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PUNCT iajs-2509 8 1 keywords keyword NOUN iajs-2509 8 2 : : PUNCT iajs-2509 8 3 nonlinear nonlinear ADJ iajs-2509 8 4 second second ADJ iajs-2509 8 5 order order NOUN iajs-2509 8 6 sturm sturm NOUN iajs-2509 8 7 - - PUNCT iajs-2509 8 8 liouville liouville NOUN iajs-2509 8 9 problems problem NOUN iajs-2509 8 10 , , PUNCT iajs-2509 8 11 the the DET iajs-2509 8 12 least least ADJ iajs-2509 8 13 square square ADJ iajs-2509 8 14 method method NOUN iajs-2509 8 15 1.introduction 1.introduction NUM iajs-2509 8 16 the the DET iajs-2509 8 17 nonlinear nonlinear ADJ iajs-2509 8 18 delay delay NOUN iajs-2509 8 19 second second ADJ iajs-2509 8 20 order order NOUN iajs-2509 8 21 eigenvalue eigenvalue NOUN iajs-2509 8 22 problems problem NOUN iajs-2509 8 23 consist consist VERB iajs-2509 8 24 of of ADP iajs-2509 8 25 delay delay NOUN iajs-2509 8 26 nonlinear nonlinear ADJ iajs-2509 8 27 ordinary ordinary ADJ iajs-2509 8 28 differential differential ADJ iajs-2509 8 29 equations equation NOUN iajs-2509 8 30 with with ADP iajs-2509 8 31 the the DET iajs-2509 8 32 boundary boundary ADJ iajs-2509 8 33 conditions condition NOUN iajs-2509 8 34 defined define VERB iajs-2509 8 35 on on ADP iajs-2509 8 36 some some DET iajs-2509 8 37 intervals interval NOUN iajs-2509 8 38 , , PUNCT iajs-2509 8 39 this this DET iajs-2509 8 40 kind kind NOUN iajs-2509 8 41 of of ADP iajs-2509 8 42 equations equation NOUN iajs-2509 8 43 has have VERB iajs-2509 8 44 many many ADJ iajs-2509 8 45 applications application NOUN iajs-2509 8 46 in in ADP iajs-2509 8 47 different different ADJ iajs-2509 8 48 scientific scientific ADJ iajs-2509 8 49 fields field NOUN iajs-2509 8 50 , , PUNCT iajs-2509 8 51 such such ADJ iajs-2509 8 52 as as ADP iajs-2509 8 53 physical physical ADJ iajs-2509 8 54 , , PUNCT iajs-2509 8 55 biological biological ADJ iajs-2509 8 56 and and CCONJ iajs-2509 8 57 engineering engineering NOUN iajs-2509 8 58 science science NOUN iajs-2509 8 59 . . 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PUNCT iajs-2509 18 1 consider consider VERB iajs-2509 18 2 the the DET iajs-2509 18 3 n n ADV iajs-2509 18 4 - - PUNCT iajs-2509 18 5 th th VERB iajs-2509 18 6 order order NOUN iajs-2509 18 7 delay delay NOUN iajs-2509 18 8 differential differential ADJ iajs-2509 18 9 equation equation NOUN iajs-2509 18 10 : : PUNCT iajs-2509 19 1 e(k e(k NOUN iajs-2509 19 2 , , PUNCT iajs-2509 19 3 f(k f(k VERB iajs-2509 19 4 ) ) PUNCT iajs-2509 19 5 , , PUNCT iajs-2509 19 6 f(k f(k VERB iajs-2509 19 7 – – PUNCT iajs-2509 19 8 1), 1), NUM iajs-2509 19 9 , , PUNCT iajs-2509 19 10 f(k f(k ADJ iajs-2509 19 11 – – PUNCT iajs-2509 19 12 m m NOUN iajs-2509 19 13 ) ) PUNCT iajs-2509 19 14 , , PUNCT iajs-2509 19 15 f f X iajs-2509 19 16 ( ( PUNCT iajs-2509 19 17 k k NOUN iajs-2509 19 18 ) ) PUNCT iajs-2509 19 19 , , PUNCT iajs-2509 19 20 f f X iajs-2509 19 21 ( ( PUNCT iajs-2509 19 22 k k X iajs-2509 19 23 – – PUNCT iajs-2509 19 24 1), 1), NUM iajs-2509 19 25 , , PUNCT iajs-2509 19 26 f f X iajs-2509 19 27 ( ( PUNCT iajs-2509 19 28 kx kx PROPN iajs-2509 19 29 – – PUNCT iajs-2509 19 30 m m NOUN iajs-2509 19 31 ) ) PUNCT iajs-2509 19 32 , , PUNCT iajs-2509 19 33 f(n f(n PROPN iajs-2509 19 34 ) ) PUNCT iajs-2509 19 35 ( ( PUNCT iajs-2509 19 36 k), k), PROPN iajs-2509 19 37 , , PUNCT iajs-2509 19 38 f(n f(n PROPN iajs-2509 19 39 ) ) PUNCT iajs-2509 19 40 ( ( PUNCT iajs-2509 19 41 k k X iajs-2509 19 42 – – PUNCT iajs-2509 19 43 m m NOUN iajs-2509 19 44 ) ) PUNCT iajs-2509 19 45 ) ) PUNCT iajs-2509 20 1 = = SYM iajs-2509 20 2 h(k h(k PROPN iajs-2509 20 3 ) ) PUNCT iajs-2509 20 4 , , PUNCT iajs-2509 21 1 k k PROPN iajs-2509 21 2   NOUN iajs-2509 21 3 [ [ X iajs-2509 21 4 a a DET iajs-2509 21 5 , , PUNCT iajs-2509 21 6 b b NOUN iajs-2509 21 7 ] ] X iajs-2509 21 8 ( ( PUNCT iajs-2509 21 9 1 1 X iajs-2509 21 10 ) ) PUNCT iajs-2509 21 11 where where SCONJ iajs-2509 21 12 e e NOUN iajs-2509 21 13 is be AUX iajs-2509 21 14 a a DET iajs-2509 21 15 given give VERB iajs-2509 21 16 function function NOUN iajs-2509 21 17 and and CCONJ iajs-2509 21 18 1 1 NOUN iajs-2509 21 19 , , PUNCT iajs-2509 21 20 2 2 PROPN iajs-2509 21 21 , , PUNCT iajs-2509 21 22   PROPN iajs-2509 21 23 , , PUNCT iajs-2509 21 24 m m PROPN iajs-2509 21 25 are be AUX iajs-2509 21 26 given give VERB iajs-2509 21 27 fixed fix VERB iajs-2509 21 28 positive positive ADJ iajs-2509 21 29 numbers number NOUN iajs-2509 21 30 called call VERB iajs-2509 21 31 the the DET iajs-2509 21 32 time time NOUN iajs-2509 21 33 delays delay NOUN iajs-2509 21 34 , , PUNCT iajs-2509 21 35 [ [ X iajs-2509 21 36 1 1 NUM iajs-2509 21 37 ] ] PUNCT iajs-2509 21 38 . . 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PUNCT iajs-2509 24 1 hence hence ADV iajs-2509 24 2 , , PUNCT iajs-2509 24 3 the the DET iajs-2509 24 4 new new ADJ iajs-2509 24 5 concepts concept NOUN iajs-2509 24 6 of of ADP iajs-2509 24 7 this this DET iajs-2509 24 8 work work NOUN iajs-2509 24 9 are be AUX iajs-2509 24 10 given give VERB iajs-2509 24 11 by by ADP iajs-2509 24 12 the the DET iajs-2509 24 13 following follow VERB iajs-2509 24 14 definition definition NOUN iajs-2509 24 15 . . PUNCT iajs-2509 25 1 definition definition NOUN iajs-2509 25 2 2.3 2.3 NUM iajs-2509 25 3 the the DET iajs-2509 25 4 delay delay NOUN iajs-2509 25 5 eigen eigen NOUN iajs-2509 25 6 - - PUNCT iajs-2509 25 7 value value NOUN iajs-2509 25 8 problem problem NOUN iajs-2509 25 9 consist consist NOUN iajs-2509 25 10 of of ADP iajs-2509 25 11 delay delay NOUN iajs-2509 25 12 ordinary ordinary ADJ iajs-2509 25 13 differential differential ADJ iajs-2509 25 14 equation equation NOUN iajs-2509 25 15 is be AUX iajs-2509 25 16 said say VERB iajs-2509 25 17 to to PART iajs-2509 25 18 be be AUX iajs-2509 25 19 nonlinear nonlinear ADJ iajs-2509 25 20 when when SCONJ iajs-2509 25 21 it it PRON iajs-2509 25 22 is be AUX iajs-2509 25 23 nonlinear nonlinear ADJ iajs-2509 25 24 with with ADP iajs-2509 25 25 respect respect NOUN iajs-2509 25 26 to to ADP iajs-2509 25 27 the the DET iajs-2509 25 28 unknown unknown ADJ iajs-2509 25 29 eigen eigen NOUN iajs-2509 25 30 - - PUNCT iajs-2509 25 31 function function NOUN iajs-2509 25 32 enter enter VERB iajs-2509 25 33 with with ADP iajs-2509 25 34 different different ADJ iajs-2509 25 35 arguments argument NOUN iajs-2509 25 36 and and CCONJ iajs-2509 25 37 their their PRON iajs-2509 25 38 derivatives derivative NOUN iajs-2509 25 39 that that PRON iajs-2509 25 40 appeared appear VERB iajs-2509 25 41 in in ADP iajs-2509 25 42 it it PRON iajs-2509 25 43 . . PUNCT iajs-2509 26 1 next next ADV iajs-2509 26 2 , , PUNCT iajs-2509 26 3 consider consider VERB iajs-2509 26 4 the the DET iajs-2509 26 5 following follow VERB iajs-2509 26 6 nonlinear nonlinear ADJ iajs-2509 26 7 delay delay NOUN iajs-2509 26 8 second second ADJ iajs-2509 26 9 order order NOUN iajs-2509 26 10 eigen eigen NOUN iajs-2509 26 11 - - PUNCT iajs-2509 26 12 value value NOUN iajs-2509 26 13 problem problem NOUN iajs-2509 26 14 : : PUNCT iajs-2509 26 15 – – PUNCT iajs-2509 26 16 ( ( PUNCT iajs-2509 26 17 p(k)f'(k p(k)f'(k X iajs-2509 26 18 ) ) PUNCT iajs-2509 26 19 ) ) PUNCT iajs-2509 26 20 ' ' PART iajs-2509 27 1 + + CCONJ iajs-2509 27 2 q(k)f(k– q(k)f(k– NOUN iajs-2509 27 3 ) ) PUNCT iajs-2509 27 4 – – PUNCT iajs-2509 27 5 h(k h(k PROPN iajs-2509 27 6 , , PUNCT iajs-2509 27 7   X iajs-2509 27 8 , , PUNCT iajs-2509 27 9 f(k– f(k– NOUN iajs-2509 27 10 ) ) PUNCT iajs-2509 27 11 ) ) PUNCT iajs-2509 28 1 = = SYM iajs-2509 28 2 0 0 PUNCT iajs-2509 28 3 ( ( PUNCT iajs-2509 28 4 2 2 NUM iajs-2509 28 5 ) ) PUNCT iajs-2509 28 6 with with ADP iajs-2509 28 7 the the DET iajs-2509 28 8 associated associated ADJ iajs-2509 28 9 conditions condition NOUN iajs-2509 28 10 : : PUNCT iajs-2509 28 11 a1f(a a1f(a PROPN iajs-2509 28 12 ) ) PUNCT iajs-2509 29 1 + + CCONJ iajs-2509 29 2 a2f a2f X iajs-2509 29 3 ' ' PUNCT iajs-2509 29 4 ( ( PUNCT iajs-2509 29 5 a a X iajs-2509 29 6 ) ) PUNCT iajs-2509 29 7 = = SYM iajs-2509 29 8 0 0 NUM iajs-2509 29 9 , , PUNCT iajs-2509 29 10 k k PROPN iajs-2509 29 11   PROPN iajs-2509 29 12 [ [ PUNCT iajs-2509 29 13 a– a– PROPN iajs-2509 29 14 , , PUNCT iajs-2509 29 15 a a PRON iajs-2509 29 16 ] ] X iajs-2509 29 17 ( ( PUNCT iajs-2509 29 18 3 3 X iajs-2509 29 19 ) ) PUNCT iajs-2509 29 20 b1f(b b1f(b PROPN iajs-2509 29 21 ) ) PUNCT iajs-2509 30 1 + + CCONJ iajs-2509 30 2 b2f b2f PRON iajs-2509 30 3 ' ' PUNCT iajs-2509 30 4 ( ( PUNCT iajs-2509 30 5 b b X iajs-2509 30 6 ) ) PUNCT iajs-2509 30 7 = = SYM iajs-2509 30 8 0 0 NUM iajs-2509 30 9 , , PUNCT iajs-2509 30 10 k k X iajs-2509 30 11   PROPN iajs-2509 30 12 [ [ PUNCT iajs-2509 30 13 b– b– ADV iajs-2509 30 14 , , PUNCT iajs-2509 30 15 b b X iajs-2509 30 16 ] ] X iajs-2509 30 17   NOUN iajs-2509 30 18   SYM iajs-2509 30 19   NOUN iajs-2509 31 1 , , PROPN iajs-2509 31 2   NOUN iajs-2509 31 3 kkf kkf NOUN iajs-2509 31 4 if if SCONJ iajs-2509 31 5 ak ak PROPN iajs-2509 31 6   VERB iajs-2509 31 7 where where SCONJ iajs-2509 31 8 , , PUNCT iajs-2509 31 9 a1 a1 PROPN iajs-2509 31 10 , , PUNCT iajs-2509 31 11 a2 a2 PROPN iajs-2509 31 12 , , PUNCT iajs-2509 31 13 b1 b1 NOUN iajs-2509 31 14 , , PUNCT iajs-2509 31 15 b2 b2 NOUN iajs-2509 31 16 , , PUNCT iajs-2509 31 17 p p X iajs-2509 31 18 , , PUNCT iajs-2509 31 19 p p X iajs-2509 31 20 ' ' PUNCT iajs-2509 31 21 and and CCONJ iajs-2509 31 22 q q NOUN iajs-2509 31 23 are be AUX iajs-2509 31 24 given give VERB iajs-2509 31 25 real real ADJ iajs-2509 31 26 continuous continuous ADJ iajs-2509 31 27 functions function NOUN iajs-2509 31 28 defined define VERB iajs-2509 31 29 on on ADP iajs-2509 31 30 the the DET iajs-2509 31 31 interval interval NOUN iajs-2509 31 32 [ [ X iajs-2509 31 33 a a X iajs-2509 31 34 , , PUNCT iajs-2509 31 35 b b NOUN iajs-2509 31 36 ] ] X iajs-2509 31 37 , , PUNCT iajs-2509 31 38 p p NOUN iajs-2509 31 39 is be AUX iajs-2509 31 40 positive positive ADJ iajs-2509 31 41 , , PUNCT iajs-2509 31 42 not not PART iajs-2509 31 43 both both DET iajs-2509 31 44 coefficients coefficient NOUN iajs-2509 31 45 in in ADP iajs-2509 31 46 one one NUM iajs-2509 31 47 condition condition NOUN iajs-2509 31 48 are be AUX iajs-2509 31 49 zero, zero, PROPN iajs-2509 31 50 > > X iajs-2509 31 51 0 0 NUM iajs-2509 31 52 is be AUX iajs-2509 31 53 the the DET iajs-2509 31 54 time time NOUN iajs-2509 31 55 delay delay NOUN iajs-2509 31 56 , , PUNCT iajs-2509 31 57 h h NOUN iajs-2509 31 58 is be AUX iajs-2509 31 59 a a DET iajs-2509 31 60 well well ADV iajs-2509 31 61 - - PUNCT iajs-2509 31 62 defined define VERB iajs-2509 31 63 nonlinear nonlinear ADJ iajs-2509 31 64 function function NOUN iajs-2509 31 65 with with ADP iajs-2509 31 66 respect respect NOUN iajs-2509 31 67 to to ADP iajs-2509 31 68 f. f. PROPN iajs-2509 31 69   PROPN iajs-2509 31 70 is be AUX iajs-2509 31 71 the the DET iajs-2509 31 72 initial initial ADJ iajs-2509 31 73 function function NOUN iajs-2509 31 74 defined define VERB iajs-2509 31 75 on on ADP iajs-2509 31 76 ] ] PUNCT iajs-2509 31 77 . . PUNCT iajs-2509 31 78 , , PUNCT iajs-2509 31 79 [ [ PUNCT iajs-2509 31 80 00 00 NUM iajs-2509 31 81 kkk kkk NOUN iajs-2509 31 82   NOUN iajs-2509 32 1 the the DET iajs-2509 32 2 problem problem NOUN iajs-2509 32 3 here here ADV iajs-2509 32 4 is be AUX iajs-2509 32 5 to to PART iajs-2509 32 6 determine determine VERB iajs-2509 32 7 the the DET iajs-2509 32 8 eigen eigen NOUN iajs-2509 32 9 - - PUNCT iajs-2509 32 10 value value NOUN iajs-2509 32 11   X iajs-2509 32 12 in in ADP iajs-2509 32 13 which which PRON iajs-2509 32 14 a a DET iajs-2509 32 15 nontrivial nontrivial ADJ iajs-2509 32 16 solution solution NOUN iajs-2509 32 17 f f PROPN iajs-2509 32 18 for for ADP iajs-2509 32 19 the the DET iajs-2509 32 20 problem problem NOUN iajs-2509 32 21 given give VERB iajs-2509 32 22 by by ADP iajs-2509 32 23 equations equation NOUN iajs-2509 32 24 ( ( PUNCT iajs-2509 32 25 2 2 NUM iajs-2509 32 26 ) ) PUNCT iajs-2509 32 27 & & CCONJ iajs-2509 32 28 ( ( PUNCT iajs-2509 32 29 3 3 X iajs-2509 32 30 ) ) PUNCT iajs-2509 32 31 occurs occur VERB iajs-2509 32 32 . . PUNCT iajs-2509 33 1 in in ADP iajs-2509 33 2 this this DET iajs-2509 33 3 case case NOUN iajs-2509 33 4   NOUN iajs-2509 33 5 is be AUX iajs-2509 33 6 said say VERB iajs-2509 33 7 to to PART iajs-2509 33 8 be be AUX iajs-2509 33 9 a a DET iajs-2509 33 10 delay delay NOUN iajs-2509 33 11 eigenvalue eigenvalue NOUN iajs-2509 33 12 and and CCONJ iajs-2509 33 13 f f PROPN iajs-2509 33 14 is be AUX iajs-2509 33 15 the the DET iajs-2509 33 16 associated associated ADJ iajs-2509 33 17 delay delay NOUN iajs-2509 33 18 eigen eigen X iajs-2509 33 19 - - PUNCT iajs-2509 33 20 function function NOUN iajs-2509 33 21 . . PUNCT iajs-2509 34 1 in in ADP iajs-2509 34 2 other other ADJ iajs-2509 34 3 words word NOUN iajs-2509 34 4 f f X iajs-2509 34 5 is be AUX iajs-2509 34 6 an an DET iajs-2509 34 7 eigen eigen NOUN iajs-2509 34 8 function function NOUN iajs-2509 34 9 for for ADP iajs-2509 34 10 the the DET iajs-2509 34 11 variable variable ADJ iajs-2509 34 12 k k PROPN iajs-2509 34 13 and and CCONJ iajs-2509 34 14 the the DET iajs-2509 34 15 nonlinear nonlinear ADJ iajs-2509 34 16 function function NOUN iajs-2509 34 17 h(k h(k PROPN iajs-2509 34 18 , , PUNCT iajs-2509 34 19   X iajs-2509 34 20 , , PUNCT iajs-2509 34 21 f(k– f(k– NOUN iajs-2509 34 22 ) ) PUNCT iajs-2509 34 23 ) ) PUNCT iajs-2509 34 24 with with ADP iajs-2509 34 25 respect respect NOUN iajs-2509 34 26 to to ADP iajs-2509 34 27 the the DET iajs-2509 34 28 eigen eigen NOUN iajs-2509 34 29 - - PUNCT iajs-2509 34 30 value value NOUN iajs-2509 34 31   NOUN iajs-2509 34 32 . . PUNCT iajs-2509 35 1 like like ADP iajs-2509 35 2 the the DET iajs-2509 35 3 linear linear ADJ iajs-2509 35 4 second second ADJ iajs-2509 35 5 order order NOUN iajs-2509 35 6 eigenvalue eigenvalue NOUN iajs-2509 35 7 problems problem NOUN iajs-2509 35 8 , , PUNCT iajs-2509 35 9 the the DET iajs-2509 35 10 problem problem NOUN iajs-2509 35 11 given give VERB iajs-2509 35 12 by by ADP iajs-2509 35 13 equations equation NOUN iajs-2509 35 14 ( ( PUNCT iajs-2509 35 15 2 2 NUM iajs-2509 35 16 ) ) PUNCT iajs-2509 35 17 & & CCONJ iajs-2509 35 18 ( ( PUNCT iajs-2509 35 19 3 3 X iajs-2509 35 20 ) ) PUNCT iajs-2509 35 21 satisfies satisfy VERB iajs-2509 35 22 the the DET iajs-2509 35 23 following follow VERB iajs-2509 35 24 remarks remark NOUN iajs-2509 35 25 , , PUNCT iajs-2509 35 26 [ [ X iajs-2509 35 27 2 2 NUM iajs-2509 35 28 ] ] PUNCT iajs-2509 35 29 . . PUNCT iajs-2509 35 30     SPACE iajs-2509 36 1 61 61 NUM iajs-2509 36 2     SPACE iajs-2509 36 3 ibn ibn PROPN iajs-2509 36 4 al al PROPN iajs-2509 36 5 - - PUNCT iajs-2509 36 6 haitham haitham PROPN iajs-2509 36 7 jour jour X iajs-2509 36 8 . . PROPN iajs-2509 36 9 for for ADP iajs-2509 36 10 pure pure ADJ iajs-2509 36 11 & & CCONJ iajs-2509 36 12 appl appl PROPN iajs-2509 36 13 . . PUNCT iajs-2509 37 1 sci sci PROPN iajs-2509 37 2 . . PROPN iajs-2509 38 1 33 33 NUM iajs-2509 38 2 ( ( PUNCT iajs-2509 38 3 4 4 NUM iajs-2509 38 4 ) ) PUNCT iajs-2509 38 5 2020 2020 NUM iajs-2509 38 6 remarks remark VERB iajs-2509 38 7 2.4 2.4 NUM iajs-2509 38 8 1 1 NUM iajs-2509 38 9 . . PUNCT iajs-2509 39 1 the the DET iajs-2509 39 2 linear linear ADJ iajs-2509 39 3 delay delay NOUN iajs-2509 39 4 operator operator NOUN iajs-2509 39 5 : : PUNCT iajs-2509 39 6 ) ) PUNCT iajs-2509 39 7 , , PUNCT iajs-2509 39 8 ( ( PUNCT iajs-2509 39 9 ) ) PUNCT iajs-2509 39 10 ( ( PUNCT iajs-2509 39 11 ) ) PUNCT iajs-2509 39 12 ( ( PUNCT iajs-2509 39 13 ) ) PUNCT iajs-2509 39 14 ( ( PUNCT iajs-2509 39 15 2 2 NUM iajs-2509 39 16 2 2 NUM iajs-2509 39 17 kqkakp kqkakp NOUN iajs-2509 39 18 dk dk PROPN iajs-2509 40 1 d d X iajs-2509 40 2 kp kp PROPN iajs-2509 40 3 dk dk PROPN iajs-2509 40 4 d d PROPN iajs-2509 40 5 l l PROPN iajs-2509 40 6   PROPN iajs-2509 40 7 where where SCONJ iajs-2509 40 8 a(x a(x NOUN iajs-2509 40 9 ) ) PUNCT iajs-2509 40 10 is be AUX iajs-2509 40 11 an an DET iajs-2509 40 12 operator operator NOUN iajs-2509 40 13 defined define VERB iajs-2509 40 14 by by ADP iajs-2509 40 15   NOUN iajs-2509 41 1   PROPN iajs-2509 41 2   NOUN iajs-2509 41 3   SYM iajs-2509 41 4   NOUN iajs-2509 41 5   NUM iajs-2509 41 6 kfkfka kfkfka VERB iajs-2509 41 7 , , PUNCT iajs-2509 41 8 is be AUX iajs-2509 41 9 selfadjoint selfadjoint NOUN iajs-2509 41 10 . . PUNCT iajs-2509 42 1 2 2 X iajs-2509 42 2 . . X iajs-2509 42 3 the the DET iajs-2509 42 4 delay delay NOUN iajs-2509 42 5 eigen eigen PROPN iajs-2509 42 6 - - PUNCT iajs-2509 42 7 functions function NOUN iajs-2509 42 8 are be AUX iajs-2509 42 9 orthogonal orthogonal ADJ iajs-2509 42 10 . . PUNCT iajs-2509 43 1 3 3 X iajs-2509 43 2 . . X iajs-2509 43 3 there there PRON iajs-2509 43 4 are be VERB iajs-2509 43 5 infinite infinite ADJ iajs-2509 43 6 number number NOUN iajs-2509 43 7 of of ADP iajs-2509 43 8 delay delay NOUN iajs-2509 43 9 eigenvalues eigenvalue VERB iajs-2509 43 10 forming form VERB iajs-2509 43 11 a a DET iajs-2509 43 12 monotone monotone NOUN iajs-2509 43 13 increasing increase VERB iajs-2509 43 14 sequence sequence NOUN iajs-2509 43 15 with with ADP iajs-2509 43 16 j j NOUN iajs-2509 43 17   NOUN iajs-2509 43 18   VERB iajs-2509 43 19 as as ADP iajs-2509 43 20 j j NOUN iajs-2509 43 21 . . VERB iajs-2509 43 22 moreover moreover ADV iajs-2509 43 23 , , PUNCT iajs-2509 43 24 the the DET iajs-2509 43 25 delay delay NOUN iajs-2509 43 26 eigen eigen NOUN iajs-2509 43 27 - - PUNCT iajs-2509 43 28 functions function NOUN iajs-2509 43 29 corresponding correspond VERB iajs-2509 43 30 to to ADP iajs-2509 43 31 the the DET iajs-2509 43 32 delay delay NOUN iajs-2509 43 33 eigenvalues eigenvalue NOUN iajs-2509 43 34 has have VERB iajs-2509 43 35 exactly exactly ADV iajs-2509 43 36 j j PROPN iajs-2509 43 37 roots root NOUN iajs-2509 43 38 on on ADP iajs-2509 43 39 the the DET iajs-2509 43 40 interval interval NOUN iajs-2509 43 41 ( ( PUNCT iajs-2509 43 42 a a DET iajs-2509 43 43 , , PUNCT iajs-2509 43 44 b b NOUN iajs-2509 43 45 ) ) PUNCT iajs-2509 43 46 . . PUNCT iajs-2509 44 1 4 4 X iajs-2509 44 2 . . X iajs-2509 44 3 the the DET iajs-2509 44 4 delay delay NOUN iajs-2509 44 5 eigen eigen PROPN iajs-2509 44 6 - - PUNCT iajs-2509 44 7 functions function NOUN iajs-2509 44 8 are be AUX iajs-2509 44 9 complete complete ADJ iajs-2509 44 10 and and CCONJ iajs-2509 44 11 normal normal ADJ iajs-2509 44 12 in in ADP iajs-2509 44 13 l2[a l2[a PROPN iajs-2509 44 14 , , PUNCT iajs-2509 44 15 b b NOUN iajs-2509 44 16 ] ] X iajs-2509 44 17 . . PUNCT iajs-2509 45 1 5 5 X iajs-2509 45 2 . . X iajs-2509 45 3 each each DET iajs-2509 45 4 delay delay NOUN iajs-2509 45 5 eigenvalue eigenvalue NOUN iajs-2509 45 6 corresponds correspond VERB iajs-2509 45 7 only only ADV iajs-2509 45 8 one one NUM iajs-2509 45 9 delay delay NOUN iajs-2509 45 10 eigen eigen X iajs-2509 45 11 - - PUNCT iajs-2509 45 12 function function NOUN iajs-2509 45 13 in in ADP iajs-2509 45 14 l2[a l2[a PROPN iajs-2509 45 15 , , PUNCT iajs-2509 45 16 b b NOUN iajs-2509 45 17 ] ] PUNCT iajs-2509 45 18 . . PUNCT iajs-2509 46 1 to to PART iajs-2509 46 2 check check VERB iajs-2509 46 3 remarks remark NOUN iajs-2509 46 4 ( ( PUNCT iajs-2509 46 5 2 2 NUM iajs-2509 46 6 - - SYM iajs-2509 46 7 4 4 NUM iajs-2509 46 8 ) ) PUNCT iajs-2509 46 9 , , PUNCT iajs-2509 46 10 see see VERB iajs-2509 46 11 [ [ X iajs-2509 46 12 3,4 3,4 NUM iajs-2509 46 13 ] ] PUNCT iajs-2509 46 14 . . PUNCT iajs-2509 47 1 3 3 X iajs-2509 47 2 . . X iajs-2509 47 3 the the DET iajs-2509 47 4 least least ADJ iajs-2509 47 5 - - PUNCT iajs-2509 47 6 square square NOUN iajs-2509 47 7 method method NOUN iajs-2509 47 8 this this DET iajs-2509 47 9 method method NOUN iajs-2509 47 10 is be AUX iajs-2509 47 11 one one NUM iajs-2509 47 12 of of ADP iajs-2509 47 13 the the DET iajs-2509 47 14 expansion expansion NOUN iajs-2509 47 15 methods method NOUN iajs-2509 47 16 that that PRON iajs-2509 47 17 used use VERB iajs-2509 47 18 to to PART iajs-2509 47 19 solve solve VERB iajs-2509 47 20 the the DET iajs-2509 47 21 linear linear ADJ iajs-2509 47 22 ( ( PUNCT iajs-2509 47 23 nonlinear nonlinear ADJ iajs-2509 47 24 ) ) PUNCT iajs-2509 47 25 differential differential ADJ iajs-2509 47 26 equations equation NOUN iajs-2509 47 27 and and CCONJ iajs-2509 47 28 equations equation NOUN iajs-2509 47 29 with with ADP iajs-2509 47 30 or or CCONJ iajs-2509 47 31 without without ADP iajs-2509 47 32 delays delay NOUN iajs-2509 47 33 , , PUNCT iajs-2509 47 34 [ [ X iajs-2509 47 35 5 5 NUM iajs-2509 47 36 , , PUNCT iajs-2509 47 37 6 6 NUM iajs-2509 47 38 ] ] PUNCT iajs-2509 47 39 . . PUNCT iajs-2509 48 1 here here ADV iajs-2509 48 2 we we PRON iajs-2509 48 3 develop develop VERB iajs-2509 48 4 this this DET iajs-2509 48 5 method method NOUN iajs-2509 48 6 to to PART iajs-2509 48 7 solve solve VERB iajs-2509 48 8 the the DET iajs-2509 48 9 problem problem NOUN iajs-2509 48 10 given give VERB iajs-2509 48 11 by by ADP iajs-2509 48 12 equations equation NOUN iajs-2509 48 13 ( ( PUNCT iajs-2509 48 14 2 2 NUM iajs-2509 48 15 ) ) PUNCT iajs-2509 48 16 & & CCONJ iajs-2509 48 17 ( ( PUNCT iajs-2509 48 18 3 3 NUM iajs-2509 48 19 ) ) PUNCT iajs-2509 48 20 . . PUNCT iajs-2509 49 1 the the DET iajs-2509 49 2 method method NOUN iajs-2509 49 3 is be AUX iajs-2509 49 4 based base VERB iajs-2509 49 5 on on ADP iajs-2509 49 6 approximating approximate VERB iajs-2509 49 7 the the DET iajs-2509 49 8 unknown unknown ADJ iajs-2509 49 9 function function NOUN iajs-2509 49 10 f f PROPN iajs-2509 49 11 as as ADP iajs-2509 49 12 a a DET iajs-2509 49 13 linear linear ADJ iajs-2509 49 14 combination combination NOUN iajs-2509 49 15 of of ADP iajs-2509 49 16 n n CCONJ iajs-2509 49 17 linearly linearly ADV iajs-2509 49 18 independent independent ADJ iajs-2509 49 19 functions function NOUN iajs-2509 49 20 { { PUNCT iajs-2509 49 21 i}n i}n NOUN iajs-2509 49 22 i=1 i=1 PROPN iajs-2509 49 23 , , PUNCT iajs-2509 49 24 that that PRON iajs-2509 49 25 is be AUX iajs-2509 49 26 write write ADJ iajs-2509 49 27 : : PUNCT iajs-2509 49 28 ) ) PUNCT iajs-2509 49 29 ( ( PUNCT iajs-2509 49 30 1 1 NUM iajs-2509 49 31 kf kf PROPN iajs-2509 49 32 n n PROPN iajs-2509 49 33 i i PRON iajs-2509 49 34 i i PROPN iajs-2509 50 1   PROPN iajs-2509 50 2   ADJ iajs-2509 50 3   NOUN iajs-2509 50 4 ( ( PUNCT iajs-2509 50 5 4 4 NUM iajs-2509 50 6 ) ) PUNCT iajs-2509 50 7 which which PRON iajs-2509 50 8 implies imply VERB iajs-2509 50 9 that that PRON iajs-2509 50 10 , , PUNCT iajs-2509 50 11 ) ) PUNCT iajs-2509 50 12 ( ( PUNCT iajs-2509 50 13 ) ) PUNCT iajs-2509 50 14 ( ( PUNCT iajs-2509 50 15 1 1 NUM iajs-2509 50 16   SYM iajs-2509 50 17   NOUN iajs-2509 50 18   X iajs-2509 50 19   NUM iajs-2509 50 20 kkf kkf NOUN iajs-2509 51 1 n n CCONJ iajs-2509 51 2 i i PRON iajs-2509 52 1 i i PRON iajs-2509 52 2 this this DET iajs-2509 52 3 approximated approximate VERB iajs-2509 52 4 solution solution NOUN iajs-2509 52 5 must must AUX iajs-2509 52 6 satisfy satisfy VERB iajs-2509 52 7 the the DET iajs-2509 52 8 boundary boundary ADJ iajs-2509 52 9 conditions condition NOUN iajs-2509 52 10 given give VERB iajs-2509 52 11 by by ADP iajs-2509 52 12 equations equation NOUN iajs-2509 52 13 ( ( PUNCT iajs-2509 52 14 3 3 NUM iajs-2509 52 15 ) ) PUNCT iajs-2509 52 16 to to PART iajs-2509 52 17 get get VERB iajs-2509 52 18 a a DET iajs-2509 52 19 new new ADJ iajs-2509 52 20 approximated approximate VERB iajs-2509 52 21 solution solution NOUN iajs-2509 52 22 . . PUNCT iajs-2509 53 1 by by ADP iajs-2509 53 2 substituting substitute VERB iajs-2509 53 3 this this DET iajs-2509 53 4 approximated approximate VERB iajs-2509 53 5 solution solution NOUN iajs-2509 53 6 into into ADP iajs-2509 53 7 equation equation NOUN iajs-2509 53 8 ( ( PUNCT iajs-2509 53 9 2 2 X iajs-2509 53 10 ) ) PUNCT iajs-2509 53 11 one one NOUN iajs-2509 53 12 can can AUX iajs-2509 53 13 get get VERB iajs-2509 53 14 : : PUNCT iajs-2509 53 15 ) ) PUNCT iajs-2509 53 16 ) ) PUNCT iajs-2509 53 17 ( ( PUNCT iajs-2509 53 18 , , PUNCT iajs-2509 53 19 , , PUNCT iajs-2509 53 20 ( ( PUNCT iajs-2509 53 21 ) ) PUNCT iajs-2509 53 22 ( ( PUNCT iajs-2509 53 23 ) ) PUNCT iajs-2509 53 24 ( ( PUNCT iajs-2509 53 25 ) ) PUNCT iajs-2509 53 26 ) ) PUNCT iajs-2509 53 27 ( ( PUNCT iajs-2509 53 28 ) ) PUNCT iajs-2509 53 29 ( ( PUNCT iajs-2509 53 30 ( ( PUNCT iajs-2509 53 31 ) ) PUNCT iajs-2509 53 32 , , PUNCT iajs-2509 53 33 , , PUNCT iajs-2509 53 34 ( ( PUNCT iajs-2509 53 35 111 111 NUM iajs-2509 53 36   NOUN iajs-2509 53 37   NOUN iajs-2509 54 1   PROPN iajs-2509 54 2   NOUN iajs-2509 54 3 kkhkkqkkpckr kkhkkqkkpckr NOUN iajs-2509 54 4 n n CCONJ iajs-2509 54 5 i i PRON iajs-2509 54 6 i i PRON iajs-2509 54 7 n n VERB iajs-2509 54 8 i i PRON iajs-2509 54 9 i i PRON iajs-2509 54 10 n n VERB iajs-2509 54 11 i i PRON iajs-2509 54 12 i i PRON iajs-2509 54 13   X iajs-2509 54 14 ( ( PUNCT iajs-2509 54 15 5 5 NUM iajs-2509 54 16 ) ) PUNCT iajs-2509 54 17 where where SCONJ iajs-2509 54 18 r r NOUN iajs-2509 54 19 is be AUX iajs-2509 54 20 the the DET iajs-2509 54 21 error error NOUN iajs-2509 54 22 in in ADP iajs-2509 54 23 the the DET iajs-2509 54 24 approximation approximation NOUN iajs-2509 54 25 of of ADP iajs-2509 54 26 equation equation NOUN iajs-2509 54 27 ( ( PUNCT iajs-2509 54 28 2 2 NUM iajs-2509 54 29 ) ) PUNCT iajs-2509 54 30 and and CCONJ iajs-2509 54 31 c c PROPN iajs-2509 54 32   NOUN iajs-2509 54 33 is be AUX iajs-2509 54 34 the the DET iajs-2509 54 35 vector vector NOUN iajs-2509 54 36 of of ADP iajs-2509 54 37 n–2 n–2 PROPN iajs-2509 54 38 elements element NOUN iajs-2509 54 39 of of ADP iajs-2509 54 40 ci ci NOUN iajs-2509 54 41 , , PUNCT iajs-2509 54 42 i=1,2,,n i=1,2,,n PROPN iajs-2509 54 43 , , PUNCT iajs-2509 54 44 [ [ X iajs-2509 54 45 7 7 NUM iajs-2509 54 46 - - SYM iajs-2509 54 47 8 8 NUM iajs-2509 54 48 ] ] PUNCT iajs-2509 54 49 thus thus ADV iajs-2509 54 50 , , PUNCT iajs-2509 54 51 to to PART iajs-2509 54 52 minimize minimize VERB iajs-2509 54 53 the the DET iajs-2509 54 54 functional functional ADJ iajs-2509 54 55 : : PUNCT iajs-2509 54 56   PROPN iajs-2509 54 57 b b NOUN iajs-2509 54 58 a a DET iajs-2509 54 59 dkckrcj dkckrcj NOUN iajs-2509 54 60 2 2 NUM iajs-2509 54 61 ) ) PUNCT iajs-2509 54 62 ) ) PUNCT iajs-2509 54 63 , , PUNCT iajs-2509 54 64 , , PUNCT iajs-2509 54 65 ( ( PUNCT iajs-2509 54 66 ( ( PUNCT iajs-2509 54 67 ) ) PUNCT iajs-2509 54 68 , , PUNCT iajs-2509 54 69 ( ( PUNCT iajs-2509 54 70   ADV iajs-2509 54 71   NOUN iajs-2509 54 72 ( ( PUNCT iajs-2509 54 73 6 6 NUM iajs-2509 54 74 ) ) PUNCT iajs-2509 54 75 put put VERB iajs-2509 54 76 0 0 NUM iajs-2509 54 77   ADJ iajs-2509 54 78   ADJ iajs-2509 54 79   PROPN iajs-2509 54 80   PROPN iajs-2509 54 81   PROPN iajs-2509 54 82 ic ic PROPN iajs-2509 54 83 jj jj PROPN iajs-2509 54 84   PROPN iajs-2509 54 85 , , PUNCT iajs-2509 54 86 i i PRON iajs-2509 54 87 = = NOUN iajs-2509 54 88 1,2,,n 1,2,,n NUM iajs-2509 54 89 , , PUNCT iajs-2509 54 90 to to PART iajs-2509 54 91 get get VERB iajs-2509 54 92 a a DET iajs-2509 54 93 system system NOUN iajs-2509 54 94 of of ADP iajs-2509 54 95 n–1 n–1 PROPN iajs-2509 54 96 nonlinear nonlinear ADJ iajs-2509 54 97 equations equation NOUN iajs-2509 54 98 with with ADP iajs-2509 54 99 n–1 n–1 PRON iajs-2509 54 100 unknowns unknown NOUN iajs-2509 54 101 which which PRON iajs-2509 54 102 can can AUX iajs-2509 54 103 be be AUX iajs-2509 54 104 solved solve VERB iajs-2509 54 105 by by ADP iajs-2509 54 106 any any DET iajs-2509 54 107 suitable suitable ADJ iajs-2509 54 108 method method NOUN iajs-2509 54 109 to to PART iajs-2509 54 110 get get VERB iajs-2509 54 111 the the DET iajs-2509 54 112 values value NOUN iajs-2509 54 113 of of ADP iajs-2509 54 114   ADJ iajs-2509 54 115 and and CCONJ iajs-2509 54 116 c c NOUN iajs-2509 54 117   NOUN iajs-2509 54 118 , , PUNCT iajs-2509 54 119 [ [ X iajs-2509 54 120 9,10 9,10 NUM iajs-2509 54 121 ] ] X iajs-2509 54 122 . . PUNCT iajs-2509 55 1 to to PART iajs-2509 55 2 check check VERB iajs-2509 55 3 this this DET iajs-2509 55 4 method method NOUN iajs-2509 55 5 , , PUNCT iajs-2509 55 6 look look VERB iajs-2509 55 7 at at ADP iajs-2509 55 8 the the DET iajs-2509 55 9 following following ADJ iajs-2509 55 10 examples example NOUN iajs-2509 55 11 : : PUNCT iajs-2509 55 12 example example NOUN iajs-2509 55 13 3.1 3.1 NUM iajs-2509 55 14 consider consider VERB iajs-2509 55 15 the the DET iajs-2509 55 16 following follow VERB iajs-2509 55 17 nonlinear nonlinear ADJ iajs-2509 55 18 delay delay NOUN iajs-2509 55 19 eigenvalue eigenvalue PROPN iajs-2509 55 20 problem problem NOUN iajs-2509 55 21 : : PUNCT iajs-2509 55 22 – – PUNCT iajs-2509 55 23 ( ( PUNCT iajs-2509 55 24 kf'(k))'+2kf(k–1)–(f2(k–1)–0.5)=0 kf'(k))'+2kf(k–1)–(f2(k–1)–0.5)=0 PROPN iajs-2509 55 25 , , PUNCT iajs-2509 55 26 k k X iajs-2509 56 1   NOUN iajs-2509 57 1 [ [ X iajs-2509 57 2 1,2 1,2 NUM iajs-2509 57 3 ] ] PUNCT iajs-2509 57 4 ( ( PUNCT iajs-2509 57 5 7 7 NUM iajs-2509 57 6 ) ) PUNCT iajs-2509 57 7 with with ADP iajs-2509 57 8 the the DET iajs-2509 57 9 associated associated ADJ iajs-2509 57 10 boundary boundary ADJ iajs-2509 57 11 conditions condition NOUN iajs-2509 57 12 : : PUNCT iajs-2509 57 13     SPACE iajs-2509 57 14 62 62 NUM iajs-2509 57 15     SPACE iajs-2509 57 16 ibn ibn PROPN iajs-2509 57 17 al al PROPN iajs-2509 57 18 - - PUNCT iajs-2509 57 19 haitham haitham PROPN iajs-2509 57 20 jour jour X iajs-2509 57 21 . . PROPN iajs-2509 58 1 for for ADP iajs-2509 58 2 pure pure ADJ iajs-2509 58 3 & & CCONJ iajs-2509 58 4 appl appl PROPN iajs-2509 58 5 . . PUNCT iajs-2509 59 1 sci sci PROPN iajs-2509 59 2 . . PROPN iajs-2509 60 1 33 33 NUM iajs-2509 60 2 ( ( PUNCT iajs-2509 60 3 4 4 NUM iajs-2509 60 4 ) ) PUNCT iajs-2509 60 5 2020 2020 NUM iajs-2509 61 1 f(1 f(1 PROPN iajs-2509 61 2 ) ) PUNCT iajs-2509 61 3 = = SYM iajs-2509 62 1 f f NOUN iajs-2509 62 2 ' ' PUNCT iajs-2509 62 3 ( ( PUNCT iajs-2509 62 4 1 1 NUM iajs-2509 62 5 ) ) PUNCT iajs-2509 62 6 , , PUNCT iajs-2509 63 1 k k PROPN iajs-2509 63 2   NOUN iajs-2509 64 1 [ [ X iajs-2509 64 2 0,1 0,1 X iajs-2509 64 3 ] ] X iajs-2509 64 4 f(2 f(2 PROPN iajs-2509 64 5 ) ) PUNCT iajs-2509 64 6 = = SYM iajs-2509 64 7 2 2 NUM iajs-2509 64 8 f f NOUN iajs-2509 64 9 ' ' PUNCT iajs-2509 64 10 ( ( PUNCT iajs-2509 64 11 2 2 NUM iajs-2509 64 12 ) ) PUNCT iajs-2509 64 13 , , PUNCT iajs-2509 65 1 k k PROPN iajs-2509 65 2   NOUN iajs-2509 66 1 [ [ X iajs-2509 66 2 1,2 1,2 NUM iajs-2509 66 3 ] ] PUNCT iajs-2509 66 4 ( ( PUNCT iajs-2509 66 5 8) 8) NUM iajs-2509 66 6 f(k f(k ADJ iajs-2509 66 7 – – PUNCT iajs-2509 66 8 1)=k 1)=k NUM iajs-2509 66 9 – – SYM iajs-2509 66 10 1 1 NUM iajs-2509 66 11 we we PRON iajs-2509 66 12 use use VERB iajs-2509 66 13 the the DET iajs-2509 66 14 least least ADJ iajs-2509 66 15 - - PUNCT iajs-2509 66 16 square square NOUN iajs-2509 66 17 method method NOUN iajs-2509 66 18 to to PART iajs-2509 66 19 solve solve VERB iajs-2509 66 20 this this DET iajs-2509 66 21 problem problem NOUN iajs-2509 66 22 . . PUNCT iajs-2509 67 1 to to PART iajs-2509 67 2 do do VERB iajs-2509 67 3 this this PRON iajs-2509 67 4 , , PUNCT iajs-2509 67 5 we we PRON iajs-2509 67 6 approximate approximate VERB iajs-2509 67 7 the the DET iajs-2509 67 8 unknown unknown ADJ iajs-2509 67 9 function function NOUN iajs-2509 67 10 y y PROPN iajs-2509 67 11 as as ADP iajs-2509 67 12 a a DET iajs-2509 67 13 polynomial polynomial NOUN iajs-2509 67 14 of of ADP iajs-2509 67 15 degree degree NOUN iajs-2509 67 16 three three NUM iajs-2509 67 17 , , PUNCT iajs-2509 67 18 that that ADV iajs-2509 67 19 is is ADV iajs-2509 67 20 , , PUNCT iajs-2509 67 21 write write VERB iajs-2509 67 22 : : PUNCT iajs-2509 67 23 1 1 NUM iajs-2509 67 24 4 4 NUM iajs-2509 67 25 1 1 NUM iajs-2509 67 26 ) ) PUNCT iajs-2509 67 27 ( ( PUNCT iajs-2509 67 28   VERB iajs-2509 67 29   NOUN iajs-2509 68 1   ADV iajs-2509 69 1 i i PRON iajs-2509 70 1 i i PRON iajs-2509 71 1 i i PRON iajs-2509 71 2 kckf kckf NOUN iajs-2509 72 1 but but CCONJ iajs-2509 72 2 this this DET iajs-2509 72 3 approximated approximate VERB iajs-2509 72 4 solution solution NOUN iajs-2509 72 5 must must AUX iajs-2509 72 6 satisfy satisfy VERB iajs-2509 72 7 the the DET iajs-2509 72 8 boundary boundary ADJ iajs-2509 72 9 conditions condition NOUN iajs-2509 72 10 given give VERB iajs-2509 72 11 by by ADP iajs-2509 72 12 equations equation NOUN iajs-2509 72 13 [ [ X iajs-2509 72 14 3.5 3.5 NUM iajs-2509 72 15 ] ] PUNCT iajs-2509 72 16 , , PUNCT iajs-2509 72 17 thus thus ADV iajs-2509 72 18 this this DET iajs-2509 72 19 approximated approximate VERB iajs-2509 72 20 solution solution NOUN iajs-2509 72 21 reduces reduce VERB iajs-2509 72 22 to to ADP iajs-2509 72 23 3 3 NUM iajs-2509 72 24 4 4 NUM iajs-2509 72 25 2 2 NUM iajs-2509 72 26 411 411 NUM iajs-2509 72 27 6 6 NUM iajs-2509 72 28 ) ) PUNCT iajs-2509 72 29 ( ( PUNCT iajs-2509 72 30 kckckcckf kckckcckf NOUN iajs-2509 72 31   X iajs-2509 72 32 from from ADP iajs-2509 72 33 which which PRON iajs-2509 72 34 , , PUNCT iajs-2509 72 35 we we PRON iajs-2509 72 36 have have VERB iajs-2509 72 37 : : PUNCT iajs-2509 72 38 3 3 NUM iajs-2509 72 39 4 4 NUM iajs-2509 72 40 2 2 NUM iajs-2509 72 41 411 411 NUM iajs-2509 72 42 ) ) PUNCT iajs-2509 72 43 1()1(6)1()1 1()1(6)1()1 PROPN iajs-2509 72 44 ( ( PUNCT iajs-2509 72 45   PROPN iajs-2509 72 46 kckckcckf kckckcckf NOUN iajs-2509 72 47 by by ADP iajs-2509 72 48 substituting substitute VERB iajs-2509 72 49 this this DET iajs-2509 72 50 approximated approximate VERB iajs-2509 72 51 solution solution NOUN iajs-2509 72 52 into into ADP iajs-2509 72 53 equation equation NOUN iajs-2509 72 54 [ [ X iajs-2509 72 55 3.4 3.4 NUM iajs-2509 72 56 ] ] PUNCT iajs-2509 72 57 , , PUNCT iajs-2509 72 58 we we PRON iajs-2509 72 59 obtain obtain VERB iajs-2509 72 60 ) ) PUNCT iajs-2509 72 61 3918()618 3918()618 NUM iajs-2509 72 62 ( ( PUNCT iajs-2509 72 63 ) ) PUNCT iajs-2509 72 64 , , PUNCT iajs-2509 72 65 , , PUNCT iajs-2509 72 66 , , PUNCT iajs-2509 72 67 ( ( PUNCT iajs-2509 72 68 2 2 NUM iajs-2509 72 69 44414441 44414441 NUM iajs-2509 72 70 kcckcckcckcckr kcckcckcckcckr PROPN iajs-2509 72 71   NOUN iajs-2509 72 72 ) ) PUNCT iajs-2509 72 73 ) ) PUNCT iajs-2509 73 1 1()1(6)1((2 1()1(6)1((2 NUM iajs-2509 74 1 3 3 NUM iajs-2509 74 2 4 4 NUM iajs-2509 74 3 2 2 NUM iajs-2509 74 4 411 411 NUM iajs-2509 74 5   PROPN iajs-2509 74 6 kckckcck kckckcck NOUN iajs-2509 74 7 ] ] PUNCT iajs-2509 74 8 2 2 NUM iajs-2509 74 9 1 1 NUM iajs-2509 74 10 ) ) PUNCT iajs-2509 74 11 ) ) PUNCT iajs-2509 75 1 1()1(6)1 1()1(6)1 NUM iajs-2509 75 2 ( ( PUNCT iajs-2509 75 3 [ [ X iajs-2509 75 4 ( ( PUNCT iajs-2509 75 5 23 23 NUM iajs-2509 75 6 4 4 NUM iajs-2509 75 7 2 2 NUM iajs-2509 75 8 411 411 NUM iajs-2509 75 9   PROPN iajs-2509 75 10 kckckcc kckckcc NOUN iajs-2509 75 11 thus thus ADV iajs-2509 75 12 , , PUNCT iajs-2509 75 13 if if SCONJ iajs-2509 75 14 we we PRON iajs-2509 75 15 minimize minimize VERB iajs-2509 75 16 the the DET iajs-2509 75 17 functional functional ADJ iajs-2509 75 18 : : PUNCT iajs-2509 75 19 dxcckrccj dxcckrccj ADJ iajs-2509 75 20 2 2 NUM iajs-2509 75 21 2 2 NUM iajs-2509 75 22 1 1 NUM iajs-2509 75 23 4141 4141 NUM iajs-2509 75 24 ) ) PUNCT iajs-2509 75 25 ) ) PUNCT iajs-2509 75 26 , , PUNCT iajs-2509 75 27 , , PUNCT iajs-2509 75 28 , , PUNCT iajs-2509 75 29 ( ( PUNCT iajs-2509 75 30 ( ( PUNCT iajs-2509 75 31 ) ) PUNCT iajs-2509 75 32 , , PUNCT iajs-2509 75 33 , , PUNCT iajs-2509 75 34 ( ( PUNCT iajs-2509 75 35   PROPN iajs-2509 75 36   NOUN iajs-2509 75 37 set set VERB iajs-2509 75 38 0 0 NUM iajs-2509 75 39 41 41 NUM iajs-2509 76 1   PROPN iajs-2509 76 2   ADJ iajs-2509 76 3   ADJ iajs-2509 77 1   PROPN iajs-2509 77 2   ADJ iajs-2509 77 3   ADJ iajs-2509 77 4   PROPN iajs-2509 77 5   ADJ iajs-2509 77 6   PROPN iajs-2509 77 7 c c PROPN iajs-2509 77 8 j j PROPN iajs-2509 77 9 c c PROPN iajs-2509 77 10 jj jj PROPN iajs-2509 77 11   X iajs-2509 77 12 to to PART iajs-2509 77 13 get get VERB iajs-2509 77 14 the the DET iajs-2509 77 15 following follow VERB iajs-2509 77 16 system system NOUN iajs-2509 77 17 of of ADP iajs-2509 77 18 nonlinear nonlinear ADJ iajs-2509 77 19 equations equation NOUN iajs-2509 77 20 : : PUNCT iajs-2509 77 21 0 0 NUM iajs-2509 77 22 ) ) PUNCT iajs-2509 77 23 ) ) PUNCT iajs-2509 77 24 , , PUNCT iajs-2509 77 25 , , PUNCT iajs-2509 77 26 , , PUNCT iajs-2509 77 27 ( ( PUNCT iajs-2509 77 28 ( ( PUNCT iajs-2509 77 29 0 0 NUM iajs-2509 77 30 ) ) PUNCT iajs-2509 77 31 ) ) PUNCT iajs-2509 77 32 , , PUNCT iajs-2509 77 33 , , PUNCT iajs-2509 77 34 , , PUNCT iajs-2509 77 35 ( ( PUNCT iajs-2509 77 36 ( ( PUNCT iajs-2509 77 37 0 0 NUM iajs-2509 77 38 ) ) PUNCT iajs-2509 77 39 ) ) PUNCT iajs-2509 77 40 , , PUNCT iajs-2509 77 41 , , PUNCT iajs-2509 77 42 , , PUNCT iajs-2509 77 43 ( ( PUNCT iajs-2509 77 44 ( ( PUNCT iajs-2509 77 45 2 2 NUM iajs-2509 77 46 2 2 NUM iajs-2509 77 47 1 1 NUM iajs-2509 77 48 41 41 NUM iajs-2509 77 49 4 4 NUM iajs-2509 77 50 2 2 NUM iajs-2509 77 51 2 2 NUM iajs-2509 77 52 1 1 NUM iajs-2509 77 53 41 41 NUM iajs-2509 77 54 1 1 NUM iajs-2509 77 55 2 2 NUM iajs-2509 77 56 2 2 NUM iajs-2509 77 57 1 1 NUM iajs-2509 77 58 41 41 NUM iajs-2509 77 59   NUM iajs-2509 77 60   ADJ iajs-2509 77 61   ADJ iajs-2509 78 1   PROPN iajs-2509 78 2   ADJ iajs-2509 78 3   ADJ iajs-2509 78 4   PROPN iajs-2509 78 5   ADJ iajs-2509 78 6   PROPN iajs-2509 78 7   X iajs-2509 78 8   PROPN iajs-2509 78 9   PUNCT iajs-2509 78 10 dxcckr dxcckr INTJ iajs-2509 79 1 c c NOUN iajs-2509 79 2 dxcckr dxcckr PROPN iajs-2509 79 3 c c PROPN iajs-2509 79 4 dxcckr dxcckr PROPN iajs-2509 79 5   PROPN iajs-2509 79 6   ADJ iajs-2509 79 7   ADJ iajs-2509 79 8   ADJ iajs-2509 79 9 solving solving NOUN iajs-2509 79 10 the the DET iajs-2509 79 11 above above ADJ iajs-2509 79 12 system system NOUN iajs-2509 79 13 by by ADP iajs-2509 79 14 any any DET iajs-2509 79 15 suitable suitable ADJ iajs-2509 79 16 method method NOUN iajs-2509 79 17 , , PUNCT iajs-2509 79 18 to to PART iajs-2509 79 19 find find VERB iajs-2509 79 20 that that SCONJ iajs-2509 79 21 the the DET iajs-2509 79 22 nontrivial nontrivial ADJ iajs-2509 79 23 solution solution NOUN iajs-2509 79 24 is be AUX iajs-2509 79 25   ADJ iajs-2509 79 26 = = X iajs-2509 79 27 2 2 NUM iajs-2509 79 28 , , PUNCT iajs-2509 79 29 c1 c1 NOUN iajs-2509 79 30 = = PROPN iajs-2509 79 31 1 1 NUM iajs-2509 79 32 and and CCONJ iajs-2509 79 33 c4 c4 NOUN iajs-2509 79 34 = = NOUN iajs-2509 79 35 0 0 X iajs-2509 79 36 . . 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PROPN iajs-2509 88 19     SPACE iajs-2509 88 20 63 63 NUM iajs-2509 88 21     SPACE iajs-2509 88 22 ibn ibn PROPN iajs-2509 88 23 al al PROPN iajs-2509 88 24 - - PUNCT iajs-2509 88 25 haitham haitham PROPN iajs-2509 88 26 jour jour X iajs-2509 88 27 . . PROPN iajs-2509 89 1 for for ADP iajs-2509 89 2 pure pure ADJ iajs-2509 89 3 & & CCONJ iajs-2509 89 4 appl appl PROPN iajs-2509 89 5 . . PUNCT iajs-2509 90 1 sci sci PROPN iajs-2509 90 2 . . PROPN iajs-2509 91 1 33 33 NUM iajs-2509 91 2 ( ( PUNCT iajs-2509 91 3 4 4 NUM iajs-2509 91 4 ) ) PUNCT iajs-2509 91 5 2020 2020 NUM iajs-2509 91 6 that that PRON iajs-2509 91 7 is be AUX iajs-2509 91 8 if if SCONJ iajs-2509 91 9 , , PUNCT iajs-2509 91 10 ) ) PUNCT iajs-2509 91 11 ( ( PUNCT iajs-2509 91 12 ) ) PUNCT iajs-2509 91 13 ( ( PUNCT iajs-2509 91 14 1 1 NUM iajs-2509 91 15 1 1 NUM iajs-2509 91 16   NOUN iajs-2509 91 17   PROPN iajs-2509 91 18   NOUN iajs-2509 91 19   NUM iajs-2509 92 1 i i PRON iajs-2509 92 2 n n VERB iajs-2509 93 1 i i PRON iajs-2509 93 2 i i PRON iajs-2509 93 3 kckf kckf VERB iajs-2509 93 4   VERB iajs-2509 93 5 then then ADV iajs-2509 93 6 f(k f(k ADJ iajs-2509 93 7 – – PUNCT iajs-2509 93 8 1 1 X iajs-2509 93 9 ) ) PUNCT iajs-2509 93 10 = = SYM iajs-2509 94 1 1 1 NUM iajs-2509 94 2 + + CCONJ iajs-2509 94 3 ( ( PUNCT iajs-2509 94 4 k k X iajs-2509 94 5 – – PUNCT iajs-2509 94 6 1 1 NUM iajs-2509 94 7 ) ) PUNCT iajs-2509 94 8 , , PUNCT iajs-2509 95 1 k k PROPN iajs-2509 95 2   NOUN iajs-2509 95 3 [ [ X iajs-2509 95 4 1,2 1,2 NUM iajs-2509 95 5 ] ] PUNCT iajs-2509 95 6 corresponding correspond VERB iajs-2509 95 7 to to ADP iajs-2509 95 8 the the DET iajs-2509 95 9 same same ADJ iajs-2509 95 10 delay delay NOUN iajs-2509 95 11 eigenvalue eigenvalue PROPN iajs-2509 95 12 . . PUNCT iajs-2509 95 13 example example NOUN iajs-2509 95 14 3.2 3.2 NUM iajs-2509 95 15 consider consider VERB iajs-2509 95 16 the the DET iajs-2509 95 17 following follow VERB iajs-2509 95 18 nonlinear nonlinear ADJ iajs-2509 95 19 delay delay NOUN iajs-2509 95 20 eigenvalue eigenvalue PROPN iajs-2509 95 21 problem problem NOUN iajs-2509 95 22 : : PUNCT iajs-2509 95 23 ] ] X iajs-2509 95 24 , , PUNCT iajs-2509 95 25 2 2 NUM iajs-2509 95 26 [ [ X iajs-2509 95 27 , , PUNCT iajs-2509 95 28 1 1 NUM iajs-2509 95 29 ) ) PUNCT iajs-2509 95 30 ( ( PUNCT iajs-2509 95 31 _ _ NOUN iajs-2509 95 32 ) ) PUNCT iajs-2509 95 33 ( ( PUNCT iajs-2509 95 34 ] ] SYM iajs-2509 95 35 2 2 NUM iajs-2509 95 36 , , PUNCT iajs-2509 95 37 0[,1 0[,1 NUM iajs-2509 95 38 ) ) PUNCT iajs-2509 95 39 2 2 NUM iajs-2509 95 40 ( ( PUNCT iajs-2509 95 41 ) ) PUNCT iajs-2509 95 42 2 2 NUM iajs-2509 95 43 ( ( PUNCT iajs-2509 95 44 ] ] X iajs-2509 95 45 , , PUNCT iajs-2509 95 46 2 2 NUM iajs-2509 95 47 [ [ NOUN iajs-2509 95 48 ) ) PUNCT iajs-2509 95 49 ) ) PUNCT iajs-2509 95 50 , , PUNCT iajs-2509 95 51 2 2 NUM iajs-2509 95 52 sin sin NOUN iajs-2509 95 53 ( ( PUNCT iajs-2509 95 54 ) ) PUNCT iajs-2509 95 55 2 2 NUM iajs-2509 95 56 ( ( PUNCT iajs-2509 95 57 2 2 NUM iajs-2509 95 58 ( ( PUNCT iajs-2509 95 59 ) ) PUNCT iajs-2509 95 60 ( ( PUNCT iajs-2509 95 61   NOUN iajs-2509 95 62   NOUN iajs-2509 95 63   PROPN iajs-2509 95 64   ADP iajs-2509 95 65   PROPN iajs-2509 95 66   PROPN iajs-2509 95 67 kff kff PROPN iajs-2509 95 68 kff kff PROPN iajs-2509 95 69 kkkfkf kkkfkf PROPN iajs-2509 95 70 2 2 NUM iajs-2509 95 71 ) ) PUNCT iajs-2509 95 72 2 2 NUM iajs-2509 95 73 ( ( PUNCT iajs-2509 95 74   PROPN iajs-2509 95 75   NUM iajs-2509 95 76 kkf kkf NOUN iajs-2509 95 77 we we PRON iajs-2509 95 78 use use VERB iajs-2509 95 79 the the DET iajs-2509 95 80 least least ADJ iajs-2509 95 81 - - PUNCT iajs-2509 95 82 square square NOUN iajs-2509 95 83 method method NOUN iajs-2509 95 84 to to PART iajs-2509 95 85 solve solve VERB iajs-2509 95 86 this this DET iajs-2509 95 87 problem problem NOUN iajs-2509 95 88 . . 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PUNCT iajs-2509 97 50 conclusions conclusion NOUN iajs-2509 97 51 from from ADP iajs-2509 97 52 this this DET iajs-2509 97 53 article article NOUN iajs-2509 97 54 we we PRON iajs-2509 97 55 can can AUX iajs-2509 97 56 conclude conclude VERB iajs-2509 97 57 the the DET iajs-2509 97 58 following following NOUN iajs-2509 97 59 : : PUNCT iajs-2509 98 1 1 1 X iajs-2509 98 2 . . X iajs-2509 98 3 the the DET iajs-2509 98 4 nonlinear nonlinear ADJ iajs-2509 98 5 second second ADJ iajs-2509 98 6 order order NOUN iajs-2509 98 7 delay delay NOUN iajs-2509 98 8 eigenvalue eigenvalue NOUN iajs-2509 98 9 problems problem NOUN iajs-2509 98 10 consist consist VERB iajs-2509 98 11 of of ADP iajs-2509 98 12 delay delay NOUN iajs-2509 98 13 nonlinear nonlinear ADJ iajs-2509 98 14 ordinary ordinary ADJ iajs-2509 98 15 differential differential ADJ iajs-2509 98 16 equations equation NOUN iajs-2509 98 17 satisfying satisfy VERB iajs-2509 98 18 the the DET iajs-2509 98 19 same same ADJ iajs-2509 98 20 properties property NOUN iajs-2509 98 21 as as ADP iajs-2509 98 22 those those DET iajs-2509 98 23 consist consist NOUN iajs-2509 98 24 of of ADP iajs-2509 98 25 delay delay NOUN iajs-2509 98 26 nonlinear nonlinear ADJ iajs-2509 98 27 ordinary ordinary ADJ iajs-2509 98 28 differential differential ADJ iajs-2509 98 29 equations equation NOUN iajs-2509 98 30 with with ADP iajs-2509 98 31 or or CCONJ iajs-2509 98 32 without without ADP iajs-2509 98 33 delay delay NOUN iajs-2509 98 34 . . PUNCT iajs-2509 99 1 2 2 X iajs-2509 99 2 . . X iajs-2509 99 3 the the DET iajs-2509 99 4 least least ADJ iajs-2509 99 5 square square ADJ iajs-2509 99 6 method method NOUN iajs-2509 99 7 has have AUX iajs-2509 99 8 been be AUX iajs-2509 99 9 developed develop VERB iajs-2509 99 10 to to PART iajs-2509 99 11 solve solve VERB iajs-2509 99 12 the the DET iajs-2509 99 13 above above ADJ iajs-2509 99 14 kind kind NOUN iajs-2509 99 15 of of ADP iajs-2509 99 16 problems problem NOUN iajs-2509 99 17 . . PUNCT iajs-2509 100 1 3 3 X iajs-2509 100 2 . . X iajs-2509 100 3 in in ADP iajs-2509 100 4 future future NOUN iajs-2509 100 5 we we PRON iajs-2509 100 6 can can AUX iajs-2509 100 7 use use VERB iajs-2509 100 8 the the DET iajs-2509 100 9 methods method NOUN iajs-2509 100 10 in in ADP iajs-2509 100 11 [ [ X iajs-2509 100 12 11 11 NUM iajs-2509 100 13 , , PUNCT iajs-2509 100 14 12 12 NUM iajs-2509 100 15 ] ] PUNCT iajs-2509 100 16 to to PART iajs-2509 100 17 solve solve VERB iajs-2509 100 18 the the DET iajs-2509 100 19 problems problem NOUN iajs-2509 100 20 of of ADP iajs-2509 100 21 this this DET iajs-2509 100 22 article article NOUN iajs-2509 100 23 . . PUNCT iajs-2509 101 1 references reference NOUN iajs-2509 101 2 1 1 NUM iajs-2509 101 3 . . PUNCT iajs-2509 102 1 koivo koivo PROPN iajs-2509 102 2 , , PUNCT iajs-2509 102 3 a. a. NOUN iajs-2509 102 4 , , PUNCT iajs-2509 102 5 estimation estimation NOUN iajs-2509 102 6 in in ADP iajs-2509 102 7 nonlinear nonlinear ADJ iajs-2509 102 8 systems system NOUN iajs-2509 102 9 with with ADP iajs-2509 102 10 transport transport NOUN iajs-2509 102 11 delay delay NOUN iajs-2509 102 12 , , PUNCT iajs-2509 102 13 1971 1971 NUM iajs-2509 102 14 , , PUNCT iajs-2509 102 15 ieee ieee NOUN iajs-2509 102 16 conference conference NOUN iajs-2509 102 17 on on ADP iajs-2509 102 18 decision decision NOUN iajs-2509 102 19 and and CCONJ iajs-2509 102 20 control control NOUN iajs-2509 102 21 . . PUNCT iajs-2509 103 1 2 2 X iajs-2509 103 2 . . X iajs-2509 103 3 bhattacharyya bhattacharyya ADJ iajs-2509 103 4 , , PUNCT iajs-2509 103 5 t. t. PROPN iajs-2509 103 6 ; ; PUNCT iajs-2509 103 7 binding bind VERB iajs-2509 103 8 , , PUNCT iajs-2509 103 9 p. p. PROPN iajs-2509 103 10 ; ; PUNCT iajs-2509 103 11 seddighi seddighi PROPN iajs-2509 103 12 , , PUNCT iajs-2509 103 13 k. k. 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