id sid tid token lemma pos iajs-2618 1 1 119 119 NUM iajs-2618 1 2 ibn ibn PROPN iajs-2618 1 3 al al PROPN iajs-2618 1 4 - - PUNCT iajs-2618 1 5 haitham haitham PROPN iajs-2618 1 6 jour jour X iajs-2618 1 7 . . PROPN iajs-2618 1 8 for for ADP iajs-2618 1 9 pure pure ADJ iajs-2618 1 10 & & CCONJ iajs-2618 1 11 appl appl PROPN iajs-2618 1 12 . . PUNCT iajs-2618 2 1 sci sci PROPN iajs-2618 2 2 . . PROPN iajs-2618 3 1 34 34 NUM iajs-2618 3 2 ( ( PUNCT iajs-2618 3 3 2 2 NUM iajs-2618 3 4 ) ) PUNCT iajs-2618 3 5 2021 2021 NUM iajs-2618 3 6 the the DET iajs-2618 3 7 galerkin galerkin ADV iajs-2618 3 8 - - PUNCT iajs-2618 3 9 implicit implicit ADJ iajs-2618 3 10 methods method NOUN iajs-2618 3 11 for for ADP iajs-2618 3 12 solving solve VERB iajs-2618 3 13 nonlinear nonlinear ADJ iajs-2618 3 14 hyperbolic hyperbolic ADJ iajs-2618 3 15 boundary boundary ADJ iajs-2618 3 16 value value NOUN iajs-2618 3 17 problem problem NOUN iajs-2618 3 18 department department PROPN iajs-2618 3 19 of of ADP iajs-2618 3 20 mathematics mathematics PROPN iajs-2618 3 21 , , PUNCT iajs-2618 3 22 college college NOUN iajs-2618 3 23 of of ADP iajs-2618 3 24 science science NOUN iajs-2618 3 25 , , PUNCT iajs-2618 3 26 mustansiriyah mustansiriyah NOUN iajs-2618 3 27 university university NOUN iajs-2618 3 28 , , PUNCT iajs-2618 3 29 baghdad baghdad PROPN iajs-2618 3 30 , , PUNCT iajs-2618 3 31 iraq iraq PROPN iajs-2618 3 32 abstract abstract NOUN iajs-2618 3 33 this this DET iajs-2618 3 34 paper paper NOUN iajs-2618 3 35 is be AUX iajs-2618 3 36 concerned concern VERB iajs-2618 3 37 with with ADP iajs-2618 3 38 finding find VERB iajs-2618 3 39 the the DET iajs-2618 3 40 approximation approximation NOUN iajs-2618 3 41 solution solution NOUN iajs-2618 3 42 ( ( PUNCT iajs-2618 3 43 apps app NOUN iajs-2618 3 44 ) ) PUNCT iajs-2618 3 45 of of ADP iajs-2618 3 46 a a DET iajs-2618 3 47 certain certain ADJ iajs-2618 3 48 type type NOUN iajs-2618 3 49 of of ADP iajs-2618 3 50 nonlinear nonlinear ADJ iajs-2618 3 51 hyperbolic hyperbolic ADJ iajs-2618 3 52 boundary boundary ADJ iajs-2618 3 53 value value NOUN iajs-2618 3 54 problem problem NOUN iajs-2618 3 55 ( ( PUNCT iajs-2618 3 56 nolhybvp nolhybvp NOUN iajs-2618 3 57 ) ) PUNCT iajs-2618 3 58 . . PUNCT iajs-2618 4 1 the the DET iajs-2618 4 2 given give VERB iajs-2618 4 3 bvp bvp PROPN iajs-2618 4 4 is be AUX iajs-2618 4 5 written write VERB iajs-2618 4 6 in in ADP iajs-2618 4 7 its its PRON iajs-2618 4 8 discrete discrete ADJ iajs-2618 4 9 ( ( PUNCT iajs-2618 4 10 di di NOUN iajs-2618 4 11 ) ) PUNCT iajs-2618 4 12 weak weak ADJ iajs-2618 4 13 form form NOUN iajs-2618 4 14 ( ( PUNCT iajs-2618 4 15 wef wef PROPN iajs-2618 4 16 ) ) PUNCT iajs-2618 4 17 , , PUNCT iajs-2618 4 18 and and CCONJ iajs-2618 4 19 is be AUX iajs-2618 4 20 proved prove VERB iajs-2618 4 21 that that SCONJ iajs-2618 4 22 it it PRON iajs-2618 4 23 has have VERB iajs-2618 4 24 a a DET iajs-2618 4 25 unique unique ADJ iajs-2618 4 26 apps app NOUN iajs-2618 4 27 , , PUNCT iajs-2618 4 28 which which PRON iajs-2618 4 29 is be AUX iajs-2618 4 30 obtained obtain VERB iajs-2618 4 31 via via ADP iajs-2618 4 32 the the DET iajs-2618 4 33 mixed mixed ADJ iajs-2618 4 34 galerkin galerkin ADJ iajs-2618 4 35 finite finite PROPN iajs-2618 4 36 element element NOUN iajs-2618 4 37 method method NOUN iajs-2618 4 38 ( ( PUNCT iajs-2618 4 39 gfe gfe NOUN iajs-2618 4 40 ) ) PUNCT iajs-2618 4 41 with with ADP iajs-2618 4 42 implicit implicit ADJ iajs-2618 4 43 method method NOUN iajs-2618 4 44 ( ( PUNCT iajs-2618 4 45 mgfeim mgfeim PROPN iajs-2618 4 46 ) ) PUNCT iajs-2618 4 47 that that PRON iajs-2618 4 48 reduces reduce VERB iajs-2618 4 49 the the DET iajs-2618 4 50 problem problem NOUN iajs-2618 4 51 to to PART iajs-2618 4 52 solve solve VERB iajs-2618 4 53 the the DET iajs-2618 4 54 galerkin galerkin PROPN iajs-2618 4 55 nonlinear nonlinear ADJ iajs-2618 4 56 algebraic algebraic ADJ iajs-2618 4 57 system system NOUN iajs-2618 4 58 ( ( PUNCT iajs-2618 4 59 gnas gnas NOUN iajs-2618 4 60 ) ) PUNCT iajs-2618 4 61 . . PUNCT iajs-2618 5 1 in in ADP iajs-2618 5 2 this this DET iajs-2618 5 3 part part NOUN iajs-2618 5 4 , , PUNCT iajs-2618 5 5 the the DET iajs-2618 5 6 predictor predictor NOUN iajs-2618 5 7 and and CCONJ iajs-2618 5 8 the the DET iajs-2618 5 9 corrector corrector NOUN iajs-2618 5 10 technique technique NOUN iajs-2618 5 11 ( ( PUNCT iajs-2618 5 12 pt pt NOUN iajs-2618 5 13 and and CCONJ iajs-2618 5 14 ct ct NUM iajs-2618 5 15 ) ) PUNCT iajs-2618 5 16 are be AUX iajs-2618 5 17 proved prove VERB iajs-2618 5 18 convergent convergent ADJ iajs-2618 5 19 and and CCONJ iajs-2618 5 20 are be AUX iajs-2618 5 21 used use VERB iajs-2618 5 22 to to PART iajs-2618 5 23 transform transform VERB iajs-2618 5 24 the the DET iajs-2618 5 25 obtained obtain VERB iajs-2618 5 26 gnas gnas NOUN iajs-2618 5 27 to to ADP iajs-2618 5 28 linear linear PROPN iajs-2618 5 29 ( ( PUNCT iajs-2618 5 30 glas glas PROPN iajs-2618 5 31 ) ) PUNCT iajs-2618 5 32 , , PUNCT iajs-2618 5 33 then then ADV iajs-2618 5 34 the the DET iajs-2618 5 35 glas glas PROPN iajs-2618 5 36 is be AUX iajs-2618 5 37 solved solve VERB iajs-2618 5 38 using use VERB iajs-2618 5 39 the the DET iajs-2618 5 40 cholesky cholesky ADJ iajs-2618 5 41 method method NOUN iajs-2618 5 42 ( ( PUNCT iajs-2618 5 43 chme chme NOUN iajs-2618 5 44 ) ) PUNCT iajs-2618 5 45 . . PUNCT iajs-2618 6 1 the the DET iajs-2618 6 2 stability stability NOUN iajs-2618 6 3 and and CCONJ iajs-2618 6 4 the the DET iajs-2618 6 5 convergence convergence NOUN iajs-2618 6 6 of of ADP iajs-2618 6 7 the the DET iajs-2618 6 8 method method NOUN iajs-2618 6 9 are be AUX iajs-2618 6 10 studied study VERB iajs-2618 6 11 . . PUNCT iajs-2618 7 1 the the DET iajs-2618 7 2 results result NOUN iajs-2618 7 3 are be AUX iajs-2618 7 4 given give VERB iajs-2618 7 5 by by ADP iajs-2618 7 6 figures figure NOUN iajs-2618 7 7 and and CCONJ iajs-2618 7 8 shown show VERB iajs-2618 7 9 the the DET iajs-2618 7 10 efficiency efficiency NOUN iajs-2618 7 11 and and CCONJ iajs-2618 7 12 accuracy accuracy NOUN iajs-2618 7 13 for for ADP iajs-2618 7 14 the the DET iajs-2618 7 15 method method NOUN iajs-2618 7 16 . . PUNCT iajs-2618 8 1 keywords keyword NOUN iajs-2618 8 2 : : PUNCT iajs-2618 8 3 nonlinear nonlinear ADJ iajs-2618 8 4 hyperbolic hyperbolic ADJ iajs-2618 8 5 boundary boundary ADJ iajs-2618 8 6 value value NOUN iajs-2618 8 7 problem problem NOUN iajs-2618 8 8 , , PUNCT iajs-2618 8 9 galekin galekin NOUN iajs-2618 8 10 finite finite PROPN iajs-2618 8 11 element element NOUN iajs-2618 8 12 method method NOUN iajs-2618 8 13 , , PUNCT iajs-2618 8 14 implicit implicit ADJ iajs-2618 8 15 method method NOUN iajs-2618 8 16 , , PUNCT iajs-2618 8 17 convergence convergence NOUN iajs-2618 8 18 , , PUNCT iajs-2618 8 19 stability stability NOUN iajs-2618 8 20 . . PUNCT iajs-2618 9 1 1 1 X iajs-2618 9 2 . . X iajs-2618 9 3 introduction introduction NOUN iajs-2618 9 4 hyperbolic hyperbolic ADJ iajs-2618 9 5 partial partial ADJ iajs-2618 9 6 differential differential NOUN iajs-2618 9 7 equations equation NOUN iajs-2618 9 8 play play VERB iajs-2618 9 9 a a DET iajs-2618 9 10 very very ADV iajs-2618 9 11 important important ADJ iajs-2618 9 12 role role NOUN iajs-2618 9 13 as as ADP iajs-2618 9 14 real real ADJ iajs-2618 9 15 life life NOUN iajs-2618 9 16 problems problem NOUN iajs-2618 9 17 in in ADP iajs-2618 9 18 many many ADJ iajs-2618 9 19 fields field NOUN iajs-2618 9 20 of of ADP iajs-2618 9 21 sciences science NOUN iajs-2618 9 22 as as ADP iajs-2618 9 23 in in ADP iajs-2618 9 24 technology technology NOUN iajs-2618 9 25 , , PUNCT iajs-2618 9 26 fluid fluid ADJ iajs-2618 9 27 dynamics dynamic NOUN iajs-2618 9 28 , , PUNCT iajs-2618 9 29 optics optic NOUN iajs-2618 9 30 , , PUNCT iajs-2618 9 31 science science NOUN iajs-2618 9 32 and and CCONJ iajs-2618 9 33 many many ADJ iajs-2618 9 34 others other NOUN iajs-2618 9 35 . . PUNCT iajs-2618 10 1 in in ADP iajs-2618 10 2 the the DET iajs-2618 10 3 past past ADJ iajs-2618 10 4 few few ADJ iajs-2618 10 5 decades decade NOUN iajs-2618 10 6 , , PUNCT iajs-2618 10 7 there there PRON iajs-2618 10 8 have have AUX iajs-2618 10 9 been be AUX iajs-2618 10 10 many many ADJ iajs-2618 10 11 researchers researcher NOUN iajs-2618 10 12 interested interested ADJ iajs-2618 10 13 in in ADP iajs-2618 10 14 their their PRON iajs-2618 10 15 study study NOUN iajs-2618 10 16 to to PART iajs-2618 10 17 solve solve VERB iajs-2618 10 18 boundary boundary ADJ iajs-2618 10 19 value value NOUN iajs-2618 10 20 problems problem NOUN iajs-2618 10 21 in in ADP iajs-2618 10 22 general general ADJ iajs-2618 10 23 and and CCONJ iajs-2618 10 24 in in ADP iajs-2618 10 25 particular particular ADJ iajs-2618 10 26 nlhbve nlhbve PROPN iajs-2618 10 27 . . PUNCT iajs-2618 11 1 many many ADJ iajs-2618 11 2 researchers researcher NOUN iajs-2618 11 3 have have AUX iajs-2618 11 4 used use VERB iajs-2618 11 5 different different ADJ iajs-2618 11 6 methods method NOUN iajs-2618 11 7 to to PART iajs-2618 11 8 solve solve VERB iajs-2618 11 9 the the DET iajs-2618 11 10 nlhbve nlhbve PROPN iajs-2618 11 11 , , PUNCT iajs-2618 11 12 smiley smiley NOUN iajs-2618 11 13 studied study VERB iajs-2618 11 14 in in ADP iajs-2618 11 15 1987 1987 NUM iajs-2618 11 16 , , PUNCT iajs-2618 11 17 was be AUX iajs-2618 11 18 used use VERB iajs-2618 11 19 eigen eigen PROPN iajs-2618 11 20 function function NOUN iajs-2618 11 21 methods method NOUN iajs-2618 11 22 to to PART iajs-2618 11 23 solve solve VERB iajs-2618 11 24 problems problem NOUN iajs-2618 11 25 of of ADP iajs-2618 11 26 nonlinear nonlinear ADJ iajs-2618 11 27 hyperbolic hyperbolic ADJ iajs-2618 11 28 value value NOUN iajs-2618 11 29 at at ADP iajs-2618 11 30 resonance resonance NOUN iajs-2618 11 31 [ [ X iajs-2618 11 32 1 1 NUM iajs-2618 11 33 ] ] PUNCT iajs-2618 11 34 . . PUNCT iajs-2618 12 1 in in ADP iajs-2618 12 2 1989 1989 NUM iajs-2618 12 3 , , PUNCT iajs-2618 12 4 chi chi NOUN iajs-2618 12 5 , , PUNCT iajs-2618 12 6 wiener wiener NOUN iajs-2618 12 7 , , PUNCT iajs-2618 12 8 and and CCONJ iajs-2618 12 9 shah shah NOUN iajs-2618 12 10 used use VERB iajs-2618 12 11 in in ADP iajs-2618 12 12 the the DET iajs-2618 12 13 exponential exponential ADJ iajs-2618 12 14 growth growth NOUN iajs-2618 12 15 of of ADP iajs-2618 12 16 solutions solution NOUN iajs-2618 12 17 of of ADP iajs-2618 12 18 nonlinear nonlinear ADJ iajs-2618 12 19 hyperbolic hyperbolic ADJ iajs-2618 12 20 equations equation NOUN iajs-2618 12 21 [ [ X iajs-2618 12 22 2 2 NUM iajs-2618 12 23 ] ] PUNCT iajs-2618 12 24 , , PUNCT iajs-2618 12 25 while while SCONJ iajs-2618 12 26 in in ADP iajs-2618 12 27 2001 2001 NUM iajs-2618 12 28 minamoto minamoto PROPN iajs-2618 12 29 used use VERB iajs-2618 12 30 the the DET iajs-2618 12 31 existence existence NOUN iajs-2618 12 32 and and CCONJ iajs-2618 12 33 demonstration demonstration NOUN iajs-2618 12 34 of of ADP iajs-2618 12 35 the the DET iajs-2618 12 36 uniqueness uniqueness NOUN iajs-2618 12 37 of of ADP iajs-2618 12 38 solutions solution NOUN iajs-2618 12 39 [ [ X iajs-2618 12 40 3 3 NUM iajs-2618 12 41 ] ] PUNCT iajs-2618 12 42 . . PUNCT iajs-2618 13 1 in in ADP iajs-2618 13 2 2004 2004 NUM iajs-2618 13 3 , , PUNCT iajs-2618 13 4 krylovas krylovas PROPN iajs-2618 13 5 , , PUNCT iajs-2618 13 6 and and CCONJ iajs-2618 13 7 čiegis čiegis VERB iajs-2618 13 8 , , PUNCT iajs-2618 13 9 used use VERB iajs-2618 13 10 the the DET iajs-2618 13 11 numerical numerical ADJ iajs-2618 13 12 asymptotic asymptotic ADJ iajs-2618 13 13 averaging averaging NOUN iajs-2618 13 14 for for ADP iajs-2618 13 15 weakly weakly ADJ iajs-2618 13 16 nonlinear nonlinear ADJ iajs-2618 13 17 hyperbolic hyperbolic ADJ iajs-2618 13 18 waves wave NOUN iajs-2618 13 19 [ [ X iajs-2618 13 20 4 4 NUM iajs-2618 13 21 ] ] PUNCT iajs-2618 13 22 . . PUNCT iajs-2618 14 1 in in ADP iajs-2618 14 2 2018 2018 NUM iajs-2618 14 3 , , PUNCT iajs-2618 14 4 ashyralyev ashyralyev CCONJ iajs-2618 14 5 and and CCONJ iajs-2618 14 6 agirseven agirseven SCONJ iajs-2618 14 7 solved solve VERB iajs-2618 14 8 nlhbve nlhbve PROPN iajs-2618 14 9 with with ADP iajs-2618 14 10 a a DET iajs-2618 14 11 time time NOUN iajs-2618 14 12 delay delay NOUN iajs-2618 14 13 [ [ X iajs-2618 14 14 5 5 NUM iajs-2618 14 15 ] ] PUNCT iajs-2618 14 16 . . PUNCT iajs-2618 15 1 ibn ibn PROPN iajs-2618 15 2 al al PROPN iajs-2618 15 3 haitham haitham PROPN iajs-2618 15 4 journal journal PROPN iajs-2618 15 5 for for ADP iajs-2618 15 6 pure pure ADJ iajs-2618 15 7 and and CCONJ iajs-2618 15 8 applied apply VERB iajs-2618 15 9 science science NOUN iajs-2618 15 10 journal journal PROPN iajs-2618 15 11 homepage homepage NOUN iajs-2618 15 12 : : PUNCT iajs-2618 15 13 http://jih.uobaghdad.edu.iq/index.php/j/index http://jih.uobaghdad.edu.iq/index.php/j/index NOUN iajs-2618 15 14 doi doi NOUN iajs-2618 15 15 : : PUNCT iajs-2618 15 16 10.30526/34.2.2618 10.30526/34.2.2618 PROPN iajs-2618 15 17 article article NOUN iajs-2618 15 18 history history NOUN iajs-2618 15 19 : : PUNCT iajs-2618 15 20 received receive VERB iajs-2618 15 21 21 21 NUM iajs-2618 15 22 , , PUNCT iajs-2618 15 23 april april PROPN iajs-2618 15 24 , , PUNCT iajs-2618 15 25 2020 2020 NUM iajs-2618 15 26 , , PUNCT iajs-2618 15 27 accepted,23,june accepted,23,june NUM iajs-2618 15 28 , , PUNCT iajs-2618 15 29 2020 2020 NUM iajs-2618 15 30 , , PUNCT iajs-2618 15 31 published publish VERB iajs-2618 15 32 in in ADP iajs-2618 15 33 april april PROPN iajs-2618 15 34 2021 2021 NUM iajs-2618 15 35 jamil jamil PROPN iajs-2618 15 36 a. a. PROPN iajs-2618 15 37 ali ali PROPN iajs-2618 16 1 al al PROPN iajs-2618 16 2 - - PUNCT iajs-2618 16 3 hawasy hawasy PROPN iajs-2618 16 4 jhawassy17@mustansiriyah.edu.iq jhawassy17@mustansiriyah.edu.iq PROPN iajs-2618 16 5 nuha nuha PROPN iajs-2618 16 6 farhan farhan PROPN iajs-2618 16 7 mansour mansour PROPN iajs-2618 17 1 hawasy20@yahoo.com hawasy20@yahoo.com PROPN iajs-2618 17 2 mailto:jhawassy17@mustansiriyah.edu.iq mailto:jhawassy17@mustansiriyah.edu.iq PROPN iajs-2618 17 3 mailto:hawasy20@yahoo.com mailto:hawasy20@yahoo.com PROPN iajs-2618 17 4 120 120 NUM iajs-2618 17 5 ibn ibn PROPN iajs-2618 17 6 al al PROPN iajs-2618 17 7 - - PUNCT iajs-2618 17 8 haitham haitham PROPN iajs-2618 17 9 jour jour X iajs-2618 17 10 . . PROPN iajs-2618 18 1 for for ADP iajs-2618 18 2 pure pure ADJ iajs-2618 18 3 & & CCONJ iajs-2618 18 4 appl appl PROPN iajs-2618 18 5 . . PUNCT iajs-2618 19 1 sci sci PROPN iajs-2618 19 2 . . PROPN iajs-2618 20 1 34 34 NUM iajs-2618 20 2 ( ( PUNCT iajs-2618 20 3 2 2 NUM iajs-2618 20 4 ) ) PUNCT iajs-2618 20 5 2021 2021 NUM iajs-2618 20 6 the the DET iajs-2618 20 7 specific specific ADJ iajs-2618 20 8 element element NOUN iajs-2618 20 9 method method NOUN iajs-2618 20 10 has have AUX iajs-2618 20 11 been be AUX iajs-2618 20 12 studied study VERB iajs-2618 20 13 by by ADP iajs-2618 20 14 several several ADJ iajs-2618 20 15 researchers researcher NOUN iajs-2618 20 16 interested interested ADJ iajs-2618 20 17 in in ADP iajs-2618 20 18 this this DET iajs-2618 20 19 field field NOUN iajs-2618 20 20 , , PUNCT iajs-2618 20 21 for for ADP iajs-2618 20 22 example example NOUN iajs-2618 20 23 , , PUNCT iajs-2618 20 24 in in ADP iajs-2618 20 25 2010 2010 NUM iajs-2618 20 26 bangerth bangerth NOUN iajs-2618 20 27 and and CCONJ iajs-2618 20 28 rannacher rannacher PROPN iajs-2618 20 29 touched touch VERB iajs-2618 20 30 on on ADP iajs-2618 20 31 galerkin galerkin PROPN iajs-2618 20 32 's 's PART iajs-2618 20 33 specific specific ADJ iajs-2618 20 34 adaptation adaptation NOUN iajs-2618 20 35 techniques technique NOUN iajs-2618 20 36 for for ADP iajs-2618 20 37 wave wave NOUN iajs-2618 20 38 equation equation NOUN iajs-2618 20 39 [ [ X iajs-2618 20 40 6 6 NUM iajs-2618 20 41 ] ] PUNCT iajs-2618 20 42 . . PUNCT iajs-2618 21 1 whereas whereas SCONJ iajs-2618 21 2 , , PUNCT iajs-2618 21 3 in in ADP iajs-2618 21 4 2017 2017 NUM iajs-2618 21 5 , , PUNCT iajs-2618 21 6 al al PROPN iajs-2618 21 7 - - PUNCT iajs-2618 21 8 haq haq PROPN iajs-2618 21 9 and and CCONJ iajs-2618 21 10 muhammad muhammad PROPN iajs-2618 21 11 discussed discuss VERB iajs-2618 21 12 numerical numerical ADJ iajs-2618 21 13 methods method NOUN iajs-2618 21 14 to to PART iajs-2618 21 15 solve solve VERB iajs-2618 21 16 lhybvp lhybvp NOUN iajs-2618 21 17 by by ADP iajs-2618 21 18 difference difference NOUN iajs-2618 21 19 method method NOUN iajs-2618 21 20 and and CCONJ iajs-2618 21 21 the the DET iajs-2618 21 22 method method NOUN iajs-2618 21 23 of of ADP iajs-2618 21 24 the the DET iajs-2618 21 25 specified specified ADJ iajs-2618 21 26 elements element NOUN iajs-2618 21 27 [ [ X iajs-2618 21 28 7 7 NUM iajs-2618 21 29 ] ] PUNCT iajs-2618 21 30 . . PUNCT iajs-2618 22 1 in in ADP iajs-2618 22 2 this this DET iajs-2618 22 3 paper paper NOUN iajs-2618 22 4 , , PUNCT iajs-2618 22 5 we we PRON iajs-2618 22 6 are be AUX iajs-2618 22 7 concerned concern VERB iajs-2618 22 8 the the DET iajs-2618 22 9 study study NOUN iajs-2618 22 10 of of ADP iajs-2618 22 11 the the DET iajs-2618 22 12 apps app NOUN iajs-2618 22 13 of of ADP iajs-2618 22 14 the the DET iajs-2618 22 15 nolhybvp nolhybvp NOUN iajs-2618 22 16 . . PUNCT iajs-2618 23 1 the the DET iajs-2618 23 2 given give VERB iajs-2618 23 3 bvp bvp PROPN iajs-2618 23 4 is be AUX iajs-2618 23 5 written write VERB iajs-2618 23 6 in in ADP iajs-2618 23 7 its its PRON iajs-2618 23 8 wef wef NOUN iajs-2618 23 9 , , PUNCT iajs-2618 23 10 and and CCONJ iajs-2618 23 11 in in ADP iajs-2618 23 12 its its PRON iajs-2618 23 13 discrete discrete ADJ iajs-2618 23 14 equation equation NOUN iajs-2618 23 15 ( ( PUNCT iajs-2618 23 16 di di NOUN iajs-2618 23 17 ) ) PUNCT iajs-2618 23 18 type type NOUN iajs-2618 23 19 . . PUNCT iajs-2618 24 1 it it PRON iajs-2618 24 2 is be AUX iajs-2618 24 3 proved prove VERB iajs-2618 24 4 to to PART iajs-2618 24 5 have have VERB iajs-2618 24 6 unique unique ADJ iajs-2618 24 7 apps app NOUN iajs-2618 24 8 . . PUNCT iajs-2618 25 1 the the DET iajs-2618 25 2 apps app NOUN iajs-2618 25 3 is be AUX iajs-2618 25 4 obtained obtain VERB iajs-2618 25 5 via via ADP iajs-2618 25 6 the the DET iajs-2618 25 7 mgfeim mgfeim NOUN iajs-2618 25 8 . . PUNCT iajs-2618 26 1 the the DET iajs-2618 26 2 problem problem NOUN iajs-2618 26 3 then then ADV iajs-2618 26 4 reduces reduce VERB iajs-2618 26 5 to to PART iajs-2618 26 6 solve solve VERB iajs-2618 26 7 the the DET iajs-2618 26 8 gnas gnas NOUN iajs-2618 26 9 , , PUNCT iajs-2618 26 10 then then ADV iajs-2618 26 11 the the DET iajs-2618 26 12 pt pt NOUN iajs-2618 26 13 and and CCONJ iajs-2618 26 14 ct ct PROPN iajs-2618 26 15 are be AUX iajs-2618 26 16 proved prove VERB iajs-2618 26 17 convergent convergent ADJ iajs-2618 26 18 and and CCONJ iajs-2618 26 19 are be AUX iajs-2618 26 20 used use VERB iajs-2618 26 21 to to PART iajs-2618 26 22 transform transform VERB iajs-2618 26 23 the the DET iajs-2618 26 24 gnas gnas NOUN iajs-2618 26 25 to to ADP iajs-2618 26 26 a a DET iajs-2618 26 27 glas gla NOUN iajs-2618 26 28 . . PUNCT iajs-2618 27 1 this this DET iajs-2618 27 2 glas glas PROPN iajs-2618 27 3 is be AUX iajs-2618 27 4 solved solve VERB iajs-2618 27 5 by by ADP iajs-2618 27 6 using use VERB iajs-2618 27 7 the the DET iajs-2618 27 8 chme chme NOUN iajs-2618 27 9 . . PUNCT iajs-2618 28 1 the the DET iajs-2618 28 2 stability stability NOUN iajs-2618 28 3 and and CCONJ iajs-2618 28 4 the the DET iajs-2618 28 5 convergence convergence NOUN iajs-2618 28 6 of of ADP iajs-2618 28 7 the the DET iajs-2618 28 8 method method NOUN iajs-2618 28 9 are be AUX iajs-2618 28 10 studied study VERB iajs-2618 28 11 . . PUNCT iajs-2618 29 1 a a DET iajs-2618 29 2 computer computer NOUN iajs-2618 29 3 program program NOUN iajs-2618 29 4 is be AUX iajs-2618 29 5 codding cod VERB iajs-2618 29 6 to to PART iajs-2618 29 7 find find VERB iajs-2618 29 8 the the DET iajs-2618 29 9 numerical numerical ADJ iajs-2618 29 10 solution solution NOUN iajs-2618 29 11 for for ADP iajs-2618 29 12 the the DET iajs-2618 29 13 problem problem NOUN iajs-2618 29 14 . . PUNCT iajs-2618 30 1 the the DET iajs-2618 30 2 results result NOUN iajs-2618 30 3 are be AUX iajs-2618 30 4 given give VERB iajs-2618 30 5 by by ADP iajs-2618 30 6 figures figure NOUN iajs-2618 30 7 , , PUNCT iajs-2618 30 8 and and CCONJ iajs-2618 30 9 are be AUX iajs-2618 30 10 shown show VERB iajs-2618 30 11 the the DET iajs-2618 30 12 efficiency efficiency NOUN iajs-2618 30 13 and and CCONJ iajs-2618 30 14 accuracy accuracy NOUN iajs-2618 30 15 for for ADP iajs-2618 30 16 the the DET iajs-2618 30 17 method method NOUN iajs-2618 30 18 which which PRON iajs-2618 30 19 is be AUX iajs-2618 30 20 highly highly ADV iajs-2618 30 21 considered consider VERB iajs-2618 30 22 in in ADP iajs-2618 30 23 this this DET iajs-2618 30 24 work work NOUN iajs-2618 30 25 . . PUNCT iajs-2618 31 1 2 2 X iajs-2618 31 2 . . X iajs-2618 31 3 description description NOUN iajs-2618 31 4 of of ADP iajs-2618 31 5 the the DET iajs-2618 31 6 nolhybvp nolhybvp NOUN iajs-2618 31 7 let let VERB iajs-2618 31 8 𝐼 𝐼 PROPN iajs-2618 31 9 = = PUNCT iajs-2618 32 1 [ [ X iajs-2618 32 2 0 0 NUM iajs-2618 32 3 , , PUNCT iajs-2618 32 4 t t PROPN iajs-2618 32 5 ] ] PUNCT iajs-2618 32 6 , , PUNCT iajs-2618 32 7 with with ADP iajs-2618 32 8 0 0 NUM iajs-2618 32 9 < < X iajs-2618 32 10 𝑇 𝑇 PROPN iajs-2618 32 11 < < X iajs-2618 32 12 ∞ ∞ PROPN iajs-2618 32 13 , , PUNCT iajs-2618 32 14 πœ“ πœ“ PROPN iajs-2618 32 15 βŠ‚ βŠ‚ PROPN iajs-2618 32 16 ℝ2 ℝ2 VERB iajs-2618 32 17 be be AUX iajs-2618 32 18 a a DET iajs-2618 32 19 bounded bounded ADJ iajs-2618 32 20 and and CCONJ iajs-2618 32 21 open open ADJ iajs-2618 32 22 region region NOUN iajs-2618 32 23 with with ADP iajs-2618 32 24 smooth smooth ADJ iajs-2618 32 25 boundary boundary NOUN iajs-2618 32 26 βˆ‚πœ“ βˆ‚πœ“ PROPN iajs-2618 32 27 , , PUNCT iajs-2618 32 28 πœ‘ πœ‘ X iajs-2618 33 1 = = PUNCT iajs-2618 33 2 πœ“ πœ“ PROPN iajs-2618 33 3 Γ— Γ— NOUN iajs-2618 33 4 𝐼 𝐼 PROPN iajs-2618 33 5 , , PUNCT iajs-2618 33 6 Οƒ Οƒ PROPN iajs-2618 33 7 = = PUNCT iajs-2618 33 8 πœ•πœ“ πœ•πœ“ PROPN iajs-2618 33 9 Γ— Γ— NOUN iajs-2618 33 10 𝐼 𝐼 PROPN iajs-2618 33 11 , , PUNCT iajs-2618 33 12 then then ADV iajs-2618 33 13 the the DET iajs-2618 33 14 nlhypvp nlhypvp NOUN iajs-2618 33 15 is be AUX iajs-2618 33 16 given give VERB iajs-2618 33 17 by by ADP iajs-2618 33 18 : : PUNCT iajs-2618 33 19 𝑀𝑑𝑑 𝑀𝑑𝑑 PROPN iajs-2618 33 20 βˆ’ βˆ’ X iajs-2618 33 21 δ𝑀 δ𝑀 ADP iajs-2618 33 22 + + CCONJ iajs-2618 33 23 𝑀 𝑀 NOUN iajs-2618 33 24 = = PUNCT iajs-2618 33 25 β„Ž( β„Ž( NOUN iajs-2618 33 26 οΏ½ οΏ½ NOUN iajs-2618 33 27 βƒ— βƒ— PROPN iajs-2618 33 28 οΏ½ οΏ½ PROPN iajs-2618 33 29 , , PUNCT iajs-2618 33 30 𝑑 𝑑 PROPN iajs-2618 33 31 , , PUNCT iajs-2618 33 32 𝑀 𝑀 ADP iajs-2618 33 33 ) ) PUNCT iajs-2618 33 34 , , PUNCT iajs-2618 33 35 in in ADP iajs-2618 33 36 πœ‘ πœ‘ PROPN iajs-2618 33 37 ( ( PUNCT iajs-2618 33 38 1 1 NUM iajs-2618 33 39 ) ) PUNCT iajs-2618 33 40 𝑀( 𝑀( NUM iajs-2618 33 41 οΏ½ οΏ½ PROPN iajs-2618 33 42 βƒ— βƒ— NOUN iajs-2618 33 43 οΏ½ οΏ½ PROPN iajs-2618 33 44 , , PUNCT iajs-2618 33 45 0 0 NUM iajs-2618 33 46 ) ) PUNCT iajs-2618 33 47 = = SYM iajs-2618 33 48 𝑀0 𝑀0 PROPN iajs-2618 33 49 ( ( PUNCT iajs-2618 33 50 οΏ½ οΏ½ PROPN iajs-2618 33 51 βƒ— βƒ— NOUN iajs-2618 33 52 οΏ½ οΏ½ PROPN iajs-2618 33 53 ) ) PUNCT iajs-2618 33 54 , , PUNCT iajs-2618 33 55 in in ADP iajs-2618 33 56 πœ“ πœ“ PROPN iajs-2618 33 57 ( ( PUNCT iajs-2618 33 58 2 2 NUM iajs-2618 33 59 ) ) PUNCT iajs-2618 33 60 𝑀𝑑( 𝑀𝑑( PROPN iajs-2618 33 61 οΏ½ οΏ½ PROPN iajs-2618 33 62 βƒ— βƒ— NOUN iajs-2618 33 63 οΏ½ οΏ½ PROPN iajs-2618 33 64 , , PUNCT iajs-2618 33 65 0 0 NUM iajs-2618 33 66 ) ) PUNCT iajs-2618 33 67 = = PRON iajs-2618 33 68 𝑀1 𝑀1 VERB iajs-2618 33 69 ( ( PUNCT iajs-2618 33 70 οΏ½ οΏ½ PROPN iajs-2618 33 71 βƒ— βƒ— NOUN iajs-2618 33 72 οΏ½ οΏ½ PROPN iajs-2618 33 73 ) ) PUNCT iajs-2618 33 74 , , PUNCT iajs-2618 33 75 in in ADP iajs-2618 33 76 πœ“ πœ“ PROPN iajs-2618 33 77 ( ( PUNCT iajs-2618 33 78 3 3 NUM iajs-2618 33 79 ) ) PUNCT iajs-2618 33 80 𝑀( 𝑀( NUM iajs-2618 33 81 οΏ½ οΏ½ PROPN iajs-2618 33 82 βƒ— βƒ— NOUN iajs-2618 33 83 οΏ½ οΏ½ PROPN iajs-2618 33 84 , , PUNCT iajs-2618 33 85 𝑑 𝑑 PROPN iajs-2618 33 86 ) ) PUNCT iajs-2618 33 87 = = SYM iajs-2618 33 88 0 0 NUM iajs-2618 33 89 , , PUNCT iajs-2618 33 90 on on ADP iajs-2618 33 91 Οƒ Οƒ PROPN iajs-2618 33 92 ( ( PUNCT iajs-2618 33 93 4 4 NUM iajs-2618 33 94 ) ) PUNCT iajs-2618 33 95 where where SCONJ iajs-2618 33 96 𝑀 𝑀 X iajs-2618 33 97 = = SYM iajs-2618 33 98 𝑀( 𝑀( NUM iajs-2618 33 99 οΏ½ οΏ½ PROPN iajs-2618 33 100 βƒ— βƒ— NOUN iajs-2618 33 101 οΏ½ οΏ½ PROPN iajs-2618 33 102 , , PUNCT iajs-2618 33 103 𝑑 𝑑 NOUN iajs-2618 33 104 ) ) PUNCT iajs-2618 33 105 ∈ ∈ PROPN iajs-2618 33 106 𝐻0 𝐻0 NOUN iajs-2618 33 107 2(πœ“ 2(πœ“ NUM iajs-2618 33 108 ) ) PUNCT iajs-2618 33 109 , , PUNCT iajs-2618 33 110 , , PUNCT iajs-2618 33 111 δ𝑀 δ𝑀 ADP iajs-2618 33 112 = = SYM iajs-2618 33 113 βˆ‘ βˆ‘ PROPN iajs-2618 33 114 πœ•2𝑀 πœ•2𝑀 ADV iajs-2618 33 115 πœ•π‘₯𝑖 πœ•π‘₯𝑖 NUM iajs-2618 33 116 2 2 NUM iajs-2618 33 117 2 2 NUM iajs-2618 33 118 𝑖=1 𝑖=1 PUNCT iajs-2618 33 119 and and CCONJ iajs-2618 33 120 β„Ž β„Ž PART iajs-2618 33 121 ∈ ∈ PROPN iajs-2618 33 122 𝐿2(πœ“ 𝐿2(πœ“ NOUN iajs-2618 33 123 ) ) PUNCT iajs-2618 33 124 is be AUX iajs-2618 33 125 a a DET iajs-2618 33 126 given give VERB iajs-2618 33 127 function function NOUN iajs-2618 33 128 . . PUNCT iajs-2618 34 1 now now ADV iajs-2618 34 2 , , PUNCT iajs-2618 34 3 let let VERB iajs-2618 34 4 𝑉 𝑉 PROPN iajs-2618 34 5 = = PROPN iajs-2618 34 6 𝐻0 𝐻0 PROPN iajs-2618 34 7 1(πœ“)= 1(πœ“)= PROPN iajs-2618 34 8 { { PUNCT iajs-2618 34 9 πœ‚:πœ‚ πœ‚:πœ‚ PROPN iajs-2618 34 10 ∈ ∈ PROPN iajs-2618 34 11 𝐻1(πœ“ 𝐻1(πœ“ NUM iajs-2618 34 12 ) ) PUNCT iajs-2618 34 13 , , PUNCT iajs-2618 34 14 πœ‚ πœ‚ X iajs-2618 34 15 = = SYM iajs-2618 34 16 0 0 NUM iajs-2618 34 17 on on ADP iajs-2618 34 18 βˆ‚πœ“ βˆ‚πœ“ PRON iajs-2618 34 19 } } PUNCT iajs-2618 34 20 , , PUNCT iajs-2618 34 21 𝑀𝑑 𝑀𝑑 VERB iajs-2618 34 22 = = SYM iajs-2618 34 23 𝑝 𝑝 NOUN iajs-2618 34 24 , , PUNCT iajs-2618 34 25 then then ADV iajs-2618 34 26 the the DET iajs-2618 34 27 wefm wefm NOUN iajs-2618 34 28 of of ADP iajs-2618 34 29 ( ( PUNCT iajs-2618 34 30 1 1 NUM iajs-2618 34 31 - - SYM iajs-2618 34 32 4 4 NUM iajs-2618 34 33 ) ) PUNCT iajs-2618 34 34 is be AUX iajs-2618 34 35 : : PUNCT iajs-2618 34 36 ⟨ ⟨ VERB iajs-2618 34 37 𝑀𝑑𝑑 𝑀𝑑𝑑 NOUN iajs-2618 34 38 , , PUNCT iajs-2618 35 1 πœ‚ πœ‚ DET iajs-2618 35 2 ⟩ ⟩ NOUN iajs-2618 35 3 + + CCONJ iajs-2618 35 4 ( ( PUNCT iajs-2618 35 5 βˆ‡π‘€ βˆ‡π‘€ ADJ iajs-2618 35 6 , , PUNCT iajs-2618 35 7 βˆ‡ βˆ‡ X iajs-2618 35 8 πœ‚ πœ‚ NOUN iajs-2618 35 9 ) ) PUNCT iajs-2618 35 10 + + CCONJ iajs-2618 35 11 ( ( PUNCT iajs-2618 35 12 𝑀 𝑀 ADP iajs-2618 35 13 , , PUNCT iajs-2618 35 14 πœ‚ πœ‚ NOUN iajs-2618 35 15 ) ) PUNCT iajs-2618 35 16 = = SYM iajs-2618 35 17 ( ( PUNCT iajs-2618 35 18 β„Ž(𝑀 β„Ž(𝑀 NOUN iajs-2618 35 19 ) ) PUNCT iajs-2618 35 20 , , PUNCT iajs-2618 35 21 πœ‚ πœ‚ NOUN iajs-2618 35 22 ) ) PUNCT iajs-2618 35 23 , , PUNCT iajs-2618 35 24 βˆ€ βˆ€ NOUN iajs-2618 35 25 πœ‚ πœ‚ X iajs-2618 35 26 ∈ ∈ PROPN iajs-2618 35 27 𝑉 𝑉 PROPN iajs-2618 35 28 are be AUX iajs-2618 35 29 on on ADP iajs-2618 35 30 𝐼 𝐼 PROPN iajs-2618 35 31 , , PUNCT iajs-2618 35 32 ( ( PUNCT iajs-2618 35 33 5 5 NUM iajs-2618 35 34 ) ) PUNCT iajs-2618 35 35 ( ( PUNCT iajs-2618 35 36 𝑀(0 𝑀(0 NOUN iajs-2618 35 37 ) ) PUNCT iajs-2618 35 38 , , PUNCT iajs-2618 35 39 πœ‚ πœ‚ NOUN iajs-2618 35 40 ) ) PUNCT iajs-2618 35 41 = = SYM iajs-2618 35 42 ( ( PUNCT iajs-2618 35 43 𝑀0 𝑀0 PROPN iajs-2618 35 44 , , PUNCT iajs-2618 35 45 πœ‚ πœ‚ NOUN iajs-2618 35 46 ) ) PUNCT iajs-2618 35 47 in in ADP iajs-2618 35 48 πœ“ πœ“ NOUN iajs-2618 35 49 , , PUNCT iajs-2618 35 50 𝑀0∈ 𝑀0∈ PROPN iajs-2618 35 51 𝑉 𝑉 PROPN iajs-2618 35 52 ( ( PUNCT iajs-2618 35 53 6 6 NUM iajs-2618 35 54 ) ) PUNCT iajs-2618 35 55 ( ( PUNCT iajs-2618 35 56 𝑝(0 𝑝(0 PROPN iajs-2618 35 57 ) ) PUNCT iajs-2618 35 58 , , PUNCT iajs-2618 35 59 πœ‚ πœ‚ NOUN iajs-2618 35 60 ) ) PUNCT iajs-2618 36 1 = = SYM iajs-2618 36 2 ( ( PUNCT iajs-2618 36 3 𝑀1 𝑀1 VERB iajs-2618 36 4 , , PUNCT iajs-2618 36 5 πœ‚ πœ‚ NOUN iajs-2618 36 6 ) ) PUNCT iajs-2618 36 7 in in ADP iajs-2618 36 8 πœ“ πœ“ NOUN iajs-2618 36 9 , , PUNCT iajs-2618 36 10 𝑀1 𝑀1 VERB iajs-2618 36 11 ∈ ∈ PROPN iajs-2618 36 12 𝐿2(πœ‘ 𝐿2(πœ‘ PROPN iajs-2618 36 13 ) ) PUNCT iajs-2618 36 14 , , PUNCT iajs-2618 36 15 ( ( PUNCT iajs-2618 36 16 7 7 X iajs-2618 36 17 ) ) PUNCT iajs-2618 36 18 definition definition NOUN iajs-2618 36 19 ( ( PUNCT iajs-2618 36 20 1),[8 1),[8 NUM iajs-2618 36 21 ] ] X iajs-2618 36 22 : : PUNCT iajs-2618 36 23 a a DET iajs-2618 36 24 point point NOUN iajs-2618 36 25 π‘ βˆ— π‘ βˆ— NOUN iajs-2618 36 26 ∈ ∈ PROPN iajs-2618 36 27 𝐷 𝐷 PROPN iajs-2618 36 28 βŠ‚ βŠ‚ PROPN iajs-2618 36 29 ℝ2 ℝ2 VERB iajs-2618 36 30 is be AUX iajs-2618 36 31 called call VERB iajs-2618 36 32 a a DET iajs-2618 36 33 fixed fix VERB iajs-2618 36 34 point point NOUN iajs-2618 36 35 of of ADP iajs-2618 36 36 the the DET iajs-2618 36 37 function function NOUN iajs-2618 36 38 𝑦 𝑦 PROPN iajs-2618 36 39 ∢ ∢ NOUN iajs-2618 36 40 𝐷 𝐷 PROPN iajs-2618 36 41 ⟢ ⟢ NOUN iajs-2618 36 42 ℝ2 ℝ2 NOUN iajs-2618 36 43 , , PUNCT iajs-2618 36 44 if if SCONJ iajs-2618 36 45 𝑦(π‘ βˆ— 𝑦(π‘ βˆ— NUM iajs-2618 36 46 ) ) PUNCT iajs-2618 36 47 = = SYM iajs-2618 36 48 π‘ βˆ— π‘ βˆ— NOUN iajs-2618 36 49 . . PUNCT iajs-2618 37 1 definition definition NOUN iajs-2618 37 2 2,[8 2,[8 NUM iajs-2618 37 3 ] ] X iajs-2618 37 4 : : PUNCT iajs-2618 37 5 a a DET iajs-2618 37 6 function function NOUN iajs-2618 37 7 𝑦 𝑦 NOUN iajs-2618 37 8 : : PUNCT iajs-2618 37 9 𝐷 𝐷 PROPN iajs-2618 37 10 βŠ‚ βŠ‚ PROPN iajs-2618 37 11 ℝ2 ℝ2 PROPN iajs-2618 37 12 ⟢ ⟢ NOUN iajs-2618 37 13 ℝ2 ℝ2 NOUN iajs-2618 37 14 is be AUX iajs-2618 37 15 called call VERB iajs-2618 37 16 contractive contractive ADJ iajs-2618 37 17 on on ADP iajs-2618 37 18 𝐷 𝐷 PROPN iajs-2618 37 19 if if SCONJ iajs-2618 37 20 for for ADP iajs-2618 37 21 each each DET iajs-2618 37 22 𝑑1 𝑑1 NOUN iajs-2618 37 23 , , PUNCT iajs-2618 37 24 𝑑2 𝑑2 PROPN iajs-2618 37 25 ∈ ∈ PROPN iajs-2618 37 26 𝐷 𝐷 NOUN iajs-2618 37 27 : : PUNCT iajs-2618 37 28 ‖𝑦(𝑑2 ‖𝑦(𝑑2 NUM iajs-2618 37 29 ) ) PUNCT iajs-2618 37 30 βˆ’ βˆ’ PRON iajs-2618 38 1 𝑦(𝑑1)β€– 𝑦(𝑑1)β€– PROPN iajs-2618 38 2 ≀ ≀ PUNCT iajs-2618 38 3 π‘Žβ€–π‘‘2 π‘Žβ€–π‘‘2 PROPN iajs-2618 38 4 βˆ’ βˆ’ ADP iajs-2618 38 5 𝑑1β€– 𝑑1β€– ADJ iajs-2618 38 6 , , PUNCT iajs-2618 38 7 where where SCONJ iajs-2618 38 8 π‘Ž π‘Ž PRON iajs-2618 38 9 ∈ ∈ NOUN iajs-2618 38 10 ( ( PUNCT iajs-2618 38 11 0,1 0,1 NUM iajs-2618 38 12 ) ) PUNCT iajs-2618 38 13 . . PUNCT iajs-2618 39 1 theorem theorem NOUN iajs-2618 39 2 ( ( PUNCT iajs-2618 39 3 3),[8 3),[8 PROPN iajs-2618 39 4 ] ] X iajs-2618 39 5 : : PUNCT iajs-2618 39 6 a a DET iajs-2618 39 7 contractive contractive ADJ iajs-2618 39 8 function function NOUN iajs-2618 39 9 𝑦 𝑦 NOUN iajs-2618 39 10 on on ADP iajs-2618 39 11 a a DET iajs-2618 39 12 complete complete ADJ iajs-2618 39 13 normed normed ADJ iajs-2618 39 14 space space NOUN iajs-2618 39 15 𝐷 𝐷 PROPN iajs-2618 39 16 has have VERB iajs-2618 39 17 a a DET iajs-2618 39 18 unique unique ADJ iajs-2618 39 19 fixed fix VERB iajs-2618 39 20 point point NOUN iajs-2618 39 21 π‘ βˆ— π‘ βˆ— NOUN iajs-2618 39 22 in in ADP iajs-2618 39 23 𝐷. 𝐷. PROPN iajs-2618 39 24 theorem theorem NOUN iajs-2618 39 25 ( ( PUNCT iajs-2618 39 26 4),[9 4),[9 NUM iajs-2618 39 27 ] ] X iajs-2618 39 28 : : PUNCT iajs-2618 39 29 let let AUX iajs-2618 39 30 { { PUNCT iajs-2618 39 31 𝑣𝑛 𝑣𝑛 VERB iajs-2618 39 32 } } PUNCT iajs-2618 39 33 be be AUX iajs-2618 39 34 a a DET iajs-2618 39 35 bounded bounded ADJ iajs-2618 39 36 sequence sequence NOUN iajs-2618 39 37 in in ADP iajs-2618 39 38 the the DET iajs-2618 39 39 space space NOUN iajs-2618 39 40 in in ADP iajs-2618 39 41 𝐿∞(πœ“ 𝐿∞(πœ“ NUM iajs-2618 39 42 ) ) PUNCT iajs-2618 39 43 . . PUNCT iajs-2618 40 1 then then ADV iajs-2618 40 2 , , PUNCT iajs-2618 40 3 there there PRON iajs-2618 40 4 exists exist VERB iajs-2618 40 5 a a DET iajs-2618 40 6 subsequence subsequence NOUN iajs-2618 40 7 { { PUNCT iajs-2618 40 8 𝑛′ 𝑛′ NOUN iajs-2618 40 9 } } PUNCT iajs-2618 40 10 and and CCONJ iajs-2618 40 11 a a DET iajs-2618 40 12 function function NOUN iajs-2618 40 13 𝑣0 𝑣0 PROPN iajs-2618 40 14 ∈ ∈ PROPN iajs-2618 40 15 𝐿∞(πœ“ 𝐿∞(πœ“ VERB iajs-2618 40 16 ) ) PUNCT iajs-2618 40 17 such such ADJ iajs-2618 40 18 that that SCONJ iajs-2618 40 19 , , PUNCT iajs-2618 40 20 in in ADP iajs-2618 40 21 𝐿∞(πœ“ 𝐿∞(πœ“ NUM iajs-2618 40 22 ) ) PUNCT iajs-2618 40 23 then then ADV iajs-2618 41 1 𝑣𝑛′ 𝑣𝑛′ NUM iajs-2618 41 2 ⟢ ⟢ NOUN iajs-2618 41 3 𝑣0 𝑣0 NOUN iajs-2618 41 4 . . PUNCT iajs-2618 42 1 121 121 NUM iajs-2618 42 2 ibn ibn PROPN iajs-2618 42 3 al al PROPN iajs-2618 42 4 - - PUNCT iajs-2618 42 5 haitham haitham PROPN iajs-2618 42 6 jour jour X iajs-2618 42 7 . . PROPN iajs-2618 42 8 for for ADP iajs-2618 42 9 pure pure ADJ iajs-2618 42 10 & & CCONJ iajs-2618 42 11 appl appl PROPN iajs-2618 42 12 . . PUNCT iajs-2618 43 1 sci sci PROPN iajs-2618 43 2 . . PROPN iajs-2618 44 1 34 34 NUM iajs-2618 44 2 ( ( PUNCT iajs-2618 44 3 2 2 NUM iajs-2618 44 4 ) ) PUNCT iajs-2618 44 5 2021 2021 NUM iajs-2618 44 6 assumptions assumption NOUN iajs-2618 44 7 2.1 2.1 NUM iajs-2618 44 8 : : PUNCT iajs-2618 44 9 ( ( PUNCT iajs-2618 44 10 i i NOUN iajs-2618 44 11 ) ) PUNCT iajs-2618 44 12 let let VERB iajs-2618 44 13 ΞΊ1 ΞΊ1 NOUN iajs-2618 44 14 and and CCONJ iajs-2618 44 15 ΞΊ2 ΞΊ2 NOUN iajs-2618 44 16 be be VERB iajs-2618 44 17 two two NUM iajs-2618 44 18 positive positive ADJ iajs-2618 44 19 constants constant NOUN iajs-2618 44 20 such such ADJ iajs-2618 44 21 that that SCONJ iajs-2618 44 22 the the DET iajs-2618 44 23 following following NOUN iajs-2618 44 24 are be AUX iajs-2618 44 25 satisfied satisfied ADJ iajs-2618 44 26 : : PUNCT iajs-2618 44 27 a) a) NUM iajs-2618 44 28 ⎹ ⎹ NUM iajs-2618 44 29 (βˆ‡ΞΌ1 (βˆ‡ΞΌ1 NOUN iajs-2618 44 30 , , PUNCT iajs-2618 44 31 βˆ‡ΞΌ2) βˆ‡ΞΌ2) NOUN iajs-2618 44 32 ⎸ ⎸ PRON iajs-2618 44 33 ≀ ≀ NOUN iajs-2618 44 34 ΞΊ1 ΞΊ1 NOUN iajs-2618 44 35 βˆ₯ βˆ₯ PROPN iajs-2618 44 36 βˆ‡ΞΌ1 βˆ‡ΞΌ1 NUM iajs-2618 44 37 βˆ₯1 βˆ₯1 PRON iajs-2618 44 38 βˆ₯ βˆ₯ PUNCT iajs-2618 44 39 βˆ‡ΞΌ2 βˆ‡ΞΌ2 PRON iajs-2618 44 40 βˆ₯1 βˆ₯1 NOUN iajs-2618 44 41 , , PUNCT iajs-2618 44 42 βˆ€ βˆ€ X iajs-2618 44 43 ΞΌ1 ΞΌ1 NOUN iajs-2618 44 44 , , PUNCT iajs-2618 44 45 ΞΌ2 ΞΌ2 NOUN iajs-2618 44 46 ∈ ∈ PROPN iajs-2618 44 47 𝑉 𝑉 PROPN iajs-2618 44 48 b b PROPN iajs-2618 44 49 ) ) PUNCT iajs-2618 44 50 ( ( PUNCT iajs-2618 44 51 βˆ‡ΞΌ βˆ‡ΞΌ PROPN iajs-2618 44 52 , , PUNCT iajs-2618 44 53 βˆ‡ΞΌ βˆ‡ΞΌ PROPN iajs-2618 44 54 ) ) PUNCT iajs-2618 44 55 β‰₯ β‰₯ PROPN iajs-2618 44 56 ΞΊ2 ΞΊ2 NOUN iajs-2618 44 57 βˆ₯ βˆ₯ PUNCT iajs-2618 44 58 βˆ‡ΞΌ βˆ‡ΞΌ PROPN iajs-2618 44 59 βˆ₯1 βˆ₯1 PRON iajs-2618 44 60 2 2 NUM iajs-2618 44 61 , , PUNCT iajs-2618 44 62 βˆ€ βˆ€ X iajs-2618 44 63 μ∈ μ∈ PROPN iajs-2618 44 64 𝑉 𝑉 PROPN iajs-2618 44 65 ( ( PUNCT iajs-2618 44 66 ii ii NOUN iajs-2618 44 67 ) ) PUNCT iajs-2618 44 68 the the DET iajs-2618 44 69 function function NOUN iajs-2618 44 70 β„Ž β„Ž PROPN iajs-2618 44 71 is be AUX iajs-2618 44 72 defined define VERB iajs-2618 44 73 on on ADP iajs-2618 44 74 πœ‘ πœ‘ DET iajs-2618 44 75 Γ— Γ— NOUN iajs-2618 44 76 ℝ ℝ PROPN iajs-2618 44 77 , , PUNCT iajs-2618 44 78 continuous continuous ADJ iajs-2618 44 79 with with ADP iajs-2618 44 80 respect respect NOUN iajs-2618 44 81 to to ADP iajs-2618 44 82 𝑀𝑗 𝑀𝑗 VERB iajs-2618 44 83 𝑛 𝑛 PRON iajs-2618 44 84 satisfies satisfie NOUN iajs-2618 44 85 the the DET iajs-2618 44 86 following following NOUN iajs-2618 44 87 : : PUNCT iajs-2618 44 88 a)|β„Ž( a)|β„Ž( ADP iajs-2618 44 89 οΏ½ οΏ½ NOUN iajs-2618 44 90 βƒ— βƒ— NOUN iajs-2618 44 91 οΏ½ οΏ½ PROPN iajs-2618 44 92 , , PUNCT iajs-2618 44 93 𝑑 𝑑 PROPN iajs-2618 44 94 , , PUNCT iajs-2618 44 95 𝑀)| 𝑀)| ADJ iajs-2618 44 96 ≀ ≀ PUNCT iajs-2618 44 97 𝛽 𝛽 PROPN iajs-2618 44 98 ( ( PUNCT iajs-2618 44 99 οΏ½ οΏ½ PROPN iajs-2618 44 100 βƒ— βƒ— NOUN iajs-2618 44 101 οΏ½ οΏ½ PROPN iajs-2618 44 102 , , PUNCT iajs-2618 44 103 𝑑 𝑑 PROPN iajs-2618 44 104 ) ) PUNCT iajs-2618 44 105 + + CCONJ iajs-2618 44 106 𝛿 𝛿 PROPN iajs-2618 44 107 |𝑀| |𝑀| PROPN iajs-2618 44 108 where where SCONJ iajs-2618 44 109 Ξ΄ Ξ΄ PROPN iajs-2618 44 110 > > X iajs-2618 44 111 0 0 PROPN iajs-2618 44 112 , , PUNCT iajs-2618 44 113 𝑀 𝑀 ADP iajs-2618 44 114 ∈ ∈ NOUN iajs-2618 44 115 πœ‘ πœ‘ NOUN iajs-2618 44 116 and and CCONJ iajs-2618 44 117 𝛽 𝛽 NOUN iajs-2618 44 118 ∈ ∈ PROPN iajs-2618 44 119 𝐿2(πœ‘ 𝐿2(πœ‘ PROPN iajs-2618 44 120 ) ) PUNCT iajs-2618 44 121 . . PUNCT iajs-2618 45 1 b b X iajs-2618 45 2 ) ) PUNCT iajs-2618 45 3 |β„Ž( |β„Ž( NOUN iajs-2618 45 4 οΏ½ οΏ½ NOUN iajs-2618 45 5 βƒ— βƒ— NOUN iajs-2618 45 6 οΏ½ οΏ½ PROPN iajs-2618 45 7 , , PUNCT iajs-2618 45 8 𝑑 𝑑 PROPN iajs-2618 45 9 , , PUNCT iajs-2618 45 10 𝑀1 𝑀1 VERB iajs-2618 45 11 ) ) PUNCT iajs-2618 45 12 βˆ’ βˆ’ ADP iajs-2618 45 13 β„Ž( β„Ž( PROPN iajs-2618 45 14 οΏ½ οΏ½ NOUN iajs-2618 45 15 βƒ— βƒ— PROPN iajs-2618 45 16 οΏ½ οΏ½ PROPN iajs-2618 45 17 , , PUNCT iajs-2618 45 18 𝑑 𝑑 PROPN iajs-2618 45 19 , , PUNCT iajs-2618 45 20 𝑀2)| 𝑀2)| PROPN iajs-2618 45 21 ≀ ≀ ADJ iajs-2618 45 22 𝐿|𝑀1 𝐿|𝑀1 NOUN iajs-2618 45 23 βˆ’ βˆ’ PROPN iajs-2618 45 24 𝑀2| 𝑀2| PROPN iajs-2618 45 25 , , PUNCT iajs-2618 45 26 where where SCONJ iajs-2618 45 27 𝐿 𝐿 PROPN iajs-2618 45 28 is be AUX iajs-2618 45 29 a a DET iajs-2618 45 30 lipchitz lipchitz NOUN iajs-2618 45 31 constant constant ADJ iajs-2618 45 32 and and CCONJ iajs-2618 45 33 𝑀1 𝑀1 VERB iajs-2618 45 34 , , PUNCT iajs-2618 45 35 𝑀2 𝑀2 NOUN iajs-2618 45 36 ∈ ∈ NOUN iajs-2618 46 1 ℝ. ℝ. PROPN iajs-2618 46 2 3 3 NUM iajs-2618 46 3 . . X iajs-2618 47 1 discretization discretization NOUN iajs-2618 47 2 of of ADP iajs-2618 47 3 the the DET iajs-2618 47 4 continuous continuous ADJ iajs-2618 47 5 equation equation NOUN iajs-2618 47 6 ( ( PUNCT iajs-2618 47 7 coe coe PROPN iajs-2618 47 8 ): ): PUNCT iajs-2618 47 9 the the DET iajs-2618 47 10 wef wef PROPN iajs-2618 47 11 of of ADP iajs-2618 47 12 ( ( PUNCT iajs-2618 47 13 ( ( PUNCT iajs-2618 47 14 5)(7 5)(7 NUM iajs-2618 47 15 ) ) PUNCT iajs-2618 47 16 ) ) PUNCT iajs-2618 47 17 is be AUX iajs-2618 47 18 discretized discretize VERB iajs-2618 47 19 by by ADP iajs-2618 47 20 using use VERB iajs-2618 47 21 the the DET iajs-2618 47 22 gfeme gfeme NOUN iajs-2618 47 23 , , PUNCT iajs-2618 47 24 let let VERB iajs-2618 47 25 πœ‘ πœ‘ PRON iajs-2618 47 26 be be AUX iajs-2618 47 27 divided divide VERB iajs-2618 47 28 into into ADP iajs-2618 47 29 sub sub NOUN iajs-2618 47 30 regions region NOUN iajs-2618 47 31 πœ‘π‘–π‘— πœ‘π‘–π‘— VERB iajs-2618 47 32 = = SYM iajs-2618 47 33 πœ“π‘– πœ“π‘– X iajs-2618 47 34 𝑛 𝑛 DET iajs-2618 47 35 Γ— Γ— NOUN iajs-2618 47 36 𝐼𝑗 𝐼𝑗 PROPN iajs-2618 47 37 𝑛 𝑛 PRON iajs-2618 47 38 , , PUNCT iajs-2618 47 39 let let AUX iajs-2618 47 40 { { PUNCT iajs-2618 47 41 πœ“π‘– πœ“π‘– ADP iajs-2618 47 42 𝑛}𝑖=1 𝑛}𝑖=1 NOUN iajs-2618 47 43 𝑁(𝑛 𝑁(𝑛 X iajs-2618 47 44 ) ) PUNCT iajs-2618 47 45 be be AUX iajs-2618 47 46 a a DET iajs-2618 47 47 triangulation triangulation NOUN iajs-2618 47 48 of of ADP iajs-2618 47 49 οΏ½ οΏ½ NOUN iajs-2618 47 50 Μ… Μ… NOUN iajs-2618 47 51 οΏ½ οΏ½ PROPN iajs-2618 47 52 and and CCONJ iajs-2618 47 53 { { PUNCT iajs-2618 47 54 𝐼𝑗 𝐼𝑗 PROPN iajs-2618 47 55 𝑛}𝑗=0 𝑛}𝑗=0 PROPN iajs-2618 47 56 be be AUX iajs-2618 47 57 a a DET iajs-2618 47 58 subdivision subdivision NOUN iajs-2618 47 59 of of ADP iajs-2618 47 60 the the DET iajs-2618 47 61 interval interval NOUN iajs-2618 47 62 𝐼 𝐼 ADP iajs-2618 47 63 Μ… Μ… NOUN iajs-2618 47 64 into into ADP iajs-2618 47 65 π‘Œ(n π‘Œ(n NOUN iajs-2618 47 66 ) ) PUNCT iajs-2618 47 67 intervals interval NOUN iajs-2618 47 68 , , PUNCT iajs-2618 47 69 then then ADV iajs-2618 47 70 𝐼𝑗 𝐼𝑗 NOUN iajs-2618 47 71 = = SYM iajs-2618 47 72 𝐼𝑗 𝐼𝑗 NOUN iajs-2618 47 73 𝑛 𝑛 PRON iajs-2618 47 74 ≔ ≔ NOUN iajs-2618 47 75 [ [ PUNCT iajs-2618 47 76 𝑑𝑗 𝑑𝑗 PART iajs-2618 47 77 𝑛 𝑛 PROPN iajs-2618 47 78 , , PUNCT iajs-2618 47 79 𝑑𝑗+1 𝑑𝑗+1 PROPN iajs-2618 47 80 𝑛 𝑛 PROPN iajs-2618 47 81 ] ] PUNCT iajs-2618 47 82 has have VERB iajs-2618 47 83 the the DET iajs-2618 47 84 same same ADJ iajs-2618 47 85 length length NOUN iajs-2618 47 86 βˆ†π‘‘ βˆ†π‘‘ PROPN iajs-2618 47 87 = = SYM iajs-2618 47 88 𝑇 𝑇 PROPN iajs-2618 47 89 π‘Œ π‘Œ PROPN iajs-2618 47 90 , , PUNCT iajs-2618 47 91 also also ADV iajs-2618 47 92 , , PUNCT iajs-2618 47 93 let let VERB iajs-2618 47 94 𝑉𝑛 𝑉𝑛 PRON iajs-2618 47 95 βŠ‚ βŠ‚ ADJ iajs-2618 47 96 v v X iajs-2618 47 97 = = SYM iajs-2618 47 98 𝐻0 𝐻0 PROPN iajs-2618 47 99 1(πœ“ 1(πœ“ NUM iajs-2618 47 100 ) ) PUNCT iajs-2618 47 101 be be AUX iajs-2618 47 102 the the DET iajs-2618 47 103 space space NOUN iajs-2618 47 104 of of ADP iajs-2618 47 105 piecewise piecewise NOUN iajs-2618 47 106 affine affine NOUN iajs-2618 47 107 functions function NOUN iajs-2618 47 108 in in ADP iajs-2618 47 109 πœ“. πœ“. PROPN iajs-2618 47 110 now now ADV iajs-2618 47 111 , , PUNCT iajs-2618 48 1 the the DET iajs-2618 48 2 discrete discrete ADJ iajs-2618 48 3 equations equation NOUN iajs-2618 48 4 ( ( PUNCT iajs-2618 48 5 des des PROPN iajs-2618 48 6 ) ) PUNCT iajs-2618 48 7 , , PUNCT iajs-2618 48 8 where where SCONJ iajs-2618 48 9 βˆ€ βˆ€ NOUN iajs-2618 48 10 πœ‚ πœ‚ X iajs-2618 48 11 ∈ ∈ PROPN iajs-2618 48 12 𝑉𝑛 𝑉𝑛 PRON iajs-2618 48 13 are be AUX iajs-2618 48 14 written write VERB iajs-2618 48 15 as as SCONJ iajs-2618 48 16 follows follow VERB iajs-2618 48 17 : : PUNCT iajs-2618 48 18 βŸ¨π‘π‘—+1 βŸ¨π‘π‘—+1 PROPN iajs-2618 48 19 𝑛 𝑛 PRON iajs-2618 48 20 βˆ’ βˆ’ PROPN iajs-2618 48 21 𝑝𝑗 𝑝𝑗 PROPN iajs-2618 48 22 𝑛 𝑛 PROPN iajs-2618 48 23 , , PUNCT iajs-2618 48 24 πœ‚ πœ‚ DET iajs-2618 48 25 ⟩ ⟩ NOUN iajs-2618 48 26 + + X iajs-2618 48 27 Ξ΄t Ξ΄t X iajs-2618 48 28 ( ( PUNCT iajs-2618 48 29 βˆ‡π‘€π‘—+1 βˆ‡π‘€π‘—+1 ADP iajs-2618 48 30 𝑛 𝑛 PRON iajs-2618 48 31 , , PUNCT iajs-2618 48 32 βˆ‡ βˆ‡ X iajs-2618 48 33 πœ‚ πœ‚ NOUN iajs-2618 48 34 ) ) PUNCT iajs-2618 49 1 + + NUM iajs-2618 49 2 Ξ΄t Ξ΄t X iajs-2618 49 3 ( ( PUNCT iajs-2618 49 4 𝑀𝑗+1 𝑀𝑗+1 NUM iajs-2618 49 5 𝑛 𝑛 PRON iajs-2618 49 6 , , PUNCT iajs-2618 49 7 πœ‚ πœ‚ NOUN iajs-2618 49 8 ) ) PUNCT iajs-2618 49 9 = = SYM iajs-2618 49 10 Ξ΄t Ξ΄t NOUN iajs-2618 49 11 ( ( PUNCT iajs-2618 49 12 β„Ž(𝑀𝑗+1 β„Ž(𝑀𝑗+1 PROPN iajs-2618 49 13 𝑛 𝑛 PRON iajs-2618 49 14 ) ) PUNCT iajs-2618 49 15 , , PUNCT iajs-2618 49 16 πœ‚ πœ‚ X iajs-2618 49 17 ) ) PUNCT iajs-2618 49 18 ( ( PUNCT iajs-2618 49 19 8) 8) NUM iajs-2618 49 20 𝑀𝑗+1 𝑀𝑗+1 ADP iajs-2618 49 21 𝑛 𝑛 PRON iajs-2618 49 22 βˆ’ βˆ’ NOUN iajs-2618 49 23 𝑀𝑗 𝑀𝑗 NOUN iajs-2618 49 24 𝑛 𝑛 NOUN iajs-2618 49 25 = = SYM iajs-2618 49 26 Ξ΄t Ξ΄t X iajs-2618 49 27 𝑝𝑗+1 𝑝𝑗+1 PROPN iajs-2618 49 28 𝑛 𝑛 PROPN iajs-2618 49 29 ( ( PUNCT iajs-2618 49 30 9 9 NUM iajs-2618 49 31 ) ) PUNCT iajs-2618 49 32 ( ( PUNCT iajs-2618 49 33 𝑀(0 𝑀(0 NOUN iajs-2618 49 34 ) ) PUNCT iajs-2618 49 35 , , PUNCT iajs-2618 49 36 πœ‚ πœ‚ NOUN iajs-2618 49 37 ) ) PUNCT iajs-2618 49 38 = = SYM iajs-2618 49 39 ( ( PUNCT iajs-2618 49 40 𝑀0 𝑀0 PROPN iajs-2618 49 41 , , PUNCT iajs-2618 49 42 πœ‚ πœ‚ NOUN iajs-2618 49 43 ) ) PUNCT iajs-2618 49 44 in in ADP iajs-2618 49 45 πœ“ πœ“ PROPN iajs-2618 49 46 ( ( PUNCT iajs-2618 49 47 10 10 NUM iajs-2618 49 48 ) ) PUNCT iajs-2618 49 49 ( ( PUNCT iajs-2618 49 50 𝑝(0 𝑝(0 PROPN iajs-2618 49 51 ) ) PUNCT iajs-2618 49 52 , , PUNCT iajs-2618 49 53 πœ‚ πœ‚ NOUN iajs-2618 49 54 ) ) PUNCT iajs-2618 49 55 = = SYM iajs-2618 49 56 ( ( PUNCT iajs-2618 49 57 𝑀1 𝑀1 VERB iajs-2618 49 58 , , PUNCT iajs-2618 49 59 πœ‚ πœ‚ NOUN iajs-2618 49 60 ) ) PUNCT iajs-2618 49 61 in in ADP iajs-2618 49 62 πœ“ πœ“ PROPN iajs-2618 49 63 ( ( PUNCT iajs-2618 49 64 11 11 NUM iajs-2618 49 65 ) ) PUNCT iajs-2618 49 66 where where SCONJ iajs-2618 49 67 , , PUNCT iajs-2618 49 68 𝑀0 𝑀0 PROPN iajs-2618 49 69 ∈ ∈ PROPN iajs-2618 49 70 𝑉 𝑉 PROPN iajs-2618 49 71 , , PUNCT iajs-2618 49 72 𝑀1 𝑀1 VERB iajs-2618 49 73 ∈ ∈ PROPN iajs-2618 49 74 𝐿2(πœ“ 𝐿2(πœ“ NOUN iajs-2618 49 75 ) ) PUNCT iajs-2618 49 76 , , PUNCT iajs-2618 49 77 and and CCONJ iajs-2618 49 78 𝑀𝑗 𝑀𝑗 ADP iajs-2618 49 79 𝑛 𝑛 NOUN iajs-2618 49 80 = = PUNCT iajs-2618 49 81 𝑀 𝑀 X iajs-2618 49 82 𝑛(π‘₯ 𝑛(π‘₯ PROPN iajs-2618 49 83 , , PUNCT iajs-2618 49 84 𝑑𝑗 𝑑𝑗 PROPN iajs-2618 49 85 𝑛 𝑛 PROPN iajs-2618 49 86 ) ) PUNCT iajs-2618 49 87 , , PUNCT iajs-2618 49 88 𝑝𝑗 𝑝𝑗 NOUN iajs-2618 49 89 𝑛 𝑛 PROPN iajs-2618 49 90 = = SYM iajs-2618 49 91 𝑝 𝑝 PROPN iajs-2618 49 92 𝑛( 𝑛( NUM iajs-2618 49 93 οΏ½ οΏ½ NOUN iajs-2618 49 94 βƒ— βƒ— PROPN iajs-2618 49 95 οΏ½ οΏ½ PROPN iajs-2618 49 96 , , PUNCT iajs-2618 49 97 𝑑𝑗 𝑑𝑗 PROPN iajs-2618 49 98 𝑛 𝑛 PROPN iajs-2618 49 99 ) ) PUNCT iajs-2618 49 100 ∈ ∈ PROPN iajs-2618 49 101 𝑉𝑛 𝑉𝑛 PROPN iajs-2618 49 102 , , PUNCT iajs-2618 49 103 βˆ€ βˆ€ X iajs-2618 49 104 𝑗 𝑗 NOUN iajs-2618 49 105 = = SYM iajs-2618 49 106 0,1 0,1 NUM iajs-2618 49 107 , , PUNCT iajs-2618 49 108 … … PUNCT iajs-2618 49 109 , , PUNCT iajs-2618 49 110 π‘Œ π‘Œ PROPN iajs-2618 49 111 βˆ’ βˆ’ PROPN iajs-2618 49 112 1 1 NUM iajs-2618 49 113 . . X iajs-2618 49 114 4 4 NUM iajs-2618 49 115 . . X iajs-2618 50 1 the the DET iajs-2618 50 2 apps app NOUN iajs-2618 50 3 of of ADP iajs-2618 50 4 the the DET iajs-2618 50 5 nlhybvp nlhybvp NOUN iajs-2618 50 6 : : PUNCT iajs-2618 50 7 to to PART iajs-2618 50 8 find find VERB iajs-2618 50 9 the the DET iajs-2618 50 10 apps app NOUN iajs-2618 50 11 οΏ½ οΏ½ PROPN iajs-2618 50 12 Μ… Μ… NOUN iajs-2618 50 13 οΏ½ οΏ½ NOUN iajs-2618 50 14 𝑛 𝑛 NOUN iajs-2618 50 15 = = SYM iajs-2618 50 16 ( ( PUNCT iajs-2618 50 17 𝑀0 𝑀0 NOUN iajs-2618 50 18 𝑛 𝑛 PROPN iajs-2618 50 19 , , PUNCT iajs-2618 50 20 𝑀1 𝑀1 VERB iajs-2618 50 21 𝑛 𝑛 ADP iajs-2618 50 22 , , PUNCT iajs-2618 50 23 … … PUNCT iajs-2618 50 24 , , PUNCT iajs-2618 50 25 π‘€π‘Œ π‘€π‘Œ NOUN iajs-2618 50 26 𝑛 𝑛 NOUN iajs-2618 50 27 ) ) PUNCT iajs-2618 50 28 for for ADP iajs-2618 50 29 the the DET iajs-2618 50 30 des des X iajs-2618 50 31 ( ( PUNCT iajs-2618 50 32 8 8 NUM iajs-2618 50 33 - - SYM iajs-2618 50 34 11 11 NUM iajs-2618 50 35 ) ) PUNCT iajs-2618 50 36 , , PUNCT iajs-2618 50 37 the the DET iajs-2618 50 38 mgfeim mgfeim NOUN iajs-2618 50 39 is be AUX iajs-2618 50 40 used use VERB iajs-2618 50 41 through through ADP iajs-2618 50 42 the the DET iajs-2618 50 43 following follow VERB iajs-2618 50 44 steps step NOUN iajs-2618 50 45 : : PUNCT iajs-2618 50 46 ( ( PUNCT iajs-2618 50 47 1 1 X iajs-2618 50 48 ) ) PUNCT iajs-2618 50 49 let let VERB iajs-2618 50 50 { { PUNCT iajs-2618 50 51 πœ‚π‘– πœ‚π‘– VERB iajs-2618 50 52 ∢ ∢ VERB iajs-2618 50 53 𝑖 𝑖 NOUN iajs-2618 50 54 = = SYM iajs-2618 50 55 1,2 1,2 NUM iajs-2618 50 56 , , PUNCT iajs-2618 50 57 … … PUNCT iajs-2618 50 58 . . PUNCT iajs-2618 51 1 𝑁 𝑁 NOUN iajs-2618 51 2 , , PUNCT iajs-2618 51 3 π‘€π‘–π‘‘β„Ž π‘€π‘–π‘‘β„Ž NOUN iajs-2618 51 4 πœ‚π‘–( πœ‚π‘–( PRON iajs-2618 51 5 οΏ½ οΏ½ PROPN iajs-2618 51 6 βƒ— βƒ— NOUN iajs-2618 51 7 οΏ½ οΏ½ PROPN iajs-2618 51 8 ) ) PUNCT iajs-2618 51 9 = = SYM iajs-2618 51 10 0 0 NUM iajs-2618 51 11 , , PUNCT iajs-2618 51 12 π‘œπ‘› π‘œπ‘› PROPN iajs-2618 51 13 βˆ‚πœ“ βˆ‚πœ“ PROPN iajs-2618 51 14 } } PUNCT iajs-2618 51 15 be be AUX iajs-2618 51 16 a a DET iajs-2618 51 17 basis basis NOUN iajs-2618 51 18 of of ADP iajs-2618 51 19 𝑉𝑛 𝑉𝑛 PROPN iajs-2618 51 20 , , PUNCT iajs-2618 51 21 and and CCONJ iajs-2618 51 22 by by ADP iajs-2618 51 23 using use VERB iajs-2618 51 24 the the DET iajs-2618 51 25 gfeme gfeme NOUN iajs-2618 51 26 , , PUNCT iajs-2618 51 27 let let VERB iajs-2618 51 28 οΏ½ οΏ½ PRON iajs-2618 51 29 Μ… Μ… VERB iajs-2618 51 30 οΏ½ οΏ½ NOUN iajs-2618 51 31 𝑛( 𝑛( NOUN iajs-2618 51 32 οΏ½ οΏ½ NOUN iajs-2618 51 33 βƒ— βƒ— PROPN iajs-2618 51 34 οΏ½ οΏ½ PROPN iajs-2618 51 35 , , PUNCT iajs-2618 51 36 𝑑𝑗 𝑑𝑗 PROPN iajs-2618 51 37 𝑛 𝑛 PROPN iajs-2618 51 38 ) ) PUNCT iajs-2618 51 39 ( ( PUNCT iajs-2618 51 40 with with ADP iajs-2618 51 41 οΏ½ οΏ½ NOUN iajs-2618 51 42 Μ… Μ… NOUN iajs-2618 51 43 οΏ½ οΏ½ NOUN iajs-2618 51 44 𝑑 𝑑 PROPN iajs-2618 51 45 𝑛( 𝑛( NUM iajs-2618 51 46 οΏ½ οΏ½ NOUN iajs-2618 51 47 βƒ— βƒ— PROPN iajs-2618 51 48 οΏ½ οΏ½ PROPN iajs-2618 51 49 , , PUNCT iajs-2618 51 50 𝑑𝑗 𝑑𝑗 PROPN iajs-2618 51 51 𝑛 𝑛 PROPN iajs-2618 51 52 ) ) PUNCT iajs-2618 51 53 = = SYM iajs-2618 51 54 οΏ½ οΏ½ PROPN iajs-2618 51 55 Μ… Μ… NOUN iajs-2618 51 56 οΏ½ οΏ½ PROPN iajs-2618 51 57 ( ( PUNCT iajs-2618 51 58 οΏ½ οΏ½ PROPN iajs-2618 51 59 βƒ— βƒ— NOUN iajs-2618 51 60 οΏ½ οΏ½ PROPN iajs-2618 51 61 , , PUNCT iajs-2618 51 62 𝑑𝑗 𝑑𝑗 PROPN iajs-2618 51 63 𝑛))be 𝑛))be NOUN iajs-2618 51 64 an an DET iajs-2618 51 65 apps app NOUN iajs-2618 51 66 of of ADP iajs-2618 51 67 ( ( PUNCT iajs-2618 51 68 8 8 NUM iajs-2618 51 69 - - SYM iajs-2618 51 70 11 11 NUM iajs-2618 51 71 ) ) PUNCT iajs-2618 51 72 such such ADJ iajs-2618 51 73 that that SCONJ iajs-2618 51 74 οΏ½ οΏ½ PROPN iajs-2618 51 75 Μ… Μ… NOUN iajs-2618 51 76 οΏ½ οΏ½ PROPN iajs-2618 51 77 𝑛( 𝑛( NOUN iajs-2618 51 78 οΏ½ οΏ½ NOUN iajs-2618 51 79 βƒ— βƒ— PROPN iajs-2618 51 80 οΏ½ οΏ½ PROPN iajs-2618 51 81 , , PUNCT iajs-2618 51 82 𝑑𝑗 𝑑𝑗 PROPN iajs-2618 51 83 𝑛 𝑛 PROPN iajs-2618 51 84 ) ) PUNCT iajs-2618 51 85 = = PUNCT iajs-2618 52 1 βˆ‘ βˆ‘ PUNCT iajs-2618 52 2 π‘Ÿπ‘˜ π‘Ÿπ‘˜ ADP iajs-2618 52 3 𝑗𝑁 𝑗𝑁 ADJ iajs-2618 52 4 π‘˜=1 π‘˜=1 X iajs-2618 52 5 πœ‚π‘– πœ‚π‘– PROPN iajs-2618 52 6 and and CCONJ iajs-2618 52 7 οΏ½ οΏ½ PROPN iajs-2618 52 8 Μ… Μ… NOUN iajs-2618 52 9 οΏ½ οΏ½ PROPN iajs-2618 52 10 𝑛( 𝑛( NOUN iajs-2618 52 11 οΏ½ οΏ½ NOUN iajs-2618 52 12 βƒ— βƒ— PROPN iajs-2618 52 13 οΏ½ οΏ½ PROPN iajs-2618 52 14 , , PUNCT iajs-2618 52 15 𝑑𝑗 𝑑𝑗 PROPN iajs-2618 52 16 𝑛 𝑛 PROPN iajs-2618 52 17 ) ) PUNCT iajs-2618 52 18 = = PUNCT iajs-2618 53 1 βˆ‘ βˆ‘ PROPN iajs-2618 53 2 π‘’π‘˜ π‘’π‘˜ ADP iajs-2618 53 3 𝑗𝑁 𝑗𝑁 ADJ iajs-2618 53 4 π‘˜=1 π‘˜=1 X iajs-2618 53 5 πœ‚π‘– πœ‚π‘– NOUN iajs-2618 53 6 βˆ€ βˆ€ X iajs-2618 53 7 πœ‚π‘– πœ‚π‘– ADP iajs-2618 53 8 ∈ ∈ PROPN iajs-2618 53 9 𝑉𝑛 𝑉𝑛 PROPN iajs-2618 53 10 , , PUNCT iajs-2618 53 11 where where SCONJ iajs-2618 53 12 π‘Ÿπ‘˜ π‘Ÿπ‘˜ PROPN iajs-2618 53 13 𝑗 𝑗 VERB iajs-2618 53 14 = = X iajs-2618 53 15 π‘Ÿπ‘˜(𝑑𝑗 π‘Ÿπ‘˜(𝑑𝑗 PRON iajs-2618 53 16 𝑛 𝑛 NOUN iajs-2618 53 17 ) ) PUNCT iajs-2618 53 18 , , PUNCT iajs-2618 53 19 and and CCONJ iajs-2618 53 20 π‘’π‘˜ π‘’π‘˜ ADP iajs-2618 53 21 𝑗 𝑗 PROPN iajs-2618 53 22 = = PUNCT iajs-2618 53 23 π‘’π‘˜(𝑑𝑗 π‘’π‘˜(𝑑𝑗 PROPN iajs-2618 53 24 𝑛 𝑛 NOUN iajs-2618 53 25 ) ) PUNCT iajs-2618 53 26 are be AUX iajs-2618 53 27 unknown unknown ADJ iajs-2618 53 28 constants constant NOUN iajs-2618 53 29 βˆ€π‘— βˆ€π‘— NOUN iajs-2618 53 30 = = SYM iajs-2618 53 31 0,1 0,1 NUM iajs-2618 53 32 , , PUNCT iajs-2618 53 33 … … PUNCT iajs-2618 53 34 , , PUNCT iajs-2618 54 1 π‘Œ π‘Œ PROPN iajs-2618 54 2 βˆ’ βˆ’ PROPN iajs-2618 54 3 1 1 NUM iajs-2618 54 4 . . PUNCT iajs-2618 55 1 ( ( PUNCT iajs-2618 55 2 2 2 X iajs-2618 55 3 ) ) PUNCT iajs-2618 55 4 using use VERB iajs-2618 55 5 the the DET iajs-2618 55 6 apps app NOUN iajs-2618 55 7 in in ADP iajs-2618 55 8 ( ( PUNCT iajs-2618 55 9 8 8 NUM iajs-2618 55 10 - - SYM iajs-2618 55 11 11 11 NUM iajs-2618 55 12 ) ) PUNCT iajs-2618 55 13 to to PART iajs-2618 55 14 get get VERB iajs-2618 55 15 , , PUNCT iajs-2618 55 16 βˆ€π‘— βˆ€π‘— NOUN iajs-2618 55 17 = = SYM iajs-2618 55 18 0,1 0,1 NUM iajs-2618 55 19 , , PUNCT iajs-2618 55 20 … … PUNCT iajs-2618 55 21 , , PUNCT iajs-2618 55 22 π‘Œ π‘Œ PROPN iajs-2618 55 23 βˆ’ βˆ’ PROPN iajs-2618 55 24 1 1 NUM iajs-2618 55 25 : : PUNCT iajs-2618 55 26 ( ( PUNCT iajs-2618 55 27 𝑀 𝑀 PROPN iajs-2618 55 28 + + CCONJ iajs-2618 55 29 ( ( PUNCT iajs-2618 55 30 Ξ΄t)2𝑄 Ξ΄t)2𝑄 NOUN iajs-2618 55 31 ) ) PUNCT iajs-2618 55 32 𝑅𝑗+1 𝑅𝑗+1 VERB iajs-2618 55 33 = = SYM iajs-2618 55 34 𝑀𝑅𝑗 𝑀𝑅𝑗 NOUN iajs-2618 55 35 + + CCONJ iajs-2618 55 36 ( ( PUNCT iajs-2618 55 37 Ξ΄t Ξ΄t NOUN iajs-2618 55 38 ) ) PUNCT iajs-2618 55 39 π‘€π‘ˆπ‘— π‘€π‘ˆπ‘— NOUN iajs-2618 56 1 + + CCONJ iajs-2618 56 2 ( ( PUNCT iajs-2618 56 3 Ξ΄t)2 Ξ΄t)2 PROPN iajs-2618 56 4 οΏ½ οΏ½ PROPN iajs-2618 56 5 βƒ—βƒ— βƒ—βƒ— PROPN iajs-2618 56 6 οΏ½ οΏ½ PROPN iajs-2618 56 7 ( ( PUNCT iajs-2618 56 8 𝑑𝑗 𝑑𝑗 PROPN iajs-2618 56 9 𝑛 𝑛 PROPN iajs-2618 56 10 , , PUNCT iajs-2618 56 11 οΏ½ οΏ½ PROPN iajs-2618 56 12 βƒ— βƒ— NOUN iajs-2618 56 13 οΏ½ οΏ½ NOUN iajs-2618 56 14 𝑇𝑅𝑗+1 𝑇𝑅𝑗+1 NOUN iajs-2618 56 15 ) ) PUNCT iajs-2618 56 16 ( ( PUNCT iajs-2618 56 17 12 12 NUM iajs-2618 56 18 ) ) PUNCT iajs-2618 56 19 π‘ˆπ‘—+1 π‘ˆπ‘—+1 NOUN iajs-2618 56 20 = = SYM iajs-2618 56 21 1 1 NUM iajs-2618 56 22 Ξ΄t Ξ΄t NOUN iajs-2618 56 23 ( ( PUNCT iajs-2618 56 24 𝑅𝑗+1 𝑅𝑗+1 ADP iajs-2618 56 25 βˆ’ βˆ’ PROPN iajs-2618 56 26 𝑅𝑗 𝑅𝑗 PROPN iajs-2618 56 27 ) ) PUNCT iajs-2618 56 28 ( ( PUNCT iajs-2618 56 29 13 13 X iajs-2618 56 30 ) ) PUNCT iajs-2618 56 31 𝑀𝑅0 𝑀𝑅0 PROPN iajs-2618 56 32 = = PUNCT iajs-2618 56 33 𝑠0 𝑠0 NOUN iajs-2618 56 34 ( ( PUNCT iajs-2618 56 35 14 14 NUM iajs-2618 56 36 ) ) PUNCT iajs-2618 56 37 π‘€π‘ˆ0 π‘€π‘ˆ0 PROPN iajs-2618 56 38 = = SYM iajs-2618 56 39 𝑠1 𝑠1 PROPN iajs-2618 56 40 ( ( PUNCT iajs-2618 56 41 15 15 NUM iajs-2618 56 42 ) ) PUNCT iajs-2618 56 43 where where SCONJ iajs-2618 56 44 , , PUNCT iajs-2618 56 45 𝑀 𝑀 PROPN iajs-2618 56 46 = = PUNCT iajs-2618 56 47 ( ( PUNCT iajs-2618 56 48 π‘šπ‘–π‘˜)𝑁×𝑁 π‘šπ‘–π‘˜)𝑁×𝑁 NOUN iajs-2618 56 49 , , PUNCT iajs-2618 56 50 π‘šπ‘–π‘˜ π‘šπ‘–π‘˜ X iajs-2618 56 51 = = SYM iajs-2618 56 52 ( ( PUNCT iajs-2618 56 53 πœ‚π‘˜ πœ‚π‘˜ ADP iajs-2618 56 54 , , PUNCT iajs-2618 56 55 πœ‚π‘– πœ‚π‘– PROPN iajs-2618 56 56 ) ) PUNCT iajs-2618 56 57 , , PUNCT iajs-2618 56 58 𝑄 𝑄 PROPN iajs-2618 56 59 = = SYM iajs-2618 56 60 ( ( PUNCT iajs-2618 56 61 π‘žπ‘–π‘˜)𝑁×𝑁 π‘žπ‘–π‘˜)𝑁×𝑁 PROPN iajs-2618 56 62 , , PUNCT iajs-2618 56 63 π‘žπ‘–π‘˜ π‘žπ‘–π‘˜ ADJ iajs-2618 56 64 = = SYM iajs-2618 56 65 ( ( PUNCT iajs-2618 56 66 βˆ‡πœ‚π‘˜ βˆ‡πœ‚π‘˜ PROPN iajs-2618 56 67 , , PUNCT iajs-2618 56 68 βˆ‡πœ‚π‘– βˆ‡πœ‚π‘– PROPN iajs-2618 56 69 ) ) PUNCT iajs-2618 56 70 , , PUNCT iajs-2618 56 71 οΏ½ οΏ½ PROPN iajs-2618 56 72 βƒ—βƒ— βƒ—βƒ— PROPN iajs-2618 56 73 οΏ½ οΏ½ PROPN iajs-2618 56 74 = = SYM iajs-2618 56 75 ( ( PUNCT iajs-2618 56 76 𝐿𝑖)𝑁×1 𝐿𝑖)𝑁×1 PROPN iajs-2618 56 77 , , PUNCT iajs-2618 56 78 𝐿𝑖 𝐿𝑖 PROPN iajs-2618 56 79 = = SYM iajs-2618 56 80 ( ( PUNCT iajs-2618 56 81 β„Ž β„Ž PROPN iajs-2618 56 82 ( ( PUNCT iajs-2618 56 83 οΏ½ οΏ½ NOUN iajs-2618 56 84 βƒ— βƒ— NOUN iajs-2618 56 85 οΏ½ οΏ½ NOUN iajs-2618 56 86 𝑇𝑅𝑗+1 𝑇𝑅𝑗+1 NOUN iajs-2618 56 87 ) ) PUNCT iajs-2618 56 88 , , PUNCT iajs-2618 56 89 πœ‚π‘– πœ‚π‘– PROPN iajs-2618 56 90 ) ) PUNCT iajs-2618 56 91 , , PUNCT iajs-2618 56 92 𝑅𝑁×1 𝑅𝑁×1 X iajs-2618 56 93 𝑗 𝑗 X iajs-2618 56 94 = = PUNCT iajs-2618 56 95 ( ( PUNCT iajs-2618 56 96 π‘Ÿ1 π‘Ÿ1 PROPN iajs-2618 56 97 𝑗 𝑗 INTJ iajs-2618 56 98 , , PUNCT iajs-2618 56 99 π‘Ÿ2 π‘Ÿ2 NOUN iajs-2618 56 100 𝑗 𝑗 NOUN iajs-2618 56 101 , , PUNCT iajs-2618 56 102 … … PUNCT iajs-2618 56 103 , , PUNCT iajs-2618 56 104 π‘Ÿπ‘ π‘Ÿπ‘ X iajs-2618 56 105 𝑗 𝑗 NOUN iajs-2618 56 106 ) ) PUNCT iajs-2618 56 107 𝑇 𝑇 PROPN iajs-2618 56 108 , , PUNCT iajs-2618 56 109 π‘ˆπ‘Γ—1 π‘ˆπ‘Γ—1 NOUN iajs-2618 56 110 𝑗 𝑗 NOUN iajs-2618 56 111 = = PUNCT iajs-2618 56 112 ( ( PUNCT iajs-2618 56 113 𝑒1 𝑒1 PROPN iajs-2618 56 114 𝑗 𝑗 INTJ iajs-2618 56 115 , , PUNCT iajs-2618 56 116 𝑒2 𝑒2 PROPN iajs-2618 56 117 𝑗 𝑗 PROPN iajs-2618 56 118 , , PUNCT iajs-2618 56 119 … … PUNCT iajs-2618 56 120 , , PUNCT iajs-2618 56 121 𝑒𝑁 𝑒𝑁 PRON iajs-2618 56 122 𝑗 𝑗 NOUN iajs-2618 56 123 ) ) PUNCT iajs-2618 56 124 𝑇 𝑇 PROPN iajs-2618 56 125 , , PUNCT iajs-2618 56 126 𝑠0 𝑠0 NOUN iajs-2618 56 127 = = SYM iajs-2618 56 128 ( ( PUNCT iajs-2618 56 129 𝑠𝑖 𝑠𝑖 PROPN iajs-2618 56 130 0)𝑁×1 0)𝑁×1 PROPN iajs-2618 56 131 , , PUNCT iajs-2618 56 132 𝑠𝑖 𝑠𝑖 PROPN iajs-2618 56 133 0 0 NUM iajs-2618 57 1 = = SYM iajs-2618 57 2 ( ( PUNCT iajs-2618 57 3 𝑀0 𝑀0 PROPN iajs-2618 57 4 , , PUNCT iajs-2618 57 5 πœ‚π‘– πœ‚π‘– PROPN iajs-2618 57 6 ) ) PUNCT iajs-2618 57 7 , , PUNCT iajs-2618 57 8 𝑠 𝑠 PROPN iajs-2618 57 9 1 1 NUM iajs-2618 57 10 = = SYM iajs-2618 57 11 ( ( PUNCT iajs-2618 57 12 𝑠𝑖 𝑠𝑖 PROPN iajs-2618 57 13 1)𝑁×1 1)𝑁×1 NUM iajs-2618 57 14 and and CCONJ iajs-2618 57 15 𝑠𝑖 𝑠𝑖 PROPN iajs-2618 57 16 1 1 NUM iajs-2618 57 17 = = SYM iajs-2618 57 18 ( ( PUNCT iajs-2618 57 19 𝑀1 𝑀1 PROPN iajs-2618 57 20 , , PUNCT iajs-2618 57 21 πœ‚π‘– πœ‚π‘– PROPN iajs-2618 57 22 ) ) PUNCT iajs-2618 57 23 , , PUNCT iajs-2618 57 24 for for ADP iajs-2618 57 25 each each DET iajs-2618 57 26 𝑖 𝑖 X iajs-2618 57 27 , , PUNCT iajs-2618 57 28 π‘˜ π‘˜ X iajs-2618 57 29 = = SYM iajs-2618 57 30 1,2 1,2 NUM iajs-2618 57 31 , , PUNCT iajs-2618 57 32 … … PUNCT iajs-2618 57 33 , , PUNCT iajs-2618 57 34 𝑁. 𝑁. PROPN iajs-2618 57 35 122 122 NUM iajs-2618 57 36 ibn ibn PROPN iajs-2618 57 37 al al PROPN iajs-2618 57 38 - - PUNCT iajs-2618 57 39 haitham haitham PROPN iajs-2618 57 40 jour jour X iajs-2618 57 41 . . PROPN iajs-2618 58 1 for for ADP iajs-2618 58 2 pure pure ADJ iajs-2618 58 3 & & CCONJ iajs-2618 58 4 appl appl PROPN iajs-2618 58 5 . . PUNCT iajs-2618 59 1 sci sci PROPN iajs-2618 59 2 . . PROPN iajs-2618 60 1 34 34 NUM iajs-2618 60 2 ( ( PUNCT iajs-2618 60 3 2 2 NUM iajs-2618 60 4 ) ) PUNCT iajs-2618 60 5 2021 2021 NUM iajs-2618 60 6 ( ( PUNCT iajs-2618 60 7 3 3 X iajs-2618 60 8 ) ) PUNCT iajs-2618 60 9 system system NOUN iajs-2618 60 10 ( ( PUNCT iajs-2618 60 11 12 12 NUM iajs-2618 60 12 - - SYM iajs-2618 60 13 15 15 NUM iajs-2618 60 14 ) ) PUNCT iajs-2618 60 15 , , PUNCT iajs-2618 60 16 is be AUX iajs-2618 60 17 gnas gnas NOUN iajs-2618 60 18 and and CCONJ iajs-2618 60 19 has have VERB iajs-2618 60 20 a a DET iajs-2618 60 21 unique unique ADJ iajs-2618 60 22 solution solution NOUN iajs-2618 60 23 . . PUNCT iajs-2618 61 1 to to PART iajs-2618 61 2 solve solve VERB iajs-2618 61 3 it it PRON iajs-2618 61 4 , , PUNCT iajs-2618 61 5 we we PRON iajs-2618 61 6 find find VERB iajs-2618 61 7 at at ADP iajs-2618 61 8 first first ADJ iajs-2618 61 9 𝑅0 𝑅0 NOUN iajs-2618 61 10 and and CCONJ iajs-2618 61 11 π‘ˆ0 π‘ˆ0 NOUN iajs-2618 61 12 from from ADP iajs-2618 61 13 solving solve VERB iajs-2618 61 14 ( ( PUNCT iajs-2618 61 15 14 14 NUM iajs-2618 61 16 ) ) PUNCT iajs-2618 61 17 and and CCONJ iajs-2618 61 18 ( ( PUNCT iajs-2618 61 19 15 15 NUM iajs-2618 61 20 ) ) PUNCT iajs-2618 61 21 respectively respectively ADV iajs-2618 61 22 , , PUNCT iajs-2618 61 23 then then ADV iajs-2618 61 24 , , PUNCT iajs-2618 61 25 the the DET iajs-2618 61 26 pt pt PROPN iajs-2618 61 27 and and CCONJ iajs-2618 61 28 the the DET iajs-2618 61 29 ct ct NOUN iajs-2618 61 30 are be AUX iajs-2618 61 31 utilized utilize VERB iajs-2618 61 32 to to PART iajs-2618 61 33 solve solve VERB iajs-2618 61 34 ( ( PUNCT iajs-2618 61 35 12 12 NUM iajs-2618 61 36 ) ) PUNCT iajs-2618 61 37 for for ADP iajs-2618 61 38 each each DET iajs-2618 61 39 𝑗 𝑗 ADJ iajs-2618 61 40 ( ( PUNCT iajs-2618 61 41 𝑗 𝑗 NOUN iajs-2618 61 42 = = SYM iajs-2618 61 43 0,1 0,1 NUM iajs-2618 61 44 , , PUNCT iajs-2618 61 45 … … PUNCT iajs-2618 61 46 , , PUNCT iajs-2618 61 47 π‘Œ π‘Œ PROPN iajs-2618 61 48 βˆ’ βˆ’ PROPN iajs-2618 61 49 1 1 NUM iajs-2618 61 50 ) ) PUNCT iajs-2618 61 51 as as SCONJ iajs-2618 61 52 follows follow VERB iajs-2618 61 53 : : PUNCT iajs-2618 61 54 in in ADP iajs-2618 61 55 the the DET iajs-2618 61 56 pt pt NOUN iajs-2618 61 57 we we PRON iajs-2618 61 58 suppose suppose VERB iajs-2618 61 59 𝑅𝑗+1 𝑅𝑗+1 ADP iajs-2618 61 60 = = SYM iajs-2618 61 61 𝑅𝑗 𝑅𝑗 PROPN iajs-2618 61 62 in in ADP iajs-2618 61 63 the the DET iajs-2618 61 64 components component NOUN iajs-2618 61 65 of of ADP iajs-2618 61 66 οΏ½ οΏ½ PROPN iajs-2618 61 67 βƒ—βƒ— βƒ—βƒ— PROPN iajs-2618 61 68 οΏ½ οΏ½ PROPN iajs-2618 61 69 in in ADP iajs-2618 61 70 the the DET iajs-2618 61 71 r.h.s r.h.s NOUN iajs-2618 61 72 of of ADP iajs-2618 61 73 ( ( PUNCT iajs-2618 61 74 12 12 NUM iajs-2618 61 75 ) ) PUNCT iajs-2618 61 76 , , PUNCT iajs-2618 61 77 then then ADV iajs-2618 61 78 it it PRON iajs-2618 61 79 turn turn VERB iajs-2618 61 80 to to ADP iajs-2618 61 81 a a DET iajs-2618 61 82 glas gla NOUN iajs-2618 61 83 , , PUNCT iajs-2618 61 84 which which PRON iajs-2618 61 85 is be AUX iajs-2618 61 86 solved solve VERB iajs-2618 61 87 to to PART iajs-2618 61 88 get get VERB iajs-2618 61 89 the the DET iajs-2618 61 90 predictor predictor NOUN iajs-2618 61 91 solution solution NOUN iajs-2618 61 92 𝑅𝑗+1 𝑅𝑗+1 ADV iajs-2618 61 93 , , PUNCT iajs-2618 61 94 then then ADV iajs-2618 61 95 in in ADP iajs-2618 61 96 the the DET iajs-2618 61 97 ct ct NOUN iajs-2618 61 98 we we PRON iajs-2618 61 99 resolve resolve VERB iajs-2618 61 100 ( ( PUNCT iajs-2618 61 101 12 12 NUM iajs-2618 61 102 ) ) PUNCT iajs-2618 61 103 with with ADP iajs-2618 61 104 setting set VERB iajs-2618 61 105 οΏ½ οΏ½ NOUN iajs-2618 61 106 Μ… Μ… NOUN iajs-2618 61 107 οΏ½ οΏ½ NOUN iajs-2618 61 108 𝑗+1 𝑗+1 NOUN iajs-2618 61 109 = = PUNCT iajs-2618 61 110 𝑅𝑗+1 𝑅𝑗+1 X iajs-2618 61 111 ( ( PUNCT iajs-2618 61 112 in in ADP iajs-2618 61 113 the the DET iajs-2618 61 114 components component NOUN iajs-2618 61 115 of of ADP iajs-2618 61 116 οΏ½ οΏ½ PROPN iajs-2618 61 117 βƒ—βƒ— βƒ—βƒ— PROPN iajs-2618 61 118 οΏ½ οΏ½ PROPN iajs-2618 61 119 of of ADP iajs-2618 61 120 the the DET iajs-2618 61 121 r.h.s r.h.s NOUN iajs-2618 61 122 of of ADP iajs-2618 61 123 it it PRON iajs-2618 61 124 ) ) PUNCT iajs-2618 61 125 to to PART iajs-2618 61 126 get get VERB iajs-2618 61 127 the the DET iajs-2618 61 128 corrector corrector NOUN iajs-2618 61 129 solution solution NOUN iajs-2618 61 130 𝑅𝑗+1 𝑅𝑗+1 ADV iajs-2618 61 131 , , PUNCT iajs-2618 61 132 finally finally ADV iajs-2618 61 133 substituting substitute VERB iajs-2618 61 134 𝑅𝑗+1 𝑅𝑗+1 NOUN iajs-2618 61 135 in in ADP iajs-2618 61 136 ( ( PUNCT iajs-2618 61 137 13 13 NUM iajs-2618 61 138 ) ) PUNCT iajs-2618 61 139 to to PART iajs-2618 61 140 get get VERB iajs-2618 61 141 π‘ˆπ‘—+1 π‘ˆπ‘—+1 NOUN iajs-2618 61 142 , , PUNCT iajs-2618 61 143 we we PRON iajs-2618 61 144 can can AUX iajs-2618 61 145 repeat repeat VERB iajs-2618 61 146 this this DET iajs-2618 61 147 procedure procedure NOUN iajs-2618 61 148 if if SCONJ iajs-2618 61 149 we we PRON iajs-2618 61 150 want want VERB iajs-2618 61 151 more more ADJ iajs-2618 61 152 than than ADP iajs-2618 61 153 one one NUM iajs-2618 61 154 time time NOUN iajs-2618 61 155 . . PUNCT iajs-2618 62 1 this this DET iajs-2618 62 2 reputation reputation NOUN iajs-2618 62 3 can can AUX iajs-2618 62 4 be be AUX iajs-2618 62 5 expressed express VERB iajs-2618 62 6 as as SCONJ iajs-2618 62 7 follows follow VERB iajs-2618 62 8 : : PUNCT iajs-2618 62 9 ( ( PUNCT iajs-2618 62 10 𝑀𝑗+1 𝑀𝑗+1 X iajs-2618 62 11 ( ( PUNCT iajs-2618 62 12 𝑙+1 𝑙+1 NOUN iajs-2618 62 13 ) ) PUNCT iajs-2618 62 14 , , PUNCT iajs-2618 62 15 πœ‚π‘– πœ‚π‘– NOUN iajs-2618 62 16 ) ) PUNCT iajs-2618 63 1 + + CCONJ iajs-2618 63 2 ( ( PUNCT iajs-2618 63 3 Ξ΄t)2(βˆ‡π‘€π‘—+1 Ξ΄t)2(βˆ‡π‘€π‘—+1 PROPN iajs-2618 63 4 ( ( PUNCT iajs-2618 63 5 𝑙+1 𝑙+1 PROPN iajs-2618 63 6 ) ) PUNCT iajs-2618 63 7 , , PUNCT iajs-2618 63 8 βˆ‡πœ‚π‘– βˆ‡πœ‚π‘– PROPN iajs-2618 63 9 ) ) PUNCT iajs-2618 63 10 + + CCONJ iajs-2618 63 11 ( ( PUNCT iajs-2618 63 12 Ξ΄t)2 Ξ΄t)2 PROPN iajs-2618 63 13 ( ( PUNCT iajs-2618 63 14 𝑀𝑗+1 𝑀𝑗+1 X iajs-2618 63 15 ( ( PUNCT iajs-2618 63 16 𝑙+1 𝑙+1 NOUN iajs-2618 63 17 ) ) PUNCT iajs-2618 63 18 , , PUNCT iajs-2618 63 19 πœ‚π‘– πœ‚π‘– NOUN iajs-2618 63 20 ) ) PUNCT iajs-2618 63 21 = = SYM iajs-2618 63 22 ( ( PUNCT iajs-2618 63 23 𝑀𝑗 𝑀𝑗 ADP iajs-2618 63 24 𝑛 𝑛 PROPN iajs-2618 63 25 , , PUNCT iajs-2618 63 26 πœ‚π‘– πœ‚π‘– NOUN iajs-2618 63 27 ) ) PUNCT iajs-2618 64 1 + + NUM iajs-2618 64 2 Ξ΄t Ξ΄t X iajs-2618 64 3 ( ( PUNCT iajs-2618 64 4 𝑝𝑗 𝑝𝑗 PROPN iajs-2618 64 5 𝑛 𝑛 PROPN iajs-2618 64 6 , , PUNCT iajs-2618 64 7 πœ‚π‘– πœ‚π‘– NOUN iajs-2618 64 8 ) ) PUNCT iajs-2618 64 9 + + CCONJ iajs-2618 64 10 ( ( PUNCT iajs-2618 64 11 Ξ΄t)2β„Ž(𝑑𝑗 Ξ΄t)2β„Ž(𝑑𝑗 ADP iajs-2618 64 12 𝑛 𝑛 PROPN iajs-2618 64 13 , , PUNCT iajs-2618 64 14 𝑀𝑗+1 𝑀𝑗+1 NUM iajs-2618 64 15 ( ( PUNCT iajs-2618 64 16 𝑙 𝑙 NOUN iajs-2618 64 17 ) ) PUNCT iajs-2618 64 18 ) ) PUNCT iajs-2618 64 19 , , PUNCT iajs-2618 64 20 πœ‚π‘– πœ‚π‘– PROPN iajs-2618 64 21 ) ) PUNCT iajs-2618 64 22 ( ( PUNCT iajs-2618 64 23 16 16 NUM iajs-2618 64 24 ) ) PUNCT iajs-2618 64 25 𝑝𝑗+1 𝑝𝑗+1 PROPN iajs-2618 64 26 ( ( PUNCT iajs-2618 64 27 𝑙+1 𝑙+1 NOUN iajs-2618 64 28 ) ) PUNCT iajs-2618 64 29 = = NOUN iajs-2618 64 30 ( ( PUNCT iajs-2618 64 31 𝑀𝑗+1 𝑀𝑗+1 X iajs-2618 64 32 ( ( PUNCT iajs-2618 64 33 𝑙+1 𝑙+1 NOUN iajs-2618 64 34 ) ) PUNCT iajs-2618 64 35 βˆ’π‘€π‘— βˆ’π‘€π‘— PROPN iajs-2618 64 36 𝑛 𝑛 PROPN iajs-2618 64 37 ) ) PUNCT iajs-2618 64 38 Ξ΄t Ξ΄t NOUN iajs-2618 64 39 ( ( PUNCT iajs-2618 64 40 17 17 NUM iajs-2618 64 41 ) ) PUNCT iajs-2618 64 42 equation equation NOUN iajs-2618 64 43 ( ( PUNCT iajs-2618 64 44 17 17 NUM iajs-2618 64 45 ) ) PUNCT iajs-2618 64 46 tells tell VERB iajs-2618 64 47 us we PRON iajs-2618 64 48 the the DET iajs-2618 64 49 iterative iterative NOUN iajs-2618 64 50 method method NOUN iajs-2618 64 51 depending depend VERB iajs-2618 64 52 on on ADP iajs-2618 64 53 just just ADV iajs-2618 64 54 𝑀𝑗+1 𝑀𝑗+1 NUM iajs-2618 64 55 ( ( PUNCT iajs-2618 64 56 𝑙+1 𝑙+1 NOUN iajs-2618 64 57 ) ) PUNCT iajs-2618 64 58 . . PUNCT iajs-2618 65 1 thus thus ADV iajs-2618 65 2 , , PUNCT iajs-2618 65 3 equation equation NOUN iajs-2618 65 4 ( ( PUNCT iajs-2618 65 5 16 16 NUM iajs-2618 65 6 ) ) PUNCT iajs-2618 65 7 is be AUX iajs-2618 65 8 reformulated reformulate VERB iajs-2618 65 9 as as ADP iajs-2618 65 10 𝑀(𝑙+1 𝑀(𝑙+1 NOUN iajs-2618 65 11 ) ) PUNCT iajs-2618 65 12 = = SYM iajs-2618 65 13 𝛿(𝑀(𝑙+1 𝛿(𝑀(𝑙+1 NOUN iajs-2618 65 14 ) ) PUNCT iajs-2618 65 15 ) ) PUNCT iajs-2618 65 16 , , PUNCT iajs-2618 65 17 where where SCONJ iajs-2618 65 18 𝑙 𝑙 NOUN iajs-2618 65 19 is be AUX iajs-2618 65 20 the the DET iajs-2618 65 21 number number NOUN iajs-2618 65 22 of of ADP iajs-2618 65 23 the the DET iajs-2618 65 24 iterations iteration NOUN iajs-2618 65 25 . . PUNCT iajs-2618 66 1 and and CCONJ iajs-2618 66 2 this this PRON iajs-2618 66 3 led lead VERB iajs-2618 66 4 us we PRON iajs-2618 66 5 to to ADP iajs-2618 66 6 the the DET iajs-2618 66 7 following follow VERB iajs-2618 66 8 theorem theorem PROPN iajs-2618 66 9 . . PUNCT iajs-2618 67 1 theorem theorem PROPN iajs-2618 67 2 ( ( PUNCT iajs-2618 67 3 5 5 NUM iajs-2618 67 4 ): ): PUNCT iajs-2618 67 5 for for ADP iajs-2618 67 6 any any DET iajs-2618 67 7 fixed fix VERB iajs-2618 67 8 point point NOUN iajs-2618 67 9 , , PUNCT iajs-2618 67 10 the the DET iajs-2618 67 11 des des X iajs-2618 67 12 ( ( PUNCT iajs-2618 67 13 8)-(11 8)-(11 NUM iajs-2618 67 14 ) ) PUNCT iajs-2618 67 15 , , PUNCT iajs-2618 67 16 and and CCONJ iajs-2618 67 17 for for ADP iajs-2618 67 18 Ξ΄ Ξ΄ PROPN iajs-2618 67 19 sufficiently sufficiently ADV iajs-2618 67 20 small small ADJ iajs-2618 67 21 , , PUNCT iajs-2618 67 22 has have VERB iajs-2618 67 23 a a DET iajs-2618 67 24 unique unique ADJ iajs-2618 67 25 solution solution NOUN iajs-2618 67 26 𝑀𝑛 𝑀𝑛 NOUN iajs-2618 67 27 = = SYM iajs-2618 67 28 ( ( PUNCT iajs-2618 67 29 𝑀0 𝑀0 PROPN iajs-2618 67 30 𝑛 𝑛 PROPN iajs-2618 67 31 , , PUNCT iajs-2618 67 32 𝑀1 𝑀1 VERB iajs-2618 67 33 𝑛 𝑛 ADP iajs-2618 67 34 , , PUNCT iajs-2618 67 35 … … PUNCT iajs-2618 67 36 . . PUNCT iajs-2618 67 37 . . PUNCT iajs-2618 68 1 , , PUNCT iajs-2618 68 2 𝑀𝑁 𝑀𝑁 PROPN iajs-2618 68 3 𝑛 𝑛 PROPN iajs-2618 68 4 ) ) PUNCT iajs-2618 68 5 and and CCONJ iajs-2618 68 6 the the DET iajs-2618 68 7 sequence sequence NOUN iajs-2618 68 8 of of ADP iajs-2618 68 9 the the DET iajs-2618 68 10 corrector corrector NOUN iajs-2618 68 11 solutions solution NOUN iajs-2618 68 12 converges converge VERB iajs-2618 68 13 on on ADP iajs-2618 68 14 ℝ. ℝ. PROPN iajs-2618 68 15 proof proof NOUN iajs-2618 68 16 : : PUNCT iajs-2618 68 17 let let VERB iajs-2618 68 18 𝑀(𝑙+1 𝑀(𝑙+1 NUM iajs-2618 68 19 ) ) PUNCT iajs-2618 68 20 = = SYM iajs-2618 68 21 ( ( PUNCT iajs-2618 68 22 𝑀0 𝑀0 NOUN iajs-2618 68 23 ( ( PUNCT iajs-2618 68 24 𝑙+1 𝑙+1 PROPN iajs-2618 68 25 ) ) PUNCT iajs-2618 68 26 , , PUNCT iajs-2618 68 27 𝑀1 𝑀1 VERB iajs-2618 68 28 ( ( PUNCT iajs-2618 68 29 𝑙+1 𝑙+1 NOUN iajs-2618 68 30 ) ) PUNCT iajs-2618 68 31 , , PUNCT iajs-2618 68 32 … … PUNCT iajs-2618 68 33 . . PUNCT iajs-2618 68 34 . . PUNCT iajs-2618 69 1 , , PUNCT iajs-2618 69 2 𝑀𝑁 𝑀𝑁 PROPN iajs-2618 69 3 ( ( PUNCT iajs-2618 69 4 𝑙+1 𝑙+1 PROPN iajs-2618 69 5 ) ) PUNCT iajs-2618 69 6 ) ) PUNCT iajs-2618 69 7 and and CCONJ iajs-2618 69 8 𝑀 𝑀 X iajs-2618 69 9 ( ( PUNCT iajs-2618 69 10 𝑙+1 𝑙+1 NOUN iajs-2618 69 11 ) ) PUNCT iajs-2618 69 12 = = SYM iajs-2618 69 13 ( ( PUNCT iajs-2618 69 14 𝑀0 𝑀0 NOUN iajs-2618 69 15 ( ( PUNCT iajs-2618 69 16 𝑙+1 𝑙+1 PROPN iajs-2618 69 17 ) ) PUNCT iajs-2618 69 18 , , PUNCT iajs-2618 69 19 𝑀1 𝑀1 VERB iajs-2618 69 20 ( ( PUNCT iajs-2618 69 21 𝑙+1 𝑙+1 NOUN iajs-2618 69 22 ) ) PUNCT iajs-2618 69 23 , , PUNCT iajs-2618 69 24 … … PUNCT iajs-2618 69 25 . . PUNCT iajs-2618 69 26 . . PUNCT iajs-2618 70 1 , , PUNCT iajs-2618 70 2 𝑀𝑁 𝑀𝑁 PROPN iajs-2618 70 3 ( ( PUNCT iajs-2618 70 4 𝑙+1 𝑙+1 PROPN iajs-2618 70 5 ) ) PUNCT iajs-2618 70 6 ) ) PUNCT iajs-2618 70 7 where where SCONJ iajs-2618 70 8 𝑀(𝑙+1 𝑀(𝑙+1 NUM iajs-2618 70 9 ) ) PUNCT iajs-2618 70 10 and and CCONJ iajs-2618 70 11 𝑀 𝑀 X iajs-2618 70 12 ( ( PUNCT iajs-2618 70 13 𝑙+1 𝑙+1 NOUN iajs-2618 70 14 ) ) PUNCT iajs-2618 70 15 are be AUX iajs-2618 70 16 solutions solution NOUN iajs-2618 70 17 of of ADP iajs-2618 70 18 equation equation NOUN iajs-2618 70 19 ( ( PUNCT iajs-2618 70 20 16 16 NUM iajs-2618 70 21 ) ) PUNCT iajs-2618 70 22 . . PUNCT iajs-2618 71 1 this this PRON iajs-2618 71 2 means mean VERB iajs-2618 71 3 , , PUNCT iajs-2618 71 4 ( ( PUNCT iajs-2618 71 5 𝑀𝑗+1 𝑀𝑗+1 X iajs-2618 71 6 ( ( PUNCT iajs-2618 71 7 𝑙+1 𝑙+1 NOUN iajs-2618 71 8 ) ) PUNCT iajs-2618 71 9 , , PUNCT iajs-2618 71 10 πœ‚π‘– πœ‚π‘– NOUN iajs-2618 71 11 ) ) PUNCT iajs-2618 71 12 + + CCONJ iajs-2618 71 13 ( ( PUNCT iajs-2618 71 14 Ξ΄t)2(βˆ‡π‘€π‘—+1 Ξ΄t)2(βˆ‡π‘€π‘—+1 PROPN iajs-2618 71 15 ( ( PUNCT iajs-2618 71 16 𝑙+1 𝑙+1 PROPN iajs-2618 71 17 ) ) PUNCT iajs-2618 71 18 , , PUNCT iajs-2618 71 19 βˆ‡ βˆ‡ X iajs-2618 71 20 πœ‚π‘– πœ‚π‘– PROPN iajs-2618 71 21 ) ) PUNCT iajs-2618 71 22 + + CCONJ iajs-2618 71 23 ( ( PUNCT iajs-2618 71 24 Ξ΄t)2 Ξ΄t)2 PROPN iajs-2618 71 25 ( ( PUNCT iajs-2618 71 26 𝑀𝑗+1 𝑀𝑗+1 X iajs-2618 71 27 ( ( PUNCT iajs-2618 71 28 𝑙+1 𝑙+1 NOUN iajs-2618 71 29 ) ) PUNCT iajs-2618 71 30 , , PUNCT iajs-2618 71 31 πœ‚π‘– πœ‚π‘– NOUN iajs-2618 71 32 ) ) PUNCT iajs-2618 71 33 = = SYM iajs-2618 71 34 ( ( PUNCT iajs-2618 71 35 𝑀𝑗 𝑀𝑗 ADP iajs-2618 71 36 𝑛 𝑛 PROPN iajs-2618 71 37 , , PUNCT iajs-2618 71 38 πœ‚π‘– πœ‚π‘– NOUN iajs-2618 71 39 ) ) PUNCT iajs-2618 72 1 + + NUM iajs-2618 72 2 Ξ΄t Ξ΄t X iajs-2618 72 3 ( ( PUNCT iajs-2618 72 4 𝑝𝑗 𝑝𝑗 PROPN iajs-2618 72 5 𝑛 𝑛 PROPN iajs-2618 72 6 , , PUNCT iajs-2618 72 7 πœ‚π‘– πœ‚π‘– NOUN iajs-2618 72 8 ) ) PUNCT iajs-2618 72 9 + + CCONJ iajs-2618 72 10 ( ( PUNCT iajs-2618 72 11 Ξ΄t)2 Ξ΄t)2 PROPN iajs-2618 72 12 ( ( PUNCT iajs-2618 72 13 β„Ž(𝑑𝑗 β„Ž(𝑑𝑗 PROPN iajs-2618 72 14 𝑛 𝑛 PROPN iajs-2618 72 15 , , PUNCT iajs-2618 72 16 𝑀𝑗+1 𝑀𝑗+1 NUM iajs-2618 72 17 ( ( PUNCT iajs-2618 72 18 𝑙 𝑙 NOUN iajs-2618 72 19 ) ) PUNCT iajs-2618 72 20 ) ) PUNCT iajs-2618 72 21 , , PUNCT iajs-2618 72 22 πœ‚π‘– πœ‚π‘– PROPN iajs-2618 72 23 ) ) PUNCT iajs-2618 72 24 ( ( PUNCT iajs-2618 72 25 18 18 NUM iajs-2618 72 26 ) ) PUNCT iajs-2618 72 27 and and CCONJ iajs-2618 72 28 ( ( PUNCT iajs-2618 72 29 𝑀 𝑀 PART iajs-2618 72 30 𝑗+1 𝑗+1 NOUN iajs-2618 72 31 ( ( PUNCT iajs-2618 72 32 𝑙+1 𝑙+1 NOUN iajs-2618 72 33 ) ) PUNCT iajs-2618 72 34 , , PUNCT iajs-2618 72 35 πœ‚π‘– πœ‚π‘– NOUN iajs-2618 72 36 ) ) PUNCT iajs-2618 72 37 + + CCONJ iajs-2618 72 38 ( ( PUNCT iajs-2618 72 39 Ξ΄t)2(βˆ‡ Ξ΄t)2(βˆ‡ NOUN iajs-2618 72 40 𝑀 𝑀 SYM iajs-2618 72 41 𝑗+1 𝑗+1 PROPN iajs-2618 72 42 ( ( PUNCT iajs-2618 72 43 𝑙+1 𝑙+1 NOUN iajs-2618 72 44 ) ) PUNCT iajs-2618 72 45 , , PUNCT iajs-2618 72 46 βˆ‡ βˆ‡ X iajs-2618 72 47 πœ‚π‘– πœ‚π‘– PROPN iajs-2618 72 48 ) ) PUNCT iajs-2618 72 49 + + CCONJ iajs-2618 72 50 ( ( PUNCT iajs-2618 72 51 Ξ΄t)2 Ξ΄t)2 PROPN iajs-2618 72 52 ( ( PUNCT iajs-2618 72 53 𝑀 𝑀 ADP iajs-2618 72 54 𝑗+1 𝑗+1 PUNCT iajs-2618 72 55 ( ( PUNCT iajs-2618 72 56 𝑙+1 𝑙+1 NOUN iajs-2618 72 57 ) ) PUNCT iajs-2618 72 58 , , PUNCT iajs-2618 72 59 πœ‚π‘– πœ‚π‘– NOUN iajs-2618 72 60 ) ) PUNCT iajs-2618 72 61 = = SYM iajs-2618 72 62 ( ( PUNCT iajs-2618 72 63 𝑀𝑗 𝑀𝑗 ADP iajs-2618 72 64 𝑛 𝑛 PROPN iajs-2618 72 65 , , PUNCT iajs-2618 72 66 πœ‚π‘– πœ‚π‘– NOUN iajs-2618 72 67 ) ) PUNCT iajs-2618 72 68 + + NUM iajs-2618 72 69 Ξ΄t Ξ΄t X iajs-2618 72 70 ( ( PUNCT iajs-2618 72 71 𝑝𝑗 𝑝𝑗 PROPN iajs-2618 72 72 𝑛 𝑛 PROPN iajs-2618 72 73 , , PUNCT iajs-2618 72 74 πœ‚π‘– πœ‚π‘– NOUN iajs-2618 72 75 ) ) PUNCT iajs-2618 72 76 + + CCONJ iajs-2618 72 77 ( ( PUNCT iajs-2618 72 78 Ξ΄t)2(β„Ž Ξ΄t)2(β„Ž ADJ iajs-2618 72 79 ( ( PUNCT iajs-2618 72 80 𝑑𝑗 𝑑𝑗 PROPN iajs-2618 72 81 𝑛 𝑛 PROPN iajs-2618 72 82 , , PUNCT iajs-2618 72 83 𝑀 𝑀 ADP iajs-2618 72 84 𝑗+1 𝑗+1 PROPN iajs-2618 72 85 ( ( PUNCT iajs-2618 72 86 𝑙 𝑙 NOUN iajs-2618 72 87 ) ) PUNCT iajs-2618 72 88 ) ) PUNCT iajs-2618 72 89 , , PUNCT iajs-2618 72 90 πœ‚π‘– πœ‚π‘– PROPN iajs-2618 72 91 ) ) PUNCT iajs-2618 72 92 ( ( PUNCT iajs-2618 72 93 19 19 NUM iajs-2618 72 94 ) ) PUNCT iajs-2618 72 95 subtracting subtract VERB iajs-2618 72 96 ) ) PUNCT iajs-2618 72 97 19 19 NUM iajs-2618 72 98 ) ) PUNCT iajs-2618 72 99 from from ADP iajs-2618 72 100 ( ( PUNCT iajs-2618 72 101 18 18 NUM iajs-2618 72 102 ) ) PUNCT iajs-2618 72 103 then then ADV iajs-2618 72 104 setting set VERB iajs-2618 72 105 πœ‚π‘– πœ‚π‘– X iajs-2618 72 106 = = SYM iajs-2618 72 107 ( ( PUNCT iajs-2618 72 108 𝑀 𝑀 PART iajs-2618 72 109 𝑗+1 𝑗+1 NOUN iajs-2618 72 110 ( ( PUNCT iajs-2618 72 111 𝑙+1 𝑙+1 NOUN iajs-2618 72 112 ) ) PUNCT iajs-2618 72 113 βˆ’ βˆ’ NOUN iajs-2618 72 114 𝑀𝑗+1 𝑀𝑗+1 NUM iajs-2618 72 115 ( ( PUNCT iajs-2618 72 116 𝑙+1 𝑙+1 NOUN iajs-2618 72 117 ) ) PUNCT iajs-2618 72 118 ) ) PUNCT iajs-2618 72 119 in in ADP iajs-2618 72 120 the the DET iajs-2618 72 121 obtained obtain VERB iajs-2618 72 122 equation equation NOUN iajs-2618 72 123 , , PUNCT iajs-2618 72 124 we we PRON iajs-2618 72 125 get get VERB iajs-2618 72 126 that that PRON iajs-2618 72 127 ( ( PUNCT iajs-2618 72 128 𝑀 𝑀 X iajs-2618 72 129 𝑗+1 𝑗+1 PUNCT iajs-2618 72 130 ( ( PUNCT iajs-2618 72 131 𝑙+1 𝑙+1 NOUN iajs-2618 72 132 ) ) PUNCT iajs-2618 72 133 βˆ’ βˆ’ NOUN iajs-2618 72 134 𝑀𝑗+1 𝑀𝑗+1 NUM iajs-2618 72 135 ( ( PUNCT iajs-2618 72 136 𝑙+1 𝑙+1 NOUN iajs-2618 72 137 ) ) PUNCT iajs-2618 72 138 , , PUNCT iajs-2618 72 139 𝑀𝑗+1 𝑀𝑗+1 X iajs-2618 72 140 ( ( PUNCT iajs-2618 72 141 𝑙+1 𝑙+1 NOUN iajs-2618 72 142 ) ) PUNCT iajs-2618 72 143 βˆ’ βˆ’ NOUN iajs-2618 72 144 𝑀𝑗+1 𝑀𝑗+1 NUM iajs-2618 72 145 ( ( PUNCT iajs-2618 72 146 𝑙+1 𝑙+1 NOUN iajs-2618 72 147 ) ) PUNCT iajs-2618 72 148 ) ) PUNCT iajs-2618 73 1 + + CCONJ iajs-2618 73 2 ( ( PUNCT iajs-2618 73 3 Ξ΄t)2(βˆ‡ Ξ΄t)2(βˆ‡ NOUN iajs-2618 73 4 𝑀 𝑀 SYM iajs-2618 73 5 𝑗+1 𝑗+1 PROPN iajs-2618 73 6 ( ( PUNCT iajs-2618 73 7 𝑙+1 𝑙+1 NOUN iajs-2618 73 8 ) ) PUNCT iajs-2618 73 9 βˆ’ βˆ’ NOUN iajs-2618 73 10 βˆ‡π‘€π‘—+1 βˆ‡π‘€π‘—+1 PROPN iajs-2618 73 11 ( ( PUNCT iajs-2618 73 12 𝑙+1 𝑙+1 NOUN iajs-2618 73 13 ) ) PUNCT iajs-2618 73 14 , , PUNCT iajs-2618 73 15 βˆ‡ βˆ‡ X iajs-2618 73 16 𝑀𝑗+1 𝑀𝑗+1 X iajs-2618 73 17 ( ( PUNCT iajs-2618 73 18 𝑙+1 𝑙+1 NOUN iajs-2618 73 19 ) ) PUNCT iajs-2618 73 20 βˆ’ βˆ’ NOUN iajs-2618 73 21 βˆ‡π‘€π‘—+1 βˆ‡π‘€π‘—+1 PROPN iajs-2618 73 22 ( ( PUNCT iajs-2618 73 23 𝑙+1 𝑙+1 NOUN iajs-2618 73 24 ) ) PUNCT iajs-2618 73 25 ) ) PUNCT iajs-2618 74 1 + + CCONJ iajs-2618 74 2 ( ( PUNCT iajs-2618 74 3 Ξ΄t)2(𝑀 Ξ΄t)2(𝑀 NOUN iajs-2618 74 4 𝑗+1 𝑗+1 PUNCT iajs-2618 74 5 ( ( PUNCT iajs-2618 74 6 𝑙+1 𝑙+1 NOUN iajs-2618 74 7 ) ) PUNCT iajs-2618 74 8 βˆ’ βˆ’ NOUN iajs-2618 74 9 𝑀𝑗+1 𝑀𝑗+1 NUM iajs-2618 74 10 ( ( PUNCT iajs-2618 74 11 𝑙+1 𝑙+1 NOUN iajs-2618 74 12 ) ) PUNCT iajs-2618 74 13 , , PUNCT iajs-2618 74 14 𝑀𝑗+1 𝑀𝑗+1 X iajs-2618 74 15 ( ( PUNCT iajs-2618 74 16 𝑙+1 𝑙+1 NOUN iajs-2618 74 17 ) ) PUNCT iajs-2618 74 18 βˆ’ βˆ’ NOUN iajs-2618 74 19 𝑀𝑗+1 𝑀𝑗+1 NUM iajs-2618 74 20 ( ( PUNCT iajs-2618 74 21 𝑙+1 𝑙+1 NOUN iajs-2618 74 22 ) ) PUNCT iajs-2618 74 23 ) ) PUNCT iajs-2618 75 1 = = SYM iajs-2618 75 2 ( ( PUNCT iajs-2618 75 3 Ξ΄t)2(β„Ž Ξ΄t)2(β„Ž ADJ iajs-2618 75 4 ( ( PUNCT iajs-2618 75 5 𝑀𝑗+1 𝑀𝑗+1 NUM iajs-2618 75 6 ( ( PUNCT iajs-2618 75 7 𝑙 𝑙 NOUN iajs-2618 75 8 ) ) PUNCT iajs-2618 75 9 ) ) PUNCT iajs-2618 75 10 βˆ’ βˆ’ ADP iajs-2618 76 1 β„Ž β„Ž PROPN iajs-2618 76 2 ( ( PUNCT iajs-2618 76 3 𝑀𝑗+1 𝑀𝑗+1 NUM iajs-2618 76 4 ( ( PUNCT iajs-2618 76 5 𝑙 𝑙 NOUN iajs-2618 76 6 ) ) PUNCT iajs-2618 76 7 ) ) PUNCT iajs-2618 76 8 , , PUNCT iajs-2618 76 9 𝑀𝑗+1 𝑀𝑗+1 X iajs-2618 76 10 ( ( PUNCT iajs-2618 76 11 𝑙+1 𝑙+1 NOUN iajs-2618 76 12 ) ) PUNCT iajs-2618 76 13 βˆ’ βˆ’ NOUN iajs-2618 76 14 𝑀𝑗+1 𝑀𝑗+1 NUM iajs-2618 76 15 ( ( PUNCT iajs-2618 76 16 𝑙+1 𝑙+1 NOUN iajs-2618 76 17 ) ) PUNCT iajs-2618 76 18 ) ) PUNCT iajs-2618 76 19 ( ( PUNCT iajs-2618 76 20 20 20 NUM iajs-2618 76 21 ) ) PUNCT iajs-2618 76 22 from from ADP iajs-2618 76 23 assumptions assumption NOUN iajs-2618 76 24 2.1 2.1 NUM iajs-2618 76 25 ( ( PUNCT iajs-2618 76 26 ib ib NOUN iajs-2618 76 27 ) ) PUNCT iajs-2618 76 28 the the DET iajs-2618 76 29 2𝑛𝑑 2𝑛𝑑 ADJ iajs-2618 76 30 and and CCONJ iajs-2618 76 31 3π‘Ÿπ‘‘ 3π‘Ÿπ‘‘ ADJ iajs-2618 76 32 terms term NOUN iajs-2618 76 33 in in ADP iajs-2618 76 34 the the DET iajs-2618 76 35 l.h.s l.h.s NOUN iajs-2618 76 36 of of ADP iajs-2618 76 37 equation equation NOUN iajs-2618 76 38 ( ( PUNCT iajs-2618 76 39 20 20 NUM iajs-2618 76 40 ) ) PUNCT iajs-2618 76 41 are be AUX iajs-2618 76 42 positive positive ADJ iajs-2618 76 43 , , PUNCT iajs-2618 76 44 and and CCONJ iajs-2618 76 45 applying apply VERB iajs-2618 76 46 assumption assumption NOUN iajs-2618 76 47 2.1 2.1 NUM iajs-2618 76 48 ( ( PUNCT iajs-2618 76 49 iib iib NOUN iajs-2618 76 50 ) ) PUNCT iajs-2618 76 51 on on ADP iajs-2618 76 52 β„Ž β„Ž PROPN iajs-2618 76 53 in in ADP iajs-2618 76 54 r.h.s r.h.s NOUN iajs-2618 76 55 of of ADP iajs-2618 76 56 equation equation NOUN iajs-2618 76 57 ( ( PUNCT iajs-2618 76 58 20 20 NUM iajs-2618 76 59 ) ) PUNCT iajs-2618 76 60 , , PUNCT iajs-2618 76 61 and and CCONJ iajs-2618 76 62 by by ADP iajs-2618 76 63 using use VERB iajs-2618 76 64 the the DET iajs-2618 76 65 cauchy cauchy ADJ iajs-2618 76 66 schwarz schwarz PROPN iajs-2618 76 67 inequality inequality NOUN iajs-2618 76 68 on on ADP iajs-2618 76 69 this this DET iajs-2618 76 70 side side NOUN iajs-2618 76 71 , , PUNCT iajs-2618 76 72 we we PRON iajs-2618 76 73 get get VERB iajs-2618 76 74 ‖𝛿 ‖𝛿 NOUN iajs-2618 76 75 ( ( PUNCT iajs-2618 76 76 𝑀𝑗+1 𝑀𝑗+1 X iajs-2618 76 77 ( ( PUNCT iajs-2618 76 78 𝑙 𝑙 NOUN iajs-2618 76 79 ) ) PUNCT iajs-2618 76 80 ) ) PUNCT iajs-2618 77 1 βˆ’ βˆ’ PROPN iajs-2618 77 2 𝛿 𝛿 ADJ iajs-2618 77 3 ( ( PUNCT iajs-2618 77 4 𝑀𝑗+1 𝑀𝑗+1 NUM iajs-2618 77 5 ( ( PUNCT iajs-2618 77 6 𝑙 𝑙 NOUN iajs-2618 77 7 ) ) PUNCT iajs-2618 77 8 ) ) PUNCT iajs-2618 78 1 β€– β€– PROPN iajs-2618 78 2 0 0 PUNCT iajs-2618 79 1 = = SYM iajs-2618 79 2 ‖𝑀𝑗+1 ‖𝑀𝑗+1 X iajs-2618 79 3 ( ( PUNCT iajs-2618 79 4 𝑙+1 𝑙+1 NOUN iajs-2618 79 5 ) ) PUNCT iajs-2618 79 6 βˆ’ βˆ’ NOUN iajs-2618 79 7 𝑀𝑗+1 𝑀𝑗+1 NUM iajs-2618 79 8 ( ( PUNCT iajs-2618 79 9 𝑙+1 𝑙+1 NOUN iajs-2618 79 10 ) ) PUNCT iajs-2618 79 11 β€– β€– PROPN iajs-2618 79 12 0 0 NUM iajs-2618 79 13 ≀ ≀ NOUN iajs-2618 79 14 πœ† πœ† PRON iajs-2618 79 15 ‖𝑀𝑗+1 ‖𝑀𝑗+1 X iajs-2618 79 16 ( ( PUNCT iajs-2618 79 17 𝑙 𝑙 NOUN iajs-2618 79 18 ) ) PUNCT iajs-2618 79 19 βˆ’ βˆ’ NOUN iajs-2618 79 20 𝑀𝑗+1 𝑀𝑗+1 NUM iajs-2618 79 21 ( ( PUNCT iajs-2618 79 22 𝑙 𝑙 X iajs-2618 79 23 ) ) PUNCT iajs-2618 79 24 β€– β€– PROPN iajs-2618 79 25 0 0 NUM iajs-2618 79 26 ( ( PUNCT iajs-2618 79 27 21 21 NUM iajs-2618 79 28 ) ) PUNCT iajs-2618 79 29 where where SCONJ iajs-2618 79 30 πœ† πœ† NOUN iajs-2618 79 31 = = X iajs-2618 79 32 ( ( PUNCT iajs-2618 79 33 Ξ΄t)2𝐿 Ξ΄t)2𝐿 NOUN iajs-2618 79 34 < < X iajs-2618 79 35 1 1 NUM iajs-2618 79 36 , , PUNCT iajs-2618 79 37 for for ADP iajs-2618 79 38 sufficiently sufficiently ADV iajs-2618 79 39 small small ADJ iajs-2618 79 40 δ𝑑. δ𝑑. PROPN iajs-2618 79 41 which which PRON iajs-2618 79 42 implies imply VERB iajs-2618 79 43 that that SCONJ iajs-2618 79 44 𝛿 𝛿 PRON iajs-2618 79 45 is be AUX iajs-2618 79 46 contractive contractive ADJ iajs-2618 79 47 , , PUNCT iajs-2618 79 48 also also ADV iajs-2618 79 49 since since SCONJ iajs-2618 79 50 { { PUNCT iajs-2618 79 51 𝑀(𝑙 𝑀(𝑙 NOUN iajs-2618 79 52 ) ) PUNCT iajs-2618 79 53 } } PUNCT iajs-2618 79 54 ∈ ∈ PROPN iajs-2618 79 55 ℝ ℝ NOUN iajs-2618 79 56 βˆ€ βˆ€ X iajs-2618 79 57 𝑙 𝑙 NOUN iajs-2618 79 58 , , PUNCT iajs-2618 79 59 that that PRON iajs-2618 79 60 𝛿(𝑀(𝑙+1 𝛿(𝑀(𝑙+1 NOUN iajs-2618 79 61 ) ) PUNCT iajs-2618 79 62 ) ) PUNCT iajs-2618 79 63 = = SYM iajs-2618 79 64 𝑀(𝑙+1 𝑀(𝑙+1 NUM iajs-2618 79 65 ) ) PUNCT iajs-2618 79 66 ∈ ∈ PROPN iajs-2618 79 67 ℝ ℝ PROPN iajs-2618 79 68 βˆ€ βˆ€ X iajs-2618 79 69 𝑙 𝑙 NOUN iajs-2618 79 70 , , PUNCT iajs-2618 79 71 𝑖. 𝑖. ADJ iajs-2618 79 72 𝑒 𝑒 ADP iajs-2618 79 73 𝛿(𝑀 𝛿(𝑀 PROPN iajs-2618 79 74 ) ) PUNCT iajs-2618 79 75 ∈ ∈ PROPN iajs-2618 79 76 ℝ ℝ PROPN iajs-2618 79 77 , , PUNCT iajs-2618 79 78 hence hence ADV iajs-2618 79 79 , , PUNCT iajs-2618 79 80 by by ADP iajs-2618 79 81 theorem theorem NOUN iajs-2618 79 82 3 3 NUM iajs-2618 79 83 the the DET iajs-2618 79 84 sequence sequence NOUN iajs-2618 79 85 { { PUNCT iajs-2618 79 86 𝑀(𝑙 𝑀(𝑙 NOUN iajs-2618 79 87 ) ) PUNCT iajs-2618 79 88 } } PUNCT iajs-2618 79 89 converges converge VERB iajs-2618 79 90 to to ADP iajs-2618 79 91 a a DET iajs-2618 79 92 point point NOUN iajs-2618 79 93 in in ADP iajs-2618 79 94 ℝ. ℝ. PROPN iajs-2618 79 95 123 123 NUM iajs-2618 79 96 ibn ibn PROPN iajs-2618 79 97 al al PROPN iajs-2618 79 98 - - PUNCT iajs-2618 79 99 haitham haitham PROPN iajs-2618 79 100 jour jour X iajs-2618 79 101 . . PROPN iajs-2618 80 1 for for ADP iajs-2618 80 2 pure pure ADJ iajs-2618 80 3 & & CCONJ iajs-2618 80 4 appl appl PROPN iajs-2618 80 5 . . PUNCT iajs-2618 81 1 sci sci PROPN iajs-2618 81 2 . . PROPN iajs-2618 82 1 34 34 NUM iajs-2618 82 2 ( ( PUNCT iajs-2618 82 3 2 2 NUM iajs-2618 82 4 ) ) PUNCT iajs-2618 82 5 2021 2021 NUM iajs-2618 82 6 5 5 NUM iajs-2618 82 7 . . PUNCT iajs-2618 83 1 stability stability NOUN iajs-2618 83 2 : : PUNCT iajs-2618 83 3 lemma lemma PROPN iajs-2618 83 4 ( ( PUNCT iajs-2618 83 5 6 6 NUM iajs-2618 83 6 ): ): PUNCT iajs-2618 83 7 if if SCONJ iajs-2618 83 8 Ξ΄ Ξ΄ PROPN iajs-2618 83 9 is be AUX iajs-2618 83 10 sufficiently sufficiently ADV iajs-2618 83 11 small small ADJ iajs-2618 83 12 , , PUNCT iajs-2618 83 13 then then ADV iajs-2618 83 14 βˆ€ βˆ€ X iajs-2618 83 15 𝑗 𝑗 NOUN iajs-2618 83 16 = = SYM iajs-2618 83 17 0,1 0,1 NUM iajs-2618 83 18 , , PUNCT iajs-2618 83 19 … … PUNCT iajs-2618 83 20 , , PUNCT iajs-2618 83 21 π‘Œ π‘Œ PROPN iajs-2618 83 22 ‖𝑀𝑗 ‖𝑀𝑗 NUM iajs-2618 83 23 𝑛 𝑛 PROPN iajs-2618 83 24 β€–1 β€–1 NOUN iajs-2618 83 25 2 2 NUM iajs-2618 83 26 ≀ ≀ NUM iajs-2618 83 27 οΏ½ οΏ½ NOUN iajs-2618 83 28 Μ… Μ… NOUN iajs-2618 83 29 οΏ½ οΏ½ NOUN iajs-2618 83 30 , , PUNCT iajs-2618 83 31 ‖𝑝𝑗 ‖𝑝𝑗 NUM iajs-2618 83 32 𝑛 𝑛 DET iajs-2618 83 33 β€–0 β€–0 NUM iajs-2618 83 34 2 2 NUM iajs-2618 83 35 ≀ ≀ NUM iajs-2618 83 36 οΏ½ οΏ½ NOUN iajs-2618 83 37 Μ… Μ… NOUN iajs-2618 83 38 οΏ½ οΏ½ PROPN iajs-2618 83 39 , , PUNCT iajs-2618 83 40 βˆ‘ βˆ‘ PUNCT iajs-2618 83 41 β€– β€– ADJ iajs-2618 83 42 𝑀𝑗+1 𝑀𝑗+1 ADP iajs-2618 83 43 𝑛 𝑛 PRON iajs-2618 83 44 βˆ’ βˆ’ NOUN iajs-2618 83 45 𝑀𝑗 𝑀𝑗 ADP iajs-2618 83 46 𝑛 𝑛 PRON iajs-2618 83 47 β€–1 β€–1 NOUN iajs-2618 83 48 2π‘Œβˆ’1 2π‘Œβˆ’1 NUM iajs-2618 83 49 𝑗=0 𝑗=0 PUNCT iajs-2618 83 50 ≀ ≀ NUM iajs-2618 83 51 οΏ½ οΏ½ PROPN iajs-2618 83 52 Μ… Μ… NOUN iajs-2618 83 53 οΏ½ οΏ½ PROPN iajs-2618 83 54 , , PUNCT iajs-2618 83 55 and and CCONJ iajs-2618 83 56 βˆ‘ βˆ‘ PUNCT iajs-2618 83 57 β€– β€– PROPN iajs-2618 83 58 𝑝𝑗+1 𝑝𝑗+1 X iajs-2618 83 59 𝑛 𝑛 PRON iajs-2618 83 60 βˆ’ βˆ’ NUM iajs-2618 83 61 𝑝𝑗 𝑝𝑗 NOUN iajs-2618 83 62 𝑛 𝑛 PRON iajs-2618 83 63 β€–0 β€–0 NOUN iajs-2618 83 64 2π‘Œβˆ’1 2π‘Œβˆ’1 NUM iajs-2618 83 65 𝑗=0 𝑗=0 PUNCT iajs-2618 83 66 ≀ ≀ NUM iajs-2618 83 67 οΏ½ οΏ½ NOUN iajs-2618 83 68 Μ… Μ… NOUN iajs-2618 83 69 οΏ½ οΏ½ PROPN iajs-2618 83 70 where where SCONJ iajs-2618 83 71 οΏ½ οΏ½ NOUN iajs-2618 83 72 Μ… Μ… NOUN iajs-2618 83 73 οΏ½ οΏ½ NOUN iajs-2618 83 74 refers refer VERB iajs-2618 83 75 to to ADP iajs-2618 83 76 a a DET iajs-2618 83 77 various various ADJ iajs-2618 83 78 constants constant NOUN iajs-2618 83 79 . . PUNCT iajs-2618 84 1 proof proof NOUN iajs-2618 84 2 : : PUNCT iajs-2618 84 3 let let VERB iajs-2618 84 4 πœ‚ πœ‚ NOUN iajs-2618 84 5 = = SYM iajs-2618 84 6 𝑝𝑗+1 𝑝𝑗+1 ADP iajs-2618 84 7 𝑛 𝑛 ADP iajs-2618 84 8 substituting substitute VERB iajs-2618 84 9 in in ADP iajs-2618 84 10 equation equation NOUN iajs-2618 84 11 ( ( PUNCT iajs-2618 84 12 8) 8) NUM iajs-2618 84 13 , , PUNCT iajs-2618 84 14 and and CCONJ iajs-2618 84 15 rewriting rewrite VERB iajs-2618 84 16 the the DET iajs-2618 84 17 first first ADJ iajs-2618 84 18 term term NOUN iajs-2618 84 19 in in ADP iajs-2618 84 20 the the DET iajs-2618 84 21 l.h.s l.h.s NOUN iajs-2618 84 22 of of ADP iajs-2618 84 23 the the DET iajs-2618 84 24 obtained obtain VERB iajs-2618 84 25 equation equation NOUN iajs-2618 84 26 , , PUNCT iajs-2618 84 27 we we PRON iajs-2618 84 28 get get VERB iajs-2618 84 29 β€– β€– ADJ iajs-2618 84 30 𝑝𝑗+1 𝑝𝑗+1 X iajs-2618 84 31 𝑛 𝑛 DET iajs-2618 84 32 β€–0 β€–0 NUM iajs-2618 84 33 2 2 NUM iajs-2618 84 34 βˆ’ βˆ’ NOUN iajs-2618 84 35 β€– β€– PROPN iajs-2618 84 36 𝑝𝑗 𝑝𝑗 PROPN iajs-2618 84 37 𝑛 𝑛 DET iajs-2618 84 38 β€–0 β€–0 NOUN iajs-2618 84 39 2 2 NUM iajs-2618 84 40 + + CCONJ iajs-2618 84 41 β€– β€– PROPN iajs-2618 84 42 𝑝𝑗+1 𝑝𝑗+1 X iajs-2618 84 43 𝑛 𝑛 PRON iajs-2618 84 44 βˆ’ βˆ’ NUM iajs-2618 84 45 𝑝𝑗 𝑝𝑗 NOUN iajs-2618 84 46 𝑛 𝑛 PRON iajs-2618 84 47 β€–0 β€–0 NOUN iajs-2618 84 48 2 2 NUM iajs-2618 84 49 + + NUM iajs-2618 84 50 Ξ΄t Ξ΄t X iajs-2618 84 51 ( ( PUNCT iajs-2618 84 52 βˆ‡π‘€π‘—+1 βˆ‡π‘€π‘—+1 ADP iajs-2618 84 53 𝑛 𝑛 PRON iajs-2618 84 54 , , PUNCT iajs-2618 84 55 βˆ‡π‘π‘—+1 βˆ‡π‘π‘—+1 NOUN iajs-2618 84 56 𝑛 𝑛 PROPN iajs-2618 84 57 ) ) PUNCT iajs-2618 85 1 + + NUM iajs-2618 85 2 Ξ΄t Ξ΄t X iajs-2618 85 3 ( ( PUNCT iajs-2618 85 4 𝑀𝑗+1 𝑀𝑗+1 NUM iajs-2618 85 5 𝑛 𝑛 PRON iajs-2618 85 6 , , PUNCT iajs-2618 85 7 𝑝𝑗+1 𝑝𝑗+1 NOUN iajs-2618 85 8 𝑛 𝑛 NOUN iajs-2618 85 9 ) ) PUNCT iajs-2618 85 10 = = SYM iajs-2618 85 11 Ξ΄t Ξ΄t NOUN iajs-2618 85 12 ( ( PUNCT iajs-2618 85 13 β„Ž(𝑀𝑗+1 β„Ž(𝑀𝑗+1 PROPN iajs-2618 85 14 𝑛 𝑛 PRON iajs-2618 85 15 ) ) PUNCT iajs-2618 85 16 , , PUNCT iajs-2618 85 17 𝑝𝑗+1 𝑝𝑗+1 PROPN iajs-2618 85 18 𝑛 𝑛 NOUN iajs-2618 85 19 ) ) PUNCT iajs-2618 85 20 ( ( PUNCT iajs-2618 85 21 22 22 NUM iajs-2618 85 22 ) ) PUNCT iajs-2618 85 23 since since SCONJ iajs-2618 85 24 , , PUNCT iajs-2618 86 1 π›₯𝑑 π›₯𝑑 PROPN iajs-2618 86 2 [ [ X iajs-2618 86 3 ( ( PUNCT iajs-2618 86 4 βˆ‡π‘€π‘—+1 βˆ‡π‘€π‘—+1 ADP iajs-2618 86 5 𝑛 𝑛 PRON iajs-2618 86 6 , , PUNCT iajs-2618 86 7 βˆ‡π‘π‘—+1 βˆ‡π‘π‘—+1 NOUN iajs-2618 86 8 𝑛 𝑛 PROPN iajs-2618 86 9 ) ) PUNCT iajs-2618 87 1 + + CCONJ iajs-2618 87 2 ( ( PUNCT iajs-2618 87 3 𝑀𝑗+1 𝑀𝑗+1 NUM iajs-2618 87 4 𝑛 𝑛 PRON iajs-2618 87 5 , , PUNCT iajs-2618 87 6 𝑝𝑗+1 𝑝𝑗+1 PROPN iajs-2618 87 7 𝑛 𝑛 NOUN iajs-2618 87 8 ) ) PUNCT iajs-2618 87 9 ] ] PUNCT iajs-2618 88 1 = = SYM iajs-2618 88 2 1 1 NUM iajs-2618 88 3 2 2 NUM iajs-2618 88 4 [ [ PUNCT iajs-2618 88 5 ( ( PUNCT iajs-2618 88 6 βˆ‡π‘€π‘—+1 βˆ‡π‘€π‘—+1 ADP iajs-2618 88 7 𝑛 𝑛 ADP iajs-2618 88 8 βˆ’ βˆ’ NOUN iajs-2618 88 9 βˆ‡π‘€π‘— βˆ‡π‘€π‘— SYM iajs-2618 88 10 𝑛 𝑛 VERB iajs-2618 88 11 , , PUNCT iajs-2618 88 12 βˆ‡π‘€π‘—+1 βˆ‡π‘€π‘—+1 VERB iajs-2618 88 13 𝑛 𝑛 ADP iajs-2618 88 14 βˆ’ βˆ’ PROPN iajs-2618 88 15 βˆ‡π‘€π‘— βˆ‡π‘€π‘— SYM iajs-2618 88 16 𝑛 𝑛 NOUN iajs-2618 88 17 ) ) PUNCT iajs-2618 88 18 + + CCONJ iajs-2618 88 19 𝑀𝑗+1 𝑀𝑗+1 NUM iajs-2618 88 20 𝑛 𝑛 PRON iajs-2618 88 21 βˆ’ βˆ’ NOUN iajs-2618 88 22 𝑀𝑗 𝑀𝑗 NOUN iajs-2618 88 23 𝑛 𝑛 PROPN iajs-2618 88 24 , , PUNCT iajs-2618 88 25 𝑀𝑗+1 𝑀𝑗+1 ADP iajs-2618 88 26 𝑛 𝑛 PRON iajs-2618 88 27 βˆ’ βˆ’ PROPN iajs-2618 88 28 𝑀𝑗 𝑀𝑗 NOUN iajs-2618 88 29 𝑛 𝑛 PROPN iajs-2618 88 30 ) ) PUNCT iajs-2618 88 31 + + CCONJ iajs-2618 88 32 ( ( PUNCT iajs-2618 88 33 βˆ‡π‘€π‘—+1 βˆ‡π‘€π‘—+1 ADP iajs-2618 88 34 𝑛 𝑛 PRON iajs-2618 88 35 , , PUNCT iajs-2618 88 36 βˆ‡π‘€π‘—+1 βˆ‡π‘€π‘—+1 ADP iajs-2618 88 37 𝑛 𝑛 PRON iajs-2618 88 38 ) ) PUNCT iajs-2618 89 1 + + CCONJ iajs-2618 89 2 ( ( PUNCT iajs-2618 89 3 𝑀𝑗+1 𝑀𝑗+1 NUM iajs-2618 89 4 𝑛 𝑛 PRON iajs-2618 89 5 , , PUNCT iajs-2618 89 6 𝑀𝑗+1 𝑀𝑗+1 NUM iajs-2618 89 7 𝑛 𝑛 PROPN iajs-2618 89 8 ) ) PUNCT iajs-2618 89 9 βˆ’ βˆ’ PROPN iajs-2618 90 1 ( ( PUNCT iajs-2618 90 2 βˆ‡π‘€π‘— βˆ‡π‘€π‘— SCONJ iajs-2618 90 3 𝑛 𝑛 VERB iajs-2618 90 4 , , PUNCT iajs-2618 90 5 βˆ‡π‘€π‘— βˆ‡π‘€π‘— PROPN iajs-2618 90 6 𝑛 𝑛 NOUN iajs-2618 90 7 ) ) PUNCT iajs-2618 90 8 βˆ’ βˆ’ PROPN iajs-2618 91 1 ( ( PUNCT iajs-2618 91 2 𝑀𝑗 𝑀𝑗 ADP iajs-2618 91 3 𝑛 𝑛 PROPN iajs-2618 91 4 , , PUNCT iajs-2618 91 5 𝑀𝑗 𝑀𝑗 ADP iajs-2618 91 6 𝑛 𝑛 PROPN iajs-2618 91 7 ) ) PUNCT iajs-2618 91 8 ] ] PUNCT iajs-2618 91 9 by by ADP iajs-2618 91 10 substituting substitute VERB iajs-2618 91 11 above above ADP iajs-2618 91 12 equality equality NOUN iajs-2618 91 13 in in ADP iajs-2618 91 14 the the DET iajs-2618 91 15 l.h.s l.h.s NOUN iajs-2618 91 16 in in ADP iajs-2618 91 17 equation equation NOUN iajs-2618 91 18 ( ( PUNCT iajs-2618 91 19 22 22 NUM iajs-2618 91 20 ) ) PUNCT iajs-2618 91 21 , , PUNCT iajs-2618 91 22 summing sum VERB iajs-2618 91 23 both both DET iajs-2618 91 24 sides side NOUN iajs-2618 91 25 of of ADP iajs-2618 91 26 the the DET iajs-2618 91 27 obtained obtain VERB iajs-2618 91 28 equality equality NOUN iajs-2618 91 29 , , PUNCT iajs-2618 91 30 for for ADP iajs-2618 91 31 𝑗 𝑗 NOUN iajs-2618 91 32 = = SYM iajs-2618 91 33 0 0 NUM iajs-2618 91 34 π‘‘π‘œ π‘‘π‘œ NOUN iajs-2618 91 35 𝑗 𝑗 NOUN iajs-2618 91 36 = = SYM iajs-2618 92 1 𝑙 𝑙 NOUN iajs-2618 92 2 βˆ’ βˆ’ PROPN iajs-2618 92 3 1 1 NUM iajs-2618 92 4 , , PUNCT iajs-2618 92 5 then then ADV iajs-2618 92 6 set set VERB iajs-2618 92 7 c c PROPN iajs-2618 92 8 = = SYM iajs-2618 92 9 max max PROPN iajs-2618 92 10 ( ( PUNCT iajs-2618 92 11 1 1 NUM iajs-2618 92 12 , , PUNCT iajs-2618 92 13 π‘˜2 π‘˜2 PROPN iajs-2618 92 14 2 2 NUM iajs-2618 92 15 ) ) PUNCT iajs-2618 92 16 , , PUNCT iajs-2618 92 17 we we PRON iajs-2618 92 18 get get VERB iajs-2618 92 19 𝑐‖ 𝑐‖ ADJ iajs-2618 92 20 𝑝𝑙 𝑝𝑙 NOUN iajs-2618 92 21 𝑛 𝑛 PROPN iajs-2618 92 22 β€–0 β€–0 NOUN iajs-2618 92 23 2 2 NUM iajs-2618 92 24 + + CCONJ iajs-2618 92 25 𝑐 𝑐 PROPN iajs-2618 92 26 βˆ‘ βˆ‘ PUNCT iajs-2618 92 27 β€– β€– PROPN iajs-2618 92 28 𝑝𝑗+1 𝑝𝑗+1 X iajs-2618 92 29 𝑛 𝑛 PRON iajs-2618 92 30 βˆ’ βˆ’ NUM iajs-2618 92 31 𝑝𝑗 𝑝𝑗 NOUN iajs-2618 92 32 𝑛 𝑛 PRON iajs-2618 92 33 β€–0 β€–0 NOUN iajs-2618 92 34 2π‘™βˆ’1 2π‘™βˆ’1 NUM iajs-2618 92 35 𝑗=0 𝑗=0 PUNCT iajs-2618 93 1 + + CCONJ iajs-2618 93 2 𝑐‖𝑀𝑙 𝑐‖𝑀𝑙 ADV iajs-2618 93 3 𝑛 𝑛 PRON iajs-2618 93 4 β€–1 β€–1 NOUN iajs-2618 93 5 2 2 NUM iajs-2618 94 1 + + CCONJ iajs-2618 94 2 𝑐 𝑐 PROPN iajs-2618 94 3 βˆ‘ βˆ‘ PUNCT iajs-2618 94 4 β€– β€– ADJ iajs-2618 94 5 𝑀𝑗+1 𝑀𝑗+1 NUM iajs-2618 94 6 𝑛 𝑛 PRON iajs-2618 94 7 βˆ’ βˆ’ NOUN iajs-2618 94 8 𝑀𝑗 𝑀𝑗 ADP iajs-2618 94 9 𝑛 𝑛 PRON iajs-2618 94 10 β€–1 β€–1 NOUN iajs-2618 94 11 2π‘™βˆ’1 2π‘™βˆ’1 NUM iajs-2618 95 1 𝑗=0 𝑗=0 PUNCT iajs-2618 96 1 ≀ ≀ NUM iajs-2618 96 2 β€– β€– ADJ iajs-2618 96 3 𝑝0 𝑝0 NOUN iajs-2618 96 4 𝑛 𝑛 DET iajs-2618 96 5 β€–0 β€–0 NOUN iajs-2618 96 6 2 2 NUM iajs-2618 96 7 + + CCONJ iajs-2618 96 8 π‘˜2 π‘˜2 PROPN iajs-2618 96 9 2 2 NUM iajs-2618 96 10 β€– β€– PROPN iajs-2618 96 11 𝑀𝑙 𝑀𝑙 X iajs-2618 96 12 𝑛 𝑛 PRON iajs-2618 96 13 β€–1 β€–1 NOUN iajs-2618 96 14 2 2 NUM iajs-2618 96 15 + + CCONJ iajs-2618 96 16 βˆ‘ βˆ‘ PROPN iajs-2618 96 17 π‘™βˆ’1 π‘™βˆ’1 PROPN iajs-2618 96 18 𝑗=0 𝑗=0 X iajs-2618 96 19 Ξ΄t Ξ΄t X iajs-2618 96 20 ( ( PUNCT iajs-2618 96 21 β„Ž(𝑀𝑗+1 β„Ž(𝑀𝑗+1 PROPN iajs-2618 96 22 𝑛 𝑛 PRON iajs-2618 96 23 ) ) PUNCT iajs-2618 96 24 , , PUNCT iajs-2618 96 25 𝑝𝑗+1 𝑝𝑗+1 PROPN iajs-2618 96 26 𝑛 𝑛 NOUN iajs-2618 96 27 ) ) PUNCT iajs-2618 96 28 ( ( PUNCT iajs-2618 96 29 23 23 NUM iajs-2618 96 30 ) ) PUNCT iajs-2618 96 31 now now ADV iajs-2618 96 32 , , PUNCT iajs-2618 96 33 using use VERB iajs-2618 96 34 the the DET iajs-2618 96 35 assumptions assumption NOUN iajs-2618 96 36 on on ADP iajs-2618 96 37 β„Ž β„Ž NOUN iajs-2618 96 38 and and CCONJ iajs-2618 96 39 then then ADV iajs-2618 96 40 by by ADP iajs-2618 96 41 the the DET iajs-2618 96 42 cauchy cauchy ADJ iajs-2618 96 43 schwarz schwarz PROPN iajs-2618 96 44 inequality inequality PROPN iajs-2618 96 45 , , PUNCT iajs-2618 96 46 to to PART iajs-2618 96 47 get get VERB iajs-2618 96 48 β”‚ β”‚ VERB iajs-2618 96 49 ( ( PUNCT iajs-2618 96 50 β„Ž(𝑀𝑗+1 β„Ž(𝑀𝑗+1 PROPN iajs-2618 96 51 𝑛 𝑛 PRON iajs-2618 96 52 ) ) PUNCT iajs-2618 96 53 , , PUNCT iajs-2618 96 54 𝑝𝑗+1 𝑝𝑗+1 PROPN iajs-2618 96 55 𝑛 𝑛 NOUN iajs-2618 96 56 ) ) PUNCT iajs-2618 96 57 β”‚ β”‚ X iajs-2618 96 58 ≀ ≀ PROPN iajs-2618 97 1 β€– β€– PROPN iajs-2618 97 2 𝛽𝑗 𝛽𝑗 NOUN iajs-2618 97 3 β€–0 β€–0 NUM iajs-2618 97 4 2 2 NUM iajs-2618 97 5 + + CCONJ iajs-2618 97 6 𝛿‖ 𝛿‖ PROPN iajs-2618 97 7 𝑀𝑗+1 𝑀𝑗+1 NUM iajs-2618 97 8 𝑛 𝑛 PRON iajs-2618 97 9 β€–1 β€–1 NOUN iajs-2618 97 10 2 2 NUM iajs-2618 97 11 + + CCONJ iajs-2618 97 12 𝛿̅‖ 𝛿̅‖ NOUN iajs-2618 97 13 𝑝𝑗+1 𝑝𝑗+1 ADP iajs-2618 97 14 𝑛 𝑛 DET iajs-2618 97 15 β€–0 β€–0 NUM iajs-2618 97 16 2 2 NUM iajs-2618 97 17 , , PUNCT iajs-2618 97 18 𝛿̅ 𝛿̅ PROPN iajs-2618 97 19 = = PUNCT iajs-2618 98 1 𝛿 𝛿 ADJ iajs-2618 98 2 + + ADJ iajs-2618 98 3 1 1 NUM iajs-2618 98 4 ( ( PUNCT iajs-2618 98 5 24 24 NUM iajs-2618 98 6 ) ) PUNCT iajs-2618 98 7 since since SCONJ iajs-2618 98 8 β€– β€– ADJ iajs-2618 98 9 𝑀𝑗+1 𝑀𝑗+1 NUM iajs-2618 98 10 𝑛 𝑛 PRON iajs-2618 98 11 β€–1 β€–1 NOUN iajs-2618 98 12 2 2 NUM iajs-2618 98 13 = = SYM iajs-2618 98 14 2β€– 2β€– PROPN iajs-2618 98 15 𝑀𝑗+1 𝑀𝑗+1 NUM iajs-2618 98 16 𝑛 𝑛 PRON iajs-2618 98 17 βˆ’ βˆ’ PROPN iajs-2618 98 18 𝑀𝑗 𝑀𝑗 ADP iajs-2618 98 19 𝑛 𝑛 PRON iajs-2618 98 20 β€–1 β€–1 NOUN iajs-2618 98 21 2 2 NUM iajs-2618 98 22 + + CCONJ iajs-2618 98 23 2β€– 2β€– NUM iajs-2618 98 24 𝑀𝑗 𝑀𝑗 ADP iajs-2618 98 25 𝑛 𝑛 PRON iajs-2618 98 26 β€–1 β€–1 NOUN iajs-2618 98 27 2 2 NUM iajs-2618 98 28 ( ( PUNCT iajs-2618 98 29 25 25 NUM iajs-2618 98 30 ) ) PUNCT iajs-2618 98 31 and and CCONJ iajs-2618 98 32 β€– β€– PROPN iajs-2618 98 33 𝑝𝑗+1 𝑝𝑗+1 X iajs-2618 98 34 𝑛 𝑛 DET iajs-2618 98 35 β€–0 β€–0 NUM iajs-2618 98 36 2 2 NUM iajs-2618 98 37 = = SYM iajs-2618 98 38 2β€– 2β€– PROPN iajs-2618 98 39 𝑝𝑗+1 𝑝𝑗+1 NUM iajs-2618 98 40 𝑛 𝑛 PRON iajs-2618 98 41 βˆ’ βˆ’ PROPN iajs-2618 98 42 𝑝𝑗 𝑝𝑗 NOUN iajs-2618 98 43 𝑛 𝑛 PRON iajs-2618 98 44 β€–0 β€–0 NOUN iajs-2618 98 45 2 2 NUM iajs-2618 99 1 + + SYM iajs-2618 99 2 2β€– 2β€– PROPN iajs-2618 99 3 𝑝𝑗 𝑝𝑗 NOUN iajs-2618 99 4 𝑛 𝑛 DET iajs-2618 99 5 β€–0 β€–0 NUM iajs-2618 99 6 2 2 NUM iajs-2618 99 7 ( ( PUNCT iajs-2618 99 8 26 26 NUM iajs-2618 99 9 ) ) PUNCT iajs-2618 99 10 substituting substitute VERB iajs-2618 99 11 ( ( PUNCT iajs-2618 99 12 25 25 NUM iajs-2618 99 13 ) ) PUNCT iajs-2618 99 14 and and CCONJ iajs-2618 99 15 ( ( PUNCT iajs-2618 99 16 26 26 NUM iajs-2618 99 17 ) ) PUNCT iajs-2618 99 18 in in ADP iajs-2618 99 19 inequality inequality NOUN iajs-2618 99 20 ( ( PUNCT iajs-2618 99 21 23 23 NUM iajs-2618 99 22 ) ) PUNCT iajs-2618 99 23 , , PUNCT iajs-2618 99 24 and and CCONJ iajs-2618 99 25 assume assume VERB iajs-2618 99 26 that that SCONJ iajs-2618 99 27 𝑑 𝑑 PROPN iajs-2618 99 28 = = SYM iajs-2618 99 29 max(2𝛿 max(2𝛿 PROPN iajs-2618 99 30 , , PUNCT iajs-2618 99 31 2𝛿̅ 2𝛿̅ NUM iajs-2618 99 32 ) ) PUNCT iajs-2618 99 33 , , PUNCT iajs-2618 99 34 to to PART iajs-2618 99 35 get get VERB iajs-2618 99 36 𝑐‖ 𝑐‖ NOUN iajs-2618 99 37 𝑝𝑙 𝑝𝑙 NOUN iajs-2618 99 38 𝑛 𝑛 PROPN iajs-2618 99 39 β€–0 β€–0 NOUN iajs-2618 99 40 2 2 NUM iajs-2618 99 41 + + CCONJ iajs-2618 99 42 ( ( PUNCT iajs-2618 99 43 𝑐 𝑐 PROPN iajs-2618 99 44 βˆ’ βˆ’ PROPN iajs-2618 99 45 𝑑δt 𝑑δt NOUN iajs-2618 99 46 ) ) PUNCT iajs-2618 99 47 βˆ‘ βˆ‘ PUNCT iajs-2618 99 48 β€– β€– PROPN iajs-2618 99 49 𝑝𝑗+1 𝑝𝑗+1 X iajs-2618 99 50 𝑛 𝑛 PRON iajs-2618 99 51 βˆ’ βˆ’ NUM iajs-2618 99 52 𝑝𝑗 𝑝𝑗 NOUN iajs-2618 99 53 𝑛 𝑛 PRON iajs-2618 99 54 β€–0 β€–0 NOUN iajs-2618 99 55 2π‘™βˆ’1 2π‘™βˆ’1 NUM iajs-2618 99 56 𝑗=0 𝑗=0 PUNCT iajs-2618 100 1 + + CCONJ iajs-2618 100 2 𝑐‖𝑀𝑙 𝑐‖𝑀𝑙 ADV iajs-2618 100 3 𝑛 𝑛 PRON iajs-2618 100 4 β€–1 β€–1 NOUN iajs-2618 100 5 2 2 NUM iajs-2618 100 6 + + CCONJ iajs-2618 100 7 ( ( PUNCT iajs-2618 100 8 𝑐 𝑐 PROPN iajs-2618 100 9 βˆ’ βˆ’ PROPN iajs-2618 100 10 𝑑δt 𝑑δt NOUN iajs-2618 100 11 ) ) PUNCT iajs-2618 100 12 βˆ‘ βˆ‘ PUNCT iajs-2618 100 13 β€– β€– ADJ iajs-2618 100 14 𝑀𝑗+1 𝑀𝑗+1 NUM iajs-2618 100 15 𝑛 𝑛 PRON iajs-2618 100 16 βˆ’ βˆ’ NOUN iajs-2618 100 17 𝑀𝑗 𝑀𝑗 ADP iajs-2618 100 18 𝑛 𝑛 PRON iajs-2618 100 19 β€–1 β€–1 NOUN iajs-2618 100 20 2π‘™βˆ’1 2π‘™βˆ’1 NUM iajs-2618 100 21 𝑗=0 𝑗=0 PUNCT iajs-2618 101 1 ≀ ≀ NUM iajs-2618 101 2 β€– β€– ADJ iajs-2618 101 3 𝑝0 𝑝0 NOUN iajs-2618 101 4 𝑛 𝑛 DET iajs-2618 101 5 β€–0 β€–0 NOUN iajs-2618 101 6 2 2 NUM iajs-2618 101 7 + + CCONJ iajs-2618 101 8 π‘˜2 π‘˜2 PROPN iajs-2618 101 9 2 2 NUM iajs-2618 101 10 ‖𝑀𝑙 ‖𝑀𝑙 PROPN iajs-2618 101 11 𝑛 𝑛 NOUN iajs-2618 101 12 β€–1 β€–1 NOUN iajs-2618 101 13 2 2 NUM iajs-2618 101 14 + + CCONJ iajs-2618 101 15 ‖𝛽‖𝑄 ‖𝛽‖𝑄 X iajs-2618 101 16 2 2 NUM iajs-2618 101 17 + + NUM iajs-2618 101 18 𝑑(Ξ΄t 𝑑(Ξ΄t NUM iajs-2618 101 19 ) ) PUNCT iajs-2618 101 20 βˆ‘ βˆ‘ PUNCT iajs-2618 101 21 ‖𝑀𝑗 ‖𝑀𝑗 NUM iajs-2618 101 22 𝑛 𝑛 PRON iajs-2618 101 23 β€–1 β€–1 NOUN iajs-2618 101 24 2 2 NUM iajs-2618 101 25 π‘™βˆ’1 π‘™βˆ’1 NUM iajs-2618 101 26 𝑗=0 𝑗=0 PROPN iajs-2618 101 27 + + CCONJ iajs-2618 101 28 𝑑(Ξ΄t 𝑑(Ξ΄t PROPN iajs-2618 101 29 ) ) PUNCT iajs-2618 101 30 βˆ‘ βˆ‘ PUNCT iajs-2618 101 31 ‖𝑝𝑗 ‖𝑝𝑗 NUM iajs-2618 101 32 𝑛 𝑛 DET iajs-2618 101 33 β€–0 β€–0 NOUN iajs-2618 101 34 2 2 NUM iajs-2618 101 35 .π‘™βˆ’1 .π‘™βˆ’1 NUM iajs-2618 101 36 𝑗=0 𝑗=0 X iajs-2618 102 1 ( ( PUNCT iajs-2618 102 2 27 27 NUM iajs-2618 102 3 ) ) PUNCT iajs-2618 102 4 now now ADV iajs-2618 102 5 , , PUNCT iajs-2618 102 6 let let VERB iajs-2618 102 7 βˆ†t βˆ†t PROPN iajs-2618 102 8 < < X iajs-2618 102 9 𝑐 𝑐 PROPN iajs-2618 102 10 𝑑⁄ 𝑑⁄ PROPN iajs-2618 102 11 then then ADV iajs-2618 102 12 the the DET iajs-2618 102 13 2𝑛𝑑 2𝑛𝑑 ADJ iajs-2618 102 14 and and CCONJ iajs-2618 102 15 4π‘‘β„Ž 4π‘‘β„Ž ADJ iajs-2618 102 16 terms term NOUN iajs-2618 102 17 in in ADP iajs-2618 102 18 the the DET iajs-2618 102 19 r.h.s r.h.s NOUN iajs-2618 102 20 of of ADP iajs-2618 102 21 ( ( PUNCT iajs-2618 102 22 27 27 NUM iajs-2618 102 23 ) ) PUNCT iajs-2618 102 24 are be AUX iajs-2618 102 25 positives positive NOUN iajs-2618 102 26 , , PUNCT iajs-2618 102 27 by by ADP iajs-2618 102 28 using use VERB iajs-2618 102 29 the the DET iajs-2618 102 30 discrete discrete ADJ iajs-2618 102 31 gronwall gronwall NOUN iajs-2618 102 32 ’s ’s PART iajs-2618 102 33 ( ( PUNCT iajs-2618 102 34 dgs dgs NOUN iajs-2618 102 35 ) ) PUNCT iajs-2618 102 36 inequality inequality NOUN iajs-2618 102 37 [ [ X iajs-2618 102 38 10 10 NUM iajs-2618 102 39 ] ] PUNCT iajs-2618 102 40 , , PUNCT iajs-2618 102 41 one one PRON iajs-2618 102 42 obtains obtain VERB iajs-2618 102 43 c(β€– c(β€– PROPN iajs-2618 102 44 𝑝𝑙 𝑝𝑙 NOUN iajs-2618 102 45 𝑛 𝑛 PROPN iajs-2618 102 46 β€–0 β€–0 NOUN iajs-2618 102 47 2 2 NUM iajs-2618 102 48 + + CCONJ iajs-2618 102 49 β€– β€– ADJ iajs-2618 102 50 𝑀𝑙 𝑀𝑙 X iajs-2618 102 51 𝑛 𝑛 DET iajs-2618 102 52 β€–1 β€–1 NOUN iajs-2618 102 53 2 2 NUM iajs-2618 102 54 ) ) PUNCT iajs-2618 102 55 ≀ ≀ PART iajs-2618 102 56 π‘Žπ‘’βˆ‘ π‘Žπ‘’βˆ‘ PROPN iajs-2618 102 57 𝑑(Ξ΄t 𝑑(Ξ΄t ADV iajs-2618 102 58 ) ) PUNCT iajs-2618 103 1 π‘™βˆ’1 π‘™βˆ’1 PROPN iajs-2618 103 2 𝑗=0 𝑗=0 PUNCT iajs-2618 104 1 = = PUNCT iajs-2618 104 2 π‘Žπ‘’π‘™π‘‘(Ξ΄t π‘Žπ‘’π‘™π‘‘(Ξ΄t PROPN iajs-2618 104 3 ) ) PUNCT iajs-2618 104 4 ≀ ≀ NOUN iajs-2618 104 5 𝑏 𝑏 PROPN iajs-2618 104 6 , , PUNCT iajs-2618 104 7 which which PRON iajs-2618 104 8 gives give VERB iajs-2618 104 9 that that PRON iajs-2618 104 10 β€– β€– ADJ iajs-2618 104 11 𝑀𝑙 𝑀𝑙 X iajs-2618 104 12 𝑛 𝑛 PRON iajs-2618 104 13 β€–1 β€–1 NOUN iajs-2618 104 14 2 2 NUM iajs-2618 104 15 ≀ ≀ NOUN iajs-2618 104 16 𝑑1 𝑑1 NOUN iajs-2618 104 17 = = PUNCT iajs-2618 104 18 𝑏 𝑏 PROPN iajs-2618 104 19 𝑐 𝑐 PROPN iajs-2618 104 20 , , PUNCT iajs-2618 104 21 and and CCONJ iajs-2618 104 22 β€– β€– PROPN iajs-2618 104 23 𝑝𝑙 𝑝𝑙 NOUN iajs-2618 104 24 𝑛 𝑛 PROPN iajs-2618 104 25 β€–0 β€–0 NUM iajs-2618 104 26 2 2 NUM iajs-2618 104 27 ≀ ≀ NOUN iajs-2618 104 28 𝑑1 𝑑1 NOUN iajs-2618 104 29 , , PUNCT iajs-2618 104 30 for for ADP iajs-2618 104 31 any any DET iajs-2618 104 32 arbitrary arbitrary ADJ iajs-2618 104 33 index index NOUN iajs-2618 104 34 𝑙. 𝑙. NOUN iajs-2618 104 35 hence hence ADV iajs-2618 104 36 , , PUNCT iajs-2618 104 37 β€– β€– PROPN iajs-2618 104 38 𝑀𝑗 𝑀𝑗 ADP iajs-2618 104 39 𝑛 𝑛 PRON iajs-2618 104 40 β€–1 β€–1 NOUN iajs-2618 104 41 2 2 NUM iajs-2618 104 42 ≀ ≀ NOUN iajs-2618 104 43 𝑑1 𝑑1 NOUN iajs-2618 104 44 and and CCONJ iajs-2618 104 45 β€– β€– PROPN iajs-2618 104 46 𝑝𝑗 𝑝𝑗 PROPN iajs-2618 104 47 𝑛 𝑛 PRON iajs-2618 104 48 β€–0 β€–0 NUM iajs-2618 104 49 2 2 NUM iajs-2618 104 50 ≀ ≀ NOUN iajs-2618 104 51 𝑑1 𝑑1 NOUN iajs-2618 104 52 , , PUNCT iajs-2618 104 53 for for ADP iajs-2618 104 54 each each DET iajs-2618 104 55 𝑗 𝑗 NOUN iajs-2618 104 56 = = SYM iajs-2618 104 57 0,1 0,1 NUM iajs-2618 104 58 , , PUNCT iajs-2618 104 59 … … PUNCT iajs-2618 104 60 . . PUNCT iajs-2618 104 61 . . PUNCT iajs-2618 105 1 , , PUNCT iajs-2618 105 2 π‘Œ π‘Œ PROPN iajs-2618 105 3 βˆ’ βˆ’ PROPN iajs-2618 105 4 1 1 X iajs-2618 105 5 . . PUNCT iajs-2618 106 1 therefore therefore ADV iajs-2618 106 2 ( ( PUNCT iajs-2618 106 3 Ξ΄t)𝑑 Ξ΄t)𝑑 NOUN iajs-2618 106 4 βˆ‘ βˆ‘ PUNCT iajs-2618 106 5 ‖𝑀𝑗 ‖𝑀𝑗 NUM iajs-2618 106 6 𝑛 𝑛 PRON iajs-2618 106 7 β€–1 β€–1 NOUN iajs-2618 106 8 2 2 NUM iajs-2618 106 9 π‘Œβˆ’1 π‘Œβˆ’1 NOUN iajs-2618 106 10 𝑗=0 𝑗=0 PUNCT iajs-2618 107 1 + + CCONJ iajs-2618 107 2 ( ( PUNCT iajs-2618 107 3 Ξ΄t)𝑑 Ξ΄t)𝑑 PART iajs-2618 107 4 βˆ‘ βˆ‘ PART iajs-2618 107 5 ‖𝑝𝑗 ‖𝑝𝑗 NUM iajs-2618 107 6 𝑛 𝑛 PRON iajs-2618 107 7 β€–1 β€–1 NOUN iajs-2618 107 8 2 2 NUM iajs-2618 107 9 π‘Œβˆ’1 π‘Œβˆ’1 NOUN iajs-2618 107 10 𝑗=0 𝑗=0 PUNCT iajs-2618 107 11 ≀ ≀ NOUN iajs-2618 107 12 2𝑑1 2𝑑1 NUM iajs-2618 108 1 𝑑 𝑑 NOUN iajs-2618 108 2 Ξ΄t Ξ΄t X iajs-2618 108 3 y y NOUN iajs-2618 108 4 = = PUNCT iajs-2618 108 5 2ct 2ct NOUN iajs-2618 108 6 = = SYM iajs-2618 108 7 οΏ½ οΏ½ PROPN iajs-2618 108 8 Μ… Μ… NOUN iajs-2618 108 9 οΏ½ οΏ½ PROPN iajs-2618 108 10 . . PUNCT iajs-2618 109 1 we we PRON iajs-2618 109 2 back back VERB iajs-2618 109 3 to to ADP iajs-2618 109 4 ( ( PUNCT iajs-2618 109 5 27 27 NUM iajs-2618 109 6 ) ) PUNCT iajs-2618 109 7 substituting substitute VERB iajs-2618 109 8 𝑙 𝑙 X iajs-2618 109 9 = = SYM iajs-2618 109 10 π‘Œ π‘Œ PROPN iajs-2618 109 11 , , PUNCT iajs-2618 109 12 the the DET iajs-2618 109 13 1st 1st NOUN iajs-2618 109 14 and and CCONJ iajs-2618 109 15 the the DET iajs-2618 109 16 3rd 3rd ADJ iajs-2618 109 17 term term NOUN iajs-2618 109 18 in in ADP iajs-2618 109 19 the the DET iajs-2618 109 20 l.h.s l.h.s NOUN iajs-2618 109 21 are be AUX iajs-2618 109 22 positives positive NOUN iajs-2618 109 23 , , PUNCT iajs-2618 109 24 then then ADV iajs-2618 109 25 we we PRON iajs-2618 109 26 use use VERB iajs-2618 109 27 the the DET iajs-2618 109 28 above above ADJ iajs-2618 109 29 results result NOUN iajs-2618 109 30 in in ADP iajs-2618 109 31 the the DET iajs-2618 109 32 r.h.s r.h.s NOUN iajs-2618 109 33 . . PUNCT iajs-2618 110 1 of of ADP iajs-2618 110 2 it it PRON iajs-2618 110 3 , , PUNCT iajs-2618 110 4 keeping keep VERB iajs-2618 110 5 in in ADP iajs-2618 110 6 mind mind NOUN iajs-2618 110 7 the the DET iajs-2618 110 8 first first ADJ iajs-2618 110 9 three three NUM iajs-2618 110 10 terms term NOUN iajs-2618 110 11 in in ADP iajs-2618 110 12 this this DET iajs-2618 110 13 side side NOUN iajs-2618 110 14 that that PRON iajs-2618 110 15 are be AUX iajs-2618 110 16 bounded bound VERB iajs-2618 110 17 ( ( PUNCT iajs-2618 110 18 from from ADP iajs-2618 110 19 the the DET iajs-2618 110 20 above above ADJ iajs-2618 110 21 steps step NOUN iajs-2618 110 22 ) ) PUNCT iajs-2618 110 23 , , PUNCT iajs-2618 110 24 to to PART iajs-2618 110 25 obtain obtain VERB iajs-2618 110 26 βˆ‘ βˆ‘ PUNCT iajs-2618 110 27 β€– β€– ADJ iajs-2618 110 28 𝑀𝑗+1 𝑀𝑗+1 NUM iajs-2618 110 29 𝑛 𝑛 PRON iajs-2618 110 30 βˆ’ βˆ’ NOUN iajs-2618 110 31 𝑀𝑗 𝑀𝑗 ADP iajs-2618 110 32 𝑛 𝑛 PRON iajs-2618 110 33 β€–1 β€–1 NOUN iajs-2618 110 34 2 2 NUM iajs-2618 110 35 π‘Œβˆ’1 π‘Œβˆ’1 NOUN iajs-2618 110 36 𝑗=0 𝑗=0 PUNCT iajs-2618 110 37 ≀ ≀ PROPN iajs-2618 110 38 οΏ½ οΏ½ PROPN iajs-2618 110 39 Μ… Μ… NOUN iajs-2618 110 40 οΏ½ οΏ½ PROPN iajs-2618 110 41 ( ( PUNCT iajs-2618 110 42 28a 28a NUM iajs-2618 110 43 ) ) PUNCT iajs-2618 110 44 βˆ‘ βˆ‘ PUNCT iajs-2618 110 45 β€– β€– PROPN iajs-2618 110 46 𝑝𝑗+1 𝑝𝑗+1 X iajs-2618 110 47 𝑛 𝑛 PRON iajs-2618 110 48 βˆ’ βˆ’ NUM iajs-2618 110 49 𝑝𝑗 𝑝𝑗 NOUN iajs-2618 110 50 𝑛 𝑛 PRON iajs-2618 110 51 β€–0 β€–0 NOUN iajs-2618 110 52 2 2 NUM iajs-2618 110 53 π‘Œβˆ’1 π‘Œβˆ’1 NOUN iajs-2618 110 54 𝑗=0 𝑗=0 PUNCT iajs-2618 110 55 ≀ ≀ PROPN iajs-2618 110 56 οΏ½ οΏ½ PROPN iajs-2618 110 57 Μ… Μ… NOUN iajs-2618 110 58 οΏ½ οΏ½ PROPN iajs-2618 110 59 ( ( PUNCT iajs-2618 110 60 28b 28b NOUN iajs-2618 110 61 ) ) PUNCT iajs-2618 110 62 6 6 NUM iajs-2618 110 63 . . PUNCT iajs-2618 110 64 convergence convergence NOUN iajs-2618 110 65 : : PUNCT iajs-2618 110 66 124 124 NUM iajs-2618 110 67 ibn ibn PROPN iajs-2618 110 68 al al PROPN iajs-2618 110 69 - - PUNCT iajs-2618 110 70 haitham haitham PROPN iajs-2618 110 71 jour jour X iajs-2618 110 72 . . PROPN iajs-2618 111 1 for for ADP iajs-2618 111 2 pure pure ADJ iajs-2618 111 3 & & CCONJ iajs-2618 111 4 appl appl PROPN iajs-2618 111 5 . . PUNCT iajs-2618 112 1 sci sci PROPN iajs-2618 112 2 . . PROPN iajs-2618 113 1 34 34 NUM iajs-2618 113 2 ( ( PUNCT iajs-2618 113 3 2 2 NUM iajs-2618 113 4 ) ) PUNCT iajs-2618 113 5 2021 2021 NUM iajs-2618 113 6 the the DET iajs-2618 113 7 following follow VERB iajs-2618 113 8 definitions definition NOUN iajs-2618 113 9 for for ADP iajs-2618 113 10 the the DET iajs-2618 113 11 functions function NOUN iajs-2618 113 12 " " PUNCT iajs-2618 113 13 almost almost ADV iajs-2618 113 14 everywhere everywhere ADV iajs-2618 113 15 on on ADP iajs-2618 113 16 i i PRON iajs-2618 113 17 " " PUNCT iajs-2618 113 18 are be AUX iajs-2618 113 19 useful useful ADJ iajs-2618 113 20 in in ADP iajs-2618 113 21 the the DET iajs-2618 113 22 proof proof NOUN iajs-2618 113 23 of of ADP iajs-2618 113 24 next next ADJ iajs-2618 113 25 theorem theorem NOUN iajs-2618 113 26 , , PUNCT iajs-2618 113 27 so so ADV iajs-2618 113 28 let let VERB iajs-2618 113 29 π‘€βˆ’ π‘€βˆ’ PROPN iajs-2618 113 30 𝑛 𝑛 PART iajs-2618 113 31 ( ( PUNCT iajs-2618 113 32 𝑑 𝑑 NOUN iajs-2618 113 33 ) ) PUNCT iajs-2618 113 34 ∢= ∢= NUM iajs-2618 113 35 𝑀𝑗 𝑀𝑗 ADP iajs-2618 113 36 𝑛 𝑛 PROPN iajs-2618 113 37 , , PUNCT iajs-2618 113 38 𝑑 𝑑 PROPN iajs-2618 113 39 ∈ ∈ PROPN iajs-2618 113 40 𝐼𝑗 𝐼𝑗 PROPN iajs-2618 113 41 𝑛 𝑛 PROPN iajs-2618 113 42 , , PUNCT iajs-2618 113 43 βˆ€ βˆ€ X iajs-2618 113 44 𝑗 𝑗 NOUN iajs-2618 113 45 = = SYM iajs-2618 113 46 0,1 0,1 NUM iajs-2618 113 47 , , PUNCT iajs-2618 113 48 … … PUNCT iajs-2618 113 49 . . PUNCT iajs-2618 114 1 , , PUNCT iajs-2618 114 2 π‘Œ π‘Œ PROPN iajs-2618 114 3 , , PUNCT iajs-2618 114 4 𝑀+ 𝑀+ VERB iajs-2618 114 5 𝑛 𝑛 DET iajs-2618 114 6 ( ( PUNCT iajs-2618 114 7 𝑑 𝑑 NOUN iajs-2618 114 8 ) ) PUNCT iajs-2618 114 9 ∢= ∢= NUM iajs-2618 114 10 𝑀𝑗+1 𝑀𝑗+1 ADP iajs-2618 114 11 𝑛 𝑛 PROPN iajs-2618 114 12 , , PUNCT iajs-2618 114 13 𝑑 𝑑 PROPN iajs-2618 114 14 ∈ ∈ PROPN iajs-2618 114 15 𝐼𝑗 𝐼𝑗 PROPN iajs-2618 114 16 𝑛 𝑛 PROPN iajs-2618 114 17 , , PUNCT iajs-2618 114 18 βˆ€ βˆ€ X iajs-2618 114 19 𝑗 𝑗 NOUN iajs-2618 114 20 = = SYM iajs-2618 114 21 0,1 0,1 NUM iajs-2618 114 22 , , PUNCT iajs-2618 114 23 … … PUNCT iajs-2618 114 24 . . PUNCT iajs-2618 115 1 , , PUNCT iajs-2618 115 2 π‘Œ π‘Œ PROPN iajs-2618 115 3 βˆ’ βˆ’ PROPN iajs-2618 115 4 1 1 NUM iajs-2618 115 5 , , PUNCT iajs-2618 115 6 𝑝+ 𝑝+ X iajs-2618 115 7 𝑛 𝑛 PROPN iajs-2618 115 8 ( ( PUNCT iajs-2618 115 9 𝑑 𝑑 NOUN iajs-2618 115 10 ) ) PUNCT iajs-2618 115 11 ∢= ∢= NUM iajs-2618 115 12 𝑝𝑗+1 𝑝𝑗+1 PROPN iajs-2618 115 13 𝑛 𝑛 NOUN iajs-2618 115 14 , , PUNCT iajs-2618 115 15 𝑑 𝑑 PROPN iajs-2618 115 16 ∈ ∈ PROPN iajs-2618 115 17 𝐼𝑗 𝐼𝑗 PROPN iajs-2618 115 18 𝑛 𝑛 PROPN iajs-2618 115 19 , , PUNCT iajs-2618 115 20 βˆ€ βˆ€ X iajs-2618 115 21 𝑗 𝑗 NOUN iajs-2618 115 22 = = SYM iajs-2618 115 23 0,1 0,1 NUM iajs-2618 115 24 , , PUNCT iajs-2618 115 25 … … PUNCT iajs-2618 115 26 . . PUNCT iajs-2618 116 1 , , PUNCT iajs-2618 116 2 π‘Œ π‘Œ PROPN iajs-2618 116 3 βˆ’ βˆ’ PROPN iajs-2618 116 4 1 1 NUM iajs-2618 116 5 , , PUNCT iajs-2618 116 6 π‘βˆ’ π‘βˆ’ NOUN iajs-2618 116 7 𝑛 𝑛 PROPN iajs-2618 116 8 ( ( PUNCT iajs-2618 116 9 𝑑𝑗 𝑑𝑗 PROPN iajs-2618 116 10 𝑛 𝑛 PROPN iajs-2618 116 11 ) ) PUNCT iajs-2618 116 12 ∢= ∢= NUM iajs-2618 116 13 𝑝𝑗 𝑝𝑗 PROPN iajs-2618 116 14 𝑛 𝑛 PROPN iajs-2618 116 15 , , PUNCT iajs-2618 116 16 𝑑𝑗 𝑑𝑗 PROPN iajs-2618 116 17 𝑛 𝑛 PRON iajs-2618 116 18 ∈ ∈ PROPN iajs-2618 116 19 𝐼𝑗 𝐼𝑗 PROPN iajs-2618 116 20 𝑛 𝑛 PROPN iajs-2618 116 21 , , PUNCT iajs-2618 116 22 βˆ€ βˆ€ X iajs-2618 116 23 𝑗 𝑗 NOUN iajs-2618 116 24 = = SYM iajs-2618 116 25 0,1 0,1 NUM iajs-2618 116 26 , , PUNCT iajs-2618 116 27 … … PUNCT iajs-2618 116 28 . . PUNCT iajs-2618 117 1 , , PUNCT iajs-2618 117 2 π‘Œ π‘Œ PROPN iajs-2618 117 3 , , PUNCT iajs-2618 117 4 also also ADV iajs-2618 117 5 , , PUNCT iajs-2618 117 6 let let VERB iajs-2618 117 7 𝑀^ 𝑀^ PRON iajs-2618 117 8 𝑛 𝑛 PRON iajs-2618 117 9 ( ( PUNCT iajs-2618 117 10 𝑑 𝑑 NOUN iajs-2618 117 11 ) ) PUNCT iajs-2618 117 12 ∢= ∢= NUM iajs-2618 117 13 𝑀𝑗 𝑀𝑗 ADP iajs-2618 117 14 𝑛 𝑛 PRON iajs-2618 117 15 be be AUX iajs-2618 117 16 an an DET iajs-2618 117 17 affine affine ADJ iajs-2618 117 18 function function NOUN iajs-2618 117 19 on on ADP iajs-2618 117 20 each each DET iajs-2618 117 21 𝐼𝑗 𝐼𝑗 PROPN iajs-2618 117 22 𝑛 𝑛 PROPN iajs-2618 117 23 , , PUNCT iajs-2618 117 24 βˆ€ βˆ€ X iajs-2618 117 25 , , PUNCT iajs-2618 117 26 𝑗 𝑗 NOUN iajs-2618 117 27 = = SYM iajs-2618 117 28 0,1 0,1 NUM iajs-2618 117 29 , , PUNCT iajs-2618 117 30 … … PUNCT iajs-2618 117 31 . . PUNCT iajs-2618 118 1 , , PUNCT iajs-2618 118 2 π‘Œ π‘Œ PROPN iajs-2618 118 3 , , PUNCT iajs-2618 118 4 and and CCONJ iajs-2618 118 5 𝑝^ 𝑝^ NOUN iajs-2618 118 6 𝑛 𝑛 PRON iajs-2618 118 7 ( ( PUNCT iajs-2618 118 8 𝑑 𝑑 NOUN iajs-2618 118 9 ) ) PUNCT iajs-2618 118 10 ∢= ∢= NUM iajs-2618 118 11 𝑝𝑗 𝑝𝑗 PROPN iajs-2618 118 12 𝑛 𝑛 PROPN iajs-2618 118 13 , , PUNCT iajs-2618 118 14 be be AUX iajs-2618 118 15 an an DET iajs-2618 118 16 affine affine ADJ iajs-2618 118 17 function function NOUN iajs-2618 118 18 on on ADP iajs-2618 118 19 each each DET iajs-2618 118 20 𝐼𝑗 𝐼𝑗 PROPN iajs-2618 118 21 𝑛 𝑛 PROPN iajs-2618 118 22 , , PUNCT iajs-2618 118 23 βˆ€ βˆ€ X iajs-2618 118 24 , , PUNCT iajs-2618 118 25 𝑗 𝑗 NOUN iajs-2618 118 26 = = SYM iajs-2618 118 27 0,1 0,1 NUM iajs-2618 118 28 , , PUNCT iajs-2618 118 29 … … PUNCT iajs-2618 118 30 . . PUNCT iajs-2618 119 1 , , PUNCT iajs-2618 119 2 π‘Œ. π‘Œ. PROPN iajs-2618 119 3 theorem theorem NOUN iajs-2618 119 4 ( ( PUNCT iajs-2618 119 5 7 7 NUM iajs-2618 119 6 ): ): PUNCT iajs-2618 119 7 the the DET iajs-2618 119 8 discrete discrete ADJ iajs-2618 119 9 solutions solution NOUN iajs-2618 119 10 π‘€βˆ’ π‘€βˆ’ NOUN iajs-2618 119 11 𝑛 𝑛 PROPN iajs-2618 119 12 ( ( PUNCT iajs-2618 119 13 𝑑 𝑑 NOUN iajs-2618 119 14 ) ) PUNCT iajs-2618 119 15 , , PUNCT iajs-2618 119 16 𝑀+ 𝑀+ VERB iajs-2618 119 17 𝑛 𝑛 PRON iajs-2618 119 18 ( ( PUNCT iajs-2618 119 19 𝑑 𝑑 NOUN iajs-2618 119 20 ) ) PUNCT iajs-2618 119 21 , , PUNCT iajs-2618 119 22 and and CCONJ iajs-2618 119 23 𝑀^ 𝑀^ NUM iajs-2618 119 24 𝑛 𝑛 DET iajs-2618 119 25 ( ( PUNCT iajs-2618 119 26 𝑑 𝑑 NOUN iajs-2618 119 27 ) ) PUNCT iajs-2618 119 28 are be AUX iajs-2618 119 29 converges converge NOUN iajs-2618 119 30 strongly strongly ADV iajs-2618 119 31 in in ADP iajs-2618 119 32 𝐿2(πœ‘ 𝐿2(πœ‘ PROPN iajs-2618 119 33 ) ) PUNCT iajs-2618 119 34 , , PUNCT iajs-2618 119 35 where where SCONJ iajs-2618 119 36 𝑛 𝑛 PROPN iajs-2618 119 37 ⟢ ⟢ NOUN iajs-2618 119 38 ∞. ∞. PROPN iajs-2618 119 39 proof proof NOUN iajs-2618 119 40 : : PUNCT iajs-2618 119 41 we we PRON iajs-2618 119 42 start start VERB iajs-2618 119 43 with with ADP iajs-2618 119 44 using use VERB iajs-2618 119 45 lemma lemma PROPN iajs-2618 119 46 ( ( PUNCT iajs-2618 119 47 6 6 NUM iajs-2618 119 48 ) ) PUNCT iajs-2618 119 49 we we PRON iajs-2618 119 50 have have VERB iajs-2618 119 51 for for ADP iajs-2618 119 52 any any DET iajs-2618 119 53 𝑗 𝑗 NOUN iajs-2618 119 54 = = SYM iajs-2618 119 55 0,1 0,1 NUM iajs-2618 119 56 , , PUNCT iajs-2618 119 57 … … PUNCT iajs-2618 119 58 . . PUNCT iajs-2618 120 1 , , PUNCT iajs-2618 120 2 π‘Œ π‘Œ PROPN iajs-2618 120 3 that that PRON iajs-2618 120 4 ‖𝑀𝑗 ‖𝑀𝑗 NUM iajs-2618 120 5 𝑛 𝑛 PRON iajs-2618 120 6 β€–1 β€–1 NOUN iajs-2618 120 7 2 2 NUM iajs-2618 120 8 ≀ ≀ NUM iajs-2618 120 9 οΏ½ οΏ½ NOUN iajs-2618 120 10 Μ… Μ… NOUN iajs-2618 120 11 οΏ½ οΏ½ PROPN iajs-2618 120 12 and and CCONJ iajs-2618 120 13 ‖𝑝𝑗 ‖𝑝𝑗 NUM iajs-2618 120 14 𝑛 𝑛 DET iajs-2618 120 15 β€–0 β€–0 NUM iajs-2618 120 16 2 2 NUM iajs-2618 120 17 ≀ ≀ NUM iajs-2618 120 18 οΏ½ οΏ½ NOUN iajs-2618 120 19 Μ… Μ… NOUN iajs-2618 120 20 οΏ½ οΏ½ NOUN iajs-2618 120 21 , , PUNCT iajs-2618 120 22 then then ADV iajs-2618 120 23 β€–π‘€βˆ’ β€–π‘€βˆ’ PROPN iajs-2618 120 24 𝑛 𝑛 PROPN iajs-2618 120 25 ‖𝐿2(𝐼,𝑉 ‖𝐿2(𝐼,𝑉 PROPN iajs-2618 120 26 ) ) PUNCT iajs-2618 120 27 2 2 NUM iajs-2618 120 28 , , PUNCT iajs-2618 120 29 β€– β€– PROPN iajs-2618 120 30 𝑀+ 𝑀+ VERB iajs-2618 120 31 𝑛 𝑛 PROPN iajs-2618 120 32 ‖𝐿2(𝐼,𝑉 ‖𝐿2(𝐼,𝑉 PROPN iajs-2618 120 33 ) ) PUNCT iajs-2618 120 34 2 2 NUM iajs-2618 120 35 , , PUNCT iajs-2618 120 36 ‖𝑀^ ‖𝑀^ NUM iajs-2618 120 37 𝑛 𝑛 PROPN iajs-2618 120 38 ‖𝐿2(𝐼,𝑉 ‖𝐿2(𝐼,𝑉 PROPN iajs-2618 120 39 ) ) PUNCT iajs-2618 120 40 2 2 NUM iajs-2618 120 41 , , PUNCT iajs-2618 120 42 β€–π‘βˆ’ β€–π‘βˆ’ INTJ iajs-2618 120 43 𝑛 𝑛 ADP iajs-2618 120 44 ‖𝐿2(πœ‘ ‖𝐿2(πœ‘ NUM iajs-2618 120 45 ) ) PUNCT iajs-2618 120 46 2 2 NUM iajs-2618 120 47 , , PUNCT iajs-2618 120 48 ‖𝑝+ ‖𝑝+ NOUN iajs-2618 120 49 𝑛 𝑛 ADJ iajs-2618 120 50 ‖𝐿2(πœ‘ ‖𝐿2(πœ‘ PROPN iajs-2618 120 51 ) ) PUNCT iajs-2618 120 52 2 2 NUM iajs-2618 120 53 , , PUNCT iajs-2618 120 54 and and CCONJ iajs-2618 120 55 ‖𝑝^ ‖𝑝^ NUM iajs-2618 120 56 𝑛‖𝐿2(πœ‘ 𝑛‖𝐿2(πœ‘ NOUN iajs-2618 120 57 ) ) PUNCT iajs-2618 120 58 2 2 NUM iajs-2618 120 59 are be AUX iajs-2618 120 60 bounded bound VERB iajs-2618 120 61 . . PUNCT iajs-2618 121 1 from from ADP iajs-2618 121 2 ( ( PUNCT iajs-2618 121 3 28a 28a NOUN iajs-2618 121 4 ) ) PUNCT iajs-2618 121 5 , , PUNCT iajs-2618 121 6 we we PRON iajs-2618 121 7 have have VERB iajs-2618 121 8 Ξ΄t Ξ΄t X iajs-2618 121 9 βˆ‘ βˆ‘ PUNCT iajs-2618 121 10 β€– β€– VERB iajs-2618 121 11 𝑀𝑗+1 𝑀𝑗+1 NUM iajs-2618 121 12 𝑛 𝑛 DET iajs-2618 121 13 βˆ’ βˆ’ NOUN iajs-2618 121 14 𝑀𝑗 𝑀𝑗 ADP iajs-2618 121 15 𝑛 𝑛 DET iajs-2618 121 16 β€–0 β€–0 NUM iajs-2618 121 17 2 2 NUM iajs-2618 121 18 π‘Œβˆ’1 π‘Œβˆ’1 NOUN iajs-2618 121 19 𝑗=0 𝑗=0 PROPN iajs-2618 121 20 ≀ ≀ PROPN iajs-2618 121 21 Ξ΄t Ξ΄t ADP iajs-2618 121 22 οΏ½ οΏ½ NOUN iajs-2618 121 23 Μ… Μ… NOUN iajs-2618 121 24 οΏ½ οΏ½ NOUN iajs-2618 121 25 ⟢ ⟢ NOUN iajs-2618 121 26 0 0 NUM iajs-2618 121 27 , , PUNCT iajs-2618 121 28 as as ADP iajs-2618 121 29 Ξ΄t Ξ΄t ADP iajs-2618 121 30 ⟢ ⟢ PROPN iajs-2618 121 31 0 0 NUM iajs-2618 121 32 , , PUNCT iajs-2618 121 33 gives give VERB iajs-2618 121 34 οΏ½ οΏ½ NOUN iajs-2618 121 35 Μ… Μ… NOUN iajs-2618 121 36 οΏ½ οΏ½ NOUN iajs-2618 121 37 + + SYM iajs-2618 122 1 𝑛 𝑛 PROPN iajs-2618 122 2 ⟢ ⟢ NOUN iajs-2618 122 3 οΏ½ οΏ½ NOUN iajs-2618 122 4 Μ… Μ… NOUN iajs-2618 122 5 οΏ½ οΏ½ NOUN iajs-2618 122 6 βˆ’ βˆ’ ADP iajs-2618 122 7 𝑛 𝑛 PRON iajs-2618 122 8 strongly strongly ADV iajs-2618 122 9 ( ( PUNCT iajs-2618 122 10 st st NOUN iajs-2618 122 11 ) ) PUNCT iajs-2618 122 12 in in ADP iajs-2618 122 13 𝐿2(𝐼 𝐿2(𝐼 PROPN iajs-2618 122 14 , , PUNCT iajs-2618 122 15 𝑉 𝑉 PROPN iajs-2618 122 16 ) ) PUNCT iajs-2618 122 17 and and CCONJ iajs-2618 122 18 then then ADV iajs-2618 122 19 in in ADP iajs-2618 122 20 𝐿2(πœ‘ 𝐿2(πœ‘ PROPN iajs-2618 122 21 ) ) PUNCT iajs-2618 122 22 . . PUNCT iajs-2618 123 1 ( ( PUNCT iajs-2618 123 2 29a 29a NUM iajs-2618 123 3 ) ) PUNCT iajs-2618 123 4 by by ADP iajs-2618 123 5 the the DET iajs-2618 123 6 same same ADJ iajs-2618 123 7 way way NOUN iajs-2618 123 8 from from ADP iajs-2618 123 9 ( ( PUNCT iajs-2618 123 10 28b 28b NOUN iajs-2618 123 11 ) ) PUNCT iajs-2618 123 12 , , PUNCT iajs-2618 123 13 we we PRON iajs-2618 123 14 get get VERB iajs-2618 123 15 that that PRON iajs-2618 123 16 , , PUNCT iajs-2618 123 17 𝑝+ 𝑝+ CCONJ iajs-2618 123 18 𝑛 𝑛 PROPN iajs-2618 123 19 ⟢ ⟢ NOUN iajs-2618 123 20 π‘βˆ’ π‘βˆ’ PROPN iajs-2618 123 21 𝑛 𝑛 PROPN iajs-2618 123 22 st st PROPN iajs-2618 123 23 in in ADP iajs-2618 123 24 𝐿2(πœ‘ 𝐿2(πœ‘ PROPN iajs-2618 123 25 ) ) PUNCT iajs-2618 123 26 ( ( PUNCT iajs-2618 123 27 29b 29b NUM iajs-2618 123 28 ) ) PUNCT iajs-2618 123 29 then then ADV iajs-2618 123 30 by by ADP iajs-2618 123 31 theorem theorem NOUN iajs-2618 123 32 3 3 NUM iajs-2618 123 33 , , PUNCT iajs-2618 123 34 there there PRON iajs-2618 123 35 exist exist VERB iajs-2618 123 36 subsequences subsequence NOUN iajs-2618 123 37 of of ADP iajs-2618 123 38 { { PUNCT iajs-2618 123 39 π‘€βˆ’ π‘€βˆ’ PROPN iajs-2618 123 40 𝑛 𝑛 PROPN iajs-2618 123 41 } } PUNCT iajs-2618 123 42 , , PUNCT iajs-2618 123 43 { { PUNCT iajs-2618 123 44 𝑀+ 𝑀+ NOUN iajs-2618 123 45 𝑛 𝑛 X iajs-2618 123 46 } } PUNCT iajs-2618 123 47 , , PUNCT iajs-2618 123 48 { { PUNCT iajs-2618 123 49 𝑀^ 𝑀^ X iajs-2618 123 50 𝑛 𝑛 PRON iajs-2618 123 51 } } PUNCT iajs-2618 123 52 ) ) PUNCT iajs-2618 123 53 , , PUNCT iajs-2618 123 54 and and CCONJ iajs-2618 123 55 of of ADP iajs-2618 123 56 ( ( PUNCT iajs-2618 123 57 { { PUNCT iajs-2618 123 58 π‘βˆ’ π‘βˆ’ NOUN iajs-2618 123 59 𝑛 𝑛 VERB iajs-2618 123 60 } } PUNCT iajs-2618 123 61 , , PUNCT iajs-2618 123 62 { { PUNCT iajs-2618 123 63 𝑝+ 𝑝+ NOUN iajs-2618 123 64 𝑛 𝑛 NOUN iajs-2618 123 65 } } PUNCT iajs-2618 123 66 , , PUNCT iajs-2618 123 67 { { PUNCT iajs-2618 123 68 𝑝^ 𝑝^ NOUN iajs-2618 123 69 𝑛 𝑛 X iajs-2618 123 70 } } PUNCT iajs-2618 123 71 , , PUNCT iajs-2618 123 72 ) ) PUNCT iajs-2618 123 73 use use VERB iajs-2618 123 74 again again ADV iajs-2618 123 75 the the DET iajs-2618 123 76 same same ADJ iajs-2618 123 77 notations notation NOUN iajs-2618 123 78 which which PRON iajs-2618 123 79 converge converge VERB iajs-2618 123 80 weakly weakly ADV iajs-2618 123 81 to to ADP iajs-2618 123 82 some some DET iajs-2618 123 83 𝑀 𝑀 NOUN iajs-2618 123 84 in in ADP iajs-2618 123 85 𝐿2(𝐼 𝐿2(𝐼 PROPN iajs-2618 123 86 , , PUNCT iajs-2618 123 87 𝑉 𝑉 PROPN iajs-2618 123 88 ) ) PUNCT iajs-2618 123 89 , , PUNCT iajs-2618 123 90 to to ADP iajs-2618 123 91 some some DET iajs-2618 123 92 𝑝 𝑝 NOUN iajs-2618 123 93 in in ADP iajs-2618 123 94 𝐿2(πœ‘ 𝐿2(πœ‘ PROPN iajs-2618 123 95 ) ) PUNCT iajs-2618 123 96 , , PUNCT iajs-2618 123 97 i.e. i.e. X iajs-2618 123 98 π‘€βˆ’ π‘€βˆ’ PROPN iajs-2618 123 99 𝑛 𝑛 PROPN iajs-2618 123 100 ⟢ ⟢ NOUN iajs-2618 123 101 𝑀 𝑀 NOUN iajs-2618 123 102 , , PUNCT iajs-2618 123 103 𝑀+ 𝑀+ VERB iajs-2618 123 104 𝑛 𝑛 PRON iajs-2618 123 105 ⟢ ⟢ NOUN iajs-2618 123 106 𝑀 𝑀 ADP iajs-2618 123 107 , , PUNCT iajs-2618 123 108 𝑀^ 𝑀^ NOUN iajs-2618 123 109 𝑛 𝑛 DET iajs-2618 123 110 ⟢ ⟢ NOUN iajs-2618 123 111 𝑀 𝑀 X iajs-2618 123 112 weakly weakly ADJ iajs-2618 123 113 in in ADP iajs-2618 123 114 𝐿2(𝐼 𝐿2(𝐼 PROPN iajs-2618 123 115 , , PUNCT iajs-2618 123 116 𝑉 𝑉 PROPN iajs-2618 123 117 ) ) PUNCT iajs-2618 123 118 π‘βˆ’ π‘βˆ’ NOUN iajs-2618 123 119 𝑛 𝑛 PROPN iajs-2618 123 120 ⟢ ⟢ PROPN iajs-2618 123 121 𝑝 𝑝 PROPN iajs-2618 123 122 , , PUNCT iajs-2618 123 123 𝑝+ 𝑝+ CCONJ iajs-2618 123 124 𝑛 𝑛 PROPN iajs-2618 123 125 ⟢ ⟢ PROPN iajs-2618 123 126 𝑝 𝑝 PROPN iajs-2618 123 127 , , PUNCT iajs-2618 123 128 𝑝^ 𝑝^ NOUN iajs-2618 123 129 𝑛 𝑛 DET iajs-2618 123 130 ⟢ ⟢ NOUN iajs-2618 123 131 𝑝 𝑝 NOUN iajs-2618 123 132 weakly weakly ADJ iajs-2618 123 133 in in ADP iajs-2618 123 134 𝐿2(πœ‘ 𝐿2(πœ‘ PROPN iajs-2618 123 135 ) ) PUNCT iajs-2618 123 136 , , PUNCT iajs-2618 123 137 n n CCONJ iajs-2618 123 138 this this DET iajs-2618 123 139 point point NOUN iajs-2618 123 140 the the DET iajs-2618 123 141 first first ADJ iajs-2618 123 142 compactness compactness NOUN iajs-2618 123 143 theorem[9 theorem[9 NOUN iajs-2618 123 144 ] ] PUNCT iajs-2618 123 145 is be AUX iajs-2618 123 146 used use VERB iajs-2618 123 147 , , PUNCT iajs-2618 123 148 to to PART iajs-2618 123 149 get get VERB iajs-2618 123 150 that that DET iajs-2618 123 151 𝑀^ 𝑀^ NOUN iajs-2618 123 152 𝑛 𝑛 DET iajs-2618 123 153 ⟢ ⟢ NOUN iajs-2618 123 154 𝑀 𝑀 ADP iajs-2618 123 155 st st PROPN iajs-2618 123 156 in in ADP iajs-2618 123 157 𝐿2(πœ‘),then 𝐿2(πœ‘),then PROPN iajs-2618 123 158 𝑀+ 𝑀+ VERB iajs-2618 123 159 𝑛 𝑛 DET iajs-2618 123 160 ⟢ ⟢ NOUN iajs-2618 123 161 𝑀 𝑀 ADP iajs-2618 123 162 and and CCONJ iajs-2618 123 163 π‘€βˆ’ π‘€βˆ’ PROPN iajs-2618 123 164 𝑛 𝑛 PROPN iajs-2618 123 165 ⟢ ⟢ NOUN iajs-2618 123 166 𝑀 𝑀 ADP iajs-2618 123 167 st st PROPN iajs-2618 123 168 in in ADP iajs-2618 123 169 𝐿2(πœ‘ 𝐿2(πœ‘ PROPN iajs-2618 123 170 ) ) PUNCT iajs-2618 123 171 . . PUNCT iajs-2618 124 1 now now ADV iajs-2618 124 2 , , PUNCT iajs-2618 124 3 let let VERB iajs-2618 124 4 { { PUNCT iajs-2618 124 5 𝑉𝑛 𝑉𝑛 NOUN iajs-2618 124 6 } } PUNCT iajs-2618 124 7 𝑛=1 𝑛=1 NOUN iajs-2618 124 8 ∞ ∞ NOUN iajs-2618 124 9 be be VERB iajs-2618 124 10 a a DET iajs-2618 124 11 sequence sequence NOUN iajs-2618 124 12 of of ADP iajs-2618 124 13 subspaces subspace NOUN iajs-2618 124 14 of of ADP iajs-2618 124 15 𝑉 𝑉 PROPN iajs-2618 124 16 , , PUNCT iajs-2618 124 17 where where SCONJ iajs-2618 124 18 𝑉𝑛 𝑉𝑛 PRON iajs-2618 124 19 is be AUX iajs-2618 124 20 as as ADV iajs-2618 124 21 defined define VERB iajs-2618 124 22 above above ADV iajs-2618 124 23 . . PUNCT iajs-2618 125 1 then then ADV iajs-2618 125 2 by by ADP iajs-2618 125 3 using use VERB iajs-2618 125 4 the the DET iajs-2618 125 5 galerkin galerkin ADJ iajs-2618 125 6 approach approach NOUN iajs-2618 125 7 , , PUNCT iajs-2618 125 8 for for ADP iajs-2618 125 9 each each DET iajs-2618 125 10 πœ‚ πœ‚ PROPN iajs-2618 125 11 ∈ ∈ PROPN iajs-2618 125 12 𝑉 𝑉 PROPN iajs-2618 125 13 , , PUNCT iajs-2618 125 14 there there PRON iajs-2618 125 15 exists exist VERB iajs-2618 125 16 a a DET iajs-2618 125 17 sequence sequence NOUN iajs-2618 125 18 { { PUNCT iajs-2618 125 19 πœ‚π‘› πœ‚π‘› NOUN iajs-2618 125 20 } } PUNCT iajs-2618 125 21 , , PUNCT iajs-2618 125 22 with with ADP iajs-2618 125 23 πœ‚π‘› πœ‚π‘› NOUN iajs-2618 125 24 ∈ ∈ NOUN iajs-2618 125 25 𝑉𝑛 𝑉𝑛 NOUN iajs-2618 125 26 for for ADP iajs-2618 125 27 each each DET iajs-2618 125 28 𝑛 𝑛 NOUN iajs-2618 125 29 , , PUNCT iajs-2618 125 30 such such ADJ iajs-2618 125 31 that that DET iajs-2618 125 32 πœ‚π‘› πœ‚π‘› NOUN iajs-2618 125 33 ⟢ ⟢ PROPN iajs-2618 125 34 πœ‚ πœ‚ PROPN iajs-2618 125 35 st st NOUN iajs-2618 125 36 in in ADP iajs-2618 125 37 𝐿2(πœ‘ 𝐿2(πœ‘ PROPN iajs-2618 125 38 ) ) PUNCT iajs-2618 125 39 . . PUNCT iajs-2618 126 1 consider consider VERB iajs-2618 126 2 that that PRON iajs-2618 126 3 πœ‰(𝑑 πœ‰(𝑑 NOUN iajs-2618 126 4 ) ) PUNCT iajs-2618 126 5 ∈ ∈ PROPN iajs-2618 126 6 𝐢2[0 𝐢2[0 PROPN iajs-2618 126 7 , , PUNCT iajs-2618 126 8 𝑇 𝑇 PROPN iajs-2618 126 9 ] ] PUNCT iajs-2618 126 10 , , PUNCT iajs-2618 126 11 for for ADP iajs-2618 126 12 which which PRON iajs-2618 126 13 πœ‰(𝑇 πœ‰(𝑇 NOUN iajs-2618 126 14 ) ) PUNCT iajs-2618 126 15 = = SYM iajs-2618 126 16 πœ‰β€²(𝑇 πœ‰β€²(𝑇 PROPN iajs-2618 126 17 ) ) PUNCT iajs-2618 126 18 = = SYM iajs-2618 126 19 0 0 NUM iajs-2618 126 20 and and CCONJ iajs-2618 126 21 πœ‰(0 πœ‰(0 PROPN iajs-2618 126 22 ) ) PUNCT iajs-2618 126 23 = = SYM iajs-2618 126 24 πœ‰β€²(0 πœ‰β€²(0 NOUN iajs-2618 126 25 ) ) PUNCT iajs-2618 126 26 β‰  β‰  PROPN iajs-2618 126 27 0 0 NUM iajs-2618 126 28 , , PUNCT iajs-2618 126 29 let let VERB iajs-2618 126 30 πœ‰π‘›(𝑑 πœ‰π‘›(𝑑 NOUN iajs-2618 126 31 ) ) PUNCT iajs-2618 126 32 continuous continuous ADJ iajs-2618 126 33 piecewise(cp piecewise(cp NOUN iajs-2618 126 34 ) ) PUNCT iajs-2618 126 35 interpolation interpolation NOUN iajs-2618 126 36 of of ADP iajs-2618 126 37 πœ‰(𝑑 πœ‰(𝑑 PROPN iajs-2618 126 38 ) ) PUNCT iajs-2618 126 39 with with ADP iajs-2618 126 40 respect respect NOUN iajs-2618 126 41 to to ADP iajs-2618 126 42 𝐼𝑗 𝐼𝑗 PROPN iajs-2618 126 43 𝑛 𝑛 NOUN iajs-2618 126 44 , , PUNCT iajs-2618 126 45 and and CCONJ iajs-2618 126 46 let let VERB iajs-2618 126 47 𝜁 𝜁 PROPN iajs-2618 126 48 = = SYM iajs-2618 126 49 πœ‚ πœ‚ PROPN iajs-2618 126 50 πœ‰(𝑑 πœ‰(𝑑 NOUN iajs-2618 126 51 ) ) PUNCT iajs-2618 126 52 , , PUNCT iajs-2618 126 53 with with ADP iajs-2618 126 54 πœπ‘› πœπ‘› ADP iajs-2618 126 55 = = PUNCT iajs-2618 126 56 πœ‚π‘› πœ‚π‘› NOUN iajs-2618 126 57 πœ‰ πœ‰ X iajs-2618 126 58 𝑛(𝑑 𝑛(𝑑 PROPN iajs-2618 126 59 ) ) PUNCT iajs-2618 126 60 , , PUNCT iajs-2618 126 61 with with ADP iajs-2618 126 62 πœβˆ’ πœβˆ’ PROPN iajs-2618 126 63 𝑛 𝑛 PRON iajs-2618 126 64 ∢= ∢= NUM iajs-2618 126 65 πœ‚π‘› πœ‚π‘› NOUN iajs-2618 126 66 πœ‰βˆ’ πœ‰βˆ’ VERB iajs-2618 126 67 𝑛(𝑑 𝑛(𝑑 PROPN iajs-2618 126 68 ) ) PUNCT iajs-2618 126 69 , , PUNCT iajs-2618 126 70 𝑑 𝑑 PROPN iajs-2618 126 71 ∈ ∈ PROPN iajs-2618 126 72 𝐼𝑗 𝐼𝑗 PROPN iajs-2618 126 73 𝑛 𝑛 NOUN iajs-2618 126 74 , , PUNCT iajs-2618 126 75 𝑗 𝑗 NOUN iajs-2618 126 76 = = SYM iajs-2618 126 77 0,1 0,1 NUM iajs-2618 126 78 , , PUNCT iajs-2618 126 79 … … PUNCT iajs-2618 126 80 . . PUNCT iajs-2618 127 1 , , PUNCT iajs-2618 127 2 π‘Œ π‘Œ PROPN iajs-2618 127 3 βˆ’ βˆ’ PROPN iajs-2618 127 4 1 1 NUM iajs-2618 127 5 , , PUNCT iajs-2618 127 6 πœ‚π‘› πœ‚π‘› NOUN iajs-2618 127 7 ∈ ∈ PROPN iajs-2618 127 8 𝑉𝑛 𝑉𝑛 PROPN iajs-2618 127 9 , , PUNCT iajs-2618 127 10 𝜁+ 𝜁+ VERB iajs-2618 127 11 𝑛 𝑛 PRON iajs-2618 127 12 ∢= ∢= NUM iajs-2618 127 13 πœ‚π‘› πœ‚π‘› NOUN iajs-2618 127 14 πœ‰+ πœ‰+ X iajs-2618 127 15 𝑛(𝑑 𝑛(𝑑 PROPN iajs-2618 127 16 ) ) PUNCT iajs-2618 127 17 , , PUNCT iajs-2618 127 18 𝑑 𝑑 PROPN iajs-2618 127 19 ∈ ∈ PROPN iajs-2618 127 20 𝐼𝑗 𝐼𝑗 PROPN iajs-2618 127 21 𝑛 𝑛 NOUN iajs-2618 127 22 , , PUNCT iajs-2618 127 23 𝑗 𝑗 NOUN iajs-2618 127 24 = = SYM iajs-2618 127 25 0,1 0,1 NUM iajs-2618 127 26 , , PUNCT iajs-2618 127 27 … … PUNCT iajs-2618 127 28 . . PUNCT iajs-2618 128 1 , , PUNCT iajs-2618 128 2 π‘Œ π‘Œ PROPN iajs-2618 128 3 βˆ’ βˆ’ PROPN iajs-2618 128 4 1 1 NUM iajs-2618 128 5 , , PUNCT iajs-2618 129 1 πœ‚π‘› πœ‚π‘› NOUN iajs-2618 129 2 ∈ ∈ PROPN iajs-2618 130 1 𝑉𝑛 𝑉𝑛 PROPN iajs-2618 130 2 , , PUNCT iajs-2618 130 3 𝜁^ 𝜁^ ADV iajs-2618 130 4 𝑛 𝑛 DET iajs-2618 130 5 ∢= ∢= NUM iajs-2618 130 6 πœ‚π‘› πœ‚π‘› NOUN iajs-2618 130 7 πœ‰ πœ‰ ADP iajs-2618 130 8 𝑛(𝑑 𝑛(𝑑 PROPN iajs-2618 130 9 ) ) PUNCT iajs-2618 130 10 , , PUNCT iajs-2618 130 11 𝑑 𝑑 PROPN iajs-2618 130 12 ∈ ∈ PROPN iajs-2618 130 13 𝐼 𝐼 PROPN iajs-2618 130 14 , , PUNCT iajs-2618 130 15 πœ‚π‘› πœ‚π‘› NOUN iajs-2618 130 16 ∈ ∈ PROPN iajs-2618 130 17 𝑉𝑛 𝑉𝑛 PROPN iajs-2618 130 18 , , PUNCT iajs-2618 130 19 setting set VERB iajs-2618 130 20 πœ‚ πœ‚ NOUN iajs-2618 130 21 = = SYM iajs-2618 130 22 πœπ‘—+1 πœπ‘—+1 X iajs-2618 130 23 𝑛 𝑛 PROPN iajs-2618 130 24 in in ADP iajs-2618 130 25 equation equation NOUN iajs-2618 130 26 ( ( PUNCT iajs-2618 130 27 8) 8) NUM iajs-2618 130 28 , , PUNCT iajs-2618 130 29 and and CCONJ iajs-2618 130 30 summing sum VERB iajs-2618 130 31 both both DET iajs-2618 130 32 sides side NOUN iajs-2618 130 33 of of ADP iajs-2618 130 34 the the DET iajs-2618 130 35 obtained obtain VERB iajs-2618 130 36 equation equation NOUN iajs-2618 130 37 for for ADP iajs-2618 130 38 𝑗 𝑗 NOUN iajs-2618 130 39 = = SYM iajs-2618 130 40 0 0 NUM iajs-2618 130 41 , , PUNCT iajs-2618 130 42 to to ADP iajs-2618 130 43 𝑗 𝑗 NOUN iajs-2618 130 44 = = SYM iajs-2618 130 45 π‘Œ π‘Œ PROPN iajs-2618 130 46 βˆ’ βˆ’ PROPN iajs-2618 130 47 1 1 NUM iajs-2618 130 48 , , PUNCT iajs-2618 130 49 then then ADV iajs-2618 130 50 using use VERB iajs-2618 130 51 discrete discrete ADJ iajs-2618 130 52 integrating integrating NOUN iajs-2618 130 53 by by ADP iajs-2618 130 54 parts part NOUN iajs-2618 130 55 ( ( PUNCT iajs-2618 130 56 dibp dibp NOUN iajs-2618 130 57 ) ) PUNCT iajs-2618 130 58 for for ADP iajs-2618 130 59 the the DET iajs-2618 130 60 1st 1st ADJ iajs-2618 130 61 term term NOUN iajs-2618 130 62 in in ADP iajs-2618 130 63 the the DET iajs-2618 130 64 l.h.s l.h.s NOUN iajs-2618 130 65 . . PUNCT iajs-2618 130 66 , , PUNCT iajs-2618 130 67 once once ADV iajs-2618 130 68 can can AUX iajs-2618 130 69 get get VERB iajs-2618 130 70 that that DET iajs-2618 130 71 125 125 NUM iajs-2618 130 72 ibn ibn PROPN iajs-2618 130 73 al al PROPN iajs-2618 130 74 - - PUNCT iajs-2618 130 75 haitham haitham PROPN iajs-2618 130 76 jour jour X iajs-2618 130 77 . . PROPN iajs-2618 131 1 for for ADP iajs-2618 131 2 pure pure ADJ iajs-2618 131 3 & & CCONJ iajs-2618 131 4 appl appl PROPN iajs-2618 131 5 . . PUNCT iajs-2618 132 1 sci sci PROPN iajs-2618 132 2 . . PROPN iajs-2618 133 1 34 34 NUM iajs-2618 133 2 ( ( PUNCT iajs-2618 133 3 2 2 NUM iajs-2618 133 4 ) ) PUNCT iajs-2618 133 5 2021 2021 NUM iajs-2618 133 6 βˆ’ βˆ’ PROPN iajs-2618 134 1 ∫ ∫ PROPN iajs-2618 134 2 ( ( PUNCT iajs-2618 134 3 π‘βˆ’ π‘βˆ’ PROPN iajs-2618 134 4 𝑛 𝑛 VERB iajs-2618 134 5 , , PUNCT iajs-2618 134 6 ( ( PUNCT iajs-2618 134 7 𝜁^ 𝜁^ X iajs-2618 134 8 𝑛 𝑛 DET iajs-2618 134 9 𝑇 𝑇 PROPN iajs-2618 134 10 0 0 NUM iajs-2618 134 11 ) ) PUNCT iajs-2618 134 12 β€² β€² NUM iajs-2618 134 13 ) ) PUNCT iajs-2618 134 14 𝑑𝑑 𝑑𝑑 ADP iajs-2618 134 15 + + NUM iajs-2618 134 16 ∫ ∫ PROPN iajs-2618 135 1 [ [ X iajs-2618 135 2 ( ( PUNCT iajs-2618 135 3 𝑇 𝑇 PROPN iajs-2618 135 4 0 0 NUM iajs-2618 135 5 βˆ‡ βˆ‡ X iajs-2618 135 6 𝑀+ 𝑀+ X iajs-2618 135 7 𝑛 𝑛 X iajs-2618 135 8 , , PUNCT iajs-2618 135 9 βˆ‡ βˆ‡ X iajs-2618 135 10 𝜁+ 𝜁+ VERB iajs-2618 135 11 𝑛 𝑛 PUNCT iajs-2618 135 12 ) ) PUNCT iajs-2618 136 1 + + CCONJ iajs-2618 136 2 ( ( PUNCT iajs-2618 136 3 𝑀+ 𝑀+ X iajs-2618 136 4 𝑛 𝑛 PRON iajs-2618 136 5 , , PUNCT iajs-2618 136 6 𝜁+ 𝜁+ VERB iajs-2618 136 7 𝑛 𝑛 PROPN iajs-2618 136 8 ) ) PUNCT iajs-2618 136 9 ] ] PUNCT iajs-2618 136 10 𝑑𝑑 𝑑𝑑 ADP iajs-2618 136 11 = = SYM iajs-2618 136 12 ∫ ∫ PROPN iajs-2618 136 13 ( ( PUNCT iajs-2618 136 14 𝑇 𝑇 PROPN iajs-2618 136 15 0 0 NUM iajs-2618 136 16 β„Ž(π‘‘βˆ’ β„Ž(π‘‘βˆ’ NOUN iajs-2618 136 17 𝑛 𝑛 PRON iajs-2618 136 18 , , PUNCT iajs-2618 136 19 𝑀+ 𝑀+ NOUN iajs-2618 136 20 𝑛 𝑛 PRON iajs-2618 136 21 ) ) PUNCT iajs-2618 136 22 , , PUNCT iajs-2618 136 23 𝜁+ 𝜁+ VERB iajs-2618 136 24 𝑛 𝑛 X iajs-2618 136 25 ) ) PUNCT iajs-2618 136 26 𝑑𝑑 𝑑𝑑 ADP iajs-2618 136 27 + + X iajs-2618 136 28 ( ( PUNCT iajs-2618 136 29 𝑝0 𝑝0 NOUN iajs-2618 136 30 𝑛 𝑛 PROPN iajs-2618 136 31 , , PUNCT iajs-2618 136 32 πœ‚π‘› πœ‚π‘› NOUN iajs-2618 136 33 ) ) PUNCT iajs-2618 136 34 πœ‰(0 πœ‰(0 NOUN iajs-2618 136 35 ) ) PUNCT iajs-2618 136 36 ( ( PUNCT iajs-2618 136 37 30 30 NUM iajs-2618 136 38 ) ) PUNCT iajs-2618 136 39 on on ADP iajs-2618 136 40 the the DET iajs-2618 136 41 other other ADJ iajs-2618 136 42 hand hand NOUN iajs-2618 136 43 , , PUNCT iajs-2618 136 44 from from ADP iajs-2618 136 45 ( ( PUNCT iajs-2618 136 46 9 9 NUM iajs-2618 136 47 ) ) PUNCT iajs-2618 136 48 , , PUNCT iajs-2618 136 49 once once ADV iajs-2618 136 50 has have VERB iajs-2618 136 51 ( ( PUNCT iajs-2618 136 52 ( ( PUNCT iajs-2618 136 53 𝑀^ 𝑀^ NOUN iajs-2618 136 54 𝑛 𝑛 X iajs-2618 136 55 ) ) PUNCT iajs-2618 136 56 β€² β€² NOUN iajs-2618 136 57 , , PUNCT iajs-2618 136 58 πœ‚π‘› πœ‚π‘› NOUN iajs-2618 136 59 ) ) PUNCT iajs-2618 136 60 ( ( PUNCT iajs-2618 137 1 πœ‰π‘›)β€² πœ‰π‘›)β€² PROPN iajs-2618 137 2 = = SYM iajs-2618 137 3 ( ( PUNCT iajs-2618 137 4 𝑝+ 𝑝+ NOUN iajs-2618 137 5 𝑛 𝑛 PROPN iajs-2618 137 6 , , PUNCT iajs-2618 137 7 πœ‚π‘› πœ‚π‘› NOUN iajs-2618 137 8 ) ) PUNCT iajs-2618 137 9 ( ( PUNCT iajs-2618 137 10 πœ‰π‘›)β€² πœ‰π‘›)β€² NUM iajs-2618 137 11 integrating integrate VERB iajs-2618 137 12 both both DET iajs-2618 137 13 sides side NOUN iajs-2618 137 14 on on ADP iajs-2618 137 15 [ [ X iajs-2618 137 16 0 0 NUM iajs-2618 137 17 , , PUNCT iajs-2618 137 18 𝑇 𝑇 PROPN iajs-2618 137 19 ] ] PUNCT iajs-2618 137 20 , , PUNCT iajs-2618 137 21 then then ADV iajs-2618 137 22 using use VERB iajs-2618 137 23 debp debp NOUN iajs-2618 137 24 for for ADP iajs-2618 137 25 the the DET iajs-2618 137 26 1st 1st ADJ iajs-2618 137 27 term term NOUN iajs-2618 137 28 in in ADP iajs-2618 137 29 the the DET iajs-2618 137 30 l.h.s l.h.s NOUN iajs-2618 137 31 . . PUNCT iajs-2618 137 32 , , PUNCT iajs-2618 137 33 to to PART iajs-2618 137 34 obtain obtain VERB iajs-2618 137 35 βˆ’ βˆ’ PROPN iajs-2618 137 36 ∫ ∫ PROPN iajs-2618 137 37 ( ( PUNCT iajs-2618 137 38 𝑀+ 𝑀+ X iajs-2618 137 39 𝑛 𝑛 PROPN iajs-2618 137 40 𝑇 𝑇 PROPN iajs-2618 137 41 0 0 NUM iajs-2618 137 42 , , PUNCT iajs-2618 137 43 πœ‚π‘› πœ‚π‘› NOUN iajs-2618 137 44 ) ) PUNCT iajs-2618 137 45 ( ( PUNCT iajs-2618 137 46 πœ‰π‘›(𝑑))′′𝑑𝑑 πœ‰π‘›(𝑑))′′𝑑𝑑 PROPN iajs-2618 137 47 = = SYM iajs-2618 137 48 ∫ ∫ PROPN iajs-2618 137 49 ( ( PUNCT iajs-2618 137 50 𝑝+ 𝑝+ NOUN iajs-2618 137 51 𝑛 𝑛 PROPN iajs-2618 137 52 𝑇 𝑇 PROPN iajs-2618 137 53 0 0 NUM iajs-2618 137 54 , , PUNCT iajs-2618 137 55 πœ‚π‘› πœ‚π‘› NOUN iajs-2618 137 56 ) ) PUNCT iajs-2618 137 57 ( ( PUNCT iajs-2618 137 58 πœ‰π‘›)′𝑑𝑑 πœ‰π‘›)′𝑑𝑑 PROPN iajs-2618 137 59 + + CCONJ iajs-2618 137 60 ( ( PUNCT iajs-2618 137 61 𝑀0 𝑀0 NOUN iajs-2618 137 62 𝑛 𝑛 PROPN iajs-2618 137 63 , , PUNCT iajs-2618 137 64 πœ‚π‘› πœ‚π‘› NOUN iajs-2618 137 65 ) ) PUNCT iajs-2618 137 66 ( ( PUNCT iajs-2618 137 67 πœ‰π‘›(0))β€² πœ‰π‘›(0))β€² X iajs-2618 137 68 ( ( PUNCT iajs-2618 137 69 31 31 NUM iajs-2618 137 70 ) ) PUNCT iajs-2618 137 71 now now ADV iajs-2618 137 72 , , PUNCT iajs-2618 137 73 since since SCONJ iajs-2618 137 74 πœ‰π‘›(𝑑 πœ‰π‘›(𝑑 NOUN iajs-2618 137 75 ) ) PUNCT iajs-2618 137 76 ⟢ ⟢ NOUN iajs-2618 137 77 πœ‰(𝑑 πœ‰(𝑑 PROPN iajs-2618 137 78 ) ) PUNCT iajs-2618 137 79 in in ADP iajs-2618 137 80 𝐢(𝐼 𝐢(𝐼 NOUN iajs-2618 137 81 ) ) PUNCT iajs-2618 137 82 βŠ‚ βŠ‚ PROPN iajs-2618 137 83 𝐿2(𝐼 𝐿2(𝐼 PROPN iajs-2618 137 84 ) ) PUNCT iajs-2618 137 85 , , PUNCT iajs-2618 137 86 πœ‚π‘› πœ‚π‘› NOUN iajs-2618 137 87 ⟢ ⟢ PROPN iajs-2618 137 88 πœ‚ πœ‚ PROPN iajs-2618 137 89 st st PROPN iajs-2618 137 90 in in ADP iajs-2618 137 91 𝐿2(𝐼 𝐿2(𝐼 PROPN iajs-2618 137 92 , , PUNCT iajs-2618 137 93 𝑉 𝑉 PROPN iajs-2618 137 94 ) ) PUNCT iajs-2618 137 95 and and CCONJ iajs-2618 137 96 in in ADP iajs-2618 137 97 𝐿2(πœ“ 𝐿2(πœ“ NOUN iajs-2618 137 98 ) ) PUNCT iajs-2618 137 99 , , PUNCT iajs-2618 137 100 then then ADV iajs-2618 137 101 , , PUNCT iajs-2618 137 102 we we PRON iajs-2618 137 103 have have AUX iajs-2618 137 104 𝜁+ 𝜁+ VERB iajs-2618 137 105 𝑛 𝑛 PROPN iajs-2618 137 106 = = PUNCT iajs-2618 137 107 πœ‚π‘› πœ‚π‘› NOUN iajs-2618 137 108 πœ‰+ πœ‰+ PUNCT iajs-2618 137 109 𝑛 𝑛 DET iajs-2618 137 110 ⟢ ⟢ NOUN iajs-2618 137 111 πœ‚ πœ‚ NOUN iajs-2618 137 112 πœ‰ πœ‰ X iajs-2618 137 113 = = SYM iajs-2618 137 114 𝜁 𝜁 PROPN iajs-2618 137 115 st st PROPN iajs-2618 137 116 in in ADP iajs-2618 137 117 𝐿2(𝐼 𝐿2(𝐼 PROPN iajs-2618 137 118 , , PUNCT iajs-2618 137 119 𝑉 𝑉 PROPN iajs-2618 137 120 ) ) PUNCT iajs-2618 137 121 and and CCONJ iajs-2618 137 122 in in ADP iajs-2618 137 123 𝐿2(πœ‘ 𝐿2(πœ‘ PROPN iajs-2618 137 124 ) ) PUNCT iajs-2618 137 125 , , PUNCT iajs-2618 137 126 πœ‚π‘› πœ‚π‘› NOUN iajs-2618 137 127 πœ‰ πœ‰ ADP iajs-2618 137 128 𝑛(0 𝑛(0 PROPN iajs-2618 137 129 ) ) PUNCT iajs-2618 137 130 ⟢ ⟢ NOUN iajs-2618 137 131 πœ‚ πœ‚ NOUN iajs-2618 137 132 πœ‰(0 πœ‰(0 PROPN iajs-2618 137 133 ) ) PUNCT iajs-2618 137 134 st st NOUN iajs-2618 137 135 in in ADP iajs-2618 137 136 𝐿2(πœ‘ 𝐿2(πœ‘ PROPN iajs-2618 137 137 ) ) PUNCT iajs-2618 137 138 , , PUNCT iajs-2618 137 139 ( ( PUNCT iajs-2618 137 140 𝜁^ 𝜁^ X iajs-2618 137 141 𝑛 𝑛 NOUN iajs-2618 137 142 ) ) PUNCT iajs-2618 137 143 β€² β€² NUM iajs-2618 138 1 = = PUNCT iajs-2618 138 2 πœ‚π‘› πœ‚π‘› NOUN iajs-2618 138 3 πœ‰ πœ‰ X iajs-2618 138 4 ′𝑛 ′𝑛 PROPN iajs-2618 138 5 ⟢ ⟢ NOUN iajs-2618 138 6 πœ‚ πœ‚ NOUN iajs-2618 138 7 πœ‰β€² πœ‰β€² X iajs-2618 138 8 = = PUNCT iajs-2618 138 9 πœ‚πœβ€² πœ‚πœβ€² X iajs-2618 138 10 st st PROPN iajs-2618 138 11 in in ADP iajs-2618 138 12 𝐿2(𝐼 𝐿2(𝐼 PROPN iajs-2618 138 13 , , PUNCT iajs-2618 138 14 𝑉 𝑉 PROPN iajs-2618 138 15 ) ) PUNCT iajs-2618 138 16 . . PUNCT iajs-2618 139 1 and and CCONJ iajs-2618 139 2 since since SCONJ iajs-2618 139 3 , , PUNCT iajs-2618 139 4 π‘‘βˆ’ π‘‘βˆ’ ADJ iajs-2618 139 5 𝑛 𝑛 PROPN iajs-2618 139 6 ⟢ ⟢ NOUN iajs-2618 139 7 𝑑 𝑑 PROPN iajs-2618 139 8 st st PROPN iajs-2618 139 9 in in ADP iajs-2618 139 10 𝐿∞(𝐼 𝐿∞(𝐼 PROPN iajs-2618 139 11 ) ) PUNCT iajs-2618 139 12 , , PUNCT iajs-2618 139 13 𝑀+ 𝑀+ VERB iajs-2618 139 14 𝑛 𝑛 PROPN iajs-2618 139 15 , , PUNCT iajs-2618 139 16 π‘€βˆ’ π‘€βˆ’ PROPN iajs-2618 139 17 𝑛 𝑛 PROPN iajs-2618 139 18 , , PUNCT iajs-2618 139 19 𝑀^ 𝑀^ PUNCT iajs-2618 139 20 𝑛 𝑛 DET iajs-2618 139 21 ⟢ ⟢ NOUN iajs-2618 139 22 𝑀 𝑀 ADP iajs-2618 139 23 st st PROPN iajs-2618 139 24 in in ADP iajs-2618 139 25 𝐿2(πœ‘ 𝐿2(πœ‘ PROPN iajs-2618 139 26 ) ) PUNCT iajs-2618 139 27 , , PUNCT iajs-2618 139 28 𝑀0 𝑀0 NOUN iajs-2618 139 29 𝑛 𝑛 PROPN iajs-2618 139 30 ⟢ ⟢ PROPN iajs-2618 139 31 𝑀0 𝑀0 PROPN iajs-2618 139 32 st st PROPN iajs-2618 139 33 in in ADP iajs-2618 139 34 𝑉 𝑉 PROPN iajs-2618 139 35 and and CCONJ iajs-2618 139 36 𝑝0 𝑝0 PROPN iajs-2618 140 1 𝑛 𝑛 PROPN iajs-2618 140 2 ⟢ ⟢ PROPN iajs-2618 140 3 𝑀1 𝑀1 PROPN iajs-2618 140 4 st st PROPN iajs-2618 140 5 in in ADP iajs-2618 140 6 𝐿2(πœ“ 𝐿2(πœ“ NOUN iajs-2618 140 7 ) ) PUNCT iajs-2618 140 8 . . PUNCT iajs-2618 141 1 now now ADV iajs-2618 141 2 , , PUNCT iajs-2618 141 3 from from ADP iajs-2618 141 4 assumptions assumption NOUN iajs-2618 141 5 β„Ž β„Ž PROPN iajs-2618 141 6 , , PUNCT iajs-2618 141 7 and and CCONJ iajs-2618 141 8 the the DET iajs-2618 141 9 above above ADJ iajs-2618 141 10 convergences convergence NOUN iajs-2618 141 11 , , PUNCT iajs-2618 141 12 one one PRON iajs-2618 141 13 can can AUX iajs-2618 141 14 passage passage VERB iajs-2618 141 15 to to ADP iajs-2618 141 16 the the DET iajs-2618 141 17 limit limit NOUN iajs-2618 141 18 in in ADP iajs-2618 141 19 ( ( PUNCT iajs-2618 141 20 30 30 NUM iajs-2618 141 21 ) ) PUNCT iajs-2618 141 22 and and CCONJ iajs-2618 141 23 in in ADP iajs-2618 141 24 ( ( PUNCT iajs-2618 141 25 31 31 NUM iajs-2618 141 26 ) ) PUNCT iajs-2618 141 27 , , PUNCT iajs-2618 141 28 to to PART iajs-2618 141 29 obtain obtain VERB iajs-2618 141 30 βˆ’ βˆ’ PROPN iajs-2618 141 31 ∫ ∫ PROPN iajs-2618 141 32 ( ( PUNCT iajs-2618 141 33 𝑝 𝑝 PROPN iajs-2618 141 34 𝑇 𝑇 PROPN iajs-2618 141 35 0 0 NUM iajs-2618 141 36 , , PUNCT iajs-2618 141 37 πœ‚ πœ‚ NOUN iajs-2618 141 38 ) ) PUNCT iajs-2618 141 39 πœ‰β€²π‘‘π‘‘ πœ‰β€²π‘‘π‘‘ VERB iajs-2618 142 1 + + NUM iajs-2618 142 2 ∫ ∫ PROPN iajs-2618 142 3 [ [ X iajs-2618 142 4 ( ( PUNCT iajs-2618 142 5 βˆ‡π‘€ βˆ‡π‘€ PROPN iajs-2618 142 6 𝑇 𝑇 PROPN iajs-2618 142 7 0 0 NUM iajs-2618 142 8 , , PUNCT iajs-2618 142 9 βˆ‡πœ‚ βˆ‡πœ‚ PROPN iajs-2618 142 10 ) ) PUNCT iajs-2618 143 1 + + CCONJ iajs-2618 143 2 ( ( PUNCT iajs-2618 143 3 𝑀 𝑀 INTJ iajs-2618 143 4 , , PUNCT iajs-2618 143 5 πœ‚ πœ‚ NOUN iajs-2618 143 6 ) ) PUNCT iajs-2618 143 7 ] ] PUNCT iajs-2618 143 8 πœ‰ πœ‰ X iajs-2618 143 9 𝑑𝑑 𝑑𝑑 ADP iajs-2618 143 10 = = PUNCT iajs-2618 143 11 ∫ ∫ PROPN iajs-2618 143 12 ( ( PUNCT iajs-2618 143 13 𝑇 𝑇 PROPN iajs-2618 143 14 0 0 NUM iajs-2618 143 15 β„Ž(𝑑 β„Ž(𝑑 PROPN iajs-2618 143 16 , , PUNCT iajs-2618 143 17 𝑀 𝑀 ADP iajs-2618 143 18 ) ) PUNCT iajs-2618 143 19 , , PUNCT iajs-2618 143 20 πœ‚ πœ‚ X iajs-2618 143 21 ) ) PUNCT iajs-2618 143 22 πœ‰ πœ‰ NOUN iajs-2618 143 23 𝑑𝑑 𝑑𝑑 ADP iajs-2618 143 24 + + X iajs-2618 143 25 ( ( PUNCT iajs-2618 143 26 𝑀1 𝑀1 VERB iajs-2618 143 27 , , PUNCT iajs-2618 143 28 πœ‚ πœ‚ NOUN iajs-2618 143 29 ) ) PUNCT iajs-2618 143 30 πœ‰(0 πœ‰(0 NOUN iajs-2618 143 31 ) ) PUNCT iajs-2618 143 32 ( ( PUNCT iajs-2618 143 33 32 32 NUM iajs-2618 143 34 ) ) PUNCT iajs-2618 143 35 and and CCONJ iajs-2618 143 36 βˆ’ βˆ’ PROPN iajs-2618 143 37 ∫ ∫ PROPN iajs-2618 143 38 ( ( PUNCT iajs-2618 143 39 𝑀 𝑀 ADP iajs-2618 143 40 𝑇 𝑇 PROPN iajs-2618 143 41 0 0 NUM iajs-2618 143 42 , , PUNCT iajs-2618 143 43 πœ‚)πœ‰β€²β€²(𝑑 πœ‚)πœ‰β€²β€²(𝑑 ADP iajs-2618 143 44 ) ) PUNCT iajs-2618 143 45 𝑑𝑑 𝑑𝑑 ADP iajs-2618 143 46 = = SYM iajs-2618 143 47 ∫ ∫ PROPN iajs-2618 143 48 ( ( PUNCT iajs-2618 143 49 𝑝 𝑝 PROPN iajs-2618 143 50 𝑇 𝑇 PROPN iajs-2618 143 51 0 0 NUM iajs-2618 143 52 , , PUNCT iajs-2618 143 53 πœ‚)πœ‰β€²(𝑑)𝑑𝑑 πœ‚)πœ‰β€²(𝑑)𝑑𝑑 PROPN iajs-2618 143 54 + + CCONJ iajs-2618 143 55 ( ( PUNCT iajs-2618 143 56 𝑀 𝑀 PART iajs-2618 143 57 0 0 NUM iajs-2618 143 58 , , PUNCT iajs-2618 143 59 πœ‚)πœ‰β€²(0 πœ‚)πœ‰β€²(0 NOUN iajs-2618 143 60 ) ) PUNCT iajs-2618 143 61 ( ( PUNCT iajs-2618 143 62 33 33 NUM iajs-2618 143 63 ) ) PUNCT iajs-2618 143 64 the the DET iajs-2618 143 65 following follow VERB iajs-2618 143 66 cases case NOUN iajs-2618 143 67 appear appear VERB iajs-2618 143 68 : : PUNCT iajs-2618 143 69 case case NOUN iajs-2618 143 70 ( ( PUNCT iajs-2618 143 71 1 1 NUM iajs-2618 143 72 ) ) PUNCT iajs-2618 143 73 : : PUNCT iajs-2618 143 74 consider consider VERB iajs-2618 143 75 πœ‰(𝑑 πœ‰(𝑑 PRON iajs-2618 143 76 ) ) PUNCT iajs-2618 143 77 ∈ ∈ PROPN iajs-2618 144 1 𝐢2[0 𝐢2[0 PROPN iajs-2618 144 2 , , PUNCT iajs-2618 144 3 𝑇 𝑇 PROPN iajs-2618 144 4 ] ] PUNCT iajs-2618 144 5 , , PUNCT iajs-2618 144 6 such such ADJ iajs-2618 144 7 that that DET iajs-2618 144 8 πœ‰(𝑇 πœ‰(𝑇 NOUN iajs-2618 144 9 ) ) PUNCT iajs-2618 144 10 = = SYM iajs-2618 144 11 πœ‰β€²(𝑇 πœ‰β€²(𝑇 PROPN iajs-2618 144 12 ) ) PUNCT iajs-2618 144 13 = = SYM iajs-2618 144 14 πœ‰(0 πœ‰(0 PROPN iajs-2618 144 15 ) ) PUNCT iajs-2618 144 16 = = SYM iajs-2618 144 17 πœ‰β€²(0 πœ‰β€²(0 NOUN iajs-2618 144 18 ) ) PUNCT iajs-2618 144 19 = = SYM iajs-2618 144 20 0 0 NUM iajs-2618 144 21 , , PUNCT iajs-2618 144 22 by by ADP iajs-2618 144 23 setting set VERB iajs-2618 144 24 πœ‰β€²(0 πœ‰β€²(0 NOUN iajs-2618 144 25 ) ) PUNCT iajs-2618 144 26 = = SYM iajs-2618 144 27 0 0 NUM iajs-2618 144 28 in in ADP iajs-2618 144 29 equation equation NOUN iajs-2618 144 30 ( ( PUNCT iajs-2618 144 31 32 32 NUM iajs-2618 144 32 ) ) PUNCT iajs-2618 144 33 and and CCONJ iajs-2618 144 34 πœ‰(0 πœ‰(0 NOUN iajs-2618 144 35 ) ) PUNCT iajs-2618 144 36 = = SYM iajs-2618 144 37 0 0 NUM iajs-2618 145 1 in in ADP iajs-2618 145 2 ( ( PUNCT iajs-2618 145 3 33 33 NUM iajs-2618 145 4 ) ) PUNCT iajs-2618 145 5 , , PUNCT iajs-2618 145 6 then then ADV iajs-2618 145 7 we we PRON iajs-2618 145 8 use use VERB iajs-2618 145 9 ibp ibp NOUN iajs-2618 145 10 for for ADP iajs-2618 145 11 the the DET iajs-2618 145 12 1st 1st ADJ iajs-2618 145 13 term term NOUN iajs-2618 145 14 of of ADP iajs-2618 145 15 each each DET iajs-2618 145 16 one one NUM iajs-2618 145 17 of of ADP iajs-2618 145 18 the the DET iajs-2618 145 19 obtained obtain VERB iajs-2618 145 20 equation equation NOUN iajs-2618 145 21 , , PUNCT iajs-2618 145 22 one one PRON iajs-2618 145 23 gets get VERB iajs-2618 145 24 respectively respectively ADV iajs-2618 145 25 ∫ ∫ PROPN iajs-2618 145 26 ( ( PUNCT iajs-2618 145 27 𝑀𝑑 𝑀𝑑 VERB iajs-2618 145 28 𝑇 𝑇 PROPN iajs-2618 145 29 0 0 NUM iajs-2618 145 30 , , PUNCT iajs-2618 145 31 πœ‚)πœ‰β€²(𝑑 πœ‚)πœ‰β€²(𝑑 PROPN iajs-2618 145 32 ) ) PUNCT iajs-2618 145 33 𝑑𝑑 𝑑𝑑 ADP iajs-2618 145 34 = = SYM iajs-2618 145 35 ∫ ∫ PROPN iajs-2618 145 36 ( ( PUNCT iajs-2618 145 37 𝑝 𝑝 PROPN iajs-2618 145 38 𝑇 𝑇 PROPN iajs-2618 145 39 0 0 NUM iajs-2618 145 40 , , PUNCT iajs-2618 145 41 πœ‚)πœ‰β€²(𝑑)𝑑𝑑 πœ‚)πœ‰β€²(𝑑)𝑑𝑑 PROPN iajs-2618 145 42 ⟹ ⟹ PUNCT iajs-2618 145 43 𝑀𝑑 𝑀𝑑 VERB iajs-2618 145 44 = = SYM iajs-2618 145 45 𝑝 𝑝 PROPN iajs-2618 145 46 , , PUNCT iajs-2618 145 47 ∫ ∫ PROPN iajs-2618 145 48 ( ( PUNCT iajs-2618 145 49 𝑀𝑑𝑑 𝑀𝑑𝑑 PROPN iajs-2618 145 50 𝑇 𝑇 PROPN iajs-2618 145 51 0 0 NUM iajs-2618 145 52 , , PUNCT iajs-2618 145 53 πœ‚ πœ‚ X iajs-2618 145 54 ) ) PUNCT iajs-2618 145 55 πœ‰ πœ‰ NOUN iajs-2618 145 56 𝑑𝑑 𝑑𝑑 ADP iajs-2618 145 57 + + NUM iajs-2618 145 58 ∫ ∫ PROPN iajs-2618 146 1 [ [ X iajs-2618 146 2 ( ( PUNCT iajs-2618 146 3 βˆ‡π‘€ βˆ‡π‘€ PROPN iajs-2618 146 4 𝑇 𝑇 PROPN iajs-2618 146 5 0 0 NUM iajs-2618 146 6 , , PUNCT iajs-2618 146 7 βˆ‡πœ‚ βˆ‡πœ‚ PROPN iajs-2618 146 8 ) ) PUNCT iajs-2618 147 1 + + CCONJ iajs-2618 147 2 ( ( PUNCT iajs-2618 147 3 𝑀 𝑀 INTJ iajs-2618 147 4 , , PUNCT iajs-2618 147 5 𝑣 𝑣 NOUN iajs-2618 147 6 ) ) PUNCT iajs-2618 147 7 ] ] PUNCT iajs-2618 147 8 πœ‰ πœ‰ X iajs-2618 147 9 𝑑𝑑 𝑑𝑑 ADP iajs-2618 147 10 = = PUNCT iajs-2618 147 11 ∫ ∫ PROPN iajs-2618 147 12 ( ( PUNCT iajs-2618 147 13 𝑇 𝑇 PROPN iajs-2618 147 14 0 0 NUM iajs-2618 147 15 β„Ž(𝑑 β„Ž(𝑑 PROPN iajs-2618 147 16 , , PUNCT iajs-2618 147 17 𝑀 𝑀 ADP iajs-2618 147 18 ) ) PUNCT iajs-2618 147 19 , , PUNCT iajs-2618 147 20 πœ‚)πœ‰ πœ‚)πœ‰ X iajs-2618 147 21 𝑑𝑑 𝑑𝑑 ADP iajs-2618 147 22 , , PUNCT iajs-2618 147 23 ( ( PUNCT iajs-2618 147 24 34 34 NUM iajs-2618 147 25 ) ) PUNCT iajs-2618 147 26 thus thus ADV iajs-2618 147 27 ( ( PUNCT iajs-2618 147 28 𝑀𝑑𝑑 𝑀𝑑𝑑 NOUN iajs-2618 147 29 , , PUNCT iajs-2618 147 30 πœ‚ πœ‚ NOUN iajs-2618 147 31 ) ) PUNCT iajs-2618 147 32 + + CCONJ iajs-2618 147 33 ( ( PUNCT iajs-2618 147 34 βˆ‡w βˆ‡w ADJ iajs-2618 147 35 , , PUNCT iajs-2618 147 36 βˆ‡πœ‚ βˆ‡πœ‚ NOUN iajs-2618 147 37 ) ) PUNCT iajs-2618 147 38 + + CCONJ iajs-2618 147 39 ( ( PUNCT iajs-2618 147 40 𝑀 𝑀 INTJ iajs-2618 147 41 , , PUNCT iajs-2618 147 42 πœ‚ πœ‚ NOUN iajs-2618 147 43 ) ) PUNCT iajs-2618 147 44 = = SYM iajs-2618 147 45 ( ( PUNCT iajs-2618 147 46 β„Ž(𝑑 β„Ž(𝑑 PROPN iajs-2618 147 47 , , PUNCT iajs-2618 147 48 𝑀 𝑀 ADP iajs-2618 147 49 ) ) PUNCT iajs-2618 147 50 , , PUNCT iajs-2618 147 51 πœ‚ πœ‚ NOUN iajs-2618 147 52 ) ) PUNCT iajs-2618 147 53 , , PUNCT iajs-2618 147 54 πœ‚ πœ‚ X iajs-2618 147 55 ∈ ∈ PROPN iajs-2618 147 56 𝑉 𝑉 PROPN iajs-2618 147 57 a. a. NOUN iajs-2618 147 58 e. e. PROPN iajs-2618 147 59 on on ADP iajs-2618 147 60 𝐼. 𝐼. PROPN iajs-2618 147 61 case case NOUN iajs-2618 147 62 ( ( PUNCT iajs-2618 147 63 2 2 NUM iajs-2618 147 64 ) ) PUNCT iajs-2618 147 65 : : PUNCT iajs-2618 147 66 consider consider VERB iajs-2618 147 67 πœ‰(𝑑 πœ‰(𝑑 PRON iajs-2618 147 68 ) ) PUNCT iajs-2618 147 69 ∈ ∈ PROPN iajs-2618 147 70 𝐷[0 𝐷[0 PROPN iajs-2618 147 71 , , PUNCT iajs-2618 147 72 𝑇 𝑇 PROPN iajs-2618 147 73 ] ] PUNCT iajs-2618 147 74 , , PUNCT iajs-2618 147 75 πœ‰(0 πœ‰(0 PROPN iajs-2618 147 76 ) ) PUNCT iajs-2618 147 77 β‰  β‰  PROPN iajs-2618 147 78 0 0 NUM iajs-2618 147 79 , , PUNCT iajs-2618 147 80 πœ‰(𝑇 πœ‰(𝑇 NOUN iajs-2618 147 81 ) ) PUNCT iajs-2618 148 1 = = SYM iajs-2618 148 2 0 0 PUNCT iajs-2618 148 3 and and CCONJ iajs-2618 148 4 use use VERB iajs-2618 148 5 ibp ibp NOUN iajs-2618 148 6 the the DET iajs-2618 148 7 first first ADJ iajs-2618 148 8 term term NOUN iajs-2618 148 9 in in ADP iajs-2618 148 10 the the DET iajs-2618 148 11 l.h.s l.h.s NOUN iajs-2618 148 12 of of ADP iajs-2618 148 13 ( ( PUNCT iajs-2618 148 14 34 34 NUM iajs-2618 148 15 ) ) PUNCT iajs-2618 148 16 , , PUNCT iajs-2618 148 17 once once ADV iajs-2618 148 18 get get VERB iajs-2618 148 19 that that PRON iajs-2618 148 20 βˆ’ βˆ’ PROPN iajs-2618 148 21 ∫ ∫ PROPN iajs-2618 148 22 ( ( PUNCT iajs-2618 148 23 𝑀𝑑 𝑀𝑑 ADV iajs-2618 148 24 𝑇 𝑇 PROPN iajs-2618 148 25 0 0 NUM iajs-2618 148 26 , , PUNCT iajs-2618 148 27 πœ‚ πœ‚ NOUN iajs-2618 148 28 ) ) PUNCT iajs-2618 148 29 πœ‰β€²π‘‘π‘‘ πœ‰β€²π‘‘π‘‘ VERB iajs-2618 149 1 + + NUM iajs-2618 149 2 ∫ ∫ PROPN iajs-2618 149 3 [ [ X iajs-2618 149 4 ( ( PUNCT iajs-2618 149 5 βˆ‡π‘€ βˆ‡π‘€ PROPN iajs-2618 149 6 𝑇 𝑇 PROPN iajs-2618 149 7 0 0 NUM iajs-2618 149 8 , , PUNCT iajs-2618 149 9 βˆ‡πœ‚ βˆ‡πœ‚ PROPN iajs-2618 149 10 ) ) PUNCT iajs-2618 150 1 + + CCONJ iajs-2618 150 2 ( ( PUNCT iajs-2618 150 3 𝑀 𝑀 INTJ iajs-2618 150 4 , , PUNCT iajs-2618 150 5 πœ‚ πœ‚ NOUN iajs-2618 150 6 ) ) PUNCT iajs-2618 150 7 ] ] PUNCT iajs-2618 150 8 πœ‰ πœ‰ X iajs-2618 150 9 𝑑𝑑 𝑑𝑑 ADP iajs-2618 150 10 = = PUNCT iajs-2618 150 11 ∫ ∫ PROPN iajs-2618 150 12 ( ( PUNCT iajs-2618 150 13 𝑇 𝑇 PROPN iajs-2618 150 14 0 0 NUM iajs-2618 150 15 β„Ž(𝑑 β„Ž(𝑑 PROPN iajs-2618 150 16 , , PUNCT iajs-2618 150 17 𝑀 𝑀 ADP iajs-2618 150 18 ) ) PUNCT iajs-2618 150 19 , , PUNCT iajs-2618 150 20 πœ‚)πœ‰ πœ‚)πœ‰ PUNCT iajs-2618 150 21 𝑑𝑑 𝑑𝑑 ADP iajs-2618 150 22 + + ADJ iajs-2618 150 23 𝑀𝑑(0 𝑀𝑑(0 NOUN iajs-2618 150 24 ) ) PUNCT iajs-2618 150 25 , , PUNCT iajs-2618 150 26 πœ‚)πœ‰(0 πœ‚)πœ‰(0 PROPN iajs-2618 150 27 ) ) PUNCT iajs-2618 150 28 ( ( PUNCT iajs-2618 150 29 35 35 NUM iajs-2618 150 30 ) ) PUNCT iajs-2618 150 31 setting set VERB iajs-2618 150 32 𝑝 𝑝 NOUN iajs-2618 150 33 = = PUNCT iajs-2618 150 34 𝑀𝑑 𝑀𝑑 NOUN iajs-2618 150 35 in in ADV iajs-2618 150 36 ( ( PUNCT iajs-2618 150 37 32 32 NUM iajs-2618 150 38 ) ) PUNCT iajs-2618 150 39 , , PUNCT iajs-2618 150 40 subtracting subtract VERB iajs-2618 150 41 the the DET iajs-2618 150 42 resulting result VERB iajs-2618 150 43 equation equation NOUN iajs-2618 150 44 from from ADP iajs-2618 150 45 ( ( PUNCT iajs-2618 150 46 35 35 NUM iajs-2618 150 47 ) ) PUNCT iajs-2618 150 48 , , PUNCT iajs-2618 150 49 to to PART iajs-2618 150 50 get get VERB iajs-2618 150 51 ( ( PUNCT iajs-2618 150 52 𝑀𝑑(0 𝑀𝑑(0 NOUN iajs-2618 150 53 ) ) PUNCT iajs-2618 150 54 , , PUNCT iajs-2618 150 55 πœ‚)πœ‰(0 πœ‚)πœ‰(0 PROPN iajs-2618 150 56 ) ) PUNCT iajs-2618 150 57 = = PUNCT iajs-2618 150 58 ( ( PUNCT iajs-2618 150 59 𝑀1 𝑀1 VERB iajs-2618 150 60 , , PUNCT iajs-2618 150 61 πœ‚)πœ‰(0 πœ‚)πœ‰(0 PROPN iajs-2618 150 62 ) ) PUNCT iajs-2618 150 63 ⟹ ⟹ PUNCT iajs-2618 150 64 ( ( PUNCT iajs-2618 150 65 𝑀𝑑(0 𝑀𝑑(0 X iajs-2618 150 66 ) ) PUNCT iajs-2618 150 67 , , PUNCT iajs-2618 150 68 πœ‚ πœ‚ X iajs-2618 150 69 ) ) PUNCT iajs-2618 150 70 = = SYM iajs-2618 150 71 ( ( PUNCT iajs-2618 150 72 𝑀1 𝑀1 VERB iajs-2618 150 73 , , PUNCT iajs-2618 150 74 πœ‚ πœ‚ NOUN iajs-2618 150 75 ) ) PUNCT iajs-2618 150 76 , , PUNCT iajs-2618 150 77 for for ADP iajs-2618 150 78 each each DET iajs-2618 150 79 πœ‚ πœ‚ NOUN iajs-2618 150 80 then then ADV iajs-2618 150 81 𝑀𝑑(0 𝑀𝑑(0 PRON iajs-2618 150 82 ) ) PUNCT iajs-2618 150 83 = = PUNCT iajs-2618 150 84 𝑀1(0 𝑀1(0 PROPN iajs-2618 150 85 ) ) PUNCT iajs-2618 150 86 . . PUNCT iajs-2618 151 1 case case NOUN iajs-2618 151 2 ( ( PUNCT iajs-2618 151 3 3 3 NUM iajs-2618 151 4 ): ): PUNCT iajs-2618 151 5 consider consider VERB iajs-2618 151 6 πœ‰(𝑑 πœ‰(𝑑 PRON iajs-2618 151 7 ) ) PUNCT iajs-2618 151 8 ∈ ∈ PROPN iajs-2618 151 9 𝐷[0 𝐷[0 PROPN iajs-2618 151 10 , , PUNCT iajs-2618 151 11 𝑇 𝑇 PROPN iajs-2618 151 12 ] ] PUNCT iajs-2618 151 13 , , PUNCT iajs-2618 151 14 with with ADP iajs-2618 151 15 πœ‰β€²(0 πœ‰β€²(0 NOUN iajs-2618 151 16 ) ) PUNCT iajs-2618 151 17 β‰  β‰  PROPN iajs-2618 151 18 0 0 NUM iajs-2618 151 19 , , PUNCT iajs-2618 151 20 πœ‰(0 πœ‰(0 NOUN iajs-2618 151 21 ) ) PUNCT iajs-2618 151 22 = = SYM iajs-2618 151 23 0 0 NUM iajs-2618 151 24 , , PUNCT iajs-2618 151 25 and and CCONJ iajs-2618 151 26 πœ‰(𝑇 πœ‰(𝑇 NOUN iajs-2618 151 27 ) ) PUNCT iajs-2618 152 1 = = SYM iajs-2618 152 2 πœ‰β€²(𝑇 πœ‰β€²(𝑇 PROPN iajs-2618 152 3 ) ) PUNCT iajs-2618 152 4 = = SYM iajs-2618 153 1 0 0 X iajs-2618 153 2 . . X iajs-2618 153 3 using use VERB iajs-2618 153 4 twice twice DET iajs-2618 153 5 the the DET iajs-2618 153 6 ibp ibp NOUN iajs-2618 153 7 for for ADP iajs-2618 153 8 the the DET iajs-2618 153 9 1st 1st ADJ iajs-2618 153 10 term term NOUN iajs-2618 153 11 in in ADP iajs-2618 153 12 the the DET iajs-2618 153 13 l.h.s l.h.s NOUN iajs-2618 153 14 . . PUNCT iajs-2618 154 1 of of ADP iajs-2618 154 2 ( ( PUNCT iajs-2618 154 3 34 34 NUM iajs-2618 154 4 ) ) PUNCT iajs-2618 154 5 , , PUNCT iajs-2618 154 6 to to PART iajs-2618 154 7 obtain obtain VERB iajs-2618 154 8 ∫ ∫ PROPN iajs-2618 154 9 ( ( PUNCT iajs-2618 154 10 𝑀 𝑀 ADP iajs-2618 154 11 𝑇 𝑇 PROPN iajs-2618 154 12 0 0 NUM iajs-2618 154 13 , , PUNCT iajs-2618 154 14 πœ‚)πœ‰β€²β€²π‘‘π‘‘ πœ‚)πœ‰β€²β€²π‘‘π‘‘ PROPN iajs-2618 154 15 + + CCONJ iajs-2618 154 16 ∫ ∫ PROPN iajs-2618 155 1 [ [ X iajs-2618 155 2 ( ( PUNCT iajs-2618 155 3 βˆ‡π‘€ βˆ‡π‘€ PROPN iajs-2618 155 4 𝑇 𝑇 PROPN iajs-2618 155 5 0 0 NUM iajs-2618 155 6 , , PUNCT iajs-2618 155 7 βˆ‡πœ‚ βˆ‡πœ‚ PROPN iajs-2618 155 8 ) ) PUNCT iajs-2618 156 1 + + CCONJ iajs-2618 156 2 ( ( PUNCT iajs-2618 156 3 𝑀 𝑀 INTJ iajs-2618 156 4 , , PUNCT iajs-2618 156 5 πœ‚)]πœ‰ πœ‚)]πœ‰ ADJ iajs-2618 156 6 𝑑𝑑 𝑑𝑑 ADP iajs-2618 156 7 = = SYM iajs-2618 156 8 ∫ ∫ PROPN iajs-2618 156 9 ( ( PUNCT iajs-2618 156 10 𝑇 𝑇 PROPN iajs-2618 156 11 0 0 NUM iajs-2618 156 12 β„Ž(𝑑 β„Ž(𝑑 PROPN iajs-2618 156 13 , , PUNCT iajs-2618 156 14 𝑀 𝑀 ADP iajs-2618 156 15 ) ) PUNCT iajs-2618 156 16 , , PUNCT iajs-2618 156 17 πœ‚)πœ‰ πœ‚)πœ‰ PUNCT iajs-2618 156 18 𝑑𝑑 𝑑𝑑 ADP iajs-2618 156 19 βˆ’ βˆ’ PROPN iajs-2618 156 20 ( ( PUNCT iajs-2618 156 21 𝑀(0 𝑀(0 NOUN iajs-2618 156 22 ) ) PUNCT iajs-2618 156 23 , , PUNCT iajs-2618 156 24 πœ‚)πœ‰β€²(0 πœ‚)πœ‰β€²(0 NOUN iajs-2618 156 25 ) ) PUNCT iajs-2618 156 26 ( ( PUNCT iajs-2618 156 27 36 36 NUM iajs-2618 156 28 ) ) PUNCT iajs-2618 156 29 rewritten rewrite VERB iajs-2618 156 30 ( ( PUNCT iajs-2618 156 31 33 33 NUM iajs-2618 156 32 ) ) PUNCT iajs-2618 156 33 , , PUNCT iajs-2618 156 34 in in ADP iajs-2618 156 35 the the DET iajs-2618 156 36 following follow VERB iajs-2618 156 37 form form NOUN iajs-2618 156 38 βˆ’ βˆ’ PROPN iajs-2618 156 39 ∫ ∫ PROPN iajs-2618 156 40 ( ( PUNCT iajs-2618 156 41 𝑝 𝑝 PROPN iajs-2618 156 42 𝑇 𝑇 PROPN iajs-2618 156 43 0 0 NUM iajs-2618 156 44 , , PUNCT iajs-2618 156 45 πœ‚)πœ‰β€²(𝑑 πœ‚)πœ‰β€²(𝑑 PROPN iajs-2618 156 46 ) ) PUNCT iajs-2618 156 47 𝑑𝑑 𝑑𝑑 ADP iajs-2618 156 48 = = SYM iajs-2618 156 49 ∫ ∫ PROPN iajs-2618 156 50 ( ( PUNCT iajs-2618 156 51 𝑀 𝑀 ADP iajs-2618 156 52 𝑇 𝑇 PROPN iajs-2618 156 53 0 0 NUM iajs-2618 156 54 , , PUNCT iajs-2618 156 55 πœ‚)πœ‰β€²β€²(𝑑)𝑑𝑑 πœ‚)πœ‰β€²β€²(𝑑)𝑑𝑑 PROPN iajs-2618 157 1 + + CCONJ iajs-2618 157 2 ( ( PUNCT iajs-2618 157 3 𝑀 𝑀 PART iajs-2618 157 4 0 0 NUM iajs-2618 157 5 , , PUNCT iajs-2618 157 6 πœ‚)πœ‰β€²(0 πœ‚)πœ‰β€²(0 NOUN iajs-2618 157 7 ) ) PUNCT iajs-2618 157 8 ( ( PUNCT iajs-2618 157 9 37 37 NUM iajs-2618 157 10 ) ) PUNCT iajs-2618 157 11 126 126 NUM iajs-2618 158 1 ibn ibn PROPN iajs-2618 158 2 al al PROPN iajs-2618 158 3 - - PUNCT iajs-2618 158 4 haitham haitham PROPN iajs-2618 158 5 jour jour X iajs-2618 158 6 . . PROPN iajs-2618 158 7 for for ADP iajs-2618 158 8 pure pure ADJ iajs-2618 158 9 & & CCONJ iajs-2618 158 10 appl appl PROPN iajs-2618 158 11 . . PUNCT iajs-2618 159 1 sci sci PROPN iajs-2618 159 2 . . PROPN iajs-2618 160 1 34 34 NUM iajs-2618 160 2 ( ( PUNCT iajs-2618 160 3 2 2 NUM iajs-2618 160 4 ) ) PUNCT iajs-2618 160 5 2021 2021 NUM iajs-2618 160 6 substituting substituting NOUN iajs-2618 160 7 ( ( PUNCT iajs-2618 160 8 37 37 NUM iajs-2618 160 9 ) ) PUNCT iajs-2618 160 10 in in ADP iajs-2618 160 11 ( ( PUNCT iajs-2618 160 12 32 32 NUM iajs-2618 160 13 ) ) PUNCT iajs-2618 160 14 , , PUNCT iajs-2618 160 15 and and CCONJ iajs-2618 160 16 using use VERB iajs-2618 160 17 πœ‰(0 πœ‰(0 NOUN iajs-2618 160 18 ) ) PUNCT iajs-2618 160 19 = = SYM iajs-2618 160 20 0 0 NUM iajs-2618 160 21 , , PUNCT iajs-2618 160 22 then then ADV iajs-2618 160 23 subtracting subtract VERB iajs-2618 160 24 the the DET iajs-2618 160 25 resulting result VERB iajs-2618 160 26 equation equation NOUN iajs-2618 160 27 from from ADP iajs-2618 160 28 ( ( PUNCT iajs-2618 160 29 36 36 NUM iajs-2618 160 30 ) ) PUNCT iajs-2618 160 31 to to PART iajs-2618 160 32 get get VERB iajs-2618 160 33 ( ( PUNCT iajs-2618 160 34 𝑀(0 𝑀(0 NOUN iajs-2618 160 35 ) ) PUNCT iajs-2618 160 36 , , PUNCT iajs-2618 160 37 πœ‚)πœ‰β€²(0 πœ‚)πœ‰β€²(0 NOUN iajs-2618 160 38 ) ) PUNCT iajs-2618 160 39 = = SYM iajs-2618 160 40 ( ( PUNCT iajs-2618 160 41 𝑀0 𝑀0 PROPN iajs-2618 160 42 , , PUNCT iajs-2618 160 43 πœ‚)πœ‰β€²(0 πœ‚)πœ‰β€²(0 NOUN iajs-2618 160 44 ) ) PUNCT iajs-2618 160 45 ⟹ ⟹ NUM iajs-2618 160 46 ( ( PUNCT iajs-2618 160 47 𝑀(0 𝑀(0 NOUN iajs-2618 160 48 ) ) PUNCT iajs-2618 160 49 , , PUNCT iajs-2618 160 50 πœ‚ πœ‚ X iajs-2618 160 51 ) ) PUNCT iajs-2618 160 52 = = SYM iajs-2618 160 53 ( ( PUNCT iajs-2618 160 54 𝑀0 𝑀0 PROPN iajs-2618 160 55 , , PUNCT iajs-2618 160 56 πœ‚ πœ‚ NOUN iajs-2618 160 57 ) ) PUNCT iajs-2618 160 58 for for ADP iajs-2618 160 59 each each DET iajs-2618 160 60 πœ‚ πœ‚ NOUN iajs-2618 160 61 , , PUNCT iajs-2618 160 62 then then ADV iajs-2618 160 63 𝑀(0 𝑀(0 NOUN iajs-2618 160 64 ) ) PUNCT iajs-2618 160 65 = = SYM iajs-2618 160 66 𝑀0(0 𝑀0(0 NOUN iajs-2618 160 67 ) ) PUNCT iajs-2618 160 68 , , PUNCT iajs-2618 160 69 thus thus ADV iajs-2618 160 70 limit limit VERB iajs-2618 160 71 point point NOUN iajs-2618 160 72 𝑀 𝑀 ADP iajs-2618 160 73 is be AUX iajs-2618 160 74 a a DET iajs-2618 160 75 solution solution NOUN iajs-2618 160 76 to to ADP iajs-2618 160 77 the the DET iajs-2618 160 78 wef wef PROPN iajs-2618 160 79 for for ADP iajs-2618 160 80 the the DET iajs-2618 160 81 coe coe PROPN iajs-2618 160 82 . . PUNCT iajs-2618 161 1 7 7 X iajs-2618 161 2 . . X iajs-2618 161 3 cholesky cholesky ADJ iajs-2618 161 4 factorization factorization NOUN iajs-2618 161 5 cholesky cholesky NOUN iajs-2618 161 6 method method NOUN iajs-2618 161 7 is be AUX iajs-2618 161 8 used use VERB iajs-2618 161 9 using use VERB iajs-2618 161 10 to to PART iajs-2618 161 11 solve solve VERB iajs-2618 161 12 glas gla NOUN iajs-2618 161 13 with with ADP iajs-2618 161 14 conditions condition NOUN iajs-2618 161 15 that that SCONJ iajs-2618 161 16 the the DET iajs-2618 161 17 coefficient coefficient NOUN iajs-2618 161 18 matrix matrix NOUN iajs-2618 161 19 𝐴 𝐴 PROPN iajs-2618 161 20 must must AUX iajs-2618 161 21 be be AUX iajs-2618 161 22 a a DET iajs-2618 161 23 symmetric symmetric ADJ iajs-2618 161 24 and and CCONJ iajs-2618 161 25 positive positive ADJ iajs-2618 161 26 definite definite NOUN iajs-2618 161 27 . . PUNCT iajs-2618 162 1 in in ADP iajs-2618 162 2 this this DET iajs-2618 162 3 method method NOUN iajs-2618 162 4 the the DET iajs-2618 162 5 matrix matrix NOUN iajs-2618 162 6 𝐴 𝐴 PROPN iajs-2618 162 7 can can AUX iajs-2618 162 8 be be AUX iajs-2618 162 9 factorized factorize VERB iajs-2618 162 10 into into ADP iajs-2618 162 11 the the DET iajs-2618 162 12 product product NOUN iajs-2618 162 13 of of ADP iajs-2618 162 14 an an DET iajs-2618 162 15 upper upper ADJ iajs-2618 162 16 triangular triangular NOUN iajs-2618 162 17 matrix matrix NOUN iajs-2618 162 18 𝐿 𝐿 NOUN iajs-2618 162 19 and and CCONJ iajs-2618 162 20 lower low ADJ iajs-2618 162 21 triangular triangular NOUN iajs-2618 162 22 matrix matrix NOUN iajs-2618 162 23 𝐿𝑇 𝐿𝑇 PROPN iajs-2618 163 1 [ [ X iajs-2618 163 2 11 11 NUM iajs-2618 163 3 ] ] PUNCT iajs-2618 163 4 , , PUNCT iajs-2618 163 5 and and CCONJ iajs-2618 163 6 𝐿 𝐿 PROPN iajs-2618 163 7 calculates calculate NOUN iajs-2618 163 8 as as SCONJ iajs-2618 163 9 follows follow VERB iajs-2618 163 10 : : PUNCT iajs-2618 164 1 π‘“π‘œπ‘Ÿ π‘“π‘œπ‘Ÿ NOUN iajs-2618 164 2 𝑖 𝑖 X iajs-2618 164 3 = = SYM iajs-2618 164 4 1,2 1,2 NUM iajs-2618 164 5 , , PUNCT iajs-2618 164 6 … … PUNCT iajs-2618 164 7 , , PUNCT iajs-2618 164 8 𝑛 𝑛 PRON iajs-2618 164 9 π‘‘β„Žπ‘’π‘› π‘‘β„Žπ‘’π‘› VERB iajs-2618 164 10 𝑙𝑖𝑖 𝑙𝑖𝑖 NOUN iajs-2618 165 1 = = SYM iajs-2618 166 1 ( ( PUNCT iajs-2618 166 2 π‘Žπ‘–π‘– π‘Žπ‘–π‘– NOUN iajs-2618 166 3 βˆ’ βˆ’ PROPN iajs-2618 166 4 βˆ‘ βˆ‘ PROPN iajs-2618 166 5 π‘™π‘˜π‘– π‘™π‘˜π‘– PROPN iajs-2618 166 6 2 2 NUM iajs-2618 166 7 π‘–βˆ’1 π‘–βˆ’1 PART iajs-2618 166 8 π‘˜=1 π‘˜=1 PROPN iajs-2618 166 9 ) ) PUNCT iajs-2618 166 10 1 1 NUM iajs-2618 166 11 2 2 NUM iajs-2618 166 12 π‘“π‘œπ‘Ÿ π‘“π‘œπ‘Ÿ NOUN iajs-2618 166 13 𝑗 𝑗 NOUN iajs-2618 166 14 = = SYM iajs-2618 166 15 𝑖 𝑖 PROPN iajs-2618 166 16 + + NOUN iajs-2618 166 17 1 1 NUM iajs-2618 166 18 , , PUNCT iajs-2618 166 19 … … PUNCT iajs-2618 166 20 , , PUNCT iajs-2618 166 21 𝑛. 𝑛. NOUN iajs-2618 166 22 π‘‘β„Žπ‘’π‘› π‘‘β„Žπ‘’π‘› VERB iajs-2618 166 23 𝑙𝑖𝑗 𝑙𝑖𝑗 NOUN iajs-2618 167 1 = = SYM iajs-2618 167 2 ( ( PUNCT iajs-2618 167 3 π‘Žπ‘–π‘— π‘Žπ‘–π‘— NOUN iajs-2618 167 4 βˆ’ βˆ’ PROPN iajs-2618 167 5 βˆ‘ βˆ‘ PROPN iajs-2618 167 6 π‘™π‘˜π‘– π‘™π‘˜π‘– PROPN iajs-2618 167 7 π‘–βˆ’1 π‘–βˆ’1 PROPN iajs-2618 167 8 π‘˜=1 π‘˜=1 PROPN iajs-2618 167 9 π‘™π‘˜π‘— π‘™π‘˜π‘— PROPN iajs-2618 167 10 ) ) PUNCT iajs-2618 167 11 / / SYM iajs-2618 167 12 𝑙𝑖𝑖 𝑙𝑖𝑖 NOUN iajs-2618 167 13 8 8 NUM iajs-2618 167 14 . . PUNCT iajs-2618 167 15 numerical numerical ADJ iajs-2618 167 16 examples example NOUN iajs-2618 167 17 : : PUNCT iajs-2618 167 18 the the DET iajs-2618 167 19 problems problem NOUN iajs-2618 167 20 in in ADP iajs-2618 167 21 the the DET iajs-2618 167 22 following follow VERB iajs-2618 167 23 examples example NOUN iajs-2618 167 24 are be AUX iajs-2618 167 25 coded code VERB iajs-2618 167 26 by by ADP iajs-2618 167 27 mat mat NOUN iajs-2618 167 28 lap lap NOUN iajs-2618 167 29 soft soft ADJ iajs-2618 167 30 . . PUNCT iajs-2618 168 1 example example NOUN iajs-2618 168 2 1 1 NUM iajs-2618 168 3 : : PUNCT iajs-2618 168 4 consider consider VERB iajs-2618 168 5 the the DET iajs-2618 168 6 following follow VERB iajs-2618 168 7 nlhbvp nlhbvp NOUN iajs-2618 168 8 : : PUNCT iajs-2618 168 9 𝑀𝑑𝑑 𝑀𝑑𝑑 NOUN iajs-2618 168 10 βˆ’ βˆ’ PROPN iajs-2618 168 11 Ξ΄ Ξ΄ NOUN iajs-2618 168 12 𝑀 𝑀 ADP iajs-2618 168 13 + + NUM iajs-2618 168 14 𝑀 𝑀 NOUN iajs-2618 168 15 = = PUNCT iajs-2618 168 16 β„Ž( β„Ž( NOUN iajs-2618 168 17 οΏ½ οΏ½ NOUN iajs-2618 168 18 βƒ— βƒ— PROPN iajs-2618 168 19 οΏ½ οΏ½ PROPN iajs-2618 168 20 , , PUNCT iajs-2618 168 21 𝑑 𝑑 PROPN iajs-2618 168 22 , , PUNCT iajs-2618 168 23 𝑀 𝑀 ADP iajs-2618 168 24 ) ) PUNCT iajs-2618 168 25 , , PUNCT iajs-2618 168 26 οΏ½ οΏ½ PROPN iajs-2618 168 27 βƒ— βƒ— NOUN iajs-2618 168 28 οΏ½ οΏ½ X iajs-2618 168 29 = = SYM iajs-2618 168 30 ( ( PUNCT iajs-2618 168 31 π‘₯ π‘₯ PROPN iajs-2618 168 32 , , PUNCT iajs-2618 168 33 𝑦 𝑦 NOUN iajs-2618 168 34 ) ) PUNCT iajs-2618 168 35 , , PUNCT iajs-2618 168 36 πœ‘ πœ‘ X iajs-2618 168 37 = = PUNCT iajs-2618 168 38 πœ“ πœ“ PROPN iajs-2618 168 39 Γ— Γ— NOUN iajs-2618 168 40 𝐼 𝐼 PROPN iajs-2618 168 41 , , PUNCT iajs-2618 168 42 πœ“ πœ“ X iajs-2618 168 43 = = SYM iajs-2618 168 44 ( ( PUNCT iajs-2618 168 45 0,1 0,1 NUM iajs-2618 168 46 ) ) PUNCT iajs-2618 168 47 Γ— Γ— NOUN iajs-2618 168 48 ( ( PUNCT iajs-2618 168 49 0,1 0,1 NUM iajs-2618 168 50 ) ) PUNCT iajs-2618 168 51 , , PUNCT iajs-2618 168 52 𝐼 𝐼 PROPN iajs-2618 168 53 = = PUNCT iajs-2618 169 1 [ [ X iajs-2618 169 2 0,1 0,1 NUM iajs-2618 169 3 ] ] PUNCT iajs-2618 169 4 𝑀( 𝑀( NUM iajs-2618 169 5 οΏ½ οΏ½ PROPN iajs-2618 169 6 βƒ— βƒ— NOUN iajs-2618 169 7 οΏ½ οΏ½ PROPN iajs-2618 169 8 , , PUNCT iajs-2618 169 9 0 0 NUM iajs-2618 169 10 ) ) PUNCT iajs-2618 169 11 = = VERB iajs-2618 169 12 π‘₯𝑦(1 π‘₯𝑦(1 NOUN iajs-2618 169 13 βˆ’ βˆ’ NOUN iajs-2618 169 14 π‘₯)(1 π‘₯)(1 X iajs-2618 169 15 βˆ’ βˆ’ PROPN iajs-2618 169 16 𝑦 𝑦 NOUN iajs-2618 169 17 ) ) PUNCT iajs-2618 169 18 , , PUNCT iajs-2618 169 19 in in ADP iajs-2618 169 20 πœ“ πœ“ PROPN iajs-2618 169 21 𝑀𝑑( 𝑀𝑑( PROPN iajs-2618 169 22 οΏ½ οΏ½ PROPN iajs-2618 169 23 βƒ— βƒ— NOUN iajs-2618 169 24 οΏ½ οΏ½ PROPN iajs-2618 169 25 , , PUNCT iajs-2618 169 26 0 0 NUM iajs-2618 169 27 ) ) PUNCT iajs-2618 170 1 = = PRON iajs-2618 170 2 𝑀1 𝑀1 VERB iajs-2618 170 3 ( ( PUNCT iajs-2618 170 4 οΏ½ οΏ½ PROPN iajs-2618 170 5 βƒ— βƒ— NOUN iajs-2618 170 6 οΏ½ οΏ½ PROPN iajs-2618 170 7 ) ) PUNCT iajs-2618 170 8 , , PUNCT iajs-2618 170 9 in in ADP iajs-2618 170 10 πœ“ πœ“ PROPN iajs-2618 170 11 𝑀( 𝑀( NUM iajs-2618 170 12 οΏ½ οΏ½ PROPN iajs-2618 170 13 βƒ— βƒ— NOUN iajs-2618 170 14 οΏ½ οΏ½ PROPN iajs-2618 170 15 , , PUNCT iajs-2618 170 16 𝑑 𝑑 PROPN iajs-2618 170 17 ) ) PUNCT iajs-2618 170 18 = = SYM iajs-2618 170 19 0 0 NUM iajs-2618 170 20 , , PUNCT iajs-2618 170 21 on on ADP iajs-2618 170 22 βˆ‘ βˆ‘ PRON iajs-2618 170 23 = = PUNCT iajs-2618 170 24 πœ•πœ“ πœ•πœ“ NOUN iajs-2618 170 25 Γ— Γ— NOUN iajs-2618 170 26 𝐼 𝐼 PROPN iajs-2618 170 27 where where SCONJ iajs-2618 170 28 β„Ž( β„Ž( NOUN iajs-2618 170 29 οΏ½ οΏ½ NOUN iajs-2618 170 30 βƒ— βƒ— PROPN iajs-2618 170 31 οΏ½ οΏ½ PROPN iajs-2618 170 32 , , PUNCT iajs-2618 170 33 𝑑 𝑑 PROPN iajs-2618 170 34 , , PUNCT iajs-2618 170 35 𝑀 𝑀 ADP iajs-2618 170 36 ) ) PUNCT iajs-2618 170 37 = = SYM iajs-2618 170 38 1 1 NUM iajs-2618 170 39 2 2 NUM iajs-2618 170 40 ( ( PUNCT iajs-2618 170 41 π‘₯𝑦 π‘₯𝑦 ADJ iajs-2618 170 42 βˆ’ βˆ’ PROPN iajs-2618 170 43 π‘₯𝑦2 π‘₯𝑦2 NOUN iajs-2618 170 44 βˆ’ βˆ’ NOUN iajs-2618 170 45 𝑦π‘₯2 𝑦π‘₯2 NOUN iajs-2618 170 46 + + X iajs-2618 170 47 π‘₯2𝑦2)βˆšπ‘π‘œπ‘ 2 π‘₯2𝑦2)βˆšπ‘π‘œπ‘ 2 X iajs-2618 171 1 [ [ X iajs-2618 171 2 1 1 NUM iajs-2618 171 3 βˆ’ βˆ’ NOUN iajs-2618 171 4 2sin(π‘₯𝑦 2sin(π‘₯𝑦 NUM iajs-2618 171 5 βˆ’ βˆ’ NOUN iajs-2618 171 6 π‘₯𝑦2 π‘₯𝑦2 NOUN iajs-2618 171 7 βˆ’ βˆ’ NOUN iajs-2618 171 8 𝑦π‘₯2 𝑦π‘₯2 NOUN iajs-2618 171 9 + + CCONJ iajs-2618 171 10 π‘₯2𝑦2)βˆšπ‘π‘œπ‘ π‘‘ π‘₯2𝑦2)βˆšπ‘π‘œπ‘ π‘‘ X iajs-2618 171 11 ] ] X iajs-2618 171 12 + + PUNCT iajs-2618 171 13 2(𝑦 2(𝑦 NUM iajs-2618 171 14 + + CCONJ iajs-2618 171 15 π‘₯ π‘₯ PROPN iajs-2618 171 16 βˆ’ βˆ’ PROPN iajs-2618 171 17 𝑦2 𝑦2 PROPN iajs-2618 171 18 βˆ’ βˆ’ PROPN iajs-2618 171 19 π‘₯2)βˆšπ‘π‘œπ‘ π‘‘ π‘₯2)βˆšπ‘π‘œπ‘ π‘‘ PROPN iajs-2618 171 20 + + CCONJ iajs-2618 171 21 ( ( PUNCT iajs-2618 171 22 π‘₯𝑦2 π‘₯𝑦2 ADV iajs-2618 171 23 βˆ’ βˆ’ NOUN iajs-2618 171 24 𝑦π‘₯2 𝑦π‘₯2 NOUN iajs-2618 171 25 βˆ’ βˆ’ NOUN iajs-2618 171 26 π‘₯𝑦 π‘₯𝑦 NOUN iajs-2618 171 27 βˆ’ βˆ’ NOUN iajs-2618 171 28 π‘₯2𝑦2 π‘₯2𝑦2 NOUN iajs-2618 171 29 ) ) PUNCT iajs-2618 171 30 𝑠𝑖𝑛2𝑑 𝑠𝑖𝑛2𝑑 NOUN iajs-2618 171 31 4√cos 4√cos NUM iajs-2618 171 32 ( ( PUNCT iajs-2618 171 33 𝑑)3⁄ 𝑑)3⁄ X iajs-2618 171 34 + + CCONJ iajs-2618 171 35 𝑀 𝑀 ADP iajs-2618 171 36 𝑠𝑖𝑛𝑀 𝑠𝑖𝑛𝑀 NOUN iajs-2618 171 37 and and CCONJ iajs-2618 171 38 the the DET iajs-2618 171 39 exact exact ADJ iajs-2618 171 40 solution solution NOUN iajs-2618 171 41 ( ( PUNCT iajs-2618 171 42 exs)of exs)of ADP iajs-2618 171 43 the the DET iajs-2618 171 44 problem problem NOUN iajs-2618 171 45 is be AUX iajs-2618 171 46 𝑀( 𝑀( NUM iajs-2618 171 47 οΏ½ οΏ½ NOUN iajs-2618 171 48 βƒ— βƒ— NOUN iajs-2618 171 49 οΏ½ οΏ½ PROPN iajs-2618 171 50 , , PUNCT iajs-2618 171 51 𝑑 𝑑 NOUN iajs-2618 171 52 ) ) PUNCT iajs-2618 171 53 = = PUNCT iajs-2618 171 54 π‘₯𝑦(1 π‘₯𝑦(1 NOUN iajs-2618 171 55 βˆ’ βˆ’ NOUN iajs-2618 171 56 π‘₯)(1 π‘₯)(1 X iajs-2618 172 1 βˆ’ βˆ’ PROPN iajs-2618 172 2 𝑦)√cos 𝑦)√cos X iajs-2618 172 3 ( ( PUNCT iajs-2618 172 4 𝑑 𝑑 NOUN iajs-2618 172 5 ) ) PUNCT iajs-2618 172 6 . . PUNCT iajs-2618 173 1 the the DET iajs-2618 173 2 mgfeim mgfeim NOUN iajs-2618 173 3 is be AUX iajs-2618 173 4 utilized utilize VERB iajs-2618 173 5 to to PART iajs-2618 173 6 solve solve VERB iajs-2618 173 7 this this DET iajs-2618 173 8 problem problem NOUN iajs-2618 173 9 with with ADP iajs-2618 173 10 = = SYM iajs-2618 173 11 9 9 NUM iajs-2618 173 12 , , PUNCT iajs-2618 173 13 π‘Œ π‘Œ PROPN iajs-2618 173 14 = = SYM iajs-2618 173 15 20 20 NUM iajs-2618 173 16 and and CCONJ iajs-2618 173 17 𝑇 𝑇 PROPN iajs-2618 173 18 = = SYM iajs-2618 173 19 1 1 NUM iajs-2618 173 20 , , PUNCT iajs-2618 173 21 the the DET iajs-2618 173 22 results result NOUN iajs-2618 173 23 are be AUX iajs-2618 173 24 shown show VERB iajs-2618 173 25 in in ADP iajs-2618 173 26 figure figure NOUN iajs-2618 173 27 1 1 NUM iajs-2618 173 28 . . PUNCT iajs-2618 174 1 ( ( PUNCT iajs-2618 174 2 a a X iajs-2618 174 3 ) ) PUNCT iajs-2618 174 4 the the DET iajs-2618 174 5 apps app NOUN iajs-2618 174 6 , , PUNCT iajs-2618 174 7 and and CCONJ iajs-2618 174 8 figure figure VERB iajs-2618 174 9 1.(b 1.(b NUM iajs-2618 174 10 ) ) PUNCT iajs-2618 174 11 shown show VERB iajs-2618 174 12 the the DET iajs-2618 174 13 exs exs PROPN iajs-2618 174 14 at at ADP iajs-2618 174 15 οΏ½ οΏ½ PROPN iajs-2618 174 16 Μ‚ Μ‚ SYM iajs-2618 174 17 οΏ½ οΏ½ NOUN iajs-2618 174 18 = = SYM iajs-2618 174 19 0.5 0.5 NUM iajs-2618 174 20 . . PUNCT iajs-2618 175 1 figure1 figure1 PROPN iajs-2618 175 2 . . PUNCT iajs-2618 176 1 ( ( PUNCT iajs-2618 176 2 a a X iajs-2618 176 3 ) ) PUNCT iajs-2618 176 4 shows show VERB iajs-2618 176 5 the the DET iajs-2618 176 6 aps ap NOUN iajs-2618 176 7 and and CCONJ iajs-2618 176 8 ( ( PUNCT iajs-2618 176 9 b b NOUN iajs-2618 176 10 ) ) PUNCT iajs-2618 176 11 shows show VERB iajs-2618 176 12 the the DET iajs-2618 176 13 exs exs PROPN iajs-2618 176 14 example example NOUN iajs-2618 176 15 2 2 NUM iajs-2618 176 16 : : PUNCT iajs-2618 176 17 consider consider VERB iajs-2618 176 18 the the DET iajs-2618 176 19 following follow VERB iajs-2618 176 20 nlhybvp nlhybvp ADJ iajs-2618 176 21 : : PUNCT iajs-2618 177 1 𝑀𝑑𝑑 𝑀𝑑𝑑 VERB iajs-2618 177 2 βˆ’ βˆ’ PROPN iajs-2618 177 3 Ξ΄ Ξ΄ NOUN iajs-2618 177 4 𝑀 𝑀 ADP iajs-2618 177 5 + + NUM iajs-2618 177 6 𝑀 𝑀 NOUN iajs-2618 177 7 = = PUNCT iajs-2618 177 8 β„Ž( β„Ž( NOUN iajs-2618 177 9 οΏ½ οΏ½ NOUN iajs-2618 177 10 βƒ— βƒ— PROPN iajs-2618 177 11 οΏ½ οΏ½ PROPN iajs-2618 177 12 , , PUNCT iajs-2618 177 13 𝑑 𝑑 PROPN iajs-2618 177 14 , , PUNCT iajs-2618 177 15 𝑀 𝑀 ADP iajs-2618 177 16 ) ) PUNCT iajs-2618 177 17 where where SCONJ iajs-2618 177 18 οΏ½ οΏ½ NOUN iajs-2618 177 19 βƒ— βƒ— X iajs-2618 177 20 οΏ½ οΏ½ NOUN iajs-2618 177 21 = = SYM iajs-2618 177 22 ( ( PUNCT iajs-2618 177 23 π‘₯ π‘₯ PROPN iajs-2618 177 24 , , PUNCT iajs-2618 177 25 𝑦 𝑦 NOUN iajs-2618 177 26 ) ) PUNCT iajs-2618 177 27 𝑀( 𝑀( NUM iajs-2618 177 28 οΏ½ οΏ½ PROPN iajs-2618 177 29 βƒ— βƒ— NOUN iajs-2618 177 30 οΏ½ οΏ½ PROPN iajs-2618 177 31 , , PUNCT iajs-2618 177 32 0 0 NUM iajs-2618 177 33 ) ) PUNCT iajs-2618 177 34 = = SYM iajs-2618 177 35 ( ( PUNCT iajs-2618 178 1 π‘₯ π‘₯ NOUN iajs-2618 178 2 βˆ’ βˆ’ NUM iajs-2618 178 3 1)(1 1)(1 NUM iajs-2618 178 4 βˆ’ βˆ’ PROPN iajs-2618 178 5 𝑦 𝑦 NOUN iajs-2618 178 6 ) ) PUNCT iajs-2618 178 7 sin(π‘₯𝑦 sin(π‘₯𝑦 NOUN iajs-2618 178 8 ) ) PUNCT iajs-2618 178 9 in in ADP iajs-2618 178 10 πœ“ πœ“ PROPN iajs-2618 178 11 127 127 NUM iajs-2618 178 12 ibn ibn PROPN iajs-2618 178 13 al al PROPN iajs-2618 178 14 - - PUNCT iajs-2618 178 15 haitham haitham PROPN iajs-2618 178 16 jour jour X iajs-2618 178 17 . . PROPN iajs-2618 179 1 for for ADP iajs-2618 179 2 pure pure ADJ iajs-2618 179 3 & & CCONJ iajs-2618 179 4 appl appl PROPN iajs-2618 179 5 . . PUNCT iajs-2618 180 1 sci sci PROPN iajs-2618 180 2 . . PROPN iajs-2618 181 1 34 34 NUM iajs-2618 181 2 ( ( PUNCT iajs-2618 181 3 2 2 NUM iajs-2618 181 4 ) ) PUNCT iajs-2618 181 5 2021 2021 NUM iajs-2618 181 6 𝑀𝑑 𝑀𝑑 PROPN iajs-2618 181 7 ( ( PUNCT iajs-2618 181 8 οΏ½ οΏ½ PROPN iajs-2618 181 9 βƒ— βƒ— NOUN iajs-2618 181 10 οΏ½ οΏ½ PROPN iajs-2618 181 11 , , PUNCT iajs-2618 181 12 0 0 NUM iajs-2618 181 13 ) ) PUNCT iajs-2618 181 14 = = PRON iajs-2618 182 1 𝑀1 𝑀1 VERB iajs-2618 182 2 ( ( PUNCT iajs-2618 182 3 οΏ½ οΏ½ PROPN iajs-2618 182 4 βƒ— βƒ— NOUN iajs-2618 182 5 οΏ½ οΏ½ PROPN iajs-2618 182 6 ) ) PUNCT iajs-2618 182 7 , , PUNCT iajs-2618 182 8 in in ADP iajs-2618 182 9 πœ“ πœ“ PROPN iajs-2618 182 10 𝑀( 𝑀( NUM iajs-2618 182 11 οΏ½ οΏ½ PROPN iajs-2618 182 12 βƒ— βƒ— NOUN iajs-2618 182 13 οΏ½ οΏ½ PROPN iajs-2618 182 14 , , PUNCT iajs-2618 182 15 𝑑 𝑑 PROPN iajs-2618 182 16 ) ) PUNCT iajs-2618 182 17 = = SYM iajs-2618 182 18 0 0 NUM iajs-2618 182 19 , , PUNCT iajs-2618 182 20 on on ADP iajs-2618 182 21 βˆ‘ βˆ‘ PRON iajs-2618 182 22 = = PUNCT iajs-2618 182 23 πœ•πœ“ πœ•πœ“ NOUN iajs-2618 182 24 Γ— Γ— NOUN iajs-2618 182 25 𝐼 𝐼 PROPN iajs-2618 182 26 β„Ž( β„Ž( NOUN iajs-2618 182 27 οΏ½ οΏ½ NOUN iajs-2618 182 28 βƒ— βƒ— NOUN iajs-2618 182 29 οΏ½ οΏ½ PROPN iajs-2618 182 30 , , PUNCT iajs-2618 182 31 𝑑 𝑑 PROPN iajs-2618 182 32 , , PUNCT iajs-2618 182 33 𝑀 𝑀 ADP iajs-2618 182 34 ) ) PUNCT iajs-2618 182 35 = = SYM iajs-2618 183 1 2(𝑦 2(𝑦 NUM iajs-2618 183 2 βˆ’ βˆ’ NOUN iajs-2618 183 3 π‘₯ π‘₯ NOUN iajs-2618 183 4 βˆ’ βˆ’ PROPN iajs-2618 183 5 𝑦2 𝑦2 NOUN iajs-2618 183 6 + + CCONJ iajs-2618 183 7 π‘₯2)βˆšπ‘’π‘‘2 π‘₯2)βˆšπ‘’π‘‘2 VERB iajs-2618 183 8 cos(π‘₯𝑦 cos(π‘₯𝑦 NOUN iajs-2618 183 9 ) ) PUNCT iajs-2618 184 1 + + CCONJ iajs-2618 184 2 ( ( PUNCT iajs-2618 184 3 1 1 NUM iajs-2618 184 4 βˆ’ βˆ’ NOUN iajs-2618 184 5 π‘₯ π‘₯ PRON iajs-2618 184 6 βˆ’ βˆ’ PROPN iajs-2618 184 7 𝑦 𝑦 SYM iajs-2618 184 8 + + NUM iajs-2618 184 9 π‘₯𝑦)βˆšπ‘’π‘‘2 π‘₯𝑦)βˆšπ‘’π‘‘2 NOUN iajs-2618 184 10 sin(xy sin(xy NOUN iajs-2618 184 11 ) ) PUNCT iajs-2618 185 1 [ [ X iajs-2618 185 2 π‘₯2 π‘₯2 NOUN iajs-2618 185 3 + + CCONJ iajs-2618 185 4 𝑦2 𝑦2 PROPN iajs-2618 185 5 βˆ’ βˆ’ PROPN iajs-2618 185 6 sin sin NOUN iajs-2618 185 7 ( ( PUNCT iajs-2618 185 8 ( ( PUNCT iajs-2618 185 9 1 1 NUM iajs-2618 185 10 βˆ’ βˆ’ NOUN iajs-2618 185 11 π‘₯ π‘₯ PRON iajs-2618 185 12 βˆ’ βˆ’ PROPN iajs-2618 185 13 𝑦 𝑦 SYM iajs-2618 185 14 + + NUM iajs-2618 185 15 π‘₯𝑦)βˆšπ‘’π‘‘2 π‘₯𝑦)βˆšπ‘’π‘‘2 NOUN iajs-2618 185 16 sin(xy sin(xy NOUN iajs-2618 185 17 ) ) PUNCT iajs-2618 185 18 ) ) PUNCT iajs-2618 185 19 ] ] PUNCT iajs-2618 186 1 + + CCONJ iajs-2618 186 2 𝑀 𝑀 PRON iajs-2618 186 3 sin sin NOUN iajs-2618 186 4 ( ( PUNCT iajs-2618 186 5 𝑀 𝑀 NOUN iajs-2618 186 6 ) ) PUNCT iajs-2618 186 7 . . PUNCT iajs-2618 187 1 and and CCONJ iajs-2618 187 2 the the DET iajs-2618 187 3 exs exs PROPN iajs-2618 187 4 is be AUX iajs-2618 187 5 𝑀( 𝑀( NUM iajs-2618 187 6 οΏ½ οΏ½ NOUN iajs-2618 187 7 βƒ— βƒ— NOUN iajs-2618 187 8 οΏ½ οΏ½ PROPN iajs-2618 187 9 , , PUNCT iajs-2618 187 10 𝑑 𝑑 PROPN iajs-2618 187 11 ) ) PUNCT iajs-2618 187 12 = = PUNCT iajs-2618 187 13 ( ( PUNCT iajs-2618 188 1 π‘₯ π‘₯ NOUN iajs-2618 188 2 βˆ’ βˆ’ NUM iajs-2618 188 3 1)(1 1)(1 NUM iajs-2618 188 4 βˆ’ βˆ’ NOUN iajs-2618 188 5 𝑦)βˆšπ‘’π‘‘2 𝑦)βˆšπ‘’π‘‘2 NOUN iajs-2618 188 6 sin(π‘₯𝑦 sin(π‘₯𝑦 NOUN iajs-2618 188 7 ) ) PUNCT iajs-2618 188 8 . . PUNCT iajs-2618 189 1 the the DET iajs-2618 189 2 mgfeim mgfeim NOUN iajs-2618 189 3 is be AUX iajs-2618 189 4 utilized utilize VERB iajs-2618 189 5 to to PART iajs-2618 189 6 solve solve VERB iajs-2618 189 7 this this DET iajs-2618 189 8 problem problem NOUN iajs-2618 189 9 with with ADP iajs-2618 189 10 = = SYM iajs-2618 189 11 9 9 NUM iajs-2618 189 12 , , PUNCT iajs-2618 189 13 π‘Œ π‘Œ PROPN iajs-2618 189 14 = = SYM iajs-2618 189 15 20 20 NUM iajs-2618 189 16 and and CCONJ iajs-2618 189 17 𝑇 𝑇 PROPN iajs-2618 189 18 = = SYM iajs-2618 189 19 1 1 NUM iajs-2618 189 20 , , PUNCT iajs-2618 189 21 the the DET iajs-2618 189 22 results result NOUN iajs-2618 189 23 are be AUX iajs-2618 189 24 shown show VERB iajs-2618 189 25 in in ADP iajs-2618 189 26 figure figure NOUN iajs-2618 189 27 2 2 NUM iajs-2618 189 28 . . PUNCT iajs-2618 190 1 ( ( PUNCT iajs-2618 190 2 a a X iajs-2618 190 3 ) ) PUNCT iajs-2618 190 4 the the DET iajs-2618 190 5 apps app NOUN iajs-2618 190 6 , , PUNCT iajs-2618 190 7 and and CCONJ iajs-2618 190 8 figure figure VERB iajs-2618 190 9 2.(b 2.(b NUM iajs-2618 190 10 ) ) PUNCT iajs-2618 190 11 shown show VERB iajs-2618 190 12 the the DET iajs-2618 190 13 exs exs PROPN iajs-2618 190 14 at at ADP iajs-2618 190 15 οΏ½ οΏ½ PROPN iajs-2618 190 16 Μ‚ Μ‚ SYM iajs-2618 190 17 οΏ½ οΏ½ NOUN iajs-2618 190 18 = = SYM iajs-2618 190 19 0.5 0.5 NUM iajs-2618 190 20 . . PUNCT iajs-2618 190 21 figure2 figure2 PROPN iajs-2618 190 22 . . PUNCT iajs-2618 191 1 ( ( PUNCT iajs-2618 191 2 a a X iajs-2618 191 3 ) ) PUNCT iajs-2618 191 4 shows show VERB iajs-2618 191 5 the the DET iajs-2618 191 6 aps ap NOUN iajs-2618 191 7 and and CCONJ iajs-2618 191 8 ( ( PUNCT iajs-2618 191 9 b b NOUN iajs-2618 191 10 ) ) PUNCT iajs-2618 191 11 shows show VERB iajs-2618 191 12 the the DET iajs-2618 191 13 exs exs PROPN iajs-2618 191 14 9 9 NUM iajs-2618 191 15 . . PUNCT iajs-2618 192 1 conclusions conclusion NOUN iajs-2618 192 2 the the DET iajs-2618 192 3 mgfeim mgfeim NOUN iajs-2618 192 4 is be AUX iajs-2618 192 5 used use VERB iajs-2618 192 6 successfully successfully ADV iajs-2618 192 7 to to PART iajs-2618 192 8 solve solve VERB iajs-2618 192 9 the the DET iajs-2618 192 10 di di NOUN iajs-2618 192 11 of of ADP iajs-2618 192 12 the the DET iajs-2618 192 13 wef wef NOUN iajs-2618 192 14 of of ADP iajs-2618 192 15 a a DET iajs-2618 192 16 certain certain ADJ iajs-2618 192 17 type type NOUN iajs-2618 192 18 of of ADP iajs-2618 192 19 nolhybvp nolhybvp NOUN iajs-2618 192 20 . . PUNCT iajs-2618 193 1 the the DET iajs-2618 193 2 existence existence NOUN iajs-2618 193 3 theorem theorem NOUN iajs-2618 193 4 of of ADP iajs-2618 193 5 a a DET iajs-2618 193 6 unique unique ADJ iajs-2618 193 7 convergent convergent NOUN iajs-2618 193 8 app app NOUN iajs-2618 193 9 is be AUX iajs-2618 193 10 proved prove VERB iajs-2618 193 11 . . PUNCT iajs-2618 194 1 the the DET iajs-2618 194 2 convergent convergent NOUN iajs-2618 194 3 of of ADP iajs-2618 194 4 the the DET iajs-2618 194 5 pt pt PROPN iajs-2618 194 6 and and CCONJ iajs-2618 194 7 ct ct NUM iajs-2618 194 8 which which PRON iajs-2618 194 9 are be AUX iajs-2618 194 10 used use VERB iajs-2618 194 11 to to PART iajs-2618 194 12 solve solve VERB iajs-2618 194 13 the the DET iajs-2618 194 14 gnas gnas NOUN iajs-2618 194 15 that that PRON iajs-2618 194 16 is be AUX iajs-2618 194 17 obtained obtain VERB iajs-2618 194 18 from from ADP iajs-2618 194 19 applying apply VERB iajs-2618 194 20 the the DET iajs-2618 194 21 mgfeim mgfeim NOUN iajs-2618 194 22 , , PUNCT iajs-2618 194 23 is be AUX iajs-2618 194 24 proved prove VERB iajs-2618 194 25 and and CCONJ iajs-2618 194 26 the the DET iajs-2618 194 27 chme chme NOUN iajs-2618 194 28 which which PRON iajs-2618 194 29 is be AUX iajs-2618 194 30 used use VERB iajs-2618 194 31 inside inside ADP iajs-2618 194 32 these these DET iajs-2618 194 33 technique technique NOUN iajs-2618 194 34 is be AUX iajs-2618 194 35 highly highly ADV iajs-2618 194 36 efficient efficient ADJ iajs-2618 194 37 for for ADP iajs-2618 194 38 solving solve VERB iajs-2618 194 39 large large ADJ iajs-2618 194 40 gas gas NOUN iajs-2618 194 41 . . PUNCT iajs-2618 195 1 the the DET iajs-2618 195 2 di di NOUN iajs-2618 195 3 of of ADP iajs-2618 195 4 the the DET iajs-2618 195 5 wef wef PROPN iajs-2618 195 6 is be AUX iajs-2618 195 7 proved prove VERB iajs-2618 195 8 itis itis NOUN iajs-2618 195 9 stable stable ADJ iajs-2618 195 10 and and CCONJ iajs-2618 195 11 convergent convergent NOUN iajs-2618 195 12 . . PUNCT iajs-2618 196 1 the the DET iajs-2618 196 2 results result NOUN iajs-2618 196 3 are be AUX iajs-2618 196 4 given give VERB iajs-2618 196 5 by by ADP iajs-2618 196 6 figures figure NOUN iajs-2618 196 7 and and CCONJ iajs-2618 196 8 show show VERB iajs-2618 196 9 the the DET iajs-2618 196 10 efficiency efficiency NOUN iajs-2618 196 11 and and CCONJ iajs-2618 196 12 accuracy accuracy NOUN iajs-2618 196 13 for for ADP iajs-2618 196 14 the the DET iajs-2618 196 15 method method NOUN iajs-2618 196 16 . . PUNCT iajs-2618 197 1 references reference NOUN iajs-2618 197 2 1 1 NUM iajs-2618 197 3 . . X iajs-2618 197 4 smiley smiley NOUN iajs-2618 197 5 , , PUNCT iajs-2618 197 6 m. m. NOUN iajs-2618 197 7 w. w. PROPN iajs-2618 197 8 eigenfunction eigenfunction NOUN iajs-2618 197 9 methods method NOUN iajs-2618 197 10 and and CCONJ iajs-2618 197 11 nonlinear nonlinear ADJ iajs-2618 197 12 hyperbolic hyperbolic ADJ iajs-2618 197 13 boundary boundary ADJ iajs-2618 197 14 value value NOUN iajs-2618 197 15 problems problem NOUN iajs-2618 197 16 at at ADP iajs-2618 197 17 resonance resonance NOUN iajs-2618 197 18 . . PUNCT iajs-2618 198 1 journal journal PROPN iajs-2618 198 2 of of ADP iajs-2618 198 3 mathematical mathematical ADJ iajs-2618 198 4 analysis analysis NOUN iajs-2618 198 5 and and CCONJ iajs-2618 198 6 applications application NOUN iajs-2618 198 7 .1987 .1987 PROPN iajs-2618 198 8 , , PUNCT iajs-2618 198 9 122(1 122(1 NUM iajs-2618 198 10 ) ) PUNCT iajs-2618 198 11 , , PUNCT iajs-2618 198 12 129 129 NUM iajs-2618 198 13 - - SYM iajs-2618 198 14 151 151 NUM iajs-2618 198 15 . . NOUN iajs-2618 199 1 2 2 NUM iajs-2618 199 2 . . X iajs-2618 199 3 chi chi PROPN iajs-2618 199 4 , , PUNCT iajs-2618 199 5 h. h. PROPN iajs-2618 199 6 ; ; PUNCT iajs-2618 199 7 poorkarimi poorkarimi PROPN iajs-2618 199 8 , , PUNCT iajs-2618 199 9 h. h. PROPN iajs-2618 199 10 ; ; PUNCT iajs-2618 199 11 wiener wiener NOUN iajs-2618 199 12 , , PUNCT iajs-2618 199 13 j j NOUN iajs-2618 199 14 ; ; PUNCT iajs-2618 199 15 shah shah PROPN iajs-2618 199 16 , , PUNCT iajs-2618 199 17 s. s. PROPN iajs-2618 199 18 m. m. PROPN iajs-2618 199 19 on on ADP iajs-2618 199 20 the the DET iajs-2618 199 21 exponential exponential ADJ iajs-2618 199 22 growth growth NOUN iajs-2618 199 23 of of ADP iajs-2618 199 24 solutions solution NOUN iajs-2618 199 25 to to ADP iajs-2618 199 26 non non ADJ iajs-2618 199 27 - - ADJ iajs-2618 199 28 linear linear ADJ iajs-2618 199 29 hyperbolic hyperbolic ADJ iajs-2618 199 30 equations equation NOUN iajs-2618 199 31 . . PUNCT iajs-2618 200 1 international international ADJ iajs-2618 200 2 journal journal PROPN iajs-2618 200 3 of of ADP iajs-2618 200 4 mathematics mathematics PROPN iajs-2618 200 5 and and CCONJ iajs-2618 200 6 mathematical mathematical ADJ iajs-2618 200 7 sciences.1989,12(3),539 sciences.1989,12(3),539 PRON iajs-2618 200 8 - - PUNCT iajs-2618 200 9 545,https://doi.org/10.1155 545,https://doi.org/10.1155 NUM iajs-2618 200 10 / / SYM iajs-2618 200 11 s0161171289000670 s0161171289000670 PROPN iajs-2618 200 12 . . PROPN iajs-2618 200 13 3 3 X iajs-2618 200 14 . . X iajs-2618 200 15 minamoto minamoto PROPN iajs-2618 200 16 , , PUNCT iajs-2618 200 17 t. t. PROPN iajs-2618 200 18 numerical numerical ADJ iajs-2618 200 19 existence existence NOUN iajs-2618 200 20 and and CCONJ iajs-2618 200 21 uniqueness uniqueness ADJ iajs-2618 200 22 proof proof NOUN iajs-2618 200 23 for for ADP iajs-2618 200 24 solutions solution NOUN iajs-2618 200 25 of of ADP iajs-2618 200 26 nonlinear nonlinear ADJ iajs-2618 200 27 hyperbolic hyperbolic ADJ iajs-2618 200 28 equations equation NOUN iajs-2618 200 29 . . PUNCT iajs-2618 201 1 journal journal NOUN iajs-2618 201 2 of of ADP iajs-2618 201 3 computational computational ADJ iajs-2618 201 4 and and CCONJ iajs-2618 201 5 applied applied ADJ iajs-2618 201 6 mathematics mathematic NOUN iajs-2618 201 7 2001 2001 NUM iajs-2618 201 8 , , PUNCT iajs-2618 201 9 135(1 135(1 NUM iajs-2618 201 10 ) ) PUNCT iajs-2618 201 11 , , PUNCT iajs-2618 201 12 7990 7990 NUM iajs-2618 201 13 . . PUNCT iajs-2618 202 1 4 4 X iajs-2618 202 2 . . X iajs-2618 202 3 krylovas krylovas PROPN iajs-2618 202 4 , , PUNCT iajs-2618 202 5 a. a. NOUN iajs-2618 202 6 ; ; PUNCT iajs-2618 202 7 čiegis čiegis PROPN iajs-2618 202 8 , , PUNCT iajs-2618 202 9 r. r. PROPN iajs-2618 202 10 a a DET iajs-2618 202 11 review review NOUN iajs-2618 202 12 of of ADP iajs-2618 202 13 numerical numerical ADJ iajs-2618 202 14 asymptotic asymptotic ADJ iajs-2618 202 15 averaging averaging NOUN iajs-2618 202 16 for for ADP iajs-2618 202 17 weakly weakly ADJ iajs-2618 202 18 nonlinear nonlinear ADJ iajs-2618 202 19 hyperbolic hyperbolic ADJ iajs-2618 202 20 waves wave NOUN iajs-2618 202 21 . . PUNCT iajs-2618 203 1 mathematical mathematical ADJ iajs-2618 203 2 modelling modelling NOUN iajs-2618 203 3 and and CCONJ iajs-2618 203 4 analysis analysis NOUN iajs-2618 203 5 2004 2004 NUM iajs-2618 203 6 , , PUNCT iajs-2618 203 7 9(3 9(3 NUM iajs-2618 203 8 ) ) PUNCT iajs-2618 203 9 , , PUNCT iajs-2618 203 10 209 209 NUM iajs-2618 203 11 - - SYM iajs-2618 203 12 222 222 NUM iajs-2618 203 13 . . PUNCT iajs-2618 204 1 doi:10.1080/13926292.2004.9637254 doi:10.1080/13926292.2004.9637254 PROPN iajs-2618 204 2 . . PROPN iajs-2618 205 1 5 5 NUM iajs-2618 205 2 . . X iajs-2618 205 3 ashyralyev ashyralyev VERB iajs-2618 205 4 , , PUNCT iajs-2618 205 5 a. a. NOUN iajs-2618 205 6 ; ; PUNCT iajs-2618 205 7 agirseven agirseven ADV iajs-2618 205 8 , , PUNCT iajs-2618 205 9 d. d. PROPN iajs-2618 205 10 bounded bound VERB iajs-2618 205 11 solutions solution NOUN iajs-2618 205 12 of of ADP iajs-2618 205 13 nonlinear nonlinear ADJ iajs-2618 205 14 hyperbolic hyperbolic ADJ iajs-2618 205 15 equations equation NOUN iajs-2618 205 16 with with ADP iajs-2618 205 17 time time NOUN iajs-2618 205 18 delay delay NOUN iajs-2618 205 19 . . PUNCT iajs-2618 206 1 electron electron PROPN iajs-2618 206 2 . . PUNCT iajs-2618 207 1 j. j. PROPN iajs-2618 207 2 differential differential PROPN iajs-2618 207 3 equations equation NOUN iajs-2618 207 4 2018[internet 2018[internet NOUN iajs-2618 207 5 ] ] X iajs-2618 207 6 , , PUNCT iajs-2618 207 7 2018(21 2018(21 NUM iajs-2618 207 8 ) ) PUNCT iajs-2618 207 9 , , PUNCT iajs-2618 207 10 1 1 NUM iajs-2618 207 11 - - SYM iajs-2618 207 12 15 15 NUM iajs-2618 207 13 , , PUNCT iajs-2618 207 14 available available ADJ iajs-2618 207 15 from from ADP iajs-2618 207 16 : : PUNCT iajs-2618 207 17 http://ejde.math.txstate.edu http://ejde.math.txstate.edu PROPN iajs-2618 207 18 . . PUNCT iajs-2618 208 1 6 6 X iajs-2618 208 2 . . X iajs-2618 208 3 bangerth bangerth NOUN iajs-2618 208 4 , , PUNCT iajs-2618 208 5 w. w. PROPN iajs-2618 208 6 ; ; PUNCT iajs-2618 208 7 geiger geiger PROPN iajs-2618 208 8 , , PUNCT iajs-2618 208 9 m. m. NOUN iajs-2618 208 10 ; ; PUNCT iajs-2618 208 11 rannacher rannacher ADV iajs-2618 208 12 , , PUNCT iajs-2618 208 13 r. r. PROPN iajs-2618 208 14 ; ; PUNCT iajs-2618 208 15 adaptive adaptive PROPN iajs-2618 208 16 galerkin galerkin ADJ iajs-2618 208 17 finite finite PROPN iajs-2618 208 18 element element NOUN iajs-2618 208 19 methods method NOUN iajs-2618 208 20 for for ADP iajs-2618 208 21 the the DET iajs-2618 208 22 wave wave NOUN iajs-2618 208 23 equation equation NOUN iajs-2618 208 24 . . PUNCT iajs-2618 209 1 computational computational ADJ iajs-2618 209 2 methods method NOUN iajs-2618 209 3 in in ADP iajs-2618 209 4 applied applied ADJ iajs-2618 209 5 mathematics mathematic NOUN iajs-2618 209 6 2010 2010 NUM iajs-2618 209 7 , , PUNCT iajs-2618 209 8 10(1 10(1 NUM iajs-2618 209 9 ) ) PUNCT iajs-2618 209 10 , , PUNCT iajs-2618 209 11 3 3 NUM iajs-2618 209 12 - - SYM iajs-2618 209 13 48 48 NUM iajs-2618 209 14 . . PUNCT iajs-2618 210 1 doi doi NOUN iajs-2618 210 2 : : PUNCT iajs-2618 210 3 10.2478 10.2478 NUM iajs-2618 210 4 / / SYM iajs-2618 210 5 cmam-2010 cmam-2010 NOUN iajs-2618 210 6 - - PUNCT iajs-2618 210 7 0001 0001 NUM iajs-2618 210 8 . . PUNCT iajs-2618 211 1 128 128 NUM iajs-2618 211 2 ibn ibn PROPN iajs-2618 211 3 al al PROPN iajs-2618 211 4 - - PUNCT iajs-2618 211 5 haitham haitham PROPN iajs-2618 211 6 jour jour X iajs-2618 211 7 . . PROPN iajs-2618 211 8 for for ADP iajs-2618 211 9 pure pure ADJ iajs-2618 211 10 & & CCONJ iajs-2618 211 11 appl appl PROPN iajs-2618 211 12 . . PUNCT iajs-2618 212 1 sci sci PROPN iajs-2618 212 2 . . PROPN iajs-2618 213 1 34 34 NUM iajs-2618 213 2 ( ( PUNCT iajs-2618 213 3 2 2 NUM iajs-2618 213 4 ) ) PUNCT iajs-2618 213 5 2021 2021 NUM iajs-2618 213 6 7 7 NUM iajs-2618 213 7 . . PUNCT iajs-2618 214 1 al al PROPN iajs-2618 214 2 - - PUNCT iajs-2618 214 3 haq haq PROPN iajs-2618 214 4 , , PUNCT iajs-2618 214 5 a. a. NOUN iajs-2618 214 6 a. a. PROPN iajs-2618 214 7 j. j. PROPN iajs-2618 214 8 m. m. PROPN iajs-2618 214 9 numerical numerical PROPN iajs-2618 214 10 methods method NOUN iajs-2618 214 11 for for ADP iajs-2618 214 12 solving solve VERB iajs-2618 214 13 hyperbolic hyperbolic ADJ iajs-2618 214 14 type type NOUN iajs-2618 214 15 problems problem NOUN iajs-2618 214 16 . . PUNCT iajs-2618 215 1 m.sc m.sc PROPN iajs-2618 215 2 . . PUNCT iajs-2618 216 1 thesis thesis NOUN iajs-2618 216 2 . . PUNCT iajs-2618 217 1 an an DET iajs-2618 217 2 - - PUNCT iajs-2618 217 3 najah najah ADJ iajs-2618 217 4 national national ADJ iajs-2618 217 5 university university NOUN iajs-2618 217 6 , , PUNCT iajs-2618 217 7 nablus nablus PROPN iajs-2618 217 8 . . PUNCT iajs-2618 217 9 2017 2017 NUM iajs-2618 217 10 . . PUNCT iajs-2618 218 1 8 8 X iajs-2618 218 2 . . PUNCT iajs-2618 219 1 al al PROPN iajs-2618 219 2 - - PUNCT iajs-2618 219 3 hawasy hawasy PROPN iajs-2618 219 4 , , PUNCT iajs-2618 219 5 j. j. PROPN iajs-2618 219 6 a. a. PROPN iajs-2618 219 7 ; ; PUNCT iajs-2618 219 8 jawad jawad PROPN iajs-2618 219 9 , , PUNCT iajs-2618 219 10 m. m. NOUN iajs-2618 219 11 a. a. NOUN iajs-2618 219 12 the the DET iajs-2618 219 13 approximation approximation NOUN iajs-2618 219 14 solution solution NOUN iajs-2618 219 15 of of ADP iajs-2618 219 16 a a DET iajs-2618 219 17 nonlinear nonlinear ADJ iajs-2618 219 18 parabolic parabolic ADJ iajs-2618 219 19 boundary boundary ADJ iajs-2618 219 20 value value NOUN iajs-2618 219 21 problem problem NOUN iajs-2618 219 22 via via ADP iajs-2618 219 23 galerkin galerkin ADJ iajs-2618 219 24 finite finite PROPN iajs-2618 219 25 elements element NOUN iajs-2618 219 26 method method VERB iajs-2618 219 27 with with ADP iajs-2618 219 28 crank crank NOUN iajs-2618 219 29 - - PUNCT iajs-2618 219 30 nicolson nicolson PROPN iajs-2618 219 31 . . PUNCT iajs-2618 219 32 ihjpas ihjpas PROPN iajs-2618 219 33 . . PUNCT iajs-2618 220 1 2018 2018 NUM iajs-2618 220 2 , , PUNCT iajs-2618 220 3 31(3 31(3 NUM iajs-2618 220 4 ) ) PUNCT iajs-2618 220 5 , , PUNCT iajs-2618 220 6 126 126 NUM iajs-2618 220 7 - - SYM iajs-2618 220 8 134 134 NUM iajs-2618 220 9 , , PUNCT iajs-2618 220 10 https://doi.org/10.30526/31.3.2002 https://doi.org/10.30526/31.3.2002 NOUN iajs-2618 220 11 . . NOUN iajs-2618 220 12 9 9 NUM iajs-2618 220 13 . . X iajs-2618 221 1 al al PROPN iajs-2618 221 2 - - PUNCT iajs-2618 221 3 hawasy hawasy PROPN iajs-2618 221 4 , , PUNCT iajs-2618 221 5 j. j. PROPN iajs-2618 221 6 a. a. PROPN iajs-2618 221 7 ; ; PUNCT iajs-2618 221 8 jasim jasim PROPN iajs-2618 221 9 , , PUNCT iajs-2618 221 10 d. d. PROPN iajs-2618 221 11 k. k. PROPN iajs-2618 222 1 the the DET iajs-2618 222 2 continuous continuous ADJ iajs-2618 222 3 classical classical ADJ iajs-2618 222 4 optimal optimal ADJ iajs-2618 222 5 control control NOUN iajs-2618 222 6 problems problem NOUN iajs-2618 222 7 for for ADP iajs-2618 222 8 triple triple ADJ iajs-2618 222 9 elliptic elliptic ADJ iajs-2618 222 10 partial partial ADJ iajs-2618 222 11 differential differential NOUN iajs-2618 222 12 equations equation NOUN iajs-2618 222 13 . . PUNCT iajs-2618 223 1 ihjpas ihjpas PROPN iajs-2618 223 2 . . PUNCT iajs-2618 224 1 2020 2020 NUM iajs-2618 224 2 , , PUNCT iajs-2618 224 3 33(1 33(1 NUM iajs-2618 224 4 ) ) PUNCT iajs-2618 224 5 , , PUNCT iajs-2618 224 6 143 143 NUM iajs-2618 224 7 - - SYM iajs-2618 224 8 151 151 NUM iajs-2618 224 9 , , PUNCT iajs-2618 224 10 doi doi NOUN iajs-2618 224 11 : : PUNCT iajs-2618 224 12 10.30526/33.1.2380 10.30526/33.1.2380 NUM iajs-2618 224 13 . . PROPN iajs-2618 224 14 10 10 NUM iajs-2618 224 15 . . PUNCT iajs-2618 225 1 holte holte PROPN iajs-2618 225 2 , , PUNCT iajs-2618 225 3 j. j. PROPN iajs-2618 225 4 m. m. PROPN iajs-2618 225 5 discrete discrete VERB iajs-2618 225 6 gronwall gronwall PROPN iajs-2618 225 7 lemma lemma PROPN iajs-2618 225 8 and and CCONJ iajs-2618 225 9 applications application NOUN iajs-2618 225 10 . . PUNCT iajs-2618 226 1 in in ADP iajs-2618 226 2 maa maa PROPN iajs-2618 226 3 - - PUNCT iajs-2618 226 4 ncs ncs PROPN iajs-2618 226 5 meeting meeting NOUN iajs-2618 226 6 at at ADP iajs-2618 226 7 the the DET iajs-2618 226 8 university university PROPN iajs-2618 226 9 of of ADP iajs-2618 226 10 north north PROPN iajs-2618 226 11 dakota dakota PROPN iajs-2618 226 12 2009 2009 NUM iajs-2618 226 13 , , PUNCT iajs-2618 226 14 24,1 24,1 NOUN iajs-2618 226 15 - - SYM iajs-2618 226 16 7 7 NUM iajs-2618 226 17 . . NOUN iajs-2618 226 18 11 11 NUM iajs-2618 226 19 . . PUNCT iajs-2618 227 1 chen chen PROPN iajs-2618 227 2 , , PUNCT iajs-2618 227 3 j. j. PROPN iajs-2618 227 4 ; ; PUNCT iajs-2618 227 5 jin jin PROPN iajs-2618 227 6 . . PUNCT iajs-2618 228 1 z. z. PROPN iajs-2618 228 2 ; ; PUNCT iajs-2618 228 3 shi shi PROPN iajs-2618 228 4 . . PUNCT iajs-2618 228 5 q. q. PROPN iajs-2618 228 6 ; ; PUNCT iajs-2618 229 1 qiu qiu PROPN iajs-2618 229 2 . . PUNCT iajs-2618 230 1 j. j. PROPN iajs-2618 230 2 ; ; PUNCT iajs-2618 230 3 liu liu PROPN iajs-2618 230 4 . . PUNCT iajs-2618 231 1 w w PROPN iajs-2618 231 2 .. .. PUNCT iajs-2618 231 3 block block NOUN iajs-2618 231 4 algorithm algorithm NOUN iajs-2618 231 5 and and CCONJ iajs-2618 231 6 its its PRON iajs-2618 231 7 implementation implementation NOUN iajs-2618 231 8 for for ADP iajs-2618 231 9 cholesky cholesky ADJ iajs-2618 231 10 factorization factorization NOUN iajs-2618 231 11 . . PUNCT iajs-2618 232 1 iccgi iccgi NOUN iajs-2618 232 2 ( ( PUNCT iajs-2618 232 3 2013 2013 NUM iajs-2618 232 4 ): ): PUNCT iajs-2618 232 5 the the DET iajs-2618 232 6 eighth eighth ADJ iajs-2618 232 7 international international ADJ iajs-2618 232 8 multi multi NOUN iajs-2618 232 9 - - NOUN iajs-2618 232 10 conference conference NOUN iajs-2618 232 11 on on ADP iajs-2618 232 12 computing compute VERB iajs-2618 232 13 in in ADP iajs-2618 232 14 the the DET iajs-2618 232 15 global global ADJ iajs-2618 232 16 information information PROPN iajs-2618 232 17 technology technology PROPN iajs-2618 232 18 china china PROPN iajs-2618 232 19 iccgi20013.232 iccgi20013.232 PROPN iajs-2618 232 20 - - PUNCT iajs-2618 232 21 236 236 NUM iajs-2618 232 22 . . PUNCT iajs-2618 233 1 isbn isbn ADJ iajs-2618 233 2 : : PUNCT iajs-2618 233 3 978 978 NUM iajs-2618 233 4 1 1 NUM iajs-2618 233 5 - - PUNCT iajs-2618 233 6 61208 61208 NUM iajs-2618 233 7 - - PUNCT iajs-2618 233 8 283 283 NUM iajs-2618 233 9 - - SYM iajs-2618 233 10 7 7 NUM iajs-2618 233 11 . . PUNCT