15 ) ) PUNCT cana-1962 254 1 + + NOUN cana-1962 254 2 β–³ β–³ X cana-1962 254 3 ( ( PUNCT cana-1962 254 4 [ [ X cana-1962 254 5 𝑀 𝑀 PROPN cana-1962 254 6 , , PUNCT cana-1962 254 7 π‘€βˆ— π‘€βˆ— PROPN cana-1962 254 8 ] ] PUNCT cana-1962 254 9 ) ) PUNCT cana-1962 254 10 ∈ ∈ PROPN cana-1962 254 11 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 254 12 ) ) PUNCT cana-1962 255 1 βˆ€ βˆ€ X cana-1962 256 1 𝑀 𝑀 PROPN cana-1962 256 2 ∈ ∈ PROPN cana-1962 256 3 𝔖. 𝔖. PROPN cana-1962 256 4 ( ( PUNCT cana-1962 256 5 ii ii NOUN cana-1962 256 6 ) ) PUNCT cana-1962 256 7 𝐺(π‘€π‘€βˆ— 𝐺(π‘€π‘€βˆ— PROPN cana-1962 256 8 ) ) PUNCT cana-1962 256 9 βˆ’ βˆ’ NOUN cana-1962 256 10 𝐺(𝑀)𝐺(π‘€βˆ— 𝐺(𝑀)𝐺(π‘€βˆ— NUM cana-1962 256 11 ) ) PUNCT cana-1962 257 1 + + NOUN cana-1962 257 2 β–³ β–³ X cana-1962 257 3 ( ( PUNCT cana-1962 257 4 𝑀 𝑀 PROPN cana-1962 257 5 ∘ ∘ PROPN cana-1962 257 6 π‘€βˆ— π‘€βˆ— PROPN cana-1962 257 7 ) ) PUNCT cana-1962 257 8 ∈ ∈ PROPN cana-1962 257 9 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 257 10 ) ) PUNCT cana-1962 257 11 βˆ€ βˆ€ X cana-1962 258 1 𝑀 𝑀 PROPN cana-1962 258 2 ∈ ∈ PROPN cana-1962 258 3 𝔖. 𝔖. PROPN cana-1962 258 4 ( ( PUNCT cana-1962 258 5 iii iii NOUN cana-1962 258 6 ) ) PUNCT cana-1962 258 7 𝐺(π‘€π‘€βˆ— 𝐺(π‘€π‘€βˆ— NOUN cana-1962 258 8 ) ) PUNCT cana-1962 258 9 βˆ’ βˆ’ PRON cana-1962 258 10 𝐺(π‘€βˆ—)𝐺(𝑀 𝐺(π‘€βˆ—)𝐺(𝑀 NOUN cana-1962 258 11 ) ) PUNCT cana-1962 258 12 + + NOUN cana-1962 258 13 β–³ β–³ X cana-1962 258 14 ( ( PUNCT cana-1962 258 15 [ [ X cana-1962 258 16 𝑀 𝑀 PROPN cana-1962 258 17 , , PUNCT cana-1962 258 18 π‘€βˆ— π‘€βˆ— PROPN cana-1962 258 19 ] ] PUNCT cana-1962 258 20 ) ) PUNCT cana-1962 259 1 ∈ ∈ PROPN cana-1962 259 2 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 259 3 ) ) PUNCT cana-1962 259 4 βˆ€ βˆ€ X cana-1962 260 1 𝑀 𝑀 PROPN cana-1962 260 2 ∈ ∈ PROPN cana-1962 260 3 𝔖. 𝔖. PROPN cana-1962 260 4 ( ( PUNCT cana-1962 260 5 iv iv NOUN cana-1962 260 6 ) ) PUNCT cana-1962 260 7 𝐺(π‘€π‘€βˆ— 𝐺(π‘€π‘€βˆ— NOUN cana-1962 260 8 ) ) PUNCT cana-1962 260 9 βˆ’ βˆ’ PRON cana-1962 260 10 𝐺(π‘€βˆ—)𝐺(𝑀 𝐺(π‘€βˆ—)𝐺(𝑀 NOUN cana-1962 260 11 ) ) PUNCT cana-1962 261 1 + + NOUN cana-1962 261 2 β–³ β–³ X cana-1962 261 3 ( ( PUNCT cana-1962 261 4 𝑀 𝑀 PROPN cana-1962 261 5 ∘ ∘ PROPN cana-1962 261 6 π‘€βˆ— π‘€βˆ— PROPN cana-1962 261 7 ) ) PUNCT cana-1962 261 8 ∈ ∈ PROPN cana-1962 261 9 𝑍(𝔖 𝑍(𝔖 NOUN cana-1962 261 10 ) ) PUNCT cana-1962 261 11 βˆ€ βˆ€ X cana-1962 262 1 𝑀 𝑀 PROPN cana-1962 262 2 ∈ ∈ PROPN cana-1962 262 3 𝔖. 𝔖. PROPN cana-1962 262 4 however however ADV cana-1962 262 5 , , PUNCT cana-1962 262 6 𝑇 𝑇 PROPN cana-1962 262 7 is be AUX cana-1962 262 8 not not PART cana-1962 262 9 an an DET cana-1962 262 10 integral integral ADJ cana-1962 262 11 domain domain NOUN cana-1962 262 12 due due ADP cana-1962 262 13 to to ADP cana-1962 262 14 it it PRON cana-1962 262 15 ’s ’ VERB cana-1962 262 16 not not PART cana-1962 262 17 prime prime ADJ cana-1962 262 18 ring ring NOUN cana-1962 262 19 . . PUNCT cana-1962 263 1 conflicts conflict NOUN cana-1962 263 2 of of ADP cana-1962 263 3 interest interest NOUN cana-1962 263 4 : : PUNCT cana-1962 263 5 " " PUNCT cana-1962 263 6 the the DET cana-1962 263 7 author author NOUN cana-1962 263 8 declares declare VERB cana-1962 263 9 that that PRON cana-1962 263 10 has have AUX cana-1962 263 11 has have VERB cana-1962 263 12 no no DET cana-1962 263 13 conflicts conflict NOUN cana-1962 263 14 of of ADP cana-1962 263 15 interest interest NOUN cana-1962 263 16 . . PUNCT cana-1962 263 17 " " PUNCT cana-1962 264 1 references reference NOUN cana-1962 264 2 [ [ X cana-1962 264 3 1 1 NUM cana-1962 264 4 ] ] X cana-1962 264 5 shakir shakir PROPN cana-1962 264 6 ali ali PROPN cana-1962 264 7 , , PUNCT cana-1962 264 8 and and CCONJ cana-1962 264 9 nadeem nadeem PROPN cana-1962 264 10 ahmed ahmed PROPN cana-1962 264 11 dar dar PROPN cana-1962 264 12 . . PUNCT cana-1962 265 1 " " PUNCT cana-1962 265 2 on*-centralizing on*-centralize VERB cana-1962 265 3 mappings mapping NOUN cana-1962 265 4 in in ADP cana-1962 265 5 rings ring NOUN cana-1962 265 6 with with ADP cana-1962 265 7 involution involution NOUN cana-1962 265 8 . . PUNCT cana-1962 265 9 " " PUNCT cana-1962 266 1 georgian georgian PROPN cana-1962 266 2 mathematical mathematical ADJ cana-1962 266 3 journal journal NOUN cana-1962 266 4 21.1 21.1 NUM cana-1962 266 5 ( ( PUNCT cana-1962 266 6 2014 2014 NUM cana-1962 266 7 ): ): PUNCT cana-1962 266 8 25 25 NUM cana-1962 266 9 - - SYM cana-1962 266 10 28 28 NUM cana-1962 266 11 . . PUNCT cana-1962 267 1 [ [ X cana-1962 267 2 2 2 NUM cana-1962 267 3 ] ] X cana-1962 267 4 bell bell NOUN cana-1962 267 5 , , PUNCT cana-1962 267 6 howard howard PROPN cana-1962 267 7 e. e. PROPN cana-1962 267 8 and and CCONJ cana-1962 267 9 wallace wallace PROPN cana-1962 267 10 s. s. PROPN cana-1962 267 11 martindale martindale PROPN cana-1962 267 12 . . PUNCT cana-1962 268 1 β€œ " PUNCT cana-1962 268 2 centralizing centralize VERB cana-1962 268 3 mappings mapping NOUN cana-1962 268 4 of of ADP cana-1962 268 5 semiprime semiprime NOUN cana-1962 268 6 rings ring NOUN cana-1962 268 7 . . PUNCT cana-1962 268 8 ” " PUNCT cana-1962 269 1 canadian canadian PROPN cana-1962 269 2 mathematical mathematical ADJ cana-1962 269 3 bulletin bulletin NOUN cana-1962 269 4 30 30 NUM cana-1962 269 5 ( ( PUNCT cana-1962 269 6 1987 1987 NUM cana-1962 269 7 ): ): PUNCT cana-1962 269 8 92 92 NUM cana-1962 269 9 101 101 NUM cana-1962 269 10 . . PUNCT cana-1962 270 1 [ [ X cana-1962 270 2 3 3 X cana-1962 270 3 ] ] X cana-1962 270 4 e. e. PROPN cana-1962 270 5 mohammadi mohammadi PROPN cana-1962 270 6 , , PUNCT cana-1962 270 7 and and CCONJ cana-1962 270 8 abdelkarim abdelkarim NOUN cana-1962 270 9 boua boua NOUN cana-1962 270 10 . . PUNCT cana-1962 271 1 " " PUNCT cana-1962 271 2 algebraic algebraic ADJ cana-1962 271 3 identities identity NOUN cana-1962 271 4 and and CCONJ cana-1962 271 5 generalized generalized ADJ cana-1962 271 6 derivations derivation NOUN cana-1962 271 7 in in ADP cana-1962 271 8 prime prime ADJ cana-1962 271 9 rings ring NOUN cana-1962 271 10 . . PUNCT cana-1962 271 11 " " PUNCT cana-1962 271 12 mathematica mathematica PROPN cana-1962 271 13 , , PUNCT cana-1962 271 14 66 66 NUM cana-1962 271 15 ( ( PUNCT cana-1962 271 16 89 89 NUM cana-1962 271 17 ) ) PUNCT cana-1962 271 18 , , PUNCT cana-1962 271 19 no no DET cana-1962 271 20 1 1 NUM cana-1962 271 21 , , PUNCT cana-1962 271 22 ( ( PUNCT cana-1962 271 23 2024 2024 NUM cana-1962 271 24 ): ): PUNCT cana-1962 271 25 105–113 105–113 NUM cana-1962 271 26 . . PUNCT cana-1962 272 1 [ [ X cana-1962 272 2 4 4 X cana-1962 272 3 ] ] X cana-1962 272 4 shuliang shuliang PROPN cana-1962 272 5 huang huang PROPN cana-1962 272 6 , , PUNCT cana-1962 272 7 " " PUNCT cana-1962 272 8 generalized generalized ADJ cana-1962 272 9 reverse reverse ADJ cana-1962 272 10 derivations derivation NOUN cana-1962 272 11 and and CCONJ cana-1962 272 12 commutativity commutativity NOUN cana-1962 272 13 of of ADP cana-1962 272 14 prime prime ADJ cana-1962 272 15 rings ring NOUN cana-1962 272 16 . . PUNCT cana-1962 272 17 " " PUNCT cana-1962 273 1 communications communication NOUN cana-1962 273 2 in in ADP cana-1962 273 3 mathematics mathematic NOUN cana-1962 273 4 27 27 NUM cana-1962 273 5 ( ( PUNCT cana-1962 273 6 2019 2019 NUM cana-1962 273 7 ) ) PUNCT cana-1962 273 8 43–50 43–50 NOUN cana-1962 273 9 . . PUNCT cana-1962 274 1 [ [ X cana-1962 274 2 5 5 NUM cana-1962 274 3 ] ] X cana-1962 274 4 muzibur muzibur PROPN cana-1962 274 5 mozumder mozumder NOUN cana-1962 274 6 , , PUNCT cana-1962 274 7 and and CCONJ cana-1962 274 8 adnan adnan PROPN cana-1962 274 9 abbasi abbasi PROPN cana-1962 274 10 , , PUNCT cana-1962 274 11 arshad arshad PROPN cana-1962 274 12 madni madni PROPN cana-1962 274 13 and and CCONJ cana-1962 274 14 wasim wasim PROPN cana-1962 274 15 ahmed ahmed PROPN cana-1962 274 16 , , PUNCT cana-1962 274 17 β€œ " PUNCT cana-1962 274 18 on on ADP cana-1962 274 19 * * PUNCT cana-1962 274 20 ideals ideal NOUN cana-1962 274 21 and and CCONJ cana-1962 274 22 derivations derivation NOUN cana-1962 274 23 in in ADP cana-1962 274 24 prime prime ADJ cana-1962 274 25 rings ring NOUN cana-1962 274 26 with with ADP cana-1962 274 27 involution involution NOUN cana-1962 274 28 ” " PUNCT cana-1962 274 29 . . PUNCT cana-1962 275 1 the the DET cana-1962 275 2 aligarh aligarh PROPN cana-1962 275 3 bulletin bulletin NOUN cana-1962 275 4 of of ADP cana-1962 275 5 mathematics mathematic NOUN cana-1962 275 6 , , PUNCT cana-1962 275 7 40(2 40(2 NUM cana-1962 275 8 ) ) PUNCT cana-1962 275 9 ( ( PUNCT cana-1962 275 10 2021 2021 NUM cana-1962 275 11 ): ): PUNCT cana-1962 275 12 77 77 NUM cana-1962 275 13 - - SYM cana-1962 275 14 93 93 NUM cana-1962 275 15 . . PUNCT cana-1962 276 1 [ [ X cana-1962 276 2 6 6 NUM cana-1962 276 3 ] ] PUNCT cana-1962 276 4 a. a. NOUN cana-1962 276 5 mamouni mamouni PROPN cana-1962 276 6 , , PUNCT cana-1962 276 7 b. b. PROPN cana-1962 276 8 nejjar nejjar PROPN cana-1962 276 9 and and CCONJ cana-1962 276 10 l. l. PROPN cana-1962 276 11 oukhtite oukhtite PROPN cana-1962 276 12 . . PUNCT cana-1962 277 1 " " PUNCT cana-1962 277 2 differential differential ADJ cana-1962 277 3 identities identity NOUN cana-1962 277 4 on on ADP cana-1962 277 5 prime prime ADJ cana-1962 277 6 rings ring NOUN cana-1962 277 7 with with ADP cana-1962 277 8 involution involution NOUN cana-1962 277 9 . . PUNCT cana-1962 277 10 " " PUNCT cana-1962 278 1 journal journal NOUN cana-1962 278 2 of of ADP cana-1962 278 3 algebra algebra PROPN cana-1962 278 4 and and CCONJ cana-1962 278 5 its its PRON cana-1962 278 6 applications application NOUN cana-1962 278 7 17(9 17(9 NOUN cana-1962 278 8 ) ) PUNCT cana-1962 278 9 ( ( PUNCT cana-1962 278 10 2018)1850163 2018)1850163 NOUN cana-1962 278 11 . . PUNCT cana-1962 279 1 [ [ X cana-1962 279 2 7 7 X cana-1962 279 3 ] ] X cana-1962 279 4 e. e. PROPN cana-1962 279 5 mohammadi mohammadi PROPN cana-1962 279 6 , , PUNCT cana-1962 279 7 and and CCONJ cana-1962 279 8 abdelkarim abdelkarim NOUN cana-1962 279 9 boua boua NOUN cana-1962 279 10 . . PUNCT cana-1962 280 1 " " PUNCT cana-1962 280 2 quotient quotient NOUN cana-1962 280 3 rings ring NOUN cana-1962 280 4 satisfying satisfy VERB cana-1962 280 5 some some DET cana-1962 280 6 identities identity NOUN cana-1962 280 7 . . PUNCT cana-1962 280 8 " " PUNCT cana-1962 281 1 cubo cubo X cana-1962 281 2 ( ( PUNCT cana-1962 281 3 temuco temuco PROPN cana-1962 281 4 ) ) PUNCT cana-1962 281 5 253 253 NUM cana-1962 281 6 ( ( PUNCT cana-1962 281 7 2023 2023 NUM cana-1962 281 8 ) ) PUNCT cana-1962 281 9 . . PUNCT cana-1962 282 1 [ [ X cana-1962 282 2 8 8 X cana-1962 282 3 ] ] X cana-1962 282 4 b. b. PROPN cana-1962 282 5 nejjar nejjar PROPN cana-1962 282 6 , , PUNCT cana-1962 282 7 et et PROPN cana-1962 282 8 al al PROPN cana-1962 282 9 . . PUNCT cana-1962 283 1 " " PUNCT cana-1962 283 2 commutativity commutativity NOUN cana-1962 283 3 theorems theorem VERB cana-1962 283 4 in in ADP cana-1962 283 5 rings ring NOUN cana-1962 283 6 with with ADP cana-1962 283 7 involution involution NOUN cana-1962 283 8 . . PUNCT cana-1962 283 9 " " PUNCT cana-1962 284 1 communications communication NOUN cana-1962 284 2 in in ADP cana-1962 284 3 algebra algebra NOUN cana-1962 284 4 45.2 45.2 NUM cana-1962 284 5 ( ( PUNCT cana-1962 284 6 2017 2017 NUM cana-1962 284 7 ): ): PUNCT cana-1962 284 8 698708 698708 NUM cana-1962 284 9 . . PUNCT cana-1962 285 1 [ [ X cana-1962 285 2 9 9 NUM cana-1962 285 3 ] ] X cana-1962 285 4 e. e. PROPN cana-1962 285 5 c. c. PROPN cana-1962 285 6 posner posner PROPN cana-1962 285 7 , , PUNCT cana-1962 285 8 " " PUNCT cana-1962 285 9 derivations derivation NOUN cana-1962 285 10 in in ADP cana-1962 285 11 prime prime ADJ cana-1962 285 12 rings ring NOUN cana-1962 285 13 . . PUNCT cana-1962 285 14 " " PUNCT cana-1962 286 1 proceedings proceeding NOUN cana-1962 286 2 of of ADP cana-1962 286 3 the the DET cana-1962 286 4 american american PROPN cana-1962 286 5 mathematical mathematical PROPN cana-1962 286 6 society society NOUN cana-1962 286 7 8.6 8.6 NUM cana-1962 286 8 ( ( PUNCT cana-1962 286 9 1957 1957 NUM cana-1962 286 10 ): ): PUNCT cana-1962 286 11 10931100 10931100 NUM cana-1962 286 12 . . PUNCT cana-1962 287 1 [ [ X cana-1962 287 2 10 10 NUM cana-1962 287 3 ] ] PUNCT cana-1962 287 4 a. a. NOUN cana-1962 287 5 boua boua NOUN cana-1962 287 6 , , PUNCT cana-1962 287 7 " " PUNCT cana-1962 287 8 study study NOUN cana-1962 287 9 of of ADP cana-1962 287 10 the the DET cana-1962 287 11 structure structure NOUN cana-1962 287 12 of of ADP cana-1962 287 13 quotient quotient NOUN cana-1962 287 14 rings ring NOUN cana-1962 287 15 satisfying satisfy VERB cana-1962 287 16 algebraic algebraic ADJ cana-1962 287 17 identities identity NOUN cana-1962 287 18 . . PUNCT cana-1962 287 19 " " PUNCT cana-1962 288 1 journal journal NOUN cana-1962 288 2 of of ADP cana-1962 288 3 algebra algebra PROPN cana-1962 288 4 and and CCONJ cana-1962 288 5 related related ADJ cana-1962 288 6 topics topic NOUN cana-1962 288 7 11.2 11.2 NUM cana-1962 288 8 ( ( PUNCT cana-1962 288 9 2023 2023 NUM cana-1962 288 10 ): ): PUNCT cana-1962 288 11 117 117 NUM cana-1962 288 12 - - SYM cana-1962 288 13 125 125 NUM cana-1962 288 14 . . PUNCT cana-1962 289 1 [ [ X cana-1962 289 2 11 11 NUM cana-1962 289 3 ] ] X cana-1962 289 4 shakir shakir PROPN cana-1962 289 5 ali ali PROPN cana-1962 289 6 , , PUNCT cana-1962 289 7 a. a. PROPN cana-1962 289 8 n. n. PROPN cana-1962 289 9 koam koam PROPN cana-1962 289 10 , , PUNCT cana-1962 289 11 m. m. NOUN cana-1962 289 12 a. a. NOUN cana-1962 289 13 ansari ansari PROPN cana-1962 289 14 β€œ " PUNCT cana-1962 289 15 on*-differential on*-differential ADJ cana-1962 289 16 identities identity NOUN cana-1962 289 17 in in ADP cana-1962 289 18 prime prime ADJ cana-1962 289 19 rings ring NOUN cana-1962 289 20 with with ADP cana-1962 289 21 involution involution NOUN cana-1962 289 22 . . PUNCT cana-1962 289 23 ” " PUNCT cana-1962 289 24 . . PUNCT cana-1962 290 1 hacettepe hacettepe PROPN cana-1962 290 2 journal journal PROPN cana-1962 290 3 of of ADP cana-1962 290 4 mathematics mathematic NOUN cana-1962 290 5 and and CCONJ cana-1962 290 6 statistics statistic NOUN cana-1962 290 7 , , PUNCT cana-1962 290 8 ( ( PUNCT cana-1962 290 9 2020 2020 NUM cana-1962 290 10 ): ): PUNCT cana-1962 290 11 1 1 NUM cana-1962 290 12 - - SYM cana-1962 290 13 8 8 NUM cana-1962 290 14 . . PUNCT cana-1962 291 1 [ [ X cana-1962 291 2 12 12 NUM cana-1962 291 3 ] ] X cana-1962 291 4 gurminder gurminder PROPN cana-1962 291 5 sandhu sandhu PROPN cana-1962 291 6 and and CCONJ cana-1962 291 7 didem didem PROPN cana-1962 291 8 camci camci PROPN cana-1962 291 9 , , PUNCT cana-1962 291 10 β€œ " PUNCT cana-1962 291 11 some some DET cana-1962 291 12 results result NOUN cana-1962 291 13 on on ADP cana-1962 291 14 prime prime ADJ cana-1962 291 15 rings ring NOUN cana-1962 291 16 with with ADP cana-1962 291 17 multiplicative multiplicative ADJ cana-1962 291 18 derivations derivation NOUN cana-1962 291 19 . . PUNCT cana-1962 291 20 ” " PUNCT cana-1962 291 21 , , PUNCT cana-1962 291 22 turk turk PROPN cana-1962 291 23 j. j. PROPN cana-1962 291 24 math math PROPN cana-1962 291 25 , , PUNCT cana-1962 291 26 44(4 44(4 NOUN cana-1962 291 27 ) ) PUNCT cana-1962 291 28 ( ( PUNCT cana-1962 291 29 2020 2020 NUM cana-1962 291 30 ) ) PUNCT cana-1962 291 31 . . PUNCT cana-1962 292 1 [ [ X cana-1962 292 2 13 13 NUM cana-1962 292 3 ] ] X cana-1962 292 4 v.s.v v.s.v ADJ cana-1962 292 5 . . PUNCT cana-1962 293 1 krishna krishna PROPN cana-1962 293 2 murty murty PROPN cana-1962 293 3 , , PUNCT cana-1962 293 4 k. k. PROPN cana-1962 293 5 chennakesavulu chennakesavulu PROPN cana-1962 293 6 , , PUNCT cana-1962 293 7 c. c. PROPN cana-1962 293 8 jaya jaya PROPN cana-1962 293 9 subba subba PROPN cana-1962 293 10 reddy reddy PROPN cana-1962 293 11 , , PUNCT cana-1962 293 12 β€œ " PUNCT cana-1962 293 13 orthogonal orthogonal ADJ cana-1962 293 14 generalized generalize VERB cana-1962 293 15 symmetric symmetric ADJ cana-1962 293 16 reverse reverse NOUN cana-1962 293 17 bi(𝜎 bi(𝜎 NOUN cana-1962 293 18 , , PUNCT cana-1962 293 19 𝜏)-derivations 𝜏)-derivation NOUN cana-1962 293 20 of of ADP cana-1962 293 21 semi semi ADJ cana-1962 293 22 prime prime ADJ cana-1962 293 23 ring ring NOUN cana-1962 293 24 . . PUNCT cana-1962 293 25 ” " PUNCT cana-1962 294 1 , , PUNCT cana-1962 294 2 communications communication NOUN cana-1962 294 3 on on ADP cana-1962 294 4 applied apply VERB cana-1962 294 5 nonlinear nonlinear ADJ cana-1962 294 6 analysis analysis NOUN cana-1962 294 7 , , PUNCT cana-1962 294 8 ( ( PUNCT cana-1962 294 9 31)1 31)1 NUM cana-1962 294 10 , , PUNCT cana-1962 294 11 2024 2024 NUM cana-1962 294 12 . . PUNCT