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PROPN iajs-3638 4 43 al al PROPN iajs-3638 4 44 - - PUNCT iajs-3638 4 45 husseiny husseiny ADJ iajs-3638 4 46 2 2 PROPN iajs-3638 4 47 1department 1department NUM iajs-3638 4 48 of of ADP iajs-3638 4 49 networks network NOUN iajs-3638 4 50 and and CCONJ iajs-3638 4 51 cyber cyber NOUN iajs-3638 4 52 security security NOUN iajs-3638 4 53 , , PUNCT iajs-3638 4 54 college college NOUN iajs-3638 4 55 of of ADP iajs-3638 4 56 engineering engineering PROPN iajs-3638 4 57 , , PUNCT iajs-3638 4 58 al al PROPN iajs-3638 4 59 - - PUNCT iajs-3638 4 60 iraqia iraqia PROPN iajs-3638 4 61 university university PROPN iajs-3638 4 62 , , PUNCT iajs-3638 4 63 baghdad baghdad PROPN iajs-3638 4 64 , , PUNCT iajs-3638 4 65 iraq iraq PROPN iajs-3638 4 66 . . PUNCT iajs-3638 5 1 1,2department 1,2department NUM iajs-3638 5 2 of of ADP iajs-3638 5 3 mathematics mathematic NOUN iajs-3638 5 4 , , PUNCT iajs-3638 5 5 college college NOUN iajs-3638 5 6 of of ADP iajs-3638 5 7 science science NOUN iajs-3638 5 8 , , PUNCT iajs-3638 5 9 university university NOUN iajs-3638 5 10 of of ADP iajs-3638 5 11 baghdad baghdad PROPN iajs-3638 5 12 , , PUNCT iajs-3638 5 13 baghdad baghdad PROPN iajs-3638 5 14 , , PUNCT iajs-3638 5 15 iraq iraq PROPN iajs-3638 5 16 . . PUNCT iajs-3638 6 1 * * PUNCT iajs-3638 6 2 corresponding correspond VERB iajs-3638 6 3 author author NOUN iajs-3638 6 4 . . PUNCT iajs-3638 7 1 received:1 received:1 PROPN iajs-3638 7 2 july july PROPN iajs-3638 7 3 2023 2023 NUM iajs-3638 7 4 accepted:4 accepted:4 X iajs-3638 7 5 september september PROPN iajs-3638 7 6 2023 2023 NUM iajs-3638 7 7 published:20 published:20 NOUN iajs-3638 7 8 january january PROPN iajs-3638 7 9 2025 2025 NUM iajs-3638 7 10 doi.org/10.30526/38.1.3638 doi.org/10.30526/38.1.3638 ADJ iajs-3638 7 11 abstract abstract NOUN iajs-3638 7 12 in in ADP iajs-3638 7 13 this this DET iajs-3638 7 14 work work NOUN iajs-3638 7 15 , , PUNCT iajs-3638 7 16 we we PRON iajs-3638 7 17 formulate formulate VERB iajs-3638 7 18 a a DET iajs-3638 7 19 mathematical mathematical ADJ iajs-3638 7 20 model model NOUN iajs-3638 7 21 of of ADP iajs-3638 7 22 the the DET iajs-3638 7 23 killer killer NOUN iajs-3638 7 24 covid-19 covid-19 PROPN iajs-3638 7 25 pandemic pandemic ADJ iajs-3638 7 26 with with ADP iajs-3638 7 27 time time NOUN iajs-3638 7 28 delay delay NOUN iajs-3638 7 29 and and CCONJ iajs-3638 7 30 some some DET iajs-3638 7 31 governmental governmental ADJ iajs-3638 7 32 measures measure NOUN iajs-3638 7 33 that that PRON iajs-3638 7 34 include include VERB iajs-3638 7 35 vaccine vaccine NOUN iajs-3638 7 36 subsidies subsidy NOUN iajs-3638 7 37 to to PART iajs-3638 7 38 understand understand VERB iajs-3638 7 39 the the DET iajs-3638 7 40 dynamic dynamic ADJ iajs-3638 7 41 behavior behavior NOUN iajs-3638 7 42 of of ADP iajs-3638 7 43 covid-19 covid-19 PROPN iajs-3638 7 44 . . PUNCT iajs-3638 8 1 for for ADP iajs-3638 8 2 the the DET iajs-3638 8 3 dynamic dynamic ADJ iajs-3638 8 4 study study NOUN iajs-3638 8 5 , , PUNCT iajs-3638 8 6 a a DET iajs-3638 8 7 new new ADJ iajs-3638 8 8 model model NOUN iajs-3638 8 9 , , PUNCT iajs-3638 8 10 𝑆𝑉1𝑉2𝐼𝑅 𝑆𝑉1𝑉2𝐼𝑅 PROPN iajs-3638 8 11 was be AUX iajs-3638 8 12 used use VERB iajs-3638 8 13 purposed purpose VERB iajs-3638 8 14 in in ADP iajs-3638 8 15 which which PRON iajs-3638 8 16 infectious infectious ADJ iajs-3638 8 17 individuals individual NOUN iajs-3638 8 18 were be AUX iajs-3638 8 19 divided divide VERB iajs-3638 8 20 into into ADP iajs-3638 8 21 five five NUM iajs-3638 8 22 sub sub NOUN iajs-3638 8 23 compartments compartment NOUN iajs-3638 8 24 . . PUNCT iajs-3638 9 1 our our PRON iajs-3638 9 2 aim aim NOUN iajs-3638 9 3 is be AUX iajs-3638 9 4 to to PART iajs-3638 9 5 construct construct VERB iajs-3638 9 6 a a DET iajs-3638 9 7 more more ADV iajs-3638 9 8 reliable reliable ADJ iajs-3638 9 9 and and CCONJ iajs-3638 9 10 realistic realistic ADJ iajs-3638 9 11 model model NOUN iajs-3638 9 12 for for ADP iajs-3638 9 13 a a DET iajs-3638 9 14 complete complete ADJ iajs-3638 9 15 mathematical mathematical ADJ iajs-3638 9 16 and and CCONJ iajs-3638 9 17 computational computational ADJ iajs-3638 9 18 analysis analysis NOUN iajs-3638 9 19 and and CCONJ iajs-3638 9 20 design design NOUN iajs-3638 9 21 of of ADP iajs-3638 9 22 different different ADJ iajs-3638 9 23 control control NOUN iajs-3638 9 24 strategies strategy NOUN iajs-3638 9 25 for for ADP iajs-3638 9 26 the the DET iajs-3638 9 27 proposed propose VERB iajs-3638 9 28 deterministic deterministic ADJ iajs-3638 9 29 model model NOUN iajs-3638 9 30 . . 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PUNCT iajs-3638 31 1 abdulkadhim abdulkadhim PROPN iajs-3638 31 2 and and CCONJ iajs-3638 31 3 al al PROPN iajs-3638 31 4 - - PUNCT iajs-3638 31 5 husseiny husseiny PROPN iajs-3638 31 6 ( ( PUNCT iajs-3638 31 7 7 7 NUM iajs-3638 31 8 ) ) PUNCT iajs-3638 31 9 studied study VERB iajs-3638 31 10 the the DET iajs-3638 31 11 global global ADJ iajs-3638 31 12 stability stability NOUN iajs-3638 31 13 and and CCONJ iajs-3638 31 14 bifurcation bifurcation NOUN iajs-3638 31 15 of of ADP iajs-3638 31 16 a a DET iajs-3638 31 17 covid-19 covid-19 PROPN iajs-3638 31 18 virus virus NOUN iajs-3638 31 19 modeling modeling NOUN iajs-3638 31 20 with with ADP iajs-3638 31 21 possible possible ADJ iajs-3638 31 22 loss loss NOUN iajs-3638 31 23 of of ADP iajs-3638 31 24 the the DET iajs-3638 31 25 immunity immunity NOUN iajs-3638 31 26 . . PUNCT iajs-3638 32 1 the the DET iajs-3638 32 2 dynamics dynamic NOUN iajs-3638 32 3 of of ADP iajs-3638 32 4 populations population NOUN iajs-3638 32 5 are be AUX iajs-3638 32 6 significantly significantly ADV iajs-3638 32 7 influenced influence VERB iajs-3638 32 8 by by ADP iajs-3638 32 9 time time NOUN iajs-3638 32 10 delays delay NOUN iajs-3638 32 11 . . PUNCT iajs-3638 33 1 the the DET iajs-3638 33 2 dynamics dynamic NOUN iajs-3638 33 3 of of ADP iajs-3638 33 4 state state NOUN iajs-3638 33 5 variables variable NOUN iajs-3638 33 6 in in ADP iajs-3638 33 7 many many ADJ iajs-3638 33 8 real real ADJ iajs-3638 33 9 - - PUNCT iajs-3638 33 10 world world NOUN iajs-3638 33 11 processes process NOUN iajs-3638 33 12 , , PUNCT iajs-3638 33 13 notably notably ADV iajs-3638 33 14 in in ADP iajs-3638 33 15 many many ADJ iajs-3638 33 16 biological biological ADJ iajs-3638 33 17 phenomena phenomenon NOUN iajs-3638 33 18 , , PUNCT iajs-3638 33 19 depend depend VERB iajs-3638 33 20 on on ADP iajs-3638 33 21 the the DET iajs-3638 33 22 phenomenon phenomenon NOUN iajs-3638 33 23 's 's PART iajs-3638 33 24 history history NOUN iajs-3638 33 25 , , PUNCT iajs-3638 33 26 or or CCONJ iajs-3638 33 27 on on ADP iajs-3638 33 28 the the DET iajs-3638 33 29 state state NOUN iajs-3638 33 30 variables variable NOUN iajs-3638 33 31 ' ' PART iajs-3638 33 32 former former ADJ iajs-3638 33 33 values value NOUN iajs-3638 33 34 , , PUNCT iajs-3638 33 35 in in ADP iajs-3638 33 36 addition addition NOUN iajs-3638 33 37 to to ADP iajs-3638 33 38 the the DET iajs-3638 33 39 phenomenon phenomenon NOUN iajs-3638 33 40 's 's PART iajs-3638 33 41 current current ADJ iajs-3638 33 42 state state NOUN iajs-3638 33 43 . . PUNCT iajs-3638 34 1 time time NOUN iajs-3638 34 2 delays delay NOUN iajs-3638 34 3 may may AUX iajs-3638 34 4 have have VERB iajs-3638 34 5 an an DET iajs-3638 34 6 impact impact NOUN iajs-3638 34 7 on on ADP iajs-3638 34 8 the the DET iajs-3638 34 9 dynamics dynamic NOUN iajs-3638 34 10 of of ADP iajs-3638 34 11 infectious infectious ADJ iajs-3638 34 12 diseases disease NOUN iajs-3638 34 13 , , PUNCT iajs-3638 34 14 as as SCONJ iajs-3638 34 15 shown show VERB iajs-3638 34 16 in in ADP iajs-3638 34 17 zuo zuo PROPN iajs-3638 34 18 et et PROPN iajs-3638 34 19 al al PROPN iajs-3638 34 20 . . PROPN iajs-3638 35 1 ( ( PUNCT iajs-3638 35 2 8) 8) NUM iajs-3638 35 3 formulated formulate VERB iajs-3638 35 4 the the DET iajs-3638 35 5 relationship relationship NOUN iajs-3638 35 6 between between ADP iajs-3638 35 7 media medium NOUN iajs-3638 35 8 coverage coverage NOUN iajs-3638 35 9 and and CCONJ iajs-3638 35 10 an an DET iajs-3638 35 11 epidemic epidemic NOUN iajs-3638 35 12 's 's PART iajs-3638 35 13 recruitment recruitment NOUN iajs-3638 35 14 and and CCONJ iajs-3638 35 15 spread spread NOUN iajs-3638 35 16 . . PUNCT iajs-3638 36 1 aekabut aekabut CCONJ iajs-3638 36 2 et et PROPN iajs-3638 36 3 al al PROPN iajs-3638 36 4 . . PROPN iajs-3638 37 1 ( ( PUNCT iajs-3638 37 2 9)investigated 9)investigate VERB iajs-3638 37 3 a a DET iajs-3638 37 4 delayed delay VERB iajs-3638 37 5 seir seir ADJ iajs-3638 37 6 epidemic epidemic NOUN iajs-3638 37 7 model model NOUN iajs-3638 37 8 in in ADP iajs-3638 37 9 which which PRON iajs-3638 37 10 the the DET iajs-3638 37 11 diseased diseased ADJ iajs-3638 37 12 and and CCONJ iajs-3638 37 13 latent latent NOUN iajs-3638 37 14 phases phase NOUN iajs-3638 37 15 are be AUX iajs-3638 37 16 contagious contagious ADJ iajs-3638 37 17 . . PUNCT iajs-3638 38 1 rasha rasha PROPN iajs-3638 38 2 m. m. PROPN iajs-3638 38 3 yaseen yaseen PROPN iajs-3638 38 4 et et PROPN iajs-3638 38 5 al al PROPN iajs-3638 38 6 . . PROPN iajs-3638 39 1 ( ( PUNCT iajs-3638 39 2 10 10 NUM iajs-3638 39 3 ) ) PUNCT iajs-3638 39 4 studied study VERB iajs-3638 39 5 the the DET iajs-3638 39 6 stability stability NOUN iajs-3638 39 7 and and CCONJ iajs-3638 39 8 hopf hopf ADJ iajs-3638 39 9 bifurcation bifurcation NOUN iajs-3638 39 10 of of ADP iajs-3638 39 11 an an DET iajs-3638 39 12 epidemiological epidemiological ADJ iajs-3638 39 13 model model NOUN iajs-3638 39 14 with with ADP iajs-3638 39 15 the the DET iajs-3638 39 16 effect effect NOUN iajs-3638 39 17 of of ADP iajs-3638 39 18 delaying delay VERB iajs-3638 39 19 the the DET iajs-3638 39 20 awareness awareness NOUN iajs-3638 39 21 programms programms NOUN iajs-3638 39 22 and and CCONJ iajs-3638 39 23 vaccination vaccination NOUN iajs-3638 39 24 : : PUNCT iajs-3638 39 25 analysis analysis NOUN iajs-3638 39 26 and and CCONJ iajs-3638 39 27 simulation simulation NOUN iajs-3638 39 28 . . PUNCT iajs-3638 40 1 zhe zhe PROPN iajs-3638 40 2 yin yin PROPN iajs-3638 40 3 et et PROPN iajs-3638 40 4 al al PROPN iajs-3638 40 5 . . PUNCT iajs-3638 41 1 ( ( PUNCT iajs-3638 41 2 11 11 NUM iajs-3638 41 3 ) ) PUNCT iajs-3638 41 4 investigated investigate VERB iajs-3638 41 5 how how SCONJ iajs-3638 41 6 an an DET iajs-3638 41 7 age age NOUN iajs-3638 41 8 - - PUNCT iajs-3638 41 9 structured structure VERB iajs-3638 41 10 seirs seir NOUN iajs-3638 41 11 model model NOUN iajs-3638 41 12 was be AUX iajs-3638 41 13 affected affect VERB iajs-3638 41 14 by by ADP iajs-3638 41 15 time time NOUN iajs-3638 41 16 delays delay NOUN iajs-3638 41 17 . . PUNCT iajs-3638 42 1 mohsen mohsen PROPN iajs-3638 42 2 et et PROPN iajs-3638 42 3 al al PROPN iajs-3638 42 4 . . PUNCT iajs-3638 43 1 ( ( PUNCT iajs-3638 43 2 12 12 NUM iajs-3638 43 3 ) ) PUNCT iajs-3638 43 4 investigated investigate VERB iajs-3638 43 5 the the DET iajs-3638 43 6 dynamics dynamic NOUN iajs-3638 43 7 of of ADP iajs-3638 43 8 a a DET iajs-3638 43 9 curfew curfew NOUN iajs-3638 43 10 technique technique NOUN iajs-3638 43 11 in in ADP iajs-3638 43 12 a a DET iajs-3638 43 13 model model NOUN iajs-3638 43 14 of of ADP iajs-3638 43 15 the the DET iajs-3638 43 16 coronavirus coronavirus NOUN iajs-3638 43 17 pandemic pandemic ADJ iajs-3638 43 18 epidemic epidemic NOUN iajs-3638 43 19 . . PUNCT iajs-3638 44 1 zizhen zizhen PROPN iajs-3638 44 2 et et PROPN iajs-3638 44 3 al al PROPN iajs-3638 44 4 . . PROPN iajs-3638 45 1 ( ( PUNCT iajs-3638 45 2 13 13 NUM iajs-3638 45 3 ) ) PUNCT iajs-3638 45 4 suggested suggest VERB iajs-3638 45 5 svirs svir NOUN iajs-3638 45 6 epidemic epidemic NOUN iajs-3638 45 7 model model NOUN iajs-3638 45 8 includes include VERB iajs-3638 45 9 a a DET iajs-3638 45 10 number number NOUN iajs-3638 45 11 of of ADP iajs-3638 45 12 delays delay NOUN iajs-3638 45 13 along along ADP iajs-3638 45 14 with with ADP iajs-3638 45 15 incidence incidence NOUN iajs-3638 45 16 and and CCONJ iajs-3638 45 17 treatment treatment NOUN iajs-3638 45 18 rates rate NOUN iajs-3638 45 19 for for ADP iajs-3638 45 20 holling holle VERB iajs-3638 45 21 type type PROPN iajs-3638 45 22 ii ii PROPN iajs-3638 45 23 . . PUNCT iajs-3638 46 1 dehingia dehingia PROPN iajs-3638 46 2 et et PROPN iajs-3638 46 3 al al PROPN iajs-3638 46 4 . . PUNCT iajs-3638 47 1 ( ( PUNCT iajs-3638 47 2 14 14 NUM iajs-3638 47 3 ) ) PUNCT iajs-3638 47 4 investigated investigate VERB iajs-3638 47 5 the the DET iajs-3638 47 6 dynamic dynamic ADJ iajs-3638 47 7 behavior behavior NOUN iajs-3638 47 8 of of ADP iajs-3638 47 9 a a DET iajs-3638 47 10 sars sar NOUN iajs-3638 47 11 - - PUNCT iajs-3638 47 12 cov-2 cov-2 NOUN iajs-3638 47 13 within within ADP iajs-3638 47 14 - - PUNCT iajs-3638 47 15 host host NOUN iajs-3638 47 16 fractional fractional ADJ iajs-3638 47 17 order order NOUN iajs-3638 47 18 model model NOUN iajs-3638 47 19 . . PUNCT iajs-3638 48 1 shurowq shurowq PROPN iajs-3638 48 2 et et PROPN iajs-3638 48 3 al al PROPN iajs-3638 48 4 . . PROPN iajs-3638 49 1 ( ( PUNCT iajs-3638 49 2 15 15 NUM iajs-3638 49 3 ) ) PUNCT iajs-3638 49 4 study study NOUN iajs-3638 49 5 of of ADP iajs-3638 49 6 the the DET iajs-3638 49 7 covid-19 covid-19 PROPN iajs-3638 49 8 epidemic epidemic NOUN iajs-3638 49 9 and and CCONJ iajs-3638 49 10 the the DET iajs-3638 49 11 bifurcation bifurcation NOUN iajs-3638 49 12 analysis analysis NOUN iajs-3638 49 13 of of ADP iajs-3638 49 14 a a DET iajs-3638 49 15 mathematical mathematical ADJ iajs-3638 49 16 model model NOUN iajs-3638 49 17 for for ADP iajs-3638 49 18 vaccination vaccination NOUN iajs-3638 49 19 . . PUNCT iajs-3638 50 1 naji naji PROPN iajs-3638 50 2 and and CCONJ iajs-3638 50 3 hussien hussien PROPN iajs-3638 50 4 ( ( PUNCT iajs-3638 50 5 16 16 NUM iajs-3638 50 6 ) ) PUNCT iajs-3638 50 7 proposed propose VERB iajs-3638 50 8 and and CCONJ iajs-3638 50 9 analyzed analyze VERB iajs-3638 50 10 the the DET iajs-3638 50 11 epidemic epidemic NOUN iajs-3638 50 12 model model NOUN iajs-3638 50 13 type type NOUN iajs-3638 50 14 of of ADP iajs-3638 50 15 seir seir NOUN iajs-3638 50 16 with with ADP iajs-3638 50 17 nonlinear nonlinear ADJ iajs-3638 50 18 incidence incidence NOUN iajs-3638 50 19 and and CCONJ iajs-3638 50 20 treatment treatment NOUN iajs-3638 50 21 rates rate NOUN iajs-3638 50 22 and and CCONJ iajs-3638 50 23 also also ADV iajs-3638 50 24 used use VERB iajs-3638 50 25 time time NOUN iajs-3638 50 26 delays delay NOUN iajs-3638 50 27 owing owe VERB iajs-3638 50 28 to to ADP iajs-3638 50 29 the the DET iajs-3638 50 30 incubation incubation NOUN iajs-3638 50 31 period period NOUN iajs-3638 50 32 . . PUNCT iajs-3638 51 1 ahmed ahmed PROPN iajs-3638 51 2 et et PROPN iajs-3638 51 3 al al PROPN iajs-3638 51 4 . . PROPN iajs-3638 52 1 ( ( PUNCT iajs-3638 52 2 17 17 NUM iajs-3638 52 3 ) ) PUNCT iajs-3638 52 4 discussed discuss VERB iajs-3638 52 5 a a DET iajs-3638 52 6 mathematical mathematical ADJ iajs-3638 52 7 model model NOUN iajs-3638 52 8 for for ADP iajs-3638 52 9 the the DET iajs-3638 52 10 dynamics dynamic NOUN iajs-3638 52 11 of of ADP iajs-3638 52 12 covid-19 covid-19 PROPN iajs-3638 52 13 pandemic pandemic NOUN iajs-3638 52 14 involving involve VERB iajs-3638 52 15 infected infected ADJ iajs-3638 52 16 immigrants immigrant NOUN iajs-3638 52 17 . . PUNCT iajs-3638 53 1 naji naji PROPN iajs-3638 53 2 and and CCONJ iajs-3638 53 3 mohsen mohsen PROPN iajs-3638 54 1 [ [ X iajs-3638 54 2 18 18 NUM iajs-3638 54 3 ] ] PUNCT iajs-3638 54 4 studied study VERB iajs-3638 54 5 the the DET iajs-3638 54 6 stability stability NOUN iajs-3638 54 7 analysise analysise NOUN iajs-3638 54 8 with with ADP iajs-3638 54 9 the the DET iajs-3638 54 10 bifurcation bifurcation NOUN iajs-3638 54 11 of of ADP iajs-3638 54 12 an an DET iajs-3638 54 13 svire svire NOUN iajs-3638 54 14 epidemic epidemic NOUN iajs-3638 54 15 model model NOUN iajs-3638 54 16 involving involve VERB iajs-3638 54 17 immigrants immigrant NOUN iajs-3638 54 18 . . PUNCT iajs-3638 55 1 ahmed ahmed PROPN iajs-3638 55 2 and and CCONJ iajs-3638 55 3 alhusseiny alhusseiny PROPN iajs-3638 55 4 ( ( PUNCT iajs-3638 55 5 19 19 NUM iajs-3638 55 6 ) ) PUNCT iajs-3638 55 7 studied study VERB iajs-3638 55 8 the the DET iajs-3638 55 9 dynamical dynamical ADJ iajs-3638 55 10 behavior behavior NOUN iajs-3638 55 11 of of ADP iajs-3638 55 12 an an DET iajs-3638 55 13 eco eco NOUN iajs-3638 55 14 - - PUNCT iajs-3638 55 15 epidemiological epidemiological ADJ iajs-3638 55 16 model model NOUN iajs-3638 55 17 involving involve VERB iajs-3638 55 18 disease disease NOUN iajs-3638 55 19 in in ADP iajs-3638 55 20 predators predator NOUN iajs-3638 55 21 and and CCONJ iajs-3638 55 22 stage stage NOUN iajs-3638 55 23 structure structure NOUN iajs-3638 55 24 in in ADP iajs-3638 55 25 prey prey NOUN iajs-3638 55 26 . . PUNCT iajs-3638 56 1 mohsen mohsen PROPN iajs-3638 56 2 and and CCONJ iajs-3638 56 3 hattaf hattaf NOUN iajs-3638 56 4 ( ( PUNCT iajs-3638 56 5 20 20 NUM iajs-3638 56 6 ) ) PUNCT iajs-3638 56 7 studied study VERB iajs-3638 56 8 the the DET iajs-3638 56 9 dynamics dynamic NOUN iajs-3638 56 10 of of ADP iajs-3638 56 11 a a DET iajs-3638 56 12 generalized generalize VERB iajs-3638 56 13 fractional fractional ADJ iajs-3638 56 14 epidemic epidemic NOUN iajs-3638 56 15 model model NOUN iajs-3638 56 16 of of ADP iajs-3638 56 17 covid-19 covid-19 PROPN iajs-3638 56 18 with with ADP iajs-3638 56 19 carrier carrier NOUN iajs-3638 56 20 effect effect NOUN iajs-3638 56 21 . . PUNCT iajs-3638 57 1 this this DET iajs-3638 57 2 study study NOUN iajs-3638 57 3 presents present VERB iajs-3638 57 4 and and CCONJ iajs-3638 57 5 evaluates evaluate VERB iajs-3638 57 6 a a DET iajs-3638 57 7 mathematical mathematical ADJ iajs-3638 57 8 model model NOUN iajs-3638 57 9 that that PRON iajs-3638 57 10 represents represent VERB iajs-3638 57 11 the the DET iajs-3638 57 12 dynamics dynamic NOUN iajs-3638 57 13 of of ADP iajs-3638 57 14 the the DET iajs-3638 57 15 covid-19 covid-19 PROPN iajs-3638 57 16 pandemic pandemic NOUN iajs-3638 57 17 's 's PART iajs-3638 57 18 delayed delay VERB iajs-3638 57 19 infection infection NOUN iajs-3638 57 20 and and CCONJ iajs-3638 57 21 includes include VERB iajs-3638 57 22 two two NUM iajs-3638 57 23 stages stage NOUN iajs-3638 57 24 of of ADP iajs-3638 57 25 vaccination vaccination NOUN iajs-3638 57 26 . . PUNCT iajs-3638 58 1 this this DET iajs-3638 58 2 paper paper NOUN iajs-3638 58 3 is be AUX iajs-3638 58 4 organized organize VERB iajs-3638 58 5 as as SCONJ iajs-3638 58 6 follows follow VERB iajs-3638 58 7 : : PUNCT iajs-3638 58 8 section section NOUN iajs-3638 58 9 2 2 NUM iajs-3638 58 10 illustrates illustrate VERB iajs-3638 58 11 the the DET iajs-3638 58 12 innovative innovative ADJ iajs-3638 58 13 coronavirus coronavirus NOUN iajs-3638 58 14 mathematical mathematical ADJ iajs-3638 58 15 modeling modeling NOUN iajs-3638 58 16 and and CCONJ iajs-3638 58 17 the the DET iajs-3638 58 18 two two NUM iajs-3638 58 19 steps step NOUN iajs-3638 58 20 of of ADP iajs-3638 58 21 immunization immunization NOUN iajs-3638 58 22 with with ADP iajs-3638 58 23 delayed delay VERB iajs-3638 58 24 infection infection NOUN iajs-3638 58 25 . . PUNCT iajs-3638 59 1 local local ADJ iajs-3638 59 2 stability stability NOUN iajs-3638 59 3 and and CCONJ iajs-3638 59 4 h.b h.b PROPN iajs-3638 59 5 . . PROPN iajs-3638 59 6 are be AUX iajs-3638 59 7 discussed discuss VERB iajs-3638 59 8 in in ADP iajs-3638 59 9 section section NOUN iajs-3638 59 10 3 3 NUM iajs-3638 59 11 . . PUNCT iajs-3638 60 1 in in ADP iajs-3638 60 2 section section NOUN iajs-3638 60 3 4 4 NUM iajs-3638 60 4 , , PUNCT iajs-3638 60 5 a a DET iajs-3638 60 6 numerical numerical ADJ iajs-3638 60 7 simulation simulation NOUN iajs-3638 60 8 is be AUX iajs-3638 60 9 utilized utilize VERB iajs-3638 60 10 to to PART iajs-3638 60 11 analyze analyze VERB iajs-3638 60 12 the the DET iajs-3638 60 13 effects effect NOUN iajs-3638 60 14 of of ADP iajs-3638 60 15 altering alter VERB iajs-3638 60 16 every every DET iajs-3638 60 17 system system NOUN iajs-3638 60 18 parameter parameter NOUN iajs-3638 60 19 . . PUNCT iajs-3638 61 1 ihjpas ihjpas PROPN iajs-3638 61 2 . . PUNCT iajs-3638 62 1 2025 2025 NUM iajs-3638 62 2 , , PUNCT iajs-3638 62 3 38 38 NUM iajs-3638 62 4 ( ( PUNCT iajs-3638 62 5 1 1 NUM iajs-3638 62 6 ) ) PUNCT iajs-3638 62 7 447 447 NUM iajs-3638 62 8 2 2 NUM iajs-3638 62 9 . . PUNCT iajs-3638 62 10 mathematical mathematical ADJ iajs-3638 62 11 model model NOUN iajs-3638 62 12 : : PUNCT iajs-3638 62 13 see see VERB iajs-3638 62 14 in(21 in(21 PROPN iajs-3638 62 15 ) ) PUNCT iajs-3638 62 16 the the DET iajs-3638 62 17 system system NOUN iajs-3638 62 18 of of ADP iajs-3638 62 19 the the DET iajs-3638 62 20 time time NOUN iajs-3638 62 21 delayed delay VERB iajs-3638 62 22 is be AUX iajs-3638 62 23 : : PUNCT iajs-3638 62 24 𝑑𝑆 𝑑𝑆 ADJ iajs-3638 62 25 𝑑𝑑 𝑑𝑑 ADP iajs-3638 62 26 = = SYM iajs-3638 62 27 𝛬 𝛬 PROPN iajs-3638 62 28 βˆ’ βˆ’ PROPN iajs-3638 62 29 𝛼𝑆 𝛼𝑆 VERB iajs-3638 62 30 1+𝑛𝑉1 1+𝑛𝑉1 NUM iajs-3638 62 31 βˆ’ βˆ’ NOUN iajs-3638 62 32 𝛽1𝑆(𝑑 𝛽1𝑆(𝑑 ADJ iajs-3638 62 33 βˆ’ βˆ’ PROPN iajs-3638 62 34 𝜏)𝐼(𝑑 𝜏)𝐼(𝑑 NOUN iajs-3638 62 35 βˆ’ βˆ’ PROPN iajs-3638 62 36 𝜏 𝜏 NOUN iajs-3638 62 37 ) ) PUNCT iajs-3638 62 38 βˆ’ βˆ’ PROPN iajs-3638 62 39 πœ‡π‘† πœ‡π‘† PROPN iajs-3638 62 40 𝑑𝑉1 𝑑𝑉1 PROPN iajs-3638 62 41 𝑑𝑑 𝑑𝑑 ADP iajs-3638 62 42 = = PUNCT iajs-3638 62 43 𝛼𝑆 𝛼𝑆 PROPN iajs-3638 63 1 1+𝑛𝑉1 1+𝑛𝑉1 NUM iajs-3638 63 2 βˆ’ βˆ’ PROPN iajs-3638 63 3 𝛾𝑉1 𝛾𝑉1 PROPN iajs-3638 63 4 βˆ’ βˆ’ PROPN iajs-3638 63 5 𝛽2𝑉1𝐼 𝛽2𝑉1𝐼 PROPN iajs-3638 63 6 βˆ’ βˆ’ PROPN iajs-3638 63 7 πœ‡π‘‰1 πœ‡π‘‰1 PROPN iajs-3638 63 8 𝑑𝑉2 𝑑𝑉2 PROPN iajs-3638 63 9 𝑑𝑑 𝑑𝑑 ADP iajs-3638 63 10 = = PUNCT iajs-3638 63 11 𝛾𝑉1 𝛾𝑉1 PROPN iajs-3638 63 12 βˆ’ βˆ’ PROPN iajs-3638 63 13 𝛽3𝑉2𝐼 𝛽3𝑉2𝐼 NOUN iajs-3638 63 14 βˆ’ βˆ’ PROPN iajs-3638 64 1 πœ‡π‘‰2 πœ‡π‘‰2 PROPN iajs-3638 64 2 𝑑𝐼 𝑑𝐼 ADJ iajs-3638 64 3 𝑑𝑑 𝑑𝑑 ADP iajs-3638 64 4 = = SYM iajs-3638 64 5 𝛽1𝑆(𝑑 𝛽1𝑆(𝑑 NOUN iajs-3638 64 6 βˆ’ βˆ’ PROPN iajs-3638 64 7 𝜏)𝐼(𝑑 𝜏)𝐼(𝑑 NOUN iajs-3638 64 8 βˆ’ βˆ’ PROPN iajs-3638 64 9 𝜏 𝜏 NOUN iajs-3638 64 10 ) ) PUNCT iajs-3638 64 11 + + CCONJ iajs-3638 64 12 𝛽2𝑉1𝐼 𝛽2𝑉1𝐼 PROPN iajs-3638 64 13 + + CCONJ iajs-3638 64 14 𝛽3𝑉2𝐼 𝛽3𝑉2𝐼 PROPN iajs-3638 64 15 βˆ’ βˆ’ PROPN iajs-3638 64 16 ( ( PUNCT iajs-3638 64 17 πœ‡ πœ‡ X iajs-3638 64 18 + + X iajs-3638 64 19 πœ‡1)𝐼 πœ‡1)𝐼 DET iajs-3638 64 20 βˆ’ βˆ’ PROPN iajs-3638 64 21 πœƒπΌ πœƒπΌ ADJ iajs-3638 64 22 𝑑𝑅 𝑑𝑅 VERB iajs-3638 64 23 𝑑𝑑 𝑑𝑑 ADP iajs-3638 64 24 = = PUNCT iajs-3638 64 25 πœƒπΌ πœƒπΌ ADJ iajs-3638 64 26 βˆ’ βˆ’ PROPN iajs-3638 64 27 πœ‡π‘… πœ‡π‘… NOUN iajs-3638 64 28 ( ( PUNCT iajs-3638 64 29 1 1 X iajs-3638 64 30 ) ) PUNCT iajs-3638 64 31 now now ADV iajs-3638 64 32 , , PUNCT iajs-3638 64 33 see(21 see(21 PROPN iajs-3638 64 34 ) ) PUNCT iajs-3638 64 35 we we PRON iajs-3638 64 36 get get VERB iajs-3638 64 37 ; ; PUNCT iajs-3638 64 38 ( ( PUNCT iajs-3638 64 39 𝑑 𝑑 NOUN iajs-3638 64 40 ) ) PUNCT iajs-3638 64 41 = = SYM iajs-3638 64 42 πœƒπΌ πœƒπΌ ADV iajs-3638 64 43 πœ‡ πœ‡ X iajs-3638 64 44 , , PUNCT iajs-3638 64 45 ( ( PUNCT iajs-3638 64 46 2 2 NUM iajs-3638 64 47 ) ) PUNCT iajs-3638 64 48 as as ADP iajs-3638 64 49 a a DET iajs-3638 64 50 result result NOUN iajs-3638 64 51 , , PUNCT iajs-3638 64 52 the the DET iajs-3638 64 53 system system NOUN iajs-3638 64 54 below below ADV iajs-3638 64 55 will will AUX iajs-3638 64 56 be be AUX iajs-3638 64 57 studied study VERB iajs-3638 64 58 rather rather ADV iajs-3638 64 59 than than ADP iajs-3638 64 60 system system NOUN iajs-3638 64 61 ( ( PUNCT iajs-3638 64 62 1 1 NUM iajs-3638 64 63 ) ) PUNCT iajs-3638 64 64 ; ; PUNCT iajs-3638 64 65 𝑑𝑆 𝑑𝑆 ADJ iajs-3638 64 66 𝑑𝑑 𝑑𝑑 ADP iajs-3638 64 67 = = SYM iajs-3638 64 68 Ξ» Ξ» PROPN iajs-3638 64 69 βˆ’ βˆ’ PROPN iajs-3638 64 70 𝛼𝑆 𝛼𝑆 NOUN iajs-3638 64 71 1+𝑛𝑉1 1+𝑛𝑉1 NUM iajs-3638 64 72 βˆ’ βˆ’ NOUN iajs-3638 64 73 𝛽1𝑆(𝑑 𝛽1𝑆(𝑑 ADJ iajs-3638 64 74 βˆ’ βˆ’ PROPN iajs-3638 64 75 𝜏)𝐼(𝑑 𝜏)𝐼(𝑑 NOUN iajs-3638 64 76 βˆ’ βˆ’ PROPN iajs-3638 64 77 𝜏 𝜏 NOUN iajs-3638 64 78 ) ) PUNCT iajs-3638 64 79 βˆ’ βˆ’ PROPN iajs-3638 64 80 πœ‡π‘† πœ‡π‘† PROPN iajs-3638 64 81 𝑑𝑉1 𝑑𝑉1 PROPN iajs-3638 64 82 𝑑𝑑 𝑑𝑑 ADP iajs-3638 64 83 = = PUNCT iajs-3638 64 84 𝛼𝑆 𝛼𝑆 PROPN iajs-3638 64 85 1+𝑛𝑉1 1+𝑛𝑉1 NUM iajs-3638 64 86 βˆ’ βˆ’ PROPN iajs-3638 64 87 𝛾𝑉1 𝛾𝑉1 PROPN iajs-3638 64 88 βˆ’ βˆ’ PROPN iajs-3638 64 89 𝛽2𝑉1𝐼 𝛽2𝑉1𝐼 PROPN iajs-3638 64 90 βˆ’ βˆ’ PROPN iajs-3638 64 91 πœ‡π‘‰1 πœ‡π‘‰1 PROPN iajs-3638 64 92 𝑑𝑉2 𝑑𝑉2 PROPN iajs-3638 64 93 𝑑𝑑 𝑑𝑑 ADP iajs-3638 64 94 = = PUNCT iajs-3638 64 95 𝛾𝑉1 𝛾𝑉1 PROPN iajs-3638 64 96 βˆ’ βˆ’ PROPN iajs-3638 64 97 𝛽3𝑉2𝐼 𝛽3𝑉2𝐼 NOUN iajs-3638 64 98 βˆ’ βˆ’ PROPN iajs-3638 64 99 πœ‡π‘‰2 πœ‡π‘‰2 PROPN iajs-3638 64 100 𝑑𝐼 𝑑𝐼 ADJ iajs-3638 64 101 𝑑𝑑 𝑑𝑑 ADP iajs-3638 64 102 = = SYM iajs-3638 64 103 𝛽1𝑆(𝑑 𝛽1𝑆(𝑑 NOUN iajs-3638 64 104 βˆ’ βˆ’ PROPN iajs-3638 64 105 𝜏)𝐼(𝑑 𝜏)𝐼(𝑑 NOUN iajs-3638 64 106 βˆ’ βˆ’ PROPN iajs-3638 64 107 𝜏 𝜏 NOUN iajs-3638 64 108 ) ) PUNCT iajs-3638 64 109 + + CCONJ iajs-3638 64 110 𝛽2𝑉1𝐼 𝛽2𝑉1𝐼 PROPN iajs-3638 64 111 + + CCONJ iajs-3638 64 112 𝛽3𝑉2𝐼 𝛽3𝑉2𝐼 PROPN iajs-3638 64 113 βˆ’ βˆ’ PROPN iajs-3638 64 114 ( ( PUNCT iajs-3638 64 115 πœ‡ πœ‡ X iajs-3638 64 116 + + X iajs-3638 64 117 πœ‡1)𝐼 πœ‡1)𝐼 DET iajs-3638 64 118 βˆ’ βˆ’ PROPN iajs-3638 64 119 πœƒπΌ πœƒπΌ ADJ iajs-3638 64 120 ( ( PUNCT iajs-3638 64 121 3 3 NUM iajs-3638 64 122 ) ) PUNCT iajs-3638 64 123 3 3 NUM iajs-3638 64 124 . . PUNCT iajs-3638 64 125 local local ADJ iajs-3638 64 126 stability stability NOUN iajs-3638 64 127 analysis analysis NOUN iajs-3638 64 128 ( ( PUNCT iajs-3638 64 129 l.s.a l.s.a NOUN iajs-3638 64 130 . . PUNCT iajs-3638 64 131 ) ) PUNCT iajs-3638 64 132 and and CCONJ iajs-3638 64 133 hopf hopf ADJ iajs-3638 64 134 bifurcation bifurcation NOUN iajs-3638 64 135 ( ( PUNCT iajs-3638 64 136 h.b h.b PROPN iajs-3638 64 137 . . PROPN iajs-3638 64 138 ) ) PUNCT iajs-3638 64 139 in in ADP iajs-3638 64 140 this this DET iajs-3638 64 141 section section NOUN iajs-3638 64 142 , , PUNCT iajs-3638 64 143 the the DET iajs-3638 64 144 l.s l.s PROPN iajs-3638 64 145 . . PROPN iajs-3638 64 146 and and CCONJ iajs-3638 64 147 h.b h.b PROPN iajs-3638 64 148 . . PROPN iajs-3638 64 149 of of ADP iajs-3638 64 150 system system NOUN iajs-3638 64 151 ( ( PUNCT iajs-3638 64 152 3 3 X iajs-3638 64 153 ) ) PUNCT iajs-3638 64 154 are be AUX iajs-3638 64 155 studied study VERB iajs-3638 64 156 . . PUNCT iajs-3638 65 1 the the DET iajs-3638 65 2 position position NOUN iajs-3638 65 3 and and CCONJ iajs-3638 65 4 quantity quantity NOUN iajs-3638 65 5 of of ADP iajs-3638 65 6 equilibrium equilibrium NOUN iajs-3638 65 7 points point NOUN iajs-3638 65 8 are be AUX iajs-3638 65 9 known know VERB iajs-3638 65 10 and and CCONJ iajs-3638 65 11 do do AUX iajs-3638 65 12 n't not PART iajs-3638 65 13 alter alter VERB iajs-3638 65 14 with with ADP iajs-3638 65 15 time time NOUN iajs-3638 65 16 delays delay NOUN iajs-3638 65 17 . . PUNCT iajs-3638 66 1 accordingly accordingly ADV iajs-3638 66 2 , , PUNCT iajs-3638 66 3 from from ADP iajs-3638 66 4 ( ( PUNCT iajs-3638 66 5 21 21 NUM iajs-3638 66 6 ) ) PUNCT iajs-3638 66 7 system system NOUN iajs-3638 66 8 ( ( PUNCT iajs-3638 66 9 3 3 X iajs-3638 66 10 ) ) PUNCT iajs-3638 66 11 have have VERB iajs-3638 66 12 six six NUM iajs-3638 66 13 e.ps e.ps PROPN iajs-3638 66 14 . . PROPN iajs-3638 66 15 , , PUNCT iajs-3638 66 16 say say VERB iajs-3638 66 17 𝐸0 𝐸0 PROPN iajs-3638 66 18 = = SYM iajs-3638 66 19 ( ( PUNCT iajs-3638 66 20 𝑆,Μ… 𝑆,Μ… NOUN iajs-3638 66 21 0,0,0 0,0,0 NOUN iajs-3638 66 22 ) ) PUNCT iajs-3638 66 23 π‘€β„Žπ‘’π‘› π‘€β„Žπ‘’π‘› VERB iajs-3638 66 24 𝛼 𝛼 NOUN iajs-3638 66 25 = = SYM iajs-3638 66 26 0 0 NUM iajs-3638 66 27 , , PUNCT iajs-3638 66 28 𝐸1 𝐸1 NOUN iajs-3638 66 29 = = SYM iajs-3638 66 30 ( ( PUNCT iajs-3638 66 31 𝑆̿ 𝑆̿ PROPN iajs-3638 66 32 , , PUNCT iajs-3638 66 33 οΏ½ οΏ½ PROPN iajs-3638 66 34 ΜΏ ΜΏ NOUN iajs-3638 66 35 οΏ½ οΏ½ PROPN iajs-3638 66 36 1 1 NUM iajs-3638 66 37 , , PUNCT iajs-3638 66 38 0,0 0,0 NOUN iajs-3638 66 39 ) ) PUNCT iajs-3638 66 40 π‘€β„Žπ‘’π‘› π‘€β„Žπ‘’π‘› VERB iajs-3638 66 41 𝛾 𝛾 ADP iajs-3638 66 42 = = SYM iajs-3638 66 43 0 0 NUM iajs-3638 66 44 , , PUNCT iajs-3638 66 45 𝐸2 𝐸2 PROPN iajs-3638 66 46 = = SYM iajs-3638 66 47 ( ( PUNCT iajs-3638 66 48 οΏ½ οΏ½ PROPN iajs-3638 66 49 Μ‚ Μ‚ SYM iajs-3638 66 50 οΏ½ οΏ½ PROPN iajs-3638 66 51 , , PUNCT iajs-3638 66 52 0,0 0,0 NOUN iajs-3638 66 53 , , PUNCT iajs-3638 66 54 𝐼 𝐼 NOUN iajs-3638 66 55 ) ) PUNCT iajs-3638 66 56 π‘€β„Žπ‘’π‘› π‘€β„Žπ‘’π‘› VERB iajs-3638 66 57 𝛼 𝛼 NOUN iajs-3638 66 58 = = NOUN iajs-3638 66 59 0 0 NUM iajs-3638 66 60 , , PUNCT iajs-3638 66 61 𝐸3 𝐸3 NOUN iajs-3638 66 62 = = SYM iajs-3638 66 63 ( ( PUNCT iajs-3638 66 64 οΏ½ οΏ½ PROPN iajs-3638 66 65 Μ† Μ† PROPN iajs-3638 66 66 οΏ½ οΏ½ PROPN iajs-3638 66 67 , , PUNCT iajs-3638 66 68 οΏ½ οΏ½ PROPN iajs-3638 66 69 Μ† Μ† NOUN iajs-3638 66 70 οΏ½ οΏ½ PROPN iajs-3638 66 71 1 1 NUM iajs-3638 66 72 , , PUNCT iajs-3638 66 73 οΏ½ οΏ½ PROPN iajs-3638 66 74 Μ† Μ† NOUN iajs-3638 66 75 οΏ½ οΏ½ PROPN iajs-3638 66 76 2 2 NUM iajs-3638 66 77 , , PUNCT iajs-3638 66 78 0 0 NUM iajs-3638 66 79 ) ) PUNCT iajs-3638 66 80 , , PUNCT iajs-3638 66 81 𝐸4 𝐸4 NOUN iajs-3638 66 82 = = SYM iajs-3638 66 83 ( ( PUNCT iajs-3638 66 84 οΏ½ οΏ½ PROPN iajs-3638 66 85 ΜƒΜƒ ΜƒΜƒ PROPN iajs-3638 66 86 οΏ½ οΏ½ PROPN iajs-3638 66 87 , , PUNCT iajs-3638 66 88 οΏ½ οΏ½ PROPN iajs-3638 66 89 ΜƒΜƒ ΜƒΜƒ PROPN iajs-3638 66 90 οΏ½ οΏ½ PROPN iajs-3638 66 91 1 1 NUM iajs-3638 66 92 , , PUNCT iajs-3638 66 93 0 0 NUM iajs-3638 66 94 , , PUNCT iajs-3638 66 95 𝐼 𝐼 NOUN iajs-3638 66 96 ) ) PUNCT iajs-3638 66 97 π‘€β„Žπ‘’π‘› π‘€β„Žπ‘’π‘› VERB iajs-3638 66 98 𝛾 𝛾 ADP iajs-3638 66 99 = = SYM iajs-3638 66 100 0 0 NUM iajs-3638 66 101 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ iajs-3638 66 102 𝐸5 𝐸5 NOUN iajs-3638 66 103 = = PUNCT iajs-3638 66 104 ( ( PUNCT iajs-3638 66 105 π‘†βˆ— π‘†βˆ— X iajs-3638 66 106 , , PUNCT iajs-3638 66 107 𝑉1 𝑉1 NOUN iajs-3638 66 108 βˆ— βˆ— NOUN iajs-3638 66 109 , , PUNCT iajs-3638 66 110 𝑉2 𝑉2 PROPN iajs-3638 66 111 βˆ— βˆ— PROPN iajs-3638 66 112 , , PUNCT iajs-3638 66 113 πΌβˆ— πΌβˆ— PROPN iajs-3638 66 114 ) ) PUNCT iajs-3638 66 115 . . PUNCT iajs-3638 67 1 it it PRON iajs-3638 67 2 is be AUX iajs-3638 67 3 well well ADV iajs-3638 67 4 known know VERB iajs-3638 67 5 that that SCONJ iajs-3638 67 6 , , PUNCT iajs-3638 67 7 the the DET iajs-3638 67 8 jacobian jacobian ADJ iajs-3638 67 9 matrix(j.m matrix(j.m NOUN iajs-3638 67 10 ) ) PUNCT iajs-3638 67 11 of of ADP iajs-3638 67 12 system system NOUN iajs-3638 67 13 ( ( PUNCT iajs-3638 67 14 3 3 NUM iajs-3638 67 15 ) ) PUNCT iajs-3638 67 16 at at ADP iajs-3638 67 17 any any DET iajs-3638 67 18 e.ps e.ps PROPN iajs-3638 67 19 . . PUNCT iajs-3638 67 20 𝐸 𝐸 PROPN iajs-3638 67 21 = = SYM iajs-3638 67 22 ( ( PUNCT iajs-3638 67 23 𝑆 𝑆 PROPN iajs-3638 67 24 , , PUNCT iajs-3638 67 25 𝑉1 𝑉1 PROPN iajs-3638 67 26 , , PUNCT iajs-3638 67 27 𝑉2 𝑉2 NOUN iajs-3638 67 28 , , PUNCT iajs-3638 67 29 𝐼 𝐼 PROPN iajs-3638 67 30 ) ) PUNCT iajs-3638 67 31 is be AUX iajs-3638 67 32 𝐽(𝐸 𝐽(𝐸 NOUN iajs-3638 67 33 ) ) PUNCT iajs-3638 67 34 = = NOUN iajs-3638 67 35 ( ( PUNCT iajs-3638 67 36 π‘Žπ‘–π‘—)4Γ—4 π‘Žπ‘–π‘—)4Γ—4 PUNCT iajs-3638 67 37 ; ; PUNCT iajs-3638 67 38 𝑖 𝑖 X iajs-3638 67 39 , , PUNCT iajs-3638 67 40 𝑗 𝑗 NOUN iajs-3638 67 41 = = ADJ iajs-3638 67 42 1,2,3,4 1,2,3,4 NUM iajs-3638 67 43 . . PUNCT iajs-3638 68 1 where where SCONJ iajs-3638 68 2 π‘Ž11 π‘Ž11 VERB iajs-3638 68 3 = = SYM iajs-3638 68 4 βˆ’ βˆ’ PROPN iajs-3638 68 5 [ [ PUNCT iajs-3638 68 6 𝛼 𝛼 PROPN iajs-3638 68 7 1+𝑛𝑉1 1+𝑛𝑉1 NUM iajs-3638 68 8 + + NUM iajs-3638 68 9 𝛽1𝐼𝑒 𝛽1𝐼𝑒 PROPN iajs-3638 68 10 βˆ’πœ†πœ βˆ’πœ†πœ NUM iajs-3638 68 11 + + CCONJ iajs-3638 68 12 πœ‡ πœ‡ X iajs-3638 68 13 ] ] X iajs-3638 68 14 , , PUNCT iajs-3638 68 15 π‘Ž12 π‘Ž12 NOUN iajs-3638 68 16 = = SYM iajs-3638 68 17 𝑛𝛼𝑆 𝑛𝛼𝑆 PROPN iajs-3638 68 18 ( ( PUNCT iajs-3638 68 19 1+𝑛𝑉1)2 1+𝑛𝑉1)2 NUM iajs-3638 68 20 , , PUNCT iajs-3638 68 21 π‘Ž14 π‘Ž14 NOUN iajs-3638 68 22 = = SYM iajs-3638 69 1 βˆ’π›½1𝑆𝑒 βˆ’π›½1𝑆𝑒 NOUN iajs-3638 69 2 βˆ’πœ†πœ βˆ’πœ†πœ PROPN iajs-3638 69 3 , , PUNCT iajs-3638 69 4 π‘Ž21 π‘Ž21 NOUN iajs-3638 69 5 = = SYM iajs-3638 69 6 𝛼 𝛼 PROPN iajs-3638 69 7 1+𝑛𝑉1 1+𝑛𝑉1 NUM iajs-3638 69 8 , , PUNCT iajs-3638 69 9 π‘Ž22 π‘Ž22 NOUN iajs-3638 69 10 = = SYM iajs-3638 69 11 βˆ’ βˆ’ PROPN iajs-3638 69 12 [ [ PUNCT iajs-3638 69 13 𝑛𝛼𝑆 𝑛𝛼𝑆 PROPN iajs-3638 69 14 ( ( PUNCT iajs-3638 69 15 1+𝑛𝑉1)2 1+𝑛𝑉1)2 NUM iajs-3638 69 16 + + CCONJ iajs-3638 69 17 𝛾 𝛾 X iajs-3638 69 18 + + PUNCT iajs-3638 69 19 𝛽2𝐼 𝛽2𝐼 ADV iajs-3638 69 20 + + CCONJ iajs-3638 69 21 πœ‡ πœ‡ X iajs-3638 69 22 ] ] X iajs-3638 69 23 , , PUNCT iajs-3638 69 24 π‘Ž24 π‘Ž24 PROPN iajs-3638 69 25 = = SYM iajs-3638 69 26 βˆ’π›½2𝑉1 βˆ’π›½2𝑉1 PROPN iajs-3638 69 27 , , PUNCT iajs-3638 69 28 π‘Ž32 π‘Ž32 NOUN iajs-3638 69 29 = = SYM iajs-3638 69 30 𝛾 𝛾 NOUN iajs-3638 69 31 , , PUNCT iajs-3638 69 32 π‘Ž33 π‘Ž33 VERB iajs-3638 69 33 = = SYM iajs-3638 69 34 βˆ’[𝛽3𝐼 βˆ’[𝛽3𝐼 PROPN iajs-3638 69 35 + + CCONJ iajs-3638 69 36 πœ‡ πœ‡ X iajs-3638 69 37 ] ] X iajs-3638 69 38 , , PUNCT iajs-3638 69 39 π‘Ž34 π‘Ž34 NOUN iajs-3638 69 40 = = SYM iajs-3638 69 41 βˆ’π›½3𝑉2 βˆ’π›½3𝑉2 PROPN iajs-3638 69 42 , , PUNCT iajs-3638 69 43 π‘Ž41 π‘Ž41 PROPN iajs-3638 69 44 = = PUNCT iajs-3638 69 45 𝛽1𝐼𝑒 𝛽1𝐼𝑒 PROPN iajs-3638 69 46 βˆ’πœ†πœ βˆ’πœ†πœ NUM iajs-3638 69 47 , , PUNCT iajs-3638 69 48 π‘Ž42 π‘Ž42 NOUN iajs-3638 69 49 = = PUNCT iajs-3638 69 50 𝛽2𝐼 𝛽2𝐼 ADV iajs-3638 69 51 , , PUNCT iajs-3638 69 52 π‘Ž43 π‘Ž43 ADJ iajs-3638 69 53 = = SYM iajs-3638 69 54 𝛽3𝐼 𝛽3𝐼 ADJ iajs-3638 69 55 , , PUNCT iajs-3638 69 56 π‘Ž13 π‘Ž13 VERB iajs-3638 69 57 = = SYM iajs-3638 69 58 π‘Ž23 π‘Ž23 NOUN iajs-3638 69 59 = = NOUN iajs-3638 69 60 π‘Ž31 π‘Ž31 NOUN iajs-3638 69 61 = = SYM iajs-3638 69 62 0 0 NUM iajs-3638 69 63 , , PUNCT iajs-3638 69 64 π‘Ž44 π‘Ž44 NOUN iajs-3638 69 65 = = SYM iajs-3638 69 66 𝛽1𝑆𝑒 𝛽1𝑆𝑒 NUM iajs-3638 69 67 βˆ’πœ†πœ βˆ’πœ†πœ PROPN iajs-3638 69 68 + + CCONJ iajs-3638 69 69 𝛽2𝑉1 𝛽2𝑉1 PROPN iajs-3638 69 70 + + CCONJ iajs-3638 69 71 𝛽3𝑉2 𝛽3𝑉2 PROPN iajs-3638 69 72 βˆ’ βˆ’ NOUN iajs-3638 69 73 ( ( PUNCT iajs-3638 69 74 πœ‡ πœ‡ ADP iajs-3638 69 75 + + X iajs-3638 69 76 πœ‡1 πœ‡1 ADJ iajs-3638 69 77 + + CCONJ iajs-3638 69 78 πœƒ πœƒ NOUN iajs-3638 69 79 ) ) PUNCT iajs-3638 69 80 ( ( PUNCT iajs-3638 69 81 4 4 NUM iajs-3638 69 82 ) ) PUNCT iajs-3638 69 83 … … PUNCT iajs-3638 69 84 . . PUNCT iajs-3638 70 1 while while SCONJ iajs-3638 70 2 its its PRON iajs-3638 70 3 associated associated ADJ iajs-3638 70 4 characteristic characteristic ADJ iajs-3638 70 5 equation(c.e equation(c.e PROPN iajs-3638 70 6 . . PUNCT iajs-3638 70 7 ) ) PUNCT iajs-3638 71 1 takes take VERB iajs-3638 71 2 the the DET iajs-3638 71 3 form form NOUN iajs-3638 71 4 ihjpas ihjpa NOUN iajs-3638 71 5 . . PUNCT iajs-3638 72 1 2025 2025 NUM iajs-3638 72 2 , , PUNCT iajs-3638 72 3 38 38 NUM iajs-3638 72 4 ( ( PUNCT iajs-3638 72 5 1 1 NUM iajs-3638 72 6 ) ) PUNCT iajs-3638 72 7 448 448 NUM iajs-3638 72 8 𝑃(πœ† 𝑃(πœ† NOUN iajs-3638 72 9 ) ) PUNCT iajs-3638 72 10 + + NUM iajs-3638 72 11 𝑄(πœ†)π‘’βˆ’πœ†πœ 𝑄(πœ†)π‘’βˆ’πœ†πœ X iajs-3638 73 1 = = SYM iajs-3638 73 2 0 0 PUNCT iajs-3638 73 3 ( ( PUNCT iajs-3638 73 4 5 5 NUM iajs-3638 73 5 ) ) PUNCT iajs-3638 73 6 here here ADV iajs-3638 73 7 𝑃(πœ† 𝑃(πœ† CCONJ iajs-3638 73 8 ) ) PUNCT iajs-3638 73 9 and and CCONJ iajs-3638 73 10 𝑄(πœ† 𝑄(πœ† NOUN iajs-3638 73 11 ) ) PUNCT iajs-3638 73 12 are be AUX iajs-3638 73 13 polynomials polynomial NOUN iajs-3638 73 14 of of ADP iajs-3638 73 15 πœ†. πœ†. NOUN iajs-3638 73 16 accordingly accordingly ADV iajs-3638 73 17 the the DET iajs-3638 73 18 l.s l.s PROPN iajs-3638 73 19 . . NOUN iajs-3638 73 20 properties property NOUN iajs-3638 73 21 of of ADP iajs-3638 73 22 system system NOUN iajs-3638 73 23 ( ( PUNCT iajs-3638 73 24 3 3 X iajs-3638 73 25 ) ) PUNCT iajs-3638 73 26 at at ADP iajs-3638 73 27 all all ADV iajs-3638 73 28 feasible feasible ADJ iajs-3638 73 29 e.ps e.ps PROPN iajs-3638 73 30 . . PROPN iajs-3638 73 31 are be AUX iajs-3638 73 32 determined determine VERB iajs-3638 73 33 by by ADP iajs-3638 73 34 the the DET iajs-3638 73 35 roots root NOUN iajs-3638 73 36 of of ADP iajs-3638 73 37 the the DET iajs-3638 73 38 above above ADJ iajs-3638 73 39 equation equation NOUN iajs-3638 73 40 for for ADP iajs-3638 73 41 all all DET iajs-3638 73 42 𝜏 𝜏 PRON iajs-3638 73 43 β‰₯ β‰₯ NOUN iajs-3638 73 44 0 0 NUM iajs-3638 73 45 . . NOUN iajs-3638 73 46 β€’ β€’ NUM iajs-3638 73 47 for for ADP iajs-3638 73 48 the the DET iajs-3638 73 49 ( ( PUNCT iajs-3638 73 50 fep fep NOUN iajs-3638 73 51 ) ) PUNCT iajs-3638 73 52 when when SCONJ iajs-3638 73 53 𝛼 𝛼 X iajs-3638 73 54 = = SYM iajs-3638 73 55 0 0 NUM iajs-3638 73 56 , , PUNCT iajs-3638 73 57 equation equation NOUN iajs-3638 73 58 ( ( PUNCT iajs-3638 73 59 4 4 NUM iajs-3638 73 60 ) ) PUNCT iajs-3638 73 61 reduces reduce VERB iajs-3638 73 62 to to PART iajs-3638 73 63 𝐽(𝐸0 𝐽(𝐸0 VERB iajs-3638 73 64 ) ) PUNCT iajs-3638 73 65 = = PUNCT iajs-3638 74 1 [ [ PUNCT iajs-3638 74 2 βˆ’πœ‡ βˆ’πœ‡ NUM iajs-3638 74 3 0 0 NUM iajs-3638 74 4 0 0 NUM iajs-3638 74 5 βˆ’π›½1𝑆̅𝑒 βˆ’π›½1𝑆̅𝑒 NOUN iajs-3638 74 6 βˆ’πœ†πœ βˆ’πœ†πœ PROPN iajs-3638 74 7 0 0 NUM iajs-3638 74 8 βˆ’(𝛾 βˆ’(𝛾 NOUN iajs-3638 74 9 + + CCONJ iajs-3638 74 10 πœ‡ πœ‡ X iajs-3638 74 11 ) ) PUNCT iajs-3638 74 12 0 0 NUM iajs-3638 74 13 0 0 NUM iajs-3638 74 14 0 0 NUM iajs-3638 74 15 𝛾 𝛾 AUX iajs-3638 74 16 βˆ’πœ‡ βˆ’πœ‡ VERB iajs-3638 74 17 0 0 NUM iajs-3638 74 18 0 0 NUM iajs-3638 74 19 0 0 NUM iajs-3638 74 20 0 0 NUM iajs-3638 74 21 𝛽1𝑆̅𝑒 𝛽1𝑆̅𝑒 NOUN iajs-3638 74 22 βˆ’πœ†πœ βˆ’πœ†πœ PROPN iajs-3638 74 23 βˆ’ βˆ’ PROPN iajs-3638 75 1 ( ( PUNCT iajs-3638 75 2 πœ‡ πœ‡ ADP iajs-3638 75 3 + + X iajs-3638 75 4 πœ‡1 πœ‡1 ADJ iajs-3638 75 5 + + CCONJ iajs-3638 75 6 πœƒ πœƒ NOUN iajs-3638 75 7 ) ) PUNCT iajs-3638 75 8 ] ] PUNCT iajs-3638 75 9 ( ( PUNCT iajs-3638 75 10 6 6 X iajs-3638 75 11 ) ) PUNCT iajs-3638 75 12 the the DET iajs-3638 75 13 ( ( PUNCT iajs-3638 75 14 c.e c.e PROPN iajs-3638 75 15 . . PROPN iajs-3638 75 16 ) ) PUNCT iajs-3638 75 17 of of ADP iajs-3638 75 18 𝐽(𝐸0 𝐽(𝐸0 X iajs-3638 75 19 ) ) PUNCT iajs-3638 75 20 is be AUX iajs-3638 75 21 : : PUNCT iajs-3638 75 22 ( ( PUNCT iajs-3638 75 23 𝛽1𝑆̅𝑒 𝛽1𝑆̅𝑒 NOUN iajs-3638 75 24 βˆ’πœ†πœ βˆ’πœ†πœ NUM iajs-3638 75 25 βˆ’ βˆ’ PROPN iajs-3638 75 26 ( ( PUNCT iajs-3638 75 27 πœ‡ πœ‡ ADP iajs-3638 75 28 + + X iajs-3638 75 29 πœ‡1 πœ‡1 ADJ iajs-3638 75 30 + + CCONJ iajs-3638 75 31 πœƒ πœƒ NOUN iajs-3638 75 32 ) ) PUNCT iajs-3638 75 33 βˆ’ βˆ’ PROPN iajs-3638 75 34 πœ†)(βˆ’πœ‡ πœ†)(βˆ’πœ‡ NOUN iajs-3638 76 1 βˆ’ βˆ’ PROPN iajs-3638 76 2 πœ†)(βˆ’(𝛾 πœ†)(βˆ’(𝛾 PROPN iajs-3638 76 3 + + CCONJ iajs-3638 76 4 πœ‡ πœ‡ NOUN iajs-3638 76 5 ) ) PUNCT iajs-3638 76 6 βˆ’ βˆ’ PROPN iajs-3638 76 7 πœ†)(βˆ’πœ‡ πœ†)(βˆ’πœ‡ NOUN iajs-3638 76 8 βˆ’ βˆ’ NOUN iajs-3638 76 9 πœ† πœ† NOUN iajs-3638 76 10 ) ) PUNCT iajs-3638 76 11 = = SYM iajs-3638 76 12 0 0 NUM iajs-3638 76 13 ( ( PUNCT iajs-3638 76 14 6a 6a NOUN iajs-3638 76 15 ) ) PUNCT iajs-3638 76 16 equation equation NOUN iajs-3638 76 17 ( ( PUNCT iajs-3638 76 18 6a 6a NOUN iajs-3638 76 19 ) ) PUNCT iajs-3638 76 20 represents represent VERB iajs-3638 76 21 the the DET iajs-3638 76 22 eigenvalues eigenvalue NOUN iajs-3638 76 23 of of ADP iajs-3638 76 24 𝐽(𝐸0 𝐽(𝐸0 NOUN iajs-3638 76 25 ) ) PUNCT iajs-3638 76 26 and and CCONJ iajs-3638 76 27 has have VERB iajs-3638 76 28 4 4 NUM iajs-3638 76 29 - - PUNCT iajs-3638 76 30 roots root NOUN iajs-3638 76 31 : : PUNCT iajs-3638 76 32 πœ†1 πœ†1 NOUN iajs-3638 76 33 = = PUNCT iajs-3638 76 34 𝛽1𝑆̅𝑒 𝛽1𝑆̅𝑒 NOUN iajs-3638 76 35 βˆ’πœ†πœ βˆ’πœ†πœ PROPN iajs-3638 76 36 βˆ’ βˆ’ PROPN iajs-3638 76 37 ( ( PUNCT iajs-3638 76 38 πœ‡ πœ‡ ADP iajs-3638 76 39 + + X iajs-3638 76 40 πœ‡1 πœ‡1 ADJ iajs-3638 76 41 + + CCONJ iajs-3638 76 42 πœƒ πœƒ NOUN iajs-3638 76 43 ) ) PUNCT iajs-3638 76 44 πœ†2 πœ†2 NOUN iajs-3638 76 45 = = PUNCT iajs-3638 76 46 βˆ’πœ‡ βˆ’πœ‡ VERB iajs-3638 76 47 πœ†3 πœ†3 NOUN iajs-3638 76 48 = = NOUN iajs-3638 77 1 βˆ’(𝛾 βˆ’(𝛾 PROPN iajs-3638 77 2 + + CCONJ iajs-3638 77 3 πœ‡ πœ‡ NOUN iajs-3638 77 4 ) ) PUNCT iajs-3638 77 5 πœ†4 πœ†4 NOUN iajs-3638 77 6 = = SYM iajs-3638 77 7 βˆ’πœ‡ βˆ’πœ‡ PROPN iajs-3638 77 8 } } PUNCT iajs-3638 77 9 ( ( PUNCT iajs-3638 77 10 6b 6b NOUN iajs-3638 77 11 ) ) PUNCT iajs-3638 77 12 now now ADV iajs-3638 77 13 , , PUNCT iajs-3638 77 14 for for ADP iajs-3638 77 15 𝜏 𝜏 NOUN iajs-3638 77 16 = = SYM iajs-3638 77 17 0 0 NUM iajs-3638 77 18 we we PRON iajs-3638 77 19 get get VERB iajs-3638 77 20 the the DET iajs-3638 77 21 eigenvalues eigenvalue NOUN iajs-3638 77 22 will will AUX iajs-3638 77 23 be be AUX iajs-3638 77 24 negative negative ADJ iajs-3638 77 25 and and CCONJ iajs-3638 77 26 fep fep NOUN iajs-3638 77 27 is be AUX iajs-3638 77 28 locally locally ADV iajs-3638 77 29 asymptotically asymptotically ADV iajs-3638 77 30 stable stable ADJ iajs-3638 77 31 ( ( PUNCT iajs-3638 77 32 l.a.s l.a.s NOUN iajs-3638 77 33 . . PUNCT iajs-3638 77 34 ) ) PUNCT iajs-3638 78 1 if if SCONJ iajs-3638 78 2 𝛽1𝑆̅ 𝛽1𝑆̅ PRON iajs-3638 78 3 < < X iajs-3638 78 4 πœ‡ πœ‡ ADP iajs-3638 78 5 + + X iajs-3638 78 6 πœ‡1 πœ‡1 ADJ iajs-3638 78 7 + + CCONJ iajs-3638 78 8 πœƒ πœƒ PROPN iajs-3638 78 9 ( ( PUNCT iajs-3638 78 10 7 7 NUM iajs-3638 78 11 ) ) PUNCT iajs-3638 78 12 now now ADV iajs-3638 78 13 , , PUNCT iajs-3638 78 14 for for ADP iajs-3638 78 15 𝜏 𝜏 PRON iajs-3638 78 16 > > X iajs-3638 78 17 0 0 NUM iajs-3638 79 1 the the DET iajs-3638 79 2 equation equation NOUN iajs-3638 79 3 ( ( PUNCT iajs-3638 79 4 6b 6b NOUN iajs-3638 79 5 ) ) PUNCT iajs-3638 79 6 , , PUNCT iajs-3638 79 7 has have VERB iajs-3638 79 8 two two NUM iajs-3638 79 9 wholly wholly ADV iajs-3638 79 10 imaginary imaginary ADJ iajs-3638 79 11 roots root NOUN iajs-3638 79 12 , , PUNCT iajs-3638 79 13 namely= namely= NUM iajs-3638 79 14 Β±π‘–πœ” Β±π‘–πœ” NOUN iajs-3638 79 15 ( ( PUNCT iajs-3638 79 16 πœ” πœ” X iajs-3638 79 17 > > X iajs-3638 79 18 0 0 NUM iajs-3638 79 19 ) ) PUNCT iajs-3638 79 20 . . PUNCT iajs-3638 80 1 by by ADP iajs-3638 80 2 substituting substitute VERB iajs-3638 80 3 πœ† πœ† PRON iajs-3638 80 4 = = SYM iajs-3638 80 5 Β±π‘–πœ” Β±π‘–πœ” NOUN iajs-3638 80 6 in in ADP iajs-3638 80 7 equation equation NOUN iajs-3638 80 8 ( ( PUNCT iajs-3638 80 9 6b 6b NUM iajs-3638 80 10 ) ) PUNCT iajs-3638 80 11 we we PRON iajs-3638 80 12 get get VERB iajs-3638 80 13 : : PUNCT iajs-3638 80 14 𝛽1𝑆̅(π‘π‘œπ‘ πœ”πœ 𝛽1𝑆̅(π‘π‘œπ‘ πœ”πœ NOUN iajs-3638 80 15 βˆ’ βˆ’ NOUN iajs-3638 80 16 π‘–π‘ π‘–π‘›πœ”πœ π‘–π‘ π‘–π‘›πœ”πœ ADJ iajs-3638 80 17 ) ) PUNCT iajs-3638 81 1 = = PUNCT iajs-3638 81 2 πœ‡ πœ‡ ADP iajs-3638 81 3 + + X iajs-3638 81 4 πœ‡1 πœ‡1 ADJ iajs-3638 81 5 + + CCONJ iajs-3638 81 6 πœƒ πœƒ PROPN iajs-3638 82 1 + + NUM iajs-3638 82 2 π‘–πœ” π‘–πœ” PRON iajs-3638 82 3 so so ADV iajs-3638 82 4 , , PUNCT iajs-3638 82 5 separating separate VERB iajs-3638 82 6 the the DET iajs-3638 82 7 real real ADJ iajs-3638 82 8 and and CCONJ iajs-3638 82 9 imagined imagined ADJ iajs-3638 82 10 parts part NOUN iajs-3638 82 11 yields yield NOUN iajs-3638 82 12 𝛽1π‘†Μ…π‘π‘œπ‘ πœ”πœ 𝛽1π‘†Μ…π‘π‘œπ‘ πœ”πœ NOUN iajs-3638 82 13 = = SYM iajs-3638 82 14 πœ‡ πœ‡ ADP iajs-3638 82 15 + + X iajs-3638 82 16 πœ‡1 πœ‡1 ADJ iajs-3638 82 17 + + CCONJ iajs-3638 82 18 πœƒ πœƒ NOUN iajs-3638 82 19 𝛽1π‘†Μ…π‘ π‘–π‘›πœ”πœ 𝛽1π‘†Μ…π‘ π‘–π‘›πœ”πœ NOUN iajs-3638 82 20 = = SYM iajs-3638 82 21 βˆ’πœ” βˆ’πœ” NOUN iajs-3638 82 22 } } PUNCT iajs-3638 82 23 ( ( PUNCT iajs-3638 82 24 8) 8) NUM iajs-3638 82 25 squaring square VERB iajs-3638 82 26 each each DET iajs-3638 82 27 equation equation NOUN iajs-3638 82 28 and and CCONJ iajs-3638 82 29 then then ADV iajs-3638 82 30 adding add VERB iajs-3638 82 31 them they PRON iajs-3638 82 32 , , PUNCT iajs-3638 82 33 we we PRON iajs-3638 82 34 get get VERB iajs-3638 82 35 that that PRON iajs-3638 82 36 πœ” πœ” ADP iajs-3638 82 37 = = X iajs-3638 82 38 βˆ“βˆšπ›½1 βˆ“βˆšπ›½1 PROPN iajs-3638 82 39 2(𝑆̅)2 2(𝑆̅)2 NUM iajs-3638 82 40 βˆ’ βˆ’ NOUN iajs-3638 82 41 ( ( PUNCT iajs-3638 82 42 πœ‡ πœ‡ ADP iajs-3638 82 43 + + X iajs-3638 82 44 πœ‡1 πœ‡1 ADJ iajs-3638 82 45 + + CCONJ iajs-3638 82 46 πœƒ)2 πœƒ)2 NOUN iajs-3638 82 47 note note VERB iajs-3638 82 48 that that SCONJ iajs-3638 82 49 , , PUNCT iajs-3638 82 50 under under ADP iajs-3638 82 51 the the DET iajs-3638 82 52 condition condition NOUN iajs-3638 82 53 ( ( PUNCT iajs-3638 82 54 7),πœ”(𝜏 7),πœ”(𝜏 NOUN iajs-3638 82 55 ) ) PUNCT iajs-3638 82 56 with with ADP iajs-3638 82 57 𝜏 𝜏 ADP iajs-3638 82 58 > > SYM iajs-3638 82 59 0 0 NUM iajs-3638 82 60 can can AUX iajs-3638 82 61 not not PART iajs-3638 82 62 be be AUX iajs-3638 82 63 real real ADJ iajs-3638 82 64 , , PUNCT iajs-3638 82 65 which which PRON iajs-3638 82 66 contradicts contradict VERB iajs-3638 82 67 with with ADP iajs-3638 82 68 the the DET iajs-3638 82 69 assumption assumption NOUN iajs-3638 82 70 . . PUNCT iajs-3638 83 1 therefore therefore ADV iajs-3638 83 2 , , PUNCT iajs-3638 83 3 the the DET iajs-3638 83 4 c.e c.e PROPN iajs-3638 83 5 . . PROPN iajs-3638 83 6 ( ( PUNCT iajs-3638 83 7 6a 6a NOUN iajs-3638 83 8 ) ) PUNCT iajs-3638 83 9 ca can AUX iajs-3638 83 10 n’t not PART iajs-3638 83 11 have have VERB iajs-3638 83 12 purely purely ADV iajs-3638 83 13 imaginary imaginary ADJ iajs-3638 83 14 root root NOUN iajs-3638 83 15 , , PUNCT iajs-3638 83 16 and and CCONJ iajs-3638 83 17 fep fep NOUN iajs-3638 83 18 is be AUX iajs-3638 83 19 l.a.s l.a.s ADJ iajs-3638 83 20 . . PUNCT iajs-3638 84 1 for for ADP iajs-3638 84 2 all all DET iajs-3638 84 3 𝜏 𝜏 DET iajs-3638 84 4 β‰₯ β‰₯ NOUN iajs-3638 84 5 0 0 NUM iajs-3638 85 1 if if SCONJ iajs-3638 85 2 the the DET iajs-3638 85 3 condition condition NOUN iajs-3638 85 4 ( ( PUNCT iajs-3638 85 5 7 7 X iajs-3638 85 6 ) ) PUNCT iajs-3638 85 7 hold hold NOUN iajs-3638 85 8 . . PUNCT iajs-3638 86 1 β€’ β€’ NUM iajs-3638 86 2 for for ADP iajs-3638 86 3 the the DET iajs-3638 86 4 ( ( PUNCT iajs-3638 86 5 sep sep PROPN iajs-3638 86 6 ) ) PUNCT iajs-3638 86 7 when when SCONJ iajs-3638 86 8 𝛾 𝛾 NOUN iajs-3638 86 9 = = SYM iajs-3638 86 10 0 0 NUM iajs-3638 86 11 , , PUNCT iajs-3638 86 12 equation equation NOUN iajs-3638 86 13 ( ( PUNCT iajs-3638 86 14 4 4 NUM iajs-3638 86 15 ) ) PUNCT iajs-3638 86 16 reduces reduce VERB iajs-3638 86 17 to to ADP iajs-3638 86 18 𝐽(𝐸1 𝐽(𝐸1 NOUN iajs-3638 86 19 ) ) PUNCT iajs-3638 87 1 = = PRON iajs-3638 87 2 [ [ PUNCT iajs-3638 87 3 βˆ’ βˆ’ PROPN iajs-3638 87 4 ( ( PUNCT iajs-3638 87 5 𝛼 𝛼 PROPN iajs-3638 87 6 1 1 NUM iajs-3638 87 7 + + NUM iajs-3638 87 8 𝑛 𝑛 PROPN iajs-3638 87 9 οΏ½ οΏ½ PROPN iajs-3638 87 10 ΜΏ ΜΏ NOUN iajs-3638 87 11 οΏ½ οΏ½ PROPN iajs-3638 87 12 1 1 NUM iajs-3638 87 13 + + CCONJ iajs-3638 87 14 πœ‡ πœ‡ X iajs-3638 87 15 ) ) PUNCT iajs-3638 87 16 𝑛𝛼𝑆̿ 𝑛𝛼𝑆̿ PROPN iajs-3638 87 17 ( ( PUNCT iajs-3638 87 18 1 1 NUM iajs-3638 87 19 + + NUM iajs-3638 87 20 𝑛 𝑛 PROPN iajs-3638 87 21 οΏ½ οΏ½ PROPN iajs-3638 87 22 ΜΏ ΜΏ NOUN iajs-3638 87 23 οΏ½ οΏ½ PROPN iajs-3638 87 24 1 1 NUM iajs-3638 87 25 ) ) PUNCT iajs-3638 87 26 2 2 NUM iajs-3638 87 27 0 0 NUM iajs-3638 87 28 βˆ’π›½1𝑆̿𝑒 βˆ’π›½1𝑆̿𝑒 NUM iajs-3638 87 29 βˆ’πœ†πœ βˆ’πœ†πœ NUM iajs-3638 87 30 𝛼 𝛼 NOUN iajs-3638 87 31 1 1 NUM iajs-3638 87 32 + + NUM iajs-3638 87 33 𝑛 𝑛 PROPN iajs-3638 87 34 οΏ½ οΏ½ PROPN iajs-3638 87 35 ΜΏ ΜΏ NOUN iajs-3638 87 36 οΏ½ οΏ½ PROPN iajs-3638 87 37 1 1 NUM iajs-3638 87 38 βˆ’ βˆ’ PROPN iajs-3638 87 39 ( ( PUNCT iajs-3638 87 40 𝑛𝛼𝑆̿ 𝑛𝛼𝑆̿ NUM iajs-3638 87 41 ( ( PUNCT iajs-3638 87 42 1 1 NUM iajs-3638 87 43 + + NUM iajs-3638 87 44 𝑛 𝑛 PROPN iajs-3638 87 45 οΏ½ οΏ½ PROPN iajs-3638 87 46 ΜΏ ΜΏ NOUN iajs-3638 87 47 οΏ½ οΏ½ PROPN iajs-3638 87 48 1 1 NUM iajs-3638 87 49 ) ) PUNCT iajs-3638 87 50 2 2 NUM iajs-3638 87 51 + + CCONJ iajs-3638 87 52 πœ‡ πœ‡ X iajs-3638 87 53 ) ) PUNCT iajs-3638 87 54 0 0 PUNCT iajs-3638 88 1 βˆ’π›½2 βˆ’π›½2 PROPN iajs-3638 88 2 οΏ½ οΏ½ PROPN iajs-3638 88 3 ΜΏ ΜΏ NOUN iajs-3638 88 4 οΏ½ οΏ½ PROPN iajs-3638 88 5 1 1 NUM iajs-3638 88 6 0 0 NUM iajs-3638 88 7 0 0 NUM iajs-3638 89 1 βˆ’πœ‡ βˆ’πœ‡ NUM iajs-3638 89 2 0 0 NUM iajs-3638 89 3 0 0 NUM iajs-3638 89 4 0 0 NUM iajs-3638 89 5 0 0 NUM iajs-3638 89 6 𝛽1𝑆̿𝑒 𝛽1𝑆̿𝑒 PROPN iajs-3638 89 7 βˆ’πœ†πœ βˆ’πœ†πœ PROPN iajs-3638 89 8 + + CCONJ iajs-3638 89 9 𝛽2 𝛽2 PROPN iajs-3638 89 10 οΏ½ οΏ½ PROPN iajs-3638 89 11 ΜΏ ΜΏ NOUN iajs-3638 89 12 οΏ½ οΏ½ PROPN iajs-3638 89 13 1 1 NUM iajs-3638 89 14 βˆ’ βˆ’ NOUN iajs-3638 89 15 ( ( PUNCT iajs-3638 89 16 πœ‡ πœ‡ ADP iajs-3638 89 17 + + X iajs-3638 89 18 πœ‡1 πœ‡1 ADJ iajs-3638 89 19 + + CCONJ iajs-3638 89 20 πœƒ πœƒ NOUN iajs-3638 89 21 ) ) PUNCT iajs-3638 89 22 ] ] PUNCT iajs-3638 89 23 ( ( PUNCT iajs-3638 89 24 9 9 X iajs-3638 89 25 ) ) PUNCT iajs-3638 89 26 the the DET iajs-3638 89 27 c.e c.e PROPN iajs-3638 89 28 . . PROPN iajs-3638 89 29 of of ADP iajs-3638 89 30 𝐽(𝐸1 𝐽(𝐸1 NOUN iajs-3638 89 31 ) ) PUNCT iajs-3638 89 32 is be AUX iajs-3638 89 33 [ [ X iajs-3638 89 34 πœ†2 πœ†2 NOUN iajs-3638 89 35 + + CCONJ iajs-3638 90 1 𝐴1πœ† 𝐴1πœ† ADJ iajs-3638 90 2 + + CCONJ iajs-3638 90 3 𝐴2][βˆ’πœ‡ 𝐴2][βˆ’πœ‡ ADV iajs-3638 90 4 βˆ’ βˆ’ PROPN iajs-3638 90 5 πœ†][𝛽1𝑆̿𝑒 πœ†][𝛽1𝑆̿𝑒 NOUN iajs-3638 90 6 βˆ’πœ†πœ βˆ’πœ†πœ PROPN iajs-3638 90 7 + + CCONJ iajs-3638 90 8 𝛽2 𝛽2 PROPN iajs-3638 90 9 οΏ½ οΏ½ PROPN iajs-3638 90 10 ΜΏ ΜΏ NOUN iajs-3638 90 11 οΏ½ οΏ½ PROPN iajs-3638 90 12 1 1 NUM iajs-3638 90 13 βˆ’ βˆ’ NOUN iajs-3638 90 14 ( ( PUNCT iajs-3638 90 15 πœ‡ πœ‡ ADP iajs-3638 90 16 + + X iajs-3638 90 17 πœ‡1 πœ‡1 ADJ iajs-3638 90 18 + + CCONJ iajs-3638 90 19 πœƒ πœƒ NOUN iajs-3638 90 20 ) ) PUNCT iajs-3638 90 21 βˆ’ βˆ’ PUNCT iajs-3638 91 1 πœ† πœ† X iajs-3638 91 2 ] ] X iajs-3638 91 3 = = SYM iajs-3638 91 4 0 0 NUM iajs-3638 91 5 ( ( PUNCT iajs-3638 91 6 10a 10a NOUN iajs-3638 91 7 ) ) PUNCT iajs-3638 91 8 where where SCONJ iajs-3638 91 9 ihjpas ihjpa NOUN iajs-3638 91 10 . . PUNCT iajs-3638 92 1 2025 2025 NUM iajs-3638 92 2 , , PUNCT iajs-3638 92 3 38 38 NUM iajs-3638 92 4 ( ( PUNCT iajs-3638 92 5 1 1 NUM iajs-3638 92 6 ) ) PUNCT iajs-3638 92 7 449 449 NUM iajs-3638 92 8 𝐴1 𝐴1 NOUN iajs-3638 92 9 = = SYM iajs-3638 92 10 𝛼 𝛼 PROPN iajs-3638 92 11 1+𝑛 1+𝑛 NUM iajs-3638 92 12 οΏ½ οΏ½ PROPN iajs-3638 92 13 ΜΏ ΜΏ NOUN iajs-3638 92 14 οΏ½ οΏ½ PROPN iajs-3638 92 15 1 1 NUM iajs-3638 92 16 + + NUM iajs-3638 92 17 𝑛𝛼 𝑛𝛼 PROPN iajs-3638 92 18 οΏ½ οΏ½ PROPN iajs-3638 92 19 ΜΏ ΜΏ PROPN iajs-3638 92 20 οΏ½ οΏ½ PROPN iajs-3638 92 21 ( ( PUNCT iajs-3638 92 22 1+𝑛𝑉1 1+𝑛𝑉1 NUM iajs-3638 92 23 ) ) PUNCT iajs-3638 92 24 2 2 NUM iajs-3638 93 1 + + NUM iajs-3638 93 2 2πœ‡ 2πœ‡ NUM iajs-3638 93 3 𝐴2 𝐴2 NOUN iajs-3638 93 4 = = PUNCT iajs-3638 93 5 πœ‡ πœ‡ X iajs-3638 93 6 ( ( PUNCT iajs-3638 93 7 𝑛𝛼 𝑛𝛼 PROPN iajs-3638 93 8 οΏ½ οΏ½ PROPN iajs-3638 93 9 ΜΏ ΜΏ PROPN iajs-3638 93 10 οΏ½ οΏ½ PROPN iajs-3638 93 11 ( ( PUNCT iajs-3638 93 12 1+𝑛 1+𝑛 NUM iajs-3638 93 13 οΏ½ οΏ½ PROPN iajs-3638 93 14 ΜΏ ΜΏ NOUN iajs-3638 93 15 οΏ½ οΏ½ NOUN iajs-3638 93 16 1 1 NUM iajs-3638 93 17 ) ) PUNCT iajs-3638 93 18 2 2 NUM iajs-3638 94 1 + + CCONJ iajs-3638 94 2 𝛼 𝛼 PROPN iajs-3638 94 3 1+𝑛 1+𝑛 NUM iajs-3638 94 4 οΏ½ οΏ½ PROPN iajs-3638 94 5 ΜΏ ΜΏ NOUN iajs-3638 94 6 οΏ½ οΏ½ PROPN iajs-3638 94 7 1 1 NUM iajs-3638 94 8 + + SYM iajs-3638 94 9 πœ‡ πœ‡ X iajs-3638 94 10 ) ) PUNCT iajs-3638 94 11 the the DET iajs-3638 94 12 equation equation NOUN iajs-3638 94 13 ( ( PUNCT iajs-3638 94 14 10a 10a NOUN iajs-3638 94 15 ) ) PUNCT iajs-3638 94 16 represents represent VERB iajs-3638 94 17 the the DET iajs-3638 94 18 eigenvalues eigenvalue NOUN iajs-3638 94 19 of of ADP iajs-3638 94 20 𝐽(𝐸1 𝐽(𝐸1 NOUN iajs-3638 94 21 ) ) PUNCT iajs-3638 94 22 and and CCONJ iajs-3638 94 23 has have VERB iajs-3638 94 24 4 4 NUM iajs-3638 94 25 - - PUNCT iajs-3638 94 26 roots root NOUN iajs-3638 94 27 : : PUNCT iajs-3638 95 1 πœ†1,2 πœ†1,2 ADJ iajs-3638 95 2 = = SYM iajs-3638 96 1 βˆ’ βˆ’ NOUN iajs-3638 96 2 𝐴1 𝐴1 PROPN iajs-3638 96 3 2 2 NUM iajs-3638 96 4 βˆ“ βˆ“ NOUN iajs-3638 96 5 1 1 NUM iajs-3638 96 6 2 2 NUM iajs-3638 96 7 √𝐴1 √𝐴1 NOUN iajs-3638 96 8 2 2 NUM iajs-3638 96 9 βˆ’ βˆ’ PROPN iajs-3638 96 10 4𝐴2 4𝐴2 NUM iajs-3638 96 11 πœ†3 πœ†3 NOUN iajs-3638 96 12 = = SYM iajs-3638 96 13 βˆ’πœ‡ βˆ’πœ‡ VERB iajs-3638 96 14 πœ†4 πœ†4 NOUN iajs-3638 96 15 = = SYM iajs-3638 96 16 𝛽1𝑆̿𝑒 𝛽1𝑆̿𝑒 NOUN iajs-3638 96 17 βˆ’πœ†πœ βˆ’πœ†πœ PROPN iajs-3638 96 18 + + CCONJ iajs-3638 96 19 𝛽2 𝛽2 PROPN iajs-3638 96 20 οΏ½ οΏ½ PROPN iajs-3638 96 21 ΜΏ ΜΏ NOUN iajs-3638 96 22 οΏ½ οΏ½ PROPN iajs-3638 96 23 1 1 NUM iajs-3638 96 24 βˆ’ βˆ’ NOUN iajs-3638 96 25 ( ( PUNCT iajs-3638 96 26 πœ‡ πœ‡ ADP iajs-3638 96 27 + + X iajs-3638 96 28 πœ‡1 πœ‡1 ADJ iajs-3638 96 29 + + CCONJ iajs-3638 96 30 πœƒ πœƒ NOUN iajs-3638 96 31 ) ) PUNCT iajs-3638 96 32 } } PUNCT iajs-3638 96 33 ( ( PUNCT iajs-3638 96 34 10b 10b NOUN iajs-3638 96 35 ) ) PUNCT iajs-3638 96 36 now now ADV iajs-3638 96 37 , , PUNCT iajs-3638 96 38 for for ADP iajs-3638 96 39 𝜏 𝜏 NOUN iajs-3638 96 40 = = SYM iajs-3638 96 41 0 0 NUM iajs-3638 96 42 we we PRON iajs-3638 96 43 get get VERB iajs-3638 96 44 all all DET iajs-3638 96 45 the the DET iajs-3638 96 46 above above ADJ iajs-3638 96 47 eigenvalues eigenvalue NOUN iajs-3638 96 48 will will AUX iajs-3638 96 49 be be AUX iajs-3638 96 50 negative negative ADJ iajs-3638 96 51 and and CCONJ iajs-3638 96 52 the the DET iajs-3638 96 53 sep sep NOUN iajs-3638 96 54 is be AUX iajs-3638 96 55 l.a.s l.a.s ADJ iajs-3638 96 56 . . PUNCT iajs-3638 97 1 if if SCONJ iajs-3638 97 2 𝛽1𝑆̿ 𝛽1𝑆̿ ADP iajs-3638 97 3 + + SYM iajs-3638 97 4 𝛽2 𝛽2 PROPN iajs-3638 97 5 οΏ½ οΏ½ PROPN iajs-3638 97 6 ΜΏ ΜΏ NOUN iajs-3638 97 7 οΏ½ οΏ½ PROPN iajs-3638 97 8 1 1 NUM iajs-3638 97 9 < < X iajs-3638 97 10 πœ‡ πœ‡ ADP iajs-3638 97 11 + + X iajs-3638 97 12 πœ‡1 πœ‡1 ADJ iajs-3638 97 13 + + CCONJ iajs-3638 97 14 πœƒ πœƒ PROPN iajs-3638 97 15 ( ( PUNCT iajs-3638 97 16 11 11 NUM iajs-3638 97 17 ) ) PUNCT iajs-3638 97 18 now now ADV iajs-3638 97 19 , , PUNCT iajs-3638 97 20 for for ADP iajs-3638 97 21 𝜏 𝜏 PRON iajs-3638 97 22 > > X iajs-3638 97 23 0 0 PUNCT iajs-3638 98 1 the the DET iajs-3638 98 2 equation equation NOUN iajs-3638 98 3 ( ( PUNCT iajs-3638 98 4 10b 10b NOUN iajs-3638 98 5 ) ) PUNCT iajs-3638 98 6 , , PUNCT iajs-3638 98 7 has have VERB iajs-3638 98 8 two two NUM iajs-3638 98 9 wholly wholly ADV iajs-3638 98 10 imaginary imaginary ADJ iajs-3638 98 11 roots root NOUN iajs-3638 98 12 , , PUNCT iajs-3638 98 13 namely= namely= NUM iajs-3638 98 14 Β±π‘–πœ” Β±π‘–πœ” NOUN iajs-3638 98 15 ( ( PUNCT iajs-3638 98 16 πœ” πœ” X iajs-3638 98 17 > > X iajs-3638 98 18 0 0 NUM iajs-3638 98 19 ) ) PUNCT iajs-3638 98 20 . . PUNCT iajs-3638 99 1 by by ADP iajs-3638 99 2 substituting substitute VERB iajs-3638 99 3 πœ† πœ† PRON iajs-3638 99 4 = = SYM iajs-3638 99 5 Β±π‘–πœ” Β±π‘–πœ” NOUN iajs-3638 99 6 in in ADP iajs-3638 99 7 equation equation NOUN iajs-3638 99 8 ( ( PUNCT iajs-3638 99 9 10b 10b NOUN iajs-3638 99 10 ) ) PUNCT iajs-3638 99 11 we we PRON iajs-3638 99 12 get get VERB iajs-3638 99 13 : : PUNCT iajs-3638 99 14 𝛽1𝑆̿(π‘π‘œπ‘ πœ”πœ 𝛽1𝑆̿(π‘π‘œπ‘ πœ”πœ PROPN iajs-3638 99 15 βˆ’ βˆ’ NOUN iajs-3638 99 16 π‘–π‘ π‘–π‘›πœ”πœ π‘–π‘ π‘–π‘›πœ”πœ ADJ iajs-3638 99 17 ) ) PUNCT iajs-3638 100 1 = = PUNCT iajs-3638 100 2 πœ‡ πœ‡ ADP iajs-3638 100 3 + + X iajs-3638 100 4 πœ‡1 πœ‡1 ADJ iajs-3638 100 5 + + CCONJ iajs-3638 100 6 πœƒ πœƒ NOUN iajs-3638 100 7 βˆ’ βˆ’ PROPN iajs-3638 100 8 𝛽2 𝛽2 PROPN iajs-3638 100 9 οΏ½ οΏ½ PROPN iajs-3638 100 10 ΜΏ ΜΏ NOUN iajs-3638 100 11 οΏ½ οΏ½ PROPN iajs-3638 100 12 1 1 NUM iajs-3638 100 13 + + NUM iajs-3638 100 14 π‘–πœ” π‘–πœ” PRON iajs-3638 100 15 so so ADV iajs-3638 100 16 , , PUNCT iajs-3638 100 17 separating separate VERB iajs-3638 100 18 the the DET iajs-3638 100 19 real real ADJ iajs-3638 100 20 and and CCONJ iajs-3638 100 21 imagined imagined ADJ iajs-3638 100 22 parts part NOUN iajs-3638 100 23 yields yield NOUN iajs-3638 100 24 𝛽1π‘†ΜΏπ‘π‘œπ‘ πœ”πœ 𝛽1π‘†ΜΏπ‘π‘œπ‘ πœ”πœ PUNCT iajs-3638 100 25 = = PUNCT iajs-3638 100 26 πœ‡ πœ‡ ADP iajs-3638 100 27 + + X iajs-3638 100 28 πœ‡1 πœ‡1 ADJ iajs-3638 100 29 + + CCONJ iajs-3638 100 30 πœƒ πœƒ NOUN iajs-3638 100 31 βˆ’ βˆ’ PROPN iajs-3638 100 32 𝛽2 𝛽2 PROPN iajs-3638 100 33 οΏ½ οΏ½ PROPN iajs-3638 100 34 ΜΏ ΜΏ NOUN iajs-3638 100 35 οΏ½ οΏ½ PROPN iajs-3638 100 36 1 1 NUM iajs-3638 100 37 𝛽1π‘†ΜΏπ‘ π‘–π‘›πœ”πœ 𝛽1π‘†ΜΏπ‘ π‘–π‘›πœ”πœ NOUN iajs-3638 100 38 = = SYM iajs-3638 100 39 βˆ’πœ” βˆ’πœ” NOUN iajs-3638 100 40 } } PUNCT iajs-3638 100 41 ( ( PUNCT iajs-3638 100 42 12 12 NUM iajs-3638 100 43 ) ) PUNCT iajs-3638 100 44 squaring square VERB iajs-3638 100 45 each each DET iajs-3638 100 46 equation equation NOUN iajs-3638 100 47 and and CCONJ iajs-3638 100 48 then then ADV iajs-3638 100 49 adding add VERB iajs-3638 100 50 them they PRON iajs-3638 100 51 , , PUNCT iajs-3638 100 52 we we PRON iajs-3638 100 53 get get VERB iajs-3638 100 54 that that PRON iajs-3638 100 55 πœ” πœ” ADP iajs-3638 100 56 = = X iajs-3638 100 57 βˆ“βˆšπ›½1 βˆ“βˆšπ›½1 NOUN iajs-3638 100 58 2(𝑆̅̅ 2(𝑆̅̅ NUM iajs-3638 100 59 ) ) PUNCT iajs-3638 100 60 2 2 NUM iajs-3638 100 61 βˆ’ βˆ’ NOUN iajs-3638 100 62 ( ( PUNCT iajs-3638 100 63 πœ‡ πœ‡ ADP iajs-3638 100 64 + + X iajs-3638 100 65 πœ‡1 πœ‡1 ADJ iajs-3638 100 66 + + CCONJ iajs-3638 100 67 πœƒ πœƒ NOUN iajs-3638 100 68 βˆ’ βˆ’ PROPN iajs-3638 100 69 𝛽2 𝛽2 PROPN iajs-3638 100 70 οΏ½ οΏ½ PROPN iajs-3638 100 71 ΜΏ ΜΏ NOUN iajs-3638 100 72 οΏ½ οΏ½ PROPN iajs-3638 100 73 1 1 NUM iajs-3638 100 74 ) ) SYM iajs-3638 100 75 2 2 NUM iajs-3638 100 76 note note VERB iajs-3638 100 77 that that SCONJ iajs-3638 100 78 , , PUNCT iajs-3638 100 79 under under ADP iajs-3638 100 80 the the DET iajs-3638 100 81 condition condition NOUN iajs-3638 100 82 ( ( PUNCT iajs-3638 100 83 11),πœ”(𝜏 11),πœ”(𝜏 NUM iajs-3638 100 84 ) ) PUNCT iajs-3638 100 85 with with ADP iajs-3638 100 86 𝜏 𝜏 ADP iajs-3638 100 87 > > SYM iajs-3638 100 88 0 0 NUM iajs-3638 100 89 can can AUX iajs-3638 100 90 not not PART iajs-3638 100 91 be be AUX iajs-3638 100 92 real real ADJ iajs-3638 100 93 , , PUNCT iajs-3638 100 94 which which PRON iajs-3638 100 95 contradicts contradict VERB iajs-3638 100 96 with with ADP iajs-3638 100 97 the the DET iajs-3638 100 98 assumption assumption NOUN iajs-3638 100 99 . . PUNCT iajs-3638 101 1 therefore therefore ADV iajs-3638 101 2 , , PUNCT iajs-3638 101 3 the the DET iajs-3638 101 4 c.e c.e PROPN iajs-3638 101 5 . . PROPN iajs-3638 101 6 ( ( PUNCT iajs-3638 101 7 10a 10a NOUN iajs-3638 101 8 ) ) PUNCT iajs-3638 101 9 ca can AUX iajs-3638 101 10 n’t not PART iajs-3638 101 11 have have VERB iajs-3638 101 12 purely purely ADV iajs-3638 101 13 imaginary imaginary ADJ iajs-3638 101 14 root root NOUN iajs-3638 101 15 , , PUNCT iajs-3638 101 16 and and CCONJ iajs-3638 101 17 sep sep PROPN iajs-3638 101 18 is be AUX iajs-3638 101 19 l.a.s l.a.s ADJ iajs-3638 101 20 . . PUNCT iajs-3638 102 1 for for ADP iajs-3638 102 2 all all DET iajs-3638 102 3 𝜏 𝜏 DET iajs-3638 102 4 β‰₯ β‰₯ NOUN iajs-3638 102 5 0 0 NUM iajs-3638 103 1 if if SCONJ iajs-3638 103 2 the the DET iajs-3638 103 3 condition condition NOUN iajs-3638 103 4 ( ( PUNCT iajs-3638 103 5 11 11 NUM iajs-3638 103 6 ) ) PUNCT iajs-3638 103 7 hold hold VERB iajs-3638 103 8 . . PUNCT iajs-3638 104 1 for for ADP iajs-3638 104 2 the the DET iajs-3638 104 3 ( ( PUNCT iajs-3638 104 4 tep tep NOUN iajs-3638 104 5 ) ) PUNCT iajs-3638 104 6 when when SCONJ iajs-3638 104 7 𝛼 𝛼 X iajs-3638 104 8 = = SYM iajs-3638 104 9 0 0 NUM iajs-3638 104 10 , , PUNCT iajs-3638 104 11 equation equation NOUN iajs-3638 104 12 ( ( PUNCT iajs-3638 104 13 4 4 NUM iajs-3638 104 14 ) ) PUNCT iajs-3638 104 15 reduces reduce VERB iajs-3638 104 16 to to ADP iajs-3638 104 17 𝐽(𝐸2 𝐽(𝐸2 PUNCT iajs-3638 104 18 ) ) PUNCT iajs-3638 105 1 = = PUNCT iajs-3638 105 2 [ [ PUNCT iajs-3638 105 3 βˆ’(𝑅1𝑒 βˆ’(𝑅1𝑒 NOUN iajs-3638 105 4 βˆ’πœ†πœ βˆ’πœ†πœ NOUN iajs-3638 105 5 + + CCONJ iajs-3638 105 6 𝑅2 𝑅2 NOUN iajs-3638 105 7 ) ) PUNCT iajs-3638 105 8 0 0 NUM iajs-3638 105 9 0 0 NUM iajs-3638 106 1 βˆ’π‘…3𝑒 βˆ’π‘…3𝑒 NOUN iajs-3638 106 2 βˆ’πœ†πœ βˆ’πœ†πœ PROPN iajs-3638 106 3 0 0 NUM iajs-3638 107 1 βˆ’(𝛾 βˆ’(𝛾 NOUN iajs-3638 107 2 + + CCONJ iajs-3638 107 3 𝛽2𝐼 𝛽2𝐼 ADV iajs-3638 107 4 + + CCONJ iajs-3638 107 5 πœ‡ πœ‡ X iajs-3638 107 6 ) ) PUNCT iajs-3638 107 7 0 0 NUM iajs-3638 107 8 0 0 NUM iajs-3638 107 9 0 0 NUM iajs-3638 107 10 𝛾 𝛾 ADP iajs-3638 107 11 βˆ’(𝛽3𝐼 βˆ’(𝛽3𝐼 PROPN iajs-3638 107 12 + + CCONJ iajs-3638 107 13 πœ‡ πœ‡ X iajs-3638 107 14 ) ) PUNCT iajs-3638 107 15 0 0 NUM iajs-3638 108 1 𝑅1𝑒 𝑅1𝑒 PROPN iajs-3638 108 2 βˆ’πœ†πœ βˆ’πœ†πœ PROPN iajs-3638 108 3 𝛽2𝐼 𝛽2𝐼 ADV iajs-3638 108 4 𝛽3𝐼 𝛽3𝐼 NUM iajs-3638 108 5 𝑅3𝑒 𝑅3𝑒 PROPN iajs-3638 108 6 βˆ’πœ†πœ βˆ’πœ†πœ PROPN iajs-3638 108 7 + + CCONJ iajs-3638 108 8 𝑅4 𝑅4 PROPN iajs-3638 108 9 ] ] X iajs-3638 108 10 ( ( PUNCT iajs-3638 108 11 13 13 NUM iajs-3638 108 12 ) ) PUNCT iajs-3638 108 13 where where SCONJ iajs-3638 108 14 , , PUNCT iajs-3638 108 15 𝑅1 𝑅1 PROPN iajs-3638 108 16 = = SYM iajs-3638 108 17 𝛽1𝐼 𝛽1𝐼 NOUN iajs-3638 108 18 , , PUNCT iajs-3638 108 19 𝑅2 𝑅2 NOUN iajs-3638 108 20 = = PUNCT iajs-3638 108 21 πœ‡ πœ‡ X iajs-3638 108 22 , , PUNCT iajs-3638 108 23 𝑅3 𝑅3 PROPN iajs-3638 108 24 = = SYM iajs-3638 108 25 𝛽1 𝛽1 PROPN iajs-3638 108 26 οΏ½ οΏ½ PROPN iajs-3638 108 27 Μ‚ Μ‚ NUM iajs-3638 108 28 οΏ½ οΏ½ PROPN iajs-3638 108 29 , , PUNCT iajs-3638 108 30 𝑅4 𝑅4 PROPN iajs-3638 108 31 = = PUNCT iajs-3638 108 32 βˆ’(πœ‡ βˆ’(πœ‡ PROPN iajs-3638 108 33 + + SYM iajs-3638 108 34 πœ‡1 πœ‡1 PROPN iajs-3638 108 35 + + CCONJ iajs-3638 108 36 πœƒ πœƒ NOUN iajs-3638 108 37 ) ) PUNCT iajs-3638 108 38 . . PUNCT iajs-3638 109 1 the the DET iajs-3638 109 2 c.e c.e PROPN iajs-3638 109 3 . . PROPN iajs-3638 109 4 of of ADP iajs-3638 109 5 𝐽(𝐸2 𝐽(𝐸2 X iajs-3638 109 6 ) ) PUNCT iajs-3638 109 7 is be AUX iajs-3638 109 8 given give VERB iajs-3638 109 9 by by ADP iajs-3638 109 10 [ [ X iajs-3638 109 11 πœ†2 πœ†2 PROPN iajs-3638 109 12 + + CCONJ iajs-3638 109 13 𝐡1πœ† 𝐡1πœ† PROPN iajs-3638 109 14 + + CCONJ iajs-3638 109 15 𝐡2 𝐡2 NOUN iajs-3638 109 16 + + CCONJ iajs-3638 110 1 ( ( PUNCT iajs-3638 110 2 𝐡3πœ† 𝐡3πœ† NOUN iajs-3638 110 3 + + PROPN iajs-3638 110 4 𝐡4)𝑒 𝐡4)𝑒 PROPN iajs-3638 110 5 βˆ’πœ†πœ][βˆ’(𝛾 βˆ’πœ†πœ][βˆ’(𝛾 NOUN iajs-3638 110 6 + + PUNCT iajs-3638 110 7 𝛽2𝐼 𝛽2𝐼 ADV iajs-3638 110 8 + + CCONJ iajs-3638 110 9 πœ‡ πœ‡ X iajs-3638 110 10 ) ) PUNCT iajs-3638 110 11 βˆ’ βˆ’ PROPN iajs-3638 110 12 πœ† πœ† X iajs-3638 110 13 ] ] X iajs-3638 111 1 [ [ X iajs-3638 111 2 βˆ’(𝛽3𝐼 βˆ’(𝛽3𝐼 NOUN iajs-3638 111 3 + + CCONJ iajs-3638 111 4 πœ‡ πœ‡ NOUN iajs-3638 111 5 ) ) PUNCT iajs-3638 111 6 βˆ’ βˆ’ PROPN iajs-3638 111 7 πœ† πœ† X iajs-3638 111 8 ] ] X iajs-3638 111 9 = = SYM iajs-3638 111 10 0 0 SYM iajs-3638 111 11 ( ( PUNCT iajs-3638 111 12 14 14 NUM iajs-3638 111 13 ) ) PUNCT iajs-3638 111 14 with with ADP iajs-3638 111 15 𝐡1 𝐡1 NOUN iajs-3638 111 16 = = PUNCT iajs-3638 111 17 𝑅2 𝑅2 NOUN iajs-3638 111 18 βˆ’ βˆ’ NOUN iajs-3638 111 19 𝑅4 𝑅4 NOUN iajs-3638 111 20 , , PUNCT iajs-3638 111 21 𝐡2 𝐡2 NOUN iajs-3638 111 22 = = SYM iajs-3638 111 23 βˆ’π‘…2𝑅4 βˆ’π‘…2𝑅4 PROPN iajs-3638 111 24 , , PUNCT iajs-3638 111 25 𝐡3 𝐡3 NOUN iajs-3638 111 26 = = SYM iajs-3638 111 27 𝑅1 𝑅1 PROPN iajs-3638 111 28 βˆ’ βˆ’ PROPN iajs-3638 111 29 𝑅3 𝑅3 NOUN iajs-3638 111 30 , , PUNCT iajs-3638 111 31 𝐡4 𝐡4 NOUN iajs-3638 111 32 = = PUNCT iajs-3638 111 33 βˆ’π‘…2𝑅3 βˆ’π‘…2𝑅3 NOUN iajs-3638 111 34 βˆ’ βˆ’ PRON iajs-3638 111 35 𝑅1𝑅4 𝑅1𝑅4 NOUN iajs-3638 111 36 . . PUNCT iajs-3638 112 1 now now ADV iajs-3638 112 2 , , PUNCT iajs-3638 112 3 when when SCONJ iajs-3638 112 4 = = PRON iajs-3638 112 5 0 0 NUM iajs-3638 112 6 , , PUNCT iajs-3638 112 7 equation equation NOUN iajs-3638 112 8 ( ( PUNCT iajs-3638 112 9 14 14 NUM iajs-3638 112 10 ) ) PUNCT iajs-3638 112 11 is be AUX iajs-3638 112 12 ; ; PUNCT iajs-3638 113 1 [ [ X iajs-3638 113 2 πœ†2 πœ†2 NOUN iajs-3638 113 3 + + CCONJ iajs-3638 113 4 ( ( PUNCT iajs-3638 113 5 𝐡1 𝐡1 NOUN iajs-3638 113 6 + + NOUN iajs-3638 113 7 𝐡3)πœ† 𝐡3)πœ† PROPN iajs-3638 113 8 + + NUM iajs-3638 113 9 𝐡2 𝐡2 NOUN iajs-3638 113 10 + + NUM iajs-3638 113 11 𝐡4][βˆ’(𝛾 𝐡4][βˆ’(𝛾 NOUN iajs-3638 113 12 + + X iajs-3638 113 13 𝛽2𝐼 𝛽2𝐼 ADV iajs-3638 113 14 + + CCONJ iajs-3638 113 15 πœ‡ πœ‡ X iajs-3638 113 16 ) ) PUNCT iajs-3638 113 17 βˆ’ βˆ’ PROPN iajs-3638 113 18 πœ†][βˆ’(𝛽3𝐼 πœ†][βˆ’(𝛽3𝐼 NOUN iajs-3638 114 1 + + X iajs-3638 114 2 πœ‡ πœ‡ NOUN iajs-3638 114 3 ) ) PUNCT iajs-3638 114 4 βˆ’ βˆ’ PROPN iajs-3638 115 1 πœ† πœ† X iajs-3638 115 2 ] ] X iajs-3638 115 3 = = SYM iajs-3638 115 4 0 0 NUM iajs-3638 115 5 ( ( PUNCT iajs-3638 115 6 15 15 NUM iajs-3638 115 7 ) ) PUNCT iajs-3638 115 8 so so ADV iajs-3638 115 9 , , PUNCT iajs-3638 115 10 either either CCONJ iajs-3638 115 11 [ [ X iajs-3638 115 12 βˆ’(𝛾 βˆ’(𝛾 NOUN iajs-3638 115 13 + + CCONJ iajs-3638 115 14 𝛽2𝐼 𝛽2𝐼 ADV iajs-3638 115 15 + + CCONJ iajs-3638 115 16 πœ‡ πœ‡ X iajs-3638 115 17 ) ) PUNCT iajs-3638 115 18 βˆ’ βˆ’ PROPN iajs-3638 115 19 πœ†][βˆ’(𝛽3𝐼 πœ†][βˆ’(𝛽3𝐼 NOUN iajs-3638 116 1 + + X iajs-3638 116 2 πœ‡ πœ‡ NOUN iajs-3638 116 3 ) ) PUNCT iajs-3638 116 4 βˆ’ βˆ’ PROPN iajs-3638 117 1 πœ† πœ† X iajs-3638 117 2 ] ] X iajs-3638 117 3 = = SYM iajs-3638 117 4 0 0 NUM iajs-3638 117 5 ( ( PUNCT iajs-3638 117 6 16a 16a NOUN iajs-3638 117 7 ) ) PUNCT iajs-3638 117 8 or or CCONJ iajs-3638 117 9 , , PUNCT iajs-3638 117 10 ihjpas ihjpas PROPN iajs-3638 117 11 . . PUNCT iajs-3638 118 1 2025 2025 NUM iajs-3638 118 2 , , PUNCT iajs-3638 118 3 38 38 NUM iajs-3638 118 4 ( ( PUNCT iajs-3638 118 5 1 1 NUM iajs-3638 118 6 ) ) PUNCT iajs-3638 118 7 450 450 NUM iajs-3638 119 1 [ [ X iajs-3638 119 2 πœ†2 πœ†2 NOUN iajs-3638 119 3 + + CCONJ iajs-3638 119 4 ( ( PUNCT iajs-3638 119 5 𝐡1 𝐡1 NOUN iajs-3638 119 6 + + NOUN iajs-3638 119 7 𝐡3)πœ† 𝐡3)πœ† PROPN iajs-3638 119 8 + + NUM iajs-3638 119 9 𝐡2 𝐡2 NOUN iajs-3638 119 10 + + NUM iajs-3638 119 11 𝐡4 𝐡4 NOUN iajs-3638 119 12 ] ] X iajs-3638 119 13 = = SYM iajs-3638 119 14 0 0 NUM iajs-3638 119 15 ( ( PUNCT iajs-3638 119 16 16b 16b PROPN iajs-3638 119 17 ) ) PUNCT iajs-3638 119 18 from from ADP iajs-3638 119 19 equation equation NOUN iajs-3638 119 20 ( ( PUNCT iajs-3638 119 21 16a 16a NOUN iajs-3638 119 22 ) ) PUNCT iajs-3638 119 23 we we PRON iajs-3638 119 24 obtain obtain VERB iajs-3638 119 25 that that DET iajs-3638 119 26 πœ†2 πœ†2 NOUN iajs-3638 119 27 = = PUNCT iajs-3638 119 28 βˆ’(𝛾 βˆ’(𝛾 PROPN iajs-3638 119 29 + + CCONJ iajs-3638 119 30 𝛽2𝐼 𝛽2𝐼 ADV iajs-3638 119 31 + + CCONJ iajs-3638 119 32 πœ‡ πœ‡ X iajs-3638 119 33 ) ) PUNCT iajs-3638 119 34 < < X iajs-3638 119 35 0 0 NUM iajs-3638 119 36 πœ†3 πœ†3 NOUN iajs-3638 120 1 = = PUNCT iajs-3638 120 2 βˆ’(𝛽3𝐼 βˆ’(𝛽3𝐼 PROPN iajs-3638 120 3 + + CCONJ iajs-3638 120 4 πœ‡ πœ‡ X iajs-3638 120 5 ) ) PUNCT iajs-3638 120 6 < < X iajs-3638 120 7 0 0 NUM iajs-3638 120 8 which which PRON iajs-3638 120 9 is be AUX iajs-3638 120 10 always always ADV iajs-3638 120 11 negative negative ADJ iajs-3638 120 12 eigenvalue eigenvalue NOUN iajs-3638 120 13 . . PUNCT iajs-3638 121 1 now now ADV iajs-3638 121 2 , , PUNCT iajs-3638 121 3 it it PRON iajs-3638 121 4 is be AUX iajs-3638 121 5 easy easy ADJ iajs-3638 121 6 to to PART iajs-3638 121 7 verify verify VERB iajs-3638 121 8 that that DET iajs-3638 121 9 𝐡1 𝐡1 NOUN iajs-3638 121 10 + + CCONJ iajs-3638 121 11 𝐡3 𝐡3 NOUN iajs-3638 121 12 > > X iajs-3638 121 13 0 0 PUNCT iajs-3638 121 14 and and CCONJ iajs-3638 121 15 𝐡2 𝐡2 NOUN iajs-3638 122 1 + + NUM iajs-3638 122 2 𝐡4 𝐡4 NOUN iajs-3638 122 3 > > X iajs-3638 122 4 0 0 PUNCT iajs-3638 123 1 under under ADP iajs-3638 123 2 the the DET iajs-3638 123 3 following follow VERB iajs-3638 123 4 sufficient sufficient ADJ iajs-3638 123 5 conditions condition NOUN iajs-3638 123 6 𝛽1 𝛽1 PROPN iajs-3638 123 7 οΏ½ οΏ½ PROPN iajs-3638 123 8 Μ‚ Μ‚ NUM iajs-3638 123 9 οΏ½ οΏ½ NOUN iajs-3638 123 10 < < X iajs-3638 123 11 𝛽1𝐼 𝛽1𝐼 NOUN iajs-3638 123 12 + + CCONJ iajs-3638 123 13 2πœ‡ 2πœ‡ NOUN iajs-3638 123 14 + + CCONJ iajs-3638 123 15 πœ‡1 πœ‡1 PROPN iajs-3638 123 16 + + CCONJ iajs-3638 123 17 πœƒ πœƒ PROPN iajs-3638 123 18 , , PUNCT iajs-3638 123 19 ( ( PUNCT iajs-3638 123 20 17a 17a X iajs-3638 123 21 ) ) PUNCT iajs-3638 124 1 πœ‡π›½1 πœ‡π›½1 PROPN iajs-3638 124 2 οΏ½ οΏ½ PROPN iajs-3638 124 3 Μ‚ Μ‚ VERB iajs-3638 124 4 οΏ½ οΏ½ PROPN iajs-3638 124 5 < < X iajs-3638 124 6 ( ( PUNCT iajs-3638 124 7 𝛽1𝐼 𝛽1𝐼 X iajs-3638 124 8 + + CCONJ iajs-3638 124 9 πœ‡)(πœ‡ πœ‡)(πœ‡ PROPN iajs-3638 124 10 + + SYM iajs-3638 124 11 πœ‡1 πœ‡1 PROPN iajs-3638 124 12 + + CCONJ iajs-3638 124 13 πœƒ πœƒ NOUN iajs-3638 124 14 ) ) PUNCT iajs-3638 124 15 . . PUNCT iajs-3638 125 1 ( ( PUNCT iajs-3638 125 2 17b 17b NUM iajs-3638 125 3 ) ) PUNCT iajs-3638 125 4 we we PRON iajs-3638 125 5 see see VERB iajs-3638 125 6 that that SCONJ iajs-3638 125 7 all all DET iajs-3638 125 8 roots root NOUN iajs-3638 125 9 of of ADP iajs-3638 125 10 equation equation NOUN iajs-3638 125 11 ( ( PUNCT iajs-3638 125 12 16b 16b PROPN iajs-3638 125 13 ) ) PUNCT iajs-3638 125 14 , , PUNCT iajs-3638 125 15 which which PRON iajs-3638 125 16 represent represent VERB iajs-3638 125 17 the the DET iajs-3638 125 18 eigenvalues eigenvalue NOUN iajs-3638 125 19 of of ADP iajs-3638 125 20 ( ( PUNCT iajs-3638 125 21 13 13 NUM iajs-3638 125 22 ) ) PUNCT iajs-3638 125 23 , , PUNCT iajs-3638 125 24 have have VERB iajs-3638 125 25 negative negative ADJ iajs-3638 125 26 real real ADJ iajs-3638 125 27 parts part NOUN iajs-3638 125 28 . . PUNCT iajs-3638 126 1 consequently consequently ADV iajs-3638 126 2 , , PUNCT iajs-3638 126 3 under under ADP iajs-3638 126 4 the the DET iajs-3638 126 5 conditions condition NOUN iajs-3638 126 6 ( ( PUNCT iajs-3638 126 7 17a-17b 17a-17b NUM iajs-3638 126 8 ) ) PUNCT iajs-3638 126 9 , , PUNCT iajs-3638 126 10 tep tep PROPN iajs-3638 126 11 is be AUX iajs-3638 126 12 l.a.s l.a.s ADJ iajs-3638 126 13 . . PUNCT iajs-3638 127 1 for for ADP iajs-3638 127 2 system system NOUN iajs-3638 127 3 ( ( PUNCT iajs-3638 127 4 3 3 NUM iajs-3638 127 5 ) ) PUNCT iajs-3638 127 6 when when SCONJ iajs-3638 127 7 𝜏 𝜏 NOUN iajs-3638 127 8 = = SYM iajs-3638 127 9 0 0 NUM iajs-3638 127 10 . . PUNCT iajs-3638 128 1 now now ADV iajs-3638 128 2 , , PUNCT iajs-3638 128 3 for for ADP iajs-3638 128 4 𝜏 𝜏 X iajs-3638 128 5 > > X iajs-3638 128 6 0 0 NUM iajs-3638 128 7 , , PUNCT iajs-3638 128 8 then then ADV iajs-3638 128 9 either either ADV iajs-3638 128 10 , , PUNCT iajs-3638 128 11 equation equation NOUN iajs-3638 128 12 ( ( PUNCT iajs-3638 128 13 16a 16a NOUN iajs-3638 128 14 ) ) PUNCT iajs-3638 128 15 which which PRON iajs-3638 128 16 is be AUX iajs-3638 128 17 always always ADV iajs-3638 128 18 negative negative ADJ iajs-3638 128 19 eigenvalue eigenvalue NOUN iajs-3638 128 20 . . PUNCT iajs-3638 129 1 or or CCONJ iajs-3638 129 2 , , PUNCT iajs-3638 130 1 [ [ X iajs-3638 130 2 πœ†2 πœ†2 NOUN iajs-3638 130 3 + + CCONJ iajs-3638 130 4 𝐡1πœ† 𝐡1πœ† PROPN iajs-3638 130 5 + + CCONJ iajs-3638 130 6 𝐡2 𝐡2 NOUN iajs-3638 130 7 + + CCONJ iajs-3638 130 8 ( ( PUNCT iajs-3638 130 9 𝐡3πœ† 𝐡3πœ† NOUN iajs-3638 130 10 + + PROPN iajs-3638 130 11 𝐡4)𝑒 𝐡4)𝑒 PROPN iajs-3638 130 12 βˆ’πœ†πœ βˆ’πœ†πœ PROPN iajs-3638 130 13 ] ] X iajs-3638 130 14 = = SYM iajs-3638 130 15 0 0 PUNCT iajs-3638 130 16 ( ( PUNCT iajs-3638 130 17 18 18 NUM iajs-3638 130 18 ) ) PUNCT iajs-3638 130 19 from from ADP iajs-3638 130 20 ( ( PUNCT iajs-3638 130 21 18 18 NUM iajs-3638 130 22 ) ) PUNCT iajs-3638 130 23 has have VERB iajs-3638 130 24 two two NUM iajs-3638 130 25 wholly wholly ADV iajs-3638 130 26 imaginary imaginary ADJ iajs-3638 130 27 roots root NOUN iajs-3638 130 28 , , PUNCT iajs-3638 130 29 namely= namely= NUM iajs-3638 130 30 Β±π‘–πœ” Β±π‘–πœ” NOUN iajs-3638 130 31 ( ( PUNCT iajs-3638 130 32 πœ” πœ” X iajs-3638 130 33 > > X iajs-3638 130 34 0 0 NUM iajs-3638 130 35 ) ) PUNCT iajs-3638 130 36 . . PUNCT iajs-3638 131 1 by by ADP iajs-3638 131 2 substituting substitute VERB iajs-3638 131 3 πœ† πœ† PRON iajs-3638 131 4 = = SYM iajs-3638 131 5 Β±π‘–πœ” Β±π‘–πœ” NOUN iajs-3638 131 6 in in ADP iajs-3638 131 7 equation equation NOUN iajs-3638 131 8 ( ( PUNCT iajs-3638 131 9 18 18 NUM iajs-3638 131 10 ) ) PUNCT iajs-3638 131 11 and and CCONJ iajs-3638 131 12 separating separate VERB iajs-3638 131 13 the the DET iajs-3638 131 14 real real ADJ iajs-3638 131 15 and and CCONJ iajs-3638 131 16 imagined imagined ADJ iajs-3638 131 17 parts part NOUN iajs-3638 131 18 yields yield NOUN iajs-3638 131 19 𝐡4π‘π‘œπ‘ πœ”πœ 𝐡4π‘π‘œπ‘ πœ”πœ NOUN iajs-3638 131 20 + + CCONJ iajs-3638 131 21 𝐡3πœ”π‘ π‘–π‘›πœ”πœ 𝐡3πœ”π‘ π‘–π‘›πœ”πœ NOUN iajs-3638 131 22 = = SYM iajs-3638 131 23 πœ” πœ” PRON iajs-3638 131 24 2 2 NUM iajs-3638 131 25 βˆ’ βˆ’ PROPN iajs-3638 131 26 𝐡2 𝐡2 NOUN iajs-3638 131 27 𝐡3πœ”π‘π‘œπ‘ πœ”πœ 𝐡3πœ”π‘π‘œπ‘ πœ”πœ PROPN iajs-3638 131 28 βˆ’ βˆ’ PROPN iajs-3638 131 29 𝐡4π‘ π‘–π‘›πœ”πœ 𝐡4π‘ π‘–π‘›πœ”πœ NOUN iajs-3638 131 30 = = SYM iajs-3638 131 31 βˆ’π΅1πœ” βˆ’π΅1πœ” PROPN iajs-3638 131 32 ( ( PUNCT iajs-3638 131 33 19 19 NUM iajs-3638 131 34 ) ) PUNCT iajs-3638 131 35 through through ADP iajs-3638 131 36 squaring square VERB iajs-3638 131 37 and and CCONJ iajs-3638 131 38 adding add VERB iajs-3638 131 39 both both DET iajs-3638 131 40 equations equation NOUN iajs-3638 131 41 in in ADP iajs-3638 131 42 equation equation NOUN iajs-3638 131 43 ( ( PUNCT iajs-3638 131 44 19 19 NUM iajs-3638 131 45 ) ) PUNCT iajs-3638 131 46 , , PUNCT iajs-3638 131 47 we we PRON iajs-3638 131 48 get get VERB iajs-3638 131 49 πœ”4 πœ”4 PROPN iajs-3638 131 50 + + CCONJ iajs-3638 131 51 β„Ž1πœ” β„Ž1πœ” PROPN iajs-3638 131 52 2 2 NUM iajs-3638 131 53 + + CCONJ iajs-3638 131 54 β„Ž2 β„Ž2 ADJ iajs-3638 131 55 = = SYM iajs-3638 131 56 0 0 NUM iajs-3638 131 57 ( ( PUNCT iajs-3638 131 58 20 20 NUM iajs-3638 131 59 ) ) PUNCT iajs-3638 131 60 further far ADV iajs-3638 131 61 , , PUNCT iajs-3638 131 62 by by ADP iajs-3638 131 63 letting let VERB iajs-3638 131 64 ℏ ℏ X iajs-3638 131 65 = = PUNCT iajs-3638 131 66 πœ”2 πœ”2 ADJ iajs-3638 131 67 in in ADP iajs-3638 131 68 equation equation NOUN iajs-3638 131 69 ( ( PUNCT iajs-3638 131 70 20 20 NUM iajs-3638 131 71 ) ) PUNCT iajs-3638 131 72 𝑔(ℏ 𝑔(ℏ PROPN iajs-3638 131 73 ) ) PUNCT iajs-3638 132 1 = = PUNCT iajs-3638 133 1 ℏ2 ℏ2 NOUN iajs-3638 133 2 + + CCONJ iajs-3638 133 3 𝑔1ℏ 𝑔1ℏ PRON iajs-3638 133 4 + + CCONJ iajs-3638 133 5 𝑔2 𝑔2 NOUN iajs-3638 133 6 = = SYM iajs-3638 133 7 0 0 NUM iajs-3638 133 8 ( ( PUNCT iajs-3638 133 9 21a 21a NUM iajs-3638 133 10 ) ) PUNCT iajs-3638 133 11 where where SCONJ iajs-3638 133 12 𝑔1 𝑔1 PROPN iajs-3638 133 13 = = PUNCT iajs-3638 133 14 𝑅2 𝑅2 NOUN iajs-3638 133 15 2 2 NUM iajs-3638 133 16 + + CCONJ iajs-3638 133 17 𝑅4 𝑅4 X iajs-3638 133 18 2 2 NUM iajs-3638 133 19 βˆ’ βˆ’ PROPN iajs-3638 133 20 ( ( PUNCT iajs-3638 133 21 𝑅1 𝑅1 PROPN iajs-3638 133 22 βˆ’ βˆ’ PROPN iajs-3638 133 23 𝑅3 𝑅3 PROPN iajs-3638 133 24 ) ) PUNCT iajs-3638 133 25 2 2 NUM iajs-3638 133 26 , , PUNCT iajs-3638 133 27 𝑔2 𝑔2 NOUN iajs-3638 133 28 = = PUNCT iajs-3638 133 29 𝑅2 𝑅2 NOUN iajs-3638 133 30 2𝑅4 2𝑅4 NUM iajs-3638 133 31 2βˆ’(𝑅2𝑅3 2βˆ’(𝑅2𝑅3 NUM iajs-3638 133 32 + + CCONJ iajs-3638 133 33 𝑅1𝑅4 𝑅1𝑅4 NOUN iajs-3638 133 34 ) ) PUNCT iajs-3638 133 35 2 2 NUM iajs-3638 133 36 . . X iajs-3638 133 37 straightforward straightforward ADJ iajs-3638 133 38 computation computation NOUN iajs-3638 133 39 shows show VERB iajs-3638 133 40 that that SCONJ iajs-3638 133 41 due due ADP iajs-3638 133 42 to to ADP iajs-3638 133 43 the the DET iajs-3638 133 44 following follow VERB iajs-3638 133 45 condition condition NOUN iajs-3638 133 46 𝑅2𝑅4 𝑅2𝑅4 NOUN iajs-3638 133 47 < < X iajs-3638 133 48 𝑅2𝑅3 𝑅2𝑅3 NOUN iajs-3638 133 49 + + CCONJ iajs-3638 133 50 𝑅1𝑅4 𝑅1𝑅4 X iajs-3638 133 51 ( ( PUNCT iajs-3638 133 52 21b 21b NOUN iajs-3638 133 53 ) ) PUNCT iajs-3638 133 54 we we PRON iajs-3638 133 55 obtain obtain VERB iajs-3638 133 56 𝑔2 𝑔2 NOUN iajs-3638 133 57 < < X iajs-3638 133 58 0 0 NUM iajs-3638 133 59 . . PUNCT iajs-3638 134 1 so so ADV iajs-3638 134 2 , , PUNCT iajs-3638 134 3 there there PRON iajs-3638 134 4 exists exist VERB iajs-3638 134 5 a a DET iajs-3638 134 6 unique unique ADJ iajs-3638 134 7 positive positive ADJ iajs-3638 134 8 root root NOUN iajs-3638 134 9 in in ADP iajs-3638 134 10 accordance accordance NOUN iajs-3638 134 11 with with ADP iajs-3638 134 12 descartes descarte NOUN iajs-3638 134 13 ' ' PART iajs-3638 134 14 rule rule NOUN iajs-3638 134 15 of of ADP iajs-3638 134 16 signs sign NOUN iajs-3638 134 17 ℏ0 ℏ0 NOUN iajs-3638 134 18 = = SYM iajs-3638 134 19 πœ”0 πœ”0 NOUN iajs-3638 134 20 2 2 NUM iajs-3638 134 21 providing provide VERB iajs-3638 134 22 equation equation NOUN iajs-3638 134 23 ( ( PUNCT iajs-3638 134 24 21a 21a NUM iajs-3638 134 25 ) ) PUNCT iajs-3638 134 26 . . PUNCT iajs-3638 135 1 which which PRON iajs-3638 135 2 is be AUX iajs-3638 135 3 , , PUNCT iajs-3638 135 4 equation equation NOUN iajs-3638 135 5 ( ( PUNCT iajs-3638 135 6 20 20 NUM iajs-3638 135 7 ) ) PUNCT iajs-3638 135 8 has have VERB iajs-3638 135 9 a a DET iajs-3638 135 10 positiveπœ”0 positiveπœ”0 NOUN iajs-3638 135 11 . . PUNCT iajs-3638 136 1 as as ADP iajs-3638 136 2 a a DET iajs-3638 136 3 result result NOUN iajs-3638 136 4 , , PUNCT iajs-3638 136 5 equation equation NOUN iajs-3638 136 6 ( ( PUNCT iajs-3638 136 7 18 18 NUM iajs-3638 136 8 ) ) PUNCT iajs-3638 136 9 has have VERB iajs-3638 136 10 at at ADV iajs-3638 136 11 least least ADV iajs-3638 136 12 two two NUM iajs-3638 136 13 roots root NOUN iajs-3638 136 14 Β±πœ”0 Β±πœ”0 NOUN iajs-3638 136 15 that that PRON iajs-3638 136 16 are be AUX iajs-3638 136 17 purely purely ADV iajs-3638 136 18 imaginary imaginary ADJ iajs-3638 136 19 and and CCONJ iajs-3638 136 20 which which PRON iajs-3638 136 21 correspond correspond VERB iajs-3638 136 22 to to ADP iajs-3638 136 23 the the DET iajs-3638 136 24 time time NOUN iajs-3638 136 25 delay delay NOUN iajs-3638 136 26 𝜏. 𝜏. PROPN iajs-3638 136 27 additionally additionally ADV iajs-3638 136 28 , , PUNCT iajs-3638 136 29 by by ADP iajs-3638 136 30 substituting substitute VERB iajs-3638 136 31 πœ”0 πœ”0 NOUN iajs-3638 136 32 in in ADP iajs-3638 136 33 ( ( PUNCT iajs-3638 136 34 19 19 NUM iajs-3638 136 35 ) ) PUNCT iajs-3638 136 36 and and CCONJ iajs-3638 136 37 solving solve VERB iajs-3638 136 38 the the DET iajs-3638 136 39 system system NOUN iajs-3638 136 40 for for ADP iajs-3638 136 41 𝜏 𝜏 NOUN iajs-3638 136 42 , , PUNCT iajs-3638 136 43 yields yield VERB iajs-3638 136 44 : : PUNCT iajs-3638 136 45 πœπ‘˜ πœπ‘˜ NOUN iajs-3638 136 46 = = SYM iajs-3638 136 47 1 1 NUM iajs-3638 136 48 πœ”0 πœ”0 NOUN iajs-3638 136 49 π‘π‘œπ‘ βˆ’1 π‘π‘œπ‘ βˆ’1 NOUN iajs-3638 136 50 ( ( PUNCT iajs-3638 136 51 𝐡4βˆ’π΅1𝐡3)πœ”0 𝐡4βˆ’π΅1𝐡3)πœ”0 NOUN iajs-3638 136 52 2βˆ’π΅2𝐡4 2βˆ’π΅2𝐡4 NUM iajs-3638 136 53 𝐡4 𝐡4 NOUN iajs-3638 136 54 2+𝐡3 2+𝐡3 NUM iajs-3638 136 55 2πœ”0 2πœ”0 NUM iajs-3638 136 56 2 2 NUM iajs-3638 136 57 + + CCONJ iajs-3638 136 58 2π‘˜πœ‹ 2π‘˜πœ‹ ADJ iajs-3638 136 59 πœ”0 πœ”0 NOUN iajs-3638 136 60 ( ( PUNCT iajs-3638 136 61 22 22 NUM iajs-3638 136 62 ) ) PUNCT iajs-3638 136 63 where where SCONJ iajs-3638 136 64 , , PUNCT iajs-3638 136 65 k k PROPN iajs-3638 136 66 = = PUNCT iajs-3638 136 67 0,1,2 0,1,2 NUM iajs-3638 136 68 , , PUNCT iajs-3638 136 69 … … PUNCT iajs-3638 136 70 . . PUNCT iajs-3638 137 1 hence hence ADV iajs-3638 137 2 we we PRON iajs-3638 137 3 get get VERB iajs-3638 137 4 the the DET iajs-3638 137 5 corresponding corresponding NOUN iajs-3638 137 6 πœπ‘˜ πœπ‘˜ ADP iajs-3638 137 7 > > X iajs-3638 137 8 0 0 NUM iajs-3638 137 9 for for ADP iajs-3638 137 10 which which DET iajs-3638 137 11 system system NOUN iajs-3638 137 12 ( ( PUNCT iajs-3638 137 13 3 3 X iajs-3638 137 14 ) ) PUNCT iajs-3638 137 15 has have VERB iajs-3638 137 16 two two NUM iajs-3638 137 17 wholly wholly ADV iajs-3638 137 18 imaginary imaginary ADJ iajs-3638 137 19 roots root NOUN iajs-3638 137 20 Β±πœ”0 Β±πœ”0 NOUN iajs-3638 137 21 . . PUNCT iajs-3638 138 1 let let VERB iajs-3638 138 2 πœ†(𝜏 πœ†(𝜏 NOUN iajs-3638 138 3 ) ) PUNCT iajs-3638 139 1 = = SYM iajs-3638 139 2 πœ‡(𝜏 πœ‡(𝜏 PROPN iajs-3638 139 3 ) ) PUNCT iajs-3638 140 1 + + CCONJ iajs-3638 140 2 π‘–πœ”(𝜏 π‘–πœ”(𝜏 PRON iajs-3638 140 3 ) ) PUNCT iajs-3638 140 4 be be VERB iajs-3638 140 5 a a DET iajs-3638 140 6 root root NOUN iajs-3638 140 7 of of ADP iajs-3638 140 8 equation equation NOUN iajs-3638 140 9 ( ( PUNCT iajs-3638 140 10 18 18 NUM iajs-3638 140 11 ) ) PUNCT iajs-3638 140 12 near near ADP iajs-3638 140 13 𝜏 𝜏 NOUN iajs-3638 140 14 = = NOUN iajs-3638 140 15 πœπ‘˜ πœπ‘˜ NOUN iajs-3638 140 16 with with ADP iajs-3638 140 17 πœ‡(πœπ‘˜ πœ‡(πœπ‘˜ NOUN iajs-3638 140 18 ) ) PUNCT iajs-3638 140 19 = = SYM iajs-3638 140 20 0 0 NUM iajs-3638 140 21 and and CCONJ iajs-3638 140 22 πœ”(πœπ‘˜ πœ”(πœπ‘˜ NUM iajs-3638 140 23 ) ) PUNCT iajs-3638 141 1 = = SYM iajs-3638 141 2 πœ”0 πœ”0 PROPN iajs-3638 141 3 . . PUNCT iajs-3638 142 1 then then ADV iajs-3638 142 2 comes come VERB iajs-3638 142 3 the the DET iajs-3638 142 4 next next ADJ iajs-3638 142 5 theorem theorem NOUN iajs-3638 142 6 : : PUNCT iajs-3638 142 7 theorem1 theorem1 X iajs-3638 142 8 . . PUNCT iajs-3638 143 1 the the DET iajs-3638 143 2 c.e c.e PROPN iajs-3638 143 3 . . PROPN iajs-3638 143 4 ( ( PUNCT iajs-3638 143 5 18 18 NUM iajs-3638 143 6 ) ) PUNCT iajs-3638 143 7 , , PUNCT iajs-3638 143 8 roots root NOUN iajs-3638 143 9 satisfy satisfy VERB iajs-3638 143 10 the the DET iajs-3638 143 11 following follow VERB iajs-3638 143 12 transversality transversality NOUN iajs-3638 143 13 requirement requirement NOUN iajs-3638 143 14 ; ; PUNCT iajs-3638 143 15 [ [ PUNCT iajs-3638 143 16 𝑑(π‘…π‘’πœ†(𝜏 𝑑(π‘…π‘’πœ†(𝜏 NOUN iajs-3638 143 17 ) ) PUNCT iajs-3638 143 18 ) ) PUNCT iajs-3638 144 1 π‘‘πœ π‘‘πœ NOUN iajs-3638 144 2 ] ] PUNCT iajs-3638 144 3 𝜏=πœπ‘˜ 𝜏=πœπ‘˜ X iajs-3638 144 4 > > X iajs-3638 144 5 0 0 PUNCT iajs-3638 144 6 ( ( PUNCT iajs-3638 144 7 23a 23a NUM iajs-3638 144 8 ) ) PUNCT iajs-3638 144 9 provided provide VERB iajs-3638 144 10 that that DET iajs-3638 144 11 ihjpas ihjpa NOUN iajs-3638 144 12 . . PUNCT iajs-3638 145 1 2025 2025 NUM iajs-3638 145 2 , , PUNCT iajs-3638 145 3 38 38 NUM iajs-3638 145 4 ( ( PUNCT iajs-3638 145 5 1 1 NUM iajs-3638 145 6 ) ) PUNCT iajs-3638 145 7 451 451 NUM iajs-3638 145 8 πœ”0 πœ”0 NOUN iajs-3638 145 9 2 2 NUM iajs-3638 145 10 > > X iajs-3638 145 11 𝐡2 𝐡2 PROPN iajs-3638 145 12 ( ( PUNCT iajs-3638 145 13 23b 23b NOUN iajs-3638 145 14 ) ) PUNCT iajs-3638 145 15 proof proof NOUN iajs-3638 145 16 . . PUNCT iajs-3638 146 1 by by ADP iajs-3638 146 2 substituting substitute VERB iajs-3638 146 3 πœ†(𝜏 πœ†(𝜏 NOUN iajs-3638 146 4 ) ) PUNCT iajs-3638 146 5 in in ADP iajs-3638 146 6 ( ( PUNCT iajs-3638 146 7 18 18 NUM iajs-3638 146 8 ) ) PUNCT iajs-3638 146 9 and and CCONJ iajs-3638 146 10 differentiating differentiate VERB iajs-3638 146 11 the the DET iajs-3638 146 12 resulting result VERB iajs-3638 146 13 equation equation NOUN iajs-3638 146 14 in in ADP iajs-3638 146 15 𝜏,we 𝜏,we PROPN iajs-3638 146 16 can can AUX iajs-3638 146 17 get get VERB iajs-3638 146 18 [ [ X iajs-3638 146 19 2πœ† 2πœ† NOUN iajs-3638 146 20 + + CCONJ iajs-3638 146 21 𝐡1 𝐡1 NOUN iajs-3638 146 22 + + CCONJ iajs-3638 147 1 𝐡3𝑒 𝐡3𝑒 PROPN iajs-3638 147 2 βˆ’πœ†πœ βˆ’πœ†πœ NUM iajs-3638 147 3 βˆ’ βˆ’ NOUN iajs-3638 148 1 𝜏(𝐡3πœ† 𝜏(𝐡3πœ† NOUN iajs-3638 148 2 + + CCONJ iajs-3638 148 3 𝐡4)𝑒 𝐡4)𝑒 PROPN iajs-3638 148 4 βˆ’πœ†πœ βˆ’πœ†πœ PROPN iajs-3638 148 5 ] ] PUNCT iajs-3638 148 6 π‘‘πœ† π‘‘πœ† ADP iajs-3638 148 7 π‘‘πœ† π‘‘πœ† NOUN iajs-3638 148 8 = = PUNCT iajs-3638 148 9 πœ†(𝐡3πœ† πœ†(𝐡3πœ† PROPN iajs-3638 149 1 + + PROPN iajs-3638 149 2 𝐡4)𝑒 𝐡4)𝑒 PROPN iajs-3638 149 3 βˆ’πœ†πœ βˆ’πœ†πœ PROPN iajs-3638 149 4 ( ( PUNCT iajs-3638 149 5 24 24 NUM iajs-3638 149 6 ) ) PUNCT iajs-3638 149 7 thus thus ADV iajs-3638 149 8 , , PUNCT iajs-3638 149 9 ( ( PUNCT iajs-3638 149 10 π‘‘πœ† π‘‘πœ† ADP iajs-3638 149 11 π‘‘πœ π‘‘πœ NOUN iajs-3638 149 12 ) ) PUNCT iajs-3638 149 13 βˆ’1 βˆ’1 NOUN iajs-3638 149 14 = = NOUN iajs-3638 150 1 2πœ†+𝐡1 2πœ†+𝐡1 NUM iajs-3638 150 2 πœ†(𝐡3πœ†+𝐡4)π‘’βˆ’πœ†πœ πœ†(𝐡3πœ†+𝐡4)π‘’βˆ’πœ†πœ NOUN iajs-3638 150 3 + + CCONJ iajs-3638 150 4 𝐡3 𝐡3 PROPN iajs-3638 150 5 πœ†(𝐡3πœ†+𝐡4 πœ†(𝐡3πœ†+𝐡4 X iajs-3638 150 6 ) ) PUNCT iajs-3638 150 7 βˆ’ βˆ’ NOUN iajs-3638 151 1 𝜏 𝜏 X iajs-3638 151 2 πœ† πœ† X iajs-3638 151 3 ( ( PUNCT iajs-3638 151 4 25 25 NUM iajs-3638 151 5 ) ) PUNCT iajs-3638 151 6 since since SCONJ iajs-3638 151 7 for for ADP iajs-3638 151 8 = = PROPN iajs-3638 151 9 𝜏0 𝜏0 PROPN iajs-3638 151 10 , , PUNCT iajs-3638 151 11 and and CCONJ iajs-3638 151 12 πœ† πœ† X iajs-3638 151 13 = = SYM iajs-3638 151 14 π‘–πœ”0 π‘–πœ”0 PROPN iajs-3638 151 15 , , PUNCT iajs-3638 151 16 we we PRON iajs-3638 151 17 have have AUX iajs-3638 151 18 got get VERB iajs-3638 151 19 [ [ PUNCT iajs-3638 151 20 π‘‘πœ† π‘‘πœ† ADP iajs-3638 151 21 π‘‘πœ π‘‘πœ NOUN iajs-3638 151 22 ] ] PUNCT iajs-3638 151 23 𝜏=𝜏0 𝜏=𝜏0 PROPN iajs-3638 151 24 βˆ’1 βˆ’1 NOUN iajs-3638 151 25 = = PUNCT iajs-3638 151 26 𝐡1 𝐡1 PROPN iajs-3638 151 27 + + PROPN iajs-3638 151 28 2π‘–πœ”0 2π‘–πœ”0 NUM iajs-3638 151 29 𝐡1πœ”0 𝐡1πœ”0 PROPN iajs-3638 151 30 2+π‘–πœ”0(πœ”0 2+π‘–πœ”0(πœ”0 NUM iajs-3638 151 31 2βˆ’π΅2 2βˆ’π΅2 NUM iajs-3638 151 32 ) ) PUNCT iajs-3638 152 1 + + CCONJ iajs-3638 152 2 𝐡3 𝐡3 PROPN iajs-3638 152 3 𝐡3πœ”0 𝐡3πœ”0 PROPN iajs-3638 152 4 2+𝑖𝐡4πœ”0 2+𝑖𝐡4πœ”0 PROPN iajs-3638 152 5 βˆ’ βˆ’ PRON iajs-3638 152 6 π‘–πœ0 π‘–πœ0 NOUN iajs-3638 152 7 πœ”0 πœ”0 NOUN iajs-3638 152 8 ( ( PUNCT iajs-3638 152 9 26 26 NUM iajs-3638 152 10 ) ) PUNCT iajs-3638 152 11 now now ADV iajs-3638 152 12 , , PUNCT iajs-3638 152 13 since since SCONJ iajs-3638 152 14 𝑠𝑖𝑔𝑛 𝑠𝑖𝑔𝑛 NOUN iajs-3638 152 15 [ [ PUNCT iajs-3638 152 16 𝑑(π‘…π‘’πœ† 𝑑(π‘…π‘’πœ† PROPN iajs-3638 152 17 ) ) PUNCT iajs-3638 152 18 π‘‘πœ π‘‘πœ NOUN iajs-3638 152 19 ] ] PUNCT iajs-3638 152 20 𝜏=𝜏0 𝜏=𝜏0 PUNCT iajs-3638 152 21 = = SYM iajs-3638 152 22 𝑠𝑖𝑔𝑛 𝑠𝑖𝑔𝑛 NOUN iajs-3638 153 1 [ [ X iajs-3638 153 2 𝑅𝑒 𝑅𝑒 PROPN iajs-3638 153 3 ( ( PUNCT iajs-3638 153 4 π‘‘πœ† π‘‘πœ† ADP iajs-3638 153 5 π‘‘πœ π‘‘πœ NOUN iajs-3638 153 6 ) ) PUNCT iajs-3638 153 7 βˆ’1 βˆ’1 NOUN iajs-3638 153 8 ] ] PUNCT iajs-3638 154 1 πœ†=π‘–πœ”0 πœ†=π‘–πœ”0 PROPN iajs-3638 154 2 ( ( PUNCT iajs-3638 154 3 27 27 NUM iajs-3638 154 4 ) ) PUNCT iajs-3638 154 5 it it PRON iajs-3638 154 6 is be AUX iajs-3638 154 7 clear clear ADJ iajs-3638 154 8 that that SCONJ iajs-3638 154 9 : : PUNCT iajs-3638 154 10 𝑅𝑒 𝑅𝑒 PROPN iajs-3638 154 11 [ [ PUNCT iajs-3638 154 12 𝐡1 𝐡1 NOUN iajs-3638 154 13 + + PROPN iajs-3638 154 14 2π‘–πœ”0 2π‘–πœ”0 NUM iajs-3638 154 15 𝐡1πœ”0 𝐡1πœ”0 PROPN iajs-3638 154 16 2+π‘–πœ”0(πœ”0 2+π‘–πœ”0(πœ”0 NUM iajs-3638 154 17 2βˆ’π΅2 2βˆ’π΅2 NUM iajs-3638 154 18 ) ) PUNCT iajs-3638 154 19 ] ] PUNCT iajs-3638 155 1 = = PUNCT iajs-3638 155 2 𝐡1 𝐡1 NOUN iajs-3638 155 3 2 2 NUM iajs-3638 155 4 + + NOUN iajs-3638 155 5 2(πœ”0 2(πœ”0 NUM iajs-3638 155 6 2βˆ’π΅2 2βˆ’π΅2 NUM iajs-3638 155 7 ) ) PUNCT iajs-3638 155 8 𝐡1 𝐡1 NOUN iajs-3638 155 9 2πœ”0 2πœ”0 NUM iajs-3638 155 10 2+(πœ”0 2+(πœ”0 NUM iajs-3638 155 11 2βˆ’π΅2 2βˆ’π΅2 NUM iajs-3638 155 12 ) ) PUNCT iajs-3638 155 13 2 2 NUM iajs-3638 155 14 , , PUNCT iajs-3638 155 15 𝑅𝑒 𝑅𝑒 PROPN iajs-3638 155 16 [ [ PUNCT iajs-3638 155 17 𝐡3 𝐡3 PROPN iajs-3638 155 18 𝐡3πœ”0 𝐡3πœ”0 PROPN iajs-3638 155 19 2+𝑖𝐡4πœ”0 2+𝑖𝐡4πœ”0 PROPN iajs-3638 155 20 ] ] PUNCT iajs-3638 155 21 = = PUNCT iajs-3638 155 22 𝐡3 𝐡3 NOUN iajs-3638 155 23 2 2 NUM iajs-3638 155 24 𝐡3 𝐡3 NOUN iajs-3638 155 25 2+𝐡4 2+𝐡4 NUM iajs-3638 155 26 2 2 NUM iajs-3638 155 27 , , PUNCT iajs-3638 155 28 𝑅𝑒 𝑅𝑒 PROPN iajs-3638 155 29 [ [ PUNCT iajs-3638 155 30 π‘–πœ0 π‘–πœ0 NOUN iajs-3638 155 31 πœ”0 πœ”0 NOUN iajs-3638 155 32 ] ] PUNCT iajs-3638 155 33 = = PUNCT iajs-3638 155 34 π‘§π‘’π‘Ÿπ‘œ π‘§π‘’π‘Ÿπ‘œ INTJ iajs-3638 155 35 . . PUNCT iajs-3638 156 1 hence hence ADV iajs-3638 156 2 , , PUNCT iajs-3638 156 3 we we PRON iajs-3638 156 4 have have AUX iajs-3638 156 5 𝑅𝑒 𝑅𝑒 VERB iajs-3638 156 6 [ [ PUNCT iajs-3638 156 7 π‘‘πœ† π‘‘πœ† ADP iajs-3638 156 8 π‘‘πœ π‘‘πœ NOUN iajs-3638 156 9 ] ] PUNCT iajs-3638 156 10 𝜏=𝜏0 𝜏=𝜏0 PROPN iajs-3638 156 11 βˆ’1 βˆ’1 NOUN iajs-3638 156 12 = = PUNCT iajs-3638 156 13 𝐡1 𝐡1 NOUN iajs-3638 156 14 2 2 NUM iajs-3638 156 15 + + NOUN iajs-3638 156 16 2(πœ”0 2(πœ”0 NUM iajs-3638 156 17 2βˆ’π΅2 2βˆ’π΅2 NUM iajs-3638 156 18 ) ) PUNCT iajs-3638 157 1 Ο†0 Ο†0 PROPN iajs-3638 157 2 + + NUM iajs-3638 157 3 𝐡3 𝐡3 NOUN iajs-3638 157 4 2 2 NUM iajs-3638 157 5 Ο†1 Ο†1 NOUN iajs-3638 157 6 where where SCONJ iajs-3638 157 7 Ο†0 Ο†0 PROPN iajs-3638 157 8 = = PROPN iajs-3638 157 9 𝐡1 𝐡1 PROPN iajs-3638 157 10 2πœ”0 2πœ”0 NUM iajs-3638 157 11 2 2 NUM iajs-3638 157 12 + + CCONJ iajs-3638 157 13 ( ( PUNCT iajs-3638 157 14 πœ”0 πœ”0 NOUN iajs-3638 157 15 2 2 NUM iajs-3638 157 16 βˆ’ βˆ’ NOUN iajs-3638 157 17 𝐡2 𝐡2 NOUN iajs-3638 157 18 ) ) PUNCT iajs-3638 157 19 2 2 NUM iajs-3638 157 20 > > SYM iajs-3638 157 21 0 0 NUM iajs-3638 157 22 , , PUNCT iajs-3638 157 23 Ο†1 Ο†1 NOUN iajs-3638 157 24 = = PUNCT iajs-3638 157 25 𝐡3 𝐡3 NOUN iajs-3638 157 26 2 2 NUM iajs-3638 157 27 + + NUM iajs-3638 157 28 𝐡4 𝐡4 NOUN iajs-3638 157 29 2 2 NUM iajs-3638 157 30 > > SYM iajs-3638 157 31 0 0 NUM iajs-3638 157 32 . . PUNCT iajs-3638 158 1 we we PRON iajs-3638 158 2 obtain obtain VERB iajs-3638 158 3 [ [ PUNCT iajs-3638 158 4 𝑑(π‘…π‘’πœ†(𝜏 𝑑(π‘…π‘’πœ†(𝜏 NOUN iajs-3638 158 5 ) ) PUNCT iajs-3638 158 6 ) ) PUNCT iajs-3638 159 1 π‘‘πœ π‘‘πœ NOUN iajs-3638 159 2 ] ] PUNCT iajs-3638 159 3 𝜏=Ο„0 𝜏=Ο„0 X iajs-3638 159 4 > > X iajs-3638 159 5 0 0 PUNCT iajs-3638 160 1 under under ADP iajs-3638 160 2 ( ( PUNCT iajs-3638 160 3 23b 23b NOUN iajs-3638 160 4 ) ) PUNCT iajs-3638 160 5 . . PUNCT iajs-3638 161 1 this this DET iajs-3638 161 2 outcome outcome NOUN iajs-3638 161 3 demonstrates demonstrate VERB iajs-3638 161 4 how how SCONJ iajs-3638 161 5 the the DET iajs-3638 161 6 roots root NOUN iajs-3638 161 7 of of ADP iajs-3638 161 8 c.e c.e PROPN iajs-3638 161 9 . . PUNCT iajs-3638 162 1 ( ( PUNCT iajs-3638 162 2 18 18 NUM iajs-3638 162 3 ) ) PUNCT iajs-3638 162 4 , , PUNCT iajs-3638 162 5 as as SCONJ iajs-3638 162 6 𝜏 𝜏 PROPN iajs-3638 162 7 passes pass VERB iajs-3638 162 8 through through ADP iajs-3638 162 9 Ο„0 Ο„0 NOUN iajs-3638 162 10 , , PUNCT iajs-3638 162 11 traverse traverse VERB iajs-3638 162 12 the the DET iajs-3638 162 13 imaginary imaginary ADJ iajs-3638 162 14 axis axis NOUN iajs-3638 162 15 from from ADP iajs-3638 162 16 left left ADJ iajs-3638 162 17 to to ADP iajs-3638 162 18 right right NOUN iajs-3638 162 19 . . PUNCT iajs-3638 163 1 as as ADP iajs-3638 163 2 a a DET iajs-3638 163 3 result result NOUN iajs-3638 163 4 , , PUNCT iajs-3638 163 5 the the DET iajs-3638 163 6 system system NOUN iajs-3638 163 7 ( ( PUNCT iajs-3638 163 8 3 3 X iajs-3638 163 9 ) ) PUNCT iajs-3638 163 10 experiences experience NOUN iajs-3638 163 11 an an DET iajs-3638 163 12 h.b h.b PROPN iajs-3638 163 13 . . PROPN iajs-3638 164 1 at at ADP iajs-3638 164 2 𝜏 𝜏 NOUN iajs-3638 164 3 = = VERB iajs-3638 164 4 Ο„0 Ο„0 NOUN iajs-3638 164 5 and and CCONJ iajs-3638 164 6 loses lose VERB iajs-3638 164 7 its its PRON iajs-3638 164 8 stability stability NOUN iajs-3638 164 9 . . PUNCT iajs-3638 165 1 for for ADP iajs-3638 165 2 the the DET iajs-3638 165 3 ( ( PUNCT iajs-3638 165 4 foep foep NOUN iajs-3638 165 5 ) ) PUNCT iajs-3638 165 6 , , PUNCT iajs-3638 165 7 equation equation NOUN iajs-3638 165 8 ( ( PUNCT iajs-3638 165 9 5 5 NUM iajs-3638 165 10 ) ) PUNCT iajs-3638 165 11 reduces reduce VERB iajs-3638 165 12 to to ADP iajs-3638 165 13 𝐽(𝐸3 𝐽(𝐸3 PUNCT iajs-3638 165 14 ) ) PUNCT iajs-3638 166 1 = = PUNCT iajs-3638 166 2 [ [ PUNCT iajs-3638 166 3 βˆ’ βˆ’ X iajs-3638 166 4 ( ( PUNCT iajs-3638 166 5 𝛼 𝛼 NOUN iajs-3638 166 6 1 1 NUM iajs-3638 166 7 + + NUM iajs-3638 166 8 𝑛 𝑛 PROPN iajs-3638 166 9 οΏ½ οΏ½ PROPN iajs-3638 166 10 Μ† Μ† NOUN iajs-3638 166 11 οΏ½ οΏ½ PROPN iajs-3638 166 12 1 1 NUM iajs-3638 166 13 + + CCONJ iajs-3638 166 14 πœ‡ πœ‡ NOUN iajs-3638 166 15 ) ) PUNCT iajs-3638 166 16 𝑛𝛼 𝑛𝛼 PROPN iajs-3638 166 17 οΏ½ οΏ½ PROPN iajs-3638 166 18 Μ† Μ† PROPN iajs-3638 166 19 οΏ½ οΏ½ PROPN iajs-3638 166 20 ( ( PUNCT iajs-3638 166 21 1 1 NUM iajs-3638 166 22 + + NUM iajs-3638 166 23 𝑛 𝑛 PROPN iajs-3638 166 24 οΏ½ οΏ½ PROPN iajs-3638 166 25 Μ† Μ† NOUN iajs-3638 166 26 οΏ½ οΏ½ PROPN iajs-3638 166 27 1 1 NUM iajs-3638 166 28 ) ) PUNCT iajs-3638 166 29 2 2 NUM iajs-3638 166 30 0 0 NUM iajs-3638 166 31 βˆ’π›½1 βˆ’π›½1 PROPN iajs-3638 166 32 οΏ½ οΏ½ PROPN iajs-3638 166 33 Μ† Μ† PROPN iajs-3638 166 34 οΏ½ οΏ½ PROPN iajs-3638 166 35 𝑒 𝑒 PROPN iajs-3638 166 36 βˆ’πœ†πœ βˆ’πœ†πœ PROPN iajs-3638 166 37 𝛼 𝛼 PROPN iajs-3638 166 38 1 1 NUM iajs-3638 166 39 + + NUM iajs-3638 166 40 𝑛 𝑛 PROPN iajs-3638 166 41 οΏ½ οΏ½ PROPN iajs-3638 166 42 Μ† Μ† NOUN iajs-3638 166 43 οΏ½ οΏ½ PROPN iajs-3638 166 44 1 1 NUM iajs-3638 167 1 βˆ’ βˆ’ PROPN iajs-3638 167 2 ( ( PUNCT iajs-3638 167 3 𝑛𝛼 𝑛𝛼 PROPN iajs-3638 167 4 οΏ½ οΏ½ PROPN iajs-3638 167 5 Μ† Μ† NOUN iajs-3638 167 6 οΏ½ οΏ½ PROPN iajs-3638 167 7 ( ( PUNCT iajs-3638 167 8 1 1 NUM iajs-3638 167 9 + + NUM iajs-3638 167 10 𝑛 𝑛 PROPN iajs-3638 167 11 οΏ½ οΏ½ PROPN iajs-3638 167 12 Μ† Μ† NOUN iajs-3638 167 13 οΏ½ οΏ½ PROPN iajs-3638 167 14 1 1 NUM iajs-3638 167 15 ) ) PUNCT iajs-3638 167 16 2 2 NUM iajs-3638 167 17 + + CCONJ iajs-3638 167 18 𝛾 𝛾 ADP iajs-3638 167 19 + + SYM iajs-3638 167 20 πœ‡ πœ‡ X iajs-3638 167 21 ) ) PUNCT iajs-3638 167 22 0 0 PUNCT iajs-3638 168 1 βˆ’π›½2 βˆ’π›½2 PROPN iajs-3638 168 2 οΏ½ οΏ½ PROPN iajs-3638 168 3 Μ† Μ† NOUN iajs-3638 168 4 οΏ½ οΏ½ PROPN iajs-3638 168 5 1 1 NUM iajs-3638 168 6 0 0 NUM iajs-3638 168 7 𝛾 𝛾 ADP iajs-3638 168 8 βˆ’πœ‡ βˆ’πœ‡ VERB iajs-3638 168 9 βˆ’π›½3 βˆ’π›½3 PROPN iajs-3638 168 10 οΏ½ οΏ½ PROPN iajs-3638 168 11 Μ† Μ† NOUN iajs-3638 168 12 οΏ½ οΏ½ PROPN iajs-3638 168 13 2 2 NUM iajs-3638 168 14 0 0 NUM iajs-3638 168 15 0 0 NUM iajs-3638 168 16 0 0 NUM iajs-3638 168 17 𝛽1 𝛽1 PROPN iajs-3638 168 18 οΏ½ οΏ½ PROPN iajs-3638 168 19 Μ† Μ† NOUN iajs-3638 168 20 οΏ½ οΏ½ PROPN iajs-3638 168 21 𝑒 𝑒 PROPN iajs-3638 168 22 βˆ’πœ†πœ βˆ’πœ†πœ PROPN iajs-3638 168 23 + + CCONJ iajs-3638 168 24 𝛽2 𝛽2 PROPN iajs-3638 168 25 οΏ½ οΏ½ PROPN iajs-3638 168 26 Μ† Μ† NOUN iajs-3638 168 27 οΏ½ οΏ½ PROPN iajs-3638 168 28 1 1 NUM iajs-3638 168 29 + + NUM iajs-3638 168 30 𝛽3 𝛽3 PROPN iajs-3638 168 31 οΏ½ οΏ½ PROPN iajs-3638 168 32 Μ† Μ† NOUN iajs-3638 168 33 οΏ½ οΏ½ PROPN iajs-3638 168 34 2 2 NUM iajs-3638 168 35 βˆ’ βˆ’ NOUN iajs-3638 168 36 ( ( PUNCT iajs-3638 168 37 πœ‡ πœ‡ ADP iajs-3638 168 38 + + X iajs-3638 168 39 πœ‡1 πœ‡1 ADJ iajs-3638 168 40 + + CCONJ iajs-3638 168 41 πœƒ πœƒ NOUN iajs-3638 168 42 ) ) PUNCT iajs-3638 168 43 ] ] PUNCT iajs-3638 168 44 ( ( PUNCT iajs-3638 168 45 28 28 NUM iajs-3638 168 46 ) ) PUNCT iajs-3638 168 47 the the DET iajs-3638 168 48 c.e c.e PROPN iajs-3638 168 49 . . PROPN iajs-3638 168 50 of of ADP iajs-3638 168 51 𝐽(𝐸3 𝐽(𝐸3 PUNCT iajs-3638 168 52 ) ) PUNCT iajs-3638 168 53 is be AUX iajs-3638 168 54 given give VERB iajs-3638 168 55 by by ADP iajs-3638 168 56 [ [ X iajs-3638 168 57 πœ†2 πœ†2 X iajs-3638 168 58 + + CCONJ iajs-3638 168 59 𝐢1πœ† 𝐢1πœ† PROPN iajs-3638 168 60 + + CCONJ iajs-3638 168 61 𝐢2][βˆ’πœ‡ 𝐢2][βˆ’πœ‡ ADJ iajs-3638 168 62 βˆ’ βˆ’ ADP iajs-3638 168 63 πœ† πœ† NOUN iajs-3638 168 64 ] ] X iajs-3638 168 65 [ [ PUNCT iajs-3638 168 66 𝛽1 𝛽1 NOUN iajs-3638 168 67 οΏ½ οΏ½ PROPN iajs-3638 168 68 Μ† Μ† NOUN iajs-3638 168 69 οΏ½ οΏ½ PROPN iajs-3638 168 70 𝑒 𝑒 PROPN iajs-3638 168 71 βˆ’πœ†πœ βˆ’πœ†πœ PROPN iajs-3638 168 72 + + CCONJ iajs-3638 169 1 𝛽2 𝛽2 PROPN iajs-3638 169 2 οΏ½ οΏ½ PROPN iajs-3638 169 3 Μ† Μ† NOUN iajs-3638 169 4 οΏ½ οΏ½ PROPN iajs-3638 169 5 1 1 NUM iajs-3638 169 6 + + NUM iajs-3638 169 7 𝛽3 𝛽3 PROPN iajs-3638 169 8 οΏ½ οΏ½ PROPN iajs-3638 169 9 Μ† Μ† NOUN iajs-3638 169 10 οΏ½ οΏ½ PROPN iajs-3638 169 11 2 2 NUM iajs-3638 169 12 βˆ’ βˆ’ NOUN iajs-3638 169 13 ( ( PUNCT iajs-3638 169 14 πœ‡ πœ‡ ADP iajs-3638 169 15 + + X iajs-3638 169 16 πœ‡1 πœ‡1 ADJ iajs-3638 169 17 + + CCONJ iajs-3638 169 18 πœƒ πœƒ NOUN iajs-3638 169 19 ) ) PUNCT iajs-3638 169 20 βˆ’ βˆ’ PUNCT iajs-3638 170 1 πœ† πœ† X iajs-3638 170 2 ] ] X iajs-3638 170 3 = = SYM iajs-3638 170 4 0 0 NUM iajs-3638 170 5 ( ( PUNCT iajs-3638 170 6 29a 29a NUM iajs-3638 170 7 ) ) PUNCT iajs-3638 170 8 ihjpas ihjpas PROPN iajs-3638 170 9 . . PUNCT iajs-3638 171 1 2025 2025 NUM iajs-3638 171 2 , , PUNCT iajs-3638 171 3 38 38 NUM iajs-3638 171 4 ( ( PUNCT iajs-3638 171 5 1 1 NUM iajs-3638 171 6 ) ) PUNCT iajs-3638 171 7 452 452 NUM iajs-3638 171 8 where where SCONJ iajs-3638 171 9 𝐢1 𝐢1 ADV iajs-3638 171 10 = = SYM iajs-3638 171 11 𝛼 𝛼 PROPN iajs-3638 171 12 1+𝑛𝑉1 1+𝑛𝑉1 NUM iajs-3638 171 13 ( ( PUNCT iajs-3638 171 14 1 1 NUM iajs-3638 171 15 + + NUM iajs-3638 171 16 𝑛 𝑛 PROPN iajs-3638 171 17 οΏ½ οΏ½ PROPN iajs-3638 171 18 Μ† Μ† NOUN iajs-3638 171 19 οΏ½ οΏ½ PROPN iajs-3638 171 20 1+𝑛𝑉1 1+𝑛𝑉1 NUM iajs-3638 171 21 ) ) PUNCT iajs-3638 172 1 + + CCONJ iajs-3638 172 2 𝛾 𝛾 X iajs-3638 172 3 + + X iajs-3638 172 4 2πœ‡ 2πœ‡ NUM iajs-3638 172 5 𝐢2 𝐢2 X iajs-3638 173 1 = = SYM iajs-3638 173 2 𝛼 𝛼 PROPN iajs-3638 173 3 1+𝑛𝑉1 1+𝑛𝑉1 NUM iajs-3638 173 4 ( ( PUNCT iajs-3638 173 5 π‘›πœ‡ π‘›πœ‡ PROPN iajs-3638 173 6 οΏ½ οΏ½ PROPN iajs-3638 173 7 Μ† Μ† NOUN iajs-3638 173 8 οΏ½ οΏ½ PROPN iajs-3638 173 9 1+𝑛𝑉1 1+𝑛𝑉1 NUM iajs-3638 173 10 + + CCONJ iajs-3638 173 11 𝛾 𝛾 X iajs-3638 173 12 + + X iajs-3638 173 13 πœ‡ πœ‡ X iajs-3638 173 14 ) ) PUNCT iajs-3638 173 15 + + CCONJ iajs-3638 173 16 πœ‡(𝛾 πœ‡(𝛾 PROPN iajs-3638 173 17 + + CCONJ iajs-3638 173 18 πœ‡ πœ‡ X iajs-3638 173 19 ) ) PUNCT iajs-3638 173 20 the the DET iajs-3638 173 21 eq eq NOUN iajs-3638 173 22 . . PUNCT iajs-3638 173 23 ( ( PUNCT iajs-3638 173 24 29a 29a NUM iajs-3638 173 25 ) ) PUNCT iajs-3638 173 26 represents represent VERB iajs-3638 173 27 the the DET iajs-3638 173 28 eigenvalues eigenvalue NOUN iajs-3638 173 29 of of ADP iajs-3638 173 30 𝐽(𝐸3 𝐽(𝐸3 PUNCT iajs-3638 173 31 ) ) PUNCT iajs-3638 173 32 and and CCONJ iajs-3638 173 33 has have VERB iajs-3638 173 34 4 4 NUM iajs-3638 173 35 - - PUNCT iajs-3638 173 36 roots root NOUN iajs-3638 173 37 : : PUNCT iajs-3638 174 1 πœ†1,2 πœ†1,2 ADJ iajs-3638 174 2 = = PUNCT iajs-3638 174 3 βˆ’ βˆ’ NOUN iajs-3638 174 4 𝐢1 𝐢1 NOUN iajs-3638 174 5 2 2 NUM iajs-3638 174 6 βˆ“ βˆ“ NOUN iajs-3638 174 7 1 1 NUM iajs-3638 174 8 2 2 NUM iajs-3638 174 9 √𝐢1 √𝐢1 NOUN iajs-3638 174 10 2 2 NUM iajs-3638 174 11 βˆ’ βˆ’ NOUN iajs-3638 174 12 4𝐢2 4𝐢2 NUM iajs-3638 174 13 πœ†3 πœ†3 NOUN iajs-3638 174 14 = = SYM iajs-3638 174 15 βˆ’πœ‡ βˆ’πœ‡ VERB iajs-3638 174 16 πœ†4 πœ†4 NOUN iajs-3638 174 17 = = SYM iajs-3638 174 18 𝛽1 𝛽1 PROPN iajs-3638 174 19 οΏ½ οΏ½ PROPN iajs-3638 174 20 Μ† Μ† NOUN iajs-3638 174 21 οΏ½ οΏ½ PROPN iajs-3638 174 22 𝑒 𝑒 PROPN iajs-3638 174 23 βˆ’πœ†πœ βˆ’πœ†πœ PROPN iajs-3638 174 24 + + CCONJ iajs-3638 174 25 𝛽2 𝛽2 PROPN iajs-3638 174 26 οΏ½ οΏ½ PROPN iajs-3638 174 27 Μ† Μ† NOUN iajs-3638 174 28 οΏ½ οΏ½ PROPN iajs-3638 174 29 1 1 NUM iajs-3638 174 30 + + NUM iajs-3638 174 31 𝛽3 𝛽3 PROPN iajs-3638 174 32 οΏ½ οΏ½ PROPN iajs-3638 174 33 Μ† Μ† NOUN iajs-3638 174 34 οΏ½ οΏ½ PROPN iajs-3638 174 35 2 2 NUM iajs-3638 174 36 βˆ’ βˆ’ NOUN iajs-3638 174 37 ( ( PUNCT iajs-3638 174 38 πœ‡ πœ‡ ADP iajs-3638 174 39 + + X iajs-3638 174 40 πœ‡1 πœ‡1 ADJ iajs-3638 174 41 + + CCONJ iajs-3638 174 42 πœƒ πœƒ NOUN iajs-3638 174 43 ) ) PUNCT iajs-3638 174 44 } } PUNCT iajs-3638 174 45 ( ( PUNCT iajs-3638 174 46 29b 29b NOUN iajs-3638 174 47 ) ) PUNCT iajs-3638 174 48 now now ADV iajs-3638 174 49 , , PUNCT iajs-3638 174 50 for for ADP iajs-3638 174 51 𝜏 𝜏 NOUN iajs-3638 174 52 = = SYM iajs-3638 174 53 0 0 NUM iajs-3638 174 54 we we PRON iajs-3638 174 55 get get VERB iajs-3638 174 56 all all DET iajs-3638 174 57 the the DET iajs-3638 174 58 above above ADJ iajs-3638 174 59 eigenvalues eigenvalue NOUN iajs-3638 174 60 will will AUX iajs-3638 174 61 be be AUX iajs-3638 174 62 negative negative ADJ iajs-3638 174 63 and and CCONJ iajs-3638 174 64 the the DET iajs-3638 174 65 foep foep NOUN iajs-3638 174 66 is be AUX iajs-3638 174 67 l.a.s l.a.s ADJ iajs-3638 174 68 . . PUNCT iajs-3638 175 1 if if SCONJ iajs-3638 175 2 the the DET iajs-3638 175 3 following follow VERB iajs-3638 175 4 condition condition NOUN iajs-3638 175 5 𝛽1 𝛽1 PROPN iajs-3638 175 6 οΏ½ οΏ½ PROPN iajs-3638 175 7 Μ† Μ† NOUN iajs-3638 175 8 οΏ½ οΏ½ PROPN iajs-3638 175 9 + + CCONJ iajs-3638 175 10 𝛽2 𝛽2 PROPN iajs-3638 175 11 οΏ½ οΏ½ PROPN iajs-3638 175 12 Μ† Μ† NOUN iajs-3638 175 13 οΏ½ οΏ½ PROPN iajs-3638 175 14 1 1 NUM iajs-3638 175 15 + + NUM iajs-3638 175 16 𝛽3 𝛽3 PROPN iajs-3638 175 17 οΏ½ οΏ½ PROPN iajs-3638 175 18 Μ† Μ† NOUN iajs-3638 175 19 οΏ½ οΏ½ PROPN iajs-3638 175 20 2 2 NUM iajs-3638 175 21 < < X iajs-3638 175 22 πœ‡ πœ‡ ADP iajs-3638 175 23 + + X iajs-3638 175 24 πœ‡1 πœ‡1 ADJ iajs-3638 175 25 + + CCONJ iajs-3638 175 26 πœƒ πœƒ PROPN iajs-3638 175 27 ( ( PUNCT iajs-3638 175 28 30 30 NUM iajs-3638 175 29 ) ) PUNCT iajs-3638 175 30 now now ADV iajs-3638 175 31 , , PUNCT iajs-3638 175 32 for for ADP iajs-3638 175 33 𝜏 𝜏 PROPN iajs-3638 175 34 > > SYM iajs-3638 175 35 0 0 PUNCT iajs-3638 175 36 suppose suppose VERB iajs-3638 175 37 that that SCONJ iajs-3638 175 38 ( ( PUNCT iajs-3638 175 39 29b 29b NUM iajs-3638 175 40 ) ) PUNCT iajs-3638 175 41 , , PUNCT iajs-3638 175 42 has have VERB iajs-3638 175 43 two two NUM iajs-3638 175 44 wholly wholly ADV iajs-3638 175 45 imaginary imaginary ADJ iajs-3638 175 46 roots root NOUN iajs-3638 175 47 , , PUNCT iajs-3638 175 48 namelyπœ† namelyπœ† NOUN iajs-3638 175 49 = = PUNCT iajs-3638 175 50 Β±π‘–πœ” Β±π‘–πœ” PROPN iajs-3638 175 51 ( ( PUNCT iajs-3638 175 52 πœ” πœ” X iajs-3638 175 53 > > X iajs-3638 175 54 0 0 NUM iajs-3638 175 55 ) ) PUNCT iajs-3638 175 56 . . PUNCT iajs-3638 176 1 by by ADP iajs-3638 176 2 substituting substitute VERB iajs-3638 176 3 πœ† πœ† PRON iajs-3638 176 4 = = SYM iajs-3638 176 5 Β±π‘–πœ” Β±π‘–πœ” NOUN iajs-3638 176 6 in in ADP iajs-3638 176 7 equation equation NOUN iajs-3638 176 8 ( ( PUNCT iajs-3638 176 9 29b 29b NUM iajs-3638 176 10 ) ) PUNCT iajs-3638 176 11 we we PRON iajs-3638 176 12 get get VERB iajs-3638 176 13 : : PUNCT iajs-3638 176 14 𝛽1 𝛽1 ADJ iajs-3638 176 15 οΏ½ οΏ½ PROPN iajs-3638 176 16 Μ† Μ† NOUN iajs-3638 176 17 οΏ½ οΏ½ PROPN iajs-3638 176 18 (π‘π‘œπ‘ πœ”πœ (π‘π‘œπ‘ πœ”πœ PROPN iajs-3638 176 19 βˆ’ βˆ’ NOUN iajs-3638 176 20 π‘–π‘ π‘–π‘›πœ”πœ π‘–π‘ π‘–π‘›πœ”πœ ADJ iajs-3638 176 21 ) ) PUNCT iajs-3638 177 1 = = PUNCT iajs-3638 177 2 πœ‡ πœ‡ ADP iajs-3638 177 3 + + X iajs-3638 177 4 πœ‡1 πœ‡1 ADJ iajs-3638 177 5 + + CCONJ iajs-3638 177 6 πœƒ πœƒ NOUN iajs-3638 177 7 βˆ’ βˆ’ PROPN iajs-3638 177 8 𝛽2 𝛽2 PROPN iajs-3638 177 9 οΏ½ οΏ½ PROPN iajs-3638 177 10 Μ† Μ† NOUN iajs-3638 177 11 οΏ½ οΏ½ PROPN iajs-3638 177 12 1 1 NUM iajs-3638 177 13 βˆ’ βˆ’ PROPN iajs-3638 177 14 𝛽3 𝛽3 PROPN iajs-3638 177 15 οΏ½ οΏ½ PROPN iajs-3638 177 16 Μ† Μ† NOUN iajs-3638 177 17 οΏ½ οΏ½ PROPN iajs-3638 177 18 2 2 NUM iajs-3638 177 19 + + NUM iajs-3638 177 20 π‘–πœ” π‘–πœ” PRON iajs-3638 177 21 so so ADV iajs-3638 177 22 , , PUNCT iajs-3638 177 23 separating separate VERB iajs-3638 177 24 the the DET iajs-3638 177 25 real real ADJ iajs-3638 177 26 and and CCONJ iajs-3638 177 27 imagined imagined ADJ iajs-3638 177 28 parts part NOUN iajs-3638 177 29 yields yield NOUN iajs-3638 177 30 𝛽1 𝛽1 PROPN iajs-3638 177 31 οΏ½ οΏ½ PROPN iajs-3638 177 32 Μ† Μ† NOUN iajs-3638 177 33 οΏ½ οΏ½ PROPN iajs-3638 177 34 π‘π‘œπ‘ πœ”πœ π‘π‘œπ‘ πœ”πœ NOUN iajs-3638 177 35 = = SYM iajs-3638 177 36 πœ‡ πœ‡ ADP iajs-3638 177 37 + + X iajs-3638 177 38 πœ‡1 πœ‡1 ADJ iajs-3638 177 39 + + CCONJ iajs-3638 177 40 πœƒ πœƒ NOUN iajs-3638 177 41 βˆ’ βˆ’ PROPN iajs-3638 177 42 𝛽2 𝛽2 PROPN iajs-3638 177 43 οΏ½ οΏ½ PROPN iajs-3638 177 44 Μ† Μ† NOUN iajs-3638 177 45 οΏ½ οΏ½ PROPN iajs-3638 177 46 1 1 NUM iajs-3638 177 47 βˆ’ βˆ’ PROPN iajs-3638 177 48 𝛽3 𝛽3 PROPN iajs-3638 177 49 οΏ½ οΏ½ PROPN iajs-3638 177 50 Μ† Μ† PROPN iajs-3638 177 51 οΏ½ οΏ½ PROPN iajs-3638 177 52 2 2 NUM iajs-3638 177 53 𝛽1 𝛽1 PROPN iajs-3638 177 54 οΏ½ οΏ½ PROPN iajs-3638 177 55 Μ† Μ† NOUN iajs-3638 177 56 οΏ½ οΏ½ PROPN iajs-3638 177 57 π‘ π‘–π‘›πœ”πœ π‘ π‘–π‘›πœ”πœ VERB iajs-3638 177 58 = = SYM iajs-3638 177 59 βˆ’πœ” βˆ’πœ” NOUN iajs-3638 177 60 } } PUNCT iajs-3638 177 61 ( ( PUNCT iajs-3638 177 62 31 31 NUM iajs-3638 177 63 ) ) PUNCT iajs-3638 177 64 squaring square VERB iajs-3638 177 65 each each DET iajs-3638 177 66 equation equation NOUN iajs-3638 177 67 and and CCONJ iajs-3638 177 68 then then ADV iajs-3638 177 69 adding add VERB iajs-3638 177 70 them they PRON iajs-3638 177 71 , , PUNCT iajs-3638 177 72 we we PRON iajs-3638 177 73 get get VERB iajs-3638 177 74 that that DET iajs-3638 177 75 πœ” πœ” ADP iajs-3638 177 76 = = PUNCT iajs-3638 177 77 βˆ“βˆšπ›½1 βˆ“βˆšπ›½1 PROPN iajs-3638 177 78 2( 2( NUM iajs-3638 177 79 οΏ½ οΏ½ PROPN iajs-3638 177 80 Μ† Μ† NOUN iajs-3638 177 81 οΏ½ οΏ½ PROPN iajs-3638 177 82 ) ) PUNCT iajs-3638 178 1 2 2 NUM iajs-3638 178 2 βˆ’ βˆ’ NOUN iajs-3638 178 3 ( ( PUNCT iajs-3638 178 4 πœ‡ πœ‡ ADP iajs-3638 178 5 + + X iajs-3638 178 6 πœ‡1 πœ‡1 ADJ iajs-3638 178 7 + + CCONJ iajs-3638 178 8 πœƒ πœƒ NOUN iajs-3638 178 9 βˆ’ βˆ’ PROPN iajs-3638 178 10 𝛽2 𝛽2 PROPN iajs-3638 178 11 οΏ½ οΏ½ PROPN iajs-3638 178 12 Μ† Μ† NOUN iajs-3638 178 13 οΏ½ οΏ½ PROPN iajs-3638 178 14 1 1 NUM iajs-3638 178 15 βˆ’ βˆ’ PROPN iajs-3638 178 16 𝛽3 𝛽3 PROPN iajs-3638 178 17 οΏ½ οΏ½ PROPN iajs-3638 178 18 Μ† Μ† NOUN iajs-3638 178 19 οΏ½ οΏ½ PROPN iajs-3638 178 20 2 2 NUM iajs-3638 178 21 ) ) PUNCT iajs-3638 178 22 2 2 NUM iajs-3638 178 23 note note VERB iajs-3638 178 24 that that SCONJ iajs-3638 178 25 , , PUNCT iajs-3638 178 26 under under ADP iajs-3638 178 27 the the DET iajs-3638 178 28 condition condition NOUN iajs-3638 178 29 ( ( PUNCT iajs-3638 178 30 30),πœ”(𝜏 30),πœ”(𝜏 NUM iajs-3638 178 31 ) ) PUNCT iajs-3638 178 32 with with ADP iajs-3638 178 33 𝜏 𝜏 ADP iajs-3638 178 34 > > SYM iajs-3638 178 35 0 0 NUM iajs-3638 178 36 can can AUX iajs-3638 178 37 not not PART iajs-3638 178 38 be be AUX iajs-3638 178 39 real real ADJ iajs-3638 178 40 , , PUNCT iajs-3638 178 41 which which PRON iajs-3638 178 42 contradicts contradict VERB iajs-3638 178 43 with with ADP iajs-3638 178 44 the the DET iajs-3638 178 45 assumption assumption NOUN iajs-3638 178 46 . . PUNCT iajs-3638 179 1 therefore therefore ADV iajs-3638 179 2 , , PUNCT iajs-3638 179 3 the the DET iajs-3638 179 4 c.e c.e PROPN iajs-3638 179 5 . . PROPN iajs-3638 179 6 ( ( PUNCT iajs-3638 179 7 29a 29a NUM iajs-3638 179 8 ) ) PUNCT iajs-3638 179 9 ca can AUX iajs-3638 179 10 n’t not PART iajs-3638 179 11 have have VERB iajs-3638 179 12 purely purely ADV iajs-3638 179 13 imaginary imaginary ADJ iajs-3638 179 14 root root NOUN iajs-3638 179 15 , , PUNCT iajs-3638 179 16 and and CCONJ iajs-3638 179 17 𝐸1 𝐸1 NOUN iajs-3638 179 18 is be AUX iajs-3638 179 19 l.a.s l.a.s ADJ iajs-3638 179 20 . . PUNCT iajs-3638 180 1 for for ADP iajs-3638 180 2 all all DET iajs-3638 180 3 𝜏 𝜏 DET iajs-3638 180 4 β‰₯ β‰₯ NOUN iajs-3638 180 5 0 0 NUM iajs-3638 181 1 if if SCONJ iajs-3638 181 2 the the DET iajs-3638 181 3 condition condition NOUN iajs-3638 181 4 ( ( PUNCT iajs-3638 181 5 30 30 NUM iajs-3638 181 6 ) ) PUNCT iajs-3638 181 7 hold hold NOUN iajs-3638 181 8 . . PUNCT iajs-3638 182 1 for for ADP iajs-3638 182 2 the the DET iajs-3638 182 3 ( ( PUNCT iajs-3638 182 4 fiep fiep PROPN iajs-3638 182 5 ) ) PUNCT iajs-3638 182 6 when when SCONJ iajs-3638 182 7 𝛾 𝛾 NOUN iajs-3638 182 8 = = SYM iajs-3638 182 9 0 0 NUM iajs-3638 182 10 , , PUNCT iajs-3638 182 11 equation equation NOUN iajs-3638 182 12 ( ( PUNCT iajs-3638 182 13 4 4 NUM iajs-3638 182 14 ) ) PUNCT iajs-3638 182 15 reduces reduce VERB iajs-3638 182 16 to to ADP iajs-3638 182 17 𝐽(𝐸4 𝐽(𝐸4 ADV iajs-3638 182 18 ) ) PUNCT iajs-3638 182 19 = = PUNCT iajs-3638 183 1 ( ( PUNCT iajs-3638 183 2 𝑑𝑖𝑗)4Γ—4 𝑑𝑖𝑗)4Γ—4 PROPN iajs-3638 183 3 ; ; PUNCT iajs-3638 183 4 𝑖 𝑖 X iajs-3638 183 5 , , PUNCT iajs-3638 183 6 𝑗 𝑗 NOUN iajs-3638 183 7 = = NOUN iajs-3638 183 8 1,2,3,4 1,2,3,4 NUM iajs-3638 183 9 here here ADV iajs-3638 183 10 𝑑11 𝑑11 NOUN iajs-3638 183 11 = = SYM iajs-3638 183 12 βˆ’[𝑅1 βˆ’[𝑅1 X iajs-3638 183 13 + + PUNCT iajs-3638 183 14 𝑅2𝑒 𝑅2𝑒 PROPN iajs-3638 183 15 βˆ’πœ†πœ βˆ’πœ†πœ PROPN iajs-3638 183 16 ] ] PUNCT iajs-3638 183 17 , , PUNCT iajs-3638 183 18 𝑑12 𝑑12 PROPN iajs-3638 183 19 = = SYM iajs-3638 183 20 𝑛𝛼 𝑛𝛼 PROPN iajs-3638 183 21 οΏ½ οΏ½ PROPN iajs-3638 183 22 ΜƒΜƒ ΜƒΜƒ PROPN iajs-3638 183 23 οΏ½ οΏ½ PROPN iajs-3638 183 24 ( ( PUNCT iajs-3638 183 25 1 1 NUM iajs-3638 183 26 + + NUM iajs-3638 183 27 𝑛 𝑛 PROPN iajs-3638 183 28 οΏ½ οΏ½ PROPN iajs-3638 183 29 ΜƒΜƒ ΜƒΜƒ NOUN iajs-3638 183 30 οΏ½ οΏ½ NOUN iajs-3638 183 31 1 1 NUM iajs-3638 183 32 ) ) PUNCT iajs-3638 183 33 2 2 NUM iajs-3638 183 34 , , PUNCT iajs-3638 183 35 𝑑14 𝑑14 NOUN iajs-3638 183 36 = = SYM iajs-3638 183 37 βˆ’π‘…3𝑒 βˆ’π‘…3𝑒 X iajs-3638 183 38 βˆ’πœ†πœ βˆ’πœ†πœ PROPN iajs-3638 183 39 , , PUNCT iajs-3638 183 40 𝑑21 𝑑21 PROPN iajs-3638 184 1 = = PUNCT iajs-3638 184 2 𝛼 𝛼 PROPN iajs-3638 184 3 1 1 NUM iajs-3638 184 4 + + NUM iajs-3638 184 5 𝑛 𝑛 PROPN iajs-3638 184 6 οΏ½ οΏ½ PROPN iajs-3638 184 7 ΜƒΜƒ ΜƒΜƒ NOUN iajs-3638 184 8 οΏ½ οΏ½ PROPN iajs-3638 184 9 1 1 NUM iajs-3638 184 10 , , PUNCT iajs-3638 184 11 𝑑22 𝑑22 NOUN iajs-3638 184 12 = = SYM iajs-3638 184 13 βˆ’ βˆ’ PROPN iajs-3638 184 14 ( ( PUNCT iajs-3638 184 15 𝑛𝛼 𝑛𝛼 PROPN iajs-3638 184 16 οΏ½ οΏ½ PROPN iajs-3638 184 17 ΜƒΜƒ ΜƒΜƒ PROPN iajs-3638 184 18 οΏ½ οΏ½ PROPN iajs-3638 184 19 ( ( PUNCT iajs-3638 184 20 1 1 NUM iajs-3638 184 21 + + NUM iajs-3638 184 22 𝑛 𝑛 PROPN iajs-3638 184 23 οΏ½ οΏ½ PROPN iajs-3638 184 24 ΜƒΜƒ ΜƒΜƒ NOUN iajs-3638 184 25 οΏ½ οΏ½ NOUN iajs-3638 184 26 1 1 NUM iajs-3638 184 27 ) ) PUNCT iajs-3638 184 28 2 2 NUM iajs-3638 185 1 + + CCONJ iajs-3638 185 2 𝛽2𝐼 𝛽2𝐼 ADV iajs-3638 185 3 + + CCONJ iajs-3638 185 4 πœ‡ πœ‡ X iajs-3638 185 5 ) ) PUNCT iajs-3638 185 6 , , PUNCT iajs-3638 185 7 𝑑24 𝑑24 PROPN iajs-3638 185 8 = = SYM iajs-3638 185 9 βˆ’π›½2 βˆ’π›½2 PROPN iajs-3638 185 10 οΏ½ οΏ½ PROPN iajs-3638 185 11 ΜƒΜƒ ΜƒΜƒ NOUN iajs-3638 185 12 οΏ½ οΏ½ PROPN iajs-3638 185 13 1 1 NUM iajs-3638 185 14 , , PUNCT iajs-3638 185 15 𝑑33 𝑑33 X iajs-3638 185 16 = = SYM iajs-3638 185 17 βˆ’(𝛽3𝐼 βˆ’(𝛽3𝐼 PROPN iajs-3638 185 18 + + CCONJ iajs-3638 185 19 πœ‡ πœ‡ X iajs-3638 185 20 ) ) PUNCT iajs-3638 185 21 , , PUNCT iajs-3638 185 22 𝑑41 𝑑41 NOUN iajs-3638 185 23 = = SYM iajs-3638 185 24 𝑅2𝑒 𝑅2𝑒 PROPN iajs-3638 185 25 βˆ’πœ†πœ βˆ’πœ†πœ PROPN iajs-3638 185 26 , , PUNCT iajs-3638 185 27 𝑑42 𝑑42 PROPN iajs-3638 185 28 = = PUNCT iajs-3638 185 29 𝛽2𝐼 𝛽2𝐼 ADV iajs-3638 185 30 , , PUNCT iajs-3638 185 31 𝑑43 𝑑43 NOUN iajs-3638 185 32 = = PUNCT iajs-3638 185 33 𝛽3𝐼 𝛽3𝐼 ADJ iajs-3638 185 34 , , PUNCT iajs-3638 185 35 𝑑44 𝑑44 NOUN iajs-3638 185 36 = = SYM iajs-3638 185 37 𝑅3𝑒 𝑅3𝑒 PROPN iajs-3638 185 38 βˆ’πœ†πœ βˆ’πœ†πœ PROPN iajs-3638 185 39 + + CCONJ iajs-3638 185 40 𝑅4 𝑅4 PROPN iajs-3638 185 41 , , PUNCT iajs-3638 185 42 𝑑13 𝑑13 ADJ iajs-3638 185 43 = = SYM iajs-3638 185 44 𝑑23 𝑑23 NOUN iajs-3638 185 45 = = NOUN iajs-3638 185 46 𝑑31 𝑑31 NOUN iajs-3638 185 47 = = NOUN iajs-3638 185 48 𝑑32 𝑑32 NOUN iajs-3638 185 49 = = PUNCT iajs-3638 185 50 𝑑34 𝑑34 NOUN iajs-3638 186 1 = = NOUN iajs-3638 186 2 0 0 PROPN iajs-3638 186 3 . . PUNCT iajs-3638 187 1 ( ( PUNCT iajs-3638 187 2 32 32 NUM iajs-3638 187 3 ) ) PUNCT iajs-3638 187 4 where where SCONJ iajs-3638 187 5 , , PUNCT iajs-3638 187 6 𝑅1 𝑅1 PROPN iajs-3638 187 7 = = SYM iajs-3638 187 8 𝛼 𝛼 PROPN iajs-3638 187 9 1+𝑛 1+𝑛 NUM iajs-3638 187 10 οΏ½ οΏ½ PROPN iajs-3638 187 11 ΜƒΜƒ ΜƒΜƒ NOUN iajs-3638 187 12 οΏ½ οΏ½ PROPN iajs-3638 187 13 1 1 NUM iajs-3638 187 14 + + CCONJ iajs-3638 187 15 πœ‡ πœ‡ ADP iajs-3638 187 16 , , PUNCT iajs-3638 187 17 𝑅2 𝑅2 ADP iajs-3638 187 18 = = SYM iajs-3638 187 19 𝛽1𝐼 𝛽1𝐼 NOUN iajs-3638 187 20 , , PUNCT iajs-3638 187 21 𝑅3 𝑅3 PROPN iajs-3638 187 22 = = SYM iajs-3638 187 23 𝛽1 𝛽1 PROPN iajs-3638 187 24 οΏ½ οΏ½ PROPN iajs-3638 187 25 ΜƒΜƒ ΜƒΜƒ PROPN iajs-3638 187 26 οΏ½ οΏ½ PROPN iajs-3638 187 27 , , PUNCT iajs-3638 187 28 𝑅4 𝑅4 PROPN iajs-3638 187 29 = = SYM iajs-3638 187 30 𝛽2 𝛽2 PROPN iajs-3638 187 31 οΏ½ οΏ½ PROPN iajs-3638 187 32 ΜƒΜƒ ΜƒΜƒ NOUN iajs-3638 187 33 οΏ½ οΏ½ PROPN iajs-3638 187 34 1 1 NUM iajs-3638 187 35 βˆ’ βˆ’ PROPN iajs-3638 187 36 ( ( PUNCT iajs-3638 187 37 πœ‡ πœ‡ ADP iajs-3638 187 38 + + X iajs-3638 187 39 πœ‡1 πœ‡1 ADJ iajs-3638 187 40 + + CCONJ iajs-3638 187 41 πœƒ πœƒ NOUN iajs-3638 187 42 ) ) PUNCT iajs-3638 187 43 . . PUNCT iajs-3638 188 1 the the DET iajs-3638 188 2 c.e c.e PROPN iajs-3638 188 3 of of ADP iajs-3638 188 4 𝐽(𝐸4 𝐽(𝐸4 PROPN iajs-3638 188 5 ) ) PUNCT iajs-3638 188 6 is be AUX iajs-3638 189 1 [ [ X iajs-3638 189 2 βˆ’(𝛽3𝐼 βˆ’(𝛽3𝐼 PROPN iajs-3638 189 3 + + CCONJ iajs-3638 189 4 πœ‡ πœ‡ NOUN iajs-3638 189 5 ) ) PUNCT iajs-3638 189 6 βˆ’ βˆ’ PROPN iajs-3638 190 1 πœ† πœ† X iajs-3638 190 2 ] ] X iajs-3638 190 3 [ [ X iajs-3638 190 4 πœ† πœ† X iajs-3638 190 5 3 3 NUM iajs-3638 190 6 + + NOUN iajs-3638 190 7 𝐷1πœ† 𝐷1πœ† PROPN iajs-3638 190 8 2 2 NUM iajs-3638 190 9 + + CCONJ iajs-3638 190 10 𝐷2πœ† 𝐷2πœ† PROPN iajs-3638 190 11 + + CCONJ iajs-3638 190 12 𝐷3 𝐷3 PROPN iajs-3638 190 13 + + CCONJ iajs-3638 190 14 ( ( PUNCT iajs-3638 190 15 𝐷4πœ† 𝐷4πœ† NOUN iajs-3638 190 16 2 2 NUM iajs-3638 190 17 + + CCONJ iajs-3638 190 18 𝐷5πœ† 𝐷5πœ† PROPN iajs-3638 190 19 + + CCONJ iajs-3638 190 20 𝐷6)𝑒 𝐷6)𝑒 PROPN iajs-3638 190 21 βˆ’πœ†πœ βˆ’πœ†πœ NUM iajs-3638 190 22 ] ] X iajs-3638 190 23 = = SYM iajs-3638 190 24 0 0 NUM iajs-3638 190 25 ( ( PUNCT iajs-3638 190 26 33 33 NUM iajs-3638 190 27 ) ) PUNCT iajs-3638 190 28 with with ADP iajs-3638 190 29 ihjpas ihjpas PROPN iajs-3638 190 30 . . PUNCT iajs-3638 191 1 2025 2025 NUM iajs-3638 191 2 , , PUNCT iajs-3638 191 3 38 38 NUM iajs-3638 191 4 ( ( PUNCT iajs-3638 191 5 1 1 NUM iajs-3638 191 6 ) ) PUNCT iajs-3638 191 7 453 453 NUM iajs-3638 191 8 𝐷1 𝐷1 NOUN iajs-3638 191 9 = = SYM iajs-3638 191 10 𝑅1 𝑅1 PROPN iajs-3638 192 1 βˆ’ βˆ’ PROPN iajs-3638 192 2 𝑑22 𝑑22 PROPN iajs-3638 192 3 βˆ’ βˆ’ PROPN iajs-3638 192 4 𝑅4 𝑅4 PROPN iajs-3638 192 5 , , PUNCT iajs-3638 192 6 𝐷2 𝐷2 PROPN iajs-3638 192 7 = = SYM iajs-3638 192 8 βˆ’(𝑑22𝑅1 βˆ’(𝑑22𝑅1 PROPN iajs-3638 192 9 + + NOUN iajs-3638 192 10 𝑅1𝑅4 𝑅1𝑅4 X iajs-3638 192 11 + + CCONJ iajs-3638 192 12 𝑑12𝑑21 𝑑12𝑑21 ADJ iajs-3638 192 13 + + CCONJ iajs-3638 192 14 𝑑23𝑑32 𝑑23𝑑32 NOUN iajs-3638 192 15 ) ) PUNCT iajs-3638 192 16 + + CCONJ iajs-3638 192 17 𝑑22𝑅4 𝑑22𝑅4 ADJ iajs-3638 192 18 , , PUNCT iajs-3638 192 19 𝐷3 𝐷3 PROPN iajs-3638 192 20 = = PUNCT iajs-3638 192 21 𝑅1𝑅4𝑑22 𝑅1𝑅4𝑑22 PROPN iajs-3638 192 22 + + CCONJ iajs-3638 192 23 𝑅4𝑑12𝑑21 𝑅4𝑑12𝑑21 PROPN iajs-3638 192 24 βˆ’ βˆ’ PROPN iajs-3638 192 25 𝑅1𝑑23𝑑32 𝑅1𝑑23𝑑32 PUNCT iajs-3638 192 26 , , PUNCT iajs-3638 192 27 𝐷4 𝐷4 NOUN iajs-3638 192 28 = = PUNCT iajs-3638 192 29 𝑅2 𝑅2 NOUN iajs-3638 192 30 βˆ’ βˆ’ PROPN iajs-3638 192 31 𝑅3 𝑅3 NOUN iajs-3638 192 32 , , PUNCT iajs-3638 192 33 𝐷5 𝐷5 PROPN iajs-3638 192 34 = = PUNCT iajs-3638 192 35 βˆ’(𝑅2𝑑22 βˆ’(𝑅2𝑑22 PROPN iajs-3638 192 36 + + CCONJ iajs-3638 192 37 𝑅1𝑅3 𝑅1𝑅3 NOUN iajs-3638 192 38 + + CCONJ iajs-3638 192 39 𝑅2𝑅4 𝑅2𝑅4 ADJ iajs-3638 192 40 ) ) PUNCT iajs-3638 192 41 + + NUM iajs-3638 192 42 𝑑22𝑅3 𝑑22𝑅3 NOUN iajs-3638 192 43 , , PUNCT iajs-3638 192 44 𝐷6 𝐷6 NOUN iajs-3638 192 45 = = PUNCT iajs-3638 192 46 𝑅1𝑅3𝑑22 𝑅1𝑅3𝑑22 PROPN iajs-3638 192 47 + + CCONJ iajs-3638 192 48 𝑅3𝑑12𝑑21 𝑅3𝑑12𝑑21 PUNCT iajs-3638 193 1 + + CCONJ iajs-3638 193 2 𝑅2𝑅4𝑑22 𝑅2𝑅4𝑑22 X iajs-3638 193 3 + + CCONJ iajs-3638 193 4 𝑅3𝑑21𝑑32 𝑅3𝑑21𝑑32 ADJ iajs-3638 193 5 βˆ’ βˆ’ PROPN iajs-3638 193 6 ( ( PUNCT iajs-3638 193 7 𝑅2𝑑12𝑑23 𝑅2𝑑12𝑑23 NOUN iajs-3638 193 8 + + NUM iajs-3638 193 9 𝑅2𝑑12𝑑23 𝑅2𝑑12𝑑23 NOUN iajs-3638 193 10 + + NUM iajs-3638 193 11 𝑅1𝑑23𝑑32 𝑅1𝑑23𝑑32 X iajs-3638 193 12 + + CCONJ iajs-3638 193 13 𝑅2𝑑232𝑑32 𝑅2𝑑232𝑑32 PROPN iajs-3638 193 14 ) ) PUNCT iajs-3638 193 15 . . PUNCT iajs-3638 194 1 now now ADV iajs-3638 194 2 , , PUNCT iajs-3638 194 3 when when SCONJ iajs-3638 194 4 = = PRON iajs-3638 194 5 0 0 NUM iajs-3638 194 6 , , PUNCT iajs-3638 194 7 equation equation NOUN iajs-3638 194 8 ( ( PUNCT iajs-3638 194 9 33 33 NUM iajs-3638 194 10 ) ) PUNCT iajs-3638 194 11 becomes become VERB iajs-3638 195 1 [ [ X iajs-3638 195 2 βˆ’(𝛽3𝐼 βˆ’(𝛽3𝐼 PROPN iajs-3638 195 3 + + CCONJ iajs-3638 195 4 πœ‡ πœ‡ NOUN iajs-3638 195 5 ) ) PUNCT iajs-3638 195 6 βˆ’ βˆ’ PROPN iajs-3638 195 7 πœ† πœ† X iajs-3638 195 8 ] ] X iajs-3638 195 9 [ [ X iajs-3638 195 10 πœ† πœ† X iajs-3638 195 11 3 3 NUM iajs-3638 195 12 + + CCONJ iajs-3638 195 13 ( ( PUNCT iajs-3638 195 14 𝐷1 𝐷1 PROPN iajs-3638 195 15 + + PUNCT iajs-3638 195 16 𝐷4)πœ† 𝐷4)πœ† PROPN iajs-3638 195 17 2 2 NUM iajs-3638 195 18 + + CCONJ iajs-3638 195 19 ( ( PUNCT iajs-3638 195 20 𝐷2 𝐷2 PROPN iajs-3638 195 21 + + PROPN iajs-3638 195 22 𝐷5)πœ† 𝐷5)πœ† PROPN iajs-3638 195 23 + + NUM iajs-3638 195 24 𝐷3 𝐷3 PROPN iajs-3638 195 25 + + CCONJ iajs-3638 195 26 𝐷6 𝐷6 NOUN iajs-3638 195 27 ] ] X iajs-3638 195 28 = = SYM iajs-3638 195 29 0 0 NUM iajs-3638 195 30 ( ( PUNCT iajs-3638 195 31 34 34 NUM iajs-3638 195 32 ) ) PUNCT iajs-3638 195 33 so so ADV iajs-3638 195 34 , , PUNCT iajs-3638 195 35 either either CCONJ iajs-3638 195 36 [ [ X iajs-3638 195 37 βˆ’(𝛽3𝐼 βˆ’(𝛽3𝐼 PROPN iajs-3638 195 38 + + CCONJ iajs-3638 195 39 πœ‡ πœ‡ NOUN iajs-3638 195 40 ) ) PUNCT iajs-3638 195 41 βˆ’ βˆ’ PROPN iajs-3638 196 1 πœ† πœ† X iajs-3638 196 2 ] ] X iajs-3638 196 3 = = SYM iajs-3638 196 4 0 0 NUM iajs-3638 196 5 ( ( PUNCT iajs-3638 196 6 35a 35a NOUN iajs-3638 196 7 ) ) PUNCT iajs-3638 196 8 or or CCONJ iajs-3638 196 9 , , PUNCT iajs-3638 197 1 [ [ X iajs-3638 197 2 πœ†3 πœ†3 NOUN iajs-3638 197 3 + + CCONJ iajs-3638 197 4 ( ( PUNCT iajs-3638 197 5 𝐷1 𝐷1 NOUN iajs-3638 197 6 + + PUNCT iajs-3638 197 7 𝐷4)πœ† 𝐷4)πœ† PROPN iajs-3638 197 8 2 2 NUM iajs-3638 197 9 + + CCONJ iajs-3638 197 10 ( ( PUNCT iajs-3638 197 11 𝐷2 𝐷2 PROPN iajs-3638 197 12 + + X iajs-3638 197 13 𝐷5)πœ† 𝐷5)πœ† PROPN iajs-3638 197 14 + + NUM iajs-3638 197 15 𝐷3 𝐷3 PROPN iajs-3638 197 16 + + CCONJ iajs-3638 197 17 𝐷6 𝐷6 NOUN iajs-3638 197 18 ] ] X iajs-3638 197 19 = = SYM iajs-3638 197 20 0 0 NUM iajs-3638 197 21 ( ( PUNCT iajs-3638 197 22 35b 35b NOUN iajs-3638 197 23 ) ) PUNCT iajs-3638 197 24 from from ADP iajs-3638 197 25 equation equation NOUN iajs-3638 197 26 ( ( PUNCT iajs-3638 197 27 35a 35a NOUN iajs-3638 197 28 ) ) PUNCT iajs-3638 197 29 then then ADV iajs-3638 197 30 πœ†3 πœ†3 NOUN iajs-3638 197 31 = = PUNCT iajs-3638 198 1 βˆ’π›½3𝐼 βˆ’π›½3𝐼 PROPN iajs-3638 198 2 βˆ’ βˆ’ NOUN iajs-3638 198 3 πœ‡ πœ‡ ADP iajs-3638 198 4 < < X iajs-3638 198 5 0 0 NUM iajs-3638 198 6 which which PRON iajs-3638 198 7 is be AUX iajs-3638 198 8 always always ADV iajs-3638 198 9 negative negative ADJ iajs-3638 198 10 eigenvalue eigenvalue NOUN iajs-3638 198 11 . . PUNCT iajs-3638 199 1 now now ADV iajs-3638 199 2 , , PUNCT iajs-3638 199 3 it it PRON iajs-3638 199 4 is be AUX iajs-3638 199 5 easy easy ADJ iajs-3638 199 6 to to PART iajs-3638 199 7 verify verify VERB iajs-3638 199 8 that that DET iajs-3638 199 9 𝐷1 𝐷1 NOUN iajs-3638 199 10 + + CCONJ iajs-3638 199 11 𝐷4 𝐷4 NOUN iajs-3638 199 12 > > X iajs-3638 199 13 0 0 PUNCT iajs-3638 200 1 and and CCONJ iajs-3638 200 2 𝐷3 𝐷3 PROPN iajs-3638 200 3 + + CCONJ iajs-3638 200 4 𝐷6 𝐷6 NOUN iajs-3638 200 5 > > X iajs-3638 200 6 0 0 PUNCT iajs-3638 201 1 under under ADP iajs-3638 201 2 the the DET iajs-3638 201 3 following follow VERB iajs-3638 201 4 sufficient sufficient ADJ iajs-3638 201 5 conditions condition NOUN iajs-3638 201 6 𝛽1 𝛽1 PROPN iajs-3638 201 7 οΏ½ οΏ½ PROPN iajs-3638 201 8 ΜƒΜƒ ΜƒΜƒ NOUN iajs-3638 201 9 οΏ½ οΏ½ PROPN iajs-3638 201 10 + + CCONJ iajs-3638 201 11 𝛽2 𝛽2 PROPN iajs-3638 201 12 οΏ½ οΏ½ PROPN iajs-3638 201 13 ΜƒΜƒ ΜƒΜƒ NOUN iajs-3638 201 14 οΏ½ οΏ½ PROPN iajs-3638 201 15 1 1 NUM iajs-3638 201 16 < < X iajs-3638 201 17 πœ‡ πœ‡ ADP iajs-3638 201 18 + + X iajs-3638 201 19 πœ‡1 πœ‡1 ADJ iajs-3638 201 20 + + CCONJ iajs-3638 201 21 πœƒ πœƒ PROPN iajs-3638 201 22 ( ( PUNCT iajs-3638 201 23 36a 36a NOUN iajs-3638 201 24 ) ) PUNCT iajs-3638 201 25 𝑛𝛼2 𝑛𝛼2 PROPN iajs-3638 201 26 οΏ½ οΏ½ PROPN iajs-3638 201 27 ΜƒΜƒ ΜƒΜƒ PROPN iajs-3638 201 28 οΏ½ οΏ½ PROPN iajs-3638 201 29 ( ( PUNCT iajs-3638 201 30 1+𝑛 1+𝑛 NUM iajs-3638 201 31 οΏ½ οΏ½ PROPN iajs-3638 201 32 ΜƒΜƒ ΜƒΜƒ NOUN iajs-3638 201 33 οΏ½ οΏ½ PROPN iajs-3638 201 34 1 1 NUM iajs-3638 201 35 ) ) PUNCT iajs-3638 201 36 3 3 NUM iajs-3638 201 37 < < X iajs-3638 201 38 ( ( PUNCT iajs-3638 201 39 𝛼 𝛼 PROPN iajs-3638 201 40 1+𝑛 1+𝑛 NUM iajs-3638 201 41 οΏ½ οΏ½ PROPN iajs-3638 201 42 ΜƒΜƒ ΜƒΜƒ NOUN iajs-3638 201 43 οΏ½ οΏ½ PROPN iajs-3638 201 44 1 1 NUM iajs-3638 201 45 + + NUM iajs-3638 201 46 𝛽1𝐼 𝛽1𝐼 NOUN iajs-3638 201 47 + + CCONJ iajs-3638 201 48 πœ‡ πœ‡ X iajs-3638 201 49 ) ) PUNCT iajs-3638 201 50 ( ( PUNCT iajs-3638 201 51 𝑛𝛼 𝑛𝛼 PROPN iajs-3638 201 52 οΏ½ οΏ½ PROPN iajs-3638 201 53 ΜƒΜƒ ΜƒΜƒ PROPN iajs-3638 201 54 οΏ½ οΏ½ PROPN iajs-3638 201 55 ( ( PUNCT iajs-3638 201 56 1+𝑛 1+𝑛 NUM iajs-3638 201 57 οΏ½ οΏ½ PROPN iajs-3638 201 58 ΜƒΜƒ ΜƒΜƒ NOUN iajs-3638 201 59 οΏ½ οΏ½ NOUN iajs-3638 201 60 1 1 NUM iajs-3638 201 61 ) ) PUNCT iajs-3638 201 62 2 2 NUM iajs-3638 201 63 + + CCONJ iajs-3638 201 64 𝛽2𝐼 𝛽2𝐼 ADV iajs-3638 201 65 + + CCONJ iajs-3638 201 66 πœ‡ πœ‡ X iajs-3638 201 67 ) ) PUNCT iajs-3638 201 68 ( ( PUNCT iajs-3638 201 69 36b 36b NUM iajs-3638 201 70 ) ) PUNCT iajs-3638 201 71 we we PRON iajs-3638 201 72 have have VERB iajs-3638 201 73 ( ( PUNCT iajs-3638 201 74 𝐷1 𝐷1 NOUN iajs-3638 201 75 + + CCONJ iajs-3638 201 76 𝐷4)(𝐷2 𝐷4)(𝐷2 X iajs-3638 202 1 + + CCONJ iajs-3638 202 2 𝐷5 𝐷5 PROPN iajs-3638 202 3 ) ) PUNCT iajs-3638 203 1 βˆ’ βˆ’ PROPN iajs-3638 203 2 ( ( PUNCT iajs-3638 203 3 𝐷3 𝐷3 PROPN iajs-3638 203 4 + + PROPN iajs-3638 203 5 𝐷6 𝐷6 NOUN iajs-3638 203 6 ) ) PUNCT iajs-3638 203 7 > > X iajs-3638 204 1 0 0 X iajs-3638 204 2 . . PUNCT iajs-3638 205 1 under under ADP iajs-3638 205 2 the the DET iajs-3638 205 3 following follow VERB iajs-3638 205 4 sufficient sufficient ADJ iajs-3638 205 5 conditions condition NOUN iajs-3638 205 6 𝑛𝛽1𝛼 𝑛𝛽1𝛼 PROPN iajs-3638 205 7 οΏ½ οΏ½ PROPN iajs-3638 205 8 ΜƒΜƒ ΜƒΜƒ ADJ iajs-3638 205 9 οΏ½ οΏ½ PROPN iajs-3638 205 10 ( ( PUNCT iajs-3638 205 11 1+𝑛 1+𝑛 NUM iajs-3638 205 12 οΏ½ οΏ½ PROPN iajs-3638 205 13 ΜƒΜƒ ΜƒΜƒ NOUN iajs-3638 205 14 οΏ½ οΏ½ PROPN iajs-3638 205 15 1 1 NUM iajs-3638 205 16 ) ) PUNCT iajs-3638 205 17 2 2 NUM iajs-3638 205 18 < < X iajs-3638 205 19 ( ( PUNCT iajs-3638 205 20 𝑛𝛼 𝑛𝛼 PROPN iajs-3638 205 21 οΏ½ οΏ½ PROPN iajs-3638 205 22 ΜƒΜƒ ΜƒΜƒ PROPN iajs-3638 205 23 οΏ½ οΏ½ PROPN iajs-3638 205 24 ( ( PUNCT iajs-3638 205 25 1+𝑛 1+𝑛 NUM iajs-3638 205 26 οΏ½ οΏ½ PROPN iajs-3638 205 27 ΜƒΜƒ ΜƒΜƒ NOUN iajs-3638 205 28 οΏ½ οΏ½ NOUN iajs-3638 205 29 1 1 NUM iajs-3638 205 30 ) ) PUNCT iajs-3638 205 31 2 2 NUM iajs-3638 205 32 + + CCONJ iajs-3638 205 33 𝛽2𝐼 𝛽2𝐼 ADV iajs-3638 205 34 + + CCONJ iajs-3638 205 35 πœ‡)𝛽2 πœ‡)𝛽2 PROPN iajs-3638 205 36 ( ( PUNCT iajs-3638 205 37 36c 36c NOUN iajs-3638 205 38 ) ) PUNCT iajs-3638 205 39 𝛽2 𝛽2 PROPN iajs-3638 205 40 οΏ½ οΏ½ PROPN iajs-3638 205 41 ΜƒΜƒ ΜƒΜƒ NOUN iajs-3638 205 42 οΏ½ οΏ½ PROPN iajs-3638 205 43 1 1 NUM iajs-3638 205 44 ( ( PUNCT iajs-3638 205 45 𝛽1 𝛽1 PROPN iajs-3638 205 46 οΏ½ οΏ½ PROPN iajs-3638 205 47 ΜƒΜƒ ΜƒΜƒ NOUN iajs-3638 205 48 οΏ½ οΏ½ PROPN iajs-3638 205 49 + + CCONJ iajs-3638 205 50 𝛽2 𝛽2 PROPN iajs-3638 205 51 οΏ½ οΏ½ PROPN iajs-3638 205 52 ΜƒΜƒ ΜƒΜƒ NOUN iajs-3638 205 53 οΏ½ οΏ½ PROPN iajs-3638 205 54 1 1 NUM iajs-3638 205 55 ) ) PUNCT iajs-3638 205 56 + + CCONJ iajs-3638 205 57 𝛽1𝛼 𝛽1𝛼 PROPN iajs-3638 205 58 οΏ½ οΏ½ PROPN iajs-3638 205 59 ΜƒΜƒ ΜƒΜƒ PROPN iajs-3638 205 60 οΏ½ οΏ½ PROPN iajs-3638 205 61 1+𝑛 1+𝑛 NUM iajs-3638 205 62 οΏ½ οΏ½ PROPN iajs-3638 205 63 ΜƒΜƒ ΜƒΜƒ NOUN iajs-3638 205 64 οΏ½ οΏ½ NOUN iajs-3638 205 65 1 1 NUM iajs-3638 205 66 < < X iajs-3638 205 67 𝛽2 𝛽2 PROPN iajs-3638 205 68 οΏ½ οΏ½ PROPN iajs-3638 205 69 ΜƒΜƒ ΜƒΜƒ PROPN iajs-3638 205 70 οΏ½ οΏ½ PROPN iajs-3638 205 71 1(πœ‡ 1(πœ‡ NUM iajs-3638 205 72 + + NUM iajs-3638 205 73 πœ‡1 πœ‡1 PROPN iajs-3638 205 74 + + CCONJ iajs-3638 205 75 πœƒ πœƒ NOUN iajs-3638 205 76 ) ) PUNCT iajs-3638 205 77 ( ( PUNCT iajs-3638 205 78 36d 36d NOUN iajs-3638 205 79 ) ) PUNCT iajs-3638 205 80 so so ADV iajs-3638 205 81 , , PUNCT iajs-3638 205 82 according accord VERB iajs-3638 205 83 to to ADP iajs-3638 205 84 routh routh PROPN iajs-3638 205 85 - - PUNCT iajs-3638 205 86 hurwitz hurwitz PROPN iajs-3638 205 87 criterion criterion NOUN iajs-3638 205 88 , , PUNCT iajs-3638 205 89 we we PRON iajs-3638 205 90 see see VERB iajs-3638 205 91 that that SCONJ iajs-3638 205 92 all all DET iajs-3638 205 93 roots root NOUN iajs-3638 205 94 of of ADP iajs-3638 205 95 equation equation NOUN iajs-3638 205 96 ( ( PUNCT iajs-3638 205 97 35b 35b NOUN iajs-3638 205 98 ) ) PUNCT iajs-3638 205 99 , , PUNCT iajs-3638 205 100 which which PRON iajs-3638 205 101 represent represent VERB iajs-3638 205 102 the the DET iajs-3638 205 103 eigenvalues eigenvalue NOUN iajs-3638 205 104 of of ADP iajs-3638 205 105 ( ( PUNCT iajs-3638 205 106 32 32 NUM iajs-3638 205 107 ) ) PUNCT iajs-3638 205 108 , , PUNCT iajs-3638 205 109 have have VERB iajs-3638 205 110 negative negative ADJ iajs-3638 205 111 real real ADJ iajs-3638 205 112 parts part NOUN iajs-3638 205 113 . . PUNCT iajs-3638 206 1 consequently consequently ADV iajs-3638 206 2 , , PUNCT iajs-3638 206 3 under under ADP iajs-3638 206 4 the the DET iajs-3638 206 5 conditions condition NOUN iajs-3638 206 6 ( ( PUNCT iajs-3638 206 7 36a)-(36d 36a)-(36d NUM iajs-3638 206 8 ) ) PUNCT iajs-3638 206 9 , , PUNCT iajs-3638 206 10 𝐸4 𝐸4 NOUN iajs-3638 206 11 is be AUX iajs-3638 206 12 l.a.s l.a.s ADJ iajs-3638 206 13 for for ADP iajs-3638 206 14 system system NOUN iajs-3638 206 15 ( ( PUNCT iajs-3638 206 16 3 3 NUM iajs-3638 206 17 ) ) PUNCT iajs-3638 206 18 when when SCONJ iajs-3638 206 19 𝜏 𝜏 NOUN iajs-3638 206 20 = = SYM iajs-3638 206 21 0 0 NUM iajs-3638 206 22 . . PUNCT iajs-3638 207 1 now now ADV iajs-3638 207 2 , , PUNCT iajs-3638 207 3 for for ADP iajs-3638 207 4 𝜏 𝜏 X iajs-3638 207 5 > > X iajs-3638 207 6 0 0 NUM iajs-3638 207 7 , , PUNCT iajs-3638 207 8 then then ADV iajs-3638 207 9 either either ADV iajs-3638 207 10 , , PUNCT iajs-3638 207 11 equation equation NOUN iajs-3638 207 12 ( ( PUNCT iajs-3638 207 13 35a 35a NOUN iajs-3638 207 14 ) ) PUNCT iajs-3638 207 15 which which PRON iajs-3638 207 16 is be AUX iajs-3638 207 17 always always ADV iajs-3638 207 18 negative negative ADJ iajs-3638 207 19 eigenvalue eigenvalue NOUN iajs-3638 207 20 . . PUNCT iajs-3638 208 1 or or CCONJ iajs-3638 208 2 , , PUNCT iajs-3638 208 3 [ [ X iajs-3638 208 4 πœ†3 πœ†3 NOUN iajs-3638 208 5 + + CCONJ iajs-3638 208 6 𝐷1πœ† 𝐷1πœ† PROPN iajs-3638 208 7 2 2 NUM iajs-3638 209 1 + + CCONJ iajs-3638 209 2 𝐷2πœ† 𝐷2πœ† PROPN iajs-3638 209 3 + + CCONJ iajs-3638 209 4 𝐷3 𝐷3 PROPN iajs-3638 209 5 + + CCONJ iajs-3638 209 6 ( ( PUNCT iajs-3638 209 7 𝐷4πœ† 𝐷4πœ† NOUN iajs-3638 209 8 2 2 NUM iajs-3638 209 9 + + CCONJ iajs-3638 209 10 𝐷5πœ† 𝐷5πœ† PROPN iajs-3638 209 11 + + CCONJ iajs-3638 209 12 𝐷6)𝑒 𝐷6)𝑒 PROPN iajs-3638 209 13 βˆ’πœ†πœ βˆ’πœ†πœ NUM iajs-3638 209 14 ] ] X iajs-3638 209 15 = = SYM iajs-3638 209 16 0 0 PUNCT iajs-3638 209 17 ( ( PUNCT iajs-3638 209 18 37 37 NUM iajs-3638 209 19 ) ) PUNCT iajs-3638 209 20 from from ADP iajs-3638 209 21 ( ( PUNCT iajs-3638 209 22 37 37 NUM iajs-3638 209 23 ) ) PUNCT iajs-3638 209 24 has have VERB iajs-3638 209 25 two two NUM iajs-3638 209 26 wholly wholly ADV iajs-3638 209 27 imaginary imaginary ADJ iajs-3638 209 28 roots root NOUN iajs-3638 209 29 , , PUNCT iajs-3638 209 30 namely= namely= NUM iajs-3638 209 31 Β±π‘–πœ” Β±π‘–πœ” NOUN iajs-3638 209 32 ( ( PUNCT iajs-3638 209 33 πœ” πœ” X iajs-3638 209 34 > > X iajs-3638 209 35 0 0 NUM iajs-3638 209 36 ) ) PUNCT iajs-3638 209 37 . . PUNCT iajs-3638 210 1 by by ADP iajs-3638 210 2 substituting substitute VERB iajs-3638 210 3 πœ† πœ† PRON iajs-3638 210 4 = = SYM iajs-3638 210 5 Β±π‘–πœ” Β±π‘–πœ” NOUN iajs-3638 210 6 in in ADP iajs-3638 210 7 equation equation NOUN iajs-3638 210 8 ( ( PUNCT iajs-3638 210 9 37 37 NUM iajs-3638 210 10 ) ) PUNCT iajs-3638 210 11 and and CCONJ iajs-3638 210 12 separating separate VERB iajs-3638 210 13 the the DET iajs-3638 210 14 real real ADJ iajs-3638 210 15 and and CCONJ iajs-3638 210 16 imagined imagined ADJ iajs-3638 210 17 parts part NOUN iajs-3638 210 18 yields yield NOUN iajs-3638 210 19 ( ( PUNCT iajs-3638 210 20 𝐷6 𝐷6 NOUN iajs-3638 210 21 βˆ’ βˆ’ PROPN iajs-3638 210 22 𝐷1πœ” 𝐷1πœ” PROPN iajs-3638 210 23 2)π‘π‘œπ‘ πœ”πœ 2)π‘π‘œπ‘ πœ”πœ NUM iajs-3638 210 24 + + CCONJ iajs-3638 210 25 𝐷5πœ”π‘ π‘–π‘›πœ”πœ 𝐷5πœ”π‘ π‘–π‘›πœ”πœ ADJ iajs-3638 210 26 = = SYM iajs-3638 210 27 𝐷1πœ” 𝐷1πœ” PROPN iajs-3638 210 28 2 2 NUM iajs-3638 210 29 βˆ’π·3 βˆ’π·3 NOUN iajs-3638 210 30 𝐷5πœ”π‘π‘œπ‘ πœ”πœ 𝐷5πœ”π‘π‘œπ‘ πœ”πœ NOUN iajs-3638 210 31 + + CCONJ iajs-3638 210 32 ( ( PUNCT iajs-3638 210 33 𝐷4πœ” 𝐷4πœ” PROPN iajs-3638 210 34 2 2 NUM iajs-3638 210 35 βˆ’ βˆ’ NOUN iajs-3638 210 36 𝐷6)π‘ π‘–π‘›πœ”πœ 𝐷6)π‘ π‘–π‘›πœ”πœ NOUN iajs-3638 210 37 = = SYM iajs-3638 210 38 πœ” πœ” SYM iajs-3638 210 39 3 3 NUM iajs-3638 210 40 βˆ’ βˆ’ NOUN iajs-3638 210 41 𝐷2πœ” 𝐷2πœ” PROPN iajs-3638 210 42 ( ( PUNCT iajs-3638 210 43 38 38 NUM iajs-3638 210 44 ) ) PUNCT iajs-3638 210 45 through through ADP iajs-3638 210 46 squaring square VERB iajs-3638 210 47 and and CCONJ iajs-3638 210 48 adding add VERB iajs-3638 210 49 both both DET iajs-3638 210 50 equations equation NOUN iajs-3638 210 51 in in ADP iajs-3638 210 52 equation equation NOUN iajs-3638 210 53 ( ( PUNCT iajs-3638 210 54 38 38 NUM iajs-3638 210 55 ) ) PUNCT iajs-3638 210 56 , , PUNCT iajs-3638 210 57 we we PRON iajs-3638 210 58 get get VERB iajs-3638 210 59 πœ”6 πœ”6 ADJ iajs-3638 210 60 + + CCONJ iajs-3638 210 61 β„Ž1πœ” β„Ž1πœ” PROPN iajs-3638 210 62 4 4 NUM iajs-3638 210 63 + + CCONJ iajs-3638 210 64 β„Ž2πœ” β„Ž2πœ” PROPN iajs-3638 210 65 2 2 NUM iajs-3638 210 66 + + NUM iajs-3638 210 67 β„Ž3 β„Ž3 NOUN iajs-3638 210 68 = = SYM iajs-3638 210 69 0 0 NUM iajs-3638 210 70 ( ( PUNCT iajs-3638 210 71 39 39 NUM iajs-3638 210 72 ) ) PUNCT iajs-3638 210 73 further far ADV iajs-3638 210 74 , , PUNCT iajs-3638 210 75 by by ADP iajs-3638 210 76 letting let VERB iajs-3638 210 77 𝜐 𝜐 X iajs-3638 210 78 = = PUNCT iajs-3638 211 1 πœ”2 πœ”2 ADJ iajs-3638 211 2 in in ADP iajs-3638 211 3 equation equation NOUN iajs-3638 211 4 ( ( PUNCT iajs-3638 211 5 39 39 NUM iajs-3638 211 6 ) ) PUNCT iajs-3638 211 7 β„Ž(𝜐 β„Ž(𝜐 PROPN iajs-3638 211 8 ) ) PUNCT iajs-3638 211 9 = = SYM iajs-3638 211 10 𝜐3 𝜐3 NOUN iajs-3638 211 11 + + CCONJ iajs-3638 211 12 β„Ž1𝜐 β„Ž1𝜐 PROPN iajs-3638 211 13 2 2 NUM iajs-3638 211 14 + + NUM iajs-3638 211 15 β„Ž2𝜐 β„Ž2𝜐 NOUN iajs-3638 211 16 + + CCONJ iajs-3638 211 17 β„Ž3 β„Ž3 NOUN iajs-3638 211 18 = = SYM iajs-3638 211 19 0 0 NUM iajs-3638 211 20 ( ( PUNCT iajs-3638 211 21 40a 40a NUM iajs-3638 211 22 ) ) PUNCT iajs-3638 211 23 where where SCONJ iajs-3638 211 24 ihjpas ihjpa NOUN iajs-3638 211 25 . . PUNCT iajs-3638 212 1 2025 2025 NUM iajs-3638 212 2 , , PUNCT iajs-3638 212 3 38 38 NUM iajs-3638 212 4 ( ( PUNCT iajs-3638 212 5 1 1 NUM iajs-3638 212 6 ) ) PUNCT iajs-3638 212 7 454 454 NUM iajs-3638 212 8 β„Ž1 β„Ž1 PROPN iajs-3638 212 9 = = PUNCT iajs-3638 212 10 𝐷1 𝐷1 NOUN iajs-3638 212 11 2 2 NUM iajs-3638 212 12 βˆ’ βˆ’ NUM iajs-3638 212 13 2𝐷2 2𝐷2 NUM iajs-3638 212 14 βˆ’ βˆ’ NOUN iajs-3638 212 15 𝐷4 𝐷4 NOUN iajs-3638 212 16 2 2 NUM iajs-3638 212 17 , , PUNCT iajs-3638 212 18 β„Ž2 β„Ž2 NOUN iajs-3638 212 19 = = SYM iajs-3638 212 20 2𝐷4𝐷6 2𝐷4𝐷6 NUM iajs-3638 212 21 + + CCONJ iajs-3638 212 22 𝐷2 𝐷2 PROPN iajs-3638 212 23 2 2 NUM iajs-3638 212 24 βˆ’ βˆ’ NOUN iajs-3638 212 25 𝐷5 𝐷5 PROPN iajs-3638 212 26 2 2 NUM iajs-3638 212 27 βˆ’ βˆ’ NOUN iajs-3638 212 28 2𝐷1𝐷3 2𝐷1𝐷3 NUM iajs-3638 212 29 , , PUNCT iajs-3638 212 30 β„Ž3 β„Ž3 NOUN iajs-3638 212 31 = = SYM iajs-3638 212 32 𝐷3 𝐷3 PROPN iajs-3638 213 1 2 2 NUM iajs-3638 213 2 βˆ’ βˆ’ NOUN iajs-3638 213 3 𝐷6 𝐷6 NOUN iajs-3638 213 4 2 2 NUM iajs-3638 213 5 . . PUNCT iajs-3638 213 6 straightforward straightforward ADJ iajs-3638 213 7 computation computation NOUN iajs-3638 213 8 shows show VERB iajs-3638 213 9 that that SCONJ iajs-3638 213 10 due due ADP iajs-3638 213 11 to to ADP iajs-3638 213 12 the the DET iajs-3638 213 13 following follow VERB iajs-3638 213 14 condition condition NOUN iajs-3638 213 15 𝐷3 𝐷3 X iajs-3638 213 16 < < X iajs-3638 213 17 𝐷6 𝐷6 PROPN iajs-3638 213 18 ( ( PUNCT iajs-3638 213 19 40b 40b NUM iajs-3638 213 20 ) ) PUNCT iajs-3638 213 21 we we PRON iajs-3638 213 22 obtain obtain VERB iajs-3638 213 23 β„Ž3 β„Ž3 NOUN iajs-3638 213 24 < < X iajs-3638 213 25 0 0 NUM iajs-3638 213 26 . . PUNCT iajs-3638 214 1 so so ADV iajs-3638 214 2 , , PUNCT iajs-3638 214 3 there there PRON iajs-3638 214 4 exists exist VERB iajs-3638 214 5 a a DET iajs-3638 214 6 unique unique ADJ iajs-3638 214 7 positive positive ADJ iajs-3638 214 8 root root NOUN iajs-3638 214 9 in in ADP iajs-3638 214 10 accordance accordance NOUN iajs-3638 214 11 with with ADP iajs-3638 214 12 descartes descarte NOUN iajs-3638 214 13 ' ' PART iajs-3638 214 14 rule rule NOUN iajs-3638 214 15 of of ADP iajs-3638 214 16 signs sign NOUN iajs-3638 214 17 𝜐0 𝜐0 NOUN iajs-3638 214 18 = = SYM iajs-3638 214 19 πœ”0 πœ”0 NOUN iajs-3638 214 20 2 2 NUM iajs-3638 214 21 providing provide VERB iajs-3638 214 22 equation equation NOUN iajs-3638 214 23 ( ( PUNCT iajs-3638 214 24 40a 40a NUM iajs-3638 214 25 ) ) PUNCT iajs-3638 214 26 . . PUNCT iajs-3638 215 1 which which PRON iajs-3638 215 2 is be AUX iajs-3638 215 3 , , PUNCT iajs-3638 215 4 equation equation NOUN iajs-3638 215 5 ( ( PUNCT iajs-3638 215 6 39 39 NUM iajs-3638 215 7 ) ) PUNCT iajs-3638 215 8 has have VERB iajs-3638 215 9 a a DET iajs-3638 215 10 positive positive ADJ iajs-3638 215 11 πœ”0 πœ”0 NOUN iajs-3638 215 12 . . PUNCT iajs-3638 216 1 as as ADP iajs-3638 216 2 a a DET iajs-3638 216 3 result result NOUN iajs-3638 216 4 , , PUNCT iajs-3638 216 5 equation equation NOUN iajs-3638 216 6 ( ( PUNCT iajs-3638 216 7 37 37 NUM iajs-3638 216 8 ) ) PUNCT iajs-3638 216 9 has have VERB iajs-3638 216 10 at at ADV iajs-3638 216 11 least least ADV iajs-3638 216 12 two two NUM iajs-3638 216 13 roots root NOUN iajs-3638 216 14 Β±πœ”0 Β±πœ”0 NOUN iajs-3638 216 15 that that PRON iajs-3638 216 16 are be AUX iajs-3638 216 17 purely purely ADV iajs-3638 216 18 imaginary imaginary ADJ iajs-3638 216 19 and and CCONJ iajs-3638 216 20 which which PRON iajs-3638 216 21 correspond correspond VERB iajs-3638 216 22 to to ADP iajs-3638 216 23 the the DET iajs-3638 216 24 time time NOUN iajs-3638 216 25 delay delay NOUN iajs-3638 216 26 𝜏. 𝜏. PROPN iajs-3638 216 27 additionally additionally ADV iajs-3638 216 28 , , PUNCT iajs-3638 216 29 by by ADP iajs-3638 216 30 substituting substitute VERB iajs-3638 216 31 πœ”0 πœ”0 NOUN iajs-3638 216 32 in in ADP iajs-3638 216 33 ( ( PUNCT iajs-3638 216 34 38 38 NUM iajs-3638 216 35 ) ) PUNCT iajs-3638 216 36 and and CCONJ iajs-3638 216 37 solving solve VERB iajs-3638 216 38 the the DET iajs-3638 216 39 system system NOUN iajs-3638 216 40 for for ADP iajs-3638 216 41 𝜏 𝜏 NOUN iajs-3638 216 42 , , PUNCT iajs-3638 216 43 yields yield VERB iajs-3638 216 44 : : PUNCT iajs-3638 216 45 πœπ‘˜ πœπ‘˜ NOUN iajs-3638 216 46 = = SYM iajs-3638 216 47 1 1 NUM iajs-3638 216 48 πœ”0 πœ”0 NOUN iajs-3638 216 49 π‘π‘œπ‘ βˆ’1 π‘π‘œπ‘ βˆ’1 NOUN iajs-3638 216 50 ( ( PUNCT iajs-3638 216 51 𝐷5βˆ’π·1𝐷4)πœ”0 𝐷5βˆ’π·1𝐷4)πœ”0 PROPN iajs-3638 216 52 4+(𝐷1𝐷6+𝐷3𝐷4βˆ’π·2𝐷5)πœ”0 4+(𝐷1𝐷6+𝐷3𝐷4βˆ’π·2𝐷5)πœ”0 PROPN iajs-3638 216 53 2βˆ’π·3𝐷6 2βˆ’π·3𝐷6 NUM iajs-3638 216 54 𝐷4 𝐷4 NOUN iajs-3638 216 55 2πœ”0 2πœ”0 NUM iajs-3638 216 56 4+(𝐷5 4+(𝐷5 PROPN iajs-3638 216 57 2βˆ’2𝐷4𝐷6)πœ”0 2βˆ’2𝐷4𝐷6)πœ”0 PROPN iajs-3638 216 58 2+𝐷6 2+𝐷6 NUM iajs-3638 216 59 2 2 NUM iajs-3638 216 60 + + CCONJ iajs-3638 216 61 2π‘˜πœ‹ 2π‘˜πœ‹ ADJ iajs-3638 216 62 πœ”0 πœ”0 NOUN iajs-3638 216 63 ( ( PUNCT iajs-3638 216 64 41 41 NUM iajs-3638 216 65 ) ) PUNCT iajs-3638 216 66 where where SCONJ iajs-3638 216 67 , , PUNCT iajs-3638 216 68 k k PROPN iajs-3638 216 69 = = PUNCT iajs-3638 216 70 0,1,2 0,1,2 NUM iajs-3638 216 71 , , PUNCT iajs-3638 216 72 … … PUNCT iajs-3638 216 73 . . PUNCT iajs-3638 217 1 hence hence ADV iajs-3638 217 2 we we PRON iajs-3638 217 3 get get VERB iajs-3638 217 4 the the DET iajs-3638 217 5 corresponding corresponding NOUN iajs-3638 217 6 πœπ‘˜ πœπ‘˜ ADP iajs-3638 217 7 > > X iajs-3638 217 8 0 0 NUM iajs-3638 217 9 for for ADP iajs-3638 217 10 which which DET iajs-3638 217 11 system system NOUN iajs-3638 217 12 ( ( PUNCT iajs-3638 217 13 3 3 X iajs-3638 217 14 ) ) PUNCT iajs-3638 217 15 has have VERB iajs-3638 217 16 two two NUM iajs-3638 217 17 wholly wholly ADV iajs-3638 217 18 imaginary imaginary ADJ iajs-3638 217 19 roots root NOUN iajs-3638 217 20 Β±πœ”0 Β±πœ”0 NOUN iajs-3638 217 21 . . PUNCT iajs-3638 218 1 let let VERB iajs-3638 218 2 πœ†(𝜏 πœ†(𝜏 NOUN iajs-3638 218 3 ) ) PUNCT iajs-3638 219 1 = = SYM iajs-3638 219 2 πœ‡(𝜏 πœ‡(𝜏 PROPN iajs-3638 219 3 ) ) PUNCT iajs-3638 220 1 + + CCONJ iajs-3638 220 2 π‘–πœ”(𝜏 π‘–πœ”(𝜏 PRON iajs-3638 220 3 ) ) PUNCT iajs-3638 220 4 be be VERB iajs-3638 220 5 a a DET iajs-3638 220 6 root root NOUN iajs-3638 220 7 of of ADP iajs-3638 220 8 equation equation NOUN iajs-3638 220 9 ( ( PUNCT iajs-3638 220 10 37 37 NUM iajs-3638 220 11 ) ) PUNCT iajs-3638 220 12 near near ADP iajs-3638 220 13 𝜏 𝜏 NOUN iajs-3638 220 14 = = NOUN iajs-3638 220 15 πœπ‘˜ πœπ‘˜ NOUN iajs-3638 220 16 with with ADP iajs-3638 220 17 πœ‡(πœπ‘˜ πœ‡(πœπ‘˜ NOUN iajs-3638 220 18 ) ) PUNCT iajs-3638 220 19 = = SYM iajs-3638 220 20 0 0 NUM iajs-3638 220 21 and and CCONJ iajs-3638 220 22 πœ”(πœπ‘˜ πœ”(πœπ‘˜ NUM iajs-3638 220 23 ) ) PUNCT iajs-3638 221 1 = = SYM iajs-3638 221 2 πœ”0 πœ”0 PROPN iajs-3638 221 3 . . PUNCT iajs-3638 222 1 then then ADV iajs-3638 222 2 comes come VERB iajs-3638 222 3 the the DET iajs-3638 222 4 next next ADJ iajs-3638 222 5 theorem theorem NOUN iajs-3638 222 6 : : PUNCT iajs-3638 222 7 theorem2 theorem2 NOUN iajs-3638 222 8 . . PUNCT iajs-3638 223 1 the the DET iajs-3638 223 2 c.e c.e PROPN iajs-3638 223 3 . . PUNCT iajs-3638 223 4 ( ( PUNCT iajs-3638 223 5 37 37 NUM iajs-3638 223 6 ) ) PUNCT iajs-3638 223 7 , , PUNCT iajs-3638 223 8 roots root NOUN iajs-3638 223 9 satisfy satisfy VERB iajs-3638 223 10 the the DET iajs-3638 223 11 following follow VERB iajs-3638 223 12 transversality transversality NOUN iajs-3638 223 13 requirement requirement NOUN iajs-3638 223 14 ; ; PUNCT iajs-3638 223 15 [ [ PUNCT iajs-3638 223 16 𝑑(π‘…π‘’πœ†(𝜏 𝑑(π‘…π‘’πœ†(𝜏 NOUN iajs-3638 223 17 ) ) PUNCT iajs-3638 223 18 ) ) PUNCT iajs-3638 224 1 π‘‘πœ π‘‘πœ NOUN iajs-3638 224 2 ] ] PUNCT iajs-3638 224 3 𝜏=πœπ‘˜ 𝜏=πœπ‘˜ X iajs-3638 224 4 > > X iajs-3638 224 5 0 0 PUNCT iajs-3638 224 6 ( ( PUNCT iajs-3638 224 7 42a 42a NOUN iajs-3638 224 8 ) ) PUNCT iajs-3638 224 9 provided provide VERB iajs-3638 224 10 that that SCONJ iajs-3638 224 11 2𝐷2 2𝐷2 PROPN iajs-3638 224 12 > > X iajs-3638 224 13 𝐷1 𝐷1 PROPN iajs-3638 224 14 2 2 NUM iajs-3638 224 15 ( ( PUNCT iajs-3638 224 16 42b 42b NOUN iajs-3638 224 17 ) ) PUNCT iajs-3638 224 18 𝐷2 𝐷2 PROPN iajs-3638 224 19 2 2 NUM iajs-3638 224 20 > > SYM iajs-3638 224 21 2𝐷1𝐷3 2𝐷1𝐷3 NUM iajs-3638 224 22 ( ( PUNCT iajs-3638 224 23 42c 42c NOUN iajs-3638 224 24 ) ) PUNCT iajs-3638 224 25 2𝐷4𝐷6 2𝐷4𝐷6 NUM iajs-3638 224 26 > > X iajs-3638 224 27 2𝐷4 2𝐷4 NUM iajs-3638 224 28 2πœ”0 2πœ”0 NUM iajs-3638 224 29 2 2 NUM iajs-3638 224 30 + + CCONJ iajs-3638 224 31 𝐷5 𝐷5 PROPN iajs-3638 224 32 2 2 NUM iajs-3638 224 33 ( ( PUNCT iajs-3638 224 34 42d 42d NOUN iajs-3638 224 35 ) ) PUNCT iajs-3638 224 36 proof proof NOUN iajs-3638 224 37 . . PUNCT iajs-3638 225 1 by by ADP iajs-3638 225 2 substituting substitute VERB iajs-3638 225 3 πœ†(𝜏 πœ†(𝜏 NOUN iajs-3638 225 4 ) ) PUNCT iajs-3638 225 5 in in ADP iajs-3638 225 6 equation equation NOUN iajs-3638 225 7 ( ( PUNCT iajs-3638 225 8 37 37 NUM iajs-3638 225 9 ) ) PUNCT iajs-3638 225 10 and and CCONJ iajs-3638 225 11 differentiating differentiate VERB iajs-3638 225 12 the the DET iajs-3638 225 13 resulting result VERB iajs-3638 225 14 equation equation NOUN iajs-3638 225 15 in in ADP iajs-3638 225 16 𝜏,we 𝜏,we PROPN iajs-3638 225 17 can can AUX iajs-3638 225 18 get get VERB iajs-3638 225 19 [ [ X iajs-3638 225 20 3πœ†2 3πœ†2 NUM iajs-3638 225 21 + + NUM iajs-3638 225 22 2𝐷1πœ† 2𝐷1πœ† NUM iajs-3638 225 23 + + NUM iajs-3638 225 24 𝐷2 𝐷2 NOUN iajs-3638 225 25 + + CCONJ iajs-3638 225 26 ( ( PUNCT iajs-3638 225 27 2𝐷4πœ† 2𝐷4πœ† NUM iajs-3638 225 28 + + NOUN iajs-3638 225 29 𝐷5)𝑒 𝐷5)𝑒 PROPN iajs-3638 225 30 βˆ’πœ†πœ βˆ’πœ†πœ NUM iajs-3638 225 31 βˆ’ βˆ’ NOUN iajs-3638 225 32 𝜏(𝐷4πœ† 𝜏(𝐷4πœ† PROPN iajs-3638 225 33 2 2 PROPN iajs-3638 225 34 + + CCONJ iajs-3638 225 35 𝐷5πœ† 𝐷5πœ† PROPN iajs-3638 225 36 + + CCONJ iajs-3638 225 37 𝐷6)𝑒 𝐷6)𝑒 PROPN iajs-3638 225 38 βˆ’πœ†πœ βˆ’πœ†πœ PROPN iajs-3638 225 39 ] ] PUNCT iajs-3638 225 40 π‘‘πœ† π‘‘πœ† ADP iajs-3638 225 41 π‘‘πœ† π‘‘πœ† ADP iajs-3638 225 42 = = SYM iajs-3638 225 43 πœ†(𝐷4πœ† πœ†(𝐷4πœ† NOUN iajs-3638 225 44 2 2 NUM iajs-3638 225 45 + + NOUN iajs-3638 225 46 𝐷5πœ† 𝐷5πœ† PROPN iajs-3638 225 47 + + CCONJ iajs-3638 225 48 𝐷6)𝑒 𝐷6)𝑒 PROPN iajs-3638 225 49 βˆ’πœ†πœ βˆ’πœ†πœ PROPN iajs-3638 225 50 ( ( PUNCT iajs-3638 225 51 43 43 NUM iajs-3638 225 52 ) ) PUNCT iajs-3638 225 53 thus thus ADV iajs-3638 225 54 , , PUNCT iajs-3638 225 55 ( ( PUNCT iajs-3638 225 56 π‘‘πœ† π‘‘πœ† ADP iajs-3638 225 57 π‘‘πœ π‘‘πœ NOUN iajs-3638 225 58 ) ) PUNCT iajs-3638 225 59 βˆ’1 βˆ’1 NOUN iajs-3638 225 60 = = SYM iajs-3638 226 1 3πœ†2 3πœ†2 NUM iajs-3638 226 2 + + NUM iajs-3638 226 3 2𝐷1πœ†+𝐷2 2𝐷1πœ†+𝐷2 NOUN iajs-3638 226 4 πœ†(𝐷4πœ†2+𝐷5πœ†+𝐷6)π‘’βˆ’πœ†πœ πœ†(𝐷4πœ†2+𝐷5πœ†+𝐷6)π‘’βˆ’πœ†πœ PUNCT iajs-3638 227 1 + + NUM iajs-3638 227 2 2𝐷4πœ†+𝐷5 2𝐷4πœ†+𝐷5 PROPN iajs-3638 227 3 πœ†(𝐷4πœ†2+𝐷5πœ†+𝐷6 πœ†(𝐷4πœ†2+𝐷5πœ†+𝐷6 SYM iajs-3638 227 4 ) ) PUNCT iajs-3638 227 5 βˆ’ βˆ’ NOUN iajs-3638 228 1 𝜏 𝜏 X iajs-3638 228 2 πœ† πœ† X iajs-3638 228 3 ( ( PUNCT iajs-3638 228 4 44 44 NUM iajs-3638 228 5 ) ) PUNCT iajs-3638 228 6 since since SCONJ iajs-3638 228 7 for for ADP iajs-3638 228 8 = = PROPN iajs-3638 228 9 𝜏0 𝜏0 PROPN iajs-3638 228 10 , , PUNCT iajs-3638 228 11 and and CCONJ iajs-3638 228 12 πœ† πœ† X iajs-3638 228 13 = = SYM iajs-3638 228 14 π‘–πœ”0 π‘–πœ”0 PROPN iajs-3638 228 15 , , PUNCT iajs-3638 228 16 we we PRON iajs-3638 228 17 have have AUX iajs-3638 228 18 got get VERB iajs-3638 228 19 [ [ PUNCT iajs-3638 228 20 π‘‘πœ† π‘‘πœ† ADP iajs-3638 228 21 π‘‘πœ π‘‘πœ NOUN iajs-3638 228 22 ] ] PUNCT iajs-3638 228 23 𝜏=𝜏0 𝜏=𝜏0 PROPN iajs-3638 228 24 βˆ’1 βˆ’1 NOUN iajs-3638 228 25 = = SYM iajs-3638 228 26 ( ( PUNCT iajs-3638 228 27 𝐷2 𝐷2 PROPN iajs-3638 228 28 βˆ’ βˆ’ PROPN iajs-3638 228 29 3πœ”0 3πœ”0 NUM iajs-3638 228 30 2 2 NUM iajs-3638 228 31 ) ) PUNCT iajs-3638 229 1 + + CCONJ iajs-3638 229 2 2𝑖𝐷1πœ”0 2𝑖𝐷1πœ”0 NUM iajs-3638 229 3 βˆ’πœ”0 βˆ’πœ”0 NOUN iajs-3638 229 4 2(πœ”0 2(πœ”0 NUM iajs-3638 229 5 2 2 NUM iajs-3638 229 6 βˆ’ βˆ’ PROPN iajs-3638 229 7 𝐷2 𝐷2 NOUN iajs-3638 229 8 ) ) PUNCT iajs-3638 230 1 + + CCONJ iajs-3638 230 2 π‘–πœ”0(𝐷1πœ”0 π‘–πœ”0(𝐷1πœ”0 PROPN iajs-3638 230 3 2 2 NUM iajs-3638 230 4 βˆ’ βˆ’ PROPN iajs-3638 230 5 𝐷3 𝐷3 PROPN iajs-3638 230 6 ) ) PUNCT iajs-3638 231 1 + + CCONJ iajs-3638 231 2 𝐷5 𝐷5 X iajs-3638 232 1 + + CCONJ iajs-3638 232 2 2𝑖𝐷4πœ”0 2𝑖𝐷4πœ”0 NUM iajs-3638 232 3 βˆ’π·5πœ”0 βˆ’π·5πœ”0 NUM iajs-3638 232 4 2 2 NUM iajs-3638 232 5 + + CCONJ iajs-3638 232 6 π‘–πœ”0(𝐷6 π‘–πœ”0(𝐷6 NOUN iajs-3638 232 7 βˆ’ βˆ’ NOUN iajs-3638 232 8 𝐷4πœ”0 𝐷4πœ”0 ADJ iajs-3638 232 9 2 2 NUM iajs-3638 232 10 ) ) PUNCT iajs-3638 233 1 βˆ’ βˆ’ NOUN iajs-3638 233 2 π‘–πœ0 π‘–πœ0 NOUN iajs-3638 233 3 πœ”0 πœ”0 NOUN iajs-3638 233 4 now now ADV iajs-3638 233 5 , , PUNCT iajs-3638 233 6 since since SCONJ iajs-3638 233 7 𝑠𝑖𝑔𝑛 𝑠𝑖𝑔𝑛 NOUN iajs-3638 233 8 [ [ PUNCT iajs-3638 233 9 𝑑(π‘…π‘’πœ† 𝑑(π‘…π‘’πœ† PROPN iajs-3638 233 10 ) ) PUNCT iajs-3638 233 11 π‘‘πœ π‘‘πœ NOUN iajs-3638 233 12 ] ] PUNCT iajs-3638 233 13 𝜏=𝜏0 𝜏=𝜏0 PUNCT iajs-3638 233 14 = = SYM iajs-3638 234 1 𝑠𝑖𝑔𝑛 𝑠𝑖𝑔𝑛 NOUN iajs-3638 234 2 [ [ X iajs-3638 234 3 𝑅𝑒 𝑅𝑒 PROPN iajs-3638 234 4 ( ( PUNCT iajs-3638 234 5 π‘‘πœ† π‘‘πœ† ADP iajs-3638 234 6 π‘‘πœ π‘‘πœ NOUN iajs-3638 234 7 ) ) PUNCT iajs-3638 234 8 βˆ’1 βˆ’1 NOUN iajs-3638 234 9 ] ] PUNCT iajs-3638 235 1 πœ†=π‘–πœ”0 πœ†=π‘–πœ”0 PROPN iajs-3638 235 2 ( ( PUNCT iajs-3638 235 3 45 45 NUM iajs-3638 235 4 ) ) PUNCT iajs-3638 235 5 it it PRON iajs-3638 235 6 is be AUX iajs-3638 235 7 clear clear ADJ iajs-3638 235 8 that that SCONJ iajs-3638 235 9 : : PUNCT iajs-3638 235 10 ihjpas ihjpas PROPN iajs-3638 235 11 . . PUNCT iajs-3638 236 1 2025 2025 NUM iajs-3638 236 2 , , PUNCT iajs-3638 236 3 38 38 NUM iajs-3638 236 4 ( ( PUNCT iajs-3638 236 5 1 1 NUM iajs-3638 236 6 ) ) PUNCT iajs-3638 236 7 455 455 NUM iajs-3638 237 1 𝑅𝑒 𝑅𝑒 VERB iajs-3638 237 2 [ [ PUNCT iajs-3638 237 3 ( ( PUNCT iajs-3638 237 4 𝐷2βˆ’3πœ”0 𝐷2βˆ’3πœ”0 NOUN iajs-3638 237 5 2)+2𝑖𝐷1πœ”0 2)+2𝑖𝐷1πœ”0 NUM iajs-3638 237 6 βˆ’πœ”0 βˆ’πœ”0 PROPN iajs-3638 237 7 2(πœ”0 2(πœ”0 NUM iajs-3638 237 8 2βˆ’π·2)+π‘–πœ”0(𝐷1πœ”0 2βˆ’π·2)+π‘–πœ”0(𝐷1πœ”0 NUM iajs-3638 237 9 2βˆ’π·3 2βˆ’π·3 NUM iajs-3638 237 10 ) ) PUNCT iajs-3638 237 11 ] ] PUNCT iajs-3638 238 1 = = PUNCT iajs-3638 238 2 2𝐷1(𝐷1πœ”0 2𝐷1(𝐷1πœ”0 NUM iajs-3638 238 3 2βˆ’π·3)βˆ’(πœ”0 2βˆ’π·3)βˆ’(πœ”0 NUM iajs-3638 238 4 2βˆ’π·2)(𝐷2βˆ’3πœ”0 2βˆ’π·2)(𝐷2βˆ’3πœ”0 PROPN iajs-3638 238 5 2 2 NUM iajs-3638 238 6 ) ) PUNCT iajs-3638 238 7 πœ”0 πœ”0 NOUN iajs-3638 238 8 2(πœ”0 2(πœ”0 NUM iajs-3638 238 9 2βˆ’π·2 2βˆ’π·2 NUM iajs-3638 238 10 ) ) PUNCT iajs-3638 238 11 2 2 NUM iajs-3638 239 1 + + NOUN iajs-3638 239 2 ( ( PUNCT iajs-3638 239 3 𝐷1πœ”0 𝐷1πœ”0 PROPN iajs-3638 239 4 2βˆ’π·3 2βˆ’π·3 NUM iajs-3638 239 5 ) ) PUNCT iajs-3638 239 6 2 2 NUM iajs-3638 239 7 𝑅𝑒 𝑅𝑒 PROPN iajs-3638 239 8 [ [ PUNCT iajs-3638 239 9 𝐷5 𝐷5 PROPN iajs-3638 239 10 + + NOUN iajs-3638 239 11 2𝑖𝐷4πœ”0 2𝑖𝐷4πœ”0 NUM iajs-3638 239 12 βˆ’π·5πœ”0 βˆ’π·5πœ”0 NOUN iajs-3638 239 13 2+π‘–πœ”0(𝐷6βˆ’π·4πœ”0 2+π‘–πœ”0(𝐷6βˆ’π·4πœ”0 NUM iajs-3638 239 14 2 2 NUM iajs-3638 239 15 ) ) PUNCT iajs-3638 239 16 ] ] PUNCT iajs-3638 240 1 = = PUNCT iajs-3638 240 2 2𝐷4(𝐷6βˆ’π·4πœ”0 2𝐷4(𝐷6βˆ’π·4πœ”0 NUM iajs-3638 240 3 2)βˆ’π·5 2)βˆ’π·5 NUM iajs-3638 240 4 2 2 NUM iajs-3638 240 5 𝐷5 𝐷5 NOUN iajs-3638 240 6 2πœ”0 2πœ”0 NUM iajs-3638 240 7 2+(𝐷6βˆ’π·4πœ”0 2+(𝐷6βˆ’π·4πœ”0 PROPN iajs-3638 240 8 2 2 NUM iajs-3638 240 9 ) ) PUNCT iajs-3638 240 10 2 2 NUM iajs-3638 240 11 𝑅𝑒 𝑅𝑒 PROPN iajs-3638 240 12 [ [ PUNCT iajs-3638 240 13 π‘–πœ0 π‘–πœ0 NOUN iajs-3638 240 14 πœ”0 πœ”0 NOUN iajs-3638 240 15 ] ] PUNCT iajs-3638 240 16 = = PUNCT iajs-3638 240 17 π‘§π‘’π‘Ÿπ‘œ π‘§π‘’π‘Ÿπ‘œ PROPN iajs-3638 240 18 hence hence ADV iajs-3638 240 19 , , PUNCT iajs-3638 240 20 we we PRON iajs-3638 240 21 have have AUX iajs-3638 240 22 𝑅𝑒 𝑅𝑒 VERB iajs-3638 240 23 [ [ PUNCT iajs-3638 240 24 π‘‘πœ† π‘‘πœ† ADP iajs-3638 240 25 π‘‘πœ π‘‘πœ NOUN iajs-3638 240 26 ] ] PUNCT iajs-3638 240 27 𝜏=𝜏0 𝜏=𝜏0 PROPN iajs-3638 240 28 βˆ’1 βˆ’1 NOUN iajs-3638 240 29 = = SYM iajs-3638 240 30 2𝐷1(𝐷1πœ”0 2𝐷1(𝐷1πœ”0 NUM iajs-3638 240 31 2βˆ’π·3)βˆ’(πœ”0 2βˆ’π·3)βˆ’(πœ”0 NUM iajs-3638 240 32 2βˆ’π·2)(𝐷2βˆ’3πœ”0 2βˆ’π·2)(𝐷2βˆ’3πœ”0 PROPN iajs-3638 240 33 2 2 NUM iajs-3638 240 34 ) ) PUNCT iajs-3638 240 35 ψ0 ψ0 ADV iajs-3638 240 36 + + CCONJ iajs-3638 240 37 2𝐷4(𝐷6βˆ’π·4πœ”0 2𝐷4(𝐷6βˆ’π·4πœ”0 NUM iajs-3638 240 38 2)βˆ’π·5 2)βˆ’π·5 NUM iajs-3638 240 39 2 2 NUM iajs-3638 240 40 ψ1 ψ1 NOUN iajs-3638 240 41 where where SCONJ iajs-3638 240 42 ψ0 ψ0 ADJ iajs-3638 240 43 = = NOUN iajs-3638 240 44 πœ”0 πœ”0 NOUN iajs-3638 240 45 2(πœ”0 2(πœ”0 NUM iajs-3638 240 46 2 2 NUM iajs-3638 240 47 βˆ’ βˆ’ PROPN iajs-3638 240 48 𝐷2 𝐷2 NOUN iajs-3638 240 49 ) ) PUNCT iajs-3638 240 50 2 2 NUM iajs-3638 241 1 + + CCONJ iajs-3638 241 2 ( ( PUNCT iajs-3638 241 3 𝐷1πœ”0 𝐷1πœ”0 PROPN iajs-3638 241 4 2 2 NUM iajs-3638 241 5 βˆ’ βˆ’ PROPN iajs-3638 241 6 𝐷3 𝐷3 PROPN iajs-3638 241 7 ) ) PUNCT iajs-3638 241 8 2 2 NUM iajs-3638 241 9 > > SYM iajs-3638 241 10 0 0 NUM iajs-3638 241 11 , , PUNCT iajs-3638 241 12 ψ1 ψ1 NOUN iajs-3638 241 13 = = SYM iajs-3638 241 14 𝐷5 𝐷5 PROPN iajs-3638 241 15 2πœ”0 2πœ”0 NUM iajs-3638 241 16 2 2 NUM iajs-3638 242 1 + + CCONJ iajs-3638 242 2 ( ( PUNCT iajs-3638 242 3 𝐷6 𝐷6 NOUN iajs-3638 242 4 βˆ’ βˆ’ PROPN iajs-3638 242 5 𝐷4πœ”0 𝐷4πœ”0 PROPN iajs-3638 242 6 2)2 2)2 NUM iajs-3638 242 7 > > SYM iajs-3638 242 8 0 0 NUM iajs-3638 242 9 . . PUNCT iajs-3638 243 1 we we PRON iajs-3638 243 2 obtain obtain VERB iajs-3638 243 3 [ [ PUNCT iajs-3638 243 4 𝑑(π‘…π‘’πœ†(𝜏 𝑑(π‘…π‘’πœ†(𝜏 NOUN iajs-3638 243 5 ) ) PUNCT iajs-3638 243 6 ) ) PUNCT iajs-3638 244 1 π‘‘πœ π‘‘πœ NOUN iajs-3638 244 2 ] ] PUNCT iajs-3638 244 3 𝜏=Ο„0 𝜏=Ο„0 X iajs-3638 244 4 > > X iajs-3638 244 5 0 0 PUNCT iajs-3638 245 1 ( ( PUNCT iajs-3638 245 2 42b42d 42b42d NUM iajs-3638 245 3 ) ) PUNCT iajs-3638 245 4 . . PUNCT iajs-3638 246 1 this this DET iajs-3638 246 2 outcome outcome NOUN iajs-3638 246 3 demonstrates demonstrate VERB iajs-3638 246 4 how how SCONJ iajs-3638 246 5 the the DET iajs-3638 246 6 roots root NOUN iajs-3638 246 7 of of ADP iajs-3638 246 8 c.e c.e PROPN iajs-3638 246 9 . . PUNCT iajs-3638 247 1 ( ( PUNCT iajs-3638 247 2 37 37 NUM iajs-3638 247 3 ) ) PUNCT iajs-3638 247 4 , , PUNCT iajs-3638 247 5 as as SCONJ iajs-3638 247 6 𝜏 𝜏 PROPN iajs-3638 247 7 passes pass VERB iajs-3638 247 8 through through ADP iajs-3638 247 9 Ο„0 Ο„0 NOUN iajs-3638 247 10 , , PUNCT iajs-3638 247 11 traverse traverse VERB iajs-3638 247 12 the the DET iajs-3638 247 13 imaginary imaginary ADJ iajs-3638 247 14 axis axis NOUN iajs-3638 247 15 from from ADP iajs-3638 247 16 left left ADJ iajs-3638 247 17 to to ADP iajs-3638 247 18 right right NOUN iajs-3638 247 19 . . PUNCT iajs-3638 248 1 as as ADP iajs-3638 248 2 a a DET iajs-3638 248 3 result result NOUN iajs-3638 248 4 , , PUNCT iajs-3638 248 5 the the DET iajs-3638 248 6 system system NOUN iajs-3638 248 7 ( ( PUNCT iajs-3638 248 8 3 3 X iajs-3638 248 9 ) ) PUNCT iajs-3638 248 10 experiences experience NOUN iajs-3638 248 11 an an DET iajs-3638 248 12 h.b h.b PROPN iajs-3638 248 13 . . PROPN iajs-3638 249 1 at at ADP iajs-3638 249 2 𝜏 𝜏 NOUN iajs-3638 249 3 = = VERB iajs-3638 249 4 Ο„0 Ο„0 NOUN iajs-3638 249 5 and and CCONJ iajs-3638 249 6 loses lose VERB iajs-3638 249 7 its its PRON iajs-3638 249 8 stability stability NOUN iajs-3638 249 9 . . PUNCT iajs-3638 250 1 for for ADP iajs-3638 250 2 the the DET iajs-3638 250 3 ( ( PUNCT iajs-3638 250 4 siep siep NOUN iajs-3638 250 5 ) ) PUNCT iajs-3638 250 6 , , PUNCT iajs-3638 250 7 equation equation NOUN iajs-3638 250 8 ( ( PUNCT iajs-3638 250 9 4 4 NUM iajs-3638 250 10 ) ) PUNCT iajs-3638 250 11 reduces reduce VERB iajs-3638 250 12 to to ADP iajs-3638 250 13 𝐽(𝐸5 𝐽(𝐸5 NUM iajs-3638 250 14 ) ) PUNCT iajs-3638 250 15 = = PUNCT iajs-3638 251 1 ( ( PUNCT iajs-3638 251 2 𝑐𝑖𝑗)4Γ—4 𝑐𝑖𝑗)4Γ—4 X iajs-3638 251 3 ; ; PUNCT iajs-3638 251 4 𝑖 𝑖 X iajs-3638 251 5 , , PUNCT iajs-3638 251 6 𝑗 𝑗 NOUN iajs-3638 251 7 = = NOUN iajs-3638 251 8 1,2,3,4 1,2,3,4 NUM iajs-3638 251 9 here here ADV iajs-3638 251 10 𝑐11 𝑐11 NOUN iajs-3638 251 11 = = PUNCT iajs-3638 251 12 βˆ’[𝑅1 βˆ’[𝑅1 X iajs-3638 251 13 + + PUNCT iajs-3638 251 14 𝑅2𝑒 𝑅2𝑒 PROPN iajs-3638 251 15 βˆ’πœ†πœ βˆ’πœ†πœ PROPN iajs-3638 251 16 ] ] PUNCT iajs-3638 251 17 , , PUNCT iajs-3638 251 18 𝑐12 𝑐12 NOUN iajs-3638 251 19 = = SYM iajs-3638 251 20 π‘›π›Όπ‘†βˆ— π‘›π›Όπ‘†βˆ— NUM iajs-3638 251 21 ( ( PUNCT iajs-3638 251 22 1 1 NUM iajs-3638 251 23 + + NUM iajs-3638 251 24 𝑛𝑉1 𝑛𝑉1 PROPN iajs-3638 251 25 βˆ—)2 βˆ—)2 ADV iajs-3638 251 26 , , PUNCT iajs-3638 251 27 𝑐14 𝑐14 VERB iajs-3638 251 28 = = PRON iajs-3638 251 29 βˆ’π‘…3𝑒 βˆ’π‘…3𝑒 NOUN iajs-3638 251 30 βˆ’πœ†πœ βˆ’πœ†πœ PROPN iajs-3638 251 31 , , PUNCT iajs-3638 251 32 𝑐21 𝑐21 VERB iajs-3638 251 33 = = SYM iajs-3638 251 34 𝛼 𝛼 PROPN iajs-3638 251 35 1 1 NUM iajs-3638 251 36 + + NUM iajs-3638 251 37 𝑛𝑉1 𝑛𝑉1 PROPN iajs-3638 251 38 βˆ— βˆ— NOUN iajs-3638 251 39 , , PUNCT iajs-3638 251 40 𝑐22 𝑐22 NOUN iajs-3638 251 41 = = SYM iajs-3638 251 42 βˆ’ βˆ’ PROPN iajs-3638 251 43 ( ( PUNCT iajs-3638 251 44 π‘›π›Όπ‘†βˆ— π‘›π›Όπ‘†βˆ— X iajs-3638 251 45 ( ( PUNCT iajs-3638 251 46 1 1 NUM iajs-3638 251 47 + + NUM iajs-3638 251 48 𝑛𝑉1 𝑛𝑉1 PROPN iajs-3638 251 49 βˆ—)2 βˆ—)2 PROPN iajs-3638 251 50 + + CCONJ iajs-3638 251 51 𝛾 𝛾 ADP iajs-3638 251 52 + + PUNCT iajs-3638 251 53 𝛽2𝐼 𝛽2𝐼 ADV iajs-3638 251 54 βˆ— βˆ— NOUN iajs-3638 251 55 + + CCONJ iajs-3638 251 56 πœ‡ πœ‡ X iajs-3638 251 57 ) ) PUNCT iajs-3638 251 58 , , PUNCT iajs-3638 251 59 𝑐24 𝑐24 NUM iajs-3638 251 60 = = PUNCT iajs-3638 251 61 βˆ’π›½2𝑉1 βˆ’π›½2𝑉1 PROPN iajs-3638 251 62 βˆ— βˆ— NOUN iajs-3638 251 63 , , PUNCT iajs-3638 251 64 𝑐32 𝑐32 NOUN iajs-3638 251 65 = = SYM iajs-3638 251 66 𝛾 𝛾 NOUN iajs-3638 251 67 , , PUNCT iajs-3638 251 68 𝑐33 𝑐33 NOUN iajs-3638 251 69 = = SYM iajs-3638 251 70 βˆ’(𝛽3𝐼 βˆ’(𝛽3𝐼 PROPN iajs-3638 251 71 βˆ— βˆ— NOUN iajs-3638 251 72 + + X iajs-3638 251 73 πœ‡ πœ‡ X iajs-3638 251 74 ) ) PUNCT iajs-3638 251 75 , , PUNCT iajs-3638 251 76 𝑐34 𝑐34 X iajs-3638 251 77 = = SYM iajs-3638 251 78 βˆ’π›½3𝑉2 βˆ’π›½3𝑉2 PROPN iajs-3638 251 79 βˆ— βˆ— NOUN iajs-3638 251 80 , , PUNCT iajs-3638 251 81 𝑐41 𝑐41 NOUN iajs-3638 251 82 = = SYM iajs-3638 251 83 𝑅2𝑒 𝑅2𝑒 NOUN iajs-3638 251 84 βˆ’πœ†πœ βˆ’πœ†πœ PROPN iajs-3638 251 85 , , PUNCT iajs-3638 251 86 𝑐42 𝑐42 NOUN iajs-3638 251 87 = = PUNCT iajs-3638 251 88 𝛽2𝐼 𝛽2𝐼 ADV iajs-3638 251 89 βˆ— βˆ— NOUN iajs-3638 251 90 , , PUNCT iajs-3638 251 91 𝑐43 𝑐43 PROPN iajs-3638 251 92 = = SYM iajs-3638 251 93 𝛽3𝐼 𝛽3𝐼 ADJ iajs-3638 251 94 βˆ— βˆ— NOUN iajs-3638 251 95 , , PUNCT iajs-3638 251 96 𝑐44 𝑐44 X iajs-3638 251 97 = = SYM iajs-3638 251 98 𝑅3𝑒 𝑅3𝑒 PROPN iajs-3638 251 99 βˆ’πœ†πœ βˆ’πœ†πœ PROPN iajs-3638 251 100 + + CCONJ iajs-3638 251 101 𝑅4 𝑅4 PROPN iajs-3638 251 102 , , PUNCT iajs-3638 251 103 𝑐13 𝑐13 NOUN iajs-3638 251 104 = = SYM iajs-3638 251 105 𝑐31 𝑐31 ADJ iajs-3638 251 106 = = NOUN iajs-3638 251 107 𝑐23 𝑐23 NOUN iajs-3638 251 108 = = SYM iajs-3638 251 109 0 0 PROPN iajs-3638 251 110 . . PUNCT iajs-3638 252 1 ( ( PUNCT iajs-3638 252 2 46 46 NUM iajs-3638 252 3 ) ) PUNCT iajs-3638 252 4 where where SCONJ iajs-3638 252 5 , , PUNCT iajs-3638 252 6 𝑅1 𝑅1 PROPN iajs-3638 252 7 = = PUNCT iajs-3638 252 8 𝛼 𝛼 PROPN iajs-3638 252 9 1 1 NUM iajs-3638 252 10 + + NUM iajs-3638 252 11 𝑛𝑉1 𝑛𝑉1 PROPN iajs-3638 252 12 βˆ— βˆ— NOUN iajs-3638 252 13 + + CCONJ iajs-3638 252 14 πœ‡ πœ‡ ADP iajs-3638 252 15 , , PUNCT iajs-3638 252 16 𝑅2 𝑅2 ADP iajs-3638 252 17 = = SYM iajs-3638 252 18 𝛽1𝐼 𝛽1𝐼 NOUN iajs-3638 252 19 βˆ— βˆ— X iajs-3638 252 20 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ PROPN iajs-3638 252 21 𝑅3 𝑅3 PROPN iajs-3638 252 22 = = SYM iajs-3638 252 23 𝛽1𝑆 𝛽1𝑆 NUM iajs-3638 252 24 βˆ— βˆ— NOUN iajs-3638 252 25 , , PUNCT iajs-3638 252 26 𝑅4 𝑅4 PROPN iajs-3638 252 27 = = SYM iajs-3638 253 1 𝛽2𝑉1 𝛽2𝑉1 PROPN iajs-3638 253 2 βˆ— βˆ— NOUN iajs-3638 253 3 + + X iajs-3638 253 4 𝛽3𝑉2 𝛽3𝑉2 ADJ iajs-3638 253 5 βˆ— βˆ— NOUN iajs-3638 253 6 βˆ’ βˆ’ PROPN iajs-3638 253 7 ( ( PUNCT iajs-3638 253 8 πœ‡ πœ‡ ADP iajs-3638 253 9 + + X iajs-3638 253 10 πœ‡1 πœ‡1 ADJ iajs-3638 253 11 + + CCONJ iajs-3638 253 12 πœƒ πœƒ NOUN iajs-3638 253 13 ) ) PUNCT iajs-3638 253 14 the the DET iajs-3638 253 15 c.e c.e PROPN iajs-3638 253 16 . . PROPN iajs-3638 253 17 of of ADP iajs-3638 253 18 𝐽(𝐸5 𝐽(𝐸5 PROPN iajs-3638 253 19 ) ) PUNCT iajs-3638 253 20 is be AUX iajs-3638 253 21 [ [ X iajs-3638 253 22 πœ†4 πœ†4 NOUN iajs-3638 253 23 + + CCONJ iajs-3638 253 24 𝐢1πœ† 𝐢1πœ† NOUN iajs-3638 253 25 3 3 NUM iajs-3638 253 26 + + NOUN iajs-3638 253 27 𝐢2πœ† 𝐢2πœ† NOUN iajs-3638 253 28 2 2 NUM iajs-3638 253 29 + + CCONJ iajs-3638 253 30 𝐢3πœ† 𝐢3πœ† ADJ iajs-3638 253 31 + + NUM iajs-3638 253 32 𝐢4 𝐢4 NOUN iajs-3638 253 33 + + CCONJ iajs-3638 253 34 ( ( PUNCT iajs-3638 253 35 𝐢5πœ† 𝐢5πœ† NOUN iajs-3638 253 36 3 3 NUM iajs-3638 253 37 + + CCONJ iajs-3638 253 38 𝐢6πœ† 𝐢6πœ† NOUN iajs-3638 253 39 2 2 NUM iajs-3638 253 40 + + NOUN iajs-3638 253 41 𝐢7πœ† 𝐢7πœ† PROPN iajs-3638 253 42 + + CCONJ iajs-3638 253 43 𝐢8)𝑒 𝐢8)𝑒 PROPN iajs-3638 253 44 βˆ’πœ†πœ βˆ’πœ†πœ NUM iajs-3638 253 45 ] ] X iajs-3638 253 46 = = SYM iajs-3638 253 47 0 0 PUNCT iajs-3638 253 48 ( ( PUNCT iajs-3638 253 49 47 47 NUM iajs-3638 253 50 ) ) PUNCT iajs-3638 253 51 ihjpas ihjpa NOUN iajs-3638 253 52 . . PUNCT iajs-3638 254 1 2025 2025 NUM iajs-3638 254 2 , , PUNCT iajs-3638 254 3 38 38 NUM iajs-3638 254 4 ( ( PUNCT iajs-3638 254 5 1 1 NUM iajs-3638 254 6 ) ) PUNCT iajs-3638 254 7 456 456 NUM iajs-3638 254 8 with with ADP iajs-3638 254 9 𝐢1 𝐢1 PROPN iajs-3638 254 10 = = SYM iajs-3638 254 11 𝑅1 𝑅1 PROPN iajs-3638 254 12 βˆ’ βˆ’ PROPN iajs-3638 255 1 ( ( PUNCT iajs-3638 255 2 𝑅4 𝑅4 PROPN iajs-3638 255 3 + + CCONJ iajs-3638 255 4 𝑐22 𝑐22 VERB iajs-3638 255 5 + + CCONJ iajs-3638 255 6 𝑐33 𝑐33 NOUN iajs-3638 255 7 ) ) PUNCT iajs-3638 255 8 , , PUNCT iajs-3638 255 9 𝐢2 𝐢2 X iajs-3638 255 10 = = PUNCT iajs-3638 255 11 ( ( PUNCT iajs-3638 255 12 𝑐22 𝑐22 NOUN iajs-3638 255 13 + + CCONJ iajs-3638 255 14 𝑐33)(𝑅4 𝑐33)(𝑅4 NOUN iajs-3638 255 15 βˆ’ βˆ’ PROPN iajs-3638 255 16 𝑅1 𝑅1 PROPN iajs-3638 255 17 ) ) PUNCT iajs-3638 255 18 + + NUM iajs-3638 255 19 𝑐22𝑐33 𝑐22𝑐33 NOUN iajs-3638 255 20 βˆ’ βˆ’ PROPN iajs-3638 255 21 ( ( PUNCT iajs-3638 255 22 𝑅1𝑅4 𝑅1𝑅4 X iajs-3638 255 23 + + CCONJ iajs-3638 255 24 𝑐12𝑐21 𝑐12𝑐21 ADJ iajs-3638 255 25 + + CCONJ iajs-3638 255 26 𝑐24𝑐42 𝑐24𝑐42 ADJ iajs-3638 255 27 + + CCONJ iajs-3638 255 28 𝑐34𝑐43 𝑐34𝑐43 NOUN iajs-3638 255 29 ) ) PUNCT iajs-3638 255 30 , , PUNCT iajs-3638 255 31 𝐢3 𝐢3 NOUN iajs-3638 255 32 = = SYM iajs-3638 255 33 ( ( PUNCT iajs-3638 255 34 𝑐22𝑐33 𝑐22𝑐33 NOUN iajs-3638 255 35 + + CCONJ iajs-3638 255 36 𝑅4𝑐22 𝑅4𝑐22 NOUN iajs-3638 255 37 + + CCONJ iajs-3638 255 38 𝑅4𝑐33 𝑅4𝑐33 NOUN iajs-3638 255 39 βˆ’ βˆ’ PROPN iajs-3638 255 40 ( ( PUNCT iajs-3638 256 1 𝑐24𝑐42 𝑐24𝑐42 NOUN iajs-3638 256 2 + + CCONJ iajs-3638 256 3 𝑐34𝑐43))𝑅1 𝑐34𝑐43))𝑅1 ADP iajs-3638 257 1 + + CCONJ iajs-3638 257 2 ( ( PUNCT iajs-3638 257 3 𝑐12𝑐21 𝑐12𝑐21 ADJ iajs-3638 257 4 βˆ’ βˆ’ PROPN iajs-3638 257 5 𝑐22𝑐33)𝑅4 𝑐22𝑐33)𝑅4 NOUN iajs-3638 257 6 + + CCONJ iajs-3638 257 7 ( ( PUNCT iajs-3638 257 8 𝑐12𝑐21 𝑐12𝑐21 ADJ iajs-3638 257 9 + + CCONJ iajs-3638 257 10 𝑐24𝑐42)𝑐33 𝑐24𝑐42)𝑐33 X iajs-3638 257 11 + + CCONJ iajs-3638 257 12 ( ( PUNCT iajs-3638 257 13 𝑐22𝑐34 𝑐22𝑐34 ADJ iajs-3638 257 14 βˆ’ βˆ’ PROPN iajs-3638 257 15 𝑐24𝑐32)𝑐43 𝑐24𝑐32)𝑐43 PROPN iajs-3638 257 16 , , PUNCT iajs-3638 257 17 𝐢4 𝐢4 NOUN iajs-3638 257 18 = = PUNCT iajs-3638 257 19 ( ( PUNCT iajs-3638 257 20 𝑐22𝑐34𝑐43 𝑐22𝑐34𝑐43 X iajs-3638 257 21 + + CCONJ iajs-3638 257 22 𝑐24𝑐33𝑐42 𝑐24𝑐33𝑐42 CCONJ iajs-3638 257 23 βˆ’ βˆ’ PROPN iajs-3638 257 24 𝑅4𝑐22𝑐33 𝑅4𝑐22𝑐33 NUM iajs-3638 257 25 βˆ’ βˆ’ NOUN iajs-3638 257 26 𝑐24𝑐32𝑐43)𝑅1 𝑐24𝑐32𝑐43)𝑅1 PRON iajs-3638 258 1 + + PUNCT iajs-3638 258 2 𝑐12𝑐21(𝑐34𝑐43 𝑐12𝑐21(𝑐34𝑐43 PROPN iajs-3638 258 3 βˆ’ βˆ’ NOUN iajs-3638 258 4 𝑐33𝑅4 𝑐33𝑅4 PROPN iajs-3638 258 5 ) ) PUNCT iajs-3638 258 6 , , PUNCT iajs-3638 258 7 𝐢5 𝐢5 NOUN iajs-3638 258 8 = = PUNCT iajs-3638 258 9 𝑅2 𝑅2 NOUN iajs-3638 258 10 βˆ’ βˆ’ PROPN iajs-3638 258 11 𝑅3 𝑅3 NOUN iajs-3638 258 12 , , PUNCT iajs-3638 258 13 𝐢6 𝐢6 NOUN iajs-3638 258 14 = = SYM iajs-3638 258 15 ( ( PUNCT iajs-3638 258 16 𝑐22 𝑐22 NOUN iajs-3638 259 1 + + CCONJ iajs-3638 259 2 𝑐33)(𝑅3 𝑐33)(𝑅3 PRON iajs-3638 259 3 βˆ’ βˆ’ NOUN iajs-3638 259 4 𝑅2 𝑅2 NOUN iajs-3638 259 5 ) ) PUNCT iajs-3638 260 1 βˆ’ βˆ’ PROPN iajs-3638 260 2 ( ( PUNCT iajs-3638 260 3 𝑅1𝑅3 𝑅1𝑅3 NOUN iajs-3638 260 4 + + CCONJ iajs-3638 260 5 𝑅2𝑅4 𝑅2𝑅4 NOUN iajs-3638 260 6 ) ) PUNCT iajs-3638 260 7 , , PUNCT iajs-3638 260 8 𝐢7 𝐢7 X iajs-3638 261 1 = = SYM iajs-3638 261 2 ( ( PUNCT iajs-3638 261 3 𝑐33(𝑐22 𝑐33(𝑐22 X iajs-3638 261 4 + + CCONJ iajs-3638 261 5 𝑅4 𝑅4 X iajs-3638 261 6 ) ) PUNCT iajs-3638 262 1 βˆ’ βˆ’ PROPN iajs-3638 262 2 𝑐24(𝑐42 𝑐24(𝑐42 NOUN iajs-3638 262 3 + + CCONJ iajs-3638 262 4 𝑐12 𝑐12 NOUN iajs-3638 262 5 ) ) PUNCT iajs-3638 262 6 βˆ’ βˆ’ NOUN iajs-3638 262 7 𝑐34𝑐43)𝑅2 𝑐34𝑐43)𝑅2 PRON iajs-3638 263 1 + + CCONJ iajs-3638 263 2 ( ( PUNCT iajs-3638 263 3 𝑅3𝑐22 𝑅3𝑐22 ADJ iajs-3638 263 4 + + CCONJ iajs-3638 263 5 𝑅4𝑐22 𝑅4𝑐22 NOUN iajs-3638 263 6 + + CCONJ iajs-3638 263 7 𝑅3𝑐33)𝑅1 𝑅3𝑐33)𝑅1 NOUN iajs-3638 263 8 + + CCONJ iajs-3638 263 9 ( ( PUNCT iajs-3638 263 10 𝑐12𝑐21 𝑐12𝑐21 ADJ iajs-3638 263 11 + + CCONJ iajs-3638 263 12 𝑐21𝑐42 𝑐21𝑐42 ADJ iajs-3638 263 13 βˆ’ βˆ’ NOUN iajs-3638 263 14 𝑐22𝑐33)𝑅3 𝑐22𝑐33)𝑅3 NOUN iajs-3638 263 15 , , PUNCT iajs-3638 263 16 𝐢8 𝐢8 NOUN iajs-3638 263 17 = = SYM iajs-3638 263 18 ( ( PUNCT iajs-3638 263 19 𝑐21𝑐32𝑐43 𝑐21𝑐32𝑐43 X iajs-3638 263 20 βˆ’ βˆ’ PROPN iajs-3638 263 21 ( ( PUNCT iajs-3638 263 22 𝑅1𝑐22 𝑅1𝑐22 PROPN iajs-3638 263 23 + + CCONJ iajs-3638 263 24 𝑐12𝑐21 𝑐12𝑐21 VERB iajs-3638 263 25 + + CCONJ iajs-3638 264 1 𝑐21𝑐42)𝑐33)𝑅3 𝑐21𝑐42)𝑐33)𝑅3 NOUN iajs-3638 264 2 + + X iajs-3638 264 3 ( ( PUNCT iajs-3638 264 4 𝑐22𝑐34𝑐43 𝑐22𝑐34𝑐43 X iajs-3638 264 5 + + PUNCT iajs-3638 264 6 𝑐24𝑐33𝑐42 𝑐24𝑐33𝑐42 PROPN iajs-3638 265 1 + + CCONJ iajs-3638 265 2 𝑐12𝑐24𝑐33 𝑐12𝑐24𝑐33 NOUN iajs-3638 265 3 βˆ’ βˆ’ NOUN iajs-3638 265 4 ( ( PUNCT iajs-3638 265 5 𝑅4𝑐22𝑐33 𝑅4𝑐22𝑐33 NOUN iajs-3638 266 1 + + SYM iajs-3638 266 2 𝑐24𝑐32𝑐43))𝑅2 𝑐24𝑐32𝑐43))𝑅2 NOUN iajs-3638 266 3 now now ADV iajs-3638 266 4 , , PUNCT iajs-3638 266 5 when when SCONJ iajs-3638 266 6 = = PRON iajs-3638 266 7 0 0 NUM iajs-3638 266 8 , , PUNCT iajs-3638 266 9 equation equation NOUN iajs-3638 266 10 ( ( PUNCT iajs-3638 266 11 47 47 NUM iajs-3638 266 12 ) ) PUNCT iajs-3638 266 13 is be AUX iajs-3638 266 14 : : PUNCT iajs-3638 266 15 [ [ X iajs-3638 266 16 πœ†4 πœ†4 NOUN iajs-3638 266 17 + + CCONJ iajs-3638 266 18 ( ( PUNCT iajs-3638 266 19 𝐢1 𝐢1 PROPN iajs-3638 266 20 + + NUM iajs-3638 266 21 𝐢5)πœ† 𝐢5)πœ† PROPN iajs-3638 266 22 3 3 NUM iajs-3638 266 23 + + CCONJ iajs-3638 266 24 ( ( PUNCT iajs-3638 266 25 𝐢2 𝐢2 X iajs-3638 266 26 + + CCONJ iajs-3638 266 27 𝐢6)πœ† 𝐢6)πœ† PROPN iajs-3638 266 28 2 2 NUM iajs-3638 266 29 + + CCONJ iajs-3638 266 30 ( ( PUNCT iajs-3638 266 31 𝐢3 𝐢3 NOUN iajs-3638 266 32 + + NUM iajs-3638 266 33 𝐢7)πœ† 𝐢7)πœ† PUNCT iajs-3638 266 34 + + NUM iajs-3638 266 35 𝐢4 𝐢4 NOUN iajs-3638 266 36 + + CCONJ iajs-3638 266 37 𝐢8 𝐢8 NOUN iajs-3638 266 38 ] ] X iajs-3638 266 39 = = SYM iajs-3638 266 40 0 0 NUM iajs-3638 266 41 ( ( PUNCT iajs-3638 266 42 48 48 NUM iajs-3638 266 43 ) ) PUNCT iajs-3638 266 44 all all DET iajs-3638 266 45 the the DET iajs-3638 266 46 eigenvalues eigenvalue NOUN iajs-3638 266 47 of of ADP iajs-3638 266 48 equation equation NOUN iajs-3638 266 49 ( ( PUNCT iajs-3638 266 50 48 48 NUM iajs-3638 266 51 ) ) PUNCT iajs-3638 266 52 will will AUX iajs-3638 266 53 be be AUX iajs-3638 266 54 present present ADJ iajs-3638 266 55 in in ADP iajs-3638 266 56 the the DET iajs-3638 266 57 left left ADJ iajs-3638 266 58 half half NOUN iajs-3638 266 59 plane plane NOUN iajs-3638 266 60 and and CCONJ iajs-3638 266 61 the the DET iajs-3638 266 62 siep siep NOUN iajs-3638 266 63 is be AUX iajs-3638 266 64 l.a.s l.a.s ADJ iajs-3638 266 65 . . PUNCT iajs-3638 266 66 of of ADP iajs-3638 266 67 system system NOUN iajs-3638 266 68 ( ( PUNCT iajs-3638 266 69 3 3 NUM iajs-3638 266 70 ) ) PUNCT iajs-3638 266 71 under under ADP iajs-3638 266 72 the the DET iajs-3638 266 73 following follow VERB iajs-3638 266 74 condition condition NOUN iajs-3638 266 75 : : PUNCT iajs-3638 266 76 2(𝛽1𝑆 2(𝛽1𝑆 NUM iajs-3638 266 77 βˆ— βˆ— NOUN iajs-3638 266 78 + + CCONJ iajs-3638 266 79 𝛽2𝑉1 𝛽2𝑉1 PROPN iajs-3638 266 80 βˆ— βˆ— NOUN iajs-3638 266 81 + + NUM iajs-3638 266 82 𝛽3𝑉2 𝛽3𝑉2 NUM iajs-3638 266 83 βˆ— βˆ— NOUN iajs-3638 266 84 ) ) PUNCT iajs-3638 266 85 < < X iajs-3638 266 86 πœ‡ πœ‡ ADP iajs-3638 266 87 + + X iajs-3638 266 88 πœ‡1 πœ‡1 ADJ iajs-3638 266 89 + + CCONJ iajs-3638 266 90 πœƒ πœƒ PROPN iajs-3638 266 91 ( ( PUNCT iajs-3638 266 92 49 49 NUM iajs-3638 266 93 ) ) PUNCT iajs-3638 266 94 now now ADV iajs-3638 266 95 , , PUNCT iajs-3638 266 96 for for ADP iajs-3638 266 97 𝜏 𝜏 X iajs-3638 266 98 > > X iajs-3638 266 99 0 0 NUM iajs-3638 266 100 , , PUNCT iajs-3638 266 101 then then ADV iajs-3638 266 102 from from ADP iajs-3638 266 103 equation equation NOUN iajs-3638 266 104 ( ( PUNCT iajs-3638 266 105 47 47 NUM iajs-3638 266 106 ) ) PUNCT iajs-3638 266 107 has have VERB iajs-3638 266 108 two two NUM iajs-3638 266 109 wholly wholly ADV iajs-3638 266 110 imaginary imaginary ADJ iajs-3638 266 111 roots root NOUN iajs-3638 266 112 , , PUNCT iajs-3638 266 113 namely namely ADV iajs-3638 266 114 πœ† πœ† PRON iajs-3638 266 115 = = SYM iajs-3638 266 116 Β±π‘–πœ” Β±π‘–πœ” NOUN iajs-3638 266 117 ( ( PUNCT iajs-3638 266 118 πœ” πœ” X iajs-3638 266 119 > > X iajs-3638 266 120 0 0 NUM iajs-3638 266 121 ) ) PUNCT iajs-3638 266 122 . . PUNCT iajs-3638 267 1 by by ADP iajs-3638 267 2 substituting substitute VERB iajs-3638 267 3 πœ† πœ† PRON iajs-3638 267 4 = = SYM iajs-3638 267 5 Β±π‘–πœ” Β±π‘–πœ” NOUN iajs-3638 267 6 in in ADP iajs-3638 267 7 equation equation NOUN iajs-3638 267 8 ( ( PUNCT iajs-3638 267 9 47 47 NUM iajs-3638 267 10 ) ) PUNCT iajs-3638 267 11 and and CCONJ iajs-3638 267 12 separating separate VERB iajs-3638 267 13 the the DET iajs-3638 267 14 real real ADJ iajs-3638 267 15 and and CCONJ iajs-3638 267 16 imaginary imaginary ADJ iajs-3638 267 17 parts part NOUN iajs-3638 267 18 , , PUNCT iajs-3638 267 19 which which PRON iajs-3638 267 20 gives give VERB iajs-3638 267 21 ( ( PUNCT iajs-3638 267 22 𝐢8 𝐢8 NOUN iajs-3638 267 23 βˆ’ βˆ’ PROPN iajs-3638 267 24 𝐢6πœ” 𝐢6πœ” PROPN iajs-3638 267 25 2)π‘π‘œπ‘ πœ”πœ 2)π‘π‘œπ‘ πœ”πœ NUM iajs-3638 267 26 + + CCONJ iajs-3638 267 27 ( ( PUNCT iajs-3638 267 28 𝐢7πœ” 𝐢7πœ” PROPN iajs-3638 267 29 βˆ’ βˆ’ NOUN iajs-3638 267 30 𝐢5πœ” 𝐢5πœ” NOUN iajs-3638 267 31 3)π‘ π‘–π‘›πœ”πœ 3)π‘ π‘–π‘›πœ”πœ NUM iajs-3638 267 32 = = SYM iajs-3638 267 33 𝐢2πœ” 𝐢2πœ” ADP iajs-3638 267 34 2 2 NUM iajs-3638 267 35 βˆ’ βˆ’ NOUN iajs-3638 267 36 πœ”4 πœ”4 NOUN iajs-3638 267 37 βˆ’ βˆ’ PROPN iajs-3638 267 38 𝐢4 𝐢4 NOUN iajs-3638 267 39 , , PUNCT iajs-3638 267 40 ( ( PUNCT iajs-3638 267 41 𝐢7πœ” 𝐢7πœ” PROPN iajs-3638 267 42 βˆ’ βˆ’ NOUN iajs-3638 267 43 𝐢5πœ” 𝐢5πœ” PROPN iajs-3638 267 44 3)π‘π‘œπ‘ πœ”πœ 3)π‘π‘œπ‘ πœ”πœ NUM iajs-3638 267 45 + + CCONJ iajs-3638 268 1 ( ( PUNCT iajs-3638 268 2 𝐢6πœ” 𝐢6πœ” PROPN iajs-3638 268 3 2 2 NUM iajs-3638 268 4 βˆ’ βˆ’ NOUN iajs-3638 268 5 𝐢8)π‘ π‘–π‘›πœ”πœ 𝐢8)π‘ π‘–π‘›πœ”πœ ADJ iajs-3638 268 6 = = SYM iajs-3638 269 1 𝐢1πœ” 𝐢1πœ” PROPN iajs-3638 269 2 3 3 NUM iajs-3638 269 3 βˆ’ βˆ’ PROPN iajs-3638 269 4 𝐢3πœ”. 𝐢3πœ”. PROPN iajs-3638 269 5 ( ( PUNCT iajs-3638 269 6 50 50 NUM iajs-3638 269 7 ) ) PUNCT iajs-3638 269 8 through through ADP iajs-3638 269 9 squaring square VERB iajs-3638 269 10 and and CCONJ iajs-3638 269 11 adding add VERB iajs-3638 269 12 both both DET iajs-3638 269 13 equations equation NOUN iajs-3638 269 14 in in ADP iajs-3638 269 15 equation equation NOUN iajs-3638 269 16 ( ( PUNCT iajs-3638 269 17 50 50 NUM iajs-3638 269 18 ) ) PUNCT iajs-3638 269 19 , , PUNCT iajs-3638 269 20 we we PRON iajs-3638 269 21 get get VERB iajs-3638 269 22 πœ”8 πœ”8 ADJ iajs-3638 269 23 + + CCONJ iajs-3638 269 24 β„Ž1πœ” β„Ž1πœ” PROPN iajs-3638 269 25 6 6 NUM iajs-3638 269 26 + + CCONJ iajs-3638 269 27 β„Ž2πœ” β„Ž2πœ” PROPN iajs-3638 269 28 4 4 NUM iajs-3638 270 1 + + CCONJ iajs-3638 270 2 β„Ž3πœ” β„Ž3πœ” PROPN iajs-3638 270 3 2 2 NUM iajs-3638 270 4 + + NUM iajs-3638 270 5 β„Ž4 β„Ž4 PROPN iajs-3638 270 6 = = SYM iajs-3638 270 7 0 0 PROPN iajs-3638 270 8 ( ( PUNCT iajs-3638 270 9 51 51 NUM iajs-3638 270 10 ) ) PUNCT iajs-3638 270 11 further far ADV iajs-3638 270 12 , , PUNCT iajs-3638 270 13 by by ADP iajs-3638 270 14 letting let VERB iajs-3638 270 15 𝜐 𝜐 X iajs-3638 270 16 = = PUNCT iajs-3638 270 17 πœ”2 πœ”2 ADJ iajs-3638 270 18 in in ADP iajs-3638 270 19 equation equation NOUN iajs-3638 270 20 ( ( PUNCT iajs-3638 270 21 51 51 NUM iajs-3638 270 22 ) ) PUNCT iajs-3638 270 23 β„Ž(𝜐 β„Ž(𝜐 PROPN iajs-3638 270 24 ) ) PUNCT iajs-3638 270 25 = = PUNCT iajs-3638 270 26 𝜐4+β„Ž1𝜐 𝜐4+β„Ž1𝜐 PROPN iajs-3638 270 27 3 3 NUM iajs-3638 270 28 + + NUM iajs-3638 270 29 β„Ž2𝜐 β„Ž2𝜐 X iajs-3638 270 30 2 2 NUM iajs-3638 270 31 + + NUM iajs-3638 270 32 β„Ž3𝜐 β„Ž3𝜐 PROPN iajs-3638 270 33 + + CCONJ iajs-3638 270 34 β„Ž4 β„Ž4 PROPN iajs-3638 270 35 = = SYM iajs-3638 270 36 0 0 NUM iajs-3638 270 37 , , PUNCT iajs-3638 270 38 ( ( PUNCT iajs-3638 270 39 52a 52a X iajs-3638 270 40 ) ) PUNCT iajs-3638 270 41 where where SCONJ iajs-3638 270 42 β„Ž1 β„Ž1 PROPN iajs-3638 270 43 = = PROPN iajs-3638 270 44 𝐢1 𝐢1 PROPN iajs-3638 270 45 2 2 NUM iajs-3638 270 46 βˆ’ βˆ’ NOUN iajs-3638 270 47 2𝐢2 2𝐢2 NUM iajs-3638 271 1 βˆ’ βˆ’ NOUN iajs-3638 271 2 𝐢5 𝐢5 VERB iajs-3638 271 3 2 2 NUM iajs-3638 271 4 , , PUNCT iajs-3638 271 5 β„Ž2 β„Ž2 NOUN iajs-3638 271 6 = = SYM iajs-3638 271 7 2𝐢5𝐢7 2𝐢5𝐢7 NUM iajs-3638 271 8 + + CCONJ iajs-3638 271 9 𝐢2 𝐢2 PROPN iajs-3638 271 10 2 2 NUM iajs-3638 271 11 + + NUM iajs-3638 271 12 2𝐢4 2𝐢4 NUM iajs-3638 271 13 βˆ’ βˆ’ NOUN iajs-3638 271 14 2𝐢1𝐢3 2𝐢1𝐢3 NUM iajs-3638 271 15 βˆ’ βˆ’ NOUN iajs-3638 271 16 𝐢6 𝐢6 NOUN iajs-3638 271 17 2 2 NUM iajs-3638 271 18 , , PUNCT iajs-3638 271 19 β„Ž3 β„Ž3 NOUN iajs-3638 271 20 = = SYM iajs-3638 271 21 2𝐢6𝐢8 2𝐢6𝐢8 NUM iajs-3638 271 22 + + NUM iajs-3638 271 23 𝐢3 𝐢3 ADJ iajs-3638 271 24 2 2 NUM iajs-3638 271 25 βˆ’ βˆ’ NOUN iajs-3638 271 26 𝐢7 𝐢7 NOUN iajs-3638 271 27 2 2 NUM iajs-3638 271 28 βˆ’ βˆ’ NOUN iajs-3638 271 29 2𝐢1𝐢4 2𝐢1𝐢4 NUM iajs-3638 271 30 , , PUNCT iajs-3638 271 31 β„Ž4 β„Ž4 PROPN iajs-3638 271 32 = = SYM iajs-3638 271 33 𝐢4 𝐢4 PROPN iajs-3638 271 34 2 2 NUM iajs-3638 271 35 βˆ’ βˆ’ NOUN iajs-3638 271 36 𝐢8 𝐢8 NOUN iajs-3638 271 37 2 2 NUM iajs-3638 271 38 . . PUNCT iajs-3638 271 39 straightforward straightforward ADJ iajs-3638 271 40 computation computation NOUN iajs-3638 271 41 shows show VERB iajs-3638 271 42 that that SCONJ iajs-3638 271 43 due due ADP iajs-3638 271 44 to to ADP iajs-3638 271 45 the the DET iajs-3638 271 46 following follow VERB iajs-3638 271 47 condition condition NOUN iajs-3638 271 48 𝐢4 𝐢4 PROPN iajs-3638 271 49 < < X iajs-3638 271 50 𝐢8 𝐢8 NOUN iajs-3638 271 51 ( ( PUNCT iajs-3638 271 52 52b 52b NOUN iajs-3638 271 53 ) ) PUNCT iajs-3638 271 54 we we PRON iajs-3638 271 55 obtain obtain VERB iajs-3638 271 56 β„Ž4 β„Ž4 PROPN iajs-3638 271 57 < < X iajs-3638 271 58 0 0 NUM iajs-3638 271 59 . . PUNCT iajs-3638 272 1 so so ADV iajs-3638 272 2 , , PUNCT iajs-3638 272 3 there there PRON iajs-3638 272 4 exists exist VERB iajs-3638 272 5 a a DET iajs-3638 272 6 unique unique ADJ iajs-3638 272 7 positive positive ADJ iajs-3638 272 8 root root NOUN iajs-3638 272 9 in in ADP iajs-3638 272 10 accordance accordance NOUN iajs-3638 272 11 with with ADP iajs-3638 272 12 descartes descarte NOUN iajs-3638 272 13 ' ' PART iajs-3638 272 14 rule rule NOUN iajs-3638 272 15 of of ADP iajs-3638 272 16 signs sign NOUN iajs-3638 272 17 𝜐0 𝜐0 NOUN iajs-3638 272 18 = = SYM iajs-3638 272 19 πœ”0 πœ”0 NOUN iajs-3638 272 20 2 2 NUM iajs-3638 272 21 providing provide VERB iajs-3638 272 22 equation equation NOUN iajs-3638 272 23 ( ( PUNCT iajs-3638 272 24 52a 52a NUM iajs-3638 272 25 ) ) PUNCT iajs-3638 272 26 . . PUNCT iajs-3638 273 1 which which PRON iajs-3638 273 2 is be AUX iajs-3638 273 3 , , PUNCT iajs-3638 273 4 equation equation NOUN iajs-3638 273 5 ( ( PUNCT iajs-3638 273 6 51 51 NUM iajs-3638 273 7 ) ) PUNCT iajs-3638 273 8 has have VERB iajs-3638 273 9 a a DET iajs-3638 273 10 positive positive ADJ iajs-3638 273 11 πœ”0 πœ”0 NOUN iajs-3638 273 12 . . PUNCT iajs-3638 274 1 as as ADP iajs-3638 274 2 a a DET iajs-3638 274 3 result result NOUN iajs-3638 274 4 , , PUNCT iajs-3638 274 5 equation equation NOUN iajs-3638 274 6 ( ( PUNCT iajs-3638 274 7 47 47 NUM iajs-3638 274 8 ) ) PUNCT iajs-3638 274 9 has have VERB iajs-3638 274 10 at at ADV iajs-3638 274 11 least least ADV iajs-3638 274 12 two two NUM iajs-3638 274 13 roots root NOUN iajs-3638 274 14 Β±πœ”0 Β±πœ”0 NOUN iajs-3638 274 15 that that PRON iajs-3638 274 16 are be AUX iajs-3638 274 17 purely purely ADV iajs-3638 274 18 imaginary imaginary ADJ iajs-3638 274 19 and and CCONJ iajs-3638 274 20 which which PRON iajs-3638 274 21 correspond correspond VERB iajs-3638 274 22 to to ADP iajs-3638 274 23 the the DET iajs-3638 274 24 time time NOUN iajs-3638 274 25 delay delay NOUN iajs-3638 274 26 𝜏. 𝜏. PROPN iajs-3638 274 27 additionally additionally ADV iajs-3638 274 28 , , PUNCT iajs-3638 274 29 by by ADP iajs-3638 274 30 substituting substitute VERB iajs-3638 274 31 πœ”0 πœ”0 NOUN iajs-3638 274 32 in in ADP iajs-3638 274 33 ( ( PUNCT iajs-3638 274 34 50 50 NUM iajs-3638 274 35 ) ) PUNCT iajs-3638 274 36 and and CCONJ iajs-3638 274 37 solving solve VERB iajs-3638 274 38 the the DET iajs-3638 274 39 system system NOUN iajs-3638 274 40 for for ADP iajs-3638 274 41 𝜏 𝜏 NOUN iajs-3638 274 42 , , PUNCT iajs-3638 274 43 yields yield VERB iajs-3638 274 44 : : PUNCT iajs-3638 274 45 πœπ‘˜ πœπ‘˜ NOUN iajs-3638 274 46 = = SYM iajs-3638 274 47 1 1 NUM iajs-3638 274 48 πœ”0 πœ”0 NOUN iajs-3638 274 49 π‘π‘œπ‘ βˆ’1 π‘π‘œπ‘ βˆ’1 NOUN iajs-3638 274 50 ( ( PUNCT iajs-3638 274 51 𝐢6βˆ’πΆ1𝐢5)πœ”0 𝐢6βˆ’πΆ1𝐢5)πœ”0 PROPN iajs-3638 274 52 6+(𝐢1𝐢7+𝐢5+𝐢3βˆ’πΆ6𝐢2βˆ’πΆ8)πœ”0 6+(𝐢1𝐢7+𝐢5+𝐢3βˆ’πΆ6𝐢2βˆ’πΆ8)πœ”0 NUM iajs-3638 274 53 4+(𝐢6𝐢4+𝐢2𝐢8βˆ’πΆ3𝐢7)πœ”0 4+(𝐢6𝐢4+𝐢2𝐢8βˆ’πΆ3𝐢7)πœ”0 PROPN iajs-3638 274 54 2βˆ’πΆ4𝐢8 2βˆ’πΆ4𝐢8 NUM iajs-3638 274 55 𝐢5 𝐢5 NOUN iajs-3638 274 56 2πœ”0 2πœ”0 NUM iajs-3638 275 1 6+(𝐢6 6+(𝐢6 NUM iajs-3638 275 2 2βˆ’2𝐢5𝐢7)πœ”0 2βˆ’2𝐢5𝐢7)πœ”0 PROPN iajs-3638 275 3 4+(𝐢7 4+(𝐢7 PROPN iajs-3638 275 4 2βˆ’2𝐢5𝐢8)πœ”0 2βˆ’2𝐢5𝐢8)πœ”0 PROPN iajs-3638 275 5 2+𝐢8 2+𝐢8 ADJ iajs-3638 275 6 2 2 NUM iajs-3638 275 7 + + NUM iajs-3638 275 8 2π‘˜πœ‹ 2π‘˜πœ‹ ADJ iajs-3638 275 9 πœ”0 πœ”0 NOUN iajs-3638 275 10 ( ( PUNCT iajs-3638 275 11 53 53 NUM iajs-3638 275 12 ) ) PUNCT iajs-3638 275 13 ihjpas ihjpa NOUN iajs-3638 275 14 . . PUNCT iajs-3638 276 1 2025 2025 NUM iajs-3638 276 2 , , PUNCT iajs-3638 276 3 38 38 NUM iajs-3638 276 4 ( ( PUNCT iajs-3638 276 5 1 1 NUM iajs-3638 276 6 ) ) PUNCT iajs-3638 276 7 457 457 NUM iajs-3638 276 8 where where SCONJ iajs-3638 276 9 , , PUNCT iajs-3638 276 10 k=0,1,2 k=0,1,2 PROPN iajs-3638 276 11 , , PUNCT iajs-3638 276 12 … … PUNCT iajs-3638 276 13 . . PUNCT iajs-3638 277 1 hence hence ADV iajs-3638 277 2 we we PRON iajs-3638 277 3 get get VERB iajs-3638 277 4 the the DET iajs-3638 277 5 corresponding corresponding NOUN iajs-3638 277 6 πœπ‘˜ πœπ‘˜ ADP iajs-3638 277 7 > > X iajs-3638 277 8 0 0 NUM iajs-3638 277 9 for for ADP iajs-3638 277 10 which which DET iajs-3638 277 11 system system NOUN iajs-3638 277 12 ( ( PUNCT iajs-3638 277 13 3 3 X iajs-3638 277 14 ) ) PUNCT iajs-3638 277 15 has have VERB iajs-3638 277 16 two two NUM iajs-3638 277 17 wholly wholly ADV iajs-3638 277 18 imaginary imaginary ADJ iajs-3638 277 19 roots root NOUN iajs-3638 277 20 Β±πœ”0 Β±πœ”0 NOUN iajs-3638 277 21 . . PUNCT iajs-3638 278 1 let let VERB iajs-3638 278 2 πœ†(𝜏 πœ†(𝜏 NOUN iajs-3638 278 3 ) ) PUNCT iajs-3638 279 1 = = SYM iajs-3638 279 2 πœ‡(𝜏 πœ‡(𝜏 PROPN iajs-3638 279 3 ) ) PUNCT iajs-3638 280 1 + + CCONJ iajs-3638 280 2 π‘–πœ”(𝜏 π‘–πœ”(𝜏 PRON iajs-3638 280 3 ) ) PUNCT iajs-3638 280 4 be be VERB iajs-3638 280 5 a a DET iajs-3638 280 6 root root NOUN iajs-3638 280 7 of of ADP iajs-3638 280 8 c.e.(47 c.e.(47 NOUN iajs-3638 280 9 ) ) PUNCT iajs-3638 280 10 near near ADP iajs-3638 280 11 𝜏 𝜏 NOUN iajs-3638 280 12 = = SYM iajs-3638 280 13 πœπ‘˜with πœπ‘˜with NOUN iajs-3638 280 14 πœ‡(πœπ‘˜ πœ‡(πœπ‘˜ NUM iajs-3638 280 15 ) ) PUNCT iajs-3638 281 1 = = SYM iajs-3638 281 2 0 0 NUM iajs-3638 281 3 and and CCONJ iajs-3638 281 4 πœ”(πœπ‘˜ πœ”(πœπ‘˜ NUM iajs-3638 281 5 ) ) PUNCT iajs-3638 281 6 = = SYM iajs-3638 281 7 πœ”0 πœ”0 PROPN iajs-3638 281 8 . . PUNCT iajs-3638 282 1 then then ADV iajs-3638 282 2 comes come VERB iajs-3638 282 3 the the DET iajs-3638 282 4 next next ADJ iajs-3638 282 5 theorem theorem NOUN iajs-3638 282 6 . . PUNCT iajs-3638 283 1 theorem3 theorem3 PROPN iajs-3638 283 2 . . PUNCT iajs-3638 284 1 the the DET iajs-3638 284 2 c.e c.e PROPN iajs-3638 284 3 . . PUNCT iajs-3638 284 4 ( ( PUNCT iajs-3638 284 5 47 47 NUM iajs-3638 284 6 ) ) PUNCT iajs-3638 284 7 , , PUNCT iajs-3638 284 8 roots root NOUN iajs-3638 284 9 satisfy satisfy VERB iajs-3638 284 10 the the DET iajs-3638 284 11 following follow VERB iajs-3638 284 12 transversality transversality NOUN iajs-3638 284 13 requirement requirement NOUN iajs-3638 284 14 ; ; PUNCT iajs-3638 284 15 [ [ PUNCT iajs-3638 284 16 𝑑(π‘…π‘’πœ†(𝜏 𝑑(π‘…π‘’πœ†(𝜏 NOUN iajs-3638 284 17 ) ) PUNCT iajs-3638 284 18 ) ) PUNCT iajs-3638 285 1 π‘‘πœ π‘‘πœ NOUN iajs-3638 285 2 ] ] PUNCT iajs-3638 285 3 𝜏=πœπ‘˜ 𝜏=πœπ‘˜ X iajs-3638 285 4 > > X iajs-3638 285 5 0 0 PUNCT iajs-3638 285 6 ( ( PUNCT iajs-3638 285 7 54a 54a NOUN iajs-3638 285 8 ) ) PUNCT iajs-3638 285 9 provided provide VERB iajs-3638 285 10 that that SCONJ iajs-3638 285 11 4πœ”0 4πœ”0 NOUN iajs-3638 285 12 6 6 NUM iajs-3638 285 13 + + NUM iajs-3638 285 14 2𝐢1 2𝐢1 NUM iajs-3638 285 15 2πœ”0 2πœ”0 NUM iajs-3638 285 16 5 5 NUM iajs-3638 285 17 + + CCONJ iajs-3638 285 18 2(𝐢2 2(𝐢2 NUM iajs-3638 285 19 2 2 NUM iajs-3638 285 20 + + NUM iajs-3638 285 21 2𝐢4)πœ”0 2𝐢4)πœ”0 NUM iajs-3638 285 22 2 2 NUM iajs-3638 285 23 + + CCONJ iajs-3638 285 24 2𝐢2𝐢3πœ”0 2𝐢2𝐢3πœ”0 NUM iajs-3638 285 25 > > PUNCT iajs-3638 286 1 6𝐢2πœ”0 6𝐢2πœ”0 NUM iajs-3638 286 2 4 4 NUM iajs-3638 286 3 + + CCONJ iajs-3638 286 4 𝐢3(𝐢1 𝐢3(𝐢1 PROPN iajs-3638 286 5 + + CCONJ iajs-3638 286 6 4 4 NUM iajs-3638 286 7 ) ) PUNCT iajs-3638 286 8 + + CCONJ iajs-3638 286 9 2𝐢2𝐢4 2𝐢2𝐢4 NUM iajs-3638 286 10 , , PUNCT iajs-3638 286 11 ( ( PUNCT iajs-3638 286 12 54b 54b NOUN iajs-3638 286 13 ) ) PUNCT iajs-3638 286 14 2𝐢5 2𝐢5 NUM iajs-3638 286 15 2πœ”0 2πœ”0 NUM iajs-3638 286 16 5 5 NUM iajs-3638 286 17 + + CCONJ iajs-3638 286 18 2𝐢6 2𝐢6 NUM iajs-3638 286 19 2πœ”0 2πœ”0 NUM iajs-3638 286 20 2 2 NUM iajs-3638 286 21 > > X iajs-3638 286 22 𝐢5𝐢7πœ”0 𝐢5𝐢7πœ”0 X iajs-3638 286 23 3 3 NUM iajs-3638 286 24 + + SYM iajs-3638 286 25 2𝐢6(𝐢7 2𝐢6(𝐢7 NUM iajs-3638 286 26 + + NUM iajs-3638 286 27 𝐢8 𝐢8 NOUN iajs-3638 286 28 ) ) PUNCT iajs-3638 286 29 . . PUNCT iajs-3638 287 1 ( ( PUNCT iajs-3638 287 2 54c 54c NOUN iajs-3638 287 3 ) ) PUNCT iajs-3638 287 4 proof proof NOUN iajs-3638 287 5 . . PUNCT iajs-3638 288 1 by by ADP iajs-3638 288 2 substituting substitute VERB iajs-3638 288 3 πœ†(𝜏 πœ†(𝜏 NOUN iajs-3638 288 4 ) ) PUNCT iajs-3638 288 5 in in ADP iajs-3638 288 6 equation equation NOUN iajs-3638 288 7 ( ( PUNCT iajs-3638 288 8 47 47 NUM iajs-3638 288 9 ) ) PUNCT iajs-3638 288 10 and and CCONJ iajs-3638 288 11 differentiating differentiate VERB iajs-3638 288 12 the the DET iajs-3638 288 13 resulting result VERB iajs-3638 288 14 equation equation NOUN iajs-3638 288 15 in in ADP iajs-3638 288 16 𝜏,we 𝜏,we PROPN iajs-3638 288 17 can can AUX iajs-3638 288 18 get get VERB iajs-3638 288 19 [ [ X iajs-3638 288 20 4πœ†3 4πœ†3 NUM iajs-3638 288 21 + + CCONJ iajs-3638 288 22 3𝐢1πœ† 3𝐢1πœ† NUM iajs-3638 288 23 2 2 NUM iajs-3638 288 24 + + CCONJ iajs-3638 288 25 2𝐢2πœ† 2𝐢2πœ† NUM iajs-3638 288 26 + + CCONJ iajs-3638 288 27 𝐢3 𝐢3 ADJ iajs-3638 288 28 + + CCONJ iajs-3638 288 29 ( ( PUNCT iajs-3638 288 30 3𝐢5πœ† 3𝐢5πœ† NUM iajs-3638 288 31 2 2 NUM iajs-3638 288 32 + + NUM iajs-3638 288 33 2𝐢6πœ† 2𝐢6πœ† NUM iajs-3638 289 1 + + CCONJ iajs-3638 289 2 𝐢7)𝑒 𝐢7)𝑒 NUM iajs-3638 289 3 βˆ’πœ†πœ βˆ’πœ†πœ NUM iajs-3638 289 4 βˆ’ βˆ’ PROPN iajs-3638 289 5 𝜏(𝐢5πœ† 𝜏(𝐢5πœ† PROPN iajs-3638 289 6 3 3 NUM iajs-3638 289 7 + + CCONJ iajs-3638 289 8 𝐢6πœ† 𝐢6πœ† NOUN iajs-3638 289 9 2 2 NUM iajs-3638 289 10 + + NOUN iajs-3638 289 11 𝐢7πœ† 𝐢7πœ† PROPN iajs-3638 289 12 + + CCONJ iajs-3638 289 13 𝐢8)𝑒 𝐢8)𝑒 PROPN iajs-3638 289 14 βˆ’πœ†πœ βˆ’πœ†πœ PROPN iajs-3638 289 15 ] ] PUNCT iajs-3638 289 16 π‘‘πœ† π‘‘πœ† ADP iajs-3638 289 17 π‘‘πœ† π‘‘πœ† NOUN iajs-3638 289 18 = = SYM iajs-3638 289 19 πœ†(𝐢5πœ† πœ†(𝐢5πœ† NOUN iajs-3638 289 20 3 3 NUM iajs-3638 290 1 + + CCONJ iajs-3638 290 2 𝐢6πœ† 𝐢6πœ† NOUN iajs-3638 290 3 2 2 NUM iajs-3638 290 4 + + NOUN iajs-3638 290 5 𝐢7πœ† 𝐢7πœ† PROPN iajs-3638 290 6 + + CCONJ iajs-3638 290 7 𝐢8)𝑒 𝐢8)𝑒 PROPN iajs-3638 290 8 βˆ’πœ†πœ. βˆ’πœ†πœ. PROPN iajs-3638 290 9 ( ( PUNCT iajs-3638 290 10 55 55 NUM iajs-3638 290 11 ) ) PUNCT iajs-3638 290 12 thus thus ADV iajs-3638 290 13 , , PUNCT iajs-3638 290 14 ( ( PUNCT iajs-3638 290 15 π‘‘πœ† π‘‘πœ† ADP iajs-3638 290 16 π‘‘πœ π‘‘πœ NOUN iajs-3638 290 17 ) ) PUNCT iajs-3638 290 18 βˆ’1 βˆ’1 NOUN iajs-3638 290 19 = = PUNCT iajs-3638 290 20 4πœ†3 4πœ†3 NUM iajs-3638 290 21 + + NOUN iajs-3638 290 22 3𝐢1πœ† 3𝐢1πœ† NUM iajs-3638 290 23 2 2 NUM iajs-3638 290 24 + + NOUN iajs-3638 290 25 2𝐢2πœ†+𝐢3 2𝐢2πœ†+𝐢3 NOUN iajs-3638 290 26 πœ†(𝐢5πœ†3+𝐢6πœ†2+𝐢7πœ†+𝐢8)π‘’βˆ’πœ†πœ πœ†(𝐢5πœ†3+𝐢6πœ†2+𝐢7πœ†+𝐢8)π‘’βˆ’πœ†πœ NOUN iajs-3638 290 27 + + CCONJ iajs-3638 290 28 3𝐢5πœ† 3𝐢5πœ† NUM iajs-3638 290 29 2 2 NUM iajs-3638 290 30 + + NOUN iajs-3638 290 31 2𝐢6πœ†+𝐢7 2𝐢6πœ†+𝐢7 NUM iajs-3638 290 32 πœ†(𝐢5πœ†3+𝐢6πœ†2+𝐢7πœ†+𝐢8 πœ†(𝐢5πœ†3+𝐢6πœ†2+𝐢7πœ†+𝐢8 NOUN iajs-3638 290 33 ) ) PUNCT iajs-3638 290 34 βˆ’ βˆ’ NOUN iajs-3638 290 35 𝜏 𝜏 X iajs-3638 290 36 πœ† πœ† X iajs-3638 290 37 ( ( PUNCT iajs-3638 290 38 56 56 NUM iajs-3638 290 39 ) ) PUNCT iajs-3638 290 40 since since SCONJ iajs-3638 290 41 for for ADP iajs-3638 290 42 = = PROPN iajs-3638 290 43 𝜏0 𝜏0 PROPN iajs-3638 290 44 , , PUNCT iajs-3638 290 45 and and CCONJ iajs-3638 290 46 πœ† πœ† X iajs-3638 290 47 = = SYM iajs-3638 290 48 π‘–πœ”0 π‘–πœ”0 PROPN iajs-3638 290 49 , , PUNCT iajs-3638 290 50 we we PRON iajs-3638 290 51 have have AUX iajs-3638 290 52 got get VERB iajs-3638 290 53 [ [ PUNCT iajs-3638 290 54 π‘‘πœ† π‘‘πœ† ADP iajs-3638 290 55 π‘‘πœ π‘‘πœ NOUN iajs-3638 290 56 ] ] PUNCT iajs-3638 290 57 𝜏=𝜏0 𝜏=𝜏0 PROPN iajs-3638 290 58 βˆ’1 βˆ’1 NOUN iajs-3638 290 59 = = NOUN iajs-3638 290 60 𝐢3 𝐢3 NOUN iajs-3638 290 61 βˆ’ βˆ’ PROPN iajs-3638 291 1 3𝐢1πœ”0 3𝐢1πœ”0 NUM iajs-3638 291 2 2 2 NUM iajs-3638 291 3 + + NOUN iajs-3638 291 4 π‘–πœ”0(2𝐢2 π‘–πœ”0(2𝐢2 NOUN iajs-3638 291 5 βˆ’ βˆ’ NUM iajs-3638 291 6 4πœ”0 4πœ”0 NUM iajs-3638 291 7 2 2 X iajs-3638 291 8 ) ) PUNCT iajs-3638 291 9 𝐢1πœ”0 𝐢1πœ”0 NOUN iajs-3638 291 10 4 4 NUM iajs-3638 291 11 + + CCONJ iajs-3638 291 12 π‘–πœ”0(πœ”0 π‘–πœ”0(πœ”0 NUM iajs-3638 291 13 4 4 NUM iajs-3638 291 14 + + NUM iajs-3638 291 15 𝐢4 𝐢4 PROPN iajs-3638 291 16 βˆ’ βˆ’ PROPN iajs-3638 291 17 𝐢2πœ”0 𝐢2πœ”0 PROPN iajs-3638 291 18 2 2 NUM iajs-3638 291 19 βˆ’ βˆ’ PROPN iajs-3638 291 20 𝐢3πœ”0 𝐢3πœ”0 PROPN iajs-3638 291 21 ) ) PUNCT iajs-3638 292 1 + + CCONJ iajs-3638 293 1 βˆ’3𝐢5πœ”0 βˆ’3𝐢5πœ”0 ADP iajs-3638 293 2 2 2 NUM iajs-3638 293 3 + + NUM iajs-3638 293 4 𝐢7 𝐢7 X iajs-3638 293 5 + + CCONJ iajs-3638 293 6 2𝑖𝐢6πœ”0 2𝑖𝐢6πœ”0 NUM iajs-3638 293 7 𝐢5πœ”0 𝐢5πœ”0 ADJ iajs-3638 293 8 4 4 NUM iajs-3638 293 9 + + CCONJ iajs-3638 293 10 π‘–πœ”0(𝐢7 π‘–πœ”0(𝐢7 CCONJ iajs-3638 293 11 + + NUM iajs-3638 293 12 𝐢8 𝐢8 NOUN iajs-3638 293 13 βˆ’ βˆ’ PROPN iajs-3638 293 14 𝐢6πœ”0 𝐢6πœ”0 NOUN iajs-3638 293 15 2 2 NUM iajs-3638 293 16 ) ) PUNCT iajs-3638 293 17 βˆ’ βˆ’ NOUN iajs-3638 294 1 π‘–πœ0 π‘–πœ0 NOUN iajs-3638 294 2 πœ”0 πœ”0 NOUN iajs-3638 294 3 now now ADV iajs-3638 294 4 , , PUNCT iajs-3638 294 5 since since SCONJ iajs-3638 294 6 𝑠𝑖𝑔𝑛 𝑠𝑖𝑔𝑛 NOUN iajs-3638 294 7 [ [ PUNCT iajs-3638 294 8 𝑑(π‘…π‘’πœ† 𝑑(π‘…π‘’πœ† PROPN iajs-3638 294 9 ) ) PUNCT iajs-3638 294 10 π‘‘πœ π‘‘πœ NOUN iajs-3638 294 11 ] ] PUNCT iajs-3638 294 12 𝜏=𝜏0 𝜏=𝜏0 PUNCT iajs-3638 294 13 = = SYM iajs-3638 295 1 𝑠𝑖𝑔𝑛 𝑠𝑖𝑔𝑛 NOUN iajs-3638 295 2 [ [ X iajs-3638 295 3 𝑅𝑒 𝑅𝑒 PROPN iajs-3638 295 4 ( ( PUNCT iajs-3638 295 5 π‘‘πœ† π‘‘πœ† ADP iajs-3638 295 6 π‘‘πœ π‘‘πœ NOUN iajs-3638 295 7 ) ) PUNCT iajs-3638 295 8 βˆ’1 βˆ’1 NOUN iajs-3638 295 9 ] ] PUNCT iajs-3638 296 1 πœ†=π‘–πœ”0 πœ†=π‘–πœ”0 PROPN iajs-3638 296 2 ( ( PUNCT iajs-3638 296 3 57 57 NUM iajs-3638 296 4 ) ) PUNCT iajs-3638 296 5 it it PRON iajs-3638 296 6 is be AUX iajs-3638 296 7 clear clear ADJ iajs-3638 296 8 that that SCONJ iajs-3638 296 9 : : PUNCT iajs-3638 296 10 𝑅𝑒 𝑅𝑒 PROPN iajs-3638 296 11 [ [ PUNCT iajs-3638 296 12 𝐢3βˆ’3𝐢1πœ”0 𝐢3βˆ’3𝐢1πœ”0 NOUN iajs-3638 296 13 2+π‘–πœ”0(2𝐢2βˆ’4πœ”0 2+π‘–πœ”0(2𝐢2βˆ’4πœ”0 NUM iajs-3638 296 14 2 2 NUM iajs-3638 296 15 ) ) PUNCT iajs-3638 296 16 𝐢1πœ”0 𝐢1πœ”0 NOUN iajs-3638 296 17 4+π‘–πœ”0(πœ”0 4+π‘–πœ”0(πœ”0 NUM iajs-3638 296 18 4+𝐢4βˆ’πΆ2πœ”0 4+𝐢4βˆ’πΆ2πœ”0 NUM iajs-3638 296 19 2βˆ’πΆ3πœ”0 2βˆ’πΆ3πœ”0 NUM iajs-3638 296 20 ) ) PUNCT iajs-3638 296 21 ] ] PUNCT iajs-3638 297 1 = = PUNCT iajs-3638 297 2 𝐢1𝐢3πœ”0 𝐢1𝐢3πœ”0 NOUN iajs-3638 298 1 3βˆ’3𝐢1 3βˆ’3𝐢1 NUM iajs-3638 298 2 2πœ”0 2πœ”0 NUM iajs-3638 298 3 5+(2𝐢2βˆ’4πœ”0 5+(2𝐢2βˆ’4πœ”0 NOUN iajs-3638 298 4 2)(πœ”0 2)(πœ”0 NUM iajs-3638 298 5 4+𝐢4βˆ’πΆ2πœ”0 4+𝐢4βˆ’πΆ2πœ”0 NUM iajs-3638 298 6 2βˆ’πΆ3πœ”0 2βˆ’πΆ3πœ”0 NUM iajs-3638 298 7 ) ) PUNCT iajs-3638 298 8 πœ”0(𝐢1 πœ”0(𝐢1 VERB iajs-3638 298 9 2πœ”0 2πœ”0 NUM iajs-3638 298 10 6+(πœ”0 6+(πœ”0 PROPN iajs-3638 298 11 4+𝐢4βˆ’πΆ2πœ”0 4+𝐢4βˆ’πΆ2πœ”0 NUM iajs-3638 298 12 2βˆ’πΆ3πœ”0 2βˆ’πΆ3πœ”0 NUM iajs-3638 298 13 ) ) PUNCT iajs-3638 298 14 2 2 NUM iajs-3638 298 15 ) ) PUNCT iajs-3638 298 16 , , PUNCT iajs-3638 299 1 𝑅𝑒 𝑅𝑒 PROPN iajs-3638 299 2 [ [ PUNCT iajs-3638 299 3 βˆ’3𝐢5πœ”0 βˆ’3𝐢5πœ”0 ADP iajs-3638 299 4 2+𝐢7 2+𝐢7 VERB iajs-3638 299 5 + + ADJ iajs-3638 299 6 2𝑖𝐢6πœ”0 2𝑖𝐢6πœ”0 NOUN iajs-3638 299 7 𝐢5πœ”0 𝐢5πœ”0 ADJ iajs-3638 299 8 4+π‘–πœ”0(𝐢7+𝐢8βˆ’πΆ6πœ”0 4+π‘–πœ”0(𝐢7+𝐢8βˆ’πΆ6πœ”0 NOUN iajs-3638 299 9 2 2 NUM iajs-3638 299 10 ) ) PUNCT iajs-3638 299 11 ] ] PUNCT iajs-3638 300 1 = = SYM iajs-3638 300 2 𝐢5𝐢7πœ”0 𝐢5𝐢7πœ”0 PROPN iajs-3638 300 3 3βˆ’3𝐢5 3βˆ’3𝐢5 NUM iajs-3638 300 4 2πœ”0 2πœ”0 NUM iajs-3638 300 5 5 5 NUM iajs-3638 300 6 + + NUM iajs-3638 300 7 2𝐢6(𝐢7+𝐢8βˆ’πΆ6πœ”0 2𝐢6(𝐢7+𝐢8βˆ’πΆ6πœ”0 NUM iajs-3638 300 8 2 2 NUM iajs-3638 300 9 ) ) PUNCT iajs-3638 300 10 𝐢5 𝐢5 VERB iajs-3638 300 11 2πœ”0 2πœ”0 NUM iajs-3638 300 12 6+(𝐢6+𝐢7βˆ’πΆ8πœ”0 6+(𝐢6+𝐢7βˆ’πΆ8πœ”0 NUM iajs-3638 300 13 2 2 NUM iajs-3638 300 14 ) ) PUNCT iajs-3638 300 15 2 2 NUM iajs-3638 300 16 , , PUNCT iajs-3638 300 17 𝑅𝑒 𝑅𝑒 PROPN iajs-3638 300 18 [ [ PUNCT iajs-3638 300 19 π‘–πœ0 π‘–πœ0 NOUN iajs-3638 300 20 πœ”0 πœ”0 NOUN iajs-3638 300 21 ] ] PUNCT iajs-3638 300 22 = = PUNCT iajs-3638 300 23 π‘§π‘’π‘Ÿπ‘œ. π‘§π‘’π‘Ÿπ‘œ. ADJ iajs-3638 300 24 hence hence ADV iajs-3638 300 25 , , PUNCT iajs-3638 300 26 we we PRON iajs-3638 300 27 have have AUX iajs-3638 300 28 𝑅𝑒 𝑅𝑒 VERB iajs-3638 300 29 [ [ PUNCT iajs-3638 300 30 π‘‘πœ† π‘‘πœ† ADP iajs-3638 300 31 π‘‘πœ π‘‘πœ NOUN iajs-3638 300 32 ] ] PUNCT iajs-3638 300 33 𝜏=𝜏0 𝜏=𝜏0 PROPN iajs-3638 300 34 βˆ’1 βˆ’1 NOUN iajs-3638 300 35 = = PUNCT iajs-3638 300 36 𝐢1𝐢3πœ”0 𝐢1𝐢3πœ”0 NOUN iajs-3638 300 37 3βˆ’3𝐢1 3βˆ’3𝐢1 NUM iajs-3638 300 38 2πœ”0 2πœ”0 NUM iajs-3638 300 39 5+(2𝐢2βˆ’4πœ”0 5+(2𝐢2βˆ’4πœ”0 NOUN iajs-3638 300 40 2)(πœ”0 2)(πœ”0 NUM iajs-3638 300 41 4+𝐢4βˆ’πΆ2πœ”0 4+𝐢4βˆ’πΆ2πœ”0 NUM iajs-3638 300 42 2βˆ’πΆ3πœ”0 2βˆ’πΆ3πœ”0 NUM iajs-3638 300 43 ) ) PUNCT iajs-3638 300 44 ψ0 ψ0 ADV iajs-3638 300 45 + + ADJ iajs-3638 300 46 𝐢5𝐢7πœ”0 𝐢5𝐢7πœ”0 PROPN iajs-3638 300 47 3βˆ’3𝐢5 3βˆ’3𝐢5 NUM iajs-3638 300 48 2πœ”0 2πœ”0 NUM iajs-3638 300 49 5 5 NUM iajs-3638 300 50 + + NUM iajs-3638 300 51 2𝐢6(𝐢7+𝐢8βˆ’πΆ6πœ”0 2𝐢6(𝐢7+𝐢8βˆ’πΆ6πœ”0 NUM iajs-3638 300 52 2 2 NUM iajs-3638 300 53 ) ) PUNCT iajs-3638 300 54 ψ1 ψ1 NOUN iajs-3638 300 55 . . PUNCT iajs-3638 301 1 where where SCONJ iajs-3638 301 2 ψ0 ψ0 ADV iajs-3638 301 3 = = SYM iajs-3638 301 4 πœ”0(𝐢1 πœ”0(𝐢1 PUNCT iajs-3638 301 5 2πœ”0 2πœ”0 NUM iajs-3638 301 6 6 6 NUM iajs-3638 302 1 + + CCONJ iajs-3638 302 2 ( ( PUNCT iajs-3638 302 3 πœ”0 πœ”0 NOUN iajs-3638 302 4 4 4 NUM iajs-3638 302 5 + + NUM iajs-3638 302 6 𝐢4 𝐢4 PROPN iajs-3638 302 7 βˆ’ βˆ’ PROPN iajs-3638 302 8 𝐢2πœ”0 𝐢2πœ”0 PROPN iajs-3638 302 9 2 2 NUM iajs-3638 302 10 βˆ’ βˆ’ PROPN iajs-3638 302 11 𝐢3πœ”0 𝐢3πœ”0 PROPN iajs-3638 302 12 ) ) PUNCT iajs-3638 302 13 2 2 NUM iajs-3638 302 14 ) ) PUNCT iajs-3638 302 15 > > X iajs-3638 302 16 0 0 NUM iajs-3638 302 17 , , PUNCT iajs-3638 302 18 ψ1 ψ1 NOUN iajs-3638 302 19 = = PUNCT iajs-3638 302 20 𝐢5 𝐢5 NOUN iajs-3638 302 21 2πœ”0 2πœ”0 NUM iajs-3638 302 22 6 6 NUM iajs-3638 302 23 + + CCONJ iajs-3638 302 24 ( ( PUNCT iajs-3638 302 25 𝐢6 𝐢6 X iajs-3638 302 26 + + X iajs-3638 302 27 𝐢7 𝐢7 NOUN iajs-3638 302 28 βˆ’ βˆ’ PROPN iajs-3638 303 1 𝐢8πœ”0 𝐢8πœ”0 PROPN iajs-3638 303 2 2)2 2)2 NUM iajs-3638 303 3 > > SYM iajs-3638 303 4 0 0 NUM iajs-3638 303 5 . . PUNCT iajs-3638 304 1 ihjpas ihjpas PROPN iajs-3638 304 2 . . PUNCT iajs-3638 305 1 2025 2025 NUM iajs-3638 305 2 , , PUNCT iajs-3638 305 3 38 38 NUM iajs-3638 305 4 ( ( PUNCT iajs-3638 305 5 1 1 NUM iajs-3638 305 6 ) ) PUNCT iajs-3638 305 7 458 458 NUM iajs-3638 305 8 we we PRON iajs-3638 305 9 obtain obtain VERB iajs-3638 305 10 [ [ PUNCT iajs-3638 305 11 𝑑(π‘…π‘’πœ†(𝜏 𝑑(π‘…π‘’πœ†(𝜏 NOUN iajs-3638 305 12 ) ) PUNCT iajs-3638 305 13 ) ) PUNCT iajs-3638 306 1 π‘‘πœ π‘‘πœ NOUN iajs-3638 306 2 ] ] PUNCT iajs-3638 306 3 𝜏=Ο„0 𝜏=Ο„0 X iajs-3638 306 4 > > X iajs-3638 306 5 0 0 PUNCT iajs-3638 307 1 under under ADP iajs-3638 307 2 ( ( PUNCT iajs-3638 307 3 54a-54b 54a-54b NOUN iajs-3638 307 4 ) ) PUNCT iajs-3638 307 5 . . PUNCT iajs-3638 308 1 this this DET iajs-3638 308 2 outcome outcome NOUN iajs-3638 308 3 demonstrates demonstrate VERB iajs-3638 308 4 how how SCONJ iajs-3638 308 5 the the DET iajs-3638 308 6 roots root NOUN iajs-3638 308 7 of of ADP iajs-3638 308 8 c.e c.e PROPN iajs-3638 308 9 . . PUNCT iajs-3638 309 1 ( ( PUNCT iajs-3638 309 2 47 47 NUM iajs-3638 309 3 ) ) PUNCT iajs-3638 309 4 , , PUNCT iajs-3638 309 5 as as SCONJ iajs-3638 309 6 𝜏 𝜏 PROPN iajs-3638 309 7 passes pass VERB iajs-3638 309 8 through through ADP iajs-3638 309 9 Ο„0 Ο„0 NOUN iajs-3638 309 10 , , PUNCT iajs-3638 309 11 traverse traverse VERB iajs-3638 309 12 the the DET iajs-3638 309 13 imaginary imaginary ADJ iajs-3638 309 14 axis axis NOUN iajs-3638 309 15 from from ADP iajs-3638 309 16 left left ADJ iajs-3638 309 17 to to ADP iajs-3638 309 18 right right NOUN iajs-3638 309 19 . . PUNCT iajs-3638 310 1 as as ADP iajs-3638 310 2 a a DET iajs-3638 310 3 result result NOUN iajs-3638 310 4 , , PUNCT iajs-3638 310 5 the the DET iajs-3638 310 6 system system NOUN iajs-3638 310 7 ( ( PUNCT iajs-3638 310 8 3 3 X iajs-3638 310 9 ) ) PUNCT iajs-3638 310 10 experiences experience NOUN iajs-3638 310 11 an an DET iajs-3638 310 12 h.b h.b PROPN iajs-3638 310 13 . . PROPN iajs-3638 311 1 at at ADP iajs-3638 311 2 𝜏 𝜏 NOUN iajs-3638 311 3 = = VERB iajs-3638 311 4 Ο„0 Ο„0 NOUN iajs-3638 311 5 and and CCONJ iajs-3638 311 6 loses lose VERB iajs-3638 311 7 its its PRON iajs-3638 311 8 stability stability NOUN iajs-3638 311 9 . . PUNCT iajs-3638 312 1 4 4 X iajs-3638 312 2 . . X iajs-3638 312 3 numerical numerical PROPN iajs-3638 312 4 simulation simulation PROPN iajs-3638 312 5 and and CCONJ iajs-3638 312 6 discussion discussion NOUN iajs-3638 312 7 numerical numerical PROPN iajs-3638 312 8 simulation simulation PROPN iajs-3638 312 9 is be AUX iajs-3638 312 10 employed employ VERB iajs-3638 312 11 in in ADP iajs-3638 312 12 this this DET iajs-3638 312 13 section section NOUN iajs-3638 312 14 to to PART iajs-3638 312 15 show show VERB iajs-3638 312 16 the the DET iajs-3638 312 17 findings finding NOUN iajs-3638 312 18 of of ADP iajs-3638 312 19 our our PRON iajs-3638 312 20 investigation investigation NOUN iajs-3638 312 21 . . PUNCT iajs-3638 313 1 in in ADP iajs-3638 313 2 this this DET iajs-3638 313 3 section section NOUN iajs-3638 313 4 , , PUNCT iajs-3638 313 5 the the DET iajs-3638 313 6 following follow VERB iajs-3638 313 7 hypothetical hypothetical ADJ iajs-3638 313 8 parameters parameter NOUN iajs-3638 313 9 have have AUX iajs-3638 313 10 been be AUX iajs-3638 313 11 chosen choose VERB iajs-3638 313 12 : : PUNCT iajs-3638 313 13 Ξ» Ξ» X iajs-3638 313 14 = = NOUN iajs-3638 313 15 0.042 0.042 NUM iajs-3638 313 16 , , PUNCT iajs-3638 313 17 𝛽1 𝛽1 NOUN iajs-3638 313 18 = = SYM iajs-3638 313 19 0.03 0.03 NUM iajs-3638 313 20 , , PUNCT iajs-3638 313 21 𝛽2 𝛽2 PROPN iajs-3638 313 22 = = SYM iajs-3638 313 23 0.02 0.02 NUM iajs-3638 313 24 , , PUNCT iajs-3638 313 25 𝛽3 𝛽3 NOUN iajs-3638 313 26 = = NOUN iajs-3638 313 27 0.01 0.01 NUM iajs-3638 313 28 , , PUNCT iajs-3638 313 29 𝛼 𝛼 X iajs-3638 313 30 = = SYM iajs-3638 313 31 0.3 0.3 NUM iajs-3638 313 32 , , PUNCT iajs-3638 313 33 𝑛 𝑛 PROPN iajs-3638 313 34 = = SYM iajs-3638 313 35 50 50 NUM iajs-3638 313 36 , , PUNCT iajs-3638 313 37 πœƒ πœƒ NOUN iajs-3638 313 38 = = SYM iajs-3638 313 39 0.1 0.1 NUM iajs-3638 313 40 , , PUNCT iajs-3638 313 41 𝛾 𝛾 NOUN iajs-3638 313 42 = = NOUN iajs-3638 313 43 0.01 0.01 NUM iajs-3638 313 44 , , PUNCT iajs-3638 313 45 πœ‡1 πœ‡1 NOUN iajs-3638 313 46 = = PROPN iajs-3638 313 47 0.03 0.03 NUM iajs-3638 313 48 , , PUNCT iajs-3638 313 49 πœ‡ πœ‡ X iajs-3638 313 50 = = PROPN iajs-3638 313 51 0.00015 0.00015 NUM iajs-3638 313 52 , , PUNCT iajs-3638 313 53 𝜏 𝜏 X iajs-3638 313 54 = = SYM iajs-3638 313 55 25.2225 25.2225 NUM iajs-3638 313 56 . . PUNCT iajs-3638 314 1 ( ( PUNCT iajs-3638 314 2 58 58 NUM iajs-3638 314 3 ) ) PUNCT iajs-3638 314 4 investigated investigate VERB iajs-3638 314 5 is be AUX iajs-3638 314 6 the the DET iajs-3638 314 7 dynamical dynamical ADJ iajs-3638 314 8 behavior behavior NOUN iajs-3638 314 9 of of ADP iajs-3638 314 10 system system NOUN iajs-3638 314 11 ( ( PUNCT iajs-3638 314 12 3 3 NUM iajs-3638 314 13 ) ) PUNCT iajs-3638 314 14 near near ADP iajs-3638 314 15 the the DET iajs-3638 314 16 siep siep NOUN iajs-3638 314 17 point point NOUN iajs-3638 314 18 when when SCONJ iajs-3638 314 19 the the DET iajs-3638 314 20 t.d t.d X iajs-3638 314 21 . . PROPN iajs-3638 314 22 is be AUX iajs-3638 314 23 increased increase VERB iajs-3638 314 24 . . PUNCT iajs-3638 315 1 for for ADP iajs-3638 315 2 the the DET iajs-3638 315 3 set set NOUN iajs-3638 315 4 of of ADP iajs-3638 315 5 parameter parameter NOUN iajs-3638 315 6 values value NOUN iajs-3638 315 7 provided provide VERB iajs-3638 315 8 by by ADP iajs-3638 315 9 ( ( PUNCT iajs-3638 315 10 58 58 NUM iajs-3638 315 11 ) ) PUNCT iajs-3638 315 12 , , PUNCT iajs-3638 315 13 the the DET iajs-3638 315 14 system system NOUN iajs-3638 315 15 ( ( PUNCT iajs-3638 315 16 3 3 X iajs-3638 315 17 ) ) PUNCT iajs-3638 315 18 is be AUX iajs-3638 315 19 numerically numerically ADV iajs-3638 315 20 solved solve VERB iajs-3638 315 21 , , PUNCT iajs-3638 315 22 and and CCONJ iajs-3638 315 23 figure figure VERB iajs-3638 315 24 1 1 NUM iajs-3638 315 25 shows show VERB iajs-3638 315 26 the the DET iajs-3638 315 27 trajectory trajectory NOUN iajs-3638 315 28 of of ADP iajs-3638 315 29 the the DET iajs-3638 315 30 system system NOUN iajs-3638 315 31 ( ( PUNCT iajs-3638 315 32 3 3 NUM iajs-3638 315 33 ) ) PUNCT iajs-3638 315 34 . . PUNCT iajs-3638 316 1 ( ( PUNCT iajs-3638 316 2 a a X iajs-3638 316 3 ) ) PUNCT iajs-3638 316 4 ( ( PUNCT iajs-3638 316 5 b b X iajs-3638 316 6 ) ) PUNCT iajs-3638 316 7 ( ( PUNCT iajs-3638 316 8 c c X iajs-3638 316 9 ) ) PUNCT iajs-3638 316 10 ( ( PUNCT iajs-3638 316 11 d d X iajs-3638 316 12 ) ) PUNCT iajs-3638 316 13 figure figure NOUN iajs-3638 316 14 1.periodic 1.periodic NUM iajs-3638 316 15 solution solution NOUN iajs-3638 316 16 near near ADP iajs-3638 316 17 ( ( PUNCT iajs-3638 316 18 siep siep NOUN iajs-3638 316 19 ) ) PUNCT iajs-3638 316 20 of of ADP iajs-3638 316 21 system system NOUN iajs-3638 316 22 ( ( PUNCT iajs-3638 316 23 3 3 NUM iajs-3638 316 24 ) ) PUNCT iajs-3638 316 25 for for ADP iajs-3638 316 26 ( ( PUNCT iajs-3638 316 27 58 58 NUM iajs-3638 316 28 ) ) PUNCT iajs-3638 316 29 ( ( PUNCT iajs-3638 316 30 a a X iajs-3638 316 31 ) ) PUNCT iajs-3638 316 32 periodic periodic ADJ iajs-3638 316 33 solution solution NOUN iajs-3638 316 34 near near ADP iajs-3638 316 35 ( ( PUNCT iajs-3638 316 36 siep siep NOUN iajs-3638 316 37 ) ) PUNCT iajs-3638 316 38 . . PUNCT iajs-3638 317 1 ( ( PUNCT iajs-3638 317 2 b),(c b),(c NOUN iajs-3638 317 3 ) ) PUNCT iajs-3638 317 4 and and CCONJ iajs-3638 317 5 ( ( PUNCT iajs-3638 317 6 d d NOUN iajs-3638 317 7 ) ) PUNCT iajs-3638 317 8 3dperiodic 3dperiodic NUM iajs-3638 317 9 solution solution NOUN iajs-3638 317 10 . . PUNCT iajs-3638 318 1 equation equation NOUN iajs-3638 318 2 ( ( PUNCT iajs-3638 318 3 58 58 NUM iajs-3638 318 4 ) ) PUNCT iajs-3638 318 5 is be AUX iajs-3638 318 6 used use VERB iajs-3638 318 7 to to PART iajs-3638 318 8 observe observe VERB iajs-3638 318 9 that that SCONJ iajs-3638 318 10 for for ADP iajs-3638 318 11 the the DET iajs-3638 318 12 provided provide VERB iajs-3638 318 13 data datum NOUN iajs-3638 318 14 , , PUNCT iajs-3638 318 15 with with ADP iajs-3638 318 16 𝜏 𝜏 NOUN iajs-3638 318 17 = = SYM iajs-3638 318 18 0 0 NUM iajs-3638 318 19 system system NOUN iajs-3638 318 20 ( ( PUNCT iajs-3638 318 21 3 3 X iajs-3638 318 22 ) ) PUNCT iajs-3638 318 23 has have VERB iajs-3638 318 24 a a DET iajs-3638 318 25 locally locally ADV iajs-3638 318 26 asymptotically asymptotically ADV iajs-3638 318 27 stable stable ADJ iajs-3638 318 28 ( ( PUNCT iajs-3638 318 29 l.a.s l.a.s NOUN iajs-3638 318 30 . . PUNCT iajs-3638 318 31 ) ) PUNCT iajs-3638 319 1 to to PART iajs-3638 319 2 siep siep NOUN iajs-3638 319 3 as as SCONJ iajs-3638 319 4 shown show VERB iajs-3638 319 5 in in ADP iajs-3638 319 6 figure figure NOUN iajs-3638 319 7 2 2 NUM iajs-3638 319 8 . . PUNCT iajs-3638 319 9 ihjpas ihjpas PROPN iajs-3638 319 10 . . PUNCT iajs-3638 320 1 2025 2025 NUM iajs-3638 320 2 , , PUNCT iajs-3638 320 3 38 38 NUM iajs-3638 320 4 ( ( PUNCT iajs-3638 320 5 1 1 NUM iajs-3638 320 6 ) ) PUNCT iajs-3638 320 7 459 459 NUM iajs-3638 320 8 figure figure NOUN iajs-3638 320 9 2 2 NUM iajs-3638 320 10 . . PUNCT iajs-3638 320 11 system system NOUN iajs-3638 320 12 ( ( PUNCT iajs-3638 320 13 3 3 X iajs-3638 320 14 ) ) PUNCT iajs-3638 320 15 trajectories trajectory NOUN iajs-3638 320 16 using use VERB iajs-3638 320 17 the the DET iajs-3638 320 18 information information NOUN iajs-3638 320 19 provided provide VERB iajs-3638 320 20 by by ADP iajs-3638 320 21 ( ( PUNCT iajs-3638 320 22 58 58 NUM iajs-3638 320 23 ) ) PUNCT iajs-3638 320 24 with with ADP iajs-3638 320 25 𝜏 𝜏 NOUN iajs-3638 320 26 = = SYM iajs-3638 320 27 0 0 NUM iajs-3638 320 28 approach approach NOUN iajs-3638 320 29 to to ADP iajs-3638 320 30 siep siep NOUN iajs-3638 320 31 . . PUNCT iajs-3638 321 1 equation equation NOUN iajs-3638 321 2 ( ( PUNCT iajs-3638 321 3 58 58 NUM iajs-3638 321 4 ) ) PUNCT iajs-3638 321 5 is be AUX iajs-3638 321 6 used use VERB iajs-3638 321 7 to to PART iajs-3638 321 8 observe observe VERB iajs-3638 321 9 that that SCONJ iajs-3638 321 10 for for ADP iajs-3638 321 11 the the DET iajs-3638 321 12 provided provide VERB iajs-3638 321 13 data datum NOUN iajs-3638 321 14 , , PUNCT iajs-3638 321 15 with with ADP iajs-3638 321 16 𝛼 𝛼 PROPN iajs-3638 321 17 = = SYM iajs-3638 321 18 0 0 NUM iajs-3638 321 19 and and CCONJ iajs-3638 321 20 𝛽1 𝛽1 NOUN iajs-3638 321 21 = = SYM iajs-3638 321 22 0.0003 0.0003 NUM iajs-3638 321 23 system system NOUN iajs-3638 321 24 ( ( PUNCT iajs-3638 321 25 3 3 X iajs-3638 321 26 ) ) PUNCT iajs-3638 321 27 has have VERB iajs-3638 321 28 a a DET iajs-3638 321 29 locally locally ADV iajs-3638 321 30 asymptotically asymptotically ADV iajs-3638 321 31 stable stable ADJ iajs-3638 321 32 ( ( PUNCT iajs-3638 321 33 l.a.s l.a.s NOUN iajs-3638 321 34 . . PUNCT iajs-3638 321 35 ) ) PUNCT iajs-3638 321 36 to to PART iajs-3638 321 37 fep fep PRON iajs-3638 321 38 as as SCONJ iajs-3638 321 39 shown show VERB iajs-3638 321 40 in in ADP iajs-3638 321 41 figure figure NOUN iajs-3638 321 42 3 3 NUM iajs-3638 321 43 . . PUNCT iajs-3638 321 44 figure figure VERB iajs-3638 321 45 3 3 NUM iajs-3638 321 46 . . PUNCT iajs-3638 321 47 system system NOUN iajs-3638 321 48 ( ( PUNCT iajs-3638 321 49 3 3 X iajs-3638 321 50 ) ) PUNCT iajs-3638 321 51 trajectories trajectory NOUN iajs-3638 321 52 using use VERB iajs-3638 321 53 the the DET iajs-3638 321 54 information information NOUN iajs-3638 321 55 provided provide VERB iajs-3638 321 56 by by ADP iajs-3638 321 57 ( ( PUNCT iajs-3638 321 58 58 58 NUM iajs-3638 321 59 ) ) PUNCT iajs-3638 321 60 with with ADP iajs-3638 321 61 𝛼 𝛼 NOUN iajs-3638 321 62 = = SYM iajs-3638 321 63 0 0 NUM iajs-3638 321 64 and and CCONJ iajs-3638 321 65 𝛽1 𝛽1 NOUN iajs-3638 321 66 = = SYM iajs-3638 321 67 0.0003 0.0003 NUM iajs-3638 321 68 approach approach NOUN iajs-3638 321 69 to to ADP iajs-3638 321 70 fep fep PROPN iajs-3638 321 71 . . PUNCT iajs-3638 322 1 equation equation NOUN iajs-3638 322 2 ( ( PUNCT iajs-3638 322 3 58 58 NUM iajs-3638 322 4 ) ) PUNCT iajs-3638 322 5 is be AUX iajs-3638 322 6 used use VERB iajs-3638 322 7 to to PART iajs-3638 322 8 observe observe VERB iajs-3638 322 9 that that SCONJ iajs-3638 322 10 for for ADP iajs-3638 322 11 the the DET iajs-3638 322 12 provided provide VERB iajs-3638 322 13 data datum NOUN iajs-3638 322 14 , , PUNCT iajs-3638 322 15 with with ADP iajs-3638 322 16 𝛾 𝛾 PROPN iajs-3638 322 17 = = SYM iajs-3638 322 18 0 0 NUM iajs-3638 322 19 , , PUNCT iajs-3638 322 20 𝛽1 𝛽1 NOUN iajs-3638 322 21 = = SYM iajs-3638 322 22 0.0003 0.0003 NUM iajs-3638 322 23 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ iajs-3638 322 24 𝛽2 𝛽2 PROPN iajs-3638 322 25 = = SYM iajs-3638 322 26 0.0002 0.0002 NUM iajs-3638 322 27 system system NOUN iajs-3638 322 28 ( ( PUNCT iajs-3638 322 29 3 3 X iajs-3638 322 30 ) ) PUNCT iajs-3638 322 31 has have AUX iajs-3638 322 32 a a DET iajs-3638 322 33 l.a.s l.a.s NOUN iajs-3638 322 34 . . PUNCT iajs-3638 323 1 to to PART iajs-3638 323 2 sep sep VERB iajs-3638 323 3 as as SCONJ iajs-3638 323 4 shown show VERB iajs-3638 323 5 in in ADP iajs-3638 323 6 figure figure NOUN iajs-3638 323 7 4 4 NUM iajs-3638 323 8 . . PUNCT iajs-3638 323 9 ihjpas ihjpas PROPN iajs-3638 323 10 . . PUNCT iajs-3638 324 1 2025 2025 NUM iajs-3638 324 2 , , PUNCT iajs-3638 324 3 38 38 NUM iajs-3638 324 4 ( ( PUNCT iajs-3638 324 5 1 1 NUM iajs-3638 324 6 ) ) PUNCT iajs-3638 324 7 460 460 NUM iajs-3638 324 8 figure figure NOUN iajs-3638 324 9 4 4 NUM iajs-3638 324 10 . . PUNCT iajs-3638 324 11 system system NOUN iajs-3638 324 12 ( ( PUNCT iajs-3638 324 13 3 3 X iajs-3638 324 14 ) ) PUNCT iajs-3638 324 15 trajectories trajectory NOUN iajs-3638 324 16 using use VERB iajs-3638 324 17 the the DET iajs-3638 324 18 information information NOUN iajs-3638 324 19 provided provide VERB iajs-3638 324 20 by by ADP iajs-3638 324 21 equation equation NOUN iajs-3638 324 22 ( ( PUNCT iajs-3638 324 23 58 58 NUM iajs-3638 324 24 ) ) PUNCT iajs-3638 324 25 with with ADP iajs-3638 324 26 𝛾 𝛾 PROPN iajs-3638 324 27 = = SYM iajs-3638 324 28 0 0 NUM iajs-3638 324 29 , , PUNCT iajs-3638 324 30 𝛽1 𝛽1 NOUN iajs-3638 324 31 = = SYM iajs-3638 324 32 0.0003 0.0003 NUM iajs-3638 324 33 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ iajs-3638 324 34 𝛽2 𝛽2 PROPN iajs-3638 324 35 = = SYM iajs-3638 324 36 0.0002 0.0002 NUM iajs-3638 324 37 approach approach NOUN iajs-3638 324 38 to to ADP iajs-3638 324 39 sep sep PROPN iajs-3638 324 40 . . PUNCT iajs-3638 325 1 we we PRON iajs-3638 325 2 talk talk VERB iajs-3638 325 3 about about ADP iajs-3638 325 4 how how SCONJ iajs-3638 325 5 the the DET iajs-3638 325 6 time time NOUN iajs-3638 325 7 delay delay NOUN iajs-3638 325 8 affects affect VERB iajs-3638 325 9 how how SCONJ iajs-3638 325 10 the the DET iajs-3638 325 11 system system NOUN iajs-3638 325 12 behaves behave VERB iajs-3638 325 13 close close ADV iajs-3638 325 14 to to ADP iajs-3638 325 15 the the DET iajs-3638 325 16 tep tep NOUN iajs-3638 325 17 . . PUNCT iajs-3638 326 1 for for ADP iajs-3638 326 2 Ο„ Ο„ PROPN iajs-3638 326 3 = = PROPN iajs-3638 326 4 10 10 NUM iajs-3638 326 5 < < X iajs-3638 326 6 Ο„0 Ο„0 NOUN iajs-3638 326 7 = = PUNCT iajs-3638 326 8 13.5 13.5 NUM iajs-3638 326 9 and and CCONJ iajs-3638 326 10 𝛼 𝛼 X iajs-3638 326 11 = = NOUN iajs-3638 326 12 0 0 NUM iajs-3638 326 13 with with ADP iajs-3638 326 14 the the DET iajs-3638 326 15 set set NOUN iajs-3638 326 16 of of ADP iajs-3638 326 17 data datum NOUN iajs-3638 326 18 in in ADP iajs-3638 326 19 equation equation NOUN iajs-3638 326 20 ( ( PUNCT iajs-3638 326 21 58 58 NUM iajs-3638 326 22 ) ) PUNCT iajs-3638 326 23 tep tep NOUN iajs-3638 326 24 is be AUX iajs-3638 326 25 still still ADV iajs-3638 326 26 l.a.s.as l.a.s.as PRON iajs-3638 326 27 shown show VERB iajs-3638 326 28 in in ADP iajs-3638 326 29 figure figure NOUN iajs-3638 326 30 5 5 NUM iajs-3638 326 31 . . PUNCT iajs-3638 326 32 figure figure NOUN iajs-3638 326 33 5 5 NUM iajs-3638 326 34 . . PUNCT iajs-3638 327 1 system system NOUN iajs-3638 327 2 ( ( PUNCT iajs-3638 327 3 3 3 X iajs-3638 327 4 ) ) PUNCT iajs-3638 327 5 trajectories trajectory NOUN iajs-3638 327 6 using use VERB iajs-3638 327 7 the the DET iajs-3638 327 8 information information NOUN iajs-3638 327 9 provided provide VERB iajs-3638 327 10 by by ADP iajs-3638 327 11 ( ( PUNCT iajs-3638 327 12 58 58 NUM iajs-3638 327 13 ) ) PUNCT iajs-3638 327 14 with with ADP iajs-3638 327 15 𝛼 𝛼 NOUN iajs-3638 327 16 = = SYM iajs-3638 327 17 0 0 NUM iajs-3638 327 18 and and CCONJ iajs-3638 327 19 Ο„ Ο„ X iajs-3638 327 20 = = SYM iajs-3638 327 21 10 10 NUM iajs-3638 327 22 approach approach NOUN iajs-3638 327 23 to to ADP iajs-3638 327 24 tep tep PROPN iajs-3638 327 25 . . PUNCT iajs-3638 328 1 now now ADV iajs-3638 328 2 , , PUNCT iajs-3638 328 3 for for ADP iajs-3638 328 4 Ο„0 Ο„0 NOUN iajs-3638 328 5 = = SYM iajs-3638 328 6 13.5 13.5 NUM iajs-3638 328 7 and and CCONJ iajs-3638 328 8 𝛼 𝛼 X iajs-3638 328 9 = = NOUN iajs-3638 328 10 0 0 NUM iajs-3638 328 11 with with SCONJ iajs-3638 328 12 the the DET iajs-3638 328 13 set set NOUN iajs-3638 328 14 of of ADP iajs-3638 328 15 data datum NOUN iajs-3638 328 16 in in ADP iajs-3638 328 17 equation equation NOUN iajs-3638 328 18 ( ( PUNCT iajs-3638 328 19 58 58 NUM iajs-3638 328 20 ) ) PUNCT iajs-3638 328 21 a a DET iajs-3638 328 22 h.b h.b PROPN iajs-3638 328 23 . . PROPN iajs-3638 328 24 occurs occur VERB iajs-3638 328 25 at at ADP iajs-3638 328 26 tep tep NOUN iajs-3638 328 27 as as SCONJ iajs-3638 328 28 shown show VERB iajs-3638 328 29 in in ADP iajs-3638 328 30 figure figure NOUN iajs-3638 328 31 6 6 NUM iajs-3638 328 32 . . PUNCT iajs-3638 329 1 ihjpas ihjpas PROPN iajs-3638 329 2 . . PUNCT iajs-3638 330 1 2025 2025 NUM iajs-3638 330 2 , , PUNCT iajs-3638 330 3 38 38 NUM iajs-3638 330 4 ( ( PUNCT iajs-3638 330 5 1 1 NUM iajs-3638 330 6 ) ) PUNCT iajs-3638 330 7 461 461 NUM iajs-3638 330 8 ( ( PUNCT iajs-3638 330 9 a a NOUN iajs-3638 330 10 ) ) PUNCT iajs-3638 330 11 ( ( PUNCT iajs-3638 330 12 b b X iajs-3638 330 13 ) ) PUNCT iajs-3638 330 14 figure figure NOUN iajs-3638 330 15 6 6 NUM iajs-3638 330 16 . . PUNCT iajs-3638 331 1 system system NOUN iajs-3638 331 2 ( ( PUNCT iajs-3638 331 3 3 3 X iajs-3638 331 4 ) ) PUNCT iajs-3638 331 5 trajectories trajectory NOUN iajs-3638 331 6 using use VERB iajs-3638 331 7 the the DET iajs-3638 331 8 information information NOUN iajs-3638 331 9 provided provide VERB iajs-3638 331 10 by by ADP iajs-3638 331 11 ( ( PUNCT iajs-3638 331 12 58 58 NUM iajs-3638 331 13 ) ) PUNCT iajs-3638 331 14 with with ADP iajs-3638 331 15 𝛼 𝛼 NOUN iajs-3638 331 16 = = SYM iajs-3638 331 17 0 0 NUM iajs-3638 331 18 and and CCONJ iajs-3638 331 19 Ο„ Ο„ X iajs-3638 331 20 = = NOUN iajs-3638 331 21 13.5 13.5 NUM iajs-3638 331 22 . . PUNCT iajs-3638 332 1 ( ( PUNCT iajs-3638 332 2 a a X iajs-3638 332 3 ) ) PUNCT iajs-3638 332 4 periodic periodic ADJ iajs-3638 332 5 solution solution NOUN iajs-3638 332 6 near near ADP iajs-3638 332 7 tep tep PROPN iajs-3638 332 8 . . PUNCT iajs-3638 333 1 ( ( PUNCT iajs-3638 333 2 b b X iajs-3638 333 3 ) ) PUNCT iajs-3638 333 4 3dperiodic 3dperiodic NUM iajs-3638 333 5 solution solution NOUN iajs-3638 333 6 . . PUNCT iajs-3638 334 1 equation equation NOUN iajs-3638 334 2 ( ( PUNCT iajs-3638 334 3 58 58 NUM iajs-3638 334 4 ) ) PUNCT iajs-3638 334 5 is be AUX iajs-3638 334 6 used use VERB iajs-3638 334 7 to to PART iajs-3638 334 8 observe observe VERB iajs-3638 334 9 that that SCONJ iajs-3638 334 10 for for ADP iajs-3638 334 11 the the DET iajs-3638 334 12 provided provide VERB iajs-3638 334 13 data datum NOUN iajs-3638 334 14 , , PUNCT iajs-3638 334 15 with with ADP iajs-3638 334 16 𝛽1 𝛽1 NOUN iajs-3638 334 17 = = SYM iajs-3638 334 18 0.0003 0.0003 NUM iajs-3638 334 19 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ iajs-3638 334 20 𝛽2 𝛽2 PROPN iajs-3638 334 21 = = SYM iajs-3638 334 22 0.0001 0.0001 NUM iajs-3638 334 23 system system NOUN iajs-3638 334 24 ( ( PUNCT iajs-3638 334 25 3 3 X iajs-3638 334 26 ) ) PUNCT iajs-3638 334 27 has have VERB iajs-3638 334 28 a a DET iajs-3638 334 29 l.a.s l.a.s NOUN iajs-3638 334 30 . . PUNCT iajs-3638 335 1 to to PART iajs-3638 335 2 foep foep VERB iajs-3638 335 3 as as SCONJ iajs-3638 335 4 shown show VERB iajs-3638 335 5 in in ADP iajs-3638 335 6 figure figure NOUN iajs-3638 335 7 7 7 NUM iajs-3638 335 8 . . PUNCT iajs-3638 335 9 figure figure VERB iajs-3638 335 10 7 7 NUM iajs-3638 335 11 . . PUNCT iajs-3638 335 12 system system NOUN iajs-3638 335 13 ( ( PUNCT iajs-3638 335 14 3 3 X iajs-3638 335 15 ) ) PUNCT iajs-3638 335 16 trajectories trajectory NOUN iajs-3638 335 17 using use VERB iajs-3638 335 18 the the DET iajs-3638 335 19 information information NOUN iajs-3638 335 20 provided provide VERB iajs-3638 335 21 by by ADP iajs-3638 335 22 equation equation NOUN iajs-3638 335 23 ( ( PUNCT iajs-3638 335 24 58 58 NUM iajs-3638 335 25 ) ) PUNCT iajs-3638 335 26 with with ADP iajs-3638 335 27 𝛽1 𝛽1 NOUN iajs-3638 335 28 = = SYM iajs-3638 335 29 0.0003 0.0003 NUM iajs-3638 335 30 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ iajs-3638 335 31 𝛽2 𝛽2 PROPN iajs-3638 335 32 = = SYM iajs-3638 335 33 0.0001 0.0001 NUM iajs-3638 335 34 approach approach NOUN iajs-3638 335 35 to to ADP iajs-3638 335 36 foep foep NOUN iajs-3638 335 37 . . PUNCT iajs-3638 336 1 we we PRON iajs-3638 336 2 talk talk VERB iajs-3638 336 3 about about ADP iajs-3638 336 4 how how SCONJ iajs-3638 336 5 the the DET iajs-3638 336 6 time time NOUN iajs-3638 336 7 delay delay NOUN iajs-3638 336 8 affects affect VERB iajs-3638 336 9 how how SCONJ iajs-3638 336 10 the the DET iajs-3638 336 11 system system NOUN iajs-3638 336 12 behaves behave VERB iajs-3638 336 13 close close ADV iajs-3638 336 14 to to ADP iajs-3638 336 15 the the DET iajs-3638 336 16 fiep fiep PROPN iajs-3638 336 17 . . PUNCT iajs-3638 337 1 for for ADP iajs-3638 337 2 Ο„ Ο„ PROPN iajs-3638 337 3 = = PROPN iajs-3638 337 4 10 10 NUM iajs-3638 337 5 < < X iajs-3638 337 6 Ο„0 Ο„0 NOUN iajs-3638 337 7 = = SYM iajs-3638 337 8 19 19 NUM iajs-3638 337 9 and and CCONJ iajs-3638 337 10 𝛾 𝛾 ADP iajs-3638 337 11 = = SYM iajs-3638 337 12 0 0 NUM iajs-3638 337 13 with with ADP iajs-3638 337 14 the the DET iajs-3638 337 15 set set NOUN iajs-3638 337 16 of of ADP iajs-3638 337 17 data datum NOUN iajs-3638 337 18 in in ADP iajs-3638 337 19 ( ( PUNCT iajs-3638 337 20 58 58 NUM iajs-3638 337 21 ) ) PUNCT iajs-3638 337 22 fiep fiep PROPN iajs-3638 337 23 is be AUX iajs-3638 337 24 still still ADV iajs-3638 337 25 l.a.s l.a.s ADJ iajs-3638 337 26 . . PUNCT iajs-3638 338 1 as as SCONJ iajs-3638 338 2 shown show VERB iajs-3638 338 3 in in ADP iajs-3638 338 4 figure figure NOUN iajs-3638 338 5 8 8 NUM iajs-3638 338 6 . . PUNCT iajs-3638 338 7 ihjpas ihjpas PROPN iajs-3638 338 8 . . PUNCT iajs-3638 339 1 2025 2025 NUM iajs-3638 339 2 , , PUNCT iajs-3638 339 3 38 38 NUM iajs-3638 339 4 ( ( PUNCT iajs-3638 339 5 1 1 NUM iajs-3638 339 6 ) ) PUNCT iajs-3638 339 7 462 462 NUM iajs-3638 339 8 figure figure NOUN iajs-3638 339 9 8 8 NUM iajs-3638 339 10 . . PUNCT iajs-3638 339 11 system system NOUN iajs-3638 339 12 ( ( PUNCT iajs-3638 339 13 3 3 X iajs-3638 339 14 ) ) PUNCT iajs-3638 339 15 trajectories trajectory NOUN iajs-3638 339 16 using use VERB iajs-3638 339 17 the the DET iajs-3638 339 18 information information NOUN iajs-3638 339 19 provided provide VERB iajs-3638 339 20 by by ADP iajs-3638 339 21 ( ( PUNCT iajs-3638 339 22 58 58 NUM iajs-3638 339 23 ) ) PUNCT iajs-3638 339 24 with with ADP iajs-3638 339 25 𝛾 𝛾 PROPN iajs-3638 339 26 = = SYM iajs-3638 339 27 0 0 NUM iajs-3638 339 28 and and CCONJ iajs-3638 339 29 Ο„ Ο„ X iajs-3638 339 30 = = SYM iajs-3638 339 31 10 10 NUM iajs-3638 339 32 approach approach NOUN iajs-3638 339 33 to to ADP iajs-3638 339 34 fiep fiep PROPN iajs-3638 339 35 . . PUNCT iajs-3638 340 1 now now ADV iajs-3638 340 2 , , PUNCT iajs-3638 340 3 for for ADP iajs-3638 340 4 Ο„0 Ο„0 NOUN iajs-3638 340 5 = = SYM iajs-3638 340 6 19 19 NUM iajs-3638 340 7 and and CCONJ iajs-3638 340 8 𝛾 𝛾 ADP iajs-3638 340 9 = = SYM iajs-3638 340 10 0 0 NUM iajs-3638 340 11 with with ADP iajs-3638 340 12 the the DET iajs-3638 340 13 set set NOUN iajs-3638 340 14 of of ADP iajs-3638 340 15 data datum NOUN iajs-3638 340 16 in in ADP iajs-3638 340 17 equation equation NOUN iajs-3638 340 18 ( ( PUNCT iajs-3638 340 19 58 58 NUM iajs-3638 340 20 ) ) PUNCT iajs-3638 340 21 a a DET iajs-3638 340 22 h.b h.b PROPN iajs-3638 340 23 . . PROPN iajs-3638 340 24 occurs occur VERB iajs-3638 340 25 at at ADP iajs-3638 340 26 fiep fiep PROPN iajs-3638 340 27 as as SCONJ iajs-3638 340 28 shown show VERB iajs-3638 340 29 in in ADP iajs-3638 340 30 figure figure NOUN iajs-3638 340 31 9 9 NUM iajs-3638 340 32 . . PUNCT iajs-3638 341 1 ( ( PUNCT iajs-3638 341 2 a a X iajs-3638 341 3 ) ) PUNCT iajs-3638 341 4 ( ( PUNCT iajs-3638 341 5 b b X iajs-3638 341 6 ) ) PUNCT iajs-3638 341 7 figure figure NOUN iajs-3638 341 8 9 9 NUM iajs-3638 341 9 . . PUNCT iajs-3638 341 10 system system NOUN iajs-3638 341 11 ( ( PUNCT iajs-3638 341 12 3 3 X iajs-3638 341 13 ) ) PUNCT iajs-3638 341 14 trajectories trajectory NOUN iajs-3638 341 15 using use VERB iajs-3638 341 16 the the DET iajs-3638 341 17 information information NOUN iajs-3638 341 18 provided provide VERB iajs-3638 341 19 by by ADP iajs-3638 341 20 ( ( PUNCT iajs-3638 341 21 58 58 NUM iajs-3638 341 22 ) ) PUNCT iajs-3638 341 23 with with ADP iajs-3638 341 24 𝛾 𝛾 PROPN iajs-3638 341 25 = = SYM iajs-3638 341 26 0 0 NUM iajs-3638 341 27 and and CCONJ iajs-3638 341 28 Ο„ Ο„ X iajs-3638 341 29 = = PROPN iajs-3638 341 30 19 19 NUM iajs-3638 341 31 . . PUNCT iajs-3638 342 1 ( ( PUNCT iajs-3638 342 2 a a X iajs-3638 342 3 ) ) PUNCT iajs-3638 342 4 periodic periodic ADJ iajs-3638 342 5 solution solution NOUN iajs-3638 342 6 near near ADP iajs-3638 342 7 fiep fiep PROPN iajs-3638 342 8 . . PUNCT iajs-3638 343 1 ( ( PUNCT iajs-3638 343 2 b b X iajs-3638 343 3 ) ) PUNCT iajs-3638 343 4 3dperiodic 3dperiodic NUM iajs-3638 343 5 solution solution NOUN iajs-3638 343 6 . . PUNCT iajs-3638 344 1 3 3 X iajs-3638 344 2 . . 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