id sid tid token lemma pos cana-736 1 1 communications communication NOUN cana-736 1 2 on on ADP cana-736 1 3 applied apply VERB cana-736 1 4 nonlinear nonlinear ADJ cana-736 1 5 analysis analysis NOUN cana-736 1 6 issn issn NOUN cana-736 1 7 : : PUNCT cana-736 1 8 1074 1074 NUM cana-736 1 9 - - PUNCT cana-736 1 10 133x 133x NUM cana-736 1 11 vol vol NOUN cana-736 1 12 31 31 NUM cana-736 1 13 no no NOUN cana-736 1 14 . . PUNCT cana-736 2 1 3s 3s NUM cana-736 2 2 ( ( PUNCT cana-736 2 3 2024 2024 NUM cana-736 2 4 ) ) PUNCT cana-736 2 5 118 118 NUM cana-736 2 6 https://internationalpubls.com https://internationalpubls.com NUM cana-736 2 7 generalized generalized ADJ cana-736 2 8 modulus modulus NOUN cana-736 2 9 of of ADP cana-736 2 10 smoothness smoothness NOUN cana-736 2 11 and and CCONJ cana-736 2 12 𝑲 𝑲 DET cana-736 2 13 βˆ’ βˆ’ PROPN cana-736 2 14 π’‡π’–π’π’„π’•π’Šπ’π’π’‚π’. π’‡π’–π’π’„π’•π’Šπ’π’π’‚π’. NOUN cana-736 2 15 bushra bushra PROPN cana-736 3 1 kadhum kadhum PROPN cana-736 3 2 awaad awaad PROPN cana-736 3 3 1 1 NUM cana-736 4 1 * * PROPN cana-736 4 2 , , PUNCT cana-736 4 3 eman eman PROPN cana-736 4 4 samer samer PROPN cana-736 4 5 bhaya bhaya PROPN cana-736 4 6 2 2 NUM cana-736 4 7 1department 1department NUM cana-736 4 8 of of ADP cana-736 4 9 mathematics mathematic NOUN cana-736 4 10 , , PUNCT cana-736 4 11 college college NOUN cana-736 4 12 of of ADP cana-736 4 13 education education NOUN cana-736 4 14 for for ADP cana-736 4 15 pure pure ADJ cana-736 4 16 sciences science NOUN cana-736 4 17 , , PUNCT cana-736 4 18 university university NOUN cana-736 4 19 of of ADP cana-736 4 20 babylon babylon PROPN cana-736 4 21 , , PUNCT cana-736 4 22 babyel babyel PROPN cana-736 4 23 , , PUNCT cana-736 4 24 iraq iraq PROPN cana-736 4 25 . . PUNCT cana-736 5 1 2department 2department NUM cana-736 5 2 of of ADP cana-736 5 3 mathematics mathematic NOUN cana-736 5 4 , , PUNCT cana-736 5 5 college college NOUN cana-736 5 6 of of ADP cana-736 5 7 education education NOUN cana-736 5 8 for for ADP cana-736 5 9 pure pure ADJ cana-736 5 10 sciences science NOUN cana-736 5 11 , , PUNCT cana-736 5 12 university university NOUN cana-736 5 13 of of ADP cana-736 5 14 babylon babylon PROPN cana-736 5 15 , , PUNCT cana-736 5 16 babyel babyel PROPN cana-736 5 17 , , PUNCT cana-736 5 18 iraq iraq PROPN cana-736 5 19 . . PUNCT cana-736 6 1 2department 2department NUM cana-736 6 2 of of ADP cana-736 6 3 mathematics mathematic NOUN cana-736 6 4 , , PUNCT cana-736 6 5 college college NOUN cana-736 6 6 of of ADP cana-736 6 7 education education NOUN cana-736 6 8 , , PUNCT cana-736 6 9 al al PROPN cana-736 6 10 zahraa zahraa PROPN cana-736 6 11 university university PROPN cana-736 6 12 for for ADP cana-736 6 13 women woman NOUN cana-736 6 14 , , PUNCT cana-736 6 15 karbala karbala PROPN cana-736 6 16 , , PUNCT cana-736 6 17 iraq iraq PROPN cana-736 6 18 . . PUNCT cana-736 7 1 bushra.k@uokerbala.edu bushra.k@uokerbala.edu PROPN cana-736 7 2 .. .. PROPN cana-736 7 3 iq iq PROPN cana-736 7 4 emanbhaya@itnet.uobabylon.edu.iq emanbhaya@itnet.uobabylon.edu.iq PROPN cana-736 7 5 emanbhaya@alzahraa.edu.iq emanbhaya@alzahraa.edu.iq NUM cana-736 7 6 article article NOUN cana-736 7 7 history history NOUN cana-736 7 8 : : PUNCT cana-736 7 9 received receive VERB cana-736 7 10 : : PUNCT cana-736 7 11 09 09 NUM cana-736 7 12 - - PUNCT cana-736 7 13 04 04 NUM cana-736 7 14 - - PUNCT cana-736 7 15 2024 2024 NUM cana-736 7 16 revised revise VERB cana-736 7 17 : : PUNCT cana-736 7 18 22 22 NUM cana-736 7 19 - - SYM cana-736 7 20 05 05 NUM cana-736 7 21 - - PUNCT cana-736 7 22 2024 2024 NUM cana-736 7 23 accepted accept VERB cana-736 7 24 : : PUNCT cana-736 7 25 09 09 NUM cana-736 7 26 - - SYM cana-736 7 27 06 06 NUM cana-736 7 28 - - PUNCT cana-736 7 29 2024 2024 NUM cana-736 7 30 abstract abstract ADJ cana-736 7 31 many many ADJ cana-736 7 32 articles article NOUN cana-736 7 33 introduced introduce VERB cana-736 7 34 about about ADP cana-736 7 35 direct direct ADJ cana-736 7 36 and and CCONJ cana-736 7 37 inverse inverse ADJ cana-736 7 38 theorems theorem NOUN cana-736 7 39 interims interim NOUN cana-736 7 40 of of ADP cana-736 7 41 ordinary ordinary ADJ cana-736 7 42 modulus modulus NOUN cana-736 7 43 of of ADP cana-736 7 44 smoothness smoothness ADJ cana-736 7 45 andk andk NOUN cana-736 7 46 - - NOUN cana-736 7 47 functional functional ADJ cana-736 7 48 . . PUNCT cana-736 8 1 here here ADV cana-736 8 2 we we PRON cana-736 8 3 shall shall AUX cana-736 8 4 define define VERB cana-736 8 5 a a DET cana-736 8 6 generalized generalized ADJ cana-736 8 7 modulus modulus NOUN cana-736 8 8 of of ADP cana-736 8 9 smoothness smoothness ADJ cana-736 8 10 andk andk NOUN cana-736 8 11 - - ADJ cana-736 8 12 functional functional ADJ cana-736 8 13 , , PUNCT cana-736 8 14 then then ADV cana-736 8 15 we we PRON cana-736 8 16 prove prove VERB cana-736 8 17 they they PRON cana-736 8 18 are be AUX cana-736 8 19 equivalent equivalent ADJ cana-736 8 20 . . PUNCT cana-736 9 1 keywords keyword NOUN cana-736 9 2 : : PUNCT cana-736 9 3 k k X cana-736 9 4 β€” β€” PUNCT cana-736 9 5 functional functional ADJ cana-736 9 6 , , PUNCT cana-736 9 7 modulus modulus ADJ cana-736 9 8 etc etc X cana-736 9 9 . . X cana-736 10 1 1 1 X cana-736 10 2 . . X cana-736 10 3 introduction introduction NOUN cana-736 10 4 many many ADJ cana-736 10 5 articles article NOUN cana-736 10 6 such such ADJ cana-736 10 7 us[1 us[1 PROPN cana-736 10 8 ] ] X cana-736 10 9 , , PUNCT cana-736 10 10 [ [ X cana-736 10 11 3 3 NUM cana-736 10 12 ] ] PUNCT cana-736 10 13 , , PUNCT cana-736 10 14 [ [ X cana-736 10 15 6 6 NUM cana-736 10 16 ] ] PUNCT cana-736 10 17 , , PUNCT cana-736 11 1 [ [ X cana-736 11 2 8],[9],[10 8],[9],[10 X cana-736 11 3 ] ] PUNCT cana-736 11 4 introduced introduce VERB cana-736 11 5 approximation approximation NOUN cana-736 11 6 theorems theorem NOUN cana-736 11 7 interims interim NOUN cana-736 11 8 of of ADP cana-736 11 9 the the DET cana-736 11 10 ordinary ordinary ADJ cana-736 11 11 modulus modulus NOUN cana-736 11 12 of of ADP cana-736 11 13 smoothness smoothness NOUN cana-736 11 14 , , PUNCT cana-736 11 15 with with ADP cana-736 11 16 ordinary ordinary ADJ cana-736 11 17 symmetric symmetric ADJ cana-736 11 18 difference difference NOUN cana-736 11 19 . . PUNCT cana-736 12 1 here here ADV cana-736 12 2 we we PRON cana-736 12 3 shall shall AUX cana-736 12 4 define define VERB cana-736 12 5 new new ADJ cana-736 12 6 symmetric symmetric ADJ cana-736 12 7 difference difference NOUN cana-736 12 8 , , PUNCT cana-736 12 9 then then ADV cana-736 12 10 we we PRON cana-736 12 11 use use VERB cana-736 12 12 it it PRON cana-736 12 13 to to PART cana-736 12 14 obtain obtain VERB cana-736 12 15 anew anew ADJ cana-736 12 16 modulus modulus NOUN cana-736 12 17 of of ADP cana-736 12 18 smoothness smoothness NOUN cana-736 12 19 and and CCONJ cana-736 12 20 𝐾 𝐾 PROPN cana-736 12 21 βˆ’ βˆ’ PROPN cana-736 12 22 π‘“π‘’π‘›π‘π‘‘π‘–π‘œπ‘›π‘Žπ‘™ π‘“π‘’π‘›π‘π‘‘π‘–π‘œπ‘›π‘Žπ‘™ NOUN cana-736 12 23 , , PUNCT cana-736 12 24 in in ADP cana-736 12 25 anew anew ADJ cana-736 12 26 quasi quasi NOUN cana-736 12 27 normed normed PROPN cana-736 12 28 spaces space NOUN cana-736 12 29 , , PUNCT cana-736 12 30 call call VERB cana-736 12 31 it it PRON cana-736 12 32 𝐿𝑝,𝛽 𝐿𝑝,𝛽 ADV cana-736 12 33 . . PUNCT cana-736 13 1 let let VERB cana-736 13 2 us we PRON cana-736 13 3 define define VERB cana-736 13 4 𝐿𝑝 𝐿𝑝 PROPN cana-736 13 5 , , PUNCT cana-736 14 1 0 0 NUM cana-736 14 2 < < X cana-736 14 3 𝑝 𝑝 X cana-736 14 4 < < X cana-736 14 5 1 1 NUM cana-736 14 6 , , PUNCT cana-736 14 7 an an PRON cana-736 15 1 [ [ X cana-736 15 2 βˆ’1,1 βˆ’1,1 NOUN cana-736 15 3 ] ] PUNCT cana-736 15 4 as as ADP cana-736 15 5 the the DET cana-736 15 6 spaces space NOUN cana-736 15 7 of of ADP cana-736 15 8 all all DET cana-736 15 9 measurable measurable ADJ cana-736 15 10 functions function NOUN cana-736 15 11 satisfies satisfie NOUN cana-736 15 12 : : PUNCT cana-736 15 13 ‖𝑓‖𝑝 ‖𝑓‖𝑝 PROPN cana-736 15 14 = = PUNCT cana-736 15 15 ( ( PUNCT cana-736 15 16 ∫ ∫ PROPN cana-736 15 17 |𝑓|𝑝1 |𝑓|𝑝1 PROPN cana-736 15 18 βˆ’1 βˆ’1 PROPN cana-736 15 19 ) ) PUNCT cana-736 15 20 1 1 NUM cana-736 15 21 𝑝 𝑝 NOUN cana-736 15 22 < < X cana-736 15 23 ∞ ∞ NUM cana-736 15 24 . . PUNCT cana-736 16 1 [ [ X cana-736 16 2 9 9 NUM cana-736 16 3 ] ] PUNCT cana-736 16 4 define define NOUN cana-736 16 5 𝐿𝑝,𝛽 𝐿𝑝,𝛽 PROPN cana-736 16 6 , , PUNCT cana-736 16 7 measurable measurable ADJ cana-736 16 8 function function NOUN cana-736 16 9 spaces space NOUN cana-736 16 10 of of ADP cana-736 16 11 functions function NOUN cana-736 16 12 𝑓 𝑓 X cana-736 16 13 with with ADP cana-736 16 14 domain domain NOUN cana-736 16 15 [ [ X cana-736 16 16 βˆ’1,1 βˆ’1,1 X cana-736 16 17 ] ] PUNCT cana-736 16 18 . . PUNCT cana-736 17 1 √(1 √(1 NOUN cana-736 17 2 βˆ’ βˆ’ NOUN cana-736 17 3 π‘₯)𝛽 π‘₯)𝛽 X cana-736 17 4 𝑓 𝑓 PROPN cana-736 17 5 ∈ ∈ PROPN cana-736 17 6 𝐿𝑃 𝐿𝑃 PROPN cana-736 17 7 , , PUNCT cana-736 17 8 it it PRON cana-736 17 9 mean mean VERB cana-736 17 10 ‖𝑓‖𝑝,𝛽 ‖𝑓‖𝑝,𝛽 VERB cana-736 18 1 = = SYM cana-736 18 2 ( ( PUNCT cana-736 18 3 ∫ ∫ PROPN cana-736 18 4 ( ( PUNCT cana-736 18 5 |𝑓(π‘₯)(1 |𝑓(π‘₯)(1 NOUN cana-736 18 6 βˆ’ βˆ’ ADP cana-736 18 7 π‘₯ π‘₯ NOUN cana-736 18 8 ) ) PUNCT cana-736 18 9 𝛽 𝛽 NOUN cana-736 18 10 2 2 NUM cana-736 18 11 ) ) PUNCT cana-736 18 12 | | ADV cana-736 18 13 𝑝 𝑝 NOUN cana-736 18 14 𝑑π‘₯ 𝑑π‘₯ VERB cana-736 18 15 1 1 NUM cana-736 18 16 βˆ’1 βˆ’1 NOUN cana-736 18 17 ) ) PUNCT cana-736 18 18 1 1 NUM cana-736 18 19 𝑝 𝑝 NOUN cana-736 18 20 , , PUNCT cana-736 18 21 = = SYM cana-736 18 22 ‖𝑓 ‖𝑓 NOUN cana-736 18 23 ( ( PUNCT cana-736 18 24 1 1 NUM cana-736 18 25 βˆ’ βˆ’ PROPN cana-736 18 26 π‘₯ π‘₯ NOUN cana-736 18 27 ) ) PUNCT cana-736 18 28 𝛽 𝛽 PROPN cana-736 18 29 2 2 NUM cana-736 18 30 β€– β€– PROPN cana-736 18 31 𝑝 𝑝 PROPN cana-736 18 32 . . PUNCT cana-736 19 1 let let VERB cana-736 19 2 𝐸𝑛(𝑓)𝑝,𝛽 𝐸𝑛(𝑓)𝑝,𝛽 VERB cana-736 19 3 , , PUNCT cana-736 19 4 best good ADJ cana-736 19 5 approximation approximation NOUN cana-736 19 6 error error NOUN cana-736 19 7 of of ADP cana-736 19 8 𝑓 𝑓 PRON cana-736 19 9 in in ADP cana-736 19 10 𝐿𝑝,𝛽 𝐿𝑝,𝛽 NOUN cana-736 19 11 , , PUNCT cana-736 19 12 using use VERB cana-736 19 13 algebraic algebraic PROPN cana-736 19 14 < < X cana-736 19 15 𝑛 𝑛 PROPN cana-736 19 16 in in ADP cana-736 19 17 polynomials polynomial NOUN cana-736 19 18 of of ADP cana-736 19 19 degree degree NOUN cana-736 19 20 𝐿𝑝,𝛽 𝐿𝑝,𝛽 NOUN cana-736 19 21 , , PUNCT cana-736 19 22 and and CCONJ cana-736 19 23 𝐸𝑛(𝑓)𝑝,𝛽 𝐸𝑛(𝑓)𝑝,𝛽 VERB cana-736 19 24 = = SYM cana-736 19 25 inf inf ADJ cana-736 19 26 π‘π‘›βˆˆπΌπ‘ƒπ‘› π‘π‘›βˆˆπΌπ‘ƒπ‘› PROPN cana-736 19 27 ‖𝑓 ‖𝑓 NOUN cana-736 19 28 βˆ’ βˆ’ NOUN cana-736 19 29 𝑝𝑛‖ 𝑝𝑛‖ NOUN cana-736 19 30 , , PUNCT cana-736 19 31 [ [ X cana-736 19 32 8 8 NUM cana-736 19 33 ] ] PUNCT cana-736 19 34 where where SCONJ cana-736 19 35 , , PUNCT cana-736 19 36 ǁℙ𝑛 ǁℙ𝑛 NOUN cana-736 19 37 is be AUX cana-736 19 38 𝑛 𝑛 DET cana-736 19 39 βˆ’ βˆ’ PROPN cana-736 19 40 1 1 NUM cana-736 19 41 , , PUNCT cana-736 19 42 degree degree NOUN cana-736 19 43 algebraic algebraic ADJ cana-736 19 44 polynomials polynomial NOUN cana-736 19 45 spaces space NOUN cana-736 19 46 . . PUNCT cana-736 20 1 communications communication NOUN cana-736 20 2 on on ADP cana-736 20 3 applied apply VERB cana-736 20 4 nonlinear nonlinear ADJ cana-736 20 5 analysis analysis NOUN cana-736 20 6 issn issn NOUN cana-736 20 7 : : PUNCT cana-736 20 8 1074 1074 NUM cana-736 20 9 - - PUNCT cana-736 20 10 133x 133x NUM cana-736 20 11 vol vol NOUN cana-736 20 12 31 31 NUM cana-736 20 13 no no NOUN cana-736 20 14 . . PUNCT cana-736 21 1 3s 3s NUM cana-736 21 2 ( ( PUNCT cana-736 21 3 2024 2024 NUM cana-736 21 4 ) ) PUNCT cana-736 21 5 119 119 NUM cana-736 21 6 https://internationalpubls.com https://internationalpubls.com X cana-736 21 7 let let VERB cana-736 21 8 us we PRON cana-736 21 9 define define VERB cana-736 21 10 the the DET cana-736 21 11 operator operator NOUN cana-736 21 12 𝐷π‘₯,𝜈,πœ‡ 𝐷π‘₯,𝜈,πœ‡ NOUN cana-736 21 13 = = SYM cana-736 21 14 ( ( PUNCT cana-736 21 15 √1 √1 ADV cana-736 21 16 βˆ’ βˆ’ PROPN cana-736 21 17 π‘₯ π‘₯ NOUN cana-736 21 18 ) ) PUNCT cana-736 21 19 βˆ’πœˆ βˆ’πœˆ NOUN cana-736 21 20 ( ( PUNCT cana-736 21 21 1 1 NUM cana-736 22 1 + + CCONJ cana-736 22 2 π‘₯)βˆ’πœ‡ π‘₯)βˆ’πœ‡ PRON cana-736 22 3 𝑑 𝑑 NOUN cana-736 22 4 𝑑π‘₯ 𝑑π‘₯ VERB cana-736 22 5 ( ( PUNCT cana-736 22 6 √(1 √(1 NOUN cana-736 22 7 βˆ’ βˆ’ PROPN cana-736 22 8 π‘₯ π‘₯ NOUN cana-736 22 9 ) ) PUNCT cana-736 22 10 ) ) PUNCT cana-736 23 1 𝜈+1 𝜈+1 PROPN cana-736 23 2 ( ( PUNCT cana-736 23 3 √(1 √(1 VERB cana-736 23 4 + + NOUN cana-736 23 5 π‘₯ π‘₯ NOUN cana-736 23 6 ) ) PUNCT cana-736 23 7 ) ) PUNCT cana-736 24 1 πœ‡+1 πœ‡+1 PROPN cana-736 24 2 𝑑 𝑑 NOUN cana-736 24 3 𝑑π‘₯ 𝑑π‘₯ NOUN cana-736 24 4 . . PUNCT cana-736 25 1 [ [ X cana-736 25 2 4 4 NUM cana-736 25 3 ] ] PUNCT cana-736 25 4 peetre peetre NOUN cana-736 25 5 𝐾 𝐾 PROPN cana-736 25 6 βˆ’ βˆ’ PROPN cana-736 25 7 π‘“π‘’π‘›π‘π‘‘π‘–π‘œπ‘›π‘Žπ‘™ π‘“π‘’π‘›π‘π‘‘π‘–π‘œπ‘›π‘Žπ‘™ NOUN cana-736 25 8 on on ADP cana-736 25 9 𝐿𝑝,𝛼 𝐿𝑝,𝛼 PROPN cana-736 25 10 and and CCONJ cana-736 25 11 𝐼𝑃𝑛is 𝐼𝑃𝑛is PROPN cana-736 25 12 defined define VERB cana-736 25 13 by by ADP cana-736 25 14 𝐾(𝑓 𝐾(𝑓 NOUN cana-736 25 15 , , PUNCT cana-736 25 16 𝛿)𝑝,𝛼 𝛿)𝑝,𝛼 NOUN cana-736 25 17 = = SYM cana-736 25 18 inf inf PROPN cana-736 25 19 π‘”βˆˆπΌπ‘ƒπ‘› π‘”βˆˆπΌπ‘ƒπ‘› PROPN cana-736 25 20 ( ( PUNCT cana-736 25 21 ‖𝑓 ‖𝑓 NOUN cana-736 25 22 βˆ’ βˆ’ NOUN cana-736 25 23 𝑔 𝑔 PROPN cana-736 25 24 ‖𝑝,𝛼 ‖𝑝,𝛼 PROPN cana-736 25 25 + + CCONJ cana-736 25 26 𝛿2β€– 𝛿2β€– PROPN cana-736 25 27 𝐷π‘₯,2,2𝑔(π‘₯ 𝐷π‘₯,2,2𝑔(π‘₯ X cana-736 25 28 ) ) PUNCT cana-736 25 29 β€– β€– PROPN cana-736 25 30 𝑝,𝛼 𝑝,𝛼 NOUN cana-736 25 31 ) ) PUNCT cana-736 25 32 , , PUNCT cana-736 26 1 [ [ X cana-736 26 2 4 4 NUM cana-736 26 3 ] ] PUNCT cana-736 26 4 . . PUNCT cana-736 27 1 let let VERB cana-736 27 2 us we PRON cana-736 27 3 introduce introduce VERB cana-736 27 4 usual usual ADJ cana-736 27 5 smoothness smoothness ADJ cana-736 27 6 modulus modulus NOUN cana-736 27 7 . . PUNCT cana-736 28 1 αΏΆ1 αΏΆ1 PROPN cana-736 28 2 ( ( PUNCT cana-736 28 3 𝑓 𝑓 PROPN cana-736 28 4 , , PUNCT cana-736 28 5 𝛿)𝑝,𝛽 𝛿)𝑝,𝛽 NOUN cana-736 28 6 = = PUNCT cana-736 28 7 sup sup NOUN cana-736 28 8 |𝑑|≀𝛿 |𝑑|≀𝛿 PROPN cana-736 28 9 β€–πœπ‘‘ β€–πœπ‘‘ PUNCT cana-736 28 10 ( ( PUNCT cana-736 28 11 𝑓 𝑓 X cana-736 28 12 ) ) PUNCT cana-736 28 13 βˆ’ βˆ’ PROPN cana-736 28 14 𝑓‖𝑝,𝛼 𝑓‖𝑝,𝛼 NOUN cana-736 28 15 . . PUNCT cana-736 29 1 ( ( PUNCT cana-736 29 2 1 1 X cana-736 29 3 ) ) PUNCT cana-736 29 4 where where SCONJ cana-736 29 5 , , PUNCT cana-736 29 6 πœπ‘‘ πœπ‘‘ PROPN cana-736 29 7 ( ( PUNCT cana-736 29 8 𝑓 𝑓 X cana-736 29 9 ) ) PUNCT cana-736 29 10 = = SYM cana-736 29 11 1 1 NUM cana-736 29 12 πœ‹(1βˆ’π‘₯2 πœ‹(1βˆ’π‘₯2 NOUN cana-736 29 13 ) ) PUNCT cana-736 29 14 cos4𝑑 cos4𝑑 PROPN cana-736 29 15 2 2 NUM cana-736 29 16 ∫ ∫ NOUN cana-736 29 17 ( ( PUNCT cana-736 29 18 2(√1 2(√1 NOUN cana-736 29 19 βˆ’ βˆ’ PROPN cana-736 29 20 π‘₯2 π‘₯2 NOUN cana-736 29 21 cos cos ADP cana-736 29 22 𝑑 𝑑 PROPN cana-736 29 23 + + ADP cana-736 29 24 π‘₯ π‘₯ PROPN cana-736 29 25 sin sin NOUN cana-736 29 26 𝑑 𝑑 X cana-736 29 27 cos cos PROPN cana-736 29 28 πœ‘ πœ‘ PROPN cana-736 29 29 + + CCONJ cana-736 29 30 √1 √1 CCONJ cana-736 29 31 βˆ’ βˆ’ PROPN cana-736 29 32 π‘₯2 π‘₯2 NOUN cana-736 29 33 ( ( PUNCT cana-736 29 34 1 1 NUM cana-736 29 35 βˆ’ βˆ’ PROPN cana-736 29 36 cos cos PROPN cana-736 29 37 𝑑 𝑑 PROPN cana-736 29 38 ) ) PUNCT cana-736 29 39 sin2 sin2 NOUN cana-736 29 40 πœ‘ πœ‘ NOUN cana-736 29 41 ) ) PUNCT cana-736 29 42 2 2 NUM cana-736 29 43 βˆ’ βˆ’ NOUN cana-736 29 44 πœ‹ πœ‹ NOUN cana-736 29 45 0 0 NUM cana-736 29 46 1 1 NUM cana-736 29 47 + + CCONJ cana-736 29 48 ( ( PUNCT cana-736 29 49 π‘₯ π‘₯ X cana-736 29 50 cos cos ADP cana-736 29 51 𝑑 𝑑 PROPN cana-736 30 1 βˆ’ βˆ’ PROPN cana-736 30 2 √1 √1 ADV cana-736 30 3 βˆ’ βˆ’ PROPN cana-736 30 4 π‘₯2 π‘₯2 PROPN cana-736 30 5 sin sin VERB cana-736 30 6 𝑑 𝑑 PROPN cana-736 30 7 cos cos PROPN cana-736 30 8 πœ‘ πœ‘ PROPN cana-736 30 9 ) ) PUNCT cana-736 30 10 2 2 NUM cana-736 30 11 ) ) PUNCT cana-736 30 12 ⬚ ⬚ PROPN cana-736 30 13 𝑓(π‘₯ 𝑓(π‘₯ PROPN cana-736 30 14 cos cos ADP cana-736 30 15 𝑑 𝑑 PRON cana-736 30 16 βˆ’ βˆ’ PROPN cana-736 30 17 √1 √1 ADV cana-736 30 18 βˆ’ βˆ’ PROPN cana-736 30 19 π‘₯2 π‘₯2 PROPN cana-736 30 20 sin sin VERB cana-736 30 21 𝑑 𝑑 PROPN cana-736 30 22 cos cos PROPN cana-736 30 23 πœ‘ πœ‘ NOUN cana-736 30 24 ) ) PUNCT cana-736 30 25 π‘‘πœ‘ π‘‘πœ‘ VERB cana-736 30 26 . . PUNCT cana-736 31 1 let let VERB cana-736 31 2 𝑓 𝑓 DET cana-736 31 3 ∈ ∈ PROPN cana-736 31 4 𝐿𝑝,𝛼 𝐿𝑝,𝛼 PROPN cana-736 31 5 , , PUNCT cana-736 31 6 . . PUNCT cana-736 32 1 put put VERB cana-736 32 2 𝑦 𝑦 NOUN cana-736 32 3 = = SYM cana-736 32 4 cos cos PROPN cana-736 32 5 𝑑 𝑑 PROPN cana-736 32 6 , , PUNCT cana-736 32 7 𝑧 𝑧 X cana-736 32 8 = = PUNCT cana-736 32 9 cos cos ADP cana-736 32 10 πœ‘ πœ‘ PROPN cana-736 32 11 , , PUNCT cana-736 32 12 in in ADP cana-736 32 13 the the DET cana-736 32 14 πœπ‘‘ πœπ‘‘ PROPN cana-736 32 15 ( ( PUNCT cana-736 32 16 𝑓 𝑓 PROPN cana-736 32 17 ) ) PUNCT cana-736 32 18 , , PUNCT cana-736 32 19 let let VERB cana-736 32 20 πœπ‘¦ πœπ‘¦ PRON cana-736 32 21 ( ( PUNCT cana-736 32 22 𝑓 𝑓 X cana-736 32 23 ) ) PUNCT cana-736 32 24 . . PUNCT cana-736 33 1 then then ADV cana-736 33 2 πœπ‘‘ πœπ‘‘ PROPN cana-736 33 3 ( ( PUNCT cana-736 33 4 𝑓 𝑓 X cana-736 33 5 ) ) PUNCT cana-736 33 6 = = SYM cana-736 33 7 4 4 NUM cana-736 33 8 πœ‹(1βˆ’π‘₯2)(1+𝑦)2 πœ‹(1βˆ’π‘₯2)(1+𝑦)2 NOUN cana-736 33 9 ∫ ∫ PROPN cana-736 33 10 𝐡𝑦(π‘₯ 𝐡𝑦(π‘₯ PROPN cana-736 33 11 , , PUNCT cana-736 33 12 𝑧 𝑧 PROPN cana-736 33 13 , , PUNCT cana-736 33 14 𝑅 𝑅 NOUN cana-736 33 15 ) ) PUNCT cana-736 33 16 1 1 NUM cana-736 33 17 βˆ’1 βˆ’1 NOUN cana-736 33 18 𝑓(𝑅 𝑓(𝑅 NOUN cana-736 33 19 ) ) PUNCT cana-736 33 20 𝑑𝑧 𝑑𝑧 NOUN cana-736 33 21 √1βˆ’π‘§2 √1βˆ’π‘§2 NOUN cana-736 33 22 , , PUNCT cana-736 33 23 [ [ X cana-736 33 24 1 1 NUM cana-736 33 25 ] ] PUNCT cana-736 33 26 . . PUNCT cana-736 34 1 where where SCONJ cana-736 34 2 , , PUNCT cana-736 34 3 𝑅 𝑅 PROPN cana-736 34 4 = = PROPN cana-736 34 5 π‘₯𝑦 π‘₯𝑦 PROPN cana-736 34 6 βˆ’ βˆ’ PROPN cana-736 34 7 π‘§βˆš1 π‘§βˆš1 NOUN cana-736 34 8 βˆ’ βˆ’ NOUN cana-736 34 9 π‘₯2 π‘₯2 NOUN cana-736 34 10 √1 √1 PART cana-736 34 11 βˆ’ βˆ’ PROPN cana-736 34 12 𝑦2 𝑦2 PROPN cana-736 34 13 , , PUNCT cana-736 34 14 𝐡𝑦(π‘₯ 𝐡𝑦(π‘₯ PROPN cana-736 34 15 , , PUNCT cana-736 34 16 𝑧 𝑧 PROPN cana-736 34 17 , , PUNCT cana-736 34 18 𝑅 𝑅 NOUN cana-736 34 19 ) ) PUNCT cana-736 34 20 = = SYM cana-736 34 21 2 2 NUM cana-736 34 22 ( ( PUNCT cana-736 34 23 √1 √1 ADV cana-736 34 24 βˆ’ βˆ’ PROPN cana-736 34 25 π‘₯2 π‘₯2 PROPN cana-736 34 26 𝑦 𝑦 PROPN cana-736 34 27 + + NUM cana-736 34 28 𝑧π‘₯√1 𝑧π‘₯√1 PROPN cana-736 34 29 βˆ’ βˆ’ PROPN cana-736 34 30 𝑦2 𝑦2 PROPN cana-736 34 31 + + CCONJ cana-736 34 32 √1 √1 CCONJ cana-736 34 33 βˆ’ βˆ’ PROPN cana-736 34 34 π‘₯2 π‘₯2 NOUN cana-736 34 35 ( ( PUNCT cana-736 34 36 1 1 NUM cana-736 34 37 βˆ’ βˆ’ NOUN cana-736 34 38 𝑦)(1 𝑦)(1 NUM cana-736 34 39 βˆ’ βˆ’ PROPN cana-736 34 40 𝑧2 𝑧2 NOUN cana-736 34 41 ) ) PUNCT cana-736 34 42 ) ) PUNCT cana-736 34 43 2 2 NUM cana-736 34 44 βˆ’ βˆ’ NOUN cana-736 34 45 ( ( PUNCT cana-736 34 46 1 1 NUM cana-736 34 47 βˆ’ βˆ’ NOUN cana-736 34 48 𝑅2 𝑅2 NOUN cana-736 34 49 ) ) PUNCT cana-736 34 50 . . PUNCT cana-736 34 51 2 2 X cana-736 34 52 . . X cana-736 34 53 auxiliary auxiliary ADJ cana-736 34 54 results result NOUN cana-736 34 55 to to PART cana-736 34 56 prove prove VERB cana-736 34 57 our our PRON cana-736 34 58 main main ADJ cana-736 34 59 theorem theorem NOUN cana-736 34 60 we we PRON cana-736 34 61 need need VERB cana-736 34 62 some some DET cana-736 34 63 results result NOUN cana-736 34 64 , , PUNCT cana-736 34 65 that that SCONJ cana-736 34 66 we we PRON cana-736 34 67 make make VERB cana-736 34 68 use use NOUN cana-736 34 69 of of ADP cana-736 34 70 them they PRON cana-736 34 71 in in ADP cana-736 34 72 our our PRON cana-736 34 73 proof proof NOUN cana-736 34 74 . . PUNCT cana-736 35 1 let let VERB cana-736 35 2 us we PRON cana-736 35 3 begin begin VERB cana-736 35 4 with with ADP cana-736 35 5 lemma lemma PROPN cana-736 35 6 .2.1 .2.1 NOUN cana-736 35 7 . . PUNCT cana-736 36 1 for for ADP cana-736 36 2 𝑧 𝑧 PROPN cana-736 36 3 = = PUNCT cana-736 36 4 cos cos PROPN cana-736 36 5 πœ‘ πœ‘ PROPN cana-736 36 6 and and CCONJ cana-736 36 7 𝑅 𝑅 PROPN cana-736 36 8 = = PUNCT cana-736 36 9 π‘₯ π‘₯ PROPN cana-736 36 10 cos cos ADP cana-736 36 11 𝑑 𝑑 PROPN cana-736 36 12 βˆ’ βˆ’ NOUN cana-736 36 13 π‘§βˆš1 π‘§βˆš1 NOUN cana-736 36 14 βˆ’ βˆ’ NOUN cana-736 36 15 π‘₯2 π‘₯2 PROPN cana-736 36 16 sin sin PROPN cana-736 36 17 𝑑 𝑑 PROPN cana-736 36 18 , , PUNCT cana-736 36 19 𝑓 𝑓 PROPN cana-736 36 20 ∈ ∈ PROPN cana-736 36 21 𝐿𝑝,𝛽[βˆ’1,1 𝐿𝑝,𝛽[βˆ’1,1 PROPN cana-736 36 22 ] ] PUNCT cana-736 36 23 , , PUNCT cana-736 36 24 we we PRON cana-736 36 25 get get VERB cana-736 36 26 ( ( PUNCT cana-736 36 27 2.1 2.1 NUM cana-736 36 28 ) ) PUNCT cana-736 36 29 β€– β€– PROPN cana-736 36 30 1 1 NUM cana-736 36 31 1 1 NUM cana-736 36 32 βˆ’ βˆ’ NOUN cana-736 36 33 π‘₯2 π‘₯2 PROPN cana-736 36 34 ∫ ∫ PROPN cana-736 36 35 ( ( PUNCT cana-736 36 36 1 1 NUM cana-736 36 37 βˆ’ βˆ’ NOUN cana-736 36 38 𝑅2)|𝑓(𝑅)| 𝑅2)|𝑓(𝑅)| ADP cana-736 36 39 𝑑𝑧 𝑑𝑧 NOUN cana-736 36 40 √1 √1 PROPN cana-736 37 1 βˆ’ βˆ’ PROPN cana-736 37 2 𝑧2 𝑧2 NOUN cana-736 37 3 1 1 NUM cana-736 37 4 βˆ’1 βˆ’1 NOUN cana-736 37 5 β€– β€– PROPN cana-736 37 6 𝑝,𝛽 𝑝,𝛽 PROPN cana-736 37 7 ≀ ≀ NUM cana-736 37 8 ∁ ∁ PROPN cana-736 38 1 |𝑓| |𝑓| NOUN cana-736 38 2 proof proof NOUN cana-736 38 3 using use VERB cana-736 38 4 simple simple ADJ cana-736 38 5 calculating calculating NOUN cana-736 38 6 of of ADP cana-736 38 7 integrals integral NOUN cana-736 38 8 we we PRON cana-736 38 9 obtain obtain VERB cana-736 38 10 the the DET cana-736 38 11 result result NOUN cana-736 38 12 in in ADP cana-736 38 13 ( ( PUNCT cana-736 38 14 2.1 2.1 NUM cana-736 38 15 ) ) PUNCT cana-736 38 16 . . PUNCT cana-736 39 1 lemma.2.2 lemma.2.2 PROPN cana-736 39 2 . . PUNCT cana-736 40 1 [ [ X cana-736 40 2 1 1 NUM cana-736 40 3 ] ] X cana-736 40 4 |𝐡𝑦(π‘₯ |𝐡𝑦(π‘₯ VERB cana-736 40 5 , , PUNCT cana-736 40 6 𝑧 𝑧 PRON cana-736 40 7 , , PUNCT cana-736 40 8 𝑅)| 𝑅)| ADV cana-736 40 9 ≀ ≀ ADJ cana-736 40 10 19(1 19(1 NUM cana-736 40 11 βˆ’ βˆ’ NOUN cana-736 40 12 𝑅2 𝑅2 NOUN cana-736 40 13 ) ) PUNCT cana-736 40 14 where where SCONJ cana-736 40 15 , , PUNCT cana-736 40 16 𝑅 𝑅 PROPN cana-736 40 17 = = PUNCT cana-736 40 18 π‘₯ π‘₯ PROPN cana-736 40 19 cos cos ADP cana-736 40 20 𝑑 𝑑 PROPN cana-736 40 21 βˆ’ βˆ’ NOUN cana-736 40 22 π‘§βˆš1 π‘§βˆš1 NOUN cana-736 40 23 βˆ’ βˆ’ NOUN cana-736 40 24 π‘₯2 π‘₯2 PROPN cana-736 40 25 sin sin PROPN cana-736 40 26 𝑑 𝑑 PROPN cana-736 40 27 , , PUNCT cana-736 40 28 𝑧 𝑧 X cana-736 40 29 = = PUNCT cana-736 40 30 cos cos ADP cana-736 40 31 πœ‘ πœ‘ PROPN cana-736 40 32 and and CCONJ cana-736 40 33 π‘₯ π‘₯ PROPN cana-736 40 34 = = SYM cana-736 40 35 cos co NOUN cana-736 40 36 πœƒ1 πœƒ1 PROPN cana-736 40 37 . . PUNCT cana-736 41 1 lemma lemma PROPN cana-736 41 2 . . PROPN cana-736 42 1 2.3 2.3 NUM cana-736 42 2 . . PUNCT cana-736 43 1 communications communication NOUN cana-736 43 2 on on ADP cana-736 43 3 applied apply VERB cana-736 43 4 nonlinear nonlinear ADJ cana-736 43 5 analysis analysis NOUN cana-736 43 6 issn issn NOUN cana-736 43 7 : : PUNCT cana-736 43 8 1074 1074 NUM cana-736 43 9 - - PUNCT cana-736 43 10 133x 133x NUM cana-736 43 11 vol vol NOUN cana-736 43 12 31 31 NUM cana-736 43 13 no no NOUN cana-736 43 14 . . PUNCT cana-736 44 1 3s 3s NUM cana-736 44 2 ( ( PUNCT cana-736 44 3 2024 2024 NUM cana-736 44 4 ) ) PUNCT cana-736 44 5 120 120 NUM cana-736 44 6 https://internationalpubls.com https://internationalpubls.com X cana-736 44 7 if if SCONJ cana-736 44 8 𝑓 𝑓 DET cana-736 44 9 ∈ ∈ PROPN cana-736 44 10 𝐿𝑝,𝛽[βˆ’1,1 𝐿𝑝,𝛽[βˆ’1,1 PROPN cana-736 44 11 ] ] PUNCT cana-736 44 12 , , PUNCT cana-736 44 13 we we PRON cana-736 44 14 get get VERB cana-736 44 15 β€– β€– ADJ cana-736 44 16 πœπ‘‘ πœπ‘‘ PROPN cana-736 44 17 ( ( PUNCT cana-736 44 18 𝑓)‖𝑝,𝛽 𝑓)‖𝑝,𝛽 PROPN cana-736 44 19 ≀ ≀ NOUN cana-736 44 20 19∁(𝑝 19∁(𝑝 NUM cana-736 44 21 ) ) PUNCT cana-736 44 22 πœ‹ πœ‹ NOUN cana-736 44 23 cos4 cos4 PROPN cana-736 44 24 𝑑 𝑑 PROPN cana-736 44 25 2 2 NUM cana-736 44 26 ‖𝑓‖𝑝,𝛽 ‖𝑓‖𝑝,𝛽 ADJ cana-736 44 27 , , PUNCT cana-736 44 28 where where SCONJ cana-736 44 29 𝑐 𝑐 X cana-736 44 30 > > X cana-736 44 31 0 0 X cana-736 44 32 . . PUNCT cana-736 45 1 proof proof NOUN cana-736 45 2 . . PUNCT cana-736 46 1 β€– β€– ADJ cana-736 46 2 πœπ‘‘ πœπ‘‘ PROPN cana-736 46 3 ( ( PUNCT cana-736 46 4 𝑓)‖𝑝,𝛽 𝑓)‖𝑝,𝛽 PROPN cana-736 46 5 = = SYM cana-736 46 6 β€– β€– PROPN cana-736 46 7 1 1 NUM cana-736 46 8 πœ‹ πœ‹ NOUN cana-736 46 9 cos4𝑑 cos4𝑑 NUM cana-736 46 10 2 2 NUM cana-736 46 11 1 1 NUM cana-736 46 12 1βˆ’π‘₯2 1βˆ’π‘₯2 NUM cana-736 46 13 ∫ ∫ PROPN cana-736 46 14 𝐡cos 𝐡cos PROPN cana-736 46 15 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-736 46 16 , , PUNCT cana-736 46 17 𝑧 𝑧 PROPN cana-736 46 18 , , PUNCT cana-736 46 19 𝑅 𝑅 NOUN cana-736 46 20 ) ) PUNCT cana-736 46 21 1 1 NUM cana-736 46 22 βˆ’1 βˆ’1 NOUN cana-736 46 23 𝑓(𝑅 𝑓(𝑅 NOUN cana-736 46 24 ) ) PUNCT cana-736 46 25 𝑑𝑧 𝑑𝑧 NOUN cana-736 46 26 √1βˆ’π‘§2 √1βˆ’π‘§2 NOUN cana-736 46 27 β€– β€– PROPN cana-736 46 28 𝑝,𝛽 𝑝,𝛽 PROPN cana-736 46 29 β€– β€– PROPN cana-736 46 30 πœπ‘‘ πœπ‘‘ PROPN cana-736 46 31 ( ( PUNCT cana-736 46 32 𝑓)‖𝑝,𝛽 𝑓)‖𝑝,𝛽 NOUN cana-736 46 33 = = SYM cana-736 46 34 1 1 NUM cana-736 46 35 πœ‹ πœ‹ NOUN cana-736 46 36 cos4 cos4 PROPN cana-736 46 37 𝑑 𝑑 PROPN cana-736 46 38 2 2 NUM cana-736 46 39 β€– β€– PROPN cana-736 46 40 1 1 NUM cana-736 46 41 1 1 NUM cana-736 46 42 βˆ’ βˆ’ NOUN cana-736 46 43 π‘₯2 π‘₯2 PROPN cana-736 46 44 ∫ ∫ PROPN cana-736 46 45 𝐡cos 𝐡cos PROPN cana-736 46 46 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-736 46 47 , , PUNCT cana-736 46 48 𝑧 𝑧 PROPN cana-736 46 49 , , PUNCT cana-736 46 50 𝑅 𝑅 NOUN cana-736 46 51 ) ) PUNCT cana-736 46 52 1 1 NUM cana-736 46 53 βˆ’1 βˆ’1 NOUN cana-736 46 54 𝑓(𝑅 𝑓(𝑅 NOUN cana-736 46 55 ) ) PUNCT cana-736 46 56 𝑑𝑧 𝑑𝑧 PROPN cana-736 46 57 √1 √1 ADV cana-736 46 58 βˆ’ βˆ’ PROPN cana-736 46 59 𝑧2 𝑧2 PROPN cana-736 46 60 β€– β€– PROPN cana-736 46 61 𝑝,𝛽 𝑝,𝛽 PROPN cana-736 46 62 β€– β€– PROPN cana-736 46 63 πœπ‘‘ πœπ‘‘ PROPN cana-736 46 64 ( ( PUNCT cana-736 46 65 𝑓)‖𝑝,𝛽 𝑓)‖𝑝,𝛽 NOUN cana-736 46 66 = = SYM cana-736 46 67 1 1 NUM cana-736 46 68 πœ‹ πœ‹ NOUN cana-736 46 69 cos4𝑑 cos4𝑑 NOUN cana-736 46 70 2 2 NUM cana-736 46 71 ( ( PUNCT cana-736 46 72 ∫ ∫ PROPN cana-736 46 73 | | ADV cana-736 46 74 1 1 NUM cana-736 46 75 1βˆ’π‘₯2 1βˆ’π‘₯2 NUM cana-736 46 76 ∫ ∫ PROPN cana-736 46 77 ∫ ∫ PROPN cana-736 46 78 𝐡cos 𝐡cos PROPN cana-736 46 79 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-736 46 80 , , PUNCT cana-736 46 81 𝑧 𝑧 PROPN cana-736 46 82 , , PUNCT cana-736 46 83 𝑅 𝑅 NOUN cana-736 46 84 ) ) PUNCT cana-736 46 85 1 1 NUM cana-736 46 86 βˆ’1 βˆ’1 NOUN cana-736 46 87 𝑓(𝑅 𝑓(𝑅 NOUN cana-736 46 88 ) ) PUNCT cana-736 46 89 𝑑𝑧 𝑑𝑧 NOUN cana-736 46 90 √1βˆ’π‘§2 √1βˆ’π‘§2 NOUN cana-736 46 91 1 1 NUM cana-736 46 92 βˆ’1 βˆ’1 NOUN cana-736 46 93 | | ADV cana-736 46 94 1 1 NUM cana-736 46 95 βˆ’1 βˆ’1 NOUN cana-736 46 96 𝑝 𝑝 NOUN cana-736 46 97 ( ( PUNCT cana-736 46 98 1 1 NUM cana-736 46 99 βˆ’ βˆ’ PROPN cana-736 46 100 π‘₯ π‘₯ NOUN cana-736 46 101 ) ) PUNCT cana-736 46 102 𝛽 𝛽 NOUN cana-736 46 103 2 2 NUM cana-736 46 104 𝑑π‘₯ 𝑑π‘₯ NOUN cana-736 46 105 ) ) PUNCT cana-736 46 106 1 1 NUM cana-736 46 107 𝑝 𝑝 NOUN cana-736 46 108 . . PUNCT cana-736 47 1 β€– β€– ADJ cana-736 47 2 πœπ‘‘ πœπ‘‘ PROPN cana-736 47 3 ( ( PUNCT cana-736 47 4 𝑓)‖𝑝,𝛽 𝑓)‖𝑝,𝛽 PROPN cana-736 47 5 ≀ ≀ NOUN cana-736 47 6 1 1 NUM cana-736 47 7 πœ‹ πœ‹ NOUN cana-736 47 8 cos4𝑑 cos4𝑑 PROPN cana-736 47 9 2 2 NUM cana-736 47 10 ( ( PUNCT cana-736 47 11 ∫ ∫ PROPN cana-736 47 12 ( ( PUNCT cana-736 47 13 1 1 NUM cana-736 47 14 1βˆ’π‘₯2 1βˆ’π‘₯2 NUM cana-736 47 15 ∫ ∫ PROPN cana-736 47 16 |𝐡cos |𝐡cos PROPN cana-736 47 17 𝑑(π‘₯ 𝑑(π‘₯ PROPN cana-736 47 18 , , PUNCT cana-736 47 19 𝑧 𝑧 NOUN cana-736 47 20 , , PUNCT cana-736 47 21 𝑅)||𝑓(𝑅)| 𝑅)||𝑓(𝑅)| ADP cana-736 47 22 1 1 NUM cana-736 47 23 βˆ’1 βˆ’1 NOUN cana-736 47 24 𝑑𝑧 𝑑𝑧 NOUN cana-736 47 25 √1βˆ’π‘§2 √1βˆ’π‘§2 NOUN cana-736 47 26 ) ) PUNCT cana-736 47 27 𝑝 𝑝 PROPN cana-736 47 28 ( ( PUNCT cana-736 47 29 1 1 NUM cana-736 47 30 βˆ’ βˆ’ PROPN cana-736 47 31 π‘₯ π‘₯ NOUN cana-736 47 32 ) ) PUNCT cana-736 47 33 𝛽 𝛽 NOUN cana-736 47 34 2 2 NUM cana-736 47 35 1 1 NUM cana-736 47 36 βˆ’1 βˆ’1 NOUN cana-736 47 37 𝑑π‘₯ 𝑑π‘₯ NOUN cana-736 47 38 ) ) PUNCT cana-736 47 39 1 1 NUM cana-736 47 40 𝑝 𝑝 NOUN cana-736 47 41 . . PUNCT cana-736 48 1 using use VERB cana-736 48 2 lemma lemma PROPN cana-736 48 3 ( ( PUNCT cana-736 48 4 2.2 2.2 NUM cana-736 48 5 ) ) PUNCT cana-736 48 6 , , PUNCT cana-736 48 7 we we PRON cana-736 48 8 get get VERB cana-736 48 9 β€– β€– ADJ cana-736 48 10 πœπ‘‘ πœπ‘‘ PROPN cana-736 48 11 ( ( PUNCT cana-736 48 12 𝑓)‖𝑝,𝛽 𝑓)‖𝑝,𝛽 PROPN cana-736 48 13 ≀ ≀ NOUN cana-736 48 14 1 1 NUM cana-736 48 15 πœ‹ πœ‹ NOUN cana-736 48 16 cos4𝑑 cos4𝑑 PROPN cana-736 48 17 2 2 NUM cana-736 48 18 ( ( PUNCT cana-736 48 19 ∫ ∫ PROPN cana-736 48 20 ( ( PUNCT cana-736 48 21 1 1 NUM cana-736 48 22 1βˆ’π‘₯2 1βˆ’π‘₯2 NUM cana-736 48 23 ∫ ∫ PROPN cana-736 48 24 19(1 19(1 NUM cana-736 48 25 βˆ’ βˆ’ NOUN cana-736 48 26 𝑅2 𝑅2 NOUN cana-736 48 27 ) ) PUNCT cana-736 48 28 1 1 NUM cana-736 48 29 βˆ’1 βˆ’1 NOUN cana-736 48 30 |𝑓(𝑅)| |𝑓(𝑅)| PROPN cana-736 48 31 𝑑𝑧 𝑑𝑧 NOUN cana-736 48 32 √1βˆ’π‘§2 √1βˆ’π‘§2 NOUN cana-736 48 33 ) ) PUNCT cana-736 48 34 𝑝1 𝑝1 NOUN cana-736 48 35 βˆ’1 βˆ’1 NOUN cana-736 48 36 ( ( PUNCT cana-736 48 37 1 1 NUM cana-736 48 38 βˆ’ βˆ’ PROPN cana-736 48 39 π‘₯ π‘₯ NOUN cana-736 48 40 ) ) PUNCT cana-736 48 41 𝛽 𝛽 NOUN cana-736 48 42 2 2 NUM cana-736 48 43 𝑑π‘₯ 𝑑π‘₯ NOUN cana-736 48 44 ) ) PUNCT cana-736 48 45 1 1 NUM cana-736 48 46 𝑝 𝑝 NOUN cana-736 48 47 . . PUNCT cana-736 49 1 using use VERB cana-736 49 2 lemma lemma PROPN cana-736 49 3 ( ( PUNCT cana-736 49 4 2.1 2.1 NUM cana-736 49 5 ) ) PUNCT cana-736 49 6 , , PUNCT cana-736 49 7 to to PART cana-736 49 8 obtain obtain VERB cana-736 49 9 β€– β€– ADJ cana-736 49 10 πœπ‘‘ πœπ‘‘ PROPN cana-736 49 11 ( ( PUNCT cana-736 49 12 𝑓)‖𝑝,𝛽 𝑓)‖𝑝,𝛽 PROPN cana-736 49 13 ≀ ≀ NOUN cana-736 49 14 19∁(𝑝 19∁(𝑝 NUM cana-736 49 15 ) ) PUNCT cana-736 49 16 πœ‹ πœ‹ NOUN cana-736 49 17 cos4 cos4 PROPN cana-736 49 18 𝑑 𝑑 PROPN cana-736 49 19 2 2 NUM cana-736 49 20 ‖𝑓‖𝑝,𝛽 ‖𝑓‖𝑝,𝛽 NOUN cana-736 49 21 . . PUNCT cana-736 50 1 lemma.2.4 lemma.2.4 PROPN cana-736 50 2 . . PUNCT cana-736 51 1 [ [ X cana-736 51 2 2 2 NUM cana-736 51 3 ] ] PUNCT cana-736 51 4 . . PUNCT cana-736 52 1 if if SCONJ cana-736 52 2 𝑦 𝑦 NOUN cana-736 52 3 ∈ ∈ NOUN cana-736 52 4 ( ( PUNCT cana-736 52 5 βˆ’1,1 βˆ’1,1 NOUN cana-736 52 6 ) ) PUNCT cana-736 52 7 , , PUNCT cana-736 52 8 and and CCONJ cana-736 52 9 π‘₯ π‘₯ PRON cana-736 52 10 ∈ ∈ PROPN cana-736 52 11 ( ( PUNCT cana-736 52 12 βˆ’1,1 βˆ’1,1 NOUN cana-736 52 13 ) ) PUNCT cana-736 52 14 , , PUNCT cana-736 52 15 for for ADP cana-736 52 16 almost almost ADV cana-736 52 17 every every PRON cana-736 52 18 then then ADV cana-736 52 19 πœπ‘¦ πœπ‘¦ PROPN cana-736 52 20 ( ( PUNCT cana-736 52 21 𝐷π‘₯,2,2𝑓 𝐷π‘₯,2,2𝑓 PROPN cana-736 52 22 ) ) PUNCT cana-736 52 23 = = SYM cana-736 52 24 𝐷π‘₯,2,2πœπ‘¦(𝑓 𝐷π‘₯,2,2πœπ‘¦(𝑓 NOUN cana-736 52 25 ) ) PUNCT cana-736 52 26 . . PUNCT cana-736 53 1 lemma.2.5.[3 lemma.2.5.[3 NOUN cana-736 53 2 ] ] PUNCT cana-736 53 3 . . PUNCT cana-736 54 1 let let VERB cana-736 54 2 𝑓 𝑓 PRON cana-736 54 3 ˊ ˊ NOUN cana-736 54 4 be be AUX cana-736 54 5 an an DET cana-736 54 6 absolutely absolutely ADV cana-736 54 7 continuous continuous ADJ cana-736 54 8 with with ADP cana-736 54 9 domain domain NOUN cana-736 54 10 [ [ X cana-736 54 11 π‘Ž π‘Ž X cana-736 54 12 , , PUNCT cana-736 54 13 𝑏 𝑏 NOUN cana-736 54 14 ] ] X cana-736 54 15 βŠ‚ βŠ‚ X cana-736 54 16 ( ( PUNCT cana-736 54 17 βˆ’1,1 βˆ’1,1 INTJ cana-736 54 18 ) ) PUNCT cana-736 54 19 , , PUNCT cana-736 54 20 with with ADP cana-736 54 21 𝐷π‘₯,2,2𝑓 𝐷π‘₯,2,2𝑓 PROPN cana-736 54 22 ∈ ∈ PROPN cana-736 54 23 𝐿1,2 𝐿1,2 PROPN cana-736 54 24 . . PUNCT cana-736 55 1 , , PUNCT cana-736 55 2 so so ADV cana-736 55 3 almost almost ADV cana-736 55 4 every every PRON cana-736 55 5 π‘₯ π‘₯ X cana-736 55 6 ∈ ∈ NOUN cana-736 55 7 ( ( PUNCT cana-736 55 8 βˆ’1,1 βˆ’1,1 NOUN cana-736 55 9 ) ) PUNCT cana-736 55 10 and and CCONJ cana-736 55 11 every every DET cana-736 55 12 𝑦 𝑦 PROPN cana-736 55 13 ∈ ∈ NOUN cana-736 55 14 ( ( PUNCT cana-736 55 15 βˆ’1,1 βˆ’1,1 NOUN cana-736 55 16 ) ) PUNCT cana-736 55 17 πœπ‘¦ πœπ‘¦ PROPN cana-736 55 18 ( ( PUNCT cana-736 55 19 𝑓 𝑓 X cana-736 55 20 ) ) PUNCT cana-736 55 21 βˆ’ βˆ’ PROPN cana-736 56 1 𝑓 𝑓 PRON cana-736 56 2 = = PUNCT cana-736 56 3 ∫ ∫ PROPN cana-736 56 4 ( ( PUNCT cana-736 56 5 1 1 NUM cana-736 56 6 βˆ’ βˆ’ NOUN cana-736 56 7 𝜈)βˆ’1(1 𝜈)βˆ’1(1 ADJ cana-736 56 8 βˆ’ βˆ’ PROPN cana-736 56 9 𝜈)βˆ’5 𝜈)βˆ’5 ADJ cana-736 56 10 ∫ ∫ NOUN cana-736 56 11 ( ( PUNCT cana-736 56 12 1 1 NUM cana-736 56 13 + + NUM cana-736 56 14 𝑒)4𝜈 𝑒)4𝜈 NOUN cana-736 56 15 1 1 NUM cana-736 56 16 𝑦 𝑦 SYM cana-736 56 17 1 1 NUM cana-736 56 18 πœπ‘’(𝐷π‘₯,2,2𝑓 πœπ‘’(𝐷π‘₯,2,2𝑓 NOUN cana-736 56 19 , , PUNCT cana-736 56 20 π‘₯)π‘‘π‘’π‘‘πœˆ π‘₯)π‘‘π‘’π‘‘πœˆ NOUN cana-736 56 21 . . PUNCT cana-736 57 1 πœπ‘¦ πœπ‘¦ PRON cana-736 57 2 ( ( PUNCT cana-736 57 3 𝑓 𝑓 X cana-736 57 4 ) ) PUNCT cana-736 57 5 βˆ’ βˆ’ PROPN cana-736 57 6 𝜏0 𝜏0 PROPN cana-736 57 7 ( ( PUNCT cana-736 57 8 𝑓 𝑓 X cana-736 57 9 ) ) PUNCT cana-736 57 10 = = SYM cana-736 58 1 βˆ’ βˆ’ PROPN cana-736 58 2 ∫ ∫ PROPN cana-736 58 3 ( ( PUNCT cana-736 58 4 1 1 NUM cana-736 58 5 βˆ’ βˆ’ NOUN cana-736 58 6 𝜈)βˆ’1(1 𝜈)βˆ’1(1 ADJ cana-736 58 7 βˆ’ βˆ’ PROPN cana-736 58 8 𝜈)βˆ’5 𝜈)βˆ’5 ADJ cana-736 58 9 ∫ ∫ NOUN cana-736 58 10 ( ( PUNCT cana-736 58 11 1 1 NUM cana-736 58 12 + + NUM cana-736 58 13 𝑒)4βˆ’1 𝑒)4βˆ’1 NOUN cana-736 58 14 𝜈 𝜈 X cana-736 58 15 𝑦 𝑦 SYM cana-736 58 16 1 1 NUM cana-736 58 17 πœπ‘’(𝐷π‘₯,2,2𝑓 πœπ‘’(𝐷π‘₯,2,2𝑓 NOUN cana-736 58 18 , , PUNCT cana-736 58 19 π‘₯)π‘‘π‘’π‘‘πœˆ π‘₯)π‘‘π‘’π‘‘πœˆ NOUN cana-736 58 20 . . PUNCT cana-736 59 1 lemma.2.6.[3 lemma.2.6.[3 NOUN cana-736 59 2 ] ] PUNCT cana-736 59 3 . . PUNCT cana-736 60 1 let let VERB cana-736 60 2 𝑓 𝑓 PRON cana-736 60 3 ˊ ˊ NOUN cana-736 60 4 be be AUX cana-736 60 5 absolutely absolutely ADV cana-736 60 6 continuous continuous ADJ cana-736 60 7 with with ADP cana-736 60 8 domain domain NOUN cana-736 60 9 [ [ X cana-736 60 10 π‘Ž π‘Ž X cana-736 60 11 , , PUNCT cana-736 60 12 𝑏 𝑏 NOUN cana-736 60 13 ] ] X cana-736 60 14 βŠ‚ βŠ‚ X cana-736 60 15 ( ( PUNCT cana-736 60 16 βˆ’1,1 βˆ’1,1 INTJ cana-736 60 17 ) ) PUNCT cana-736 60 18 , , PUNCT cana-736 60 19 with with ADP cana-736 60 20 𝐷π‘₯,2,2𝑓(π‘₯ 𝐷π‘₯,2,2𝑓(π‘₯ NOUN cana-736 60 21 ) ) PUNCT cana-736 60 22 ∈ ∈ PROPN cana-736 60 23 𝐿1,2 𝐿1,2 PROPN cana-736 60 24 . . PUNCT cana-736 61 1 so so ADV cana-736 61 2 almost almost ADV cana-736 61 3 every every PRON cana-736 61 4 π‘₯ π‘₯ X cana-736 61 5 ∈ ∈ NOUN cana-736 61 6 ( ( PUNCT cana-736 61 7 βˆ’1,1 βˆ’1,1 NOUN cana-736 61 8 ) ) PUNCT cana-736 61 9 and and CCONJ cana-736 61 10 every every DET cana-736 61 11 𝑑 𝑑 PROPN cana-736 61 12 ∈ ∈ PROPN cana-736 61 13 ( ( PUNCT cana-736 61 14 βˆ’πœ‹ βˆ’πœ‹ ADJ cana-736 61 15 , , PUNCT cana-736 61 16 πœ‹ πœ‹ NOUN cana-736 61 17 ) ) PUNCT cana-736 61 18 . . PUNCT cana-736 62 1 communications communication NOUN cana-736 62 2 on on ADP cana-736 62 3 applied apply VERB cana-736 62 4 nonlinear nonlinear ADJ cana-736 62 5 analysis analysis NOUN cana-736 62 6 issn issn NOUN cana-736 62 7 : : PUNCT cana-736 62 8 1074 1074 NUM cana-736 62 9 - - PUNCT cana-736 62 10 133x 133x NUM cana-736 62 11 vol vol NOUN cana-736 62 12 31 31 NUM cana-736 62 13 no no NOUN cana-736 62 14 . . PUNCT cana-736 63 1 3s 3s NUM cana-736 63 2 ( ( PUNCT cana-736 63 3 2024 2024 NUM cana-736 63 4 ) ) PUNCT cana-736 63 5 121 121 NUM cana-736 63 6 https://internationalpubls.com https://internationalpubls.com X cana-736 63 7 πœπ‘‘ πœπ‘‘ PROPN cana-736 63 8 Μ­ Μ­ PROPN cana-736 63 9 ( ( PUNCT cana-736 63 10 𝑓 𝑓 X cana-736 63 11 ) ) PUNCT cana-736 63 12 βˆ’ βˆ’ PROPN cana-736 64 1 𝑓 𝑓 PRON cana-736 64 2 = = PUNCT cana-736 64 3 ∫ ∫ PROPN cana-736 64 4 ( ( PUNCT cana-736 64 5 sin sin NOUN cana-736 64 6 𝜈 𝜈 X cana-736 64 7 2 2 NUM cana-736 64 8 ) ) PUNCT cana-736 64 9 βˆ’1 βˆ’1 NOUN cana-736 64 10 ( ( PUNCT cana-736 64 11 cos cos ADP cana-736 64 12 𝜈 𝜈 X cana-736 64 13 2 2 NUM cana-736 64 14 ) ) PUNCT cana-736 64 15 βˆ’9 βˆ’9 NOUN cana-736 64 16 ∫ ∫ PROPN cana-736 64 17 πœπ‘’(𝐷π‘₯,2,2𝑓 πœπ‘’(𝐷π‘₯,2,2𝑓 PROPN cana-736 64 18 , , PUNCT cana-736 64 19 π‘₯ π‘₯ NOUN cana-736 64 20 ) ) PUNCT cana-736 64 21 sin sin NOUN cana-736 64 22 𝑒 𝑒 PROPN cana-736 64 23 2 2 NUM cana-736 64 24 𝜈 𝜈 NOUN cana-736 64 25 0 0 NUM cana-736 64 26 𝑑 𝑑 NOUN cana-736 64 27 0 0 NUM cana-736 64 28 ( ( PUNCT cana-736 64 29 cos cos ADP cana-736 64 30 𝑒 𝑒 PROPN cana-736 64 31 2 2 NUM cana-736 64 32 ) ) PUNCT cana-736 64 33 9 9 NUM cana-736 64 34 π‘‘π‘’π‘‘πœˆ π‘‘π‘’π‘‘πœˆ NOUN cana-736 64 35 . . PUNCT cana-736 65 1 πœπ‘‘ πœπ‘‘ PROPN cana-736 65 2 ( ( PUNCT cana-736 65 3 𝑓 𝑓 PROPN cana-736 65 4 ) ) PUNCT cana-736 65 5 βˆ’ βˆ’ NOUN cana-736 65 6 πœπœ‹ πœπœ‹ NOUN cana-736 65 7 2 2 NUM cana-736 65 8 ( ( PUNCT cana-736 65 9 𝑓 𝑓 X cana-736 65 10 ) ) PUNCT cana-736 65 11 = = SYM cana-736 65 12 βˆ’ βˆ’ PROPN cana-736 65 13 ∫ ∫ PROPN cana-736 65 14 ( ( PUNCT cana-736 65 15 sin sin NOUN cana-736 65 16 𝜈 𝜈 X cana-736 65 17 2 2 NUM cana-736 65 18 ) ) PUNCT cana-736 65 19 βˆ’1 βˆ’1 NOUN cana-736 65 20 ( ( PUNCT cana-736 65 21 cos cos ADP cana-736 65 22 𝜈 𝜈 X cana-736 65 23 2 2 NUM cana-736 65 24 ) ) PUNCT cana-736 65 25 βˆ’9 βˆ’9 NOUN cana-736 65 26 ∫ ∫ PROPN cana-736 65 27 πœπ‘’(𝐷π‘₯,2,2𝑓 πœπ‘’(𝐷π‘₯,2,2𝑓 PROPN cana-736 65 28 , , PUNCT cana-736 65 29 π‘₯ π‘₯ NOUN cana-736 65 30 ) ) PUNCT cana-736 65 31 πœ‹ πœ‹ NOUN cana-736 65 32 𝜈 𝜈 X cana-736 65 33 𝑑 𝑑 X cana-736 65 34 πœ‹ πœ‹ PROPN cana-736 65 35 2 2 NUM cana-736 65 36 sin sin NOUN cana-736 65 37 𝑒 𝑒 X cana-736 65 38 2 2 NUM cana-736 65 39 ( ( PUNCT cana-736 65 40 cos cos ADP cana-736 65 41 𝑒 𝑒 PROPN cana-736 65 42 2 2 NUM cana-736 65 43 ) ) PUNCT cana-736 65 44 9 9 NUM cana-736 65 45 π‘‘π‘’π‘‘πœˆ π‘‘π‘’π‘‘πœˆ NOUN cana-736 65 46 . . PUNCT cana-736 66 1 lemma.2.7 lemma.2.7 PROPN cana-736 66 2 . . PUNCT cana-736 67 1 if if SCONJ cana-736 67 2 𝑓 𝑓 DET cana-736 67 3 ∈ ∈ PROPN cana-736 67 4 𝐿𝑝,𝛽 𝐿𝑝,𝛽 NOUN cana-736 67 5 , , PUNCT cana-736 67 6 then then ADV cana-736 67 7 𝐸𝑛(𝑓)𝑝,𝛽 𝐸𝑛(𝑓)𝑝,𝛽 VERB cana-736 67 8 ≀ ≀ ADJ cana-736 67 9 ∁(𝑝 ∁(𝑝 NOUN cana-736 67 10 ) ) PUNCT cana-736 67 11 1 1 NUM cana-736 67 12 𝑛2 𝑛2 PROPN cana-736 67 13 ‖𝐷(𝑓)‖𝑝,𝛽 ‖𝐷(𝑓)‖𝑝,𝛽 PROPN cana-736 67 14 . . PUNCT cana-736 68 1 proof proof NOUN cana-736 68 2 . . PUNCT cana-736 69 1 by by ADP cana-736 69 2 using use VERB cana-736 69 3 𝐸𝑛(𝑓)𝑝,𝛽 𝐸𝑛(𝑓)𝑝,𝛽 NOUN cana-736 69 4 ≀ ≀ ADJ cana-736 69 5 ‖𝑓 ‖𝑓 NOUN cana-736 69 6 βˆ’ βˆ’ VERB cana-736 69 7 𝑄‖𝑝,𝛽 𝑄‖𝑝,𝛽 PROPN cana-736 69 8 , , PUNCT cana-736 69 9 where where SCONJ cana-736 69 10 𝑄 𝑄 PRON cana-736 69 11 is be AUX cana-736 69 12 algebraic algebraic ADV cana-736 69 13 polynomial polynomial ADJ cana-736 69 14 of of ADP cana-736 69 15 degree degree NOUN cana-736 69 16 less less ADJ cana-736 69 17 than than ADP cana-736 69 18 𝑛 𝑛 PRON cana-736 69 19 βˆ’ βˆ’ PROPN cana-736 69 20 1 1 NUM cana-736 69 21 , , PUNCT cana-736 69 22 and and CCONJ cana-736 69 23 𝑄(π‘₯ 𝑄(π‘₯ NUM cana-736 69 24 ) ) PUNCT cana-736 69 25 = = SYM cana-736 69 26 1 1 NUM cana-736 69 27 π›Ύπ‘š π›Ύπ‘š NUM cana-736 69 28 ∫ ∫ PROPN cana-736 70 1 𝑇2;cos 𝑇2;cos PROPN cana-736 70 2 𝑑(𝑓 𝑑(𝑓 NOUN cana-736 70 3 , , PUNCT cana-736 70 4 π‘₯ π‘₯ NOUN cana-736 70 5 ) ) PUNCT cana-736 70 6 πœ‹ πœ‹ NOUN cana-736 70 7 0 0 NUM cana-736 71 1 ( ( PUNCT cana-736 71 2 sin sin NOUN cana-736 71 3 π‘šπ‘‘ π‘šπ‘‘ ADP cana-736 71 4 2 2 NUM cana-736 71 5 sin sin NOUN cana-736 71 6 𝑑 𝑑 X cana-736 71 7 2 2 NUM cana-736 71 8 ) ) PUNCT cana-736 71 9 2π‘ž+4 2π‘ž+4 NOUN cana-736 71 10 sin5 sin5 NOUN cana-736 71 11 𝑑𝑑𝑑 𝑑𝑑𝑑 NOUN cana-736 71 12 where where SCONJ cana-736 71 13 π›Ύπ‘š π›Ύπ‘š ADP cana-736 71 14 = = SYM cana-736 71 15 ∫ ∫ PROPN cana-736 71 16 ( ( PUNCT cana-736 71 17 sin sin NOUN cana-736 71 18 π‘šπ‘‘ π‘šπ‘‘ ADP cana-736 71 19 2 2 NUM cana-736 71 20 sin sin NOUN cana-736 71 21 𝑑 𝑑 X cana-736 71 22 2 2 NUM cana-736 71 23 ) ) PUNCT cana-736 71 24 2π‘ž+4 2π‘ž+4 NOUN cana-736 71 25 sin5 sin5 NOUN cana-736 71 26 𝑑𝑑𝑑 𝑑𝑑𝑑 VERB cana-736 71 27 πœ‹ πœ‹ NOUN cana-736 71 28 0 0 NUM cana-736 71 29 , , PUNCT cana-736 71 30 then then ADV cana-736 71 31 𝐸𝑛(𝑓)𝑝,𝛽 𝐸𝑛(𝑓)𝑝,𝛽 VERB cana-736 71 32 ≀ ≀ ADJ cana-736 71 33 ‖𝑓 ‖𝑓 NOUN cana-736 71 34 βˆ’ βˆ’ PROPN cana-736 71 35 1 1 NUM cana-736 71 36 π›Ύπ‘š π›Ύπ‘š NOUN cana-736 71 37 ∫ ∫ PROPN cana-736 71 38 𝑇2;cos 𝑇2;cos PROPN cana-736 71 39 𝑑(𝑓 𝑑(𝑓 NOUN cana-736 71 40 , , PUNCT cana-736 71 41 π‘₯ π‘₯ NOUN cana-736 71 42 ) ) PUNCT cana-736 71 43 πœ‹ πœ‹ NOUN cana-736 71 44 0 0 NUM cana-736 72 1 ( ( PUNCT cana-736 72 2 sin sin NOUN cana-736 72 3 π‘šπ‘‘ π‘šπ‘‘ ADP cana-736 72 4 2 2 NUM cana-736 72 5 sin sin NOUN cana-736 72 6 𝑑 𝑑 X cana-736 72 7 2 2 NUM cana-736 72 8 ) ) PUNCT cana-736 72 9 2π‘ž+4 2π‘ž+4 NOUN cana-736 72 10 sin5 sin5 NOUN cana-736 72 11 𝑑𝑑𝑑‖ 𝑑𝑑𝑑‖ NOUN cana-736 72 12 𝑝,𝛽 𝑝,𝛽 NOUN cana-736 72 13 . . PUNCT cana-736 73 1 = = PUNCT cana-736 73 2 β€– β€– PROPN cana-736 73 3 1 1 NUM cana-736 73 4 π›Ύπ‘š π›Ύπ‘š NOUN cana-736 73 5 ∫ ∫ PROPN cana-736 73 6 ( ( PUNCT cana-736 73 7 sin sin NOUN cana-736 73 8 π‘šπ‘‘ π‘šπ‘‘ ADP cana-736 73 9 2 2 NUM cana-736 73 10 sin sin NOUN cana-736 73 11 𝑑 𝑑 X cana-736 73 12 2 2 NUM cana-736 73 13 ) ) PUNCT cana-736 73 14 2π‘ž+4 2π‘ž+4 NOUN cana-736 73 15 sin5 sin5 NOUN cana-736 73 16 𝑑𝑑𝑑 𝑑𝑑𝑑 VERB cana-736 73 17 πœ‹ πœ‹ NOUN cana-736 73 18 0 0 NUM cana-736 73 19 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-736 73 20 ) ) PUNCT cana-736 74 1 βˆ’ βˆ’ PROPN cana-736 75 1 1 1 NUM cana-736 75 2 π›Ύπ‘š π›Ύπ‘š NOUN cana-736 75 3 ∫ ∫ PROPN cana-736 75 4 𝑇(𝑓 𝑇(𝑓 NUM cana-736 75 5 ) ) PUNCT cana-736 75 6 ( ( PUNCT cana-736 75 7 sin sin VERB cana-736 75 8 π‘šπ‘‘ π‘šπ‘‘ ADP cana-736 75 9 2 2 NUM cana-736 75 10 sin sin NOUN cana-736 75 11 𝑑 𝑑 X cana-736 75 12 2 2 NUM cana-736 75 13 ) ) PUNCT cana-736 75 14 2π‘ž+4 2π‘ž+4 NOUN cana-736 75 15 sin5 sin5 NOUN cana-736 75 16 𝑑𝑑𝑑 𝑑𝑑𝑑 VERB cana-736 75 17 πœ‹ πœ‹ NOUN cana-736 75 18 0 0 NUM cana-736 75 19 β€– β€– PROPN cana-736 75 20 𝑝,𝛽 𝑝,𝛽 PROPN cana-736 75 21 𝐸𝑛(𝑓)𝑝,𝛽 𝐸𝑛(𝑓)𝑝,𝛽 VERB cana-736 75 22 ≀ ≀ ADJ cana-736 75 23 ∁(𝑝)‖𝑓 ∁(𝑝)‖𝑓 PROPN cana-736 76 1 βˆ’ βˆ’ PROPN cana-736 77 1 𝑇‖𝑝,𝛽. 𝑇‖𝑝,𝛽. PROPN cana-736 77 2 using use VERB cana-736 77 3 the the PRON cana-736 77 4 β€–πœπ‘‘ β€–πœπ‘‘ PROPN cana-736 77 5 ( ( PUNCT cana-736 77 6 𝑔 𝑔 NOUN cana-736 77 7 ) ) PUNCT cana-736 77 8 βˆ’ βˆ’ PROPN cana-736 77 9 𝑔 𝑔 PROPN cana-736 77 10 ‖𝑝,𝛽 ‖𝑝,𝛽 ADV cana-736 77 11 ≀ ≀ PROPN cana-736 77 12 ∁(𝑝 ∁(𝑝 NOUN cana-736 77 13 ) ) PUNCT cana-736 77 14 1 1 NUM cana-736 77 15 cos4𝑑 cos4𝑑 PROPN cana-736 77 16 2 2 NUM cana-736 77 17 𝑑2 𝑑2 PROPN cana-736 77 18 ‖𝐷π‘₯,2,2𝑔 ‖𝐷π‘₯,2,2𝑔 PROPN cana-736 77 19 β€– β€– PROPN cana-736 77 20 𝑝,𝛽 𝑝,𝛽 PROPN cana-736 77 21 , , PUNCT cana-736 77 22 [ [ X cana-736 77 23 2 2 NUM cana-736 77 24 ] ] PUNCT cana-736 77 25 . . PUNCT cana-736 78 1 to to PART cana-736 78 2 obtain obtain VERB cana-736 78 3 𝐸𝑛(𝑓)𝑝,𝛽 𝐸𝑛(𝑓)𝑝,𝛽 VERB cana-736 78 4 ≀ ≀ ADJ cana-736 78 5 ∁(𝑝 ∁(𝑝 NOUN cana-736 78 6 ) ) PUNCT cana-736 78 7 1 1 NUM cana-736 78 8 𝑛2 𝑛2 NOUN cana-736 78 9 ‖𝐷(𝑓)‖𝑝,𝛽 ‖𝐷(𝑓)‖𝑝,𝛽 PROPN cana-736 78 10 . . PUNCT cana-736 79 1 3 3 X cana-736 79 2 . . X cana-736 79 3 the the DET cana-736 79 4 main main ADJ cana-736 79 5 result result NOUN cana-736 79 6 after after SCONJ cana-736 79 7 we we PRON cana-736 79 8 define define VERB cana-736 79 9 our our PRON cana-736 79 10 new new ADJ cana-736 79 11 modulus modulus NOUN cana-736 79 12 of of ADP cana-736 79 13 smoothness smoothness NOUN cana-736 79 14 , , PUNCT cana-736 79 15 and and CCONJ cana-736 79 16 π‘˜ π‘˜ DET cana-736 79 17 βˆ’ βˆ’ PROPN cana-736 79 18 π‘“π‘’π‘›π‘π‘‘π‘–π‘œπ‘›π‘Žπ‘™. π‘“π‘’π‘›π‘π‘‘π‘–π‘œπ‘›π‘Žπ‘™. NOUN cana-736 79 19 we we PRON cana-736 79 20 shall shall AUX cana-736 79 21 relate relate VERB cana-736 79 22 then then ADV cana-736 79 23 to to ADP cana-736 79 24 others other NOUN cana-736 79 25 . . PUNCT cana-736 80 1 that that PRON cana-736 80 2 is is ADV cana-736 80 3 we we PRON cana-736 80 4 shall shall AUX cana-736 80 5 introduce introduce VERB cana-736 80 6 the the DET cana-736 80 7 theorem theorem NOUN cana-736 80 8 : : PUNCT cana-736 80 9 communications communication NOUN cana-736 80 10 on on ADP cana-736 80 11 applied apply VERB cana-736 80 12 nonlinear nonlinear ADJ cana-736 80 13 analysis analysis NOUN cana-736 80 14 issn issn NOUN cana-736 80 15 : : PUNCT cana-736 80 16 1074 1074 NUM cana-736 80 17 - - PUNCT cana-736 80 18 133x 133x NUM cana-736 80 19 vol vol NOUN cana-736 80 20 31 31 NUM cana-736 80 21 no no NOUN cana-736 80 22 . . PUNCT cana-736 81 1 3s 3s NUM cana-736 81 2 ( ( PUNCT cana-736 81 3 2024 2024 NUM cana-736 81 4 ) ) PUNCT cana-736 81 5 122 122 NUM cana-736 81 6 https://internationalpubls.com https://internationalpubls.com X cana-736 81 7 theorem.3.1 theorem.3.1 PROPN cana-736 81 8 . . PUNCT cana-736 82 1 for for ADP cana-736 82 2 𝑓 𝑓 DET cana-736 82 3 ∈ ∈ PROPN cana-736 82 4 𝐿𝑝,𝛽 𝐿𝑝,𝛽 NOUN cana-736 82 5 , , PUNCT cana-736 82 6 0 0 PUNCT cana-736 82 7 < < X cana-736 82 8 𝑝 𝑝 X cana-736 82 9 < < X cana-736 82 10 1 1 NUM cana-736 82 11 , , PUNCT cana-736 82 12 we we PRON cana-736 82 13 have have VERB cana-736 82 14 ∁1(𝑝 ∁1(𝑝 NUM cana-736 82 15 ) ) PUNCT cana-736 83 1 𝐾(𝑓 𝐾(𝑓 NOUN cana-736 83 2 , , PUNCT cana-736 83 3 𝛿)𝑝,𝛽 𝛿)𝑝,𝛽 VERB cana-736 83 4 ≀ ≀ ADJ cana-736 83 5 αΏΆ(𝑓 αΏΆ(𝑓 NOUN cana-736 83 6 , , PUNCT cana-736 83 7 𝛿)𝑝,𝛽 𝛿)𝑝,𝛽 VERB cana-736 83 8 ≀ ≀ NUM cana-736 83 9 ∁2(𝑝 ∁2(𝑝 NUM cana-736 83 10 ) ) PUNCT cana-736 83 11 1 1 NUM cana-736 83 12 cos4𝛿 cos4𝛿 PROPN cana-736 83 13 2 2 NUM cana-736 83 14 𝐾(𝑓 𝐾(𝑓 NOUN cana-736 83 15 , , PUNCT cana-736 83 16 𝛿)𝑝,𝛽 𝛿)𝑝,𝛽 NOUN cana-736 83 17 . . PUNCT cana-736 84 1 where where SCONJ cana-736 84 2 , , PUNCT cana-736 84 3 ∁1(𝑝 ∁1(𝑝 NUM cana-736 84 4 ) ) PUNCT cana-736 84 5 and and CCONJ cana-736 84 6 ∁2(𝑝 ∁2(𝑝 NUM cana-736 84 7 ) ) PUNCT cana-736 84 8 are be AUX cana-736 84 9 positive positive ADJ cana-736 84 10 constants constant NOUN cana-736 84 11 depending depend VERB cana-736 84 12 on on ADP cana-736 84 13 p p NOUN cana-736 84 14 only only ADV cana-736 84 15 . . PUNCT cana-736 85 1 proof proof NOUN cana-736 85 2 . . PUNCT cana-736 86 1 let let VERB cana-736 86 2 ℙ𝑛be ℙ𝑛be PROPN cana-736 86 3 the the DET cana-736 86 4 set set NOUN cana-736 86 5 of of ADP cana-736 86 6 𝑛 𝑛 DET cana-736 86 7 th th NOUN cana-736 86 8 degree degree NOUN cana-736 86 9 algebraic algebraic ADJ cana-736 86 10 polynomials polynomial NOUN cana-736 86 11 we we PRON cana-736 86 12 shall shall AUX cana-736 86 13 prove prove VERB cana-736 86 14 ( ( PUNCT cana-736 86 15 3.1 3.1 NUM cana-736 86 16 ) ) PUNCT cana-736 87 1 β€–πœπ‘‘ β€–πœπ‘‘ NUM cana-736 87 2 ( ( PUNCT cana-736 87 3 𝑔 𝑔 NOUN cana-736 87 4 ) ) PUNCT cana-736 87 5 βˆ’ βˆ’ PROPN cana-736 87 6 𝑔‖𝑝,𝛽 𝑔‖𝑝,𝛽 ADJ cana-736 87 7 ≀ ≀ NUM cana-736 87 8 ∁(𝑝 ∁(𝑝 NOUN cana-736 87 9 ) ) PUNCT cana-736 87 10 1 1 NUM cana-736 87 11 cos4 cos4 PROPN cana-736 87 12 𝑑 𝑑 NOUN cana-736 87 13 2 2 NUM cana-736 87 14 𝑑2 𝑑2 PROPN cana-736 87 15 ‖𝐷π‘₯,2,2𝑔‖ ‖𝐷π‘₯,2,2𝑔‖ PROPN cana-736 87 16 𝑝,𝛽 𝑝,𝛽 NOUN cana-736 87 17 , , PUNCT cana-736 87 18 where where SCONJ cana-736 87 19 , , PUNCT cana-736 87 20 ∁(𝑝 ∁(𝑝 NOUN cana-736 87 21 ) ) PUNCT cana-736 87 22 is be AUX cana-736 87 23 constant constant ADJ cana-736 87 24 depend depend NOUN cana-736 87 25 on on ADP cana-736 87 26 p p NOUN cana-736 87 27 only only ADV cana-736 87 28 . . PUNCT cana-736 88 1 if if SCONJ cana-736 88 2 0 0 NUM cana-736 88 3 < < X cana-736 88 4 𝑑 𝑑 X cana-736 88 5 ≀ ≀ NUM cana-736 88 6 πœ‹ πœ‹ NOUN cana-736 88 7 2 2 NUM cana-736 89 1 , , PUNCT cana-736 89 2 then then ADV cana-736 89 3 lemma.2.6.implies lemma.2.6.implies PROPN cana-736 89 4 𝐼1 𝐼1 NOUN cana-736 89 5 = = SYM cana-736 89 6 β€–πœπ‘‘ β€–πœπ‘‘ X cana-736 89 7 ( ( PUNCT cana-736 89 8 𝑔 𝑔 NOUN cana-736 89 9 ) ) PUNCT cana-736 89 10 βˆ’ βˆ’ PROPN cana-736 89 11 𝑔‖𝑝,𝛽. 𝑔‖𝑝,𝛽. NOUN cana-736 89 12 = = SYM cana-736 89 13 β€–βˆ« β€–βˆ« PROPN cana-736 89 14 ( ( PUNCT cana-736 89 15 sin sin NOUN cana-736 89 16 𝜈 𝜈 X cana-736 89 17 2 2 NUM cana-736 89 18 ) ) PUNCT cana-736 89 19 βˆ’1 βˆ’1 NOUN cana-736 89 20 ( ( PUNCT cana-736 89 21 cos cos ADP cana-736 89 22 𝜈 𝜈 X cana-736 89 23 2 2 NUM cana-736 89 24 ) ) PUNCT cana-736 89 25 βˆ’9 βˆ’9 NOUN cana-736 89 26 ∫ ∫ PROPN cana-736 89 27 πœπ‘’(𝐷π‘₯,2,2𝑔 πœπ‘’(𝐷π‘₯,2,2𝑔 PROPN cana-736 89 28 , , PUNCT cana-736 89 29 π‘₯ π‘₯ NOUN cana-736 89 30 ) ) PUNCT cana-736 89 31 sin sin NOUN cana-736 89 32 𝑒 𝑒 PROPN cana-736 89 33 2 2 NUM cana-736 89 34 𝜈 𝜈 NOUN cana-736 89 35 0 0 NUM cana-736 89 36 𝑑 𝑑 NOUN cana-736 89 37 0 0 NUM cana-736 89 38 ( ( PUNCT cana-736 89 39 cos cos ADP cana-736 89 40 𝑒 𝑒 PROPN cana-736 89 41 2 2 NUM cana-736 89 42 ) ) PUNCT cana-736 89 43 9 9 NUM cana-736 89 44 π‘‘π‘’π‘‘πœˆβ€– π‘‘π‘’π‘‘πœˆβ€– ADP cana-736 89 45 𝑝,𝛽 𝑝,𝛽 NOUN cana-736 89 46 . . PUNCT cana-736 90 1 applying apply VERB cana-736 90 2 lemma lemma PROPN cana-736 90 3 .2.3 .2.3 PUNCT cana-736 91 1 [ [ X cana-736 91 2 β€– β€– X cana-736 91 3 πœπ‘‘ πœπ‘‘ PROPN cana-736 91 4 ( ( PUNCT cana-736 91 5 𝑓)‖𝑝,𝛽 𝑓)‖𝑝,𝛽 PROPN cana-736 91 6 ≀ ≀ NOUN cana-736 91 7 19∁(𝑝 19∁(𝑝 NUM cana-736 91 8 ) ) PUNCT cana-736 91 9 πœ‹ πœ‹ NOUN cana-736 91 10 cos4𝑑 cos4𝑑 NOUN cana-736 91 11 2 2 NUM cana-736 91 12 ‖𝑓‖𝑝,𝛽 ‖𝑓‖𝑝,𝛽 NOUN cana-736 91 13 ] ] PUNCT cana-736 91 14 , , PUNCT cana-736 91 15 we we PRON cana-736 91 16 get get VERB cana-736 91 17 𝐼1 𝐼1 ADJ cana-736 91 18 ≀ ≀ NUM cana-736 91 19 ∫ ∫ PROPN cana-736 91 20 ( ( PUNCT cana-736 91 21 sin sin NOUN cana-736 91 22 𝜈 𝜈 X cana-736 91 23 2 2 NUM cana-736 91 24 ) ) PUNCT cana-736 91 25 βˆ’1 βˆ’1 NOUN cana-736 91 26 ( ( PUNCT cana-736 91 27 cos cos ADP cana-736 91 28 𝜈 𝜈 X cana-736 91 29 2 2 NUM cana-736 91 30 ) ) PUNCT cana-736 91 31 βˆ’9 βˆ’9 NOUN cana-736 91 32 ∫ ∫ PROPN cana-736 91 33 πœπ‘’(𝐷π‘₯,2,2𝑔 πœπ‘’(𝐷π‘₯,2,2𝑔 PROPN cana-736 91 34 , , PUNCT cana-736 91 35 ) ) PUNCT cana-736 91 36 sin sin VERB cana-736 91 37 𝑒 𝑒 PROPN cana-736 91 38 2 2 NUM cana-736 91 39 𝜈 𝜈 NOUN cana-736 91 40 0 0 NUM cana-736 91 41 𝑑 𝑑 NOUN cana-736 91 42 0 0 NUM cana-736 92 1 ( ( PUNCT cana-736 92 2 cos cos ADP cana-736 92 3 𝑒 𝑒 PROPN cana-736 92 4 2 2 NUM cana-736 92 5 ) ) PUNCT cana-736 92 6 9 9 NUM cana-736 92 7 π‘‘π‘’π‘‘πœˆ. π‘‘π‘’π‘‘πœˆ. NOUN cana-736 92 8 𝐼1 𝐼1 VERB cana-736 92 9 ≀ ≀ NOUN cana-736 93 1 ∁(𝑝)‖𝐷π‘₯,2,2𝑔‖ ∁(𝑝)‖𝐷π‘₯,2,2𝑔‖ PROPN cana-736 93 2 𝑝,𝛽 𝑝,𝛽 PROPN cana-736 93 3 ∫ ∫ PROPN cana-736 93 4 ( ( PUNCT cana-736 93 5 sin sin NOUN cana-736 93 6 𝜈 𝜈 X cana-736 93 7 2 2 NUM cana-736 93 8 ) ) PUNCT cana-736 93 9 βˆ’1 βˆ’1 NOUN cana-736 93 10 ( ( PUNCT cana-736 93 11 cos cos ADP cana-736 93 12 𝜈 𝜈 X cana-736 93 13 2 2 NUM cana-736 93 14 ) ) PUNCT cana-736 93 15 βˆ’9 βˆ’9 NOUN cana-736 93 16 ∫ ∫ PROPN cana-736 93 17 sin sin NOUN cana-736 93 18 𝑒 𝑒 PROPN cana-736 93 19 2 2 NUM cana-736 93 20 𝜈 𝜈 NOUN cana-736 93 21 0 0 NUM cana-736 93 22 𝑑 𝑑 NOUN cana-736 93 23 0 0 NUM cana-736 94 1 ( ( PUNCT cana-736 94 2 cos cos ADP cana-736 94 3 𝑒 𝑒 PROPN cana-736 94 4 2 2 NUM cana-736 94 5 ) ) PUNCT cana-736 94 6 5 5 NUM cana-736 94 7 π‘‘π‘’π‘‘πœˆ. π‘‘π‘’π‘‘πœˆ. NOUN cana-736 94 8 the the DET cana-736 94 9 inequality inequality NOUN cana-736 94 10 ∫ ∫ PROPN cana-736 94 11 ( ( PUNCT cana-736 94 12 sin sin NOUN cana-736 94 13 𝜈 𝜈 X cana-736 94 14 2 2 NUM cana-736 94 15 ) ) PUNCT cana-736 94 16 βˆ’1 βˆ’1 NOUN cana-736 94 17 ( ( PUNCT cana-736 94 18 cos cos ADP cana-736 94 19 𝜈 𝜈 X cana-736 94 20 2 2 NUM cana-736 94 21 ) ) PUNCT cana-736 94 22 βˆ’9 βˆ’9 NOUN cana-736 94 23 ∫ ∫ PROPN cana-736 94 24 sin sin NOUN cana-736 94 25 𝑒 𝑒 PROPN cana-736 94 26 2 2 NUM cana-736 94 27 𝜈 𝜈 NOUN cana-736 94 28 0 0 NUM cana-736 94 29 𝑑 𝑑 NOUN cana-736 94 30 0 0 NUM cana-736 95 1 ( ( PUNCT cana-736 95 2 cos cos ADP cana-736 95 3 𝑒 𝑒 PROPN cana-736 95 4 2 2 NUM cana-736 95 5 ) ) PUNCT cana-736 95 6 5 5 NUM cana-736 95 7 π‘‘π‘’π‘‘πœˆ π‘‘π‘’π‘‘πœˆ NOUN cana-736 95 8 ≀ ≀ PUNCT cana-736 95 9 ∁(𝑝)𝑑2 ∁(𝑝)𝑑2 PROPN cana-736 95 10 . . PUNCT cana-736 96 1 for for ADP cana-736 96 2 0 0 NUM cana-736 96 3 < < X cana-736 96 4 𝑑 𝑑 PROPN cana-736 96 5 ≀ ≀ NUM cana-736 96 6 πœ‹ πœ‹ NOUN cana-736 96 7 2 2 NUM cana-736 96 8 , , PUNCT cana-736 96 9 we we PRON cana-736 96 10 obtain obtain VERB cana-736 96 11 𝐼1 𝐼1 NOUN cana-736 96 12 ≀ ≀ NUM cana-736 96 13 ∁(𝑝)𝑑2‖𝐷π‘₯,2,2𝑔‖ ∁(𝑝)𝑑2‖𝐷π‘₯,2,2𝑔‖ NOUN cana-736 96 14 𝑝,𝛽 𝑝,𝛽 NOUN cana-736 96 15 ≀ ≀ NUM cana-736 96 16 ∁(𝑝 ∁(𝑝 NOUN cana-736 96 17 ) ) PUNCT cana-736 96 18 1 1 NUM cana-736 96 19 cos4𝑑 cos4𝑑 PROPN cana-736 96 20 2 2 NUM cana-736 96 21 𝑑2 𝑑2 PROPN cana-736 96 22 ‖𝐷π‘₯,2,2𝑔‖ ‖𝐷π‘₯,2,2𝑔‖ PROPN cana-736 96 23 𝑝,𝛽 𝑝,𝛽 PROPN cana-736 96 24 . . PUNCT cana-736 97 1 if if SCONJ cana-736 97 2 πœ‹ πœ‹ PRON cana-736 97 3 2 2 NUM cana-736 97 4 ≀ ≀ NUM cana-736 97 5 𝑑 𝑑 X cana-736 97 6 < < X cana-736 97 7 πœ‹ πœ‹ NOUN cana-736 97 8 , , PUNCT cana-736 97 9 so so ADV cana-736 97 10 by by ADP cana-736 97 11 lemma lemma PROPN cana-736 97 12 ( ( PUNCT cana-736 97 13 2.6 2.6 NUM cana-736 97 14 ) ) PUNCT cana-736 97 15 . . PUNCT cana-736 98 1 [ [ X cana-736 98 2 πœπ‘‘ πœπ‘‘ X cana-736 98 3 ( ( PUNCT cana-736 98 4 𝑓 𝑓 PROPN cana-736 98 5 ) ) PUNCT cana-736 98 6 βˆ’ βˆ’ NOUN cana-736 98 7 πœπœ‹ πœπœ‹ NOUN cana-736 98 8 2 2 NUM cana-736 98 9 ( ( PUNCT cana-736 98 10 𝑓 𝑓 X cana-736 98 11 ) ) PUNCT cana-736 98 12 = = SYM cana-736 99 1 βˆ’ βˆ’ PROPN cana-736 99 2 ∫ ∫ PROPN cana-736 99 3 ( ( PUNCT cana-736 99 4 sin sin NOUN cana-736 99 5 𝜈 𝜈 X cana-736 99 6 2 2 NUM cana-736 99 7 ) ) PUNCT cana-736 99 8 βˆ’1 βˆ’1 NOUN cana-736 99 9 ( ( PUNCT cana-736 99 10 cos cos ADP cana-736 99 11 𝜈 𝜈 X cana-736 99 12 2 2 NUM cana-736 99 13 ) ) PUNCT cana-736 99 14 βˆ’9 βˆ’9 NOUN cana-736 99 15 ∫ ∫ NOUN cana-736 99 16 πœπ‘’(𝐷π‘₯,2,2𝑓 πœπ‘’(𝐷π‘₯,2,2𝑓 NOUN cana-736 99 17 ) ) PUNCT cana-736 99 18 πœ‹ πœ‹ NOUN cana-736 99 19 𝜈 𝜈 X cana-736 99 20 𝑑 𝑑 X cana-736 99 21 πœ‹ πœ‹ PROPN cana-736 99 22 2 2 NUM cana-736 99 23 sin sin NOUN cana-736 99 24 𝑒 𝑒 X cana-736 99 25 2 2 NUM cana-736 99 26 ( ( PUNCT cana-736 99 27 cos cos ADP cana-736 99 28 𝑒 𝑒 PROPN cana-736 99 29 2 2 NUM cana-736 99 30 ) ) PUNCT cana-736 99 31 9 9 NUM cana-736 99 32 π‘‘π‘’π‘‘πœˆ. π‘‘π‘’π‘‘πœˆ. NOUN cana-736 99 33 ] ] X cana-736 99 34 we we PRON cana-736 99 35 get get VERB cana-736 99 36 𝐼2 𝐼2 NOUN cana-736 99 37 = = PUNCT cana-736 100 1 β€–πœπ‘‘ β€–πœπ‘‘ PUNCT cana-736 100 2 ( ( PUNCT cana-736 100 3 𝑓 𝑓 X cana-736 100 4 ) ) PUNCT cana-736 100 5 βˆ’ βˆ’ NOUN cana-736 100 6 πœπœ‹ πœπœ‹ NOUN cana-736 100 7 2 2 NUM cana-736 100 8 ( ( PUNCT cana-736 100 9 𝑓)β€– 𝑓)β€– NOUN cana-736 100 10 𝑝,𝛽 𝑝,𝛽 NOUN cana-736 100 11 = = PROPN cana-736 100 12 β€–βˆ« β€–βˆ« PROPN cana-736 100 13 ( ( PUNCT cana-736 100 14 sin sin NOUN cana-736 100 15 𝜈 𝜈 X cana-736 100 16 2 2 NUM cana-736 100 17 ) ) PUNCT cana-736 100 18 βˆ’1 βˆ’1 NOUN cana-736 100 19 ( ( PUNCT cana-736 100 20 cos cos ADP cana-736 100 21 𝜈 𝜈 X cana-736 100 22 2 2 NUM cana-736 100 23 ) ) PUNCT cana-736 100 24 βˆ’9 βˆ’9 NOUN cana-736 100 25 ∫ ∫ PROPN cana-736 100 26 πœπ‘’(𝐷π‘₯,2,2𝑔 πœπ‘’(𝐷π‘₯,2,2𝑔 PROPN cana-736 100 27 , , PUNCT cana-736 100 28 π‘₯ π‘₯ NOUN cana-736 100 29 ) ) PUNCT cana-736 100 30 πœ‹ πœ‹ NOUN cana-736 100 31 𝜈 𝜈 X cana-736 100 32 𝑑 𝑑 X cana-736 100 33 πœ‹ πœ‹ PROPN cana-736 100 34 2 2 NUM cana-736 100 35 sin sin NOUN cana-736 100 36 𝑒 𝑒 X cana-736 100 37 2 2 NUM cana-736 100 38 ( ( PUNCT cana-736 100 39 cos cos ADP cana-736 100 40 𝑒 𝑒 PROPN cana-736 100 41 2 2 NUM cana-736 100 42 ) ) PUNCT cana-736 100 43 9 9 NUM cana-736 100 44 π‘‘π‘’π‘‘πœˆβ€– π‘‘π‘’π‘‘πœˆβ€– ADP cana-736 100 45 𝑝,𝛽 𝑝,𝛽 NOUN cana-736 100 46 . . PUNCT cana-736 101 1 communications communication NOUN cana-736 101 2 on on ADP cana-736 101 3 applied apply VERB cana-736 101 4 nonlinear nonlinear ADJ cana-736 101 5 analysis analysis NOUN cana-736 101 6 issn issn NOUN cana-736 101 7 : : PUNCT cana-736 101 8 1074 1074 NUM cana-736 101 9 - - PUNCT cana-736 101 10 133x 133x NUM cana-736 101 11 vol vol NOUN cana-736 101 12 31 31 NUM cana-736 101 13 no no NOUN cana-736 101 14 . . PUNCT cana-736 102 1 3s 3s NUM cana-736 102 2 ( ( PUNCT cana-736 102 3 2024 2024 NUM cana-736 102 4 ) ) PUNCT cana-736 102 5 123 123 NUM cana-736 102 6 https://internationalpubls.com https://internationalpubls.com X cana-736 102 7 applying apply VERB cana-736 102 8 lemma lemma PROPN cana-736 102 9 . . PUNCT cana-736 103 1 2.3 2.3 NUM cana-736 104 1 [ [ X cana-736 104 2 β€– β€– ADJ cana-736 104 3 πœπ‘‘ πœπ‘‘ PROPN cana-736 104 4 ( ( PUNCT cana-736 104 5 𝑓)‖𝑝,𝛽 𝑓)‖𝑝,𝛽 PROPN cana-736 104 6 ≀ ≀ NOUN cana-736 104 7 19∁(𝑝 19∁(𝑝 NUM cana-736 104 8 ) ) PUNCT cana-736 104 9 πœ‹ πœ‹ NOUN cana-736 104 10 cos4𝑑 cos4𝑑 NOUN cana-736 104 11 2 2 NUM cana-736 104 12 ‖𝑓‖𝑝,𝛽 ‖𝑓‖𝑝,𝛽 NOUN cana-736 104 13 ] ] PUNCT cana-736 104 14 , , PUNCT cana-736 104 15 we we PRON cana-736 104 16 get get VERB cana-736 104 17 𝐼2 𝐼2 NOUN cana-736 104 18 ≀ ≀ NOUN cana-736 105 1 ∁(𝑝)‖𝐷π‘₯,2,2𝑔‖ ∁(𝑝)‖𝐷π‘₯,2,2𝑔‖ PROPN cana-736 105 2 𝑝,𝛽 𝑝,𝛽 PROPN cana-736 105 3 ∫ ∫ PROPN cana-736 105 4 ( ( PUNCT cana-736 105 5 sin sin NOUN cana-736 105 6 𝜈 𝜈 X cana-736 105 7 2 2 NUM cana-736 105 8 ) ) PUNCT cana-736 105 9 βˆ’1 βˆ’1 NOUN cana-736 105 10 ( ( PUNCT cana-736 105 11 cos cos ADP cana-736 105 12 𝜈 𝜈 X cana-736 105 13 2 2 NUM cana-736 105 14 ) ) PUNCT cana-736 105 15 βˆ’9 βˆ’9 NOUN cana-736 105 16 ∫ ∫ PROPN cana-736 105 17 sin sin NOUN cana-736 105 18 𝑒 𝑒 PROPN cana-736 105 19 2 2 NUM cana-736 105 20 πœ‹ πœ‹ NOUN cana-736 105 21 𝜈 𝜈 X cana-736 105 22 𝑑 𝑑 X cana-736 105 23 πœ‹ πœ‹ NOUN cana-736 105 24 2 2 NUM cana-736 105 25 ( ( PUNCT cana-736 105 26 cos cos ADP cana-736 105 27 𝑒 𝑒 PROPN cana-736 105 28 2 2 NUM cana-736 105 29 ) ) PUNCT cana-736 105 30 5 5 NUM cana-736 105 31 π‘‘π‘’π‘‘πœˆ. π‘‘π‘’π‘‘πœˆ. NOUN cana-736 105 32 assume assume VERB cana-736 105 33 πœ‹ πœ‹ PUNCT cana-736 105 34 2 2 NUM cana-736 105 35 ≀ ≀ NUM cana-736 105 36 𝑑 𝑑 ADP cana-736 105 37 < < X cana-736 105 38 πœ‹ πœ‹ NOUN cana-736 105 39 , , PUNCT cana-736 105 40 so so ADV cana-736 105 41 ∫ ∫ PROPN cana-736 105 42 ( ( PUNCT cana-736 105 43 sin sin NOUN cana-736 105 44 𝜈 𝜈 X cana-736 105 45 2 2 NUM cana-736 105 46 ) ) PUNCT cana-736 105 47 βˆ’1 βˆ’1 NOUN cana-736 105 48 ( ( PUNCT cana-736 105 49 cos cos ADP cana-736 105 50 𝜈 𝜈 X cana-736 105 51 2 2 NUM cana-736 105 52 ) ) PUNCT cana-736 105 53 βˆ’9 βˆ’9 NOUN cana-736 105 54 ∫ ∫ PROPN cana-736 105 55 sin sin NOUN cana-736 105 56 𝑒 𝑒 PROPN cana-736 105 57 2 2 NUM cana-736 105 58 πœ‹ πœ‹ NOUN cana-736 105 59 𝜈 𝜈 X cana-736 105 60 𝑑 𝑑 X cana-736 105 61 πœ‹ πœ‹ NOUN cana-736 105 62 2 2 NUM cana-736 105 63 ( ( PUNCT cana-736 105 64 cos cos ADP cana-736 105 65 𝑒 𝑒 PROPN cana-736 105 66 2 2 NUM cana-736 105 67 ) ) PUNCT cana-736 105 68 5 5 NUM cana-736 105 69 π‘‘π‘’π‘‘πœˆ π‘‘π‘’π‘‘πœˆ NOUN cana-736 105 70 ≀ ≀ PROPN cana-736 105 71 ∁(𝑝 ∁(𝑝 NOUN cana-736 105 72 ) ) PUNCT cana-736 105 73 1 1 NUM cana-736 105 74 cos4 cos4 PROPN cana-736 105 75 𝑑 𝑑 NOUN cana-736 105 76 2 2 NUM cana-736 105 77 , , PUNCT cana-736 105 78 it it PRON cana-736 105 79 the the PRON cana-736 105 80 follows follow VERB cana-736 105 81 that that SCONJ cana-736 105 82 ( ( PUNCT cana-736 105 83 3.2 3.2 NUM cana-736 105 84 ) ) PUNCT cana-736 105 85 𝐼2 𝐼2 NOUN cana-736 105 86 ≀ ≀ NUM cana-736 105 87 ∁(𝑝 ∁(𝑝 NOUN cana-736 105 88 ) ) PUNCT cana-736 105 89 1 1 NUM cana-736 105 90 cos4𝑑 cos4𝑑 NUM cana-736 105 91 2 2 NUM cana-736 105 92 ‖𝐷π‘₯,2,2𝑔‖ ‖𝐷π‘₯,2,2𝑔‖ NOUN cana-736 105 93 𝑝,𝛽 𝑝,𝛽 VERB cana-736 105 94 ≀ ≀ NUM cana-736 105 95 ∁(𝑝 ∁(𝑝 NOUN cana-736 105 96 ) ) PUNCT cana-736 105 97 1 1 NUM cana-736 105 98 cos4𝑑 cos4𝑑 NUM cana-736 105 99 2 2 NUM cana-736 105 100 𝑑2‖𝐷π‘₯,2,2𝑔‖ 𝑑2‖𝐷π‘₯,2,2𝑔‖ NOUN cana-736 105 101 𝑝,𝛽 𝑝,𝛽 NOUN cana-736 105 102 . . PUNCT cana-736 106 1 since since SCONJ cana-736 106 2 β€–πœπ‘‘ β€–πœπ‘‘ X cana-736 106 3 ( ( PUNCT cana-736 106 4 𝑔 𝑔 NOUN cana-736 106 5 ) ) PUNCT cana-736 106 6 βˆ’ βˆ’ PROPN cana-736 106 7 𝑔‖𝑝,𝛽 𝑔‖𝑝,𝛽 NOUN cana-736 106 8 ≀ ≀ NOUN cana-736 106 9 β€–πœπ‘‘ β€–πœπ‘‘ PUNCT cana-736 106 10 ( ( PUNCT cana-736 106 11 𝑔 𝑔 NOUN cana-736 106 12 ) ) PUNCT cana-736 106 13 βˆ’ βˆ’ NOUN cana-736 106 14 πœπœ‹ πœπœ‹ NOUN cana-736 106 15 2 2 NUM cana-736 106 16 ( ( PUNCT cana-736 106 17 𝑔)β€– 𝑔)β€– ADJ cana-736 106 18 𝑝,𝛽 𝑝,𝛽 NOUN cana-736 106 19 + + CCONJ cana-736 106 20 β€–πœπœ‹ β€–πœπœ‹ NUM cana-736 106 21 2 2 NUM cana-736 106 22 ( ( PUNCT cana-736 106 23 𝑔 𝑔 NOUN cana-736 106 24 ) ) PUNCT cana-736 106 25 βˆ’ βˆ’ NOUN cana-736 106 26 𝑔‖ 𝑔‖ NOUN cana-736 106 27 𝑝,𝛽 𝑝,𝛽 NOUN cana-736 106 28 using use VERB cana-736 106 29 ( ( PUNCT cana-736 106 30 3.2)and 3.2)and NUM cana-736 106 31 ( ( PUNCT cana-736 106 32 3.1)we 3.1)we NOUN cana-736 106 33 obtain obtain VERB cana-736 106 34 , , PUNCT cana-736 106 35 for for ADP cana-736 106 36 0 0 NUM cana-736 106 37 < < X cana-736 106 38 𝑑 𝑑 X cana-736 106 39 ≀ ≀ NUM cana-736 106 40 πœ‹ πœ‹ NOUN cana-736 106 41 2 2 NUM cana-736 106 42 , , PUNCT cana-736 106 43 that that SCONJ cana-736 106 44 β€–πœπ‘‘ β€–πœπ‘‘ NUM cana-736 106 45 ( ( PUNCT cana-736 106 46 𝑔 𝑔 NOUN cana-736 106 47 ) ) PUNCT cana-736 106 48 βˆ’ βˆ’ PROPN cana-736 106 49 𝑔‖𝑝,𝛽 𝑔‖𝑝,𝛽 ADJ cana-736 106 50 ≀ ≀ NUM cana-736 106 51 ∁(𝑝 ∁(𝑝 NOUN cana-736 106 52 ) ) PUNCT cana-736 106 53 1 1 NUM cana-736 106 54 cos4 cos4 PROPN cana-736 106 55 𝑑 𝑑 NOUN cana-736 106 56 2 2 NUM cana-736 106 57 𝑑2 𝑑2 PROPN cana-736 106 58 ‖𝐷π‘₯,2,2𝑔‖ ‖𝐷π‘₯,2,2𝑔‖ PROPN cana-736 106 59 𝑝,𝛽 𝑝,𝛽 NOUN cana-736 106 60 . . PUNCT cana-736 107 1 for for ADP cana-736 107 2 πœ‹ πœ‹ NOUN cana-736 107 3 2 2 NUM cana-736 107 4 < < X cana-736 107 5 𝑑 𝑑 X cana-736 107 6 ≀ ≀ NUM cana-736 107 7 πœ‹ πœ‹ NOUN cana-736 107 8 , , PUNCT cana-736 107 9 we we PRON cana-736 107 10 proved prove VERB cana-736 107 11 inequality inequality NOUN cana-736 107 12 ( ( PUNCT cana-736 107 13 3.1 3.1 NUM cana-736 107 14 ) ) PUNCT cana-736 107 15 for for ADP cana-736 107 16 0 0 NUM cana-736 107 17 < < X cana-736 107 18 𝑑 𝑑 PROPN cana-736 107 19 ≀ ≀ NUM cana-736 107 20 πœ‹. πœ‹. ADJ cana-736 107 21 𝜏cos 𝜏co NOUN cana-736 107 22 𝑑(𝑔 𝑑(𝑔 ADJ cana-736 107 23 ) ) PUNCT cana-736 108 1 = = SYM cana-736 108 2 𝜏cos 𝜏co NOUN cana-736 108 3 βˆ’π‘‘(𝑔 βˆ’π‘‘(𝑔 NOUN cana-736 108 4 , , PUNCT cana-736 108 5 ) ) PUNCT cana-736 108 6 , , PUNCT cana-736 108 7 let let VERB cana-736 108 8 𝑓 𝑓 DET cana-736 108 9 ∈ ∈ PROPN cana-736 108 10 𝐿𝑝,𝛽 𝐿𝑝,𝛽 NOUN cana-736 108 11 and and CCONJ cana-736 108 12 0 0 NUM cana-736 108 13 ≀ ≀ NUM cana-736 108 14 |𝑑| |𝑑| ADP cana-736 108 15 ≀ ≀ NOUN cana-736 109 1 𝛿 𝛿 PRON cana-736 109 2 < < X cana-736 109 3 πœ‹. πœ‹. NOUN cana-736 109 4 so so ADV cana-736 109 5 for for ADP cana-736 109 6 any any DET cana-736 109 7 𝑔 𝑔 PROPN cana-736 109 8 ∈ ∈ PROPN cana-736 109 9 𝐿𝑝,𝛽 𝐿𝑝,𝛽 NOUN cana-736 109 10 . . PUNCT cana-736 110 1 using use VERB cana-736 110 2 lemma lemma PROPN cana-736 110 3 2.3 2.3 NUM cana-736 110 4 we we PRON cana-736 110 5 get get VERB cana-736 110 6 β€–πœπ‘‘ β€–πœπ‘‘ NUM cana-736 110 7 ( ( PUNCT cana-736 110 8 𝑓 𝑓 X cana-736 110 9 ) ) PUNCT cana-736 110 10 βˆ’ βˆ’ ADP cana-736 110 11 𝑓‖𝑝,𝛽 𝑓‖𝑝,𝛽 ADV cana-736 110 12 ≀ ≀ NUM cana-736 110 13 ∁(𝑝)(β€–πœπ‘‘ ∁(𝑝)(β€–πœπ‘‘ NOUN cana-736 110 14 ( ( PUNCT cana-736 110 15 𝑓 𝑓 DET cana-736 110 16 βˆ’ βˆ’ PROPN cana-736 110 17 𝑔 𝑔 NOUN cana-736 110 18 , , PUNCT cana-736 110 19 π‘₯)‖𝑝,𝛽 π‘₯)‖𝑝,𝛽 NOUN cana-736 110 20 + + SYM cana-736 110 21 β€–πœπ‘‘ β€–πœπ‘‘ PUNCT cana-736 110 22 ( ( PUNCT cana-736 110 23 𝑔 𝑔 NOUN cana-736 110 24 ) ) PUNCT cana-736 110 25 βˆ’ βˆ’ ADP cana-736 110 26 𝑔‖𝑝,𝛽 𝑔‖𝑝,𝛽 ADJ cana-736 110 27 + + CCONJ cana-736 110 28 ‖𝑔 ‖𝑔 ADJ cana-736 110 29 βˆ’ βˆ’ NOUN cana-736 110 30 𝑓‖𝑝,𝛽 𝑓‖𝑝,𝛽 NUM cana-736 110 31 ) ) PUNCT cana-736 110 32 . . PUNCT cana-736 111 1 β€–πœπ‘‘ β€–πœπ‘‘ PUNCT cana-736 111 2 ( ( PUNCT cana-736 111 3 𝑓 𝑓 X cana-736 111 4 ) ) PUNCT cana-736 111 5 βˆ’ βˆ’ ADP cana-736 111 6 𝑓‖𝑝,𝛽 𝑓‖𝑝,𝛽 ADV cana-736 111 7 ≀ ≀ NUM cana-736 111 8 ∁(𝑝 ∁(𝑝 NOUN cana-736 111 9 ) ) PUNCT cana-736 111 10 1 1 NUM cana-736 111 11 cos4𝑑 cos4𝑑 NUM cana-736 111 12 2 2 NUM cana-736 111 13 ‖𝑓 ‖𝑓 NOUN cana-736 111 14 βˆ’ βˆ’ NOUN cana-736 111 15 𝑔‖𝑝,𝛽 𝑔‖𝑝,𝛽 PART cana-736 111 16 + + CCONJ cana-736 111 17 β€–πœπ‘‘(𝑔 β€–πœπ‘‘(𝑔 X cana-736 111 18 ) ) PUNCT cana-736 111 19 βˆ’ βˆ’ PROPN cana-736 111 20 𝑔‖𝑝,𝛽. 𝑔‖𝑝,𝛽. NOUN cana-736 111 21 by by ADP cana-736 111 22 using(3.1 using(3.1 NOUN cana-736 111 23 ) ) PUNCT cana-736 111 24 , , PUNCT cana-736 111 25 to to PART cana-736 111 26 get get VERB cana-736 111 27 β€–πœπ‘‘ β€–πœπ‘‘ X cana-736 111 28 ( ( PUNCT cana-736 111 29 𝑓 𝑓 X cana-736 111 30 ) ) PUNCT cana-736 111 31 βˆ’ βˆ’ ADP cana-736 111 32 𝑓‖𝑝,𝛽 𝑓‖𝑝,𝛽 ADV cana-736 111 33 ≀ ≀ NUM cana-736 111 34 ∁(𝑝 ∁(𝑝 NOUN cana-736 111 35 ) ) PUNCT cana-736 111 36 1 1 NUM cana-736 111 37 cos4 cos4 PROPN cana-736 111 38 𝑑 𝑑 NOUN cana-736 111 39 2 2 NUM cana-736 111 40 ‖𝑓 ‖𝑓 NOUN cana-736 111 41 βˆ’ βˆ’ NOUN cana-736 111 42 𝑔‖𝑝,𝛽 𝑔‖𝑝,𝛽 NOUN cana-736 111 43 + + NUM cana-736 111 44 𝑑2‖𝐷π‘₯,2,2𝑔‖ 𝑑2‖𝐷π‘₯,2,2𝑔‖ NOUN cana-736 111 45 𝑝,𝛽 𝑝,𝛽 NOUN cana-736 111 46 . . PUNCT cana-736 112 1 to to PART cana-736 112 2 prove prove VERB cana-736 112 3 the the DET cana-736 112 4 second second ADJ cana-736 112 5 inequality inequality NOUN cana-736 112 6 , , PUNCT cana-736 112 7 assume assume VERB cana-736 112 8 function function NOUN cana-736 112 9 𝑔𝛿 𝑔𝛿 NOUN cana-736 112 10 = = SYM cana-736 112 11 1 1 NUM cana-736 112 12 𝒦(𝛿 𝒦(𝛿 NUM cana-736 112 13 ) ) PUNCT cana-736 112 14 ∫ ∫ PROPN cana-736 112 15 ( ( PUNCT cana-736 112 16 sin sin NOUN cana-736 112 17 𝜈 𝜈 X cana-736 112 18 2 2 NUM cana-736 112 19 ) ) PUNCT cana-736 112 20 βˆ’1 βˆ’1 NOUN cana-736 112 21 ( ( PUNCT cana-736 112 22 cos cos ADP cana-736 112 23 𝜈 𝜈 X cana-736 112 24 2 2 NUM cana-736 112 25 ) ) PUNCT cana-736 112 26 βˆ’9 βˆ’9 NOUN cana-736 112 27 ∫ ∫ PROPN cana-736 112 28 πœπ‘’(𝑓 πœπ‘’(𝑓 PROPN cana-736 112 29 , , PUNCT cana-736 112 30 π‘₯ π‘₯ NOUN cana-736 112 31 ) ) PUNCT cana-736 112 32 sin sin NOUN cana-736 112 33 𝑒 𝑒 PROPN cana-736 112 34 2 2 NUM cana-736 112 35 𝜈 𝜈 NOUN cana-736 112 36 0 0 NUM cana-736 112 37 𝛿 𝛿 NOUN cana-736 112 38 0 0 NUM cana-736 113 1 ( ( PUNCT cana-736 113 2 cos cos ADP cana-736 113 3 𝑒 𝑒 PROPN cana-736 113 4 2 2 NUM cana-736 113 5 ) ) PUNCT cana-736 113 6 9 9 NUM cana-736 113 7 π‘‘π‘’π‘‘πœˆ π‘‘π‘’π‘‘πœˆ NOUN cana-736 113 8 , , PUNCT cana-736 113 9 communications communication NOUN cana-736 113 10 on on ADP cana-736 113 11 applied apply VERB cana-736 113 12 nonlinear nonlinear ADJ cana-736 113 13 analysis analysis NOUN cana-736 113 14 issn issn NOUN cana-736 113 15 : : PUNCT cana-736 113 16 1074 1074 NUM cana-736 113 17 - - PUNCT cana-736 113 18 133x 133x NUM cana-736 113 19 vol vol NOUN cana-736 113 20 31 31 NUM cana-736 113 21 no no NOUN cana-736 113 22 . . PUNCT cana-736 114 1 3s 3s NUM cana-736 114 2 ( ( PUNCT cana-736 114 3 2024 2024 NUM cana-736 114 4 ) ) PUNCT cana-736 114 5 124 124 NUM cana-736 114 6 https://internationalpubls.com https://internationalpubls.com X cana-736 114 7 where where SCONJ cana-736 114 8 𝒦(𝛿 𝒦(𝛿 SYM cana-736 114 9 ) ) PUNCT cana-736 114 10 = = SYM cana-736 114 11 ∫ ∫ PROPN cana-736 114 12 ( ( PUNCT cana-736 114 13 sin sin NOUN cana-736 114 14 𝜈 𝜈 X cana-736 114 15 2 2 NUM cana-736 114 16 ) ) PUNCT cana-736 114 17 βˆ’1 βˆ’1 NOUN cana-736 114 18 ( ( PUNCT cana-736 114 19 cos cos ADP cana-736 114 20 𝜈 𝜈 X cana-736 114 21 2 2 NUM cana-736 114 22 ) ) PUNCT cana-736 114 23 βˆ’9 βˆ’9 NOUN cana-736 114 24 ∫ ∫ PROPN cana-736 114 25 πœπ‘’(𝑓 πœπ‘’(𝑓 AUX cana-736 114 26 ) ) PUNCT cana-736 114 27 sin sin NOUN cana-736 114 28 𝑒 𝑒 PROPN cana-736 114 29 2 2 NUM cana-736 114 30 𝜈 𝜈 NOUN cana-736 114 31 0 0 NUM cana-736 114 32 𝛿 𝛿 NOUN cana-736 114 33 0 0 NUM cana-736 114 34 ( ( PUNCT cana-736 114 35 cos cos ADP cana-736 114 36 𝑒 𝑒 PROPN cana-736 114 37 2 2 NUM cana-736 114 38 ) ) PUNCT cana-736 114 39 9 9 NUM cana-736 114 40 π‘‘π‘’π‘‘πœˆ. π‘‘π‘’π‘‘πœˆ. NOUN cana-736 114 41 let let VERB cana-736 114 42 0 0 NUM cana-736 114 43 < < X cana-736 115 1 𝛿 𝛿 X cana-736 115 2 < < X cana-736 115 3 πœ‹ πœ‹ X cana-736 115 4 2 2 NUM cana-736 115 5 , , PUNCT cana-736 115 6 then then ADV cana-736 115 7 ( ( PUNCT cana-736 115 8 3.3 3.3 NUM cana-736 115 9 ) ) PUNCT cana-736 115 10 ∁(𝑝)𝛿2 ∁(𝑝)𝛿2 PROPN cana-736 115 11 ≀ ≀ NOUN cana-736 115 12 𝒦(𝛿 𝒦(𝛿 NUM cana-736 115 13 ) ) PUNCT cana-736 115 14 ≀ ≀ PUNCT cana-736 116 1 ∁(𝑝)𝛿2 ∁(𝑝)𝛿2 NOUN cana-736 116 2 . . PUNCT cana-736 117 1 applying apply VERB cana-736 117 2 lemma.2.3 lemma.2.3 PROPN cana-736 117 3 , , PUNCT cana-736 117 4 to to PART cana-736 117 5 get get VERB cana-736 117 6 ‖𝑔𝛿‖𝑝,𝛽 ‖𝑔𝛿‖𝑝,𝛽 ADJ cana-736 117 7 ≀ ≀ NUM cana-736 117 8 1 1 NUM cana-736 117 9 𝒦(𝛿 𝒦(𝛿 NUM cana-736 117 10 ) ) PUNCT cana-736 117 11 ∫ ∫ PROPN cana-736 117 12 ( ( PUNCT cana-736 117 13 sin sin NOUN cana-736 117 14 𝜈 𝜈 X cana-736 117 15 2 2 NUM cana-736 117 16 ) ) PUNCT cana-736 117 17 βˆ’1 βˆ’1 NOUN cana-736 117 18 ( ( PUNCT cana-736 117 19 cos cos ADP cana-736 117 20 𝜈 𝜈 X cana-736 117 21 2 2 NUM cana-736 117 22 ) ) PUNCT cana-736 117 23 βˆ’9 βˆ’9 NOUN cana-736 117 24 ∫ ∫ PROPN cana-736 117 25 β€–πœπ‘’(𝑓)‖𝑝,𝛽 β€–πœπ‘’(𝑓)‖𝑝,𝛽 NUM cana-736 117 26 sin sin NOUN cana-736 117 27 𝑒 𝑒 PROPN cana-736 117 28 2 2 NUM cana-736 117 29 𝜈 𝜈 NOUN cana-736 117 30 0 0 NUM cana-736 117 31 𝛿 𝛿 NOUN cana-736 117 32 0 0 NUM cana-736 118 1 ( ( PUNCT cana-736 118 2 cos cos ADP cana-736 118 3 𝑒 𝑒 PROPN cana-736 118 4 2 2 NUM cana-736 118 5 ) ) PUNCT cana-736 118 6 9 9 NUM cana-736 118 7 π‘‘π‘’π‘‘πœˆ π‘‘π‘’π‘‘πœˆ NOUN cana-736 118 8 ≀ ≀ PROPN cana-736 118 9 ∁(𝑝 ∁(𝑝 NOUN cana-736 118 10 ) ) PUNCT cana-736 118 11 1 1 NUM cana-736 118 12 cos4𝛿 cos4𝛿 PROPN cana-736 118 13 2 2 NUM cana-736 118 14 ‖𝑓‖𝑝,𝛽 ‖𝑓‖𝑝,𝛽 NOUN cana-736 118 15 . . PUNCT cana-736 119 1 that that PRON cana-736 119 2 is be AUX cana-736 119 3 𝑔𝛿 𝑔𝛿 PRON cana-736 119 4 ∈ ∈ PROPN cana-736 119 5 𝐿𝑝,𝛽 𝐿𝑝,𝛽 PROPN cana-736 119 6 . . PUNCT cana-736 120 1 put put VERB cana-736 120 2 𝑔 𝑔 NOUN cana-736 120 3 = = SYM cana-736 120 4 βˆ’ βˆ’ PROPN cana-736 120 5 ∫ ∫ PROPN cana-736 120 6 ( ( PUNCT cana-736 120 7 1 1 NUM cana-736 120 8 βˆ’ βˆ’ NOUN cana-736 120 9 𝑦2)βˆ’3π‘₯ 𝑦2)βˆ’3π‘₯ NUM cana-736 120 10 0 0 NUM cana-736 121 1 ∫ ∫ NOUN cana-736 122 1 ( ( PUNCT cana-736 122 2 1 1 NUM cana-736 122 3 βˆ’ βˆ’ PROPN cana-736 122 4 𝑧2)21 𝑧2)21 PROPN cana-736 122 5 𝑦 𝑦 NOUN cana-736 122 6 ( ( PUNCT cana-736 122 7 𝑓(𝑧 𝑓(𝑧 PROPN cana-736 122 8 ) ) PUNCT cana-736 122 9 βˆ’ βˆ’ PROPN cana-736 122 10 π‘š1 π‘š1 NOUN cana-736 122 11 π‘š2 π‘š2 PROPN cana-736 122 12 ) ) PUNCT cana-736 122 13 𝑑𝑧𝑑𝑦 𝑑𝑧𝑑𝑦 NOUN cana-736 122 14 , , PUNCT cana-736 122 15 where where SCONJ cana-736 122 16 π‘š1 π‘š1 NOUN cana-736 122 17 = = SYM cana-736 122 18 ∫ ∫ PROPN cana-736 122 19 ( ( PUNCT cana-736 122 20 1 1 NUM cana-736 122 21 βˆ’ βˆ’ PROPN cana-736 122 22 𝑧2)2𝑓(𝑧)𝑑𝑧 𝑧2)2𝑓(𝑧)𝑑𝑧 NOUN cana-736 122 23 1 1 NUM cana-736 122 24 βˆ’1 βˆ’1 NOUN cana-736 122 25 , , PUNCT cana-736 122 26 π‘š2 π‘š2 NOUN cana-736 122 27 = = SYM cana-736 122 28 ∫ ∫ PROPN cana-736 122 29 ( ( PUNCT cana-736 122 30 1 1 NUM cana-736 122 31 βˆ’ βˆ’ PROPN cana-736 122 32 𝑧2)21 𝑧2)21 PROPN cana-736 122 33 βˆ’1 βˆ’1 NOUN cana-736 122 34 𝑑𝑧. 𝑑𝑧. X cana-736 122 35 𝐷π‘₯,2,2𝑔(π‘₯ 𝐷π‘₯,2,2𝑔(π‘₯ PROPN cana-736 122 36 ) ) PUNCT cana-736 122 37 = = SYM cana-736 123 1 𝑓 𝑓 DET cana-736 123 2 βˆ’ βˆ’ PROPN cana-736 123 3 π‘š1 π‘š1 NOUN cana-736 123 4 π‘š2 π‘š2 PROPN cana-736 123 5 , , PUNCT cana-736 123 6 𝐷π‘₯,2,2𝑔(π‘₯ 𝐷π‘₯,2,2𝑔(π‘₯ PROPN cana-736 123 7 ) ) PUNCT cana-736 123 8 = = SYM cana-736 123 9 ( ( PUNCT cana-736 123 10 √1 √1 ADV cana-736 123 11 βˆ’ βˆ’ PROPN cana-736 123 12 π‘₯ π‘₯ NOUN cana-736 123 13 ) ) PUNCT cana-736 123 14 βˆ’2 βˆ’2 NOUN cana-736 123 15 ( ( PUNCT cana-736 123 16 1 1 NUM cana-736 123 17 + + NUM cana-736 123 18 π‘₯)βˆ’2 π‘₯)βˆ’2 VERB cana-736 123 19 𝑑𝑔 𝑑𝑔 VERB cana-736 123 20 𝑑π‘₯ 𝑑π‘₯ VERB cana-736 123 21 ( ( PUNCT cana-736 123 22 √1 √1 ADV cana-736 123 23 βˆ’ βˆ’ PROPN cana-736 123 24 π‘₯ π‘₯ NOUN cana-736 123 25 ) ) PUNCT cana-736 123 26 3 3 NUM cana-736 123 27 ( ( PUNCT cana-736 123 28 √1 √1 PROPN cana-736 123 29 + + CCONJ cana-736 123 30 π‘₯ π‘₯ X cana-736 123 31 ) ) PUNCT cana-736 123 32 3 3 NUM cana-736 123 33 𝑑𝑔 𝑑𝑔 NOUN cana-736 123 34 𝑑π‘₯ 𝑑π‘₯ VERB cana-736 123 35 , , PUNCT cana-736 123 36 we we PRON cana-736 123 37 have have AUX cana-736 123 38 𝑔𝛿 𝑔𝛿 VERB cana-736 123 39 = = SYM cana-736 123 40 1 1 NUM cana-736 123 41 𝒦(𝛿 𝒦(𝛿 NUM cana-736 123 42 ) ) PUNCT cana-736 123 43 ∫ ∫ PROPN cana-736 123 44 ( ( PUNCT cana-736 123 45 sin sin NOUN cana-736 123 46 𝜈 𝜈 X cana-736 123 47 2 2 NUM cana-736 123 48 ) ) PUNCT cana-736 123 49 βˆ’1 βˆ’1 NOUN cana-736 123 50 ( ( PUNCT cana-736 123 51 cos cos ADP cana-736 123 52 𝜈 𝜈 X cana-736 123 53 2 2 NUM cana-736 123 54 ) ) PUNCT cana-736 123 55 βˆ’9 βˆ’9 NOUN cana-736 123 56 ∫ ∫ PROPN cana-736 123 57 πœπ‘’(𝐷π‘₯,2,2𝑔 πœπ‘’(𝐷π‘₯,2,2𝑔 PROPN cana-736 123 58 , , PUNCT cana-736 123 59 π‘₯ π‘₯ NOUN cana-736 123 60 ) ) PUNCT cana-736 123 61 sin sin NOUN cana-736 123 62 𝑒 𝑒 PROPN cana-736 123 63 2 2 NUM cana-736 123 64 𝜈 𝜈 NOUN cana-736 123 65 0 0 NUM cana-736 123 66 𝛿 𝛿 NOUN cana-736 123 67 0 0 NUM cana-736 123 68 ( ( PUNCT cana-736 123 69 cos cos ADP cana-736 123 70 𝑒 𝑒 PROPN cana-736 123 71 2 2 NUM cana-736 123 72 ) ) PUNCT cana-736 123 73 9 9 NUM cana-736 123 74 π‘‘π‘’π‘‘πœˆ π‘‘π‘’π‘‘πœˆ NOUN cana-736 123 75 + + NUM cana-736 123 76 π‘š1 π‘š1 NOUN cana-736 123 77 π‘š2 π‘š2 NOUN cana-736 123 78 . . PUNCT cana-736 124 1 using use VERB cana-736 124 2 lemma.2.6.we lemma.2.6.we NOUN cana-736 124 3 get get VERB cana-736 124 4 ( ( PUNCT cana-736 124 5 3.4 3.4 NUM cana-736 124 6 ) ) PUNCT cana-736 124 7 𝑔𝛿 𝑔𝛿 NOUN cana-736 124 8 = = SYM cana-736 124 9 1 1 NUM cana-736 124 10 𝒦(𝛿 𝒦(𝛿 NUM cana-736 124 11 ) ) PUNCT cana-736 124 12 ( ( PUNCT cana-736 124 13 πœπ›Ώ(𝑔 πœπ›Ώ(𝑔 NOUN cana-736 124 14 ) ) PUNCT cana-736 124 15 βˆ’ βˆ’ PROPN cana-736 124 16 𝑔 𝑔 X cana-736 124 17 ) ) PUNCT cana-736 124 18 + + NUM cana-736 124 19 π‘š1 π‘š1 NOUN cana-736 124 20 π‘š2 π‘š2 NOUN cana-736 124 21 . . PUNCT cana-736 125 1 applying apply VERB cana-736 125 2 the the DET cana-736 125 3 operator operator NOUN cana-736 125 4 𝐷π‘₯,2,2𝑔 𝐷π‘₯,2,2𝑔 NOUN cana-736 125 5 for for ADP cana-736 125 6 ( ( PUNCT cana-736 125 7 3.4 3.4 NUM cana-736 125 8 ) ) PUNCT cana-736 125 9 and and CCONJ cana-736 125 10 lemm2.4 lemm2.4 NOUN cana-736 125 11 , , PUNCT cana-736 125 12 ( ( PUNCT cana-736 125 13 πœπ‘¦(𝐷π‘₯,2,2𝑓 πœπ‘¦(𝐷π‘₯,2,2𝑓 PROPN cana-736 125 14 , , PUNCT cana-736 125 15 ) ) PUNCT cana-736 125 16 = = SYM cana-736 125 17 𝐷π‘₯,2,2πœπ‘¦(𝑓 𝐷π‘₯,2,2πœπ‘¦(𝑓 NOUN cana-736 125 18 ) ) PUNCT cana-736 125 19 . . PUNCT cana-736 125 20 ) ) PUNCT cana-736 125 21 , , PUNCT cana-736 125 22 we we PRON cana-736 125 23 get get VERB cana-736 125 24 𝐷π‘₯,2,2𝑔𝛿(π‘₯ 𝐷π‘₯,2,2𝑔𝛿(π‘₯ NOUN cana-736 125 25 ) ) PUNCT cana-736 125 26 , , PUNCT cana-736 125 27 𝑔𝛿(π‘₯ 𝑔𝛿(π‘₯ PUNCT cana-736 125 28 ) ) PUNCT cana-736 125 29 = = SYM cana-736 125 30 1 1 NUM cana-736 125 31 𝒦(𝛿 𝒦(𝛿 NUM cana-736 125 32 ) ) PUNCT cana-736 125 33 ( ( PUNCT cana-736 125 34 𝐷π‘₯,2,2𝑔(πœπ›Ώ 𝐷π‘₯,2,2𝑔(πœπ›Ώ PROPN cana-736 125 35 ) ) PUNCT cana-736 125 36 βˆ’ βˆ’ PROPN cana-736 125 37 𝐷π‘₯,2,2𝑔(π‘₯ 𝐷π‘₯,2,2𝑔(π‘₯ PROPN cana-736 125 38 ) ) PUNCT cana-736 125 39 ) ) PUNCT cana-736 126 1 + + NUM cana-736 126 2 π‘š1 π‘š1 NOUN cana-736 126 3 π‘š2 π‘š2 PROPN cana-736 126 4 . . PUNCT cana-736 127 1 𝐷π‘₯,2,2𝑔𝛿(π‘₯ 𝐷π‘₯,2,2𝑔𝛿(π‘₯ NOUN cana-736 127 2 ) ) PUNCT cana-736 127 3 = = SYM cana-736 127 4 1 1 NUM cana-736 127 5 𝒦(𝛿 𝒦(𝛿 NUM cana-736 127 6 ) ) PUNCT cana-736 127 7 ( ( PUNCT cana-736 127 8 πœπ›Ώ(𝐷π‘₯,2,2𝑔 πœπ›Ώ(𝐷π‘₯,2,2𝑔 NOUN cana-736 127 9 , , PUNCT cana-736 127 10 π‘₯ π‘₯ NOUN cana-736 127 11 ) ) PUNCT cana-736 127 12 βˆ’ βˆ’ PROPN cana-736 128 1 𝑓 𝑓 X cana-736 128 2 ) ) PUNCT cana-736 128 3 βˆ’ βˆ’ PROPN cana-736 128 4 π‘š1 π‘š1 NOUN cana-736 128 5 π‘š2 π‘š2 PROPN cana-736 128 6 + + CCONJ cana-736 128 7 π‘š1 π‘š1 NOUN cana-736 128 8 π‘š2 π‘š2 PROPN cana-736 128 9 . . PUNCT cana-736 129 1 ( ( PUNCT cana-736 129 2 3.5 3.5 NUM cana-736 129 3 ) ) PUNCT cana-736 129 4 𝐷π‘₯,2,2𝑔𝛿(π‘₯ 𝐷π‘₯,2,2𝑔𝛿(π‘₯ NOUN cana-736 129 5 ) ) PUNCT cana-736 129 6 = = SYM cana-736 129 7 1 1 NUM cana-736 129 8 𝒦(𝛿 𝒦(𝛿 NUM cana-736 129 9 ) ) PUNCT cana-736 129 10 ( ( PUNCT cana-736 129 11 πœπ›Ώ(𝑓 πœπ›Ώ(𝑓 NOUN cana-736 129 12 ) ) PUNCT cana-736 129 13 βˆ’ βˆ’ PROPN cana-736 130 1 𝑓 𝑓 X cana-736 130 2 ) ) PUNCT cana-736 130 3 . . PUNCT cana-736 131 1 therefore therefore ADV cana-736 131 2 , , PUNCT cana-736 131 3 by by ADP cana-736 131 4 lemmas lemma NOUN cana-736 131 5 . . PUNCT cana-736 131 6 2.3 2.3 NUM cana-736 131 7 . . PUNCT cana-736 132 1 and and CCONJ cana-736 132 2 2.4 2.4 NUM cana-736 132 3 , , PUNCT cana-736 132 4 also also ADV cana-736 132 5 𝑔𝛿 𝑔𝛿 VERB cana-736 132 6 ∈ ∈ NOUN cana-736 132 7 𝐿𝑝,𝛽. 𝐿𝑝,𝛽. NOUN cana-736 132 8 then then ADV cana-736 132 9 using use VERB cana-736 132 10 ( ( PUNCT cana-736 132 11 3.5 3.5 NUM cana-736 132 12 ) ) PUNCT cana-736 132 13 equality equality NOUN cana-736 132 14 and and CCONJ cana-736 132 15 inequality inequality NOUN cana-736 132 16 ( ( PUNCT cana-736 132 17 3.3 3.3 NUM cana-736 132 18 ) ) PUNCT cana-736 132 19 , , PUNCT cana-736 132 20 we we PRON cana-736 132 21 get get VERB cana-736 132 22 communications communication NOUN cana-736 132 23 on on ADP cana-736 132 24 applied apply VERB cana-736 132 25 nonlinear nonlinear ADJ cana-736 132 26 analysis analysis NOUN cana-736 132 27 issn issn NOUN cana-736 132 28 : : PUNCT cana-736 132 29 1074 1074 NUM cana-736 132 30 - - PUNCT cana-736 132 31 133x 133x NUM cana-736 132 32 vol vol NOUN cana-736 132 33 31 31 NUM cana-736 132 34 no no NOUN cana-736 132 35 . . PUNCT cana-736 133 1 3s 3s NUM cana-736 133 2 ( ( PUNCT cana-736 133 3 2024 2024 NUM cana-736 133 4 ) ) PUNCT cana-736 133 5 125 125 NUM cana-736 133 6 https://internationalpubls.com https://internationalpubls.com X cana-736 133 7 ( ( PUNCT cana-736 133 8 3.6 3.6 NUM cana-736 133 9 ) ) PUNCT cana-736 133 10 ‖𝐷π‘₯,2,2𝑔𝛿 ‖𝐷π‘₯,2,2𝑔𝛿 ADJ cana-736 133 11 ( ( PUNCT cana-736 133 12 π‘₯)β€– π‘₯)β€– INTJ cana-736 133 13 𝑝,𝛽 𝑝,𝛽 NOUN cana-736 133 14 ≀ ≀ NUM cana-736 133 15 ∁(𝑝 ∁(𝑝 NOUN cana-736 133 16 ) ) PUNCT cana-736 133 17 1 1 NUM cana-736 133 18 𝛿2 𝛿2 PROPN cana-736 133 19 β€–πœπ›Ώ(𝑓 β€–πœπ›Ώ(𝑓 NOUN cana-736 133 20 ) ) PUNCT cana-736 133 21 βˆ’ βˆ’ ADP cana-736 133 22 𝑓‖𝑝,𝛽 𝑓‖𝑝,𝛽 NOUN cana-736 133 23 . . PUNCT cana-736 134 1 then then ADV cana-736 134 2 using(3.6 using(3.6 PROPN cana-736 134 3 ) ) PUNCT cana-736 134 4 , , PUNCT cana-736 134 5 to to PART cana-736 134 6 get get VERB cana-736 134 7 ( ( PUNCT cana-736 134 8 3.7 3.7 NUM cana-736 134 9 ) ) PUNCT cana-736 134 10 ‖𝐷π‘₯,2,2𝑔𝛿(π‘₯)β€– ‖𝐷π‘₯,2,2𝑔𝛿(π‘₯)β€– PUNCT cana-736 135 1 𝑝,𝛽 𝑝,𝛽 VERB cana-736 135 2 ≀ ≀ NUM cana-736 135 3 ∁(𝑝 ∁(𝑝 NOUN cana-736 135 4 ) ) PUNCT cana-736 135 5 1 1 NUM cana-736 135 6 𝛿2 𝛿2 PROPN cana-736 135 7 αΏΆ(𝑓 αΏΆ(𝑓 PROPN cana-736 135 8 , , PUNCT cana-736 135 9 𝛿)𝑝,𝛽. 𝛿)𝑝,𝛽. PROPN cana-736 135 10 and and CCONJ cana-736 135 11 ( ( PUNCT cana-736 135 12 3.8 3.8 NUM cana-736 135 13 ) ) PUNCT cana-736 135 14 ‖𝑓 ‖𝑓 NOUN cana-736 135 15 βˆ’ βˆ’ ADP cana-736 136 1 𝑔𝛿‖𝑝,𝛽 𝑔𝛿‖𝑝,𝛽 VERB cana-736 136 2 ≀ ≀ NUM cana-736 136 3 1 1 NUM cana-736 136 4 𝒦(𝛿 𝒦(𝛿 NUM cana-736 136 5 ) ) PUNCT cana-736 136 6 ∫ ∫ PROPN cana-736 136 7 ( ( PUNCT cana-736 136 8 sin sin NOUN cana-736 136 9 𝜈 𝜈 X cana-736 136 10 2 2 NUM cana-736 136 11 ) ) PUNCT cana-736 136 12 βˆ’1 βˆ’1 NOUN cana-736 136 13 ( ( PUNCT cana-736 136 14 cos cos ADP cana-736 136 15 𝜈 𝜈 X cana-736 136 16 2 2 NUM cana-736 136 17 ) ) PUNCT cana-736 137 1 βˆ’9 βˆ’9 NOUN cana-736 137 2 βˆ«β€–π‘“ βˆ«β€–π‘“ PUNCT cana-736 137 3 βˆ’ βˆ’ NOUN cana-736 137 4 πœπ‘’(𝑓)‖𝑝,𝛽 πœπ‘’(𝑓)‖𝑝,𝛽 NOUN cana-736 137 5 sin sin VERB cana-736 137 6 𝑒 𝑒 PROPN cana-736 137 7 2 2 NUM cana-736 137 8 𝜈 𝜈 NOUN cana-736 137 9 0 0 NUM cana-736 137 10 𝛿 𝛿 NOUN cana-736 137 11 0 0 NUM cana-736 137 12 ( ( PUNCT cana-736 137 13 cos cos ADP cana-736 137 14 𝑒 𝑒 PROPN cana-736 137 15 2 2 NUM cana-736 137 16 ) ) PUNCT cana-736 137 17 9 9 NUM cana-736 137 18 π‘‘π‘’π‘‘πœˆ π‘‘π‘’π‘‘πœˆ NOUN cana-736 137 19 ≀ ≀ NOUN cana-736 137 20 αΏΆ(𝑓 αΏΆ(𝑓 PROPN cana-736 137 21 , , PUNCT cana-736 137 22 𝛿)𝑝,𝛽 𝛿)𝑝,𝛽 NOUN cana-736 137 23 . . PUNCT cana-736 138 1 for for ADP cana-736 138 2 0 0 NUM cana-736 138 3 < < X cana-736 138 4 𝛿 𝛿 PROPN cana-736 138 5 ≀ ≀ NUM cana-736 138 6 πœ‹ πœ‹ NOUN cana-736 138 7 2 2 NUM cana-736 138 8 , , PUNCT cana-736 138 9 we we PRON cana-736 138 10 have have AUX cana-736 138 11 proved prove VERB cana-736 138 12 that that PRON cana-736 138 13 𝐼(𝛿 𝐼(𝛿 VERB cana-736 138 14 ) ) PUNCT cana-736 138 15 = = SYM cana-736 138 16 ‖𝑓 ‖𝑓 NOUN cana-736 138 17 βˆ’ βˆ’ NOUN cana-736 139 1 𝑔𝛿‖𝑝,𝛽 𝑔𝛿‖𝑝,𝛽 PROPN cana-736 140 1 + + CCONJ cana-736 140 2 𝛿2‖𝐷π‘₯,2,2𝑔𝛿(π‘₯)β€– 𝛿2‖𝐷π‘₯,2,2𝑔𝛿(π‘₯)β€– PROPN cana-736 140 3 𝑝,𝛽 𝑝,𝛽 PROPN cana-736 140 4 ≀ ≀ NUM cana-736 140 5 ∁(𝑝)αΏΆ(𝑓 ∁(𝑝)αΏΆ(𝑓 NOUN cana-736 140 6 , , PUNCT cana-736 140 7 𝛿)𝑝,𝛽 𝛿)𝑝,𝛽 NOUN cana-736 140 8 . . PUNCT cana-736 141 1 for for ADP cana-736 141 2 πœ‹ πœ‹ NOUN cana-736 141 3 2 2 NUM cana-736 141 4 < < X cana-736 141 5 𝛿 𝛿 PROPN cana-736 141 6 ≀ ≀ ADJ cana-736 141 7 πœ‹ πœ‹ NOUN cana-736 141 8 , , PUNCT cana-736 141 9 we we PRON cana-736 141 10 get get VERB cana-736 141 11 𝛿2 𝛿2 NOUN cana-736 141 12 < < X cana-736 141 13 πœ‹2 πœ‹2 NOUN cana-736 141 14 . . PUNCT cana-736 141 15 1 1 NUM cana-736 141 16 , , PUNCT cana-736 141 17 and and CCONJ cana-736 141 18 , , PUNCT cana-736 141 19 1 1 NUM cana-736 141 20 < < X cana-736 141 21 πœ‹ πœ‹ X cana-736 141 22 2 2 NUM cana-736 141 23 , , PUNCT cana-736 141 24 𝐾(𝑓 𝐾(𝑓 PROPN cana-736 141 25 , , PUNCT cana-736 141 26 𝛿)𝑝,𝛽 𝛿)𝑝,𝛽 VERB cana-736 141 27 ≀ ≀ NUM cana-736 141 28 πœ‹2 πœ‹2 NOUN cana-736 141 29 ( ( PUNCT cana-736 141 30 ‖𝑓 ‖𝑓 NOUN cana-736 141 31 βˆ’ βˆ’ ADP cana-736 142 1 𝑔1‖𝑝,𝛽 𝑔1‖𝑝,𝛽 NUM cana-736 142 2 + + CCONJ cana-736 142 3 12‖𝐷π‘₯,2,2𝑔1(π‘₯)β€– 12‖𝐷π‘₯,2,2𝑔1(π‘₯)β€– NUM cana-736 142 4 𝑝,𝛽 𝑝,𝛽 NOUN cana-736 142 5 ) ) PUNCT cana-736 142 6 using use VERB cana-736 142 7 ( ( PUNCT cana-736 142 8 3.7 3.7 NUM cana-736 142 9 ) ) PUNCT cana-736 142 10 π‘Žπ‘›π‘‘(3.8),to π‘Žπ‘›π‘‘(3.8),to NOUN cana-736 142 11 obtain obtain VERB cana-736 142 12 𝐾(𝑓 𝐾(𝑓 NOUN cana-736 142 13 , , PUNCT cana-736 142 14 𝛿)𝑝,𝛽 𝛿)𝑝,𝛽 VERB cana-736 142 15 ≀ ≀ NUM cana-736 142 16 ∁(𝑝 ∁(𝑝 NOUN cana-736 142 17 ) ) PUNCT cana-736 142 18 αΏΆ(𝑓 αΏΆ(𝑓 PROPN cana-736 142 19 , , PUNCT cana-736 142 20 1)𝑝,𝛽 1)𝑝,𝛽 NUM cana-736 142 21 ≀ ≀ NOUN cana-736 142 22 αΏΆ(𝑓 αΏΆ(𝑓 NOUN cana-736 142 23 , , PUNCT cana-736 142 24 1)𝑝,𝛽 1)𝑝,𝛽 NUM cana-736 142 25 . . PUNCT cana-736 143 1 conclusions conclusion NOUN cana-736 144 1 k k PROPN cana-736 144 2 functional functional ADJ cana-736 144 3 and and CCONJ cana-736 144 4 modulus modulus NOUN cana-736 144 5 of of ADP cana-736 144 6 smoothness smoothness NOUN cana-736 144 7 are be AUX cana-736 144 8 equivalent equivalent ADJ cana-736 144 9 on on ADP cana-736 144 10 weighted weight VERB cana-736 144 11 spaces space NOUN cana-736 144 12 . . PUNCT cana-736 145 1 references reference NOUN cana-736 145 2 . . PUNCT cana-736 146 1 [ [ X cana-736 146 2 1 1 NUM cana-736 146 3 ] ] X cana-736 146 4 m.k.potapov m.k.potapov NOUN cana-736 146 5 and and CCONJ cana-736 146 6 f.m f.m PROPN cana-736 146 7 . . PROPN cana-736 146 8 berisha berisha PROPN cana-736 146 9 , , PUNCT cana-736 146 10 approximation approximation NOUN cana-736 146 11 of of ADP cana-736 146 12 classes class NOUN cana-736 146 13 of of ADP cana-736 146 14 functions function NOUN cana-736 146 15 defined define VERB cana-736 146 16 by by ADP cana-736 146 17 a a DET cana-736 146 18 generalized generalized ADJ cana-736 146 19 k k NOUN cana-736 146 20 - - PUNCT cana-736 146 21 th th VERB cana-736 146 22 modulus modulus NOUN cana-736 146 23 of of ADP cana-736 146 24 smoothness smoothness NOUN cana-736 146 25 , , PUNCT cana-736 146 26 east east NOUN cana-736 146 27 j.approx.4(1998),no.2,217 j.approx.4(1998),no.2,217 NOUN cana-736 146 28 - - PUNCT cana-736 146 29 241.mr 241.mr PROPN cana-736 146 30 1638345 1638345 NUM cana-736 146 31 . . PUNCT cana-736 147 1 [ [ X cana-736 147 2 2 2 NUM cana-736 147 3 ] ] X cana-736 147 4 m.k.potapov m.k.potapov NOUN cana-736 147 5 and and CCONJ cana-736 147 6 f.m f.m PROPN cana-736 147 7 . . PROPN cana-736 147 8 berisha berisha PROPN cana-736 147 9 , , PUNCT cana-736 147 10 on on ADP cana-736 147 11 the the DET cana-736 147 12 connection connection NOUN cana-736 147 13 between between ADP cana-736 147 14 the the DET cana-736 147 15 best good ADJ cana-736 147 16 approximation approximation NOUN cana-736 147 17 by by ADP cana-736 147 18 algebraic algebraic ADJ cana-736 147 19 polynomials polynomial NOUN cana-736 147 20 and and CCONJ cana-736 147 21 the the DET cana-736 147 22 modulus modulus NOUN cana-736 147 23 of of ADP cana-736 147 24 smoothness smoothness NOUN cana-736 147 25 of of ADP cana-736 147 26 order order NOUN cana-736 147 27 r r NOUN cana-736 147 28 , , PUNCT cana-736 147 29 journal journal NOUN cana-736 147 30 of of ADP cana-736 147 31 mathematical mathematical ADJ cana-736 147 32 sciences sciences PROPN cana-736 147 33 vol.155,no.1,2008 vol.155,no.1,2008 NOUN cana-736 147 34 . . PUNCT cana-736 148 1 [ [ X cana-736 148 2 3 3 X cana-736 148 3 ] ] X cana-736 148 4 m.k m.k PROPN cana-736 148 5 . . PROPN cana-736 148 6 potapov potapov PROPN cana-736 148 7 andv.m andv.m PROPN cana-736 148 8 . . PUNCT cana-736 149 1 fedorov fedorov PROPN cana-736 149 2 , , PUNCT cana-736 149 3 𝑂teoremakh 𝑂teoremakh PROPN cana-736 149 4 dzheksona dzheksona PROPN cana-736 149 5 dlya dlya PROPN cana-736 149 6 obobshchennogo obobshchennogo PROPN cana-736 149 7 modulya modulya NOUN cana-736 149 8 gladkosti gladkosti NOUN cana-736 149 9 , , PUNCT cana-736 149 10 trudy trudy PROPN cana-736 149 11 mat mat NOUN cana-736 149 12 . . PUNCT cana-736 150 1 inst.steklov inst.steklov X cana-736 150 2 . . PUNCT cana-736 151 1 172(1985 172(1985 NUM cana-736 151 2 ) ) PUNCT cana-736 151 3 , , PUNCT cana-736 151 4 291 291 NUM cana-736 151 5 - - SYM cana-736 151 6 298,355.mr 298,355.mr NUM cana-736 151 7 86m:41012 86m:41012 NUM cana-736 151 8 . . PUNCT cana-736 152 1 [ [ X cana-736 152 2 4 4 X cana-736 152 3 ] ] PUNCT cana-736 152 4 n.sh.berisha n.sh.berisha PROPN cana-736 152 5 and and CCONJ cana-736 152 6 f.m f.m PROPN cana-736 152 7 . . PROPN cana-736 152 8 berisha berisha PROPN cana-736 152 9 , , PUNCT cana-736 152 10 approximation approximation NOUN cana-736 152 11 classes class NOUN cana-736 152 12 of of ADP cana-736 152 13 functions function NOUN cana-736 152 14 defined define VERB cana-736 152 15 by by ADP cana-736 152 16 operators operator NOUN cana-736 152 17 of of ADP cana-736 152 18 differention differention NOUN cana-736 152 19 or or CCONJ cana-736 152 20 operators operator NOUN cana-736 152 21 of of ADP cana-736 152 22 generalized generalized ADJ cana-736 152 23 translation translation NOUN cana-736 152 24 by by ADP cana-736 152 25 means mean NOUN cana-736 152 26 of of ADP cana-736 152 27 al al PROPN cana-736 152 28 gebraic gebraic PROPN cana-736 152 29 polynomials polynomial NOUN cana-736 152 30 , , PUNCT cana-736 152 31 int int NOUN cana-736 152 32 . . PUNCT cana-736 153 1 journal journal PROPN cana-736 153 2 of of ADP cana-736 153 3 math math NOUN cana-736 153 4 . . PUNCT cana-736 154 1 analysis analysis NOUN cana-736 154 2 6(2012 6(2012 NUM cana-736 154 3 ) ) PUNCT cana-736 154 4 , , PUNCT cana-736 154 5 no.55,2709 no.55,2709 NOUN cana-736 154 6 . . PUNCT cana-736 155 1 [ [ X cana-736 155 2 5 5 NUM cana-736 155 3 ] ] PUNCT cana-736 155 4 β€” β€” PUNCT cana-736 155 5 , , PUNCT cana-736 155 6 𝑂 𝑂 PROPN cana-736 155 7 priblizhenii priblizhenii ADJ cana-736 155 8 algebraicheskimi algebraicheskimi PROPN cana-736 155 9 mnogochlemi mnogochlemi NOUN cana-736 155 10 Ο… Ο… PROPN cana-736 155 11 integral integral PROPN cana-736 155 12 noi noi PROPN cana-736 155 13 metrike metrike PROPN cana-736 155 14 sΟ…esom sΟ…esom PROPN cana-736 155 15 yakobi yakobi PROPN cana-736 155 16 , , PUNCT cana-736 155 17 vestnik vestnik PROPN cana-736 155 18 moskov moskov PROPN cana-736 155 19 . . PUNCT cana-736 156 1 univ.ser.i univ.ser.i ADJ cana-736 156 2 mat mat NOUN cana-736 156 3 . . PUNCT cana-736 156 4 mekh mekh PROPN cana-736 156 5 . . PUNCT cana-736 157 1 ( ( PUNCT cana-736 157 2 1983 1983 NUM cana-736 157 3 ) ) PUNCT cana-736 157 4 , , PUNCT cana-736 157 5 no no INTJ cana-736 157 6 . . PUNCT cana-736 158 1 4,43 4,43 NUM cana-736 158 2 - - PUNCT cana-736 158 3 52 52 NUM cana-736 158 4 . . PUNCT cana-736 159 1 mr mr PROPN cana-736 159 2 84i:41008 84i:41008 PROPN cana-736 159 3 . . PUNCT cana-736 160 1 [ [ X cana-736 160 2 6 6 NUM cana-736 160 3 ] ] X cana-736 160 4 p.l p.l PROPN cana-736 160 5 . . PROPN cana-736 160 6 butzer butzer PROPN cana-736 160 7 , , PUNCT cana-736 160 8 r.l r.l PROPN cana-736 160 9 . . PROPN cana-736 160 10 stens stens PROPN cana-736 160 11 , , PUNCT cana-736 160 12 and and CCONJ cana-736 160 13 m m VERB cana-736 160 14 .wehrens .wehrens ADJ cana-736 160 15 , , PUNCT cana-736 160 16 higher high ADJ cana-736 160 17 order order NOUN cana-736 160 18 moduli modulus NOUN cana-736 160 19 of of ADP cana-736 160 20 continuity continuity NOUN cana-736 160 21 based base VERB cana-736 160 22 on on ADP cana-736 160 23 the the DET cana-736 160 24 jacobi jacobi PROPN cana-736 160 25 translation translation NOUN cana-736 160 26 operator operator NOUN cana-736 160 27 and and CCONJ cana-736 160 28 best good ADJ cana-736 160 29 approximation approximation NOUN cana-736 160 30 , , PUNCT cana-736 160 31 c.r.math.rep.acad.sci c.r.math.rep.acad.sci NOUN cana-736 160 32 canada canada PROPN cana-736 160 33 2 2 NUM cana-736 160 34 ( ( PUNCT cana-736 160 35 1980),no.2,83 1980),no.2,83 NUM cana-736 160 36 - - PUNCT cana-736 160 37 88 88 NUM cana-736 160 38 . . PUNCT cana-736 161 1 mr mr PROPN cana-736 161 2 81a:41040 81a:41040 NUM cana-736 161 3 . . PUNCT cana-736 162 1 [ [ X cana-736 162 2 7 7 NUM cana-736 162 3 ] ] SYM cana-736 162 4 s.pawelke s.pawelke NUM cana-736 162 5 , , PUNCT cana-736 162 6 ein ein PROPN cana-736 162 7 satz satz PROPN cana-736 162 8 Ο…om Ο…om PROPN cana-736 162 9 jacksonschen jacksonschen PROPN cana-736 162 10 typ typ PROPN cana-736 162 11 fur fur NOUN cana-736 162 12 algebraische algebraische NOUN cana-736 162 13 polynome polynome PROPN cana-736 162 14 , , PUNCT cana-736 162 15 acta acta PROPN cana-736 162 16 sci sci PROPN cana-736 162 17 . . PROPN cana-736 162 18 math.(szeged math.(szege VERB cana-736 162 19 ) ) PUNCT cana-736 162 20 33(1972),no 33(1972),no NUM cana-736 162 21 . . PROPN cana-736 162 22 34 34 NUM cana-736 162 23 , , PUNCT cana-736 162 24 323 323 NUM cana-736 162 25 - - SYM cana-736 162 26 336 336 NUM cana-736 162 27 . . PUNCT cana-736 163 1 mr mr PROPN cana-736 163 2 58β‹• 58β‹• PROPN cana-736 163 3 1848 1848 NUM cana-736 163 4 . . PUNCT cana-736 164 1 [ [ X cana-736 164 2 8 8 NUM cana-736 164 3 ] ] PUNCT cana-736 164 4 z. z. PROPN cana-736 164 5 ditzian ditzian PROPN cana-736 164 6 , , PUNCT cana-736 164 7 v. v. ADP cana-736 164 8 totik totik PROPN cana-736 164 9 , , PUNCT cana-736 164 10 β€œ " PUNCT cana-736 164 11 k k ADJ cana-736 164 12 - - ADJ cana-736 164 13 functional functional ADJ cana-736 164 14 and and CCONJ cana-736 164 15 weighted weight VERB cana-736 164 16 moduli modulus NOUN cana-736 164 17 of of ADP cana-736 164 18 smoothness smoothness NOUN cana-736 164 19 , , PUNCT cana-736 164 20 ” " PUNCT cana-736 164 21 j. j. PROPN cana-736 164 22 approx approx PROPN cana-736 164 23 . . PUNCT cana-736 165 1 theory theory NOUN cana-736 165 2 , , PUNCT cana-736 165 3 63(1990 63(1990 NUM cana-736 165 4 ) ) PUNCT cana-736 165 5 , , PUNCT cana-736 165 6 pp pp ADP cana-736 165 7 3 3 NUM cana-736 165 8 - - SYM cana-736 165 9 29 29 NUM cana-736 165 10 . . PUNCT cana-736 166 1 [ [ X cana-736 166 2 9 9 NUM cana-736 166 3 ] ] PUNCT cana-736 166 4 z. z. PROPN cana-736 166 5 ditzian ditzian PROPN cana-736 166 6 , , PUNCT cana-736 166 7 v. v. ADP cana-736 166 8 totik totik NOUN cana-736 166 9 , , PUNCT cana-736 166 10 β€œ " PUNCT cana-736 166 11 moduli modulus NOUN cana-736 166 12 of of ADP cana-736 166 13 smoothness smoothness NOUN cana-736 166 14 , , PUNCT cana-736 166 15 ” " PUNCT cana-736 166 16 new new PROPN cana-736 166 17 york york PROPN cana-736 166 18 : : PUNCT cana-736 166 19 springer springer NOUN cana-736 166 20 – – PUNCT cana-736 166 21 verlag verlag PROPN cana-736 166 22 . . PUNCT cana-736 167 1 ( ( PUNCT cana-736 167 2 1987 1987 NUM cana-736 167 3 ) ) PUNCT cana-736 167 4 . . PUNCT cana-736 168 1 [ [ X cana-736 168 2 10 10 NUM cana-736 168 3 ] ] X cana-736 168 4 kadham kadham PROPN cana-736 168 5 , , PUNCT cana-736 168 6 s.m s.m PROPN cana-736 168 7 . . PROPN cana-736 168 8 acute acute PROPN cana-736 168 9 interstitial interstitial ADJ cana-736 168 10 pneumonia pneumonia NOUN cana-736 168 11 image image NOUN cana-736 168 12 enhancement enhancement NOUN cana-736 168 13 using use VERB cana-736 168 14 fuzzy fuzzy ADJ cana-736 168 15 partial partial ADJ cana-736 168 16 transforms transform NOUN cana-736 168 17 . . PUNCT cana-736 169 1 appl appl PROPN cana-736 169 2 geomat geomat PROPN cana-736 169 3 16 16 NUM cana-736 169 4 , , PUNCT cana-736 169 5 35–39 35–39 NUM cana-736 169 6 ( ( PUNCT cana-736 169 7 2024 2024 NUM cana-736 169 8 ) ) PUNCT cana-736 169 9 . . PUNCT cana-736 170 1 https://doi.org/10.1007/s12518-023-00509-8 https://doi.org/10.1007/s12518-023-00509-8 X cana-736 170 2 https://doi.org/10.1007/s12518-023-00509-8 https://doi.org/10.1007/s12518-023-00509-8 NUM cana-736 171 1 communications communication NOUN cana-736 171 2 on on ADP cana-736 171 3 applied apply VERB cana-736 171 4 nonlinear nonlinear ADJ cana-736 171 5 analysis analysis NOUN cana-736 171 6 issn issn NOUN cana-736 171 7 : : PUNCT cana-736 171 8 1074 1074 NUM cana-736 171 9 - - PUNCT cana-736 171 10 133x 133x NUM cana-736 171 11 vol vol NOUN cana-736 171 12 31 31 NUM cana-736 171 13 no no NOUN cana-736 171 14 . . PUNCT cana-736 172 1 3s 3s NUM cana-736 172 2 ( ( PUNCT cana-736 172 3 2024 2024 NUM cana-736 172 4 ) ) PUNCT cana-736 172 5 126 126 NUM cana-736 172 6 https://internationalpubls.com https://internationalpubls.com X cana-736 173 1 [ [ X cana-736 173 2 11 11 NUM cana-736 173 3 ] ] X cana-736 173 4 mustafa mustafa PROPN cana-736 173 5 , , PUNCT cana-736 173 6 m.a m.a PROPN cana-736 173 7 . . PROPN cana-736 173 8 , , PUNCT cana-736 173 9 kadham kadham PROPN cana-736 173 10 , , PUNCT cana-736 173 11 s.m s.m PROPN cana-736 173 12 . . PROPN cana-736 173 13 , , PUNCT cana-736 173 14 abbass abbass PROPN cana-736 173 15 , , PUNCT cana-736 173 16 n.k n.k PROPN cana-736 173 17 . . PROPN cana-736 173 18 et et PROPN cana-736 173 19 al al PROPN cana-736 173 20 . . PUNCT cana-736 174 1 a a DET cana-736 174 2 novel novel ADJ cana-736 174 3 fuzzy fuzzy ADJ cana-736 174 4 m m NOUN cana-736 174 5 - - PUNCT cana-736 174 6 transform transform NOUN cana-736 174 7 technique technique NOUN cana-736 174 8 for for ADP cana-736 174 9 sustainable sustainable ADJ cana-736 174 10 ground ground NOUN cana-736 174 11 water water NOUN cana-736 174 12 level level NOUN cana-736 174 13 prediction prediction NOUN cana-736 174 14 . . PUNCT cana-736 175 1 appl appl PROPN cana-736 175 2 geomat geomat PROPN cana-736 175 3 16 16 NUM cana-736 175 4 , , PUNCT cana-736 175 5 9–15 9–15 PROPN cana-736 175 6 ( ( PUNCT cana-736 175 7 2024 2024 NUM cana-736 175 8 ) ) PUNCT cana-736 175 9 . . PUNCT cana-736 176 1 https://doi.org/10.1007/s12518-022-00486-4 https://doi.org/10.1007/s12518-022-00486-4 X cana-736 176 2 https://doi.org/10.1007/s12518-022-00486-4 https://doi.org/10.1007/s12518-022-00486-4 NUM